Supplier Selection and Order Allocation Problem Modeling with the Aim of Comparing Incremental Discounts Versus Wholesale Discounts by Using GA and NSGA Algorithms

Supply chain includes all the stage which are involved in satisfying customer's need directly & indirectly. Supply chain not only includes the suppliers, but also involves transportation, stores, retailers & customers, too. The manufacturing firms should decrease the extra costs of supply chain to retain their competitive position. For example, it should be better to outsource parts & services which are not strategic goods of manufacturer. When a firm decides outsourcing, its main challenge is supplier selection problem ‎[1]. Selecting of suppliers is one of the main parts in supply chain as well as it has turned to a strategic decision in supply chain during recent years. In manufacturing industries, procurement of raw materials includes approximately 70 % of the manufacturing costs. In such a condition, firm's purchasing department plays an important role in cost decrement and supplier selection is an important task in purchasing management ‎[2].

Supplier selection problem is two types:

(1) Supplier selection without capacity restrictions. i.e. It can satisfy all the buyer’s needs.

(2) Supplier selection with capacity restrictions. It means that, the supplier can’t satisfy all the buyer’s needs. So, buyer has to supply some of his requirements by another supplier ‎[3].

In supplier selection problem, we are facing with a multi-dimensional problem. Thus, researchers use single or mixed multi-criteria decision-making techniques for supplier selection problem.

Amid et al. used a weighted multi-objective fuzzy model with three objectives including price and lead-time minimization and quality maximization for order allocating while each supplier offer various quantity discount ‎[4, 5]. Have used goal programming approach & Fuzzy TOPSIS for supplier selection. The relationship closeness, quality, delivery ability, guarantee & expire date criteria have been used in their research. Network analysis process and mixed integer linear programming have been applied by Liao and Kao to investigate quantitative & qualitative criteria in supplier selection. They have determined the optimal order allocation to each supplier with the aim of maximizing purchase value, minimizing consuming budget & the least failure rate. 14 criteria have been investigated in 4 clusters; profit, opportunity, cost & risk ‎[6]. Demirtas & Üstün applied a fuzzy approach for supplier selection in a washing machine company among three candidates ‎[7]. A multi-criteria intuitionistic fuzzy group decision making has been exploited by Kilincci & Onal ‎[8], 2011 to rank 5 supplier. Boran et al. ‎[9] proposed an AHP-based approaches for supplier evaluation and they used quality, serving level, innovation & management and financial conditions as selection criteria. Bruno et al. ‎[10] have presented a two-layer model for supplier selection. They proposed a multi-objective mixed integer programming model. Shahroodi & Hassani ‎[11] suggested a mathematical model to select suppliers through using DEA integrated approach and wholesale ownership cost with a case study in construction value chain of Irankhodro industry. They have introduced the most efficient supplier with the least wholesale cost & have presented solutions for achieving to efficiency by the other part makers. Tabriz & Azar ‎[12] have presented a fuzzy decision making process model for strategic supplier selection. The supplier selection & order allocation for the selected suppliers is very complicated when quantity discounts are considered in problem. We can find supplier selection problem & order allocating simultaneously in the researches of Yücel and Güneri ‎[2], Sadrian and Yoon ‎[13]. Recently some other studies like Perić and Babić ‎[14], investigated supplier selection & order allocating problem through using Fuzzy AHP hybrid approach & multi-objective linear programming. Kannan et al. ‎[15] have used a Fuzzy simulation based Fuzzy TOPSIS method for proper suppliers selection by evaluation of 4 criteria; operational strategy, service quality, innovation & risk. Zouggari and Benyoucef ‎[16] have investigated the problem of dynamic supplier selection. Sivrikaya et al. ‎[17] have investigated the problem of supplier selection in textile industry they have used two phases that consist of fuzzy AHP and goal programming. Ayhan and Kilic ‎[18] have presented a two stage approach for supplier section problem, in the first stage, the relative weight of each criterion for each type of item are determined via F-AHP technique and in second stage these output are used as inputs in the MILP model to determine the supplier selection. In this research 4 criteria namely, price, quality, delivery time performance, and after sales performance are used for altenatives evaluation. Like the other research, they applied wholesale discount in their model. New approaches for supply chain coordination problem are developed by Arshinder & Deshmukh ‎[21, 20] and Cardenas & Barron ‎[22]. A multi-objective supplier selection and order allocation problem with fuzzy objective are studied by Kazemi et al. ‎[23].

As it can be seen, former researches mostly consider wholesale discount in their studies and not only don't apply other type of discount, but also there isn't any research that compares the results of these two types of discount. So one of the main contribution of this study is investigating and modeling of incremental discount and also comparing the result of them toward wholesale discount.

The main differences of this study than previous research is:

(1) As much as the authors are aware, this study for the first time considers the incremental discount in the supplier selection problem.

(2) Unlike previous researches that just solve the model just for small size problem, in this work by dealing with large size problem, we have investigated the efficiency of exact solution according to time consuming.   

(3) In previous studies, Weighted Method have been used to solve the Multi-Objective optimization problem, however we applied a non-dominated solution for solving the problem.

(4) In spite of previous studies, this is the first work that used meta-heuristic algorithms (e.g. NSGA II) to solve multi-objective optimization problems.

This research tries to investigate the supplier selection problem while there are some restrictions on supplier’s capacity, quality and so on as well as a supplier can’t satisfy all of the buyer’s demand lonely. So, in this research, a two-stage approach has been used for modeling & supplier selection Problem Solving as well as allocating order to them in quantitative discount existing condition for various levels. In the first stage, an integrated fuzzy AHP- approach & extent analysis have been used for determining criteria weight. In second stage, the multi-objective problem has changed to a single-objective problem through weight method as well as single-objective modeling & problem solving have been done in two incremental & wholesale discounts mode.

Research structure is as follow; in the second section, a brief description on the used methods by the research such as calculating fuzzy numbers (the reason for fuzzy numbers will be explained), introducing extent analysis method, introducing weight methods have been presented. Then in the third section, problem mathematical modeling in partial & wholesale discounts existing condition for order quantities has been presented. In forth section, model performance in two partial & wholesale discounts mode has been compared by a numerical example to investigate the accuracy of the presented model. Finally, in the fifth section, conclusion & recommendations for future studies have been presented.

In AHP method, a cut of 1 to 9 discrete numerical scale is used for decision making on a criterion priority to other criteria while, fuzzy numbers or lingual values are being used in fuzzy AHP. When fuzzy numbers are being used in AHP technique, its way of solving will differ from the way of solving definite AHP. One of the most common methods for solving fuzzy AHP is extent analysis method which has been presented by Chang ‎[17]. In this method, the “extent” is being valued by fuzzy numbers. The value of fuzzy compound degree can be achieved by the resulted fuzzy values from extent analysis for each criterion as follow;

In supplier selection problem, it is assumed that X={${x1,x2,…,xn}$} represents alternatives set as objective set as well as U={${u1,u2,…,um}$} represents set of supplier selection criteria as the goal set. According to extent analysis method, at first each of objective are chosen and the extent analysis is being done for each $g_i$  ideal, respectively. Therefore, m is the extent analysis value for each measurable objective which is represents as follow;

Where All $M_{g_i}^j$ are triangular fuzzy numbers. Generally, extent analysis method can be summarized in below steps;

First stage: Calculating fuzzy compound extent value according to the $i^{th}$ objective.

$s_i=\sum_{j=1}^m\quad m_{g_i}^j×{[\sum_{i=1}^n\quad\sum_{j=1}^m\quad M_{g_i}^j]}^{-1}$ (6)

The below equations represent the way of calculating ${[\sum_{i=1}^n\quad\sum_{j=1}^m\quad m_{g_i}^j]}^{-1}$ & $\sum_{j=1}^m\quad m_{g_i}^j$

$\sum_{j=1}^m\quad m_{g_i}^j=(\sum_{j=1}^m\quad l_j, \sum_{j=1}^m\quad m_j, \sum_{j=1}^m\quad u_j)$ (7)

${[\sum_{i=1}^n\quad\sum_{j=1}^m\quad m_{g_i}^j]}^{-1}=(\frac{1}{\sum_{i=1}^n\quad u_i}, \frac{1}{[\sum_{i=1}^n\quad m_i}, (\frac{1}{\sum_{i=1}^n\quad l_i})$ (8)

Second stage: Determining the possibility of $M_2=(l_2,m_2,u_2 )≥M_1=(l_1,m_1,u_1)$.

$V(M_2≥M_1)=\begin{cases}1, & \text{if}m_2≥m_1\\0, & \text{if}l_1≥u_2\\\frac{l_1-u_2}{(m_2-u_2)-(m_1-l_1)} , otherwise\end{cases}$ (9)

To compare, $M_1$ & $M_2$, the values of $V(M_1≥M_2)$ & $V(M_2≥M_1)$ should be calculated.

Third stage: Determining objective weight vector.

$\acute{W}=((\acute{d}(A_1), \acute{d}(A_2 ),…,\acute{d}(A_n ))^T, \\A_i (i=1,2,…,n),  \acute{d}(A_i )=min⁡V (s_i≥s_k ) \\k=1,2,…,n; k≠n$ (10)

Fourth stage: Determining normalized weight vector.

$W=((d(A_1 ),(d(A_2 ),…,(d(A_n))^T$ (11)

This research tries to investigate supplier selection problem in two partial & wholesale discounts mode for order quantity. So, in this section two model have been presented for the possibility of existing partial & wholesale discount.

5.1 Mathematical model with the possibility of existing partial discount for order quantity

Supplier selection problem is a problem with multiple criteria. The considered criteria in this research are cost, quality & delivery time, respectively. These criteria are of the most important criteria which are used in the related researches to supplier selection as well as their application in various researches during recent years has been presented in table (1) in the appendixes. So, these three criteria have been used in this research as the main criteria in selecting proper supplier. Multi-objective integer linear programming modeling in the special partial discount condition for order quantity which has not been presented in the former researches has been presented in this section. Partial discount for order quantity is that the price is different for each determined quantity interval as well as wholesale price is being measured aggregately while in wholesale discount, all of the purchased quantities are being calculated by the related prices to the same price interval.

The model variables & parameters are being described before modeming the problem.

Variables;

$X_{ij}$ : Number of the purchased units from $i^{th}$ supplier in $j^{th}$ price level.

$Y_{ij}$: Zero & one variable for $i^{th}$ supplier $j^{th}$ price level.

Parameters;

$P_{ij}$: Product price for the $i^{th}$ supplier in the $j^{th}$ level.

$V_{ij}$: The length of the order interval from the $i^{th}$ supplier in the $j^{th}$ price level.

D: the demand of whole period.

$M_i$: The price levels of the $i^{th}$ supplier.

$C_i$: The capacity of the $i^{th}$ supplier.

$F_i$: The percentage of delayed items for the $i^{th}$ supplier.

$Q_i$: Quality of the $i^{th}$ supplier items.

n: Numbers of the suppliers.

$MinZ_1=\sum_{i=1}^n\quad p_ix_{i,1}+p_{i,2}\bullet(x_{i,2})+   +p_{i,m}\bullet(x_{i,m})$ (14)

$MaxZ_2=\sum_{i=1}^n\quad Q_i \sum_{j=1}^{mi}\quad x_{i,j}$ (15)

$MinZ_3=\sum_{i=1}^n\quad F_i\sum_{j=1}^{mi}\quad x_{i,j}$ (16)

S.t.

$\sum_{i=1}^n\quad\sum_{j=1}^{mi}\quad x_{i,j}≥D$ (17)

$\sum_{j=1}^{mi}\quad x_{i,j}≤ C_i   i=1,2,...,n$ (18)

$V_{i,j}Y_(i,j)≤x_{i,j}, i=1,...,n, j=1,..., m$ (19)

$x_{i,j}≤v_{i,j}\bullet Y_{i,j-1}   i=1,2,...,n, j=1,2,..., m_i$ (20)

$\sum_{j=1}^m\quad y_{ij}≤m,     ∀iϵ i=1,…,n$ (21)

$Y_{i,j}=0 or 1      i=1,2,…,n,  j=1,2,…$ (22)

$x_{i,j}≥0 , Integer    i=1,2,…,n    , j=1,2,…,m_i$ (23)

In the above model, the (14), (15) & (16) relations which represent objective functions are price minimizing, quality maximizing & lead time minimizing in delivery, respectively. The (17) & (18) relations are limitations for demand & capacity, respectively. The (19) & (20) relations also, are the related limitations to partial discount. Finally, the (21) & (22) relations represent the limitation of variables being zero & one as well as being positive.

5.2 Mathematical model with the possibility of existing wholesale discount for order quantity

In this part, a mathematical model is presented with the wholesale discount existence for order quantity. Since, in wholesale discount unlike partial discount all the purchased items are being purchased with the related price to the same interval; here, instead the parameter of $\acute{V}_{i,j}$ order quantity interval length the parameter of $\acute{V}_{i,j}$ order quantity upper limit is used. So, mathematical model of supplier selection problem & allocating optimum order to them in wholesale discount condition will be as follow;

New parameter;

$\acute{V}_{i,j}$: Upper limit of order interval from the $i_{th}$ supplier in the $j_{th}$ price level

$MinZ1=\sum_{i=1}^n p_{i,1} x_{i,1} +p_{i,2}∙(x_{i,2} )+⋯+p_{i,m}∙(x_{i,m} )$ (24)

$Max Z2 =\sum_{i=1}^n Q_i   \sum_{j=1}^{mi} x_{i,j} $ (25)

$MinZ3=\sum_{i=1}^n F_i  \sum_{j=1}^{mi} x_{i,j} $ (26)

S.t.

$\sum_{i=1}^n \sum_{j=1}^{mi} x_{i,j}≥D$ (27)

$\sum_{j=1}^{mi} x_{i,j}≤ C_i     i=1,2,….,n$ (28)

$\acute{v}_{i,j} Y_{i,j}≤x_{i,j},       i=1,…,n, j=1,…,m$ (29)

$x_{i,j}≤\acute{v}_{i,j}∙Y_{i,j}     i=1,2,…,n    , j=1,2,…,m_i $ (30)

$\sum_{j=1}^m y_{ij} ≤1 ,        ∀iϵ i=1,…,n$ (31)

$Y_{i,j}=0 or 1      i=1,2,…,n  , j=1,2,…$ (32)

$x_{i,j}≥0     i=1,2,…,n    , j=1,2,…,m_i$ (33)

All limitations of the above model are the same as the former model except the (29) & (31) which are for wholesale discount.

5.3 Normalization

In multi-objective optimization problem, when we have different objective functions with different scales, normalization of objective functions, play an important role in ensuring the consistency of optimal solutions.

There are different approaches for normalization and one of the simplest (but appropriate) approaches is to optimize each of the objectives individually first, then divide each objective by those optimum values and finally sum up all normalized terms as one objective function

Initially, each objective function is optimized separately and the negative ideal solution (worst solution) and positive ideal solution (best solution) of them are found. Since the values of objective function vary in different scales equation 34 and 35 is used for normalize the objective functions.

For minimization objective function

$f_i^N=\frac{NIS_f-f}{NIS_f-PIS_f}$ (34)

For maximization objective function

$f_i^N=\frac{f-NIS_f}{PIS_f-NIS_f }$ (35)

where, $f_i^N$ is the normalized value of the ith objective function, NIS is negative ideal solution of objective function and PIS is best solution or positive ideal solution at the end model is changed to a single-objective function by summing up all weighted functions as shown in Eq. (36).

$MAX(f)=\sum_{i=1}^n w_i f_i^N$ (36)

Srinivas and Deb ‎[24] have instructed NSGA method for optimization multi objective problems. This algorithm use Darwin's principle of natural selection for to find the formula or optimal solution. But NSGA algorithm is highly sensitive to share fitness and other parameters. So for the second version of NSGA algorithm called NSGA-II was introduced in ‎[25] by Deb et al. in the NSGA algorithm some of answer selected by using tournament selection from answer of each generation. note in this method the answer that there isn’t any answer better than it is best answer and has more point. Answers ranked and sorted based on how many answers there are better than them ‎[26]. Fitness allocated to answer based on their ranks and not overcome of other answers. In the first, rank of answer and second, compaction distance are criteria for selecting in NSGA –II algorithm. The answer is more favorable that rank answer is less and has more compaction distance. The not overcome answer that obtained archived and sorted as elite.

Table 10. Parameters levels of algorithms

Table 11. Taguchi results to adjust parameter’s NSGA algorithm

7.1 Adjusting parameter’s algorithms

In this study used Taguchi method to adjust the parameters of the proposed algorithms. In this way at the first different levels of parameters be determined. Related parameters and values of GA& NSGA II algorithms are given in Table 10.

It is available of the value of parameters after running using the signal to noise. This graph is available for GA&NSGA algorithms that is showed in Figures 2 and 3.

At the end of each replication, a final answer has calculated by using weights that are obtained from fuzzy AHP method. The GA and NSGA run for 10 times and averaged for value can be seen in table 11 the final answer of Pareto solution is shown in Z.

Table 12. Taguchi results to adjust parameter’s GA algorithm

figure_2.png

Figure 2. Signal to noise for GA algorithm

figure_3.png

Figure 3. Signal to noise for NSGA algorithm

Table 13. Best value for parameters of GA and NSGA algorithms

Table 14. Value of GA and NSGA

Table 15. GA and NSGA error parentage

Appropriate values for the parameters of algorithm NSGAII and GA have been reported in Table 13.

Then the given example has solved by using the optimal value is obtained that has shown in table 14. Compared the obtained value with the optimal value is known as a very low percentage of errors, especially in the NSGA algorithm that this reflects the accurate of algorithms are used. Table 15 Shows value of errors.

Now given that accuracy of algorithms is evaluated then a big size problem has solved in both wholesale and practical discount. This problem can be seen in appendix 2. In this problem number of supplier is 35. The best solution for each of discount is shown in Table 16.

Table 16. Best value for big problem