It is proposed to use fuzzy similarity in fuzzy decision-making approach to deal with the supplier selection problem in supply chain system. According to the concept of fuzzy TOPSIS earlier methods use closeness coefficient which is defined to determine the ranking order of all suppliers by calculating the distances to both fuzzy positive-ideal solution (FPIS) and fuzzy negative-ideal solution (FNIS) simultaneously. In this paper we propose a new method by doing the ranking using similarity. New proposed method can do ranking with less computations than original fuzzy TOPSIS. We also propose three different cases for selection of FPIS and FNIS and compare closeness coefficient criteria and fuzzy similarity criteria. Numerical example is used to demonstrate the process. Results show that the proposed model is well suited for multiple criteria decision-making for supplier selection. In this paper we also show that the evaluation of the supplier using traditional fuzzy TOPSIS depends highly on FPIS and FNIS, and one needs to select suitable fuzzy ideal solution to get reasonable evaluation.
The overall objective of supplier selection process is to reduce purchase risk, maximize overall value to the purchaser, and build the closeness and long-term relationships between buyers and suppliers [
Previously, many methods have been proposed to solve the supplier selection problem, some of the popular ones being the linear weighting methods (LW) [
Rest of the paper is organized as follows. Section
Fuzzy logic is a logic that allows vagueness, imprecision, and uncertainty. Zadeh (1965) [
Wei and Chen introduced a fuzzy similarity measure [
Chen [
A generalized trapezoidal fuzzy number
The membership function
Assume that we have two generalized trapezoidal fuzzy numbers
Now, for example, addition
Notice that these operations for positive trapezoidal fuzzy numbers follows the same arithmetic rules as also earlier used in concept of using fuzzy TOPSIS, for example, in [
Degree of similarity between two generalized fuzzy numbers can be computed as
Notice that in normalized form where height equals one (
In this paper we follow basically the same process as given by Chen et al. [
Here we consider the importance weights of various criteria and the ratings of qualitative criteria as linguistic variables. Because linguistic assessments merely approximate the subjective judgement of decision-maker, we are using linear trapezoidal membership functions to capture the vagueness of these linguistic assessments [ a set of a set of a set on a set of performance ratings of
Linguistic variables for importance weight of each criterion.
Linguistic variables for ratings.
Assume that a decision group has
To avoid complexity of mathematical operations in a decision process, the linear scale transformation is used to transform the various criteria scales into comparable scales. The set of criteria can be divided into benefit criteria (the larger the rating, the greater the preference) and cost criteria (the smaller the rating, the greater the preference). Therefore, the normalized fuzzy decision matrix can be represented as
Considering the importance of each criterion, the weighted normalized fuzzy decision matrix can now be expressed as
Let fuzzy positive-ideal solution be
Next in method described in [
Approval status table.
Assessment status | |
---|---|
Do not recommend | |
Recommend with high risk | |
Recommend with low risk | |
Approved | |
Approved and preferred |
According to Table If If If If If
So basically assessment status from Table
Form a committee of decision-makers, and then identify the evaluation criteria.
Choose the appropriate linguistic variables for the importance weight of the criteria and the linguistic ratings for suppliers.
Aggregate the weight of criteria to get the aggregated fuzzy weight
Construct the fuzzy decision matrix and the normalized fuzzy decision matrix.
Construct weighted normalized fuzzy decision matrix.
Determine FPIS.
Calculate the similarity of each supplier from FPIS by calculating the similarity matrix, and then average similarity value for each supplier.
according to the average similarity value we can get the assessment status of each supplier and determine the ranking order of all suppliers.
Next we use numerical example to compare fuzzy TOPSIS and proposed method using fuzzy similarity and also to test how three different FPIS and FNIS criteria are effecting the results. We use the same numerical example which was given in [ profitability of supplier ( relationship closeness ( technological capability ( conformance quality ( conflict resolution (
The proposed method is applied to solve the problem, the computational procedure is summarized as follows:
Three decision-makers use the linguistic weighting variables shown in Figure
Importance weight of criteria from three decision-makers.
Criteria | Decision-maker | ||
H | H | H | |
VH | VH | VH | |
VH | VH | H | |
H | H | H | |
H | H | H |
Three decision-makers use the linguistic rating variables shown in Figure
Ratings of the five suppliers by three decision-makers under five criteria.
Criteria | Supplier | Decision-makers | ||
MG | MG | MG | ||
G | G | G | ||
VG | VG | G | ||
G | G | G | ||
MG | MG | MG | ||
MG | MG | VG | ||
VG | VG | VG | ||
VG | G | G | ||
G | G | MG | ||
MG | G | G | ||
G | G | G | ||
VG | VG | VG | ||
VG | VG | G | ||
MG | MG | G | ||
MG | MG | MG | ||
G | G | G | ||
G | VG | VG | ||
VG | VG | VG | ||
G | G | G | ||
MG | MG | G | ||
G | G | G | ||
VG | VG | VG | ||
G | VG | G | ||
G | G | VG | ||
MG | MG | MG |
Convert the linguistic evaluations in Tables
Normalize the fuzzy-decision matrix.
Build the weighted normalized fuzzy-decision matrix as in Table
Weighted normalized fuzzy decision matrix.
(0.35, 0.48, 0.56, 0.72) | (0.4, 0.63, 0.8, 1) | (0.49, 0.7, 0.74, 0.9) | (0.49, 0.64, 0.64, 0.81) | (0.49, 0.64, 0.64, 0.81) | |
(0.49, 0.64, 0.64, 0.81) | (0.64, 0.81, 1, 1) | (0.56, 0.78, 0.93, 1) | (0.49, 0.7, 0.74, 0.9) | (0.56, 0.72, 0.8, 0.9) | |
(0.49, 0.7, 0.75, 0.9) | (0.56, 0.75, 0.87, 1) | (0.49, 0.76, 0.86, 1) | (0.56, 0.72, 0.8, 0.9) | (0.49, 0.66, 0.7, 0.9) | |
(0.49, 0.64, 0.64, 0.81) | (0.4, 0.66, 0.77, 0.9) | (0.35, 0.58, 0.68, 0.9) | (0.49, 0.64, 0.64, 0.81) | (0.49, 0.66, 0.7, 0.9) | |
(0.35, 0.48, 0.56, 0.72) | (0.4, 0.66, 0.77, 0.9) | (0.35, 0.52, 0.65, 0.8) | (0.35, 0.53, 0.59, 0.81) | (0.35, 0.48, 0.56, 0.72) |
Determine FPIS and FNIS. Now according to three different criteria we get three different FPIS and FNIS which are given in Table
Fuzzy positive-ideal solutions and fuzzy negative-ideal solutions according to different criteria.
(0.9, 0.9, 0.9, 0.9) | (1, 1, 1, 1) | (1, 1, 1, 1) | (0.9, 0.9, 0.9, 0.9) | (0.9, 0.9, 0.9, 0.9) | |
(0.35, 0.35, 0.35, 0.35) | (0.4, 0.4, 0.4, 0.4) | (0.35, 0.35, 0.35, 0.35) | (0.35, 0.35, 0.35, 0.35) | (0.35, 0.35, 0.35, 0.35) | |
(1, 1, 1, 1) | (1, 1, 1, 1) | (1, 1, 1, 1) | (1, 1, 1, 1) | (1, 1, 1, 1) | |
(0, 0, 0, 0) | (0, 0, 0, 0) | (0, 0, 0, 0) | (0, 0, 0, 0) | (0, 0, 0, 0) | |
(0.49, 0.69, 0.75, 0.9) | (0.64, 0.81, 1, 1) | (0.56, 0.78, 0.93, 1) | (0.56, 0.72, 0.8, 0.9) | (0.56, 0.72, 0.8, 0.9) | |
(0.35, 0.48, 0.56, 0.72) | (0.4, 0.63, 0.77, 0.9) | (0.35, 0.52, 0.65, 0.8) | (0.35, 0.53, 0.59, 0.81) | (0.35, 0.48, 0.56, 0.72) |
Calculate the similarity of each supplier by using weighted normalized fuzzy decision matrix and FPIS by calculating the similarity matrix, and then average similarity value for each supplier, as in Tables
Similarities between FPIS and weighted normalized fuzzy decision matrix.
Average | ||||||
---|---|---|---|---|---|---|
0.54 | 0.56 | 0.61 | 0.67 | 0.67 | 0.61 | |
0.69 | 0.73 | 0.68 | 0.69 | 0.74 | 0.70 | |
0.69 | 0.66 | 0.64 | 0.74 | 0.68 | 0.68 | |
0.67 | 0.56 | 0.51 | 0.67 | 0.68 | 0.62 | |
0.54 | 0.56 | 0.48 | 0.57 | 0.54 | 0.54 | |
0.46 | 0.56 | 0.61 | 0.58 | 0.58 | 0.56 | |
0.58 | 0.73 | 0.68 | 0.61 | 0.65 | 0.65 | |
0.61 | 0.66 | 0.64 | 0.65 | 0.59 | 0.63 | |
0.58 | 0.56 | 0.51 | 0.58 | 0.59 | 0.56 | |
0.46 | 0.56 | 0.48 | 0.48 | 0.46 | 0.49 | |
0.81 | 0.79 | 0.86 | 0.88 | 0.88 | 0.84 | |
0.9 | 1 | 1 | 0.95 | 1 | 0.97 | |
1 | 0.93 | 0.95 | 1 | 0.93 | 0.96 | |
0.90 | 0.8 | 0.79 | 0.88 | 0.93 | 0.86 | |
0.81 | 0.8 | 0.76 | 0.80 | 0.78 | 0.79 |
Comparison of closeness coefficient (
Supplier | Class | Class | Class | Class | Class | Class | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
0.496 | III | 0.608 | IV | 0.632 | IV | 0.555 | III | 0.364 | II | 0.842 | V | |
0.642 | IV | 0.701 | IV | 0.728 | IV | 0.648 | IV | 0.884 | V | 0.969 | V | |
0.622 | IV | 0.682 | IV | 0.715 | IV | 0.630 | IV | 0.808 | V | 0.961 | V | |
0.511 | III | 0.617 | IV | 0.643 | IV | 0.564 | III | 0.414 | III | 0.859 | V | |
0.404 | III | 0.539 | III | 0.570 | III | 0.487 | III | 0.015 | I | 0.790 | IV |
According to the average similarity we get the ranking
In Table
Having pointed this one also should notice that this approval status Table
Approval status table.
Similarity value | Assessment status |
---|---|
Do not recommend | |
Recommend with high risk | |
Recommend with low risk | |
Approved | |
Approved and preferred |
From this approval status Table
In this paper a new approach is proposed to deal with the supplier selection problem in an uncertain environment. This problem to be solved is MCDM problem. In conventional MCDM methods, the ratings and the weights of attributes must be known precisely [