Formulate a 0-1 integer programming model for this problem Solve this model by using the computer
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The Answer to my last question regarding the formulation of the problem is:
The Reliance Manufacturing Company produces an aircraft part. The company can produce the part entirely at a flexible work center with multiple computerized machines. The company has four work centers, all of which are different because they were purchased at different times. Each work center has a single operator; however, the company’s operators have different skill levels, resulting in different levels of daily output and product quality. The following tables show the average daily output and an average number of defects per day for each of the company’s five operators who are capable of producing the aircraft part:
Table 1
|
Operator |
Average Daily Output per Machine |
|||
|
A |
B |
C |
D |
|
|
1 |
18 |
20 |
21 |
17 |
|
2 |
19 |
15 |
22 |
18 |
|
3 |
20 |
20 |
17 |
19 |
|
4 |
24 |
21 |
16 |
23 |
|
5 |
22 |
19 |
21 |
21 |
Table 2
|
Operator |
Average Number of Defects per Machine |
|||
|
A |
B |
C |
D |
|
|
1 |
0.3 |
0.9 |
0.6 |
0.4 |
|
2 |
0.8 |
0.5 |
1.1 |
0.7 |
|
3 |
1.1 |
1.3 |
0.6 |
0.8 |
|
4 |
1.2 |
0.8 |
0.6 |
0.9 |
|
5 |
1.0 |
0.9 |
1.0 |
1.0 |
The company wants to determine which operator to assign to each machine to maximize daily output and keep the percentage of defect to less than 4%.
- Formulate a 0-1 integer programming model for this problem
- Solve this model by using the computer
SOLUTION:
- Decision Variable
- Define 0 - 1 variables related to the Operator for Output at each Machine as :
Xij = 1 if Operator i is selected to operate Machine j
= 0 if Operator i is not selected to operate Machine j
- Assume 0 – 1 variables related to the Operator for defects at each Machine as :
Yij = 1 if Machine j is selected for operator i
= 0 if Machine j is not selected for operator i
i = Operator 1, 2, 3, 4, 5
j = Machine A, B, C, D
- Objective Function
The objected Function for the given model will be
Z =
Maximize Z = 18 X1A + 20 X1B + 21 X1C + 17 X1D + 19 X2A + 15 X2B + 22 X2C + 18 X2D + 20 X3A + 20 X3B + 17 X3C + 19 X3D + 24 X4A + 21 X4B + 16 X4C + 23 X4D + 22 X5A + 19 X5B + 21 X5C + 21 X5D
- Constraints are:
Subject to Total Defects ≤ 4% x Total Production (Output Machine)
0,3 X1A + 0,9 X1B + 0,6 X1C + 0,4 X1D + 0,8 X2A + 0,5 X2B + 1,1 X2C + 0,7 X2D + 1,1X3A + 1,3 X3B + 0,6 X3C + 0,8 X3D + 1,2 X4A + 0,8 X4B + 0,6 X4C + 0,9 X4D + 1 X5A + 0,9 X5B + 1 X5C + 1 X5D ≤ 0,04 . Z
X1A + X1B + X1C + X1D ≤ 1 (Operator 1 Constraint)
X2A + X2B + X2C + X2D ≤ 1 (Operator 2 Constraint)
X3A + X3B + X3C + X3D ≤ 1 (Operator 3 Constraint)
X4A + X4B + X4C + X4D ≤ 1 (Operator 4 Constraint)
X5A + X5B + X5C + X5D ≤ 1 (Operator 5 Constraint)
Y1A + Y2A + Y3A + Y4A + Y5A = 1 (Machine 1 Constraint)
Y1B + Y2B + Y3B + Y4B + Y5B = 1 (Machine 2 Constraint)
Y1C + Y2C + Y3C + Y4C + Y5C = 1 (Machine 3 Constraint)
Y1D + Y2D + Y3D + Y4D + Y5D = 1 (Machine 4 Constraint)
X are variable 0 or 1.
Y is variable 1.
I hope the answer can help in solving this problem, have a nice day..
Regards,
HHn
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