OPTIMAL INVENTORY CONTROL POLICY OF A HYBRID …

OPTIMAL INVENTORY CONTROL POLICY OF A

HYBRID MANUFACTURING-REMANUFACTURING

SYSTEM USING A FUZZY LINEAR PROGRAMMING

BY

MR. KITTIPHAN NUAMCHIT

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF MASTER OF

ENGINEERING (LOGISTICS AND SUPPLY CHAIN SYSTEMS

ENGINEERING)

SIRINDHORN INTERNATIONAL INSTITUTE OF TECHNOLOGY

THAMMASAT UNIVERSITY

ACADEMIC YEAR 2018

COPYRIGHT OF THAMMASAT UNIVERSITY

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Thesis Title OPTIMAL INVENTORY CONTROL POLICY

OF A HYBRID MANUFACTURING-

REMANUFACTURING SYSTEM USING A

FUZZY LINEAR PROGRAMMING

Author Mr. Kittiphan Nuamchit

Degree Master of Engineering Program (Logistics and

Supply Chain Systems Engineering)

Faculty/University Sirindhorn International Institute of Technology/

Thammasat University

Thesis Advisor Associate Professor Navee Chiadamrong, Ph.D.

Academic Years 2018

ABSTRACT

Remanufacturing is one of the product recovery techniques to bring used or end

of life products back to like-new products. Production planning of inventory control

policies in the hybrid manufacturing/remanufacturing system with different

prioritizations (manufacturing vs remanufacturing) is investigated in this study. For this

production planning problem, the imprecision of customer demand, related operating

costs, number of returned used components and all lead time and production times are

uncertain and it would complicate the planning of the production and inventory control.

As a result, the fuzzy set theory is employed due to the presence of the imprecise

information and the Fuzzy Linear Programming (FLP) is applied to optimize the model

under these uncertainties. The proposed approach maximizes the most likely value of

the profit, minimizes the risk of obtaining a lower profit, and maximizes the possibility

of obtaining a higher profit for each production planning policy. The result shows that

the Priority-To-Remanufacturing (PTR) policy shows a higher profit than the policy of

Priority-To-Manufacturing (PTM) and FLP can help decision makers to be well aware

of the risks and effects of uncertainties towards their plans. As a result, they can prepare

in advance for such scenarios.

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Keywords: Hybrid manufacturing/remanufacturing, Fuzzy linear programming, Fuzzy

optimization, Fuzzy lead time and production times.

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ACKNOWLEDGEMENTS

I would like to take this section to be opportunity to show my gratitude for the

generousness of those who have helped making this research become possible.

First of all, I would like to show my gratitude to Assoc. Prof. Dr. Navee

Chiadamrong, my thesis adviser for his opportunity, patience, suggestions,

encouragement, and valuable guidance throughout my Master study and research. It has

been an admirable experience working under his supervision. Besides, I am particularly

grateful to the members of examination committee, Assoc. Prof. Dr. Pisal Yenradee,

Assoc. Prof. Dr. Teeradej Wuttipornpun, and Dr. Panitan Kewcharoenwong for their

useful comments and suggestions, which have enlarged my research perspectives. In

addition, I would like to reflect my appreciation to all my teachers in the curriculum of

industrial engineering (Bachelor Degree) especially Asst.Prof.Dr. Suchada Rianmora

to advise me to continue Master Degree at SIIT, and Logistics and Supply Chain

Systems Engineering (Master Degree) for their kindness and willing to provide useful

knowledge to all students.

Special thanks are given to all SIIT staffs for their kind support and guidance. I

thank my fellow course mates for their fruitful suggestion and encouragement in all

those years. Lastly, I am grateful to my beloved parents and other family members for

their extreme patience, understanding, and continuous spiritual support.

Mr. Kittiphan Nuamchit

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TABLE OF CONTENTS

Page

ABSTRACT (2)

ACKNOWLEDGEMENTS (4)

LIST OF TABLES (8)

LIST OF FIGURES (9)

LIST OF SYMBOLS/ABBREVIATIONS (10)

CHAPTER 1 INTRODUCTION 1

1.1 Problem Statement 1

1.2 Objectives of the thesis 2

1.3 Overview of the thesis 2

CHAPTER 2 REVIEW OF LITERATURE 4

CHAPTER 3 METHODOLOGY 7

3.1 Deterministic condition using MILP 7

3.2 Stochastic condition using FLP 8

3.2.1 Model the uncertainty data with triangular (possibility) distribution 8

3.2.2 Develop three new crisp objective functions of multi objective linear

programming (MOLP) 9

3.2.3 Defuzzification method 10

3.2.4 Specify the linear membership functions for three new objective

functions, and then convert the auxiliary MOLP problem into an

equivalent linear programming model using the fuzzy decision method

10

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3.2.4.1 Find the lower bound and the upper bound of each objective 11

3.2.4.2 Find the corresponding linear membership function of each objective

function 11

3.2.4.3 Find the maximum overall satisfaction 12

CHAPTER 4 A CASE STUDY WITHOUT LEAD TIME UNCERTAINTY 13

4.1 Priority-To-Remanufacturing (PTR) 14

4.2 Priority-To-Manufacturing (PTM) 14

4.3 Other relevant information 15

4.4 LP formulation (Deterministic case) 17

4.4.1 Parameters 17

4.4.1.1 Production planning parameters (units) 17

4.4.1.2 Cost parameters ($) 18

4.4.2 Decision Variables 18

4.4.3 Objective Function 18

4.5 Fuzzy Linear Programming (FLP) 22

4.6 Results and Discussion 26

4.6.1 Deterministic case with linear programming model 26

4.6.2 Stochastic case with the Fuzzy Linear Programming (FLP) 27

4.6.3 Comparison between the deterministic and the best stochastic results 32

CHAPTER 5 A CASE STUDY WITH LEAD TIME UNCERTAINTY 34

5.1 LP formulation (Deterministic case) 34

5.1.1 Parameter 34

5.1.1.1 Production planning parameters (units) 34

5.1.1.2 Cost parameters ($) 35

5.1.2 Decision Variables 35

5.1.3 Objective Function 35

5.2 Result and Discussion 46

5.2.1 Stochastic case with the Fuzzy Linear Programming (FLP) 46

5.2.2 Comparison between the deterministic and the best stochastic results 53

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CHAPTER 6 CONCLUSIONS 54

6.1 Conclusion 54

6.2 Further Study 55

REFERENCES 56

APPENDICES 58

APPENDIX A 59

APPENDIX B 62

BIOGRAPHY 70

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LIST OF TABLES

Tables Page

4.1 Cost Structure. 16

4.2 PTR and PTM results. 27

4.3 Upper and lower boundary of the PTR policy without lead time uncertainty. 28

4.4 Upper and lower boundary of the PTM policy without lead time uncertainty. 28

4.5 Result of the sensitivity analysis of the PTR policy

without lead time uncertainty. 29

4.6 Result of the sensitivity analysis of the PTM policy

without lead time uncertainty. 31

4.7 Result comparison. 33

5.1 Upper and lower boundary of the PTR policy with lead time uncertainty. 477

5.2 Upper and lower boundary of the PTM policy with lead time uncertainty. 47

5.3 Result of the sensitivity analysis of the PTR policy with lead time uncertainty. 48

5.4 Result of the sensitivity analysis of the PTM policy with lead time uncertainty. 50

5.5 Example of value in case 1 and case 2. 51

5.6 Result comparison. 53

A.1 Customer demand, amount of returned component arrival, and production

capacity in each week. 59

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LIST OF FIGURES

Figures Page

3.1 Method of approach. 7

3.2 Triangular possibility distribution of 𝑥. 8

3.3 Strategy for maximizing the imprecise objective function. 9

3.4 Linear membership function of 𝑧1 and 𝑧3. 12

3.5 Linear membership function of 𝑧2. 12

4.1 General flowchart of the studied system. 13

4.2 Flow diagram of the PTR and the PTM policies. 15

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LIST OF SYMBOLS/ABBREVIATIONS

Symbols/Abbreviations Terms

SIIT Sirindhorn International Institute of

Technology

FLP Fuzzy Linear Programming

PLP Possibilistic Linear Programming

PTM Priority-To-Manufacturing

PTR Priority-To-Remanufacturing

UK United Kingdom

AHP Analytic Hierarchical Process

MRP Material Requirements Planning

MILP Mixed-Integer Linear Programming

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CHAPTER 1

INTRODUCTION

1.1 Problem Statement

Many products such as electrical apparatus, toner cartridges, cellular phones,

and automotive parts require remanufacturing for recovering components as raw

materials to produce finished products (Sundin, 2004). Because of limited resources,

remanufacturing of these used products is significantly growing in many countries.

According to the Carbon Trust news, remanufacturing in the UK had a market potential

of up to £5.6 billion, with benefits to improve business revenues, margins, and stability

of supply (Smith-Gillespie, 2014).

Remanufacturing includes the processes of disassembly, cleaning, inspection,

and sorting of used products before reassembling to new parts or new products. It can

be divided into two groups. First, a stand-alone remanufacturing system is concerned

with used products that are supplied for all demands. Second, a hybrid

manufacturing/remanufacturing system can be used when remanufacturing is required

to work along with the existing manufacturing processes where new components may

only be required when there is a shortage of old or used components.

In remanufacturing processes, the uncertainty of quantity, timing, and quality

of receiving used products dramatically increases the difficulty of production planning

and inventory management. Planners need to make decisions on how many new

components or how many used products are required and especially on when such

orders should be placed. Moreover, customer demand and related operating costs are

very much subject to uncertainty. Customer demand normally varies with time due to

errors in forecasting, or lack of information while certain operating costs are difficult

to be fixed in advance. To handle such issues, a certain approach can be used to optimize

such problems under the uncertainty. Instead of using a typical linear programming

where the uncertainty is disregarded, Fuzzy Linear Programming (FLP) is one of the

approaches that can be used to incorporate fuzzy data and optimize a problem. Our

main contribution of this study is to introduce the Fuzzy Linear Programming (FLP)

model to optimize the profit of the production and inventory planning in a hybrid

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manufacturing/remanufacturing system can then recommend the best policy with

optimal operating parameters between the Priority-To-Remanufacturing (PTR) and

Priority-To-Manufacturing (PTM) in an uncertain environment, especially with the lead

time and production times’ uncertainty. This time uncertainty can complicate the

computation and has not been much incorporated in the model solution from the past

researches. A sensitivity analysis is then carried out with many scenarios to investigate

the effects of each policy under the different levels of decision weight from the

pessimistic, most likely or optimistic conditions. The result would provide a possible

range of profit that might occur from implementing each policy in each scenario, which

indicates possible benefits and hidden risk from the policy that might exist.

1.2 Objectives of the thesis

The objectives of this thesis are:

• To introduce Fuzzy linear programming (FLP) with uncertainty environment

to hybrid manufacturing/remanufacturing production planning and inventory

control problem.

• To observe performance of prioritization in an inventory control policy of the

hybrid manufacturing/remanufacturing system when used deterministic and

stochastic model.

• To construct sensitivity analysis with many scenarios to investigate the effects

of each policy under the different levels of weight for pessimistic, most likely

or optimistic condition

1.3 Overview of the thesis

There are sixth chapters in this thesis, which are as follows:

Chapter 1 is the introduction part, which includes problem statement,

objectives, and overview of the thesis.

Chapter 2 presents a literature review. Related researches are discussed to create

more understanding about the remanufacturing processes, hybrid

manufacturing/remanufacturing system, and the fuzzy theory.

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Chapter 3 shows the methodology for solving the stochastic model based on

the Fuzzy Linear Programming model.

Chapter 4 presents the results with the sensitivity analysis and compares the

results between the deterministic and stochastic cases in a hybrid

manufacturing/remanufacturing under an uncertain demand, production capacity,

related operating cost, and amount of returned components arrival every week.

Chapter 5 presents the result with sensitivity analysis and compares the results

between the deterministic and stochastic case in a hybrid

manufacturing/remanufacturing under uncertainty in production lead time, return

component lead time, and ordering lead time in addition to other uncertain is as

presented in Chapter 4.

Chapter 6 presents the conclusion and possible further study.

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CHAPTER 2

REVIEW OF LITERATURE

Lund (1985) illustrated remanufacturing as “an industrial process in which

worn-out products are restored to like-new condition”. The remanufacturing industry

boomed during World War II because of insufficient resources. The automotive

(industrial) sector has the most experience in remanufacturing (Sundin, 2004).

Remanufacturing is a product recovery option. There are many recovery options such

as reusing, repairing, refurbishing, and recycling (Ijomah, Bennett, & Pearce, 1999).

Remanufacturing is a direct form of reuse that combines returned products, to resell

them again like new products. The remanufacturing process can be separated into

disassembling the returned products, cleaning disassemble parts, replacing or repairing

any worn or damaged components, testing the quality of products, updating some parts

of electro-electronic products, and reassembling the products. Products that pass

remanufacturing testing can be considered to have the quality of “like-new” products

(Abbey, Meloy, Blackburn, & Guide, 2015). According to Li, González, & Zhu, (2009),

remanufacturing is normally managed under two business strategies: combined and

dedicated models. The combined model is mostly applied in European countries.

Remanufacturing and manufacturing are combined in the same line as a hybrid process.

In contrast, the dedicated model is mostly applied in North America. It is considered a

standalone remanufacturing line.

For production planning and inventory control problems, attention is placed on

different inventory policies such as periodic and continuous reviews. There are many

complicated characteristics of remanufacturing such as the uncertain quality of returns

and the need to balance the returns and new components with customer demand.

Inderfurth (2004) found that the optimal inventory control policy for the hybrid

manufacturing/remanufacturing system under strictly proportional costs and revenues

is the order-up-to policy. The system is constructed in a single-period with stochastic

returns of used products and customer demand, to substitute remanufactured products

with manufactured products when a shortage occurs. Wang (2011) investigated the

optimal production strategy for short life cycle products with stochastic returned

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products and customer demand. The objective was to minimize the total costs of this

system. The optimal total costs are incurred when applying a mixed strategy, which is

a mixture of remanufacturing, manufacturing, and disposal simultaneously. A

significant decrease in the total cost can be obtained by setting the suitable values of

return products and of manufactured products.

With the complication of inventory control policies, the best settings for the

operating parameters are required to obtain the maximum profit of the system.

Optimization is carried out by either analytical or heuristics models. The analytical

models always give global optimal solutions but take a longer running time. The

heuristics optimization may provide only local optimal solutions, which can be useable

solutions, with a shorter running time than the analytical models. Selection of suitable

solving method is a trade-off between solution quality and computational time. In

addition, analytical models need the simplification of assumptions and provide static

results. Without the uncertainty environment, solutions are applied in real-world

problems. Hence, the fuzzy logic is introduced to take care of uncertainties and to

provide results that, can reflect real-world problems. As a result, the Fuzzy Linear

Programming (FLP) can provide practical optimal results under realistic circumstances

(Amid, Ghodsypour, & O'Brien, 2009).

In practice, input data or related parameters for production planning and control

problems are imprecise/fuzzy, such as costs of operations, customer demand, and

number of returned products, delivery lead time, and production lead time. These

imprecise/fuzzy data occur because some information is incomplete or uncontrollable.

These problems cannot be solved and optimized by a traditional mathematical

analytical model such as linear programming because it only operates under

deterministic circumstances. To incorporate the uncertainty, Zimmerman (1978)

introduced the fuzzy set theory into traditional linear programming problems. His study

considered linear programming problems with fuzzy goals and constraints. Zadeh

(1977) presented the theory of possibility, which is related to the theory of fuzzy sets

by applying the concept of a possibility distribution as a fuzzy restriction, acting as an

elastic constraint. In addition, Buckley (1989) presented Possibilistic Linear

Programming (PLP) in a standard form with no equality constraints. Then, Ozgen &

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Gulsun (2014) applied a two-phase PLP combined with the fuzzy Analytical

Hierarchical Process (AHP) to optimize multi-objective linear programming. There are

very few research papers that have applied fuzzy lead time in their problems while

solving with the fuzzy mathematical model. For example, Diaz-Madroflero, Mula, and

Jiménez (2015) applied a fuzzy multi-objective integer linear programming to a model

of Material Requirements Planning (MRP) problem with the fuzzy lead time. By

incorporating different possibilities of lead times to the crisp MRP, the result showed

that combining the possibility of existence of the lead times with MRP model, decision

makers would know their risk when the uncertainty of lead times is considered. This

research gap will be further explored in this study.

This study applies the Fuzzy Linear Programming (FLP) approach for solving

the production planning and inventory control policy in a hybrid

manufacturing/remanufacturing with imprecise forecast demand, related operating

costs, number of returned components, ordering lead time of new components, delivery

lead time of returned components, as well as the production lead time of manufacturing

and remanufacturing processes. Taking care of the fuzzy ordering and production times

is a new research area in which it has not been much investigated in the past. The

proposed approach can help capture the consequence of these times’ variation with

other variations from other uncertainties by attempting to find the most likely value of

the imprecise total profits, minimize the risk of obtaining a lower profit, and maximize

the possibility of obtaining higher profits for each policy simultaneously. This approach

can provide realistic results, which are better than a typical deterministic approach as it

can handle fuzzy data and recommend a range of possible optimal objective values. The

result can help decision makers to be aware of possible outcomes in all scenarios from

the most likely, optimistic, and pessimistic cases. Therefore, decision makers can well

prepare themselves in advance from such scenarios.

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CHAPTER 3

METHODOLOGY

Two optimization solving methods are introduced to measure their

performances on a hybrid manufacturing/remanufacturing system in an uncertain

environment. The system is modeled with both deterministic and stochastic conditions,

using Mixed-Integer Linear Programming (MILP) and Fuzzy Linear Programming

(FLP), respectively. Figure 3.1 shows the approach for each solving method.

Figure 3.1 Method of approach.

3.1 Deterministic condition using MILP

The system is modeled, and the profit of the model is optimized using MILP.

There is no uncertainty in this situation. As a result, the ideal solution from this method

is be used as a benchmark for a comparison.

Fuzzy in lead time include

- Production lead time

- Order new components

lead time

- Delivery lead time of

returned components

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3.2 Stochastic condition using FLP

The FLP method can handle a problem with uncertainty. An uncertain

environment in this case contains uncertain customer demand in each week, related

operating costs, and number of arriving returned components in each week and all

related timing. The FLP converts these uncertain/fuzzy data to crisp data. There are

four main steps to solve the FLP:

3.2.1 Model the uncertainty data with triangular (possibility) distribution

Figure 3.2 presents the triangular distribution of an uncertain coefficient x =

(xP, xM, xO). In general, the triangular distribution is based on the three important data

values as follows:

a) The pessimistic value (𝒙𝑷) that has the lowest possibilistic degree for the

set of available values (possibilistic degree = 0).

b) The most likely value (𝒙𝑴) that has the highest possibilistic degree for the

set of available values (possibilistic degree = 1).

c) The optimistic value (𝒙𝑶) that has the lowest possibilistic degree for the set

of available values (possibilistic degree = 0).

Figure 3.2 Triangular possibility distribution of ��.

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3.2.2 Develop three new crisp objective functions of multi objective linear

programming (MOLP)

The imprecise objective function in this model has a triangular distribution z =

(zP, zM, zO). This imprecise objective function is defined by three important points

(zP, 0), (zM, 1), and (zO, 0). The imprecise objective can be maximized by pushing

three important points to the right. Because the vertical coordinate of important points

is fixed at either 1 or 0, only three horizontal coordinates are considered. Solving the

imprecise objective requires simultaneously maximizing zP, zM, and zO. Instead of

simultaneously maximizing zP, zM, and zO, the proposed approach maximizes the profit

zM, minimizes the range of the profit of ( zM- zP) and, maximizes the range of the profit

of (zO- zM). This proposed approach involves maximizing the most likely value of the

imprecise total profit zM, minimizing the risk of obtaining a lower profit ( zM- zP), and

maximizing the possibility of obtaining a higher profit (zO- zM). Figure 3.3 presents

the strategy for maximizing the imprecise objective functions.

Figure 3.3 Strategy for maximizing the imprecise objective function.

As presented in Figure 3.3, the possibility distribution A is preferred to the

possibility distribution B . The results for three new crisp objective functions are

presented as follows:

Max 𝑧1 = 𝑧𝑀 (3.1)

Min 𝑧2 = (𝑧𝑀- 𝑧𝑃) (3.2)

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Max 𝑧3 = (𝑧𝑂- 𝑧𝑀) (3.3)

Equation (3.1) to Equation (3.3) are equivalent to simultaneously maximizing

the most likely value of the total profit, minimizing the risk of obtaining a lower profit

(area A of the possibility distribution in Figure 3.3), and maximizing the possibility of

obtaining a higher profit (area B of the possibility distribution in Figure 3.3).

3.2.3 Defuzzification method

1. Convert uncertainty constraints into crisp constraints using the weighted

average method: To defuzzification uncertainty on demand, production capacity, and

amount of returned components.

We consider the situation where the number of returned products, customer

demand, and production capacity are uncertain and have the triangular distribution with

the most likely and least possible values. The problem is to obtain crisp numbers for

the uncertainty of the number of returned products, customer demand, and production

capacity by applying the weighted average method to convert them into crisp values,

where 𝑤1, 𝑤2, and 𝑤3 denote the weight of the pessimistic, most likely, and optimistic

cases respectively. The weights 𝑤1, 𝑤2, and 𝑤3 can be determined by the experience of

decision makers and 𝑤1 + 𝑤2 + 𝑤3 = 1.

2. Convert uncertainty constraints into crisp constraints using the fuzzy

ranking concept: To defuzzification uncertainty on lead time.

We consider the situation where the lead times of delivery of the returned

products, ordering new components, and production processes are uncertain under the

triangular distribution. This step is to obtain crisp constraints from the lead time’s

variability. The fuzzy ranking concept is applied to convert them into crisp constraints.

3.2.4 Specify the linear membership functions for three new objective functions,

and then convert the auxiliary MOLP problem into an equivalent linear

programming model using the fuzzy decision method

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3.2.4.1 Find the lower bound and the upper bound of each objective

To convert the auxiliary MOLP problem into an equivalent single-goal linear

programming problem, the fuzzy decision method from Bellman & Zadeh (1970), and

the Zimmermann (1978) fuzzy programming method are employed. The Negative Ideal

Solution (NIS) and Positive Ideal Solution (PIS) of three objective functions from step

two are required. Three new crisp objective functions of the multi objective linear

programming can be stated as follows:

𝑧1𝑃𝐼𝑆 = max 𝑧𝑀, 𝑧1

𝑁𝐼𝑆 = min 𝑧𝑀 (3.4)

𝑧2𝑃𝐼𝑆 = min (𝑧𝑀- 𝑧𝑃), 𝑧2

𝑁𝐼𝑆 = max (𝑧𝑀- 𝑧𝑃) (3.5)

𝑧3𝑃𝐼𝑆 = max (𝑧𝑂- 𝑧𝑀), 𝑧3

𝑁𝐼𝑆 = min (𝑧𝑂- 𝑧𝑀) (3.6)

3.2.4.2 Find the corresponding linear membership function of each objective

function

The corresponding linear membership function for each objective function is

defined by

𝑓1(𝑧1) =

{

1 , 𝑧1 < 𝑧1

𝑃𝐼𝑆,𝑧1−𝑧1

𝑁𝐼𝑆

𝑧1𝑃𝐼𝑆−𝑧1

𝑁𝐼𝑆 , 𝑧1𝑁𝐼𝑆 ≤ 𝑧1 ≤ 𝑧1

𝑃𝐼𝑆,

0 , 𝑧1 > 𝑧1𝑁𝐼𝑆,

(3.7)

𝑓2(z2) =

{

1 , 𝑧2 < 𝑧2

𝑃𝐼𝑆,𝑧2𝑁𝐼𝑆−𝑧2

𝑧1𝑁𝐼𝑆−𝑧2

𝑃𝐼𝑆 , 𝑧2𝑃𝐼𝑆 ≤ 𝑧2 ≤ 𝑧2

𝑁𝐼𝑆,

0 , 𝑧2 > 𝑧2𝑁𝐼𝑆

(3.8)

𝑓3(𝑧3) =

{

1 , 𝑧3 < 𝑧3

𝑃𝐼𝑆,𝑧3−𝑧3

𝑁𝐼𝑆

𝑧3𝑃𝐼𝑆−𝑧3


𝑃𝐼𝑆,

0 , 𝑧3 > 𝑧3𝑁𝐼𝑆.

(3.9)

Each linear membership function can be obtained by using a case study in

Chapter 4 and Chapter 5 to specify an imprecise objective value in an interval (0-1).

Figure 3.4 shows a graph of the linear membership functions for Equation (3.7) and

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Equation (3.9). Figure 3.5 shows a graph of the linear membership functions for

Equation (3.8).

Figure 3.4 Linear membership function of 𝑧1 and 𝑧3.

Figure 3.5 Linear membership function of 𝑧2.

3.2.4.3 Find the maximum overall satisfaction

The fuzzy decision method from Bellman & Zadeh (1970), and Zimmermann

(1978) is used to formulate a single goal linear programming model, which is minimax

of the satisfaction values from the three above mentioned objective functions.

Max λ

Subject to

λ ≤ 𝑓𝑖(𝑧𝑖) , i = 1,2,3

0 ≤ λ ≤ 1

i = number of linear membership functions of each objective.

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CHAPTER 4

A CASE STUDY WITHOUT LEAD TIME UNCERTAINTY

For the case study of the production planning and inventory control

optimization in a hybrid manufacturing/remanufacturing in an uncertain environment,

Figure 4.1 shows a flow diagram of the system. It requires two types of components for

production: new components and returned components. Returned components indicate

the used parts that are returned from customers. Returned components use the reorder

cycle policy to control their inventory, as returned components come in a batch at the

beginning of each week. A certain percentage of returned components need to be

disposed of to prevent excess inventory. Accepted returned components are then kept

in a Returned Component Inventory (RCI), pending for the remanufacturing processes.

It is assumed that the remanufacturing time of the returned components is zero week.

However, the remanufacturing cost per unit can be more expensive than the

manufacturing cost per unit depending the quality of the returned components as poor-

quality components could incur more expense to produce.

There is also a New Component Inventory (NCI), which is also reviewed every

week. Lead time for ordering the new components is constant at one week. Upon

arrival, these new components are kept in the NCI before they are used in the

manufacturing processes. One calendar week is considered for a manufacturing week.

Finished products from manufacturing and remanufacturing processes are then kept in

a Finished Product Inventory (FPI), waiting for the customer demand.

Figure 4.1 General flowchart of the studied system.

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Customer demand arrives every week. Lost sales cost incurs when there are not

enough products in the FPI. Otherwise, customers buy the finished products from the

FPI. For finished products, the quality from manufacturing and remanufacturing is

considered to be the same. When the level of inventory on the FPI is decreased, the

system requests for the production. The purpose is to fill back the inventory level of the

FPI. The beginning quantity of finished product inventory is set to be equal to the FPI

target inventory level, which is one of the decision variables in this case study. Two

inventory control policies (Priority-To-Remanufacturing vs Priority-To-

Manufacturing) are experimented here to decide which process (remanufacturing vs

manufacturing) has a higher priority. As shown in Figure 4.2, the operation flow of

each policy is shown as follows:

4.1 Priority-To-Remanufacturing (PTR)

For the PTR policy, returned components are assigned with a higher priority

than new components unless they are not available. Manufacturing production is

provoked only when there are not enough components in the RCI. Decision variables

in this policy include Disposal Rate (disR), Target Inventory Level of NCI (TinvN),

and Target Inventory Level of FPI (TinvF). Every week returned components are

disposed of depending on disR. Then, new components are ordered up to TinvN in

every review cycle (a week). The initial inventory level of NCI is set to be equal to

TinvN. When finished products are sent to customers, upstream components are pulled

to replenish the taken products by filling the FPI back to TinvF.

4.2 Priority-To-Manufacturing (PTM)

Opposite to PTR policy, new components are assigned with a higher priority

than returned components. Returned components are supplied for production only when

there is not enough of new components. Inventories are regulated by the decision

variables, similar to PTR.

Ref. code: 25615822042262GRA

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Figure 4.2 Flow diagram of the PTR and the PTM policies.

The system is inspected for one year or 50 weeks with 5 days a week and 8

hours a day, there are 40 hours a week. Table 4.1 presents the cost structure. Other

relevant information is described below.

4.3 Other relevant information

1. The maximum levels of the target inventory for new component

inventory and finished product inventory depend on the maximum of the production

capacity vary under the normal distribution in an uncertain environment. The mean and

standard deviation of the production capacity is 150 and 20 units per week, respectively.

2. The overall customer service level in this case study is set to at least 85%

3. The returned component ratio is set at 0.6: This is the ratio of arriving

returned components per total customer demand. Both arriving returned components

and total customer demand fluctuate under the normal distribution in an uncertain

environment. The mean and standard deviation for the returned components is 60 and

10 units per week, and 100 and 20 units per week for customer demand.

4. A supplier has the ability to send new components, but not more than

Ref. code: 25615822042262GRA

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70 units per week.

5. The holding cost is 40 %per year for a unit )i.e., new component cost,

returned component preparation cost, finished product cost. It is assumed that the

finished product holding costs per unit of both manufacturing (𝐹𝐻𝑚𝑦 ) and

remanufacturing (𝐹𝐻𝑟��) products are the same based on the average unit cost value

from both types of product.

6. The symbol ”⁓“refers to ambiguous data that are determined to be fuzzy

in this study.

Table 4.1 Cost Structure.

Cost parameters Notation Cost ($ per unit)

Disposal cost of returned component 𝑅𝐷𝑢 0

New component cost 𝑁𝐶�� (26,30,35)

Returned component preparation cost 𝑅𝑃�� (4,5,7)

Manufacturing cost 𝑀𝑀�� (9,10,13)

Remanufacturing cost 𝑅𝑀�� ((9*(2- y), (10*(2- y)),

(13*(2-y)) *

Returned component holding cost 𝑅𝐻�� (1.6,2,2.4) per year

New component holding cost 𝑁𝐻�� (10,12,15) per year

Finished product holding cost 𝐹𝐻�� (9.5,11,14) per year

Lost sales cost 𝐿𝑆�� (45,50,58)

Sales price 𝑃𝑟𝑖𝑐𝑒𝑢 (45,50,58)

Remark: * y is the yield of returned components, which is set equal to

(0.3,0.65,1). For example, the remanufacturing cost of the most likely case is calculated

as (10*(2-0.65)) = $ 13.5 per unit.

Ref. code: 25615822042262GRA

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4.4 LP formulation (Deterministic case)

The mathematical model presented in this case study is formulated as the linear

programming model. Notations and the formulation of the analytical model are

presented below where t refers to the time in weeks, ranging from 1 to 50.

4.4.1 Parameters

4.4.1.1 Production planning parameters (units)

𝑅𝑒𝐴𝑡 = Arriving returned components in week 𝑡

𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡 = Returned components (disposed of) in week 𝑡

𝑅0𝑡 = Returned components accepted to inventory in week 𝑡

𝑅𝐶𝐼𝑡 = Ending inventory level in the Returned Component Inventory (RCI) in

week 𝑡

𝑅1𝑡 = Returned components sent to remanufacturing in week 𝑡

𝑅2𝑡 = Finished products from remanufacturing in week 𝑡

𝐹𝑃𝐼𝑟𝑡 =Ending inventory level in Finished Product Inventory (FPI) from

remanufacturing in week 𝑡

𝑅3𝑡 = Finished products from remanufacturing sent to customer in week 𝑡

𝑜𝑟𝑑𝑒𝑟𝑡 = New components ordered in week 𝑡

𝑁𝐶𝐼𝑡 = Ending inventory level in the New Component Inventory (NCI) in week 𝑡

𝑀1𝑡 = New components sent to manufacturing in week 𝑡

𝑀2𝑡 = Finished products from manufacturing in week 𝑡

𝐹𝑃𝐼𝑚𝑡 = Ending inventory level in the Finished Product Inventory (FPI) from

manufacturing in week 𝑡

𝑀3𝑡 = Finished products from manufacturing sent to customer in week 𝑡

𝐹𝑃𝐼𝑡 = Ending inventory level in the Finished Product Inventory (FPI) in week 𝑡

𝐷𝑡 = Customer demand in week 𝑡

𝐿𝑆𝑡 = Lost sales in week 𝑡

𝑃𝐶𝑡 = Production capacity to produces a product in week 𝑡

Ref. code: 25615822042262GRA

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4.4.1.2 Cost parameters ($)

𝑅𝐷 = Total disposal cost of returned component

𝐿𝑆 = Total lost sales cost

𝑅𝑀𝑀 = Total remanufacturing and manufacturing cost

𝑁𝐶 = Total new component cost

𝑅𝑃 = Total returned component cost

𝐶𝐻 = Total component holding cost

𝐹𝐻 = Total finished product holding cost

𝑇𝐶 = Total costs

𝑅𝑒𝑣𝑒𝑛𝑢𝑒 = Total income from selling finished products

𝑃𝑟𝑜𝑓𝑖𝑡 = Total income after deducting total costs

4.4.2 Decision Variables

𝑇𝑖𝑛𝑣𝑁 = Target inventory level in NCI (unit)

𝑇𝑖𝑛𝑣𝐹 = Target inventory level in FPI (unit)

𝑑𝑖𝑠𝑅𝑡 = Disposal rate in week 𝑡 (%)

4.4.3 Objective Function

Maximize 𝑃𝑟𝑜𝑓𝑖𝑡 = 𝑅𝑒𝑣𝑒𝑛𝑢𝑒 - 𝑇�� (4.1)

subject to

inventory balance constraints:

𝑅𝑒𝐴�� = 𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡 + 𝑅0𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.2)

𝑅2𝑡= 𝑅1𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.3)

𝑀2𝑡 = {0 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1 𝑀1𝑡−1 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50

(4.4)

𝑅3𝑡 + 𝑀3𝑡 + 𝐿𝑆𝑡 = 𝐷�� 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.5)

𝐹𝑃𝐼𝑡 = 𝐹𝑃𝐼𝑟𝑡 + 𝐹𝑃𝐼𝑚𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.6)

𝑃𝐶�� ≥ 𝑅1𝑡 + 𝑀1𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.7)

Ref. code: 25615822042262GRA

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𝑅𝐶𝐼𝑡= {𝑅0𝑡 − 𝑅1𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1 𝑅𝐶𝐼𝑡−1 + 𝑅0𝑡 − 𝑅1𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50

(4.8)

𝑁𝐶𝐼𝑡= {𝑇𝑖𝑛𝑣𝑁 −𝑀1𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1 𝑁𝐶𝐼𝑡−1 + 𝑜𝑟𝑑𝑒𝑟𝑡−1 −𝑀1𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50

(4.9)

𝐹𝑃𝐼𝑟𝑡={𝑅2𝑡 − 𝑅3𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1 𝐹𝑃𝐼𝑟𝑡−1 + 𝑅2𝑡 − 𝑅3𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50

(4.10)

𝐹𝑃𝐼𝑚𝑡={𝑇𝑖𝑛𝑣𝐹 − 𝑀3𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1 𝐹𝑃𝐼𝑚𝑡−1 +𝑀2𝑡 − 𝑀3𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50

(4.11)

Where 𝑅𝑒𝐴��, and 𝐷�� are uncertainty coefficients with the triangular

distribution .The objective function (constraint (4.1)) is to maximize the profit.

Constraint (4.2) and Constraint (4.3) are inventory balance constraints. For Constraint

(4.2), some of the returned components are disposed of depending on disposal rate in

every week, and the rest is sent to the Returned Component Inventory (RCI). Constraint

(4.3) ensures that the quantity of returned components sent to remanufacturing

processes are the same quality as the finished products coming out of the

remanufacturing processes within the same week and the remanufacturing time is

negligible. Constraint (4.4) states that the manufacturing lead time is one week .For

Constraint (4.5), amount of customer demand is satisfied by the finished products at

FPI, otherwise, some of lost sales will occur. For Constraint (4.6), total units of finished

products in manufacturing and remanufacturing are combined in the FPI. Constraint

(4.7) describes the constraint of production capacity for remanufacturing and

manufacturing, in which the number of components sent to the manufacturing process

must be less than or equal to the production capacity. Constraints (4.8) to (4.11)

describe the constraints for returned components, new components, and finished

products for remanufacturing and manufacturing. The ending inventory is equal to the

previous ending inventory (week) plus the incoming inventory minus the outgoing

inventory. In the first week, the ending inventory is equal to the incoming inventory

minus the outgoing inventory.

Ref. code: 25615822042262GRA

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The PTR constraints:

𝑅𝐶𝐼𝑡={

0 𝑖𝑓 𝐷�� ≥ 𝑅0𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1

0 𝑖𝑓 𝐷�� ≥ 𝑅0𝑡 + 𝑅𝐶𝐼𝑡−1 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50𝑎𝑛𝑦 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡

(4.12)

𝑅0𝑡 = {𝐷�� 𝑖𝑓 𝐷�� < 𝑅𝑒𝐴�� 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡

𝑅𝑒𝐴�� 𝑖𝑓 𝑅𝑒𝐴�� < 𝐷�� 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.13)

𝑅1𝑡 = {𝐷�� 𝑖𝑓 𝐷�� < 𝑅0𝑡 + 𝑅𝐶𝐼𝑡−1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡𝑎𝑛𝑦 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡

(4.14)

Where 𝑅𝑒𝐴��,and 𝐷�� are uncertain coefficients with the triangular distribution. Constraints (4.12) and (4.14) describe the policies of the PTR. All of the returned

components in RCI are sent to remanufacturing when the customer demand is greater

than or equal the accepted returned components plus the number of the previous week’s

ending inventory in RCI. Otherwise the number of returned components sent to

remanufacturing is equal to the customer demand.

The PTM constraints:

𝑁𝐶𝐼𝑡 = {

0 𝑖𝑓 𝐷�� ≥ 𝑇𝑖𝑛𝑣𝑁 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1

𝑜𝑟𝑑𝑒𝑟𝑡−1 𝑖𝑓 𝐷�� ≥ 𝑁𝐶𝐼𝑡−1 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50𝑎𝑛𝑦 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡

(4.15)

𝑀1𝑡= {𝐷�� 𝑖𝑓 𝐷�� < 𝑁𝐶𝐼𝑡−1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡𝑎𝑛𝑦 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡

(4.16)

Where 𝐷�� are uncertain coefficients with the triangular distribution. Constraints

(4.15) and (4.16) describe the policies of the PTM, where the number of new

components in the NCI that is sent to manufacturing is equal to the customer demand

when it is less than the number of new components. Otherwise, all of the new

components are sent to manufacturing .The ending level of inventory in the NCI is

equal to the incoming orders from the previous week when the customer demand is

greater than the previous period’s ending inventory level .

Ref. code: 25615822042262GRA

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Economic constraints:

𝑅𝐷 = ∑ 𝑅𝐷_𝑢 ∗ 𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.17)

𝐿�� = ∑ 𝐿𝑆_𝑢 ∗ 𝐿𝑆��𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.18)

𝑅𝑀�� = ∑ 𝑅𝑀_�� ∗ 𝑅1𝑡𝑡 + ∑ 𝑀𝑀_�� ∗ 𝑀1𝑡𝑡 + 𝑀𝑀�� ∗ 𝑇𝑖𝑛𝑣𝐹 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.19)

𝑁�� = ∑ 𝑁𝐶�� ∗ (𝑜𝑟𝑑𝑒𝑟𝑡 + 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑁𝐶𝐼 + 𝑇𝑖𝑛𝑣𝐹) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 𝑡 (4.20)

𝑅�� = ∑ 𝑅𝑃_�� ∗ 𝑅0𝑡𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.21)

𝐶�� =

(𝑅𝐻𝑦

50

∗(𝑅01+𝑅𝐶𝐼1))

2+∑ (

𝑅𝐻𝑦

50

∗(𝑅0𝑡+ 𝑅𝐶𝐼𝑡−1+ 𝑅𝐶𝐼𝑡))

50𝑡=2

2+(𝑁𝐻𝑦

50

∗(𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑁𝐶𝐼+𝑁𝐶𝐼1))

2+

∑(𝑁𝐻𝑦

50

∗(𝑜𝑟𝑑𝑒𝑟𝑡−1+𝑁𝐶𝐼𝑡−1+𝑁𝐶𝐼𝑡))

2

50𝑡=2 (4.22)

𝐹�� = (𝐹𝐻_𝑦

50

∗ (𝑅21 + 𝐹𝑃𝐼𝑟1)) /2 + ∑ (

𝐹𝐻_𝑦

50

∗ (𝑅2𝑡 + 𝐹𝑃𝐼𝑟𝑡−1 + 𝐹𝑃𝐼𝑟𝑡)) /2 +

50𝑡=2

(𝐹𝐻_𝑦

50

∗ (𝑀21 + 𝑇𝑖𝑛𝑣𝐹 + 𝐹𝑃𝐼𝑚1)) /2 + ∑ (

𝐹𝐻_𝑦

50

∗ (𝑀2𝑡 + 𝐹𝑃𝐼𝑚𝑡−1 +

50𝑡=2

𝐹𝑃𝐼𝑚𝑡)) /2 (4.23)

𝑇�� = 𝑅�� + 𝐿�� + 𝑅𝑀�� + 𝑁�� + 𝑅�� + 𝐶�� + 𝐹�� (4.24)

𝑅𝑒𝑣𝑒𝑛𝑢𝑒 = ∑ 𝑃𝑟𝑖𝑐𝑒_𝑢 ∗ (𝑅3𝑡 +𝑀3𝑡)𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.25)

𝑃𝑟𝑜𝑓𝑖𝑡 = 𝑅𝑒𝑣𝑒𝑛𝑢𝑒 −𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡𝑠 (4.26)

Where 𝐿𝑆��, 𝑅𝑀_��, 𝑀𝑀_��, 𝑁𝐶��, 𝑅𝑃_��, 𝑁𝐻_��, 𝑅𝐻_��, 𝐹𝐻_��, and 𝑃𝑟𝑖𝑐𝑒_𝑢 are

uncertain coefficients with the triangular distribution. Constraints (4.17) to (4.23)

describe the cost parameters to calculate the total cost (TC). The TC consists of the

Disposal Cost of Returned Component (RD), Lost Sales Cost (LS), Remanufacturing

and Manufacturing Cost (RMM), New Component Cost (NC), Returned Component

Preparation Cost (RP), Component Holding Cost (CH), and Finished Product Holding

Cost (FH). All holding costs are calculated based on the average level of inventory .

Constraint (4.26) states that the profit is calculated by the revenue (Constraint (4.25))

Ref. code: 25615822042262GRA

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minus total costs (Constraint (4.24)).

Decision Variables:

𝑜𝑟𝑑𝑒𝑟𝑡 = 𝑇𝑖𝑛𝑣𝑁 – 𝑁𝐶𝐼𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.27)

𝑅1𝑡 + 𝑀1𝑡 = 𝑇𝑖𝑛𝑣𝐹 − 𝐹𝑃𝐼𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.28)

𝑑𝑖𝑠𝑅𝑡 = 100 ∗ (𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡 𝑅𝑒𝐴��⁄ ) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.29)

For Equation (4.27), the order of new components depends on 𝑇𝑖𝑛𝑣𝑁 .Equation

(4.28) shows that the total number of new and returned components sent to production

equal the number of finished products. Equation (4.29) describes the disposal rate,

which is the percentage of disposed of components over the returned components.

4.5 Fuzzy Linear Programming (FLP)

We develop the fuzzy linear programming method to solve this problem under

uncertainty by developing three new crisp objective functions of the multi-objective

linear programming (MOLP) to replace Equations (3.1) - (3.3).

Max 𝑧1 = 𝑃𝑟𝑜𝑓𝑖𝑡𝑀 = 𝑅𝑒𝑣𝑒𝑛𝑢𝑒𝑀 − 𝑇𝐶𝑀 =𝑅𝑒𝑣𝑒𝑛𝑢𝑒𝑀-𝑅𝐷𝑀+𝐿𝑆𝑀+𝑅𝑀𝑀𝑀+𝑁𝐶𝑀 +𝑅𝑃𝑀+𝐶𝐻𝑀+𝐹𝐻𝑀 (4.30)

Min 𝑧2 = 𝑃𝑟𝑜𝑓𝑖𝑡𝑃−𝑀 = 𝑅𝑒𝑣𝑒𝑛𝑢𝑒𝑃−𝑀 − 𝑇𝐶𝑃−𝑀 = 𝑅𝑒𝑣𝑒𝑛𝑢𝑒𝑃−𝑀 − 𝑅𝐷𝑃−𝑀+𝐿𝑆𝑃−𝑀+ 𝑅𝑀𝑀𝑃−𝑀 +𝑁𝐶𝑃−𝑀+𝑅𝑃𝑃−𝑀+𝐶𝐻𝑃−𝑀+𝐹𝐻𝑃−𝑀 (4.31)

Max 𝑧3= 𝑃𝑟𝑜𝑓𝑖𝑡𝑀−𝑂 = 𝑅𝑒𝑣𝑒𝑛𝑢𝑒𝑀−𝑂 − 𝑇𝐶𝑀−𝑂

= 𝑅𝑒𝑣𝑒𝑛𝑢𝑒𝑀−𝑂 − 𝑅𝐷𝑀−𝑂+𝐿𝑆𝑀−𝑂+𝑅𝑀𝑀𝑀−𝑂 +𝑁𝐶𝑀−𝑂+𝑅𝑃𝑀−𝑂+𝐶𝐻𝑀−𝑂+𝐹𝐻𝑀−𝑂 (4.32)

As the demand of customers, quantity of returned components, and production

capacity, a defuzzification method using the weighted average method is applied to

solve the imprecise data. Constraint (4.2), (4.5), (4.7), (4.12), (4.13), (4.14), (4.15), and

(4.16) need to be transformed to crisp constraints as follows:

𝑤1𝑅𝑒𝐴𝑡𝑃+𝑤2𝑅𝑒𝐴𝑡

𝑀+𝑤3𝑅𝑒𝐴𝑡𝑂

= 𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡 + 𝑅0𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.33)

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𝑅3𝑡 + 𝑀3𝑡 + 𝐿𝑆𝑡 = 𝑤1𝐷𝑡𝑃+ 𝑤2𝐷𝑡

𝑀+ 𝑤3𝐷𝑡𝑂 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.34)

𝑤1𝑃𝐶𝑡

𝑃 + 𝑤2𝑃𝐶𝑡𝑀 + 𝑤3𝑃𝐶𝑡

𝑂 ≥ 𝑅1𝑡 + 𝑀1𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (4.35)

𝑅𝐶𝐼𝑡= {

0 𝑖𝑓 𝑤1𝐷𝑡𝑃 + 𝑤2𝐷𝑡

𝑀 + 𝑤3𝐷𝑡𝑂 ≥ 𝑅0𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1


𝑀 + 𝑤3𝐷𝑡𝑂 ≥ 𝑅0𝑡 + 𝑅𝐶𝐼𝑡−1𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50

𝑎𝑛𝑦 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡

(4.36)

𝑅0𝑡=

{

𝑤1𝐷𝑡

𝑃 + 𝑤2𝐷𝑡𝑀 + 𝑤3𝐷𝑡

𝑂

𝑖𝑓 𝑤1𝐷𝑡𝑃 + 𝑤2𝐷𝑡

𝑀 + 𝑤3𝐷𝑡𝑂 < 𝑤1𝑅𝑒𝐴𝑡

𝑃 + 𝑤2𝑅𝑒𝐴𝑡𝑀 +𝑤3𝑅𝑒𝐴𝑡

𝑂 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡

𝑤1𝑅𝑒𝐴𝑡𝑃 + 𝑤2𝑅𝑒𝐴𝑡

𝑀 + 𝑤3𝑅𝑒𝐴𝑡𝑂

𝑖𝑓 𝑤1𝑤1𝑅𝑒𝐴𝑡𝑃 + 𝑤2𝑅𝑒𝐴𝑡

𝑀 + 𝑤3𝑅𝑒𝐴𝑡𝑂 < 𝑤1𝐷𝑡



(4.37)

𝑅1𝑡= {

𝑤1𝐷𝑡𝑃 + 𝑤2𝐷𝑡

𝑀 + 𝑤3𝐷𝑡𝑂

𝑖𝑓𝑤1𝐷𝑡𝑃 + 𝑤2𝐷𝑡

𝑀 + 𝑤3𝐷𝑡𝑂 < 𝑅0𝑡 + 𝑅𝐶𝐼𝑡−1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡


(4.38)

𝑁𝐶𝐼𝑡=

{


𝑀 + 𝑤3𝐷𝑡𝑂 ≥ 𝑇𝑖𝑛𝑣𝑁 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1

𝑜𝑟𝑑𝑒𝑟𝑡−1 𝑖𝑓 𝑤1𝐷𝑡𝑃 + 𝑤2𝐷𝑡

𝑀 + 𝑤3𝐷𝑡𝑂 ≥ 𝑁𝐶𝐼𝑡−1 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50


(4.39)

𝑀1𝑡=

{𝑤1𝐷𝑡


𝑂 𝑖𝑓 𝑤1𝐷𝑡𝑃 + 𝑤2𝐷𝑡

𝑀 + 𝑤3𝐷𝑡𝑂 < 𝑁𝐶𝐼𝑡−1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡


(4.40)

Where 𝑤1 + 𝑤2 + 𝑤3 = 1 and the weights of 𝑤1, 𝑤2, and 𝑤3 can be assigned by the

knowledge and experience of decision makers.

The linear membership functions for three new objective functions are specified

and converted the auxiliary MOLP problem into an equivalent linear programming

model by the fuzzy decision method.

To find the Negative Ideal Solution (NIS) and the Positive Ideal Solution (PIS)

of each objective, the fuzzy decision method as presented in Equations (3.4) to (3.6) is

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24

introduced to solve MOLP by finding the corresponding linear membership function of

each objective function.

𝑧1𝑃𝐼𝑆 = max 𝑃𝑟𝑜𝑓𝑖𝑡𝑀, 𝑧1

𝑁𝐼𝑆 = min 𝑃𝑟𝑜𝑓𝑖𝑡𝑀 (4.41)

𝑧2𝑃𝐼𝑆 =min 𝑃𝑟𝑜𝑓𝑖𝑡𝑃−𝑀, 𝑧2

𝑁𝐼𝑆 = max 𝑃𝑟𝑜𝑓𝑖𝑡𝑃−𝑀 (4.42)

𝑧3𝑃𝐼𝑆 = max 𝑃𝑟𝑜𝑓𝑖𝑡𝑀−𝑂, 𝑧3

𝑁𝐼𝑆 = min 𝑃𝑟𝑜𝑓𝑖𝑡𝑀−𝑂 (4.43)

Equations (3.7) to (3.9) are applied to the corresponding linear membership

functions for each objective function in this case study as follows:

𝑓1(𝑧1) =

{

1 , 𝑧1 < 𝑧1

𝑃𝐼𝑆,

𝑧1 − 𝑧1𝑁𝐼𝑆

𝑧1𝑃𝐼𝑆 − 𝑧1


𝑃𝐼𝑆,

0 , 𝑧1 > 𝑧1𝑁𝐼𝑆,

𝑓2(z2) =

{

1 , 𝑧2 < 𝑧2

𝑃𝐼𝑆,

𝑧2𝑁𝐼𝑆 − 𝑧2



𝑁𝐼𝑆,

0 , 𝑧2 > 𝑧2𝑁𝐼𝑆,

𝑓3(𝑧3) =

{

1 , 𝑧3 < 𝑧3

𝑃𝐼𝑆,




𝑃𝐼𝑆,

0 , 𝑧3 > 𝑧3𝑁𝐼𝑆.

Find the maximum overall satisfaction:

Max λ

Subject to

λ ≤ 𝑓𝑖(𝑧𝑖), where i = 1,2,3

i = number of linear membership function of each objective.

0 ≤ λ ≤ 1

𝑤1𝑅𝑒𝐴𝑡𝑃+𝑤2𝑅𝑒𝐴𝑡

𝑀+𝑤3𝑅𝑒𝐴𝑡𝑂

= 𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡 + 𝑅0𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 ,

𝑅2𝑡 = 𝑅1𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

𝑅3𝑡 + 𝑀3𝑡 + 𝐿𝑆𝑡 = 𝑤1𝐷𝑡𝑃+ 𝑤2𝐷𝑡

𝑀+ 𝑤3𝐷𝑡𝑂 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 ,

𝐹𝑃𝐼𝑡 = 𝐹𝑃𝐼𝑟𝑡 + 𝐹𝑃𝐼𝑚𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

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𝑤1𝑃𝐶𝑡𝑃 + 𝑤2𝑃𝐶𝑡

𝑀 + 𝑤3𝑃𝐶𝑡𝑂 ≤ 𝑅1𝑡 + 𝑀1𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

𝑅𝐶𝐼𝑡={𝑅0𝑡 − 𝑅1𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1, 𝑅𝐶𝐼𝑡−1 + 𝑅0𝑡 − 𝑅1𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50,

𝑁𝐶𝐼𝑡= {𝑇𝑖𝑛𝑣𝑁 − 𝑀𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1, 𝑁𝐶𝐼𝑡−1 + 𝑜𝑟𝑑𝑒𝑟𝑡−1 − 𝑀1𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50,

𝐹𝑃𝐼𝑟𝑡= {𝑅2𝑡 − 𝑅3𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1, 𝐹𝑃𝐼𝑟𝑡−1 + 𝑅2𝑡 − 𝑅3𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50,

𝐹𝑃𝐼𝑚𝑡= {𝑇𝑖𝑛𝑣𝐹 − 𝑀3𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1, 𝐹𝑃𝐼𝑚𝑡−1 +𝑀2𝑡 − 𝑀3𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50,

For the PTR

𝑅𝐶𝐼𝑡= {


𝑀 + 𝑤3𝐷𝑡𝑂 ≥ 𝑅0𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1,


𝑀 + 𝑤3𝐷𝑡𝑂 ≥ 𝑅0𝑡 + 𝑅𝐶𝐼𝑡−1𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50,

𝑎𝑛𝑦 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

𝑅0𝑡=

{

𝑤1𝐷𝑡


𝑂



𝑃 + 𝑤2𝑅𝑒𝐴𝑡𝑀 +𝑤3𝑅𝑒𝐴𝑡

𝑂 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,



𝑖𝑓 𝑤1𝑤1𝑅𝑒𝐴𝑡𝑃 + 𝑤2𝑅𝑒𝐴𝑡




𝑅1𝑡= {



𝑖𝑓𝑤1𝐷𝑡𝑃 + 𝑤2𝐷𝑡

𝑀 + 𝑤3𝐷𝑡𝑂 < 𝑅0𝑡 + 𝑅𝐶𝐼𝑡−1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,


For the PTM

𝑁𝐶𝐼𝑡=

{


𝑀 + 𝑤3𝐷𝑡𝑂 ≥ 𝑇𝑖𝑛𝑣𝑁 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1,


𝑀 + 𝑤3𝐷𝑡𝑂 ≥ 𝑁𝐶𝐼𝑡−1 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50,


𝑀1𝑡=

{𝑤1𝐷𝑡



𝑀 + 𝑤3𝐷𝑡𝑂 < 𝑁𝐶𝐼𝑡−1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,


𝑅𝐷= ∑ 𝑅𝐷_𝑢 ∗ 𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

𝐿�� = ∑ 𝐿𝑆_𝑢 ∗ 𝐿𝑆��𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

𝑅𝑀�� = ∑ 𝑅𝑀_�� ∗ 𝑅1𝑡𝑡 + ∑ 𝑀𝑀_�� ∗ 𝑀1𝑡𝑡 +𝑀𝑀�� ∗ 𝑇𝑖𝑛𝑣𝐹 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

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𝑁�� = ∑ 𝑁𝐶�� ∗ (𝑜𝑟𝑑𝑒𝑟𝑡 + 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑁𝐶𝐼 + 𝑇𝑖𝑛𝑣𝐹) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡, 𝑡

𝑅�� = ∑ 𝑅𝑃_�� ∗ 𝑅0𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡𝑡 ,

𝐶�� = (𝑅𝐻_𝑦

50

∗ (𝑅01 + 𝑅𝐶𝐼1)) /2 + ∑ (

𝑅𝐻_𝑦

50

∗ (𝑅0𝑡 + 𝑅𝐶𝐼𝑡−1 + 𝑅𝐶𝐼𝑡))

50𝑡=2 /2 +

(𝑁𝐻_𝑦

50

∗(𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑁𝐶𝐼 + 𝑁𝐶𝐼1)) /2 + ∑ (

𝑁𝐻_𝑦

50

∗(𝑜𝑟𝑑𝑒𝑟𝑡−1 + 𝑁𝐶𝐼𝑡−1 + 𝑁𝐶𝐼𝑡)) /

50𝑡=2

2,

𝐹�� = (𝐹𝐻_𝑦

50

∗ (𝑅21 + 𝐹𝑃𝐼𝑟1)) /2 + ∑ (

𝐹𝐻_𝑦

50

∗ (𝑅2𝑡 + 𝐹𝑃𝐼𝑟𝑡−1 + 𝐹𝑃𝐼𝑟𝑡)) /2 +

50𝑡=2

(𝐹𝐻_𝑦

50

∗ (𝑀21 + 𝑇𝑖𝑛𝑣𝐹 + 𝐹𝑃𝐼𝑚1)) /2 + ∑ (

𝐹𝐻_𝑦

50

∗ (𝑀2𝑡 + 𝐹𝑃𝐼𝑚𝑡−1 +

50𝑡=2

𝐹𝑃𝐼𝑚𝑡)) /2,

𝑇�� = 𝑅�� + 𝐿�� + 𝑅𝑀�� + 𝑁�� + 𝑅�� + 𝐶�� + 𝐹��,

𝑅𝑒𝑣𝑒𝑛𝑢𝑒 = ∑ 𝑃𝑟𝑖𝑐𝑒_𝑢 ∗ (𝑅3𝑡 +𝑀3𝑡)𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

𝑃𝑟𝑜𝑓𝑖𝑡 = 𝑅𝑒𝑣𝑒𝑛𝑢𝑒 −𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡𝑠 .

4.6 Results and Discussion

4.6.1 Deterministic case with linear programming model

An LP model is built and solved by IBM ILOG CPLEX Optimization Studio

software. The deterministic results of the PTR and the PTM policies are shown in Table

4.2.

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Table 4.2 PTR and PTM results.

Policy

PTR PTM

Decision

Variables

Target inventory of new

components (TinvN) (units) 47 40

Target inventory of finished

products (TinvF) (units) 92 167

Number of disposed returned

components (units) 0 0

Cost

Parameters

Lost sales cost (LS) $ 0 $ 0

New Component Cost (NC) $ 62,970.00 $ 66,210.00

Returned Component Cost (RP) $ 15,380.00 $ 15,005.00

Remanufacturing and

Manufacturing Cost (RMM) $ 62,046.00 $ 62,184.00

Total Component Holding Cost

(CH) $ 390.52 $ 501.18

Finished Product Holding Cost

(FH) $ 751.52 $ 593.34

Total costs (TC) $ 141,538.04 $ 144,493.52

Revenue $ 256,400.00 $ 256,400.00

Profit $ 114,861.96 $ 111,906.48

Table 4.2 compares the solutions obtained from these two policies. The result

shows that the PTR policy is better than the PTM policy under the deterministic case,

because the PTR policy can generate a higher profit than the PTM policy. From the

result, the PTR policy and the PTM policy do not dispose of any returned components,

but the PTM policy has a higher cost from ordering new components (NC). However,

it should be noted that the difference of the profits from these two policies is close, with

less than a 4% gap.

4.6.2 Stochastic case with the Fuzzy Linear Programming (FLP)

With the lower bound and the upper bound of each objective (Constraints (4.41)

to (4.43)), Tables 4.3-4.4 show the Negative Ideal Solution (NIS) and the Positive Ideal

Solution (PIS) of each objective of the PTR and the PTM policies without lead time

uncertainty, respectively.

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Table 4.3 Upper and lower boundary of the PTR policy without lead time uncertainty.

PTR

MOLP Profit ($)

𝑧1𝑃𝐼𝑆 = max 𝑃𝑟𝑜𝑓𝑖𝑡𝑀 128,923.48

𝑧1𝑁𝐼𝑆 = min 𝑃𝑟𝑜𝑓𝑖𝑡𝑀 45,359.18

𝑧2𝑃𝐼𝑆 = min 𝑃𝑟𝑜𝑓𝑖𝑡𝑃−𝑀 52,925.274

𝑧2𝑁𝐼𝑆 = max 𝑃𝑟𝑜𝑓𝑖𝑡𝑃−𝑀 83,719.564

𝑧3𝑃𝐼𝑆 = max 𝑃𝑟𝑜𝑓𝑖𝑡𝑀−𝑂 86,436.134

𝑧3𝑁𝐼𝑆 = min 𝑃𝑟𝑜𝑓𝑖𝑡𝑀−𝑂 50,253.914

Table 4.4 Upper and lower boundary of the PTM policy without lead time uncertainty.

PTM

MOLP Profit ($)


𝑧1𝑁𝐼𝑆 = min 𝑃𝑟𝑜𝑓𝑖𝑡𝑀 0





According to Constraints (4.33) to (4.40) of the fuzzy linear programming

model, the process of defuzzification imprecise customer demand, number of arriving

returned components, and production rate with the weighted average method is subject

to the pattern of weight allocation. Different patterns of weight allocation can have an

impact on the obtained solution. As a result, a sensitivity analysis of this weight

allocation should be carried out among pessimistic, most likely, and optimistic cases to

investigate the impact of such weight allocation on the overall satisfaction of the linear

membership functions of each objective function. For a demonstration, a sensitivity

analysis is performed in 4 scenarios by varying the weights of pessimistic, most likely,

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and optimistic values. Tables 4.5-4.6 present the results of the sensitivity analysis for

both the PTR and the PTM policies without lead time uncertainty.

Table 4.5 Result of the sensitivity analysis of the PTR policy without lead time

uncertainty.

Scenario I Scenario II Scenario III Scenario IV

P M O P M O P M O P M O

33% 33% 33% 80% 10% 10% 10% 80% 10% 10% 10% 80%

Decision Variables

Target

inventory of

new

components

(TinvN)

(units)

197 171 124 29

Target

inventory of

finished

products

(TinvF)

(units)

91 171 92 195

Number of

disposed

returned

components

(units)

6 0 5 3

Results

Overall

satisfaction 49.52% 34.68% 49.87% 41.67%s

Z1 ($) 110,130 74,335 114,900 80,182

Z2 ($) 68,470 65,389 68,362 70,887

Z3 ($) 68,171 62,800 68,298 68,572

Maximum

the most

likely value

of profit ($)

110,130 74,335 114,900 80,182

Minimize the

risk of

obtaining

lower profit

($)

41,660 8,946 46,538 9,295

Maximize of

the

possibility of

obtaining a

higher profit

($)

178,301 137,135 183,198 148,754

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For example, in Table 4.5 with Scenario I, equal weights (33%) are assigned to

the pessimistic, most likely, and optimistic values. It is found that the overall

satisfaction is 49.52%, and the outcomes of the maximum most likely value of profit,

minimum possibility of obtaining a lower profit, and maximum of the risk of obtaining

a higher profit are $110,130, $41,660, and $178,301, respectively.

The result of this sensitivity analysis also shows that Scenario III, which set the

weight of the most likely value at 80% and the weight of optimistic and pessimistic

values at 10%, gives the highest overall satisfaction of 49.87%. Scenario III also gives

the highest possible value of the most likely profit, obtaining the highest possibility of

a higher profit and the highest profit from the risk of obtaining a lower profit.

However, the profit generated from Scenario IV, which is supposed to be the

highest as it is the most optimistic case, is not the highest. This is due to the fact that

the amount of optimistic demand is too high for the manufacturing process. As a result,

a large shortage cost needs to be paid.

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Table 4.6 Result of the sensitivity analysis of the PTM policy without lead time

uncertainty.



33% 33% 33% 80% 10% 10% 10% 80% 10% 10% 10% 80%

Decision Variables

Target

inventory of

new

components

(TinvN)

(units)

70 53 65 29

Target

inventory of

finished

products

(TinvF)

(units)

197 171 176 195

Number of

disposed

returned

components

(units)

1,729 0 1,523 91

Results

Overall

satisfaction 52.65% 36.04% 52.23% 38.74%

Z1 ($) 67,215 63,508 67,325 71,635

Z2 ($) 67,215 63,298 67,325 71,628

Z3 ($) 66,481 61,180 66,372 68,881

Maximum

the most

likely value

of profit ($)

67,215 63,508 67,325 71,635

Minimize the

risk of

obtaining

lower profit

($)

0 210 0 7

Maximize of

the

possibility of

obtaining a

higher profit

($)

133,696 124,688 133,697 140,516

For example, in Table 4.6 with Scenario I, equal weights (33%) are assigned to

the pessimistic, most possible, and optimistic values. It is found that the overall

satisfaction is 52.65%, and the outcomes of maximum most possible value of profit,

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minimum possibility of obtaining a lower profit, and maximum of the risk of obtaining

a higher profit are $67,215, $0, and $67,215, respectively.

The best result of the PTM policy is from setting the weight of optimistic value

at 80% and 10% at both pessimistic and the most likely values (Scenario IV). Opposite

to the PTR policy, the profit generated from Scenario IV (the optimistic case) is higher

than from Scenario I. This is because it turns out to use less amount of new components

and use more returned components than the case of setting equal weight to all three

value (scenario I). This is due to the fact that the PTM policy is require to use the new

components first but the new components take longer time to arrive than the returned

components. As a result, to prevent shortage, the arriving returned components would

be selected for the production.

As for a comparison between the two policies, it was found that the PTR policy

can outperform the PTM policy in terms of profit in all objective functions and in all

scenarios.

4.6.3 Comparison between the deterministic and the best stochastic results

Table 4.7 compares the results between the optimal LP result under a

deterministic condition and the best FLP result under a more realistic stochastic

condition. As seen in Table 4.7, the most likely profits that are generated from the

uncertain circumstance are lower as compared to the profits under no uncertainty as the

overall degree of satisfaction is lower. This is because the result was recommended

from the scenario that yields the lowest degree of satisfaction among the three

objectives, which are maximizing the most likely profit, minimizing the risk obtaining

a lower profit, and maximizing the possibility of obtaining a higher profit. The PTR

policy under Scenario III and the PTM policy under Scenario IV show the best result

among experimental scenarios, whereas The PTR policy generates a higher profit than

the PTM policy in all cases. With the range of profit from each scenario obtained from

FLP, decision makers are well prepared for any expected outcome. This is an important

feature for decision makers to build the production planning and inventory control

subject to an uncertain environment.

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Table 4.7 Result comparison.

The PTR Policy

Priority-To-Remanufacturing

policy (PTR)

Linear

Programming

(LP) model

Fuzzy Linear Programming (FLP)

model from Scenario III

Objective function λ Max z Max λ

(overall degree of satisfaction) 100% 49.87%

z (Profit) $ 114,861.96 ($46,538, $114,900, $183,198)

The PTM Policy

Priority-To-Manufacturing

policy (PTR)

Linear

Programming

(LP) model


model from Scenario IV



z (Profit) $ 111,906.48 ($7, $71,635, $140,516)

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CHAPTER 5

A CASE STUDY WITH LEAD TIME UNCERTAINTY

For this case study, the uncertainty of production and lead times is incorporated

in to the model in addition to other remaining uncertain factors. As a result, the returned

components arrive in a batch at the beginning of each week with varying lead time from

zero to one week following the triangular distribution. The remanufacturing time of the

returned components also varies from zero to one week following the triangular

distribution. The ordering lead time of the new components varies from one to two

weeks. The uncertainty of the production lead time, also varied from 1 to 2 weeks.

According to the uncertainty of the lead time, some of the constraints need to be

changed. Details of the model formulation and solving method can be presented as

follows:

5.1 LP formulation (Deterministic case)

5.1.1 Parameter

5.1.1.1 Production planning parameters (units)

𝑅𝑒𝐴�� = Arriving returned components in week ��

𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡 = Returned components (disposed of) in week 𝑡

𝑅0𝑡 = Returned components accepted to inventory in week 𝑡

𝑅𝐶𝐼𝑡 = Ending inventory level in the Returned Component Inventory (RCI) in

week 𝑡

𝑅1�� = Returned components sent to remanufacturing in week ��

𝑅2𝑡 = Finished products from remanufacturing in week 𝑡

𝐹𝑃𝐼𝑟𝑡 = Ending inventory level in Finished Product Inventory (FPI) from

remanufacturing in week 𝑡

𝑅3𝑡 = Finished products from remanufacturing sent to customer in week 𝑡

𝑜𝑟𝑑𝑒𝑟�� = New components ordered in week ��

𝑁𝐶𝐼𝑡 = Ending inventory level in the New Component Inventory (NCI) in week 𝑡

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𝑀1�� = New components sent to manufacturing in week ��

𝑀2𝑡 = Finished products from manufacturing in week 𝑡

𝐹𝑃𝐼𝑚𝑡 = Ending inventory level in the Finished Product Inventory (FPI) from

manufacturing in week 𝑡

𝑀3𝑡 = Finished products from manufacturing sent to customer in week 𝑡

𝐹𝑃𝐼𝑡 = Ending inventory level in the Finished Product Inventory (FPI) in week 𝑡

𝐷𝑡 = Customer demand in week 𝑡

𝐿𝑆𝑡 = Lost sales in week 𝑡

𝑃𝐶𝑡 = Production capacity to produces a product in week 𝑡

5.1.1.2 Cost parameters ($)

𝑅𝐷 = Total disposal cost of returned component

𝐿𝑆 = Total lost sales cost

𝑅𝑀𝑀 = Total remanufacturing and manufacturing cost

𝑁𝐶 = Total new component cost

𝑅𝑃 = Total returned component cost

𝐶𝐻 = Total component holding cost

𝐹𝐻 = Total finished product holding cost

𝑇𝐶 = Total costs

𝑅𝑒𝑣𝑒𝑛𝑢𝑒 = Total income from selling finished products

𝑃𝑟𝑜𝑓𝑖𝑡 = Total income after deducting total costs

5.1.2 Decision Variables

𝑇𝑖𝑛𝑣𝑁 = Target inventory level in NCI (unit)

𝑇𝑖𝑛𝑣𝐹 = Target inventory level in FPI (unit)

𝑑𝑖𝑠𝑅𝑡 = Disposal rate in week 𝑡 (%)

5.1.3 Objective Function

Maximize 𝑃𝑟𝑜𝑓𝑖𝑡 = 𝑅𝑒𝑣𝑒𝑛𝑢𝑒 - 𝑇�� (5.1)

subject to

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inventory balance constraints:

𝑅𝑒𝐴�� = 𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡 + 𝑅0𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.2)

𝑅2𝑡 = 𝑅1�� 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.3)

𝑀2𝑡 = {0 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1 𝑀1��−1 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50

(5.4)

𝑅3𝑡 + 𝑀3𝑡 + 𝐿𝑆𝑡 = 𝐷�� 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.5)

𝐹𝑃𝐼𝑡 = 𝐹𝑃𝐼𝑟𝑡 + 𝐹𝑃𝐼𝑚𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.6)

𝑃𝐶�� ≥ 𝑅1𝑡 + 𝑀1𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.7)

𝑅𝐶𝐼𝑡 = {𝑅0𝑡 − 𝑅1�� 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1 𝑅𝐶𝐼𝑡−1 + 𝑅0𝑡 − 𝑅1�� 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50 (5.8)

𝑁𝐶𝐼𝑡 = {𝑇𝑖𝑛𝑣𝑁 − 𝑀1𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1 𝑁𝐶𝐼𝑡−1 + 𝑜𝑟𝑑𝑒𝑟��−1 − 𝑀1𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50

(5.9)

𝐹𝑃𝐼𝑟𝑡 = {𝑅2𝑡 − 𝑅3𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1 𝐹𝑃𝐼𝑟𝑡−1 + 𝑅2𝑡 − 𝑅3𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50

(5.10)

𝐹𝑃𝐼𝑚𝑡 = {𝑇𝑖𝑛𝑣𝐹 − 𝑀3𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1 𝐹𝑃𝐼𝑚𝑡−1 +𝑀2𝑡 − 𝑀3𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50

(5.11)

where 𝑅𝑒𝐴��, 𝐷��, and �� are uncertain coefficients with the triangular distribution.

The objective function (constraint (5.1)) is to maximize the profit. Constraint (5.2) and

Constraint (5.3) are inventory balance constraints. For Constraint (5.2), some of the

returned components are disposed of depending on disposal rate in every week, and the

rest is sent to the Returned Component Inventory (RCI). Constraint (5.3) ensures that

the quantity of returned components sent to remanufacturing processes are the same

quality as the finished products coming out of the remanufacturing processes within the

same week and the remanufacturing time is negligible. Constraint (5.4) states that the

manufacturing lead time is one week. For Constraint (5.5), customer demand is satisfied

by the finished products, otherwise, lost sales occur. For Constraint (5.6), total units of

finished products in remanufacturing and manufacturing are combined in the FPI.

Constraint (5.7) describes the constraint of production capacity for remanufacturing and

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manufacturing, in which the number of components sent to the manufacturing process

must be less than or equal to the production capacity. Constraints (5.8) to (5.11)

describe the constraints for returned components, new components, and finished

products for remanufacturing and manufacturing. The ending inventory is equal to the

previous ending inventory (week) plus the incoming inventory minus the outgoing

inventory. In the first week, the ending inventory is equal to the incoming inventory

minus the outgoing inventory.

The PTR constraints:

𝑅𝐶𝐼𝑡={

0 𝑖𝑓 𝐷�� ≥ 𝑅0𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1

0 𝑖𝑓 𝐷�� ≥ 𝑅0𝑡 + 𝑅𝐶𝐼𝑡−1 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50𝑎𝑛𝑦 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡

(5.12)

𝑅0𝑡 = {𝐷�� 𝑖𝑓 𝐷�� < 𝑅𝑒𝐴�� 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡

𝑅𝑒𝐴�� 𝑖𝑓 𝑅𝑒𝐴�� < 𝐷�� 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.13)

𝑅1�� = {𝐷�� 𝑖𝑓 𝐷�� < 𝑅0𝑡 + 𝑅𝐶𝐼𝑡−1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡𝑎𝑛𝑦 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡

(5.14)

where 𝑅𝑒𝐴��, and 𝐷�� are uncertain coefficients with the triangular distribution.

Constraints (5.12) and (5.14) describe the policies of the PTR. All of the returned

components in RCI are sent to remanufacturing when the customer demand is greater

than or equal the accepted returned components plus the number of the previous week’s

ending inventory in RCI. Otherwise the number of returned components sent to

remanufacturing is equal to the customer demand.

The PTM constraints:

𝑁𝐶𝐼𝑡 = {

0 𝑖𝑓 𝐷�� ≥ 𝑇𝑖𝑛𝑣𝑁 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1

𝑜𝑟𝑑𝑒𝑟𝑡−1 𝑖𝑓 𝐷�� ≥ 𝑁𝐶𝐼𝑡−1 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50𝑎𝑛𝑦 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡

(5.15)

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𝑀1�� = {𝐷�� 𝑖𝑓 𝐷�� < 𝑁𝐶𝐼𝑡−1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 𝑎𝑛𝑦 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡

(5.16)

where 𝐷�� are uncertain coefficients with the triangular distribution. Constraints

(5.15) and (5.16) describe the policies of the priority-to-manufacturing, where the

number of new components in the NCI that is sent to manufacturing is equal to the

customer demand when it is less than the number of new components. Otherwise, all

of the returned components are sent to manufacturing. The ending inventory in the NCI

is equal to the incoming orders from the previous week when the customer demand is

greater than the previous period’s ending inventory.

Economic constraints:

𝑅𝐷 = ∑ 𝑅𝐷_𝑢 ∗ 𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.17)

𝐿�� = ∑ 𝐿𝑆_𝑢 ∗ 𝐿𝑆��𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.18)

𝑅𝑀�� = ∑ 𝑅𝑀_�� ∗ 𝑅1��𝑡 + ∑ 𝑀𝑀_�� ∗ 𝑀1��𝑡 + 𝑀𝑀�� ∗ 𝑇𝑖𝑛𝑣𝐹 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.19)

𝑁�� = ∑ 𝑁𝐶�� ∗ (𝑜𝑟𝑑𝑒𝑟𝑡 + 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑁𝐶𝐼 + 𝑇𝑖𝑛𝑣𝐹) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 𝑡 (5.20)

𝑅�� = ∑ 𝑅𝑃_�� ∗ 𝑅0𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡𝑡 (5.21)

𝐶�� = (𝑅𝐻_𝑦

50

∗ (𝑅01 + 𝑅𝐶𝐼1)) /2 + ∑ (

𝑅𝐻_𝑦

50


50𝑡=2 /2 +

(𝑁𝐻_𝑦

50


𝑁𝐻_𝑦

50


50𝑡=2

2 (5.22)

𝐹�� = (𝐹𝐻_𝑦

50

∗ (𝑅21 + 𝐹𝑃𝐼𝑟1)) /2 + ∑ (

𝐹𝐻_𝑦

50

∗ (𝑅2𝑡 + 𝐹𝑃𝐼𝑟𝑡−1 + 𝐹𝑃𝐼𝑟𝑡)) /2 +

50𝑡=2

(𝐹𝐻_𝑦

50

∗ (𝑀21 + 𝑇𝑖𝑛𝑣𝐹 + 𝐹𝑃𝐼𝑚1)) /2 + ∑ (

𝐹𝐻_𝑦

50

∗ (𝑀2𝑡 + 𝐹𝑃𝐼𝑚𝑡−1 +

50𝑡=2

𝐹𝑃𝐼𝑚𝑡)) /2 (5.23)

𝑇�� = 𝑅�� + 𝐿�� + 𝑅𝑀�� + 𝑁�� + 𝑅�� + 𝐶�� + 𝐹�� (5.24)

𝑅𝑒𝑣𝑒𝑛𝑢𝑒 = ∑ 𝑃𝑟𝑖𝑐𝑒_𝑢 ∗ (𝑅3𝑡 +𝑀3𝑡)𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.25)

𝑃𝑟𝑜𝑓𝑖𝑡 = 𝑅𝑒𝑣𝑒𝑛𝑢𝑒 −𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡𝑠 (5.26)

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where 𝐿𝑆��, 𝑅𝑀_��, 𝑀𝑀_��, 𝑁𝐶��, 𝑅𝑃_��, 𝑁𝐻_��, 𝑅𝐻_��, 𝐹𝐻_��, and 𝑃𝑟𝑖𝑐𝑒_𝑢 are

uncertain coefficients with the triangular distribution. Constraints (5.17) to (5.23)

describe the cost parameters to calculate the total cost (TC). The TC consists of the

Disposal Returned Components Cost (𝑅𝐷), Lost Sales Cost (𝐿𝑆), Remanufacturing and

Manufacturing Cost (𝑅𝑀𝑀), New Component Cost (𝑁𝐶), Returned Component

Preparation Cost (𝑅𝑃), Component Holding Cost (𝐶𝐻), and Finished Product Holding

Cost (𝐹𝐻). All holding costs are calculated based on the average inventory level.

Constraint (5.26) states that the profit is calculated by the revenue (Constraint (5.25))

minus total costs (Constraint (5.24)).

Decision Variables:

𝑜𝑟𝑑𝑒𝑟𝑡 = 𝑇𝑖𝑛𝑣𝑁 – 𝑁𝐶𝐼𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.27)

𝑅1�� + 𝑀1�� = 𝑇𝑖𝑛𝑣𝐹 − 𝐹𝑃𝐼𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.28)

𝑑𝑖𝑠𝑅𝑡 = 100 ∗ (𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡 𝑅𝑒𝐴𝑡⁄ ) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.29)

For Equation (5.27), the order of new components depends on 𝑇𝑖𝑛𝑣𝑁. Equation

(5.28) shows that the total number of new and returned components sent to production

is equal to the number of finished products. Equation (5.29) describes the disposal rate,

which is the percentage of disposed of components over the returned components.


We develop the fuzzy linear programming method to solve this problem under

uncertainty by developing three new crisp objective functions of the multi-objective

linear programming (MOLP) to replace Equations (3.1) - (3.3)

Max 𝑧1 = 𝑃𝑟𝑜𝑓𝑖𝑡𝑀

= 𝑅𝑒𝑣𝑒𝑛𝑢𝑒𝑀 − 𝑇𝐶𝑀

= 𝑅𝑒𝑣𝑒𝑛𝑢𝑒𝑀-𝑅𝐷𝑀+𝐿𝑆𝑀+𝑅𝑀𝑀𝑀+𝑁𝐶𝑀+𝑅𝑃𝑀+𝐶𝐻𝑀+𝐹𝐻𝑀 (5.30)

Min 𝑧2 = 𝑃𝑟𝑜𝑓𝑖𝑡𝑃−𝑀

= 𝑅𝑒𝑣𝑒𝑛𝑢𝑒𝑃−𝑀 − 𝑇𝐶𝑃−𝑀

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= 𝑅𝑒𝑣𝑒𝑛𝑢𝑒𝑃−𝑀 −

𝑅𝐷𝑃−𝑀+𝐿𝑆𝑃−𝑀+𝑅𝑀𝑀𝑃−𝑀+𝑁𝐶𝑃−𝑀+𝑅𝑃𝑃−𝑀+𝐶𝐻𝑃−𝑀+𝐹𝐻𝑃−𝑀 (5.31)

Max 𝑧3= 𝑃𝑟𝑜𝑓𝑖𝑡𝑀−𝑂

= 𝑅𝑒𝑣𝑒𝑛𝑢𝑒𝑀−𝑂 − 𝑇𝐶𝑀−𝑂

= 𝑅𝑒𝑣𝑒𝑛𝑢𝑒𝑀−𝑂 −

𝑅𝐷𝑀−𝑂+𝐿𝑆𝑀−𝑂+𝑅𝑀𝑀𝑀−𝑂+𝑁𝐶𝑀−𝑂+𝑅𝑃𝑀−𝑂+𝐶𝐻𝑀−𝑂+𝐹𝐻𝑀−𝑂 (5.32)

As the demand of customers, quantity of returned components, and production

capacity are fuzzy, a defuzzification method using the weighted average method is

applied to solve the imprecise data. Constraints (5.2), (5.5), (5.7), (5.12), (5.13), (5.14),

(5.15), and (5.16) need to be transformed to crisp constraints as follows:

𝑤1𝑅𝑒𝐴��𝑃+ 𝑤2𝑅𝑒𝐴��

𝑀+ 𝑤3𝑅𝑒𝐴��𝑂= 𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡 + 𝑅0𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.33)

𝑅3𝑡 + 𝑀3𝑡 + 𝐿𝑆𝑡 = 𝑤1𝑑𝑡𝑃+ 𝑤2𝑑𝑡

𝑀+ 𝑤3𝑑𝑡𝑂 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.34)


𝑀 + 𝑤3𝑃𝐶𝑡𝑂 ≥ 𝑅1𝑡 + 𝑀1𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.35)

𝑅𝐶𝐼𝑡=

{


𝑀 + 𝑤3𝐷𝑡𝑂 ≥ 𝑅0𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1


𝑀 + 𝑤3𝐷𝑡𝑂 ≥ 𝑅0𝑡 + 𝑅𝐶𝐼𝑡−1 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50


(5.36)

𝑅0𝑡=

{

𝑤1𝐷𝑡


𝑂



𝑃 + 𝑤2𝑅𝑒𝐴𝑡𝑀 + 𝑤3𝑅𝑒𝐴𝑡




𝑖𝑓 𝑤1𝑅𝑒𝐴𝑡𝑃 + 𝑤2𝑅𝑒𝐴𝑡




(5.37)

𝑅1��= {




𝑀 + 𝑤3𝐷𝑡𝑂 < 𝑅0𝑡 + 𝑅𝐶𝐼𝑡−1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡


(5.38)

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𝑁𝐶𝐼𝑡=

{






(5.39)

𝑀1��=

{𝑤1𝐷𝑡





(5.40)

where 𝑤1 + 𝑤2 + 𝑤3 = 1 and the weights of 𝑤1, 𝑤2, and 𝑤3 can be assigned by the

knowledge and experience of decision makers.

As the production lead time, ordering lead time, and returned component

delivery time are fuzzy, a defuzzification method using the fuzzy ranking concept is

applied to solve these imprecise data. Constraints (5.3), (5.4), (5.9), and (5.33) need to

be transformed to crisp constraints as follows:

For constraint (5.3):

A case when the remanufacturing lead time is 1 week

𝑅21 ≤ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.41)

𝑅2𝑡 ≤ 𝑅1𝑡−1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.42)

A case when the remanufacturing lead time is negligible (assuming 0 week)

𝑅2𝑡 ≤ 𝑅1𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.43)

Constraints (5.41) and (5.42) show the case when the remanufacturing

processing time is 1 week. As a result, in the first week, there is no finished product

from the remanufacturing processes. Constraint (5.43) shows the case when the

remanufacturing processing time is negligible the quantity of returned components sent

to remanufacturing processes are the same quality as the finished products coming out

of the remanufacturing processes within the same week.

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A case when the manufacturing processing time is 2 weeks

𝑀2𝑡 ≤ {0 0 𝑀1𝑡−2

𝑤ℎ𝑒𝑟𝑒 𝑡 = 1 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2

𝑤ℎ𝑒𝑟𝑒 𝑡 = 3 𝑡𝑜 𝑡 = 50 (5.44)

A case when the manufacturing processing time is 1 week.

𝑀2𝑡 ≤ {0 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1 𝑀1𝑡−1 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50

(5.45)

Constraint (5.44) shows the case when the manufacturing processing time is 2

weeks. As a result, there is no finished product from the manufacturing processes on

week 1 and week 2 because manufacturing process take 2 weeks to produce. Constraint

(5.45) shows the case when the manufacturing lead time is one week.

For constraint (5.9)

A case when the order lead time is 2 weeks

𝑁𝐶𝐼𝑡 ≤ {

𝑇𝑖𝑛𝑣𝑁 − 𝑀11 𝑁𝐶𝐼1 − 𝑀12 𝑁𝐶𝐼𝑡−1 + 𝑜𝑟𝑑𝑒𝑟𝑡−2 − 𝑀1𝑡

𝑤ℎ𝑒𝑟𝑒 𝑡 = 1 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2

𝑤ℎ𝑒𝑟𝑒 𝑡 = 3 𝑡𝑜 𝑡 = 50 (5.46)

Case when order lead time is 1 week

𝑁𝐶𝐼𝑡 ≤ {𝑇𝑖𝑛𝑣𝑁 − 𝑀1𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1 𝑁𝐶𝐼𝑡−1 + 𝑜𝑟𝑑𝑒𝑟𝑡−1 − 𝑀1𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50

(5.47)

Constraint (5.46) shows the case when the order lead time is 2 weeks. So, there

is no new components on week 1 and week 2 since the ordering process takes 2 weeks

to delivery. Constraint (5.47) shows the case when the order lead time is 1 week. So,

new components from the ordering process in week 1 will arrive in week 2.


A case when the returned components lead time is 1 week

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𝑅01 <= 0 (5.48)

𝑤1𝑅𝑒𝐴𝑡−1𝑃 + 𝑤2𝑅𝑒𝐴𝑡−1

𝑀 + 𝑤3𝑅𝑒𝐴𝑡−1𝑂 ≥ 𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡 + 𝑅0𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50

(5.49)

A case when the returned components lead time is negligible (assuming 0 week).

𝑤1𝑅𝑒𝐴𝑡𝑃+ 𝑤2𝑅𝑒𝐴𝑡

𝑀+ 𝑤3𝑅𝑒𝐴𝑡𝑂 ≥ 𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡 + 𝑅0𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 (5.50)

Constraints (5.48) and (5.49) show the worst case that return components’ lead

time is 1 week. As a result, in the first week, there is no returned components since it

takes one week to be delivered. Constraint (5.50) shows the case when returned

components lead time is zero, some of the returned components are disposed of

depending on disposal rate in every week, and the rest is sent to the Returned

Component Inventory (RCI).

The linear membership functions for three new objective functions are

specified, and converted the auxiliary MOLP problem into an equivalent linear

programming model by the fuzzy decision method.

To find the Negative Ideal Solution (NIS) and the Positive Ideal Solution (PIS)

of each objective, the fuzzy decision method as presented in Equations (3.4) to (3.6) is

introduced to solve MOLP by finding the corresponding linear membership function of

each objective function.

𝑧1𝑃𝐼𝑆 = max 𝑃𝑟𝑜𝑓𝑖𝑡𝑀 , 𝑧1

𝑁𝐼𝑆 = min 𝑃𝑟𝑜𝑓𝑖𝑡𝑀 (5.51)

𝑧2𝑃𝐼𝑆 = min 𝑃𝑟𝑜𝑓𝑖𝑡𝑃−𝑀 , 𝑧2

𝑁𝐼𝑆 = max 𝑃𝑟𝑜𝑓𝑖𝑡𝑃−𝑀 (5.52)

𝑧3𝑃𝐼𝑆 = max 𝑃𝑟𝑜𝑓𝑖𝑡𝑀−𝑂, 𝑧3

𝑁𝐼𝑆 = min 𝑃𝑟𝑜𝑓𝑖𝑡𝑀−𝑂 (5.53)

Equations (5.7) to (5.9) are applied to the corresponding linear membership

functions for each objective function in this case study as follows:

𝑓1(𝑧1) =

{

1 , 𝑧1 < 𝑧1

𝑃𝐼𝑆,




𝑃𝐼𝑆,

0 , 𝑧1 > 𝑧1𝑁𝐼𝑆,

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𝑓2(z2) =

{

1 , 𝑧2 < 𝑧2

𝑃𝐼𝑆,




𝑁𝐼𝑆,

0 , 𝑧2 > 𝑧2𝑁𝐼𝑆,

𝑓3(𝑧3) =

{

1 , 𝑧3 < 𝑧3

𝑃𝐼𝑆,




𝑃𝐼𝑆,

0 , 𝑧3 > 𝑧3𝑁𝐼𝑆.

Find the maximum overall satisfaction:

Max λ

Subject to

λ ≤ 𝑓𝑖(𝑧𝑖), where i = 1,2,3

i = number of linear membership function of each objective.

0 ≤ λ ≤ 1

𝑅01 <= 0,

𝑤1𝑅𝑒𝐴𝑡−1𝑃 + 𝑤2𝑅𝑒𝐴𝑡−1

𝑀 + 𝑤3𝑅𝑒𝐴𝑡−1𝑂 ≥ 𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡 + 𝑅0𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50,

𝑤1𝑅𝑒𝐴𝑡𝑃+ 𝑤2𝑅𝑒𝐴𝑡

𝑀+ 𝑤3𝑅𝑒𝐴𝑡𝑂 ≥ 𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡 + 𝑅0𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

𝑅21 ≤ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

𝑅2𝑡 ≤ 𝑅1𝑡−1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

𝑅2𝑡 ≤ 𝑅1𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

𝑀2𝑡 ≤ {0 0 𝑀1𝑡−2

𝑤ℎ𝑒𝑟𝑒 𝑡 = 1, 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2,

𝑤ℎ𝑒𝑟𝑒 𝑡 = 3 𝑡𝑜 𝑡 = 50,

𝑀2𝑡 ≤ {0 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1, 𝑀1𝑡−1 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50,

𝑅3𝑡 + 𝑀3𝑡 + 𝐿𝑆𝑡 = 𝑤1𝑑𝑡𝑃+ 𝑤2𝑑𝑡

𝑀+ 𝑤3𝑑𝑡𝑂 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 ,

𝐹𝑃𝐼𝑡 = 𝐹𝑃𝐼𝑟𝑡 + 𝐹𝑃𝐼𝑚𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,


𝑀 + 𝑤3𝑃𝐶𝑡𝑂 ≥ 𝑅1𝑡 + 𝑀1𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

𝑅𝐶𝐼𝑡 = {𝑅0𝑡 − 𝑅1𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1, 𝑅𝐶𝐼𝑡−1 + 𝑅0𝑡 − 𝑅1𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50,

𝑁𝐶𝐼𝑡 ≤ {

𝑇𝑖𝑛𝑣𝑁 − 𝑀11 𝑁𝐶𝐼1 − 𝑀12 𝑁𝐶𝐼𝑡−1 + 𝑜𝑟𝑑𝑒𝑟𝑡−1 − 𝑀1𝑡

𝑤ℎ𝑒𝑟𝑒 𝑡 = 1, 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2,

𝑤ℎ𝑒𝑟𝑒 𝑡 = 3 𝑡𝑜 𝑡 = 50,

Ref. code: 25615822042262GRA

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𝑁𝐶𝐼𝑡 ≤ {𝑇𝑖𝑛𝑣𝑁 − 𝑀1𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1, 𝑁𝐶𝐼𝑡−1 + 𝑜𝑟𝑑𝑒𝑟𝑡−1 − 𝑀1𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50,

𝐹𝑃𝐼𝑟𝑡 = {𝑅2𝑡 − 𝑅3𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1, 𝐹𝑃𝐼𝑟𝑡−1 + 𝑅2𝑡 − 𝑅3𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50,

𝐹𝑃𝐼𝑚𝑡 = {𝑇𝑖𝑛𝑣𝐹 − 𝑀3𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1, 𝐹𝑃𝐼𝑚𝑡−1 +𝑀2𝑡 − 𝑀3𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50,

For the PTR

𝑅𝐶𝐼𝑡=

{


𝑀 + 𝑤3𝐷𝑡𝑂 ≥ 𝑅0𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1,


𝑀 + 𝑤3𝐷𝑡𝑂 ≥ 𝑅0𝑡 + 𝑅𝐶𝐼𝑡−1 𝑤ℎ𝑒𝑟𝑒 𝑡 = 2 𝑡𝑜 𝑡 = 50,


𝑅0𝑡=

{

𝑤1𝐷𝑡


𝑂



𝑃 + 𝑤2𝑅𝑒𝐴𝑡𝑀 + 𝑤3𝑅𝑒𝐴𝑡




𝑖𝑓 𝑤1𝑅𝑒𝐴𝑡𝑃 + 𝑤2𝑅𝑒𝐴𝑡




𝑅1𝑡= {




𝑀 + 𝑤3𝐷𝑡𝑂 < 𝑅0𝑡 + 𝑅𝐶𝐼𝑡−1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,


For PTM

𝑁𝐶𝐼𝑡=

{






𝑀1𝑡=

{𝑤1𝐷𝑡





𝑅𝐷= ∑ 𝑅𝐷_𝑢 ∗ 𝑑𝑖𝑠𝑝𝑜𝑠𝑒𝑡𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

𝐿�� = ∑ 𝐿𝑆_𝑢 ∗ 𝐿𝑆��𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

𝑅𝑀�� = ∑ 𝑅𝑀_�� ∗ 𝑅1𝑡𝑡 + ∑ 𝑀𝑀_�� ∗ 𝑀1𝑡𝑡 + 𝑀𝑀�� ∗ 𝑇𝑖𝑛𝑣𝐹 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

𝑁�� = ∑ 𝑁𝐶�� ∗ (𝑜𝑟𝑑𝑒𝑟𝑡 + 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑁𝐶𝐼 + 𝑇𝑖𝑛𝑣𝐹) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡, 𝑡

𝑅�� = ∑ 𝑅𝑃_�� ∗ 𝑅0𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡𝑡 ,

Ref. code: 25615822042262GRA

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𝐶�� = (𝑅𝐻_𝑦

50

∗ (𝑅01 + 𝑅𝐶𝐼1)) /2 + ∑ (

𝑅𝐻_𝑦

50


50𝑡=2 /2 +

(𝑁𝐻_𝑦

50


𝑁𝐻_𝑦

50


50𝑡=2

2,

𝐹�� = (𝐹𝐻𝑟_𝑦

50

∗(𝑅21 + 𝐹𝑃𝐼𝑟1)) /2 + ∑ (

𝐹𝐻𝑟_𝑦

50

∗(𝑅2𝑡 + 𝐹𝑃𝐼𝑟𝑡−1 + 𝐹𝑃𝐼𝑟𝑡)) /

50𝑡=2

2 + (𝐹𝐻𝑚_𝑦

50

∗(𝑀21 + 𝑇𝑖𝑛𝑣𝐹 + 𝐹𝑃𝐼𝑚1)) /2 + ∑ (

𝐹𝐻𝑚_𝑦

50

∗(𝑀2𝑡 + 𝐹𝑃𝐼𝑚𝑡−1 +

50𝑡=2

𝐹𝑃𝐼𝑚𝑡)) /2,

𝑇�� = 𝑅�� + 𝐿�� + 𝑅𝑀�� + 𝑁�� + 𝑅�� + 𝐶�� + 𝐹��,

𝑅𝑒𝑣𝑒𝑛𝑢𝑒 = ∑ 𝑃𝑟𝑖𝑐𝑒_𝑢 ∗ (𝑅3𝑡 +𝑀3𝑡)𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡,

𝑃𝑟𝑜𝑓𝑖𝑡 = 𝑅𝑒𝑣𝑒𝑛𝑢𝑒 −𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡𝑠 .

5.2 Result and Discussion

5.2.1 Stochastic case with the Fuzzy Linear Programming (FLP)

With the lower bound and the upper bound of each objective (Constraints (4.41)

to (4.43)), Tables 4.3-4.4 show the Negative Ideal Solution (NIS) and the Positive Ideal

Solution (PIS) of each objective of the PTR and the PTM policies with lead time

uncertainty, respectively.

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Table 5.1 Upper and lower boundary of the PTR policy with the lead time

uncertainty.

PTR

MOLP Profit ($)


𝑧1𝑁𝐼𝑆 = min 𝑃𝑟𝑜𝑓𝑖𝑡𝑀 45,359.18





Table 5.2 Upper and lower boundary of the PTM policy with the lead time uncertainty.

PTM

MOLP Profit ($)


𝑧1𝑁𝐼𝑆 = min 𝑃𝑟𝑜𝑓𝑖𝑡𝑀 0





According to Constraints (5.33) to (5.40) of the Fuzzy Linear Programming

model, the process of defuzzification of imprecise customer demand, number of

arriving returned components, and production capacity with the weighted average

method is subject to the pattern of weight allocation. Different patterns of weight

allocation can have an impact on the obtained solution. As a result, a sensitivity analysis

of this weight allocation should be carried out among pessimistic, most likely, and

optimistic cases to investigate the impact of such weight allocation on the overall

satisfaction of the linear membership functions of each objective function. For a

demonstration, a sensitivity analysis is performed in 4 scenarios by varying the weights

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of pessimistic, most likely, and optimistic values. Tables 5.3-5.4 present the results of

the sensitivity analysis for both the PTR and the PTM policies with the lead time

uncertainty.

Table 5.3 Result of the sensitivity analysis of the PTR policy with lead time

uncertainty.



33% 33% 33% 80% 10% 10% 10% 80% 10% 10% 10% 80%

Decision Variables

Target

inventory of

new

components

(TinvN)

(units)

43 171 41 39

Target

inventory of

finished

products

(TinvF)

(units)

196 171 192 204

Number of

disposed

returned

components

(units)

4 8 3 0

Results

Overall

satisfaction 53.69% 37.58% 53.21% 38.16%

Z1 ($) 103,420 77,369 97,824 77,270

Z2 ($) 68,011 64,014 68,165 73,017

Z3 ($) 67,275 62,165 67,122 70,427

Maximum

the most

likely value

of profit ($)

103,420 77,369 97,824 77,270

Minimize the

risk of

obtaining

lower profit

($)

35,409 13,355 29,659 4,253

Maximize of

the

possibility of

obtaining a

higher profit

($)

170,695 139,534 164,946 147,697

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For example, in Table 5 of the PTR policy with Scenario I, equal weights (33%)

are assigned to the pessimistic, most likely, and optimistic values. It is found that the

overall satisfaction is 53.69%, and the outcomes of the maximum most likely value of

profit, minimum possibility of obtaining a lower profit, and maximum of the risk of

obtaining a higher profit are $103,420, $35,409, and $170,695, respectively.

The result of this sensitivity analysis also shows that Scenario I, which sets the

weight of the most likely value at 33% and the weight of optimistic and pessimistic

values at 33%, gives the highest overall satisfaction of 53.69%. Scenario I also gives

the highest possible value of the most likely profit, the profit from the risk of obtaining

a lower profit, as well as the profit from maximum of the risk of obtaining a higher

profit.

In this scenario, the system adjusts itself by avoiding the high cost of shortage

from keeping high target inventory of finished product and disposing only a few units

of returned components. However, when the weight of the optimistic value is set at 80%

and 10% at both pessimistic and the most likely values (scenario IV), it is shown in

results that the optimistic value of end customer demand could lead to too high demand

to be able to produce by the limited production capacity and causes quite high shortage

cost. This would eventually deteriorate the profit of the system.

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Table 5.4 Result of the sensitivity analysis of the PTM policy with lead time

uncertainty.



33% 33% 33% 80% 10% 10% 10% 80% 10% 10% 10% 80%

Decision Variables

Target

inventory of

new

components

(TinvN)

(units)

64 52 58 34

Target

inventory of

finished

products

(TinvF)

(units)

197 171 199 208

Number of

disposed

returned

components

(units)

1,315 0 1,088 95

Overall

satisfaction 53.12% 43.42% 52.71% 38.34%

Z1 ($) 67,720 63,961 67,852 72,528

Z2 ($) 67,720 63,961 67,852 72,528

Z3 ($) 66,630 61,744 66,500 69,900

Maximum

the most

likely value

of profit ($)

67,720 63,961 67,852 72,528

Minimize the

risk of

obtaining

lower profit

($)

0 0 0 0

Maximize of

the

possibility of

obtaining a

higher profit

($)

134,350 125,705 134,352 142,428

For example, in Table 6 of the PTM policy with Scenario I, equal weights (33%)

are assigned to the pessimistic, most possible, and optimistic values. It is found that the

overall satisfaction is 53.12%, and the outcomes of maximum most possible value of

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profit, minimum possibility of obtaining a lower profit, and maximum of the risk of

obtaining a higher profit are $67,720, $0, and $134,350, respectively.

The best result of the PTM policy is from setting the weight of the optimistic

values at 80% and 10% at both pessimistic and the most likely value (Scenario IV).

Opposite to the PTR policy, the profit generated from Scenario IV (the optimistic case)

is higher than the profit from Scenario I (equal weight). This is because it turns out to

use less amount of new components and use more returned components than the case

of setting equal weight to all three values (Scenario I). This is due to the fact that the

PTM policy is required to use the new components first but the new components take

longer time to arrive with the fuzzy delivery time than the returned components. As a

result, to prevent shortage, the arriving returned components would be selected for the

production.

As for a comparison between the two policies, it was found that the PTR policy

can outperform the PTM policy in terms of profit in all objective functions and in all

scenarios. In addition, the profits of the PTM policy during the pessimistic condition in

all scenarios are found to be as low as 0 whereas the lowest profit of the PTR policy is

$4,253 the lowest profit of its best scenario could be up to $35,409. This can

demonstrate the superiority of the PTR policy, even during the pessimistic condition.

Focusing on the effect of fuzzy lead time and production time, the actual amount of

inventory is needed to calculate at all points based on the worst scenario in each period

according to the defuzzification method by the fuzzy ranking concept. For

demonstration, the current values of related data in week 3 is assumed and shown in

Table 5.5:

Table 5.5 Example of value in case 1 and case 2.

NCI2 Order1 Order2 M13

Case1 10 3 5 3

Case2 10 10 5 3

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For Case1, according to constraints (5.10) and (5.11), when the pessimistic lead

time is 2 weeks, the amount of new component inventory in week 3 can be calculated

as follows:

𝑁𝐶𝐼𝑡 ≤ 𝑁𝐶𝐼𝑡−1 + 𝑂𝑟𝑑𝑒𝑟𝑡−2 − 𝑀1𝑡

𝑁𝐶𝐼3 ≤ 𝑁𝐶𝐼2 + 𝑂𝑟𝑑𝑒𝑟1 − 𝑀13

So, 𝑁𝐶𝐼3 ≤ 10 units

However, when the optimistic and the most likely lead time is 1 week, the

amount of new component inventory in week 3 can be calculated as follows:



So, 𝑁𝐶𝐼3 ≤ 12

As a result, the actual amount of new component would be 10 units from the

pessimistic lead time period as it is the worst case.

For Case2, according to constraints (5.10) and (5.11). If the pessimistic lead

time is 2 weeks, the amount of new component inventory in week 3 can be calculated

as follows:



So, 𝑁𝐶𝐼3 ≤ 17

If the optimistic and most likely lead time is 1 week, the amount of new

component inventory at week 3 can be calculated as follows:



So, 𝑁𝐶𝐼3 ≤ 12

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As a result, the actual amount of new components would be 12 as it is the worst

case. In addition, it was also found that the possibility to obtain the lower profit of PTM

policy can be as low as zero or even make a loss as its costs may dramatically increase

under the pessimistic scenario. This indicates the possible hidden risk from the policy.

5.2.2 Comparison between the deterministic and the best stochastic results

Table 5.6 compares the results between the optimal LP result under a

deterministic condition and the best FLP result under a more realistic stochastic

condition. As seen in Table 5.6, the most likely profit that are generated from the

uncertain circumstance is lower as compared to the profit under no uncertainty as the

overall degree of satisfaction is lower. This is because the result was recommended

from the scenario that yields the lowest degree of satisfaction among the three

objectives, which are maximizing the most likely value of profit, minimizing the risk

obtaining a lower profit, and maximizing the possibility of obtaining a higher profit.

The PTR policy under scenario I and the PTM policy under Scenario IV. show the best

result among experimental scenarios. However, the PTR policy generates a higher

profit than the PTM policy in all cases. With the range of profit from each scenario

obtained from FLP, decision makers are well prepared for any expected outcome. This

is an important feature for decision makers to build the production planning and

inventory control subject to an uncertain environment.

Table 5.6 Result comparison

The PTR Policy

Priority-To-Remanufacturing

policy (PTR)

Linear

Programming

(LP) model

Fuzzy Linear Programming

(FLP) model from Scenario I



z (Profit) $ 114,861.96 ($35,409, $103,420, $170,695)

The PTM Policy

Priority-To-Manufacturing

policy (PTR)

Linear

Programming

(LP) model

Fuzzy Linear Programming

(FLP) model from Scenario IV



z (Profit) $ 111,906.48 ($0, $72,528, $142,428)

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CHAPTER 6

CONCLUSIONS

6.1 Conclusion

Production planning and inventory control is intermediate planning for finding

suitable levels of disposal units, production units, lost sales, inventory, and stock outs

to meet uncertainty in the forecast demand and fluctuation of production lead time,

order lead time, returned component delivery time, related operating costs, number of

returned components, and production capacity. This study finds the optimal inventory

control policy of a hybrid manufacturing/remanufacturing system under two priority

policies, Priority-To-Remanufacturing (PTR) and Priority-To-Manufacturing (PTM).

Fuzzy linear programming (FLP) is introduced to optimize and find the optimal result

in this uncertain environment. It maximizes the most possible value of the imprecise

profit, minimizes the risk of obtaining a lower profit, and maximize the possibility of

obtaining a higher profit by pushing the three prominent points with the triangular

distribution towards the right (as profit maximization). Results from this deterministic

case show that the PTR policy is slightly better than the PTM policy. However, the

results from the stochastic case show that the PTR policy is more stable and lot better

than the PTM policy in every scenario.

In addition, it was also found that the possibility to obtain the lower profits of

PTM policy can be as low as zero or even make a loss as its costs are quite high due to

longer lead times than the PTR policy especially under the pessimistic scenario. This

indicates possible hidden risk from the policy. As shown in the case study, the proposed

method can be used to solve most real-world planning problems involving imprecise

parameters through an interactive decision-making process. The proposed method

constitutes a systematic framework that facilitates the decision-making process,

enabling a decision maker to interactively modify the imprecise data until a satisfactory

solution is found. Notably, the optimal goal values using the LP approach are imprecise

since the forecasted demand, related operating costs, the number of returned

components, ordering lead time, production lead time, and returned components’

delivery time are always imprecise in nature. This brings the results closer to real-world

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problems. Decision makers should have some knowledge in advance to prepare and

take necessary action for future uncertainty.

This study focuses on multi-periods with a single product from a hybrid

manufacturing/remanufacturing problem. It also provides information on different

policies in response to any designed variations. Additionally, the approach also

considers the actual tradeoff between the PTM and the PTR policies in each

circumstance. This lets us know how each policy would be beneficial or vulnerable

depending on uncertain conditions.

The main limitation of our case study is the assumption of the triangular

distribution that represents imprecise data. Decision makers should generate and obtain

appropriate distributions based on true judgment and historical resources. Future

researchers can also explore different levels of the relative importance of individual

goals and different types of distributions, to make their models to be better suited to

their practical applications.

6.2 Further Study

Further study can be extended to incorporate other uncertain parameters such as

defect rate, unit cost and etc. to make the system more dynamic and realistic. Different

methods of the Fuzzy Linear Programming (FLP) including Weight Max-Min and

Weight additive methods, which can incorporate the fuzzy constraint objective in

addition to the main objective can be considered. Also, in the part of the case study, the

hybrid method between the PTR policy and the PTM policy where the advantages of

each method can be utilized is another possibility to be explored. This result can be

beneficial to the model implementation and to the decision makers, who can have an

early warning or can be well prepared for any unexpected outcome.

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APPENDICES

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APPENDIX A

RAW DATA

Raw data of customer demand, amount of returned components arrival, and

production capacity is shown in this section.

Table A.1 Customer demand, amount of returned component arrival, and production

capacity in each week.

Week Demand (��)

units

Returned component arrival

(𝑅𝑒 ��) units

Production capacity

(𝑃��) units

1 (74,92,106) (48,53,58) (107,133,154)

2 (55,68,79) (46,51,56) (80,99,115)

3 (61,76,88) (41,46,51) (88,110,128)

4 (82,102,118) (70,78,86) (119,148,171)

5 (96,120,138) (65,72,79) (139,174,200)

6 (95,118,136) (41,46,51) (138,171,197)

7 (60,75,87) (72,80,88) (87,109,126)

8 (81,101,117) (52,58,64) (117,146,170)

9 (66,82,95) (48,53,58) (96,119,138)

10 (111,138,159) (44,49,54) (161,200,231)

11 (100,124,143) (67,74,81) (145,180,207)

12 (96,119,137) (84,93,102) (139,173,199)

13 (91,113,130) (60,67,74) (132,164,189)

14 (79,98,113) (46,51,56) (115,142,164)

15 (68,85,98) (31,34,37) (99,123,142)

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16 (76,95,110) (62,69,76) (110,138,160)

17 (87,108,125) (59,66,73) (126,157,181)

18 (97,121,140) (43,48,53) (141,175,203)

19 (62,77,89) (61,68,75) (90,112,129)

20 (93,116,134) (75,83,91) (135,168,194)

21 (78,97,112) (59,65,72) (113,141,162)

22 (58,72,83) (49,54,59) (84,104,120)

23 (107,133,153) (49,54,59) (155,193,222)

24 (72,89,103) (67,74,81) (104,129,149)

25 (100,125,144) (55,61,67) (145,181,209)

26 (71,88,102) (52,58,64) (103,128,148)

27 (105,131,151) (59,66,73) (152,190,219)

28 (82,102,118) (56,62,68) (119,148,171)

29 (72,90,104) (54,60,66) (104,131,151)

30 (85,106,122) (58,64,70) (123,154,177)

31 (104,130,150) (68,75,83) (151,189,218)

32 (104,129,149) (57,63,69) (151,187,216)

33 (72,89,103) (31,34,37) (104,129,149)

34 (79,98,113) (47,52,57) (115,142,164)

35 (84,104,120) (55,61,67) (122,151,174)

36 (68,85,98) (50,55,61) (99,123,142)

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37 (64,79,91) (49,54,59) (93,115,132)

38 (76,95,110) (24,27,30) (110,138,160)

39 (56,69,80) (50,56,62) (81,100,116)

40 (83,103,119) (70,78,86) (120,149,173)

41 (108,134,155) (56,62,68) (157,194,225)

42 (88,110,127) (43,48,53) (128,160,184)

43 (108,134,155) (46,51,56) (157,194,225)

44 (76,94,109) (78,87,96) (110,136,158)

45 (68,84,97) (69,77,85) (99,122,141)

46 (100,125,144) (56,62,68) (145,181,209)

47 (74,92,106) (47,52,57) (107,133,154)

48 (71,88,102) (59,66,73) (103,128,148)

49 (94,117,135) (86,96,106) (136,170,196)

50 (87,108,125) (57,63,69) (126,157,181)

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APPENDIX B

EXAMPLE OF CPLEX CODE

This section show example of Cplex code in the PTR deterministic case.

Step 1: Cplex model file for PTR in deterministic case.

/*********************************************

* OPL 12.6.0.0 Model

* Author: Admin

* Creation Date: Dec 15, 2017 at 5:38:51 PM

********************************************/

int p = 50;

range p1 = 1..p;

range p2 = 1..2;

int ReA[p1] = ...;

int demand[p1] = ...;

int demandW[p1] = ...;

int M = 10000;

int MC[p1] = ...;

// variables for inventory balance constraints

dvar float+ dispose[p1];

dvar int+ R0[p1];

dvar int+ order[p1];

dvar float+ RCI[p1]; // Returned Component Inventory (RCI)

dvar float+ NCI[p1]; // New Component Inventory (NCI)

dvar float+ R1[p1];

dvar int+ M1[p1];

dvar float+ R2[p1];

dvar float+ M2[p1];

dvar float+ FPIR[p1]; // Finished Product Inventory from remanufacturing

dvar float+ FPIM[p1]; // Finished Product Inventory from manufacturing

dvar float+ FPI[p1]; // Finished Product Inventory (FPI)

dvar float+ R3[p1];

dvar float+ M3[p1];

dvar float+ LS[p1];

dvar boolean priBi[p1][p2];

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dvar boolean priBy[p1][p2];

dvar boolean priBr[p1][p2];

dvar float+ sumDisp;

dvar float+ sumOrder;

dvar float+ sumRCI;

dvar float+ sumNCI;

dvar float+ sumFPIR;

dvar float+ sumFPIM;

dvar float+ sumRMM;

// Cost parameter

// Cost parameter

int RD_u = 0;

int NC_u = 30;

int RP_u = 5;

int RM_u = 10;

int MM_u = 10;

int LS_u = 50;

int Price_u = 50;

float RH_y = 2;

float NH_y = 12;

int FH_y = 11;

float q = 0.65;

// decision variables

dvar float+ disR[p1]; // Disposal Rate (disR)

dvar int+ TinvN; // Target Inventory of New Components (TinvN)

dvar int+ TinvF; // Target Inventory of Finished Product (TinvF)

// variables for financial constraints

dvar float+ Sales; // revenue

dvar float+ TotalCost; // total costs

dvar float+ dispC; // total returned component disposal cost (RD)

dvar float+ LSC; // Total lost sales cost (LS)

dvar float+ RemanuandManuCost; // Total remanufacturing and manufacturing cost

(RMM)

dvar float+ RemanuandManuCost_R[p1];

dvar float+ NewComponentCost; // Total new component cost (NC)

dvar float+ ReturnComponentCost; // Total returned component cost (RP)

dvar float+ ComponentHoldingCost; // Total component holding cost (CH)

dvar float+ FinishHoldingCost; // Total finished product holding cost (FH)

dvar float+ ComponentHoldingCost_R[p1];

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dvar float+ ComponentHoldingCost_M[p1];

dvar float+ FinishHoldingCost_R[p1];

dvar float+ FinishHoldingCost_M[p1];

dvar float Profit;

// objective function

maximize Profit;

subject to {

// inventory balance constraints

flow1: forall (i in p1)

ReA[i] == dispose[i] + R0[i];

flow2: RCI[1] == R0[1] - R1[1];

flow3: forall (i in p1: i > 1)

RCI[i] == RCI[i-1] + R0[i] - R1[i];

// initial inventory = target inventory

flow4: NCI[1] == TinvN - M1[1];

//Machine capacity

test1: forall(i in p1:i>1)

R1[i] + M1[i] <= MC[i];

test2: forall(i in p1)

max(i in p1)(MC[i]) >= TinvF;

test3: forall(i in p1)

max(i in p1)(MC[i]) >= TinvN;


NCI[i] == NCI[i-1] + order[i-1] - M1[i];

// priority policy for pulling returned components from RCI

priR1: demand[1] - R0[1] >= M*(priBi[1][1] - 1);

demand[1] - R0[1] <= M*(1 - priBi[1][2]);

priR2: forall (i in p1: i > 1)

{

demand[i] - R0[i] - RCI[i-1] >= M*(priBi[i][1] - 1);

demand[i] - R0[i] - RCI[i-1] <= M*(1 - priBi[i][2]);

}

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priR3: forall (i in p1)

{

RCI[i] >= M*(priBi[i][1] - 1);

RCI[i] <= M*(priBi[i][2]);

}


{

R1[i] - demand[i] >= M*(priBi[i][2] - 1);

R1[i] - demand[i] <= M*(priBi[i][1]);

}


priBi[i][1] == 1 - priBi[i][2];


R0[i] == ReA[i] * priBr[i][1] + (demand[i] * priBr[i][2]);


priBr[i][1] == 1 - priBr[i][2];

forall(i in p1)

order[i]<= 70;

// no finished products from production at period 1

flow6: M2[1] == 0;

// remanufacturing time is negligible


R2[i] == R1[i];

// manufacturing time = 1 period


M2[i] == M1[i-1];

flow9: FPIR[1] == R2[1] - R3[1];


FPIR[i] == FPIR[i-1] + R2[i] - R3[i];

flow11: FPIM[1] == TinvF + 0 - M3[1];

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//TinvF <= 38;


FPIM[i] == FPIM[i-1] + M2[i] - M3[i];


R3[i] + M3[i] + LS[i] == demand[i];


R3[i] + M3[i] >= demandW[i];

//Extra Flow15


{

demand[i] - R2[i] - M2[i] >= M*(priBy[i][1]-1);

demand[i] - R2[i] - M2[i] <= M*(1-priBy[i][2]);

}

forall (i in p1)

{

LS[i] >= M*(priBy[i][1]-1);

LS[i] <= M*(1-priBy[i][2]);

}

forall (i in p1)

priBy[i][1] == 1 - priBy[i][2];

// Customer Satisfy 85% per year

//sum(i in p1)(R3[i] + M3[i]) >= sum(i in p1)demandW[i];

// Decision variables

main1: forall (i in p1)

order[i] == TinvN - NCI[i];

main2: forall (i in p1)

FPI[i] == FPIR[i] + FPIM[i];

forall (i in p1)

TinvF >= FPI[i];

main3: R3[1] + M3[1] <= TinvF + M2[1];

forall (i in p1: i > 1)

R3[i] + M3[i] <= FPI[i-1] + R2[i] + M2[i];

Main4: forall (i in p1)

disR[i] == dispose[i] / ReA[i] * 100;

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// summation of units for checking

sumDisp == sum(i in p1) dispose[i];

sumOrder == sum(i in p1) order[i];

sumRCI == sum(i in p1) RCI[i];

sumNCI == sum(i in p1) NCI[i];

sumFPIR == sum(i in p1) FPIR[i];

sumFPIM == sum(i in p1) FPIM[i];

sumRMM == sum(i in p1) (R1[i] + M1[i]);

// Financial constraints

dispC == sum(i in p1) RD_u * dispose[i];

LSC == sum(i in p1) LS_u * LS[i];

NewComponentCost == sum(i in p1) NC_u * order[i] + NC_u * TinvN +

NC_u * TinvF;

ReturnComponentCost == sum(i in p1) RP_u * R0[i];

RemanuandManuCost == sum(i in p1) RemanuandManuCost_R[i] + sum(i in

p1) MM_u * M1[i] + MM_u * TinvF;

forall (i in p1)

RemanuandManuCost_R[i] == (RM_u + (1-q)*10)*R1[i];

ComponentHoldingCost_M[1] == ((NH_y/50)*(TinvN + NCI[1])/2);

forall (i in p1: i>1)

ComponentHoldingCost_M[i] == ((NH_y/50)*(NCI[i-1] + NCI[i] + order[i-

1])/2);

ComponentHoldingCost_R[1] == ((RH_y/50)*(R0[1] + RCI[1])/2);


ComponentHoldingCost_R[i] == ((RH_y/50)*(R0[i] + RCI[i-1] + RCI[i])/2);

ComponentHoldingCost == sum(i in p1) (ComponentHoldingCost_R[i] +

ComponentHoldingCost_M[i]);

FinishHoldingCost_M[1] == (FH_y/50)*(M2[1] + TinvF + FPIM[1])/2;


FinishHoldingCost_M[i] == (FH_y/50)*(M2[i] + FPIM[i-1] + FPIM[i])/2;

FinishHoldingCost_R[1] == (FH_y/50)*(R2[1] + FPIR[1])/2;


FinishHoldingCost_R[i] == (FH_y/50)*(R2[i] + FPIR[i-1] + FPIR[i])/2;

FinishHoldingCost == sum(i in p1) (FinishHoldingCost_R[i] +

FinishHoldingCost_M[i]);

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Sales == (sum (i in p1) (R3[i] + M3[i])*Price_u);

TotalCost == dispC + LSC + RemanuandManuCost +

NewComponentCost + ReturnComponentCost + ComponentHoldingCost +

FinishHoldingCost;

// Worst case of Profit

Profit >= 0;

Profit == Sales - TotalCost;

}

Step 2: Cplex data file for PTR in deterministic case.

/*********************************************

* OPL 12.6.0.0 Data

* Author: Admin

* Creation Date: Dec 16, 2017 at 6:05:20 PM

*********************************************/

SheetConnection sheet("PTR_DATA_NEW.xlsx");

//read data from the excel file

//normal

/*ReA from SheetRead(sheet,"Test!E2:E51");

demand from SheetRead(sheet,"Sheet1!C4:C53");

demandW from SheetRead(sheet,"Sheet2!I4:I53");

MC from SheetRead(sheet,"MC!B2:B51");*/

//max

//old

/*ReA from SheetRead(sheet,"Test!I2:I51");

//new

//ReA from SheetRead(sheet,"Test!P2:P51");

demand from SheetRead(sheet,"Sheet2!D4:D53");

demandW from SheetRead(sheet,"Sheet2!M4:M53");

MC from SheetRead(sheet,"MC!C2:C51");*/

//min

//old

ReA from SheetRead(sheet,"Test!J2:J51");

//new

//ReA from SheetRead(sheet,"Test!N2:N51");

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demand from SheetRead(sheet,"Sheet2!B4:B53");

demandW from SheetRead(sheet,"Sheet2!L4:L53");

MC from SheetRead(sheet,"MC!A2:A51");

//Test

/*ReA from SheetRead(sheet,"Test!E1:E50");

demand from SheetRead(sheet,"Sheet2!Q4:Q53");

demandW from SheetRead(sheet,"Sheet2!S4:S53");

MC from SheetRead(sheet,"Sheet2!P4:P53");*/

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BIOGRAPHY

Name Mr. Kittiphan Nuamchit

Date of Birth December 11, 1992

Education 2016: Bachelor of Engineering (Industrial Engineering)

Sirindhorn International Institute of Technology

Thammasat University

Ref. code: 25615822042262GRA

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