Using interactive multiobjective methods to solve DEA problems with value judgements
Introduction
Data envelopment analysis (DEA) is a nonparametric frontier estimation methodology based on linear programming for measuring the relative efficiency of a set of comparable decision making units (DMUs) that posse shared functional goals. Travares [1] in his extensive bibliography report of DEA has found a plethora number of DEA publications, last counted at 3203, of which there were 1259 journal papers published and over 50 books written up on DEA. The growing popularity of DEA over the years is reflected by the many enhancements to the original methodologies of Charnes, Cooper and Rhodes (CCR) [2] and Banker, Charnes and Coopers (BCC) [3] and an increasing number of successful DEA applications in the industry.
The original DEA model does not include a decision maker (DM)'s preference structure or value judgements while measuring relative efficiency, with no or minimal input from the DM. Value judgements are defined by [4] as “logical constructs, incorporated within an efficiency assessment study, reflecting the DM's preferences in the process of assessing efficiency”. To incorporate DM's preference information in DEA, various techniques have been proposed such as the goal and target setting models of [5], [6], [7], [8] and weight restrictions models including imposing bounds on individual weights [9], assurance region [10], restricting composite inputs and outputs, weight ratios and proportions [11], and the cone ratio concept by adjusting the observed input–output levels or weights to capture value judgement to belong to a given closed cone [12], [13]. Alternative approaches include [14] whose model adopts unobserved DMUs, derived from pareto-efficient observed DMUs, and which incorporate value judgements; [15] also integrates preference information into a modified DEA formulation, while [16] uses hypothetical DMUs to represent preference information. However, all the above-mentioned techniques would require prior articulated preference knowledge from the DM, which in most cases can be subjective and difficult to obtain.
In manufacturing or service organizations, decision making can become more complex and often inherently uncertain, more so due to multiple attributes and conflicting objectives. Multiobjective programming methods such as multiple objective linear programming (MOLP) are techniques used to solve such multiple criteria decision making (MCDM) problems. An appealing method to incorporate preference information, without necessary prior judgement or target setting, is the use of an interactive decision making technique that encompasses both DEA and MOLP. Golany [5] first proposed an interactive model combining both of these approaches where the DM will allocate a set of level of inputs as resources and be able to select the most preferred set of level of outputs from alternative points on the efficient frontier. Post and Spronk [17] combined the use of DEA and interactive multiple goal programming where preference information are incorporated interactively with the DM by adjusting the upper and lower feasible boundaries of the input and output levels. Then, [18] showed that there are synergies from both DEA and MOLP, and showed that the DEA formulation is structurally similar to the reference point approach of the MOLP formulation. Further papers by [19], [20], [21] introduced the concept of value efficiency analysis (VEA) that effectively incorporate preference information in DEA.
In a similar vein, the aims of the paper are to establish an equivalence model between DEA and MOLP, and show how a DEA problem can be solved interactively without any prior judgement by transforming it into an MOLP formulation.
The main interactive multiobjective methods in the literature such as the tradeoff method of [22], Tchebychev method by [23], the step method by [24], the gradient projection method by [25] and the reference point methods by [26], [27], [28] would be used to solve DEA problems.
The combination of several interactive methods in the solution process has also been used previously like for example [29], where for the same preferences (reference levels), different methods based on reference point approach or classification are used. In this same line, Kaliszewski [30] insists in the necessity to combine different interactive methods under a common methodology. Furthermore, relationships between reference point techniques and local tradeoffs are analysed in [31]. There, relations among different types of information requested from the DM (e.g., reference points and local tradeoffs) are studied and such preferences are found which would produce the same solution, starting from the same previous solution. They can be regarded as equivalent pieces of information in the sense that they produce the same solution.
To carry out our purpose, we will use PROMOIN [32], an interactive multiobjective programming software. The DM can then interactively search along the efficient frontier to locate his most preferred solution (MPS). As all the Pareto optimal solutions are equally desirable and cannot be ordered completely, a DM can express his preferences in relation to conflicting objectives and is able to identify the most preferred one among them, also known as the MPS. Once the MPS is determined, better estimates of efficiency are calculated with more precise resource allocation and target levels that incorporate the DM's value judgements.
The remainder of the paper is organized as follows. The next section provides a brief description of the DEA techniques and introduces the basic DEA models. In the subsequent sections, the equivalence model between DEA and MOLP, and the general formulation for our proposed model is shown. The next section describes briefly the various interactive multiobjective programming methods. An application on the performance measurement of retail banks in the UK is examined with an analysis of results comparing the various interactive MOLP methods. Concluding remarks are provided in the last section.
Section snippets
DEA models
DEA is a linear programming method developed by [2] as a performance measurement technique for evaluating the relative efficiencies of a set of comparable organizational units or DMUs.
Suppose an organization has n DMUs , produces s outputs denoted by , the th output of for and consumes m inputs denoted by , the th input of for . The formulation of the CCR DEA primal model is given by
Equivalence between DEA and MOLP
In a DEA model, an efficiency score is generated for a DMU by maximizing outputs with limited inputs, or minimizing inputs with desired or fixed outputs, or simultaneously maximizing outputs and minimizing inputs. Either way, this can be regarded as a kind of multiple objective optimization problem. In this section, the theoretical considerations of combining MOLP and DEA are presented, and the equivalence between the output-orientated DEA dual formulation and the minimax formulation in MOLP
Interactive multiobjective programming
Interactive multiobjective programming methods constitute techniques that allow the DM to search for different solutions along the efficient frontier, so that the DM can reach the MPS. At each stage, the current solution is adapted to the structure of preferences of the DM. It can be said that an interactive method is designed to drive the DM towards his MPS, or at least, to a good solution, in the sense that it is acceptable by the DM. For this reason, interactive methods are powerful tools
Empirical application: assessment of retail banks in the UK
The UK retail bank industry, specifically seven major retail banks, is examined to demonstrate the interactive approach to search for the MPS on the efficient frontier. The data set is obtained from [49] through a study on using DEA and the ER approach for performance measurement of UK retail banks (Table 1).
For the DEA formulation, the reference set consists of seven DMUs, and three inputs and three outputs are considered. The DMUs are homogenous comparable major banks in the UK including
Conclusion
The paper establishes the equivalence relationship between the output-orientated DEA dual models and minimax reference point approach of MOLP, showing how a DEA problem can be solved interactively without any prior judgements by transforming it into an MOLP formulation. This provides the basis to apply interactive techniques in MOLP to solve DEA problems and further locate the MPS along the efficient frontier for each DMU. Moreover, it effectively allows evaluating past performances and
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