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 integer LPP, a bound obtained by the branch and bound procedure is associated with a feasible point of the integer problem. The total number of possible sequences for processing 5 jobs on 3 machines is 5^3. In a two-dimensional LP solution, the objective function can assume the same values at two or more distinct extreme Points. x1 x2 x3 1 is a basic feasible solution for the system of equations x1 x2 x3 2x1 x2 x3 =5.
For a queuing model the expected
number of customers are determined by
where and denote the arrival and service
rates, respectively.

2. A project consists of eight activities with the following relevant information: <img src='./qimages/8324-2a.jpg'>

Draw the PERT network. Compute the slack for each activity and determine the critical path.
Obtain the dual of the following LPP: Minimize z =2x1 3x2
subject to x1 x2 6
2x1 x2 7
x1 4x2 8
x1,x2 0.
Your dual variable. must contain one unrestricted

3. Use the dual simplex method to solve the following LPP
Minimize z =2x1 x2
subject to 3x1 x2 3
4x1 3x2 6
x1 2x2
x,x2 0 A book-binder has one printing press, one binding machine and seven books for
processing .on these machines. The time
required for printing and binding operations
for different books are shown below: <img src='./qimages/8324-3b.jpg'>

Find the optimum sequence of processing of books in order to minimize the total time required to process all the books. Also, write the other optimum sequences, if they exist.

4. A firm assembles and sells two different types of outboard motors, A and using four resources. The Production Process is described as follows <img src='./qimages/8324-4a.jpg'>

Type A units bring in a profit of 90 each
and type units, RS 60 each. What should
be the optimum product mix The average rate of arrivals at a service center is 30 per hour. At present, there is one cashier who on an average attends to 45 customers per hour. The center proprietor estimates that each extra minute of system process time per customer means a loss of RS 0·50. An assistant can be provided to the cashier and in that case the service unit can deal with 75 customers per hour. The wage rate of the assistant is RS 15 per hour. Is it worth employing an assistant Give
reasons.

5. Find the initial basic feasible solution of the following transportation problem using VAM method <img src='./qimages/8324-5a.jpg'>

Also, check the optimality of the solution obtained. A particular item has demand of 3000 units per year. The cost of 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. Determine:
The economic lot size

(ii) The number of orders per year .

(iii) The time between orders

6. Solve the following integer programming problem using branch and bound method
Minimize z =3x1 2.5 X2 subject to
x1 2x2 20
3x1 2x2 50
Xl,x2 0 and integers. Three Customs officers separately check the luggage of the passengers at an airport. The passengers arrive at an average rate of five per hour. The time a Customs officer spends with a passenger is exponentially
distributed, with mean service time
24 minutes. Find the probability that all the
Customs officers are idle. Also, find the probability that there are exactly
2 customers in the queue.

7. A company manufactures 30 items per
day. The sale of these items depends upon
the demand which has the following
distribution <img src='./qimages/8324-7a.jpg'>

Using the following random numbers, estimate the shortage/surplus of items per day for the next 10 days:
5,23,44,69,37,28,51,88,13,35 Solve the following assignment problem:
<img src='./qimages/8324-43.jpg'>