Supplier Selection Analysis Using Minmax  Multi Choice Goal Programming Model


CAUCHY – Jurnal Matematika Murni dan Aplikasi 
Volume 7(1) (2021), Pages 97-104 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: July 16, 2021 Reviewed: September 02, 2021 Accepted: November 01, 2021 
DOI: https://doi.org/10.18860/ca.v7i1.12944   

Supplier Selection Analysis Using Minmax  Multi Choice Goal 
Programming Model 

Novi Rustiana Dewi1, Eka Susanti2, Bambang Suprihatin3, Endro Setyo Cahyono4, 

Anggun Permata5, Nurul Fadhila Yanita6  

1,2,3,4,5,6 Department of Mathematics, Universitas Sriwijaya 

 

Email: novirustiana@unsri.ac.id, eka_susanti@mipa.unsri.ac.id, 
endrosetyo_c@yahoo.co.id, bambangs@unsri.ac.id 

 

ABSTRACT  

Production control, inventory and distribution is an important factor in trading activities. These 
three factors are discussed in a system called Supply Chain Management (SCM). Procurement of 
goods from a company or trading business related to suppliers. In some cases, there are several 
suppliers that can be assessed by considering certain factors. In certain cases, the data from 
several factors that are considered are uncertainty, so the fuzzy approach can be used. The 
MINMAX Multi Choice Goal Programming model can be used to solve fuzzy supplier selection 
problems with linear membership function. It can be applied to selecting supplier of Brastagi 
Oranges. There are four suppliers, namely Jaya, Mako, Baros.  Gina. There are three factor to 
consider, cost, quality and delivery. The decision maker selects the best supplier for ordering 
17000 kg Brastagi oranges. The results, the best supplier is Gina with an order quantity of 10000 
kg and Mako with a total order of 7000 kg. 
 
Keywords: fuzzy; MINMAX multi choice goal programming; supply chain management; supplier 
selection 

INTRODUCTION 

Supply chain management has three main components, namely the process of 
obtaining suppliers of raw materials, the process of changing raw materials into finished 
products and the product distribution process. The first stage in the supply chain is 
supplier selection. Selection of suppliers aims to get products with good quality and 
competitive prices. Supplier selection is related to the process of procuring goods to meet 
customer demands. price and quality, time of delivery is a consideration in supplier’s 
selection, especially for perishable products. Fruit is a type of product that does not last 
long if not stored in the refrigerator. 

Research related to supply chains with application in various fields and solutions 
have been carried out with several approaches. The application of fuzzy TOPSIS in 
supplier selection was introduced by [1]. The fuzzy approach is also used by [2] in the 
selection of suppliers in manufacturing companies. The application of the supply chain 
concept to inventory control and supplier selection for planning new product production 
in several planning horizons was carried out by [3]. Discussion of supply chain problems 

https://doi.org/10.18860/ca.v7i1.12944
mailto:novirustiana@unsri.ac.id
mailto:eka_susanti@mipa.unsri.ac.id
mailto:endrosetyo_c@yahoo.co.id
mailto:bambangs@unsri.ac.id


Supplier Selection Analysis Using Minmax  Multi Choice Goal Programming Model 

Novi Rustiana Dewi 98 

by considering price, supply and demand factors is carried out by [4] and an efficient 
Lagrangian relaxation algorithm is proposed to solve the model. A discussion of 
bioethanol supply chain network problems with a robust approach was introduced [5]. A 
deterministic approach to solving the supply chain problem of food product distribution 
is discussed by [6]. The application of the mix integer programming model to the 
distribution and supply chain problems of liquid helium is given by [7]. The research of 
the [8] is combines the concepts of siting, inventory and routing in the supply chain. 

There are two main studies related to the supplier selection model to be used, 
namely the concept of fuzzy and fuzzy goal programming. The Goal Programming (GP) 
model is used in problems with several objectives to be achieved simultaneously. The GP 
model with fuzzy numbers is called the Fuzzy Goal Programming (FGP) model. The 
concept of FGP with random variables was introduced by [9]. Fuzzy and probabilistic 
approaches to the FGP model are discussed by [10]. Completion of the FGP model with a 
genetic algorithm is discussed by [11]. Research [12] uses a multi-choice goal 
programming model to determine energy renewal facilities. [13] used the FGP model in 
production planning. The choice of waste transportation mode using the FGP model was 
introduced by [14]. The application of the Weighted Goal Programming model in the 
urban planning process is given by [15]. The application of the GP model in capital 
management is given by [16]. The use of the FGP model in transportation problems with 
several modes of transportation is given by [17]. 

The research that has been mentioned is the implementation of the supply chain 
concept to supplier, inventory and distribution components. This research will discuss 
the problem of selecting suppliers of Brastagi oranges using MINMAX Multi Choice Goal 
Programming models (Minmax MCGP). The research focus is on component suppliers. 
This research is a basic research by developing the MINMAX Multi Choice Goal 
Programming introduced by [2]. In [2], the fuzzy number used is the trapezoid fuzzy 
number by considering the factors of price, quality and technology offered. In this study, 
price, quality and time of delivery are considering. Linear membership function is used to 
define these tree factor. 
 

METHODS 

The steps for completing the supplier selection using the MINMAX MCGP method are: 
1. Data Collection and Description 

The data used in this study is primary, consist of data on the purchase with the 

parameters of cost, quality and delivery. The data collection period is from 18 

February to 18 March 2020.  

2. Determine the fuzzy triangular membership value for the goal of price, quality and 

delivery. Following are given fuzzy membership functions for the respective three 

goals, in order of price, quality and timeliness of delivery which are formulated based 

on the data in step 1. The restriction value of variable 𝑐, 𝑘, 𝑑 is determined based on 

the data in step 1. 

 

𝜇(𝑐) = {

1,           𝑐 ≤ 7800

1 − [
(𝐶−𝑆𝐿1(𝑐))

𝑆𝐿2(𝑐)−𝑆𝐿1(𝑐)
] , 7800 ≤ 𝑐 ≤ 10000

0,              𝑐 ≥ 10000

           (1) 

 



Supplier Selection Analysis Using Minmax  Multi Choice Goal Programming Model 

Novi Rustiana Dewi 99 

𝜇(𝑘) = {

1,             𝑘 ≥ 100.
𝑘

𝑆𝐿2(𝑘)
,      0 < 𝑘 ≤ 100.

0,                 𝑘 ≤ 0.

        (2) 

 

𝜇(𝑑) = {

1,             𝑑 ≥ 100.
𝑑

𝑆𝐿2(𝑑)
,          0 < 𝑑 ≤ 100.

0,                 𝑑 ≤ 0.

        (3) 

Where  

𝜇(𝑐) is the membership function for the cost. 
𝜇(𝑘) is the membership function for the quality. 
𝜇(𝑑) is membership function for delivery 
𝑘 is the percentage of average supplier quality. 
𝑆𝐿1(𝑐) is Satisfaction Level lower bound for the unit cost. 
𝑆𝐿2(𝑐) is Satisfaction Level upper bound for the unit cost. 
𝑆𝐿2(𝑘) is Satisfaction Level upper bound for the unit quality. 
𝑆𝐿2(𝑑)is Satisfaction Level upper bound for the unit delivery. 

 

3. The MINMAX MCGP model formulation based on the membership function values 

defined in Step 2. The following is the MINMAX MCGP model introduced by [2]. 

Min 𝐷 

Subject to 

𝐷 ≥ 𝛼𝑖 𝑑𝑖
+ + 𝛽𝑖 𝑑𝑖

−, 𝑖 = 1,2, … 𝑚,   

𝐷 ≥ 𝛿𝑖 (𝑒𝑖
+ + 𝑒𝑖

−), 𝑖 = 1,2, … 𝑚,          (4) 

𝜇(𝑥𝑖 )  − 𝑑𝑖
+ + 𝑑𝑖

− = 𝑦𝑖 ,        𝑖 = 1,2, … 𝑚,           

𝑦𝑖 − 𝑒𝑖
+ + 𝑒𝑖

− = 𝑔𝑖,𝑚𝑎𝑥 ,   𝑖 = 1,2, … , 𝑚,  
𝑔𝑖,𝑚𝑖𝑛 ≤ 𝑦𝑖 ≤ 𝑔𝑖,𝑚𝑎𝑥 , 𝑖 = 1,2, … , 𝑚,  

𝑑𝑖
+, 𝑑𝑖

−, 𝑒𝑖
+, 𝑒𝑖

− ≥ 0, 𝑖 = 1,2, … , 𝑚, 
 

where 

𝐷   : the deviation variable of the objective function 
𝛼𝑖 and 𝛽𝑖 : weight of the positive deviation penalty in the objective function 
𝑑𝑖

+
 and 𝑑𝑖

−  : positive and negative deviation of the objective function 

𝛿𝑖   : the sum of the deviation in the objective function 
𝑒𝑖

+
 and 𝑒𝑖

− : positive and negative deviation on |𝑦𝑖 − 𝑔𝑖,𝑚𝑎𝑥 |. 
𝑦𝑖    : continuous variable with a range of interval value 
𝑔𝑖,𝑚𝑖𝑛 and 𝑔𝑖,𝑚𝑎𝑥 : minimum and maximum  𝑦𝑖 value 
𝜇(𝑥𝑖 )   : membership function for the supplier to i 
 

4. Completion of the model obtained in step (4) uses Lingo 13.0 software 

5. Analyses and conclusion 

  



Supplier Selection Analysis Using Minmax  Multi Choice Goal Programming Model 

Novi Rustiana Dewi 100 

RESULTS AND DISCUSSION  

This research discusses supplier selection problem of citrus fruits for the type of Brastagi 
oranges. The data used are primary data with a data collection period of 30 ordering 
periods. The research was conducted at a fruit shop in Palembang . The following is given 
the research data. 
 

Table 1. Ordering the Data for Each Supplier 

No 
Supplie
r Name 

Ordering delivery 

On 
time 
deliv
ery 

Price offered 

Prece
ntage 

of 
qualit
y (%) 

Date Month Date Month   
Cost 

(@kg) 
Total  

1 Jaya 21 Feb 21 Feb √ - 8500 45900000 80 

2 Mako 21 Feb 21 Feb √ - 8000 43200000 85 

3 Baros 22 Feb 24 Feb - √ 8500 45900000 80 

4 Gina 22 Feb 22 Feb √ - 9000 48600000 95 

5 Jaya 23 Feb 23 Feb √ - 8500 45900000 85 

6 Mako 24 Feb 24 Feb √ - 8500 45900000 90 

7 Baros 25 Feb 25 Feb √ - 8000 43200000 80 

8 Mako 25 Feb 25 Feb √ - 9000 48600000 90 

9 Gina 26 Feb 26 Feb √ - 9000 48600000 90 

10 Mako 27 Feb 28 Feb - √ 8000 43200000 85 

11 Jaya 27 Feb 27 Feb √ - 8500 45900000 85 

12 Baros 28 Feb 28 Feb √ - 8000 43200000 85 

13 Mako 29 Feb 1 Maret - √ 9000 48600000 85 

14 Gina 29 Feb 29 Feb √ - 9500 51300000 90 

15 Jaya 1 Maret 2 Maret - √ 8500 45900000 85 

16 Mako 1 Maret 1 Maret √ - 9000 48600000 95 

17 Baros 2 Maret 2 Maret √ - 9000 48600000 80 

18 Mako 3 Maret 3 Maret √ - 9000 48600000 85 

19 Gina 4 Maret 4 Maret √ - 9500 51300000 85 

20 Jaya 5 Maret 6 Maret - √ 9000 48600000 85 

21 Baros 5 Maret 5 Maret √ - 9000 48600000 80 

22 Mako 6 Maret 6 Maret √ - 9000 48600000 90 

23 Gina 7 Maret 7 Maret √ - 9500 51300000 95 

24 Jaya 8 Maret 10 Maret - √ 9000 48600000 80 

25 Baros 8 Maret 8 Maret √ - 9000 48600000 85 

26 Gina 9 Maret 9 Maret √ - 9500 51300000 95 

27 Mako 9 Maret 9 Maret √ - 9000 48600000 90 

28 Jaya 10 Maret 12 Maret - √ 8500 45900000 85 

29 Baros 10 Maret 10 Maret √ - 9000 48600000 80 

30 Gina 11 Maret 11 Maret √ - 9500 51300000 85 

31 Jaya 12 Maret 13 Maret - √ 9000 48600000 80 

32 Mako 13 Maret 13 Maret √ - 9000 48600000 85 

33 Jaya 13 Maret 14 Maret - √ 9000 48600000 90 



Supplier Selection Analysis Using Minmax  Multi Choice Goal Programming Model 

Novi Rustiana Dewi 101 

34 Gina 14 Maret 14 Maret √ - 9500 51300000 90 

35 Baros 15 Maret 16 Maret - √ 8500 45900000 85 

36 Mako 15 Maret 15 Maret √ - 9000 48600000 90 

37 Gina 16 Maret 16 Maret √ - 9500 51300000 90 
38 Jaya 16 Maret 18 Maret - √ 8500 45900000 85 

39 Mako 17 Maret 17 Maret √ - 9000 48600000 90 

40 Gina 19 Maret 18 Maret - √ 9000 48600000 95 

41 Baros 19 Maret 19 Maret √ - 8500 45900000 90 

42 Jaya 19 Maret 20 Maret - √ 8500 45900000 80 

43 Mako 19 Maret 21 Maret - √ 8500 45900000 80 

44 Gina 20 Maret 20 Maret √ - 9000 48600000 90 

45 Jaya 20 Maret 22 Maret - √ 8000 43200000 80 

46 Baros 21 Maret 21 Maret √ - 8500 45900000 90 

47 Mako 21 Maret 22 Maret - √ 8500 45900000 85 

(Source : PD Wibowo, 21 februari until Maret 2020)  

Table 1 can determine the percentage of on-time delivery, the variable price offered, and 
the varying percentage of quality citrus in good condition with the total of all oranges sent 
by the supplier. The price value of each supplier is obtained by adding up each price in 
purchases divided by the number of investments, determined the average value for each 
data cost, quality, and timeliness. The calculation results are given in Table 2 below. 
 

Table 2.  Value Percentage Criteria from Four Suppliers 

Supplier 𝒙𝒊 Cost (Rp) Quality (%) Delivery (%) 
Total Order 

(kg) 

Jaya 𝒙𝟏 8625 83,33 25,00 64800 

Mako 𝒙𝟐 8750 87,50 71,43 75600 

Baros 𝒙𝟑 8600 83,50 80,00 54000 

Gina 𝒙𝟒 9318 90,91 90,91 59400 

 

Determined the degree of membership for the level of satisfaction of the Decision Maker 
(DM) of each goal using (1), (2), (3). The calculation results are given in Table 3 below: 
 

Table 3. Degree of Membership for DM Satisfaction Level of Each Goal 
Decision Lowest  Highest 

𝒄: Cost > 10000 8465.4 8243.6 8021.8 7800 
SL(c), Satisfaction Level c 0 0,7 0,8 0,9 1 

𝒌 : Kualitas 0 40 60 80 100 
SL(k), Satisfaction Level k 0 0,4 0,6 0,8 1 

𝒅 : Ketepatan Waktu 0 40 60 80 100 
SL(d), Satisfaction Level d 0 0,4 0,6 0,8 1 

 

The value of the level of satisfaction is in the interval [0,1]. Based on Table 3, it is known 
that for the lowest decision value, DM gives a satisfaction level value of 0. For the highest 
decision value, DM gives a satisfaction level value 1. The level of satisfaction for each goal 
of cost, quality, and time delivery is determined based on equations (1), (2), and (3). The 
results are given in Table 4 below. 
 



Supplier Selection Analysis Using Minmax  Multi Choice Goal Programming Model 

Novi Rustiana Dewi 102 

Table 4. Membership Function Value for Each Goal 

Supplier 
Amount Of 

Order 
Cost Quality Delivery 

Jaya 𝒙𝟏 0,625 0,83 0,25 

Mako 𝒙𝟐 0,568 0,88 0,71 

Baros 𝒙𝟑 0,636 0,84 0,8 

Gina 𝒙𝟒 0,31 0,91 0,91 

Average 0,53 0,865 0,6675 

Maximum Value 0,636 0,91 0,91 

 

The lower bound for the price goal is determined based on the average price value 
multiplied by the minimum order. The upper price is the product of the maximum value 
of the price times the maximum order. The same calculation is done for quality goals and 
on time delivery. We obtained a lower bound and an upper bound for the goal value of 
price, quality and on time delivery respectively 28876,5; 48081,6; 
49013,2; 70308;   36045; 68796. The formulation of the MINMAX MCGP model (4) the 
problem of supplier’s selection of Brastagi Oranges with a maximum order quantity for 
each supplier of 10000 kg, minimum order of 15000 kg and a maximum of 17000 kg is 
given as follows. 
 
Minimum  D 

Subject to 

𝐷 ≥ 3𝑑1
+ + 𝑑1

−  

𝐷 ≥ 𝑒1
+ + 𝑒1

− ; 𝐷 ≥ 𝑑2
+ + 5𝑑2

−   

𝐷 ≥ 𝑒2
+ + 𝑒2

−; 𝐷 ≥ 𝑑3
+ + 3𝑑3

− ; 𝐷 ≥ 𝑒3
+ + 𝑒3

−       (5) 

0,625𝑥1 + 0,568𝑥2 + 0,636𝑥3 + 0,31𝑥4 − 𝑑1
+ + 𝑑1

− = 𝑦1   
𝑦1 − 𝑒1

+ + 𝑒1
− = 48081,6  

28876,5 ≤ 𝑦1 ≤ 48081,6   
0,83𝑥1 + 0,88𝑥2 + 0,84𝑥3 + 0,91𝑥4 − 𝑑2

+ + 𝑑2
− = 𝑦2  

𝑦2 − 𝑒2
+ + 𝑒2

− = 70308  
49013,2 ≤ 𝑦2 ≤ 70308  
0,25𝑥1 + 0,71𝑥2 + 0,8𝑥3 + 0,91𝑥4 − 𝑑3

+ + 𝑑3
− = 𝑦3  

𝑦3 − 𝑒3
+ + 𝑒3

− = 68796  
36045 ≤ 𝑦3 ≤ 68796  
𝑥1 ≤ 10000; 𝑥2 ≤ 10000 ; 𝑥3 ≤ 10000 ; 𝑥4 ≤ 10000   
𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 ≥ 15000 ; 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 ≤ 17000  
𝑑1

+, 𝑑1
−, 𝑒1

+, 𝑒1
−, 𝑦1, 𝑦2, 𝑦3 ≥ 0  

 

Solving the linear model (5) uses LINGO 13 software and the solution is obtained in Table 
5 below. 
 
 
 
 
  



Supplier Selection Analysis Using Minmax  Multi Choice Goal Programming Model 

Novi Rustiana Dewi 103 

Table 5. MINMAX MCGP Model Solution for Citrus Fruit Supplier Selection 
No Variable Value 
1. 𝑥1 0 
2. 𝑥2 7000 
3. 𝑥3 0 
4. 𝑥4 10000 
5. 𝑦1 39463.88 
6. 𝑦2 28876.50 
7. 𝑦3 36764.17 
8. 𝐷1

+ 0 
9. 𝐷1

− 32387.88 
10. 𝑒1

+ 0 
11. 𝑒1

− 8617.725 
12. 𝐷2

+ 0 
13. 𝐷2

− 13616.5 
14. 𝑒2

+ 0 
15. 𝑒2

− 39919.50 
16. 𝐷3

+ 0 
17. 𝐷3

− 22694.17 
18. 𝑒3

+ 0 
19. 𝑒3

− 32031.83 
20. 𝐷 68082.50 

 

In Table 5, for a maximum total order of 17000 kg, an order is recommended for 𝑥2 
(Supplier Mako) and 𝑥4 (Supplier Gina). The values of 𝑦1 (Aspiration Rate G1) = 39463.88, 
𝑦2 (Aspiration Rate G2) = 28876.50, 𝑦3 (Aspiration Rate G3) = 36764.17, and other 
deviations are given in Table 5. The values of 𝑥1, 𝑥2, 𝑥3, and 𝑥4 are 0, 7000, 0, 10000, 
respectively. It can be concluded that the order for selecting the best supplier is Supplier 
Gina with an order quantity of 10000 kg, Supplier Mako with an order quantity of 7000 
kg. 
 

CONCLUSIONS 

the results obtained the best supplier for orders of a maximum of 17000 kg are Gina 
Supplier with a total order of 1000 kg of Brastagi oranges and Mako supplier with a 
maximum order of 7000 kg. The best supplier order is obtained by looking at the 
difference in the value of the deviation from the target for each goal of price, quality and 
delivery. The difference in goal value results in a different order of supplier selection. 

ACKNOWLEDGMENTS 

This Research is supported by Universitas Sriwijaya through Sains Teknologi dan Seni 
(SATEKS) Research Scheme with the number of the research assignment contract number 
0163.177/UN9/SB3.LPPM.PT/2020. 

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