b) The initial simplex table of a LPP in the maximization form
(no artificial variables are required) is as follows
i)
ii)
y1
y2
y3
y4
y5
y6
x5
4
9
7
10
1
0
x6
1
1
3
40
0
1
0
-12 -20 -18 -40
0
0
Write (P) and (D).
Solve (P) by the Simplex method and hence find the
optimal solution of the dual problem.
(5)
MCA IV (Fourth) Semester Examination 2013-14
Course Code: MCA407
Operation Research
Time: 3 Hours
a)
b)
c)
d)
e)
8.
A harbour has single dock to unload the containers from the
incoming ships. The arrival rate of ships at the harbour
follows poisson distribution and the unload time for the ships
follows exponential (negative) distribution and hence, the
service rate also follows the poisson distribution. The arrival
rate and the service rate are 8 ships per week and 14 ships per
week respectively. Find the following(10)
Utilization of the dock.
Average number of waiting ships in the queue.
Average number of waiting ships in the system.
Average waiting time per ship in the queue.
Average waiting time per ship in the system.
The annual demand for a component is 7,200 units. The
carrying cost is Rs.500 units per year, the ordering cost is
Rs.1500 per order and the shortage cost is Rs.2000 units per
year. Find the optimal values of economic order quantity,
maximum inventory, maximum shortage quantity, cycle time
(t), inventory period (t1) and , shortage period (t2).
(10)
Max. Marks: 70
Note: Attempt six questions in all. Q. No. 1 is compulsory.
1.
a)
7.
Paper ID: 0874407
b)
c)
d)
e)
Answer any five of the following (limit your answer to 50
words).
(4x5=20)
Derive the EOQ formula for the purchase model without
shortage.
Distinguish between breakdown maintenance and preventive
maintance.
Discuss the similarity between transportation and assignment
problem.
Solve the linear programming problem
Max Z = 2 x1  3x2  4 x3  x4  7 x5  5x6
Subject to
x1  x2  1
x3  x 4  1
x5  x 6  1
x1  x3  x5  1
x 2  x 4  x6  1
x1 , x2 , x3 , x4 , x5 , x6  0.
Solve the following LPP without using the Simplex method
Max Z = 4 x1  5x2  11x3  2 x4
Subject to 21x1  7 x2  3x3  10 x4  210
x1 , x2 , x3 , x4  0.
f)
g)
h)
What are the reasons for stocking items in inventory?
List and explain different types of costs in inventory system.
Define the slack, surplus and artificial variable in linear
programing.
2.
3. a)
b)
Machine A costs `9,000. Annual operating costs are `200 for
the first year and then increases by `2,000 every year.
Determine the best age at which to replace the machine. If the
optimum replacement policy is followed, what will be the
average yearly cost of owning and operating the machine?
Machine B costs `10000. Annual operating costs are `400 for
the first year, and then increases by `800 every year. You now
have machine A, which is one year old. Should you replace it
with B? If so when? Assume that resale value for both the
machines are negligible.
(10)
In an unbalanced transportation problem sometimes there are
penalties for unsatisfied demand, to reflect the failure of the
supplier to meet the required demand. Consider the problem
Let the penalty costs per unit of unsatisfied demand by 6,4
and 2 respectively.
W1 W2 W3 Supply
F1
1
5
6
90
F2
3
2
3
10
F3
2
6
1
20
Demand 60 50
50
for destinations which will have short supply and by how
many units?
(5)
A supplier ships 100 units of a product every Monday. The
purchase cost of the product is `60 per unit. The cost of
ordering and transportation from the supplier is `150 per
order. The Cost of carrying inventory is estimated at 15% per
year of the purchase cost. Find the lot size that will minimize
the cost of the system. Also determine the optimum cost. (5)
b) Use the Simplex method to solve the following
Max Z=  2 x1  x2
Subject to 3x1  x2  3
4 x1  3x2  x3  6
x1  2x2  x4  3
x1 , x2 , x3 , x4  0.
5. a) Consider the following assignment problem (AP)
J1
J2
J3
J4
P1
8
26
17
11
P2
12
28
4
26
P3
18
19
18
5
P4
18
16
24
10
i)
Solve the above problem to find the minimum cost
assignment.
ii)
What will be the optimum assignment if it is given
that job J4 cannot be assigned to person P1.
iii)
Find the optimum assignment if the person P2 is sure
to do the job J4.
(5)
b) Use the Simplex method to solve the following problem. If
the problem has more than one optimal solution find out all
of them.
Max Z = x1  x2
Subject to
x1  x2  8
2x1  x2  10
(5)
x1 , x2  0.
6. a)
4. a) Write the dual of the following linear programming problem
Max Z = 2 x1  x2  x3
2 x1  x2  x3  8
Subject to
 x1  x3  1
x1  2 x2  3x3  9
x3 is unrestricted, x1 , x2  0.
(5)
(5)
Use the dual simplex method to check that the following LPP
is infeasible
Max Z =  x1
Subject to
x1  x2  3
 x1  x2  4
(5)
x1 , x2  0.