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In today's competitive world, it is very important for organizations to select suppliers according to price, quality, satisfactory service and timely delivery. Since a considerable portion of production costs is associated with purchasing raw materials from suppliers, selection of the right suppliers and allocation of optimal order quantities plays an important role in the success of an organization. So far, extensive research has been conducted in the context of supplier selection, and multicriteria decisionmaking techniques are the common approach used to select the appropriate option. Recently, some studies in the context of supplier selection considered variable assumptions like quantity discount possibility. So, the aim of this study is modeling the supplier selection problem based on incremental and wholesale discounts and comparing the results of them like best selected supplier(s) and optimal order allocation to them. And solve a big problem with GA and NSGA algorithms and comparing with each other’s for this an integrated threestage approach has been proposed by combining fuzzy AHP and Extended Analysis Method for the supplier selection problem and develop a GA&NSGA algorithms. Finally, the performance of the proposed approach and proposed algorithms has been appraised by numerical examples.
supplier selection, fuzzy AHP method, discount, weighting method, GA and NSGAII
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 multidimensional problem. Thus, researchers use single or mixed multicriteria decisionmaking techniques for supplier selection problem.
Amid et al. used a weighted multiobjective fuzzy model with three objectives including price and leadtime 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 multicriteria intuitionistic fuzzy group decision making has been exploited by Kilincci & Onal [8], 2011 to rank 5 supplier. Boran et al. [9] proposed an AHPbased 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 twolayer model for supplier selection. They proposed a multiobjective 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 & multiobjective 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 FAHP 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 multiobjective 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 MultiObjective optimization problem, however we applied a nondominated solution for solving the problem.
(4) In spite of previous studies, this is the first work that used metaheuristic algorithms (e.g. NSGA II) to solve multiobjective 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 twostage 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 multiobjective problem has changed to a singleobjective problem through weight method as well as singleobjective 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 this section, before modeling & problem solving, a brief description on the used techniques in the research & their way of calculating has been presented. In this part, at first the way of calculating fuzzy numbers by some decision makers has been presented; then, the extent analysis method has been introduced; finally, a brief description has been presented on the weighed method which its weights are being calculated by extent analysis method. The way of calculating fuzzy numbers in group decision making mode In fact, in a group decision making a group of individual comment on a special issue. In such a condition, the way of determining final resulted conclusion from the done decisions by the decision makers is important. Thus, this section tries to represent the way of calculating resulted conclusion from group decision making. It should be point out that the done decisions by the decision makers are being represented lingual. Therefore, the predetermined triangular fuzzy numbers have been used for each of lingual variables. The way of calculating the resulted fuzzy numbers from group decision making is follow; At first, each of the decision makers are being requested to fill out criteria comparison questionnaire (these decision makers can compare the criteria individually or by group & in interaction with each other). This research has considered that each of decision makers have compared the criteria independently. Equations (1), (2) & (3) are being used for achieving to a fuzzy number resulted from n decision makers while decision makers do compare the criteria independently [16].
Figure (1) illustrates display of triangular fuzzy numbers. In this figure, the horizontal axis indicates triangular fuzzy numbers include three l, m & u values as well as the vertical axis indicates the related membership function to horizontal axis values and is calculated according to equation (4).
Figure 1. Triangular fuzzy numbers display
$l_{ij}={min}_k (l_{ijk})$ (1)
$m_{ij}={(\prod_{k=1}^n m_{ijk} )}^{(1/n)}$ (2)
$u_{ij}={max}_k (u_{ijk})$ (3)
In above equations, $l_{ij}$ , $m_{ij}$ & $u_{ij}$ are equivalent with the left value, average value & right value of resulted triangular fuzzy number through group decision making by k number for $i^{th}$ value of the $j^{th}$ supplier, respectively. K index represents the made decision by of $k^{th}$ decision maker.
$μ_M ̃ =\left\{\begin{array}{rl}0, x<l\\(xl)/(ml), l≤x≤m,\\(ux)/(um),m≤x≤u,\\0, x≥u\end{array} \right.$ (4)
In relation (4), $μ_M ̃ $ indicates multivalue membership function by x values.
The fuzzy numbers are intuitively & easily useable to represent decision maker’s qualitative assessments.
Each fuzzy number can be indicated by its right & left values of membership degree as follow:
$M ̃=(M^{l(y)},M^{r(y)})=(l+(ml)y, u+(mu)y), yϵ[0,1]$ (5)
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 $M_2=(l_2,m_2,u_2 )≥M_1=(l_1,m_1,u_1)$.
$V(M_2≥M_1)=\begin{cases}1, & \text{if}m_2≥m_1\\0, & \text{if}l_1≥u_2\\\frac{l_1u_2}{(m_2u_2)(m_1l_1)} , otherwise\end{cases}$ (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.
$\acute{W}=((\acute{d}(A_1), \acute{d}(A_2 ),…,\acute{d}(A_n ))^T, \\A_i (i=1,2,…,n), \acute{d}(A_i )=minV (s_i≥s_k ) \\k=1,2,…,n; k≠n$ (10)
Fourth stage: Determining normalized weight vector.
$W=((d(A_1 ),(d(A_2 ),…,(d(A_n))^T$ (11)
The weight method is one of the most applicable methods in modeling & solving multiobjective optimization problem in spite of being easy. In this method, a weight is considered for the problem various objective functions. This weight represents the importance & relativepriority of the objectives to each other. Since, the supplier selection problem & allocating order to them is of multiobjective optimization problems; thus, this research tries to use weight method for modeling & solving it. Notable point in weight method is the way of calculating the weights. So, the extent analysis method is used for determining relative weights of criteria to each other. After calculating relative weights of criteria to each other, the multiobjective linear programming modeling & solving it is done by the weight method. In the below part the way of changing the multiobjective problem to singleobjective problem through using weight method has been represented.
$Max(Min) f_i (x), i=1,2,…,n\\S.t.\\g_i (x)≤or≥b_i, i=1,2,…,n\\x_i≥0.$ (12)
$Max(Min)w_1.f_1(x)+w_2.f_2(x)+…+w_n.f_n(x)\\S.t.\\g_i (x)≤or≥b_i, i=1,2,…,n\\x_i≥0$ (13)
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. Multiobjective 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)
$x_{i,j}≤v_{i,j}\bullet Y_{i,j1} i=1,2,...,n, j=1,2,..., m_i$ (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 multiobjective 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
$f_i^N=\frac{NIS_ff}{NIS_fPIS_f}$ (34)
For maximization objective function
$f_i^N=\frac{fNIS_f}{PIS_fNIS_f }$ (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 singleobjective function by summing up all weighted functions as shown in Eq. (36).
$MAX(f)=\sum_{i=1}^n w_i f_i^N$ (36)
In this part, the implementation stages of supplier selection problem as well as their step by step solving to investigate the presented model validity is presented though giving a numerical example. In this example, three decision makers (to do group decision making), three supplier wilt limited capacity & the presenter of partial & wholesale discount for order quantity have been considered. The considered criteria in this example are cost, quality & lead time. Table 2 indicates the complete related information to price, quality & delivery as well as productive capacity for the three suppliers. The implementation steps of the recommended model for our example are as follow Table 2.
Table 2. Problem information
productive capacity 
Percentage lead time in delivery 
quality 
Purchase interval 
cost 

16000 
0.1 
80 
[04000] 
15 
Supplier 1 
[4001 8000] 
14.5 

[800116000] 
14 

15000 
0.15 
70 
[03000] 
17 
Supplier 2 
[3001 10000] 
16.5 

[1000115000] 
16 

17000 
0.3

95 
[05000] 
13 
Supplier 3 
[5001 11000] 
12.5 

[1100117000] 
12 
In this section, at first a questionnaire is prepared for the experts. This questionnaire includes questions through which the ecision makers are requested to represent their viewpoints on the criteria. Table (3) indicates an example of the filled questionnaire by the decision makers.
Table 3. The pairwise comparison by the experts
lead time in delivery 
Quality (Q) 
Cost (C) 
First decision maker 
Very very imporant 
Equal importance 
1 
Cost (C) 
Very very imporant 
1 

Quality(Q) 
1 


lead time 
In the next section, after filling the questionnaire by the experts the lingual values are changed to corresponding fuzzy numbers with them according to table (2) in the appendix. Table (5) indicates related fuzzy numbers to resulted lingual values from decision makers’ viewpoints. In this table C, Q & F represent cost, quality & lead time, respectively as well as D1, D2 & D3 represent decision maker 1, 2 & 3, respectively.
Table 4. Lingual values & their corres
Importance 
values 
The same importance 
(1,1,1) 
more Important portion 
(2/3,1,3/2) 
More Important 
(3/2,2,5/2) 
More & more important 
(5/2,3,7/2) 
Completely Important 
(7/2,4,9/2) 
Table 5. Corresponding fuzzy numbers with each decision makers’ preferences
D1 
D2 
D3 

C/Q 
(1,1,1) 
(5/2,3,7/2) 
(1,1,1) 
C/F 
(5/2,3,7/2) 
(1,1,1) 
(3/2,2,5/2) 
Q/F 
(5/2,3,7/2) 
(2/7,1/3,2/5) 
(3/2,2,5/2) 
In the next section, resulted values of decision makers are being replaced with a fuzzy number through using (1), (2) & (3) equations.
Table (6) indicates these values.
Table 6. Fuzzy values of resulted mean from three decision makers
Lij 
Mij 
Uij 

C/Q 
1 
1.44 
3.5 
C/F 
1 
1.96 
3.5 
Q/F 
0.28 
1.26 
3.5 
Table (7) indicates resulted triangular numbers from paired comparisons.
Table 7. Paired comparisons values resulted from table (6)
C 
Q 
F 

C 
(1,1,1) 
(1,1.44,3.5) 
(1,1.96,3.5) 
Q 
(1/3.51/1.44.,1/1) 
(1,1,1) 
(0.28,1.26,3.5) 
F 
(1/3.5,1/1.96,1/1) 
(1/3.5,1/1.26,1/0.28) 
(1,1,1) 
$S_{cost}= (3, 4.4, 8)\bigotimes(\frac{1}{19.071}, \frac{1}{11.42}, \frac{1}{6.137})=(0.157, 0.385, 1.303)\\S_{quality}=(1.56, 2.954,5.5)\bigotimes(\frac{1}{19.071}, \frac{1}{11.42}, \frac{1}{6.137})=(0.82,0.258,0.896)\\S_{delivery}=(1.571, 4.068, 5.57)\bigotimes(\frac{1}{19.071}, \frac{1}{11.42}, \frac{1}{6.137})=(0.082,0.356,0.9078)\\V(S_{cost}≥S_{quality})=1, V(S_{quality}≥S_{cost})=0.854\\V(S_{cost}≥S_{delivery})=1, V(S_{delivery}≥S_{cost})=0.963\\V(S_{quality}≥S_{delivery})=0.893, V(S_{delivery}≥S_{quality})=1\\\acute{d}(cost)=min(1,1)=1\\\acute{d}(Quality)=min(0.893,854)=0.854\\\acute{d}(Delivery)=min(0.963,1)=0.963\\\acute{W}=(1,0.854,0.963)\\$
The normalized resulted weight is as follow:
WG= (0.36, 0.3, 0.34)
Second step: Changing multiobjective optimizing model to singleobjective model.
In this section, after determining the weight significance of each objective functions, multiobjective linear programming model is changed to singleobjective model. The singleobjective model is as follow for the example of partial & wholesale discount mode;
So in this section, we intend to normalize objective functions according to above mentioned approach. For the first step, we solve the single objective optimization problem for each objective function, to obtain the optimum value of them. Table (8), show the optimum value of objective functions.
Table 8. PIS and NIS for z1 to z3
Function 
PIS 
NIS 
Z1(min is best) 
249000 
313000 
Z2(max is best) 
1855000 
1450000 
Z3(min is best) 
22 
55.5 
$MAX Z=0.36*((313000(15x_11+14.5x_12+14x_13+17x_21+16.5x_22+16x_23+13x_31+12.5x_32+12x_33 )))/(313000249000)+0.3*((80 (x_11+x_12+x_13 )+70 (x_21+x_22+x_23 )+95(x_31+x_32+x_33 ))1450000)/(18550001450000)+0.34*(55.5(0.001(x_11+x_12+x_13 )+0.0015(x_21+x_22+x_23 )+0.003(x_31+x_32+x_33 )))/(55.522)$
S.t.
$x_{11}+x_{12}+x_{13}+x_{21}+x_{22}+x_{23}+x_{31}+x_{32}+x_{33}=20000\\x_{11}+x_{12}+x_{13}≤16000\\x_{21}+x_{22}+x_{23}≤15000\\x_{31}+x_{32}+x_{33}≤17000\\4000*Y_{11}x_{11}≤0\\x_{11}4000≤0\\4000*y_{12}x_{12}≤0\\x_{12}4000*Y_{11}≤0\\8000*y_{13}x_{13}≤0\\x_{13}8000*Y_{12}≤0\\3000*y_{21}x_{21}≤0\\x_{21}3000≤0\\7000*y_{22}x_{22}≤0\\x_{22}7000*Y_{21}≤0\\ 5000*y_{23}x_{23}≤0\\x_{23}5000*Y_{22}≤\\5000*y_{31}x_{31}≤0\\x_{31}5000≤0\\6000*y_{32}x_{32}≤0\\x_{32}6000*Y_{31}≤0\\6000*y_{33}x_{33}≤0\\x_{33}6000*Y_{32}≤0\\Y_{11}+Y_{12}+Y_{13}≤3\\Y_{21}+Y_{22}+Y_{23}≤3\\Y_{31}+Y_{32}+Y_{32}≤3\\Y_{i,j}=0,1 , i=1,2,…,n j=1,2,….,m_i\\x_{ij}≥0, i=1,2,…,n , j=1,2,….,m_i, x_{ij}=INTEGER$
$x_{11}+x_{12}+x_{13}+x_{21}+x_{22}+x_{23}+x_{31}+x_{32}+x_{33}=20000\\x_{11}+x_{12}+x_{13}≤16000\\x_{21}+x_{22}+x_{23}≤15000\\x_{31}+x_{32}+x_{33}≤17000\\x_{11}4000*Y_{11}≤0\\4001*Y_{12}x_{12}≤0\\x_{12}8000*Y_{12}≤0\\8001*y_{13}x_{13}≤0\\x_{13}16000*Y_{13}≤0\\x_{21}3000*Y_{21}≤0\\3001y_{22}x_{22}≤0\\x_{22}10000*Y_{22}≤0\\10001*y_{23}x_{23}≤0\\x_{23}15000*Y_{23}≤\\x_{31}5000*Y_{31}≤0\\5001y_{32}x_{32}≤0\\x_{32}11000*Y_{31}≤0\\11001*y_{33}x_{33}≤0\\x_{33}16000*Y_{32}≤0\\Y_{11}+Y_{12}+Y_{13}≤1\\Y_{21}+Y_{22}+Y_{23}≤1\\Y_{31}+Y_{32}+Y_{32}≤1\\Y_{i,j}=0,1 ,i=1,2,…,n j=1,2,….,m_i\\x_{ij}≥0,i=1,2,…,n ,j=1,2,….,m_i, x_{ij}=INTEGER$
Third step: Running the model & finding the optimum solution.
The optimum values have been achieved through running the two above mentioned model in GAMS software (Table 8);
As it can be seen in above Table, 1 & 3 supplies have been selected for demand supply in both partial & wholesale discount mode. But, way of order allocating is different in both modes. As it has been stated, in partial discount mode each order interval has its own special price as well as only the values of this interval will have this discount While in wholesale discount, all of the related order to the mentioned supplier are sold with the price of this order quantity in the same interval. Generally, the results indicate that both discounts have selected the same suppliers for order allocation as well as the order quantity for them is the same. The only difference in these two discounts is the way of calculating order wholesale price. This has been shown well in table (9) in the part of calculating the first objective function value (ZI). The cause of the second & third objective functions being the same in both kinds of the discounts is order quantity being the same for suppliers is both kinds of discount.
Table 9. Optimum values of objective functions & resulted variables from model solving
Variables 
Partial discount 
Normalized value 
Wholesale discount 
Normalized value 
Z1 
257000 
1 
256002 
0.89059375 
Z2 
1855000 
1 
1779985 
0.814777778 
Z3 
54 
0.045 
43.998 
0.343343284 
X11 
3000 

0 

X12 
0 

0 

X13 
0 

8001 

X21 
0 

0 

X22 
0 

0 

X23 
0 

0 

X31 
5000 

0 

X32 
6000 

0 

X33 
6000 

11999 

0.36*Z1+0.3*Z2+0.34*Z3 

0.675 

0.682 
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 NSGAII 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
levels 
Symbols 
parameters 
algorithms 
0.40.60.8 
P_{c} 
Crossover rate 
NSGAII 
0.10.30.5 
P_{m} 
Mutation rate 

2050100 
pop 
Pop size 

100200300 
It 
iteration 

0.40.60.8 
P_{c} 
Crossover rate 
GA 
0.10.30.5 
P_{m} 
Mutation rate 

2050100 
pop 
Pop size 

100200300 
It 
iteration 
Table 11. Taguchi results to adjust parameter’s NSGA algorithm
Wholesale discount 
Practical discount 
It 
pop 
P_{m} 
P_{c} 
Number of experiments 
0.60199 
0.624179 
100 
20 
0.1 
0.4 
1 
0.65644 
0.635072 
200 
50 
0.3 
0.4 
2 
0.66566 
0.667806 
300 
100 
0.5 
0.4 
3 
0.64461 
0.642662 
300 
50 
0.1 
0.6 
4 
0.66679 
0.644928 
100 
100 
0.3 
0.6 
5 
0.64456 
0.620905 
200 
20 
0.5 
0.6 
6 
0.66269 
0.663939 
200 
100 
0.1 
0.8 
7 
0.61330 
0.645692 
300 
20 
0.3 
0.8 
8 
0.66295 
0.61057 
100 
50 
0.5 
0.8 
9 
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
Wholesale discount 
Practical discount 
It 
pop 
P_{m} 
P_{c} 
Number of experiment 
0.67001 
0.65481 
100 
20 
0.1 
0.4 
1 
0.67522 
0.67045 
200 
50 
0.3 
0.4 
2 
0.67599 
0.67332 
300 
100 
0.5 
0.4 
3 
0.66974 
0.67007 
300 
50 
0.1 
0.6 
4 
0.67318 
0.66916 
100 
100 
0.3 
0.6 
5 
0.67621 
0.66679 
200 
20 
0.5 
0.6 
6 
0.67036 
0.66691 
200 
100 
0.1 
0.8 
7 
0.67531 
0.66982 
300 
20 
0.3 
0.8 
8 
0.67525 
0.66855 
100 
50 
0.5 
0.8 
9 
Figure 2. Signal to noise for GA algorithm
Figure 3. Signal to noise for NSGA algorithm
Table 13. Best value for parameters of GA and NSGA algorithms
Value of parameter (wholesale discount) 
Value of parameter (practical discount) 
Symbol 
parameters 
algorithms 
0.6 
0.4 
P_{c} 
Crossover rate 
NSGAII 
0.5 
0.1 
P_{m} 
Mutation rate 

100 
100 
pop 
Pop size 

200 
300 
It 
iteration 

0.4 
0.6 
P_{c} 
Crossover rate 
GA 
0.5 
0.3 
P_{m} 
Mutation rate 

20 
100 
pop 
Pop size 

200 
300 
It 
iteration 
Table 14. Value of GA and NSGA
NSGA for practical 
GA for practical 
GA for wholesale 
NSGA for wholesale 

3586 
3555 
8838 
8093 
X1 
60 
93 
100 
5 
X2 
16355 
16361 
11067 
11904 
X3 
259070 
259238 
258236 
256235 
Z1 
1844805 
1845205 
1765405 
1778670 
Z2 
52.7400 
52.7775 
42.189 
43.812 
Z3 
20000 
20000 
20000 
20000 
X1+ X2+ X3 
0.668 
0.667 
0.677 
0.681 
Normalized SAW 
Table 15. GA and NSGA error parentage
GA Error Parentage 
NSGA Error Parentage 
Value of GA 
Value of NSGAII 
Optimal Normalized value 

1.18 
1.03 
0.667 
0.668 
0.675 
Practical discount 
0.73 
1.02 
0.677 
0.681 
0.682 
Wholesale discount 
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
NSGA for practical 
GA for practical 
GA for wholesale 
NSGA for wholesale 

880 
8014 
7394 
11633 
X1 
1006 
4434 
13000 
4531 
X2 
3237 
5379 
1438 
5201 
X3 
5766 
2234 
1201 
2949 
X4 
15315 
5259 
2292 
10530 
X5 
5839 
8769 
12166 
9995 
X6 
5731 
11293 
625 
10840 
X7 
880 
8014 
14413 
11027 
X8 
1006 
4434 
1347 
332 
X9 
3237 
5379 
2560 
3200 
X10 
5766 
2234 
3205 
3158 
X11 
15315 
5259 
1048 
8246 
X12 
5839 
8769 
5377 
100 
X13 
5731 
11293 
9618 
10872 
X14 
880 
8014 
15693 
4211 
X15 
1006 
4434 
6371 
343 
X16 
3237 
5379 
164 
5087 
X17 
5766 
2234 
3643 
254 
X18 
15315 
5259 
795 
8334 
X19 
5839 
8769 
8461 
157 
X20 
5731 
11293 
7654 
4509 
X21 
880 
8014 
1701 
14972 
X22 
1006 
4434 
5622 
758 
X23 
3237 
5379 
15579 
14945 
X24 
5766 
2234 
4695 
412 
X25 
15315 
5259 
3342 
6445 
X26 
5839 
8769 
3919 
1877 
X27 
5731 
11293 
12405 
3704 
X28 
880 
8014 
2853 
1657 
X29 
1006 
4434 
1080 
9449 
X30 
3237 
5379 
9008 
4638 
X31 
5766 
2234 
7672 
5914 
X32 
15315 
5259 
9213 
5326 
X33 
3256842 
8769 
304 
9817 
X34 
16473804 
11293 
4143 
4581 
X35 
231 
3143148 
3173353 
3120360 
Z1 
200001 
16316710 
15979768 
16281954 
Z2 
0.65 
254 
243 
247 
Z3 
880 
200004 
200001 
200004 
$\sum x_{i}$ 
1006 
0.550 
0.682 
0.729 
Normalized SAW 
In this research step by step has been used for solving the supplier selection problem & determining order quantity to them in the condition which both partial & wholesale discounts are valid for order quantity. A threestage approach has been used for modeling & problem solving. In the first stage, the criteria’s weight has been determined by an integrated AHP method & extent analysis. In the second stage, supplier selection problem modeming has been done by multiobjective integer linear programming technique which has been solved by weight method and the third stage has developed a GA and NSGA algorithms then a big problem has solved with algorithms and compared with each other. This research innovation is 1 presenting mathematical model for supplier selection problem in the partial discount exiting condition for order quantity; 2 comparing & investigating the role of partial & wholesale discount in the way of selecting optimum supplier & allocating order to them optimally. In the following, supplier selection problem has been investigated through giving a numerical example as well as results have indicated that both discounts select the same suppliers. However, the way of allocating order to selected supplier & way of calculating order wholesale price are different in wholesale & partial discount.
The recommendations for future studies are:
1. Using other methods to solve multiobjective problem & comparing its results with this research results.
2. In this research. It has been assumed that the decision makers evaluate the criteria independently, although it can be seen that some of the decision makers affect by the others. So, this such condition can be investigated & be investigated with the result of the mentioned condition.
Finally, another objective functions can be added to the problem & their results to be analyzed.
Table 1. Supplier selection criteria for 1966 to 2013

Dicson 
Evans 
Shipley 
Ellram 
Weber et.al 
Tam and tummada 
Pi and low 
Chen et.al 
In and chen 
Wang et.al 
Perić 
Lee et al. 
Yücel and Güneri 
Selection criteria 




Price (Cost) 
√ 
√ 
√ 
√ 
√ 
√ 
√ 
√ 
√ 

Product Quality 
√ 
√ 
√ 
√ 
√ 
√ 
√ 
√ 
√ 
√ 

√ 

OnTime Delivery 
√ 
√ 
√ 
√ 
√ 
√ 


√ 

Warranty And Claims 
√ 




After Sales Service 
√ 
√ 




Technical Support/Expertise 
√ 




Attitude 
√ 




Total Service Quality 
√ 
√ 




Training Aids 
√ 




Performance History 
√ 
√ 
√ 




Financial Stability 
√ 
√ 
√ 
√ 




Location 
√ 
√ 




Labor Relations 
√ 




Relationship Closeness 
√ 
√ 




Management And Organization 
√ 
√ 




Conflict/Problem Solving Capability 
√ 
√ 
√ 




Communication System 
√ 
√ 




Respond To Customer Request 




Technical Capability 
√ 
√ 
√ 




Production Capability 
√ 
√ 




Packaging Capability 
√ 




Operational Controls 
√ 




Amount Of Past Business 
√ 




Reputation And Position In Industry 
√ 
√ 
√ 
√ 
√ 




Reciprocal Managements 
√ 




Impression 
√ 




Business Attempt 
√ 




Maintainability 
√ 




Reliability 










√ 


Size 
√ 
√ 
√ 



productive capacity 
Percentage lead time in delivery 
quality 
Purchase interval 
cost 

16000 
0.1 
80 
[04000] 
15 
S1 
[4001 8000] 
14.5 

[800116000] 
14 

15000 
0.15 
70 
[03000] 
17 
S 2 
[3001 10000] 
16.5 

[1000115000] 
16 

17000 
0.3 
95 
[05000] 
13 
S 3 
[5001 11000] 
12.5 

[1100117000] 
12 

12000 
0.33 
97 
[02500] 
17 
S 4 
[2501 5500] 
16.5 

[550112000] 
16 

14000 
0.15 
65 
[04500] 
14 
S 5 
[4501 9000] 
13.5 

[900114000] 
13 

12500 
0.12 
85 
[03500] 
15.5 
S 6 
[3501 7000] 
14.5 

[700112500] 
14 

16000 
0.05 
98 
[05000] 
18 
S 7 
[5001 8000] 
17.5 

[800116000] 
17 

16000 
0.07 
83 
[03000] 
15.25 
S 8 
[3001 9000] 
15 

[900116000] 
14.75 

16000 
0.11 
95 
[05000] 
19 
S 9 
[5001 12000] 
18.5 

[1200116000] 
18 

16500 
0.25 
60 
[05500] 
13 
S 10 
[5501 9000] 
12 

[900116500] 
11 

13500 
0.12 
90 
[02250] 
18.5 
S 11 
[2251 9500] 
18 

[950113500] 
17.5 

17000 
0.18 
85 
[06000] 
15.5 
S 12 
[6001 10500] 
15 

[1050117000] 
14.5 

145000 
0.11 
75 
[04000] 
17.5 
S 13 
[4001 7000] 
17 

[700114500] 
16.5 

12000 
0.15 
68 
[03250] 
13 
S 14 
[3251 9500] 
12.5 

[950112000] 
11.5 

16500 
0.17 
75 
[05000] 
15.25 
S 15 
[5001 10500] 
15 

[1050116500] 
14 

15000 
0.09 
80 
[04250] 
18 
S 16 
[4251 7250] 
17.5 

[725115000] 
17 

14500 
0.10 
85 
[05500] 
19 
S 17 
[5501 9500] 
18.25 

[950114500] 
17.75 
productive capacity 
Percentage lead time in delivery 
quality 
Purchase interval 
cost 

15550 
0.11 
96 
[03500] 
20 
S 18 
[3501 10000] 
19 

[1000115550] 
19.5 

15000 
0.21 
70 
[03250] 
16 
S 19 
[3251 7550] 
15.5 

[755115000] 
14.5 

15500 
0.15 
84 
[04250] 
18.5 
S 20 
[4250 9500] 
18 

[950115500] 
17.5 

18000 
0.23 
60 
[06200] 
13 
S 21 
[6201 11000] 
12.5 

[1100118000] 
12 

15000 
0.09 
85 
[03250] 
21 
S 22 
[3250 8500] 
19 

[850115000] 
17 

14500 
0.13 
73 
[03000] 
16.25 
S 23 
[3001 9000] 
16 

[900114500] 
15.5 

16500 
0.07 
84 
[04500] 
18 
S 24 
[4501 9500] 
17.5 

[950116500] 
17 

16250 
0.10 
76 
[04300] 
15.75 
S 25 
[4301 9550] 
15 

[955116250] 
14.25 

14500 
0.12 
78 
[05250] 
16.75 
S 26 
[5251 9500] 
16.5 

[950114500] 
15.75 

14000 
0.08 
86 
[03500] 
20 
S 27 
[3501 8500] 
19.75 

[850114000] 
18.5 

15000 
0.12 
82 
[04000] 
19 
S 28 
[4001 7500] 
18.25 

[750115000] 
17.75 

15000 
0.18 
72 
[04200] 
14 
S 29 
[4201 9750] 
13.5 

[975115000] 
13.25 

18000 
0.12 
78 
[05500] 
15.5 
S 30 
[5501 12000] 
14.5 

[1200118000] 
13.5 

18000 
0.12 
78 
[05300] 
16 
S 31 
[5301 9500] 
15.25 

[950115000] 
14.5 

15000 
0.10 
82 
[027500] 
17.5 
S 32 
[27501 6000] 
17 

[600115500] 
16.25 

15500 
0.09 
85 
[05400] 
19.25 
S 33 
[5401 9750] 
19 

[975115000] 
18.75 

15000 
0.05 
96 
[04250] 
18.75 
S 34 
[4250 9800] 
18 

[980114000] 
17.25 

14000 
0.08 
90 
[01500] 
19.5 
S 35 
[1501 7500] 
19.25 

[750112500] 
18.75 
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