A fuzzy computational approach for selecting interdependent projects using prioritized criteria

Abstract

This paper presents a fuzzy computational approach for selecting project portfolio by combining fuzzy logic, Quality Function Deployment (QFD) and Genetic algorithm (GA) approaches with the consideration of prioritized selection criteria as per objectives of the organization to make decisions effectively with incomplete and ambiguous information to help in portfolio selection. This approach addresses the issues of the uncertainty of experts in selecting projects, prioritizing criteria before initiating project selection process and evaluating the number of interdependent projects for their maximal values. It completes the task in three stages. Firstly, it involves interaction with experts to extract fuzzy input about the benefits of organization and selection criteria for selecting a project portfolio. The second stage requires the application of fuzzy QFD to prioritize criteria before deciding the project portfolio. In this stage, the paper contributes a method for using fuzzy values in a distinct way for obtaining priorities of selection criteria. The final stage evaluates the candidate projects concurrently based on top priority selection criteria by considering interrelation among projects by proposing a distinct fitness function of GA. The validity of the proposed approach is demonstrated by an example that considers three experts, three objectives of the organization and four selection criteria.

Keywords

Fuzzy quality function development (FQFD)genetic algorithm project portfolio management project portfolio selection quality function development (QFD)

1 Introduction

Selecting a suitable strategic project that is worth of cost, time, and return is a challenging task in project portfolio selection and management. Several researchers invested many efforts by proposing different methods for selecting a strategic project for an organization [1 –6]. Most researchers highlighted cooperative efforts including internal capacities and external chances in addition to strategic resources for choosing a set of projects leading to maximum return over investment (ROI) for an organization and leverage of strategic resources [4 , 8].

Danila [9] defines portfolio management as “selecting an investment from a list of candidate investments to maximize some objectives without violating constraints”. Most of portfolio management models employ five steps, namely, Project Analysis, Project categorization, Project Evaluation, Project Selection and filtering, Monitoring. These steps are considered as necessary steps for developing a project portfolio model. Many approaches have been proposed for each step in the portfolio management process and summarized in study [1].

The portfolio selection requires comparing the projects on a set of selection criteria for obtaining an optimized set of projects. The top projects are evaluated on a given set of criteria for its selection. The projects are prioritized as per given importance of selection criteria. Carvalho et al. [10] divided various prioritization methods into different categories like financial methods, statistics models, mathematical models, fuzzy logic, decision tree models, scoring models and multiple criteria decision models [1 , 11] and highlighted strengths and weaknesses of these methods for valid selection of strategic projects.

In this paper, we review significant research on selecting a set of strategic projects in the project portfolio management process. A fuzzy computational approach is proposed based upon Fuzzy Logic, Quality Function Development (QFD)and Genetic Algorithm (GA) for addressing the limitation of existing methods in selecting projects for an organization. This paper presents a fuzzy computational approach for selecting project portfolio by combining fuzzy logic, Quality Function Deployment (QFD) and Genetic algorithm (GA) approaches with the consideration of prioritized selection criteria as per objectives of the organization to make decisions effectively with incomplete and ambiguous information to help in portfolio selection. The proposed approach is analyzed using a numerical example.

Rest of the paper is structured as follows: Section 2 provides the analysis of related work to identify the research gaps in the field of project portfolio selection. Fundamental concepts of tools and techniques used in the proposal of a hybrid approach in this work are provided in Section 3. For bridging research gaps, a hybrid approach is proposed using fuzzy logic integrated with QFD and Genetic Algorithm and presented its working Section 4. Section 5 provides a numerical example for illustrating the operation of the proposed approach. Section 6 presents the findings of the proposed research work followed by the scope for future work.

2 Related work

Numerous researchers highlighted the need for project portfolio that must fulfil the objective of the organization [12 –14]. The most common requirements include Alignment with organization objectives, Maximizing the business value, Balancing of resources, Suitable to organization’s cultural environment, Directly or indirectly leads to cash flow, Efficiently use the resources, Projects should contribute to short term as well as long-term development of the organization.

Therefore, different organization implement different methods for selecting an appropriate set of projects that meet the objectives of the organization by utilizing limited resources. Many studies demonstrated that the right combination of projects is very crucial for achieving a successful portfolio selection [12 –16].

Researchers in the field proposed different frameworks for selecting the right combination of projects based on a given set of criteria. These frameworks either employ a single method or a hybrid method for selecting strategic projects. The most commonly used single methods include weighted average, fuzzy-AHP [17, 18], fuzzy DEA [19], multi-objective algorithms [20], or goal programming [21]. Hybrid methods involve the employment of more than one methods for selecting the project. For example, use of DEA and AHP [22], fuzzy AHP and TOPSIS [23], Fuzzy DEA and TOPSIS [24], Fuzzy-QFD and FDA [8] or QFD and AHP [23].

The recent studies are mainly proposed by emphasizing in selecting the best combination of strategic projects. But, most studies ignored to recognize the best criteria and filter them for selecting projects. These studies used the given set of criteria in the selection phase without a rigorous analysis of the criteria and prioritizing them as per their importance to the organization.

For example, Tavana et al. [24] introduced a hybrid model for selecting project using a combined hybrid approach of DEA, TOPSIS and integer programming. Their proposed hybrid approach is used to rank IT projects on a given set of selection criteria (Provided by the concerned organization as per their importance). The authors did not provide any directions for determining selection criteria. Similarly, Ghapanchi et al. [19] proposed a hybrid approach of Fuzzy DEA to select a project portfolio. The authors highlighted the need for performing an analysis of selection criteria to recognize the essential criteria properly. A comprehensive analysis of studies for project selection indicates that most of the studies have ignored to identify and filter selection criteria in selecting the appropriate combination of strategic projects in portfolio [4 , 25]. Most studies used the criteria as supplied by the organization in their selection phase [16 , 24]. Some of the studies mentioned [17 , 26] prioritized selection criteria using AHP method but ignored to consider uncertainty or interdependency of projects.

Table 1
Summary of related studies

Study /factors Focus A ^* B ^ C ^*

Bardhan et al. [27] Prioritizing a portfolio No No Yes

Lin and Chen [26] A fuzzy strategic alliance selection framework for supply chain partnering Yes Yes No

Conka et al. [22] Project portfolio selection Yes No Yes

Huang et al. [17] A fuzzy AHP application in government-sponsored R&D project selection Yes Yes No

Chen and Cheng [28] Project selection No Yes No

Wang et al. [32] Project portfolio selection Yes Yes No

Ghapanchi et al. [19] Project portfolio selection No Yes Yes

Karsak et al. [7] Framework to prioritize design requirements Yes Yes No

Tavana et al. [24] Project portfolio selection No Yes Yes

Jimenez et al. [21] Project portfolio selection No Yes No

Liu et al. [33] Making group decisions No Yes No

Saleh et al. [31] Replacement of Medical Equipment Yes No Yes

Lim et al. [30] Stochastic quality-cost optimization No Yes No

He et al. [29] Refinement of Kano model as importance-frequency Kano (IF-Kano) model for project design Yes Yes No

Our approach Project portfolio selection Yes Yes Yes

Study /factors	Focus	A ^*	B ^**	C ^***
Bardhan et al. [27]	Prioritizing a portfolio	No	No	Yes
Lin and Chen [26]	A fuzzy strategic alliance selection framework for supply chain partnering	Yes	Yes	No
Conka et al. [22]	Project portfolio selection	Yes	No	Yes
Huang et al. [17]	A fuzzy AHP application in government-sponsored R&D project selection	Yes	Yes	No
Chen and Cheng [28]	Project selection	No	Yes	No
Wang et al. [32]	Project portfolio selection	Yes	Yes	No
Ghapanchi et al. [19]	Project portfolio selection	No	Yes	Yes
Karsak et al. [7]	Framework to prioritize design requirements	Yes	Yes	No
Tavana et al. [24]	Project portfolio selection	No	Yes	Yes
Jimenez et al. [21]	Project portfolio selection	No	Yes	No
Liu et al. [33]	Making group decisions	No	Yes	No
Saleh et al. [31]	Replacement of Medical Equipment	Yes	No	Yes
Lim et al. [30]	Stochastic quality-cost optimization	No	Yes	No
He et al. [29]	Refinement of Kano model as importance-frequency Kano (IF-Kano) model for project design	Yes	Yes	No
Our approach	Project portfolio selection	Yes	Yes	Yes

*^AFiltering & Prioritizing of selection criteria method; **^BUncertainty in decision making; ***^CInterdependency of projects.

Some of the studies considered interdependence of projects while selecting a project in portfolio in deterministic environment [27]. Whereas, some researchers focused stochastic environments but ignored interdependencies of the projects [17 , 28]. Such studies are not widely used in selecting projects.

Another issue that has also been highlighted in the literature is uncertainty in the strategic projects. The need for simultaneous consideration of uncertainty and interdependencies of the projects is ignored in most studies. Some researchers only focused on uncertainty but ignored interdependency aspects while selecting projects [17 , 28]. Whereas, some researchers focused on project inter-dependency projects assuming deterministic environment. However, in real-life, there exist uncertainty in projects. Therefore, simultaneous consideration of uncertainty and interdependencies of projects is highly needed for improving the applicability of such project selection framework in practice [8 , 19].

Several researchers have also identified the need of prioritizing selecting criteria and optimizing list of resources to fulfil specific objectives in various domains, like Replacement of Medical Equipment, Stochastic quality-cost optimization etc. For example, He et al. [29] proposed a refinement of Kano model as importance-frequency Kano (IF-Kano) model. In this model, they focused on identifying critical CRs and incorporating CRs into product design as principle factors for product development. Whereas, Lim et al. [30] focused on stochastic quality-cost optimization using QFD and MOGA for conflicting objectives of participants. However, the authors have not considered on interdependency concepts. They have also not focused vague importance participants’ intensions as criteria for stochastic quality-cost optimization. Similarly, Saleh et al. [31] applied Quality Function Deployment and Genetic Algorithm for Replacement of Medical Equipment. Their proposed model, QFD - GA, was developed to prioritize the medical equipment for replacement process taking into account a set of criteria; besides, the prioritized list is optimized according to the available budget of the hospital to maximize the number of replaced devices.

Table 1 summarizes the comparison of related work in the field of portfolio selection and other fields.

To address this issue along with the need of identifying and filtering selection criteria before selection phase, this paper proposes an approach that utilizes the hybrid approach of fuzzy logic, QFD and genetic algorithm for identifying, filtering selection criteria and selecting a right combination of strategic projects to meet the overall objective of the organizations for maximizing their business value.

3 Preliminaries of the tools and techniques

This section presents the basics of the techniques used in this paper. It introduces QFD, fuzzy logic, genetic algorithms and their usage in selecting the right combination of strategic projects based on identified and prioritized selection criteria.

3.1 Quality function deployment (QFD) approach

QFD approach is considered as one of the most commonly used strategies for managing quality, initially proposed for manufacturing systems [34]. This approach derives linear and structured guidelines for transforming the client’s requirements into design specifications while designing new products and services.

Chan and Wu [35] defined QFD as “a system to assure that customer needs to drive product design and production process”. QFD approach consists of the development of four matrices in the house of quality (HoQ) [36] as presented in Fig. 3.1.

These matrices provide the design specifications of the product in terms of their relative significance and of target values that have to be achieved during the development of the product design. These matrices of HoQ enable to derive design specification from client needs [37]. The important steps for forming HoQ are described below [34].

Step 1: Identify the client needs (CNs) as WHATs, represented as label A in Fig. 3.1. This matrix lists the required benefits of the client. Matrix B represents the relative importance or priorities of CNs.

Step 2: Finalize the design specification or engineering characteristics as HOWs, represented as label C in Fig. 3.1.

Step 3: Prepare relationship matrix (D). This matrix provides the degree of impact of each WHAT on each HOW.

Step 4: Formulate correlation matrix (F). This matrix provides a correlation among various HOWs.

Step 5: Calculate relative weights. This step computes the relative weights of HOWs, labelled as E in Fig. 3.1. These weights represent the priority of each design specification during product designing process and are computed as per Equation 1. $\begin{matrix} W (HOW)_{i} = R (HOW)_{i 1} \times I (WHAT)_{i} + \dot{.} . . . + R (HOW)_{in} \times I (WHAT)_{i} \end{matrix}$ (1)

Where, W(HOW) gives the weights of i^th HOW, R(HOW) gives relationship of i^th HOW with n^th WHAT and I(WHAT) represents importance of i^th WHAT.

3.2 Fuzzy logic

Mostly, experts find uncertainties in taking their decisions in real-life problems [34]. One possible solution to handle uncertainty in real-life problems is the use of probability theory. The probability theory applies random law to handle uncertainty and inaccuracy in real-life problems. This theory provides good results for the stochastic nature of decisional analysis during the decision making process. However, it is found that it cannot measure the uncertainty stemming from human behaviour as human behaviour is neither stochastic nor random.

To tackle the difficulty of such behaviour of uncertainty, fuzzy logic has been proposed [38]. Most of the conventional tools generally consider the result as a bivalent logic (on/off, true/false, yes/no). But, in real-life problems and human thought processes are not solved using bivalent logic [39]. To address the issue of bivalent logic, Zadeh [38] proposed fuzzy logic based on fuzzy sets, that defines a set of objects having no clear-cut or predefined boundary between the objects that are or are not members of the set. The fundamental concept of fuzzy logic is based on the membership function. Membership function defines that each element in a set is associated with a value indicating to what degree the element is a member of the set. It results in outcome within the range [0,1]. Where, 0 and 1, indicate the minimum and maximum degree of membership respectively. The intermediate values show degrees of partial membership.

Several fuzzy membership functions have been proposed to represent different fuzzy numbers. The triangular fuzzy membership function is the most commonly used [7, 35] as these fuzzy numbers are easy to manage from the computational view. A fuzzy number is represented in the form of triplet A (L_min, M_Mid, R_Max), where, L_Min and R_Max represent lower and upper bound of fuzzy numbers and M_Mid gives the nearest fit element. These numbers are utilized to quantify linguistic data. For example, consider a linguistic variable set S (VL; L; M; H; VH) for expressing opinions on a group of attributes (VL - very low, L - low, M - medium, H - high, VH - very high). This linguistic variable set S can be quantified by using fuzzy numbers as VL -(0, 1, 2); L-(2, 3, 4); M-(4, 5, 6); H-(6, 7,8); VH-(8, 9, 10) and presented in Fig. 2.

Fig. 1

House of Quality (HoQ) [36].

Fig. 2

Linguistic scale of importance.

The linguistic variable L indicates that the expert’s assessment contains elements of grade L_Min - 2 up to a grade R_Max - 4, with a maximum degree of membership in M_Mid - 3.

3.3 Hybrid fuzzy-QFD approach

Recently, several hybrid approaches have been proposed in different fields by integrating fuzzy logic and QFD in converting client’s needs to design proposals and prioritizing their needs [8 , 40]. The integration of fuzzy logic with QFD approach help to address the issue of uncertainty in QFD operations. Taking advantages of integrating these two approaches, Fung et al. [40] proposed a hybrid system that incorporates the principles of Fuzzy logic, QFD, and AHP for determining design targets. Wang [41] suggested a fuzzy outranking approach that prioritizes HOWs. Similarly, Shen et al. [42] suggested a procedure based on fuzzy logic for examining the sensitivity of the ranking of HOWs to the de-fuzzification technique and degree of fuzziness of fuzzy numbers. The main objective of the above-cited studies is to prioritize HOWs.

Therefore, considering the benefits of integrating fuzzy logic and QFD for handling uncertainty in QFD operations, we propose a hybrid methodology for prioritizing selection criteria for selecting the strategic projects at different levels in the different organization. This kind of integration has got limited attention previously in the field of project selection in the portfolio.

In this paper, the basic approach of HoQ remains the same where their roles have been redefined as per the current problem of prioritizing selection criteria in project selection. Traditionally, the QFD approach consists of the identification of client needs and their relative significance for determining design specification and finding their weights. Whereas, in this paper, WHATs represent client’s needs and HOWs represent various selection criteria used in selecting the project in the portfolio. The impact of WHATs and HOWs is given in fuzzy numbers that are further quantized using triangular fuzzy membership functions. Following the approach of QFD, the outcome of this hybrid approach is a set of prioritized selection criteria for further utilization in selecting appropriate project into the portfolio. The details are provided in Section 4.

3.4 Genetic algorithm

Genetic algorithm (GA) is one of the most commonly used algorithms for searching an optimized solution in solving optimization problems [4, 20]. This algorithm applies a metaheuristic approach that involves random initialization of solutions and iterates through the solutions to find the best solution under a given set of constraints, after performing some specified set of operations over target solutions. It consists of representing a gene by a real number or binary number; a chromosome is a set of genes. A population is a set of chromosomes generated in different generations using different operations. Generally, GA involves operations of selection, crossover and mutation on the chromosome of a population to form a new population of next-generation [43, 44].

The selection operation is applied to select the chromosomes having higher fitness for the next generation and leads to more probability of reproducing high fitness solutions. Crossover operation consisting of interchanging two chromosomes for generating two new chromosomes for the population. Several types of crossover operations have been proposed in the literature like one-point, two-point, multi-point and uniform [45, 46]. Mutation operation consists of changing a chromosome into a new chromosome by inverting randomly selected genes of chromosomes with a given mutation rate.

The overall working of GA is depicted in Fig. 3, indicating the use of these operations for selecting the best solution for a given problem. The working of GA can be explained in following steps.

Defining the fitness function

Initialing population randomly

Crossover initial chromosomes

Mutate chromosomes

Select chromosomes to create the population with higher fitness

Evaluate the population

Fig. 3

Working of Genetic algorithm [4].

The above-cited steps are iterated for different generations in GA to find an optimal solution. In this paper, we used GA to find an optimal set of projects from a given set of projects using filtered selection criteria. Here, a gene is represented as a binary number of 0 or 1, representing the selection and non-selection of a project in the project portfolio. Each chromosome in the population is represented using a sequence of 0’s, and 1’s giving a set of projects selected in a portfolio with a fitness value. The genetic operations are performed on the population to find optimal chromosome as the best solution of project selection problem under a given set of constraints optimized by GA. The use of GA for selecting an optimal project portfolio is described in Section 4.

4 The proposed hybrid approach

This paper proposes an approach that involves the integration of fuzzy logic, QFD and a meta-heuristic genetic algorithm. The proposed approach enables the selection of the strategic project in portfolio by analyzing input of selection criteria for meeting their specific requirements as provided by the experts of the field, filtering the selection criteria and evaluating the projects for achieving organization objective to the maximum extent. To complete the task of project selection efficiently and effectively, the working of proposed work is divided into three stages as described below and presented in Fig. 4.

Fig. 4

The proposed approach.

4.1 Stage 1 –Formulating the problem

This stage involves defining the client needs (WHATs) and selection criteria (HOWs) as input from the experts in the field. The WHATs are assigned importance score as per the requirement of the organization. There exist many research studies highlighting different WHATs and HOWs and their impact on each other. The user can analyze the existing literature for identifying WHATs and HOWs. Identified WHATs and HOWs can be further defined through experts for a specific organization. Some important HOWs as selection criteria and WHATs as organization objectives highlighted by various researchers in the literature are presented in Tables 2 and 3 respectively.

Table 2
Possible WHATs

Possible WHATs Studies

Contribution to strategic goals [47]

Maximize portfolio value [48]

User acceptance [49]

Risk minimization [50]

Overall performance [49]

Managing stakeholders [51]

Possible WHATs	Studies
Contribution to strategic goals	[47]
Maximize portfolio value	[48]
User acceptance	[49]
Risk minimization	[50]
Overall performance	[49]
Managing stakeholders	[51]

Table 3

Possible HOWs

Possible HOWs	Studies
Alignment with strategic goals	[52]
Net profit	[51]
Project risk	[53]
Support of higher authorities	[54]
Implementation cost	[55]
ROI of project	[50]
Complexity of the project	[56]
Interdependency of projects	[56]

These WHATs and HOWs can be refined/added by the experts to meet the specific requirements of an organization. The outcome of this stage consists of a list of WHATs, HOWs and their impact as provided by the experts to fulfil the requirement of the organization. The refined WHATs and HOWs are further fed to the next stage.

4.2 Stage 2 - Filtering selection criteria using fuzzy QFD

This stage involves developing HoQ by populating corresponding matrices for computing the relative score of HOWs based on fuzzy input provided by the experts. The fuzzy numbers are quantized to their crisp values by using triangular membership function. The motivation for using triangular membership function is its common use for solving similar problems and proved to be very useful for taking a decision based on subjective and imprecise information [1, 4].

After getting the refined WHATs and HOWs, the experts provide the relative importance of WHATs as per goals of the organization in terms of fuzzy values. In this paper, we use fuzzy values as fuzzy numbers like VL, L, M, H, and VH explained in Section 3.2. The fuzzy values are further converted to crisp values for filling the values on matrix B depicted in Fig. 3.1. We can receive an assessment of the relative importance of WHATs from more than one experts. In the case of multiple experts, the aggregated values of fuzzy values from different experts are used to populate into matrix B. Let us assume that I (i₁, i₂, i₃) be fuzzy values input by the experts. The input values of the relative importance of m WHATs from p experts are aggregated as per Equation 2. $I_{n} = \frac{1}{p} \sum_{k = 1}^{m} \sum_{E = 1}^{p} i_{k, E}$ (2) Where, p is number of experts determining the relative importance of m WHATs.

Further, the impact of WHATs on HOWs is determined by the experts in terms of fuzzy values. We can receive an assessment of the impact of WHATs on HOWs from more than one experts. In the case of multiple experts, fuzzy input values from different experts are aggregated and used for populating matrix D depicted in Fig. 3.1. Assume, O_i,j,k represents the opinion of relationship between i^th WHAT with j^th HOW by k^th expert. The fuzzy input values of m WHATs on n HOWs from p experts are aggregated as per Equation 3. ${OAgg}_{i, j} = \frac{1}{p} \sum_{k = 1}^{p} \sum_{i = 1}^{m} \sum_{j = 1}^{n} O_{i, j, k}$ (3) Where p, m, n represents the number of experts, number of WHATs and number of HOWs respectively.

The weighted fuzzy priority of each HOWs (representing criteria for selecting project in the portfolio) is computing as per Equation 4 using values from Eqs. 2 and 3. ${PFuzzy}_{j} = \frac{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {OAgg}_{i, j} \times I_{i}}{\sum_{k = 1}^{m} i_{k}}$ (4)

The absolute value of each HOWs is de-fuzzified as per Equation 5. ${PAbsolute}_{j} = \frac{i_{1} + i_{2} + i_{3}}{3}$ (5)

The relative priority of each HOWs is computed using Equation 6. ${PRelative}_{j} = \frac{{PAbsolute}_{j}}{\sum_{j = 1}^{n} {PAbsolute}_{j}}$ (6)

The prioritized HOWs as criteria for selecting a project in the portfolio is provided as input to the next phase of the process for selecting the project. Determining priority of criteria for selecting a project in this study addresses a critical gap in the existing research.

4.3 Stage 3 - Evaluating project portfolio

This stage applies GA for evaluating the projects in the portfolio to maximize the objective function of the organization on the basis of top priority criteria provided in stage 2. The project selection problem is converted into a population of chromosomes having the binary gene. Different genetic operations are performed on the candidate population. An optimal solution is obtained after iterating through many generations in GA. The optimal solution corresponds to the best solution for selecting a given number of projects in a portfolio having a maximum value of an organization’s objective as a fitness function in GA. The project selection problem can be formulated in GA as follows.

Assume, there exist P projects to be evaluated and selecting L projects in portfolio based on decision variables d_i. d_i takes the value of 1 or 0 in selecting a project indicating selection or rejection of project in portfolio represented as D = (d₁, d₂, . . . , d_p). Let w_j be the priority of the expert on selection criterion j (j = 1,2,...,J) when evaluating alternative project P_i (i = 1,2,...,P). c_ij be the rank of project i(i = 1,2,...,I) on selection criterion j (j = 1,2,...,J).

If we assume that projects are independent of each other, then the fitness function of GA to be maximized can be represented as per Equation 7. $Obj = Max (\sum_{i = 1}^{P} \sum_{j = 1}^{J} (W_{j} \times C_{ij}) \times d_{i})$ (7) such that $\sum_{i = 1}^{P} d_{i} = L$

Where, P, J and L represent the number of projects to be evaluated, number of selection criteria and number of projects to be selected in portfolio.

However, in real-life scenarios, the projects in the portfolio are independent of each other. There exists interaction among the projects in terms of different selection criteria. Ignoring interactions among projects may result in an undesirable output that can be used in practical situations such as R & D project selection. Therefore, the effect of interaction among projects must be considered while selecting portfolio projects. For brevity, assume d_j (S_m) be the interaction value in a combination of m projects in terms of selection criterion j (j = 1,2,...,J) and S_m (m = 1,2,..., M) be a combination of m projects. The fitness function in GA by considering interaction among portfolio projects can be represented in Equation 8. $\begin{matrix} Obj = Max (\sum_{i = 1}^{P} \sum_{j = 1}^{J} (W_{j} \times C_{ij}) \times d_{i} + \\ {sum}_{j = 1}^{J} \sum_{m = 1}^{M} (w_{j} \times (d_{j} \times S_{k})) (\sum_{i = 1}^{Z} c_{ij}) \prod_{i = 1}^{Z} d_{i}) \end{matrix}$ (8) such that $\sum_{i = 1}^{P} d_{i} = L$ ; and d_i = 0, 1 ;

Where, Z gives the number of variables with interaction effects. The first component of Equation 8 gives individual effects of all individual projects p_i (i = 1,2,...,P). The second component is the related interaction effect of different projects. Particularly, interactive effects d_j (S_m) can be observed as additional effects on project portfolio p that contains a combination of at least m projects.

5 Numerical analysis of the proposed approach

Most significant studies in the field are evaluated using a numerical example. Therefore, in this section, we illustrate the proposed hybrid approach for project portfolio selection based on prioritized criteria. Simultaneous use of prioritized section criteria along with uncertainty and interaction among portfolio projects are the main contributions of this work that is mostly ignored in prior research. The proposed hybrid approach is implemented in Python, and programs are executed on Laptop having Intel (R) Core (TM) i3-2330M CPU 2.20GHz and RAM size of 4 GB. We explain working of the proposed approach stage-wise in the following sub-sections.

5.1 Stage 1

For a demonstration of the proposed hybrid approach, we assume that there are three experts designated for providing their opinions for different WHATs, HOWs and relationship among them in terms of fuzzy values. Also, suppose three experts finalize four WHATs (A1 to A4) as benefits of the organization corresponding to A-matrix of HoQ, depicted in Fig. 3.1. Further, experts provide a list of refined HOWs and relationship between WHATs and HOWs corresponding to matrices C and D of HoQ.

5.2 Stage 2

Suppose, the experts provide input values on WHATs and their importance for fulfilling the objective of the organization as presented in Table 4.

Table 4
Analysis values of experts for WHATs

A matrix (WHATs) Expert 1 Expert 2 Expert 3

A1 L VL H

A2 H VH H

A3 L H M

A4 H M VH

A matrix (WHATs)	Expert 1	Expert 2	Expert 3
A1	L	VL	H
A2	H	VH	H
A3	L	H	M
A4	H	M	VH

We convert fuzzy values provided by the experts to fuzzy numbers using triangular membership function as described in Section 3.2. The opinions of experts are aggregated to get a weighted assessment of each WHAT, presented in Table 5 using Equation 2.

Table 5

Aggregated fuzzy opinion assessment of WHATs

A matrix (WHATs)	Expert 1	Expert 2	Expert 3	Aggregate opinion assessment
A1	(2,3,4)	(0,1,2)	(6,7,8)	(2.67,3.67,4.67)
A2	(6,7,8)	(8,9,10)	(6,7,8)	(6.67,7.67,8.67)
A3	(2,3,4)	(6,7,8)	(4,5,6)	(4.00,5.00,6.00)
A4	(6,7,8)	(4,5,6)	(8,9,10)	(6.00,7.00,8.00)

Further, experts are asked to identify HOWs as criteria for selecting a project portfolio (C1-C5) corresponding to C matrix of HoQ. The experts also provide their judgement for relationship between WHATs and HOWs for populating matrix D of HOQ depicted in Fig. 3.1. Suppose, Tables 6 –8 presents the fuzzy input of three experts for relation between WHATs on HOWs in terms of correlation and their converted values to fuzzy numbers.

Table 6

Fuzzy Relationship between WHATs and HOWs - opinion of Expert 1

WHATs /HOWs	B1	B2	B3	B4	B5	B1	B2	B3	B4	B5
A1	H	VH	M	L	VL	(6,7,8)	(8,9,10)	(4,5,6)	(2,3,4)	(0,1,2)
A2	M	H	L	VL	H	(4,5,6)	(6,7,8)	(2,3,4)	(0,1,2)	(6,7,8)
A3	VH	M	VL	L	M	(8,9,10)	(4,5,6)	(0,1,2)	(2,3,4)	(4,5,6)

Table 7

Fuzzy Relationship between WHATs and HOWs - opinion of Expert 2

WHATs /HOWs	B1	B2	B3	B4	B5	B1	B2	B3	B4	B5
A1	M	H	H	M	L	(4,5,6)	(6,7,8)	(6,7,8)	(4,5,6)	(2,3,4)
A2	H	M	VL	H	M	(6,7,8)	(4,5,6)	(0,1,2)	(6,7,8)	(4,5,6)
A3	M	H	M	VL	H	(4,5,6)	(6,7,8)	(4,5,6)	(0,1,2)	(6,7,8)
A4	VL	VL	H	L	H	(0,1,2)	(0,1,2)	(6,7,8)	(2,3,4)	(6,7,8)

Table 8

Fuzzy Relationship between WHATs and HOWs - opinion of Expert 3

WHATs/HOWs	B1	B2	B3	B4	B5	B1	B2	B3	B4	B5
A1	VH	M	H	VL	M	(8,9,10)	(4,5,6)	(6,7,8)	(0,1,2)	(4,5,6)
A2	H	VH	M	L	VH	(6,7,8)	(8,9,10)	(4,5,6)	(2,3,4)	(8,9,10)
A3	M	H	H	M	H	(4,5,6)	(6,7,8)	(6,7,8)	(4,5,6)	(6,7,8)
A4	M	M	VL	H	M	(4,5,6)	(4,5,6)	(0,1,2)	(6,7,8)	(4,5,6)

Correlation between each WHAT with HOW is aggregated from different experts and aggregated values (using Equation 3) are presented in Table 9.

Table 9

Aggregated fuzzy values of correlation assessment of different experts

WHATs /HOWs	B1	B2	B3	B4	B5
A1	(6.00,7.00,8.00)	(6.00,7.00,8.00)	(5.33,6.33,7.33)	(2.00,3.00,4.00)	(2.00,3.00,4.00)
A2	(5.33,6.33,7.33)	(6.00,7.00,8.00)	(2.00,3.00,4.00)	(2.67,3.67,4.67)	(6.00,7.00,8.00)
A3	(5.33,6.33,7.33)	(5.33,6.33,7.33)	(3.33,4.33,5.33)	(2.00,3.00,4.00)	(5.33,6.33,7.33)
A4	(1.33,2.33,3.33)	(2.00,3.00,4.00)	(4.00,5.00,6.00)	(4.00,5.00,6.00)	(6.00,7.00,8.00)

This is followed by computing the weight of each HOWs as per Equation 3 and presented in Table 10.

Table 10

Aggregated fuzzy number weights of HOWs

HOWs	Aggregated weights
B1	(4.18,5.24,6.28)
B2	(5.24,6.28,4.62)
B3	(6.28,4.62,5.66)
B4	(4.62,5.66,6.68)
B5	(5.66,6.68,3.36)

The fuzzy numbers corresponding to weights of HOWs presented in Table 10 are de-fuzzified to obtain absolute values weight values of HOWs using Equation 4 and absolute weight is computed using Equation 5. The relative weight values are computed using Equation 6 for assigning priority to different HOWs as criteria for selecting project portfolio as per their relative weight is shown in Table 11.

Table 11

Priorities of HOWs as criteria for project portfolio selection

HOWs	Absolute weight (De-fuzzified values)	Relative and final weight	Priority Assigned
B1	5.232808683	0.206427309	II
B2	5.653586447	0.223026429	III
B3	4.404333373	0.173745065	IV
B4	3.822123433	0.150777661	V
B5	6.236549321	0.246023536	I

5.3 Stage 3

The identified top priority criteria are further provided to stage 2 for usage with GA to evaluate different projects in the portfolio to maximize the fitness function. The resultant solution provided by GA using various genetic operators is the best solution leading to a maximum return to the organization. The detailed working of GA is described in Section 3.4.

For demonstration purpose in this study, we assume that there are five candidate projects as D (d₁, …d₅). We are supposed to select any three projects in our portfolio on the basis of top priority identified selection criteria. We assume, experts identify three top priority selection criteria SC_i (i=1, 2, 3). The experts determine the effect of each SC_i on each project D_j and values are presented in Table 12 (Assumed values for illustration purpose only).

Table 12
Illustrative input data

SC Priority D1 D2 D3 D4 D5

SC1 1 0.33 0.24 0.65 0.28 0.41

SC2 3 0.8 0.52 0.57 0.45 0.62

SC3 6 0.55 0.12 0.32 0.83 0.54

SC	Priority	D1	D2	D3	D4	D5
SC1	1	0.33	0.24	0.65	0.28	0.41
SC2	3	0.8	0.52	0.57	0.45	0.62
SC3	6	0.55	0.12	0.32	0.83	0.54

In this work, the absolute priority is converted into relative priority by using Equation 9. ${Priority}_{i} = \frac{{Priority}_{i}}{\sum_{j = 1}^{3} {Priority}_{j}}$ (9)

The transformed values are presented in Table 13.

Table 13

Transformed input data

SC	Priority	D1	D2	D3	D4	D5
SC1	0.1	0.33	0.24	0.65	0.28	0.41
SC2	0.3	0.8	0.52	0.57	0.45	0.62
SC3	0.6	0.55	0.12	0.32	0.83	0.54

The objective function without considering interacting among the projects can be represented using Equation 7GA as below. $\begin{matrix} Obj = Max ([(0.1 \times 0.33) + (0.3 \times 0.8) + \\ (0.6 \times 0.55)] d_{1} + [(0.1 \times 0.24) + \\ (0.3 \times 0.52) + (0.6 \times 0.12)] d_{2} + \\ [(0.1 \times 0.65) + (0.3 \times 0.57) + (0.6 \times 0.12)] d_{3} + [(0.1 \times 0.28) + \\ (0.3 \times 0.45) + (0.6 \times 0.83)] d_{4} + \\ [(0.1 \times 0.41) + (0.3 \times 0.62) + (0.6 \times 0.54)] d_{5})) \end{matrix}$ (10) such that $\sum_{i = 1}^{5} d_{i} = 3$

This leads to the selection of D₁, D₄ and D₅ projects with maximum fitness value of 1.815 using GA having best solution as (1,0,0,1,1) without considering interaction among portfolio projects.

But, there is interaction among projects in portfolio for different selection criteria in real scenarios. For illustration purpose, suppose experts provide interaction among various projects on the basis of selection criteria as shown in Table 14.

Table 14

Interaction values of portfolio projects

SC	D ₁ D ₂	D ₁ D ₃	D ₁ D ₄	D ₁ D ₅	D ₂ D ₃	D ₂ D ₄	D ₂ D ₅	D ₃ D ₄	D ₃ D ₅	D ₄ D ₅
SC1	0.6	0.24	0	0	0.8	0	0	0	0.25	0
SC2	0.2	0	0	0.1	0	0.69	0	0.4	0	0.53
SC3	0	0	0.25	0.1	–0.3	–0.2	0	0	–0.2	0.13

From Table 14, it can be observed that there exist six interaction of project in terms of selection criteria SC3 as {D₁D₄}, {D₁D₅}, {D₂D₃}, {D₂D₄}, {D₃D₅} and {D₄D₅} interaction values as 0.25, 0.1, -0.3, -0.2, -0.2 and 0.13 respectively.

The objective function by considering interacting among the projects can be represented using Equation 10 as below. $\begin{matrix} Obj = Max ([(0.1 \times 0.33) + (0.3 \times 0.8) + (0.6 \times 0.55)] d_{1} + [(0.1 \times 0.24) + \\ (0.3 \times 0.52) + (0.6 \times 0.12)] d_{2} + \\ [(0.1 \times 0.65) + (0.3 \times 0.57) + \\ (0.6 \times 0.12)] d_{3} + [(0.1 \times 0.28) + \\ (0.3 \times 0.45) + (0.6 \times 0.83)] d_{4} + \\ [(0.1 \times 0.41) + (0.3 \times 0.62) + (0.6 \times 0.54)] d_{5} + 0.1 \times [(0.33 + 0.24) \times 0.6 d_{1} d_{2} + (0.33 + 0.65) \times 0.24 d_{1} d_{3} + \\ (0.24 + 0.65) \times 0.8 d_{2} d_{3} + (0.65 + 0.41) \times 0.25 d_{3} d_{5}] + 0.3 \times [(0.8 + 0.52) \times 0.2 d_{1} d_{2} + (0.8 + 0.45) \times 0.1 d_{1} d_{5} + (0.52 + 0.45) \times 0.69 d_{2} d_{4} + (0.57 + 0.45) \times 0.4 d_{3} d_{4} + (0.45 + 0.62) \times 0.53 d_{4} d_{5}] + 0.6 \times [(0.55 + 0.83) \times 0.25 d_{1} d_{4} + \\ (0.55 + 0.54) \times 0.1 d_{1} d_{5} + (0.12 + 0.32) \times (- 0.3) d_{2} d_{3} + \\ (0.12 + 0.83) \times (- 0.2) d_{2} d_{4} + (0.32 + 0.54) \times (- 0.2) d_{3} d_{5} + (0.83 + 0.54) \times 0.13 d_{4} d_{5}]) \end{matrix}$ (11) $\begin{matrix} Obj = Max ([(0.603 \times d_{1} + 0.252 \times d_{2} + 0.428 \times d_{3} + 0.661 \times d_{4} + \\ 0.551 \times d_{5}) + \end{matrix}$

$\begin{matrix} 0.1 \times [(0.33 + 0.24) \times 0.6 d_{1} d_{2} + \\ (0.33 + 0.65) \times 0.24 d_{1} d_{3} + (0.24 + 0.65) \times 0.8 d_{2} d_{3} + (0.65 + 0.41) \times 0.25 d_{3} d_{5}] + 0.3 \times [(0.8 + 0.52) \times 0.2 d_{1} d_{2} + \\ (0.8 + 0.45) \times 0.1 d_{1} d_{5} + (0.52 + 0.45) \times 0.69 d_{2} d_{4} + \\ (0.57 + 0.45) \times 0.4 d_{3} d_{4} + (0.45 + 0.62) \times 0.53 d_{4} d_{5}] + \\ 0.6 \times [(0.55 + 0.83) \times 0.25 d_{1} d_{4} + (0.55 + 0.54) \times 0.1 d_{1} d_{5} + \\ (0.12 + 0.32) \times (- 0.3) d_{2} d_{3} + (0.12 + 0.83) \times (- 0.2) d_{2} d_{4} + \\ (0.32 + 0.54) \times (- 0.2) d_{3} d_{5} + (0.83 + 0.54) \times 0.13 d_{4} d_{5}]) \end{matrix}$ (12) such that $\sum_{i = 1}^{5} d_{i} = 3$

In this way, putting the values of interaction among the projects, the above fitness function can be optimized to obtain the best solution as project portfolio using GA based on top priority selection criteria similar to illustration for optimization without considering the interaction of projects described above.

The above-cited example for analysis of the proposed hybrid approach demonstrates its capability to prioritize criteria for selecting a project portfolio by simultaneously considering uncertainty and interdependency among the projects during the project selection process.

6 Conclusion

Simultaneous consideration of uncertainty, prioritization of selection criteria and concurrent evaluation of projects considering the interaction among the projects are ignored by most of the research in the field. In order to address these issues, this paper presents a hybrid approach for selecting a project portfolio by combining many approaches. The proposed approach integrates fuzzy logic, QFD and Genetic algorithm to address the issue of uncertainty of experts in selecting projects, prioritizing criteria before initiating project selection process and evaluating the number of projects for maximal values taking interaction among projects into consideration respectively. The proposed approach completes the task in three stages. In the first stage, it involves interaction with experts to extract fuzzy input about the benefits of organization and selection criteria for selecting a project portfolio. The second stage involves the application of fuzzy QFD to prioritize criteria before selecting a project portfolio. In this stage, the paper contributes a method for using fuzzy values in a distinct way for obtaining priorities of selection criteria. The final stage evaluates the candidate projects concurrently based on top priority selection criteria by considering interrelation among projects by proposing a distinct fitness function of GA.

An illustrative example is provided to demonstrates the working of the proposed approach, considering three experts, three objectives of the organization and four selection criteria. Fitness function is derived for both the cases with interaction and non-interaction among projects of the portfolio. It is demonstrated that concurrent evaluation of interacting projects on basis of top priority selection criteria derived from uncertain input of experts can lead to a maximum gain to organizations. In this study, we focus on proposing a fuzzy computational approach by integrating fuzzy, QFD and GA for addressing ignored issues in the literature. In this paper, we use de-fuzzification for computing absolute values, that can lead to loss of information. This exercise can be repeated using fuzzy operations in our future work. Our future work will also focus on refining the proposed methods of the approach and apply it for selecting strategic projects by government and private organizations of different levels.

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A fuzzy computational approach for selecting interdependent projects using prioritized criteria

Abstract

Keywords

1 Introduction

2 Related work

3.1 Quality function deployment (QFD) approach

3.4 Genetic algorithm

Table 2 Possible WHATs Possible WHATs Studies Contribution to strategic goals [47] Maximize portfolio value [48] User acceptance [49] Risk minimization [50] Overall performance [49] Managing stakeholders [51]

5.1 Stage 1

5.2 Stage 2

Table 4 Analysis values of experts for WHATs A matrix (WHATs) Expert 1 Expert 2 Expert 3 A1 L VL H A2 H VH H A3 L H M A4 H M VH

Table 12 Illustrative input data SC Priority D1 D2 D3 D4 D5 SC1 1 0.33 0.24 0.65 0.28 0.41 SC2 3 0.8 0.52 0.57 0.45 0.62 SC3 6 0.55 0.12 0.32 0.83 0.54

References

Table 2
Possible WHATs

Possible WHATs Studies

Contribution to strategic goals [47]

Maximize portfolio value [48]

User acceptance [49]

Risk minimization [50]

Overall performance [49]

Managing stakeholders [51]

Table 4
Analysis values of experts for WHATs

A matrix (WHATs) Expert 1 Expert 2 Expert 3

A1 L VL H

A2 H VH H

A3 L H M

A4 H M VH

Table 12
Illustrative input data

SC Priority D1 D2 D3 D4 D5

SC1 1 0.33 0.24 0.65 0.28 0.41

SC2 3 0.8 0.52 0.57 0.45 0.62

SC3 6 0.55 0.12 0.32 0.83 0.54