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Fixed cost allocation in two-stage system using DEA from a noncooperative view

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Abstract

This paper uses the data envelopment analysis (DEA) approach to solve the issue of allocating fixed costs among a set of decision-making units (DMUs) with two-stage network structure. Fixed cost allocation is a prominent issue that is encountered by organizations, and it is one of the most important applications of DEA. It exists not only in single-stage systems but in two-stage network systems, such as bank systems in real situations, which comprises the deposits process and lending process. However, branch banks often compete with one another out of self-interest. The objective of this paper is to design a fair allocation scheme for two-stage network systems and considering the noncooperative game relationship among DMUs. The idea of satisfaction degree and noncooperative game theory is integrated into the proposed allocation model. In the noncooperative framework, we define a DMU’s payoff as the product of two substage satisfaction degree, and every member is noncooperative and selfishly seeks to maximize its own payoff. The allocation plan to DMUs and substages is determined by the final competition payoff. A real case is analyzed to illustrate the applicability of the proposed approaches.

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Acknowledgements

The research is supported by National Natural Science Foundation of China (No. 71871223, 72088101, 72071072, 71631008, 71790615,72071072), Innovation‐Driven Planning Foundation of Central South University (2019CX041); the Fundamental Research Funds for the Central Universities of Central South University (No. 2021zzts0028).

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Authors and Affiliations

  1. School of Business, Central South University, Changsha, 410083, Hunan, China

    Qingxian An, Ping Wang & Zongrun Wang

  2. School of Business Administration, Hunan University, Changsha, 410082, Hunan, China

    Honglin Yang

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  1. Qingxian An
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Correspondence to Honglin Yang or Zongrun Wang.

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Appendix: Proof of Theorem 2

Appendix: Proof of Theorem 2

Theorem 2

The final competition payoff result is a Pareto solution.

Proof

Suppose that the final competition payoff result \(\left( {\varphi _{1}^{{t{\text{*}}}} ,\varphi _{2}^{{t{\text{*}}}} ,~\varphi _{3}^{{t{\text{*}}}} ,~ \ldots ,~\varphi _{n}^{{t{\text{*}}}} } \right)\) is not a Pareto solution, the corresponding allocation plan is \(\left[ {\left( {R_{1}^{{1{\text{*}}}} ,~R_{1}^{{2{\text{*}}}} } \right),~\left( {R_{2}^{{1{\text{*}}}} ,~R_{2}^{{2{\text{*}}}} } \right),~ \ldots ,~\left( {R_{g}^{{1{\text{*}}}} ,~R_{g}^{{2{\text{*}}}} } \right),~ \ldots ,~\left( {R_{n}^{{1{\text{*}}}} ,~R_{n}^{{2{\text{*}}}} } \right)} \right]\). Assume \(DMU_{{j0}}\) can decrease the allocation of stage 1, that is, \(R_{{{\text{j}}0}}^{1} < R_{{{\text{j}}0}}^{{1{\text{*}}}}\), at the same time, the payoff value of \(DMU_{{j0}}\) increases. However, in order to allocate the overall fixed cost R, the fixed cost allocated to other DMUs will increase, that is, \(\mathop \sum \limits_{{j = 1,j \ne j0}}^{n} R_{j} > \mathop \sum \limits_{{j = 1,j \ne j0}}^{n} \left( {R_{{\text{j}}}^{{1{\text{*}}}} + R_{{\text{j}}}^{{2{\text{*}}}} } \right)\), and the payoff value of a certain DMU decrease.

Therefore, this assumption is not valid. T here exists no DMUs can improve its payoff value without reducing other DMUs payoff. And then the final competition payoff result is a pareto solution.

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An, Q., Wang, P., Yang, H. et al. Fixed cost allocation in two-stage system using DEA from a noncooperative view. OR Spectrum 43, 1077–1102 (2021). https://doi.org/10.1007/s00291-021-00643-y

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