Topic – 1 Linear Programming Problem (LPP) Formulation
1. A firm manufactures two types of electric items A and B, which can make a profit of rupees 20 per unit of A and rupees 30 per unit of B. Each unit of A requires 3 motors and 2 transformers and each unit of B requires 2 motors and 4 transformers. The total supply of these per month is restricted to 210 motors and 300 transformers. Type B is an export model requiring a voltage stabilizer which has a supply restricted to 65 units per month. Formulate the linear programming problem to determine maximum profit. [30, 60; 2400] 2. A firm manufactures manufactures two sizes of headache headache pills A and B. Size A contains 2 grains of aspirin, aspirin, 5 grains of bicarbonate and 1 grain of codeine; Size B contains 1 grain of aspirin, 8 grains of bicarbonate and 6 grains of codeine. It has been found by users that it requires atleast 12 grains of aspirin, 74 grains of bicarbonate and 24 grains of codeine for providing the immediate effects. Formulate the linear programming problem to determine the least number of pills a patient should have to get immediate relief. [2, 8; 10] 3.
PRODUCT ALLOCATION PROBLEM :
A company has three allocation departments (weaving,
processing and packing) with capacity to produce three different types of colthes namely suitings, shirtings and woollens yielding a profit of Rs. 2, Rs. 4 and Rs. 3 respectively. One meter of suiting requires 3 minutes in weaving, 2 minutes in processing and 1 minute in packing. Similarly one meter of shirting requires 4 minutes in weaving, 1 minutes in processing and 3 minute in packing. One meter of woollen requires 3 minutes in each department. In a week, total run time of each department is 60, 40 and 80 hours for weaving, processing and packing respectively. Formulate the linear programming problem to find the product mix to maximize the profit. 4. A carpenter makes chairs and tables, for which he uses two machines M1 and M2. A chair require 2 hours of working on M1 and 6 hours of working on M2, while for a table he works for 4 hours on M1 and 2 hours on M2. If one working day be of 8 hours, represent it as a linear programming problem. Given that he earns Rs. 30 on each chair and Rs 50 on each table, which he produces and the shop has 2 machines of type M1 and 3 of type M2. 5. A company is involved in the production of two items (X and Y). The resources need to produce X and Y are twofold, namely machine time for automatic processing and craftsman time for hand finishing. The table below gives the number of minutes required for each item:
Item
LPP
Machine time
Craftsman time
X
13
20
Y
19
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The company has 40 hours of machine time available in the next working week but only 35 hours of craftsman time. Machine time is costed at £10 per hour worked and craftsman time is costed at £2 per hour worked. Both machine and craftsman idle times incur no costs. The revenue received for each item produced (all production is sold) is £20 for X and £30 for Y. The company has a specific contract to produce 10 items of X per week for a particular customer. (a) Formulate the problem of deciding how much to produce per week as a linear program. (b) Solve this linear program graphically 6. A firm is engaged in producing two products, A and B. each unit of product A requires 2 kg of raw materials and 4 labour hours for processing, whereas each unit of B requires 3 kg of raw materials and 3 hours of labour, of the same type. Every week, the firm has an availability of 60 kg of raw material and 96 labour hours. One unit of product A sold yields Rs 40 and one unit of product B yields Rs 35 as profit. Formulate this problem as a linear programming problem 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 that there is no marketing constraints so that all the items produced can be sold. 7. The Agricultural Research Institute suggested to a farmer to spread out atleast 4800 kg of a special phosphate fertilizer and not less than 7200 kg of a special nitrogen fertilizer to raise productivity of crops in his fields. There are two sourses of obtaining these mixuters A and B. both of these are available in bags weighing 100 kg each and they cost Rs 40 and Rs 24 respectively. Mixture A contains phosphate and nitrogen equivalent of 20 kg and 80 kg respectively, while mixture B contains these ingredients equivalent to 50 kg each. Write this as a LPP to determine how many bags of each type the farmer should buy in order to obtain the required fertilizer at minimum cost. 8. DIET PROBLEM : A diet is being prepared for the University of Health Management. The objective is to feed the students at the least cost, but the diet must have between 1,800 and 3,600 calories. No more than 1,400 calories can be starch, and no fewer than 400 can be protein. The varied diet is to be made of two foods: A and B . Food A costs $0.75 per pound and contains 600 calories, 400 of which are protein and 200 starch. No more than two pounds of food A can be used per resident. Food B costs $0.15 per pound and contains 900 calories, of which 700 are starch, 100 are protein, and 100 are fat. a. Write the equations representing this information. b. Solve the problem graphically for the amounts of each food that should be used. 9. Do Problem 3 with the added constraint that not more than 150 calories shall be fat and that the price of food has escalated to $1.75 per pound for food A and $2.50 per pound for food B.
10. M ARKETI NG PROBLEM : The PQR stone company sells stone secured from any of the three adjacent quarries. The stone sold by the company must conform to the following specifications: Material X equal to 30 %; material Y equal to or less than 40 % and material Z between 30 % and 40 %. LPP
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Stone from quarry A costs Rs. 100 per tonne and has the following properties: Material X : 20 %; material Y : 60 % and material Z : 20 % Stone from quarry B costs Rs. 120 per tonne and has the following properties: Material X : 40 %; material Y : 30 % and material Z : 30 % Stone from quarry C costs Rs. 150 per tonne and has the following properties: Material X : 10 %; material Y : 40 % and material Z : 50 % Formulate the above LPP to minimise cost per tonne. 11. I NVESTMENT PROBLEM: A retired person wants to invest up to an amount of Rs. 30,000 in fixed income securities. His broker recommends investing in two bonds: Bond A yielding 7 % and bond B yielding 10 %. After some consideration, he decides to invest at most Rs 12,000 in bond B and at least Rs 6,000 in Bond A. he also wants the amount invested in Bond A to be at least equal to the amount invested in Bond B. what should the broker recommend if the investor wants to maximize his return on investment? Solve graphically. 12. M E D I A P ROB L E M : The marketing Department of Everest Company has collected information on the problem of advertising for its products. This relates to the advertising media available, the number of families expected to be reached with each alternative, cost per advertisement, the maximum availability of each medium and the expected exposure of each one (measured as the relative value of one advertisement in each of the media). The information is given as under: Expected Maximum No. of families Cost/ad. exposure Advertising media availability to cover (Rs) (units) (no. of times) TV (30 sec.) Radio (15 sec.) Sunday edition (1/4 page) Magazine (1 page)
3000
8000
8
80
7000
3000
30
20
5000
4000
4
50
2000
3000
2
60
Other information and requirements: The advertising budget is Rs. 70,000
Atleast 40,000 families should be covered. At least 2 insertions be given in Sunday Edition but not more than 4 advertisements should be given on the TV. Formulate this as a LPP. The company’s objective is to maximize the expected exposure. 13. A company makes two kinds of leather belts. Belt A is a high quality belt, and belt B is of lower quality. The respective profits are Rs 4 and Rs 3 per belt. Each belt of type A requires twice as much time as a belt of type B, and if all belts were of type B, the
LPP
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company could make 1000 belts per day. The supply of leather is sufficient for inly 800 belts per day (both A and B combined). Belt A requires a fancy buckle and only 400 buckles per day are available. There are only 700 buckles a day available for belt B. determine the optimal product mix. 14. Old hens can be bought for RS. 2 each and young one Rs 5 each. The old hens lay 3 eggs per week and young ones, 5 per week, each being worth 30 paisa. A hen costs Re 1 per week to feed. If I have only Rs. 80 to spend for hens, how many of each king should I buy, assuming that I cannot house more than 20 hens, assuming that I can not house more than 20 hens. Write a mathematical model of the problem. 15. Designing a diet - A dietician wants to design a breakfast menu for certain hospital patients. The menu is to include two items A and B. Suppose that each ounce of A provides 2 units of vitamin C and 2 units of iron and each ounce of B provides 1 unit of vitamin C and 2 units of iron. Suppose the cost of A is 4¢/ounce and the cost of B is 3¢/ounce. If the breakfast menu must provide at least 8 units of vitamin C and 10 units of iron, how many ounces of each item should be provided in order to meet the iron and vitamin C requirements for the least cost? What will this breakfast cost? 16. Bicycle factories - A small business makes 3-speed and 10-speed bicycles at two different factories. Factory A produces 16 3-speed and 20 10-speed bikes in one day while factory B produces 12 3-speed and 20 10-speed bikes daily. It costs $1000/day to operate factory A and $800/day to operate factory B. An order for 96 3-speed bikes and 140 10-speed bikes has just arrived. How many days should each factory is operated in order to fill this order at a minimum cost? What is the minimum cost? 17. Michigan Polar Products makes downhill and cross-country skis. A pair of downhill skis requires 2 man-hours for cutting, 1 man-hour for shaping and 3 man-hours for finishing while a pair of cross-country skis requires 2 man-hours for cutting, 2 manhours for shaping and 1 man-hour for finishing. Each day the company has available 140 man-hours for cutting, 120 man-hours for shaping and 150 man-hours for finishing. How many pairs of each type of ski should the company manufacture each day in order to maximize profit if a pair of downhill skis yields a profit of $10 and a pair of cross-country skis yields a profit of $8? 18. A biscuit manufacturing company plans to produce two types of biscuits, one with a round shape and another with a square shape. The following resources are used in manufacturing the biscuits, (i) Raw material, of which daily availability is 150 kg. (ii) Machinery, of which daily availability is 25 machine hours. (iii) Labour, of which daily availability is 40 man-hours. The resources used are shown in Table 1. If the unit profit of round and square biscuits is Rs 3.00 and Rs 2.00 respectively, how many round and square biscuits should be produced to maximize total profit ? Table 4.1: Resources Used Requirement / unit Daily availability Resources Round Square Raw material Machine Man-power
LPP
100
115
1500 gms
10
12
720 min
3
2
240 min
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19. Rahul Ads, an advertising company is planning a promotional campaign for the client's product, i.e., sunglasses. The client is willing to spend Rs. 5 lakhs. It was decided to limit the campaign media to a weekly magazine, a daily newspaper and TV advertisement. The product is targeted at middle-aged men and women, and the following data was collected (Table 4.2). Table 4.2: Data Collected
Media Weekly magazine Daily newspaper TV ad
Cost per advertisement
Expected viewers
30000
115000
45000
205000
125000
700000
The client is interested to spend only Rs. 1 lakh on the ads in the weekly magazine which expecting a viewership of a minimum of 21 lakh people in the case of the television advertising. Maximize the viewers to the advertisements.
Lpp by lingo LP_Examples.html LPP
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http://www.utdallas.edu/~scniu/OPRE-6201/documents/LP2-
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