Practice Questions For Lpp[1] 4v686h

PRACTICE QUESTIONS FOR IIPM UGP06-09 STUDENTS TOPIC: LINEAR PROGRAMMING PROBLEM (1) Ashok Chemicals Company manufactures two chemicals A and B which are sold to the manufacturers of soaps and detergents. On the basis of next month’s demand, the management has decided that the total production of chemicals A and B should be at least 350 kilograms. Moreover a major customer’s order for 125 kilograms of product A must also be supplied. Product A requires two hours of processing time per kilogram, and product B requires one hour of processing time per kilogram. For the coming month, 600 hours of processing time are available. The company wants to meet the above requirements at minimum total production cost. The production costs are Rs.2 and Rs.3 per kilo of product A and B respectively. The company wants to determine the optimal product mix and the total minimum cost. 1. Formulate the above as a linear programming problem. 2. Solve the problem with the graphical method. 3. Does the problem have any alternative solutions? (2) A tour operator is planning a charter trip to a hill resort. The cost of tour package includes the fare for round trip, surface transportation, food, stay, and site seeing. The trip is restricted to 200 persons and as per past experience, there will be no difficulty in getting 200 persons for the trip. The problem of the tour operator is to determine the best number of Deluxe, Standard and Economy tour packages for this charter tour. These three plans differ according to the facilities offered to the customers in of quality of accommodation, food plan, and other tour options. The following table summarises the estimated prices and corresponding expenses for the travel agent. Prices and costs for tour packages per person. Variable

Tour Plan

Price (Rs.)

X1 X2 X3

Deluxe Standard Economy

10000 7000 6500

Hotel cost (Rs.) 3000 2200 1900

Meal & other exp. (Rs.) 4750 2500 2200

In deciding the numbers of persons in each category the following constraints must be taken into . 1. At least 10% packages must be of the deluxe type. 2. At least 35% but not more than 70% must be in the standard type. 3. At least 30% must be of the economy type. 4. The maximum number of seats for the deluxe package in the aircraft is restricted to 60. 5. The hotel requires that at least 120 persons should be on the deluxe and standard packages put together. The travel agent wishes to determine the number of packages to offer in each category so as to maximize his profit. A. Formulate the above as a linear programming problem in three variables. B. Restate the problem in of two variables taking advantage of the fact that the total number of packages is 200. Solve the restated problem graphically as a two variable problem to determine the gross contribution. If the cost of hiring aircraft is Rs. 200000, what is the highest possible profit?

(3) A firm uses three machines in the manufacture of three products. Each unit of product A requires 3 hours on machine I, 2 hours on machine II and 3 hours on machine III. Each unit of product B requires 4 hours on machine I, one hour on machine II and 3 hours on machine III. Each unit of product C requires 2 hours on each of the machines. The contribution of the three products A, B and C is Rs.30, Rs.40 and Rs.35 per unit respectively. The firm desires to maximise the contribution. The machine hours available on three machines I, II and III are 90, 54 and 93 respectively. A. Formulate the above as a linear programming problem. B. Using the Simplex method, obtain the optimal solution. C. Which of the three products shall not be produced by the firm? (4) 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.3 and Rs.4 from these varieties. Each belt of type A requires twice as much time as belt B, and, if all belts were of type B, the company could make 1000 per day. The supply of leather is sufficient for only 800 per day and both the belts require same amount of leather. Belt A requires a fancy buckle of which only 400 are available per day and an ordinary buckle needed for B type is available to the extent of 700 per day. Formulate this as a linear programming problem and solve it using (a) graphical method and (b) simplex algorithm. (5) An animal feed company must produce 200 kg of a mixture per day consisting of ingredients ‘A’ and ‘B’. ‘A’ costs Rs.3 per kg and ‘B’ Rs.8 per kg. No more than 80 kgs of ‘A’ can be used and at least 60 kgs of ‘B’ must be used. Using simplex algorithm, find how much of each ingredient should be used if the company wants to minimise the cost. (6)

Solve the following linear programming problem graphically: Maximise 6X1+14X2 subject to: 5X1+4X2>=60 3X1+7X2<=84 X1+2X2>=18 X1<=24 X1, X2>=0

(7) Maximise Z= 3X1+ 4X2+ X3 Subject to: X1 + 2X2 + 3X3 <= 90 2X1 + X2 + X3<= 60 3X1 + X2 + 2X3 <= 80 X1, X2, X3 >=0 (8)

Solve graphically Maximize 6X1 + 20X2 Subject to: 2X1 + X2 <= 32 3X1 + 4X2 <= 80 X1 >= 8 X2 >= 10

(9) Solve the following linear programming Problem graphically Minimise Z = 8X1 + 10 X2 Subject to: 8X1 + 4X2 >= 150 3X1 + 9X2 >= 100 X1, X2 >= 0 (10)

(11)

Solve the following LPP graphically: Maximise 20X1+10X2 subject to: X1+2X2<=40 3X1+X2>=30 4X1+3X2>=60 X1, X2>=0 Minimise 5X1+3X2 subject to: 2X1+4X2<=12 2X1+2X2=10 5X1+2X2>=10 X1, X2 >=0

(12) A scrap metal dealer has received an order from a customer for at least 2000 kgs of scrap metal. The customer requires that at least 1000 kgs of a high quality metal called ‘Alpha’ must be present in the total scrap to be supplied, since he wants to use this ‘Alpha’ for producing downstream products of high quality. Further the customer has stipulated that if the total scrap supplied contains more that 175 kgs of a metal that he deems unfit for commercial use, he will not accept the order. The dealer can purchase scrap from two different suppliers ‘A’ and ‘B’ in unlimited quantities with the following percentage composition (by weight) of Alpha and unfit scrap. Supplier A Alpha Unfit Scrap

25% 5%

Supplier B 75% 10%

Supplier A and B are willing to supply the scrap at Rs 1 and Rs 4 per kg respectively, as per these compositions. Formulate this as a linear programming problem to minimise the cost of scrap purchase. (13) Solve the question No. 12 above graphically