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.     

A multiple criteria ranking procedure based on distance between partial preorders

In this paper, a multiple criteria ranking procedure based on distance between partial preorders is proposed. This method consists of two phases. In the first phase, the decision maker is asked to rank alternatives with a preorder (complete or partial) for each criterion and provide complete or partial linear information about the relative importance (weights) of the criteria. In the second phase, we introduce a distance procedure to aggregate the above individual rankings into a global ranking (a partial preorder). An algorithm for the aggregation procedure is proposed, followed by a numerical illustration.     

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.     

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.     

Decision risk analysis for an interval TOPSIS method

TOPSIS is a multi-attribute decision making (MADM) technique for ranking and selection of a number of externally determined alternatives through distance measures. When the collected data for each criterion is interval and the risk attitude for a decision maker is unknown, we present a new TOPSIS method for normalizing the collected data and ranking the alternatives. The results show that the decision maker with different risk attitude ranks the different alternatives.     

Optimal Criterion Weights in Repetitive Multicriteria Decision-Making

In repetitive judgmental discrete decision-making with multiple criteria, the decision maker usually behaves as if there is a set of appropriate criterion weights such that the decisions chosen are based on the weighted sum of all the criteria. Many different procedures for estimating these implied criterion weights have been proposed. Most of these procedures emphasize the preference trade-off among the multiple criteria of the decision maker, and thus the criterion weights obtained are not directly related to the hit ratio of matching decisions. Based on past data, statistical discriminant analysis can be used to determine the implied criterion weights that would reflect the past decisions. The most interesting performance measure is the hit ratio. In this work, we use the integer linear goal-programming technique to determine optimal criterion weights which minimize the number of misclassification of decisions. The linear goal-programming formulation has m constraints and m + k + 1 variables, where m is the number of cases and k is the number of criteria. Empirical study is done by using two different procedures on the actual past admission data of an M.B.A. programme. The hit ratios of the different procedures are compared.     

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.     

Extension of VIKOR method for decision making problem with interval numbers

The VIKOR method was developed for multi-criteria optimization of complex systems. It determines the compromise ranking list and the compromise solution obtained with the initial (given) weights. This method focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria. It introduces the multi-criteria ranking index based on the particular measure of “closeness” to the “ideal” solution. The aim of this paper is to extend the VIKOR method for decision making problems with interval number. The extended VIKOR method’s ranking is obtained through comparison of interval numbers and for doing the comparisons between intervals, we introduce α as optimism level of decision maker. Finally, a numerical example illustrates and clarifies the main results developed in this paper.     

Aggregation of utility-based individual preferences for group decision-making

Multi-attribute utility theory (MAUT) elicits an individual decision maker’s preferences for single attributes and develops a utility function by mathematics formulation to add up the preferences of the entire set of attributes when assessing alternatives. A common aggregation method of MAUT for group decisions is the simple additive weighting (SAW) method, which does not consider the different preferential levels and preferential ranks for individual decision makers’ assessments of alternatives in a decision group, and thus seems too intuitive in achieving the consensus and commitment for group decision aggregation. In this paper, the preferential differences denoting the preference degrees among different alternatives and preferential priorities denoting the favorite ranking of the alternatives for each decision maker are both considered and aggregated to construct the utility discriminative values for assessing alternatives in a decision group. A comparative analysis is performed to compare the proposed approach to the SAW model, and a satisfaction index is used to investigate the satisfaction levels of the final two resulting group decisions. In addition, a feedback interview is conducted to understand the subjective perceptions of decision makers while examining the results obtained from these two approaches for the second practical case. Both investigation results show that the proposed approach is able to achieve a more satisfying and agreeable group decision than that of the SAW method.     

A visual interactive approach for multiple criteria decision making with monotone utility functions

We consider the problem of choosing the best of a set of alternatives where each alternative is evaluated on multiple criteria. We develop a visual interactive approach assuming that the decision maker (DM) has a general monotone utility function. The approach partitions the criteria space into nonoverlapping cells. The DM uses various graphical aids to move between cells and to further manipulate selected cells with the goal of creating cells that have ideal points less preferred than an alternative. When the DM identifies such cells, all alternatives in those cells are eliminated from further consideration. The DM may also compare pairs of alternatives. The approach terminates with the most preferred alternative of the DM.     

Strong reciprocity and strong consistency in pairwise comparison matrix with fuzzy elements

The decision making problem considered in this paper is to rank n alternatives from the best to the worst, using the information given by the decision maker in the form of an \(n\times n\) pairwise comparison matrix. Here, we deal with pairwise comparison matrices with fuzzy elements. Fuzzy elements of the pairwise comparison matrix are applied whenever the decision maker is not sure about the value of his/her evaluation of the relative importance of elements in question. We investigate pairwise comparison matrices with elements from abelian linearly ordered group (alo-group) over a real interval. The concept of reciprocity and consistency of pairwise comparison matrices with fuzzy elements have been already studied in the literature. Here, we define stronger concepts, namely the strong reciprocity and strong consistency of pairwise comparison matrices with fuzzy intervals as the matrix elements (PCF matrices), derive the necessary and sufficient conditions for strong reciprocity and strong consistency and investigate their properties as well as some consequences to the problem of ranking the alternatives.     

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.     

Ordinal regression revisited: Multiple criteria ranking using a set of additive value functions

We present a new method, called UTA^GMS, for multiple criteria ranking of alternatives from set A using a set of additive value functions which result from an ordinal regression. The preference information provided by the decision maker is a set of pairwise comparisons on a subset of alternatives A^R ⊆ A, called reference alternatives. The preference model built via ordinal regression is the set of all additive value functions compatible with the preference information. Using this model, one can define two relations in the set A: the necessary weak preference relation which holds for any two alternatives a, b from set A if and only if for all compatible value functions a is preferred to b, and the possible weak preference relation which holds for this pair if and only if for at least one compatible value function a is preferred to b. These relations establish a necessary and a possible ranking of alternatives from A, being, respectively, a partial preorder and a strongly complete relation. The UTA^GMS method is intended to be used interactively, with an increasing subset A^R and a progressive statement of pairwise comparisons. When no preference information is provided, the necessary weak preference relation is a weak dominance relation, and the possible weak preference relation is a complete relation. Every new pairwise comparison of reference alternatives, for which the dominance relation does not hold, is enriching the necessary relation and it is impoverishing the possible relation, so that they converge with the growth of the preference information. Distinguishing necessary and possible consequences of preference information on the complete set of actions, UTA^GMS answers questions of robustness analysis. Moreover, the method can support the decision maker when his/her preference statements cannot be represented in terms of an additive value function. The method is illustrated by an example solved using the UTA^GMS software. Some extensions of the method are also presented.     

An extension of TOPSIS for group decision making

An extension of TOPSIS (technique for order performance by similarity to ideal solution), a multi-attribute decision making (MADM) technique, to a group decision environment is investigated. TOPSIS is a practical and useful technique for ranking and selection of a number of externally determined alternatives through distance measures. To get a broad view of the techniques used, we provide a few options for the operations, such as normalization, distance measures and mean operators, at each of the corresponding steps of TOPSIS. In addition, the preferences of more than one decision maker are internally aggregated into the TOPSIS procedure. Unlike in previous developments, our group preferences are aggregated within the procedure. The proposed model is indeed a unified process and it will be readily applicable to many real-world decision making situations without increasing the computational burden. In the final part, the effects of external aggregation and internal aggregation of group preferences for TOPSIS with different computational combinations are compared using examples. The results have demonstrated our model to be both robust and efficient.     

Decision making in multiobjective optimization aided by the multicriteria tournament decision method

This paper proposes a new method for multicriteria analysis, named Multicriteria Tournament Decision (MTD). It provides the ranking of alternatives from best to worst, according to the preferences of a human decision-maker (DM). It has some positive aspects such as: it has a simple algorithm with intuitive appeal; it involves few input parameters (just the importance weight of each criterion).The helpfulness of MTD is demonstrated by using it to select the final solution of multiobjective optimization problems in an a posteriori decision making approach. Having at hand a discrete approximation of the Pareto front (provided by a multiobjective evolutionary search algorithm), the choice of the preferred Pareto-optimal solution is performed using MTD.A simple method, named Gain Analysis method (GAM), for verifying the existence of a better solution (a solution associated to higher marginal rates of return) than the one originally chosen by the DM, is also introduced here. The usefulness of MTD and GAM methods is confirmed by the suitable results shown in this paper.     

A new integrated DEA model for finding most BCC-efficient DMU

In many applications of widely recognized technique, DEA, finding the most efficient DMU is desirable for decision maker. Using basic DEA models, decision maker is not able to identify most efficient DMU. Amin and Toloo [Gholam R. Amin, M. Toloo, Finding the most efficient DMUs in DEA: an improved integrated model. Comput. Ind. Eng. 52 (2007) 71–77] introduced an integrated DEA model for finding most CCR-efficient DMU. In this paper, we propose a new integrated model for determining most BCC-efficient DMU by solving only one linear programming (LP). This model is useful for situations in which return to scale is variable, so has wider range of application than other models which find most CCR-efficient DMU. The applicability of the proposed integrated model is illustrated, using a real data set of a case study, which consists of 19 facility layout alternatives.     

UTA-poly and UTA-splines: Additive value functions with polynomial marginals

Additive utility function models are widely used in multiple criteria decision analysis. In such models, a numerical value is associated to each alternative involved in the decision problem. It is computed by aggregating the scores of the alternative on the different criteria of the decision problem. The score of an alternative is determined by a marginal value function that evolves monotonically as a function of the performance of the alternative on this criterion. Determining the shape of the marginals is not easy for a decision maker. It is easier for him/her to make statements such as “alternative a is preferred to b”. In order to help the decision maker, UTA disaggregation procedures use linear programming to approximate the marginals by piecewise linear functions based only on such statements. In this paper, we propose to infer polynomials and splines instead of piecewise linear functions for the marginals. In this aim, we use semidefinite programming instead of linear programming. We illustrate this new elicitation method and present some experimental results.     

Decision making in multiobjective optimization aided by the multicriteria tournament decision method

This paper proposes a new method for multicriteria analysis, named Multicriteria Tournament Decision (MTD). It provides the ranking of alternatives from best to worst, according to the preferences of a human decision-maker (DM). It has some positive aspects such as: it has a simple algorithm with intuitive appeal; it involves few input parameters (just the importance weight of each criterion).The helpfulness of MTD is demonstrated by using it to select the final solution of multiobjective optimization problems in an a posteriori decision making approach. Having at hand a discrete approximation of the Pareto front (provided by a multiobjective evolutionary search algorithm), the choice of the preferred Pareto-optimal solution is performed using MTD.A simple method, named Gain Analysis method (GAM), for verifying the existence of a better solution (a solution associated to higher marginal rates of return) than the one originally chosen by the DM, is also introduced here. The usefulness of MTD and GAM methods is confirmed by the suitable results shown in this paper.     

Ranking residential properties by a multicriteria single price model

This study describes an application of the multicriteria single price model (Ballestero) to the ranking of alternatives. By a generalization of the original model, the equilibrium set of alternatives can be characterized from the viewpoints, respectively, of the demander and the supplier, and from that the efficiency index can be calculated. We demonstrate how, in a state of equilibrium, the two viewpoints result inevitably in inverse orders of ranking. In contrast with other proposals for full ranking of alternatives, the method used in the present study (i) assumes a moderate attitude on the part of the decision maker towards risk, with a robust axiomatic basis; (ii) assigns weights to the criteria independently of which alternative is being evaluated and the attitude (optimistic or pessimistic) of the decision maker; (iii) produces a cardinal hierarchy of the alternatives and not just an ordinal one. The model is illustrated by a sample of residential properties in the city of Valencia, Spain.     

基于灰色关联法和HA算子的Pythagorean模糊群决策方法

针对Pythagorean模糊群决策问题,提出一种基于Pythagorean模糊混合平均算子的决策方法。首先,提出一种基于Pythagorean模糊信息及其运算法则的Pythagorean模糊混合平均算子;其次,构建一种基于最大熵模型的属性位置权重定权方法,同时根据灰色关联方法提出一种属性客观权重计算方法,进而获得Pythagorean模糊混合平均算子的定权方法;利用Pythagorean模糊混合平均算子对单决策者信息进行融合,通过Pythagorean模糊加权平均算子对各专家信息进行融合,并依据得分函数与精确函数进行排序择优;最后,通过一个算例说明该方法的有效性和可行性。