Linear Programming 1

Example: 2.9: (Media selection) A producer is willing to spend Rs. 20,000 to produce a new model if pressure cooker through advertisement. He wishes to advertise in daily newspaper, radio and prime time television. The cost effectiveness of advertising depends on its exposure to housewives. Advertising Media

Cost per unit (Rs.)

Estimated Number of Housewives Exported to each Advertising Unit

Newspaper

200

2,000

Radio

400

4,000

Television

1,000

8,000

The unit wishes that at least 5,000 housewives should exposed to T.V. advertising. Also the expense on newspaper advertising must not exceed Rs.5,0000. Formulate the problem as a linear programme problem.

Example: 2.12: A research laboratory has two melts of A & B alloy to make up a new mix. The compositions of metals are as under.

Melt X1 X2

Composite (Parts)

A

B

2

1

1

1

To make the new mix, at least 12 kgs of A and 8 kgs of B is needed. Melt x1 costs Rs.20 per kg. While melt x2 costs Rs. 25 per kg. Formulate the problem as LPP to minimize cost.

Example: 2.13: A large factory has following requirements for supervisors. Period

Time

Minimum Number of Supervisors Required

1

6 A.M.

10 A.M.

3

2

10 A.M.

2 P.M.

6

3

2 P.M.

6 P.M.

10

4

6 P.M.

10 P.M.

6

5

10 P.M.

2 A.M.

10

6

2 A.M.

6 A.M.

5

Supervisors report to the factory at the beginning of each period and work for 8 consecutive hours. The factory wants to determine the minimum number of supervisors to be employed so that there will be sufficient number of supervisors available for each period. Formulate the problem as LPP.

Example: 2.16: (Product composition and Mix) A manufacturer wants to market a new fertilizer produced from a mixture of two ingredients I & II. The compositions of the two ingredients are as follows: (Per Kg)

Ingredients

Bonemeal

Nitrogen

Lime

Phosphate

I

20%

30%

40%

10%

II

40%

35%

40%

05%

The manufacturer insists that the fertilizer must contain at least 30 % meal, 20 % nitrogen and 15 % phosphate. The cost of ingredients is Rs. 30 per kg for I and Rs. 25 per kg for II. Formulate the problem as LPP for the quantities of the ingredients to be mixed per kg to minimized cost.

Example: 2.27: Solve graphically the following linear programming problem: Minimize Z = 3x1+5x2 Subject to -3x1 + 4x2 = 12;

2x1-x2 >-2

2x1 + 3x2 > 12;

x1 < 4; x2 > 2 and X1, x2

>0

Example: 2.28: A company manufacturing animal feed must produce 500 kgs of a mixture daily. The mixture consists two ingredients G and G2. Ingredient G1 costs Rs. 5 per kg and ingredient G2 costs Rs. 8 per kg. Nutrient considerations require that the feed contains not more that 400 kgs of G1 and minimum of 200 kgs of G2. Find the mixture of two ingredients that would minimises the cost of the feed.

Example: 2.29: An oil refinery has to decide upon the optimal mix of two possible blending processes. The inputs and outputs per production run are as follows.

Out put Process

Input A

B

A

B

I

5

3

5

8

II

4

5

4

4

The maximum amount of crude A and B are 200 units and 150 units respectively. Market requirements show that at least 100 units of gasoline x and 80 units of Gasoline Y must be produced. Profits per production run from process I & process II are Rs. 4,000 and Rs. 5,000 respectively. Determine the optimum production runs of each process to maximize the profit.

Example: 3.0: A firm is engaged in producing two products A and B. Its costs ad selling price for both the products and relevant data are given below. Per Unit a) Selling price b) Materials c) Wages: Dept A Dept B Dept C

Product A Rs. 200 Rs. 45

Product B Rs. 240 Rs. 50

8 hrs @ Rs 2/ hr 10 hrs @ Rs. 2.25/ hr 4 hrs @ Rs. 2.5/ hr

10 hrs @ Rs. 2/hr 6 hrs @ Rs. 2.25/hr 12 hrs @ Rs 2.5/ hr

d) Variable costs = Rs. 6.50 Fixed costs = Rs 285,000 p.a No. of employees: Dept A = 20 Dept B = 15 Dept C = 18 No. of hours/employee/week = 40 in each department No. of weeks per annum = 50 (1) Formulate the given problem as a linear programming problem, and solve graphically to find:a) Production that will maximize the contribution margin of the firm b) The amount of contribution margin and profit attainable per year. (2) Do you observe any constraint that is redundant? If yes, which one is it?