A DEA approach for fair allocation of common revenue
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:
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.
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2021, European Journal of Operational ResearchCitation Excerpt :Cook and Zhu (2005) extended their approach to develop a practical allocation approach. Jahanshahloo, Hosseinzadeh Lotfi, Shoja, and Sanei (2004) and Jahanshahloo, Hosseinzadeh Lotfi, and Moradi (2005) employed simple formulas to achieve efficiency invariance. However, the method proposed by Jahanshahloo et al. (2004) renders the allocation plan in total dependence on inputs, while the method proposed by Jahanshahloo et al. (2005) renders the allocation plan in total dependence on outputs.
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2015, Computers and Industrial EngineeringCitation Excerpt :Korhonen and Syrjänen (2004) developed a resource-allocation model for finding an equitable allocation plan using DEA and MOLP. Jahanshahloo, Hosseinzadeh Lotfi, and Moradi (2005a) presented a method for fairly allocating a fixed output among DMUs without solving any linear program while keeping the efficiency scores unchanged. Amirteimoori and Shafiei (2006) proposed a DEA-based method for equitably removing a fix resource from all the DMUs and ensuring that the efficiency of units before and after reduction remains unchanged.
Allocating fixed resources and setting targets using a common-weights DEA approach
2013, Computers and Industrial EngineeringCitation Excerpt :Their aim is to maximize the total output values of DMUs by allocating the fixed resources and they assumed that DMUs are able to change production in production possibility set (PPS). Jahanshahloo, Hosseinzadeh Lotfi, and Moradi (2005) presented a method for allocating a fixed output in a fair way among DMUs without solving any linear program. Amirteimoori and Shafiei (2006) proposed a DEA-based method for removing a fix resource from all DMUs in a fair way such that the efficiency of units before and after reduction remains unchanged.