Theory and Methodology

An interactive procedure for multiple attribute group decisionmaking with incomplete information: Range-based approach

Soung Hie Kim a,*, Sang Hyun Choi a, Jae Kyeong Kim b

a Department of Management Information Systems, Korea Advanced Institute of Science and Technology,

Graduate School of Management, 207-43 Cheongryangri, Dongdaemun, Seoul 130-012, South Koreab 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 range-

typed 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 singledecision maker (DM) to consider all relevant aspects of a problem. Therefore, many organizations employgroup members in decision making. Moving from single DM's setting to group members' setting introducesa great deal of complexity into the analysis. A group decision making process is usually understood to bethe process of reducing di�erent individual preferences among objects in a given set to a single collectivepreference, or group preference (Kim and Ahn, 1997). Furthermore, when group members provide only

European Journal of Operational Research 118 (1999) 139±152www.elsevier.com/locate/orms

* Corresponding author. Tel.: 0082 29583611; fax: 0082 29583604; e-mail: shkim@dss.kaist.ac.kr

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

incomplete information, a selection is not generally made in a single step. For example, a DM can notprovide exact estimations of attribute weights or he/she is not willing or able to specify the preferences indetail. The reasons why a DM provides only incomplete information are that (1) a decision should be madeunder time pressure and lack of data, (2) many of the attributes are intangible or non-monetary becausethey re¯ect social and environmental impacts, (3) a DM has limited attention and information processingcapabilities (Weber, 1987; Park et al., 1996).

There are many methods in the ®eld of decision analysis to ®nd an optimal or preferred solution ofgroup decision making problem (Ramanathan and Ganesh, 1994). Among them, only a few studies haveemployed 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 thesuggested procedure is as follows: each group member who can give only incomplete information, wants tocompare his/her information to other group members'. Rather than that a selection is made based on initialincomplete information only, each group member wishes to make his/her incomplete information be aconcrete or complete one. When each group member's utility information is very di�erent and it is di�cultto ®nd an agreed or common information range, then the alternative(s) selected directly from the disagreedinformation can not represent well the group's utility, even if any aggregating method is used. Instead it ispreferable to interactively modify each group member's incomplete utility information to get a group'saggregated information.

Our procedure represents utility information as a linear range, because it can be calculated from theincomplete utility information with ease. Range type makes it e�ective and e�cient to show utility infor-mation to group members. Furthermore range-typed utility information makes it easy to compare eachgroup member's utility information with group's one and to aggregate each group member's utility in-formation into a group's one.

The characteristics of the suggested procedure are as follows: (1) Each group member is asked to expresshis/her preference in relation to an additive value model with incomplete preference statements, (2) A utilityrange is calculated based on each group member's incomplete information. To get the group's utility, apreference aggregation method is described. (3) An interactive procedure is provided to help the groupreach a consensus. It helps each group member to modify or complete his/her utility with ease compared togroup's utility range. (4) Theoretic models are formally described for establishing group's pairwise domi-nance 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 groupdecision making and the burden of group members.

The rest of this paper is organized as follows. Section 2 reviews a number of approaches for MAGDMand their characteristics. Section 3 de®nes the problem of group decision making. Section 4 develops theproposed procedure and an illustrative example is included in Section 5. Summary and further researcharea are discussed in Section 6.

2. Literature reviews of group decision making

Group decision making methods considering multiple attributes involve weighted aggregation of dif-ferent individual preferences to obtain a single collective preference. This subject has received a great deal ofattention from researchers in many disciplines. There have been research e�orts to represent group pref-erence as additive individual value (utility) functions under some conditions (Anandaligam, 1989; Dyer andSarin, 1979; Keeney and Kirkwood, 1975; Kenemy and Snell, 1962; Salo, 1995).

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

Harsanyi (1955) presented the theory for an additive cardinal preference aggregation rule consistent withvon Neumann and Morgenstern rationality axioms. Keeney and Kirkwood (1975) have speci®ed su�cientconditions for a group cardinal social welfare function whose arguments are the individual utility functionsof group members to have weighted additive form. The assessment of group members' importance weightsinvolves 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 bycombining the individual preference models into an additive value or utility function because of its ro-bustness, transparency, and ease of development (Keeney and Rai�a, 1976). Although each approach hasits own characteristics, there may be some drawbacks that make itself di�cult to apply to real worldproblem. The reasons are because they require more burden in eliciting group members' parameter in-formation, make fewer e�orts to provide more advanced information about solutions, and interact withtheir 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 thecontext of an evolving value representation. He suggests strong or weak dominance relations and in the caseof weak dominance, the results are presented to the group members and additional preferences are elicitedfor 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 utilityvalue of alternatives) is precisely or numerically assessed by group members but they are willing to provideonly 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 singlestep and some additional information is required to get a ®nal selection. From this point of view, an in-teractive procedure is required for MAGDM support.

3. Problem statement

The MAGDM models are characterized by the following components:· A � faigi�1;M : a ®nite set of M possible alternatives.· I � figi�1;N : a set of indices of N attributes.· K � fkgk�1;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.· wk

i is the set of kth DM's trade-o� weights associated with attribute i. For example, when three DMs areinvolved and three attributes are considered, w1 P w2 P w3 is induced only when wk

1 P wk2 P wk

3 , for 8k.· wk is the importance weight associated with group member k.· uk

i (a): individual k's incomplete utility information for the consequences of selecting alternative a whenattribute i is given.

· uGi (a): the aggregated group's utility value associated with attribute i of selecting alternative a.

· W � fUw;Riwi � 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 relativeimportance 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�, uk

i �a�g 2 Uki .

· w�a0; a� � fW ;Ug, where �a0; a� is a pair of alternatives to be evaluated, and U � fU ki gi�1;N ;k�1;K .

· X: the set of collecting dominance relations between the alternatives, X � A� A; for example, �a0; a� 2 Xmeans 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

S.H. Kim et al. / European Journal of Operational Research 118 (1999) 139±152 141

attributes into a unique attribute called a utility function. In this paper, the attributes are assumed to beadditive independent which leads to an additive multiattribute utility function. Individual k's aggregatedvalue or expected utility associated with alternative am is given as follows:

V k am� � � f uk1 am� �; . . . ; uk

n am� �ÿ � �XN

i�1

wki uk

i am� �: �1�

Eq. (1) is an expected utility under certainty, but it can be expanded into uncertain situation withoutdi�culty (Park et al., 1996). The main purpose of approaches for single decision making under multipleattributes is that (1) we calculate expected value or expected utility about each alternative, (2) we comparethe magnitude of each expected utilities among alternatives, and that (3) we specify dominance relation-ships between alternatives. When a DM gives incomplete information (for example, attribute i is twiceimportant as much as attribute j), that information can be considered as linear constraints. However, if theinformation 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 constructpreference order of alternatives through pairwise dominance relationship due to the fact that commonattribute weights should be applied. The pairwise comparison problem with incomplete information hasbeen formulated and solved by many researchers (Fishburn, 1965; Ko¯er et al., 1984; Kmietowicz andPearman, 1984; Pearman and Kmietowicz, 1986; Park and Kim, 1997).

The main problem of group decision making is this: How can many individuals' preferences be com-bined to yield a collective choice? Various procedures have been proposed to accomplish this problem, all ofwhich di�er from each other in many respects and su�er from getting the complete information aboutattributes weights and utilities of alternatives. This paper suggests a procedure using individual incompleteinformation 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 eachstep is given at the following subsections. Since a selection is not made in a single step with incompleteinformation, 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

methodStep 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 inequalitiesand/or rankings (examples of such linear inequalities can be found at Section 5 of this paper or at

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

(Park et al., 1996; Park and Kim, 1997). Such linear partial information is denoted by incomplete infor-mation in this research. The constraint set W(a0, a) constructed by group members' information plays amajor role to calculate group members' utility range or boundary. Note that group members have to becognitively comfortable, while group members provide their knowledge about attributes' weights and utilityvalues of alternatives. Thus, providing a description on W(a0, a) by incomplete information makes groupmembers feel comfortable.

The type or form of W(a0, a) that can be provided by group members are linear-inequality-typed in-formation. For simple illustration, we present some forms of Uw 2 W�a0; a�. The other, Ui, can be similarlydealt 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 aiwj}.F4. an interval form: {ai6wi6 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� � f �V 1�am�; . . . ; V K�am�� �XK

k�1

wkXN

i�1

wki uk

i �am�: �2a�

Assuming that DMs consider common attributes and use group's trade-o� weights, gives us a simplerformula of a group's utility as Eq. (2b).

VG�am� �XK

k�1

wkXN

i�1

wiuki �am� �

XN

i�1

wi

XK

k�1

wkuki �am�: �2b�

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

min z�a0; a� � VG�a0� ÿ VG�a� �XN

i�1

wi

XK

k�1

wk uki �a0� ÿ uk

i �a�� �

;

s:t: w�a0; a�:�3�

This pairwise dominance concept is analogous to the dominance concept under single decision makingsituation. However, the group's pairwise dominance value function in Eq. (3) is an intractable nonlinearform, 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 forgroup members to assess cardinal values to utility of an alternative. Instead they feel easy to give incompleteinformation about weights of attributes and utility values, W(a0, a). Each group member's utility range ofeach alternative can be calculated by maximizing and minimizing each group member's utility value underconstraint sets Uk

i (Ì W(a0, a)). These computation processes are run for each attribute. Then, all groupmembers' utility values can be represented by ranges within 0 and 1. Representing individual utility valuesby ranges makes it easy to aggregate individual utilities into a group's utility. Interactive process makeseach group member compare his/her utility with others and modify his/her utility to get an acceptable level

S.H. Kim et al. / European Journal of Operational Research 118 (1999) 139±152 143

of common preference. In such an interactive process, range-typed utility representation is more e�cientthan other representation methods.

Therefore, we use a group members' aggregated utility range of an alternative instead of cardinal orpoint values. If group's utility is represented by ranges, the next step will be to ®nd a dominance relation ofalternatives.

As the way of combining individual utility range into a common range of the group's preferences, thegroup 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 rangeand agreed range. For example, we consider two group members' utility ranges of alternative am on at-tribute 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 minUk

iuk

i �a�;mink maxUki

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 allgroup members' ranges, �mink minUk

iuk

i �a�;maxk maxUki

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¯ictbetween group members. If the di�erence is large, then it is unrealistic to elicit a dominance relation fromthe con¯icting information. Instead, we suggest an interactive process for each group member to reconcilehis/her incomplete utility information to make a consensus judgment for the alternative. A process tocompromise 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 eachgroup member's information has a common basis. The di�erence between agreed range and total range isde®ned as the di�erence between the width of both ranges in this research. The width of group's agreedrange is de®ned by �mink maxUk

iuk

i �a� ÿmaxk minUki

uki �a��. Similarly, the width of total range is de®ned

by �maxk maxUki

uki �a� ÿmink minUk

iuk

i �a��. Then we de®ne vi(a) as the ratio of the width of agreed rangeto that of total range. The vi(a) measures a group's degree of consensus for attribute i and alternative a. Thevalue of vi(a) indicates also whether group's agreed range exists or not. In order to ensure the groupmembers have a consensus, the following three interactive processes are suggested according to the value ofvi(a).

Fig. 1. Group's aggregated ranges.

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

Case 1: vi(a)6 0. This case implies that there is a con¯ict and group utility has no agreed range like case 1in Fig. 2. In this case, we represent to each group member a graphical display of his/her utility range andagreed/total range with his/her utility equations. Then each group member is demanded to modify hisinformation 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 boundaryof total range. The value di means a threshold of group consensus for attribute i. This di is given by groupmembers, considering the importance and characteristics of problem. This case is represented in case 2 ofFig. 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 relatedwith 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 agreedrange.

If the interaction with group members does not result in an agreed range with a vi(a) which is greaterthan or equal to di, one has two options; One is to decrease di, and the other is to exclude members whoseutilities are equal to maxk maxUk

iuk

i �a� or mink minUki

uki �a�; and are not willing to modify their infor-

mation. 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 groupmembers.

4.5. Pairwise dominance conditions for group utility range

In order to compare alternatives, we suggest group's pairwise dominance conditions using the group'sutility ranges. The conditions for strict dominance are extended from the single decision making context tothe group decision making situation considering group's utility ranges.

Theorem 1. Assume that the incomplete information of utility values for each attribute are functionallyindependent and group's utility values are represented by group's total range. Then the su�cient conditionrequired that a0 dominates strictly a for the group, is

minX

i

wi mink

minUk

i

uki �a0� ÿmax

kmax

Uki

uki �a�

" #P 0: �4�

Proof. In formula (4), mink minUki

uki �a0� means the lowest value of aggregated group's utility of alternative

a0 on attribute i and maxk maxUki

uki �a� means the highest value of aggregated group's utility of alternative a

on attribute i. It follows that minP

i wi uGi �a0� ÿ uG

i �a�� �

P 0 because

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

S.H. Kim et al. / European Journal of Operational Research 118 (1999) 139±152 145

Xi

wi uGi �a0� ÿ uG

i �a�� �

PX

i

wi mink

minUk

i

uki �a0� ÿmax

kmax

Uki

uki �a�

" #:

Therefore, if Eq. (4) is satis®ed,P

i wi uGi �a0� ÿ uG

i �a�� � � 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 The-orem 2 is about group's agreed range. The group's agreed range, that is, the range of uG

i (a), is representedby �maxk min uk

i �a�;mink max uki �a��. It can be brie¯y represented by �min uG

i �a�;max uGi �a��.

Theorem 2. If the incomplete information of utility values for each attribute are functionally independent andgroup's utility values can be represented by the group's agreed range, then the su�cient condition required thata0 dominates strictly a for the group, is

minX

i

wi maxk

minUk

i

uki �a0� ÿmin

kmax

Uki

uki �a�

" #P 0: �5�

Proof. Formula (5) implies minP

i wi min uGi �a0� ÿmax uG

i �a�� �

P 0 by changing the notation. That meansPi wi min uG

i �a0� ÿmax uGi �a�

� �P 0. In other words,

minX

i

wi uGi �a0� ÿ uG

i �a�� �

PX

i

wi min uGi �a0� ÿmax uG

i �a�� �

;

whereP

i wi uGi �a0� ÿ uG

i �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 re-duced in the interactive process in Section 4.4, but Theorem 1 shows a stronger dominance relation thanTheorem 2. Because, if alternative a0 is strictly dominant than a at total range, a0 is strictly dominant than aat agreed range. But the reverse condition is not always true. Though these group's pairwise dominancerules are seemingly good, strict dominance (SD) relations between alternatives rarely happen because SDrelation under group decision making context analogous to Theorem 1 or Theorem 2 does not alwayshappen in the real world.

Instead of SD, weak dominance (WD) relation is usually used between alternatives in group decisionmaking problems. WD means that we are to try to select the alternative that minimizes `regrets'. HoweverWD on total range is meaningless, because the density of the aggregated group's utility value uG

i (a) in thetotal range is not uniform. That means WD relation may be determined by an extreme value of a speci®cgroup member, if total range is used. By applying basic logic of WD condition under single decision makingcontext to group decision making situation, we can suggest a WD condition for group's agreed range likefollowing Theorem 3.

Theorem 3. Let Zmin�a0; a� � minP

i wi �maxk minUki

uki �a0� ÿmink maxUk

iuk

i �a�� and assume that group'sutility value uG

i (a) is a random variable with one-dimensional interval space of group's agreed range. Furtherassume the probability distribution function f(uG

i (a)) of uGi (a) is symmetric about a vertical axis through

uGi (a)� l(a), where l(a)� (min uG

i (a) + max uGi (a))/2.

146 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: �6�

Proof. The group's agreed range for alternative a, (maxkmin uki (a), minkmax uk

i (a)) can be brie¯y representedby (min uG

i (a), max uGi (a)). Then Zmin�a0; a� � min

Pi wi min uG

i �a0� ÿmax uGi �a�

� �and Zmin�a; a0� �

minP

i wi �min uGi �a� ÿmax uG

i �a0��. From l�a� � �min uGi �a� �max uG

i �a��=2, min uGi �a� � 2 l�a�ÿ

max uGi �a�. Similarly min uG

i �a0� � 2 l�a0� ÿmax uGi �a0�. Then,

Zmin�a0; a� ÿ Zmin�a; a0�� min

Xi

wi min uGi �a0� ÿmax uG

i �a�� �ÿmin

Xi

wi min uGi �a� ÿmax uG

i �a0�� �

� 2l�a0� �minX

i

wi ÿmax uGi �a0� ÿmax uG

i �a�� �ÿ 2l�a� ÿmin

Xi

wi ÿmax uGi �a� ÿmax uG

i �a0�� �

� 2�l�a0� ÿ l�a��P 0:

That is, EU�a0� ÿ EU�a�P 0. Therefore, we can conclude that alternative a0 weakly dominates alternative aif Zmin�a0; a�P Zmin�a; a0�. h

This condition means that the alternative having the higher positioned range in utility value is preferredto lower range in order to minimize the degree of group's regret. The WD condition can be used as a certaindecision 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 ontheorems in Section 4.5.

Step 1: (Computing individual utility ranges) Solve the following LPs to calculate individual utilityranges of each alternative on each attribute with group members' incomplete information:

min =max uki �a�;

s:t: U ki � w�a0; a�:

Step 2: (Preference aggregation) Get a group's utility range of each alternative on each attribute by totalrange and agreed range from individual utility ranges:

Group's agreed range � maxk

min uki �a�;min

kmax uk

i �a�� �

;

Group's total range � mink

min uki �a�;max

kmax uk

i �a�� �

:

Step 3: (Interaction for specifying group members' preferences) Interact with group members by eachattribute to re®ne their utility information to reduce the gap between agreed range and total range. Thespeci®c processes are suggested in Section 4.4.

Step 4: (Finding dominance relations) Find dominance relations with theorems in Section 4.5 by solvingLPs 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 torevise his/her previous assessments, then go to step 1. Otherwise, stop with the result in Step 4.

S.H. Kim et al. / European Journal of Operational Research 118 (1999) 139±152 147

The above interactive procedure can be used in the context that the group has same attributes andimprecise/incomplete attributes' weights.

5. Paper submission example

This section illustrates features of interactive procedure for group decision making in the context of apaper submission problem. After completing a manuscript, author(s) usually confront the problem whichjournal is appropriate for the manuscript. If more than two authors are involved, the problem becomes aMAGDM 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 in-formation 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 importantthan acceptability and acceptability is more important than compatibility. Kim feels that reputation is moreimportant than acceptability and acceptability is as important but no more than two times as important ascompatibility. Choi and Kim set the threshold value of group consensus at 0.5 for all attributes for sim-plicity.

Step 1: Under constraints on each member's utility values, we calculate the minimum and maximumvalues 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 0.26 u2(a1)6 0.7 u3(a1) P 0.6

u1(a2) P 0.7 u2(a3) P 0.5 u3(a2) P 0.7

u1(a3)6 u1(a1) u2(a2)6 u2(a1) u3(a3) P 0.5

u1(a1)6 u1(a2) u2(a1) ÿ u2(a2)6 u2(a3) ÿ u2(a1) u3(a3)6 u3(a2)

u2(a1)6 u2(a3)

Table 2

Kim's utility information

Reputation Acceptability Compatibility

u1(a1) P 0.8 0.56 u2(a1)6 0.8 0.66 u3(a2)6 0.9

u1(a2)6 0.7 0.26 u2(a2)6 0.6 0.46 u3(a3)6 0.8

2u1(a3)6 u1(a1) u2(a3) P u2(a1) u3(a1) P 2u3(a3)

u2(a3) P u2(a2)

Table 3

The computed values group members' utility range

Members Choi Kim

Attributes 1 2 3 1 2 3

a1 (0.5, 1) (0.2, 0.7) (0.6, 1) (0.8, 1) (0.5, 0.8) (0.8, 1)

a2 (0.7, 1) (0, 0.7) (0.7, 1) (0, 0.7) (0.2, 0.6) (0.6, 0.9)

a3 (0, 0.9) (0.5, 1) (0.5, 1) (0, 0.5) (0.5, 1) (0.4, 0.8)

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

Step 2: As shown in Table 4, the group's utility ranges represented by total and agreed range are ob-tained 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'' meansthere 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 acceptableconsensus.

Step 3: The procedure interacts with group members by each attribute. But, it is not necessary to getmore information about compatibility attribute because the agreed ranges of all alternatives on the at-tribute are within the acceptable bound of total ranges. For the reputation attribute, Choi and Kim areasked to modify his information related with alternative a1 and alternative a2 because they have providedinformation 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 shownin Fig. 3 except a shaded area. Referred to the graphic display, he modi®es the set of incomplete infor-mation 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 hisincomplete information referring group's utility information. After those modi®cations, go to step 1 with

Table 4

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

Attributes Reputation Acceptability Compatibility

Agreed Total Agreed Total Agreed Total

a1 (0.8, 1) (0.5, 1) (0.5, 0.7) (0.2, 0.8) (0.8, 1) (0.6, 1)

vi(a1) 0.4 0.33 0.5

a2 No (0, 1) (0.2, 0.6) (0, 0.7) (0.7, 0.9) (0.6, 1)

vi(a2) 0 0.57 0.5

a3 (0, 0.5) (0, 0.9) (0.5, 1) (0.5, 1) (0.5, 0.8) (0.4, 1)

vi(a3) 0.56 1.0 0.5

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

S.H. Kim et al. / European Journal of Operational Research 118 (1999) 139±152 149

modi®ed information on the reputation attribute and calculate modi®ed group's utility ranges. At secondinteraction, same processes for the acceptability attribute are performed. The result is shown in Table 5. Wecan 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 constraintsof attributes' weights; w1 P w2; w2 P w3; w26 2w3, and w1 � w2 � w3 � 1. Solving 6 LPs yields X �f�a1; a3�; �a2; a3�g because Zmin�1; 2� � ÿ0:2; Zmin�1; 3� � 0:036; Zmin(2, 1)�ÿ0.4, Zmin(2, 3)� 0.033,Zmin(3, 1)�ÿ1, Zmin(3, 2)�ÿ1. If group members are not able to provide further information, then theymay 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 weakdominance 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.

Table 5

Group's utility ranges at ®nal interaction

Attributes Reputation Acceptability Compatability

Agreed Total Agreed Total Agreed Total

a1 (0.8, 1) (0.6, 1) (0.3, 0.7) (0.2, 0.8) (0.8, 1) (0.6, 1)

vi(a1) 0.5 0.67 0.5

a2 (0.6, 1) (0.27, 1) (0.2, 0.5) (0, 0.53) (0.7, 0.9) (0.6, 1)

vi(a2) 0.55 0.57 0.5

a3 (0, 0.5) (0, 0.5) (0.5, 1) (0.5, 1) (0.5, 0.8) (0.4, 1)

vi(a3) 0.56 1.0 0.5

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

In this example, a few interaction are performed for illustrative convenience but more interaction will bedone if we set a higher value on di. And the fact that a little information are modi®ed at each interactionmeans that sensitivity analysis is helpful for calculating modi®ed group's utility ranges. Sensitivity analysisdemands a reduced computation time. The interactive process helps the group seek a consensus. Fig. 4shows 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 eachattribute. If di becomes 1, agreed range becomes closer to total range, that is, group's information should beclose to an identical information state. If di becomes 0, a group's decision making can be made only ifagreed ranges exist. So a larger di value demands much burden of group members, but the result of groupdecision making represents well the group's preference. And a smaller di value is helpful to speed up theinteraction process, but the result can not represent well the group's preference. So di determines thetradeo� 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 informa-tion. An interactive modi®cation procedure is described for each member to make a group consensus byinteractively modifying his/her incomplete information to be a concrete or complete one. To help such aprocedure, we used a range-typed representation method for utility value. Range-typed utility informationmakes it easy to compare each group member's utility information with group's one and to aggregate eachgroup member's utility information into a group's one. We formally described theoretic models for es-tablishing group's pairwise dominance relations with group's utility ranges by using a separable linearprogramming technique.

Group decision making needs diverse situations, such as a speedy decision or a decision demandinggroup's time-consuming negotiation. Our procedure can handle this tradeo� among the time, the quality ofgroup decision making and the burden of group members.

A promising research area is to develop a group decision support system (GDSS) for interactive pref-erence elicitation from group members and an automated solving tool because the number of LPs to besolved increases in multiplicative fashion as the number of alternatives, attributes, or participating groupmembers increase.

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