Fixed cost allocation for two-stage systems with cooperative relationship using data envelopment analysis

https://doi.org/10.1016/j.cie.2020.106534Get rights and content

Highlights

  • A fixed cost allocation model considering cooperation among DMUs is built.

  • Each DMU and its sub-stages are proved to be efficient after allocation.

  • A nucleolus-based method is extended to two-stage systems.

  • The row generation procedure is used to simplify calculation in this paper.

Abstract

Allocating fixed cost among a group of entities is becoming increasingly important in management. Numerous studies have addressed this issue in single-stage systems based on data envelopment analysis (DEA). However, these studies frequently ignored the internal structure of systems, and in many real applications, enterprises with multiple stage processes cooperate with one another. Taking this issue into account in the allocation process, we approach fixed-cost allocation issues of two-stage systems by considering a cooperative relationship among decision making units (DMUs). We integrate cooperative game theory and the DEA methodology to generate a unique and fair allocation plan. The results confirm that each DMU can maximize its relative efficiency to one by a series of optimal variables after the fixed cost allocation. Based on these results, a unique nucleolus solution can be generated through a feasible computation algorithm. Finally, we apply the proposed approach to a random dataset and an empirical commercial bank application in China.

Introduction

Fixed cost, from the perspective of accounting, refers to the cost that can be kept constant regardless of increasing or decreasing business volume in a certain period of time. It is typically encountered in real life when certain agents use a common platform, such as the advertisement expenditure of a manufacture across its retailers (Cook & Kress, 1999), the cost of common communication cables among users (Beasley, 2003), and the transportation cost allocation (Sun, Rangarajan, Karwan, & Pinto, 2015) on a given route. For decision makers, the problems lie on how to allocate the fixed cost to various users in an equitable manner.

Recently, data envelopment analysis (DEA) has been proposed to address the fixed cost allocation and resource sharing issues (Cook and Kress, 1999, Cook and Zhu, 2005, Li et al., 2020). DEA is an effective non-parametric approach for evaluating the relative efficiency of homogenous decision making units (DMUs), especially with multiple inputs and outputs (An et al., 2020, An et al., 2020, Charnes et al., 1978, Li et al., 2019). The DEA approach is first introduced by Charnes et al. (1978), who proposed a linear mathematical programming model (i.e., CCR). Then, Banker et al., 1984, Cook and Seiford, 2009 proposed various DEA models. DEA is effective and successful for allocating the fixed cost because of the following reasons. First, fixed cost allocation issues often refer to multiple inputs and outputs. Second, various users that share a common platform are independent and homogenous, i.e., each user can be viewed as a DMU. Finally, decision makers typically consider each user’s performance before and after allocation.

DEA-based allocation methods can be divided into several groups by different standards. First, from an efficiency perspective, most allocation methods are based on two principles: (1) Efficiency invariance principle, which refers to that the efficiency of DMUs remains unchanged before and after allocation. It was initially proposed by Cook and Kress (1999), who firstly applied the DEA technique to realize an allocation plan based on the principles of invariance and Pareto-minimality. Subsequently, Jahanshahloo, Lotfi, Shoja, and Sanei (2004) demonstrated that the computation of Cook and Kress (1999)’s method is difficult, and then introduced an approach based on efficiency invariance and a common set of weight principles. A drawback of Cook and Kress (1999)’s approach, as proposed by Cook and Zhu (2005), is its failure to determine the cost allocation plan among DMUs directly, Then, Cook and Zhu (2005) extended the results of Cook and Kress (1999) into various orientation DEA models. In a subsequent work, Lin (2011) improved Cook and Zhu (2005) approach and set fixed cost targets according to the amounts of sharing cost. Based on previous allocation approach, Lin and Chen (2016) suggested a new approach that can enable the super CCR efficiency scores of all DMUs to remain unchanged after allocation. Jahanshahloo, Sadeghi, and Khodabakhshi (2017) presented two equitable approaches based on the efficiency invariance principle and a common set of weight. (2) The efficiency-maximization principle, which means that the sum or average efficiency of each DMU should be maximized after allocation. This approach was first illustrated by Beasley (2003), who obtained a unique allocation plan by maximizing the average efficiency of all DMUs. Amirteimoori and Kordrostami (2005) proved that the Beasley (2003)’s approach is infeasible in certain cases. However, Jahanshahloo et al. (2017) demonstrated that it is always feasible. Li, Yang, Liang, and Hua (2009) observed a fact that, in certain cases, the fixed cost is a complement of other cost input, instead of an independent factor. Amirteimoori and Tabar (2010) obtained fixed cost allocation plan by minimizing total and maximal deviation and considered output targets simultaneously. Lotfi et al. (2012) applied a common-weights DEA method to fixed cost allocation when efficiencies are taken into account. Similarly, based on DEA common-weight evaluation framework, Chu and Jiang (2019) proposed an allocation approach based on the utility. Li, Yang, Chen, Dai, and Liang (2013) investigated the effect of fixed cost on each DMU and proved that there exist cost allocation plans making each DMU efficient. Si et al. (2013) examined the concept of the equity of the proportional sharing method and investigated the relationship between the extended method of proportional sharing and other DEA-based allocation methods (i.e. Beasley). Du, Cook, Liang, and Zhu (2014) established a method on the basis of the cross-efficiency concept by considering competitive and cooperative situations. Khodabakhshi and Aryavash (2014) proposed that the allocation must be directly proportional or inversely proportional to the elements (inputs and outputs). Lin and Chen (2017) introduced a global modified additive DEA (MAD) model to allocate fixed cost by optimizing the global MAD-efficiency. Recently, Li, Zhu, and Liang (2019) suggested a new non-egoistic principle, which means that DMU should proposes an allocation plan that punish itself to guarantee the acceptability. These approaches are based on efficiency maximization principle, but some DMUs may not achieve the common technological level, then Ding, Chen, Wu, and Wei (2018) proposed an approach considering technological heterogeneity.

Second, from the view of game theory, a reasonable allocation plan in cooperative and competitive scenarios is significant. Nakabayashi and Tone (2006) introduced a social dilemma called the “egoist’s dilemma” and studied its properties by using DEA and cooperative game theory. Du et al. (2014) showed that cooperative and competitive situations exist among DMUs, and dealt with cost allocation problems using DEA cross-efficiency approach. Yang and Zhang (2015), from the perspective of cooperative game, designed characteristic functions based on DEA and proposed a modified Shapley value to solve the resource allocation problem. Sun, Fu, Ji, and Zhong (2017) discussed schemes for allocating emission permits among a group of companies. They integrated DEA and game theory to construct various emission allocation models. Zhang, Wang, Qi, and Wu (2018) combined game theory and DEA approach to solve the problem of transmission cost allocation. Li, Li, Emrouznejad, Liang, and Xie (2019) considered the game relations in the allocation process, then they proposed a cooperative game approach and utilized the nucleolus as a solution to the cooperative game. Li, Zhu, and Liang (2018) solved the allocation problem by considering the competitive and cooperative relationships among DMUs. They developed a DEA-game cross-efficiency method to generate a unique and fair allocation plan.

Third, considering the internal structure of systems, fixed cost allocation problems exist both in single-stage and two-stage systems. Most of the abovementioned studies only investigated the allocation of fixed cost among single-stage systems, but ignored the internal structure of systems by considering them as “black boxes” (Yu, Chen, & Bo, 2016). Recently, studies that address the resource allocation issues based on network DEA have been conducted by several researchers. However, few studies on fixed cost allocation in a network environment are investigated. For example, Bi, Ding, Luo, and Liang (2011) generated a resource allocation and target setting plan for each production unit by opening the black box. Xiong, Wu, An, Chu, and Liang (2018) proposed a new DEA approach to allocate resources in a bidirectional interactive parallel system. Yu et al. (2016) intended to solve allocation issues in two-stage systems, they presented a two-stage network DEA-based model that considers the internal structure of the operational process and concept of cross-efficiency. However, their approach can’t guarantee a unique allocation plan. Ding, Zhu, Zhang, and Liang (2019) dealt with the fixed cost allocation problem for a general two-stage network structure, by introducing the concepts of satisfaction degree and fairness degree. Zhu, Zhang, and Wang (2019) treated the fixed cost as an additional input factor shared in two-stage DMUs and proposed three allocation procedures based on different objectives in reality. Chu, Wu, Chu, and Zhang (2019) considered the competition between the DMU’s two stages in fixed cost allocation problems and regarded these two kinds of stages as two unions, then the allocation plan is calculated by satisfaction degree bargaining model. An, Wang, Emrouznejad, and Hu (2020) proposed a fixed cost allocation approach for two stage system based on efficiency invariance principle, which consider the internal relationship of two-stage DMUs. Li, Zhu, and Chen (2019) adopt the DEA methodology to determine relative efficiency while considering internal structure and possible allocation cost. Furthermore, their method can guarantee a unique allocation plan with considering the operation sizes of all DMUs and sub-stages.

For an overview of the existing approaches, fixed cost allocation of single-stage systems may involve cross efficiency, target setting, and game theory. However, approaches in the two-stage system are still in the early research stage. In this paper, we approach the fixed cost allocation issue by considering the cooperative game relationship among a set of DMUs. Obviously, solutions based on game theory are acceptable to the majority of DMUs from the perspective of fairness. Based on the above-mentioned considerations, this paper focuses on the basic two-stage systems, where all outputs from the first stage are inputs to the second stage (Kao and Hwang, 2008), and proposes an allocation plan for each DMU and its sub-stages. In the distribution process, we use the additive efficiency evaluation method to evaluate DMUs with a two-stage structure. Furthermore, we demonstrate that each DMU and its sub-stages can be efficient after allocation. However, there maybe multiple allocation plans satisfying efficient principle after allocation. Thus, based on these principles and the cooperative relationship among a set of DMUs, a nucleolus-based method is proposed to formulate the allocation plan through a series of computation algorithm. Finally, the proposed approach is applied to a random dataset and an empirical commercial bank application. In summary, the main contribution of the existing literature is as follows. First, this paper considers the external cooperative game relationship among a set of two-stage DMUs while making a fixed cost allocation plan. Most previous studies incorporated game theory into fixed cost allocation issue in single-stage systems and ignored external relationship among two-stage DMUs. Second, we demonstrate that each DMU and its sub-stages will be efficient after allocation. Third, the nucleolus-based method is acceptable to each DMU. Thus, this paper extends the fixed cost allocation approach into the two-stage system.

The remainder of this paper is organized as follows. Section 2 introduces the classical DEA models and efficiency measurement method with fixed cost in the single-stage and two-stage systems. Then, we propose the nucleolus-based method in the two-stage system when a set of DMUs are in cooperative relationship in Section 3 and use a numerical example to verify. Afterward, the proposed approach is applied to an empirical application of commercial banks in Section 4. Finally, conclusions and directions for future research are provided in Section 5.

Section snippets

Preliminaries

This section introduces the efficiency evaluation of a single DMU and all DMUs with considering fixed cost. The evaluation object not only include the single-stage DMU, but also the two-stage DMU. Then, we prove that all DMUs that are involved in the two-stage systems can be efficient after allocation.

Cooperative game DEA approach

To embody the cooperative relationship among DMUs, we propose a cooperative game DEA approach to solve the fixed cost allocation issues in two-stage systems. Cooperative and competitive relationships not only exist among a group of DMUs, but also in two substages. Liang et al., 2008 considered the non-cooperative relationship between two stages, whereas Li et al. (2019) approached the cooperative relationship among DMUs in the single-stage system. Here, we not only consider cooperative

Applications

This section demonstrates the proposed nucleolus-based allocation method to Commercial banks. We consider a two-stage commercial bank in Guizhou Province, China that consists of 27 branches (Li et al., 2019). In the current age of instantaneous information, enterprises show increased attention to information construction, such as commercial banks. In this paper, the cost of commercial banks is incurred from the information service system, which is extremely expensive for these banks. The cost

Conclusions

Fixed cost allocation is commonly encountered in social life, such as manufacturing enterprise, banks, or universities, and several of these organizations have two-stage network structures. Apparently, cooperative and competitive relationships exist among DMUs when allocating cost. Few studies have considered the cooperative relation among single stage DMUs during fixed cost allocation. Furthermore, this relationship also exists in two-stage systems. Therefore, this paper proposes a

CRediT authorship contribution statement

Qingxian An: Conceptualization, Supervision, Funding acquisition, Writing - review & editing. Ping Wang: Methodology, Software, Investigation, Writing - original draft. Shasha Shi: Supervision, Writing - review & editing.

Acknowledgement

The research is supported by National Natural Science Foundation of China (71871223; 71631008), Innovation‐Driven Planning Foundation of Central South University (2019CX041). Major Project for National Natural Science Foundation of China (71790615; 71991465; 71991483). The Key Consulting Project of the Chinese Academy of Engineering (2020-XZ-053-06; 2020-XY-36).

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