Guidelines: Answer all four questions in the books provided to you. Each problem

ID: 364232 • Letter: G

Question

Guidelines: Answer all four questions in the books provided to you. Each problem carries 25 points. Write only the formulation. I think problem 4 is the most difficult one. Follow the main steps of formulating an LP type of problem: 1. 2. 3. Define Variables-5 points Define the Objective Function-10 points Define the Constraints-10 points Question 1. Assignment type problem. An electronics firm produces electronic equipment, which it supplies to various electrical manufacturers. Quality control records indicate that different employees produce different numbers of defective items. The average number of defects produced by each employee for each of five components is given in Table 1 below: Table 1. Average Number of Defects per Employee Component 25 25 28 14 11 27 1715 2914 24 26 30 14 28 18 17 16 18 28 22 18 16 16 27 19 23 21 19 29 Formulate the problem as an LP that will minimize the total average number of defects produced by the firm each month. Define the variables, objective function and constraints. Question 2. Transshipment type problem. You are given the following transshipment problem. The supply is depicted at auto plants 1 and 2, there are two transshipment centers at nodes 3 and 4 and demand is depicted at the three dealers at nodes 5, 6,7. As you may have noticed, the transshipment centers are connected to each other and cars can be transported from center 3 to center 4 as necessary. Also the dealerships at 5 and 7 can receive cars frorm dealer 6. The transfer center 3 has a capacity of 1500 autos. The transfer center 4 has a capacity of 1600 autos. The corresponding travel costs are depicted at each roadway link in Figure 1 below: a) 20 points-Formulate the problem as an LP, b) 5 points -Assume that the auto plant one (1) has increased its capability to produce an additional 100 autos to a total of 1500 autos. Show the change in the constraints because of this extra capability of auto plant 1

Explanation / Answer

Question 1:

Decision Variable:

Let,

Xij = Assignment of ith employee for producing jth component

i = 1, 2, 3, 4, 5, and 6 for the six employees

j = A, B, C, D, and E for six components

Objective Function:

The objective of assignment model is to minimize number of defects by assigning employees for producing components.

Let, dij represents number of defects produced by assigning ith employee to produce jth component

The objective function is given as follows:

Max Z = (dij x Xij)

Max Z = 25X1A + 25X1B + 28X1C +…+ 19X6D + 29X6E

Subject To:

Assignment of employees to components:

X1j <= 1 Assignment of Employee 1

X2j <= 1 Assignment of Employee 2

…

X6j <= 1 Assignment of Employee 6

Assignment of Components to employees:

XiA = 1 Assignment of component A

XiB = 1 Assignment of component B

…

XiE = 1 Assignment of component E

Non-negative constraint:

Xij >= 0

Question 3:

Decision Variable:

Let, Xi = number of volunteers in shift i,

where i = 1, 2, 3, 4, 5, and 6 for 8-hours shift starting at 00:00, 04:00, 08:00, 12:00, 16:00, and 20:00 respectively.

Hours Scheduled

Hours off

Shift Starting at

00:00

04:00

08:00

12:00

16:00

20:00

D.V

Time Slots

Hours Scheduled for Shifts

Min. required

00.00 - 04.00

04.00 - 08.00

08.00 - 12.00

12.00 - 16.00

16.00 - 20.00

20.00 - 24.00

Objective Function:

Objective is to minimize the total number of employees required:

Minimize Z = X1 + X2 + X3 + X4 + X5 + X6

Subject To:

Minimum number of employees required in slot 00:00 to 04:00

X1 + X6 >= 4

Minimum number of employees required in slot 04.00 - 08.00

X1 + X2 >= 8

Minimum number of employees required in slot 08.00 - 12.00

X2 + X3 >= 10

Minimum number of employees required in slot 12.00 - 16.00

X3 + X4 >= 7

Minimum number of employees required in slot 16.00 - 20.00

X4 + X5 >= 12

Minimum number of employees required in slot 20.00 - 24.00

X5 + X6 >= 4

Non-negative constraint

Xi >= 0