A multiple criteria approach to two-stage data envelopment analysis

Highlights

• A new approach for evaluating and ranking sustainable suppliers.

• A two-stage DEA approach for multiple criteria sustainable processes analysis.

• Model provides improved discrimination power to traditional two-stage DEA models.

• More effective sustainability assessment of the DMUs is achieved using the model.

Abstract

Two-stage data envelopment analysis (DEA) models are commonly used in the evaluation and benchmarking of sustainable operations and processes across multiple research fields. To date, however, little attention has been given to the unrealistic weight distribution and weak discrimination power in the modeling and evaluation of the two-stage sustainable operations when using two-stage DEA models. In order to overcome this methodological weakness, we use the multiple criteria DEA (MCDEA) approach in the evaluation of the two-stage processes. The outcome is a multiple criteria two-stage DEA model which yields more realistic weights for the inputs and outputs and thus has better discrimination power than traditional two-stage DEA models. The developed model is tested and validated by assessing the sustainable design performances of a sample of car product designs.

Introduction

Two-stage data envelopment analysis (DEA) models are used to evaluate the efficiency of the processes which are made up of two separate and distinct stages. The first stage of such a two-stage process consumes some inputs and produces some outputs. The outputs of the first stage, which are called the intermediate measures, are then used as the inputs of the second stage. The main difficulty in the modeling and measurement of the efficiency of the two-stage processes arises from the presence of the intermediate measures since larger values of these measures represent the better performance of the first stage and worse performance of the second stage.

Traditional DEA models use a “black-box” approach in measuring the efficiency of the activities and do not relate the sources of the inefficiency of the processes to their different stages. However, this approach is limited in its measurement of the efficiencies of activities which are consist of two stages (or sub-processes) where the outputs of one stage are the inputs of another stage (Chen et al., 2012). By applying the traditional black-box approach to the measurement of the efficiency of a two-stage process it is not always possible to track the sources of the inefficiencies. However, by using a two-stage DEA model to measure the overall efficiency as the combination of two separate efficiency ratios, i.e. efficiency ratio of stage 1 and efficiency ratio of stage 2, it is possible to identify not only the overall efficiency of the activities, but also the efficiency status of their sub-processes. This is, clearly, more informative than simply measuring overall efficiency as the ratio of the final outputs of the whole system to its inputs.

Moreover, by opening the black box and applying the two-stage DEA models, optimization of the decision making units (DMUs) and activities can be achieved by adopting different scenarios. It can be reached by: (i) simultaneously optimizing the efficiencies of the stages 1 and 2 of the two-stage process, or (ii) optimizing the efficiency of the stage 1 as the more important and the leader stage first, and then optimizing the less important and the follower stage 2, or (iii) optimizing the efficiency of the stage 2 first and then the stage 1. These alternative scenarios are not possible with the traditional DEA models since each activity and process is not represented by the combination of its sub-processes.

Seiford and Zhu (1999) introduced one of the first two-stage DEA models and these authors used this to evaluate the efficiency of the US commercial banks by measuring their overall efficiency as a result of the two sub-processes of the marketability and profitability. However, the efficiency of each step was measured by a separate DEA model and thus did not reflect the potential conflict between the two stages. Following this initial approach, the focus shifted to the measurement of both the efficiency of each of the steps and also the overall efficiency in an integrated model. To achieve this, Liang et al. (2006) applied the concepts of the cooperative and non-cooperative games from the game theory literature to measure the efficiency of the DMUs through a two-stage process. In the non-cooperative context, the efficiency score of the more important stage (the leader) is measured and optimized first, and then the efficiency score of the less important stage (the follower) is measured subject to keeping the efficiency of the leader stage unchanged. In this cooperative approach, the efficiency scores of the both stages are simultaneously measured and optimized. It should be noted that both the cooperative and non-cooperative models developed in Liang et al. (2006) are non-linear models and are solved as parametric linear programs.

Liang et al. (2008) went on to develop a further set of linear cooperative and non-cooperative two-stage DEA models. In order to create a linear cooperative model, Liang and his colleagues defined the overall efficiency as the product of the efficiency ratios of the stages 1 and 2. This allows for the linear formulation for the cooperative model since the intermediate measures exist in the denominator of the efficiency ratio of the first stage and in the numerator of the efficiency ratio of the second stage. By multiplying the efficiency ratios of the stages 1 and 2, the intermediate measures are omitted and the overall efficiency is maximized by maximizing only one efficiency ratio. Kao and Hwang (2008) also used a similar approach whereby the product of the efficiencies of the stages was treated as the overall efficiency, and this allowed them to maximize the efficiencies of the stages simultaneously.

Chen et al. (2010) took the discussion further by highlighting that, due to the existence of intermediate measures, defining improvement targets for the inefficient DMUs is a challenging task. As a result, these authors developed an approach to determine the benchmark efficient DMUs for the inefficient DMUs within a two-stage structure. Chen et al. noted that, although many of the two-stage DEA models in the literature are able to measure the efficiency scores of the DMUs, they do not provide any information as to where the DEA efficiency frontier is located. Therefore, the projection of the inefficient DMUs onto the efficiency frontier is not possible. Using the model developed by Kao and Hwang (2008) as an example and employing two different tests, Chen et al. showed that the projections defined by this model do not yield efficient target DMUs for the inefficient DMUs.

Chen et al. (2009) also developed an additive efficiency decomposition to measure the efficiency of the two-stage processes under both constant returns to scale (CRS) and variable returns to scale (VRS) assumptions. In a subsequent review of the existing two-stage DEA models, Cook et al. (2010) showed that all the approaches found within the literature can be categorized as using either non-cooperative or cooperative game concepts. Li et al. (2012) extended the work of Kao and Hwang, 2008, Liang et al., 2008 by considering some other inputs entered into the second stage which were additional to the intermediate measures. They argued that, due to the existence of the additional inputs to the second stage, application of the product of the efficiency of the two stages will lead to a non-linear program. In order to find the global optimal solutions, they converted the non-linear models to parametric linear models. Wang et al. (2014) also developed the additive model proposed by Chen et al. (2009) by considering undesirable outputs and applied this approach to an efficiency evaluation of the Chinese commercial banks.

However, from this overview of the extensive literature of the two-stage DEA models one common issue emerges. Current models do not consider the lack of discrimination power and unrealistic weights distribution among input, intermediate measure and output factors. This is not a new issue in the sense that it has been raised in the literature relating to the traditional one-stage (black box) DEA models where different solutions have been suggested. However, it is posited that the arguments made in the literature relating to one stage DEA models (where the focus is on improving discrimination power and gaining more realistic inputs and outputs weights) are also applicable in the two-stage DEA models.

As a result, the main focus of our paper is to make a link between the approaches suggested and applied in the original black box DEA models literature to improve the discrimination power and weighting system of the two-stage DEA models. At the same time, since the two-stage DEA models are frequently used in sustainability assessment of different types of DMUs or processes, our developed procedure will contribute to a better and more effective application of the two-stage DEA models in this very important area.

It will also be appreciated that the two problems of lack of discrimination and unrealistic weights distribution are inter-related and thus, as noted by Li and Reeves (1999) they are usually found simultaneously. The weak discrimination and consequential identification of many DMUs as ‘efficient’ takes place when the number of DMUs in the evaluation sample is not large enough relative to the number of evaluation factors (inputs and outputs). On the other hand, unrealistic weight distribution refers to a situation where some DMUs gain high efficiency scores by attaching unrealistically large weight(s) (i.e. importance) and/or unrealistically small weight(s) to a single factor(s) (Li and Reeves, 1999, Ghasemi et al., 2014). These problems are usually dealt with by using weight restriction techniques such as those suggested by Charnes et al. (1990) (cone ratio), Thompson et al. (1990) (assurance region) and Wong and Beasley (1990) who suggest using experts’ judgments to restrict the feasible region of the weights. In all of these cases, this can be done by adding lower and upper bounds on the weights in the form of some additional constraints.

One main criticism on these approaches is, however, their need for prior information on the weights which are collected as human subjective judgments. Moreover, application of the weight restrictions, especially absolute weight restrictions, need a careful consideration before their use since they might lead to incorrect relative efficiency evaluation of the DMUs and this, in turn, can lead to the conflict with economic principles of the efficiency evaluation (Førsund, 2013).

Therefore, we do not suggest using weight restriction techniques in two-stage DEA models and instead apply the multiple criteria DEA (MCDEA) model approach to tackle both these two problems (lack of discrimination and unrealistic weights). The MCDEA model was introduced by Li and Reeves (1999) and, as we will demonstrate, can deal with both the interrelated problems discussed above without the need to the human subjective judgments.

Unlike the classical DEA models which measure the efficiency of each DMU by finding the best possible weights to maximize its own efficiency score, in MCDEA some other alternative objective functions are also used. Each of the objective functions is called a criterion to be optimized. To complete the efficiency analysis in the original MCDEA model developed by Li and Reeves (1999), three separate models are run. In each run of the model, one of the objective functions is optimized. However, in two recent papers, Bal et al., 2010, Ghasemi et al., 2014 developed a unified model to optimize all the objective functions simultaneously. In the context of the development of this approach (which will be outlined in greater detail later in this paper), some applications of MCDEA in environmental assessments can be seen in evaluating the efficiency of the renewable energy technologies (San Cristóbal, 2011), measuring sustainable energy index of the Asia-pacific economic cooperation (APEC) members (Hatefi and Torabi, 2010), and assessing the environmental impacts of the large construction projects (Zhao et al., 2006).

In summary, therefore, the main focus of our paper is to develop a multiple criteria framework for a two-stage DEA model. The combined model will be more effective than the basic two-stage DEA models when they are applied in environmental evaluations and in process-related sustainability assessments. We will subsequently demonstrate an application of the developed model in assessing the sustainable design in the vehicle industry.

The remainder of this paper is organized as follows. In Section ‘Proposed model’, the multiple criteria two-stage DEA model is developed. In Section ‘Application’, the developed model is tested by assessing the sustainable products design in the vehicle industry. Section ‘Concluding remarks’ concludes the paper.