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ANALYTICAL PERFORMANCE MODELING OF SUPPLY CHAIN NETWORKS By SANDEEP SRIVATHSAN Bachelor of Engineering University of Madras Madras, India 2002 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE December, 2004 ii ANALYTICAL PERFORMANCE MODELING OF SUPPLY CHAIN NETWORKS Thesis Approved: Dr. Manjunath Kamath Thesis Adviser Dr. Allen. C. Schuermann Dr. Ricki. G. Ingalls Dr. A. Gordon Emslie Dean of the Graduate College ii i ACKNOWLEDGEMENTS My obeisance to all upadhyayas – respects to all learned teachers. The world is made up of three classes of people. 1. The majority who do not know what is happening. 2. Many watch things happen 3. A few make things happen. My teachers at OSU belong to the third type of people. It is difficult to express my gratitude for the amount of help that my advisor, Dr. Manjunath Kamath, rendered in every respect, for being patient and for giving me an opportunity to carry out this research work as well as giving various valuable suggestions throughout my stay here as a Master's student. I owe a lot to him for providing advice regarding the course selection, moral support and last but not the least for proof reading this thesis. I would also like to thank the committee members, Dr. Allen Schuermann and Dr. Ricki Ingalls, for providing some valuable information and insights into my work. I would like to thank Dr. Allen Schuermann for helping me with suggestions in understanding of various simulation aspects and for continuing as a committee member even after his retirement. I would like to thank Dr. Ricki Ingalls for sharing his work experience with his students and thereby creating an interest in me for supply chains. I would also take this opportunity to thank all my friends in Center for Computer Integrated Manufacturing Enterprises (CCiMe) team for their support at various stages of iv my thesis. I would like to thank Sarath Kureti and Ananth Krishnamoorthy for helping me with the simulation models and Mohan Chinnaswamy for helping me with some queueing details. I appreciate the help that I received from Parthiban Dhananjeyan, Karthik Ayodhiramanujan and Uma Maheshwar Chalavadi in developing my C programs for the analytical model. Finally, I would like to dedicate this thesis to my parents, Sri. S. Srivathsan and Smt. Janaki Srivathsan, my sister Ms. Jaisree Srivathsan, and all other close relatives for their love and support. I owe a lot to them for constantly motivating me with a lot of advice each and every week over phone. But for their constant encouragement, support and guidance, the work would have faced problems time and again. v LIST OF CONTENTS INTRODUCTION........................................................................................................................................1 1.1 ANALYTICAL MODELS BASED ON QUEUEING..........................................................................3 1.2 OUTLINE OF THE THESIS ................................................................................................................4 LITERATURE REVIEW............................................................................................................................5 2.1 PRODUCTION  INVENTORY NETWORKS....................................................................................5 2.1.1 MaketoOrder Production Inventory Networks ...........................................................................6 2.1.2 MaketoStock Production Inventory Networks ............................................................................6 2.1.3 Hybrid Production Inventory Networks ......................................................................................10 2.2 SUPPLY CHAIN NETWORKS.........................................................................................................11 STATEMENT OF RESEARCH................................................................................................................13 3.1 RESEARCH OBJECTIVES ...............................................................................................................13 3.2 RESEARCH SCOPE AND LIMITATIONS ......................................................................................14 3.3 RESEARCH CONTRIBUTIONS.......................................................................................................14 RESEARCH APPROACH........................................................................................................................15 4.1 METHODOLOGY............................................................................................................................15 4.1.1 The Parametric Decomposition (PD) Approach.........................................................................15 4.1.2 Modeling Approach ....................................................................................................................17 4.1.3 Validation...................................................................................................................................17 4.2 PERFORMANCE MEASURES.........................................................................................................19 4.3 SUPPLY CHAIN NETWORK CONFIGURATIONS........................................................................19 4.3.1 A Divergent Supply Chain Network Configuration.....................................................................19 vi 4.3.2 Convergent Supply Chain Network Configurations....................................................................20 4.3.3 Combination ConvergentDivergent Supply Chain Network Configuration ..............................21 DIVERGENT CONFIGURATION WITH TWO RETAILERS ............................................................22 5.1 SYSTEM DESCRIPTION..................................................................................................................22 5.2 QUEUEING MODEL OF THE DIVERGENT CONFIGURATION .................................................23 5.2.1 Approximate Solution of the Queueing Model ............................................................................24 5.3 NUMERICAL EXPERIMENTS ........................................................................................................29 5.4 SUMMARY OF RESULTS................................................................................................................29 CONVERGENT CONFIGURATION WITH TWO MANUFACTURERS..........................................32 6.1 SYSTEM DESCRIPTION..................................................................................................................32 6.2 QUEUEING MODEL OF THE CONVERGENT CONFIGURATION WITH TWO MANUFACTURERS..............................................................................................................................34 6.2.1 Approximate Solution of the Queueing Model ............................................................................34 6.3 NUMERICAL EXPERIMENTS ........................................................................................................38 6.4 SUMMARY OF RESULTS................................................................................................................38 CONVERGENT CONFIGURATION WITH TWO SUPPLIERS.........................................................41 7.1 SYSTEM DESCRIPTION..................................................................................................................41 7.2 QUEUEING MODEL OF THE CONVERGENT CONFIGURATION WITH TWO SUPPLIERS...43 7.2.1 Approximate Solution of the Queueing Model ............................................................................43 7.3 NUMERICAL EXPERIMENTS ........................................................................................................48 7.4 SUMMARY OF RESULTS................................................................................................................49 CONVERGENT–DIVERGENT CONFIGURATION.............................................................................51 8.1 SYSTEM DESCRIPTION..................................................................................................................51 vi i 8.2 QUEUEING MODEL OF THE CONVERGENTDIVERGENT CONFIGURATION.....................52 8.2.1 Approximate Solution for the Queueing Model...........................................................................53 8.3 NUMERICAL EXPERIMENTS ........................................................................................................61 8.4 SUMMARY OF RESULTS................................................................................................................62 CONCLUSIONS AND FUTURE RESEARCH........................................................................................64 9.1 RESEARCH SUMMARY..................................................................................................................64 9.2 Research Contributions .................................................................................................................65 9.3 FUTURE RESEARCH.......................................................................................................................66 9.3.1 Extensions to the Divergent Configuration.................................................................................66 9.3.2 Extensions to Convergent Configuration 1 .................................................................................68 9.3.3 Extensions to the Convergent Configuration 2 ...........................................................................69 REFERENCES...........................................................................................................................................70 RESULTS FOR THE DIVERGENT CONFIGURATION.....................................................................74 RESULTS FOR THE CONVERGENT CONFIGURATION WITH TWO MANUFACTURERS ....99 RESULTS FOR THE CONVERGENT CONFIGURATION WITH TWO SUPPLIERS.................124 RESULTS FOR THE CONVERGENTDIVERGENT NETWORK...................................................135 SIMULATION MODEL..........................................................................................................................141 A5.1 THE SIMULATION LOGIC.........................................................................................................141 A5.2 DETERMINATION OF DISTRIBUTION PARAMETERS FOR SIMULATION.......................148 A5.2.1 Erlang Distribution ................................................................................................................148 A5.2.2 Exponential Distribution ........................................................................................................148 A5.2.3 HyperExponential Distribution .............................................................................................149 vi ii DETERMINATION OF THE WARMUP PERIOD AND RUN LENGTH FOR SIMULATION EXPERIMENTS.......................................................................................................................................151 A6.1 INTRODUCTION........................................................................................................................151 A6.2 WELCH’S TECHNIQUE TO DETERMINE WARMUP PERIOD..............................................152 A6.3 DETERMINATION OF RUN LENGTH.......................................................................................153 ix LIST OF FIGURES FIGURE 1.1 SUPPLY CHAIN NETWORK ..............................................................................................1 FIGURE 2.1 SINGLESTAGE MAKETOSTOCK SYSTEM WITH A DELAY NODE....................9 FIGURE 4.1 DIVERGENT CONFIGURATION.....................................................................................20 FIGURE 4.2 CONVERGENT CONFIGURATION 1 .............................................................................20 FIGURE 4.3 CONVERGENT CONFIGURATION 2 .............................................................................21 FIGURE 4.4 CONVERGENTDIVERGENT CONFIGURATION.......................................................21 FIGURE 5.1 DIVERGENT CONFIGURATION.....................................................................................23 FIGURE 6.1 CONVERGENT CONFIGURATION 1 .............................................................................33 FIGURE 7.1 CONVERGENT CONFIGURATION 2 .............................................................................42 FIGURE 7.2 SPLITTING AND MERGING AT THE INPUT STORE.................................................45 FIGURE 8.1 CONVERGENTDIVERGENT SUPPLY CHAIN NETWORK......................................52 FIGURE A5.1 SIMULATION MODEL .................................................................................................144 FIGURE A5.2 RETAILER 1 SUBMODEL...........................................................................................145 FIGURE A5.3 RETAILER 2 SUBMODEL...........................................................................................146 FIGURE A5.4 MANUFACTURER SUBMODEL................................................................................147 FIGURE A6.1 PLOT OF TIME IN SYSTEM........................................................................................154 FIGURE A6.2 PLOT OF TIME IN SYSTEM (CONTINUED)............................................................154 FIGURE A6.3 PLOT OF TIME IN SYSTEM (CONTINUED)............................................................155 x FIGURE A6.4 PLOT OF TIME IN SYSTEM (CONTINUED)............................................................155 FIGURE A6.5 PLOT OF TIME IN SYSTEM (CONTINUED)............................................................156 FIGURE A6.6 PLOT OF TIME IN SYSTEM (CONTINUED)............................................................156 FIGURE A6.7 PLOT OF TIME IN SYSTEM (CONTINUED)............................................................157 FIGURE A6.8 PLOT OF TIME IN SYSTEM (CONTINUED .............................................................157 FIGURE A6.9 PLOT OF TIME IN SYSTEM (CONTINUED)............................................................158 FIGURE A6.10 PLOT OF TIME IN SYSTEM (CONTINUED)..........................................................158 xi LIST OF TABLES TABLE 4.1: SIMULATION PARAMETERS.........................................................................................18 TABLE 4.2: SCV AND DISTRIBUTIONS..............................................................................................18 TABLE 5.1: EXPERIMENTAL DESIGN FOR NUMERICAL COMPARISON.................................29 TABLE 5.2: SUMMARY OF RESULTS .................................................................................................30 TABLE 6.1: EXPERIMENTAL DESIGN FOR NUMERICAL COMPARISON.................................38 TABLE 6.2: SUMMARY OF RESULTS .................................................................................................39 TABLE 7.1: EXPERIMENTAL DESIGN FOR NUMERICAL COMPARISON................................48 TABLE 7.2: SUMMARY OF RESULTS .................................................................................................49 TABLE 8.1: EXPERIMENTAL DESIGN FOR NUMERICAL COMPARISON.................................61 TABLE 8.2: SUMMARY OF RESULTS .................................................................................................62 xi i NOTATION Ni random variable representing the number of orders at retailer i Nsi random variable representing the number of orders at the supplier side of manufacturer i Nmi random variable representing the number of orders in queue or being processed at manufacturer i Nni random variable representing the number of orders at manufacturer i (backorders + in queue + in process) Npij random variable representing the number of orders at manufacturer j corresponding to the pending customer orders at retailer i Ntmirj random variable representing the number of orders in transit from manufacturer i to retailer j Brmpij random variable representing the number of orders that are backordered at the supplier side of manufacturer j that belong to retailer i ir λ order arrival rate at retailer i j im r λ order arrival rate from retailer i at manufacturer j xi ii mi λ order arrival rate at manufacturer i misj λ order arrival rate from manufacturer i at supplier j i s λ order arrival rate at supplier i mi τ mean processing time at manufacturer i i ρ utilization of manufacturer i 2 cari SCV of the arrival process at retailer i 2 cspij SCV of the order arrival process at manufacturer j after splitting at retailer i 2 cmergei SCV of the combined order arrival process at manufacturer i after merging of arrival streams 2 cami SCV of the arrival process at manufacturer i after accounting for supplier delay 2 csmi SCV of the service time distribution at manufacturer i 2 carmi SCV of the arrival stream at manufacturer i that consist of orders that find a part in the input store 2 cdi SCV of the arrival stream at manufacturer i that consist of satisfied backorders at the input store Sri basestock level at retailer i I ri random variable representing the inventory level at retailer i xi v Bri random variable representing the number of backorders at retailer i Fri fill rate at retailer i Smi basestock level at manufacturer i Imi random variable representing the inventory level at the output store of manufacturer i Bmi random variable representing the number of backorders at the output store of manufacturer i Fmi fill rate at the output store of manufacturer i S rmi basestock level at the input store of manufacturer i I rm i random variable representing the inventory level at the input store of manufacturer i Brmi random variable representing the backorder at the input store of manufacturer i Frmi fill rate at the input store of manufacturer i pmirj proportion of orders at manufacturer i that belong to retailer j pmisj proportion of orders from manufacturer i that go to supplier j mirj τ expected transit time from manufacturer i to retailer j tmirj ρ average number of items in the transit from manufacturer i to retailer j xv smi τ expected aggregate delay in delivering raw material at the supplier side of manufacturer i tsmi ρ average number of orders at the supplier stage of manufacturer i simj τ mean transit time from supplier i to the input store at manufacturer j lti τ mean leadtime delay at supplier i plti probability of stockout at supplier i xv i GLOSSARY OF TERMS The definitions of selected technical terms are included in this glossary to clarify their intended meaning and usage. Oneforone replenishment The consumption of an item triggers an immediate replenishment order for that item. This is a special case of the (s, S) inventory policy, namely, the (S1, S) policy. Basestock level This value determines the maximum inventory that can be stored at a particular stage. Fill rate This value gives the probability that an order is satisfied instantaneously by items in stock. SCV The Squared Coefficient of Variation (SCV) is defined as the ratio of the variance of a random variable to the square of its mean. 1 CHAPTER 1 INTRODUCTION A supply chain network consists of nodes representing suppliers, production points, warehouses, distributors, retailers and end customers. An example supply chain network is illustrated in Figure 1.1 (adapted from Cohen and Lee 1988). A supply chain network represents a collection of organizations that are involved in processes related to the production of end items. It aims at attaining customer satisfaction and maximizing profit for all organizations belonging to the supply chain. So as a whole, the various constituents of the supply chain act as a closelyknit structure. Figure 1.1 Supply Chain Network Performance evaluation involves the development and solution of analytical or simulation models for determining the values of the performance measures that can be Raw Material Vendors Intermedia te Product Plants Final Product Pla nts Distribution Centers Warehouse s Customer Zon es 2 expected from a given set of decisions (e.g., those related to capacity and inventory levels). Performance evaluation tools aid system designers and operations managers in making some key decisions, while keeping in mind the goals of the company (Suri et al. 1993). Analytical performance evaluation tools are typically based on modeling techniques such as Markov chains, stochastic Petri nets, and queueing networks (Viswanadham and Narahari 1992). Simulation models mimic the detailed operations of a system by means of a computer program and hence, they need more detailed information for modeling. Simulation model development and model execution could be time consuming. On the other hand, analytical models like queueing networks describe the actual system based on mathematical relationships. These models need certain simplifying assumptions to be made (e.g., first come first serve (FCFS) queueing discipline) and the results obtained are generally less accurate compared to simulation results. However, these models yield results more quickly and “are appropriate for rapid and rough cut analysis (Suri et al. 1993).” In fact, analytical and simulation models can be used in tandem to analyze and design complex systems. For example, analytical models can be used to reduce a large set of alternatives, and the remaining few alternatives can be studied in detail using simulation models (Suri et al. 1993; Leung and Suri 1990). The development of performance evaluation and optimization models for supply chain networks is an active area of research. Only a few studies have dealt with the development of analytical performance evaluation models for supply chain networks. The focus of this research was on developing queueing network models of supply chain networks using Whitt’s parametric decomposition approach (Whitt 1983, 1994). The 3 approach used in this research was similar to the approach developed by Sivaramakrishnan (1998) in modeling productioninventory networks. The approach used by Sivaramakrishnan has been extended to include supply chain network features such as supply, transportation and distribution (Srivathsan et al. 2004). 1.1 ANALYTICAL MODELS BASED ON QUEUEING Queueing models of manufacturing systems have been developed since the 1950’s. Some of the literature in this field includes the works of Jackson (1957), Segal and Whitt (1989) and text books and handbook chapters that focus on queueing models of manufacturing systems including Buzacott and Shanthikumar (1993), Viswanadham and Narahari (1992) and Suri et al. (1993). Queueing models have often ignored the effect of stocking (the holding of planned inventory). On the other hand, inventory models have ignored the effect of capacity and congestion issues and concentrated mainly on the inventory policy, order quantity, etc. (Sivaramakrishnan 1998). Models of productioninventory networks which extend standard queueing models to include planned inventories by considering capacity/congestion issues and inventory issues, simultaneously, came to the forefront with the works of Buzacott and Shanthikumar (1993), Lee and Zipkin (1992), Sivaramakrishnan (1998), Sivaramakrishnan and Kamath (1996) and Zipkin (1995). These models make assumptions like the oneforone replenishment policy (where for each item consumed, an order is immediately placed for one item), orders for a single item or orders for batches of constant size, unlimited supply of raw materials, etc. Buzacott and Shanthikumar (1993) have modeled the singlestage systems extensively, while Lee and Zipkin (1992), Sivaramakrishnan (1998), Sivaramakrishnan and Kamath (1996) and 4 Zipkin (1995) concentrated on multistage systems. In particular, Sivaramakrishnan (1998) dealt extensively with tandem maketostock systems, and later extended his work to model feedforward type maketostock networks. Modeling a supply chain network includes the modeling of capacity and inventory issues. However, additional features like supply, distribution and transportation have to be included to yield more useful models. Hence, this research has extended the previous work on productioninventory networks to model supply chain networks by including such features. 1.2 OUTLINE OF THE THESIS The rest of the document is structured as follows. Chapter 2 presents a literature review of the work done in modeling productioninventory networks and supply chain networks. Chapter 3 presents the research statement. It includes the research goals, objectives, limitations and contributions to the field of modeling supply chain networks using queueing theory. Chapter 4 presents the research approach. It includes the methodology, the list of performance measures that will be evaluated and the validation method. Chapter 5 presents the analytical model of a divergent network consisting of one manufacturer with a finished goods store and two retailers. Chapter 6 presents the analytical modeling of the convergent network with two manufacturers and one retailer. Chapter 7 presents the analytical model of the convergent network with two suppliers. Chapter 8 presents the analytical model of a convergentdivergent network, which is a combination of the above three networks. Finally, Chapter 9 presents the conclusions and the scope for future research. 5 CHAPTER 2 LITERATURE REVIEW This chapter presents a review of the literature on the analytical performance modeling of productioninventory networks and supply chain networks. Section 2.1 deals with the modeling of productioninventory networks. Queueing models have been mainly used to model the following types of productioninventory networks, namely, maketoorder systems, maketostock systems and hybrid systems (combination of maketoorder and maketostock systems). This section summarizes the work related to these three types of productioninventory networks. Section 2.2 deals with the literature on performance evaluation of supply chain networks. 2.1 PRODUCTION  INVENTORY NETWORKS When manufacturing systems are considered along with the holding of planned inventories, the resulting network is a productioninventory network. These systems may be classified as maketoorder, maketostock, and a hybrid of these two based on the inventory policy. Research related to the modeling of these networks is presented in this section. 6 2.1.1 MaketoOrder Production Inventory Networks In a maketoorder system, the constituents wait for the customer to place an order. Once an order is placed, the constituents of the system start interacting with each other through the flow of material as well as information about customer requirements and the final product is manufactured. In a pure maketoorder system, no inventory of finished products or intermediate items is maintained. A pure maketoorder system is exactly what a queueing model represents. For a comprehensive review of singlestage as well as network models, the reader is referred to Suri et al. (1993). Detailed coverage of queueing models of production networks can also be found in text books such as Buzacott and Shanthikumar (1993) and Viswanadham and Narahari (1992). Whitt (1983, and 1994) and Segal and Whitt (1989) describe the popular parametric decomposition approach based on twomoment queueing approximations to solve general production network models. 2.1.2 MaketoStock Production Inventory Networks In a maketostock system, the final product is manufactured and sent to the retail store in anticipation of customer demand. As customer orders arrive, the retailer sells from the stock of finished product inventory held at the retail store. As the inventory level falls to a predetermined value, the retailer orders for the product from the manufacturer so that he is in the best position to satisfy future customer demand. Svoronos and Zipkin (1991) modeled a multiechelon inventory system where demand occurred at the lowest hierarchy of the system called the leaf, and each leaf placed an order at its predecessor and so on till the demand reached the central depot. The central depot received raw materials from an outside source assumed to have infinite 7 capacity. They analyzed the singlestage location with Poisson demand arrivals and onefor one replenishment policy with a basestock level S. The consumption of an item immediately triggers a replenishment order. The steadystate behavior of the system was characterized by the random variables I, B and K where I is the inventory, B is the number of backorders and K = (S–I+B) is the number of outstanding orders at a particular stage. They applied the results (expected inventory and expected backorders) of the singlestage problem, recursively, starting from the highestlevel echelon to analyze the complete network. Lee and Zipkin (1992) developed a model for tandem queues with planned inventories. They assumed Poisson demand arrivals, mutually independent exponential service times, and a oneforone replenishment policy where the system was controlled by a stationary demand pull or basestock policy. The basestock level for stage j was specified by Sj. The arrival of customer demand consumed a finished product from the output store, if available, and a corresponding order was placed at the previous stage for replenishment. If the finished product was not available, then the order waited for the part to arrive from the previous stage and the same process was assumed to occur at all stages. They assumed that the queue at the processing stage to be infinite. After processing, the part either moved to the output store at that stage or to the next stage depending on the demand. Any backorder at the instant of process completion immediately released the part to the next stage or to the customer to fulfill the backorder. When the basestock level is zero at all stages, the system behaves like a maketoorder system, and if only certain stages have a basestock level of zero, then the system behaves as a combination of maketoorder and maketostock systems. Lee and Zipkin (1992) captured congestion 8 measures like expected inventory and expected backorders using the M/M/1 queueing model, and made use of the approximation developed by Svoronos and Zipkin (1991). Buzacott and Shanthikumar (1993) analyzed a singlestage maketostock system in great detail. They used a production authorization card (PA card) concept to model the maketostock system. They assumed that as a product was manufactured and sent to a retail store, a tag was attached to it and this tag was converted into a production authorization card when the product was delivered to the customer. They modeled a single machine with unit demand and backlogging by assuming that if the finished product inventory is not available when a customer order arrived, the order is backlogged and that each customer placed an order for only one finished product. They considered the retail store to be full initially and at that instant, there were Z tags in the system. ( ) ( ) ( ) Therefore, ( ) { ( ), } ( ) {0, ( ) } ( ) {0, ( )} B t C t N t C t Min N t Z B t Max N t Z I t Max Z N t + = = = − = − where I(t) is the inventory level or the number of finished products, B(t) is the number of customer orders backlogged, C(t) is the number of PA cards available, and N(t) is the number of jobs in the system, all at time t. As a result of Equation 3.4, it can be seen that by studying the process N(t), I(t), B(t) and C(t) can be derived. The maketostock system was modeled using an M/M/1 queue and under steadystate condition, they found out the expected inventory and expected number of backorders. They also presented the respective results for the GI/G/1 (3.1) (3.2) (3.3) (3.4) 9 model. They also modeled singlestage maketostock systems with lost sales, interrupted demand, bulk demand, machine failures and yield losses. Sivaramakrishnan (1998) modeled an Mstage tandem maketostock system that was controlled by a stationary demandpull or basestock policy. He assumed that the setup times were included in the processing times, and that if finished goods were available at the output store, then the order was satisfied and an order for the item would be triggered to replenish the finished item as per the oneforone replenishment policy. If parts were available at stage M1, then replenishment took place. Otherwise the order was backlogged at stage M1 and an order was placed at the previous stage and so on. The arrival of a part at any stage could be an item directly after production at the previous stage in the case of a backorder or an item from the output store of the previous stage in the case of replenishment orders. At the first stage, the order went directly to process as unlimited supply of raw material was assumed. A delay node was used in the model mainly to “capture the upstream delay experienced by an order when there was no part in the output store of the previous stage (Sivaramakrishnan 1998).” The singlestage model with a delay node is as shown in Figure 2.1. Figure 2.1 Singlestage Maketostock System with a Delay Node Output store Demand Processing Stage Backorder Delay 10 Sivaramakrishnan (1998) developed a new decomposition framework for the analysis of the above system using results of M/M/1 and M/M/∞ queueing systems and obtained the results for the expected inventory and expected backorders at each stage of the tandem network. He also presented the corresponding results for a maketostock tandem network with general arrivals and general service times by extending the parametric decomposition approach (Whitt 1983, 1994) to maketostock systems. He extended the tandem model to include (1) multiple servers at a stage, (2) batch service, (3) limited supply of raw material, (4) multiple part types, and (5) service interruptions due to machine failure. Sivaramakrishnan (1998) also extended his approach to tandem networks with feedback and feedforward networks. Karaesmen et al. (2002) assumed that the interarrival time and the processing times are geometrically distributed and modeled the system with advance order information, where the orders are received well in advance of the time when the items covered by the orders are required to be delivered. They analyzed the basestock policy (S, L) with a focus on optimization and performance evaluation of the Geo/Geo/1 maketo stock queue, where L is the release leadtime. They went on to obtain the expressions for expected backlog and expected inventory. 2.1.3 Hybrid Production Inventory Networks Nguyen (1995) analyzed the problem of setting the basestock levels in a production system that produced both maketoorder and maketostock products. She made the following assumptions: 1) the demand that cannot be satisfied is lost and 2) the production of maketostock items is based on a basestock policy with a basestock level for each product. She derived the productform steadystate distributions for the above 11 network under the assumptions that 1) each station operates under first in first out (FIFO) service discipline, 2) all processing times and interdemand times follow exponential distribution, 3) all products have the same mean processing time, and 4) the contribution from maketoorder jobs to relative traffic intensity at the workstation is less than one. She proposed approximations for the basestock levels based on heavy traffic analysis of queueing networks. Nguyen (1998) presented an algorithm for setting the basestock levels for 1) FIFO service priority, 2) priority service for maketoorder products, and 3) priority service for maketostock products. 2.2 SUPPLY CHAIN NETWORKS Cohen and Lee (1988) developed an analytical model for integrated productiondistribution systems to predict the impact of alternative strategies. They decomposed the network into submodels and optimized the submodels based on certain control parameters which serve as links between the submodels. Lee and Billington (1993) and Ettl et al. (2000) both focused on “capturing the interdependence of basestock levels at different stores (Ettl et al. 2000).” Further, both used models with limited capacity. While Lee and Billington (1993) focused on only performance evaluation and assumed stationary demand, Ettl et al. (2000) focused on optimization as well and considered nonstationary demand. Raghavan and Viswanadham (2001) and Viswanadham and Raghavan (1999, 2001, and 2002) developed highlevel models of a supply chain network using a variety of tools like Petri nets, series parallel graphs, and queueing theory. With regard to queueing models, their focus was on the use of forkjoin approximations to compute the mean and variance of the departure process in a supply chain network. They presented 12 simple approximations for the case of deterministic arrivals and Normal service time distributions. In summary, it was observed that research on developing analytical performance evaluation models of supply chain networks was very limited. The preliminary work done as part of this research had revealed that productioninventory network models could be extended to model supply chain networks (Srivathsan et al. 2004). This was the focus of the thesis research effort. 13 CHAPTER 3 STATEMENT OF RESEARCH The overall goals of this research were (i) to develop queueing models of “building block” type configurations of supply chain networks and solve them using Whitt’s parametric decomposition approach and (ii) to lay a foundation for the development of a queueingbased rapid analysis tool for the performance evaluation of supply chain networks. 3.1 RESEARCH OBJECTIVES The objectives of this research were as follows. Objective 1: To perform a review of the literature related to analytical modeling of productioninventory networks and supply chain networks. Objective 2: To model a divergent supply chain network configuration with one manufacturer supplying two retail locations. Objective 3: To model two convergent supply chain network configurations, one with two manufacturers supplying one retailer and the other with two suppliers supplying a manufacturer. 14 Objective 4: To model a combination convergentdivergent supply chain network configuration with three suppliers supplying two manufacturers, who in turn provide finished goods to three retailers. The details of the supply chain network configurations are contained in Section 4.3. Modeling involves developing the analytical model and validating it using simulation (see Section 4.1.3). 3.2 RESEARCH SCOPE AND LIMITATIONS The scope of this thesis was limited by the following assumptions: 1. Assembly operations and material handling issues were not considered. 2. Each order was for a single item or a batch of items with a constant batch size. 3. Only limited “building block” type configurations were studied (see Section 4.3). 3.3 RESEARCH CONTRIBUTIONS The purpose of this thesis was to contribute to the development of a queueingbased rapid performance analysis tool for supply chain networks. The following contributions have resulted from this work. 1. Development of queueingbased models of “building block” type supply chain network configurations. 2. Preliminary work on combining the “building block” configurations to model larger supply chain networks. 15 CHAPTER 4 RESEARCH APPROACH This chapter briefly describes the approach that was taken to successfully complete the research work. It also includes a list of important supply chain network performance measures that were addressed in this research, and the specific supply chain network configurations that were included in this study. 4.1 METHODOLOGY In this research, the supply chain network was modeled using queueing network models based on the twomoment framework. Whitt’s (1983, and 1994) parametric decomposition (PD) approach was then used to solve the network model. The PD approach is summarized in the next section. 4.1.1 The Parametric Decomposition (PD) Approach In the 1980’s, Whitt defined a new modeling ideology highlighted by the PD approach. According to Whitt, “a natural alternative to an exact analysis of an approximate model is an approximate analysis of an exact model (Whitt 1983).” The PD approach is a very comprehensive method of analyzing a network and uses only the first two moments of both the interarrival and service times. This approach is the basis of 16 a software package developed by Whitt, on behalf of AT&T, called the Queueing Network Analyzer (QNA). The PD approach for open queueing networks consists of two main steps: 1) analyzing the nodes and the interaction between the nodes to obtain the means and the squared coefficients of variation (SCV) of the interarrival times and 2) obtaining the node and system performance measures based on GI/G/1 or GI/G/m approximations. 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 (1957) 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 (1983). 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 and GI/G/m queues provided by Whitt (1983, 1993). The expected waiting time at each node is calculated from the results provided and 17 the expected queue length is obtained using Little’s law (Little 1961). Whitt (1983) also explains how several other node and network measures can be calculated. 4.1.2 Modeling Approach The approach developed by Sivaramakrishnan (1998) has been extended to model a productioninventory network by adding supply, distribution and transportation features. The results provided by Sivaramakrishnan (1998) and Buzacott and Shanthikumar (1993) were used to model the manufacturer stage. In the case of raw material supply and transportation, delay nodes were used to represent the supply leadtimes and transit times. The delay node was modeled using the M/G/∞ queue (as a result, the number in transit distribution is given by the Poisson distribution), which was similar to the approach of Sivaramakrishnan (1998). The only difference being that Sivaramakrishnan had used the delay node to capture the upstream delay experienced by an order when there was no part in the output store of the previous stage. The splitting and merging approximations presented by Whitt (1983) were extensively used to model a single node supplying multiple retail stores, and multiple nodes supplying a single retail store, respectively. 4.1.3 Validation Each supply chain network configuration that was modeled was simulated using Arena 7.0 software (Kelton et al. 2002). The parameters used for all the simulation experiments are presented in Table 4.1. The warmup period was determined by the application of Welch’s procedure (Welch 1983). Further details regarding the application of Welch’s procedure are contained in Appendix A6. 18 Table 4.1: Simulation Parameters Number of Replications Warmup period (time units) Run time (time units) 10 10,000 100,000 The parameters of the supply chain network configuration, namely, basestock levels; variability of interarrival, processing and transit times; utilization; and probabilities (if any)  were varied systematically to cover a wide range of scenarios. Table 4.2 gives the levels of variability that were tested in all the experimental configurations. The details about the distributions used and the procedure to estimate their parameters for running the simulation model are given in Appendix A6. The various values of the parameters used for experimentation are summarized in tables at the end of each chapter. Table 4.2: SCV and Distributions SCV Distribution 0.25 4stage Erlang 1 Exponential 2.25 2stage Hyperexponential For each scenario, the analytical results were compared with steadystate simulation estimates to evaluate the accuracy of the analytical results. Relative percentage error was used as an indication of the accuracy of the analytical model whenever it was appropriate. 100% simulation estimate Relative percentage error = (analytical result  simulation estimate) ⋅ As per Whitt’s (1993) suggestion, when the analytical and simulation estimates of a performance measure were small, i.e., less than one, the criterion that was used to 19 evaluate the accuracy of the analytical model was the absolute difference and not the relative percentage error. 4.2 PERFORMANCE MEASURES The performance measures that were evaluated were the expected inventory, expected backorders and the fill rate at the various stages of the supply chain network. The expected inventory at any stage is the average number of items in the store at that stage. The expected backorder at any stage is the average number of unsatisfied orders waiting at the stage. The fill rate is the probability that an order will be satisfied immediately and this depends on the instantaneous availability of inventory at the stage. This arrivaltime probability is approximated by the steadystate probability that the inventory level is not zero. 4.3 SUPPLY CHAIN NETWORK CONFIGURATIONS There are various configurations of a supply chain network, such as the serial network, convergent network, divergent network and convergentdivergent network. The following supply chain network “building block” configurations were studied in this thesis. 4.3.1 A Divergent Supply Chain Network Configuration In the divergent supply chain network configuration that was considered (see Figure 4.1), a single manufacturer produced the finished product and shipped it to two retailers. 20 Figure 4.1 Divergent Configuration 4.3.2 Convergent Supply Chain Network Configurations In the first convergent supply chain network configuration (see Figure 4.2), two manufacturers produced the same finished product and shipped it to a single retailer. Figure 4.2 Convergent Configuration 1 In the second convergent supply chain network configuration (see Figure 4.3), two suppliers supplied the same raw material to a single manufacturer. The case where the suppliers had different leadtimes was also modeled. M Manufacturer Retailers D1 D2 M1 Manufacturers Retailer D1 M1 21 Figure 4.3 Convergent Configuration 2 4.3.3 Combination ConvergentDivergent Supply Chain Network Configuration In the convergentdivergent supply chain network configuration (see Figure 4.4), three suppliers supplied two manufacturers who produced the same finished product and shipped it to three different retailers. The case where the manufacturers had different service time parameters was also modeled. Figure 4.4 ConvergentDivergent Configuration M1 D1 D2 M2 Manufacturers Retailers D3 S1 S2 Suppliers S3 M Manufacturer D S1 Suppliers S2 22 CHAPTER 5 DIVERGENT CONFIGURATION WITH TWO RETAILERS In this chapter, an analytical model for the divergent configuration with one manufacturer and two retailers with general demand interarrival times and general service times at the manufacturer is presented. The approach developed by Sivaramakrishnan (1998) was extended by adding distribution and transportation features. In order to model the general demand arrival processes, the traffic variability equations for the superposition of arrivals (Whitt 1983) were used. The remainder of the chapter is organized as follows. Section 5.1 gives a description of the system under study and the assumptions made. The mathematical procedure is explained in Section 5.2 and numerical results are presented in Section 5.3. Section 5.4 presents a summary of the results and future scope for improving the accuracy of the analytical model. 5.1 SYSTEM DESCRIPTION The system modeled has one manufacturer with a finished goods store and two retailers (see Figure 5.1). The finished goods store and the retail stores operate under a basestock policy with oneforone replenishment. The demand interarrival times at each of the retailers as well as the service times at the manufacturer follow a general distribution. The arrival of a demand consumes a finished product from the retailer and 23 causes an order to be placed at the manufacturer immediately. The transit time to ship the finished product from the output store of the manufacturer to the retailer was modeled using a delay node. The manufacturer was modeled by a singleserver queueing system. Figure 5.1 Divergent Configuration Ample supply of raw material was assumed. Furthermore, it was assumed that the raw material is released as and when an order is placed at the manufacturer. If the finished product is not available at the time of arrival of the customer order, then demand is backordered. When a finished product is produced, it is used to satisfy an outstanding backorder, if any; otherwise it is stored. 5.2 QUEUEING MODEL OF THE DIVERGENT CONFIGURATION The manufacturer was modeled by a singleserver queue and the two transit operations to the two retailers were modeled using delay nodes. Each delay node was modeled as an M/G/∞ queue. Manufacturer Retailer 1 Retailer 2 Output Store of Manufacturer External Demand External Demand Part Flow Replenishment+ Backorders Transit Time Transit Time 24 5.2.1 Approximate Solution of the Queueing Model When a customer order arrives at a retailer, if parts are available, the order is satisfied immediately and a replenishment order is sent to manufacturer 1. If a part is available in the output store at manufacturer 1, then it is immediately shipped to the retailer who placed the order. So any order waiting at either of the retailers is reflected in the parts in transit to the retailer or in manufacturer’s queue in case of any backorder at manufacturer 1. The distribution of the number in system can be obtained using the fact that if there are n orders at retailer i, k may be in transit and the remaining (nk) will be at manufacturer 1 and assuming that the random variables Ntm ri 1 and N pi1 are stochastically independent. This is an approximation because the network model does not satisfy the productform conditions and includes an approximation for merging in the arrival process at manufacturer 1 (Sivaramakrishnan 1998; Suri et al. 1993). ( ) ( ) ( ) 1 1 0 P N n P Nt k P N n k i p i n k r m i − = ⋅ = = = Σ= Since the demand arrival processes at the retailers and the service processes at manufacturer 1 are general, the wellknown Kraemer and LangenbachBelz (1976) formula for the GI/G/1 queue together with Little’s law (Little 1961) can be used to calculate the expected number of orders in manufacturer system 1. The arrival process at manufacturer 1 is a superposition of the demand arrival processes at the retailers. To calculate the SCV of this combined arrival process at retailer i, an approximation for the superposition of two arrival streams is used as shown in the following equations (Whitt 1983). 1 2 1 ρ1 = (λ r + λ r ) ⋅τ m (5.1) (5.2) 25 1 2 1 2 / 1 1 c w c w i k ari i k r r merge − + ⋅ = ⋅Σ λ Σλ ((( ) ) (( ) )) 1 2 2 1 2 /( ) /( ) 1 1 1 1 2 1 2 1 2 2 cmerge = w ⋅ λr λr + λr ⋅ car + λr λr + λr ⋅ car + − w (( ) ( )) 1 2 2 1 2 1 1 1 1 1 1 2 2 cmerge = w ⋅ pm r ⋅ car + pm r ⋅ car + − w where [ ( ) ( )] 1 1 2 1 1 4 1 1 1 − w = + ⋅ − ρ ⋅ ν − and [ 2 2 ] 1 1 1 1 1 2 − ν = pm r + pm r The expected number of orders at manufacturer system 1 can now be calculated using the Kraemer and LangenbachBelz (1976) formula and Little’s law (Little 1961). 1 2 2 1 2 1 2 ( ) (1 ) [ ] 1 1 1 ρ ρ ρ + + ⋅ − = ⋅ merge sm m c c E N g where, ≥ + ⋅ − − ⋅ − < ⋅ ⋅ + − ⋅ − ⋅ − = ; 1 ( 4 ) (1 ) ( 1) exp ; 1 3 ( ) (2 (1 ) (1 ) ) exp 2 2 2 2 1 2 2 2 1 2 2 1 1 1 1 1 1 1 1 1 merge merge sm merge merge merge sm merge c c c c c c c c g ρ ρ ρ The probability distribution for the number of backorders present in manufacturer system 1 is given by the following expression (Buzacott and Shanthikumar 1993) for a GI/G/1 maketostock system. ( ) ⋅ − ⋅ ≥ − ⋅ = = = (1 ) + − ; 1 (1 ) ; 0 ( 1) 1 1 1 1 1 1 1 1 l l P N l m m l S S m ρ σ σ ρ σ where, ( [ ] )/ [ ] 1 E Nm1 1 E Nm1 σ = −ρ An order at a retailer will be reflected in manufacturer queue only when there is a backorder at manufacturer 1. An order at manufacturer 1 is actually an “aggregate” order; (5.10) (5.11) (5.3) (5.4) (5.5) (5.6) (5.8) (5.7) (5.9) 26 the identity of the order (i.e., the retailer number) is lost after merging of the order arrival streams at manufacturer 1. Hence, a “disaggregation” procedure was used to find the distribution of the number of orders pertaining to retailer i at manufacturer 1, which is obtained by conditioning on the number of backorders in manufacturer system 1. The number of orders belonging to retailer i in manufacturing stage 1, given the number of backorders at the finished goods store, follows the Binomial distribution. ( ) 1 1 1 1 1 2 (  ) l r m r r p m pm r p r l P N r N l i ⋅ ⋅ − = = = where ( / ( )) pm1r1 r1 r1 r2 = λ λ +λ and ( / ( )) pm1r2 r2 r1 r2 = λ λ +λ are the proportions of orders from retailers 1 and 2, respectively. To derive the unconditional probability that a backorder at manufacturer 1 belongs to retailer i, the above value is multiplied by the probability of backorders and then summed over all values of l (backorders) greater than or equal to r. Σ ∞ = = = = = ⋅ = l r P Np r P Np r Nm l P Nm l i i ( ) (  ) ( ) 1 1 1 1 For retailer 1, ≥ − ⋅ − ⋅ ⋅ ⋅ + − = − ⋅ ⋅ ⋅ − ⋅ = = + + − ; 1 (1 ) (1 ) (1 ) ; 0 (1 ) (1 ) ( ) ( 1) 1 ( 1) 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2 1 1 2 11 r p p r p p P N r r m r r r S m r S m r S m r p m m m σ σ ρ σ ρσ σ σ σ ρ (5.12) (5.13) (5.14) 27 Similarly for retailer 2, ≥ − ⋅ − ⋅ ⋅ ⋅ + − ⋅ = − ⋅ ⋅ ⋅ ⋅ − = = + + − ; 1 (1 ) (1 ) (1 ) ; 0 (1 ) (1 ) ( ) ( 1) 1 ( 1) 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 21 r p p r p p P N r r m r r r S m r S m r S m r p m m m σ σ ρ σ ρ σ σ ρ σ σ Recall that the transit time was modeled using a delay node. Hence, the distribution of the number in transit is approximated using the M/G/∞ formula (Sivaramakrishnan 1998) as follows ! ( ) 1 1 1 k e P Nt k k tm r m r i tm ri i ρ ρ ⋅ = = − where, tm1ri m1 m1ri ρ = λ ⋅τ The distribution of the number in system (obtained using equation (5.1)) can be used to find the distribution of the inventory and backorders at the retailers. It can be seen that if there are n orders in the system, then the inventory level is given by (S – n), provided n is less than S (S is the basestock level). Similarly, there are n backorders in the system if there are (S + n) orders in the system, provided S is less than n. From these relations, the probability distributions for the inventory level and backorders at retailer i are obtained using the following relations (Sivaramakrishnan 1998) i i i i i r i r r r k r i P I k P N S k k S P I P N S k ( ) ( ) ; 1, 2, , ( 0) ( ) 0 = = = − = K + = = = Σ ∞ = ( ) ( ) ; 1, 2,K ( 0) ( ) 0 = = = + = = = = Σ= P B k P N k S k P B P N k i i ri i r i r S k r i (5.15) (5.18) (5.19) (5.16) (5.17) 28 The expected inventory, expected backorder and fill rate at retailer i are given by [ ] ( ) 1 E I n P I n i ri i r S n r = ⋅ = Σ= [ ] ( ) 0 E B n P B n i ri n r = ⋅ = Σ ∞ = ( ) 1 F P I n ri i i S n r r = =Σ= At manufacturer stage 1, the results provided by Buzacott and Shanthikumar (1993) for a GI/G/1 maketostock system are used to arrive at the expected inventory, expected backorders and fill rate. The expected number of backorders at manufacturer 1 is given by Buzacott and Shanthikumar (1993), (1 ) [ ] 1 1 1 1 1 σ ρ σ − ⋅ = S m E Bm The probability distribution of the inventory level at manufacturer 1 is given by Buzacott and Shanthikumar (1993), − = ⋅ − ⋅ = − ⋅ = = ≈ − − − 1 1 1 1 1 1 ; (1 ) ; 1, 2, , 1 ; 0 ( ) 1 ( 1) 1 1 1 ( 1) 1 1 m m S n S m i S i S i P I n m m ρ ρ σ σ ρ σ K The expected inventory at the output store of manufacturer 1 is given by Σ= = ⋅ = 1 1 1 0 [ ] ( ) S m n E I m n P I m n (5.23) (5.25) (5.20) (5.21) (5.22) (5.24) 29 The fill rate at manufacturer 1 is given by ( ) 1 1 1 1 F P I n S m n m m = =Σ= 5.3 NUMERICAL EXPERIMENTS The accuracy of the analytical model was tested by comparing the analytical results to simulation estimates for a wide range of parameter values. The experimental design for comparison is shown in Table 5.1. Table 5.1 shows the different parameters that were varied and the values that were used for different experiments. Table 5.1: Experimental Design for Numerical Comparison Parameter Levels Level Values Arrival Rate Pair ( r1 λ ,λr2 ) 2 ( 1, 1 ) and ( 1.25, 0.75 ) Basestock level 3 2, 4, 8 (same for the retailers and manufacturer) Transit time distribution 2 Unif ( 1, 5 ) and Unif ( 4, 8 ) Utilization 2 80% and 90% Interarrival Distribution at Retailer 1 3 Erlang , Exponential and Hyperexponential Interarrival Distribution at Retailer 2 3 Erlang , Exponential and Hyperexponential Service time Distribution at manufacturer 1 3 Erlang , Exponential and Hyperexponential The analytical and simulation results for the various experiments are contained in Appendix A1. 5.4 SUMMARY OF RESULTS A total of 648 experiments were used to evaluate the accuracy of the analytical model. The analytical method gives excellent results (relative percentage error < 10%) in (5.26) 30 cases where the interarrival distribution is exponential and the service time distribution is exponential. The analytical results are fairly consistent with the simulation estimates when the interarrival and service times are general. Table 5.2 shows the accuracy of the approximations developed. The table shows the proportion of the results that fall within a certain error range. The absolute relative error was used to check the accuracy for the expected inventory and expected backorders whereas the minimum of the absolute relative error and the absolute difference between the analytical and simulation values was used for the fill rates due to the fact that the value of fill rate is always less than one and sometimes even less than 0.5. In a case where the values are small, the criteria to be tested will be the absolute difference and not the absolute relative error as suggested by Whitt (1993). Table 5.2: Summary of Results Percentage of Results within the Error Range (%) Error Range Expected Inventory Expected Backorders Fill Rate < 0.25 96.5 94.9 99.9 < 0.20 94.2 92.4 99.9 < 0.15 86.9 88.6 99.3 < 0.10 79.2 80.8 95.8 < 0.05 67.0 61.2 83.6 From the detailed results presented in Appendix A1, it can be seen that the analytical model does not perform well (relative error percentage greater than 15%) when the SCV is high. Some possible sources of error are presented below. The use of M/G/∞ queue to model the delay node seems to be a major area of concern. This is because even when the interarrival time at the delay node is general, it is 31 assumed to be Poisson. The problem here is that there are not any analytical results available for the distribution of the number in the system for a GI/G/∞ queue. It can be seen from the result tables in Appendix A1 that when the SCV of interarrival process is different for the two retailers, there is a discrepancy in the results. It can be seen that the analytical model gives the same value of a performance measure, whether it is the expected inventory or expected backorder or fill rate, for both the retailers. The analytical results in such a case agree closely with the simulation results when the arrival process is Poisson. The analytical method overestimates when the interarrival time is Erlang and underestimates when the interarrival time is Hyperexponential. The reason for this was the fact that the analytical model was not able to account for the difference in variability while computing the individual retailer performance measures. After computing the distribution of the “aggregate” orders at the manufacturer, the “disaggregation” procedure to find the distribution of orders corresponding to a particular retailer was based on a “proportion of orders” belonging to a retailer (See Equation 5.12).While calculating these proportions, the model used only the arrival rates and not the SCVs (See Equation 5.12). It would be reasonable to expect the analytical results to improve if SCVs are included in the computation of “proportions.” This is suggested as an area for future research. 32 CHAPTER 6 CONVERGENT CONFIGURATION WITH TWO MANUFACTURERS In this chapter, a convergent network with two manufacturers supplying a single retailer is considered. The demand interarrival and service times at the manufacturers are considered to be general. The splitting approximation for traffic variability was used to compute the SCV of the order arrival process at each of the manufacturers. The next section, Section 6.1, gives a complete description of the system under study and the various assumptions made. The mathematical procedure is explained in Section 6.2 and numerical experiments are discussed in Section 6.3. Section 6.4 presents a summary of the results and future scope for improving the accuracy of the analytical model. 6.1 SYSTEM DESCRIPTION A maketostock convergent network with two manufacturers, each with a finished goods store, and a retailer (see Figure 6.1) was considered. The finished goods stores and the retail store operate under a basestock policy with oneforone replenishment. The demand interarrival times at the retailer as well as the service times at both the manufacturers follow a general distribution. The arrival of a demand consumes a finished product from the retailer and causes an instantaneous order to be 33 placed at one of the manufacturers based on a fixed probability. The finished product from the output store of a manufacturer is shipped to the retailer to satisfy the order for replenishment and the output store places a replenishment order at the manufacturing stage. As before, the transit time to ship finished products from the output store of the manufacturer to the retailer was modeled using a delay node. The manufacturer was modeled by a singleserver queueing system. Figure 6.1 Convergent Configuration 1 An ample supply of raw material was assumed. Also it was assumed that the raw material is released as and when an order is placed at a manufacturer. If the finished product is not available at the time of arrival of the customer order, then demand is backordered. When a finished product arrives, it is used to satisfy an outstanding backorder, if any; otherwise it is stored. Retailer Manufacturer 2 Manufacturer 1 Output Store of Manufacturer 2 Output Store of Manufacturer 1 Part Flow Replenishment + Backorders Transit Time Transit Time 34 6.2 QUEUEING MODEL OF THE CONVERGENT CONFIGURATION WITH TWO MANUFACTURERS Each manufacturer is modeled by a singleserver queue and the two transit operations to retailer 1 from the manufacturers are modeled using delay nodes. Each delay node is modeled as an M/G/∞ queue. 6.2.1 Approximate Solution of the Queueing Model When a customer order arrives at retailer 1, if parts are available, the order is satisfied immediately and a replenishment order is sent to one of the manufacturers depending on a probabilistic split. It is assumed this split probability is known. If a part is available in the output store at manufacturer i, then it is immediately shipped to retailer 1. So any order waiting at retailer 1 is reflected in the items in transit or in a manufacturer’s queue in case of backorder at that manufacturer. The distribution of the number in system can be obtained by using the fact that if there are n orders in the system, if k are at manufacturer 1 then the remaining (n  k) will be in manufacturer 2. Of the k orders at manufacturer 1, j may be in transit from manufacturer 1 and the remaining (k – j) in manufacturing stage 1, in case of backorder at the output store of manufacturer 1. Similarly of the (n  k) orders at manufacturer 2, l of them may be in transit and the remaining in manufacturing stage 2. The distribution of the number of orders in the system can be obtained by assuming that the random variables representing the orders for the manufacturers are stochastically independent. Similarly, the distribution of the number of orders for a manufacturer can be obtained by assuming that the random variables representing the orders in transit and the random variable representing the orders in the manufacturer queue are stochastically independent. The above approach is 35 an approximation because the network model does not satisfy the productform conditions and includes an approximation for splitting in the arrival process at retailer 1 (Sivaramakrishnan 1998; Suri et al. 1993). The probability distribution for the number of orders at retailer 1 is given by ( ) ( ) ( ) 1 2 0 P N 1 n P N k P N n n k n k n − = ⋅ = = = Σ= The probability distribution for the number of orders at manufacturer i is given by ( ) ( ) ( ) 1 1 0 P N k P Nt j P N k j i i p i k j r m n − = ⋅ = = = Σ= As before, the Kraemer and LangenbachBelz (1976) formula and Little’s law (Little 1961) were used to calculate the expected number of orders at each of the manufacturers. To compute the SCV of the order arrival process at manufacturer i, the approximations for the splitting of an arrival stream are used as shown in the following equations (Whitt 1983). c q c p c p c q sp ar sp ar = ⋅ + = ⋅ + ( ) ( ) 2 2 2 2 12 1 11 1 where, p is the probability that an order from retailer 1 goes to manufacturer 1 q is the probability that an order from retailer 1 goes to manufacturer 2 The expected number of orders at manufacturer system i is given by i sp sm i i m i i i i c c E N g ρ ρ ρ + + ⋅ − = ⋅ 2 ( ) (1 ) [ ] 2 2 2 1 where, = ⋅ ; i = 1, 2 i ri m i ρ λ τ (6.1) (6.2) (6.5) (6.6) (6.3) (6.4) 36 and ≥ + ⋅ − − ⋅ − < ⋅ ⋅ + − ⋅ − ⋅ − = ; 1 ( 4 ) (1 ) ( 1) exp ; 1 3 ( ) (2 (1 ) (1 ) ) exp 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 i i i i i i i i sp sp sm i sp sp i sp sm i sp i c c c c c c c c g ρ ρ ρ The probability distribution of the number of orders at manufacturing stage i is given by ( ) ⋅ − ⋅ ≥ − ⋅ = = = (1 ) + − ; 1 1 ( ) ; 0 ( 1) m m P N m mi mi i m S i i i S i i m ρ σ σ ρ σ where, ( [ ] ) / [ ] i E Nmi i E Nmi σ = −ρ The probability distribution of the number of orders in transit from manufacturer i to retailer 1 is given by ! ( ) 1 1 1 j e P Nt j j m r tmir tmir i ρ ρ ⋅ = = − where, tmi r1 mi mi r1 ρ = λ ⋅τ The distribution of the number in system (obtained using equation (6.1)) can be used to find the distribution of the inventory and backorders as in Section 5.2.1. 1 1 1 1 1 ( ) ( ) ; 1, 2, , ( 0) ( ) 1 0 1 r r r r k r P I k P N S k k S P I P N S k = = = − = K + = = = Σ ∞ = ( ) ( ) ; 1, 2 , K ( 0 ) ( ) 1 1 1 1 1 0 1 = = = + = = = = Σ= P B k P N k S k P B P N k r r S k r r (6.7) (6.8) (6.10) (6.11) (6.9) (6.12) (6.13) 37 The expected inventory, expected backorders and fill rate at retailer 1 are given by [ ] ( ) 1 1 1 1 E I n P I r n S n r r = ⋅ = Σ= [ ] ( ) 1 1 0 E B n P B r n n r = ⋅ = Σ ∞ = ( ) 1 1 1 1 F P I n S r n r r = = Σ= At each manufacturer, the results provided by Buzacott and Shanthikumar (1993) for a GI/G/1 maketostock system were used to arrive at the expected inventory, expected backorders and fill rate. The expected backorders at the output store of manufacturer i is given by (1 ) [ ] i S i i m mi i E B σ ρ σ − ⋅ = The probability distribution of the inventory level at manufacturer i from (Buzacott and Shanthikumar 1993), − = ⋅ − ⋅ = ⋅ = = ≈ − − − i i mi mi i i m m S n i i i S i i m n S n S n P I n 1 ; (1 ) ; 1, 2, , ; 0 ( ) ( 1) ( 1) ρ ρ σ σ ρ σ K The expected inventory at the output store of manufacturer i is given by Σ= = ⋅ = mi i i S n E Im n P Im n 0 [ ] ( ) The fill rate for the output store of manufacturer i is given by ( ) 1 F P I n mi i i S n m m = =Σ= (6.17) (6.19) (6.20) (6.14) (6.15) (6.16) (6.18) 38 6.3 NUMERICAL EXPERIMENTS The accuracy of the analytical model was tested by comparing the analytical results to simulation estimates for a wide range of parameter values. The experimental design for comparison is shown in Table 6.1. Table 6.1: Experimental Design for Numerical Comparison Parameter Levels Level Values Arrival Rate (λr1 ) 1 1 Split to Manufacturer ( pm1r1 , pm2 r1 ) 2 ( 50%, 50%) and ( 75%, 25% ) Basestock level 3 2, 4, 8 (same for the retailer and manufacturers) Transit time distribution 2 Unif ( 1, 5 ) and Unif ( 4, 8 ) Utilization 2 80% and 90% Interarrival distribution at retailer1 3 Erlang , Exponential and Hyperexponential Service time distribution at manufacturer 1 3 Erlang , Exponential and Hyperexponential Service time distribution at manufacturer 2 3 Erlang , Exponential and Hyperexponential The analytical and simulation results for the various experiments are contained in Appendix A2 6.4 SUMMARY OF RESULTS A total of 648 experiments were used to evaluate the accuracy of the analytical model. The results show that the analytical method performs very well for almost all cases. The analytical results are consistent with the simulation estimates when the interarrival and service times are general. The relative percentage error was less than 10% in approximately 9 out of 10 experiments on an average. Table 6.2 shows the accuracy of the approximations developed. The table shows the proportion of the results that fall 39 within a certain error range. The absolute relative error was used to check the accuracy for the expected inventory and expected backorders whereas the minimum of the absolute relative error and the absolute difference between the analytical and simulation values was used for the fill rates due to the fact that the value of fill rate is always less than one and sometimes even less than 0.5. In a case where the values are small, the criteria to be tested will be the absolute relative difference and not the absolute error as suggested by Whitt (1993). Table 6.2: Summary of Results Percentage of Results within the Error Range (%) Error Range Expected Inventory Expected Backorders Fill Rate < 0.25 98.3 98.5 99.9 < 0.20 96.6 97.6 99.9 < 0.15 93.9 92.5 99.7 < 0.10 88.6 83.6 98.4 < 0.05 72.0 61.5 89.8 The results show that the analytical method performs very well for almost all cases. This is due to the fact that the approximation for queue length distribution in a GI/G/1 queue is known to yield good results. This model performs better than the divergent configuration because of the following reasons. In the divergent network, since the order that is backordered at the manufacturer could have originated from either of the retailers, a conditional probability (the disaggregation procedure) had to be used to obtain the probability distribution of the number of orders belonging to retailer i that are backordered at the manufacturer. Whereas, in the case of the convergent network with two manufacturers, it is known that the order that is backordered at the manufacturer belongs to retailer 1 only. Therefore the 40 queue length distribution of the GI/G/1 queue could be directly used to find the probability distribution of the number of orders from retailer 1 that get backordered at either of the manufacturers. 41 CHAPTER 7 CONVERGENT CONFIGURATION WITH TWO SUPPLIERS In this chapter, a convergent configuration with two suppliers supplying a single manufacturer is considered. The demand arrival and manufacturing service processes are considered to be general. Approximations for splitting and merging are used to take into account the delay due to nonavailability of part at the input store of the manufacturer. The next section, Section 7.1, gives a complete description of the configuration under study and the various assumptions made. The approximate solution procedure is explained in Section 7.2, and numerical experiments are presented in Section 7.3. Section 7.4 presents a summary of the results and future scope for improving the accuracy of the analytical model. 7.1 SYSTEM DESCRIPTION A convergent configuration with two suppliers and a manufacturer with an input store as well as an output store (see Figure 7.1) was considered. The input store and the output store operate under a basestock policy with oneforone replenishment. The demand interarrival times at the manufacturer as well as the service times at the manufacturer follow a general distribution. The arrival of a demand consumes a finished 42 product from the output store of the manufacturer and causes an order to be sent to the manufacturing stage. For each order, the manufacturing stage needs a part from the input store. The input store of the manufacturer in turn places an order at one of the suppliers. The time needed to transport the materials ordered from a supplier to the input store was modeled using a delay node. The manufacturer was modeled by a singleserver queueing system. Figure 7.1 Convergent Configuration 2 When the supplier receives an order, if a part is readily available at the supplier, it reaches the input store of the manufacturer after a transit delay. In case of stockout at the supplier, the order experiences a leadtime delay to acquire the raw material in addition to the transit delay. Transit Time Transit Time Lead Time Lead Time Part Flow Supplier 1 Supplier 2 Input Store Manufacturer Demand Output Store Replenishment + Backorders 43 7.2 QUEUEING MODEL OF THE CONVERGENT CONFIGURATION WITH TWO SUPPLIERS The manufacturer was modeled by a singleserver queue and the supplier operations to the manufacturer were modeled using delay nodes. Each delay at the supplier side was modeled as an M/G/∞ queue. 7.2.1 Approximate Solution of the Queueing Model The model behaves like the other models discussed in earlier Chapters with the orders proceeding from the output store of the manufacturer to the manufacturing stage to the input store of the manufacturer and finally to the supplier. So any order arriving at manufacturer 1 is reflected in the call for replenishment at either manufacturing stage queue 1 or in the supplier node in case of any backorder at that input store of manufacturer 1. The distribution of the number in manufacturer 1 can be obtained using the fact that if there are n orders in manufacturer 1, k may be in manufacturing stage 1 and the remaining (n  k) will be at the supplier “node”, if there is a backorder at the input store of manufacturer1. The distribution of the number of orders in the system can be obtained by assuming that the random variables representing the orders in manufacturing stage 1 and the random variable representing the orders at the supplier node are stochastically independent. As before, this is an approximation because the network model does not satisfy the productform conditions (Suri et al. 1993). The probability distribution for the number of orders at manufacturer 1 is given by ( ) ( ) ( ) 1 1 1 0 P N n P N k P Brm n k n k m n − = ⋅ = = = Σ= (7.1) 44 The delay at the supplier end which includes both the leadtime delay and transit delay was modeled as a single delay node (see Figures 7.1 and 7.2). The approximations for splitting and merging are applied to modify the SCV of the arrival process to take into account the delay (if any) in obtaining the raw material (Whitt 1983). From Figure 7.2, it can be seen that the arrival at manufacturer 1 splits into two streams – one which finds the raw material at the input store of manufacturer 1 and the second one where the raw material is not available at the input store and a backorder is placed at the supplier stage. So the splitting approximation (Whitt 1983) is used to find the variability of the component that forms part of the arrival at manufacturer 1. 1 1 1 1 2 2 1 carm = Frm ⋅ cmerge + − Frm The other component of arrival process is the fraction of the departure from the supplier node. As the supplier node is modeled as an M/G/∞ queue, the departure process from the supplier node is Poisson. As seen from the Figure 7.2, this departure process splits into two, and one of them forms the input to manufacturer 1. It is a known result that the splitting of a Poisson stream gives rise to Poisson streams. So 2 cd1 and 2 cd2 are both equal to one. The merging approximation is used to arrive at the variability of the arrival process at manufacturer 1. 2 2 2 1 1 2 1 1 cam = (1− Frm ) ⋅ cd + Frm ⋅ carm (7.2) (7.3) 45 Figure 7.2 Splitting and Merging at the Input Store The expected number of orders at manufacturer system 1 is given by (Kraemer and LangenbachBelz 1976) and Little’s law (Little 1961). 1 2 2 1 2 1 1 2 ( ) (1 ) [ ] 1 1 1 ρ ρ ρ + + ⋅ − = ⋅ am sm m c c E N g where, 1 m 1 m 1 ρ = λ ⋅τ ≥ + ⋅ − − ⋅ − < ⋅ ⋅ + − ⋅ − ⋅ − = ; 1 ( 4 ) (1 ) ( 1) exp ; 1 3 ( ) (2 (1 ) (1 ) ) exp 2 2 2 2 1 2 2 2 1 2 2 1 1 1 1 1 1 1 1 1 am am sm am am am sm am c c c c c c c c g ρ ρ ρ (7.4) (7.5) (7.6) Manufacturer stage Raw Material Store Supplier Stage cd 2 ca 2 cam c 2 d2 2 ca 2 (Frm1) carm 2 ca 2 (Frm1) (Frm1) 1 – Frm1 Output Store cd1 2 Backorders + Replenishment Orders carm 2 46 The probability distribution of the number of orders at manufacturing stage 1 is given by ( ) ⋅ − ⋅ ≥ − = = ≈ (1 ) − ; 1 1 ; 0 ( 1) 1 1 1 1 1 k k P N k m k ρ σ σ ρ where, σ 1 = (E[Nm1 ] − ρ1 ) / E[Nm1 ] The distribution of the number in system (obtained using equation (7.1)) can be used to find the distribution of the inventory and backorders as in Section 5.2.1. 1 1 1 1 1 1 1 ( ) ( ) ; 1, 2, , ( 0) ( ) 0 m n m m m k m n P I k P N S k k S P I P N S k = = = − = K + = = = Σ ∞ = ( ) ( ) ; 1, 2,K ( 0 ) ( ) 1 1 1 1 1 1 0 = = = + = = = = Σ= P B k P N k S k P B P N k m n m S k m n m The expected inventory, expected backorder and fill rate at output store of manufacturer 1 are given by [ ] ( ) 1 1 1 1 E I n P I m n S n m m = ⋅ = Σ= [ ] ( ) 1 1 0 E B n P B m n n m = ⋅ = Σ ∞ = ( ) 1 1 1 1 F P I n S m n m m = = Σ= The order from the input store can be routed to supplier 1 or to supplier 2 based on specified probabilities. If raw material is available at the supplier, the order is immediately shipped and arrives at the input store after a transit delay. If there is a (7.9) (7.10) (7.11) (7.12) (7.13) (7.7) (7.8) 47 stockout at the supplier, then the order experiences a leadtime delay in addition to the transit delay. A single delay node was used to model the transit and leadtime delays at the supplier. Thus, the expected aggregate delay experienced at the supplier side is given by (( ) ) (( ) ) sm1 pm1s1 plt1 lt1 s1m1 pm1s2 plt 2 lt 2 s2m1 τ = ⋅ ⋅τ +τ + ⋅ ⋅τ +τ The probability distribution of the number of orders in the supplier delay node is given by ( ) ! 1 1 1 j e P N j j s tms tsm ρ ρ ⋅ = = − where, tms1 m1 sm1 ρ = λ ⋅τ The probability distribution of the inventory level and backorder level at the input store of manufacturer 1 is given by ( ) ( ) { } { } 1 1 1 1 1 1 P B i P N S i P I i P N S i rm s rm rm s rm = = = + = = = − The expected inventory, expected backorder and fill rate at the input store of manufacturer 1 are given by [ ] ( ) 1 1 1 1 E I n P I rm n S n rm rm = ⋅ = Σ= [ ] ( ) 1 1 0 E B n P B rm n n rm = ⋅ = Σ ∞ = ( ) 1 1 1 1 F P I n S rm n rm rm = = Σ= (7.14) (7.17) (7.16) (7.15) (7.18) (7.19) (7.20) 48 7.3 NUMERICAL EXPERIMENTS The accuracy of the analytical model was tested by comparing the analytical results to simulation estimates for a wide range of parameter values. The experimental design for comparison is shown in Table 7.1. Table 7.1 shows the different parameters that were varied and the values that were used for different experiments. Table 7.1: Experimental Design for Numerical Comparison Parameter Levels Level Values Arrival Rate (λm1 ) 1 1 Split to supplier ( p m1 s 1 , pm1s2 ) 1 (75%, 25%) Probability of leadtime delay (plt1, plt2) 2 (20% and 40%) and (30% and 30%) Basestock level of output store 3 2, 4, 8 (see note) Basestock level of input store 1 8 Leadtime delay (Expo (lt1), Expo(lt2)) 1 (Expo (2), Expo (4)) and (Expo (3), Expo (3)) Transit time distribution 2 Unif (1, 5) and Unif (4, 8) Utilization 2 80% and 90% Interarrival Distribution at manufacturer 1 3 Erlang , Exponential and Hyperexponential Service time Distribution at manufacturer 1 3 Erlang , Exponential and Hyperexponential Note: In one half of the experiments, the suppliers were identical, i.e., they had the same leadtime of Expo (3) and probability of stock out of 30%. In this case, the basestock levels used were 2, 4 and 8. In the other half of the experiments, the suppliers were asymmetrical, i.e., the leadtimes were Expo (2) and Expo (4) for the two suppliers and the corresponding probabilities of stockout were 20% and 40%. In this case, basestock levels used were 2 and 4. The analytical and simulation results for the various experiments are contained in Appendix A3 49 7.4 SUMMARY OF RESULTS A total of 180 experiments were used to evaluate the accuracy of the analytical model. The results show that there is scope for improvement in the analytical model. A majority of the error are found in the supplier side (See tables in Appendix A3). Table 7.2 shows the accuracy of the approximations developed. The table shows the proportion of the results that fall within a certain error range. The absolute relative error was used to check the accuracy for the expected inventory and expected backorders whereas the minimum of the absolute relative error and the absolute difference between the analytical and simulation values was used for the fill rates due to the fact that the value of fill rate is always less than one and sometimes even less than 0.5. In a case where the values are small, the criteria to be tested will be the absolute difference and not the absolute error as used by Whitt (1993). Table 7.2: Summary of Results Percentage of Results within Error Range (%) Error Range Expected Inventory Expected Backorders Fill Rate < 0.25 91.7 82.9 100.0 < 0.20 91.7 80.1 91.7 < 0.15 82.4 78.2 89.8 < 0.10 80.1 71.8 76.9 < 0.05 63.4 56.5 56.0 The detailed results shown in Appendix A3 show that there is room for improvement. The supplier side delays are first aggregated and then modeled using an M/G/∞ queue. The approximation works reasonably well when the interarrival process is Poisson and a majority of larger errors occur when the interarrival process is Erlang or Hyper 50 exponential. The approximation used for the delay model is insensitive to the variability in the interarrival process. In the result tables in Appendix A3, this can be shown by the fact that the analytical model gives the same output for the input store irrespective of the interarrival distribution, whereas the simulation model is sensitive to changes in the variability of the arrival processes. A majority of the error can be attributed to the use of the M/G/∞ queue. As mentioned in Chapter 5, twomoment approximations for GI/G/∞ queue are not available. 51 CHAPTER 8 CONVERGENT–DIVERGENT CONFIGURATION In this chapter, a convergentdivergent network with three retailers, two manufacturers and three suppliers is considered. This chapter shows how the solution procedures of the building blocks can be used to build a solution procedure for a larger network. The arrival and service processes are considered to be general. The remainder of the chapter is organized as follows. Section 8.1 gives a description of the system under study and the various assumptions made. An approximation procedure is explained in Section 8.2 and numerical experiments are defined in Section 8.3. This chapter concludes with a summary of the numerical results in Section 8.4. 8.1 SYSTEM DESCRIPTION A maketostock supply chain network with three retailers, two manufacturers (each having an input store and an output store) and three suppliers (see Figure 8.1) was considered. The retailers, input stores and the output stores operate under a basestock policy with oneforone replenishment. The interarrival times at the retailers as well as the service times at the manufacturers follow a general distribution. The arrival of a demand consumes a finished product from the retailer and causes an order for replenishment at the output store of the manufacturer. The finished product from the 52 output store of the manufacturer is shipped to the retailer to satisfy the order for replenishment and the output store places an order for replenishment to its manufacturing stage. The time needed to ship finished products from the output store to the retailer was modeled using a delay node. For every order, the manufacturing stage consumes a part from its input store. This causes the manufacturing stage to place an order at the supplier stage. If the raw material is readily available at the supplier, it reaches the input store after a transit delay. If the raw material is not available at the supplier, then the order experiences a leadtime delay in addition to the transit delay. Figure 8.1 ConvergentDivergent Supply Chain Network 8.2 QUEUEING MODEL OF THE CONVERGENTDIVERGENT CONFIGURATION The manufacturer is modeled by a singleserver queue and the supply operations are modeled using delay nodes (M/G/∞ queue). M1 D1 D2 M2 Manufacturers Retailers D3 S1 S2 Suppliers S3 53 8.2.1 Approximate Solution for the Queueing Model The convergentdivergent network can be seen as a combination of the building blocks that have been modeled in the previous three chapters. Retailer 2 along with manufacturers 1 and 2 forms a convergent network with two manufacturers. There are two divergent networks. One is the combination of retailers 1 and 2 with manufacturer 1, and the other is a combination of retailers 2 and 3 with manufacturer 2. Similarly, there are two cases of convergent network with two suppliers. One is the combination of manufacturer 1 with suppliers 1 and 2, and the other is a combination of manufacturer 2 with suppliers 2 and 3. The network model can be solved by first considering the split at retailer 2, i.e. the orders arriving at retailer 2 that go to either of the two manufacturers. Once the splitting approximation is used to get the SCVs of the arrival streams from retailer 2 to either of the manufacturers, the merging approximation can then be used to find the SCV for the arrival process at the manufacturers. The approximations for the calculation of the SCVs are as follows. First, the splitting approximations (Whitt 1983) are applied at retailer 2. c q c p c p c q sp ar sp ar = ⋅ + = ⋅ + ( ) ( ) 2 2 2 2 22 2 21 2 where, p is the probability that an order from retailer 2 goes to manufacturer 1 q is the probability that an order from retailer 2 goes to manufacturer 2 (8.1) (8.2) 54 The arrival rate at the manufacturers are given by 2 3 2 2 2 1 1 1 2 1 3 2 3 2 2 2 2 1 2 1 1 1 m r m r m m r m r m r m r r m r r m r r m r q p λ λ λ λ λ λ λ λ λ λ λ λ λ λ = + = + = = ⋅ = ⋅ = The proportion of orders at manufacturer i that belong to retailer j is given by 2 3 2 2 2 2 2 2 2 1 2 1 1 1 1 1 1 1 1 1 m r m r m r m m r m r m r m r m m r p p p p p p = − = = − = λ λ λ λ Now, the approximation for the superposition of two arrival streams (Whitt 1983) can be applied at both manufacturer 1 and manufacturer 2 as shown below. (( ) ) ( ) 2 2 2 2 2 1 2 2 1 2 ( ) ( ) 1 ( ) 1 2 2 2 22 2 3 3 1 1 1 1 1 2 21 c w p c p c w c w p c p c w merge m r sp m r ar merge m r ar m r sp = ⋅ ⋅ + ⋅ + − = ⋅ ⋅ + ⋅ + − where, [1 4 (1 )2 ( 1)] 1 − wi = + ⋅ − ρ i ⋅ ν i − ( ) ( 2 2 ) 1 2 2 2 1 1 2 2 2 3 1 1 1 2 − − = + = + m r m r m r m r p p p p ν ν and i mi mi ρ = λ ⋅τ (8.13) (8.15) (8.9) (8.3) (8.17) (8.4) (8.5) (8.6) (8.7) (8.8) (8.10) (8.11) (8.12) (8.14) (8.16) (8.18) 55 The splitting and merging approximations are now applied to the 2 mergei c to take into account the delay at the supplier as shown in the Section 7.2.1. The SCV of arrivals at manufacturer i is obtained as follows: carm i Frmi cmerge i Frmi 2 = ⋅ 2 + 1 − 2 (1 ) 2 2 cami Frmi cdi F carmi = − ⋅ + ⋅ where 2 cdi is 1 (departure process from an M/G/∞ queue). Since the demand arrival processes at the retailers are general, the Kraemer and LangenbachBelz (1976) formula and Little’s law (Little 1961) are used once again to calculate the expected number of orders in the manufacturer system. The expected number of orders at manufacturer i is given by i am sm i i m i i i i c c E N g ρ ρ ρ + + ⋅ − = ⋅ 2 ( ) (1 ) [ ] 2 2 2 where ≥ + ⋅ − − ⋅ − < ⋅ ⋅ + − ⋅ − ⋅ − = ; 1 ( 4 ) (1 ) ( 1) exp ; 1 3 ( ) (2 (1 ) (1 ) ) exp 2 2 2 2 2 2 2 2 2 i i i i i i i i am am sm i am am i am sm i am i c c c c c c c c g ρ ρ ρ The probability distribution of the number of orders in queue or being processed at manufacturer i is given by ( ) ⋅ − ⋅ ≥ − = = ≈ (1 ) − ; 1 1 ; 0 ( 1) n n P N n n i i i i mi ρ σ σ ρ where, ( [ ] )/ [ ] i E Nmi i E Nmi σ = −ρ (8.19) (8.20) (8.22) (8.21) (8.23) (8.24) 56 The network behaves in much the same way as the previous configurations; the orders proceed from the retailers to the output store of the manufacturers and then to the manufacturing stage after consuming a part at the input store of the manufacturers. Finally, the orders go from the manufacturer to the suppliers. So an order at the retailer can be found either in transit from a manufacturer or at the manufacturer or at the supplier stage. So if there are n orders in the system, then k of them will be present in transit from the manufacturer, l will be present at the manufacturer and the remaining (n– k– l) will be present in the supplier stage. The probability distribution for the number of orders at retailer j is given by Σ Σ Σ Σ Σ Σ Σ Σ Σ = = = = = = = = = = = = ⋅ = ⋅ = − − = = = ⋅ = − = = = ⋅ = ⋅ = − − = = = ⋅ = ⋅ = − − = = = ⋅ = ⋅ = − − n l l k m r p rmp n n n l l k m r p rmp n l l k m r p rmp n l l k m r p rmp P N n P Nt k P N l P B n k l P N n P N n P N n n P N n P Nt k P N l P B n k l P N n P Nt k P N l P B n k l P N n P Nt k P N l P B n k l 0 0 3 22 1 0 2 21 0 0 22 0 0 21 0 0 1 ( ) ( ) ( ) ( ) ( ) ( 1) ( 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 32 32 2 2 22 22 1 2 21 21 1 1 11 11 where, N2i is the random variable that represents the number of orders from retailer 2 that go to manufacturer i The probability distribution of the number of orders in transit from manufacturer i to retailer j is given by ! ( ) k e P Nt k k t m r mirj tmirj i j ρ ρ ⋅ = = − where, tmirj rjmi mirj ρ = λ ⋅τ (8.25) (8.30) (8.31) (8.26) (8.27) (8.28) (8.29) 57 The probability distribution of orders in manufacturing stage j corresponding to pending orders at retailer i is given by the formulas as used for the diverging network (see Equations 5.14 and 5.15). ≥ − ⋅ − ⋅ ⋅ ⋅ + − = − ⋅ ⋅ ⋅ − ⋅ = = + + − ; 1 (1 ) (1 ) (1 ) ; 0 (1 ) (1 ) ( ) ( 1) 1 ( 1) 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2 1 1 2 11 l p p l p p P N l l m r l l S S m r S m r p m m r m m σ σ ρ σ ρ σ σ σ σ ρ ≥ − ⋅ − ⋅ ⋅ + − = − ⋅ ⋅ ⋅ − ⋅ = = + + − ; 1 (1 ) (1 ) (1 ) ; 0 (1 ) (1 ) ( ) ( 1) 1 ( 1) 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 21 l p p l p p P N l l m r l l S S m r S m r p m m r m m σ σ ρ σ ρ σ σ σ σ ρ ≥ − ⋅ − ⋅ ⋅ ⋅ + − = − ⋅ ⋅ ⋅ − ⋅ = = + + − ; 1 (1 ) (1 ) (1 ) ; 0 (1 ) (1 ) ( ) ( 1) 2 ( 1) 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 3 2 2 3 22 l p p l p p P N l l m r l l S S m r m r p m m r m Sm σ σ ρ σ ρ σ σ σ σ ρ ≥ − ⋅ − ⋅ ⋅ ⋅ + − = − ⋅ ⋅ ⋅ − ⋅ = = + + − ; 1 (1 ) (1 ) (1 ) ; 0 (1 ) (1 ) ( ) ( 1) 2 ( 1) 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 32 l p p l p p P N l l m r l l S S m r S m r p m m r m m σ σ ρ σ ρ σ σ σ σ ρ The input store of manufacturer i was modeled first before obtaining the probability distribution of the number of orders that are backordered at the input store of manufacturer j that belong to retailer i Brmpij . The raw material order from a manufacturer is routed probabilistically to the suppliers. If raw material is available at a supplier, the order is immediately shipped and arrives at the manufacturer’s input store after a transit delay. If the raw materials are not (8.32) (8.35) (8.33) (8.34) 58 available at the supplier, then the order experiences a leadtime delay in addition to the transit delay. Thus, the expected aggregate delay experienced at the supplier side is given by (( ) ) (( ) ) (( ) ) (( ) ) 2 2 2 2 2 2 2 2 3 3 3 3 2 1 1 1 1 1 1 1 1 2 2 2 2 1 sm m s lt lt s m m s lt lt s m sm m s lt lt s m m s lt lt s m p p p p p p p p τ τ τ τ τ τ τ τ τ τ = ⋅ ⋅ + + ⋅ ⋅ + = ⋅ ⋅ + + ⋅ ⋅ + The probability distribution of the number of orders at the supplier side of manufacturer i is given by ; 1, 2 ! { ) = ⋅ = = − i n e P N n n tsm s i tsmi i ρ ρ where, = ⋅ ; i =1, 2 tsmi mi smi ρ λ τ The probability distribution of the inventory level and backorder level at the input store of manufacturer i is given by ( ) ( ) ( ) ( ) ( ) = + = ∞ = = = = = = = − = Σ= ; 1, 2, , ; 0 ; 1, 2, , 0 K K P N S n n P N n n P B n P I n P N S n n S i i rmi i i i i i i s rm S n s rm rm s rm rm The expected inventory, expected backorder and fill rate at input store of manufacturer i are given by [ ] ( ) 1 E I n P I n S n = ⋅ = Σ= [ ] ( ) 1 E B n P B n i rmi n rm = ⋅ =Σ ∞ = ( ) 1 F P I n i rm i i rm S n rm = = Σ= (8.40) (8.39) (8.38) (8.42) (8.43) (8.44) (8.36) (8.41) (8.37) 59 The probability distribution of the number of orders that are backordered at the input store of manufacturer j that belong to retailer i can be obtained by using an approach that is similar to the one used for calculating the probability distribution of the number of orders in manufacturing stage j corresponding to pending orders at retailer i (See Chapter 5). The probability distribution of the number of orders belonging to retailer i that are in the input store of manufacturer j, given the number of backorders at the input store of manufacturer j follows the Binomial distribution. 1 ( 1) 1 ( 1) 1 ( 1) 1 ( 1) 32 2 2 3 2 2 22 2 2 2 2 3 21 1 1 2 1 1 11 1 1 1 1 2 1 { 1 } 1 { 1 } 1 { 1 } 1 { 1 } n n m r n rmp rm m r n n m r n rmp rm m r n n m r n rmp rm m r n n m r n rmp rm m r p p n n P B n B n p p n n P B n B n p p n n P B n B n p p n n P B n B n − − − − ⋅ ⋅ = = = ⋅ ⋅ = = = ⋅ ⋅ = = = ⋅ ⋅ = = = To derive the unconditional probability that the backorder at the input store of manufacturer j belongs to retailer i, the above conditional probability is multiplied probability of backorder at the input store of manufacturer j and summed over all values of n (backorders) greater than or equal to n1. { } { 1 } { } 1 P B n P B n B n P B n ij ij j rmj n n rm rmp rmp = ⋅ = = = = Σ ∞ = The probability distribution of the inventory level and backorder level at retailer i is given by ( ) ( ) P(B n) P(N S n) P I n P N S n r i r i i i = = = + = = = − (8.49) (8.50) (8.45) (8.46) (8.47) (8.48) (8.51) 60 The expected inventory, expected backorder and fill rate at retailer i are given by [ ] ( ) 1 E I n P I n i ri i r S n r = ⋅ = Σ= [ ] ( ) 1 E B n P B n i ri n r = ⋅ =Σ ∞ = ( ) 1 F P I n i ri i r S n r = = Σ= The order at manufacturer i can be found either at the manufacturing stage or in the backorder queue at the input store of manufacturer i. So, if there are n orders at the manufacturer i, j are present in manufacturing stage and the remaining (nj) are in the backorder queue at the input store of manufacturer i. The probability distribution for the number of orders at manufacturer i is given by ( ) ( ) ( ) 0 P N n P N j P B n j i i rmi n j m n − = ⋅ = = = Σ= The probability distribution of the inventory level and backorder level at manufacturer i is given by ( ) ( ) P(B n) P(N S n) P I n P N S n i i i i m n m n = = = + = = = − The expected inventory, expected backorder and fill rate at manufacturer i are given by [ ] ( ) 1 E I n P I n i mi i m S n m = ⋅ =Σ= [ ] ( ) 1 E B n P B n i mi n m = ⋅ =Σ ∞ = (8.55) (8.52) (8.53) (8.54) (8.56) (8.58) (8.59) (8.57) 61 ( ) 1 F P I n i m i i m S n m = = Σ= 8.3 NUMERICAL EXPERIMENTS The accuracy of the analytical model was tested by comparing the analytical results to simulation estimates for a wide range of parameter values. The design of experiments is shown in Table 8.1. Table 8.1 shows the different parameters that were varied and also gives the values that were used for different experiments. Table 8.1: Experimental Design for Numerical Comparison Parameter Levels Level Values Arrival Rates ( λ1, λ2, λ3) 2 ( 1, 1, 1 ) and ( 0.5, 2, 0.5 ) Split to manufacturer ( p, q ) 1 ( 50%, 50% ) Split to supplier ( p1, q1 ) 1 ( 50%, 50% ) Probability of stockout 1 30% Leadtime delay 1 Expo ( 3 ) Basestock level at retailer 1 4 Basestock level at manufacturer 1 6 ( for both input and output store ) Transit time distribution 1 Unif ( 1, 5 ) Utilization 2 80% and 90% (see note) Interarrival Distribution at the retailers 3 Erlang , Exponential and Hyperexponential Service time Distribution at the manufacturers 3 Erlang , Exponential and Hyperexponential Note: In one set of experiments, the manufacturers had different utilizations, i.e., manufacturer 1 had 90% utilization and manufacturer 2 had 80% utilization. In all other experiments, the manufacturers had the same utilization level. (8.60) 62 The analytical and simulation results for the various experiments are contained in Appendix A4. 8.4 SUMMARY OF RESULTS A total of 45 experiments were used to evaluate the accuracy of the analytical model. The analytical model seems to give results which are very comparable with the simulation estimates. However, the supplier side poses a concern as the approximations used are prone to errors when the interarrival time is general. The reasons are discussed in Section 9.3.3. Table 8.2 shows the accuracy of the approximations developed for this configuration. The table shows the proportion of the results that fall within a certain value of error. The relative percentage error was used to check the accuracy for the expected inventory and expected backorders whereas the minimum of the relative error and the absolute difference between the analytical and simulation values was used for the fill rates due to the fact that the value of fill rate is always less than one and sometimes even less than 0.5. Table 8.2: Summary of Results Percentage of Results within Error Range (%) Error Range Expected Inventory Expected Backorders Fill Rate < 0.25 94.6 73.7 100.0 < 0.20 94.6 68.9 100.0 < 0.15 94.3 61.6 97.8 < 0.10 92.1 52.1 91.4 < 0.05 76.2 41.3 49.8 The main source of errors is the use of M/G/∞ queue to model delay nodes. As mentioned in the previous chapters, better results can be expected if the approximations 63 for the delay node could be improved. This problem is worse in the convergentdivergent network because of the repeated use of the M/G/∞ queue to model delay nodes at the supplier and retailer (transit) sides. 64 CHAPTER 9 CONCLUSIONS AND FUTURE RESEARCH In this chapter, a summary of the research carried out in this thesis effort, and some directions for future research are provided. The chapter is organized as follows. Section 9.1 provides a summary of the research that has been completed and provides the conclusions. Section 9.2 summarizes the contributions made by the successful completion of the research, and Section 9.3 provides suggestions for future research. 9.1 RESEARCH SUMMARY The main objective of this research was to develop analytical models of the ‘building block type’ configurations of a supply chain network and to apply these analytical models to model a somewhat complex supply chain network. In Chapter 5, the first ‘building block type’ configuration, namely, the divergent network was modeled for a general arrival process and general service times. The numerical results showed that the approximations were quite accurate when the arrival process was Poisson and the service times were exponential. The analytical results were fairly consistent with the simulation estimates when the interarrival and service times were general. The errors were in the generally acceptable range (< 20%) for more than 95% of the cases. In Chapter 6, the second configuration, namely, a convergent network with two suppliers was modeled and the numerical results showed that the approximations yielded excellent results. In 65 Chapter 7, the last of the ‘building block type’ configurations, namely, a convergent network with two suppliers was modeled. The results showed that the approximation performed reasonably well in several cases. However, there was a problem with the way the supplier delay node was modeled, and as a result the analytical model could not adequately capture the variability of the arrival process. This is discussed further in Section 9.3.2. In Chapter 8, a convergentdivergent network was modeled and the results showed that the building block approach worked well in modeling a fairly complicated network. The problem with the delay node approximation was still apparent in this configuration, and this is an area that needs to be researched further. The detailed results are organized in several tables and are presented in Appendices A1 through A4. 9.2 Research Contributions Previous research work in performance evaluation of supply chain networks mainly focused on the optimization part, although queueing models were also explored. In general, queueing models focus on capacity and congestion issues and often ignore planned inventories. With the works of Lee and Zipkin (1992), Zipkin (1995), Buzacott and Shanthikumar (1993), Sivaramakrishnan (1998) and Sivaramakrishnan and Kamath (1996), performance modeling of productioninventory systems has started gaining attention. Sivaramakrishnan (1998) primarily modeled multistage maketostock productioninventory systems. This research effort has focused on extending the models developed by Sivaramakrishnan (1998) to supply chain networks by including key activities like supply, transportation and distribution. The specific contributions of this research in the field of analytical queueing models are as follows. 66 1. Developed queueingbased models of “building block” type supply chain network configurations. 2. Demonstrated the usefulness of modeling “building block” configurations for developing queueing models of larger supply chain networks. 3. Laid the foundation for the development of a rapid analysis tool for supply chain networks. 4. Extended the popular parametric decomposition approach based on twomoment approximations to model supply chain networks. 9.3 FUTURE RESEARCH One of the key issues that has to be addressed is the use of the M/G/∞ system to model the delays, be it transit or leadtime delay. When the interarrival times are general, it was seen, in the third and last building block configuration, that the use of an M/G/∞ model to approximate a GI/G/∞ system did not work that well. The need for this approximation arose because of the lack of system length distribution type results for the GI/G/∞ system. Hence, obtaining better approximations for a GI/G/∞ system is critical to the improvement of some of the analytical models developed. All the models used in this study assumed the oneforone replenishment policy, and this is the policy that has been widely used in the literature. Modeling other inventory policies such as the (s, S) policy would be another interesting research topic from a practical viewpoint. 9.3.1 Extensions to the Divergent Configuration In the experiments for the divergent configuration, it was found that when the demand arrival processes had different variability at the two retailers, the analytical 67 model was not able to account for the difference in variability while computing the individual retailer performance measures. After computing the distribution of the “aggregate” orders at the manufacturer, the “disaggregation” procedure to find the distribution of orders corresponding to a particular retailer was based on a “proportion of orders” belonging to a retailer (See Equation 5.12). The expression used for the proportion of orders from a retailer to a manufacturer used only on the first moment (or rate) of the interarrival distribution. This made the disaggregation scheme insensitive to the differences in variability of the interarrival processes. Two heuristic extensions are suggested to address this issue. In the first extension, the arrival rate at a retailer is multiplied by the respective interarrival SCV and the proportions are computed using this product as shown in Equations 9.1 and 9.2. These new proportions would then be used in the calculations for the conditional probability distribution of the number of orders belonging to retailer i in the manufacturing stage, given the number of backorders at the finished goods store which is given by a Binomial distribution (See Equation 5.12), and the rest of the calculations would remain the same. ( 2 ) ( 2 ) 2 1 1 2 2 1 1 1 1 r ar r ar r ar m r c c c p ⋅ + ⋅ ⋅ = λ λ λ and 1 2 1 1 pm r = 1− pm r In the second extension, the arrival rate is multiplied by the sum of the arrival SCV at that retailer and the service SCV at the manufacturer and the proportions are calculated as follows. ( ( )) ( ( )) ( ) 2 2 2 2 2 2 1 1 1 2 2 2 1 1 1 1 1 r ar sm r ar sm r ar sm m r c c c c c c p ⋅ + + ⋅ + ⋅ + = λ λ λ (9.1) (9.2) (9.3) (9.4) 68 1 2 1 1 pm r = 1− pm r The above extensions were based on the observation that the interarrival and service time SCVs appear in the expected number in queue expressions along with the arrival rate. 9.3.2 Extensions to Convergent Configuration 1 In convergent configuration 1, it was assumed that the order at the retailer is split probabilistically, and that the routing probability is specified. But this may not be true in reality. In many scenarios, the availability of the part may play a role in the selection of the manufacturer. Thus, the existing model could be modified to consider a case where the retailer can check the availability of parts at the output stores of the manufacturers and then place an order at the store which has parts. If both the stores had parts or did not have parts, then the order could be sent to either manufacturer randomly. To solve this network, an iterative procedure is suggested. The probability of split can be expressed in terms of the fill rates of the manufacturers. It is known that the probability that an order is satisfied immediately is given by the fill rate. The proportion of orders going to a manufacturer depends on part availability at both. There are three different scenarios. They are 1) the part is available at manufacturer 1 and not at manufacturer 2 or vice versa, 2) the part is available at both manufacturers and the order is placed at a manufacturer with equal probability, and 3) the part is not available at either manufacturer and the order is placed with equal probability. 1 2 0.5 1 m m F F and order placed with manufacturer Part is available at both manufacturers P = ⋅ ⋅ (1 ) 2 1 Fm1 Fm2 and not at manufacturer Part is available at manufacturer P = ⋅ − (9.5) (9.6) 69 0.5 (1 ) (1 ) 1 m1 m2 F F and order placed with manufacturer Part is not available at both manufacturers P = ⋅ − ⋅ − The probability of ordering from manufacturer 1 would then be the sum of the three probabilities in equations 9.5 through 9.7. But, now the fill rates are needed to compute this probability. An iterative scheme which starts with a 5050 split and continues till the split probabilities stabilize is suggested. 9.3.3 Extensions to the Convergent Configuration 2 In the experiments for convergent configuration 2, it was found that the results for the supplier node performed reasonably well when the interarrival process at the node was Poisson. However, when the interarrival process was either Erlang or Hyperexponential, the results deviated significantly from the simulation estimates. This is due to the following reasons. The supplier side delays are first aggregated and then modeled using an M/G/∞ queue. In the result tables in Appendix A3, it can be seen that the analytical model gives the same output for the input store irrespective of the interarrival distribution, whereas the simulation model is sensitive to changes in the variability of the interarrival times. A majority of the error can be attributed to the use of the M/G/∞ queue. 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(1995), “Processing Networks with Planned Inventories: Tandem Queues with Feedback,” European Journal of Operational Research, 80, 344349. 31. Adan, I., “Basic Concepts from Probability Theory,” Class Notes, http://www.win.tue.nl/~iadan/sdp/h1.pdf, last viewed on December 9, 2004. 74 APPENDIX A1 RESULTS FOR THE DIVERGENT CONFIGURATION The experimental results for the divergent configuration with two retailers and one manufacturer are presented in a series of tables in this appendix. The experiments were conducted by fixing the basestock level, transit time from the manufacturer to retailer and utilization at particular values and varying the variability of interarrival and service times. The experimentation was performed in accordance with the design discussed in Table 5.1. In the results table the column titled SCV has three parameters, the SCV of the interarrival time at retailer 1, the SCV of the interarrival time at retailer 2, and the service SCV at the manufacturer. It is given as (SCVretailer1, SCVretailer2, SCVmanufacturer). Table A1.1: TwoRetailer Case: Results for basestock level of 2, transit time of 3, λ1= λ2= 1 and ρ = 80% SCV Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim (0.25, 0.25, 0.25) 0.218 0.077 0.218 0.077 0.785 0.756 1.440 1.236 1.440 1.237 0.446 0.320 0.175 0.130 0.175 0.130 0.585 0.686 (0.25,0.25,1.00) 0.175 0.057 0.175 0.056 0.624 0.621 1.916 1.712 1.916 1.710 1.481 1.309 0.142 0.097 0.142 0.096 0.424 0.501 (0.25, 0.25, 2.25) 0.136 0.044 0.136 0.044 0.532 0.570 2.830 2.675 2.830 2.671 3.387 3.260 0.110 0.076 0.110 0.076 0.332 0.425 (0.25, 1.00, 0.25) 0.194 0.065 0.194 0.202 0.684 0.674 1.664 1.446 1.664 1.667 0.939 0.846 0.157 0.111 0.157 0.162 0.484 0.518 (0.25,1.00,1.00) 0.161 0.050 0.161 0.165 0.585 0.590 2.181 1.970 2.181 2.190 2.040 1.943 0.130 0.086 0.130 0.133 0.385 0.421 (0.25, 1.00, 2.25) 0.129 0.041 0.129 0.137 0.517 0.555 3.115 2.908 3.115 3.119 3.972 3.848 0.104 0.071 0.104 0.110 0.317 0.376 (0.25, 2.25, 0.25) 0.169 0.056 0.169 0.339 0.606 0.627 2.022 1.728 2.022 2.228 1.706 1.552 0.137 0.096 0.137 0.138 0.406 0.418 (0.25,2.25,1.00) 0.144 0.046 0.144 0.300 0.548 0.571 2.576 2.281 2.576 2.784 2.864 2.722 0.117 0.078 0.117 0.121 0.348 0.363 (0.25, 2.25, 2.25) 0.120 0.038 0.120 0.257 0.501 0.539 3.531 3.282 3.531 3.786 4.823 4.773 0.097 0.065 0.097 0.104 0.301 0.334 (1.00, 0.25, 0.25) 0.194 0.201 0.194 0.065 0.684 0.674 1.664 1.668 1.664 1.444 0.939 0.845 0.157 0.162 0.157 0.112 0.484 0.518 (1.00, 0.25,1.00) 0.161 0.166 0.161 0.050 0.585 0.592 2.181 2.176 2.181 1.955 2.040 1.917 0.130 0.134 0.130 0.087 0.385 0.422 (1.00, 0.25,2.25) 0.129 0.137 0.129 0.041 0.517 0.554 3.115 3.116 3.115 2.914 3.972 3.856 0.104 0.110 0.104 0.071 0.317 0.375 (1.00, 1.00, 0.25) 0.177 0.182 0.177 0.182 0.629 0.614 1.891 1.902 1.891 1.896 1.429 1.432 0.143 0.147 0.143 0.147 0.429 0.414 (1.00, 1.00, 1.00) 0.150 0.154 0.150 0.153 0.560 0.564 2.430 2.426 2.430 2.430 2.560 2.551 0.121 0.124 0.121 0.123 0.360 0.362 (1.00, 1.00, 2.25) 0.123 0.127 0.123 0.128 0.507 0.531 3.375 3.406 3.375 3.389 4.507 4.533 0.100 0.103 0.100 0.104 0.307 0.331 (1.00, 2.25, 0.25) 0.159 0.166 0.159 0.322 0.581 0.582 2.221 2.183 2.221 2.454 2.124 2.153 0.128 0.133 0.128 0.131 0.381 0.347 (1.00, 2.25, 1.00) 0.137 0.140 0.137 0.282 0.534 0.539 2.796 2.795 2.796 3.074 3.318 3.441 0.111 0.113 0.111 0.114 0.334 0.313 (1.00, 2.25, 2.25) 0.116 0.120 0.116 0.245 0.494 0.514 3.763 3.817 3.763 4.117 5.295 5.562 0.094 0.097 0.094 0.099 0.294 0.295 (2.25, 0.25, 0.25) 0.169 0.340 0.169 0.056 0.606 0.625 2.022 2.235 2.022 1.736 1.706 1.570 0.137 0.139 0.137 0.096 0.406 0.416 (2.25,0.25,1.00) 0.144 0.298 0.144 0.045 0.548 0.569 2.576 2.813 2.576 2.303 2.864 2.775 0.117 0.120 0.117 0.078 0.348 0.361 (2.25, 0.25, 2.25) 0.120 0.258 0.120 0.038 0.501 0.538 3.531 3.796 3.531 3.286 4.823 4.788 0.097 0.104 0.097 0.066 0.301 0.334 (2.25, 1.00, 0.25) 0.159 0.319 0.159 0.164 0.581 0.579 2.221 2.469 2.221 2.189 2.124 2.170 0.128 0.130 0.128 0.133 0.381 0.345 (2.25, 1.00, 1.00) 0.137 0.284 0.137 0.142 0.534 0.543 2.796 3.049 2.796 2.763 3.318 3.391 0.111 0.115 0.111 0.114 0.334 0.315 (2.25, 1.00, 2.25) 0.116 0.246 0.116 0.120 0.494 0.517 3.763 4.116 3.763 3.793 5.295 5.531 0.094 0.099 0.094 0.097 0.294 0.296 (2.25, 2.25, 0.25) 0.145 0.299 0.145 0.297 0.550 0.552 2.556 2.810 2.556 2.810 2.822 3.019 0.117 0.121 0.117 0.120 0.350 0.295 (2.25, 2.25, 1.00) 0.128 0.266 0.128 0.265 0.515 0.520 3.162 3.468 3.162 3.463 4.068 4.393 0.104 0.107 0.104 0.106 0.315 0.275 (2.25, 2.25, 2.25) 0.110 0.235 0.110 0.236 0.484 0.506 4.148 4.425 4.148 4.450 6.075 6.406 0.089 0.095 0.089 0.095 0.284 0.268 Expected Inventory Expected Backorders Fill Rate Retailer 1 Retailer 2 Manufacturer Retailer 1 Retailer 2 Manufacturer Retailer 1 Retailer 2 Manufacturer 75 Table A1.2: TwoRetailer Case: Results for basestock level of 2, transit time of 6, λ1= λ2= 1 and ρ = 80% SCV Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim (0.25, 0.25, 0.25) 0.017 0.000 0.017 0.000 0.785 0.756 4.240 4.159 4.240 4.162 0.446 0.320 0.015 0.000 0.015 0.000 0.585 0.686 (0.25,0.25,1.00) 0.014 0.000 0.014 0.000 0.624 0.621 4.754 4.656 4.754 4.655 1.481 1.309 0.012 0.000 0.012 0.000 0.424 0.501 (0.25, 0.25, 2.25) 0.011 0.000 0.011 0.000 0.532 0.570 5.704 5.634 5.704 5.625 3.387 3.260 0.009 0.000 0.009 0.000 0.332 0.425 (0.25, 1.00, 0.25) 0.015 0.000 0.015 0.016 0.684 0.674 4.485 4.379 4.485 4.483 0.939 0.846 0.013 0.000 0.013 0.014 0.484 0.518 (0.25,1.00,1.00) 0.013 0.000 0.013 0.013 0.585 0.590 5.033 4.922 5.033 5.040 2.040 1.943 0.011 0.000 0.011 0.012 0.385 0.421 (0.25, 1.00, 2.25) 0.010 0.000 0.010 0.011 0.517 0.555 5.996 5.866 5.996 5.993 3.972 3.848 0.009 0.000 0.009 0.009 0.317 0.376 (0.25, 2.25, 0.25) 0.013 0.000 0.013 0.111 0.606 0.627 4.866 4.673 4.866 5.007 1.706 1.552 0.012 0.000 0.012 0.040 0.406 0.418 (0.25,2.25,1.00) 0.011 0.000 0.011 0.099 0.548 0.571 5.443 5.236 5.443 5.582 2.864 2.722 0.010 0.000 0.010 0.035 0.348 0.363 (0.25, 2.25, 2.25) 0.009 0.000 0.009 0.084 0.501 0.539 6.421 6.245 6.421 6.616 4.823 4.773 0.008 0.000 0.008 0.030 0.301 0.334 (1.00, 0.25, 0.25) 0.015 0.016 0.015 0.000 0.684 0.674 4.485 4.486 4.485 4.377 0.939 0.845 0.013 0.014 0.013 0.000 0.484 0.518 (1.00, 0.25,1.00) 0.013 0.013 0.013 0.000 0.585 0.592 5.033 5.023 5.033 4.906 2.040 1.917 0.011 0.012 0.011 0.000 0.385 0.422 (1.00, 0.25,2.25) 0.010 0.011 0.010 0.000 0.517 0.554 5.996 5.988 5.996 5.872 3.972 3.856 0.009 0.010 0.009 0.000 0.317 0.375 (1.00, 1.00, 0.25) 0.014 0.015 0.014 0.014 0.629 0.614 4.728 4.737 4.728 4.727 1.429 1.432 0.012 0.013 0.012 0.013 0.429 0.414 (1.00, 1.00, 1.00) 0.012 0.012 0.012 0.012 0.560 0.564 5.292 5.282 5.292 5.289 2.560 2.551 0.010 0.011 0.010 0.011 0.360 0.362 (1.00, 1.00, 2.25) 0.010 0.010 0.010 0.010 0.507 0.531 6.260 6.295 6.260 6.271 4.507 4.533 0.008 0.009 0.008 0.009 0.307 0.331 (1.00, 2.25, 0.25) 0.012 0.013 0.012 0.106 0.581 0.582 5.074 5.034 5.074 5.233 2.124 2.153 0.011 0.012 0.011 0.038 0.381 0.347 (1.00, 2.25, 1.00) 0.011 0.011 0.011 0.093 0.534 0.539 5.670 5.674 5.670 5.885 3.318 3.441 0.009 0.010 0.009 0.033 0.334 0.313 (1.00, 2.25, 2.25) 0.009 0.009 0.009 0.081 0.494 0.514 6.656 6.709 6.656 6.959 5.295 5.562 0.008 0.008 0.008 0.029 0.294 0.295 (2.25, 0.25, 0.25) 0.013 0.111 0.013 0.000 0.606 0.625 4.866 5.015 4.866 4.681 1.706 1.570 0.012 0.039 0.012 0.000 0.406 0.416 (2.25,0.25,1.00) 0.011 0.099 0.011 0.000 0.548 0.569 5.443 5.613 5.443 5.257 2.864 2.775 0.010 0.035 0.010 0.000 0.348 0.361 (2.25, 0.25, 2.25) 0.009 0.084 0.009 0.000 0.501 0.538 6.421 6.624 6.421 6.248 4.823 4.788 0.008 0.031 0.008 0.000 0.301 0.334 (2.25, 1.00, 0.25) 0.012 0.104 0.012 0.013 0.581 0.579 5.074 5.256 5.074 5.042 2.124 2.170 0.011 0.037 0.011 0.011 0.381 0.345 (2.25, 1.00, 1.00) 0.011 0.093 0.011 0.011 0.534 0.543 5.670 5.851 5.670 5.635 3.318 3.391 0.009 0.033 0.009 0.010 0.334 0.315 (2.25, 1.00, 2.25) 0.009 0.081 0.009 0.009 0.494 0.517 6.656 6.960 6.656 6.685 5.295 5.531 0.008 0.029 0.008 0.008 0.294 0.296 (2.25, 2.25, 0.25) 0.011 0.098 0.011 0.097 0.550 0.552 5.422 5.613 5.422 5.613 2.822 3.019 0.010 0.035 0.010 0.035 0.350 0.295 (2.25, 2.25, 1.00) 0.010 0.087 0.010 0.086 0.515 0.520 6.044 6.294 6.044 6.288 4.068 4.393 0.009 0.031 0.009 0.031 0.315 0.275 (2.25, 2.25, 2.25) 0.009 0.077 0.009 0.077 0.484 0.506 7.046 7.265 7.046 7.294 6.075 6.406 0.008 0.028 0.008 0.028 0.284 0.268 Expected Inventory Expected Backorders Fill Rate Retailer 2 Retailer 1 Retailer 2 Manufacturer Retailer 1 Manufacturer Retailer 1 Retailer 2 Manufacturer 76 Table A1.3: TwoRetailer Case: Results for basestock level of 4, transit time of 3, λ1= λ2= 1 and ρ = 80% SCV Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim (0.25, 0.25, 0.25) 1.287 1.100 1.287 1.102 2.459 2.495 0.347 0.126 0.347 0.124 0.120 0.053 0.635 0.785 0.635 0.787 0.889 0.947 (0.25,0.25,1.00) 1.159 0.971 1.159 0.969 1.911 1.949 0.542 0.284 0.542 0.286 0.768 0.631 0.579 0.712 0.579 0.711 0.701 0.761 (0.25, 0.25, 2.25) 0.961 0.784 0.961 0.785 1.508 1.647 1.143 0.913 1.143 0.911 2.363 2.258 0.487 0.588 0.487 0.588 0.534 0.606 (0.25, 1.00, 0.25) 1.227 1.037 1.227 1.246 2.136 2.146 0.423 0.182 0.423 0.420 0.391 0.322 0.609 0.753 0.609 0.615 0.785 0.817 (0.25,1.00,1.00) 1.093 0.898 1.093 1.109 1.750 1.780 0.695 0.425 0.695 0.689 1.205 1.112 0.549 0.667 0.549 0.554 0.637 0.667 (0.25, 1.00, 2.25) 0.915 0.743 0.915 0.938 1.438 1.558 1.361 1.098 1.361 1.373 2.893 2.795 0.464 0.560 0.464 0.472 0.503 0.555 (0.25, 2.25, 0.25) 1.131 0.954 1.131 1.339 1.839 1.908 0.601 0.301 0.601 0.803 0.939 0.808 0.567 0.702 0.567 0.475 0.673 0.689 (0.25,2.25,1.00) 1.008 0.820 1.008 1.202 1.585 1.639 0.958 0.660 0.95
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Title  Analytical Performance Modeling of Supply Chain Networks 
Date  20041201 
Author  Srivathsan, Sandeep 
Keywords  queueing; supply chain; parametric decomposition; planned inventory; basestock 
Department  Industrial Engineering & Management 
Document Type  
Full Text Type  Open Access 
Abstract  The purpose of this study was to develop queueingbased analytical models of buildingblock configurations of supply chain networks based on the parametric decomposition approach, and to use these models to develop the model of a more complex supply chain network. Existing queueing models of productioninventory networks were extended by including supply operations, transportation and distribution. Different buildingblock configurations such as a divergent network, a convergent network with two manufacturers, and a convergent network with two suppliers were modeled. In modeling a convergentdivergent configuration, which was a combination of the buildingblock configurations, the analytical expressions from the models of individual configurations were used. Validation of the analytical models was done by comparing analytical results to simulation estimates obtained using Arena simulation models. The analytical models were found to yield reasonably accurate results and the relative percentage errors were well within the permissible error of 20% for a majority of the cases tested. This research also showed that the buildingblock approach works well when applied to larger and more complicated networks. This work has laid the foundation for the development of rapid analysis tools for supply chain networks, and has successfully extended the popular parametric decomposition approach based on twomoment approximations to model supply chain networks. 
Note  Thesis 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  ANALYTICAL PERFORMANCE MODELING OF SUPPLY CHAIN NETWORKS By SANDEEP SRIVATHSAN Bachelor of Engineering University of Madras Madras, India 2002 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE December, 2004 ii ANALYTICAL PERFORMANCE MODELING OF SUPPLY CHAIN NETWORKS Thesis Approved: Dr. Manjunath Kamath Thesis Adviser Dr. Allen. C. Schuermann Dr. Ricki. G. Ingalls Dr. A. Gordon Emslie Dean of the Graduate College ii i ACKNOWLEDGEMENTS My obeisance to all upadhyayas – respects to all learned teachers. The world is made up of three classes of people. 1. The majority who do not know what is happening. 2. Many watch things happen 3. A few make things happen. My teachers at OSU belong to the third type of people. It is difficult to express my gratitude for the amount of help that my advisor, Dr. Manjunath Kamath, rendered in every respect, for being patient and for giving me an opportunity to carry out this research work as well as giving various valuable suggestions throughout my stay here as a Master's student. I owe a lot to him for providing advice regarding the course selection, moral support and last but not the least for proof reading this thesis. I would also like to thank the committee members, Dr. Allen Schuermann and Dr. Ricki Ingalls, for providing some valuable information and insights into my work. I would like to thank Dr. Allen Schuermann for helping me with suggestions in understanding of various simulation aspects and for continuing as a committee member even after his retirement. I would like to thank Dr. Ricki Ingalls for sharing his work experience with his students and thereby creating an interest in me for supply chains. I would also take this opportunity to thank all my friends in Center for Computer Integrated Manufacturing Enterprises (CCiMe) team for their support at various stages of iv my thesis. I would like to thank Sarath Kureti and Ananth Krishnamoorthy for helping me with the simulation models and Mohan Chinnaswamy for helping me with some queueing details. I appreciate the help that I received from Parthiban Dhananjeyan, Karthik Ayodhiramanujan and Uma Maheshwar Chalavadi in developing my C programs for the analytical model. Finally, I would like to dedicate this thesis to my parents, Sri. S. Srivathsan and Smt. Janaki Srivathsan, my sister Ms. Jaisree Srivathsan, and all other close relatives for their love and support. I owe a lot to them for constantly motivating me with a lot of advice each and every week over phone. But for their constant encouragement, support and guidance, the work would have faced problems time and again. v LIST OF CONTENTS INTRODUCTION........................................................................................................................................1 1.1 ANALYTICAL MODELS BASED ON QUEUEING..........................................................................3 1.2 OUTLINE OF THE THESIS ................................................................................................................4 LITERATURE REVIEW............................................................................................................................5 2.1 PRODUCTION  INVENTORY NETWORKS....................................................................................5 2.1.1 MaketoOrder Production Inventory Networks ...........................................................................6 2.1.2 MaketoStock Production Inventory Networks ............................................................................6 2.1.3 Hybrid Production Inventory Networks ......................................................................................10 2.2 SUPPLY CHAIN NETWORKS.........................................................................................................11 STATEMENT OF RESEARCH................................................................................................................13 3.1 RESEARCH OBJECTIVES ...............................................................................................................13 3.2 RESEARCH SCOPE AND LIMITATIONS ......................................................................................14 3.3 RESEARCH CONTRIBUTIONS.......................................................................................................14 RESEARCH APPROACH........................................................................................................................15 4.1 METHODOLOGY............................................................................................................................15 4.1.1 The Parametric Decomposition (PD) Approach.........................................................................15 4.1.2 Modeling Approach ....................................................................................................................17 4.1.3 Validation...................................................................................................................................17 4.2 PERFORMANCE MEASURES.........................................................................................................19 4.3 SUPPLY CHAIN NETWORK CONFIGURATIONS........................................................................19 4.3.1 A Divergent Supply Chain Network Configuration.....................................................................19 vi 4.3.2 Convergent Supply Chain Network Configurations....................................................................20 4.3.3 Combination ConvergentDivergent Supply Chain Network Configuration ..............................21 DIVERGENT CONFIGURATION WITH TWO RETAILERS ............................................................22 5.1 SYSTEM DESCRIPTION..................................................................................................................22 5.2 QUEUEING MODEL OF THE DIVERGENT CONFIGURATION .................................................23 5.2.1 Approximate Solution of the Queueing Model ............................................................................24 5.3 NUMERICAL EXPERIMENTS ........................................................................................................29 5.4 SUMMARY OF RESULTS................................................................................................................29 CONVERGENT CONFIGURATION WITH TWO MANUFACTURERS..........................................32 6.1 SYSTEM DESCRIPTION..................................................................................................................32 6.2 QUEUEING MODEL OF THE CONVERGENT CONFIGURATION WITH TWO MANUFACTURERS..............................................................................................................................34 6.2.1 Approximate Solution of the Queueing Model ............................................................................34 6.3 NUMERICAL EXPERIMENTS ........................................................................................................38 6.4 SUMMARY OF RESULTS................................................................................................................38 CONVERGENT CONFIGURATION WITH TWO SUPPLIERS.........................................................41 7.1 SYSTEM DESCRIPTION..................................................................................................................41 7.2 QUEUEING MODEL OF THE CONVERGENT CONFIGURATION WITH TWO SUPPLIERS...43 7.2.1 Approximate Solution of the Queueing Model ............................................................................43 7.3 NUMERICAL EXPERIMENTS ........................................................................................................48 7.4 SUMMARY OF RESULTS................................................................................................................49 CONVERGENT–DIVERGENT CONFIGURATION.............................................................................51 8.1 SYSTEM DESCRIPTION..................................................................................................................51 vi i 8.2 QUEUEING MODEL OF THE CONVERGENTDIVERGENT CONFIGURATION.....................52 8.2.1 Approximate Solution for the Queueing Model...........................................................................53 8.3 NUMERICAL EXPERIMENTS ........................................................................................................61 8.4 SUMMARY OF RESULTS................................................................................................................62 CONCLUSIONS AND FUTURE RESEARCH........................................................................................64 9.1 RESEARCH SUMMARY..................................................................................................................64 9.2 Research Contributions .................................................................................................................65 9.3 FUTURE RESEARCH.......................................................................................................................66 9.3.1 Extensions to the Divergent Configuration.................................................................................66 9.3.2 Extensions to Convergent Configuration 1 .................................................................................68 9.3.3 Extensions to the Convergent Configuration 2 ...........................................................................69 REFERENCES...........................................................................................................................................70 RESULTS FOR THE DIVERGENT CONFIGURATION.....................................................................74 RESULTS FOR THE CONVERGENT CONFIGURATION WITH TWO MANUFACTURERS ....99 RESULTS FOR THE CONVERGENT CONFIGURATION WITH TWO SUPPLIERS.................124 RESULTS FOR THE CONVERGENTDIVERGENT NETWORK...................................................135 SIMULATION MODEL..........................................................................................................................141 A5.1 THE SIMULATION LOGIC.........................................................................................................141 A5.2 DETERMINATION OF DISTRIBUTION PARAMETERS FOR SIMULATION.......................148 A5.2.1 Erlang Distribution ................................................................................................................148 A5.2.2 Exponential Distribution ........................................................................................................148 A5.2.3 HyperExponential Distribution .............................................................................................149 vi ii DETERMINATION OF THE WARMUP PERIOD AND RUN LENGTH FOR SIMULATION EXPERIMENTS.......................................................................................................................................151 A6.1 INTRODUCTION........................................................................................................................151 A6.2 WELCH’S TECHNIQUE TO DETERMINE WARMUP PERIOD..............................................152 A6.3 DETERMINATION OF RUN LENGTH.......................................................................................153 ix LIST OF FIGURES FIGURE 1.1 SUPPLY CHAIN NETWORK ..............................................................................................1 FIGURE 2.1 SINGLESTAGE MAKETOSTOCK SYSTEM WITH A DELAY NODE....................9 FIGURE 4.1 DIVERGENT CONFIGURATION.....................................................................................20 FIGURE 4.2 CONVERGENT CONFIGURATION 1 .............................................................................20 FIGURE 4.3 CONVERGENT CONFIGURATION 2 .............................................................................21 FIGURE 4.4 CONVERGENTDIVERGENT CONFIGURATION.......................................................21 FIGURE 5.1 DIVERGENT CONFIGURATION.....................................................................................23 FIGURE 6.1 CONVERGENT CONFIGURATION 1 .............................................................................33 FIGURE 7.1 CONVERGENT CONFIGURATION 2 .............................................................................42 FIGURE 7.2 SPLITTING AND MERGING AT THE INPUT STORE.................................................45 FIGURE 8.1 CONVERGENTDIVERGENT SUPPLY CHAIN NETWORK......................................52 FIGURE A5.1 SIMULATION MODEL .................................................................................................144 FIGURE A5.2 RETAILER 1 SUBMODEL...........................................................................................145 FIGURE A5.3 RETAILER 2 SUBMODEL...........................................................................................146 FIGURE A5.4 MANUFACTURER SUBMODEL................................................................................147 FIGURE A6.1 PLOT OF TIME IN SYSTEM........................................................................................154 FIGURE A6.2 PLOT OF TIME IN SYSTEM (CONTINUED)............................................................154 FIGURE A6.3 PLOT OF TIME IN SYSTEM (CONTINUED)............................................................155 x FIGURE A6.4 PLOT OF TIME IN SYSTEM (CONTINUED)............................................................155 FIGURE A6.5 PLOT OF TIME IN SYSTEM (CONTINUED)............................................................156 FIGURE A6.6 PLOT OF TIME IN SYSTEM (CONTINUED)............................................................156 FIGURE A6.7 PLOT OF TIME IN SYSTEM (CONTINUED)............................................................157 FIGURE A6.8 PLOT OF TIME IN SYSTEM (CONTINUED .............................................................157 FIGURE A6.9 PLOT OF TIME IN SYSTEM (CONTINUED)............................................................158 FIGURE A6.10 PLOT OF TIME IN SYSTEM (CONTINUED)..........................................................158 xi LIST OF TABLES TABLE 4.1: SIMULATION PARAMETERS.........................................................................................18 TABLE 4.2: SCV AND DISTRIBUTIONS..............................................................................................18 TABLE 5.1: EXPERIMENTAL DESIGN FOR NUMERICAL COMPARISON.................................29 TABLE 5.2: SUMMARY OF RESULTS .................................................................................................30 TABLE 6.1: EXPERIMENTAL DESIGN FOR NUMERICAL COMPARISON.................................38 TABLE 6.2: SUMMARY OF RESULTS .................................................................................................39 TABLE 7.1: EXPERIMENTAL DESIGN FOR NUMERICAL COMPARISON................................48 TABLE 7.2: SUMMARY OF RESULTS .................................................................................................49 TABLE 8.1: EXPERIMENTAL DESIGN FOR NUMERICAL COMPARISON.................................61 TABLE 8.2: SUMMARY OF RESULTS .................................................................................................62 xi i NOTATION Ni random variable representing the number of orders at retailer i Nsi random variable representing the number of orders at the supplier side of manufacturer i Nmi random variable representing the number of orders in queue or being processed at manufacturer i Nni random variable representing the number of orders at manufacturer i (backorders + in queue + in process) Npij random variable representing the number of orders at manufacturer j corresponding to the pending customer orders at retailer i Ntmirj random variable representing the number of orders in transit from manufacturer i to retailer j Brmpij random variable representing the number of orders that are backordered at the supplier side of manufacturer j that belong to retailer i ir λ order arrival rate at retailer i j im r λ order arrival rate from retailer i at manufacturer j xi ii mi λ order arrival rate at manufacturer i misj λ order arrival rate from manufacturer i at supplier j i s λ order arrival rate at supplier i mi τ mean processing time at manufacturer i i ρ utilization of manufacturer i 2 cari SCV of the arrival process at retailer i 2 cspij SCV of the order arrival process at manufacturer j after splitting at retailer i 2 cmergei SCV of the combined order arrival process at manufacturer i after merging of arrival streams 2 cami SCV of the arrival process at manufacturer i after accounting for supplier delay 2 csmi SCV of the service time distribution at manufacturer i 2 carmi SCV of the arrival stream at manufacturer i that consist of orders that find a part in the input store 2 cdi SCV of the arrival stream at manufacturer i that consist of satisfied backorders at the input store Sri basestock level at retailer i I ri random variable representing the inventory level at retailer i xi v Bri random variable representing the number of backorders at retailer i Fri fill rate at retailer i Smi basestock level at manufacturer i Imi random variable representing the inventory level at the output store of manufacturer i Bmi random variable representing the number of backorders at the output store of manufacturer i Fmi fill rate at the output store of manufacturer i S rmi basestock level at the input store of manufacturer i I rm i random variable representing the inventory level at the input store of manufacturer i Brmi random variable representing the backorder at the input store of manufacturer i Frmi fill rate at the input store of manufacturer i pmirj proportion of orders at manufacturer i that belong to retailer j pmisj proportion of orders from manufacturer i that go to supplier j mirj τ expected transit time from manufacturer i to retailer j tmirj ρ average number of items in the transit from manufacturer i to retailer j xv smi τ expected aggregate delay in delivering raw material at the supplier side of manufacturer i tsmi ρ average number of orders at the supplier stage of manufacturer i simj τ mean transit time from supplier i to the input store at manufacturer j lti τ mean leadtime delay at supplier i plti probability of stockout at supplier i xv i GLOSSARY OF TERMS The definitions of selected technical terms are included in this glossary to clarify their intended meaning and usage. Oneforone replenishment The consumption of an item triggers an immediate replenishment order for that item. This is a special case of the (s, S) inventory policy, namely, the (S1, S) policy. Basestock level This value determines the maximum inventory that can be stored at a particular stage. Fill rate This value gives the probability that an order is satisfied instantaneously by items in stock. SCV The Squared Coefficient of Variation (SCV) is defined as the ratio of the variance of a random variable to the square of its mean. 1 CHAPTER 1 INTRODUCTION A supply chain network consists of nodes representing suppliers, production points, warehouses, distributors, retailers and end customers. An example supply chain network is illustrated in Figure 1.1 (adapted from Cohen and Lee 1988). A supply chain network represents a collection of organizations that are involved in processes related to the production of end items. It aims at attaining customer satisfaction and maximizing profit for all organizations belonging to the supply chain. So as a whole, the various constituents of the supply chain act as a closelyknit structure. Figure 1.1 Supply Chain Network Performance evaluation involves the development and solution of analytical or simulation models for determining the values of the performance measures that can be Raw Material Vendors Intermedia te Product Plants Final Product Pla nts Distribution Centers Warehouse s Customer Zon es 2 expected from a given set of decisions (e.g., those related to capacity and inventory levels). Performance evaluation tools aid system designers and operations managers in making some key decisions, while keeping in mind the goals of the company (Suri et al. 1993). Analytical performance evaluation tools are typically based on modeling techniques such as Markov chains, stochastic Petri nets, and queueing networks (Viswanadham and Narahari 1992). Simulation models mimic the detailed operations of a system by means of a computer program and hence, they need more detailed information for modeling. Simulation model development and model execution could be time consuming. On the other hand, analytical models like queueing networks describe the actual system based on mathematical relationships. These models need certain simplifying assumptions to be made (e.g., first come first serve (FCFS) queueing discipline) and the results obtained are generally less accurate compared to simulation results. However, these models yield results more quickly and “are appropriate for rapid and rough cut analysis (Suri et al. 1993).” In fact, analytical and simulation models can be used in tandem to analyze and design complex systems. For example, analytical models can be used to reduce a large set of alternatives, and the remaining few alternatives can be studied in detail using simulation models (Suri et al. 1993; Leung and Suri 1990). The development of performance evaluation and optimization models for supply chain networks is an active area of research. Only a few studies have dealt with the development of analytical performance evaluation models for supply chain networks. The focus of this research was on developing queueing network models of supply chain networks using Whitt’s parametric decomposition approach (Whitt 1983, 1994). The 3 approach used in this research was similar to the approach developed by Sivaramakrishnan (1998) in modeling productioninventory networks. The approach used by Sivaramakrishnan has been extended to include supply chain network features such as supply, transportation and distribution (Srivathsan et al. 2004). 1.1 ANALYTICAL MODELS BASED ON QUEUEING Queueing models of manufacturing systems have been developed since the 1950’s. Some of the literature in this field includes the works of Jackson (1957), Segal and Whitt (1989) and text books and handbook chapters that focus on queueing models of manufacturing systems including Buzacott and Shanthikumar (1993), Viswanadham and Narahari (1992) and Suri et al. (1993). Queueing models have often ignored the effect of stocking (the holding of planned inventory). On the other hand, inventory models have ignored the effect of capacity and congestion issues and concentrated mainly on the inventory policy, order quantity, etc. (Sivaramakrishnan 1998). Models of productioninventory networks which extend standard queueing models to include planned inventories by considering capacity/congestion issues and inventory issues, simultaneously, came to the forefront with the works of Buzacott and Shanthikumar (1993), Lee and Zipkin (1992), Sivaramakrishnan (1998), Sivaramakrishnan and Kamath (1996) and Zipkin (1995). These models make assumptions like the oneforone replenishment policy (where for each item consumed, an order is immediately placed for one item), orders for a single item or orders for batches of constant size, unlimited supply of raw materials, etc. Buzacott and Shanthikumar (1993) have modeled the singlestage systems extensively, while Lee and Zipkin (1992), Sivaramakrishnan (1998), Sivaramakrishnan and Kamath (1996) and 4 Zipkin (1995) concentrated on multistage systems. In particular, Sivaramakrishnan (1998) dealt extensively with tandem maketostock systems, and later extended his work to model feedforward type maketostock networks. Modeling a supply chain network includes the modeling of capacity and inventory issues. However, additional features like supply, distribution and transportation have to be included to yield more useful models. Hence, this research has extended the previous work on productioninventory networks to model supply chain networks by including such features. 1.2 OUTLINE OF THE THESIS The rest of the document is structured as follows. Chapter 2 presents a literature review of the work done in modeling productioninventory networks and supply chain networks. Chapter 3 presents the research statement. It includes the research goals, objectives, limitations and contributions to the field of modeling supply chain networks using queueing theory. Chapter 4 presents the research approach. It includes the methodology, the list of performance measures that will be evaluated and the validation method. Chapter 5 presents the analytical model of a divergent network consisting of one manufacturer with a finished goods store and two retailers. Chapter 6 presents the analytical modeling of the convergent network with two manufacturers and one retailer. Chapter 7 presents the analytical model of the convergent network with two suppliers. Chapter 8 presents the analytical model of a convergentdivergent network, which is a combination of the above three networks. Finally, Chapter 9 presents the conclusions and the scope for future research. 5 CHAPTER 2 LITERATURE REVIEW This chapter presents a review of the literature on the analytical performance modeling of productioninventory networks and supply chain networks. Section 2.1 deals with the modeling of productioninventory networks. Queueing models have been mainly used to model the following types of productioninventory networks, namely, maketoorder systems, maketostock systems and hybrid systems (combination of maketoorder and maketostock systems). This section summarizes the work related to these three types of productioninventory networks. Section 2.2 deals with the literature on performance evaluation of supply chain networks. 2.1 PRODUCTION  INVENTORY NETWORKS When manufacturing systems are considered along with the holding of planned inventories, the resulting network is a productioninventory network. These systems may be classified as maketoorder, maketostock, and a hybrid of these two based on the inventory policy. Research related to the modeling of these networks is presented in this section. 6 2.1.1 MaketoOrder Production Inventory Networks In a maketoorder system, the constituents wait for the customer to place an order. Once an order is placed, the constituents of the system start interacting with each other through the flow of material as well as information about customer requirements and the final product is manufactured. In a pure maketoorder system, no inventory of finished products or intermediate items is maintained. A pure maketoorder system is exactly what a queueing model represents. For a comprehensive review of singlestage as well as network models, the reader is referred to Suri et al. (1993). Detailed coverage of queueing models of production networks can also be found in text books such as Buzacott and Shanthikumar (1993) and Viswanadham and Narahari (1992). Whitt (1983, and 1994) and Segal and Whitt (1989) describe the popular parametric decomposition approach based on twomoment queueing approximations to solve general production network models. 2.1.2 MaketoStock Production Inventory Networks In a maketostock system, the final product is manufactured and sent to the retail store in anticipation of customer demand. As customer orders arrive, the retailer sells from the stock of finished product inventory held at the retail store. As the inventory level falls to a predetermined value, the retailer orders for the product from the manufacturer so that he is in the best position to satisfy future customer demand. Svoronos and Zipkin (1991) modeled a multiechelon inventory system where demand occurred at the lowest hierarchy of the system called the leaf, and each leaf placed an order at its predecessor and so on till the demand reached the central depot. The central depot received raw materials from an outside source assumed to have infinite 7 capacity. They analyzed the singlestage location with Poisson demand arrivals and onefor one replenishment policy with a basestock level S. The consumption of an item immediately triggers a replenishment order. The steadystate behavior of the system was characterized by the random variables I, B and K where I is the inventory, B is the number of backorders and K = (S–I+B) is the number of outstanding orders at a particular stage. They applied the results (expected inventory and expected backorders) of the singlestage problem, recursively, starting from the highestlevel echelon to analyze the complete network. Lee and Zipkin (1992) developed a model for tandem queues with planned inventories. They assumed Poisson demand arrivals, mutually independent exponential service times, and a oneforone replenishment policy where the system was controlled by a stationary demand pull or basestock policy. The basestock level for stage j was specified by Sj. The arrival of customer demand consumed a finished product from the output store, if available, and a corresponding order was placed at the previous stage for replenishment. If the finished product was not available, then the order waited for the part to arrive from the previous stage and the same process was assumed to occur at all stages. They assumed that the queue at the processing stage to be infinite. After processing, the part either moved to the output store at that stage or to the next stage depending on the demand. Any backorder at the instant of process completion immediately released the part to the next stage or to the customer to fulfill the backorder. When the basestock level is zero at all stages, the system behaves like a maketoorder system, and if only certain stages have a basestock level of zero, then the system behaves as a combination of maketoorder and maketostock systems. Lee and Zipkin (1992) captured congestion 8 measures like expected inventory and expected backorders using the M/M/1 queueing model, and made use of the approximation developed by Svoronos and Zipkin (1991). Buzacott and Shanthikumar (1993) analyzed a singlestage maketostock system in great detail. They used a production authorization card (PA card) concept to model the maketostock system. They assumed that as a product was manufactured and sent to a retail store, a tag was attached to it and this tag was converted into a production authorization card when the product was delivered to the customer. They modeled a single machine with unit demand and backlogging by assuming that if the finished product inventory is not available when a customer order arrived, the order is backlogged and that each customer placed an order for only one finished product. They considered the retail store to be full initially and at that instant, there were Z tags in the system. ( ) ( ) ( ) Therefore, ( ) { ( ), } ( ) {0, ( ) } ( ) {0, ( )} B t C t N t C t Min N t Z B t Max N t Z I t Max Z N t + = = = − = − where I(t) is the inventory level or the number of finished products, B(t) is the number of customer orders backlogged, C(t) is the number of PA cards available, and N(t) is the number of jobs in the system, all at time t. As a result of Equation 3.4, it can be seen that by studying the process N(t), I(t), B(t) and C(t) can be derived. The maketostock system was modeled using an M/M/1 queue and under steadystate condition, they found out the expected inventory and expected number of backorders. They also presented the respective results for the GI/G/1 (3.1) (3.2) (3.3) (3.4) 9 model. They also modeled singlestage maketostock systems with lost sales, interrupted demand, bulk demand, machine failures and yield losses. Sivaramakrishnan (1998) modeled an Mstage tandem maketostock system that was controlled by a stationary demandpull or basestock policy. He assumed that the setup times were included in the processing times, and that if finished goods were available at the output store, then the order was satisfied and an order for the item would be triggered to replenish the finished item as per the oneforone replenishment policy. If parts were available at stage M1, then replenishment took place. Otherwise the order was backlogged at stage M1 and an order was placed at the previous stage and so on. The arrival of a part at any stage could be an item directly after production at the previous stage in the case of a backorder or an item from the output store of the previous stage in the case of replenishment orders. At the first stage, the order went directly to process as unlimited supply of raw material was assumed. A delay node was used in the model mainly to “capture the upstream delay experienced by an order when there was no part in the output store of the previous stage (Sivaramakrishnan 1998).” The singlestage model with a delay node is as shown in Figure 2.1. Figure 2.1 Singlestage Maketostock System with a Delay Node Output store Demand Processing Stage Backorder Delay 10 Sivaramakrishnan (1998) developed a new decomposition framework for the analysis of the above system using results of M/M/1 and M/M/∞ queueing systems and obtained the results for the expected inventory and expected backorders at each stage of the tandem network. He also presented the corresponding results for a maketostock tandem network with general arrivals and general service times by extending the parametric decomposition approach (Whitt 1983, 1994) to maketostock systems. He extended the tandem model to include (1) multiple servers at a stage, (2) batch service, (3) limited supply of raw material, (4) multiple part types, and (5) service interruptions due to machine failure. Sivaramakrishnan (1998) also extended his approach to tandem networks with feedback and feedforward networks. Karaesmen et al. (2002) assumed that the interarrival time and the processing times are geometrically distributed and modeled the system with advance order information, where the orders are received well in advance of the time when the items covered by the orders are required to be delivered. They analyzed the basestock policy (S, L) with a focus on optimization and performance evaluation of the Geo/Geo/1 maketo stock queue, where L is the release leadtime. They went on to obtain the expressions for expected backlog and expected inventory. 2.1.3 Hybrid Production Inventory Networks Nguyen (1995) analyzed the problem of setting the basestock levels in a production system that produced both maketoorder and maketostock products. She made the following assumptions: 1) the demand that cannot be satisfied is lost and 2) the production of maketostock items is based on a basestock policy with a basestock level for each product. She derived the productform steadystate distributions for the above 11 network under the assumptions that 1) each station operates under first in first out (FIFO) service discipline, 2) all processing times and interdemand times follow exponential distribution, 3) all products have the same mean processing time, and 4) the contribution from maketoorder jobs to relative traffic intensity at the workstation is less than one. She proposed approximations for the basestock levels based on heavy traffic analysis of queueing networks. Nguyen (1998) presented an algorithm for setting the basestock levels for 1) FIFO service priority, 2) priority service for maketoorder products, and 3) priority service for maketostock products. 2.2 SUPPLY CHAIN NETWORKS Cohen and Lee (1988) developed an analytical model for integrated productiondistribution systems to predict the impact of alternative strategies. They decomposed the network into submodels and optimized the submodels based on certain control parameters which serve as links between the submodels. Lee and Billington (1993) and Ettl et al. (2000) both focused on “capturing the interdependence of basestock levels at different stores (Ettl et al. 2000).” Further, both used models with limited capacity. While Lee and Billington (1993) focused on only performance evaluation and assumed stationary demand, Ettl et al. (2000) focused on optimization as well and considered nonstationary demand. Raghavan and Viswanadham (2001) and Viswanadham and Raghavan (1999, 2001, and 2002) developed highlevel models of a supply chain network using a variety of tools like Petri nets, series parallel graphs, and queueing theory. With regard to queueing models, their focus was on the use of forkjoin approximations to compute the mean and variance of the departure process in a supply chain network. They presented 12 simple approximations for the case of deterministic arrivals and Normal service time distributions. In summary, it was observed that research on developing analytical performance evaluation models of supply chain networks was very limited. The preliminary work done as part of this research had revealed that productioninventory network models could be extended to model supply chain networks (Srivathsan et al. 2004). This was the focus of the thesis research effort. 13 CHAPTER 3 STATEMENT OF RESEARCH The overall goals of this research were (i) to develop queueing models of “building block” type configurations of supply chain networks and solve them using Whitt’s parametric decomposition approach and (ii) to lay a foundation for the development of a queueingbased rapid analysis tool for the performance evaluation of supply chain networks. 3.1 RESEARCH OBJECTIVES The objectives of this research were as follows. Objective 1: To perform a review of the literature related to analytical modeling of productioninventory networks and supply chain networks. Objective 2: To model a divergent supply chain network configuration with one manufacturer supplying two retail locations. Objective 3: To model two convergent supply chain network configurations, one with two manufacturers supplying one retailer and the other with two suppliers supplying a manufacturer. 14 Objective 4: To model a combination convergentdivergent supply chain network configuration with three suppliers supplying two manufacturers, who in turn provide finished goods to three retailers. The details of the supply chain network configurations are contained in Section 4.3. Modeling involves developing the analytical model and validating it using simulation (see Section 4.1.3). 3.2 RESEARCH SCOPE AND LIMITATIONS The scope of this thesis was limited by the following assumptions: 1. Assembly operations and material handling issues were not considered. 2. Each order was for a single item or a batch of items with a constant batch size. 3. Only limited “building block” type configurations were studied (see Section 4.3). 3.3 RESEARCH CONTRIBUTIONS The purpose of this thesis was to contribute to the development of a queueingbased rapid performance analysis tool for supply chain networks. The following contributions have resulted from this work. 1. Development of queueingbased models of “building block” type supply chain network configurations. 2. Preliminary work on combining the “building block” configurations to model larger supply chain networks. 15 CHAPTER 4 RESEARCH APPROACH This chapter briefly describes the approach that was taken to successfully complete the research work. It also includes a list of important supply chain network performance measures that were addressed in this research, and the specific supply chain network configurations that were included in this study. 4.1 METHODOLOGY In this research, the supply chain network was modeled using queueing network models based on the twomoment framework. Whitt’s (1983, and 1994) parametric decomposition (PD) approach was then used to solve the network model. The PD approach is summarized in the next section. 4.1.1 The Parametric Decomposition (PD) Approach In the 1980’s, Whitt defined a new modeling ideology highlighted by the PD approach. According to Whitt, “a natural alternative to an exact analysis of an approximate model is an approximate analysis of an exact model (Whitt 1983).” The PD approach is a very comprehensive method of analyzing a network and uses only the first two moments of both the interarrival and service times. This approach is the basis of 16 a software package developed by Whitt, on behalf of AT&T, called the Queueing Network Analyzer (QNA). The PD approach for open queueing networks consists of two main steps: 1) analyzing the nodes and the interaction between the nodes to obtain the means and the squared coefficients of variation (SCV) of the interarrival times and 2) obtaining the node and system performance measures based on GI/G/1 or GI/G/m approximations. 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 (1957) 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 (1983). 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 and GI/G/m queues provided by Whitt (1983, 1993). The expected waiting time at each node is calculated from the results provided and 17 the expected queue length is obtained using Little’s law (Little 1961). Whitt (1983) also explains how several other node and network measures can be calculated. 4.1.2 Modeling Approach The approach developed by Sivaramakrishnan (1998) has been extended to model a productioninventory network by adding supply, distribution and transportation features. The results provided by Sivaramakrishnan (1998) and Buzacott and Shanthikumar (1993) were used to model the manufacturer stage. In the case of raw material supply and transportation, delay nodes were used to represent the supply leadtimes and transit times. The delay node was modeled using the M/G/∞ queue (as a result, the number in transit distribution is given by the Poisson distribution), which was similar to the approach of Sivaramakrishnan (1998). The only difference being that Sivaramakrishnan had used the delay node to capture the upstream delay experienced by an order when there was no part in the output store of the previous stage. The splitting and merging approximations presented by Whitt (1983) were extensively used to model a single node supplying multiple retail stores, and multiple nodes supplying a single retail store, respectively. 4.1.3 Validation Each supply chain network configuration that was modeled was simulated using Arena 7.0 software (Kelton et al. 2002). The parameters used for all the simulation experiments are presented in Table 4.1. The warmup period was determined by the application of Welch’s procedure (Welch 1983). Further details regarding the application of Welch’s procedure are contained in Appendix A6. 18 Table 4.1: Simulation Parameters Number of Replications Warmup period (time units) Run time (time units) 10 10,000 100,000 The parameters of the supply chain network configuration, namely, basestock levels; variability of interarrival, processing and transit times; utilization; and probabilities (if any)  were varied systematically to cover a wide range of scenarios. Table 4.2 gives the levels of variability that were tested in all the experimental configurations. The details about the distributions used and the procedure to estimate their parameters for running the simulation model are given in Appendix A6. The various values of the parameters used for experimentation are summarized in tables at the end of each chapter. Table 4.2: SCV and Distributions SCV Distribution 0.25 4stage Erlang 1 Exponential 2.25 2stage Hyperexponential For each scenario, the analytical results were compared with steadystate simulation estimates to evaluate the accuracy of the analytical results. Relative percentage error was used as an indication of the accuracy of the analytical model whenever it was appropriate. 100% simulation estimate Relative percentage error = (analytical result  simulation estimate) ⋅ As per Whitt’s (1993) suggestion, when the analytical and simulation estimates of a performance measure were small, i.e., less than one, the criterion that was used to 19 evaluate the accuracy of the analytical model was the absolute difference and not the relative percentage error. 4.2 PERFORMANCE MEASURES The performance measures that were evaluated were the expected inventory, expected backorders and the fill rate at the various stages of the supply chain network. The expected inventory at any stage is the average number of items in the store at that stage. The expected backorder at any stage is the average number of unsatisfied orders waiting at the stage. The fill rate is the probability that an order will be satisfied immediately and this depends on the instantaneous availability of inventory at the stage. This arrivaltime probability is approximated by the steadystate probability that the inventory level is not zero. 4.3 SUPPLY CHAIN NETWORK CONFIGURATIONS There are various configurations of a supply chain network, such as the serial network, convergent network, divergent network and convergentdivergent network. The following supply chain network “building block” configurations were studied in this thesis. 4.3.1 A Divergent Supply Chain Network Configuration In the divergent supply chain network configuration that was considered (see Figure 4.1), a single manufacturer produced the finished product and shipped it to two retailers. 20 Figure 4.1 Divergent Configuration 4.3.2 Convergent Supply Chain Network Configurations In the first convergent supply chain network configuration (see Figure 4.2), two manufacturers produced the same finished product and shipped it to a single retailer. Figure 4.2 Convergent Configuration 1 In the second convergent supply chain network configuration (see Figure 4.3), two suppliers supplied the same raw material to a single manufacturer. The case where the suppliers had different leadtimes was also modeled. M Manufacturer Retailers D1 D2 M1 Manufacturers Retailer D1 M1 21 Figure 4.3 Convergent Configuration 2 4.3.3 Combination ConvergentDivergent Supply Chain Network Configuration In the convergentdivergent supply chain network configuration (see Figure 4.4), three suppliers supplied two manufacturers who produced the same finished product and shipped it to three different retailers. The case where the manufacturers had different service time parameters was also modeled. Figure 4.4 ConvergentDivergent Configuration M1 D1 D2 M2 Manufacturers Retailers D3 S1 S2 Suppliers S3 M Manufacturer D S1 Suppliers S2 22 CHAPTER 5 DIVERGENT CONFIGURATION WITH TWO RETAILERS In this chapter, an analytical model for the divergent configuration with one manufacturer and two retailers with general demand interarrival times and general service times at the manufacturer is presented. The approach developed by Sivaramakrishnan (1998) was extended by adding distribution and transportation features. In order to model the general demand arrival processes, the traffic variability equations for the superposition of arrivals (Whitt 1983) were used. The remainder of the chapter is organized as follows. Section 5.1 gives a description of the system under study and the assumptions made. The mathematical procedure is explained in Section 5.2 and numerical results are presented in Section 5.3. Section 5.4 presents a summary of the results and future scope for improving the accuracy of the analytical model. 5.1 SYSTEM DESCRIPTION The system modeled has one manufacturer with a finished goods store and two retailers (see Figure 5.1). The finished goods store and the retail stores operate under a basestock policy with oneforone replenishment. The demand interarrival times at each of the retailers as well as the service times at the manufacturer follow a general distribution. The arrival of a demand consumes a finished product from the retailer and 23 causes an order to be placed at the manufacturer immediately. The transit time to ship the finished product from the output store of the manufacturer to the retailer was modeled using a delay node. The manufacturer was modeled by a singleserver queueing system. Figure 5.1 Divergent Configuration Ample supply of raw material was assumed. Furthermore, it was assumed that the raw material is released as and when an order is placed at the manufacturer. If the finished product is not available at the time of arrival of the customer order, then demand is backordered. When a finished product is produced, it is used to satisfy an outstanding backorder, if any; otherwise it is stored. 5.2 QUEUEING MODEL OF THE DIVERGENT CONFIGURATION The manufacturer was modeled by a singleserver queue and the two transit operations to the two retailers were modeled using delay nodes. Each delay node was modeled as an M/G/∞ queue. Manufacturer Retailer 1 Retailer 2 Output Store of Manufacturer External Demand External Demand Part Flow Replenishment+ Backorders Transit Time Transit Time 24 5.2.1 Approximate Solution of the Queueing Model When a customer order arrives at a retailer, if parts are available, the order is satisfied immediately and a replenishment order is sent to manufacturer 1. If a part is available in the output store at manufacturer 1, then it is immediately shipped to the retailer who placed the order. So any order waiting at either of the retailers is reflected in the parts in transit to the retailer or in manufacturer’s queue in case of any backorder at manufacturer 1. The distribution of the number in system can be obtained using the fact that if there are n orders at retailer i, k may be in transit and the remaining (nk) will be at manufacturer 1 and assuming that the random variables Ntm ri 1 and N pi1 are stochastically independent. This is an approximation because the network model does not satisfy the productform conditions and includes an approximation for merging in the arrival process at manufacturer 1 (Sivaramakrishnan 1998; Suri et al. 1993). ( ) ( ) ( ) 1 1 0 P N n P Nt k P N n k i p i n k r m i − = ⋅ = = = Σ= Since the demand arrival processes at the retailers and the service processes at manufacturer 1 are general, the wellknown Kraemer and LangenbachBelz (1976) formula for the GI/G/1 queue together with Little’s law (Little 1961) can be used to calculate the expected number of orders in manufacturer system 1. The arrival process at manufacturer 1 is a superposition of the demand arrival processes at the retailers. To calculate the SCV of this combined arrival process at retailer i, an approximation for the superposition of two arrival streams is used as shown in the following equations (Whitt 1983). 1 2 1 ρ1 = (λ r + λ r ) ⋅τ m (5.1) (5.2) 25 1 2 1 2 / 1 1 c w c w i k ari i k r r merge − + ⋅ = ⋅Σ λ Σλ ((( ) ) (( ) )) 1 2 2 1 2 /( ) /( ) 1 1 1 1 2 1 2 1 2 2 cmerge = w ⋅ λr λr + λr ⋅ car + λr λr + λr ⋅ car + − w (( ) ( )) 1 2 2 1 2 1 1 1 1 1 1 2 2 cmerge = w ⋅ pm r ⋅ car + pm r ⋅ car + − w where [ ( ) ( )] 1 1 2 1 1 4 1 1 1 − w = + ⋅ − ρ ⋅ ν − and [ 2 2 ] 1 1 1 1 1 2 − ν = pm r + pm r The expected number of orders at manufacturer system 1 can now be calculated using the Kraemer and LangenbachBelz (1976) formula and Little’s law (Little 1961). 1 2 2 1 2 1 2 ( ) (1 ) [ ] 1 1 1 ρ ρ ρ + + ⋅ − = ⋅ merge sm m c c E N g where, ≥ + ⋅ − − ⋅ − < ⋅ ⋅ + − ⋅ − ⋅ − = ; 1 ( 4 ) (1 ) ( 1) exp ; 1 3 ( ) (2 (1 ) (1 ) ) exp 2 2 2 2 1 2 2 2 1 2 2 1 1 1 1 1 1 1 1 1 merge merge sm merge merge merge sm merge c c c c c c c c g ρ ρ ρ The probability distribution for the number of backorders present in manufacturer system 1 is given by the following expression (Buzacott and Shanthikumar 1993) for a GI/G/1 maketostock system. ( ) ⋅ − ⋅ ≥ − ⋅ = = = (1 ) + − ; 1 (1 ) ; 0 ( 1) 1 1 1 1 1 1 1 1 l l P N l m m l S S m ρ σ σ ρ σ where, ( [ ] )/ [ ] 1 E Nm1 1 E Nm1 σ = −ρ An order at a retailer will be reflected in manufacturer queue only when there is a backorder at manufacturer 1. An order at manufacturer 1 is actually an “aggregate” order; (5.10) (5.11) (5.3) (5.4) (5.5) (5.6) (5.8) (5.7) (5.9) 26 the identity of the order (i.e., the retailer number) is lost after merging of the order arrival streams at manufacturer 1. Hence, a “disaggregation” procedure was used to find the distribution of the number of orders pertaining to retailer i at manufacturer 1, which is obtained by conditioning on the number of backorders in manufacturer system 1. The number of orders belonging to retailer i in manufacturing stage 1, given the number of backorders at the finished goods store, follows the Binomial distribution. ( ) 1 1 1 1 1 2 (  ) l r m r r p m pm r p r l P N r N l i ⋅ ⋅ − = = = where ( / ( )) pm1r1 r1 r1 r2 = λ λ +λ and ( / ( )) pm1r2 r2 r1 r2 = λ λ +λ are the proportions of orders from retailers 1 and 2, respectively. To derive the unconditional probability that a backorder at manufacturer 1 belongs to retailer i, the above value is multiplied by the probability of backorders and then summed over all values of l (backorders) greater than or equal to r. Σ ∞ = = = = = ⋅ = l r P Np r P Np r Nm l P Nm l i i ( ) (  ) ( ) 1 1 1 1 For retailer 1, ≥ − ⋅ − ⋅ ⋅ ⋅ + − = − ⋅ ⋅ ⋅ − ⋅ = = + + − ; 1 (1 ) (1 ) (1 ) ; 0 (1 ) (1 ) ( ) ( 1) 1 ( 1) 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2 1 1 2 11 r p p r p p P N r r m r r r S m r S m r S m r p m m m σ σ ρ σ ρσ σ σ σ ρ (5.12) (5.13) (5.14) 27 Similarly for retailer 2, ≥ − ⋅ − ⋅ ⋅ ⋅ + − ⋅ = − ⋅ ⋅ ⋅ ⋅ − = = + + − ; 1 (1 ) (1 ) (1 ) ; 0 (1 ) (1 ) ( ) ( 1) 1 ( 1) 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 21 r p p r p p P N r r m r r r S m r S m r S m r p m m m σ σ ρ σ ρ σ σ ρ σ σ Recall that the transit time was modeled using a delay node. Hence, the distribution of the number in transit is approximated using the M/G/∞ formula (Sivaramakrishnan 1998) as follows ! ( ) 1 1 1 k e P Nt k k tm r m r i tm ri i ρ ρ ⋅ = = − where, tm1ri m1 m1ri ρ = λ ⋅τ The distribution of the number in system (obtained using equation (5.1)) can be used to find the distribution of the inventory and backorders at the retailers. It can be seen that if there are n orders in the system, then the inventory level is given by (S – n), provided n is less than S (S is the basestock level). Similarly, there are n backorders in the system if there are (S + n) orders in the system, provided S is less than n. From these relations, the probability distributions for the inventory level and backorders at retailer i are obtained using the following relations (Sivaramakrishnan 1998) i i i i i r i r r r k r i P I k P N S k k S P I P N S k ( ) ( ) ; 1, 2, , ( 0) ( ) 0 = = = − = K + = = = Σ ∞ = ( ) ( ) ; 1, 2,K ( 0) ( ) 0 = = = + = = = = Σ= P B k P N k S k P B P N k i i ri i r i r S k r i (5.15) (5.18) (5.19) (5.16) (5.17) 28 The expected inventory, expected backorder and fill rate at retailer i are given by [ ] ( ) 1 E I n P I n i ri i r S n r = ⋅ = Σ= [ ] ( ) 0 E B n P B n i ri n r = ⋅ = Σ ∞ = ( ) 1 F P I n ri i i S n r r = =Σ= At manufacturer stage 1, the results provided by Buzacott and Shanthikumar (1993) for a GI/G/1 maketostock system are used to arrive at the expected inventory, expected backorders and fill rate. The expected number of backorders at manufacturer 1 is given by Buzacott and Shanthikumar (1993), (1 ) [ ] 1 1 1 1 1 σ ρ σ − ⋅ = S m E Bm The probability distribution of the inventory level at manufacturer 1 is given by Buzacott and Shanthikumar (1993), − = ⋅ − ⋅ = − ⋅ = = ≈ − − − 1 1 1 1 1 1 ; (1 ) ; 1, 2, , 1 ; 0 ( ) 1 ( 1) 1 1 1 ( 1) 1 1 m m S n S m i S i S i P I n m m ρ ρ σ σ ρ σ K The expected inventory at the output store of manufacturer 1 is given by Σ= = ⋅ = 1 1 1 0 [ ] ( ) S m n E I m n P I m n (5.23) (5.25) (5.20) (5.21) (5.22) (5.24) 29 The fill rate at manufacturer 1 is given by ( ) 1 1 1 1 F P I n S m n m m = =Σ= 5.3 NUMERICAL EXPERIMENTS The accuracy of the analytical model was tested by comparing the analytical results to simulation estimates for a wide range of parameter values. The experimental design for comparison is shown in Table 5.1. Table 5.1 shows the different parameters that were varied and the values that were used for different experiments. Table 5.1: Experimental Design for Numerical Comparison Parameter Levels Level Values Arrival Rate Pair ( r1 λ ,λr2 ) 2 ( 1, 1 ) and ( 1.25, 0.75 ) Basestock level 3 2, 4, 8 (same for the retailers and manufacturer) Transit time distribution 2 Unif ( 1, 5 ) and Unif ( 4, 8 ) Utilization 2 80% and 90% Interarrival Distribution at Retailer 1 3 Erlang , Exponential and Hyperexponential Interarrival Distribution at Retailer 2 3 Erlang , Exponential and Hyperexponential Service time Distribution at manufacturer 1 3 Erlang , Exponential and Hyperexponential The analytical and simulation results for the various experiments are contained in Appendix A1. 5.4 SUMMARY OF RESULTS A total of 648 experiments were used to evaluate the accuracy of the analytical model. The analytical method gives excellent results (relative percentage error < 10%) in (5.26) 30 cases where the interarrival distribution is exponential and the service time distribution is exponential. The analytical results are fairly consistent with the simulation estimates when the interarrival and service times are general. Table 5.2 shows the accuracy of the approximations developed. The table shows the proportion of the results that fall within a certain error range. The absolute relative error was used to check the accuracy for the expected inventory and expected backorders whereas the minimum of the absolute relative error and the absolute difference between the analytical and simulation values was used for the fill rates due to the fact that the value of fill rate is always less than one and sometimes even less than 0.5. In a case where the values are small, the criteria to be tested will be the absolute difference and not the absolute relative error as suggested by Whitt (1993). Table 5.2: Summary of Results Percentage of Results within the Error Range (%) Error Range Expected Inventory Expected Backorders Fill Rate < 0.25 96.5 94.9 99.9 < 0.20 94.2 92.4 99.9 < 0.15 86.9 88.6 99.3 < 0.10 79.2 80.8 95.8 < 0.05 67.0 61.2 83.6 From the detailed results presented in Appendix A1, it can be seen that the analytical model does not perform well (relative error percentage greater than 15%) when the SCV is high. Some possible sources of error are presented below. The use of M/G/∞ queue to model the delay node seems to be a major area of concern. This is because even when the interarrival time at the delay node is general, it is 31 assumed to be Poisson. The problem here is that there are not any analytical results available for the distribution of the number in the system for a GI/G/∞ queue. It can be seen from the result tables in Appendix A1 that when the SCV of interarrival process is different for the two retailers, there is a discrepancy in the results. It can be seen that the analytical model gives the same value of a performance measure, whether it is the expected inventory or expected backorder or fill rate, for both the retailers. The analytical results in such a case agree closely with the simulation results when the arrival process is Poisson. The analytical method overestimates when the interarrival time is Erlang and underestimates when the interarrival time is Hyperexponential. The reason for this was the fact that the analytical model was not able to account for the difference in variability while computing the individual retailer performance measures. After computing the distribution of the “aggregate” orders at the manufacturer, the “disaggregation” procedure to find the distribution of orders corresponding to a particular retailer was based on a “proportion of orders” belonging to a retailer (See Equation 5.12).While calculating these proportions, the model used only the arrival rates and not the SCVs (See Equation 5.12). It would be reasonable to expect the analytical results to improve if SCVs are included in the computation of “proportions.” This is suggested as an area for future research. 32 CHAPTER 6 CONVERGENT CONFIGURATION WITH TWO MANUFACTURERS In this chapter, a convergent network with two manufacturers supplying a single retailer is considered. The demand interarrival and service times at the manufacturers are considered to be general. The splitting approximation for traffic variability was used to compute the SCV of the order arrival process at each of the manufacturers. The next section, Section 6.1, gives a complete description of the system under study and the various assumptions made. The mathematical procedure is explained in Section 6.2 and numerical experiments are discussed in Section 6.3. Section 6.4 presents a summary of the results and future scope for improving the accuracy of the analytical model. 6.1 SYSTEM DESCRIPTION A maketostock convergent network with two manufacturers, each with a finished goods store, and a retailer (see Figure 6.1) was considered. The finished goods stores and the retail store operate under a basestock policy with oneforone replenishment. The demand interarrival times at the retailer as well as the service times at both the manufacturers follow a general distribution. The arrival of a demand consumes a finished product from the retailer and causes an instantaneous order to be 33 placed at one of the manufacturers based on a fixed probability. The finished product from the output store of a manufacturer is shipped to the retailer to satisfy the order for replenishment and the output store places a replenishment order at the manufacturing stage. As before, the transit time to ship finished products from the output store of the manufacturer to the retailer was modeled using a delay node. The manufacturer was modeled by a singleserver queueing system. Figure 6.1 Convergent Configuration 1 An ample supply of raw material was assumed. Also it was assumed that the raw material is released as and when an order is placed at a manufacturer. If the finished product is not available at the time of arrival of the customer order, then demand is backordered. When a finished product arrives, it is used to satisfy an outstanding backorder, if any; otherwise it is stored. Retailer Manufacturer 2 Manufacturer 1 Output Store of Manufacturer 2 Output Store of Manufacturer 1 Part Flow Replenishment + Backorders Transit Time Transit Time 34 6.2 QUEUEING MODEL OF THE CONVERGENT CONFIGURATION WITH TWO MANUFACTURERS Each manufacturer is modeled by a singleserver queue and the two transit operations to retailer 1 from the manufacturers are modeled using delay nodes. Each delay node is modeled as an M/G/∞ queue. 6.2.1 Approximate Solution of the Queueing Model When a customer order arrives at retailer 1, if parts are available, the order is satisfied immediately and a replenishment order is sent to one of the manufacturers depending on a probabilistic split. It is assumed this split probability is known. If a part is available in the output store at manufacturer i, then it is immediately shipped to retailer 1. So any order waiting at retailer 1 is reflected in the items in transit or in a manufacturer’s queue in case of backorder at that manufacturer. The distribution of the number in system can be obtained by using the fact that if there are n orders in the system, if k are at manufacturer 1 then the remaining (n  k) will be in manufacturer 2. Of the k orders at manufacturer 1, j may be in transit from manufacturer 1 and the remaining (k – j) in manufacturing stage 1, in case of backorder at the output store of manufacturer 1. Similarly of the (n  k) orders at manufacturer 2, l of them may be in transit and the remaining in manufacturing stage 2. The distribution of the number of orders in the system can be obtained by assuming that the random variables representing the orders for the manufacturers are stochastically independent. Similarly, the distribution of the number of orders for a manufacturer can be obtained by assuming that the random variables representing the orders in transit and the random variable representing the orders in the manufacturer queue are stochastically independent. The above approach is 35 an approximation because the network model does not satisfy the productform conditions and includes an approximation for splitting in the arrival process at retailer 1 (Sivaramakrishnan 1998; Suri et al. 1993). The probability distribution for the number of orders at retailer 1 is given by ( ) ( ) ( ) 1 2 0 P N 1 n P N k P N n n k n k n − = ⋅ = = = Σ= The probability distribution for the number of orders at manufacturer i is given by ( ) ( ) ( ) 1 1 0 P N k P Nt j P N k j i i p i k j r m n − = ⋅ = = = Σ= As before, the Kraemer and LangenbachBelz (1976) formula and Little’s law (Little 1961) were used to calculate the expected number of orders at each of the manufacturers. To compute the SCV of the order arrival process at manufacturer i, the approximations for the splitting of an arrival stream are used as shown in the following equations (Whitt 1983). c q c p c p c q sp ar sp ar = ⋅ + = ⋅ + ( ) ( ) 2 2 2 2 12 1 11 1 where, p is the probability that an order from retailer 1 goes to manufacturer 1 q is the probability that an order from retailer 1 goes to manufacturer 2 The expected number of orders at manufacturer system i is given by i sp sm i i m i i i i c c E N g ρ ρ ρ + + ⋅ − = ⋅ 2 ( ) (1 ) [ ] 2 2 2 1 where, = ⋅ ; i = 1, 2 i ri m i ρ λ τ (6.1) (6.2) (6.5) (6.6) (6.3) (6.4) 36 and ≥ + ⋅ − − ⋅ − < ⋅ ⋅ + − ⋅ − ⋅ − = ; 1 ( 4 ) (1 ) ( 1) exp ; 1 3 ( ) (2 (1 ) (1 ) ) exp 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 i i i i i i i i sp sp sm i sp sp i sp sm i sp i c c c c c c c c g ρ ρ ρ The probability distribution of the number of orders at manufacturing stage i is given by ( ) ⋅ − ⋅ ≥ − ⋅ = = = (1 ) + − ; 1 1 ( ) ; 0 ( 1) m m P N m mi mi i m S i i i S i i m ρ σ σ ρ σ where, ( [ ] ) / [ ] i E Nmi i E Nmi σ = −ρ The probability distribution of the number of orders in transit from manufacturer i to retailer 1 is given by ! ( ) 1 1 1 j e P Nt j j m r tmir tmir i ρ ρ ⋅ = = − where, tmi r1 mi mi r1 ρ = λ ⋅τ The distribution of the number in system (obtained using equation (6.1)) can be used to find the distribution of the inventory and backorders as in Section 5.2.1. 1 1 1 1 1 ( ) ( ) ; 1, 2, , ( 0) ( ) 1 0 1 r r r r k r P I k P N S k k S P I P N S k = = = − = K + = = = Σ ∞ = ( ) ( ) ; 1, 2 , K ( 0 ) ( ) 1 1 1 1 1 0 1 = = = + = = = = Σ= P B k P N k S k P B P N k r r S k r r (6.7) (6.8) (6.10) (6.11) (6.9) (6.12) (6.13) 37 The expected inventory, expected backorders and fill rate at retailer 1 are given by [ ] ( ) 1 1 1 1 E I n P I r n S n r r = ⋅ = Σ= [ ] ( ) 1 1 0 E B n P B r n n r = ⋅ = Σ ∞ = ( ) 1 1 1 1 F P I n S r n r r = = Σ= At each manufacturer, the results provided by Buzacott and Shanthikumar (1993) for a GI/G/1 maketostock system were used to arrive at the expected inventory, expected backorders and fill rate. The expected backorders at the output store of manufacturer i is given by (1 ) [ ] i S i i m mi i E B σ ρ σ − ⋅ = The probability distribution of the inventory level at manufacturer i from (Buzacott and Shanthikumar 1993), − = ⋅ − ⋅ = ⋅ = = ≈ − − − i i mi mi i i m m S n i i i S i i m n S n S n P I n 1 ; (1 ) ; 1, 2, , ; 0 ( ) ( 1) ( 1) ρ ρ σ σ ρ σ K The expected inventory at the output store of manufacturer i is given by Σ= = ⋅ = mi i i S n E Im n P Im n 0 [ ] ( ) The fill rate for the output store of manufacturer i is given by ( ) 1 F P I n mi i i S n m m = =Σ= (6.17) (6.19) (6.20) (6.14) (6.15) (6.16) (6.18) 38 6.3 NUMERICAL EXPERIMENTS The accuracy of the analytical model was tested by comparing the analytical results to simulation estimates for a wide range of parameter values. The experimental design for comparison is shown in Table 6.1. Table 6.1: Experimental Design for Numerical Comparison Parameter Levels Level Values Arrival Rate (λr1 ) 1 1 Split to Manufacturer ( pm1r1 , pm2 r1 ) 2 ( 50%, 50%) and ( 75%, 25% ) Basestock level 3 2, 4, 8 (same for the retailer and manufacturers) Transit time distribution 2 Unif ( 1, 5 ) and Unif ( 4, 8 ) Utilization 2 80% and 90% Interarrival distribution at retailer1 3 Erlang , Exponential and Hyperexponential Service time distribution at manufacturer 1 3 Erlang , Exponential and Hyperexponential Service time distribution at manufacturer 2 3 Erlang , Exponential and Hyperexponential The analytical and simulation results for the various experiments are contained in Appendix A2 6.4 SUMMARY OF RESULTS A total of 648 experiments were used to evaluate the accuracy of the analytical model. The results show that the analytical method performs very well for almost all cases. The analytical results are consistent with the simulation estimates when the interarrival and service times are general. The relative percentage error was less than 10% in approximately 9 out of 10 experiments on an average. Table 6.2 shows the accuracy of the approximations developed. The table shows the proportion of the results that fall 39 within a certain error range. The absolute relative error was used to check the accuracy for the expected inventory and expected backorders whereas the minimum of the absolute relative error and the absolute difference between the analytical and simulation values was used for the fill rates due to the fact that the value of fill rate is always less than one and sometimes even less than 0.5. In a case where the values are small, the criteria to be tested will be the absolute relative difference and not the absolute error as suggested by Whitt (1993). Table 6.2: Summary of Results Percentage of Results within the Error Range (%) Error Range Expected Inventory Expected Backorders Fill Rate < 0.25 98.3 98.5 99.9 < 0.20 96.6 97.6 99.9 < 0.15 93.9 92.5 99.7 < 0.10 88.6 83.6 98.4 < 0.05 72.0 61.5 89.8 The results show that the analytical method performs very well for almost all cases. This is due to the fact that the approximation for queue length distribution in a GI/G/1 queue is known to yield good results. This model performs better than the divergent configuration because of the following reasons. In the divergent network, since the order that is backordered at the manufacturer could have originated from either of the retailers, a conditional probability (the disaggregation procedure) had to be used to obtain the probability distribution of the number of orders belonging to retailer i that are backordered at the manufacturer. Whereas, in the case of the convergent network with two manufacturers, it is known that the order that is backordered at the manufacturer belongs to retailer 1 only. Therefore the 40 queue length distribution of the GI/G/1 queue could be directly used to find the probability distribution of the number of orders from retailer 1 that get backordered at either of the manufacturers. 41 CHAPTER 7 CONVERGENT CONFIGURATION WITH TWO SUPPLIERS In this chapter, a convergent configuration with two suppliers supplying a single manufacturer is considered. The demand arrival and manufacturing service processes are considered to be general. Approximations for splitting and merging are used to take into account the delay due to nonavailability of part at the input store of the manufacturer. The next section, Section 7.1, gives a complete description of the configuration under study and the various assumptions made. The approximate solution procedure is explained in Section 7.2, and numerical experiments are presented in Section 7.3. Section 7.4 presents a summary of the results and future scope for improving the accuracy of the analytical model. 7.1 SYSTEM DESCRIPTION A convergent configuration with two suppliers and a manufacturer with an input store as well as an output store (see Figure 7.1) was considered. The input store and the output store operate under a basestock policy with oneforone replenishment. The demand interarrival times at the manufacturer as well as the service times at the manufacturer follow a general distribution. The arrival of a demand consumes a finished 42 product from the output store of the manufacturer and causes an order to be sent to the manufacturing stage. For each order, the manufacturing stage needs a part from the input store. The input store of the manufacturer in turn places an order at one of the suppliers. The time needed to transport the materials ordered from a supplier to the input store was modeled using a delay node. The manufacturer was modeled by a singleserver queueing system. Figure 7.1 Convergent Configuration 2 When the supplier receives an order, if a part is readily available at the supplier, it reaches the input store of the manufacturer after a transit delay. In case of stockout at the supplier, the order experiences a leadtime delay to acquire the raw material in addition to the transit delay. Transit Time Transit Time Lead Time Lead Time Part Flow Supplier 1 Supplier 2 Input Store Manufacturer Demand Output Store Replenishment + Backorders 43 7.2 QUEUEING MODEL OF THE CONVERGENT CONFIGURATION WITH TWO SUPPLIERS The manufacturer was modeled by a singleserver queue and the supplier operations to the manufacturer were modeled using delay nodes. Each delay at the supplier side was modeled as an M/G/∞ queue. 7.2.1 Approximate Solution of the Queueing Model The model behaves like the other models discussed in earlier Chapters with the orders proceeding from the output store of the manufacturer to the manufacturing stage to the input store of the manufacturer and finally to the supplier. So any order arriving at manufacturer 1 is reflected in the call for replenishment at either manufacturing stage queue 1 or in the supplier node in case of any backorder at that input store of manufacturer 1. The distribution of the number in manufacturer 1 can be obtained using the fact that if there are n orders in manufacturer 1, k may be in manufacturing stage 1 and the remaining (n  k) will be at the supplier “node”, if there is a backorder at the input store of manufacturer1. The distribution of the number of orders in the system can be obtained by assuming that the random variables representing the orders in manufacturing stage 1 and the random variable representing the orders at the supplier node are stochastically independent. As before, this is an approximation because the network model does not satisfy the productform conditions (Suri et al. 1993). The probability distribution for the number of orders at manufacturer 1 is given by ( ) ( ) ( ) 1 1 1 0 P N n P N k P Brm n k n k m n − = ⋅ = = = Σ= (7.1) 44 The delay at the supplier end which includes both the leadtime delay and transit delay was modeled as a single delay node (see Figures 7.1 and 7.2). The approximations for splitting and merging are applied to modify the SCV of the arrival process to take into account the delay (if any) in obtaining the raw material (Whitt 1983). From Figure 7.2, it can be seen that the arrival at manufacturer 1 splits into two streams – one which finds the raw material at the input store of manufacturer 1 and the second one where the raw material is not available at the input store and a backorder is placed at the supplier stage. So the splitting approximation (Whitt 1983) is used to find the variability of the component that forms part of the arrival at manufacturer 1. 1 1 1 1 2 2 1 carm = Frm ⋅ cmerge + − Frm The other component of arrival process is the fraction of the departure from the supplier node. As the supplier node is modeled as an M/G/∞ queue, the departure process from the supplier node is Poisson. As seen from the Figure 7.2, this departure process splits into two, and one of them forms the input to manufacturer 1. It is a known result that the splitting of a Poisson stream gives rise to Poisson streams. So 2 cd1 and 2 cd2 are both equal to one. The merging approximation is used to arrive at the variability of the arrival process at manufacturer 1. 2 2 2 1 1 2 1 1 cam = (1− Frm ) ⋅ cd + Frm ⋅ carm (7.2) (7.3) 45 Figure 7.2 Splitting and Merging at the Input Store The expected number of orders at manufacturer system 1 is given by (Kraemer and LangenbachBelz 1976) and Little’s law (Little 1961). 1 2 2 1 2 1 1 2 ( ) (1 ) [ ] 1 1 1 ρ ρ ρ + + ⋅ − = ⋅ am sm m c c E N g where, 1 m 1 m 1 ρ = λ ⋅τ ≥ + ⋅ − − ⋅ − < ⋅ ⋅ + − ⋅ − ⋅ − = ; 1 ( 4 ) (1 ) ( 1) exp ; 1 3 ( ) (2 (1 ) (1 ) ) exp 2 2 2 2 1 2 2 2 1 2 2 1 1 1 1 1 1 1 1 1 am am sm am am am sm am c c c c c c c c g ρ ρ ρ (7.4) (7.5) (7.6) Manufacturer stage Raw Material Store Supplier Stage cd 2 ca 2 cam c 2 d2 2 ca 2 (Frm1) carm 2 ca 2 (Frm1) (Frm1) 1 – Frm1 Output Store cd1 2 Backorders + Replenishment Orders carm 2 46 The probability distribution of the number of orders at manufacturing stage 1 is given by ( ) ⋅ − ⋅ ≥ − = = ≈ (1 ) − ; 1 1 ; 0 ( 1) 1 1 1 1 1 k k P N k m k ρ σ σ ρ where, σ 1 = (E[Nm1 ] − ρ1 ) / E[Nm1 ] The distribution of the number in system (obtained using equation (7.1)) can be used to find the distribution of the inventory and backorders as in Section 5.2.1. 1 1 1 1 1 1 1 ( ) ( ) ; 1, 2, , ( 0) ( ) 0 m n m m m k m n P I k P N S k k S P I P N S k = = = − = K + = = = Σ ∞ = ( ) ( ) ; 1, 2,K ( 0 ) ( ) 1 1 1 1 1 1 0 = = = + = = = = Σ= P B k P N k S k P B P N k m n m S k m n m The expected inventory, expected backorder and fill rate at output store of manufacturer 1 are given by [ ] ( ) 1 1 1 1 E I n P I m n S n m m = ⋅ = Σ= [ ] ( ) 1 1 0 E B n P B m n n m = ⋅ = Σ ∞ = ( ) 1 1 1 1 F P I n S m n m m = = Σ= The order from the input store can be routed to supplier 1 or to supplier 2 based on specified probabilities. If raw material is available at the supplier, the order is immediately shipped and arrives at the input store after a transit delay. If there is a (7.9) (7.10) (7.11) (7.12) (7.13) (7.7) (7.8) 47 stockout at the supplier, then the order experiences a leadtime delay in addition to the transit delay. A single delay node was used to model the transit and leadtime delays at the supplier. Thus, the expected aggregate delay experienced at the supplier side is given by (( ) ) (( ) ) sm1 pm1s1 plt1 lt1 s1m1 pm1s2 plt 2 lt 2 s2m1 τ = ⋅ ⋅τ +τ + ⋅ ⋅τ +τ The probability distribution of the number of orders in the supplier delay node is given by ( ) ! 1 1 1 j e P N j j s tms tsm ρ ρ ⋅ = = − where, tms1 m1 sm1 ρ = λ ⋅τ The probability distribution of the inventory level and backorder level at the input store of manufacturer 1 is given by ( ) ( ) { } { } 1 1 1 1 1 1 P B i P N S i P I i P N S i rm s rm rm s rm = = = + = = = − The expected inventory, expected backorder and fill rate at the input store of manufacturer 1 are given by [ ] ( ) 1 1 1 1 E I n P I rm n S n rm rm = ⋅ = Σ= [ ] ( ) 1 1 0 E B n P B rm n n rm = ⋅ = Σ ∞ = ( ) 1 1 1 1 F P I n S rm n rm rm = = Σ= (7.14) (7.17) (7.16) (7.15) (7.18) (7.19) (7.20) 48 7.3 NUMERICAL EXPERIMENTS The accuracy of the analytical model was tested by comparing the analytical results to simulation estimates for a wide range of parameter values. The experimental design for comparison is shown in Table 7.1. Table 7.1 shows the different parameters that were varied and the values that were used for different experiments. Table 7.1: Experimental Design for Numerical Comparison Parameter Levels Level Values Arrival Rate (λm1 ) 1 1 Split to supplier ( p m1 s 1 , pm1s2 ) 1 (75%, 25%) Probability of leadtime delay (plt1, plt2) 2 (20% and 40%) and (30% and 30%) Basestock level of output store 3 2, 4, 8 (see note) Basestock level of input store 1 8 Leadtime delay (Expo (lt1), Expo(lt2)) 1 (Expo (2), Expo (4)) and (Expo (3), Expo (3)) Transit time distribution 2 Unif (1, 5) and Unif (4, 8) Utilization 2 80% and 90% Interarrival Distribution at manufacturer 1 3 Erlang , Exponential and Hyperexponential Service time Distribution at manufacturer 1 3 Erlang , Exponential and Hyperexponential Note: In one half of the experiments, the suppliers were identical, i.e., they had the same leadtime of Expo (3) and probability of stock out of 30%. In this case, the basestock levels used were 2, 4 and 8. In the other half of the experiments, the suppliers were asymmetrical, i.e., the leadtimes were Expo (2) and Expo (4) for the two suppliers and the corresponding probabilities of stockout were 20% and 40%. In this case, basestock levels used were 2 and 4. The analytical and simulation results for the various experiments are contained in Appendix A3 49 7.4 SUMMARY OF RESULTS A total of 180 experiments were used to evaluate the accuracy of the analytical model. The results show that there is scope for improvement in the analytical model. A majority of the error are found in the supplier side (See tables in Appendix A3). Table 7.2 shows the accuracy of the approximations developed. The table shows the proportion of the results that fall within a certain error range. The absolute relative error was used to check the accuracy for the expected inventory and expected backorders whereas the minimum of the absolute relative error and the absolute difference between the analytical and simulation values was used for the fill rates due to the fact that the value of fill rate is always less than one and sometimes even less than 0.5. In a case where the values are small, the criteria to be tested will be the absolute difference and not the absolute error as used by Whitt (1993). Table 7.2: Summary of Results Percentage of Results within Error Range (%) Error Range Expected Inventory Expected Backorders Fill Rate < 0.25 91.7 82.9 100.0 < 0.20 91.7 80.1 91.7 < 0.15 82.4 78.2 89.8 < 0.10 80.1 71.8 76.9 < 0.05 63.4 56.5 56.0 The detailed results shown in Appendix A3 show that there is room for improvement. The supplier side delays are first aggregated and then modeled using an M/G/∞ queue. The approximation works reasonably well when the interarrival process is Poisson and a majority of larger errors occur when the interarrival process is Erlang or Hyper 50 exponential. The approximation used for the delay model is insensitive to the variability in the interarrival process. In the result tables in Appendix A3, this can be shown by the fact that the analytical model gives the same output for the input store irrespective of the interarrival distribution, whereas the simulation model is sensitive to changes in the variability of the arrival processes. A majority of the error can be attributed to the use of the M/G/∞ queue. As mentioned in Chapter 5, twomoment approximations for GI/G/∞ queue are not available. 51 CHAPTER 8 CONVERGENT–DIVERGENT CONFIGURATION In this chapter, a convergentdivergent network with three retailers, two manufacturers and three suppliers is considered. This chapter shows how the solution procedures of the building blocks can be used to build a solution procedure for a larger network. The arrival and service processes are considered to be general. The remainder of the chapter is organized as follows. Section 8.1 gives a description of the system under study and the various assumptions made. An approximation procedure is explained in Section 8.2 and numerical experiments are defined in Section 8.3. This chapter concludes with a summary of the numerical results in Section 8.4. 8.1 SYSTEM DESCRIPTION A maketostock supply chain network with three retailers, two manufacturers (each having an input store and an output store) and three suppliers (see Figure 8.1) was considered. The retailers, input stores and the output stores operate under a basestock policy with oneforone replenishment. The interarrival times at the retailers as well as the service times at the manufacturers follow a general distribution. The arrival of a demand consumes a finished product from the retailer and causes an order for replenishment at the output store of the manufacturer. The finished product from the 52 output store of the manufacturer is shipped to the retailer to satisfy the order for replenishment and the output store places an order for replenishment to its manufacturing stage. The time needed to ship finished products from the output store to the retailer was modeled using a delay node. For every order, the manufacturing stage consumes a part from its input store. This causes the manufacturing stage to place an order at the supplier stage. If the raw material is readily available at the supplier, it reaches the input store after a transit delay. If the raw material is not available at the supplier, then the order experiences a leadtime delay in addition to the transit delay. Figure 8.1 ConvergentDivergent Supply Chain Network 8.2 QUEUEING MODEL OF THE CONVERGENTDIVERGENT CONFIGURATION The manufacturer is modeled by a singleserver queue and the supply operations are modeled using delay nodes (M/G/∞ queue). M1 D1 D2 M2 Manufacturers Retailers D3 S1 S2 Suppliers S3 53 8.2.1 Approximate Solution for the Queueing Model The convergentdivergent network can be seen as a combination of the building blocks that have been modeled in the previous three chapters. Retailer 2 along with manufacturers 1 and 2 forms a convergent network with two manufacturers. There are two divergent networks. One is the combination of retailers 1 and 2 with manufacturer 1, and the other is a combination of retailers 2 and 3 with manufacturer 2. Similarly, there are two cases of convergent network with two suppliers. One is the combination of manufacturer 1 with suppliers 1 and 2, and the other is a combination of manufacturer 2 with suppliers 2 and 3. The network model can be solved by first considering the split at retailer 2, i.e. the orders arriving at retailer 2 that go to either of the two manufacturers. Once the splitting approximation is used to get the SCVs of the arrival streams from retailer 2 to either of the manufacturers, the merging approximation can then be used to find the SCV for the arrival process at the manufacturers. The approximations for the calculation of the SCVs are as follows. First, the splitting approximations (Whitt 1983) are applied at retailer 2. c q c p c p c q sp ar sp ar = ⋅ + = ⋅ + ( ) ( ) 2 2 2 2 22 2 21 2 where, p is the probability that an order from retailer 2 goes to manufacturer 1 q is the probability that an order from retailer 2 goes to manufacturer 2 (8.1) (8.2) 54 The arrival rate at the manufacturers are given by 2 3 2 2 2 1 1 1 2 1 3 2 3 2 2 2 2 1 2 1 1 1 m r m r m m r m r m r m r r m r r m r r m r q p λ λ λ λ λ λ λ λ λ λ λ λ λ λ = + = + = = ⋅ = ⋅ = The proportion of orders at manufacturer i that belong to retailer j is given by 2 3 2 2 2 2 2 2 2 1 2 1 1 1 1 1 1 1 1 1 m r m r m r m m r m r m r m r m m r p p p p p p = − = = − = λ λ λ λ Now, the approximation for the superposition of two arrival streams (Whitt 1983) can be applied at both manufacturer 1 and manufacturer 2 as shown below. (( ) ) ( ) 2 2 2 2 2 1 2 2 1 2 ( ) ( ) 1 ( ) 1 2 2 2 22 2 3 3 1 1 1 1 1 2 21 c w p c p c w c w p c p c w merge m r sp m r ar merge m r ar m r sp = ⋅ ⋅ + ⋅ + − = ⋅ ⋅ + ⋅ + − where, [1 4 (1 )2 ( 1)] 1 − wi = + ⋅ − ρ i ⋅ ν i − ( ) ( 2 2 ) 1 2 2 2 1 1 2 2 2 3 1 1 1 2 − − = + = + m r m r m r m r p p p p ν ν and i mi mi ρ = λ ⋅τ (8.13) (8.15) (8.9) (8.3) (8.17) (8.4) (8.5) (8.6) (8.7) (8.8) (8.10) (8.11) (8.12) (8.14) (8.16) (8.18) 55 The splitting and merging approximations are now applied to the 2 mergei c to take into account the delay at the supplier as shown in the Section 7.2.1. The SCV of arrivals at manufacturer i is obtained as follows: carm i Frmi cmerge i Frmi 2 = ⋅ 2 + 1 − 2 (1 ) 2 2 cami Frmi cdi F carmi = − ⋅ + ⋅ where 2 cdi is 1 (departure process from an M/G/∞ queue). Since the demand arrival processes at the retailers are general, the Kraemer and LangenbachBelz (1976) formula and Little’s law (Little 1961) are used once again to calculate the expected number of orders in the manufacturer system. The expected number of orders at manufacturer i is given by i am sm i i m i i i i c c E N g ρ ρ ρ + + ⋅ − = ⋅ 2 ( ) (1 ) [ ] 2 2 2 where ≥ + ⋅ − − ⋅ − < ⋅ ⋅ + − ⋅ − ⋅ − = ; 1 ( 4 ) (1 ) ( 1) exp ; 1 3 ( ) (2 (1 ) (1 ) ) exp 2 2 2 2 2 2 2 2 2 i i i i i i i i am am sm i am am i am sm i am i c c c c c c c c g ρ ρ ρ The probability distribution of the number of orders in queue or being processed at manufacturer i is given by ( ) ⋅ − ⋅ ≥ − = = ≈ (1 ) − ; 1 1 ; 0 ( 1) n n P N n n i i i i mi ρ σ σ ρ where, ( [ ] )/ [ ] i E Nmi i E Nmi σ = −ρ (8.19) (8.20) (8.22) (8.21) (8.23) (8.24) 56 The network behaves in much the same way as the previous configurations; the orders proceed from the retailers to the output store of the manufacturers and then to the manufacturing stage after consuming a part at the input store of the manufacturers. Finally, the orders go from the manufacturer to the suppliers. So an order at the retailer can be found either in transit from a manufacturer or at the manufacturer or at the supplier stage. So if there are n orders in the system, then k of them will be present in transit from the manufacturer, l will be present at the manufacturer and the remaining (n– k– l) will be present in the supplier stage. The probability distribution for the number of orders at retailer j is given by Σ Σ Σ Σ Σ Σ Σ Σ Σ = = = = = = = = = = = = ⋅ = ⋅ = − − = = = ⋅ = − = = = ⋅ = ⋅ = − − = = = ⋅ = ⋅ = − − = = = ⋅ = ⋅ = − − n l l k m r p rmp n n n l l k m r p rmp n l l k m r p rmp n l l k m r p rmp P N n P Nt k P N l P B n k l P N n P N n P N n n P N n P Nt k P N l P B n k l P N n P Nt k P N l P B n k l P N n P Nt k P N l P B n k l 0 0 3 22 1 0 2 21 0 0 22 0 0 21 0 0 1 ( ) ( ) ( ) ( ) ( ) ( 1) ( 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 32 32 2 2 22 22 1 2 21 21 1 1 11 11 where, N2i is the random variable that represents the number of orders from retailer 2 that go to manufacturer i The probability distribution of the number of orders in transit from manufacturer i to retailer j is given by ! ( ) k e P Nt k k t m r mirj tmirj i j ρ ρ ⋅ = = − where, tmirj rjmi mirj ρ = λ ⋅τ (8.25) (8.30) (8.31) (8.26) (8.27) (8.28) (8.29) 57 The probability distribution of orders in manufacturing stage j corresponding to pending orders at retailer i is given by the formulas as used for the diverging network (see Equations 5.14 and 5.15). ≥ − ⋅ − ⋅ ⋅ ⋅ + − = − ⋅ ⋅ ⋅ − ⋅ = = + + − ; 1 (1 ) (1 ) (1 ) ; 0 (1 ) (1 ) ( ) ( 1) 1 ( 1) 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2 1 1 2 11 l p p l p p P N l l m r l l S S m r S m r p m m r m m σ σ ρ σ ρ σ σ σ σ ρ ≥ − ⋅ − ⋅ ⋅ + − = − ⋅ ⋅ ⋅ − ⋅ = = + + − ; 1 (1 ) (1 ) (1 ) ; 0 (1 ) (1 ) ( ) ( 1) 1 ( 1) 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 21 l p p l p p P N l l m r l l S S m r S m r p m m r m m σ σ ρ σ ρ σ σ σ σ ρ ≥ − ⋅ − ⋅ ⋅ ⋅ + − = − ⋅ ⋅ ⋅ − ⋅ = = + + − ; 1 (1 ) (1 ) (1 ) ; 0 (1 ) (1 ) ( ) ( 1) 2 ( 1) 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 3 2 2 3 22 l p p l p p P N l l m r l l S S m r m r p m m r m Sm σ σ ρ σ ρ σ σ σ σ ρ ≥ − ⋅ − ⋅ ⋅ ⋅ + − = − ⋅ ⋅ ⋅ − ⋅ = = + + − ; 1 (1 ) (1 ) (1 ) ; 0 (1 ) (1 ) ( ) ( 1) 2 ( 1) 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 32 l p p l p p P N l l m r l l S S m r S m r p m m r m m σ σ ρ σ ρ σ σ σ σ ρ The input store of manufacturer i was modeled first before obtaining the probability distribution of the number of orders that are backordered at the input store of manufacturer j that belong to retailer i Brmpij . The raw material order from a manufacturer is routed probabilistically to the suppliers. If raw material is available at a supplier, the order is immediately shipped and arrives at the manufacturer’s input store after a transit delay. If the raw materials are not (8.32) (8.35) (8.33) (8.34) 58 available at the supplier, then the order experiences a leadtime delay in addition to the transit delay. Thus, the expected aggregate delay experienced at the supplier side is given by (( ) ) (( ) ) (( ) ) (( ) ) 2 2 2 2 2 2 2 2 3 3 3 3 2 1 1 1 1 1 1 1 1 2 2 2 2 1 sm m s lt lt s m m s lt lt s m sm m s lt lt s m m s lt lt s m p p p p p p p p τ τ τ τ τ τ τ τ τ τ = ⋅ ⋅ + + ⋅ ⋅ + = ⋅ ⋅ + + ⋅ ⋅ + The probability distribution of the number of orders at the supplier side of manufacturer i is given by ; 1, 2 ! { ) = ⋅ = = − i n e P N n n tsm s i tsmi i ρ ρ where, = ⋅ ; i =1, 2 tsmi mi smi ρ λ τ The probability distribution of the inventory level and backorder level at the input store of manufacturer i is given by ( ) ( ) ( ) ( ) ( ) = + = ∞ = = = = = = = − = Σ= ; 1, 2, , ; 0 ; 1, 2, , 0 K K P N S n n P N n n P B n P I n P N S n n S i i rmi i i i i i i s rm S n s rm rm s rm rm The expected inventory, expected backorder and fill rate at input store of manufacturer i are given by [ ] ( ) 1 E I n P I n S n = ⋅ = Σ= [ ] ( ) 1 E B n P B n i rmi n rm = ⋅ =Σ ∞ = ( ) 1 F P I n i rm i i rm S n rm = = Σ= (8.40) (8.39) (8.38) (8.42) (8.43) (8.44) (8.36) (8.41) (8.37) 59 The probability distribution of the number of orders that are backordered at the input store of manufacturer j that belong to retailer i can be obtained by using an approach that is similar to the one used for calculating the probability distribution of the number of orders in manufacturing stage j corresponding to pending orders at retailer i (See Chapter 5). The probability distribution of the number of orders belonging to retailer i that are in the input store of manufacturer j, given the number of backorders at the input store of manufacturer j follows the Binomial distribution. 1 ( 1) 1 ( 1) 1 ( 1) 1 ( 1) 32 2 2 3 2 2 22 2 2 2 2 3 21 1 1 2 1 1 11 1 1 1 1 2 1 { 1 } 1 { 1 } 1 { 1 } 1 { 1 } n n m r n rmp rm m r n n m r n rmp rm m r n n m r n rmp rm m r n n m r n rmp rm m r p p n n P B n B n p p n n P B n B n p p n n P B n B n p p n n P B n B n − − − − ⋅ ⋅ = = = ⋅ ⋅ = = = ⋅ ⋅ = = = ⋅ ⋅ = = = To derive the unconditional probability that the backorder at the input store of manufacturer j belongs to retailer i, the above conditional probability is multiplied probability of backorder at the input store of manufacturer j and summed over all values of n (backorders) greater than or equal to n1. { } { 1 } { } 1 P B n P B n B n P B n ij ij j rmj n n rm rmp rmp = ⋅ = = = = Σ ∞ = The probability distribution of the inventory level and backorder level at retailer i is given by ( ) ( ) P(B n) P(N S n) P I n P N S n r i r i i i = = = + = = = − (8.49) (8.50) (8.45) (8.46) (8.47) (8.48) (8.51) 60 The expected inventory, expected backorder and fill rate at retailer i are given by [ ] ( ) 1 E I n P I n i ri i r S n r = ⋅ = Σ= [ ] ( ) 1 E B n P B n i ri n r = ⋅ =Σ ∞ = ( ) 1 F P I n i ri i r S n r = = Σ= The order at manufacturer i can be found either at the manufacturing stage or in the backorder queue at the input store of manufacturer i. So, if there are n orders at the manufacturer i, j are present in manufacturing stage and the remaining (nj) are in the backorder queue at the input store of manufacturer i. The probability distribution for the number of orders at manufacturer i is given by ( ) ( ) ( ) 0 P N n P N j P B n j i i rmi n j m n − = ⋅ = = = Σ= The probability distribution of the inventory level and backorder level at manufacturer i is given by ( ) ( ) P(B n) P(N S n) P I n P N S n i i i i m n m n = = = + = = = − The expected inventory, expected backorder and fill rate at manufacturer i are given by [ ] ( ) 1 E I n P I n i mi i m S n m = ⋅ =Σ= [ ] ( ) 1 E B n P B n i mi n m = ⋅ =Σ ∞ = (8.55) (8.52) (8.53) (8.54) (8.56) (8.58) (8.59) (8.57) 61 ( ) 1 F P I n i m i i m S n m = = Σ= 8.3 NUMERICAL EXPERIMENTS The accuracy of the analytical model was tested by comparing the analytical results to simulation estimates for a wide range of parameter values. The design of experiments is shown in Table 8.1. Table 8.1 shows the different parameters that were varied and also gives the values that were used for different experiments. Table 8.1: Experimental Design for Numerical Comparison Parameter Levels Level Values Arrival Rates ( λ1, λ2, λ3) 2 ( 1, 1, 1 ) and ( 0.5, 2, 0.5 ) Split to manufacturer ( p, q ) 1 ( 50%, 50% ) Split to supplier ( p1, q1 ) 1 ( 50%, 50% ) Probability of stockout 1 30% Leadtime delay 1 Expo ( 3 ) Basestock level at retailer 1 4 Basestock level at manufacturer 1 6 ( for both input and output store ) Transit time distribution 1 Unif ( 1, 5 ) Utilization 2 80% and 90% (see note) Interarrival Distribution at the retailers 3 Erlang , Exponential and Hyperexponential Service time Distribution at the manufacturers 3 Erlang , Exponential and Hyperexponential Note: In one set of experiments, the manufacturers had different utilizations, i.e., manufacturer 1 had 90% utilization and manufacturer 2 had 80% utilization. In all other experiments, the manufacturers had the same utilization level. (8.60) 62 The analytical and simulation results for the various experiments are contained in Appendix A4. 8.4 SUMMARY OF RESULTS A total of 45 experiments were used to evaluate the accuracy of the analytical model. The analytical model seems to give results which are very comparable with the simulation estimates. However, the supplier side poses a concern as the approximations used are prone to errors when the interarrival time is general. The reasons are discussed in Section 9.3.3. Table 8.2 shows the accuracy of the approximations developed for this configuration. The table shows the proportion of the results that fall within a certain value of error. The relative percentage error was used to check the accuracy for the expected inventory and expected backorders whereas the minimum of the relative error and the absolute difference between the analytical and simulation values was used for the fill rates due to the fact that the value of fill rate is always less than one and sometimes even less than 0.5. Table 8.2: Summary of Results Percentage of Results within Error Range (%) Error Range Expected Inventory Expected Backorders Fill Rate < 0.25 94.6 73.7 100.0 < 0.20 94.6 68.9 100.0 < 0.15 94.3 61.6 97.8 < 0.10 92.1 52.1 91.4 < 0.05 76.2 41.3 49.8 The main source of errors is the use of M/G/∞ queue to model delay nodes. As mentioned in the previous chapters, better results can be expected if the approximations 63 for the delay node could be improved. This problem is worse in the convergentdivergent network because of the repeated use of the M/G/∞ queue to model delay nodes at the supplier and retailer (transit) sides. 64 CHAPTER 9 CONCLUSIONS AND FUTURE RESEARCH In this chapter, a summary of the research carried out in this thesis effort, and some directions for future research are provided. The chapter is organized as follows. Section 9.1 provides a summary of the research that has been completed and provides the conclusions. Section 9.2 summarizes the contributions made by the successful completion of the research, and Section 9.3 provides suggestions for future research. 9.1 RESEARCH SUMMARY The main objective of this research was to develop analytical models of the ‘building block type’ configurations of a supply chain network and to apply these analytical models to model a somewhat complex supply chain network. In Chapter 5, the first ‘building block type’ configuration, namely, the divergent network was modeled for a general arrival process and general service times. The numerical results showed that the approximations were quite accurate when the arrival process was Poisson and the service times were exponential. The analytical results were fairly consistent with the simulation estimates when the interarrival and service times were general. The errors were in the generally acceptable range (< 20%) for more than 95% of the cases. In Chapter 6, the second configuration, namely, a convergent network with two suppliers was modeled and the numerical results showed that the approximations yielded excellent results. In 65 Chapter 7, the last of the ‘building block type’ configurations, namely, a convergent network with two suppliers was modeled. The results showed that the approximation performed reasonably well in several cases. However, there was a problem with the way the supplier delay node was modeled, and as a result the analytical model could not adequately capture the variability of the arrival process. This is discussed further in Section 9.3.2. In Chapter 8, a convergentdivergent network was modeled and the results showed that the building block approach worked well in modeling a fairly complicated network. The problem with the delay node approximation was still apparent in this configuration, and this is an area that needs to be researched further. The detailed results are organized in several tables and are presented in Appendices A1 through A4. 9.2 Research Contributions Previous research work in performance evaluation of supply chain networks mainly focused on the optimization part, although queueing models were also explored. In general, queueing models focus on capacity and congestion issues and often ignore planned inventories. With the works of Lee and Zipkin (1992), Zipkin (1995), Buzacott and Shanthikumar (1993), Sivaramakrishnan (1998) and Sivaramakrishnan and Kamath (1996), performance modeling of productioninventory systems has started gaining attention. Sivaramakrishnan (1998) primarily modeled multistage maketostock productioninventory systems. This research effort has focused on extending the models developed by Sivaramakrishnan (1998) to supply chain networks by including key activities like supply, transportation and distribution. The specific contributions of this research in the field of analytical queueing models are as follows. 66 1. Developed queueingbased models of “building block” type supply chain network configurations. 2. Demonstrated the usefulness of modeling “building block” configurations for developing queueing models of larger supply chain networks. 3. Laid the foundation for the development of a rapid analysis tool for supply chain networks. 4. Extended the popular parametric decomposition approach based on twomoment approximations to model supply chain networks. 9.3 FUTURE RESEARCH One of the key issues that has to be addressed is the use of the M/G/∞ system to model the delays, be it transit or leadtime delay. When the interarrival times are general, it was seen, in the third and last building block configuration, that the use of an M/G/∞ model to approximate a GI/G/∞ system did not work that well. The need for this approximation arose because of the lack of system length distribution type results for the GI/G/∞ system. Hence, obtaining better approximations for a GI/G/∞ system is critical to the improvement of some of the analytical models developed. All the models used in this study assumed the oneforone replenishment policy, and this is the policy that has been widely used in the literature. Modeling other inventory policies such as the (s, S) policy would be another interesting research topic from a practical viewpoint. 9.3.1 Extensions to the Divergent Configuration In the experiments for the divergent configuration, it was found that when the demand arrival processes had different variability at the two retailers, the analytical 67 model was not able to account for the difference in variability while computing the individual retailer performance measures. After computing the distribution of the “aggregate” orders at the manufacturer, the “disaggregation” procedure to find the distribution of orders corresponding to a particular retailer was based on a “proportion of orders” belonging to a retailer (See Equation 5.12). The expression used for the proportion of orders from a retailer to a manufacturer used only on the first moment (or rate) of the interarrival distribution. This made the disaggregation scheme insensitive to the differences in variability of the interarrival processes. Two heuristic extensions are suggested to address this issue. In the first extension, the arrival rate at a retailer is multiplied by the respective interarrival SCV and the proportions are computed using this product as shown in Equations 9.1 and 9.2. These new proportions would then be used in the calculations for the conditional probability distribution of the number of orders belonging to retailer i in the manufacturing stage, given the number of backorders at the finished goods store which is given by a Binomial distribution (See Equation 5.12), and the rest of the calculations would remain the same. ( 2 ) ( 2 ) 2 1 1 2 2 1 1 1 1 r ar r ar r ar m r c c c p ⋅ + ⋅ ⋅ = λ λ λ and 1 2 1 1 pm r = 1− pm r In the second extension, the arrival rate is multiplied by the sum of the arrival SCV at that retailer and the service SCV at the manufacturer and the proportions are calculated as follows. ( ( )) ( ( )) ( ) 2 2 2 2 2 2 1 1 1 2 2 2 1 1 1 1 1 r ar sm r ar sm r ar sm m r c c c c c c p ⋅ + + ⋅ + ⋅ + = λ λ λ (9.1) (9.2) (9.3) (9.4) 68 1 2 1 1 pm r = 1− pm r The above extensions were based on the observation that the interarrival and service time SCVs appear in the expected number in queue expressions along with the arrival rate. 9.3.2 Extensions to Convergent Configuration 1 In convergent configuration 1, it was assumed that the order at the retailer is split probabilistically, and that the routing probability is specified. But this may not be true in reality. In many scenarios, the availability of the part may play a role in the selection of the manufacturer. Thus, the existing model could be modified to consider a case where the retailer can check the availability of parts at the output stores of the manufacturers and then place an order at the store which has parts. If both the stores had parts or did not have parts, then the order could be sent to either manufacturer randomly. To solve this network, an iterative procedure is suggested. The probability of split can be expressed in terms of the fill rates of the manufacturers. It is known that the probability that an order is satisfied immediately is given by the fill rate. The proportion of orders going to a manufacturer depends on part availability at both. There are three different scenarios. They are 1) the part is available at manufacturer 1 and not at manufacturer 2 or vice versa, 2) the part is available at both manufacturers and the order is placed at a manufacturer with equal probability, and 3) the part is not available at either manufacturer and the order is placed with equal probability. 1 2 0.5 1 m m F F and order placed with manufacturer Part is available at both manufacturers P = ⋅ ⋅ (1 ) 2 1 Fm1 Fm2 and not at manufacturer Part is available at manufacturer P = ⋅ − (9.5) (9.6) 69 0.5 (1 ) (1 ) 1 m1 m2 F F and order placed with manufacturer Part is not available at both manufacturers P = ⋅ − ⋅ − The probability of ordering from manufacturer 1 would then be the sum of the three probabilities in equations 9.5 through 9.7. But, now the fill rates are needed to compute this probability. An iterative scheme which starts with a 5050 split and continues till the split probabilities stabilize is suggested. 9.3.3 Extensions to the Convergent Configuration 2 In the experiments for convergent configuration 2, it was found that the results for the supplier node performed reasonably well when the interarrival process at the node was Poisson. However, when the interarrival process was either Erlang or Hyperexponential, the results deviated significantly from the simulation estimates. This is due to the following reasons. The supplier side delays are first aggregated and then modeled using an M/G/∞ queue. In the result tables in Appendix A3, it can be seen that the analytical model gives the same output for the input store irrespective of the interarrival distribution, whereas the simulation model is sensitive to changes in the variability of the interarrival times. A majority of the error can be attributed to the use of the M/G/∞ queue. 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(1995), “Processing Networks with Planned Inventories: Tandem Queues with Feedback,” European Journal of Operational Research, 80, 344349. 31. Adan, I., “Basic Concepts from Probability Theory,” Class Notes, http://www.win.tue.nl/~iadan/sdp/h1.pdf, last viewed on December 9, 2004. 74 APPENDIX A1 RESULTS FOR THE DIVERGENT CONFIGURATION The experimental results for the divergent configuration with two retailers and one manufacturer are presented in a series of tables in this appendix. The experiments were conducted by fixing the basestock level, transit time from the manufacturer to retailer and utilization at particular values and varying the variability of interarrival and service times. The experimentation was performed in accordance with the design discussed in Table 5.1. In the results table the column titled SCV has three parameters, the SCV of the interarrival time at retailer 1, the SCV of the interarrival time at retailer 2, and the service SCV at the manufacturer. It is given as (SCVretailer1, SCVretailer2, SCVmanufacturer). Table A1.1: TwoRetailer Case: Results for basestock level of 2, transit time of 3, λ1= λ2= 1 and ρ = 80% SCV Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim (0.25, 0.25, 0.25) 0.218 0.077 0.218 0.077 0.785 0.756 1.440 1.236 1.440 1.237 0.446 0.320 0.175 0.130 0.175 0.130 0.585 0.686 (0.25,0.25,1.00) 0.175 0.057 0.175 0.056 0.624 0.621 1.916 1.712 1.916 1.710 1.481 1.309 0.142 0.097 0.142 0.096 0.424 0.501 (0.25, 0.25, 2.25) 0.136 0.044 0.136 0.044 0.532 0.570 2.830 2.675 2.830 2.671 3.387 3.260 0.110 0.076 0.110 0.076 0.332 0.425 (0.25, 1.00, 0.25) 0.194 0.065 0.194 0.202 0.684 0.674 1.664 1.446 1.664 1.667 0.939 0.846 0.157 0.111 0.157 0.162 0.484 0.518 (0.25,1.00,1.00) 0.161 0.050 0.161 0.165 0.585 0.590 2.181 1.970 2.181 2.190 2.040 1.943 0.130 0.086 0.130 0.133 0.385 0.421 (0.25, 1.00, 2.25) 0.129 0.041 0.129 0.137 0.517 0.555 3.115 2.908 3.115 3.119 3.972 3.848 0.104 0.071 0.104 0.110 0.317 0.376 (0.25, 2.25, 0.25) 0.169 0.056 0.169 0.339 0.606 0.627 2.022 1.728 2.022 2.228 1.706 1.552 0.137 0.096 0.137 0.138 0.406 0.418 (0.25,2.25,1.00) 0.144 0.046 0.144 0.300 0.548 0.571 2.576 2.281 2.576 2.784 2.864 2.722 0.117 0.078 0.117 0.121 0.348 0.363 (0.25, 2.25, 2.25) 0.120 0.038 0.120 0.257 0.501 0.539 3.531 3.282 3.531 3.786 4.823 4.773 0.097 0.065 0.097 0.104 0.301 0.334 (1.00, 0.25, 0.25) 0.194 0.201 0.194 0.065 0.684 0.674 1.664 1.668 1.664 1.444 0.939 0.845 0.157 0.162 0.157 0.112 0.484 0.518 (1.00, 0.25,1.00) 0.161 0.166 0.161 0.050 0.585 0.592 2.181 2.176 2.181 1.955 2.040 1.917 0.130 0.134 0.130 0.087 0.385 0.422 (1.00, 0.25,2.25) 0.129 0.137 0.129 0.041 0.517 0.554 3.115 3.116 3.115 2.914 3.972 3.856 0.104 0.110 0.104 0.071 0.317 0.375 (1.00, 1.00, 0.25) 0.177 0.182 0.177 0.182 0.629 0.614 1.891 1.902 1.891 1.896 1.429 1.432 0.143 0.147 0.143 0.147 0.429 0.414 (1.00, 1.00, 1.00) 0.150 0.154 0.150 0.153 0.560 0.564 2.430 2.426 2.430 2.430 2.560 2.551 0.121 0.124 0.121 0.123 0.360 0.362 (1.00, 1.00, 2.25) 0.123 0.127 0.123 0.128 0.507 0.531 3.375 3.406 3.375 3.389 4.507 4.533 0.100 0.103 0.100 0.104 0.307 0.331 (1.00, 2.25, 0.25) 0.159 0.166 0.159 0.322 0.581 0.582 2.221 2.183 2.221 2.454 2.124 2.153 0.128 0.133 0.128 0.131 0.381 0.347 (1.00, 2.25, 1.00) 0.137 0.140 0.137 0.282 0.534 0.539 2.796 2.795 2.796 3.074 3.318 3.441 0.111 0.113 0.111 0.114 0.334 0.313 (1.00, 2.25, 2.25) 0.116 0.120 0.116 0.245 0.494 0.514 3.763 3.817 3.763 4.117 5.295 5.562 0.094 0.097 0.094 0.099 0.294 0.295 (2.25, 0.25, 0.25) 0.169 0.340 0.169 0.056 0.606 0.625 2.022 2.235 2.022 1.736 1.706 1.570 0.137 0.139 0.137 0.096 0.406 0.416 (2.25,0.25,1.00) 0.144 0.298 0.144 0.045 0.548 0.569 2.576 2.813 2.576 2.303 2.864 2.775 0.117 0.120 0.117 0.078 0.348 0.361 (2.25, 0.25, 2.25) 0.120 0.258 0.120 0.038 0.501 0.538 3.531 3.796 3.531 3.286 4.823 4.788 0.097 0.104 0.097 0.066 0.301 0.334 (2.25, 1.00, 0.25) 0.159 0.319 0.159 0.164 0.581 0.579 2.221 2.469 2.221 2.189 2.124 2.170 0.128 0.130 0.128 0.133 0.381 0.345 (2.25, 1.00, 1.00) 0.137 0.284 0.137 0.142 0.534 0.543 2.796 3.049 2.796 2.763 3.318 3.391 0.111 0.115 0.111 0.114 0.334 0.315 (2.25, 1.00, 2.25) 0.116 0.246 0.116 0.120 0.494 0.517 3.763 4.116 3.763 3.793 5.295 5.531 0.094 0.099 0.094 0.097 0.294 0.296 (2.25, 2.25, 0.25) 0.145 0.299 0.145 0.297 0.550 0.552 2.556 2.810 2.556 2.810 2.822 3.019 0.117 0.121 0.117 0.120 0.350 0.295 (2.25, 2.25, 1.00) 0.128 0.266 0.128 0.265 0.515 0.520 3.162 3.468 3.162 3.463 4.068 4.393 0.104 0.107 0.104 0.106 0.315 0.275 (2.25, 2.25, 2.25) 0.110 0.235 0.110 0.236 0.484 0.506 4.148 4.425 4.148 4.450 6.075 6.406 0.089 0.095 0.089 0.095 0.284 0.268 Expected Inventory Expected Backorders Fill Rate Retailer 1 Retailer 2 Manufacturer Retailer 1 Retailer 2 Manufacturer Retailer 1 Retailer 2 Manufacturer 75 Table A1.2: TwoRetailer Case: Results for basestock level of 2, transit time of 6, λ1= λ2= 1 and ρ = 80% SCV Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim (0.25, 0.25, 0.25) 0.017 0.000 0.017 0.000 0.785 0.756 4.240 4.159 4.240 4.162 0.446 0.320 0.015 0.000 0.015 0.000 0.585 0.686 (0.25,0.25,1.00) 0.014 0.000 0.014 0.000 0.624 0.621 4.754 4.656 4.754 4.655 1.481 1.309 0.012 0.000 0.012 0.000 0.424 0.501 (0.25, 0.25, 2.25) 0.011 0.000 0.011 0.000 0.532 0.570 5.704 5.634 5.704 5.625 3.387 3.260 0.009 0.000 0.009 0.000 0.332 0.425 (0.25, 1.00, 0.25) 0.015 0.000 0.015 0.016 0.684 0.674 4.485 4.379 4.485 4.483 0.939 0.846 0.013 0.000 0.013 0.014 0.484 0.518 (0.25,1.00,1.00) 0.013 0.000 0.013 0.013 0.585 0.590 5.033 4.922 5.033 5.040 2.040 1.943 0.011 0.000 0.011 0.012 0.385 0.421 (0.25, 1.00, 2.25) 0.010 0.000 0.010 0.011 0.517 0.555 5.996 5.866 5.996 5.993 3.972 3.848 0.009 0.000 0.009 0.009 0.317 0.376 (0.25, 2.25, 0.25) 0.013 0.000 0.013 0.111 0.606 0.627 4.866 4.673 4.866 5.007 1.706 1.552 0.012 0.000 0.012 0.040 0.406 0.418 (0.25,2.25,1.00) 0.011 0.000 0.011 0.099 0.548 0.571 5.443 5.236 5.443 5.582 2.864 2.722 0.010 0.000 0.010 0.035 0.348 0.363 (0.25, 2.25, 2.25) 0.009 0.000 0.009 0.084 0.501 0.539 6.421 6.245 6.421 6.616 4.823 4.773 0.008 0.000 0.008 0.030 0.301 0.334 (1.00, 0.25, 0.25) 0.015 0.016 0.015 0.000 0.684 0.674 4.485 4.486 4.485 4.377 0.939 0.845 0.013 0.014 0.013 0.000 0.484 0.518 (1.00, 0.25,1.00) 0.013 0.013 0.013 0.000 0.585 0.592 5.033 5.023 5.033 4.906 2.040 1.917 0.011 0.012 0.011 0.000 0.385 0.422 (1.00, 0.25,2.25) 0.010 0.011 0.010 0.000 0.517 0.554 5.996 5.988 5.996 5.872 3.972 3.856 0.009 0.010 0.009 0.000 0.317 0.375 (1.00, 1.00, 0.25) 0.014 0.015 0.014 0.014 0.629 0.614 4.728 4.737 4.728 4.727 1.429 1.432 0.012 0.013 0.012 0.013 0.429 0.414 (1.00, 1.00, 1.00) 0.012 0.012 0.012 0.012 0.560 0.564 5.292 5.282 5.292 5.289 2.560 2.551 0.010 0.011 0.010 0.011 0.360 0.362 (1.00, 1.00, 2.25) 0.010 0.010 0.010 0.010 0.507 0.531 6.260 6.295 6.260 6.271 4.507 4.533 0.008 0.009 0.008 0.009 0.307 0.331 (1.00, 2.25, 0.25) 0.012 0.013 0.012 0.106 0.581 0.582 5.074 5.034 5.074 5.233 2.124 2.153 0.011 0.012 0.011 0.038 0.381 0.347 (1.00, 2.25, 1.00) 0.011 0.011 0.011 0.093 0.534 0.539 5.670 5.674 5.670 5.885 3.318 3.441 0.009 0.010 0.009 0.033 0.334 0.313 (1.00, 2.25, 2.25) 0.009 0.009 0.009 0.081 0.494 0.514 6.656 6.709 6.656 6.959 5.295 5.562 0.008 0.008 0.008 0.029 0.294 0.295 (2.25, 0.25, 0.25) 0.013 0.111 0.013 0.000 0.606 0.625 4.866 5.015 4.866 4.681 1.706 1.570 0.012 0.039 0.012 0.000 0.406 0.416 (2.25,0.25,1.00) 0.011 0.099 0.011 0.000 0.548 0.569 5.443 5.613 5.443 5.257 2.864 2.775 0.010 0.035 0.010 0.000 0.348 0.361 (2.25, 0.25, 2.25) 0.009 0.084 0.009 0.000 0.501 0.538 6.421 6.624 6.421 6.248 4.823 4.788 0.008 0.031 0.008 0.000 0.301 0.334 (2.25, 1.00, 0.25) 0.012 0.104 0.012 0.013 0.581 0.579 5.074 5.256 5.074 5.042 2.124 2.170 0.011 0.037 0.011 0.011 0.381 0.345 (2.25, 1.00, 1.00) 0.011 0.093 0.011 0.011 0.534 0.543 5.670 5.851 5.670 5.635 3.318 3.391 0.009 0.033 0.009 0.010 0.334 0.315 (2.25, 1.00, 2.25) 0.009 0.081 0.009 0.009 0.494 0.517 6.656 6.960 6.656 6.685 5.295 5.531 0.008 0.029 0.008 0.008 0.294 0.296 (2.25, 2.25, 0.25) 0.011 0.098 0.011 0.097 0.550 0.552 5.422 5.613 5.422 5.613 2.822 3.019 0.010 0.035 0.010 0.035 0.350 0.295 (2.25, 2.25, 1.00) 0.010 0.087 0.010 0.086 0.515 0.520 6.044 6.294 6.044 6.288 4.068 4.393 0.009 0.031 0.009 0.031 0.315 0.275 (2.25, 2.25, 2.25) 0.009 0.077 0.009 0.077 0.484 0.506 7.046 7.265 7.046 7.294 6.075 6.406 0.008 0.028 0.008 0.028 0.284 0.268 Expected Inventory Expected Backorders Fill Rate Retailer 2 Retailer 1 Retailer 2 Manufacturer Retailer 1 Manufacturer Retailer 1 Retailer 2 Manufacturer 76 Table A1.3: TwoRetailer Case: Results for basestock level of 4, transit time of 3, λ1= λ2= 1 and ρ = 80% SCV Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim Anal Sim (0.25, 0.25, 0.25) 1.287 1.100 1.287 1.102 2.459 2.495 0.347 0.126 0.347 0.124 0.120 0.053 0.635 0.785 0.635 0.787 0.889 0.947 (0.25,0.25,1.00) 1.159 0.971 1.159 0.969 1.911 1.949 0.542 0.284 0.542 0.286 0.768 0.631 0.579 0.712 0.579 0.711 0.701 0.761 (0.25, 0.25, 2.25) 0.961 0.784 0.961 0.785 1.508 1.647 1.143 0.913 1.143 0.911 2.363 2.258 0.487 0.588 0.487 0.588 0.534 0.606 (0.25, 1.00, 0.25) 1.227 1.037 1.227 1.246 2.136 2.146 0.423 0.182 0.423 0.420 0.391 0.322 0.609 0.753 0.609 0.615 0.785 0.817 (0.25,1.00,1.00) 1.093 0.898 1.093 1.109 1.750 1.780 0.695 0.425 0.695 0.689 1.205 1.112 0.549 0.667 0.549 0.554 0.637 0.667 (0.25, 1.00, 2.25) 0.915 0.743 0.915 0.938 1.438 1.558 1.361 1.098 1.361 1.373 2.893 2.795 0.464 0.560 0.464 0.472 0.503 0.555 (0.25, 2.25, 0.25) 1.131 0.954 1.131 1.339 1.839 1.908 0.601 0.301 0.601 0.803 0.939 0.808 0.567 0.702 0.567 0.475 0.673 0.689 (0.25,2.25,1.00) 1.008 0.820 1.008 1.202 1.585 1.639 0.958 0.660 0.95 



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