Optimization of customer satisfaction using fuzzy analytic hierarchy process method

. Today's competitive environment is characterized by accelerated product renewal, the rapid development of production, information and communication technologies, the computerization of many aspects of economic activity, and changing customer requirements. In such a competitive context, the customer has become the key factor in the manufacturing process, requiring products designed to fully meet his expectations at competitive prices. The question of customer satisfaction, and therefore the quality of the product, has become a major concern for companies. In this context, the purpose of this study is to help a company specialized in producing vehicles, and that aims to launch a new vehicle model, classifying and prioritizing the customers’ requirements, and thus determining which of the 6 available models meets the customers’ expectations the most, using a fuzzy AHP approach. This study was based on 5 Customer Requirements (CR) that represent the main characteristics of a vehicle : Price (CR1), Fuel Consumption (CR2), Security (CR3), Comfort (CR4), and Design (CR5). A sample of 5 customers was considered for this study. The results showed that price (CR1) and security (CR3) had the highest priorities among the customers’ requirements. Based on these results, we were able to classify the 6 vehicle models and determine which of these models meet the customers’ requirements the most, using the Analytic Hierarchy Process method (AHP).


Introduction
In today's industrial environment, the quality of the product must be taken into consideration right from the design phase, by integrating the customer's expectations and requirements into the initial product definition, as it is practically impossible to rectify a design error halfway through the manufacturing process.It is estimated that 70% of costs resulting from non-quality are due to poor design.In addition, the design of the product is practically responsible for 75% of product's costs.By the end of the design phase, almost 75% of the cost has already been predetermined, so cost-cutting opportunities are only available for the remaining 25%.In this context, the notion of "design for quality" appeared, and many tools and methods were developed over the years in order to take into account the customer's needs and requirements.In order to understand and identify the customers' needs and requirements even better, fuzzy logic and the AHP method are an important tool.
This paper aims to study the customers' requirements in order to optimize the customer satisfaction, using the fuzzy AHP method.The rest of the paper is organized as follows : in Section 2 we present a literature review on the AHP method and the fuzzy AHP method.In Section 3 we present the data used in our study.In Section 4 we analyze and discuss the results of our study.In Section 5 we provide a conclusion.

Theoretical method 2.1 Analytic Hierarchy Process method (AHP)
The Analytic Hierarchy Process method is a multi-criteria decision analysis method that was firstly established by Saaty in 1978 [1,2,3].This method can help making an overall decision by breaking down a complex problem into a multi-level hierarchical structure of goal, criteria, and alternatives.The top level of the hierarchy describes the goal of the problem, while the intermediate level denotes the factors or the criteria of the problem.Meanwhile, the bottom level contains the solutions or the alternatives considered [4].The AHP method performs pairwise comparisons to establish the relative importance of the variables at each level of the hierarchy, and evaluates each of the alternatives at the hierarchy's lowest level, in order to obtain the best decision among the available alternatives [5].The AHP method is very effective in the decision making, especially when there is subjectivity.It is very well suited to solving problems where the decision criteria can be hierarchically sorted into sub-criteria.The main steps of the AHP method are as follows [6,7,8] : The first step is to structure the problem and decompose it in a hierarchy that includes the goal we're trying to reach, the criteria of the problem and the available alternatives, as shown in Fig 1 [9].The next step is to establish the pairwise comparison matrix.The decision makers evaluate the relative importance of each criterion over the other criteria, using Saaty's scale in Table 1 [10,11].Let C = [cij]nxn be the pairwise comparison matrix for n criteria, where cij is the relative importance of criterion i over criterion j, and its reciprocal, 1/cij, is equal to the relative importance of criterion j over criterion i, cji.
In most cases, there are multiple experts or customers who express their judgments or opinions on the criteria, thus there are multiple comparison matrices, who will be aggregated to one collective comparison matrix, by calculating the average value of each element of the matrix.
In the AHP method, some judgments in the comparison matrix can be in conflict.For instance, let's suppose that an expert judges that criterion 1 is more important than criterion 2 and a lot more important than criterion 3. Logically, criterion 2 should be more important than criterion 3.But if the expert judges that criterion 2 is less important than criterion 3, then the judgments between the criteria 1, 2 and 3 are in conflict.
Without this step, weights can still be obtained, therefore it is overlooked by some researchers.However, it's necessary to examine the consistency of the matrix, using a consistency ratio CR that enables us to check the degree of consistency or inconsistency of the matrix.The consistency ratio may be calculated using equation (1) [12] : CI is the consistency index that can be estimated using equation (2) (λmax is the principal Eigen value and n is the number of parameters).RI is the random index that may be estimated using Table 2 proposed by Saaty (n is the number of parameters).The matrix is considered consistent if CR is inferior to 0.1.If CR is superior to 0.1, the matrix needs revision and the experts need to re-compare and re-evaluate the criteria.

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If the matrix is consistent, the next step is to rank and prioritize the criteria.The values of each element in the pairwise comparison matrix are divided by the total value of its column, to obtain a normalized matrix [13].The weight wk of each criterion expressed in percentage is the average value of all the elements of the corresponding criterion's row in the normalized matrix.The bigger the weight, the more important is the criterion.Now that the criteria are classified and prioritized, we can proceed to ranking and classifying the alternatives.The same process and steps are applied, the only difference is that the alternatives are classified for each criterion.Therefore there will be n comparison matrices.The output of this process is the preference index pi k of the alternative i for the criterion k.The preference index of the alternatives may be then calculated using equation (3).

Fuzzy AHP method (FAHP)
Despite the convenience of the AHP method in handling both qualitative and quantitative criteria of multi-criteria decision making problems based on decision makers judgments, vagueness and fuzziness that exist in many decision-making problems can contribute to some imprecise judgments of decision makers in conventional AHP approaches, which is why the fuzzy AHP approach was introduced.
The fuzzy AHP method can be viewed as an advanced analytic method developed from the traditional AHP method.Many researchers who have studied the fuzzy AHP method, which is the extension of Saaty's theory, have provided evidence and confirmed that it shows relatively more sufficient description of these kind of decision making processes compared to the traditional AHP method.

Applications of FAHP
Numerous fuzzy AHP methods and applications have been proposed by various researchers in the literature.
For example, Chang [14] developed a methodology that evaluates the performance of airports.They used the gray statistics method to select the criteria, and the FAHP method to determine the weights of the criteria.Finally, they adopted a fuzzy synthetic and TOPSIS approach for the ranking of the airports' performance.
Mosase [15] compared the accuracy of the AHP and the FAHP methods in determining the appropriate location of rain water harvesting in South East District of Botswana.He eventually concluded that even though the AHP method is widely used in the decision analysis, it can't model the uncertainties inherent in the criteria and the confidence of the decision maker.

FAHP process
The fuzzy AHP process is more or less the same as the AHP process described before, except for 3 differences : • 1 st difference : To handle the imprecision in the AHP method, the exact crisp numbers are replaced with fuzzy numbers that represent the linguistic expressions in fuzzy AHP.This enables us to tolerate the vague judgments by assigning membership degrees to exact numbers, in order to describe to what extent these numbers belong to a certain expression.• 2 nd difference : Before establishing the fuzzy pairwise comparison matrix, the decision makers evaluate the relative importance of each criterion over the other criteria.The Saaty's scale used in this evaluation is different with fuzzy numbers, as shown in Table 3. • 3 rd difference : At the end of the fuzzy AHP process, we obtain the fuzzy weights wi = (li, mi, hi) of the criteria and the alternatives.These fuzzy weights are defuzzified using equation (4).

Case study and data
The purpose of this study is to help a company specialized in producing vehicles, and that aims to launch a new vehicle model, classifying and prioritizing the customers' requirements, and thus determining which of the available models meets the customers' expectations the most, using a fuzzy AHP approach.This study was based on 5 Customer Requirements (CR) that represent the main characteristics of a vehicle, there are then 5 criteria in our AHP problem : Price (CR1), Fuel Consumption (CR2), Security (CR3), Comfort (CR4), and Design (CR5).
A sample of 5 customers was considered for this study.These 5 customers represent the decision makers in our AHP problem.The vehicles company has 6 models available.Therefore there are 6 alternatives in our AHP problem.

Hierarchy structure
After defining the problem and its main goal, and the chosen criteria as well as the alternatives, the hierarchy structure was established in
In our study, we have n = 5 parameters.Therefore according to Saaty's random index value in Table 2, RI = 1.12.
The principal Eigen value of our matrix is λmax = 5.417.Therefore, using equation ( 2), the consistency index is CI = The consistency ratio is inferior to 0.1, therefore the comparison matrix is considered consistent, and we can normalize the comparison matrix.The resultant normalized matrix is shown in Table 5.

Conclusion :
The most important criterion for the customers is the price of the vehicle, followed by the security of the vehicle and the fuel consumption.On the other hand, the design of the vehicle is not very important for the customers.

Alternatives classification
Now that the criteria are classified and prioritized, we can proceed to ranking and classifying the alternatives, this time using a simple AHP approach.As mentioned earlier, the alternatives are classified for each criterion, therefore there will be n = 5 comparison matrices.The output of the matrix of criterion k is the preference index pi k of the alternative i for the criterion k.
The customers are asked to make a pairwise comparison of the alternatives for each criterion, based on Saaty's scale in Table 1.After aggregating their evaluations and their respective matrices, we were able to establish the pairwise comparison matrices of the alternatives for each criterion, and determine the preference index pi k of the alternatives.The results for each criterion are shown in Tables 7, 8

Conclusion
In this paper, we tried to analyze and study the customers' requirements, and then determine and identify which of the available vehicle models meets the customers' needs and requirements the most, in order to optimize the customers' satisfaction, using a fuzzy AHP approach.First of all, a literature review of the AHP and fuzzy AHP methods was necessary in order to present the methodological process of our study.Then, we identified the data and parameters of our study : we determined a sample of customers to achieve a market research, which was crucial in our study, as well as the characteristics and requirements of the vehicle to be considered.After the data and the parameters were all set, we were able to conduct a market research, and based on its results, we were able to classify and prioritize the customers' requirements.The results showed that price and security had the highest priorities among the customers' requirements.Based on these results, we were able to classify the 6 vehicle models and determine which of these models meet the customers' requirements the most.

Fig 2 .
The first level of the hierarchy E3S Web of Conferences 469, 00073 (2023) ICEGC'2023 https://doi.org/10.1051/e3sconf/202346900073corresponds to the main goal of the problem, the second level to the criteria of the problem, and the third level to the alternatives, as explained above.

Table 6 .
Priorities of the criteria.
Based on these results, we can determine the preference index pi of the alternatives using equation (3), and thus prioritize the alternatives.The results are shown in Table12.

Table 12 .
Priorities of the alternatives.Based on the market research and on the customers' evaluations, and after the AHP study that was conducted, we conclude that the best alternative is the first one, therefore the vehicle model that meets the customers' requirements the most is model number 1, followed closely by model number 6.