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.

A non-critical activity cannot have zero slack.

A balanced transportation model may not have any feasible solution.

In queuing theory, if the arrival occurs according to the Poisson process, the inter-arrival time is exponential.

An increase in the set-up cost decreases the economic order quantity.

An unrestricted primal variable will have the effect of yielding an equality dual constraint.

A company produces two products X and Y. The products are produced and sold on a weekly basis. The weekly production of X cannot exceed 25 and for the production is limited to 35. The company employs maximum of 60 workers. One unit of product X requires 2 workers working for a week, while one unit of product Y requires 1 worker working for a week for production. Profit on X is RS 60 and on Y is RS 40. Formulate the problem for maximizing the profit as a Linear Programming Problem and solve it using graphical method.

A bank plans to open a single server drive-in banking facility in a particular centre. It is estimated that 28 customers will arrive each hour on an average. If, on an average, it requires 2 minutes to process a customer transaction, determine:

The probability that the system is idle.

The average time a customer spends in the system.

Listed in the table below are the activities and sequencing requirements necessary for the completion of a project:

Activity Predecessor Duration (in weeks)
A 6
B A 24
C A 6
D A 12
E A 9
F 18
G B,F 12
H G 24

Draw a network diagram for the project.

Use CPM,to find the critical path and the duration for the completion of the project.

Compute EOQ and the total variable costs for the following available data on stock of items:

Annual demand 5000 units,
Unit price RS 20,
Ordering cost per order RS 16,
Inventory carrying charge 20%.

An equipment needs five repair jobs which have to be assigned to five machines. The estimated time (in hours) that each machine takes to complete the repair job is given below:

Machines J1 J2 J3 J4 J5
M1 7 5 9 8 11
M2 9 12 7 11 10
Mg 8 5 4 6 9
M4 7 3 6 9 5
M5 4 6 7 5 11

Assuming that each machine can be
assigned to only one job, determine this assignment.

Seven jobs are to be processed on two machines A and B in the order A B. Each machine can process only one job at a time. The processing time (in hours) of the jobs on the machines are as follows:

Job I II III IV V VI VII
Machine A 10 12 13 7 14 5 16
Machine B 15 11 8 9 6 7 16

Suggest optimal sequence of processing the jobs and the total elapsed time.

The details of a project on market survey of households is given below:

Activities Immediate predecessor(s) Dur
ation (in days)
Code Description
A Plan Survey 4
B Hire Personnel A 6
C Design Questionnaire A 11
D Train Personnel B 8
E Select Households C,D 5
F Print Questionnaire C 6
G Conduct Survey E,F 16
H Analyze Results G 6

Prepare a Gantt chart for the project. Also, find the total time taken.

Using Vogel's Approximation method, find the initial basic feasible solution of the following transportation problem:

Plant Warehouse Availability

WI W2 W3

P1 3 5 1 70
P2 3 4 6 90
P3 1 6 2 140
Requirement 80 70 150

Also, find the optimal solution.

Solve the following integer Linear Programming Problem by Branch and Bound method

Maximize z =2x1 3x2
subject to 5x1 7x2 35
4x1 9x2 36
x1,x2 and integers.

Find the dual of the following Linear Programming Problem:

Maximize z =2x1 3x2
subject to
x1 x2 6
2x1 x2
x1 4x2 8
x2 x1 is unrestricted.

The demand per day for an item occurs according to the following probability distribution function:

Demand Probability
0 0.2
1 0·3
2 0.4
3 0.1

By using the following random numbers, determine the demands in each of the first five days:

64,27,43,8,91

A petrol filling station has three pumps. The cars are served on first come first served basis. The petrol filling station can accommodate at most 4 cars waiting (total 7 cars in the filling station) at one time. The arrival pattern is Poisson with a mean of 1 car per minute during the peak hours. The service time is exponential with mean 6 minutes. Find the probability that all the pumps are idle. Also, find the expected number of customers in the queue.