Uncertain RUSSEL data envelopment analysis model: A case study in iranian banks

Abstract

Data Envelopment Analysis (DEA) is recognized as a robust analytical tool extensively utilized in measuring the relative efficiency of a group of decision-making units (DMUs) with multiple inputs and outputs. The DEA models require inputs and outputs equipped with precise information. However, in real-world situations, inputs and outputs may be unstable and complicated, thus unable to be accurately measured. This problem resulted in the investigation of uncertain DEA models. The RUSSELL model was studied in this paper in an uncertain environment where uncertain inputs and outputs were belief degree-based uncertainty, useful for the cases for which no historical information of an uncertain event is available. As the solution method, the uncertain RUSSELL model was converted to a crisp form using two approaches of expected value model and expected value and dependent chance-constrained model separately. Finally, an applied example regarding the Iranian banking system was presented to document the proposed models.

Keywords

Data envelopment analysis Uncertainty theory New product RUSSELL model Iranian Bank

1 Introduction

As a strong analytical tool, data envelopment analysis (DEA) is a method for evaluating the relation efficiency of decision-making units (DMUs), originally developed by Charens et al. 7 within a printed-paper named CCR. They extended the nonparametric method introduced by Farrell 21 to gauge DMUs with multiple inputs and outputs.

Afterward, Banker et al. 3 introduced the BCC model. In addition to CCR and BCC, there are several models that discuss DEA from several perspectives: RAM 14, slack-adjust 38, additive model 1, SBM model 38, and FDH model 19. In addition to the linear programming problems, there also exist nonlinear ones in DEA. Färe and Lovell 22 introduced an alternative way for radial models with input/output orientations, the so-called RUSSELL measure (RM). The RM model is monotonous, units invariant, and economically well interpretable (see 15 for some basic properties), all of which are DEA basic models.

In classical DEA models, DMUs are evaluated by considering input and output values in order to measure rational efficiency as compared to different DMUs; eventually, the measure to which rational efficiency belong is obtained (0,1).

The first DEA models expected that inputs and outputs be measured by precise values. However, in numerous situations such as when producing a system or in preparation process, the banking system, insurance industry and financial service system, inputs and outputs are unstable and complicated and, therefore, cannot be accurately measured. Consequently, many analysts attempted to model DEA with completely different questioning hypotheses. The possible hypothesis is the earliest principles which may be used to build stochastic DEA models. Sengupta 39 summed up the stochastic DEA model utilizing expected value. Moreover, Banker 5 consolidated applied mathematics elements underneath DEA in order to develop a statistical method. Several papers utilized chance-constrained programming to DEA so as to introduce stochastic varieties to information 13, 14, 36. Fuzzy outlook is another theory in which the hypothesis has been incorporated to cope with the uncertainty in DEA. United with the DEA innovator, Cooper et al. 16, 17 introduced a technique to deal with inaccurate information such as moderate data, adjectival data and ratio moderate data in DEA. Furthermore, Kao and Liu 25 designed a technique to discover the membership function of fuzzy performance marks when each input and output are fuzzy numbers. Entani et al. 20 proposed an interval potency DEA model by pessimistic and idealistic values. Several researchers have introduced the possibility measure into DEA 44, 23, 27. As confirmed by several studies, human uncertainty does not come with the same fuzziness. Because of the shortages of Fuzzy Theory, Liu 30 introduced Uncertain Theory and refined it in 2010 as an understandable mathematical structure for confronting uncertainty in data which serves as a strong alternative to the probability theory when one has to restrict the information in the face of insufficient trusted data. The belief degree function is associated with an underlying concept of this theory built according to the experts’ opinion.

Optimization problems, including uncertain data, can be even more interesting and realistic in uncertain environments with uncertain values for parameters and even variables. To tackle such problems with uncertain parameters, any approach based on randomness, fuzzy theory, stochastic programming, probability theory, and so on can be applied in the face of historical information of the parameters. In such cases, the uncertain manner of the problem is estimated from the historical data as the probability function, random number, fuzzy number, etc. On the other hand, for cases in which no historical information for an uncertain event exists, uncertainty theory based on belief degree has been applied to solve the problem. This uncertainty theory can be explained by a simple example. Consider a bridge and strength. At first, it is assumed that no destructive experiment is allowed for the bridge. Thus, there is no sample regarding the strength of the bridge. In this case, no statistical methods exists for estimating its probability distribution. Therefore, there is choice but to invite bridge engineers to evaluate the belief degrees about the bridge’s strength. Some basic concepts of the belief degree-based uncertainty theory will be explained in Section 2, and a complete study of this topic can be found in Liu 30.

The belief degree depends heavily on personal knowledge (even including preferences) concerning the event. When the personal knowledge changes, the belief degree changes as well. Different people may produce different belief degrees. The question is which belief degree is correct, a question which may be answered as follows: All belief degrees are wrong, but some are useful. A belief degree becomes “correct” only when it is close enough to the frequency of the indeterminate quantity which, however, does not usually occur. Numerous surveys demonstrated that human beings usually estimate a much wider range of values than the object actually takes. This human conservatism makes the belief degrees deviate far from the frequency. Thus, all belief degrees are wrong compared with the frequency. Nevertheless, it is undeniable that these belief degrees are indeed helpful for decision-making. Moreover, as determined by the choice of α, there is a risk that DMUs would not be efficient even when their condition is satisfied.

Wen et al. 41 applied the uncertain theory for the first time to rewrite the DEA model in uncertainty condition and then published a paper on the sensitivity and stability of the additive model in terms of uncertainty. Wen et al. 42 also introduced a new Additive model with uncertain inputs and outputs. Additionally, Wen et al. 43 developed the DEA model with uncertainty index ranking for criteria. Liu et al. 29 also published a paper to evaluate DMU with uncertain inputs and outputs. These studies are nevertheless insufficient for describing programming with uncertain data 43, 34, 35, 28, 10, 46, 18, 9, 45, and new models are often introduced to create a new method.

One of the main aims of radial models is to measure the efficiency of a DMU by a scalar measure ranging between zero (the worst) and one (the best). The optimal objective value (θ ^*) is called the radial efficiency of the DMU. Simultaneously, the optimal solution reveals the existence, if any, of excesses in inputs and shortfalls in outputs (called slacks). A DMU with full radial efficiency, θ ^* = 1, and with no slacks in any optimal solution is called efficient. Otherwise, the DMU has a disadvantage against the DMUs in its reference set. Therefore, in discussing total efficiency, both radial efficiency and slacks are important.

This article proposed a scalar measure RUSSELL of efficiency in DEA. In contrast to the previous model measures based on the proportional reduction (enlargement) of input (output) vectors that do not take slacks into account, the RUSSELL deals directly with input excess and output shortfall. Although the additive model 41 has the (weighted) sum of slacks as its objective and can discriminate between efficient and inefficient DMUs, it has no means to gauge the depth of inefficiency per se. None of the earlier works discussed non-radial-based RUSSELL with uncertain variables. Also, in real world, a decrease (increase) in inputs (outputs) at the same time is impossible in most cases, and thus it is useful to apply the non-radial model rather than the radial model.

The present paper assumed inputs and outputs to be uncertain variables in RUSSELL model and then, to deal with uncertainty downside, recommend two completely different models to crisp this new model. Afterward, the uncertain RUSSELL model is applied to an Iranian Bank to see how the target customer accepts a replacement-banking product before promotion and announcement phase.

The paper proceeds as follows: In Section 2, some preliminary knowledge of uncertainty theory and basic notions of some DEA models are reviewed. In Section 3, several new uncertain DEA models are introduced and their new structures are verified. In Section 4, the crisp equivalents of the model are presented. Finally, a practical example of the Iranian banking system with the uncertain RUSSELL model is introduced.

2 Preliminaries

Here, discuss basic concept and present uncertain variables. Let Γ be a nonempty set, and L an σ-algebra over Γ. Each element Λ ∈ L is called an event. A set function $M {\land} \in [0, 1]$ is known as an uncertain measure if it satisfies the following three axioms:

$M {\land} = 1$ for the universal set Γ

$M {\land} + M {\land^{c}} = 1$ for any event Λ

For every countable subadditive of events, {∧ _i} we have $M {\cup_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M {Λ_{i}}$ .

Definition 1 [30]. The set function $M$ called an uncertain measure if it contents the duality, normality, and subadditivity axioms.

The uncertain measure has the following attributes:

$M {\emptyset} = 0$

$0 \leq M {\land} \leq 1$ for any event Λ

$M {Λ_{1}} \leq M {Λ_{2}}$ for any events Λ ₁ ⊂ Λ ₂.

The triplet $(Γ, L, M)$ called an uncertainty space. In order to define product uncertain measure, Liu 31 proposed the fourth axiom as follow:

Let $(Γ_{k}, L_{k}, M_{k})$ be uncertainty space for k = 1, 2, . . . , then the product uncertain measure $M$ is an uncertain measure satisfying $M {\prod_{k = 1}^{\infty} Λ_{k}} = Λ_{k = 1}^{\infty} M_{k} {Λ_{k}}$ .

Definition 2 [30]. An uncertain variable is a measurable function ξ from an uncertainty space $(Γ, L, M)$ to the set of real numbers, i.e., for any Borel set B of real numbers, the set ${ξ \in B} = {γ \in Γ | ξ (γ) \in B}$ is an event.

An uncertainty distribution function is used to characterize an uncertain variable and is defined as follows 30: $φ (x) = M {ξ \leq x} \forall x, x \in R$ (1)

Theorem 1 [37]. A function $Φ (x) : K \to$ [0,1] is an uncertainty distribution if and only if it is a monotone increasing function except Φ (x) ≡0 and Φ (x) ≡1

Example 1. the uncertain variable with linear uncertainty distribution is defined as follow: $φ (x) = {\begin{matrix} 0 x \leq a \\ \frac{(x - a)}{(b - a)} a \leq x \leq b \\ 1 x \geq b . \end{matrix}$ (2) for convenience, it is denoted in the paper by $ξ \sim L (a, b)$ where a < b.

Example 2. The zigzag uncertainty distribution is an uncertain variable ξ is shown with $Z (a, b, c)$ expressed as follows:

$φ (x) = {\begin{matrix} 0 x \leq a \\ \frac{(x - a)}{2 (b - a)} a \leq x \leq b \\ \frac{(x + c - 2 b)}{2 (c - b)} b \leq x \leq c \\ 1 x \geq c \end{matrix}$ (3) where a, b, c are real numbers with a < b < c.

Definition 3 An uncertainty distribution φ is said to be regular if its inverse function φ ^-1 (α)exists and is unique for each α ∈ (0, 1), for instance the uncertain distributions are called in (2), (3).

Fig.1

Inverse linear uncertainty distribution liu [30].

Example 3. The inverse uncertainty distribution of linear uncertain variable $L (a, b)$ is Φ ^-1 (α) = (1 - α) a + αb (Figure 1) $Φ^{- 1} (α)$

Example 4. The inverse uncertainty distribution of zigzag uncertain variable Z (a, b, c) is $Φ^{- 1} (α) = {\begin{matrix} (1 - 2 α) a + 2 α b, & α < 0.5 \\ (2 - 2 α) b + (2 α - 1) c, & α \geq 0.5 \end{matrix}$ (4)

Definition 4 [33]. The uncertain variables ξ ₁, ξ ₂, . . . , ξ _n are considered to be independent if $M {\cap_{i = 1}^{n} (ξ_{i} \in B_{i})} = ⋁_{n}^{i = 1} M {ξ_{i} \in B_{i}}$ for any Borel sets B ₁, B ₂, …, B _n.

Definition 5 [33]. The uncertain variable ξ ₁, ξ ₂, . . . , ξ _n are independent if and only if $M {\cup_{i = 1}^{n} (ξ_{i} \in B_{i})} = \lor_{i = 1}^{n} M {(ξ_{i} \in B_{i})}$ for any Borel sets B ₁, B ₂, . . . , B _n.

Theorem 2 [33]. Let ξ₁, ξ₂, . . . , ξ_n be independent uncertain variables, and f₁, f₂ . . . f_n measurable functions. Then f₁ (ξ_1,) , f₂ (ξ_2,) , . . . , f_n (ξ_n,) are independent uncertain variables.

Theorem 3 [32]. Let ξ₁, ξ₂, . . . , ξ_n be variables with independent uncertainty with regular uncertainty distributions φ_1,φ_2, … , φ_nrespectively. If f is a strictly increasing function, so ξ = f (ξ₁, ξ₂, . . . , ξ_n) is an uncertain variable with inverse uncertain distribution $ψ^{- 1} {= f (φ}_{1}^{- 1} (α), φ_{2}^{- 1} (α), \dots, φ_{n}^{- 1} (α))$ .

Theorem 4 [32]. Let ξ₁, ξ₂, . . . , ξ_n be independent uncertain variables with regular uncertainty distributions φ_1,φ_2, … , φ_nrespectively. If f is a strictly increasing with respect to ξ₁, ξ₂, . . . , ξ_m and strictly decreasing with respect to ξ_m+1, ξ_m+2, . . . , ξ_n, then ξ = f (ξ₁, ξ₂, . . . , ξ_n) is an uncertain variable with an inverse uncertainty distribution φ^-1 (α) = $f (φ_{1}^{- 1} (α)$ , …, $φ_{m}^{- 1}$ (α), $φ_{m + 1}^{- 1}$ (1 - α), …, $φ_{n}^{- 1}$ (1 - α)).

Definition 6 [30]. If ξ be an uncertain variable. So the expected value of ξ is defined by $E [ξ] = \int_{0}^{+ \infty} M (ξ \geq x) dx - \int_{- \infty}^{0} (ξ \leq x) dx$ provided that at least one of the two integral is finite.

For the couple uncertain variable and distribution (ξ, φ) have some formulas explained as follow: $E ξ = \int_{0}^{+ \infty} (1 - φ (x)) dx - \int_{- \infty}^{0} φ (x) dx$

Example 5. Let $ξ \sim L (a, b)$ be a variable with linear uncertainty. Then its inverse is showed with Φ ^-1 (α) = (1 - α) a + αb, and its expected value is $E [ξ] = \int_{0}^{1} ((1 - α) a + α b) d α = \frac{a + b}{2}$

Example 6. The expected value for variable ξ with zigzag uncertain distribution is defined as fallow: $E [ξ] = \frac{a + 2 b + c}{4}$

Theorem 5 [32]. Let ξ₁, ξ₂, … ξ_n be independent uncertain variables with regular uncertainty distributions φ_1,φ_2, … , φ_n, respectively. If f is a strictly increasing with respect to ξ₁, ξ₂, … ξ_m and strictly decreasing with respect to ξ_m+1, ξ_m+2, … ξ_n, thenξ = f (ξ₁, ξ₂, … ξ_n) has an expected value $\begin{matrix} E [ξ] = \int_{0}^{1} f (φ_{1}^{- 1} (α), \dots, φ_{m}^{- 1} (α), φ_{m + 1}^{- 1} (1 - α), \\ \dots, φ_{n}^{- 1} (1 - α) d α \end{matrix}$

Theorem 6 [32]. Let ξ and η be independent uncertain variables with finite expected value. Then for any real numbers a and b have, E [aξ + bη] = aE [ξ] + bE [η].

Theorem 7 [32]. Let ξ₁, ξ₂, … ξ_n be independent uncertain variables with regular uncertainty distributions φ_1,φ_2, … , φ_n, respectively. If the function f (ξ₁, ξ₂, … ξ_n) is a strictly increasing with respect to ξ₁, ξ₂, … ξ_m and strictly decreasing with respect to ξ_m+1, ξ_m+2, … ξ_n, then $μ {f (ξ_{1,} ξ_{2,} \dots, ξ_{n}) \leq 0} \geq α$ if and only if $\begin{matrix} f (φ_{1}^{- 1} (α), \dots . φ_{m}^{- 1} (α), φ_{m + 1}^{- 1} (1 - α), \\ \dots, φ_{n}^{- 1} (1 - α)) \leq 0 \end{matrix}$

Theorem 8 [32]. Let ξ be an uncertain variable with regular uncertainty distribution Φ. Then $E [ξ] = \int_{0}^{1} φ^{- 1} (α) d α$ consider a set of DMU _k (k = 1, . . . , n) with m positive inputs (x _ik, i = 1, . . . , m) and s positive outputs (y _rk, r = 1, . . . , s). Now for evaluate DMU_o = (x_o, y_o), the nonlinear programming RM model will be formulated as follows: $min_{θ_{i}, φ_{r}, λ_{k}} \frac{1}{m + s} (\sum_{i = 1}^{m} θ_{i} + \sum_{r = 1}^{s} \frac{1}{φ_{r}})$ (5) s.t $\sum_{k = 1}^{n} λ_{k} x_{ik} \leq θ_{r} x_{io}, θ_{i} \leq 1, i = 1, . . ., m$ (6) $\sum_{k = 1}^{n} λ_{k} y_{rk} \geq φ_{r} y_{ro}, φ_{r} \geq 1, r = 1, . . ., s$ (7) $λ_{k} \geq 0 k = 1, . . ., n$ (8) $\sum_{k = 1}^{n} λ_{k} = 1 k = 1, . . ., n$ (9)

Thus, described by (5)-(9), the mentioned model is associated with the technology set

$\begin{matrix} T = {(x, y) \in ℝ^{m} \times ℝ^{s} | \sum_{k = 1}^{n} λ_{k} x_{k} \leq x, \\ \sum_{k = 1}^{n} λ_{k} y_{k} \geq y, λ \geq 0, \sum_{k = 1}^{n} λ_{k} = 1} \end{matrix}$ (10)

Since $\sum_{k = 1}^{n} λ_{k} = 1$ , the above model is variable returns to scale (VRS). If condition (9) is excluded from the constraints of the model and also from the set T, the RM model will correspond with constant returns to scale (CRS).

To explain the RM model, assume that the optimal solution vectors of (5)-(9) are θ ^*, φ ^* and λ ^* and marked as follows:

$\begin{matrix} {\hat{x}}_{i} & : = \sum_{k = 1}^{n} x_{ik} λ_{k}^{*} (i = 1, \dots, m) and \\ {\hat{y}}_{r} & : = \sum_{k = 1}^{n} y_{rk} λ_{k}^{*} (r = 1, \dots, s) \end{matrix}$ (11)

The image of (x _o, y _o) on the efficient frontier of T is given in (11) with unit $(\hat{x}, \hat{y})$ ; this unit (x _o, y _o) is compared to unit $(\hat{x}, \hat{y})$ as a figurative benchmark. Because the efficiency value is a number between (0, 1], we will be able to put the values $\frac{{\hat{x}}_{i}}{x_{io}} \leq 1$ and $\frac{y_{ro}}{{\hat{y}}_{r}} \leq 1$ as the partial efficiencies of the input i and the output r, respectively. Finally, because of a binding inequalities in (6) and (7) (i.e. holding with equalities at the optimal solution), we have $θ_{i}^{*} = \frac{{\hat{x}}_{i}}{x_{io}} and \frac{1}{φ_{r}^{*}} = \frac{y_{ro}}{{\hat{y}}_{r}}$ (12)

Therefore, the optimal value of (5) is the arithmetic mean of the partial efficiencies of the inputs and outputs which is a number between (0, 1]. If all its partial efficiencies are equal to one, the unit (x _o, y _o) is considered as efficient.

Theorem 9 [24]. The dual Russell measure (RM) model introduce as follows:

$\begin{matrix} max_{σ, u_{r}, v_{i}, t_{r}} - σ & + \sum_{r = 1}^{s} u_{r} Y_{ro} - \sum_{i = 1}^{m} v_{i} X_{io} \\ + 1 - \frac{1}{m + s} \sum_{r = 1}^{s} F (t_{r}) \end{matrix}$ (13) s.t $\sum_{r = 1}^{s} u_{r} y_{rk} - \sum_{i = 1}^{m} v_{i} x_{ik} \leq σ, k = 1, \dots, n$ (14) $v_{i} \geq \frac{1}{m + s} \frac{1}{x_{io}}, i = 1, \dots, m$ (15) $u_{r} \geq \frac{1}{m + s} \frac{t_{r}}{y_{ro}} \geq 0, r = 1, \dots, s$ (16)

Where $F (t_{r}) = {(1 - \sqrt{t_{r}})}^{2}$ (17)

Remark 1. Considering that, the Model (13)-(17) is also the Lagrange dual of the original RM-V Model (5)-(9). It can be taken by utilizing the sense of intermix functions (see e.g. [6]) and the standard nonlinear programming techniques. Now we are using the primal-dual property as follow:

Property 1. Let (θ^*, φ^*) and (σ^*, u^*, v^*, t^*)be optimal solutions for the primal and dual RM-V Models (5)-(9) and (13)-(17) for (x_o, y_o), respectively. Then the following implications hold:

if $θ_{i}^{*} < 1, then v_{i}^{*} x_{io} = \frac{1}{m + s}$ ;

if $φ_{i}^{*} > 1$ , then $u_{r}^{*} y_{ro} = \frac{t_{r}^{*}}{m + s}$ $= \frac{1}{m + s} = \frac{1}{(φ_{r}^{*})^{2}}$ , $\sqrt{t_{r}^{*}} = \frac{1}{φ_{r}^{*}} = ψ_{r}^{*}$ , (leading to $0 < t_{r}^{*} < 1$ and $0 < F (t_{r}^{*}) < 1)$ ;

if $φ_{r}^{*} = 1$ , then $t_{r}^{*} = 1, F (t_{r}^{*}) = 0$

In addition, the set of all $(x, y) \in ℝ^{m + s}$ such that $\sum_{r = 1}^{s} u_{r}^{*} y_{r} - \sum_{i = 1}^{m} v_{i}^{*} x_{i} = σ^{*}$ forms a supporting hyperplane to the technology set T at the projection $(\hat{x}, \hat{y})$ of (x_o, y_o) defined by (10) and (11) and

$\begin{matrix} σ^{*} & = max_{k} {\sum_{r = 1}^{s} u_{r}^{*} y_{rk} - \sum_{i = 1}^{m} v_{i}^{*} x_{ik}} \\ = max_{(x, y) \in T} {\sum_{r = 1}^{s} u_{r}^{*} y_{r} - \sum_{i = 1}^{m} v_{i}^{*} x_{i}} \end{matrix}$ (18)

Property 2. (Efficiency) Let (σ ^*, u ^*, v ^*, t ^*) be an optimal solution of the dual RM-V Model (13)-(17) for an efficient DMU (x _o, y _o),i.e. optimal solutions of the primal RM-V Model (5)-(9) satisfy $θ_{i}^{*} = 1$ (i = 1, . . . , m) and $φ_{r}^{*} = 1 (r = 1, \dots, s)$ . Then

$\begin{matrix} \sum_{r = 1}^{s} u_{r}^{*} y_{ro} - \sum_{i = 1}^{m} v_{i}^{*} x_{io} = σ^{*} and t_{r}^{*} = 1, \\ F (t_{r}^{*}) = 0 r = 1, \dots, s \end{matrix}$ (19) as a consequence, for each optimal (σ ^*, u ^*, v ^*, 1) there exists an α ₀ > 0 such that for any α ≥ α ₀ also (ασ ^*, αu ^*, αv ^*, 1) is an optimal solution.

One of the main purposes of a DEA study is to project the inefficient DMUs onto production frontiers. There are three directions: one called input-oriented aims to reduce the amounts of input as much as possible while keeping at least the present output levels; the other, called output-oriented, maximizes output levels under at most the present input consumption; and the third one, represented by the Additive, SBM and RUSSELL, the so-called non-radial model, deals with the input excesses and output shortfalls simultaneously while jointly maximizing both. If achievement of efficiency, or failure to do so, is the only topic of interest, then these different models will all yield the same result insofar as technical and mix inefficiency is concerned. However, when DMUo cannot decrease (increase) all inputs (outputs) with the same ratio, the non-radial models are suitable.

3 Uncertain DEA model

The RUSSELL model requires inputs and outputs equipped with precise data. Nevertheless, in real-world situations, inputs and outputs may be unstable and complicated and, therefore, cannot be measured in an accurate manner. Consequently, this conflict results in the investigation of uncertain DEA models.

Decision-makers in real-word situations make their decisions in the indeterminacy state. To model indeterminacy, there exist two mathematical systems, one the probability theory [26] and the other the uncertainty theory [30]. If there exist frequency in phenomena, the probability theory is employed; otherwise, the uncertain theory can be a powerful technique for resolving the drawback with no sample, using the personal belief degree. For this purpose, skilled consultants and experts should be invited to measure the belief degree.

Throughout this approach, we aimed to introduce a RUSSELL model with uncertain inputs and outputs referred to as the uncertain RUSSELL model. First, new symbols and notations are presented:

${\tilde{x}}_{k} = ({\tilde{x}}_{1 k}, {\tilde{x}}_{2 k}, . . ., {\tilde{x}}_{mk})$ : the uncertain inputs vector of DMU_k, k = 1, 2, . . . , n ;

φ _ik (x) : the uncertainty distribution of ${\tilde{x}}_{ik}$ , k = 1, 2, . . . , n, i = 1, 2, . . . , m ;

${\tilde{y}}_{k} = ({\tilde{y}}_{1 k}, {\tilde{y}}_{2 k}, . . ., {\tilde{y}}_{rk})$ : the uncertain outputs vector of DMU_k, k = 1, 2, . . . , n .

ψ _rk (x): the uncertainty distribution of ${\tilde{y}}_{rk}, k = 1, 2,$ . . . , n, r = 1, 2, . . . , s ;

α: is a predetermined confidence level;

$M$ :the uncertainty measure expressed in section 2;

Second the uncertain RUSSELL model expresses as fallow: $\max_{{σ, u}_{r} {, v}_{i}, t_{r}} - σ + \sum_{r = 1}^{s} u_{r} {\tilde{y}}_{ro} - \sum_{i = 1}^{m} v_{i} {\tilde{x}}_{io} + 1 - \frac{1}{m + s} \sum_{r = 1}^{s} {F (t}_{r})$ s.t $\sum_{r = 1}^{s} u_{r} {\tilde{y}}_{rk} - \sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ik} \leq σ, k = 1, \dots, n$ $v_{i} \geq \frac{1}{m + s} \frac{1}{{\tilde{x}}_{io}}, i = 1, \dots, m$ (20) $u_{r} \geq \frac{1}{m + s} \frac{t_{r}}{{\tilde{y}}_{ro}} \geq 0, r = 1, \dots, s$

Where ${F (t}_{r}) = 1 - \sqrt{t_{r}})^{2}$

To encounter with the uncertainty problem in the uncertain RUSSELL model, we introduced two models to crisp it: (1) expected value combined with the chance-constrained model, and (2) expected value model using uncertain programming techniques.

4 Crisp equivalents of the model

4.1 expected value combined with chance-constrained model

Liu [30] proposed the expected value operator of the uncertain variable with the chance-constrained model as follow: $θ = max_{u, v} E [\frac{v^{t} {\tilde{y}}_{o}}{u^{t} {\tilde{x}}_{o}}]$ s.t $\begin{matrix} M {v^{t} {\tilde{y}}_{k} \leq u^{t} {\tilde{x}}_{k}} \geq α k = 1, \dots, n \\ u \geq 0 \\ v \geq 0 \end{matrix}$ (21) in which α ∈ (0.5, 1]

Definition 7. A vector (u, v) ≥ 0 is called a feasible solution to the uncertain programming Model (21) if $M {v^{t} {\tilde{y}}_{k} \leq u^{t} {\tilde{x}}_{k}} \geq α for k = 1, \dots, n$

Definition 8. The feasible solution (u ^*, v ^*) is called an expected optimal solution to the uncertain programming model (21) if $E [\frac{v^{* T} {\tilde{y}}_{o}}{u^{* T} {\tilde{x}}_{o}}] \geq E [\frac{v^{T} {\tilde{y}}_{o}}{u^{T} {\tilde{x}}_{o}}]$ for any solution (u,v)

Definition 9. The greater the optimal objective value is, the more efficient DMU_o is ranked.

According to Model (21), the crisp equivalent of the uncertain Russell model is provided below in Theorem (10). $\begin{matrix} max_{σ, u_{r}, v_{i}, t_{r}} E [- σ + \sum_{r = 1}^{s} u_{r} {\tilde{y}}_{ro} - \sum_{i = 1}^{m} v_{i} {\tilde{x}}_{io} \\ + 1 - \frac{1}{m + s} \sum_{r = 1}^{s} F (t_{r})] \end{matrix}$ s.t $μ {\sum_{r = 1}^{s} u_{r} {\tilde{y}}_{rk} - \sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ik} \leq σ} \geq α k = 1, \dots, n$ (22) $μ {(m + s) v_{i} {\tilde{x}}_{io} \geq 1} \geq α i = 1, \dots, m$ $μ {(m + s) u_{r} {\tilde{y}}_{ro} \geq t_{r}} \geq α r = 1, \dots, s$

Where $F (t_{r}) = {(1 - \sqrt{t_{r}})}^{2}$

Theorem 10. Assume ${\tilde{X}}_{i 1}, {\tilde{X}}_{i 2}, . . ., {\tilde{X}}_{in}$ are independent uncertain inputs with uncertainty distribution φ_i1, φ_i2, . . . , φ_in for each i, i = 1, 2, . . . , m, and ${\tilde{y}}_{r 1}, {\tilde{y}}_{r 2}, . . ., {\tilde{y}}_{rn}$ are independent uncertain outputs with uncertainty distribution Ψ_r1, Ψ_r1, . . . , Ψ_rn for each r, r = 1, 2, . . . , s Then, the uncertain programming Model (22) is equivalent to the following model: $\begin{matrix} max_{σ, u_{r}, v_{i}, t_{r}} - σ + \sum_{r = 1}^{s} u_{r} \int_{0}^{1} ψ_{ro}^{- 1} (α) d α - \sum_{i = 1}^{m} v_{i} \\ \int_{0}^{1} φ_{io}^{- 1} (1 - α) d α + 1 - \frac{1}{m + s} \sum_{r = 1}^{s} F (t_{r}) \end{matrix}$ s.t $\sum_{r = 1}^{s} u_{r} ψ_{rk}^{- 1} (α) - \sum_{i = 1}^{m} v_{i} φ_{ik}^{- 1} (1 - α) \leq σ k = 1, \dots, n$ (23) $(m + s) v_{i} φ_{io}^{- 1} (1 - α) \geq 1 i = 1, \dots, m$ $(m + s) u_{r} ψ_{ro}^{- 1} (1 - α) \geq t_{r} r = 1, \dots, s$

Where $F (t_{r}) = {(1 - \sqrt{t_{r}})}^{2}$

Proof: First, according to Theorem (6), the objective function in (22) is rewritten as follows: $E [- σ + \sum_{r = 1}^{s} u_{r} {\tilde{y}}_{ro} - \sum_{i = 1}^{m} v_{i} {\tilde{x}}_{io} + 1 - \frac{1}{m + s} \sum_{r = 1}^{s} F (t_{r})]$ $\begin{matrix} = E [- σ] + E [\sum_{r = 1}^{s} u_{r} {\tilde{y}}_{ro}] - E [\sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ro}] \\ + E [1] - \frac{1}{m + s} E [\sum_{r = 1}^{s} F (t_{r})] \end{matrix}$ $\begin{matrix} = - σ + \sum_{r = 1}^{s} u_{r} E [{\tilde{y}}_{ro}] - \sum_{i = 1}^{m} v_{i} E [{\tilde{x}}_{ro}] + 1 \\ - \frac{1}{m + s} \sum_{r = 1}^{s} F (t_{r}) \end{matrix}$

The function $[\sum_{i = 1}^{s} v_{i} E [{\tilde{x}}_{io}]]$ is strictly decreasing with respect to ${\tilde{x}}_{k}$ , and the function $[\sum_{r = 1}^{s} u_{r} E [{\tilde{y}}_{ro}]]$ is strictly increasing with respect to ${\tilde{y}}_{k}$ for each k, as it follows from Theorems (4) and (5). $\begin{matrix} F^{- 1} (α) = - σ + \sum_{r = 1}^{s} u_{r} \int_{0}^{1} ψ_{ro}^{- 1} (α) d α - \sum_{i = 1}^{m} \\ v_{i} \int_{0}^{1} φ_{io}^{- 1} (1 - α) d α + 1 - \frac{1}{m + s} \sum_{r = 1}^{s} F (t_{r}) \end{matrix}$

Then, according to Theorem (6): $\begin{matrix} \begin{matrix} E [- σ + \sum_{r = 1}^{s} u_{r} {\tilde{y}}_{ro} - \sum_{i = 1}^{m} v_{i} {\tilde{x}}_{io} + 1 - \frac{1}{m + s} \sum_{r = 1}^{s} F (t_{r})] = \\ - σ + \sum_{r = 1}^{s} u_{r} \int_{0}^{1} ψ_{ro}^{- 1} (α) d α - \sum_{i = 1}^{m} v_{i} \int_{0}^{1} φ_{io}^{- 1} (1 - α) d α \\ + 1 - \frac{1}{m + s} \sum_{r = 1}^{s} F (t_{r}) \end{matrix} \end{matrix}$

and the objective value has proved. According to Theorems (5) and (7), the chance-constraint will be proved.□

4.2 Expected value model using uncertain programming techniques

Lio and Liu [29] presented the expected value model of uncertain programming which optimizes the expected value of an objective function under the expected constraints as follows: $θ = max_{u, v} E [\frac{v^{t} {\tilde{y}}_{o}}{u^{t} {\tilde{x}}_{o}}]$ s.t $\begin{matrix} E [\frac{v^{t} {\tilde{y}}_{k}}{u^{t} {\tilde{x}}_{k}}] \leq 1 k = 1, \dots, n \\ u \geq 0 \\ v \geq 0 \end{matrix}$ (24)

Definition 10 (Efficiency) DMUo is said to be efficient if and only if θ can achieve 1, where θ is the optimal value of (24).

The uncertain Model (24) can be applied to determine the efficiency of DMU_o by evaluating the optimal value of the model. According to Model (24), the crisp equivalent of uncertain Russell model is provided below in Theorem (11). Here, we formulate uncertain RUSSELL model as follows: $\begin{matrix} max_{σ, u_{r}, v_{i}, t_{r}} E [- σ + \sum_{r = 1}^{s} u_{r} {\tilde{y}}_{ro} - \sum_{i = 1}^{m} v_{i} {\tilde{x}}_{io} + \\ 1 - \frac{1}{m + s} \sum_{r = 1}^{s} F (t_{r})] \end{matrix}$ s.t $E {\sum_{r = 1}^{s} u_{r} {\tilde{y}}_{rk} - \sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ik} - σ} \leq 0 k = 1, \dots, n$ (25) $E {1 - (m + s) v_{i} {\tilde{x}}_{io}} \leq 0 i = 1, \dots, m$ $E {t_{r} - (m + s) u_{r} {\tilde{y}}_{ro}} \leq 0 r = 1, \dots, s$

Where $F (t_{r}) = {(1 - \sqrt{t_{r}})}^{2}$

Theorem 11. Assume ${\tilde{x}}_{i 1}, {\tilde{x}}_{i 2}, . . ., {\tilde{x}}_{in}$ are independent uncertain inputs with uncertainty distribution φ_i1, φ_i2, . . . , φ_in for each i, i = 1, 2, . . . , m, and ${\tilde{y}}_{r 1}, {\tilde{y}}_{r 2}, . . ., {\tilde{y}}_{rn}$ are independent uncertain outputs with uncertainty distribution ψ_r1, ψ_r2, . . . , ψ_rn for each r, r = 1, 2, . . . , s. Then the uncertain programming Model (25) is equivalent to the following model: $\begin{matrix} max_{σ, u_{r}, v_{i}, t_{r}} - σ + \sum_{r = 1}^{s} u_{r} \int_{0}^{1} ψ_{ro}^{- 1} (α) d α - \\ \sum_{i = 1}^{m} v_{i} \int_{0}^{1} φ_{io}^{- 1} (1 - α) d α + 1 - \frac{1}{m + s} \sum_{r = 1}^{s} F (t_{r}) \end{matrix}$ s.t $\sum_{r = 1}^{s} u_{r} \int_{0}^{1} ψ_{rk}^{- 1} (α) d α - \sum_{i = 1}^{m} v_{i} \int_{0}^{1} φ_{ik}^{- 1} (1 - α) d α \leq σ k = 1, \dots, n$ (26) $(m + s) v_{i} \int_{0}^{1} φ_{io}^{- 1} (1 - α) d α \geq 1 i = 1, \dots, m$ $(m + s) u_{r} \int_{0}^{1} ψ_{ro}^{- 1} (1 - α) d α \geq t_{r} r = 1, \dots, s$

Where $F (t_{r}) = {(1 - \sqrt{t_{r}})}^{2}$

Where φ _1o, φ _2o, . . . , φ _mo and ψ _1o, ψ _2o, . . . , ψ _ro are the regular uncertainty distributions of ${\tilde{x}}_{1 o}, {\tilde{x}}_{2 o}, . . ., {\tilde{x}}_{mo}$ and ${\tilde{y}}_{1 o}, {\tilde{y}}_{2 o}, . . ., {\tilde{y}}_{ro}$ , respectively.

Proof: According to Theorems (4), (5) and (6) theorem is proved.□

Corollary 1. If the ${\tilde{x}}_{io}$ , ${\tilde{y}}_{ro}$ , ${\tilde{x}}_{ik}$ and ${\tilde{y}}_{rk}$ are the independent uncertain variables of the form $L (a, b)$ with a, b ∈ R and a < b, then the crisp equivalent of Model (26) can be written as shown in Model (29): $\begin{matrix} max_{σ, u_{r}, v_{i}, t_{r}} - σ + \sum_{r = 1}^{s} u_{r} (\frac{a_{ro} + b_{ro}}{2}) \\ - \sum_{i = 1}^{m} v_{i} (\frac{a_{io} + b_{io}}{2}) + 1 - \frac{1}{m + s} \sum_{r = 1}^{s} F (t_{r}) \end{matrix}$ s.t

$\begin{matrix} \sum_{r = 1}^{s} u_{r} (\frac{a_{rk} + b_{rk}}{2}) - \sum_{i = 1}^{m} v_{i} (\frac{a_{ik} + b_{ik}}{2}) \leq σ \\ k = 1, \dots, n \end{matrix}$ (27) $(m + s) v_{i} (\frac{a_{io} + b_{io}}{2}) \geq 1 i = 1, \dots, m$ $(m + s) u_{r} (\frac{a_{ro} + b_{ro}}{2}) \geq t_{r} r = 1, \dots, s$

Where $F (t_{r}) = {(1 - \sqrt{t_{r}})}^{2}$

Proof: According to Theorem (8) $\begin{matrix} E [a ξ] & = \int_{0}^{1} a φ^{- 1} (1 - α) d α \\ = a \int_{0}^{1} φ^{- 1} (α) d α = aE [ξ] \end{matrix}$

Now, according to Theorem (6), rewrite (25) as follows: $\begin{matrix} max_{σ, u_{r}, v_{i}, t_{r}} - σ + \sum_{r = 1}^{s} u_{r} E [{\tilde{y}}_{ro}] - \sum_{i = 1}^{m} v_{i} E [{\tilde{x}}_{io}] \\ + 1 - \frac{1}{m + s} \sum_{r = 1}^{s} F (t_{r}) \end{matrix}$ s.t $\sum_{r = 1}^{s} u_{r} E [{\tilde{y}}_{rk}] - \sum_{i = 1}^{m} v_{i} E [{\tilde{x}}_{ik}] - σ \leq 0 k = 1, \dots, n$ (28) $1 - (m + s) v_{i} E [{\tilde{x}}_{io}] \leq 0 i = 1, \dots, m$ $t_{r} - (m + s) u_{r} E [{\tilde{y}}_{ro}] \leq 0 r = 1, \dots, s$

Where $F (t_{r}) = (1 - \sqrt{t_{r}})^{2}$

From the linearity property of expected value operator, the crisp equivalent of uncertain RUSSELL model in Model (28) can be formulated as follows: $\begin{matrix} max_{σ, u_{r}, v_{i}, t_{r}} - σ + \sum_{r = 1}^{s} u_{r} (\frac{a_{ro} + b_{ro}}{2}) \\ - \sum_{i = 1}^{m} v_{i} (\frac{a_{io} + b_{io}}{2}) + 1 - \frac{1}{m + s} \sum_{r = 1}^{s} F (t_{r}) \end{matrix}$ s.t $\begin{matrix} \sum_{r = 1}^{s} u_{r} (\frac{a_{rk} + b_{rk}}{2}) - \sum_{i = 1}^{m} v_{i} (\frac{a_{ik} + b_{ik}}{2}) \\ \leq σ k = 1, \dots, n \end{matrix}$ $(m + s) v_{i} (\frac{a_{io} + b_{io}}{2}) \geq 1 i = 1, \dots, m$ $(m + s) u_{i} (\frac{a_{ro} + b_{ro}}{2}) \geq t_{r} r = 1, \dots, s$ (29)

Where $F (t_{r}) = (1 - \sqrt{t_{r}})^{2}$

According to Definition 9, the greater the optimal objective value of the corresponding optimal solution of (22), the more efficiently the DMU_o is ranked. However, efficiency cannot be determined by the optimal value of (22). Nevertheless, as for Definition 10, the uncertain Model (25) can be applied to determine the efficiency of DMU_o by evaluating the optimal value of the model.□

5 Practical Example

In this section, we applied the RUSSELL uncertain models to an example of an Iranian bank to see how the target customer accepts a new banking product before the promotion and announcement phase. To this end, 64 branch managers (banking experts) were selected to answer to an eight-item questionnaire (Table 1) to produce the linear uncertain variable $ξ \sim L (a, b)$ to test the newly implemented models (Table 2).

Table 1
Input and output questions

input question

To what percent do you predict the acceptability of this product would be for customers?

To what percent does this product have competitive power with the rivals?

To what percent does this product have the ability of WOM (word-of-mouth)?

Resources that this product managed to grab is a number between ... and...

output question

To what percent will you predict the success of this product?

To what percent do you think this product will attract the customers?

To what percent will this product could grab rival banks’ resources?

The loan which must be paid for the resources of this product is a number between... and ...

input question
To what percent do you predict the acceptability of this product would be for customers?
To what percent does this product have competitive power with the rivals?
To what percent does this product have the ability of WOM (word-of-mouth)?
Resources that this product managed to grab is a number between ... and...
output question
To what percent will you predict the success of this product?
To what percent do you think this product will attract the customers?
To what percent will this product could grab rival banks’ resources?
The loan which must be paid for the resources of this product is a number between... and ...

Table 2

DMUs with four uncertain inputs and four uncertain outputs

DMU_i	Inputs				Outputs
	1	2	3	4	1	2	3	4
1	$L$ (60,70)	$L$ (60,70)	$L$ (60,70)	$L$ (20,25)	$L$ (50,70)	$L$ (70,80)	$L$ (40,50)	$L$ (14,17.5)
2	$L$ (70,80)	$L$ (60,70)	$L$ (40,50)	$L$ (50,120)	$L$ (50,70)	$L$ (50,60)	$L$ (40,50)	$L$ (35,84)
3	$L$ (80,90)	$L$ (70,80)	$L$ (70,80)	$L$ (50,80)	$L$ (80,90)	$L$ (80,90)	$L$ (80,90)	$L$ (35,56)
4	$L$ (70,100)	$L$ (50,100)	$L$ (40,100)	$L$ (1,3)	$L$ (50,100)	$L$ (70,100)	$L$ (80,100)	$L$ (.7,2.1)
5	$L$ (10,15)	$L$ (10,15)	$L$ (5,10)	$L$ (.3,.5)	$L$ (10,20)	$L$ (10,15)	$L$ (5,10)	$L$ (.21,.35)
6	$L$ (30,60)	$L$ (30,40)	$L$ (30,50)	$L$ (30,40)	$L$ (30,50)	$L$ (30,40)	$L$ (30,60)	$L$ (21,28)
7	$L$ (50,100)	$L$ (50,100)	$L$ (30,100)	$L$ (1,2)	$L$ (50,100)	$L$ (30,100)	$L$ (30,100)	$L$ (.7,1.4)
8	$L$ (70,90)	$L$ (60,70)	$L$ (30,60)	$L$ (70,80)	$L$ (70,90)	$L$ (70,90)	$L$ (20,30)	$L$ (49,56)
9	$L$ (30,70)	$L$ (40,70)	$L$ (40,80)	$L$ (20,50)	$L$ (20,60)	$L$ (30,70)	$L$ (20,50)	$L$ (14,35)
10	$L$ (80,90)	$L$ (60,80)	$L$ (40,80)	$L$ (7,10)	$L$ (50,80)	$L$ (50,70)	$L$ (40,70)	$L$ (4.9,7)
11	$L$ (5,10)	$L$ (10,20)	$L$ (10,15)	$L$ (1.5,2.5)	$L$ (10,15)	$L$ (10,20)	$L$ (10,20)	$L$ (1.05,1.7)
12	$L$ (10,30)	$L$ (17,49)	$L$ (11,35)	$L$ (20,30)	$L$ (15,20)	$L$ (13,20)	$L$ (34,51)	$L$ (14,21)
13	$L$ (20,40)	$L$ (20,30)	$L$ (20,40)	$L$ (10,15)	$L$ (20,30)	$L$ (10,20)	$L$ (10,15)	$L$ (7,10.5)
14	$L$ (50,80)	$L$ (70,80)	$L$ (50,60)	$L$ (20,30)	$L$ (50,90)	$L$ (60,80)	$L$ (1,50)	$L$ (14,21)
15	$L$ (70,100)	$L$ (70,100)	$L$ (80,100)	$L$ (10,30)	$L$ (80,100)	$L$ (90,100)	$L$ (70,100)	$L$ (7,21)
16	$L$ (60,80)	$L$ (80,100)	$L$ (70,80)	$L$ (60,70)	$L$ (20,70)	$L$ (80,90)	$L$ (80,90)	$L$ (42,49)
17	$L$ (20,30)	$L$ (30,50)	$L$ (10,30)	$L$ (10,30)	$L$ (30,60)	$L$ (10,30)	$L$ (10,20)	$L$ (7,21)
18	$L$ (50,70)	$L$ (40,50)	$L$ (20,50)	$L$ (10,30)	$L$ (20,70)	$L$ (30,80)	$L$ (20,50)	$L$ (7,21)
19	$L$ (20,40)	$L$ (10,20)	$L$ (60,90)	$L$ (10,30)	$L$ (20,50)	$L$ (20,40)	$L$ (10,20)	$L$ (7,21)
20	$L$ (50,80)	$L$ (30,60)	$L$ (30,50)	$L$ (30,60)	$L$ (30,70)	$L$ (50,80)	$L$ (30,60)	$L$ (21,42)
21	$L$ (30,40)	$L$ (40,50)	$L$ (30,40)	$L$ (20,35)	$L$ (50,60)	$L$ (20,30)	$L$ (10,15)	$L$ (14,24.5)
22	$L$ (10,50)	$L$ (10,50)	$L$ (10,30)	$L$ (10,35)	$L$ (35,45)	$L$ (10,50)	$L$ (10,50)	$L$ (7,10.5)
23	$L$ (30,50)	$L$ (10,20)	$L$ (20,30)	$L$ (5,10)	$L$ (50,70)	$L$ (30,50)	$L$ (10,20)	$L$ (3.5,7)
24	$L$ (80,100)	$L$ (90,100)	$L$ (80,100)	$L$ (5,30)	$L$ (80,100)	$L$ (90,100)	$L$ (70,100)	$L$ (3.5,21)
25	$L$ (10,20)	$L$ (10,15)	$L$ (10,20)	$L$ (5,10)	$L$ (20,25)	$L$ (5,15)	$L$ (10,20)	$L$ (3.5,7.5)
26	$L$ (20,30)	$L$ (10,15)	$L$ (1,20)	$L$ (5,10)	$L$ (20,30)	$L$ (10,20)	$L$ (5,10)	$L$ (3.5,7.5)
27	$L$ (20,30)	$L$ (5,10)	$L$ (10,20)	$L$ (4,8)	$L$ (10,20)	$L$ (10,20)	$L$ (10,20)	$L$ (2.8,5.6)
28	$L$ (60,70)	$L$ (50,60)	$L$ (60,70)	$L$ (5,10)	$L$ (60,70)	$L$ (50,60)	$L$ (50,60)	$L$ (3.5,7)
29	$L$ (60,80)	$L$ (40,60)	$L$ (30,50)	$L$ (3,7)	$L$ (60,80)	$L$ (60,80)	$L$ (40,60)	$L$ (2.1,4.9)
30	$L$ (1,60)	$L$ (10,50)	$L$ (1,90)	$L$ (20,50)	$L$ (10,40)	$L$ (20,50)	$L$ (20,60)	$L$ (14,35)
31	$L$ (10,30)	$L$ (10,15)	$L$ (20,30)	$L$ (1,2)	$L$ (20,30)	$L$ (10,20)	$L$ (10,20)	$L$ (.7,1.4)
32	$L$ (40,50)	$L$ (90,100)	$L$ (40,50)	$L$ (10,20)	$L$ (1,10)	$L$ (40,50)	$L$ (40,50)	$L$ (7,14)
33	$L$ (10,40)	$L$ (5,20)	$L$ (5,20)	$L$ (5,20)	$L$ (10,50)	$L$ (10,30)	$L$ (5,10)	$L$ (3.5,14)
34	$L$ (70,90)	$L$ (60,70)	$L$ (50,60)	$L$ (50,70)	$L$ (60,80)	$L$ (40,50)	$L$ (30,40)	$L$ (35,49)
35	$L$ (80,90)	$L$ (70,80)	$L$ (80,90)	$L$ (25,35)	$L$ (85,95)	$L$ (70,80)	$L$ (50,60)	$L$ (17.5,24.)
36	$L$ (50,60)	$L$ (70,80)	$L$ (40,50)	$L$ (20,30)	$L$ (40,60)	$L$ (40,50)	$L$ (30,40)	$L$ (14,21)
37	$L$ (1,20)	$L$ (1,20)	$L$ (1,10)	$L$ (1,5)	$L$ (1,30)	$L$ (10,30)	$L$ (10,10)	$L$ (.7,3.5)
38	$L$ (40,70)	$L$ (30,50)	$L$ (10,50)	$L$ (10,15)	$L$ (50,80)	$L$ (20,50)	$L$ (30,50)	$L$ (7,10.5)
39	$L$ (50,70)	$L$ (10,50)	$L$ (30,80)	$L$ (40,70)	$L$ (50,70)	$L$ (30,80)	$L$ (30,80)	$L$ (28,49)
40	$L$ (50,100)	$L$ (50,100)	$L$ (70,100)	$L$ (12,15)	$L$ (50,100)	$L$ (50,100)	$L$ (30,100)	$L$ (8.4,10.5)
41	$L$ (90,100)	$L$ (60,100)	$L$ (80,100)	$L$ (20,50)	$L$ (90,100)	$L$ (90,100)	$L$ (80,100)	$L$ (14,35)
42	$L$ (5,10)	$L$ (5,10)	$L$ (5,10)	$L$ (1,2)	$L$ (5,10)	$L$ (1,10)	$L$ (1,10)	$L$ (.7,1.4)
43	$L$ (10,95)	$L$ (10,80)	$L$ (10,80)	$L$ (14,40)	$L$ (10,95)	$L$ (10,100)	$L$ (10,70)	$L$ (9.8,28)
44	$L$ (1,90)	$L$ (1,60)	$L$ (1,80)	$L$ (60,75)	$L$ (1,80)	$L$ (1,70)	$L$ (1,60)	$L$ (42,52.5)
45	$L$ (50,90)	$L$ (50,90)	$L$ (50,90)	$L$ (2,20)	$L$ (50,80)	$L$ (50,90)	$L$ (50,90)	$L$ (1.4,14)
46	$L$ (40,60)	$L$ (80,90)	$L$ (50,60)	$L$ (54,70)	$L$ (50,70)	$L$ (30,50)	$L$ (70,80)	$L$ (37.8,49)
47	$L$ (1,60)	$L$ (1,50)	$L$ (1,70)	$L$ (1,50)	$L$ (1,100)	$L$ (1,50)	$L$ (1,50)	$L$ (1,35)
48	$L$ (40,60)	$L$ (40,50)	$L$ (1,10)	$L$ (1,3)	$L$ (50,70)	$L$ (60,70)	$L$ (10,20)	$L$ (.7,2.1)
49	$L$ (1,5)	$L$ (5,10)	$L$ (1,5)	$L$ (.5,1)	$L$ (5,10)	$L$ (1,5)	$L$ (1,5)	$L$ (.35,.7)
50	$L$ (30,40)	$L$ (20,30)	$L$ (20,30)	$L$ (2,5)	$L$ (40,50)	$L$ (30,40)	$L$ (20,30)	$L$ (1.4,3.5)
51	$L$ (1,4)	$L$ (1,2)	$L$ (1,5)	$L$ (1,10)	$L$ (1,5)	$L$ (1,7)	$L$ (1,4)	$L$ (.7,7)
52	$L$ (99,100)	$L$ (20,30)	$L$ (50,60)	$L$ (25,30)	$L$ (50,80)	$L$ (50,80)	$L$ (20,30)	$L$ (17.5,21)
53	$L$ (50,60)	$L$ (20,30)	$L$ (20,50)	$L$ (30,50)	$L$ (50,70)	$L$ (20,50)	$L$ (20,40)	$L$ (21,35)
54	$L$ (20,30)	$L$ (40,50)	$L$ (20,30)	$L$ (10,20)	$L$ (40,50)	$L$ (50,60)	$L$ (30,40)	$L$ (7,14)
55	$L$ (10,30)	$L$ (70,90)	$L$ (30,40)	$L$ (1,10)	$L$ (40,50)	$L$ (10,30)	$L$ (10,20)	$L$ (.7,7)
56	$L$ (1,20)	$L$ (10,20)	$L$ (1,10)	$L$ (5,20)	$L$ (1,10)	$L$ (1,10)	$L$ (30,50)	$L$ (3.5,14)
57	$L$ (50,70)	$L$ (30,70)	$L$ (30,50)	$L$ (30,50)	$L$ (40,60)	$L$ (40,60)	$L$ (30,60)	$L$ (21,35)
58	$L$ (80,100)	$L$ (60,100)	$L$ (95,100)	$L$ (5,30)	$L$ (95,100)	$L$ (99,100)	$L$ (70,100)	$L$ (3.5,21)
59	$L$ (1,10)	$L$ (1,20)	$L$ (1,10)	$L$ (20,25)	$L$ (1,20)	$L$ (1,30)	$L$ (1,15)	$L$ (14,17.5)
60	$L$ (80,100)	$L$ (10,20)	$L$ (5,10)	$L$ (.1,1)	$L$ (20,30)	$L$ (40,50)	$L$ (10,15)	$L$ (.07,.7)
61	$L$ (60,100)	$L$ (80,100)	$L$ (70,100)	$L$ (50,100)	$L$ (70,100)	$L$ (50,100)	$L$ (80,100)	$L$ (35,70)
62	$L$ (10,50)	$L$ (10,70)	$L$ (10,80)	$L$ (50,100)	$L$ (10,50)	$L$ (10,70)	$L$ (10,65)	$L$ (35,70)
63	$L$ (40,70)	$L$ (30,50)	$L$ (10,50)	$L$ (10,15)	$L$ (50,80)	$L$ (20,50)	$L$ (30,50)	$L$ (7,10.5)
64	$L$ (99,100)	$L$ (99,100)	$L$ (99,100)	$L$ (99,100)	$L$ (99,100)	$L$ (99,100)	$L$ (99,100)	$L$ (35,70)

Results revealed that 37 branches supported this new product as an attractive use case, indicating a 57% success probability for the new product before the announcement phase. In other words, this new method assisted managers in making a proper decision to implement the new banking product. As a result of classical banking system, in order to employ a new product in branches, first a cost-benefit analysis and must be performed; if result is positive; it will be directly announced in branches while the customers’ feedback on the product is not considered. In several cases, they solely performed the cost-benefit analysis phase and then, in practice, customers did not accept the product, thereby wasting a considerable amount of money in the banking system.

Results for Model (23) showed that 37 branches supported this new product as an attractive use case, showing a 57% success likelihood for the product before the announcement phase (Table 3).

Table 3

The result of evaluating the efficiency with model (23) (α= 0.5)

DMU1	DMU2	DMU3	DMU4	DMU5	DMU6	DMU7	DMU8
0.9126	0.9160	1	1	1	0.9160	1	1
inefficiency	inefficiency	Efficiency	Efficiency	efficiency	inefficiency	efficiency	efficiency
DMU9	DMU10	DMU11	DMU12	DMU13	DMU14	DMU15	DMU16
1	0.8855	1	0.8891	0.7271	0.8623	1	1
efficiency	inefficiency	Efficiency	Inefficiency	inefficiency	inefficiency	efficiency	efficiency
DMU17	DMU18	DMU19	DMU20	DMU21	DMU22	DMU23	DMU24
1	0.8947	0.8613	1	1	0.8919	1	0.9716
efficiency	inefficiency	inefficiency	efficiency	efficiency	inefficiency	efficiency	inefficiency
DMU25	DMU26	DMU27	DMU28	DMU29	DMU30	DMU31	DMU32
0.9316	0.8634	1	0.9238	1	0.8995	1	0.7291
inefficiency	inefficiency	efficiency	Inefficiency	efficiency	inefficiency	efficiency	inefficiency
DMU33	DMU34	DMU35	DMU36	DMU37	DMU38	DMU39	DMU40
0.9056	0.8549	0.9336	0.8209	1	1	1	0.9293
inefficiency	inefficiency	inefficiency	Inefficiency	efficiency	efficiency	efficiency	inefficiency
DMU41	DMU42	DMU43	DMU44	DMU45	DMU46	DMU47	DMU48
1	0.8238	0.9230	0.9052	0.9431	1	1	1
efficiency	efficiency	inefficiency	Inefficiency	inefficiency	efficiency	efficiency	efficiency
DMU49	DMU50	DMU51	DMU52	DMU53	DMU54	DMU55	DMU56
1	1	1	1	1	1	1	1
efficiency	efficiency	efficiency	Efficiency	efficiency	efficiency	efficiency	inefficiency
DMU57	DMU58	DMU59	DMU60	DMU61	DMU62	DMU63	DMU64
0.9180	1	1	1	1	0.9143	1	1
inefficiency	efficiency	efficiency	Efficiency	efficiency	inefficiency	efficiency	efficiency

This results for Model (26) is presented in Table 4. According to the CPU times reported in Table 6, it can be observed that Model (23) is run in 140 seconds depending on the combination of confidence levels, while Model (26) is run in 6665 seconds depending on the combination of confidence levels. It seems that Model (23) has the best performance in terms of CPU running time in big data, whereas Model (26) has the worst performance.

Table 4

The result of evaluating the efficiency with model (26) (α= 0.5)

DMU1	DMU2	DMU3	DMU4	DMU5	DMU6	DMU7	DMU8
0.9126	0.9160	1	1	1	0.9160	1	1
inefficiency	inefficiency	Efficiency	Efficiency	efficiency	inefficiency	efficiency	efficiency
DMU9	DMU10	DMU11	DMU12	DMU13	DMU14	DMU15	DMU16
1	0.8855	1	0.8891	0.7271	0.8623	1	1
efficiency	inefficiency	Efficiency	Inefficiency	inefficiency	inefficiency	efficiency	efficiency
DMU17	DMU18	DMU19	DMU20	DMU21	DMU22	DMU23	DMU24
1	0.8947	0.8613	1	1	0.8919	1	0.9716
efficiency	inefficiency	inefficiency	efficiency	efficiency	inefficiency	efficiency	inefficiency
DMU25	DMU26	DMU27	DMU28	DMU29	DMU30	DMU31	DMU32
0.9316	0.8634	1	0.9238	1	0.8995	1	0.7291
inefficiency	inefficiency	efficiency	Inefficiency	efficiency	inefficiency	efficiency	inefficiency
DMU33	DMU34	DMU35	DMU36	DMU37	DMU38	DMU39	DMU40
0.9056	0.8549	0.9336	0.8209	1	1	1	0.9293
inefficiency	inefficiency	inefficiency	Inefficiency	efficiency	efficiency	efficiency	inefficiency
DMU41	DMU42	DMU43	DMU44	DMU45	DMU46	DMU47	DMU48
1	0.8238	0.9230	0.9052	0.9431	1	1	1
efficiency	efficiency	inefficiency	Inefficiency	inefficiency	efficiency	efficiency	efficiency
DMU49	DMU50	DMU51	DMU52	DMU53	DMU54	DMU55	DMU56
1	1	1	1	1	1	1	1
efficiency	efficiency	efficiency	Efficiency	efficiency	efficiency	efficiency	inefficiency
DMU57	DMU58	DMU59	DMU60	DMU61	DMU62	DMU63	DMU64
0.9180	1	1	1	1	0.9143	1	1
inefficiency	efficiency	efficiency	Efficiency	efficiency	inefficiency	efficiency	efficiency

The results of Lio and Liu’s [29] radial model demonstrated that 15 branches support this new product. The new product is an unattractive use case, indicating a 23% success likelihood for the product before the announcement phase (Table 5). It can be seen that the model is run in 812 seconds depending on the combination of confidence levels.

Table 5

The result of evaluating the efficiency with Lio and Liu (2017) radial model (α= 0.5)

DMU1	DMU2	DMU3	DMU4	DMU5	DMU6	DMU7	DMU8
0.6200	0.4126	0.4126	1	0.9461	0.4733	1	07912
inefficiency	inefficiency	inefficiency	efficiency	inefficiency	inefficiency	efficiency	inefficiency
DMU9	DMU10	DMU11	DMU12	DMU13	DMU14	DMU15	DMU16
1	0.8501	0.7608	0.7075	0.6657	1	0.6934	0.5937
efficiency	inefficiency	inefficiency	inefficiency	inefficiency	efficiency	inefficiency	Inefficiency
DMU17	DMU18	DMU19	DMU20	DMU21	DMU22	DMU23	DMU24
0.6739	0.5710	1	0.5323	0.8921	0.6939	0.6843	0.7411
inefficiency	inefficiency	efficiency	inefficiency	inefficiency	inefficiency	inefficiency	Inefficiency
DMU25	DMU26	DMU27	DMU28	DMU29	DMU30	DMU31	DMU32
0.5552	0.7109	0.6816	0.8383	0.6330	0.7634	1	1
inefficiency	inefficiency	inefficiency	inefficiency	inefficiency	inefficiency	efficiency	efficiency
DMU33	DMU34	DMU35	DMU36	DMU37	DMU38	DMU39	DMU40
0.7456	0.5988	0.5786	0.6737	1	0.7186	0.4601	0.8562
inefficiency	inefficiency	inefficiency	Inefficiency	efficiency	inefficiency	inefficiency	inefficiency
DMU41	DMU42	DMU43	DMU44	DMU45	DMU46	DMU47	DMU48
0.5381	0.9406	0.6849	1	0.8747	0.6707	1	1
inefficiency	inefficiency	inefficiency	efficiency	inefficiency	inefficiency	efficiency	efficiency
DMU49	DMU50	DMU51	DMU52	DMU53	DMU54	DMU55	DMU56
0.8598	0.6229	1	0.7736	0.4953	0.4851	1	1
inefficiency	inefficiency	efficiency	inefficiency	inefficiency	inefficiency	efficiency	efficiency
DMU57	DMU58	DMU59	DMU60	DMU61	DMU62	DMU63	DMU64
0.4745	0.7449	0.9456	1	0.4613	0.7936	0.7186	0.4224
inefficiency	inefficiency	inefficiency	efficiency	inefficiency	inefficiency	inefficiency	inefficiency

Table 6

The compare of ranking of DMUs in three models (α= 0.5)

DMU_i	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
rank with model 23	49	46	1	1	1	46	1	1	1	56	1	55	64	58	1	1
CPU time(seconds)	3	2	2	2	3	2	2	2	2	2	2	2	2	3	2	2
rank with model 26	49	46	1	1	1	46	1	1	1	56	1	55	64	58	1	1
CPU time(seconds)	97	88	82	82	122	96	82	115	75	108	116	116	88	150	82	82
rank with model
Lio and Liu	48	63	63	1	16	59	1	26	1	23	29	36	45	1	38	50
CPU time(seconds)	15	20	18	9	14	14	7	15	7	8	10	12	16	8	13	15
DMU_i	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
rank with model 23	1	53	59	1	1	54	1	38	41	57	1	43	1	52	1	63
CPU time(seconds)	2	2	2	2	3	2	2	4	2	2	2	2	2	2	2	4
rank with model 26	1	53	59	1	1	54	1	38	41	57	1	43	1	52	1	63
CPU time(seconds)	89	124	89	116	170	116	89	190	95	82	89	102	84	103	96	175
rank with model
Lio and Liu	42	52	1	54	19	37	40	32	53	35	41	24	46	28	1	1
CPU time(seconds)	22	20	6	10	11	21	11	14	12	18	13	13	11	13	8	6
DMU_i	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48
rank with model 23	50	60	49	62	1	1	1	42	1	61	44	51	39	1	1	1
CPU time(seconds)	3	2	3	2	2	2	2	2	2	2	2	2	3	2	2	2
rank with model 26	50	60	49	62	1	1	1	42	1	61	44	51	39	1	1	1
CPU time(seconds)	130	111	144	123	116	82	82	96	82	122	122	89	135	91	124	90
rank with model
Lio and Liu	30	49	51	43	1	33	61	22	55	18	39	1	20	44	1	1
CPU time(seconds)	17	14	15	13	11	8	16	15	18	9	19	6	12	10	6	10
DMU_i	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63	64
rank with model 23	1	1	1	1	1	1	1	1	45	1	1	1	1	48	1	1
CPU time(seconds)	3	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
rank with model 26	1	1	1	1	1	1	1	1	45	1	1	1	1	48	1	1
CPU time(seconds)	130	89	123	82	123	82	82	89	90	82	110	83	83	123	83	82
rank with model
Lio and Liu	21	47	1	27	56	57	1	1	58	31	17	1	60	25	33	62
CPU time(seconds)	11	17	8	23	12	13	9	5	13	12	12	6	15	14	10	23

In other words, this new method helps managers make a rational decision in implementing the new banking product. In the classical banking system, for a new product to be employed in branches, a cost-benefit analysis process must first be performed and if the result is positive, it would be directly announced in branches, while no attention is paid to the customer feedback about the attractiveness of the product. In several cases, branches solely performed the cost-benefit analysis whereas later, in practice, customers did not accept the product which resulted in the failure of the new product with huge cost. Finally, we attempted to show the ranking of all branches in the three models (Table 6).

Given the similarity of the results obtained by the two Models (23) and (26), it can be concluded that Definitions 9 and 10 are equivalent for both the models. Therefore, it can be claimed that the efficiency of a DMU in Model (26) has a direct relationship to the maximum value of Model (23) corresponding to that DMU. Table 6 summarizes the comparison results between the proposed Models (23) and (26) and the Lio and Liu’s [30] radial model. The results of Lio and Liu’s [29] radial model show that 15 branches are efficient, representing that the product is inappropriate for the announcement phase. This is precisely the disadvantage of uncertain radial models compared with the uncertain non-radial models described in this paper. Since BANK CEOs (chief executive officer) cannot decrease (increase) all inputs (outputs) with the same ratio with regard to this product and in the real world in general, the results of the uncertain radial models confuse decision-makers when evaluating newly designed products.

6 Conclusions

In this paper, we aimed to explain an uncertain RUSSELL model with inherent complexity for uncertain models. Due to the complexity of the new model, we recommended two models and compared them with the existing model. These models attempted to crisp the new uncertain model. Finally, an applicable example regarding the Iranian banking system was proposed to document the new models. For this purpose, we selected 64 branch managers (bank experts) to respond to an eight-item questionnaire to examine the success probability for a new product before the announcement phase. This method assisted bank CEOs in choosing the product which might be interesting to the costumer. Further studies can find a new method for saving cost in bank products before promotion and announcement phase.

As a future study direction, the proposed variance instead of the expected value in objective functions with uncertain variables can be taken into account. Also, the problem of this paper with normal type uncertain variables can be studied.

Footnotes

Acknowledgments

The authors are grateful of the editors and reviewers of the journal for their helpful and constructive comments that improved the quality of the paper.

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