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.
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:
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.
Compartment  Weight and capacity(in Tonnes)  Space capacity(in cubic meters) 
Front  10  6800 
Center  16  8700 
Rear  8  5300 
Cargo  Weight (Tonnes)  Volume (Cubic meter)  Profit£(Tonnes) 
C1  18  480  310 
C2  15  650  380 
C3  23  580  350 
C4  12  390  285 
10 M
1(b)
Solve the following problem using graphical methos. Maximize Z=2x_{1}+3x_{2}
Subject to 2x_{1}+x_{2}≤6
x_{1}x_{2}≥3
x_{1},x_{2}≥0
Subject to 2x_{1}+x_{2}≤6
x_{1}x_{2}≥3
x_{1},x_{2}≥0
10 M
2(a)
Solve the following linear programming problem using simplex method.
Maximize Z=6000x_{1}+4000x_{2}
Subject to 4x_{1}+3x_{2}≤360
2x_{1}+x_{22x1+3x2≤300 x12≥0}
Maximize Z=6000x_{1}+4000x_{2}
Subject to 4x_{1}+3x_{2}≤360
2x_{1}+x_{22x1+3x2≤300 x12≥0}
12 M
2(b)
Solve by dual simplex method the following problem.
Minimize Z=2x_{1}+2x_{2}+4x_{3}
Subject to 2x_{1}+3x_{2}+5x_{3}≥2
3x_{1}+x_{2}+7x_{3}≤3
x_{1}+4x_{2}+6x_{3}≤5
x_{1}, x_{2},x_{3}≥0
Minimize Z=2x_{1}+2x_{2}+4x_{3}
Subject to 2x_{1}+3x_{2}+5x_{3}≥2
3x_{1}+x_{2}+7x_{3}≤3
x_{1}+4x_{2}+6x_{3}≤5
x_{1}, x_{2},x_{3}≥0
8 M
3(a)
A product is produced by four factories A,B,C & D. The unit production counts in them are A50 units; B70 units; C30 units and D50 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.
Determine the extent of deliveries from each factory to each of the stores so that the total production and transportation cost is minimum.
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 M_{1},M_{2},M_{3}&M_{4} are to be installed in a machine shop. There are five vacant places A, B, C, D & E. Because of limited place, machine M_{2} Cannot be placed at C and M_{3} cannot be placed at A. C_{ij} , the assignment cost of machine i to place j in dollars is shown below.
Find the optimum assignment schedule.
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=20000x_{1}+30000x_{2}
Subject to 2x_{1}+x_{2}≤6 ;
x_{1}+2x_{2}≤8 ;
x_{1}x_{2}≤1 ;
x_{1}≤2
x_{1},x_{2}≥0 and are integers.
Maximize Z=20000x_{1}+30000x_{2}
Subject to 2x_{1}+x_{2}≤6 ;
x_{1}+2x_{2}≤8 ;
x_{1}x_{2}≤1 ;
x_{1}≤2
x_{1},x_{2}≥0 and are integers.
20 M
5(a)
A project schedule has the following characteristics:
i) Construct the network and compute E & L for each event.
Find the critical path and project duration.
Activity  Time (Weeks)  Activity  Time (Weeks) 
12  4  56  4 
13  1  57  8 
24  1  68  1 
34  1  78  2 
35  6  810  5 
49  5  910  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 selfservice 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.
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 

10 M
7(b)
Solve the following game by the graphical method.
Player A 

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
i) Optimal sequence as per SPT rule
ii) Completion time of the jobs
iii) Mean flow time
iv) Avg. In process inventory.
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
Determine a sequence of these jobs that will minimize the total elapsed time. Also find the idle time for both the machines.
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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