JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD MCA II OPERATIONS RESEARCH Previous Question Paper (October/November - 2022 - R13)
Answer any five questions. All questions carry equal marks.
Main
Mr. Raju, a retired Govt, officer, has recently received his retirement benefits. He is contemplating as to how much he should invest in various alternatives open to him so as to maximise ROI. The relevant data about the investment, return and no. of years blocked is as follows:
| Investment | Return | No. of Years | Risk |
|---|---|---|---|
| Govt. Securities | 6% | 12 | 1 |
| Company deposits | 13% | 3 | 3 |
| Time deposits | 10% | 5 | 2 |
| Equity shares | 20% | 6 | 5 |
| House construction | 25% | 10 | 1 |
Mr. Raju wants to invest to maximise ROI so that the risk is not more than 4 and funds should not be blocked for more than 15 years. He should necessarily invest at least 28% in house construction. Formulate the data as an LPP.
Answer
To formulate the investment problem as a Linear Programming Problem (LPP), we define the decision variables as the fraction of the total retirement fund allocated to each investment alternative:
Let x1 = fraction invested in Govt. Securities
x2 = fraction invested in Company deposits
x3 = fraction invested in Time deposits
x4 = fraction invested in Equity shares
x5 = fraction invested in House construction
The objective is to maximize the total Return on Investment (ROI):
Subject to the following constraints:
- Total investment constraint: x1 + x2 + x3 + x4 + x5 = 1
- Risk constraint (weighted average risk ≤ 4): 1x1 + 3x2 + 2x3 + 5x4 + 1x5 ≤ 4
- Time constraint (weighted average years ≤ 15): 12x1 + 3x2 + 5x3 + 6x4 + 10x5 ≤ 15
- Minimum investment in house construction: x5 ≥ 0.28
- Non-negativity constraints: x1, x2, x3, x4, x5 ≥ 0
Use graphical method to solve the following LPP. Examine whether the LPP has any alternative solution, other than you found:
Subject to:
Answer
To solve the given Linear Programming Problem (LPP) graphically, we first identify the feasible region defined by the constraints:
- Constraint x1 ≥ 5: This is a vertical line passing through x1 = 5.
- Constraint 2x2 ≥ 7, which simplifies to x2 ≥ 3.5: This is a horizontal line passing through x2 = 3.5.
- Constraint 4x1 + 6x2 ≤ 24: To plot this, find the intercepts. If x1 = 0, x2 = 4. If x2 = 0, x1 = 6. The line passes through (0, 4) and (6, 0).
Examine the feasibility: The constraint x1 ≥ 5 requires x1 to be at least 5. The constraint 4x1 + 6x2 ≤ 24, when x1 is at its minimum of 5, implies 4(5) + 6x2 ≤ 24, which simplifies to 20 + 6x2 ≤ 24, or 6x2 ≤ 4, meaning x2 ≤ 0.66. However, the constraint x2 ≥ 3.5 requires x2 to be at least 3.5. Since x2 cannot be simultaneously ≤ 0.66 and ≥ 3.5, there is no region that satisfies all constraints simultaneously.
Conclusion: The feasible region is empty. Therefore, the LPP has no solution, and consequently, no alternative solutions exist.
Find the optimal solution to the following transportation problem using VAM-MODI method each cell value being the unit cost.
| D1 | D2 | D3 | D4 | D5 | Supply | |
|---|---|---|---|---|---|---|
| C1 | 35 | 41 | 28 | 16 | 20 | 285 |
| C2 | 14 | 21 | 28 | 30 | 15 | 145 |
| C3 | 45 | 18 | 17 | 29 | 26 | 165 |
| Demand | 125 | 125 | 100 | 100 | 175 |
Answer
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The Indian Cricket captain Rahul Dravid has to decide first five batting positions from the following five batsmen whose average runs in various positions, estimated from the past records is given below. Help him in assigning the batting order. If another batsman Yuvraj Singh with the following batting order positions and corresponding record of average runs, is also considered, find the optimum assignment (He never played in the V position and treat that it is risky). If Yuvraj is to be retained in the team then which of the current five batsmen will be out? What is the change in number of runs now?
Answer
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There are six jobs, each of which must go through machines A, B and C. Processing time (in hours) are given in the following table. Find the sequence that minimizes the total elapsed time required to complete the following tasks.
| Job | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Machine A | 12 | 10 | 9 | 14 | 7 | 9 |
| Machine B | 7 | 6 | 6 | 5 | 4 | 4 |
| Machine C | 6 | 5 | 6 | 4 | 2 | 4 |
Answer
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A truck-owner finds from his past experience that the maintenance costs are Rs. 200 for the first year and then increase by Rs. 2,000 every year. The cost of truck type A is Rs. 9,000. Determine the best age at which to replace the truck. If the optimum replacement is followed what will be the average yearly cost of owning and operating the truck? Truck type B costs Rs. 20,000. Annual operating costs are Rs. 400 for the first year and then increase by Rs. 800 every year. The truck owner has now the truck type A which is one year old. Should it be replaced by B type, and if so, when?
Answer
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In a railway marshalling yard, goods trains arrive at a rate of 30 trains per day. Assuming that the inter-arrival time follows an exponential distribution and the service time (the time taken to dump a train) distribution is also exponential with an average of 36 minutes; calculate:
(a) Expected queue size (line length)
(b) Probability that the queue size exceeds 10.
If the input of trains increases to an average of 33 per day, what will be the change in (a) and (b).
Answer
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Monthly demand for an item is 200 units. Ordering cost is Rs. 350, inventory carrying charge is 24% of the purchase price per year. The purchase prices are P1 = Rs. 10 for Q1 < 500; P2 = Rs. 9.25 for 500 ≤ Q2 < 750 and P3 = Rs. 8.75 for Q3 ≥ 750. Determine optimum purchase quantity. If the order cost is reduced to Rs. 100 per order, compute the optimum purchase quantity.
Answer
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