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VTU Mechanical Engineering (Semester 7)
Operations Research
May 2016
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1(a) A cargo plane has 3 compartments for storing cargo: front, center and rear. These compartments have the following limits on both weight and space.
 Compartment Weight and capacity(in Tonnes) Space capacity(in cubic meters) Front 10 6800 Center 16 8700 Rear 8 5300
Furthermore, the weight of the cargo in the cargo in the respective compartments must be the same proportion of that comaprtment's weight capacity to maintain the balance of the plane. The following four cargoes are available for shipment on the next flight:
 Cargo Weight (Tonnes) Volume (Cubic meter) Profit£(Tonnes) C1 18 480 310 C2 15 650 380 C3 23 580 350 C4 12 390 285
Any proportion of these cargoes can be accepted. The objective is to determine how much of each cargo C1,C2,C3,and C4 should be accepted and how to distribute each among the compartments so that the total profit for the flight is maximized. Formulate the above problem as a linear program.
10 M
1(b) Solve the following problem using graphical methos. Maximize Z=2x1+3x2
Subject to 2x1+x2≤6
x1-x2≥3
x1,x2≥0
10 M

2(a) Solve the following linear programming problem using simplex method.
Maximize Z=6000x1+4000x2
Subject to 4x1+3x2≤360
2x1+x22x1+3x2≤300
x12≥0
12 M
2(b) Solve by dual simplex method the following problem.
Minimize Z=2x1+2x2+4x3
Subject to 2x1+3x2+5x3≥2
3x1+x2+7x3≤3
x1+4x2+6x3≤5
x1, x2,x3≥0
8 M

3(a) A product is produced by four factories A,B,C & D. The unit production counts in them are A-50 units; B-70 units; C-30 units and D-50 units. These factories supply the product to four stores, demand of which are 25,35,105 and 20 units respectively. Unit transport costin Rupees from each stores is given below.
 1 2 3 4 A 2 4 6 11 B 10 8 7 5 C 13 3 9 12 D 4 6 8 3

Determine the extent of deliveries from each factory to each of the stores so that the total production and transportation cost is minimum.
12 M
3(b) Four new machines M1,M2,M3&M4 are to be installed in a machine shop. There are five vacant places A, B, C, D & E. Because of limited place, machine M2 Cannot be placed at C and M3 cannot be placed at A. Cij , the assignment cost of machine i to place j in dollars is shown below.
 A B C D E M1 4 6 10 5 6 M2 7 4 - 5 4 M3 - 6 9 6 2 M4 9 3 7 2 3

Find the optimum assignment schedule.
8 M

4 Solve the following using Gomory's cutting plane algorithm.
Maximize Z=20000x1+30000x2
Subject to 2x1+x2≤6 ;
x1+2x2≤8 ;
x1-x2≤1 ;
x1≤2
x1,x2≥0 and are integers.
20 M

5(a) A project schedule has the following characteristics:
 Activity Time (Weeks) Activity Time (Weeks) 1-2 4 5-6 4 1-3 1 5-7 8 2-4 1 6-8 1 3-4 1 7-8 2 3-5 6 8-10 5 4-9 5 9-10 7

i) Construct the network and compute E & L for each event.
Find the critical path and project duration.
12 M
5(b) What are the characteristics of a project? Also define the PERT and crashing cost.
8 M

6(a) Define five operating characteristics of queueing system.
10 M
6(b) A self-service store employs one cashier at its counter. Nine customers arrive on a average every 5 minutes while the cashier can serve 10 customers in 5 minutes. Assuming Poisson distribution for arrival rate and exponential distribution for service time, find
i) Average no.of customers
ii) Average no.of customers in the queue.
iii) Average time a customer spends in the system.
iv) Average time a customer waits before being served.
10 M

7(a) Reduce the following game by dominance and find the game value.

Player A
Player B
 1 2 3 4 1 3 2 4 0 2 3 4 2 4 3 4 2 4 0 4 0 4 0 8
10 M
7(b) Solve the following game by the graphical method.
Player A
Player  B
 1 2 3 4 1 3 3 4 0 2 5 4 3 7
10 M

8(a) Six jobs A, B,C, D, E & F have arrived at one time to be processed on a single machine. Assuming that no new jobs arrive thereafter, determine
 Job A B C D E F Processing Time  (in minutes) 7 6 8 4 3 5

i) Optimal sequence as per SPT rule
ii) Completion time of the jobs
iii) Mean flow time
iv) Avg. In process inventory.
8 M
8(b) There are seven jobs, each of which has to go through the machines A &B in the order AB. Processing times in hours are given as
 Job 1 2 3 4 5 6 7 Machine A 3 12 15 6 10 11 9 Machine B 8 10 10 6 12 1 3

Determine a sequence of these jobs that will minimize the total elapsed time. Also find the idle time for both the machines.
12 M

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