Assignment I

Linear Programming Problems

Q1.    A firm is engaged in producing two products P1 and P2. Each unit of product P1 requires 2 kg of raw material and 4 labour hours for processing, where as each unit of product P2 requires 5 kg of raw material and 3 hours of labour of the same type . every week the firm has the availability of 50 kg of raw material and 60 labour hours. One unit of product P1 sold earn profit of Rs, 20 & one unit of product P2 sold gives Rs. 30 as profit. Formulate this problem as LPP to determine as to how many units of each of the products should be produced per week so that the firm can earn maximum profit, assume all units produced cab be sold in the market.

Ans:  Maximize (Z) = 20x₁+30x₂

                Subject to constraints

          2x₁+5x₂≤50, 4x₁+3x₂≤60

          Non Negative Restrictions x₁, x₂≥0

Q2     Vitamins A and B are found in two different foods F1 and F2. One unit of food F1 contains 2 units of vitamin A and 5 units of vitamin B. One unit of food F2 contains 4 units of vitamin A and 2 units of vitamin B. one unit of food F1 and F2 cost Rs. 10 and 12.50 respectively. The minimum daily requirement (for a person) of vitamin A and B is 40 and 50 units respectively. Assuming that anything in excess of daily minimum requirement of vitamin A and B is not harmful. Find out the optimal minimum of food F1 and F2 at the minimum cost which meets the daily minimum requirement of vitamin A and B. Formulate this as a linear programming problem.

Ans:  Minimise (Z) = 10x₁+12.5x₂

                Subject to constraints

          2x₁+4x₂≥40, 4x₁+3x₂≥50

          Non Negative Restrictions x₁, x₂≥0

Q3.    A Marketing manager wishes to allocate his annual advertising budget of Rs. 20,000 in two media A & B. The unit cost a message in media A is Rs. 1000 and in media B is Rs. 1,500. Media A is monthly magazine and not more then one insertion is desired in the issue. Al least five message should appear in media B. The expected effective audience for one message in media A is 40,000 and for Media B is 50,000. Formulate it and solve graphically.

Ans:  Maximize (Z) = 40,000x₁+50,000x₂

                Subject to constraints

          1000x₁+1500x₂≤20,000, x₁≤1, x₂≥5

          Non Negative Restrictions x₁, x₂≥0

          Z=6,66,667 X₁=0, X₂=40/3

Q4     A company produces two types of pen, say A & B. Pen A is a Superior quality and Pen B is lower Quality. Profit on pens A & B is Rs. 5 and 3 per pen respectively. Raw material required for each pen A is twice as that for pen B. The supply of raw material is sufficient only for 1000 pens of type B per day. Pen A requires a special clip and only 400 such clips are available per day. For pen B only 700 Clips are available per day. Find graphically the product mix so that the company can make maximum profit.

 Ans: Maximize (Z) = 5x₁+3x₂

                Subject to constraints

          2x₁+x₂≤1000, x₁≤400, x₂≤700

          Non Negative Restrictions x₁, x₂≥0

          Z=150 X₁=700, X₂=2850

Q5     A dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5,760 to invest and has space for almost 20 items. A fan cost him Rs. 360 and sewing machine Rs. 240. His expectation is that he can sell a fair at profit of Rs. 22 & a sewing machine at a profit of Rs. 18. Assuming that he can sell all the items that can buy, how should be invest his money in order to maximize his profit. Formulate it as a LPP and then use the Graphical method to solve it.

Ans:  Maximize (Z) = 22x₁+18x₂

                Subject to constraints

          360x₁+240x₂≤5760, x₁+x₂≤20

          Non Negative Restrictions x₁, x₂≥0

          Z=392, x₁=8, x₂=12

Q6     A rubber co. is engaged in producing three different types of tyres A, B and C. These three different tyres are produced at the company’s two different production capacities. In a normal eight hour working day plant 1 produces 100, 200 and 200 type of tyres of A, B, C respectively. Plant II produces 120, 120, 400 type of tyres of ABC respectively. The monthly demand of A B & C is 5000, 6000, 14,000 units resp. The daily cost of operation of plant I II is Rs. 5000 and Rs. 7000 respectively. Find the minimum no. of days of operation per month at two different plants to min. the total cost while meeting the demand.

Ans:  Minimize  (Z) = 22x₁+18x₂

                Subject to constraints

          100x₁+120x₂≥5000,  200x₁+120x₂≥6000,  200x₁+400x₂≥14,000

Non Negative Restrictions x₁, x₂≥0

          Z=2,75,000, x₁=20, x₂=25

Q7     A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 calories. Two foods A and B are available at a cost of Rs.4/- and Rs. 3/- per unit respectively. If one unit of A contains 200 units of vitamins , 1 unit of mineral and 40 calories and one unit of food B contain 100 units of vitamins, 2 units of minerals and 40 calories, find by graphic method, what combination of foods be used to have least cost?

Ans:  Minimize  (Z) = 4x₁+3x₂

                Subject to constraints

          200x₁+100x₂≥4000,  x₁+2x₂≥50,  40x₁+40x₂≤1400

Non Negative Restrictions x₁, x₂≥0

          Z=110 x₁=5, x₂=30

Q8     A firm makes product x and y and has a total production of capacity of 9 tones per day x and y requiring the same production capacity. The firm has a permanent contract to supply at least 2 tons of x and at least 3 tones of y per day to another company. Each tonne of x requires 20 machine hours production time and each tonne of y requires 50 machine hours production time, the daily maximum possible number of machine hours is 360. All the firms output can be sold, and the profit made is Rs. 80 per tonne of x and Rs. 120 per tonne of y. It is required to determine the production schedule for maximum profit and to calculate the profit.

Ans:  Maximize (Z) = 80x₁+120x₂

                Subject to constraints

          x₁+x₂≤ 9, x₁≥2, x₂≥3, 20x₁+50x₂≤360

          Non Negative Restrictions x₁, x₂≥0

          Z=960, x₁=3, x₂=6

Q9     Old hens can be bought at Rs. 2 each and young ones at Rs. 5 each. The ole hens lay 3 eggs per week and the young ones lay 5 eggs per week, each egg being worth 30 Paise. A hen costs Rs. 1 per week to feed. Mr. Amit has only Rs. 80 to spend for hens. How many of each kind should Mr. Amit buy to give a profit of at least Rs. 6 per week, assuming that Mr. Amit cannot have more than 20 hens. Solve the linear Programming problem graphically.

Ans:  Maximize (Z) = 0.3(3x₁+5x₂)-1(x₁+x₂)=- 0.1x₁+.5x₂

                Subject to constraints

          2x₁+5x₂≤ 80,       x₁+x₂≤ 20,  -0.1x₁+5x₂≥6

          Non Negative Restrictions x₁, x₂≥0

          Z=8, x₁=0, x₂=16

Q10   The manager of an oil refinery must decide on the optimal mix of two possible blending processes of which the inputs and outputs per production run are as follow

Process	Input(units)		Output(units)
	Crude A	Crude B	Gasoline X	Gasoline Y
1 2	5 4	3 5	5 4	8 4

          The maximum amount available for crude A and B is 200 units and 150 units respectively. Market requirement show that at least 100 units of gasoline X and 80 Units of gasoline Y must be produced . the profits per production run for process 1 and process 2 are Rs. 300 and Rs. 400 respectively. Solve the LPP by Graphical Method.

Ans:  Maximize (Z) = 300x₁+400x₂

                Subject to constraints

          5x₁+4x₂≤200, 3x₁+5x₂≤150

5x₁+4x₂≥100, 3x₁+5x₂≥80

          Non Negative Restrictions x₁, x₂≥0

Z=180000/13, x₁=400/13, x₂=150/13

Q11   A farmer is engaged in breeding pigs. The pigs are fed on various products grown on the farm. Because of the need to ensure nutrient constituents, it is necessary to buy additional one or two products, which we shall call A and B. The nutrient constituents (vitamins and protein) in each of the product are given below:

Nutrient Constituents	Nutrient in the product		Minimum requirement of nutrient constituents
X	36	6	108
Y	3	12	36
Z	20	10	100

          Product A costs Rs. 20 per unit and Product B cost Rs. 40 per unit Determine how much of products A and B must be Purchased so as to provide the pigs nutrients not less tan the minimum required, at the lowest possible cost. Solve graphically.

Ans:  Minimize  (Z) = 20x₁+40x₂        Subject to constraints 36x₁+6x₂≥108,  3x₁+12x₂≥36,  20x₁+10x₂≥100 Non Negative Restrictions x₁, x₂≥0

          Z=160 x₁=4, x₂=2

Q12   A company produces two types of hats. Each hat of the first type requires twice as much labour time as the second type. If all hats are of the second type only, the company can produce a total of 500 hats a day. The market limits daily sales of first and second types to 150 and 250 hats. Assuming that the profits per hat are Rs. 8 for type I and Rs. 5 for type II. Formulate the problem as a linear programming model in order to determine the number of hats to be produced of each type so as to maximize the profit.

Ans:  Maximize (Z) = 8x₁+5x₂                Subject to constraints  2x₁+x₂≤500, x₁≤150, x₂≤250

          Non Negative Restrictions x₁, x₂≥0

Q13   A toy company manufactures two types of doll, a basic version- doll A and a deluxe version- doll B. Each doll of type B takes twice as long to produce as one of type A, and the company would have time to make a maximum of 2000 per day the supply of plastic is sufficient to produce 1500 dolls per day (both A and B combined). The deluxe version requires a fancy dress of which there are only 600 per day available. If the company makes a profit of Rs. 3.00  and Rs. 5.00 per doll, respectively on doll A and B, then how many of each doll should be produced per day in order to maximize the total profit. Formulate this problem.         

Ans:  Maximize (Z) = 3x₁+5x₂                Subject to constraints  x₁+2x₂≤2000,

x₁+2x₂≤1500, x₂≤600   Non Negative Restrictions x₁, x₂≥0

Q14   A farmer has 100 acre farm. He can sell all tomatoes, lettuce, or radishes he can raise. The price he can obtain is Re. 1.00 per kg for tomatoes, Rs. 0.75 a head for lettuce and Rs. 2.00 per kg for radishes. The average yield per acre is 2,000 kg of tomatoes, 3000 head of lettuce, and 1000 kgs of radishes. Fertilizer is available at Rs. 0.50 per kg and the amount required per acre is 100 kgs each for tomatoes and lettuce, and 50 kgs for radishes. Labour required for sowing, cultivating and harvesting per acre is 5 man-days for tomatoes and radishes, and 6 man-days for lettuce. A total of 400 man-days of labour are available at Rs. 20.00 per man-day. Formulate this problem as a linear programming model to maximize the farmer total profit.

Ans:  Maximize (Z) = 1850x₁+2080x₂+1875x₃   Subject to constraints  x₁+x₂+x₃≤100, 5x₁+6x₂+5x₃≤400,

Non Negative Restrictions x₁, x_2, x₃≥0

Q15   An agriculturist has a farm with 126 acres. He produces Radish, Muttar and Potato. Whatever he raises is fully sold in the market, he gets Rs.5 for radish per kg., Rs. 4 for Muttar per kg. and Rs. 5 for Potato per kg. the average yield is 1,500 kg. of Radish per acre, 1800 kg. of Matter per acre and 1200 kg. of Potato per acre. To produce each 100 kg. of Radish and Muttar and to produce each 80 Kg. of potato, a sum of Rs. 12.50 has to be used for manure. Labour required for each acre to raise the crop is 6 man-days for Radish and Potato each and 5 man-days for Muttar. A total of 500 man-days of labour a a rate of Rs. 40 per man day are available. Formulate this as a linear programming model to maximize the agriculturists total profit.  

Ans:  Maximize (Z) = 7072.5x₁+6775x₂+5572.5x₃       

Subject to constraints  x₁+x₂+x₃≤125, 6x₁+5x₂+6x₃≤500,

Non Negative Restrictions x₁, x_2, x₃≥0