Procedure to Solve Network DEA Based on a Virtual Gap Measurement Model

NetworkDEAmodels assess production systems that contain a set of network-structured subsystems. Each subsystemhas input and output measures from and to the external network and has intermediate measures that link to other subsystems. Most published studies demonstrate how to employ DEA models to establish network DEA models. Neither static nor dynamic network DEA models adjust the links. This paper applies the virtual gap measurement (VGM) model to construct a mixed integer program to solve dynamic network DEA problems. The mixed integer program sets the total numbers of “as-input” and “as-output” equal to the total number of links in the objective function. To obtain the best-practice efficiency, each DMU determines a set of weights for inputs, outputs, and links.The links are played either “as-input” or “as-output.” Input and as-input measures reduce slack, whereas output and as-output measures increase slacks to attain their target on the production frontier.


Introduction
Data envelopment analysis (DEA) models are used to measure the relative efficiency of each decision-making unit (DMU) relative to its peers regarding multiple input indices versus multiple output indices.Fewer input values and more output values are desired to improve the model's aggregated performance score.The DMU assigns a set of weights to the indices to obtain the best-practice performance score.The drawback of these models is the omission of the internal processes between inputs and outputs.
Seiford and Zhu [1] developed a DEA approach for evaluating US commercial banks in a two-stage process characterized by profitability and marketability.Zhu [2] applied the same two-stage process to the Fortune Global 500 companies.Sexton and Lewis [3] studied the performance of Major League Baseball in a two-stage process.The above twostage DEA papers are among the first to address formally the links between the two internal stages and the link categories being distinguished and discussed; these are the origin of network DEA.For a two-stage model, the links' role implies a reduction in the first-stage outputs, thereby reducing the efficiency of that stage but increasing the efficiency of the second stage.A number of DEA studies have been developed in an attempt to address this type of conflict.For instance, Lewis and Sexton [4] used the network DEA approaches of Färe and Whittaker [5] and Färe and Grosskopf [6,7] to compute the efficiency scores of subprocesses."Link" cannot be adjusted freely in radial models, which adjust the inputs and outputs by the efficiency scores in a twostage process.For these models, the entire system efficiency cannot be improved by adjusting links; see Kao and Hwang [8] and Lewis and Sexton [4]."Link" applied in nonradial models (SBM), as has been discussed in recent years.Tone and Tsutsui [9] introduced a network DEA and categorized links into two types of "fixed links" and "free links.""Free links" means that the intermediate items are adjustable or discretionary, not to change direction, and each DMU can be increased or decreased from the observed one and is free to assign each individual link to one of the three characteristics: as-input, as-output, or nondiscretionary so that entire system efficiency could be maximized."Fixed links" means that the intermediate items are nonadjustable or nondiscretionary.The linking activities are kept unchanged that cannot improve the efficiency of the entire system.In other words, the intermediate products are beyond the control of DMUs.

Mathematical Problems in Engineering
The recent extension of network DEA development to dynamic DEA does not represent the effect of carry-over activities (links) between two consecutive terms (nodes).Tone and Tsutsui [10] and Kao [11] introduced the dynamic DEA model using the nonradial and radial DEA models, respectively.These researchers express the outputs from term "" as being carried over to term " + 1," which means that the carry-over can be regarded as one type of link.Tone and Tsutsui [12] categorized carry-over into four types: desirable (good), undesirable (bad), discretionary (free), and nondiscretionary (fixed).The discretionary link is the same as the free link in network DEA.Kao [13] considered general multistage systems in which exogenous inputs are consumed in addition to intermediate products.Cheng and Gao [14] proposed a matrix-type network model which uses data in input-output tables; it is tested and can be feasible in evaluating the relative performance.There are many articles illustrating the application of free links and fixed links.Lozano [15] relaxed the constraints for both the fixed-link and the freelink cases, thus enhancing the discriminating power of the model.This study is to present the performance assessment of the individual processes of an external efficiency model.Avkiran [16] illustrated dynamic network data envelopment analysis (DN-DEA) in commercial banking with emphasis on testing robustness.Huang et al. [17] proposed a two-stage network model with bad outputs and supper efficiency (US-NSBM).Empirical comparisons show that the US-NSBM may be promising and practical for taking the nonperforming loans into account and being able to rank all samples.
Decreasing the volumes of inputs and "as-input" links and increasing the volumes of outputs and "as-output" links would improve the entire system efficiency of DMU  .Tone and Tsutsui [10] introduced a slack-based measurement (SBM) ex-post approach (adjusted score) and a 0-1 mixed integer fractional program (MIP) to address the discretionary slacks.The MIP model is, in fact, a nonlinear program.Therefore, a 0-1 binary decision variable   is assigned to free link  at term , transforming the process into a SBM model.These links introduced objective function in the MIP model introduced by Tone and Tsutsui [10], which is the ratio of inputs and as-input links to output and as-output links.The symbol free is the total number of free links.We suggest the upper bounds of the furthest right summations in the numeration and denomination should be replaced by free − and free + , respectively.The nonradial model SBM has the advantage of determining the slack on each input, output, and free link.However, this model requires the total number of as-input and as-output free links to generate the aggregate efficiency score.
The current paper adopts the virtual gap measurement (VGM) model introduced by Liu and Huang [18].The prime form of VGM is to seek the minimum virtual gap instead of the maximal efficiency score.The obtained optimal values of dual variables are used to compute the final efficiency score.The prime and dual models are well defined and explainable and the analysis is reliable.The contribution of this paper is to solve the network DEA problem raised in Tone and Tsutsui [10].We employ a VGM two-phase procedure and linear integer restrictions, which were developed by Cook and Zhu [19].The researchers adopted linear integer restrictions to capture the nonlinear expression, without actually having to specify it directly in the optimization model.This paper introduces a two-phase approach to solve the problem.Phase-I is a mixed integer program model to partition the links into two sets: as-input and as-output.Phase-II is a linear program model to determine the slack of each input, as-input link, output, and as-output link.The best-practice aggregated efficiency of DMU  in the entire system was obtained.The slacks of inputs, outputs, and free links are obtainable.The VGM model is presented in Section 2. The proposed two-phase performance evaluation model is presented in Section 3.Because the uniqueness of the optimal solution is important, we present an experiment on this subject in Section 4. Managerial insights are introduced in Section 5. We conclude this paper in the last section.

Virtual Gap Measurement Model
A set of DMUs  = {1, 2, . . ., } uses a production technology that transforms a set of inputs  = {1, 2, . . ., } into a set of outputs  = {1, 2, . . ., }.Let the notions   and   be the nonnegative volumes of inputs and outputs of DMU  .The VGM DEA model is to measure the maximum efficiency score of DMU  .The multiplier (dual) form of the model depicts the objective function as minimizing the virtual gap (Δ *  ) between virtual-input (∑ ∈   V  ) and virtual-output (∑ ∈     ).Each DMU in set  alternatively acts as DMU  .Let the symbol "" denote the commensurate virtual unit that is used for virtual gap, virtual-input, and virtual-output.The first set of constraints ensures all DMUs have a nonnegative virtual gap.The vectors V = (V 1 , V 2 , . . ., V  ) and  = ( 1 ,  2 , . . .,   ) are the decision variables of the weights to be assigned to the inputs and outputs.The second and third constraints define that each weighted input and output have a lower bound   , a constant value with unit of .These assign dual variables   ,   , and   to the three constraints.The envelopment (prime) model of VGM is [M1]: V   ≥ 0,  ∈ ; (5) The dual to model [M1] can be expressed as [M2]: ≥ 0,  ∈ ; The decision variable   denotes the weight of DMU  .  and   denote the slacks of th input and th output, respectively.The objective function (7) expresses the maximum summation of improvement ratios of inputs and outputs.Set   = 1 temporarily for computation convenient, and it will not affect the optimal solutions.The decision variable with a superscript " * " denotes its optimal value.The following equation defines the constant value of ; it is equal to the reciprocal of the maximum virtual-inputs of DMUs: Thus, the optimal solutions are normalized as Evaluating different DMU  's, one may directly compare their weights, virtual gap, virtual-input, and virtual-output vectors.
According to (1), the ensuing equation ( 15) existed.It is obvious that the minimum virtual gap Δ *  is equivalent to  *  .Referring to (16), this result ensures the nearest improvement target is found.The maximum total of improvement ratios in (16) ensures the improvement target is located on the envelopment.
3. Proposed Network Structure of VGM + denote the volumes of the th input measure and the th output measure at subprocess ℎ, respectively.Let  ℎ  and  ℎ  be the slack of the th input and the th output at subprocess ℎ, respectively.

Links.
Each subprocess may have links to other subprocesses.Let (ℎ, ) denote the link between subprocesses ℎ and , ℎ > .Let  (ℎ,) denote the set of link measures on link (ℎ, ). (ℎ,) + denotes the volume of the th link in set  (ℎ,) .Each DMU alternatively acts as the DMU  that is under evaluation.The volume of link  on link (ℎ, ),  (ℎ,)  , could be increased or decreased with a slack  (ℎ,)  to improve the efficiency of DMU  as well.
In Phase-I, we introduce the mixed binary integer virtual gap-based measurement model [M3] to partition the links into two subsets: as-input and as-output.
In (17), the objective function maximizes the total improvement ratios of each input, output, and link of DMU  .The decision variable  ℎ  is the weight of DMU  at subprocess ℎ.The set of left-side of inequalities ( 18)∼( 22) is the efficient frontier with respect to DMU  .The right-side of ( 18) is the improved th input at subprocess ℎ located on the frontier.The right-side of ( 19) is the improved th output at subprocess ℎ located on the frontier. denotes a considerably large constant.When the decision variable  (ℎ,)  = 0, (20) becomes effective and ( 21) becomes ineffective and this constrain could be ignored, and the th measurement on link  (ℎ,) is treated as "as-input" with respect to the network.Conversely, when  (ℎ,)  = 1, (20) becomes ineffective and ( 21) becomes effective, and the th measurement on link  (ℎ,) is treated as "asoutput" with respect to the network.When  (ℎ,)  = 0, (27) and (28) derived from ( 20) and ( 22) would ensure the improved th as-input at link (ℎ, ) projects on the frontier, respectively, to subprocesses ℎ and  with the same value, When  (ℎ,)  = 1, the two equations ( 29) and (30) derived from ( 21) and ( 22) would ensure the deteriorated th asoutput at link (ℎ, ) projects on the frontier, respectively, to subprocesses ℎ and  with the same value,  (ℎ,)  + Therefore, a single constraint (22) would replace the two cases above (Chen et al. [20]).
Proposition 1.If it is an as-output direction for subprocess ℎ, then it must be an as-input direction for subprocess .
Proof.When  (ℎ,)  = 0, equations ( 20) and ( 22) are partitioned to (27) and (28) which is explained in prior section.If (28) is multiplied by −1 on each side of the equation, due to the fact that For (27), from  ℎ  perspective, the links are as-input.For (31), from    perspective, the links are as-output.With respect to (27) and (31), the direction is from subprocess  to subprocess ℎ.
Solve [M3] to obtain the optimal solutions of the binary integer variables.If  (ℎ,) *  = 0, then assign index  to the set "as-input" that regards the entire system.Conversely, if  (ℎ,) *  = 1, then assign index  to the set "as-output" that regards the entire system.On each link (ℎ, ), the set of links on  (ℎ,) is then partitioned into two subsets, The objective function (32) expresses the maximum summation of improvement ratios of inputs, outputs, as-inputs, and as-outputs.Set   = 1 for convenient computation and it would be adjusted according to a normalization process.Constraints (33), (34), (35), and (36) ensure the modified values of inputs, outputs, as-inputs, and as-outputs would project on the efficient frontier, respectively.Constraints (37) and ( 38) ensure the modified as-input and as-output links would project on the efficient frontier (Chen et al. [20]).Assign dual variables V ℎ  ,  ℎ  ,  (ℎ,)−  ,  (ℎ,)+  ,  (ℎ,)−  , and  (ℎ,)+  to the functional constraints (33)∼(38), respectively.The transformed dual form [M5] is shown below.
Replace the coefficient   in VGM models by , which is expressed as (57).Therefore, all the upper bound of normalized values of virtual-input plus virtual-as-input of DMUs is 1.Furthermore, all of the upper bound of normalized values of virtual-output plus virtual-as-output of DMUs is 1, as well.
Then, the optimal solutions are normalized as Evaluating different DMU  's, one may directly compare their weights, virtual gap, virtual-input, virtual-as-input, virtualas-output, and virtual-output vectors.According to (43), the ensuing equation (65) existed.It is obvious that the minimum virtual gap "Δ *  " is equivalent to  (II) *

𝑜
. The maximum efficiency score of the entire network could be computed as (66).).For (66), it represents not only the maximum efficient score of the entire system but also the slope of the line from DMU  to origin.From (65), if the virtual gap is zero, the optimal efficient score is 1.We define the line with a slope equal to 1 to be the frontier.A larger virtual gap will entail a smaller slop and lower efficiency.Figure 1   x 2 ij , i ∈ I 2 z (2,3) dj , d ∈ D (2,3) z (1,2)  dj , d ∈ D (1,2)   z (1,3)  dj , d ∈ D (1,3)   y 3 rj , r ∈ R 3 x 3 ij , i ∈ I 3 x 1 ij , i ∈ I 1 y 1 rj , r ∈ R 1 gap diagram is (0.5, 0.3).For this DMU  , the virtual gap is 0.2 (0.5 minus 0.3).In order to improve the DMU  efficiency, the virtual gap needs to be decreased.
The network contains a set of subprocesses (nodes), .The nodes are assigned ordinal numbers 1, 2, 3, ..., ‖‖.Let  denote the set of network links.There are  homogeneous DMUs in set , named DMU 1 , DMU 2 , ..., and DMU  , which are randomly processed by the subprocesses in set .3.1.1.Inputs and Outputs.At each subprocess ℎ, there is a set of input measures  ℎ that flow into the network and a set of output measures  ℎ that flow out of the network.For DMU  in set , let  ℎ  ∈ R  ℎ + and  ℎ  ∈ R  ℎ 3.1.Network Structure.
depicts the DMU  performance on the virtual gap diagram.The slope of DMU 1 is 1; it is located on the efficiency frontier, indicating high efficiency.DMU 2 slope is 0.6, indicating lower efficiency; its location on the virtual

Table 1 :
The dataset of indices.

Table 2 :
Partitions of links, as-input and as-output.DMU  Binary decision variables for the five measures on links Sets of as-input and as-output on the three links  (1,2) *

Table 3 :
Slacks of the indices of every DMU  .
"-" means the decision variable is not in model [M4].