LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS
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SIXTH SEMESTER – APRIL 2007
ST 6601 – OPERATIONS RESEARCH
Date & Time: 18/04/2007 / 9:00 – 12:00 Dept. No. Max. : 100 Marks
SECTION A
Answer all questions. (10×2=20)
- Define: Basic Feasible Solution.
- Mention any two limitations in solving a LPP by graphical method.
- Express the following LPP in its standard form:
Maximize Z = 3x1 + 2x2
subject to x1 + x2 ≤ 2; 5x1 + 4x2 ≥4; x1,x2 ≥ 0;
- What is the difference between Big-M and Two Phase method?
- What is an unbalanced Transportation Problem?
- Comment on the following statement:
“Assignment problem is a particular case of Transportation problem”.
- Define: Activity and Node with reference to network analysis.
- Draw the network, given the following precedence relationships:
Event : 1 2,3 4 5 6 7
Preceded by: — 1 2,3 3 4,5 5,6
- Verify whether saddle point exists for the following game:
Player B
Player A B1 B2 B3
A1 5 8 1
A2 10 16 13
A3 25 22 20
- Give the Laplace criteria for making decisions under uncertainty.
SECTION B
Answer any FIVE questions. (5×8=40)
- Show graphically that the maximum or minimum values of the objective function for the following problem are same:
Maximize (Minimize) Z = 5x1 + 3x2
Subject to: x1 + x2 ≤ 6;
2x1 + 3x2 ≥ 3;
x1 ≥ 3;
x2 ≥ 3;
x1,x2 ≥ 0;
- Use Big M- method to maximize Z = 2x1 + 3x2 subject to the constraints:
x1 + 2x2 ≤ 4; x1 + x2 =3; x1,x2 ≥ 0
- Explain the MODI algorithm to obtain an optimum solution for a transportation problem.
- Solve the following assignment problem.
1 | 2 | 3 | 4 | 5 | |
A | 3 | 8 | 2 | 10 | 3 |
B | 8 | 7 | 2 | 9 | 7 |
C | 6 | 4 | 2 | 7 | 5 |
D | 8 | 4 | 2 | 3 | 5 |
E | 9 | 10 | 6 | 9 | 10 |
- Explain the Floyd’s method of finding the shortest route between any two given nodes.
- Draw the network based on the following information and determine the critical path.
Activity Duration (in days) Activity Duration (in days)
(1,2) 3 (3,4) 3
(1,3) 1 (3,7) 10
(1,4) 15 (4,5) 10
(1,6) 7 (4,7) 22
(2,3) 8 (5,6) 5
(2,5) 10 (5,7) 12
(6,7) 7
- A farmer wants to decide which of the three crops he should plant on his 100 acre farm. The profit from each is dependent on the rainfall during the growing season. The farmer has categorized the amount of rainfall as high, medium and low. His estimated profit for each crop is shown in the table below:
Rainfall Crop A Crop B Crop C
High 8000 3500 5000
Medium 4500 4500 5000
Low 2000 5000 4000
If the farmer wishes to plant only one crop, decide which should be his ‘best
Crop’ using Maximin, Savage and Hurwicz (α = 0.5) rule
- Use simplex method to find the inverse of the following matrix
A =
SECTION C
Answer any TWO questions. (2×20=40)
- a.) Explain the mathematical formulation of a Linear Programming Problem.
b.) Use two phase method to verify that there does not exist a feasible solution
to the following LPP:
Maximize z = 2x1 + 3x2 + 5x3 subject to
3x1 + 10x2 +5x3 ≤ 15;
33x1 – 10x2 + 9x3 ≤ 33;
x1 + 2x2 + x3 ≥ 4;
x1 , x2 , x3 ≥ 0.
- Obtain an optimum solution to the following transportation problem where the entries denote the unit cost of transporting commodities from source to destination:
From↓To→ | I | II | III | IV | Supply |
A | 2 | 3 | 11 | 7 | 6 |
B | 1 | 0 | 6 | 1 | 1 |
C | 5 | 8 | 15 | 9 | 10 |
Demand | 7 | 5 | 3 | 2 |
Find initial solution by least cost method and Vogle’s method and use the best
among them as the starting solution.
- a.) Solve the following traveling salesman problem so as to minimize the total
distance traveled:
To
From A B C D E
A — 20 4 10 —
B 20 — 5 — 10
C 4 5 — 6 6
D 10 — 6 — 20
E — 10 6 20 —
- Explain the calculations in PERT to identify the critical path of a project.
- ) Consider the following data that gives the distance between pairs of cities
1, 2…,6:
Route: (1,2) (1,3) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5)
Distance: 5 1 1 5 2 2 1 3 Route: (3,6) (4,6) (5,6)
Distance: 4 4 3
Use Dijkstra’s algorithm to find the shortest route between the cities
i.) 1 and 4 ii.) 1 and 6.
b.) Solve graphically the 2 x 4 game whose pay-off matrix is given below
Player B
Player A 1 3 11 7
8 5 2 5
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