Enhancing technical proposal evaluation in consultant selection in Department of Water Resources and Irrigation, Nepal: a fuzzy analytic hierarchy process and fuzzy TOPSIS

2.1. Triangular fuzzy number (TFN)

Uncertain and vague data in decision-making problems can be effectively addressed through the application of the Fuzzy Set Theory, pioneered by Zadeh (1965) [17]. A fuzzy set comprises membership functions that encapsulate degrees of membership using real numbers within the [0, 1] interval. Membership values of zero and one signify absence and full membership, respectively. Values between zero and one denote varying degrees of membership. Additionally, translating linguistic terms into fuzzy numbers presents a promising approach to mitigate vagueness and ambiguities. Among the diverse shapes of fuzzy numbers, the triangular fuzzy number stands out as the most prevalent and simplest form. Comprising triplets (L, M, U), where L represents the minimum likely value, M denotes the probable value, and U signifies the maximum possible value, it offers a structured representation of fuzzy events.

2.2. Fuzzy analytic hierarchy process (FAHP)

The Analytic Hierarchy Process (AHP) method, introduced by Satty [18], is a quantitative technique utilized for multi-criteria decision making, initially proposed by Satty [19]. An extension of this conventional approach is the Fuzzy Analytic Hierarchy Process (FAHP), which serves as a decision-making tool tailored for analyzing complex problems featuring multiple criteria and alternatives. While the conventional AHP method has its merits, it may not wholly capture the nuances of human decision-making, as decision-makers often prefer to provide interval judgments rather than singular numeric values [3]. Fuzzy AHP addresses this constraint by enabling decision-makers to articulate their judgments using linguistic terms alongside corresponding fuzzy numbers, such as “equal importance (1, 1, 1)”, “moderate importance (2, 3, 4)”, “strong importance (4, 5, 6)”, “very strong importance (6, 7, 8)”, or “extreme importance (9, 9, 9)”. Pairwise comparisons within fuzzy AHP are conducted using fuzzy numbers, with aggregation performed through fuzzy arithmetic. This framework accommodates uncertainty and vagueness in decision-making processes, rendering fuzzy AHP particularly advantageous in scenarios where precise numerical data may be scarce or challenging to obtain.

2.3. Fuzzy technique for order preference by similarity to ideal solution (FTOPSIS)

TOPSIS, a Multi-Criteria Decision Making (MCDM) method pioneered by Hwang and Yoon [20]; stands out as a robust framework for decision-making processes. When TOPSIS integrates fuzzy numbers to capture uncertain or imprecise information, it morphs into a fuzzy technique. Fuzzy TOPSIS extends the classical TOPSIS method by embracing linguistic terms or fuzzy numbers to represent weights, criteria, or performance scores. This augmentation empowers decision-makers to adeptly navigate subjective judgments and uncertainties. TOPSIS introduces an index termed “similarity to the positive-ideal solution (PIS)” and “remoteness from the negative-ideal solution (NIS)”. Leveraging these indices, the method identifies an alternative with the maximum similarity to the ideal solution.

Fig. 1Hybrid approach framework

2.4. Proposed hybrid fuzzy AHP- fuzzy TOPSIS approach

As shown in framework (Fig. 1), a review of the literature was conducted in order to specify the criteria and sub-criteria for the selection of consultants in DWRI. The weightage of each criteria and sub-criteria was defined using MCDA tool Fuzzy AHP method. The Questionnaire was prepared for expert survey through which weightage was computed. Linguistic Rating of alternatives w.r.t criteria was done for ranking of criteria by Fuzzy TOPSIS method. The study data in the form of qualitative was transformed into quantitative data followed by fuzzy AHP and fuzzy TOPSIS method.

4.1. Fuzzy AHP results

Pair-wise comparison matrices obtained from the respondents were combined using the geometric mean approach at each hierarchy level to obtain the corresponding consensus pair-wise comparison matrices. Each of these matrices were then translated into the corresponding largest eigen value problem and were solved to find the normalized and unique priority weights for each criterion. All the experts’ ratings were categorized and pairwise comparison matrices were made. Their Consistency Index and Consistency Ratios were calculated and checked.

Each of the criteria and sub-criteria under main criteria were subjected to confirm the consistent by the consistency ratio less than 0.1 obtained from 28 experts. Then, the weightage was aggregated and turned into fuzzy weights as proposed by (Buckley, 1985) [21]. The local weightages for sub-criteria under main criteria were obtained as the result from the aggregated data calculation. The average global weights of all criteria calculated from experts’ rating are presented below in Table 2. So, in this section the above frequency of consistent data was subjected to aggregated fuzzified pair-wise comparison matrix followed by defuzzification, normalization and following result were obtained: the weightage of the main criteria and local and global weightage of sub-criteria under each criterion.

From Table 2, it was observed that the five highest weighted sub-criteria for standing list were: experience in related field, experience in similar projects, experience in similar geographical area, Technical Approach and Methodology and experience in region and language; whereas the five lowest weighted sub-criteria were: organization and staffing, general experience of firm, Training approach and methodology, comments on TOR, training program and seminar, and regular staff.

4.2. Fuzzy TOPSIS results

Here, the model practiced in DWRI representing a national RFP, comprising four main criteria (C1, C2, C3, C4), involving four consultants: Consultant 1 (AC-1), Consultant 2 (AC-2), Consultant 3 (AC-3), and Consultant4 (AC-4) was shortlisted for the study based on the varying nature of work and the level of competition. Samples of consultant evaluations were organized, and linguistic ratings for the alternatives were collected from the same three members of the consultant evaluation committee as Decision Makers (DM) by briefing them on expected output as shown in Table 7.

These ratings were combined into a seven-point scale using respective triangular fuzzy numbers and represented in Table 8.

The criteria were categorized into Benefit Function (where larger values are preferable) for all alternatives neglecting the Cost Function. Then, the combined and normalized fuzzy decision matrix was shown in Table 9 and Table 10 respectively.

These normalized matrices were multiplied by the average weights of all criteria obtained from fuzzy AHP to generate a Weighted Normalized Fuzzy Decision Matrix as in Table 11. To calculate the Closeness Coefficient, the FPIS (Fuzzy Positive Ideal Solution) and FNIS (Fuzzy Negative Ideal Solution) were determined. FPIS represents the maximum value among all alternatives, while FNIS represents the minimum value.

Distance ($d^{*}$) and sum of distance ($S{i}^{*}$) from FPIS & Distance ($d^{-}$) and sum of distance ($S{i}^{-}$) from FNIS were calculated and presented in Table 13 and 14 respectively.

Finally, the alternatives were ranked based on their closeness coefficient as shown in Table 14.

4.3. Rank comparison of fuzzy TOPSIS with current practices

From the DWRI model, the ranking of consultant as per current practices was obtained. The equivalence ranking for the Fuzzy AHP was generated on the basis of the deviations implied by the weightage obtained over actual practices. The ranking between the Fuzzy TOPSIS, equivalence Fuzzy AHP and actual practice were compared which is shown in Table 15.

The rankings from the Fuzzy TOPSIS and Fuzzy AHP methods are consistent, with consultant 2 (AC-2) being ranked the highest in both cases. This indicates that Consultant 2 (AC-2) is perceived as the most favorable option according to both evaluation methods. Consultant 1 (AC-1) performs well according to the Current Practices ranking, indicating that it is currently favored based on existing evaluation methods. The discrepancy between the rankings of Consultant 1 (AC-1) and Consultant 2 (AC-2) between the Fuzzy TOPSIS/Fuzzy AHP methods and Current Practices suggests a potential for improvement in the current evaluation practices, as the methods employing Fuzzy TOPSIS and Fuzzy AHP may provide a more robust and objective evaluation of consultants.