Directional distance functions and slacks-based measures of efficiency: Some clarifications

This note clarifies issues concerning the relationship between directional distance functions and slacks-based measures of efficiency from our earlier paper.     

Cross-efficiency evaluation with directional distance functions

This paper extends the cross-efficiency evaluation for use with directional distance functions. Cross-efficiency evaluation has been developed with oriented Data Envelopment Analysis (DEA) models, so the extension proposed here is aimed at providing a peer-evaluation of decision making units (DMUs) based on measures that account for the inefficiency both in inputs and in outputs simultaneously. We explore the duality relations regarding the models of directional distance functions and define the cross-efficiencies on the basis of the equivalences with some fractional programming problems. Finally, we address in this new context the problem with the alternate optima for the weights and propose some models that implement different alternative secondary goals.     

Variations on the theme of slacks-based measure of efficiency in DEA

In DEA, there are typically two schemes for measuring efficiency of DMUs; radial and non-radial. Radial models assume proportional change of inputs/outputs and usually remaining slacks are not directly accounted for inefficiency. On the other hand, non-radial models deal with slacks of each input/output individually and independently, and integrate them into an efficiency measure, called slacks-based measure (SBM). In this paper, we point out shortcomings of the SBM and propose four variants of the SBM model. The original SBM model evaluates efficiency of DMUs referring to the furthest frontier point within a range. This results in the hardest score for the objective DMU and the projection may go to a remote point on the efficient frontier which may be inappropriate as the reference. In an effort to overcome this shortcoming, we first investigate frontier (facet) structure of the production possibility set. Then we propose Variation I that evaluates each DMU by the nearest point on the same frontier as the SBM found. However, there exist other potential facets for evaluating DMUs. Therefore we propose Variation II that evaluates each DMU from all facets. We then employ clustering methods to classify DMUs into several groups, and apply Variation II within each cluster. This Variation III gives more reasonable efficiency scores with less effort. Lastly we propose a random search method (Variation IV) for reducing the burden of enumeration of facets. The results are approximate but practical in usage.     

Assessing eco-efficiency with directional distance functions

Eco-efficiency is a matter of concern at present that is receiving increasing attention in political, academic and business circles. Broadly speaking, this concept refers to the ability to create more goods and services with less impact on the environment and less consumption of natural resources, thus involving both economic and also ecological issues. In this paper we propose the use of directional distance functions and Data Envelopment Analysis techniques to assess eco-efficiency. More specifically, we show how these functions can be used to compute a wide range of indicators representing different objectives regarding economic and ecological performance. This methodological approach is applied to a sample of Spanish olive-growing farms to illustrate its great potential to provide policymakers and farm managers with sound information as a basis for strategic decision making. We also suggest further avenues to explore in this burgeoning line of research.     

Network DEA: A slacks-based measure approach

Traditional DEA models deal with measurements of relative efficiency of DMUs regarding multiple-inputs vs. multiple-outputs. One of the drawbacks of these models is the neglect of intermediate products or linking activities. After pointing out needs for inclusion of them to DEA models, we propose a slacks-based network DEA model, called Network SBM, that can deal with intermediate products formally. Using this model we can evaluate divisional efficiencies along with the overall efficiency of decision making units (DMUs).     

Profit,Directional Distance Functions,and Nerlovian Efficiency

The directional technology distance function is introduced, given an interpretation as a min-max, and compared with other functional representations of the technology including the Shephard input and output distance functions and the McFadden gauge function. A dual correspondence is developed between the directional technology distance function and the profit function, and it is shown that all previous dual correspondences are special cases of this correspondence. We then show how Nerlovian (profit-based) efficiency measures can be computed using the directional technology distance function.     

Generating functions and duality for integer programs

We consider the integer program P→max c^′x|Ax=y;xNⁿ . Using the generating function of an associated counting problem, and a generalized residue formula of Brion and Vergne, we explicitly relate P with its continuous linear programming (LP) analogue and provide a characterization of its optimal value. In particular, dual variables λR^m have discrete analogues zC^m, related in a simple manner. Moreover, both optimal values of P and the LP obey the same formula, using z for P and |z| for the LP. One retrieves (and refines) the so-called group-relaxations of Gomory which, in this dual approach, arise naturally from a detailed analysis of a generalized residue formula of Brion and Vergne. Finally, we also provide an explicit formulation of a dual problem P^*, the analogue of the dual LP in linear programming.     

A slacks-based measure of efficiency in data envelopment analysis

In this paper, we will propose a slacks-based measure (SBM) of efficiency in Data Envelopment Analysis (DEA). This scalar measure deals directly with the input excesses and the output shortfalls of the decision making unit (DMU) concerned. It is units invariant and monotone decreasing with respect to input excess and output shortfall. Furthermore, this measure is determined only by consulting the reference-set of the DMU and is not affected by statistics over the whole data set. The new measure has a close connection with other measures proposed so far, e.g., Charnes–Cooper–Rhodes (CCR), Banker–Charnes–Cooper (BCC) and the Russell measure of efficiency. The dual side of this model can be interpreted as profit maximization, in contrast to the ratio maximization of the CCR model. Numerical experiments show its validity as an efficiency measurement tool and its compatibility with other measures of efficiency.     

Lower bounds on the minimum average distance of binary codes

Let β(n,M) denote the minimum average Hamming distance of a binary code of length n and cardinality M. In this paper we consider lower bounds on β(n,M). All the known lower bounds on β(n,M) are useful when M is at least of size about 2ⁿ⁻¹/n. We derive new lower bounds which give good estimations when size of M is about n. These bounds are obtained using a linear programming approach. In particular, it is proved that lim_n→∞β(n,2n)=5/2. We also give a new recursive inequality for β(n,M).     

Local differentiability of distance functions

Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets for which the distance function is continuously differentiable everywhere on an open ``tube' of uniform thickness around . Here a corresponding local theory is developed for the property of being continuously differentiable outside of on some neighborhood of a point . This is shown to be equivalent to the prox-regularity of at , which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation. Additional characterizations are provided in terms of being locally of class or such that is convex around for some 0$">. Prox-regularity of at corresponds further to the normal cone mapping having a hypomonotone truncation around , and leads to a formula for by way of . The local theory also yields new insights on the global level of the Clarke-Stern-Wolenski results, and on a property of sets introduced by Shapiro, as well as on the concept of sets with positive reach considered by Federer in the finite dimensional setting.

     

Efficiency analysis, shortage functions, arbitrage, and martingales

This paper shows that standard tools of efficiency analysis, directional distance functions, can be used to characterize the investment-returns technology. That ability to characterize the investment-returns technology and fundamental duality relationships imply that directional distance functions can be used to detect the presence of an arbitrage, to value financial assets in the absence of an arbitrage lying in the span of the market and to place bounds on the no-arbitrage values of assets lying outside the span of the market.     

Duality,efficiency computations and interpretations in imprecise DEA

Data envelopment analysis (DEA) has proven to be a useful tool for assessing efficiency or productivity of organizations which is of vital practical importance in managerial decision making. While DEA assumes exact input and output data, the development of imprecise DEA (IDEA) broadens the scope of applications to efficiency evaluations involving imprecise information which implies various forms of ordinal and bounded data possibly or often occurring in practice. The primary purpose of this article is to characterize the variable efficiency in IDEA. Since DEA describes a pair of primal and dual models, also called envelopment and multiplier models, we can basically consider two IDEA models: One incorporates imprecise data into envelopment model and the other includes the same imprecise data in multiplier model. The issues of rising importance are thus the relationships between the two models and how to solve them. The groundwork we will make includes a duality study, which makes it possible to characterize the efficiency solutions from the two models and link with the efficiency bounds and classifications that some of the published IDEA studies have done. The other purposes are to present computational aspects of the efficiency bounds and how to interpret the efficiency solutions. The computational method developed here extends the previous IDEA method to effectively incorporate a more general form of strict ordinal data and partial orders in its framework, which in turn overcomes some drawbacks of the previous approaches. The interpretation of the resulting efficiency is also important but we have never seen it before.     

Firm and industry level profit efficiency analysis using absolute and uniform shadow prices

We discuss the nonparametric approach to profit efficiency analysis at the firm and industry levels in the absence of complete price information. Two new insights are developed. First, we measure profit inefficiency in monetary terms using absolute shadow prices. Second, we evaluate all firms using the same input–output prices. This allows us to aggregate firm-level profit inefficiencies to the overall industry inefficiency. Besides the measurement of profit losses, the presented approach enables one to recover absolute price information from quantity data. We conduct a series of Monte Carlo simulations to study the performance of the proposed approach in controlled production environments.     

A conditional directional distance function approach for measuring regional environmental efficiency: Evidence from UK regions

This paper, by using conditional directional distance functions as introduced by Simar and Vanhems [J. Econometrics 166 (2012) 342–354] modifies the model by Färe and Grosskopf [Eur. J. Operat. Res. 157 (2004) 242–245] and examines the link between regional environmental efficiency and economic growth. The proposed model using conditional directional distance functions incorporates the effect of regional economic growth on regions’ environmental efficiency levels. The results from UK regional data reveal a negative relationship between regions’ GDP per capita and environmental inefficiency up to a certain GDP per capita level. After that level it appears that the relationship becomes positive. As an overall result the regional environmental inefficiency-GDP per capita relationship appears to have a ‘U’ shape form.     

On the duality between the profit and the indirect distance functions in production theory

Färe and Primont [3] stated, without proofs, duality results between the profit and the indirect distance functions in production theory. The aim of this note is to state these duality results in precise terms and to prove them.     

由方向导数表述的对偶问题

本文利用扰动函数的方向导数引进凸规划的一个新的对偶问题，并证明了相应的对偶性．     

Subdifferentiability and superdifferentiability of distance functions

We obtain necessary and sufficient conditions for the subdifferentiability and superdifferentiability (in the Dem'yanov-Rubinov sense) of the distance in an arbitrary norm from a point to a set for the finitedimensional case. The geometric structure of the subdifferential and the superdifferential is described. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 530–542, April, 1997. Translated by N. K. Kulman     

Contributions of Professor William W. Cooper in Operations Research and Management Science

Over a long and remarkably productive career, Professor William W. (Bill) Cooper has made many pioneering contributions to Operations Research and Management Science (OR/MS), with notable forays into the areas of (a) linear and non-linear programming, (b) goal programming, (c) chance-constrained programming, (d) data envelopment analysis, and (e) manpower planning, among others. His legendary partnership with Abraham Charnes has provided results whose connections go back to the 18th century, bearing on problems conceived but left unsolved by Laplace and Gauss. We document cross-fertilizing links among Bill Cooper’s multiple research focuses, and their impacts on other researchers. A trace of his work discloses a web of influence that has produced a wide range of advances in OR/MS by those who follow in his footsteps, representing a productive tour de force that shows no sign of abating.