A multi-objective supplier selection and order allocation through incremental discount in a fuzzy environment

Abstract

It is generally believed that the process of supplier selection plays a critical role in the purchasing management. To improve the performance of a supply chain network, it is essential to build a strategic and a strong relationships. As such, all firms should select the best suppliers by applying appropriate methods through different selection criteria. An appropriate supplier reduces all purchasing costs as well as increases customer satisfaction to improve the final product and strengthen corporate competitiveness. Due to the natural uncertainty of this dilemma, most of recent works show a great deal of interest in applying uncertainty approaches. The main innovation of this paper is to develop a new multi-objective model for both supplier selection and order allocation operations considering incremental discount in a fuzzy environment. The proposed model considers the material cost with incremental discount and the transportation cost, holding costs along with its control and interest as well as the possibility of payment, brought back, and replacement costs, simultaneously, for the first time in this research area. Based on the proposed fuzzy model, the Zimmermann fuzzy approach is used in order to covert the model in a single objective form. Accordingly, a Genetic Algorithm (GA) is applied to solve the proposed problem. Based on the multi-objective optimization proposed, a Non-dominated Sorting GA (NSGA-II) is also employed to solve the developed model through the multi-objective assessment methodologies. Finally, a comprehensive evaluation and discussion based on the results are provided to reveal the performance of developed methodology.

Keywords

Supplier selection order allocation incremental discount zimmermann fuzzy approach genetic algorithm NSGA-II

1 Introduction

There is a great deal of interest in choosing the right suppliers due to its key role in all supply chain systems. The rich literature of supplier selection shows that its importance through the success of corporates is undeniable. This need to find an appropriate supplier is motivated by both industrial and academic practitioners during last decades, significantly. A well-designed network of supplier selection provides many benefits for all components of networks. It can reduce the cost of purchasing raw materials as well as the lead time of the ordered goods. Another advantage is to improve the quality of products. However, it also increases the competitiveness of the company [1]. Taking all of this into account, the supplier’s choice is not only the most important duty of the purchasing department for a supplier selection network but it also can be considered as a complicated multi-objective optimization problem with different conflicting objectives and several real-life constraints [2]. Based on this motivation, this study proposes a new multi-objective model for both supplier selection and order allocation operations considering incremental discount in a fuzzy environment.

As it is obvious, the choice of suppliers is a type of Multiple Criteria Decision Making (MCDM) modelling influenced by several factors based on the environment, final products and economic situation of producer [3]. In this regard, Dickson [4] identified 23 criteria which can be considered by purchasing managers in various issues of supplier’s choices. These criteria show that generally, there are more than one objective function in addition to minimizing total costs which can be considered into the optimization model such as maximizing quality and minimizing delivery time and other important factors as objective functions. These limitations alter the nature of supplier selection into a complicated multi-objective optimization problem [5]. In this case, the choice of supplier selection is very difficult due to different objective functions which should be considered simultaneously [6]. On the other hand, from real-world situations, there is always a conflict between these goals. Hence, the main goal would be to find a tradeoff between the main criteria of supplier selection. The MCDM techniques are efficient tools to evaluate a set of options, accordingly. In terms of purchaser conditions, all criteria can be weighted satisfactorily [7]. From another point of view, the role of unpredictable events during the process of supplier selection is undeniable. Actually, most of input information are not definitive, completely [8]. As such, there are many other limitations with some criteria which are defined by vague phrases like “very top quality” or “too low price”. Therefore, it is difficult to handle such ambiguities by using deterministic models. In regards to these difficulties, uncertainty tools via fuzzy or stochastic programming approaches are needed to better design a supplier selection network to get closer for real-world applications [3]. These reasons motivate several authors to employ the fuzzy sets theory as one of well-known tools of handling uncertainty in this area [3 –8].

The main contributions of this study consider several contents in the area of supplier selection included but not limited to the purchasing price of a product with incremental discounts, shipping costs, inspection, storage, and the interest rates at the final cost of the product. Another innovation of this paper is to formulate the problem based on the fuzzy theory to be more practical. In addition to this development, another goal of the proposed model is to maximize the level of services provided from the suppliers by considering the possibility of refund, replace, or purchase on installments. These issues leads to consider other dimensions of the supplier selection problem and ultimately more accurate selections for the final suppliers. Overall, the main limitations of the proposed model are the acceptable quality level, the minimum order of allocation, the storage, the supplier capacity, and the acceptable trust. Based on the literature, there is no similar study concerning all aforementioned items, simultaneously, in an integrated manner.

This study proposes a new multi-objective supplier selection problem with different parameters including the quality, the interest rate, the vehicle capacity, the product price, the discount rate, the order amount, the supplier capacity, the storage space, the acceptable trust, the taking refund, the replacement possibility, the installment purchase, the quality control and the warehousing cost. Furthermore, the demand is estimated by the fuzzy theory. The dependent variables of proposed methodology include the intended suppliers and the amount of the purchase. Taking all of this into account, a new multi-objective optimization model is proposed. To solve this complex problem, the Genetic Algorithm (GA) for its single objective form and a Non-dominated Sorting GA (NSGA-II) to perform a multi-objective evaluation are utilized. In brief, the main contributions of this paper can be summarized as follows:

A new multi-objective model is proposed for the supplier selection and order allocation through incremental discounts in a fuzzy environment.

The Zimmermann fuzzy methodology is applied to control the natural uncertainty of the model.

The proposed problem is solved by a GA for its single objective form and an NSGA-II is employed to do a multi-objective evaluation of developed problem.

The rest of this research is summarized as follows. Section 2 investigates and reviews the relevant works. Section 3 develops a modeling approach to formulate the proposed problem. Section 4 applies the proposed fuzzy approach. Section 5 introduces GA and NSGA-II to solve numerical instances. Finally, the conclusion and future directions are presented in Section 6.

2 Literature review

Academically, the literature of supplier selection is very rich in using of different mathematical programming approaches and soft computing techniques including heuristics and metaheuristics to solve such complicated models [9, 10]. To the best of our knowledge, the first article in this field was published in the early 1950s [11]. Since then, many studies were conducted on the supplier selection topic. One of the first studies refers to Dickson [4] who concluded that three factors including of standard quality, timely delivery of goods, and performance history were essential for choosing the right suppliers, properly.

In one of primary review studies in this research area, Weber et al., [12] reviewed 74 articles between 1961– 1990 related to the supplier selection and related criteria. They declared that the study of Dickson [4] in 1966 was to provide a broader view of the criteria from both aspects of academics and supply chain practitioners to consider the importance of choosing suppliers. They also argued that the supplier selection decisions were complicated due to different criteria which should be taken into account. The main purpose of this review was to identify the most important criteria and indicators. They came to the conclusion that the application of multi-objective programming techniques could be useful in the field of supplier selection due to conflicting objectives among criteria. They also notified that due to natural uncertainty in such systems, both randomness and fuzziness are needed to be explored more in future works. Later in 2000, another review work was conducted by Degraeve et al., [13], where different mathematical models in the field, the use of ownership total cost as the basis for comparing these models were suggested. Later in 2007, Aissaoui et al. [11] did a comprehensive overview for the area of supplier selection models and order lot sizing. Their study covered the relevant works from 1950 to 2006. They also completed the previous review studies by considering the entire procurement process and adding the internet-based environments to the main criteria. They focused their attention on the final selection of suppliers to identify the best group of suppliers as well as to assign the orders among the system to meet their needs. Another main recommendation of their work was to apply high-efficient solution algorithms to solve complicated models in this research area.

Generally, applying MCDM techniques and considering fuzzy operators are the main sectors from the literature [14 –18]. As may being indicated from studies published during the last decade, there are a few review works to explore the main features of MCDM techniques and fuzzy numbers to better present the literature gaps. In 2010, Ho et al., [18] provided the important works from 2000 to 2008. Their main findings were that among MCDM techniques, TOPSIS was utilized in most of studies. As such, the main evaluation criteria involved the total cost and the quality of final products. They also notified that fuzzy sets would be an interesting tool in future studies. Among several fuzzy approaches, the Zimmermann fuzzy technique can be utilized more in this area. Later in 2013, Chai et al., [19] in another review probed the application of all decision making models in the field of supplier selection from 2008 to 2012. They found 26 decision making techniques from three perspectives, i.e., MCDM, mathematical programming approaches and artificial intelligence techniques including popular heuristics and metaheuristics such as GA. Their main contributions of future works are mainly interested in developing new solution algorithms to solve such complicated models with different factors and objectives. At the last but not the least, from a recent study, Simić et al., [20] in 2017 provided a 50-year review of supplier selection based on aforementioned reviews. They revealed that how fuzzy set theory and hybrid solution algorithms can support this research area for the future works. In conclusion, the future trends of supplier selection from aforementioned review papers consider multi-objective optimization models via uncertainty tools as well as adding more factors about the components of a supplier selection network such as the discount rate, the order amount, the acceptable trust, the taking refund, the replacement possibility, the installment purchase and the quality control [17].

To better identify the recent advances in this research area and to show that how this work may fill the literature gaps, a brief explanation about relevant studies during the last decade is provided here. In 2009, Boran et al. [21] proposed a combination of the TOPSIS technique with fuzzy sets to select appropriate suppliers in a group decision making environment. As such, Wang et al. [22] offered the fuzzy hierarchical TOPSIS methodology and solved it by the choice of supplier problem. In another research, Awasthi et al. [23] provided a solution algorithm based on heuristics to the supplier selection problem considering the amount of demand under uncertainty. In 2010, Bhattacharya et al. [24] presented a hybrid approach of AHP and QFD to rank and to select the candidate suppliers in a multi-criteria environment and value-added supply chain framework. As such, Wu et al., [25] presented a fuzzy multi-objective decision-making model for the choice of supplier by considering different risk factors. In another interesting work, Liao and Kao [26], by using the Taguchi loss function as well as the AHP technique along with ideal planning approach, presented an integrated optimization model for the supplier selection problem. In 2011, Jiuping and Fang [27] introduced a new optimization model to formulate the supplier selection problem considering two levels of fuzzy sets i.e., the biphasic environment. Using a Particle Swarm Optimization (PSO) algorithm was applied to solve their NP-hard problem. In another different work in this year, Rezaei and Davoodi [28] proposed two multi-objective mixed integer nonlinear models for a multi-period and lot-sizing supplier selection problem by adding multi-products and multi-suppliers. Another innovation for selecting suppliers in this year was proposed by Aksoy and Ozturk [29]. They developed an optimization approach based on linear programming for the selection of suppliers and their performance evaluation in an environment where the Just-In-Time (JIT) system is being implemented. At the last but not the least in this year, Hassanzadeh Amin et al. [30] presented a two-phase methodology to address the choice of supplier and assignment order. In the first phase of their algorithm, they utilized the fuzzy SWOT technique for the final selection of the best suppliers. After that in the second phase, by using the fuzzy linear programming model, they determined the optimal purchasing amount from each supplier as the main output of their research.

In 2012, Zouggari and Benyousef [31] used two decision making phases for the group supplier selection problem. Therefore, they chosen suppliers and the optimal order allocation to each one. In the first phase, a fuzzy AHP was used with four classes to select suppliers, and in the second phase, they used a fuzzy TOPSIS technique in their decision making. They considered some criteria (i.e., price, quality, and delivery) and used Fuzzy TOPSIS to evaluate the optimal order regarding the selected suppliers. In another similar research, Lee et al., [32] offered a mixed integer programming model to formulate the problem of determining order allocation of goods as a multi-period and multi-product through several suppliers by considering incremental discounts. They ultimately performed an effective GA to solve the problem properly in real sizes. Their main goal was to minimize the total costs including the ordering cost, the storage, the buying and shipping under conditions where the lack of inventory was not allowed in the system. Their results revealed the efficiency and effectiveness of their proposed solution methodology.

In 2013, Arikan [33] utilized a fuzzy linear multi-objective programming model to select supplier with multiple sources. The main contributions of this optimization model was to minimize the total cost and to maximize the service quality as well as to maximize the goods amounts delivered on time simultaneously. Another innovation was to solve this fuzzy multi-objective model by the Zimmermann fuzzy methodology. In 2014, Ghadimi and Heavey [34] evaluated the sustainability criteria for a supplier selection problem in the industry of medical device. Their main innovation was to propose an Efficient Fuzzy Inference System (EFIS) to quantify the registered data based on each sub-criteria. In another interesting work, Li et al., [35] innovated a fuzzy inhomogeneous multi-attribute decision making approach to solve an outsourcing supplier selection problem. The main aim of the model was to select which supplier should be outsourced to better optimize the total cost of network.

In 2015, Azadnia et al., [36] introduced a coordinated method to formulate a rule-based weighted fuzzy along with fuzzy analytical hierarchy process though a multi-objective programming approach. This formulation was used to model a sustainable supplier selection and order allocation problem. In regards to outsourcing supplier selection, Wan et al., [37] proposed a new intuitionistic fuzzy linear programming approach to optimize their proposed two-stage logistic network. In a similar study, Torabi et al., [38] developed a bi-objective scenario-based possibilistic mixed integer programming model to cover both fortified suppliers with their backup. In another different study, Mazdeh et al. [39] considered the lot sizing impact in a supplier selection framework. They applied a simple single solution heuristic to solve the developed model due to its high complexity to solve large-scale test problems. In a recent research, Nourmohamadi Shalke et al. [40] proposed a sustainable supplier selection strategy considering the quantity discount for the first time. They offered a TOPSIS methodology to address their proposed problem. More recently, Cheraghalipour and Farsad [41] developed a bi-objective supplier selection and order allocation through quantity discount. The first objective was the total cost, while the second one was to cover the environmental emissions.

Based on the literature review presented and to the best of our knowledge, there is no similar study to consider the purchasing price of a product with incremental discounts, shipping costs, inspection, storage, and interest rates at the final cost of the product. In addition to minimize the total cost, other objectives aim to maximize the level of services provided from the suppliers by considering the possibility of refund, replace, or purchase on installments. It should be noted that no existing study has treated these four objectives functions simultaneously. The model is also in a fuzzy environment and the main characteristics of proposed problem are provided in the following section.

3 Proposed problem

Since human beings engage in business from all levels of their life by the tools of quantitative or qualitative terms or both, the importance and necessity of purchasing and choosing the right supplier are one of the important sectors of human life especially in particular of the producers. Choosing a suitable supplier in regards to the organization’s limitations and conditions is one of the significant activities of the purchasing management. Based on this motivation, the main goal of the selection process is to reduce purchasing risk as well as to maximize the total benefits for the buyer along with the extension, the closeness, and the creation of a lasting relationship between the buyer and the supplier.

From the recent works as mentioned in the literature review, there is much attention to uncertainty for designing a supplier selection network. Regarding the supplier’s risks, there are customer-related risks which are uncertain demands. Randomness of customer demands may result in failure of business or any financial distress. Therefore, the demand risk is one of important sectors to design a supplier selection network which should be considered when deciding on the supplier’s choice.

One of the key issues in the supplier selection is the needs and benefits of consideration among a large number of selection criteria. Accordingly, this possibility is always existed that there is no supplier that can meet all the selection criteria at the best candidate. In majority of cases, while a supplier may be the best with respect to one criterion, another supplier can meet the best in another criterion. Furthermore, regarding an organization that has adopted a strategy for producing quality products, the price issue is less important than product quality. On the other hand, such organizations may use prompt response and flexible production. These reasons indicate that the supplier’s response will be very impressive. In conclusion, as a supplier analysis, there is required an interaction between the selection criteria and the ability of the suppliers, comprehensively.

3.1 Assumptions

This research innovates the costs of shipping and warehousing, interest rates, to select the supplier and order allocation in the supply chain with a fuzzy approach through a multi-objective mathematical model simultaneously. Based on this new development, the main assumptions and suppositions of proposed model are as follows.

The proposed mathematical formulation can be classified as a fuzzy multi-objective programming approach.

There are only one type of products a variety of different suppliers.

There are different types of vehicles with their capacity and cost.

The proposed model covers both variable and fix shipping costs.

Interest rates are not fixed for all suppliers.

Triangular fuzzy numbers are supposed in the model.

Shortage from any supplier is not allowed.

Similar to many earlier studies, each supplier uses only one kind of transportation means.

Based on discount supposition, all suppliers can use incremental discounts.

3.2 Notations

All notations utilized in the proposed model including sets, parameters and decision variables are as follows.

Sets:

n Total number of suppliers

m_i The number of discounts provided by the supplier i

Parameters:

P_ij The price offered by the ith supplier at the discount level j

>K_i (%) The product quality percentage provided by the ith supplier

r_i (%) Percentage of trust on the ith supplier

D Demand for the product determined by the buyer

G Warehouse capacity determined by the buyer

C_i Production capacity of ith supplier

y_ij It gets 1, if the ith supplier is at the discount level j; 0 otherwise

b_ij The jth price level for the ith variable

f_i Percentage of the returned goods of the ith supplier

F Acceptable return upper limit on the buyer’s side

e_i Percentage of distrust to the ith supplier

E The upper limit of acceptable distrust

a_i Interest rate for the ith supplier

d_i The delay time until the delivery of the goods for the ith supplier

s The cost of goods inspection

h The cost of storing goods

u_i The capacity of the vehicle used by the ith supplier

t_i The fixed shipping cost of the ith supplier

T_i The variable shipping cost of the ith supplier

W₁ The value of returning the goods from the buyer’s point of view

W₂ The value of the exchanging the goods from the buyer’s point of view

W₃ The value of buying on installment from the buyer’s point of view

Decision variables:

V_i It gets 1, if the ith supplier is able to retrieve goods, 0 otherwise.

Q_i It gets 1, if the ith supplier is able to replace the goods with other suppliers, 0 otherwise.

Z_i It gets 1, if the ith supplier is able to buy on installment, 0 otherwise.

x_ij The number of purchased items from ith supplier at jth discount level

3.3 Problem formulation

Here, the considered multi-objective mathematical formulation is given as follows: $min z_{1} = \sum_{i = 1}^{n} \sum_{j = 1}^{m_{i}} ((P_{ij} (x_{ij} - b_{ij - 1})$ $+ \sum_{k = 1}^{j - 1} P_{ik} (b_{ik} - b_{ik - 1})) y_{ij}) * (1 + a_{i})^{- d_{i}}$ $+ \sum_{i = 1}^{n} \sum_{j = 1}^{m_{i}} x_{ij} (s + h) + \sum_{i = 1}^{n} \sum_{j = 1}^{m_{i}}$ $(⌈ \frac{x_{ij}}{u_{i}} ⌉ * t_{i} + x_{ij} . T_{i})$ (1) $max z_{2} = \sum_{i = 1}^{n} \sum_{j = 1}^{m_{i}} (w_{1} V_{i} + w_{2} Q_{i} + w_{3} Z_{i})$ (2) $max z_{3} = \sum_{i = 1}^{n} \sum_{j = 1}^{m_{i}} k_{i} x_{ij} \leq$ (3) $max z_{4} = \sum_{i = 1}^{n} \sum_{j = 1}^{m_{i}} r_{i} x_{ij}$ (4)s.t.: $\sum_{i = 1}^{n} \sum_{j = 1}^{m_{i}} x_{ij} \geq D$ (5) $\sum_{i = 1}^{n} \sum_{j = 1}^{m_{i}} x_{ij} \leq g$ (6) $\sum_{i = 1}^{n} \sum_{j = 1}^{m_{i}} x_{ij} \leq C_{i} i = 1, . . ., n$ (7) $y_{ij} {\begin{matrix} = 0, & if x_{ij} = 0 \\ = 1, & if x_{ij} > 0 \end{matrix} \forall i = 1, 2, . . ., n, \forall j = 1, 2, . . ., m_{i}$ (8) $b_{ij - 1} . y_{ij} < x_{ij} \leq b_{ij} . y_{ij} \forall i = 1, 2, . . ., n,$ $\forall j = 1, 2, . . ., m_{i}$ (9) $\sum_{j = 1}^{m_{i}} y_{ij} \leq 1 i = 1, 2, . . ., n$ (10) $\sum_{i = 1}^{n} \sum_{j = 1}^{m_{i}} f_{i} . x_{ij} \leq F$ (11) $\sum_{i = 1}^{n} \sum_{j = 1}^{m_{i}} e_{i} . x_{ij} \leq E$ (12) $(\sum_{i = 1}^{n} \sum_{j = 1}^{m_{i}} ((P_{ij} (x_{ij} - b_{ij - 1}) + \sum_{k = 1}^{j - 1} P_{ik} (b_{ik}$ $- b_{ik - 1})) y_{ij}) * (1 + a_{i})^{- d_{i}} + \sum_{i = 1}^{n} \sum_{j = 1}^{m_{i}} x_{ij} (s + h)$ $+ \sum_{i = 1}^{n} \sum_{j = 1}^{m_{i}} (⌈ \frac{x_{ij}}{u_{i}} ⌉ * t_{i} + x_{ij} * T_{i})) \leq B$ (13) $V_{i}, Z_{i}, Q_{i} \in {0, 1}, \forall i = 1, 2, . . ., n$ (14) $x_{ij} \geq 0 \forall i = 1, 2, . . ., n, \forall j = 1, 2, . . ., m_{i}$ (15)

The proposed model has four different objective functions. From Equation (1), the first objective function minimizes the cost borne by the buyer including the price of the product in addition to the incremental discount and the interest rate for the purchaser on the installment, the costs of the control and maintenance along with the shipping costs. Equation (2) presents the second objective function to maximize the level of services that suppliers provide for the buyers. The services in this model include the goods recall, the exchange of goods, and the possibility of installment purchase. Each of these items has a weight as a value from the buyer’s perspective. Equation (3) represents the maximizing of third objective function which relates to the quality of the goods. This objective function denotes that the buyer aims to improve the quality of his purchased products to the extent that the customer’s satisfaction may be improved. This issue depends on the quality level of suppliers. The last objective function as the fourth one is shown by Equation (4). This objective aims to maximize the level of trust to suppliers. This objective function also aligns with the buyer’s desire. This issue depends on the supplier’s trust-ability and loyalty to the words. Obviously, regarding this objective function, any supplier that provides a higher level of trust, its corresponding r_j would be higher and which could lead to more sales of its products.

Since the shortage is not allowable in the proposed model, Equation (5) ensures that all demand must be met. This means that the total number of products purchased from each supplier should be greater than or equal to the buyer’s demand. Additionally, the demand is under uncertainty and valued by fuzzy sets. Equation (6) indicates that the buyer can buy in most of cases along with the whole capacity of the warehouse. Since suppliers have the ability to produce a number of each product depending on the labor, tools and equipment, hours of work and so on. As such, Equation (7) limits the purchase according to the supplier’s production capacity. If a supplier is unable to meet all the requirements of the buyer, then the rest of the buyer’s orders will be provided with by other suppliers. Since each supplier provides some discounts, it is necessary to include a discount limitation in the model to determine the purchase value of the ith supplier at the level of discount j. To do this end, Equation (8) states that there is a relation between the parameter of y_ij and our decision variable (x_ij). In this case, if we have bought from the ith supplier at the jth discount level, then the related binary variable (y_ij) of the same supplier with that discount level is 1; otherwise, i.e., in the absence of buying from the supplier i at the discount level j, the corresponding y_ij variable is 0. If y_ij gets 1, the Equation (8) is validated by the same supplier and discount level to ensure that the amount purchased from the ith supplier is true at the desired discount level or not. By another point of view, if it is true, the buying is done at that discount level. It is assumed that the lower bound for the first interval is equal or more than 0 (b_i0 ≥ 0). Finally, the limitations of the discounts provided in the Equation (9) is checked. Since only a buyer can buy at least one discount level from every supplier, the Equation (10) will guarantee this fact. With this limitation, it is allowed that at most one of y_ij values for each supplier i becomes 1. In other words, in case of buying from the ith supplier, the purchase will include only one discount level. Equation (11) means that the buyer does not want the number of crash products found in packages purchased from different suppliers to be higher than some certain rate. With regard to the purchase record history of each supplier, a certain percentage of mistrust is considered for each supplier. Therefore, the trust Equation (12) is quite similar to the quality constraint. Considerably, Equation (13) represents that the total cost the buyer pays to different suppliers can be at most equal to the purchaser’s budget. Finally, two last constraints as Equations (14) and (15) are support the feasible values of decision variables.

4 Proposed fuzzy model

Recent years have seen a great deal of interest in applying fuzzy sets for modeling different real-world applications [42 –46]. In this section, first of all, a general fuzzy multi-objective model for the proposed supplier selection problem is introduced. After that some appropriate operators based on the Zimmermann’s method for this decision-making problem are discussed and evaluated. Finally, the proposed solution algorithm and a numerical example are provided.

Here, a linear type of proposed multi-objective model can be represented as follows: Find the vector x as the main decision variable of proposed model that is translated as x^T = [x₁, x₂, . . . x_n]. The main goal is to minimize the objective function of Z_k and to maximize objective function of Z_l [3]: $Z_{k} = \sum_{i = 1}^{n} c_{ki} x_{i}, k = 1, 2, . . ., p$ $Z_{l} = \sum_{i = 1}^{n} c_{li} x_{i}, l = p, p + 1, p + 2, . . ., q$ $x \in X_{d}, X_{d} = {x | g (x) = \sum a_{ri} x_{i} \leq b_{r}$ $r = 1, 2, . . ., m, x > 0}$ (16)

The last row relates to the problem constraints, and c_ki, c_li, a_ri and b_r are crisp or fuzzy values. In 1978, Zimmermann solved a set of problems such the model presented in Equation (16) using linear fuzzy programming. This author formulated a fuzzy linear program by separating each Z_j objective function into its maximum ( $Z_{j}^{+}$ ) and minimal ( $Z_{j}^{-}$ ) values with solving the problem given in Equation (17): $\begin{matrix} Z_{j}^{+} = max Z_{k}, x \in X_{a}, Z_{k}^{-} = min Z_{k}, x \in X_{d} \\ Z_{l}^{+} = max Z_{l}, x \in X_{d}, Z_{l}^{-} = min Z_{l}, x \in X_{a} \end{matrix}$ (17)

$Z_{k}^{-}$ and $Z_{l}^{+}$ are obtained through a multi-objective problem, so that the problem is solved each time by considering only one of the objective functions and the same problem constraints; x ∈ X_d means that the answers should be true within constraints, while x ∈ X_a is the set of all the optimal solutions obtained by solving the objective functions, individually. Given that for any objective function Z_j, its value changes linearly between $Z_{j}^{-}$ and $Z_{j}^{+}$ , it can be considered a fuzzy number with linear membership function μ_zj(x) [42] as shown in Fig. 1.

Fig.1

Objective function as a fuzzy number: Fig. (a) is to minimize the objective function Z_k (negative objectives), (b) is to maximize the objective function Z_l (positive objectives).

As stated before, a linear programming problem given in Equation (16) with fuzzy objectives and fuzzy constraints can be represented as given in Equation (18) [43]. Accordingly, find the x vector so that: $\begin{matrix} Z_{k}^{\sim} = \sum_{i = 1}^{n} c_{ki} x_{i} \leq Z_{k}^{0}, k = 1, 2, . . ., p \\ Z_{l}^{\sim} = \sum_{i = 1}^{n} c_{li} x_{i} \leq Z_{l}^{0}, l = p + 1, p + 2, . . ., q \\ s . t . : \\ g_{i}^{\sim} (x) = \sum_{i = 1}^{n} a_{ri} x_{i} \leq \sim b_{r}, r = 1, 2, . . ., h \\ g_{p} (x) = \sum_{i = 1}^{n} a_{pi} x_{i} \leq \sim b_{p}, p = h + 1, h + 2, . . ., m \end{matrix}$ (18)

In this model, the sign shows fuzzy environments. The symbol ≤ ∼ in the set of constraints means ≤ fuzzy and refers to “necessarily smaller or equal to” [44]. As such, Z_k^. and Z_l^. are the levels that the decision maker intends to achieve.

Assuming that membership functions are preference-based or linear satisfaction [45], the linear membership for the minimum objectives (i.e. Z_k) and the maximum objectives (i.e. Z_l) are defined as relations presented in Equations (19) and (20). Note that p and q are the total number of minimum and maximum objectives, respectively. $μ_{z_{k}} (X) = {\begin{matrix} 1 for Z_{k} \leq Z_{k}^{-} \\ (ZK+ (x) - Zk (x)) / (zK+ (x) - Zk- (x)) for Z_{k}^{-} \\ \leq Z_{k} (x) \leq Z_{k}^{+}, \\ k = 1, 2, . . ., p \\ 0 for Z_{k} \geq Z_{k}^{+} \end{matrix}$ (19) $μ_{z_{l}} (X) = {\begin{matrix} 1 for Z_{l} \geq Z_{l}^{+} \\ (Zl (x) - Zl- (x)) / (zl+ (x) - Zl- (x)) for Z_{l}^{-} \\ \leq Z_{l} (x) \leq Z_{l}^{+}, \\ l = 1, 2, . . ., \\ 0 for Z_{l} \leq Z_{l}^{-} \end{matrix}$ (20)

A linear membership function for fuzzy constraints is represented as given in Equation (21): $μ_{gr} (x) = {\begin{matrix} 1 g_{r} (x) \leq b_{r} \\ (1 - g_{r} (x) - b_{r}) / d_{r} b_{r} \leq g_{r} (x) \leq b_{r} \\ + d_{r}, r = 1, 2, . . ., h \\ 0 g_{r} (x) \geq b_{r} + d_{r} \end{matrix}$ (21) where d_r are, according to Fig. 2, constant values that indicate allowed violation from the limits of the rth inequality.

Fig.2

Linear membership function for fuzzy constraints.

4.1 Decision-machining operators:

First, the applied max-min operator is discussed, here. The used formulation was introduced by Zimmermann [46] between 1987 and 1993 for solving fuzzy multi-objective problems. After that the weighted additive operator is proposed, which enables the decision maker to assign different weights in regards to different criteria.

In the fuzzy programming modeling, using Zimmermann’s method, a fuzzy solution will be obtained by union among all fuzzy sets, including fuzzy objectives or fuzzy constraints. The fuzzy solution for all fuzzy objectives and h fuzzy limitation can be presented as follows: $μ_{D} (X) = {{⋂_{j = 1}^{g} μ_{Z_{j}} (X)} ⋂ {⋂_{r = 1}^{h} μ_{g_{r}} (X)}}$ (22)

Accordingly, the optimal answer (x*) is obtained through the relation presented in Equation (23): $μ_{D} (x^{*}) = max_{x \in X_{d}} μ_{D} (x) = max_{x \in X_{d}} min$ $[min_{j = 1, 2, . . ., q} μ_{Z_{j}} (x), min_{r = 1, 2, . . ., h} μ_{g_{r}} (x)]$ (23)

To reach the optimal solution (x*) in the above model, we solve its equivalent i.e., the deterministic model given in Equation (24) [47]: $\begin{matrix} Maximize λ \\ st : \\ λ \leq μ_{Z_{j}} (x), j = 1, 2, . . ., q \\ λ \leq μ_{g_{r}} (x), r = 1, 2, . . ., h \\ g_{p} (x) \leq b_{p} (x), p = 1, 2, . . ., h + 1, h + 2, . . ., m \end{matrix}$ (24) where μ_D (x) and μ_Zj (x) and μ_gr (x) respectively represent the members of the solution functions, the objective functions, and the constraints. In this solution, the relationship between objective functions and constraints in a fuzzy environment is completely symmetric [48]. In other words, in this definition of fuzzy decision, there is no difference between fuzzy objectives and fuzzy constraints. Therefore, depending on the supplier’s selection, conditions and situations in which fuzzy objectives and fuzzy restrictions do not have the same importance for the supplier, should be taken into consideration. In conclusion, the weighted additive model can be used to solve this problem as follows:

The weighted additive model has been widely used in vector objective optimization problems; the basic concept is to employ a solitary utility function that expresses the overall decision-maker’s function in order to extract the relative importance of the criteria [49, 50]. In this case, by multiplying each membership function of fuzzy goals in their respective weights, and then adding results to each other, there will be a linear weighted utility function. The weighted additive model was applied in many earlier studies [51] as suggested in Equation (25): $\begin{matrix} μ_{D} (x) = \sum_{i = 1}^{q} w_{j} μ_{Z_{j}} (x) + \sum_{r = 1}^{h} β_{r} μ_{g_{r}} (x) \\ \sum_{j = 1}^{q} w_{j} + \sum_{r = 1}^{h} β_{r} = 1, w_{j}, β_{r} \geq 0 \end{matrix}$ (25) where w_j and β_r are weight coefficients that represent the relative importance of fuzzy objectives and fuzzy constraints. In order to solve the above model, a deterministic linear programming model is constructed that is conceptually equivalent to this model. In the subject of presented literature review, two deterministic linear programming models are introduced that are equivalent to this fuzzy model. The first model was utilized in many earlier studies [43 –50] as described as relation given in Equation (26) and the second model is investigated by Equation (27). The numerical example of this study using the model presented in Equation (27) will be translated into a linear programming problem and solved by the proposed metaheuristic and exact solver. $\begin{matrix} max λ \\ st : \\ w_{j} λ \leq μ_{Z_{j}} (x), j = 1, 2, . . ., q \\ β_{r} λ \leq μ_{g_{r}} (x), r = 1, 2, . . ., h \\ g_{p} (x) \leq b_{p}, p = h + 1, h + 2, . . ., m \\ λ \in [0, 1], j = 1, 2, . . ., q and r = 1, 2, . . ., h \end{matrix}$ (26) $\begin{matrix} \sum_{j = 1}^{q} w_{j} + \sum_{r = 1}^{h} β_{r} = 1 \\ x_{i} \geq 0, i = 1, 2, . . ., n and λ \in [0, 1] \end{matrix}$ $\begin{matrix} max \sum_{j = 1}^{q} w_{j} λ_{j} + \sum_{r = 1}^{h} β_{r} γ_{r} \\ s . t . \\ λ_{j} \leq μ_{Zj} (X), j = 1, 2, . . ., q \\ γ_{r} \leq μ_{gr} (X), r = 1, 2, . . ., h \end{matrix}$ $g_{p} (X) \leq b_{p}, p = h + 1, . . ., m$ (27) $λ \in [0, 1], j = 1, 2, . . ., q and r = 1, 2, . . ., h$ $\sum_{j = 1}^{q} w_{j} + \sum_{r = 1}^{h} β_{r} = 1$ $X_{i} \geq 0, i = 1, 2, . . ., n and λ \in [0, 1]$

4.2 Designation of solution algorithm

Generally speaking, the proposed problem can be solved by using following steps.

Step 1. Design a supplier selection model in accordance with the buyer and supplier’s criteria and constraints.

Step 2. Solve the problem of selecting multi-objective suppliers by considering just one objective function. We do this for all objective functions, in order to obtain the best values for the objective functions.

Step 3. Using the results of step two, determine the corresponding values for each objective.

Step 4. From step three, for each objective function, we find a lower and a higher limit in accordance with the set of solutions for each solution. For the jth objective function (Z_j) as given in Equation (17), Z_j^- and Z_j⁺ indicates the upper and lower limits, respectively.

Step 5. We obtain membership functions for the objective functions and fuzzy constraints in regards to given Equations (19) to (21).

Step 6. Formulate a crisp model equivalent to fuzzy optimization problem through the fifth step and decision-maker preferences based on fuzzy weighted additive decision making.

Step 7. Obtain the optimal vector x^*, which is the efficient solution to the multi-objective provider selection problem with the decision makers’ preferences (i.e., the very initial problem).

4.3 Numerical example

Here, a hypothetical example is provided to better describe the model and to implement the solution algorithm illustrated earlier.

To better understand the model, it is assumed that three suppliers should be considered. The price of each product by each supplier is at three levels as given in Table 1. The rest of parameters in the problem are set. As such, the shopping criteria include cost, service level, quality, and trust level. Note that the demand restrictions, the storage capacity, and the supplier capacity, the acceptable quality of product, the acceptable trust to supplier and the budget should also be met. These parameters are given in Table 2. Notably, w₁, w₂ and w₃ are respectively 0.4, 0.2 and 0.4. As such, the rest of parameters including s, h, g, F, E, D and B are respectively 3, 2, 850, 30, 35, 800 and 71000.

Table 1
Price levels provided by each supplier

Supplier (i) b _i0 p _i1 b _i1 p _i2 b _i2 p _i3

1 0 78 100 75 200 70

2 0 120 130 105 230 91

3 0 99 110 95 210 90

Supplier (i)	p _i1	b _i1	p _i2	b _i2	p _i3
1	78	100	75	200	70
2	120	130	105	230	91
3	99	110	95	210	90

Table 2

Problem parameters

Supplier	c _i	a _i	d _i	u _i	t _i	T _i	k _i	r _i	f _i	e _i	V _i	Q _i	Z _i
(i)		(%)					(%)	(%)	(%)	(%)
1	400	4	2	100	10	3	80	80	3	6	1	1	1
2	600	3	3	80	5	1	93	70	5	5	0	0	0
3	400	4	1	90	6	2	85	90	4	2	0	1	1

Therefore, the proposed multi-objective linear formulation for this numerical example is shown in Equation (28): $min z_{1} = 72.072 x_{11} y_{11} + 69.3 x_{12} y_{12} - 277.2 y_{12}$ $+ 64.68 x_{13} y_{13} + 1202 y_{13} + 109.8 x_{21} y_{21} + 96.075$ $x_{22} y_{22} + 1784.25 y_{22} + 83.265 x_{23} y_{23} + 4730.55 y_{23}$ $+ 95.139 x_{31} y_{31} + 91.295 x_{32} y_{32} + 422.83 y_{23}$ $+ 86.49 x_{33} y_{33} + 1431.89 y_{33} + 8.1 (x_{11} + x_{12} + x_{13})$ $+ 6.062 (x_{21} + x_{22} + x_{23}) + 7.066 (x_{31} + x_{32} + x_{33})$ $max z_{2} = x_{11} + x_{12} + x_{13} + 0.6 (x_{31} + x_{32} + x_{33})$ $max z_{3} = 0.8 (x_{11} + x_{12} + x_{13}) + 0.93 (x_{21} + x_{22} + x_{23})$ $+ 0.85 (x_{31} + x_{32} + x_{33}) max z_{4} = 0.8 (x_{11} + x_{12}$ $+ x_{13}) + 0.7 (x_{21} + x_{22} + x_{23}) + 0.9 (x_{31} + x_{32} + x_{33})$ $st :$ $x_{11} + x_{12} + x_{13} + x_{21} + x_{22} + x_{23} + x_{31} + x_{32} + x_{33} ≅ 800$ $x_{11} + x_{12} + x_{13} + x_{21} + x_{22} + x_{23} + x_{31} + x_{32} + x_{33} \leq 800$ $x_{11} + x_{12} + x_{13} \leq 400 x_{21} + x_{22} + x_{23} \leq 600$ $x_{31} + x_{32} + x_{33} \leq 400$ (28) $\begin{matrix} y_{ij} = {\begin{matrix} 0 \\ 1 \end{matrix} \begin{matrix} if x_{ij} = 0 \\ if x_{ij} > 0, i = 1, 2, 3, j = 1, 2, 3 \end{matrix} \\ 0 < x_{11} < 100 y_{11}, 100 y_{12} < x_{12} < 200 y_{12}, \\ 200 y_{13} < x_{13}, \\ 0 < x_{21} < 130 y_{21}, 130 y_{22} < x_{22} < 130 y_{22}, \\ 230 y_{23} < x_{23}, \\ 0 < x_{31} < 110 y_{31}, 110 y_{32} < x_{32} < 210 y_{32}, \\ 210 y_{33} < x_{33}, \\ y_{11} + y_{12} + y_{13} \leq 1, \\ y_{21} + y_{22} + y_{23} \leq 1, \\ y_{31} + y_{32} + y_{33} \leq 1, \\ 0.03 (x_{11} + x_{12} + x_{13}) + 0.05 (x_{21} + x_{22} + x_{23}) \\ + 0.04 (x_{31} + x_{32} + x_{33}) \leq 30, \\ 0.06 (x_{11} + x_{12} + x_{13}) + 0.05 (x_{21} + x_{22} + x_{23}) \\ + 0.02 (x_{31} + x_{32} + x_{33}) \leq 35, \end{matrix}$ $\begin{matrix} (72.072 x_{11} y_{11} + 69.3 x_{12} y_{12} - 277.2 y_{12} \\ + 64.68 x_{13} y_{13} + 1202 y_{13} \\ + 109.8 x_{21} y_{21} + 96.075 x_{22} y_{22} + 1784.25 y_{22} \\ + 83.265 x_{23} y_{23} + 4730.55 y_{23} \\ + 95.139 x_{31} y_{31} + 91.295 x_{32} y_{32} + 422.83 y_{23} \\ + 86.49 x_{33} y_{33} + 1431.89 y_{33} \\ + 8.1 (x_{11} + x_{12} + x_{13}) + 6.062 (x_{21} + x_{22} + x_{23}) \\ + 7.066 (x_{31} + x_{32} + x_{33}) \leq 71000 \end{matrix}$

The data set for lower and upper bound values of the objective functions and a fuzzy number for demand are presented in Table 3.

Table 3

Data sets for membership functions

	μ= 0	μ= 1	μ= 0
Z₁ (Cost)	–	69168.5	71000.11
Z₂ (Service Level)	590.8	640	–
Z₃ (Quality Level)	660	674.27	–
Z₄ (Trust Level)	663.6	690.5	–
Demand	700	800	850

As mentioned earlier, the four objective functions i.e., Z₁, Z₂, Z₃ and Z₄ are cost, service, quality, and trust respectively, and x_ij as the main decision variable is the number of units purchased from the ith supplier at the jth discount level. Regarding the fuzzy objective functions and the demand constraints for the above problem, we use linear membership function as pointed out from steps 2 to 5. The data set for lower and upper bound values of the objective functions and a fuzzy number for demand are presented in Table 3. As such, the membership functions for the four objective functions and the demand constraint are in the form of relations to Equations (29) to (33) and Figs. 3 to 7 (i.e., Step 5):

Fig.3

Membership function for cost objective function.

Fig.4

Membership function for service objective function.

Fig.5

Membership function for quality objective function.

Fig.6

Membership function for trust objective.

Fig.7

Membership function for demand constraint.

μ_{Z_{1}} = {\begin{matrix} 1 \\ (71000.11 - Z_{1}) \\ / 1831.61 \\ 0 \end{matrix} \begin{matrix} Z_{1} \leq 69168.5 \\ 69168.5 \leq Z_{1} \\ \leq 71000.11 \\ Z_{1} \geq 71000.11 \end{matrix}

(29)

μ_{Z_{2}} = {\begin{matrix} 1 \\ (Z_{2} - 590.8) \\ / 49.2 \\ 0 \end{matrix} \begin{matrix} Z_{2} \geq 640 \\ 590.8 \leq Z_{2} \\ \leq 640 \\ Z_{2} \leq 590.8 \end{matrix}

(30)

μ_{Z_{3}} = {\begin{matrix} 1 \\ (Z_{3} - 674.27) \\ / 14.27 \\ 0 \end{matrix} \begin{matrix} Z_{3} \geq 674.27 \\ 660 \leq Z_{3} \\ \leq 674.27 \\ Z_{3} \leq 660 \end{matrix}

(31)

μ_{Z_{4}} = {\begin{matrix} 1 \\ (Z_{4} - 663.6) \\ / 26.9 \\ 0 \end{matrix} \begin{matrix} Z_{4} \geq 690.5 \\ 663.6 \leq Z_{4} \\ \leq 690.5 \\ Z_{4} \leq 663.6 \end{matrix}

(32)

μ_{g_{d}} (x) = {\begin{matrix} \frac{d (x) - 700}{100} 700 \leq d (x) \leq 800 \\ \frac{850 - d (x)}{50} 700 \leq d (x) \leq 800 \\ 0 d (x) \leq 700 and d (x) \geq 850 \end{matrix}

(33)

If we assume that the decision maker considers the relative importance weights for objectives and constraints to be the same, the model comes in the following single-objective and deterministic form as given in Equation (34):

$\begin{matrix} max z \\ st : \\ z \leq (\frac{71000 - z_{1}}{1831.61}) \\ z \leq (\frac{z_{2} - 590.8}{49.2}) \\ z \leq (\frac{z_{3} - 660}{14.27}) \\ z \leq (\frac{z_{4} - 663.6}{26.9}) \\ z \leq (\frac{850 - z_{5}}{50}) \\ z \leq (\frac{z_{5} - 700}{100}) \\ z_{5} = x_{11} + x_{12} + x_{13} + x_{21} + x_{22} + x_{23} + x_{31} + \\ x_{32} + x_{33} \\ x_{11} + x_{12} + x_{13} + x_{21} + x_{22} + x_{23} + x_{31} + x_{32} \\ + x_{33} ≅ 800 \\ x_{11} + x_{12} + x_{13} + x_{21} + x_{22} + x_{23} + x_{31} + x_{32} \\ + x_{33} \leq 850 \end{matrix}$ $\begin{matrix} x_{11} + x_{12} + x_{13} \leq 400 \\ x_{21} + x_{22} + x_{23} \leq 600 \\ x_{31} + x_{32} + x_{33} \leq 400 \end{matrix}$ $\begin{matrix} y_{ij} = {\begin{matrix} 0 \\ 1 \end{matrix} \begin{matrix} if x_{ij} = 0 \\ if x_{ij} > 0, i = 1, 2, 3, j = 1, 2, 3 \end{matrix} \\ 0 < x_{11} < 100 y_{11}, 100 y_{12} < x_{12} < 200 y_{12}, \\ 200 y_{13} < x_{13}, \\ 0 < x_{21} < 130 y_{21}, 130 y_{22} < x_{22} < 130 y_{22}, \\ 230 y_{23} < x_{23}, \\ 0 < x_{31} < 110 y_{31}, 110 y_{32} < x_{32} < 210 y_{32}, \\ 210 y_{33} < x_{33}, \\ y_{11} + y_{12} + y_{13} \leq 1, \\ y_{21} + y_{22} + y_{23} \leq 1, \\ y_{31} + y_{32} + y_{33} \leq 1, \\ 0.03 (x_{11} + x_{12} + x_{13}) + 0.05 (x_{21} + x_{22} + x_{23}) \\ + 0.04 (x_{31} + x_{32} + x_{33}) \leq 30, \\ 0.06 (x_{11} + x_{12} + x_{13}) + 0.05 (x_{21} + x_{22} + x_{23}) \\ + 0.02 (x_{31} + x_{32} + x_{33}) \leq 35, \end{matrix}$ $\begin{matrix} (72.072 x_{11} y_{11} + 69.3 x_{12} y_{12} - 277.2 y_{12} \\ + 64.68 x_{13} y_{13} + 1202 y_{13} + 109.8 x_{21} y_{21} \\ + 96.075 x_{22} y_{22} + 1784.25 y_{22} + 83.265 x_{23} y_{23} \\ + 4730.55 y_{23} + 95.139 x_{31} y_{31} + 91.295 x_{32} y_{32} \\ + 422.83 y_{23} + 86.49 x_{33} y_{33} + 1431.89 y_{33} \\ + 8.1 (x_{11} + x_{12} + x_{13}) + 6.062 (x_{21} \\ + x_{22} + x_{23}) + 7.066 (x_{31} + x_{32} + x_{33}) \leq 71000 \end{matrix}$ (34)

The solution that is obtained after the problem solving with the help of exact solver i.e. Lingo is as follows: $\begin{matrix} x_{13} = 400 \\ x_{12} = 8 \\ x_{33} = 400 \end{matrix}$

The values of the objective function with regard to this solution are as follows:

The value of the cost objective function = 70095.39

The value of the service objective function = 640

The value of the quality objective function = 667.44

Value of the trust objective function = 6.685

Required amount = 808

5 Applying GA and NSGA-II to solve the proposed problem

Due to the complexity of the proposed model especially in large-scale networks, it is necessary to solve the problem by metaheuristics. As mentioned before, the literature of supplier selection problems is very rich in using metaheuristics [47 –56]. One of successful algorithms which has shown its performance in this research area is the GA. The GA is revealed its high-efficiency in other optimization problems such as protection of facilities [47], logistic network design [49] and constraint optimization problems [57] and so on. Its wonderful history is a motivation for us to apply GA and its multi-objective version, namely, NSGA-II to solve the proposed problem. The encoding and decoding of the proposed problem is as the same of Xu and Fang [27]. Therefore, readers can look at their paper to understand how the proposed algorithm can be applied for the considered model.

In this section, first of all, the proposed algorithms will be tuned by a Response Surface Method (RSM) to increase its efficiency. After that the results of GA will be checked by an exact solver for the single-objective form of problem to validate its performance. Finally, a multi-objective version of GA as a Non-dominated Sorting GA (NSGA-II) will be applied to solve the large samples by using multi-objective assessment metrics. Note that since both metaheuristics are well-known and were utilized in many earlier studies, readers can refer to them to have more description of algorithms’ steps [47 –52].

5.1 Parameter tuning

Experiment design is a sequence of experiments in which the desired changes are applied to the input process or system variables so that we can see and determine the causes of the variation in the solution [53]. In fact, the purpose of the experiments design is to determine the variables that have the greatest impact on y, to determine the position of the controllable variables x, where y is always close to the nominal value, to determine the position of the controllable variables x such that y always be close to the desired face value, to determine the position of the controllable variables x so that y changeability be small, or to determine the position of the controllable variables x so that the effects of uncontrolled variables be minimized. This is the main structure to design a set of experiments such the used treatments for tuning of metaheuristics [54].

In the design of experiments, the selection of factors and their levels, selection of the solution variable and the design of trials should be done correctly so that the obtained results meet the demands as satisfactory as possible [55, 56]. From another point of view, different experimental designs have been identified and investigated, each of which has its own application. Here, the Response Surface Method (RSM) is employed for tuning of applied algorithms i.e., GA and NSGA-II.

To identify this methodology, note that if the result of a process (y) is influenced by several variables (x), the objective function is defined as y = f (x₁, x₂, . . . , x_n) + ɛ so that ɛ is the observed error in the y solution. If the proposed mathematical model would be considered as a transformed function as E (y) = f (x₁, x₂, . . . , x_n) = φ, after that the procedure that is specified by φ = f (x₁, x₂, . . . , x_n) is called the solution procedure [57]. Based on this supposition in RSM, finding the appropriate approximation for the actual relationship between y and the set of independent variables can be the main goal of optimization in this methodology [58]. If the solution is well modelled by a linear function of the independent variables, then the approximation function would be the first order model as given in Equation (35), and if there is a curvature in system, then higher order polynomials such as the second order model as given in Equation (36) would be utilized. $y = β_{0} + B_{1} x_{1} + B_{2} x_{2} + . . . + B_{k} x_{k} + ɛ$ (35) $y = β_{0} + \sum_{i = 1}^{k} B_{i} x_{i} + \sum_{i = 1}^{k} B_{ii} x_{i}^{2} + \sum_{i} \sum_{j} B_{ii} x_{i} x_{j} + ɛ$ (36)

The least squares method can be used to estimate the model’s parameters and the objective function. After the estimation and recognition of the response procedure, we must find such levels of the variables that result in an optimal response solution. To do this end, the steepest ascend method for a maximization case can be utilized. As such, the steepest descent method would be suggested for a minimization case problem. The main goal is to move in a path with a maximum gradient in the direction that maximizes the increase or decrease for the amount of objective function. In the following, this methodology is applied for employed algorithms i.e., GA and NSGA-II to solve the proposed model, satisfactorily.

As mentioned earlier, to achieve the best values of the algorithms’ parameters, first of all, the experimental design approach is utilized and after that the effects of the parameters on the obtained solutions have been measured. At the last but not the least, to optimize the estimated response function and using the applied RSM, a combination of effective factors is employed by this study.

Additionally, since there may be a curvature in the range of search solutions, the Central Composite Design (CCD) is selected from a partial factorial, 2^k - p design with four central points. This implementation is applied to tune the algorithms. Each parameter of algorithms is a factor in this methodology. So, both GA and NSGA-II have four factors (k = 4). Accordingly, each factor has three levels i.e. low, medium, and high shown with (-1), (0) and (+1), respectively. Overall, Table 4 presents all aforementioned details of algorithms’ calibration.

Table 4

Search domain and parameters levels

Algorithm	Parameter	Range	Low	Medium	High
			(-1)	(0)	(+1)
	nPop	100– 200	100	150	200
	P _c	0.4– 0.8	0.4	0.6	0.8
GA	P _m	0.05– 0.2	0.05	0.125	0.2
	nIt	50– 100	50	75	100
	nPop	50– 100	50	75	100
	P _c	0.4– 0.8	0.4	0.6	0.8
NSGA-II	P _m	0.05– 0.2	0.05	0.125	0.2
	nIt	50– 100	50	75	100

In the central composite design, two parameters must be considered: (1) the distance α of the pivots from the center of the design; and (2) the number of central performances. The value of the parameter α depends on the spherical property of the design. Since the target area is cubic, the central composite face-centered design is used with α= 1 [59].

To implement the proposed procedure, the central composite face-centered design with 2^k-1 factor points, 2^k pivot points and 5 central points for all three algorithms with its related factors have been considered. An approximation of response variables denoted as R_GA and R_NSGA - II are reported from Equations (37) and (38), respectively. $\begin{matrix} R_{GA} = 2.69623 * pm - 1.0425 e - 005 * npop * \\ max it + 0.00156688 * npop * pc + 0.00796833 * \\ npop * pm + 0.00038875 * max it * pc \\ + 0.00345667 * max it * pm - 1.57625 * pc * pm \\ + 1.13586 e - 006 * npop * npop + 1.38234 e \\ - 005 * max it * max it + 1.11724 * pc * pc \\ - 12.2241 * pm * pm \end{matrix}$ (37) $\begin{matrix} R_{NSG - II} = 1.94653 - 0.008741 * npop \\ + 0.00851993 * maxit + 0.571275 * pc - 2.6983 * \\ pm + 7.374 e - 0.006 * npop * max it - 8.225 e \\ - 0.005 * npop * pc - 0.00239333 * npop * pm \\ - 0.0001185 * max it * pc + 0.0002156 * max it * \\ pm + 1.12717 * pc * pm + 5.5746 e - 0.005 \\ * npop * npop - 6.007 e - 0.005 * max it * max it \\ - 0.552218 * pc * pc + 6.89356 * pm * pm \end{matrix}$ (38)

In the continuation of proposed methodology, Tables 5 and 6 reveals the results of analysis of variance implemented from Minitab software. These outputs indicate that both regressions are suitable for implementation of RSM. To do this end and to achieve the optimal output of the input parameters of the algorithms, regression relations as given in Equations (37) and (38) within the range of those input parameters are obtained using illustrated methodology. From Figs. 8 and 9, the red values are set values of the parameters. In conclusion, the optimal levels of input parameters of proposed algorithms i.e. GA and NSGA-II are given in Table 7.

Table 5

Results of Analysis of variance for revealing R_GA

Source	df	Seq SS	Adj SS	Adj MS	F	P value
Regression	14	0.053558	0.053558	0.003826	6	0.001
Linear	4	0.008569	0.010041	0.00251	3.94	0.024
npop	1	0.004484	0.000228	0.000228	0.36	0.56
Maxit	1	0.000465	0.000039	0.000039	0.06	0.809
Pc	1	0.001373	0.004873	0.004873	7.64	0.015
Pm	1	0.002247	0.00677	0.00677	10.61	0.006
Square	4	0.014381	0.014381	0.003595	5.64	0.006
Npop*npop	1	0.000445	0.000021	0.000021	0.03	0.859

Table 6

Source	df	Seq SS	Adj SS	Adj MS	F	P value
Regression	14	0.01951	0.01951	0.001394	1.24	0.344
Linear	4	0.004149	0.012623	0.003156	2.82	0.066
Npop	1	0.000101	0.003126	0.003126	2.79	0.117
Maxit	1	0.000766	0.00297	0.00297	2.65	0.126
Pc	1	0.000855	0.000855	0.000855	0.76	0.397
Pm	1	0.002428	0.00678	0.00678	6.05	0.027
Square	4	0.010174	0.010174	0.002544	2.27	0.113
Total	1	0.002757	0.003139	0.003139	2.8	0.116

Fig.8

The tuned parameters along with their best levels in GA.

Fig.9

Tuned parameters levels in NSGA-II.

Table 7

Calibrated algorithms’ results

Algorithm	Parameter	Optimum value
GA	Npop	100
	P _c	0.8
	P _m	0.2
	Nit	50
NSGA-II	Npop	100
	P _c	0.4
	P _m	0.2
	Nit	50

5.2 Validation of GA in small sizes with an exact solver

To generate the test problems, the range of parameters of proposed model is given in Table 8 as our data set.

Table 8
Parameters distribution values of proposed mathematical model

Parameter Distribution Parameter Distribution

function function

P _ij Unifrnd(50,200) k _i Unifrnd(0.5,0.9)

a _i Unifrnd(0.2,0.6) f _i Unifrnd(0.03,0.1)

d _i Unifrnd(0,12) e _i Unifrnd(0.04,0.11)

u _i Unifrnd(50,300) v _i Unifrnd(0,1)

t _i Unifrnd(4,15) q _i Unifrnd(0,1)

T _i Unifrnd(1,5) z _i Unifrnd(0,1)

r _j Unifrnd(0.5,0.9) c _i Unifrnd(300,1000)

Parameter	Distribution	Parameter	Distribution
P _ij	Unifrnd(50,200)	k _i	Unifrnd(0.5,0.9)
a _i	Unifrnd(0.2,0.6)	f _i	Unifrnd(0.03,0.1)
d _i	Unifrnd(0,12)	e _i	Unifrnd(0.04,0.11)
u _i	Unifrnd(50,300)	v _i	Unifrnd(0,1)
t _i	Unifrnd(4,15)	q _i	Unifrnd(0,1)
T _i	Unifrnd(1,5)	z _i	Unifrnd(0,1)
r _j	Unifrnd(0.5,0.9)	c _i	Unifrnd(300,1000)

Regarding the generated instances, the computational results of GA’s performance with Zimmermann fuzzy approach are revealed by considering the mentioned numerical example that is comparable with an exact solver implemented by to the Lingo software. Table 9 provides optimal results from both approaches. Due to randomization of GA, during 10 run times, the best, the worst and the average of solution as well as the standard deviation of results are provided to be reliable. From the table, it can be revealed that there is a little difference between the best and the worst solutions and generally the algorithm shows robust behavior. In addition to this analysis, the convergence diagram of proposed GA for the best case is also shown in Fig. 10.

Table 9

Comparison of GA for a small instance with an exact solver

Algorithm		x ₁₃	x ₂₁	x ₃₃	$z_{1}^{*}$	$z_{2}^{*}$	$z_{3}^{*}$	$z_{4}^{*}$	Cost
Exact		400	8	400	70095.39	640	667.44	685.6	0.49
GA	Best	398	12	397	70168	636.2	667.01	684.1	0.45412
	Worst	401	9	399	70246	632.6	658.52	679.5	0.39
	Average	–	–	–	70214.8	634.04	661.916	681.34	0.41564
	SD	–	–	–	49.259	11.811	14.27	2.3152	0.0322

Fig.10

Convergence diagram of Zimmermann fuzzy method implemented by GA.

5.3 Solving the large-scale instances by NSGA-II

Here, the applied NSGA-II is implemented to solve the large test problems. Regarding the proposed multi-objective model, the best solution is a set of candidate solutions as a Pareto optimal frontier. Due to four objective functions, it is very difficult to find a solution which can dominate all other reached solution. Therefore, the non-dominated solutions can be a group of solutions. To see the description of about this algorithm and multi-objective optimization, recent papers are suggested to read [49 , 60– 62]. The size of generated test problems are given in Table 10. Accordingly, by changing the size of parameters, 10 various experimental problems are created and accordingly, Table 11 gives the results of algorithm based on the solution time, Diversification Metric (DM) [55], Spread of Non-dominance Solutions (SNS) [53] and MID [62]. Note that the higher value of DM and SNS are more desirable. Conversely, the lower value of solution time and MID brings the better capability of applied algorithm. To see more description about these well-known assessment metrics of Pareto optimal frontier, recent papers are recommended for interested readers such as [60 –64].

Table 10
Different states of the size of instances in large-scale instances

No. n m E F D G B

1 5 3 400 300 800 900 100000

2 5 5 400 400 800 1000 100000

3 5 5 300 300 800 950 90000

4 5 6 400 300 800 1100 100000

5 6 6 250 250 800 1100 100000

6 7 6 400 350 700 1100 100000

7 7 7 350 350 650 1200 110000

8 8 7 500 450 650 1200 130000

9 8 8 400 400 650 1200 120000

10 9 8 450 500 600 1300 110000

No.	n	m	E	F	D	G	B
1	5	3	400	300	800	900	100000
2	5	5	400	400	800	1000	100000
3	5	5	300	300	800	950	90000
4	5	6	400	300	800	1100	100000
5	6	6	250	250	800	1100	100000
6	7	6	400	350	700	1100	100000
7	7	7	350	350	650	1200	110000
8	8	7	500	450	650	1200	130000
9	8	8	400	400	650	1200	120000
10	9	8	450	500	600	1300	110000

Table 11

Obtained assessment indexes from NSGA-II in different states

No.	Time	DM	SNS	Num. of	MID
	(second)			non-dominated
				solutions
1	337.5048	1.04E+05	735.0114	20	3.63E+04
2	322.3536	9.23E+04	1.20E+03	15	9.97E+04
3	321.826	6.46E+04	887.6549	14	1.57E+04
4	322.1705	1.07E+11	1.71E+09	10	2.03E+04
5	322.3655	1.07E+11	1.71E+09	12	2.13E+11
6	321.7159	4.96E+09	5.30E+07	15	9.85E+09
7	320.071	1.35E+11	9.63E+08	16	2.70E+11
8	319.575	3.22E+04	469.1396	18	3.05E+04
9	322.2345	5.07E+11	5.71E+09	22	2.36E+10
10	320.8807	2.77E+11	7.97E+09	18	2.36E+10

To see the non-dominated solution based on four objective functions, the convergence diagram for the first test problem is shown in Fig. 11. Due to dimensional limitations for graphical view, 3-dimensional figures are considered in four sub-figures. In each of them, one objective has been removed to be fixed. In conclusion, form the results provided in Table 11, we can see that the applied NSGA-II can solve all test problems properly and reveal its high-efficiency to solve the proposed problem.

Fig.11

Convergence diagram of NSGA-II to reveal the non-dominated solutions.

6 Discussion and conclusion

As already mentioned, supplier selection plays a key role in supply chain management and in today’s global competitive market. This issue companies pay a lot of attention to choose the right supplier. Needless to say, it reduces the cost of purchase, improves the quality of the final product and services and so on. However, choosing the right supplier is a multi-criteria decision-making problem involving both qualitative and quantitative criteria and many of these criteria may oppose each other so the decision making process becomes more complicated. This research has tried to offer a new multi-objective model for selecting the best suppliers and their right order allocation where suppliers can offer general, incremental, and general-incremental discounts.

This multi-criteria model aimed to minimize costs, by considering discounts, maximize the quality of products by considering the capacity limitations, demands, discount, returns, and budget. It also tries to offer flexible solutions by using fuzzy approaches to solve the model. Therefore, Zimmermann fuzzy approach was used to solve the model. In the small size study problem, we used an exact solver by Lingo software to solve the model. Due to the single-objective nature of the Zimmermann fuzzy approach, the GA was employed to solve the small instances. In large-scale test problems, as a multi-objective optimization with four objective functions, the NSGA-II as a successful metaheuristic was applied by employing a number of efficient assessment metrics of Pareto optimal frontier to evaluate its performance to solve the problem.

There are several new avenues as the continuation of this work. From the aspect of proposed mathematical model, several real-life impacts such as the sustainability factors to cover the environmental and social objectives can be suggested. Multi-stage stochastic programming approach can be applied along with other stochastic parameters by considering fuzzy sets. As such, different fleet sizes of vehicle routing operations to reduce the shipment costs and the fuel consumption can be ordered. Nowadays, there are many metaheuristics which have been proposed recently such as Red Deer Algorithm [6], Keshtel Algorithm [1] and Social Engineering Optimizer [58] etc. Generally, this chance even with low possibility always exists for a new metaheuristic to better solve an optimization problem in comparison with current ones. Therefore, one of main future lines of this study is to compare the results of GA and NSGA-II with other novel metaheuristic algorithms as suggested [63 –65].

Footnotes

Acknowledgments

This work was financially supported by National Natural Science Foundation of China under Grant No. 51775238.

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A multi-objective supplier selection and order allocation through incremental discount in a fuzzy environment

Abstract

Keywords

1 Introduction

2 Literature review

3 Proposed problem

3.1 Assumptions

3.2 Notations

3.3 Problem formulation

4.3 Numerical example

Table 1 Price levels provided by each supplier Supplier (i) b i0 p i1 b i1 p i2 b i2 p i3 1 0 78 100 75 200 70 2 0 120 130 105 230 91 3 0 99 110 95 210 90

5.1 Parameter tuning

Footnotes

Acknowledgments

References

Table 1
Price levels provided by each supplier

Supplier (i) b _i0 p _i1 b _i1 p _i2 b _i2 p _i3

1 0 78 100 75 200 70

2 0 120 130 105 230 91

3 0 99 110 95 210 90