The Data Envelopment Analysis (DEA) method is applied to evaluate the performance of Decision-Maker Units (DMUs). In this method, outputs are the services provided by an organization, and inputs are the facilities delivered to DMUs to enable better products and services. DMU efficiency is calculated by comparing the input and output indices across DMUs. One of the most critical DEA models is the CCR. Cooper and Rhodes first presented this model in [1].

This model is used for measuring the efficiency of an observed DMU, which is the organization to be evaluated by the ratio, a linear combination of outputs per linear combination of inputs, i.e., how well a DMU can convert its inputs into its outputs. Consider n DMUs each using m inputs, Xj = (x1j, x2j, … , xmj), to produce s output Yj = (y1j, y2j, … , ysj). Supposed weights of inputs and outputs are shown respectively in the form of v (v1, … , vm) and u(u1, … , us); thus, relative efficiency is defined as follows:

Nonlinear model of REP is transformed to a linear model as follows [1], [2]:

In some cases, indices are time-varying. In other words, they receive different amounts at different times. Because indices must be fixed in traditional DEA, models with variable data cannot be solved using traditional DEA methods, such as the CCR model [3–6]. The objective of this work is to propose a model and method for estimating efficiency and ranking such DMUs. Therefore, first, the best and worst times for each DMU are calculated, and then the efficiencies at the indices associated with the best and worst times are estimated. This paper is organized as follows. We introduce different methods in Section 2. Section 3 presents a method for estimating the relative efficiency and relative inefficiency of each DMU at the fixed time with respect to itself at other times. In Section 4, a method is delivered for finding the best and worst times. Section 5 introduces our methodology for solving the problem, based on indices for the best and worst times. In Section 6, a method for ranking DMUs is presented. A numerical example is given in Section 7, and Section 8 indicates the conclusion.

Suppose yrP(t) is a given level of rth output of pth DMU at the time of t, likewise, xiP(t) is a given level of ith input of P.th DMU at the time, t. There are m inputs, s outputs, n DMUs, and the domain of t is the interval [a, b]. We want to estimate the relative efficiency of DMUp with respect to other DMUS. The calculating methods can be estimated for efficiency at different times. We define efficiency at time t.

Definition 1. It is said that DMUp is efficient at time t if the optimal solution to the model below is 1. And it is efficient when maxθ(t) equals 1.

The problems with the definition mentioned above are summarized as follows: Each DMU has the best efficiency at a time that may differ from that of the other DMUs. So, comparing DMUs at a fixed time is neither just nor logical. For example, suppose the indices of three DMU are as follows.

On the first day, the efficient DMU is DMU1; on the second and third days, the results are DMU2 and DMU3, respectively. Therefore, according to the above method, each of the three DMUs is efficient. Still, under the condition that the indices associated with DMU 1 are considered the first and second days for DMU 2, and the third day for DMU 3, it results in DMU 1 being inefficient and DMU 2 and 3 being efficient. Considering the summation of indices can be another method for estimating the efficient DMU.

Definition 2. It is said that DMUp is fully efficient if the optimal solution to the following model is 1.

Total efficiency can't be convenient, because it is not considered a relation between indices; moreover, one DMU may be total efficient, but it is not efficient at any time. To clarify the subject, consider the following example: Consider three DMUs with one input and one output, as shown in the table below:

It is seen that on the first and second days, DMU1 is efficient; on the third and fourth days, DMU2 is efficient; and throughout the period, DMU3 is efficient, although it never reached the efficient frontier.

It seems that, to estimate DMUs' efficiency, it is convenient that the DMUs be compared at the best times and likewise at the worst times. Attention: the best time for each DMU differs from that of other DMUs. Likewise, the worst times for DMUs differ. Now we propose a method based on the best and worst times for each DMU. The method is to compare the indices associated with the best time of each DMU with those of other DMUs. In the same way, indices associated with the worst time of each DMU are compared with indices associated with the best time of other DMUs. This method is summarized into four steps.

Step 1. It is the relative efficiency of a DMU at a given time with respect to other times for the same DMU. Step 2. Is finding the best and worst times for all of the DMUs.

Step 3. Is finding the efficiency of each DMU based on the best and the worst times.

Step 4. Is ranking DMUs. In the parts below, each one of the above-mentioned steps is explained .

Let yrP(t) and xiP(t) be respectively the given levels of rth output and ith input of DMUP at time t. In other words, indices of DMUP are variable in the interval of [a, b]. There are m inputs, s outputs, and the domain of t is the interval of [a, b]. We suppose that yrP(t) and xiP(t) are continuous and nonnegative functions. In this part, we suggest a method for estimating the relative efficiency of a DMU at a fixed time vs. the same DMU at other times. It means any time is considered a DMU. We define DT(P,t0) as a DMU with inputs and outputs . Likewise, we define . Thus the relative efficiency DT (P,t0) vs. Deterministic Time D)T((P) is defined as follows:

Because of

If it continues on the interval [a, b], we can write:

Thus, we have:

By substituting instead of , the above model is transformed as follows:

Thus, by putting and in FCCRT, the following model will result:

The Above model is the same CCR model, with an infinite number of DMUs. We profess that the optimum solution of the model below is an upper bound of the optimum solution of Categorical and Combinatorial Representation Theory (CCRT):

Theorem 1. Suppose that (u ∗, v ∗, θp) and ( are respectively optimum solutions of CCRT, BCCRT models. Therefore

Proof: The logic below shows that (u*, v*) is a feasible solution to BCCRT.

Therefore, we will have:

Theorem 2. Suppose it is the optimum solution of the BCCRT model and

Therefore, it is an optimal solution of the CCRT model.

Proof: Suppose is a value of optimum associated with the objective function of the CCRT model, then according to Theorem 1 . Also, because f (t ∗) is less than or equal to zero, for each t we have:

Thus , it is a feasible solution of the CCRT model, and as a result, we have . Therefore, it is clear that it is an optimum solution of the CCRT model. Now we present an Algorithm for solving the CCRT model by using the linear model of BCCRT.

In this part, we present a method for obtaining the most efficient and the least efficient DT(P). Suppose we want to estimate the time at which it has maximum efficiency in a bank. One of the input indices can be employees' scores; likewise, investment at short and long times can be listed as output indices. The above indices are changed each time, whereas we have only the values of the indices for a finite days. Therefore, for each index that is variable, we fit a piecewise linear function to the data. In other words, we make a function by drawing a line between each pair of adjacent points. However, we make a continuous function for each index; it is variable, but actually, the variation of indices, in the length of day, is almost very little. Therefore, we can estimate the relative efficiency of a DMU in a time interval of a day. In other words, we divide the time interval [a, b] into small parts with length Δ, and we estimate relative efficiency at partition points. Now we present an algorithm for obtaining the best and the worst member of DT (P).

Now we want to calculate the efficiency of DMUp vs. all of the DMUs. We use the CCR model once when indices are associated with best-DT (k) and once separately at worst-DT (k) for each DMUk. The efficiency of the DMUs will be specified by the following definitions.

Definition 3. Suppose that, for each DMUk, the indices have been considered as best-DT(k). Also, suppose DMUp is the unit under evaluation. Therefore, the optimal value of the objective function of the CCR model is called the relative efficiency at best times and is denoted by θP(best).

Definition 4. Suppose for each DMUk, indices are associated with worst-DT (k). Also, suppose DMUp is the unit under evaluation. Therefore, the optimal value of the objective function of the CCR model is called the relative efficiency at worst times and is denoted by θP(worst). Thus, by solving CCR models according to the above-mentioned definitions, we obtain θP(best) and θP(worst) for each DMU. Now we define an efficiency range for DMUp.

Definition 5. The range of efficiency of the Pth unit is defined as follows:

RP = min(θP(best), θP(worst)),

max(θP(best), θP(worst)).

Definition 6. If Rp is equal to [1,1], then DMUp is efficient (E). But if Rp is equal to [a<1,1] then DMUp is a Weakly Efficient unit (WE) and otherwise it is inefficient (IE).

For ranking the DMUs, efficient units (E), WE units, and inefficient units (IE) are placed respectively. Ranking efficient DMUs (E) will be done by the following models.

In which are respectively output and input indices associated with worst-DT (p), and are respectively output and input indices associated with best-DT (k). Rs (P) is not equal to one because the indices of DMUp are not on the efficiency frontier. Note that if Rs(P) is greater than Rs(q), then DMUp is more efficient than DMUq. Ranking WE units will be done according to min (θP(best), θP(worst) ), and finally, the ranking of inefficient units was ordered according to the mean of RP.

We perform the above method at twenty banks. We consider four inputs (area, archaism, employee score, and equipment) and five outputs (Loan savings, other deposits, loan current, long-term investment, and short-term investments). Area, archaism, and equipment are constant, but other indices are variable. Amounts of indices have been specified in thirty-eight months, and at the end of each month. For each unit, a fixed piecewise linear variable index is used. Thus, we have six curves for each unit and a total of one hundred twenty curves. We select the best (worst) member of DT (P) by setting ∆=1. It means the criterion of time is one day. Table 1 shows data associated with DMU one. The constant indices of DMU1 are eight, three thousand, five hundred eighty one million and five hundred fifty two thousand three hundred thirty seven point of three respectively for archaism, area, and equipment.

The piecewise linear functions of indices of DMU1 have been shown in Fig. 1. Functional indices of employees' score, associated with twenty DMUs, have been shown in Fig. 2. We apply the algorithm to find the best (worst) member of DT (P) for any DMU. Table 2 shows the best times and the worst times of DMUs in thirty-eight months or one thousand one hundred fifty-seven days. Now, indices are obtained at the best and worst times using piecewise linear functions. Then we solve the CCR model using best- and worst-time DMU scores, and thereafter rank the DMUs and solve the RSCCR model (See Table 3).

In this paper, we analyzed a new DEA in which some indices are time-dependent.

For the calculation of the relative efficiency of the units, we selected indices in the following forms: 1) Indices associated with the best time of each unit are selected. The best time of a unit is the time at which the unit is in its best position. On the other hand, if we suppose indices associated with different times of a unit make the virtual units, then the virtual unit associated with the best time is efficient against all of the virtual units. In fact, we can say that in this form, indices are selected by the own DMU. Eventually, the efficiency of units is calculated at best times. 2) Indices associated with the worst time of each unit are selected. The worst time for a unit is the time that it is in the worst position. On the other hand, the virtual unit associated with the worst time is inefficient against all of the virtual units. In fact, in this form, the indices of a DMU are selected by the other DMUs. Thus, the efficiency of units is calculated at worst times, and 3) Indices associated with the unit under evaluation are selected at best times, and indices associated with other units are selected at worst times. This form is applied for ranking.

Finally, for each unit, we have a range of efficiency that has been obtained at Form 1 and Form 2. Therefore, units are divided into efficient units, WE units, and inefficient units. Efficient units are efficient at both forms. WE units are only efficient in one form. Inefficient units are inefficient in both forms. For ranking, efficient units are ranked first according to Form 3. The next time, WE units are ranked according to the minimum efficiency associated with form 1 or form 2. At the end of time, inefficient units are ranked according to the mean of efficiency associated with form 1 or form 2.

The authors declare that they have no conflict of interest regarding the publication of this manuscript.

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

This research received no external funding.