Decision Support
Interactive Multiple Criteria Decision Making based on preference driven Evolutionary Multiobjective Optimization with controllable accuracy

https://doi.org/10.1016/j.ejor.2011.07.013Get rights and content

Abstract

We present an approach to interactive Multiple Criteria Decision Making based on preference driven Evolutionary Multiobjective Optimization with controllable accuracy.

The approach relies on formulae for lower and upper bounds on coordinates of the outcome of an arbitrary efficient variant corresponding to preference information expressed by the Decision Maker. In contrast to earlier works on that subject, here lower and upper bounds can be calculated and their accuracy controlled entirely within evolutionary computation framework. This is made possible by exploration of not only the region of feasible variants – a standard within evolutionary optimization, but also the region of infeasible variants, the latter to our best knowledge being a novel approach within Evolutionary Multiobjective Optimization.

To illustrate how this concept can be applied to interactive Multiple Criteria Decision Making, two algorithms employing evolutionary computations are proposed and their usefulness demonstrated by a numerical example.

Highlights

► Human preferences drive evolutionary computations for interactive MCDM processes. ► Decision making processes are based on lower and upper bounds on outcomes. ► Evolutionary computations explore regions of feasible and infeasible variants. ► Two evolutionary algorithms are proposed and a numerical example is solved.

Introduction

Evolutionary computations have been in focus of science community already for considerable time. The general idea of Evolutionary Multiobjective Optimization (EMO) is to derive discrete approximations of the sets of efficient variants related to Multiple Criteria Decision Making (MCDM) problems by means of evolutionary computations.

As a rule, such approximations consist of feasible variants. Assuming that all criteria are of the type “the more, the better”, approximations can be regarded, conventionally, as lower approximations of the set of efficient variants. Books by Deb, 2001, Coello Coello et al., 2002, Talbi, 2009 provide state-of-the-art surveys of methods and algorithms which follow this idea.

Once a lower approximation to an MCDM problem is derived, the Decision Maker (DM) can search over it for an element which he or she regards as the most preferred variant. This means, however, that the variant selected as the most preferred one is in general not necessarily efficient. To make such a selection justifiable, lower approximations should be tight, i.e. they should be close, in a sense, to the set of efficient variants. EMO algorithms with plausible numerical behavior with respect to tightness of lower approximations have been proposed (Deb, 2001, Coello Coello et al., 2002, Talbi, 2009), but no formal (as opposed to empirical) mechanisms to ensure tightness of lower approximations have been indicated, which is the consequence of the fact that in general the approximated set, i.e. the set of efficient outcomes, is not known.

Producing populous approximations may be prohibitive for large-scale or computing intensive MCDM problems. Moreover, populous lower approximations can be, from the decision making standpoint, greatly redundant and thus represent a vain computing effort. To cope with these drawbacks, it has been recently proposed to combine (hybridize) evolutionary computations and interactive MCDM methods. This is done by incorporating iteratively the DM’s partial preferences, as soon as they become available in the course of interactive decision making process, into evolutionary computations to produce lower approximations which represent fairly the set of efficient variants, but only in regions of variants, which are of the DM’s interest at the current stage of the decision process (cf. Sakawa and Kato, 2002, Phelps and Köksalan, 2003, Deb and Sundar, 2006, Thiele et al., 2009, Chaudhuria and Deb, 2010).1 In this work we follow that line of research.

However, an open fundamental question of EMO is “how tight is the current lower approximation?” When EMO gets hybridized with MCDM this question becomes “how tight is the current lower approximation in the regions of variants, which are of the DM’s interest at the current stage of the decision process?” The question is indeed fundamental, for if the current lower approximation is tight (in a sense of some formal mechanism) in the region of the DM’s interest, then no further evolutionary computations are needed, but if otherwise, computations should be continued. EMO, aiming to provide lower approximation of the whole set of efficient outcomes, does not provide for an answer to this question.

Therefore in this work we take an alternative course to the existing EMO-based approaches to MCDM problems. Namely, we use of the very old mathematical concept of approximating quantities both from below (by lower bounds) and from above (by upper bounds). This in fact is a basic concept of optimization, all primal–dual optimization schemes make use of it to assess the optimal value. We apply this concept to assess efficient outcomes (i.e. vectors of criteria values of efficient feasible variants).

More specifically, we present formulae for calculating lower bounds and upper bounds on coordinates of the outcome of an arbitrary efficient variant corresponding to preference information expressed by the DM via a preference carrier. Specific forms of preference carriers are discussed in Section 4. Given an instance of the preference carrier, the corresponding outcome can be represented, for each coordinate, by any number between the calculated lower and the upper bound (e.g. by the middle point of the range). If accuracy of such a representation is found by the DM unsatisfactory, it can be improved at the cost of some extra computing effort focused on that instance of preference carrier only. In that way computing effort necessary to support the decision process is fully controlled by the requirements of the DM with respect to accuracy of outcome assessment, and thus, from the decision making stand point, is never redundant.

In contrast to the earlier works on that subject (Kaliszewski, 2004, Kaliszewski, 2006a, Kaliszewski, 2006b), in this work we show that lower and upper bounds can be calculated and their accuracy controlled entirely within evolutionary computation framework. However, as shown below, this requires an active exploration of regions of feasible as well as infeasible variants. To our best knowledge it is a novel approach within EMO, for in all EMO algorithms proposed thus far, whenever newly generated elements fail the feasibility test they are immediately either modified towards feasibility or dropped.

It is worth observing that the proposed approach applies also when outcomes are calculated not as functional mappings of variants but are just outputs of “black box” decision support systems (this is exactly the case where EMO approach has a clear advantage over the exact optimization methods). Such situations occur e.g. as identified in Miroforidis (2010), in hypermarkets where information to the managerial level comes from complex decision support systems dedicated to operational units. In engineering the necessity to feed optimization models with data from complex simulation systems has been realized early. Examples from electromagnetic engineering can be found in Di Barba (2010) and from aviation engineering in Vankan and Kesseler, 2006, Kesseler and van Houten, 2007.

Section snippets

An outline

Decision making, whether in engineering, economic or social domain, calls for multi aspect deliberations. The field of Multiple Criteria Decision Making (where “criteria” stands for “aspects”) provides methodologies and supporting tools to cope with technical complexities of decision making. For a wide range of “complex” decision problems, where because of scale, data bulk, and/or intricate framing a formal model is requested, efficient variants, and among them the most preferred variant (the

Definitions and notation

In this section we define all the basic formal notions and technical tools we use in our development, namely the underlying model for MCDM, elements of the model whose derivation we are interested in (efficient and weakly efficient elements of a set), the standard MCDM dominance relation, EMO related constructs – lower and upper shells, and an optimization problem used here as the main vehicle to select variants satisfying, in a sense, the DM preferences.

Let x denote a (decision) variant, X a

Direct and indirect preference carriers

In this section we address the crucial issue to decision making processes, namely the issue of a “protocol” (preference carrier) by which the DMs can communicate with the underlying model to secure its support in his or her endeavor to select the most preferred variant.

Most MCDM interactive methods reduce in fact to implicit search (implicit enumeration) of the set of efficient outcomes in the quest for the outcome the DM declares he or she prefers most (cf. e.g. Wierzbicki, 1999, Miettinen,

Parametric bounds on efficient outcomes

In this section we show how to calculate bounds on weakly efficient outcome components, which, if sufficiently tight, can represent weakly efficient outcomes to MCDM problems without the necessity to derive them explicitly.

Below by fτ we denote a weakly efficient outcome corresponding to a weakly efficient variant x, i.e. fτ = f(x), where x is designated by

  • selecting an instance of direction of concessions τ,

  • assuming that x is a solution of optimization problem (8) for λ related to τ by formula

MCDM with preference activated EMO and preference adjusted accuracy

To derive SL one can use any EMO algorithm (cf. Michalewicz, 1996, Deb, 2001, Coello Coello et al., 2002, Hanne, 2007) satisfying or modified appropriately to satisfy condition (2).

To cope with derivation of SU observe first that since the definition of upper shell involves N, SU is not a suitable construct for evolutionary computations. A more suitable one, referring to SL instead of N, namely upper approximation (of N) AU, is obtained by replacing condition (4) in the definition of upper

An illustrative example

To illustrate the versatility of our development we simulate the decision process for a managerial model steaming from hypermarket management problem conducted with the help of algorithms PDAE/M and EPO (Miroforidis, 2010). The model contains three objective functions and concerns three operational units to be managed.

The model has the following form:f1(x)=40(x1)0.35-x1-x2-x3,f2(x)=0.1e-f1(x)700x2+0.3e-f1(x)500x3,f3(x)=200(x1)0.35,s.t.i=13xi120,x127,x220,x320,and all the objective

Deriving variants corresponding to the most preferred assessment

The most preferred variant (a variant which corresponds to the most preferred outcome assessment) is derived from set {x  X0L(SL, τ)  f(x)  U(AU, τ)}, so its outcome components satisfy the same bounds as components of the most preferred outcome assessment. Below we show that set {x  X0L(SL, τ)  f(x)  U(AU, τ)} is not empty. Moreover, we show that this set contains at least one element of SL used to calculate L(SL, τ).

We assume that SL is not empty and filtered.

Lemma 1

Set {x  X0L(SL, τ)  f(x)  U(AU, τ)} is not

Discussion, concluding remarks and directions for further research

The development presented in the paper builds on established MCDM features:

  • interactivity of decision making processes,

  • flexibility of preferences expressing (either in holistic or in atomistic manner).

It is novel by the very fact that it at the same time:

  • offers a conceptually simple framework for the DM to express his or her preferences,

  • enables explicit approximate calculations within (multiple criteria) decision making processes with assured accuracy,

  • provides for a method to drive EMO

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