Validating absolute weight bounds in Data Envelopment Analysis (DEA) models

The Charnes, Cooper and Rhodes (CCR) DEA model and its linear forms maximise the efficiency of the assessed decision making unit (DMU) and, at the same time, the ratio of this efficiency to the maximum efficiency taken across all the DMUs, the latter naturally always being equal to one. It has been shown recently that, in the presence of absolute weight bounds, these models may not maximise the ratio of these efficiencies, a fact that may cause problems with the interpretation and use of the optimal primal and dual solutions. For example, an inefficient DMU may have greater efficiency than its target unit for some weights. This paper investigates the problem in greater detail; it shows that, in the linear DEA model maximising the total virtual output of the assessed DMU, the problem occurs only if upper bounds are imposed on the output weights. A similar result is established for the model that minimises the total virtual input.     

Parametric importance of criteria and intervals of value tradeoff uncertainty in the analysis of multicriteria problems

Multicriteria decision-making problems under bounded (above, below, or from both sides) continuous or discrete criteria are considered. Methods for comparing variants of solutions using the information accumulated in the form of interval estimates of replacing the values of some criteria by the values of others (such replacements are called tradeoffs; in other words, this can be considered as a compensation of the deterioration of some criteria by improving the values of others) are proposed along with simple consistency conditions of such information. The issue of constructing the set of nondominated variants is discussed.     

Dominance and potential optimality in multiple criteria decision analysis with imprecise information

This paper discusses multiple criteria models of decision analysis with finite sets of alternatives. A weighted sum of criteria is used to evaluate the performance of alternatives. Information about the weights is assumed to be in the form of arbitrary linear constraints. Conditions for checking dominance and potential optimality of decision alternatives are presented. In the case of testing potential optimality, the proposed appoach leads to the consideration of a couple of mutually dual linear programming problems. The analysis of these problems gives valuable information for the decision maker. In particular, if a decision alternative is not potentially optimal, then a mixed alternative dominating it is defined by a solution to one of the LP problems. This statement generalizes similar results known for some special cases. The interpretation of the mixed alternative is discussed and compared to its analogue in a data envelopment analysis context.     

The explicit role of weight bounds in models of data envelopment analysis

In this paper, we suggest that weight bounds used in models of data envelopment analysis (DEA) can be assessed using production trade-offs between inputs and outputs. This development is based on a transformation of the envelopment DEA model to a special form, which exhibits an explicit link between weight bounds and production trade-offs. This allows us to specify the guidelines to the construction of weight bounds, which ensure that the envelopment form has a clear economic meaning. In this development, the radial efficiency measure retains its standard meaning as a technologically realistic improvement factor. This contrasts with the methods based on value judgements for which this traditional meaning of efficiency becomes void. We show that the exact economic meaning of the same weight bounds depends on the unit under the assessment and the orientation (input-minimization or output-maximization) of the model. The suggested approach is valid under the assumptions of constant and variable returns to scale.     

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.     

Potential optimality in multicriterial optimization

The relation between Pareto, Slater, Geoffrion, and potential optimality is investigated for basic classes of value functions in multicriterial optimization problems.     

An extended maximin approach for decision analysis with uncontrollable factors

In this paper an extension to the maximin approach to decision analysis in the presence of uncontrollable factors is proposed. This extension is based on the assumption that probabilities of consequences are known. Using the language of stochastic dominance, one decision alternative is preferred to another if the cumulative distribution function of the first alternative dominates that of the second in some area of low value consequences. This approach is an extension of a standard lexicographic maximin procedure to a case in which decision alternatives are characterised by arbitrary, including continuous, sets of consequences. Applications of the suggested approach to an ‘attack–defence’ type game and to the problems of location of public facilities are discussed.     

Local and global returns to scale in performance measurement

In data envelopment analysis, the type of local returns to scale (RTS) exhibited by a technically efficient unit indicates whether an increase or reduction of the scale of operations could improve the productivity of the unit. One of the approaches to testing RTS is based on the comparison of the efficiency of the unit in specially constructed reference technologies. It has been suggested that this approach is equally suitable for convex and non-convex, including the free disposal hull, technologies. In this paper, we construct examples that show that this suggestion in the case of non-convex technologies is not correct. We show that the type of RTS obtained by this approach is not a local, but global, characteristic of the technology, as it indicates the direction to the most productive scale size of the unit. In non-convex technologies, the local and global classifications of RTS are generally different.     

Set choice problems with incomplete information about the preferences of the decision maker

We consider the problem of selecting a predetermined number of objects from a given finite set. It is assumed that the preferences of the decision maker on this set are only partially known. Our solution approach is based on the notions of optimal and non-dominated subsets. The properties of such subsets and the objects they contain are investigated. The implementation of the developed approach is discussed and illustrated by various examples.     

Decision making under uncertainty with unknown utility function and rank-ordered probabilities

We consider the ranking of decision alternatives in decision analysis problems under uncertainty, under very weak assumptions about the type of utility function and information about the probabilities of the states of nature. Namely, the following two assumptions are required for the suggested method: the utility function is in the class of increasing continuous functions, and the probabilities of the states of nature are rank-ordered. We develop a simple analytical method for the partial ranking of decision alternatives under the stated assumptions. This method does not require solving optimization programs and is free of the rounding errors.