- 1. Department of Mathematics Brainware Group of Institutions Assignment-I (ME 705C) 1. Using graphical method to solve the following LPP : Maximize 3 2 subject to 2 2 3 4 12 , 0 z x y x y x y x y 2. Solve the following LPP by simplex method: Maximize 3 subject to 3 2 3 2 2 2 , , 0 u x y z x y z x y z x y z 3. Solve the following LPP by Big M method : 1 2 3 1 2 3 1 2 3 1 2 3 2 9 4 2 5 3 2 4 , , 0 Max z x x x Subject to x x x x x x x x x 4. Find all the basic feasible solutions of the system 1 2 3 1 2 3 2 4 11, 3 5 14, x x x x x x 5. A manufacturer of patent medicine is preparing a production plant on medicines A and B. There are sufficient ingredients available to make 20000 bottles of A and 40000 bottles of B,but there are only 45000 bottles into which either of the medicines can be put. Furthermore it takes three hours to prepare enough material to fill 1000 bottles of A, it takes one hours to prepare enough material to fill 1000 bottles of B and there are 66 hours available for this operation.The profit is Rs. 8.00 per bottle of A and Rs. 7.00 per bottle of B. Formulate this as a linear programming problem to maximize the profit. 6. Distinguish Between Transportation Problem and Assignment Problem. 7. Find the optimal solution and the corresponding cost of transportation in the following transportation problem D1 D2 D3 D4 Supply O1 19 20 50 10 7 O2 70 30 40 60 9 O3 40 8 70 20 18 Demand 5 8 7 14
- 2. Department of Mathematics Brainware Group of Institutions 8. The Head of the department has five jobs A, B, C, D, E and five sub-ordinates V, W, X, Y, Z. The number of hours each sub-ordinates would take to perform each job is as follows: V W X Y Z A 3 5 10 15 8 B 4 7 15 18 8 C 8 12 20 20 12 D 5 5 8 10 6 E 10 10 15 25 10 How would the jobs be allocated to minimize the total time. 9. A firm makes two types of furniture – chairs and tables. The profit for each product as calculated by the accounting department is Rs. 20 per chair and Rs.30 per table. Both products are to be processed on three machines M1, M2, M3. The time required in hours by each product and total time available in hours per week on each machine is as follows: Machine Chair Table Available Time(hrs) M1 3 3 36 M2 5 2 50 M3 2 6 60 i) Give a mathematical formulation to this linear programming problem. ii) Use the graphical method to solve this problem. 10. The owner of a small machine shop has four machinists available to assign to jobs for the days. Five jobs are offered with expected profit for the each machinist on each job as given in the table. Find the assignment of Machinists to jobs that will result in a maximum profit. A B C D E 1 62 78 50 101 82 2 71 84 61 73 59 3 87 92 111 71 81 4 48 64 87 77 80
- 3. Department of Mathematics Brainware Group of Institutions 11. Deduce the expression expected number of customers in the system and expected number of customer waiting in the plane(i.e. queue length) 12. A project schedule has the following characteristics 1-2 1-3 2-4 3-4 3-5 4-9 5-6 5-7 6-8 7-8 8-10 9-10 4 1 1 1 6 5 4 8 1 2 5 7 Draw the project network and find the critical path. 13. A harbour has single dock to unload the containers from the incoming ships. The arrival rate of ships at the harbour follows Poisson distribution and the unloading time for the ships follows exponential distribution. The arrival rate and service rate are 8 ships per week and 14 ships per week, respectively. Find the following: i) Utilization of the dock. ii) Average number of waiting ships in the queue. iii) Average number of waiting ships in the system iv) Average waiting time per ship in the queue. v) Average waiting time per ship in the system.