LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
FIFTH SEMESTER – NOVEMBER 2012
MT 5507/MT 5504 – OPERATIONS RESEARCH
Date : 06/11/2012 Dept. No. Max. : 100 Marks
Time : 9:00 – 12:00
PART – A
Answer any ALL questions: (10 x 2 = 20 Marks)
- Define the following: (i) Basic solution (ii) Basic feasible solution
- Express the following linear programming problem into standard form:
Maximize
Subject to
3
- What is a transportation problem?
- Give the mathematical formulation of an assignment problem.
- Define a pure strategy in game theory.
- Define a saddle point.
- Define a spanning tree in a network.
- Define a critical path in a network.
- What is the Economic order quantity?
- Differentiate the deterministic and the probabilistic demand inventory models.
PART – B
Answer any FIVE questions: (5 x 8 = 40 Marks)
- Use the graphical method to solve the following linear programming problem.
Minimize
Subject to
- Solve the following LPP by dual simplex method.
Maximize
Subject to
- Determine an initial basic feasible solution to the following transportation problem by Vogel Approximation Method.
Available
Requirement 6 10 12 15
- Solve the following assignment problem.
Jobs
I II III
Men
- Solve the following game graphically
- A project consists of a series of tables labeled A, B, …, H, I with the following relationships (W < X, Y means X&Y cannot start until W is completed; X, Y < W means W cannot start until both X&Y are completed). With this notations construct the network diagram having the following constraints: A < D, E; B, D < F; C < G; B < H; F, G < I
- Determine the critical path of the following network.
- A particular item has a demand of quantity 9000 units/year. The cost of the one procurement is Rs.100 and the holding cost per unit is Rs.2.40 per year. The replacement is instantaneous and no shortages are allowed. Determine
- the economic lot size
- the number of orders per year
- the time between orders
PART – C
Answer any TWO questions: (2 x 20 = 40 Marks)
- a) Find the optimal solution for the following transportation problem using MODI method.
D1 D2 D3 D4 Supply
| S1 | 19 | 30 | 50 | 10 | 7 |
| S2 | 70 | 30 | 40 | 60 | 9 |
| S3 | 40 | 8 | 70 | 20 | 18 |
| Demand | 5 | 8 | 7 | 14 | 34 |
- Use the penalty (Big-M) method to solve the following LP problem. (10)
Minimize
Subject to
- a) Define the Total float, free float and Independent float. (6)
- b) The following indicates the details of the activities of a project.
The durations are in days. (14)
| Activities | TO | TM | TP |
| 1 – 2 | 4 | 5 | 6 |
| 1 – 3 | 8 | 9 | 11 |
| 1 – 4 | 6 | 8 | 12 |
| 2 – 4 | 2 | 4 | 6 |
| 2 – 5 | 3 | 4 | 6 |
| 3 – 4 | 2 | 3 | 4 |
| 4 – 5 | 3 | 5 | 8 |
- Draw the network
- Find the critical path
- Find the mean and standard deviation of the project completion time
- a) Reduce the following game to game and hence find the optimum strategies and the value
of the game. (12)
Player B
| I | II | III | IV | |
| I | 3 | 2 | 4 | 0 |
| II | 3 | 4 | 2 | 4 |
| III | 4 | 2 | 4 | 0 |
| IV | 0 | 4 | 0 | 8 |
Player A
- b) Solve the following unbalanced assignment problem of minimizing total time for doing all the jobs. (8)
| Jobs | |||||
| Operators | 1 | 2 | 3 | 4 | 5 |
| 1 | 6 | 2 | 5 | 2 | 6 |
| 2 | 2 | 5 | 8 | 7 | 7 |
| 3 | 7 | 8 | 6 | 9 | 8 |
| 4 | 6 | 2 | 3 | 4 | 5 |
| 5 | 9 | 3 | 8 | 9 | 7 |
| 6 | 4 | 7 | 4 | 6 | 8 |
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