ALLOCATION OF FINITE BUFFER CAPACITY TO
PART TYPES FOR MAXIMIZING PROFITS IN
SERIAL LINES
By
YOUSUFF ZAMAN HABmULLAHKHAN
Bachelor of Engineering
University of Madras
Madras, India
1997
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
1n partial fulfillment of
the requirements for
the degree of
MASTER OF SCIENCE
May, 2000
ALLOCAnON OF FINITE BUFFER CAPACITY TO
PART TYPES FOR MAXIMIZING PROFITS IN
SERIAL LINES
Thesis Approved:
Thesis Advisor
__M~~~l{~
__f'i\Ar bQ 1 J ~AQ oo.i).n..... ~__
II
ACKNOWLEDGEMENTS
In the name of Allah, Most Gracious, Most Merciful
This thesis is an important milestone in my academic career. I feel obligated to
acknowledge the help of some people without whom this work would not have been
completed.
First and foremost, I wish to express my greatest thanks, deepest appreciation,
sincere gratitude to my family, especially my father Mr. Habibullah Khan and my mother
Mrs. Iqbal Begum. I thank them for supporting me and giving me the opportunity to
pursue my graduate studies. Moreover, I will not forget their love, encouragement, warm
emotions, and sacrifice to make me a better person in all aspects of life.
I also thank my advisor, Dr. David B. Pratt for his continuous guidance, his
encouragement during frustrating periods, and his compliments during the productive
periods. His continuous support kept my motivation at the hi.ghest level during the
course of this research. My great appreciation is also extended to my committee
members Dr. Manjunath Kamath and Dr. Michael H. Branson. Their valuable comments
and suggestions had a significant impact on the quality of this research. I also want to
thank Dr. Allen C. Schuermann for his help with the simulation software, ARENA.
I would also like to thank the School of Industrial Engineering and Management
at Oklahoma State University for the financial support during the period of August 1998May
1999.
iii
t.
TABLE OF CONTENTS
Chapter Page
I. THE PROBLEM AND ITS SETTING 1
INTRODUCTION 1
BUFFERS " 1
PERFORMANCE MEASURES 2
PROBLEM 3
DEFINlTION OF TERMINOLOGy 3
n. LITERATURE REVIEW 5
INTRODUCfION 5
LITERATURE REVIEW ON THE BUFFER CAPACITY 6
LITERATURE REVIEW ON DUE DATES AssIGNMENTS AND TARDINESS PENALTY 7
CONCLUSION 10
III. RESEARCH GOAL AND OBJECTIVES 12
RESEARCH GOAL 12
REsEARCH OBJECTIVE 12
METHODOLOGY 13
AsSUMPTIONS 13
IV. RESEARCH METHODOLOGy 15
SYSTEM DESCRIPTION 15
ORDER ACCEPTANCE LOGIC 18
ORDER SEQUENCING LOGIC 18
EXPERIMENTAL FACTORS 19
COST STRUCTURE 21
SIMULATION VERIFICATION AND VALIDATION 24
SIMULATION MODEL 25
SIMULATION CHARACTERISTICS , 25
DETERMINATION OF RUN LENGTH 26
V. RESULTS 27
VI. CONCLUSIONS AND FUTURE RESEARCH 39
BIBLIOGRAPHY 42
APPENDIX 1: BLOCK DIAGRAM OF THE SYSTEM 44
APPENDIX 2: SUM OF THE YEARS DIGIT METHOD CALCULATION 45
IV
APPENDIX 3: WARM UP PERIOD DETERMINATION PROCEDURE . 46
APPENDIX 4: DATA POINTS FOR THE GRAPHS , 49
APPENDIX 5: EXAMPLE CALCULAnON FOR THE COST STRUCTURE 52
v
Table
LIST OF TABLES
Page
I  EFFECT OF DIFFERENT RUN LENGTHS AND WARM UP PERIOD ON THE RAnON OF HALF LENGTH
OVER THE MEAN 26
II  TABLE OF MEANS WITH VARYING NUMBER OF
PARTS , 27
vi
LIST OF FIGURES
Figure Page
1: THE SIXSTATION PRODUCTION LINE SYSTEM 16
2: MODELING TARDINESS PENALTY COST 22
3: ANOYA TEST RESULTS 28
4: DUNCAN TEST RESULT FOR THE EXPERIMENTAL FACTOR PART TYPE 28
5: DUNCAN TEST RESULT FOR THE EXPERIMENTAL FACTOR DUE DATE : 29
6: DUNCAN TEsT RESULT FOR THE EXPERIMENTAL FACTOR VARIANCE ~ 29
7: AVERAGE TOTAL PROFIT FOR V.(\R.=O.25, J>rF=1, B~=;'60 32
8: AYERAGE TOTAL PROFIT FOR VAR=O.25, ~1, BUFFER=90 32
9: AVERAGE TOTAL PROFIT FOR VAR=O.25, ~2, BUFFER=60 33
10: AVERAGE TOTAL PROFIT FOR VAR=<l.25, PrF=2, BUFFER=90 33
11: AVERAGE TOTAL PROFIT FOR V AR=l.OO, ~1, BUFFER=60 : 34
12: AVERAGE TOTAL PROFIT FOR VAR=l.OO, PTf;=1, BUFFBR=90 34
13: AVERAGE TOTAL PROFIT FOR VAR=I.00, PTF=2, BUFFER=60 35
14: AVERAGE TOTAL PROFIT FOR VAR=l.OO, PTF=2, BUFFER=90 35
15: AVERAGE TOTAL PROFIT FOR VAR=2.00, PTF=l, BUFFER=60 36
16: AVERAGE TOTAL PROFIT FOR VAR=2.00, PTF=1, BUFFER=90 36
17: AVERAGE TOTAL PROFIT FOR VAR=2.00, PTF=2,. BUFFER=60 37
18: AVERAGE TOTAL PROFIT FOR VAR=2.00, PTF=2,.BUFFER=60 37
19: WARM UP PERIOD GRAPH 48
Vll
CHAPTER I
THE PROBLEM AND ITS SETIING
Introduction
Significant pressure from competitors has forced manufacturers to review their
present manufacturing/management techniques, such as JustInTime (TIT). TIT is both a
philosophy and a set of techniques (Vollmann et aI., 1992). The ultimate objectives of
them philosophy are to obtain zero inventory, zero leadtime, zero failures, zero
disturbances, zero waste, and a flow process. These objectives lead to routine execution
ofschedule day in and day out (VoHmann et aI., 1992).
TIT systems are pull systems wherein parts are produced in upstream departments
whenever there is demand for those parts in the downstream departments. A pull system
is employed to minimize inprocess inventory and to enable all processes to know
accurate timing and required quantity (Monden, 1983).
Buffers
Buffers are included in most production systems to maintain product flow in the
presence of variation. One way buffer capacities can be established is by determining the
number of parts that can be accommodated in a given fmite space based on the part(s)
dimensions, the maximum number of parts can. then be calculated. The variability in
demand for different part types can be handled by varying the capacity allocation for each
part type within the buffer space. One disadvantage ofthis approach is that the total
number of parts that can be handled may vary with varying dimensions of parts.
Another way to determine buffer capacity is to declare the maximum'number of
part(s) that must be accommodated. The finite space requirement is then calculated. The
advantage ofthis approach is that each part type has a finite amount of space available
within the buffer. The disadvantage of this approach is that it is difficult to tradeoff
allocated buffer spaces for part types with varying priority factors i.e., the capacity of part
types with a high priority factor will be limited by their fixed buffer space.
One of the objectives of the nT philosophy is to reduce the buffer capacity. This
can be accomplished by reducing the floor space devoted to the inprocess inventory or
by reducing the number of parts. In this research, we define the buffer capacity as the
maximum number ofparts that can be accommodated in a finite space. The situation
wherein the actual capacity is less than the desired capacity will be overcome by
allocating the buffer space to the part types with higher priority factors. This aspect of
buffer space allocation is discussed with an example in Chapter N.
Performance Measures
Perfonnance measures can be cIassijied into two broad categories. The first
category is timebased perfonnance measures, such as those based on job completion
time, tardiness, earliness and deviation from due dates. The second category is monetary
measures, such as profit per order. In this research, the primary performance measure
2
will be a monetary performance measure. that is, the accumulated total profit rate.
Deviation from due dates is directly linked to a penalty; Ifthe product is delivered to the
customer on time, then no lateness penalty is incurred. If the product is delivered to the
customer after the expected due date, then a lateness penalty is applied. If the product is
completed before the due date, the product is stored as finished goods inventory and
inventory holding cost is applied until the due date. Holding costs are also applied to
WIP inventories as they proceed through the manufacturing process. Since the cost
performance of a system may be sensitive to the cost structure. timebased performance
measures will be considered as secondary performance measures to balance the effects
the cost structure will have on the system performance.
Problem
This research models a system with fmite buffer capacity where different part
types compete for capacity. Therefore, the objective ofthis research is to determine the
optimum number ofpart types to produce and allocation of space for a finite buffer
capacity pull system to maximize profits for the manufacturer.
Definition of Terminology
We define here some ofthe key terminology used in this research.
Lead Time: Lead time for a job is defmed as the time difference between the
completion time ofjob and the order arrival time of the job.
Throughput: Throughput is defined as the number ofparts produced for different
part types in a month.
3
Due date: Due date is the expected date ofdelivery of a part(s) to a customer(s).
Lateness Penalty: Lateness penalty is the penalty to be applied when a part(s) is
delivered to a customer(s) after the expected due date.
Holding Cost: Holding cost is the cost applied when a part is completed before the
due date and is held until its delivery.
WIP Holding Cost: WIP Holding cost is the cost incurred due to WIP inventories
as they proceed through the manufacturing process.
4
CHAPTER II
LITERATURE REVIEW
Introduction
In this chapter, a review of related literature is presented. This review focuses on
research involving flT production systems. Although, the success of a ill system
depends on factors throughout an entire organization, this research studies its application
to the shop floor. The shop floor aspect of a flT system involves better vendor
scheduling; reduction in leadtime and inprocess inventory; and better quality control.
The inprocess inventory aspect of the flT literature is reviewed, as it is relevant to the
research. Controlling the buffer capacity and the number of kanbans that are used in a
system can reduce the inprocess inventory. The literature review is divided into two
sections. First, the problem of detennining the buffer capacity is considered.
A second major issue in this type of research is consideration of the
characteristics of the system to study. In many cases, results vary based on the system
characteristics. Therefore, the second part of the literature review considers different
experimental design factors.
5
Literature Review on the Buffer C pacJty
A buffer is a space where WIP or finished parts are stored. A buffer is placed
between two machines and is used to store the finished goods from an upstream machine,
which will be subsequently used as "raw materials" for the downstream machine. In this
research, we define the buffer capacity as the maximum number of parts that can be
accommodated in a finite space.
Leisten (1990) analyzes a static deterministic flowshop problem using heuristics
for various buffer conditions, namely, unlimited buffers, finite buffers, and no
intermediate storage. He finds that heuristics do not provide good results when job
passing is allowed. Job passing is a situation wherein jobs are not processed in the same
sequence at every machine center and therefore some jobs may overtake other jobs.
Koulamas et a1. (1987) calculate the optimal buffer size for a twostage machining
process to maximize the profit rate under varying cutting speeds and tool replacement
intervals. The processing times are deterministic in nature. The optimal buffer space is
defmed as the one necessary to keep the critical machine running when there is a tool
change on the non~tical machine. The result show that the unit price increases as the
tool variability and/or the penalty cost increases.
So and Pinault (1988) propose a method for allocating buffer storage in a single
product pull system. Each machine center has two buffers, one in front of it (input
material buffer) and another behind it (output material buffer). Although the inputbuffer
and outputbuffer may correspond to the same physical buffer, they are logically treated
as different buffers. So and Pinault decompose their system into individual MIMII
stations with bulk service. The authors conclude that the performance (i.e., average
6
percentage ofdemand backlogged) oftheir model will hold good if the perfonnanoe
parameter is less than 0.05.
Berkley (1993) analyzed the change in relative perfonnance ofthe first come first
senre (FCFS) and shortest processing time (SPT) sequencing rules with changes in
processing time variability and station input buffer capacities in a singlecard kanban
system. Using an example system~ Berkley found that while FCFS has greater average
production rates when processing times are nonnal and input buffer capacities are large.
8PT has greater average production rates when processing times are exponential and
input buffer capacities are small. The maximum input buffer capacity used in Berkley's
research was ten containers.
Aligina (1996) extended the work ofBerkley (1993) by incorporating other
sequencing rules, such as earliest due date (EDD) and critical ratio (CR) and studied their
effect under the same set of conditions. Aligina suggested the need for an algoritbm~
which will provide the optimum number ofpart types for a finite buffer capacity in a
single card kanban system. This research will pursue this research question.
Literature Review on Due Dates Asslg.nments and Tardlne s Penalty
The objective ofthis section is to help in designing the shop structure that will be
studied in this research. In this research~ dlue date assignment is considered endogenous
in nature. Endogenous implies that the due. dates are set internally by the scheduler as
each job arrives on the basis ofjob characteristics, shop status infonnation~ and an
estimate of the job flow time (Cheng and Gupta, 1989). One such due date assignment
method is the Total Work Content (TWK) method. Ragatz and Mabert (1984) define due
7
dates using TWK. as the sum ofthe arrival time plus the product of allowance factor (k)
and total job processing time. Mathematically, due dates can be represented as
/I
DDj =aj +kLPI/
iI
where;
DDj: is the due date ofjob j;
aj: is the arrival time ofjob j;
k: is the allowance factor;
n: is the number ofoperations for job j; and
Pif is the processing time of operation i for job j.
Ragatz and Mabert (1988) used three levels for due date assignment, namely tight,
medium, and loose. The three levels ofdue date tightness are set such that, when the
FCFS dispatching rule is used,. the number of tardy jobs will be 20%, 10%, and 5% for
tight, medium, and loose due dates, respectively. AbuSuleiman (1998) studied the job
shop environment and used allowance factors of9, 6, and 3 to generate the tight,
medium, and loose due dates, respectively.
Ragatz and Mabert (1988) consider average total cost per period as the primary
performance measure. The total cost consists of late delivery cost (penalty tardiness cost)
and holding cost. The penalty tardiness cost is estimated as total work content per time
period late implying that the penalty tardiness cost is directly proportional to the work
content. According to the defmition ofthe TWK method, due date allowance is directly
proportional to work content. Therefore, penalty tardiness is directly proportional to the
due date allowance. Thus, penalty tardiness becomes a function ofjob value and absolute
tardiness. Holding cost consists of both WIP and fmished goods inventory cost implying
8
that the cost ofholding the raw materials before station one is zero. The ratio between
penalty tardiness cost and holding cost in these studies was 1:20. Ahmed and Fisher
(1992) followed the same cost structure.
Kawtummachi et aI. (1997) applied metascheduling methods in an automated
flowshop. The objective oftheir study was to minimize the total cost. The tardiness
penalty cost is represented as the cost of overtime. WorkInProcess (WIP) cost is
proportional to the product of the number ofjobs in the system and the average holding
time. Inventory cost is calculated as being proportional to the product of average
inventory of a job and inventory time.
The inventory and penalty cost that have been used in the above literature can be
expressed in the following generic fonn:
where;
Ij: the inventory cost for job j;
Vj: the value ofjob j;
~: the time job j spent in the system; and
p. =f (Y. d·  DD·) J J' ~ J
where;
Pj: is the tardiness penalty ofjob j;
dj: the time job j departed the syste~; and
DDj: is the due date ofjob j.
AbuSuleiman (1998) models the tardiness penalty as a function ofjob value 8I)d
relative tardiness (tardiness divided by leadtime). This is a major shift from the
traditional method of modeling the tardiness penalty as a function ofjob value and
9
absolute tardiness. In this research, we will follow Abu.Suleiman'sllpproach to model
the tardiness penalty oost. This issue will be discussed in detail in Chapter IV.
Koulamas et aI. (1987) defines profit rate as the ratio of the difference between
the selling price per job and the total cost per job to the total processing time.
Mathematically the profit rate per job is shown as:
PAj =(Sj  TCj) / tj
where;
PAf is the profit per job j;
Sr is the selling price for job j;
TCj : is the total cost incurred for job j; (The total cost incurred is the sum of
inventory holding cost and penalty cost.) and
tj: is the total processing time per job j.
Different authors have used different performance measures to evaluate the fmite
buffer space. Most of these performance measures are timebased. This research will
evaluate the problem of buffer space allocation based on monetary performance
measures.
Conclusion
The literature review provides insight into various factors affecting this research.
The cost structure, which will form an integral part of this research and will determine the
total profit generated was also reviewed. This research will provide operational
guidelines, which will seek to maximize the profit for a manufacturer by determining the
10
optimum number of simultaneously processed part types and the allocation ofspace to
those part types in a finite buffer capacity system.
11
CBAPTERID
RESEARCH GOAL AND OBJECTIVES
Research Goal
The primary goal of this research was to detennine the optimum number of
simultaneously processed part types in a finite buffer capacity manufacturing system to
maximize profits for the part type(s) manufacturer. The motivation for this research was
the fact that the current literature does not adequately consider the case of multiple
products Wldergoing different operations while competing for finite buffer capacity.
Research Objectives
The primary objectives ofthis research are:
• to review current journals, books, and articles on finite buffer allocation
methodology,
• to determine a set of experimental factors and their appropriate levels to assess the
importance of the main factors on plant perfonnance,
• to develop a simulation model using the simulation software package ARENA
(Kelton et al., 1998) to execute the experimental design,
• to perfonn statistical analysis on the simulation results, and
12
• to develop a set ofoperational guidelines for manufacturers to detennine the optimum
number ofpart types, which will maximize profits.
Methodology
The methodology used to accomplish the research objectives was:
1. To model a six station serial production system capable of handling ten different part
types.
2. To develop the simulation model, and to execute the experimental design using the
simulation software package ARENA (Kelton et ai, 1998). The process of
veri:ficatio~and validation will be conducted to determine the correctness ofthe
model. The characteristics of the experiments namely the run length, number of
replications, and warmup period will be determined based on the pilot runs. This
issue will be discussed in detail in Chapter N.
3. To conduct the experimental design runs and perfonn statistical analysis of the
simulation results obtained.
4. To develop conclusions and recommendations based upon the results obtained from
the statistical analysis while simultaneously developing a set of operational guidelines
for manufacturers to allocate fmite buffer space.
5. To document the research.
6. To identify areas of future research.
Assumptions
The following assumptions were made for this research.
13
1. Raw materials are available as required and spac,e is not a problem for storing the
finished goods.
2. Orders occur for a single part type with the size oforders sampled from unifonn
distribution with a minimum value of 1 and a maximum value of 18. The interarrival
time for all the orders is exponentially distributed with a mean of 144.
3. The different part types occupy the same amount of space within the buffers.
4. Materialhandling time is zero.
5. All time units are in minutes.
6. Decisions regarding acceptance or rejection ofan incoming order once taken cannot
be revoked.
7. Machine breakdowns are not considered.
8. No scrap or rework is taken into account.
9. Setup time for each part type is negligible.
10. Time value ofmoney is included in the holding cost and penalty cost factors.
11. The processing cost, i.e., tools, worker's pay, and overheads are the same regardless
ofthe buffer allocation procedure and therefore need not be modeled.
14
CHAPTER IV
RESEARCH MEmODOLOGY
System Description
Simulation is the evaluation tool used in this research. A sixstation production line
model is developed using the simulation software package ARENA (Kelton et aI., 1998)
(see Figure 1). The system is similar to and based upon the systems discovered in the
literature survey. All the stations have a single buffer between them i.e., the output
buffer for the upstream station becomes the input buffer for the downstream station and
so on. Buffer space is one of the experimental factors and consists of two levels namely,
60 and 90 parts. Therefore, in an empty system the buffer space in front of the first
machine has a capacity of 60 or 90 parts based on the experimental factor level chosen
implying available capacity is fixed. In a nonempty system, the buffer space will be the
difference between its capacity (Le., 60 or 90 parts) and space occupied by the existing
parts. Buffers present in front of other stations have ample space for workinprocess and
finished goods. The other buffers have infinite capacity. It is assumed that there is ample
space for storing the finished goods.
Orders arrive for parts in batches with the time between orders exponentially
distributed with a mean of 144 minutes. An order is always for a single part type but it
may be for quantity greater than one part. The order quantity for each order is generated
15
0\
Part Type Part Waiting Excess
Quantity Area Buffer
For
Orders Received
Figure 1: The SuStatlon Production Line System
from a discrete unifonn distribution with a minimum value of I and a maximum value of
18. The system can simultaneously process a maximum of"n" different part types. the
parameter "n" is an experimental factor, which will be examined at three levels, namely,
2, 6, and 10. This implies that when th.e experimental factor for maximum number of
part types is set at 2, then only the two part types with the highest priorities will be
generated. The system in ARENA for part type level =lOis shown in Appendix 1.
Processing time for each part type at each station is independent and identically
distributed according to a normal distribution with a mean ofsixteen. The variance of
processing time, which is one of the experimental factors, will be considered at three
levels namely 0.25, LOa, and 2.00. All times are expressed in the same time units (i.e.,
minutes). The average batch size, buffer size and tqe average interarrival times for the
parts are fixed quantities. The probability for an. order generated of a particular part type
is determined using the sum ofthe years digits method refer to Appendix 2. The
objective is to achieve higher volume of lower margin parts and lower vol~e of high
margin parts. In other words, higher margin parts have less frequent orders (special
orders) and lower margin parts have more frequent orders (commodities). The generality
ofthe distribution is as follows:
Discrete distribution (part type 1, cumulative probability, part type 2, cumulative
probability ...part type n, cumulative probability)
For a system processing two part types, the probability is
Disc(l, II3, 2, I),
For a system processing six part types, the probability is
Disc(l, 1/21,2,3/21,3,6/21,4, 10/21,5, 15/21,6, 1), and
For a system processing ten part types, the probability is
17 !I I~
Disc(l, 1/55, 2, 3/55, 3, 6/55,4,10/55, S, 15/55,6, 21/55, 7,28/55,8, 36/55,9,45155,
10,1).
All orders reside in an area called "Waiting Area for Orders Received". Once
each day the decision to accept or reject waiting orders (i.e., release it to the shop or
choose not to accept the order) is taken. This decision is taken by considering the present
shop conditions i.e., the available space in the finite buffer (buffer preceding machine 1)
and by taking into account the previously accepted orders that have not yet completed
machine 1 processing.
Order Acceptance Logic
Whenever an order is to be accepted, preference is given to orders with high
priority with respect to accumulated total profit rate potential and for those, which will
not violate the available space with full orders (order splitting is not allowed). For
example, if the available space is 2, and an order for 3 parts with the highest priority is
evaluated, it will be rejected. Once an order is accepted, a due date for that order is set.
The due date applies to all parts in the order.
Order Sequencing Logic
A hybrid logic is used for shuffling the queue. Initially, fmt in first out (FIFO)
within the priority factor will be used to schedule the flow of parts. For example, in a
system comprising oftwo part types, part type 1 will be scheduled ahead of part type 2.
Orders from any prior day's acceptance decisions may remain in the buffer at machine 1
at the time the current day's acceptance decisions are being evaluated. If an order for a
18
I·
part type has crossed its due date then the priority is given to that order to minimize the
penalty cost. The order sequencing logic rules are applicable only for scheduling of
orders for machineI. Downstream machines always use the FIFO rule.
Materialhandling time is assumed to be zero implying that there occurs an
instantaneous transfer of orders from an input buffer to the processing station and, after
completion of work on all parts within an ordert from the processing station to the output
buffer. The decision regarding acceptance or rejection of an order from a customer is
taken once every 1440 minutes (24 hours). This is done to avoid disturbances in the
production system caused by frequent modification of available orders. Orders arriving
between decision points are held until a decision is made. Ottce a decision is takent it
cannot be reversed. A station can produce parts ofvarious part type mixes. Set up time
is assumed to be zero.
Experimental Factors
The effect of the following factors on cost performance will be considered:
1. Buffer sizet
2. Due date allowance factor (k)t
3. Penalty tightness factor (ptf)t
4. Variation of processing timet and
5. Maximum number of part types.
Buffer size consists oftwo levels namely 60 and 90. With respect to Figure 1 this
implies that the first buffer in front ofthe first machine (machine I buffer) has a capacity
of 60 or 90 parts based on the experimental factor level chosen. Due date allowance
19

factor levels are 3, 6, and 9. Two levels of penalty tightness factors are considered,
namely, ptf=1 and ptf=2 (a factor related to lateness, defined later in this section). The
mean processing time per part per station IS fixed at 16.0. The variance of the processing
time is analyzed at 3 levels namely, 0.25, 1.00, and 2.00. The three levels of number of
part types are 2,6, and 10. For each of these combinations, the average total profit is
detennined experimentally. A total of 108 experimental combinations are examined.
The focus ofthis research is a preliminary investigation of the importance of the main
factors.
Due dates are set using the Total Work Content method (TWK). The value of the
constant k is chosen to be 3, 6, and 9 for low, medium, and high due dates, respectively.
As discussed in the literature review, tardiness penalty cost and inventory holding cost
have been modeled in the following generic forms:
p. = .cr\!. d·  DD·) and J J.\ y J' 1 J ,
In this research, the inventory carrying cost will follow the same generic fonn
mentioned above. The tardiness penalty is considered a function ofjob value and relative
tardiness. Relative tardiness will be modeled with respect to leadtime Therefore, the
tardiness penalty cost will be modeled as:
Pj =f(Vj, «dj  DDj)/(DDj  aj»), and
Ij =f(Vj,~)
where;
Pj: is the tardiness penalty ofjob j;
Vf the value ofjob j;
dj: the time job j departed the system;
20
DDj : due date ofjob j;
af the order arrival time ofjob j;
If the inventory cost for job j; and
tj: the time job j spent in the system.
Figure 2 illustrates how the tardiness penalty cost is modeled. Two levels of
penalty tightness factor (ptf) are used; 1 and 2. If the penalty tightness is set to 1, a job
will incur a tardiness penalty cost equal to its selling price if it is late for the period ofits
leadtime. Similarly, a job will incur a tardiness penalty cost equal to its selling price if
its lateness is twice its leadtime when ptf is set to 2.
AbuSuleiman's (1998) approach is used to determine the job value, overhead
expenses, and profits from each job. The raw material cost of a job j (Rj) is initially
asswned and follows the generic form
Raw material cost = 1000  100*(part type  1)
where part type = 1,2,... 10
The value added after each processing step is 5 % ofits raw material cost. The profits for
different part types are set individually. AbuSuleiman had arbitrarily chosen the various
percentages described above but asswned them to be the representative ofrealistic
scenarios. This research also assumes the same. .'
Cost Structure
The perfonnance measure in this research is total profit and is defined as:
11
ATPR= ~ ~PAj and
11 iI
21
Selling Price
ITardiness Penalty Cost I
Due Date I Due date +(Lead Time * ptf) Delivery Date
Figure 2: Modeling Tardiness Penalty Cost
where;
ATPR: is the accumulated total profit rate based on experimental processing time;
PAj: is the profit rate per job j;
n: number of part types;
Sf is the selling price for job j (selling price is 1.5 • the total value ofthe job);
TCj: is the total cost incurred for job j (the total cost incurred is the sum of
inventory holding cost and penalty cost); and
~: is the expected total processing time for job j.
Two types ofcosts are considered in this research, namely, the inventory holding
cost and the penalty cost. The inventory holding cost per job is defined as follows:
dj
1J = JHV·J dt
rj
where;
22
Ij: is the inventory holding cost for job j.
H: is the holding cost factor,
Vr is the value ofjob j,
rj: is the release time for job j (the time at which job j is released to the shop
floor), and
dj: is the time job j departed the system.
Since the system under study is discrete in nature, the above integration can be
expressed as follows (AbuSuleiman, 1998):
n+1
IJ· = "L.JHV·1, J. (tloJ.  t.....\, J.)
iI
where;
Ij: is the inventory holding cost for job j,
H: is the holding cost factor,
Vi,j: is the value ofjobj before being processed on machine i, and
ti,j: is the time at which job j leaves machine i, !oj = rj.
Here VI, j is the cost of raw material for job j (Rj). The storage area where jobs
wait until their due date is modeled as machine number (n+1). The holding c,ost factor is
set arbitrarily as 0.01% of the selling price of the order. The value of the holding cost
factor does not affect the generality ofthe study that is conducted.
Penalty cost is the second type of cost and is defined as follows:
where;
Pf is the penalty cost for job j,
23
Pj: is the penalty cost factor for job j,
dj: the time job j departed the system, and
DDf is the due date ofjob j.
Since the tardiness penalty is proportional to the job's leadtime, penalty cost
factor pj is calculated as follows:
where;
Pj: is the penalty cost factor for job j,
Sj: is the selling price of the job j,
ptf: is the level of penalty, when ptf=l, the penalty cost is equal to the selling
price if a job tardiness is equal to its leadtime,
DDj: is the due date ofjob j, and
aj: is the arrival time ofjob j.
Simulation Verification and VaUdation
Verification is the process of ensuring that the ARENA model behaves in the way
it is intended according to the modeling assumptions made (Kelton et aI., 1998).
Animation is an effective tool to perfonn the verification process. The path ofthe entities
is traced as they progress through the system. This ensures that the entities go through the
proper sequence of events and proper assigpment ofattribute values. The implementation
of priority factors into the model can be observed using the animation technique.
Validation is the process of ensuring that the model behaves consistent with the
real system. Since there is no existing real system that can be used to compare the
24
J
?
I
simulation results, the validation process cannot be conducted in this manner. Instead,
the process ofparts accounting, wherein all the parts that entered the system were
accounted for, as accepted, workinprocess, and processed parts, is used to validate the
model.
Simulation Model
The ARENA simulation software package is used to simulate the sixstage
production line. All the job's attributes are assigned as soon as the job enters the system.
This is done to ensure that the jobs in different simulation scenarios have the same
attributes based on consistent use of the random numbers. The flow chart of the flow of
parts is shown in Appendix 1. A disk copy of the ARENA model is available from the
author or from the School ofIndustrial Engineering and Management at Oklahoma State
University.
Simulation Characteristics
The characteristics essential to ensure good simulation results are run length,
number ofreplication, and warmup period. Pilot runs were conducted to determine the
above characteristics. The system is started in an "empty and idle" condition. The period
until the system reaches steady state is known as the warmup period. To detennine the
warmup period Welch's procedure (Law and Kelton, 1991) is used. This method
consists of several steps that are described below. The number ofreplications is set at 7
(arbitrarily chosen within the recommended range of 5 to 10). It is detennined that the
2S
)
•
system reaches its steady state at around 65,000 time units (refer Appendix3 for
additional information).
Determination of Run Length
The cycle time for a part type I order is collected after the wann up period. A
confidence interval (el) with a halfwidth less than or equal to 5% of the mean is desired.
Three different combinations ofronlength and warmup period were considered. The
number ofreplicatlons is set at 7. The results are summarized in the Table I below.
TABLE!
Effects of Different Run Lengths and Warm Up Period on the Ratio ofHalfLength over
the Mean
Note: SDStandard Deviation
Since all values tested satisfied the confidence interval, the value (Le., 65000), which led
to warm being approximately 10% of the run length (rule of thumb) is chosen. The warm
up period graph is shown in Appendix 3.
26
5•
I,.I
;
1)
J
•
,L
CHAPTER V
RESULTS
As discussed previously, 108 different system configurations were studied. The
primary performance measure of this research is accumulated total profit rate. For each
experiment, ATPR is determined by knowing the raw material cost, processing cost, and
the selling price. The results can be summarized by the following table:
TABLE II
Table of Means with Varying Number of Parts
No. of Parts Profit Mean
2 41148
6 28581
10 17108
ANOVA tests were performed using the SAS software to determine the
statistical significance of the main factors. For each experiment, the ATPR for individual
replication were found and used. The General Linear Model (GLM) procedure in SAS
produced an F value and a probability value (pr) for each of the main factors. The F
value is the ratio produced by dividing the Mean Square for the model by the Mean
Square for Error. For a confidence interval of95%, i.e., with an alpha (ex:) value of 0.05,
27
5•
•
•
;
•:
~
)

.1:4' latu~, II., " ,.. 2r
SoUr'Cr OF . s.. 01 Squar.. ....n Squa!" f ValIM ~, :> F
lI011eL I 13'$643"1$3.003000D 'I~SUlSlI.I2!l4OOO '211.20 O.DOG'
Error 7a' lOO300273375.1'1OODO 13a27071O.1SIOeOO .
Corr,c',d rlul 75' .2:1,S164l13222l.1 .
A·SqIl.r, C.Y. ROot lIS( . '1'RlW1TS; ....n
0.S7"5~ 40.032'1 1 1sa7•S:KIIHOD 21114~ ;nll74.a
Sourel . OF Type I U II.." Iquarl F valu, '.r • F
'AATTYPE 2 YZ.,3i40003.2.alOOOO 3MaGOOO' .10720000 211.37 0.000'
00 2 ~'''52a303lO.770'OOOO 2422G15'H.3UOOOOQ '10:.3 0.000'
VAIl 2 2IOS...a:z.50,q37OD ·,302Q431.2SDS.1OO 0.'7 0.37'.
TIGHT I 1271'54~'4.:tN1OOOO . '27.,5412".311JOOOO IS." 0.0001
IUf'SIZE I 11l1l114l1:Jl2. 151 ,'3000 ""'40382.'IiI'~ 1.111 0.00211 . '.
Figure 3: Anova Test Results
IOTE: T~l' , ••, control. ,ft. tYPI I c..,a~llonwl.e arrlr ~te, no, ,~,
,"pe,~"twll' error 'ate
~r If ....nl 2 3
Cr1Uul "lange 2027 :213a
·...an. ""ll 'II' ._ lett,,. a,.. IIOt UI"1hconUy .,Uarwnt.
5•
A
I
...." "II PNlTTY'.!
41148 252 2
21511 Z52 •
•7101 2S2 ID
Figure 4: Duncan Test Result for the Experime~talFactor Part Type
28
Till SAl 'Ylt..
IIOTE: Til" t.~t cOIl'nllI tile type t ~."iI_". arnl,. ran•. riot tlMi
Ixpe,.i..ntwL.. ImI,. ,.atl
NwlII,. 0'...... 2 3
, Critical i .... 202T 21.34
11I...1 .LUI th.1 1.1 'litter 1"1 IIOt ILgniftCUlUy lUUIMlnt •.
Duncllll Cll"Ollp1na.
A
•
·C
lIan
3M..,
:12417
17H2
N 110
2S2 I'
252 •
252 .3
Figure 5: Duncan Test Result for the Experimental Factor Due Date:
. '1: 41 SltUrdlY, lI,y I, ,'" :10
)••
ouncln" lIultipl. Rlngt T••t '0,. wlriabl.: PROFITS
IOTE: ThL. t ••t cOIltrol. tnt t,p. I caepar1.onw1•••rror Mltl, not th•
• x,.r~nt1l1a. Ir.nlr rlt.
Alp/lt O.llS Itf 747 IISE, .3427E1
..be,. 0' .....,. 2 :I
Critical Ringe 2027 21,.
~.., Group1llO ...... II Y~
A 211775 252 2
A
A 2070 252 ,
A
A *12 252 0.2'
Figure 6: Duncan Test Result for the Experimental Factor Variance
29
the probability of acceptance ofthe main factors is 0.05. A difference between the Pr
value and the F value greater than 0.05 for a factor implies that the experimental factor is
insignificant.
Figure 3 displays the result ofthe ANOYA test. It is observed that the Pr value
for the experimental factor "variance" yielded a value of0.3794, which is greater than
0.05 implying that variance is insignificant for this system. The other experimental
factors namely, part type, due date allowance factor, penalty tightness factor, and buffer
size yielded a Pr value lesser than 0.05 implying that these were statistically significant
factors. High order interactions (i.e., twoway and greater) were not considered in this
research. Preliminary analysis showed all the 2way and 3way interaction were
significant. By not including the subsequent higher level interactions as part of the model
in SAS, the degrees of freedom for these terms are pooled into the Error term. There is
risk in this approach since the significance of high order interactions may be lost;
however, the focus of this research is a preliminary investigation of the importance of the
main factors.
In addition to the above test, Duncan's multiplerange tests are also conducted
with the same alpha (oe) value of0.05 to group the factor levels within each experimental
factor. Since each mean faUs into a different group for each factor except variance (refer
to Figure 4,5, and 6), there are significant differences between the means. For example,
in Figure 4, the different levels ofthe number ofpart types fall in different grouping
levels namely, A, B, and C implying that these three levels ofnumber of part types are
significantly different from one another. This interpretation holds good for Figure 5. The
means with the different letter implies that levels of the above said experimental factors
are significantly different. It is found that there is no significant difference between the
30
).
!....

means for different levels of variance (refer Figure 6). The means with the same letter
implies that different levels of the experimental factor, variance are not significantly
different. From the Anova test, the Ptfand the buffer size .results confum. the obvious,
i.e., there is a significant difference between the two means.
The results can be consolidated using graphs, which provide a summary ofthe
performance of the system with respect to ATPR under varying due date levels and
numbers of part types. The data points for these graphs are shown in Appendix 4. Figure
7 displays the ATPR generated under different due dates and number ofpart types but
under the same variance, penalty tightness factor, and buffer level. It is observed that for
a given number of part types, having tighter due date level (Le., 9) generated more ATPR.
This can be seen by observing the third bar in each group. Moreover, maximum ATPR is
generated when fewer part types are used. The frrst group ofbars is greater than their
respective bar in other groups. It is shown by the Duncan's test ofmeans that the
different levels ofthe experimental factors were statistically significantly different.
Therefore the condition ofmaximum ATPR is used to determine the part type level
which generated maximum profits (refer to Figure 7).
The only differenc·e between Figure 7 and Figure 8 is the change in the buffer size
level. By comparing Figure 7 and Figure 8, it is observed that having a bigger buffer size
generated greater ATPR since each of the respective bars is taller except in cases when
the ptf. The only difference between Figure 7 and Figure 9 is the change in the penalty
tightness factor level. By comparing Figure 7 and Figure 9, it is observed by looking at
the first bar in each group that greater ATPR was generated for a due date tightness level
of3 and by having a higher penalty tightness factor (i.e., 2). It is observed that maximum
31
).
~
•t
.Va~O.25; Ptt=1, Bl!ffer:60 .
50000
45000
40000 .
35000
~ 30000 e 25000
.a., 20000
15000
10000
5000o
~: 0"1 g,
fir ~Q' ~ ..
. "" ". tv" r#~" rtf'" ~4 .",,"{J. ~4,~
~'l;~ ~~ ~~~
Figure 7: Average Tota~ Pr.oflt for Var=O.25, Ptf::1, Buffer=60.
,.
60000
.50000 .
.!! 40000
.o; . .30000' . '
a.. 20000 :
10000
O'
Figure 8 ~ Average Total Profit for Var=O.25, Ptf=1, Butfer=90
32
.)
.
Var::Q.25, Pt1=2, Buffer::;60 '.' . . .. '. . ' ..
60000
50000
."... 40000·
~.... 30000
Q. 20000
10000
a
<:f/' ~<J' ~<:;fOJ . ~<(, <:>~j:J ~<:;fOJ <:>~' ~<:I ~<:(O)
Pt' JI" gf~" ~j:J' .jJ" IJ' '. ~~~ ~<:). ~<:),
,,4"A..~ ,,~ . ,,~,,4 <..~ . ~~q; ~/ ~~qt
~ ~ ~ ~.~ ~ , ..' , «~ «~ Q..'ti. q'ti .q,llf Q..'ti . <t1;~ ~ttJ.~ qttJ.~
. " .. .part Types and cue Date Level
Figure 9: Average Total Profit for Var=0.25, Ptf=2, Butfer=60
........"
60000
5000d
en 40000· ~ .30000
Q.; 20000
10000
Var.=O.25, ,Ptf2, BufferdO·
Figure 10: Average Total ProUt for Var=O.25, Pt1=2, Buffer::90
33
L
Var::1.00, Ptf=1, Buffer=60
. ..
FIgure 11:: Average T~tal Profit for V8r=1.0~, .Ptt=1, Buffer::6«;)
II
.
I
. Var::1.0~ Pttc1, Suffer::90 ". . .' .. . ""
6000p.
50000·
.e.en.. 40000, 30000
c.. 20000
·10000
o
1"'1' ;o/$' r#OJ 1"'1' Jp~, /, j) f)J
<:)v· <:)'V <:)v. ~:::r <:)<J' <Jv <o'V <J'J.' <J<J'
JI." ./.I"'" ../5V' ,,?' ..i" ,jJ' ~(;:), '. ~(;:), ~(;:),
"'.¢ ",~VT ",~0' A..~ ",~<fi ~4 "'~( ~4' ",4' ':.
«."I;~' «."I;~ «.."I;~ «."I;~ q,1" <ll;~· q,'IJ.~ q,fb.~ q,fb.~ .
Part Types and Due 'Date Level
Figure 12:: Average Total Profit for Var=1..00, Ptf=1, Buffer::90
34

V~r::1 .•00, Ptt::2,. Buffer=60
Flgu~e 13: Average Total Profit for Var=1.90~ Ptf=2, Buffe~'60 . . . . '. .
Var=1.tio, Ptf=2; BUfler::90
• • • f 60000
50000
Ie 40000
=e= 30000.
a. 20000
10000
o
. if'~.p ifOJ
<J <J <J
rt"" At'" Al!q,.
,,~~ ,,~"" ,,~""
«~{o «~{o <?'l;~
Figure 14:: Average Total Profit for Var=1.00, Ptf=2, Buffer=90 .
35

. .
Figure 15: ~verage Totai Profit for Vai=2.•00, Ptf=1', Buff~'r=SO
.,",, ,
....
. .
60000
··6000Q
.I.n.'· 40000 " ..
~ .. 30000
et .20000' ..
10000
O'
. <:)<1' Q"<;:;<::f' <:)<::f~ <:)~ <:)~ <:)<:f~ <:)~ <:)<::f'
rtfo/' ~~/~/~ /'/'~' ~<':)~ ~<':). ~<':)' AJ ~ ~ #
,,~q ~'" . *'" ?:",~" ~.<...~"'C ,,4 ".f ,,~
<l..'lT~ <l..'ti <llf ~'ti <l..tt; q,'ti <l~~ <i.'Ir~ q,'b'~
Part Types and Due Date Level
fi.gure 16:: Avera.ge To~1 Profit for Var=2.00, Ptf=1, Buffer::90
36
60000
. 50000.
! 40000
~... 30000
n. 20000'
10000
o
Var=2.90,·Ptf=2, Butfer::60
:Figure 17: Average Total Profit 1!Jr y~r::~.OO, Ptf=2, .8utfer=60
.....
60qO.O
. .
50000
. 1J" ~oooo·
, ~ 30000 ....
n. 20000
10000
{)
. .
. ·.VB~2.00, Ptf=2, Buffer:;:90
Figure 18: Average Total Profit for·Va~2.00, Ptf=2, '8uffer::90
37
ATPR is generated when fewer part types are used. The break up of the ATPR with
respect to different part types indicated that higher numbered (but lower priority) part
types generated more ATPR than the lower part types. For example, in an experimental
configuration involving 6 part types the higher part types (i.e., 4,5,6) generated greater
ATPR than the lower part types (i.e., 1,2,3) although lower part types generated more
profit than the higher part types with respect to profit per part (i.e., profit margin). This
can be attributed to the experimental assumption of increasing the probability of
generation of orders with respect to the part types. In other words higher margin parts
have less frequent orders (special orders) and lower margin parts have more frequent
orders (commodities). These trends hold good for all the experimental combinations.
It is observed that processing fewer part types with tighter due date levels, higher
penalty tightness factors, and larger buffer sizes generated maximum ATPR. This result
was expected due to the fact that the high margin part types were being processed. An
increased buffer size provides additional opportunity for higher margin part types to be
accepted and it also reduces the number of part types to be rejected.
The only surprise is the fact that although higher numbered part types (i.e., 4,5,
and 6) generated more profits than the lower part types (i.e., 1, 2, and 3) maximum ATPR
is not generated by the higher part type experimental factor levels (6 and 10). This can be
attributed to the loose raw material cost structure wherein the percent difference between
the highest numbered part type and the lowest numbered part type is 90 %.
It can be concluded that processing fewer numbers ofpart types with tighter due
date levels, higher penalty tightness factors, and larger buffer size generates maximum
ATPR.
38
CHAPTER VI
CONCLUSIONS AND FUTURE RESEARCH
This chapter concludes this research report by presenting the conclusions and
future directions of this research. A simulation model is developed using the simulation
software, ARENA. It is found that four of the experimental factors namely, due date
allowance factor, buffer size, penalty tightness factor, and number of part types were
significant with respect to ATPR.
This research has also identified the experimental settings that will maximize
profit. It is observed that processing fewer numbers of part types with tighter due date
levels, higher penalty tightness factors, and larger buffer size generated maximum ATPR.
This finding is important for part type manufacturers as it provides operational guidelines
to improve the profits generated by a manufacturing system. For an upstart
manufacturing firm whose initial aim is to generate maximum ATPR, the operational
guidelines provided are iliat they should use lesser number of part types, looser due dates
level, higher penalty tightness level, and a higher buffer size to maximize profits. For an
established manufacturing firm, which wants to use a smaller buffer size (say 60) due to
implementation of a pull system like nT, the operational guidelines provided are that
they should use looser due date level, higher tightness level, and low number of part
types to maximize ATPR. From the research point of view, to further generalize the
results of this research these experimental factors should be further analyzed to determine
39
their effectiveness on the system under additional operational parameters such as
machine breakdowns, and arrival of rush orders.
Another important finding ofthis research is that variation in processing time
(denoted by the factor, variance) was not a significant factor in this study. Variance is
relatively small compared to due dates level to have any effect on the system. It is
possible the higher levels ofvariation relative to the mean (highest level in this study was
2/16 =12.5 %) with tighter du~ date factors might indicate significance. Higher levels of
variation could be the focus of additional studies in this area.
Future Research
Some possible directions for future research are given below:
• In this research, no penalty cost was attached to the. orders that are rejected. In a
more general situation, penalty cost for the order rejected can be considered during
the decision making process. If some part types 1 were rejected due to unavailability
of buffer space, it might have reduced the ATPR generated, thereby providing an
opportunity for different levels of part types to generate maximum profits
individually.
• A more complicated flow shop system may be considered~ for example, a system with
machine failures and rework. This will increase the inventory holding cost thereby
cutting down on the ATPR generated.
• The decision to accept or reject an order is based on the profit rate generated per part,
which is assumed to be known apriori. Instead a more complicated profitability
factor can be incorporated by considering the actual profit rate per part by considering
40
the penalty cost and inventory holding cost generated at the completion of
manufacturing ofeach part.
• Rather than processing orders in batches at each station, they could be split into parts
and subsequently processed at individual stations. This will smooth the flow since
the parts are transferred individually.
• Increasing the numerical value ofthe variance ofthe processing time. This could
bring about the significance ofthe variance as an experimental factor.
• Since virtually all the main factors were significant, additional studies including high
order interactions is recommended.
• In this research, the percent difference between the raw material cost ofthe higher
profit rate per part generating part type (i.e., I) and that of the lower profit rate per
part generating part type (i.e., 10) was 90%. This percent difference is due to the
assumptions made for the initial raw material cost. Changing this percent difference
might bring about a change in the final results.
• Further research can be conducted by considering the possibility of investigating two
parts where one has the highest priority (part which generates comparatively
maximum profit rate per part) and another has the lowest priority (part which
generates comparatively minimum profit rate per part).
41
BIBLIOGRAPHY
AbuSuleiman, A., 1998, "Job shop scheduling: A quantified sequencing rule for
improving system performance under diversified operational parameters,"
Unpublished Masters Thesis, Oklahoma State University.
Ahmed, 1, and Fisher, W.W., 1992, "Due dates assignment,job order release, and
sequencing interaction in job shop scheduling," Decision Sciences, 23, 633647.
Aligina, 8.B., 1996, "Investigation of the effect of buffer capacity and sequencing rules in
a single card kanban system," Masters Creative Component, Oklahoma State
University.
Berkley, B. J., 1993, "Effect of buffer capacity and sequencing rules on singlecard
kanban system performance. " International Journal ofProduction Research,
31(12),28752893.
Cheng, T.C.E., and Gupta, M.C., 1989, "Survey of scheduling research involving due
date determination decisions," European Journal ofOperational Research, 38,
156166.
Kawtummachi, R., Yanagawa, Y., Ohashi, K., and Miyazaki, S., 1997, "Scheduling in an
automated flow shop to minimize cost: Backwardmeta scheduling method,"
International Journal ofProduction Economics, 49,225235.
Kelton, W.D., Sadowski, R.P., and Sadowski, D.A., "Simulation with ARENA," WCB
McGrawHill, New York, NY, 1998.
42
Kiviat, PJ., R. Villanueva, and H. Markowitz, ''The SIMSCRIPT n Programming
Language," Prentice Hall, 1969. '
Koulamas, C. P., Lambert, B. K., and Smith, M. L., 1987, "Optimal machining conditions
and buffer space size for the twostage case," International Journal 0/Production
Research, 25(03), 327336.
Law and Kelton, Simulation Modeling and Analysis, Second Edition, McGrawHill,
1991.
Leisten, R., 1990, "Flowshop sequencing problems with limited buffer storages,"
International Journal o/Production Research, 28(11), 20852100.
Monden, y" 1983, Toyota Production System, Industrial Engineering and Management
I
Press, Norcross, Ga.
Pritsker, A.B., "Introduction to Simulation and SLAMII," John Wiley and Sons, 1995.
Ragatz, G. L., and Mabert, V, A., 1984, "A simulation analysis of due date assignment
rules," Journal ofOperations Management, 5(1), 1984.
Ragatz, G. L., and Mabert, V. A., 1988, "An evaluation oforder release mechanisms in a
jobshop environment," Decision Sciences, 19, 167189.
So, K. C., and Pinault, S. C., 1988, "Allocating buffer storages in a pull system,"
International Journal ofProduction Research, 26(12), 19591980.
Vollmann, T. E., Berry, W.L., and Whybark, D.C., 1992, Manufacturing Planning and
Control, John Wiley and Sons.
43
Appendixl: The Block Diagram of the System
Arrange Part types
in ascending order
Select part types
depending on
availability of space
No Space Space
Assign
attributes
lfdue date>
No TNOW
Yes
Attribute = 100,000
ISelect low value of Attribute I • Parts processed from machine 1 through
machine 6
NO
Penalty time =Processing
time  due date
Assign penalty cost and total cost
:Determine Profit
I Depart I
44
Appendix2: Sum Of The Yean Digit Method Calculation
The generality of the distribution is as follows:
Discrete distribution (part type 1, cumulative probability part type 2, cumulative
probability ... part type n, cumulative probability)
(i) For a system processing two part types, the sum ofthe two numbers is 3 (i.e., 1 and 2),
therefore the probability is
Disc (part type 1, (part type l/sum ofthe two numbers), part type 2, (part type 2/sum of
the two numbers»
i.e., Disc(l, 1(3,2, 1)
(il) For a system processing six part types, the sum ofthe six numbers is 21 (i.e.,
1,2,3,4,5, and 6), therefore the probability is
Disc (part type 1, (part type l/sum of the six numbers), part type 2, (part type, 2/sum of
the six numbers) ... part type 6, (part type 6/sum ofthe six numbers»
i.e., Disc(1, 1/21,2,3/21,3,6/21,4, 10/21,5, 15/21,6, 1), and
(ill) For a system processing ten part. types, the sum of the ten numbers is 55 (Le.,
1,2,3,4,5,6,7,8,9, and 10), therefore the probability is
Disc (part type 1, (part type l/sum of the six numbers), part type 2, (part type 2/sum of
the six numbers)... part type 10, (part type 10/sum of the six numbers»
i.e., Disc(1, 1/55,2,3/55,3,6/55,4,10/55,5, 15/55,6,21/55, 7, 28/55,8,36/55,9,45,55,
10,1).
'.
4S
Appendix3: Warm Up Period Determination Procedure
Step 1: Simulation runs were made for the worst conditions for seven replications each of
1,000,000 (Xn) number ofobservations (arbitrary chosen). The timeinsystem ofpart
type 1 order is recorded as single observation data (Yij, j=l,2,...,7; i=l,2, ... ,Xn).
Step 2: For the 7 replications, the average timeinsystem (Yij) ofpart type 1 order is
determined using the formula:
7 Y = LYij /n, n = 7 and i= 1,2,...,Xn.
Ij )1
Xn = number of observations.
Step 3: To smooth out the high frequency oscillation in the time in system measure, the
moving average method is used. The window (w) ofthe moving average is a positive
integer such that w;S; (Xn / 2). The bigger the window values the smoother the curve.
Based on the selected window, the moving average values (Vi(W» are calculated as
follows:
2w+1
2i +1
ifi=W+l,.. " mw
ifi=l,....w
Step 4: Plot YiCw) where i= l~, ...(Xn  w) for different window sizes, w. Window size
of200 is used to determine the truncation point. Microsoft Excel spreadsheet software is
46
used to plot the graph. In the graph (refer Appendix3), the xaxis represents the total
time and the yaxis represents the timeinsystem for part type 1 order. It is determined
that the system reaches its steady state at around 65,000 time units.
47
8'17
Time In System for Part Type 1
~
~
.;;.J..
\.C.
~
»a
~
e:l
tD
:cSo = ~... »
'tl
l':I'"
0 (JI ~
(:) (:) (:)
0 0 0
m m m
+ + +
0 0 0
0 N w
1.22E+04
2.23E+04
3.28E+04
4.37E+04
S.34E+04
6.41E+04
7.S5E+04
8.67E+04
9.65E+04
1.06E+05
1.15E+05
1.26E+05
1.36E+05
t 2. 1.45E+05
lD .
t 1.54E+05
i
• 1.64E+05
1.74E+05
1.83E+05
1.9SE+05
2.06E+05
2.16E+05
2.27E+05
2.38E+05
2.48E+05
2.61E+05
2.71E+05
2.82E+05
2.91E+05
~ N N
c.n (:) (JI
0 0 0 m m m
+ + +
0 0 0
w w w
Appendix4: Data Points for the Graphs
Part types D.D.level Var.level ptf level Buffer Size ATPR
2 3 0.25 1 60 13468.29
2 6 0.25 1 60 39222.43
2 9 0.25 1 60 45763.43
6 3 0.25 1 60 10751.17
6 6 0.25 1 60 31416.14
6 9 0.25 1 60 36669.43
10 3 0.25 1 60 6246.771
10 6 0.25 1 60 16921.57
10 9 0.25 1 60 19616.43
2 3 0.25 1 90 11328.1
2 6 0.25 1 90 44747.29
2 9 0.25 1 90 53236.57
6 3 0.25 1 90 9142.314
6 6 0.25 1 90 35970
6 9 0.25 1 90 42789.57
10 3 0.25 1 90 4838.771
10 6 0.25 1 90 19137.14
10 9 0.25 1 90 22750
2 3 0.25 2 60 32237.57
2 6 0.25 2 60 45114.43
2 9 0.25 2 60 48385
6 3 0.25 2 60 25816.29
6 6 0.25 2 60 36148.86
6 9 0.25 2 60 38775.43
10 3 0.25 2 60 13985.86
10 6 0.25 2 60 19323.43
10 9 0.25 2: 60 20671
2 3 0.25 2 90 35813.43
·2 6 0.25 2 90 52523.14
2 9 0.25 2 90 56767.71
6 3 0.25 2 90 28801.86
6 6 0.25 2 90 42216
6 9 0.25 2 90 45625.57
10 3 0.25 2 90 15315
10 6 0.25 2 90 22464.43
10 9 0.25 2 90 24270.86
49
Part types D.D.level Var.level Ptf.level Buffer Size ATPR
2 3 1 1 60 14407.86
2 6 1 1 60 39801.29
2 9 1 1 60 46282]1
6 3 1 1 60 11662.54
6 6 1 1 60 32006.71
6 9 1 1 60 37191.57·
10 3 1 1 60 6384.529
10 6 1 1 60 16942.57
10 9 1 1 60 19610.14
2 3 1 1 90 10450.26
2 6 1 1 90 44338
2 9 1 1 90 52949.43
6 3 1 1 90 8504.614
6 6 1 1 90 35661.86
6 9 1 1 90 42564.43
10 3 1 1 90 4943.614
10 6 1 1 90 19220
10 9 1 1 90 22797.86
2 3 1 2 60 32887.71
2 6 1 2 60 45584.57
2 9 1 2 60 48825.14
6 3 1 2 60 26458.57
6 6 1 2 60 36630.57
6 9 1 2 60 39223.29
10 3 1 2 60 14037.71
10 6 1 2 60 19317.14
10 9 1 2 60 20651
2 3 1 2 90 35289.57
2 6 1 2 90 52233.57
2 9 1 2 90 56539.86
6 3 1 2 90 28412.57
6 6 1 2 90 41991.57
6 9 1 2, 90 45442.86
10 3 1 2 90 15372.57
10 6 1 2 90 22510.71
10 9 1 2 90 24299.86
50
.,
Part types D.D.level Var. level ptf level Buffer Size ATPR
2 3 2 1 60 14625.14
2 6 2 1 60 39662.29
2 9 2 1 60 46036.71
6 3 2 1 60 11734.67
6 6 2 1 60 31834
6 9 2 1 60 36949.57
10 3 2 1 60 5988.557
10 6 2 1 60 16824.86
10 9 2 1 60 19580.57
2 3 2 1 90 11431.07
2 6 2 1 90 44843.86
2 9 2 1 90 53253.86
6 3 2 1 90 9357.771
6 6 2 1 90 36133.57
6 9 2 1 90 42862.71
10 3 2 1 90 4235.229
10 6 2 1 90 19049.71
10 9 2 1 90 22791.86
2 3 2 2 60 35837.71
2 6 21 2 60 45353
2 9 2 2 60 48540.57
6 3 2 2 60 26348
6 6 2 2 60 36397.86
6 9 2 2 60 38955.29
10 3 2 2 60 13880.29
10 6 2 2 60 19298.8611
10 9 2 2 60 20676.57
2 3 2 2 90 35837.71
2 6 2 2 90 52544.43
2 9 2 2 90 56749.29
6 3 2 2 90 28902.86
6 6 2 2i 90 42290.86
6 9 2 2 90 45655.43
10 3 2 2 90 15076.86
10 6 2 2 90 22484.29
10 9 2 2 90 24355.29
51
Appendix5: Example Calculation for the Cost Structure
Consider a part that goes through 3 operations. The initial raw material cost is $ 1000
and the value added after each operation is 5 %. The job arrives at time 0 and an
allowance factor of 9 is used. The average processing time per operation is 1.5 hrs.
n
Due Date = DDj = aj +k:LPIj
j.1
=0 + 9*(3*1.5)
=40.5 hrs.
Selling Price = 1.5 * the total value of the job
= 1.5*(Value 12 + 1.05*(value 12) + 1.05*(value 13) + 1.05*(value 14»
= $ 6465.19
Selling Price (excluding cost of raw material cost) =$ 5465.19
Holding Cost Factor =0.01/100 • 1736.44
= 0.546519
n+l
Inventory Cost =I = ~ J ~BY .I,J. (1'I,J.  LII. J.)
II
=0.1736 ·«1000*1.5) + (1.05*1000*1.5) +(1.052 *1000*1.5»
= $ 2584.36
Penalty time =Pj (max [OA  DDj])
=0
Penalty time factor =6465.19 / 2* 40.5
= 79.82
Penalty cost = Penalty time factor * Penalty time
52
=0
Total cost = Inventory cost + Penalty cost
=$ 2584.36
Profit =(Selling Price  Total cost)/total Processing time
= (6465.19  2584.36)/4.5
=$ 862.41
53
VITA
'v Yousuff Zaman Habibullahkhan
Candidate for the degree of
Master of Science
Thesis: AllOCATION OF FINITE BUFFER CAPACITY TO PART TYPES FOR
MAXIMIZING PROFITS IN SERIAL UNES
Major field: Industrial Engineering and Management
Biographical:
Personal Data: Born in Madras, India, January 27, 1976, the son of Mr. Habibullah Khan
and Mrs. Iqbal, Begum
Education: Graduated from Gill Adarsh Matriculation Higher Secondary School,
Madras, India, in May 1993; received Bachelor of Engineering degree in
Mechanical Engineering from University of Madras, Madras, India in May
1997; completed the requirement for the Master of Science degree in
Industrial Engineering and Management at Oklahoma State University in
December 1999.
Experience: Graduate Teaching Assistant, Oklahoma State University, from August
1998 to May 1999. Production Engineer, Ingersoll Products Company,
Chicago, Illinois, from June 1999 to Present.
Affiliations: Institute For Operations Research and Management Science (INFORMS).