Roll No. :
Total No. of Questions : 08 Total No. of Pages : 03
M.Tech.(ME) (Sem.-1)
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OPTIMIZATION TECHNIQUES
Subject Code : MME-501 M.Code : 38202
Time : 3 Hrs. Max. Marks : 100
INSTRUCTIONS TO CANDIDATES :
- Attempt any FIVE questions out of EIGHT questions.
- Each question carries TWENTY marks.
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Q1. a) Explain the important characteristics of the industrial situation to which I.P. method can be successfully applied. Illustrate application of these technique with a suitable example.
b) A company sells two different products A and B the company makes a profit of Rs. 40 and Rs. 30 on two products respectively. They are produced by a common production process and are sold in two different markets. The production process has a capacity of 30,000 man hours. It takes 3 hours to produce a unit of A and 1 hour to produce a unit of B. The maximum number of Units of A and B that can be sold in the market are 8000 and 12,000 respectively. Formulate the above as linear programming model.
Q2. A person require 10, 12 and 12 units of chemical A, B and C respectively for his garden. A liquid product contains 5, 2 and 1 unit of A, B and C respectively per jar. A dry product contains 1, 2 and 4 units of A, B and C per carton. If the liquid product sells for Rs. 3 per jar and the dry product sells for Rs. 2 per carton. How many of these should be purchased to minimize the cost and meet the requirement.
Q3. Obtain the initial solution by VAM and optimal solution by MODI method for the transportation problem shown below :
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| A | B | C | Supply | |
|---|---|---|---|---|
| W1 | 5 | 4 | 6 | 65 |
| W2 | 7 | 4 | 7 | 42 |
| Demand W3 | 8 | 6 | 7 | 43 |
Q4. Five machine are to be located in the machine shop. There are five possible locations in which the machine can be located. Cij the cost of placing i in place j is given in the table below.
Place
| 1 | 2 | 3 | 4 | 5 | |
|---|---|---|---|---|---|
| 1 | 15 | 10 | 25 | 25 | 10 |
| 2 | 1 | 8 | 10 | 20 | 2 |
| Machine 3 | 8 | 9 | 17 | 20 | 10 |
| 4 | 14 | 10 | 25 | 27 | 15 |
| 5 | 10 | 8 | 25 | 27 | 12 |
It is required to place the machines at suitable place so as to minimize the total cost.
a) Formulate an L.P. model to find an optimal assignment.
b) Solve the problem by assignment technique of L.P.
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Q5. Consider the following table :
| Activity | Predecessor Activity | Times in Weeks | ||
|---|---|---|---|---|
| tm | to | tp | ||
| A | - | 2 | 3 | 10 |
| B | A | 1 | 2 | 3 |
| C | B | 1 | 2 | 3 |
| D | A | 4 | 6 | 14 |
| E | B | 4 | 5 | 12 |
| F | C | 3 | 4 | 5 |
| G | D,E | 1 | 1 | 7 |
a) Draw the network diagram path.
b) Find the critical path and variance of each event.
c) What is the probability that the project will be completed in 16 weeks?
Q6. a) What is dynamic programming? Write step by step procedure to solve a general problem by D.P.
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b) Write the algorithm for Fibonacci search method.
Q7. Solve the following game by the principle of dominance :
Player B
| Player A | I | II | III | IV | V | VI |
|---|---|---|---|---|---|---|
| 1 | 4 | 3 | 0 | 2 | 1 | 1 |
| 2 | 4 | 3 | 1 | 3 | 2 | 2 |
| 3 | 4 | 3 | 7 | –1 | 1 | 2 |
| 4 | 4 | 3 | 4 | –5 | 2 | 2 |
| 5 | 4 | 3 | 3 | –2 | 2 | 2 |
Q8. Find the minimum of the function using the Newton-Raphson method with the starting point ?1 = 0.1. Use ? = 0.01 for checking the convergence.
f (?) = 0.65 - 0.75 1+ ?2 - 0.65? tan-1 1 ?
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