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Performance Measurement in the Health Care Sector: a Leader-Follower Network DEA Model based on Double Frontier Analysis

Measuring the performance of laboratories as one of the most significant areas of healthcare plays a key role in the quality of laboratories management. In this paper, we consider a three-stage network comprised of a leader and two followers in respect to the additional desirable and undesirable inputs and outputs. The suggested model simulates the internal structure of a diagnostic lab (the pre-test, the test and the post-test). The criteria for evaluation are achieved by using the Fuzzy Delphi technique. Due to the social, economic and environmental impacts of health care systems, the significance of sustainability criteria is obvious in the case study indicators. We utilize the non-cooperative approach multiplicative model to measure the efficiency of the overall system and the performances of decision-making units (DMUs) from both the optimistic and pessimistic views. The non-cooperative models from these view cannot be converted into linear models. Therefore, a heuristic method is suggested to convert the nonlinear models into linear models. Finally, after obtaining the efficiencies based on the double-frontier view, the DMUs are ranked and classified into three clusters by the k-means algorithm.


Introduction
Currently, medical diagnostic laboratories (MDLs) are a significant part of the health organization in the real world. The MDL performs a vital role in the arenas of healthcare, diagnosis, maintenance and prevention of different illnesses. Based on global standards, the share of MDLs in the healthcare market is about 5.6 percent, which represents a significant contribution. Due to the economic, social and environmental situations of the MDLs, there is an increasing public demand for better-quality performance, reduced functional costs and improved quality of such organizations. Efficacy as the main pillar of growth, is one of the most commonly used mechanisms for evaluating the performance 3

Literature Review
Currently, efficiency measurement is an extremely crucial issue towards a better understanding of problems in a system and planning for future improvement [4]. DEA is one of the most important and appropriate approaches for measuring the performance of DMUs [5]. DEA goals are to measure efficiency and present the efficient and inefficient units. Efficient units generate the maximal amount of outputs from the minimal amount of inputs and their efficiency equates to (1). These units form the efficiency frontier and the other units which have an efficiency ranging from (0 to 1) are identified as inefficient units. The DEA has such capacities, where each DMU, can be considered in segregation and specifies the efficiency appraisal on the basis of the distance of this unit till the efficiency frontier [6].
In his initial DEA task, Farrell [7], which was later, expensed by Charnes et al. [8], was reputedly known CCR or the (Charnes-Cooper-Rhodes DEA Model). Then, Banker et al. [9] developed the DEA and proposed BCC or the (Banker-Charnes-Cooper Model). DEA is a non-parametric linear programming model to evaluate the performance for a set of homogeneous DMUs with several inputs and outputs [10]. In classical DEA models, such as the CCR and BCC models, we do not consider the intermediate measures of DMUs or internal operations of DMUs and measure the efficiencies of DMUs as a "Black Box" [11]. In actual fact, the efficiency analysis in this method is carried out by the initial inputs and the final outputs. Ignoring the internal structure of systems leads to the classical DEA models where important information and segregation between the efficient units cannot be presented [12]. To overcome the problem, Fare and Grosskopf [13] suggested a network DEA (NDEA) model. In the NDEA models, the internal structure of systems and internal interactions of DMUs are taken under consideration until there is an increase in the accuracy of efficiencies. Kao [14] categorized the network models into three sets, namely, series, parallel and hybrid. Kao stated that, when activities in a system are protracted in respect to each other, the system is of a series structure; and whenever activities are in a parallel form alongside each other, the system has a parallel structure. Similarly, when there is a hybrid condition between the series and parallel aspects, a hybrid mode is engaged. The NDEA approach can simulate networks and then estimate the efficiencies [3, 4 15]. The internal structure of systems can simulate with the sub-DMUs either in series or in parallel.
Thus, the systems with series and parallel structures are two very significant areas in NDEA [15]. For the parallel structure, the sum of the inputs or outputs of all stages are considered as the inputs or outputs of the whole structure, but in the series structure the inputs of first stage and the outputs of last stage are the inputs and outputs of the whole structure [16]. The general efficiency of the parallel and series structures are measured by the multiplicative and additive methods, respectively [17]. In recent years, the efficiency evaluation of the multi-stage is one of the most important topics in NDEA and the parallel and series structures are used by many researchers.
kao [14] proposed a "closed system" with a series structure to be taken under consideration for intermediate measures, but without any additional input or output in each stage; whereas, Yu and Lin [18] used the NDEA to measure service effectiveness and technical efficiency. Kao [4] utilized the NDEA approach to evaluate the overall efficiency of the network with multi-stage and additional inputs. Cook et al. [19] used the network DEA for the evaluation of efficiency. In the network structure, the sub-DMUs have desirable or undesirable outputs.
The role of undesirable factors has been extremely crucial in NDEA, in the recent years, Liu et al. [20] used the clustering techniques and defined this part as one of the four Main parts of NDEA. For the first time, Fare et al. [21] took the undesirable factors under consideration to evaluate the efficiency in DEA models. Lu and Lo [22][23][24][25] categorized the methods for working with undesirable outputs in network DEA in the following three modes: 1-The first mode is to ignore the undesirable outputs which are done in order to simplify the models actually. 2-The second mode is to measure distances in such a manner, so as to limit the expansion of the undesired output or that the undesired output is modeled as a nonlinear network DEA model. 3-The third mode is to consider the undesired output as a desired input, or to employ the negative sign as a desirable output, or that a decrease in conversion is applied to them. Over the past few years, Wang et al. [26] and Wu et al. [27] contemplated on the role of undesired factors in manufacturing processes and utilized the NDEA to measure efficiency. In recent years, the evolution of unfavorable features has led to the use of undesirable factors for the generation of favorable aspects. For example, in a new approach, Wu et al. [28] considered an interactive network consisting of two stages, where the first stage inserts the 5 undesirable outputs to the second stage and ultimately, the second stage produces the desirable output and in actual fact, has utilized the undesirable outputs for production.
In recent years, NDEA models have undergone development and models combining this science, with the game theory branch have been rendered [29]. Li et al. [30] presented a network DEA model with two-stage, a part which holds a more significant viewpoint is named as "leader" and the other part is named as "follower". The performance of the leader part is maximized to the optimum and thus, the performance of the follower part is secured by keeping a constant performance in the part of the leader. This exemplary, is designated as a Stackelberg Game. In another research, An et al. [31] considered a network in two stages in an interactive mode and compared the efficiency of this network in a cooperative and non-cooperative (leader-follower mode). In yet another research by Zhou et al. [32] who dealt with evaluating the efficiency performance of a multi-stage network in the black box and non-cooperative (leader-follower) mode, comparing their results with each other. A research performed by Du et al. [33] in the grounds of leader-follower, can also be designated, in which they studied a parallel structure in the cooperative and non-cooperative modes. On the basis of the abovementioned facts, the main difference between the black box and the network approach is summarized in the internal correlations of systems. Some of the previous researchers have studied on Bi-level programming under different situations as follows: Maiti and Roy [34] designed the model on bi-level programming for Stackelberg game under intuitionistic fuzzy environment. Maiti and Roy [35] offered and solved multi-choice stochastic bi-level programming problem in cooperative nature via fuzzy programming method; and Roy and Maiti [36,37] studied on bi-level programming with multi-choice for Stackelberg game under fuzzy and stochastic environments.
DEA with a double-frontier contemplates on two frontiers to compute the efficiency for each DMU.
One is the efficient frontier and the other, the inefficient one. The efficiency calculated by the efficient frontier is called the optimistic efficiency, whereas, the efficiency computed by the inefficient frontier is known as the pessimistic efficiency [38]. In the optimistic view, each DMU, along with a set of efficient units, which form the efficient frontier, are compared. In the pessimistic view, each DMU, together with a set of inefficient units, forming the inefficient frontier are taken to comparison [30]. The value of the optimistic view is less than or equal to (1); whereas, the efficiency 6 of the pessimistic view is greater or equivalent to (1). The optimistic and pessimistic efficiency values are exactly equivalent to (1), if the DMU under evaluation, is placed respectively on the efficient or inefficient frontier [39,40]. Though, when calculating the optimistic efficiency, the nearer the DMU proves to be to the efficiency frontier, the more desirable, whereas, in the case of computing the pessimistic efficiency, the further the distance of the DMU, the better and has an additional desirability. In fact, the double-frontier views each DMU from two outlooks and any conclusion which implies to only one of these perspectives, shall be one-sided and inadequate [41,42]. For the first time, Doyle et al. [43] computed the efficiency of DMUs from the optimistic and pessimistic views. In recent years, many researchers have utilized the double-frontier for evaluating efficiency and stated various approaches towards calculating an overall performance; in this concern tasks in relative to Wang and Chin can be designated, who proposed a numerical measure for a general geometrical mean efficiency [44][45][46][47].
Producing a sustainable product as a practical way to minimize the environmental impacts of a product is the significant way for reaching sustainability [48]. In order to assess eco-efficient performance, Chen et al. [49] recently introduced the concept of "sustainable performance". This concept explains how achieve the desired output or lower undesired output in the process of production. Sustainability assessment is not limited to environmental criteria. Accordingly, three categories of sustainability factors (social, economic, and environmental) are presented in the literature. Determining sustainability goals requires some knowledge and comprehending of the current level of sustainability. This can be achieved through sustainability assessments taking into account all three factors of sustainability "economic, social, and environmental" [50]. Since the evaluation of a system involves a wide range of economic, social and environmental indicators, this leads to complex multi-criteria decision-making problems. A possible way to simplify the assessment is to define the concept of sustainability and to determine the importance of economic, social and environmental indicators [51] . Extensive research has conducted on methods and applications of DEA, but these efforts have focused mainly on assessing of DMUs in area of engineering. More recently, researchers have used the DEA to evaluate system performance by considering sustainability factors. However, many of these studies cover only environmental and economic aspects, but social 7 dimension has neglected as a dimension of sustainability [52,53]. Table 1 reviews the studies which have applied the game theory methods in DEA. The last row of Table 1 presents characteristics of the current paper.
DEA has proposed as a theoretical framework for performance analysis, but its application in the field of health care has very low. As a summarization, contributions of this research are as below:  A three-stage network is taken under consideration in regard to the additional desirable and undesirable inputs and outputs.
 We simulate a MDL with three level (pre-test, test and post-test) which allowed us to obtain important information about the causes of inefficiency and efficiency of laboratory units.
 We consider sustainability criteria (economic, social and environmental) to appraise the performance of MDLs, thus helping to improve the social, economic, and environmental problems of laboratories.
 The criteria for evaluation are gotten by utilizing the Fuzzy Delphi method  A Double Frontier Approach is utilized to evaluate efficiency, in order to make results more realistic.
 k-means algorithm based on the double-frontier view is suggested to determine the efficient and inefficient points of the network.
 A heuristic technique is suggested to turn non-linear models into linear ones  Implementation of the suggested model on an authentic example.

Methodology
In this study, the methodology is formed in four parts. In First part, the data and variables are gathered According to the library studies, interview and observation by Fuzzy Delphi technique. In Second part, a network DEA method is explain to evaluate the efficiency of units According to the doublefrontier viewpoint. In Third part, a heuristic method is suggested to turn non-linear models into linear ones. Lastly, k-means algorithm based on the double-frontier view is suggested to determine the efficient and inefficient points of the network. In Fig. 1, the methodology is shown as below.  Table 3.
In the following, the effective criteria of the three MDL processes (pre-test, test, post-test) were gained from the Fuzzy Delphi technique. The conceptual model along with the descriptions of the criteria, is sent to the expert members. In the following, the amount of their agreement with criteria is taken. Given the offered options and the linguistic variables defined in the questionnaire, the results shown in Table 4  In Table 4, the defuzzification operations and the triangular fuzzy average are calculated by Formulas  Table 5.
In the last column of Table 5, expert members of the expert group have gotten agreement in totally criteria, except " available pace for service, number of false tests, waste weight", due to that the difference between the first and second round is lower than the threshold of very low (0.1). Therefore, we poll to continue for only three criteria. At third round, a third questionnaire Including three criteria was planned. According the prior point of opinion of each specialist and their degree of change with the view of other experts, the questionnaire was again given to the expert members. The fuzzy analysis of the results shown in Table 6. Table 6 shows, the difference between the second and third rounds is lower than the threshold of very low (0.1). According to Cheng et al.
[62], the consensus stops at this round. In the Table 5 Table 7 shows effective final indicators for measuring the efficiency of MDLs by Fuzzy Delphi technique.

Model Description
We study a set of n homogeneous DMUs that are shown by j DMU (j=1,..., n), and each j DMU has three-stages with a complex internal structure as shown in Fig. 2  Researchers in efficiency analysis are likely to use input-oriented models, due to three major reasons.
Firstly, because demand is on the growth and estimating demand is an intricate matter. Secondly, managers have more control over inputs than outputs. Thirdly, this model reflects the primary goals of policymakers, based on being responsible in responding to the requirements of people and units must reduce costs, or else, limit the use of resources. Thereby, in this research we utilize the input-oriented model. In accordance with Korhonen and Luptacik [63], we signify the undesirable output in the models with a negative mark as a desirable output. Also, we consider the undesirable input as a desirable input with a negative mark. Based on the opinions of managers, we will consider the first 11 stage (reception unit), the second stage (sampling and testing unit) and the third stage (test results unit) in the roles of the "leader", "first follower" and "second follower", respectively. Thence, we demonstrate the optimistic and pessimistic efficiencies of the leader's stage with The optimum efficiency has been demonstrated with the symbol (*) in the Model (1).
, , , , , , , , , ; 1,2, , ; 1,2, , ; 1,2, , ; 1,2, , ;  (2) and have defined the efficiency of the leader stage from the pessimistic view according to the following:  1  2  1  2  3  4  1  2  3  4   1  1 2  2 1  1 2  2 , , , , , , , , , ; 1,2, , ; 1,2, , ; 1,2, , ; 1,2, , ; The maximal optimistic and minimal pessimistic efficiencies of the leader stage ( L* o θ and L* o φ ) is gotten respectively, from models 2 and 3. To calculate the efficiency of the followers we will consider the second and third stages as one stage and gain the efficiency of the follower stage. Kao and Hwang [16] used the multiplicative approach to measure the overall efficiency of a series structure. We then define the efficiencies of the second and third stages of a system shown in Fig.2 , , , , , , , , , ; 1,2, , ; 1,2, , ; 1,2, , ; 1,2, , ;   (3.4) We, based on the tasks of Wang and Chin [39], define the overall efficiency according to the doublefrontier in formula (3.5) according to the following: So we performed the modeling of the network shown in Fig 2, from three approaches (the optimistic, pessimistic and double-frontier). In the next, we utilize the k-means algorithm to cluster the DMUs into several groups based on the double-frontier view and these results are shown in the case study section.

Model Solution
Two exploratory approaches are proposed for the optimistic and pessimistic views relatively. It is for the first time that we have developed an exploratory approach, in relevance with the pessimistic perspective or condition, which, with the best of our findings, we had not been able to perform the modeling, as to this conceptual approach under pessimistic conditions, till date. Similarly, we have also implemented an exploratory approach from the optimistic standpoint, in relative to the leaderfollower concept. Thereby, the exploratory approach from the pessimistic angle is proposed in this paper. Models (4) and (5) cannot be converted into linear models because of the additional inputs and outputs in the stages. Thus, we propose the heuristic approach given hereunder for solving models (4) and (5).

A heuristic technique to solve the optimistic view
The objective function of model (4) We take "M" as a larger value and ε  as a similar optimistic approach, as a step size and of very small value. Moreover, 1F-min o φ is the minimal pessimistic efficiency of the first follower stage, and from the following model its value is capable of being computed.

Model 10
The entire variables are non-negative in the model (10). The said models have attained a minimum efficiency of the first follower stage, on condition that, the efficiency of the stages is more than (1).  (11), we convert model (5) to model (12) as follows: It should be noted that similar to the optimistic approach in the model (12) φ (k ) in Fig. 1 from the pessimistic view. It should be observed that, we have tested our proposed method in two modes and each time have taken one stage into consideration as variables. Given that at this point, the efficiency of a stage is unique, hence, the results of these two methods are outstandingly in approximation to each other and in order to explain, we have broached one of these two conditions to describe our above approach.  Fig. 3.

Case Study Description
To evaluate the efficiency of the MDL, environmental, social and economic criteria are considered.
Cost of consumables, income from admission, MDL profit and staff wage are introduced as economic criteria. Staff safety as social criteria is the costs that each service center payments for dangers that exist and accidents which happen in workplace. This criterion is considered to protect the health of the 21 staff, patients and other clients, as well as protecting the environment. The variables are shown in Table 8 as follows: We have many variables of inputs and outputs. Therefore, Tables 9 and 10 provide the data in 2017.
The inputs are shown in Table 9 and the outputs are rendered in Table 10. Thereby, we measure the efficiency of the structure as shown in Fig. 3 from both, the optimistic and pessimistic views.
According the opinions of the experts, we consider the step size in the models ε=0.01  , M=3 and ε =0.001. We implemented our heuristic approach, for the two models (4 and 5). The values gained for the first follower are shown in the Table 11 as follows: By studying the values of 1 k , we found the optimal values of network shown in Fig. 2 to occur when the first follower is at their maximum and minimum values. For this reason, all values of 1 k vanish. Table 12 shows the efficiencies based on the optimistic, pessimistic and double-frontier views.
In Table 12, the maximum and minimum mean efficiency of the course is shown in gray and green respectively. The second column of Table 12 show that the efficiency scores of period 22 is the highest with a total efficiency of 1 and the efficiency scores of period 12 is the lowest with a total efficiency of 0.228176, from the optimistic view. From the third, fourth and fifth columns of Table   12, we note that the average optimistic efficiency values of the leader, first follower and second follower are of the following values respectively, (0.73, 0.99, 0.82). It should be noted that, the closer the efficiency value is closer to (1), from the optimistic standpoint and the farther the efficiency value is from (1), from the pessimistic standpoint, the DMU proves to have a better condition. The sixth column of  Table 13 as bellow: The third and sixth columns of Table 13 demonstrate the clustering of the DMUs by k-means method based on double-frontier view that we have explained in the Section (3). Table 13 reports that units 7, 18, 19, 22 are located in the first cluster. Units 1, 2, 3, 5, 6, 8, 9, 11, 13, 16,

Conclusions
The The efficiency evaluation of a network opens the "black box" and deliberates on the internal structures and innermost interactions of the system. The black box method ignores the internal structure of systems and this task lead to inaccuracy in the results. This paper presents the modeling method and solution for evaluating the efficiency of a complex system with additional inputs and undesirable outputs. We tried to pay attention to the intra-system activities using the proposed model.
In this study, we simulated a three-stage series structure of a MDL in a real-world. This network model of three processes (The pre-test process, The test process, The post-test process). The pretesting process contain of the reception unit. Also, the testing process consists of the sampling and since the activities of an enterprise such as MDLs is not sectional over a period of time, but is continuous activity, therefore, the cross-sectional efficiency assessment cannot provide a realistic answer to the performance of laboratories. Therefore, network analysis in dynamic mode is also recommended.
[        Fig. 1 The methodology of research