Checking the Consistency of Solutions in Decision-Making Problems with Multiple Weighted Agents

Checking the Consistency of Solutions in Decision-Making Problems with Multiple Weighted Agents

Domenico Maisano (Department of Management and Production Engineering (DIGEP), Politecnico di Torino, Turin, Italy) and Luca Mastrogiacomo (Department of Management and Production Engineering (DIGEP), Politecnico di Torino, Turin, Italy)
Copyright: © 2018 |Pages: 20
DOI: 10.4018/IJDSST.2018010103
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A decision-making problem diffused in various practical contexts is that of aggregating multi-agent judgements into a consensus ordering, in the case the agents' importance is expressed through a set of weights. A crucial point in this aggregation is that the consensus ordering well reflects the input data, i.e., agents' judgements and importance. The scientific literature encompasses several aggregation techniques, even if it does not include a versatile tool for a quantitative assessment and comparison of their performance. The aim of this paper is introducing a new indicator (p), which allows to verify the degree of consistency between consensus ordering and input data. This indicator is simple, intuitive and independent from the aggregation technique in use; for this reason, it can be applied to a variety of practical contexts and used to compare the results obtained through different aggregation techniques, when applied to a specific problem. The description is supported by various application examples.
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A very general decision-making problem is that of aggregating multi-agent judgments, concerning different alternatives, into a consensus ordering. The adjective “consensus” indicates that this ordering should reflect agents’ judgements as much as possible, even in the presence of divergences. Summarizing, the problem is characterized by the following elements (see the scheme in Figure 1):

  • A set of alternatives to be prioritized (a, b, c, d, e, ...);

  • A set of decision-making agents1 (D1, D2, …, DM) expressing their opinion on the alternatives, through various possible forms (e.g., paired-comparison judgements, evaluations/measurements on ordinal/interval/ratio scales, linear/partial preference orderings, etc.);

  • An importance hierarchy of agents, which is usually expressed through a set of weights or importances (r1, r2, …, rM), defined on a ratio scale by a team of experts (Vora et al., 2014; Ngan et al., 2016);

  • A consensus ordering of the alternatives, which represents the output of the problem.

Figure 1.

Input/Output data concerning the general problem of the aggregation of multi-agent judgements on a set of alternatives, into a consensus ordering


The problem of interest is quite old and has been studied in various fields, stimulating the development of a variety of aggregation techniques (Von Neumann & Morgenstern, 1944; Fine & Fine, 1974; Fishburn, 1974; Hwang & Lin, 1987; Keeney & Raiffa, 1993). For example, in the field of social choice and voting theory, the authors recall the method by Condorcet and that by Borda (Borda, 1781; Franceschini et al., 2007); in the field of multicriteria decision making, the Electre (Figueira et al., 2005), Promethee (Brans & Mareschal, 2005) or AHP (Saaty, 1980) methods, in the field of the Internet intelligent agents, that by Yager (2001), etc.

Each of these techniques has its pro and contra; based on this consideration, an interesting question may arise: For a generic decision-making problem, how could the best aggregation technique(s) be identified? It is probably impossible to answer this question, since the “true” solution for a generic problem is not known a priori (Figueira et al., 2005; Cook, 2006). Nevertheless, the fact remains that one technique may be more or less appropriate than one other depending on: (i) the practical purpose of the aggregation (for example, isolating the best alternative or a limited number of excellent alternatives, excluding the worst alternatives, defining a complete ranking, etc.), (ii) the form in which the agents’ judgments and/or the importance hierarchy are expressed, and (iii) the ability to encourage the involvement and participation of decision-makers in constructing a shared solution (Zopounidis & Pardalos, 2010).

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