The different DMUs can operate together. These units can be service organizations such as banks or manufacturing plants. Senior managers may want to divide a fixed cost among DMUs in many situations. This fixed cost allocation plan can be based on different strategies. One of these strategies is based on the efficiency invariance principle. This strategy states that the efficiency of DMUs will not change before and after the fixed cost allocation process. We now review recent studies on fixed cost allocation using Data Envelopment Analysis (DEA). Fixed cost allocation is one of the important issues in many organizations, including banks, commercial enterprises, and industrial firms. It helps managers take a fair perspective on the organization they manage and prevent the overall waste of resources. Additionally, fixed costs play a key role in decision-making processes related to pricing and determining profitability across different industries. Proper allocation of these costs among various Decision-Making Units (DMUs) is essential, such as allocating advertising costs among retailers and distributing health resources and equipment upgrades.

In this area, a group of studies has been developed based on the fundamental work of Cook and Kress [1], which introduce two main assumptions: “efficiency invariance” and “pareto-minimal.” The efficiency invariance principle states that after allocating fixed costs among DMUs, the efficiency of these units should not change compared to the situation before the allocation. A fixed or common cost is imposed on all DMUs in many DEA applications. The goal is to allocate these costs among different DMUs so each DMU bears its share. In Cook and Kress’s [1] proposed approach, several computational issues arise after fairly allocating shared costs that require solving linear programming problems. However, Jahanshahloo et al. [2] provided a simple model that resolves these problems easily without solving linear programming problems, thus proposing a straightforward method for cost allocation without computational complexity. This method can be applied to constant and variable returns to scale (CRS and VRS) technologies. It should be noted that in both Cook and Kress’s methods [1] and the method proposed in this article, the principle of “pareto-minimal” is not preserved. Subsequently, Cook and Zhu [3] extended the work of Cook and Kress [1] for various models. Lin [4], [5] also suggested modifications based on the approach of Cook and Zhu [3], although the changes were not very significant. Nevertheless, these studies emphasize the importance of fixed cost allocation and address concerns in this process, presenting algorithms and theorems that help ensure the correct allocation of costs.

Beasley [6] introduced an approach called efficiency maximization. This approach aims to improve the average efficiency of DMUs. This approach is beneficial in environments where balanced and simultaneous allocation among various DMUs is necessary. Beasley's approach, by optimizing efficiency, can aid in the optimal allocation of resources and prevent unfair allocation. Si et al. [7] developed methods for cost allocation using common weight and proportional sharing. This approach can be applied in systems seeking equitable cost allocation among DMUs. This model helps ensure that cost allocation minimizes cost gaps between units and maintains fairness in allocation. Mostafaei [8], in his research, particularly in the field of fixed cost allocation in practical applications like public services and production processes, addresses the problem of allocating shared costs using DEA. He introduced a new method for allocating fixed costs to DMUs using DEA. Due et al. [9] introduced an approach based on cross-efficiency. In this approach, cost allocation is conducted in such a way that the efficiency consideration of each unit directly impacts the efficiency of other units. Lin and Chen [10] used DEA to address resource and fixed cost allocation issues.

The researchers proposed a new sharing model in which fixed resources and targets are divided among DMUs. This model also includes two corresponding algorithms that can generate a unique allocation for each DMU. Furthermore, the proposed method can be extended to CRS and VRS technologies. Jahanshahloo et al. [11] proposed two important approaches for solving the fixed cost allocation problem in line with efficiency invariance and weight set principles. These methods are instrumental in fixed cost allocation in DEA applications. Li et al. [12] also addressed target setting and resource allocation issues, considering the principles of efficiency invariance and common weights in allocating resources and targets between DMUs. They proposed a new mechanism incorporating these principles in allocating multiple resources, particularly in situations involving multiple targets across units. Zhu et al. [13] proposed DEA models for measuring fixed cost performance in two-stage systems. This approach carefully considers input and output factors to allocate the best cost. This approach is suitable for complex systems with multi-stage processes where resources must be meticulously allocated. Li et al. [14] extended the traditional fixed cost allocation issue to systems with a two-stage network structure. Using DEA, researchers assess the relative efficiency of DMUs and make efficient cost allocation possible under a set of common weights. This approach allows DMUs to maximize efficiency by choosing different allocations and relative weights. Due [9] to the diverse allocations existing in the efficient set, allocation programs are optimized with a focus on operational unit sizes, and a minimax model and an algorithm are provided to reduce deviations between efficient allocation and size. An et al. [15] proposed an efficiency-based approach to solve fixed cost allocation issues in two-stage systems, extending this model to general systems. This approach is particularly important in scenarios where decision-making systems involve multiple stages or levels of operation. Utilizing this model in cooperative and non-cooperative systems can help optimize the allocation of costs while also maintaining system efficiency. Chu. et al. [16] expanded the FCA method by incorporating the "full efficiency" principle, offering new fixed-cost allocation approaches in two-stage structures. They propose a range of possible allocations and models to consider competition between two-stage DMUs under a centralized framework. Leader-follower models and the concept of union satisfaction degree have also been utilized to provide stable and acceptable allocations. Dai et al. [17] introduced a two-stage incentive approach for allocating shared revenues or fixed costs.

The proposed method suggests cross-efficiency DEA for evaluating DMUs and incentive allocation, providing nonlinear allocation models and simple equations for global optimization. Practical aspects of these methods include improved performance and efficiency in information asymmetry decision-making environments. This approach motivates sub-units to enhance their performance. An et al. [18] identified shortcomings in Dai et al.'s [17] approach and proposed two alternative incentive mechanisms for allocating shared revenues or fixed costs. These mechanisms are designed under informational symmetry and asymmetry conditions and establish incentive productivity criteria. These mechanisms are tested on real data from a Chinese company based on the efficiency ratio. Chu et al. [19] emphasized fairness in allocation and presented a multi-objective model for cost allocation that considers the needs and preferences of DMUs. This model is particularly important when multiple stakeholders with different preferences are involved in decision-making. This approach can create a fair allocation that responds to the diverse needs and desires of DMUs. Yang et al. [20] examined discrepancies in cost allocation and sought to minimize the deviation between individual efficiency and total preferences. This approach can improve coordination among DMUs, make cost allocation mathematically optimal, and reduce disagreements. Zhang et al. [21] introduced aggressive game strategies for cost allocation in a decentralized environment. This approach is particularly useful in systems where DMUs operate independently. Utilizing game theory, this approach seeks agreements among DMUs to allocate costs to ensure fairness and efficiency.

Additionally, this approach demonstrates that after cost allocation, the average efficiencies converge towards the cross-efficiency of the aggressive game. Yang et al. [20] introduce a new DEA-based fixed cost allocation method that balances individual efficiency assurance goals and collective priority objectives simultaneously. This approach involves constructing a Priority Value Loss (PVL) index, which accurately measures the effects of priority considerations. Moreover, our generalized fixed cost allocation strategy minimizes PVL and provides a prioritized evaluation process for selecting the final allocation plan.

Value efficiency analysis is needed to apply the DMs (senior managers of companies) opinion in the DEA. Halmé et al. [22] created a DEA-based operational method that incorporates DM’s preference information to identify the most preferred input-output configuration. In this study, they develop a necessary procedure and theory for integrating preference information in a novel way within the context of DMU efficiency analysis. The efficiency of DMUs is defined based on the essence of DEA, which is complemented with preference information from the DM regarding the desired structure of inputs and outputs. The proposed method starts by helping the Decision-Maker (DM) identify the most preferred combination of input and output among efficient DMUs in DEA. Then, assuming that the Most Preferred Solution (MPS) maximizes the DM's underlying objective function (unknown), the indifference curve of the function at that point is approximated by its potential tangent hyperplanes. Finally, efficiency scores for each DMU are calculated, and inefficient units are compared with those with a similar value to the preferred ones. The resulting efficiency scores are optimistic approximations of the real scores. This method and the resulting efficiency scores immediately apply to solving practical problems. In value efficiency analysis, the efficiency of units is calculated based on a new efficiency frontier. In this way, the DM selects several units as the units with the highest efficiency (MPS units). If the MPS units are observed, we call them the Most Perfect Units (MPUs). These Units have the best performance among the DMUs. For example, in evaluating a bank branch complex, some banks have the best performance from a management point of view, and based on value efficiency analysis, the efficiency of other branches is obtained based on the efficiency of these units. In the process of value efficiency analysis, the value efficiency frontier is introduced instead of the efficiency frontier, and an efficiency value called value efficiency corresponding to each unit is obtained. Similarly, we can perform value efficiency analysis based on economic efficiency instead of technical efficiency [23]. Halmé. et al [22] proposed a non-convex value efficiency analysis and its application to bank branch. Gerami [24] proposed an interactive procedure for improving the estimate of value efficiency in DEA. Gerami et al. [25] propose a novel geometric interpretation for value efficiency when plugging it into radial and non-radial DEA models under VRS technology. Gerami et al. [26] proposed a generalized inverse DEA model based on value efficiency for firm restructuring.

This paper mainly introduces the MPS units and presents value efficiency evaluation models in VRS technology. We also present a new fixed-cost allocation approach based on the efficiency invariance strategy and value efficiency analysis. We illustrate our models by using a case study to evaluate commercial banks. Based on the presented approach, we can obtain a fair allocation for banks.

The rest of the paper is organized as follows: in the second section, we present the basics of fixed cost allocation with the efficiency invariance strategy of DMUs and value efficiency analysis. The third section presents an algorithm for providing a fixed cost allocation plan based on value efficiency. In the fifth section, we use the presented algorithm to allocate a fixed cost among DMUs that operate under a single management. In the fifth section, we present the results of the presented models.

Let n DMU as DMUj, j = 1, … , n. These DMUs use of vectors Xj = (x1j, … , xmj) ∈ R+ for producting vectors Yj = (y1j, … , ysj) ∈ R+. In this section, we briefly describe Cook and Zhu's method [3] based on the efficiency invariance principle for the issue of fixed cost allocation. Assume that the fixed does not change. According to this method, the efficiency score of all DMUs does not change before and after the fixed cost allocation process. We consider the unit under evaluation to be DMUo = (Xo, Yo). We proposed the BCC model under VRS technology and output-oriented as follows.

min ∑i=1 vi xio + vo, s. t. ∑i=1 vi xij − ∑r=1 ur yrj + vo ≥ 0, j = 1, … , n, ∑r=1 ur yro = 1, vi ≥ 0, i = 1, … , m, ur ≥ 0, r = 1, … , s, vo. is free in sign.

In Model (1), vi, i = 1, … , m, ur, r = 1, … , s are the weights of input and output components, respectively the Model's Dual (1) is as follows.

φo = max φo,

s. t. ∑j=1 λjo xij ≤ xio, i = 1, … , m,

∑j=1 λjo yrj ≥ φoyro, r = 1, … , s,

∑j=1 λjo = 1,

λjo ≥ 0, j = 1, … , n.

In Model (2), λjo shows the intensity variable. he φo is xxpansion variable of outputs.

Definition 1. DMU0 is efficient under VRS if φo = 1. Otherwise, it is inefficient.

In the allocation plan, we want to allocate the fixed cost allocation as R amount among the DMUj, j = 1, … , n. uppose the cost allocated to DMUj, j = 1, … , n is as Rj, j = 1, … , n. e put ∑j=1 Rj = R. e Consider the cost allocated to each DMU as a new input. In this case, Model (1) in the evaluation of DMU0 is as follows:

min ∑i=1 vi xio + vo + vm+1Ro, m s (3) s. t. ∑i=1 vi xij − ∑r=1 ur yrj + vo + vm+1Rj ≥ 0, j = 1, … , n, ∑r=1 ur yro = 1, ∑j=1 Rj = R,

vi ≥ 0, i = 1, … , m, Rj, j = 1, … , n,

ur ≥ 0, r = 1, … , s, vo is free in sign.

We consider vm+1 as the weight of the new input, namely Rj, j = 1, … , n. These variables are constant. When Model (3) is a linear programming, we will have a feasible non-allocation plan by considering vm+1 = 0. e put vm+1 > 0. The dual corresponding to Model (3) is as follows.

CRA∗ CRA

φo = max φo ,

s. t. ∑j=1 λjo xij ≤ xio, i = 1, … , m,

∑j=1 λjo Rj ≤ Ro, n (4) ∑j=1 λjo yrj ≥ φoyro, r = 1, … , s,

∑j=1 λjo = 1,

λjo ≥ 0, Rj ≥ o, j = 1, … , n.

Cook and Kress [1] presented the efficiency invariance principle. his principle states that the value of the Model's Objective Function (1) is equal to the value of the Model's Objective Function (3), or realizing the efficiency invariance principle when we solve the Model (3) using the simplex method the variable vm+1 must be out of the basis [2].

Definition 2. Metavariable vm+1 remains out of the basis n; namely t, the reduced cost is nonnegative. Then we will have

c v − z v = c v − cBB A ≥ 0

m+1 m+1 m+1

Then

n ∗ n ∗

Ro − ∑j=1 λjo Rj ≥ 0 or ∑j=1 λjo Rj ≤ Ro.

The variables λjo, j = 1, … , n, are the optimal dual variables of the Model (3) [1]. For the values of the Objective Functions of Models (1) and (3) to be equal, the second constraint in Model (4) must be redundant, meaning that λjo, j = 1, … , n in Relation (5) must be the optimal solution of Model (2). Then, Rj, j = 1, … , n should satisfy in the ∑j=1 λjo Rj ≤ Ro. This allocation is not unique. If the fixed cost is distributed among the inefficient DMUs completely in any proportion, then efficiency scores do not change, and the assumption of invariance is established [1]. Then, they defined another condition under the title of the Input Pareto-Minimality, which is as follows:

Definition 3. The input Pareto-minimality in fixed-cost allocation means that no cost can be transferred from one DMU to another without violating the invariance.

To meet efficiency invariance and the input pareto-minimality principle, the constraint ∑j=1 λjo Rj = Ro must be in place for all inefficient DMUs. Suppose (φo, λjo, j = 1, … , n) is an optimal solution of Model (2), then we define M = {j|λjo > 0 in the optimal solution of model (2)}. This set is used as a reference set for DMUo.

The other dual variables are equal to zero according to complementary slackness. Lin and Chen [10] showed that this economic interpretation of pareto-minimality for the equality constraints is unsuitable, and they considered it a practical feasibility assumption. his equation makes exceptions for the possible inefficiency (non-zero slack) from the cost allocation plane. Then we put ∑j∈M λjo Rj = Ro. his constraint ensures that the cost allocation is not completely distributed among inefficient DMUs. Put NE shows the set includes the index corresponding to the inefficient DMUs. Suppose we have an appropriate cost allocation plan of the

∗ ∗ n

form Rj, j = 1, … , n that ∑j∈M λjo Rj = Ro and λjo > 0, j ∈ M. Also, we must have ∑j=1 Rj = R. Considering the above, we seek to present a fair allocation plan based on value efficiency analysis.

In value efficiency analysis, the efficiency of DMUs is obtained based on a set of units called MPUs. These units are the units that, in the opinion of the manager, have the best performance. For this purpose, we evaluate DMUo based on Model (2). Suppose (φo, λjo, j = 1, … , n) is an optimal solution of Model (2). Almé et al. [22] proposed a Model (5) to calculate the value efficiency of DMUo as follows.

va∗ va

φo = max φo ,

s. t. ∑j=1 μjo xij ≤ xio, i = 1, … , m,

n va

∑j=1 μjo yrj − φo yro ≥ yro, r = 1, … , s, n (5) ∑j=1 μjo = 1, μjo, is free in sign if λjo > 0,

μjo ≥ 0, if λjo = 0.

va∗

Definition 4. DMU0 is value efficient under VRS if φo = 0. Otherwise it is inefficient.

The value efficiency score of DMU0 is defined as va∗.

1+φo

Suppose we show MPU units as MP. The set can be MP ⊆ M or selected from the set of observed DMUs based on the manager's opinion. The Model (5) is converted as follows.

va∗ va

φo = max φo ,

s. t. ∑j=1 μjo xij ≤ xio, i = 1, … , m,

n va

∑j=1 μjo yrj − φo yro ≥ yro, r = 1, … , s, n (6) ∑j=1 μjo = 1, μjo, is free in sign if j ∈ MP,

μjo ≥ 0, if j ∉ MP.

We first solve Model (2) for each of the DMUs. Suppose that the set of inefficient units is represented as NE. Suppose (φq, λjq, q ∈ NE) is an optimal solution of Model (2) for DMUq, q ∈ NE, e construct the reference set corresponding to the inefficient units (DMUq, q ∈ NE) as follows:

Mq = {j|λjq > 0 in the optimal solution of model (2) for DMUq, q ∈ NE}.

We can develop an algorithm for a fixed cost allocation plane based on the value efficiency analysis. For this purpose, we assume that (φ ̂ , μ ̂ , q ∈ NE) is an optimal solution of Model (6). so we put R ̅ = , e put MP ⊆

q jq q

Mq, q ∈ NE. To find the optimal fixed cost allocation, we propose the following model.

min ∑j=1|Rj − R|,

s. t. ∑j∈MP μ ̂jq Rj = Rq, q ∈ NE, ∑j=1 Rj = R,

Rj ≥ o, j = 1, … , n.

Model (7) is nonlinear because of its objective function. e can replace it with a linear objective function. Or example min . The proposed algorithm is as follows:

Step 1. Initially, we form a set MPq for q ∈ NE.

Step 2. Solve Model (6) and obtain (φ ̂q, μ ̂jq, q ∈ NE) as an optimal solution

Step 3. Solve Model (7) and obtain the fair fixed cost allocation plane.

A fixed cost allocation plan has a unique solution if the following linear equation system has a unique solution.

∑j∈MP μ ̂jq Rj = Rq, q ∈ NE,

∑j=1 Rj = R.

A bove linear equation system has |NE| + 1 equations and n variables.

We now present a numerical example previously used by Cook and Kress [1] to demonstrate the fixed cost allocation plan presented in this paper. Table 2 illustrates 12 DMUs that each have three inputs and two outputs. First, we solve Model (2) for each DMU to form the set NE.

According to column 7 in Table 2, DMUs 4, 5, 6, 8, 9, and 12 are efficient, and DMUs 1, 2, 3, 7, 10, and 11 are inefficient. Column 8 of Table 2 shows the set of DMUs in the reference set corresponding to each DMU. We assume that we have a cost fixed equal to 100 for allocation; to obtain the optimal allocation plan, we solve the Model (6) for inefficient DMUs. For this purpose, we form a set MP for inefficient DMUs, namely MPUs units, e set

MP1 = {DMU6, DMU8, DMU9, DMU12},MP2 = {DMU4, DMU5, DMU6, DMU8},

MP3 = {DMU5, DMU6, DMU12}, MP7 = {DMU5, DMU6}, MP10 = {DMU12}, MP11 = {DMU9}.

Table 3 shows the results of Model (6) for inefficient units. e obtain a cost allocation by the last column of Table 3. It can be easily seen that this cost allocation satisfies invariance and pareto-minimalitytions in Cook and Kress [1]. The valve valuesiency scores of DMUs are the same as before and after allocating fixed costs to DMUs according to Tables 2 and 3. Table 3 shows the results of Model (6) for inefficient units. e obtain a cost allocation by the last column of Table 3. This cost allocation can easily satisfy the invariance and pareto-minimalityonditions in Cook and Kress [1]. The efficiency scores of DMUs are measured before and after allocating fixed costs to DMUs according to Tables 2 and 3.

Now, for sensitivity analysis, we change the MPUs units corresponding to each of the inefficient units as follows:

MP1 = {DMU6, DMU8, DMU9, DMU12},MP2 = {DMU4, DMU5, DMU6, DMU8},

MP3 = {DMU4, DMU5, DMU6, DMU8}, MP7 = {DMU4, DMU5, DMU6, DMU8},

MP10 = {DMU6, DMU8, DMU9, DMU12},MP11 = { DMU6, DMU8, DMU9, DMU12}.

In this case, the cost allocated to the units is as follows.

∗ ∗ ∗ ∗ ∗ ∗ ∗

R1 = 0, R2 = 3.0684, R3 = 27.9827, R4 = 0, R5 = 11.8561, R6 = 0, R7 = 56.691,

∗ ∗ ∗ ∗ ∗

R8 = 0, R9 = 0, R10 = 0, R11 = 0.4019, R12 = 0.

And the value efficiency scors are as

va∗ va∗ va∗ va∗ va∗ va∗

φ1 = 0.7825, φ2 = 0.9453, φ3 = 0.7867, φ4 = 1, φ5 = 1, φ6 = 1,

va∗ va∗ va∗ va∗ va∗ va∗

φ7 = 0.678, φ8 = 1, φ9 = 1, φ10 = 0.7557, φ11 = 0.3324, φ12 = 1.

As can be seen, the allocated cost is allocated to both efficient and inefficient DMUs.

Now, we propose the results of fixed cost allocation based on the input-oriented model. According to column 7 in Table 4, DMUs 4, 5, 6, 8, 9, 11, and 12 are efficient, and DMUs 1, 2, 3, 7, and 10 are inefficient. Column 8 of Table 2 shows the set of DMUs in the reference set corresponding to each of the DMUs of the input-oriented model under VRS technology.

As we said before, we assume that we have a cost fixed equal to 100 for allocation; to obtain the optimal allocation plan, we solve the BCC model in the input-oriented for inefficient DMUs. For this purpose, we form set MP for inefficient DMUs, namely MPUs units, e set

MP1 = {DMU4, DMU8, DMU9},MP2 = {DMU4, DMU8},MP3 = {DMU5, DMU8, DMU9},

MP7 = {DMU4, DMU5}, MP10 = {DMU9}, MP11 = {DMU9}.

Table 5 shows the value efficiency of the BCC DEA model, which results in the input orientation for inefficient units. e obtain a cost allocation by the last column of Table 5. It can be easily seen that this cost allocation satisfies the invariance and pareto-minimality conditions in Cook and Kress [1]. The values efficiency scores of DMUs are the same before and after allocating fixed costs to DMUs according to Tables 4 and 5.

This paper proposes a DEA approach to cost allocation problems based on the value efficiency analysis. We can incorporate the DMs opinion into the fixed cost allocation process using value efficiency analysis. We proposed an algorithm for a fixed cost allocation plane using VRS technology. e also presents a new approach for fixed cost allocation based on the efficiency invariance strategy and value efficiency analysis. One of the strengths of the presented approach is that based on the DM's preferred information, we can allocate the fixed cost among all DMUs, including efficient and inefficient DMUs. We illustrate our models by providing a numerical example. e can extend the presented algorithm to obtain a unique cost allocation scheme. e can also develop models for the two-stage network structure in DEA.

The authors declare that there is no conflict of interest regarding the publication of this paper.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.