BE10-R3: APPLIED OPERATIONS RESEARCH

NOTE:
1. Answer question 1 and any FOUR questions from 2 to 7.
2. Parts of the same question should be answered together and in the same
sequence.
Time: 3 Hours Total Marks: 100

1.
a) Formulate the following Diet problem as a linear programming problem. One of the
interesting problems in linear programming is that of balanced diet. Dieticians tell us that
a balanced diet must contain certain quantities of nutrients such as calories, minerals
and vitamins etc. Suppose that we are asked to find out the food that should be
recommended from a large number of alternative sources of these nutrients so that the
total cost of food satisfying minimum requirements of balanced diet is the lowest.

The medical experts and dieticians tell us that are necessary for an adult to consume at
least 75 g. of fats and 300 g. of carbohydrates daily. The following table gives the food
items (which are readily available in market), their analysis, and the cost.

Table
Food Food value (gms.) per 100 gm. Cost per kg.
(Rs.)
Proteins Fat Carbohydrates
1 8.0 1.5 35.0 1.00
2 18.0 15.0 − 3.00
3 16.0 4.0 7.0 4.00
4 4.0 20.0 2.5 2.00
5 5.0 8.0 40.0 1.50
6 2.5 − 25.0 3.00

Minimum
Daily 75 85 300 −
Requirements

b) We have five jobs, each of which has to go through the machines A and B in the order
AB. Processing times are given in the table below:

Processing-times in hours

job Ai BI
1 5 2
2 1 6
3 9 7
4 3 8
5 10 4

Determine a sequence of these jobs that will minimize the total elapsed time T.

c) There is congestion on the platform of a railway station. The trains arrive at the rate of
30 trains per day. The waiting time for any train to hump is exponentially distributed with
an average of 36 minutes. Calculate the following:

BE10-R3 Page 1 of 5 January, 2006

 i) The mean queue size.
ii) The probability that queue size exceeds 9 (obtain the expression).
d) Draw a network diagram for the following set of operations represented by separate
letters.
Operations POST-OPR
A precedes B, C
B precedes D, E
C precedes D
D precedes F
E precedes G
F precedes G
e) A TV repairman finds that the time spent on his jobs has an exponential distribution with
mean 30 minutes. If he repairs sets in the order in which they come in, and if the arrival
of sets is approximately Poisson with an average of 10 per 8 hour day, what is the
repairman's expected idle time each day? How many jobs are ahead of the average set
just brought in?
f) Show that if dominance occurs in the pay off matrix of a 2× 2 game, then there is a
saddle point. Is the converse true?
g) Write the dual of the problem
Max. z = 2x1 + 3x2,+ x3,
4x1 + 3x2 +x3 = 6,
x1+2x2 + 5x3 = 4, x1, x2, x3, >0.
(7x4)

2.
a) Solve the following linear programming problem by simplex method.
Max. z = x1 + x2 + 3x3 − x4
s.t. x1 + 2x2 + 3x3 = 15,
2x1 + x2 + 5x3 =20,
x1 + 2x2 + x3 + x4 = 10, x1, x2, x3, x4 > 0
b) Reduce the following transportation problem to an assignment problem and solve it.
Distances (in km) are given following table:
Depot buses required
a b c
Terminal A 6 10 15 2
B 4 6 16 2
C 12 5 8 1
buses available 1 1 3
Make an allocation so that total distance traveled is minimum.
(9+9)

3.
a) Find the optimal sequence for processing 4 jobs A, B, C, D on four Machines A1, A2, A3,
A4 in the order A1 A2 A3 A4. Processing times are as given below:

Processing times (aij) in hours

Job/Machine A1(ai1) A2(ai2) A3(ai3) A4(ai4)
A 15 5 4 14
B 12 2 10 12
C 13 3 6 15
D 16 0 3 19

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b) Calculate the various time estimates for the network in fig.

(9+9)

4.
a) A telephone exchange has two long distance operators. The telephone company finds
that during the peak load, long distance calls arrive in Poisson fashion at an average
rate of 15 per hour. The length of service on these calls is approximately exponentially
distributed with mean length 5 minuets. What is the probability that a subscriber will have
to wait for his long distance call during the peak hours of the day? If subscribers wait and
are serviced in turn, what is the expected waiting time?
b) Use the concept of dominance of dominance to solve the game

B
I II III IV
I 3 2 4 0
II 3 4 2 4
A III 4 2 4 0
(9+9) IV 0 4 0 8
5.
a) Solve the following problem using dynamic programming.
n
Minimize z =
J= 1
∑ y 2j ,
Subject to the constraints
n

∏j= 1
y j = b, y j ≥ 0 for all j.

b) Consider the inventory system with the following data in usual notations:
R =1000 units/year, I =0.30, P= Re. 0.50 per unit
C3 =Rs.10.00, L= 2=years (lead time).
Determine:
i) Optimal order quantity
ii) Reorder point
iii) Minimum average cost
(9+9)

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6.
a) There are three parties who supply the following quantities of coal and three consumers
who require the coal as follows:

Party 1 14 tons Consumer A 6 tons

Party 2 12 tons Consumer B 10 tons
Party 3 5 tons Consumer C 15 tons
Total 31 tons

A B C
1 6 8 4

2 4 9 3

3 1 2 6

The cost matrix is as shown here.

Find the schedule of a transportation policy, which minimizes the cost.
b) Average time taken by an operator on a specific machine is tabulated below. The
management is considering replacing one of the old machines by a new one and the
estimated time for operation by each operator on the new machine is also indicated.

Machines
operators 1 2 3 4 5 6 new
A 10 12 8 10 8 12 11
B 9 10 8 7 8 9 10
C 8 7 8 8 8 6 8
D 12 13 14 14 15 14 11
E 9 9 9 8 8 10 9
F 7 8 9 9 9 8 8

i) Find out an allocation of operators to old machines to achieve a minimum

operation time.
ii) Reset the problem with the new machine and find out the allocation of the
operator to each machine and comment on whether it is advantageous to replace
an old machine to achieve education in operating time only.
iii) How will the operators be allocated to the machines after replacement?
(9+9)

7.
a) A firm can backorder, if out of stock, the demands of its customers. The given facts are
as follows:
Total annual demand D = 100 units
Ordering Cost O = Rs. 10 per order
Price of the item P = Rs. 20 per unit
Inventory carrying cost I = 20%
Penalty cost of backordering K = Rs. 5 per unit per year
Determine the optimum order size ad the amount backordered for each cycle on the
basis of above information.

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b) Three manufacturers A, B and C are competing with each other. The transition-matrix
given below gives the probabilities that customers will move from one manufacture to
other in any month. Interpret the matrix in terms of
a) retention and loss and
b) retention and gain.

Transition Matrix
To A B C
From
A 0.7 0.1 0.2

B 0.1 0.8 0.1

C 0.2 0.1 0.7

(9+9)

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