Fuzzy AHP and Modified Fuzzy TOPSIS based Supplier Selection Model

The methods of AHP and Fuzzy AHP provide supports for decision making process, go through normalization procedure, produce different values for decision criteria weights and finally determine decision result. Interestingly both the method produces same decision result in various cases. The model for supplier selection showed by Foriborz Jolai (2011) based on AHP with TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) is a matter of modification replacing by Fuzzy AHP and Modified Fuzzy TOPSIS methods in order to introduce ‘Fuzzification’ and ‘Defuzzification’ which is not available in existing model. The proposed model is verified with an illustrative example and comparing the results generated by both the existing and proposed model in consensus decision making. Keywords: AHP, Fuzzy, MCDM, TOPSIS, Supplier Selection DOI: 10.7176/JIEA/10-4-07 Publication date: October 30 th 2020


Introduction
Fuzzy logic is a tool used to control various processes ranging from medical diagnosis to engineering process control. This tool is used as mode of reasoning underlying approximate not exact. Fuzzy logic is a mode of reasoning that deals with approximate not precise. Let's say, 'Usually rose is red'. Here 'usually' is a fuzzy proportion of how many times snow is seen white and how many times not-neither all the time nor too few times to say. This is called vagueness that is quantified between 'no times it is seen red' and 'each and every time it is seen red' considering 0 and 1 the two extreme levels respectively. The value of 'usually' may reside between 0 and 1 e.g. 0.67. A fuzzy set can be written if we like to describe four different flowers' redness. Let's say we have four flour F1, F2, F3 and F4 and their redness are fuzzy quantified as 0.45, 0.55, 0, 0.6 respectively and with set notation it can be written as {(F1, 0.45), (F2, 0.55), (F3, 0), (F4, 0.6)}. This process of fuzzification and fuzzy quantification can also be done using triangular fuzzy numbers like (0, 0.25, 0.5), (0.25, 0.5, 0.75) and (0.5, 0.75, 1.0) for describing 'non red', 'slightly red' and 'full red'. In doing such, we need proper linguistics for proper description. We have flexibility to define the linguistics and fuzzy numerical values for the linguistics on case basis. This is suitably applied in Multi Criteria Decision Making (MCDM) by fuzzy quantification using predefined linguistics for different criteria and normalization using some particular mathematical method. Such a method is Analytical Hierarchical Method or AHP in short. A manager needs to make decision when he has some options available like investing in a project A considering criteria c1 and c2 or investing in project B with consideration of same criteria. The manager may have options to decide with possible return r1 and r2 in project A and B respectively with weight values of the criteria of c1 and c2 are w1 and w2 respectively. Such a case can be resolved making a decision tree. If there is options available to project A and B of return possibilities rA and rB respectively with the condition to the investment variation in sub sectors of sA and sB then the decision for maximizing the project return considering the objective criteria can be achieved using Linear Programming. Obviously this is a deterministic approach.
AHP is the first method that deals with Multi Criteria Decision Making (MCDM) introduced by Thomas Saaty (1970) which generates decision from stochastic data. Fuzzy Logic introduced by Lofti A Jadeh (1988) is another decision making tool that can quantify the vagueness. Fuzzy Logic became a powerful tool to deal with the non deterministic cases for determining results. Fuzzy MCDM, combination of Fuzzy Logic and MCDM method, showed its role for decision making became a more preferable method by the decision makers later. A lot more work have been done on both these methods to apply in various disciplines for implementation in decision making problems.
In an extensive case study of selection of ERP solution providers, decision makers' choices were fed into AHP, Fuzzy AHP and ANP methods. They found the same selection of solution provider among several as per assessors' choice of preference and the result determined. A crucial job is the supplier selection in tendering process through a consensus decision making. The assessment of different members of a bidder selection committee can have different level of preferences and assessment for different criteria. In a consensus decision making, a group of decision makers may submit individual choices of preference against various selection criteria. These preference levels can be quantified according to a particular scale with proper linguistics and fed into a mathematical method like AHP. The resulting value may not be same if these choice levels are fed into another mathematical process like Fuzzy AHP. But we are concerned about the desired selection of particular bidder whether it is same for the both the methods or not in special case of Jolai Model (2011) for supplier selection. Jolai's mode is a three phase model discussed in next section.
The aim of this research is to incorporate Fuzzy TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) method in Jolai Model for supplier selection which is based on AHP and Fuzzy AHP. The aim also includes the verification of the result generated by the proposed model with existing AHP and AHP TOPSIS model. The objective is to feed a set of preference for the decision criteria and supplier choices and determine the results of mathematical simulation for verification.

Supplier Selection Model
Foriborz Jolai (2011) presented his model as a three phase model where Phase 0 is for primary selection of bidders meeting the basic requirements, Phase 1 is for determining weights of decision criteria and Phase 2 is for final selection of bidder according to the decision criteria weights and preferences of individual bidder assessed by the decision makers. This model is shown in Figure 1 whereas Figure 2 depicts the same model with substitution of Fuzzy based methods by AHP based methods.
We have got the mathematical modelling from Jolai model (2011) described here in this section. For evaluating suppliers, the decision criteria could be suited for the process of evaluation of bidders. These are: On-time delivery (C1), Closeness of relationship with supplier (C2), Supplier product/service quality (C3), Supplier operational capability (C4), Price/cost (C5). Fuzzy pair wise decision criteria evaluation matrix using all above criteria (C1~C5) are formed in Table 1 and generalized in Table 2 The decision makers place their preferences for decision criteria weight determination using the scale shown in Table 3 which are substituted in a matrix as shown in Table 1 or Table 2. For normalization of such a matrix, eq 1 can be applied in order to form the normalized matrix after substituting the triangular fuzzy numbers as per ratings by Decision Makers using Linguistics in Table 3.
Where K = number of decision makers, = is the normalized matrix. The normalized matrix is further simplified into a row matrix using eq. 2. = ( , , … , ) / (2) Using the fuzzy geometric mean technique, the above row matrix can be transformed into fuzzy weight matrix of Journal of Information Engineering and Applications www.iiste.org ISSN 2224-5782 (print) ISSN 2225-0506 (online) Vol.10, No.4, 2020 59 .

= . ( ⨁ ⨁ … ⨁ ) = . ∑
(3) These above equations eq. (1), (2) and (3) are belonged to Phase 1 of Jolai Model depicted in Figure 1 and lets start to describe Phase 2 computation mathematics of same model. Since (a, b, c) be a triangular fuzzy number, the graded mean integration method represents The linear scale normalization formulas are used for transformation of the various criteria from linguistic variables to equivalent fuzzy numeric values according to scales into normalized values of matrix which is to be used for normalized fuzzy decision making.
= ∑ . ( , ) , = 1,2, … , = , = 1,2, … , In this stage, evaluation of alternative bidders for different criteria is the major important task which is the core job function of this particular model. Table 4 is the tool which describes the linguistic variable and their fuzzy equivalent numeric.   Vol.10, No.4, 2020 method in Phase for evaluating the suppliers. The input to the proposed model is nothing but the Fuzzy ratings or Fuzzy values as described in the introduction whereas in the older method there was no scope of Fuzzy input. Now, let us have an illustrative example.

Illustrative Example
In a case of procurement, four bidders (B1, B2, B3 & B4) have been primarily selected for final selection of one particular bidder for awarding the contract. In this procurement process, three members committee is considered for decision making. The decision makers are DM1, DM2 and DM3. Firstly, they put their preference weights for each decision criteria according to linguistics and their equivalent numeric in Table 3 then substitute in pair wise comparison matrix shown in Table 5, Table 6 and Table 7. The values of and for the criteria C1 to C5 are calculated using eq 1 and summarized in Table 9. The data set used for illustration is taken from Jolai (2011) illustration.
In phase 2, the decision makers are supposed to put the grading for primarily selected four bidders (B1, B2, B3 & B4) to determine the ranking value of each bidder to finally select a single one. The bidders are evaluated as per assessments of decision makers in the same way in fuzzy comparison matrixes. The decision makers' ratings are shown in Table 10, Table 13, Table 16, Table 19 and Table 22 for criteria C1, C2, C3, C4 and C5 respectively using the scale described in Table 4. Three decision makers' ratings are quantified in Table 11, Table 14, Table 17, Table  20 and Table 23 which are then aggregated in Table 12, Table 15, Table 18, Table 21 and Table 24 respectively.
Each bidder's aggregated fuzzy ratings obtained are summarized in Table 25 for each criteria C1 ~ C5. After that Table 25 is normalized and formed Table 26 and Table 26 to determine the values of and which will lead to values of and that has provided the value for and . Hence finally the ranking of the bidders have been obtained with .
The result of bidders' ranking and selected bidder with highest rank shown in Table 31 is determined through the model described in Figure 1. Now using the Jolai Model with AHP and TOPSIS method in Figure 2, we have got the aggregated pair wise comparison matrix for criteria C1~C5 is shown in Table 32. The matrix in Table 32 is normalized by dividing each element by the each respective column sum and a new normalized matrix in Table 34 is obtained. The row averages of Table 34 are mentioned in the same table in additional row and also listed in Table  33 for comparing the Wj values determined in Table 9 of Fuzzy based approach. The comparison of both the approach results are graphically shown in Figure 3, Figure 4 and Figure 5 and found least difference between Fuzzy MCDM and AHP normalization results in determining .
For Phase 2 calculation in Figure 2 model, we have considered the same ratings of the decision maker for four selected bidders whose were primarily selected which are shown in Table 10, Table 13, Table 16, Table 19, Table  22 and the aggregated ratings for all criteria of them for evaluation are in Table 34. Each value of Table 26 Table 34. New matrix in Table 34 is further normalized dividing each element by the respective column sum and taking the row averages determined the values of AHP Ci. AHP Ci values are converted into AHP CCi using eq. 10 and Ranking values of AHP Ri using eq. 11. Finally AHP and Fuzzy Ranking values are compared with each other as in Table 35 and Table 31

Conclusions and Recommendations
Foriborz Jolai (2011) showed the application of Fuzzy MCDM with Fuzzy TOPSIS method very successfully. In this research paper, we have substituted AHP and AHP TOPSIS method by Fuzzy MCDM and Fuzzy TOPSIS method respectively in Forborz Jolai's model for supplier selection and compared the result for the same set of input values both the cases. According to Jolai Model using Fuzzy MCDM and its mathematical outline, the result shows that Bidder B2 is mostly preferred by decision makers and Bidder B3 is the least preference by the decision makers as in Figure 10. Fuzzy MCDM has been applied instead of AHP to method fit into Jolai Model, differences in selection values have been found but same bidder is selected in this case too i.e. Bidder B2 is the selection result of both AHP and Fuzzy MCDM method both. If we like to select one bidder among many, we should select one with highest ranking or top ranked bidder. According to AHP method, Bidder B2 is selected and also the same selection (B2) is seen using Fuzzy MCDM method incorporated in Jolai Model i.e. Fuzzzy MCDM has successfully produce same result as produced by AHP method. Besides Jolai Model computes the values of Di+ and Di-which defines a particular range where optimized selection is resided and then the ranking is determined though human assessments can be varied and fluctuated in greater range. The strong feature of proposed model can clearly be stated that it generates a reliable selection result of supplier and it is independent of available mathematical methods incorporated. Hence proposed model proves its versatility.