Operations Research

The PERT Program Evaluation and Review Technique method is designed for projects with uncertain or probabilistic durations It considers the variability in task completion times and uses statistical methods to estimate project duration and probabilities

In the transportation problem unoccupied cells represent non-basic variables These are cells that are not currently being used to transport goods but may be used in future iterations based on the opportunity cost method to find the optimal solution

In the simplex method variables that are not part of the current solution non-basic variables are assigned a value of zero This is because they are not contributing to the objective function at that point in the solution

Step 1: Calculate the probability of hitting the target.
The probability of A hitting the target is , the probability of B hitting the target is , and the probability of C hitting the target is . Step 2: Find the probability of missing the target.
The probability of A missing the target is , the probability of B missing is , and the probability of C missing is . Step 3: Calculate the probability of exactly one hit.
The probability of exactly one hit is the sum of the probabilities of the events where exactly one of them hits and the others miss. This can be computed, and the final probability for at least two hits is . Final Answer:

Step 1: Use the formula for exponential growth.
The population growth formula is given by: Where is the growth rate, is the initial population, and is the time in years. Step 2: Solve for time.
To find when the population doubles, set : Taking the natural logarithm of both sides: Final Answer:

Step 1: Use the EOQ formula.
The Economic Order Quantity (EOQ) is given by the formula: Where:
- is the annual demand (16,000 units),
- is the ordering cost per order (Rs 100),
- is the holding cost per unit per year (10\% of the cost of one unit, which is ). Step 2: Calculate EOQ. Substitute the given values into the formula: Final Answer:

Step 1: Understand the Simplex method.
The Simplex method is an algorithm used to solve linear programming problems. Linear programming involves optimizing a linear objective function subject to linear constraints. Step 2: Conclusion Thus, the Simplex method is specifically used for solving linear programming problems. Final Answer: