Operations Research Answers

2. A firm produces three products A, B, and C, each of which passes through three departments: Fabrication, Finishing and Packaging. Each unit of product A requires 3, 4 and 2 hours; a unit of B requires 5, 4 and 4 hours while each unit of product C requires 2, 4, 5 hours respectively in the three departments. Every day, 60 hours are available in fabrication department, 72 hours in the finishing department and 100 hours in the packaging department. If the unit contribution of product A is ETB 5, of product B is ETB 10, and of product C is ETB 8, determine the number of units of each of the products, which should be made each day to maximize the total contribution. Also determine if any capacity would remain unutilised.

a. Write the formulation for this linear program.

b. Solve the Linear programming problem using simplex method.

2. A firm produces three products A, B, and C, each of which passes through three departments: Fabrication, Finishing and Packaging. Each unit of product A requires 3, 4 and 2 hours; a unit of B requires 5, 4 and 4 hours while each unit of product C requires 2, 4, 5 hours respectively in the three departments. Every day, 60 hours are available in fabrication department, 72 hours in the finishing department and 100 hours in the packaging department. If the unit contribution of product A is ETB 5, of product B is ETB 10, and of product C is ETB 8, determine the number of units of each of the products, which should be made each day to maximize the total contribution. Also determine if any capacity would remain unutilised.

a. Write the formulation for this linear program.

b. Solve the Linear programming problem using simplex method.

1. A dealer wishes to purchase a number of fans and sewing machines. He has only 5760 ETB to invest and has space for almost 20 items. A fan costs him 360 ETB and sewing machine 240 ETB. His expectation is that he can sell a fan at profit of 22 ETB and a sewing machine at a profit of 18 ETB. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit. (Note: ETB = Ethiopian Birr)

a. Formulate it as a linear programming problem (LPP)

b. Use the graphical method to solve it.

A dog owner wants to minimize the cost of buying dog food for his bullmastiffs. Two types of dog

food are available mixed with three kinds of minerals: calcium, phosphorous, and potassium. The per

kilo cost of dog food A and dog food B are P 20.00 and P 16.00, respectively, whereas the minimum

requirement of calcium, phosphorous and potassium are 75, 50, and 30 units, respectively. Also, in

one kilogram of dog food A, 5 units of calcium, 2 units of phosphorous, and 3 units of potassium are

mixed. Again, in one kilogram of dog food B, 4, 6, 3 units of calcium, phosphorous, and potassium,

respectively are mixed. How many of each type of dog food must be bought to minimize the cost?

A dog owner wants to minimize the cost of buying dog food for his bullmastiffs. Two types of dog

food are available mixed with three kinds of minerals: calcium, phosphorous, and potassium. The per

kilo cost of dog food A and dog food B are P 20.00 and P 16.00, respectively, whereas the minimum

requirement of calcium, phosphorous and potassium are 75, 50, and 30 units, respectively. Also, in

one kilogram of dog food A, 5 units of calcium, 2 units of phosphorous, and 3 units of potassium are

mixed. Again, in one kilogram of dog food B, 4, 6, 3 units of calcium, phosphorous, and potassium,

respectively are mixed. How many of each type of dog food must be bought to minimize the cost?

A farmer can plant up to eight hectares of land with rice and corn. He can earn P 5,000.00 for

every hectare he plants with rice, and P 3,000.00 for every hectare he plants with corn. His use of a

necessary fertilizer is limited by the Credit Cooperative Policy of 10 gallons for his entire 8-hectare

land. Rice requires 2 gallons of fertilizer for every hectare planted, and corn just 1 gallon per hectare.

Find the farmer’s maximum profit

A dog owner wants to minimize the cost of buying dog food for his bullmastiffs. Two types of dog

food are available mixed with three kinds of minerals: calcium, phosphorous, and potassium. The per

kilo cost of dog food A and dog food B are P 20.00 and P 16.00, respectively, whereas the minimum

requirement of calcium, phosphorous and potassium are 75, 50, and 30 units, respectively. Also, in

one kilogram of dog food A, 5 units of calcium, 2 units of phosphorous, and 3 units of potassium are

mixed. Again, in one kilogram of dog food B, 4, 6, 3 units of calcium, phosphorous, and potassium,

respectively are mixed. How many of each type of dog food must be bought to minimize the cost?

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

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 onmachine (M1)

, 2 hours on machine (M2)

 and 2 hours on machine (M3)

 and Belt C requires 5hours on machine (M2)

and 4 hours on machine (M3)

. There are 8 hours of time per dayavailable on machine M1

, 10 hours of time per day available on machine M2

 and 15 hoursof 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)

Determine the range of feasibility, optimality and insignificance

d)

Interpret the shadow price

1.    Strong steel company operates two steel mills with different production capacities. Mill I can produces 1000 tons per day of AAA steel, 3000 tons per day of AA steel , 5000 tons of per day of A steel. Mill F can produce 2000 tons per day of each grade of steel. The company has made a contract with the construction firm to provide 24,000 tons of AAA steel, 32,000 tons of AA steel and 40,000 tons of A steel. The cost of running mill I is $1,400 per day and mill F is $ 1000 per day. (2 pt each)

a)  Formulate linear programming model?

b)  How many amount of mill I and F should be produced? Solve by using graphical method? & Simplex Solution Method