1. Which of the following statements are true and which are false Give a short proof or a counter-example in support of your answer.

In an inventory model, the optimum order quantity decreases with the increase in the shortage cost.

The arrival rate in the Poisson distribution equals the mean of the exponential inter-arrival time.

A non-critical activity cannot have zero total float.

In LPP, the dual of the dual is primal.

The optimal solution to an integer linear programming problem can be found by rounding off the optimal solution of its LP relaxation.

A firm plans to purchase at least 200 quintals of scrap containing high quality metal X and low quality metal Y. It is decided that the scrap to be purchased must contain at least 100 quintals of X metal and not more than 35 quintals of Y metal. The firm can purchase the scrap from two suppliers A and B in unlimited quantities. The percentage of X and Y metals in terms of weight in the scrap supplied by suppliers A and B is given below:

<img src='./qimages/10429-2a.jpg'>

The price of scrap is RS 200 per quintal and that of is RS 400 per quintal. Formulate the problem as LP model and solve it to determine the quantities that the firm should buy from two suppliers so as to minimize total purchase cost.

A company sells 2000 units of a product in a year. Each unit costs RS 125 to the company. The vendor supplying the product charges RS 150 for each order demanded by the company. The inventory carrying cost is estimated at 20% of unit value. What is the total cost of the existing inventory policy Determine the economic order quantity and the number of orders per year so as to minimize the total annual inventory cost.

Consider the transportation problem having the following cost and requirement table:

<img src='./qimages/10429-3a.jpg'>

Find an initial basic feasible solution using Vogel's method and North-West corner method, respectively. Compare both the solutions to find which one is better and hence find the optimum solution of the transportation problem.

Customers arrive at a booking office window being manned by a single individual at a rate of 25 per hour. The time required to serve a customer has exponential distribution with a mean of 120 seconds. Find the average waiting time of a customer. Also determine the probability that the server is idle in the system.

A plant assembles PCs through 9 interlinked stages according to the precedence process as given below:

<img src='./qimages/10429-4a.jpg'>

Draw the network diagram and hence find
the critical path and the optimal duration for
completing the project.

Obtain the dual of the following LPP:

Maximize z =2x1 3x2
subject to x1 x2 6
x1 x2 7
x1 4x2 8
x1 x2 is unrestricted
Your dual must have at least one
unrestricted variable.

Find a sequence that minimizes the total elapsed time required to complete the following tasks on two machines:

<img src='./qimages/10429-5a.jpg'>

Also, write other optimal sequences, that may exist.

Solve the following assignment problem:

<img src='./qimages/10429-5b.jpg'>

A company manufactures 30 items per day. The sale of these items depends upon demand which has the following distribution:

<img src='./qimages/10429-6a.jpg'>

Using the following random numbers, estimate the shortage/surplus of items per day for the next 10 days:
10,99,65,99,95,01,79,11,16,20

Use the dual-simplex method to solve the following LPP
Minimize z =2x1 x2
subject to
3x1 x2 3
4x1 3x2 6
x1 2x2 3
X1, x2 0.

Solve the following integer linear programming problem using Branch and Bound method:

Maximize z 3x1 5x2
subject to
2x1 4x2 25
x1 8
2x2 10
x1, x2 0.

A purchase manager has decided to place order for a minimum quantity of 500 numbers of a particular item in order to get a discount of 10%. From the past records, it was found that in the last year, 8 orders each of size 200 units were placed. The ordering cost is RS 500 per order, inventory carrying cost is 40% of the inventory value and the price of the item is RS 400 per unit. Determine an ordering policy for the purchase manager.