Formulation

Formulation 1. Production Allocation problem, A firm manufactures two types of products A& B and sell them at a profit of Rs 2 on type A and Rs 3 on type B. Each product is processed on two machines G and H. Type A requires one minute of processing time on G and two minutes on H, type B requires one minute on G and one on H, the machine G is available for not more than 6 hour 40 minutes while machine His available for 10 hours during any work day. Formulate the problem as LPP. 2. Dieticians tell us that a balance diet must contain quantities of nutrients such as calories, minerals. Vitamins The medical experts and dieticians tell that is necessary for an adult to consume 75g of proteins, 85g of fats, 300g of carbohydrates daily. The following table gives the food items, analysis, and their respective cost. Construct this problem as LPP to find out the food that should be recommended from a large number of Alternative sources of these nutrients so that the total cost of food satisfying minimum requirements of balanced diet is the lowest. 3. A firm manufactures 3 products A, B and C. The profits are Rs 3,Rs2, &Rs4 respectively. The firm has two machines and below is given the required processing time in minutes for each machine on each product. A Machine D E

4 2

Products B C 3 2

5 4

Machines D and E have 2,000 and 2,500 machine-minutes respectively. The firm must manufacture 100 A’s, 200B’s and 50c’s but not more than 150 A’s. Formulate an LPP to maximize profit. 4) The manager of an oil refinery must decide on the optimum mix of 2 possible blending processes of which the inputs and production runs are as follows. Input Out put Proces Crude A Crude B Gasoline X Gasoline Y s 1 6 4 6 9 2 5 6 5 5 The maximum amounts available of crudes A and B are 250 units and 200 units respectively. Market demand shows that at least 150 units of gasoline X and 130units of gasoline Y must be produced. The profits per production run from process 1 and 2 are Rs 4 and Rs 5 respectively. Write the mathematical form of this problem for maximizing the profit. 5)

A manufacturer produces 2 types of models M1 and M2 .Each M1 model requires 4 hrs of grinding and 2 hrs of polishing, where as each M2 Model requires 2hrs of grinding and 5hrs of polishing. The manufacturer has 2 grinders and 3 polishers. Each grinder works for 40 hrs per week and each polisher works for 60 hrs per week. Profit on M1 model is Rs 3, and on M2 model is Rs 4. What we produced in a week is sold in the market.

How should the manufacturer should allocate his production capacity to the two types of models so that he may Make the maximum profit. Write the mathematical form of the given problem. 6. An animal feed company must produce 200 Ibs of a mixture containing ingredients X1, X2,.X1 costs Rs 3 per Ib and X2 costs Rs 8 per Ib. Not more than 80 Ibs of X1 can be used and miximum quantity to be used for X2 is 60 Ibs. Formulate the mathematical form to find how much of each X1,X2 should be used if the company wants to minimize the cost. 7. Old hens can be bought for Rs 2 each but young hens cost Rs 5 each. The old hen lay 3 eggs per week each being worth 30 paise. A hen cost Rs 1 per week to feed. If have only Rs 80 to spend for hens, how many of each kind should buy to give a profit of more than Rs 6 per week, assuming that I can’t house more than 20 hens. Write the mathematical form of the given problem. 8. A manufacturer produces two types of models M1 and M2. Each M1 model requires 4 hrs of grinding and 2 hrs of polishing where as each M2 model requires 2 hrs of grinding and 5 hrs of polishing. The manufacturer has 2 grinders and 3 polisher Each grinder works for 40 hrs per week and each polisher works for 60 hrs per week. Profit on M1 model is Rs 3 and on M2 model is Rs 4. What is produced in a week is sold in the market. How Should the manufacturer should allocate his production capacity to the two types of models. So that he may make the maximum profit write the mathematical form of the given problem 9. A whole-sale dealer deals in two kinds A and B,say, of mixtures of nuts. Each kg. of mixture A contains 60 gms of almonds, 30 gms of cashew nuts and 30 gms of hazel nuts. Each kg. of mixture B contains 30 gms of almonds, 60 gms of cashew nuts and 180 gms of hazel nuts. The remainder of both mixtures is peanuts. The dealer is contemplating to use mixtures A and B to make a bag, which will contain at least 240 gms of almonds, 300 gms of cashew nuts and 540 gms of hazel nuts. Mixture A costs Rs. 8.00 per kg. and mixture B costs Rs. 12.00 per kg. Assuming that mixtures A and B are uniform, formulate the problem as an L.P.P. Solve the problem by graphical method 10. A manufacturer of leather belts makes three types of belts A, B and C which are processed on three machines M1 , M2 and M3 . Belt A requires 2 hours on machine M1 and 3 hours on machine on M3 . Belt B requires 3 hours on machine M1 , 2 hours on machine M2 and 2 hours on machine M3 and Belt C requires 5 hours on machine M2 and 4 hours on machine M3. There are 8 hours of time per day available on machine M1 , 10 hours of time per day available on machine M2 and 15 hours of time per day available on machine M3. The profit gained from belt A is Rs. 3.00 per unit, from belt B is Rs. 5.00 per unit, from belt C is Rs. 4.00 per unit. Formulate the problem as L.P.P. Solve the problem by Simplex Method 11. A local travel agent is planning a charter trip to a major sea resort. The eight day/seven night package includes the fare for round-trip travel, surface transportation, boarding and loading and selected four options. The charter trip is restricted to 200 persons and past experience indicates that there will not be any problem for getting 200 persons. The problem for the travel agent is to determine the number of Deluxe, Standard and Economy tour packages to offer for this charter. These three plans differ according to seating and service for the flight, quality of accommodations, meat plans and tour options. The following table summarizes the estimated

prices for the three packages and the corresponding expenses for the travel agent. The travel agent has hired an aircraft for the flat fee of Rs. 2,00,000 for the entire trip. Prices and Costs for Tour Packages per person

Packages offered

Price(Rs.)

Hotel Costs(Rs.)

Deluxe Standard Economy

10,000 7,000 6,500

3,000 2,200 1,900

Meals & Other Expenses (Rs.) 4,750 2,500 2,200

In planning the trip, the following considerations must be taken into account: (i) At least 10 per cent of the packages must be of the deluxe type. (ii) At least 35 per cent but no more than 70 per cent must be of the standard type. (iii) At least 30 per cent must be of the economy type. (iv) The maximum number of deluxe packages available in any aircraft is restricted to 60. (v) The hotel desires that at least 120 of the tourists should be on the deluxe and standard packages together. The travel agent wishes to determine the number of packages to offer in each type so as to maximize the total profit. (a) Formulate the problem as a linear programming problem. (b) Restate the above linear programming problem in terms of two decision variables , taking advantage of the fact that 200 packages will be sold. Find the optimum solution using graphical method for the restated linear programming problem and interpret your results.

12. A manufacturer of wooden articles produces tables and chairs, which require two types of inputs mainly, they being wood and labour. The manufacturers knows that for a table 3 units of wood and 1 unit of labour are required while for a chair they are 2 units each. The profit from each table is Rs.20 while it is Rs.16 for each chair. The total available resources for the manufacturer are 150 units wood and 75 units of labour. The manufacturer wants to maximize his profit by distributing his resources for tables and chairs. Formulate the problem mathematically. Also solve the problem by graphical method. 13. A firm can manufacture three types of cloth namely and C. Three types of wool are required for it –red, green and blue. One unit length of type A cloth needs 2yards of red wool and 3yards of blue wool one unit length of type B cloth needs 3yards of red, 2yards of green and 4yards of blue wool while one unit length of type C needs 5yards of green wool and 4yards of blue wool. The firm has a stock of 8yards of red wool, 10yards of green wool and 15yards of blue wool. The income obtained by the firm from one unit length of cloth of type A is Rs.3,of the type B is Rs.5 and that of the type C is Rs.4.How should the firm allocate the available material so as to maximize total income from the finished cloths? Formulate the linear programming problem.

14. A person wants to decide the constituents of a diet, which will fulfil his daily requirements of proteins, fats and carbohydrates at the minimum cost. The choice is to be made from four different types of food. The yields per unit of these foods are given in the table below:. YIELD PER UNIT Food type Proteins Fats Carbohydrates Cost per unti(RS) 1 3 2 6 45 2 4 2 4 40 3 8 7 7 85 4 6 5 4 65 Minimum 800 200 700 Requirement Formulate linear programming model for the problem

15. An advertising company wishes to plan its advertising strategy in three different media- television, radio and magazine. The purpose of advertising is to reach as large number of potential customers as possible. The following data are availableTV RADIO MAGAZINE Cost of an advertising unit Rs.30000 Rs.20000 Rs.25000 No. Of potential customers 2,00,000 6,00,000 2,50,000 Reached per unit No. Of female customers 1,50,000 4,00,000 1,20,000 Reached per unit The Co. wants to spend to spend not more than Rs. 4,50,000 on ads. The following requirements must be met: (i) The no. of advertising unit on TV and Radio each should be between 5 and 12. (ii) At least 10,00,000 exposures take place among female customers. (iii) Advertising on magazines be limited to Rs. 2,50,000 (iv) At least 5 ad units be bought on magazine. Formulate a mathematical model for LP.