Use branch and bound method to solve the following LPP:

Minimize: Z = 4x₁ + 3x₂

Subject to the constraints:

 and are integers.

A road transport company has one reservation clerk on duty at a time. He handles information of bus schedules and make reservations. Consumers arrive at a rate of 8 per hour and the clerk can service 12 customers on an average per hour.

i) What is the average number of customers waiting for the service of the clerk?

ii) What is the average time a customer has to wait before getting service?

) A road transport company has one reservation clerk on duty at a time. He handles information of bus schedules and make reservations. Consumers arrive at a rate of 8 per hour and the clerk can service 12 customers on an average per hour.

On an average 96 patients per 24-hour a day require the services of an emergency clinic. Also on an average, a patient requires 10 minutes of active attention. Assume that the facility can handle only one emergency at a time. Suppose that it costs the clinic ₹ 100 per patient treated to obtain an average servicing time of 10 minutes, and that each minute of decrease in this average time would cost ₹ 10 per patient treated. How much would have to budgetted by the clinic to decrease the average size of the queue from 1x1/3 patients to 1/2 a patient?

i) Construct PERT network.

ii) Find the critica

Determine the EOQ and the optimal number of orders placed in a year.

ii) Determine the optimum production lot size and the average duration of the production run.

 A manufacturing company needs 2500 units of a particular item every year. The company buys it at the rate of ₹ 30 per unit. The order processing cost for this item is estimated at ₹ 15 and the cost of carrying a item in stock comes to about ₹ 4 per year. The company can manufacture this item internally. In that case it saves 20% of the price of the product. However, it estimates a set-up cost of ₹ 250 per production run. The annual production rate would be 4800 units. However, the inventory holding costs remain unchanged.

The time taken (in hours) by five different machines for completing five different jobs is given below:

Find the optimal assignment.

In a factory, there are six jobs to perform each of which should go through two machines A and B in the order A B. The processing timings (in hours) for the jobs are given here. You are required to determine the sequence for performing the jobs that would minimize the total elapsed time T, what is the value of T?

 Given  units,  units,  units,  units,  units and  units. Is it an optimal solution to the transportation problem:

If not, modify it to obtain a better feasible solution.

Use dual simplex method to solve the following LPP:
Minimize:

Subject to the constraints:

Use two-phase method to solve the following LPP:
Maximize:

Subject to the constraints:

Give the dual of the following LPP:
Minimize:

Subject to the constraints:



andx₃ is unrestricted.

) Use simplex method to solve the following LPP:
Maximize:

Subject to the constraints:



and .

Use the graphical method to solve the following LPP:
Maximize:

Subject to the constraints:


$

and .

A firm produces there product A, B and C. It uses two types of raw materials I and II of which 5000 and 7500 units respectively are available. The raw material requirements per unit of the products are given below:

The minimum demand of the three products is 600, 650 and 500 units respectively. Assuming the profits per unit of A, B and C as ₹ 50, ₹50 and ₹80 respectively. Formulate the problem as LPP model in order to determine the number of units of each product which will maximize the profit.

Which of the following statements are true? Give reasons for your answers. 
i) If an item is ordered frequently, then the risk of running out of stock is least.
ii) In deterministic queuing model, arrival rate must not exceed the service rate.
iii) The critical path of a project network represents the minimum time needed to complete the project.
iv) A necessary and sufficient condition for a basic feasible solution for a minimization LPP to be optimum is that all .
v) If dual has an unbounded solution, primal has a feasible solution.

 Alice wants to use the Digital Signature algorithm for signing messages. She chooses , ,  and . Alice wants to sign the message . She chooses the secret value . Explain the procedure that Alice will use for computing the signature. What information will she send Bob? 

Alice wants to use the ElGamal digital signature scheme with public parameters , , secret value  and . She wants to sign the message  and send it to Bob. She chooses  as the secret value. Explain the procedure that Alice will use for computing the signature of the message. What information will she send Bob? 

Explain how Bob will decrypt the message. 
9) a) Solve the discrete logarithm problem  using Baby-Step, Giant-Step algorithm. 

 Alice wants to send Bob the message . She chooses . How will she compute the cipher text? What information does she send to Bob?