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This paper extends the works by Olesen and Petersen (2003), Russell and Schworm (2006) and Cooper et al. (2007) about describing the efficient frontier of a production possibility set by the intersection of a finite number of closed halfspaces, in several ways. First, we decompose the efficient frontier into a smallest number of convex polyhedrons, or equivalently into a new class of efficient faces, called maximal efficient faces (MEFs). Second, we show how to identify all MEFs even if full dimensional efficient faces do not exist. Third, by applying the MEF decomposition to various real-world data sets, we demonstrate the validity of the MEF decomposition and how it can contribute to the DEA literature. Finally, we illustrate how to use the identified MEFs in practice. (C) 2012 Elsevier B.V. All rights reserved.

Since Briec developed a family of the least distance based inefficiency measures satisfying weak monotonicity over weakly efficient frontier, the existence of a least distance based efficiency measure satisfying strong monotonicity on the strongly efficient frontier is still an open problem. This paper gives a negative answer to the open problem and its relaxed open problem. Modifying Briec's inefficiency measures gives an alternative solution to the relaxed open problem, that can be used for theoretical and practical applications. © The Operations Research Society of Japan.

This study discusses nine desirable properties that a measure of technical efficiency (TE) needs to satisfy from the perspective of production economics and optimization. Seven data envelopment analysis (DEA) models are theoretically compared from a viewpoint of nine TE criteria. All the seven DEA models suffer from a problem of multiple projections even though a unique projection for efficiency comparison is one of the nine desirable properties. Furthermore, all the DEA models violate the property on aggregation of inputs and outputs. Thus, the seven DEA models do not satisfy all desirable TE properties. In addition, the comparison provides us with the following guidelines: (a) The additive model violates all desirable TE properties. (b) Russell measure and SBM (=ERGM) perform as well as RAM as a non-radial measure. If we are interested in strict monotonicity, the two models outperform the other DEA models including RAM. In contrast, if we are interested in translation invariance, RAM is better than Russell measure and SBM (=ERGM). (c) The radial measures (CCR and BCC) have the property of linear homogeneity. (d) The CCR model is useful for measuring a frontier shift among different periods. (e) If a data set contains a negative value, RAM becomes a DEA model to handle the negative value because it has the property of translation invariance. After examining the desirable TE properties, this study proposes a new approach to deal with an occurrence of multiple projections. The proposed approach includes a test to examine an occurrence of multiple projections, a mathematical expression of a projection set, and a selection process of a unique reference set as the largest one covering all the possible reference sets. (C) 2008 Elsevier B.V. All rights reserved.

Economic implications of congestion have been recently discussed in many DEA (data envelopment analysis) studies. In addition, several previous research efforts have explored a theoretical linkage between returns to scale (RTS) and the concept of congestion, because the two economic concepts are closely connected to each other. Tone and Sahoo [Tone, K., Sahoo, B.K., 2004. Degree of scale economies and congestion: A unified DEA approach. European Journal of Operational Research 158, 755-772] have published the theoretical linkage in this journal. All of the previous studies, including their research (2004), assume a unique optimal solution in the investigation on DEA-based congestion. When multiple solutions occur in DEA-based congestion measurement, the economic implications of congestion obtained from the previous research are all problematic from both theoretical and practical perspectives. To deal with the issue, this study explores how to deal with the occurrence of multiple solutions in the DEA-based congestion measurement. This study proposes a new approach for the congestion measurement and theoretically compares the proposed approach with Tone and Salloo (2004). (C) 2008 Elsevier B.V. All rights reserved.

A matrix balancing problem and an eigenvalue problem are transformed into two minimum-norm point problems whose difference is only a norm. The matrix balancing problem is solved by scaling algorithms that are as simple as the power method of the eigenvalue problem. This study gives a proof of global convergence for scaling algorithms and applies the algorithm to Analytic Hierarchy process (AHP), which derives priority weights from pairwise comparison values by the eigenvalue method (EM) traditionally. Scaling algorithms provide the minimum X square estimate from pairwise comparison values. The estimate has properties of priority weights such as right-left symmetry and robust ranking that are not guaranteed by the EM.

This research theoretically explores the measurement of RTS (Returns to Scale) under a possible occurrence of multiple solutions in DEA (Data Envelopment Analysis). In this study, the occurrence of multiple solutions is classified into Type I and Type II. Type I is an occurrence of multiple solutions in a reference set. Type 11 is an occurrence of multiple solutions on a supporting hyperplane passing on the reference set. Both Types I and 11 are very well known among DEA researchers, but previous research has not sufficiently explored a simultaneous occurrence of Type I and Type 11 in the RTS measurement. The two types of multiple solutions influence a degree of RTS in the DEA measurement. Such a quantitative issue on RTS is examined from the perspective of the Type I and Type 11 problems. To deal with such difficulties, a new linear programming approach is proposed to identify all efficient DMUs (Decision Making Units) that consist of a reference set, even if multiple solutions occur on the reference set. Based upon the research result, we can identify when multiple solutions of Type I and/or Type 11 occur on the RTS measurement and how to deal with such difficulties. Our research result makes it possible to measure a degree of scale economies (RTS) under the simultaneous occurrence of Type I and Type II. (c) 2006 Elsevier B.V. All rights reserved.

in aggregation for data envelopment analysis (DEA), a jointly measured efficiency score among inputs and outputs is desirable in performance analysis. A separate treatment between output-oriented efficiency and input-oriented efficiency is often needed in the conventional radial DEA models. Such radial measures usually need to measure both that a current performance attains an efficiency frontier and that all the slacks are zero on optimality. In the analytical framework of the radial measure, Russell measure is proposed to deal with such a difficulty. A major difficulty associated with the Russell measure is that it is modeled by a nonlinear programming formulation. Hence, a conventional linear programming algorithm, usually applied for DEA, cannot solve the Russell measure. This study newly proposes a reformulation of the Russell measure by a second-order cone programming (SOCP) model and applies the primal-dual interior point algorithm to solve the Russell measure. (c) 2006 Elsevier B.V. All rights reserved.

This research theoretically explores the measurement of returns to scale (RTS), using a non-radial DEA (data envelopment analysis) model. A range-adjusted measure (RAM) is used as a representative of such non-radial models. Historically, a type of RTS has been discussed within an analytical framework of radial models. The radial-based RTS measurement is replaced by the non-radial RAM/RTS measurement in this study. When discussing the non-radial RAM/RTS measurement, this study finds a problem of multiple projections that cannot be found in the radial measurement. In this research, a new linear programming approach is proposed to identify all efficient DMUs (decision making units) on a reference set. The important feature of the proposed approach is that it can deal with a simultaneous occurrence of (a) multiple reference sets, (b) multiple supporting hyperplanes and (c) multiple projections. All of the three difficulties are handled by the proposed RAM/RTS measurement. In particular, we discuss both when the three different types of multiple solutions occur on the RAM/RTS measurement and how to deal with such difficulties. Our research results make it possible to measure not only the type of RTS but also the magnitude of RTS in the RAM measurement. (c) 2005 Elsevier B.V. All rights reserved.

In the typical type of ANP with a matrix U evaluating alternatives by criteria and a matrix W evaluating criteria by alternatives in the so-called supermatrix S, W is often said to be unstable. Here, we propose a method to revise W into a stable (W) over tilde and to calculate the weights of criteria and alternatives at the same time under the revised supermatrix S. Our method is formulated as an optimization problem based on Bayes Theorem which T.L. Saaty claimed to be included in ANP scheme. Concurrent Convergence method developed by Kinoshita and Nakanishi also intends to be correct W, but this method includes some contradictions. We prove that our method has no such contradiction. We introduce some eigenvalue problems, which give a lower bound of our optimal value and their special cases coincide with our problem. Furthermore, we clear what perturbations of W preserve weights of criteria and alternatives to be invariant under the concept of inactive alternatives.

Two different types of inputs (variable inputs and quasi-fixed inputs) are incorporated into an analytical framework of dynamic data envelopment analysis (DEA). A unique feature of the quasi-inputs is that those are considered as outputs at the current period, while being treated as inputs at the next period. The dynamic DEA can measure interdependency among consecutive periods. This study incorporates the concept of returns to scale into the dynamic DEA. (C) 2003 Elsevier B.V. All rights reserved.

A mutual evaluation system deals with a decision making problem including the feedback structure between alternatives and criteria. By incorporating decision makers' intuitive judgments from alternatives to criteria into overall evaluations for alternatives, the mutual evaluation system may be effective to make a consensus among the decision makers of the problem. An analyzing tool of the mutual evaluation system, Concurrent Convergent Method (CCM) by Kinoshita and Nakanishi, has practical advantage. This paper introduces an overall weight vector for criteria into CCM. By using the overall weight vector not for alternatives but for criteria, we demonstrate irrationality of CCM under the Pareto principle.

This study discusses the mathematical structure of the dominant AHP and the concurrent convergence method which were originally developed by Kinoshita and Nakanishi. They introduced a new concept of a regulating alternative into an analyzing tool for a simple evaluation problem with a criterion set and an alternative set. Although the original idea of the dominant AHP and the concurrent convergence method is unique, the dominant AHP and the concurrent convergence method are not sufficiently analyzed in mathematical theory. This study shows that the dominant AHP consists of a pair of evaluation rules satisfying a certain property of overall evaluation vectors, This study also shows that the convergence of concurrent convergence method is guaranteed theoretically.

Saaty proposes the analyzing methods for AHP using the principal eigenvector of the comparison matrix, and for ANP (analytic network process) using the limiting process method of the powers of the supermatrix. These methods are based on the irreducibility of the evaluation matrices. We develop the unified method solving both ABP and ANP based on Frobenius min-max theorem. Further this method is extended to the solving method of the general (not necessarily irreducible) evaluation matrix, by the graphic decomposition process.

The eigenvalue method (EM), that is to find the principal eigenvector of a pairwise comparison matrix, is widely used and known to be practical in Analytic Hierarchy Process (AHP). However, the validity of EM has never been fully proved. In this article, we present an equilibrium model and four optimization models to show the logical justification for using EM in AHP. By introducing two concepts, self-evaluation and non-self-evaluation, into AHP, the fundamental theorem (Frobenius' Theorem) for EM is interpreted as two optimization problems. From these two concepts, a noncooperative game with a pairwise comparison matrix is also formulated and its equilibrium solution is the principal eigenvector. We propose two discrepancy indices between self-evaluation and non-self-evaluation and formulate four discrepancy-minimization problems. An optimal solution for two minimization problems among them is equal to the principal eigenvector.

This study explores mathematical properties of Least Absolute Value Estimation (LAVE) with Serial Correlation (SC), focusing upon its formulation, algorithm, multiple solutions, and global optimality conditions. The SC is expressed by a first-order autocorrelated disturbance which is incorporated into LAVE. The resulting LAVE-SC estimation is discussed from the perspectives of mathematical programming.

Given a point set P of the general dimension we present a recursive algorithm for finding the sphere with the smallest radius which contains all points of P. For given point sets Q(1), ..., Q(l) we extend the algorithm so that it finds l concentric spheres with the smallest sum of radii such that each sphere covers the corresponding point set. © 1993 JJIAM Publishing Committee.

We consider a class of convex min-max problems including the minimum norm point problem and the minimum covering sphere problem. We generalize Sekitani and Yamamoto's recursive algorithms for these two problems and show that the unified algorithm is valid for a class of convex min-max problems under two assumptions.