Operations Research Answers

A firm uses three machines in the manufacture of three products. Each unit of product A requires 3 hours on machine I, two hours on machine II, and one hour on machine III. While each unit of product B requires four hours on machine I, one hour on machine II, and three hours on machine III. While each unit of product C requires two hours on each of the three machines. The contribution margin of the three products is birr 30, birr 40 and birr 35 per unit respectively. The machine hours available on the three machines are 90, 54, and 93 respectively. a. Formulate the above problem as a linear programing model b. Obtain optimal solution to the problem by using the simplex method. Which of the three products shall not by produced by the firm? Why? c. Calculate the unused capacity if any.

Question No. 1

Messers A BC & Co has recently received an order to prepare 20000 units of their regular product which needs two types of material that is material A and Material B. To produce one unit of finished good 2.5 units of material A are required and 2 units of B are required. Each unit of material A is available for 25 Rs. Per unit and B is available for 30 Rs. Per unit. 15% of their purchase price is required for transportation. Two types of labor is also applied. Labour A uses 2 hours for one unit and labour 2 uses 4 hours for one unit. All types of labors are paid @ 25 Rs. Per hour and FOH is applied @ 20Rs. Per hour. Required: - If company applies 30% margin what price do you suggest for these units?

According to the model: 

4x1+x2+8x3+5x4+2x5>= 12000 

x2+x4+2x5+3x6>=18000 

F(c)=0,1x1+0,2x2+0,2x3+0,3x4+0,4x5+0x6

Build a dual model, solve using the graphical method 

A trading company buys and sells 10000 bottles of pain-balm every year. The company's cost of placing an order of pain-balm is $100. The holding cost per bottle on inventory is $0.30.

To determine the optimum order quantity and inventory cycle time for the pain-balm bottles.

How many orders should be placed each year?

Write the dual of the following LPP

Minimise Z = 16x₁ + 9x₂ + 21x₃

Subject to the constraints:

x₁ + x₂ + x₃ = 16

2x₁ + x₂ + x₃ ≥ 12

x₁ , x₂ ≥ 0

x₃ - unrestricted.

1.) Rina needs at least 48 units of protein, 60 units of carbohydrates, and 50 units of fat each month. From each kilogram of food A, she receives 2 units of protein, 4 units of carbohydrates, and 5 units of fats. Food b contains 3 units of protein, 3 units of carbohydrates, and 2 units of fats. If food A costs Php110 per kilogram and food B costs Php 90 per kilogram. How many kilograms of each food should Rina buy each month to keep costs at a minimum?

Which of the following statements are true? Give a short proof or a counter example in 

support of your answer.

 (i) For any two square matrices A and B, AB = BA.

 (ii) If the following table is obtained in the intermediate stage while solving an LPP by the Simplex method, then the LPP has an unbounded solution: 

____________________________

. -1 -2 0 0 0

____________________________

1 x₁  1 2 -1 0 1

0 x₄  0 3 -1 1 2

____________________________

. 0 4 -1 0 1

____________________________

(iii) The number of basic variables in a feasible solution of a transportation problem with m sources and n destinations is mn.

iv) An optimal assignment of the assignment problem with cost matrix C is also an optimal assignment of the assignment problem with cost matrix C^t

 (v) (1,2) is an optimal solution to the following LPP: 

 Max Z = 2x₁ + 4x₂ subject to 

 x₁ + 2x₂ ≤ 5

 x₁ + x₂ ≤ 4

 x₁, x₂ ≥ 0

Solve the (4x3) game with pay off matrix.

A=

8 5 8

8 6 5

7 4 5

6 5 6

 At each stage, clearly explain the steps involved.

Write the dual of the following LPP after reducing it to canonical form.

Min Z = 3x₁ + 4x₂ + 3x₃

Subject to

2x₁+4x₂ =12

5x₁+3x₃ ≥11

6x₁+ x₂ ≥ 8

x₁,x_2,x₃≥0