BEST SELECTION OF PROJECT PORTFOLIO USING FUZZY AHP AND FUZZY TOPSIS

Choosing the optimal portfolio for the project is one of the most important and strategic decisions in most organizations, especially project-based organizations. The issue of the project selection is a periodic activity in order to choose the appropriate and optimal portfolio from the proposed projects and ongoing projects within the organization which can meet organizational goals without waste of the resources and taking into account certain constraints. Consequently, the aim of this paper is to select the best project portfolio by using fuzzy AHP and fuzzy TOPSIS methods. The forgoing methods have been used in a case study, and the result and data have been evaluated from different points of view.


INTRODUCTION
In today's competitive environment only organizations that properly utilize existing opportunities with respect to their constraints can survive; and others will inevitably be condemned to fall over the time.The use of situations, prior to recognizing situations, requires full understanding of the strengths and weaknesses, abilities and limitations.One of the ways that can help organizations to choose the best positions is the theme of project portfolio management.Project portfolio management by collecting information from different parts of the organization, and collecting project information help managers to make the right decision.Therefore, one of the important issues in project management is the best selection of project portfolio, (Rabieh & Faddaie, 2015).A project portfolio or portfolio is a set of projects, plans or operations that are grouped together to make effective management of activities and to achieve strategic goals, (Ravanshadnia & Jahromi, 2015).Project portfolio has been created to achieve one or more corporate strategy and goals, and may consist of a set of past, present, scheduled and future portfolio (PMI).The issue of selecting a project and its related activities is one of the important activities in many organizations, especially contracting companies and project-based construction companies.In selecting a project portfolio, the most important issue that is to be asked is whether the portfolio should include what projects.The more choosing the lightweight portfolio projects, the more likely the missions of the organization will be (Alinezhad & Simiari, 2013).
Decision making problems are one of the most important issues in all sciences.Fan, Ma and Zhang (2002) and Chuu (2009) stated that human beings are faced with issues of decision making that basically involves choosing the most-preferred alternatives from a limited set of alternatives to obtain certain-predefined objectives.It is the process to find the best alternative from all feasible alternatives, (Bashiri & Badri, 2010).Decisions in the public and private sector decision-making often involve the assessment and ranking of available alternatives or decision options based on multicriteria, (Hwang & Yoon, 1995).Multi criteria decision making (MCDM) in the field is one of the most widely used methods, (Yeh & Chang, 2008).
The MCDM goal is to choose the best alternative from some mutually exclusive options regarding different criteria decided by the decision maker.Zadeh (1965) presented fuzzy set theory to cope with the ambiguity of human thought.Fuzzy set allows partial membership unlike the classical set.Classical set theory concepts are made based on the member or non-member.There is a clear, sharp, and crisp boundary exists between a member and non-member in this set, so the classical set theory cannot describe many realworld applications (Ertu rul & Karaka o lu, 2009).Zadeh (1965) proposed to use values ranging from 0 to 1 for showing the membership of the objects in a fuzzy set.Complete non membership is represented by 0, and complete membership as 1.Values between 0 and 1 represent intermediate degrees of membership."Not very clear", "probably so", "very likely", these terms of expression can be heard very often in daily life and their commonality is that they are more or less tainted with uncertainty.With different daily decision making problems of diverse intensity, the results can be misleading if the fuzziness of human decision making is not taken into account, (Tsaur, Chang, & Yen, 2002).Fuzzy sets theory providing a more widely frame than classic sets theory, has been contributing to capability of reflecting real world (Ertu rul & Tu , 2007).Fuzzy sets and fuzzy logic are powerful mathematical tools for modeling: uncertain systems in industry, nature and humanity; and facilitators for common-sense reasoning in decision making in the absence of complete and precise information.Their role is significant when applied to complex phenomena not easily described by traditional mathematical methods, especially when the goal is to find a good approximate solution.

FUZZY SET THEORY
Definition 1.Let U be the universe of discourse and u a generic element of U, then U ={u}.A fuzzy subset Ã, defined in U, is: where ( ) A u µ is designated as membership function or membership grade (also designated as degree of compatibility or degree of truth) of u in Ã.The membership function associates with each element u, of U, a real number ( ) A u µ , in the interval [0, 1], (Simões-Marques, Ribeiro, & Gameiro-Marques, 2000).Definition 2. Dubois and Prade (1978) proposed the triangular fuzzy number and defined it as follows: A is a fuzzy number, if A is normal and convex, (Liu & Wu, 2007).A triangular fuzzy numbers can be expressed as M = (l, m, u), where l m u, in which l m u.In the fuzzy event, parameters (l, m, u) are the smallest, promising, and the largest possible value, correspondingly.Equation 3 describes the triangular fuzzy number membership function M, when l=m=u, it is a nonfuzzy number by agreement as shown in Figure 1, (Önüt, Kara, & Efendigil, 2008).
Definition 3. The main operational laws for two triangular fuzzy numbers M 1 and M 2 are as follows, (Kaufmann & Gupta, 1991). ) Definition 4. A variable with values of word or sentences in an artificial language is defined as linguistic variable, (Wang, Cheng, & Huang, 2009).After recognition of the cardinality of the linguistic terms, linguistic and semantics terms must be arranged.There are various possibilities to fulfil this work.Considering all the terms distributed on a scale involves directly supplying the term set where the total order is defined, (Herrera & Martínez, 2001).For instance, a set of seven terms can be presented in Table 1.
be the set of decision makers, and The fuzzy group decision problem can be concisely expressed as matrix format: 1 2 where k ij x and k j w are linguistic variables that can be shown by fuzzy numbers, (Mahdavi, Mahdavi-Amiri, Heidarzade, & Nourifar, 2008).
Step 1. Identify the evaluation criteria that are connected to system goals.
Step 2. Develop alternatives to achieve the systems goals (alternatives generation).
Step 3. Evaluate alternatives in terms of criteria (the values of the criterion functions are fuzzy).
Step 4. Identify the weights of criteria and the weights of decision makers based on fuzzy AHP method (the values of the weights can be crisp or fuzzy).− Establish pairwise comparison matrix of the criteria using Triangular Fuzzy Numbers.A 9point scale was used to describe the relative importance of criteria with respect to the goal, (Tsaur et al., 2002) as shown in Table 2. − Convert to crisp value.Each fuzzy evaluation of every criterion has to be converted into crisp value.It was done as in the following step.The relative weight of all evaluation criteria are still in the form of triangular fuzzy number and need to defuzzify using Center of Gravity, (Chou, Chang, & Shen, 2008).

( ) ( )
− Finally, in order to effectively compare the relative importance of the evaluation criteria, the defuzzified priority values are normalized using: Best selection of project portfolio using fuzzy AHP and fuzzy TOPSIS 6 JEMC, VOL. 8, NO. 1, 2018, 3-10 − Consistency ratio (CR) is required to determine whether the weight assign by the decision maker is correct or not.CR< 0.1 indicates consistent judgment in pairwise comparisons.CR is calculated using equation 7 and 8 (Ahmad, Kasim, & Rajoo, 2016).
Step 5. Construct fuzzy decision matrix.Each k ij x ĩs supposed fuzzy number in the fuzzy decision matrix.
Step 8.The weighted normalized decision matrix is calculated as following: where w j is the weight of the i th attribute or criterion, and 1 1.
Step 9. Verify the positive ideal and negative ideal solution: where I is associated with benefit criteria, and J is associated with cost criteria.
Step 10.The separation measures are calculated utilizing the n-dimensional Euclidean distance.The separation measure of each alternative from the ideal solution is presented as: ( ) Similarly, the separation from the negative ideal solution is given as: ( ) Step 11.The relative closeness to the ideal solution is calculated.
( ) Step 8.The alternatives' ranking orders are determined according to the closeness coefficient, and the best alternative is selected between a set of feasible alternatives.

EMPIRICAL STUDY
This study was performed in Qazvin municipality in three phases: Identifying the evaluation criteria (effective factors for selecting the best project portfolio), identifying the weights of criteria (effective factors' weights) and finally prioritizing of the projects.Five projects (alternatives) are evaluated using the linguistic variables as follows: Building under construction of the municipality, Rajaie bridge, Nokhbegan footbridge, Aqueduct construction and park construction.At first stage an expert team of five members from deputy of construction was formed.
At the second stage using fuzzy Delphi method with its five stages, the criteria and the sub criteria (Effective factors) based on expert viewpoints were extracted as tabulated in Table 3. and Table 4. based on the linguistic variables.
Next, the expert team determines weight importance of the criteria by using fuzzy AHP method in step 4 as shown in Table 5.Five projects (alternatives) are evaluated using the linguistic variables by five DMs whose weight are equal under these 32 attributes, as listed in Table 5. Linguistic evaluations are converted into triangular fuzzy numbers to construct a fuzzy decision matrix (shown in Table 4).
Regarding the Table 4, and the continuation of step six of the proposed algorithm, alternatives were classified by Eq. ( 18) as follows in Table 6.The alternative + li C with the highest value would be considered as the first rank and the lowest value is represented as the last rank.Therefore, the ranking order of five projects will be as follows: 2 3 5 1

. A A A A A
≈ > > > So, A 2 and A 3 Rajaie Bridge and Nokhbegan Footbridge are the best projects among the five projects, and A 4 is the worst project.

CONCLUSION
The work complexity and the interference of many parameters affecting the success factors of the projects on one hand, and the dependencies and effects that the organization's projects manipulate on each other make the topic of the project portfolio very important.This paper extracted some effective factors for selecting the best project portfolio through fuzzy Delphi method and consequently determined weights of the criteria by using fuzzy AHP method.Fuzzy TOPSIS method was then used to evaluate municipality projects by a group of experts by linguistic terminologies.The results showed that the "Impact of resources required for the new project on the resources of ongoing projects" criterion was more important than other criteria.According to the proposed methodology, Rajaie Bridge and Nokhbegan Footbridge were selected as the best among the five ongoing projects.

Table 1 :
Linguistic variables for the ratings

Table 2 :
Saaty's Crips Scale and Fuzzified Scale for Pairwise Comparison

Table 3 :
Average views of experts in comparing general factors with each other

Table 4 :
The triangular fuzzy numbers decision matrix

Table 5 :
Effective component weights using fuzzy AHP

Table 6 :
The relative closeness to the ideal solution