Technical efficiency estimation with multiple inputs and multiple outputs using regression analysis

Regression and linear programming provide the basis for popular techniques for estimating technical efficiency. Regression-based approaches are typically parametric and can be both deterministic or stochastic where the later allows for measurement error. In contrast, linear programming models are nonparametric and allow multiple inputs and outputs. The purported disadvantage of the regression-based models is the inability to allow multiple outputs without additional data on input prices. In this paper, deterministic cross-sectional and stochastic panel data regression models that allow multiple inputs and outputs are developed. Notably, technical efficiency can be estimated using regression models characterized by multiple input, multiple output environments without input price data. We provide multiple examples including a Monte Carlo analysis.     

An extended transportation problem: a DEA-based approach

The transportation problem is an important network structured linear programming problem that arises in several contexts and that has received a great deal of attention in the literature. The existing transportation problems are limited to consideration unit of shipping cost or profit along an arc. However, in many real applications, various attributes are usually considered in a transportation problem. The current paper, proposes an extension to this problem in the presence of multiple in-commensurate inputs and outputs for each arc. The concept of relative efficiency is defined for each possible transportation plan. Two linear programming models are proposed to determine the transportation plan with the maximum efficiency. A numerical example is used to illustrate the applicability of the approach.     

Solving Bilevel Linear Programs Using Multiple Objective Linear Programming

We present an algorithm for solving bilevel linear programs that uses simplex pivots on an expanded tableau. The algorithm uses the relationship between multiple objective linear programs and bilevel linear programs along with results for minimizing a linear objective over the efficient set for a multiple objective problem. Results in multiple objective programming needed are presented. We report computational experience demonstrating that this approach is more effective than a standard branch-and-bound algorithm when the number of leader variables is small.     

An extended slacks-based measure model for data envelopment analysis with negative data

Data envelopment analysis (DEA) is a non-parametric approach based on linear programming that has been widely applied for evaluating the relative efficiency of a set of homogeneous decision-making units (DMUs) with multiple inputs and outputs. The original DEA models use positive input and output variables that are measured on a ratio scale, but these models do not apply to the variables in which negative data can appear. However, with the widespread use of interval scale data and undesirable data, the emphasis has been directed towards the simultaneous consideration of the positive and negative data in DEA models. In this paper, using the slacks-based measure, we propose an extended model to evaluate the efficiency of DMUs, even if some variables are measured on an interval scale and some on a ratio scale. Moreover, the extended model allows for the presence of all interval-scale variables, which are capable of taking both negative and positive values.     

A game theoretic approach to measuring efficiency

Correspondence with a new mathematical programming definition for efficiency as proposed by Charnes, Cooper and Rhodes (CCR) is established by means of game theoretical models. Contact with all of the CCR results is also maintained so that their results extend to our new game theoretic interpretations. The latter proceeds by means of a family of games related to a linear programming problem. The games-to-programming relations which we establish also open new possibilities for further relations between families of games and linear programs.     

Multiobjective data envelopment analysis

Data envelopment analysis (DEA) is popularly used to evaluate relative efficiency among public or private firms. Most DEA models are established by individually maximizing each firm's efficiency according to its advantageous expectation by a ratio. Some scholars have pointed out the interesting relationship between the multiobjective linear programming (MOLP) problem and the DEA problem. They also introduced the common weight approach to DEA based on MOLP. This paper proposes a new linear programming problem for computing the efficiency of a decision-making unit (DMU). The proposed model differs from traditional and existing multiobjective DEA models in that its objective function is the difference between inputs and outputs instead of the outputs/inputs ratio. Then an MOLP problem, based on the introduced linear programming problem, is formulated for the computation of common weights for all DMUs. To be precise, the modified Chebychev distance and the ideal point of MOLP are used to generate common weights. The dual problem of this model is also investigated. Finally, this study presents an actual case study analysing R&D efficiency of 10 TFT-LCD companies in Taiwan to illustrate this new approach. Our model demonstrates better performance than the traditional DEA model as well as some of the most important existing multiobjective DEA models.     

DEA models for supply chain efficiency evaluation

An appropriate performance measurement system is an important requirement for the effective management of a supply chain. Two hurdles are present in measuring the performance of a supply chain and its members. One is the existence of multiple measures that characterize the performance of chain members, and for which data must be acquired; the other is the existence of conflicts between the members of the chain with respect to specific measures. Conventional data envelopment analysis (DEA) cannot be employed directly to measure the performance of supply chain and its members, because of the existence of the intermediate measures connecting the supply chain members. In this paper it is shown that a supply chain can be deemed as efficient while its members may be inefficient in DEA-terms. The current study develops several DEA-based approaches for characterizing and measuring supply chain efficiency when intermediate measures are incorporated into the performance evaluation. The models are illustrated in a seller-buyer supply chain context, when the relationship between the seller and buyer is treated first as one of leader-follower, and second as one that is cooperative. In the leader-follower structure, the leader is first evaluated, and then the follower is evaluated using information related to the leader's efficiency. In the cooperative structure, the joint efficiency which is modelled as the average of the seller's and buyer's efficiency scores is maximized, and both supply chain members are evaluated simultaneously. Non-linear programming problems are developed to solve these new supply chain efficiency models. It is shown that these DEA-based non-linear programs can be treated as parametric linear programming problems, and best solutions can be obtained via a heuristic technique. The approaches are demonstrated with a numerical example.     

Interval efficiency measures in data envelopment analysis with imprecise data

The conventional data envelopment analysis (DEA) measures the relative efficiencies of a set of decision making units (DMUs) with exact values of inputs and outputs. For imprecise data, i.e., mixtures of interval data and ordinal data, some methods have been developed to calculate the upper bound of the efficiency scores. This paper constructs a pair of two-level mathematical programming models, whose objective values represent the lower bound and upper bound of the efficiency scores, respectively. Based on the concept of productive efficiency and the application of a variable substitution technique, the pair of two-level nonlinear programs is transformed to a pair of ordinary one-level linear programs. Solving the associated pairs of linear programs produces the efficiency intervals of all DMUs. An illustrative example verifies the idea of this paper. A real case is also provided to give some interpretation of the interval efficiency. Interval efficiency not only describes the real situation in better detail; psychologically, it also eases the tension of the DMUs being evaluated as well as the persons conducting the evaluation.     

Convex quadratic relaxations for mixed-integer nonlinear programs in power systems

This paper presents a set of new convex quadratic relaxations for nonlinear and mixed-integer nonlinear programs arising in power systems. The considered models are motivated by hybrid discrete/continuous applications where existing approximations do not provide optimality guarantees. The new relaxations offer computational efficiency along with minimal optimality gaps, providing an interesting alternative to state-of-the-art semidefinite programming relaxations. Three case studies in optimal power flow, optimal transmission switching and capacitor placement demonstrate the benefits of the new relaxations.     

Generalized S-Lemma and strong duality in nonconvex quadratic programming

On the basis of a new topological minimax theorem, a simple and unified approach is developed to Lagrange duality in nonconvex quadratic programming. Diverse generalizations as well as equivalent forms of the S-Lemma, providing a thorough study of duality for single constrained quadratic optimization, are derived along with new strong duality conditions for multiple constrained quadratic optimization. The results allow many quadratic programs to be solved by solving one or just a few SDP’s (semidefinite programs) of about the same size, rather than solving a sequence, often infinite, of SDP’s or linear programs of a very large size as in most existing methods.     

UTA-poly and UTA-splines: Additive value functions with polynomial marginals

Additive utility function models are widely used in multiple criteria decision analysis. In such models, a numerical value is associated to each alternative involved in the decision problem. It is computed by aggregating the scores of the alternative on the different criteria of the decision problem. The score of an alternative is determined by a marginal value function that evolves monotonically as a function of the performance of the alternative on this criterion. Determining the shape of the marginals is not easy for a decision maker. It is easier for him/her to make statements such as “alternative a is preferred to b”. In order to help the decision maker, UTA disaggregation procedures use linear programming to approximate the marginals by piecewise linear functions based only on such statements. In this paper, we propose to infer polynomials and splines instead of piecewise linear functions for the marginals. In this aim, we use semidefinite programming instead of linear programming. We illustrate this new elicitation method and present some experimental results.     

Successive linear programming for ratio goal problems

This paper develops two effective methods for solution of the nonlinear and non-convex programming problems of multiple ratio goal models. They are based on Charnes and Cooper's convergence theorem for an associated sequence class of linear programs. The Charnes-Cooper results are extended to develop two alternate algorithms and implementations effectively employing new information en route to solution to achieve significant savings in computation time over methods not taking advantage of such information. Possible uses for other applications than the computed examples or for other model classes are also indicated.     

Collision computation of moving bodies

In this paper, an explicit mathematical representation of n-dimensional bodies moving in translation along general trajectories is derived. This representation is used to find out if two moving bodies are going to collide. An optimization problem is developed for finding the time and location of collision. We consider the special cases of linear and piece-wise linear trajectories. The collision in this case can be obtained by solving a linear program or a sequence of linear programs, respectively. The problem of finding the collision time and location of several moving bodies is cast as an integer programming problem. A comprehensive simulation study shows that this approach requires much lesser computation time when compared with the current approach of finding the collision between all pairs of bodies.     

On the measurement of technical efficiency in the public sector

Existing measures of technical inefficiency obtained through linear programming models in the public sector do not properly control for environmental variables that affect production. It will be shown that the consequences of not controlling for these fixed factors are biased estimates of technical efficiency. This paper extends the mathematical programming approach to frontier estimation known as Data Envelopment Analysis to allow for environmental variables. This modified model will be then contrasted with the existing model that purportedly controls for exogeneous factors to measure public sector efficiency with simulated data. The results provide evidence that the existing Data Envelopment Analysis model will overestimate the level of technical inefficiency and that the modified model developed in this paper does a better job controlling for exogenous factors. The modified model is also applied to analyze the technical efficiency of school districts.     

Linear programming via a quadratic penalty function

We use quadratic penalty functions along with some recent ideas from linearl ₁ estimation to arrive at a new characterization of primal optimal solutions in linear programs. The algorithmic implications of this analysis are studied, and a new, finite penalty algorithm for linear programming is designed. Preliminary computational results are presented.Research supported by grant No. 11-0505 from the Danish Natural Science Research Council SNF.     

A parallel algorithm of multi-variant evolutionary synthesis of nonlinear models

A parallel algorithm has been proposed for solving the problem of construction of nonlinear models (mathematical expressions, functions, algorithms, and programs) using given experimental data, set of variables, basic functions and operations. The designed algorithm of multivariant evolutionary synthesis of nonlinear models includes linear representation of a chromosome, modular operations in decoding of a genotype into a phenotype for interpreting a chromosome as a sequence of instructions, and a multi-variant method for presenting a set of models (expressions) using a single chromosome. A sequential version of the algorithm is compared with a standard genetic programming (GP) algorithm and a Cartesian genetic programming (CGP) one. The algorithm proposed was shown to excel the GP and CGP algorithms both in the time required for search for a solution (more than by an order of magnitude in most cases) and in the probability of finding a given function (model). Experiments have been carried out on parallel supercomputer systems, and estimates of the efficiency of the parallel algorithm offered have been obtained; the estimates demonstrate linear acceleration and scalability.     

Target setting in data envelopment analysis using MOLP

Data envelopment analysis (DEA) and multiple objective linear programming (MOLP) can be used as tools in management control and planning. The existing models have been established during the investigation of the relations between the output-oriented dual DEA model and the minimax reference point formulations, namely the super-ideal point model, the ideal point model and the shortest distance model. Through these models, the decision makers’ preferences are considered by interactive trade-off analysis procedures in multiple objective linear programming. These models only consider the output-oriented dual DEA model, which is a radial model that focuses more on output increase. In this paper, we improve those models to obtain models that address both inputs and outputs. Our main aim is to decrease total input consumption and increase total output production which results in solving one mathematical programming model instead of n models. Numerical illustration is provided to show some advantages of our method over the previous methods.     

Integrating DEA-oriented performance assessment and target setting using interactive MOLP methods

Data envelopment analysis (DEA) and multiple objective linear programming (MOLP) are tools that can be used in management control and planning. Whilst these two types of model are similar in structure, DEA is directed to assessing past performances as part of management control function and MOLP to planning future performance targets. This paper is devoted to investigating equivalence models and interactive tradeoff analysis procedures in MOLP, such that DEA-oriented performance assessment and target setting can be integrated in a way that the decision makers’ preferences can be taken into account in an interactive fashion. Three equivalence models are investigated between the output-oriented dual DEA model and the minimax reference point formulations, namely the super-ideal point model, the ideal point model and the shortest distance model. These models can be used to support efficiency analysis in the same way as the conventional DEA model does and also support tradeoff analysis for setting target values by individuals or groups. A case study is conducted to illustrate how DEA-oriented efficiency analysis can be conducted using the MOLP methods and how such performance assessment can be integrated into an interactive procedure for setting realistic target values.     

Linear complementarity problems and multiple objective programming

An equivalence is demonstrated between solving a linear complementarity problem with general data and finding a certain subset of the efficient points of a multiple objective programming problem. A new multiple objective programming based approach to solving linear complementarity problems is presented. Results on existence, uniqueness and computational complexity are included.     

An extended assignment problem considering multiple inputs and outputs

The existing assignment problems for assigning n jobs to n individuals are limited to the considerations of cost or profit incurred by each possible assignment. However, in real applications, various inputs and outputs are usually concerned in an assignment problem, such as a general decision-making problem. This paper develops a procedure for resolving assignment problems with multiple incommensurate inputs and outputs for each possible assignment. The concept of the relative efficiency in using various resources, instead of cost or profit, is adopted for each possible assignment of the problem. Data envelopment analysis (DEA) is employed in this paper to measure the efficiency of one assignment relative to that of the others according to a set of decision-making units. A composite efficiency index, consisting of two kinds of relative efficiencies under different comparison bases, is defined to serve as the performance measurement of each possible assignment in the problem formulation. A mathematical programming model for the extended assignment problem is proposed, which is then expressed as a classical integer linear programming model to determine the assignments with the maximum efficiency. A numerical example is used to demonstrate the approach.