Advances in Operations Research

Volume 2014 (2014), Article ID 954219, 8 pages

http://dx.doi.org/10.1155/2014/954219

## A Hybrid Grey Relational Analysis and Nondominated Sorting Genetic Algorithm-II for Project Portfolio Selection

Department of Industrial Management, Islamic Azad University, Semnan Branch, Semnan, Iran

Received 27 June 2014; Revised 24 November 2014; Accepted 30 November 2014; Published 24 December 2014

Academic Editor: Konstantina Skouri

Copyright © 2014 Farshad Faezy Razi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Project selection and formation of an optimal portfolio of selected projects are among the main challenges of project management. For this purpose, several factors and indicators are simultaneously examined considering the terms and conditions of the decision problem. Obviously, both qualitative and quantitative factors may influence the formation of a portfolio of projects. In this study, the projects were first ranked using grey relational analysis to form an optimal portfolio of projects and to create an expert system for the final project selection. Because of the fuzzy nature of the environmental risk of each project, the environmental risk was predicted and analyzed using the fuzzy inference system and failure mode and effect analysis based on fuzzy rules. Then, the rank and risk of each project were optimized using a two-objective zero-one mathematical programming model considering the practical constraints of the decision problem through the nondominated sorting genetic algorithm-II (NSGA-II). A case study was used to discuss the practical methodology for selecting a portfolio of projects.

#### 1. Introduction

Project selection is among important issues in industrial management, industrial engineering, and governmental, nonprofit, and commercial organizations [1]. The selection of the best portfolio or project to achieve full satisfaction in an organization has been considered in previous studies [2]. The project selection process can be defined as follows: it is started by continuous collecting, analyzing, and judging the available information on the project leading to project selection considering the factors influencing the selection process [3]. The project portfolio selection is a multicriteria decision problem which considers multicriteria quantitative and qualitative factors simultaneously [4]. In the multicriteria decision-making model, the solution may already exist and therefore the purpose is to select the best solution from the available solution set. This class of decision problems is called multicriteria decision models. On the other hand, the solution may be unknown. In this case, the purpose is to find the optimal Pareto solution of the problem in the continuous or discrete space [5]. Such decision models are called multiple objective decision-making models. The multicriteria decision models are formed based on utility theory and human pressures in dealing with the behavior of max finder [6]. In 1945, John Newman published his famous book* Theory of Games and Economic Behavior* and proposed a mathematical theory for game theory-based economic and social organizations. This provided the ground for developing multiple attribute decision-making (MADM) models in the decision theory [7]. In general, MADM models are designed based on one of the philosophical approaches of choice, rank, description, sort, design, and portfolio [8]. In this study, the choice, rank, and design approaches were combined to form a portfolio of projects. According to this approach, the projects were ranked through the grey relational analysis in MADM literature. Then, the environmental risk of the project was analyzed and predicted by a fuzzy inference system. Thereafter, a two-objective zero-one mathematical programming model was designed to optimize the risk and rank of each project considering the constraints governing the optimal decision problem and the design philosophy in the multiple objective decision-making literature. The Pareto solution of the model was obtained using the nondominated sorting genetic algorithm-II (NSGA-II). The paper proceeds as follows: the literature is reviewed in Section 2. Section 3 examines the grey relational analysis method. Fuzzy inference system is described in Section 4. NSGA-II is introduced in Section 5. A new approach for selecting the portfolio of projects is presented in Section 6. A case study and conclusions are presented in Sections 7 and 8, respectively.

#### 2. Literature Review

Zarei et al. (2009) developed an expert system for portfolio selection. The proposed system analyzed the technical risk and return on investment. In this model, the preferences are weighted and then the optimal portfolio is clustered through the rough set theory [9]. Lin and Liu proposed a portfolio optimization model based on Markowitz linear programming model for project portfolio selection using the minimum swap size. The optimal Pareto solution of the model was obtained by a genetic algorithm [10]. Doerner et al. presented a multiobjective integer programming model for optimal portfolio selection using the ant colony optimization algorithm [11]. Martínez-Lorente et al. considered both qualitative and quantitative objectives to optimize the portfolio of projects. In this study, the path analysis was used to analyze the qualitative objectives [12]. Bilbao-Terol et al. considered the social responsibility to select the optimal portfolio. In this approach, enterprises do not invest in activities neglecting ethical standards. Obviously, the portfolio is completed through the assets observing the ethical standards. For this purpose, a measure called social responsibility attractiveness was used [13]. Eshlaghy and Razi proposed a -mean algorithm-based grey relational analysis model for project portfolio selection. In this model, the projects are first clustered through the -mean algorithm; then, each cluster is ranked using the grey relational analysis. Finally, the Pareto solutions of rank and risk are analyzed by the genetic algorithm [14]. In another study, Razi et al. clustered projects using the -mean fuzzy algorithm and then analyzed the clusters using the grey relational analysis. In this study, the project risk analysis was carried out through a fuzzy inference system [15]. Huang et al. designed a model based on the semivariance index to invest in a portfolio of real estate assets considering the risk preference to optimize the portfolio of real estate assets. In the second stage, the Pareto optimal solution of the model was analyzed using the bee colony algorithm. The modeling approach in this study is based on the salesman network model [16].

#### 3. Grey Relational Analysis

In 1982, Deng published the first paper on the grey system theory entitled “The Control of the Grey Systems” and then the grey system theory was introduced [17]. Briefly, the basic idea of grey theory is as follows: the overall picture of the system is imagined considering the partial or limited information about a system. This methodology deals with uncertain, incomplete, and poor problems. As one of the main features of the grey system theory, this theory can provide satisfactory outputs using relatively low information and the high variability in the criteria. Like the fuzzy theory, the grey theory is an effective mathematical model for solving uncertain and ambiguous problems [18]. There are many different systems in the real world; each of them has its own components and subsystems. To recognize a system, the relations between the components as well as the structure of the system should be identified in addition to understanding the components. If the completely known and unknown information of a system is, respectively, shown by white and black colors, the information on most systems in nature is not white (well known) or black (unknown), but it is a mixture of both colors, that is, grey information. Such systems are called grey systems. The main characteristic of grey systems is incomplete information. The aim of the grey systems theory and its applications is to create a bridge between the social sciences and natural sciences. Grey color means the deficiency of information and uncertainty [19]. The grey relational analysis includes the following steps.

*Formation of a Grey Relation.* When the performance measurement units for different indicators are different, it is likely that the effects of some parameters are ignored. Furthermore, when some performance indicators have a wide range, this may happen. In addition, the performance indicators with different objectives or directions may lead to inaccurate results. Thus, it is necessary to convert all performance values of an alternative to comparative series through a process similar to normalization process. In grey systems theory, this process is called the formation of grey relations. In a multicriteria decision-making problem with alternatives and indexes, the th alternative is shown by in which is the performance value of the index for the alternative . can be converted into the comparative series using one of the following equations:
Equation (1) is used for “the bigger, the better” indexes while (2) is used for “the smaller, the better” indexes. Equation (3) is used for the case where “values closer to the optimal value of are better” [20].

*The Reference Target Series.* Once the grey relations were formed using (1), (2), or (3), all performance values are located in the range . In the case where the value of generated by the grey relation creation process is equal to 1 or closer to 1 than the value of any alternative, the performance of the index in the alternative is better than other alternatives. Thus, the alternative for which all performance values are equal to 1 is the best alternative. In this study, the reference series is defined as . Accordingly, it searches for an alternative whose comparative series is closer to this target series [21].

*Grey Relational Coefficient.* The grey relational coefficient is used to determine the proximity of to . Higher grey relational coefficient, closer to . The grey relational coefficient is calculated using (4), where represents the gray relational coefficient between and . The coefficient of determination is used to expand or limit the domain of the grey relational coefficient [22]

*Grey Relational Rank.* Once all grey relational coefficients, , were calculated, the grey relational rank can be calculated using
Equation (5) represents the grey relational rank, , between and . In fact, (5) shows the correlation between the reference target series and the comparative series in which is the weight of index . is usually dependent on the judgment of the decision-maker or the structure of problem. In addition, . As mentioned earlier, the reference series shows the best achievable performance of each index in the comparative series. Therefore, the comparative series with the highest grey relational rank with the reference series has the highest similarity with the reference target series. Thus, this is the best choice [23].

#### 4. Fuzzy Inference System

Fuzzy inference system provides a systematic process to convert a knowledge base to a nonlinear mapping. This is why the knowledge-based systems (fuzzy systems) are used in engineering and decision-making applications [24]. Mamdani and Assilian used fuzzy inference systems to control a steam engine and boiler combination using a combination of linguistic control rules and the experience of human operators [25]. A fuzzy system has the following components:(i)a fuzzifier to convert the numerical values of the variables into a fuzzy set,(ii)a fuzzy rules base as a set of “if then” rules,(iii)a fuzzy inference engine to convert inputs to outputs through a series of actions,(iv)a defuzzifier to convert the fuzzy output into a crisp number [26].

In this study, the fuzzy inference system described in Figure 1 is used to analyze the environmental risk for each project. As shown in Figure 1, the factors constituting the environmental risk of each project are analyzed by failure mode and effect analysis based on three factors, S, O, and D.