St. Joseph’s College of Commerce (Autonomous)
End Semester Examination – April 2011
MIB – II Semester
Operations Research
Time: 3 hours Marks: 100
SECTION – A
I Answer ALL questions. Each carries TWO marks. (2 x 10 = 20)
- Show the feasible region for the following equation 3x-2y<=6
- What is degeneracy in transportation problem?
- A computer centre has three expert programmers. The center wants three application programs to be developed. After a study the time estimates in minutes required by experts for the job was found. Assign the programmers to the Application programs. What is the minimum time for the completion of all the programs.
Programmers
| 120 | 100 | 80 |
| 80 | 90 | 110 |
| 110 | 140 | 120 |
Application
Programs
- Explain the terms ‘ zero sum two players game’ and saddle point.
- What is meant by critical activities? Why is it necessary to know about them?
- Explain the term Float and its types.
- What is a queuing model? Explain balking and jockeying.
- What is the role of Simulation in OR?
- With the help of an example show how allocations can be in North West Corner Rule.
- How do you identify multiple optimal solution in Simplex Method.
SECTION – B
II Answer any FOUR questions. Each carries FIVE marks. (4 x 5 = 20)
- A rubber company is engaged in producing three different types of tyres A,B,C . These three different tyres are produced at the company’s two different production capacities. In a normal 8 hours working day plant 1 produces 100, 200 and 200 type of tyres of A , B, C respectively. Plant 2 produces 120, 120, 400 type of tyres of ABC respectively. The monthly demand of A,B and C is 5000, 6000, 14000 units resp. The daily cost of operation of Plant 1 and 2 is Rs.5000 and Rs.7000 resp. Find the minimum number of days of operation per month at two different plants to minimize the total cost while meeting the demand. Formulate the LPP.
- Use graphical method to solve the following Linear Programming Problem.
Maximize Z= 80x + 120y
Subject to constraints
x + y <= 9
x>=2
y>=3
20x +50y <=360
Where x, y >=0
- A steel company is concerned with the problem of distributing imported ore from three ports to four steel mills. The supplies of ore arriving at ports are:
| Port | P | Q | R |
| Tonnes per week | 20,000 | 28000 | 16000 |
The Demand at the steel mills is as follows
| Steel mills | A | B | C | D |
| Tonnes per week | 10,000 | 18,000 | 22,000 | 24,000 |
Transportation cost is Rs5 per tone per Km. The distance between the ports and the steel mills is given below.
| A | B | C | D | |
| P | 50 | 60 | 100 | 50 |
| Q | 80 | 40 | 70 | 50 |
| R | 90 | 70 | 30 | 50 |
Calculate a transportation plan using least cost method and interpret the solution.
- Solve the following game by dominance principle.
| Firm B à
Firm A
|
B1 | B2 | B3 | B4 |
| A1 | 35 | 65 | 25 | 5 |
| A2 | 30 | 20 | 15 | 0 |
| A3 | 40 | 50 | 0 | 10 |
| A4 | 55 | 60 | 10 | 15 |
- Construct the network diagram for the following project. Through forward pass and backward pass identify the critical path and expected project completion time. If indirect cost per week is Rs.160 , find the expected cost of the project.
| Activity | Immediate
Predecessor |
Duration
(week) |
Cost (Rs.) |
| A | – | 5 | 900 |
| B | – | 2 | 460 |
| C | A | 3 | 810 |
| D | C | 4 | 865 |
| E | C | 2 | 1130 |
| F | B | 4 | 1180 |
| G | D | 7 | 1800 |
| H | E,F | 6 | 390 |
- State any three areas for the application of OR techniques in financial management. How do they improve the performance of an organization.
SECTION – C
III Answer any THREE questions. Each carries FIFTEEN marks. (3×15 = 45)
- Write the dual of the following LPP and solve the dual problem by Simplex Method.
Minimize Z= 24x + 30y
Subject to
2x + 3y >= 10
4x + 9y >= 15
6x + 6y >= 20
Where x, y >=0
- Solve the following Transportation problem using Vogel’s approximation method and MODI method. What is the minimum transportation cost? Does the problem have multiple solution? ( The numbers in the table indicate the transportation cost per unit.)
| Stores ->
Warehouses
|
S1 | S2 | S3 | S4 | Supply |
| W1 | 2 | 4 | 6 | 11 | 50 |
| W2 | 10 | 8 | 7 | 5 | 70 |
| W3 | 13 | 3 | 9 | 12 | 30 |
| W4 | 4 | 6 | 8 | 3 | 35 |
| Demand | 25 | 35 | 105 | 20 |
- (a) Consider a problem assigning 4 clerks to 4 tasks. The time (in hrs) required to complete the tasks are given below.
| Tasks ->
Clerks |
A | B | C | D |
| I | 4 | 7 | 5 | 6 |
| II | – | 8 | 7 | 4 |
| III | 3 | – | 5 | 3 |
| IV | 6 | 6 | 4 | 2 |
Clerk II cannot be assigned to Task A and Clerk III cannot be assigned to task B. Find all the optimum assignment schedules.
(b) Solve the following game by graphic method . The payoff matrix has been prepared for Player A.
| Player B | |||||
| Player A | I | II | III | IV | |
| I | 45 | 35 | 70 | 50 | |
| II | 40 | 50 | 35 | 30 | |
(7+8)
- At Dr.Shetty’s clinic patients arrive at an average of 6 patient’s per hour. The clinic is attended to by Dr. Shetty himself. Some patients require only the required prescription. Some come for minor checkup while some others require thorough inspection for the diagnosis. This takes the doctor 6 minutes per patient on an average. It can be assumed that arrivals follow a poisson distribution and doctor’s inspection time follows an exponential distribution . Determine:
- The percentage of time that a patient can walk to the doctor without having to wait.
- The average number of patients in the system.
- The average number of patients in the Queue.
- The average waiting time of a patient in the system
- The average time the doctor is free on a 7 hours working day.
- What are the essential characteristics of Operations Research? Mention different phases in an operations research study. Point out its limitations if any.
SECTION – D
IV Case Study – Compulsory Question. (15 marks)
- (a) The activities of a project are tabulated below.
| Time duration (weeks) | |||
| Activity | Optimistic | Most likely | Pessimistic |
| 1-2 | 1 | 3 | 5 |
| 2-3 | 1 | 4 | 7 |
| 2-4 | 1 | 3 | 5 |
| 2-5 | 5 | 8 | 11 |
| 3-6 | 2 | 4 | 6 |
| 4-6 | 5 | 6 | 7 |
| 5-7 | 4 | 5 | 6 |
| 6-7 | 1 | 3 | 5 |
- Draw the network diagram
- Find the expected duration of the project.
- What is the probability that the project will be completed in 15 weeks.?
- What is the probability that the project will not be completed in 17 weeks?
(b) A bakery keeps stock of a popular brand of cake . Previous experience shows the daily demand pattern for the item with associated probabilities, as given below:
| Daily demand (number) | 0 | 10 | 20 | 30 | 40 | 50 |
| Probability | 0.01 | 0.20 | 0.15 | 0.50 | 0.12 | 0.02 |
Use the following sequence of random numbers to simulate the demand for next 10 days.
Random Numbers: 25, 39, 65, 76, 12, 05, 73, 89, 19, 49
Also estimate the daily average demand for the cakes on the basis of simulated data.
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