NETWORK DATA ENVELOPMENT ANALYSIS WITH TWO-LEVEL MAXIMIN STRATEGY

. Network data envelopment analysis (NDEA), one of the most important branches of recent DEA developments, has been developed for examining the decision making units (DMUs) of a system with complex and internal component divisions. In this study we apply a maximin strategy to network DEA at two levels. At the individual DMU level, we evaluate the system’s performance by maximizing the minimum of the divisions efficiencies, which is based on the weak-link approach. At the all DMUs level, we evaluate the system’s performance by maximizing the minimum of the DMUs’ efficiencies, which is based on the maximin ratio efficiency model. With such two-level maximin strategy, we propose the two-level maximin NDEA model to evaluate efficiencies of all divisions as well as all DMUs at the same time. The model will provide unique and unbiased efficiency scores for all divisions in a system and improve incomparable efficiency scores and weak discrimination power of traditional DEA models. In addition, we discuss the cross efficiency evaluation based on the two-level maximin NDEA model. The proposed models are applied to the efficiency evaluation of supply chains for illustrations.


Introduction
Data envelopment analysis (DEA), initiated by Charnes et al. [8], is a nonparametric approach for measuring the relative efficiency of independent peer decision making units (DMU) incorporating multiple inputs and outputs [12].Generally, DEA estimates the efficiency of a DMU by calculating the ratio of its weighted sum of inputs and weighted sum of outputs through a set of weights.In its basic form, DEA allows each DMU under evaluation to arbitrarily choose its favorable weights for inputs and outputs such that it obtains its optimally maximized efficiency score.For its effectiveness in identifying the best-practice frontier and ranking the DMUs, DEA has been applied to many activities in many sectors for various purposes [2,29,35], such as agricultures [1,40,50], banks [19,33,55], and supply chains [22,51,53].
In conventional DEA, DMUs are considered as a whole unit, in that component structures are generally ignored, and the performance of a DMU is assumed to be a function of the chosen inputs and outputs.However, most systems are composed of many divisions operating interdependently via the intermediate products that are created by some divisions and consumed by some others within the system.Ignoring the operations of the component divisions may cause misleading results in efficiency evaluation [5].Network DEA models are thus developed to address such issue, in which efficiencies of competent divisions and the whole system are evaluated simultaneously [18,27].In the network DEA models, three kinds of system structures are often investigated, including series, parallel and mixed structures [28].The series structure refers to several processes are linked by intermediate measures in sequence.For example, the operations of a bank are divided into two serial processes to measure profitability and marketability [44].The parallel structure is composed of parallel processes that operate independently.University is an example of a parallel system, where the individual stages are the departments that operate parallel and separately inside the university [28].The mixed structure is more complex than the series and parallel ones, which is a mixture of serial and parallel structures [26].
In network DEA, the efficiency is a multi-dimensional measure, as one has to consider the efficiency of the component divisions as well as the overall system efficiency.For comparable component divisions across all DMUs, divisional efficiencies can be defined as the ratio of the weighted sum of inputs and weighted sum of outputs.According to whether the overall system or the divisional efficiencies is given priority for optimization, two broad approaches can be classified, namely the top-down approach and the bottom-up approach [45].In the top-down approach, the overall system efficiency is optimized first, and then the divisional efficiencies are obtained from offspring from the optimal solution that maximizes the system efficiency.This approach mainly includes the additive efficiency decomposition method [9,13] and the relational model [24,27].On the contrary, in the bottom-up approach, the divisional efficiencies are measured first and the system efficiency is achieved ex post.Representative methods of the bottom-up approach are the additive aggregation method [3,21], multiplicative aggregation method [34,54], the min-max method [17] and the weak-link method [15,16,32,43].The weak-link approach, first presented by Despotis et al. [16], is inspired by the weak-link notion in supply chains and the maximal flow-minimal cut problem in networks.In the weak-link approach, the system efficiency is obtained by maximizing the minimum of the stage efficiencies.This approach provides unique and unbiased efficiency scores of the divisions and identifies adequately the source of system inefficiency.
On the other hand, a prominent characteristic of the traditional DEA models is that allows each DMU to select the most desirable weights in calculating its efficiency score.Therefore, in the classical DEA model, this flexible weight selection prevents DMUs from comparing the efficiencies under the same baseline.Further, weak discrimination is often found in classical DEA models, since many of the DMUs are estimated as efficient.To overcome the above mentioned limitations, common set of weight (CSW) DEA models have been developed by many researchers.The common weights method indicates that each DMU applies the same benchmark for computing efficiency.Several methods are proposed for finding CSW including separation vector [11,31], cross efficiency [4,41], ideal point method [23,30], goal programming [20,37], and evaluation of a subset of units [39,46,48,49].The maximin efficiency ratio model is a significant method for evaluation of DMUs.In the maximin efficiency ratio model, CSW is determined by maximizing the minimum efficiency ratio across a set of all DMUs [48,49].This method not only can surmount the issues in classical DEA models which include incomparable efficiency scores and weak discrimination power but also can find the minimum efficiency unit between all DMUs.
In the field of DEA, the property of units invariance indicates that efficiency results are independent of the units in which the observed inputs and outputs are measured so long as the units are the same for every DMU [7], is a desirable property of an ideal efficiency measure [38].Lovell and Pastor [36] discussed that Charnes, Cooper and Rhodes's CCR and Banker, Charnes and Cooper' BCC models are units invariant with respect to the radial measure, but not to the slack component.Cooper et al. [14] claimed the range adjusted measure (RAM) model is units invariant.Tone [47] noted that the slacks-based measure (SBM) model satisfies the property of units invariant.The property of units invariance is considered important in designing the measures in DEA.We will discuss units invariance of the proposed models in the study.
Almost all previous studies on the maximin efficiency ratio model focused on the single-stage system, only one exception is Wu et al. [52].Wu et al. [52] developed a maximin efficiency multistage supply chain model which considers the internal structure of DMUs to assess the supply chain performance.In their approach, the overall efficiency of the supply chain is defined as a weighted sum efficiency of its individual divisions.Although they looked into the internal stages of the DMUs, they did not consider the efficiency of the minimum efficiency stage in multi-stage system.In this study, we establish a two-level maximin network data envelopment analysis (NDEA) model for the efficiency evaluation of network production processes.The proposed model combines weak-link approach which can identify adequately the minimum efficiency stage inside the DMU and maximin efficiency ratio model which can find the minimum efficiency unit between all DMUs.Thus, our model exhibits an advantage in enabling decision makers to identify the stages that is the minimum efficiency stage in the minimum efficiency unit and effectively improve the performance of these systems.Moreover, the proposed model can provide unique and unbiased efficiency scores for the divisions.Otherwise, this model can surmount incomparable efficiency scores and weak discrimination power.Then, we extend cross efficiency evaluation method to the multi-stage system based on the two-level maximin NDEA model.The new cross efficiency evaluation method does not require secondary objective functions and can provide unique efficiency results.
Main contributions of this study are as follows.First, we are the first to develop a two-level maximin NDEA model.In the depicted approach, we explicitly provide the measures of overall efficiency and stage efficiencies.In addition, we discuss units invariance of the two-level maximin NDEA model.Last, we develop cross efficiency model based on the two-level maximin NDEA model for network system, which is firstly to extend the cross efficiency evaluation to a network system.
The remainder of the paper is organized as follows.Section 2 introduces the CCR ratio model, maximin efficiency ratio model, and cross efficiency evaluation method.Section 3 develops the two-level maximin NDEA model and cross efficiency model based on the two-level maximin NDEA model for the efficiency evaluation of the network system.Section 4 applies these models to an empirical study with 8 three-stage supply chains.Finally, the last section concludes the paper.

The CCR ratio model
Suppose that there are  DMUs, and each DMU  ( = 1, . . ., ) products  outputs   ( = 1, . . ., ) by  inputs   ( = 1, . . ., ).For any given DMU  , its CCR ratio efficiency is defined as a ratio of the weighted sum of outputs to weighted sum of inputs [8]: where   and   are the weights given respectively to the -th input and the -th output.The CCR ratio efficiency can be formulated by the following model.
2) is equivalent to the following linear programming after the Charnes-Cooper transformation [6].

The maximin efficiency ratio model
The maximin efficiency ratio was proposed to enable DEA analysis along with a subset of DMUs [52].The maximin efficiency ratio model [48] can be given as the following.

Max min Σ
It is not easy to derive an optimal solution by transforming the nonlinear model (2.4) into a linear one.Fortunately, Bolzano (bisection) search procedure can be used to get an approximate solution for model (2.4).

Cross efficiency evaluation in DEA
The cross efficiency method measures each DMU through self-evaluation and peer-evaluation two processes.For a DMU  , its self-evaluation efficiency score is computed by using model (2.3).

Model development
A serial multi-stage structure is one of the most representatives network systems [25].We next present the formulation of the new model under the serial multi-stage structure.In a general series multi-stage production system, several processes are connected in series, as shown in Figure 1.In this system, any process ( = 1, . . ., ) utilizes exogenous inputs   ( = 1, . . ., ;  = 1, . . ., ) for the succeeding process to use.For the first process, there are no intermediate products from other processes are utilized for the first process, and for the last process  there are no intermediate products are generated for other processes.The notation that will be used throughout the paper is listed in Table 1.
Typically, the ratio efficiencies of process  ( = 1, . . ., ) of DMU  are defined as follows: where

The two-level maximin NDEA model
In our model, the maximin strategy will be applied at two levels.At the individual DMU level, the system efficiency of each DMU in the model is defined as the minimum of the stage efficiencies.Formally, the system efficiency of is defined as where    is defined in (3.1).At the all DMU level, we evaluate all the DMUs simultaneously by using the maximin efficiency ratio model for multistage as follows.

Max min
which is further transformed into a parametric linear programming with parameter : We provide the following bisection algorithm to solve the model (3.5).
Step 1. Set  = 1 and denote )︁ as an optimal solution to model (3.5).The set of  processes is then divided into two groups  1 and  2 as follows: If the set  2 is empty,  2 = ∅ or | 2 | = 0, then the procedure exits; otherwise go to Step 2.
Step 2. Set  =  + 1, solve the following general model:  )︁ be the solution to the problem using the maximin principle.The DEAefficiency of  process and DMU  can be denoted as follows: The bisection algorithm has the following characteristics.)︁ is an optimal solution of model (3.6).Therefore, we have Thus, the below inequation is obtained.
The bisection algorithm converges.
In the field of DEA, the property of units invariance, which is in fact an application of a general mathematical property known as "dimensionless" [36], is that efficiency results is independent of the units in which the observed inputs and outputs are measured so long as the units are the same for every DMU [7].Units invariance is a desirable property [42].The following theorem shows the proposed model (3.3) ,  ()  ≥ 0,  = 1, . . ., ;  = 1, . . ., ;  = 1, . . ., . Let , model (3.8) can be converted to the following model: Comparing model (3.3) and model (3.9), we find that the constraints have changed.Therefore, the solutions to model (3.9) are different from that of model (3.3) and the model is not units invariant.
To obtain the unit invariance property, we replace the constraint Σ   =1  ()  .This solution is a feasible solution to model (3.10).Thus, the model given in (3.10) has feasible solution.
The model (3.10) is units invariance.However, the model (3.10) cannot gain a common set of weight.To surmount incomparable efficiency scores and weak discrimination power, we next present cross efficiency model based on the two-level maximin NDEA model.)︁ be the solution to the problem using the maximin principle for DMU  .The DEA efficiencies of  process and DMU  are Then, the peer-evaluation efficiencies given to DMU  when DMU  is under evaluation corresponding to the  process are defined as follows: In this manner, the cross efficiency scores of  process and DMU  are defined as follows: Note that all obtained efficiency scores    ( = 1, . . ., ) are unique.This is due to that all efficiency scores    ( = 1, . . ., ) are directly determined in the objective function in our modeling optimization.This is different from those secondary goals in DEA cross efficiency evaluation methods that are optimized for obtaining a set of optimal weights for the next efficiency score calculation.Then,    and   can be unique.Thus, model (3.10) can obtain the unique cross efficiency.

Cross efficiency model based on the two-level maximin NDEA model
The new models for parallel structure are similar to serial structures.In a parallel production system, at the individual DMU level, we define the system efficiency as the minimum of the subsystem efficiencies.Then, at the all DMU level, we evaluate all the DMUs simultaneously by using the maximin efficiency ratio model.For network structure, by utilizing dummy processes, a network system can be represented by a series structure where each stage in the series is of a parallel structure composed of a set of processes [24].Therefore, the proposed approach can be applied to any network structure.

A numerical illustration
In this section, we use the dataset of supply chain systems from Wu et al. [52] to illustrate the usefulness and the validity of the proposed models.The supply chain system is a network system that can be divided into three processes: the supplier, the manufacturer, and the retailer.The supplier consumes various inputs such as labor ( 11 ) and operating cost ( 12 ) to produce revenue ( 11 ), which then becomes the input cost to the downstream manufacturer.The manufacturer utilizes manufacturing cost ( 21 ) and lead time ( 22 ) to absorb fill rate ( 21 ) and quantity of products ( 22 ).Products are then shipped to the retailer.The retailer bear inventory cost ( 31 ) and backorders ( 32 ) in inventory to earn profits ( 31 ).
Table 2 reports this dataset.The results obtained by the model (3.3) and the supply chain model of Wu et al. [52] are provided in Table 3.
In Table 3, the second and fifth columns report the efficiencies of the supply chain, the supplier, the manufacturer, and the retailer which are obtained by the model (3.3).The sixth to ninth columns describe the efficiencies of the supply chain, the supplier, the manufacturer, and the retailer which are got by the Wu et al.'s method.From Table 3, it can be concluded that the overall efficiency scores from the model (3.3) are all much less than those from Wu et al.'s method.This is due to the fact that the overall efficiency scores from the model (3.3) are the efficiencies of the weak-link of the supply chain and those from Wu et al.'s method are the weighted sum efficiency of stages.These differences indicate that the efficiency of the supply chain systems may be overestimated due to applying the weighted sum model.
Note that our model can show the bottleneck in the production process that is critical to the total efficiency of the system.For example, the overall efficiency of DMU 3 (0.8153) is higher than this of DMU 5 (0.7722) in Wu et al.'s method.However, the overall efficiency of DMU 3 (0.5918) is lower than this of DMU 5 (0.6657) in our method.This is attributed to the efficiency scores of the second stage and the third stage of DMU 3 is significantly higher than those of DMU 5 , while the efficiency of the first stage of DMU 3 is lower than this of DMU 5 .This infers that the efficiency of partial stages conceals the overall inefficiency in the weighted sum model.
The cross efficiency scores are reported in Table 4.It is shown that the efficiencies of the supply chain which are obtained by the model (3.3).This finding indicates that the cross efficiency model has stronger discriminating power in ranking DMUs and is able to produce more representative results.

Figure 1 .
Figure 1.General multi-stage series production system.

Table 1 .
Notation used in the paper.

Table 2 .
Data for the example.

Table 3 .
Results for two-level maximin NDEA model approach.

Table 4 .
Results for cross efficiency model based on the two-level maximin NDEA model.