Page1 PRACTICE QUESTIONS TOPIC: LINEAR PROGRAMMING PROBLEM SIMPLEX ALGORITHM: (Read the hints and comments in italics appearing in brackets) These questions have been collected from various sources. Broadly they are from various CA final question papers, MMS / MBA final question papers, IIPM Mumbai Sem exams, from the book of Prof. N D Vohra, Oxford University website, some American university websites and those created by Prof. Madhusudan Sohani for his students. For checking your solutions, please use the web page link at: http://www.zweigmedia.com/realworld/simplex.html While using this link, please study the problem given under example to see how a problem is to be fed. 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 simplex method. 3. Does the problem have any alternative solutions? (A minimisation problem of mixed constraints) 2. 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. (Try doing a sensitivity analysis also of this problem!) 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? 3. Solve the following Linear Programming problem using simplex method. Maximise Z = 2X 1 + 4X 2 2X 1 + X 2 <= 18 3X 1 + 2X 2 >= 30 X 1 + 2X 2 =26 X 1, X 2 >= 0 (Mixed constraints, alternative solutions!) 4. A company makes two kinds of leather belts. Belt A is a high quality belt and belt B is of lower
Page2 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 simplex algorithm. (Also try graphically!) 5. Maximise 3X 1 +5X 2 +4X 3 subject to: 2X 1 +3X 2 <=8 2X 2 +5X 3 <=10 3X 1 +2X 2 +4X 3 <=15 X i >=0 6. 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. (Can you simplify and reduce one constraint? Try solving with and without simplifying) 7. For a company engaged in manufacturing three products viz. X, Y and Z, the available data are given in the three tables below. Table 1 Minimum Sales requirements Product Minimum Sales requirement X 10 Y 20 Z 30 Table 2 Operations, Processing time (hours) and Availability of time (Hours per week) Operation Products Time X Y Z available 1 1 2 2 200 2 2 1 1 220 3 3 1 2 180 Table 3 Profit per unit Product Profit per unit X 10 Y 15
Page3 Z 8 Advise the company on the optimal product mix per week and calculate the weekly profit. (Simplify to remove three constraints!) 8. Maximise Z= 22X 1 + 30X 2 + 25X 3 2X 1 + 2X 2 <= 100 2X 1 + X 2 + X 3 <= 100 X 1 + 2X 2 + 2X 3 <= 100 X 1, X 2, X 3 >= 0 (Degeneracy! Refer the famous book on LP by N Paul Loomba) 9. Minimise Z = X 1 + 4X 2 + 3X 4 X 1 + 2X 2 - X 3 + X 4 >=3-2X 1 - X 2 + 4X 3 + X 4 >= 2 X 1, X 2, X 3, X 4 >= 0 (How many iterations did you get?) 10. Using Simplex method, solve the following LPP Maximize: Z = 2X 1 + 4X 2 2X 1 + X 2 <= 18 3X 1 + 2X 2 >= 30 X 1 + 2X 2 = 25 X 1, X 2 >= 0 (Similar to Q-3) 11. Maximise Z= 3X 1 + 4X 2 + X 3 X 1 + 2X 2 + 3X 3 <= 90 2X 1 + X 2 + X 3 <= 60 3X 1 + X 2 + 2X 3 <= 80 X 1, X 2, X 3 >=0 (No trick here! Direct application of Simplex) 12. State and solve the dual of the following linear programming problem.(primal problem is not required to be solved) Minimise Z = 8X 1 + 10 X 2 8X 1 + 4X 2 >= 150 3X 1 + 9X 2 >= 100 X 1, X 2 >= 0 13. Minimise Z=60X 1 + 40X 2 + 80X 3 3X 1 + 2X 2 + X 3 >= 2 4X 1 + X 2 + 3X 3 >= 4 2X 1 + 2X 2 + 2X 3 >= 3 X 1, X 2, X 3 >= 0
Page4 14. Considering the minimum values of variables given below, modify the problem into a two constraint problem and then solve it by Simplex Algorithm. Hence obtain the solution to the original problem as given. Maximize 6X 1 + 20X 2 2X 1 + X 2 <= 32 3X 1 + 4X 2 <= 80 X 1 >= 8 X 2 >= 10 (This is a precursor to Q-7) 15. Solve the following Linear Programming problem using simplex method. Maximise 8X 1 4X 2 4X 1 + 5X 2 <= 20 -X 1 + 3X 2 >= - 23 X 1 >= 0 X 2 unrestricted (Look up ND Vohra!) 16. Minimise 5X 1 +3X 2 subject to: 2X 1 +4X 2 <=12 2X 1 +2X 2 =10 5X 1 +2X 2 >=10 X 1, X 2 >=0 (All types of constraints here!) 17. Solve the following LPP using simplex algorithm: Maximise 20X 1 +10X 2 subject to: X 1 +2X 2 <=40 3X 1 +X 2 >=30 4X 1 +3X 2 >=60 X 1, X 2 >=0 18. Minimise X 1 + 4X 2 X 1 + X 2 >= 2000 X 1 + 3X 2 >= 4000 X 1 + 2X 2 <= 3500 (All variables taking only positive values) (Does this question appear in a different form elsewhere?) 19. Minimise 16X 1 + 9X 2 + 9X 3 4X 1 + 3X 2 + 3X 3 >= 5 2X 1 + X 2 X 3 >= 2 (Try solving dual of this problem also for an interesting twist!!) 20. 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.
Page5 Supplier A Supplier B Alpha 25% 75% Unfit Scrap 5% 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 and solve using Simplex (From one of the CA final papers) 21. Maximise p=4a+2b+3c+5d subject to: 2a+3b+4c+2d=300 8a+b+c+5d=300 a, b, c, d >=0 (Oxford University) 22. Maximise p=3a+b+3c subject to: 2a+b+c<=2 a+2b+3c<=5 2a+2b+c<=6 a, b, c >=0 (Oxford University but nothing new here!) 23. Maximise p=13a+5b-12c subject to: 2a+b+2c<=5 3a+3b+c>=7 2a+5b+4c=10 a, b, c >=0 (Oxford University) 24. Maximise X 1 + 2X 2 + 3X 3 - X 4 X 1 +2X 2 +3X 3 =15 2X 1 +X 2 +5X 3 =20 X 1 +2X 2 +X 3 +X 4 =10 X i >=0 (Care! How many artificial variables do you actually need??) 25. Maximise Z= -X 2 +3X 3-2X 5 subject to: X 1 +3X 2 -X 3 +2X 5 =7-2X 2 +4X 3 +X 4 =12-4X 2 +3X 3 +8X 5 +X 6 =10 X i >=0 (Care! How many artificial variables do you actually need??) 26. An agricultural company located in Maharashtra owns land in a village. In that village it has totally three farms, which it wishes to cultivate. The total produce of each farm is limited by both the amount of available usable irrigable land and the quantity of water available. The data are given below. Farm Usable Land (acres) Available Water (acrefeet) 1 400 600 2 600 800 3 300 375
Page6 The crops being considered on these farms include sugarcane, cotton, and corn. These crops differ primarily in their expected net return per acre and their consumption of water. In addition, the company has set a maximum quota for the total acreage that can be devoted to each of these crops. The data is shown below. Crop Maximum quota Water consumption acre-feet per acre Net return in hundred Rupees per acre Sugarcane 600 3 1000 Cotton 500 2 750 Corn 325 1 250 On account of the limited water available for irrigation, the company will not be able to use all its irrigable land for planting crops. To ensure equity between three farms, it has been agreed that each farm will plant the same proportion of its available irrigable land. For example, if farm 1 plants 200 of its 400 acres, then farm 2 must plant 300 of its 600 acres, while farm 3 plants 150 of its 300 acres. However, any combination of the crops may be grown at any of the farms. The job facing the company is to plan how many acres to devote to each crop on these farms while satisfying the given restrictions. The objective is to maximise the total net return of the company. (This is a challenging question) 27. A production firm manufacturers 2 products and has three sub-processes involved in the manufacturing, viz. Stamping, Assembly and Painting. It can manufacture either one or any or both the products, called as product-1 and product-2. The profits from both the products and respective capacities of these products are given below in a table. These capacities are for manufacturing a single product. The firm can manufacture any combination of these products within the overall capacity. You are required to suggest an optimal product mix for the firm to maximise the total profit. Per week capacity Process Product-1 Product-2 Stamping 50 75 Assembly 40 80 Painting 90 45 Profit per unit 150 120 28. A coffee shop at the Dubai international airport is open for all 24 hours on all days, whereas the waiters get their weekly off by rotation. A waiter should join duty at any of the six times in a day, that is at midnight, at 4 am, at 8 am, at 12 noon, at 4 pm and at 8 pm as decided by the management and work for 8 hours thereafter. The management has estimated that the minimum number of waiters required during various 4 hour time-slots in a day is as under: From To Number of waiters needed 00.00 hrs 04.00 hrs 20
Page7 04.00 hrs 08.00 hrs 30 08.00 hrs 12.00 hrs 42 12.00 hrs 16.00 hrs 45 16.00 hrs 20.00 hrs 67 20.00 hrs 00.00 hrs 53 Formulate this as a linear programming problem for minimising the waiters salaries. The waiters are paid at the same rate whenever they join in a day. Also write the first simplex tableau for this LPP. 29. IT Transports Limited provides tourist vehicles of 3 types 20 seater vans, 8 seater big cars and 5 seater small cars. These seating capacities are in addition to the driver who has not been counted. The company has 4 vehicles of twenty seater type, 10 of eight seater cars and 20 of five seater cars. These vehicles have to be used to transport employees of their client company from their residences to their offices and back. All the residences are in the same housing colony. The offices are at two different places the Head Office and the Branch Office. Each vehicle plies only one round trip per day, i.e. residence to office in the morning, and office to residence in the evening. Each day 180 officers need to be transported on Route I (from residence to Head Office and back) and 40 officers need to taken on Route-II (from residence to Branch Office and back). The route wise cost per round trip for each type of vehicle is given below. You are required to formulate this information as a linear programming problem to minimise the total cost of hiring vehicles for the client company with the constraints as given. (From one of the CA final papers) Cost in Rupees per round trip Route-I (Residence to HO and back) Route-II (Residence to Branch and back) 20 seater vans 8 seater cars 5 seater cars 600 400 300 500 300 200 For a online lecture on Simplex Algorithm by an IIT Professor, please visit youtube.com or click the following link: There are some more similar lectures on youtube. http://www.youtube.com/watch?v=qxls3cyg8to&playnext=1&list=ple31a60e415425926