Abstract

There are various methods regarding project selection in different fields. This paper deals with an actual application of construction project selection, using two aggregation operators. First, the opinion of experts is used in a model of group decision making called intuitionistic fuzzy TOPSIS (IFT). Secondly, project evaluation is formulated by dynamic intuitionistic fuzzy weighted averaging (DIFWA). Intuitionistic fuzzy weighted averaging (IFWA) operator is utilized to aggregate individual opinions of decision makers (DMs) for rating the importance of criteria and alternatives. A numerical example for project selection is given to clarify the main developed result in this paper.

1. Introduction

Project selection and project evaluation involve decisions that are critical in terms of the profitability, growth, and survival of project management organizations in the increasingly competitive global scenario. Such decisions are often complex, because they require identification, consideration, and analysis of many tangible and intangible factors [1].

There are various methods regarding project selection in different fields. Project selection problem has attracted great endeavor by practitioners and academicians in recent years. One of the major fields that have been applied to this problem is mathematical programming, especially Mix-Integer Programming (MIP), since the problems comprise selection of projects while other aspects are considered using real-value variables [2]. For instance, a MIP model is developed by [3] to conquer Research and Development (R&D) portfolio selection.

Multicriteria decision making (MCDM) is a modeling and methodological tool for dealing with complex engineering problems [4]. Many mathematical programming models have been developed to address project-selection problems. However, in recent years, MCDM methods have gained considerable acceptance for judging different proposals. The objective of Mohanty’s [5] study was to integrate the multidimensional issues in an MCDM framework that may help decision makers to develop insights and make decisions. They computed weight of each criterion and then assessed the projects by doing technique for order preference by similarity to ideal solution algorithm (TOPSIS) [6]. To select R&D project, the application of the fuzzy analytical network process (ANP) and fuzzy cost analysis has been used by some researchers [7]. In their studies, triangular fuzzy numbers (TFNs) are used to prefer one criterion over another by using a pairwise comparison with the fuzzy set theory, where the weight of each criterion in the format of triangular fuzzy numbers is calculated [7]. The project selection problem was presented through a methodology which is based on the analytic hierarchy process (AHP) for quantitative and qualitative aspects of a problem [8]. It assists the measuring of the initial viability of industrial projects. The study shows that industrial investment company should concentrate its efforts in development of prefeasibility studies for a specific number of industrial projects which have a high likelihood of realization [8].

Multiattribute decision making (MADM) is the other applied approach in which criteria are mostly defined in qualitative scale and the decision is made with respect to assigned weights using some methods, such as PROMETHEE [9, 10]. To have more comprehensive study on MADM methods in this field, readers are referred to [11–15].

The rest of the paper is organized as follows. Section 2 provides materials and methods, mainly fuzzy set theory (FST) and intuitionistic fuzzy set (IFS). The IFT and DIFWA are introduced in Section 3. How the proposed model is used in an actual example is explained in Section 4. Finally, the conclusions are provided in the final section.

2. Materials and Methods

2.1. FST

Zadeh (1965) introduced the fuzzy set theory (FST) to deal with the uncertainty due to imprecision and vagueness. A major contribution of this theory is capability of representing vague data; it also allows mathematical operators and programming to be applied to the fuzzy domain. An FS is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership function, which assigns to each object a grade of membership ranging between zero and one [16, 17].

A tilde “~” will be placed above a symbol if the symbol represents an FST. A TFN 𝑀 is shown in Figure 1. A TFN is denoted simply as (𝑙/𝑚,𝑚/𝑢) or (𝑙,𝑚,𝑢). The parameters 𝑙,𝑚 and 𝑢 (𝑙≤𝑚≤𝑢), respectively, denote the smallest possible value, the most promising value, and the largest possible value that describe a fuzzy event. The membership function of TFN is as follows.


Figure 1 
A TFN 𝑀 .

Each TFN has linear representations on its left and right side, such that its membership function can be defined as 𝜇𝑥𝑀=⎧⎪⎪⎪⎨⎪⎪⎪⎩0,𝑥<𝑙,𝑥−𝑙𝑚−𝑙,𝑙≤𝑥≤𝑚,𝑢−𝑥𝑢−𝑚,𝑚≤𝑥≤𝑢,0,𝑥>𝑢.(2.1)

A fuzzy number can always be given by its corresponding left and right representation of each degree of membership as in the following: 𝑀𝑀=𝑙(𝑦),𝑀𝑟(𝑦)[]=(𝑙+(𝑚−𝑙)𝑦,𝑢+(𝑚−𝑢)𝑦),𝑦∈0,1,(2.2) where 𝑙(𝑦) and 𝑟(𝑦) denote the left side representation and the right side representation of a fuzzy number (FN), respectively. Many ranking methods for FNs have been developed in the literature. These methods may provide different ranking results, and most of them are tedious in graphic manipulation requiring complex mathematical calculation [18].

While there are various operations on TFNs, only the important operations used in this study are illustrated. If we define two positive TFNs (𝑙1,𝑚1,𝑢1) and (𝑙2,𝑚2,𝑢2), then,.(𝑙1,𝑚1,𝑢1)+(𝑙2,𝑚2,𝑢2)=(𝑙1+𝑙2,𝑚1+𝑚2,𝑢1+𝑢2),(𝑙1,𝑚1,𝑢1)∗(𝑙2,𝑚2,𝑢2)=(𝑙1∗𝑙2,𝑚1+𝑚2,𝑢1∗𝑢2)(𝑙1,𝑚1,𝑢1)+𝑘=(𝑙1∗𝑘𝑚1∗𝑘,𝑢1∗𝑘),where𝑘>0(2.3)

2.2. Basic Concept of IFS

The application of IFS method within the overall goal to select the best project has been described. IFSs introduced by Atanassov [19] are an extension of the classical FST, which is a suitable way to deal with vagueness. IFSs have been applied to many areas such as medical diagnosis [20–22], decision-making problems [23–46], pattern recognition [47–52], supplier selection [53, 54], enterprise partners selection [55], personnel selection [56], evaluation of renewable energy [57], facility location selection [58], web service selection [59], printed circuit board assembly [60], and management information system [61].

The following briefly introduces some necessary introductory concepts of IFS. IFS 𝐴 in a finite set 𝑋 can be written as [19] 𝐴=⟨𝑥,𝜇𝐴(𝑥),𝑣𝐴(𝑥)⟩∣𝑥∈𝑋,where𝜇𝐴(𝑥),𝑉𝐴[](𝑥)∶𝑋⟶0,1(2.4) are membership function and nonmembership function, respectively, such that 0≤𝜇𝐴𝑉(𝑥)𝐴(𝑥)≤1.(2.5) A third parameter of IFS is 𝜋𝐴(𝑥), known as the intuitionistic fuzzy index or hesitation degree of whether 𝑥 belongs to 𝐴 or not: 𝜋𝐴(𝑥)=1−𝜇𝐴(𝑥)−𝑉𝐴(𝑥).(2.6) It is obviously seen that for every 𝑥∈𝑋0≤𝜋𝐴(𝑥)≤1ifthe𝜋𝐴(𝑥).(2.7)

If it is small, knowledge about 𝑥 is more certain. If 𝜋𝐴(𝑥) is great, knowledge about 𝑥 is more uncertain. Obviously, when 𝜇𝐴(𝑥)=1−𝑣𝐴(𝑥)𝜇𝐴(𝑥)=1−𝑣(𝑥)(2.8) for all elements of the universe, the ordinary FST concept is recovered [60].

Let 𝐴 and 𝐵 be IFSs of the set 𝑋, then multiplication operator is defined as follows [19]: 𝐴𝜇𝐵=𝐴(𝑥)⋅𝜇𝐵(𝑥),𝑣𝐴(𝑥)+𝑣𝐵(𝑥)−𝑣𝐴(𝑥)⋅𝑣𝐵(𝑥)∣𝑥∈𝑋.(2.9)

3. Intuitionistic Fuzzy TOPSIS (IFT) and Dynamic Intuitionistic Fuzzy Weighted Averaging (DIFWA) Methods

3.1. IFT

It should be mentioned here that the presented approach mainly utilizes the IFT method presented in [53, 56, 57] to handle a project selection problem with six projects and six criteria. In the current paper we validate the method in an actual context and show this method applicability with an extensive set of selection criteria. The IFT method is a suitable way to deal with MCDM problem in intuitionistic fuzzy environment (IFE). Let 𝐴={𝐴1,𝐴2,…,𝐴𝑚} be a set of alternatives and let 𝑋={𝑋1,𝑋2,…,𝑋𝑛} be a set of criteria, the procedure for IFT method has been conducted in eight steps presented as follows.

Step 1. Determine the weights of importance of DMs.
In the first step, we assume that decision group contains𝑙={𝑙1,𝑙2,…,𝑙𝑛} DMs. The importances of the DMs are considered as linguistic terms. These linguistic terms were assigned to IFN. Let 𝐷𝑘=[𝜇𝑘,𝑣𝑘,𝜋𝑘] be an intuitionistic fuzzy number for rating of 𝑘th DM.Then the weight of 𝑘th DM can be calculated as 𝜆𝑘=𝜇𝑘+𝜋𝑘𝜇𝑘/𝜇𝑘+𝑣𝑘∑𝑙𝑘=1𝜇𝑘+𝜋𝑘𝜇𝑘/𝜇𝑘+𝑣𝑘,where𝜆𝑘∈[],0,1𝑙𝑘=1𝜆𝑘=1.(3.1)

Step 2. Determine intuitionistic fuzzy decision matrix (IFDM).
Based on the weight of DMs, the aggregated intuitionistic fuzzy decision matrix (AIFDM) was calculated by applying intuitionistic fuzzy weighted averaging (IFWA) operator Xu [62]. In group decision-making process, all the individual decision opinions need to be fused into a group opinion to construct AIFDM.
Let 𝑅(𝑘)=(𝑟(𝑘)𝑖𝑗)𝑚×𝑛 be an IFDM of each DM. 𝜆={𝜆1,𝜆2,𝜆3,…,𝜆𝑙} is the weight of DM. Consider 𝑟𝑅=𝑖𝑗𝑚×𝑛,(3.2) where 𝑟𝑖𝑗=IFWA𝜆𝑟𝑖𝑗(1),𝑟𝑖𝑗(2),…,𝑟𝑖𝑗(𝑙)=𝜆1𝑟𝑖𝑗(1)𝜆2𝑟𝑖𝑗(2)𝜆3𝑟𝑖𝑗(3)⋯𝜆𝑙𝑟𝑖𝑗(𝑙)=1−𝑙𝑘=11−𝜇𝑖𝑗(𝑘)𝜆𝑘,𝑙𝑘=1𝑣𝑖𝑗(𝑘)𝜆𝑘,𝑙𝑘=11−𝜇𝑖𝑗(𝑘)𝜆𝑘−𝑙𝑘=1𝑣𝑖𝑗(𝑘)𝜆𝑘.(3.3)

Step 3. Determine the weights of the selection criteria.
In this step, all criteria may not be assumed to be of equal importance. 𝑊 represents a set of grades of importance. In order to obtain 𝑊, all the individual DM opinions for the importance of each criteria need to be fused. Let 𝑤𝑗(𝑘)=(𝜇𝑗(𝑘),𝑣𝑗(𝑘),𝜋𝑗(𝑘)) be an IFN assigned to criterion 𝑋𝑗 by the 𝑘th DM.
The weights of the criteria can be calculated as follows: 𝑤𝑗=IFWA𝜆𝑤𝑗(1),𝑤𝑗(2),…,𝑤𝑗(𝑙)=𝜆1𝑤𝑗(1)𝜆2𝑤𝑗(2)𝜆3𝑤𝑗(3)⋯𝜆𝑙𝑤𝑗(𝑙)=1−𝑙𝑘=11−𝜇𝑗(𝑘)𝜆𝑘,𝑙𝑘=1𝑣𝑗(𝑘)𝜆𝑘,𝑙𝑘=11−𝜇𝑗(𝑘)𝜆𝑘−𝑙𝑘=1𝑣𝑗(𝑘)𝜆𝑘.(3.4)
Thus, a vector of criteria weight is obtained: 𝑊=[𝑤1,𝑤2,𝑤3,…,𝑤𝑗], where 𝑤𝑗=(𝜇𝑗,𝑣𝑗,𝜋𝑗)(𝑗=1,2,…,𝑛).

Step 4. Construct the aggregated weighted IFDM.
In Step 4, the weights of criteria (𝑊) and the aggregated IFDM are determined to the aggregated weighted IFDM which is constructed according to the following definition [19]: 𝑅𝜇=𝑅𝑊=𝑖𝑗,𝑣𝑖𝑗=𝑥,𝜇𝑖𝑗⋅𝜇𝑗,𝑣𝑖𝑗+𝑣𝑗−𝑣𝑖𝑗⋅𝑣𝑗,𝜋𝑖𝑗=1−𝑣𝑖𝑗−𝑣𝑗−𝜇𝑖𝑗⋅𝜇𝑗+𝑣𝑖𝑗⋅𝑣𝑗.(3.5)𝑅′ is a matrix composed with elements IFNs, 𝑟𝚤𝑗=(𝜇𝑖𝑗,𝑣𝑖𝑗,𝜋𝑖𝑗)(𝑖=1,2,…,𝑚;𝑗=1,2,…,𝑛).

Step 5. Determine intuitionistic fuzzy positive and negative ideal solution.
In this step, the intuitionistic fuzzy positive ideal solution (IFPIS) and intuitionistic fuzzy negative ideal solution (IFNIS) have to be determined. Let 𝐽1 and 𝐽2 be benefit criteria and cost criteria, respectively. 𝐴∗ is IFPIS and 𝐴− is IFNIS. Then 𝐴∗ and 𝐴− are equal to 𝐴∗=𝑟1∗,𝑟2∗,…,𝑟𝑛∗,𝑟𝑗∗=𝜇𝑗∗,𝑣𝑗∗,𝜋𝑗∗𝐴,𝑗=1,2,…,𝑛,−=𝑟1−,𝑟2−,…,𝑟𝑛−,𝑟𝑗−=𝜇𝑗−,𝑣𝑗−,𝜋𝑗−,𝑗=1,2,…,𝑛,(3.6) where 𝜇𝑗∗=max𝑖𝜇𝑖𝑗𝑗∈𝐽1,min𝑖𝜇𝑖𝑗𝑗∈𝐽2,𝑣∗𝑗=min𝑖𝑣𝑖𝑗𝑗∈𝐽1,max𝑖𝑣𝑖𝑗𝑗∈𝐽2,𝜋𝑗∗=1−max𝑖𝜇𝑖𝑗−min𝑖𝑣𝑖𝑗𝑗∈𝐽1,1−min𝑖𝜇𝑖𝑗−max𝑖𝑣𝑖𝑗𝑗∈𝐽2,𝜇𝑗−=min𝑖𝜇𝑖𝑗𝑗∈𝐽1,max𝑖𝜇𝑖𝑗𝑗∈𝐽2,𝑣𝑗−=max𝑖𝑣𝑖𝑗𝑗∈𝐽1,min𝑖𝑣𝑖𝑗𝑗∈𝐽2,𝜋𝑗−=1−min𝑖𝜇𝑖𝑗−max𝑖𝑣𝑖𝑗𝑗∈𝐽1,1−max𝑖𝜇𝑖𝑗−min𝑖𝑣𝑖𝑗𝑗∈𝐽2.(3.7)

Step 6. Determine the separation measures between the alternative.
Separation between alternatives on IFS, distance measures proposed by Atanassov [63], Szmidt and Kacprzyk [64], and Grzegorzewski [65] including the generalizations of Hamming distance, Euclidean distance and their normalized distance measures can be used. After selecting the distance measure, the separation measures, 𝑆∗𝑖 and 𝑆−𝑖, of each alternative from IFPIS and IFNIS, are calculated: 𝑆∗𝑖=12𝑛𝑗=1||𝜇𝑖𝑗−𝜇𝑗∗||+||𝑣𝑖𝑗−𝑣𝑗∗||+||𝜋𝑖𝑗−𝜋𝑗∗||,𝑆−𝑖=12𝑛𝑗=1||𝜇𝑖𝑗−𝜇𝑗−||+||𝑣𝑖𝑗−𝑣𝑗−||+||𝜋𝑖𝑗−𝜋𝑗−||.(3.8)

Step 7. Determine the final ranking.
In the final step, the relative closeness coefficient of an alternative 𝐴𝑖 with respect to the IFPIS 𝐴∗ is defined as follows: 𝐶∗𝑖=𝑆−𝑖𝑆∗𝑖+𝑆−𝑖,where0≤𝐶∗𝑖≤1.(3.9) The alternatives were ranked according to descending order of 𝐶∗𝑖’s score.

3.2. DIFWA

The DIFWA method, proposed by Xu and Yager [33], is a suitable way to deal with problem in IFE. The procedure for DIFWA method has been given as follows.

Step 1. Utilize the DIFWA operator 𝑟𝑖𝑗=DIFWA𝜆(𝑡)𝑟𝑖𝑗𝑡1,𝑟𝑖𝑗𝑡2,…,𝑟𝑖𝑗𝑡𝑝=1−𝑝𝑘=11−𝜇𝑟𝑖𝑗(𝑡𝑘)𝜆(𝑡𝑘),𝑝𝑘=1𝑣𝜆(𝑡𝑘)𝑟𝑖𝑗𝑡𝑘,𝑝𝑘=11−𝜇𝑟𝑖𝑗(𝑡𝑘)𝜆(𝑡𝑘)−𝑝𝑘=1𝑣𝜆(𝑡𝑘)𝑟𝑖𝑗𝑡𝑘.(3.10) to aggregate all the intuitionistic fuzzy matrix 𝑅(𝑡𝑘)=(𝑟𝑖𝑗(𝑡𝑘))𝑚×𝑛(𝑘=1,2,…,𝑝) into a complex IFDM: 𝑟𝑅=𝑖𝑗𝑚×𝑛,where𝑟𝑖𝑗=𝜇𝑖𝑗,𝑣𝑖𝑗,𝜋𝑖𝑗,𝜇𝑖𝑗=1−𝑝𝑘=11−𝜇𝑟𝑖𝑗(𝑡𝑘)𝜆(𝑡𝑘),𝑣𝑖𝑗=𝑝𝑘=1𝑣𝜆(𝑡𝑘)𝑟𝑖𝑗𝑡𝑘,𝜋𝑖𝑗=𝑝𝑘=11−𝜇𝑟𝑖𝑗(𝑡𝑘)𝜆(𝑡𝑘)−𝑝𝑘=1𝑣𝜆(𝑡𝑘)𝑟𝑖𝑗𝑡𝑘,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚.(3.11)

Step 2. Define 𝛼+=(𝛼+1,𝛼+2,…,𝛼+𝑚)𝑇and𝛼−=(𝛼−1,𝛼−2,…,𝛼−𝑚)𝑇 as the IFPIS and the IFNIS, respectively, where 𝛼+=(1,0,0)(𝑖=1,2,…,𝑚) are the 𝑚 largest IFNs and 𝛼−=(0,1,0)(𝑖=1,2,…,𝑚) are the m smallest IFNs. Furthermore, for convenience of depiction, we denote the alternative 𝑥𝑖(𝑖=1,2,…,𝑛) by 𝑥𝑖=(𝑟𝑖1,𝑟𝑖2,…,𝑟𝑖𝑚)𝑇,𝑖=1,2,…,𝑛.

Step 3. Calculate the distance between the alternative 𝑥𝑖 and the IFIS and the distance between the largest native 𝑥𝑖 and the IFNIS, respectively: 𝑑𝑥𝑖,𝛼+=𝑚𝑗=1𝑤𝑗𝑑𝑟𝑖𝑗,𝛼+𝑗=12𝑚𝑗=1𝑤𝑗||𝜇𝑖𝑗||+||𝑣−1𝑖𝑗||+||𝜋−0𝑖𝑗||=1−02𝑚𝑗=1𝑤𝑗1−𝜇𝑖𝑗+𝑣𝑖𝑗+𝜋𝑖𝑗=12𝑚𝑗=1𝑤𝑗1−𝜇𝑖𝑗+𝑣𝑖𝑗+1−𝜇𝑖𝑗−𝑣𝑖𝑗=𝑚𝑗=1𝑤𝑗1−𝜇𝑖𝑗,𝑑𝑥𝑖,𝛼−=𝑚𝑗=1𝑤𝑗𝑑𝑟𝑖𝑗,𝛼−𝑗=12𝑚𝑗=1𝑤𝑗||𝜇𝑖𝑗||+||𝑣−0𝑖𝑗||+||𝜋−1𝑖𝑗||=1−02𝑚𝑗=1𝑤𝑗1−𝜇𝑖𝑗−𝑣𝑖𝑗+𝜋𝑖𝑗=12𝑚𝑗=1𝑤𝑗1+𝜇𝑖𝑗−𝑣𝑖𝑗+1−𝜇𝑖𝑗−𝑣𝑖𝑗=𝑚𝑗=1𝑤𝑗1−𝑣𝑖𝑗,(3.12) where 𝑟𝑖𝑗=(𝜇𝑖𝑗,𝑣𝑖𝑗,𝜋𝑖𝑗),𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚.

Step 4. Calculate the closeness coefficient of each alternative: 𝑐𝑥𝑖=𝑑𝑥𝑖,𝛼−𝑑𝑥𝑖,𝛼+𝑥+𝑑𝑖,𝛼−,𝑖=1,2,…,𝑛.(3.13) Since 𝑑𝑥𝑖,𝛼+𝑥+𝑑𝑖,𝛼−=𝑚𝑗=1𝑤𝑗1−𝜇𝑖𝑗+𝑚𝑗=1𝑤𝑗1−𝑣𝑖𝑗=𝑚𝑗=1𝑤𝑗2−𝜇𝑖𝑗−𝑣𝑖𝑗=𝑚𝑗=1𝑤𝑗1+𝜋𝑖𝑗,(3.14) then (3.13) can be rewritten as 𝑐𝑥𝑖=∑𝑚𝑗=1𝑤𝑗1−𝑣𝑖𝑗∑𝑚𝑗=1𝑤𝑗1−𝜋𝑖𝑗,𝑖=1,2,…,𝑛.(3.15)

Step 5. Rank all the alternatives 𝑥𝑖(1,2,…,𝑛) according to the closeness coefficients 𝑐(𝑥𝑖)(1,2,…,𝑛), the greater the value 𝑐(𝑥𝑖), the better the alternative 𝑥𝑖.

4. Case Study

In this section, we will describe how an IFT method was applied via an example of selection of the most appropriate projects. Criteria to be considered in the selection of projects are determined by the expert team from a construction group. In our study, we employ six evaluation criteria. The attributes which are considered here in assessment of 𝑃𝑖(𝑖=1,2,…,6) are (1)𝐶1 is benefit and (2)𝐶2,…,𝐶6 are cost. The committee evaluates the performance of projects 𝑃𝑖(𝑖=1,2,…,6) according to the attributes 𝐶𝑗(𝑗=1,2,…,6), respectively. Criteria are mainly considered as follows(i)net present value (𝐶1),(ii)quality (𝐶2),(iii)duration (𝐶3),(iv)contractor’s rank (𝐶4),(v)contractor’s technology (𝐶5),(vi)contractor’s economic status (𝐶6).

Therefore, one cost criterion, 𝐶1 and five benefit criteria, 𝐶2,…,𝐶6 are considered. After preliminary screening, six projects 𝑃1,𝑃2,𝑃3,𝑃4,𝑃5, and 𝑃6 remain for further evaluation. A team of four DMs such as DM1, DM2, DM3, and DM4 has been formed to select the most suitable project.

Now utilizing the proposed IFT to prioritize these construction projects, the following steps were taken.

Degree of the DMs on group decision, shown in Table 1, and linguistic terms used for the ratings of the DMs and criteria, as Table 2, respectively.

Table 1 
Linguistic term for rating DMs.
Table 2 
The importance of DMs and their weights.

Construct the aggregated IFDM based on the opinions of DMs and the linguistic terms shown in Table 3.

Table 3 
Linguistic terms for rating the alternatives.

The ratings given by the DMs to six projects were shown in Table 4.

Table 4 
The ratings of the projects.

The aggregated IFDM based on aggregation of DMs’ opinions was constructed as follows: 𝐶1𝐶2𝐶3𝐴𝑅=1𝐴2𝐴3𝐴4𝐴5𝐴6⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦𝐶(0.80,0.08,0.12)(0.69,0.20,0.11)(0.76,0.12,0.12)(0.68,0.20,0.12)(0.78,0.11,0.11)(0.74,0.13,0.13)(0.82,0.07,0.11)(0.79,0.10,0.11)(0.79,0.10,0.11)(0.83,0.16,0.1)(0.75,0.14,0.11)(0.70,0.19,0.11)(0.55,0.38,0.07)(0.42,0.52,0.06)(0.64,0.40,0.06)(0.75,0.13,0.12)(0.69,0.19,0.12)(0.75,0.13,0.12)4𝐶5𝐶6×𝐴1𝐴2𝐴3𝐴4𝐴5𝐴6⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(0.80,0.09,0.11)(0.78,0.11,0.11)(0.69,0.20,0.11)(0.78,0.11,0.11)(0.69,0.21,0.10)(0.75,0.13,0.12)(0.84,0.05,0.11)(0.84,0.05,0.11)(0.84,0.05,0.11)(0.81,0.08,0.11)(0.82,0.07,0.11)(0.85,0.05,0.10)(0.55,0.33,0.12)(0.54,0.33,0.13)(0.40,0.54,0.06)(0.75,0.13,0.12)(0.85,0.05,0.10)(0.78,0.11,0.11)(4.1)

The linguistic terms shown in Table 5 were used to rate each criterion. The importance of the criteria represented as linguistic terms was shown in Table 6.

Table 5 
The linguistic terms for the importance of the criteria.
Table 6 
The importance weight of the criteria.

The opinions of DMs on criteria were aggregated to determine the weight of each criterion: 𝑊{𝑋1,𝑋2,𝑋3,𝑋4,𝑋5,𝑋6}=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(0,71,0.19,0.10)(0,90,0.00,0.10)(0,65,0.27,0.80)(0,78,0.11,0.11)(0,80,0.10,0.10)(0,67,0.24,0.9)𝑇.(4.2) After the weights of the criteria and the rating of the projects were determined, the aggregated weighted IFDM was constructed as follows: 𝐶1𝐶2𝐶3𝑅=𝐴1𝐴2𝐴3𝐴4𝐴5𝐴6⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦𝐶(0.57,0.26,0.18)(0.62,0.20,0.18)(0.49,0.36,0.19)(0.48,0.35,0.17)(0.70,0.11,0.19)(0.48,0.37,0.15)(0.58,0.25,0.17)(0.70,0.10,0.19)(0.51,0.34,0.14)(0.59,0.32,0.09)(0.70,0.14,0.19)(0.45,0.41,0.14)(0.39,0.50,0.11)(0.38,0.52,0.10)(0.42,0.56,0.02)(0.53,0.30,0.17)(0.62,0.19,0.19)(0.49,0.36,0.15)4𝐶5𝐶6×𝐴1𝐴2𝐴3𝐴4𝐴5𝐴6⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(0.62,0.19,0.19)(0.62,0.20,0.18)(0.46,0.39,0.15)(0.61,0.21,0.18)(0.55,0.29,0.16)(0.50,0.34,0.16)(0.66,0.16,0.19)(0.67,0.15,0.18)(0.56,0.28,0.16)(0.63,0.18,0.19)(0.57,0.16,0.18)(0.57,0.28,0.15)(0.43,0.40,0.17)(0.43,0.40,0.17)(0.27,0.65,0.08)(0.59,0.23,0.19)(0.70,0.14,0.18)(0.52,0.33,0.15)(4.3)

The net present value is cost criteria 𝑗1={𝑋1}, and quality, duration, contractor’s rank, contractor’s technology, and contractor’s economic status are benefit criteria 𝑗1={𝑋2,𝑋3,𝑋4,𝑋5}.

Then IFPIS and IFNIS were provided as follows: 𝐴∗𝐴={(0.59,0.25,0.16),(0.71,0.10,0.19),(0.51,0.34,0.15),(0.66,0.15,0.18),(0.68,0.14,0.18),(0.57,0.28,0.15)},−={(0.39,0.5,0.11),(0.38,0.5,0.12),(0.42,0.56,0.02),(0.43,0.4,0.17),(0.43,0.4,0.17),(0.27,0.65,0.08)}.(4.4) Negative and positive separation measures based on normalized Euclidean distance for each project and the relative closeness coefficient were calculated in Table 7.

Table 7 
Separation measures and the relative closeness coefficient of each project.

Six projects were ranked according to descending order of 𝐶∗𝑖’s. The result score is always the bigger the better. As visible in Table 6, project 3 has the largest score, and project 5 has the smallest score of the six projects which is ranked in the last pace. The projects were ranked as 𝑃3 >𝑃4 >𝑃6 >𝑃1> 𝑃2 > 𝑃5. Project 3 was selected as appropriate project among the alternatives.

In the second part, we utilize the proposed DIFWA to prioritize these construction projects, and the following steps were taken.

First, utilize the DIFWA to aggregate all the IFDM 𝑅(𝑡𝑘) into a complex IFDM 𝑅: 𝐶1𝐶2𝐶3𝑅=𝐴1𝐴2𝐴3𝐴4𝐴5𝐴6⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦𝐶(0.57,0.26,0.18)(0.62,0.20,0.18)(0.49,0.36,0.19)(0.48,0.35,0.17)(0.70,0.11,0.19)(0.48,0.37,0.15)(0.58,0.25,0.17)(0.70,0.10,0.19)(0.51,0.34,0.14)(0.59,0.32,0.09)(0.70,0.14,0.19)(0.45,0.41,0.14)(0.39,0.50,0.11)(0.38,0.52,0.10)(0.42,0.56,0.02)(0.53,0.30,0.17)(0.62,0.19,0.19)(0.49,0.36,0.15)4𝐶5𝐶6×𝐴1𝐴2𝐴3𝐴4𝐴5𝐴6⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(0.62,0.19,0.19)(0.62,0.20,0.18)(0.46,0.39,0.15)(0.61,0.21,0.18)(0.55,0.29,0.16)(0.50,0.34,0.16)(0.66,0.16,0.19)(0.67,0.15,0.18)(0.56,0.28,0.16)(0.63,0.18,0.19)(0.57,0.16,0.18)(0.57,0.28,0.15)(0.43,0.40,0.17)(0.43,0.40,0.17)(0.27,0.65,0.08)(0.59,0.23,0.19)(0.70,0.14,0.18)(0.52,0.33,0.15)(4.5) Denote the IFIS, IFNIS, and the alternatives by 𝛼+=((1,0,0),(1,0,0),(1,0,0))𝑇,𝛼−=((0,1,0),(0,1,0),(0,1,0))𝑇,(4.6) and calculate the closeness coefficient of each alternative: 𝐶𝑃1𝑃=0.622,𝐶2𝑃=0.618,𝐶3𝐶𝑃=0.671,4𝑃=0.650,𝐶5𝑃=0.447,𝐶6=0.633.(4.7) Rank all the projects according to the closeness coefficients.

The projects were ranked as 𝐶(𝑃3)>𝐶(𝑃4)>𝐶(𝑃6)>𝐶(𝑃1)>𝐶(𝑃2)>𝐶(𝑃5). The greater value of 𝐶(𝑋𝑖), the better alternative; thus the best alternative is also project 3.

5. Conclusion

The IFT and DIFWA have been emphasized in this paper which occurs in construction projects evaluation. In the evaluation process, the ratings of each project, given with intuitionistic fuzzy information, were represented as IFNs. The IFWA operator was used to aggregate the rating of DM. In project selection problem the project’s information and performance are usually uncertain. Therefore, the decision makers are unable to express their judgment on the project with crisp value, and the evaluations are very often expressed in linguistic terms. IFT and DIFWA are suitable ways to deal with MCDM because it contains a vague perception of DMs’ opinions. An actual life example in construction sector was illustrated, and finally the result is as follows Among 6 construction projects with respect to 6 criteria, after using these two methods, the best one is project 3 and project 4, project 6, project 1, project 2, project 5 will follow it, respectively. The presented approach not only validates the methods, as it was originally defined in Boran and Xu in a new application field that was the evaluation of construction projects, but also considers a more extensive list of benefit and cost-oriented criteria, suitable for construction project selection. Finally, the IFT and DIFWA methods have capability to deal with similar types of the same situations with uncertainty in MCDM problems such as ERP software selection and many other areas.

Acknowledgment

The author is very grateful to the anonymous referees for their constructive comments and suggestions that led to an improved version of this paper.