Development of Subcontractor Selection Models Using Fuzzy and AHP Methods in the Apparel Industry Supply Chain


 The subcontractor selection decision is inherently a multicriterion problem. It is a decision of strategic importance for companies. The nature of this decision is usually complex and unstructured. Management science techniques might be helpful tools for solving these kinds of decision-making problems. In this research, the fuzzy logic method and the analytic hierarchy process were applied for the selection of suitable subcontractors in an apparel supply chain.
 In general, many factors, such as quality level, price offer, and delivery delay, were considered to determine the most suitable and reliable subcontractors that fit the company's strategy. This survey is carried out using the database of an apparel company manufacturing denim products.


Introduction
Nowadays subcontractors' categorization, selection, and performance evaluation are decisions of strategic importance for companies. Global competition, mass customization, high customer expectations, and harsh economic conditions are forcing companies to rely on external subcontractors to contribute a larger portion of parts, materials, and assemblies to fi nished products and to manage a growing number of processes and functions that were once controlled internally. The literature suggests that many studies are interested in this topic, one of them focused on supplier selection and evaluation using the multiple criteria decision-making model (MCDM) according to the concept TOPSIS based on the closeness coeffi cient, and as a result, this model gave the ranking of supplier and evaluation status of all suppliers [1]. Other research studies have used the fuzzy supplier selection algorithm, based on the ranking of the supplier, and found that the used model is an easy and realistic approach for supplier selection. It gives a concrete result by recording the purchasing experts' experiences and processes these with fuzzy logic arithmetic [2]. Maurizio et al. have used a fuzzy approach to determine the rank of the supplier, their attempts based on the fuzzy suitability index to determine the fi nal ranking, and they approved that the fuzzy approach is able to deal with linguistic variables [3]. Felix et al. have developed their research according to the model fuzzy analytic hierarchy process (AHP), they combined the fuzzy method with the AHP method to clarify the fuzzy. As a result, the used model is proved to be simple, less time taking, and having a less computational expense. They found that the use of fuzzy AHP does not involve cumbersome mathematical operation and it is easy to handle the multiattribute decision-making problems like global supplier selection. They found that the combined fuzzy-AHP has the ability to capture the vagueness of human thinking style and effectively solve multiattribute decision-making problems [4]. In addition, Sharon et al. have described a decision model that incorporates a decision maker's subjective assessments and applies fuzzy arithmetic operators to manipulate and quantify these assessments. This model treated the subjectivity in linguistic terms [5]. Some other research studies have used an integrated approach of fuzzy multiattribute utility theory and multiobjective programming for rating and selecting the best suppliers and allocating the optimum order quantities; however, this study integrates fuzzy AHP, fuzzy TOPSIS, and fuzzy MOLP to solve the problem of supplier selection and order allocation. At fi rst, they have used the fuzzy AHP to calculate the relative weights of supplier selection criteria; then, they have used the fuzzy TOPSIS for ranking of suppliers according to the selected criteria. Finally, the weights of the criteria and ranks of suppliers were incorporated into the MOLP model to determine the optimal order quantity from each supplier [6]. In the construction fi eld, many research studies have used fuzzy logic for the selection and evaluation of subcontractors [7][8][9]. Kumar et al. have used a fuzzy goal programming approach for solving the vendor selection problem with multiple objectives, in which some of the parameters are fuzzy and they showed that this approach has the capability to handle realistic situations in a fuzzy environment and provides a better decision tool for the vendor selection decision in a supply chain [10]. Many researchers have used the fuzzy logic to predict the hydrophobic nature of knitted fabrics [11] and others have used the fuzzy logic for modeling the residual bagging behaviors of denim fabric [12]. Weber and Current have developed their research according to the model multiobjective programming [13]. In the work of Soukoup, he used the payoff matrix model that allows defi ning several scenarios of the future behavior of suppliers [14]. Ellram used a vendor profi le analysis model to choose the best supplier [15]. Roodhooft and Konings have used a method based on the total cost to classify the supplier in descending order [16]. Hinkle et al. have used a cluster analysis model, which allows to group the suppliers according to the scores obtained [17]. In the fi eld of supply chain management and, in particular, in the selection and evaluation of textile subcontractors, the majority of articles in the literature are concerned with the selection of the supplier. Given the importance of outsourcing in the textile sector, we have focused on this theme and tried to model the selection of subcontractors in a clothing supply chain for specifi c orders by using two methods, namely, the AHP and the fuzzy logic.

Data set used
This work was carried out in a company specialized in manufacturing Denim products employing 350 persons with an annual production of 800,000 pieces. This company operates as an ordering party for several subcontractors and as a manufacturer of several items (pants, jackets, skirts, etc.) in small and medium orders for different international brands requiring high-quality level, good price, and short-time delivery.
In this study, a database composed of nine subcontractors of the company was used, and each subcontractor is characterized by four parameters, namely, daily production, late delivery, second choice ratio, and subcontractor price. The variation range of the dataset is indicated in Table 1.
As a sample, 30 production orders were used, and each production order is characterized by quantity, delivery delay, accepted defects ratio, and price per piece ( Table 2).

Fuzzy logic modeling
The fuzzy logic is an extension of crisp logic, which was fi rst proposed by Lotfi Zadeh [18]. In crisp logic, like binary logic, variables are true or false, 1 or 0. In fuzzy logic, a fuzzy set contains elements with only partial membership ranging from 0 to 1 to defi ne the uncertainty of classes that do not have clearly defi ned boundaries. For each input and output variable of a fuzzy inference system, the fuzzy sets are created by dividing the universe of discourse into several subregions, named in linguistic terms (high, medium, low, etc.).

Fuzzy logic steps:
Step 1: Defi ne the linguistic variables and terms Linguistic variables are the input or output variables of the system whose values are words or sentences from a natural language instead of numerical values.

Step 2: Develop membership functions
Membership functions are used in the fuzzifi cation and defuzzifi cation steps of a fuzzy logic system to map the nonfuzzy input values to fuzzy linguistic terms and vice versa. A membership function is used to quantify a linguistic term. There are different forms of membership functions such as triangular, trapezoidal, piecewise linear, Gaussian, or singleton [19] (Figure 19).

Step 4: Defuzzification
Fuzzy rule-based systems evaluate linguistic if-then rules using fuzzification, inference, and composition procedures. They produce fuzzy results that usually have to be converted into crisp output by defuzzification.

Step 3: Fuzzy inference
The fuzzy rules deduce knowledge about the state of the system according to the linguistic qualifications provided in the fuzzification stage.

Structuring a decision problem and the selection of criteria
The first step in AHP is to develop a graphical representation of the problem in terms of the overall goal, the criteria, and the decision alternatives ( Figure 2).

Developing a pairwise comparison matrix for each criterion
Step 1: Construct a pairwise comparison matrix (n*n) for criteria with respect to objective. The weights of the criteria should be calculated by using a pairwise comparison between criteria by applying Saaty's scales ranging from l to 9 (Table 3, 4).
Step 2: Normalize the resulting matrix: Sum the values in each column of the pairwise matrix.
Step 3: Divide each element in the matrix by its column total to generate a normalized pairwise matrix (Table 5).
Step 4: Calculate the row averages "Vs" of the normalized pairwise matrix, a weights vector is obtained:

Developing a pairwise comparison matrix for each alternative for each criterion
Step 1: Construct a pairwise comparison matrix (n*n) for criteria with respect to objective. The weights of the alternatives should be calculated by using a pairwise comparison between alternatives for each criterion by applying Saaty's scales ranging from l to 9. The alternative in the row is being compared to the alternative in the column.
The next table shows the comparison matrix for each alternative for each criterion: for criteria C1 (Table 6).
Step 2: Normalizing the resulting matrix: Sum the values in each column of the pairwise matrix.

Analytic hierarchy process
The process of hierarchical analysis is a structured technique for organizing and analyzing complex decisions, based on mathematics and psychology. It was developed by Thomas L. Saaty in 1970 [20]. It is used worldwide for a wide variety of decision-making processes whether government decisions, business, industry, health, shipbuilding, or education [21].

AHP process
The principal steps of the AHP algorithm are as follows: -Structure a decision problem and criteria selection.
-Develop a pairwise comparison matrix for each criterion.
-Develop a pairwise comparison matrix for each alternative for each criterion.
-Calculate the coefficient consistency.
-Obtain an overall relative score for each alternative.    The value of RI is related to the dimension of the matrix and will be extracted from the table (Table 8).
The CR has shown that a CR of 0.10 or less is acceptable to continue the AHP analysis. If the CR is greater than 0.10, it is necessary to revise the judgments to locate the cause of the inconsistency and correct it.

2.2.5.
Obtaining an overall relative score for each alternative: Step 1: A matrix of solutions was created. For each alternative, the average value of each criterion for each alternative was taken (Table 9).
Tsij: It represents the average vector for each alternative for each criterion.
Step 2: Determines the fi nal ratings of the subcontractors: multiply the matrix of the solution and the average vector.

Numerical example
In this numerical example, we applied the fuzzy logic and the AHP model to choose the best subcontractors for a production order.

Application of fuzzy logic method
Step 1: Defi nition of inputs and outputs The commands are taken as input and the subcontractors are taken as output into our system.
Step 3: Divide each element in the matrix by its column total to generate a normalized pairwise matrix (Table 7).
Step 4: Calculate the row averages "Ts" of the normalized pairwise matrix, a weights vector is obtained

Calculate the coeffi cient consistency
To validate the results of the AHP, the consistency ratio (CR) is calculated using the following equation: In which the consistency index (CI) is, in turn, measured through the following equation: With: ˄max: maximum eigenvalue of the matrix   Step 3: Fuzzy inference: In this step, we have entered the fuzzy rules that connect the subsets of inputs and outputs. This step is to determine the relationships between the input set and the outputs. Table 12 shows the relationships between inputs and outputs (H: high; M: medium; L: low) ( Table 12).
The operators used in this case are the "Mamdani"-type operators. So, the inference engine will be like this ( Figure 5).  Step 2: Develop membership functions Before creating the membership function, the classes of each input and output variable were determined in collaboration with an expert in industrial planning as already indicated in the tables (Table 10,      Step 4: Defuzzification In this step, the center of gravity method was used for defuzzification. Figure 6 shows the rules window under Matlab where we can choose our inputs for the system to predict results ( Figure 6).

Step 5: Model validation
To evaluate the fuzzy system, a dataset composed of 10 samples was used. Each input data is characterized by the order number, quantity, delivery delay, accepted defects ratio, and price per piece. Then, these data were tested in a real case. It should be noted that these orders, which are already assigned to well-chosen subcontractors, are selected from those which have been successfully performed. The selected subcontractors are called desired results, as shown in Figure  7. Afterward, these data were tested in our fuzzy logic system. The obtained results are mentioned in the same figure as the calculated results. Since each subcontractor is defined, the fuzzy model will calculate and deduct four outputs (daily production, the late delivery, the second choice ratio, and subcontractor price). To test the reliability of our system, the correlation between the desired and the calculated values for the three output parameters is established as plotted in Figure 7.
According to Figure 7, it is shown that we have obtained good correlation results. The correlation coefficients are 0.807, 0.702, 0.823, and 0.871 for, respectively, the daily production, the late delivery, the second choice defects ratio, and the subcontractor   Step 2: The AHP process is determined in Figure 8.
Step 3: The criteria matrix of decisions based on Saaty's scale is given as follows (Table 17).
Step 5: Develop a pairwise comparison matrix for each subcontractor for the criterion: quality (Table 19).
Step 6: Normalize the resulting matrix (Table 20). price. Our model was tested for a production order with the values of the following parameters summarized in Table 13.
The parameters of the subcontractors are used in Table 14.
The result was indicated in Table 15.

Application of AHP method
Step 1: Define the criteria for subcontractor selection for a production order with the following parameters (Table 16).  Acceptable results are obtained since the coeffi cient consistency is equal to 0.1.
Step 11: Develop a pairwise comparison matrix for each subcontractor for the criterion: delivery delay (Table 25).
Step 13: To validate the results, the CR was calculated as follows (Table 27).
Acceptable results are obtained since the coeffi cient consistency is equal to 0.08 less than 0.1.
Step 14: Develop a pairwise comparison matrix for each subcontractor for the criterion: price/piece (Table 28).
Step 16: To validate the results, the CR was calculated as follows (Table 30).
Acceptable results are obtained since the coeffi cient consistency is equal to 0.04 less than 0.1.
Step 7: To validate the results, the CR was calculated as follows (Table 21).
The coeffi cient consistency is equal to 0.05, so the results are considered good.
Step 8: Develop a pairwise comparison matrix for each subcontractor for the criterion: capacity (Table 22).
Step 10: To validate the results, the CR was calculated as follows (Table 24).

Subcontractors
Subcontractors rank for the production order Step 17: Obtain an overall relative score for each alternative by creating a matrix of solutions. For each subcontractor, the average value of each criterion for each subcontractor was taken (Table 31).
Step 15: The fi nal ratings of the suppliers were determined by multiplying the matrix of the solution and the average vector (Vs) (Figure 9).     From this fi gure, subcontractor number 4 represents the best choice to treat the production order.

Results and discussion
To evaluate the performance of the method AHP and fuzzy logic in predicting the best choice of the subcontractor, a dataset composed of 30 samples was used. Each input data is characterized by the order number, quantity, delivery delay, accepted defects ratio, and production order price per piece. Then, these data were tested in a real case. It should be noted that these orders, which are given to well-defi ned subcontractors, are selected from those which have been successful in the choice that was made. Table 32 presents the rank of the solution adopted by the company in the list of solutions determined by both the AHP method and the fuzzy logic.    these results, we can conclude that both methods are efficient, but, in our case, the AHP method was more efficient than the fuzzy logic method. Figures 11 and 12 give more details.
From Figure 10, we can conclude that the found results are very acceptable. The percentage of a coincidence for the AHP method with the choice of the company is equal to 87%, as for the fuzzy logic method, this percentage is about 77%. Based on

Conclusions
In this work, we have used two methods for the selection of subcontractors in a Denim manufacturing company. Indeed, the fi rst one was the fuzzy logic and represents a tool to understand practically the similarity, the preferences, and the uncertainty in the inference systems. The second one is the AHP, and it is very important to make a decision or to evaluate several options in situations where no possibility is perfect.
In our case study, it has been proved that the AHP method is more effi cient than the fuzzy logic method for the selection of the best subcontractors. Indeed, this interpretation is based on the coincidence percentage between the obtained solutions using the developed models and those corresponding to the best choice made by the managers in the company. Therefore, it can be concluded that the AHP model and the fuzzy logic method are feasible for predicting and selecting subcontractors in the supply chain of our Denim clothing company.