A Two-Stage Bi-Objective Data Envelopment Analysis Problem

. This paper proposes a novel two-stage bi-objective Data Envelopment Analysis (DEA) model. The objectives considered are maximization of overall efficiency and maximization of labor efficiency. The existing literature has not yet proposed any multi-objective DEA model till date. However, such model is in demand because most of the real life problems consist of multiple stages. The proposed model has also considered dissimilar intermediate weights. The proposed model has been applied to a multi-stage insurance problem with two input variables, two intermediate variables and two output variables in order to establish the effectiveness of the proposed model. The proposed model is an algorithmically complex problem and therefore a hybrid Multi-Objective Genetic Algorithm has been applied in order to solve the proposed model.


Introduction
The application of Data Envelopment Analysis (DEA) is a popular way of measuring efficiency of different problems as evident from the existing literature. In particular, Network Data Envelopment Analysis (NDEA) has drawn significant attention of researchers recently. Till date, the existing literature has proposed single objective network DEA problems in general although most of the real life problems where NDEA can be applied have multiple objectives. Thus, the investigation in to multi-objective NDEA problem is in demand. Towards this direction, this paper contributes to the existing literature on DEA in the following ways -1) by proposing a two-stage DEA with dissimilar intermediate weights, 2) by considering two objectives. Thus, this paper has proposed a multi-objective two-stage DEA problem, emphasizing the internal structure of DEA models. Network structure can be of three typesseries structure, parallel structure and a mix of these two structures. This paper has emphasized on NDEA with two stages with series structure whose basic structure is shown in Figure 1.

Literature Review
DEA was first proposed by Charnes et al. [1] in the form of CCR model followed by the BCC model by Banker et al. [2]. These two models were very simple models with no mention of internal structure. Later, Fӓre and Grosskopf [3] proposed network DEA (NDEA) which considered internal structure of DEA models by considering intermediate stages. There is significant number of two-stage models in the existing literature. Some of them are the research studies of Alder et al. [4], Tsolas [5], Lim and Zhu [6]. There are multiplicative decomposition approaches for DEA such as the research studies of Kao and Hwang [7] and Kao [8]. For a two-stage model, if the first stage has some inputs and the second stage has some additional inputs, then the aggregation approach may be applicable instead of decomposition approach. Such works include the research studies of Lozano [9], Chen and Zhu [10], Lu et al. [11]. However, none of the previously proposed model has considered multi-objective scenario with the DEA models and none has considered dissimilar intermediate weights. This paper has considered both of these approaches, thereby, filling the gap of research in the existing literature. Some DEA researchers have also applied Genetic Algorithm (GA) like the research studies of González et al. [12], Pendharkar [13], Mozaffari et al. [14]. However, since the current has considered Multi-Objective problem, thus, Multi-Objective Genetic Algorithm has been applied.

Proposed DEA Model
Based on the DEA model as shown in Figure 1, the mathematical formulation for the proposed two-stage DEA is given in Figure 2. This model, converted to linearized form, is shown in Figure 3. The weights are represented by , , , W is the weight of output j from stage 1 and j W is the weight of input to stage 2. The data as shown in Section 4 from a practical example can help to calculate these weights.

Case Study and Proposed Multi-Objective Genetic Model
This paper has uses a similar case study as used by Kao and Huang [15]. The data have been simulated with the help of the data in the case study of Kao and Huang [15]. The data is provided in Table 1. This is a case study based on the data as collected from a total of 16 insurance companies. The input for the first stage are number of employees (L), operating expenses ( 1 X ), insurance expenses ( Table 1. This proposed model has been solved by a Multi-Objective Genetic Algorithm which is shown in Figure 4. The structure of each chromosome is shown in Figure 5.

Results and Discussion
The Multi-Objective GA has been implemented in Matlab 2014b. The crossover and mutation probabilities are 0.7 and 0.3 respectively. The population size and the number of iterations have been taken to be 100 and 100 respectively. The final results of the proposed model and that obtained from the model of Kao and Huang [15] are shown in Table 2. Table 2 shows different rankings for the two methods. However, the results for the proposed method are similar to those for the approach of Kao and Huang [15] In order to verify whether the proposed method is providing any similar results, Table 3 shows the results for Spearman's rank correlations which can find associations between two sets of ranks. Table 3 shows high positive association between the proposed approach and the approach by Kao and Huang [15].      The two objectives as considered are maximization of overall efficiency and maximization of labor efficiency. The proposed approach has been applied to a case study on different insurance companies. The results of the proposed approach have been compared with those of an existing approach. The association among these two shows that the two approaches provide similar results.