Weighted linear goal programming approach for maximising profit in industry

(1)

WEIGHTED LINEAR GOAL PROGRAMMING APPROACH FOR MAXIMISING PROFIT IN INDUSTRY

by

Moath (Mohd Khair) Alluwaici (1632122093)

A dissertation submitted in partial fulfillment of the requirements for the degree of Master of Science (Engineering Mathematics)

Institute of Engineering Mathematics UNIVERSITI MALAYSIA PERLIS

2017

(2)

i

DEDICATION OF THE THESIS

Dedicated to my

Beloved Family (Mother, Father Wife, Children, Sister and my Brothers)

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(3)

ii

THESIS DECLARATION FORM UNIVERSITI MALAYSIA PERLI DECLARATION OF THESIS

Author’s full name : Moath (Mohd Khair) Alluwaici

Date of birth : 31/07/1982

Title: Weighted Linear Goal Programming Approach for Maximising Profit in Industry

Academic Session : 2016

I hereby declare that the thesis becomes the property of Universiti Malaysia Perlis (UniMAP) and to be placed at the library of UniMAP. This thesis is classified as:

CONFIDENTIAL (Contains confidential information under the Official Secret Act 1972)*

RESTRICTED (Contains restricted information as specified by the organization where research was done)*

OPEN ACCESS I agree that my thesis is to be made immediately available as hard copy or on-line open access (full text)

I, the author, give permission to the UniMAP to reproduce this thesis in whole or in part for the purpose of research or academic exchange only (except during a period of years, if so requested above).

Certified by:

ــــــــــــــــــــــــــــــــــــــــــــــــــــ ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ SIGNATURE SIGNATURE OF SUPERVISOR

K420800 Dr. Ahmad Kadri Junoh

(NEW IC NO. / PASSPORT No.) NAME OF SUPERVISOR Date: 31/10/2016 Date: 31/10/2016

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(4)

iii

ACKNOWLEDGMENT

ميحرلا نمحرلا الله مسب

First of all I would like to thank Almighty Allah for granting me health, strength and giving me the opportunity to complete my thesis.

Undertaking this Master has been a truly life changing experience for me and it would not have been possible to do without the support and guidance that I received from many people.

First and foremost I’m deeply grateful to my dear supervisor Dr. Ahmad Kadri Junoh for his endless insights, supports, and organized guidance throughout the journey to complete my thesis. Your assistance and encouragements are highly appreciated. It has been an honor and privilege to be his Master student. He has taught me, both consciously and un-consciously, how good thesis and analysis are done. I appreciate all his contributions of time, ideas, and kindness to make my Master experience productive and stimulating. The joy and enthusiasm he has for his research was contagious and motivational for me, even during tough times in the Master pursuit. I am also thankful for the excellent example he has provided as a successful man, researcher, engineer and Mathematics senior lecturer.

My upmost appreciation and gratitude goes to my beloved brother SakharAl-Aluwaici who I love dearly for his constant support and motivations throughout my studying. He is the backbone of my life and my gratitude to him is beyond imaginations.

I would like to extend my appreciation and gratitude to my family, my wife and children for standing by my side and for always believing in me all times. Their presence in my

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(5)

iv

life has made things easier and encouraged me to study hard and gave me the resilience and strength to never give up when things went extremely hard. Thank you all.

I would like to acknowledge the kind assistance and friend Adel Mohammed Mohammed Al-Dhahebi, master student at IMK, UniMAP. Since first I met him, he has been of great help. His acknowledgement of the place, as well as his research skills enabled him to guide me through various obstacles. His assistance is highly appreciated.

Last but not least, I would like to express my gratitude to the Institute of Engineering Mathematics department as a whole staff and lecturers at Universiti Malaysia Perlis for providing the perfect research environment and for their kind assistant and friendly management.

Thank you so much!

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(6)

v

TABLE OF CONTENTS

Content Page No.

DEDICATION OF THE THESIS I

THESIS DECLARATION FORM II

ACKNOWLEDGMENT III

TABLE OF CONTENTS V

LIST OF FIGURES VIII

LIST OF TABLES IX

LIST OF ABBREVIATIONS X

ABSTRAK XI

ABSTRACT XII

CHAPTER 1 INTRODUCTION 1

1.1 OVERVIEW 1

1.2 PROBLEM STATEMENT 4

1.3 OBJECTIVES 5

1.4 SCOPE 5

1.5 RESEARCH CONTRIBUTION 6

1.6 SUMMARY OF METHODOLOGY 6

1.7 THESIS ORGANIZATION 7

CHAPTER 2 LITERATURE REVIEW 9

2.1 INTRODUCTION 9

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(7)

vi

2.2 BACKGROUND 9

2.3 LINEAR PROGRAMMING 10

2.4 GOAL PROGRAMMING (GP) 17

2.4.1 Weighted Goal Programming 22

2.5 SENSITIVITY TEST ANALYSIS 29

2.6 SUMMARY OF THE LITERATURE REVIEW 32

CHAPTER 3 METHODOLOGY 33

3.1 INTRODUCTION 33

3.2 METHODOLOGY FLOWCHART 34

3.3 DATA ACQUISITION 36

3.4.1 Mathematical Modeling 42

3.4.2 Summary of the Linear Programming Model 45

3.5 WEIGHTED GOAL PROGRAMMING 47

3.5.1 Weighted Goal Programming Mathematical Model 49

3.5.2 Summary of weighted goal programming model 52

3.6 SENSITIVITY TEST 54

CHAPTER 4 RESULTS AND DISCUSSION 55

4.1 INTRODUCTION 55

4.3 WEIGHTED GOAL PROGRAMMING 57

4.4 SENSITIVITY TEST 60

4.5 T-TEST 64

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(8)

vii

CHAPTER5 CONCLUSION 65

5.1 INTRODUCTION 65

5.2 RECOMMENDATIONS 67

5.3 FUTURE RESEARCH 67

REFERENCES 68

APPENDICES 73

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(9)

viii

LIST OF FIGURES

Figure Page No.

Figure 3.1: Overall flowchart of the research 35 Figure 4.1: Distribution Results of Sensitivity Test for LP and WGP Models 63 Figure 4.2: T-Test of LP and WGP profits 64

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(10)

ix

LIST OF TABLES

Table Page No.

Table2.1: The general structure of GP model 23

Table 3.1: The Sample of the acquired data from Al-Wedyan Company 36

Table 3.2: Number of boxes produced monthly in the company with the profit for 10 months Table 3.3: The monthly amount of production (planned and actual) in Al-Wedyan Company Table 3.4: The produced items distributed over the different containers 40

Table 3.5: The lower production limit for each produced item 41

Table 4.1: Comparison between achieved values for products and the constraints. 56

Table 4.2: The WGP model results 58

Table 4.3: The WGP model deviational variables results 58

Table 4.4: The LP model sensitivity test results 61

Table 4.5: The WGP model sensitivity test results 62

Table 4.6: The LP and WGP resultant profits 63

Table 4.7: Two-Sample T for LP Vs WGP 64 37

39

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(11)

x

LIST OF ABBREVIATIONS LP Linear Programming

WGP Weighted Goal Programming

GP Goal Programming

T2FLR Type-2 Fuzzy Liner Regression GDP Gross Domestic Product

GHG Green House Gases GCC Gulf Corporation Council PVC PolyVinyl Chloride SAR Saudi Arabia Reyal

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(12)

xi

Pendekatan Goal Programming Berpemberat untuk Mengmaksimumkan Keuntungan dalam Industri

ABSTRAK

Pada masa kini, dalam dunia yang berkompetitif dalam pasaran dan globalisasi pengoptimuman bagi proses perancangan bajet merupakan kreteria penting yang mesti dipertimbangkan dalam industri untuk menghasilkan produk berkualiti tinggi pada waktu yang sama memaksimumkan keuntungan. Fungsi permodelan matematik adalah penting untuk memeriksa proses pembuatan keputusan untuk memperolehi penyelesaian alternatif yang optimum yang munasabah bagi permasalahan tertentu dalam proses perancangan pengeluaran. Di antara masalah bagi bagi proses perancangan pengeluaran adalah untuk memaksimumkan keuntungan bagi menghasilkan produk dengan kos yang minimum.

Dalam kajian ini dua formulasi model bagi Pengaturcaraan Linear (PL) dan Pengaturcaraan Goal Berpemberat (PGB) dibangunkan untuk mengoptimumkan dan mensimulasikan keuntungan bagi Syarikat Pemakanan Al-Wedyan. Model yang dicadangkan telah dibangunkan dengan menggunakan perisian LINGO dan berdasarkan kepada data yang betul yang telah dipungut daripada industry. Ujian analisis kepekaan telah dilakukan untuk membandingkan keupayaan bagi keputusan untuk kedua-dua model. Melalui keputusan PGB telah menghasilkan keuntungan tertinggi diikuti oleh model PL.

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(13)

xii

Weighted Linear Goal Programming Approach for Maximising Profit in Industry

ABSTRACT

Nowadays, in the competitive world of marketing and globalization optimizing the budgetary planning process is an essential criterion that is considered by industries in order to produce high quality products while maximizing the profit. The role of mathematical modeling is significant for examining the decision-making process to obtain an optimal possible alternative solution to a particular problem in the production planning process. Among the major production planning process problems is to maximize the profit by producing products with minimum cost. In this study, two formulations of Linear Programming (LP) and Weighted Goal Programming (WGP) models were developed to optimize and to simulate the profit of Al-Wedyan Company Food Ltd. The proposed models are developed through the use LINGO software and based on the real data gained from the aforementioned industry. The sensitivity test analysis was carried to compare the performances for both models results. Throughout the results, the WGP produced the highest profit followed by the LP.

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(14)

1

CHAPTER 1

INTRODUCTION

1.1 Overview

Mathematical models such as linear programming, weighted goal programming and other multi decision-making methods were exploited to examine production process and budgetary management systems. Nevertheless, quality and cost have gained substantial consideration by various manufacturers in order to attract more consumers and further promote the image of their brands. The role of mathematical modeling is significant for analyzing the decision-making process and provides decision makers with more options and choices so they can examine and chose the optimal possible solution to a particular problem in the production planning process with respect to some constraints. In recent years, numerous mathematical models have been developed to solve multi-objectives based decision-making problems (Abichandani, Torabi Basu, & Benson, 2015).

Moreover, multi objective programming methods were introduced over the past two decades into operational research with main goal to focus on decision-making problems that possess several conflicting objectives (Marler & Arora, 2004).

One of the main process that is involved in any industry is the planning process.

Planning is a very important process that ensures the smoothness of all the successive

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(15)

2

processes of any production system. Also, planning is very crucial to decision makers in producing the optimal size of production that maximize the overall profits.

There are short term and long term production planning to get the best results based on optimal plan. A linear optimization model and weighted goal programming are important mathematical tools used to obtain the optimal production and decisions making in various fields. In the linear programming the constraints serve as a boundary for the solution of the space. Meanwhile the solution must satisfy the constraints to which at least one constraint is violated to produce feasible solution that is optimal. The weighted goal programming is extended from the linear programming and it is used to obtain the best solution of the model.

This study aims to examine budgetary manufacturing process through the use of mathematical models, which are the linear programming and weighted goal programming for Al-Wedyan Company for Food Ltd Jeddah. In addition, the study aims to compare the profit obtained from the developed models with the industry’s actual profit. The sensitivity test analysis is also employed to examine the optimal solutions of linear and weighted goal programming by selecting different parameters. In addition, manufacturers are also concerns about the budgetary of the process production as well as the productivity and quality of their products and services. This study is crucial as it allows the industry to increase the production number of specific products based on pre-requests of customers and at the same time to maintain the quality and productivity and to obtain optimal profit while minimizing the cost.

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(16)

3

Thus, linear programming model and weighted goal programming model are formulated in this study for improving production number of specific products in Al- Wedyan Company. Weighted Goal Programming model has a popular consideration and acceptance as applied technique for obtaining the best effective solution decision for looking the multi objectives problems. Therefore, weighted goal programming is extremely useful for decision making analysis in operations research, for instance planning production, solving multi objectives decision problems, financial marketing decisions, design and quality control.

Another important technique developed in this study is linear programming.

Linear programming is a mathematical modeling, which was first introduced in the last 1947 by Dantzig for solving linear problems. Linear programming is an important tool that can be utilized in engineering fields, mathematical optimizations, optics, truss structures and etc. In this project, it is used in order to obtain the minimum and maximum products number in Al-Wedyan Company. Moreover, linear programming always deals with optimization problems. In addition, linear programming is essential optimization technique due to that the problems found in operational research and optimization algorithm problems can be expressed in linear program function problem.

It is crucial in the decision-making processes and offer high positivity role to solve various problems.

The study also touches on the sensitivity analysis that refers to the critical parameters which are likely to influence the results of the models (Linear and Weighted models). The importance of sensitivity test comes from the fact that it serves as a

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(17)

4

validation technique for the model and to provide suitable guide for future research. In the other hand, the sensitivity analysis is concerns with examining whether an output’s uncertainty of a system of a mathematical model can produce optimal outcomes while changing or modifying specific parameters such as constraints and weights of the linear and weight goal models respectively (Saltelli, 2002; Saltelli et al., 2008).

1.2 Problem Statement

Nowadays, the competition among world industries and international marketing is significantly increasing. Therefore, due to this intensive competition, cost has gained substantial consideration by various manufacturers in order to attract more consumers and further improve the image of their brands. In big manufacturing companies and industries, certain limitations exist such as to increase or decrease the production number of certain products based on pre-determined targets. Another limitation is the in-ability of industry to assess and control the budgetary manufacturing process based on predetermined measures. Therefore, the industries are in need for a mathematical modeling system that can be utilized specifically to improve the budgetary planning of the profit.

Among the problem statement are the followings:

i. The necessity to examine budgetary process of the Al-Wedyan Company for Food Ltd - Jeddah, Saudi Arabia which is significant to meet the expectation and demands of the management and to maximize the profit.

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(18)

5

ii. Limited numbers of pre-determined products are produced by Al-Wedyan Company each month which create significant demands to certain products to be maximized.

iii. The inability of the industry to successfully integrate budgetary manufacturing process with all production stages which creates defects on some products and contribute to low profit and productivity.

1.3 Objectives

This study aims to meet the following objectives:

i. To develop Linear Programming (LP) model that examine the budgetary production process and maximize the profit of Al-Wedyan Company.

ii. To develop Weighted Goal Programming (WGP) model for assessing budgetary manufacturing process and improve the overall profit of Al- Wedyan Company.

iii. To evaluate the outputs of both LP and WGP models.

1.4 Scope

This study emphasizes the use of Weighted Goal Programming (WGP) model as well as the Linear Programming (LP) model to examine budgetary manufacturing process and improve the production profit of Al-Wedyan Company for Food ltd Jeddah,

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(19)

6

Saudi Arabia. The scope of this study is limited to the budgetary manufacturing process of Al-Wedyan Company.

1.5 Research contribution

The overall contribution of this study to the industry can be summarized as to improve budgetary production process and facilitate the process of maximizing the profit of the factory. In addition, the study will provide optimal solution to maximize the profit of production process.

By developing LP and WGP, the number of products in the industry process production can be controlled to obtain the most ideal numbers and consequently create satisfaction and trust between costumers and industries.

1.6 Summary of Methodology

The methodology of this study is divided into three sections where the first section illustrates the development of LP model. Meanwhile, the second section concerns about the development of WGP model for the purpose of assessing the budgetary production process and improving the profit of industry.

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(20)

7

It also focuses on the best solutions for maximizing the overall profit of Al- Wedyan Company. Finally, sensitivity analysis is used to provide assessment toward the sensitiveness of the outputs of both models LP and WGP.

1.7 Thesis Organization

This thesis comprises of five chapters as follows:

Chapter 1 concerns about the general description including introduction, problem statements, objectives, and the scopes. It also addresses the contribution of this study, summary of methodology and finally thesis organization.

Chapter 2 presents the literature review, which contains the background of the research and briefly explained about the Linear Programming (LP) and Weighted Goal Programming (WGP). This chapter illustrates the related studies on the applications of LP and WGP in industries. The chapter ended up with the sensitivity analysis.

Chapter 3 illustrates the methods utilized in this study and it contains introduction, flow chart of methodology, data collection, regression analysis, LP and WGP models.

Chapter 4 addresses the results and discussion and the overall findings of this study.

The results of the mathematical models LP and WGP are illustrated in this chapter.

Finally, the findings of sensitivity test analysis are presented in this chapter as well.

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(21)

8

Chapter 5 presents the final chapter of the thesis and it involves the conclusion of this study. It also brings about the recommendations as well as suggestions for future research.

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(22)

9

CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

In the previous chapter, background on the role of mathematical models Liner Programming (LP) and Weighted Goal Programming (WGP) to optimize profit during manufacturing process of Al-Wedyan Company was presented. Meanwhile, this chapter attempts to present the relevant literatures about the LP and WGP models. The literature serves as the initial step toward the design and development of the models. Toward the end of the chapter, LP was discussed in details before moving to WGP and sensitivity test analysis. The scope of this literature review is limited to applications of LP and WGP models in industry.

2.2 Background

Planning is essential criteria in manufacturing process, which describes allocation of specific limited resources into the process of producing products that match with the requirements of customers. In today’s competitive world industries are striving to produce production plans with minimal total production cost and maximum possible

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(23)

10

revenues from producing a certain amount of raw materials while meeting customer demand. For the past decades, optimizations of budgetary planning have gained wide consideration and interests of many researchers in the fields of control systems, chemical and physical systems, planning and management systems, productions, transportation, mathematical models, financial systems and many engineering related designs and systems (Sahinidis, 2004).

The assessment of budgetary manufacturing process and the production process are very crucial criteria that must be realized by manufacturers in order to obtain optimality in terms of production number and quality of productivity, cost minimization and maximization of revenues. Utilizing mathematical models to assess the budgetary manufacturing process and promote quality of productivity is significant. According to Astolfi (2006), optimizations is a procedure undertaken to obtain the best possible results based on certain conditions and circumstances. In all procedures of optimizations, the aim is whether to maximize a benefit or to minimize an effort. The benefit or the effort can be formulated as a function of certain design variables.

2.3 Linear Programming

Linear programming can be defined as a mathematical technique, which can be utilized for achieving the maximum or minimum linear function value under specific settings and constraints. Nonetheless, it is a dominant technique for providing allocation optimality of resources scares as well as maximizing the profit. Dantzig firstly developed linear programming in 1947 for solving linearity problems. It has proven the

s i tem

is pr ot ec ted by

or igi nal

c opy

right

(24)

11

effectiveness of employing LP method due its capability to solve various problems for instance planning of productivity, resources allocating, control of inventory, advertisement and to attain the most reliable accurate decisions and to obtain optimum productivity.

In addition, LP is useful for selecting the desirable patterns effectively among several variables in the planning production (Edwards, Malekzadeh, & Yisa, 2001). It is employed for optimizing the dependent variable subjected to a particular a set of related variables that are independent in a relationship that is linear. The values obtained represent the values of the variables that are dependent which subjected to the independent variable of the solving problem. Meanwhile, the dependent variable can be set to represent the objective function that describes particular variable such as sales, distance, profit and costs. In this case the independent variable is known as unknown variable value which is determined by the solving problem (Mehdipoor, Sadr-ol- ashraafi, & Karbaasi, 2006).

LP is essential optimization technique due to the fact that the problems found in operational research and optimization algorithm problems can be expressed in linear program function problem. It is crucial in the decision-making processes and offer high positivity role to solve various problems. LP is an effective optimization tool that can be applied to solve various problems from diverse fields such as telecommunication, transportation, energy, production, scheduling of airline crew, flows of network (Adedayo, Ojo, & Obamiro, 2006; Anderson, Sweeney, Williams, Camm, & Cochran, 2015; Chaharsooghi & Jafari, 2007).

s i tem

is pr ot ec ted by

or igi nal

c opy

right