JNTU
JNTU B.Tech III OPERATIONS RESEARCH Previous Question Paper (January/February - 2021 - R16)
Answer any five questions. All questions carry equal marks.
Main
Solve the LPP by simplex method
Maximize Z = 3x1+5x2+4x3
Subject to:
x1, x2, x3≥0
Answer
To solve the given Linear Programming Problem (LPP) using the Simplex method, we first convert the inequality constraints into equalities by introducing slack variables s1, s2, and s3.
The standard form is:
Subject to:
Initial Simplex Table:
| Basic | x1 | x2 | x3 | s1 | s2 | s3 | Ratio |
|---|---|---|---|---|---|---|---|
| s1 | 0 | 2 | 3 | 1 | 0 | 0 | 18/2=9 |
| s2 | 0 | 2 | 5 | 1 | 0 | 0 | 18/2=9 |
| s3 | 3 | 2 | 4 | 0 | 0 | 1 | 25/2=12.5 |
| Z-C | -3 | -5 | -4 | 0 | 0 | 0 |
Choosing x2 as the entering variable (most negative in Z-C row) and s1 as the leaving variable (minimum ratio):
| Basic | x1 | x2 | x3 | s1 | s2 | s3 | Ratio |
|---|---|---|---|---|---|---|---|
| x2 | 0 | 1 | 1.5 | 0.5 | 0 | 0 | 1.5/1.5=1 |
| s2 | 0 | 0 | 2 | -1 | 1 | 0 | 2/2=1 |
| s3 | 3 | 0 | 1 | -1 | 0 | 1 | 1/1=1 |
| Z-C | -3 | 0 | 3.5 | 2.5 | 0 | 0 |
Choosing x1 as the entering variable and s3 as the leaving variable:
| Basic | x1 | x2 | x3 | s1 | s2 | s3 |
|---|---|---|---|---|---|---|
| x2 | 0 | 1 | 1.5 | 0.5 | 0 | 0 |
| s2 | 0 | 0 | 2 | -1 | 1 | 0 |
| x1 | 1 | 0 | 0.33 | -0.33 | 0 | 0.33 |
| Z-C | 0 | 0 | 4.5 | 1.5 | 0 | 1 |
Since all values in the Z-C row are non-negative, the optimal solution is reached. The values are x1 = 8.33, x2 = 9, x3 = 0. The maximum value of Z = 3(8.33) + 5(9) + 4(0) = 25 + 45 = 70.
Explain the OR models with examples.
Answer
Operations Research (OR) models are mathematical or logical representations of real-world systems used to analyze complex decision-making problems. They are broadly categorized as follows:
- Mathematical Models: These use mathematical symbols and relationships to describe a system. For example, Linear Programming (LP) models are used to optimize resource allocation, such as determining the product mix to maximize profit under constraints like labor and raw materials.
- Simulation Models: These mimic the behavior of a system over time using computer programs. For example, a queuing model can simulate the arrival and service times of customers at a bank to determine the optimal number of tellers required to minimize waiting time.
- Heuristic Models: These are rule-of-thumb approaches used when an optimal solution is computationally too expensive. For example, the Nearest Neighbor algorithm is used to find a near-optimal solution for the Traveling Salesman Problem.
- Deterministic vs. Probabilistic Models: Deterministic models assume all parameters are known with certainty, such as the Transportation Model for logistics. Probabilistic models incorporate uncertainty, such as Inventory Control models that account for variable demand using probability distributions.
Discuss the scope for OR being used in business.
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Find the basic feasible solution with cost using Vogel’s approximation method. Also find the optimal solution.
| D1 | D2 | D3 | D4 | supply | |
|---|---|---|---|---|---|
| O1 | 1 | 2 | -2 | 3 | 70 |
| O2 | 2 | 4 | 0 | 1 | 38 |
| O3 | 1 | 2 | -2 | 5 | 32 |
| Demand | 40 | 28 | 30 | 32 |
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Find the sequence that minimizes the total time required in performing the following jobs on three machines in the order A-B-C as shown in the below table. Also, find the total elapsed time. Also find the idle time on each machines.
| Machine/Job | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| A | 8 | 3 | 7 | 2 | 5 | 1 |
| B | 3 | 4 | 5 | 2 | 1 | 6 |
| C | 8 | 7 | 6 | 9 | 10 | 9 |
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A machine M1 costing Rs 9000 has a maintenance cost of Rs 200 in the first year of its operation which rises by Rs 2000 in each of the successive years. Assuming that the machine replacement can be done only at the end of a year, determine the best age at which the machined be replaced. There is an offer to replace the machine M1 which is a year old, by another machine M2 which costs Rs 8000. The expected maintenance costs and resale price in different years are provided.
What is replacement? Describe some important replacement situations and replacement policies.
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Should the machine M1 be replaced by machine M2? If so when?
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Obtain the optimal strategies for both persons and the value of the game for zero-sum two-person game whose payoff matrix is given below by dominance principle:
| 3 | 2 | 4 | 0 |
| 2 | 4 | 4 | 2 |
| 4 | 2 | 4 | 0 |
| 0 | 4 | 0 | 8 |
Define the following terms: i) Rectangular games ii) Saddle point iii) Payoff matrix.
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Refer to the provided matrix for dominance calculation.
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A particular item has a demand of 9000 units/year. The cost of one procurement is Rs 100 and the holding cost per unit is Rs 2.40 per year. The replacement is instantaneous and no shortages are allowed.
EOQ
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Number of orders
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Lead time
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Total cost
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A self-service store employs one cashier at its counter. Nine customers arrive on an average every 5 minutes while the cashier can serve 10 customers in 5 minutes. Assuming Poisson distribution for arrival rate and exponential distribution for service rate.
Average number of customers in the system
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Average number of customers in the queue
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Average time a customer spends in the system
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Average time a customer waits before being served
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Probability that the queue is empty.
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