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MODELING INVENTORY INFORMATION VISIBILITY IN SUPPLY CHAIN NETWORKS By SANDEEP SRIVATHSAN Bachelor of Engineering University of Madras Chennai, Tamil Nadu, India May 2002 Master of Science Oklahoma State University Stillwater, Oklahoma, USA December 2004 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY May, 2012 COPYRIGHT c By SANDEEP SRIVATHSAN May, 2012 MODELING INVENTORY INFORMATION VISIBILITY IN SUPPLY CHAIN NETWORKS Dissertation Approved: Dr. Manjunath Kamath Dissertation advisor Dr. Ramesh Sharda Dr. Ricki Ingalls Dr. Tieming Liu Dr. Balabhaskar Balasundaram Dr. Sheryl A. Tucker Dean of the Graduate College iii TABLE OF CONTENTS Chapter Page 1 INTRODUCTION 1 1.1 Information Sharing in Supply Chains . . . . . . . . . . . . . . . . . 4 1.2 Performance Evaluation and Performance Optimization Models . . . 5 1.3 Motivation for the Proposed Research . . . . . . . . . . . . . . . . . . 6 1.3.1 ProblemStatement . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Outline of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . 7 2 LITERATURE REVIEW 9 2.1 Literature on Value of Information Sharing . . . . . . . . . . . . . . . 9 2.2 Modeling ProductionInventory and Supply Chain Networks . . . . . 14 2.2.1 Modeling ProductionInventory Networks . . . . . . . . . . . . 14 2.2.2 Modeling Supply Chain Networks and its Constituents . . . . 20 2.3 Summary of the Literature Review . . . . . . . . . . . . . . . . . . . 23 3 RESEARCH STATEMENT 25 3.1 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Research Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Research Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4 RESEARCH APPROACH 30 4.1 ResearchMethodology . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.1.1 Markov Chain Approach . . . . . . . . . . . . . . . . . . . . . 31 4.1.2 Parametric Decomposition Approach . . . . . . . . . . . . . . 32 iv 4.2 PerformanceMeasures . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3 Numerical Validation Procedure . . . . . . . . . . . . . . . . . . . . . 36 5 CTMC MODELS OF THE 1R/2P SCN CONFIGURATION 37 5.1 1R/2P SCN Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.2 CTMC Model for Poisson Arrivals and Exponential Processing Times 38 5.3 CTMCModel of 1R/2P SCN with HiVis . . . . . . . . . . . . . . . . 44 5.4 CTMCModel of the 1R/2P SCN withMedVis . . . . . . . . . . . . . 48 5.5 CTMCModel of the 1R/2P SCN with LoVis . . . . . . . . . . . . . . 53 5.6 CTMC Model of the 1R/2P SCN with NoVis . . . . . . . . . . . . . 58 5.7 Value of Information Sharing . . . . . . . . . . . . . . . . . . . . . . . 63 5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6 QUEUEING MODELS OF THE 1R/2P SCN CONFIGURATION WITH POISSON ARRIVALS AND EXPONENTIAL PROCESSING TIMES 71 6.1 Queueing Model of the 1R/2P SCN Configuration with HiVis . . . . 71 6.1.1 Validation of the M/M/2 Approximation of the 1R/2P SCN Configuration with HiVis . . . . . . . . . . . . . . . . . . . . . 76 6.1.2 Queueing Model of SCN with Lower Levels of Information Sharing 79 6.2 An Asymmetric 1R/2P SCN Configuration . . . . . . . . . . . . . . . 92 6.2.1 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7 QUEUEING MODELS OF THE 1R/2P SCN CONFIGURATION WITH GENERAL INTERARRIVAL AND PROCESSING TIME DISTRIBUTIONS 105 7.1 Queueing Model of the General 1R/2P SCN Configuration with HiVis 105 7.1.1 Validation of the GI/G/2 Approximation . . . . . . . . . . . . 107 v 7.2 Queueing Model of a General 1R/2P SCN Configuration with NoVis . 108 7.2.1 QueueingModel . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.3 Effect of Interarrival Time and Processing Time SCVs on the Value of Information Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 8 ANALYTICALMODELS OF THE 2R/2P SCN CONFIGURATION122 8.1 2R/2P SCN Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.2 Analytical Model for Poisson Arrivals and Exponential Processing Times123 8.2.1 Queueing Model of the 2R/2P SCN Configuration with HiVis 124 8.2.2 Validation of the M/M/2 Approximation . . . . . . . . . . . . 129 8.3 Queueing Model of the 2R/2P SCN Configuration with Lower Levels of Information Sharing (MedVis and LoVis) . . . . . . . . . . . . . . 135 8.4 Analytical Model for General InterArrival and Processing Time Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 8.4.1 Validation of the GI/G/2 Approximation . . . . . . . . . . . . 143 8.4.2 Insights fromthe AnalyticalModel . . . . . . . . . . . . . . . 144 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 9 MODELING TRANSIT TIMES IN SCNs WITH INVENTORY INFORMATION SHARING 151 9.1 Analytical Model of 1R/2P SCN Configuration with Transit Delay . . 151 9.1.1 Effect of Transit Delay on the Performance of the Retail Stores in the 1R/2P Configuration . . . . . . . . . . . . . . . . . . . 154 9.2 Analytical Model of the 2R/2P SCN Configuration with Transit Delays 156 9.2.1 Effect of Transit Delays on the Performance of the Retail Stores in the 2R/2P Configuration . . . . . . . . . . . . . . . . . . . 159 9.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 vi 10 CONCLUSIONS 165 10.1 Summary of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 10.2 Research Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 167 10.3 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 BIBLIOGRAPHY 171 A DETERMINATION OF THE PARAMETERS FOR SIMULATION EXPERIMENTS 179 A.1 Determination of the Warmup Period using Welch’s Method . . . . . 180 A.2 Determination of the Run Length . . . . . . . . . . . . . . . . . . . . 181 B CTMC MODEL RESULTS FOR THE 1R/2P SCN CONFIGURATION 182 B.1 Validation of the CTMC Model of 1R/2P SCN Configuration with HiVis182 B.2 Validation of the CTMC Model of 1R/2P SCN Configuration with MedVis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 B.3 Validation of the CTMC Model of 1R/2P SCN Configuration with LoVis185 B.4 CTMC Model Results for 1R/2P SCN Configuration with NoVis . . . 187 C QUEUEING RESULTS FOR THE 1R/2P SCN CONFIGURATION WITH POISSON ARRIVALS AND EXPONENTIAL PROCESSING TIMES 188 C.1 Validation of the M/M/2 Approximation . . . . . . . . . . . . . . . . 188 C.2 Validation of the Modified M/M/2 Model . . . . . . . . . . . . . . . . 191 C.3 Validation of the M/G/2 based model for the 1R/2P SCN with Heterogeneous Production Facilities . . . . . . . . . . . . . . . . . . . . . 194 D QUEUEING RESULTS FOR THE SCN CONFIGURATION 1R/2P: GENERAL INTERARRIVAL AND PROCESSING TIME DISvii TRIBUTIONS 199 E QUEUEING RESULTS FOR THE 2R/2P SCN CONFIGURATION218 E.1 Results for the Validation of the M/M/2 based Queueing Model for the 2R/2P SCN Configuration . . . . . . . . . . . . . . . . . . . . . . 218 E.2 Results for the Validation of the Modified M/M/2 Model for the 2R/2P SCN Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 E.3 Results for the Validation of the GI/G/2 based Model for the 2R/2P SCN Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 F QUEUEING RESULTS FOR SCN CONFIGURATIONS WITH TRANSIT TIME 274 F.1 Validation of the Analytical Model for the 1R/2P SCN Configuration with Transit Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 F.2 Validation of the Analytical Model for the 2R/2P SCN Configuration with Transit Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 viii LIST OF TABLES Table Page 3.1 Order Routing Policy for SCN with Low Level of Inventory Information Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Order Routing Policy for SCN with Medium Level of Inventory Information Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 Order Routing Policy for SCN with High Level of Inventory Information Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.1 Rate Balance Equations for the Reduced CTMC Model of the 1R/2P SCN with HiVis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Rate Balance Equations of the Reduced CTMC Model of the 1R/2P SCN withMedVis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.3 Rate Balance Equations of the Reduced CTMC Model of the 1R/2P SCN with LoVis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.4 Rate Balance Equations of the CTMC Model of the 1R/2P SCN with NoVis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.1 Experiments for the 1R/2P SCN Configuration under Poisson Arrivals and Exponential Processing Times . . . . . . . . . . . . . . . . . . . 77 6.2 Dependence of π0, 0 on the Basestock Level and Production Facility Utilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.3 Experiments for the Asymmetric 1R/2P SCN Configuration . . . . . 98 6.4 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 ix 7.1 1R/2P SCN Experiments for General Interarrival and Processing Time Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 7.2 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 8.1 Experiments for the 2R/2P SCN Configuration under Poisson Arrivals and Exponential Processing Times . . . . . . . . . . . . . . . . . . . 129 8.2 Experiments for 2R/2P SCN configuration under General Interarrival and Processing Time Distributions . . . . . . . . . . . . . . . . . . . 143 8.3 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 9.1 Experiments for 1R/2P SCN Configuration with Transit Delay . . . . 153 9.2 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 9.3 Experiments for the 2R/2P SCN Configuration with Transit Delay . . 159 9.4 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 B.1 Validation of CTMC Model of SCN with HiVis (Production Facility) 183 B.2 Validation of CTMC Model of SCN with HiVis (Retail Store) . . . . 183 B.3 Validation of CTMC Model of SCN with MedVis (Production Facility) 184 B.4 Validation of CTMC Model of SCN with MedVis (Retail Store) . . . 185 B.5 Validation of CTMC Model of SCN with LoVis (Production Facility) 186 B.6 Validation of CTMC Model of SCN with LoVis (Retail Store) . . . . 186 B.7 Exact Results for the SCN with NoVis (Production Facility) . . . . . 187 B.8 Exact Results for the SCN with NoVis (Retail Store) . . . . . . . . . 187 C.1 Fill Rate and Expected Number of Backorders at the Retail Store . . 189 C.2 Expected Inventory Level and Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 C.3 Fill Rate and Expected Number of Backorders at a Production Facility 190 x C.4 Expected Inventory Level and Expected Time to Fulfill a Backorder at a Production Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 C.5 Expected Time Spent by an Order at the Retail Store and a Production Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 C.6 Fill Rate and Expected Number of Backorders at the Retail Store . . 192 C.7 Expected Inventory Level and Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 C.8 Fill Rate and Expected Number of Backorders at a Production Facility 193 C.9 Expected Inventory Level and Expected Time to Fulfill a Backorder at a Production Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 C.10 Expected Time Spent by an Order at the Retail Store and a Production Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 C.11 PerformanceMeasures at the Retail Store . . . . . . . . . . . . . . . 195 C.12 Fill Rate and Expected Number of Backorders at the Production Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 C.13 Expected Inventory Level and Expected Time to Fulfill a Backorder at the Production Facilities . . . . . . . . . . . . . . . . . . . . . . . . . 197 C.14 Expected Time Spent by an Order at the Production Facilities and Utilizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 D.1 Fill Rate & Expected Number of Backorders at the Retail Store . . . 199 D.2 Expected Inventory Level & Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 D.3 Fill Rate & Expected Number of Backorders at a Production Facility 207 D.4 Expected Inventory Level & Expected Time to Fulfill a Backorder at a Production Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 D.5 Expected Time Spent by an Order at the Retail Store and a Production Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 xi E.1 Fill Rate & Expected Inventory Level at Retail Store 1 . . . . . . . . 219 E.2 Expected Number of Backorders & Expected Time to Fulfill a Backorder at Retail Store 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 E.3 Fill Rate & Expected Inventory Level at Retail Store 2 . . . . . . . . 219 E.4 Expected Number of Backorders & Expected Time to Fulfill a Backorder at Retail Store 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 E.5 Expected Time Spent by an Order at Retail Stores 1 and 2 . . . . . . 220 E.6 Fill Rate & Expected Inventory Level at a Production Facility . . . . 220 E.7 Expected Number of Backorders & Expected Time to Fulfill a Backorder at a Production Facility . . . . . . . . . . . . . . . . . . . . . . . 221 E.8 Expected Time Spent by an Order at a Production Facility . . . . . . 221 E.9 Fill Rate & Expected Inventory Level at Retail Store 1 . . . . . . . . 222 E.10 Expected Number of Backorders & Expected Time to Fulfill a Backorder at Retail Store 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 E.11 Fill Rate & Expected Inventory Level at Retail Store 2 . . . . . . . . 223 E.12 Expected Number of Backorders & Expected Time to Fulfill a Backorder at Retail Store 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 E.13 Expected Time Spent by an Order at Retail Stores 1 and 2 . . . . . . 224 E.14 Fill Rate & Expected Inventory Level at a Production Facility . . . . 224 E.15 Expected Number of Backorders & Expected Time to Fulfill a Backorder at a Production Facility . . . . . . . . . . . . . . . . . . . . . . . 224 E.16 Expected Time Spent by an Order at a Production Facility . . . . . . 224 E.17 Fill Rate and Expected Inventory Level at Retail Store 1 (λr 1, λr 2) = (1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 E.18 Fill Rate and Expected Inventory Level at Retail Store 1 (λr 1, λr 2) = (2, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 xii E.19 Expected Number of Backorders and Expected Time to Fulfill a Backorder at Retail Store 1 (λr 1, λr 2) = (1, 1) . . . . . . . . . . . . . . . . 233 E.20 Expected Number of Backorders and Expected Time to Fulfill a Backorder at Retail Store 1 (λr 1, λr 2) = (2, 1) . . . . . . . . . . . . . . . . 237 E.21 Fill Rate and Expected Inventory Level at Retail Store 2 (λr 1, λr 2) = (1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 E.22 Fill Rate and Expected Inventory Level at Retail Store 2 (λr 1, λr 2) = (2, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 E.23 Expected Number of Backorders and Expected Time to Fulfill a Backorder at Retail Store 2 (λr 1, λr 2) = (1, 1) . . . . . . . . . . . . . . . . 249 E.24 Expected Number of Backorders and Expected Time to Fulfill a Backorder at Retail Store 2 (λr 1, λr 2) = (2, 1) . . . . . . . . . . . . . . . . 252 E.25 Expected Time Spent by an Order at Retail Stores 1 and 2 (λr 1, λr 2) = (1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 E.26 Expected Time Spent by an Order at Retail Stores 1 and 2 (λr 1, λr 2) = (2, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 E.27 Fill Rate and Expected Inventory Level at a Production Facility (λr 1, λr 2) = (1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 E.28 Fill Rate and Expected Inventory Level at a Production Facility (λr 1, λr 2) = (2, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 E.29 Expected Number of Backorders and Expected Time to Fulfill a Backorder at a Production Facility (λr 1, λr 2) = (1, 1) . . . . . . . . . . . . 268 E.30 Expected Number of Backorders and Expected Time to Fulfill a Backorder at a Production Facility (λr 1, λr 2) = (2, 1) . . . . . . . . . . . . 270 E.31 Expected Time Spent by an Order at a Production Facility (λr 1, λr 2) = (1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 xiii E.32 Expected Time Spent by an Order at a Production Facility (λr 1, λr 2) = (2, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 F.1 Performance Measures at the Retail Store (Basestock settings  (2, 4) and (4, 2)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 F.2 Performance Measures at the Retail Store (Basestock settings  (3, 6) and (6, 3)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 F.3 Performance Measures at a Production Facility (Basestock settings  (2, 4) and (4, 2)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 F.4 Performance Measures at a Production Facility (Basestock settings  (3, 6) and (6, 3)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 F.5 Performance Measures at Retail Store 1 (Basestock setting  (3, 6, 3)) 279 F.6 Performance Measures at Retail Store 1 (Basestock setting  (3, 3, 6)) 280 F.7 Performance Measures at Retail Store 2 (Basestock setting  (3, 6, 3)) 280 F.8 Performance Measures at Retail Store 2 (Basestock setting  (3, 3, 6)) 281 F.9 Performance Measures at a Production Facility (Basestock setting  (3, 6, 3)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 xiv LIST OF FIGURES Figure Page 1.1 Generic Supply Chain Network [54] . . . . . . . . . . . . . . . . . . . 1 5.1 1R/2P SCN Configuration . . . . . . . . . . . . . . . . . . . . . . . . 38 5.2 CTMCModel of the 1R/2P SCN with HiVis . . . . . . . . . . . . . . 45 5.3 Reduced CTMC Model of the 1R/2P SCN with HiVis . . . . . . . . . 46 5.4 CTMCModel of the 1R/2P SCN withMedVis . . . . . . . . . . . . . 49 5.5 Reduced CTMC Model of the 1R/2P SCN with MedVis . . . . . . . 50 5.6 CTMCModel of the 1R/2P SCN with LoVis . . . . . . . . . . . . . . 54 5.7 Reduced CTMC of the 1R/2P SCN with LoVis . . . . . . . . . . . . 55 5.8 CTMC Model of the 1R/2P SCN with NoVis (p = 0.5) . . . . . . . . 58 5.9 Fill Rate at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . 64 5.10 Expected Number of Backorders at the Retail Store . . . . . . . . . . 64 5.11 Expected Inventory Level at the Retail Store . . . . . . . . . . . . . . 65 5.12 Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . 65 5.13 Expected Time Spent by an Order at the Retail Store . . . . . . . . . 66 5.14 Fill Rate at a Production Facility . . . . . . . . . . . . . . . . . . . . 66 5.15 Expected Number of Backorders at a Production Facility . . . . . . . 67 5.16 Expected Inventory Level at a Production Facility . . . . . . . . . . . 67 5.17 Expected Time to Fulfill a Backorder at a Production Facility . . . . 68 5.18 Expected Time Spent by an Order at a Production Facility . . . . . . 68 6.1 Queueing Representation of the 1R/2P SCN Configuration . . . . . . 73 6.2 Queueing Representation of the M/M/2 Approximation . . . . . . . . 73 xv 6.3 Fill Rate at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . 77 6.4 Expected Number of Backorders at the Retail Store . . . . . . . . . . 78 6.5 Expected Inventory Level at the Retail Store . . . . . . . . . . . . . . 78 6.6 Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . 79 6.7 Expected Time Spent by an Order at the Retail Store . . . . . . . . . 79 6.8 Fill Rate at a Production Facility . . . . . . . . . . . . . . . . . . . . 80 6.9 Expected Number of Backorders at a Production Facility . . . . . . . 80 6.10 Expected Inventory Level at a Production Facility . . . . . . . . . . . 81 6.11 Expected Time to Fulfill a Backorder at a Production Facility . . . . 81 6.12 Expected Time Spent by an Order at a Production Facility . . . . . . 82 6.13 ω(ρ, S) Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.14 Fill Rate at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . 87 6.15 Expected Number of Backorders at the Retail Store . . . . . . . . . . 88 6.16 Expected Inventory Level at the Retail Store . . . . . . . . . . . . . . 88 6.17 Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . 89 6.18 Expected Time Spent by an Order at the Retail Store . . . . . . . . . 89 6.19 Fill Rate at a Production Facility . . . . . . . . . . . . . . . . . . . . 90 6.20 Expected Number of Backorders at a Production Facility . . . . . . . 90 6.21 Expected Inventory Level at a Production Facility . . . . . . . . . . . 91 6.22 Expected Time to Fulfill a Backorder at a Production Facility . . . . 91 6.23 Expected Time Spent by an Order at a Production Facility . . . . . . 92 6.24 Fill Rate at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . 99 6.25 Expected Number of Backorders at the Retail Store . . . . . . . . . . 99 6.26 Expected Inventory Level at the Retail Store . . . . . . . . . . . . . . 100 6.27 Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . 100 6.28 Expected Time Spent by an Order at the Retail Store . . . . . . . . . 101 6.29 Utilization at the Production Facilities . . . . . . . . . . . . . . . . . 101 xvi 6.30 Fill Rate at the Production Facilities . . . . . . . . . . . . . . . . . . 102 6.31 Expected Number of Backorders at the Production Facilities . . . . . 102 6.32 Expected Inventory Level at the Production Facilities . . . . . . . . . 103 6.33 Expected Time to Fulfill a Backorder at the Production Facilities . . 103 6.34 Expected Time Spent by an Order at the Production Facilities . . . . 104 7.1 Fill Rate at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . 112 7.2 Expected Number of Backorders at the Retail Store . . . . . . . . . . 113 7.3 Expected Inventory Level at the Retail Store . . . . . . . . . . . . . . 114 7.4 Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . 115 7.5 Expected Time Spent by an Order at the Retail Store . . . . . . . . . 116 7.6 Fill Rate at a Production Facility . . . . . . . . . . . . . . . . . . . . 117 7.7 Expected Number of Backorders at a Production Facility . . . . . . . 118 7.8 Expected Inventory Level at a Production Facility . . . . . . . . . . . 119 7.9 Expected Time to Fulfill a Backorder at a Production Facility . . . . 120 7.10 Expected Time Spent by an Order at a Production Facility . . . . . . 121 8.1 Supply Chain Instance . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.2 Exact Representation of the 2R/2P SCN Structure . . . . . . . . . . 125 8.3 Aggregate Representation of the 2R/2P SCN Structure . . . . . . . . 125 8.4 Fill Rate at Retail Store 1 . . . . . . . . . . . . . . . . . . . . . . . . 130 8.5 Expected Number of Backorders at Retail Store 1 . . . . . . . . . . . 130 8.6 Expected Inventory Level at Retail Store 1 . . . . . . . . . . . . . . . 131 8.7 Expected Time to Fulfill a Backorder at Retail Store 1 . . . . . . . . 131 8.8 Expected Time Spent by an Order at Retail Store 1 . . . . . . . . . . 132 8.9 Fill Rate at Retail Store 2 . . . . . . . . . . . . . . . . . . . . . . . . 132 8.10 Expected Number of Backorders at Retail Store 2 . . . . . . . . . . . 133 8.11 Expected Inventory Level at Retail Store 2 . . . . . . . . . . . . . . . 133 xvii 8.12 Expected Time to Fulfill a Backorder at Retail Store 2 . . . . . . . . 134 8.13 Expected Time Spent by an Order at Retail Store 2 . . . . . . . . . . 134 8.14 Fill Rate at Retail Store 1 . . . . . . . . . . . . . . . . . . . . . . . . 137 8.15 Expected Number of Backorders at Retail Store 1 . . . . . . . . . . . 137 8.16 Expected Inventory Level at Retail Store 1 . . . . . . . . . . . . . . . 138 8.17 Expected Time to Fulfill a Backorder at Retail Store 1 . . . . . . . . 138 8.18 Expected Time Spent by an Order at Retail Store 1 . . . . . . . . . . 139 8.19 Fill Rate at Retail Store 2 . . . . . . . . . . . . . . . . . . . . . . . . 139 8.20 Expected Number of Backorders at Retail Store 2 . . . . . . . . . . . 140 8.21 Expected Inventory Level at Retail Store 2 . . . . . . . . . . . . . . . 140 8.22 Expected Time to Fulfill a Backorder at Retail Store 2 . . . . . . . . 141 8.23 Expected Time Spent by an Order at Retail Store 2 . . . . . . . . . . 141 8.24 Fill Rate at Retail Store 1 . . . . . . . . . . . . . . . . . . . . . . . . 144 8.25 Expected Number of Backorders at Retail Store 1 . . . . . . . . . . . 145 8.26 Expected Inventory Level at Retail Store 1 . . . . . . . . . . . . . . . 145 8.27 Expected Time to Fulfill a Backorder at Retail Store 1 . . . . . . . . 146 8.28 Expected Time Spent by an Order at Retail Store 1 . . . . . . . . . . 146 8.29 Fill Rate at Retail Store 2 . . . . . . . . . . . . . . . . . . . . . . . . 147 8.30 Expected Number of Backorders at Retail Store 2 . . . . . . . . . . . 147 8.31 Expected Inventory Level at Retail Store 2 . . . . . . . . . . . . . . . 148 8.32 Expected Time to Fulfill a Backorder at Retail Store 2 . . . . . . . . 148 8.33 Expected Time Spent by an Order at Retail Store 2 . . . . . . . . . . 149 8.34 Effect of Processing Time SCV on Fill Rate . . . . . . . . . . . . . . 149 8.35 Effect of Interarrival Time SCV on Fill Rate . . . . . . . . . . . . . . 150 9.1 Fill Rate at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . 154 9.2 Expected Number of Backorders at the Retail Store . . . . . . . . . . 155 9.3 Expected Inventory Level at the Retail Store . . . . . . . . . . . . . . 155 xviii 9.4 Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . 156 9.5 Expected Time Spent by an Order at the Retail Store . . . . . . . . . 156 9.6 Fill Rate at Retail Store 1 . . . . . . . . . . . . . . . . . . . . . . . . 160 9.7 Expected Number of Backorders at Retail Store 1 . . . . . . . . . . . 160 9.8 Expected Inventory Level at Retail Store 1 . . . . . . . . . . . . . . . 161 9.9 Expected Time to Fulfill a Backorder at Retail Store 1 . . . . . . . . 161 9.10 Expected Time spent by an Order at Retail Store 1 . . . . . . . . . . 162 9.11 Fill Rate at Retail Store 2 . . . . . . . . . . . . . . . . . . . . . . . . 162 9.12 Expected Number of Backorders at Retail Store 2 . . . . . . . . . . . 163 9.13 Expected Inventory Level at Retail Store 2 . . . . . . . . . . . . . . . 163 9.14 Expected Time to Fulfill a Backorder at Retail Store 2 . . . . . . . . 164 9.15 Expected Time spent by an Order at Retail Store 2 . . . . . . . . . . 164 A.1 Plot of Time in System . . . . . . . . . . . . . . . . . . . . . . . . . . 181 xix CHAPTER 1 INTRODUCTION A supply chain network (SCN) is a network of firms that work together to supply the end products to the customer with a focus on both customer satisfaction and profitability of all firms. The nodes or firms involved in a SCN may be raw material suppliers, production facilities where the raw material is converted into finished products, warehouses that store the finished products, distribution centers that deliver the finished products to the retailers and retailers who satisfy the end customer demand. An example SCN is illustrated in Figure 1.1. Suppliers Production Facilities Distribution Centers Retailers Figure 1.1: Generic Supply Chain Network [54] The globalization and rapid growth of ebusiness has meant that the customers have more choices of “suppliers” for any product that they need. As a result, stockout situation at a store may result in a customer choosing a competitor’s product. A 1 study by Corsten and Gruen [15] focused on a common problem in SCNs, namely, the shelf stockout rate, and its effect on the behavior of the end customer. According to their article, several trade associations and joint tradeindustry bodies have sponsored and released major reports on the stockout rate for fastmoving consumer goods. The average stockout rate at the store for all 40 studies was found to be 8.3%. Corsten and Gruen [15] also looked at consumer reactions to shelf stockouts using 29 studies in 20 different countries. The results show that about 31% of the consumers switch the store, 15% delay their purchase, 19% substitute the product with the same brand, 26% switch the brand, while 9% of the consumers do not purchase the item and contribute to lost sales. As a result, a key focus of the firms in a SCN is to reduce the shelf stockout rate and hence, improve end customer satisfaction. Supply chain visibility among the SCN partners is a key area of research and is seen as a panacea to the stockout problem. Supply chain visibility refers to the real time transparency in the supply chain facilitated through builtin information systems that allow the participants in the SCN to keep track of semifinished/finished goods that may be in store or in transit. The global supply chain benchmark report published by Aberdeen group in 2006 [18] found that 79% of the large enterprises that were surveyed reported that lack of supply chain visibility is a critical concern. 51% of the enterprises that were surveyed identified supply chain visibility among their top three concerns. In addition, more than 77% of the enterprises were prepared to spend heavily to attain supply chain visibility. With the advances in information technology, sharing demand, capacity, and inventory information among the partners in the SCN has become quicker and cheaper [12]. There have been studies (e.g., Graves [24], Zipkin [63], Gavirneni et al. [21]) that have focused on the effects of information sharing. While most of these studies showed that information sharing has benefits, a few (e.g., Graves [24]) argued that there are no added benefits of sharing information with the upstream firms for the situations they modeled. More details are presented 2 in Section 2.1. In this research, our focus is on inventory information sharing or inventory visibility. Inventory visibility allows companies to be informed about their partner’s instock inventory and intransit inventory in order to make their supply chain as effective as possible. In a typical SCN, demand and sales information flows from the downstream firms (e.g., retailers) to the upstream firms (e.g., suppliers), while there is flow of material in the opposite direction. There can also be other information exchanges that can occur between the various firms in the SCN depending on the amount of collaboration between the partners in the SCN. The order status information can flow from the upstream firms to the downstream partners, while there can be twoway communication between firms when it comes to sharing inventory information and production plan information [41]. Several critical decisions need to be taken while designing and operating a SCN. These decisions include facility locations, production capacities, the safety stock to be held at each location, and the basestock level at each location. Performance evaluation tools aid decision makers in making these decisions during the design and operation phases of a SCN. In addition, these models provide insight into the working of the SCN and would help in gaining a better understanding of the dynamics of the system (SCN). This research focused on analytical performance evaluation models of supply chain networks and has developed models that can explicitly capture inventory visibility in order to study the effects of sharing inventory information on the SCN performance. Performance evaluation tools are generally used to measure key performance measures of a system (e.g., average response time, fill rate, expected inventory, and expected backorders in the case of a SCN) for any given set of parameter values (e.g., interarrival and service time parameters, basestock levels, etc.) through the development and solution of analytical and simulation models. Performance evaluation tools 3 aid system designers and operations managers in making some key decisions, while keeping in mind the goals of the company [55]. The analytical performance evaluation tools are typically based on modeling techniques such as Markov chains, stochastic Petri nets, and queueing networks [57]. Simulation models can also be used for performance evaluation, but need more detailed information for modeling and more time for model development and model execution phases. Analytical models yield results more quickly and “are appropriate for rapid and rough cut analysis” [55]. In fact, analytical and simulation models have been used in tandem to analyze and design complex systems. For example, analytical models can be used to reduce a large set of design alternatives, and the remaining few alternatives can be studied in detail using simulation models [55]. The development of performance evaluation and optimization models for supply chain networks is an active area of research (see e.g., Ettl et al. [19], Raghavan and Viswanadham [49], Dong and Chen [17], and Srivathsan [53]). However, very few performance modeling studies have explicitly addressed the issue of modeling inventory visibility in a SCN. The focus of this research will be on developing such performance evaluation models of SCNs. 1.1 Information Sharing in Supply Chains Information sharing is believed to be a key component in the success of a SCN. Research on studying the effects of information sharing have dealt with demand information (e.g., [24], [21]), inventory information (e.g., [63], [20]), and a combination of demand and inventory information (e.g., [13], [12], [45]). Li et al. [41] presented a review of ten different models used to study the value of information sharing in supply chains. Lee and Whang [39] described the inventory, sales, demand forecast, order status, and production schedule information that can be shared and also discussed how and why such information is shared along with industry examples. While most of these studies revealed that there are benefits of information sharing, a few studies 4 including Graves [24] showed that sharing demand information with upstream firms did not add value to their SCN model setup. The studies that consider inventory information sharing mostly used simulation models. Some of the research studies that focused on developing queueing models with inventory visibility include Zipkin [63] and Armony and Plambeck [4]. There is also considerable research that focuses on the effect of information distortion. The bullwhip effect, the phenomenon in which there is an increase in the mean and variability of the demand process as we proceed from the retailer to the supplier stage in a SCN, occurs due to distorted demand information. For example, Lee et al. [38] is a frequently cited study on the bullwhip effect in a SCN. 1.2 Performance Evaluation and Performance Optimization Models The common analytical performance modeling tools that are used in performance evaluation include queueing models, Markov chain models, and stochastic Petri nets. Analytical models based on queueing theory have been developed for performance evaluation of production networks since the early 1950s. Prior to the 1990s, the literature on stochastic models of production networks primarily considered capacity, congestion and reliability issues, and did not explicitly model the presence of planned inventory. Similarly, the early literature on inventory theory did not consider capacity and congestion issues explicitly. Inventory theory models have been developed since the 1910s. The earliest work in this field includes the EOQ model [27] and its various extensions. Some of the recent text books that focus on inventory theory are Bramel and SimchiLevi [10], Zipkin [65], Nahmias [46], and Axsater [5]. Since the 1990s, there has been a growing interest among researchers in developing analytical models of production networks which consider capacity, congestion and planned inventory issues in a unified manner. Recent work on modeling productioninventory networks includes models of singlestage systems developed by Buzacott 5 and Shanthikumar [11] and models of multistage productioninventory networks developed by Lee and Zipkin [40], Sivaramakrishnan [51], Sivaramakrishnan and Kamath [52] and Zipkin [64]. A few research efforts also developed performance optimization models for SCNs. These include the studies of Cohen and Lee [14] and Lee and Billington [37]. More recently, queueing models have been used in tandem with optimization models to take into account the stochastic components in the SCN. For example, Ettl et al. [19] combined optimization under service level constraints along with a queueing model to support strategic level decision making in large SCNs. 1.3 Motivation for the Proposed Research In the development of queueingbased performance evaluation models of productioninventory and supply chain networks, the common assumption about routing of orders from downstream stages to upstream stages is that the routing probabilities are fixed and not dependent on state information such as inventory levels. The routing based on fixed probabilities is also called Bernoulli routing. This routing policy allows the placement of an order at a production facility even when it is facing a stockout situation. Such a routing policy is appropriate for the development of models for cases with no inventory visibility. In such a situation, the order may be placed based on historical information (say, for example, the percentage of orders placed by a downstream firm at an immediate upstream firm). In cases where the downstream firm has visibility of the immediate upstream firm’s inventory, a routing policy in which orders are placed based on item availability is certainly a possibility and perhaps desirable. Similarly, when the retail stores share their inventory information with their immediate upstream “suppliers”, then the latter can choose to satisfy an order from the retail store that has the least net inventory level. These situations pose very interesting modeling scenarios that have not been studied much in the queueing literature. In addition, there is a need to study the value in sharing inventory information among 6 the various firms in the SCN, and also to quantify the value of such information sharing. When there is value in sharing inventory information among firms in the SCN, then there is also a need to find the sensitivity of the benefits to different SCN parameters such as basestock levels, production capacities, and variability in service and arrival processes. 1.3.1 Problem Statement With the advancements in the field of information technology, information sharing among firms within a SCN has become easier. Information sharing to provide inventory visibility has gained the attention of practitioners and academic researchers. There is a need to develop analytical models that can explicitly model inventory visibility in productioninventory and supply chain networks. Such models will not only give us better insight into the value of inventory visibility, but also enable us to develop performance optimization tools to maximize the value of inventory visibility by choosing the right combination of system parameter values. 1.4 Outline of the Dissertation The rest of the document is structured as follows. Chapter 2 presents a review of the literature on information sharing and performance modeling of productioninventory and supply chain networks. Chapter 3 presents the research goals, objectives, and contributions. Chapter 4 presents the research approach. Chapter 5 presents the Markov chain models that were developed for a SCN with one retail store and two production facilities at different levels of inventory information sharing. Chapter 6 presents queueingbased analytical models that were developed for a SCN with one retail store and two production facilities under Poisson arrivals and exponential processing times, while Chapter 7 presents the analytical models for the case of general interarrival and processing time distributions. Chapter 8 presents the analytical 7 models that were developed for a SCN configuration with two retail stores and two production facilities, while Chapter 9 extends the analytical models to include transit times in a SCN with inventory information sharing. Chapter 10 presents the research contributions and highlights the scope for future research. 8 CHAPTER 2 LITERATURE REVIEW This chapter presents a detailed review of the literature on the study of value of information sharing in supply chains and the analytical performance modeling of production networks with planned inventory. Section 2.1 presents a review of the published studies that focused on the value of information sharing in SCNs. Section 2.2 summarizes the literature on modeling productioninventory and supply chain networks. Section 2.3 provides a summary of the literature review and identifies some gaps in the literature, which form the basis for the research conducted. 2.1 Literature on Value of Information Sharing Information sharing is increasingly seen as a contributing factor to the success of a SCN. Research efforts have focused on finding if there is any value in information sharing, in assessing the actual value of information sharing, and in identifying factors influencing the value of information sharing. Studies have considered various types of information that can be shared, namely supplier status, inventory levels, demand forecasts, price, schedule and capacity information [41]. Graves [24] studied a singleitem inventory system with a nonstationary demand process that behaved like a random walk. Exponentialweighted moving average was used to obtain the mean square forecast of the demand and deterministic leadtime was assumed. An adaptive basestock policy where the basestock is adjusted based on changes in demand forecast was proposed. The safety stock required for nonstationary demand was found to be much more than that for stationary demand. 9 The relationship between lead time and safety stock was found to be convex in the case of nonstationary demand indicating that more safety stock is required with increasing lead time. This singleitem model was then extended to a singleitem, multistage system and the upstream demand was found to be nonstationary with the same form as the downstream demand process. So the adaptive basestock policy was applied to the upstream stage and the results showed that the bullwhip effect cannot be mitigated by sharing more information to the upstream stage. Zipkin [63] studied the performance of a multiitem productioninventory system with a production facility under the firstcomefirstserved (FCFS) policy and longestqueue (LQ) policy. The production facility had a finite capacity and ample raw material supply. The processor at the production facility was assumed to be perfectly flexible with no setup costs. All products were assumed to be symmetric (i.e., same demand rates, processing times, etc.). The demands for the products were assumed to be Poisson and the processing times followed a general distribution. Each product was assumed to follow a basestock policy with identical basestock levels. Each demand for a product at a store resulted in the consumption of the product from the store and a resultant replenishment order being placed at the production facility. Backorders were allowed in the system. In the case of the FCFS policy, these replenishment orders waited at the processor’s queue and were processed on a firstcomefirstserved basis. In the case of the LQ policy, the processor at the production facility used the inventory information and serviced the product that had the smallest net inventory level. Preemption was allowed in the case of the LQ policy and ties were resolved randomly. The sum of the standard deviations of number of outstanding orders of type i (σi) over all products (σ = M i=1 σi) was used to study the performance of the system as it captured “the gross behavior of performance over a fairly wide range of systems, and no other measure of comparable simplicity did so” [63]. A closed form expression for σ was developed in the case of FIFO policy. In the case of LQ policy, 10 an approximation for σ was developed. Numerical experiments suggested that the LQ policy performed better than the FIFO policy by about 20% in some cases. The difference in performance was the greatest when the number of product types was large, and at small SCVs. The difference was small at low utilizations, increased till 90% utilization and vanished in heavy traffic. Cachon and Fisher [12] studied the impact of sharing demand and inventory data in a supply chain with one supplier, N identical retailers and a stationary stochastic demand. Costs were associated with holding inventory and backorders. The traditional information policy that did not use shared information was compared against a full information policy that exploited the shared information. For both the models, it was assumed that the supplier’s orders are always received at the retailer after a constant lead time. A stockout at a supplier caused a resulting replenishment delay for the retailer. In the traditional information policy, the retailers and suppliers were assumed to follow (Rr, nQr) and (Rs, nQs) policies, respectively. The supplier was assumed to allocate inventory to each retailer based on a batch priority allocation. As per this allocation, if retailer i ordered b batches, then the first batch was given priority b, the second batch was given priority b1, and so on. All the batches were assumed to be placed in a shipment queue based on decreasing priority order with ties broken randomly. In the full information model, the supplier could improve its order quantity decisions as well as allocation decisions based on the demand and inventory information from the retailer. The decision of allocation of batches could be improved as batches could be allocated based on the inventory position at the retailer in the period the batch is shipped as against the period it was ordered. Based on their comparison, it was found that supply chain costs were 2.2% lower on average with full information policy than with the traditional policy. Gavirneni et al. [21] studied the periodic review inventory control problem in a twoechelon supply chain with a retailer and a supplier. The sequence of events 11 during each period started with the supplier deciding on its production quantity for the period followed by the retailer realizing its customer demand. On satisfying the demand, the retailer placed an order with the supplier if its inventory level fell below the reorder point s. The order was assumed to be satisfied at the beginning of the next period. An order not satisfied by the supplier was assumed to be satisfied from some other supplier with no lead time. The (s, S) inventory policy was assumed to be optimal for the retailer who incurs a fixed plus linear ordering cost, linear holding and backorder costs, while the supplier incurred a linear holding and backorder cost. They studied three different models which differ in the amount of information shared. In the first case, there was no information sharing and the supplier followed a naive approach, assuming that the retailer demand followed an i.i.d. process. In the second model, the supplier knew the number of periods that had elapsed since the last order from the retailer as well as that the retailer was using the (s, S) inventory policy. In the third model, the supplier knew the number of units that had been sold by the retailer since the last order. In all three models, the optimal order upto level was computed via simulationbased optimization using infinitesimal perturbation analysis. It was shown that the third model had the least cost with the first model having the highest cost, showing that full information sharing is beneficial. Gavirneni [20] studied a periodic review inventory control problem in a supply chain with one capacitated supplier who supplies a single product to multiple identical retailers. The customer demands were assumed to be i.i.d. During each period, the sequence of events started with the retailer reviewing its inventory and placing an order with the supplier. The supplier responded by using the available retailer inventory information to satisfy as many demands as possible. The retailer received the supplier shipments and satisfied the customer demand. Costs were associated with holding inventory and penalties were imposed for unsatisfied demand. The author studied this system under three different levels of cooperation. In the first level 12 (no cooperation), the only information available to the supplier was the demand from the retailers. If the total demand was less than the capacity of the supplier, then all retailers’ demands were satisfied. If there were orders from the retailers in excess of the supplier capacity, the supplier was assumed to use the lexicographic allocation scheme. This scheme “ranked the consumers in the order of their importance (independent of their order quantities) and the demands were satisfied in that order” [20]. In the second level of cooperation, the supplier received the current retailer inventory level in addition to the demand. The supplier could allocate its capacity based on this information in such a way that the retailer with a larger inventory level received smaller shipments and the one with smaller inventory level received larger shipments. The third level of cooperation extended the second level model to include the possibility of transfer of inventory from one retailer to another. The research used a simulation model to search for an optimal target inventory level and optimal cost. The computational results showed that the third level of coordination resulted in the least cost and the benefits of cooperation in this supply chain decreased with increase in supplier capacity, increase in number of retailers, decrease in penalty cost, and decrease in demand variance. Armony and Plambeck [4] studied the effect of duplicate orders on a manufacturer’s estimation of demand rate and customer’s sensitivity to delay, and decisions on capacity investment. They considered a SCN where a manufacturer sold its products through two distributors. The customer demand arrival process at each distributor was assumed to be a Poisson process and each demand was assumed to be for a single product. Each distributor followed oneforone replenishment policy. When one of the distributors faced a stockout situation, then an arriving customer had the option to immediately obtain the finished goods from the other distributor, provided the other distributor had inventory. When both the distributors were out of stock, then the customer could place an order with both distributors. When the product was 13 delivered by one of the distributors, then the customer canceled the duplicate order placed at the other distributor. The customers were assumed to be impatient, leading to cancellations of outstanding orders after a waiting time that was exponentially distributed. They assumed that the manufacturer had knowledge of the basestock policy used by the distributors, which would help in the inference about the inventory level and the number of outstanding orders at the distributors. They obtained the maximum likelihood estimators for the demand rate, the reneging rate, and the probability that a customer would place a duplicate order when made to wait. Li et al. [41] presented a review of about 10 different information sharing models. These included the works of Zipkin [63], Gavirneni et al. [21], Gavirneni [20], Graves [24], Moinzadeh [45], Chen [13], Kulp [36], Cachon and Fisher [12], Schouten et al. [50], and Bourland et al. [9]. From the literature review presented here, it can be seen that only Armony and Plambeck [4] and Zipkin [63] considered the development of queueingbased performance evaluation models. 2.2 Modeling ProductionInventory and Supply Chain Networks 2.2.1 Modeling ProductionInventory Networks Performance evaluation models of productioninventory networks consider capacity, variability, and inventory issues in a unified manner. Productioninventory network models are typically used to analyze the performance of maketostock systems, in which planned inventory is maintained for finished products and semifinished products at the intermediate stages. Gavish and Graves ([22], [23]) studied a singlestage, singleproduct maketostock production facility with Poisson demand. They focused on finding a control policy that minimizes the expected cost per unit time using an M/D/1 queueing system [22] as well as an M/G/1 queueing system [23]. The optimal decision policy for both the models was found to be a twocriticalnumber policy characterized by the parameters 14 (Q∗, Q∗∗), “such that the server is turned off when the queue length is first reduced to Q∗, and is turned on when the queue length first reaches Q∗∗” [22]. Graves and Keilson [23] studied a oneproduct, onemachine problem under Poisson demand and exponentially distributed order size. They focused on minimizing the system cost that included setup cost, inventory holding cost, and backorder cost using a spatially constrained Markov process model. The work of Svoronos and Zipkin [56] does not explicitly model capacity issues, but forms the basis for models developed by Zipkin [40] and others. Svoronos and Zipkin [56] modeled a multiechelon inventory system with Poisson demand arrivals at the lowest hierarchy of the system called the leaf. Each demand reached the central depot through instantaneous replenishment orders to preceding stages. The central depot was assumed to have infinite raw material supply from an outside source. Each location was considered to be a node and each arc connecting a node with its predecessor was treated as a transit system (representing transportation or production activities). Parts are assumed to be processed sequentially and the behavior of each transit system was assumed to be independent of the demands and orders in the inventory system. The transit systems for all the arcs on the path from the outside source node to the leaf node were assumed to be mutually independent. Svoronos and Zipkin [56] applied the results of the singlelocation problem recursively, starting from the highestlevel echelon to analyze the complete network. Buzacott and Shanthikumar [11] presented results for general singlestage systems with unit demand and backlogging using the production authorization (PA) card concept, where a tag attached to an item is converted into a PA card when demand consumes the item. Using the number of outstanding orders in the stage at time t, K(t), the total number of tags in the system, S, the inventory level at time t, I(t), and the number of customer orders backordered at time t, B(t), can be obtained using equations (2.1) and (2.2). Since the PA card represents an outstanding order 15 at a stage, the number of PA cards available at time t, C(t), can be obtained using equation (2.3). It can be seen from equations (2.1) through (2.3) that by studying the process K(t), we can derive I(t), B(t) and C(t). I(t) = max[0, S − K(t)] (2.1) B(t) = max[0,K(t) − S] (2.2) C(t) = min[S,K(t)] (2.3) The distribution of the number of outstanding orders in the system (K) is the same as the number of customers in a GI/G/1 queue because of the assumption of infinite raw material supply. The expected inventory level and the expected number of backorders can be obtained using equations (2.4) and (2.5). E[I] = S n=0 n P(I = n) (2.4) E[B] = E[I] + E[K] − S (2.5) Buzacott and Shanthikumar [11] also modeled singlestage systems with lost sales, interrupted demand, bulk demand, machine failures and yield losses. The analytical models for multistage tandem productioninventory networks developed by Lee and Zipkin [40], Sivaramakrishnan [51], Liu et al [43], and Srivathsan [53] have many common assumptions, such as oneforone replenishment with backordering permitted at each stage, no limit on WIP queues at a stage and ample raw material supply at the first stage [30]. The approaches developed by Sivaramakrishnan [51], Liu et al. [43], and Srivathsan [53] are based on the parametric decomposition approach ([59], [61]) and share a common solution structure as explained in Kamath and Srivathsan [30]. 16 Sivaramakrishnan and Kamath [52] modeled an Mstage tandem maketostock system by decomposing it into individual stages where each stage is made up of the manufacturing resource at that stage and a delay node to “capture the upstream delay experienced by an order when there was no part in the output store of the previous stage” [51]. Sivaramakrishnan [51] modeled the manufacturing unit as a GI/G/1 queue and the delay node as an M/G/∞ queue. Sivaramakrishnan [51] extended the tandem model to include (i) multiple servers at a stage, (ii) batch service, (iii) limited supply of raw material, (iv) multiple part types, and (v) service interruptions due to machine failure. Sivaramakrishnan [51] also extended his approach to tandem networks with feedback and feedforward networks. Liu et al. [43] modeled a multistage tandem manufacturing/supply network with general interarrival and service processes. The tandem network was modeled as a system of inventory queues and then the overall inventory in the network was optimized with servicelevel constraints. A decomposition scheme was proposed, in which a semifinished product from a stage is moved into the input buffer of the downstream stage, whenever the job queue (material queue at the input buffer of node i + backorder queue at node i) is increased by one. If the inventory store at stage i1 is empty, then the request for the semifinished part is backordered. As a result, the job queue or outstanding orders queue at stage i consists of the material queue comprising of the semifinished products from the upstream stage and a backorder queue. In this approach, the backorder distribution is used to account for any delay due to stockout at stage i1. Lee and Zipkin [40] modeled a tandem productioninventory network with Poisson demand arrivals, and exponential service times at all stages. The multiechelon system developed by Svoronos and Zipkin [56] was used to analyze the tandem system by assuming that stage i behaved like an M/M/1 queue [40] and the sojourn times was exponentially distributed with mean 1/(μi  λ), where λ is the demand rate, 17 and μi is the processing rate at stage i. The sojourn times at all the stages were considered to be independent. Such an effective lead time decomposition approach is similar to Whitt’s parametric decomposition approach [59]. The approach developed by Svoronos and Zipkin [56] that employed phase type approximations (for more details, refer to Theorem 2.2.8 in Neuts [47]) was then used to solve the above approximate system. Zipkin [64] extended this work to model tandem productioninventory networks with feedback. For a detailed comparison of the models developed by Lee and Zipkin [40], Sivaramakrishnan [51], Liu et al. [43] and Srivathsan [53], the reader is referred to Kamath and Srivathsan [30], wherein the similarities and differences between the approaches are clearly documented along with the results of an extensive numerical study to gain better insight into the performance prediction capability of the different approximations. Nguyen [48] analyzed the problem of setting the basestock levels in a production system that produced both maketoorder and maketostock products with lost sales. Nguyen [48] derived the productform steadystate distribution for the above network under the assumptions that each station operated under a FIFO service discipline, all processing times and interdemand times were exponential, and all products had the same mean processing time. Nguyen [48] proposed approximations for the basestock levels based on heavy traffic analysis of queueing networks. Karaesmen et al. [31] assumed that the interarrival times and the processing times are geometrically distributed and modeled the system with advance order information for contract suppliers. They analyzed the basestock policy (S, L) with a focus on optimization and performance evaluation of the Geo/Geo/1 maketostock queue, where L is the release leadtime, which was used to “regulate the timing of material release into the manufacturing stage” [31]. Benjaafar et al. [8] examined the effect of product variety on the inventory cost 18 in a finite capacity maketostock productioninventory network where the products were manufactured in bulk and shared a common manufacturing facility. The production facility was assumed to comprise of a batching stage and a processing stage. The production stage was viewed as a GI/G/1 queue whose arrival process was the superposition of the order arrivals for the individual products. The key findings of the study were that the total cost increased linearly with the number of products, and that the rate of increase depended on the system parameters such as demand and processing time variability and capacity levels. Benjaafar et al. [7] studied the effect of inventory pooling in productioninventory networks with n locations having identical costs. The production facility was assumed to have finite capacity and supply lead times were assumed to be endogenous. They studied the sensitivity of the cost benefits from inventory pooling to system parameters such as service levels, demand and service time variability, and structure of the productioninventory network. The different structures of the network that were studied include (i) single production facility with Poisson arrivals and exponential service times and two sets of priorities  FIFO and the longest queue first policies, (ii) single production facility with service level constraints, (iii) single production facility with nonMarkovian demand and general service times, (iv) systems with multiple production facilities, and (v) systems with multiple production stages. For the system with multiple production facilities, they considered three different scenarios  (a) inventory pooling without capacity pooling, (b) inventory pooling with capacity pooling, and (c) capacity pooling without inventory pooling. In the first scenario, they considered a single inventory location supplied by n production facilities, with a demand from stream i resulting in a replenishment order at production facility i. They assumed a Markovian system and each facility was modeled as an independent M/M/1 queue. In the second scenario, both the inventory and capacity were assumed to be pooled. Under Markovian assumptions, the production system was modeled as 19 an M/M/n queue. In the third scenario, the production facilities are consolidated, while the inventory locations were distinct. The Markovian system with n identical facilities was modeled as an M/M/n multiclass FIFO production system. The results of the study showed that the benefit of inventory pooling decreased with utilization, while the benefit of capacity pooling increased with utilization. 2.2.2 Modeling Supply Chain Networks and its Constituents Performance evaluation models of SCNs consider the supply, transportation and distribution operations in a SCN in addition to the capacity, congestion, and inventory issues. These models are typically used to analyze the flow of information and goods between the various stores in a SCN. Cohen and Lee [14] developed an analytical model for integrated productiondistribution systems by decomposing the network into submodels such as the material control submodel, production submodel, stockpile inventory submodel, and the distribution submodel. The submodels were optimized based on certain control parameters which served as links between the submodels. These control parameters included lot sizes, reorder points and safety stocks. Lee and Billington [37] and Ettl et al. [19] focused on developing models to capture the interdependence of basestock levels at different stores in a SCN. Lee and Billington [37] study addressed the needs of the manufacturing managers at Hewlett Packard in managing the material flows in their decentralized supply chains. The study focused on computing the mean and variance of the replenishment lead time for every stockkeeping unit (SKU) at a site. Lee and Billington [37] then used this information to compute the required basestock level to attain a target service level for each SKU at that site. On the other hand, Ettl et al. [19] used a combination of analytical and queueing models to support strategic decision making for large SCNs with nonstationary demand. They considered a oneforone replenishment policy, 20 while ignoring order size at each store and other operational aspects of inventory management in a SCN. They developed an optimization model with service level constraints and used a queueing model to obtain the order queue distribution at each stage. Raghavan and Viswanadham [49] used forkjoin approximations to compute the mean and variance of departure processes at nodes in a SCN. They presented simple approximations for the case of deterministic arrivals and normally distributed service times. Dong [16] and Dong and Chen [17] used the PA card concept of Buzacott and Shanthikumar [11] to model an integrated logistics system with (s, S) inventory policy, where each orders had a fixed lot size. They made use of the expressions provided by Buzacott and Shanthikumar [11] for a system with batch arrivals to find the distribution of number of orders at the manufacturing stage and then used this to find the stockout probability and fill rate. Srivathsan [53] extended the performance evaluation models for productioninventory networks developed by Sivaramakrishnan [51] to model a generic supply chain network where each manufacturer was assumed to have an input store for raw material storage and an output store for finished product storage. An example network with three suppliers, two manufacturers and three retailers was used to illustrate the performance prediction capability of the approximations. While Sivaramakrishnan [51] used the delay block to link successive nodes in the network, Srivathsan [53] used the backorder distribution to establish the link. With oneforone replenishment, an end customer order at the retailer triggered an instantaneous replenishment order at each of the upstream stores. Each manufacturer was modeled as a single server queue, and the lead time delays at the raw material suppliers and the transportation delays in the network were approximated by an M/G/∞ model. Ayodhiramanujan [6] developed integrated analytical models that addressed capacity, congestion, and inventory issues simultaneously in warehouse systems. The 21 research effort focused on developing queueing network models for a sharedserver system and an orderpicking system. The former is an inventory store with a server performing both the storage and retrieval operations. In the orderpicking system, the configuration of the unitload that is stored (pallets) is different from that which is retrieved (cases). Ayodhiramanujan [6] also extended these models to include multisever cases. An integrated model was also developed to demonstrate the applicability of these two key building blocks in developing endtoend models of warehouse systems. Jain and Raghavan [29] considered a productioninventory system with a manufacturing plant and a warehouse. The warehouse inventory was modeled as the input control mechanism for the manufacturing plant, which was modeled as a singlestage discretetime queueing system with an infinite waiting line. The warehouse was assumed to follow a basestock policy with oneforone replenishment and a production authorization card was assumed to be attached to each finished good. The customer order arrival process was assumed to be Poisson and the manufacturer was assumed to process orders from the warehouse at fixed discrete time slots. The queueing models were embedded into two optimization models, the first of which focused on minimizing the long run expected total cost per unit time (comprising of holding cost at warehouse and backorder cost). The second optimization model focused on minimizing the long run expected total cost per unit time (comprising of the holding cost alone) subject to service level constraints based on the probability of backorders. Arda and Hennet [3] considered an enterprise network where the endproduct manufacturer had several potential suppliers for components. The supply system with random arrivals of customer orders and random supplier delivery times was modeled as a queueing system. The manufacturer was assumed to follow a basestock inventory policy. Demand at the manufacturer was assumed to follow a Poisson process and was of unit size. The manufacturer was assumed to follow a (S1, S) 22 inventory policy and placed an order with supplier i with a probability αi as per a Bernoulli splitting process. An optimization model with an objective of minimizing the average cost of the manufacturer (expressed as the sum of mean inventory holding cost and mean backordering cost) was used to obtain the Bernoulli probabilities and the optimal basestock value. Each supplier was assumed to follow an exponential service time and FIFO discipline. They modeled each supplier as an M/M/1 queue as the order arrival process at the supplier was Poisson because of the Bernoulli splitting process. They showed that the problem at hand was hard as the convexity of the objective function was not guaranteed. They presented optimal solutions for the maketoorder system by solving the Lagrangian of the relaxed problem. In the case of the maketostock system, due to the complexity of the objective function, the problem was decomposed into two parts. In the first part, they considered the Bernoulli parameters as the decision variables, while considering the basestock level as zero (maketoorder system). The values of the Bernoulli variables from the first part were then used as an input in the second part. The decision variable in this case was the basestock level, which was computed using a discrete version of the newsvendor problem. 2.3 Summary of the Literature Review This chapter presented a detailed review of literature on modeling information sharing in a supply chain network as well as performance evaluation models of productioninventory and supply chain networks. Based on the literature review, we note the following. Queueing models of productioninventory networks have focused on considering capacity, variability, and inventory issues in a unified manner and included the works of Graves and Kielson [25], Lee and Zipkin [40], Buzacott and Shanthikumar [11], Sivaramakrishnan [51], Liu et al. [43] and a few others. The ability of queueing models 23 to explicitly consider inventory issue has been considerably strengthened by the works of Sivaramakrishnan [51] on modeling productioninventory networks, Srivathsan [53] on modeling supply chain networks, and Ayodhiramanujan [6] on modeling warehouse operations. Other research efforts that focused on performance evaluation of SCNs include Dong [16], Dong and Chen [17], and Jain and Raghavan [29]. The research efforts that focused on performance optimization of SCNs included Ettl et al. [19], Lee and Billington [37], and Cohen and Lee [14]. The review of the literature on modeling inventory information sharing in a SCN showed that there are only a few efforts on developing performance evaluation models of SCNs in this context. Armony and Plambeck [4] and Zipkin [63] focused on performance evaluation using queuing models. The former focused on modeling the effect of duplicate orders on SCN performance, while the latter modeled a productioninventory network with multiple products produced by a single production facility. We concluded that there is a need for more research in explicitly modeling upstream inventory information sharing within SCN performance evaluation models. The availability of a richer set of performance evaluation models will enable us to better predict the value of sharing upstream inventory information in a SCN and understand the complex dynamics resulting from decisions related to production capacities, maximum inventory levels, and order placement policies. 24 CHAPTER 3 RESEARCH STATEMENT The overall goals of this research were (i) to develop analytical performance evaluation models that consider inventory information sharing between SCN firms, and (ii) to study the value of inventory information sharing and identify SCN conditions under which the benefits of inventory information sharing are significant. 3.1 Research Objectives In most analytical performance evaluation models of SCNs, it is assumed that a downstream firm places an order at one of its immediate upstream “suppliers” using a Bernoulli routing policy. As per this policy, orders from a retail store would be routed to say the production facilities based on fixed probabilities (based on preference or historical information) and this would allow an order to be placed at a production facility facing a stockout situation even when there is inventory at the other facility. This routing is appropriate for SCNs with no inventory information sharing. On the other hand, if a downstream firm has information about the net inventory level at all of its immediate upstream firms, orders from the downstream firm could possibly be placed in such a way that this situation can be averted, thereby reducing the backorders and decreasing the stockout rate. The models developed in this research consider inventory information sharing in one direction  upstream stores sharing information with the immediate downstream stores. An analytical model that addresses inventory information sharing in a SCN can be used to quantify the value of information sharing and provide insight into the 25 sensitivity of the value of inventory information sharing to different SCN parameters (interarrival and service time parameters, basestock levels, etc.). The model could then be used to identify the ranges of the various SCN parameters where the benefit of inventory information sharing is significant. The specific objectives of this research are as follows. Objective 1: To perform a thorough investigation of the literature related to (i) the value of information sharing in a SCN, and (ii) the analytical modeling of productioninventory and supply chain networks. Objective 2: To develop analytical models that can capture the effect of inventory information sharing on the performance of a twoechelon SCN comprising of one retail store that can order items from one of two upstream production facilities, each with its own inventory store and to study the benefits of inventory information sharing in this context. As discussed in the beginning of this section, this objective focused on modeling the effect of inventory information sharing on the performance of a supply chain. We first considered a SCN with one retail store and two production facilities each with its own inventory store. Henceforth, we will refer to this twoechelon configuration as “1R/2P.” For this case, we considered three levels of inventory information sharing and developed analytical models for these three levels. The performance measures from these analytical models were compared with the corresponding measures for the SCN with no visibility (henceforth referred to as NoVis) to study the value of inventory information sharing. The three levels of inventory information sharing and the assumed routing policy for each level are presented next. Low level of Inventory Information Sharing (LoVis) In a SCN with minimum level of inventory visibility, the retail store is assumed to have information about the presence of inventory or backorders at the production facilities. In such cases, an order routing policy can be adopted where the placement 26 of orders at a production facility with backorders can be avoided when the other facility has inventory (see Table 3.1). Table 3.1: Order Routing Policy for SCN with Low Level of Inventory Information Sharing Net Inventory Level at Production Facility 1 Net Inventory Level at Production Facility 2 Order Routing Policy ≥ 1 ≥ 1 Order routed with equal probability ≥ 1 ≤ 0 Order routed to production facility 1 ≤ 0 ≥ 1 Order routed to production facility 2 ≤ 0 ≤ 0 Order routed with equal probability Medium level of Inventory Information Sharing (MedVis) In a SCN with medium level of inventory visibility, the retail store is assumed to also have information about the number of backorders at the individual production facilities. When both production facilities are backordered, the order is routed to the facility with the shortest backorder queue. The order routing policy in such a scenario can be modified as shown in Table 3.2. Table 3.2: Order Routing Policy for SCN with Medium Level of Inventory Information Sharing Net Inventory Level at Production Facility 1 (i) Net Inventory Level at Production Facility 2 (j) Condition Order Routing Policy ≥ 1 ≥ 1 Order routed with equal probability ≥ 1 ≤ 0 Order routed to production facility 1 ≤ 0 ≥ 1 Order routed to production facility 2 ≤ 0 ≤ 0 i < j Order routed to production facility 1 i > j Order routed to production facility 2 i = j Order routed with equal probability High level of Inventory Information Sharing (HiVis) In a SCN with a high level of inventory visibility, the retail store is assumed to have information about the number of items in stock as well as the number of backorders 27 at the individual production facilities. In such cases, the order routing policy can be modified as shown in Table 3.3. Table 3.3: Order Routing Policy for SCN with High Level of Inventory Information Sharing Net Inventory Level at Production Facility 1 (i) Net Inventory Level at Production Facility 2 (j) Condition Order Routing Policy ≥ 1 ≥ 1 i > j Order routed to production facility 1 i < j Order routed to production facility 2 i = j Order routed with equal probability ≥ 1 ≤ 0 Order routed to production facility 1 ≤ 0 ≥ 1 Order routed to production facility 2 ≤ 0 ≤ 0 i < j Order routed to production facility 1 i > j Order routed to production facility 2 i = j Order routed with equal probability Objective 3: To develop analytical models that can capture the effect of inventory information sharing on the performance of a twoechelon SCN comprising of two retail stores that can order items from one of two production facilities, each with its own inventory store and to study the benefits of inventory information sharing in this context. We considered a SCN with two retail stores that place orders with one of two production facilities. The production facilities share their inventory information with the retail stores. Henceforth, we will refer to this twoechelon configuration as “2R/2P.” For this case, we considered the three levels of information sharing defined in Objective 2 and developed analytical models for these levels. Objective 4: To extend the models developed in Objectives 3 and 4 to include intransit inventory. 28 3.2 Research Scope The scope of this research effort was limited by the following assumptions. 1. There is no limit on the size of the WIP and backorder queues. 2. Each production facility has a singlestage with a single server. Each demand is for one unit of the product. 3. As the SCN configurations modeled are considered to be of the building block type, only twoechelon SCN structures will be modeled. 3.3 Research Contributions The contributions of this research effort are listed below. 1. Development of analytical queueing models that can explicitly model inventory information sharing from upstream stores to downstream stores in supply chain networks. 2. Understanding the benefits of sharing inventory information and developing insights into SCN configurations for which these benefits are significant. 3. Evaluating the significance of each incremental piece of information that becomes available in the SCN. 4. An offshoot of this research is the potential to develop good approximations for the wellknown shortest queue problem. 29 CHAPTER 4 RESEARCH APPROACH This chapter explains the overall research methodology and the various modeling approaches that were used to achieve the research objectives outlined in the previous chapter. Section 4.1 presents the research methodology, and then describes the Markov chain approach followed by the parametric decomposition approach. Section 4.2 presents a list of performance measures that were used along with their definitions and their significance in a SCN context. Section 4.3 explains the validation procedure used for the analytical models. 4.1 Research Methodology This research effort involved the development of analytical performance evaluation models. A standard methodology for such developmental research was employed. First, we modeled the SCN configurations with Poisson arrivals at the retail store(s) and exponential processing times at the production facilities as these conditions make the models more analytically tractable. For this case, we initially used the Markov chain approach. If the approach did not yield a closedform solution, we then focused on the development of approximate queueing models based on the characteristics of the model under consideration. We then relaxed the exponential assumption and considered SCN configurations with general interarrival and processing time distributions using the twomoment framework. Whitt’s ([59], [61]) parametric decomposition (PD) method was used to solve the resulting queueing models. Each analytical model developed was validated by comparing its results to equivalent simulation esti 30 mates. The objective for such a comparison was to determine the regions of the model parameter space where the accuracy of the analytical results was good or deemed acceptable. A design of experiments approach was used to ensure adequate coverage of the parameter space from the perspective of model usage in practice and to control the number of numerical experiments that had to be conducted. The numerical experimentation had the added benefit of providing insights into the system behavior under different parameter settings. The results of the numerical investigation were used to develop additional corrections or enhancements to improve the accuracy of the analytical models. Section 4.1.1 summarizes the Markov chain approach, and Section 4.1.2 briefly describes the PD approach. 4.1.1 Markov Chain Approach Continuous Time Markov Chain (CTMC) models have been widely used to develop performance evaluation models of discrete event systems. The advantage of CTMC models is that they can yield exact solutions under exponential or Markovian assumptions. Also, because the CTMC models are based on detailed state information, they are preferred when statebased decisions have to be modeled. For example, the modeling of visibilitybased routing policies in SCNs requires detailed state level information and could be easily modeled using the CTMC approach. Both transient and steadystate analysis can be performed using a CTMC model [44]. In the case of performance evaluation, steadystate analysis is done to compute the longrun performance measures. This involves the solution of the rate balance equations, which could be an issue in the case of an infinite state space. A standard method in such cases is to express all state probabilities in terms of the probability of one particular state and to use the total probability equation to find this probability. However, this is possible if the expression involving the sum of the probabilities can be simplified to yield a closedform expression. If simplification is not possible, then 31 we need to find ways (e.g., limit the number of backorders) to truncate the state space in order to yield a finitestate CTMC which can then be solved numerically. Another strategy to solve the balance equations is to use the structure of the CTMC to identify similarities between the state transitions in that CTMC and other CTMC models for which closedform solutions are known. For instance, while solving the CTMC model of the SCN under the Bernoulli routing policy, we were able to identify transitions patterns in the CTMC model that resembled the state transitions in an M/M/1 queuing model (see Section 5.6). Such similarities may allow us to guess a solution, which can be verified by substituting the guessed solution into the balance equations. If there is symmetry in a multituple system, we might be able to combine states (e.g., (i, j) combined with (j, i)) to reduce the size of the state space. The reduced CTMC can sometimes simplify the solution of the original CTMC. This reduced CTMC model can be solved using the rate balance equations or by identifying patterns in the chain. Once the CTMC is solved and the limiting probabilities are obtained, we can use a reverse mapping to obtain the limiting probabilities of the original CTMC model. Other methods that can be used to solve the CTMC models include the differenceequation technique and the method of generating functions. For more details on these methods, the reader is referred to Medhi [44]. 4.1.2 Parametric Decomposition Approach In the 1980s, Whitt [59] defined a new modeling ideology highlighted by the parametric decomposition (PD) approach. According to Whitt, “a natural alternative to an exact analysis of an approximate model is an approximate analysis of an exact model” [59]. The PD approach is a very comprehensive method for analyzing a queueing network and uses only the first two moments of both the interarrival and 32 service times. This approach formed the basis for a software package developed by Whitt, called the Queueing Network Analyzer (QNA) [59]. The PD approach for open queueing networks consists of two main steps: 1) analyzing the nodes and the interaction among the nodes to obtain the mean and the squared coefficient of variation (SCV = variance / mean2) of the interarrival times at each node, and 2) obtaining the node and system performance measures based on GI/G/1 or GI/G/m approximations ([59], [61]). Analyzing nodes: In a network, nodes interact with each other because of customer movement and these interactions can be approximately captured by the flow parameters, namely, the rates and variability parameters of the arrival processes at the nodes. The total arrival rate at each node is obtained using the traffic rate equations, which represent the conservation of flow. The utilizations of each of the nodes are calculated to check for stability of the system. The system is said to be stable if all utilizations are strictly less than one. This part of the analysis is similar to the approach introduced by Jackson [28] in solving open networks and involves no approximations. The approximations come into the picture while calculating the variability parameters related to the flow, namely, the SCVs of the interarrival times. The SCVs are calculated using the traffic variability equations, which involve approximations for the basic network operations, which are a) flow through a node, b) merging of flow, and c) splitting of flow. These approximations can be found in Whitt ([59], [61]). Calculating node and system performance measures: The nodes are treated as stochastically independent. The performance measures at each node can be calculated from the results available for the GI/G/1 ([59], [34]) and GI/G/m queues ([60]). The expected waiting time at each node is calculated from the results provided and the expected queue length is obtained using Little’s law [42]. Whitt [59] also explains how several other node and network measures can be calculated. 33 In our research, there is a need to incorporate statelevel details while modeling the SCN under visibilitybased routing policies. The twomoment framework is wellsuited for queueing network type models and it could be challenging to accommodate statelevel details in the analytical model developed. A strategy that is sometimes used involves the use of phasetype distributions to represent general distributions. The feature or subsystem that needs to be modeled in detail (e.g., the order routing or splitting process at the retailer) is studied in isolation. The phasetype approach enables CTMC type modeling because the phases are exponential stages. To analyze a feature or submodel in detail, the states in the Markov chain embedded at an instant of the feature need to be identified. The solution of this Markov chain results in the stationary probability vector, which can be used in obtaining the twomoment approximations for the feature or subsystem. Such an approach can be employed when closedform expressions exist for the Markov chain with Poisson arrivals and exponential processing times. A good example is the development of twomoment approximations for a forkjoin configuration [35]. In our research, we did not consider the phasetype approach because an exact solution to the CTMC model under Poisson arrivals and exponential distributions was not possible (discussed in Chapter 5). 4.2 Performance Measures The performance measures that were computed at each stage of the SCN are fill rate, expected number of backorders, expected inventory level, the expected time to fulfill a backorder and expected time spent by an order. • Fill rate is the probability that an order will be satisfied immediately and this depends on the availability of inventory at the stage. The definition of fill rate suggests that as the fill rate increases, the stockout rate decreases and the relationship between them is given by stockout rate = 1  fill rate. In our analytical approach, we approximate the fill rate by the ready rate, which is the 34 probability that there is inventory in the system. It should be noted that for Poisson arrivals, fill rate and ready rate will be the same because of the PASTA (Poisson Arrivals See Time Averages) principle [62]. • The expected number of backorders at any stage is the average number of unsatisfied orders at the stage. The significance of this measure is that a high value could indicate a potential for loss of customer goodwill and sales. • The expected inventory level at any stage is the average number of items in the store at that stage. The expected inventory level is a paradoxical measure as its high value would increase the inventory holding cost, while its low value could be an indicator of lower fill rates and higher backorder levels. • The expected time to fulfill a backorder at a stage is the average time that an order has to wait before being satisfied at a store given that the store is facing a stockout situation. It has to be noted that the expected time to fulfill an order can be considerably lower than the expected time to fulfill a backorder. This is because, majority of the orders could be satisfied instantaneously, while orders that are backordered could take significantly longer time to complete. Thus, this is a measure of the experience of a customer who faces a stockout situation. • The expected time spent by an order is defined as the sum of the average time that an order spends at that stage and the average time spent (as a product) in the output store at that stage. This measure is important as it is not a desirable situation to have an item sitting in stock for long periods as this would increase the holding cost. 35 4.3 Numerical Validation Procedure Each SCN configuration that was modeled in our research effort was also simulated using a model developed in Arena 11.0 software [32]. The warmup period was determined using Welch’s procedure [58] (See Appendix A). The number of independent replications was set to 10. The parameters of the various SCN configurations, namely, basestock levels; variability of interarrival and processing times; utilization; and probabilities (if any)  were varied systematically using a design of experiments approach to cover a wide range of scenarios. For each scenario, the analytical results were compared with steadystate simulation estimates to evaluate the accuracy of the analytical results. As mentioned in Whitt [60], the two standard ways to measure the accuracy are absolute difference and relative percentage error. As Whitt [60] contends, neither procedure is appropriate for a wide range of values. When the performance measure values are themselves small (e.g., less than 0.5), the absolute difference seems to be appropriate. Whitt [60] is of the opinion that the quality of the approximations is satisfactory “if either the absolute difference is below a critical threshold or the relative percentage error is below another critical threshold” [60]. Recently, another approach, namely, normalized error has been used to evaluate the accuracy of queueing approximations ([35], [54]). Hence, we designed the following approach to evaluate the accuracy of the analytical results. For the fill rate (bounded by 1), we used the absolute difference, simulation − analytical, expressed as a percentage. For the expected inventory level and the expected number of backorders, we used the normalized percentage error given by 100 (simulation − analytical)/basestock level. For the expected time to fulfill a backorder and the expected time spent by an order, we used the relative percentage error defined as 100 (simulation − analytical)/simulation. This scheme allowed us to perform an overall analysis of the quality of the analytical results. 36 CHAPTER 5 CTMC MODELS OF THE 1R/2P SCN CONFIGURATION This chapter presents the CTMC models of the 1R/2P SCN configuration under three levels of inventory information sharing as defined in Section 3.1. Section 5.1 presents the description of the SCN structure used in the study. The state definition used in the CTMC models is presented in Section 5.2. Sections 5.3, 5.4 and 5.5 present the CTMC models for the SCN with high (HiVis), medium (MedVis) and low (LoVis) levels of information sharing, respectively. Section 5.6 presents the CTMC model for the SCN with no information sharing (NoVis). Section 5.7 presents the results of the study on the benefits of inventory information sharing on the SCN performance. Finally, some concluding remarks about the CTMC models are presented in Section 5.8. For the sake of simplicity, we will refer to inventory information sharing as simply information sharing. 5.1 1R/2P SCN Structure We consider a twoechelon SCN with one retail store and two production facilities each with its own output store to stock finished products. Each store in the SCN is assumed to operate under a basestock control policy with oneforone replenishment. Both the production facilities are assumed to have the same basestock level (S). The basestock level at the retail store is R. Each customer order is assumed to be for a single unit of the finished product. The demand interarrival times at the retail store as well as the processing times at the production facilities follow general distributions, but only the exponential case is considered in this chapter. The arrival 37 of a demand consumes a finished product at the retail store if available and causes an order for replenishment to be placed at one of the two production facilities (see Figure 5.1). If available, a finished product from the output store of the production facility is instantaneously sent to the retail store and the output store sends an order for replenishment to its processing stage. As noted in Chapter 1, extensions to include transit delays will be presented in Chapter 9. We assume that the processing stage has a single server. An order can join the WIP queue or be processed as soon as it is received. There are no limits on the number of backorders at either production facility. It has to be noted that when the basestock levels at the two production facilities are the same, the HiVis routing policy closely resembles the shortest queue problem studied in the literature (e.g. [33], [26], and [2]). Production Facility 1 Retail Store Production Facility 2 Output Store 2 Output Store 1 Customer Demand Figure 5.1: 1R/2P SCN Configuration 5.2 CTMC Model for Poisson Arrivals and Exponential Processing Times This section presents the statespace definition for the CTMC model used to study the SCN with or without information sharing. This section also provides the details on obtaining the performance measures for the SCNs with information sharing (LoVis, 38 MedVis and HiVis). The following assumptions were made in developing the CTMC models. 1. The demand/order arrival process is Poisson with rate λ. 2. The processing times at both the production facilities follow an exponential distribution with rate μ (the two production facilities are identical). 3. The transportation time from the production facility to the retail store is not modeled (relaxed later). Ignoring the transportation delay between the production facilities and the retail store means that there is no need to consider the number of orders at the retail store while defining the state of the SCN (see explanation below). As a result, the state of the SCN at time t is defined by X(t) = {i, j}, where i and j are nonnegative integers representing the number of orders at production facilities 1 and 2, respectively. The details about the net inventory levels at the various stores in the SCN can be obtained from the state definition as follows. When the number of orders at a production facility is less than the basestock level, the output store at the production facility will have inventory and the inventory level will be given by (S  x), where x = i for production facility 1 and x = j for production facility 2. Similarly, when the number of orders at a production facility exceeds the basestock level, the output store is backordered. The number of backorders at the production facility will be given by (x  S). The net inventory level at a production facility will be given by (S  x). This value will be positive in the presence of inventory and negative in the presence of backorders. As the transportation delay is ignored, when the number of orders at each production facility does not exceed the basestock level of its output store (i.e. 0 ≤ i, j ≤ S), the inventory level at the retail store will be equal to its basestock level (R). When 39 at least one of the production facilities is backordered, the inventory level at the retail store will be equal to its basestock level less the sum of the backorders at the production facilities and the number of orders (backorders + replenishment orders) at the retail store is given by (i − S)+ + (j − S)+. The retail store will be backordered only when the sum of the backorders at the two production facilities exceeds the basestock level at the retail store (i.e., (i − S)+ + (j − S)+ > R). The net inventory level at the retail store will be [R − (i − S)+ − (j − S)+]. To better understand the above discussion, let us consider a case with R = S = 3. If the number of orders at production facility 1 is 1, and the number of orders at production facility 2 is 1, then the current state of the SCN is (1, 1). The net inventory level at the two production facilities will be 2 (i.e., 31), indicating that there is inventory at both the production facilities. As a result, the number of orders at the retail store is zero and the net inventory level is 3 (i.e., [3−(1−3)+−(1−3)+]). If the number of orders at production facility 1 is 2, and the number of orders at production facility 2 is 4, then the current state of the SCN is (2, 4). In this case the net inventory level at production facility 1 is 1 (i.e. 32), indicating that there is one unit of inventory at production facility 1. The net inventory level at production facility 2 is 1 (i.e., 34), indicating that there is one backorder at production facility 2. As a result, number of orders (replenishment in this case) at the retail store is 1 (because of the backorder at production facility 2) and the net inventory level is 2 (i.e., [3 − (2 − 3)+ − (4 − 3)+]). If the number of orders at both the production facilities is 5, the current state of the SCN is (5, 5). In this case the net inventory level at both the production facilities is 2 (i.e. 35), indicating that there are two backorders at each production facility. The number of orders at the retail store is 4 (sum of the backorders at the two production facilities) and the net inventory level is 1 (i.e., [3−(5 − 3)+−(5 − 3)+]), indicating one backorder at the retail store. 40 The following notations are used in this chapter. Model Parameters λ Order arrival rate at retail store R, S Basestock levels at retail store and output stores of the production facilities, respectively Model Variables (Upper case letters are random variables) Nr Number of orders at the retail store Np Number of orders at a production facility ρ Utilization of a production facility Ir, Ip Inventory levels at the retail store and output store of a production facility Br, Bp Number of backorders at the retail store and output store of a production facility fr, fp fill rates at the retail store and output store of a production facility Wbr, Wbp time to fulfill a backorder at the retail store and the output store of a production facility Tr, Tp time spent by an order at the retail store and a production facility including its output store The CTMC models for the different levels of information sharing have some common properties that are listed below. 1. The state space of the CTMC is symmetrical as the two production facilities are identical (same basestock level and processing rate). This means that the transition rates into (out of) the states (i, j) and (j, i) are the same except when i = j. This observation allowed us to combine the states (i, j) and (j, i) and 41 reduce the state space of the CTMC model. 2. The patterns present in the reduced CTMC models for the SCNs with information sharing (Figures 5.3, 5.5, and 5.7) could not be exploited to yield a closed form solution. As a result, the CTMC models were solved by limiting the maximum number of backorders at the retail store to M. 3. Once we numerically solved for πi,j by limiting the maximum number of backorders at the retail store, we computed the key performance measures for a production facility as follows. The fill rate at a production facility is simply the probability that an order from the retail store is satisfied immediately as shown in equation (5.1). fp = S−1 i=0 S−1 j=0 πi,j + S+ R+M i=S S−1 j=0 πi,j + S−1 i=0 S+ R+M j=S πi,j (5.1) The rest of the performance measures for a production facility were obtained using expressions (5.2) and (5.5). E[Ip] = S i=0 S+ R+M j=0 (S − i)πi, j (5.2) The expected backorder at a store can be calculated using the fundamental expression relating the expected inventory level, expected number of orders and the basestock level as shown in equation (5.3). E[Bp] = E[Ip] + E[Np] − S = S i=0 S+ R+M j=0 (S − i)πi, j + S+ R+M i=0 S+ R+M j=0 iπi, j − S (5.3) 42 Now, using Little’s law we have E[Wbp] = E[Bp] (1 − fp)λ/2 (5.4) E[Tp] = S+R+M i=0 S+R+M j=0 iπi, j λ/2 + E[Ip] λ/2 (5.5) 4. The arrival process at the retail store is Poisson and as per the PASTA principle [62], the ready rate can be used to calculate the fill rate. Note that an arriving customer order at the retail store would see the number of orders at the two production facilities. The fill rate at the retail store was obtained by calculating the probability that the total number of orders at the two production facilities is less than the sum of their basestock levels and that of the retail store, i.e. i + j < 2S + R. The other performance measures at the retail store can be obtained using expressions (5.7) through (5.11). fr = i, j∈A1 πi, j (5.6) where A1 = {i, j : 0 ≤ i + j ≤ 2S + R − 1} E[Ir] = S−1 i=0 S−1 j=0 Rπi, j + i, j∈A2 R − (i − S)+ − (j − S)+ πi,j (5.7) where A2 = {i, j : 1 ≤ (i − S)+ + (j − S)+ ≤ R} E[Nr] = i, j∈A3 (i − S)+ + (j − S)+ πi,j (5.8) where A3 = {1 ≤ (i − S)+ + (j − S)+ < ∞} 43 E[Br] = E[Ir] + E[Nr] − R (5.9) E[Wbr] = E[Br] λ(1 − fr) (5.10) E[Tr] = E[Ir] λ + E[Nr] λ (5.11) Next, we present the CTMC models for all three levels of information sharing followed by the no information sharing case. Sections 5.3, 5.4 and 5.5 present the CTMC models of the SCN with high (HiVis), medium (MedVis) and low (LoVis) levels of information sharing, respectively. Section 5.6 presents the CTMC model of the SCN with no information sharing (NoVis). 5.3 CTMC Model of 1R/2P SCN with HiVis In the HiVis case, it is assumed that each production facility shares complete information about the number of items in stock as well as the number of backorders. In the presence of such detailed information, a routing policy presented in Table 3.3 can be adopted. Based on the state definition and the routing policy, we obtain the symmetric CTMC model shown in Figure 5.2. The symmetry in Figure 5.2 is exploited to obtain the reduced CTMC presented in Figure 5.3. The rate balance equations based on the different state transitions that are possible in the reduced CTMC model are presented in Table 5.1. 44 /2 μ μ μ /2 μ μ μ μ μ μ μ μ μ 2, S2 1, S1 0, S μ μ μ μ μ μ 1, S2 0, S1 μ μ μ μ 1, S 0, S+1 2, S1 μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ /2 /2 /2 /2 0, 0 1, 0 0, 1 2, 0 1, 1 0, 2 3, 0 2, 1 1, 2 0, 3 S, 0 S1, 1 S2, 2 μ μ μ μ μ μ μ μ 4, 0 3, 1 /2 2, 2 1, 3 0, 4 μ μ μ μ μ μ μ S1, 0 S2, 1 μ μ μ μ S+1, 0 S, 1 S1, 2 μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ /2 /2 /2 /2 S1, S1 S, S1 S1, S S+1, S1 S, S S1, S+1 S+1, S S, S+1 /2 S+1, S+1 μ μ μ μ μ μ Figure 5.2: CTMC Model of the 1R/2P SCN with HiVis 45 2μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ 2μ μ μ μ μ 2μ μ 0, 0 1, 0 2, 0 1, 1 3, 0 2, 1 S, 0 S1, 1 S2, 2 μ μ μ μ μ 4, 0 3, 1 2, 2 μ μ μ μ μ S1, 0 S2, 1 μ μ μ μ S+1, 0 S, 1 S1, 2 μ 2μ μ μ μ 2μ μ S1, S1 S, S1 S+1, S1 S, S S+1, S S+1, S+1 μ Figure 5.3: Reduced CTMC Model of the 1R/2P SCN with HiVis 46 Table 5.1: Rate Balance Equations for the Reduced CTMC Model of the 1R/2P SCN with HiVis State State Transitions Rate Balance Equation i = j = 0 μ i, j i+1, j πi,j = μπi+1,j λ i ≥ 1, j ≥ 1, i = j μ 2μ i, j1 i, j i+1, j πi,j = λπi,j−1+μπi+1,j (λ+2μ) i = 1, j = 0 μ 2μ μ i1, j i, j i+1, j i, j+1 πi,j = λπi−1,j+μπi+1,j+2μπi,j+1] (λ+μ) i ≥ 2, j = 0 μ μ μ i1, j i, j i+1, j i, j+1 πi,j = μ[πi+1,j+πi,j+1] (λ+μ) i ≥ 2, j ≥ 1, i = j + 1 μ 2μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λ[πi,j−1+πi−1,j]+μπi+1,j+2μπi,j+1 (λ+2μ) i ≥ 3, j ≥ 1, i = j + 2 μ μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λπi,j−1+μ[πi+1,j+πi,j+1] (λ+2μ) 47 The reduced CTMC model revealed some state transition patterns, but we could not exploit these patterns to arrive at a closedform solution. The different approaches that were explored to obtain a closedform solution included the solution to the system of rate balance equations, differenceequation techniques and the method of generating functions. Hence, we fixed the maximum number of backorders at the retail store to 1,000 and numerically solved the CTMC. By fixing the maximum number of backorders, we make sure that the state space is finite. Further, a careful examination of the CTMC model shows that all the states communicate and that the CTMC model is irreducible. Hence, the CTMC is positive recurrent and has a steady state solution. The CTMC model was validated by comparing its numerical solution to the estimates obtained from an Arena simulation model of the 1R/2P SCN with HiVis, as shown in Tables B.1 and B.2. The numerical experiments show that the analytical results match the simulation results, thus confirming the validity of the CTMC model. 5.4 CTMC Model of the 1R/2P SCN with MedVis In this section, we present the CTMC model of the 1R/2P SCN with medium level of inventory information sharing. It is assumed that the information about the number of backorders at the individual production facilities is available at the retail store. In the presence of the backorder information, the routing policy presented in Table 3.2 can be adopted. The symmetric CTMC model for this case is shown in Figure 5.4 and the reduced CTMC model is presented in Figure 5.5. In Figures 5.4 and 5.5, the transitions shown in blue are common to the HiVis CTMC model and the MedVis CTMC model, and those shown in red are the additional transitions for the MedVis CTMC model. The transition rates marked in red when the transitions are blue indicate that the transitions are common to both the HiVis and MedVis models, but the rates have changed in the case of the MedVis model. The rate balance equations based on the 48 different state transitions that are possible in the reduced CTMC model are presented in Table 5.2. /2 /2 μ /2 /2 /2 /2 /2 /2 /2 /2 /2 μ μ μ μ μ μ μ μ μ 2, S2 1, S1 0, S μ μ μ μ 1, S2 0, S1 μ μ μ μ 2, S1 1, S 0, S+1 /2 /2 μ /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 μ /2 μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ /2 /2 /2 /2 /2 /2 0, 0 1, 0 0, 1 2, 0 1, 1 0, 2 3, 0 2, 1 1, 2 0, 3 S, 0 S1, 1 S2, 2 μ μ μ μ μ μ μ μ 4, 0 3, 1 2, 2 1, 3 0, 4 μ μ μ μ μ μ S1, 0 S2, 1 μ μ μ μ S+1, 0 S, 1 S1, 2 /2 μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ /2 /2 /2 /2 S1, S1 S, S1 S1, S S+1, S1 S, S S1, S+1 S+1, S S, S+1 /2 S+1, S+1 μ μ /2 /2 μ μ μ μ μ μ Figure 5.4: CTMC Model of the 1R/2P SCN with MedVis 49 2μ μ /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 μ μ μ μ μ μ μ μ μ μ μ μ μ 2μ μ μ μ μ 2μ μ /2 0, 0 1, 0 2, 0 1, 1 3, 0 2, 1 S, 0 S1, 1 S2, 2 μ μ μ μ μ 4, 0 3, 1 2, 2 μ μ μ μ S1, 0 S2, 1 μ μ μ μ S+1, 0 S, 1 S1, 2 /2 μ 2μ μ μ μ 2μ μ S1, S1 S, S1 S+1, S1 S, S S+1, S S+1, S+1 μ /2 /2 μ μ Figure 5.5: Reduced CTMC Model of the 1R/2P SCN with MedVis 50 Table 5.2: Rate Balance Equations of the Reduced CTMC Model of the 1R/2P SCN with MedVis State State Transitions Rate Balance Equation i = j = 0 μ i, j i+1, j πi,j = μπi+1,j λ i = 1, j = 0 /2 /2 μ 2μ /2 μ i1, j i, j 2, 0 1, 1 πi,j = λπi−1,j+μπi+1,j+2μπi,j+1 (λ+μ) 2 ≤ i ≤ S−1, j = 0 /2 /2 μ μ /2 μ i1, j i, j i+1, j i, j+1 πi,j = λ 2 πi−1,j+μ[πi+1,j+πi,j+1] (λ+μ) i = S, j = 0 μ μ /2 μ i1, j i, j i+1, j i, j+1 πS,0 = λ 2 πS−1,0+μ[πS+1,0+πS,1] (λ+μ) S + 1 ≤ i < ∞, j = 0 μ μ μ i1, j i, j i+1, j i, j+1 πi,j = μ[πi,j+1+πi+1,j ] (λ+μ) 1 ≤ i ≤ S −1, i = j /2 μ 2μ i, j1 i, j 2, 1 πi,j = λ 2 πi,j−1+μπi+1,j (λ+2μ) S ≤ i < ∞, i = j μ 2μ i, j1 i, j 2, 1 πi,j = λπi,j−1+μπi+1,j (λ+2μ) 2 ≤ i ≤ S − 1, j = i − 1 /2 /2 μ 2μ μ μ i, j1 /2 i1, j i, j i+1, j i, j+1 πi,j = λ 2 πi,j−1+λπi−1,j+μπi+1,j+2μπi,j+1] (λ+2μ) S ≤ i < ∞, j = i−1 μ 2μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λ[πi,j−1+πi−1,j]+μπi+1,j+2μπi,j+1] (λ+2μ) Continued on next page 51 Table5.2 – continued from previous page State State Transitions Rate Balance Equation 1 ≤ j ≤ S − 3, i ≥ j + 2, μ 2μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λ 2 [πi,j−1+πi−1,j]+μ[πi+1,j+πi,j+1] (λ+2μ) i = S, 1 ≤ j ≤ S−2 μ μ μ μ i, j1 /2 i1, j i, j i+1, j i, j+1 πi,j = λπi,j−1+λ 2 πi−1,j+μ[πi+1,j+πi,j+1] (λ+2μ) S + 1 ≤ i < ∞, 1 ≤ j ≤ S − 1 μ μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λπi,j−1+μ[πi,j+1+πi+1,j ] (λ+2μ) The state transition patterns identified in the reduced CTMC model could not be exploited to obtain a closedform solution. As before, we explored the following approaches to obtain a closedform solution including the solution to the system of rate balance equations, differenceequation techniques and the method of generating functions. Hence, we fixed the maximum number of backorders at the retail store to 1,000 and numerically solved the CTMC. By fixing the maximum number of backorders, we make sure that the state space is finite. Further, a careful examination of the CTMC model shows that all the states communicate. Hence, the CTMC is irreducible and positive recurrent. This ensures that steady state solution exists for the reduced CTMC model. The CTMC model was validated by comparing its numerical solution to estimates from an Arena simulation model of the 1R/2P SCN with MedVis, as shown in Tables B.3 and B.4. The numerical experiments show that the analytical results match the simulation results, thus confirming the validity of the CTMC model. 52 5.5 CTMC Model of the 1R/2P SCN with LoVis In this case, the retail store is assumed to have the minimum amount of inventory information (presence or absence of inventory) from the production facilities. The order routing policy presented in Table 3.1 can be adopted in this case. The symmetric CTMC model for this case is presented in Figure 5.6, while the reduced CTMC model is presented in Figure 5.7. In Figures 5.6 and 5.7, the transitions and rates shown in red indicate that these are common for the MedVis and LoVis models. Also, the transitions shown in blue are common to the CTMC models corresponding to all three levels of information sharing. The additional transitions and rates that are specific to the CTMC model of SCN with LoVis are shown in purple in Figures 5.6 and 5.7. The rates shown in purple when the transitions are blue indicate that the transitions are common for all three levels of information sharing, but the rate has changed for the LoVis case. The rate balance equations based on the different state transitions that are possible in the reduced CTMC model are presented in Table 5.3. 53 /2 μ /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 μ 2 /2 μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ /2 /2 /2 /2 /2 /2 0, 0 1, 0 0, 1 2, 0 1, 1 0, 2 3, 0 2, 1 1, 2 0, 3 S, 0 S1, 1 S2, 2 μ μ μ μ μ μ μ μ 4, 0 3, 1 2, 2 1, 3 0, 4 μ μ μ μ μ μ S1, 0 S2, 1 μ μ μ μ S+1, 0 S, 1 S1, 2 /2 /2 /2 μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ /2 /2 /2 /2 S1, S1 S, S1 S1, S S+1, S1 S, S S1, S+1 S+1, S S, S+1 /2 S+1, S+1 μ μ /2 /2 μ /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 μ μ μ μ μ μ μ μ μ 2, S2 1, S1 0, S μ μ μ μ 1, S2 0, S1 μ μ μ μ 2, S1 1, S 0, S+1 /2 /2 μ μ μ μ μ μ Figure 5.6: CTMC Model of the 1R/2P SCN with LoVis 54 2μ /2 μ /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 μ μ μ μ μ μ μ μ μ μ μ μ μ 2μ μ μ μ μ 2μ μ /2 0, 0 1, 0 2, 0 1, 1 3, 0 2, 1 S, 0 S1, 1 S2, 2 μ μ μ μ μ 4, 0 3, 1 2, 2 μ μ μ μ S1, 0 S2, 1 μ μ μ μ S+1, 0 S, 1 S1, 2 /2 /2 μ 2μ μ μ μ 2μ μ S1, S1 S, S1 S+1, S1 S, S S+1, S S+1, S+1 μ /2 /2 μ μ Figure 5.7: Reduced CTMC of the 1R/2P SCN with LoVis 55 Table 5.3: Rate Balance Equations of the Reduced CTMC Model of the 1R/2P SCN with LoVis State State Transitions Rate Balance Equation i = j = 0 μ i, j i+1, j πi,j = μπi+1,j λ i = 1, j = 0 /2 /2 μ 2μ μ i1, j i, j i+1, j i, j+1 πi,j = λπi−1,j+μπi+1,j+2μπi,j+1] (λ+μ) 2 ≤ i ≤ S−1, j = 0 /2 /2 μ μ μ /2 i1, j i, j i+1, j i, j+1 πi,j = λ 2 πi−1,j+μ[πi,j+1+πi+1,j ] (λ+μ) i = S, j = 0 μ i, j μ μ i1, j i+1, j i, j+1 /2 πi,j = λ 2 πi−1,j+μ[πi,j+1+πi+1,j ] (λ+μ) S + 1 ≤ i < ∞, j = 0 μ μ μ i, j i1, j i+1, j i, j+1 πi,j = μ[πi,j+1+πi+1,j ] (λ+μ) 1 ≤ i < ∞, i = S, i = j μ /2 2μ i, j1 i, j i+1, j πi,j = λ 2 πi,j−1+μπi+1,j ] (λ+2μ) i = j = S μ 2μ i, j1 i, j i+1, j πi,j = λπi,j−1+μπi+1,j ] (λ+2μ) 2 ≤ i < ∞, i = S, i = S+1, j = i−1 /2 /2 μ 2μ μ μ i, j1 /2 i1, j i, j i+1, j i, j+1 πi,j = λ 2 πi,j−1+λπi−1,j+μπi+1,j+2μπi,j+1 (λ+2μ) Continued on next page 56 Table5.3 – continued from previous page State State Transitions Rate Balance Equation i = S, j = S − 1 μ 2μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λ[πi,j−1+πi−1,j]+μπi+1,j+2μπi,j+1 (λ+2μ) i = S + 1, j = S /2 /2 μ 2μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λ[πi,j−1+πi−1,j]+μπi+1,j+2μπi,j+1 (λ+2μ) i ≤ S − 1, 1 ≤ j ≤ S − 3, i ≥ j + 2, and S+ 1 ≤ j < ∞, i ≥ j + 2 /2 /2 μ μ μ μ i, j1 /2 /2 i1, j i, j i+1, j i, j+1 πi,j = λ 2 [πi,j−1+πi−1,j]+μ[πi,j+1+πi+1,j ] (λ+2μ) j = S, i ≥ j + 2 /2 /2 μ μ μ μ i, j1 /2 i1, j i, j i+1, j i, j+1 πi,j = λ 2 πi−1,j+λπi,j−1+μ[πi,j+1+πi+1,j ] (λ+2μ) i = S, 1 ≤ j ≤ S−2 μ μ μ μ i, j1 /2 i1, j i, j i+1, j i, j+1 πi,j = λ 2 πi−1,j+λπi,j−1+μ[πi,j+1+πi+1,j ] (λ+2μ) S +1 ≤ i < ∞, 1 ≤ j ≤ S −1, i ≥ j +2 μ μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λπi,j−1+μ[πi,j+1+πi+1,j ] (λ+2μ) As before, the state transition patterns identified in the reduced CTMC model could not be exploited to obtain a closedform solution. We fixed the maximum number of backorders at the retail store to 1,000 and numerically solved the CTMC. All states in the CTMC model communicate and the CTMC model is positive recurrent, thus confirming the existence of steady state. The CTMC model was validated by comparing its numerical solution to estimates 57 from an Arena simulation model of the 1R/2P SCN with MedVis, as shown in Tables B.5 and B.6. The numerical experiments show that the analytical results match the simulation results, thus confirming the validity of the CTMC model. 5.6 CTMC Model of the 1R/2P SCN with NoVis In this case, we assumed that the retail store places an order with production facility 1 with a fixed probability (p). Because of symmetry, p is 0.5. The corresponding CTMC model is shown in Figure 5.8. The rate balance equations based on the different state transitions that are possible are presented in Table 5.4. /2 /2 μ μ μ μ /2 μ μ μ μ μ μ μ μ /2 /2 /2 /2 /2 /2 /2 /2 /2 0, 0 1, 0 0, 1 2, 0 1, 1 0, 2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ /2 /2 /2 /2 /2 /2 /2 S1, 0 S2, 1 S3, 2 S, 0 S1, 1 S2, 2 S+1, 0 S, 1 S1, 2 μ μ μ μ μ μ /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ /2 /2 /2 /2 /2 /2 /2 2, S3 1, S2 0, S1 2, S2 1, S1 0, S 2, S1 1, S 0, S+1 μ μ μ μ μ μ Figure 5.8: CTMC Model of the 1R/2P SCN with NoVis (p = 0.5) 58 Table 5.4: Rate Balance Equations of the CTMC Model of the 1R/2P SCN with NoVis State State Transitions Rate Balance Equation i = j = 0 μ μ /2 /2 i, j i+1, j i, j+1 πi,j = μ[πi+1,j+πi,j+1] λ i ≥ 1, j = 0 μ μ /2 μ /2 /2 i1, j i, j i+1, j i, j+1 πi,j = λ 2 πi−1,j+μ[πi+1,j+πi,j+1] (λ+μ) i = 0, j ≥ 1 /2 μ μ μ /2 /2 i, j1 i, j i+1, j i, j+1 πi,j = λ 2 πi,j−1+μ[πi,j+1+πi+1,j ] (λ+μ) i ≥ 1, j ≥ 1 /2 /2 /2 /2 μ μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λ 2 [πi,j−1+πi−1,j]+μ[πi,j+1+πi+1,j ] (λ+2μ) To solve the CTMC model for the NoVis case, we used the following approach. The demand arrival process at the retail store is split into two streams according to fixed probabilities (as per the Bernoulli routing policy), each stream representing the demand arrival process at the two production facilities. Thus, the order arrival process at each production facility is a Poisson process with rate λ/2. Since the server at each production facility has an exponential processingtime distribution with mean μ, each production facility can be modeled as an independent M/M/1 queue. Let πi, j be the steadystate joint probability that the number of orders at production facilities 1 and 2 are i and j, respectively. The joint probability, πi, j , is a product of the marginal probabilities (steady state solution for the M/M/1 queue) and is given 59 by the expression in (5.12). πi, j = πi · πj = (1 − ρ)ρi(1 − ρ)ρj = (1 − ρ)2ρi+j (5.12) Note that ρ is the utilization of a production facility and is given by λ 2μ The above solution satisfies the rate balance equations shown in Table 5.4, thereby confirming the validity of the solution. Hence, the steadystate distribution for the CTMC model of a symmetric 1R/2P SCN with no information sharing has a productform. Using this distribution, the expressions for the performance measures at a production facility as well as the retail store can be derived. Since the production facilities are symmetric, we present the expressions for only one production facility. The fill rate, the expected number of backorders, the expected inventory level, the expected time to fulfill a backorder, and the expected time spent by an order at a production facility were obtained using expressions (5.13) through (5.17). fp = S−1 i=0 ∞ j=0 πi, j = S−1 i=0 ∞ j=0 (1 − ρ)2 ρi+j = (1 − ρ) (1 − ρ) ∞ j=0 ρj S−1 i=0 ρi = 1− ρS (5.13) 60 E[Ip] = S i=0 ∞ j=0 (S − i)πi, j = (1 − ρ)2 S ∞ k=0 ρk + (S − 1) ∞ k=1 ρk + (S − 2) ∞ k=2 ρk + · · · + ∞ k=S−1 ρk = (1 − ρ)2 S 1 − ρ + (S − 1) ρ 1 − ρ + (S − 2) ρ2 1 − ρ + · · · + ρS−1 1 − ρ = S − ρ − ρS + ρS+1 1 − ρ (5.14) E[Bp] = E[Ip] + E[Np] − S = S − ρ − ρS + ρS+1 1 − ρ + λ 2μ − λ − S = ρS+1 1 − ρ (5.15) E[Wbp] = E[Bp] (λ/2) (1 − fp) = ρS+1 (λ/2) (1 − ρ) ρS = ρ (λ/2) (1 − ρ) (5.16) E[Tp] = E[Time in facility] + E[Time in inventory store] = 1 μ − (λ/2) + E[Ip] λ/2 = 1 μ − (λ/2) + 2 S − ρ − ρS + ρS+1 λ (1 − ρ) (5.17) The reasoning presented in Section 5.2 to obtain the number of orders at the retail store was used to obtain the performance measures at the retail store. The fill rate, 61 the expected inventory level, the expected number of backorders, the expected time to fulfill a backorder, and the expected time spent by an order at the retail store can be obtained by using expressions (5.18) through (5.23). fr = i, j∈A1 πi, j = (1 − ρ)2 1 + 2ρ + · · · + (S + 1) ρS + SρS+1 + (S − 1) ρS+2 + · · · + ρ2S + (1 − ρ)2 S k=1 2ρS+k + S l=1 2ρS+l+1 + · · · + S m=1 2ρS+R+m−2 + (1 − ρ)2 2ρ2S+1 + 3ρ2S+2 + · · · + Rρ2S+R−1 = (1 − ρ)2 S x1=0 (1 + x1) ρx1 + S x2=1
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Title  Modeling Inventory Information Visibility in Supply Chain Networks 
Date  20120501 
Author  Srivathsan, Sandeep 
Keywords  basestock policy, Continuous Time Markov Chain (CTMC), inventory visibility, queueing model, supply chain network 
Department  Industrial Engineering & Management 
Document Type  
Full Text Type  Open Access 
Abstract  The stockout rate at a store has become a serious concern for firms in a supply chain network (SCN). This situation can lead to loss in customer good will and market share. Information sharing among the SCN partners is seen as a strategy to address this problem and we studied inventory visibility or the sharing of inventory information among the supply chain constituents. This research focused on developing performance evaluation models of two "buildingblock" type SCN configurations (one retail store and two production facilities, and two retail stores and two production facilities) under three levels of inventory visibility (LoVis, MedVis and HiVis). Our general approach was to first model the SCN under Poisson arrivals and exponential processing times. For this case, we initially used Continuous Time Markov Chain models. If this model did not yield a closedform solution, then we developed approximate queueing models. We then extended these models to general interarrival and processing time distributions. We also studied the value of information sharing by comparing the results for the SCN configurations with and without information sharing. In our research we have developed analytical models for two SCN configurations under three levels of information sharing. We solved the CTMC models numerically by fixing the maximum number of backorders. An M/M/2 based approximate queueing model for the SCN with HiVis was found to serve as a good approximation and provided useful bounds for the LoVis and MedVis cases. We then obtained correction factors based on analytical inferences and empirical observations to develop a modified M/M/2 model, which improved the performance prediction capability of the M/M/2 based model for the SCN with LoVis and MedVis. Our study on the value of information sharing showed that there is significance at lower basestock levels and higher utilizations at the production facilities. The value was also found to be more sensitive to processing time variability than to interarrival time variability. We also developed analytical models for configurations with asymmetric production facilities under Poisson arrivals and exponential processing times. As an extension, we considered transit delays in the analytical models developed. Extensive numerical experiments indicated that the analytical models developed in our study yield accurate results (e.g., 90% of the results within a 15% error range) over a wide range of SCN parameter values when compared with simulation estimates. 
Note  Dissertation 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  MODELING INVENTORY INFORMATION VISIBILITY IN SUPPLY CHAIN NETWORKS By SANDEEP SRIVATHSAN Bachelor of Engineering University of Madras Chennai, Tamil Nadu, India May 2002 Master of Science Oklahoma State University Stillwater, Oklahoma, USA December 2004 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY May, 2012 COPYRIGHT c By SANDEEP SRIVATHSAN May, 2012 MODELING INVENTORY INFORMATION VISIBILITY IN SUPPLY CHAIN NETWORKS Dissertation Approved: Dr. Manjunath Kamath Dissertation advisor Dr. Ramesh Sharda Dr. Ricki Ingalls Dr. Tieming Liu Dr. Balabhaskar Balasundaram Dr. Sheryl A. Tucker Dean of the Graduate College iii TABLE OF CONTENTS Chapter Page 1 INTRODUCTION 1 1.1 Information Sharing in Supply Chains . . . . . . . . . . . . . . . . . 4 1.2 Performance Evaluation and Performance Optimization Models . . . 5 1.3 Motivation for the Proposed Research . . . . . . . . . . . . . . . . . . 6 1.3.1 ProblemStatement . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Outline of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . 7 2 LITERATURE REVIEW 9 2.1 Literature on Value of Information Sharing . . . . . . . . . . . . . . . 9 2.2 Modeling ProductionInventory and Supply Chain Networks . . . . . 14 2.2.1 Modeling ProductionInventory Networks . . . . . . . . . . . . 14 2.2.2 Modeling Supply Chain Networks and its Constituents . . . . 20 2.3 Summary of the Literature Review . . . . . . . . . . . . . . . . . . . 23 3 RESEARCH STATEMENT 25 3.1 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Research Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Research Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4 RESEARCH APPROACH 30 4.1 ResearchMethodology . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.1.1 Markov Chain Approach . . . . . . . . . . . . . . . . . . . . . 31 4.1.2 Parametric Decomposition Approach . . . . . . . . . . . . . . 32 iv 4.2 PerformanceMeasures . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3 Numerical Validation Procedure . . . . . . . . . . . . . . . . . . . . . 36 5 CTMC MODELS OF THE 1R/2P SCN CONFIGURATION 37 5.1 1R/2P SCN Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.2 CTMC Model for Poisson Arrivals and Exponential Processing Times 38 5.3 CTMCModel of 1R/2P SCN with HiVis . . . . . . . . . . . . . . . . 44 5.4 CTMCModel of the 1R/2P SCN withMedVis . . . . . . . . . . . . . 48 5.5 CTMCModel of the 1R/2P SCN with LoVis . . . . . . . . . . . . . . 53 5.6 CTMC Model of the 1R/2P SCN with NoVis . . . . . . . . . . . . . 58 5.7 Value of Information Sharing . . . . . . . . . . . . . . . . . . . . . . . 63 5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6 QUEUEING MODELS OF THE 1R/2P SCN CONFIGURATION WITH POISSON ARRIVALS AND EXPONENTIAL PROCESSING TIMES 71 6.1 Queueing Model of the 1R/2P SCN Configuration with HiVis . . . . 71 6.1.1 Validation of the M/M/2 Approximation of the 1R/2P SCN Configuration with HiVis . . . . . . . . . . . . . . . . . . . . . 76 6.1.2 Queueing Model of SCN with Lower Levels of Information Sharing 79 6.2 An Asymmetric 1R/2P SCN Configuration . . . . . . . . . . . . . . . 92 6.2.1 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7 QUEUEING MODELS OF THE 1R/2P SCN CONFIGURATION WITH GENERAL INTERARRIVAL AND PROCESSING TIME DISTRIBUTIONS 105 7.1 Queueing Model of the General 1R/2P SCN Configuration with HiVis 105 7.1.1 Validation of the GI/G/2 Approximation . . . . . . . . . . . . 107 v 7.2 Queueing Model of a General 1R/2P SCN Configuration with NoVis . 108 7.2.1 QueueingModel . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.3 Effect of Interarrival Time and Processing Time SCVs on the Value of Information Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 8 ANALYTICALMODELS OF THE 2R/2P SCN CONFIGURATION122 8.1 2R/2P SCN Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.2 Analytical Model for Poisson Arrivals and Exponential Processing Times123 8.2.1 Queueing Model of the 2R/2P SCN Configuration with HiVis 124 8.2.2 Validation of the M/M/2 Approximation . . . . . . . . . . . . 129 8.3 Queueing Model of the 2R/2P SCN Configuration with Lower Levels of Information Sharing (MedVis and LoVis) . . . . . . . . . . . . . . 135 8.4 Analytical Model for General InterArrival and Processing Time Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 8.4.1 Validation of the GI/G/2 Approximation . . . . . . . . . . . . 143 8.4.2 Insights fromthe AnalyticalModel . . . . . . . . . . . . . . . 144 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 9 MODELING TRANSIT TIMES IN SCNs WITH INVENTORY INFORMATION SHARING 151 9.1 Analytical Model of 1R/2P SCN Configuration with Transit Delay . . 151 9.1.1 Effect of Transit Delay on the Performance of the Retail Stores in the 1R/2P Configuration . . . . . . . . . . . . . . . . . . . 154 9.2 Analytical Model of the 2R/2P SCN Configuration with Transit Delays 156 9.2.1 Effect of Transit Delays on the Performance of the Retail Stores in the 2R/2P Configuration . . . . . . . . . . . . . . . . . . . 159 9.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 vi 10 CONCLUSIONS 165 10.1 Summary of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 10.2 Research Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 167 10.3 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 BIBLIOGRAPHY 171 A DETERMINATION OF THE PARAMETERS FOR SIMULATION EXPERIMENTS 179 A.1 Determination of the Warmup Period using Welch’s Method . . . . . 180 A.2 Determination of the Run Length . . . . . . . . . . . . . . . . . . . . 181 B CTMC MODEL RESULTS FOR THE 1R/2P SCN CONFIGURATION 182 B.1 Validation of the CTMC Model of 1R/2P SCN Configuration with HiVis182 B.2 Validation of the CTMC Model of 1R/2P SCN Configuration with MedVis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 B.3 Validation of the CTMC Model of 1R/2P SCN Configuration with LoVis185 B.4 CTMC Model Results for 1R/2P SCN Configuration with NoVis . . . 187 C QUEUEING RESULTS FOR THE 1R/2P SCN CONFIGURATION WITH POISSON ARRIVALS AND EXPONENTIAL PROCESSING TIMES 188 C.1 Validation of the M/M/2 Approximation . . . . . . . . . . . . . . . . 188 C.2 Validation of the Modified M/M/2 Model . . . . . . . . . . . . . . . . 191 C.3 Validation of the M/G/2 based model for the 1R/2P SCN with Heterogeneous Production Facilities . . . . . . . . . . . . . . . . . . . . . 194 D QUEUEING RESULTS FOR THE SCN CONFIGURATION 1R/2P: GENERAL INTERARRIVAL AND PROCESSING TIME DISvii TRIBUTIONS 199 E QUEUEING RESULTS FOR THE 2R/2P SCN CONFIGURATION218 E.1 Results for the Validation of the M/M/2 based Queueing Model for the 2R/2P SCN Configuration . . . . . . . . . . . . . . . . . . . . . . 218 E.2 Results for the Validation of the Modified M/M/2 Model for the 2R/2P SCN Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 E.3 Results for the Validation of the GI/G/2 based Model for the 2R/2P SCN Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 F QUEUEING RESULTS FOR SCN CONFIGURATIONS WITH TRANSIT TIME 274 F.1 Validation of the Analytical Model for the 1R/2P SCN Configuration with Transit Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 F.2 Validation of the Analytical Model for the 2R/2P SCN Configuration with Transit Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 viii LIST OF TABLES Table Page 3.1 Order Routing Policy for SCN with Low Level of Inventory Information Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Order Routing Policy for SCN with Medium Level of Inventory Information Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 Order Routing Policy for SCN with High Level of Inventory Information Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.1 Rate Balance Equations for the Reduced CTMC Model of the 1R/2P SCN with HiVis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Rate Balance Equations of the Reduced CTMC Model of the 1R/2P SCN withMedVis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.3 Rate Balance Equations of the Reduced CTMC Model of the 1R/2P SCN with LoVis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.4 Rate Balance Equations of the CTMC Model of the 1R/2P SCN with NoVis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.1 Experiments for the 1R/2P SCN Configuration under Poisson Arrivals and Exponential Processing Times . . . . . . . . . . . . . . . . . . . 77 6.2 Dependence of π0, 0 on the Basestock Level and Production Facility Utilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.3 Experiments for the Asymmetric 1R/2P SCN Configuration . . . . . 98 6.4 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 ix 7.1 1R/2P SCN Experiments for General Interarrival and Processing Time Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 7.2 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 8.1 Experiments for the 2R/2P SCN Configuration under Poisson Arrivals and Exponential Processing Times . . . . . . . . . . . . . . . . . . . 129 8.2 Experiments for 2R/2P SCN configuration under General Interarrival and Processing Time Distributions . . . . . . . . . . . . . . . . . . . 143 8.3 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 9.1 Experiments for 1R/2P SCN Configuration with Transit Delay . . . . 153 9.2 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 9.3 Experiments for the 2R/2P SCN Configuration with Transit Delay . . 159 9.4 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 B.1 Validation of CTMC Model of SCN with HiVis (Production Facility) 183 B.2 Validation of CTMC Model of SCN with HiVis (Retail Store) . . . . 183 B.3 Validation of CTMC Model of SCN with MedVis (Production Facility) 184 B.4 Validation of CTMC Model of SCN with MedVis (Retail Store) . . . 185 B.5 Validation of CTMC Model of SCN with LoVis (Production Facility) 186 B.6 Validation of CTMC Model of SCN with LoVis (Retail Store) . . . . 186 B.7 Exact Results for the SCN with NoVis (Production Facility) . . . . . 187 B.8 Exact Results for the SCN with NoVis (Retail Store) . . . . . . . . . 187 C.1 Fill Rate and Expected Number of Backorders at the Retail Store . . 189 C.2 Expected Inventory Level and Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 C.3 Fill Rate and Expected Number of Backorders at a Production Facility 190 x C.4 Expected Inventory Level and Expected Time to Fulfill a Backorder at a Production Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 C.5 Expected Time Spent by an Order at the Retail Store and a Production Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 C.6 Fill Rate and Expected Number of Backorders at the Retail Store . . 192 C.7 Expected Inventory Level and Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 C.8 Fill Rate and Expected Number of Backorders at a Production Facility 193 C.9 Expected Inventory Level and Expected Time to Fulfill a Backorder at a Production Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 C.10 Expected Time Spent by an Order at the Retail Store and a Production Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 C.11 PerformanceMeasures at the Retail Store . . . . . . . . . . . . . . . 195 C.12 Fill Rate and Expected Number of Backorders at the Production Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 C.13 Expected Inventory Level and Expected Time to Fulfill a Backorder at the Production Facilities . . . . . . . . . . . . . . . . . . . . . . . . . 197 C.14 Expected Time Spent by an Order at the Production Facilities and Utilizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 D.1 Fill Rate & Expected Number of Backorders at the Retail Store . . . 199 D.2 Expected Inventory Level & Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 D.3 Fill Rate & Expected Number of Backorders at a Production Facility 207 D.4 Expected Inventory Level & Expected Time to Fulfill a Backorder at a Production Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 D.5 Expected Time Spent by an Order at the Retail Store and a Production Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 xi E.1 Fill Rate & Expected Inventory Level at Retail Store 1 . . . . . . . . 219 E.2 Expected Number of Backorders & Expected Time to Fulfill a Backorder at Retail Store 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 E.3 Fill Rate & Expected Inventory Level at Retail Store 2 . . . . . . . . 219 E.4 Expected Number of Backorders & Expected Time to Fulfill a Backorder at Retail Store 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 E.5 Expected Time Spent by an Order at Retail Stores 1 and 2 . . . . . . 220 E.6 Fill Rate & Expected Inventory Level at a Production Facility . . . . 220 E.7 Expected Number of Backorders & Expected Time to Fulfill a Backorder at a Production Facility . . . . . . . . . . . . . . . . . . . . . . . 221 E.8 Expected Time Spent by an Order at a Production Facility . . . . . . 221 E.9 Fill Rate & Expected Inventory Level at Retail Store 1 . . . . . . . . 222 E.10 Expected Number of Backorders & Expected Time to Fulfill a Backorder at Retail Store 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 E.11 Fill Rate & Expected Inventory Level at Retail Store 2 . . . . . . . . 223 E.12 Expected Number of Backorders & Expected Time to Fulfill a Backorder at Retail Store 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 E.13 Expected Time Spent by an Order at Retail Stores 1 and 2 . . . . . . 224 E.14 Fill Rate & Expected Inventory Level at a Production Facility . . . . 224 E.15 Expected Number of Backorders & Expected Time to Fulfill a Backorder at a Production Facility . . . . . . . . . . . . . . . . . . . . . . . 224 E.16 Expected Time Spent by an Order at a Production Facility . . . . . . 224 E.17 Fill Rate and Expected Inventory Level at Retail Store 1 (λr 1, λr 2) = (1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 E.18 Fill Rate and Expected Inventory Level at Retail Store 1 (λr 1, λr 2) = (2, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 xii E.19 Expected Number of Backorders and Expected Time to Fulfill a Backorder at Retail Store 1 (λr 1, λr 2) = (1, 1) . . . . . . . . . . . . . . . . 233 E.20 Expected Number of Backorders and Expected Time to Fulfill a Backorder at Retail Store 1 (λr 1, λr 2) = (2, 1) . . . . . . . . . . . . . . . . 237 E.21 Fill Rate and Expected Inventory Level at Retail Store 2 (λr 1, λr 2) = (1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 E.22 Fill Rate and Expected Inventory Level at Retail Store 2 (λr 1, λr 2) = (2, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 E.23 Expected Number of Backorders and Expected Time to Fulfill a Backorder at Retail Store 2 (λr 1, λr 2) = (1, 1) . . . . . . . . . . . . . . . . 249 E.24 Expected Number of Backorders and Expected Time to Fulfill a Backorder at Retail Store 2 (λr 1, λr 2) = (2, 1) . . . . . . . . . . . . . . . . 252 E.25 Expected Time Spent by an Order at Retail Stores 1 and 2 (λr 1, λr 2) = (1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 E.26 Expected Time Spent by an Order at Retail Stores 1 and 2 (λr 1, λr 2) = (2, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 E.27 Fill Rate and Expected Inventory Level at a Production Facility (λr 1, λr 2) = (1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 E.28 Fill Rate and Expected Inventory Level at a Production Facility (λr 1, λr 2) = (2, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 E.29 Expected Number of Backorders and Expected Time to Fulfill a Backorder at a Production Facility (λr 1, λr 2) = (1, 1) . . . . . . . . . . . . 268 E.30 Expected Number of Backorders and Expected Time to Fulfill a Backorder at a Production Facility (λr 1, λr 2) = (2, 1) . . . . . . . . . . . . 270 E.31 Expected Time Spent by an Order at a Production Facility (λr 1, λr 2) = (1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 xiii E.32 Expected Time Spent by an Order at a Production Facility (λr 1, λr 2) = (2, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 F.1 Performance Measures at the Retail Store (Basestock settings  (2, 4) and (4, 2)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 F.2 Performance Measures at the Retail Store (Basestock settings  (3, 6) and (6, 3)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 F.3 Performance Measures at a Production Facility (Basestock settings  (2, 4) and (4, 2)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 F.4 Performance Measures at a Production Facility (Basestock settings  (3, 6) and (6, 3)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 F.5 Performance Measures at Retail Store 1 (Basestock setting  (3, 6, 3)) 279 F.6 Performance Measures at Retail Store 1 (Basestock setting  (3, 3, 6)) 280 F.7 Performance Measures at Retail Store 2 (Basestock setting  (3, 6, 3)) 280 F.8 Performance Measures at Retail Store 2 (Basestock setting  (3, 3, 6)) 281 F.9 Performance Measures at a Production Facility (Basestock setting  (3, 6, 3)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 xiv LIST OF FIGURES Figure Page 1.1 Generic Supply Chain Network [54] . . . . . . . . . . . . . . . . . . . 1 5.1 1R/2P SCN Configuration . . . . . . . . . . . . . . . . . . . . . . . . 38 5.2 CTMCModel of the 1R/2P SCN with HiVis . . . . . . . . . . . . . . 45 5.3 Reduced CTMC Model of the 1R/2P SCN with HiVis . . . . . . . . . 46 5.4 CTMCModel of the 1R/2P SCN withMedVis . . . . . . . . . . . . . 49 5.5 Reduced CTMC Model of the 1R/2P SCN with MedVis . . . . . . . 50 5.6 CTMCModel of the 1R/2P SCN with LoVis . . . . . . . . . . . . . . 54 5.7 Reduced CTMC of the 1R/2P SCN with LoVis . . . . . . . . . . . . 55 5.8 CTMC Model of the 1R/2P SCN with NoVis (p = 0.5) . . . . . . . . 58 5.9 Fill Rate at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . 64 5.10 Expected Number of Backorders at the Retail Store . . . . . . . . . . 64 5.11 Expected Inventory Level at the Retail Store . . . . . . . . . . . . . . 65 5.12 Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . 65 5.13 Expected Time Spent by an Order at the Retail Store . . . . . . . . . 66 5.14 Fill Rate at a Production Facility . . . . . . . . . . . . . . . . . . . . 66 5.15 Expected Number of Backorders at a Production Facility . . . . . . . 67 5.16 Expected Inventory Level at a Production Facility . . . . . . . . . . . 67 5.17 Expected Time to Fulfill a Backorder at a Production Facility . . . . 68 5.18 Expected Time Spent by an Order at a Production Facility . . . . . . 68 6.1 Queueing Representation of the 1R/2P SCN Configuration . . . . . . 73 6.2 Queueing Representation of the M/M/2 Approximation . . . . . . . . 73 xv 6.3 Fill Rate at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . 77 6.4 Expected Number of Backorders at the Retail Store . . . . . . . . . . 78 6.5 Expected Inventory Level at the Retail Store . . . . . . . . . . . . . . 78 6.6 Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . 79 6.7 Expected Time Spent by an Order at the Retail Store . . . . . . . . . 79 6.8 Fill Rate at a Production Facility . . . . . . . . . . . . . . . . . . . . 80 6.9 Expected Number of Backorders at a Production Facility . . . . . . . 80 6.10 Expected Inventory Level at a Production Facility . . . . . . . . . . . 81 6.11 Expected Time to Fulfill a Backorder at a Production Facility . . . . 81 6.12 Expected Time Spent by an Order at a Production Facility . . . . . . 82 6.13 ω(ρ, S) Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.14 Fill Rate at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . 87 6.15 Expected Number of Backorders at the Retail Store . . . . . . . . . . 88 6.16 Expected Inventory Level at the Retail Store . . . . . . . . . . . . . . 88 6.17 Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . 89 6.18 Expected Time Spent by an Order at the Retail Store . . . . . . . . . 89 6.19 Fill Rate at a Production Facility . . . . . . . . . . . . . . . . . . . . 90 6.20 Expected Number of Backorders at a Production Facility . . . . . . . 90 6.21 Expected Inventory Level at a Production Facility . . . . . . . . . . . 91 6.22 Expected Time to Fulfill a Backorder at a Production Facility . . . . 91 6.23 Expected Time Spent by an Order at a Production Facility . . . . . . 92 6.24 Fill Rate at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . 99 6.25 Expected Number of Backorders at the Retail Store . . . . . . . . . . 99 6.26 Expected Inventory Level at the Retail Store . . . . . . . . . . . . . . 100 6.27 Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . 100 6.28 Expected Time Spent by an Order at the Retail Store . . . . . . . . . 101 6.29 Utilization at the Production Facilities . . . . . . . . . . . . . . . . . 101 xvi 6.30 Fill Rate at the Production Facilities . . . . . . . . . . . . . . . . . . 102 6.31 Expected Number of Backorders at the Production Facilities . . . . . 102 6.32 Expected Inventory Level at the Production Facilities . . . . . . . . . 103 6.33 Expected Time to Fulfill a Backorder at the Production Facilities . . 103 6.34 Expected Time Spent by an Order at the Production Facilities . . . . 104 7.1 Fill Rate at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . 112 7.2 Expected Number of Backorders at the Retail Store . . . . . . . . . . 113 7.3 Expected Inventory Level at the Retail Store . . . . . . . . . . . . . . 114 7.4 Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . 115 7.5 Expected Time Spent by an Order at the Retail Store . . . . . . . . . 116 7.6 Fill Rate at a Production Facility . . . . . . . . . . . . . . . . . . . . 117 7.7 Expected Number of Backorders at a Production Facility . . . . . . . 118 7.8 Expected Inventory Level at a Production Facility . . . . . . . . . . . 119 7.9 Expected Time to Fulfill a Backorder at a Production Facility . . . . 120 7.10 Expected Time Spent by an Order at a Production Facility . . . . . . 121 8.1 Supply Chain Instance . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.2 Exact Representation of the 2R/2P SCN Structure . . . . . . . . . . 125 8.3 Aggregate Representation of the 2R/2P SCN Structure . . . . . . . . 125 8.4 Fill Rate at Retail Store 1 . . . . . . . . . . . . . . . . . . . . . . . . 130 8.5 Expected Number of Backorders at Retail Store 1 . . . . . . . . . . . 130 8.6 Expected Inventory Level at Retail Store 1 . . . . . . . . . . . . . . . 131 8.7 Expected Time to Fulfill a Backorder at Retail Store 1 . . . . . . . . 131 8.8 Expected Time Spent by an Order at Retail Store 1 . . . . . . . . . . 132 8.9 Fill Rate at Retail Store 2 . . . . . . . . . . . . . . . . . . . . . . . . 132 8.10 Expected Number of Backorders at Retail Store 2 . . . . . . . . . . . 133 8.11 Expected Inventory Level at Retail Store 2 . . . . . . . . . . . . . . . 133 xvii 8.12 Expected Time to Fulfill a Backorder at Retail Store 2 . . . . . . . . 134 8.13 Expected Time Spent by an Order at Retail Store 2 . . . . . . . . . . 134 8.14 Fill Rate at Retail Store 1 . . . . . . . . . . . . . . . . . . . . . . . . 137 8.15 Expected Number of Backorders at Retail Store 1 . . . . . . . . . . . 137 8.16 Expected Inventory Level at Retail Store 1 . . . . . . . . . . . . . . . 138 8.17 Expected Time to Fulfill a Backorder at Retail Store 1 . . . . . . . . 138 8.18 Expected Time Spent by an Order at Retail Store 1 . . . . . . . . . . 139 8.19 Fill Rate at Retail Store 2 . . . . . . . . . . . . . . . . . . . . . . . . 139 8.20 Expected Number of Backorders at Retail Store 2 . . . . . . . . . . . 140 8.21 Expected Inventory Level at Retail Store 2 . . . . . . . . . . . . . . . 140 8.22 Expected Time to Fulfill a Backorder at Retail Store 2 . . . . . . . . 141 8.23 Expected Time Spent by an Order at Retail Store 2 . . . . . . . . . . 141 8.24 Fill Rate at Retail Store 1 . . . . . . . . . . . . . . . . . . . . . . . . 144 8.25 Expected Number of Backorders at Retail Store 1 . . . . . . . . . . . 145 8.26 Expected Inventory Level at Retail Store 1 . . . . . . . . . . . . . . . 145 8.27 Expected Time to Fulfill a Backorder at Retail Store 1 . . . . . . . . 146 8.28 Expected Time Spent by an Order at Retail Store 1 . . . . . . . . . . 146 8.29 Fill Rate at Retail Store 2 . . . . . . . . . . . . . . . . . . . . . . . . 147 8.30 Expected Number of Backorders at Retail Store 2 . . . . . . . . . . . 147 8.31 Expected Inventory Level at Retail Store 2 . . . . . . . . . . . . . . . 148 8.32 Expected Time to Fulfill a Backorder at Retail Store 2 . . . . . . . . 148 8.33 Expected Time Spent by an Order at Retail Store 2 . . . . . . . . . . 149 8.34 Effect of Processing Time SCV on Fill Rate . . . . . . . . . . . . . . 149 8.35 Effect of Interarrival Time SCV on Fill Rate . . . . . . . . . . . . . . 150 9.1 Fill Rate at the Retail Store . . . . . . . . . . . . . . . . . . . . . . . 154 9.2 Expected Number of Backorders at the Retail Store . . . . . . . . . . 155 9.3 Expected Inventory Level at the Retail Store . . . . . . . . . . . . . . 155 xviii 9.4 Expected Time to Fulfill a Backorder at the Retail Store . . . . . . . 156 9.5 Expected Time Spent by an Order at the Retail Store . . . . . . . . . 156 9.6 Fill Rate at Retail Store 1 . . . . . . . . . . . . . . . . . . . . . . . . 160 9.7 Expected Number of Backorders at Retail Store 1 . . . . . . . . . . . 160 9.8 Expected Inventory Level at Retail Store 1 . . . . . . . . . . . . . . . 161 9.9 Expected Time to Fulfill a Backorder at Retail Store 1 . . . . . . . . 161 9.10 Expected Time spent by an Order at Retail Store 1 . . . . . . . . . . 162 9.11 Fill Rate at Retail Store 2 . . . . . . . . . . . . . . . . . . . . . . . . 162 9.12 Expected Number of Backorders at Retail Store 2 . . . . . . . . . . . 163 9.13 Expected Inventory Level at Retail Store 2 . . . . . . . . . . . . . . . 163 9.14 Expected Time to Fulfill a Backorder at Retail Store 2 . . . . . . . . 164 9.15 Expected Time spent by an Order at Retail Store 2 . . . . . . . . . . 164 A.1 Plot of Time in System . . . . . . . . . . . . . . . . . . . . . . . . . . 181 xix CHAPTER 1 INTRODUCTION A supply chain network (SCN) is a network of firms that work together to supply the end products to the customer with a focus on both customer satisfaction and profitability of all firms. The nodes or firms involved in a SCN may be raw material suppliers, production facilities where the raw material is converted into finished products, warehouses that store the finished products, distribution centers that deliver the finished products to the retailers and retailers who satisfy the end customer demand. An example SCN is illustrated in Figure 1.1. Suppliers Production Facilities Distribution Centers Retailers Figure 1.1: Generic Supply Chain Network [54] The globalization and rapid growth of ebusiness has meant that the customers have more choices of “suppliers” for any product that they need. As a result, stockout situation at a store may result in a customer choosing a competitor’s product. A 1 study by Corsten and Gruen [15] focused on a common problem in SCNs, namely, the shelf stockout rate, and its effect on the behavior of the end customer. According to their article, several trade associations and joint tradeindustry bodies have sponsored and released major reports on the stockout rate for fastmoving consumer goods. The average stockout rate at the store for all 40 studies was found to be 8.3%. Corsten and Gruen [15] also looked at consumer reactions to shelf stockouts using 29 studies in 20 different countries. The results show that about 31% of the consumers switch the store, 15% delay their purchase, 19% substitute the product with the same brand, 26% switch the brand, while 9% of the consumers do not purchase the item and contribute to lost sales. As a result, a key focus of the firms in a SCN is to reduce the shelf stockout rate and hence, improve end customer satisfaction. Supply chain visibility among the SCN partners is a key area of research and is seen as a panacea to the stockout problem. Supply chain visibility refers to the real time transparency in the supply chain facilitated through builtin information systems that allow the participants in the SCN to keep track of semifinished/finished goods that may be in store or in transit. The global supply chain benchmark report published by Aberdeen group in 2006 [18] found that 79% of the large enterprises that were surveyed reported that lack of supply chain visibility is a critical concern. 51% of the enterprises that were surveyed identified supply chain visibility among their top three concerns. In addition, more than 77% of the enterprises were prepared to spend heavily to attain supply chain visibility. With the advances in information technology, sharing demand, capacity, and inventory information among the partners in the SCN has become quicker and cheaper [12]. There have been studies (e.g., Graves [24], Zipkin [63], Gavirneni et al. [21]) that have focused on the effects of information sharing. While most of these studies showed that information sharing has benefits, a few (e.g., Graves [24]) argued that there are no added benefits of sharing information with the upstream firms for the situations they modeled. More details are presented 2 in Section 2.1. In this research, our focus is on inventory information sharing or inventory visibility. Inventory visibility allows companies to be informed about their partner’s instock inventory and intransit inventory in order to make their supply chain as effective as possible. In a typical SCN, demand and sales information flows from the downstream firms (e.g., retailers) to the upstream firms (e.g., suppliers), while there is flow of material in the opposite direction. There can also be other information exchanges that can occur between the various firms in the SCN depending on the amount of collaboration between the partners in the SCN. The order status information can flow from the upstream firms to the downstream partners, while there can be twoway communication between firms when it comes to sharing inventory information and production plan information [41]. Several critical decisions need to be taken while designing and operating a SCN. These decisions include facility locations, production capacities, the safety stock to be held at each location, and the basestock level at each location. Performance evaluation tools aid decision makers in making these decisions during the design and operation phases of a SCN. In addition, these models provide insight into the working of the SCN and would help in gaining a better understanding of the dynamics of the system (SCN). This research focused on analytical performance evaluation models of supply chain networks and has developed models that can explicitly capture inventory visibility in order to study the effects of sharing inventory information on the SCN performance. Performance evaluation tools are generally used to measure key performance measures of a system (e.g., average response time, fill rate, expected inventory, and expected backorders in the case of a SCN) for any given set of parameter values (e.g., interarrival and service time parameters, basestock levels, etc.) through the development and solution of analytical and simulation models. Performance evaluation tools 3 aid system designers and operations managers in making some key decisions, while keeping in mind the goals of the company [55]. The analytical performance evaluation tools are typically based on modeling techniques such as Markov chains, stochastic Petri nets, and queueing networks [57]. Simulation models can also be used for performance evaluation, but need more detailed information for modeling and more time for model development and model execution phases. Analytical models yield results more quickly and “are appropriate for rapid and rough cut analysis” [55]. In fact, analytical and simulation models have been used in tandem to analyze and design complex systems. For example, analytical models can be used to reduce a large set of design alternatives, and the remaining few alternatives can be studied in detail using simulation models [55]. The development of performance evaluation and optimization models for supply chain networks is an active area of research (see e.g., Ettl et al. [19], Raghavan and Viswanadham [49], Dong and Chen [17], and Srivathsan [53]). However, very few performance modeling studies have explicitly addressed the issue of modeling inventory visibility in a SCN. The focus of this research will be on developing such performance evaluation models of SCNs. 1.1 Information Sharing in Supply Chains Information sharing is believed to be a key component in the success of a SCN. Research on studying the effects of information sharing have dealt with demand information (e.g., [24], [21]), inventory information (e.g., [63], [20]), and a combination of demand and inventory information (e.g., [13], [12], [45]). Li et al. [41] presented a review of ten different models used to study the value of information sharing in supply chains. Lee and Whang [39] described the inventory, sales, demand forecast, order status, and production schedule information that can be shared and also discussed how and why such information is shared along with industry examples. While most of these studies revealed that there are benefits of information sharing, a few studies 4 including Graves [24] showed that sharing demand information with upstream firms did not add value to their SCN model setup. The studies that consider inventory information sharing mostly used simulation models. Some of the research studies that focused on developing queueing models with inventory visibility include Zipkin [63] and Armony and Plambeck [4]. There is also considerable research that focuses on the effect of information distortion. The bullwhip effect, the phenomenon in which there is an increase in the mean and variability of the demand process as we proceed from the retailer to the supplier stage in a SCN, occurs due to distorted demand information. For example, Lee et al. [38] is a frequently cited study on the bullwhip effect in a SCN. 1.2 Performance Evaluation and Performance Optimization Models The common analytical performance modeling tools that are used in performance evaluation include queueing models, Markov chain models, and stochastic Petri nets. Analytical models based on queueing theory have been developed for performance evaluation of production networks since the early 1950s. Prior to the 1990s, the literature on stochastic models of production networks primarily considered capacity, congestion and reliability issues, and did not explicitly model the presence of planned inventory. Similarly, the early literature on inventory theory did not consider capacity and congestion issues explicitly. Inventory theory models have been developed since the 1910s. The earliest work in this field includes the EOQ model [27] and its various extensions. Some of the recent text books that focus on inventory theory are Bramel and SimchiLevi [10], Zipkin [65], Nahmias [46], and Axsater [5]. Since the 1990s, there has been a growing interest among researchers in developing analytical models of production networks which consider capacity, congestion and planned inventory issues in a unified manner. Recent work on modeling productioninventory networks includes models of singlestage systems developed by Buzacott 5 and Shanthikumar [11] and models of multistage productioninventory networks developed by Lee and Zipkin [40], Sivaramakrishnan [51], Sivaramakrishnan and Kamath [52] and Zipkin [64]. A few research efforts also developed performance optimization models for SCNs. These include the studies of Cohen and Lee [14] and Lee and Billington [37]. More recently, queueing models have been used in tandem with optimization models to take into account the stochastic components in the SCN. For example, Ettl et al. [19] combined optimization under service level constraints along with a queueing model to support strategic level decision making in large SCNs. 1.3 Motivation for the Proposed Research In the development of queueingbased performance evaluation models of productioninventory and supply chain networks, the common assumption about routing of orders from downstream stages to upstream stages is that the routing probabilities are fixed and not dependent on state information such as inventory levels. The routing based on fixed probabilities is also called Bernoulli routing. This routing policy allows the placement of an order at a production facility even when it is facing a stockout situation. Such a routing policy is appropriate for the development of models for cases with no inventory visibility. In such a situation, the order may be placed based on historical information (say, for example, the percentage of orders placed by a downstream firm at an immediate upstream firm). In cases where the downstream firm has visibility of the immediate upstream firm’s inventory, a routing policy in which orders are placed based on item availability is certainly a possibility and perhaps desirable. Similarly, when the retail stores share their inventory information with their immediate upstream “suppliers”, then the latter can choose to satisfy an order from the retail store that has the least net inventory level. These situations pose very interesting modeling scenarios that have not been studied much in the queueing literature. In addition, there is a need to study the value in sharing inventory information among 6 the various firms in the SCN, and also to quantify the value of such information sharing. When there is value in sharing inventory information among firms in the SCN, then there is also a need to find the sensitivity of the benefits to different SCN parameters such as basestock levels, production capacities, and variability in service and arrival processes. 1.3.1 Problem Statement With the advancements in the field of information technology, information sharing among firms within a SCN has become easier. Information sharing to provide inventory visibility has gained the attention of practitioners and academic researchers. There is a need to develop analytical models that can explicitly model inventory visibility in productioninventory and supply chain networks. Such models will not only give us better insight into the value of inventory visibility, but also enable us to develop performance optimization tools to maximize the value of inventory visibility by choosing the right combination of system parameter values. 1.4 Outline of the Dissertation The rest of the document is structured as follows. Chapter 2 presents a review of the literature on information sharing and performance modeling of productioninventory and supply chain networks. Chapter 3 presents the research goals, objectives, and contributions. Chapter 4 presents the research approach. Chapter 5 presents the Markov chain models that were developed for a SCN with one retail store and two production facilities at different levels of inventory information sharing. Chapter 6 presents queueingbased analytical models that were developed for a SCN with one retail store and two production facilities under Poisson arrivals and exponential processing times, while Chapter 7 presents the analytical models for the case of general interarrival and processing time distributions. Chapter 8 presents the analytical 7 models that were developed for a SCN configuration with two retail stores and two production facilities, while Chapter 9 extends the analytical models to include transit times in a SCN with inventory information sharing. Chapter 10 presents the research contributions and highlights the scope for future research. 8 CHAPTER 2 LITERATURE REVIEW This chapter presents a detailed review of the literature on the study of value of information sharing in supply chains and the analytical performance modeling of production networks with planned inventory. Section 2.1 presents a review of the published studies that focused on the value of information sharing in SCNs. Section 2.2 summarizes the literature on modeling productioninventory and supply chain networks. Section 2.3 provides a summary of the literature review and identifies some gaps in the literature, which form the basis for the research conducted. 2.1 Literature on Value of Information Sharing Information sharing is increasingly seen as a contributing factor to the success of a SCN. Research efforts have focused on finding if there is any value in information sharing, in assessing the actual value of information sharing, and in identifying factors influencing the value of information sharing. Studies have considered various types of information that can be shared, namely supplier status, inventory levels, demand forecasts, price, schedule and capacity information [41]. Graves [24] studied a singleitem inventory system with a nonstationary demand process that behaved like a random walk. Exponentialweighted moving average was used to obtain the mean square forecast of the demand and deterministic leadtime was assumed. An adaptive basestock policy where the basestock is adjusted based on changes in demand forecast was proposed. The safety stock required for nonstationary demand was found to be much more than that for stationary demand. 9 The relationship between lead time and safety stock was found to be convex in the case of nonstationary demand indicating that more safety stock is required with increasing lead time. This singleitem model was then extended to a singleitem, multistage system and the upstream demand was found to be nonstationary with the same form as the downstream demand process. So the adaptive basestock policy was applied to the upstream stage and the results showed that the bullwhip effect cannot be mitigated by sharing more information to the upstream stage. Zipkin [63] studied the performance of a multiitem productioninventory system with a production facility under the firstcomefirstserved (FCFS) policy and longestqueue (LQ) policy. The production facility had a finite capacity and ample raw material supply. The processor at the production facility was assumed to be perfectly flexible with no setup costs. All products were assumed to be symmetric (i.e., same demand rates, processing times, etc.). The demands for the products were assumed to be Poisson and the processing times followed a general distribution. Each product was assumed to follow a basestock policy with identical basestock levels. Each demand for a product at a store resulted in the consumption of the product from the store and a resultant replenishment order being placed at the production facility. Backorders were allowed in the system. In the case of the FCFS policy, these replenishment orders waited at the processor’s queue and were processed on a firstcomefirstserved basis. In the case of the LQ policy, the processor at the production facility used the inventory information and serviced the product that had the smallest net inventory level. Preemption was allowed in the case of the LQ policy and ties were resolved randomly. The sum of the standard deviations of number of outstanding orders of type i (σi) over all products (σ = M i=1 σi) was used to study the performance of the system as it captured “the gross behavior of performance over a fairly wide range of systems, and no other measure of comparable simplicity did so” [63]. A closed form expression for σ was developed in the case of FIFO policy. In the case of LQ policy, 10 an approximation for σ was developed. Numerical experiments suggested that the LQ policy performed better than the FIFO policy by about 20% in some cases. The difference in performance was the greatest when the number of product types was large, and at small SCVs. The difference was small at low utilizations, increased till 90% utilization and vanished in heavy traffic. Cachon and Fisher [12] studied the impact of sharing demand and inventory data in a supply chain with one supplier, N identical retailers and a stationary stochastic demand. Costs were associated with holding inventory and backorders. The traditional information policy that did not use shared information was compared against a full information policy that exploited the shared information. For both the models, it was assumed that the supplier’s orders are always received at the retailer after a constant lead time. A stockout at a supplier caused a resulting replenishment delay for the retailer. In the traditional information policy, the retailers and suppliers were assumed to follow (Rr, nQr) and (Rs, nQs) policies, respectively. The supplier was assumed to allocate inventory to each retailer based on a batch priority allocation. As per this allocation, if retailer i ordered b batches, then the first batch was given priority b, the second batch was given priority b1, and so on. All the batches were assumed to be placed in a shipment queue based on decreasing priority order with ties broken randomly. In the full information model, the supplier could improve its order quantity decisions as well as allocation decisions based on the demand and inventory information from the retailer. The decision of allocation of batches could be improved as batches could be allocated based on the inventory position at the retailer in the period the batch is shipped as against the period it was ordered. Based on their comparison, it was found that supply chain costs were 2.2% lower on average with full information policy than with the traditional policy. Gavirneni et al. [21] studied the periodic review inventory control problem in a twoechelon supply chain with a retailer and a supplier. The sequence of events 11 during each period started with the supplier deciding on its production quantity for the period followed by the retailer realizing its customer demand. On satisfying the demand, the retailer placed an order with the supplier if its inventory level fell below the reorder point s. The order was assumed to be satisfied at the beginning of the next period. An order not satisfied by the supplier was assumed to be satisfied from some other supplier with no lead time. The (s, S) inventory policy was assumed to be optimal for the retailer who incurs a fixed plus linear ordering cost, linear holding and backorder costs, while the supplier incurred a linear holding and backorder cost. They studied three different models which differ in the amount of information shared. In the first case, there was no information sharing and the supplier followed a naive approach, assuming that the retailer demand followed an i.i.d. process. In the second model, the supplier knew the number of periods that had elapsed since the last order from the retailer as well as that the retailer was using the (s, S) inventory policy. In the third model, the supplier knew the number of units that had been sold by the retailer since the last order. In all three models, the optimal order upto level was computed via simulationbased optimization using infinitesimal perturbation analysis. It was shown that the third model had the least cost with the first model having the highest cost, showing that full information sharing is beneficial. Gavirneni [20] studied a periodic review inventory control problem in a supply chain with one capacitated supplier who supplies a single product to multiple identical retailers. The customer demands were assumed to be i.i.d. During each period, the sequence of events started with the retailer reviewing its inventory and placing an order with the supplier. The supplier responded by using the available retailer inventory information to satisfy as many demands as possible. The retailer received the supplier shipments and satisfied the customer demand. Costs were associated with holding inventory and penalties were imposed for unsatisfied demand. The author studied this system under three different levels of cooperation. In the first level 12 (no cooperation), the only information available to the supplier was the demand from the retailers. If the total demand was less than the capacity of the supplier, then all retailers’ demands were satisfied. If there were orders from the retailers in excess of the supplier capacity, the supplier was assumed to use the lexicographic allocation scheme. This scheme “ranked the consumers in the order of their importance (independent of their order quantities) and the demands were satisfied in that order” [20]. In the second level of cooperation, the supplier received the current retailer inventory level in addition to the demand. The supplier could allocate its capacity based on this information in such a way that the retailer with a larger inventory level received smaller shipments and the one with smaller inventory level received larger shipments. The third level of cooperation extended the second level model to include the possibility of transfer of inventory from one retailer to another. The research used a simulation model to search for an optimal target inventory level and optimal cost. The computational results showed that the third level of coordination resulted in the least cost and the benefits of cooperation in this supply chain decreased with increase in supplier capacity, increase in number of retailers, decrease in penalty cost, and decrease in demand variance. Armony and Plambeck [4] studied the effect of duplicate orders on a manufacturer’s estimation of demand rate and customer’s sensitivity to delay, and decisions on capacity investment. They considered a SCN where a manufacturer sold its products through two distributors. The customer demand arrival process at each distributor was assumed to be a Poisson process and each demand was assumed to be for a single product. Each distributor followed oneforone replenishment policy. When one of the distributors faced a stockout situation, then an arriving customer had the option to immediately obtain the finished goods from the other distributor, provided the other distributor had inventory. When both the distributors were out of stock, then the customer could place an order with both distributors. When the product was 13 delivered by one of the distributors, then the customer canceled the duplicate order placed at the other distributor. The customers were assumed to be impatient, leading to cancellations of outstanding orders after a waiting time that was exponentially distributed. They assumed that the manufacturer had knowledge of the basestock policy used by the distributors, which would help in the inference about the inventory level and the number of outstanding orders at the distributors. They obtained the maximum likelihood estimators for the demand rate, the reneging rate, and the probability that a customer would place a duplicate order when made to wait. Li et al. [41] presented a review of about 10 different information sharing models. These included the works of Zipkin [63], Gavirneni et al. [21], Gavirneni [20], Graves [24], Moinzadeh [45], Chen [13], Kulp [36], Cachon and Fisher [12], Schouten et al. [50], and Bourland et al. [9]. From the literature review presented here, it can be seen that only Armony and Plambeck [4] and Zipkin [63] considered the development of queueingbased performance evaluation models. 2.2 Modeling ProductionInventory and Supply Chain Networks 2.2.1 Modeling ProductionInventory Networks Performance evaluation models of productioninventory networks consider capacity, variability, and inventory issues in a unified manner. Productioninventory network models are typically used to analyze the performance of maketostock systems, in which planned inventory is maintained for finished products and semifinished products at the intermediate stages. Gavish and Graves ([22], [23]) studied a singlestage, singleproduct maketostock production facility with Poisson demand. They focused on finding a control policy that minimizes the expected cost per unit time using an M/D/1 queueing system [22] as well as an M/G/1 queueing system [23]. The optimal decision policy for both the models was found to be a twocriticalnumber policy characterized by the parameters 14 (Q∗, Q∗∗), “such that the server is turned off when the queue length is first reduced to Q∗, and is turned on when the queue length first reaches Q∗∗” [22]. Graves and Keilson [23] studied a oneproduct, onemachine problem under Poisson demand and exponentially distributed order size. They focused on minimizing the system cost that included setup cost, inventory holding cost, and backorder cost using a spatially constrained Markov process model. The work of Svoronos and Zipkin [56] does not explicitly model capacity issues, but forms the basis for models developed by Zipkin [40] and others. Svoronos and Zipkin [56] modeled a multiechelon inventory system with Poisson demand arrivals at the lowest hierarchy of the system called the leaf. Each demand reached the central depot through instantaneous replenishment orders to preceding stages. The central depot was assumed to have infinite raw material supply from an outside source. Each location was considered to be a node and each arc connecting a node with its predecessor was treated as a transit system (representing transportation or production activities). Parts are assumed to be processed sequentially and the behavior of each transit system was assumed to be independent of the demands and orders in the inventory system. The transit systems for all the arcs on the path from the outside source node to the leaf node were assumed to be mutually independent. Svoronos and Zipkin [56] applied the results of the singlelocation problem recursively, starting from the highestlevel echelon to analyze the complete network. Buzacott and Shanthikumar [11] presented results for general singlestage systems with unit demand and backlogging using the production authorization (PA) card concept, where a tag attached to an item is converted into a PA card when demand consumes the item. Using the number of outstanding orders in the stage at time t, K(t), the total number of tags in the system, S, the inventory level at time t, I(t), and the number of customer orders backordered at time t, B(t), can be obtained using equations (2.1) and (2.2). Since the PA card represents an outstanding order 15 at a stage, the number of PA cards available at time t, C(t), can be obtained using equation (2.3). It can be seen from equations (2.1) through (2.3) that by studying the process K(t), we can derive I(t), B(t) and C(t). I(t) = max[0, S − K(t)] (2.1) B(t) = max[0,K(t) − S] (2.2) C(t) = min[S,K(t)] (2.3) The distribution of the number of outstanding orders in the system (K) is the same as the number of customers in a GI/G/1 queue because of the assumption of infinite raw material supply. The expected inventory level and the expected number of backorders can be obtained using equations (2.4) and (2.5). E[I] = S n=0 n P(I = n) (2.4) E[B] = E[I] + E[K] − S (2.5) Buzacott and Shanthikumar [11] also modeled singlestage systems with lost sales, interrupted demand, bulk demand, machine failures and yield losses. The analytical models for multistage tandem productioninventory networks developed by Lee and Zipkin [40], Sivaramakrishnan [51], Liu et al [43], and Srivathsan [53] have many common assumptions, such as oneforone replenishment with backordering permitted at each stage, no limit on WIP queues at a stage and ample raw material supply at the first stage [30]. The approaches developed by Sivaramakrishnan [51], Liu et al. [43], and Srivathsan [53] are based on the parametric decomposition approach ([59], [61]) and share a common solution structure as explained in Kamath and Srivathsan [30]. 16 Sivaramakrishnan and Kamath [52] modeled an Mstage tandem maketostock system by decomposing it into individual stages where each stage is made up of the manufacturing resource at that stage and a delay node to “capture the upstream delay experienced by an order when there was no part in the output store of the previous stage” [51]. Sivaramakrishnan [51] modeled the manufacturing unit as a GI/G/1 queue and the delay node as an M/G/∞ queue. Sivaramakrishnan [51] extended the tandem model to include (i) multiple servers at a stage, (ii) batch service, (iii) limited supply of raw material, (iv) multiple part types, and (v) service interruptions due to machine failure. Sivaramakrishnan [51] also extended his approach to tandem networks with feedback and feedforward networks. Liu et al. [43] modeled a multistage tandem manufacturing/supply network with general interarrival and service processes. The tandem network was modeled as a system of inventory queues and then the overall inventory in the network was optimized with servicelevel constraints. A decomposition scheme was proposed, in which a semifinished product from a stage is moved into the input buffer of the downstream stage, whenever the job queue (material queue at the input buffer of node i + backorder queue at node i) is increased by one. If the inventory store at stage i1 is empty, then the request for the semifinished part is backordered. As a result, the job queue or outstanding orders queue at stage i consists of the material queue comprising of the semifinished products from the upstream stage and a backorder queue. In this approach, the backorder distribution is used to account for any delay due to stockout at stage i1. Lee and Zipkin [40] modeled a tandem productioninventory network with Poisson demand arrivals, and exponential service times at all stages. The multiechelon system developed by Svoronos and Zipkin [56] was used to analyze the tandem system by assuming that stage i behaved like an M/M/1 queue [40] and the sojourn times was exponentially distributed with mean 1/(μi  λ), where λ is the demand rate, 17 and μi is the processing rate at stage i. The sojourn times at all the stages were considered to be independent. Such an effective lead time decomposition approach is similar to Whitt’s parametric decomposition approach [59]. The approach developed by Svoronos and Zipkin [56] that employed phase type approximations (for more details, refer to Theorem 2.2.8 in Neuts [47]) was then used to solve the above approximate system. Zipkin [64] extended this work to model tandem productioninventory networks with feedback. For a detailed comparison of the models developed by Lee and Zipkin [40], Sivaramakrishnan [51], Liu et al. [43] and Srivathsan [53], the reader is referred to Kamath and Srivathsan [30], wherein the similarities and differences between the approaches are clearly documented along with the results of an extensive numerical study to gain better insight into the performance prediction capability of the different approximations. Nguyen [48] analyzed the problem of setting the basestock levels in a production system that produced both maketoorder and maketostock products with lost sales. Nguyen [48] derived the productform steadystate distribution for the above network under the assumptions that each station operated under a FIFO service discipline, all processing times and interdemand times were exponential, and all products had the same mean processing time. Nguyen [48] proposed approximations for the basestock levels based on heavy traffic analysis of queueing networks. Karaesmen et al. [31] assumed that the interarrival times and the processing times are geometrically distributed and modeled the system with advance order information for contract suppliers. They analyzed the basestock policy (S, L) with a focus on optimization and performance evaluation of the Geo/Geo/1 maketostock queue, where L is the release leadtime, which was used to “regulate the timing of material release into the manufacturing stage” [31]. Benjaafar et al. [8] examined the effect of product variety on the inventory cost 18 in a finite capacity maketostock productioninventory network where the products were manufactured in bulk and shared a common manufacturing facility. The production facility was assumed to comprise of a batching stage and a processing stage. The production stage was viewed as a GI/G/1 queue whose arrival process was the superposition of the order arrivals for the individual products. The key findings of the study were that the total cost increased linearly with the number of products, and that the rate of increase depended on the system parameters such as demand and processing time variability and capacity levels. Benjaafar et al. [7] studied the effect of inventory pooling in productioninventory networks with n locations having identical costs. The production facility was assumed to have finite capacity and supply lead times were assumed to be endogenous. They studied the sensitivity of the cost benefits from inventory pooling to system parameters such as service levels, demand and service time variability, and structure of the productioninventory network. The different structures of the network that were studied include (i) single production facility with Poisson arrivals and exponential service times and two sets of priorities  FIFO and the longest queue first policies, (ii) single production facility with service level constraints, (iii) single production facility with nonMarkovian demand and general service times, (iv) systems with multiple production facilities, and (v) systems with multiple production stages. For the system with multiple production facilities, they considered three different scenarios  (a) inventory pooling without capacity pooling, (b) inventory pooling with capacity pooling, and (c) capacity pooling without inventory pooling. In the first scenario, they considered a single inventory location supplied by n production facilities, with a demand from stream i resulting in a replenishment order at production facility i. They assumed a Markovian system and each facility was modeled as an independent M/M/1 queue. In the second scenario, both the inventory and capacity were assumed to be pooled. Under Markovian assumptions, the production system was modeled as 19 an M/M/n queue. In the third scenario, the production facilities are consolidated, while the inventory locations were distinct. The Markovian system with n identical facilities was modeled as an M/M/n multiclass FIFO production system. The results of the study showed that the benefit of inventory pooling decreased with utilization, while the benefit of capacity pooling increased with utilization. 2.2.2 Modeling Supply Chain Networks and its Constituents Performance evaluation models of SCNs consider the supply, transportation and distribution operations in a SCN in addition to the capacity, congestion, and inventory issues. These models are typically used to analyze the flow of information and goods between the various stores in a SCN. Cohen and Lee [14] developed an analytical model for integrated productiondistribution systems by decomposing the network into submodels such as the material control submodel, production submodel, stockpile inventory submodel, and the distribution submodel. The submodels were optimized based on certain control parameters which served as links between the submodels. These control parameters included lot sizes, reorder points and safety stocks. Lee and Billington [37] and Ettl et al. [19] focused on developing models to capture the interdependence of basestock levels at different stores in a SCN. Lee and Billington [37] study addressed the needs of the manufacturing managers at Hewlett Packard in managing the material flows in their decentralized supply chains. The study focused on computing the mean and variance of the replenishment lead time for every stockkeeping unit (SKU) at a site. Lee and Billington [37] then used this information to compute the required basestock level to attain a target service level for each SKU at that site. On the other hand, Ettl et al. [19] used a combination of analytical and queueing models to support strategic decision making for large SCNs with nonstationary demand. They considered a oneforone replenishment policy, 20 while ignoring order size at each store and other operational aspects of inventory management in a SCN. They developed an optimization model with service level constraints and used a queueing model to obtain the order queue distribution at each stage. Raghavan and Viswanadham [49] used forkjoin approximations to compute the mean and variance of departure processes at nodes in a SCN. They presented simple approximations for the case of deterministic arrivals and normally distributed service times. Dong [16] and Dong and Chen [17] used the PA card concept of Buzacott and Shanthikumar [11] to model an integrated logistics system with (s, S) inventory policy, where each orders had a fixed lot size. They made use of the expressions provided by Buzacott and Shanthikumar [11] for a system with batch arrivals to find the distribution of number of orders at the manufacturing stage and then used this to find the stockout probability and fill rate. Srivathsan [53] extended the performance evaluation models for productioninventory networks developed by Sivaramakrishnan [51] to model a generic supply chain network where each manufacturer was assumed to have an input store for raw material storage and an output store for finished product storage. An example network with three suppliers, two manufacturers and three retailers was used to illustrate the performance prediction capability of the approximations. While Sivaramakrishnan [51] used the delay block to link successive nodes in the network, Srivathsan [53] used the backorder distribution to establish the link. With oneforone replenishment, an end customer order at the retailer triggered an instantaneous replenishment order at each of the upstream stores. Each manufacturer was modeled as a single server queue, and the lead time delays at the raw material suppliers and the transportation delays in the network were approximated by an M/G/∞ model. Ayodhiramanujan [6] developed integrated analytical models that addressed capacity, congestion, and inventory issues simultaneously in warehouse systems. The 21 research effort focused on developing queueing network models for a sharedserver system and an orderpicking system. The former is an inventory store with a server performing both the storage and retrieval operations. In the orderpicking system, the configuration of the unitload that is stored (pallets) is different from that which is retrieved (cases). Ayodhiramanujan [6] also extended these models to include multisever cases. An integrated model was also developed to demonstrate the applicability of these two key building blocks in developing endtoend models of warehouse systems. Jain and Raghavan [29] considered a productioninventory system with a manufacturing plant and a warehouse. The warehouse inventory was modeled as the input control mechanism for the manufacturing plant, which was modeled as a singlestage discretetime queueing system with an infinite waiting line. The warehouse was assumed to follow a basestock policy with oneforone replenishment and a production authorization card was assumed to be attached to each finished good. The customer order arrival process was assumed to be Poisson and the manufacturer was assumed to process orders from the warehouse at fixed discrete time slots. The queueing models were embedded into two optimization models, the first of which focused on minimizing the long run expected total cost per unit time (comprising of holding cost at warehouse and backorder cost). The second optimization model focused on minimizing the long run expected total cost per unit time (comprising of the holding cost alone) subject to service level constraints based on the probability of backorders. Arda and Hennet [3] considered an enterprise network where the endproduct manufacturer had several potential suppliers for components. The supply system with random arrivals of customer orders and random supplier delivery times was modeled as a queueing system. The manufacturer was assumed to follow a basestock inventory policy. Demand at the manufacturer was assumed to follow a Poisson process and was of unit size. The manufacturer was assumed to follow a (S1, S) 22 inventory policy and placed an order with supplier i with a probability αi as per a Bernoulli splitting process. An optimization model with an objective of minimizing the average cost of the manufacturer (expressed as the sum of mean inventory holding cost and mean backordering cost) was used to obtain the Bernoulli probabilities and the optimal basestock value. Each supplier was assumed to follow an exponential service time and FIFO discipline. They modeled each supplier as an M/M/1 queue as the order arrival process at the supplier was Poisson because of the Bernoulli splitting process. They showed that the problem at hand was hard as the convexity of the objective function was not guaranteed. They presented optimal solutions for the maketoorder system by solving the Lagrangian of the relaxed problem. In the case of the maketostock system, due to the complexity of the objective function, the problem was decomposed into two parts. In the first part, they considered the Bernoulli parameters as the decision variables, while considering the basestock level as zero (maketoorder system). The values of the Bernoulli variables from the first part were then used as an input in the second part. The decision variable in this case was the basestock level, which was computed using a discrete version of the newsvendor problem. 2.3 Summary of the Literature Review This chapter presented a detailed review of literature on modeling information sharing in a supply chain network as well as performance evaluation models of productioninventory and supply chain networks. Based on the literature review, we note the following. Queueing models of productioninventory networks have focused on considering capacity, variability, and inventory issues in a unified manner and included the works of Graves and Kielson [25], Lee and Zipkin [40], Buzacott and Shanthikumar [11], Sivaramakrishnan [51], Liu et al. [43] and a few others. The ability of queueing models 23 to explicitly consider inventory issue has been considerably strengthened by the works of Sivaramakrishnan [51] on modeling productioninventory networks, Srivathsan [53] on modeling supply chain networks, and Ayodhiramanujan [6] on modeling warehouse operations. Other research efforts that focused on performance evaluation of SCNs include Dong [16], Dong and Chen [17], and Jain and Raghavan [29]. The research efforts that focused on performance optimization of SCNs included Ettl et al. [19], Lee and Billington [37], and Cohen and Lee [14]. The review of the literature on modeling inventory information sharing in a SCN showed that there are only a few efforts on developing performance evaluation models of SCNs in this context. Armony and Plambeck [4] and Zipkin [63] focused on performance evaluation using queuing models. The former focused on modeling the effect of duplicate orders on SCN performance, while the latter modeled a productioninventory network with multiple products produced by a single production facility. We concluded that there is a need for more research in explicitly modeling upstream inventory information sharing within SCN performance evaluation models. The availability of a richer set of performance evaluation models will enable us to better predict the value of sharing upstream inventory information in a SCN and understand the complex dynamics resulting from decisions related to production capacities, maximum inventory levels, and order placement policies. 24 CHAPTER 3 RESEARCH STATEMENT The overall goals of this research were (i) to develop analytical performance evaluation models that consider inventory information sharing between SCN firms, and (ii) to study the value of inventory information sharing and identify SCN conditions under which the benefits of inventory information sharing are significant. 3.1 Research Objectives In most analytical performance evaluation models of SCNs, it is assumed that a downstream firm places an order at one of its immediate upstream “suppliers” using a Bernoulli routing policy. As per this policy, orders from a retail store would be routed to say the production facilities based on fixed probabilities (based on preference or historical information) and this would allow an order to be placed at a production facility facing a stockout situation even when there is inventory at the other facility. This routing is appropriate for SCNs with no inventory information sharing. On the other hand, if a downstream firm has information about the net inventory level at all of its immediate upstream firms, orders from the downstream firm could possibly be placed in such a way that this situation can be averted, thereby reducing the backorders and decreasing the stockout rate. The models developed in this research consider inventory information sharing in one direction  upstream stores sharing information with the immediate downstream stores. An analytical model that addresses inventory information sharing in a SCN can be used to quantify the value of information sharing and provide insight into the 25 sensitivity of the value of inventory information sharing to different SCN parameters (interarrival and service time parameters, basestock levels, etc.). The model could then be used to identify the ranges of the various SCN parameters where the benefit of inventory information sharing is significant. The specific objectives of this research are as follows. Objective 1: To perform a thorough investigation of the literature related to (i) the value of information sharing in a SCN, and (ii) the analytical modeling of productioninventory and supply chain networks. Objective 2: To develop analytical models that can capture the effect of inventory information sharing on the performance of a twoechelon SCN comprising of one retail store that can order items from one of two upstream production facilities, each with its own inventory store and to study the benefits of inventory information sharing in this context. As discussed in the beginning of this section, this objective focused on modeling the effect of inventory information sharing on the performance of a supply chain. We first considered a SCN with one retail store and two production facilities each with its own inventory store. Henceforth, we will refer to this twoechelon configuration as “1R/2P.” For this case, we considered three levels of inventory information sharing and developed analytical models for these three levels. The performance measures from these analytical models were compared with the corresponding measures for the SCN with no visibility (henceforth referred to as NoVis) to study the value of inventory information sharing. The three levels of inventory information sharing and the assumed routing policy for each level are presented next. Low level of Inventory Information Sharing (LoVis) In a SCN with minimum level of inventory visibility, the retail store is assumed to have information about the presence of inventory or backorders at the production facilities. In such cases, an order routing policy can be adopted where the placement 26 of orders at a production facility with backorders can be avoided when the other facility has inventory (see Table 3.1). Table 3.1: Order Routing Policy for SCN with Low Level of Inventory Information Sharing Net Inventory Level at Production Facility 1 Net Inventory Level at Production Facility 2 Order Routing Policy ≥ 1 ≥ 1 Order routed with equal probability ≥ 1 ≤ 0 Order routed to production facility 1 ≤ 0 ≥ 1 Order routed to production facility 2 ≤ 0 ≤ 0 Order routed with equal probability Medium level of Inventory Information Sharing (MedVis) In a SCN with medium level of inventory visibility, the retail store is assumed to also have information about the number of backorders at the individual production facilities. When both production facilities are backordered, the order is routed to the facility with the shortest backorder queue. The order routing policy in such a scenario can be modified as shown in Table 3.2. Table 3.2: Order Routing Policy for SCN with Medium Level of Inventory Information Sharing Net Inventory Level at Production Facility 1 (i) Net Inventory Level at Production Facility 2 (j) Condition Order Routing Policy ≥ 1 ≥ 1 Order routed with equal probability ≥ 1 ≤ 0 Order routed to production facility 1 ≤ 0 ≥ 1 Order routed to production facility 2 ≤ 0 ≤ 0 i < j Order routed to production facility 1 i > j Order routed to production facility 2 i = j Order routed with equal probability High level of Inventory Information Sharing (HiVis) In a SCN with a high level of inventory visibility, the retail store is assumed to have information about the number of items in stock as well as the number of backorders 27 at the individual production facilities. In such cases, the order routing policy can be modified as shown in Table 3.3. Table 3.3: Order Routing Policy for SCN with High Level of Inventory Information Sharing Net Inventory Level at Production Facility 1 (i) Net Inventory Level at Production Facility 2 (j) Condition Order Routing Policy ≥ 1 ≥ 1 i > j Order routed to production facility 1 i < j Order routed to production facility 2 i = j Order routed with equal probability ≥ 1 ≤ 0 Order routed to production facility 1 ≤ 0 ≥ 1 Order routed to production facility 2 ≤ 0 ≤ 0 i < j Order routed to production facility 1 i > j Order routed to production facility 2 i = j Order routed with equal probability Objective 3: To develop analytical models that can capture the effect of inventory information sharing on the performance of a twoechelon SCN comprising of two retail stores that can order items from one of two production facilities, each with its own inventory store and to study the benefits of inventory information sharing in this context. We considered a SCN with two retail stores that place orders with one of two production facilities. The production facilities share their inventory information with the retail stores. Henceforth, we will refer to this twoechelon configuration as “2R/2P.” For this case, we considered the three levels of information sharing defined in Objective 2 and developed analytical models for these levels. Objective 4: To extend the models developed in Objectives 3 and 4 to include intransit inventory. 28 3.2 Research Scope The scope of this research effort was limited by the following assumptions. 1. There is no limit on the size of the WIP and backorder queues. 2. Each production facility has a singlestage with a single server. Each demand is for one unit of the product. 3. As the SCN configurations modeled are considered to be of the building block type, only twoechelon SCN structures will be modeled. 3.3 Research Contributions The contributions of this research effort are listed below. 1. Development of analytical queueing models that can explicitly model inventory information sharing from upstream stores to downstream stores in supply chain networks. 2. Understanding the benefits of sharing inventory information and developing insights into SCN configurations for which these benefits are significant. 3. Evaluating the significance of each incremental piece of information that becomes available in the SCN. 4. An offshoot of this research is the potential to develop good approximations for the wellknown shortest queue problem. 29 CHAPTER 4 RESEARCH APPROACH This chapter explains the overall research methodology and the various modeling approaches that were used to achieve the research objectives outlined in the previous chapter. Section 4.1 presents the research methodology, and then describes the Markov chain approach followed by the parametric decomposition approach. Section 4.2 presents a list of performance measures that were used along with their definitions and their significance in a SCN context. Section 4.3 explains the validation procedure used for the analytical models. 4.1 Research Methodology This research effort involved the development of analytical performance evaluation models. A standard methodology for such developmental research was employed. First, we modeled the SCN configurations with Poisson arrivals at the retail store(s) and exponential processing times at the production facilities as these conditions make the models more analytically tractable. For this case, we initially used the Markov chain approach. If the approach did not yield a closedform solution, we then focused on the development of approximate queueing models based on the characteristics of the model under consideration. We then relaxed the exponential assumption and considered SCN configurations with general interarrival and processing time distributions using the twomoment framework. Whitt’s ([59], [61]) parametric decomposition (PD) method was used to solve the resulting queueing models. Each analytical model developed was validated by comparing its results to equivalent simulation esti 30 mates. The objective for such a comparison was to determine the regions of the model parameter space where the accuracy of the analytical results was good or deemed acceptable. A design of experiments approach was used to ensure adequate coverage of the parameter space from the perspective of model usage in practice and to control the number of numerical experiments that had to be conducted. The numerical experimentation had the added benefit of providing insights into the system behavior under different parameter settings. The results of the numerical investigation were used to develop additional corrections or enhancements to improve the accuracy of the analytical models. Section 4.1.1 summarizes the Markov chain approach, and Section 4.1.2 briefly describes the PD approach. 4.1.1 Markov Chain Approach Continuous Time Markov Chain (CTMC) models have been widely used to develop performance evaluation models of discrete event systems. The advantage of CTMC models is that they can yield exact solutions under exponential or Markovian assumptions. Also, because the CTMC models are based on detailed state information, they are preferred when statebased decisions have to be modeled. For example, the modeling of visibilitybased routing policies in SCNs requires detailed state level information and could be easily modeled using the CTMC approach. Both transient and steadystate analysis can be performed using a CTMC model [44]. In the case of performance evaluation, steadystate analysis is done to compute the longrun performance measures. This involves the solution of the rate balance equations, which could be an issue in the case of an infinite state space. A standard method in such cases is to express all state probabilities in terms of the probability of one particular state and to use the total probability equation to find this probability. However, this is possible if the expression involving the sum of the probabilities can be simplified to yield a closedform expression. If simplification is not possible, then 31 we need to find ways (e.g., limit the number of backorders) to truncate the state space in order to yield a finitestate CTMC which can then be solved numerically. Another strategy to solve the balance equations is to use the structure of the CTMC to identify similarities between the state transitions in that CTMC and other CTMC models for which closedform solutions are known. For instance, while solving the CTMC model of the SCN under the Bernoulli routing policy, we were able to identify transitions patterns in the CTMC model that resembled the state transitions in an M/M/1 queuing model (see Section 5.6). Such similarities may allow us to guess a solution, which can be verified by substituting the guessed solution into the balance equations. If there is symmetry in a multituple system, we might be able to combine states (e.g., (i, j) combined with (j, i)) to reduce the size of the state space. The reduced CTMC can sometimes simplify the solution of the original CTMC. This reduced CTMC model can be solved using the rate balance equations or by identifying patterns in the chain. Once the CTMC is solved and the limiting probabilities are obtained, we can use a reverse mapping to obtain the limiting probabilities of the original CTMC model. Other methods that can be used to solve the CTMC models include the differenceequation technique and the method of generating functions. For more details on these methods, the reader is referred to Medhi [44]. 4.1.2 Parametric Decomposition Approach In the 1980s, Whitt [59] defined a new modeling ideology highlighted by the parametric decomposition (PD) approach. According to Whitt, “a natural alternative to an exact analysis of an approximate model is an approximate analysis of an exact model” [59]. The PD approach is a very comprehensive method for analyzing a queueing network and uses only the first two moments of both the interarrival and 32 service times. This approach formed the basis for a software package developed by Whitt, called the Queueing Network Analyzer (QNA) [59]. The PD approach for open queueing networks consists of two main steps: 1) analyzing the nodes and the interaction among the nodes to obtain the mean and the squared coefficient of variation (SCV = variance / mean2) of the interarrival times at each node, and 2) obtaining the node and system performance measures based on GI/G/1 or GI/G/m approximations ([59], [61]). Analyzing nodes: In a network, nodes interact with each other because of customer movement and these interactions can be approximately captured by the flow parameters, namely, the rates and variability parameters of the arrival processes at the nodes. The total arrival rate at each node is obtained using the traffic rate equations, which represent the conservation of flow. The utilizations of each of the nodes are calculated to check for stability of the system. The system is said to be stable if all utilizations are strictly less than one. This part of the analysis is similar to the approach introduced by Jackson [28] in solving open networks and involves no approximations. The approximations come into the picture while calculating the variability parameters related to the flow, namely, the SCVs of the interarrival times. The SCVs are calculated using the traffic variability equations, which involve approximations for the basic network operations, which are a) flow through a node, b) merging of flow, and c) splitting of flow. These approximations can be found in Whitt ([59], [61]). Calculating node and system performance measures: The nodes are treated as stochastically independent. The performance measures at each node can be calculated from the results available for the GI/G/1 ([59], [34]) and GI/G/m queues ([60]). The expected waiting time at each node is calculated from the results provided and the expected queue length is obtained using Little’s law [42]. Whitt [59] also explains how several other node and network measures can be calculated. 33 In our research, there is a need to incorporate statelevel details while modeling the SCN under visibilitybased routing policies. The twomoment framework is wellsuited for queueing network type models and it could be challenging to accommodate statelevel details in the analytical model developed. A strategy that is sometimes used involves the use of phasetype distributions to represent general distributions. The feature or subsystem that needs to be modeled in detail (e.g., the order routing or splitting process at the retailer) is studied in isolation. The phasetype approach enables CTMC type modeling because the phases are exponential stages. To analyze a feature or submodel in detail, the states in the Markov chain embedded at an instant of the feature need to be identified. The solution of this Markov chain results in the stationary probability vector, which can be used in obtaining the twomoment approximations for the feature or subsystem. Such an approach can be employed when closedform expressions exist for the Markov chain with Poisson arrivals and exponential processing times. A good example is the development of twomoment approximations for a forkjoin configuration [35]. In our research, we did not consider the phasetype approach because an exact solution to the CTMC model under Poisson arrivals and exponential distributions was not possible (discussed in Chapter 5). 4.2 Performance Measures The performance measures that were computed at each stage of the SCN are fill rate, expected number of backorders, expected inventory level, the expected time to fulfill a backorder and expected time spent by an order. • Fill rate is the probability that an order will be satisfied immediately and this depends on the availability of inventory at the stage. The definition of fill rate suggests that as the fill rate increases, the stockout rate decreases and the relationship between them is given by stockout rate = 1  fill rate. In our analytical approach, we approximate the fill rate by the ready rate, which is the 34 probability that there is inventory in the system. It should be noted that for Poisson arrivals, fill rate and ready rate will be the same because of the PASTA (Poisson Arrivals See Time Averages) principle [62]. • The expected number of backorders at any stage is the average number of unsatisfied orders at the stage. The significance of this measure is that a high value could indicate a potential for loss of customer goodwill and sales. • The expected inventory level at any stage is the average number of items in the store at that stage. The expected inventory level is a paradoxical measure as its high value would increase the inventory holding cost, while its low value could be an indicator of lower fill rates and higher backorder levels. • The expected time to fulfill a backorder at a stage is the average time that an order has to wait before being satisfied at a store given that the store is facing a stockout situation. It has to be noted that the expected time to fulfill an order can be considerably lower than the expected time to fulfill a backorder. This is because, majority of the orders could be satisfied instantaneously, while orders that are backordered could take significantly longer time to complete. Thus, this is a measure of the experience of a customer who faces a stockout situation. • The expected time spent by an order is defined as the sum of the average time that an order spends at that stage and the average time spent (as a product) in the output store at that stage. This measure is important as it is not a desirable situation to have an item sitting in stock for long periods as this would increase the holding cost. 35 4.3 Numerical Validation Procedure Each SCN configuration that was modeled in our research effort was also simulated using a model developed in Arena 11.0 software [32]. The warmup period was determined using Welch’s procedure [58] (See Appendix A). The number of independent replications was set to 10. The parameters of the various SCN configurations, namely, basestock levels; variability of interarrival and processing times; utilization; and probabilities (if any)  were varied systematically using a design of experiments approach to cover a wide range of scenarios. For each scenario, the analytical results were compared with steadystate simulation estimates to evaluate the accuracy of the analytical results. As mentioned in Whitt [60], the two standard ways to measure the accuracy are absolute difference and relative percentage error. As Whitt [60] contends, neither procedure is appropriate for a wide range of values. When the performance measure values are themselves small (e.g., less than 0.5), the absolute difference seems to be appropriate. Whitt [60] is of the opinion that the quality of the approximations is satisfactory “if either the absolute difference is below a critical threshold or the relative percentage error is below another critical threshold” [60]. Recently, another approach, namely, normalized error has been used to evaluate the accuracy of queueing approximations ([35], [54]). Hence, we designed the following approach to evaluate the accuracy of the analytical results. For the fill rate (bounded by 1), we used the absolute difference, simulation − analytical, expressed as a percentage. For the expected inventory level and the expected number of backorders, we used the normalized percentage error given by 100 (simulation − analytical)/basestock level. For the expected time to fulfill a backorder and the expected time spent by an order, we used the relative percentage error defined as 100 (simulation − analytical)/simulation. This scheme allowed us to perform an overall analysis of the quality of the analytical results. 36 CHAPTER 5 CTMC MODELS OF THE 1R/2P SCN CONFIGURATION This chapter presents the CTMC models of the 1R/2P SCN configuration under three levels of inventory information sharing as defined in Section 3.1. Section 5.1 presents the description of the SCN structure used in the study. The state definition used in the CTMC models is presented in Section 5.2. Sections 5.3, 5.4 and 5.5 present the CTMC models for the SCN with high (HiVis), medium (MedVis) and low (LoVis) levels of information sharing, respectively. Section 5.6 presents the CTMC model for the SCN with no information sharing (NoVis). Section 5.7 presents the results of the study on the benefits of inventory information sharing on the SCN performance. Finally, some concluding remarks about the CTMC models are presented in Section 5.8. For the sake of simplicity, we will refer to inventory information sharing as simply information sharing. 5.1 1R/2P SCN Structure We consider a twoechelon SCN with one retail store and two production facilities each with its own output store to stock finished products. Each store in the SCN is assumed to operate under a basestock control policy with oneforone replenishment. Both the production facilities are assumed to have the same basestock level (S). The basestock level at the retail store is R. Each customer order is assumed to be for a single unit of the finished product. The demand interarrival times at the retail store as well as the processing times at the production facilities follow general distributions, but only the exponential case is considered in this chapter. The arrival 37 of a demand consumes a finished product at the retail store if available and causes an order for replenishment to be placed at one of the two production facilities (see Figure 5.1). If available, a finished product from the output store of the production facility is instantaneously sent to the retail store and the output store sends an order for replenishment to its processing stage. As noted in Chapter 1, extensions to include transit delays will be presented in Chapter 9. We assume that the processing stage has a single server. An order can join the WIP queue or be processed as soon as it is received. There are no limits on the number of backorders at either production facility. It has to be noted that when the basestock levels at the two production facilities are the same, the HiVis routing policy closely resembles the shortest queue problem studied in the literature (e.g. [33], [26], and [2]). Production Facility 1 Retail Store Production Facility 2 Output Store 2 Output Store 1 Customer Demand Figure 5.1: 1R/2P SCN Configuration 5.2 CTMC Model for Poisson Arrivals and Exponential Processing Times This section presents the statespace definition for the CTMC model used to study the SCN with or without information sharing. This section also provides the details on obtaining the performance measures for the SCNs with information sharing (LoVis, 38 MedVis and HiVis). The following assumptions were made in developing the CTMC models. 1. The demand/order arrival process is Poisson with rate λ. 2. The processing times at both the production facilities follow an exponential distribution with rate μ (the two production facilities are identical). 3. The transportation time from the production facility to the retail store is not modeled (relaxed later). Ignoring the transportation delay between the production facilities and the retail store means that there is no need to consider the number of orders at the retail store while defining the state of the SCN (see explanation below). As a result, the state of the SCN at time t is defined by X(t) = {i, j}, where i and j are nonnegative integers representing the number of orders at production facilities 1 and 2, respectively. The details about the net inventory levels at the various stores in the SCN can be obtained from the state definition as follows. When the number of orders at a production facility is less than the basestock level, the output store at the production facility will have inventory and the inventory level will be given by (S  x), where x = i for production facility 1 and x = j for production facility 2. Similarly, when the number of orders at a production facility exceeds the basestock level, the output store is backordered. The number of backorders at the production facility will be given by (x  S). The net inventory level at a production facility will be given by (S  x). This value will be positive in the presence of inventory and negative in the presence of backorders. As the transportation delay is ignored, when the number of orders at each production facility does not exceed the basestock level of its output store (i.e. 0 ≤ i, j ≤ S), the inventory level at the retail store will be equal to its basestock level (R). When 39 at least one of the production facilities is backordered, the inventory level at the retail store will be equal to its basestock level less the sum of the backorders at the production facilities and the number of orders (backorders + replenishment orders) at the retail store is given by (i − S)+ + (j − S)+. The retail store will be backordered only when the sum of the backorders at the two production facilities exceeds the basestock level at the retail store (i.e., (i − S)+ + (j − S)+ > R). The net inventory level at the retail store will be [R − (i − S)+ − (j − S)+]. To better understand the above discussion, let us consider a case with R = S = 3. If the number of orders at production facility 1 is 1, and the number of orders at production facility 2 is 1, then the current state of the SCN is (1, 1). The net inventory level at the two production facilities will be 2 (i.e., 31), indicating that there is inventory at both the production facilities. As a result, the number of orders at the retail store is zero and the net inventory level is 3 (i.e., [3−(1−3)+−(1−3)+]). If the number of orders at production facility 1 is 2, and the number of orders at production facility 2 is 4, then the current state of the SCN is (2, 4). In this case the net inventory level at production facility 1 is 1 (i.e. 32), indicating that there is one unit of inventory at production facility 1. The net inventory level at production facility 2 is 1 (i.e., 34), indicating that there is one backorder at production facility 2. As a result, number of orders (replenishment in this case) at the retail store is 1 (because of the backorder at production facility 2) and the net inventory level is 2 (i.e., [3 − (2 − 3)+ − (4 − 3)+]). If the number of orders at both the production facilities is 5, the current state of the SCN is (5, 5). In this case the net inventory level at both the production facilities is 2 (i.e. 35), indicating that there are two backorders at each production facility. The number of orders at the retail store is 4 (sum of the backorders at the two production facilities) and the net inventory level is 1 (i.e., [3−(5 − 3)+−(5 − 3)+]), indicating one backorder at the retail store. 40 The following notations are used in this chapter. Model Parameters λ Order arrival rate at retail store R, S Basestock levels at retail store and output stores of the production facilities, respectively Model Variables (Upper case letters are random variables) Nr Number of orders at the retail store Np Number of orders at a production facility ρ Utilization of a production facility Ir, Ip Inventory levels at the retail store and output store of a production facility Br, Bp Number of backorders at the retail store and output store of a production facility fr, fp fill rates at the retail store and output store of a production facility Wbr, Wbp time to fulfill a backorder at the retail store and the output store of a production facility Tr, Tp time spent by an order at the retail store and a production facility including its output store The CTMC models for the different levels of information sharing have some common properties that are listed below. 1. The state space of the CTMC is symmetrical as the two production facilities are identical (same basestock level and processing rate). This means that the transition rates into (out of) the states (i, j) and (j, i) are the same except when i = j. This observation allowed us to combine the states (i, j) and (j, i) and 41 reduce the state space of the CTMC model. 2. The patterns present in the reduced CTMC models for the SCNs with information sharing (Figures 5.3, 5.5, and 5.7) could not be exploited to yield a closed form solution. As a result, the CTMC models were solved by limiting the maximum number of backorders at the retail store to M. 3. Once we numerically solved for πi,j by limiting the maximum number of backorders at the retail store, we computed the key performance measures for a production facility as follows. The fill rate at a production facility is simply the probability that an order from the retail store is satisfied immediately as shown in equation (5.1). fp = S−1 i=0 S−1 j=0 πi,j + S+ R+M i=S S−1 j=0 πi,j + S−1 i=0 S+ R+M j=S πi,j (5.1) The rest of the performance measures for a production facility were obtained using expressions (5.2) and (5.5). E[Ip] = S i=0 S+ R+M j=0 (S − i)πi, j (5.2) The expected backorder at a store can be calculated using the fundamental expression relating the expected inventory level, expected number of orders and the basestock level as shown in equation (5.3). E[Bp] = E[Ip] + E[Np] − S = S i=0 S+ R+M j=0 (S − i)πi, j + S+ R+M i=0 S+ R+M j=0 iπi, j − S (5.3) 42 Now, using Little’s law we have E[Wbp] = E[Bp] (1 − fp)λ/2 (5.4) E[Tp] = S+R+M i=0 S+R+M j=0 iπi, j λ/2 + E[Ip] λ/2 (5.5) 4. The arrival process at the retail store is Poisson and as per the PASTA principle [62], the ready rate can be used to calculate the fill rate. Note that an arriving customer order at the retail store would see the number of orders at the two production facilities. The fill rate at the retail store was obtained by calculating the probability that the total number of orders at the two production facilities is less than the sum of their basestock levels and that of the retail store, i.e. i + j < 2S + R. The other performance measures at the retail store can be obtained using expressions (5.7) through (5.11). fr = i, j∈A1 πi, j (5.6) where A1 = {i, j : 0 ≤ i + j ≤ 2S + R − 1} E[Ir] = S−1 i=0 S−1 j=0 Rπi, j + i, j∈A2 R − (i − S)+ − (j − S)+ πi,j (5.7) where A2 = {i, j : 1 ≤ (i − S)+ + (j − S)+ ≤ R} E[Nr] = i, j∈A3 (i − S)+ + (j − S)+ πi,j (5.8) where A3 = {1 ≤ (i − S)+ + (j − S)+ < ∞} 43 E[Br] = E[Ir] + E[Nr] − R (5.9) E[Wbr] = E[Br] λ(1 − fr) (5.10) E[Tr] = E[Ir] λ + E[Nr] λ (5.11) Next, we present the CTMC models for all three levels of information sharing followed by the no information sharing case. Sections 5.3, 5.4 and 5.5 present the CTMC models of the SCN with high (HiVis), medium (MedVis) and low (LoVis) levels of information sharing, respectively. Section 5.6 presents the CTMC model of the SCN with no information sharing (NoVis). 5.3 CTMC Model of 1R/2P SCN with HiVis In the HiVis case, it is assumed that each production facility shares complete information about the number of items in stock as well as the number of backorders. In the presence of such detailed information, a routing policy presented in Table 3.3 can be adopted. Based on the state definition and the routing policy, we obtain the symmetric CTMC model shown in Figure 5.2. The symmetry in Figure 5.2 is exploited to obtain the reduced CTMC presented in Figure 5.3. The rate balance equations based on the different state transitions that are possible in the reduced CTMC model are presented in Table 5.1. 44 /2 μ μ μ /2 μ μ μ μ μ μ μ μ μ 2, S2 1, S1 0, S μ μ μ μ μ μ 1, S2 0, S1 μ μ μ μ 1, S 0, S+1 2, S1 μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ /2 /2 /2 /2 0, 0 1, 0 0, 1 2, 0 1, 1 0, 2 3, 0 2, 1 1, 2 0, 3 S, 0 S1, 1 S2, 2 μ μ μ μ μ μ μ μ 4, 0 3, 1 /2 2, 2 1, 3 0, 4 μ μ μ μ μ μ μ S1, 0 S2, 1 μ μ μ μ S+1, 0 S, 1 S1, 2 μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ /2 /2 /2 /2 S1, S1 S, S1 S1, S S+1, S1 S, S S1, S+1 S+1, S S, S+1 /2 S+1, S+1 μ μ μ μ μ μ Figure 5.2: CTMC Model of the 1R/2P SCN with HiVis 45 2μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ 2μ μ μ μ μ 2μ μ 0, 0 1, 0 2, 0 1, 1 3, 0 2, 1 S, 0 S1, 1 S2, 2 μ μ μ μ μ 4, 0 3, 1 2, 2 μ μ μ μ μ S1, 0 S2, 1 μ μ μ μ S+1, 0 S, 1 S1, 2 μ 2μ μ μ μ 2μ μ S1, S1 S, S1 S+1, S1 S, S S+1, S S+1, S+1 μ Figure 5.3: Reduced CTMC Model of the 1R/2P SCN with HiVis 46 Table 5.1: Rate Balance Equations for the Reduced CTMC Model of the 1R/2P SCN with HiVis State State Transitions Rate Balance Equation i = j = 0 μ i, j i+1, j πi,j = μπi+1,j λ i ≥ 1, j ≥ 1, i = j μ 2μ i, j1 i, j i+1, j πi,j = λπi,j−1+μπi+1,j (λ+2μ) i = 1, j = 0 μ 2μ μ i1, j i, j i+1, j i, j+1 πi,j = λπi−1,j+μπi+1,j+2μπi,j+1] (λ+μ) i ≥ 2, j = 0 μ μ μ i1, j i, j i+1, j i, j+1 πi,j = μ[πi+1,j+πi,j+1] (λ+μ) i ≥ 2, j ≥ 1, i = j + 1 μ 2μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λ[πi,j−1+πi−1,j]+μπi+1,j+2μπi,j+1 (λ+2μ) i ≥ 3, j ≥ 1, i = j + 2 μ μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λπi,j−1+μ[πi+1,j+πi,j+1] (λ+2μ) 47 The reduced CTMC model revealed some state transition patterns, but we could not exploit these patterns to arrive at a closedform solution. The different approaches that were explored to obtain a closedform solution included the solution to the system of rate balance equations, differenceequation techniques and the method of generating functions. Hence, we fixed the maximum number of backorders at the retail store to 1,000 and numerically solved the CTMC. By fixing the maximum number of backorders, we make sure that the state space is finite. Further, a careful examination of the CTMC model shows that all the states communicate and that the CTMC model is irreducible. Hence, the CTMC is positive recurrent and has a steady state solution. The CTMC model was validated by comparing its numerical solution to the estimates obtained from an Arena simulation model of the 1R/2P SCN with HiVis, as shown in Tables B.1 and B.2. The numerical experiments show that the analytical results match the simulation results, thus confirming the validity of the CTMC model. 5.4 CTMC Model of the 1R/2P SCN with MedVis In this section, we present the CTMC model of the 1R/2P SCN with medium level of inventory information sharing. It is assumed that the information about the number of backorders at the individual production facilities is available at the retail store. In the presence of the backorder information, the routing policy presented in Table 3.2 can be adopted. The symmetric CTMC model for this case is shown in Figure 5.4 and the reduced CTMC model is presented in Figure 5.5. In Figures 5.4 and 5.5, the transitions shown in blue are common to the HiVis CTMC model and the MedVis CTMC model, and those shown in red are the additional transitions for the MedVis CTMC model. The transition rates marked in red when the transitions are blue indicate that the transitions are common to both the HiVis and MedVis models, but the rates have changed in the case of the MedVis model. The rate balance equations based on the 48 different state transitions that are possible in the reduced CTMC model are presented in Table 5.2. /2 /2 μ /2 /2 /2 /2 /2 /2 /2 /2 /2 μ μ μ μ μ μ μ μ μ 2, S2 1, S1 0, S μ μ μ μ 1, S2 0, S1 μ μ μ μ 2, S1 1, S 0, S+1 /2 /2 μ /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 μ /2 μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ /2 /2 /2 /2 /2 /2 0, 0 1, 0 0, 1 2, 0 1, 1 0, 2 3, 0 2, 1 1, 2 0, 3 S, 0 S1, 1 S2, 2 μ μ μ μ μ μ μ μ 4, 0 3, 1 2, 2 1, 3 0, 4 μ μ μ μ μ μ S1, 0 S2, 1 μ μ μ μ S+1, 0 S, 1 S1, 2 /2 μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ /2 /2 /2 /2 S1, S1 S, S1 S1, S S+1, S1 S, S S1, S+1 S+1, S S, S+1 /2 S+1, S+1 μ μ /2 /2 μ μ μ μ μ μ Figure 5.4: CTMC Model of the 1R/2P SCN with MedVis 49 2μ μ /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 μ μ μ μ μ μ μ μ μ μ μ μ μ 2μ μ μ μ μ 2μ μ /2 0, 0 1, 0 2, 0 1, 1 3, 0 2, 1 S, 0 S1, 1 S2, 2 μ μ μ μ μ 4, 0 3, 1 2, 2 μ μ μ μ S1, 0 S2, 1 μ μ μ μ S+1, 0 S, 1 S1, 2 /2 μ 2μ μ μ μ 2μ μ S1, S1 S, S1 S+1, S1 S, S S+1, S S+1, S+1 μ /2 /2 μ μ Figure 5.5: Reduced CTMC Model of the 1R/2P SCN with MedVis 50 Table 5.2: Rate Balance Equations of the Reduced CTMC Model of the 1R/2P SCN with MedVis State State Transitions Rate Balance Equation i = j = 0 μ i, j i+1, j πi,j = μπi+1,j λ i = 1, j = 0 /2 /2 μ 2μ /2 μ i1, j i, j 2, 0 1, 1 πi,j = λπi−1,j+μπi+1,j+2μπi,j+1 (λ+μ) 2 ≤ i ≤ S−1, j = 0 /2 /2 μ μ /2 μ i1, j i, j i+1, j i, j+1 πi,j = λ 2 πi−1,j+μ[πi+1,j+πi,j+1] (λ+μ) i = S, j = 0 μ μ /2 μ i1, j i, j i+1, j i, j+1 πS,0 = λ 2 πS−1,0+μ[πS+1,0+πS,1] (λ+μ) S + 1 ≤ i < ∞, j = 0 μ μ μ i1, j i, j i+1, j i, j+1 πi,j = μ[πi,j+1+πi+1,j ] (λ+μ) 1 ≤ i ≤ S −1, i = j /2 μ 2μ i, j1 i, j 2, 1 πi,j = λ 2 πi,j−1+μπi+1,j (λ+2μ) S ≤ i < ∞, i = j μ 2μ i, j1 i, j 2, 1 πi,j = λπi,j−1+μπi+1,j (λ+2μ) 2 ≤ i ≤ S − 1, j = i − 1 /2 /2 μ 2μ μ μ i, j1 /2 i1, j i, j i+1, j i, j+1 πi,j = λ 2 πi,j−1+λπi−1,j+μπi+1,j+2μπi,j+1] (λ+2μ) S ≤ i < ∞, j = i−1 μ 2μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λ[πi,j−1+πi−1,j]+μπi+1,j+2μπi,j+1] (λ+2μ) Continued on next page 51 Table5.2 – continued from previous page State State Transitions Rate Balance Equation 1 ≤ j ≤ S − 3, i ≥ j + 2, μ 2μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λ 2 [πi,j−1+πi−1,j]+μ[πi+1,j+πi,j+1] (λ+2μ) i = S, 1 ≤ j ≤ S−2 μ μ μ μ i, j1 /2 i1, j i, j i+1, j i, j+1 πi,j = λπi,j−1+λ 2 πi−1,j+μ[πi+1,j+πi,j+1] (λ+2μ) S + 1 ≤ i < ∞, 1 ≤ j ≤ S − 1 μ μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λπi,j−1+μ[πi,j+1+πi+1,j ] (λ+2μ) The state transition patterns identified in the reduced CTMC model could not be exploited to obtain a closedform solution. As before, we explored the following approaches to obtain a closedform solution including the solution to the system of rate balance equations, differenceequation techniques and the method of generating functions. Hence, we fixed the maximum number of backorders at the retail store to 1,000 and numerically solved the CTMC. By fixing the maximum number of backorders, we make sure that the state space is finite. Further, a careful examination of the CTMC model shows that all the states communicate. Hence, the CTMC is irreducible and positive recurrent. This ensures that steady state solution exists for the reduced CTMC model. The CTMC model was validated by comparing its numerical solution to estimates from an Arena simulation model of the 1R/2P SCN with MedVis, as shown in Tables B.3 and B.4. The numerical experiments show that the analytical results match the simulation results, thus confirming the validity of the CTMC model. 52 5.5 CTMC Model of the 1R/2P SCN with LoVis In this case, the retail store is assumed to have the minimum amount of inventory information (presence or absence of inventory) from the production facilities. The order routing policy presented in Table 3.1 can be adopted in this case. The symmetric CTMC model for this case is presented in Figure 5.6, while the reduced CTMC model is presented in Figure 5.7. In Figures 5.6 and 5.7, the transitions and rates shown in red indicate that these are common for the MedVis and LoVis models. Also, the transitions shown in blue are common to the CTMC models corresponding to all three levels of information sharing. The additional transitions and rates that are specific to the CTMC model of SCN with LoVis are shown in purple in Figures 5.6 and 5.7. The rates shown in purple when the transitions are blue indicate that the transitions are common for all three levels of information sharing, but the rate has changed for the LoVis case. The rate balance equations based on the different state transitions that are possible in the reduced CTMC model are presented in Table 5.3. 53 /2 μ /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 μ 2 /2 μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ /2 /2 /2 /2 /2 /2 0, 0 1, 0 0, 1 2, 0 1, 1 0, 2 3, 0 2, 1 1, 2 0, 3 S, 0 S1, 1 S2, 2 μ μ μ μ μ μ μ μ 4, 0 3, 1 2, 2 1, 3 0, 4 μ μ μ μ μ μ S1, 0 S2, 1 μ μ μ μ S+1, 0 S, 1 S1, 2 /2 /2 /2 μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ /2 /2 /2 /2 S1, S1 S, S1 S1, S S+1, S1 S, S S1, S+1 S+1, S S, S+1 /2 S+1, S+1 μ μ /2 /2 μ /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 μ μ μ μ μ μ μ μ μ 2, S2 1, S1 0, S μ μ μ μ 1, S2 0, S1 μ μ μ μ 2, S1 1, S 0, S+1 /2 /2 μ μ μ μ μ μ Figure 5.6: CTMC Model of the 1R/2P SCN with LoVis 54 2μ /2 μ /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 μ μ μ μ μ μ μ μ μ μ μ μ μ 2μ μ μ μ μ 2μ μ /2 0, 0 1, 0 2, 0 1, 1 3, 0 2, 1 S, 0 S1, 1 S2, 2 μ μ μ μ μ 4, 0 3, 1 2, 2 μ μ μ μ S1, 0 S2, 1 μ μ μ μ S+1, 0 S, 1 S1, 2 /2 /2 μ 2μ μ μ μ 2μ μ S1, S1 S, S1 S+1, S1 S, S S+1, S S+1, S+1 μ /2 /2 μ μ Figure 5.7: Reduced CTMC of the 1R/2P SCN with LoVis 55 Table 5.3: Rate Balance Equations of the Reduced CTMC Model of the 1R/2P SCN with LoVis State State Transitions Rate Balance Equation i = j = 0 μ i, j i+1, j πi,j = μπi+1,j λ i = 1, j = 0 /2 /2 μ 2μ μ i1, j i, j i+1, j i, j+1 πi,j = λπi−1,j+μπi+1,j+2μπi,j+1] (λ+μ) 2 ≤ i ≤ S−1, j = 0 /2 /2 μ μ μ /2 i1, j i, j i+1, j i, j+1 πi,j = λ 2 πi−1,j+μ[πi,j+1+πi+1,j ] (λ+μ) i = S, j = 0 μ i, j μ μ i1, j i+1, j i, j+1 /2 πi,j = λ 2 πi−1,j+μ[πi,j+1+πi+1,j ] (λ+μ) S + 1 ≤ i < ∞, j = 0 μ μ μ i, j i1, j i+1, j i, j+1 πi,j = μ[πi,j+1+πi+1,j ] (λ+μ) 1 ≤ i < ∞, i = S, i = j μ /2 2μ i, j1 i, j i+1, j πi,j = λ 2 πi,j−1+μπi+1,j ] (λ+2μ) i = j = S μ 2μ i, j1 i, j i+1, j πi,j = λπi,j−1+μπi+1,j ] (λ+2μ) 2 ≤ i < ∞, i = S, i = S+1, j = i−1 /2 /2 μ 2μ μ μ i, j1 /2 i1, j i, j i+1, j i, j+1 πi,j = λ 2 πi,j−1+λπi−1,j+μπi+1,j+2μπi,j+1 (λ+2μ) Continued on next page 56 Table5.3 – continued from previous page State State Transitions Rate Balance Equation i = S, j = S − 1 μ 2μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λ[πi,j−1+πi−1,j]+μπi+1,j+2μπi,j+1 (λ+2μ) i = S + 1, j = S /2 /2 μ 2μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λ[πi,j−1+πi−1,j]+μπi+1,j+2μπi,j+1 (λ+2μ) i ≤ S − 1, 1 ≤ j ≤ S − 3, i ≥ j + 2, and S+ 1 ≤ j < ∞, i ≥ j + 2 /2 /2 μ μ μ μ i, j1 /2 /2 i1, j i, j i+1, j i, j+1 πi,j = λ 2 [πi,j−1+πi−1,j]+μ[πi,j+1+πi+1,j ] (λ+2μ) j = S, i ≥ j + 2 /2 /2 μ μ μ μ i, j1 /2 i1, j i, j i+1, j i, j+1 πi,j = λ 2 πi−1,j+λπi,j−1+μ[πi,j+1+πi+1,j ] (λ+2μ) i = S, 1 ≤ j ≤ S−2 μ μ μ μ i, j1 /2 i1, j i, j i+1, j i, j+1 πi,j = λ 2 πi−1,j+λπi,j−1+μ[πi,j+1+πi+1,j ] (λ+2μ) S +1 ≤ i < ∞, 1 ≤ j ≤ S −1, i ≥ j +2 μ μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λπi,j−1+μ[πi,j+1+πi+1,j ] (λ+2μ) As before, the state transition patterns identified in the reduced CTMC model could not be exploited to obtain a closedform solution. We fixed the maximum number of backorders at the retail store to 1,000 and numerically solved the CTMC. All states in the CTMC model communicate and the CTMC model is positive recurrent, thus confirming the existence of steady state. The CTMC model was validated by comparing its numerical solution to estimates 57 from an Arena simulation model of the 1R/2P SCN with MedVis, as shown in Tables B.5 and B.6. The numerical experiments show that the analytical results match the simulation results, thus confirming the validity of the CTMC model. 5.6 CTMC Model of the 1R/2P SCN with NoVis In this case, we assumed that the retail store places an order with production facility 1 with a fixed probability (p). Because of symmetry, p is 0.5. The corresponding CTMC model is shown in Figure 5.8. The rate balance equations based on the different state transitions that are possible are presented in Table 5.4. /2 /2 μ μ μ μ /2 μ μ μ μ μ μ μ μ /2 /2 /2 /2 /2 /2 /2 /2 /2 0, 0 1, 0 0, 1 2, 0 1, 1 0, 2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ /2 /2 /2 /2 /2 /2 /2 S1, 0 S2, 1 S3, 2 S, 0 S1, 1 S2, 2 S+1, 0 S, 1 S1, 2 μ μ μ μ μ μ /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ /2 /2 /2 /2 /2 /2 /2 2, S3 1, S2 0, S1 2, S2 1, S1 0, S 2, S1 1, S 0, S+1 μ μ μ μ μ μ Figure 5.8: CTMC Model of the 1R/2P SCN with NoVis (p = 0.5) 58 Table 5.4: Rate Balance Equations of the CTMC Model of the 1R/2P SCN with NoVis State State Transitions Rate Balance Equation i = j = 0 μ μ /2 /2 i, j i+1, j i, j+1 πi,j = μ[πi+1,j+πi,j+1] λ i ≥ 1, j = 0 μ μ /2 μ /2 /2 i1, j i, j i+1, j i, j+1 πi,j = λ 2 πi−1,j+μ[πi+1,j+πi,j+1] (λ+μ) i = 0, j ≥ 1 /2 μ μ μ /2 /2 i, j1 i, j i+1, j i, j+1 πi,j = λ 2 πi,j−1+μ[πi,j+1+πi+1,j ] (λ+μ) i ≥ 1, j ≥ 1 /2 /2 /2 /2 μ μ μ μ i, j1 i1, j i, j i+1, j i, j+1 πi,j = λ 2 [πi,j−1+πi−1,j]+μ[πi,j+1+πi+1,j ] (λ+2μ) To solve the CTMC model for the NoVis case, we used the following approach. The demand arrival process at the retail store is split into two streams according to fixed probabilities (as per the Bernoulli routing policy), each stream representing the demand arrival process at the two production facilities. Thus, the order arrival process at each production facility is a Poisson process with rate λ/2. Since the server at each production facility has an exponential processingtime distribution with mean μ, each production facility can be modeled as an independent M/M/1 queue. Let πi, j be the steadystate joint probability that the number of orders at production facilities 1 and 2 are i and j, respectively. The joint probability, πi, j , is a product of the marginal probabilities (steady state solution for the M/M/1 queue) and is given 59 by the expression in (5.12). πi, j = πi · πj = (1 − ρ)ρi(1 − ρ)ρj = (1 − ρ)2ρi+j (5.12) Note that ρ is the utilization of a production facility and is given by λ 2μ The above solution satisfies the rate balance equations shown in Table 5.4, thereby confirming the validity of the solution. Hence, the steadystate distribution for the CTMC model of a symmetric 1R/2P SCN with no information sharing has a productform. Using this distribution, the expressions for the performance measures at a production facility as well as the retail store can be derived. Since the production facilities are symmetric, we present the expressions for only one production facility. The fill rate, the expected number of backorders, the expected inventory level, the expected time to fulfill a backorder, and the expected time spent by an order at a production facility were obtained using expressions (5.13) through (5.17). fp = S−1 i=0 ∞ j=0 πi, j = S−1 i=0 ∞ j=0 (1 − ρ)2 ρi+j = (1 − ρ) (1 − ρ) ∞ j=0 ρj S−1 i=0 ρi = 1− ρS (5.13) 60 E[Ip] = S i=0 ∞ j=0 (S − i)πi, j = (1 − ρ)2 S ∞ k=0 ρk + (S − 1) ∞ k=1 ρk + (S − 2) ∞ k=2 ρk + · · · + ∞ k=S−1 ρk = (1 − ρ)2 S 1 − ρ + (S − 1) ρ 1 − ρ + (S − 2) ρ2 1 − ρ + · · · + ρS−1 1 − ρ = S − ρ − ρS + ρS+1 1 − ρ (5.14) E[Bp] = E[Ip] + E[Np] − S = S − ρ − ρS + ρS+1 1 − ρ + λ 2μ − λ − S = ρS+1 1 − ρ (5.15) E[Wbp] = E[Bp] (λ/2) (1 − fp) = ρS+1 (λ/2) (1 − ρ) ρS = ρ (λ/2) (1 − ρ) (5.16) E[Tp] = E[Time in facility] + E[Time in inventory store] = 1 μ − (λ/2) + E[Ip] λ/2 = 1 μ − (λ/2) + 2 S − ρ − ρS + ρS+1 λ (1 − ρ) (5.17) The reasoning presented in Section 5.2 to obtain the number of orders at the retail store was used to obtain the performance measures at the retail store. The fill rate, 61 the expected inventory level, the expected number of backorders, the expected time to fulfill a backorder, and the expected time spent by an order at the retail store can be obtained by using expressions (5.18) through (5.23). fr = i, j∈A1 πi, j = (1 − ρ)2 1 + 2ρ + · · · + (S + 1) ρS + SρS+1 + (S − 1) ρS+2 + · · · + ρ2S + (1 − ρ)2 S k=1 2ρS+k + S l=1 2ρS+l+1 + · · · + S m=1 2ρS+R+m−2 + (1 − ρ)2 2ρ2S+1 + 3ρ2S+2 + · · · + Rρ2S+R−1 = (1 − ρ)2 S x1=0 (1 + x1) ρx1 + S x2=1 



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