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A generalized decision model for naval weapon procurement

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A generalized decision model for naval weapon procurement multi-attribute decision making
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Chang, Jin O
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Best selection method
Saw
Topsis
Sensitivity analysis
Ahp
Utility theory
Hierarchy of attributes
Dissertations, Academic -- Industrial Engineering -- Doctoral -- USF   ( lcsh )
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government publication (state, provincial, terriorial, dependent)   ( marcgt )
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theses   ( marcgt )
non-fiction   ( marcgt )

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Summary:
ABSTRACT: For any given reason, every year many countries spend a lot of money purchasing at least one weapon. Due to the secret character of the military, the decision process for specific weapon procurement is shrouded. Moreover, there are several funds loss cases due to mistakes in weapon contractions. Weapon procurement requires very large amounts of money which comes from tax payers. Therefore, an effort to reduce a possible monetary loss is needed. A decision process based on an analytic model can present a better chance to decision makers for better weapon decisions. In general, weapon procurement decision is a multi criteria environment. Decision making in such environments is defined as Multi-Criteria Decision Making (MCDM). MCDM is broadly classified into two areas: Multi-Attribute Decision Making (MADM) and Multi-Objective Decision Making (MODM). MADM methods are used for selecting an alternative from a small explicit list of alternatives.MODM methods are used for designing problems involving an infinite number of alternatives implicitly defined by mathematical constraints. This research is intended to be used by the South Korean Navy when there is a need to select one weapon type among several candidate types. Therefore, MADM methods are used in this research.Many researches for developing an analytical model for better decision-making have been done. However, there is no research for a generalized weapon procurement decision model that is easy to implement. For this reason, whenever there is a need for weapon procurement decision, the Navy has to spend a lot of effort in determining the best weapon. These efforts can be reduced with a generalized model that is proposed in this research for naval weapon procurement. MADM methods determine alternatives ranking orders and the highest ranked alternative is the best one. Various MADM methods are used in computing the alternatives ranking scores.
Thesis:
Thesis (Ph.D.)--University of South Florida, 2005.
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Includes bibliographical references.
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by Jin O. Chang.
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Document formatted into pages; contains 149 pages.
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Includes vita.

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A Generalized Decision Model for Naval Weapon Procurement: Multi-Attribute Decision Making by Jin O Chang A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Industrial and Ma nagement System Engineering College of Engineering University of South Florida Major Professor: Michael Weng, Ph.D. William Miller, Ph.D. Tapas Das, Ph.D. Ram M. Pendyala, Ph.D. A.N.V. Rao, Ph.D. Date of Approval: March 22, 2005 Keywords: best selection method, SAW, TOPSIS, sensitivity analysis, AHP, utility theory, hierarchy of attributes Copyright 2005, Jin O Chang

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Acknowledgments I would like to express my appreciation to my thesis advisor, committee chairman and friend, Dr. Michael Weng, for his careful readi ng of the manuscript and valuable feedback. Without his constructive critic ism and advice, this thesis would have not progressed as smoothly as it did throughout its completion. I w ould also like to thank the other committee members: Dr. William Miller, Dr. Tapas K. Da s, Dr. Ram M. Pendyala, and Dr. A.N.V. Rao for a number of valuable comments and suggestions. I would especially like to th ank my parents Soonhyuck Cha ng and Kumja Lee, my wife Malran Lee and my two sons Hankil and Hanhae for their love, encouragement and support in the pursuit of this degree. Wit hout their understanding, none of this would be possible. To them this thesis is dedicated.

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i Table of Contents List of Tables iii List of Figures vi Abstract vii Chapter One. Introduction 1 1.1 Problem Definition 1 1.2 A Generalized Decision Model for Weapon Procurement 4 1.3 Research Objective and Contributions 7 1.4 Thesis Overview 8 Chapter Two. Literature Review 9 2.1 Introduction 9 2.2 MADM Methods 10 2.2.1 General Steps for MADM 11 2.2.2 Noncompensatory Methods 14 2.2.3 Compensatory Methods 15 2.2.4 Analytic Hierarchy Process (AHP) 39 2.2.5 Utility Theory for Decision Making 47 2.2.6 The Delphi Method 52 2.3 Research on Similar Problems Related to this Problem 55 2.4 Summary 61 Chapter Three. Problem Statement and Methodology 64 3.1 Introduction 64 3.2 Problem Statement 64 3.3 Best Selection Method (BSM) 65 3.3.1 A Numerical Example 71 3.3.2 Comparison with Current MADM Methods 72 3.4 Sensitivity Analysis 73 3.4.1 Current Basis and Allowable Ranges 75 3.4.2 A Numerical Example 76 3.5 Summary 79 Chapter Four. Construction of A Hier archical Structure 81 4.1 Introduction 81 4.2 Operational Performance 83 4.3 Readiness on Time 85 4.4 Technical Merits 88 4.5 Cost Effectiveness 89 4.6 Sustainment 90

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i Table of Contents List of Tables iii List of Figures vi Abstract vii Chapter One. Introduction 1 1.1 Problem Definition 1 1.2 A Generalized Decision Model for Weapon Procurement 4 1.3 Research Objective and Contributions 7 1.4 Thesis Overview 8 Chapter Two. Literature Review 9 2.1 Introduction 9 2.2 MADM Methods 10 2.2.1 General Steps for MADM 11 2.2.2 Noncompensatory Methods 14 2.2.3 Compensatory Methods 15 2.2.4 Analytic Hierarchy Process (AHP) 39 2.2.5 Utility Theory for Decision Making 47 2.2.6 The Delphi Method 52 2.3 Research on Similar Problems Related to this Problem 55 2.4 Summary 61 Chapter Three. Problem Statement and Methodology 64 3.1 Introduction 64 3.2 Problem Statement 64 3.3 Best Selection Method (BSM) 65 3.3.1 A Numerical Example 71 3.3.2 Comparison with Current MADM Methods 72 3.4 Sensitivity Analysis 73 3.4.1 Current Basis and Allowable Ranges 75 3.4.2 A Numerical Example 76 3.5 Summary 79 Chapter Four. Construction of A Hier archical Structure 81 4.1 Introduction 81 4.2 Operational Performance 83 4.3 Readiness on Time 85 4.4 Technical Merits 88 4.5 Cost Effectiveness 89 4.6 Sustainment 90

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iii List of Tables Table 1.1 A Rank for Defense Expenditures in 2002 1 Table 1.2 Military Funds Loss Cases in South Korea 3 Table 1.3 Comparisons between MODM and MADM 5 Table 2.1 Data for Evaluation of Fighters 19 Table 2.2 Three Sets of Preference Rankings 28 Table 2.3 Determination of Outranking Relationship 36 Table 2.4 The Net Concordance and Discordance Inde xes of the Alternatives in the K 38 Table 2.5 The Fundamental Scales 41 Table 2.6 Average Random Consistency Index (RI) 44 Table 2.7 Comparison of the First Level of Attr ibutes with Respect to Satisfy a Good Fighter 45 Table 2.8 Attribute Weights 46 Table 2.9 Example of Pairwise Comparison Matrix with Respect to Each Attribute 46 Table 2.10 Decision Matrix for Ten Cars 48 Table 2.11 The Final Alternatives’ Rank Scores 52 Table 2.12 A Sample Questionnaire of the Delphi Method 54 Table 2.13 Summary of Similar Researches 57 Table 2.14 Comparison of MADM Methods 62 Table 2.15 Comparisons Within Several Weighting Methods 63 Table 3.1 Example for an Extreme Alternative 65 Table 3.2 ijr and ijv for the Extreme Alternative Problem 69

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iv Table 3.3 Ranking Scores for the Extreme Alternative Problem 71 Table 3.4 Ranking Scores for the Example fr om the Three MADM Methods 72 Table 3.5 Ranking Scores for the Fight er Selection Problem 72 Table 3.6 The Alternative Values and Ranking Information 76 Table 3.7 Critical Values for 3A in terms of ) ( ) (2 3A V A V 78 Table 3.8 Critical Values for 4 2 1, A A A and 5 A in terms of ) ( ) (3iA V A V 79 Table 4.1 Principles of Weapon Procurement 81 Table 4.2 Indexation of Readiness of Weapons 87 Table 4.3 Indexation of Readiness of Supporting Systems 87 Table 4.4 Indexation of Technology Acquisition 89 Table 4.5 Indexation of Logistics 91 Table 4.6 Indexation of Depot Maintenance 92 Table 4.7 Indexation of Field Maintenance 93 Table 5.1 Weights for the Five Fi rst-Level Attributes 98 Table 5.2 Weights for the Two Second-Level Attri butes of Operational Performance 99 Table 5.3 Weights for the Three Second-Level A ttributes of Readiness on Time 100 Table 5.4 Weights for the Two Second-Level A ttributes of Technical Merits 100 Table 5.5 Weights for the Two Second-Level Attributes of Cost Effectiveness 101 Table 5.6 Weights for the Three Second-Leve l Attributes of Sustainment 101 Table 5.7 Weights for the Third Level Attributes 102 Table 5.8 Attribute Weightings for the Best Weapon Procurement 104 Table 6.1 Data for Regular Operational Performances 108 Table 6.2 Data for Readiness on Time 108 Table 6.3 Data for Technical Merits 109

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v Table 6.4 Data for Cost Effectiveness 109 Table 6.5 Data for Logistics and Maintenance 110 Table 6.6 2004 Defense Company Rankings 111 Table 6.7 2000 Defense Company Rankings 111 Table 6.8 Data for Reliability 112 Table 6.9 Data for Evaluation of Submarines 113 Table 6.10 Alternatives’ Normalized and We ighted Normalized Values 114 Table 6.11 Ranking Scores for the Submarine Selection Problem 115 Table 6.12 Critical Values for 1A in terms of) ( ) (2 1A V A V 115 Table 6.13 Data for Evaluation of Submarines with Modified Weight Values 117 Table 6.14 Ranking Scores Based on Modi fied Attribute Weights 118 Table 6.15 Ranking Scores Based on Modifi ed Attribute Weights (when56 041w) 118 Table 7.1 Alternative Ranking Scores for Different and 122

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vi List of Figures Figure 1.1 Ratio Defense Expenditure to Government Budget and GDP 2 Figure 2.1 A Taxonomy of MADM Methods 15 Figure 2.2 A Hierarchy for Fighter Evaluation 16 Figure 2.3 Weight Assessment for Fighter Evaluation 20 Figure 2.4 Euclidean Distances to Positive-Ideal and Negative-Ideal Solution in Two Dimensional Space 23 Figure 2.5 A Digraph for Eight Alternatives 30 Figure 2.6 The Kernel of Figure 2.5 31 Figure 2.7 The Kernel of the Example Problem 37 Figure 2.8 Graphic Representation of Indifference Curve 49 Figure 2.9 Utility Functions for Car Selection Example 51 Figure 2.10 Two Variance from the First and Second Round in the Delphi Method 53 Figure 4.1 A Hierarchy for Best Weapon Selection 82 Figure 4.2 Hierarchical Structure of Operational Performance 83 Figure 4.3 Hierarchical Structure of Readiness on Time 86 Figure 4.4 Hierarchical Structure of Technical Merit 88 Figure 4.5 Hierarchical Structure of Cost Effectiveness 90 Figure 4.6 Hierarchical Structure of Sustainment 91 Figure 4.7 A Hierarchy of Attributes For the Best Weapon Procurement 95 Figure 5.1 Weight Assessments for Best Weapon Procurement 103

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vii A Generalized Decision Model for Naval Weapon Procurement: Multi-Attribute Decision Making Jin O Chang ABSTRACT For any given reason, every year many countri es spend a lot of money purchasing at least one weapon. Due to the secret character of the military, the decision process for specific weapon procurement is shrouded. Moreover, there are several funds loss cas es due to mistakes in weapon contractions. Weapon procurement requires very large amounts of money which comes from tax payers. Therefore, an effort to reduce a possible monetary loss is needed. A decision process based on an analytic mode l can present a better chance to decision makers for better weapon decisions. In general, weapon procurement decision is a multi criteria environment. Decision making in such environm ents is defined as Multi-Criteria Decision Making (MCDM). MCDM is broadl y classified into two areas: Multi-Attribute Decision Making (MADM) and Multi-Objective Decision Making (MODM). MADM methods are used for selecting an al ternative from a sma ll explicit list of alternatives. MODM methods are used for design ing problems involving an infinite number of alternatives implicitly defined by mathematical cons traints. This research is intended to be used by the South Korean Navy when there is a need to select one weapon type among several candidate types. Therefore, MADM me thods are used in this research. Many researches for developing an analyti cal model for better decision-making have been done. However, there is no research for a generalized weapon procurement decision

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viii model that is easy to implement. For this reason, whenever there is a need for weapon procurement decision, the Navy has to spend a lot of effort in determining the best weapon. These efforts can be reduced with a generalized model that is proposed in this research for naval weapon procurement. MADM methods determine alternatives’ ranking orders and the highest ranked alternative is the best one. Various MADM methods are used in computing the alternative’s ranking scores. However, there is no MADM met hod which can compensate individual values for an overall value. Our new MADM model can compensate for that. We also provide a sensitivity analysis to the solutions obtained by the proposed model. This new model is applied to a real problem in the South Korean Navy.

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1 Chapter One Introduction 1.1 Problem Definition For most countries, the portion of the nationa l budget allocated to defense spending is a significant value. In 2002, 14 billion dollars of South Korea’s budget was spent on national defense. And this was ranked 11th in the world. Table 1.1 shows the rank based on national defense money spent in 2002 (MND, 2004). Table 1.1 A Rank for Defense Expenditures in 2002 Rank Nation Expenditure (million $) 1 U.S.A 399,900 2 Russia 65,000 3 China 47,000 4 Japan 42,000 5 England 38,000 6 France 29,000 7 Germany 24,000 8 Saudi Arabia 21,000 9 Italy 19,000 10 India 15,000 11 South Korea 14,000 For South Korea, this expend iture is due to it s political and geogr aphical situation. South Korea is still in conflict with North Korea. Also it is surrounded by many countries that

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2 have a strong military power, such as China, Japan and Russia. However, there is a trend in the world to try to reduce war expenditures to provid e more benefits for its citizens. South Korea could not be an exception in that. In fact, as shown in figure 1.1, there was a continuous decrease in defense expenditure ra tio to government and GDP (Kwon, 2003). Figure 1.1 Ratio Defense Expenditure to Government Budget and GDP Due to its situation, it is very important fo r South Korea to keep its military strength. Reducing costs is the best way to maintain abil ity when there is not much increase or decrease in budget. Reducing cost can be obtained by scien tific and systematic management. Therefore, research for the scientific management for the military is required.

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3 South Korea spends millions of dollars fo r weapon procurements. And there are many funds lost due to managers’ mistakes during weapon procurement decisions. These factors do the opposite of reducing costs. Therefore, efforts to reduce the loss caused by these factors can play a role to reduce costs in weapon procurement and saved money can be used in other important areas. Kim (2000), the author of “Arms Procurement Decision Making”, analyzed the defense funds loss and presen ts the reasons in Table 1.2. Table 1.2 Military Funds Loss Cases in South Korea (Source: Kim, 2000) Reasons for the loss Weapon sy stem Funds loss (million $) Excessive payments for the weapons K-1 tank, UH-60 hello, K200 Armored Vehicle 132 Excessive transaction fees UH-60 hello, P3-C 48.7 Money exchange late loss M-60, M-16, Howitzer 263 Total 443.7 In general, the public are restricted in accessing data for weapon procurement decisions for a military secrete purpose. This access is only possible for only few authorized people. It is impossible even to look at the items for other peoples. Every year, ther e are requirements for new weapon procurements dependant to a new st rategy or a replacement for a life cycle ended weapons. Because of the secret nature the milit ary, the decision process for specific weapon procurement is shrouded. In general, weapon procurement requires large sums of money. Therefore, decision making in weapon procur ement can be easily affected by political pressures. Kim (2000) presents an interaction of political factors in a decision process changed decisions regardless of weapon performance and its cost effectiveness. In many cases, those decisions changed by politicians lost a lot of money.

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4 Kim (2000) also points out the lack of a ny systematic decision support system in weapon procurements and suggests that with a systematic decision suppo rt system for weapon procurement those problems can be resolved. 1.2 A Generalized Decision Model for Weapon Procurement A generalized decision model for weapon proc urement is defined as follows: it is a scientific and systematic model designed to help senior officers of South Korean Navy (hereafter called SKN) to make the best decision for weapon pr ocurement with a quantitative ranking score. To be scientific and systematic a model must be suppo rted by an analytical procedure and consistent for the same problem for every calculation. In the SKN Regulation Book 2 (hereafter called NR 2), there are seve ral criteria for selecting weapon. Our model considers these criteria and presents quantitative rank inform ation of the candidate weapons to a Decision Makers (DM). Multi Criteria Decision Making (MCDM) is broadly classified in to two categories: Multiple Attribute Decision Making (MADM) and Multiple Objective Decision Making (MODM) (Yoon, 1980). MADM methods are used for selecting one alternative from a small, explicit list of altern atives, while MODM methods are us ed for designing a problem involving an infinite number of altern atives implicitly defined by math ematical constraints. Table 1.3 displays the comparisons between MODM and MADM.

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5 Table 1.3 Comparisons between MODM and MADM MODM MADM Goal Finding optimal solution by using mathematical model under some constraints Can take several alternatives Find only one alternative among several of candidate ones Each alternative has a same level of identification Objective Function A function has decision variables representing amounts of acquisition of each alternative No such objective function Attributes Functions as constraints Functi ons as giving alternative’s numerical scores Supporting research Yoon (1980), Yoon and Hwang (1995), Pomerol and Romero (2000), Saaty (1986) To understand the difference between these tw o methods, let us consider two example problems. Hall et al (1992) presents a mode l for making project funding decisions at the National Cancer Institute (hereafter called as NCI). NCI has funded a series of studies for reducing smoking prevalence. DMs of NCI want maximize the budget ability to fund the states with the most highly evaluated proposals. In addition to budget availability, DMs have to consider political pressure as well. Hall et al. (1992) in troduce a preference function and propose a decision model such as Max 8 10 1 9 1 x x s.t. 17 7 7 3 68 1 x x : minimum number of preference points 37 6 5 8 98 1 x x : maximum available budget and some other constraints.

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6 In this example, DMs want to maximize NCI’s budget ability by supporting as many as they can while meeting constraints. This is a typical MODM problem. Let us assume that NCI can only support one proposal. In addition, DMs of the NCI want to review every proposal, regardless of th eir rank. From the first assumption that NCI can only support one proposal, the abov e model can be rewritten as Max 8 10 1 9 1 x x s.t. 17 7 7 3 68 1 x x 37 6 5 8 98 1 x x i x xi i i } 1 0 { 18 1. This model will allow only one proposal selection. However, DMs can not see other proposals that are not selected. Therefore, th e second assumption that DMs want to review each proposal is not met by the MODM method. MADM method presents a set of alternative ranks based on their ranking scores. Based on this rank information, DMs can select the best alternative that has th e highest ranking score. In addition, DMs can see all other alternatives ’ ranking scores. An alternative ranking score is computed by a given set of attributes that ar e used for constraints in the MOMD method. A detailed procedure for computi ng an alternative ranking score is presented in Chapter 2. If the two methods are different, various st udies for mutual improvement have been done. For example, Saaty (1986), Hughes (19 86) and Rahman (2003) propose that MODM method can be simplified by using MADM method.

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7 In this research, we assume that the anal ysis is of a single weapon type selection problem. In other words, the SKN can not sele ct multiple weapon systems. In addition, we assume that DMs want to review each candida te weapon regardless of their ranking orders. Based on these two assumptions, the appropria te solution belongs to MADM method. 1.3 Research Objective and Contributions The objective of this research is to devel op a new decision model that can identify the proper best alternative in the presence of extreme alternatives, which may be caused by possible political pressures, on weapon procurem ent decision. The existing MADM methods in the open literature can not handl e such extreme alternatives. The contributions of this research are summarized as follows. 1. Since the SKN does not have any hierarchy of attributes for evaluating weapon systems, the suggested hierarchical stru cture can be used for every weapon procurement decision. 2. Since the current MADM methods only consid er either an altern ative overall ranking score or individual attribute values, they cannot address any polit ical pressure during weapon procurement decision. However, our new decision model can address this pressure by compensating an alternative overa ll ranking score for its individual attribute values. 3. Proposed sensitivity analysis can present what-if analysis for both the SKN and weapon suppliers. 4. Our new model can be used for any other non-military decision process especially when a decision can be easily affected by some political powers.

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8 1.4 Thesis Overview The rest of this research is organized into six chapters. Chapter 2 reviews the methodologies as well as the relevant researches to our problem. In chapter 3, a new MADM method is suggested with numerical example probl ems. A sensitivity analysis to the solutions obtained by the proposed model is also presented in this chapter. In Chapter 4, we develop a hierarchy of attribute for evalua ting weapon systems. In Chapter 5, attribute we ights are given to this hierarchy by using the AHP method. In Chapter 6, our new model is applied into a real problem in the SKN. The conclusion and fu rther research is followed in Chapter 7.

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9 Chapter Two Literature Review 2.1 Introduction Every other year, SKN procur es at least one weapon eith er for improving its defense ability or replacing its old weapons It ranges from thousands of dollars worth to billions of weapon systems. SKN selects a weapon among several candidate weapons in terms of various requirements defined as Requirement of Operati onal capability (ROC), co st effectiveness and political situations. Each consideration is c onflicting and sometimes requires compensation with each other (e.g., better pe rformance weapon is usually expe nsive than worse so that in order to save money we may need to choose one th at is not best in performance). In other word, DMs deal with decision problems that involv e multiple and usually conflicting criteria. MADM procedures can be applied to a wide ra nge of human choices, from the professional to the managerial to the political. Pomerol and Romero (2000) present the historical background of MADM. From a scientific view point, th e research into economics which took place at the end of the nineteenth century and the beginning of the twentieth is one of the sources of inspiration for the MADM. At that time economists were beginning to look for links between the behavior of economic agents and the economy itself. One of the basi c factors governing beha vior, applying to both producer and consumer, is the way choices are ma de in consumption and production. Later this is developed as a consumer theory with utility function. By 1960, multi-criterion analysis was acquiring its own vocabulary and problem form ulations (i.e., the problem of choosing

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10 alternatives in the presence of multiple criteria called attribut es in MADM). In 1976, Keeney and Raiffa proposed Multi Attribute Utility Theory (MAUT). Pomerol and Romero classify scholars into two groups: one group is the supporte rs of the utility and the other is pragmatists using other methods like AHP, TOPSIS, and ELEC TRE. The latter group’ s methods are called MADM methods. MADM refers to making preference decision s (e.g., evaluation, prioritization, and selection) over the available alternatives that are characterized by multiple, usually conflicting, attributes (Hwang and Yoon, 1981). MADM methods are management decision aids used in evaluating competing alternatives defined by multiple attributes. Starr and Zeleny (1977), Zionts (1978), and Yoon and Hwang (1995), and S aaty and Vargas (2000) are representative researchers in MADM area. In this chapter, we present numerous MADM methods as well as researches done in decision making problems for both military and non-military areas. 2.2 MADM Methods MADM methodology tries to obtain a meaningf ul index from multidimensional data to evaluate competing alternativ es. Pioneering surveys on MADM methods were carried out by MacCrimmon (1973). Since then many methods have been developed by researchers in disciplines as diverse as management science, economics, psychometrics, marketing research, applied statistics, and decision theory. All MADM methods can be classified compensatory or noncompensatory, ordinal or cardinal, and quantita tive or qualitative. In this section, general steps for MADM and several MADM methods are discussed.

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11 2.2.1 General Steps for MADM An analysis begins by defining attributes that can measure the relevant goal of accomplishments. These attributes are set up by constructing a problem structure. Then alternatives are contrasted over the chosen attributes. Often a ll attributes are not of equal importance to the DM. Thus, the rendering of appr opriate weights among at tributes is of prime concern to the DM. Suppose that there are two types of attributes: qualitative and quantitative. Also each quantitative attribute has a different unit of measurement (e.g., number of people and amounts of dollars). We need a homogenous data type for a DM to compare each alternative. Homogenous data sets can be obtained through the normalization procedure. There are three possible ways of defining attributes: re viewing literatures, using possible documents, and asking experts’ opi nions (Keeney and Raiffa, 1976). The Delphi technique and the AHP method are wide ly used for the latter purpose. Pardee (1969) suggests that a desirable list of attributes should be complete and exhaustive, contain mutually exclusive items, and be restricted to performance attributes of the highest degree of importance. Again, attributes are developed as a re sult of constructing a problem structure. Weights represent the relative importance of each attribute with respect to an overall goal. Therefore, we can define that weights that can play a key role in MADM problems. Moreover, the weights themselves can be useful information to those concerned with the program or project management, since they indicat e what the DM is most concerned about in a quantitative way (Edwards and Newman, 1982). So metimes, the weights themselves are useful tools for management purposes (Chang, 1997). A DM may use either an ordinal or a cardina l scale to express hi s or her preference among attributes. Although it is us ually easier for a DM to assign weights by an ordinal scale,

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12 most MADM methods require cardi nal weights. Cardinal weights are normalized to sum to 1, that is 1jw, where jwrepresents weight of the jth attribute The simplest way of assessing weights is to a rrange the attributes in a simple rank order, listing the most important attribute first and th e least important attribute last. When 1 is assigned to the most important attribute, and n (t he number of attributes at hand) to the least important, the cardinal weights can be obtained from one of the following formulas (Stillwell et al, 1981): n k k j js s w11 1 (2.1) n k k j js n s n w11 1 (2.2) where js is the rank of the jth attribute. Equation 2.1 is called as rank reciprocal weight method, while the Equation 2.2 is called as rank sum weight method (Yoon and Hwang, 1995). If attributes are tied in the ranking, their mean ranking can be used. Ranking all attributes at the same time may place a heavy cognitive burden on the DM. Therefore, a method by which a complete ranking can be obtained from a set of pairwise judgments is the preferred approach (M orris, 1964; Saaty and Vargas, 2000). Attribute ratings are normalized to eliminate computational problems caused by differing measurement units in a decision matrix. It is not always necessary but is essential for

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13 many compensatory MADM methods. Normaliza tion aims at obtaining comparable scales, which allow inter-attribute as well as intra-attribute comparisons. Consequently, normalized ratings have dimensionless units and, the larger the rating becomes, the more preference it has. There are two types of normalization me thods: linear and vector normalizations. Linear normalization is simple procedure that divides the ratings of a certain attribute by its maximum value. The normalized value of ijx is given as attribute cost for value a is en wh / 1 1/ attribute benefit for value a is when /* ij j ij ij j ij ijx x x x x x r (2.3) where ijx is the response of alternative ion attribute j *jx is the maximum value of the j th benefit attribute, jxis the minimum value of the j th cost attribute, and ijr is the normalized value of ijx. Vector normalization divides the rating of each attribute by its norm, so that each normalized rating of ijx can be calculated as attribute cost for value a is when ) / 1 ( / 1 / 1 attribute benefit for value a is when ,1 2 1 2ij m i ij ij ij m i ij ij ijx x x x x x r (2.4)

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14 Saaty and Vargas (1993) show that there is only a minor difference between these two normalization methods by using simulation re sults. The proposed model uses the linear normalization method. 2.2.2 Noncompensatory Methods Yoon and Hwang (1995) present the taxonomy of 13 methods as shown in Figure 2.1 and they classify methods in the bottom box under Major Class of Method as compensatory methods and others as noncompensatory method. A compensatory or noncompensatory distin ction is made on the basis of whether advantages of one attribute can be traded for disadvantages of another or not. A choice strategy is compensatory if trade-offs among attri bute values are permitted, otherwise it is noncompensatory. Noncompensatory methods are relatively easy to facilitate compar ed to compensatory methods. However, this approach considers onl y one attribute at a ti me and can miss overall good alternatives. For example, when we cons ider buying a new car, there might be several important factors to be consid ered like cost, gas mileage, and performance measure. Suppose that we have three cars (A1, A2, and A3, respectively) for consideration. A1 and A2 may have one outstanding attribute. While A3 does not, but in overall score it is first. Noncompensatory may lose A3 for selection (for mo re detail, see Yoon and Hwang, 1995).

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15 Figure 2.1 A Taxonomy of MADM Met hods (Source: Yoon and Hwang (1995)) Weapon procurement decision should not be si mply decided because of its monetary big scale and huge impact to th e country. Therefore, for this research compensatory methods should be applied. 2.2.3 Compensatory Methods In compensatory methods, all attributes ar e considered to make a final decision. For example, let us consider a military fighter select ion decision. A country decides to reinforce its air force by purchasing sophisticated jet fighter s. Five competing models are available for purchase from the market. The huge acquisition co st and long-term impact on national security force the acquisition officers to make circum spect decisions. They proceed to generate MADM N o information Information on Environment Information on Attribute Dominance Maximin Maximax Conjunctive Method Disjunctive Method Pessimistic O p timistic Standard Level Ordinal Cardinal Salient Feature of Information Type of Information from Decision Makers Major Class of Method Lexicographic Method Elimination by Aspect Simple Additive Weighting Weighted Product TOPSIS ELECTRE Median Ranking Method AHP

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16 selection criteria by way of a goal hierarchy. The hierarchy for a good fighter is shown in the following figure. Figure 2.2 A Hierarchy for Fighter Ev aluation (Source: Yoon and Hwang (1995)) DMs want a good fighter and they define that a good fighter is dependent on four important factors (mechanical performance, handling quality, serviceability, and economic merit). Let us define factors under good fighter as attributes. Then trad e-offs among attribute values should be done in compensatory method. There may be several cases such as: one fighter may have the highest speed but not as good in other attributes one fighter may have overall good score but not ranked fi rst in any attributes, and so on. If this situations happen, how DM can decide to purchase which fight er. There are several compensatory MADM methods called “Simple Additive Weight (SAW)”, “TOPSIS”, “ELECTRE”, and “AHP” method. Good Fighter Mechanical Performance Handling Quality Serviceability Economic Merit Top Speed Operating Altitude Maximum Payload Ferry Range Maneuver ability Survivabi lity Reliabili ty Maintaina bility Purchasing Cost Operating Cost

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17 In the SAW method, each alternative’s ra nking score is obtained by adding all attributes’ scores. Formally the value of an a lternative in the SAW method can be expressed as m j ij j j in i x v w A V1, 2 1 ), ( ) ( (2.5) where ) (iA V is the value function of alternative iA, and jwis j th attribute weights, and ) (ij jx v is the value of response of alternative ion attribute j Through the normalization process, each incommensurable attribute becomes a pseudo-value function which allows direct addition among attributes. Th e value of alternative iA can be rewritten as m j ij j in i r w A V1, 2 1 ) ( (2.6) where ijr is the comparable scale of ijx, which can be obtained by Equation 2.3. The underlying assumption of th e SAW method is that attr ibutes are preferentially independent. Less formally, this means that the contribution of an indi vidual attribute to the total score is independent of other attribute va lues. Therefore, DM’s preference regarding the value of one attribute is not influenced in any way by the va lues of the other attributes (Fishburn, 1976). Fortunately, studies (Edw ards, 1977; Farmer, 1987) show that the SAW method yields extremely clos e approximations to “true” va lue functions even when independence among attributes does not exactly hold. In addition to the preference independe nce assumption, the SAW has a required characteristic for weights. Th at is, the SAW presumes that weights are proportional to the

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18 relative value of a unit change in each attribute’s value f unction (Hobbs, 1980). For instance, let us consider a value f unction with two attributes:2 2 1 1v w v w V By setting the amount of V constant, we can deri ve the relationship of 1 2 2 1/ / v v w w This relationship indicates that if 1w=0.33 and 2w=0.66, the DM must be indifferent to the trade between 2 units of 1vand 1 unit of 2v. This is the same as utility function’s ma rginal utility (MU) and marginal rate of substitution (MRS) (S her and Pinola, 1981). Let us apply this ASW into the example probl em in the previous figure 2.2. Recall that there are ten attributes and five alternatives. Four attributes (i.e., mechanical performance, handling quality, serviceability, and economic merit) are under ove rall goal (i.e., selection of good fighter) and they have four or two sub-attr ibutes (i.e., top speed, operating altitude, maximum payload, ferry range, maneuverability, survivability, reliab ility, maintainability, purchasing cost, and operating cost). Henceforth first level of attribute is represented as iX standing first level of ith attribute. From the second level of attribute, the nu mber of subscript ciphers represents the level of its attribute. For example, top speed which is the first subattribute is represented as 11X and if there is sub-sub-attrib utes than it can be written as jX11 and means j th attribute under top speed. There are five alternatives and represented as4 3 2 1, ,A A A A, and5A. Henceforth alternative is represented as iA that means ith alternative. Table 2.1 in the following page shows data for evaluation of these fighters.

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19 Table 2.1 Data for Evaluation of Fi ghters (Source: Yoon and Hwang, 1995) Alternatives Attribute Weight A1 A2 A3 A4 A5 1. Mechanical performance 1.1 Top speed (Mach) 0.20 2.0 2.0 2.5 2.0 1.8 1.2 Operating altitude (1,000 ft) 0.04 60 50 60 50 50 1.3 Maximum payload (1,000 lbs) 0.04 23 20 18 20 21 1.4 Ferry range (NM) 0.12 1,900 2,000 3,500 2,400 2,300 2. Handling quality 2.1 Maneuverability(*) 0.09 7 8 8 9 9 2.2 Survivability (*) 0.21 8 9 7 8 8 3. Serviceability 3.1 Reliability (*) 0.12 8 7 9 8 8 3.2 Maintainability (*) 0.08 9 7 8 7 7 4. Economic merit 4.1 Purchasing cost ($M/ea) 0.06 4.5 5.0 6.5 5.5 5.0 4.2 Operating cost ($1,000/year) 0.04 90 90 100 80 70 Note that weights are assumed given in this table and units are from a 10-point scale, from 1 (worst) to 10 (best). The normalized decision ma trix from the above data by Equation 2.3 is given as 00 1 90 0 91 0 83 0 72 0 88 0 82 0 87 0 83 0 80 0 70 0 69 0 78 0 00 1 00 1 78 0 90 0 87 0 83 0 80 0 78 0 00 1 00 1 00 1 80 0 X X X X X 5 4 3 2 1 42 41 13 12 11 A A A A A where columns of 31 22 21 14, ,X X X X, and 32X are not shown here. 41X and 42X are cost attributes. Therefore, their normalized values of ijr are computed as

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20 10 , 2 1 ; 5 4 3 2 1 1 1 j i x x rj ij ij, (2.7) where jx is the smallest value of ijx Other values of ijr are calculated by Equation 2.3 and detailed calculation processes ar e not present here. For the pur pose of graphical view, weights of each attribute are shown in the following figure. Figure 2.3 Weight Assessmen t for Fighter Evaluation The value of alternative A1 is then computed by SAW as below. 10 1 1) (j ij jr w A V, = 0.2(.8) + 0.04(1.0) + . + 0.04(.78) = 0.8396. w(1.0) w1 (0.4) w2 (0.3) w3 (0.2) w4 (0.1) w11 ( 0.5 ) w12 ( 0.1 ) w13 ( 0.1 ) w14 ( 0.3 ) w21 ( 0.3 ) w22 ( 0.7 ) w31 ( 0.6 ) w32 ( 0.4 ) w41 ( 0.6 ) w42 ( 0.4 ) 0.20 0.04 0.04 0.12 0.09 0.21 0.12 0.08 0. 60 0.04

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21 The other alternatives have values of ) (2A V=0.8274, ) (3A V =0.8953, ) (4A V=0.8400, and ) (5A V =0.8323. The preference order is 2 5 1 4 3, , A A A A A where 3A is the first rank and 2A is the last. Even though this method has eas y computational mer it, it has somewhat of a weakness, that it may not consider extreme data: some of alternatives may have higher overall values because of some extreme high values in some attri butes but low scores in the other attributes. It is the same problem that we have when we use mean value itself in statistics. In addition to this problem, to be able to use the ASW met hod we should have weight information. In the SAW method, addition among attribut e values was allowed only after the different measurement units were transformed into a dimensionless scale by a normalization process. However, this transformation is no t necessary if attributes are connected by multiplication (see Brauers (2001) for more detailed example). When we use multiplication among attribute values, the weights become expone nts associated with each attribute value: a positive power for benefit attributes, and a negative power for cost attributes. Formally, the value of alternative iA is given by , 2 1 ) (1n i x A Vm j w ij ij (2.8) Because of the exponent property, this method re quires that all ratings be greater than 1. For instance, when an attribute has fractional ratings, all ratings in that attribute are multiplied by m10 to meet this requirement. Since there is no fractional number in our example in Figure 2.2, we can plug all the ijx values in the previous table 2.1 into Equation 2.15 and values of

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22 alternatives become such as:1716 8 ) (1 A V,0707 8 ) (2 A V, 7689 8 ) (3 A V ,2565 8 ) (4 A V, and 1485 8 ) (5 A V The final rank order is 2 5 1 4 3, , A A A A A and this rank is the same as previous SAW method. Saaty and Vargas (2000) show that multip licative and additive syntheses are related analytically through the a pproximation as below: ij i w ij w ij w ijx w x x xi i ilog exp log exp log exp ij i i ij i ij ix w w x w x w1 log 1. (2.9) Therefore, we can say that there is no difference between SAW method and weight product method from a point of final rank order. A MADM problem with m alternatives that are evaluated by n attributes may be viewed as a geometric system with m point s in the n-dimensional space. Hwang and Yoon (1981) develop the Technique for Order Preferen ce by Similarity to Id eal Solution (TOPSIS) based on the concept that the chosen alternativ e should have the shortest distance from the positive ideal solution and the longest distance from the negative-ideal solution. This principle is also suggested by Zeleny (1982) and Hall (1989). Loerch et al (1998) apply this method to their research. Recently this method is enriched by Y oon (1987) and Hwang et al. (1993). TOPSIS starts from the concept of an ideal solution. An ideal solu tion is defined as a collection of ideal levels (or ratings) in all attr ibutes considered. However, the ideal solution is usually unattainable or infeasible. Then to be as close as possible to such and ideal solution is the rational of human choice (Yoon and Hwa ng, 1995). Coombs (1958, 1964) also claimed that there is an ideal level of attributes for a lternatives of choice and that the DM’s utilities

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23 decrease monotonically when an alternative moves away from this ideal (or utopia) point (Yu, 1985). Formally the positive-id eal solution is denoted as * 1 *, , ,n jx x x A (2.10) where *jx is the best value for the jth attribute among all availa ble alternatives. While the negative-ideal solution is composed of all worst attribute ratings attainable. Figure 2.4 Euclidean Distances to Positive-Ideal and Negative-Ideal Solutions in TwoDimensional Space (Yoon and Hwang, 1981) The negative-ideal solution is given as

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24 n jx x x A, , ,1 (2.11) where jx is the worst value for the jth attribute among all available alternatives. Figure 2.4 graphically shows the two ideal solutions. For example, consider two alternatives1A and 2A with respect to two benefit attributes in Figure 2.4. 1A is the closest to *A but 2A is the farthest from A TOPSIS defines an index called similarity (or relative closeness) to the positive-ideal solution by combining the proximity to the posi tive-ideal solution and the remoteness from the negative-ideal solution. Then the method chooses an alterna tive with the maximum similarity to the positive-ideal solution. TOPSIS assumes th at each attribute takes either monotonically increasing or monotonically decrea sing utility. That is, the larg er the attribute outcome, the greater the preference for benefi t attributes and the less the pref erence for cost attributes. The method is presented as a series of successive steps: Step 1. Calculate normalized ratings. We use th e ideal normalization is used for computing ijr which is given as Equation 2.3. Step 2. Calculate weighted normalized ratings. Th e weighted normalized value is calculated as m j n i r w vij j ij, 1 ; , 1 , (2.12) where jw is the weight of the jth attribute. Step 3. Identify positive-ideal and negative-ideal solutions.

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25 n i J j v J j v v v v v Aij i ij i n j, 1 min max , ,2 1 * 2 1 (2.13) n i J j v J j v v v v v Aij i ij i n j, 1 max min , ,2 1 2 1 (2.14) where 1J is a set of benefit attributes and 2J is a set of cost attributes. Step 4. Calculate separation meas ures. The separation (distance) between alternatives can be measured by the n-dimensional Euclidean distance The separation of each alternative from the positive-ideal solution, *A is then given by , 1 ,1 2 *n i v v Sn j j ij i (2.15) Similarly, the separation from the negative-ideal solution, A is given by , 1 ,1 2n i v v Sn j j ij i (2.16) Step 5. Calculate similarities to positive-ideal solution. n i S S S A Vi i i i, 1 / ) (* (2.17)

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26 Note that 1 ) ( 0 iA V where 0 ) ( iA V when A Ai, and 1 ) (iA V when *A Ai. Step 6. Rank preference order. Choos e an alternative with the maximum *iC in descending order. Let us solve the previous exampl e problem given in the Figure 2.2. Step 1. Normalization We use normalized decision matrix under tabl e 2.2 and show all the ratings that were abbreviated. The decision matrix is shown as: 00 1 90 0 0.778 889 0 88 0 82 0 0.778 889 0 70 0 69 0 0.889 1 78 0 90 0 0.778 778 0 78 0 00 1 1 889 0 889 0 1 657 0 91 0 83 0 72 0 889 0 1 686 0 87 0 83 0 80 0 778 0 889. 0 1 78 0 00 1 00 1 1 889 0 571 0 87 0 83 0 80 0 889 0 778 0 543 0 00 1 00 1 80 0 X X X X X 5 4 3 2 1 42 41 32 31 22 21 14 13 12 11A A A A A X X X X X Step 2. Weighted Normaliza tion. The weights of (0.2, 0.04,0.04) from the table 2.2 are multiplied with each column of the normalized rating matrix: 04 0 054 0 0.062 107 0 035 0 049 0 0.062 107 0 028 0 041 0 0.071 12 0 031 0 054 0 0.062 093 0 031 0 06 0 0.08 107 0 187 0 09 0 079 0 036 0 033 0 144 0 187 0 09 0 082 0 035 0 033 0 16 0163 0 080 0 12 0 031 0 04 0 2 0 21 0 080 0 069 0 035 0 033 0 16 0 187 0 070 0 065 0 04 0 04 0 16 0 X X X X X 5 4 3 2 1 42 41 32 31 22 21 14 13 12 11A A A A A X X X X X Note: Since we already normalized by using Equati on 2.3, there is no cost attributes in the above matrix. Therefore, we only need to consider be nefit attributes.

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27 Step 3. Positive ideal solution and negative ideal solutions are *A = (0.2, 0.04, 0.04, 0.12, 0.09, 0.21, 0.12, 0.08, 0.06, 0.04) A = (0.144, 0.033, 0.031, 0.065, 0.07, 0.163, 0.093, 0.062, 0.041, 0.028) Step 4. The separation measures from *A are computed first: 10 1 2 1 *1j j j Av v S = 2 / 1 2 204 0 031 0 2 0 16 00.075 Separation measures from *A of all alternatives are * * *5 4 3 2 1, , ,A A A A AS S S S S = (0.076, 0.074, 0.055, 0.064, 0.077) The separation measures from A are computed as 10 1 2 11j j j Av v S = 2 / 1 2 2028 0 031 0 144 0 16 00.043 All separation measures from are

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28 5 4 3 2 1, , ,A A A A AS S S S S = (0.043, 0.053, 0.084, 0.043, 0.041) Step 5. Similarities to positive ideal solution are computed as ) /( ) (1 1 1* 1 A A AS S S A V = 0.043/(0.076+0.043) = 0.361 All similarities to the positive ideal solution are ) ( ), ( ), ( ), ( ), (5 4 3 2 1A V A V A V A V A V = (0.361, 0.414, 0.607, 0.396, 0.349). Step 6. Preference rank. Based on the descending order of ) (iA V the preference order is given as 5 1 4 2 3, , A A A A A which selects alternative 3 fighter to purchase. Three se ts of preference rankings are shown at the following table. Table 2.2 Three Sets of Preference Rankings *S S ) (iA V Fighter Value Rank Value Rank Value Rank A1 0.076 4 0.043 3.5 0.361 4 A2 0.074 3 0.053 2 0.414 2 A3 0.055 1 0.084 1 0.607 1 A4 0.064 2 0.043 3.5 0.396 3 A5 0.077 5 0.041 5 0.349 5 The idea of this method can be summarized as follows: an alternativ e has to be close to the positive ideal solution and this represents a pr eference of the shorter di stance; an alternative

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29 has to be located far away fr om the negative ideal solution wh ich prefers longer distance; the final rank is based on the compen sation of these two distances. We observed that this idea is exactly the same as variance information in the statistics. The detailed discussions are presented in Chapter 3. Like problems found from the ASW method, this TOPSIS method also requires the weight information. Moreover, this method has weakness that it only considers distance from the ideal solutions and hence does not consider the alternative’s overall values. However, DMs may want to see an individual alternative’s overall scores in addition to the distance information. The ELECTRE (Elimination et choix traduisant la ralit) method is originated from Roy (1971) in the late 1960s. Since then Nijkamp and van De lft (1977) and Voogd (1983) have developed this method to its pres ent state. The method dichotomi zes preferred alternatives and nonpreferred ones by establishing outranking rela tionships. This method is most popular in Europe, especially among the French-speaking community. When a DM feels that A is better than B, then it is defined that A outranks B and the notation is (A R B) or (A B). For the utility theory, the SAW method, and the AHP method, the transitive assumption (i.e., if A is better th an B and B is better than C, then A must be better than C) is important. However, this ELECTRE method does not need this transitive assumption. Therefore, the relation of (A R B) and (B R C) do not nece ssarily imply (A R C). The outranking relationships are determined by concordance and discordance indexes (Yoon and Hwang, 1995). Figure 2.5 shows one example of a relations hip of preferred al ternatives in the ELECTRE method.

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30 Figure 2.5 A Digraph for Eight Alternatives (Yoon and Hwang, 1995) Nine outranking relationships fr om this figure are given as: (A1 A2), (A2 A3), (A3 A8), (A4 A2), (A5 A4), (A5 A7), (A6 A3), (A7 A4), and (A8 A6). When a directed path begins in a node and comes back to this very node, this path is called as a cycle. All nodes in a cycle are considered to have an equi valent preference. In the above figure, A3 A8 A6 A3 is a cycle. The kernel (or core) of an acyclic digraph is a reduced set of nodes that is preferred to the set of nodes that do not be long to the kernel. Kernel (K) is defined as a set of preferred alternatives by ELECTRE. The K should satisfy the following two conditions: 1. Each node in K is not outranked by any other node in K. 2. Every node not in K is outranke d by at least one node in K. Figure 2.6 shows the Kernel of this example. A3 A6 A1 A2 A4 A5 A7 A8

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31 Figure 2.6 The Kernel of Figure 2.5 The set of preferred alternativ es defined by the kernel is K= 5 2 1, ,A A A. The ELECTRE method formulates concordance and discordance indexes in order to obtain outranking relationships, and renders a set of preferred alte rnatives by forming a kernel. Concordance and discordance i ndexes can be viewed as meas urements of satisfaction and dissatisfaction that a DM feels on choosi ng one alternative over the other. Let us imply this ELECTRE method into the same problem given in the Figure 2.2 and follow each step given by Yoon and Hwang (1995). For the convenience, we rewrite the weighted normalization matrix such as below. One thing that we change is the subs cripts of attributes and this is done for a notational convenience (i.e., iXis used instead of ijX). K A3 A6 A8 A7 A4 A1 A2 A5

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32 04 0 054 0 0.062 107 0 035 0 049 0 0.062 107 0 028 0 041 0 0.071 12 0 031 0 054 0 0.062 093 0 031 0 06 0 0.08 107 0 187 0 09 0 079 0 036 0 033 0 144 0 187 0 09 0 082 0 035 0 033 0 16 0163 0 080 0 12 0 031 0 04 0 2 0 21 0 080 0 069 0 035 0 033 0 16 0 187 0 070 0 065 0 04 0 04 0 16 0 X X X X X 5 4 3 2 1 10 9 8 7 6 5 4 3 2 1A A A A A X X X X X Step 1. Construct concordance and discordan ce sets. For each pair of alternatives pA andqA ) 5 , 1 ( q p the set of attributes is divided in to these two distinct subsets. The concordance set, which is composed of all attributes for which alternative pA is preferred to alternativeqA can be written as qj pjv v j q p C ) ( (2.18) where pjv is the weighted normalized rating of alternative pA with respect to the j th attribute. In other words, ) ( q p C is the collection of attributes where pA is better than or equal toqA The complement of) ( q p C which is called the discordance set, contains all attributes for which pA is worse thanqA This can be written as qj pjv v j q p D ) (. (2.19) Note that ) ( q p C is not equal to ) ( q p D when tied ratings exist. The concordance and discordance sets are obtained as

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33C(1,2) = {1, 2, 3, 7, 8, 9, 10} D(1,2) = {4, 5, 6} C(1,3) = {2, 3, 6, 8, 9, 10} D(1,3) = {1, 4, 5, 7} C(1,4) = {1, 2, 3, 6, 7, 8, 9} D(1,4) = {4, 5, 10} C(1,5) = {1, 2, 3, 6, 7, 8, 9} D(1,5) = {4, 5, 10} C(2,1) = {1, 4, 5, 6, 10} D(2,1) = {2, 3, 7, 8 ,9} C(2,3) = {3, 5, 6, 9, 10} D(2,3) = {1, 2, 4, 7, 8} C(2,4) = {1, 2, 3, 6, 8, 9} D(2,4) = {4, 5, 7, 10} C(2,5) = {1, 2, 6, 8, 9} D(2,5) = {3, 4, 5, 7, 10} C(3,1) = {1, 2, 4, 5, 7} D(3,1) = {3, 6, 8, 9, 10} C(3,2) = {1, 2, 4, 5, 7, 8} D(3,2) = {3, 6, 9, 10} C(3,4) = {1, 2, 4, 7, 8} D(3,4) = {3, 5, 6, 9, 10} C(3,5) = {1, 2, 4, 7, 8} D(3,5) = {3, 5, 6, 9, 10} C(4,1) = {1, 4, 5, 6, 7, 10} D(4,1) = {2, 3, 8, 9} C(4,2) = {1, 2, 3, 4, 5, 7, 8, 10} D(4,2) = {6, 9} C(4,3) = {3, 5, 6, 9, 10} D(4,3) = {1, 2, 4, 7, 8} C(4,5) = {1, 2, 4, 5, 6, 7, 8} D(4,5) = {3, 9, 10} C(5,1) = {4, 5, 6, 7, 10} D(5,1) = {1, 2, 3, 8, 9} C(5,2) = {2, 3, 4, 5, 7, 8, 9,10} D(5,2) = {1, 6} C(5,3) = {3, 5, 6, 9, 10} D(5,3) = {1, 2, 4, 7, 8} C(5,4) = {2, 3, 5, 6, 7, 8, 9, 10} D(5,4) = {1, 4} Step 2. Compute concordance and Discorda nce Indexes. The relative power of each concordance set is measured by means of th e concordance index. The concordance index pqC represents the degree of confiden ce in the pairwise judgments of) (q pA A The concordance index of ) ( q p Cis defined as *j j pqw C (2.20) where *j are attributes contained in the concordance set ) ( q p C The concordance indexes of this example are

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34 12C= 0.58 13C =0.47 14C=0.75 15C =0.75 21C=0.66 23C =0.44 24C=0.63 25C =0.59 31C =0.57 32C =0.65 34C =0.56 35C =0.56 41C=0.78 42C=0.73 43C =0.44 45C =0.86 51C =0.58 52C =0.59 53C =0.44 54C =0.68 where 12C= 0.58 was obtained from 10 9 8 7 3 2 1 12* *w w w w w w w w Cj j = 0.2+0.04+0.04+0.12+0.08+0.06+0.04 = 0.58. The discordance index, on the othe r hand, measures the powers of ) ( q p D The discordance index of) ( q p D, which represents the degree of disagreement in) (q pA A can be defined as j qj pj j qj pj pqv v v v D (2.21) where j are attributes that are contai ned in the discordance set ) ( q p D The discordance indexes of this example are

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35 12D=0.425 13D =0.648 14D=0.500 15D =0.457 21D=0.575 23D =0.667 24D=0.594 25D =0.530 31D =0.352 32D =0.333 34D =0.331 35D =0.337 41D=0.500 42D=0.406 43D =0.669 45D =0.367 51D =0.543 52D =0.470 53D =0.663 54D =0.633 where 12D= 0.58 was obtained from j qj pj j qj pjv v v v D 12 10 1 2 1 26 16 25 15 24 14 j j jv v v v v v v v 087 0 21 0 187 0 08 0 07 0 069 0 065 0 4253 0 087 0 037 0 Step 3. Outranking Relationships. The dominance relationship of alternative pA over alternative qAbecomes stronger with a higher concordance index pqCand a lower discordance indexpqD. The method defines that pAoutranks qAwhen C Cpqand D Dpq, where Cand Dare the averages of pqCand pqD, respectively. For this problem,

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36 62 0 20 68 0 58 0 C and 50 0 20 633 0 425 0 D Table 2.3 Determination of Outranking Relationship pqC Is ) (C Cpq? C=0.62 pqD Is ) (D Dpq? D=0.50 Is ) (q pA A ? 12C=0.58 No 12D=0.43 Yes No 13C=0.47 No 13D=0.65 No No 14C=0.75 Yes 14D=0.50 No No 15C=0.75 Yes 15D=0.46 Yes Yes 21C=0.66 Yes 21D=0.57 No No 23C=0.44 No 23D=0.67 No No 24C=0.63 Yes 24D=0.59 No No 25C=0.59 No 25D=0.53 No No 31C=0.57 No 31D=0.35 Yes No 32C=0.65 Yes 32D=0.33 Yes Yes 34C=0.56 No 34D=0.33 Yes No 35C=0.56 No 35D=0.34 Yes No 41C=0.78 Yes 41D=0.50 No No 42C=0.73 Yes 42D=0.41 Yes Yes 43C=0.44 No 43D=0.67 No No 45C=0.86 Yes 45D=0.37 Yes Yes 51C=0.58 No 51D=0.54 No No 52C=0.59 No 52D=0.47 Yes No 53C=0.44 No 53D=0.66 No No 54C=0.68 Yes 54D=0.63 No No Table 2.3 illustrates the determination of outranking relations hips. Four outranking relationships are obtained: (5 1A A ), (2 3A A ), (2 4A A ), and (5 4A A ). The Kernel of this problem is shown in Figure 2.7.

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37 Figure 2.7 The Kernel of the Example Problem Yoon and Hwang (1995) state that a weakness of ELECTRE might lie in its use of the critical threshold values of C and D These values are rather arb itrary, although their impact upon the ultimate result may be significant. They also notice that there is no rank order for alternatives that are inside of K. They introduce the net outranking relationship into the ELECTRE method to address these problems. By su ing this relationship they can transform the current ELECTRE’s ordinal rank into cardinal ra nk, and hence DMs can see the preference between alternatives among in the K. Complementary ELECTRE defines the net concordance index pC which measures the degree to which the domi nance of alternative pA over competing alternatives exceeds the dominance of competing alternatives over pA Similarly, the net discordance index pD K A5 A2 A1 A3 A4

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38 measures the relative w eakness of alternative pA with respect to other alternatives. These net indexes are mathematically denoted as m p k k kp m p k k pk pC C C1 1, and (2.22) m p k k kp m p k k pk pD D D1 1. (2.23) Obviously, an alternative pA has a greater preference with a higher pC and a lower pD Hence the final selection should satisfy th e condition that its ne t concordance index should be at a maximum and its net discordance index at a minimum. If both these conditions are not satisfied, the altern ative that scores the highest average rank can be selected as the final solution. For our example problem, the net concordance and discorda nce indexes are shown in Table 2.4. Table 2.4 The Net Concordance and Discordan ce Indexes of the Alternatives in the K Net concordance index value rank Net discordance index value rank Final rank 1C -0.04 3 1D 0.06 3 3 3C 0.55 1 3D -1.29 1 1 4C 0.19 2 4D -0.12 2 2

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39 Based on this complementary method, we can present alternatives ranking order which are in the Kernel. However, this complement ary method causes that the Kernel of ELECTRE method is not required: we can present all th e alternatives ranking order followed by this complementary method, and hence there is no need for us to define the Kernel. Moreover, both the ELECTRE and the complementary ELECTRE me thod do not consider an overall ranking score of each alternative. 2.2.4 Analytic Hierarchy Process (AHP) Previous Methods assume that attributes’ we ights are already given. Therefore, there was no need to assign weights for each attribute. However, most of real life decision problems are different from this assumption. Therefore, we need to assign each at tributes’ weight with one of following three methods: the AHP method, the Delphi technique, and the Utility theory. These three methods are also used for a problem not only with quantitative data, but also with qualitative data. In 1980, Saaty presented the AHP method. Th is method is widely used for many different areas such as political, economic, so ciology, and even in medical areas because of these superiorities: 1) This method can handl e both quantitative and qual itative data at the same time; 2) This method uses the eigenvect or and eigenvalue property and this property presents a computational merit; 3) Saaty already proved the advantages of this method with many case studies; and 4) This method gives le ss cognitive burden to DM s compared to the other two methods. The AHP method has two important theoreti cal backgrounds: the fundamental scale, and eigenvector and eigenvalue property. The Saat y’s fundamental 1-9 scal e has its origin on the Weber-Fechner’s sensation (response) equation (i.e., 0 log a b s a M, where M

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40 denotes the sensation and s the stimulus) (Fechner, 1966). While the noticeable ratio stimulus increases geometrically, the response to that stimulus increases arithmetically. In making pairwise comparisons, nearest integer approximation from the fundamental scales are used. This scale has been validated for effectiveness, not only in many appli cations by a number of people, but also through theoreti cal justification of what scale one must use in the comparison of homogeneous elements (Saaty and Vargas, 2 000). The upper limit of 9 is defined following Miller (1956)’s “Magic al number theory”. Alternatives are compared by DMs with resp ect to those fundamental scales of Table 2.5. For example, if a DM decides that alternative iA is strongly important than alternativejA then he or she assigns the value of 5 into the corresponding cell of a decision matrix. From these pairwise comparisons, a decision matrix is composed. This decision matrix is defined as A, or A = ) (ija where ija denotes the number which indicates the preference strength of an alternative iA over an alternative jA By the reciprocal prop erty of the AHP (i.e., ji ija a/ 1 ), the matrix A has the form 1 / 1 / 1 1 / 1 12 1 2 12 1 12 n n n na a a a a a A If a DM’s judgment is consistent over all th e comparisons, then there is a transitivity of the preferred relationship such as jk ij ika a a These two properties (i.e., reciprocal and transitivity) are important assu mptions in this AHP method.

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41 Table 2.5 The Fundamental Sc ales (Saaty and Vargas, 2000) Intensity of importance Definition Explanation 1 Equal Importance Two activities contribute equally to the objective 2 Weak 3 Moderate importance Experience and judgment slightly favor one activity over another 4 Moderate plus 5 Strong importance Experience and judgment strongly favor one activity over another 6 Strong plus 7 Very strong or demonstrated importance An activity is favored very strongly over another; its dominance demonstrated in practice 8 Very, very strong 9 Extreme importance The evidence favori ng one activity over another is of the highest possible order of affirmation An obvious case of the consistent matrix is one in which the comparisons are based on exact measurements: the weights nw w, ,1 are already known, and hence n j i w w aj i ij, 1 (2.24) And thus ik k j j i jk ija w w w w a a

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42 Also, of course, ij j i i j jia w w w w a 1 / 1 Note that the notation iw used in above development is diffe rent from previous example: in previous example, iw denotes the relative importance of each attribute; while, iw used in equation 2.24 denotes the absolute importance of an alternativeiA Let us consider this paradigm case further. The matrix e quation of the homogeneous equation of n i y x ai n j i ij, 1 1 (2.25) is denoted by y x A where nx x x ,1 and ny y y ,1 From the equation 2.24, we obtain n j i w w ai j ij, 1 1 and consequently n j i j ijn i n w w a1, 1 1 or

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43 n j i j ijn i nw w a1, 1 which is equivalent to nw Aw (2.26) There is an infinite number of ways to derive the vector of priorities from that matrix. But emphasis on consistency leads to the eigenvalue formulationnw Aw such as n n n n n n n nw w w n w w w w w w w w w w w w w w w w w w w w w 2 1 2 1 2 1 2 2 2 1 2 1 2 1 1 1 It is known that matrix A=) (ija is said to be consistent if and only if its principal eigenvalue is equal to n. The sum of the eigenvalues of a matrix is equal to its trace (i.e., the sum of its diagonal elements). In this case the trace of A is equal to n. However, since a DM is a human, he or she cannot give the precise values of j iw w /, but only an estimate. Therefore, Saaty replaces max for the n, and nw Aw becomes w Awmax (2.27)

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44 where max is the largest or principal eigenvalue of matrix A. Saaty defines the difference between max and n as a Consistency Index (CI). CI is calculated as 1max n n CI. (2.28) Saaty and his colleagues generated an averag e random index (RI) for matrices of order 1-15 using a sample size of 100 at Oak Ridge National Laboratory developed average random consistency index (for more deta il, see Saaty,1980). RI increases as the order of the matrix increases and is shown in the following table. Table 2.6 Average Random Consistency Index (RI) n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 RI 0 0 .58 .90 1.12 1.24 1.32 1.41 1.45 1.49 1.49 1.51 1.56 1.57 1.59 Consistency Ratio (CR) is used to check the consistency of comparisons and is computed by CR = CI / RI. (2.29) The less value of CR represents the more consis tent. Saaty suggests that we have a consistency if CR is less than 0.1. If CR is more than 0.1, that comparison is considered as inconsistent and should be excluded to calculate weight because th at DM is considered to have no rationality.

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45 A pairwise comparison is simple and conve nient for both DM and analyst. Moreover, any qualitative data can be easily handled. Howe ver, because pairwise comparison is done by human being, there can be any inconsistency or irrational response. AHP method uses this powerful pairwise comparison and solves any po ssible human’s irrational responses by using CR. To show the procedure of a hierarchical co mposition of priorities, let us imply example problem in the Figure 2.2 with the assump tion that no weights are assigned yet. Step 1. Proceed with pairwise judgments for the first level attribute. Each questionnaire given to DMs is designed to compare two attributes at a time under the consideration that DMs need to achieve a goal and they need to decide which attribute or alternative is more important and how much. For example, a DM may compare mech anical performance a nd handling quality in terms of achieving a good fighter, and between m echanical performance and serviceability, and so on. Let us assume that we did pairwise compar isons then the pairwise judgment matrix is as in Table 2.7. Table 2.7 Comparison of First Level of Attributes with Respect to Satisfying a Good Fighter Mechanical performance Handling quality Serviceability Economic Merit Mechanical performance 1 4 6 7 Handling quality 1/4 1 3 4 Serviceability 1/6 1/3 1 2 Economic Merit 1/7 1/4 1/2 1 From Table 2.7, the attribute’s weig hts are obtained as in Table 2.8.

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46 Table 2.8 Attribute Weights Mechanical performance Handling quality Serviceability Economic Merit weight 0.62 0.22 0.10 0.06 max = 4.1, C.R. = 0.03 Since C.R. < 0.1, we can consider this comparison is consistent and hence acceptable for being used to present weight information. Step 2. Precede the pairwise comparison for the second level of attributes in terms of the first level of attributes. If there is some lower level of attributes, we need do the same step until we reach the very bottom level of attribute. Th e process is same and not present here. Step 3. Develop a decision matrix based on these pairwise comparisons and compute each alternative’s relative importance with respect to e ach attribute. Since the alternative’s value can be obtained from the lower level of attributes we only need to compute based on the lowest level of attributes. In our example, we have ten lowest attributes. Therefore, we need to develop ten individual pa irwise judgment matrices. Table 2. 9 shows one of these ten matrices. Table 2.9 Example of Pairwise Comparison Matrix with Respect to Each Attribute With respect to maintainability A1 A2 A3 A4 A5 Eigenvector A1 1 1/3 1/2 1/2 7 0.140 A2 3 1 1 2 9 0.343 A3 2 1 1 1 7 0.259 A4 2 1/2 1 1 7 0.226 A5 1/7 1/9 1/7 1/7 1 0.031 max =5.11, C.I. = 0.03, C.R. = 0.03

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47 If the information is quantitative such as each fighter’s top speed, there is no need for pairwise comparison. Instead, the normalization procedures are used (see Section 2.2.1 and 2.2.2). Step 4. Compute alternatives’ cardinal rank scor es by synthesizing all attributes’ values. For the synthesis, we can use additive or multiplicat ive function. However, there is no difference between these two methods (see Equation 2.9). Since a different weight is used for our example in compare to other methods, we do not present the alternative rank scores and rank order. 2.2.5 Utility Theory for Decision Making Utility theory has its origin on consumer be havior in Microeconomics. The behavior of a consumer in the market (i.e., what choice a consumer makes) is in fluenced by numerous factors, including individual preference and purch asing power such as b udget availability. The relationship between the amount of commodities a nd/or services that an individual consumes and the satisfaction called utility derived from th em can be likened to the relationship between inputs and output in production (Sher and Pinol a, 1981). The alternative value function in the utility theory is given by j ij j ix U w A V ) ( ) (, (2.30) where ) (ijx U is a utility value for an ijx 1 ) ( 0 ijx U 0 jw and 1jw. Suppose that we need to buy a car from the market and we decide a car in terms of price (in thousands of dollars), comfort and fu el consumption (in miles per gallon), these are

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48 called attributes. Let us assume that we have ten alternatives, which means ten car types are available to buy from the market, and three at tributes namely price, comfort, and fuel consumption. The detailed data are given as Table 2.10. Table 2.10 Decision Matrix for Ten Cars (Source: Pomerol et al. 2000) Price (K$) 1X Comfort 2X Miles per gallon 3X 1A 70 10 14 2A 60 9 20 3A 50 9 18 4A 45 8 22 5A 40 9 16 6A 40 7 14 7A 30 6 20 8A 30 7 22 9A 20 5 24 10A 20 4 26 Note thatiA is the car type i in the market and a comfort is quantified based on a 10point scale, from 1 (worst) to 10 (best). Utility theorists assume that there is indifference curve and any two points on this curve ar e indifferent in terms of DM’s satisfaction. This indifferent relationship is written such as) , ( ) , (k k k i i iz y x z y x where the parenthesis represents a set of consumption and elements in the parenthesis are value from2 1,X X, and 3X respectably. Because of the dimensional restriction for draw ing, we only present two-dimensional graphs with two values from 1X and 2X in the following figure. This indifference relationship is the core of the utility theory because DMs can de termine utility function as well as attribute weights based on this assumption.

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49 Figure 2.8 Graphic Representation of Indifference Curve We assume that DMs in the SKN decide attr ibute weights and alternative values with certainty. In other words, they do not answer such as “I pr efer twice A than B with 90% confidence”. Therefore, our problem is deterministic and this figure is for a deterministic problem case. However, if an uncertainty of pref erences exists, we need to add a probability to Equation 2.30 (for more detail, see Keeney and Raiffa, 1976). Like other MADM methods, ut ility theory depends on expe rts’ opinion for determining the attribute weight and alte rnative utility. Pomerol and Ro mero (2000) show the decision procedure for determining the attr ibute weights and the value of ) (ij jx U given in the previous car selection problem. This procedure is as follows: Although we are in the discrete case, we can fictitiously consider that we are reasoning under ] 26 14 [ ] 10 4 [ ] 70 20 [ X such as the topological assump tion. This is equivalent to 2X 1X Indifference Curve ix jx iy jy ) (i iy x ) (j jy x

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50 saying that any triplet of X represents a possible choice. We proceed to the dialogue stage. One protagonist is the analys t (A) and the other the DM. We start off with 0 ) 70 (1 U and 1 ) 20 (2 U, because 70 is the most not preferred value under attribute 1X while 20 is best on that attribute. A. What gas per mileage can you assign for this question mark in (45,8, ?) to make yourself feel indifferent to (20,7,20) in terms of satisfaction? This question is simplified as “What gas per mileage can you give to ma ke you indifferent between (20,7,20) and (45,8,?)”. DM. I would be indifferent be tween (20,7,20) and (45,8,26). A. Where does the price have to lie so that (20,7,20) (?,8,26) and (?,7,20) (70,8,26)? DM. 50. A. Where does the price lie such that (20,7,20) (?,8.5,26) and (?,7,20) (50,8.5,26)? DM. 40. A. Where does the price lie such that (50,7,20) (?,8.5,26) and (?,7,20) (70,8.5,26)? DM. 60. With more intermediate points from this c onversation, we can construct the cure of 1U 2U and 3U (see Figure 2.8). We can note that 1 3 3 2 2 1 1) 14 ( ) 4 ( ) 20 ( ) 14 4 20 ( w U w U w U w V and likewise 2) 14 10 70 (w V and 3) 26 4 70 ( w V To determine ) 14 4 20 ( V we can ask DM what value of comfort (70,?,14) is indiffere nt to (20,4,14). If the respons e is 9, this wi ll lead to the equation: ) 9 (2 2 1U w w or 2 19 0w w since 9 0 ) 9 (2 Ufrom the Figure 2.8.

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51 Figure 2.9 Utility Functions for Car Selection Example (Source: Pomerol and Romero, 2000) Similarly, we ask the question: for what consumption is (70,4,?) indifferent to (20,4,14)? If the response is 26, this will lead to: ) 26 (3 3 1U w w or 3 1w w Finally since we have13 2 1 w w w then 32 03 1 w w and36 02 w, which completes the determination of V. The final alternatives’ rank sc ores are given as Table 2.11.

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52 Table 2.11 The Final Alternatives’ Rank Scores (Source: Pomerol et al. 2000) Price 1U Comfort 2U Fuel consumption (miles/gallon) 3U ) (iA V 1A 70 0 10 1 14 0 0.36 2A 60 0.25 9 0.9 20 0.35 0.516 3A 50 0.5 9 0.9 18 0.25 0.564 4A 45 0.62 8 0.75 22 0.5 0.628 5A 40 0.75 9 0.9 16 0.1 0.596 6A 40 0.75 7 0.58 24 0.75 0.688 7A 30 0.9 6 0.4 20 0.35 0.544 8A 30 0.9 7 0.58 22 0.5 0.656 9A 20 1 5 0.2 24 0.75 0.632 10A 20 1 4 0 26 1 0.64 Since this procedure requires DMs to dete rmine one value based on all other attribute values are given, DMs must consider all attribute values at the same time. In contrast to this utility theory, the AHP method asks DMs to comp are only two attributes at a time. Therefore, this procedure gives lots of cognitive burden to DMs compared to the AHP method especially when the problem size becomes big. Moreover, Bard (1992) shows that there is no significant difference between these two method if and only if the same questions are given to the same DMs. Therefore, we decide not to use the utility theory for attribute weighting decision, but to use AHP method. 2.2.6 The Delphi Method In the Delphi method, DMs are directly aske d about each attribute’s relative weight by using questionnaires. An analys t continues surveyi ng until his or her desired variance is

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53 achieved. Figure 2.10 shows an ex ample that the second survey has a small variance compared to the first survey. Figure 2.10 Two Variances from the First and Second Round in the Delphi Method (Source: Dalkey at el. 1972) Therefore, DMs tend to move to majority opini ons regardless of the quality of opinions. Moreover, the quality of result s is dependent on the quality of the questionnaires. Table 2.12 shows a sample questionnaire wh ich is used in this method. The Delphi method repeats questions to th e DMs until the errors between answers goes to allowable ranges predefined by analysts. This method shows the survey results as well as the variance of the result. DM may change his or he r answer by looking at the common opinions as well as his or her deviation from them. This is the way to decrease errors in the Delphi method. However, by showing other’s opinions to DMs, and asking them to answer again can force

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54 each DM to move to a common idea. From a sta tistical perspective, the mean value does not always represent the best information. Table 2.12 A Sample Questionnaire of the De lphi Method (Source: Da lkey at el. 1972) Based on these two weaknesses (i.e., depende nce on the quality of questionnaire and preference of mean valu e) of this Delphi method, we decide this method is not proper for our research.

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55 2.3 Research on Similar Problems Related to this Problem Research in decision making for weapon procurement has been done using two different approaches: weapon selection among seve ral similar candidate weapons and decisions of budget allocation for weapon procurements. The la tter approach is an overall and strategic plan which covers all weapons procurements, while the first approach ca n be classified as a tactical and specific weapon proc urement decision. If the latter presents a proposed weapon set for procurement, the first approach suggests only one weapon. Budget allocation typically requires optimization techniques because the solu tion varies depending on various combination sets of weapons. The determination of one weapon does not require optimization techniques because of the assumption that the candidate weapons are already fit within budget and one type of weapon is required. Various research efforts in decision making models for both military and non-military areas have been reviewed. Tabl e 2.13 displays the main appro aches along with the strengths and weaknesses of these efforts. Kim (1987) develops a model called W eapon Acquisition Support System (WASS). This model is aimed at helping DMs to deci de between developing and buying in terms of weapon procurement. Therefore, we suggest th at this model can be placed before our model because our model is aimed at how we can buy a best weapon. Loerch et al. (1998) use Corps Battle Anal yzer (CORBAN), which is an U.S. army combat simulation model. CORBAN develops a response surface model and this model is used as an objective function of an army weapon procurement budget allocation problem. The formulation of their model is summarized as fo llows: maximize force effectiveness subject to budget ceiling, production limitations, force st ructure requirements, and other decision constraints. In this case the decision variable s are the quantities of each weapon procured in

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56 each year. This model assists the army leader ship in evaluating and prioritizing competing weapon system alternatives during the proces s of building the army budget. This model evaluates two different combat scenarios with tw o terms (i.e., long and short term) and presents an optimal weapon procurement set. The strength of Loerch et al (1998)’s model is that it provides an optimal weapon combination for the army as they prepare fo r any combat. Since their model depends on combat simulation, DMs can have confidence in their weapons without actual war testing. However, because their model’s emphasis is on combat operations, it does not work well as a generalized weapon procurement decision model. Moreover, their model requires sophisticated combat simulation model which can provide all detailed combat information. Hall et al. (1992) present a project funding decision model at the National Cancer Institute (NCI). They use the Delphi method to construct a de cision structur e as well as attribute’s weight. The preliminary results select the highest twenty proposals and assigns each a rank score. These eight proposals are added in to a decision model as decision variables. Through a maximization model under specified constr aints, DMs select pr oposals to be funded. Like other decision making proble ms, NCI is subject to some pol itical pressures. Therefore, a preference function is developed so that d ecision makers’ preferences can function as a constraint. Brauers (2001) develops a fighter decision model for Belgian Air force. His model is categorized as a MADM method. Hi s model has five attributes (i.e ., opinion of the Secretary of Defense, an increase of employment, deficit of balance of payments, a fighter price, and risk related with contraction. He uses Delphi method for developi ng attributes and weights.

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57 Table 2.13 Summary of Similar Researches Method Area Research (Author, Year) Main approaches Strengths Weaknesses MODM Military Loerch et al. (1998) -Allocating army budget for weapon procurement Response surface model by using combat simulation Use as an objective function Solve optimization problem for a budget allocation for army weapon procurements -Improvement of a combat performance based on the best weapons combination Need a sophisticate combat simulation model which can give DMs detailed data Not enough to be a general decision model Nonmilitary Hall et al. (1992) NCI funds allocation decision (funds for reducing smoking rate researches) Develop a rank function of candidate researches based on quality of that Applied as an objective function in maximization model Compensate qualities of researches with DMs preferences -Atypical MODM process therefore it is not proper to be used for our research MADM Military Brauers (2001) Rankings of candidate fighters for Belgium Air force Use a multiplicative form as an alternative value function Exclude a normalization procedure Since additive function is no different from multiplicative form, his approach can be modified more easy Nonmilitary Dyer (1990) Analyze flaw of AHP method Focus on AHP’s arbitrary ranking method Propose a possible solution for AHP method (Applied to our research) Propose a possible solution without any detailed procedures Bard (1992) A comparison between AHP and Utility theory Presented with a numerical problem An analysis based on one example Not enough to be used as an generalized comparison Hughes (1986), Saaty (1986), Rahman (2003) Some approaches to develop a GLM by using MADM approaches Possible to reduce a computational requirement of MODM More applicable budget allocating types of problems than our problem definition

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58 In MADM method, each alternative is assigne d values with respect to all attributes which can be obtained by summation of all at tribute values. This function is called an alternative value function. Since each attribute is different from other attributes in terms of its relative importance, the value f unction has a form of product su m of each attribute score and the attribute’s weight such as Equation 2.6. Brauers presents multiplicative form of alternative value function given by i i iC B A V ) ( (2.31) with g g ij g ix B g = DMs preference for attribute g g = weight of attribute g iB= sum of benefit attribute value of alternative i, and k k k ik ix C iC= sum of cost attribute value of alternative i. This model was seen to be an improvement, since normalization is not required. However, this normalization step is less cumbersome with computer advancements and its omission is not suggested. By us ing normalization procedure, Brau ers’ model can be simplified to Equation 2.6. Let us show the simplification procedures as follows. In addition to the reasoning of Equation 2.9, we suggest that ther e is no reason to use two different types of

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59 weights for one attribute. Two different weights can be combined as one by some a proper method such as AHP. Braurers’s model has two weights for one attribute, therefore, each benefit and cost value function can be changed into a form given by gw g ij ix B. (2.32) kw k ik ix C. (2.33) Through normalization, we do not need to divi de into two equations (i.e., benefits and costs attributes). By this reason, we can modify Braurers’ alternative va lue function, which is Equation 2.32, into a form as j w ij ijx A V) ( ) ( (2.34) By the reasoning of Equation 2.9, the E quation 2.31 has a form of Equation 2.6. In addition to the computational complexity of Braurers’ model, th is model only allows five attributes, which is not sufficient for all the important factors in weapon procurement decision making. Moreover, his model does not consider extreme alternatives which can be covered in our model. Dyer (1990) reviews the AHP method and shows this method’s weakness of rank reversal by using an example problem. A rank re versal may occur when a new alternative is added or deleted from the candidate list; we disc uss this problem in more detail in Chapter 3.

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60 He states that the fundamental problem of th e AHP method is its subj ective approach from DMs. In other words, attribute weights and alternatives’ ranking are dependent on DMs’ subjective opinions. Dyer proposes two possible methods to avoi d this rank reversal problem: using a absolute measurement and using an empirical method such as utility theory. Since utility theory has computational and cognitive difficulty when a problem size becomes large, we suggest using his first suggestion. This is applied into our res earch as an indexation procedure which is a categorization of each attribute; we explain this inde xation procedure for more detail in Chapter 4. Additionally, Dyer also does not consider any extreme alternatives. Chapter 3 presents the definition of the extreme alte rnative as well as our solution for that. Bard (1992) compares AHP method and utili ty theory by using an example problem. He uses the exact same approach presented before for both AHP and utility theory. The example problem is composed of twelve attributes and three alternatives. By chance, the results of AHP and utility theory give almost the sa me ranking scores. Theoretically speaking, the AHP weights and the MAUT (multi-attribute utility theory) scaling constants measure different phenomena, and hence, cannot be given the same interpretation (Kamenentzky, 1982). Bard analyzes the reason for the same so lutions as follows: the same questions are given to each DM for both AHP and utility theory therefore their respon ses would be the same even if they are answering for two different question types (i.e ., pairwise comparisons for AHP method, and determining indifferent score of each attributes based on ot her attribute are given as maximum values for utility theory). Based on Bard’s analysis, we decided to use AHP method for assigning attribute weights even though this method has a possible ra nking reversal weakness. Both utility theory and AHP rely on experts’ opinion. However, th e AHP method is for both computational and

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61 cognitive simpler for DMs. Due to weakness of the AHP, we should only use AHP method to decide attributes weights. For the alternat ive ranking scores, we will use an indexation procedure which is an absolute measurement. Even if MADM methods are not proper fo r MODM problems, these MADM methods can help MODM problems become easier to so lve. Hughes (1986), Saaty (1986), and Rahman (2003) present some approaches to develop a GL M (general linear model) for MODM problem by using MADM approaches. Their approaches depend on experts or DMs opinions which are also used to decide attribute weights. Theref ore, DMs can avoid analyzing empirical data to figure out the coefficient of each decision variable called as alternatives in MADM problems. 2.4 Summary In Chapter 2, we have shown various M ADM methods as well as weighting methods. Even though, these weighting methods are clas sified as MADM methods, since these methods are applied into our research only for defining we ights, we define thes e methods as weighting methods. In this chapter, we su mmarize this literature review in two ways: one is for MADM methods and the other is for the weighting methods. The following table provides comparisons within MADM methodologies in terms of strengths and weaknesses. From this table, we identify that the alternative ranking order is different between the SAW met hod and other two methods. We s uggest this difference occurs because the SAW method uses ove rall value of each al ternative, while the other two methods use different information (i.e., TOPSIS use the distance information and ELECTRE uses outranking information). However, none of thes e methods can be used without a weighting method because they require weight information.

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62 We provide three different weighting methods and the Table 2.14 and 2.15 provide comparisons among these methods in term s of advantages and disadvantages. Table 2.14 Comparison of MADM Methods Method Rank order (from example problem) Strengths Weaknesses SAW 5 1 4 2 3, , ,A A A A A A Simple calculation procedure Present overall alternatives’ rank scores Individual data information is not given by this method TOPSIS 2 5 1 4 3, , ,A A A A A Variance information is available by using two ideal solutions Information how close to the ideal solution Overall rank scores are not given by this method When a 5 0*iC, this method can choose an alternative which has an overall good score but individually bad scores ELECTRE 1 4 3, A A A Transitive assumption is not required -Do not present all alternatives’ rank scores As Bard’s (1992) suggestion, we assume that there is no difference among these three methods in terms of weighting information. Ho wever, the AHP method can give the smallest cognitive burdens to a DM. With this merit, AHP method is applied fo r a various decision making procedure specially when there are larg e number of qualitative attributes. A weapon procurement decision making contains a large am ounts of qualitative attri butes, therefore, AHP method is used for determining attribute weights. The one thing that we have not found in this literature review is a consideration of extreme alternatives. An extreme alternative is defined as an alternative which has an overall good score but some poor individua l scores. In other words, th is alternative may have a good overall score due to some extremely high scores in some attributes and in other attributes it has

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63 a very low score. The navy does not want this type of weapon. Therefore we introduce a new alternative ranking method which can compensate alternative’s overall ranking scores and individual attribute’s values. This new method is explained in detail in Chapter 3. Table 2.15 Comparisons Within Several Weighting Methods Methods Descriptions Strengths Weaknesses Utility theory Empirical modeling procedure Also use experts’ opinion for a qualitative problem structure Can present a preference function called utility function An utility function can be used as an objective function in MODM environment When a problem size becomes big, this method gives more cognitive burden than AHP method For that reason, right decision is more challenge than AHP method Delphi technique Use experts opinion with several times of interviewing or surveying Gives DM a chance to see what other DMs opinions are and how his/her opinion is different from them Relatively convenient than utility theory because an analyst dose not need any conditional types of question used in utility theory Good quality of results Asking several times for the same problem by showing other DMs opinions can forces DMs to move into median or mean values which does not need to be best solution AHP method Pairwise comparison is used Eigenvector and eigenvalue approach are used By using pairwise comparison, this method has less cognitive burdens than other two methods By using consistence index, any irrational response can be filtered to determine weights Still subjective as other two methods because this method is also dependents of experts opinion Another thing that we have f ound in this literature review is that there is no general decision model for weapon procurement. Based on these two things that we have found our model can be defined as uniqueness as opposed to any other MADM methods.

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64 Chapter Three Problem Statement and Methodology 3.1 Introduction There are a given number of alternative weapon systems, from which one weapon system will be selected to procure, based on a given set of attributes. The problem related to weapon procurement is complex. A MADM mode l is developed for naval weapon procurement decision. This model is expect ed to give DMs a better wea pon selection. There are several MADM methods but none of them provides a so lution in terms of co mpensating individual values for an overall value. This research suggests a new MADM me thod which can provide an alternative ranking score by compensating thes e two values. We also provide a sensitivity analysis to the solutions obt ained by the proposed model. 3.2 Problem Statement An extreme alternative is defi ned as an alternative which has an overall good score but some poor individual scores. In other words, this alternative may ha ve a good score due to some extremely high scores in some attributes and in other attributes it has very low scores. For example, let us consider two alternatives 1A and 2A, where 1A is considered as an extreme alternative. Table 3.1 presents attribut es values for thes e two alternatives.

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65 Table 3.1 Example for an Extreme Alternative Attributes 1X 2X 3X 4X 5X Weights (jw) 0.6 0.1 0.1 0.1 0.1 1jw 1A 1000 1 1 1 1 Alternatives 2A 200 200 200 200 200 Note that in this example, all attributes are assumed to be benefit attributes. For candidate weapons given in Table 3.1, SKN probably does not want to procure1A. As we described in the previous chapter, there ar e several MADM methods designed to determine alternative rank scores. However, there is no MADM method which can consider both an alternative’s overall rank scor e and individual attribute valu es. Most MADM methods would select 1A to be the best alternative, which will be seen in Section 3.3. This ranking result is evidently not appropriate as alternative 2A is clearly better than1A. Therefore, the development of a new method that can deal with such extreme alternatives is well justified. 3.3 Best Selection Method (BSM) Our new MADM method calle d BSM can present an a lternative rank score by compensating overall rank score for individual attribute values. Therefore, DMs can avoid selecting an extreme alternative which is bad. To be able to compensate an overall score for individual attribute values, we need to have tw o types of value functions: one is for an overall value and the other is fo r an individual value. From a statistical point of vi ew, there are two types of info rmation that we can use for overall and individual scores: mean and variance. The mean is computed by

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66 ix n x1 (3.1) where x is the mean of random variable X from n samples. The sample variance is generally computed by 2 21 1 x x n si. (3.2) Let us compare mean and the SAW method first. If we define thatij j jr w x then mathematical term of SAW method is gi ven as Equation 2.6. When we multiple by m / 1 on both sides, then Equation 2.6 can be written as m j j ix m A V m11 ) ( 1. (3.3) Since we assume that all alternatives should be c onsidered with respect to all the attributes of a problem structure, multiplication by m / 1 does not affect the final rank score in SAW method. By this reasoning, we ca n rewrite Equation 3.3 as m j j ix m A V11 ) (. (3.4) Equation 3.4 has the same mathem atical term as mean given by (3.1). Therefore, we can use SAW rank scores to represent mean information.

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67 In comparison between TOPSIS and variance, we note that both e quations have same function (i.e., sum of individual data point from a constant value): j jv v x and ,* are constant. Therefore, we can say that TO PSIS is an arithmetic combinati on of variances. We also know that arithmetic combination of variance is al so variance. Therefore, we can use TOPSIS as information of variance of alternatives. However, TOPSIS is a non-linear function and this causes mathematical computational difficulty. Therefore, an effort making a linear function is required. Modification to a linear function can be accomplished by defining new ideal solutions: called natural positive solution 1 and natural negative solution 0. In TOPSIS, the ranking score of an alternative is close to 1 if it is close to the positive ideal solution *A and close to 0 if it is close toA as computed by Equation 2.17. However, if we replace our two solutions for TOPSIS’s id eal solutions, Equation 2.17 can be simplified as a linear function given by m j ij ir A V11 ) (. (3.5) This modification is possible because of the following reasons: 1. Since ijr ranges from 0 to 1, we can de fine two new ideal solutions as 1 and 0 instead of *A and A 2. Because 1 is a replacement for a positive ideal solution, by minimizing the deviation from 1 we can choose an alternative which has maximum closeness to the positive ideal solution which is the same as the TOPSIS.

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68 3. Maximizing the deviation from the negative ideal solution is the same as minimizing the deviation from the positive ideal soluti on, and hence we do not need to use two deviation terms used in TOPSIS. For Equation 3.5, there is an absolute value term. However, this term causes mathematical computational difficulty and hence should be removed. Since 0 1 ijr for all i and j ,0 1 ijr Therefore, Equation 3.5 can be rewritten as m j ij ir A V11 ) (. (3.6) From Equation 3.6, we notify that as 0 ) ( 1 i iA V A and as m A V Ai i ) ( 0. Up to now, it is defined that ) (iA V =1 as the best, and ) (iA V =0 as the worst. In addition, m j ij m j ij m j ij m j ijr m r m r m r1 1 1 1max min min ) 1 ( min. That is, m j ijr1) 1 ( min is equivalent to m j ijr1max. Therefore, we modi fy Equation 3.6 as m j ij ir m A V11 ) (. (3.7) Now Equation 3.7 can present an alternat ive ranking scores such that when 1 ) ( 1 i iA V A and when 0 ) ( 0 i iA V A We define Equation 3.7 as an individual value function (IV).

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69 In addition to simplifying the TOPSIS, th ere is one weakness in this method for addressing an extreme alternative: TOPSIS uses ijv (see Equation 2.12) an d this value is not proper to measure how an altern ative has individually good attri bute values. For example, let us consider the same example shown in Table 3.1. The following table presents both ijr and ijv for this example. Table 3.2 ijr and ijv for the Extreme Alternative Problem Attributes 1X 2X 3X 4X 5X Weights (jw) 0.6 0.1 0.1 0.1 0.1 1jw 1A jx1 1000 1 1 1 1 2A jx2 200 200 200 200 200 1A jr1 1 0.005 0.005 0.005 0.005 jr1=1.02 2A jr2 0.2 1 1 1 1 jr2=4.20 1A jv1 0.6 0.0005 0.0005 0.0005 0.0005 jv1=0.602 2A jv2 0.12 0.1 0.1 0.1 0.1 jv2=0.52 From this table, one can see that 2A has individually better attribute values than1A, for 4 of 5 attributes. However, since 2 2 2 1 11) ( ) ( j jv v v vand 2 2 2 1 11) ( ) (j jv v v v for 5 ~ 2 j, the TOPSIS presents 1A as a better alternative and this is not a proper solution in terms of individual attribute values The detailed results are presen ted in Table 3.4. In contrast to the TOPSIS, the IV uses ijr and hence can present 2A as a better alternative in terms of individual attribute values This is computed as

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70 204 0 5 02 1 5 1 ) (5 1 1 1 j jr A V, 84 0 5 2 4 5 1 ) (5 1 2 2 j jr A V. As we can see, 2A has higher rank score than 1A. Therefore, using ijr is more proper than using ijv to measure how individua lly good attribute values. By compensating mean type of value for varian ce type of value, we can compensate an alternative’s overall ranking score for individual attribute values. As explained earlier in this section, we use the value from the SAW met hod representing an altern ative’s overall ranking score. For the variance type of informa tion, we use our new function of IV. To compensate two different values, each va lue must have same scale factors. For example, if one value ranges from 0 to 100 while the other value ranges 1 to 2, the former value absorbs the latter value. The SA W method ranges from 0 to 1: since 11 m j jw and Max (ijr ) =1, therefore, m j ij j ir w A V1) ( ranges from 0 to 1. And we al ready showed that IV ranges from 0 to 1. Since these two values (i.e., the SAW functi on and IV) can be adde d, the sum of these two values can present information of comp ensating an alternative’s overall score and individual attribute values. In BSM, the alternative ranking score is given by m j ij j ir m w A V11 2 1 ) (. (3.8)

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71 The ranking score from the BSM also ranges from 0 to 1. 3.3.1 A Numerical Example From Table 3.2 and the BSM ranking function, we can compute an alternative ranking score shown in Table 3.3. The value of 1A is computed as 5 1 5 1 15 1 2 1 ) (j ij j ij jr r w A V 204 0 602 0 2 1 5 02 1 602 0 2 1 =0.403, likewise 680 0 ) (2 A V. Table 3.3 Ranking Scores for th e Extreme Alternative Problem Alternatives Values from the SAW method IV values BSM ranking scores Ranking 1A 0.602 0.204 0.403 2 2A 0.520 0.840 0.680 1 In this example, 1A can have better score in terms of an overall value that is given by the SAW method while individuall y bad scores presented by the IV. In other words, even though 2A has lower overall score by SAW, sin ce this alternative has individually good attribute values, 2A can be ranked first in our method. Th erefore, we can a void selecting an extreme alternative.

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72 3.3.2 Comparison with Current MADM Methods The most common MADM methods are th e SAW, TOPSIS, and ELECTRE methods. However, as explained in the previous chap ter, the ELECTRE method does not provide all alternatives’ ranking scores. For this reason, in this research we compare the BSM with the SAW and TOPSIS methods. First let us consider the extreme alternative case which is shown in previous section. Table 3.4 shows each ranking scores as well as ranking orders by each method. Table 3.4 Ranking Scores for the Exam ple from the Three MADM Methods Alternatives SAW Ranking TOPSIS Ranking BSM Ranking 1A 0.602 1 0.707 1 0.403 2 2A 0.520 2 0.293 2 0.680 1 From this example, the SAW and TOPSIS are shown to be inappropriate in the preference of an extr eme alternative. Table 3.5 Ranking Scores for th e Fighter Selection Problem BSM Alternative ASW Rank TOPSIS Rank ASW IV BSM Rank 1A 0.840 2 0.361 4 0.840 0.868 0.854 2 2A 0.827 5 0.414 2 0.827 0.820 0.823 5 3A 0.895 1 0.607 1 0.865 0.873 0.883 1 4A 0.840 2 0.396 3 0.840 0.844 0.842 4 5A 0.832 4 0.349 5 0.832 0.857 0.845 3

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73 Let us consider another example problem shown in Table 2.1 which does not include any extreme alternatives. Table 3. 5 contains all five alternatives ranking scores as well as ranking information by SAW, TOPSIS and BSM. Fr om this example, all three methods present 3A as the best alternative because this alte rnative has an overall good score as well as individual good attribute values. 3.4 Sensitivity Analysis DMs may want to see if the ranking order will change if some attribute value changes. For instance, in the previous fighter example 2Ais ranked second. If one of the ten attributes values for this fighter changes, this altern ative may improve its rank from the second to the first. Sensitivity analysis is concerned with how outcomes change when inputs changes. In this research, the sensitivity analysis is define d as follows: how changes in an attribute’s value affect the current ranking. To do this, it is neces sary to consider the proposed alternative value function (henceforth called as BSM function) presented in the previous section. The alternative ranking is determined in terms of each altern ative’s ranking value calculated by Equation 3.8. Ther efore, if we calculate th e difference between any two alternatives’ ranking values, this difference can be considered as a critical value for the case of a ranking change with each alternative. For exam ple, let us consider any two alternatives,iA andkA which has a ranking relationship of ) ( ) (k iA V A V This relationship is denoted by 0 1 1 2 1 ) ( ) (1 1 1 1 m j kj m j kj j m j ij m j ij j k ir m r w r m r w A V A V (3.9)

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74 From this example, we can consider tw o possible critical values: changes in iA such that ) ( ) (k iA V A V and changes in kA such that ) ( ) (k iA V A V To be able to process the sensitivity analysis, we change one attribute’s value of some alternative at a time and let all other attribute values remain the same. To be able to change the ranking of iA and kA the value of ) ( ) (k iA V A V should be at least zero and this is given by 0 1 1 2 1 ) ( ) (1 1 1 1 m j kj m j kj j m j ij m j ij j k ir m r w r m r w A V A V (3.10) Let c klr be the critical attribute value such that when the current klr is changed toc klr iA and kA will have the same rank. Since all other attr ibutes’ values remain the same, the critical value c klr can be computed by 0 1 2 1 1 2 1 ) ( c kl c kl l m l j kj m l j kj j ir m r w r m r w A V, and when we solve this equation for c klr then m l j kj m l j kj j i l c klr m r w A V mw m r 1 ) ( 2 1 (3.11) If we are interested in a critical value in iA called c ilr it is easily computed by switching iA andkA

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75 3.4.1 Current Basis and Allowable Ranges Since ijr ranges from 0 to 1, c klr is valid only within this ra nge. Therefore, the allowable range for c klr is defined as 1 0 c klr (3.12) However, since ijr is computed as the proportion between ijx and *jx or jx, if the value of *jx and jx are changed, all ijr has to be computed again (see Equation 2.3). Therefore, to be able to remain the current ra nking scores valid, we need to make sure that *jx and jx are not changed. Therefore, the current basis can be defined as Set that for 1 and 1 i k k r rkj ij (3.13) For example, for the fighter example, the curren t basis can be defined by (3.13) as follows: Current basis = 542 141 132 331 222 314 314 113 311 and , , , , r r r r r r r r r If critical values fall in the range given by (3.12), the ranking change within the two alternatives is possible. On th e other hand, the case of out of range does not provide a chance of ranking change by sensitivity analysis. Based on the critical values and the allowable ranges, the sensitivity analysis of this research provide s two possible scenarios fo r any two alternatives iA and kA

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76 Scenario 1 (changes in kA ). If ) ( ) (k iA V A V then there are three cases: 1. ] 1 0 [ c klr Then ) ( ) (k iA V A V for all ) 0 [c kl klr r and ) ( ) (k iA V A V for all ] 1 (c kl klr r 2. ] 1 0 [ c klr Then sensitivity analysis does not apply. 3. klr Current basis. Then sensitivity analysis does not apply. Scenario 2 (changes in iA ). If ) ( ) (k iA V A V then there are three cases: 1. ] 1 0 [ c ilr Then ) ( ) (k iA V A V for all ) 0 [c il ilr r and ) ( ) (k iA V A V for all ] 1 (c il ilr r 2. ] 1 0 [ c ilr Then sensitivity analysis does not apply. 3. ilr Current basis. Then sensitivity analysis does not apply. The next section provides a numerical ex ample of this sensitivity analysis. 3.4.2 A Numerical Example Let us consider the fighter selection probl em. Table 3.6 shows th e alternative ranking values and ranking orders. Table 3.6 The Alternative Values and Ranking Information Alternative BSM ranking scores Rank 1A 0.854 2 2A 0.823 5 3A 0.883 1 4A 0.842 4 5A 0.845 3

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77 Recall the normalized decision matrix used to calculate each altern ative ranking score. Weight: 0.2 0.04 0.04 0.12 0.09 0.21 0.12 0.08 0.06 0.04 1 90 0 0.778 889 0 88 0 82 0 0.778 889 0 70 0 69 0 0.889 1 78 0 90 0 0.778 778 0 78 0 00 1 1 889 0 889 0 1 657 0 91 0 83 0 72 0 889 0 1 686 0 87 0 83 0 80 0 778 0 889 0 1 78. 0 1 1 1 889 0 571 0 87 0 83 0 80 0 889 0 778 0 543 0 1 1 80 0 X X X X X 5 4 3 2 1 42 41 32 31 22 21 14 13 12 11A A A A A X X X X X From this matrix, the current basis is define d as previous, i.e., cu rrent basis equals to 542 141 132 331 222 314 314 113 311 and , , , , r r r r r r r r r Sensitivity Analysis forjr3 in terms of ) ( ) (3iA V A V is as follows. Since DMs are only interested in the first ranked alternative (i.e, 3A ), every alternative has to be co mpared with respect to the firs t ordered alternative. For this reason, the considerable change in 3A is loosing its firs t ranking order. If 3A loose its first order position, the only possible alternative which can be ranked first is 2A. In other words, intermediate ranking orde rs are not important to DMs. Therefore, only 1A and 3A are considered for the sensitivity analysis of jr3 Let us solve each critical value of c jr3 which provide the condition that ) ( ) (2 3A V A V cr312 for ) ( ) (1 3A V A V is computed by Equation 3.11. ) 10 1 ( ) ( 2 1 10 1012 3 12 3 1 12 312j j j j j cr r w A V w r = 574 0 ) 773 0 855 0 ( ) 854 0 ( 2 1 4 0 10

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78 Since cr312 = 0.574, if ) 574 0 0 [312 r then ) ( ) (1 3A V A V and if ] 1 574 0 (312 r then ) ( ) (1 3A V A V The critical value of c ijx can be computed by solving Equation 2.3 for c ijx, that is given by attribute cost for value a is en wh attribute benefit for value a is when ** ij j c ij ij j c ij c ijx x r x x r x (3.14) By Equation 3.14, the critical value of 312x (denoted as cx312) can be computed as 12 312 312* x r xc c = 0.574 60 = 34.44 (1,000 ft). If 3A has more than 34,440 ft of operating altitude, this alternative can keep its first rank order. In other words, if 3A has its operating altitude le ss than 34,440 ft, then this alternative can loose its first rank order, and 1A can be ranked first. Through the same procedure, the rest c jr3 are computed as in the following table. Table 3.7 Critical Values for 3A in terms of ) ( ) (2 3A V A V Attributes ) ( j 11X 12X 13X 14X 21X 22X 31X 32X 41X 42X c jr3 c.b. 0.574 0.354 c.b. 0.575 0.585 c.b. 0.557 0.317 0.274

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79 Note that c.b. denotes that ijr Current basis and hence no sensitivity analysis is applied for this value. Sensitivity analysis for 4 2 1, ,A A A and 5 A in terms of ) ( ) (3iA V A Vis as follows. Since the only first rank order can be chosen, it is reasonable to consider the case of being improved as rank one. For this reason, 4 2 1, ,A A A and 5 A should be considered in terms of ). ( ) (3iA V A V Table 3.8 shows critical values for 4 2 1, ,A A A and 5 A in terms of ) ( ) (3iA V A V. Table 3.8 Critical Values for 4 2 1, ,A A A and 5 A in terms of ) ( ) (3iA V A V Attributes ) ( j 11X 12X 13X 14X 21X 22X 31X 32X 41X 42X c jr1 1.002 1.433 c.b. 0.818 1.097 1.084 1.164 c.b. c.b. 1.211 c jr2 1.203 1.693 1.733 1.12 1.525 c.b. 1.327 1.449 1.655 1.643 c jr4 1.077 1.424 1.464 1.064 1.438 1.157 1.267 1.240 1.340 1.474 c jr5 0.9812 1.390 1.470 1.013 1.412 1.142 1.245 1.121 1.390 c.b. From Table 3.8, it is shown that 1A can be ranked first if 114r can be increased more than 0.818 (this alternative has current value of 0.543). However, the results show that the sensitivity analysis do es not apply in many ijr for this example. 3.5 Summary In this chapter, we present a new ranki ng score method called BS M with a sensitivity analysis. By introducing this method, we expect that DMs can choose an alternative which has an overall good ranking score as well as individual good attribute values. In addi tion to this

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80 method, the provided sensitivity analysis can give what if analysis for both SKN and weapon companies.

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81 Chapter Four Construction of A Hierarchical Structure 4.1 Introduction Like Saaty and Vargas’s (2000) suggestion, th e most creative task in making a decision is to determine what factors to include in a hi erarchical problem struct ure. SKN follows NR 2 for a weapon procurement decision. However, there is no problem structure which can be used for analytic decision model. Th erefore, there must be a work for developing a problem structure which can be used for an analytic decision model. NR 2 contains five principles for weapon procurement decision and these are shown in the following table. Table 4.1 Principles of Weapon Procurement (Source: NR 2) Principles Explanation Operational Performance Maintain the level of performance to meet the operational requirements Readiness on Time Weapons should be ready on time for a specific purpose Technical Merits Try to get technologies from weapon procurement if it is come from other countries. Give some priority for the domestic products Cost Effectiveness Acquire the best products with the lowest price Sustainment Keep the required performance within the life From Table 4.1, we can construct hierarchi cal problem structure wi th five attributes. This problem structure is graphi cally displayed in Figure 4.1. In this figure, we can see that

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82 there are five level 1 attributes which are defined as “P rinciples for weapon procurement” by NR 2. Figure 4.1 A Hierarchy for Best Weapon Selection The indexation is defined as a categorizati on of an individual qualitative attribute to present each alternative’s value of the attribute. For example, we can categorize an attribute into five indexes such as outstanding, above average, average, below average, and unsatisfactory. With this categorization, we assign value of 5, 4, 3, 2, or 1 to each index. Based on this index, each alternative’s va lue can be determined (e.g., if an alternative is classified as an outstanding, the value of 5 is assigned to th e alternative for that attribute value). We use these indexes for assigning alternatives’ values fo r qualitative attributes. In this chapter, we develop a more detailed problem stru cture with indexation procedures. Alternatives Attributes Best weapon selection Readiness on Time Technical Merits Cost Effectivene ss Sustainment Weapon 1 Weapon 2 Weapon n Operational Performanc e Goal

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83 4.2 Operational Performance SKN defines operational performance as “maint aining the level of pe rformance to meet the naval operational requirements” (NR 2). Nava l operations can be classified into two types: An actual naval warfare operation and peace-time regular operation. It is difficult to decide which operation is more important. However, during a peaceful time, people may consider regular operations (e.g., sea patr ol, ensuring freedom of the seas so that merchant ships can bring the vital raw materials into Korea, collecting information surrounded in Korea, and so on) as more important. In contrast to this, act ual combat effectiveness can be considered as more important than effectiveness in regular operations if DMs stre ss on an actual warfare. Therefore, an operational performance is meas ured in terms of sum of two sub attributes’ scores: a combat operational performance and a regular operational performance. Figure 4.2 shows a hierarchical structure of an operational performance. Figure 4.2 Hierarchical Structur e of Operational Performance The sub-attribute items under a regular operational perfor mance are weapon dependant. That is because each weapon has different purpo ses and hence should be evaluated differently Operational Performance Combat Operational Performance Regular Operational Performance FER obtained from war game Weapon-dependent

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84 in terms of each purpose. For example, the purpos e of the missile is to hit an opponent’s object, while radar has a purpose to detect an opponent’s obj ect. In terms of differ ent purposes of these two weapon types, the measure of effectiveness is defined differe ntly. For example, the attack range, attack precision, and pe netration are appropriate measur e of effectiveness for missile and the detection range, accuracy and performances regarding electrical warfare are good for radar. Therefore, detailed stru ctures under a regular operational performance can be defined according to weapon type. Combat effectiveness, which can m easure the performance of weapon on naval warfare, can not be tested unless there is a r eal war. However, a simulation game called War Game is widely used in many countries on a real combat’s behalf for several purposes such as training people in case of a combat, evaluating task forces in terms of real warfare, and improving commanders’ tactical abilities. Loerch et al. (1998) propose a Fractional Exch ange Ratio (FER) as a measure of combat effectiveness. FER is computed by (note that red represents enemy force and blue represents friendly force) i i i i i i i i i ib b a r r a / / (4.1) where = FER otherwise 0, combat a joins i force blue when 1ib

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85 otherwise 0, combat a joins i force red when 1ir ia = remaining availability of blue or red force i after the combat, id = damage rate of blue or red force i after the combat, i di 1 0, i id a 1, and } 1 0 { i ir b In Equation 4.1, 0 id means that a blue or red force idoes not have any damage while 1 id represents entire loss. However, FER is only applied for a unit combat system such as a ship, a submarine, or an aircraft. That is because a component weapon system (e.g., gunnery, missile, radar, sonar, and radio) can not be measured separately from its host system (i.e, a unit weapon system). 4.3 Readiness on Time The terminology of readiness on time is de fined as weapons should be ready on time for a specific purpose by NR 2. The attribute of readiness on time is composed of three sub attributes as follows. 1. Readiness of operators. Operator s should be trained. Therefor e, training time as well as number of people of both trainers and trainees should be considered. 2. Readiness of weapons. If any supplier can not meet demand (i.e., be on time, and to meet amounts of weapons), that company can not be attractive to DMs. Being on time represents that weapons shoul d be prepared for use before or at the time when SKN requires. SKN requires a certain number of weapons to be able for use and it is represented by the at tribute of meeting amounts of weapons.

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86 3. Readiness of supporting systems. Supportin g systems can be defined such as department of administration, logistics, and maintenance. If SKN can use current supporting systems to operate a candidate w eapon, the weapon should have advantages under this attribute. In many cases, more efforts than buying a weapon itself are required to develop supporting systems. These three factors are shown as sub-attributes of readiness on time in Figure 4.3. Figure 4.3 Hierarchical Stru cture of Readiness on Time Sub-attributes of readiness of operators are cost attributes. Therefore, value of each alternative should be pl ugged as a form of 1/ijx as explained in Section 2.2.1. Data for readiness of weapons and supporti ng systems are both qual itative and hence an indexation process is required. Table 4.2 and 4.3 shows, respectably, indexation of readiness of weapons and supporting systems. Readiness on Time Readiness of Operators Readiness of weapons Readiness of Supporting Systems Training Time Number of People required

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87 Table 4.2 Indexation of Readiness of Weapons Scale Criteria Scores Outstanding All two factors can be met before due date 5 Above Average Weapons are ready but not all of them 4 Average Weapons can be ready shortly after due date but with all of weapons 3 Below Average Weapons can be ready shortly after due date but still not meeting the amounts of weapons 2 Unsatisfactory Preparing weapons take longer time than SKN’s expectation 1 Note that the two factors are being on time, and meeting the amounts of weapons. If SKN can use current supporting systems for operating any candidate weapon, that weapon has an advantage of smaller cost requ ired in building a supporting system. This advantage is considered by the attribute of readine ss of supporting systems. Table 4.3 Indexation of Readiness of Supporting Systems Scale Criteria Scores Outstanding Current three supporting systems are available to use for using a candidate weapon 5 Above Average Two of three current supporting systems are available to use for using a candidate weapon 4 Average One of three current suppor ting system is available to use for using a candidate weapon 3 Below Average None of systems are available to use, but these three systems can be built with no difficulty 2 Unsatisfactory None of systems are av ailable to use, and building these three systems are challenging 1 Note that the three supporting systems are ad ministration, logistics, and maintenance.

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88 4.4 Technical Merits NR 2 defines technical merits as “Try to get technologies from weapon procurement if it is coming from other countries. However, some priorities are given to the domestic products. One should also consider giving a benefit on co ntributions toward a domestic economic. By this definition, two sub attributes under the attr ibute of technical merits are presented in the following figure. Figure 4.4 Hierarchical Structure of Technical Merit Percentage of domestic components used is provided by each candidate company. This attribute is considered as a benefit attribute since the more domestic components are used, the better for Korean economics. Korea has recently developed and hence there are not enough technol ogies available for not only developing weapons, but also producing a competitive product in world wide markets. For this reason, Korean governme nt considers obtaini ng technologies from outside of Korea as an important strategy. SKN classifies th e degree of technology acquisition into three categories: core technologies, important technologies, and general technologies. The indexation of acquisition of technol ogies is presented in Table 4.4. Technical Merits Percentage of domestic Components usage Technology Acquisition

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89 Table 4.4 Indexation of Technology Acquisition Scale Criteria Scores Excellent condition Core technologies can be obtained 4 Very good condition Important technologies can be obtained 3 Good condition General technologies can be obtained 2 Poor condition No technologies can be obtained 1 SKN defines these four categories us ed in the above table as follows. 1. A core technology is a very important te chnology which has a critical impact on domestic economies especially for a dom estic weapon industry. The classification about what should be the core technologies is decided be fore weapon procurement is issued by SKN. 2. An important technology is not considered as a core technology, but still considered as an important technology. This technology is also classified by SKN before the weapon procurement decision is processed. 3. A general technology is neither critical nor important. However, if a company can give us any technologies which Ko rea lacks, this is good for both domestic companies and the SKN. If any contract cannot present any technologies to Korea, this weapon has a value of one on this attribute. 4.5 Cost Effectiveness There are two types of costs that are involved in weapon procurement: the first acquisition cost and operational costs. To show how these two different costs affect decision making, let us consider two extreme cases.

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90 1. Low acquisition price, but high operational cost. In this case, the weapon cannot be attractive to buyers. 2. Reasonable operational cost, but high acquisition cost. In this case, DM can hesitate to decide to buy the weapon. Therefore, a trade off between acquisition and operational cost is nece ssary. The attribute of cost effectiveness is designed for this tradeoff. Figure 4.5 shows corresponding hierarchical structure. Figure 4.5 Hierarchical Structure of Cost Effectiveness 4.6 Sustainment To sustain means to supply with necessi ties or provide for. SKN defines three important factors in the sustainment of equipmen t as logistics, maintenance, and reliability. These three factors are presented in Figure 4.6. Cost Effectiveness First Acquisition Cost Operational Cost

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91 Figure 4.6 Hierarchical Structure of Sustainment Web dictionary defines logistics as “hand ling an operation that involves providing labor and materials be supplied as needed” (http:// www.wordwebonline.com). From a user point of view, good logistics means that any supplement a nd maintenance parts sh ould be ready within an expected period of time. And how the logi stics works depends on supplier types. For military contracts, there are typically three diffe rent types of suppliers: a domestic company, a foreign company, and a foreign government. The latt er one is defined as Foreign Military Sale (FMS) by NR 2. The table below shows an indexation of logistics. Table 4.5 Indexation of Logistics Scale Criteria Scores Best supplier type Supplement and parts are distributed by a domestic company 3 Normal supplier type Supplement and parts are distributed by a foreign company 2 Worst supplier type Supplement and parts are distributed by the contraction type of FMS 1 Sustainment Logistics Maintenance Reliability Depot Maintenance Field Maintenance Reliability of Company Reliability of Weapon

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92 A domestic company can be considered as the best supplier in terms of a quick response time. In addition to this response time SKN can have various advantages such as small transport costs, no affects from foreign ex change rate, and increasi ng rate of employment by contracting a domestic company. A contract with a foreign company can lead to long respons e time and high costs because all products are transported to Korea from a foreign country. However, this contraction type is better than the FMS. FMS is a cont raction between two countries. A supplying country needs to get an approval from its Congress to export products to other countries. For this reason, it normally takes more than a year to receive a product supplied by FMS contraction. In contrast to FMS, direct cont raction between SKN and a foreign company does not need an approval from Congress. Therefore, SKN pref er a contraction with a foreign company compared to FMS. SKN operates two types of maintenance syst ems: field and depot maintenance. Field maintenance is generally rou tine maintenance and is conducte d by individual operators. Depot maintenance is conducted by special technicians and most of maintenan ces are repeated more than once per year. Both field and depot maintena nces include pre-maintenance as well as after failure maintenance. Table 4.6 shows an indexation of depot maintenance. Table 4.6 Indexation of Depot Maintenance Scale Criteria Scores Best depot maintenance SKN can conduct a depot maintenance 3 Normal depot maintenance Depot maintenance needs to be conducted by a domestic company 2 Worst depot maintenance Depot maintenance needs to be conducted by a foreign company 1

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93 Depot maintenance is composed of wo rks beyond weapon operator’s abilities. Moreover, it is a big job such as an overhaul an d hence requires a faci lity with many special tools. This maintenance is conducted either by SKN or a private company. From a SKN point of view, navy facility is the best and a fore ign company is the worst in terms of depot maintenance. Since individual weapon operato rs conduct field maintenance, how they feel in terms of easiness can be important criteri a. If operators feel difficulty in maintaining a weapon, this weapon can not be considered as good in terms of field maintenance an d vice versa. Table 4.7 presents indexation of field maintenance. The decision regarding which scale should be assigned to each candidate weapon is done by consensus from operators. Table 4.7 Indexation of Field Maintenance Scale Criteria Scores Outstanding Very easy for conducting field maintenance 5 Above Average Easy for conducting field maintenance 4 Average Commonly difficult for conducting field maintenance 3 Below Average Difficult for conducting field maintenance 2 Unsatisfactory Very difficult for conducting field maintenance 1 Reliability is a characteristic of an item, expressed by the probability that the item will perform its required function under given conditions for a stated time interval (Birolini, 1999). Normally reliability is considered only for a product. Even though SKN buys a best weapon which has best performance in terms of reliabi lity, if a company whic h produces the weapon is unstable and disappeared in the market, SKN can not maintain the weapon anymore. For this reason, we expand reliabili ty into each company.

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94 SKN defines mean time between failures (M TBF) as an index for representing weapon’s reliability. Since a longer MTBF is more appropriate for SKN, this index is considered as a benefit attribute. SKN obtains this information from each candidate company. Company ranking information can be obtai ned by reliable periodicals such as DefenseNews, Business.com and Fortune.com Since high ranking is represented as low number index, this index is considered as a cost attribute (i.e., a value of each alternative has a form of 1/ijx ). In addition to these two reliab ilities, warranty condition can play an important role in terms of reliability. Even if a weapon has a short MTBF, but the company can offer a good warranty condition, SKN can buy the weapon offere d from this company. From a company’s point of view, a good warranty condition can in crease its competition ability. From a SKN point of view, a good warranty condition can be c onsidered as a merit. However, an indexation for this condition is difficult. Therefore, we assume that each company offers the same conditions in terms of warranty condition. This assumption is based on the reasoning that (1) SKN requires a certain limit of warranty conditions and each comp any has to meet the limit to be eligible as a candidate comp any, (2) companies will try to lower their weapon’s prices to increase their competition edge, and (3) consequently, companies are unable to offer better warranty condition than required by SKN. 4.7 Summary Since SKN follows NR 2 for weapon procur ement decision, this problem structure can be considered as non creative works. However, si nce there is no structur ed procedure like this hierarchical structure, the ri ght decision of sele cting the best wea pon has been always challenging for SKN. We expect th is hierarchical structure can be used as a general structure

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95 for a weapon procurement decision. Figure 4.7 shows a hierarchy of attri butes for the weapon procurement. This hierarchical structure has three levels of attributes: five first-level attributes, twelve second-level at tributes, and six third-level attributes. Figure 4.7 A Hierarchy of Attributes For the Best Weapon Procurement BSM function presents alte rnatives’ raking scores in terms of this hierarchical structure. However, as presented in the previous example, weights should be assigned into all attributes. The attribute weighting for this problem st ructure is followed in the next chapter.

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96 Chapter Five Attribute Weighting 5.1 Introduction Since all attributes do not have the same im portance, we need to assign a degree of importance to each attribute. As explained earlie r, these are called weights. The weights are the core of compensation methods. In most cases, it is difficult to determine relative importance among attributes, especially for qualitative ones. In this research, th e AHP method is used, based on the comparison results shown in Section 2.4 The AHP method uses pairwise comparisons from Experts. In this research, Experts are defined as SKN senior offi cers. The sample size of n is computed by Equation 5.1 (Lee and Park, 1995). Computation results are not pres ented in detail due to military secrets. pq Z NB pq NZ n2 2 / 2 2 2 / (5.1) In this equation, Nis the population size, B the significance level, and p and q are expectation values of population proportion, i.e., p represents proportion of seni or officers in the SKN and q the other SKN personal. The sample size is calculated as 50 and mailing interviews will be used. A survey form is composed of two parts. Part 1 is composed of questionnaires for personal and general parts which aims to help an individual officer f eels comfortable and can

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97 answer with ease for questionnaires given in part 2. Part 2 is an actual pairwise comparison designed to assign attribute weights. These tw o parts of survey forms are presented in Appendix 1. As explained in Chapter 2, Saaty (1980) develops the CR presen ted in Equation 2.29 and suggests that the answer is not consistent if CR is more than 0.1 and such answers should be excluded in calculatin g an attribute weight. However, th is suggestion has no support for the value of CR=0.1. Sin (1988) suggests using a value of 0.2 instead of 0.1 is more practical than Saaty’s suggestion because we can increase the number of answers which can be used to calculate an attribute we ight. By following Sin’s suggestion, we define the critical value for consistency as 0.2. The actual calculation is computed by the weighting program developed by using C++ (Chang, 1997). Since this program is not used in Visual C program, we modify this program to be used in Visual C program. We pr esent this modified program in Appendix 2. We obtain total 450 pairwise comparison matrices (i.e., 9 matrices for each DM and we have fifty DMs; therefore,450 50 9 ). Among these matrices, 34 matrices are excluded in weighting computation due to2 0 CR. Weight for each attribute is computed by the following steps. Step 1. From each decision matrix, we obtain i ndividual DM’s weighting values. This follows the five steps in Section 2.2.4. Step 2. Each decision matrix presents fifty or fewer weighting values depend on the state of2 0 CR. Step 3. From these weighting values, each attribute’s mean weight is computed. Step 4. Attribute weights are defined as this mean values. We also compute 95% confiden tial intervals for each attribute weight. This interval is presented as a half width with an attribut e weight value. Half width is computed by

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98 Half width = n S n S Z 96 1975 0 (5.1) where S is a standard deviation of the n weight values whose 2 0 CR. Note that since 30 n for all matrices whose2 0 CR, we use Z-distribution in stead of t-distribution. Detailed results are described in the following sections. 5.2 Weights for the First Level Attributes Among the fifty decision matrices for five fi rst-level attributes, f ourteen matrices are excluded in computing weights due to2 0 CR. Table 5.1 shows the weights of these five first-level attributes. From this table, sustainment and operati onal performance are drawn as important attributes while readiness on time is considered as less important decision factor with respect to the best weapon procurement. Table 5.1 Weights for the Five First-Level Attributes Attributes with respect to best weapon selection Results Operational performances Readiness on Time Technical merits Cost effectiveness Sustainment Total Mean (weights) 0.294 0.073 0.177 0.126 0.330 1 Standard deviation 0.127 0.044 0.057 0.054 0.087 Half width 0.042 0.015 0.019 0.018 0.029

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99 5.3 Weights for the Second Level Attributes Operational performance has two second level attributes: combat and regular operational performance. In this case, since we ha ve two attributes that we need to compare, only one comparison is required to determine we ights. In addition, when we compare two attributes, we have CI= 0 because n max for n=2 (see Equation 2.28). Due to CI= 0, 0 CR (see Equation 2.29). Therefore, we can co mpute mean value for each of the fifty decision matrices. Table 5.2 s hows weights for combat and re gular operational performance with respect to operational pe rformance. In this table, co mbat operational performance is considered as two times important th an regular operational performance. Table 5.2 Weights for the Two Second-Level Attributes of Operational Performance Attributes with respect to operational performance Results Combat operational performance Regular operational performance Total Mean (weights) 0.693 0.307 1 Standard deviation 0.192 0.192 Half width 0.053 0.053 Readiness of operators, weapon and s upporting systems are three second-level attributes of readiness on time. In this case, eleven decision matrices are excluded in computing weights due to2 0 CR. Table 5.3 shows weights fo r these three second-level attributes. From this table, read iness of supporting system is cons idered as more important than the other two attributes.

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100 Table 5.3 Weights for the Three Second-Le vel Attributes of Readiness on Time Attributes with respect to readiness on time Results Readiness of operators Readiness of weapons Readiness of supporting systems Total Mean (weights) 0.283 0.309 0.408 1 Standard deviation 0.105 0.184 0.198 Half width 0.033 0.058 0.062 Table 5.4 shows weights for the two secondlevel attributes unde r the attribute of technical merits (i.e., percentage of domes tic components usage and technology acquisitions). In this case, as in Table 5.2, all of fifty deci sion matrices are used fo r computing weights. In this table, the attribute of t echnology acquisitions is considered as more important decision factor than the attribute of pe rcentage of domestic components usage with respect to technical merits. Table 5.4 Weights for the Two Second-Le vel Attributes of Technical Merits Attributes with respect to technical merits Results Percentage of domestic components usage Technology Acquisitions Total Mean (weights) 0.375 0.625 1 Standard deviation 0.216 0.216 Half width 0.060 0.060 Table 5.5 shows weights for the first ac quisition costs and operational costs with respect to cost effectiveness. From this tabl e, one can see that operational costs are more important than the first acquisition costs.

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101 Table 5.5 Weights for the Two Second-Leve l Attributes of Cost Effectiveness Attributes with respect to cost effectiveness Results First acquisition costs Operational costs Total Mean (weights) 0.341 0.659 1 Standard deviation 0.217 0.217 Half width 0.060 0.060 Logistics, maintenance, and reliability are second-level attributes which compose the attribute of sustainment. In this case, nine matrices are excluded in computing weights due to2 0 CR. Table 5.6 presents weights for these thr ee second-level attributes. In this table, logistics is shown as the most important at tribute with respect to sustainment. Table 5.6 Weights for the Three Sec ond-Level Attributes of Sustainment Attributes with respect to sustainment Results Logistics Maintenance Reliability Total Mean (weights) 0.429 0.341 0.230 1 Standard deviation 0.174 0.123 0.144 Half width 0.055 0.039 0.045 5.4 Weights for the Third Level Attributes Three of the second-level attributes (i.e., readiness of operators maintenance, and reliability) have their sub-attributes (i.e., third level attributes). Weights for these third level attributes are presented in the following table.

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102 Table 5.7 Weights for the Third Level Attributes Second level attributes Third level attributes Weights Standard deviation Half width Training Time 0.530 0.170 0.047 Readiness of operators Number of people required 0.470 0.170 0.047 Field maintenance 0.617 0.193 0.053 Maintenance Depot maintenance 0.383 0.193 0.053 Reliability of weapon 0.540 0.173 0.048 Reliability Reliability of weapon company 0.460 0.173 0.048 From this table, there is no big difference between training time and the number of people required in terms of wei ghts. Field maintenance appears to be about two times more important than depot maintenance. Weapon’s re liability is considered as slightly more important than company’s reliability. 5.5 Summary Up to now, weights of the same level attr ibutes having the same upper level attribute sum to one as shown in the prev ious tables. However, a final weight should be computed in terms of its upper level attribute’ s weight and hence all the botto m level attributes can sum to one. A tree structure is used to obtain the final weights (Yoon and Hwang, 1995). The final weights for attributes at each twig of the tree of Figure 4.7 are obtained by multiplying through the branches. Figure 5.1 shows the entire weight assessment process.

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103 Figure 5.1 Weight Assessments for Best Weapon Procurement Based on the above figure, we present attribut e weightings for our problem structure in Table 5.8. In this table, numbers prefixed in each attribute are the same as subscripts in Figure 5.1. There are fifteen bottom level attributes in th is table. These attribut es are assumed to be independent with each other. This is the basic assumption of MADM methods (see Yoon and Hwang (1995), Saaty (1980) for more detail). In this table, combat ope rational performance is determined as the most important attribute wh ile training time and nu mber of people required are the least important attributes.

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104 Table 5.8 Attribute Weightings fo r the Best Weapon Procurement Attribute (weight) Weight 1. Operational Performances 1.1 Combat operational performances 0.204 1.2 Regular operational performances 0.090 2. Readiness on Time 2.1 Readiness of operators 2.1.1 Training time 0.011 2.1.2 Number of people required 0.010 2.2 Readiness of weapons 0.029 2.3 Readiness of supporting systems 0.023 3. Technical Merits 3.1 Percentage of domestic components usages 0.066 3.2 Technology acquisitions 0.111 4. Cost Effectiveness 4.1 First acquisition costs 0.043 4.2 Operational costs 0.083 5. Sustainment 5.1 Logistics 0.142 5.2 Maintenance 5.2.1 Field maintenance 0.069 5.2.2 Depot maintenance 0.043 5.3 Reliability 5.3.1 Reliability of weapons 0.041 5.3.2 Reliability of company 0.035 Total 1 In the next chapter, we apply this hierarchical problem structure into the real problem in the SKN with the BSM. This application will validate the BSM.

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105 Chapter Six Case Study 6.1 Introduction In 1998, South Korea considered procuring a fe w submarines from Russia. At that time, Russia had borrowed approximately 0.18 billion do llars from South Korea and could not return that money. Instead, Russia wanted to re fund the debts with th eir military weapons. The type Kilo submarine was offered to South Korea by Russia for this reason. The payment condition for this submarine was consid ered 30% from the debts and 70% from cash ( Dongailbo 8.3.2000). Since the price of this submarine was 0.2 billion dollars ( Chosunilbo 12.21.2004), South Korea could buy this submarin e for only 0.14 billion dollars. This price was less than half price of th e type 214 submarine made by Germany. Type 214 submarine was considered as the same type of type Kilo s ubmarine and its price wa s 0.3 billion dollars ( Shindonga July 2001). Since Russia could return their debts and South Korea could obtain important weapon systems for a good price, this offer was attr active for the South Korean government. During this time, SKN was going to devel op submarine power and already had some submarines from Germany. Therefore, all the supporting and operatin g systems were setup with respect to German submarines. In order to be able to operate this Russian submarine, SKN would have to expend a lot of effort to c onstruct all the supporting and operating systems, which could be considered a double invest. Therefore, type Kilo submarine was not an

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106 attractive plan for the SKN. As a result, SKN did not agree with the Government’s plan. Instead, SKN asked the Government to procure t ype 214 submarines for the following reasons. 1. Many military weapon analysis organizations such as Military review and Naval technology have data that clearly show that type 214 has better performance measures than type Kilo submarines do. 2. Type 214 submarine has advantages in term s of logistics and maintenance because SKN has used similar type submarines from Germany. Therefore, SKN can use current logistics and maintenance systems. However, if type Kilo submarines are used, SKN has to construct all these sy stems and costs would rise. 3. Germany suggests giving core technology (i .e., submarine design technology) to the SKN if their submarines are accepted by the SKN. However, Russia does not suggest this core technology transfer. As explained by the above reasons, type 214 submarine has many advantages over type Kilo submarine. Therefore, the Government cancelled the plan of procuring type Kilo submarines. However, since this submarine coul d be obtained for a good price, the offer from Russia was attractive to the South Korean Government. In this chapter, we compute these two subm arines’ ranking scores by the BSM in terms of two cases. 1. The SKN’s view point. In this case, weight s in Table 5.8 are used in computing these two submarines ranking scores because these weights came from the SKN. 2. The Government’s view point. In this case, we ights that are artificially assigned for the purpose of aiming to represent Government’s intention are used in computing these two ranking scores.

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107 Since type 214 submarine is reviewed as a more proper decision for the SKN, a good decision model must select this submarine as the best alternative in both cases. The BSM can select type 214 submarines as the best altern ative in both cases while current MADM methods can not. This result can be considered as a justification for our new model, the BSM. 6.2 Data Collection for Evaluating Submarines Before we compute alternative ranking scores, we need to collect data for each alternative with respect to all th e attributes. In this section, data for these two submarine types are collected. Various sources of information such as Military review and Naval technology are used for this purpose. For the qualitative data, we use indexation tables presented in Chapter 4. 6.2.1 Data for Operational Performance As shown in Table 5.8, combat operational pe rformance is the most important attribute. However, there is a restriction in collecting data for this attribute (i.e ., conducting actual war game for this research is not allowed due to m ilitary secrete purpose). Therefore, responses for this attribute are assumed to be equal be tween type 214 and type Kilo submarine. Regular operational performance can be m easured by submarine speed, diving depth, cruse range, attack ability, and mission endurance Table 6.1 presents these performance data for both submarines. Note that Naval technology is an internet resource for navy ship technology, information on naval projects, conferences, exhi bitions and suppliers as well as a detailed manufacturer directory.

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108 Table 6.1 Data for Regular Oper ational Performances (Source: Naval technology ) Regular operational performance measures (units) Type Kilo Submarine Type 214 Submarine Maximum submerged speed (knots) 17 20 Maximum surface cruise range (NM) 6,000 12,000 Maximum submerged cruise range (NM) 400 420 Maximum diving depth (meters) 300 400 Attack ability (anti submar ine, anti surface, and an ti air) Similar Similar Mission endurance (days) 45 50 6.2.2 Data for Readiness on Time From interviews with some SKN submarin e officers, we obtain that there is no significant difference between the two submarines in terms of training time and number of people required. For the readiness of weapons, both submarines can be considered as outstanding (see Table 4.2). This consideration is possible because they can be supplied within the time required by the SKN. Therefore, score 5 is given to both alternatives. By the reasoning in the previous section and Table 4.3, type 214 and type Kilo submarines are classified respectably as outstan ding and unsatisfactory in terms of readiness of supporting systems. Therefore, score 5 and 1 ar e given to type 214 and type Kilo submarine respectably. Table 6.2 presents data for readiness on time. Table 6.2 Data for Readiness on Time Attributes Type Kilo Subm arine Type 214 Submarine Training time No difference No difference Number of people required No difference No difference Readiness of weapons 5 5 Readiness of supporting systems 1 5

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109 6.2.3 Data for Technical Merits and Cost Effectiveness Howaldtswerke Deutsche Werft (HDW) is a German company which builds and exports type 214 submarines. HDW offered that they can give submarin e design technology for their submarines. This technology is very important, especially for developing a new type of submarine. Therefore, type 214 submarine can be considered as an excellent condition in terms of technology acquisition (see Ta ble 4.4). However, type Kilo submarine does not have any technical merits and hence is classified as poor condition for the same attribute. Therefore, score 4 and 1 are given to type 214 and type Kilo submarine, respectively. The following table shows the responses of these two submarines on the attribute of technical merits. Since no domestic components ar e used in both alternatives, value 0 is given for the percentage of domestic component usage. Table 6.3 Data for Technical Merits Attributes Type Kilo Subm arine Type 214 Submarine Percentage of domestic component usage 0 0 Technology acquisition 1 4 Table 6.4 shows data for cost effectivene ss. First acquisition co sts are explained in Section 6.1. However, we could not obtain data for the operational costs for military secretes. Table 6.4 Data for Cost Effectiveness (Source: Shin-donga July 2001) Attributes Type Kilo Subm arine Type 214 Submarine First acquisition costs (million dollars) 140 300 Operational costs NA NA

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110 6.2.4 Data for Sustainment Both companies of type 214 and type Kilo submarines can be considered as normal suppliers in terms of logistics (see Table 4.5). Therefore, score 2 is given to both alternatives on the attribute of logistics. For the depot maintenance, type 214 and type Kilo submarin es are classified respectively as the best and worst alternatives for the following reason. Type 214 submarines are supposed to be built and maintained by a Korean domestic company based on technologies given by HDW, while type Kilo submarines should be sent to Russia for the depot maintenance. Therefore, score 3 and 1 are given to type 214 and type Kilo submarines respectively with respect to the attribute of depot maintenance. Type 214 submarine can be classified as the above average in terms of field maintenance because of the following reasons. SKN already has some submarines from Germany and they are not much different than type 214 submarines in terms of field maintenance. Therefore, operators can maintain this submarine with ease. Contrast to type 214 submarine, type Kilo is consider ed as below average in terms of field maintenance because SKN has never used Ru ssian submarines before. Therefore, based on Table 4.7, score 4 and 2 are gi ven to type 214 and type Kilo submarines respectively for the attribute of field maintenance. Table 6.5 shows data for logistics and maintenance. Table 6.5 Data for Logistics and Maintenance Attributes Type Kilo Subm arine Type 214 Submarine Logistics 2 2 Depot maintenance 1 3 Field maintenance 2 4

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111 Since 1991, Defense News has published the Defense News Top 100, a ranking and report about the world's leading defense companie s. The highlight of th is report is the annual list of the world's top 100 defense companies based on defense revenu es. Table 6.6 shows 2004 defense company rankings. In th is table, one can see that T hyssenKrupp Werften (i.e., the parents company of HDW) is ranked 39th. However, Rosvoorouzhenie (i.e., Russian state owned company which makes type Kilo submarin es) has not been ranked in 100 ranking list since 2000. Table 6.7 shows Rosvoorouzhenie’s last ranking within 100 ranking list. Table 6.6 2004 Defense Company Rankings (Source: Defense News ) Rank Company Country 2002 Rank 2003 Defense Revenue (million $) 2003 Total Revenue (million $) Percent of Revenue from Defense (%) 2002 Defense Revenue (million $) 1 Lockheed Martin U.S. 1 30,097.0 31,824.0 94.6 23,337.0 2 Boeing U.S. 2 27,360.0 50,500.0 54.2 22,033.0 3 Northrop Grumman U.S. 5 18,700.0 26,200.0 71.4 12,278.1 4 BAE Systems U.K. 4 17,159.0 22,359.3 76.7 15,036.4 39 ThyssenKrupp Werften Germany NR 1,110.0 6,152.9 18.0 955.0 Table 6.7 2000 Defense Company Rankings (Source: Defense News ) Rank Company Countr y 1998 rank 1999 Defense Revenue (million $) 1999 Total Revenue (million $) Percent of Revenue from Defense (%) 1999 Net Income (million $) 1 Lockheed Martin U.S. 1 17,800.00 25,500.00 69.80 382.00 2 Boeing U.S. 2 16,250.00 58,000.00 28 1,120.00 3 BAE Systems U.K. 4 15,200.00 19,400.00 78.4 491.5 4 Raytheon Co. U.S. 3 14,489.00 19,841.00 73 404 12 Rosvoorouzhenie Russia 14 2,830.00 2,830.00 100 NA

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112 Since Rosvoorouzhenie is not listed in th e 2004 company ranking list, we arbitrary define this company’s ranking as 101st in the ranking list. 101st ranking is the first ranking out of 100 ranking list. Data for reliability of weapons are not available to obtain for the military secrets. However, within the military this data can be obtained and hence can be applied for computing alternative ranking orders. In this case, we assume that there is no difference between two alternatives in terms of reliability of weapons. Table 6.8 shows the responses of these two alternatives on the attribute of reliability. Table 6.8 Data for Reliability Attributes Type Kilo Subm arine Type 214 Submarine Reliability of weapons NA NA Reliability of companies 101 39 6.2.5 Summary Let us define two alternatives: 1A for type 214 submarine and 2A for type Kilo submarine. The values of these alternatives on each attribute described in previous sections can be summarized in Table 6.9. From Table 6.9, values for combat operati onal performance, operational costs, and the reliability of weapon are given as 1 for both alte rnatives because these values are not available to obtain at this time. In addition, it is know n that there is no diffe rence between the two alternatives in terms of attack ability, training time, and the number of people required. Therefore, values for these attributes are given as 1 for both alternatives. Note that training time, number of people re quired, first acquisiti on costs, operational costs, and reliability of company are cost attrib utes and the others are benefit attributes. Their normalized values are computed in the following section.

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113 Table 6.9 Data for Evaluation of Submarines Alternatives Attribute (weight) Weight Type 214 (1A) Type Kilo (2A) 1. Operational Performances 1.1 Combat operational performances 0.204 1 1 1.2 Regular operational performances 1.2.1 Maximum submerged speed (knots) 0.015 20 17 1.2.2 Maximum surface cruise range (NM) 0.015 12,000 6,000 1.2.3 Maximum submerged cruise range (NM) 0.015 420 400 1.2.4 Maximum diving depth (meters) 0.015 400 300 1.2.5 Attack ability 0.015 1 1 1.2.6 Mission endurance (days) 0.015 50 45 2. Readiness on Time 2.1 Readiness of operators 2.1.1 Training time 0.011 1 1 2.1.2 Number of people required 0.010 1 1 2.2 Readiness of weapons 0.029 5 5 2.3 Readiness of supporting systems 0.023 5 1 3. Technical Merits 3.1 Percentage of domestic components usages 0.066 0 0 3.2 Technology acquisitions 0.111 4 1 4. Cost Effectiveness 4.1 First acquisition costs (million $) 0.043 300 140 4.2 Operational costs 0.083 1 1 5. Sustainment 5.1 Logistics 0.142 2 2 5.2 Maintenance 5.2.1 Field maintenance 0.069 4 2 5.2.2 Depot maintenance 0.043 3 1 5.3 Reliability 5.3.1 Reliability of weapons 0.041 1 1 5.3.2 Reliability of company 0.035 39 101 Total 1 6.3 Alternatives Rankings Based on SKN’s View Points From Table 6.9, we can compute alternat ives’ normalized and weighted normalized values shown in Table 6.10. Each alternative ranking score is computed based on Table 6.10. Following the definition in Section 6.2.5, j jr x1 1, and jv1 represent the values of type 214 submarines for an attribute j Like wise,j jr x2 2, and jv2 represent the values of type Kilo submarines. Subscript numbers of each attribute are the same as numbers prefixed in each attribute in Table 6.9.

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114 Table 6.10 Alternatives’ Normalized and Weighted Normalized Values Alternative values (ijx) Alternative’s normalized values (ijr) Alternative’s weighted normalized values (ijv) Attributes (jX) Weights (jw) jx1 jx2 jr1 jr2 jv1 jv2 11X 0.204 1 1 1 1 0.204 0.204 121X 0.015 20 17 1 0.85 0.015 0.013 122X 0.015 12,000 6,000 1 0.5 0.015 0.008 123X 0.015 420 400 1 0.952 0.015 0.014 124X 0.015 400 300 1 0.75 0.015 0.011 125X 0.015 1 1 1 1 0.015 0.015 126X 0.015 50 45 1 0.9 0.015 0.014 211X 0.011 1 1 1 1 0.011 0.011 212X 0.010 1 1 1 1 0.010 0.010 22X 0.029 5 5 1 1 0.029 0.029 23X 0.023 5 1 1 0.2 0.023 0.005 31X 0.066 0 0 0 0 0 0 32X 0.111 4 1 1 0.25 0.111 0.028 41X 0.043 300 140 0.167 1 0.020 0.043 42X 0.083 1 1 1 1 0.083 0.083 51X 0.142 2 2 1 1 0.142 0.142 521X 0.069 4 2 1 0.5 0.069 0.035 522X 0.043 3 1 1 0.333 0.043 0.014 531X 0.041 1 1 1 1 0.041 0.041 532X 0.035 39 101 1 0.386 0.035 0.014 Total 1 18.467 14.622 0.911 0.732 Alternatives’ ranking scores ar e computed from Table 6.10. Table 6.11 presents the two alternatives’ ranking scores computed by thr ee MADM methods (i.e., SAW, TOPSIS, and the BSM). From this table, one can se e all three MADM methods select 1A as the best alternative. Therefore, we can say that SKN’s opposition to the Government’s plan about type Kilo submarine is reasonable.

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115 Table 6.11 Ranking Scores for the Submarine Selection Problem BSM Alternatives ASW Rank TOPSIS Rank IV BSM Rank Type 214 submarine (1A) 0.911 1 0.812 1 0.923 0.917 1 Type Kilo submarine (2A) 0.732 2 0.188 2 0.731 0.732 2 There are three data that we could not obtain (i.e., data for combat operational performance, operational costs, and reliability of weapon). Even if 1A is considered to have better values for these attributes, since no exact data are available, we need to present what if analysis for these attribute values. This anal ysis can be done by the sensitivity analysis explained in Chapter 3. An a lternative ranking score (i.e.,) (iA V ) is defined as the BSM score for this sensitivity analysis. Since ) ( ) (2 1A V A V and each 1 ijr for i1 and 2, and j 11, 42, and 531, the ranking change is only possible w ithin the changes (i.e., decrea se in these three attribute values) in1A. This is the scenario 2 of Section 3.4.1. c cr r142 111, and cr1531 are computed by Equation 3.11. These critical valu es are presented in Table 6.12. Table 6.12 Critical Values for 1A in terms of) ( ) (2 1A V A V Attributes (jX) 11X 42X 531X c jr1 -0.462 -1.792 -3.080

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116 From Table 6.12, since c jr1allowable range (see Equation 3.12), sensitivity analysis does not apply for these attribute values. In other words, regardless of how bad scores 1A has for these attributes, this alternative can be ranked first in terms of sensitivity analysis. 6.4 Alternative Rakings From the Government’s View Points To represent the Government’s intention, wei ght values can be modified such as in the following table. In this case, we arbitrarily a ssume that all attribute weights are equal except the attribute of first acquisition cost. That is because the Government considers that the first acquisition cost is much more important than any other decisi on factors, even though they do not know exactly what factors are the most important. Table 6.13 shows the case that the Government considers the first acquisition cost is important as much a bout 40% of the entire decision factors.

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117 Table 6.13 Data for Evaluation of Submar ines with Modified Weight Values Alternative values (ijx) Alternative’s normalized values (ijr) Alternative’s weighted normalized values (ijv) Attributes (jX) Weights (jw) jx1 jx2 jr1 jr2 jv1 jv2 11X 0.03 1 1 1 1 0.204 0.204 121X 0.03 20 17 1 0.85 0.015 0.013 122X 0.03 12,000 6,000 1 0.5 0.015 0.008 123X 0.03 420 400 1 0.952 0.015 0.014 124X 0.03 400 300 1 0.75 0.015 0.011 125X 0.03 1 1 1 1 0.015 0.015 126X 0.03 50 45 1 0.9 0.015 0.014 211X 0.03 1 1 1 1 0.011 0.011 212X 0.03 1 1 1 1 0.010 0.010 22X 0.03 5 5 1 1 0.029 0.029 23X 0.03 5 1 1 0.2 0.023 0.005 31X 0.03 0 0 0 0 0 0 32X 0.03 4 1 1 0.25 0.111 0.028 41X 0.43 300 140 0.167 1 0.020 0.043 42X 0.03 1 1 1 1 0.083 0.083 51X 0.03 2 2 1 1 0.142 0.142 521X 0.03 4 2 1 0.5 0.069 0.035 522X 0.03 3 1 1 0.333 0.043 0.014 531X 0.03 1 1 1 1 0.041 0.041 532X 0.03 39 101 1 0.386 0.035 0.014 Total 1 18.467 14.622 0.911 0.732 Based on this assumption, we can compute the two alternatives ranking scores as shown in Table 6.14.

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118 Table 6.14 Ranking Scores Based on Modified Attribute Weights BSM Alternatives ASW Rank TOPSIS Rank IV BSM Rank Type 214 submarine (1A) 0.741 2 0.175 2 0.923 0.832 1 Type Kilo submarine (2A) 0.839 1 0.825 1 0.731 0.785 2 As the results shown in Table 6.14, only the BSM selects 1A as the best alternative and others do not. The reason how the BSM can select 1A as the best alternative is that this alternative has better IV value than2A. However, since the difference of these IV values is not as big as in the extreme alternative problem in Section 3.3.1 (see Table 3.3), the BSM will not select 1A as the best when55 041 w. One example of this case is shown in Table 6.15. Table 6.15 Ranking Scores Based on Modified Attribute Weights (when56 041 w) BSM Alternatives ASW Rank TOPSIS Rank IV BSM Rank Type 214 submarine (1A) 0.675 2 0.111 2 0.923 0.799 2 Type Kilo submarine (2A) 0.873 1 0.889 1 0.731 0.802 1 The results shown in Table 6.15 can be explained by the following reason: even if 2A does have very few good attribute values compared to1A, each value’s gap is not as serious as in the extreme alternative case. However, when th ere is a political pressure such as in Table 6.13, only the BSM can work properly in terms of best weapon decisions.

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119 6.5 Summary From this submarine selection problem, one can see how seriously a DM can affect the final decision. A DM can be any person with po wer such as a political leader, a group of DMs who decide a final decision, and so on. We sh owed that the current MADM methods did not work properly under this political power. However, the BSM could avoid this political pressure by compensating an overall value for individual attribute values.

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120 Chapter Seven Summary and Further Research 7.1 Summary At the end of the nineteenth century and the beginning of the twentiet h, the research for better decision making has begun by applying ec onomic theory into human decision process. By 1960, MCDM acquired its own vocabular y (Pomerol and Romero, 2000). From 1975, numerous researches were made in MCDM area s. During this time, MCDM was divided into two different areas (i.e., MODM and MADM). Fi shburn (1970), and Keeney and Raiffa (1976) are representative researcher s in MODM area. Zionts (1978) Saaty (1980), Yoon (1980), and Zeleny (1982) are representative researchers in MADM area. By 1985, these methods had been recognized with many countries cont ributing (Pomerol and Romero, 2000). In this research, we reviewed many M ADM methods and found that there is a drawback in the current methods (i.e., lack of ability addressing political pressures which can be obstacles for the best weapon procurements ). Therefore, the idea of compensating an alternative’s overall value for its individual attribute values is suggested for overcoming this drawback. This idea is based on the following reasons: DMs can change an alternative’s overall value by changing some weights but can not change alternatives’ attribute values. For compensating these two values, the conc ept of two statistics (i.e., mean and variance) was introduced. Then we found that the SAW method and the TOPSIS can be used for representing mean and variance type of in formation, respectively. However, since the TOPSIS use a non-linear function, we developed a new value function, which is linear and is

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121 called IV. Based on the concept of compensating mean type value for a variance type value, the SAW and IV are added and the sum is used as an alternative ranking score. This new method is referred to as the BSM. The BSM can compensate an alternative’s overall value for its individual attribute values. To further strengthen the proposed BSM, we presented a sensitivity analysis for what-if analysis for both consumers and suppliers. Com putation results on several numerical examples indicate that the BSM can work properly as a generalized decision making model for naval weapon procurement, even when there are any political pressures th at lead to extreme alternatives. We expect this method can also work well in other decision making situations, especially when the decision can be easil y affected by some political powers. 7.2 Further Research The BSM value function has the same wei ght for both overall and individual value functions (i.e., 0.5 for both the SAW and IV). However, one can ask that 0.5 may or may not be the best weight. And this question can be answered by determining two parameter values, and shown in Equation 7.1. This equation ca me from the BSM value function with the consideration of different weights for the SAW and IV. m j ij j ir m w A V1) ( 0 and 1 (7.1) From this equation, alternatives rank can be changed by assigning different values of and For example, let us consider the same probl em in Table 3.1. There are two alternatives 1A and 2A, and 1A is considered as an extreme alte rnative because it has overall good score

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122 due to one extremely high attrib ute value. In other words, 2A is ranked second by current MADM methods even if it has i ndividually good attribute values. However, these alternatives have different ranking scores as well as different rank or der by Equation 7.1 with different and The table shows their ranking scores as well as rank orde r with respect to different and Table 7.1 Alternative Ranking Scores for Different and 1 0.9 0.886 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 0.114 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ) (1A V 0.602 0.562 0.557 0.522 0.483 0.443 0.403 0.363 0.323 0.284 0.244 0.204 ) (2A V 0.520 0.552 0.557 0.584 0.616 0.648 0.680 0.712 0.744 0.776 0.808 0.840 1A 1 1 1 2 2 2 2 2 2 2 2 2 Rank 2A 2 2 1 1 1 1 1 1 1 1 1 1 From this table, one can see that when >0.1143, ) ( ) (2 1A V A V However, determining these parameter values is another decision process, which can be difficult. One possible way is to ask DMs’ opi nions. This can be done either by an individual interview or brainstorming. This method can be useful because the two parameter values might be depend on weapon procurement environment.

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123 References Ahn, Youngsoo (2000). A Study on the Development Di rection of the Acquisition Management Work for the Weapon System s of the Defense Investment Project Unpublished Master thesis. University of Hannam. Korea. 13-14. Birolini, Alessandro (1999). Reliability Engineering Springer. New York. 2-3. Bard, Jonathan F. (1992). A Comparison of the Analytic Hierarchy Process With Multiattribute Utility Theory: A Case Study IIE Transactions. 24 111-121. Brauers, Willem K. (2001). The Multiplicative Representati on for Multiple Objectives Optimization with an Applica tion for Arms Procurement Naval Research Logistics. 49 327340. Business.com (http://www.business.com). A directory contains more than 400,000 listings within 65,000 industry, prod uct and service subcategories. Chang, Jin (1997). A Study on The Systematic Safety Management of Naval Ship Master Thesis. Advanced Institute of Milita ry Science and Technology. Seoul Korea. 79-81. Chosunilbo (http://www.chosun.com). Daily newspaper published in South Korea. Coombs, C. H. (1958). On the use of inconsistency of preferences in psychological measurement Journal of Experimental Psychology. 55 1-7. Coombs, C. H. (1964). A Theory of Data Wiley. New York. Dalkey, Norman C. (1972). Studies in the Quality of Life : Delphi and Decision-Making Lexington Books. London. 55-80. DefenseNews (http://www.defenseNews.com). Defense News provides the global defense community with the latest insight and news analysis on defense programs, policy, business and technology. Dongailbo (http://www.donga.com). Daily newspaper published in South Korea. Dyer, James S. (1990). Remarks on the Analytic Hierarchy Process Management Science. 36 249-258.

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124 Edwards, W. (1977). Use of Multiattribute utility measurement for social decision makin g. Wiley. New York. 247-276. Edwards, W., and Newman, J.R. (1982). Multiattribute Evaluation Sage University Paper series on Quantitative Applications in the Social Sciences. CA. Farmer, T.A. (1987). Testing the robustness of multia ttribute utility theory in an applied setting Decision Sciences. 18 178-193. Fechner, G. (1966). Elements of Psychophysics Translated by Helmut E. Adler, Holt, Rinehart, and Winston. New York. Fishburn, Peter C. (1970). Utility Theory For Decision Making John Wiley & Sons. New York. 1-15. Hall, A.D. (1989). Metasystems Methodology: A New Synthesis and Unifications Research Pergamon Press. Oxford. Hall, Nicholas G., John C. Hershey, Larry G. Kessler and R. Craig Stotts (1992). A Model For Making Project Funding Decisi ons At the National Cancer Institute Operational Research. 40 1040-1052. Hobbs, B.F. (1980). A comparison of weighting methods in power plant citing. Decision Sciences 11 725-737. Hughes, Warren R. (1986). Deriving Utilities Using the Analytic Hierarchy Process Socio-Econ. Plann. Sci. 20 393-395. Hwang, C. L., and Yo on, K., (1981). Multiple Attribute Decision Making: Methods and Applications Berlin/Heidelberg/New York. Springer-Verlag. Hwang, C.L., Lai, U.J., and Liu, T.Y. (1993). A new approach for multiple objective decision making Computers and Operation Research. 20 889-899. Kamenentzky, R.D., (1982). The relationship between th e AHP and the additive value function Decision Science. 13 702-713. Keeney, Ralph L. and Howard Faiffa (1976). Decisions with Multiple Objectives: Preferences and Value Tradeoffs John Wiley & Sons, New York. 219-344. Kim, Jong-Ha (2000). Arms Procurement Decision Making: Principle, Problem, and Alternative Korea. 20-50. Kim, Tae-Un (1987). Decision support system for th e weapon system acquisition. Master thesis Korea Advanced Institute of Science and Technology. Seoul Korea. Kwon, H.C. (2003). Trends of defense expenditure ratio to government and GDP Hankyoreh News. South Korea.

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125 Korean navy regulation rule book 2 (2003). Lee, K.O. and Park, J.W. (1995). Sampling Theory KNOU Press. Seoul, Korea. Loerch, Andrew G., Robert R. K oury, and Daniel T. Maxwell (1998). Value Added Analysis for Army Equipment Modernization Naval Research Logistics. 46 233-253. MacCrimmon K.R. (1973). An overview of multiple objective decision making in Multiple criteria decision making University of South Caroli na Press, Columbia, USA. McAnarney, D.K. (1987). Multiple Attribute Decision Making Methods: A Comparative Study Unpublished master’s thesis, Kansas Sate University. Miller, G. A. (1956). The magical number seven plus or minus two: some limits on out capacity for processing information Psychological Rev. 63 81-97. Military review (http://www. militaryreview.com). A web re source for military weapons published in South Korea. Morris, W.T. (1964). The analysis of management decisions Homewood, IL: Irwin. Nijkamp P., and Van Delft, A. (1977). Multi-Criteria Analysis and Regional Decision Making Marinus Nijhoff. Leiden, the Netherlands. MND (South Korean Ministry of National Defense) (2004). A rank for defense expenditures in 2002 A public link data, www.mnd.go.kr. Naval technology (http://www.naval-technol ogy.com). Established in 1972, SPG Media PLC is an international business-to-business media company providing world-class controlled circulation magazines, Internet reference portals and business conferences and forums. Pardee, E.S. (1969). Measurement and Evalustion of Transportation System Effectiveness RAND Memorandum RM-5869-DOT. Pomerol, Jean-Charles and Sergio Barba-Romero (2000). Multicriterion Decision in Management: Principles and Practice. Kluwer Academic Publishers. Boston. 2-6, 145-170. Rahman, A. (2003). Multi-attribute utility analysisa major decision aid technique Nuclear Energy. 42 87-93. Roy, B. (1971). Problems and Methods with Multiple Objective Functions Mathematical Programming. 1 239-266. Saaty, Thomas L. (1980). The Analytic Hierarchy Proces s: Planning, Priority Setting, Resource Allocation McGraw-Hill. New York. 1-35. Saaty, Thomas L. (1986). Exploring Optimization Through Hierarchies and Ratio Scales Socio-Econ. Plann. Sci. 20 355-360.

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126 Saaty, Thomas L. and Luis G. Vargas (2001). Models, Methods, Concepts & Applications of the Analytic Hierarchy Process Kluwer Academic Pub lishers. Boston. 1-45, 305-315. Sin, Myung-Ho (1988). Application of the AHP method for a MODM Airforce Research (South Korea). 3 119-120. Shin-donga (http://shindonga.donga.com). A monthly news magazine published in South Korea. Starr, M.K., Zeleny M. (1977). Multiple Criteria Decision Making, Time Study in Management Sciences North-Holland. Amsterdam. Stevens, S. S. (1957). On the Psychophysical Law Psychological Reviews. 64 153-181. Stillwell, W.G., Seaver, D.A ., and Edwards, W. (1981). A comparison of weight approximation techniques in mult iattribute utilit y decision making Organizational Behavior and Human Performance 28 62-77. Sher, William and Rudy Pinola (1981). Microeconomic Theory North Holland. New York. 185-232. Voogd, H. (1983). Multicriteria Evaluation for Urban and Regional Planning Pion. London. Weber, E. H. (1978). The sense of touch. Academic Press for Experimental Psychology Society. New York. Yeh, J-M, C Lin, B Kreng and J-Y Gee (1995). A Modified Procedure for Synthesising Ratio Judgements in the Analytic Hierarchy Process Journal of the Operational Research Society. 50 867-873. Yoon, K. Paul and Ching-Lai Hwang (1995). Multiple Attribute Decision Making Sage Publications. Iowa City.2-69. Yoon, Kwangsun (1980). Systems Selection By Multiple Attribute Decision Making Ph.D. Dissertation. Kansas State University. 8-11. Yoon, K. (1987). A reconciliation among discre te compromise situations Journal of Operational Research Society. 38 277-286. Yu, P.L. (1985). Multiple Criteria Decision Making: Concepts, Techniques and Extensions Plenum. New York. Zionts, S. (1978). Multiple Criteria Problem Solving Springer. Zeleny, M. (1982). Multiple Criteria Decision Making McGraw-Hill. New York.

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127 Appendices

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128 Appendix A. Survey Forms for Pairwise Comparisons Part 1 This survey form is designed to determin e weights for the decision model of naval weapon procurement. Your sincere answers are ve ry important to develop a successful decision model for naval weapon procurement. This mode l is expected to help the South Korean navy choose the best weapon in terms of performance as well as cost s. Please read each question carefully and give your answers. 1. General questions a. What is your current rank? _________________ b. How many years have you been in the Navy? ____________ years c. Have you ever worked for weapon procurements? _____ Yes _____ No 2. What is your opinion regarding the cu rrent weapon procurement system? a. It is a very proper system b. It is a proper system c. It should be improved d. It should be improved immediately 3. If you think that the current weapon procuremen t system is required to be improved, what is the most important problem that you are considering? a. No, there is no need to be improvement. b. There is no generalized weapon procuremen t decision model that can help decision makers decide best weapon. c. Not enough experts are in the Navy who can decide best weapons. d. There is a political pressure which can obstruct best weapon selection. e. The decision procedures are not open to the public. Part 2 to be continued!

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129 Appendix A. (Continued) Part 2 The criteria below show how to compare two factors at a time. For example, if you think that factor A is very strongly more important than f actor B, you should mark on the number 7 placed in A-side. The table right below the criteria shows this example. Criteria for pairwise comparisons Intensity of importance Definition Explanation 1 Equal Importance Two activities contribute equally to the objective 3 Moderate importance Experience and judgment slightly favor one activity over another 5 Strong importance Experience and judgment strongly favor one activity over another 7 Very strong or demonstrated importance An activity is favored very strongly over another; its dominance demonstrated in practice 9 Extreme importance The evidence fa voring one activity over another is of the highest possible order of affirmation *** Intermediate scales such as 2, 4, 6, and 8 are possible to use! Example The case that you think that factor A (Economic) is very strongly more important than factor B (Education) in terms of allocating budget. Factor (A) Relative Importance Factor (B) Economic 9 7 5 3 1 3 5 7 9 Education A is more important B is more important

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130 Appendix A. (Continued) 1. Among the five principles (i.e., operationa l performances, readiness on time, technical merits, cost effectiveness, and sustainment) which affect the deci sion for best weapon selection, please determine a relative importance between each principle in terms of the best weapon selection. Factor (A) Relative Importance Factor (B) Operational performance 9 7 5 3 1 3 5 7 9 Readiness on Time Operational performance 9 7 5 3 1 3 5 7 9 Technical merits Operational performance 9 7 5 3 1 3 5 7 9 Cost effectiveness Operational performance 9 7 5 3 1 3 5 7 9 Sustainment Readiness on Time 9 7 5 3 1 3 5 7 9 Technical merits Readiness on Time 9 7 5 3 1 3 5 7 9 Cost effectiveness Readiness on Time 9 7 5 3 1 3 5 7 9 Sustainment Technical merits 9 7 5 3 1 3 5 7 9 Cost effectiveness Technical merits 9 7 5 3 1 3 5 7 9 Sustainment Cost effectiveness 9 7 5 3 1 3 5 7 9 Sustainment A is more important B is more important

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131 Appendix A. (Continued) 2. Among the two factors (i.e., combat operatio nal performances and regular operational performances) which compose the principle of operational performance, please determine a relative importance between thes e two factors in terms of w eapon’s operational performance. Factor (A) Relative Importance Factor (B) Combat operational performances 9 7 5 3 1 3 5 7 9 Regular operational performances A is more important B is more important 3. Among the three factors (i.e., readiness of operators readiness of weapon s, and readiness of supporting systems) which compose the principl e of readiness on time, please determine a relative importance between these three f actors in terms of readiness on time. Factor (A) Relative Importance Factor (B) Readiness of operators 9 7 5 3 1 3 5 7 9 Readiness of weapons Readiness of operators 9 7 5 3 1 3 5 7 9 Readiness of supporting systems Readiness of supporting systems 9 7 5 3 1 3 5 7 9 Readiness of supporting systems A is more important B is more important

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132 Appendix A. (Continued) 4. Among the two factors (i.e., percentage of domestic components usage and technology acquisitions) which compose the principle of technical merits, please determine a relative importance between these two factors in terms of technical merits. Factor (A) Relative Importance Factor (B) Percentage of domestic components usage 9 7 5 3 1 3 5 7 9 Technology acquisitions A is more important B is more important 5. Among the two factors (i.e., first acquisition costs and operational costs) which compose the principle of cost effectiveness, please dete rmine a relative importance between these two factors in terms of cost effectiveness. Factor (A) Relative Importance Factor (B) First acquisition costs 9 7 5 3 1 3 5 7 9 Operational costs A is more important B is more important

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133 Appendix A. (Continued) 6. Among the three factors (i.e., logistics, ma intenance, and reliability) which compose the principle of sustainment, please determine a relative importance between these three factors in terms of sustainment. Factor (A) Relative Importance Factor (B) Logistics 9 7 5 3 1 3 5 7 9 Maintenance Logistics 9 7 5 3 1 3 5 7 9 Reliability Maintenance 9 7 5 3 1 3 5 7 9 Reliability A is more important B is more important 7. Among the two sub-factors (i .e., training time and number of people required) which compose the factor of readiness of operators, please determine a relative importance between these two sub-factors in term s of readiness of operators. Factor (A) Relative Importance Factor (B) Training time 9 7 5 3 1 3 5 7 9 Number of people required A is more important B is more important

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134 Appendix A. (Continued) 8. Among the two sub-factors (i.e., depot mainte nance and field maintenance) which compose the factor of maintenance, please determine a relative importance between these two subfactors in terms of maintenance. Factor (A) Relative Importance Factor (B) Depot maintenance 9 7 5 3 1 3 5 7 9 Field maintenance A is more important B is more important 9. Among the two sub-factors (i.e., reliability of weapon and supplying company) which compose the factor of reliability, please dete rmine a relative importance between these two sub-factors in terms of maintenance. Factor (A) Relative Importance Factor (B) Reliability of weapons 9 7 5 3 1 3 5 7 9 Reliability of company A is more important B is more important

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135 Appendix B. C++ Weight Computation Program #include #include #include #include #include #define MAX 16 float RI[MAX]={0, 0, 0, 0.58, 0.90, 1.12, 1.24, 1.32, 1.41, 1.45, 1.49, 1.51, 1.48, 1.56, 1.57, 1.59}; float matrix[MAX][MAX]; float row_rate[MAX]; int mtrx_size; FILE *in; // *************************************************************/ void printmatrix(){ int i,j; for(i=1; i<= mtrx_size; i++) for (j=1; j<=mtrx_size; j++) scanf("%f",&matrix[i][j]); printf("\n\n1. Given matrix is as follows !\n\n"); for(i=1; i<=mtrx_size; i++){ for (j=1; j<=mtrx_size; j++) printf("%6.2f", matrix[i][j]); printf("\n"); } } //**************************************************************/

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136 Appendix B. (Continued) void value_added_vector(){ float row_mul[MAX]; float row_nsqr[MAX]; float sum = 0.0; int i, j; for (i=1; i<=mtrx_size; i++){ row_mul[i]=1; for (j=1; j<= mtrx_size; j++) row_mul[i] *= matrix[i][j]; } for (i=1; i<=mtrx_size; i++){ row_nsqr[i] = pow(row_mul[i], (float) 1.0/mtrx_size); sum +=row_nsqr[i]; } printf("\n2. Value added Vector is as follows \n\n"); for (i=1; i<=mtrx_size; i++){ row_rate[i] = row_nsqr[i] / sum; printf("%7.3f", row_rate[i]); } } //**************************************************************/ void cal_cr(){ float prod[MAX] = {0.0, }; float con_did[MAX]; float sum1 = 0.0; float con_idx; float ci;

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137 Appendix B. (Continued) float cr; int i, j; for(i=1; i<=mtrx_size; i++) for (j=1; j<=mtrx_size; j++) prod[i] += matrix[i][j] row_rate[j]; for(i=1; i<=mtrx_size; i++){ con_did[i] = prod[i]/row_rate[i]; sum1 += con_did[i]; } printf("\n\n3. Consistency : "); con_idx=sum1/mtrx_size; ci= (con_idx mtrx_size)/(mtrx_size-1); cr=ci/RI[mtrx_size]; printf("%7.2f", cr); } //**************************************************************/ main(int argc, char* argv[]){ int i,j; /*if((in = fopen(argv[1], "rt")) == NULL){ printf("\n\nError opening the input file !!!\n"); exit(0); }*/ //cin>>mtrx_size; scanf("%d", &mtrx_size); printmatrix(); value_added_vector(); cal_cr();

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138 Appendix B. (Continued) //fclose(in); }

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About the Author Jin O Chang received a B.S. degree in Mana gement Science in 1992 from South Korea Naval Academy. He was then commissioned an ensign and worked as a naval officer until 1995. In 1995, he was selected as a navy scholarsh ip student for a Master degree. In 1997, he received a M.S. degree in Aero Space Engineeri ng from Advance Institute of Military Science and Technology, Seoul, Korea. After that he return ed to the navy and worked in various areas. In 2002, he was selected as a student to be sent abroad on government support and joined the Ph.D. Program in the Department of Industria l and Management System Engineering at the University of South Florida. His research interests include deci sion modeling, scheduling, statistical analysis, mathematical programming and simulation.


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datafield ind1 8 ind2 024
subfield code a E14-SFE0001025
035
(OCoLC)62752352
SFE0001025
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FHM
c FHM
049
FHMM
090
T56 (Online)
1 100
Chang, Jin O.
2 245
A generalized decision model for naval weapon procurement
h [electronic resource] :
b multi-attribute decision making /
by Jin O. Chang.
260
[Tampa, Fla.] :
University of South Florida,
2005.
502
Thesis (Ph.D.)--University of South Florida, 2005.
504
Includes bibliographical references.
516
Text (Electronic thesis) in PDF format.
538
System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
500
Title from PDF of title page.
Document formatted into pages; contains 149 pages.
Includes vita.
520
ABSTRACT: For any given reason, every year many countries spend a lot of money purchasing at least one weapon. Due to the secret character of the military, the decision process for specific weapon procurement is shrouded. Moreover, there are several funds loss cases due to mistakes in weapon contractions. Weapon procurement requires very large amounts of money which comes from tax payers. Therefore, an effort to reduce a possible monetary loss is needed. A decision process based on an analytic model can present a better chance to decision makers for better weapon decisions. In general, weapon procurement decision is a multi criteria environment. Decision making in such environments is defined as Multi-Criteria Decision Making (MCDM). MCDM is broadly classified into two areas: Multi-Attribute Decision Making (MADM) and Multi-Objective Decision Making (MODM). MADM methods are used for selecting an alternative from a small explicit list of alternatives.MODM methods are used for designing problems involving an infinite number of alternatives implicitly defined by mathematical constraints. This research is intended to be used by the South Korean Navy when there is a need to select one weapon type among several candidate types. Therefore, MADM methods are used in this research.Many researches for developing an analytical model for better decision-making have been done. However, there is no research for a generalized weapon procurement decision model that is easy to implement. For this reason, whenever there is a need for weapon procurement decision, the Navy has to spend a lot of effort in determining the best weapon. These efforts can be reduced with a generalized model that is proposed in this research for naval weapon procurement. MADM methods determine alternatives ranking orders and the highest ranked alternative is the best one. Various MADM methods are used in computing the alternatives ranking scores.
590
Adviser: Dr. M. Weng.
653
Best selection method.
Saw.
Topsis.
Sensitivity analysis.
Ahp.
Utility theory.
Hierarchy of attributes.
0 690
Dissertations, Academic
z USF
x Industrial Engineering
Doctoral.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.1025