Performance appraisement of supplier selection in construction company with Fuzzy AHP,Fuzzy TOPSIS,and DEA: A case study based approach

3.1.1 Step 1: Identification of parameters

In order to choose the most crucial criteria from the questionnaire survey, the Fuzzy Delphi Method is used [40].

Optimal values for each criterion are obtained based on the opinion by the expert committee. The resulting comment must be in the set of possible denoted by S _C range from 1 to 10. The scores for each criterion is C _i (denoted as most pessimistic value) and E _v (denoted as most optimistic value) obtained from Equations (2) [41]. C i = ( l k i , h k i ) , i ∈ S C .(1) E v = ( l k v , h k v ) , v ∈ S C .(2) where expert k rated the factors like l k i ( l k v ) is the pessimistic index of factor i(v) and h k i ( h k v ) is the optimistic index of factor i(v). For each factor i(v) the triangular fuzzy number for the most pessimistic index and the most optimistic index are identified. The minimum value l l i ( h l i ) and l l v ( h l v ), the geometric mean l l m ( h l m ) and l l m ( h l m ), and the maximum value l l u ( h l u ) and l l u ( h l u ), of the experts’ opinions on the most pessimistic index and most optimistic index are obtained, i.e., l _i = ( l l i , l m i , l u i ), l _v = ( l l v , l l v , l l v ) and h _i = ( h l i , h m i , h u i ), h _v = ( h l v , h m v , h u v ). The consensus significance value of each factor is computed after examining the expert consensuses. The overlap portion of l _i and h _i known as the “Gray Zone” (Fig. 2) in which the experts’ opinions on each factor are examined.

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Fig. 2

Gray zone.

The following equations are used to obtain the consensus significance value of the factor i, sⁱ. If l ⁱ and h ⁱ do not overlap (there is no gray area), then the factor’s consensus significance value is mentioned. S i = l m i + h m i 2 .(3)

If a gray zone is present and the gray zone interval value of g ⁱ (g ⁱ = l u i – h u i ) is less than the interval value of l ⁱ and h ⁱ ( d i = h m i - l m i ) i.e., g ⁱ  ⩽ d ⁱ , the consensus significance value of the factor is computed by Equations (5). F i ( p ) = { ∫ p { min [ l i ( p ) , h i ( p ) ] } dp } , i ∈ S(4) S i = { p / max μ Fi ( p ) , i ∈ S(5) where (p) is the area of the intersection of l ⁱ and h ⁱ while s ⁱ is the cognition value (p) with the highest degree of membership μ^* of the intersection of l ⁱ and h ⁱ .

If gi > di and there is a gray area, there is a significant difference in opinion among the experts. Repeat the aforementioned steps until convergence is attained. To determine the consensus significance value of factor v, s ^v , the same process is used. Pick a factor from the list of factors. Select factor i(v) if sⁱ ≥T_C (s^v ≥T_E),which is a threshold value set by experts based on the mean of all sⁱ (s^v) and is greater than or equal to the consensus significance value of the factor i(v).

3.2.1 Step 2: Fuzzy AHP

The weight vectors of the five risk variables are calculated using Chang’s extent analysis [42] approach for Fuzzy AHP. To compute the weights of the risk factors, Fuzzy AHP is employed Equation (6). The decision-makers who identify the flaws in the prototype model are part of the Fuzzy AHP technique. Assume that U={u₁, u₂,  . . . ., u_n} is a target set and that X={x₁, x₂,  . . . , x_n} is a target set. Every object is taken and an extent analysis is done for each target in accordance with the extent analysis approach. As a result, ‘m’ extent analysis values for each object ‘n’ are obtained by the subsequent steps:

M ij 1 ˜ , M ij 2 ˜ , … , M ij j ˜ where all the M ij j (i = 1,2 . . . .,n; j = 1,2 . . . .,m) are TFNs

The following are the steps in extent analysis:

Step 1 Equation (6) is used to determine the fuzzy synthetic extent value in relation to the i^th item S i ˜ = ∑ j = 1 m M ij j ˜ ⊗ [ ∑ i = 1 n ∑ j = 1 m M ij j ˜ ](6) for a specific matrix, the fuzzy calculation operation of m extent analysis values is carried out to produce

∑ j = 1 m M ij j ˜ and ∑ i = 1 n ∑ j = 1 m M ij j ˜ as shown in Equations (8) as evaluated in previous studies [43, 44] ∑ j = 1 m M ij j ˜ = [ ∑ j = 1 m l j , ∑ j = 1 m m j , ∑ j = 1 m u j ](7) ∑ i = 1 n ∑ j = 1 m M ij j ˜ = ( ∑ i = 1 n ∑ j = 1 m l ij , ∑ i = 1 n ∑ j = 1 m m ij , ∑ i = 1 n ∑ j = 1 m u ij )(8) where M ij j ˜ (i = 1,2, . . . .,n; j = 1,2, . . . .,m) values and the inverse of the above vector are computed as shown in Equation (9) [ ∑ i = 1 n ∑ j = 1 m M ij j ˜ ] - 1 = ( 1 ∑ i = 1 n ∑ j = 1 m u ij , 1 ∑ i = 1 n ∑ j = 1 m m ij , 1 ∑ i = 1 n ∑ j = 1 m l ij ) .(9) with regard to i^th object, the value of the fuzzy synthetic extent is defined, and S1, S2, and so forth are computed.

Step 2 As M ˜ 1 and M ˜ 2 are two TFNs, the degree of possibility of M ˜ 2 ⩽ M ˜ 1 is defined as V ( M 2 ˜ ⩾ M 1 ˜ = sup y ⩾ x [ min μ M ˜ 1 ( x ) , min μ M ˜ 2 ( y ) ] and can be phrased similarly as follows:

V ( M 2 ˜ ⩾ M 1 ˜ ) = μ ( d ) = { 1 , if m 2 ⩾ m 1 0 , if l 2 ⩾ u 2 l 1 - u 2 ( ( m 2 - u 2 ) - ( m 1 - u 1 ) ) otherwise(10) where d is the ordinate of the highest intersection point D between μ M ˜ 1 and μ M ˜ 2 as shown in Fig. 3 [45]

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Fig. 3

The intersection between M 1 ˜ and M 2 ˜ .

To compare M 1 ˜ and M 2 ˜ , both values of V ( M 2 ˜ ≥ M 2 ˜ ) and V( M 1 ˜ ≥ M 2 ˜ ) are needed. d’ ( A i ) = minV ( S i ˜ ⩾ S k ˜ )(11)

Step 3 provides the formulas needed to compare two triangular fuzzy numbers (frequently adopted indecision problems), and Step 3 calculates the degree of possibility.

Step 4 the probability that a convex fuzzy number is bigger than k convex fuzzy numbers M i ˜ can be expressed as follows:

V ( M ˜ ⩾ M 1 ˜ , M 2 ˜ , . . . , M k ˜ )=min V( M ˜ ⩾ M i ˜ ), where i = 1, 2,..., k. Assume that d’ ( A i ) = min V ( S i ˜ ⩾ S k ˜ ) for k = 1, 2,..., k≠i. Then the weight vector is given by W’ = ( ( d’ ( A 1 ) , d’ ( A 2 ) , … , d’ ( A n ) ) T(12) where A_i (i = 1, 2,..., n) are n elements.

Step 5 Via normalization, the normalized weight vectors are W = ( ( d ( A 1 ) , d ( A 2 ) , … , d ( A n ) ) T(13)

Where W is a non-fuzzy number.

The pair-wise assessment method can be used to generate the weight vector of risk factors. The decision-makers assess the weight vector risk factors using the linguistic variables in Table 1 [49]. It is necessary to examine the comparison matrix’s consistency ratio. This is accomplished by using the graded mean integration method, and the following equation 14 defuzzifies the fuzzy numbers into crisp numbers.

P ( M ˜ ) = M = m 1 + 4 m 2 + m 3 6 .(14)

Each matrix value is defuzzified. The consistency ratio is calculated and tested to check whether it is less than or equal to 0.1.

The closeness co-efficient is utilized to order the errors using fuzzy TOPSIS [47]. The decision-makers assess the evaluations of alternatives in relation to criteria using linguistic factors. The linguistic scale fuzzy ratings for the five criteria—Supplier connection, Supplier accessibility, Risk and Safety, Customer service, and SC performance—are shown in Table 2 [48]. The following formula can be used to determine how K experts rated each choice in relation to each criterion:

x ij ˜ = 1 k(15) where X ij K ˜ is the rating of the k^th decision maker for the i^th alternative with respect to j^th criterion.

After getting the weights of the criteria and fuzzy ratings of options with regard to five criteria, the fuzzy multi-criteria decision-making problem is described as follows in matrix format: D = [ X 11 ˜ X 12 ˜ ⋯ X 1 n ˜ X 21 ˜ X 22 ˜ ⋯ X 2 n ˜ ⋮ ⋮ ⋮ ⋮ X m 1 ˜ X m 2 ˜ ⋯ X m ˜ ](16) W = [ w 1 , w 2 , … , wn ] , where j = 1 , 2 , 3 … , n(17)

The linguistic variables are described by triangular fuzzy numbers: X ij ˜ = ( a ij , b ij , c ij ) . The several criteria scales are transformed into a comparable scale using the linear scale transformation. The derived normalized fuzzy decision matrix (R) R ˜ = [ r ij ] ˜(18) where r ˜ , is i^th and j^th elements of decision matrix r ˜ = ( a ij ˜ c j * , b ij ˜ c j * , c ij ˜ c j * ) , j ∈ B ;(19) r ˜ = ( a j - c ij , b j - b ij , c j - a ij ) , j ∈ C ;(20) Where, C is the set of cost criteria and B is the benefit criteria c j * = max i c ij If j ∈ B ;(21) a j - = min i a ij If j ∈ C ;(22)

The aforementioned normalization technique keeps the characteristic that the ranges of normalized triangular fuzzy numbers belong to (0, 1). The weighted normalized fuzzy decision matrix is created taking into account the varying importance of each criterion: V ˜ = [ c ] m × n , i = 1 , 2 , … , m ; j = 1 , 2 , … , n(23) Where, v ˜ = r ij ˜ ( · ) d ( C j )(24)

The weighted normalized fuzzy decision matrix indicates that, the elements v ij ˜ ∀ i, j are closed intervals and normalized as positive triangular fuzzy integers (0, 1). Finally, the fuzzy positive-ideal solution (FPIS, A*) and fuzzy negative-ideal solution (FPIS, A) are given as follows: A * = ( v 1 * ˜ , v 2 , * ˜ … … v n * ˜ ) ,(25) A - = ( v 1 - ˜ , v 2 , - ˜ … … v n - ˜ ) ,(26) Where v j * ˜ = ( 1 , 1 , 1 ) and v 1 - ˜ = ( 0 , 0 , 0 ) , j = 1 , 2 , … , n(27)

The distance of each alternative from A^* and A^- can be currently calculated as: d i * = ∑ j = 1 n d ( v ij ˜ , v j * ˜ ) , i = 1 , 2 … , m(28) d i - = ∑ j = 1 n d ( v ij ˜ , v j - ˜ ) , i = 1 , 2 … , m(29) where d(i, j) is the distance measurement between two fuzzy numbers calculating with the following formula d ( ρ , τ ) = 1 3 [ ( ρ 1 - τ 1 ) 2 + ( ρ 2 - τ 2 ) 2 + ( ρ 3 - τ 3 ) 2 ](30)

A closeness coefficient (CC _j ) is defined to establish the ranking order of all alternatives once the d i * and d i - of each alternative A_i (i = 1, 2... m) are calculated. The closeness coefficient of each alternative is calculated as C C j = d j - ( d j * + d j - )(31)

The alternative suppliers are ranked in order of importance based on the closeness co-efficient value. These suggestions are sent to the decision-making authorities so as to can choose the supplier. Up until the choice of the optimum supplier selection, the actions in the aforementioned stages are repeated.

3.3.1 Step 4: Data Envelopment Analysis (DEA)

According to the constant returns to scale (CRS) assumption, the authors R. D. Banker et al. [49] created the CCR-(DEA), and Banker, Charnes, and Cooper [50] created the BCC-DEA model according to the variable returns to scale (VRS) assumption. The Decision Making Units (DMU) that are on the efficiency frontier in DEA, which is not a parametrical method, are rated as being moderately efficient and given the letter “1”. In sustainable supplier selection using DEA, DMUs refer to the Decision-Making Units, which represent the suppliers being evaluated. These DMUs are assessed based on their inputs and outputs to determine their relative efficiency and performance in sustainable practices. DEA helps identify suppliers that demonstrate efficient resource utilization and sustainable output generation, aiding decision-makers in selecting suppliers aligned with sustainable goals. Based on the CRS assumption, the CCR-DEA model posits that all DMUs function at the ideal scale. Equations are employed to formulate the input-oriented CCR-DEA model that was used in this investigation. The mathematical language and formulation of the DEA approach are briefly described below,

Let Y _i be the appropriate vector of the outputs for the DMU I and let X _i be the vector of inputs for the DMU _i, therefore, let’s assume that X _O and Y _O represent the input and output vectors of the DMU₀ whose efficiency is being examined. The following linear programme can then be solved to determine the effectiveness of the DMU₀: Minimize : θ , s . t ∑ λ i X i ⩽ θ X 0 ∑ λ i Y i ⩽ X 0 λ ⩾ 0(32) where λ _i is the weight given to DMU_i to take over DMU₀ and θ is the efficiency of the DMU being investigated DMU₀. The efficiency of the DMUs can be determined by solving the above-mentioned linear programme (Equation (32)). It be supposed to be noted that the maximum value of the efficiency is θ = 1.

The number of quantitative factors (3 different types of occurrences) and the number of qualitative factors (5 factors) are taken into consideration for the system’s inputs and outputs, respectively. To meet the theoretical requirements of the DEA, some alterations are done to the data that was initially acquired. Section 4 gives a detailed explanation of these adjustments and changes made to the study’s model.

4.2.1 Selected factors by using FDM

Important criterions were selected from group of experts by questionnaire survey. In order to determine the consensus significance value ((s ^v )  ^a ), FDM is applied to the criteria. The results of FDM for CRs are shown in Table 3. with the TE for CRs set at 6, five of the 13 CRs are shortlisted. From FDM, the selected CRs are depicted in Table 3.

From Table 3, five criterion were shortlisted for supplier selection namely, Supplier relationship (C1); Accessibility of the supplier (C2); Risk and Safety (C3); Customer service (C4); SC performance (C5)

Criteria and sub-criteria from the Delphi approach have been identified from the previous step. The second step is to find pair wise comparisons between each criterion in order to specify the best from expert opinion. This Fuzzy AHP analysis used to rank and prioritize the alternative potential suppliers for M-Sand. To conclude the weight vector of five risk factors namely Fuzzy AHP is utilized namely, Supplier relationship (C1); Accessibility of the supplier (C2); Risk and Safety (C3); Customer service (C4); SC performance (C5) are considered. The expert committee offers their thoughts on the risk elements and how they relate to the possible failure modes. Based on the linguistic values given in Table 1 and the risk factors provided in Table 3, a paired comparison matrix has been created. Using equations (11), the weight vectors for each risk factor are calculated and given below in Table 4.

Fuzzy TOPSIS method can be utilized to determine the final evaluation of the entire weighted criterion obtained from Fuzzy AHP method. M-Sand supplier alternatives choice matrix that has been weighted and normalized ought to be computed. The TOPSIS method’s result has been obtained by the following procedure. Create a group of three decision-makers by generating all feasible options (A1, A2, A3, and A4) and identifying the various evaluation criteria’s (C1, C2, C3, C4 and C5) (DMs1, DMs2, DMs3). Decision-makers use a group of applied criteria from criteria to assess how well the literature and desires align [51]. Decide proper linguistic terms proposed for the importance weights of the criteria. Make linguistic judgments for alternatives with respect to weight of criteria expressed as TFN. It is shown in Table 5 [52],

By aggregating the weight of criteria, determine the aggregated fuzzy weight of the criterion by taking into account the experts’ judgment on the criterion to obtain combined fuzzy evaluations of the alternatives. Using Table 6, a normalized fuzzy decision matrix has been created. From Table 7 the normalized fuzzy decision matrix is used to create a weighted normalized fuzzy decision matrix (Table 8).The computations used to derive the fuzzy PIS Z+(FPIS) and fuzzy NIS Z–(FNIS) are displayed below (Table 9) [53]. Determine how far possible alternatives or suppliers are from FNIS and FPIS in relation to each of the criteria in Table 10 [54]. Determine each option’s closeness coefficient (CCr) as shown in Table 1.

From Table 12 by relating the Closeness coefficient results of four alternatives, it is decided that A2 is the most preferred supplier than the other alternative suppliers and A4 is the least preferred supplier.

Using AHP model, the main criteria for selecting the suppliers were identified with weight for each alternatives. For finding out the efficiency scores for each alternatives, DEA model were used. In Table 13, the final weights were displayed. Thus, the qualitative data utilized as output in DEA namely cost, quality, and delivery time were transformed into quantitative data. The super-efficiency scores of suppliers are derived through DEA by using qualitative characteristics as outputs and a quantitative component as an input.

With the input-oriented CCR-DEA analysis, the study evaluated the suppliers’ performance. The super-efficiency scores from DEA are displayed in Table 13. Out of the efficient suppliers, four were evaluated in this study. The 50 companies chose these suppliers because they could create most of the products for the least amount of money. Using the super-efficiency scores, it is possible to order the efficient suppliers.

Using the super-efficiency scores, it is possible to order the efficient suppliers. Even if their input costs rise, suppliers with efficiency scores higher than “1” might still be efficient. The other vendors were not productive. A4 received the lowest efficiency rating with a score of 72.8 %. This means that even if this supplier’s cost were to drop by 20.2%, the output would remain the same. Supplier A1 had the highest performance with 109.4% followed by A3 with 104.55%.

Finally the AHP results indicate that A2 is the best supplier followed by A1 and A2 as shown in Table 7. The TOPSIS results, however, indicate that A2 is the best supplier followed by A3 and A1 as shown in Table 12. In general, the comparison matrices demonstrate high consistency, with a consistency ratio consistently below 10%.