決策球模式與其應用

行政院國家科學委員會補助專題研究計畫

□ 成 果 報 告

■期中進度報告

決策球模式及其運用

Decision Ball Models and Applications

計畫類別：■ 個別型計畫 □ 整合型計畫

計畫編號：

NSC 97-2221-E-009-104-MY3

執行期間：

97 年 8 月 1 日 至 98 年 7 月 31 日

計畫主持人：

黎漢林

共同主持人：

計畫參與人員：

馬麗菁、柯宇謙、黃曜輝

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執行單位：國立交通大學資訊管理研究所

行政院國家科學委員會專題研究計畫期中報告

決策球模式及其運用

Decision Ball Models and Applications

計 畫 編 號：97-2221-E-009-104-MY3

執 行 期 限：97 年 8 月 1 日至 98 年 7 月 31 日

主 持 人：黎漢林 國立交通大學資訊管理研究所

計畫參與人員：馬麗菁、柯宇謙、黃曜輝、

Abstract

Many decision-making or choice problems in Marketing incorporate preferences. How to assist decision makers in understanding the decision context and improving inconsistencies in judgments are two important issues in ranking choices. This study develops a decision-making framework based on the screening, ordering, and choosing phases. Two optimization models and a Decision Ball model are proposed to assist decision makers in improving inconsistencies and observing relationships among alternatives. By examining a Decision Ball, a decision maker can observe ranks of and similarities among alternatives, and iteratively adjust preferences and improve inconsistencies thus to achieve a more consistent and informed decision.

1. Introduction

Many decision-making or choice problems in Marketing incorporate preferences (Liechty et al, 2005; Horsky et al. 2006; Gilbride and Allenby, 2006). Keeney (2002) identified 12 important mistakes frequently made that limit one’s ability in making good value judgments, in which “not understanding the decision context” and “failure to use consistency checks in assessing value trade-offs” are two critical mistakes. Hence, how to assist decision makers in understanding the decision context and adjusting inconsistencies in judgments are two important issues in ranking choices.

There is evidence that decision makers’ preferences are often influenced by the visual background information (e.g., Simonson and Tversky 1992; Tversky and Simonson, 1993; Seiford and Zhu, 2003). From marketing it is known from consumer choice theories that context impacts the choices consumers make (Seiford and Zhu, 2003). For example, a product may appear attractive against a background of less attractive alternatives and unattractive when compared to more attractive alternatives (Simonson and Tversky, 1992). Visual representations can simplify and aggregate complex information into meaningful pattern, assist people in comprehending their environment, and allow for simultaneous perception of parts as well as a perception of interrelations between parts (Maruyama, 1986; Meyer, 1991; Sullivan, 1998). Hence, how to provide visual aids to help decision makers make a more informed decision is the first issue addressed by this study.

Ranking alternatives incorporating preferences is a popular issue in decision-making. One common format for expressing preferences is to use pairwise comparisons, which forces one to make a direct choice of one object over another when compariing two objects, rather than requiring one to comparing all objects simultaneously (Cook et al., 2005). For example, in sports competitions, such as tennis, football and baseball, pairwise rankings are the typical input (Hochbaum and Levin, 2006). Several methods have been proposed (e.g., Saaty, 1980; Jensen, 1984; Genest and Rivest, 1994) to rank alternatives in pairwise comparisons fashion. However, inconsistencies are not unexpected, as making value judgments is difficult (Keeney, 2002). The ranks different methods yield do not vary much when the decision makers’ preferences are consistent. But, if a preference matrix is highly inconsistent, different ranking methods may produce wildly different priorities and rankings. Hence, how to help the decision makers to detect and improve those inconsistencies thus to make a more reliable decision is the second issue addressed here.

Multicriteria decision makers tend to use screening, ordering and choosing phases to find a preference (Brugha, 2004). They tend to make little effort in the first phase as they screen out clearly unwanted alternatives, use somewhat more effort in the second phase as they try to put a

preference order on the remaining alternatives, and reach the highest effort in the final phase when making a choice between a few close alternatives.

This study develops a decision-making framework based on these three phases. Preferences in pairwise comparison fashion are adopted in the choosing phase. Two optimization models and a Decision Ball model are proposed to assist decision makers in improving inconsistency and observing relationships among alternatives. By examining Decision Balls, a decision maker can iteratively adjust preferences and improve inconsistencies thus to achieve a more consistent decision. The proposed approach can be extensively applied in Marketing. Possible applications are the selection of promotion plans, decisions regarding product sourcing, choice of marketing channels, evaluation of advertising strategy, research of customer behavior …etc.

The reasons why this study uses a sphere model instead of a traditional 2-dimensional plane or a 3-dimensional cube model are described as follows. A 2-dimensional plane model cannot depict three points that do not obey the triangular inequality (i.e. the total length of any two edges must be larger than the length of the third edge) neither can it display four points that are not on the same plane. For instance, as illustrated in Figure 1, consider three points, Q1, Q2, Q3,

where the distance between Q1Q2, Q2Q3, and Q1Q3 are 3, 1, and 6, respectively, as shown in

Figure 1(b). It is impossible to show their relationships by three line segments on a 2-dimensional plane, as shown in Figure 1(a). If there are four points, Q1, Q2, Q3, and Q4, which are not on the

same plane, as shown in Figure 1(c), it is impossible to present these four points on a 2-dimensional plane too. In addition, a sphere model is also easier for a decision maker to observe than a 3-dimensional cube model because the former exhibits alternatives on the surface of a sphere rather than inside the cube.

This paper is organized as follows. Section 2 reviews the relevant literature. Section 3 sets the three-phase decision making framework, including the screening, ordering and choosing phases. Section 4 proposes a weight-approximation model and a Decision Ball model to support a decision maker to filter out poor alternatives in the ordering phase. Section 5 develops an optimization model which can assist a decision maker in improving inconsistencies in preferences, and provides three methods to allow a decision maker to iteratively adjust his preferences in the choosing phase. Sections 4 and 5 form the main theoretical part of this paper; therefore, readers only interested in the application of proposed approach can skip these two sections. Section 6 uses an example to demonstrate the whole decision process.

2. Relevant Literature

Several visualization approaches have been developed to provide visual aids to support decision-making process. For instance, Li (1999) used deduction graphs to treat decision

problems associated with expanding competence sets. Jank and Kannan (2005) proposed a spatial multinomial model of customer choice to assist firms in understanding how their online customers’ preferences and choices vary across geographical markets. Kiang (2001) extended a self-organizing map (SOM) (Kohonen, 1995) network to classify decision groups by neural network techniques. Many studies (Kruskal, 1964; Borg and Groenen, 1997; Cox and Cox, 2000) adopted Multidimensional scaling (MDS), which is widely used in Marketing, to provide a visual representation of similarities among a set of alternatives. For instance, Desarbo and Jedidi (1995) proposed a new MDS method to spatially represent preference intensity collected over consumers’ consideration sets. However, most of conventional visualization approaches are incapable of detecting and improving the decision makers’ inconsistent preferences. Gower (1977), Genest and Zhang (1996) proposed a powerful graphical tool, the so-called Gower Plot, to detect the inconsistencies in decision maker’s preferences on a 2-dimensional plane. Nevertheless, the Gower plots do not provide suggestions about how to improve those inconsistencies either.

A pairwise-comparison ranking problem can be provided with magnitude of the degree of preference, intensity ranking; or in terms of ordinal preferences only, preference ranking. These are sometimes referred to also as cardinal versus ordinal preference (Hochbaum and Levin, 2006). Many studies (Saaty, 1980; Saaty and Vargas, 1984; Hochbaum and Levin, 2006; etc.) use multicriteria decision making approaches to find a consistent ranking at minimum error. However, conventional eigenvalue approaches cannot treat preference matrix with incomplete judgments. And, most of them focus on adjusting cardinal or ordinal inconsistencies instead of adjusting both cardinal and ordinal inconsistencies simultaneously. Li and Ma (2006)(2007) developed goal programming models which can treat incomplete judgments and improve cardinal and ordinal inconsistencies simultaneously. However, the ranks of and similarities among alternatives can be displayed.

This study cannot only improve cardinal and ordinal inconsistencies simultaneously but provide visual aids to decision makers. They can observe ranks of and similarities among alternatives, and iteratively adjust their preferences to achieve a more consistent decision.

3. Setting the Decision-Making Framework

The proposed decision-making framework is illustrated by the screening, ordering, and choosing phases as listed below:

(i) The screening phase: the decision maker tries to screen out clearly unwanted alternatives. The decision maker specifies upper and/or lower bounds of attributes to screen out poor alternatives.

(ii) The ordering phase: the decision maker tries to put a preference order on the remaining alternatives.

z The decision maker roughly specifies partial order of alternatives.

z An optimization model and a Decision Ball model are developed to assist decision maker in calculating and viewing ranks of and similarities among alternatives. z The decision maker filters out poor alternatives according to the information

displayed on the Decision Ball.

(iii) The choosing phase: the decision maker tries to make a final choice among a few alternatives. There are four steps in this phase, including specifying pairwise-comparison preferences, detecting and improving inconsistencies, adjusting preferences, and determining the best alternatives.

z Specifying pairwise-comparison preferences. Decision maker has to make more sophisticated comparisons for the remaining alternatives in this phase. Pairwise-comparison fashion, like analytical hierarchy process (AHP; Saaty, 1980), is adopted here because it is good for choosing phase (Brugha, 2004).

z Detecting and improving inconsistencies. Because inconsistent preferences may result in unreliable rank order, significant inconsistencies should be modified to obtain a more consistent solution. An optimization model is proposed to assist decision maker in detecting and improving inconsistencies. After inconsistencies have been reduced, the ranks of and similarities among alternatives are calculated and displayed on a Decision Ball.

z Adjusting preferences. According to the information displayed on the Decision Ball, the decision maker can iteratively adjust his preferences and see the corresponding changes on the Decision Ball.

z Determining the best alternatives. Decision maker makes the final choice with the assistance of the Decision Ball.

The detailed explanations about the ordering and choosing phases are illustrated in the following two sections.

4. The models for ordering phase

Consider a set of alternatives A = {A1, A2, …, An} for solving a choice problem, where the

decision maker selects m criteria to fulfill. The values of criteria c1, …, cm for alternative Ai are expressed as ci,k, for k = 1,…, m. All criterion values are assumed to be continuous data. Denote C

= [c_i,_k]_n×m as the criterion matrix of the decision problem. Denote c and k ck as the lower

and upper bounds of the criterion value of ck, respectively. The value of c and k c can be k

criterion value of ck. The score function in this study is assumed to be in an additive form because it is the most commonly used form in practice and more understandable for the decision maker (Belton and Stewart, 2002). Denote Sias the score value of an alternative Ai. An additive score function of an alternative Ai (ci,1, ci,2, …, ci,m) is defined as below:

∑

= − − = m k k k k k i k i c c c c w S 1 , ) (w , (1)

where (i) wk is the weight of criterion k, wk ≥ 0, ∀k and

∑

= = m k k w 1 1. w=(w₁,w₂,...,w_m) is a weight vector, (ii) 0≤Si(w)≤1. In order to make sure that all weights of criteria and scores

of alternatives are positive, a criterion ci,k with cost feature (i.e., a DM likes to keep it as small as

possible) is transferred from ci,k to (ck −ci,k) in advance.

Following the score function, the dissimilarity function of reflecting the dissimilarity between alternatives Ai and Aj is defined as

∑

= − − = m k k k k j k i k j i c c c c w 1 , , , | | ) (w δ , (2)

where 0≤

δ

_i,j(w)≤1 and

δ

_i,j(w)=

δ

_j,i(w). Clearly, if ci,k = cj,kfor all k then

δ

_i,j(w)= 0. In the ordering phase, a decision maker has to roughly specify partial order of alternatives. If the decision maker prefers Aito Aj, denoted asAi Aj, score of Ai should be higher than that of

Aj(Si> Sj). However, there may be some inconsistent preferences. For instance, a decision maker may specifyA_i A_j,A_j A_kand A_k A_i. A binary variable ti,jis used to record the inconsistent relationship between Ai and Aj: if Ai Aj and Si > Sj, then ti,j =0; otherwise, ti,j = 1. A weight approximation model for ordering phase is developed as follows:

Model 1 (Weight approximation model for ordering phase)

} {wk Min

∑∑

= = n i n j j i t 1 1 , s.t.

∑

= − − = m k k k k k i k i c c c c w S 1 , ) (w , ∀ , i (3) 1 1 =

∑

= m k k w , (4) j i j i S Mt S ≥ +

ε

− _, , ∀A_i A_j, (5)

, 0, , w k w w w_k ≤ _k ≤ _{k k} ≥ ∀ (6) } 1 , 0 { ,j∈ i

u , M is a large value,

ε

is a tolerable error. (7) The objective of Model 1 is to minimize the sum of ti,j. Expressions (3) and (4) are from the definition of an additive score function (1). Expression (5) indicates that if A_i A_j

andS_i ≥S_j +ε, then ti,j =0; otherwise, ti,j = 1, where ε and M are a computational precision and a large value which can be normally set as 10−6and 10 , respectively. Denote 6 w and _k

k

w as the lower and upper bound of wk, which could be set by the decision maker as in Expression (6). From (1) and (2), the score Si of alternative Ai and dissimilarity δ between i,j alternative Ai and Aj can be calculated based on the results of Model 1.

A Decision Ball model is then constructed to display all alternatives Ai in A = {A1, A2, …,

An} on the surface of a hemisphere. A non-metric multidimensional scaling technique is adopted here to provide a visual representation of the dissimilarities among alternatives. The arc length between two alternatives is used to represent the dissimilarity between them, e.g., the larger the difference, the longer the arc length. However, because the arc length is monotonically related to the Euclidean distance between two points and both approximation methods make little difference to the resulting configuration (Cox and Cox, 1991), the Euclidean distance is used here for simplification.

In addition, the alternative with a higher score is designed to be closer to the North Pole so that alternatives will be located on the concentric circles in the order of score from top view. For the purpose of comparison, we define an ideal alternativeA_*, where A_* =A_*(c₁,c₂,...,c_m) and S_* =1.A_*is designed to be located at the north pole with coordinate(x_*,y_*,z_*)= (0, 1, 0).

The following propositions are deduced:

Proposition 1 The relationship between δ_i,*(w) (the dissimilarity between Ai and A*)) and

Si(w) is expressed as δi,*(w)=1−Si(w). <Proof>

∑

∑

= = − − − − = − − = m k m k k k k k i k k k k k k k i k i c c c c c c w c c c c w 1 1 , , ,* ) ( ) ( | | ) (w δ ) ( 1 ) ) ( ) ( ( 1 1 , w i m k m k k k k k i k k k k k k S c c c c w c c c c w = − − − − − − =

∑

∑

= =

j i,

δ = 0 then di,j = 0 and if δi,j = 1 then di,j = 2, where 2 is used because the distance

between the north pole and equator is 2 when radius = 1. Denote the coordinates of an alternative Ai on a ball as (xi, yi, zi). The relationship between yi and Si is expressed as

Proposition 2 y_i =2S_i −S_i2. <Proof> Since di2_,*=(xi −0)2 +(yi −1)2 +(zi −0)2 =2δi2_,*=2(1−Si)2,

it is clear y_i =2S_i −S_i2. Clearly, if Si = 1 then yi = 1; if Si = 0, then yi = 0.

Based on the non-metric multidimensional scaling technique, denote dˆ_i,j as a

monotonic transformation of

δ

_i,j satisfying following condition: if

δ

_i,j <

δ

_p,q , then q

p j i d

dˆ_, < ˆ _, . The coordinate (xi, yi, zi) of alternative Ai all i can be calculated by the following Decision Ball model:

Model 2 (A Decision Ball Model)

} , , {xMin _i y_iz_i

∑∑

_{= >} − n i n i j j i j i d d 1 2 , , ˆ ) ( s.t. y_i =2S_i −S_i2, ∀i, (8) q p j i q p j i d dˆ, ≤ ˆ , −

ε

, ∀

δ

, <

δ

, , (9) j i z z y y x x d_i2_,j =( _i − _j)2 +( _i − _j)2 +( _i − _j)2, ∀, , (10) i z y xi + i + i =1, ∀ 2 2 2 , (11) 1 , 1≤ ≤ − x_i z_i , 0≤yi ≤1, ∀i,

ε

is a tolerable error. (12)

The objective of Model 2 is to minimize the sum of squared differences between di,jand

j i

dˆ, . Expression (8) is from Proposition 2, where the alternative with a higher score is designed

to be closer to the North Pole. Expression (9) is the monotonic transformation from

δ

_i,j todˆ_i,j. All alternatives are graphed on the surface of the northern hemisphere (11)(12).

Model 2 is a nonlinear model, which can be solved by some commercialized optimization software, such as Global Solver of Lingo 9.0, to obtain an optimum solution. One restriction of this model is the running time that may considerably increase when the number of alternatives becomes large because the time complexity of Model 2 is n2. This model has good performance

when the number of alternatives less than 10. However, in this case of alternatives more than 10, some classification techniques, like k-means (MacQueen,1967) for instance, can be used to reduce the solving time by dividing alternatives into several groups. The coordinates of group centers are calculated first. Then, these group centers are treated as anchor points. The coordinates of alternatives can be obtained by calculating dissimilarity between alternatives and anchor points. Thus, all alternatives can be displayed on the Decision Ball within tolerable time.

According to the information displayed on the Decision Ball, the decision maker can select better alternatives into the next phase.

5. The models for choosing phase

In this phase, the decision maker has to make more sophisticated comparisons for the remaining alternatives. Pairwise comparisons are adopted here (Brugha, 2004). For some

i and j pairs, assume a decision maker can specify pi,j, the ratio of the score of Ai to that of Aj, which is expressed as pi,j = ij j i e S S , × , (13)

where Si is the score of Ai and e_i,j is a multiplicative term accounting for inconsistencies, as illustrated in the Analytic Hierarchy Process (AHP) (Saaty, 1980). It is assumed that pi,j = 1/pj,i. If the decision maker cannot specify the ratio for a specific pair i and j then p_i,j =φ. Denote P =[p_i,j]_n×n as a n×n preference matrix. P is incomplete if there is anyp_i,j =

φ

. Pis perfectly consistent if ei,j=1 for all i,j (i.e. pi,j = Si/Sj for all i, j). P is ordinally inconsistent (intransitive) if for some i, j, k∈ {1, 2, 3, …, n}there exists pi,j > 1, pj,k > 1, but pi,k < 1. P is cardinally inconsistent if for some i, j, k∈ {1, 2, 3, …, n} there exists p_i,k ≠ p_i,j×p_j,k (Genest and Zhang, 1996).

If P is complete and ordinal consistent, all Ai can be ranked immediately. However, if there is ordinal or highly cardinal inconsistency, these inconsistenciesshould be improved before ranking because significant inconsistencies may result in unreliable rank order.

An optimization model, developed by a goal-programming optimization technique, is developed to assist decision maker in detecting and improving inconsistencies. In order to reduce the ordinal inconsistency, a binary variable ui,jis used to record if the preference pi,j, specified by

= 1; otherwise, ui,j= 0. A variable

α

i,j, defined as the difference between pi,j and Si/Sj, is used to indicate the degree of cardinal inconsistency of pi,j: the larger the value of

α

_i,j, the higher the cardinal inconsistency. The inconsistencies improving model is formulated as below:

Model 3 (Inconsistencies improving model )

} { Min k w M×Obj1 Obj+ 2 Obj1 =

∑∑

= > n i n i j j i u 1 , Obj2 =

∑∑

= > n i n i j j i 1 , α

s.t. ( −1)×( i,j −1)+ × i,j ≥ , for all where i,j ≠ and i,j ≠1

j i p p j i, u M p S S

_ε

_φ

, (14) 1 where all for , 0 , ≥ = × + − − Si Sj M ui j i,j pi,j , (15) p, , , i, j, S S j i j i j i − ≤

α

∀ (16)

∑

= − − = m k k k k k i k i c c c c w S 1 , ) (w , i∀ , (17) 1 1 =

∑

= m k k w , (18) , k k k w w w ≤ ≤ wk ≥0,∀k, (19)

u_i,j ∈{0,1}, M is a large value,

ε

is a tolerable error. (20) This model tries to improve ordinal and cardinal inconsistencies simultaneously. The first objective (Obj1) is to achieve ordinal consistency by minimizing the number of preferences

(i.e.,p_i,_j) being reversed. Constraint (14) means: when pi,j ≠

φ

and pi,j ≠1, ui,j = 0, if (i)

) 1 ( and ) 1 ( > _i,j > j i p S S or (ii) ( <1)and( _i,j <1) j i p S S

; and otherwise ui,j = 1. A tolerable

positive number

ε

is used to avoid =1

j i

S S

ui,j = 0; otherwise ui,j = 1. The second objective (Obj2) is to reduce cardinal consistency by

minimizing the

α

_i,j values, i.e. to minimize the difference between j i

S S

and p_i,_j. Since

ordinal consistency (Obj1) is more important than cardinal consistency (Obj2), Obj1 is multiplied by a large value M in the objective function. Constraints (17) and (18) come from Notation 1. Constraint (19) sets the upper and lower bound of weights. An improved complete preference

matrix can be obtained as P’ = [p_i',_j]_n×_n, where

j i j i S S p' = , if pi,j =

φ

or ui,j= 1; otherwise, j i j i p p , ' , = .

Model 3 is a nonlinear model, which can be converted into the following linear mixed 0-1 program: } { Min k w M×Obj1 Obj+ 2 Obj1 =

∑∑

= > n i n i j j i u 1 , Obj2 =

∑∑

= > n i n i j j i 1 , α

s.t. (Si −Sj)×(pi,j −1)+M ×ui,j ≥

ε

, for alli, j wherepi,j ≠

φ

and pi,j ≠1, (21) 1 where all for , , ≤ − ≤ × = ×

−M u_ij S_i S_j M u_i,j i,j p_i,j , (22) Sj×pi,j −

α

i,j ≤Si ≤ Sj×pi,j +

α

i,j, ∀i, j, (23)

(17) ~ (20),

where (21), (22) and (23) are converted from (14), (15) and (16) respectively.

After the weight vector, (w1, w2, …, wm), is found,

∑

= − − = m k k k k k i k i c c c c w S 1 , ) (w and

∑

= − − = m k k k k j k i k j i c c c c w 1 , , , | | ) (w

δ can be calculated. All alternatives are shown on a Decision Ball by Model 2.

According to the information visualized on the Decision Ball, the decision maker can iteratively adjust his preferences by the following ways:

to the North Pole so that a decision maker can see the rank order by the location of alternative: the higher the latitude, the higher the score. If the decision maker would like to adjust a preference order, from A1≺A3 to A1 A3 for instance, a constraint

ε

+

≥ ₃

1

S S will be added into Model 3.

(ii) Adjusting dissimilarity. The distance between two alternatives on a Decision Ball implies the dissimilarity between them: the larger the dissimilarity, the longer the distance. Therefore, if a decision maker observes the Decision Ball and decides to adjust the dissimilarity relationship, from

δ

_1,3(w)<

δ

_1,2(w) to

δ

_1,3(w)>

δ

_1,2(w) for example, a constraint

δ

1,3(w)>

δ

1,2(w) (i,e. +

ε

− − ≥ − −

∑

∑

= = m k _{k k} k k k m k _{k k} k k k c c c c w c c c c w 1 , 2 , 1 1 , 3 , 1 | | | | ) will

be added into Model 3.

(iii) Adjusting preference matrix. A decision maker can choose to adjust the preference matrix directly. The value of pi,j in Model 3 will be modified according to the change in the

preference matrix.

Solving Model 3 yields a new set of weights, and an adjusted Decision Ball will be displayed. The decision maker can iteratively adjust his preferences until he feels no adjustments have to be made. A final choice can be made with the assistance of a resulting Decision Ball.

6. Application to choice data: selection of a store location

Example 1 (Selection of a store location)

The choice of a store location has a profound effect on the entire business life of a retail operation. Suppose a manager of a convenience store in Taiwan who needs to select a store location from a list of 43 spots A = {A1, …, A43}. The manager sets four criteria to fulfill: (c1)

sufficient space, (c2) high population density, (c3) heavy traffic, and (c4) low cost. Store size is

measured in square feet. The number of people who live within a one-mile radius is used to calculate population density. The average number of vehicle traffic passing the spot per hour is adopted to evaluate the volumes of traffic. Cost is measured by monthly rent. The criteria values of 43 candidate locations are listed in the criterion matrix C1, as shown in Table 1.

The manager would like to rank choices incorporating his personal preferences. The manager can rank these choices by the following three phases:

Phase 1 – the screening phase

The manager tries to screen out clearly unwanted alternatives by setting upper or lower bound of each criterion. He sets the minimum space required to be 800 square feet, the minimum population density to be 700, the minimal traffic to be 400, and the maximum rental fee to be

5000. That is, c₁= 800, c₂ = 700, c₃= 400 and c₄ = 5000. The values of c₁, c₂ , c₃ and

4

c can be set as the maximum values of c1, c2, c3 and minimum value of c4, i.e. c1= 1500,

2

c = 1260, c₃ =780, and c₄ = 3100. After filtering out alternatives with criterion values exceeding these boundaries, only 23 choices {A3, A4, A6, A7, A8, A11, A13, A15, A17, A18, A21, A23, A24,

A25, A26, A29, A31, A32, A34, A37, A40, A42, A43} are remaining for the next phase.

Phase 2 – the ordering phase

The decision maker roughly specifies partial order of alternatives. He specifies A3 A7,

A7 A37, A15 A8, A17 A6, A31 A25 and A42 A40. The minimum weight of each criterion is set as

k

w = 0.01 for all k by the decision maker. Applying Model 1 to these preference relationships yields w = {w1, w2, w3, w4} = {0.21, 0.43, 0.01, 0.35}, t15,8 =1, and the

rest of ti,j = 0. The objective value is 1. The variable t15,8 = 1 indicates the preference relationship

A15 A8 should be reversed. When checking criterion matrix in Table 1, all criterion values of A8

are better than or equal to those of A15 which makes A15 A8 impossible; therefore, the

relationship between A15 and A8 is reversed.

The score of alternatives can be calculated according to Expression (1), where S3 = 0.54,

S4 = 0.10, S6 = 0.33, S7 = 0.54, S8 = 0.71, S11 = 0.29, S13 = 0.59, S15 = 0.36, S17 = 0.53, S18 = 0.31, S21

= 0.30, S23 = 0.30, S24 = 0.45, S25 = 0.22, S26 = 0.39, S29 = 0.23, S31 = 0.22, S32 = 0.42, S34 = 0.46, S37

= 0.39, S40 = 0.31, S42 = 0.34, S43 = 0.24. The dissimilarity between alternatives can also be

calculated according to Expression (2).

Applying Model 2 to this example yields coordinates of alternatives. The resulting Decision Ball is displayed in Figure 2. Because the alternative with a higher score is designed to be closer to the North Pole, the order of alternatives can be read by the latitudes of alternative: the higher the latitude, the higher the score. The order of top ten alternatives is A8 A13 A3 A7

A17 A34 A24 A32 A37 A26. In addition, the distance between two alternatives represents

the dissimilarity between them: the longer the distance, the larger the dissimilarity. For instance, the dissimilarity between A26 and A37 is smaller than that of between A37 and A7.

Based on the information provided on the Decision Ball, assume the decision maker decides to select the top eight alternatives to make more sophisticated comparisons. That is, only A8, A13, A3, A7, A17, A34, A24 and A32 are remaining for the next phase.

Phase 3 – the choosing phase

In the choosing phase, the manager uses pairwise comparisons to express preferences among pairs of choices in preference matrix R1, as listed in Table 2. Because the manager is

unable to make comparison among some spots, the relationships p3,34, p7,17, p8,24, p13,34 are left

blank, which means R1 is incomplete. The preference matrix R1 is ordinally inconsistent because there is an intransitive relationship among A3, A8 and A32. That is, A3 is preferred to A8 (p3,8 > 1),

and A8 is preferred to A32 (p8,32 > 1); however, A32 is preferred to A3 (p3,32 < 1). R1 is also cardinally inconsistent. For instance, there exists p3,8 = 1.6, p8,13 = 2.5; but, p3,13 = 2 (1.6 × 2.5 =

4, that is p_3,8×p_8,13 ≠ p_3,13).

Applying Model 3 to the example yields Obj1= 1, Obj2= 3.91, u3,8 = 1 and the rest

of ui,j = 0, (w1, w2, w3, w4) = (0.04, 0.19, 0.06, 0.71), (S3, S7, S8, S13, S17, S24, S32, S34) = (0.55,

0.55, 0.78, 0.27, 0.39, 0.40, 0.74, 0.51). The variable u3,8 = 1 implies that the value of p3,8 is

suggested to be changed from p3,8 >1 to p3,8 <1 (i.e. from A3 A8 to A3 ≺A8) to improve

ordinal inconsistency. The values of unspecified preferences can be computed as p3,34 =

34 3

S S

= 1.08, , p7,17, = 1.41, p8,24 = 1.93, and p13,34 = 0.76. The corresponding Decision Ball is

shown in Figure 3. The order of alternatives is A8 A32 A3 A7 A34 A24 A17.

According to the information observed on the Decision Ball, the decision maker can iteratively adjust his preferences. Suppose he would like to adjust a preference order from A7

A34 to A34 A7. A constraint S34 ≥S7 +

ε

is added into Model 3. Solving Model 3 yields

Obj1= 3, Obj2= 3.96, u3,8 = u7,34 = u17,24 = 1 and the rest of ui,j = 0, (w1, w2, w3, w4) = (0.01, 0.13,

0.17, 0.69), (S3, S7, S8, S13, S17, S24, S32, S34) = (0.53, 0.50, 0.76, 0.27, 0.44, 0.40, 0.71, 0.51). In

order to satisfy the relationship A34 A7, the relationship between A17 and A24 has to be reversed

(u17,24 = 1). Applying Model 2 to this result yields a new set of coordinates. An adjusted Decision

Ball is displayed in Figure 4. On this Decision Ball, the latitude of A34 is higher than that of A7.

By seeing the relationships of alternatives displayed on the Decision Ball in Figure 4, the decision maker would like to adjust some dissimilarity relationships between alternatives. His adjustment is that the dissimilarity between A3 and A8 is larger than that of between A7 and A8. A

constraint +

ε

− − ≥ − −

∑

∑

= = m k _{k k} k k k m k _{k k} k k k c c c c w c c c c w 1 , 8 , 7 1 , 8 , 3 | | | |

is added into Model 3. Solving Model 3

again yields Obj1= 5, Obj2= 4.33, u3,8 = u7,34 = u17,24 = u3,7 = u8,32 = 1 and the rest of ui,j = 0, (w1,

w2, w3, w4) = (0.01, 0.04, 0.19, 0.76), (S3, S7, S8, S13, S17, S24, S32, S34) = (0.51, 0.53, 0.74, 0.19,

0.39, 0.36, 0.78, 0.53). This result shows that in addition to rank reversal of A3 and A8, A7 and A34,

A17 and A24 (u3,8 = u7,34 = u17,24 =1), the relationship between A3 and A7, A8 and A32 are suggested to

be reversed to satisfy the adjustment of dissimilarity. A corresponding Decision Ball is depicted in Figure 5.

Suppose the decision maker stops further adjustment. The decision maker can make a final decision based on the Decision Ball in Figure 5. From the latitude of alternatives, the decision maker can tell the rank of choices as A32 A8 A34 A7 A3 A17 A24 A13. The best choice is

A32. The dissimilarity between alternatives can be read by the distance between them. For instance,

the dissimilarity between A3 and A34 is the smallest because the distance between them is the

shortest. That is, if A32 , A8 and A34 are not available, A3 as well as A7 will be a good choice.

It is important to notice that A3 is more similar to A34 than A7 is but A34 A7 A3. This kind

of relationship is possible. For instance, comparing with three alternatives A, B, C with benefit criterion values (5, 5, 5), (4, 4, 6) and (3, 5, 5), given equal weight and c_k = 0 and c_k =10 for

k = 1…3. The scores of three alternatives are SA = 0.5, SB = 0.47, and SC = 0.43. The dissimilarities

between alternatives are

δ

_A,B =0.1,

δ

_B,C =0.1 and

δ

_A,C =0.067. It is obvious that A B C but C is more similar to A than B is because

δ

_A,C<

δ

_A,B.

Example 1 was solved by Global Solver of Lingo 9.0 [20] on a Pentium 4 personal computer. The running time was less than 3 minimums for three phases totally.

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Table 1 Criterion Matrix C1 of Example 1

C1 C2 C3 C4 C1 C2 C3 C4

Store

Size Population Traffic

Rental Fee

Store

Size Population Traffic

Rental Fee A1 1600 580 320 3200 A23 960 750 650 3900 A2 390 680 450 2900 A24 860 1100 550 4350 A3 850 1140 550 4000 A25 866 810 550 4400 A4 1000 750 440 5000 A26 1058 750 450 3500 A5 900 840 450 5500 A27 998 1100 750 5200 A6 1000 900 500 4400 A28 665 900 650 3900 A7 1500 840 450 3800 A29 1055 800 450 4600 A8 800 1260 600 3500 A30 1008 900 650 5100 A9 755 700 400 1800 A31 1100 850 520 4950 A10 1400 600 500 4800 A32 885 720 420 3100 A11 1100 720 480 4000 A33 750 780 185 2800 A12 700 800 450 4800 A34 1205 880 580 3950 A13 1300 1250 650 4950 A35 1900 400 280 3000 A14 1250 1500 800 6800 A36 680 1500 950 5200 A15 800 900 420 3900 A37 920 780 480 3400 A16 820 500 450 3200 A38 1204 1200 550 5300 A17 1000 1200 780 4600 A39 580 1000 850 5500 A18 1300 720 420 4200 A40 850 960 520 4500 A19 950 700 330 3500 A41 565 665 380 2500 A20 1550 550 390 4100 A42 980 920 650 4400 A21 850 780 480 3800 A43 810 810 520 4200 Alternative Alternative

Table 2 Preference matrix R1 of Example 1 pi,j A3 A7 A8 A13 A17 A24 A32 A34 A3 1.4 1.6 2 1.2 2 0.5 A7 0.5 1.5 2 0.5 2 A8 2.5 2 1.2 1.5 A13 0.6 0.6 0.8 A17 0.5 0.5 0.7 A24 0.5 A32 2 A34

A8 A3 A17 A24 A34 A7 A13 A32 A37 A26 A18 A11 A23 A6 A42 A15 A40 A43 A21 A25 A4 A31 A29 A13 A7 A34 A3 A24 A17 A32 A8

Figure 2 The resulting Decision Ball after Phase 2

Figure 3 The resulting Decision Ball after Phase 3 A13 A7 A34 A3 A24 A17 A32 A8 A17 A13 A24 A8 A7 A34 A3 A32

Figure 4 The resulting Decision Ball after adjusting A34 A7

Figure 5 The resulting Decision Ball after adjusting

δ

_3,8 >

δ

_7,8 Q1 ₃ Q2 1 Q3 6 6 1 3 (a) (b) (c)

Figure 1 Advantage of a sphere model (a) Display line segments on a 2-D plane (b) Display curves on a sphere (c) Display four points that are not on the same plane

Q1

Q2 Q3 Q2

Q3

Q1