# An Integrated Decision Support Model Based on BWM and Fuzzy-VIKOR Techniques for Contractor Selection in Construction Projects

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Best-Worst Method

#### 2.2. VIKOR Method

#### 2.3. Fuzzy Set Theory

- Lack of sufficient information about competencies of contractors so that experts are obliged in some cases to surmise about them.
- Ambiguousness of the decision-maker about his understanding of the selection process.
- Complexity, lack of clarity, or incompleteness in project’s specifications which makes it hard for experts to decide on the sufficiency of contractors’ skills and capabilities for construction of the project.
- Uncertainty in the project’s information: the complete information is utilized by experts or decision-makers whereas incomplete information has yet to be processed.
- Indeterminate conditions of companies: strengths and weaknesses of companies, alongside with the lack of credible information about their abilities, makes it hard for the decision-makers to state their views quantitatively with respect to project criteria.

**Definition**

**1**.

**Definition**

**2**.

## 3. Methodology

**Phase 1**- Introduction of the problem: in this phase, alternatives and the criteria for contractor assessment are determined.
**Phase 2**- Weighting: in this phase, the weight or influence of the members of the decision-making group and the weights of assessment criteria are determined.
**Phase 3**- Ranking: in this phase, the performances of alternatives with respect to each criterion are determined and then, final ranks of alternatives are specified.

#### 3.1. Best-Worst Method (BWM)

**Step 1.**Determine the set of decision criteria.

**Step 2.**Determine the best and the worst criteria.

**Step 3.**Determine preferences of the best criterion over all other criteria, using a number between 1 and 9, and establish the best-to-others vector. The elements of this vector denoted by a

_{Bj}, represents the importance of the best criterion over criterion j.

**Step 4.**Determine preferences of all other criteria over the worst criterion using an integer of 1 to 9 and establish the others-to-worst vector, elements of which are denoted by a

_{jW}that represents the importance of criterion j over the worst criterion.

**Step 5.**Find optimized weights (w

^{*}

_{1}, w

^{*}

_{2}, …, w

^{*}

_{n}) and ${\xi}^{{L}^{*}}$ by solving the following linear model:

**Definition**

**3**.

_{j}[85]. Allowable thresholds of input-based consistency ratio are obtained from Table 3 of [85] with respect to the number of criteria and the scale used in the BWM (Table 3).

#### Number of Decision-Makers

#### 3.2. Fuzzy-VIKOR Method

_{i}with respect to criterion C

_{j}. Then, the steps of this method can be summarized as follows:

**Step 1**. Determine the best and the worst fuzzy values for each criterion, and denote them respectively by ${\tilde{f}}_{j}^{+}=\left({l}_{j}^{+},{m}_{j}^{+},{u}_{j}^{+}\right)$ and ${\tilde{f}}_{j}^{-}=\left({l}_{j}^{-},{m}_{j}^{-},{u}_{j}^{-}\right)$ for all j = 1,2,…,n. If the criterion is positive or benefit one, then,

**Step 2**. Calculate fuzzy normalized differences, ${\tilde{d}}_{ij},i=1,\dots ,m,j=1,\dots ,n$ as:

**Step 3**. Calculate ${\tilde{S}}_{i}=\left({S}_{i}^{l},{S}_{i}^{m},{S}_{i}^{u}\right)$ and ${\tilde{R}}_{i}=\left({R}_{i}^{l},{R}_{i}^{m},{R}_{i}^{u}\right)$ for all alternative (I = 1, …, m) as:

**Step 4**. Calculate ${\tilde{Q}}_{i}=\left({Q}_{i}^{l},{Q}_{i}^{m},{Q}_{i}^{u}\right)$ values for all alternatives as:

**Step 5**. Defuzzify the values, ${\tilde{S}}_{i}$, ${\tilde{R}}_{i}$, and ${\tilde{Q}}_{i}$ by the rule of second weighted mean according to Equation (9) and obtain crisp values, S, R, and Q.

**Step 6**. Rank the alternative by sorting out crisp values S, R, and Q and prepare three lists of ranking.

**Step 7**. Determine an alternative as a compromise solution with the best value (minimum) of Q provided that these two conditions are satisfied:

**Condition**

**1**.

^{(m)}are respectively the alternatives with the first, second and last rankings in the Q-list. DQ is 0.25 for values of m not more than 4 [87,88].

**Condition**

**2**.

- Alternative A′ and A″ if only Condition 2 is not satisfied.
- Alternatives A′, A″, …, A
^{(H)}, if Condition 1 is not satisfied, where A^{(H)}is the last alternative with which Condition 1 is not satisfied i.e., Q(A^{(H)}) − Q(A′) < DQ for maximum H.

## 4. Illustrative Example

#### 4.1. Criteria

#### 4.2. Criteria Weights

#### 4.3. Decision Matrix

#### 4.4. Ranking the Alternatives

**Condition**

**1**.

**Condition**

**2**.

#### 4.5. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Criteria | sc01 | sc02 | sc03 | sc04 | sc05 | sc06 | sc07 | sc08 | sc09 | sc10 | sc11 | sc12 | sc13 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Criteria Weights | 0.1184 | 0.0635 | 0.0296 | 0.0381 | 0.1847 | 0.0291 | 0.0581 | 0.0799 | 0.0574 | 0.0818 | 0.1156 | 0.1020 | 0.0417 | |

Alternatives | A | (60,60,60) | (84.55,84.55,84.55) | (20,20,20) | (0.833,0.967,1) | (70,70,70) | (0.7,0.867,0.967) | (0.7,0.9,1) | (0.7,0.9,1) | (0.5,0.7,0.9) | (0.9,1,1) | (782000000000,782000000000,782000,000,000) | (100,100,100) | (0,0,0) |

B | (60,60,60) | (36.25,36.25,36.25) | (25,25,25) | (0.767,0.933,1) | (50,50,50) | (0.9,1,1) | (0.5,0.7,0.9) | (0.5,0.7,0.9) | (0.3,0.5,0.7) | (0.633,0.833,0.967) | (1077195211299,1077195211299,1077195211299) | (72,72,72) | (100,100,100) | |

C | (60,60,60) | (70.45,70.45,70.45) | (10,10,10) | (0.367,0.567,0.767) | (60,60,60) | (0.333,0.5,0.667) | (0.3,0.5,0.7) | (0.5,0.7,0.9) | (0.5,0.7,0.9) | (0.4,0.567,0.733) | (717135210855,717135210855,717135210855) | (45,45,45) | (100,100,100) | |

D | (48,48,48) | (78.05,78.05,78.05) | (20,20,20) | (0.433,0.633,0.833) | (65,65,65) | (0.367,0.567,0.767) | (0.5,0.7,0.9) | (0.3,0.5,0.7) | (0.5,0.7,0.9) | (0.3,0.5,0.7) | (823639532214,823639532214,823639532214) | (100,100,100) | (0,0,0) | |

f^{~+} | (60,60,60) | (84.55,84.55,84.55) | (25,25,25) | (0.833,0.967,1) | (70,70,70) | (0.9,1,1) | (0.7,0.9,1) | (0.7,0.9,1) | (0.5,0.7,0.9) | (0.9,1,1) | (717135210855,717135210855,717135210855) | (100,100,100) | (100,100,100) | |

f^{~-} | (48,48,48) | (36.25,36.25,36.25) | (10,10,10) | (0.367,0.567,0.767) | (50,50,50) | (0.333,0.5,0.667) | (0.3,0.5,0.7) | (0.3,0.5,0.7) | (0.3,0.5,0.7) | (0.3,0.5,0.7) | (1077195211299,1077195211299,1077195211299) | (45,45,45) | (0,0,0) | |

d^{~}_{ij} = (f^{~+} − x^{~}_{ij})/(u^{+}_{j} − l^{-}_{j}) | A | (0,0,0) | (0,0,0) | (0.333,0.333,0.333) | (−0.264,0,0.264) | (0,0,0) | (−0.1,0.199,0.45) | (−0.429,0,0.429) | (−0.429,0,0.429) | (−0.667,0,0.667) | (−0.143,0,0.143) | (0.18,0.18,0.18) | (0,0,0) | (1,1,1) |

B | (0,0,0) | (1,1,1) | (0,0,0) | (−0.264,0.054,0.368) | (1,1,1) | (−0.15,0,0.15) | (−0.286,0.286,0.714) | (−0.286,0.286,0.714) | (−0.333,0.333,1) | (−0.096,0.239,0.524) | (1,1,1) | (0.509,0.509,0.509) | (0,0,0) | |

C | (0,0,0) | (0.292,0.292,0.292) | (1,1,1) | (0.104,0.632,1) | (0.5,0.5,0.5) | (0.349,0.75,1) | (0,0.571,1) | (−0.286,0.286,0.714) | (−0.667,0,0.667) | (0.239,0.619,0.857) | (0,0,0) | (1,1,1) | (0,0,0) | |

D | (1,1,1) | (0.135,0.135,0.135) | (0.333,0.333,0.333) | (0,0.528,0.896) | (0.25,0.25,0.25) | (0.199,0.649,0.949) | (−0.286,0.286,0.714) | (0,0.571,1) | (−0.667,0,0.667) | (0.286,0.714,1) | (0.296,0.296,0.296) | (0,0,0) | (1,1,1) | |

w_{j}.d^{~}_{ij} | A | (0,0,0) | (0,0,0) | (0.01,0.01,0.01) | (−0.01,0,0.01) | (0,0,0) | (−0.003,0.006,0.013) | (−0.025,0,0.025) | (−0.034,0,0.034) | (−0.038,0,0.038) | (−0.012,0,0.012) | (0.021,0.021,0.021) | (0,0,0) | (0.042,0.042,0.042) |

B | (0,0,0) | (0.064,0.064,0.064) | (0,0,0) | (−0.01,0.002,0.014) | (0.185,0.185,0.185) | (−0.004,0,0.004) | (−0.017,0.017,0.041) | (−0.023,0.023,0.057) | (−0.019,0.019,0.057) | (−0.008,0.02,0.043) | (0.116,0.116,0.116) | (0.052,0.052,0.052) | (0,0,0) | |

C | (0,0,0) | (0.019,0.019,0.019) | (0.03,0.03,0.03) | (0.004,0.024,0.038) | (0.092,0.092,0.092) | (0.01,0.022,0.029) | (0,0.033,0.058) | (−0.023,0.023,0.057) | (−0.038,0,0.038) | (0.02,0.051,0.07) | (0,0,0) | (0.102,0.102,0.102) | (0,0,0) | |

D | (0.118,0.118,0.118) | (0.009,0.009,0.009) | (0.01,0.01,0.01) | (0,0.02,0.034) | (0.046,0.046,0.046) | (0.006,0.019,0.028) | (−0.017,0.017,0.041) | (0,0.046,0.08) | (−0.038,0,0.038) | (0.023,0.058,0.082) | (0.034,0.034,0.034) | (0,0,0) | (0.042,0.042,0.042) |

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**Figure 4.**Solving linear BWM model by Excel Solver to obtain the main criteria weights for decision-maker 01. * The Ksi shows to what extent the results are reliable, the closer the Ksi to zero the better.

Reference. | Year | Values Type | prequalification | CS | Sub-CS | CPE ^{2} | Criteria Importance Determination Method | |||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Crisp | Fuzzy | Semantic | IFS ^{1} | AHP | WAM ^{3} | Delphi | ANP | ANN ^{4} | CBR ^{5} | SC ^{6} | QMIV ^{7} | Entropy | TOPSIS | FLT ^{8} | FST ^{9} | GT&M ^{10} | G-AHP ^{11} | QFD ^{12} | EJM ^{13} | Client ^{14} | IE ^{15} | SWARA ^{16} | PR ^{17} | BWM ^{18} | ||||||

[52] | 1990 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[32] | 1994 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

[33] | 1996 | ✓ | ✓ | |||||||||||||||||||||||||||

[53] | 1997 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

[26] | 1999 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

[15] | 2000 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[27] | 2000 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

[36] | 2001 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

[28] | 2001 | ✓ | ✓ | |||||||||||||||||||||||||||

[16] | 2001 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[54] | 2002 | ✓ | ✓ | ✓ | ✓ | ✓ | ||||||||||||||||||||||||

[37] | 2002 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[6] | 2002 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[38] | 2003 | ✓ | ✓ | |||||||||||||||||||||||||||

[17] | 2004 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

[25] | 2004 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[40] | 2004 | ✓ | ✓ | |||||||||||||||||||||||||||

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[55] | 2005 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

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[43] | 2005 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

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[3] | 2006 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[45] | 2007 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[29] | 2007 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[20] | 2008 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

[47] | 2008 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[56] | 2008 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[46] | 2008 | ✓ | ✓ | |||||||||||||||||||||||||||

[57] | 2009 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

[14] | 2009 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

[58] | 2009 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

[48] | 2009 | ✓ | ✓ | |||||||||||||||||||||||||||

[49] | 2010 | ✓ | ✓ | |||||||||||||||||||||||||||

[22] | 2012 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

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[21] | 2012 | ✓ | ✓ | ✓ | ✓ | ✓ | ||||||||||||||||||||||||

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[4] | 2013 | ✓ | ✓ | |||||||||||||||||||||||||||

[62] | 2013 | ✓ | ✓ | |||||||||||||||||||||||||||

[18] | 2015 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[23] | 2016 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[50] | 2016 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[63] | 2016 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

[19] | 2017 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[64] | 2017 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

[2] | 2017 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

[65] | 2017 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

[66] | 2018 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[51] | 2018 | ✓ | ✓ | ✓ | ✓ | ✓ | ||||||||||||||||||||||||

[24] | 2018 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ||||||||||||||||||||||

[67] | 2018 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

[8] | 2019 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||

[9] | 2019 | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||||

[68] | 2020 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||||||

This paper | ✓ | ✓ | ✓ | ✓ | ✓ |

^{1}Intuitionistic fuzzy sets,

^{2}contractor performance evaluation,

^{3}weighted average method,

^{4}artificial neural network,

^{5}case-based reasoning,

^{6}survey conducting,

^{7}questionnaire and mean impact value method,

^{8}fuzzy linguistic terms,

^{9}fuzzy set theory,

^{10}graph theory and matrix method,

^{11}group-AHP,

^{12}quality function deployment,

^{13}expert’s judgment method,

^{14}determining the importance of time and cost by the client,

^{15}interview with experts,

^{16}stepwise weights assessment ratio analysis,

^{17}preference relation,

^{18}best-worst method.

Reference. | Year | Ranking Method | ||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

AHP | BVD ^{19} | TOPSIS | VIKOR | PROMETHEE | MAUT ^{20} | ANP | ANN | CBR | CA ^{21} | COPRAS ^{22} | MAA ^{23} | HLR ^{24} | MCPM ^{25} | DEA ^{26} | DWA ^{27} | DF-EDAS ^{28} | ELECTRE ^{29} | ER ^{30} | OFASPT ^{31} | WBS ^{32} | CSM ^{33} | T 2FSM ^{34} | FST | FNN ^{35} | GNN ^{36} | GT&M | G-AHP | QFD | PERT ^{37} | GRA ^{38} | LR ^{39} | MDA ^{40} | MOORA ^{41} | QBPR ^{42} | ZDIM ^{43} | SVM ^{44} | WASPAS-SVNS ^{45} | BFPM ^{46} | ||

[52] | 1990 | ✓ | ||||||||||||||||||||||||||||||||||||||

[32] | 1994 | ✓ | ||||||||||||||||||||||||||||||||||||||

[33] | 1996 | ✓ | ||||||||||||||||||||||||||||||||||||||

[53] | 1997 | ✓ | ||||||||||||||||||||||||||||||||||||||

[26] | 1999 | ✓ | ||||||||||||||||||||||||||||||||||||||

[15] | 2000 | ✓ | ||||||||||||||||||||||||||||||||||||||

[27] | 2000 | ✓ | ||||||||||||||||||||||||||||||||||||||

[36] | 2001 | ✓ | ||||||||||||||||||||||||||||||||||||||

[28] | 2001 | ✓ | ||||||||||||||||||||||||||||||||||||||

[16] | 2001 | ✓ | ||||||||||||||||||||||||||||||||||||||

[54] | 2002 | ✓ | ||||||||||||||||||||||||||||||||||||||

[37] | 2002 | ✓ | ||||||||||||||||||||||||||||||||||||||

[6] | 2002 | ✓ | ||||||||||||||||||||||||||||||||||||||

[38] | 2003 | ✓ | ✓ | |||||||||||||||||||||||||||||||||||||

[17] | 2004 | ✓ | ||||||||||||||||||||||||||||||||||||||

[25] | 2004 | ✓ | ||||||||||||||||||||||||||||||||||||||

[40] | 2004 | ✓ | ||||||||||||||||||||||||||||||||||||||

[42] | 2005 | ✓ | ||||||||||||||||||||||||||||||||||||||

[55] | 2005 | ✓ | ||||||||||||||||||||||||||||||||||||||

[41] | 2005 | ✓ | ||||||||||||||||||||||||||||||||||||||

[43] | 2005 | ✓ | ||||||||||||||||||||||||||||||||||||||

[44] | 2006 | ✓ | ||||||||||||||||||||||||||||||||||||||

[3] | 2006 | |||||||||||||||||||||||||||||||||||||||

[45] | 2007 | ✓ | ||||||||||||||||||||||||||||||||||||||

[29] | 2007 | ✓ | ||||||||||||||||||||||||||||||||||||||

[20] | 2008 | ✓ | ||||||||||||||||||||||||||||||||||||||

[47] | 2008 | ✓ | ||||||||||||||||||||||||||||||||||||||

[56] | 2008 | ✓ | ||||||||||||||||||||||||||||||||||||||

[46] | 2008 | ✓ | ||||||||||||||||||||||||||||||||||||||

[57] | 2009 | ✓ | ||||||||||||||||||||||||||||||||||||||

[14] | 2009 | ✓ | ||||||||||||||||||||||||||||||||||||||

[58] | 2009 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||||||||||||

[48] | 2009 | ✓ | ✓ | |||||||||||||||||||||||||||||||||||||

[49] | 2010 | ✓ | ✓ | |||||||||||||||||||||||||||||||||||||

[22] | 2012 | ✓ | ✓ | |||||||||||||||||||||||||||||||||||||

[59] | 2012 | ✓ | ||||||||||||||||||||||||||||||||||||||

[21] | 2012 | ✓ | ✓ | |||||||||||||||||||||||||||||||||||||

[60] | 2012 | ✓ | ||||||||||||||||||||||||||||||||||||||

[61] | 2013 | ✓ | ||||||||||||||||||||||||||||||||||||||

[4] | 2013 | ✓ | ||||||||||||||||||||||||||||||||||||||

[62] | 2013 | ✓ | ✓ | |||||||||||||||||||||||||||||||||||||

[18] | 2015 | ✓ | ||||||||||||||||||||||||||||||||||||||

[23] | 2016 | ✓ | ||||||||||||||||||||||||||||||||||||||

[50] | 2016 | ✓ | ||||||||||||||||||||||||||||||||||||||

[63] | 2016 | ✓ | ||||||||||||||||||||||||||||||||||||||

[19] | 2017 | ✓ | ||||||||||||||||||||||||||||||||||||||

[64] | 2017 | ✓ | ✓ | |||||||||||||||||||||||||||||||||||||

[2] | 2017 | |||||||||||||||||||||||||||||||||||||||

[65] | 2017 | ✓ | ||||||||||||||||||||||||||||||||||||||

[66] | 2018 | ✓ | ||||||||||||||||||||||||||||||||||||||

[51] | 2018 | ✓ | ||||||||||||||||||||||||||||||||||||||

[24] | 2018 | ✓ | ✓ | |||||||||||||||||||||||||||||||||||||

[67] | 2018 | ✓ | ✓ | ✓ | ||||||||||||||||||||||||||||||||||||

[8] | 2019 | ✓ | ||||||||||||||||||||||||||||||||||||||

[9] | 2019 | ✓ | ||||||||||||||||||||||||||||||||||||||

[68] | 2020 | ✓ | ||||||||||||||||||||||||||||||||||||||

This paper | ✓ | ✓ |

^{19}Best value determination,

^{20}Multiattribute utility theory,

^{21}cluster analysis,

^{22}complex proportional assessment,

^{23}multiattribute analysis,

^{24}Hodges-Lehmann rule,

^{25}multicriteria prospect model,

^{26}data envelopment analysis,

^{27}dimensional weighting aggregation,

^{28}dynamic fuzzy evaluation-based on distance from average solution,

^{29}elimination et choix traduisant la realité,

^{30}evidential reasoning approach,

^{31}ordering feasible alternatives of solutions in terms of preferability technique,

^{32}web-based system,

^{33}compromise solution method,

^{34}type-2 fuzzy set model,

^{35}fuzzy neural network,

^{36}genetic-neural network,

^{37}program evaluation and review technique,

^{38}grey relational analysis,

^{39}logistic regression,

^{40}multivariate discriminant analysis,

^{41}multiobjective optimization on the basis of ratio analysis,

^{42}quality-based performance rating,

^{43}Zeleny’s displaced ideal model,

^{44}support vector machine,

^{45}weight aggregated sum product assessment-single valued neutrosophic Set,

^{46}Bayesian fuzzy prospect model.

**Table 3.**Allowable thresholds for input-based consistency ratio [85].

Scale | Criteria Number | ||||||
---|---|---|---|---|---|---|---|

3 | 4 | 5 | 6 | 7 | 8 | 9 | |

3 | 0.1667 | 0.1667 | 0.1667 | 0.1667 | 0.1667 | 0.1667 | 0.1667 |

4 | 0.1121 | 0.1529 | 0.1898 | 0.2206 | 0.2527 | 0.2577 | 0.2683 |

5 | 0.1354 | 0.1994 | 0.2306 | 0.2546 | 0.2716 | 0.2844 | 0.2960 |

6 | 0.1330 | 0.1990 | 0.2643 | 0.3044 | 0.3144 | 0.3221 | 0.3262 |

7 | 0.1294 | 0.2457 | 0.2819 | 0.3029 | 0.3144 | 0.3251 | 0.3403 |

8 | 0.1309 | 0.2521 | 0.2958 | 0.3154 | 0.3408 | 0.3620 | 0.3657 |

9 | 0.1359 | 0.2681 | 0.3062 | 0.3337 | 0.3517 | 0.3620 | 0.3662 |

Alternative | Bid Price (Rials) |
---|---|

A | 782,000,000,000 |

B | 1,077,195,211,299 |

C | 717,135,210,855 |

D | 823,639,532,214 |

Decision-Maker | Preference of the Best Criterion over Others | |||||||
---|---|---|---|---|---|---|---|---|

Best Criterion | MC1 | MC2 | MC3 | MC4 | MC5 | MC6 | ||

DM01 | MC2 | 2 | 1 | 7 | 3 | 4 | 8 | |

DM02 | MC2 | 2 | 1 | 6 | 6 | 7 | 9 | |

DM03 | MC2 | 2 | 1 | 5 | 4 | 3 | 5 | |

Preference of Other Criteria over the Worst Criterion | ||||||||

Worst Criterion | MC1 | MC2 | MC3 | MC4 | MC5 | MC6 | ||

DM01 | MC6 | 9 | 9 | 5 | 6 | 7 | 1 | |

DM02 | MC6 | 9 | 9 | 4 | 5 | 8 | 1 | |

DM03 | MC6 | 4 | 5 | 2 | 3 | 3 | 1 | threshold |

DM01 | $C{R}_{j}^{I}$ | 0.1786 | 0.0179 | 0.4821 | 0.1786 | 0.3571 | 0.0000 | 0.3337 |

DM02 | 0.1250 | 0.0000 | 0.2083 | 0.2917 | 0.6528 | 0.0000 | ||

DM03 | 0.1500 | 0.0000 | 0.2500 | 0.3500 | 0.2000 | 0.0000 |

Decision-Maker | Preference of the Best Criterion over Others | |||||
---|---|---|---|---|---|---|

Best Criterion | sc01 | sc02 | sc03 | sc04 | ||

DM01 | sc02 | 1 | 1 | 3 | 2 | |

DM02 | sc01 | 1 | 3 | 5 | 5 | |

DM03 | sc01 | 1 | 3 | 4 | 2 | |

Preference of Other Criteria over the Worst Criterion | ||||||

Worst Criterion | sc01 | sc02 | sc03 | sc04 | ||

DM01 | sc03 | 3 | 3 | 1 | 2 | |

DM02 | sc04 | 7 | 6 | 4 | 1 | |

DM03 | sc03 | 4 | 2 | 1 | 3 | threshold |

DM01 | $C{R}_{j}^{I}$ | 0.0000 | 0.0000 | 0.0000 | 0.1667 | 0.2681 |

DM02 | 0.1000 | 0.6500 | 0.7500 | 0.0000 | ||

DM03 | 0.0000 | 0.1667 | 0.0000 | 0.1667 |

Decision-Maker | Preference of the Best criterion over Others | ||||||
---|---|---|---|---|---|---|---|

Best Criterion | sc05 | sc06 | sc07 | sc08 | sc09 | ||

DM01 | sc05 | 1 | 6 | 4 | 2 | 3 | |

DM02 | sc05 | 1 | 5 | 4 | 4 | 5 | |

DM03 | sc05 | 1 | 4 | 3 | 2 | 3 | |

Preference of Other Criteria over the Worst Criterion | |||||||

Worst Criterion | sc05 | sc06 | sc07 | sc08 | sc09 | ||

DM01 | sc06 | 7 | 1 | 4 | 6 | 3 | |

DM02 | sc06 | 8 | 1 | 7 | 7 | 5 | |

DM03 | sc06 | 4 | 1 | 2 | 3 | 2 | threshold |

DM01 | $C{R}_{j}^{I}$ | 0.0333 | 0.0000 | 0.3333 | 0.2000 | 0.1000 | 0.3062 |

DM02 | 0.1500 | 0.0000 | 1.1500 | 1.1500 | 1.0000 | ||

DM03 | 0.0000 | 0.0000 | 0.1667 | 0.1667 | 0.1667 |

Decision-Maker | Preference of the Best Criterion over Others | |||||||
---|---|---|---|---|---|---|---|---|

Best Criterion | MC1 | MC2 | MC3 | MC4 | MC5 | MC6 | ||

DM01 | MC2 | 2 | 1 | 7 | 3 | 4 | 8 | |

DM02 | MC2 | 2 | 1 | 6 | 6 | 7 | 9 | |

DM03 | MC2 | 2 | 1 | 5 | 4 | 3 | 5 | |

Preference of Other Criteria over the Worst Criterion | ||||||||

Worst Criterion | MC1 | MC2 | MC3 | MC4 | MC5 | MC6 | ||

DM01 | MC6 | 9 | 9 | 3 | 6 | 6 | 1 | |

DM02 | MC6 | 9 | 9 | 4 | 5 | 4 | 1 | |

DM03 | MC6 | 4 | 5 | 2 | 2 | 3 | 1 | threshold |

DM01 | $C{R}_{j}^{I}$ | 0.1786 | 0.0179 | 0.2321 | 0.1786 | 0.2857 | 0.0000 | 0.3337 |

DM02 | 0.1250 | 0.0000 | 0.2083 | 0.2917 | 0.2639 | 0.0000 | ||

DM03 | 0.1500 | 0.0000 | 0.2500 | 0.1500 | 0.2000 | 0.0000 |

Decision-Maker | Preference of the Best Criterion over Others | |||||
---|---|---|---|---|---|---|

Best Criterion | sc01 | sc02 | sc03 | sc04 | ||

DM01 | sc02 | 1 | 1 | 3 | 2 | |

DM02 | sc01 | 1 | 3 | 5 | 5 | |

DM03 | sc01 | 1 | 3 | 4 | 2 | |

Preference of Other Criteria over the Worst Criterion | ||||||

Worst Criterion | sc01 | sc02 | sc03 | sc04 | ||

DM01 | sc03 | 3 | 3 | 1 | 2 | |

DM02 | sc04 | 7 | 3 | 2 | 1 | |

DM03 | sc03 | 4 | 2 | 1 | 3 | threshold |

DM01 | $C{R}_{j}^{I}$ | 0.0000 | 0.0000 | 0.0000 | 0.1667 | 0.2681 |

DM02 | 0.1000 | 0.2000 | 0.2500 | 0.0000 | ||

DM03 | 0.0000 | 0.1667 | 0.0000 | 0.1667 |

Decision-Maker | Preference of the Best criterion over Others | ||||||
---|---|---|---|---|---|---|---|

Best Criterion | sc05 | sc06 | sc07 | sc08 | sc09 | ||

DM01 | sc05 | 1 | 6 | 4 | 2 | 3 | |

DM02 | sc05 | 1 | 5 | 4 | 4 | 5 | |

DM03 | sc05 | 1 | 4 | 3 | 2 | 3 | |

Preference of Other Criteria over the Worst Criterion | |||||||

Worst Criterion | sc05 | sc06 | sc07 | sc08 | sc09 | ||

DM01 | sc06 | 7 | 1 | 3 | 6 | 3 | |

DM02 | sc06 | 8 | 1 | 2 | 2 | 2 | |

DM03 | sc06 | 4 | 1 | 2 | 3 | 2 | threshold |

DM01 | $C{R}_{j}^{I}$ | 0.0333 | 0.0000 | 0.2000 | 0.2000 | 0.1000 | 0.3062 |

DM02 | 0.1500 | 0.0000 | 0.1500 | 0.1500 | 0.2500 | ||

DM03 | 0.0000 | 0.0000 | 0.1667 | 0.1667 | 0.1667 |

Decision-Maker | Weight of DM | MC1 | MC2 | MC3 | MC4 | MC5 | MC6 |
---|---|---|---|---|---|---|---|

DM01 | 0.3333 | 0.2380 | 0.3807 | 0.0680 | 0.1586 | 0.1190 | 0.0357 |

DM02 | 0.5000 | 0.2685 | 0.4381 | 0.0895 | 0.0895 | 0.0767 | 0.0377 |

DM03 | 0.1667 | 0.2161 | 0.3798 | 0.0864 | 0.1081 | 0.1441 | 0.0655 |

Group average | 0.2496 | 0.4093 | 0.0818 | 0.1156 | 0.1020 | 0.0417 |

Decision-Maker | Weight of DM | sc01 | sc02 | sc03 | sc04 |
---|---|---|---|---|---|

DM01 | 0.3333 | 0.3514 | 0.3514 | 0.1081 | 0.1892 |

DM02 | 0.5000 | 0.5590 | 0.2174 | 0.1304 | 0.0932 |

DM03 | 0.1667 | 0.4655 | 0.1724 | 0.1034 | 0.2586 |

Group average | 0.4742 | 0.2545 | 0.1185 | 0.1528 |

Decision-Maker | Weight of DM | sc05 | sc06 | sc07 | sc08 | sc09 |
---|---|---|---|---|---|---|

DM01 | 0.3333 | 0.4110 | 0.0548 | 0.1233 | 0.2466 | 0.1644 |

DM02 | 0.5000 | 0.4962 | 0.0763 | 0.1527 | 0.1527 | 0.1221 |

DM03 | 0.1667 | 0.3971 | 0.0882 | 0.1471 | 0.2206 | 0.1471 |

Group average | 0.4513 | 0.0711 | 0.1419 | 0.1953 | 0.1404 |

Main Criteria | Main Criteria Weights | Subcriteria | Local Weights | Global Weights | Rank |
---|---|---|---|---|---|

MC1 | 0.2496 | sc01 | 0.4742 | 0.1184 | 2 |

sc02 | 0.2545 | 0.0635 | 7 | ||

sc03 | 0.1185 | 0.0296 | 12 | ||

sc04 | 0.1528 | 0.0381 | 11 | ||

MC2 | 0.4093 | sc05 | 0.4513 | 0.1847 | 1 |

sc06 | 0.0711 | 0.0291 | 13 | ||

sc07 | 0.1419 | 0.0581 | 8 | ||

sc08 | 0.1953 | 0.0799 | 6 | ||

sc09 | 0.1404 | 0.0574 | 9 | ||

MC3 | 0.0818 | sc10 | 1.0000 | 0.0818 | 5 |

MC4 | 0.1156 | sc11 | 1.0000 | 0.1156 | 3 |

MC5 | 0.1020 | sc12 | 1.0000 | 0.1020 | 4 |

MC6 | 0.0417 | sc13 | 1.0000 | 0.0417 | 10 |

Linguistic Variable | Abbreviation | TFN |
---|---|---|

Very Good | VG | (0.9,1,1) |

Good | G | (0.7,0.9,1) |

Medium Good | MG | (0.5,0.7,0.9) |

Fair | F | (0.3,0.5,0.7) |

Medium Poor | MP | (0.1,0.3,0.5) |

Poor | P | (0,0.1,0.3) |

Very Poor | VP | (0,0,0.1) |

Criteria | sc01(+) | sc02(+) | sc03(+) | sc04(+) | sc05(+) | sc06(+) | sc07(+) | sc08(+) | sc09(+) | sc10(+) | sc11(-) | sc12(+) | sc13(+) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

^{Weights} | 0.1184 | 0.0635 | 0.0296 | 0.0381 | 0.1847 | 0.0291 | 0.0581 | 0.0799 | 0.0574 | 0.0818 | 0.1156 | 0.1020 | 0.0417 | |

_{Alternatives} | ||||||||||||||

A | 60 | 84.55 | 20 | (0.833,0.967,1) | 70 | (0.7,0.867,0.967) | (0.7,0.9,1) | (0.7,0.9,1) | (0.5,0.7,0.9) | (0.9,1,1) | 7.82 × 10^{11} | 100 | 0 | |

B | 60 | 36.25 | 25 | (0.767,0.933,1) | 50 | (0.9,1,1) | (0.5,0.7,0.9) | (0.5,0.7,0.9) | (0.3,0.5,0.7) | (0.633,0.833,0.967) | 1.08 × 10^{12} | 72 | 100 | |

C | 60 | 70.45 | 10 | (0.367,0.567,0.767) | 60 | (0.333,0.5,0.667) | (0.3,0.5,0.7) | (0.5,0.7,0.9) | (0.5,0.7,0.9) | (0.4,0.567,0.733) | 7.17 × 10^{11} | 45 | 100 | |

D | 48 | 78.05 | 20 | (0.433,0.633,0.833) | 65 | (0.367,0.567,0.767) | (0.5,0.7,0.9) | (0.3,0.5,0.7) | (0.5,0.7,0.9) | (0.3,0.5,0.7) | 8.24 × 10^{11} | 100 | 0 |

**Table 17.**Calculations of fuzzy parameters: ${\tilde{S}}_{i},{\tilde{R}}_{i}$ and ${\tilde{Q}}_{i}$.

${\tilde{\mathit{S}}}_{\mathit{i}}$ | ${\tilde{\mathit{R}}}_{\mathit{i}}$ | ${\tilde{\mathit{Q}}}_{\mathit{i}}$ | |
---|---|---|---|

A | (−0.05,0.078,0.205) | (0.042,0.042,0.042) | (−0.186,0,0.186) |

B | (0.335,0.496,0.633) | (0.185,0.185,0.185) | (0.595,0.806,1) |

C | (0.215,0.395,0.533) | (0.102,0.102,0.102) | (0.219,0.443,0.638) |

D | (0.233,0.419,0.562) | (0.118,0.118,0.118) | (0.289,0.517,0.716) |

${\tilde{S}}^{*}$ | (−0.05,0.078,0.205) | ||

${\tilde{R}}^{*}$ | (0.042,0.042,0.042) | ||

${S}^{*l}$ | −0.050 | ${R}^{*l}$ | 0.042 |

${S}^{\circ u}$ | 0.633 | ${R}^{\circ u}$ | 0.185 |

Alternatives | Parameters | Rankings (with Respect to) | ||||
---|---|---|---|---|---|---|

S_{i} | R_{i} | Q_{i} | S_{i} | R_{i} | Q_{i} | |

A | 0.0778 | 0.0417 | 0.0000 | 1 | 1 | 1 |

B | 0.4900 | 0.1847 | 0.8018 | 4 | 4 | 4 |

C | 0.3846 | 0.1020 | 0.4357 | 2 | 2 | 2 |

D | 0.4081 | 0.1184 | 0.5100 | 3 | 3 | 3 |

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**MDPI and ACS Style**

Naghizadeh Vardin, A.; Ansari, R.; Khalilzadeh, M.; Antucheviciene, J.; Bausys, R. An Integrated Decision Support Model Based on BWM and Fuzzy-VIKOR Techniques for Contractor Selection in Construction Projects. *Sustainability* **2021**, *13*, 6933.
https://doi.org/10.3390/su13126933

**AMA Style**

Naghizadeh Vardin A, Ansari R, Khalilzadeh M, Antucheviciene J, Bausys R. An Integrated Decision Support Model Based on BWM and Fuzzy-VIKOR Techniques for Contractor Selection in Construction Projects. *Sustainability*. 2021; 13(12):6933.
https://doi.org/10.3390/su13126933

**Chicago/Turabian Style**

Naghizadeh Vardin, Aziz, Ramin Ansari, Mohammad Khalilzadeh, Jurgita Antucheviciene, and Romualdas Bausys. 2021. "An Integrated Decision Support Model Based on BWM and Fuzzy-VIKOR Techniques for Contractor Selection in Construction Projects" *Sustainability* 13, no. 12: 6933.
https://doi.org/10.3390/su13126933