An Interactive Approach for Choosing the Best of a Set of Alternatives

In this paper we address the problem of choosing the most preferred alternative among a large number of alternatives where each alternative is defined by multiple criteria. We assume that the decision maker has a quasiconcave utility function. We develop an exact approach that combines the ideas that have appeared in the literature regarding the use of different types of dummy alternatives in conjunction with real alternatives. Our experimental results indicate that the new approach is comparable to leading existing approaches.     

Interactive outranking approaches for multicriteria decision-making problems with imprecise information

PROMETHEE is a powerful method, which can solve many multiple criteria decision making (MCDM) problems. It involves sophisticated preference modelling techniques but requires too much a priori precise information about parameter values (such as criterion weights and thresholds). In this paper, we consider a MCDM problem where alternatives are evaluated on several conflicting criteria, and the criterion weights and/or thresholds are imprecise or unknown to the decision maker (DM). We build robust outranking relations among the alternatives in order to help the DM to rank the alternatives and select the best alternative. We propose interactive approaches based on PROMETHEE method. We develop a decision aid tool called INTOUR, which implements the developed approaches.     

Inferring an ELECTRE TRI Model from Assignment Examples

Given a finite set of alternatives, the sorting problem consists in the assignment of each alternative to one of the pre-defined categories. In this paper, we are interested in multiple criteria sorting problems and, more precisely, in the existing method ELECTRE TRI. This method requires the elicitation of parameters (weights, thresholds, category limits,...) in order to construct the Decision Maker's (DM) preference model. A direct elicitation of these parameters being rather difficult, we proceed to solve this problem in a way that requires from the DM much less cognitive effort. We elicit these parameters indirectly using holistic information given by the DM through assignment examples. We propose an interactive approach that infers the parameters of an ELECTRE TRI model from assignment examples. The determination of an ELECTRE TRI model that best restitutes the assignment examples is formulated through an optimization problem. The interactive aspect of this approach lies in the possibility given to the DM to revise his/her assignment examples and/or to give additional information before the optimization phase restarts.     

A hierarchical interactive method for ranking alternatives with multiple qualitative criteria

In this study we deal with the problem of finding the most preferred composite ranking of a set of alternatives evaluated using a large number of criteria having a hierarchical structure. The criteria may be qualitative or quantitative. The decision maker evaluates alternatives using each criterion at the lowest (basic) level. That information is then used to construct the generalized correlation matrix to describe interdependencies between the criteria. The correlation matrix and the criterion hierarchy are the basic information used in the approach. Our interactive approach is designed to help the decision maker find the most preferred aggregation of the kth level criteria, which produces the criteria at the (k + 1)st level. As the final result of the aggregation we obtain the strength of the preference matrix for the criterion at the highest level. By means of that matrix, we produce the final ranking of the alternatives using the Bowman and Colantoni (1973) model. The approach is easy to implement and convenient to use. We have implemented an experimental version of it on an Apple III microcomputer. The graphical colour display is used as an aid in finding the most preferred aggregation. An illustrative example is provided.     

Robust ordinal regression for value functions handling interacting criteria

We present a new method called UTA^GMS–INT for ranking a finite set of alternatives evaluated on multiple criteria. It belongs to the family of Robust Ordinal Regression (ROR) methods which build a set of preference models compatible with preference information elicited by the Decision Maker (DM). The preference model used by UTA^GMS–INT is a general additive value function augmented by two types of components corresponding to “bonus” or “penalty” values for positively or negatively interacting pairs of criteria, respectively. When calculating value of a particular alternative, a bonus is added to the additive component of the value function if a given pair of criteria is in a positive synergy for performances of this alternative on the two criteria. Similarly, a penalty is subtracted from the additive component of the value function if a given pair of criteria is in a negative synergy for performances of the considered alternative on the two criteria. The preference information elicited by the DM is composed of pairwise comparisons of some reference alternatives, as well as of comparisons of some pairs of reference alternatives with respect to intensity of preference, either comprehensively or on a particular criterion. In UTA^GMS–INT, ROR starts with identification of pairs of interacting criteria for given preference information by solving a mixed-integer linear program. Once the interacting pairs are validated by the DM, ROR continues calculations with the whole set of compatible value functions handling the interacting criteria, to get necessary and possible preference relations in the considered set of alternatives. A single representative value function can be calculated to attribute specific scores to alternatives. It also gives values to bonuses and penalties. UTA^GMS–INT handles quite general interactions among criteria and provides an interesting alternative to the Choquet integral.     

Interactive procedure for a multiobjective stochastic discrete dynamic problem

Multiple objectives and dynamics characterize many sequential decision problems. In the paper we consider returns in partially ordered criteria space as a way of generalization of single criterion dynamic programming models to multiobjective case. In our problem evaluations of alternatives with respect to criteria are represented by distribution functions. Thus, the overall comparison of two alternatives is equivalent to the comparison of two vectors of probability distributions. We assume that the decision maker tries to find a solution preferred to all other solutions (the most preferred solution). In the paper a new interactive procedure for stochastic, dynamic multiple criteria decision making problem is proposed. The procedure consists of two steps. First, the Bellman principle is used to identify the set of efficient solutions. Next interactive approach is employed to find the most preferred solution. A numerical example and a real-world application are presented to illustrate the applicability of the proposed technique.     

Robust multi-criteria ranking with additive value models and holistic pair-wise preference statements

We consider a problem of ranking alternatives based on their deterministic performance evaluations on multiple criteria. We apply additive value theory and assume the Decision Maker’s (DM) preferences to be representable with general additive monotone value functions. The DM provides indirect preference information in form of pair-wise comparisons of reference alternatives, and we use this to derive the set of compatible value functions. Then, this set is analyzed to describe (1) the possible and necessary preference relations, (2) probabilities of the possible relations, (3) ranges of ranks the alternatives may obtain, and (4) the distributions of these ranks. Our work combines previous results from Robust Ordinal Regression, Extreme Ranking Analysis and Stochastic Multicriteria Acceptability Analysis under a unified decision support framework. We show how the four different results complement each other, discuss extensions of the main proposal, and demonstrate practical use of the approach by considering a problem of ranking 20 European countries in terms of 4 criteria reflecting the quality of their universities.     

Classifying efficient alternatives in SMAA using cross confidence factors

Stochastic multicriteria acceptability analysis (SMAA) is a family of methods for aiding multicriteria group decision making. These methods are based on exploring the weight space in order to describe the preferences that make each alternative the most preferred one. The main results of the analysis are rank acceptability indices, central weight vectors and confidence factors for different alternatives. The rank acceptability indices describe the variety of different preferences resulting in a certain rank for an alternative; the central weight vectors represent the typical preferences favouring each alternative; and the confidence factors measure whether the criteria data are sufficiently accurate for making an informed decision.In some cases, when the problem involves a large number of efficient alternatives, the analysis may fail to discriminate between them. This situation is revealed by low confidence factors. In this paper we develop cross confidence factors, which are based on computing confidence factors for alternatives using each other’s central weight vectors. The cross confidence factors can be used for classifying efficient alternatives into sets of similar and competing alternatives. These sets are related to the concept of reference sets in Data Envelopment Analysis (DEA), but generalized for stochastic models. Forming these sets is useful when trying to identify one or more most preferred alternatives, or suitable compromise alternatives. The reference sets can also be used for evaluating whether criteria need to be measured more accurately, and at which alternatives the measurements should be focused. This may cause considerable savings in measurement costs. We demonstrate the use of the cross confidence factors and reference sets using a real-life example.     

DIS-CARD: a new method of multiple criteria sorting to classes with desired cardinality

In this paper, we present a new preference disaggregation method for multiple criteria sorting problems, called DIS-CARD. Real-life experience indicates the need of considering decision making situations in which a decision maker (DM) specifies a desired number of alternatives to be assigned to single classes or to unions of some classes. These situations require special methods for multiple criteria sorting subject to desired cardinalities of classes. DIS-CARD deals with such a problem, using the ordinal regression approach to construct a model of DM’s preferences from preference information provided in terms of exemplary assignments of some reference alternatives, together with the above desired cardinalities. We develop a mathematical model for incorporating such preference information via mixed integer linear programming (MILP). Then, we adapt the MILP model to two types of preference models: an additive value function and an outranking relation. Illustrative example is solved to illustrate the methodology.     

An interactive approach for multiobjective decision making

We develop an interactive approach for multiobjective decision-making problems, where the solution space is defined by a set of constraints. We first reduce the solution space by eliminating some undesirable regions. We generate solutions (partition ideals) that dominate portions of the efficient frontier and the decision maker (DM) compares these with feasible solutions. Whenever the decision maker prefers a feasible solution, we eliminate the region dominated by the partition ideal. We then employ an interactive search method on the reduced solution space to help the DM further converge toward a highly preferred solution. We demonstrate our approach and discuss some variations.     

An Interactive Multiple Criteria Approach to Ranking Alternatives

In this paper we are concerned with ranking various orderings of a set of alternatives to a composite order as a multiple criteria problem. The orderings (called preference orderings) can be real preference orderings or any natural orderings. The objective is to find the most preferred order of the decision maker using the preference orderings as criteria.In principle, the problem can be formulated as a multiple objective linear programming problem using the model of Bowman and Colantoni and then solved with the interactive method proposed by Zionts and Wallenius. However, the fact that we are dealing with integer variables prohibits us from applying this approach as such. We discuss the problem formulation and propose a modified approach to that of Zionts and Wallenius for solving the problem.     

Distributed preference relations for multiple attribute decision analysis

In this paper, we propose a new pairwise comparison approach called distributed preference relation (DPR) to simultaneously signify preferred, non-preferred, indifferent, and uncertain degrees of one alternative over another on a set of grades, which is more versatile for elicitation of preference information from a decision maker than multiplicative preference relation, fuzzy preference relation (FPR) and intuitionistic FPR. In a DPR matrix on a set of alternatives, each element is a distribution recording the preferred, non-preferred, indifferent, and uncertain degrees of one alternative over another using a set of grades. To facilitate the comparison of alternatives, we define a score matrix based on a DPR matrix using the given score values of the grades. Its additive consistency is constructed, analysed, and compared with the additive consistency of FPRs between alternatives. A method for comparing two interval numbers is then employed to create a possibility matrix from the score matrix, which can generate a ranking order of alternatives with possibility degrees. A problem of evaluating strategic emerging industries is investigated using the approach to demonstrate the application of a DPR matrix to modelling and analysing a multiple attribute decision analysis problem.     

Adapting and refining in multi-criteria decision-making

This paper describes the implementation of a Structured Methodology for Direct-Interactive Structured-Criteria (DISC) Multi-Criteria Decision-Making (MCDM), an eight-stage nomological adjusting cycle of activities that shape the information used to make a decision, requiring it be accessible, differentiable, abstractable, understandable, verifiable, measurable, refinable and usable. It shows, in a major IT strategic investment case, that Structured DISC MCDM provides a robust model that can be used for deep and serious consideration of multi-criteria decisions by a group of decision-makers over a long period. The paper describes the case as it moves through stages of the adjusting cycle and shows that, after completing the cycle, it reverses and becomes an adapting process, starting with refining the information. Refining is shown to be more extensive than previously understood, and to cover ‘alternatives & scores’, ‘criteria & weights’ and ‘set of alternatives’. Next the form of measurement is adapted. As the number of alternatives are reduced it can become more appropriate to directly compare the two or three most preferred alternatives relative to one another rather than objectively. Finally the criteria tree can be adapted using a ‘magnifying glass’ approach. This confines the evaluation to that part of the criteria tree in which the difference between a few preferred alternatives is mainly emphasised.     

Multicriterion analysis of a vegetation management problem using ELECTRE II

A brief review of the ELECTRE II technique based on a strong and a weak ordinal outranking relationship is given. This technique, which is applied to alternative vegetation management schemes, leads to a complete ranking of the alternatives by means of three thresholds for concordance conditions and two thresholds for discordance conditions. Specifically, six alternative schemes are evaluated with respect to seven criteria, leading to the determination of a preferred system. A sensitivity analysis indicates that the ELECTRE II ranking is fairly robust with respect to parameter changes for the conditions of the case study.     

Multicriteria Modelling of Repair Contract Based on Utility and ELECTRE I Method with Dependability and Service Quality Criteria

A DM faces a choice among several alternatives of repair contract for a system. Each alternative of a repair contract implies specific results regarding the following characteristics or criteria: response time, quality service, dependability and related cost. This problem has been analysed through a multicriteria decision model. The model is based on the ELECTRE method combined with utility functions. Main theoretical aspects and practical implications are presented, including a numerical application.     

The assessment of the information quality with the aid of multiple criteria analysis

The virtual business work flow depends on the information quality (IQ) which is essential attribute of information. The IQ depends strongly on organization of the information system (IS) and how the information is processed. In our approach we incorporate the four-aspect representation of IQ: (1) intrinsic, (2) contextual, (3) representational, and (4) accessibility. These four-aspects are divided into several criteria at the next level of hierarchy. The weights, representing the relative importance of criteria, have been assessed by pair-wise comparisons made by group of experts. Based on discussion with experts, six alternative strategies, that could be used for improving the IQ, were designed. For each given criterion, the group of subjects revealed the opinion about the level of achievement of every alternative. The set of scores, assigned to the alternative by different subjects, formed the discrete distribution that is used for a comparison of alternatives with the aid of stochastic dominances. In analogy to the Electre I methodology, the simple algorithm for the aggregate evaluation of analyzed alternatives was proposed. The benefits of proposed approach were demonstrated in a case study of the semiconductor industry. The results of our study suggest, that in case of matured company, the external strategies, that point out to the information exchange and strategic networked alliance with customers and suppliers, are preferred to the internal ones. The latter ones might be of greater importance for the new set up or for a young company.     

Implementing stochastic multicriteria acceptability analysis

Stochastic multicriteria acceptability analysis (SMAA) is a family of methods for aiding multicriteria group decision making in problems with inaccurate, uncertain, or missing information. These methods are based on exploring the weight space in order to describe the preferences that make each alternative the most preferred one, or that would give a certain rank for a specific alternative. The main results of the analysis are rank acceptability indices, central weight vectors and confidence factors for different alternatives. The rank acceptability indices describe the variety of different preferences resulting in a certain rank for an alternative, the central weight vectors represent the typical preferences favouring each alternative, and the confidence factors measure whether the criteria measurements are sufficiently accurate for making an informed decision.     

Rank reversals in multi-criteria decision analysis with statistical modelling of ratio-scale pairwise comparisons

In multi-criteria decision analysis, the overall performance of decision alternatives is evaluated with respect to several, generally conflicting decision criteria. One approach to perform the multi-criteria decision analysis is to use ratio-scale pairwise comparisons concerning the performance of decision alternatives and the importance of decision criteria. In this approach, a classical problem has been the phenomenon of rank reversals. In particular, when a new decision alternative is added to a decision problem, and while the assessments concerning the original decision alternatives remain unchanged, the new alternative may cause rank reversals between the utility estimates of the original decision alternatives. This paper studies the connections between rank reversals and the potential inconsistency of the utility assessments in the case of ratio-scale pairwise comparisons data. The analysis was carried out by recently developed statistical modelling techniques so that the inconsistency of the assessments was measured according to statistical estimation theory. Several type of decision problems were analysed and the results showed that rank reversals caused by inconsistency are natural and acceptable. On the other hand, rank reversals caused by the traditional arithmetic-mean aggregation rule are not in line with the ratio-scale measurement of utilities, whereas geometric-mean aggregation does not cause undesired rank reversals.     

A geometric averaging procedure for constructing supertransitive approximation to binary comparison matrices

The usefulness of encoding the fuzzy evaluations of alternatives and the importance weights of criteria, in a multiple objective decision problem through binary comparison matrices (or pairwise judgment matrices) is receiving considerable attention. The methodology for identifying the best alternative in a given decision problem involves the computation of the principal eigenvectors of the binary comparison matrices. The eigenvectors transform the fuzzy evaluations of the importance of the criteria and the ratings of the alternatives into a ratio scale. A difficulty that is often experienced in using this approach in practice, is the inconsistency of the binary evaluations. This paper proposes a simple averaging procedure to construct a supertransitive approximation to a binary comparison matrix, where inconsistency is a problem. It is further suggested that such an adjustment might be necessary to more closely reflect the inherent fuzziness of the evaluations contained in a binary comparison matrix. The procedure is illustrated by means of examples.