THE INTEGRATED ECONOMIC PRODUCTION
QUANTITY MODEL FOR INVENTORY
AND QUALITY
by
THARAT ITTHARAT, B.Sc, M.Sc.
A DISSERTATION
IN
INDUSTRIAL ENGINEERING
Submitted to the Graduate Faculty
of Texas Tech University in
Partial Fulfillment of
the Requirements for
the Degree of
DOCTOR OF PHILOSOPHY
Approved
Co^^airperson of the Com^ilftee
Co-Chairper^elToftiie Committee
Accepted
Dean of the Graduate School
December, 2004
ACKNOWLEDGEMENTS
I gratefully acknowledge all the people who gave me support and help during my
Ph.D. program and my life during the past years of my stay in America, although the
words here are too limited to express my sincere thanks.
I would like to express my gratitude to my advisors, Dr. Elliot J. Montes and Dr.
Mario G. Beruvides, for their inspiring and encouraging way to guide me to a deeper
understanding of knowledgable work, and their invaluable comments during the whole
work with this dissertation. I would also like to acknowledge Dr. Milton L. Smith, Dr.
James Bums, and Dr. Hong C. Zhang for serving as my committee members and their
helpful suggestions. My special appreciation is extended to Dr. Montes who has
patiently listened to and always assisted my questions. It is not often that one finds an
advisor that always finds the time for listening to the little problems throughout the
course of this research.
I am deeply indebted to my parents, Charan and Pontip Ittharat, for their constant
support over the years. Last, but not least, I would like to thank my parents and my
girlfriend, for their ever-loving support and understanding during the years of my studies.
n
TABLE OF CONTENTS
ACKNOWLEDGEMENTS
ii
ABSTRACT
vi
LIST OF TABLES
vii
LIST OF FIGURES
ix
CHAPTER
I. INTRODUCTION
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
History and Background of the Cost of Quality and Inventory
Problem Statement
Research Questions
General Hypotheses
Research Purpose
Research Objectives
Relevance of this Study
1.7.1 Theoretical Research Needs
1.7.2 Practical Research Needs
1.7.3 Benefits of this Research
1.8 Research Outputs and Outcomes
II. LITERATURE SURVEY
1
3
5
5
6
6
6
7
7
7
8
9
2.1 Introduction
2.2 Theories and Historical Background in Cost of Quality
2.2.1 Six Primary Theories inCost of Quality
2.2.2 Opportunity Costs and Hidden Costs
2.3 Theories and Historical Background in Inventory
2.3.1 Notation
2.3.2 Economic Order/Production Quantity (EOQ/EPQ) Models
2.3.3 Inventory Policies with Defective Items
2.3.4 Inventory Policies with Imperfect Process
2.3.5 Inventory Policies with Quality Costs
2.3.6 Inventory Policies with Repair/Rework and Warranty
2.3.7 Inventory Policies with Stochastic Demand with Remanufacturing
2.4 Conclusions
2.5 Theoretical Model
in
9
11
11
13
14
15
17
19
25
30
34
39
46
48
2.5.1 Definition of Reference Variable
III. METHODOLOGY
49
51
3.1 Introduction
3.2 Research Design
3.2.1 Type of Research
3.2.2 Research Hypotheses
3.2.3 Research Environment
3.3 Research Method and Instrument
3.3.1 Research Method
3.3.2 Research Instrument
3.4 Data Collection and Treatment
3.4.1 Data Collection
3.4.2 Data Treatment
3.5 Research Constraints and Limitations
3.6 Final Remarks and Conclusions
IV. THE INTEGRATED EPQ MODELS WITH THE COST OF QUALITY
4.1 Introduction
4.2 EPQ Models Associated with Quality Costs
4.2.1 Model 1
4.2.2 Model 2
4.2.3 Model 3
4.2.4 Model 4
4.2.5 Model 5
4.2.6 Model 6
4.2.7 Model 7
4.2.8 Model 8
51
52
52
53
54
59
59
60
60
60
61
62
63
64
64
64
66
79
72
75
78
81
84
87
V. STATISTICAL AND RESULT ANALYSIS
91
5.1 Introduction
5.2 Result Validation
5.3 ECQPQ Sensitivity Calculation Example
5.4 Variable Selection
5.5 Generating the Data for Statistical Tests
5.5.1 Numerical values for test problems
5.6 Statistical Analysis
5.6.1 The optimal lot size differences
5.6.2 The total annual cost differences
5.7 Statistical Analysis (Hypothesis Tests)
5.7.1 Hypothesis 1
91
92
93
98
102
103
105
105
107
109
109
IV
5.7.2 Hypothesis 2
5.8 Data and Graphical Interpretations
5.8.1 Effect of Holding Cost
5.8.2 Effect of Setup Cost
5.8.3 Effect of Production Rate
5.8.4 Effect of Rework Rate
5.8.5 Effect of Defective Proportion
5.8.6 Effect of Customer Defective Proportion
5.9 Sensitivity Analysis
5.9.1 Summary of Sensitivity Analysis
VI. CONCLUSIONS, CONTRIBUTIONS, AND FUTURE RESEARCH
6.1 Conclusions
6.2 Contributions
6.3 Future Research
6.3.1 Multiple Product Types
6.3.2 The Capacity Constraint
6.3.3 The Function of Setup Cost
REFERENCES
Ill
113
113
115
116
117
119
120
122
135
137
137
138
139
140
140
140
142
APPENDIX
A.
B.
C.
D.
E.
MATHEMATICA SOFTWARE CODE
MICROSOFT VISUAL STUDIO.NET CODE
SENSITIVITY ANALYSIS DATA FROM EXAMPLE
TESTING DATA
THE OUTPUT DATA
147
165
197
210
215
ABSTRACT
Determining the optimal production lot sizing has been widely used by the
classical economic production quantity (EPQ) model. However, the analysis for finding
an EPQ has several weaknesses which lead many researchers to make extensions in
several aspects on the original EPQ model. The cost of quality is one of good aspects to
be added to the EPQ model since there are a lot of costs incurred such as prevention,
appraisal, failure, warranty (products retumed fi'om customer), inspection, and rework
costs. The integration of cost of quality and EPQ should be able to link and classify each
cost of quality in practical way of inventory management. This paper deals with the
finite production inventory model integrated with quality costs for a single product
imperfect manufacturing system. This problem assumes that the product quality is not
always perfect unlike the traditional EPQ model. The defect rate is considered as a
proportion of the production rate, and defective items are reworked at some cost either
before, or after sales (products retumed by the customer). The prevention, appraisal, and
inspection costs have somewhat inverse relationships to the defective rate. The
replacement rate from products retumed by the customer is also considered to be another
random variable with known failure rate in the field.
The purpose of this research is to investigate the quality cost factors in the
economic production quantity inventory model in order to find the optimal lot size. The
objective is to develop mathematical models in order to minimize the annual total cost of
inventory and quality.
VI
LIST OF TABLES
2.1
Summary of the inventory model associated with quality costs
47
5.1
Effect of Errors in the rework rate on 70C* and Q*
95
5.2
The samples of the randomly generated problems
103
5.3
Variables and their values
104
5.4
Thenumberof differences in g * between model 1 and other models
106
5.5
The number of differences in TOC^ between model 1 and other models
108
5.6
The optimal lot size from test problem#4 when varying x
110
5.7
The optimal total annual cost from test problem#4 when varying x
112
5.8
Error on iQ* when variable C changes
123
5.9
Error on TOC* when variable C changes
124
5.10
Error on 2 * when variable F changes
125
5.11
Error on TOC* when variable F changes
126
5.12
Error on g * when variable 7/changes
126
5.13
Error on TOC* when variable i/changes
127
5.14
Error on Q* when variable AT changes
128
5.15
Error on TOC* when variable iT changes
128
5.16
Error on 2 * when variable P changes
129
5.17
Error on TOC* when variable P changes
130
5.18
Error on g * when variable i? changes
131
5.19
Error on TOC* when variable 7? changes
131
5.20
Error on g * when variables changes
132
vii
5.21
Error on TOC* when variable x changes
133
5.22
Error on Q* when variable 7changes
133
5.23
Error on TOC* when variable 7changes
134
5.24
The effects on Q* and TOC* when parameters change
vui
136
LIST OF FIGURES
2.1
On-hand inventory of defective items
35
2.2
Nye and etal. (2001)'Model
41
2.3
Theoretical model of this study
49
3.1
Problem Description
55
3.2
Possible combinations of cost of quality curves
61
4.1
On-hand inventory of non-defective items for Model 1
66
4.2
On-hand inventory of non-defective items for Model 2
69
4.3
On-hand inventory of non-defective items for Model 3
72
4.4
On-hand inventory of non-defective items for Model 4
75
4.5
On-hand inventory of non-defective items for Model 5
78
4.6
On-hand inventory of non-defective items for Model 6
81
4.7
On-hand inventory of non-defective items for Model 7
84
4.8
On-hand inventory of non-defective items for Model 8
87
5.1
Search solution programming
92
5.2
Effect of errors on TOC(QV
96
5.3
Effect of errors on TOC(Q^) without variable C
96
5.4
Effect of errors on TOC(QV without variables C A F, andy
97
5.5
Effect of errors on (g*)
97
5.6
Effect of errors on (2*) without variables P, A x, and A:
98
5.7
Relationship between different levels of P and 2 *
100
5.8
Relationship between different levels of i? and 2 *
100
ix
5.9
Relationship between different levels of x and g *
100
5.10
Relationship between different levels of 7and Q*
101
5.11
Relationship between different levels of AT and Q*
101
5.12
Relationship between different levels of//and 0 *
101
5.13
The relationships ofQ* andx
110
5.14
The relationships of TOC* andx
112
5.15
The relationships of g * a n d / /
114
5.16
The relationships of r o c * and//
114
5.17
The relationships ofQ'^a.ndK
115
5.18
The relationships of r o c * a n d / :
116
5.19
The relationships of g * and P
117
5.20
The relationships of r o c * and P
117
5.21
The relationships of e * audi?
118
5.22
The relationships of r o c * audi?
118
5.23
The relationships of g * andx
119
5.24
The relationships of r o c * and X
120
5.25
The relationships of g * and 7.
121
5.26
The relationships of r o c * and 7
121
5.27
Error on Q* when variable C changes
123
5.28
Error on TOC* when variable C changes
125
5.29
Error on g * when variable F changes
125
5.30
Error on TOC* when variable P changes
126
X
5.31
Error on g * when variable//changes
127
5.32
Error on TOC* when variable//changes
127
5.33
Error on 2 * when variable AT changes
128
5.34
Error on TOC* when variable/: changes
129
5.35
Error on 2 * when variable P changes
130
5.36
Error on TOC^ when variable P changes
130
5.37
Error on Q* when variable/? changes
131
5.38
Error on TOC* when variable R changes
132
5.39
Error on 2 * when variables changes
132
5.40
Error on TOC* when variable x changes
133
5.41
Error on Q* when variable 7changes
134
5.42
Error on g * when variable P changes
134
XI
CHAPTER I
INTRODUCTION
1.1 Historv and Background of the cost of quality and inventory
The cost of quality (COQ) is a tool for companies to evaluate and improve the
performance in terms of cost and profit for years, and COQ is also an increasingly
important issue in the debates over quality. Traditional thinking assumed that as quality
improves, costs increase. That is, to improve quality, more testing and rigorous
inspection would be needed using more sophisticated monitoring equipment and
personnel. Today, however, the costs associated with poor quality are considered to be
more significant than previously acknowledged.
Quality costs are generally categorized into costs of prevention, appraisal, internal,
and extemal failure by Feigenbaum (1956). In the application of the Taguchi loss
function (1989), intemal and extemal failure costs are considered as part of the "loss to
society." In manufacturing environments, the most visible costs of intemal failures are
rework and scrap, which are usually available from the standard cost accounting system.
Furthermore, the extemal failure cost includes warranty cost, replacement cost, and repair
cost for retum items from customers. However, another variable, opportunity cost/loss,
plays an important role in COQ as shown in the case study of Sandoval-Chavez and
Bemvides(1998).
Another significant area involving COQ is inventory control, especially in
manufacturing systems. The control of inventory is a problem common to all
organizations in any sector of the economy. One of the major reasons for having
inventory is to enable an organization to buy, produce, or sell items in economic lot sizes.
There are two types of economic lot sizes, the Economic Order Quantity (EOQ) and the
Economic Production Quantity (EPQ).
Both the EOQ and EPQ models presented in text books are widely used by
practitioners as decision-making tool for the control of inventory. From Osteryoung's
(1986) survey of companies, he concluded that, in practice, the assumptions necessary to
justify the use of these EOQ models are rarely met. Ideally, all products are 100%
conforming, but it's almost impossible in real practice to obtain no defective items.
Therefore, the cost of defects, inspection, and rework should be considered in the EOQ
and EPQ inventory model. Those costs are part of the cost of quality (COQ), and have
relationship to the COQ in terms of prevention, appraisal, and failure costs as stated by
Feigenbaum (1956).
As stated earlier, COQ and inventory management have the relationships a
tremendous influence on the ultimate cost of a product, because they handle the
production costs and total flow of materials in an organization. Both the COQ and
inventory control are responsible for the planning, acquisition, storage, inspection,
movement, and control from raw materials to final products.
Both COQ and EOQ/EPQ have the same objective function which is to minimize
cost of manufacturing systems. All companies need to pursue the goal of the highest
output with the lowest input. However, previous researches in the COQ and EOQ/EPQ
are rarely linked together. Thus, this research is to develop and investigate the
relationship of the cost of quality (COQ) and the economic production quantity (EPQ)
both in theory and practical ways under the production system.
1.2 Problem Statement
Although many researchers claim to assess quality costs, often researchers
measure only what is visible in terms of quality, thus understating the true cost of poor
quality. Research in the area of the actual costs of prevention, appraisal, intemal failures,
extemal failures, and opportunity loss appears to be very limited. The integration of
COQ and EPQ should be able to link and classify each cost of quality in terms of
inventory management. In this research, the economic production quantity (EPQ) is
mostly used to study the relationship of COQ and inventory management.
The EPQ model has been widely used for more than three decades as an important
tool to control the inventory since the EPQ is powerful to help practitioners and engineer
to make a decision. However, the EPQ model did not represent the real world problem
in some situations. Regardless of such an acceptance, the analysis for finding an
economic production quantity has several weaknesses. The obvious is the number of
unrealistic assumptions which lead many researchers to make extensions in several
aspects of the original EPQ model. The cost of quality is one good aspect to be added to
the EPQ model since there are several costs incurred in the real world practice such as
cost of defect, inspection, and warranty. Many studies of cost of quality are discussed in
terms of several factors and hidden costs in quality control such as prevention cost.
appraisal cost, failure cost, inspection cost, lost of goodwill cost, loss sale cost, rework
cost, defect cost, equipment cost, machinery cost, storage cost, and labor cost.
There are a few articles which represent the inventory model with cost of quality
as stated earlier. Assume one scenario which the produced items are not always perfect,
and these defective items have to be reworked or scrapped in other production lines.
Furthermore, if the number of inspection stations is too few, that the investment of
prevention and appraisal are too low, the cost of defects (failure costs) might go high
because the defective items have already gone to the next station. On the other hand, if
the number of inspection stations is too many, the cost of quality might go high. The
failure cost can be distinguished to intemal and extemal costs such as the scraps, rework,
or warranty product. Furthermore, the opportunity costs/losses which are the loss of
goodwill and under/over production capacity are considered. Various intriguing
questions are brought to this point as follows:
Do the assumptions of EPQ and inventory model still hold?
What is the equilibrium point of investment cost of quality and cost of
inventory?
What are the affects of all quality costs in the inventory model?
-
What are the hidden costs in this scenario?
-
How tight of quality control policies should be applied?
These interesting problems and questions will lead to the study of inventory
models with cost of quality. This scenario resembles the real world practice which
results in questions leading to the new area of research.
1.3 Research Questions
The main questions of this research are as follows:
1. What are the variables in an inventory model in terms of the COQ?
2. How do we modify the classical EPQ mathematic model to include the cost of
quality?
3. What is the effect of COQ variables such as setup, inspection, defect, and
warranty costs in EPQ models that are based on minimizing total cost?
4. How do we classify the COQ in terms of the inventory management?
5. What is the optimal lot size of the EPQ model when the COQ frame work is
incorporated?
1.4 General Hypotheses
The general hypotheses for this research are as follows:
1. It is believed that the production lot size from the traditional economic
production quantity (EPQ) approach has the unequal value from those defined
by the new inventory model which considers the cost of quality.
2. The total cost of the traditional EPQ approach has the unequal value from the
new inventory model which considers the cost of quality.
3. The optimal production lot sizes for different quality conformance levels are
different. According to Porteus (1986), when the quality conformance level
decreases, the optimal lot size level increases.
4. The minimum total costs at the each quality conformance level are different.
1.5 Research Purpose
The purpose of this research is to extend the previous research in the economic
producfion quantity model by employing the knowledge of cost of quality such as
prevention, appraisal, failure, and opportunity costs in order to determine the optimal lot
size. Addifionally this research will examine the trade-offs between investments to
reduce the defect and failure costs, and the other operafing costs in order to find the
optimal investment point. Furthermore, the expected result will present
recommendafions on how to use this model in theorefical and pracfical ways.
1.6 Research Objectives
The objectives of this research are: (1) to investigate the effect of cost of quality
in the inventory model; (2) to determine the factors and costs which involve in this
scenario not only the general quality control and inventory costs, but also the hidden and
indirect costs; (3) to develop mathematical models under the behavior of cost of quality
and inventory model; (4) to determine the best quality policy, quantity policy, and
equilibrium point of investment and benefit in this scenario.
1.7 Relevance of this study
This research is relevant to both industry and academic. In the first part of this
research, the inventory models associated with cost of quality are defined, which the
characteristics are relied on the real world practice. Second, the statistical tests for EPQ
and new EPQ associated with COQ are performed.
1.7.1 Theoretical Research Needs
Sandoval-Chavez and Bemvides (1997) summarized several COQ models to be 6
major models. However, the COQ behavior which is based on inventory view has not
discovered yet. Thus, the investigafion of new approach of COQ curves will be needed
and explored in this research based on minimizing the total cost.
1.7.2 Practical Research Needs
The new mathematical inventory models are developed in this research project in
the resemble character to the real industry. The new defined cost elements of inventory
and quality, incorporation of EPQ approach, and its quantitative model will be provided
in this research.
1.7.3 Benefits of this Research
The benefits of this research are (1) presents literature review based on the cost of
quality behavior and inventory model which incorporates to quality perspectives, (2)
develops new mathematical inventory models associated with the cost of quality, and (3)
presents a theoretical, quantitative research approach for statistical tests of the cost of
quality in the inventory model.
1.8 Research Outputs and Outcomes
The outcomes of this research will be as follows:
1. The reviewed of the inventory models associated with cost of quality in
several ways such as classifying inspection, defect, and warranty costs in the
inventory models.
2. The extension of mathematical EPQ inventory models with considering the
cost of quality.
3. The optimal production lot size of the new EPQ which associates with COQ
based on minimizing the total cost.
4. The significant factors for the new EPQ model with considering COQ.
CHAPTER II
LITERATURE SURVEY
2.1 Introduction
Currently, the cost of quality is one of the most important tools in industries since
this tool has been widely used for more than six decades. The first time the term quality
costing appeared was in the 1930s (Crocket, 1935; Miner, 1933), but until the 1950s there
was no systematic approach for quality costing, as quality costs were considered to be
only the scrap, rework and the cost of running the quality department. The first attempts
to categorize the quality costs were made by Juran (1951) and Feigenbaum (1956).
During that time, quality costs were classified into three main categories: prevention,
appraisal and failure.
There are many attempts to improve prevention, appraisal and failure (PAF)
model in other perspectives as well such as design cost, intangible cost, and tangible cost
in PAF quality model. Dalghaard et al. (1992) introduced another classification of the
quality costs. They classified them as visible and invisible costs. As its name suggests,
invisible costs are the costs due mainly to the loss of goodwill and additional costs
incurred due to intemal inefficiencies. The cost due to intemal inefficiencies is a cost that
has not been studied in detail so far, and it will be examined later in this paper. Finally,
Kume (1985) explained the importance of hidden quality costs and their importance.
Since there are many categories of hidden costs in several areas, one of the important
areas conceming hidden cost is inventory in industries. The meanings of each category
in the COQ are as follows:
1. Prevention Cost - These are expenditures that prevent failures from occurring
and include employee quality training, process control, quality engineering, and quality
improvement projects.
2. Appraisal Cost ~ These are the costs incurred to evaluate the quality of
products. Examples of appraisal items are incoming inspection, testing, quality audits,
and evaluation of stock.
3. Intemal Failure Cost — Costs incurred as a result of defects prior to shipment to
a customer are classified as intemal failures. Some typical intemal failure items are scrap,
rework, downtime and overtime.
4. Extemal Failure Cost — Extemal failures are defects that are found after
shipment is made to the customer. These costs may include warranty, allowances,
retumed materiel, customer complaints, product liability lawsuits, and customer service.
5. Opportunity Cost/Loss -Mostly, opportunity costs/losses are intangible costs.
These costs may include customer satisfaction, goodwill of company, cost of lost sales,
undemtilization of capacity, and any cost which can not be defined as prevention,
appraisal, or failure costs.
In the inventory point of view, the traditional Economic Order Quantity (EOQ)
and Economic Production Quantity (EPQ) inventory models have not been included the
cost of quality. The neglect of quality cost in inventory model might be presented the
inaccuracy of the mathematical model. The inventory model with cost of quality has
10
been recently discovered, so there are just a few research papers which presented this
research area. Therefore, the research in this area is still needed to analyze and discover.
The quality-inventory relafionship is linked by somewhat measurement of cost of quality
and cost in inventory. To understand the basic definitions and history of quality and
inventory, it is also necessary to review the previous research from other researchers.
Therefore, the history and literature survey are presented in this secfion.
2.2 Theories and Historical Background in Cost of Ouality
There are many research papers in the cost of quality. However, according to
Sandoval-Chavez and Bemvides (1998), there are six primary theories in the costs related
to quality as follows: Juran's Model, Lesser's Classification, Prevention-AppraisalFailure Model, The economics of quality. Business Management and the COQ, and
Juran's Model Revised.
2.2.1 Six Primary Theories in Cost of Ouality
For the first cost of quality model, Juran (1951) described the cost of poor quality
as "the sum of all costs that would disappear if there were no quality problems" and
presented the analogy that poor quality and its related costs are "gold in mine". Juran
influenced the companies to minimize the non value added costs and waste that are
associated with poor quality. Also, Juran emphasized the need for quantification of the
quality costs in order of importance and the potential benefits of their reduction to be
shown. Quality costs could be used for the assessment of the quality control system and
11
progress made by the improvement process. Juran made the first step, describing the
importance of the quality costs, but it was not clear how to reduce them.
Feigenbaum (1956) classified the quality costs in the prevention, appraisal, and
failure categories. They emphasized the importance of the quality cost measurement and
reporting to the top management in order to influence the company's interest in quality
improvement. Furthermore, Feigenbaum (1956) showed that investment in prevenfion
costs could resuh in the reduction of appraisal and failure costs.
The business management and quality cost have been studied. "A management
perspective quality economics is more important than quality cosf was first introduced
by Kume (1985). Kume (1985) demonstrates the different strategies in specify products,
which the example from real world practice shows that a management perspective quality
economics is more important than quality cost. Kume presented some principles of
quality economics: (1) Minimum quality cost does not necessarily mean maximum profit,
(2) Minimum quality cost does not necessarily mean minimum product cost, (3) Losses
due to failure cannot be calculated only by failure cost, (4) The cost of marketing
research should be included in prevention cost, (5)Quality of design cannot be evaluated
by quality cost, and (6) The important thing about prevention and appraisal cost is not the
total, but the way the money is used. Kume also showed the importance of several
hidden costs which incurred beside the traditional prevention, appraisal, and failure costs
of quality. Kume presented an approach that the important thing is "the way of
prevention and appraisal activities are carried out, not the amount to spend on those
activities."
12
2.2.2 Opportunity Costs and Hidden Costs
Using PAF and opportunity costs to determine the accurate COQ has been
successfully used by Carr (1992). Carr also presented the management perspective like
Kume (1985), but Carr's research paper presented more applicable way in the real world.
Cost of quality was used as a tool in Carr's research paper to help manager to line up the
problem in each department at Xerox Company. The key factors in improvement the cost
of quality are the definifion and classification of quality which are costs of conformance,
cost of nonconformance, and lost opportunifies (opportunity costs).
According to Sandoval-Chavez and Beruvides (1998), they presented the using
opportunity costs to determine the cost of quality in a case study. They also presented the
strategic and economic importance of opportunity factors for a continuous-process
industry which located along the US-Mexican border. This research paper has been
accomplished with collecting data from the real company for six months. They classified
four costs of quality: prevention, appraisal, failure, and opportunity costs. They identify
costs of opportunity factors as poor delivery service, inadequate material handling, and
installed capacity undemtilization. Finally, they presented an empirical model that
expressed cost of quality (COQ) as a function of prevention, appraisal, failure, and
opportunity expenses. The final from this collecting data for six months showed that the
opportunity costs in COQ are necessary, and should not be ignored in the cost of quality
model.
The distinction between quality cost and quality loss is first presented by Giakatis
et al. (2001). The authors believe that a distinction must be made between quality costs
13
and quality losses. Instead of only considering the total quality costs, it would be better
for a company first to make a distinction between quality costs and quality losses and
then to try to reduce quality losses. The informative between difference quality costs and
quality losses is that the former adds value, while the latter does not add value and
sometimes reduces value. Giakafis et al. provided the sequence of steps in quality cost
reducfion including hidden costs in manufacturing loss and design loss.
2.3 Theories and Historical Background in Inventory
From the production inventory point of view, when a company orders too much
material, inventory tums and order-related costs are reduced while carrying and storagerelated costs are increased. Conversely, customer service level increases, while handling
costs decrease. Determining how much inventory to order is a fairly straightforward
decision. Any number of techniques can provide very precise order quantities. Knowing
when and how to apply the trade-offs and when to use which technique is the key to
ordering the correct quantity. "Correct order quantities" is defined as the quantity which
satisfies all of the company's various targets such as minimizing total costs. It is up to the
planner to determine which order quantity is most reasonable according to the business
and operating perspectives given the specific objectives to be achieved. This assessment
does not come from using techniques but from a knowledge and understanding of current
business conditions and management inventory targets. As business conditions and
inventory targets change, the planner may need to change the materials, parts, assembly,
and finished goods order quantity techniques as well. The inventory system needs to
14
support the planner's choices in this respect. The application of order quanfity techniques
typically varies based on the planning category within which the particular part or part
category falls.
The inventory model can be disfinguished by the character of demand. There are
two types of demand. If demand and lead time are treated as constants, they are called
deterministic demand. If they are treated as random variables, they are called
probabilisfic or stochastic demand which is discussed in section 2.3.7. Most of research
papers in this literature survey secfion are in the deterministic inventory model.
2.3.1 Notation
This section defines the notations used in the quality and inventory models.
A = the percentage of setup time between regular and rework production
Bj = the percentage of products retumed by the customer which is processed at rework
production.
B2 = the percentage of time "t" to scrap the customer retum product
C = unit production cost ($/item)
c = unit rework cost per item of imperfect quality ($/item)
D = demand rate in units per unit time
P = cost of customer retum (Cost of disposal, shipment, and penalty)
/ / = holding cost of perfect items per unit per year
h = holding cost of imperfect items being reworked per unit per year
h] == holding cost of imperfect items from products retumed by the customer
15
/ - inspecfion cost incurred with each inspection or cost of inspections in each unit
K = setup cost ($/time)
K} = quality investment cost = K/x
n = sample size of inspections
P = production rate in units per unit time
p = producfion rate of imperfect items in units per unit time (p = Px)
q^ on hand inventory level
qj = on hand inventory level at time ti
q2 = on hand inventory level at time ti
qs = on hand inventory level at fime between time ti and t2
Q = size of production mn
g * = optimal size of production run
R = rework rate of imperfect items in units per unit time
S] = unit scrap cost per item of imperfect quality ($/item)
iS'2 = the shortage cost per item per unit time ($/item/unit time)
t = unit time in one cycle
ti = unit time in periods i (i = 1, 2, 3, ....)
TC = total cost
TOC = the total cost per cycle (total annual cost) = TC/time in one cycle
TOC^ = the minimum total cost per cycle (total annual cost)
W= defect rate of defective items from end customers in units per unit time (W = DY)
X = the proportion or percentage of defective items from regular production (x is between
16
Otol)
7 = the proportion or percentage of defecfive items after distributing to end customers (Y
is between 0 to 1)
2.3.2 Economic Order/Producfion Ouantity (EOQ/EPQ) Models
In the deterministic demand inventory model, researchers have obtained the
classical economic order quantity (EOQ) and economic production quantity (EPQ) for
years. Both EOQ and EPQ are generally used to find the opfimal order quantity in order
to minimize the total inventory cost. The difference of EOQ and EPQ is about the
received lot. The EOQ model assumes that the entire order for an item is received into
inventory at one given time, while the EPQ model assumes that the item are produced
and added into inventory gradually rather than all at the same time like EOQ model.
This section discusses the EOQ and EPQ model which are widely discussed by
many textbooks and researchers such as Tersine (1994). The logic used by the EOQ
equation has good surface validity. Minimum total costs are achieved at the point where
the cost to purchase and order material match the cost to carry it. That is, carrying cost
for material may be incurred up to the point where it becomes more economical to place
another order. In the EPQ model, purchase cost will be replaced by production cost and
ordering cost is replaced by setup cost. The setup cost is the cost of the time required to
prepare the equipment or work station to do the job.
However, both EOQ and EPQ share many assumptions together. The conditions
under which the EOQ/EPQ equafions may be used are as follows:
17
1. The part demand can be extended accurately over the part's planning horizon.
Demand rate is constant and deterministic.
2. The production rate is known, uniform, and continuous.
3. Order, production, carrying and setup costs can be accurately determined for the
part, and are known and fixed.
4. The unit variable cost does not depend on the replenishment quantity.
5. The cost factors do not change appreciably with time.
6. The replenishment lead time is zero.
7. No shortages/stockouts are allowed.
8. For EOQ, the entire order quantity is delivered at the same time, and for EPQ,
items are produced and added to the inventory gradually rather than all at once.
9. The item is treated entirely independently of other items.
10. There are sufficient space, capacity, and capital to procure the desired quantity.
EOQ model:
TCiQ) = C D ^ ^ ^ ^
(2.1)
TC{Q*) = CD + HQ*
(2.3)
EPQ model:
rcie^,CD^^^"S!lzdl
K^J
Q
18
2P
(2.4)
e«= r"'^
TCiQ'^) = C / ) + ^ ^ * ^ ^
(2.5)
"^^
(2.6)
There are many areas of inventory models for almost forty years. Those research
areas have been discovered in several ways in order to make the model closed to the real
world pracfice. One of the interesfing areas is the integrated inventory and quality
aspects. These aspects give rise to many altemafives of the inventory models. To
distinguish the different types of works done in this area, we propose that the literatures
be divided major categories based on the extension areas as follows:
Inventory policies with defective items.
Inventory policies with imperfect process.
Inventory policies with quality costs.
Inventory policies with repair/rework and warranty.
Inventory policies with stochastic demand with remanufacturing and rework.
2.3.3 Inventory policies with defective items
By EOQ/EPQ assumptions in previous section, stockouts are not permitted in the
EOQ/EPQ models. However, Shih (1980) shows that the defective items can affect the
stockout in EOQ/EPQ model. Usually, by traditional definition, stockout or shortage
occurs when demand exceeds the amount of inventory on hand. It is not, however, the
only way stockout could occur. Stockout can be occurred by the unexpected presence of
defective product in inventory. The unknown and undiscovered defective items in an
19
accepted lot would reduce the amount of inventory on hand which is below the optimal
order quantity that already calculated before. This leads to the stockout problem since
the EOQ neglects the effect of defective items in the on hand inventory. Shih (1980)
considers average inventory by considering the time which defective items are presented.
When defective items are presented in a lot, the average inventory carried in an inventory
cycle depends on when those defective items are found and removed from the inventory.
Defective items can be sorted out either by inspecting all items on receipt of a lot, or by
inspecfing each item when it is brought out for sale. Shih (1980) not only extends EOQ
with defective items, but also single-period inventory model with probabilistic percentage
of defective items in probabilistic model.
Osteryoung (1986) also think that, in practice, the assumptions necessary to
justify the use of EOQ models are rarely met. To provide mathematical models that more
closely conform to real-world inventories and respond to the factors that contribute to
inventory costs, the models must be altered or extended. Many researchers changed and
added parameters and variables in the traditional EOQ model. For instance, Esrock
(1986) discussed reducing setup times. Porteus (1985) also examined the setup time with
tradeoff between the investment costs needed to reduce the setup cost and the operating
costs identified in the EOQ model. Schwaller (1988) also presented the EOQ model with
inspection costs. The model focused on the impact of inspection costs on lot size when a
known proportion of defectives which must be removed are presented. Schwaller (1988)
had some interesting assumptions as follows:
Demand is assumed constant for each of the model considered.
20
End units are inspected when received.
-
Receipt may be instantaneous or at a constant rate unfil the EOQ value is
realized.
-
The supplier does not charge for defective items that are discarded after
inspection.
Schwaller (1988) created three models to obtain the annual cost functions and
optimal quantities. Those three models are Model I: Instantaneous receipt with
replenishment upon depletion; Model II: Non-instantaneous receipt with replenishment
upon depletion; Model III: Instantaneous receipt with backlogging. For each of the three
models, the total annual cost function will include the ordering, carrying, and inspection
costs. For other strategies, Larson (1989) presented several new EOQ modifications.
The models recognize that a purchasing or procuring entity has several alternatives for
both (1) inspection of incoming materials and (2) handling of defective material. This
research paper shows three ways of inspection plans: no inspection, 100% inspection, and
sampling plan. The objective function is to minimize the total cost of EOQ for each
inspection plan.
An extension of the EOQ production model (EPQ) based on damage costs is
presented and the relevant costs of damage and their effects on stockholding costs are
discussed by Chyr et al. (1989). This model added damage cost to the traditional EPQ
model. By definifion of damage cost in this research paper, materials, work-in-process,
and finished goods in stock might lead to damage costs arising from breakage, quality,
and pilferage. These costs might be incurred only at a single point in time, but their value
21
may be very high. Besides, damage costs may arise many times a year, and although
they are difficult to determine, they can not be ignored. Damage costs in this model
measured by the average annual damage rate which depends on annual total inventory
level. When damage occurs, companies may lose sales opportunities through stockouts.
A new EPQ formula including damage costs is developed and a comparison between the
conventional concept and the new concept based on annual total cost is made. The
analysis shows that, if the costs of damage to stock are taken into account, the computed
EPQ is smaller, and the conventional EPQ is not the opfimum solution.
The effect of dynamic process quality control on the economic of production and
the total system cost is studied by Goyal and Gunasekaran (1989). A mathematical
model is presented for estimating the economic investment in quality (EIQ) and the
economic production quantity (EPQ) in a multi-stage production-inventory system. The
basic criterion considered for the determination of EIQ and EPQ is the minimization of
total costs. The feature of this research is to consider the aspect of dynamic process
quality control, namely, 'quality at the source', by monitoring the quality of the product as
the process continues and stopping the process if it goes out of control. Goyal and
Gunasekaran (1989) point out some interesting assumptions which are the inventorycarrying cost is directly proportional to the investment in inventory; number of reset-ups
is inversely proportional to the investment in quality improvement program.
Recently, the EOQ and EPQ inventory models with considering defective items
are presented by Salameh and Jaber (2000). This research paper hypothesizes a
production/inventory situation where items, received or produced, are not of perfect
22
quality. Items of imperfect quality; not necessarily defective; could be used in another
producfion/inventory situafion, that is, less restrictive process and acceptance control.
This paper extends the tradifional EPQ/EOQ model by accounting for imperfect quality
items when using the EPQ/EOQ formulae. This paper also considers the issue that poorquality items are sold as a single batch by the end of the 100% screening process. A
mathematical model is developed and numerical examples are provided to illustrate the
solution procedure. The actual mathemafic model is based on EOQ with imperfect
quality items which received from the vendors. The objective function is to maximize
the profit of this inventory system.
The resuh of Salameh and Jaber (2000) shows that the economic lot size quantity
tends to increase as the average percentage of imperfect quality items increase. This also
shows the contradiction with other research that the reducing the lot size quantity as the
average percentage of imperfect quality items increase.
The traditional EOQ and EPQ model have been extensively studied and
continually modified in many ways as discuss in previous section. Another way to
extend EOQ and EPQ models is to include upstream and downstream of the business
chain including intermediary firm. The early paper survey of inspection and two echelon
inventory model with intermediary firm is Chen and Min (1991). Intermediary firms are
economic agents that purchase from mostly small and numerous independent producers
and sell to other firms or to the public. Chen and Min (1991) invesfigates how
intermediary firms can optimally determine both selling quantity and purchasing price of
a product. By incorporating the special stmcture of intermediary firms' environments and
23
by modifying the conventional economic order quantity (EOQ) model accordingly, the
authors provide opfimal decision rules regarding the selling quantity and purchasing price
for intermediary firms under profit maximizafion. Goyal et al. (1993) also extends EPQ
in different ways which are in the area of lot size for muhiple items in a muUilevel
manufacturing system. The objective of Goyal et al. (1993) is to minimize the total
system cost, consisting of the following cost elements: set-up cost, in-process inventory
carrying cost, owing to processing of products, in-process inventory-carrying cost, while
waiting for batches, and finished product inventory-carrying cost.
One extension of Chen and Min (1991) works in two echelon model is Huang
(2002). Huang (2002) presented the EOQ for the integration vendor-buyer cooperative
inventory model (two echelons) for items with imperfect quality. This research paper
develops a model to determine an optimal integrated vendor-buyer inventory policy. The
objective is to minimize the total joint annual costs incurred by the vendor and the buyer.
The current model in this article extends the integrated vendor-buyer inventory model by
accounting for imperfect quality items. This model considers a simple and practical
situation where the delivery quantity to the buyer is identical at every replenishment. The
expected annual integrated total cost functions for buyer and vendor are derived, and an
analytic solution procedure is proposed to determine the optimal policy.
The inventory policies with defective items have been applied to many areas
including fashionable items such as clothes and shoes. According to Hariga and Azaiez
(2001), the primary purpose in this research paper is to determine optimal ordering
policies of style goods (fashionable items) in the presence of defective units, where the
24
vendor has to handle both types of stock, first and second class items, at a primary and
secondary market respectively. The authors also provide tools for the vendor to select the
appropriate arrangement to handle defective units. The authors start by investigating the
case where the vendor has no control on the prices in each of the markets. Then, they
consider the integrated inventory-pricing policies to help the vendor decide on selling
prices of both classes of products. They consider this problem as the single-period
framework since they make assumptions the fashionable items that have to be sold during
a short period of time.
2.3.4 Inventory policies with imperfect process
The primary focus for the paper in this section is to incorporate imperfections in
the production process into the classical EOQ/EPQ models. Recent research analyzing
the relationship between production lot sizes and imperfect production processes have
largely centered around two key issues: process deterioration and yield, and machine
breakdowns and repairs. Porteus (1986) was perhaps the first to model the relationship
between production lot size and process deterioration. While the imperfect process
occurs, defective items are produced. Product quality, however, is not always perfect and
actually depends on the state of the production process, which may shift from an "incontrol" state to an "out-of-control" state and produce defective items (Lee and
Rosenblatt, 1986). When the production process was in an "in-control" state, items
produced maybe of high or of perfect quality. As time goes on, the process may
deteriorate and produce some defective items.
25
Porteus (1986) presented the EOQ with imperfect production processes.
Production processes will produce defective items while processes are "out of control".
While producing a single unit of the product, the producfion process (machine) becomes
"out of control", and begins to produce defective products, with probability a where a>0.
After the process goes out of control, h remains in this state while processing the
remainder of the lot. This assumption parallels the inspection policy suggested by Hall
(1983) whereby only the first and the last pieces of the lot are inspected. Let Q denote
the lot size, S = ordering cost, and a'=\-a. Portues' modified EOQ model has the
following total cost function {TC (Q)}:
rC(0 = « + i S ^ , r f _ f ^ ^ i l z i ^
(2.7)
This research uses approximations when assuming that a is close to zero.
l n a ' « - a / a ' , a / a ' « ^ , (a')^ ^ e^'"^'^^ ^ { ( l n a 0 2 f
^3.8)
Porteus obtains an approximate total cost per unit fime of
TC{Q) = ^
+ ^{H + cda)
(2.9)
Thus, an approximate optimal lot size is
e*=j-^
(2.10)
V / / + cda
Porteus (1986) uses these total cost and lot size approximations to derive some
properties of the adapted model and to demonstrate the benefits of producing in lot sizes
smaller than those suggested by a traditional EOQ model.
26
Lee and Rosenblatt (1986) also studied the effects of an imperfect producfion
process on the optimal producfion cycle time. The system is assumed to deteriorate
during the producfion process and produce some proportion of defective items. However,
Lee and Rosenblatt (1986) assumed that the duration of the "in-control" state is a random
variable with an exponenfial failure time distribufion. Both of Portues (1986) and Lee
and Rosenblatt (1986) concluded that managers should use a smaller lot size or
production mn times since these lead to fewer defective items. The optimal producfion
cycle is derived, and is shown to be shorter than that of the classical EOQ model. The
analysis is extended to the case where the defective rate is a function of the set-up cost,
for which the set-up cost level and the production cycle time are jointly optimized. The
case where the deterioration process is dynamic in its nature is also considered in this
paper.
Guo and Lewis (1994) also presented the EPQ with imperfect production
processes. This research is extended from Porteus (1986) and showed more accurate
ways to calculate the production lot sizing. Moreover, the rework cost has been included
in this model. The lot sizing technique under this study is the EPQ model with d as the
constant demand rate, P as the finite production rate (P>d), K as the setup cost, and H as
the holding cost per unit per unit time. Each of these parameters is assumed to be strictly
positive. While producing a single unit of the product, the production process (machine)
becomes "out of control", and begins to produce defective products, with probability a
where a > 0. Each defective unit costs an additional c to correct. They assume that
defectives are corrected instantaneously, once discovered. Let Q denote the lot size and
27
a'=l-a. Then the expected number of defectives in a lot size Q is
Q-a'(l-a'^)/a.
Hence, the equation would be:
Total cost per unit time (f (Q)) = Set up cost + Holding Cost + (Rework cost/unit * the
expected number of defectives per lot * the number of lots per unit time)
Or,
/(e).^,^,,rf^^'''-'0-^'°).„i,„,e^(P-d)/p
Q
1
aQ
(2.11)
After differentiating the above equation by Q, the opfimal lot size Q* satisfies:
^—-\2
(Q a'
Ina' + a-a'
) = Sd
(2.12)
a
After applying Porteus' approximation from equation (2.10), then f(Q) equation will be:
f(Q) = — + — {He + dca), where 0 = (P-d)/P
(2.13)
In this paper they refer to the EPQ model without quality consideration in section
2.3.2 as the standard EPQ model, to equation (2.11) as the exact EPQ model, and to
equation (2.13) as the approximate EOQ model. The total costs per unit time (f(Q) under
the approximate and exact methods increase as a increases, and when a is close to zero,
the total costs incurred with the exact and approximate solutions are fairly close.
However, as a increases, the difference between the approximate and exact
solutions becomes quite large. They also show that the optimal lot size is always smaller
than that of the standard EPQ model, and the defective probability significanfiy affects
the optimal lot size.
28
According to Lee and Park (1991), they extended the imperfect production
process on the EPQ model with two kinds of cost: reworking and warranty costs. During
the producfion run the process is assumed to deteriorate, so defective items are produced
and sold. The defecfive products are reworked at some cost before being shipped, or if
passed to the customer, incur a much larger warranty cost.
Another area of imperfect processes from Lee and Srinivasan (2001) considered
the unreliable production facility. This means that the facility is assumed to deteriorate
while it is in operation with an increasing failure rate. The re-setup and maintenance
machine have been applied to restore the facility to its original condition. However, Lee
and Srinivasan (2001) did not consider the stock out inventory, which meant the demand
during the stock out period will be assumed as the loss sale. A production run is inifiated
as soon as the inventory level drops down to zero, and it will continue until the inventory
reaches a predetermine level (S). If the facility fails during the production run, it is
minimally repaired and put back to the process. The facility is set aside for preventive
maintenance every N production runs. The authors show how to specify both the
inventory level S and the number of preventive maintenance N carried out. The objective
is to determine the optimal control policy (S, N) that minimizes the average cost of
operating the facility per unit time. This model is very useful to analyze the tradeoffs
involved in balancing preventive maintenance durations and costs against the cost of lost
sales, setup costs, and holding costs.
29
2.3.5 Inventory policies with quality costs
Most research indicates that the relafionship between inventory and quality is
important. In 1956, Feigenbaum stated that "a certain hidden and non-productive plant
exists to rework and repair defects and retums, and if quality is improved, this hidden
plant would be available for increased producfivity". From this statement, we can see
that the related quality costs are involved with producfivity directly. This can lead to the
production in inventory model especially in economic productivity model (EPQ).
The relafionship of the setup cost (Quality cost) and EOQ model is first discussed
by Schonberger (1982). Schonberger (1982) illustrated the tradeoffs associated with
decreasing the setup cost in the classical EOQ model. As Schonberger (1982) and Hall
(1983) make clear, the benefits of reducing the setup cost transcend the benefits identified
in the EOQ model. Setup time is defined as the time it takes to go from the production of
the last food piece of a prior run to the first food piece of a new production run. There
are many benefits of setup time reduction. These benefits include small-lot production
capabilities which yield savings in storage, handling, and inventory carrying costs;
reduced lead times; increased quality; increased flexibility; and increased capacity.
However, many of these benefits have only been expressed qualitatively, so it would be
hard to add some qualitative benefits into the mathematical model. Presently, there are
relatively many models which attempt to justify setup time reduction in quantitative
terms. Related works in which the impact of reducing setup costs on the economic order
quantity (EOQ) model have been studied and presented by several authors. According to
Porteus (1985), the investing to reduce the setup cost and inventory cost are tied up by
30
discussing at the investment cost of quality in the related inventory costs such as
operafing cost, setup cost, and holding cost. The goal of this paper is to begin to provide
such a framework. The framework developed identifies only one aspect of the advantages
of reducing setups, namely reduced inventory related operating costs. The approach
taken in this paper introduces an investment cost associated with changing the setup level
and adds a per unit time amortization of this cost to the other costs identified in the
standard EOQ model.
Later in 1993, Trevino et al. (1993) presented the mathematical model for the
economic justification of setup time reduction. Trevino et al. (1993) stated that "Setup
cost, not only includes changing fixtures, dies, and /or tooling, but also tear down,
cleanup, inspection, trial mns, and any material handling, administrative work, idle time,
etc. that occurs between the production of good parts. The objective of this mathematical
model is to define a total relative cost function which incorporates most of the cost
elements from the models previously from other authors, and the cost of quality.
Moreover, a methodology has been developed to determine a continuous function relating
setup time reduction to investment cost. This investment cost covers costs for personnel,
time, training, and equipment. Any benefits resulting from a reduction in setup time are
quantified by considering inventory carrying cost, storage cost, setup cost, and quality
cost. Setup cost in this model is a function of five basic factors: demand, lot size setup
operator burden rate, number of setup operators, and setup time. The results based on the
application of the justification model, and the following generalizations has been made
by Trevino etal. (1993)
31
-
Investment in setup time reducfion is not always justified based on one
product and one type of setup for a particular machine.
-
Large increases in demand, part cost or expected number of defectives per lot
are necessary before additional investments can be justified for setup
reduction.
-
Reducing lot size with reducfions in setup time may increase the optimal
percentage of setup time reduction.
-
The relafionship between percentage of setup time reduction and investment is
exponential.
-
Changes in cost per square foot of storage have no impact on the optimal
percentage of setup time reduction.
Reductions in lot size can be economically justified with reductions in setup
time.
Recently, the research on the set-up time reduction in EPQ is presented by Kreng
and Wu (2000). In this research, two analysis models have been developed to decide
simultaneously the optimal lot size and the optimal set-up time reduction ratio in an EPQ
environment without back order. This research paper also studies EPQ system consisting
of both single item and multiple items. The set-up time reduction ratio is used as
decision variable under various cases of demand in the EPQ model. In such an EPQ
model, very few studies have involved how the set-up reduction rate affects the lot size
and total operational cost. In order to consider the total operation cost, this study
attempts to find the optimal set-up reduction rate to acquire an economic production
32
quantity in the case of a single item as well as multiple items. The primary contribution
of this study is to model the increasing effective capacity in response to demand change
by considering the optimal set-up reduction rate and lot size. The authors assume that the
set-up cost is linearly related to the set-up fime. While the related costs are available, the
production-inventory system can determine the optimal set-up fime reducfion ratio that
the system could minimize the total annual cost.
The relationships of inventory and quality costs are not clear in terms of quality
improvement, so Porteus (1986) begins to address the benefits of improved quality
control. With the model that is postulated, improved output quality (percentage of units
produced that meet specifications) can be achieved simply by reducing the lot size since
Schonberger (1982) stated that reducing setup cost can also improve output quality,
because it further reduces the optimal lot size. Porteus (1986) has introduced a model
that shows a significant relationship between quality and lot size which is shown in
previous section. For situations in which this relationship is valid, taking it into account
results in reducing the lot size and decreasing the fraction of defective units. The model
procedure is "while producing a lot, the process can go 'out of control' with a given
probability each time it produces another item. Once out of control, the process produces
defective units throughout its production of the current lot. The system incurs an extra
cost for rework and related operations for each defective piece that it produces. Thus,
there is an incentive to produce smaller lots, and have a smaller fraction of defective
units." The paper also introduces three options for investing in quality improvements:
33
reducing the probability that the process moves out of control; reducing setup costs; and
simultaneously using the two previous options.
According to Porteus (1990), lot sizes should be reduced to compensate for poor
quality if no effective inspection is possible. This note introduces an inspecfion delay
time, measured in units produced after an inspection is made until results are known. If
the inspection delay is negligible, then the problem reduces essentially to two separate lot
sizing problems: the classical EOQ lot sizing problem and an inspection lot sizing
problem. If the delay is great, then only one inspection should be made and the lot size is
as given by Porteus (1986).
2.3.6 Inventory policies with repair/rework and warranty
In this section of the survey, the EOQ and EPQ models are discussed. The earlier
work in the inventory policies and inspect are presented in the previous section. The
earliest work of the inventory model with rework is Goyal and Gunasekaran (1989),
which presented the effect of dynamic process quality control on the economics of
production. One of Goyal and Gunasekaran (1989) approaches is focused on the re-setup
time and cost that leads to repair and rework research papers in early 2000s.
Hayek and Salameh (2001) clearly presented EPQ with defective rate, rework rate,
and repair cost. This paper extended the EPQ model which studies the effect of imperfect
quality items on the finite production model. When production stops, defective items are
assumed to be reworked at a constant rate. The percentage of imperfect quality items is
considered to be a random variable with a known probability density function (uniform
34
distribufion). The producfion rate is linearly affected to the defective items which depend
on the proportional value of defect in the production system. The demand for the inifial
perfect items and the perfect items being reworked is confinuous during the cycle. The
optimal operafing policy that minimizes the total inventory cost per unit time for the
finite production model under the effect of imperfect quality is derived where shortages
are allowed and backordered.
Figure 2.1 On-hand inventory of defective items.
Hayek's model uses on-hand inventory to develop a mathemafical model by using
each t period to construct the holding costs. Then all costs including rework and
defective items are in function of quantity (Q). Finally, the expected values of optimum
cost and lot sizing are determined.
35
Another interesting area to integrate the inventory system is warranty of products.
Almost all products whether sold directly to the customer or to a producer for assembly
into a consumer product now. carry a warranty of some kind. Warranty is an important
element of marketing new products as better warranty signals higher product quality and
provides greater assurance to customers. Servicing warranty involves additional costs to
the manufacturer and this cost depends on product reliability and warranty terms.
Product reliability is influenced by the decisions made during the design and
manufacturing of the product. This implies that warranty can be viewed as a link to
integrate the different stages of manufacturing - design, engineering, production,
marketing, and post sale service - in an effective manner. As such warranty is very
important in the context of new products. Recently research articles, emphasize the
growing importance of this subject to both consumer and producer. Objective
determinations of warranty costs will help manufacturers plan operations more
effectively since an accurate knowledge of warranty costs allows more accurate profit
expectations which may, in tum, lead to unanticipated marketing advantages. As same as
other production systems, most inventory models in the past are also neglected the
warranty cost and reserve items in order to pursue the optimum inventory model in the
economic production quantity (EPQ).
The first research paper which concems on the warranty cost is from Menke
(1969). Menke (1969) considered objective methods of calculating warranty reserve
funds for the expensed warranty for non-repairable products where an explicit warranty is
in force. It is proposed that warranty costs be treated as manufacturing costs and be
36
included in the final price of a product to the extent the product pricing structure will
allow. Menke (1969) also stated several vital questions from manufacturer as follows:
- How much product cost increase is required to cover the risk and can the pricing
structure absorb all the added costs for warranty reserves?
- If not, how much are warranty claims going to cost?
- Too little reserve results in unexpected reduction of profits; too much is liable to
make sales price noncompetifive with resultant dilufion of sales volume and profit. Then
how much reserve should manufacturers prepare?
Wang (2001) is another research paper which extended the Porteus (1986) model.
This research paper addressed the imperfect processes which produced defective items.
Moreover, this research concerns on both rework and warranty costs. The objective of
this paper is to determine the production lot size while minimizing the total cost per unit
of time. Various cases are presented one of which is Porteus' model.
Wang assumes that at the beginning of the production cycle, the production
process is "in control". Then while producing t h e / unit item, the production process
shifts from the "in control" state to the "out of control" state which begins to produce
defective item with probability a^ = P(M=j), where Mis the total number of items needed
to produce the first defective item from the beginning of the production run. Let^j =
P(M>j). That is, Aj is the probability that the produced items are defective-free is larger
thanj. The domain of Aj is {0, 1, 2,...} and that 1 = Ao> Ai > A2 >.... This means the
process reliability Aj decreases with the number of produced items j . It is obvious that
aj = Aj-i - Aj with domain {1,2,....}. Let
37
Sr = cost of a defective item before sale ($/unit),
kn
dQ
is the number of items sold at the end of a producfion in a lot size Q
\x\= the smallest integer not smaller than x.
T = production mn time in a cycle, where T = Q/P
Ci = unit inspection cost of evaluating the quality of the product, and
Sw = cost of a defective items after sale ($/unit).
Then, the total cost per unit of time consists of the usual setup and holding costs
plus the product inspection cost, the defective item cost includes the reworked cost before
sale, and the reworked cost after sale becomes
rc(2,{^,) = ^ + ^ ( i - | ) + ^ * | | t ( ^ ^ . _ , - 4 ) [ ( A : „ - / + i)5„+(e-^„)5j|
+ | ; ( ^ ^ . . - ^ , ) ( e - / + l)5,+ | ; c i f o r Q = l , 2 , 3 , . . .
y=A„+l
(2.14)
7=^„+l J
The optimal production lot size Q can be obtained by differentiating the above
equation by Q. However, it is difficult to obtain Q so this research creates an algorithm
which is used to compute Q .
According to Murthy and Djamaludin (2001), their research paper presented a
warranty management system and an integration of warranty with quality and
manufacturing to assist in decision making at the different stages and discusses the
elements of the system and the management of information.
38
2.3.7 Inventory policies with stochastic demand with remanufacturing
From research in previous sections, they mostly consider only EOQ or EPQ
models, which is deterministic demand. Another research area in inventory is pointed out
to stochastic demand.
Boucher (1984) has pointed out the important part of work-in-process (WIP), and
by not including WIP inventories, previous existing models neglect one of the most
significant cost justifications for setup reduction investment. In group technology
production, there is a traceable relationship between lot size and work-in-process
inventory. This paper explores this relationship and describes an economic lot sizing
model appropriate to group technology. The model minimizes the sum of setup cost and
work-in-process and finished goods carrying cost with stochastic demand.
Karmarkar (1987) developed a model of a manufacturing operation that captures
WIP costs. The total lead time taken to manufacture a product is usually an important
consideration. Long lead times impose costs due to higher work-in-process inventory,
increased uncertainty about requirements, larger safety stocks and poorer performance to
due dates. Traditional lot sizing models ignore lead time related costs, although there are
systematic relationships between lot sizes and lead times. In this paper, the relationships
between lot sizing, manufacturing lead times and work-in process inventories are
illustrated through standard queuing models which investigate congestion phenomena
and the resulting effect on waiting times. Subsequently, the implications for lot sizing
decisions are briefly discussed. These ideas are most applicable to manufacturing
facilities which exhibit substantial queuing and where batching is a realistic option. The
39
opfimal batch size could be determined, which minimized the sum of setup and inventory
costs.
Latest work on work-in-process (WIP) inventory with stochastic demand is Nye
et al. (2001). Since setup time reduction in manufacturing operations is widely
recognized to provide significant benefits in areas such as cost, agility and quality. This
paper not only uses the investment in setup fime reduction as the main goal, but also
includes the queuing behavior to predict Work-In-Process (WIP) holding costs as a
function of batch size and setup times. This paper uses both an M/G/1 queuing model to
predict WIP levels for holding costs calculation, and also determines the optimal
investment in setup reduction. The WIP holding cost is very important factor to
determine the total cost in the model since other papers neglected this cost, so the result
in the real world which is not included the WIP holding cost seems to be inaccurate. The
objective function of this paper is to minimize the expected total cost per period of the
single manufacturing system which is included the WIP holding cost and investment of
setup reduction cost in the objective function. Furthermore, this paper also determines
the setup time and batch size as the decision variables of the objective function since
these two variables are the functions of WIP levels.
According to Nye et al. (2001) article, they represent the basic one servermanufacturing cell. A server processes work one batch at a time, with each batch
consisting of a number of identical units as shown in Figure 2.2. Batches of work enter
the manufacturing cell and, if the server is busy, must wait in a queue. Upstream
operations are aggregated as an n arrival process to this cell, and downstream operations
40
are ignored by assuming that once a batch is completed it leaves the cell and has no more
effect upon it. As each batch enters the server, it causes a new setup of the server. Once
the setup is finished, processing of the batch begins. The server processes each unit in
the batch sequenfially, and once complete, the batch leaves the system. Setups are
assumed to require a fixed amount of time for each batch, but that time can be reduced by
investing in setup reduction.
The arrival of jobs in a manufacturing system of this paper is Poisson process.
Processing times are assumed stochastic and area described with a general service time
distribution. From these two assumptions, this system can be represented as an M/G/1
type queue, whose steady-state flow time is known analytically.
Operating Cost
Investment Level
Economic
Model
•
Setup Time
WIP Level
©
D
Batches Enter
Queue
Batches Leave
Server
Figure 2.2 Nye and et al. (2001)' Model
The factory overhead costs are assumed to be fixed with respect to changes in
setup time and batch size, and thus do not affect optimization problem. Similarly, under
41
the assumption of fixed demand, raw material costs, scrap, and rework levels are
invariant under changes to setup times and batch sizes.
Setup costs in this problem are fixed although the setup time reducfion is varied.
On the other hand, we can say that the setup costs will not be proportional to setup time.
Even though batch sizes may change, the unit processing time is fixed, so the total
processing time per period is also fixed. Then demand in this problem is assumed to be
constant per period. The only holding cost in this manufacturing cell is WIP hold cost
since the finished good holding cost is not counted for this manufacturing cell and
considered as very small when compares to the WIP holding cost. The investment in
setup reduction function is originally assumed to be linear function, and the interest rate
of investment cost is also considered.
Another inventory area which uses the same queuing model approach to
determine the manufacturing time is the impact of response time on retailer inventory.
When retailers receiving items from a manufacturer carry inventory to meet customer
demand, as items are sold, a retailer orders new items to replenish the inventory. Once an
order is placed, there is a time taken for the items to be delivered to the retailer. This
time is the manufacturing response time. It includes processing, production, and delivery
times. These different components of time can result in response times that are long and
uncertain since it is stochastic demand inventory system. This research paper develops a
queuing model for analyzing how manufacturing response time affects the inventory
needed at retailers to meet demand. The model accounts for variability in response times
and allows for products to be delivered to a retailer in a different sequence than they were
42
ordered. Simple equafions are derived for the average inventory in terms of demand and
response time parameters.
An important assumpfion underlying most of the inventory models in the
literature is that the lot ordered will not contain any defective units. In reality, this is
often not tme. These defective units could be a resuh of imperfect production of the
suppliers, pilferage, and/or damage in transit. The presence of defective units in orders
would have an impact on the on-hand inventory level, the number of shortages and the
frequency of orders in such system. Models that take into account the possibility of
having defective items in the lot ordered are thus important for effective control of such
systems.
First paper with stochastic demand with defective is Shih (1980). Shih (1980)
analyzed a single-period inventory model with random demand, zero lead time and no
ordering cost, similar to the newsboy problem, but allowing for defective units in the lots
purchased. Shih (1980) analyzed both the cases in which the percentage of defective
units is a constant and a random variable with known probability distribution.
Later in 1987, Moinzadeh and Lee (1987) presented the stochastic demand
models with defective units and nonzero replenishment lead times. This article deals
with a continuous-review inventory system with Poisson demand arrivals and constant
reorder time. Items in reorder lots may not be of perfect quality. Upon arrival of an
order, the items are inspected and defective units are discarded. If the demand is not
satisfied, the backorder supply is allowed. They also study the operating characteristics
of such an inventory system. Both exact and approximate procedures are presented. The
43
performance of the approximation scheme is evaluated by comparing the costs of the best
ordering policy obtained by the approximation and those obtained by the exact model, as
well as the respective average backorder levels.
The problem of determining optimal inventory levels in a repair/rework
environment characterized by stochastic demand, stochastic lead times, and multi-item
inventories are very complex task. Consequently many current solution methods for
determining optimal stocking quantities are based on the simplifying assumption that
parameters are known deterministically. Although sensitivity analysis has been
performed on inventory models in stochastic environments, an area of research that has
not been adequately addressed is the effectiveness and sensitivity of various inventory
models and related parameters to a stochastic repair/rework environment.
The stochastic demand with defective and rework/replacement the item is
presented by Chow (1992). This article uses the practical assembly process as a sample
to analyze. Test operations are often introduced to ensure product quality. After having
been rejected by a test operation due to a bad component, a product might be sent to a
rework station for component replacement. Three different policies are identified. Under
the first policy, all bad components will be replaced by untested ones, and therefore, the
product must be retested. The second policy will replace all bad components with good
ones if available. In this case, no addifional test is required. However, if the number of
bad components is greater than the inventory level, all components (both good and bad)
will be replaced by untested ones. Then the product is sent for retest and good
components are placed in inventory. The third policy always replaces the bad ones with
44
good ones. It is assumed that all (good) replacement components come from an
independent source. If there are not enough good components for replacement, the
product must wait. The paper investigates all three policies, using stochastic models. The
performance of a policy is dependent on yield, test time, product configuration, and
production demand. A good choice should consider the tradeoff between producfion lead
time and inventory cycle.
The practical problem in the real world has been invesfigated. One of those is to
determine the effectiveness of inventory stocking methodologies in repair/rework
operations at a United States Army Depot which is accomplished with Humphrey et al.
(1998). Case study information for the research is obtained from historical data as well
as through dialogue with personnel at the depot to identify existing methodologies and
unique cost structures. These data directly support a case study for comparing altemative
inventory models. The primary objectives of this study are twofold. First, they seek to
perform a robustness study on the performance of existing inventory stocking policies in
stochastic repair/rework situations. To achieve this objective, they experimentally isolate
and vary key modeling parameters and make use of computer simulation to evaluate
several performance measures (including total inventory cost, backorder delays, and the
percentage of items backordered). Second, they seek to provide, based on the robustness
study, a systematic approach for determining near optimal inventory stocking policies in
stochastic repair/rework environments.
Van Der Laan et al. (1999) also develop a stochastic demand inventory model
from the real world problem because their research in the area of production planning and
45
inventory control with remanufacturing was initiated by the US manufacturer of
photocopiers. This article considers production planning and inventory control in
systems where manufacturing and remanufacturing operations occur simultaneously.
Typical character for these hybrid systems is that both the output of the manufacturing
process and the output of the remanufacturing process can be used to fulfill customer
demands. The authors consider a relatively simple hybrid system, related to a single
component durable product. For this system, they present a methodology to analyze a
"PUSH" control strategy (in which all retumed products are remanufactured as early as
possible) and a "PULL" control strategy (in which all retumed products are
remanufactured as late as is convenient). The main contributions of this paper are:
i) To compare traditional systems without remanufacturing to push and to pull
controlled systems with remanufacturing, and
(ii) To derive managerial insights into the inventory related effects of rework and
remanufacturing.
2.4 Conclusions
Several important inventory models with cost of quality consideration are
discussed in this chapter. Moreover, major costs of quality and EOQ/EPQ approaches
have been discussed in this study. The summary of the inventory models based on
EOQ/EPQ approach associated with quality costs are shown in Table 2.1. According to
Table 2.1, these EOQ/EPQ approaches still have some points which might be extended to
46
a better inventory model. For example, the consideration of the prevention, appraisal
failure, and opportunity costs in inventory model.
Table 2.1 Summary of the inventory model associated with quality costs
Inventory policies with
Inventory policies
Inventory poUcies
Inventory policies with
defective and inspection
with imperfect
w/ investment and
repair/rework and
process
setup time in quality
warranty
Shih (1980)
Porteus (1986)
Schonberger (1982)
Goyal and Gunasekaran
Osteryoung (1986)
Lee and Rosenblatt
Porteus (1985)
(1989)
Esrock (1986)
(1986)
Trevino etal. (1993)
Chow (1992)
Schwaller (1988)
Guo and Lewis(1994)
Goyal etal. (1993)
Humphrey etal. (1998)
Chyr etal. (1989)
Lee and Park (1991)
Kreng and Wu (2000)
Van Der Laan et al. (1999)
Porteus (1990)
Lee and Srinivasan
Murthy and Djamaludin
Chen and Min (1991)
(2001)
(2001)
Salameh and Jaber (2000)
Hayek and Salameh (2001)
Hariga and Azaiez (2001)
Wang (2001)
Huang (2002)
Here we can raise questions like,
1. Can we develop the new mathematical inventory model which associates all
quality costs including opportunity cost in one model under the EPQ approach?
2. How can we integrate quality costs in the EPQ model?
3. How can we classify inventory cost under the definition of COQ?
4. Will the inventory cost impact the behavior or character of COQ curves?
47
In the next chapter, the invesfigafion of effects from cost of quality in inventory is
discussed. The lot sizing policy that minimizes the total cost inventory for the finite
production model under the effect of cost of quality is determined. Furthermore, an
analysis, interpretafion, and experimental design of these effects will be discussed in
terms of the cost of quality model as well.
2.5 Theoretical Model
In this research study, we divide the research into two concepts.
1. The cost of quality(COQ) concept
2. The inventory model: economic production quantity (EPQ) concept
According to research of Sandoval-Chavez and Beruvides (1997), there are six
primary theories related to the cost of quality: (1) Juran's Model, (2) Leser's
Classification, (3) Prevention-Appraisal-Failure Model, (4) The Economics of Quality, (5)
Business Management and the COQ, and (6) Juran's Revised Model. The costs of
quality (COQ) definitions and variables have been discussed in these theories.
This research conducts seek to determine the mathematical EPQ inventory model
integrated with cost of quality. Although the economic production quantity (EPQ)
approach associated with defective, inspection, failure, and warranty costs have been
mentioned earlier in the literature, none of them has provided the complete associated
cost of quality' variables in the inventory models which resemble the industry. One of
the important costs of quality to be added is opportunity cost, which Sandoval-Chavez
and Bemvides (1998) found to be a major cost in the company. The theoretical model of
48
this study consists of the link in COQ and inventory concepts together as shown in Figure
2.3.
Prevention
cost
Production
cost
/ Appraisal
/
cost
Holding
cost
COQ
/
Failure
cost
\
Opportunity
cost
Figure 2.3 Theoretical model of this study
2.5.1 Definition of Reference Variable
This section presents the classification of dependent and independent variables
which are used in this research study. Gay (1987) stated that "A variable is a concept or
characteristics that can assume any one of a range of values".
The dependent variable is usually the primary of interest to the researcher because
the goal is to study, explain, or predict the variability in this kind of variable. The
dependent variables of this research study are the optimal production quantity (EPQ) and
the minimum total system cost.
49
The independent variable influences other variables and accounts for the
variations in the dependent variable. In this research, we manipulate the independent
variables and then observe the effect on the dependent variables. Since we use the
concept of the cost of quality (COQ) and economic production quantity (EPQ), the
independent variables in terms of inventory concept would be investment, setup, defect,
inspection, warranty, and holding costs. In the meantime, the independent variables in
terms of COQ concept would be prevenfion, appraisal, failure, and opportunity costs.
50
CHAPTER III
METHODOLOGY
3.1 Introduction
The analysis for finding an EPQ has several weaknesses which lead many
researchers to extend in several aspects of the original EPQ model. The previous chapter
discussed other models for obtaining a solufion to a tradifional Economic Producfion
Quantity (EPQ) that integrates the inventory approach with some quality aspects.
However, the research papers discussed in chapters 1 and 2 do not cover many quality
aspects. Specifically, the cost of quality is one aspect which could be added to the EPQ
model since there are many costs incurred such as prevention, appraisal, failure, rework,
inspection, and warranty costs. Hence, the main objective of this research is to develop
mathematical models in order to minimize the expected total cost of inventory and
quality related model. We also determine the best quality and ordering quantity policies
based on the total system cost.
Leedy (1993) initiated a research methodology which is one way to solve
problems. This research methodology, as seen below, will be followed in these
procedures.
1. Research Design
a. Type of research
b. Research hypothesis
c. Research environment
51
2. Data Collection and Treatment
a. Data collection
b. Data treatment
c. Data measurement
d. Limitation
3. Data Analysis and interpretation
a. Data Analysis: mathematical and statistical models
b. Data interpretation: theoretical and practical interpretations
4. Research Constraints
3.2 Research Design
This section provides a plan for this research. The following issues are addressed
in this section: type of research, research focus, research hypotheses, research
environment, research method, and research instrument.
3.2.1 Type of Research
This research paper is conducted with many mathematical models. Basically, this
research can be classified as an applied and quantitative study, since the inventory model
integrated with cost of quality will be conducted. Thus, the optimal lot size of this new
inventory model has been discussed, and from now on, it will be called "Economic Cost
of Quality Production Quantity (ECQPQ). Moreover, this research investigates the
52
relationship between inventory costs and quality costs based on minimizing the total costs
of the system.
3.2.2 Research Hypotheses
Basically, we determine lot sizing from the EPQ model, so it is good to study the
different values of lot sizing and quality of conformance level in the production process.
Hence, this research paper has two main hypotheses. The purpose of these hypotheses is
to investigate and verify the relationships of traditional economic production quantity
(EPQ) and economic cost of quality-production quantity (ECQPQ) which is identified in
the mathematical models.
3.2.2.1 Hypothesis 1
Let QC[ equal the conformance level at process / of a product, and Q / be the lot
sizing level which results from QC^ 2i\. process /, and if
QC, > QCy for i, j = 1, 2, 3,...., n (i ^ j) then the Ho and Hi are as follows:
Ho: Q\ < Q^j
HJ:Q\>Q^J
In the null hypothesis, the optimal lot size from ECQPQ model should be
decreased when the quality level increases. This hypothesis based on the compensafion
between producfion, setup, defect, and holding costs of the problem.
53
3.2.2.2 Hypothesis 2
Let Qd is the conformance level at process i of a product, and PC*i is the total cost
which results from QC, at process i, and if
QC\ > QCj for i, j = 1, 2, 3,...., n (i 7^ j) then the Ho and Hj are as follows:
Ho: TC\ < TC*j
Hj: TC\ > TC)
This hypothesis follows to the hypothesis 1 since the minimum total cost has the
same direction with the optimal lot size when the quality level changes.
3.2.3 Research Environment
3.2.3.1 Problem Description
In this research problem, the manufacturer produces only a single product. From
Figure 3.1, raw material has been bought from other companies. There are 5 main
stations in this factory: Production, Inspection, Shipping, Rework, and Scrap stations.
Raw materials and items flow by the arrow in Figure 3.1.
54
Shipping station
^f
Raw
Materials
Producfion
Processes
^
w
Good Final
Products
Customers
w
^r
Inspection
Processes
<—
V
Scrap
-
--•
4--
Failure at
customers
ir
V
Replaced
Defective
Items
1r
Rework
Processes
EOQ
EPO
^
Figure 3.1 Problem Description
Procedures for each step are as follows:
Raw materials go to the production processing station.
-
All products from the producfion station go to the inspection station.
Acceptable finished items from the inspection station are sent to the shipping
station as good final products.
Defective finished items from the inspection station are sent to the rework station
to be reworked into good final products.
-
Non-reworkable finished items from the inspecfion station are sent to the scrap
station as destmctive items.
55
Items from the rework station are sent to the shipping station as good final
products.
-
Acceptable final products from the shipping stafion are shipped to end customers.
-
If final products are broken or defective when received by the end customers,
those defective items will be sent back to the manufacturer as reworkable or scrap
items. Furthermore, acceptable final products are then sent to the end customers.
In contrast to the assumptions made by the traditional EPQ model, this problem
assumes that the product quality is not always perfect. This environment can be any type
of products as long as it fits the scenario and assumptions stated in the next section.
3.2.3.2 Assumptions
1. The demand rate is known, constant, and continuous.
2. The lead time is known and constant.
3. Items are produced and added to the inventory gradually rather than all at once as
in the EOQ.
4. Stockouts are not permitted; since demand and lead time are known.
5. The item is a single product; it does not interact with any other inventory items.
6. Defects are produced in the regular process only (No defective items are produced
in the rework process).
7. The production rate of perfect items is always greater than or equal to the sum of
the demand rate and the rate at which defective items are produced.
56
8. There are 3 inspection policies: No inspecfion, 100% Inspecfion with rework, and
Sampling inspecfion with rework.
9. All inspecfions occur after regular production.
10. This is a single line production which implies that the defective items are
reworked on the same machine as the regular process.
11. The rework process is performed after finishing the regular process.
12. The rework rate is always slower than regular production rate due to varying
difficulty levels from defecfive items.
13. The total good items produced in order to meet the demand are from the regular
production and rework processes.
14. The defective items under warranty are sent to scrap or rework and then replaced
for the end customers.
15. If all defective items under warranty are reworked, these items will be reworked
in the same period as defective items from the regular process.
16. If all defective items under warranty are replaced, these items will be added to the
demand rate.
17. There is no return-item time constraint for reworked or replaced items under
warranty.
18. The prevention and appraisal costs are positively related to the level of system
quality, while the failure costs are negatively related to the level of system quality.
Furthermore, the opportunity costs can be defined positively and negatively with
the system quality level.
57
3.2.3.3 Variables in terms of "cost of quality" (COQ) in the mathematical model
The cost of quality is one good aspect to be added to the EPQ model since there
are many costs incurred such as prevention, appraisal, failure, warranty, inspection, and
rework costs. This paper deals with the finite production inventory model integrated with
quality costs for a single product imperfect manufacturing system. We define each
variable in terms of COQ in order to help us interpret the data for quality improvement.
The lists of variables/cost parameters in this research paper are classified as follows:
1. Prevention Costs
•
The amount of investment [Ki]
2. Appraisal Costs:
•
Inspection costs [i]
•
Setup cost/time for machine [K]
3. Internal failure costs
•
Cost of rework (from the producfion process and customer) [c]
•
Cost of scrap (from the production process and products retumed by
customer) [Si]
4. External Failure costs
•
Retum Products (penalty cost) [F]
5. Opportunity costs
•
Backlogging cost [S2]. When the manufacturer can not be able to
satisfy demand due to defects.
58
•
Cost of machines delay and downfime caused by defectives [K]
(model 8). This is penalty cost from stopping or adjusting the
machines.
3.3 Research Method and Instmment
3.3.1 Research Method
The research methods have been divided in six main sections.
1. Classifying each inventory cost and all possible costs of quality
2. Classifying each cost in the ECQPQ model as the cost of quality such as
prevention, appraisal, intemal failure, extemal failure, and opportunity costs.
3. Developing mathematical ECQPQ models for integrated inventory and quality
costs.
4. Solving optimal lot size which minimizes the total cost in each ECQPQ model
by using Mathematica 5.0 Software.
5. Validating the solutions from Mathematica 5.0 Software by programming a
search function to find the optimal lot sizes (Microsoft Visual Studio.Net).
6. Validating optimal solutions with randomly generated problems
7. Using the lot sizes and total costs from the ECQPQ models after performing
the randomly generated problem in order to do statistical test and compare
results with the tradifional EPQ models.
8. Generafing the example problem in the several cases, and performing the
statistical and result analysis.
59
3.3.2 Research Instrument
The instruments used in this research are the following:
1. Mathemafics software: Maple and Mathemafica version 5.0.
2. Stafistical tools: SAS and Minitab.
3. Personal computer (IBM compafible), and
4. Associated software: Visual Basic, Visual.net, and MS Office
3.4 Data Collection and Treatment
Leedy (1993) stated that, "The data dictate the research methodology. And data
play a crucial role in conducting research". Hence, this secfion explains the data
collection and treatment procedures.
3.4.1 Data Collection
Data can be obtained in two main ways: from given parameters, given variable
values, and mathematical models. The parameters and variable values are given in
reasonable terms according to prevention, appraisal, failure, and opportunity costs
discussed in chapter 1, section 2.2, and section 3.2.3.3. The data from the mathematical
models of this research are measured and observed directly from developing the
mathematical models and We substitute the values of each parameter and variable. Then
these data will be implemented in the cost of quality terms. For instance, the prevention
and appraisal costs (P+A) are high, while the failure cost (F) is low, and the P+A costs
60
are low, while the F cost is high, etc. Figure 3.2 shows the different patterns of cost of
quality.
Cost
Cost
% Conform
% Conform
Cost
Cost
% Conform
% Conform
Figure 3.2 Possible combinations of cost of quality curves
3.4.2 Data Treatment
When the data have been collected, it is important to consider how to deal with
data. Previous section demonstrated how to collect the data in three ways. Statistical
methods such as the hypothesis test, t-test, normality test, and nonparametric methods
will be used to test data. Moreover, a regression analysis and analysis of variance
(ANOVA) will be used to treat the data in the design of experiment including sensitivity
data analysis.
61
3.5 Research Constraints and Limitations
The general limitafions of this research have been broadly discussed in chapter 1.
However, there are some limitations which have not been specifically addressed. In fact,
it is hard to conclusively mention all of the constraints and limitations, but this section
will address the main points which are additional from chapter 1.
1. Only one product is studied in this research because this is the limitation of
the original EPQ model. Basically, all assumptions have to follow the EPQ
model, for example, demand is deterministic.
2. The combinations of cost curves shown in Figure 3.2 are limited. It is
difficuh to exactly show all combinafion curves as there are thousands of
combination curves. However, this research points out the importance of
combination curves.
3. Some parameters and variables values which are used are given since we did
not have the actual values from real world experience. However, we have
given all variables according to reasonable interpretation.
4. The definifions of COQ such as prevenfion, appraisal, failure, and opportunity
costs may be different among companies since each company has its own
characterisfics. Hence, the COQ definitions for each company will be unique.
62
3.6 Final Remarks and Conclusions
This research can be defined in 2 stages:
1. This paper deals with the finite production inventory model (based on EPQ) integrated
with quality costs for a single product imperfect manufacturing systems. The defect rate
is considered to be a random variable with a known probability density funcfion.
Defective items are reworked at some cost either before or after sales (warranty). The
prevention, appraisal, and inspection costs have an inverse relationship to the defect rate.
The replacement rate as related to the warranty is also considered to be another random
variable with a known failure rate. The objective of this research is to develop
mathematical models in order to minimize the expected total cost of inventory and
quality related models. Moreover, we will determine the best quality and ordering
quantity policies based on the total system cost.
2. Then Mathematica Software is used as a tool to solve for solutions, and Microsoft
Visual Studio.Net is used for programming the search solution to validate the results from
Mathematica Software.
3. After finishing the first two stages, the statistical tests for the optimal lot sizes and
minimum total costs between traditional EPQ and ECQPQ are performed (as shown in
chapter 5). Finally, the sensitivity analysis will be used to investigate how the output of a
model will be influenced by changes or errors in the input parameters.
63
CHAPTER IV
THE INTEGRATED EPQ MODELS WITH THE COST OF QUALITY
4.1 Introduction
In the previous chapter, the procedures for obtaining and constmcfing an
Economic Production Quantity (EPQ) mathemafical model that integrates the cost of
quality are discussed. This chapter invesfigates different mathematical models in several
scenarios.
The objective of this research is to develop mathematical models to minimize the
expected total cost of inventory and quality related model. Initially, the manufacturer
must define all costs (such as the cost of production, holding cost, setup cost, inspection
cost, etc.), production characteristics, and all capabilities of the production process.
These have to be accurate because these variables will directly affect the production
quantity and total cost. The various EPQ models integrated with the cost of quality are
shown in section 4.2.
4.2 EPQ models associated with the quality costs
This paper deals with a finite production inventory model integrated with quality
costs for a single product imperfect manufacturing system. The defect rate is considered
as a variable of known proportions, and defective items are reworked at some costs either
before or after sales (product retumed by the customer). The prevenfion, appraisal, and
inspecfion costs have somewhat inverse relationships to the defecfive rate. The
64
replacement rate from warranty is also considered to be another variable with a known
failure rate.
The mathemafical models for optimal production lot size in this research can be
classified as follows:
-
Model 1: There are products retumed by the customer, with no inspection and no
rework.
-
Model 2: There are products retumed by the customer, with no inspecfion.
However, the additional demand from the products retumed by the customer is
satisfactory in regular and rework processes.
-
Model 3: There are products retumed by the customer, with no inspection.
However, the additional demand from the products retumed by the customer is
satisfactory in a rework process.
Model 4: There are products retumed by the customer, which require 100%
inspection. However, the additional demand from the products retumed by the
customer is satisfactory in regular processes.
Model 5: There are products retumed by the customer, which undergo sampling
inspection (proportion inspection) when the lot is accepted. However, the
additional demand from products retumed by the customer is processed in regular
production only. There is no rework process in this case.
Model 6: There are products retumed by the customer, which require 100%
inspection. However, the additional demand from the products retumed by the
customer is satisfactory by a ratio of regular and rework processes.
65
-
Model 7: Machine Delay/Downtimes effect: There are products returned by the
customer, with 100% inspection. However, the additional demand from the
products retumed by the customer is satisfactory in regular process.
-
Model 8: There are products retumed by the customer, and 100% inspecfion.
However, the additional demand from the products retumed by the customer is
satisfactory in regular processes.
4.2.1 Model 1
Model 1: There are products returned by the customer, with no inspection and no
rework. (This model is the same as the traditional EPQ model when there is an
additional demand from products returned by customer.)
On hand
Inventory
qi
P-D-w
X ^
X ^
Time
Figure 4.1: On-hand inventory of non-defective items for Model 1
66
The production rate of imperfect items:
W = D(x^Y) (external)
(4.1)
The producfion rate of good items is always greater than or equal to the sum of the
demand rate and the rate at which defective items are produced. So we must have:
P> D+W
(4.2)
Time "ti" needed to build up "qi" units of items:
h=
^
P-D-W
(4.3)
Time "t2" needed to consume the maximum on-hand inventory "qi"
t,=-^^
(4.4)
Time t needed to consume all units Q at demand rate plus defects:
t =-^-
(4.5)
D+W
Inventory level during production cycles:
q, = {P-D-W){^)
(4.6)
The relevant costs per cycle appropriate to this model are as follows:
•
Production cost of all items = CQ
•
Fixed cost (set up cost) = K
•
Cost of quality investment = Kj =^ K/x
•
Holding cost: the holding costs should include that of all produced items,
defective and non-defective.
o / / = Holding cost of perfect and imperfect items (per item per unit time)
67
•
o
// = Holding cost of reworked items (No h in this model)
o
hi = Holding cost of imperfect items from customers (No hi in this model)
Cost per defect passed forward to customers (scrap and penalty costs) =
SxQ(x+Y) + FQ(x+Y)
Holding cost = H 1A , g/2
•
(4.7)
Total cost (TC) would be:
TC(Q)=H 1A I g/2 + (CQ) + (K) + (Ki) + SiQ(x+Y) + FQ(x+Y)
(4.8)
Solving for the average cost per cycle, we get
TOC(Q)-(TC(Q)/t)
(4.9)
We substitute t, ti, t2, and qi from previous equations to equation (4.8) and (4.9) to
get TOC(Q). Mathemathica Software is used to solve for the minimize total cost
(TOC(Q)) with respect to quantity (Q). Moreover, the convex function proof is also
accomplished with Mathematica Software. Finally, the opfimal quantity lot size (Q"^)
equation is
Q*=
V2J-(K+K,)P(D+W)
^ ^
'^ ^
7H(D-P+W)
68
(4.10)
4.2.2 Model 2
Model 2: There are products returned by the customer, with no inspection.
However, the additional demand from the products returned by the customer is
satisfactory in regular and rework processes.
q2
On hand
Inventory
R-D-B,W
q3
qi
X
X
Time
Figure 4.2: On-hand inventory of non-defective items for Model 2
The production rate of imperfect items:
p = Px (intemal) and there is nop in this case
W = D(x+Y) (external)
The production rate of good items is always greater than or equal to the sum of the
demand rate and the rate at which defective items are produced. So we must have:
P>Px + D
Time "?;" needed to build up "qi" units of items:
Time "/'2" needed to build up "qs": reworking the defective items produced.
69
(4.11)
(4.12)
t,=
^
P-D-{\-B,)W
(4.13)
/,=[5,(x + r ) ^ ] = ( ^ ) ( f )
K
(4.14)
UK
Bi is the percentage of products retumed by the customer which is processed at rework
production.
Time "r^" needed to consume the maximum on-hand inventory "q2"
t,=-^^
(4.15)
Time t needed to consume all units Q at demand rate plus defects:
t =.t^+t^+t^
(4.16)
Inventory level during production cycles:
q,={P-D-{\~B,)W){^)
^2=^1+^3
q,={R-D-W){t^)
The relevant costs per cycle appropriate to this model are as follows.
•
Production cost of all items = CQ
•
Repair cost of all defective items = cQ(x-^Y)
•
Fixed cost (set up cost) = K
•
Cost of quality investment = Kj = K/x
•
Holding cost: the holding costs should include that of all produced items,
defective and non-defective.
70
(4.17)
(4-18)
(4.19)
•
o
H- Holding cost of perfect and imperfect items (per item per unit time)
o
/? = Holding cost of reworked items
o
hi = Holding cost of imperfect items from end customers
Cost per defect passed forward to customers (Cost of scraps and penalty) =
S,Q(x+Y) + FQ(x+Y)
•
Holding cost = H g/l I ( g l + g 2 > 2 I ^2^3
2
2
2
•
Total cost (TC) would be:
TC(Q)=H
g/l , ( ? l + g 2 > 2 I <l2h
2
2
2
hB,QY
2
^hMLf + h^t^
HRt,
2 '
+ (CQ) + (cqs) + (K) + (Kj)
S,Q(x+Y) + FQ(x+Y)
(4.20)
Solving for the average cost per cycle, we get
TOC(Q) = (TC(Q) /1)
(4.21)
We substitute t, tj, t2, ts, qj, q2, and qj from previous equations to equation (4.20)
and (4.21) to get TOC(Q). Then Mathemathica Software is used to solve for the
minimize total cost (TOC(Q)) with respect to quantity (Q). Moreover, the convex
function proof is also accomplished with Mathematica Software. Finally, the optimal
quantity lot size (Q*) equation is
{Q*= ( 7 2 ^ 0 ' (K+Ki)PR(D+W))
/
(^(-D'HR+Bj'PW' (H(R-W)+hW) +
B,DW(B,hPW+H(2PR-B,PW-2RW))+
D'R(H(P-W-2BiW)+Bih,PY)))}
71
,^ ^2)
4.2.3 Model 3
Model 3: There are products returned by the customer, with no inspection.
However, the additional demand from the products returned by the customer is
satisfactory in a rework process.
On hand
Inventory
q2
R-D-w
q3
qi
<
>^<-
X
Time
-*
^
Figure 4.3: On-hand inventory of non-defective items for Model 3
The production rates of imperfect items are shown in equation 4.11 and 4.12.
Time "fy" needed to build up "qT units of items:
Time "^2" needed to build up "^3": reworking the defective items produced.
h=
^
(4.23)
P-D
R
DR
Time "^3" needed to consume the maximum on-hand inventory ''qi
72
(4.24)
_^2
h - ^
(4.25)
Time t needed to consume all units Q at demand rate plus defects:
t^t^^t^^t^
(4.26)
Inventory level during production cycles:
^i=(P-/))(^)
(4.27)
^2=^1+^3
(4.28)
q^={R-D~^W){t^)
(4.29)
The relevant costs per cycle appropriate to this model are as follows.
•
Production cost of all items = CQ
•
Repair cost of all defective items = cQ(x-^Y)
•
Fixed cost (set up cost) = K
•
Cost of quality investment = Kj = K/x
•
Holding cost: the holding costs should include that of all produced items,
defective and non-defective.
o / / = Holding cost of perfect and imperfect items (per item per unit time)
•
o
h = Holding cost of reworked items
o
hj = Holding cost of imperfect items from end customers
Cost per defect passed forward to customers (Cost of scraps and penalty) =
SjQ(x+Y) ^ FQfx+Y)
73
kQY
Holding cost == H g/l I (gl+g2V2 I Qlh
•
hRt,
Total cost (TC) would be
TC(Q)=H
1A
2
I ( ^ 1 + ^ 2 ^ 2 I ^2^3
2
2
+^ t ,
+^t,
+ (CQ) + (cQx) +(cQY) + (K) +
(4.30)
(K,)+ FQ(x+Y) + S]Q(x+Y)
Solving for the average cost per cycle, we get
TOC(Q) = (TC(Q)/t)
(4.31)
We substitute t, ti, t2, ts, qi, q2, and qs from previous equations to equation (4.30)
and (4.31) to get TOC(Q). Mathemathica Software is used to solve for the minimize total
cost (TOC(Q)) with respect to quantity (Q). Moreover, the convex function proof is also
accomplished with Mathematica Software. Finally, the optimal quantity lot size (Q*)
equation is
{Q*= (V27D'(K+KJ)PR-)
/
(Ar(HP(R-W)'W' + D ' H R ( P R + 2 W ( - R + W ) )
+DPW(hRW+H(2R' -3RW+W' ) ) + D ' R ' (-H+hl Y ) ) ) }
74
(4.32)
4.2.4 Model 4
Model 4: There are products returned by the customer, which require 100%
inspection. However, the additional demand from the products returned by the
customer is satisfactory in regular processes.
q2
On hand
Inventory
R-D-w
q3
P-D-p-W
X
•<
to
Time
t
-•
•
Figure 4.4: On-hand inventory of non-defective items for Model 4
The production rate of imperfect items:
p = Px (intemal)
(4.33)
W = DY(extQmal)
(4.34)
Time "?y" needed to build up "^y" units of items:
Time "^2" needed to build up "^5": reworking the defective items produced.
A = — ' - " —
'
P-p-D-W
75
(4.35)
( Pxt, ^
^2
=
(4.36)
Time "^5" needed to consume the maximum on-hand inventory "q2'
t,=-^^—
(4.37)
Time t needed to consume all units Q at demand rate plus defects:
Q
t=-^—
D+W
(4.38)
^ '
q^={P-p-D-W)\^
(4.39)
^2=^1+^3
(4.40)
Inventory level during production cycles:
q,={R-D-W){t,)
(4.41)
The relevant costs per cycle appropriate to this model are as follows.
•
Production cost of all items = CQ
•
Repair cost of all defective items = cQ(x)
•
Fixed cost (set up cost) = K
•
Cost of quality investment = Kj = K/x
•
Inspection cost = Q*i
•
Holding cost: the holding costs should include that of all produced items,
defective and non-defective.
o H= Holding cost of perfect and imperfect items (per item per unit time)
o
h = Holding cost of reworked items
76
o
•
hi- Holding cost of imperfect items from end customers
Cost per defect passed forward to customers (Cost of scraps and penalty) = SiQfY)
+ FQ(Y)
, Hpt,t, ^ hRt, ^
Holding cost - H gl^ I ( g l + g 2 K I ^2^3
2
2
2
•
Total cost (TC) would be:
TC(Q)=H
g/l , (gl+?2)^2 . ^2^3
1
2
1
2
^ t
+^ t ,
+ (CQ) + (cQx) + (K) + (KI)
2
+FQ(Y) + S,Q(Y)
(4.42)
Solving for the average cost per cycle, we get
TOC(Q) = (TC(Q)/t)
(4.43)
We substitute t, tj, t2, ts, qi, q2, and qs from previous equations to equation (4.42)
and (4.43) to get TOC(Q). Mathemathica Software is used to solve for the minimize total
cost (TOC(Q)) with respect to quantity (Q). Moreover, the convex function proof is also
accomplished with Mathematica Software. Finally, the optimal quantity lot size (Q*)
equation is
{Q*= (V27(K+K,)P-R(D+W))
/
(^(-HR(D(P-2p,)-(P-p,)^+(P-2pJW)2HPR(D-P+p, +W)x+P' (D(h-H)+H(R-W)+
hW)x^))}
77
(4.45)
4.2.5 Model 5
Model 5: There are products returned by the customer, which undergo proportion
inspection when the lot is accepted. However, the additional demand from products
returned by the customer is processed in regular production only. There is no
rework process in this case.
On hand
Inventory
P-D-W
q3
P-s-D-W
X
Time
<
•
Figure 4.5: On-hand inventory of non-defective items for Model 5
The producfion rates of imperfect items are shown in equafion 4.11 and 4.12.
Time "^y" needed to build up "^y" units of items:
Time "^2" needed to build up "95": reworking the defective items produced (However,
there is no rework in this case).
(4.46)
P-D-W
78
^2=0
(4.47)
Time " / j " needed to consume the maximum on-hand inventory "q2"
h=^^~
D+W
(4.48)
^
^
Time t needed to consume all units Q at demand rate plus defects;
D+W
(4.49)
Inventory level during production cycles:
q^=(P-D-W){^)
(4.50)
^2=^1+^3
(4.51)
^3=0
(4.52)
The relevant costs per cycle appropriate to this model are as follows.
•
Production cost of all items = CQ
•
Fixed cost (set up cost) = K
•
Cost of quality investment = Kj = K/x
•
Holding cost: the holding costs should include that of all produced items,
defective and non-defective.
o H= Holding cost of perfect and imperfect items (per item per unit time)
•
o
h = Holding cost of reworked items
o
h] = Holding cost of imperfect items from end customers
Cost per defect passed forward to customers (Cost of scraps and penalty) =
Si(Q)(x) + SjQfY) + F(Q)(x) + FQY
79
g/l
I gl^2
•
Holding cost = H
•
Total cost (TC) would be:
TC(Q)=H
1A , g/2
+ (CQ) + ni + (K) + (K,)+ Si(Q)(x) + S,Q(Y) + F(Q)(x) +
FQY
(4.53)
Solving for the average cost per cycle, we get
TOC(Q) = (TC(Q)/t)
(4.54)
We substitute t, ti, t?, ts, qj, q2, and qs from previous equations to equation (4.53)
and (4.54) to get TOC(Q). Mathemathica Software is used to solve for the minimize total
cost (TOC(Q)) with respect to quantity (Q). Moreover, the convex function proof is also
accomplished with Mathematica Software. Finally, the optimal quantity lot size (Q"^)
equation is
V2^-(K+Ki+ni) * P(D+W)
7H(D
- P + W)
80
(4.55)
4.2.6 Model 6
Model 6: There are products returned by the customer, which require 100%
inspection. However, the additional demand from the products returned by the
customer is satisfactory by a ratio of regular and rework processes.
On hand
Inventory
q3
P-D-p-W
X
t.
•<
t
Time
Figure 4.6: On-hand inventory of non-defecfive items for Model 6
The production rate of imperfect items:
p = Px (intemal)
(4.56)
W ^ DY {external)
(4.57)
Time "^y" needed to build up "q'y" units of items:
Time "?/' needed to build up "95": reworking the defecfive items produced.
f, =
'
?i
P-p-D-W
81
(4.58)
PxtA
^2 =
R J
fB^QY^
+
(4.59)
R
Time "^j" needed to consume the maximum on-hand inventory "q2"
t,^-^^D+W
(4.60)
Time t needed to consume all units Q at demand rate plus defects
t=
Q
^
D+W
(4.61)
Inventory level during production cycles:
f r\\
q,=(P-P-D-W)
Q
(4.62)
^2=^1+^3
(4.63)
q,=iR-D-W)(t,)
(4.64)
The relevant costs per cycle appropriate to this model are as follows.
•
Production cost of all items = CQ
•
Repair cost of all defective items = cQ(x)+cQBiY
•
Fixed cost (set up cost) = K
•
Cost of quality investment = Kj = K/x
•
Inspection cost = Q*i
•
Holding cost: the holding costs should include that of all produced items,
defective and non-defective.
o / / = Holding cost of perfect and imperfect items (per item per unit time)
o
h = Holding cost of reworked items
82
o
hi- Holding cost of imperfect items from end customers
Cost per defect passed forward to customers (Cost of scraps and penalty) = S}Q(Y)
+ FQ(Y)
Holding cost = H
•
g / l 1. (^1+^2)^2 .1 ^2^3
2
2
2
I Hpt.t, ^ hRt, ^
2
2 '
Total cost (TC) would be:
TC(Q)=H
1A
2
I ( g l + ? 2 > 2 I ^2^3
2
2
HpJ,
hB,QY
+- ^ t , + A J ^ ^
HRt,
+_^^^
,^^,
, ^ ,
+ (CQ) + (cQx) +
(cQBiY) + Qi + (K) + (Ki)+ FQ(Y) + S,Q(Y)
(4.65)
Solving for the average cost per cycle, we get
TOC(Q) = (TC(Q)/t)
(4.66)
We substitute t, tj, t2, ts, qi, q2, and qs from previous equations to equation (4.65)
and (4.66) to get TOC(Q). Mathemathica Software is used to solve for the minimize total
cost (TOC(Q)) with respect to quantity (Q). Moreover, the convex function proof is also
accomplished with Mathematica Software. Finally, the optimal quantity lot size (Q*)
equation is
{Q*= (^/27(K+K,)P'R(D+W))
/
(Ar(PW(B,h,RY+hP(x+BiY)') +
D (P (B,h,RY+hP(x+B,Y)') -H(-2p,R+P'(x+BiY)'+
PR(1+2X+2B,Y)))+H(P,R(PI+2W)+P'(-W(X+B,Y)' +
R(l+x+B, Y)')-PR(W+2(pi+PiX+Wx+B, (p,+W)Y)))))}
83
(4.67)
4.2.7 Model 7
Model 7: Machine Delay/Downtimes effect: There are products returned by the
customer, with 100% inspection. However, the additional demand from the
products returned by the customer is satisfactory in regular process.
i
On hand
Inventory
D+w
Re-setup Processes or
/
i i q4
/ /
^3
qi
q2
Machine Adjustment
/
R-D-W
UAI
\
D+W
/ P-D-p-W
/p-D-p-w
^
^
^
^
t,
w
^
t2
F ^
%
W'
u
t3
Time
•
t
^
W
Figure 4.7: On-hand inventory of non-defective items for Model 7
The production rates of imperfect items are shown in equation 4.56 and 4.57.
Time "/y" needed to build up "^y" units of items:
Time "^2" needed to setup machine:
Time "^3" needed to build up ''qs'': reworking the defective items produced.
/. =
1
^1
p_p^D-jY
84
(4.68)
_ J*f
t^=A*ti
-
=
^
(4.69)
D+W
A = the percentage of setup time between regular and rework production
^Pxt,^
h=
(4.70)
V R J
Time "?/' needed to consume the maximum on-hand inventory "q2"
U=
<li
(4.71)
D+W
Time t needed to consume all units Q at demand rate plus defects;
Q
D+W
(4.72)
q,=iP-p-D-W)
(4.73)
t =
Inventory level during production cycles:
^2=^1-^5
(4.74)
^ 4 = ^ 2 + ^3
(4.75)
q,^{R-D-W){t,)
(4.76)
q,=iD + W)iA*t,)
(4.77)
Setup cost equation has the inverse function with the percentage of setup time as follows:
^ = 100 +
1400
\ +A
The relevant costs per cycle appropriate to this model are as follows.
•
Production cost of all items = e g
85
(4.78)
•
Repair cost of all defective items = cQ(x)
•
Set up cost = K
•
Cost of quality investment = K] = K/x
•
Inspection cost = Q*i
•
Holding cost: the holding costs should include that of all produced items,
defective and non-defective.
o / / = Holding cost of perfect and imperfect items (per item per unit time)
•
o
h = Holding cost of reworked items
o
hj = Holding cost of imperfect items from end customers
Cost per defect passed forward to customers (Cost of scraps and penalty) = SiQ(Y)
+ FQ(Y)
Holding cost = H g/l , {^1+^2%
2
2
•
Hpt,
2 '
I (g2+?4>3 I ^4^
2
2
hRt,
2 '
Total cost (TC) would be:
TC(Q)=H
9/l , ( g l + g 2 X 2 I ( g 2 + g 4 > 3 , ^4^4
2
2
2
2
+ (K) + (Ki)+ FQ(Y) + S,Q(Y)
^t,+^t,+(CQ)
^
2
•' '
2
+ (cQx) + Qi
'^
(4.79)
Solving for the average cost per cycle, we get
TOC(Q) = (TC(Q) /1)
(4.80)
We substitute t, ti, t2, ts, t^, qi, q2, qs, q4, and qs from previous equations to
equation (4.79) and (4.80) to get TOC(Q). Mathemathica Software is used to solve for
the minimize total cost (TOC(Q)) with respect to quantity (Q). Moreover, the convex
86
funcfion proof is also accomplished with Mathematica Software. Finally, the opfimal
quantity lot size (QV equation is
{ Q = (V27-(1500 + K,+ A(100+K,)) P'R ( D + W ) ) /
(^^((l+A) (-HR((p - P)^+ D(2p - P) + (2p - P)W) +
(4.81)
2HPR(D + AD + P - P + W + AW)x - P'(D(h - H) +
H(R-W) + hW)x')))}
4.2.8 Model 8
Model 8: There are products returned by the customer, and 100% inspection.
However, the additional demand from the products returned by the customer is
satisfactory in regular processes.
On hand
Inventory
R-D-w
q3
Time
q4
t4
>
<
X
•
t2
^M
ts
>^<
•
t
Figure 4.8: On-hand inventory of non-defective items for Model 8
87
The production rates of imperfect items are shown in equation 4.56 and 4.57.
Time ' 7 / ' needed to build up " 9 / ' units of items:
Time "^2" needed to build up "q-j": reworking the defective items produced.
^1
^1 =
(4.82)
P-p-D-W
C D^t \
Pxt
(4.83)
V 'R J
Time "^j" needed to consume the maximum on-hand inventory "q2"
?2
(4.84)
h= D+W
Time t needed to consume all units Q at demand rate plus defects:
t=
Q
D+W
(4.85)
Inventory level during production cycles:
q,={P-p-D-W)
'Q^
^4
(4.86)
K^ J
qj
=qi+q3
(4.87)
q^^{R-D-W){t,)
(4.88)
q,={P-p-D-W%
(4.89)
The relevant costs per cycle appropriate to this model are as follows.
•
Productioncost of all items = e g
•
Repair cost of all defective items = cQ(x)
•
Fixed cost (set up cost) = K
88
•
Cost of quality investment = Kj = K/x
•
Inspection cost = Q*i
•
Holding cost: the holding costs should include that of all produced items,
defective and non-defective.
•
o
H= Holding cost of perfect and imperfect items (per item per unit fime)
o
h = Holding cost of reworked items
o
h} = Holding cost of imperfect items from end customers
Cost per defect passed forward to customers (Cost of scraps and penalty) = SiQ(Y)
+ FQ(Y)
•
Shortage cost = '^2*^4(^+^5)
•
Holding cost
H g/i , Jqi^qiyi
•
, ^ih
2
^
'
2
'
2
2
Total cost (TC) would be:
TC(Q) =
H
<1A I (^1+^2)^2
1
1
I ^2^3
^(^L±M(A+/3) + ^ . , + ^ ^ ^ i ^ ^ i ± ^ + ^ M ^ r + (CQ)
1
+ (cQx) + Qi + (K) + (K,)+ FQ(Y) + SjQ(Y)
*B2 is the percentage of time "t" to scrap the customer return product
*S2 is the shortage cost per item per unit fime ($/item/unit time)
*q4 is the backorder level (units)
89
(4.90)
Solving for the average cost per cycle, we get
TOC(Q) = (TC(Q) /1)
(4.91)
We substitute t, ti, t?, ts, qi, q2, and qs from previous equations to equation (4.90)
and (4.91) to get TOC(Q). Mathemathica Software is used to solve for the minimize total
cost (TOC(Q)) with respect to quantity (Q). Moreover, the convex function proof is also
accomplished with Mathematica Software. Finally, the optimal quantity lot size (Q*)
equation is
{Q=(>r(P'(R(D-P+p,+W)(2D^(K+K,)-(P-p,)q4^(H+S,)2(K+K 1 )(P-pi) W+2(K+Ki )W' +2D(K+K, )(-P+p, +2 W))2HPq4' (D-P+p, +W)x+P'q4' (D(h-H)+H(R-W)+hW)x')))
/
(4.92)
(^^(-(D-P+p,+W)'(D(H(P-2p,)R+2HPRx+(-h+H)P'x')H(p,R(p, +2 W)-PR(2p, +W+2(pi +W)x)+P' (-Wx' +R(1 +x)' ))P'(hWx'+B2h,RY))))}
90
CHAPTER V
STATISTICAL AND RESULT ANALYSIS
5.1 Introduction
In Chapter IV, the economic cost of quality production quantity (ECQPQ)
mathemafical models are presented. Mathematica Software is used to find the optimal
solufions for all of the models. Moreover, validafion of the solufions obtained from
Mathematica software and the data are performed to ensure that the results are accurate.
For validafion, the models were coded in Microsoft Visual Studio.Net (Appendix B).
According to section 3.3, a stafistical analysis will be fully completed after the additional
mathematical ECQPQ (the complete model) has been performed.
In this chapter, we would like to assess the performance of the ECQPQ
comparison with traditional Economic Production Quantity (EPQ). Hence, the
hypothesis tests (as described in section 3.2.2) and result analysis will be performed in
this chapter as well. Since the solutions of the models may depend on several factors
such as production rate, setup cost, holding cost, etc., we seek to identify the significance
of the selected factors and analyze the relative performance of the model. Once
significant variables are identified from the sensitivity analysis, we will further
investigate their effects on the model performance. It should be noted that some
notafions such as "/// will be used for hypothesis tesfing in this chapter" are defined
differently in this chapter in order to follow the convenfional notations used in stafisfical
analysis.
91
5.2 Resuh Validation
Resuh validation is a necessary step in this research since we can not rely only on
Mathemafica Software. Validating all solufions from Mathemafica Software ensures that
the results will not be wasted. For validafion, the models were coded in Microsoft Visual
Studio.Net (Appendix B) and run on a Pentium4, 1.8 GHz personal computer.
p
j'laa
• M
i\^
;- W
V
••
TOO
[ ^^^
'^ i
IDC4
|KQ
^
: fl
JKEO
n
ift.i
¥
^os
«
w
1
T.>;i
1«
5i4iJ
J«
nun
i
OUi
1
Figure 5.1 Search solution programming
Figure 5.1 represents the search solution programming for single and multiple
runs. Basically, we just put the numerical parameters in the space for single run while the
multiple mn needs the data in text file by putting one problem in one line and one space
92
for separating the parameter values. The algorithm to use to write this program works
like search solutions. Since total annual cost (TOC) is the funcfion of the production lot
size (Q), then we vary Q from 1 unfil we get the tuming point of the total annual cost
value. In fact, we would get the minimum total annual cost at that tuming point or the
point before. However, we have to prove the total annual cost (TOC) function as a
convex funcfion first; otherwise, we can not create this search solufion program. By
definition, if a function has the second derivative (f"(x)) > 0 for all x, then that funcfion
will be a convex fimction.
5.3 ECQPQ Sensitivity Calculation Example
This section shows how to calculate the sensifivity analysis. This example is
based on model 5 in section 4.2.5 and numerical value in section 5.4. The purpose of this
section is to examine the issue of how sensitive the annual cost function is to errors in the
calculation of Production rate (P), Demand rate (D), Rework Rate (R), the percentage of
intemal defects (x), the percentage of extemal defects (y), unit production cost (C), unit
rework cost (c), inspection cost (i), cost of customers retum (F), setup cost (K), sampling
size (n), and all holding costs (H, hi, and h). Sensitivity analysis determines how the
output of a model will be influenced by changes or errors in the input parameters. Since
the EPQ is a deterministic model and all parameters are estimated until the actual value
data in the future have been collected, a sensitivity analysis is necessary to test the model
to know how errors of estimated values could affect quantity decisions and the total
annual costs.
93
The error factor is the percentage value which deviates from the actual value. The
formulation of error factors is as follows:
Error Factor = Z, = (Estimated Value)/(Actual Value)
Where / is any of the above parameters and there are no errors in the other
parameters. The example of rework rate error factors {Zj^, errors in g*, and errors in the
total annual costs are calculated in Table 5.1. All error factors (Z^), errors in g*, and
errors in the total annual costs are in Appendix C. When all the error factors are equal to
1, the total annual cost TOC (Q*) error fraction is zero. Basically, this error factor (Zj)
will translate the factor error such as holding cost, setup cost, etc. into their impact on
total annual cost and optimal lot size. For example, if the production cost (C) has an
error of 30%, it results in only a 19.1% increase over the total annual cost while there is
no change in the optimal lot size. This analysis will help to avoid under/over estimations
of the numerical value for each factor. Figure 5.2-5.6 shows the effect of errors on the
total annual cost (TOC(Q*)) and optimal lot size (Q*).
94
Table 5.1 Effect of Errors in the rework rate on TOCfQ"^) and Q*
Error Factor
Error in TOC(Q*),
Error in Q*,
(ZR)
(%)
(%)
0.1
0.15
-1.62
0.2
0.10
-1.17
0.3
0.07
-0.78
0.4
0.04
-0.45
0.5
0.02
-0.19
0.6
0.00
-0.03
0.7
-0.01
0.10
0.8
-0.01
0.13
0.9
-0.01
0.10
1
0.00
0.00
1.1
0.02
-0.19
1.2
0.04
-0.42
1.3
0.07
-0.75
1.4
0.10
-1.14
1.5
0.14
-1.59
1.6
0.19
-2.11
1.7
0.24
-2.66
1.8
0.30
-3.28
1.9
0.37
-3.96
2
0.44
-4.71
Error in total annual cost (%)
TOC{Q),,„,,,-TOC{Q%,„„^,,
100
TOC{Q*),,,„^„,
Error in Q* = Production Quantity Error (%)
95
\i:i)Actual
(e*)
V^
'Estimated
Estimated
(5.1)
He 1 HA
(5.2)
TOC errors Chart
-•—D
-
R
800 00
700 00 ^
Ia>
600.00 500.00 -
_3
^ww.ww
«
-^K—y
\
—1—0
\
1
I
Z>
3
F
\
200.00 100.00 -
__-•<__ X
1
<
<
Ann nn _
teT
O
O
^
^
^
K
V
n
0 00 -
(3
0.5
1
1.5
2
25
Error Factor = Estimated/Actual value
—a— h
Figure 5.2 Effect of errors on TOC(Q*)
TOC errors Chart w/o Variable C
0.5
1
1.5
2
Error Factor = Estimated/Actual value
Figure 5.3 Effect of errors on TOC(QV without variable C
96
Total annual cost errors w/o Variable C, D, F, and Y
25.00
H—I
—K
n
H
1
1.5
2.5
Error Factor
Figure 5.4 Effect of errors on TOC(Q^) without variables C, A F, andy
Error on Q"
-•—R
•»—y
250.00
C
c
200.00
X
I
•
F
+
n
— H
hi
h
-t^P
D
2.5
-x~~x
-^—K
Figure 5.5 Effect of errors on (g*)
97
Error on Q* without Var P, D, x, and K
-R
70.00
-y
60.00
c
2. 50.00
X
c
UJ 40.00
X
1
*
o
•
F
+
n
e
S
30.00
10.00
I 20.00
::ss
—
-H
-hi
0.00
2.5
— i > - -h
Figure 5.6 Effect of errors on (g*) without variables P, D, x, and k
5.4 Variables Selection
This section discusses the importance of the chosen factors based on an
understanding of tradifional EPQ. In tradifional EPQ as discussed in secfion 2.3.2, the
factors which effect the optimal lot size (g*) are setup cost and holding cost.
The variables selection in this secfion is based on tradifional EPQ and sensitivity
analysis for ECQPQ models which is discussed in section 5.3. The data in the sensitivity
analysis in Appendix C shows that the producfion rate, rework rate, defective rate, setup
cost, and holding cost directly effect the opfimal lot size (g*).
Thus, Model 4 (secfion 4.2.4) is used to test the relafionships of each factor with
respect to optimal lot size (2*) based on the numerical example below. We choose
Model 4 to show the relafionship because this model shows all factors used by other
98
models (from Model 2 to Model 8) use. In this numerical example, we vary the
production rate (P), rework rate(7?), defective percentage of production process (x) and at
customer hand (Y), setup cost (X), and holding cost (//). All other variables that are not
included are assumed to be constants as shown in the numerical example.
Numerical Example
A manufactured product has a constant demand rate of 1,200 units/year. The
machine used to manufacture this item has a production rate of 1,600 units/year. The
production cost per item is $100. The machine setup cost is $1,500. The holding cost per
unit is $20/year. The percentage of imperfect quality items produced by the manufacturer
is 10. The percentage of products retumed by the customer is 5. The defecfive items are
reworked at a rate of 100 units/year. The repair cost per defecfive item is $15. The
holding cost per unit of the items being reworked is $22/year.
C = $100/unit, c = $15/unit, i = $l/unit, F = $150/unit, K = $1,500, H = $20/unit/year,
h = $22/unit/year, hi = $40/unit/year, P = 1,600 units/year, D = 1,200 units/year, R = 100
units/year, x = 10%, and Y = 5%
The graphical representafions of the variafions of each factor (production rate,
rework rate, defective percentage of production process and at customer hand, setup cost,
and holding cost) with respect to optimal lot size based on Model 4 are as follows:
99
Relationship b e t w e e n different levels of P and Q*
4000
«• 3000
5
2000
a
1000
0
1500
2000
2500
3000
3500
P (unit/time)
Figure 5.7 Relationship between different levels of P and Q*
Relationship between different levels of R and Q'
_
^
3
*
3090
3080
3070
3060
3050
3040
3030
80
100
120
140
160
180
200
220
R (Unit/time)
Figure 5.8 Relationship between different levels of i? and Q
Relationship b e t w e e n d i f f e r e n t levels of x a n d Q*
4000
Figure 5.9 Relationship between different levels of x and g*
100
Relationship between different level of Y and Q*
J
c
^
C?
3500
3400
3300
3200
3100
3000
0.0375
0.0625
0.0875
0.1125
Y
Figure 5.10 Relafionship between different levels of Fand Q
Relationship between different level of K and Q*
-^
'E
^
O
5000
4000
3000
2000
1000
0
1000
1500
2000
2500
3000
3500
K($)
Figure 5.11 Relationship between different levels of Z;^ and Q
Relationship between different level of Hand Q*
(units)
4000
3000
a
1000
2000
0
15
20
25
30
35
40
45
H($/item)
Figure 5.12 Relationship between different levels of/f and Q
101
Based on Figure 5.7-5.12, the relationships of each factor {P, R, x, Y, K, and H)
and optimal lot size (g*) clearly show a significant difference at each level with respect
to opfimal lot size. Thus, the production rate (P), rework rate(i?), defective percentage of
production process (x) and at customer hand (7), setup cost (/Q, and holding cost (//) are
the factors of choice.
According to this numerical example, it shows that the optimal lot size (g*) from
traditional EPQ is not resemble to the results from the ECQPQ which integrates defecfive
and other quality costs. This shows that the traditional EPQ is not representative with
respect to cost of quality. Furthermore, if cost of quality such as defects is not considered
in the traditional EPQ inventory model, the actual lot size produced is not accurate. This
means the manufacturers have not been able to produce the good item in order to satisfy
demand since there are defects from the process.
5.5 Generating the data for statistical tests
A statistics test requires many samples to generate an effective statistical analysis.
In stafisfics, if numerous data are generated, the test will be more accurate. However, it
costs fime and money to generate a great deal of data. Thus, this research generates 144
random data to test the hypothesis. The data samples are shown in Table 5.2. All other
parameters that are not varied are assumed to be constant.
102
Table 5.2 The samples of the randomly generated problems
Variable Variable Variable
Variable
ZU2
Zl,3
Zi^n
Z2,l
Z2,2
Z2,3
Z2,n
Z3.1
Z3,2
Z3,3
Z3,n
Z4,l
Z4,2
Z4,3
Z4,n
Z5.I
Z5,2
Z5,3
Zsn
Problem
144
144,1
144,2
Zl44,n
•144,3
Let Zi^i = the numerical value of variable 1 in problem 1.
Zi,2 = the numerical value of variable 2 in problem 1.
Z2,i = the numerical value of variable 1 in problem 2., and so on.
5.5.1 Numerical values for test problems
Fixed variables and their values:
-
Demand rate (/)) = 1,000 unit/time
103
-
Production cost (C) = $100/item
-
Producfion cost for rework items (c) = $15/item
-
Penalty cost for customer retums (F) = $150/item
Variables:
In previous section (section 5.4), we clearly discussed the variable selecfion.
Hence, the production rate (P), rework rate(P), defective percentage of production
process (x) and at customer hand (Y), setup cost (K), and holding cost (//) are varied to
generate data. The values for these variables are presented in Table 5.3.
Table 5.3 Variables and their values
Variables
Unit
Levels
Low (0) Med (1) High (2)
1500
Production rate [P]
5,000
1.25*P Unit/time
0.25*P
Rework rate [R]
The % of defects from production [x]
5%
The % of defects at end customers [Y]
2.5%
Setup cost [K\
Holding cost of perfect items [H]
Holding cost of rework items [h]
Holding cost of defects from end customers [hi]
104
Unit/time
-
15%
10%
50
500
5000
$
0.1*C
0.5*C
5*C
$
1.1*H
-
$
2*H
$
The total problems would be 2*2*2*2*3*3 = 144 test problems, and they are in
Appendix D. These problems will be used to test all models (8 models) which are
discussed in section 4.2.
5.6 Statistical Analvsis
5.6.1 The optimal lot size differences
From 144 test problems seen in Appendix D, we generate the difference between
Model 1 (traditional EPQ) and Model 2, 3, 4, 5, 6, 7, and 8 (ECQPQs). Then we will
have 7 pairs to compare as follows:
P a i r - t e S t # l : Q EPQ(Model}) V S Q ECQPQfModel 2)
P a i r - t e S t # 2 : Q EPQ(Modell)
VS Q
ECQPQfModel3)
P a i r - t e s t # 3 : Q EPQfModell)
V S Q ECQPQfModel 4)
P a i r - t e s t # 4 : Q EPQ(Modell)
VS Q
ECQPQfModel5)
P a i r - t e S t # 5 : Q EPQ(ModeU) V S Q ECQPQfModel 6)
P a i r - t e S t # 6 : Q EPQ(Modell)
V S Q ECQPQfModel 7)
P a i r - t e s t # 7 : Q EPQfModell)
VS Q
ECQPQfModel8)
The optimal lot size (Q*) and differences in Q* for each problem are presented in
Appendix E. The purpose is to compare the optimal lot size from the traditional EPQ and
the new EPQ (let modified EPQ = ECQPQ). We have to do this in order to verify that
the results from different mathematical models should not give the same optimal lot size.
Using the basic statistical comparison, the numbers of the optimal lot size differences
105
between each pair are shown in Table 5.4. Please note that the g*; = the opfimal lot size
of model 1, Q^'modified = the optimal lot size of model 2, 3, 4, 5, 6, 7, or 8.
Table 5.4 The number of differences in g * between model 1 and other models
Total is 144 problems
Q*j < Q*modified
Q^l - Q*modified
Q*l > Q*modifted
Model 1 VS Model 2
0
0
144
Model 1 VS Model 3
0
0
144
Model 1 VS Model 4
0
0
144
Model 1 VS Model 5
115
29
0
Model 1 VS Model 6
0
0
144
Model 1 VS Model 7
108
0
36
Model 1 VS Model 8
126
0
18
The results in Table 5.4 show that the optimal lot size of model 1 is more than the
optimal lot size of model 2, 3, 4, and 6. This is exactly what we expect since the model 1
has not taken care of defective items, so the retum products from customers are more
than other models. However, there are 115, 108, and 126 problems which optimal lot
size of model 1 is less than the optimal lot size of model 5, 7, and 8 respectively. For
model 5, the optimal lot size is less than model 1 cause of more fixed cost (inspecfion
cost). This can be explained that the proportion inspecfion (lot is accepted) is not useful
if the penalty cost is more than the production cost. For model 7 and 8, the optimal lot
106
size is less than model 1 cause of more fixed cost and backorder considerafions
respectively.
5.6.2 The total annual cost differences
From 144 test problems seen in Appendix D, we generate the difference between
Model 1 (tradifional EPQ) and Model 2, 3, 4, 5, 6, 7, and 8 (ECQPQs). Then we will
have 7 pairs to compare as follows:
.
*
*
Pair-teSt#l: TOC EPQ(Modell) V S TOC ECQPQfModel 2)
-
Pair-teSt#2: TOC
EPQfModellJ^S TOC
ECQPQfModel 3)
Pair-teSt#3: TOC
EPQfModelljyS
ECQPQ(Model 4)
Pair-test#4: TOC
EPQ(Modell) V S TOC
ECQPQ(Model 5)
Pair-teSt#5: TOC
EPQ(ModelJ) V S TOC
ECQPQfModel 6)
Pair-teSt#6: TOC
EPQ(ModeU) V S TOC
ECQPQfModel 7)
Pair-test#7: TOC
EPQ(ModeU) V S TOC
ECQPQfModel 8)
TOC
The minimum annual total cost (TOC*) and differences in TOC* for each
problem are presented in Appendix E. The purpose is to compare the minimum annual
total cost from the tradifional EPQ and the new EPQ (let the new EPQ = ECQPQ). We
have to do this in order to verify that the results from different mathematical models will
give the different annual total cost. Moreover, we need to see which models will perform
well based on this problem characteristics and minimum total cost. Using the basic
statistical comparison, the numbers of the total annual cost differences between each pair
107
are shown in Table 5.5. Please note that the TOC*; = the optimal lot size of model 1,
TOC^modified = the optimal lot size of model 2, 3, 4, 5, 6, 7, or 8.
Table 5.5 The number of differences in TOC* between model 1 and other models
Total is 144 problems
TOC*, < TOC Modified
TOC *, = TOC Modified
Model 1 VS Model 2
20
0
124
Model 1 VS Model 3
4
0
140
Model 1 VS Model 4
0
0
144
Model 1 VS Model 5
144
0
0
Model 1 VS Model 6
8
0
136
Model 1 VS Model 7
0
0
144
Model 1 VS Model 8
0
0
144
TOC *, > TOC *,„odified
The results in Table 5.5 show that the total annual cost of model 1 is more than
the optimal lot size of model 2, 3, 4, 6, 7, and 8. This is exactly what we expect since the
model 1 has not taken care of defective items, so the retum products from customers lead
to higher total annual cost since the penalty cost in these problems is set to 150% higher
than producfion cost. However, the total annual cost of model 5 is higher than the total
annual cost of model 1 cause of additional cost from inspecfion.
The special cases which the total annual cost of model lis less than other models
are shown in model 2, 3, and 6. In these special cases, the holding cost and setup cost
108
(fixed cost) are very high, so we can conclude that the model 1 will be getting better
when the penalty cost (cost of defects) is a lot less than the holding and setup costs.
5.7 Statistical Analysis (Hypothesis tests)
From hypothesis statements in secfion 1.4 and 3.2., the decision-making
procedure about the hypothesis is called hypothesis testing. This is one of the most
useful aspects of statistical inference, since many types of decision problems can be
formulated as hypothesis-testing problems. This section explains how to conduct
statistical tests for the hypothesis 1 and 2.
5.7.1 Hypothesis 1
Recall hypothesis 1 from section 3.2.2.1
Let gCi equal the conformance level at process / of a product, and g / be the lot sizing
level which results from QC\ at process /, and if
gCi > gCj for i, j = 1, 2, 3,...., n (i ^ j) then the Ho and Hi are as follows:
Ho:Q*i<Q)
Hi:Q\>Q*j
In the null hypothesis, the optimal lot size from EPQ models should be decreased
when the quality level increases. This hypothesis based on the compensation between
production, setup, defect, and holding costs of the problem.
Since x = the percentage of defective items so Qd will be (l'Xi)*100 =
Conformance level at process /.
109
Table 5.6 The opfimal lot size from test problem#4 when varying x
Q*
#4
Q*
Q*
Q*
Q*
Q*
Model 5
Model 6
Q*
Q*
QC(i)
X
%
Model 1 Model 2
Model 3 Model 4
Model 7 Model 8
0.025
97.5
3788
3554
3189
3642
3790
3504
4111
4194
0.05
95
2823
2610
2226
2604
2826
2505
2893
2992
0.075
92.5
2432
2214
1793
2149
2436
2067
2365
2463
0.1
90
2225
1994
1530
1879
2230
1807
2060
2150
0.125
87.5
2106
1855
1348
1695
2112
1631
1861
1937
0,15
85
2039
1764
1212
1560
2046
1500
1722
1782
The relationships of Q* and x
Model 1
Model 2
Model 3
Model 4
^Hh- Model 5
-•— Model 6
-I— Model 7
— Model 8
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Figure 5.13 The relationships of g * andx
From Figure 5.13, all of the models have a different starting point, but the value
of Q on all these models start out decreasing exponentially as the value of x increase until
110
X = 0.05; subsequently, the declination of the value of g become less and less as x
increase until each model comes to a hah atx = 0.15.
The variable x is in the function of g*. Thus, the relationships of g * and x are
exist which shows in graphical representation in Figure 5.13. Based on Figure 5.13 and
sensitivity analysis in section 5.9, we can conclude that as decreasing the conformance
level (increasing the x value), the optimal lot size will be decreased. The explanation is
that model 1 and 5 virtually has more demand than other models since the defective items
do not catch and take off before. Hence, those defect items pass by customers and
certainly retum back to be additional demand. Finally, the null hypothesis is rejected
since when the QCi decreases, g*,- also decreases.
5.7.2 Hypothesis 2
Recall hypothesis 2 from section 3.2.2:
Let gCi is the conformance level at process i of a product, and TC [ is the total cost
which resuhs from gCj at process i, and if
gCi > gCj for i, j = 1, 2, 3,...., n (i ^j) then the HQ and Hj are as follows:
Ho: TC\ < TC*,
HJ: TC*i > TC)
In the null hypothesis, the optimal total annual cost from EPQ models should be
decreased when the quality level (conformance level) increases. Since x = the percentage
of defective items so Qd will be (l-XiJ'^lOO = Conformance level at process /.
Ill
Table 5.7 The optimal total annual cost from test problem#4 when varying x
#4
TOC*
TOC*
TOC*
TOC*
TOC*
TOC*
TOC*
TOC*
Model 3
Model 4
Model 5
Model 6
Model 7
Model 8
QC(i)
X
%
Model 1 Model 2
0.025
97.5 124239.42 120341.32 115339.85 119292.23 124246.35 119745.31 118441.04 117772.99
0.05
95 127591.41 119051.89 113310.50 116402.46 127600.92 116728.61 116238.96 115332.13
0.075
92,5 132984.17 119650.42 113177.65 115359.24 132995.47 115630.11 115544.45 114486.98
0.1
90 139155.90 120816.38 113612.16 114907.28 139168.53 115145.69 115309.51 114151.41
0.125
87.5 145789.26 122217.35 114283.22 114731.81 145802.90 114948.35 115276.75 114052.85
0.15
85 152761.67 123724.24 115065.44 114711.67 152776.06 114912.25 115348.96 114086.02
Relationships of TOC* andx
160000.00
-•— Model 1
150000.00
-»— Model 2
S
*
140000.00
O
130000.00
Model 3
-^— Model 4
•¥ik— Model 5
-•— Model 6
120000.00
H— Model 7
110000.00
Model 8
Figure 5.14 The relationships of TOC* and x
On model 1, 2, and 5, from the starting point until they reach x = 0.05, the TOC*
values increase slightly. From x = 0.05, the TOC* on model 1, 2, and 3 increase almost
consistently and reach to a halt at x = 0.15.
112
On model 2, 3, 4, 6, 7, and 8, the TOC* value decrease constantly until x = 0.05.
Subsequentiy, as the value of x increases, the declination of TOC* value becomes less
and less on these models until they hah at x = 0.15.
Basically, model 1 and 5 have the most increase rate since the defective items do
not catch and fix before ship to customers. Hence, the penalty cost would incur in the
most in model 1 and 5. This also can be explained by the difference in inspection
programs. Model 1, 2, and 5 carry the zero inspection policy, while model 4, 6, 7, and 8
use 100%) inspection policy. Thus, when more defects in process, the total cost will be
increased in model 1, 2, and 5 cause of higher penalty costs. In this particular problem
(problem#4), the penalty cost is set to be higher than production cost.
5.8 Data and Graphical Interpretations
In this section, the graph representations of test problems show the relationships
of the optimal lot size (g*) and the optimal total annual cost (TOC*), and varying
parameters. The trends of g * and TOC* are investigated based on the traditional EPQ
assumptions. For example, the more setup cost leads to more production quantity, while
the more holding cost leads to less production quantity since the traditional EPQ model in
section 2.3.2 shows the compensation of setup cost and holding cost.
5.8.1 Effect of holding cost
We use the problem number 1, 2, and 3 to show the change of holding cost
parameter (H). In Figure 5.15 and 5.16, we plot the optimal lot size fg*^ and total annual
113
cost (TOC) as a function of holding cost (H). What is interesting about Figure 5.15 and
5.16 is that the optimal lot sizes decrease as the values of holding costs increase, and the
optimal annual costs increase as the values of holding costs increase. This suggests that
the holding cost increases, the manufacturer should produce less to avoid a big storage
cost in the total annual cost.
Relationships of H and Q^*
Model]
-' — ModeI2
-)K—Model3
-•— Model4
H— Model5
-•—Model6
500
—
Model?
—
Models
H ($/unit)
Figure 5.15 The relationships of g* and H
Relationships of TOC* and H
Model 1
Model2
^
•^K— Model3
120000
-•— Model4
o
^
110000
H—Models
100000
•—Model6
- — Model?
—
Figure 5.16 The relationships of TOC* and H
114
Models
5.8.2 Effect of setup cost
We use the problem number 1, 4, and 7 to show the change of setup cost
parameter (K). In Figure 5.17 and 5.18, we plot the optimal lot size fg*) and total annual
cost (TOC*) as a function of setup cost (K). What is interesting about Figure 5.17 and
5.18 is that the optimal lot sizes increase as the values of setup costs increase, and the
optimal annual costs increase as the values of setup costs increase. This suggests that the
setup cost increases, the manufacturer should produce bigger lot size in order to keep the
less number of cycles.
Relationships of Q* and K
10000
Model 1
8000
I
a
Model2
6000
- Model3
4000
- Model4
-ModeI5
2000
- Model6
0
5000
Figure 5.17 The relationships of g* and K
115
-Model?
-Models
Relationships of TOC* and K
-C: Model 1
>-~Model2
SK—ModeI3
• — Model4
H—Models
• — Model6
Model?
110000
•—Models
100000
Figure 5.18 The relationships of TOC* and K
5.8.3 Effect of production rate
We use the problem number 1 and 73 to show the change of production rate
parameter (P). In Figure 5.19 and 5.20, we plot the optimal lot size (Q*) and total annual
cost (TOC*) as a function of production rate (P). What is interesting about Figure 5.19
and 5.20 is that the optimal lot sizes decrease as the values of production rate increase,
and the optimal annual costs increase as the values of production rates increase. This
suggests that the production rate increases, the machine produces the item to satisfy
demand in the shorter time. This is also the consequence to avoid higher holding cost, so
the result is smaller lot size.
116
Relationships of Q* andP
Model 1
1400
1200
Model2
• ^ 1 ^ Model3
-•— Model4
H— Models
1500
5000
-•— Model6
^— Model?
P (Unit/time)
Models
Figure 5.19 The relationships of g* and P
Relationships of TOC* and P
125000
Modell
120000
•^1^- Model2
115000
-•— Model3
110000
H— Model4
105000
-•— Models
100000
ISOO
5000
P (Unit/time)
—
Model6
—
Model?
-•— Models
Figure 5.20 The relationships of TOC* and P
5.8.4 Effect of rework rate
We use the problem number 1 and 37 to show the change of rework rate
parameter (R). In Figure 5.21 and 5.22, we plot the optimal lot size (Q*) and total annual
cost (TOC*) as a function of rework rate (R). What is interesting about Figure 5.21 and
5.22 is that the optimal lot sizes stay constant as the values of rework rate increase, and
the optimal annual costs slightly increase or decrease as the values of production rates
117
increase depending on the particular model. This suggests that the rework rate increases,
the results would not change much because the number of defects (only 5%) is just a little
comparing with the optimal lot size. Thus, the rework rate is less significant in these test
problems. However, the rework rate (R) will have more effect when the values of 7? is
less than 25%) of producfion rate P since the rework rate is so slow in this case, then the
manufacturer needs to produce less in order to avoid holding cost of items.
The relationships of Q* and R
H
1000
950
900
Modell
§. 850
Q> 800
?50
^
^
Model2
4
4
Model3
~^
•
•
Model4
-9K—Models
-•— Model6
?00
3?5
?50
1125
1500
18?5
2250
H— Model?
Models
R (Units/time)
Figure 5.21 The relationships of g* and R
The relationships of TOC* and R
125000 n
120000 -
m
—•—Modell
M
—«—Model2
m
^
* 115000 -
Model3
' .
O 110000 105000 100000 -
()
Model4
— ^ 1 ^ Models
W
,
3?5
750
1125
1500
1875
MOUCIO
2250
Models
R (units/time)
Figure 5.22 The relafionships of TOC* and R
118
5.8.5 Effect of defective proportion Cx)
We use the problem number 1 and 19 to show the change of defective proportion
parameter (x). In Figure 5.23 and 5.24, we plot the optimal lot size (Q*) and total annual
cost (TOC*) as a funcfion of defective proportion (x). What is interesfing about Figure
5.23 and 5.24 is that the optimal lot sizes decrease as the values of defective proportion
increase, and the optimal annual costs increase as the values of defective proportion
increase. This suggests that the defective proportion increases, the machine produces
more defects. Thus, to avoid more defects and rework process (since rework process is
slower than regular process), the optimal lot size will decrease. The total annual cost is
also higher because the holding cost of defects, penalty and rework costs are incurred.
However, the results are not in this trend in every problem. For example, if setup cost
(K) is relatively very high comparing to the total penalty cost (FQx) and holding cost of
defective items, the optimal lot size (Q*) will increase while the x value increases.
Graphical presentation of Q* andx
Modell
Model2
•^1^— Model3
-•— Model4
H — Models
-•— Mode 16
0.15
0.05
—
Model?
—
Models
Figure 5.23 The relafionships of g* and x
119
Relationships of TOC* andx
150000
Modell
140000
S
X - Model2
130000
-^If- Models
120000
-•— Mode 14
«•
O
H
- I — Models
110000
100000
-•— ModeI6
0.05
Model?
0.15
- — Models
Figure 5.24 The relationships of TOC* and x
5.8.6 Effect of customer defective proportion (Y)
We use the problem number 1 and 10 to show the change of customer defective
proportion parameter (Y). In Figure 5.25 and 5.26, we plot the optimal lot size (Q*) and
total annual cost (TOC*) as a function of customer defective proportion (Y). What is
interesting about Figure 5.25 and 5.26 is that the optimal lot sizes increase and decrease
as the values of customer defective proportion increase, and the optimal annual costs
increase as the values of customers defective proportion increase. This suggests that the
customer defective proportion increases, the production has more additional demand.
However, each model has the different production characteristics and assumption to take
care of additional demand from customer return items. The total annual cost increase
when the customer defective proportion increases cause of higher in the holding cost of
defects, penalty and rework costs, while the total annual costs from model 2, 3, and 6 are
decreased. This is because the model 2, 3, and 6 have taken care of defects from
120
customer in rework process which the cost of rework is lower than the cost of regular
process.
Relationships of Q* and Y
1500
1400
1300
_ 1200
Modell
Model2
I 1100
D
a
- Models
1000
900
800
?00
600
500
- Model4
-Models
w - Moaelo
"
0.025
iviuuti /
0,1
Figure 5.25 the relationships of g* and Y
Relationships of TOC*and Y
Modell
• Model2
145000
•9K— Models
135000
•#— Model4
5 125000
O
^ 115000
H—Models
-•— Model6
Model?
—
105000
01
0.025
Figure 5.26 the relationships of TOC* and Y
111
Models
5.9 Sensitivity Analysis
This section shows the results of sensitivity analysis from 144 test problems
which we generate in section 5.5 and appendix D. The calculation of sensitivity analysis
is presented in section 5.3. Based on variable selection and inventory model
interpretation, there are 8 variables which are used to test the sensitivity analysis. Those
8 variables are production cost (C), penalty cost (F), holding cost (//), setup cost (/Q,
production rate (P), rework rate (i?), percent of intemal defects (x), and percent of
extemal defects (percent of broken down items at customers) (Y). For all parameters
except production rate (P) and rework rate (i?), we vary 4 levels at 50%, 75%, 125%, and
150% of original parameters, or we can imply that we decrease value at 25% and 50%
and increase values at 25% and 50% respectively. For production rate (P) and rework
rate (R), we vary at 20%), 90%, 110%, and 120% of original parameters cause of
limitation in feasible solution and assumption.
Tables of error on the optimal lot size (g*) show the value of ((g*vao' - Q'^onginai)/
g*ong/w)*100, while Tables of error on total annual cost (TOC*) show the value of
{{TOC*,ary-TOC*ongimj)/TOC*onginad''lOO. For Calculation cxamplc for Table 5.8, fiTSt,
the average of the optimal lot size from 144 test problems in each model is calculated for
each parameter change (such as 0.5C, 0.75C, C, 1.25C and 1.5C). Then using these
average values to find the error on optimal lot size as shown earlier of this paragraph.
Those values in each column in Table 5.8 represent the percentage of error on the optimal
lot size (g*) in each model. For calculation of the percentage total annual cost error, the
122
procedures are the same, but we need to use the optimal total annual cost values instead
of the optimal lot size values.
Table 5.8 Error on g* when variable C changes
% ERROR IN Q*
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Model 7
Model 8
0.5*C
0.75*C
41.43
15.47
41.42
15.47
41.42
15.48
41.42
15.48
41.42
15.47
41.43
15.48
41.42
15.47
26.96
10.07
C
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.25*C
-10.56
-10.56
-10.55
-10.56
-10.56
-10.55
-10.56
-6.91
1.5*C
-18.35
-18.35
-18.35
-18.35
-18.35
-18.35
-18.35
-12.04
" C " E F F E C T O F E R R O R ON Q
_
?
Di
§
g
*
50.00
40.00
30.00
20.00
10.00
0.00
-10.00
-20.00
-30.00
: ^
^
:^
^
:9)&
MODEL
Figure 5.27 Error on g * when variable C changes
For Figure 5.27 explanation example, the variable C with zero change, Q*
percentage error of all model will also be zero. The variable C with 50 percent change
will cause the percentage error in g* to change in the following model; model 1, 2, 3, 4,
5, 6, and 7 increase to 41.43, and model 8 increases to 26.96. This can be concluded that
123
the backorder consideration in model 8 has less effect for the optimal lot size error when
production cost changes.
Table 5.9 Error on TOC* when variable C changes
% ERROR IN
TOC*
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Model 7
Model 8
0,5*C
0.75*C
-37.57
-18.50
-41.13
-20.21
-36.54
-17.90
-40.49
-19.91
-37.56
-18.49
-40.29
-19.79
-41.24
-20.35
-40.85
-20.27
C
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.25*C
18.13
19.76
17.42
19.48
18.13
19.35
20.00
20.09
1,5*C
36.00
39.21
34.51
38.65
35.99
38.37
39.75
40.06
For Figure 5.28 explanation example, the variable C with zero change, TOC*
percentage error of all model will also be zero. The variable C with 50 percent change
will cause the percentage error in TOC* to change in the following model; model 1
decreases to -37.57; model 2 decreases to -41.13; model 3 decreases to -36.54; model 4
decreases to -40.49; model 5 decreases to -37.56; model 6 decreases to -40.29; model 7
decreases to -41.24; and model 8 decreases to -40.85. For all variable C change, TOC*
has pretty much the same error for all models.
124
C" EFFECT OF ERROR ON TOC*
60.00
g
40.00
g
20.00
i
0.00
•
^
•
^
^^•^=—5IF
=*
•
«
-
-X-
"5^
^
—•—0.5*C
—•—0.?S*C
c
w
*
•
—>^-1.25*C
-20.00
g ^0.00
-60.00
4
5
MODEL
Figure 5.28 Error on TOC* when variable C changes
Table 5.10 Error on g * when variable /^changes
% ERROR IN Q*
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Model 7
Model 8
0.5*F
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.75*F
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
F
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.25*F
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.5*F
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
"F" EFFECT OF ERROR ON Q*
50.00
^
40.00
-•—0.5*F
2'
30.00
•»~0.?S*F
I
20.00
F
10.00
V- 1,2S*F
w
^
0.00 -|-H«—^-«—r—afi-
-5K-
*
-5R—I—SR-
•«—1.5*F
-10.00
4
5
MODEL
Figure 5.29 Error on g * when variable F changes
125
Table 5.11 Error on TOC* when variable F changes
% ERROR IN
TOC*
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Model 7
Model 8
0.5*F
0.75*F
-7.67
-3.83
-3.14
-1.57
-6.64
-3.32
-3.12
-1.56
-7.67
-3.83
-3.07
-1.53
-3.33
-1.67
-3.65
-1.83
F
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.25*F
3.83
1.57
3.32
1.56
3.83
1.53
1.67
1.83
1.5*F
7.67
3.14
6.64
3.12
7.67
3.07
3.33
3.65
F" EFFECT OF ERROR ON TOC
25.00
5.00
tu
-5.00
*
*—0.5*F
15.00
RROR
?
0.?5*F
-X-—F
•^l^l.25*F
CJ
-15.00
-•—1.5*F
-25.00
Figure 5.30 Error on TOC* when variable Fchanges
Table 5.12 Error on g * when variable //"changes
% ERROR IN Q*
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Model 7
Model 8
0.75*H
0.5*H
15.47
41.43
15.47
41.42
15.48
41.42
15.48
41.42
15.47
41.42
15.48
41.43
15.47
41.42
10.07
26.96
126
H
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.25*H
-10.56
-10.56
-10.55
-10.56
-10.56
-10.55
-10.56
-6.91
1.5*H
-18.35
-18.35
-18.35
-18.35
-18.35
-18.35
-18.35
-12.04
" H " EFFECT OF ERROR ON Q*
^0 on -,
? 40.00 O 20.00 i
0.00 ^ ' . . ' . . ' ^ "•" ; . , - " T ,
*y -20.00 - JK—__j4t
m
ik
£
&
-40.00 1
2
3
4
5
6
-•—0.5*H
^^--ii
0.75*H
^Cr^"~^
t
•
-
—H
^l^l.25*H
7
—•—1.5*H
8
MODEL
Figure 5.31 Error on g * when variable //changes
Table 5.13 Error on TOC* when variable //changes
% ERROR IN
TOC*
0.5*H
0.75*H
H
1.25*H
1.5*H
Model 1
-6.74
-3.08
0.00
Model 2
-8.24
-3.77
0.00
2.72
3.32
5.17
6.32
Model 3
-8.71
0.00
3.51
6.68
Model 4
-7.89
-6.75
-3.98
-3.61
0.00
0.00
3.18
2.72
6.05
0.00
Model 5
Model 6
Model 7
-8.26
-6.44
Model 8
-2.69
-3.09
-3.78
-2.95
0.00
3.33
2.59
5.18
6.34
4.94
-1.19
0.00
1.01
1.89
" H " EFFECT O F ERROR ON TOC*
*—0.5*H
0.75*H
->e—H
•^l^l.25*H
•#—1.5*H
MODEL
Figure 5.32 Error on TOC* when variable //changes
127
Table 5.14 Error on g * when variable A^ changes
% ERROR IN Q*
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Model 7
Model 8
0.5*K
0.75*K
-29.29
-13.39
-29.29
-13.40
-29.28
-13.40
-29.29
-13.40
-29.15
-13.34
-29.29
-13.39
-24.84
-11.47
-29.29
-13.40
K
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.25*K
11.81
11.81
11.81
11.80
11.76
11.81
10.21
11.80
1.5*K
22.48
22.47
22.48
22.47
22.39
22.48
19.51
22.47
K'' EFFECT O F ERROR ON Q^
30.00
^_^ 20.00
0.5*K
10.00
0.00
q
-10.00
S
* -20.00
-30.00
-40.00
0.75*K
K
1.25*K
I.5*K:
MODEL
Figure 5.33 Error on g * when variable K changes
Table 5.15 Error on TOC* when variable /[T changes
% ERROR IN
TOC*
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Model 7
Model 8
0.75*K
0.5*K
-3.08
-6.74
-3.77
-8.24
-3.98
-8.71
-3.61
-7.89
-3.07
-6.72
-3.78
-8.26
-2.50
-5.42
-1.94
-4.23
128
K
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.25*K
2.72
3.32
3.51
3.18
2.71
3.33
2.23
1.71
1.5*K
5.17
6.32
6.68
6.05
5.16
6.34
4.26
3.25
K'» EFFECT OF ERROR ON TOC*
20.00
^
Q
2
10.00
0.00
jir^*
••—0.5*K
^
0.?5*K
PJ
U
-10.00
o
^
•^I^I.2S*K:
-20.00
1
2
3
4
5
•«—I.S*K
6
MODEL
Figure 5.34 Error on TOC* when variable ^ changes
Table 5.16 and Figure 5.35 represent data and trends of sensitivity analysis
when variable P changes respectively. However at 20% decreasing of F (0.8*P),
we will not have a feasible solution for model 1, 2, 3, and 5 since it is violated the
assumption that the production rate has to be greater than demand rate and
defective rate.
Table 5.16 Error on g* when variable P changes
% ERROR IN Q*
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Model 7
Model 8
0.8*P
N/A
N/A
N/A
40.65
N/A
25.94
35.05
22.92
0.9*P
19.93
12.86
6.11
11.38
19.93
8.57
6.63
6.95
129
P
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.1*P
-8.83
-6.93
-4.01
-6.41
-8.84
-5.25
-2.59
-4.14
1.2*P
-14.10
-11.38
-6.87
-10.60
-14.10
-8.84
-3.63
-6.93
"P" EFFECT OF ERROR ON Q
08*P
0.9*P
P
1.1*P
1.2*P
MODEL
Figure 5.35 Error on g* when variable P changes
Table 5.17 Error on TOC* when variable P changes
% ERROR IN
TOC*
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Model 7
Model 8
0.8*P
N/A
N/A
N/A
-4.41
N/A
-3.74
-3.81
-0.65
0.9*P
-1.98
-1.96
-1.25
-1.74
-1.98
-1.53
-1.19
-0.25
P
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.1*P
1.36
1.42
0.96
1.27
1.37
1.15
0.63
0.18
1.2*P
2.39
2.50
1.72
2.26
2.40
2.05
0.96
0.32
"P" EFFECT OF ERROR ON TOC*
g'
g
15.00 1
10.00 5.00 -
S
0.00 -
•— — • —
M
^ ••••A
-^ .,-—*•••—•
,....%.
1
., - - * - - , -
u
PJ
*
^—
~ ,
"W
*
B
'&r
.
• —
—•— U,o r
0.9*P
p
W ..1 1 +p
-#—1.2*P
g -10.00 -15,00 1
2
3
4
5
6
7
8
MODEL
Figure 5.36 Error on TOC* when variable F changes
130
Table 5.18 Error on g* when variable R changes
% ERROR IN Q*
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Model 7
Model 8
0.8*R
0.00
-0.03
-0.05
-0.05
0.00
-0.06
-1.05
-0.02
0.9*R
0.00
-0.01
-0.02
-0.02
0.00
-0.03
-0.48
-0.01
R
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.1*R
0.00
0.01
0.02
0.02
0.00
0.03
0.39
0.01
1.2*R
0.00
0.02
0.04
0.03
0.00
0.05
0.73
0.01
"R" EFFECT OF ERROR ON Q*
15.00
,„^
10.00
-0.8*R
^
5 00
0.9*R
g
PJ
0.00
-5.00
-1.1*R
*a
-10.00
-1.2*R
ei
n
R
-15.00
4
5
MODEL
Figure 5.37 Error on g* when variable R changes
Table 5.19 Error on TOC* when variable R changes
% ERROR IN
TOC*
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Model 7
Model 8
0.8*R
0.00
-0.25
0.01
0.01
0.00
0.01
0.25
0.00
0.9*R
0.00
-0.11
0.01
0.00
0.00
0.01
0.11
0.00
131
R
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.1*R
0.00
0.09
0.00
0.00
0.00
0.00
-0.09
0.00
1.2*R
0.00
0.16
-0.01
-0.01
0.00
-0.01
-0.17
0.00
"R" EFFECT OF ERROR ON TOC*
15.00
10.00
5.00
0,00
-5.00
-10.00
-15.00
MODEL
Figure 5.38 Error on TOC* when variable R changes
Table 5.20 Error on g* when variable x changes
% ERROR IN Q*
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Model 7
Model 8
0.5*x
29.85
34.14
44.23
38.81
29.72
38.83
44.08
43.24
0.75*x
10.51
12.42
16.62
14.42
10.45
14.43
16.38
16.00
X
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.25*x
-6.22
-8.05
-11.45
-9.78
-6.18
-9.77
-11.03
-10.74
1.5*x
-9.79
-13.60
-19.97
-16.96
-9.71
-16.95
-19.01
-18.49
"x" EFFECT O F ERROR ON Q'
60.00
-
40.00
^
20.00
g
0.00
- 0 'i+v
0.75*x
- ^
X
-1 25*x
-1.5*x
C^ -20.00
-40.00
4
5
MODEL
Figure 5.39 Error on g * when variable x changes
132
Table 5.21 Error on TOC* when variable x changes
% ERROR IN
TOC*
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Model 7
Model 8
0.5*x
-0.89
7.19
9.84
9.87
-0.90
10.36
5.65
3.53
0.75*x
-0.60
2.27
3.46
3.60
-0.60
3.78
1.99
1.20
X
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.25*x
1.79
-1.04
-2.12
-2.34
1.79
-2.47
-1.23
-0.69
1.5*x
4.29
-1.41
-3.49
-3.99
4.30
-4.20
-2.06
-1.09
x" EFFECT OF ERROR ON T O C
20.00
TOC* ERROR(=
^
15.00
10.00
5.00
0.00
-5.00
-10.00
4
5
MODEL
Figure 5.40 Error on TOC* when variable x changes
Table 5.22 Error on g * when variable /changes
% ERROR IN Q*
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Model 7
Model 8
0.5*Y
-4.54
1.22
5.33
-3.84
-4.54
0.13
-2.48
-2.98
0.75*Y
-2.35
0.58
2.55
-1.97
-2.36
0.08
-1.28
-1.52
133
Y
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.25*Y
2.58
-0.53
-2.34
2.10
2.58
-0.09
1.39
1.57
L5*Y
5.45
-1.07
-4.54
4.32
5.44
-0.25
2.90
3.22
' V EFFECT OF ERROR ON Q^
—•—- 0 . 5 * Y
0.?5*Y
Y
1 7<;*Y
-1.5*Y
Figure 5.41 Error on g * when variable /changes
Table 5.23 Error on TOC* when variable /changes
% ERROR IN
TOC*
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Model 7
Model 8
0.5*Y
0.75*Y
-4.82
-2.43
-6.05
-3.05
-1.55
-0.76
-5.15
-2.60
-4.82
-2.43
-3.01
-5.95
-5.67
-2.86
-3.15
-6.22
Y
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.25*Y
2.47
3.10
0.72
2.65
2.47
3.07
2.91
3.21
1.5*Y
5.02
6.30
1.46
5.40
5.02
6.27
5.86
6.55
Y" EFFECT OF ERROR ON TOC*
O.S*Y
0.?5*Y
Y
1.25*Y
1.5*Y
MODEL
Figure 5.42 Error on g * when variable F changes
134
5.9.1 Summary of sensitivity analvsis
Based on 144 test problems from 8 models in Appendix D, the sensitivity analysis
was then employed to know how changes in the optimal lot size (g*) and the optimal
total annual cost (TOC*) from variations in the various parameters of the model. One
obvious strange effect of variables x and /shows both increase and decrease based on
model characteristics. This can be explained for percentage of production defects (x) that
if trend of "x" is based on the compensation between setup cost (K) and total penalty cost
(FQx) with holding cost of both good and defective items. For example, if setup cost (K)
is relatively very high comparing to total penalty cost (FQx) and holding cost of both
good and defective items, the total cost (TOC*) will decrease while the x value increases
on all models. On the other hand, if setup cost (K) is relatively very high comparing to
total penalty cost (FQx) and holding cost of both good and defective items, the total cost
(TOC*) will decrease while the x value increases the x value increases as shown in
Figure 5.23. For the variable "Y", the increase or decrease trends on the optimal lot size
(g*) is based on problem characteristics, which is how to handle the defective items from
customers. Model 2, 3, and 6 handles the defects from customers by rework process
which is slower than regular process. Moreover, the cost of rework is also a lot lower
than regular production cost.
Basically from the results in the previous section, the variables production cost
(C), holding cost (//), and setup cost(^ are still the most sensitive in all models, so these
are variables which we have to be careful in terms of prediction and estimation.
However, the other variables such as production rate, percent of defects inside and
135
outside factory are also significant factors which we should not neglect. Otherwise, the
errors will be occurred.
Finally, it was concluded with the aid of marginal analyses that, depending on the
parameter being increased, the direction of the concomitant increase (+) or decrease (-),
where in some cases the direction of change depends on the particular data (±). Table
5.34 shows the effect on g * and TOC* when parameters change.
Table 5.24 The effects on g* and TOC* when parameters change
Parameter increased
Effect
Effect on
ong*
TOC*
Production cost (C)
(-)
(+)
Penalty cost (F)
(-)
(+)
Production rate (P)
(-)
(+)
Rework rate (R)
(+)
(+)
Setup & investment costs (K, Kj)
(-h)
(+)
Holding cost (//, h, and h/)
(-)
(+)
Proportion of defect (x) (Model 1 and 5)
(+)
Proportion of defect (x) (Model 2, 3, 4, 6, 7, and 8)
(-)
(-)
Proportion of defect (/) (Model 1,4, 5, 7, and 8)
(4-)
(+)
Proportion of defect (/) (Model 2,3, and 6)
136
(+)
CHAPTER VI
CONCLUSIONS, CONTRIBUTIONS, AND FUTURE RESEARCH
6.1 Conclusions
In previous chapters, we have presented production characteristics and procedure
to develop an economic production quantity mathematical model which integrates with
cost of quality.
This research incorporated the integrated method and imperfect items into the
inventory model. Moreover, additional costs such as costs of inspection and customer
retums were considered in the traditional economic production quantity (EPQ) as an
extended model. This approach has not been considered in previous research. The total
annual cost function has been derived. The research also indicated that the annual total
cost function possesses convexities that can derive an analytic solution procedure to
determine the optimal production quantity. Mathematica Software V5.0 were used as a
tool to find the solutions, and Microsoft Visual Studio.Net 2003 search solution
programming was used to validate the results from Mathematica in order to ensure that
the solutions are correct.
Thus, this research shows that we can not rely solely on traditional economic
production quantity (EPQ) since the EPQ is the approach which compensates for the
setup cost and holding cost of the item. However, the cost of quality, such as cost of
defects, inspection, etc. is another interesting aspect to be added in the traditional EPQ
137
model. For example, the problem of defective items and how they can influence
production quantity in order to minimize the total costs should be investigated.
The statistical analysis also showed that the optimal production quantity and total
annual cost of the extended model (economic cost of quality production quantity:
ECQPQ) are different from the traditional EPQ. A sensitivity analysis was employed to
show change in the optimal production quantity and total annual cost from variations in
the various parameters of the model. The results of statistical and sensitivity analysis are
discussed in Chapter 5.
By developing the economic cost of quality production quantity model, a firm
will leam something about the interrelatedness of parts of its operation, and may enjoy
improved performance as well. The greatest challenge of implementation is in
estimating/forecasting the full range of parameters needed to mn the models. The impact
of quality in economic quantity model cuts across departments within the firm, such as
purchasing, quality control, production, and planning. Furthermore, the results of this
mathematic model can be used in making decisions which affect the firm's vendors and
customers. Thus, an integrated effort within the firm and between the firm and its
vendors is required for full implementation/interpretation of parameters and problem
characters in the economic cost of quality production quantity model.
6.2 Contributions
This research provides significant contributions to the development of production,
quality, and inventory systems in the economic production quantity approach. From a
138
practical standpoint, the economic cost of quality production quantity model (ECQPQ) is
useful to help a manufacturer to produce the right amount of items in order to minimize
the total cost or to maximize the profit. The ECQPQ provides an excellent methodology
for manufacturers to effectively produce the item in a cost-effective manner especially
when the defective proportion is presented in the process since it is almost impossible to
produce zero defects. From a theoretical standpoint, this research shows how to extend
the well known economic production quantity (EPQ) in many different ways. The
ECQPQ provides more robustness to the model than the traditional EPQ.
Moreover, this research also uses Mathematica Software as a new tool to solve for
solutions, and MS Visual Studio.net to validate the results. By using new software and
programming, the practitioners and researchers will save time and money to get the
results. Finally, this research should be a good start for researchers and practitioners to
develop and extend the traditional EPQ in a way which corresponds to the actual
production process. Under various manufacturing conditions, the proposed ECQPQ
provides the manufacturer with opportunity to extend the traditional EPQ in order to
forecast accurate production quantity, which thereby minimizes the total cost.
6.3 Future Research
There are several aspects of this research that can be extended to incorporate more
realistic constraints. In general, the direction of future research should consist of
incorporating more realistic constraints and assumptions that will allow the economic
139
cost of quality production quantity model to be implemented in a real worid situation. In
the following sections, we will discuss possible research topics.
6.3.1 Multiple product types
One assumption of the traditional EPQ is "the item is a single product". To make
the research more practical, we can relax this assumption and assume that the same
machine can produce multiple product types. However, the basic problem is finding an
accurate estimation for all parameters such as the demand for multiple items, the
production rate for each product type, the holding cost for each product type, etc.
6.3.2 The capacity constraint
The current research regarding the ECQPQ does not consider the capacity of the
firm such as the holding space capacity and working machine time limit. To better
represent a real world environment, we might want to consider these capacities in the
ECQPQ. We can assume that the holding space has limited capacity such that only fixed
amount of item can be used. We may also consider the holding cost as a function of
space capacity.
6.3.3 The function of setup cost
The traditional EPQ was originally an approach which compensates for the setup
cost and holding cost of the item. The traditional EPQ considers setup cost as a constant.
To make the research more realistic, we can consider setup cost as a function of time.
140
For example, the setup cost is the cost of the time required to prepare the machine to do a
job. On the other hand, we can say that longer machine setup time will ensure better
quality in the process. However, the exact function has to be estimated by experience
workers/planners.
141
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146
APPENDIX A
MATHEMATICA SOFTWARE CODE
147
Model
1
w = D(x+Y)
D1+W'
P-Dl-W
t2
\ DI + W / '
ql=
Q
(P-D1-W)*(-^);
TOC[Q_] =
/ql* tl
ql* t2\
H* I—
+—
j + (C1*Q) +K +K1+ ( F * Q * x ) +
a
(F*Q*Y) + ( S l * Q * x ) + (S1*Q*Y)) / t )
//FullSinplify
aQTOC[Q] / / F u l l S i n p l i f y
2D1 (K + Kl) P + DlHQ^ + 2 (K + KI) PW + HQ^ (-P+W)
2PQ2
Solve[dgTOC[Q] == 0 , Q] / / R a l l S i n p l i f y
rr
j Q^
^^
r
Q^
'^
^2 ^TK+'Kiyp'"(DiTw)~ ^
____,
-\^
^(Dl-P+W)
^
V2~ ^TK+^')~P~(DiTwr
^fR ( D I - P + W)
D[TOC[Q],{Q,2}]//FullSimplify
2 (K + Kl) (Dl+W)
P=1600;Dl=1200;W=60;p=160;R=1000;x=0.1;Y=0.05;Cl=100;c=15;i
l;F=150;K=1500;Kl=15000;h=22;hl=40;H=20;Sl=15;
Solve[aQTOC[Q] == 0 , Q] / / F u l l S i i t p l i f y
{{Q^-3127.86},{Q^3127.86}}
Q=3127
3127
TOC[Q]
170478
Model
148
2
W = D(x+Y)
Q
t =
Dl+W'
ql
tl =
P-Dl- W
/B1*W
i-2 -
I DI
R #
/
q2
t3 =
' Dl+W'
q l =: ( P - D l --W) *
ii)'
q2 == q l + q 3 ;
q3 == ( R - D l --W) * ( t 2 ) ;
TOC[Q_] =
ql*tl
((-(
(ql+q2) * t2
+
2
h*R*t2*t2
q2* t 3 \
+
hl*Bl*Q*Y*t
+
2
2
/
2
_
+ (C1*Q) + ( c * q 3 ) +K +K1+ ( F * Q * x ) +
: S l * Q * x ) + (F*Q*Y) + (S1*Q*Y)) / t j
//FullSinplify
aQTOC[Q] / / F u l l S i n p l i f y
1
, ^.3
,r. ,..
..-.. T. T T ^ 2 . , .
. - , 2 T^^2,.^2
(-Dl^
(2
(K + Kl)
P+ HQ^) R + .Bl"PQ'^W" (H (R-W) + hW) +
2 D12 p Q2 R
BIDIQ^W
BI
DI Q^ ( B l h P W + H ( 2 P R - B 1 P W - 2RW)) +
Dl^R ( - 2 (K + Kl) PW+HQ^ ( P - W - 2 B 1 W ) + B l h l P Q " Y))
S o l v e [dQTOC[Q] ==0, Q] / / F u l l S i n p l i f y
149
| | Q ^
- | V 2 V D l ^ (K + Kl) PR (Dl + W)
(V (-Dl^HR+Bl^PViP (H (R-W) + hW) +
BIDIW ( B l h P W + H ( 2 P R - B 1 P W - 2 R W ) ) +
Dl^R (H ( P - W - 2 B 1 W ) + B l h l P Y ) ) ) j ,
Q-> (^2" V D l ^ (K + Kl) P R (Dl + W) )
/
(V ( - D l ^ H R + B l ^ P W ^ (H (R-W) +hW) +
BlDlW (BlhPW+H ( 2 P R - B 1 P W - 2 R W ) ) +
Dl^R (H ( P - W - 2B1W) + B l h l P Y ) ) ) j l
D[TOC [Q] , { Q , 2 } ] / / F u l l S i m p l i f y
2 (K + Kl) (Dl+W)
P=1600;Dl=120 0;W=60;p=160;R=1000;x=0-l;Y=0.05;Cl=100;c=15;i:
l;F=150;K=1500;Kl=15000;h=22;hl=40;H=20;Sl=15;Bl=0.5;
aQTOC[Q] / / F u l l S i n p l i f y
2.079x10'^
2.73829Q2
S o l v e [aQ TOC [Q] ==0, Q] / / F u l l S i q p l i f y
{{0^-2 755.42},{Q^2 755.42}}
Q=2756
2756
TOC [Q]
151362.
150
Model
w=
t=
3
D(x+Y)
t l + t 2 + t:3;
tl=
ql
P-Dl '
W*Q
t2 =
' D1*R '
t3 =
" DI'
q l =: ( P - D l )
•(I)'-
c^ =: q l + q 3 ;
q3 == ( R - D l - W) * ( t 2 ) ;
TOC[Q ]
/ ql *t l
( q l + q2) * t 2
q2 * t 3 \ h l * Q * Y * t l
h*R* t 2 * t 2
H*
+
+
1 +
+
V 22
2
2
/
2
2
/
2
2
(C1*Q) + ( c * Q * x ) + ( c * Q * Y ) +K + K1+ ( F * Q * x ) + (F*Q*Y) +
(Sl*Q*x) + (Sl*Q*Y)j / t )
+
//FullSinplify
aQTOC[Q] / / F u l l S i n p l i f y
(HPQ^ (R-W)^W^ + D1^HQ^R (PR+ 2W (-R+W)) +
2 D 1 P Q 2 R (D1R+ (R-W) W)
DIPQ^W ( h R W + H (2R^ 3RW + W^)) - Dl^ R^ (2 (K+KI) P + Q^ (H - h i Y) ) )
S o l v e [aQ TOC [Q] == 0 , Q] / / F u l l S i i t p l i f y
V T V D I ^ (K + Kl) P R 2 j / ( V ( H P (R-W)^W^ + D1^HR ( P R + 2 W (-R+W)) 4
DIPW (hRW+H ( 2 R ^ - 3 R W + W ^ ) ) + Dl^ R^ (-H + h i Y) ) ) | ,
Q-^ (-/2 V " D I 3 ( K + K l ) P R 2 J
/ ( V ( H P ( R - W) ^ W^ + D 1 ^ HR ( P R + 2 W (-R + W) )
DIPW (hRW+H (2R^ 3RW+W^)) + Dl^ R^ (-H + h l Y ) ) ) | }
D[TOC[Q],{Q,2}]//FullSimplify
151
2D1^ (K + Kl) R
Q3 (D1R+ (R-W) W)
P=1600;Dl=1200;W=60;p=160;R=1000;x=0.1;Y=0.05;Cl=100;c=15;i:
l;F=150;K=1500;Kl=15000;h=22;hl=40;H=20;Sl=15;
S o l v e [aQ TOC [Q] == 0 , Q] / / R a l l S i i t p l i f y
{(Q-^-2 3 7 4 . 4 5 } , { Q ^ 2 3 7 4 . 4 5 } }
0=2374.74
TOC [Q]
2374.74
161488.
152
Model
4
Q
Dl+W'
ql
tl
t2:
P-pl-Dl- W
/P*x * tl \
"i
)r
R
q2
Dl+W
q l = ( P - p l - D l - W ) * (Q- ) ;
P
q2 = q l + q 3 ;
q3= ( R - D l - W ) * ( t 2 ) ;
TOC[Q_] =
//
/ ql * t l
(ql+ q2) * t2
q2* tS v H* p i * t l * t l
( r ( ^ - ^ — 2 — ^ - ^ ) ^ — 2 — ^
h * R * t 2 * t2
2
+
(C1*Q) + (c*Q*x) +K + K1+ (Q*i) + (F*Q*Y) + (S1*Q*Y)) / t) / /
FullSinplify
aQTOC[Q] //EVillSiiiplify
1
.2
TT . T .
. V ^ 2
(-R (2D1 (K + Kl) P"
+T^.,D1H
(P-^ „2- pi)
Q" +
2P2Q2R
2 (K+Kl) P^ W+HQ^ ( - ( P - p l ) ^ + ( P - 2 p l ) W)) 2HPQ^R (DI- P + p l + W) x+ P^Q^ (DI (h-H) +H (R-W) + hW) x^)
Solve [aQ TOC [Q] == 0, Q] / / R i l l S i i t p l i f y
153
| | Q ^
-[AA2
V (K + Kl) P2R (Dl + W) j
/
( V ( - H R (DI ( P - 2 p l ) - ( P - p l ) ^ + ( P - 2 p l ) W)
2 H P R ( D l - P + pl+W) x + P^ (DI ( h - H ) +H (R-W) +hW) x ^ ) ) } ,
| Q ^ [^2
V (K+Kl) p2R (Dl+W) j
/
( V ( - H R (DI ( P - 2 p i ) - ( P - p l ) ^ + ( P - 2 p l ) W)
2 H P R ( D l - P + pl+W) x + P ^ (DI ( h - H ) +H (R-W) + hW) x^)) j l
D[TOC[Q],{Q,2}]//FullSimplify
2 (K + Kl) (Dl+W)
P=1600;Dl=1200;W=60;pl=160;R=1000;x=0.1;Y=0.05;Cl=100;c=l;i
l;F=150;K=1500;Kl=15000;h=22;hl=40;H=20;Sl=0;
Solve[aQTOC[Q] == 0 , Q] / / F u l l S i i t p l i f y
{{Q^-3118.63},{0^3118.63}}
Q=3119
3119
TOC[Q]
151933.
154
Model
5
W = D(x+Y)
Dl + W
P - Dl - W
t 2 = ( ^ ^ ) ;
\ Dl + W /
q l = ( P - D l - W) * ( —] ;
TOC[Q_] =
/ql*tl
ql*t2 \
H* I+~
1 + (C1*Q) + K+ K l + n * i + ( F * Q * x ) +
a
(F*Q*Y) + (Sl*Q*x) + (S1*Q*Y)) / t ) / /
aQTOC[Q] / / F u l l S i n p l i f y
2 D 1 (K + K l + i n ) P + D l H Q ^ + 2 ( K + K l + i n )
FullSimplify
PW+HQ^(-P+W)
2PQ2
Solve[dQTOC[Q] == 0 , Q] / /
rr
FullSimplify
'/~2 V ^ T K 7 1 Q T i ' n j ~ P ~ ( ' D r + w T ^
Q^
^^
r
Q^
^
'
A/H
( D l - P +W)
^
/ 2 ~ \ ^ T K + KlTi~n)~P~(lDr+wT -^ ^
VH
( D l - P + W)
^-^
D[TOC[Q],{Q,2}]//FullSimplify
2 (K+ Kl + i n ) (Dl + W)
P=1600;Dl=1200;W=60;p=160;R=1000;x=0.1;Y=0.05;Cl=100;c=15;n:
20;i=l;F=150;K=1500;Kl=15000;h=22;hl=40;H=20;Sl=15;
Solve[dQTOC[Q] == 0 , Q] / /
{{Q^-3129.76},{0^3129
FullSimplify
76}}
155
Q=3129.76
3129.76
TOC [Q]
170486.
156
Model
6
Q
Dl+W
tl =
t2
ql
P - p i - Dl - W
P*x*tl\
B1*Q*Y
R
q2
t 3 = ~-^
;
Dl+W
Q
ql= (P-pl-Dl-W) * (— ) ;
P
q2=ql+q3;
q3= ( R - D l - W ) * ( t 2 ) ;
TOC[Q ]
/ql*tl
(ql+q2) *t2
q2* t 3 \
H*pl*tl*tl
H*
+
+
+
+
\
2
2
2
/
2
hl*Bl*Q*Y*tl
h*R* t 2 * t2
+
+
2
2
(C1*Q) + ( c * Q * x ) + ( c * Q * B l * Y ) +K + K1+ ( Q * i ) + (F*Q* Y) +
(Sl*Q*Y)j / t j / / F u l l S i n p l i f y
aQTOC[Q] / / F u l l S i n p l i f y
2P2Q2R
(PW ( - 2 K P R - 2 K 1 P R + Q ^ ( B l h l R Y + h P (x + B l Y ) ^ ) ) +
HQ^ ( p l R ( p l + 2W) + P^ (-W ( x + B l Y ) ^ + R (1 + x + BlY)^)
PR (W+ 2 ( p l + p l x + W x + B l ( p l +W) Y)) ) Dl ( 2 K P ^ R + 2 K 1 P ^ R + Q ^ ( - P ( B l h l R Y + h P ( x + B l Y ) ^ ) •
H ( - 2 p l R + P^ ( x + B l Y ) ^ + PR (1 + 2 X + 2 B 1 Y ) ) ) ) )
Solve[aQTOC[Q] == 0, Q] / / F u l l S i n p l i f y
157
{|Q^
-(AA2
V (K + Kl) p2R (Dl + W) I /
(V(PW ( B l h l R Y + h P ( x + B l Y ) ^ ) + Dl (P ( B l h l R Y + h P ( x + B l Y ) ^ )
H ( - 2 p l R + P^ ( x + B l Y ) ^ f PR (1 + 2 X + 2 B 1 Y ) ) ) +
H ( p l R ( p l f 2W) + P^ (-W (X + B1Y)^ + R ( 1 + x + B l Y ) ^ )
PR (W+ 2 ( p l + p l x + Wx-^Bl ( p l + W) Y) ) ) ) ) } ,
|Q^
[42
V (K+Kl) p2R (Dl + W) j / (V (PW ( B l h l R Y + h P ( x + B l Y ) ^ ) +
Dl (P ( B l h l R Y + h P ( x + B l Y ) ^ ) H ( - 2 p l R + P^ (X+B1Y)^+ PR (1+ 2 X + 2 B 1 Y ) ) ) +
H ( p l R ( p l + 2W) +P^ (-W (X+B1Y)^ + R (1 + x + B l Y ) ^ ) P R (W+ 2 ( p l + p l x + W x + B l ( p l + W) Y ) ) ) )
D[TOC[Q],{Q,2}]//FullSimplify
2 (K + Kl) (Dl+W)
P=5000;Dl=1200;W=2 5;pl=25;R=62 5 0 ; x = 0 . 0 5 ; Y = 0 . 0 2 5 ; C l = 1 0 0 ; c = 1 5 ;
i=l;F=150;K=50;Kl=1000;h=550;hl=1000;H=100;Sl=15;Bl=0-5;
S o l v e [dQ TOC [Q] == 0 , Q] / / F u l l S i n p l i f y
{{Q^-171.054},{0^171.054}}
0=171.054
TOC[Q]
171.054
144736.
158
Model
t-
°
7
-
(Dl+W) '
tl-
*
(P - Dl - W -P)
-
t 2 = A* t l ;
q5
t 2 - (Dl + W) '
P*x*tl
t3 R
t4=
^
•
D1+w'
ql= (P-Dl-W-p)
* ( - ) ;
P
q2 = q l - q 5 ;
q4 = q2 + q 3 ;
q3= ( R - D l - W ) * t 3 ;
q5 = q l - q 2 ;
q5= (Dl+W) * A * t l ;
1400
K = 100 +
;
1 +A
TOC[QJ =
H*
\
ql*tl
+
+
(ql+c^)*t2
V 22
H*p*tl*tl
+
((^ + ( ^ ) * t 3
22
h*R*t3*t3
+
2
+
q4*t4
2
+
/
+
(C1*Q) + ( c * Q * x ) +K+K1+ (Q*i) + (F*Q*Y) + (S1*Q*Y))
RiOSdirplify
159
/t//
aQTOC[Q]
//FullSinplify
"on
A N n 2 . ^ n ( ^ ^ ( 2 (1500+K1 + A ( 1 0 0 + K1)) R - (1 + A) h o V ) +
2 (1+A) P-^^ Q^ R
Dl ((2 (1500+K1 + A ( 1 0 0 + K1)) P^ + (1 +A) H ( - 2 p + P) Q^) R +
2 ( l + A ) ^ H P Q ^ R x + (1+A) ( - h + H ) P^Q^x^) + (1+A) HQ^
(-R ( ( p - P ) ^ + ( 2 p - P ) W) + 2 P R ( p - P + W+AW) x - P ^ (R-W) x^))
Solve[aQTOC[Q] == 0 , Q] / /
FullSinplify
{ { Q - ^ - - / 2 ~ V - (1500 + Kl + A (100+ K1)) P^R (Dl + W)
/
( V ( ( l + A) (-HR ( ( p - P ) ^ + Dl ( 2 p - P ) + ( 2 p - P ) W) +
2 H P R (D1 +AD1 + P - P + W +AW) x P^ (Dl ( h - H ) +H (R-W) +hW) x^) ) ) } ,
|Q-4
42
V - ( 1 5 0 0 + K l + A (100+ K1) ) p2 R (Dl+W)
/
( V ( ( l + A) (-HR ( ( p - P ) ^ + Dl ( 2 p - P ) + ( 2 p - P ) W) +
2 H P R (D1 +AD1 + P - P +W +AW) x P^ (Dl ( h - H ) +H (R-W) +hW) x ^ ) ) ) | j
D [TOC [01 , { 0 / 2 } ] / / F u l l S i m p l i f y
2 (1500 + K l + A (100 + KI) ) (Dl + W)
(1+A) Q3
P=1600;Dl=1200;W=60;p=160;R=1000;x=0,l;Y=0.05;Cl=100;c=15;ii
l;F=150;Kl=15000;h=22;hl=40;H=20;Sl=15;A=0.2;
aQTOC[Q] / /
1.8226
FullSiitplify
2.0496x10
Solve[aQTOC[Q] == 0 , Q] / / F u l l S i m p l i f y
{{Q^-3353.43},{0^3353.43}}
160
Model
8
Dl + W
tl
t3 =
ql
P - p i - Dl - W
X ** t X
t l\
/ P
F ** X
R
q2
Dl+W
t 4 = —^^
;
Dl+W
q4
t5
^
P-pl-Dl-W
Q
ql= (P-pl-Dl-W) * ( - j
P
q2 = q l + q S ;
-q4;
q3= ( R - D l - W ) * ( t 2 ) ;
q4= (Dl+W) * t 4 ;
q4 = ( P - p l - D l - W ) * t 5 ;
TOC[Q_] =
ql*tl
((H.(
(ql+q2)*t2
q2*t3
2
2
2
H*pl* (tl+t5) * (tl+t5)
/
hl*B2*Q*Y*t
+
2
+
2
h*R*t2*t2
S 2 * q 4 * ( t 4 + t5)
+
^ ^
+ (C1*Q) +
2
2
( c * Q * x ) +K+K1+ (Q*i) + (F*Q*Y) + ( S l * Q * Y ) j / t j / /
FullSinplify
161
aQTOC[Q] / / F u l l S i n p l i f y
1 [ H p l (Dl + W)
2 I
P2
2 K (Dl + W)
Q2
2 K 1 (Dl + W)
Q2
( P - p l ) q4^ S2
Q2 ( D l - P + p l + W)
2 h (Dl + 1/^) (DIQ + P ( - Q + q 4 ) +Q ( p i + W)) x2
QR ( D l - P + p l + W )
h (Dl + W) (DIQ + P ( - Q + q 4 ) +Q ( p l + W))2 y?
Q 2 R ( D l - P + p l + W) 2
1
_ n J. rr A\
P2QR ( D l - P + p l + W)
( ( P - p l ) R ( D l - P + p l + W) +
2 P R (Dl - P + p i + W) X + P^ (Dl - R + W) x^) ) +
(H (D1Q+ P (-Q + q4) + Q ( p i + W) ) ^ ( (P - p i ) R (Dl - P + p i + W) +
2 P R ( D l - P + p l + W ) x + P ^ ( D l - R + W) x^)) /
P^Q^R ( D l - P + p l + W ) ^ ) + B 2 h l Y
Solve[aQTOC[Q] == 0, Q] / / F u l l S i i t p l i f y
162
{ { Q ^ - ( V (P^ (R ( D l - P + p l + W) (2D1^ ( K + K l )
( P - p l ) q4^ (H+ S2) - 2 (K+ KI) ( P - p l ) W +
2 (K+Kl) W^ + 2 D 1 ( K + K l ) ( - P + p l + 2W)) 2 H P q 4 ^ R ( D l - P + p l + W) x +
P^q4^ (Dl ( h - H ) +H (R-W) +hW) x ^ ) ) ) /
(V (- ( D l - P + p l + W ) ^ (Dl (H ( P - 2 p i ) R +
2 H P R X + ( - h + H) P^x^) H ( p l R ( p l + 2W) - PR ( 2 p l + W + 2 ( p l + W) x) + P^
(-Wx^ +R (1 + x ) ^ ) ) - P^ (hWx^ + B 2 h l R Y ) ) ) ) } ,
{Q-> (V (P^ (R ( D l - P + p l + W) (2D1^ (K+Kl) ( P - p l ) q4^ (H+ S2) - 2 (K+ KI) ( P - p l ) W +
2 (K+KI) W^+2D1 (K+KI)
(-P+pl+2W))-
2 H P q4^ R (Dl - P + p i + W) x +
P^q4^ (Dl ( h - H ) +H (R-W) +hW) x^) )) /
(V (- ( D l - P + p l + W ) ^ (Dl (H ( P - 2 p l ) R +
2 H P R X + ( - h + H) P^x^) H ( p l R ( p l + 2W) - PR ( 2 p l + W + 2 ( p l + W) x) +
P^ (-Wx^ + R ( 1 + x ) ^ ) ) - P ^ (hWx^ + B 2 h l R Y ) ) ) ) } }
D[TOC[Q],{0,2}]//FullSimplify
1
(R (Dl - P + p i + W)
Q 3 R ( D l - P + p l +W)2
(2D1^ (K+ KI) - ( P - p l ) q4^ (H + S2) - 2 (K+ KI) ( P - p l ) W
2 (K+ KI) W^+ 2 D 1 (K+ KI) ( - P + p i + 2 W) ) - 2 HP q4^ R
( D l - P + p l + W ) x + P^q4^ (Dl ( h - H ) +H (R-W) + hW) x^)
163
9q4T0C[Q] / / F u l l S i t t p l i f y
^
f ( D l + W) f
Q ( D l - P + p l + W)2 [
[
( P - p l ) q 4 S 2 ( D l - P + p l + W)
Dl+W
h P ( D I Q + P (-Q + q4) +Q ( p l + W ) ) x^
1
R
PR (Dl+W)
(H (D1Q+ P (-Q + q4) +Q ( p l + W) ) ( ( P - p i ) R (Dl - P + pl+W) +
2 P R (Dl - P + p i + W) X + P^ (Dl - R + W) x^) ,
Solve[aq4T0C[Q] == 0 , q4] / /
FullSinplify
{ { q 4 ^ (Q ( D l - P + p l + W ) (-H ( P - p l ) R ( D l - P + p l + W) - 2 H P R
( D l - P + p l + W ) x + P^ (Dl ( h - H ) +H (R-W) + hW) x^) ) /
(P ( ( P - p l ) R (H+S2) ( D l - P + p l + W) + 2 H P R
( D l - P + p l + W ) X - P^ (Dl ( h - H ) +H (R-W) + hW) x^)) }}
P=1500;Dl=1000;W=25;pl=75;R=375;x=0.05;Y=0.025;Cl=100;c=l;i:
l;F=150;K=50;Kl=10 0 0 ; h = l l ; h l = 2 0;H=10;Sl=0;S2=30;B2=0.1;
S o l v e [ a g TOC[Q] == 0 , Q] / / F u l l S i n p l i f y
{{Q-^-44 6 . 8 02} . ( 0 ^ 4 4 6 . 8 0 2 } }
0=186.402
TOC[0]
q4
186.402
157979.
5.64855
164
APPENDIX B
MICROSOFT VISUAL STUDIO.NET CODE
165
Model 1
Private Sub Buttonl_Click(ByVal sender As System.Object, ByVal e As
System.EventArgs) Handles Buttonl.Click
Dim fTOC As Double
Dim Q As Double
Dim PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, n, h, hi, H2, SI
As Double
PI = Val(TextBoxl.Text) '1600
Dl = Val (TextBox2 .Text) U 2 0 0
W = Val(TextBox3.Text) '60
p = Val(TextBox4.Text) '160
R = Val(TextBoxS.Text) '1000
X = Val(TextBox6.Text) '0.1
Y = Val(TextBox7.Text) '0.05
CI = Val(TextBoxB.Text) '100
c = Val(TextBox9.Text) '15
i = Val(TextBoxlO.Text) '1
F = Val(TextBoxll.Text) '150
K = Val(TextBoxl2.Text) '1500
KI = Val(TextBoxl3.Text) '15000
SI = Val(TextBoxl4.Text) '15
h = Val(TextBoxie.Text) '22
hi = Val(TextBoxl7.Text) '40
H2 = Val(TextBoxlB.Text) '20
'Q = Val(TextBoxl9.Text)
Dim tocl, toc2 As Double
Dim countl As Integer
Dim max, min As Integer
countl = 1
tocl = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, h,
hi, H2, SI)
countl = 100
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, h,
hi, H2, SI)
While tocl > toc2
TextBoxl9.Text = countl
countl += 100
tocl = toc2
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K,
KI, h, hi, H2, SI)
End While
166
max = countl
If (countl
200) < 1 Then
min = 1
Else
min = countl - 2 00
End If
tocl = TOC(min, PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, h,
hi, H2, SI)
toc2 = TOC(min + 1, PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI,
h, hi, H2, SI)
countl = min + 1
While tocl > toc2
TextBoxl9.Text = countl
countl += 1
tocl = toc2
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K,
KI, h, hi, H2, SI)
End While
fTOC = TOC(countl
KI, h, hi, H2, SI)
1, PI, Dl, W, p, R, x, Y, CI, c, i, F, K,
TextBoxl9.Text = countl
MessageBox.Show(fTOC)
1
End Sub
Private Sub Contextiyienul_Popup (ByVal sender As System.Object, ByVal
e As System.EventArgs)
End Sub
Public Function TOC(ByVal Q As Double, ByVal PI As Double, ByVal Dl
As Double, ByVal W As Double, ByVal p As Double, ByVal R As Double,
ByVal x As Double, ByVal Y As Double, ByVal CI As Double, ByVal c As
Double, ByVal i As Double, ByVal F As Double, ByVal K As Double, ByVal
KI As Double, ByVal h As Double, ByVal hi As Double, ByVal H2 As Double,
SI as Double) As Double
Dim ft, ftl, ft2, fql As Double
Dim ffunl, ffun2 As Double
Dim f u n d As New Forml
Dim func2 As New T0C2
Dim func3 As New T0C3
fql = func3.ql(Pl, Dl, W, Q)
ftl = func3.tl(fql, PI, Dl, W)
ft2 = t2(fql, W, Dl)
ft = funcl.t(Q, Dl, W)
ffunl = funl(H2, fql, ftl, ft2)
ffun2 = fun2(Cl, Q, K, KI, F, x, Y, si)
167
TOC = (ffunl + ffun2) / ft
End Function
' t function use TOCl
' tl function use T0C2
' ql function use T0C3
Public Function t2(ByVal fql As Double, ByVal W As Double, ByVal Dl
As Double) As Double
t2 = (fql / (Dl + W))
End Function
Public Function funl(ByVal H2 As Double, ByVal fql As Double, ByVal
ftl As Double, ByVal ft2 As Double) As Double
funl = ((H2 * (((fql * ftl) / 2) + ((fql * ft2) / 2))))
End Function
Public Function fun2(ByVal CI As Double, ByVal Q As Double, ByVal K
As Double, ByVal KI As Double, ByVal F As Double, ByVal x As Double,
ByVal Y As Double, ByVal SI As Double) As Double
fun2 = ((CI * Q) + K + KI + (F * Q * x) + (F * Q * Y) + (SI * Q
* x) + (SI * Q * Y))
End Function
168
Model 2
Private Sub Buttonl_Click(ByVal sender As System.Object, ByVal e As
System.EventArgs) Handles Buttonl.Click
Dim fTOC As Double
Dim Q As Double
Dim PI, Dl, W, p, R, X, Y, CI, c, i, F, K, KI, n, h, hi, H2, SI,
BI As Double
PI = Val(TextBoxl.Text) '1600
Dl = Val(TextBox2-Text) '1200
W = Val(TextBox3.Text) '60
p = Val(TextBox4.Text) '160
R = Val(TextBoxB.Text) '1000
X = Val(TextBox6.Text) '0.1
Y = Val(TextBox7.Text) '0.05
CI = Val(TextBoxB.Text) '100
c = Val(TextBox9.Text) '15
i = Val(TextBoxlO.Text) '1
F = Val(TextBoxll.Text) '150
K = Val(TextBoxl2.Text) '1500
KI = Val(TextBoxl3.Text) '15000
SI = Val(TextBoxl4.Text)
BI = Val(TextBoxlS.Text)
h = Val(TextBoxie-Text) '22
hi = Val(TextBoxlV.Text) '40
H2 = Val(TextBoxlB.Text) '20
'Q = Val(TextBoxl9.Text)
Dim tocl, toc2 As Double
Dim countl As Integer
Dim max, min As Integer
countl = 1
tocl = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F. K, KI, h,
hi, H2, SI, BI)
countl = 100
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, h,
hi, H2, SI, BI)
Vmile tocl > toc2
TextBoxl9.Text = countl
countl += 100
tocl = toc2
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K,
KI, h, hi, H2, SI, BI)
End While
169
max = countl
If (countl
200) < 1 Then
min = 1
Else
min = countl
200
End If
tocl = TOC{min, PI, Dl, W, p, R, x, Y, CI, c, i, F. K, KI, h,
hi, H2, SI, BI)
toc2 = TOC(min + 1, Pi, Dl, W, p, R, x, Y, CI, c, i, F, K, KI,
h, hi, H2, SI, BI)
countl = min + 1
While tocl > toc2
TextBoxl9.Text = countl
countl += 1
tocl = toc2
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K,
KI, h, hi, H2, SI, BI)
End While
fTOC = TOC(countl
KI, h, hi, H2, SI, BI)
1, PI, Dl, W, p, R, x, Y, CI, c, i, F. K,
TextBoxl9-Text = countl
MessageBox.Show(fTOC)
1
End Sub
Private Sub ContextMenul_Popup(ByVal sender As System.Object, ByVal
e As System.EventArgs)
End Sub
Public Function TOC(ByVal Q As Double, ByVal PI As Double, ByVal Dl
As Double, ByVal W As Double, ByVal p As Double, ByVal R As Double,
ByVal X As Double, ByVal Y As Double, ByVal CI As Double, ByVal c As
Double, ByVal i As Double, ByVal F As Double, ByVal K As Double, ByVal
KI As Double, ByVal h As Double, ByVal hi As Double, ByVal H2 As Double,
ByVal SI As Double, ByVal BI As Double) As Double
Dim ft, ftl, ft2, ft3, fql, fq2, fq3 As Double
Dim ffunl, ffun2, ffun3 As Double
Dim f u n d As New Forml
Dim func2 As New T0C2
fql = ql(Pl, Dl, W, Q)
ftl = t K f q l , PI, Dl, W)
ft2 = t2 (BI, W, Dl, Q, R)
fq3 = func2.q3{R, Dl, W, ft2)
fq2 = funci.q2(fql, fq3)
ft = funcl.t(Q, Dl, W)
ft3 - funcl.t3(fq2, Dl, W)
170
ffunl = func2.funl(H2, fql, ftl, fq2, ft2, ft3)
ffun2 = fun2(hl, BI, Q, Y, ft, h, R, ft2)
ffun3 = fun3(Cl, Q, fq3, c, x, K, KI, i, F, Y, SI)
TOC = (ffunl + ffun2 + ffun3) / ft
End Function
Public Function tl(ByVal ql As Double, ByVal PI As Double, ByVal Dl
As Double, ByVal W As Double) As Double
tl = ql / (PI
Dl
W)
End Function
Public Function t2(ByVal BI As Double, ByVal W As Double, ByVal Dl
As Double, ByVal Q As Double, ByVal R As Double) As Double
t2 = ((BI * W / Dl) * (Q / R))
End Function
Public Function ql(ByVal PI As Double, ByVal Dl As Double, ByVal W
As Double, ByVal Q As Double) As Double
ql = (PI
Dl
W) * (Q / PI)
End Function
Public Function fun2(ByVal hi As Double, ByVal BI As Double, ByVal
Q As Double, ByVal Y As Double, ByVal ft As Double, ByVal h As Double,
ByVal R As Double, ByVal ft2 As Double) As Double
fun2 = ( (hi * BI * Q '^ Y * ft) / 2) + ( (h * R * ft2 * ft2) / 2)
End Function
Public Function fun3(ByVal CI As Double,
fq3 As Double, ByVal c As Double, ByVal x As
ByVal KI As Double, ByVal i As Double, ByVal
Double, ByVal SI As Double) As Double
fun3 = (CI * Q) + (c * fq3) + K + KI
Y)
End Function
Private Sub MenuIteml_Click(ByVal sender
As System.EventArgs)
MessageBox.Show("help")
End Sub
171
ByVal Q As Double, ByVal
Double, ByVal K As Double,
F As Double, ByVal Y As
+ (F * Q * Y) + (SI * Q *
As System.Object, ByVal e
Model 3
Private Sub Buttonl_Click(ByVal sender As System.Object, ByVal e As
System.EventArgs) Handles Buttonl.Click
Dim fTOC As Double
Dim Q As Double
Dim PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, n, h, hi, H2, SI
As Double
PI = Val(TextBoxl.Text) '1600
Dl = Val(TextBox2-Text) '1200
W = Val(TextBox3.Text) '60
p = Val(TextBox4.Text) '160
R = VaKTextBoxS .Text) '1000
X = Val(TextBox6.Text) '0.1
Y = Val(TextBox7.Text) '0.05
CI = Val(TextBoxB-Text) '100
c = Val(TextBox9.Text) '15
i = Val(TextBoxlO.Text) '1
F - Val(TextBoxll.Text) '150
K = Val(TextBoxl2-Text) '1500
KI = Val(TextBoxl3.Text) '15000
SI = Val(TextBoxl4.Text) '15
h = Val(TextBoxie-Text) '22
hi = Val(TextBoxl7.Text) '40
H2 = Val(TextBoxlB.Text) '20
'Q = Val(TextBoxl9.Text)
Dim tocl, toc2 As Double
Dim countl As Integer
Dim max, min As Integer
countl = 1
tocl = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, h,
hi, H2, SI)
countl = 100
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, h,
hi, H2, SI)
V^hile tocl > toc2
TextBoxl9.Text = countl
countl += 100
tocl = toc2
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K,
KI, h, hi, H2, SI)
End While
max = countl
172
If (countl - 200) < 1 Then
min = 1
Else
min = countl
200
End If
tocl = TOC(min, PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, h,
hi, H2, SI)
toc2 = TOC(min + 1, PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI,
h, hi, H2, SI)
countl = min + 1
While tocl > toc2
TextBoxl9.Text = countl
countl += 1
tocl = toc2
toc2 = TOC{countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K,
KI, h, hi, H2, SI)
End While
fTOC = TOC(countl
KI, h, hi, H2, SI)
1, PI, Dl, W, p, R, x, Y, CI, c, i, F, K,
TextBoxl9.Text = countl
MessageBox.Show(fTOC)
1
End Sub
Private Sub ContextMenul_Popup(ByVal sender As System.Object, ByVal
e As System.EventArgs)
End Sub
Public Function TOC(ByVal Q As Double, ByVal PI As Double, ByVal Dl
As Double, ByVal W As Double, ByVal p As Double, ByVal R As Double,
ByVal X As Double, ByVal Y As Double, ByVal CI As Double, ByVal c As
Double, ByVal i As Double, ByVal F As Double, ByVal K As Double, ByVal
KI As Double, ByVal h As Double, ByVal hi As Double, ByVal H2 As Double,
ByVal SI As Double) As Double
Dim ft, ftl, ft2, ft3, fql, fq2, fq3 As Double
Dim ffunl, ffun2, ffun3 As Double
fql = ql(Pl, Dl, Q)
ft2 = t2(W, Q, Dl, R)
fq3 = q3(R, Dl, W, ft2)
fq2 = q2(fql, fq3)
ftl = tl(fql, PI, Dl)
ft3 = t3(fq2, Dl)
ft = t (ftl, ft2, ft3)
ffunl = funl{H2, fql, ftl, fq2, ft2, ft3)
ffun2 = fun2(hl, Q, Y, ftl, h, R, ft2)
ffun3 = fun3(Cl, Q, c, K, KI, F, x, Y, SI)
173
TOC = (ffunl + ffun2 + ffun3) / ft
End Function
Public Function t(ByVal ftl As Double, ByVal ft2 As Double, ByVal
ft3 As Double) As Double
t = ftl + ft2 + ft3
End Function
Public Function tl(ByVal fql As Double, ByVal PI As Double, ByVal
Dl As Double) As Double
tl = fql / (PI
Dl)
End Function
Public Function t2(ByVal W As Double, ByVal Q As Double, ByVal Dl
As Double, ByVal R As Double) As Double
t2 = ((W * Q) / (Dl * R))
End Function
Public Function t3(ByVal fq2 As Double, ByVal Dl As Double) As
Double
t3 = fq2 / Dl
End Function
Public Function ql(ByVal PI As Double, ByVal Dl As Double, ByVal Q
As Double) As Double
ql = (PI
Dl) * (Q / PI)
End Function
Public Function q2(ByVal fql As Double, ByVal fq3 As Double) As
Double
q2 - fql + fq3
End Function
Public Function q3(ByVal R As Double, ByVal Dl As Double, ByVal W
As Double, ByVal ft2 As Double) As Double
q3 = (R
Dl
W) * ft2
End Function
Public Function funl(ByVal H2 As Double, ByVal fql As Double, ByVal
ftl As Double, ByVal fq2 As Double, ByVal ft2 As Double, ByVai ft3 As
Double) As Double
funl = ((H2 * (((fql * ftl) / 2) + (((fql + fq2) * ft2) / 2) +
((fq2 * ft3) / 2))))
End Function
Public Function fun2(ByVal hi As Double, ByVal Q As Double, ByVal Y
As Double, ByVal ftl As Double, ByVal h As Double, ByVal R As Double,
ByVal ft2 As Double) As Double
fun2 = ((hi '^ Q * Y * ftl) / 2) + ((h * R * ft2 * ft2) / 2)
End Function
Public Function fun3(ByVal CI As Double, ByVal Q As Double, ByVal c
As Double, ByVal K As Double, ByVal KI As Double, ByVal F As Double,
ByVal X As Double, ByVal Y As Double, ByVal SI As Double) As Double
fun3 = ( (CI * Q) + (c * Q * x) + (c * Q * Y) + K + KI + (F * Q
* x) + (F * Q * Y) + (SI ^ Q * x) + (SI * Q * Y) )
End Function
174
Model 4
Private Sub Buttonl_Click(ByVal sender As System.Object, ByVal e As
System.EventArgs) Handles Buttonl.Click
Dim fTOC As Double
Dim Q As Double
Dim PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, n, h, hi, H2, SI
As Double
Dim MIDparent As New T0C2
PI = Val(TextBoxl.Text) '1600
Dl = Val(TextBox2.Text) '1200
W = Val(TextBox3.Text) '60
p = Val(TextBox4.Text) '160
R = VaKTextBoxS. Text) '1000
X = Val(TextBox6.Text) '0.1
Y = Val(TextBox7.Text) '0.05
CI = Val(TextBoxB.Text) '100
c = Val(TextBox9.Text) '15
i = Val(TextBoxlO.Text) '1
F = Val(TextBoxll.Text) '150
K = Val(TextBoxl2.Text) '1500
KI = Val(TextBoxl3.Text) '15000
SI = Val{TextBoxl4.text)
h = Val(TextBoxie.Text) '22
hi = Val(TextBoxlV.Text) '40
H2 = Val(TextBoxlB.Text) '20
'Q = Val(TextBoxl9.Text)
Dim tocl, toc2 As Double
Dim countl As Integer
Dim max, min As Integer
countl = 1
tocl = MIDparent.TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F,
K, KI, h, hi, H2, SI)
countl = 100
toc2 = MIDparent.TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F,
K, KI, h, hi, H2, SI)
While tocl > toc2
TextBoxl9.Text = countl
countl += 100
tocl = toc2
toc2 = MIDparent.TOC(countl, PI, Dl, W, p, R, x, Y, CI, c,
i, F. K, KI, h, hi, H2, SI)
End While
175
max = countl
If (countl - 200) < 1 Then
min = 1
Else
min = countl - 200
End If
tocl = MIDparent.TOC(min, PI, Dl, W, p, R, x, Y, CI, c, i, F, K,
KI, h, hi, H2, SI)
toc2 = MIDparent.TOC(min + 1, PI, Dl, W, p, R, x, Y, CI, c, i,
F, K, KI, h, hi, H2, SI)
countl = min + 1
While tocl > toc2
TextBoxl9.Text = countl
countl += 1
tocl = toc2
toc2 = MIDparent.TOC(countl, PI, Dl, W, p, R, x, Y, CI, c,
i, F, K, KI, h, hi, H2, SI)
End While
fTOC = MIDparent.TOC(countl
i, F, K, KI, h, hi, H2, SI)
TextBoxl9.Text = countl
MessageBox.Show(fTOC)
1, PI, Dl, W, p, R, x, Y, CI, c,
1
End Sub
Private Sub ContextMenul_Popup(ByVal sender As System.Object, ByVal
e As System.EventArgs)
End Sub
Public Function TOC(ByVal Q As Double, ByVal PI As Double, ByVal Dl
As Double, ByVal W As Double, ByVal p As Double, ByVal R As Double,
ByVal X As Double, ByVal Y As Double, ByVal CI As Double, ByVal c As
Double, ByVal i As Double, ByVal F As Double, ByVal K As Double, ByVai
KI As Double, ByVal h As Double, ByVal hi As Double, ByVal H2 As Double,
ByVal SI As Double) As Double
Dim ft, ftl, ft2, ft3, fql, fq2, fq3 As Double
Dim ffunl, ffun2, ffun3, ffun4 As Double
fql = ql(Pl, Dl, W, p, Q)
ftl = tl(fql, PI, Dl, W, p)
ft2 = t2(ftl, Q, PI, R, Y, x)
fq3 = q3(R, Dl, W, ft2)
fq2 = q2(fql, fq3)
ft = t(Q, Dl, W)
ft3 = t3(fq2, Dl, W)
ffunl = funl(H2, fql, ftl, fq2, ft2, ft3)
ffun2 = fun2(H2, p, ftl, hi, W, ft, h, R, ft2)
ffun4 = fun3(Cl, Q, c, x, K, KI, i, F, Y, SI)
176
TOC = (ffunl + ffun2 + ffun3 + ffun4) / ft
End Function
Public Function t(ByVal Q As Double, ByVal Dl As Double, ByVal W As
Double) As Double
t = Q / (Dl+W)
End Function
Public Function tl(ByVal ql As Double, ByVal PI As Double, ByVal Dl
As Double, ByVal W As Double, ByVal p As Double) As Double
tl = ql / (PI
p
Dl - W)
End Function
Public Function t2(ByVal ftl As Double, ByVal Q As Double, ByVal PI
As Double, ByVal R As Double, ByVal Y As Double, ByVal x As Double) As
Double
t2 = ( (PI * X * ftl) / R) '+ ( (Y * Q) / P.)
End Function
Public Function t3 (ByVal q2 As Double, ByVal Dl As Double, ByVal W
As Double) As Double
t3 = q2 / (Dl + W)
End Function
Public Function ql(ByVal PI As Double, ByVal Dl As Double, ByVal W
As Double, ByVal p As Double, ByVal Q As Double) As Double
ql = (PI
p
Dl
W) * (Q / PI)
End Function
Public Function q2(ByVal fql As Double, ByVal fq3 As Double) As
Double
q2 = fql + fq3
End Function
Public Function q3(ByVal R As Double, ByVal Dl As Double, ByVal W
As Double, ByVal ft2 As Double) As Double
q3 = (R
Dl
W) * (ft2)
End Function
Public Function funl(ByVal H2 As Double, ByVal fql As Double, ByVal
ftl As Double, ByVal fq2 As Double, ByVal ft2 As Double, ByVal ft3 As
Double) As Double
funl = H2 * (((fql * ftl) / 2) + (((fql + fq2) * ft2) / 2) +
((fq2 * ft3) / 2))
End Function
Public Function fun2(ByVal H2 As Double, ByVal p As Double, ByVal
ftl As Double, ByVal hi As Double, ByVal W As Double, ByVal ft As
Double, ByVal h As Double, ByVal R As Double, ByVal ft2 As Double) As
Double
fun2 = (((H2 * p * ftl * ftl) / 2) + ((h * R * ft2 * ft2) / 2))
' ( (hi -^ W ' ft -^ ft) / 2) +
End Function
177
Public Function fun3(ByVal CI As Double, ByVal Q As Double, ByVal
As Double, ByVal x As Double, ByVal K As Double, ByVal KI As Double,
ByVal i As Double, ByVal F As Double, ByVal Y As Double, ByVal SI As
Double) As Double
fun3 = (CI * Q) + (c * Q * x) + K + KI + (Q * i) + (F * Q * Y
+ (SI * Q * Y)
End Function
178
Model 5
Private Sub Buttonl_Click(ByVal sender As System.Object, ByVal e As
System.EventArgs) Handles Buttonl.Click
Dim fTOC As Double
Dim Q As Double
Dim PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, n. Pa, h, hi, H2
As Double
Dim MIDparent As New Forml
PI = Val(TextBoxl.Text) '1600
Dl = Val(TextBox2.Text) '1200
W = Val(TextBox3.Text) '60
p = Val(TextBox4.Text) '160
R = Val(TextBoxB.Text) '1000
X = Val(TextBox6.Text) '0.1
Y = Val(TextBox7.Text) '0.05
CI - Val(TextBoxB.Text) '100
c = Val(TextBox9.Text) '15
i = Val(TextBoxlO.Text) '1
F = Val(TextBoxll.Text) '150
K = Val(TextBoxl2.Text) '1500
KI = Val(TextBoxl3.Text) '15000
n = Val(TextBoxl4.Text) '20
Pa = Val(TextBoxlB.Text) '0.1
h = Val(TextBoxl6.Text) '22
hi = Val(TextBoxlV.Text) '40
H2 = Val(TextBoxlB.Text) '20
'Q = Val(TextBoxl9.Text)
Dim tocl, toc2 As Double
Dim countl As Integer
Dim max, min As Integer
countl = 1
tocl = MIDparent.TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F,
K, KI, n. Pa, h, hi, H2)
countl = 100
toc2 = MIDparent.TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F,
K, KI, n, Pa, h, hi, H2)
While tocl > toc2
TextBoxl9.Text = countl
countl += 100
tocl = toc2
toc2 = MIDparent.TOC(countl, PI, Dl, W, p, R, x, Y, CI, c,
i, F, K, KI, n. Pa, h, hi, H2)
End While
max = countl
179
If (countl
200) < 1 Then
min = 1
Else
min = countl
200
End If
tocl = MIDparent.TOC(min, PI, Dl, W, p, R, x, Y, CI, c, i, F, K,
KI, n. Pa, h, hi, H2)
toc2 = MIDparent.TOC(min + 1, Pi, Dl, W, p, R, x, Y, CI, c, i,
F. K, KI, n. Pa, h, hi, H2)
countl = min + 1
While tocl > toc2
TextBoxl9.Text = countl
countl += 1
tocl = toc2
toc2 = MIDparent.TOC(countl, PI, Dl, W, p, R, x, Y, CI, c,
i, F, K, KI, n. Pa, h, hi, H2)
End While
fTOC = MIDparent.TOC(countl
i, F. K, KI, n. Pa, h, hi, H2)
TextBoxl9.Text = countl
MessageBox.Show(fTOC)
1, PI, Dl, W, p, R, x, Y, CI, c,
1
End Sub
Public Function TOC(ByVal Q As Double, ByVal PI As Double, ByVal Dl As
Double, ByVal W As Double, ByVal p As Double, ByVal R As Double, ByVal
X As Double, ByVal Y As Double, ByVal CI As Double, ByVal c As Double,
ByVal i As Double, ByVal F As Double, ByVal K As Double, ByVal KI As
Double, ByVal n As Double, ByVal Pa As Double, ByVal h As Double, ByVal
hi As Double, ByVal H2 As Double) As Double
Dim ft, ftl, ft2, ft3, fql, fq2, fq3 As Double
Dim ffunl, ffun2, ffun3, ffun4 As Double
fql = ql(Pl, Dl, W, p, Q)
fq3 = q3 (PI, Q, p, Dl, R, W)
fq2 = q2(fql, fq3)
ft = t(Q, Dl, W)
ftl = t K f q l , PI, Dl, W, p)
ft2 = t2 (p, Q, PI, R)
ft3 = t3(fq2, Dl, W)
ffunl = funl(H2, fql, ftl, ft2)
ffun2 = fun2(H2, Q, Dl, p, PI, W, fq3, ft3)
ffun3 = fun3(hl, W, ft, h, R, ft2)
ffun4 = fun4(Cl, Q, c, x, K, KI, n, i. Pa, F, Y)
TOC = (ffunl + ffun2 + ffun3 + ffun4) / ft
End Function
Public Function t(ByVal Q As Double, ByVal Dl As Double, ByVal W As
Double) As Double
180
t = Q / (Dl+W)
End Function
Public Function tl(ByVal ql As Double, ByVal PI As Double, ByVal Dl
As Double, ByVal W As Double, ByVal p As Double) As Double
tl = ql / (PI
Dl
W
p)
End Function
Public Function t2(ByVal p As Double, ByVal Q As Double, ByVal PI
As Double, ByVal R As Double) As Double
t2 = (p * Q) / (PI * R)
End Function
Public Function t3 (ByVal q2 As Double, ByVal Dl As Double, ByVal W
As Double) As Double
t3 = q2 / (Dl + W)
End Function
Public Function ql(ByVal PI As Double, ByVal Dl As Double, ByVal W
As Double, ByVal p As Double, ByVal Q As Double) As Double
ql = (PI
Dl - W - p) * (Q / PI)
End Function
Public Function q2(ByVal fql As Double, ByVal fq3 As Double) As
Double
q2 = fql + fq3
End Function
Public Function q3(ByVal PI As Double, ByVal Q As Double, ByVal p
As Double, ByVal Dl As Double, ByVal R As Double, ByVal W As Double) As
Double
q3 = ( ( (p * Q) / PI)
( (p * Q * Dl) / (PI * R) )
( (p * Q * W)
/ (PI * R)))
End Function
Public Function funl(ByVal H2 As Double, ByVal fql As Double, ByVal
ftl As Double, ByVal ft2 As Double) As Double
funl = H2 * (((fql * ftl) / 2) + ((fql * ft2) / 2))
End Function
Public Function fun2(ByVal H2 As Double, ByVal Q As Double, ByVal
Dl As Double, ByVal p As Double, ByVal PI As Double, ByVal W As Double,
ByVal fq3 As Double, EyVal ft3 As Double) As Double
fun2 = H2 * (((Q
( (Dl * Q) / PI)
( (W * Q) / PI)
( (p * Q)
/ PI)) + fq3) / 2) * ft3
End Function
Public Function fun3(ByVal hi As Double, ByVal W As Double, ByVal
ft As Double, ByVal h As Double, ByVal R As Double, ByVal ft2 As Double)
As Double
fun3 = (((hi * W * (ft * ft)) / 2) + ((h * R * (ft2 * ft2)) /
2))
End Function
181
Public Function fun4(ByVal CI As Double, ByVal Q As Double, ByVal c
As Double, ByVal x As Double, ByVal K As Double, ByVal KI As Double,
ByVal n As Double, ByVal i As Double, ByVal Pa As Double, ByVal F As
Double, ByVal Y As Double) As Double
fun4 = (CI * Q) + (c * Q * x) + K + KI + (n * i) + (i * (1 Pa) ) * (Q
n) + (F * Q * Y)
End Function
182
Model 6
Private Sub Buttonl_Click(ByVal sender As System.Object, ByVal e As
System.EventArgs) Handles Buttonl.Click
Dim fTOC As Double
Dim Q As Double
Dim PI, Dl, W, p, R, X, Y, CI, c, i, F, K, KI, n, h, hi, H2, SI,
BI As Double
PI = Val(TextBoxl.Text) '1600
Dl = Val(TextBox2.Text) '1200
W = Val(TextBox3.Text) '60
p = Val(TextBox4.Text) '160
R = Val(TextBoxB.Text) '1000
X = Val(TextBox6.Text) '0.1
Y = Val(TextBoxV.Text) '0.05
CI = Val(TextBoxB.Text) '100
c = Val(TextBox9.Text) '15
i = VaKTextBoxlO-Text) '1
F = Val(TextBoxll.Text) '150
K = Val(TextBoxl2.Text) '1500
KI = Val(TextBoxl3.Text) '15000
SI = Val(TextBoxl4.Text) '15
BI = Val(TextBoxlB.Text)
h = Val(TextBoxl6.Text) '22
hi = Val(TextBoxlV.Text) '40
H2 = Val(TextBoxlB.Text) '20
'Q = Val(TextBoxl9.Text)
Dim tocl, toc2 As Double
Dim countl As Integer
Dim max, min As Integer
countl = 1
tocl = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, h,
hi, H2, SI, BI)
countl = 100
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, h,
hi, H2, SI, BI)
While tocl > toc2
TextBoxl9.Text = countl
countl += 100
tocl = toc2
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K,
KI, h, hi, H2, SI, BI)
End While
max = countl
183
If (countl
200) < 1 Then
min = 1
Else
min = countl
200
End If
tocl = TOC(min, PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, h,
hi, H2, SI, BI)
toc2 = TOC(min + 1, PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI,
h, hi, H2, SI, BI)
countl = min + 1
While tocl > toc2
TextBoxl9.Text = countl
countl += 1
tocl = toc2
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F. K,
KI, h, hi, H2, SI, BI)
End While
fTOC = TOC(countl
KI, h, hi, H2, SI, BI)
1, Pi, Dl, W, p, R, x, Y, CI, c, i, F, K,
TextBoxl9.Text = countl
MessageBox.Show(fTOC)
1
End Sub
Private Sub ContextMenul_Popup(ByVal sender As System.Object, ByVal
e As System.EventArgs)
End Sub
Public Function TOC(ByVal Q As Double, ByVal PI As Double, ByVal Dl
As Double, ByVal W As Double, ByVal p As Double, ByVal R As Double,
ByVal X As Double, ByVal Y As Double, ByVal CI As Double, ByVal c As
Double, ByVal i As Double, ByVal F As Double, ByVal K As Double, ByVal
KI As Double, ByVal h As Double, ByVal hi As Double, ByVal H2 As Double
ByVal SI As Double, ByVal BI As Double) As Double
Dim ft, ftl, ft2, ft3, fql, fq2, fq3, fq4 As Double
Dim ffunl, ffun2, ffun3 As Double
ft = t(Q, Dl, W)
fql = ql(Pl, p, Dl, W, Q)
ftl = t K f q l , PI, p, Dl, W)
ft2 = t2(Pl, X, ftl, R, BI, Q, Y)
fq3 = q3(R, Dl, W, ft2)
fq2 = q2(fql, fq3)
ft3 = t3(fq2, Dl, W)
ffunl = funl(H2, ftl, fql, fq2, ft2, ft3)
184
ffun2 = fun2(H2, p, ftl, hi, BI, Q, Y, h, R, ft2)
ffun3 = fun3(Cl, Q, c, K, KI, i, F, x, Y, SI)
TOC = (ffunl + ffun2 + ffun3) / ft
End Function
Public Function t (ByVal Q As Double, ByVal Dl As Double, ByVal W As
Double) As Double
t = Q / (Dl+W)
End Function
Public Function tl(ByVal fql As Double, ByVal PI As Double, ByVal p
As Double, ByVal Dl As Double, ByVal W As Double) As Double
tl = (fql) / (PI
p
Dl - W)
End Function
Public Function t2(ByVal PI As Double, ByVal x As Double, ByVal ftl
As Double, ByVal R As Double, ByVal BI As Double, ByVal Q As Double,
ByVal Y As Double) As Double
t2 = ((PI * X * ftl) / R) + ((BI * Q * Y) / R)
End Function
Public Function t3(ByVal fq2 As Double, ByVal Dl As Double, ByVal W
As Double) As Double
t3 = fq2 / (Dl + W)
End Function
Public Function ql(ByVal PI As Double, ByVal p As Double, ByVal Dl
As Double, ByVal W As Double, ByVal Q As Double) As Double
ql = ((PI
p
Dl
W) * (Q / PI))
End Function
Public Function q2(ByVal fql As Double, ByVal fq3 As Double) As
Double
q2 = fql + fq3
End Function
•Public Function q3(ByVal fq2 As Double, ByVal fq4 As Double) As
Double
q3 = fql + fq4
'End Function
Public Function q3(ByVal R As Double, ByVal Dl As Double, ByVal W
As Double, ByVal ft2 As Double) As Double
q3 = (R
Dl
W) * ft2
End Function
Public Function funl(ByVal H2 As Double, ByVal ftl As Double, ByVal
fql As Double, ByVal fq2 As Double, ByVal ft2 As Double, ByVal ft3 As
Double) As Double
funl = ((H2 * (((fql * ftl) / 2) + (((fql + fq2) * ft2) / 2) +
((fq2 * ft3) / 2))))
End Function
185
Public Function fun2(ByVal H2 As Double, ByVal p As Double, ByVal
ftl As Double, ByVal hi As Double, ByVal BI As Double, ByVal Q As
Double, ByVal Y As Double, ByVal h As Double, ByVal R As Double, ByVal
ft2 As Double) As Double
fun2 = ((H2 * p * ftl * ftl) / 2) + ((hi * BI * Q * Y * ftl) /
2) + ((h * R * ft2 * ft2) / 2)
End Function
Public Function fun3(ByVal CI As Double, ByVal Q As Double, ByVal c
As Double, ByVal K As Double, ByVal KI As Double, ByVal i As Double,
ByVal F As Double, ByVal x As Double, ByVal Y As Double, ByVal SI As
Double) As Double
fun3 = ( (CI * Q) + (c * Q * x) + K + KI + (Q *^ i) + (F * Q * Y)
+ (SI * Q * Y))
End Function
186
Model 7
Private Sub Buttonl_Click(ByVal sender As System.Object, ByVal e As
System.EventArgs) Handles Buttonl.Click
Dim fTOC As Double
Dim Q As Double
Dim PI, Dl, W, p, R, X, Y, CI, c, i, F, KI, n, h, hi, H2, SI, A
As Double
PI = Val(TextBoxl.Text) '1600
Dl = Val(TextBox2.Text) '1200
W = VaKTextBox3.Text) '60
p = Val(TextBox4.Text) '160
R = Val(TextBoxB.Text) '1000
X = Val(TextBox6.Text) '0.1
Y = Val(TextBoxV.Text) '0.05
CI = Val(TextBoxB-Text) '100
c = Val(TextBox9.Text) '15
i = VaKTextBoxlO. Text) '1
F = Val(TextBoxll.Text) '150
KI = Val(TextBoxl3.Text) '15000
SI = Val(TextBoxl4.Text) '15
A = Val(TextBoxlB.Text)
h = Val(TextBoxie.Text) '22
hi = Val(TextBoxlV.Text) '40
H2 = VaKTextBoxlB.Text) '20
'Q = Val(TeztBoxl9.Text)
Dim tocl, toc2 As Double
Dim countl As Integer
Dim max, min As Integer
countl = 1
tocl = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, KI, h,
hi, H2, SI, A)
countl = 100
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, KI, h,
hi, H2, SI, A)
While tocl > toc2
TextBoxl9-Text = countl
countl += 100
tocl = toc2
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, KI,
h, hi, H2, SI, A)
End While
187
max = countl
If (countl
200) < 1 Then
min = 1
Else
min = countl
2 00
End If
tocl = TOC(min, PI, Dl, W, p, R, x, Y, CI, c, i, F, KI, h, hi,
H2, SI, A)
toc2 = TOC(min + 1, PI, Dl, W, p, R, x, Y, CI, c, i, F, KI, h,
hi, H2, SI, A)
countl = min + 1
While tocl > toc2
TextBoxl9.Text = countl
countl += 1
tocl = toc2
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, KI,
h, hi, H2, SI, A)
End While
fTOC = TOC(countl - 1, PI, Dl, W, p, R, x, Y, CI, c, i, F, KI,
h, hi, H2, SI, A)
TextBoxl9.Text = countl
MessageBox.Show(fTOC)
1
End Sub
Private Sub Form2_Load(ByVal sender As System.Object, ByVal e As
System.EventArgs) Handles MyBase.Load
TextBoxl.Text = " 1 6 0 0 "
TextBox2.Text = " 1 2 0 0 "
TextBox3.Text = " 6 0 "
TextBox4.Text = " 1 6 0 "
TextBoxB.Text = " 1 0 0 0 "
TextBox6.Text = " 0 . 1 "
TextBoxV.Text = " O . O B "
TextBoxB.Text = " 1 0 0 "
TextBox9.Text = " 1 5 "
TextBoxlO.Text : 11 -] 11
TextBoxll.Text : " I B O "
'TextBoxl2.Text = " 1 5 0 0 "
TextBoxl3.Text = "IBOOO"
TextBoxl4.Text - "IB"
TextBoxlB.Text = "0.2"
TextBoxl6.Text = "22"
TextBoxlV.Text = "40"
TextBoxlB.Text = "20"
188
TextBoxl9.Text = "0 "
End Sub
Private Sub Button2_Click(ByVal sender As System.Object, ByVal e As
System.EventArgs) Handles Button2.Click
TextBoxl.Text =
TextBox2.Text =
TextBox3.Text =
TextBox4.Text =
TextBoxB.Text =
TextBox6.Text =
TextBoxV.Text =
TextBoxB.Text =
TextBox9.Text =
TextBoxlO.Text =
TextBoxll.Text =
TextBoxlB.Text =
TextBoxl3.Text =
TextBoxl4.Text =
TextBoxl6.Text
TextBoxlV.Text
TextBoxlB.Text
TextBoxl9.Text
End Sub
11 n
II 11
II II
It It
Private Sub ContextMenul_Popup(ByVal sender As System.Object, ByVal
e As System.EventArgs)
End Sub
Public Function TOC(ByVal Q As Double, ByVal PI As Double, ByVal Dl
As Double, ByVal W As Double, ByVal p As Double, ByVal R As Double,
ByVal X As Double, ByVal Y As Double, ByVal CI As Double, ByVal c As
Double, ByVal i As Double, ByVal F As Double, ByVal KI As Double, ByVal
h As Double, ByVal hi As Double, ByVal H2 As Double, ByVal SI As Double,
ByVal A As Double) As Double
Dim ft, ftl, ft2, ft3, ft4, fql, fq2, fq3, fq4, fqS As Double
Dim ffunl, ffun2, ffun3 As Double
ft = t(Q, Dl, W)
fql = ql(Pl, p, Dl, W, Q)
ftl = t K f q l , PI, p, Dl, W)
ft2 = t2(A, ftl)
ft3 = t3(Pl, X, ftl, R)
fq3 = q3(R, Dl, W, ft3)
ft4 = t4(fq3, Dl, W)
fqS ==qB(Dl, W, A, ftl)
fq2 = q2(fql, fqB)
fq4 = q4(fq2, f q3)
ffunl = funl(H2, ftl, fql, fq2, ft2
ffun2 = fun2(H2, p, ftl, h, R, ft3)
ffun3 = fun3(CI, Q, c. A, KI, i, F,
189
ft3, fq4, ft4)
Y, SI)
TOC = (ffunl + ffun2 + ffun3) / ft
End Function
Public Function t(ByVal Q As Double, ByVal Dl As Double, ByVal W As
Double) As Double
t = Q / (Dl+W)
End Function
Public Function tl(ByVal fql As Double, ByVal PI As Double, ByVal p
As Double, ByVal Dl As Double, ByVal W As Double) As Double
tl = (fql) / (PI
Dl
W
p)
End Function
Public Function t2(ByVal A As Double, ByVal ftl As Double) As
Double
t2 = A * ftl
End Function
Public Function t3(ByVal PI As Double, ByVal x As Double, ByVal ftl
As Double, ByVal R As Double) As Double
t3 = (PI * X * ftl) / R
End Function
Public Function t4(ByVal fq3 As Double, ByVal Dl As Double, ByVal W
As Double) As Double
t4 = fq3 / (Dl + W)
End Function
Public Function ql(ByVal PI As Double, ByVal p As Double, ByVal Dl
As Double, ByVal W As Double, ByVal Q As Double) As Double
ql = ((PI
p
Dl
W) * (Q / PI) )
End Function
Public Function q2(ByVal fql As Double, ByVal fqB As Double) As
Double
q2 = fql
fqB
End Function
Public Function q3(ByVal R As Double, ByVal Dl As Double, ByVal W
As Double, ByVal ft3 As Double) As Double
q3 = (R
Dl
W) * ft3
End Function
Public Function q4(ByVal fq2 As Double, ByVal fq3 As Double) As
Double
q4 = fq2 + fq3
End Function
Public Function qB(ByVal Dl As Double, ByVal W As Double, ByVal A
As Double, ByVal ftl As Double) As Double
qB = ( D l + W ) * A * ftl
End Function
190
Public Function funl(ByVal H2 As Double, ByVal ftl As Double, ByVal
fql As Double, ByVal fq2 As Double, ByVal ft2 As Double, ByVal ft3 As
Double, ByVal fq4 As Double, ByVal ft4 As Double) As Double
funl = ((H2 * (((fql * ftl) / 2) + (((fql + fq2) * ft2) / 2) +
(((fq2 + fq4) * ft3) / 2) + ((fq4 * ft4) / 2))))
End Function
Public Function fun2(ByVal H2 As Double, ByVal p As Double, ByVal
ftl As Double, ByVal h As Double, ByVal R As Double, ByVal ft3 As
Double) As Double
fun2 = ((H2 * p * ftl * ftl) / 2) + ((h * R * ft3 * ft3) / 2)
End Function
Public Function fun3(ByVal CI As Double, ByVal Q As Double, ByVal c
As Double, ByVal A As Double, ByVal KI As Double, ByVal i As Double,
ByVal F As Double, ByVal x As Double, ByVal Y As Double, ByVal SI As
Double) As Double
Dim K As Double
K = 100 + (1400 / (1 + A))
fun3 = ((CI * Q) + (c * Q * x) + K + KI + (Q * i) + (F * Q * Y)
+ (SI * Q * Y))
End Function
191
Model 8
Private Sub Buttonl_Click(ByVal sender As System.Object, ByVal e As
System.EventArgs) Handles Buttonl.Click
Dim fTOC As Double
Dim Q As Double
Dim PI, Dl, W, p, R, X, Y, CI, c, i, F, K, KI, n, h, hi, H2, SI,
B2, S2 As Double
PI = Val(TextBoxl.Text) '1600
Dl = Val(TextBox2.Text) '1200
W = Val(TextBox3.Text) '60
p = Val(TextBox4.Text) '160
R = Val(TextBoxB.Text) '1000
X = Val(TextBox6.Text) '0.1
Y = Val(TextBoxV.Text) '0.05
CI = Val(TextBoxB.Text) '100
c = Val(TextBox9.Text) '15
i = VaKTextBoxlO.Text) '1
F = Val(TextBoxll.Text) '150
K = Val(TextBoxl2.Text) '1500
KI - Val(TextBoxl3.Text) '15000
SI = Val(TextBoxl4.Text) '15
B2 = VaKTextBoxlB.Text)
h = Val(TextBoxie.Text) '22
hi = Val(TextBoxlV.Text) '40
H2 = Val(TextBoxlB.Text) '20
S2 = Val(TextBox2 0.Text) '30
'Q = Val(TextBoxl9.Text)
Dim tocl, toc2 As Double
Dim countl As Integer
Dim max, min As Integer
count1=1
tocl = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, h,
hi, H2, SI, B2, S2)
countl = 100
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, h,
hi, H2, SI, B2, S2)
While tocl > toc2
TextBoxl9.Text = countl
countl += 100
tocl = toc2
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K,
KI, h, hi, H2, SI, B2, S2)
End While
192
max = countl
If (countl
200) < 1 Then
min = 1
Else
min = countl
200
End If
tocl = TOC(min, PI, Dl, W, p, R, x, Y, CI, c, i, F, K, KI, h,
hi, H2, SI, B2, S2)
toc2 = TOC(min + 1, Pi, Dl, W, p, R, x, Y, CI, c, i, F, K, KI,
h, hi, H2, SI, B2, S2)
countl = min + 1
While tocl > toc2
TextBoxl9.Text = countl
countl += 1
tocl = toc2
toc2 = TOC(countl, PI, Dl, W, p, R, x, Y, CI, c, i, F, K,
KI, h, hi, H2, SI, B2, S2)
End While
fTOC = TOC(countl
KI, h, hi, H2, SI, B2, S2)
1, PI, Dl, W, p, R, x, Y, CI, c, i, F, K,
TextBoxl9.Text = countl
MessageBox.Show(fTOC)
1
End Sub
Private Sub Form2_Ijoad(ByVal sender As System.Object, ByVal e As
System.EventArgs) Handles MyBase.Load
TextBoxl.Text = "1600"
TextBox2.Text = "1200"
TextBox3.Text = "60"
TextBox4.Text = "160"
TextBoxB.Text = "1000"
TextBox6.Text = "0.1"
TextBoxV.Text = "O.OB"
TextBoxB.Text = "100"
TextBox9.Text = "IB"
TextBoxlO.Text = "1"
TextBoxll.Text = "IBO"
TextBoxl2.Text = "1500"
TextBoxl3.Text = "ISOOO"
TextBoxl4.Text = "IB"
TextBoxlB.Text = "0.1"
TextBox2 0.Text = "30"
193
TextBoxl6.Text
TextBoxlV.Text
TextBoxlB.Text
TextBoxl9.Text
=
=
=
=
"22"
"40"
"20"
"0"
End Sub
Private Sub Button2_Click(ByVal sender As System.Object, ByVal e As
System.EventArgs) Handles Button2.Click
TextBoxl.Text = ""
TextBox2.Text = ""
TextBox3.Text = ""
TextBox4.Text = ""
TextBoxB.Text = ""
TextBox6.Text = ""
TextBoxV.Text = ""
TextBoxB.Text = ""
TextBox9.Text = ""
TextBoxlO.Text = ""
TextBoxll.Text = ""
TextBoxl2.Text = ""
TextBoxl3-Text = ""
TextBoxl4.Text = ""
TextBox2 0.Text = ""
TextBoxl6.Text
TextBoxlV.Text
TextBoxlB-Text
TextBoxl9.Text
End Sub
=
=
=
=
""
""
""
""
Private Sub ContextMenul_Popup(ByVal sender As System.Object, ByVal
e As System.EventArgs)
End Sub
Public Function TOC(ByVal Q As Double, ByVal PI As Double, ByVal Dl
As Double, ByVal W As Double, ByVal p As Double, ByVal R As Double,
ByVal X As Double, ByVal Y As Double, ByVal CI As Double, ByVal c As
Double, ByVal i As Double, ByVal F As Double, ByVal K As Double, ByVal
KI As Double, ByVal h As Double, ByVal hi As Double, ByVal H2 As Double,
ByVal SI As Double, ByVal B2 As Double, ByVal S2 As Double) As Double
Dim ft, ftl, ft2, ft3, ft4, ftB, fql, fq2, fq3, fq4 As Double
Dim ffunl, ffun2, ffun3 As Double
ft = t(Q, Dl, W)
fq4 = q4(Q, PI, Dl, W, p, R, x, h, H2, S2)
fql = ql(Pl, p, Dl, W, Q, fq4)
ftl = t K f q l , PI, p, Dl, W)
ft2 = t2(PI, X, ftl, R)
fq3 = q3(R, Dl, W, ft2)
fq2 = q2(fql, fq3)
ft3 = t3(fq2, Dl, W)
ft4 = t4(fq4, Dl, W)
194
ftB = tB(fq4, PI, p, Dl, W)
ffunl = funl(H2, ftl, fql, fq2, ft2, ft3)
ffun2 = fun2(H2, p, ftl, ftB, hi, B2, Q, Y, ft, h, R, ft2, S2,
fq4, ft4)
ffun3 = fun3(Cl, Q, c, K, KI, i, F, x, Y, SI)
TOC = (ffunl + ffun2 + ffun3) / ft
End Function
Public Function t(ByVal Q As Double, ByVal Dl As Double, ByVal W As
Double) As Double
t = Q / (Dl+W)
End Function
Public Function tl(ByVal fql As Double, ByVal PI As Double, ByVal p
As Double, ByVal Dl As Double, ByVal W As Double) As Double
tl = (fql) / (PI - p
Dl
W)
End Function
Public Function t2(ByVal PI As Double, ByVal x As Double, ByVal ftl
As Double, ByVal R As Double) As Double
t2 = ((PI * X * ftl) / R)
End Function
Public Function t3(ByVal fq2 As Double, ByVal Dl As Double, ByVal W
As Double) As Double
t3 = fq2 / (Dl + W)
End Function
Public Function t4(ByVal fq4 As Double, ByVal Dl As Double, ByVal W
As Double) As Double
t4 = fq4 / (Dl + W)
End Function
Public Function tB(ByVal fq4 As Double, ByVal PI As Double, ByVal p
As Double, ByVal Dl As Double, ByVal W As Double) As Double
t5 = fq4 / (PI
p
Dl - W)
End Function
Public Function qKByVal PI As Double, ByVal p As Double, ByVai Dl
As Double, ByVal W As Double, ByVal Q As Double, ByVal fq4 As Double)
As Double
ql = ((PI
p
Dl
W) * (Q / PI)
fq4)
End Function
Public Function q2(ByVal fql As Double, EyVal fq3 As Double) As
Double
q2 = fql + fq3
End Function
Public Function q3(ByVal R As Double, ByVal Dl As Double, ByVal W
As Double, ByVal ft2 As Louble) As Double
q3 = (R - Dl
W) * ft2
End Function
195
Public Function q4(ByVal Q As Double, ByVal PI As Double, ByVal Dl
As Double, ByVal W As Double, ByVal p As Double, ByVal R As Double,
ByVal X As Double, ByVal h As Double, ByVal H2 As Double, ByVal S2 As
Double) As Double
Dim uptemp, downtemp As Double
uptemp = Q * (Dl
PI + p + w) "^ (-H2 * (PI
p) * R * (Dl
PI
+ p + W)
2 * H2 * PI * R * (Dl
PI + p + W) * X -f- (PI * P I ) * (Dl *
(h
H2) + H2 * (R
W) + h * W) * (x * X ) )
downtemp = PI * ((Pi
p) * R * (H2 + S 2 ) * (Dl - PI + p + W) +
2 * H2 * PI * R * (Dl
PI + p + W) * X
H2) +
( P I * P I ) * (Dl * (h
H2 * (R
W) + h * W) * (x * x) )
q4 = uptemp / downtemp
End Function
Public Function funl(ByVal H2 As Double, ByVal ftl As Double, ByVal
fql As Double, ByVal fq2 As Double, ByVal ft2 As Double, ByVal ft3 As
Double) As Double
funl = ((H2 * (((fql * ftl) / 2) + (((fql + fq2) * ft2) / 2) +
((fq2 * ft3) / 2))))
End Function
Public Function fun2(ByVal H2 As Double, ByVal p As Double, ByVal
ftl As Double, ByVal ftB As Double, ByVal hi As Double, ByVal B2 As
Double, ByVal Q As Double, ByVal Y As Double, ByVal ft As Double, ByVal
h As Double, ByVal R As Double, ByVal ft2 As Double, ByVal S2 As Double,
ByVal fq4 As Double, ByVal ft4 As Double) As Double
fun2 = (((H2 * p * (ftl + ftB) * (ftl + ftS)) / 2) + ((hi * B2
* Q * Y * ft) / 2) + ((h * R * ft2 * ft2) / 2) + ((S2 * fq4 * (ft4 *
ftS)) / 2))
End Function
Public Function fun3(ByVal Cl As Double, ByVal Q As Double, ByVal c
As Double, ByVal K As Double, ByVal KI As Double, ByVal i As Double,
ByVal F As Double, ByVal x As Double, ByVal Y As Double, ByVal SI As
Double) As Double
(c
(F
Y)
i)
fun3 = ((Cl * Q)
+ K + KI
(Q
(SI * Q * Y) )
End Function
196
APPENDIX C
SENSITIVITY ANALYSIS DATA
197
Table C. 1. Error factor of P.
Error Factor of
TOC* error,
Q* error,
(P)
(%)
(%)
0.1
9.21
-50.89
0.2
8.43
-48.68
0.3
7.61
-46.15
0.4
6.76
-43.20
0.5
5.85
-39.69
0.6
4.88
-35.43
0.7
3.83
-30.14
0.8
2.70
-23.29
0.9
1.44
-13.93
1
0.00
0.00
1.1
-1.72
24.00
1.2
-4.01
82.36
1.3
n/a (P<D+W)
n/a (P<D+W)
1.4
n/a (P<D+W)
n/a (P<D+W)
1.5
n/a (P<D+W)
n/a (P<D+V\/)
1.6
n/a (P<D+W)
n/a (P<D+W)
1.7
n/a (P<D+W)
n/a (P<D+W)
1.8
n/a (P<D+W)
n/a (P<D+W)
1.9
n/a (P<D+W)
n/a (P<D+W)
2
n/a (P<D+W)
n/a (P<D+W)
198
Table C.2. Error factor of D.
Error Factor of
TOC* error,
Q* error,
(D)
(%)
(%)
0.1
0.2
-
0.3
0.4
0.5
0.6
0.7
0.8
17.93567358
174.6346216
0.9
8.619323736
33.74472231
1
0
0
1.1
-7.436059935
-17.0185125
1.2
-13.83425277
-27.67132186
1.3
-19.37370011
-35.14127964
1.4
-24.20781821
-40.69503085
1.5
-28.45997354
-45.14452744
1.6
-32.22809418
-48.68463787
1.7
-35.59008954
-51.67262098
1.8
-38.60836257
-54.17343293
1.9
-41.33332989
-56.34946411
2
-43.80609481
-58.23319259
199
Table C.3. Error factor of R.
Error Factor of
TOC* error,
Q* error,
(R)
(%)
(%)
0.1
0.145602
-1.623903865
0.2
0.104145
-1.169210783
0.3
0.068881
-0.779473855
0.4
0.039883
-0.454693082
0.5
0.017213
-0.194868464
0.6
0.000919
-0.032478077
0.7
-0.00896
0.097434232
0.8
-0.01241
0.129912309
0.9
-0.00942
0.097434232
1
0
0
1.1
0.015837
-0.194868464
1.2
0.038053
-0.422215005
1.3
0.066601
-0.746995778
1.4
0.10142
-1.136732705
1.5
0.142437
-1.591425788
1.6
0.18957
-2.111075024
1.7
0.242722
-2.663202338
1.8
0.30179
-3.280285807
1.9
0.36666
-3.96232543
2
0.437211
-4.709321208
200
Table C.4. Error factor of x.
Error Factor of
TOC* error,
Q* error,
(X)
(%)
(%)
0.1
10.26005
-79.66872361
0.2
0.612964
-46.44365054
0.3
-1.60695
-26.1773303
0.4
-2.04968
-18.7398506
0.5
-1.94662
-14.90743748
0.6
-1.64527
-11.82202014
0.7
-1.264
-8.834037025
0.8
-0.8503
-5.878531991
0.9
-0.42552
-2.923026957
1
0
0
1.1
0.421088
2.89054888
1.2
0.835135
5.781097759
1.3
1.24089
8.606690484
1.4
1.637842
11.36732705
1.5
2.025894
14.09548555
1.6
2.405175
16.79116596
1.7
2.775943
19.42189022
1.8
3.138517
22.02013641
1.9
3.493248
24.58590451
2
3.840496
27.08671647
201
Table C.5. Error factor of y.
Error Factor of
TOC* error,
Q* error,
(y)
(%)
(%)
0.1
125.914532
-19.68171484
0.2
50.8960641
-17.47320559
0.3
28.7898878
-13.64079247
0.4
18.2381803
-10.39298474
0.5
12.0568544
-7.729782397
0.6
7.99466538
-5.586229295
0.7
5.12034054
-3.832413121
0.8
2.97879174
-2.338421565
0.9
1.32119712
-1.071776551
1
0
0
1.1
-1.07786707
0.941864242
1.2
-1.97401879
1.753816174
1.3
-2.73087395
2.468333875
1.4
-3.37859424
3.117895421
1.5
-3.93921469
3.702500812
1.6
-4.42920927
4.222150049
1.7
-4.86114173
4.676843131
1.8
-5.24476113
5.099058136
1.9
-5.58774872
5.488795063
2
-5.89623862
5.846053914
202
Table C.6. Error factor of C.
Error Factor of
TOC* error,
Q* error,
(C)
(%)
(%)
0.1
746.1470321
0
0.2
331.6209032
0
0.3
193.4455269
0
0.4
124.3578387
0
0.5
82.90522579
0
0-6
55.27015053
0
0.7
35.53081105
0
0.8
20.72630645
0
0.9
9.211691755
0
1
0
0
1.1
-7.536838708
0
1.2
-13.81753763
0
1.3
-19.13197518
0
1.4
-23.68720737
0
1.5
-27.63507526
0
1.6
-31.08945967
0
1.7
-34.13744591
0
1.8
-36.84676702
0
1.9
-39.27089643
0
2
-41.4526129
0
203
Table C.7. Error factor of c.
Error Factor of
TOC* error,
(c)
Q* error,
(%)
0.1
11.19220548
0
0.2
4.974313548
0
0.3
2.901682903
0
0.4
1.86536758
0
0.5
1.243578387
0
0.6
0.829052258
0
0.7
0.532962166
0
0.8
0.310894597
0
0.9
0.138175376
0
1
0
0
1.1
-0.113052581
0
1.2
-0.207263064
0
1.3
-0.286979628
0
1.4
-0.355308111
0
1.5
-0.414526129
0
1.6
-0.466341895
0
1.7
-0.512061689
0
1.8
-0.552701505
0
1.9
-0.589063446
0
2
-0.621789193
0
204
Table C.8. Error factor of F.
Error Factor of
TOC* error,
Q* error,
(f=)
(%)
(%)
0.1
55.96102741
0
0.2
24.87156774
0
0.3
14.50841451
0
0.4
9.326837902
0
0.5
6.217891934
0
0.6
4.14526129
0
0.7
2.664810829
0
0.8
1.554472984
0
0.9
0.690876882
0
1
0
0
1.1
-0.565262903
0
1.2
-1.036315322
0
1.3
-1.434898139
0
1.4
-1.776540553
0
1.5
-2.072630645
0
1.6
-2.331709475
0
1.7
-2.560308444
0
1.8
-2.763507526
0
1.9
-2.945317232
0
2
-3.108945967
0
205
Table C.9. Error factor of K.
Error Factor of
r o c * error,
Q* error,
(K)
(%)
(%)
0.1
19.21
216.21
0.2
10.98
123.58
0.3
7.34
82.56
0.4
5.16
58.10
0.5
3.68
41.41
0.6
2.59
29.10
0.7
1.73
19.52
0.8
1.05
11.79
0.9
0.48
5.39
1
0.00
0.00
1.1
-0.41
-4.64
1.2
-0.77
-8.70
1.3
-1.09
-12.31
1.4
-1.38
-15.49
1.5
-1.63
-18.35
1.6
-1.86
-20.95
1.7
-2.07
-23.32
1.8
-2.26
-25.46
1.9
-2.44
-27.44
2
-2.60
-29.30
206
Table CIO. Error factor of H.
Error Factor of
TOC* error,
Q* error,
(H)
(%)
(%)
0.1
12.00
-57.39
0.2
6.53
-42.38
0.3
4.21
-32.15
0.4
2.88
-24.49
0.5
2.01
-18.42
0.6
1.38
-13.45
0.7
0.91
-9.29
0.8
0.54
-5.75
0.9
0.24
-2.70
1
0.00
0.00
1.1
-0.21
2.37
1.2
-0.38
4.45
1.3
-0.53
6.37
1.4
-0.66
8.05
1.5
-0.78
9.58
1.6
-0.88
11.01
1.7
-0.97
12.28
1.8
-1.06
13.48
1.9
-1.13
14.55
2
-1.20
15.56
207
Table C. 11. Error factor of h,.
Error Factor of
TOC* error,
Q* error,
(h1)
(%)
(%)
0.1
10.80
-54.86
0.2
5.81
-39.56
0.3
3.72
-29.52
0.4
2.53
-22.18
0.5
1.76
-16.40
0.6
1.20
-11.95
0.7
0.79
-8.18
0.8
0.47
-5.03
0.9
0.21
-2.34
1
0.00
0.00
1.1
-0.18
2.01
1.2
-0.33
3.83
1.3
-0.46
5.42
1.4
-0.57
6.85
1.5
-0.67
8.12
1.6
-0.76
9.29
1.7
-0.83
10.33
1.8
-0.90
11.30
1.9
-0.97
12.18
2
-1.02
13.02
208
Table C.12. Error factor of h.
Error Factor of
TOC* error,
Q* error,
(h)
(%)
(%)
0.1
2.24
-20.17
0.2
1.06
-10.65
0.3
0.63
-6.66
0.4
0.41
-4.42
0.5
0.28
-2.99
0.6
0.19
-2.05
0.7
0.12
-1.33
0.8
0.07
-0.78
0.9
0.03
-0.36
1
0.00
0.00
1.1
-0.03
0.29
1.2
-0.05
0.52
1.3
-0.07
0.71
1.4
-0.08
0.91
1.5
-0.09
1.07
1.6
-0.11
1.20
1.7
-0.12
1.33
1.8
-0.13
1.43
1.9
-0.13
1.53
2
-0.14
1.62
209
APPENDIX D
TESTING DATA
210
Table D.l. The 144 test problems.
C = 100, c = 15, D = 1000, F = 15, i = 1, and S, = 0
#
P
w
X
y
K
K1
h
h1
H
1500
25
P
75
R
1
375
0.05
0.025
50
1000
11
20
10
2
1500
25
75
375
0.05
0.025
50
1000
55
100
50
3
1500
25
75
375
0.05
0.025
50
1000
550
1000
500
4
1500
25
75
375
0.05
0.025
500
10000
11
20
10
5
1500
25
75
375
0.05
0.025
500
10000
55
100
50
6
1500
25
75
375
0.05
0.025
500
10000
550
1000
500
7
1500
25
75
375
0.05
0.025
5000
100000
11
20
10
8
1500
25
75
375
0.05
0.025
5000
100000
55
100
50
9
1500
25
75
375
0.05
0.025
5000
100000
550
1000
500
0.05
0.1
50
1000
11
20
10
10
1500
100
75
375
11
1500
100
75
375
0.05
0.1
50
1000
55
100
50
12
1500
100
75
375
0.05
0.1
50
1000
550
1000
500
13
1500
100
75
375
0.05
0.1
500
10000
11
20
10
14
1500
100
75
375
0.05
0.1
500
10000
55
100
50
550
1000
500
15
1500
100
75
375
0.05
0.1
500
10000
16
1500
100
75
375
0.05
0.1
5000
100000
11
20
10
0.1
5000
100000
55
100
50
5000
100000
550
1000
500
50 333.33333
11
20
10
50
17
1500
100
75
375
0.05
18
1500
100
75
375
0.05
0.1
225
375
0.15
0.025
19
1500
25
20
1500
25
225
375
0.15
0.025
50 333.33333
55
100
21
1500
25
225
375
0.15
0.025
50 333.33333
550
1000
500
11
20
10
22
1500
25
225
375
0.15
0.025
500 3333.3333
23
1500
25
225
375
0.15
0.025
500 3333.3333
55
100
50
550
1000
500
24
25
1500
1500
25
225
375
0.15
0.025
500 3333.3333
25
225
375
0.15
0.025
5000 33333.333
11
20
10
55
100
50
26
1500
25
225
375
0.15
0.025
5000 33333.333
27
1500
25
225
375
0.15
0.025
5000 33333.333
550
1000
500
0.1
50 333.33333
11
20
10
100
50
28
1500
100
225
375
0.15
29
1500
100
225
375
0.15
0.1
50 333.33333
55
30
1500
100
225
375
0.15
0.1
50 333.33333
550
1000
500
11
20
10
31
1500
100
225
375
0.15
0.1
500 3333.3333
32
1500
100
225
375
0.15
0.1
500 3333.3333
55
100
550
1000
50
500
33
34
35
36
1500
1500
1500
1500
100
100
100
100
225
225
225
225
375
0.15
0.1
500 3333.3333
375
0.15
0.1
5000 33333.333
11
20
10
0.1
5000 33333.333
55
100
50
5000 33333.333
550
1000
500
1000
11
20
10
375
375
0.15
0.15
0.1
37
1500
25
75
1875
0.05
0.025
50
38
25
75
1875
0.05
0.025
50
1000
55
100
50
1500
550
1000
500
39
40
1500
1500
41
1500
42
1500
25
25
25
25
75
75
75
75
1875
1875
1875
1875
0.05
0.025
50
1000
0.05
0.025
500
10000
11
20
10
500
10000
55
100
50
550
1000
500
0.05
0.05
0.025
0.025
211
500
10000
Table D.l. Continued.
#
P
w
Y
K
K1
1500
25
R
p
75 1875
X
43
0.05
0.025
5000
100000
11
20
10
44
1500
25
75 1875
0.05
0.025
5000
100000
55
100
50
45
1500
25
75
1875
0.05
0.025
5000
100000
550
1000
500
46
1500
100
75 1875
0.05
0.1
50
1000
11
20
10
47
1500
100
75 1875
0.05
0.1
50
1000
55
100
50
48
1500
100
75 1875
0.05
0.1
50
1000
550
1000
500
49
1500
100
75 1875
0.05
0.1
500
10000
11
20
10
50
1500
100
75 1875
0.05
0.1
500
10000
55
100
50
51
1500
100
75 1875
0.05
0.1
500
10000
550
1000
500
52
1500
100
75 1875
0.05
0.1
5000
100000
11
20
10
53
1500
100
75 1875
0.05
0.1
5000
100000
55
100
50
54
1500
100
75 1875
0.05
0.1
5000
100000
550
1000
500
55
1500
25
225
1875
0.15
0.025
50 333.33333
11
20
10
56
1500
25
225 1875
0.15
0.025
50 333.33333
55
100
50
57
1500
25
225
1875
0.15
0.025
50 333.33333
550
1000
500
58
1500
25
225 1875
0.15
0.025
500 3333.3333
11
20
10
h
h1
H
59
1500
25
225
1875
0.15
0.025
500 3333.3333
55
100
50
60
1500
25
225
1875
0.15
0.025
500 3333.3333
550
1000
500
61
1500
25
225
1875
0.15
0.025
5000 33333.333
11
20
10
62
1500
25
225
1875
0.15
0.025
5000 33333.333
55
100
50
225 1875
0.15
0.025
5000 33333.333
550
1000
500
11
20
10
63
1500
25
64
1500
100
225
1875
0.15
0.1
50 333.33333
65
1500
100
225
1875
0.15
0.1
50 333.33333
55
100
50
550
1000
500
66
1500
100
225
1875
0.15
0.1
50 333.33333
67
1500
100
225
1875
0.15
0.1
500 3333.3333
11
20
10
500 3333.3333
55
100
50
500
68
1500
100
225
1875
0.15
0.1
69
1500
100
225
1875
0.15
0.1
500 3333.3333
550
1000
1500
100
225
1875
0.15
0.1
5000 33333.333
11
20
10
100
50
1000
500
70
71
72
73
74
75
1500
100
225
1875
0.15
0.1
5000 33333.333
55
1500
100
225
1875
0.15
0.1
5000 33333.333
550
11
5000
5000
5000
25
25
25
250 1250
0.05
0.025
50
1000
20
10
250 1250
0.05
0.025
50
1000
55
100
50
250
0.05
0.025
50
1000
550
1000
500
11
20
10
1250
76
5000
25
250 1250
0.05
0.025
500
10000
77
5000
25
250 1250
0.05
0.025
500
10000
55
100
50
550
1000
500
78
79
5000
5000
25
250 1250
0.05
0.025
500
10000
25
250 1250
0.05
0.025
5000
100000
11
20
10
100000
55
100
50
550
1000
500
5000
25
250 1250
0.05
0.025
5000
81
5000
25
250 1250
0.05
0.025
5000
100000
82
5000
100
250 1250
0.05
0.1
50
1000
11
20
10
100
50
1000
500
80
83
84
5000
5000
100
100
250
1250
250 1250
0.05
0.1
50
1000
55
0.05
0.1
50
1000
550
212
Table D.l. Continued.
#
P
w
p
R
X
Y
K
K1
h
85
5000
100
250
1250
0.05
0.1
500
10000
11
20
10
50
H
h1
86
5000
100
250
1250
0.05
0.1
500
10000
55
100
87
5000
100
250
1250
0.05
0.1
500
10000
550
1000
500
11
20
10
88
5000
100
250
1250
0.05
0.1
5000
100000
89
5000
100
250
1250
0.05
0.1
5000
100000
55
100
50
5000
100000
550
1000
500
90
5000
100
250
1250
0.05
0.1
91
5000
25
750
1250
0.15
0.025
50 333.33333
11
20
10
0.025
50 333.33333
55
100
50
1000
500
92
5000
25
750
1250
0.15
93
5000
25
750
1250
0.15
0.025
50 :333.33333
550
94
5000
25
750
1250
0.15
0.025
500 :3333.3333
11
20
10
50
95
5000
25
750
1250
0.15
0.025
500 :3333.3333
55
100
96
5000
25
750
1250
0.15
0.025
500 3333.3333
550
1000
500
11
20
10
97
5000
25
750
1250
0.15
0.025
5000 33333.333
98
5000
25
750
1250
0.15
0.025
5000 33333.333
55
100
50
5000 33333.333
550
1000
11
20
500
10
5000
25
750
1250
0.15
0.025
100
5000
100
750
1250
0.15
0.1
50 333.33333
101
5000
100
750
1250
0.15
0.1
50 333.33333
55
100
50
1000
500
99
102
103
5000
5000
100
100
750
750
1250
0.15
0.1
1250
0.15
0.1
50 333.33333
500 3333.3333
550
11
20
10
55
100
50
104
5000
100
750
1250
0.15
0.1
500 3333.3333
105
5000
100
750
1250
0.15
0.1
500 3333.3333
550
1000
500
11
20
10
106
5000
100
750
1250
0.15
0.1
5000 33333.333
107
5000
100
750
1250
0.15
0.1
5000 33333.333
55
100
50
5000 33333.333
550
1000
500
10
108
109
5000
5000
110
5000
111
5000
112
5000
100
750
1250
0.15
0.1
25
250
6250
0.05
0.025
50
1000
11
20
0.025
50
1000
55
100
50
1000
500
25
25
25
250
250
250
6250
6250
6250
0.05
0.05
0.025
50
1000
550
0.05
0.025
500
10000
11
20
10
10000
55
100
50
113
5000
25
250
6250
0.05
0.025
500
25
250
6250
0.05
0.025
500
10000
1000
5000
550
500
114
11
20
10
115
5000
25
250
6250
0.05
0.025
5000
100000
5000
25
250
6250
0.05
0.025
5000
100000
55
100
50
116
100000
550
1000
500
11
20
10
117
5000
25
250
6250
0.05
0.025
5000
0.1
50
1000
0.1
501
1000
55
100
50
1000
500
118
5000
100
250
6250
0.05
119
5000
100
250
6250
0.05
120
5000
100
100
250
250
6250
6250
0.05
0.05
0.1
0.1
5C1
500
1000
550
10000
11
20
10
100
50
121
5000
100
0.05
0.1
10000
5000
6250
500
122
250
55
0.05
0.1
550
1000
500
100
6250
10000
5000
250
500
123
0.05
0.1
11
10
6250
100000
5000
100
5000
124
250
20
0.05
0.1
55
100
50
100
6250
100000
5000
250
5000
125
500
126
5000
100
250
550
1000
6250
0.05
0.1
213
5000
100000
Table D.l. Continued.
#
P
w
127
5000
128
5000
129
130
X
y
25
R
p
750 6250
0.15
25
750 6250
0.15
5000
25
750 6250
5000
25
750 6250
h1
H
K1
h
0.025
50 333.33333
11
20
10
0.025
50 333.33333
55
100
50
0.15
0.025
50 333.33333
550
1000
500
0.15
0.025
500 3333.3333
11
20
10
50
K
131
5000
25
750 6250
0.15
0.025
500 3333.3333
55
100
132
5000
25
750 6250
0.15
0.025
500 3333.3333
550
1000
500
133
5000
11
20
10
134
25
750
6250
0.15
0.025
5000 33333.333
5000
25
750 6250
0.15
0.025
5000 33333.333
55
100
50
135
5000
25
750 6250
0.15
0.025
5000 33333.333
550
1000
500
136
5000
100
750 6250
0.15
0.1
50 333.33333
11
20
10
137
5000
100
750 6250
0.15
0.1
50 333.33333
55
100
50
1000
500
10
138
5000
100
750 6250
0.15
0.1
5000
100
750 6250
0.15
0.1
50 333.33333
500 3333.3333
550
139
11
20
0.1
500 3333.3333
55
100
50
1000
500
140
141
142
143
144
5000
5000
5000
5000
5000
100
750 6250
0.15
100
750 6250
0.15
0.1
500 3333.3333
550
100
750 6250
0.15
0.1
5000 33333.333
11
20
10
50
500
100
750 6250
0.15
0.1
5000 33333.333
55
100
100
750
0.15
0.1
5000 33333.333
550
1000
6250
214
APPENDIX E
THE OUTPUT DATA
215
Table E. 1. The output data for model 1, 2, 3, and 4.
Model 1
(Traditional EPQ)
Problem
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
Q*
TOC^
893
399
126
2823
1262
399
8926
3992
1262
1017
455
144
3217
1439
455
10173
4550
1439
645
288
91
2039
912
288
6448
2884
912
758
339
107
2398
1072
339
7583
3391
1072
893
399
126
2823
1262
399
122122.8
125248.9
137477.1
127591.41
137477.1
176145.7
144884.6
176145.7
298426.8
143248.8
146183
157660.4
148381.7
157660.4
193955.1
164613.2
193955.1
308729.1
149740.8
151467.7
158222.6
152761.7
158222.5
179583.2
162314.5
179583.2
247131.5
173138.8
174701
180811.5
175871.5
180811.5
200134.7
184513.1
200134.7
261240
122122.8
125248.9
137477.1
127591.4
137477.1
176145.7
Model 2
Model 4
Model 3
TOC*
TOC^
TOC
825 113137.7
369 116518.5
117 129743.2
2610 119051.9
1167 129743.1
369 171562.9
8254 137754.3
3691 171562.9
1167 303808.7
802 132588.4
359 136311.5
113 150875.1
2535 139101.3
1134 150875
359 196928.9
8018 159697.3
3586 196928.9
1134 342564
558 120231.5
249 122228.2
79 130038.1
1764 123724.2
789 130038.1
249 154735.3
5577 134769.1
2494 154735.2
789 232834.5
540 140057.3
142252.7
241
150840.2
76
143897.8
1706
150840.1
763
177996
241
156042.2
5396
177995.9
2413
263870.1
763
114523.1
826
117902.3
369
131120.5
117
120434.4
2611
131120.5
1168
172919.9
369
216
704
315
100
2226
996
315
7039
3148
996
594
266
84
1878
840
266
5940
2656
840
383
171
54
1212
542
171
3832
1714
542
328
147
46
1037
464
147
3280
1467
464
705
315
100
2229
997
315
107310
110740.2
124157.9
113310.5
124157.7
166587.8
132285.8
166587.8
300763.5
111552.5
115352.4
130216.1
118199.7
130216.1
177219.4
139220.2
177219.3
325856.5
111383.6
113488.3
121721.2
115065.41
121721.2
147755.7
126708.4
147755.7
230084
114869.8
117180.9
126221.7
118912.7
126221.2
154809.2
131697.6
154809.1
245212
107306
110731.4
124130
113298.1
124129.9
166499.9
824
368
116
2604
1165
368
8236
3683
1165
929
416
131
2939
1314
416
9294
4157
1314
493
221
70
1560
698
221
4934
2206
698
556
248
79
1757
786
248
5555
2484
786
824
369
117
2607
1166
369
110751.1
113981.7
126618.7
116402.5
126618.5
166579.6
134273.6
166579.6
292947.7
130910.3
133982.4
145999.1
136284.4
145999
183999
153278.5
183999
304165.4
111267.7
113236.5
120937.4
114711.7
120937.3
145289.7
125602.3
145289.6
222298.3
131593.1
133469.5
140809.6
134875.6
140809.4
164020.2
145255.7
164020.2
237419.1
110748.9
113976.7
126602.7
116395.3
126602.6
166529.3
Table E.l. Continued.
Model 1
(Traditional EPQ)
Problem
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
Q*
TOC*
8926 144884.6
3992 176145.7
1262 298426.8
1017 143248.8
146183
455
144 157660.4
3217 148381.7
1439 157660.4
455 193955.1
10173 164613.2
4550 193955.1
1439 308729.1
645 149740.8
288 151467.7
91 158222.6
2039 152761.7
912 158222.5
288 179583.2
6448 162314.5
2884 179583.2
912 247131.5
758 173138.8
174701
339
107 180811.5
2398 175871.5
1072 180811.5
339 200134.7
7583 184513.1
3391 200134.7
261240
1072
536 123803.4
240 129006.9
76 149360.7
1696 132905.9
758 149360.7
213725
240
5363 161690.5
213725
2398
758 417262.9
560 145187.3
250 150517.5
79 171367.3
Model 2
Q*
Model 4
Model 3
TOC^
Q*
8258 139127.7
3693 172919.9
1168 305101.3
803 135756.8
359 139473.1
114 154010
2540 142257.8
1136 154009.9
359 199979.1
8032 162815.9
3592 199979.1
1136 345346.4
560 124091.9
250 126081.9
79 133865.9
1769 127573
791 133865.9
250 158481.1
5595 138581.3
2502 158481.1
791 236321.2
543 146296.1
243 148477.5
77 157010.4
1717 150112.1
768 157010.4
243 183993.8
5430 162179.4
2428 183993.8
768 269322.7
509 116048.7
228 121527.9
72 142960.2
1610 125633.5
720 142960.2
228 210735.4
5093 155943.5
2278 210735.4
720 425059.1
493 137247.2
221 143296.1
70 166957.2
217
TOC^
7049
3153
997
596
267
84
1886
843
267
5964
2667
843
386
172
55
1219
545
172
3856
1725
545
331
148
47
1047
468
148
3311
1481
468
474
212
67
1498
670
212
4736
2118
670
436
195
62
132246.5
166499.9
300485.5
111540
115324.6
130128.4
118160.4
130128.3
176941.6
139096
176941.5
324978.2
111372.9
113464.4
121646
115031.6
121645.6
147516.8
126601.6
147516.7
229328.3
114852.2
117141.6
126097
118857.1
126096.9
154416.1
131521.8
154416
243969.1
108659.2
113757.2
133698.5
117577.3
133698.5
196758.4
145778.5
196758.4
396171.1
112665.8
117841.8
138088.8
TOC*
134251.1
8243
166529.2
3686
292788.4
1166
130907.6
930
133976.3
416
145979.8
132
136275.7
2942
145979.8
1316
183938
416
153251.2
9305
183938
4161
303972.5
1316
111255.6
497
113209.2
222
120851.3
70
114673.1
1572
120851.2
703
145017.2
222
125480.5
4972
145017.2
2224
221436.8
703
131578.3
561
133436.6
251
140705.3
79
134829
1774
140705.2
793
163690.6
251
145108.4
5610
163690.6
2509
236377
793
112274.7
520
117388.7
233
137392.7
74
121220.6
1645
137392.2
736
200649.2
233
149509.9
5203
200649.1
2327
400684.8
736
132670.4
544
137917.9
243
158444.2
77
Table E.l. Continued.
Model 1
(Traditional EPQ)
TOC^
Problem
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
1771
792
250
5600
2505
792
343
153
49
1085
485
153
3432
1535
485
357
160
51
1130
506
160
3575
1599
506
536
240
76
1696
758
240
5363
2398
758
560
250
79
1771
792
250
5600
2505
792
154511.5
171367.2
237300
183997.5
237299.8
445797.1
150968.9
154213.8
166907.2
156645.2
166906.3
207044
174595.2
207043.7
333969.4
174556
177869.8
190832.9
180352.9
190832.2
231822.9
198684.5
231822.9
361447
123803.4
129006.9
149360.7
132905.9
149360.7
213725
161690.5
213725
417262.9
145187.3
150517.5
171367.3
154511.5
171367.2
237300
183997.5
237299.8
445797.1
Model 2
Model 4
Model 3
TOC^
TOC^
Q*
1561 147828.7
698 166957.1
221 241779.8
4935 181290.4
2207 241779.7
698 478389.4
311 124896.7
139 128478.8
44 142490.3
983 131162.9
440 142490.3
139 186798.6
3109 150978.1
1390 186798.6
440 326913.8
301 146929.2
135 150859
43 166231.3
953 153803.6
426 166230.5
135 214839.5
3014 175542.2
1348 214839.5
426 368554.7
509 116464.4
228 121943.3
72 143374.5
1611 126048.8
720 143374.5
228 211145.9
5093 156357
2278 211145.9
720 425457.7
494 138198.4
144246
221
70 167902.2
1561 148777.7
698 167902.1
221 242709.1
4936 182232.3
2207 242709
698 479269.3
126
218
1379
617
195
4361
1950
617
261
117
37
826
369
117
2613
1168
369
242
108
34
766
342
108
2421
1083
342
474
212
67
1498
670
212
4737
2119
670
436
195
62
1380
617
195
4364
1951
617
121720.3
138088.4
202113.7
150353.3
202113.7
404579.6
112178.1
115265
127339.4
117578
127339.4
165522.1
134653.8
165522.1
286266.4
115532.9
118663.7
130910.6
121009.7
130910.2
169637.1
138328.8
169636.9
292101.6
108658.5
113755.5
133692.9
117574.7
133692.9
196740.6
145770.5
196740.6
396115.1
112663
117835.7
138069.4
121711.7
138069.1
202052.7
150326
202052.7
404386.5
Q*
1721
770
243
5441
2433
770
314
140
44
993
444
140
3140
1404
444
328
147
46
1038
464
147
3284
1469
464
520
233
74
1645
736
233
5203
2327
736
544
243
77
1721
770
243
5442
2434
770
TOC
141850
158444.2
223354.2
170878.6
223354.1
428617.4
112177.4
115270.5
127370.2
117588.2
127369.5
165630.1
134698.7
165629.8
286619.5
132643
135817.3
148234.4
138195.8
148233.6
187497.6
155755.1
187497.5
311661.1
112274.3
117387.7
137389.7
121219.3
137389.2
200639.6
149505.7
200639.5
400654.7
132669.9
137916.9
158440.8
141848.5
158440.8
223343.5
170873.8
223343.4
428583.5
Table E.l. Continued.
Model 1
(Traditional EPQ)
Problem
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
Q*
TOC*
343 150968.9
153 154213.8
49 166907.2
156645.2
485 166906.3
207044
153
3432. 174595.2
153f> 207043.7
48f; 333969.4
357
174556
160 177869.8
51I 190832.9
1130 180352.9
506 190832.2
160 231822.S
3575 198684.f
1599 231822.SI
506
36144':^
1085
Model 2
Q*
Model 4
Model 3
TOC*
126055.5
129636.5
143643.7
132319.8
143643.7
187938.3
310S 152128.9
1391 187938.3
44C) 328010.2
302 148802.3
135 152729.7
43 168092.9
954 155672.6
427 168092.1
135 216672.3
3016 177398.3
1349 216672.3
427 370296.f
311
139
44
983
440
13S
219
Q*
262
117
37
827
370
117
2615
116S>
37C)
242\
10^\
34
767
343
108
2425
1085
343
TOC*
112176
115260.1
127324
117571.1
127324
165473.3
134632
165473.3
286112.1
115529
118655.1
130883.4
120997.4
130882.7
169550.7
138290
169550.2
291827.3
Q*
314
141
44
994
445
141
3143
1406
445
325
14-;'
Ati
1040
46^i
147
3287
1470
465
TOC*
112175.1
115265.3
127354
117580.9
127353.1
165578.2
134675.5
165577.9
286455.7
132640.4
135811.4
148216.2
138187.5
148215.2
187439.4
155729.1
187439.4
311477
Table E.2. The output data for model 5, 6, 7, and 8
Model 5
F'roblem
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
Q*
Model 6
TOC*
903 122152.8
404 125315.9
128 137688.7
2826 127600.9
1264 137498.3
400
176213
8927 144887.6
3992 176152.4
1263 298448.1
1029 143276.9
460 146245.8
146 157859.1
3221 148390.6
1440 157680.4
456 194018.3
10175
164616
4550 193961.4
1439 308749.1
665 149785.7
298
151568
94 158539.6
2046 152776.1
915 158254.7
289 179684.9
6450
162319
2885 179593.4
912 247163.8
783 173179.4
350 174791.7
111 181098.3
2406 175884.5
1076 180840.6i
340 200226.^/
7585 184517.:>
3392 200143.^)
1073 261269.:I
903 122152.!I
404 125315.*^
128 137688.7
2826 127600.9
1264 137498.3
400 17621 3
Q*
Model 8
Model 7
TOC*
792 110854.2
354 114212.3
112 127347.8
2505 116728.6
1120 127347.8
354 168885.9
7923
135305
3543 168885.8
1120 300240.6
788 131356.8
352 134980.7
111 149156.2
2492 137696-2
1114
149156
352 193982.3
7879 157743.1
3524 193982.3
1114 335735.3
474 111331.2
212 113378.3
67 121385,9
1500 114912.2
671 121385.8
212
146708
4745 126236.6
2122 146707.9
671 226783.4
471 131867.3
210 134082.7
67 142748.6,
1488 135742.'J
665 142748.:\
210 170151.(5
4705 147997.JI
2104 170151-5
665 256808. 1
793 110850.9
355 114204.8
112 127323.9
2509 116717.9
1122 127323.9
355 168810.1
220
Q*
TOC*
1335
111875
597 116494.7
189 134565.4
2893
116239
1294 126252.9
409 165423.5
8617 132268.1
3854 162095.1
1219 278766.3
1467
132074
656 136584.5
207 154227.7
3180 136334.8
1422 146111.8
450 184355.6
9472 151984.6
4236
181106
1340 295016.8
1048 113129.3
469
117399
148 134100.4
1722
115349
770 122362.4
244
149796
4651 124998.6
2080 143939.6
658
218029
1180 133368.4
528 137439.2
167 153362.^
1938 135484.7'
867 142171.^\
274 168327.:i
5235 144684.^)
2341 162743.'7
740 233382.5
1401 111697.5
627 116097.8
198 133310.3
3037 115854.2
1358 125392.6
430 162702 9
Q*
TOC*
946
569
298
110412.6
111922
115369.2
115332.1
120105.1
131006.2
130888.9
145982.4
180454.5
130599.2
132102.8
135900.9
135300.4
140055.2
152066
150166.9
165202.9
203184.3
111069.9
112212.2
116269.7
114086
117698.3
130529.2
123623.9
135047
175621.7
131422.8
132620
137061
134337.2
138123.1
152166.6
143553.4
155525.2
199934.9
110411.3
111921
115369.1
115327.8
120102.1
131005.8
2992
1799
941
9461
5688
2976
1062
628
309
3360
1986
977
10625
6281
3090
563
310
119
1782
979
377
5634
3097
1192
626
331
121
1979
1048
382
6257
3314
1207
947
569
298
2994
1799
941
Table E.2. Continued.
Model 5
Problem
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
Model 6
Model 7
TOC*
Q*
TOC*
Q*
8927 144887.6
7933 135271.1
3992 176152,4
3548 168810.1
1263 298448.1
1122 300001.2
1029 143276.9
790 131347.6
460 146245.8
353
134960
112 149090.6
146 157859.1
2499 137666.9
3221 148390.6
1118 149090.6
1440 157680.4
353 193775.3
456 194018.3
7904 157650.5
10175
164616
3535 193775.2
4550 193961.4
1118 335080.5
1439 308749.1
478 111317.4
665 149785.7
214 113347.5
298
151568
68 121288.7
94 158539.6
1513 114868.7
2046 152776.1
677 121288.6
915 158254.7
214 146400.3
289 179684.9
4785 126099.1
6450
162319
2140 146400.3
2885 179593.4
677 225810.8
912 247163.8
476 131845.1
783 173179.4
213
134033
350 174791.7
67 142591.4
111 181098,3
1507 135672.4
2406 175884.5
674 142591.2
1076 180840.6
213 169654.6
340 200226.7
4764 147775.6
7585 184517.2
2131 169654.6
3392 200143.S
674 255236.^
1073 261269.2
I
512
11234C)
543 123853.2
5
i
229
117534.^
243 129118.:
72
13785^\
77 149713.]I
1620 121426.f)
1698 132921,iI
724 137853.5
759 149396. I
229 202107.5
240
21383'7
5122 150162.2
5363 161695.5
2291 202107.6
2399 213736.2
724 405297.1
758 417298.3
511 132942.3
567 145238.3
229
138526
253 150631.6
72 160367 5
80 171728.l|
221
Model 8
TOC*
Q*
Q*
9467
9047
131122
5689
4046 159532.5
2977
1279 270662,7
1063
1536 131911.2
628
687 136220.3
309
217 153076,1
3362
3329 135981,8
1987
1489 145322.5
977
471 181859,4
9914 150933,2 10633
4434 178754.8
6283
3090
1402 287581.8
566
1135 112865.6
310
508 116809.4
119
161 132235.9
1789
1865 114915.8
981
834 121393,8
377
264 146733.3
5658
5036 123828.8
3101
2252 141323.9
1192
712 209757,6
629
1258 133164,8
332
563 136983.9
178
151923
121
1989
2066 135150.2
1049
924 141423.6
292 165962.4
382
6290
5580 143781.6
3318
2496
160724
789 226995.5
1207
607
1012 111652.''
635 115997.^J
388
292
201 132993.'7
3076 115757. [ 1921
1376 125175.5 1227
435 162016.5
923
6075
9162 130832.9
4097 158886.1 3881
1296 268618.6 2919
1441 132140.3
635
644 136732 6
405
204
154656
303
TOC*
130875.2
145972.9
180453.5
130597,5
132101.7
135900.8
135295,2
140051.7
152065,6
150150.4
165191.7
203183,1
111063.8
112209.2
116269.5
114066.8
117688.8
130528.5
123563.2
135017
175619.6
131415.9
132616.8
137060.8
134315.2
138112.9
152166
143483.5
155493.2
199932.8
111681
113683.8
115512.4
119343
125676.4
131458.9
143572.4
163600.5
181886.4
132061.9
134122.3
136060.3
Table E.2. Continued.
Model 5
Problem
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
Model 6
TOC*
Q*
1773 154527.8
793 171403.5
251 237414,6
5601 184002.6
2505 237311.3
792 445833.4
354 151053.1
158 154402.2
50 167502.1
1089 156672.2
487 166966,7
154 207234.8
3433 174603.7
1535 207062.9
485
334030
369
174642
165 178062.2
52 191440.7
1134 180380.5
507 190893.9
160 232018.2
3576 198693.3
1599 231842.4
506 361508.7
543 123853.2
243 129118.3
77 149713.1
1698 132921.8
759 149396,1
240
213837
5363 161695.5
2399 213736.2
758 417298.3
567 145238,2
253 150631.6i
80
1773
793
251
5601
2505
792
171728.1
154527.J]
171403.;j
237414.(5
184002.(5
237311.:I
445833.^^
Model 7
Q*
1617
723
229
5114
2287
723
309
138
44
978
437
138
3091
1382
437
309
138
44
976
436
138
3085
1380
436
512
229
72
1620
724
229
5123
2291
724
512
229
72
1618
723
229
5116
2288
723
TOC*
142710
160367.2
229435.2
173598.1
229435,1
447847
112217.1
115359.2
127650.5
117713,7
127650.1
166517.3
135095.6
166517.2
289425.9
132808.5
136187.3
149404.6
138719.1
149403.9
191198.2
157410.1
191198.2
323363.7
112339.3
117533
137849.4
121424.8
137848.8
202092.5
150155.6
202092.5>
405250.7^
132940.f]
13852:I
160354.?I
142704.:]
160354..5
229394.'?
173580.1
229394.8
447719.7
222
Model 8
Q*
TOC*
3123
136478.4
1397 146432.9
442 185370,9
9303
152412.3
4160 182062.2
1316 298040,7
113354
984
440 117901.5
139 135689.5
1617 115718.1
723 123187.8
229 152406.2
4367 125995,5
1953 146168,8
618 225078.5
1006 133940.2
450 138717.8
142 157405.8
1652 136423.9
739 144271.5
234 174968.5
147221,3
4461
1995 168415.4
251318
631
1494 111476.1
668 115602.8
131745
211
3239 115374.3
1448 124319.6
458 159309.8
9647 129692.8
4314 156336.5
1364 260556.2
131951.S)
1518
136311.^I
679
15336^\
215
13607()
3290
145519.3
1471
465 182483.5
151196
9800
4383 179342.6
1386 289440.5
Q*
2009
1282
957
6352
4055
3025
377
245
159
1192
776
502
3768
2454
1588
394
256
163
1245
809
514
3938
2558
1626
607
388
292
1921
1227
923
6075
3881
2919
635
405
303
2009
1282
957
6352
4055
3025
TOC*
139925.8
146441.4
152570
164793,6
185397.8
204778.3
111760.4
112876.7
114622.2
116269.6
119799.5
125319.5
130528.8
141691.5
159147.3
132216.6
133372.3
135262.1
136847.2
140502.1
146478
151490.5
163048.4
181945.9
111680.7
113683.6
115512.4
119342.2
125675.8
131458.8
143569.8
163598.5
181886
132061.6
134122.1
136060.3
139924.8
146440.7
152569.9
164790.6
185395.6
204777.9
Table E,2. Continued.
Model 6
Model 5
Problem
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
Q*
TOC*
354 151053.1
158 154402,2
50 167502,1
1089 156672.2
487 166966.7
154 207234.8
3433 174603.7
1535 207062.9
485J 334030
369
174642
165 178062.2
52 191440,7
1134 180380.5
507 190893.9
160 232018.2
3576 198693.3
1599 231842.^[
506 361508.'1
Model 7
TOC*
Q*
309 112214.4
138 115353.3
44 127631.4
979 117705.2
438 127631.2
138 166457.6
1081
Q*
3095
1384
438)
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138
44
977
437
138
3090
1382
437
135068.8
166457.3
289236.5
132804.2
136177.6
149373.6
138705.4
149373.1
191101.1
157366.7
191101
323056.2
223
484
153
1776
794
251
4797
2145
678
1107
49f
156
1818
813
257
4909
2195
694
Model 8
TOC*
113024.6
117164.8
133359.9
115177
121977.7
148579.7
124534.1
142900.9
214744.4
133587.5
137929,1
154911,9
135844.5
142976.1
170871.7
145656.7
164916.7
240254.2
Q*
377
246
159
1192
776
502
3771
2455
1588
394
256
162
1246i
809
514
3941
2558
1626
TOC*
111759.1
112875.8
114622,1
116265.5
119796.9
125319.2
130515.9
141683.3
159146.3
132215.1
133371.4
135262
136842.6
140499.2
146477,7
151476
163039.3
181944.8