Potential Optimality of Pareto Optima

AbstractIn this paper the notion of potential optimality without an assumption that a value function exists is used to investigate multicriterial optimization problems. Our results show that the notions of potential optimality and strong Pareto optimality (weak Pareto optimality, properly Pareto optimality) are equivalent for special forms of objective functions which are increasing with respect to strong Pareto relation (weak Pareto relation)

Returns to scale in convex production technologies

The notion of returns to scale (RTS) is well-established in data envelopment analysis (DEA). In the variable returns-to-scale production technology, the RTS characterization is closely related to other scale characteristics, such as the scale elasticity, most productive scale size (MPSS), and global RTS types indicative of the direction to MPSS. In recent years, a number of alternative production technologies have been developed in the DEA literature. Most of these technologies are polyhedral, and hence are closed and convex sets. Examples include technologies with weakly disposable undesirable outputs, models with weight restrictions and production trade-offs, technologies that include several component production processes, and network DEA models. For most of these technologies, the relationship between RTS and other scale characteristics has remained unexplored. The theoretical results obtained in this paper establish such relationships for a very large class of closed convex technologies, of which polyhedral technologies are an important example

DEA models with production trade-offs and weight restrictions

There is a large literature on the use of weight restrictions in multiplier DEA models. In this chapter we provide an alternative view of this subject from the perspective of dual envelopment DEA models in which weight restrictions can be interpreted as production trade-offs. The notion of production trade-offs allows us to state assumptions that certain simultaneous changes to the inputs and outputs are technologically possible in the production process. The incorporation of production trade-offs in the envelopment DEA model, or the corresponding weight restrictions in the multiplier model, leads to a meaningful expansion of the model of production technology. The efficiency measures in DEA models with production trade-offs retain their traditional meaning as the ultimate and technologically realistic improvement factors. This overcomes one of the known drawbacks of weight restrictions assessed using other methods. In this chapter we discuss the assessment of production trade-offs, provide the corresponding theoretical developments and suggest computational methods suitable for the solution of the resulting DEA models

Optimal weights in DEA models with weight restrictions

According to a conventional interpretation of a multiplier DEA model, its optimal weights show the decision making unit under the assessment, denoted DMUo, in the best light in comparison to all observed DMUs. For multiplier models with additional weight restrictions such an interpretation is known to be generally incorrect (specifically, if weight restrictions are linked or nonhomogeneous), and the meaning of optimal weights in such models has remained unclear. In this paper we prove that, for any weight restrictions, the optimal weights of the multiplier model show DMUo in the best light in comparison to the entire technology expanded by the weight restrictions. This result is consistent with the fact that the dual envelopment DEA model benchmarks DMUo against all DMUs in the technology, and not only against the observed DMUs. Our development overcomes previous concerns about the use of weight restrictions of certain types in DEA models and provides their rigorous and meaningful interpretation

A linear programming approach to efficiency evaluation in nonconvex metatechnologies

The notions of metatechnology and metafrontier arise in applications of data envelopment analysis (DEA) in which decision making units (DMUs) are not sufficiently homogeneous to be considered as operating in the same technology. In this case, DMUs are partitioned into different groups, each operating in the same technology. In contrast, the metatechnology includes all DMUs and represents all production possibilities that can in principle be achieved in different production environments. Often, the metatechnology cannot be assumed to be a convex set. In such cases benchmarking a DMU against the common metafrontier requires implementing either an enumeration algorithm and solving a linear program at each of its steps, or solving an equivalent mixed integer linear program. In this paper we show that the same task can be accomplished by solving a single linear program. We also show that its dual can be used for the returns-to-scale characterization of efficient DMUs on the metafrontier

A linear programming approach to efficiency evaluation in nonconvex metatechnologies

The notions of metatechnology and metafrontier arise in applications of data envelopment analysis (DEA) in which decision making units (DMUs) are not sufficiently homogeneous to be considered as operating in the same technology. In this case, DMUs are partitioned into different groups, each operating in the same technology. In contrast, the metatechnology includes all DMUs and represents all production possibilities that can in principle be achieved in different production environments. Often, the metatechnology cannot be assumed to be a convex set. In such cases benchmarking a DMU against the common metafrontier requires implementing either an enumeration algorithm and solving a linear program at each of its steps, or solving an equivalent mixed integer linear program. In this paper we show that the same task can be accomplished by solving a single linear program. We also show that its dual can be used for the returns-to-scale characterization of efficient DMUs on the metafrontier

Nonparametric production technologies with weakly disposable inputs

In models of production theory and efficiency analysis, the inputs and outputs are assumed to satisfy some form of disposability. In this paper, we consider the assumption of weak input disposability. It states that any activity remains feasible if its inputs are simultaneously scaled up in the same proportion. As suggested in the literature, the Shephard technology incorporating weak input disposability could be used to evaluate the effect of input congestion. We show that the Shephard technology is not convex and therefore introduces bias in evaluation of congestion. To address this, we develop an alternative convex technology whose use in the evaluation of congestion removes the noted bias. We undertake a further axiomatic investigation and obtain a range of production technologies, all of which exhibit weak input disposability but are based on different, progressively relaxed, convexity assumptions. Apart from the evaluation of input congestion, such technologies should also be useful in applications in which some inputs are closely related or are overlapping, and therefore satisfy only the weak input disposability assumption incorporated in the new models

On single-stage DEA models with weight restrictions

The literature on data envelopment analysis (DEA) often employs multiplier models that incorporate very small (theoretically infinitesimal) lower bounds on the input and output weights. Computational problems arising from the solution of such programs are well known. In this paper we identify an additional theoretical problem that may arise if such bounds are used in a multiplier model with weight restrictions. Namely, we show that the use of small lower bounds may lead to the identification of an efficient target with negative inputs. We suggest a corrected model that overcomes this problem

Solving DEA models in a single optimization stage: Can the non-Archimedean infinitesimal be replaced by a small finite epsilon?

Single-stage DEA models aim to assess the input or output radial efficiency of a decision making unit and potential mix inefficiency in a single optimization stage. This is achieved by incorporating the sum of input and output slacks, multiplied by a small (theoretically non-Archimedean infinitesimal) value epsilon in the envelopment model or, equivalently, by using this value as the lower bound on the input and output weights in the dual multiplier model. When this approach is used, it is common practice to select a very small value for epsilon. This is based on the expectation that, for a sufficiently small epsilon, the radial efficiency and optimal slacks obtained by solving the single-stage model should be approximately equal to their true values obtained by the two separate optimization stages. However, as well-known, selecting a small epsilon may lead to significant computational inaccuracies. In this paper we prove that there exists a threshold value, referred to as the effective bound, such that, if epsilon is smaller than this bound, the solution to the single-stage program is not approximate but precise (exactly the same as in the two-stage approach), provided there are no computational errors

Consistent weight restrictions in data envelopment analysis

It has recently been shown that the incorporation of weight restrictions in models of data envelopment analysis (DEA) may induce free or unlimited production of output vectors in the underlying production technology, which is expressly disallowed by standard production assumptions. This effect may either result in an infeasible multiplier model with weight restrictions or remain undetected by normal efficiency computations. The latter is potentially troubling because even if the efficiency scores appear unproblematic, they may still be assessed in an erroneous model of production technology. Two approaches to testing the existence of free and unlimited production have recently been developed: computational and analytical. While the latter is more straightforward than the former, its application is limited only to unlinked weight restrictions. In this paper we develop several new analytical conditions for a larger class of unlinked and linked weight restrictions