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A new multicriteria decision making method for the selection of construction contractors using interval valued fuzzy set
BMC Research Notes volumeÂ 17, ArticleÂ number:Â 113 (2024)
Abstract
Objective
This article introduces a novel approach called Digital Weighted Multi Criteria Decision Making (DWMCDM) that employs interval valued fuzzy sets to select the best contractor for building projects. The contractor is chosen based on the prequalification and bid evaluation phases. In the first phase, the distance between the actual and required skills of the significant criteria is determined, and it is then converted into digital weighted distances to identify the maximum number of criteria related to the specific project of each contractor. The second step ranks the best contractor based on the bid price and digital weighted distances.
Results
The suggested technique integrates the prequalification and bid review phases to address project award delays and other restrictions. Finally, a realworld application is addressed to demonstrate the applicability of the proposed approach to any type of interval valued fuzzy inputs.
Introduction
To maintain the dignity, competitiveness and scarcity of the work, the contractors who quote the lowest bid price remain in business. The selection of the lowest bidder is one of the main reasons for project delivery problem, when the contractor faced with a shortage of work, they desperately quoted a lowest bid price simply to stay in business with the expectation of be compensated by complaints [1, 2]. It includes multiple performance assessment criteria, Project quality, inefficient labor, ontime project delivery, and other nonprice related aspects are impacted by the higher bid value. As a result, the project assignment should involve a number of choices regarding things like cost, quality, and other aspects. The performance of a project could be jeopardized without a sufficient and precise approach of choosing the best suitable contractor [3]. Any construction project's key issue is the delay that develops throughout the project, which extends the project's timeline [4].
The mathematical criteria must be included in the tender specifications whenever it is necessary to translate price bids into scores for conjunction with technical qualities of the proposal, such as quality or customer preferences [5]. The requirements could be in opposition, as in the case of customers who desire both quality and affordability. The decisionmaking process is complicated by conflicting criteria [6]. In actuality, the contractor prequalification phase (examination of predetermined criteria) and the bid evaluation phase (determination of minimum bid price) are assigned to the project. The prequalification is then the topic of a thorough examination to determine the current status of each contractor's management, technical, and financial capabilities [4].
The invention of mathematical and computational tools to assist the arbitrary evaluation of the performance criterion for decisionmaking is the subject of the operations research subfield known as multiplecriteria decision making (MCDM) [7]. Making decision in contractor selection includes multiple performance assessment criteria, both qualitative and quantitative and it is the main issue in supply chain management system. Numerous methods and algorithms are described in the literature; a few of these include the Fuzzy Analytical Hierarchy Process (FAHP) [8], Data Envelopment Analysis (DEA), MultiAttribution Decision Making (MADM), and MultiObjective Decision Making (MODM), as well as the TOPSIS and VIKOR models, which are used for supplier selection [9], performance evaluation [10], and project management [11]. In general, a multiple criteria decision making (MCDM) process consists of a group of decision makers, a number of admissible alternatives, and a number of criteria. Various decision making algorithms are discussed in the literature, see [12,13,14,15,16,17,18,19,20,21,22], references therein.
The human preference model is unclear in many realworld situations, and decisionmakers may be reluctant or unable to give precise numerical values to comparative judgments [9]. Fuzzy logic is built on natural language and functions similarly to how people think. To face the factors of uncertain and imprecise information in real life situation, many theories have been developed, e.g. fuzzy sets [17, 23], approximation theory [24,25,26] etc. Intuitionistic fuzzy sets (IFS) was introduced by [27] to assign non membership grades and it was studied by [28, 29]. Neutrosophic fuzzy set (NFS) was introduced in 1998 by [30] to assign incomplete, indeterminate, and inconsistent information.
In decisionmaking management, when historical data is either unavailable or insufficient, expert opinion is the only source of knowledge that may be used to make a decision [15]. Fuzzy sets cannot give adequate explanations as the evaluation criteria, decision makers, and personal judgments vary. In order to clear up the ambiguity in this process, fuzzy sets with interval values IVFS are added [12, 31]. Decisionmakers express their thoughts using language scales, and with the aid of an algorithm, they are further translated into interval valued fuzzy numbers with the help of different membership functions. MCDM technique has several qualities, numerous criteria, and a number of decisionmakers. Decision makers must aid in the evaluation of ncontractors and all contractors in relation to each criterion. Our proposed method identifies the best contractor from a set of ncontractors and all contractors must be evaluated against to each criterion with the help of decision makers.
Methodology
Criteria for contractor selection
The main source of criteria which includes the related subcriteria for assessing the pre qualification status of construction companies [32] given in Table 1.
Preliminaries
Definition 1: [23] A fuzzy set A is a universe of discourse X characterized by a membership function \(\mu_{A} (x)\)â€‰â†’â€‰[0, 1], where \(\mu_{A} (x)\), âˆ€ xâ€‰âˆˆâ€‰X, represents the degree of truth of x in fuzzy set A.
Definition 2: [7] Let X be an universal set with cardinality n. Let [0, 1] be the set of all closed sub intervals of the interval [0, 1] and elements of this set are denoted by uppercase letters. If Mâ€‰âˆˆâ€‰[0, 1], then it can be represented as Mâ€‰=â€‰[M_{L}, M_{U}], where M_{L} and M_{U} are the lower and upper limits of M. An IntervalValued Fuzzy Set (IVFS) A in X is given by \(A = \left\{ {\;(x_{i\;,} M_{A} (x_{i} ):x_{i} \in X\;} \right\}\) Where \(M_{A} :\;X \to \;[0,\,\,1]\), M_{A} (x_{i}) denote the degree of membership of the element x to the set A and 0â€‰â‰¤â€‰M_{AL}â€‰â‰¤â€‰M_{AU}â€‰â‰¤â€‰1.
Algorithm
In this section, we proposed the Digital Weighted Multi Criteria Decision Making (DWMCDM) for contractor selection using IVFS. Suppose S is a set of skills of contractor, P is a set of projects and C is a set of contractors. Let Q: C \(\to\) S is an interval valued fuzzy relation from the set contractors to the skills and R: S \(\to\) P is an interval valued fuzzy relation from the set of skills to the projects. The main steps of our proposed method are as follows:

1. Decision makers convey their opinions on the actual performance levels of the contractors' competencies using linguistic scales that are transformed into an intervalvalued fuzzy membership functions.

2. Determine the necessary system assessment standards that are related to the scope of the projects.

3. Based on the project requirements, calculate the difference or deviation between the contractor criterions expected interval and observed interval.

4. Applying digital weights to each criterion will allow you to find the digital weighted distance score values.

5. Determine the digital weighted distance scores, which satisfies all required criteriaâ€™s, and confirm that all other values are 0.

6. Enter the price quotation matrix B for each project.

7. The bid values are set in accordance with the cost reduction criteria.

8. The value of the final score is equal to the corresponding product values of the above average weighted digital distance scores (in step6) and the reciprocal of the project bid price.
Steps involved in algorithm
The proposed method involves the following steps:
Step 1: The ratings of decisionmakers contractorsskills relation matrix Q for each criterion in terms of the interval valued fuzzy set (QL_{ij}, QU_{ij}), where iâ€‰=â€‰1, 2, 3, â€¦, m and jâ€‰=â€‰1, 2, 3, â€¦, n. In order to find the actual and necessary gap between the skills of the significant construction projects criteria's, decision makers translate their judgments about the criterion expressed in terms of language scales into an intervalvalued fuzzy membership function.
Step 2: Create an intervalvalued fuzzy relation matrix R referred to the skillsprojects relation matrix using the anticipated ratings of the decisionmakers under each criterion(RL_{ij}, RU_{ij}), where iâ€‰=â€‰1, 2, 3, â€¦, m and jâ€‰=â€‰1, 2, 3,â€¦, n.
Step 3: The distance matrix (see Table 6) relations Dâ€‰=â€‰[C_{ik}]_{mxp}, where.
C_{ik}â€‰=â€‰QU_{ij}â€”RL_{ji}, jâ€‰=â€‰1, 2, 3, â€¦, n and iâ€‰=â€‰1, 2, 3, â€¦, m
Step 4: Applying the corresponding weights, the digital weighted distance matrix (see Table 7).
Wâ€‰=â€‰[Z_{ik}] _{mxp,}
where \(Z_{ik} = \left\{ {\begin{array}{*{20}c} {1000000\quad if\;0.01 \le {\text{C}}_{{{\text{ik}}}} \le 0.05} \\ {1100000\quad if\;0.06 \le {\text{C}}_{{{\text{ik}}}} \le 0.1,} \\ {1110000\quad if\;0.11 \le {\text{C}}_{{{\text{ik}}}} \le 0.15,} \\ {1111000\quad if\;0.16 \le {\text{C}}_{{{\text{ik}}}} \le 0.20,} \\ {1111100\quad if\;0.21 \le {\text{C}}_{{{\text{ik}}}} \le 0.25,} \\ \begin{gathered} 1111110\quad if\;0.26 \le {\text{C}}_{{{\text{ik}}}} \le 0.30 \hfill \\ 1111111\quad if\;0.31 \le {\text{C}}_{{{\text{ik}}}} \le 0.35 \hfill \\ \end{gathered} \\ {0\quad \quad otherwise} \\ \end{array} } \right.\;\)
Step 5: Calculate the sum of digital weighted distance score matrix (see Table 8) for each element.
where \(S_{ik} = \sum\limits_{j = 1}^{n} {Z_{ij} }\)
Step 6: Identify and select the digital weighted distance scores, which satisfies all required criteriaâ€™s (see Table 9) and make it that all other values are zero.
Step 7: Enter the bid price matrix B (see Table 10) for the corresponding project.
Step 8: The reciprocal values (see Table 11) of the bid prices are selected in order to minimize cost criterion.
Step 9: Determine the final score matrix F (see Table 12) by multiplying the corresponding above average digital weighted distance score matrix and the reciprocal values of the bid prices.
Application
Let us say there are six contractors. The universal set Câ€‰=â€‰{C_{1}, C_{2},â€¦,C_{6}} represents construction industry contractors with strong industrial backgrounds. The set Sâ€‰=â€‰{S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}} stand for firm capacity, material quality, management, experience, financial stability, and workplace safety respectively,. The set Pâ€‰=â€‰{P_{1}, P_{2}, P_{3}} where P_{1}, P_{2} and P_{3} stand for three types of residential buildings respectively. Interval valued fuzzy numbers and corresponding linguistic variables are listed in Table 2.
Step 1: Using the ratings of field experts for each contractor under each criterion in terms of linguistic scales are listed in Table 3, an interval valued fuzzy relation matrix Q called a contractorsskills relation matrix is determined. This matrix is then converted into an interval valued fuzzy set (QL_{ij}, QU_{ij}).The committee's findings are listed in Table 4.
STEP 2: Construction of the interval valued fuzzy relation matrix R, which contains the field expert expected skills, where R may be either an expert matrix or any set of needed relation matrix. The recruiter's priority throughout the selection process for the project Pi may be on a selection of few skills, such as S_{2}, S_{4}, S_{5}, and S_{6}, whereas the other skills in the set S may not be a priority. As a result, each project Pi's priority for the skills differs, and the field expert's recommendations are expressed in terms of an intervalvalued fuzzy set (RL_{ij}, RU_{ij}) in Tables 4, 5, 6, 7, 8, 9, 10, 11, 12.
From the above final score values (bold values), it is clear that the Residential TypeI projectP1, will be given to the contractor C4, Residential TypeII projectP2will be given to the contractor C2, and for Residential TypeIII projectP3, no contactor satisfied all the four criteria.
Results and discussion
In this paper, an IVFSbased Digital Weighted Multi Criteria Decision Making (DWMCDM) was proposed. It could identify the best contractor from a set of ncontractors in the construction industry contractor selection.First, the decisionmakers express their thoughts about the skills of contractors using language scales, and with the aid of an algorithm for decision making, further translated into interval valued fuzzy sets and it was processed by the digital weighted operation, digital weighted fuzzy scores were generated in pre qualification stage. Then the second stage input the bid price values were set in accordance with the cost reduction criteria. Finally, ranks the best contractor based on the bid price and digital weighted distance scores using proposed algorithm for decision making.
Conclusion
Our proposed DWMCDM method for contractor selection increases the performance of the construction industry and lowers the risk of project failure for the customer based on the following five factors (i) Choosing the project wise suitable criteria, rather than considering all criteria (ii) Digital weights helps to identify the best contractors based on number of criteria satisfied, rather than the highest possible scores by identifying the positive distance between each expected and actual interval. (iii) The contractor selection depends on scores in phaseI but not on lowest bid price (iv) Prequalification phase and the bid evaluation phase which will overcome the delay in project award and other constraints (v) Suitable for any kind of interval valued fuzzy sets. Moreover, our contractor selection method suggests that it can be used as a framework that is active for assessing actual contractor selection. We suggest that this study be expanded to include situations in which the problem's data has extensions from other interval valued fuzzy sets.
Limitations
The length of the interval is quite small, and there is no direct measurement in any construction contractors in this study. All potential cases of fuzzy intervals are taken on assumptions. Due to time restrictions, this research is limited to initiatives involving the construction of buildings, but it is more broadly relevant. Decision experts of an industry are required to take for their opinion about the criteria affecting the contractor selection in such as construction system, contractor comparison, project allotment, etc., For simplicity, we evaluated a limited sample size and a small number of contractors for the case study; however, large sample can be taken with Maple implementation.
Availability of data and materials
This article contains all of the information that was created or examined throughout the investigation.
Abbreviations
 DWMCDM:

Digital Weighted Multi Criteria Decision Making
 MCDM:

Multiplecriteria decision making
 FAHP:

Fuzzy Analytical Hierarchy Process
 DEA:

Data Envelopment Analysis
 MADM:

MultiAttribution Decision Making
 MODM:

MultiObjective Decision Making
 IFS:

Intuitionistic fuzzy Sets
 NFS:

Neutrosophic fuzzy Sets
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The authors are grateful to the editor and anonymous referees for their valuable suggestions that helped to strengthen the work.
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N. S. Nithya is involved in algorithm creation. S. Thota is involved in writing and suggestion of sample computations, and P. Shanmugasundaram and Laxmi Rathour are involved in verification and validation of the algorithm and computations. All authors read and approved the final manuscript.
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Nithya, N.S., Thota, S., Rathour, L. et al. A new multicriteria decision making method for the selection of construction contractors using interval valued fuzzy set. BMC Res Notes 17, 113 (2024). https://doi.org/10.1186/s1310402406769w
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DOI: https://doi.org/10.1186/s1310402406769w