Group decision making with incomplete information: a dominance and quasi‐optimality volume‐based approach using Monte‐Carlo simulation

• DOI: http://doi.org/10.1111/itor.12315
• Published date: 01 January 2019
• Date: 01 January 2019

Intl. Trans. in Op. Res. 26 (2019) 318–339

DOI: 10.1111/itor.12315

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RESEARCH

Group decision making with incomplete information:

a dominance and quasi-optimality volume-based approach

using Monte-Carlo simulation

Paula Sarabandoa,LuisC.Dias

band Rudolf Vetscherac

aDepartment of Mathematics, INESC Coimbra and Viseu Polytechnic Institute, Campus Polit´

ecnico de Viseu, 3504-510

Viseu, Portugal

bINESC Coimbra, Faculty of Economics, University of Coimbra,Av Dias da Silva 165, 3004-512 Coimbra, Portugal

cDepartment of Business Administration, University of Vienna,Oskar Morgenstern Platz 1, A-1090 Vienna, Austria

E-mail: psarabando@estv.ipv.pt [Sarabando]; lmcdias@fe.uc.pt [Dias]; rudolf.vetschera@univie.ac.at [Vetschera]

Received 18 September 2015; receivedin revised form 22 February 2016; accepted 7 May 2016

Abstract

In this paper, we present a comprehensive framework for multiattribute group decision making considering

that neither information about individual preferences nor the importance of individual decision makers for

the group is available in exact form. We study several different forms of incomplete preference information,

including a ranking of attribute weights, ranking of values of alternatives in each attribute, and ranking of

value differences. Our approach is based on relative volumes in parameter space and allows for probabilistic

statementsabout different results, including optimality or quasi-optimality of alternatives, or relationsbetween

alternatives.

Keywords:multiple criteria analysis; group decisions; incomplete information; ordinal information; additive model

1. Introduction

Complex and important decision problems in organizations often require the consideration of

multiple attributes, and are frequently made by groups of experts or decision makers (DMs), who

are able to analyze the problem from different perspectives. Over the last decades, multiattribute

group decisions have thus evolved into a major area of interest in decision analysis (Hwang and

Lim, 1987; Belton and Stewart, 2002). Despite several decades of research, there are still many

challenges to the effective support of DMs in a multiattribute group decision context. Perhaps the

most critical issue in this context is the amount and type of information thatgroup members need to

provide in order to reacha group decision. Even single DMs, who do not operate in a groupcontext,

sometimes have difficulties in specifying the preference information needed to solve a multicriteria

C

2016 The Authors.

International Transactionsin Operational Research C

2016 International Federation ofOperational Research Societies

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P. Sarabando et al. / Intl. Trans.in Op. Res.26 (2019) 318–339 319

decision problem exactly (Damart et al., 2007). This problem is even more severein a group context,

where aggregation of opinions across group members might require additional information, for

example, about the importance of each member’s opinion for the group decision.

Existing literature on group decision analysis (for multiattribute problems and in more general

settings) thus uses a wide spectrum of different types of information to be provided by group

members. One end of this spectrum is formed by models of social choice theory, which only

require group members to specify an ordinal ranking of alternatives and thus pose very weak

requirements on their ability to provide information. The downside of this approach is the well-

known impossibility theorem by Arrow (1951), which indicates that it is not possible to find an

aggregation of rankings that produces a transitive and complete group ranking and simultaneously

fulfills four reasonable conditions (universal domain, Pareto optimality, independence of irrelevant

alternatives, and nondictatorship).

At the other end of the spectrum, many approaches to multiattribute group decision making

require group members to specify a complete multiattribute value function (Watson and Buede,

1987) or multiattribute utility function (Keeney and Raiffa, 1976) (the difference between these

two functions is not relevant for our work), and then combine these individual functions in a

group utility function (Keeney and Kirkwood, 1975; Dyer and Sarin, 1979; Keeney and Nau,

2011; Keeney, 2013). It can be shown (Keeney, 1976) that a group utility function, which additively

combines weighted individual utility functions of each DM, fulfills axioms analogous to those of

Arrow (1951). However, specification of these individual utility functions is already a complex task

for the group members, and the additional specification of member weights further adds to this

complexity.

The approach, which we pursue in this paper, attempts to cover a middle ground between these

two ends of the spectrum by requiring less information from group members than the complete

specification of cardinal utilities, and at the same time avoiding (as far as possible) violation of

Arrow’s axioms. Our approach builds upon models of decision making under incomplete (partial,

imprecise) information (Weber, 1987; Dias and Cl´

ımaco, 2000; Salo and H¨

am¨

al¨

ainen, 2001). These

methods allow DMs to specify information on preferenceparameters only imprecisely, for example,

in the form of linear constraints that can be defined directly (Kirkwood and Sarin, 1985; Hazen,

1986; Park and Kim, 1997) or indirectly via a comparison of alternatives(Greco et al., 2008), rather

than as exact values. In particular, our approach is similar to stochastic multicriteria acceptability

analysis (SMAA) (Lahdelma et al., 1998) that utilizes incomplete preference informationto provide

probabilistic statements about possible relations between alternatives as well as their ranks.

Methods for decision making under incomplete information have been proposed as extensionsto

many different methods for multicriteria decision making. The original SMAA method (Lahdelma

et al., 1998) builds upon additive multiattribute utility functions, and it has been extended later

to many other multicriteria decision-making methods (Tervonen and Figueira, 2008). Likewise,

different methods to aggregate individual evaluations based on various multicriteria methods to

a group evaluation exist. Such aggregation can take place at the level of preference parameters

or at the level of evaluations of alternatives (Vetschera, 1990). In the first case, a multicriteria

method is applied at the group level using aggregated parameters elicited from the entire group, or

computed from individual parameters. In the second case, group members first evaluate alternatives

individually using some multiattribute method, and then the evaluations are aggregated across

members. Both approaches could be applied to different multicriteria methods, leading to a variety

C

2016 The Authors.

International Transactionsin Operational Research C

2016 International Federation of OperationalResearch Societies