Operations Research  May 2015
Mechanical Engineering (Semester 7)
TOTAL MARKS: 80
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.
1 (b) Explain Bellman's principle of optimality. (4 marks)
1 (c) Explain the applications of linear programming. (4 marks)
1 (d) What do you understand by a queue? Give some applications of queuing theory. (4 marks)
1 (e) Explain Johnson's alorithm. (4 marks)
1 (f) Explain degeneracy in the transportation problem. (4 marks)
2 (a) A manufacture has five lathes and three milling machines in his workshop and produces an assembly that consists of 2 units of part A and three units of parts B. The processing time for each part on the two types of machines is given below.
In order to maintain the uniform work load on the two types of machines, the manufacturer has a policy that no type of machine should run more than 40 minutes per day longer than other machine. Formulte the problem as 1.7 problem if the objective is to produce the maximum number of assemblies in any 8 how working day.
Part  Processing time in minutes on  
Lathe  Milling  
A  10  18 
B  25  12 
2 (b) Solve the following L.P. Problem by simplex method.
Minimize: Z=X_{1}3X_{2}+3X_{3}
Subject to:
3X_{1}X_{2}+2X_{3}≤7
2X_{1}+4X_{2}≥12
4X_{1}+3X_{2}+8X_{3}≤10
X_{1}, X_{2}, X_{3} ≥0. (10 marks)
3 (a) Use two phase method to
Maximize
Z=5X_{1}+3X_{2}
Subject to 2X_{1}+X_{2} ≤1
X_{1}+4X_{2}≥6
X_{1}, X_{2} ≥0 (10 marks)
3 (b) A company has a team of four salesmen and there are four districts where the company wants to start its business. After taking into account the capabilities of salesmen and the nature of districts, the company estimates that the profit per day in rupees for each salesman in each district is as below.
Find the assignment of salesmen to various districts which will yeild maximum profit.
District  
Slesmen 

4 (a) There are five jobs, each of which is to be processed through three machines A, B and C in the order ABC. Processing times in hours are:
Determine the optimum sequence for the five jobs and minimum elapsed time. Also find the idle time for the three machines and waiting time for the jobs.
Job  A  B  C 
1  3  4  7 
2  8  5  9 
3  7  1  5 
4  5  2  6 
5  4  3  10 
4 (b) Solve the following problem using Big M method,
Minimize
12x_{1}+20X_{2}
Subject to 6X_{1}+8X_{2}≥100
7X_{1}+12X_{2}≥120
X_{1}, X_{2} ≥ 0. (10 marks)
5 (b) Solve the following transportation problem where the cell entries are the unit costs.
D1  D2  D3  D4  D5  Available  
O1  68  35  4  74  15  18 
O2  57  88  91  3  8  17 
O3  91  60  75  45  60  19 
O4  52  53  24  7  82  13 
O5  51  18  82  13  7  15 
Required  16  18  20  14  14  82/82 
5 (b) The cost of a machine is Rs 6,100 and its scrap value is Rs. 100. The maintenance costs found from experience are as follows:
When should the machine be replaced?
Year  1  2  3  4  5  6  7  8 
Maintenance Cost (Rs)  100  250  400  600  900  1200  1600  2000 
6 (a) Solve the following L.P. problem by dynamic programming approach.
Maximum
Z=50X_{1}+100X_{2}
Subjected to
10X_{1}+5X_{2}≤2500
4X_{1}+10X_{2}≤2000
X_{1}+3/2 X_{2}≤450
X_{1}, X_{2}≤0. (10 marks)
6 (b) A particular item has demand of 9,000 units / year. The cost of one procurement is Rs. 100 and the holding cost per unit is Rs. 2.40 per year. The replacement is instantaneous and no shortage are allowed. Determine
i) The economic lot size
ii) the number of orders per year.
iii) the time between orders.
iv) the total cost per if the cost of one unit is Re. 1. (10 marks)
7 (a) Reduce the following game by dominance and find the game value:
Player B  
Player A 

7 (b) A bakery keeps stocks of a popular brand of cake. Daily demand based on past experience is given below:
Consider the following sequence of random numbers:
48, 78, 09, 51, 56, 77, 15, 14, 68 and 09
i) Using the sequence, simulate the demand for next 10 days.
ii) Find the stock situation if the owner of the bakery decides to make 35 cakes every day. Also estimate the daily average demand for the cakes on the basis of the simulated data.
Daily demand  0  15  25  35  45  50 
Probability  0.01  0.15  0.20  0.50  0.12  0.02 
Attempt any five:
1 (a) Briefly explain the Monte  Carlo Simulation with suitable example.(4 marks)