An interactive procedure for multiple attribute group decision making ...

European Journal of Operational Research 118 (1999) 139±152

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Theory and Methodology

An interactive procedure for multiple attribute group decision making with incomplete information: Range-based approach Soung Hie Kim

a,*

, Sang Hyun Choi a, Jae Kyeong Kim

b

a

b

Department of Management Information Systems, Korea Advanced Institute of Science and Technology, Graduate School of Management, 207-43 Cheongryangri, Dongdaemun, Seoul 130-012, South Korea Department of Management Information Systems, Kyonggi University, San 94-6, Yiui, Paldal, Suwon, Kyonggi 442-760, South Korea Received 8 December 1997; accepted 22 July 1998

Abstract This paper presents an interactive procedure for solving a multiple attribute group decision making (MAGDM) problem with incomplete information. The main properties of the procedure are: (1) Each decision maker is asked to express his/her preference in relation to an additive value model with incomplete preference statements. (2) A rangetyped representation method for utility is used. The range-typed utility representation makes it easy to compare each group member's utility information with a group's one and to aggregate each group member's utility information into a group's one. Utility range is calculated from each group member's incomplete information. (3) An interactive procedure is provided to help the group reach a consensus. It helps each group member to modify or complete his/her utility with ease compared to group's utility range. (4) We formally describe theoretic models for establishing group's pairwise dominance relations with group's utility range by using a separable linear programming technique. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Decision analysis; Multiple attribute group decision making; Incomplete information; Utility range

1. Introduction The increasing complexity of the socio-economic environment makes it less and less possible for single decision maker (DM) to consider all relevant aspects of a problem. Therefore, many organizations employ group members in decision making. Moving from single DM's setting to group members' setting introduces a great deal of complexity into the analysis. A group decision making process is usually understood to be the process of reducing di€erent individual preferences among objects in a given set to a single collective preference, or group preference (Kim and Ahn, 1997). Furthermore, when group members provide only

*

Corresponding author. Tel.: 0082 29583611; fax: 0082 29583604; e-mail: [email protected]

0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 3 0 9 - 9

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incomplete information, a selection is not generally made in a single step. For example, a DM can not provide exact estimations of attribute weights or he/she is not willing or able to specify the preferences in detail. The reasons why a DM provides only incomplete information are that (1) a decision should be made under time pressure and lack of data, (2) many of the attributes are intangible or non-monetary because they re¯ect social and environmental impacts, (3) a DM has limited attention and information processing capabilities (Weber, 1987; Park et al., 1996). There are many methods in the ®eld of decision analysis to ®nd an optimal or preferred solution of group decision making problem (Ramanathan and Ganesh, 1994). Among them, only a few studies have employed imprecise preference models in group decision making settings so far (Salo, 1995; Kim and Ahn, 1997). This paper suggests an interactive procedure for solving a multiple attribute group decision making (MAGDM) problem based on each group member's incomplete information. The basic idea behind the suggested procedure is as follows: each group member who can give only incomplete information, wants to compare his/her information to other group members'. Rather than that a selection is made based on initial incomplete information only, each group member wishes to make his/her incomplete information be a concrete or complete one. When each group member's utility information is very di€erent and it is dicult to ®nd an agreed or common information range, then the alternative(s) selected directly from the disagreed information can not represent well the group's utility, even if any aggregating method is used. Instead it is preferable to interactively modify each group member's incomplete utility information to get a group's aggregated information. Our procedure represents utility information as a linear range, because it can be calculated from the incomplete utility information with ease. Range type makes it e€ective and ecient to show utility information to group members. Furthermore range-typed utility information makes it easy to compare each group member's utility information with group's one and to aggregate each group member's utility information into a group's one. The characteristics of the suggested procedure are as follows: (1) Each group member is asked to express his/her preference in relation to an additive value model with incomplete preference statements, (2) A utility range is calculated based on each group member's incomplete information. To get the group's utility, a preference aggregation method is described. (3) An interactive procedure is provided to help the group reach a consensus. It helps each group member to modify or complete his/her utility with ease compared to group's utility range. (4) Theoretic models are formally described for establishing group's pairwise dominance relations with group's utility range. (5) The methodology is based on only linear programming (LP) models under functionally independent condition, which is regarded as a realistic condition (Park and Kim, 1997). (6) Furthermore, the procedure can handle the tradeo€ of decision making time, the quality of group decision making and the burden of group members. The rest of this paper is organized as follows. Section 2 reviews a number of approaches for MAGDM and their characteristics. Section 3 de®nes the problem of group decision making. Section 4 develops the proposed procedure and an illustrative example is included in Section 5. Summary and further research area are discussed in Section 6. 2. Literature reviews of group decision making Group decision making methods considering multiple attributes involve weighted aggregation of different individual preferences to obtain a single collective preference. This subject has received a great deal of attention from researchers in many disciplines. There have been research e€orts to represent group preference as additive individual value (utility) functions under some conditions (Anandaligam, 1989; Dyer and Sarin, 1979; Keeney and Kirkwood, 1975; Kenemy and Snell, 1962; Salo, 1995).

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Harsanyi (1955) presented the theory for an additive cardinal preference aggregation rule consistent with von Neumann and Morgenstern rationality axioms. Keeney and Kirkwood (1975) have speci®ed sucient conditions for a group cardinal social welfare function whose arguments are the individual utility functions of group members to have weighted additive form. The assessment of group members' importance weights involves addressing questions of trading-o€ utility to one individual against utility to another individual. Unfortunately, there is no entirely satisfactory procedure for making these tradeo€s. A lot of studies have employed that cardinal representations of group preferences have been built by combining the individual preference models into an additive value or utility function because of its robustness, transparency, and ease of development (Keeney and Rai€a, 1976). Although each approach has its own characteristics, there may be some drawbacks that make itself dicult to apply to real world problem. The reasons are because they require more burden in eliciting group members' parameter information, make fewer e€orts to provide more advanced information about solutions, and interact with their users. Only a few studies have employed imprecise models in group decision setting. Recently, Salo (1995) develops an interactive approach for the aggregation of group members' preference judgments in the context of an evolving value representation. He suggests strong or weak dominance relations and in the case of weak dominance, the results are presented to the group members and additional preferences are elicited for further analysis. However, he used only an interval of incomplete information with cardinal values. In the research of MAGDM, the parameter value information (i.e., weights of attributes and utility value of alternatives) is precisely or numerically assessed by group members but they are willing to provide only the incomplete information because of time pressure and lack of knowledge or data (Park and Kim, 1997). With the imprecisely identi®ed information, however, a selection is not generally made in a single step and some additional information is required to get a ®nal selection. From this point of view, an interactive procedure is required for MAGDM support. 3. Problem statement The MAGDM models are characterized by the following components: A ˆ fai giˆ1;M : a ®nite set of M possible alternatives. I ˆ figiˆ1;N : a set of indices of N attributes. K ˆ fkgkˆ1;K : a set of indices of K group members that participate in the decision making. wi is the set of group's trade-o€ weights associated with attribute i. wki is the set of kth DM's trade-o€ weights associated with attribute i. For example, when three DMs are involved and three attributes are considered, w1 P w2 P w3 is induced only when wk1 P wk2 P wk3 , for 8k. · wk is the importance weight associated with group member k. · uki (a): individual k's incomplete utility information for the consequences of selecting alternative a when attribute i is given. · uGi (a): the aggregated group's utility value associated with attribute i of selecting alternative a. · W ˆ fUw ; Ri wi ˆ 1; wi P 0g: the set of constraints or all possible values on the attributes' weights, w 2 W, where Uw is a set derived from group members' incomplete information regarding the relative importance of attributes. · Uki : the set of constraints on the utilities obtained by kth group member's incomplete information for the consequences when attribute i is given, fuki …a0 †, uki …a†g 2 Uik . · w…a0 ; a† ˆ fW ; U g, where …a0 ; a† is a pair of alternatives to be evaluated, and U ˆ fUik giˆ1;N ;kˆ1;K . · X: the set of collecting dominance relations between the alternatives, X  A  A; for example, …a0 ; a† 2 X means that alternative a0 is at least as preferred as alternative a. In multiple attribute single decision making situation, one usually considers a set of alternatives, which is evaluated by a family of attributes. A classical evaluation of alternatives leads to the aggregation of all · · · · ·

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attributes into a unique attribute called a utility function. In this paper, the attributes are assumed to be additive independent which leads to an additive multiattribute utility function. Individual k's aggregated value or expected utility associated with alternative am is given as follows: N ÿ
X V k …am † ˆ f uk1 …am †; . . . ; ukn …am † ˆ wki uki …am †: …1† iˆ1

Eq. (1) is an expected utility under certainty, but it can be expanded into uncertain situation without diculty (Park et al., 1996). The main purpose of approaches for single decision making under multiple attributes is that (1) we calculate expected value or expected utility about each alternative, (2) we compare the magnitude of each expected utilities among alternatives, and that (3) we specify dominance relationships between alternatives. When a DM gives incomplete information (for example, attribute i is twice important as much as attribute j), that information can be considered as linear constraints. However, if the information about attributes weights and utilities of alternatives is incompletely identi®ed at the same time, the form of expected utility (value) becomes nonlinear form. When information about attribute weights and utilities is speci®ed incompletely, we can construct preference order of alternatives through pairwise dominance relationship due to the fact that common attribute weights should be applied. The pairwise comparison problem with incomplete information has been formulated and solved by many researchers (Fishburn, 1965; Ko¯er et al., 1984; Kmietowicz and Pearman, 1984; Pearman and Kmietowicz, 1986; Park and Kim, 1997). The main problem of group decision making is this: How can many individuals' preferences be combined to yield a collective choice? Various procedures have been proposed to accomplish this problem, all of which di€er from each other in many respects and su€er from getting the complete information about attributes weights and utilities of alternatives. This paper suggests a procedure using individual incomplete information to overcome these diculties. 4. An interactive group decision making procedure 4.1. Overall procedure An overall interactive procedure is suggested for MAGDM problem. The speci®c description of each step is given at the following subsections. Since a selection is not made in a single step with incomplete information, White et al. (1982) have proposed a framework of interactive procedure for single MADM. Slightly modi®ed the procedure of their framework is: Step 1: Calculate individual utility range of each alternative on each attribute. Step 2: Get group's utility range of each alternative on each attribute using a preference aggregation method Step 3: Interact with group members by each attribute to re®ne their utility information to get a group's common range. Step 4: Find strict or weak dominance relation based on pairwise comparison. Step 5: If at least one group member doesn't satisfy the result and wants to revise his/her previous assessments, then go to Step 1. Otherwise, stop with the result in the Step 4. 4.2. Incomplete information Incomplete (imprecise or partial) parameter information can be taken by the form of linear inequalities and/or rankings (examples of such linear inequalities can be found at Section 5 of this paper or at

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(Park et al., 1996; Park and Kim, 1997). Such linear partial information is denoted by incomplete information in this research. The constraint set W(a0 , a) constructed by group members' information plays a major role to calculate group members' utility range or boundary. Note that group members have to be cognitively comfortable, while group members provide their knowledge about attributes' weights and utility values of alternatives. Thus, providing a description on W(a0 , a) by incomplete information makes group members feel comfortable. The type or form of W(a0 , a) that can be provided by group members are linear-inequality-typed information. For simple illustration, we present some forms of Uw 2 W…a0 ; a†. The other, Ui , can be similarly dealt as in Uw . Uw can be constructed by the following forms, for i ¹ j: F1. a weak ranking: {wi P wj }. F2. a strict ranking: {wi ÿ wj P ai }. F3. a ranking with multiples: {wi P ai wj }. F4. an interval form: {ai 6 wi 6 ai + ei }. F5. a ranking of di€erences: {wi ÿ wj P wk ÿ wl }, for j ¹ k ¹ l, where {ai } and {ei } are non-negative constants. For the more detail about this, please refer Park and Kim (1997). 4.3. Aggregation of individual group members' utility range The group's utility function associated with alternative am has the following form: VG …am † ˆ f …V 1 …am †; . . . ; V K …am †† ˆ

K X

wk

kˆ1

N X wki uki …am †:

…2a†

iˆ1

Assuming that DMs consider common attributes and use group's trade-o€ weights, gives us a simpler formula of a group's utility as Eq. (2b). VG …am † ˆ

K N N K X X X X wk wi uki …am † ˆ wi wk uki …am †: kˆ1

iˆ1

iˆ1

…2b†

kˆ1

Group's consensus is built by pairwise dominance relationship between alternatives. The group's pairwise comparison with incomplete information can be formulated as Eq. (3). min

z…a0 ; a† ˆ VG …a0 † ÿ VG …a† ˆ

N K X X   wi wk uki …a0 † ÿ uki …a† ; iˆ1

kˆ1

…3†

0

s:t: w…a ; a†: This pairwise dominance concept is analogous to the dominance concept under single decision making situation. However, the group's pairwise dominance value function in Eq. (3) is an intractable nonlinear form, so we consider another aggregation method. When we evaluate the group's utility value of an alternative by Eqs. (2a) and (2b) it is very dicult for group members to assess cardinal values to utility of an alternative. Instead they feel easy to give incomplete information about weights of attributes and utility values, W(a0 , a). Each group member's utility range of each alternative can be calculated by maximizing and minimizing each group member's utility value under constraint sets Uki (Ì W(a0 , a)). These computation processes are run for each attribute. Then, all group members' utility values can be represented by ranges within 0 and 1. Representing individual utility values by ranges makes it easy to aggregate individual utilities into a group's utility. Interactive process makes each group member compare his/her utility with others and modify his/her utility to get an acceptable level

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of common preference. In such an interactive process, range-typed utility representation is more ecient than other representation methods. Therefore, we use a group members' aggregated utility range of an alternative instead of cardinal or point values. If group's utility is represented by ranges, the next step will be to ®nd a dominance relation of alternatives. As the way of combining individual utility range into a common range of the group's preferences, the group members may attempt to reconcile di€erences of opinion by searching for consensus judgments (Salo, 1995). In this research, we use two types of group's utility range which are referred to as total range and agreed range. For example, we consider two group members' utility ranges of alternative am on attribute i, then the group's agreed range and total range are depicted as in Fig. 1. The group's agreed range is calculated by ®nding the intersection of all group members' ranges, …maxk minUik uki …a†; mink maxUik uki …a††. It means the overlapped or intersected range of utility they can take and each group member agrees on the agreed range. The total range, calculated by ®nding the union of all group members' ranges, …mink minUik uki …a†; maxk maxUik uki …a†† represents a possible extent of group member's preferences of the alternative. The width of the individual utility range represents a measure for the imprecision of the individual preference. The di€erence between total range and agreed range implies a measure of the preference con¯ict between group members. If the di€erence is large, then it is unrealistic to elicit a dominance relation from the con¯icting information. Instead, we suggest an interactive process for each group member to reconcile his/her incomplete utility information to make a consensus judgment for the alternative. A process to compromise the disagreement of group members' utility is described in the next section. 4.4. Interactive process to reduce the gap between total range and agreed range After obtaining group's agreed range and total range, the di€erence is checked to con®rm that each group member's information has a common basis. The di€erence between agreed range and total range is de®ned as the di€erence between the width of both ranges in this research. The width of group's agreed range is de®ned by …mink maxUik uki …a† ÿ maxk minUik uki …a††. Similarly, the width of total range is de®ned by …maxk maxUik uki …a† ÿ mink minUik uki …a††. Then we de®ne vi (a) as the ratio of the width of agreed range to that of total range. The vi (a) measures a group's degree of consensus for attribute i and alternative a. The value of vi (a) indicates also whether group's agreed range exists or not. In order to ensure the group members have a consensus, the following three interactive processes are suggested according to the value of vi (a).

Fig. 1. Group's aggregated ranges.

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Case 1: vi (a) 6 0. This case implies that there is a con¯ict and group utility has no agreed range like case 1 in Fig. 2. In this case, we represent to each group member a graphical display of his/her utility range and agreed/total range with his/her utility equations. Then each group member is demanded to modify his information compared with other group members' information. Case 2: 0 < vi (a) < di . This means that the agreed range exists, but is not within an acceptable boundary of total range. The value di means a threshold of group consensus for attribute i. This di is given by group members, considering the importance and characteristics of problem. This case is represented in case 2 of Fig. 2. Each group member is demanded to modify his/her information on the boundary of total range. Some guidelines may be helpful for a group member to re®ne his/her information. Guidelines are related with the increase or decrease of utility value per alternative and attribute. Case 3: vi (a) P di . This means that the agreed range is within an acceptable boundary of the total range. If all the alternatives satisfy this condition, the next step is to verify the dominance relation using the agreed range. If the interaction with group members does not result in an agreed range with a vi (a) which is greater than or equal to di , one has two options; One is to decrease di , and the other is to exclude members whose utilities are equal to maxk maxUik uki …a† or mink minUik uki …a†; and are not willing to modify their information. The second option is not an extreme attitude, but a technique to reach a group to have a consensus. This process is repeated until ®nding an agreed range within an acceptable boundary of remaining group members. 4.5. Pairwise dominance conditions for group utility range In order to compare alternatives, we suggest group's pairwise dominance conditions using the group's utility ranges. The conditions for strict dominance are extended from the single decision making context to the group decision making situation considering group's utility ranges. Theorem 1. Assume that the incomplete information of utility values for each attribute are functionally independent and group's utility values are represented by group's total range. Then the sucient condition required that a0 dominates strictly a for the group, is " # X min wi min min uki …a0 † ÿ max max uki …a† P 0: …4† i

k

Uik

k

Uik

Proof. In formula (4), mink minUik uki …a0 † means the lowest value of aggregated group's utility of alternative …a† means the highest a0 on attribute i and maxk maxUik ukiP   value of aggregated group's utility of alternative a on attribute i. It follows that min i wi uGi …a0 † ÿ uGi …a† P 0 because

Fig. 2. Three cases of group's utility range after two group members' preferences.

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X i





wi uGi …a0 † ÿ uGi …a† P

X i

"

#

wi min min uki …a0 † ÿ max max uki …a† : k

k

Uik

Uik

 P  Therefore, if Eq. (4) is satis®ed, i wi uGi …a0 † ÿ uGi …a† ˆ VG …a0 † ÿ VG …a† P 0 and we can conclude that a0 dominates strictly a for the group. Here, uGi (a) represents the aggregated group's utility value of alternative a on attribute i. h Theorem 1 shows a dominance condition using the group's total range. In contra, the following Theorem 2 is about group's agreed range. The group's agreed range, that is, the range of uGi (a), is represented by …maxk min uki …a†; mink max uki …a††. It can be brie¯y represented by …min uGi …a†; max uGi …a††. Theorem 2. If the incomplete information of utility values for each attribute are functionally independent and group's utility values can be represented by the group's agreed range, then the sucient condition required that a0 dominates strictly a for the group, is min

X

"

#

wi max min k

i

Uik

uki …a0 †

ÿ min max k

Uik

uki …a†

P 0:

…5†

 P  G 0 G w min u …a † ÿ max u …a† P 0 by changing the notation. That means Proof. Formula (5) implies min i i i i   P G 0 G w min u …a † ÿ max u …a† P 0. In other words, i i i i min

X i

P

X     wi uGi …a0 † ÿ uGi …a† P wi min uGi …a0 † ÿ max uGi …a† ; i





where i wi uGi …a0 † ÿ uGi …a† is the group's aggregated value di€erence between alternative a0 and a. So, VG …a0 † ÿ VG …a† P 0 and a0 dominates strictly a for group. h To construct pairwise dominance relationships requires …2KN ‡ 1†M…M ÿ 1† LPs for solving Eq. (4) and …2KN ‡ 1†M…M ÿ 1† LPs for solving Eq. (5). The di€erence between total range and agreed range is reduced in the interactive process in Section 4.4, but Theorem 1 shows a stronger dominance relation than Theorem 2. Because, if alternative a0 is strictly dominant than a at total range, a0 is strictly dominant than a at agreed range. But the reverse condition is not always true. Though these group's pairwise dominance rules are seemingly good, strict dominance (SD) relations between alternatives rarely happen because SD relation under group decision making context analogous to Theorem 1 or Theorem 2 does not always happen in the real world. Instead of SD, weak dominance (WD) relation is usually used between alternatives in group decision making problems. WD means that we are to try to select the alternative that minimizes `regrets'. However WD on total range is meaningless, because the density of the aggregated group's utility value uGi (a) in the total range is not uniform. That means WD relation may be determined by an extreme value of a speci®c group member, if total range is used. By applying basic logic of WD condition under single decision making context to group decision making situation, we can suggest a WD condition for group's agreed range like following Theorem 3. P Theorem 3. Let Zmin …a0 ; a† ˆ min i wi ‰maxk minUik uki …a0 † ÿ mink maxUik uki …a†Š and assume that group's utility value uGi (a) is a random variable with one-dimensional interval space of group's agreed range. Further assume the probability distribution function f(uGi (a)) of uGi (a) is symmetric about a vertical axis through uGi (a) ˆ l(a), where l(a) ˆ (min uGi (a) + max uGi (a))/2.

S.H. Kim et al. / European Journal of Operational Research 118 (1999) 139±152

If Zmin …a0 ; a† P Zmin …a; a0 †; then a0 dominates weakly a:

147

…6†

k uki (a)) can be brie¯y represented Proof. The group's agreed range for alternative a, (maxk min P ui (a), minGk max G G 0 0 ui (a)). Then Zmin …a ; a† ˆ min i wi min ui …a † ÿ max uGi …a† and Zmin …a; a0 † ˆ by (min P ui (a), max G min i wi ‰min ui …a† ÿ max uGi …a0 †Š. From l…a† ˆ …min uGi …a† ‡ max uGi …a††=2, min uGi …a† ˆ 2 l…a†ÿ max uGi …a†. Similarly min uGi …a0 † ˆ 2 l…a0 † ÿ max uGi …a0 †. Then,

Zmin …a0 ; a† ÿ Zmin …a; a0 † X  X    wi min uGi …a0 † ÿ max uGi …a† ÿ min wi min uGi …a† ÿ max uGi …a0 † ˆ min i

0

ˆ 2l…a † ‡ min

X

i



wi ÿ max

i

uGi …a0 †

ÿ max

uGi …a†



ÿ 2l…a† ÿ min

X i

  wi ÿ max uGi …a† ÿ max uGi …a0 †

0

ˆ 2‰l…a † ÿ l…a†Š P 0: That is, EU …a0 † ÿ EU …a† P 0. Therefore, we can conclude that alternative a0 weakly dominates alternative a if Zmin …a0 ; a† P Zmin …a; a0 †. h This condition means that the alternative having the higher positioned range in utility value is preferred to lower range in order to minimize the degree of group's regret. The WD condition can be used as a certain decision rule when we don't get enough information to get a SD relation from group members.

4.6. Interactive procedure for group decision making with incomplete information An interactive procedure is suggested to ®nd out dominance relations between alternatives based on theorems in Section 4.5. Step 1: (Computing individual utility ranges) Solve the following LPs to calculate individual utility ranges of each alternative on each attribute with group members' incomplete information: min = max uki …a†; k s:t: Ui  w…a0 ; a†: Step 2: (Preference aggregation) Get a group's utility range of each alternative on each attribute by total range and agreed range from individual utility ranges:   Group's agreed range ˆ max min uki …a†; min max uki …a† ;  Group's total range ˆ

k

k

 min min uki …a†; max max uki …a† : k

k

Step 3: (Interaction for specifying group members' preferences) Interact with group members by each attribute to re®ne their utility information to reduce the gap between agreed range and total range. The speci®c processes are suggested in Section 4.4. Step 4: (Finding dominance relations) Find dominance relations with theorems in Section 4.5 by solving LPs of formula (4), (5), and (6). Step 5: (Appraisal of the result) If at least one group member doesn't satisfy the result and wants to revise his/her previous assessments, then go to step 1. Otherwise, stop with the result in Step 4.

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The above interactive procedure can be used in the context that the group has same attributes and imprecise/incomplete attributes' weights. 5. Paper submission example This section illustrates features of interactive procedure for group decision making in the context of a paper submission problem. After completing a manuscript, author(s) usually confront the problem which journal is appropriate for the manuscript. If more than two authors are involved, the problem becomes a MAGDM problem. For an illustrative example, it is considered three attributes; reputation, acceptability, and compatibility, two authors; Choi and Kim, and three alternatives; a1 , a2 , and a3 . The preference information of two members' are given by the incomplete information as shown in Tables 1 and 2. In the context of journal selection under consideration, Choi feels that reputation is more important than acceptability and acceptability is more important than compatibility. Kim feels that reputation is more important than acceptability and acceptability is as important but no more than two times as important as compatibility. Choi and Kim set the threshold value of group consensus at 0.5 for all attributes for simplicity. Step 1: Under constraints on each member's utility values, we calculate the minimum and maximum values by solving 36 LP problems on each alternative and attribute as shown Table 3. Table 1 Choi's utility information Reputation

Acceptability

Compatibility

u1 (a1 ) P 0.5 u1 (a2 ) P 0.7 u1 (a3 ) 6 u1 (a1 ) u1 (a1 ) 6 u1 (a2 )

0.2 6 u2 (a1 ) 6 0.7 u2 (a3 ) P 0.5 u2 (a2 ) 6 u2 (a1 ) u2 (a1 ) ÿ u2 (a2 ) 6 u2 (a3 ) ÿ u2 (a1 ) u2 (a1 ) 6 u2 (a3 )

u3 (a1 ) P 0.6 u3 (a2 ) P 0.7 u3 (a3 ) P 0.5 u3 (a3 ) 6 u3 (a2 )

Reputation

Acceptability

Compatibility

u1 (a1 ) P 0.8 u1 (a2 ) 6 0.7 2u1 (a3 ) 6 u1 (a1 )

0.5 6 u2 (a1 ) 6 0.8 0.2 6 u2 (a2 ) 6 0.6 u2 (a3 ) P u2 (a1 ) u2 (a3 ) P u2 (a2 )

0.6 6 u3 (a2 ) 6 0.9 0.4 6 u3 (a3 ) 6 0.8 u3 (a1 ) P 2u3 (a3 )

Table 2 Kim's utility information

Table 3 The computed values group members' utility range Members

Choi

Attributes

1

2

3

Kim 1

2

3

a1 a2 a3

(0.5, 1) (0.7, 1) (0, 0.9)

(0.2, 0.7) (0, 0.7) (0.5, 1)

(0.6, 1) (0.7, 1) (0.5, 1)

(0.8, 1) (0, 0.7) (0, 0.5)

(0.5, 0.8) (0.2, 0.6) (0.5, 1)

(0.8, 1) (0.6, 0.9) (0.4, 0.8)

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149

Table 4 Group's utility ranges of each alternative and attribute at ®rst interaction Attributes

Reputation

Acceptability

Agreed a1 vi (a1 ) a2 vi (a2 ) a3 vi (a3 )

(0.8, 1) No (0, 0.5)

0.4 0 0.56

Total

Agreed

(0.5, 1)

(0.5, 0.7)

(0, 1)

(0.2, 0.6)

(0, 0.9)

(0.5, 1)

Compatibility

0.33 0.57 1.0

Total

Agreed

(0.2, 0.8)

(0.8, 1)

(0, 0.7)

(0.7, 0.9)

(0.5, 1)

(0.5, 0.8)

Total 0.5 0.5 0.5

(0.6, 1) (0.6, 1) (0.4, 1)

Step 2: As shown in Table 4, the group's utility ranges represented by total and agreed range are obtained from individual utility ranges in Table 3. The cells with vi …a† value above the threshold value, 0.5, imply the agreed range is within an acceptable boundary of total range and cell marked by ``No'' means there is no agreed range and it implies they have no consensus for alternative a2 on attribute 1. Then, according to three cases suggested in Section 4.4, di€erent procedures are run until getting the acceptable consensus. Step 3: The procedure interacts with group members by each attribute. But, it is not necessary to get more information about compatibility attribute because the agreed ranges of all alternatives on the attribute are within the acceptable bound of total ranges. For the reputation attribute, Choi and Kim are asked to modify his information related with alternative a1 and alternative a2 because they have provided information on the boundary of total range. The graphic display which includes Choi's utility ranges, agreed ranges, total ranges, and utility constraints on reputation attribute are represented to him as shown in Fig. 3 except a shaded area. Referred to the graphic display, he modi®es the set of incomplete information as constraints in the shaded area of Fig. 3. At the same time, the graphic display with Kim's utility ranges is presented to Kim. He modi®es his incomplete information referring group's utility information. After those modi®cations, go to step 1 with

Fig. 3. Graphic display of Choi's value at ®rst interaction.

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Table 5 Group's utility ranges at ®nal interaction Attributes

Reputation

Acceptability

Agreed a1 vi (a1 ) a2 vi (a2 ) a3 vi (a3 )

(0.8, 1) (0.6, 1) (0, 0.5)

0.5 0.55 0.56

Total

Agreed

(0.6, 1)

(0.3, 0.7)

(0.27, 1)

(0.2, 0.5)

(0, 0.5)

(0.5, 1)

Compatability

0.67 0.57 1.0

Total

Agreed

(0.2, 0.8)

(0.8, 1)

(0, 0.53)

(0.7, 0.9)

(0.5, 1)

(0.5, 0.8)

Total 0.5 0.5 0.5

(0.6, 1) (0.6, 1) (0.4, 1)

modi®ed information on the reputation attribute and calculate modi®ed group's utility ranges. At second interaction, same processes for the acceptability attribute are performed. The result is shown in Table 5. We can get a consensus with all vi …a† values above the threshold value, 0.5. Step 4: For constructing dominance relations for group, we use Formula (4) with following constraints of attributes' weights; w1 P w2 ; w2 P w3 ; w2 6 2w3 , and w1 ‡ w2 ‡ w3 ˆ 1. Solving 6 LPs yields X ˆ f…a1 ; a3 †; …a2 ; a3 †g because Zmin …1; 2† ˆ ÿ0:2; Zmin …1; 3† ˆ 0:036; Zmin (2, 1) ˆ ÿ0.4, Zmin (2, 3) ˆ 0.033, Zmin (3, 1) ˆ ÿ1, Zmin (3, 2) ˆ ÿ1. If group members are not able to provide further information, then they may choose a1 as the most preferred alternative by applying the pairwise WD condition of Theorem 3. Step 5: Stop with the result, a1 > a2 > a3 , because Choi and Kim satisfy with the result using weak dominance condition. Note that the above result comes from SD and WD relations. However, if we apply only SD relation, a1 and a2 are indi€erent.

Fig. 4. Comparative graphic display between ®rst and ®nal iteration.

S.H. Kim et al. / European Journal of Operational Research 118 (1999) 139±152

151

In this example, a few interaction are performed for illustrative convenience but more interaction will be done if we set a higher value on di . And the fact that a little information are modi®ed at each interaction means that sensitivity analysis is helpful for calculating modi®ed group's utility ranges. Sensitivity analysis demands a reduced computation time. The interactive process helps the group seek a consensus. Fig. 4 shows that the degree of consensus for group's agreed range becomes wider as the interaction goes on. The value di implies the threshold ratio value of the width of agreed range to that of total range for each attribute. If di becomes 1, agreed range becomes closer to total range, that is, group's information should be close to an identical information state. If di becomes 0, a group's decision making can be made only if agreed ranges exist. So a larger di value demands much burden of group members, but the result of group decision making represents well the group's preference. And a smaller di value is helpful to speed up the interaction process, but the result can not represent well the group's preference. So di determines the tradeo€ between the quality of group decision making and the burden of group members. 6. Conclusions In this paper, we presented an interactive procedure for MAGDM problem with incomplete information. An interactive modi®cation procedure is described for each member to make a group consensus by interactively modifying his/her incomplete information to be a concrete or complete one. To help such a procedure, we used a range-typed representation method for utility value. Range-typed utility information makes it easy to compare each group member's utility information with group's one and to aggregate each group member's utility information into a group's one. We formally described theoretic models for establishing group's pairwise dominance relations with group's utility ranges by using a separable linear programming technique. Group decision making needs diverse situations, such as a speedy decision or a decision demanding group's time-consuming negotiation. Our procedure can handle this tradeo€ among the time, the quality of group decision making and the burden of group members. A promising research area is to develop a group decision support system (GDSS) for interactive preference elicitation from group members and an automated solving tool because the number of LPs to be solved increases in multiplicative fashion as the number of alternatives, attributes, or participating group members increase.

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