LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
FIFTH SEMESTER – APRIL 2012
MT 5507/MT 5504 – OPERATIONS RESEARCH
Date : 30-04-2012 Dept. No. Max. : 100 Marks
Time : 9:00 – 12:00
PART – A
Answer ALL questions: (10×2=20 marks)
- A firm plans to purchase at least 200 quintals of scrap containing high quality metal X and low quality metal Y. It decides that the scrap to be purchased must contain at least 100 quintals of X and not more than 35 quintals of Y. 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 A and B is given below:
Metals | Supplier A | Supplier B |
X | 25% | 75% |
Y | 10% | 20% |
The price of A’s scrap is Rs.200 per quintal and that of B is Rs. 400 per quintal. The firm wants to determine the quantities that it should buy from the two suppliers so that total cost is minimized. Construct a LP model.
- Give at least four different contrasting properties of primal and dual of general LP problems
- What is meant by unbalanced transportation problem?
- Name four different methods to solve an assignment problem.
- Consider the game with the following payoff table:
Player B | ||
Player A | B_{1} | B_{2} |
A_{1} | 2 | 6 |
A_{2} | -2 | λ |
Show that the game is strictly determinable, whatever λ may be.
- When no saddle point is found in a payoff matrix of a game, how do we find the value of the game?
- Give an example of weighted graph which is not a tree. Find two different minimal spanning trees of the weighted graph you constructed.
- Why is in PERT network each activity time assume a Beta-distribution?
- What are the basic information required for an efficient control of inventory?
- What is Economic Order Quantity (EOQ) concept?
PART – B
Answer any FIVE questions: (5×8=40 marks)
- Using Simplex Method, solve the following Problem and give your comments on the solution: Maximize Z= 6x_{1}+4x_{2}, Subject to: x_{1}+x_{2}£ 5, x_{2}≥ 8, x_{1}, x_{2}≥ 0.
- Prove that dual of the dual is the primal.
Tasks | |||||
Clerks | A | B | C | D | |
1 | 4 | 7 | 5 | 6 | |
2 | – | 8 | 7 | 4 | |
3 | 3 | – | 5 | 3 | |
4 | 6 | 6 | 4 | 2 |
- Consider a problem of assigning four clerks to four tasks. The time (hours) required to complete the task is given below:
Clerk 2 cannot be assigned to task A and clerk 3 cannot be assigned to task B. Find all the optimum assignment schedules.
- A company has three production facilities S_{1}, S_{2} and S_{3} with production capacity of 7, 9 and 18 units (in 100s) per week of a product, respectively. These units are to be shipped to four warehouses D_{1}, D_{2}, D_{3} and D_{4} with requirement of 5, 6, 7 and 14 units (in 100s) per week, respectively. The transportation costs (in rupees) pre unit between factories to warehouses are given in the table below:
D_{1} | D_{2} | D_{3} | D_{4} | Capacity | |
S_{1} | 19 | 30 | 50 | 10 | 7 |
S_{2} | 70 | 30 | 40 | 60 | 9 |
S_{3} | 40 | 8 | 70 | 20 | 18 |
Demand | 5 | 8 | 7 | 14 | 34 |
Use North-West Corner Method to find an initial basic feasible solution to the transportation problem.
- A soft drink company calculated the market share of two products against its major competitor having three products and found out the impact of additional advertisement in any one of its products against the other
Company B | |||
Company A | B_{1} | B_{2} | B_{3} |
A_{1} | 6 | 7 | 15 |
A_{2} | 20 | 12 | 10 |
What is the best strategy for the company as well as the competitor? What is the payoff obtained by the company and the competitor in the long run? Use graphical method to obtain the solution.
- Solve the following game after reducing it to a 2×2 game
Player B | |||
Player A | B_{1} | B_{2} | B_{3} |
A_{1} | 1 | 7 | 2 |
A_{2} | 6 | 2 | 7 |
A_{3} | 5 | 1 | 6 |
- An assembly is to be made from two parts X and Y. Both parts must be turned on a lathe and Y must be polished whereas X need not be polished. The sequence of activities together with their predecessors is given below.
Activity | Description | Predecessor Activity |
A | Open work order | – |
B | Get material for X | A |
C | Get material for Y | A |
D | Turn X on lathe | B |
E | Turn Y on lathe | B, C |
F | Polish Y | E |
G | Assemble X and Y | D, F |
H | Pack | G |
Draw a network diagram of activities for the project.
- The production department for a company requires 3600 kg of raw material for manufacturing a particular item per year. It has been estimated that the cost of placing an order is Rs. 36 and the cost of carrying inventory is 25% of the investment in the inventories. The price is Rs. 10 per kg. Calculate the optimal lot size, optimal order cycle time, minimum yearly variable inventory cost and minimum yearly total inventory cost.
PART – C
Answer any TWO questions: (2×20=40 marks)
- (a) Use graphical method and solve the LPP: Maximize Z= 15x_{1}+10x_{2}, Subject to: 4x_{1}+6x_{2}£ 360,
3x_{1}£ 180, x_{2}≥ 8, 5x_{2}£ 200, x_{1}, x_{2}≥ 0. (7 marks)
(b) Use simplex method to solve the LPP: Maximize Z= 3x_{1}+5x_{2}+4x_{3}, Subject to: 2x_{1}+3x_{2}£ 8,
2x_{2}+5x_{3}£ 10, 3x_{1}+2x_{2}+4x_{3}£ 15, x_{1}, x_{2}, x_{3}≥ 0. (13 marks)
- (a) ABC limited has three production shops supplying a product to five warehouses. The cost of production varies from shop to shop and cost of transportation form one shop to a warehouse also varies. Each shop has a specific production capacity and each warehouse has certain amount of requirement. The costs of transportation are given below:
Warehouse | |||||||
Shop | I | II | III | IV | V | Supply | |
A | 6 | 4 | 4 | 7 | 5 | 100 | |
B | 5 | 6 | 7 | 4 | 8 | 125 | |
C | 3 | 4 | 6 | 3 | 4 | 175 | |
Demand | 60 | 80 | 85 | 105 | 70 | 400 |
The cost of manufacturing the product at different production shops is given by:
Shop | Variable Cost CCost | Fixed Cost |
A | 14 | 7000 |
B | 16 | 4000 |
C | 15 | 5000 |
Find the optimal quantity to be supplied from each shop to different warehouses at minimum total cost. (10 marks)
(b) Two firms A and B make smart phones and tablets. Firm A can make either 150 smart phones in a
week or an equal number of tablets, and make a profit of Rs. 400 per smart phone and Rs. 300 per
tablet. Firm B can, on the other hand, make either 300 smart phones, or 150 smart phones and 150
tablets, or 300 tablets per week. It also has the same profit margin on the two products as A. Each
week there is a market of 150 smart phones and 300 tablets and the manufacturers would share
market in the proportion in which they manufacture a particular product. Write the payoff matrix
of A per week. Obtain graphically A’s and B’s optimal strategies and value of the game.
(10 marks)
- A small project is composed of 7 activities whose time estimates are listed in the table below. Activities are identified by their beginning(i) and ending(j) node numbers.
Activity | Estimated Duration (weeks) | ||
(i–j) | Optimistic | Most Likely | Pessimistic |
1-2 | 1 | 1 | 7 |
1-3 | 1 | 4 | 7 |
1-4 | 2 | 2 | 8 |
2-5 | 1 | 1 | 1 |
3-5 | 2 | 5 | 14 |
4-6 | 2 | 5 | 8 |
5-6 | 3 | 6 | 15 |
- Draw the network of the activities in the project (5 marks)
- Find the expected duration and variance for each activity. (5 marks)
- What are the expected project length and critical path? (5 marks)
- Calculate the standard deviation of the project length. (5 marks)
- (a) The annual demand of a product is 10,000 units. Each unit costs Rs.100 if orders are placed in
quantities below 200 units but for orders of 200 or above the price is Rs. 95. The annual inventory
holding costs is 10% of the value of the item and the ordering cost is Rs. 5 per order. Find the
economic lot size. (8 marks)
(b) A dealer supplies you the following information with regard to a product dealt in by her: Annual
demand=10000 units; Ordering Cost=Rs 10/order; Price=Rs.20/unit; Inventory carrying cost=20%
of the value of the inventory per year. The dealer is considering the possibility of allowing some
backorder to occur. She has estimated that the annual cost of backordering will be 25% of the
value of inventory.
- What should be the optimum number of units of the product she should buy in one lot?
- What quantity of the product should be allowed to be backordered, if any?
- What would be the maximum quantity of inventory at any time of the year?
- Would you recommend her to allow backordering? If so, what would be the annual cost saving by adopting the policy of backordering? (12 marks)
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