DEA models for the explicit maximisation of relative efficiency

It has recently been demonstrated that incorporating weight bounds and other non-homogeneous restrictions in DEA models may lead to underestimation of the maximum relative efficiency of decision making units. This paper suggests a way of avoiding this by replacing the objective function in DEA models by the relative efficiency of the assessed unit and converting the resulting models to linear forms. An alternative approach based on incorporating weight restrictions in the recently introduced maximin DEA model is also considered. It is shown that imposing weight bounds in the maximin model is equivalent to imposing bounds on ratios of individual weights.     

Maximin Programming,Fuzzy Linear Programming and Multi-Criteria Decision Making

Mathematical programming models present an approach to solving multicriteria decision problems. This paper compares two such models. One based on fuzzy set theory, the other relying on utility functions and the maximin decision criteria. The two models have identical form. The paper also illustrates the relationship between fuzzy set theory and multi-criteria decision making.     

Selecting preferred solutions in the minimax approach to dynamic programming problems under flexible constraints

Dynamic Programming is a powerful approach to the optimization of sequential or multistage decision processes, e.g., in planning or in system control. In this paper, we consider both theoretical and algorithmic issues in sequential decision processes under flexible constraints. Such processes must attain a given goal within some tolerance. Tolerances or preferences also apply to the values the decision variables may take or on the action chosen at each step. Such problems boil down to maximin optimization. Unfortunately, this approach suffers from the so-called “drowning effect” (lack of discrimination) and the optimality principle of dynamic programming is not always verified. In this context, we introduce a general framework for refined minimax optimization procedures in order to compare and select preferred alternatives. This framework encompasses already introduced methods such as LexiMin and DiscriMin, but it allows their extension to the comparison of vectors of unequal lengths. We show that these refined comparisons restore compatibility with the optimality principle, and that classical algorithms can be adapted to compute such preferred solutions, by exploiting existing results on idempotent semirings.     

On the Role of Mixed Strategies in Elementary Decision Analysis and Related Decision-Support-System Treatments

We first demonstrate that mixed strategies are relevant in decision analysis for a maximin decision-maker quite apart from any game-theory considerations. This rectifies the apparent misconception that results from MS/OR textbooks which discuss mixed strategies only in the game-theory setting. Next we show an example of an implementable mixed strategy, by which we mean a mixed decision strategy which does not require randomization for its implementation. This application is to portfolio construction.     

Robust optimization analysis for multiple attribute decision making problems with imprecise information

This paper considers ranking decision alternatives under multiple attributes with imprecise information on both attribute weights and alternative ratings. It is demonstrated that regret results from the decision maker??s inadequate knowledge about the true scenario to occur. Potential optimality analysis is a traditional method to evaluate alternatives with imprecise information. The essence of this approach is to identify any alternative that outperforms the others in its best-case scenario. Our analysis shows that potential optimality analysis is optimistic in nature and may lead to a significant loss if an unfavorable scenario occurs. We suggest a robust optimization analysis approach that ranks alternatives in terms of their worst-case absolute or relative regret. A robust optimal alternative performs reasonably well in all scenarios and is shown to be desirable for a risk-concerned decision maker. Linear programming models are developed to check robust optimality.     

A preference disaggregation decision support system for financial classification problems

Classification is one of the most extensively studied problems in the fields of multivariate statistical analysis, operations research and artificial intelligence. Decisions involving a classification of the alternative solutions are of major interest in finance, since several financial decision problems are best studied by classifying a set of alternative solutions (firms, loan applications, investment projects, etc.) in predefined classes. This paper proposes an alternative approach to the classical statistical methodologies that have been extensively used for the study of financial classification problems. The proposed methodology combines the preference disaggregation approach (a multicriteria decision aid method) with decision support systems. More specifically, the FINancial CLASsification (FINCLAS) multicriteria decision support system is presented. The system incorporates a plethora of financial modeling tools, along with powerful preference disaggregation methods that lead to the development of additive utility models for the classification of the considered alternatives into predefined classes. An application in credit granting is used to illustrate the capabilities of the system.     

Ranking decision making units by imposing a minimum weight restriction in the data envelopment analysis

It has been widely recognized that data envelopment analysis (DEA) lacks discrimination power to distinguish between DEA efficient units. This paper proposes a new methodology for ranking decision making units (DMUs). The new methodology ranks DMUs by imposing an appropriate minimum weight restriction on all inputs and outputs, which is decided by a decision maker (DM) or an assessor in terms of the solutions to a series of linear programming (LP) models that are specially constructed to determine a maximin weight for each DEA efficient unit. The DM can decide how many DMUs to be retained as DEA efficient in final efficiency ranking according to the requirement of real applications, which provides flexibility for DEA ranking. Three numerical examples are investigated using the proposed ranking methodology to illustrate its power in discriminating between DMUs, particularly DEA efficient units.     

Assessing the relative efficiency of decision making units using DEA models with weight restrictions

In models of data envelopment analysis (DEA), an optimal set of input and output weights is generally assumed to represent the assessed decision making unit (DMU) in the best light in comparison to all the other DMUs. The paper shows that this may not be correct if absolute weight bounds or some other weight restrictions are added to the model. A consequence may be that the model will underestimate the relative efficiency of DMUs. The incorporation of weight restrictions in a maximin DEA model is suggested. This model can be further converted to more operational forms, which are similar to the classical DEA models.     

A methodology for analyzing decision networks, based on information theory

This paper assumes the organization as a distributed decision network. It proposes an approach based on application and extension of information theory concepts, in order to analyze informational complexity in a decision network, due to interdependence between decision centers.Based on this approach, new quantitative concepts and definitions are proposed in order to measure the information in a decision center, based on Shannon entropy and its complement in possibility theory, U uncertainty. This approach also measures the quantity of interdependence between decision centers and informational complexity of decision networks.The paper presents an agent-based model of organization as a graph composed of decision centers. The application of the proposed approach is in analyzing and assessing a measure to the organization structure efficiency, based on informational communication view. The structure improvement, analysis of information flow in organization and grouping algorithms are investigated in this paper. The results obtained from this model in different systems as distributed decision networks, clarifies the importance of structure and information distribution sources effect’s on network efficiency.     

Interactive multiple objective programming subject to flexible constraints

An interactive decision support system is introduced which aids in solving multiple objective programming problems subject to strict and flexible constraints. Integral part is an extension of a well-known fuzzy sets approach evaluating possible solutions by their degrees of membership to objectives and constraints. This approach is linked to classical multiple objective programming models. If the decision maker cannot determine membership functions a priori the system suggests functions dependent on the given information and interactive modifications are allowed.     

Generalized Lagrange multiplier technique for nonlinear programming

Our aim here is to present numerical methods for solving a general nonlinear programming problem. These methods are based on transformation of a given constrained minimization problem into an unconstrained maximin problem. This transformation is done by using a generalized Lagrange multiplier technique. Such an approach permits us to use Newton's and gradient methods for nonlinear programming. Convergence proofs are provided, and some numerical results are given.     

Generalized Maximin Solutions in Multicriteria Bargaining

In this paper we address bargaining games where the agents have to take into account different criteria to value the decisions. We propose the class of generalized maximin solutions, as the natural extension for these games of the maximin solutions in conventional bargaining. In order to refine this solution concept, we define a multicriteria lexicographic partial ordering and present the class of generalized leximin solutions as those that are nondominated with respect to this relation. We establish some properties of these solutions and characterize them as solutions of multicriteria problems. The research of the authors is partially supported by the Spanish Ministry of Science and Technology projects BFM2002-11282-E and BEC2003-03111.     

A multiple criteria ranking method based on game cross-evaluation approach

The application of Data Envelopment Analysis (DEA) as an alternative multiple criteria decision making (MCDM) tool has been gaining more attentions in the literatures. Doyle (Organ. Behav. Hum. Decis. Process. 62(1):87?C100, 1995) presents a method of multi-attribute choice based on an application of DEA. In the first part of his method, the straightforward DEA is considered as an idealized process of self-evaluation in which each alternative weighs the attributes in order to maximize its own score (or desirability) relative to the other alternatives. Then, in the second step, each alternative applies its own DEA-derived best weights to each of the other alternatives (i.e., cross-evaluation), then the average of the cross-evaluations that get placed on an alternative is taken as an index of its overall score. In some cases of multiple criteria decision making, direct or indirect competitions exist among the alternatives, while the factor of competition is usually ignored in most of MCDM settings. This paper proposes an approach to evaluate and rank alternatives in MCDM via an extension of DEA method, namely DEA game cross-efficiency model in Liang, Wu, Cook and Zhu (Oper. Res. 56(5):1278?C1288, 2008b), in which each alternative is viewed as a player who seeks to maximize its own score (or desirability), under the condition that the cross-evaluation scores of each of other alternatives does not deteriorate. The game cross-evaluation score is obtained when the alternative??s own maximized scores are averaged. The obtained game cross-evaluation scores are unique and constitute a Nash equilibrium point. Therefore, the results and rankings based upon game cross-evaluation score analysis are more reliable and will benefit the decision makers.     

How to allocate hard candies fairly

We consider the problem of allocating a finite number of indivisible items to two players with additive utilities. We design a procedure that looks for all the maximin allocations and makes repeated use of an extension of the Adjusted Winner, an effective procedure that deals with divisible items, to find new candidate solutions, and to suggest which items should be assigned to the players.     

Constructive decision theory

We sketch a theory of decision that allows us to construct both goals and degrees of belief. Before choosing an action, we create and weight goals. We represent our beliefs about the consequences of each action by constructing a belief function on the set of possible consequences. Then we combine the beliefs and goals to see the value that will be secured and the value that will be excluded by each action. This approach is more constructive than Bayesian decision theory, because as Savage's problem of small worlds teaches us, that theory assumes preferences that antedate deliberation.     

Correlation coefficient of dual hesitant fuzzy sets and its application to multiple attribute decision making

A dual hesitant fuzzy set (DHFS) consists of two parts, that is, the membership hesitancy function and the nonmembership hesitancy function, supporting a more exemplary and flexible access to assign values for each element in the domain, and can handle two kinds of hesitancy in this situation. It can be considered as a powerful tool to express uncertain information in the process of group decision making. Therefore, we propose a correlation coefficient between DHFSs as a new extension of existing correlation coefficients for hesitant fuzzy sets and intuitionistic fuzzy sets and apply it to multiple attribute decision making under dual hesitant fuzzy environments. Through the weighted correlation coefficient between each alternative and the ideal alternative, the ranking order of all alternatives can be determined and the best alternative can be easily identified as well. Finally, a practical example of investment alternatives is given to demonstrate the practicality and effectiveness of the developed approach.     

Multicriteria modelling for a repair contract problem based on utility and the ELECTRE I method

Each alternative for a repair contract implies a specific responsetime and related cost. The response time is associated withthe commitment of repair time, based on the contract. A decisionmaker chooses the best alternative taking into account the systemperformance and the cost of the contract. This problem has beenanalysed through a multicriteria decision model. This decisionmodel supports decision makers in the determination of the bestcombination of contracting conditions. The decision model proposedallows the decision maker to quantify the consequences of anaction taking into account two basic criteria: the cost of thecontract and the system performance. Two different decisionmodels have been built to support decision makers. These decisionmodels are based on different multicriteria approaches. Thefirst, reported in a previous paper, is based on the multiattributeutility theory (MAUT). The model presented in this paper isbased on the ELECTRE I method combined with utility functions.The paper presents the main theoretical aspects related to bothapproaches and practical implications related to model building.A numerical application is presented in order to illustratethe use of the decision model.     

Single valued neutrosophic cross-entropy for multicriteria decision making problems

A single valued neutrosophic set (SVNS) is an instance of a neutrosophic set, which give us an additional possibility to represent uncertainty, imprecise, incomplete, and inconsistent information which exist in real world. It would be more suitable to apply indeterminate information and inconsistent information measures. In this paper, the cross entropy of SVNSs, called single valued neutrosophic cross entropy, is proposed as an extension of the cross entropy of fuzzy sets. Then, a multicriteria decision-making method based on the proposed single valued neutrosophic cross entropy is established in which criteria values for alternatives are SVNSs. In decision making process, we utilize the single-valued neutrosophic weighted cross entropy between the ideal alternative and an alternative to rank the alternatives corresponding to the cross entropy values and to select the most desirable one(s). Finally, a practical example of the choosing problem of suppliers is provided to illustrate the application of the developed approach.     

Environmental impact assessment using the evidential reasoning approach

Environmental impact assessment (EIA) problems are often characterised by a large number of identified environmental factors that are qualitative in nature and can only be assessed on the basis of human judgments, which inevitably involve various types of uncertainties such as ignorance and fuzziness. So, EIA problems need to be modelled and analysed using methods that can handle uncertainties. The evidential reasoning (ER) approach provides such a modelling framework and analysis method. In this paper the ER approach will be applied to conduct EIA analysis for the first time. The environmental impact consequences are characterized by a set of assessment grades that are assumed to be collectively exhaustive and mutually exclusive. All assessment information, quantitative or qualitative, complete or incomplete, and precise or imprecise, is modelled using a unified framework of a belief structure. The original ER approach with a recursive ER algorithm will be introduced and a new analytical ER algorithm will be investigated which provides a means for using the ER approach in decision situations where an explicit ER aggregation function is needed such as in optimisation problems. The ER approach will be used to aggregate multiple environmental factors, resulting in an aggregated distributed assessment for each alternative policy. A numerical example and its modified version are studied to illustrate the detailed implementation process of the ER approach and demonstrate its potential applications in EIA.     

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.