1 Introduction

Banks play a direct role in the economic growth of every country; and a study of banks, as the chief financial system is of particular importance. Data envelopment analysis (DEA) has absorbed numerous researchers, as a technique, to assess the efficiency or performance of banks. Sherman and Gold (1985) were the initial researchers to evaluate the efficiency of a savings bank branches and utilized DEA for this assessment. There are many papers available in the grounds of the banking industry, by the means of DEA techniques. In noting, that these studies are related to the branches of banks of a country or that these assessments are comparisons of banks between several countries, are categorized into two groups. Similarly, varied review studies have also been executed in the sphere of financial services within the grounds of the DEA; and papers present and published in this respect, have been divided into various classifications based on the assessment perspective. Kaffash and Marra (2017) revised 620 papers in this arena and presented three main categories in the field of financial services, referred to, as the banking industry, the insurance industry and money market funds.

The first DEA Model, reputedly known as the CCR Model was debated by Charnes et al. (1978). Not only does this technique have the capabilities of evaluating the efficiency of DMUs with multiple inputs and outputs, but has the ability to render a “target” for every inefficient DMU. Comparing Data Envelopment Analysis and Multiple Objective Linear Programming Problem, Joro et al. (1998) mentioned that in both methods the purpose is characterizing efficient facets and identifying projection points as targets for inefficient units. The target point is the coordinate of the projection point of the DMU under assessment and lies on the efficiency frontier. Target points indicate the amount of changes required for the inputs and outputs of the inefficient DMUs to perform efficiently. To find the benchmark, many of the existing models, namely, slack based measurement (SBM) or range adjusted method (RAM) are designed; Cooper et al. (2000). These models find the maximum distance from the efficiency frontier, thus they introduce the furthest target. While, if the closest target is introduced, the inefficient DMU can reach the efficiency frontier with the minimum changes in inputs and outputs.

Finding the closest target has been one of the important issues in DEA literature. Frei and Harker (1999) applied the Euclidean distance to find the closest efficient point on the supporting hyperplane. However, as the introduced point is projected on the supporting hyperplane and not on the efficiency frontier, the introduced targets may not belong to the production possibility set. Coelli (1998) proposed a multi-stage method to find the closest target using radial models. Gonzalez and Alvarez (2001) suggested minimizing the sum of input contractions in order to reach the efficient frontier. Cherchye and Van Puyenbroeck (2001) also used oriented measures and the least distance combination. To find closest targets, Portela et al. (2003) used Qhull, which was proposed by Barber et al. (1996) that determines all the facets of a polyhedron. Meanwhile, Lozano and Villa (2005) proposed a method which determines a sequence of targets. Briec (1998) introduced a family of least distance, based on the inefficiency measure. Baek and Lee (2009) proposed a weighted least distance as the efficiency measure in DEA and stated that, their approach introduces not only the closest benchmark, but also it is a well-defined efficiency measure. However, using a counter example, Pastor and Aparicio (2010), showed that the measure introduced by Baek and Lee (2009) does not satisfy strong monotonicity on the strong efficient frontier. Cook and Seiford (2009) in their DEA survey, classified different papers on determining DEA benchmarks based on the least distance measure. Additionally, Jahanshahloo et al. (2012a) used minimum distance for evaluating group performance of DMUs. Moreover, Jahanshahloo et al. (2012b) suggested a Bi-level Linear Programming problem to obtain the closest targets from the strong efficient frontier.

Ando et al. (2012) explained that least-distance measure based on Hölder norms does not satisfy weak monotonicity and they suggested transforming the original definition of Hölder distance functions. Fukuyama et al. (2014) discussed about least distance p-norm efficiency satisfying strong monotonicity. Meanwhile, Amirteimoori and kordrostami (2010) offering an efficiency measure based on the Euclidean distance, proposed a super-efficiency model which not only ranks the DMUs, but also introduces closest targets. Besides Aparicio et al. (2007) proposed a model for finding the closest target by considering the set of Pareto-efficient points, which dominate the DMU under the assessment; and they stated that the proposed model can be applied instead of the classical additive model. Their method has been used numerously in the literature. With respect to this method, Cook et al. (2017) developed an approach for benchmarking DMUs in groups that experience similar circumstances. Moreover, Ruiz and Sirvent (2011) determined the technical and allocative components of the overall profit efficiency. Ruiz and Sirvent (2016) also developed a common framework for benchmarking and ranking the DMUs. In addition, Aparicio et al. (2017a, b, c) introduced a method based on Bi-level Linear Programming to attain the least distance for DEA oriented models. Furthermore, Aparicio (2016) proposes a method to measure the productivity change of decision making units based on the calculation of the least distance to the Pareto-efficient frontier. Their method always leads to feasible solutions. Additionally, Ruiz and Sirvent (2016) developed a common best practice frontier for benchmarking and ranking the DMUs. Aparicio et al. (2014) applying the principle of least action (PLA), decomposed technical inefficiency and showed the usefulness of the proposed approach evaluating 28 international airlines. Recently, Aparicio (2016) reviewed published studies of determining the least distance in DEA and classified contributions from methodological point of view. Besides, Aparicio et al. (2017a, b, c) proposed an input-oriented weighted additive model for estimating technical inefficiency based on the PLA.

Evaluating the DMUs, using DEA models, the weights are chosen in such a manner that the evaluated DMU achieves its best performance. This freedom in selecting the weights leads to achieving inappropriate efficiency measures, as the weight for some inputs or outputs may be considered so small that these inputs and outputs cannot influence the evaluation; Thanassoulis (1995). One of the suggestions to tackle this problem is to consider weight restrictions. To review some methods of weight restriction in DEA, one can refer to Allen et al. (1997), Thanassoulis et al. (2004) and Cooper et al. (2011). Moreover, symmetric weight restriction was first proposed by Dimitrov and Sutton (2009). Jahanshahloo et al. (2011) used symmetric weights as a secondary goal in DEA cross-efficiency evaluation. Podinovski (1999), has extensively discussed relative efficiency, in the presence of weight restrictions. Also Jahanshahloo et al. (2005), applied a goal programming technique in order to avoid infeasibility and introduced a feasible interval for DEA models with weight restrictions. Assurance regions were first proposed by Thompson et al. (1986) and Thompson (1990). Later, inspired by that, Cook et al. (1996) ranked the candidates in preferential voting. Thanassoulis et al. (2004) expressed that for considering the decision-makers and experts ‘opinions in evaluating and also reflecting on the significance of inputs and outputs as to each other, it is necessary to use weight restrictions. By applying the opinions of experts in the model, the achieved results are more to the satisfaction of the decision makers; therefore, weight restriction is of particular importance from the managerial viewpoint. Podinovski (2004a) and Podinovski (2005) mentioned that there may be judgment between the inputs or outputs in production technology. Which appear as trade-offs in an envelopment form of DEA models and as weight restrictions in the multiplier form. Following the proposed principles by Banker et al. (1984), Podinovski (2004a) introduced the production possibility set (PPS) with trade-off. One of the main problems of this PPS was that, the targets obtained by the models that were generated by this extended PPS were sometimes in the negative region. To tackle this problem, Podinovski (2007) proposed a three-level method for calculating non-dominated targets. Subsequently, Davoodi and Zhiani Rezai (2014) proposed a new condition which leads to a new PPS, so as to deal with the problem of placing the points in the negative region and then regarding the new proposed PPS, they introduced a two-level method for finding target points. Also Atici and Podinovski (2015) discussed about specializing DMUs using production trade-offs in DEA and applied their approach for assessment of agricultural farms in Turkey.

Adding weight restriction to the multiplier form of the model may cause some problems, such as infeasibility or attaining dominated targets. Thanassoulis and Allen (1998) discussed tackling the problem of target points and introducing a non-dominated target. Moreover, Podinovski (2004b) argued over calculating the relative efficiency in the presence of non-homogeneous weight restrictions and the redundancy of certain types of weight restriction in DEA models. Furthermore, Podinovski (2016) expressed that, in presence of any type of weight restriction, the interpretation of the proposed target in the multiplier form of the model is the same as the target point in the envelopment form and the benchmark is obtained according to all the DMUs including the observed DMUs and the generated virtual DMUs.

Moreover, Ruiz et al. (2015) proposed a model for finding the closest target by considering the preferences of experts and adding weight restrictions to the problem. Following their research, Ramon et al. (2016) discussed about benchmarking in the presence of weight restrictions. They stated that, if weight restrictions represent production trade-offs, the introduced benchmarks are achievable. But if the weight restrictions are based on value judgments or preferences, the target point may not necessarily be attainable.

In DEA models, the target point of each inefficient DMU is always on the efficient frontier of the production possibility set on which the model is defined. Attention should be paid to the fact that, the production possibility set may alter with the addition of tradeoffs. This change always moves towards enlarging the previous PPS; that is, with the supplement of weight constraints, the PPS may get enlarged. Hence, in models, where the PPS is generated in the presence of weight constraints, the target point for the DMU under evaluation are on the efficient frontier of the new PPS. This projection point can be a point of the prior PPS or in the exterior of this PPS. Since, the addition of weight constraints in this paper, generates a new PPS in which all the points and the efficient frontier are accessible. Thereby, the benchmark attained from the proposed model is available for each inefficient DMU.

In this paper, a model for seeking the closest target in the presence of weight constraints is proposed. The weight constraints imposed in the model is a result of the tradeoffs. The difference between the proposed model and that of the model of Ramon et al. (2016) can be explained by the disparity of the objective function and the constraints enforced on the configuration of the model. Moreover, by utilizing Theorem 2, it has been proven that, the proposed model, always ensures the shortest path to the efficient frontier. The presented target point is in the extended production possibility set, it is non-dominated and therefore, is on the efficiency frontier and the target point is obtainable. In addition to which, a numerical example also illustrates that the proposed model in comparison to the Ramon et al. (2016) model, offers less amount of deviation between the inputs/outputs of the DMU under evaluation and the target point.

The paper is written as follow: In the second section, a conventional approach to finding the closest target and weight restrictions concept on DEA models are discussed. The proposed model is expressed in the third section and it is also compared with the previous models. In the fourth section, the proposed model is administered for evaluating one of Iranian Commercial Bank. The results are mentioned in section five.

2 Preliminaries

2.1 Finding the closest target

Let us assume that there are n DMUs each of which uses m inputs to produce s output. The jth input vector is illustrated by Xj = (x1j,…,xmj) and the jth output vector is illustrated by Yj = (y1j,…,ysj) for j = 1,…,n and we suppose that the input and output vectors are non-negative.

By accepting the five principles: feasibility of observed data, convexity, Free disposability, constant returns to scale (CRS) and minimum of interpolation of PPS, the production possibility set is as follows (Charnes et al. 1978):

$$ T_{CRS} = \left\{ {(X,Y)|X \ge \sum\limits_{j = 1}^{n} {\lambda_{j} X_{j} ,Y \le \sum\limits_{j = 1}^{n} {\lambda_{j} Y_{j} ,\lambda \ge 0} } } \right\} $$
(1)

By maximizing the total amount of input or output slacks, many DEA models actually calculate the maximum distance of the DMU under evaluation from efficiency frontier and therefore the farthest target is introduced.

Considering the Pareto-efficient points which dominate DMUp, Aparicio et al. (2007) presented a model for finding the closest target. In fact. By applying the L1-distance norm, Aparicio et al. (2007) presented the following Mixed-Integer linear programming problem:

$$ \begin{aligned} & (mADD)\begin{array}{*{20}c} {Min} & {\sum\limits_{i = 1}^{m} {s_{ip}^{ - } } + } \\ \end{array} \sum\limits_{r = 1}^{s} {s_{rp}^{ + } } \\ & \begin{array}{*{20}l} {s.t.} \hfill & {\sum\limits_{j \in E} {\lambda_{j} x_{ij} } = x_{ip} - s_{i}^{ - } ,} \hfill & {i = 1, \ldots ,m.} \hfill & {(2.1)} \hfill \\ {} \hfill & {\sum\limits_{j \in E} {\lambda_{j} y_{rj} } = y_{rp} + s_{r}^{ + } ,} \hfill & {r = 1, \ldots ,s.} \hfill & {(2.2)} \hfill \\ {} \hfill & {\sum\limits_{r = 1}^{s} {u_{r} } y_{rj} - \sum\limits_{i = 1}^{m} {v_{i} x_{ij} + d_{j} = 0,} } \hfill & {j \in E} \hfill & {(2.3)} \hfill \\ {} \hfill & {u_{r} \ge 1,} \hfill & {r = 1, \ldots ,s.} \hfill & {(2.4)} \hfill \\ {} \hfill & {v_{i} \ge 1,} \hfill & {i = 1, \ldots ,m.} \hfill & {(2.5)} \hfill \\ {} \hfill & {d_{j} \le M\gamma_{j} ,} \hfill & {j \in E} \hfill & {(2.6)} \hfill \\ {} \hfill & {\lambda_{j} \le M(1 - \gamma_{j} ),} \hfill & {j \in E} \hfill & {(2.7)} \hfill \\ {} \hfill & \gamma_{j} \in \left\{ {0,1} \right\}, \hfill & {j \in E} \hfill & {(2.8)} \hfill \\ {} \hfill & \lambda ,d,\mu ,s^{ - } ,s^{ + } \ge 0 \hfill \\ \hfill \\ \end{array} \\ \end{aligned} $$
(2)

E is the set of efficient DMUs.

DMUp is efficient if and only if the objective function of the preceding model equals zero; whereas, if the objective function replaces with \( \sum\nolimits_{i = 1}^{m} {\frac{{s_{ip}^{ - } }}{{x_{ip} }}} + \sum\nolimits_{r = 1}^{s} {\frac{{s_{rp}^{ + } }}{{y_{rp} }}} \) and also the constraints \( \begin{array}{*{20}c} {u_{r} y_{rp} \ge 1} & {r = 1, \ldots ,s} \\ \end{array} \) and \( \begin{array}{*{20}c} {v_{i} x_{ip} \ge 1} & {i = 1, \ldots ,m} \\ \end{array} \) replaces with constraints (2.4) and (2.5), the efficiency score will become unit invariant.

2.2 Weight restriction and trade-off

Utilizing weight restrictions, increases the ability to distinguish the performance of DMUs in DEA models. Weight restrictions are added to the multiplier form of models and they can be divided into four categories, based on whether they are homogeneous or non-homogeneous and linked or unlinked. On the other hand, there may be trade-offs among the inputs and outputs of DMUs. Podinovski (2005) mentioned that trade-off is a statement that changes of some of inputs or outputs are technologically possible, without affecting remaining inputs and outputs. For example, the first input can decrease at the most amount ‘A’, as long as the second input is increased to ‘B’’. Such judgments in technology appear as trade-offs in an envelopment form of DEA models and as weight restrictions in the multiplier form. Weight restrictions which are constructed by trade-off will extended the PPS and are producible.

In assuming that, the vector (Pt, Qt) illustrates the value of changes in inputs and outputs. Podinovski (2004a) identified the following six principles to define the PPS, in the presence of trade-offs:

  1. (A1)

    Feasibility of observed data. (Xj, Yj) ∊ T for any j = 1,…,n.

  2. (A2)

    Convexity. The set T is convex.

  3. (A3)

    Free disposability. If (X, Y) ∊ T and X ≤ X′, Y ≥ Y′ then(X′, Y′) ∊ T.

  4. (A4)

    Feasibility of trade-offs. If (X, Y) ∊ T then for any trade-off in the form (Pt, Qt) and any πt ≥ 0 the unit (X + πtPt, Y + πtQt) ∊ T, provided X + πtPt ≥ 0 and Y + πtQt ≥ 0.

  5. (A5)

    Proportionality. For (X, Y) ∊ T and any α ≥ 0, (αX, αY) ∊ T.

  6. (A6)

    Closedness: The set T is closed.

Regarding these principles, the minimal PPS was proposed as follows:

$$ T_{CRS - To} = \left\{ {\begin{array}{*{20}l} {(X,Y)|X \ge 0,Y \ge 0,X = \sum\limits_{j = 1}^{n} {\lambda_{j} X_{j} } + \sum\limits_{t = 1}^{k} {\pi_{t} P_{t} } + d,} \hfill \\ {Y = \sum\limits_{j = 1}^{n} {\lambda_{j} Y_{j} } + \sum\limits_{t = 1}^{k} {\pi_{t} Q_{t} } - e,\lambda \in R_{ + }^{n} ,\pi \in R_{ + }^{k} ,d \in R_{ + }^{m} ,e \in R_{ + }^{s} } \hfill \\ \end{array} } \right\} $$
(3)

The proceeding PPS may also include a negative area. To fix this problem, Davoodi and Zhiani Rezai (2014), presented a new PPS which has the non-negative condition, in addition to the previous principles; and they introduced the PPS in presence of weight restrictions as follows:

$$ T^{'} = \left\{ {\begin{array}{*{20}l} {(X,Y)|X \ge 0,Y \ge 0,X \ge \sum\limits_{j = 1}^{n} {\lambda_{j} X_{j} } + \sum\limits_{t = 1}^{k} {\pi_{t} P_{t} } ,Y \le \sum\limits_{j = 1}^{n} {\lambda_{j} Y_{j} } + \sum\limits_{t = 1}^{k} {\pi_{t} Q_{t} } } \hfill \\ {\sum\limits_{j = 1}^{n} {\lambda_{j} X_{j} } + \sum\limits_{t = 1}^{k} {\pi_{t} P_{t} } \ge 0,\sum\limits_{j = 1}^{n} {\lambda_{j} Y_{j} } + \sum\limits_{t = 1}^{k} {\pi_{t} Q_{t} } \ge 0,\lambda \in R_{ + }^{n} ,\pi \in R_{ + }^{k} } \hfill \\ \end{array} } \right\} $$
(4)

and proved that this set is the minimal PPS which satisfies the principles (A1)–(A6).

3 Proposed model

In this section, a model for finding the closest target in the presence of weight restrictions is proposed in the first sub-section. Then, the proposed model is compared with the previous models in the second sub-section.

3.1 A model for finding the closest target in the presence of weight restrictions

The less the difference between the inputs and outputs of the DMU under evaluation and the presented target is, the DMU can reach the efficiency frontier with fewer amount of changes. In considering the experts’ opinions in modeling too, makes the achieved results more desirable for the experts. In this section, we present a model for finding the closest target when there are weight restrictions based on trade-offs. Consider the following homogenous and unlinked weight restrictions:

$$ \begin{aligned} \sum\limits_{r = 1}^{s} {u_{r} a_{rt} \le } 0,\quad t = 1, \ldots ,N, \hfill \\ \sum\limits_{i = 1}^{m} {v_{i} b_{it} } \ge 0,\quad t = N + 1, \ldots ,L. \hfill \\ \end{aligned} $$

The trade-off vectors of the above constraint are Pt = 0, Qt = (a1t,…, ast); t = 1,…,N and Pt = (b1t,…, bmt), Qt = 0; t = N + 1,…, L, respectively. Considering the following trade-off matrix.

$$ \left[ {\begin{array}{*{20}c} {a_{1t} , \ldots ,a_{st} } & 0 \\ 0 & {b_{1t} , \ldots ,b_{mt} } \\ \end{array} } \right]_{L \times (s + m)} $$

The above constraints can be written as follows:

$$ \sum\limits_{r = 1}^{s} {u_{r} Q_{rt} } - \sum\limits_{i = 1}^{m} {v_{i} P_{it} } \le 0,\quad t = 1, \ldots ,L. $$

Regarding T′, the additive model with the existence of trade-offs among the input or output vectors, as mentioned above, is as follows:

$$ \begin{aligned} & \gamma^{*}_{p} = \begin{array}{*{20}c} {Max} & {\sum\limits_{i = 1}^{m} {s_{i}^{ - } } } \\ \end{array} + \sum\limits_{r = 1}^{s} {s_{r}^{ + } } \\ & \begin{array}{*{20}l} {s.t.} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} } x_{ij} + \sum\limits_{t = 1}^{L} {\mu_{t} P_{t} + s_{i}^{ - } = x_{ip} ,} } \hfill & {i = 1, \ldots ,m.} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} } y_{rj} + \sum\limits_{t = 1}^{L} {\mu_{t} Q_{t} - s_{r}^{ + } = y_{rp} ,} } \hfill & {r = 1, \ldots ,s.} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} } x_{ij} + \sum\limits_{t = 1}^{L} {\mu_{t} P_{t} \ge 0,} } \hfill & {i = 1, \ldots ,m.} \hfill \\ {} \hfill & \begin{aligned} \sum\limits_{j = 1}^{n} {\lambda_{j} } y_{rj} + \sum\limits_{t = 1}^{L} {\mu_{t} Q_{t} \ge 0,} \hfill \\ \lambda ,\mu^{'} ,S^{ + } ,S^{ - } \ge 0 \hfill \\ \end{aligned} \hfill & {r = 1, \ldots ,s.} \hfill \\ \end{array} \\ \\ \end{aligned} $$
(5)

The dual of model (5) is:

$$ \begin{aligned} \begin{array}{*{20}l} Min & {VX_{p} - UY_{p} }& \\ {s.t.}&&\\ & {VX_{j} - UY_{j} - W^{\prime \prime } X_{j} - W^{\prime } Y_{j} \ge 0,} &\quad {j = 1, \ldots ,n.} \\ {} & {VP_{t} - UQ_{t} - W^{\prime \prime } P_{t} - W^{\prime } Q_{t} \ge 0,} &\quad {t = 1, \ldots ,L.} \\ {} & {u_{r} \ge 1,} &\quad {r = 1, \ldots ,s.} \\ {} & {v_{i} \ge 1,} &\quad {i = 1, \ldots ,m.} \\ {} & {w^{{^{\prime } }} ,w^{\prime \prime } \ge 0} & {} \\ \end{array} \end{aligned} $$
(6)

Solving model (5), if the objective function equals zero, it means that the evaluated DMU in the extended technology (production technology in the presence of weight restriction or trade-off) is strongly efficient.

Theorem 1

Suppose that Dp is the set of the Pareto-efficient points of T′ which dominateDMUp. Then (X, Y) ∊ Dp if and only if there exists:

$$ \begin{array}{*{20}l} {\lambda_{j} ,d_{j} \ge 0,} \hfill & {\gamma_{j} \in \left\{ {0,1} \right\}} \hfill & {j \in E} \hfill \\ {\mu_{t} ,d_{t}^{\prime } \ge 0} \hfill & {\gamma_{t}^{'} \in \left\{ {0,1} \right\}} \hfill & {t = 1, \ldots ,L.} \hfill \\ {u_{r} \ge 1;s_{r}^{ + } \ge 0;w_{r}^{\prime } \ge 0} \hfill & {} \hfill & {r = 1, \ldots ,s.} \hfill \\ {v_{i} \ge 1;s_{i}^{ - } \ge 0;w_{i}^{\prime \prime } \ge 0} \hfill & {} \hfill & {i = 1, \ldots ,m.} \hfill \\ \end{array} $$

such that

$$ \begin{array}{*{20}c} {X = \sum\limits_{j \in E} {\lambda_{j} x_{j} } + \sum\limits_{t = 1}^{L} {\mu_{t} P_{t} } } & {} & {(a.1)} \\ {Y = \sum\limits_{j \in E} {\lambda_{j} y_{j} } + \sum\limits_{t = 1}^{L} {\mu_{t} Q_{t} } } & {} & {(a.2)} \\ {\sum\limits_{j \in E} {\lambda_{j} x_{ij} } + \sum\limits_{t = 1}^{L} {\mu_{t} P_{t} } = x_{ip} - s_{i}^{ - } ,} & {i = 1, \ldots ,m.} & {(a.3)} \\ {\sum\limits_{j \in E} {\lambda_{j} y_{rj} } + \sum\limits_{t = 1}^{L} {\mu_{t} Q_{t} } = y_{rp} + s_{r}^{ + } ,} & {r = 1, \ldots ,s.} & {(a.4)} \\ {\sum\limits_{j = 1}^{n} {\lambda_{j} } x_{j} + \sum\limits_{t = 1}^{L} {\mu_{t} P_{t} } \ge 0} & {i = 1, \ldots ,m.} & {(a.5)} \\ {\sum\limits_{j = 1}^{n} {\lambda_{j} } y_{j} + \sum\limits_{t = 1}^{L} {\mu_{t} Q_{t} } \ge 0} & {r = 1, \ldots ,s.} & {(a.6)} \\ {\sum\limits_{r = 1}^{s} {u_{r} } y_{rj} - \sum\limits_{i = 1}^{m} {v_{i} x_{ij} + \sum\limits_{r = 1}^{s} {w_{r}^{\prime } y_{rj} } + \sum\limits_{i = 1}^{m} {w_{i}^{\prime \prime } x_{ij} } + d_{j} = 0,} } & {j \in E} & {(a.7)} \\ {d_{j} \le M\gamma_{j} ,} & {j \in E} & {(a.8)} \\ {\lambda_{j} \le M(1 - \gamma_{j} ),} & {j \in E} & {(a.9)} \\ {\sum\limits_{r = 1}^{s} {u_{r} Q_{rt} - \sum\limits_{i = 1}^{m} {v_{i} P_{it} + \sum\limits_{i = 1}^{m} {w_{i}^{\prime \prime } P_{it} } + } \sum\limits_{r = 1}^{s} {w_{r}^{\prime } Q_{rt} } + d_{t}^{\prime } } = 0} & {t = 1, \ldots ,L.} & {(a.10)} \\ {d_{t}^{\prime } \le M\gamma_{t}^{\prime } ,} & {t = 1, \ldots ,L.} & {(a.11)} \\ {\mu_{t} \le M\left( {1 - \gamma_{t}^{\prime } } \right),} & {t = 1, \ldots ,L.} & {(a.12)} \\ \end{array} $$

M is a large positive scalar and E denotes the set of strong efficient DMUs.

Proof

See the “Appendix A”.□

Theorem (1) shows the set of points dominating DMUp which lies on the strong efficient frontier in the presence of trade-offs or weight restrictions. In other words, Dp is the set of strong efficient points dominating DMUp in the extended PPS.

Definition 1

Least L1-distance measure of the point \( \bar{X} \) from the closed Set S is as follows:

$$ LD_{{\bar{X}:S}} = Min\left\{ {\left\| {X - \bar{X}} \right\|_{1} ;X \in S} \right\} $$

Regarding Theorem (1) and the above mentioned definition, we suggest the following model for finding the closest target in the presence of homogenous and unlinked weight restrictions:

$$ \begin{aligned} & Z_{p} = \begin{array}{*{20}c} {Min} & {\sum\limits_{i = 1}^{m} {s_{i}^{ - } } } \\ \end{array} + \sum\limits_{r = 1}^{s} {s_{r}^{ + } } \\ & \begin{array}{*{20}c} {s.t.} & {\sum\limits_{j \in E} {\lambda_{j} x_{ij} } + \sum\limits_{t = 1}^{L} {\mu_{t} P_{t} } = x_{ip} - s_{i}^{ - } ,} & {i = 1, \ldots ,m.} & {(7.1)} \\ {} & {\sum\limits_{j \in E} {\lambda_{j} y_{rj} } + \sum\limits_{t = 1}^{L} {\mu_{t} Q_{t} } = y_{rp} + s_{r}^{ + } ,} & {r = 1, \ldots ,s.} & {(7.2)} \\ {} & {\sum\limits_{j = 1}^{n} {\lambda_{j} } x_{j} + \sum\limits_{t = 1}^{L} {\mu_{t} P_{t} } \ge 0} & {i = , \ldots ,m.} & {(7.3)} \\ {} & {\sum\limits_{j = 1}^{n} {\lambda_{j} } y_{j} + \sum\limits_{t = 1}^{L} {\mu_{t} Q_{t} } \ge 0} & {r = 1, \ldots ,s.} & {(7.4)} \\ {} & {\sum\limits_{j = 1}^{n} {\lambda_{j} } y_{rj} - \sum\limits_{i = 1}^{m} {v_{i} x_{ij} + \sum\limits_{r = 1}^{s} {w_{r}^{\prime } y_{rj} } + \sum\limits_{i = 1}^{m} {w_{i}^{\prime \prime } x_{ij} } + d_{j} = 0,} } & {j \in E} & {(7.5)} \\ {} & {d_{j} \le M\gamma_{j} ,} & {j \in E} & {(7.6)} \\ {} & {\lambda_{j} \le M(1 - \gamma_{j} ),} & {j \in E} & {(7.7)} \\ {} & {\gamma_{j} \in \left\{ {0,1} \right\},} & {j \in E} & {(7.8)} \\ {} & {\sum\limits_{r = 1}^{s} {u_{r} Q_{rt} - \sum\limits_{i = 1}^{m} {v_{i} P_{it} + \sum\limits_{i = 1}^{m} {w_{i}^{\prime \prime } P_{it} } + } \sum\limits_{r = 1}^{s} {w_{r}^{\prime } Q_{rt} } + d_{t}^{\prime } } = 0,} & {t = 1, \ldots ,L} & {(7.9)} \\ {} & {d_{t}^{'} \le M\gamma_{t}^{\prime } ,} & {t = 1, \ldots ,L.} & {(7.10)} \\ {} & {\mu_{t} \le M(1 - \gamma_{t}^{\prime } ),} & {t = 1, \ldots ,L.} & {(7.11)} \\ {} & {\gamma_{t}^{'} \in \left\{ {0,1} \right\},} & {t = 1, \ldots ,L.} & {(7.12)} \\ {} & {u_{r} \ge 1,} & {r = 1, \ldots ,s.} & {(7.13)} \\ {} & {v_{i} \ge 1,} & {i = 1, \ldots ,m.} & {(7.14)} \\ {} & {\lambda ,d,d^{\prime } ,\mu ,S^{ - } ,S^{ + } ,w^{\prime } ,w^{\prime \prime } \ge 0} & {} & {(7.15)} \\ \end{array} \\ \end{aligned} $$
(7)

Of which, M is a large positive scalar and E is the set of efficient DMUs.

Theorem 2

If the optimal value of the objective function of model (7) equals \( Z_{p}^{*} \) then the least L1-distance of DMUp from the strong efficient frontier DP is equal to \( Z_{p}^{*} \).

Proof

See the “Appendix B”.□

Theorem 3

DMUp is efficient if and only if the objective function score of model (7) equals zero.

Proof

See the “Appendix C”.□

Model (7) is a mixed integer linear programming problem (MIP) which introduces a non-dominated target which lies on the efficient frontier of the extended PPS for each DMU under the assessment.

The constraints (7.1)–(7.4) represent all the points of the extended PPS which dominate DMUp. (7.5) and (7.9) are the constraints that correspond to the multiplier form of the additive model with trade-offs which were explained earlier in models (5) and (6).

If λj > 0, then based on (7.7) γj = 0 and based on (7.6) dj = 0 which means \( \sum\nolimits_{r = 1}^{s} {u_{r} y_{rj} } + \sum\nolimits_{i = 1}^{m} {v_{i} x_{ij} + \sum\nolimits_{r = 1}^{s} {w_{r}^{\prime } y_{rj} } + \sum\nolimits_{i = 1}^{m} {w_{i}^{\prime \prime } x_{ij} } = 0,} \).

Therefore, if DMUj plays any role in producing a target, it is binding on the hyperplane. Thus, the feasible region of model (7) for each evaluated DMU introduces a facet of the extended PPS which the target point and all the points participating in producing the target, belong to this facet.

To find the closest target, the L1-distance norm has been used, and by minimizing the summation of input and output slacks to Dp (the set of strongly efficient points which dominate DMUp), the closest target is obtained for each observed DMU. By “closest” here, we mean the minimum changes of inputs and outputs of a DMU to reach the efficient frontier. It is worth mentioning that, instead of using the norm 1 for calculating the distance, the other norms such as the infinity norm (Chebyshev norm), or L2-norm can be used. The overall, model (7) looks for a non-dominated point in which the evaluated DMU has the minimum distance from the strong efficient frontier in the PPS in the presence of weight restrictions. Note that weight restrictions are constructed by production trade-offs and any point of the extended PPS is technologically producible. Theorem (1) ensures that the proposed target lies on the Pareto-efficient frontier.

Lemma 1

Suppose that the weight restrictions are consistent, then model (7) is always feasible.

Proof

See the “Appendix D”.□

Lemma 2

The optimal value of the objective function of model (5) is an upper bound of the optimal value of the objective function of model (7) (\( Z_{p}^{*} \le \gamma_{p}^{*} \)).

Proof

See the “Appendix E”.□

If the objective function of model (7) replaces by \( \sum\nolimits_{i = 1}^{m} {\frac{{s_{ip}^{ - } }}{{x_{ip} }}} + \sum\nolimits_{r = 1}^{s} {\frac{{s_{rp}^{ + } }}{{y_{rp} }}} \) and we put the constraints \( \begin{array}{*{20}c} {u_{r} y_{rp} \ge 1} & {r = 1, \ldots ,s} \\ \end{array} \) and \( \begin{array}{*{20}c} {v_{i} x_{ip} \ge 1} & {i = 1, \ldots ,m} \\ \end{array} \) instead of the constraints \( \begin{array}{*{20}c} {u_{r} \ge 1} & {r = 1, \ldots ,s} \\ \end{array} \) and \( \begin{array}{*{20}c} {v_{i} \ge 1} & {i = 1, \ldots ,m} \\ \end{array} \), the efficiency score will turn to unit invariant. It is obvious that it has been assumed that the inputs and outputs are strictly positive.

If the objective function is maximized, then the optimal solution of model (7) is equivalent to model (5) or an additive model in the presence of trade-offs or weight restrictions. Therefore, the proposed model can be used as an additive model in the presence of trade-off and the proposed target shows the least changes in inputs and outputs for evaluated DMU to reach the efficiency frontier.

Solving model (7), coordinates of the target point are as follows:

$$ \begin{aligned} \bar{X} & = \sum\limits_{j \in E} {\lambda_{j}^{*} x_{j} } + \sum\limits_{t = 1}^{L} {\mu_{t}^{*} P_{t} } \\ \bar{Y} & = \sum\limits_{j \in E} {\lambda_{j}^{*} y_{j} } + \sum\limits_{t = 1}^{L} {\mu_{t}^{*} Q_{t} } \\ \end{aligned} $$

where \( \lambda_{j}^{*} ;\; j \in E \) and \( \mu_{t}^{*} ;\; t = 1, \ldots , L \) are the optimal solutions of model (7).

Model (7) is a multi-objective linear programming problem, which is converted into a single-objective linear programming problem, by utilizing weighted sum approach and the L1-norm. The objective function can be solved by other multi-objective resolving methods, for example the Lexicographic Method, in which each of the components of the objective function of model (7) is minimized according to priority; and hence, it is possible to prioritize increasing or decreasing of indexes as changing them needs less effort.

3.2 A comparison of the proposed model with the previous models

In considering the experts’ opinions, Ruiz et al. (2015) and Ramon et al. (2016) presented models for finding the closest target. In this sub-section the result of their methods is compared with the proposed model in this paper.

Taking into account the expert preferences, Ruiz et al. (2015) proposed a model to find the closest target. In their proposed model, they didn’t consider the requirement of trade-offs which result from added weight restrictions, which is mentioned in model (5) and model (6) in this paper. To clarify the difference between Ruiz et al. (2015) and our proposed model, consider the following example. Figure 1 is based on the data presented by Podinovski (2004a), for 3 observed DMUs with one input and two outputs. Trade-off vectors are P1 = 0, Q = (1,− 1) and P2 = 0, Q = (− 2, 1).

Fig. 1
figure 1

“The PPS with and without trade-offs” —Podinovski (2004a)

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The dark hatched region represents the original PPS. Considering trade-off between outputs or consequent weight restrictions i.e. u1 − u2 ≤ 0 and − 2u1 + u2 ≤ 0, the light dotted area is added to PPS (extended PPS). The extended PPS is based on trade-offs, so DMUB and DMUD are technologically producible. Hence target of DMU1 is generated by DMU2 and DMUB, but Ruiz et al. (2015) model introduces only DMU2 as a reference. Table 1 shows the results of the proposed model for the proceeding example.

Table 1 Results of the proposed model using data of Podinovski (2004a)
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The results of model (7) are illustrated in Table 1. The sixth column depicts the objective function score of model (7) which represents the least distance using L1-norm. In comparison with the fifth column, which depicts the maximum distance, the DMU under evaluation will reach the efficiency frontier with fewer amount of changes by model (7). Comparing the coordinate of the target point in the seventh column with the coordinate of the evaluated DMU, shows that input of DMU1 needs to be decreased to 87.5 which means \( s_{1}^{ - } = 12.5 \) and input of DMU3 needs to be decreased to 86.11 which means \( s_{1}^{ - } = 13.89 \). Theorem (2) guarantees that these are the minimum amount of changes each DMU needs to lies on the efficiency frontier. The last column shows the introduced reference set.

Later Ramon et al. (2016) recommended an approach that identifies the best practice regarding weight restrictions. They mentioned that if weight restrictions are based on value judgments or preferences, targets are not necessarily attainable. To compare their model with the proposed model in this paper, suppose 7 DMUs which consume 2 inputs to produce 2 outputs. Trade-off vector between inputs and outputs is P = (− 4, 1), Q = (− 4, 1), consequently the weight restrictions in dual form are u2 ≤ 4u1 and 4v1 ≤ v2. Table 2 shows the minimum distance of Ramon et al. (2016) and the proposed model, using L1-norm.

Table 2 Results of the proposed model versus Ramon et al. (2016)
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The last two columns of Table 2, illustrates the target point of each inefficient DMU by utilizing both models. The views denoted by Ramon et al. (2016) and this paper differs. Ramon et al. (2016) mentioned that if weight restrictions are based on value judgments, then all points of the extended PPS are not necessarily producible and they suggested a model which provides attainable targets. However in this paper, weight restrictions represent production trade-offs and the extended production possibility is producible. A comparison between the sixth and seventh columns demonstrates that, the proposed model of this paper, presents the minimal distance of the unit under evaluation to the extended PPS. The purpose of the abovementioned example is to state the dissimilarity of these two models.

4 Empirical example

Banks play a critical role in the enhancement and economic development of every country. They pursue to improve their performance, so as to secure a better share in the market and as a result, increment in profits. Today, an evaluation of performance in banking industry is one of the most crucial activities of the bank managers. Moreover, an introduction of a target pattern to an inefficient branch assists in a sounder understanding and recognition of the weak points of that branch, as well as a perceptive as to the policies of other branches, from which it can benefit, so as to gain an improvement in their performance. In this context, the introduction of a model which offers the minimum modifications for the inefficient branch is of specific importance.

In accordance with the evaluation indexes, Berger and Humphrey (1997), divided the studies into two main categories, these being the production and intermediation approaches, where, in the first approach, banks are considered as service supplying units; and the employees are treated as inputs and the services rendered to clienteles as outputs. Whereas, in the second approach, banks are considered as financial intermediaries, between those holding savings and investors who collect funds and provide loans, including other assets. Paradi et al. (2018), in their review paper denoted another approach, called the profitability approach. In addition to which, Ouenniche and Carrales (2018), classified the research literature into six categories and considered the commercial banks of UK as intermediation agents. These banks were surveyed throughout a period of 29 years. Eventually, they proposed a computation of efficiency, utilizing slacks-based measures, in place of radial models. Moreover, Fethi and Pasiouras (2010) or Ouenniche et al. (2017) can be mentioned, for gaining more information, in the sphere of selecting inputs and outputs.

It must be observed that in this paper, a part of the shares of the bank, under study, is governmental; and the bank branches have been assessed from both, the intermediation and production perspective. Aspects, such as, indexes relative to staff and facilities are entailed in the production viewpoint, whereas, indexes relating to interest received and or interest paid are contemplated as indexes, where a bank is considered as a mediator in the evaluation.

In this section, the proposed model has been taken into contemplation for the evaluation of 66 branches of one of the Iranian banks. The data in relative to the branches of one of a commercial bank in Iran and is from the year 1394 AH equivalent to March 2015–2016.

In considering trade-offs and weight restrictions which incorporate technological judgment, the least distance to the Pareto Efficient Frontier for each branch is calculated by means of Model (7) and ultimately, the minimum modifications necessary, in terms of the total inputs and outputs, are assessed for each branch.

Staff privileges, Interests paid and arrears are considered as input indicators, whereas, facilities, total deposits, Interests received, bank charges received and other sources are output indexes. The selected indicators for evaluating bank branches are briefly described hereunder:

  1. I1.

    Staff Privileges: This is an index which comes to hand by combining the quantitative and qualitative indices of manpower present in a branch of a bank. The amount of manpower, level of training involved, the organization’s post, work experience, educational degree and its relativity with the occupation or profession, are factors which are combined and their normalized weighted sum retrieves the staff privileges.

  2. I2.

    Interests Paid: Banks are forced to tolerate the payment of costs for securing their proceeds or investments, in order to compensate their deficiency of capital. The resources collected, comprise of bank accounts of individuals and legal entities, for which the bank has to pay to its clienteles interest for these deposits. This amount depends on the shelf-life of the client’s resources.

  3. I3.

    Arrears: One of the main duties of banks is to grant facilities or loans to legal or authentic clients. Normally, banks attain guarantees for loan installments from customers. But in some cases, all or part of the loans paid by the client and receiver of loan are not refundable. This is a demand of the bank from the client and in this respect banks endure heavy damages and costs.

  4. O1.

    Facilities: The amount of loan banks render to authentic or legal clients.

  5. O2.

    Total Deposits: This comprises of an aggregate of current deposits, Gharz-Al-HassanehFootnote 1 deposits (without interest) and short and long-term deposits. These are held by authentic or legal persons in banks on a basis of deposits, enabling banks to utilize these deposits in the way of investments or granting of loans, in addition to which, of having a manner to receive profits.

  6. O3.

    Interests received: Banks grant loans to their clientele. The various types of loans are entitled to distinctive bank interest or dividends. The customer is usually compelled to pay this also (profit or interest), when the principal loan is being paid to the bank. The amount of total interests which banks receive from their clientele for these loans are called Interests received.

  7. O4.

    Bank Charges Received: In addition to the absorption of resources and granting facilities, banks also perform services, such as, remittance, sale of bonds, bank guarantees and …. For these services, they receive service or bank charges from their customers.

  8. O5.

    Other Sources: Moreover, in addition to the total deposits secured in banks by authentic and legal clients, governmental organizations and non-governmental organizations also hold individual accounts to secure their resources, apart from classical deposits. Or some accounts are present, for which clientele receive services, such as deposits for housing and…. The total sum of these resources is known as other sources.

The mean, median, minimum, maximum and standard deviation of input and output indices, the coefficient variation and skewness of 66 branches of the bank under study has been illustrated in Table 3.

Table 3 Statistical analysis of input and output indices
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The second and third rows of Table 3 demonstrate the mean and median of each one of the input and output indices respectively. According to the second row, the minimum mean relates to the staff privileges with a value of 9 and the maximum is relevant to facilities, holding a value of 63,736,310,123. The third row indicates that fifty percent of the data in the staff privileges rating is less than 7 and fifty percent of the facilities are more than 46,143,100,258. The maximum and minimum datum indicates that the total data is within a range of \( (2.67, 3 4 7 8 1 2000000) \). The standard deviation of each index has been shown in the sixth row. The lower the value, the better, as it represents less dispersion amongst the data. Since the values of inputs and outputs are not equivalent in terms of scale or unit of measurement, in order to compare the dispersion of indices, the variance coefficient has been taken into assistance. As observed in 7th row of Table 3, the third input (arrears), the third output (interests received), the fifth output (other resources) and the fourth output (bank charges received) have the maximum coefficient variance respectively. The high value in the third output in respect to the coefficient of variation represents that there are branches, which have numerous dissimilarities in terms of arrears in regards to each other. Similarly, due to the mission-oriented criteria of some branches, which provide high yielding loans and thereby, the interests received by them are much higher than other branches; whereas, some, due to the fact of being located in a specific geographical region, such as, the iron market, offer more customer services, resulting in receiving an extremely higher amount of bank charges in comparison to others; as well as other sources, which special organizations known as governmental and or non-governmental organizations, have brought about in banks. This occurs usually for some exclusive bank branches. Hence, in concern with other jurisdictions, some branches are significantly dissimilar in comparison to the others.

In utilizing the skewness coefficient, the symmetrical deviation of the entire inputs and outputs has been illustrated in the last row of Table 3. This shows that all the inputs and outputs of the bank branches under study do not have a normal distribution.

The correlation coefficient between all the indices is calculated in two’s and is rendered in Table 4. The correlation coefficient holds between − 1 and + 1. This entails the fact that, however much the correlation coefficient between two indices is closer to + 1 signifies that the increase or decrease of one, causes an increase or decrease of the other; whereas as the proximity of this value to − 1 indicates an alternative or contradictory change of the two indices relative to it, meaning that, an increase causes a decrease and vice versa. For example, the correlation coefficient between the first input and first output as well as the second, show that an increment in staff privileges has a greater impact on the increase of facilities and the total sum of deposits, or in view of the values relevant to the correlation of the second input and outputs, it can be comprehended that, an increase in the interests paid is extremely effective as to the increase of the total sum of deposits, facilities and the interests received respectively.

Table 4 Correlation coefficient of each of inputs and outputs
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The trade-offs of the bank under study are as given below: Each bank branch can raise its facilities to the minimum of 1.2 units, instead of receiving other sources which are equivalent to a unit. Instead of resorting to total deposits, each branch of a bank can raise its other sources by 1.02 units. Likewise, each branch, in place of interest received can raise its bank charges rate by the minimum of 1.2 units; and similarly, by decreasing the interest paid by 1.12 units, the arrears is liable to being raised by a unit.

Thereby, the following weight restrictions are considered:

$$ \begin{aligned} 1.2u_{1} \le u_{5} , \hfill \\ 1.02u_{5} \le u_{2} , \hfill \\ 1.2u_{4} \le u_{3} , \hfill \\ 1.12v_{2} \le v_{3} . \hfill \\ \end{aligned} $$

Which vi denotes the relative importance of ith input for i = 1,…,3 and ur denotes the relative importance of rth output for r = 1,…,4.

With due attention to the above-mentioned weight restrictions, evaluation of 66 bank branches has taken place by utilizing the proposed Model. The second and fifth columns show Z*P that is the value of the proposed Model (7) which indicates the least variations in the total inputs and outputs for the branches under evaluation to reach the Pareto Efficient Frontier. The third and sixth columns represent the maximum changes required by the evaluated branch to reach the efficiency frontier which is obtained from Additive model in the presence of weight restrictions.

As can be observed, the values of the third and sixth columns are larger than the values of the second and fifth columns. This means that the proposed model suggests less modifications for the total inputs and outputs in order to make the inefficient branch efficient. For example for branch3, the Additive model that calculates the maximum distance using the Euclidean norm, suggest 68,052,920,000 changes. Meanwhile, the proposed model states these changes to be 3,317,292,300 for the values of the total inputs and outputs. This is the reason as to why the proposed model seeks the lowest possible reduction in the total inputs and outputs. The values of Table 5 for inefficient branches are shown in Fig. 2. as can be noted, in majority of the branches, variations in respect to the total modifications required in the input and output indices, in order to reach the efficient frontier, is extremely outstanding, between the additive model in the presence of weight restrictions and the proposed model, this suggests the superiority and advantage of Model (7).

Table 5 Result of the proposed model
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Fig. 2
figure 2

Maximum distance to the pareto efficient frontier versus minimum distance of the proposed model

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The values of Table 5 show the distance to the efficient frontier and has a reverse connection with the efficient values. In other words, the lower the distance to the efficiency frontier, the efficiency of the unit under evaluation is better. \( Z_{P}^{*} \) for Branches 2, 5, 12, 14, 19, 24, 28, 35, 48, 49, 65, 57 and 56 equates to zero, that is, these branches lies on the efficient frontier and are efficient. Considering that all the values of the second and fifth columns for the inefficient branches are smaller than the values of the third and sixth columns, the efficiency of the branches with the proposed model would be better than the calculated efficiency with the additive model.

Table 6 demonstrates the coordinates of the target points represented by the proposed model for eight of the inefficient branches.

Table 6 Target Point represented by the proposed model for some inefficient branches
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Model (7), in addition to attaining the least distance to the efficient frontier and introducing a target point, it also introduces a reference set for each branch, so that each branch would be aware of the fact that, if it wanted to be efficient, it could follow the target pattern of which certain branches. Given that the views of the managers have been added to the model in terms of weight restrictions and similarly, Podinovski (2004a) has stated that between weight restrictions and trade-off there is a direct connection. The weight restrictions applied to assessing the bank branches have led to increase in the production possibility set which the generated DMUs, are called trade-offs.

Table 7 illustrates the reference set introduced by the proposed model for some inefficient branches. For example, B.3 can be implemented with the slightest modifications by modeling branches 12, 14, 48, trade-off 1 and 3 in order to gain efficiency. It must be noted that since the weight restrictions are based on trade-offs and technological judgments, the target points are attainable.

Table 7 Reference set introduced by the proposed model for some inefficient branches
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Table 8 demonstrates the statistical analyzes relative to 13 efficient branches. A comparison of the two Tables 3 and 8 shows that the average of staff privileges is lower amongst the efficient branches, than the entire branches. Likewise, the average for the efficient branches for each of the output indexes is higher than the average of the output index for the total society. In addition, the domain of the modification of indexes, in the efficient branches, is lower than that, of the entire society.

Table 8 Statistical analysis of efficient branches
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5 Conclusion

In taking account of the opinions of decision-makers in evaluation, the results increasingly approach the satisfaction level of decision-makers. Thus, weight restriction is of significant importance in managerial perspective. However, by introducing a model which needs less changes in inputs and outputs, the inefficient DMU finds the closest path to reaches the efficiency frontier. In this paper, so as to evaluate the efficiency of 66 branches of one of the commercial banks of Iran a model for finding the closest target in the presence of weight restrictions which are constructed by trade-offs, is presented. With the help of the proposed model, for each inefficient branch, a path has been introduced. This can reach the extended efficiency frontier with the minimum of modifications. Theorems guarantee that the benchmark introduced by the proposed model introduces the closest target on the extended efficiency frontier and the target point is non-dominated. To secure the distance, the L1-norm has been used and if increasing or decreasing of a particular index is under the consideration of the manager, other multi-objective methods can be employed to solve the problems. Despite the presence of an extensive literature, as to the assessment of banks with DEA techniques, few studies have been performed, to achieve the closest target, in evaluating the branches of banks. In this research, the bank branches were assessed, from the intermediation and production perspectives. It is proposed that in future researches, the efficiency assessment of bank branches throughout a country, and banks of several other countries, should be conducted by other assessment perspectives in order to secure the closest target. Furthermore, designing similar models considering the variable returns to scale is suggested, in different fields such as healthcare. Additionally, a survey to seek the closest target, in the presence of trade-offs, when some of the indexes are undesirable, has been proposed as future studies.