Operations Research Answers

Max. Z = GH¢1x1 + GH¢2x2

Subject to:

x1 + x2 ≤ 20

2x1 + x2 ≤ 30

x1 ≤ 25

x1, x2 ≥ 0

(i)Construct the graph of this linear programming problem.

(ii) After constructing the graph in (i), how would you describe the third constraint

Question 3

a) Consider the minimization of this LPP

Min. Z = GH¢3x1 + GH¢2x2

Subject to:

6x1 + 4x2 ≤ 24

x1 ≤ 3

x1, x2 ≥ 0

(i) Graph the constraints and the objective function. Use ‘Z’ of GH¢ 12, and GH¢ 8

(ii) Comment on your graph

1.A solicitor’s firm employs typists on hourly piece-rate basis for their daily work. There are five typists and their charges and speed are different. According to an earlier understanding, only one job is given to one typist and the typist is paid for a full hour even when s/he works for a fraction of an hour. Find the least cost allocation for the following data:

Typist

Rate/hour (Birr)

Number of Pages Typed/hour

Job

No. of Pages

A

5

12

P

199

B

6

14

Q

175

C

3

8

R

145

D

4

10

S

298

E

4

11

T

178

Assume that the typist is to be paid @ birr 5 per hour, the elements of the cost matrix are obtained as follows. To illustrate, if typist A is given job P, he would require 199/12 =16 hours and, hence, be paid for 17 hours. This results in a cost of Birr 85(17 X 5) for this combination.

1.Using the following cost matrix, determine

(a) Optimal job assignment, and

(b) The cost of assignments.

Machinist

Job

1

2

3

4

5

A

10

3

3

2

8

B

9

7

8

2

7

C

7

5

6

2

4

D

3

5

8

2

4

E

9

10

9

6

10

A firm manufactures three different types of hand calculators and classifies them as small, medium, and large according to their calculating capabilities. The three types hone production requirements given by the following table

Small medium large

Electronic circuit component 5 7 10

Assembly time (n) 1 3 4

Cases 1 1 1

The firm has methyl limit of 90.000 circuit components, 30.000 hours of labor and 9000 cases. If the profit is birr 6 for the small, birr 13 for the medium and birr 20 for the large calculator, then

How many of each should be produced to yield maximum profit

1.     Determine an initial basic feasible solution to the following transportation problem by using (a) NWCM, (b) LCM. And (c) VAM. (b) Based the NWCM solution, carryout a stepping stone method of post optimality analysis to arrive at the optimum solution.

Source

D₁

D₂

D₃

D₄

Supply

A

11

13

17

14

250

B

16

18

14

10

300

C

21

24

13

10

400

Demand

200

225

275

200

1.     Determine an initial basic feasible solution to the following transportation problem by using (a) the least cost method, and (b) Vogel’s approximation method. Based the initial basic feasible solution that is relatively small conduct a modified distribution method to determine the optimum solution to the problem.

Source

Destinations

Supply

D₁

D₂

D₃

D₄

S₁

1

2

1

4

30

S₂

3

3

2

1

50

S₃

4

2

5

9

20

Demand

20

40

30

10

1.     JOY leather, a manufacturer of leather Products, makes three types of belts A, B and C which are processed on three machines M₁, M₂ and M₃. Belt A requires 2 hours on machine (M₁₎ and 3 hours on machine (M₂₎ and 2 hours on machine (M₃). Belt B requires 3 hours on machine (M₁₎, 2 hours on machine (M₂₎ and 2 hour on machine (M₃₎ and Belt C requires 5 hours on machine (M₂₎ and 4 hours on machine (M₃₎. There are 8 hours of time per day available on machine M₁, 10 hours of time per day available on machine M₂ and 15 hours of time per day available on machine M₃. The profit gained from belt A is birr 3.00 per unit, from Belt B is birr 5.00 per unit, from belt C is birr 4.00 per unit. What should be the daily production of each type of belt so that the profit is maximum?

a)     Formulate the problem as LPM

b)    Solve the LPM using simplex algorithm.

c)     Interpret the shadow prices

1.     Sony, a television company, has three major departments for the manufacture of its two models, A and B. The monthly capacities are given as follows:

Per Unit Time Requirement (hours)

Model A

Model B

Hours Available this Month

Department I

4.0

2.0

1,600

Department II

2.5

1.0

1,200

Department III

4.5

1.5

1,600

The marginal profit per unit from model A is Birr 400 and that of model B is Birr 100. Assuming that the company can sell any quantity of either product due to favorable market conditions, determine the optimum output for both the models, the highest possible profit for this month and the slack time in the three departments.

Required

a)     Formulate the problem as LPM

b)     Solve the LPM using graphical method.

5. Consider the following transportation problem:

Destination

Company A B C Supply

1 5 1 7 10

2 6 4 6 80

3 3 2 5 15

Demand 75 20 50

Since there is not enough supply, some of the demands at these destinations may not be satisfied. Suppose these are penalty costs for every unsatisfied demand unit which are given by 5, 3, and 2 for destination A, B, and C respectively.

1. Determine how many units and from which source needs to be shipped using the following methods: (6pts)

A) NWC

B) LCM

C) VAM

2. Find the minimum distribution cost (5pts)