A DEA approach for fair allocation of common revenue

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Abstract

An issue of considerable importance, how to allocate a common revenue in an equitable manner across a set of competing entities. This paper introduces a new approach to obtaining allocation common revenue on all decision making units (DMUs) in such a way that the relative efficiency is not changed. In this method for determining allocation common revenue dose not need to solving any linear programming. A numerical example is provided to illustrate the results of the analysis.

Introduction

Data envelopment analysis (DEA) is a methodology that has been used to evaluate the efficiency of entities which are responsible for utilizing resources to obtain outputs of interest. It has been used to evaluate activities as varied as schools, bank branches or sales outlets. It computes a scalar measure of efficiency and determines efficient levels of inputs and outputs for the organizations under evaluation.

There are many applications of this technique, one of the important applications is equitable allocation of shared costs to all DMUs or fair allocation of common revenue of each DMU. To provide a practical setting within which to investigate this issue, we refer to the recent papers by Cook et al. [1], [2] and Jahanshahloo et al. [3], [4].

For example, in a industry, when increase the price of a product, manager receives a revenue. We show how this revenue can be allocated in a equitable way and determine the share of each DMU in such a way that the relative efficiency is not change. Jahanshahloo et al. [4] proposed a manner to do it.

In this paper, we show that the suggested method is very simple and characterize fair allocations and the re-evaluated efficiencies would remain unchanged. For determining allocation of revenue dose not need to solving any linear programming and by one simple ratio this distribution is calculated.

The paper is structured as follows. Section 2 is an introduction to the CCR model [5]. Section 3 discuss the concept of fair allocation of common revenue. Section 4 numerical example is solved. Finally, in Section 5 the conclusion are put forward.

Section snippets

Preliminaries

Assume that there are n DMUs to be evaluated, which we associate with points Xj=(x1j,…,xmj)T and Yj=(y1j,…,ytj)T that represent (m×1) and (t×1) vectors of observed input and output values, respectively, for each DMUj (j=1,…,n), where superscript T represents the transpose. It is assumed that all inputs and outputs are non-negative and that at least one input and one output of each DMU is strictly positive. We begin with the CCR [5] model:Minimizeθ0s.t.j=1nλjxij⩽θ0xi0,i=1,…,m,j=1nλjyrj⩾yr0,

The fair allocation of a common revenue

In this section we want to allocate a common revenue to each DMU such that this distribution be fair and the relative efficiency is not change. Also output pareto-maximality will be guaranteed. As defined in [4], a revenue allocation is output pareto-maximality if no revenue can be transferred from one DMU to another without violating the invariance principle.

In this section, we consider two cases.

Numerical example

Consider the 28 DMUs with three input and three output as defined by Table 1. These data originally has been reported by Charnes et al. [7] which consist of 28 Chinese cities in 1983. There are three outputs (gross industrial output value, profit and taxes, and retail sales) and three inputs (labor, working funds, and investment).

Suppose that we want to distribute $2,000,000 among these cities. Here, the objective of management is to distribute these revenues in a such a way that the relative

Conclusion

An issue of considerable importance, we want to know how to allocate a common revenue in an equitable manner across a set of competing entities. In this paper, a method for fair allocation of a common revenue is provided. Without solving any linear programming, this method determine share of revenue each DMU for the simple cases and general forms. Also the ideas presented, preserves invariancy and pareto-maximality. The method can be extended to BCC model [6], and standard other DEA models.

References (7)

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