1.1 Motivation and background
Decision-making problems involve both quantitative and non-quantitative factors. The non-quantitative factors are not usually well defined or are subjectively determined by the decision-maker. Such factors cannot be included in the mathematical models while the quantitative factors are modeled as multiple objective linear programming (MOLP). The coefficients in MOLP may obtainable, well defined, or not sensitive to the final solution. An example of MOLP may be projects of government investment, in which the minimization objective functions (inputs) may be manpower, machines, construction costs, operation costs, other controllable costs and uncontrollable costs while the maximization objective functions (outputs) may be revenues, rate of population growth, growth of economic improvement.
Project selection problems have received substantial attention in recent decades (Martino, 1995). This research concerns the problem of selection and evaluation of collective projects from a feasible set of projects. There are many difficulties associated with the evaluation problems of collective projects, such as multiple conflicting objectives, non-quantitative objects, and the enormous number of possible combinations. In this paper, each subset of the projects is treated as a single portfolio, and evaluated against a relative production technology. Many researchers have proposed the evaluation and selection of projects in a portfolio (Oral et al., 1991; Cook & Green, 2000; Linton et al., 2002). It is desired to establish the portfolios of projects that can be justified as making the best use of available resources. It involves the evaluation, from a larger set of projects, of each portfolio to be undertaken. The problem discussed here falls firmly into the multiple criteria decision-making (MCDM) arena.
In MCDM, there are a number of alternatives among which a decision-maker must decide. Each alternative is described by its performance according to certain criteria, attributes, or objectives. Stewart (1996) defines a criterion as being a particular point of view according to which alternatives may be assessed and rank-ordered. An attribute is a particular feature of the alternative with which a numerical measure can be associated. An objective is a specific direction of preference defined in terms of an attribute. The aim of MCDM is to provide support to a decision-maker in making the best choice among alternatives, and to
Data envelopment analysis (DEA) is a robust and valuable methodology for frontier estimation (Charnes et al., 1978). Based on mathematical programming techniques, it is particularly suited to estimating multiple input and output production correspondence. In the last two decades, DEA has become a popular method for analyzing the efficiency of various organization units (Norman & Stoker, 1991) which differ both in the quantities of inputs they consume and in the outputs they produce, and does not require any subjective or economic parameters (weights, prices, etc.). Many studies have been concerned with the efficiency of production. It is clear that DEA is now playing a wider role in management science. In particular, DEA approaches have assumed important status within the toolkits of investigators concerned with MCDM (Joro et al., 1998).
It is worthwhile to identify the role of our problem in the related academic studies.
DEA and MCDM are two related techniques that have received considerable attention in the OR/MS literature. Many papers have proposed to analyze the links between DEA and MCDM (Belton & Vickers, 1993; Stewart, 1996; Joro et al., 1998; Sarkis, 2000). The success of DEA in the area of performance evaluation, with formal analogies between DEA and MCDM, has led some authors to propose DEA as a tool for MCDM (Doyle & Green, 1993; Stewart, 1994;
Bouyssou, 1999; Liu et al., 2000). There appears, nevertheless, to be little interaction between these two sub-fields, despite the fact that they address rather similar problems. In general, the aim of DEA is not to select one optimal decision-making unit (DMU), but rather to separate efficient DMUs from inefficient ones and to indicate the ‘efficient peers’ for each inefficient DMU. The MCDM and DEA formulations coincide (although their ultimate aims may still differ) if we view inputs and outputs as criteria or attributes for evaluating DEA, with minimized inputs and maximized outputs as associated objectives (Belton & Vickers, 1993).
Many researchers have discussed the project selection problems in various forms.
Bunch et al. (1989) apply DEA additive model to solve the problems, Oral et al. (1991) depart from the DEA CCR model and propose a rather complex multi-stage collective evaluation and selection model, which is called the OKL point. Cook & Green (2000) follow the OKL point to solve the resource-constrained project selection problem by using mixed-integer programming.
1.2 Problem definition
Suppose a set of K candidate project proposals numbered k = 1, … , K is somehow to be evaluated and selected. Project k consumes amounts of aik, i = 1, … , m resources to produce crk, r = 1, … , s products. A portfolio comprises a subset of the K feasible projects is denoted by P = (w1, … , wK), where wk = 1 if the kth project belongs to portfolio P and wk = 0 otherwise. Let Ω denote the set of all feasible portfolios where:
Ω = {P = (w1, … , wK) | wk = 0 or 1, k = 1, … , K.}. (1.1)
Let n be the number of total possible portfolios in set Ω under evaluation, n =||Ω|| = 2K. It is assumed that the projects are neither synergistic nor interfering, and all portfolios are supportable since resource constraints are absent for a decision maker. If both projects were selected, the outputs produced would be the sum of their respective outputs, and so as the input resources used. The correspondence set of DMUs is:
ΩD = {DMUP = (y1P, … , ysP, x1P, … , xmP) | P∈Ω}, (1.2)
where yrP = cr1w1 + … + crKwK, r = 1, … , s, and xiP = ai1w1 + … + aiKwK, i = 1, … , m. Then, the collective evaluation problem is modeled as following multiple objective binary integer linear programming (MOBILP):
Maximize yrP = cr1w1 + … + crKwK, r = 1, … , s. (M1)
Minimize xiP = ai1w1 + … + aiKwK, i = 1, … , m.
Subject to P∈Ω.
For solving model (M1), some different methods are proposed in Keeney & Raiffa (1976) and Steuer (1986). Difficulties arise due to disagreement between various interested parties concerning its form and detail. Instead of considering optimization of the criteria, a DEA-based approach circumvents these difficulties by allowing each portfolio to evaluate itself relative to all portfolios under consideration. DEA is intended to identify efficient portfolios, to characterize inefficient portfolios, and to assess from where inefficiencies arise.
However, DEA methodology is computationally intensive, requiring the solution of n mathematical programs when analyzing a data set that comprises n DMUs. As discussed in Ali (1990; 1992; 1994), identification of efficient and inefficient DMUs without solving a DEA program is very useful in streamlining the solution of DEA computations. In this study, we present mathematical properties to characterize the inherent relationships between
efficiency of portfolios and data of projects. By using the output-input ratio of individual project, efficient and inefficient portfolios are identified prior to the DEA program. The frontier of the pre-identified efficient portfolios is developed as a filter and is used to characterize inefficient portfolios from the class of candidate efficiencies. Inefficiency of portfolios is identified with portfolios that lie within the DEA frontier. The case-based computer systems use linear programming (LP) with a small problem size to rapidly identify a large number of inefficient portfolios. Then, the remaining portfolios are evaluated by using DEA programs to identify efficient units and measure the stability of each efficient unit to rank all efficient units for the decision aim.
A large number of alternatives would be ruled out from final decision. There are many ways to use the solution of our method to obtain the final decision under the consideration of non-quantitative factors, such as follows: (i) Compare the super-efficiencies of all the efficient portfolios, (ii) Sensitivity analysis on the coefficients so that a specific extremely efficient portfolio becomes inefficient, and (iii) Sensitivity analysis on the coefficients so that a particular inefficient portfolio becomes efficient. Therefore, the effort for making the final decision is significant reduced.
The literatures of sensitivity analysis of DEA deal with only change values of input and/or output of one particular efficient DMU while the other DMUs are hold fixed, or change data of all efficient DMUs simultaneously according some given rules. To investigate the stability of each efficient portfolio with respect to the coefficients of a specific project, the super-efficiency measure could not satisfy our requirements, since the portfolio consists of some projects. For a specific efficient portfolio, we are considering the stability of the portfolio while we are changing data of some portfolios through changing the coefficients in the inputs and outputs of a particular project. When all the stability measures are obtained, they are helpful to the final decision maker to possess the fine comparison of efficient portfolios.
1.3 Objectives of the research
Without predetermined the weights of the objectives, we use DEA to measure the efficiency and stability of each portfolio. The objective is to select and rank portfolios that are efficient in terms of the characteristic of DEA. The difficulty of the DEA analysis may spend more effort on computations while the number of portfolios (DMUs) tends to be large. In our
problem, the total number of alternatives is 2K, and it could be doubled when we added one more project to the MOBILP (M1). If use the conventional DEA model to assess each portfolio against the 2K portfolios, one needs to solve a linear programming models with 2K variables and (m+s) constraints. For instance, if K equals to 30, one needs a linear programming software package with the capacity to accommodate the 230 variables. It may reach the capacity of existing software and the personal computers. The problem with K value beyond 30 would not be solved. The computation time is the other issue has to be conquered.
In our experiment, for the case K=24, we spent more than one day to have final solution.
We develop an efficient method to identify the efficient portfolios for MOBILP with single minimization (input) and minimization (output) problem. One does not need to employ linear programming to obtain the solution. For the MOBILP with multiple minimization and maximization objective functions, an efficient and effective process for identifying inefficient portfolios is proposed to reduce the computation prior to the DEA programs, and identifying some efficient portfolios whose frontier is used to implement the filtering algorithm.
Therefore, all of the efficient portfolios and their correspondence efficiency measures are obtained by using the proposed process.
The inputs and outputs of each portfolio are respectively obtained from the sum of the input and output of the selected projects. Hence, the changes of any one coefficient in (M1) would change a half number of the total portfolios that contain the changed project. The efficiency measures of those portfolios may be changed while the coefficient is perturbed. For instance, if the coefficient, say aik, is changed, all the portfolios with wk=1 are changed respectively while the other half portfolios are remain unchanged. Our purpose concerns the perturbation of coefficients, aik and crk, of project k in an interested efficient portfolio to preserve its efficiency. We are considering the stability of an extremely efficient portfolio while we are changing the inputs and outputs of some portfolios through changing the coefficients of objective functions of a particular project (binary decision variable). The sensitivity analysis for the coefficients is modeled as a non-linear programming whose optimal values yield a stability region of an extremely efficient portfolio. Sufficient and necessary conditions are provided for upward variations of aik and downward variations of crk for a specific project such that an extremely efficient portfolio remains efficiency. A technique using linear programming to approximate the optimal solution to the non-linear programming also proposed.
1.4 Organization of the dissertation
The second chapter reviews the related literature in MCDM, DEA and its sensitivity analysis. Chapter three introduces an efficient process for constructing efficient frontier. The output-input ratio analysis for quickly identify dominated portfolios are proposed. Then, a filtering algorithm is used to solve the MOBILP (M1). Chapter four proposes the sensitivity analysis for DEA models. Non-linear models are proposed for finding the stability regions of efficient portfolios with respect to the data changed in project. The method that uses linear programming model to approximate non-linear programming stability model is also provided.
Conclusion and discussion are presented in chapter five. The structure of this study is illustrated in Figure 1.