Operations Research
Section 1
- Explain the following concepts (3×5=15)
- Degeneracy in simplex
- Updating in crashing
- Hungarian method in assignment
- Decision making under risk
- PERT vs CPM
- 2.      A company produces two models of electronic products using Resistors, Capacitors and Chips. The following table gives the entire Technological and other details in this regard:                                  Â
|
Resource |
Unit resource requirement |
Maximum Availability |
|
|
Model 1 |
Model 2 |
||
| Resistor
Capacitor Chips |
2 2 0 |
3 1 4 |
1200 1000 800 |
| Unit profit (Rs). |
3 |
4 |
|
After formulating the above problem as a Linear Programming Problem the following optimal Simplex Solution table is obtained.
| Profit
Coefficient |
Basis
Variables |
Solution
Values |
3 |
4 |
0 |
0 |
0 |
|
|
X1 |
X2 |
S1 |
S2 |
S3 |
||||
|
C |
X |
B |
||||||
|
3 0 4 |
X1 S3 X2 |
450 400 100 |
1 0 0 |
0 0 1 |
-1/4 -2 1/3 |
¾ 2 -1/2 |
0 1 0 |
|
|
Z=Rs.1750 |
Z |
3 |
4 |
5/4 |
¼ |
0 |
||
|
= C-Z |
0 |
0 |
-5/4 |
-1/4 |
0 |
|||
(i)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Is the solution optimum? Why? (2 marks)
(ii)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Determine the worth of each resource.(1 mark)
(iii)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Which resources are scarce and which are abundant (1 mark).
(iv)Â Â Â Â Â Â Â Â Â Â Â Â Â Determine the range of the applicability of the shadow prices for each resource. (5 marks)
(v)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â If the available number of chips is reduced to 350 units, will you be able to determine the new optimum solution directly from the given information? Explain.(2 marks)
(vi)Â Â Â Â Â Â Â Â Â Â Â Â Â There is a plan to launch a new product represented by x3 which would consume 4 units each of all resources. It would give a contribution of 8 Rs. Should it be manufactured? Why? (2 marks)
(vii)Â Â Â Â Â Â Â Â Â Â Â Â A company BMS Ltd is willing to provide a capacitor at 1/5 Rs per capacitor. Should we accept the offer. Why (2 marks)
Section 2 (Solve any 3. 10 marks each)
- A firm does animal breeding. Various products & nutrients are fed to animals. A & B are nutrients.
               Nutrient Constituents  Quantity per unit                            Minimum requirement
                                                                               A                            B
1Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 36Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 6Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 108
2Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 3Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 12Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 36
               3                                                             20                          10                          100
Cost per unit (Rs.)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 20Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 40
Determine minimum cost at which minimum requirement of nutrition can be achieved. Solve graphically.
- A small project is composed of 7 activities whose time estimates are listed in the table below. Activities are identified by their beginning (i) and ending (j) node numbers.
|
Activity (I – j) |
Estimated Duration (in week) |
||
|
Optimistic |
Most likely |
pessimistic |
|
|
1-2 |
1 |
1 |
7 |
|
1-3 |
1 |
4 |
7 |
|
1-4 |
2 |
2 |
8 |
|
2-5 |
1 |
1 |
1 |
|
3-5 |
2 |
5 |
14 |
|
4-6 |
2 |
5 |
8 |
|
5-6 |
3 |
6 |
15 |
(i)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Draw the project network
(ii)               Find the expected duration and variance for each activity. What is the expected project length?
(iii)Â Â Â Â Â Â Â Â Â Â Â Â Â Â What is the probability that the project will be completed?
(a)Â Â Â At least 4 weeks earlier than the expected time.
(b)Â Â Not more than 4 weeks later than expected time.
(iv)Â Â Â Â Â Â Â Â Â Â Â Â Â If the project due date is 19 weeks, what is the probability of not meeting the due date?
|
Given : |
Z: |
0.50 |
0.67 |
1.00 |
1.33 |
2.00 |
|
Prob: |
0.3085 |
0.2514 |
0.1587 |
0.0918 |
0.0228 |
- Transportation unit cost matrix –
| Â |
W1 |
W2 |
W3 |
W4 |
Capacity |
|
P1 |
5 |
6 |
9 |
11 |
40 |
|
P2 |
11 |
9 |
14 |
13 |
30 |
|
P3 |
15 |
18 |
26 |
20 |
30 |
|
Demand |
5 |
15 |
35 |
45 |
|
Find IFS by VAM and Optimal solution by MODI method. Does degeneracy occur during any stage of the solution? If yes, comment on it.
- A. The payoffs under some circumstances are given. Find EMV and EOL (4 marks)
| Event | EI | E2 | E3 | E4 | |
| Event | Probability | ||||
| Strong demand |
0.2 |
800 |
600 |
500 |
250 |
| Weak demand |
0.3 |
450 |
780 |
400 |
158 |
| Poor demand |
0.5 |
830 |
190 |
600 |
450 |
- Make a decision tree and help the company choose a correct strategy (6 marks)
A company makes Pen, pencils and eraser. All these of them have different payoffs under excellent, average and poor demand. Probability of excellent, average and poor demand are 0.3, 0.4, 0.3 respectively. Payoffs for pen, pencil and eraser under excellent, average and poor demand are as follows
Pen:- 10000, 8000, 5000
Pencil:- 11000, 9000, 3000
Eraser:- 12000, 8500, 4500
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