DEA models for resource reallocation and production input/output estimation

Abstract

This paper discusses the “inverse” data envelopment analysis (DEA) problem with preference cone constraints. An inverse DEA model can be used for a decision making unit (DMU) to estimate its input/output levels when some or all of its input/output entities are revised, given its current DEA efficiency level. The extension of introducing additional preference cones to the previously developed inverse DEA model allows the decision makers to incorporate their preferences or important policies over inputs/outputs into the production analysis and resource allocation process. We provide the properties of the inverse DEA problem through a discussion of its related multi-objective and weighted sum single-objective programming problems. Numerical examples are presented to illustrate the application procedure of our extended inverse DEA model. In particular, we demonstrate how to apply the model to the case of a local home electrical appliance group company for its resource reallocation decisions.

Introduction

In the past few years, the data envelopment analysis (DEA) approach [1], [3], [4], [5], [8], [9], [13], [14] has become increasingly popular in the practice and research of efficiency analysis. A great number and variety of DEA applications and research have led to many new developments in concepts and methodologies related to the DEA-efficiency analysis. One can see this from extensive surveys provided by [2], [6], [9]. DEA contributes certain noteworthy advantages and distinctive characteristics to efficiency analysis, yet it is restrained in various aspects that necessitate improvement and further research, as commonly discussed in the literature.

Recently, Wei et al. [11] proposed, for the first time, an “inverse” DEA model for short-term input and output estimation. The basic idea is to extend the concept of the inverse optimization problem to the DEA context. In a general inverse optimization problem, we are interested in either adjusting constraint parameters such that a given feasible solution becomes optimal, or varying constraint parameters such that the optimal objective value remains unchanged. As the parameters in a DEA optimization model are actual quantities of inputs and outputs given by each concerned decision making unit (DMU), its inverse optimization problem can be naturally considered for input/output analysis under the DEA framework. The inverse DEA model discusses the problems of determining the best possible output for a given input level under the condition that the optimal objective value of the original DEA model remains the same. The objective value in a DEA model is the efficiency index of a DMU, which is determined by and, thus, reflects the existing technical structure or efficiency level of the concerned DMU. Under normal circumstances, the internal technical structure of a DMU should not change dramatically in a short term. Therefore, the inverse DEA model can be used to deal with resource reallocation problems without lowering the efficiency level. In an inverse DEA problem, the following questions are addressed. Among a group of DMUs, if a DMU attempts to increase or decrease some of its input levels while maintaining its relative efficiency position among the group, what are the changes to the outputs this DMU would expect? Or the other way around, if a DMU wants to increase or decrease some of its output levels while maintaining its efficiency position, how much additional or reduced resources (inputs) are needed? In [11], the inverse DEA problem is transformed into and solved as a multi-objective programming problem. It is also shown that in some special cases, the inverse DEA problem can be simplified as a single-objective linear programming problem. The work by Wei et al. is apparently of significance. Firstly, the inverse DEA model implies a new avenue for DEA applications, i.e. production analysis or production planning. At present, various DEA models are mainly used for relative technical efficiency measure and analysis. The physical quantities of inputs and outputs associated with the concerned DMUs are given fixed for parametric representation of the efficiency model. In the existing literature, the problem of resource reallocation is dealt with only for the purpose of guiding an inefficient DMU shifting along the direction of a projected ray from its current position onto the frontier. The DEA model has not been used as a kind of conditioned production model for a DMU to deal with certain choices or optimal choices of inputs and/or outputs, given its standing efficiency level [11]. Secondly, as also indicated in [11], the inverse DEA adds in a new and important class of application problems for research on the inverse optimization problem.

In this paper, we extend the result of [11] to the case of the generalized cone ratio DEA model [12], [14]. In other words, we consider the inverse DEA problem with ratio cone structure. The concepts of ratio cone [5], [14] are introduced to represent decision makers' preferences or pre-determined policies regarding the relative importance of different input and/or output entities. Such a preference cone is particularly important in short-term production planning or resource reallocation as it more closely reflects the management reality. In the process of short-term resource planning, decision makers do often regard some input/output entities as being more valuable or preferable than others. Or frequently it is the case that the availability of some inputs and the production of some outputs are subject to certain restrictions. With the preference cones, it becomes possible to incorporate such important information in the production planning process. In comparison with the previous inverse DEA model in [11], the inverse DEA model with cone structure provides a more effective means for the tasks of input/output estimation and resources planning. With the concept of preference cone, we can extend the Pareto solution in the inverse DEA model to the non-dominated solution, and discuss the resource allocation problem, given the decision makers' preference on inputs and outputs. Incorporating the ratio cone structure into the inverse DEA model provides additional advantages in supporting resource reallocation and production planning decisions but it also involves additional complexity in model mathematics. In this paper, we investigate the important properties of this extended inverse DEA model and provide some useful results for its real applications. We show that when the preference cones are linear convex, we can study the corresponding production possibility set and the production frontier under the DEA framework. We also show that under certain conditions, our model can be converted to the regular inverse DEA models reported in [11]. Numerical examples are included to illustrate the process of how to apply the model in real applications.

When considering production input and output estimation, a common approach is to employ various parametric forms of production functions or input/output distance functions, including the Cobb–Douglas, the constant elasticity of substitution (CES) and the translog function, to name just a few. The inverse DEA model presents an alternative means under a different framework. One main departure point is that the conventional approaches rely on DMU's existing production technology, which is obtained using the functional mapping of all given or technologically feasible combinations of inputs and outputs. The DEA model is based on an efficiency framework that is most favorable to each DMU's relative efficiency assessment, i.e. to achieve the maximal relative efficiency index value. The efficiency index value each DMU can achieve under the corresponding DEA model is determined by the inherent technological structure or strength of this unit, which is reflected by the multipliers in the model. When utilizing the inverse DEA model for production analysis, we assume that this technological structure is unchanged, or in other words, the relative efficiency index given by the DEA model remains constant. For a DMU which is classified DEA efficient, it is reasonable to require that any further production adjustments should guarantee that it remains in an efficient position after the adjustment. For a DMU which is DEA inefficient, an unchanging efficiency level implies that, in a short-term, there is no dramatic variation in its internal technological structure, and whatever alteration in input or output should ensure at least its present efficiency level. Some additional advantages of using the inverse DEA model for the tasks of production analysis or resource reallocation are:

• it can be naturally used for multiple input/output production without pre-assigned weights;

• it can be used for production input/output estimation and planning without knowing the true form of production function;

• the decision makers' preferences can be incorporated into the production analysis, especially by using the extended inverse DEA model proposed in this paper;

• the inverse DEA model is related to multi-objective programming or single-objective linear programming, which are well structured and studied with well-developed theories and useful results.

The rest of the paper proceeds as follows. The following section introduces and discusses the properties of the inverse DEA model with cone structure. We show how the inverse DEA problem can be transformed to and solved by a multi-objective programming problem. When the preference cones are all polyhedral, the problem can be handled as a single-objective linear programming problem, for which a numerical example is included for illustration. Section 3 focuses on how to apply the inverse DEA model to the resource reallocation problem. We consider the case of a local group company manufacturing various home appliances. The company has a number of member companies belonging to different types of industries, including electrical, electronic, plastic, fabricated metal product and printing industry. We describe the procedure for allocating the additional amount of resources to its member DMUs in different industries to maximize the overall benefit of the group company, which is presented by a single-objective function. We also discuss the relationship between the related single-objective optimization problem and the multi-objective inverse DEA model. The last section summarizes the results of the paper and points out the areas for further research.