Assessing most productive scale size with fuzzy stochastic data envelopment analysis

Abstract

Classic data envelopment analysis (DEA) is a linear programming method for evaluating the relative efficiency of decision making units (DMUs) that uses multiple inputs to produce multiple outputs. In the classic DEA model inputs and outputs of DMUs are deterministic, while in the real world, are often fuzzy, random, or fuzzy-random. Many researchers have proposed different approaches to evaluate the relative efficiency with fuzzy and random data in DEA.

In many studies, the most productive scale size (mpss) of decision making units has been estimated with fuzzy and random inputs and outputs. Also, the concept of fuzzy random variable is used in the DEA literature to describe events or occurrences in which fuzzy and random changes occur simultaneously.

This paper has proposed the fuzzy stochastic DEA model to assess the most productive scale size of DMUs that produce multiple fuzzy random outputs using multiple fuzzy random inputs with respect to the possibility-probability constraints. For solving the fuzzy stochastic DEA model, we obtained a nonlinear deterministic equivalent for the probability constraints using chance constrained programming approaches (CCP). Then, using the possibility theory the possibilities of fuzzy events transformed to the deterministic equivalents with definite data. In the final section, the fuzzy stochastic DEA model, proposed model, has been used to evaluate the most productive scale size of sixteen Iranian hospitals with four fuzzy random inputs and two fuzzy random outputs with symmetrical triangular membership functions.

Keywords

Data envelopment analysis fuzzy-random most productive scale size hospitals

1 Introduction

Data envelopment analysis (DEA) is a non-parametric approach for evaluating the relative efficiency of decision making units (DMUs) that utilizes multiple inputs to generate multiple outputs. Charnes et al. [1] presented the first DEA model to evaluate the relative efficiency of DMUs, known as the CCR model. This model assumes that DMUs have constant return to scale (CRT) in their inputs and outputs. That is, by multiplying the inputs by a constant amount, the outputs increase or decrease in the same proportion. Six years later, Banker and his colleagues [30] expanded the CCR model to the state of variable returns to scale (VRS), so their model was called the BCC model.

One of the advantages of the DEA method is that it is not sensitive to the unit of measurement and the inputs and outputs can have different units. Also, it is a management method that measures the efficiency of DMUs relatively. One of the weaknesses of the DEA method is that if one of the inputs or outputs of DMUs changes, the efficiency of the DMUs will change. Banker [29] proposed a new model with fractional objective function for determining the most productive scale size (mpss) in DMUs. The most productive scale size is a benchmark that determines how resources are used to achieve maximum outputs. In addition, the most productive scale size is a criterion that indicates an organization has been successful in producing a product or providing services to customers or not.

Jahanshahloo and Khodabakhshi [12] have proposed an input-output orientation model to estimate mpss as a special case of returns to scale with deterministic data. Lee [9] proposed a multi-objective mathematical program with the constraints of data envelopment analysis approach to set an efficient target that display a trade-off between the most productive scale size benchmark and a potential demand fulfillment benchmark. Moghaddas and Ghasemi [39] have computed the most productive scale size in the data envelopment analysis method with real and integer value data.

The most productive scale size (mpss) has been studied by many researchers, for example, you can refer to Eslami et al. [32], Sahoo et al. [6], and Koushki [11]. In the classic DEA model, it is assumed that the inputs and outputs of DMUs are deterministic, while in the real world problems, the inputs and outputs of the organizations under study are often fuzzy, random, or fuzzy and random simultaneously. Khodabakhshi [21] has applied the approach of stochastic DEA to estimate mpss in decision making units (DMUs). Behzadi and Mirbolouki [20] have used a linear form of stochastic CCR to evaluate the efficiency of 20 branches of an Iranian bank with stochastic inputs and outputs by introducing symmetric error structure.

The main advantage of working with random data in DEA is that the efficiency of DMUs can be predicted in the future. Minh and Khanh [25] utilize the approach of chance constrained DEA with two models A and B for decomposing of provincial productivity growth in Vietnamese agriculture in the period 1995-2007 into two components: technological progress and efficiency change. Dibachi [13] introduced a stochastic multiplicative DEA model to estimate the most productive scale size in DMUs. Kheirollahi et al. [14] have developed an input relaxation model in stochastic data to identify the congestion of Iranian hospitals with chance constrained programming approaches in data envelopment analysis (DEA).

Just as the random variable is used in the probability theory to describe random changes, in the possibility theory proposed by Zadeh [19], the fuzzy variable is used to describe some uncertainties and ambiguities in some real situations. Sengupta [17] was the first to consider fuzzy changes in both the objective function and constraints in the fuzzy DEA model. Peykani et al. [28] have introduced a fuzzy DEA method to evaluate the efficiency and ranking of the Tehran stock exchange. Kheirollahi et al. [15] develop an input relaxation model to identify the congestion of hospitals. They utilize the possibility approach to obtain deterministic equivalents of the fuzzy models. To become more familiar with fuzzy DEA models refer to Ebrahimnejad and Amani [3], Lertworasirikul et al. [35], Ruiz and Sirvent [18], and Guo and Tanaka [27].

Khodabakhshi, Gholami, and Kheirollahi [22] have used an additive model to provide an alternative approach for estimating returns to scale (RTS) in both stochastic and fuzzy inputs and outputs in DEA. This article uses the CCP approaches and the fuzzy possibility approach to obtain the deterministic equivalents of stochastic and fuzzy models, respectively. Since in the real world, fuzzy and random uncertainties sometimes occur together, the inputs and outputs of DMUs are defined as fuzzy-random variables to indicate uncertainty. Kwakernaak [16] first defined the fuzzy random variable to explain the events in which fuzzy and random changes occur, simultaneously. Tavana et al. [23] provided three DEA models to evaluate the radial efficiency of DMUs with fuzzy random inputs and outputs and Poisson, uniform, and normal distributions. Nasseri et al. [34] present the fuzzy stochastic DEA model when the outputs of decision making units (DMUs) are undesirable.

There has been a lot of research on DEA with the data of fuzzy and random together, for instance, see Liu and Liu [38], Yaghoubi et al. [5], Gholami [26], Tavana et al. [24], Ebrahimnejad et al. [2], Ebrahimnejad, Nasseri, Gholami [4]. In this article, we intend to estimate the most productive scale size (mpss) of decision making units (DMUs) using the fuzzy stochastic DEA model with fuzzy random inputs and outputs. Therefore, this paper proposes the probability-possibility approach for solving the mpss fuzzy stochastic DEA model. We obtained a nonlinear deterministic equivalent for the probability constraints using CCP approaches. Then, using the possibility theory proposed by Zadeh [19], the possibilities of fuzzy events transformed to deterministic equivalents.

The remainder of the article has been compiled as follows: In the next section, the classic DEA model for estimating the most productive scale size and its definition are presented. In Section 3, the most productive scale size is developed in fuzzy stochastic DEA, and its deterministic equivalent is also obtained. In section 4, the possibility approach and using this approach to find the deterministic equivalent of the fuzzy model are described. Section 5 presents an application to illustrate the proposed model. Section 6 concludes the paper and suggests future research directions.

2 Preliminaries

We assume that there are n decisions-making units, DMU_j, j = 1, 2, . . . , n, each of these decision making units (DMUs) generates s nonnegative different outputs y_rj (r = 1, 2, . . . , s), using m nonnegative different inputs x_ij (i = 1, 2, . . . , m). One of the basic models used to evaluate DMUs efficiency is the most productive scale size model presented by Jahanshahloo and Khodabakhshi [12]. This model which evaluates most productive scale size in DMU_o is as follows: $\begin{matrix} Maximize φ - θ + ɛ (\sum_{i = 1}^{m} s_{i}^{-} + \sum_{r = 1}^{s} s_{r}^{+}) \\ subject . to \\ \sum_{j = 1}^{n} λ_{j} x_{ij} + s_{i}^{-} = θ x_{io}, i = 1, 2, \dots, m, \\ \sum_{j = 1}^{n} λ_{j} y_{rj} - s_{r}^{+} = φ y_{ro}, r = 1, 2, \dots, s, \\ \sum_{j = 1}^{n} λ_{j} = 1, \\ s_{i}^{-}, s_{r}^{+}, λ_{j} \geq 0; i = 1, 2, \dots, m, r = 1, 2, \dots, \\ s, j = 1, 2, \dots, n . \end{matrix}$ (1) In above model, the symbol j = o nominates one of the DMU_j, j = 1, 2, . . . , n, as the DMU_o to be evaluated relative to other DMUs. Since φ = θ = 1, λ_o = 1 and λ_j = 0 (j ≠ o) is the feasible solution of model (1), wherein the amount of the objective function is equal to zero, the optimal value of the objective function is always nonnegative. In other words, we always have φ^* - θ^* ≥ 0 or φ^* ≥ θ^*. The term “ɛ > 0” in the objective objective of model (1) is a non-Archimedean element which is smaller than any positive real number that is used in order to optimize slacks and ensure that all slacks are considered in the optimal amount.

Definition 1 (Most productive scale size): DMU_o, decision-making unit under evaluation, is most productive scale size if and only if the two following conditions are satisfied for model (1):

φ^* = θ^*

$s_{i}^{- *} = s_{r}^{+ *} = 0$ , for i = 1, 2, . . . , m and r = 1, 2, . . . , s.

In this paper, the symbol * is used to determine an optimal solution. DMU_o is weakly most productive scale size if only condition (1) is satisfied.

3 Most productive scale size model with fuzzy random variables

In this section, the data of inputs and outputs of decision making units (DMUs), which are definite numbers in conventional DEA models, are replaced by fuzzy random data. Now, we use model (1) which is used to evaluate most productive scale size (mpss) with real inputs and outputs, to estimate mpss with fuzzy random inputs and outputs. The DEA models with fuzzy random inputs and outputs have been used by many researchers to evaluate the efficiency of DMUs. For more information about fuzzy random variables can refer to, Punyangarm et al. [36] and Qin and Liu [33].

Here, following Punyangarm et al. [36] suppose that ${\tilde{\hat{x}}}_{j} = ({\tilde{\hat{x}}}_{1 j}, {\tilde{\hat{x}}}_{2 j}, \dots, {\tilde{\hat{x}}}_{mj})^{T}$ and ${\tilde{\hat{y}}}_{j} = ({\tilde{\hat{y}}}_{1 j}$ , ${\tilde{\hat{y}}}_{2 j}, \dots, {\tilde{\hat{y}}}_{sj})^{T}$ represent the fuzzy random inputs and outputs victors of DMU_j,j = 1, 2, …, n, respectively. Also, following [24] and [23] suppose ${\tilde{\hat{x}}}_{ij} = (x_{ij}^{m}, x_{ij}^{α}, x_{ij}^{β})$ , i = 1, 2, . . . , m, and ${\tilde{\hat{y}}}_{rj} =$ $(y_{rj}^{m}, y_{rj}^{α}, y_{rj}^{β})$ , r = 1, 2, . . . , s, are triangular fuzzy numbers that $x_{ij}^{m}$ and $y_{rj}^{m}$ are random variables with normal distribution and be shown with $x_{ij}^{m} \sim ({\bar{x}}_{ij}, σ_{ij}^{2})$ and $y_{rj}^{m} \sim ({\bar{y}}_{rj}, σ_{rj}^{2})$ . So, ${\bar{x}}_{ij}$ , ${\bar{y}}_{rj}$ and $σ_{ij}^{2}$ , $σ_{rj}^{2}$ are the means and the variances of $x_{ij}^{m}$ and $y_{rj}^{m}$ , respectively. Also, $x_{ij}^{α}$ , $y_{rj}^{α}$ and $x_{ij}^{β}$ , $y_{rj}^{β}$ are the left and right arms of triangular fuzzy random numbers ${\tilde{\hat{x}}}_{ij}$ and ${\tilde{\hat{y}}}_{rj}$ , respectively.

The chance constrained programming approaches developed by Cooper et al. [37] is a stochastic optimization approach for solving optimization problems with uncertain parameters. Based on CCP approaches and the theory of possibility of fuzzy events as the principal techniques, we propose the possibility-probability of the most productive scale size of model (1) as follows:

$\begin{matrix} Maximize φ - θ \\ subject . to \\ π (P (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} \leq θ {\tilde{\hat{x}}}_{io}) \geq α_{i}) \geq δ_{i}, i = 1, 2, \dots, m, \\ π (P (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} \geq φ {\tilde{\hat{y}}}_{ro}) \geq β_{r}) \geq {\bar{δ}}_{r}, r = 1, 2, \dots, s, \\ \sum_{j = 1}^{n} λ_{j} = 1, \\ θ, φ unrestricted; λ_{j} \geq 0; j = 1, 2, \dots, n . \end{matrix}$ (2) In the above model, π is the possibility measure from P (A_i) to [0,1], in which P (A_i) is the set of all subsets of A_i and P is the probability of an accidental event. Also α_i and β_r are pre-specified minimum probability and δ_i and ${\bar{δ}}_{r}$ are pre-specified acceptable values of possibility in which α_i, δ_i, β_r and ${\bar{δ}}_{r}$ are values between 0 and 1 which determine the significance level of the constraints.

Definition 2 (Fuzzy stochastic mpss): DMU_o, decision making unit under evaluation, is fuzzy stochastic most productive scale size (fsmpss) if and only if the two-following conditions for model (3) are satisfied:

φ^* = θ^*

For all optimal solutions, slack values are all zeros.

By shifting

θ {\tilde{\hat{x}}}_{io}

and

φ {\tilde{\hat{y}}}_{ro}

to the left side of inequalities in the constraints of model (2) have:

$\begin{matrix} Maximize φ - θ \\ subject . to \\ π (P (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io}) \leq 0) \geq α_{i}) \geq δ_{i}, (a) i = 1, 2, \dots, m, \\ π (P (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro} \geq 0) \geq β_{r}) \geq {\bar{δ}}_{r}, (b) r = 1, 2, \dots, s, \\ \sum_{j = 1}^{n} λ_{j} = 1, \\ θ, φ unrestricted; λ_{j} \geq 0; j = 1, 2, \dots, n . \end{matrix}$ (3)

Now, to obtain a deterministic equivalent of the chance-constrained of inputs (a) in model (3) do the following:

$\begin{matrix} P (\frac{\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io} - E (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io})}{\sqrt{V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io})}} \\ \leq \frac{0 - E (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io})}{\sqrt{V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io})}}) \geq α_{i}, (a 1) \end{matrix}$ We define the variable Z_i as follows:

$\begin{matrix} Z_{i} = \frac{\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io} - E (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io})}{\sqrt{V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io})}}, \\ i = 1, 2, \dots, m, \end{matrix}$ Whereas E (Z_i) =0 and V (Z_i) =1. Therefore Z_i, i = 1, 2, . . . , m, is the standard normal random variable. Therefore, (a1) can be written as follows: $\begin{matrix} P (Z_{i} \leq \frac{- E (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io})}{\sqrt{V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io})}}) \geq α_{i}, \\ i = 1, 2, \dots, m, (a 2) \end{matrix}$ From inequality (a2) have $\begin{matrix} Φ (\frac{- E (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io})}{\sqrt{V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io})}}) \geq α_{i}, \\ i = 1, 2, \dots, m, (a 3) \end{matrix}$

In inequality (a3) have $\begin{matrix} [σ_{i}^{I} (θ, λ)] 2 = V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io}) = \\ \sum_{k \neq o} \sum_{j \neq o} λ_{k} λ_{j} Cov ({\tilde{\hat{x}}}_{ik}, {\tilde{\hat{x}}}_{ij}) + 2 (λ_{o} - θ) \\ \sum_{k \neq o} λ_{k} Cov ({\tilde{\hat{x}}}_{ik}, {\tilde{\hat{x}}}_{io}) + {(λ_{o} - θ)}^{2} V ({\tilde{\hat{x}}}_{io}) \end{matrix}$ Also, in inequality (a3), Φ is the standard normal cumulative distribution function, so it is invertible and its inverse is called Φ^-1. So we can write (a3) as follows:

$\begin{matrix} - E (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io}) \geq Φ^{- 1} (α_{i}) σ_{i}^{I} (θ, λ), \\ i = 1, 2, \dots, m, (a 4) \end{matrix}$

As we’ve mentioned before that $E ({\tilde{\hat{x}}}_{ij}) = {\tilde{\bar{x}}}_{ij}, i = 1, 2, \dots, m,$ using the summation property of the mathematical expectation (E) in which for any two random variable X and Y and any two real numbers a and b have E (aX + bY) = aE (X) + bE (Y), inequality (a4) will be written as follows: $\begin{matrix} \sum_{j = 1}^{n} λ_{j} {\tilde{\bar{x}}}_{ij} - θ {\tilde{\bar{x}}}_{io} + Φ^{- 1} (α_{i}) σ_{i}^{I} (θ, λ) \leq 0, \\ i = 1, 2, \dots, m, (a 5) \end{matrix}$

Similar to what we did for the inputs constraints of model (3), for the output constraints in model (3) have: $\begin{matrix} P (\frac{\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro} - E (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro})}{\sqrt{V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro})}} \\ \leq \frac{0 - E (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro})}{\sqrt{V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro})}}) \geq β_{r}, (b 1) \end{matrix}$

Again, the variable Z_r will be defined as follows: $\begin{matrix} Z_{r} = \frac{\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro} - E (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro})}{\sqrt{V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro})}}, \\ r = 1, 2, \dots, s, \end{matrix}$

Then, E (Z_r) =0 and V (Z_r) =1. So, Z_r, r = 1, 2, . . . , s, is the standard normal random variable. Therefore, inequality (b1) can be written as below: $\begin{matrix} P (Z_{r} \geq \frac{- E (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro})}{\sqrt{V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro})}}) \geq β_{r}, \\ r = 1, 2, \dots, s, (b 2) \end{matrix}$

Using P (Z_r ≥ z) =1 - P (Z_r ≤ z), inequality (b2) will be converted to

$\begin{matrix} 1 - P (Z_{r} \leq \frac{- E (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro})}{\sqrt{V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro})}}) \geq β_{r}, \\ r = 1, 2, \dots, s, (b 3) \end{matrix}$

Consequently, have $\begin{matrix} Φ (\frac{- E (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro})}{\sqrt{V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro})}}) \leq 1 - β_{r}, \\ r = 1, 2, \dots, s, (b 4) \end{matrix}$

Again, since Φ is the standard normal cumulative distribution function, so it has an inverse and it’s inverse is called Φ^-1. Thus, we can write inequality (b4) as below: $\begin{matrix} \frac{- E (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro})}{\sqrt{V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro})}} \leq Φ^{- 1} (1 - β_{r}), \\ r = 1, 2, \dots, s, (b 5) \end{matrix}$

Then $\begin{matrix} E (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro}) + Φ^{- 1} (1 - β_{r}) σ_{r}^{o} (φ, λ) \\ \geq 0, r = 1, 2, \dots, s, (b 6) \end{matrix}$

In the above inequality, we have $\begin{matrix} [σ_{r}^{o} (φ, λ)]^{2} = V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro}) = \\ \sum_{k \neq o} \sum_{j \neq o} λ_{k} λ_{j} Cov ({\tilde{\hat{y}}}_{rk}, {\tilde{\hat{y}}}_{rj}) + 2 (λ_{o} - φ) \\ \sum_{k \neq o} λ_{k} Cov ({\tilde{\hat{y}}}_{rk}, {\tilde{\hat{y}}}_{ro}) + {(λ_{o} - φ)}^{2} V ({\tilde{\hat{y}}}_{ro}) \end{matrix}$

Again, we assume $E ({\tilde{\hat{y}}}_{rj}) = {\tilde{\bar{y}}}_{rj}, r = 1, 2, \dots, s,$ using the summation property of the mathematical expectation (E), inequality (b6) is written as follows: $\begin{matrix} \sum_{j = 1}^{n} λ_{j} {\tilde{\bar{y}}}_{rj} - φ {\tilde{\bar{y}}}_{ro} + Φ^{- 1} (1 - β_{r}) σ_{r}^{o} (φ, λ) \geq 0, \\ r = 1, 2, \dots, s, (b 7) \end{matrix}$

Using inequalities (a5) and (b7), model (3) will be converted as follows: $\begin{matrix} Maximize φ - θ \\ subject . to \\ π (\sum_{j = 1}^{n} λ_{j} {\tilde{\bar{x}}}_{ij} - θ {\tilde{\bar{x}}}_{io} + Φ^{- 1} (α_{i}) σ_{i}^{I} (θ, λ) \leq 0) \geq α_{i}) \\ \geq δ_{i}, i = 1, 2, \dots, m, \\ π (\sum_{j = 1}^{n} λ_{j} {\tilde{\bar{y}}}_{rj} - φ {\tilde{\bar{y}}}_{ro} + Φ^{- 1} (1 - β_{r}) σ_{r}^{o} (φ, λ) \geq 0 .) \\ \geq β_{r}) \geq {\bar{δ}}_{r}, r = 1, 2, \dots, s, \\ \sum_{j = 1}^{n} λ_{j} = 1, \\ [σ_{i}^{I} (θ, λ)]^{2} = V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io}) \\ = \sum_{k \neq o} \sum_{j \neq o} λ_{k} λ_{j} Cov ({\tilde{\hat{x}}}_{ik}, {\tilde{\hat{x}}}_{ij}) + 2 (λ_{o} - θ) \\ \sum_{k \neq o} λ_{k} Cov ({\tilde{\hat{x}}}_{ik}, {\tilde{\hat{x}}}_{io}) + {(λ_{o} - θ)}^{2} V ({\tilde{\hat{x}}}_{io}) \\ [σ_{r}^{o} (φ, λ)]^{2} = V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro}) \\ = \sum_{k \neq o} \sum_{j \neq o} λ_{k} λ_{j} Cov ({\tilde{\hat{y}}}_{rk}, {\tilde{\hat{y}}}_{rj}) + 2 (λ_{o} - φ) \\ \sum_{k \neq o} λ_{k} Cov ({\tilde{\hat{y}}}_{rk}, {\tilde{\hat{y}}}_{ro}) + {(λ_{o} - φ)}^{2} V ({\tilde{\hat{y}}}_{ro}) \\ θ, φ unrestricted; λ_{j} \geq 0; j = 1, 2, \dots, n . \end{matrix}$ (4)

We let $V ({\tilde{\hat{x}}}_{ij}) = {\tilde{a}}_{ij}^{2}$ and $V ({\tilde{\hat{y}}}_{rj}) = {\tilde{b}}_{rj}^{2}$ , therefore ${\tilde{a}}_{ij}$ and ${\tilde{b}}_{rj}$ indicate the fuzzy standard deviation of fuzzy input and output variables, respectively. Also, we put $ζ_{ij} = \frac{{\tilde{\hat{x}}}_{ij} - E ({\tilde{\hat{x}}}_{ij})}{\sqrt{V ({\tilde{\hat{x}}}_{ij})}}$ and ${\bar{ζ}}_{rj} = \frac{{\tilde{\hat{y}}}_{rj} - E ({\tilde{\hat{y}}}_{rj})}{\sqrt{V ({\tilde{\hat{y}}}_{rj})}},$ so ζ_ij and ${\bar{ζ}}_{rj}$ that are wrong statements have the standard normal distribution. From above relations we conclude that the statements of ${\tilde{a}}_{ij} ζ_{ij}$ and ${\tilde{b}}_{rj} {\bar{ζ}}_{rj}$ show the error of fuzzy inputs and outputs, respectively, therefore we can write ${\tilde{\hat{x}}}_{ij} = {\tilde{\bar{x}}}_{ij} + {\tilde{a}}_{ij} ζ_{ij}$ and ${\tilde{\hat{y}}}_{rj} = {\tilde{\bar{y}}}_{rj} + {\tilde{b}}_{rj} {\bar{ζ}}_{rj}$ .

Since $V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{x}}}_{ij} - θ {\tilde{\hat{x}}}_{io}) = (\sum_{j = 1}^{n} λ_{j} {\tilde{a}}_{ij} - θ {\tilde{a}}_{io})^{2} = [σ_{i}^{I} (θ, λ)]^{2}$ and $V (\sum_{j = 1}^{n} λ_{j} {\tilde{\hat{y}}}_{rj} - φ {\tilde{\hat{y}}}_{ro}) = (\sum_{j = 1}^{n} λ_{j} {\tilde{b}}_{rj} - φ {\tilde{b}}_{ro})^{2} = [σ_{r}^{o} (φ, λ)]^{2},$ model (4) will be written as follow:

$\begin{matrix} Maximize φ - θ \\ subject . to \\ π (\sum_{j = 1}^{n} λ_{j} ({\tilde{\bar{x}}}_{ij} + Φ^{- 1} (α_{i}) {\tilde{a}}_{ij}) . \\ . - θ ({\tilde{\bar{x}}}_{io} + Φ^{- 1} (α_{i}) {\tilde{a}}_{io}) \leq 0) \\ \geq δ_{i}, i = 1, 2, \dots, m, \\ π (\sum_{j = 1}^{n} λ_{j} ({\tilde{\bar{y}}}_{rj} + Φ^{- 1} (1 - β_{r}) {\tilde{b}}_{rj}) . \\ . - φ ({\tilde{\bar{y}}}_{ro} + Φ^{- 1} (1 - β_{r}) {\tilde{b}}_{ro}) \geq 0) \geq {\bar{δ}}_{r}, \\ r = 1, 2, \dots, s, \\ θ, φ unrestricted; λ_{j} \geq 0; j = 1, 2, \dots, n . \end{matrix}$ (5)

If we put ${\tilde{x}}_{ij} = {\tilde{\bar{x}}}_{ij} + Φ^{- 1} (α_{i}) {\tilde{a}}_{ij}, i = 1, 2, \dots, m,$ and ${\tilde{y}}_{rj} = {\tilde{\bar{y}}}_{rj} + Φ^{- 1} (1 - β_{r}) {\tilde{b}}_{rj}, r = 1, 2, \dots, s,$ model (5) will be written as follows: $\begin{matrix} Maximize φ - θ \\ subject . to \\ π (\sum_{j = 1}^{n} λ_{j} {\tilde{x}}_{ij} - θ {\tilde{x}}_{io} \leq 0) \geq δ_{i}, i = 1, 2, \dots, m, \\ π (\sum_{j = 1}^{n} λ_{j} {\tilde{y}}_{rj} - φ {\tilde{y}}_{ro} \geq 0) \geq {\bar{δ}}_{r}, r = 1, 2, \dots, s, \\ θ, φ unrestricted; λ_{j} \geq 0; j = 1, 2, \dots, n . \end{matrix}$ (6) In next section, we will introduce the possibility theory to solve model (6) in which ${\tilde{x}}_{ij}, i = 1, 2, \dots, m,$ and ${\tilde{y}}_{rj}, r = 1, 2, \dots, s,$ are trapezoidal fuzzy numbers.

4 The possibility approach

Professor Zadeh [19] provided the possibility theory based on the fuzzy sets theory in mathematics. The possibility theory has been developed in mathematics and in other branches of sciences by many researchers. For finding a good reference on the possibility theory you can refer to Dubois and Prade [10]. The possibility theory as a new mathematical subject analyzes many vague and inaccurate concepts in real-world problems. Zadeh defined the possibility theory as a base for the theory of fuzzy sets as the measure theory has been introduced as a base for the probability theory. He proposed the fuzzy variable with the possibility distribution as the random variable is associated with the probability distribution.

In a fuzzy programming problem, each coefficient is a fuzzy variable and each constraint is a fuzzy event or fuzzy accident. Therefore, using the possibility theory, the possibilities of fuzzy events will be determined. However, to make a reasonable efficiency comparison of decision making units (DMUs), the possibility levels of constraints in (6), should be set at the same level. The following lemma, introduced by [35] is useful to find the optimal solution of the possibility model (6) in which inputs and outputs variables are trapezoidal fuzzy numbers.

Lemma 1: Suppose ${\tilde{a}}_{1}, {\tilde{a}}_{2}, . . ., {\tilde{a}}_{n}$ be fuzzy variables with normal and convex membership functions. Also, suppose that $(.)_{α_{j}}^{L}$ and $(.)_{α_{j}}^{U}$ be the lower and upper bounds of the α-level set of ${\tilde{a}}_{i}, i = 1, 2, . . ., m,$ respectively. Then, for three possibility levels α₁, α₂ and α₃ with 0 ≤ α₁, α₂, α₃ ≤ 1, have:

$π ({\tilde{a}}_{1} + {\tilde{a}}_{2} + . . . + {\tilde{a}}_{n} \leq b) \geq α_{1}$ if and only if $({\tilde{a}}_{1})_{α_{1}}^{L} + ({\tilde{a}}_{2})_{α_{1}}^{L} + . . . + ({\tilde{a}}_{n})_{α_{1}}^{L} \leq b .$

$π ({\tilde{a}}_{1} + {\tilde{a}}_{2} + . . . + {\tilde{a}}_{n} \geq b) \geq α_{2}$ if and only if $({\tilde{a}}_{1})_{α_{2}}^{U} + ({\tilde{a}}_{2})_{α_{2}}^{U} + . . . + ({\tilde{a}}_{n})_{α_{2}}^{U} \geq b .$

$π ({\tilde{a}}_{1} + {\tilde{a}}_{2} + . . . + {\tilde{a}}_{n} = b) \geq α_{3}$ if and only if $({\tilde{a}}_{1})_{α_{3}}^{L} + ({\tilde{a}}_{2})_{α_{3}}^{L} + . . . + ({\tilde{a}}_{n})_{α_{3}}^{L} \leq b and$

$({\tilde{a}}_{1})_{α_{3}}^{U} + ({\tilde{a}}_{2})_{α_{3}}^{U} + . . . + ({\tilde{a}}_{n})_{α_{3}}^{U} \geq b .$

According to lemma 1, and Liu [8], for a trapezoidal fuzzy number ${\tilde{r}}_{i} = [({\tilde{r}}_{i})_{0}^{L}, ({\tilde{r}}_{i})_{1}^{L}, ({\tilde{r}}_{i})_{1}^{U}, ({\tilde{r}}_{i})_{0}^{U}], i = 1, 2, . . ., m,$ and any given possibility level α,0 ≤ α ≤ 1, the following expressions are confirmed:

$π ({\tilde{r}}_{1} + {\tilde{r}}_{2} + . . . + {\tilde{r}}_{n} \leq b) \geq α$ if and only if

$(1 - α) [({\tilde{r}}_{1})_{0}^{L} + ({\tilde{r}}_{2})_{0}^{L} + . . . + ({\tilde{r}}_{n})_{0}^{L}] + α [({\tilde{r}}_{1})_{1}^{L} + ({\tilde{r}}_{2})_{1}^{L} + . . . + ({\tilde{r}}_{n})_{1}^{L}] \leq b .$

$π ({\tilde{r}}_{1} + {\tilde{r}}_{2} + . . . + {\tilde{r}}_{n} \geq b) \geq α$ if and only if

$(1 - α) [({\tilde{r}}_{1})_{0}^{U} + ({\tilde{r}}_{2})_{0}^{U} + . . . + ({\tilde{r}}_{n})_{0}^{U}] + α [({\tilde{r}}_{1})_{1}^{U} + ({\tilde{r}}_{2})_{1}^{U} + . . . + ({\tilde{r}}_{n})_{1}^{U}] \geq b .$

$π ({\tilde{r}}_{1} + {\tilde{r}}_{2} + . . . + {\tilde{r}}_{n} = b) \geq α$ if and only if

Therefore, when inputs and outputs of DMUs are trapezoidal fuzzy numbers, by adding the slacks of the inputs and outputs constraints, simultaneously, model (6) becomes as follows: $\begin{matrix} Maximize φ - θ + ɛ (\sum_{i = 1}^{m} s_{i}^{-} + \sum_{r = 1}^{s} s_{r}^{+}) \\ Subject to \\ (1 - δ_{i}) (\sum_{j = 1}^{n} λ_{j} {\tilde{x}}_{ij} - θ {\tilde{x}}_{io})_{0}^{L} \\ + α_{i} (\sum_{j = 1}^{n} λ_{j} {\tilde{x}}_{ij} - θ {\tilde{x}}_{io})_{1}^{L} + s_{i}^{-} = 0, \\ i = 1, \dots, m \\ (1 - {\bar{δ}}_{r}) (\sum_{j = 1}^{n} λ_{j} {\tilde{y}}_{rj} - φ {\tilde{y}}_{ro})_{0}^{U} \\ + {\bar{δ}}_{r} (\sum_{j = 1}^{n} λ_{j} {\tilde{y}}_{rj} - φ {\tilde{y}}_{ro})_{1}^{U} \\ - s_{r}^{+} = 0, r = 1, \dots, s \\ \sum_{j = 1}^{n} λ_{j} = 1, \\ θ, φ unrestricted; s_{i}^{-}, s_{r}^{+}, λ_{j} \geq 0, \\ i = 1, \dots, m; r = 1, \dots, s; j = 1, \dots, n . \end{matrix}$ (7)

Definition 3 (Fuzzy mpss): According to the fuzzy model (7), DMU_o, the DMU under evaluation, is fuzzy most productive scale size (fmpss) if and only if (iff) the two-following conditions are satisfied:

φ^* = θ^*

$s_{i}^{- *} = s_{r}^{+ *} = 0$ , for i = 1, 2, . . . , m and r = 1, 2, . . . , s.

5 Application

Hospitals, as the main organizations providing health care services, have a special role and importance in the health economy. Paying attention to the health sector has led to the social, political, and cultural development of all societies and plays an important role in the infrastructure of various sections of society. Increasing the costs of the healthcare sector in Iran, and expanding technology, and increasing the population, make it necessary to pay attention to economic concepts in this area.

The non-parametric DEA method, unlike the parametric methods, does not require a production function such as the Cobb Douglas production function and can evaluate the relative efficiency of homogeneous decision making units (DMUs) without assigning prior weights to input and output variables. While the parametric methods for evaluating the performance of a unit apply multiple inputs to generate only one output, the nonparametric methods such as DEA evaluate the relative efficiency of DMUs that apply multiple inputs to generate multiple outputs. However, the appropriate choice of input and output variables of DMUs has an important role in evaluating their performance with DEA approach. Also, an issue to consider in using DEA method is the relationship between the number of DMUs and the number of inputs and outputs. Dyson et al. [31] have proposed that the number of DMUs should be greater than or equal to twice the product of the number of input and output variables. Also, Golany and Roll [7], have recommended that the number of DMUs should be at least twice the summation of the number of inputs and outputs.

In this section, we use the resent model (model (7)) to evaluate the most productive scale size of sixteen Iranian hospitals with four fuzzy random inputs and two fuzzy random outputs. As the above references have mentioned, the evaluating of sixteen hospitals with this number of inputs and outputs is satisfactory. The data of these four fuzzy random inputs and two fuzzy random outputs are presented in Table 1. These fuzzy random inputs and outputs have symmetrical triangular membership functions that are denoted by (X ; d), in which X as a random variable with normal distribution function, $X \sim N (\bar{x}, σ^{2})$ , is the center and d is the left and right spread of the fuzzy random variable of (X ; d) that demonstrate the degree of fuzziness. If d = 0, the fuzzy random variable (X ; d) converted to the normal random variable X.

In this article, the inputs are doctors working hours (I1), nurses working hours (I2), technical workers working hours (I3) and office staff’s working hours (I4). The outputs are total medical insurance points for outpatients (O1) and total medical insurance points for inpatients (O2). Let α_i = β_r = 0.95, i = 1, 2, 3, 4 and r = 1, 2, then Φ^-1 (α_i) =1.645 and Φ^-1 (1 - β_r) = -1.645 and $δ_{i} = {\bar{δ}}_{r} = 0.75$ , i = 1, 2, 3, 4 and r = 1, 2. The results of using model (7), which is equivalent to the fuzzy-random model (3) to determine most productive scale size (mpss) sixteen hospitals at the possibility level $δ_{i} = {\bar{δ}}_{r} = 0.75$ , i = 1, 2, 3, 4 and r = 1, 2 are presented in Table 2.

Table 2
The results of using model (7) with fuzzy random data of Table 1

Hospitals φ^* - θ^* φ^* θ^* $s_{1}^{- }$ $s_{2}^{- }$ $s_{3}^{- }$ $s_{4}^{- }$ $s_{1}^{+ }$ $s_{2}^{+ }$

H1 0 1.00 1.00 0 0 0 0 0 0

H2 0 1.00 1.00 0 0 0 0 0 0

H3 1.15 2.38 1.23 1109.5 0 1547.9 0 0 0

H4 0.65 2.68 2.03 79.4 723.7 0 0 0 0

H5 0 1.00 1.00 0 0 0 0 0 0

H6 0.30 1.18 0.88 5.57 405.1 0 455.9 0 0

H7 0 1.00 1.00 0 0 0 0 0 0

H8 2.68 4.226 1.546 10.035 1212.6 683.16 0 0 0

H9 0.02 0.92 0.90 0 1936 151.8 0 0 0

H10 0.11 0.77 0.66 338.7 0 220.2 304.4 0 84.6

H11 0.03 0.71 0.68 0 1112.2 89.62 0 0 0

H12 0 0.76 0.76 0 0 0 220.9 0 0

H13 0.07 0.70 0.63 162.8 0 82.72 223.96 0 0

H14 0.17 1.33 1.16 0 2680.5 3009.7 0 0 0

H15 0 1.00 1.00 0 0 0 0 0 0

H16 0 1.00 1.00 0 0 0 0 0 0

Hospitals	φ^* - θ^*	φ^*	θ^*	$s_{1}^{- *}$	$s_{2}^{- *}$	$s_{3}^{- *}$	$s_{4}^{- *}$	$s_{2}^{+ *}$
H1	0	1.00	1.00	0	0	0	0	0
H2	0	1.00	1.00	0	0	0	0	0
H3	1.15	2.38	1.23	1109.5	0	1547.9	0	0
H4	0.65	2.68	2.03	79.4	723.7	0	0	0
H5	0	1.00	1.00	0	0	0	0	0
H6	0.30	1.18	0.88	5.57	405.1	0	455.9	0
H7	0	1.00	1.00	0	0	0	0	0
H8	2.68	4.226	1.546	10.035	1212.6	683.16	0	0
H9	0.02	0.92	0.90	0	1936	151.8	0	0
H10	0.11	0.77	0.66	338.7	0	220.2	304.4	84.6
H11	0.03	0.71	0.68	0	1112.2	89.62	0	0
H12	0	0.76	0.76	0	0	0	220.9	0
H13	0.07	0.70	0.63	162.8	0	82.72	223.96	0
H14	0.17	1.33	1.16	0	2680.5	3009.7	0	0
H15	0	1.00	1.00	0	0	0	0	0
H16	0	1.00	1.00	0	0	0	0	0

According to the results presented in Table 2, the optimal solution of model (7) for Hospitals H1, H2, H5, H7, H15 and H16 is φ^* - θ^* = 0, φ^* = θ^* = 1 and $s_{i}^{- *} = s_{r}^{+ *} = 0$ , for i = 1, 2, 3, 4 and r = 1, 2. Therefore, by Definition 3, these hospitals are fuzzy random most productive scale size. This means that these hospitals have produced maximum output with available inputs. Also, these six hospitals can be a benchmark for other hospitals for improving their performance. As well as, the results presented in Table 2 show that the optimal solution for Hospital H12, using the fuzzy model (7), is φ^* - θ^* = 0, φ^* = θ^* = 0.76, $s_{1}^{- *} = s_{2}^{- *} = s_{3}^{- *} = 0$ , $s_{4}^{- *} = 220.9$ , $s_{1}^{+ *} = 0$ and $s_{2}^{+ *} = 0$ . Based on Definition 3, Hospital H12 is not fuzzy random most productive scale size. However, this hospital has a good performance in comparison to the hospitals that are not fuzzy random mpss and it’s performance is very close to fuzzy random mpss.

Table 1

Fuzzy random data of four inputs and two outputs of sixteen hospitals

Hospitals	I1	I2	I3	I4	O1	O2
H1	(N(995,1),17)	(N(6205,9),262)	(N(1375,4),35)	(N(2629,4),49)	(N(4127,9),83)	(N(1678,1),17)
H2	(N(917,1),15)	(N(5898,9),248)	(N(1379,4),37)	(N(2047,4),37)	(N(3721,9),77)	(N(1277,1),13)
H3	(N(3178,4),91)	(N(10049,100),349)	(N(3615,9),85)	(N(3511,4),61)	(N(2706,4),27)	(N(2051,4),20)
H4	(N(813,1),14)	(N(5833,9),233)	(N(1124,4),34)	(N(1730,4),27)	(N(2176,4),21)	(N(1538,1),15)
H5	(N(1236,4),21)	(N(8639,81),309)	(N(2486,9),56)	(N(4990,4),92)	(N(5220,9),105)	(N(20426,144),204)
H6	(N(1146,4),11)	(N(7610,81),289)	(N(1600,4),30)	(N(3589,4),59)	(N(3517,9),65)	(N(1856,1),19)
H7	(N(705,1),13)	(N(5600,64),200)	(N(1557,4),27)	(N(3623,4),63)	(N(2352,4),24)	(N(20606,121),206)
H8	(N(2871,4),89)	(N(11524,100),404)	(N(2880,9),51)	(N(2452,4),52)	(N(1755,4),18)	(N(1664,1),17)
H9	(N(1089,1),20)	(N(8998,81),228)	(N(1730,9),40)	(N(2823,4),53)	(N(4412,9),89)	(N(2334,1),24)
H10	(N(2032,4),52)	(N(9383,81),243)	(N(2421,9),47)	(N(4454,4),84)	(N(5386,9),111)	(N(2080,1),21)
H11	(N(1414,1),24)	(N(10468,100),263)	(N(2140,16),57)	(N(3649,4),69)	(N(5735,9),118)	(N(2691,1),27)
H12	(N(1967,4),29)	(N(11260,100),272)	(N(2759,9),69)	(N(3178,4),48)	(N(6079,9),125)	(N(2804,1),29)
H13	(N(1851,1),28)	(N(9880,100),250)	(N(2335,4),55)	(N(4570,4),90)	(N(5893,9),113)	(N(2495,1),26)
H14	(N(3100,4),102)	(N(15649,121),479)	(N(5487,9),127)	(N(2940,4),55)	(N(5248,4),103)	(N(3692,4),37)
H15	(N(5016,9),176)	(N(18010,100),480)	(N(4008,9),98)	(N(3567,4),60)	(N(7800,9),131)	(N(4582,4),47)
H16	(N(1924,4),35)	(N(12682,100),452)	(N(2490,9),63)	(N(2975,4),53)	(N(6040,9),123)	(N(3396,4),35)

Also, according to Table 2, the optimal solution for Hospital H13 is φ^* - θ^* = 0.07, φ^* = 0.70, θ^* = 0.63 and $s_{1}^{- *} = 162.8$ , $s_{2}^{- *} = 0$ , $s_{3}^{- *} = 82.72$ , $s_{4}^{- *} = 223.96$ , $s_{1}^{+ *} = 0$ and $s_{2}^{+ *} = 0$ . By referring to Definition 3, Hospital H13 is not fuzzy random most productive scale size and it’s performance is very close to fuzzy random mpss in comparison to hospitals that are not fuzzy random mpss.

As we can see from Table 2, the optimal solution of model (7) with data of Table 1 for Hospital H8 is φ^* - θ^* = 2.68, φ^* = 4.226, θ^* = 1.546 and $s_{1}^{- *} = 10.035$ , $s_{2}^{- *} = 1212.6$ , $s_{3}^{- *} = 683.16$ , $s_{4}^{- *} = 0$ , $s_{1}^{+ *} = 0$ and $s_{2}^{+ *} = 0$ . So, both two conditions of Definition 3 aren’t satisfied. Therefore, Hospital H8 is not fuzzy random mpss. In this evaluation, this hospital with the amount of the objective function of 2.68 has the worst performance among 16 hospitals.

Considering Table 2, φ^* - θ^* = 1.15, φ^* = 2.38, θ^* = 1.23 and $s_{1}^{- *} = 1109.5$ , $s_{2}^{- *} = 0$ , $s_{3}^{- *} = 1547.9$ , $s_{4}^{- *} = 0$ , $s_{1}^{+ *} = 0$ and $s_{2}^{+ *} = 0$ , is the optimal solution of model (7) for Hospital H3. As mentioned before, this hospital is not fuzzy random most productive scale size (mpss). The amount of the objective function of φ^* - θ^* = 1.15 in the second column of Table 2 shows that the performance of Hospital H3 is only better than Hospital H8 with φ^* - θ^* = 2.68.

For the rest of the hospitals φ^* ≠ θ^*, so they aren’t fuzzy random most productive scale size. Since six hospitals that are fuzzy random most productive scale size in terms of using inputs to generate outputs are in optimal status, they can be a benchmark for improving the performance of ten other hospitals. Comparing φ^* - θ^* in the second column of Table 2 shows that Hospitals H9, H11, H13, H10, H14, H6, H4, H3 and H8 perform better, respectively, although all of these hospitals aren’t fuzzy random most productive scale size. Hospitals that aren’t fuzzy random most productive scale size can improve the initial value of their inputs, ${\tilde{x}}_{ij}; i = 1, 2, 3, 4$ , and their outputs ${\tilde{y}}_{rj}, r = 1, 2$ , to the new values of ${\hat{x}}_{ij}$ and ${\hat{y}}_{rj}$ , by the following two formulas: ${\hat{x}}_{ij} = θ^{*} - s_{i}^{- *} (8)$ and ${\hat{y}}_{rj} = φ^{*} [(1 - {\bar{δ}}_{r}) ({\tilde{y}}_{rj})_{0}^{U} + {\bar{δ}}_{r} ({\tilde{y}}_{rj})_{1}^{U}] + s_{r}^{+ *} (9)$

In above two equations, have ${\tilde{x}}_{ij} = {\tilde{\bar{x}}}_{ij} + Φ^{- 1} (α_{i}) {\tilde{a}}_{ij}$ and ${\tilde{y}}_{rj} = {\tilde{\bar{y}}}_{rj} + Φ^{- 1} (1 - β_{r}) {\tilde{b}}_{rj},$ . For example, the results presented in Table 1 show that Hospital H3 is not fuzzy random most productive scale size, but this hospital will be converted to fuzzy random most productive scale size if it changes its inputs and outputs as follows: $\begin{matrix} {\hat{x}}_{13} = 1.23 \times [(1 - 0.75) \times 3090 + 0.75 \times 3178] \\ - 1109.5 ≅ 2772 \end{matrix}$ $\begin{matrix} {\hat{x}}_{23} = 1.23 \times [(1 - 0.75) \times 9717 + 0.75 \times 10049] \\ - 0 ≅ 12258 \end{matrix}$ $\begin{matrix} {\hat{x}}_{33} = 1.23 \times [(1 - 0.75) \times 3530 + 0.75 \times 3615] \\ - 1547.9 ≅ 2872 \end{matrix}$ $\begin{matrix} {\hat{x}}_{43} = 1.23 \times [(1 - 0.75) \times 3453 + 0.75 \times 3511] \\ - 0 ≅ 4301 \end{matrix}$ and $\begin{matrix} {\hat{y}}_{13} = 2.38 \times [(1 - 0.75) \times 2730 + 0.75 \times 2706] \\ + 0 ≅ 6455 \end{matrix}$ $\begin{matrix} {\hat{y}}_{23} = 2.38 \times [(1 - 0.75) \times 2068 + 0.75 \times 2051] \\ + 0 ≅ 4892 . \end{matrix}$

The above results show that to improve the performance of Hospital H3, the first and third inputs should be reduced and the second and fourth inputs should be increased. Also, two outputs of this hospital should be increased with relatively large amount. Ultimately, using Equations (9), the optimal amount of input and output variables can be obtained for other non mpss hospitals to improve their performance.

6 Conclusion

Data envelopment analysis (DEA) is a nonparametric mathematical programming method for evaluating the relative efficiency of homogeneous decision making units (DMUs) that use similar inputs to produce similar outputs. The classic DEA method evaluates the efficiency of decision making units with deterministic data, while in real-world problems the input and output data are often vague and inaccurate. Previously, the most productive scale size (mpss) of DMUs was evaluated with random input and output variables using the chance constrained programming approaches.

In this article, we proposed a fuzzy random DEA model to evaluate the most productive scale size of DMUs with fuzzy random inputs and outputs. Then the chance constrained programming approaches and the possibility method are used to obtain the corresponding deterministic equivalent of the fuzzy-random DEA model of the most productive scale size. In the application section (Section 6), the fuzzy-random DEA model (Proposed model) is applied to estimate the most productive scale size of sixteen Iranian hospitals using four inputs to produce two outputs with fuzzy random data. The results presented in Table 2 show that Hospitals H1, H2, H5, H7, H15 and H16 are fuzzy random mpss with proposed model and other DMUs or hospitals aren’t fuzzy random mpss. Therefore, these six hospitals can be utilized as benchmark to improve the performance of other hospitals. Hospital H12, is not fuzzy random most productive scale size. However, this hospital has a good performance in comparison to the hospitals that are not fuzzy random mpss and it’s performance is very close to fuzzy random mpss. Also, Hospital H3 with the objective function of 2.68 has the worst performance among sixteen hospitals.

Identifying the congestion of inputs of DMUs or the ranking of DMUs with fuzzy random inputs and outputs is suggested to researchers for further research in the future. Also, estimating return to scale (RTS) with fuzzy random inputs and outputs in DEA is recommended to interested researchers.

References

Charnes

, Cooper

W.W.

and Rhodes

, Measuring the Efficiency of Decision Making Units, European Journal of Operational Research 2(6) (1978), 429–444.

Ebrahimnejad

, Tavana

, Nasseri

and Gholami

, A New Method for Solving Dual DEA Problems with Fuzzy Stochastic Data, International Journal of Information Technology and Decision Making 18(1) (2019), 147–170.

Ebrahimnejad

and Amani

, Fuzzy data envelopment analysis in the presence of undesirable outputs with ideal points, Complex and Intelligent Systems 7(2021), 379–400.

Ebrahimnejad

, Nasseri

S.H.

and Gholami

, Fuzzy stochastic data envelopment analysis with application to NATO enlargement, RAIRO Operations Research 53(2) (2019), 705–721.

Yaghoubi

, Maghsoud

and Safi Samghabadi

, A New Dynamic Random Fuzzy DEA Model to Predict Performance of Decision Making Units, Journal of Optimization in Industrial Engineering 9(20) (2015), 75–90.

Sahoo

B.K.

, Khoveyni

, Eslami

and Chaudhury

, Returns to scale and most productive scale size in DEA with negative data, European Journal of Operational Research 255(2) (2016), 545–558.

Golany

and Roll

, An application procedure for DEA, Omega 17(3) (1989), 237–250.

Liu

, Uncertain Programming, A Wiley-Interscience Publication, New York, 1999.

Lee

C.Y.

, Most productive scale size versus demand fulfillment: A solution to the capacity dilemma, European Journal of Operational Research 248(3) (2016), 954–962.

10.

Dubois

and Prade

, Possibility Theory: An Approach to Computerized Processing of Uncertainty, Plenum Press, New York, 1988.

11.

Koushki

, Performance Measurement and Productivity Management in Production Units with Network Structure by Identification of the Most Productive Scale Size Pattern, International Journal of Supply and Operations Management 5(4) (2018), 379–395.

12.

Jahanshahloo

G.R.

and Khodabakhshi

, Using input-output orientation model for determining most productive scale size in DEA, Applied Mathematics and Computation 146(2) (2003), 849–855.

13.

Dibachi

, Stochastic multiplicative DEA for estimating most productive scale size, Theory of Approximation and Applications 12(1) (2018), 93–115.

14.

Kheirollahi

, Karami Matin

, Mahboubi

and Mirzaei Alavijeh

, Chance constrained input relaxation to congestion in stochastic DEA, An application to Iranian hospitals, Global Journal of Health Science 7(4) (2015), 151–160.

15.

Kheirollahi

, Hessari

, Charles

and Chawshini

, An input relaxation model for evaluating congestion in fuzzy DEA, Croatian Operational Research Review 8(22) (2017), 391–408.

16.

Kwakernaak

, Fuzzy random variables-I: Definitions and theorems, Information Science 15(1) (1978), 1–29.

17.

Sengupta

J.K.

, A fuzzy systems approach in data envelopment analysis, Computers and Mathematics with Applications 24(8-9) (1992), 259–266.

18.

Ruiz

J.L.

and Sirvent

, Fuzzy cross-efficiency evaluation: a possibility approach, Fuzzy Optimization and Decision Making 16(1) (2017), 111–126.

19.

Zadeh

L.A.

, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1(1978), 3–28.

20.

Behzadi

M.H.

and Mirbolouki

, Symmetric Error Structure in Stochastic DEA, International Journal Industrial Mathematics 4(4) (2012), 335–343.

21.

Khodabakhshi

, Estimating most productive scale size in stochastic data envelopment analysis, Economic Modelling 26(5) (2009), 968–973.

22.

Khodabakhshi

, Gholami

and Kheirollahi

, An additive model approach for estimating returns to scale in imprecise data envelopment analysis, Applied Mathematical Modelling 34(5) (2010), 1247–1257.

23.

Tavana

, Khanjani Shiraz

, Hatami-Marbini

, Agrell

P.J.

and Paryab

, Chance-constrained DEA models with random fuzzy inputs and outputs, Knowledge-Based Systems 52 (2013), 32–52.

24.

Tavana

, Khanjani Shiraz

, Hatami-Marbini

, Agrell

P.J.

and Paryab

, Fuzzy stochastic data envelopment analysis with application to base realignment and closure (BRAC), Expert Systems with Applications 39(15) (2012), 12247–12259.

25.

Minh

and Khanh

, A chance-constrained data envelopment analysis approach to problem provincial productivity growth in Vietnamese agriculture from 1995 to 2007, Open Journal of Statistics 1(3) (2011), 217–235.

26.

Gholami

, A Novel Approach for Solving Fuzzy Stochastic Data Envelopment Analysis Model in the Presence of Undesirable Outputs, Fuzzy Optimization and Modelling Journal 2(1) (2021), 22–36.

27.

Guo

and Tanaka

, Fuzzy DEA: A perceptual evaluation method, Fuzzy Sets and Systems 119(1) (2001), 149–160.

28.

Peykani

, Mohammadi

, Rostamy-Malkhalifeh

and Hosseinzadeh Lotfi

, Fuzzy data envelopment analysis approach for ranking of stocks with an application to Tehran stock exchange, Advances in Mathematical Finance and Applications 4(1) (2019), 31–43.

29.

Banker

R.D.

, Estimating most productive scale size using data envelopment analysis, European Journal of Operational Research 17(1) (1984), 35–44.

30.

Banker

R.D.

, Charnes

and Cooper

W.W.

, Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science 30(9) (1984), 1078–1092.

31.

Dyson

R.G.

, Allen

, Camanho

A.S.

, Podinovski

V.V.

, Sarrico

C.S.

and Shale

E.A.

, Pitfalls and protocols in DEA, European Journal of Operational Research 132(2) (2001), 245–259.

32.

Eslami

, Khodabakhshi

, Jahanshahloo

G.R.

, Hosseinzadeh Lotfi

and Khoveyni

, Estimating most productive scale size with imprecise chance constrained input–output orientation model in data envelopment analysis, Computers and Industrial Engineering 63(1) (2012), 254–261.

33.

Qin

and Liu

Y.K.

, A new data envelopment analysis model with fuzzy random inputs and outputs, Korean Society for Computational and Applied Mathematics, Journal of Applied Mathematics and Computing 33(1) (2010), 327–356.

34.

Nasseri

S.H.

, Ebrahimnejad

and Gholami

, Fuzzy Stochastic Data Envelopment Analysis with Undesirable Outputs and its Application to Banking Industry, International Journal of Fuzzy Systems 20(1) (2018), 534–548.

35.

Lertworasirikul

, Fanga

S.C.

, Joines

J.A.

and Nuttle

H.L.W.

, Fuzzy data envelopment analysis (DEA): a possibility approach, Fuzzy Sets and Systems 139(2) (2003), 379–394.

36.

Punyangarm

, Yanpirat

, Charnsethikul

and Lertworasirikul

, A Case of Constant Returns to Scale in Fuzzy Stochastic Data Envelopment Analysis: Chance-Constrained Programming and Possibility Approach, Thailand Statistician 6(1) (2008), 75–90.

37.

Cooper

W.W.

, Deng

, Huang

Z.M.

and Li

S.X.

, Change constrained programming approaches to technical efficiencies and inefficiencies in stochastic data envelopment analysis, Journal of the Operational Research Society 53(12) (2002), 1347–1356.

38.

Liu

Y.K.

and Liu

, On minimum-risk problems in fuzzy random decision systems, Computers and Operations Research, 32(2) (2005), 257–283.

39.

Moghaddas

and Ghasemi

M.Z.

, Estimating most productive scale size in DEA with real and integer value data, International Journal Industrial Mathematics 5(2) (2014), 107–113.