APPLICATION
Akyama? Engineering & Construction Firm in the construction sector in Aksaray, Turkey is planning on purchasing heat and noise insulation materials for a new project. Firm management is desiring to make a new buying plan based on three criteria from five different suppliers that it worked in the past period. The business top manager wants to be underaffected by delays during product delivery due to Covidl9. Therefore, the delivery time of the products in the last 8 months has been examined, and ‘OnTime Delivery Performance Level of the Product (%)’ has been determined as the first criteria for each supplier. In order to reduce the negative effects of the human and environmental health of the material to be purchased, the criteria of ‘Green Product Quality' and ‘Occupational Safety and Occupational Health Practices’ applied by suppliers due to the epidemic have been scored as *%’. Information about criteria and unit price of the material to be purchased from suppliers are given in Table 7.2.
The business has detected that the demand for the products will be 450 at least and has budgeted $26,000 in total for the material to be purchased from the suppliers. Besides, according to previous period information of the business, as per agreements with suppliers, at least 100 from Supplier1, at least 130 from Supplier2, and at most 170 from Supplier5 are purchased. A minimum of 140 and a maximum of 180 from Supplier3, a minimum of 155, and a maximum of 165 from Supplier4 can be purchased. The number of materials to be purchased from each supplier is given in Table 7.3.
Firm management is planning on increasing the amount to be purchased from Supplier3, Supplier4 and Supplier5 according to the buying plan. The tolerance values for these suppliers are 15, 20, 30, respectively. The business has allocated an
TABLE 7.2
Performance Information of Suppliers
Max Z,: OnTime Delivery Performance Level of the Product (%) 
Max Z_{2}: Green Product Quality (%) 
Max Z,: Work Safety and Labor Health (%) 
Product Unit Cost (25 m^{2}/$) 

Supplier1 
75 
80 
90 
35 
Supplier2 
60 
75 
80 
30 
Supplier3 
80 
70 
70 
38 
Supplier4 
85 
70 
75 
42 
Supplier5 
70 
80 
100 
40 
TABLE 7.3
The Number of Materials to be Purchased from Suppliers
Suppliers 
Minimum 
Maximum 
Supplier1 
100 
 
Supplier2 
130 
 
Supplier3 
140 
180 
Supplier4 
155 
165 
Supplier5 
 
170 
additional budget, which is $4000. According to the information given above, the Multiobjective Sustainable Supplier Selection problem (SSSP) of the business is created as follows:
Subject to (PI)
Step 1: In this step, the positive ideal solution set is determined. Solutions determined for each goal function in (PI) are given below.
The ideal solution set of (PI) is obtained as Z^{l}* = {51,895; 52,410; 57,905}. Step 2: In this step, compromised solutions are determined by depending on Z^{1}* = {51,895; 52,410; 57,905}. Namely, the solution is made forGCM (for p = 1) and Compromise Programming (CP) (forp = »). While determining the longest distance for p =1, the longest distance is completely dominated for p = °o.
i. forp = 7; (PI) is organized as follows.
In order global objective function to be minimum, (0.004256х, + 0.003969x_{2}+0.004086x_{3}+0.004269x.i+0.004602x_{5}) should be maximum. According to this, the model below is obtained.
Subject to (PI.la)
The variable values obtained from the solution of (PI.la) are specified as X = 102; x_{2} = 160; x_{3} 140; x_{4} = 155; x_{5} = 146; Max F = 2.99771. Accordig to these variable values, x" =(102; 160; 140; 155; 146) ve Z^{11} =(51,775; 52,410; 7,905) are obtained. Moreover, Min G: [3(2.997717 )J is 0.002283. Accordingly, the goal functions are very close to their positive ideal solutions.
ii. forp = °o; (PI) is organized as follows.
Subject to (PI.lb)
The variable values and objective function values specified from the solution of (PI.lb) are obtained as = (114; 146; 140; 155; 14 and Z'^{00} = (51,835; 52,320; 57,865). Besides. Min d„ = 0.001717 shows that objective functions are close to their own positive ideal solutions.
Step 3: Similar to Step 1, find the PIS under the constraints w'ith maximum tolerances. Maximum amounts to be purchased from Supplier3, Supplier4 and Supplier5 are demanded to be increased as 15, 20 and 30, respectively. Due to these increases, the budget has been increased $4000. According to these information, (PI) is organized as follows.
Subject to (P2)
In (P2), solutions made for each objective function are given below.
Depending on these solutions, PIS set of (P2) are given below.
Step 4: In this step, by using Z°* ={60,475; 62,395; 68,565}, Global Criterion method and Compromise Proramming solutions are determined.
i. for p = 1, (P2) is organized as follows.
Subject to (P2.1a)
The variable values obtained from the solution of (P2.la) are determined as x, = 104; x_{2} = 291; x_{3} = 140; x_{4} = 155; x_{5} = 145; MaxF = 2.988625. According to these variable values, Min r/_{:} = [3(2.988625 )J = 0.011375. Based on this information, x^{01} =(104; 291; 140; 155; 145) and Z^{01} = (59,785; 62.395; 68.565) are obtained.
ii. For p = (P2) is organized as follows.
Subject to (P2.1b)
The variable values and objective function values determined from the solution of (P2.1b) are дг°~ =(158; 228; 140; 155; 145) and Z^{0o}° = (60,055; 61,990; 68,385). Min d„ = 0.006945 is obtained.
Step 5: In this step, the fuzzy linear programming solution is made for each objective function. At this stage; firstly, ideal solution sets obtained from Steps 1 and 3 are used. These are Z^{1}* ={51,895; 52,410; 57,905} and Z°*= {60,475; 62,395; 68,565}. Membership functions are defined according to positive ideal solutions (Figures 7.37.8).
Membership function for Z,(x);
Membership function for Z_{2}(a);
FIGURE 7.2 Fuzzy membership function for Z, (.v).
FIGURE 7.3 Fuzzy membership function forZ_{2}(x).
FIGURE 7.4 Fuzzy membership function for Z,(x)
FIGURE 7.5 Membership function for budget constraint.
FIGURE 7.6 Fuzzy membership function for constraint g,,(x)
FIGURE 7.8 Fuzzy membership function for constraint gio(*) Membership function for Z_{3}(;r);
Membership functions are then created for constraint functions. Membership function for budget constraint;
Membership function for g_{6}(x);
Membership function for g_{s}(x);
Membership function for g_{10} (x);
Using fuzzy membership functions, the problem is arranged mathematically as follows.
Subject to (P3)
The results obtained from the Fuzzy Linear Programming solution for each objective function (P3) are given below'.
For Max Z,(.r);
For Max Z_{2} (.v);
For Max Z, (a);
Based on these results; a' = max(0.604895;0.5;0.5) = 0.604895 is chosen. Step 6: Final compromise solution steps for a* = 0.604895 are given below. Firstly (PI), if the membership functions in step 5 are arranged fora* cut = 0.604895. the mathematical model named (P4) is obtained.
Subject to (P4)
Positive ideal solutions for (P4) are determined at first.
Max Z (a) igin;
Max Z_{2} (a) igin;
Max Zi(a) igin;
Based on these results, positive ideal solution set of (P4) is (54925; 56375; 62125). By using positive ideal solution set;
i. For p = 1;
Subject to (4.1a)
From the solution of (P4.1a), X = 100; x_{2} = 215; дг_{3} = 140; x_{4} = 155 ve x_{5} = 145 variable values are obtained. According to this, each objective function is obtained as Z = 54,925; Z =56,375; Z = 62,125. Min G.(33) = 0 indicates that all objectives realize in their own positive ideal solution.
ii. For /) = “
Subject to (4.1b)
The variable values obtained from the solution of (4.1b) are x, = 100; x_{2} = 215; x_{3} = 140; x_{4} = 155; x_{5} = 145. On the other hand, each objective function value is obtained as Z, = 54,925;Z_{2} = 56,375; Z_{3} = 62,125. Each objective function value of (P4.1b) is designated as = 0. Thereby, each objective has realized in its own positive ideal solution.
With the Fuzzy GCM algorithm consisting of six steps, variable values and goal function values obtained in each step are given in Table 7.4.
It is possible to analyze the results in Table 7.4 by dividing them into three groups.
i. In Step1, a positive ideal solution set for (PI) has been designated. These values indicate the best performance of each objective function. Due to the fact that objective function values realize in different values of the variables, there is no feasible solution in this step. Actually, non dominated solutions can be determined using these information. By using the information in Step1, nondominated solutions table is given in Table 7.5.
The values in Table 7.5 include the results that Global Criterion Method and CP will give in terms of nondominated solution. This situation is
TABLE 7.4
Fuzzy Global Criterion Method Results
Step1 
Z" = {51.895; 52,410; 57.905} 
No feasible solution 
Step2 
Z" =(51.775; 52,41ft 57.905) 
x" =(102; 160; 140; 155; 146) 
Z'" = (51,835; 52,320; 57.S65) 
*'“ = (114; 146; 140; 155; 145) 

Step3 
Z°‘ ={60,475; 62,395; 68.565} 
No feasible solution 
Step4 
Z^{01} ={59,785; 62,395; 68.565} 
ЛГ^{01} = (104; 291; 140; 155; 145) 
Z°" = (60,055;61.990; 68,385) 
лг°~ = (158; 228; 140; 155; 145) 

Step5 
Г ={55,195; 57,425; 63,245} 
Fuzzy LP Solutions 
Step6 
Г ={54,925; 56,375; 62,125} 
x" =(100; 215; 140; 155; 145) 
Z' ={54.925; 56.375; 62.125} 
л:' = (100; 215; 140; 155; 145) 

Z" ={54,925; 56,375; 62,125} 
*“ = (100: 215; 140; 155; 145) 
TABLE 7.5
NonDominated Solution of (PI)
z, 
z_{2} 
Z_{s} 

51.895 
52.230 
57.825 

X^{1}' 
51.775 
52.410 
57.905 
3* X 
51.775 
52.410 
57.905 
explainable by using the information in Steps1 and 2. Firstly, (%) value of each objective function has been calculated based on the results in Step1.
These rates show the average % value of the purchased material. In other words, they show the average performance value of each objective. The calculations can be made for Table 7.5 (%) transformation of nondominated solutions are given in Table 7.6.
Ranges of each objective function from the results in Table 7.6 that realizes in terms of Global Criterion Method and CP are determined as 73.75356
From the information from Step2,
for p = 1 (Global Criterion Method),
TABLE 7.6
Nondominated Solutions in Terms of Percentage
z, 
Z 2 
Zi 

X^{1}* 
74.34814 
74.82808 
82.84384 
х~* 
73.75356 
74.658125 
82.48575 
X^{5}* 
73.75356 
74.65812 
82.48575 
These results have realized in lower limits of % ranges of non dominated solutions.
For p = ~ (CP).
The % values obtained in terms of CP has been within the % ranges of the nondominated solutions and higher than the lower limits. Two solution results are an expected situation in terms of distance family. The solution obtained for p = ~ is dominate the results given by P= 1
ii. The difference in these steps stems from increasing right hand side values of gi(.v), gb(x), gs(x) and gio(J0 constraints in accordance with the information coming from the decision maker. The values designated in these steps are given in Table 7.4. It is possible to make similar calculations and comments given above for Steps 3 and 4.
iii. In Steps 5 and 6, the decision is made according to the positive ideal solutions obtained from Steps 1 and 3. From the fuzzy linear programming perspective in Step5, firstly, membership functions were created for each goal function using Z^{S} ={51895; 52410; 57905} and Z°* ={60,475; 62,395; 68,565} afterwards, fuzzy model was established by defining membership functions for constraint functions. In Step5, the established model is solved for each objective function, and the most satisfactory one is preferred among these solutions. In Step6, firstly, by using cc cut determined from a = max*a{, right hand side values of giU), ge(x), gg(x) and gioOO constraint functions have been assigned.
In (P4), each goal function is realized at the same value of same decision variables. According to Lai and Hwang (1994), this solution type in MODM problems are called optimal solution. Therefore, the solutions obtained from (P4.la) and (P4.1b) are optimal solutions. When these results are taken into consideration, it is seen that positive ideal solution set and compromised solutions are equal to each other in the problem. Objective function values for these three situations have been calculated below as %.
Accordingly, when 100, 215, 140, 155 and 145 material are purchased respectively, ontime delivery performance mean of products are calculated as 72.74834%, green product quality mean is 74.66887%, and Work Safety and Labor Health average performance of the suppliers are 82.48575%.