Fixed Cost Allocation Research Articles

Investment allocation in an adjustment-cost production technology framework for two-stage network structures

Effective resource allocation can assist organizations in utilizing infinite resources, thereby improving efficiency and productivity. For an organization, investment is an essential production resource, and how to appropriately allocate investment is one of the most important decisions they need to make. This study aims to rationally allocate investment across each stage of two-stage network structure under two scenarios. We first propose general investment allocation model with adjustment costs to efficiently allocate investments across each stage. Then, given a fixed total investment, we use the DEA-based fixed cost allocation method to achieve the optimal allocation across stages, and we also incorporate two cooperative and noncooperative relationships between two stages. Additionally, capital stock efficiency is defined to analyze the relationship between optimal and actual capital stocks, clarifying the degree to which the optimal capital stock is reached. Finally, the proposed approach is applied to 14 A-share listed companies in China, and suggestions on the optimal amount of investment allocation and capital stock required are provided to facilitate organizational production.

Fixed cost allocation based on a data envelopment analysis aggressive game approach

The literature on organizational strategy and production operations suggests that establishing a shared platform would be beneficial for addressing competition and mitigating risks. However, the creation of a common platform inevitably incurs expenditures and costs, prompting the natural question of how to equitably distribute the total fixed cost among all participating units. While the fixed cost allocation (FCA) problem has been explored in the context of data envelopment analysis (DEA), this paper addresses a distinct decentralized environment wherein there is no central decision-maker, and each individual decision-making unit (DMU) pursues a self-interested strategy to minimize its allocated cost share. In tackling the challenge of proposing an acceptable allocation plan within this decentralized setting, we initially improve the non-egoistic principle by considering the perspective of operational size. This improvement mandates that each DMU bears the maximum fixed cost per unit of operational size in its own allocation proposal. The improved non-egoistic principle implicitly addresses the platform constraint and fairness concerns, while each DMU remains committed to maximizing other DMUs’ cost shares and minimizing their own performance efficiency. Additionally, we introduce an aggressive game process to amalgamate all DMUs’ allocation proposals into the DEA efficiency analysis. A DEA aggressive game approach and its corresponding computational algorithm are developed to facilitate consensus and determine the final FCA plan. Mathematically, we prove that all average post-allocation efficiencies, incorporating fixed costs, converge to the same aggressive game cross-efficiency for each DMU in the iterative algorithm. To the best of our knowledge, this paper represents the first study that employs an aggressive strategy in the FCA problem to minimize efficiency scores, while existing literature predominantly focuses on maximizing or maintaining efficiency scores. Finally, we apply the proposed approach to a classical numerical example and an empirical study involving nine independent truck fleets to showcase its characteristics and efficacy.

Allocating the fixed cost as a complementary input in two-stage system: A DEA approach

The existing research on fixed cost allocation (FCA) in two-stage system data envelopment analysis (DEA) models typically regards the fixed cost as an additional input for each decision-making unit (DMU). However, these models overlook the prevalent real-world scenario where fixed costs act as complementary inputs in actual production processes. This study proposes a general two-stage network DEA model that incorporates the fixed cost as a complementary input to optimize the allocation scheme. First, we construct a functional relationship between the efficiency scores of DMUs and their allocated fixed costsusing a modified super-efficiency DEA model, which effectively classifies DMUs and solves infeasible solutions within the variable returns to scale (VRS) framework. Then, we propose a fair and unique allocation model based on fairness and efficiency maximization principles. Specifically, we allocate fixed costs based on the operational scale for inelastic DMUs while equitably increasing the efficiency scores for elastic DMUs. Finally, a numerical example and an empirical study on subsidy allocation among 30 provinces demonstrate the rationality and acceptability of our approach.

Fixed cost allocation using asymmetrical core-Nash bargaining DEA game

This paper introduces cooperative game theory into the data envelopment analysis (DEA) method for building a novel fixed cost allocation (FCA) approach from group and individual consciousness. We start by acknowledging the divergences of decision-making units (DMUs) and propose an asymmetrical Nash bargaining DEA game, where DMUs’ bargaining powers are defined through the functions of cross-efficiency and operation scale. However, the asymmetrical Nash bargaining solution is proposed from an individual perspective, without providing the theoretical support of full cooperation. To narrow this gap, we further introduce the core concept of a coalitional cooperative game into the built asymmetrical Nash bargaining DEA game, resulting in the asymmetrical core-Nash bargaining DEA game. The asymmetrical core-Nash bargaining solution is based on the common weights, with the properties of stability, fairness, Pareto-efficiency, and invariance to affine transformation. Furthermore, our proposed approach inherently ensures the uniqueness of the fixed cost scheme. Finally, numerical analysis and case study are provided to show the concrete application of the new method.

A generalized data envelopment analysis approach for fixed cost allocation with preference information

Exploring a commonly acceptable fixed cost allocation (FCA) scheme is a challenging yet significant task, since one may contend with conflicting objectives. In this paper, we propose a new data envelopment analysis based FCA technique. The proposed approach aims to balance individual efficiency guarantees and collective preference objectives simultaneously. Particularly, a preference value loss (PVL) index is explicitly constructed to quantify the impact of preference considerations. Moreover, our generalized FCA strategy guarantees a minimal PVL and provides a reasonable and realizable lexicographic evaluation procedure to single out the final FCA scheme. Interestingly, the efficiency invariance and efficiency variance approaches from the prior literature are special cases of our proposed approach. Finally, the proposed approach is validated with a numerical example and an empirical application to bank maintenance fee distribution.

Fixed cost allocation based on data envelopment analysis from inequality aversion perspectives

In many real-world cases, fixed cost allocation frequently occurs in constructing the common platform of an organization: all divisions related to the platform must share the platform's fixed cost. Addressing the fixed cost allocation issue has become a vital topic, and numerous approaches have been used, including data envelopment analysis (DEA). However, inequality aversion, one vital concept of fairness concern, is frequently ignored in the previous publications. This paper is devoted to guaranteeing fairness for all decision-making units (DMUs) in allocation solutions from DEA. Firstly, we discuss how inequality impacts DMUs by proposing the concept of “fairness concern disutility” and using various parameters to reflect the negative impact of an unfair allocation plan on DMUs. Secondly, we discuss fixed cost allocation from three perspectives: optimistic, pessimistic, and mixed. Moreover, we give algorithms to obtain a unique, optimal solution and propose theorems to better understand our model. Lastly, we use a numerical example to illustrate the proposed models and theorems before applying our approach to address a fixed allocation issue in a commercial bank. Our analysis results show that the proposed approaches can guarantee fairness in fixed cost allocation.

Efficiency improvement and balance in fixed cost allocation: A trade-off approach based on DEA

Fixed cost allocation (FCA) is a problem that decision-makers usually encounter in practice. The existing FCA studies based on data envelopment analysis (DEA) have not recognized the trade-off between improving efficiency and balancing FCA result and failed to provide an approach to generate an equilibrium solution between the two goals. In this study, under DEA common-weight evaluation mechanism, we propose an approach for centralized FCA using several principles featuring fairness in the situation where the fixed cost is regarded as a complement of one of the current inputs. It first considers the principle of making equal-proportional efficiency improvements for all the decision-making units (DMUs) in FCA. Then, we define a measure, called “balanced indicator”, to measure the balance of a FCA result. Finally, a bi-objective model is built that considers both the objectives of maximizing the proportional efficiency improvement of the DMUs and improving the balance of the FCA result. A bargaining-game-like model and a trade-off algorithm are provided to generate the final FCA result. Our research contributes by elucidating the trade-off between enhancing efficiency and achieving balance in FCA result. It also makes a distinct contribution by proposing a bargaining-game-like model and a trade-off algorithm to generate a FCA result that is an equilibrium solution between the two goals and ensures no DMU is allocated zero cost. Finally, we demonstrate the superiority of our approach by comparing it with a previous representative approach and applying it to a case study of China National Heavy Duty Truck Group.

DEA-based proportional-sharing fixed cost allocation considering bi-objective optimization

DEA-based proportional-sharing fixed cost allocation considering bi-objective optimization

Calculation and Allocation of Atmospheric Environment Governance Cost in the Yangtze River Economic Belt of China

Atmospheric environment governance requires necessary cost input. Only by accurately calculating regional atmospheric environment governance cost and scientifically allocating it within a region can the operability and realization of the coordinated governance of the regional environment be ensured. Firstly, based on the consideration of avoiding the technological regression of decision-making units, this paper constructs a sequential SBM-DEA efficiency measurement model and solves the shadow prices of various atmospheric environmental factors, that is, their unit governance costs. Secondly, combined with the emission reduction potential, the total regional atmospheric environment governance cost can be calculated. Thirdly, the Shapley value method is modified to calculate the contribution rate of each province to the whole region, and the equitable allocation scheme of the atmospheric environment governance cost is obtained. Finally, with the goal that the allocation scheme based on the fixed cost allocation DEA (FCA-DEA) model converges with the fair allocation scheme based on the modified Shapley value, a modified FCA-DEA model is constructed to achieve the efficiency and fairness of the allocation of atmospheric environment governance cost. The calculation and allocation of the atmospheric environmental governance cost in the Yangtze River Economic Belt in 2025 verify the feasibility and advantages of the models proposed in this paper.

Green finance policy coupling effect of fossil energy use rights trading and renewable energy certificates trading on low carbon economy: Taking China as an example

In the context of low carbon economy development, controlling fossil energy consumption and promoting development in renewable energy are important paths for energy transition in various countries. Taking China as an example, this paper investigates the green finance policy coupling effect of fossil energy use rights trading (FET), renewable energy certificates trading (RECT), and renewable energy subsidies (RES) on the low carbon economy using a dynamic recursive computable general equilibrium (CGE) model. The main findings of this paper are as follows. (1) When using a game equity fixed-cost allocation model (Game-EFCAM) to allocate fossil energy use rights quotas among sectors, allocations differ in only a few sectors compared to the base period, and most sectors have relatively stable allocations compared to the base period. (2) The implementation of FET and RECT has a negative impact on per capita GDP. In the short term, RES can mitigate the negative impact of FET and RECT on per capita GDP, but eventually, the government needs to guide the enterprise innovation to improve the industrial structure to realize the double benefits of green finance policy. (3) The implementation of FET and RECT limits fossil energy consumption and also reduces the production possibility frontier of the secondary industries with higher levels of fossil energy consumption. The implementation of FET and RECT increases the proportions of tertiary industry and renewable energy. (4) The implementation of FET and RECT reduces carbon emissions. In other words, the implementation of FET and RECT mitigates the negative environmental externality. RES can enhance the carbon reduction efforts of FET and RECT. This research not only provides policy analysis for the development of the low carbon economy in China, but also gives a reference for other developing countries’ low carbon economy transition.

Fixed cost allocation in two-stage system: A data-driven approach from the perspective of fairness concern

Fixed cost allocation is the common issue for decision maker in real case, especially in the two-stage system such as banks, insurance companies, schools and other enterprises. Due to the complexity of two-stage system, fixed cost allocation in two-stage system between two stages is always worthy of investigation. Numerous studies have emerged to address the issue on the basis of data envelopment analysis (DEA). However, fairness concern is frequently ignored in those studies. Fairness concern is regarded as one of the most important factors guiding human behaviour in daily life. This paper studies how fairness concern impacts allocating fixed cost between two stages among all decision making units (DMUs). On the basis of cooperation and conflict between two stages, we investigate the fixed cost allocation under non-cooperative mode and cooperative mode respectively, and minimize the maximal fairness concern disutility by proposing max–min model. Further, we give some theorems to illustrate our proposed models and give algorithms for models to obtain the unique allocation plan under different modes. Lastly, a numerical example is presented, and we apply our approaches to address fixed cost allocation issue of a commercial bank which has 27 branches. Results shows that our proposed approaches can obtain an equitable allocation plan as much as possible compared with previous studies.

Measuring environmental efficiency of the EU based on a DEA approach with fixed cost allocation under different decision goals

Reasonably measure the environmental efficiency is a significant step towards a greener European Union (EU). This paper proposes several data envelopment analysis (DEA) approaches to evaluate the environmental efficiency of EU countries, considering the allocation of fixed environmental costs and three different decision objectives for each member state. Empirical results for the 28 EU member states from 2013 to 2019 show that the efficiency scores of each member remained broadly stable. Spearman’s analysis indicates that the efficiency ranking under economic priority and win–win policy is significantly positively correlated, and the ranking under environment priority is significantly negatively correlated with the rankings under the other two policies. The EU as a whole is most efficient under a win–win strategy, but there are significant differences among member states. Member states that joined the EU earlier have higher efficiency values under economic priority. These results can help the EU and Member states understand strengths and weaknesses and implement appropriate policy measures.

Allocating a fixed cost as a reduction of outputs in a non-cooperative decision-making environment: A data envelopment analysis approach

The fixed cost generated from constructing a common platform by all units needs to be allocated among these units in a reasonable way in many management practices. As market information is often incomplete, data envelopment analysis (DEA) approaches have been successfully used in fixed cost allocation problems. The fixed cost was treated as an input measure in previous studies. However, it has no effect on the original inputs and outputs when making the allocation decision after an operating cycle. In this case, it can be regarded as a reduction of outputs rather than an input. This paper developed a new DEA approach based fixed cost allocation model in a decentralized decision-making environment that treats the fixed cost as a reduction of outputs. Unlike the existing approaches addressing efficiencies, the proposed model considers the cost itself in the allocation process. An iterative procedure is then presented to derive the average game cost allocation scheme. A classical numerical example from the prior literature is given to show the applicability of the proposed model. The simulation results show that the developed approach can obtain a unique allocation scheme.

Individual rationality and overall fairness in fixed cost allocation: An approach under DEA cross-efficiency evaluation mechanism

This paper develops a new fixed cost allocation (FCA) approach that considers individual rationality and fairness using the data envelopment analysis (DEA) cross-efficiency evaluation. We propose two FCA principles from individual rationality: a novel self-lowest principle and the efficient after FCA principle. Under the precondition that the individual rationality principles are satisfied, we suggest determining the final FCA result based on two overall fairness principles: the lower bound principle and the cross-efficiency Pareto-optimality principle. A multi-objective FCA model is built following these principles. An algorithm and some rules are used to solve the multi-objective model and obtain the final cross-efficiency evaluation and FCA result. Our study contributes by explicitly showing the design of individual rationality and overall fairness in FCA using the DEA cross-efficiency evaluation mechanism. Finally, we compare the suggested approach with some previous ones using a numerical example and illustrate its usefulness through a truck fleet case study.

Allocation of carbon emission quotas for China's provincial public buildings based on principles of equity and efficiency

In accordance with China's commitments to reduce greenhouse gas emissions under the Paris Agreement, it is important to establish a fair and effective carbon emission allocation mechanism; this would help to achieve China's 2030 “carbon peak” target. Here, the fixed cost allocation model (FCAM) was used to allocate carbon emission quotas to public buildings in China in 2030, based on the principles of equity and efficiency. By comparing these provincial allocation results with those obtained using zero-sum data envelope analysis (ZSG-DEA), optimal carbon emission quotas were proposed for the public building sectors of thirty Chinese provinces. The results revealed large differences, with the public buildings of Guangdong having the highest quota (11.39%) and those of Xinjiang having the lowest (0.08%). Additionally, Heilongjiang, Inner Mongolia, and Liaoning were revealed to be under high pressure with regard to the reduction of emissions from their public buildings; Heilongjiang was predicted to require the highest reduction, exhibiting a need to reduce 34 million tons of emissions. These findings will help provide new ideas for relevant policy makers to formulate fair and effective carbon allowance allocation plans.

Second order cone programming formulation of the fixed cost allocation in DEA based on Nash bargaining game

<p style='text-indent:20px;'>A vital issue in many organizations is the fair allocation of fixed cost among its subsets. In this paper, using data envelopment analysis, first we study fixed cost allocation based on both additive and multiplicative efficiency decompositions in the cooperative context for a two-stage structure in the presence of exogenous inputs and outputs. A conic relaxation formulation of multiplicative decomposition is given. Then, fixed cost allocation based on the leader-follower paradigm are presented. In the sequel, for allocating a fair fixed cost between the stages, using the results of the leader-follower model, we present the nonlinear Nash bargaining game model that independent of the efficiency score of each unit, allocates fixed cost to the stages. The nonlinear model is reformulated as a second order cone program which is an imporvement over the parametric linear models in the literature. Finally, two examples are used to illustrate the proposed models and compare their results with the existing models.</p>

Frontier-based incentive mechanisms for allocating common revenues or fixed costs

In practice, allocating common revenues or fixed costs is a common practice in multi-unit organizations to motivate subunits to improve performance. Recently, Dai et al. (2021, EJOR) developed an incentive approach for allocating common revenues or fixed costs. We identify two shortcomings of their approach and propose two frontier-based incentive mechanisms for allocating common revenues or fixed costs. First, the incentive common revenue allocation (ICRA) and incentive fixed cost allocation (IFCA) are conceptualized. On the basis of ICRA and IFCA, the generalized incentive efficiency measures that satisfy strong monotonicity and self-excluded in technology are identified to evaluate subunit performances. To seek for incentive efficiency measures, several ex-post technologies with frontier change rate (FCR) are considered. Different technologies are formulated for the information asymmetry situation, in which FCR is not available to subunits when they select production mixes, and the information symmetry situation, in which FCR is available to subunits when they select production mixes. We elucidate that the formulated technologies are applicable to derive corresponding incentive efficiency measures. ICRA and IFCA are then obtained via the concept of efficiency proportion. Subsequently, the mechanisms for information asymmetry situation and information symmetry situation are summarized, and several theorems and interesting observations of the proposed mechanisms are reported. Finally, we illustrate our proposed mechanisms for allocating common revenues or fixed costs using data from a home appliance enterprise in China.

Water pollutant discharge permit allocation based on DEA and non-cooperative game theory

Against the background of the ecological civilization system reform in the new era, the appropriate allocation of water pollutant discharge permits is an important policy for controlling the amount of wastewater discharge. Traditional allocation methods have disadvantages, such as high additional costs, an unfair allocation scheme, and market distortion. In the present study, a fixed-cost allocation model based on data envelopment analysis (DEA) and the Nash non-cooperative game theory is employed to allocate water pollutant discharge permits of totally 31 provinces in China from 2008 to 2017. The allocation scheme considers environmental efficiency. The results demonstrate regional differences in the allocation of water pollutant discharge permits. The eastern region has abundant allocations. The northeastern and central regions have insufficient allocations. Besides, the western region has a significant shortage of allocations. It indicates the higher the utilization efficiency of the water pollutant discharge permits, the higher the region's sustainable development is. Based on the analysis, we propose guidelines for industrial wastewater discharge reduction.

Design of energy use rights trading policy from the perspective of energy vulnerability

This paper takes Sichuan Province (China) as an example to improve the design of energy use rights trading policy (ERTP) with the goal of reducing regional energy vulnerability. The following contributions are made to improve the ERTP: (1) The first step in designing the ERTP is to set the total energy consumption target, and this paper formulates the total energy consumption of Sichuan Province in 2025 with reference to China's 14th Five-Year Plan. (2) This paper adopts the historical egalitarianism, economic egalitarianism, efficiency allocation model and the Game-Equity Fixed Cost Allocation Model (Game-EFCAM) to allocate energy use quotas at the sectoral level. Therein, Game-EFCAM is an allocation model which considers both equity and efficiency. (3) In designing the trading mechanism, this paper exogenizes the price of energy use rights within the quotas and endogenizes (determined by the market) the price of energy use rights outside the quotas. Moreover, this paper adopts a recursive dynamic computable general equilibrium (CGE) model to simulate the effects of different allocation schemes and economic development levels on energy vulnerability. The results show that regional energy vulnerability is minimized under the allocation results of Game-EFCAM. This paper not only improves the ERTP, but also provides a reference for other countries in energy dilemmas.

Allocating a fixed cost across decision-making units with undesirable outputs: A bargaining game approach

Allocating a fixed cost among a set of peer decision-making units (DMUs) is one of the most important applications of data envelopment analysis. However, almost all existing studies have addressed the fixed cost allocation (FCA) problem within a traditional framework while ignoring the existence of undesirable outputs. Undesirable outputs are neither scarce in various production activities in real world applications nor trivial in efficiency evaluation and subsequent decision making. Motivated by this observation, this article attempts to explicitly extend the traditional FCA problem to situations in which DMUs are necessarily involved with undesirable outputs. To this end, we first investigate the efficiency evaluation of DMUs considering undesirable outputs based on the joint weak disposability assumption. Then, flexible FCA schemes are considered to revisit the efficiency evaluation process. The results show that feasible allocation schemes exist such that all DMUs can be simultaneously efficient. Furthermore, we define the comprehensive satisfaction degree and develop a satisfaction degree bargaining game approach to determine a unique FCA scheme. Finally, the proposed approach is tested with an empirical study of banking activities based on real conditions.