Ann Oper Res DOI 10.1007/s10479-010-0819-6
Genetic algorithm-based multi-criteria project portfolio selection Lean Yu · Shouyang Wang · Fenghua Wen · Kin Keung Lai
© Springer Science+Business Media, LLC 2010
Abstract Project portfolio selection is one of the most important decision-making problems for most organizations in project management and engineering management. Usually project portfolio decisions are very complicated when project interactions in terms of multiple selection criteria and preference information of decision makers (DMs) in terms of the criteria importance are taken into consideration simultaneously. In order to solve this complex decision-making problem, a multi-criteria project portfolio selection problem considering project interactions in terms of multiple selection criteria and DMs’ preferences is first formulated. Then a genetic algorithm (GA)-based nonlinear integer programming (NIP) approach is used to solve the multi-criteria project portfolio selection problem. Finally, two illustrative examples are presented for demonstration and verification purposes. Experimental results obtained indicate that the GA-based NIP approach can be used as a feasible and effective solution to multi-criteria project portfolio selection problems. Keywords Multi-criteria decision making · Nonlinear integer programming · Genetic algorithm · Project portfolio selection · Project interactions · Preference 1 Introduction Project portfolio selection is an active research topic in the field of project management and engineering management (Aaker and Tyebjee 1978; Mavrotas et al. 2006; Fox et al. 1984; L. Yu () · S. Wang MADIS, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China e-mail:
[email protected] F. Wen Research Center for Financial Engineering and Financial Management, School of Economics and Management, Changsha University of Science and Technology, Changsha 410114, China K.K. Lai Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
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Peng et al. 2008a; Bouyssou et al. 2006; Carraway and Schmidt 1991; Dickinson et al. 2001; Kuei et al. 1994; Medaglia et al. 2007a, 2007b; Santhanam and Kyparisis 1995; Golabi 1987; Kleinmuntz and Kleinmuntz 1999; Kleinmuntz 2007; Ewing et al. 2006; Golabi et al. 1981; Talias 2007; Stummer et al. 2003; Liesio 2006; Cooper et al. 1999; Liesio et al. 2007; Stummer et al. 2009; Henriksen and Palocsay 2008). Usually, selecting a small subset of projects from a bigger project set as a project portfolio on the basis of multiple selection criteria is a typical multi-criteria decision-making (MCDM) problem in both public organizations (Medaglia et al. 2007b; Golabi 1987; Kleinmuntz and Kleinmuntz 1999; Kleinmuntz 2007; Ewing et al. 2006) and industrial firms (Mavrotas et al. 2006; Golabi et al. 1981; Talias 2007; Stummer et al. 2003). Project portfolio selection is often a complex decision if the project interactions in terms of multiple selection criteria and information of preferences of decision-makers are taken into account, particularly in the presence of a large number of projects (Liesio 2006). For this purpose, some flexible decision analytic models and decision support systems (DSS) (Stummer et al. 2009; Henriksen and Palocsay 2008) are used to provide effective decision support for decisionmakers (DMs), though simple and understandable decision models are more readily applicable and likely to be accepted easily by most practitioners (Cooper et al. 1999). Due to the importance of project portfolio selection in project management and engineering management, many approaches (most with successful practical applications) have been proposed to solve the project portfolio selection problem (Aaker and Tyebjee 1978; Mavrotas et al. 2006; Fox et al. 1984; Peng et al. 2008a; Bouyssou et al. 2006; Carraway and Schmidt 1991; Dickinson et al. 2001; Kuei et al. 1994; Medaglia et al. 2007a, 2007b; Santhanam and Kyparisis 1995; Golabi 1987; Kleinmuntz and Kleinmuntz 1999; Kleinmuntz 2007; Ewing et al. 2006; Golabi et al. 1981; Talias 2007; Stummer et al. 2003; Liesio 2006; Cooper et al. 1999; Liesio et al. 2007; Stummer et al. 2009; Henriksen and Palocsay 2008). For example, Aaker and Tyebjee (1978) applied a quadratic 0–1 programming approach to select interdependent R&D projects. Similarly, Mavrotas et al. (2006) combined multi-criteria decision analysis (MCDA) with 0–1 integer programming (IP) for project prioritization under policy restrictions. Fox et al. (1984) constructed an economic model that included benefit interactions (called present value interactions) between R&D projects and obtained some interesting results about project portfolio selection. Carraway and Schmidt (1991) proposed an improved discrete dynamic programming (DDP) algorithm to allocate resources among interdependent projects. Medaglia et al. (2007a) proposed a multiobjective evolutionary approach for linearly constrained projects selection under uncertainty. In particular, Golabi et al. (1981) proposed an additive multi-criteria value model for project portfolio selection problems. In particularly, they extended the multi-attribute value theory (MAVT) (Roy 1996; Keeney and Raiffa 1976), which is a general framework for obtaining multicriteria values of explicitly defined alternatives, to project portfolio selection problems and gave some assumptions under which the portfolio’s overall value has an additive form. Using such additive value representations, the solution to the optimal portfolio subject to resource constraints could be obtained from an integer linear programming (ILP) problem. This approach has been used by the U.S. Department of Energy (DOE) to select a portfolio of solar energy projects. Similarly, Kleinmuntz and Kleinmuntz (1999) and Kleinmuntz (2007) also applied the additive model proposed in Golabi et al. (1981) to assist capital budgeting in over 750 hospitals and healthcare organizations in the United States. They found that the additive model resulting in an ILP problem was particularly useful for implementing sensitivity analysis because the optimal project portfolios are readily obtained for different model parameters. In addition, Stummer and Heidenberger (2003) proposed a three-phase approach including multi-objective integer linear programming (MOILP) for
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R&D project portfolio selection and reported results of implementing a decision support system (DSS) based on this model in an industrial enterprise. Similarly, Stummer et al. (2009) and Henriksen and Palocsay (2008) developed some multi-criteria DSS for evaluating and selecting different projects and obtained some interesting results about project portfolio selection. In accordance with the above analyses, it can be found that the existing approaches for project portfolio selection did not handle project interactive effects in terms of different selection criteria, nor did they consider preference of decision-makers in terms of the importance of selection criteria. Actually, in project portfolio selection problem, if a project is selected in conjunction with other projects, it may have positive or negative interactive effects in terms of a specific selection criterion. If one does not consider project interactive effects based on different selection criteria, the decision process may yield an undesirable outcome (Carlsson and Fuller 1995). Usually, Interaction is a kind of action that occurs as two or more objects have an effect upon one another and it can be measured by some statistical methods such as Analysis of Variance (ANOVA) (Cox 1984). Furthermore, the existing approaches are computationally applicable only up to 30 projects if interactions among the projects are considered, since the resulting MOILP problem is solved by examining all possible portfolios. For these reasons, project selection decisions are often subjected to criticism especially in cases of public decision-making. In order to solve the above issues, this paper tries to formulate a multi-criteria project portfolio selection problem considering project interactions in terms of multiple criteria and DMs’ preference information based on criteria importance. However, the multi-criteria project portfolio selection problem involving project interactions and DMs’ preferences are definitely more complex than the simple project selection process with several independent projects. In order to solve the complex multi-criteria project portfolio selection problem, this paper proposes a typical evolutionary algorithm (EA)—genetic algorithm (GA) to devise a feasible and effective solution. The main motivation of this study is (1) to formulate a multi-criteria project portfolio selection problem considering the project interactions in terms of multiple selection criteria and DMs’ preferences in terms of criteria importance and (2) to apply GA to devise a feasible and efficient solution to the proposed multi-criteria project portfolio selection problem. The remainder of this study is organized as follows. In Sect. 2, a multi-criteria project portfolio selection problem with project interactions and DMs’ preferences is formulated in detail. In Sect. 3, an efficient GA-based solution is presented to the proposed multi-criteria project portfolio selection problem. For illustration and demonstration purpose, two numerical examples are given and computational results and analyses are reported in Sect. 4. In Sect. 5, some concluding remarks are drawn.
2 Formulation of multi-criteria project portfolio selection problem In project portfolio selection, a decision-maker is often faced with the problem of selecting a small subset of projects from a bigger set of projects based on a set of selection criteria. This process is known as the multi-criteria decision making (MCDM) (Korhonen et al. 1992; Stewart 1992; Ostermark 1997) process. Generally speaking, there are two types in MCDM problems: multi-attribute decision making (MADM) problem and multi-objective decision making (MODM) problem. Usually, the MADM problem can be solved by the multiattribute utility theory (MAUT), while the MODM problem can be solved by multiobjective mathematical programming (MOMP) such as dynamic programming (Carraway
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and Schmidt 1991) and quadratic programming (Peng et al. 2008b). In accordance with this classification, multi-criteria project portfolio selection is seen as a typical MADM problem in terms of the characteristics of project portfolio selection. Accordingly the MAUT theory can be used to solve the multi-criteria project portfolio selection problem. For illustration purpose, the multi-criteria portfolio selection problem can be presented below. Suppose there are I projects to be evaluated and selected, and decision variable xi denotes whether proposed project ai (i = 1, 2, . . . , I ) is included in the portfolio (xi = 1) or not (xi = 0). Thus a project portfolio can be represented by the value of x = (x1 , x2 , . . . , xI ). Let wj be the preference degree of decision makers on criterion j (j = 1, 2, . . . , J ) when evaluating alternative project ai (i = 1, 2, . . . , I ), and cij be the score or value of project i (i = 1, 2, . . . , I ) on selection criterion j (j = 1, 2, . . . , J ). If there are no interactions among projects, the project portfolio selection model can be formulated by using a standard multiattribute additive method based on the multi-attribute utility theory (MAUT) (Korhonen et al. 1992; Stewart 1992; Ostermark 1997), which is shown below. ⎧ ⎪ ⎪ ⎪ Max ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ s.t. ⎪ ⎪ ⎪ ⎪ ⎩
V=
J I i=1
I
wj cij xi
j =1
(1)
xi = M
i=1
xi = {0, 1}
where V represents the total utility values (or total effects) of a subset of selected projects and M is the number of the selected projects in terms of a set of selection criteria. In the project portfolio selection model shown above in (1), the involved projects are usually assumed to be independent, i.e., there are no interactions among the projects. However, there may be some interactive effects among projects in real situations, in terms of different selection criteria. Furthermore, not considering interactions among projects may lead to an undesirable outcome, as pointed out in Carlsson and Fuller (1995). Therefore, decision results based on (1) might be impracticable and accordingly the interactive effects among projects should be incorporated into the decision modeling. For convenience of description, let dj (Sk ) be the value of interactive effects in a combination of k projects in terms of selection criterion j (j = 1, 2, . . . , J ) and Sk (k = 1, 2, . . . , K) be a combination of k projects. Accordingly, the multi-criteria project portfolio selection problem with project interactions in terms of multiple selection criteria can be formulated in the following form: ⎧ ⎪ ⎪ ⎪ Max ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ s.t. ⎪ ⎪ ⎪ ⎪ ⎩
V=
J I i=1
I
j =1
xi = M
wj cij xi +
J K j =1 k=1
wj (dj (Sk ))
L i=1
cij
L
xi
i=1
(2)
i=1
xi = {0, 1}
where L is the number of variables with interactive effects, the first sum of the objective function in (2) gathers individual effects of all individual projects ai (i = 1, 2, . . . , I ), as identical to (1), and the second sum of the objective function in (2) concerns interactive
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effects of different projects. In particular, interactive effects dj (Sk ) can be seen as additional effects if project portfolio x contains a combination of at least k projects. In the above multi-criteria project portfolio selection problem shown in (2), it is easy to see that the multi-criteria project portfolio selection model with consideration of project interactions and DMs’ preferences is a typical multi-criteria 0–1 nonlinear integer programming (NIP) problem. Meantime, it is a typical pseudo Boolean optimization (PBO) problem (Boros and Hammer 2002), which is often NP-hard. Obviously, the decision-making process involving project interactions and DMs’ preferences are definitely more complex than a portfolio of independent projects without interactions. In order to solve the NIP problem in (2), many methods, such as bound and branch algorithm (BBA) and dynamic programming (DP) method (Hammer and Rudeanu 1968) are proposed. However, the computational process of these methods is often too complicated especially in the case where a large number of projects need to be evaluated for selection purpose. For this reason, this paper proposes an intelligent solution to the NIP problems by introducing evolutionary algorithms (EAs) elaborated in the next section. It should be noticed that if interactive effects among projects are not considered, the 0–1 nonlinear integer programming (NIP) problem in (2) becomes a standard 0–1 integer programming (IP) problem, as shown in (1), which is easily solved by a standard binary integer programming (BIP) algorithm. In this sense, the 0–1 NIP problem in (2) can be seen as the generalized form of a standard 0–1 IP problem in (1). That is, the standard 0–1 IP problem is a special case of the 0–1 NIP problem.
3 GA-based optimization approach to 0–1 nonlinear integer programming For the nonlinear integer programming (NIP) problem presented in (2), some commonly used optimization methods such as bound and branch algorithm (BBA) and dynamic programming (DP) method (Hammer and Rudeanu 1968) are difficult to be used to solve the multi-criteria project selection problem with project interactions and DMs’ preferences. For this reason, this paper proposes evolutionary algorithms (EAs) to solve the 0–1 NIP for project portfolio selection problem. EAs are a class of generic population-based meta-heuristic optimization algorithms. Usually EAs use some mechanisms inspired by biological evolution: reproduction, mutation, recombination, natural selection and survival of the fittest, for solving optimization problems. Candidate solutions to the optimization problem play the role of individuals in a population, and the objective function determines the environment within which the solutions “live”. Usually, EAs include four typical algorithms: genetic algorithm (GA), genetic programming (GP), evolutionary programming (EP) and evolutionary strategy (ES). Of the four types, GA is the most popular EA. One seeks the solution for a problem in the form of strings of numbers (traditionally binary, although the best representations are usually those that reflect something about the problem being solved, which are normally not binary), virtually always applying recombination operators such as crossover in addition to selection and mutation. For this reason, GA is often used in optimization problems (Goldberg 1989; Abiyev and Menekay 2007). Compared with tabu search, GA is less problem-dependent and provides a high probability of reaching the global optimum. In comparison with dynamic programming, GA allows users to get sub-optimal solution while dynamic programming cannot, which is very important for some business decision optimization problems. Due to these advantages, we use the popular GA to perform project portfolio selection problem (Yu et al. 2006).
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GA, also known as a meta-heuristic search algorithm for solutions of optimization problems, begins with a random initial solution and attempts to find the best solution under some criteria and conditions. In GA, a gene is represented by a real number or binary bit; a chromosome is a set of genes. A population is a set of chromosomes produced in different generations. Usually GA learning is carried out by using GA operators—selection, crossover and mutation of a population (Goldberg 1989). The main task in the selection operation is to choose the best chromosomes from the population for the next generation and the aim of selection is to choose population members with higher fitness and greater probability of reproduction. Crossover operation interchanges two chromosomes to create two new chromosomes for the population. The main aim of crossover operation is to give the chance to children to differ from their parents and to wish that some of the children could be closer to the optimal destination than their parents. There are some forms of crossover: one-point, two-point, multipoint and uniform. When chromosomes are very long, the use of one or two point crossover operation may lead to undesirable results and the learning of parameter values takes more computational time (Abiyev and Menekay 2007). Mutation operation changes a chromosome into a new chromosome by inverting randomly selected genes of chromosomes with a given mutation rate. Similar to crossover operation, the aim of mutation operation is to give more chances to offspring to increase the opportunity of obtaining optimal solution. More details about these operators can be found in Goldberg (1989). Using the above selection, crossover and mutation operators, we can easily solve the above 0–1 nonlinear integer programming, as shown in (2). Generally the GA for 0–1 nonlinear integer programming consists of the following steps, as illustrated in Fig. 1. As can be seen from Fig. 1, it is easy to find that the GA-based optimization approach includes six main steps. Taking project portfolio selection as an example, some detailed illustrations are presented below. Fig. 1 Flowchart of GA-based optimization approach to nonlinear integer programming problem
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(a) Define the fitness function. The fitness function is an important factor in GA learning, which determines what a GA should optimize. Usually the fitness function defines a score which gives each chromosome the probability to be chosen for breeding or to live. In the optimization problem, fitness is often determined by the value of objective function. In this paper we use the objective function shown in (2) as the fitness function of GA. (b) Reproduce initial population randomly. Usually a population comprises a set of chromosomes often represented in an array of chromosomes, while a chromosome is composed of many genes. In the mathematical optimization problem, a gene corresponds to a variable xi , and a chromosome corresponds to a solution represented in a set of genes x = (x1 , x2 , . . . , xm ) if m variables exist. In the 0–1 nonlinear integer programming problem, a gene can be easily represented by two generic binary codes, 0 or 1. In the project portfolio selection problem (xi = 0) means that the project (variable) is not included in the portfolio and (xi = 1) means the project will be included in the portfolio. The top right part of Fig. 1 shows a population with m chromosomes, where each chromosome includes several genes. Note that in the present form the chromosome is identical to the gene due to the specific 0–1 integer programming problem. (c) Apply crossover operation to initial chromosomes. As earlier mentioned, the main goal of crossover operation is to generate different offspring chromosomes to obtain a more optimal solution than their parents. Offspring chromosomes are created by some crossover techniques such as one-point, two-point and multi-point crossover operations. A so-called one-point crossover technique is employed in the paper, which randomly selects a crossover point within the chromosome. Then parent chromosomes are interchanged at this point to produce new offspring chromosomes, as shown in middle-right part of Fig. 1. (d) Perform mutation operation in chromosomes with a small mutation rate. A mutation operation is a method to create a new chromosome from another chromosome in the population. The main goal of the mutation operation is to prevent the GA from converging too quickly in a small area of the search space. Usually the position performing mutation operation is a random point in chromosomes. The new chromosome is generated based on randomly changing the gene at a bit from “0” to “1” and vice versa. In middle-right part of Fig. 1, “bit inversion” indicates a mutation point chosen to mutate with a certain probability. (e) Use selection operation to create the population with higher fitness. After performing crossover and mutation operations, a new population is created and the next operation will select the best chromosomes with the highest fitness value by means of the roulette wheel approach. Thereby, the chromosomes are allocated to the search space on a roulette wheel proportional to their fitness and thus the fittest chromosome is more likely to be selected. (f) Evaluate the final chromosome. In the final step, the final chromosome will be evaluated to confirm whether it is a best solution. If yes, then the optimized results are obtained. If no, then the reproduction-crossover-mutation-selection step will be repeated until a certain number of generations, a defined fitness goal or a convergence criterion of the population is reached. In the ideal case, all chromosomes of the last generation have the same genes representing the optimal solution (Yu et al. 2006). Using the above steps, the GA-based optimization approach can perform the 0–1 nonlinear integer programming (NIP) for multi-criteria project portfolio selection. A distinct advantage of GA-based optimization technique for NIP problems is that GA can avoid an
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explosion in branching on the tree which is a big limitation of the Branch and Bound Algorithm (BBA). In order to illustrate and verify the GA-based approach for multi-criteria project portfolio selection, two numerical examples are presented in the next section.
4 Numerical examples In this section, two numerical examples are presented to illustrate GA-based approach to 0–1 nonlinear integer programming (NIP) for multi-criteria project portfolio selection. The first example has only 5 variables in the project portfolio selection problem, as it is used to illustrate the detailed process of using GA for project selection. The second example is a 50-variable project selection problem, which is used to verify the effectiveness of the proposed GA-based optimization approach. Note that the GA-based approach for NIP problem is implemented by Matlab2006a software package and all programs are run on IBM ThinkPad T60 notebook computer with CPU of Pentium IV 1.66 GHz and random-access memory (RAM) size of 512 MB. 4.1 A illustrative 5-variable example The main purpose of 5-variable numerical example is to illustrate the detailed implementation process of applying GA to select a subset of projects from a bigger set of projects. For comparison purpose, we make two different calculations, with and without consideration of project interactions among involved projects based on different selection criteria, which is illustrated in (1) and (2) in Sect. 2. Meantime, in two cases the preference information about different projects on a specific criterion is also included in the project portfolio selection problem. Suppose there are five alternative projects A = {a1 , a2 , a3 , a4 , a5 }, the objective is to find the two best projects from the given five projects in terms of three criteria J = {1: return, 2: risk, 3: feasibility of projects}. The original data about these projects are presented in Table 1. In Table 1, project a5 has effects of c51 = 0.78, c52 = 0.42, and c53 = 0.70 on three selection criteria j = 1, 2, and 3 respectively. Preference wj represents the preference degree in terms of criteria j (j = 1, 2, 3) as determined by decision makers. Usually the preference value can be seen as the degree of importance of a criterion. For convenience of computation, they are often normalized into the range of [0, 1]. In the paper the following formula is used to normalize the preference. wj wj = J
(3)
j =1 wj
Table 1 Initial data of five different projects
Criteria
Preference
Projects (i)
(j )
(wj )
a1
a2
a3
a4
a5
1: Return
3
0.33
0.27
0.56
0.44
0.78
2: Risk
1
0.65
0.48
0.89
0.51
0.42
3: Feasibility
4
0.50
0.75
0.48
0.68
0.70
Ann Oper Res Table 2 Normalized data for five different projects
Criteria
Preference
Projects (i)
(j )
(wj )
a1
a2
a3
a4
a5
1: Return
0.375
0.33
0.27
0.56
0.44
0.78
2: Risk
0.125
0.65
0.48
0.89
0.51
0.42
3: Feasibility
0.500
0.50
0.75
0.48
0.68
0.70
Table 3 Interaction values among projects in terms of criteria Criteria
Project pairs (Sk )
(j )
a1 a2
1: Return 2: Risk 3: Feasibility
a1 a3
a1 a4
a1 a5
a2 a3
a2 a4
a2 a5
a3 a4
a3 a5
a4 a5
0.20
0.45
0.40
0.65
0.45
0.35
0.55
0.40
0.60
0.55
−0.15
−0.25
−0.14
−0.15
−0.20
−0.05
−0.10
−0.15
−0.25
−0.15
0
0
0
0
−0.42
0
0
0
−0.80
0.35
Using the above normalization formula, Table 1 can be transformed into Table 2, as shown. In the above discussions, the interactions among these projects are not considered. However, in many practical project portfolio selection problems, there are some interactive effects among the projects, which can be measured by analysis of variance (ANOVA) (Cox 1984). For example, if two projects, saying a1 and a5 , are selected simultaneously, it will lead to return increase and risk decrease according to the mean-variance principle of the traditional portfolio theory (Markowitz 1952). However, the feasibility is uncertain. It may either increase or decrease, depending upon properties of projects. For illustration purpose, interactive effects among the projects are also given in Table 3. From Table 3, it is easy to see the interactions among projects. Taking criterion 3, i.e., Feasibility, as an illustrative example, there are three interactive project pairs S1 = {a1 , a4 }, S2 = {a2 , a4 }, and S3 = {a4 , a5 }, with interactive effects of d3 (S1 ) = 0.35, d3 (S2 ) = −0.42 and d3 (S3 ) = −0.80. With values of interactions among the projects, the current main task is to compare the project portfolio selection process under two different conditions, i.e., with and without the consideration of the interactive effects among the projects. First of all, the project portfolio selection is performed without considering interactive effects among projects. Using the multi-attribute utility theory (MAUT), the objective function of (1) can be computed as V = [(0.375 × 0.33) + (0.125 × 0.65) + (0.500 × 0.50)] x1 + [(0.375 × 0.27) + (0.125 × 0.48) + (0.500 × 0.75)] x2 + [(0.375 × 0.56) + (0.125 × 0.89) + (0.500 × 0.48)] x3 + [(0.375 × 0.44) + (0.125 × 0.51) + (0.500 × 0.68)] x4 + [(0.375 × 0.78) + (0.125 × 0.42) + (0.500 × 0.70)] x5 = 0.455x1 + 0.53625x2 + 0.56125x3 + 0.56875x4 + 0.695x5
(4)
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Accordingly, the optimization problem in terms of (1) for project portfolio selection can be formulated as ⎧ Max V = 0.455x1 + 0.53625x2 + 0.56125x3 + 0.56875x4 + 0.695x5 ⎪ ⎪ ⎨ s.t. x1 + x2 + x3 + x4 + x5 = 2 (5) ⎪ ⎪ ⎩ xi = {0, 1}, i = 1, 2, . . . , 5 Using binary integer programming (BIP) algorithm, optimal solution x = (0, 0, 0, 1, 1) with the largest objective value (V = 1.26375) can be easily obtained. However, the above case is only the simplest situation. If interactive effects among projects are considered, the problem will become complicated. Therefore, the subsequent task is to implement project portfolio selection process with consideration of interactive effects among projects. In terms of the multi-attribute utility theory (MAUT) and interactive effects among projects, the objective function V can be calculated by using (2), represented by V = 0.455x1 + 0.53625x2 + 0.56125x3 + 0.56875x4 + 0.695x5 + 0.375 (0.20)(0.33 + 0.27)x1 x2 + (0.45)(0.33 + 0.56)x1 x3 + (0.4)(0.33 + 0.44)x1 x4 + (0.65)(0.33 + 0.78)x1 x5 + (0.45)(0.27 + 0.56)x2 x3 + (0.35)(0.27 + 0.44)x2 x4 + (0.55)(0.27 + 0.78)x2 x5 + (0.40)(0.56 + 0.44)x3 x4 + (0.60)(0.56 + 0.78)x3 x5 + (0.55)(0.44 + 0.78)x4 x5 − 0.125 (0.15)(0.65 + 0.48)x1 x2 + (0.25)(0.65 + 0.89)x1 x3 + (0.14)(0.65 + 0.51)x1 x4 + (0.15)(0.65 + 0.42)x1 x5 + (0.20)(0.48 + 0.89)x2 x3 + (0.05)(0.48 + 0.51)x2 x4 + (0.10)(0.48 + 0.42)x2 x5 + (0.15)(0.89 + 0.51)x3 x4 + (0.25)(0.89 + 0.42)x3 x5 + (0.15)(0.51 + 0.42)x4 x5 + 0.5 (0.35)(0.50 + 0.69)x1 x4 + (−0.42)(0.75 + 0.68)x2 x4 + (−0.80)(0.68 + 0.70)x4 x5 = 0.455x1 + 0.53625x2 + 0.56125x3 + 0.56875x4 + 0.695x5 + 0.0238x1 x2 + 0.1021x1 x3 + 0.3035x1 x4 + 0.2505x1 x5 + 0.1058x2 x3 − 0.5136x2 x4 + 0.2053x2 x5 + 0.1238x3 x4 + 0.2606x3 x5 − 0.2972x4 x5
(6)
In terms of (2), the optimization problem considering project interactions can be formulated as ⎧ Max V = 0.455x1 + 0.53625x2 + 0.56125x3 + 0.56875x4 + 0.695x5 + 0.0238x1 x2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 0.1021x1 x3 + 0.3035x1 x4 + 0.2505x1 x5 + 0.1058x2 x3 − 0.5136x2 x4 ⎪ ⎪ ⎨ + 0.2053x2 x5 + 0.1238x3 x4 + 0.2606x3 x5 − 0.2972x4 x5 (7) ⎪ ⎪ ⎪ ⎪ s.t. x1 + x2 + x3 + x4 + x5 = 2 ⎪ ⎪ ⎪ ⎪ ⎩ xi = {0, 1}, i = 1, 2, . . . , 5
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Fig. 2 Optimal solution of the 5-variable example using the GA-based optimization approach
As can be seen from (5) and (7), the standard 0–1 integer programming (IP) problem in (5) has become a typical 0–1 nonlinear integer programming (NIP) problem shown in (7). Using the GA-based optimization algorithm presented in Sect. 3, it is easy to obtain the optimal solution x = (0, 0, 1, 0, 1) with the largest objective value (V = 1.51685), as illustrated in Fig. 2. Comparing (5) with (7), it should be noticed that there is a distinct difference between project portfolio selection with and without consideration of project interactions. The main difference reflects two-fold. On the one hand, the optimal solution is different. When project interactions are not considered, the optimal solution x = (0, 0, 0, 1, 1). Conversely, the optimal solution x = (0, 0, 1, 0, 1) when project interactions are considered, On the other hand, the values of the two objective function have an obvious difference and the difference between them is 0.2531 (i.e., 1.51685 − 1.26375 = 0.2531). These two differences indicate that project interactions are very important for project portfolio selection because they can affect the final results. In the 5-variable numerical example presented in this subsection, interactive effects between only two projects are tested. Actually interactive effects between two projects can be extended to the three or more projects in accordance with representation of (2), which is shown in Sect. 2. Thus interactions among multiple projects should be considered. Furthermore, there are a large number of projects and selection criteria in practical problems. Therefore more variables need to be addressed in practical applications, but the issue of computational efficiency needs to be considered in the case of large-scale problems. For this purpose, the next subsection presents a 50-variable example with consideration of interactions among multiple projects to verify computational efficiency of the proposed GA-based optimization approach.
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4.2 Two 50-variable examples In this subsection, two 50-variable examples with consideration of interactions among multiple projects is presented. The main purpose of selecting 50 variables for project portfolio selection is to make the example closer to practical applications and also to verify computational efficiency of the proposed GA-based optimization approach relative to the traditional nonlinear optimizer. Suppose there are 50 projects and 20 evaluation criteria, and decision makers need to select 10 best projects to formulate a project portfolio. Without loss of generality, interactions between only two or three projects are considered. Similar to Sect. 4.1, it is assumed to have the following optimization problems, as shown in (8). ⎧ Max V = 0.6554x1 + 0.3625x2 + 0.6124x5 + 0.6878x9 + 0.9511x10 + 0.2342x12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 0.4556x15 + 0.4554x18 + 0.5626x19 + 0.1258x25 + 0.8751x31 ⎪ ⎪ ⎪ ⎪ ⎪ + 0.5432x32 + 0.4344x35 + 0.8765x40 + 0.6567x41 + 0.7622x43 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 0.1878x45 + 0.8787x46 + 0.5436x48 + 0.4566x49 + 0.2344x50 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 0.3757x1 x2 + 0.4533x1 x5 + 0.4344x1 x9 + 0.6756x1 x12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − 0.8132x1 x19 + 0.9342x1 x32 + 0.6578x1 x7 x35 + 0.4527x2 x25 ⎪ ⎪ ⎪ ⎪ ⎪ + 0.3544x2 x5 x43 −0.6636x2 x48 + 0.3465x2 x49 + 0.5528x2 x50 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 0.4056x3 x6 + 0.6068x3 x31 + 0.3242x3 x40 + 0.5643x3 x41 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ + 0.4324x3 x49 + 0.4465x4 x25 − 0.1258x5 x32 + 0.2589x5 x43 (8) ⎪ + 0.5455x9 x31 + 0.3434x10 x35 + 0.4838x15 x50 + 0.1451x18 x27 x41 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 0.1542x19 x50 − 0.2089x25 x30 x35 − 0.7883x31 x48 + 0.6542x32 x49 ⎪ ⎪ ⎪ ⎪ ⎪ + 0.0551x32 x39 x45 + 0.1041x35 x50 + 0.1589x40 x42 x46 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 0.5429x43 x48 − 0.0211x43 x49 + 0.2524x45 x46 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 0.1557x46 x48 x50 + 0.5359x48 x49 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − 0.8743x48 x49 x50 − 0.3214x49 x50 ⎪ ⎪ ⎪ ⎪ ⎪ 50 ⎪ ⎪ ⎪ ⎪ s.t. xi = 10 ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎩ xi = {0, 1}, i = 1, 2, . . . , 50 Using the GA-based optimization approach presented in Sect. 3, it is easy to obtain the optimal solution x = (1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0) to the 50-variable project portfolio selection problem, as illustrated in Fig. 3. To compare the computational efficiency between the proposed GA-based optimization approach and the traditional nonlinear optimizer, another 50-variable project portfolio selection problem is given in (9). In this comparison, a typical nonlinear optimizer, ILOG CPLEX barrier optimizer (ILOG)1 is used. As is known to all, GA is a meta-heuristic algorithm and CPLEX barrier optimizer is an exact method for mixed integer quadratic programming (MIQP). For this reason of MIOP, the thrice terms in (8) should be removed for 1 ILOG CPLEX, ILOG software online: http://www.ilog.com/products/cplex/.
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Fig. 3 Optimal solution of the 50-variable example using the GA-based optimization approach
comparison purpose. Accordingly, another 50-variable project portfolio selection problem can be formulated, as shown in (9). More details about nonlinear optimizer, ILOG CPLEX barrier optimizer, can be found in ILOG. ⎧ ⎪ ⎪ Max V = 0.6554x1 + 0.3625x2 + 0.6124x5 + 0.6878x9 + 0.9511x10 + 0.2342x12 ⎪ ⎪ ⎪ ⎪ + 0.4556x15 + 0.4554x18 + 0.5626x19 + 0.1258x25 + 0.8751x31 ⎪ ⎪ ⎪ ⎪ ⎪ + 0.5432x32 + 0.4344x35 + 0.8765x40 + 0.6567x41 + 0.7622x43 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 0.1878x45 + 0.8787x46 + 0.5436x48 + 0.4566x49 + 0.2344x50 ⎪ ⎪ ⎪ ⎪ ⎪ + 0.3757x1 x2 + 0.4533x1 x5 + 0.4344x1 x9 + 0.6756x1 x12 ⎪ ⎪ ⎪ ⎪ ⎪ − 0.8132x1 x19 + 0.9342x1 x32 + 0.6578x1 x35 + 0.4527x2 x25 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 0.3544x2 x43 − 0.6636x2 x48 + 0.3465x2 x49 + 0.5528x2 x50 ⎪ ⎪ ⎪ ⎪ ⎪ + 0.4056x3 x6 + 0.6068x3 x31 + 0.3242x3 x40 + 0.5643x3 x41 ⎪ ⎪ ⎨ + 0.4324x3 x49 + 0.4465x4 x25 − 0.1258x5 x32 + 0.2589x5 x43 (9) ⎪ ⎪ ⎪ + 0.5455x9 x31 + 0.3434x10 x35 + 0.4838x15 x50 + 0.1451x18 x41 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 0.1542x19 x50 − 0.2089x25 x35 − 0.7883x31 x48 + 0.6542x32 x49 ⎪ ⎪ ⎪ ⎪ ⎪ + 0.0551x32 x45 + 0.1041x35 x50 + 0.1589x40 x46 + 0.5429x43 x48 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − 0.0211x43 x49 + 0.2524x45 x46 + 0.1557x46 x50 + 0.5359x48 x49 ⎪ ⎪ ⎪ ⎪ ⎪ − 0.8743x48 x50 − 0.3214x49 x50 ⎪ ⎪ ⎪ ⎪ ⎪ 50 ⎪ ⎪ ⎪ ⎪ s.t. xi = 10 ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎩ xi = {0, 1}, i = 1, 2, . . . , 50
Ann Oper Res Table 4 Comparison of computational time for GA-based approach and CPLEX barrier optimizer
Number of
Number of
Computational time
alternative projects
quadratic terms
(seconds) GA
CPLEX
20
10
3.24
2.18
40
20
5.05
4.22
50
40
10.57
11.93
65
90
22.39
27.16
80
200
39.76
56.34
100
500
67.73
80.81
150
1000
89.34
133.25
200
2000
102.05
198.67
In order to find some evidence on computational performance of GA-based optimization approach in large-scale instances, a set of testing problems for selecting 10 best projects from some alternative projects with 20 selection criteria were performed. The number of alternative projects varied from 20 to 200 and the number of quadratic terms in the objective function varied from 10 to 2000. The project value cij is a randomly generated number between 0 and 1 in terms of each criterion. The interactive effect values for project pairs in terms of each selection criteria are also some randomly generated numbers between −0.5 to 0.5. Accordingly computational results are reported in Table 4. As can be seen from Table 4, several interesting conclusions can be found. First of all, with the increase of the number of projects and quadratic terms, the computational time are increased proportionally for both GA-based optimization approach and CPLEX barrier optimizer (ILOG). Second, when there is a small number of alternative projects and quadratic terms, the traditional nonlinear optimization technique, e.g., CPLEX barrier optimizer (ILOG), is better than the GA-based optimization approach. The main reason is that GA-based optimization is a stochastic optimization approach, which may affect computational efficiency when a small-scale problem is encountered. Third, with increase of the number of alternative projects and quadratic terms, the extent of computational time improvement in GA-based optimization approach becomes larger and larger relative to the traditional nonlinear optimizer. This implies the GA-based optimization approach is a feasible solution to 0–1 nonlinear integer programming problem. Finally, computational results obtained show that the proposed GA-based optimization approach is computationally efficient relative to the traditional nonlinear optimization approach. This indicates GA-based optimization approach can be used as an effective and feasible solution to other selection problems, such as stock selection and location selection problems.
5 Conclusions and future directions Due to multiple conflicting selection criteria and the difficulty in obtaining information on DMs’ preferences and interactive effects among different projects, project portfolio selection is a rather challenging task. In this study, a 0–1 nonlinear integer programming model is formulated for modeling the project portfolio selection problem with consideration of
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multiple selection criteria, preference information of decision makers and project interactions. Meantime a GA-based optimization technique is proposed to solve the 0–1 nonlinear integer programming for project portfolio selection. Using the GA-based optimization technique, the nonlinear integer programming for multi-criteria project portfolio selection problem is easily solved. Three illustrative examples demonstrate the effectiveness of the proposed GA-based optimization approach. A potential limitation of the GA-based optimization technique concerns some disadvantages of GA. Although GA is a distinct meta-heuristic algorithm to solve an optimization problem, sometimes the GA method can obtain the best value only after some generations and this value is only an approximation of the global solution. Furthermore, there is still no consensus on the number of generations required to be performed to obtain the global solution. In addition, GA-based approaches with reproduction- crossover-mutation-selection process usually produce many invalid chromosomes which cannot satisfy the constraints. Although there are some strategies to remove the invalid solution, it will affect the computational efficiency of the proposed GA-based method. Another limitation is that the application is restricted to some decision problems with all necessary data available. However, in practical cases, many projects did not provide necessary data for evaluation and selection decision. Future research can be done in at least two directions. The first direction is related to explore some other methods to obtain preference information of decision makers and interactive effects among projects in addition to current Delphi method (Stummer et al. 2003). Another research direction is to extend the current optimization framework into a group decision-making optimization framework. We will look into the issues in the future research. Acknowledgements Authors would like to express their sincere appreciation to the guest editors and the four independent referees in making valuable comments and suggestions to the paper. Their comments have improved the quality of the paper immensely. This work is supported by grants from the National Science Fund for Distinguished Young Scholars (NSFC No. 71025005), National Natural Science Foundation of China (NSFC No. 90924024, 70971013), the Knowledge Innovation Program of Chinese Academy of Sciences (CAS), Hunan Province Natural Science Fund for Distinguished Young Scientists (No. 09JJ010), and The Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities. The authors gratefully acknowledge the support of K.C. Wong Education Foundation, Hong Kong.
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