JNTU
JNTU B.Tech III OPERATIONS RESEARCH Previous Question Paper (February - 2022 - R18)
Answer any five questions
All questions carry equal marks
Main
Define Operations Research and discuss its characteristics.
Answer
Operations Research (OR) is the application of scientific methods, techniques, and tools to problems involving the operations of systems so as to provide those in control of the operations with optimum solutions to the problems. It is essentially a systematic, quantitative approach to decision-making.
The key characteristics of Operations Research include:
- System Orientation: It considers the organization as a whole rather than focusing on individual components, ensuring that decisions in one area do not negatively impact others.
- Interdisciplinary Team Approach: It utilizes the expertise of specialists from various fields (e.g., mathematics, statistics, engineering, economics) to analyze complex problems.
- Scientific Method: It follows a rigorous process of observation, problem definition, model formulation, experimentation, and validation.
- Mathematical Modeling: It relies on constructing mathematical models that represent real-world situations to predict the effect of different courses of action.
- Optimization: The primary objective is to find the best possible solution (maximum profit or minimum cost) given the constraints.
Solve the following linear programming problem by graphical method.
Maximize Z = 5x1 + 3x2
Subject to:
and x1, x2 ≥ 0
Answer
To solve the LPP by the graphical method, we first determine the boundary lines for the constraints:
- Constraint 1: 3x1 + 5x2 = 15. If x1=0, x2=3; if x2=0, x1=5. Points: (0, 3) and (5, 0).
- Constraint 2: 5x1 + 2x2 = 10. If x1=0, x2=5; if x2=0, x1=2. Points: (0, 5) and (2, 0).
The feasible region is bounded by the vertices (0,0), (2,0), the intersection point of the two lines, and (0,3). To find the intersection:
Subtracting the equations: 19x1 = 20, so x1 = 20/19 ≈ 1.05. Substituting back: 5(20/19) + 2x2 = 10, resulting in x2 = 45/19 ≈ 2.37.
Evaluating Z = 5x1 + 3x2 at the vertices:
- (0,0): Z = 0
- (2,0): Z = 5(2) + 3(0) = 10
- (0,3): Z = 5(0) + 3(3) = 9
- (20/19, 45/19): Z = 5(20/19) + 3(45/19) = (100 + 135) / 19 = 235 / 19 ≈ 12.37
The maximum value of Z is 12.37 at x1 = 1.05 and x2 = 2.37.
Use big M method to solve the following LPP.
Maximize Z = 3x1 - x2
s.t:
and x1, x2 ≥ 0
Answer
Sign in to view the complete answer paper.
A steel company has three open-hearth furnaces and five rolling mills. The transportation cost (Rs. per quintal) for shipping steel from furnaces to rolling mills is shown in the following table. Determine the optimal basic feasible solution by Vogel’s Approximation method.
| M1 | M2 | M3 | M4 | M5 | Capacity | |
|---|---|---|---|---|---|---|
| F1 | 4 | 2 | 3 | 2 | 6 | 8 |
| F2 | 5 | 4 | 5 | 2 | 1 | 12 |
| F3 | 6 | 5 | 4 | 7 | 3 | 14 |
| Requirement | 4 | 4 | 6 | 8 | 8 |
Answer
Sign in to view the complete answer paper.
Solve the following travelling salesman problem so as to minimize the cost per cycle:
| From/To | A | B | C | D | E |
|---|---|---|---|---|---|
| A | - | 3 | 6 | 2 | 3 |
| B | 3 | - | 5 | 2 | 3 |
| C | 6 | 5 | - | 6 | 4 |
| D | 2 | 2 | 6 | - | 6 |
| E | 3 | 3 | 4 | 6 | - |
Answer
Sign in to view the complete answer paper.
There are seven jobs, each of which has to go through the machines A and B in the order AB. Processing times in hours are given below:
| Job | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Machine A | 3 | 12 | 15 | 6 | 10 | 11 | 9 |
| Machine B | 8 | 10 | 10 | 6 | 12 | 1 | 3 |
Determine a sequence of these jobs that will minimize the total elapsed time T. Also find idle time for machines A and B.
Answer
Sign in to view the complete answer paper.
The following mortality rates have been observed for a certain type of light bulbs in an installation with 1,000 bulbs:
| End of week | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Prob. of failure | 0.09 | 0.25 | 0.49 | 0.85 | 0.97 | 1.00 |
If a bulb fails in service, it costs Rs.3 to replace but if all the bulbs are replaced in the same operation, it can be done for only Rs.0.70 a bulb. It is proposed to replace all bulbs at fixed intervals. (a) What is the best interval between group replacements? (b) At what group replacement price per bulb, would a policy of strictly individual replacement become preferable to the adopted policy?
Answer
Sign in to view the complete answer paper.
Explain the following: i) Two-person zero-sum game, ii) Pure strategy, iii) Saddle point.
Answer
Sign in to view the complete answer paper.
Use graphical method to solve the following game and find the value of the game.
| Player A/B | B1 | B2 | B3 | B4 |
|---|---|---|---|---|
| A1 | 2 | 2 | 3 | -2 |
| A2 | 4 | 3 | 2 | 6 |
Answer
Sign in to view the complete answer paper.
A drive-in bank window has a mean service time of 2 minutes, while the customers arrive at a rate of 20 per hour. Assuming that these represent rates with a Poisson distribution, determine:
a) The proportion the teller will be idle,
b) How long a customer will wait before reaching the server,
c) What fraction of customers will have to wait in line?
Answer
Sign in to view the complete answer paper.