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Page no. 1 ASSIGNMENT ON LINEAR PROGRAMMING (2009-2010) QN. 1 A furniture dealer deals in only two items tables and chair. He has Rs 5000 to invest and a space to store at most 60 pieces. A table costs him Rs 250and a chair Rs 50.he can sell a table at a profit of Rs 50 and the chair at the profit of Rs 15. Assuming that he can sell all the items he buys how should he invest his money in order to maximize his profit? Ans Number of chairs 10, Number of tables 50. QN. 2 A housewife wishes to mix two types of foods x and y in such a way that the mixture contains at least 10 units of vitamin A , 12 units of vitamin B and 8 units of vitamin C . The vitamin contents of one Kg food is given below. Vitamin A Vitamin B Vitamin c Food X 1 2 3 Food Y 2 2 1 One Kg of food X costs Rs 6 and one Kg of food Y costs Rs 10 using L.p. find the least cost of the mixture that will provide the diet. Ans X= 2, Y= 4. QN. 3 A manufacturer makes two types of cups A and B. Three machines are required to manufacturer the cups and the time in minutes required by each is given below. MACHINES Type of cup I II III A 12 18 6 B 6 0 9 Each machine is available for a maximum period of 6 hours per day. If the profit on each cup is 75 paisa and on B it is 50 paisa, show that 15 cups of type a and 30 cups of type B should be manufactured per day to get the maximum profit. QN. 4 A company manufactures two articles A and B. There are two departments through which these articles are processed: (1) assembly (2) finishing departments. The maximum capacity of the first department is 60 hours a week and that of other department is 48 hours a week. The production of each article A requires 4 hours in assembly and 2 hours in finishing and that of each unit of B requires 2 hours in assembly and 4 hours in finishing. If the profit is Rs 6 for each unit of A and Rs 8 for each unit of B, find the number of units of A and B to be produced per week in order to have the maximum profit. Ans: A 12 articles; B 6 articles; Max profit Rs 120. QN. 5 A factory owner wants to purchase two types of machines, A and B for his factory. The machine A requires an area of 1,000 sq mts and 12 skilled men for running it and its daily output is 50 units, whereas the machine B requires 12 00 sq mts area and 8 skilled men, and its daily output is 40 units. If an area of 7600 sq mts and 72 skilled men be available to operate machine, how many machines of each type should be bought to maximize the output? Ans: Machine A : 4 Machine B : 3; Maximum output is 320 Units. QN. 6 A dealer wishes to purchase a number of fans and sewing machines. He has only Rs 5,760 to invest and has a space for at most 20 items. A fan costs him Rs 360 and a sewing machine Rs 240.He expects to sell a fan at a profit of Rs 22 and a sewing machine at a profit of Rs 18. Assuming that he can sell all the items that he buys, how should he invest his money to maximize his profit? Solve graphically and maximize the profit. Ans: 8 Fans; 12 sewing machines; Maximum profit=Rs 392. QN.7 A farmer has a supply of chemical fertilizer of type 1 that contains 10% nitrogen and 6% phosphoric acid and type 2 fertilizer which contains 5% nitrogen and 10% phosphoric acid. After testing soil conditions of the field, it is found that at least 14 Kg of nitrogen and 14 Kg of phosphoric acid are required for good crop. The fertilizer type 1 costs Rs 2 per Kg and the type 2 costs Rs 3 per kg. How many kilograms of each type of fertilizer should be used to meet the requirement and the cost be minimum? Page No 2 Ans: Type 1= 100 Kg; type 2= 80 kg; Minimum cost= Rs440. QN (8) A manufacturer produces two type of steel trunks. He has two machines A and B. The first type of trunk requires 3 hours on machine A and 3 hours on machine B. The second type requires 3 hours on machine A and 2 hours on machine B. Machines A and B can work at most 18 hours and 15 hours per day respectively. He earns a profit of Rs 30 and Rs 25 of the first type and second type respectively. How many trunks of each type must he make each day to make maximum profit? Ans:3 trunks of each type Max.Profit=Rs.120 QN: (9) A producer has 30 and 17 units of labour and capital respectively which he can use to produce two types of goods Xand Y. To produce one unit of X, 2 units of labour and 3 units of capital are required. Similarly, 3 units of labour and 1 unit of capital are required to produce one unit of Y. If X and Y are priced at Rs 100 and Rs 120 per unit respectively, how should the producer use his resources to maximize the total revenue? Solve the problem graphically. Ans: X=3units ; Y= 8units; Maximum revenue= 1,260. QN: (10) Two tailors A and B earn Rs 150 and Rs 200 per day respectively. A can stitch 6 shirts and 4 pants per day while B can stitch 10 shirts and 4 pants per day. Form a linear programming problem to minimize the labour cost to produce at least 60 shirts and 32 pants. Ans: Minimize Z = 150X +200Y subject to 3X+ 5Y≥ 30 X+Y ≥ 8 X, Y≥ 0 QN: (11) A diet for a sick person must contain at least 4,000 units of vitamin C, 50 units of minerals and 1,400 calories. Two foods X and Y are available at cost of Rs 4 and Rs 3 per unit respectively. One unit of the food X contains 200 units of vitamins, 1 unit of minerals and 40 calories, whereas one unit of food Y contains 100 units of vitamins, 2 units of minerals and 40 calories. Find what combination of X & Y should be used to have least cost, satisfying the requirements. Ans: X=5units and Y=30 units. QN: (12) A small firm manufactures items A and B. The total number of items A and B that it can manufacture is at the most 24. Item A takes one hour and B takes only half an hour. The maximum time available per day is 16 hours. If the profit on one unit of item A be Rs 300 and one unit of item B Rs 160, how many of each type of item be produced to maximize the profit? Solve the problem graphically. Ans: A: 8 items; B: 16 items; Maximum profit=4,960. QN: (13) A brick manufacturer has two depots, A and B, with stocks 0f 30,000 and 20,000 bricks respectively. He receives orders from three buildings P,Q and R 15,000, 20,000 and 15,000 bricks respectively. The cost of transporting 1000 bricks to the builders from the depots (in Rs) is given below. TO Transportation cost per 1000 bricks (In Rs) FROM P Q R A 40 20 20 B 20 60 40 How should the manufacturer fulfill the orders to keep the cost of transportation minimum? Formulate the linear programming problem and solve it graphically. Ans: Minimize Z= 40X- 20Y+ 1500 subject to the constraints Page no.3 X+Y ≥ 15, X ≤ 15, Y ≤ 20, X+Y≤ 30, X ≥ 0, Y ≥ 0 Where 1 unit of brick = 1000 bricks. QN: (14) A company produces soft drinks has a contract which requires that a minimum of 80 units of chemical A and 60 units of chemical B are to go in each bottle of drink. The Chemicals are available in a prepared mix from two different suppliers. Supplier X has a mix of 4units of A and 2 units of B that costs Rs 10 and the supplier Y has a mix of 1 unit of A and 1 unit of B that costs Rs 4. How many mixes from X and Y should the company purchase to honour contract requirement and yet minimize the cost. Ans: Minimum cost= Rs 260, Mix of type A=10 units, Mix of type B= 40 units. QN: (15) An airplane can carry a maximum of 200 passengers. A profit of Rs 400 is made on each first class ticket and a profit of Rs 300 is made on each economy class ticket. The airline reserves at least 20 seats for first class. However, at least 4 times as many passengers prefer to travel by economy class to the first class. Determine how many each type of tickets must be sold in order to maximize the profit for the airline. What is the maximum profit? Ans: Max. Profit = 64000,First class tickets=40, Economy class tickets = 160 QN (16) A medical company has factories at two places, A and B. From these places, supply is made to each of its three agencies situated at P,Q and R. the monthly requirements of the agencies are, respectively 40,40 and 50 packets of the medicines, while the production capacity of Factories ,A and B are 60and 70 packets, respectively. The transportation cost per packet from the factories to the agencies is given below. Transportation cost per packet (in Rs.) FROM A B TO P 5 4 Q 4 2 R 3 5 How many packets from each factory are transported to each agency so that the cost of transportation is minimum? Also, find the minimum cost. Ans: Minimum cost = Rs 400, from A 10 packets, 0 packets, 50 packets to P, Q, R respectively. From B 30 packets, 40 packets, and 0 packets to P, Q, R respectively. QN: (17) A small manufacturer has employed 5 skilled men and 10 semi- skilled men and makes an article in two qualities, a deluxe model and an ordinary model. The making of a deluxe model requires 2 hours of work by a skilled man and 2 hours of work by semi- skilled man. The ordinary model requires 1 hour by skilled man and 3 hours by semi- skilled man. By union rule, no man may work for more than 8 hours per day. The manufacturer gains Rs 15 on a deluxe model and Rs 10 on an ordinary model. How many of each type should be made in order to maximize the total daily profit? Ans: Maximum profit = Rs 350, Deluxe model = 10, Ordinary model = 20. QN:(18) A dealer whishes to purchase a number of fans and sewing machines. He has only Rs 5,760 to invest and has a space for at most 20 items. A fan and a sewing machine cost Rs 360 and Rs 240 respectively. He can sell a fan at a profit of Rs 22 and a sewing machine at a profit of Rs 18. Assuming that he can sell whatever he buys, how he invest his money in order to maximize his profits. Translate this problem into a linear programming problem and solve it graphically. Ans: The dealer should buy 8 fans and 12 sewing machines to get maximum profit which is Rs 392. Q(19).A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 units of calories. Two foods A and B are available at a cost of Rs 5 and Rs 4 per unit respectively. One unit of the food A contains 200 units of vitamins, 1 unit of minerals and 40 units of calories, while one unit of food B contains 100 units of vitamins, tow units of minerals and 40 units of calories. Find what combination of foods A and B should be used to have least cost, but it must satisfy the requirements of the sick person. Form the question as LPP and solve it graphically. Ans: Minimum value of Z is 145 i.e. at x=5, y=30 Q(20) An aeroplane can carry a maximum of 200 passengers. A profit of Rs.400 is made on each first class ticket and a profit of Rs. 300 is made on each second class ticket. The airline reserves at least 20 seats for first class. However, at least four times as many passengers prefer to travel by second class than by first class. Determine how many tickets of each type must be sold to maximize profit for the airline. Form an L.P.P. and solve it graphically. Ans: 1st class ticket sold= 40 2nd class tickets sold= 160 Q(21) A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1 costs Rs.4 per unit and F2 costs Rs.6 per unit. One unit of food F1 contains 3 units of Vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of Vitamin A and 3 units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutrition requirements.
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