GUJARAT TECHNOLOGICAL UNIVERSITY
BE- VIIth SEMESTER–EXAMINATION – MAY/JUNE- 2012
Subject code: 171901
Subject Name: Operation Research
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 (a) Explain significance of any two assumptions of Linear Programming
A small fabrication industry is faced with a problem of scheduling
production and subcontracting for three products A, B and C. Each
product requires casting, machining and assembly operations. Casting
operation for product A and B can be subcontracted but product C
requires special tooling hence it can not be subcontracted. Each unit of
product A, B and C requires 6, 10 and 8 minutes of casting time in the
foundry shop of a company. Machining times per unit of products A, B
and C are 6, 3 and 8 minutes while assembly times are 3, 2 and 2 minutes
respectively. The time available per week in foundry, machining and
assembly shop are 8000, 12000 and 10000 minutes respectively. If
product A, B and C are produced completely in the company, the overall
profits per unit of product are Rs. 700, Rs. 1000 and Rs. 1100
respectively. When castings are obtained from subcontractors, the profit
per unit of product A and B are Rs. 500 and 900 respectively. Formulate
above problem as LPP so as to maximize the profit for company by
scheduling its production and subcontracting.
(b) Solve the following LPP using Simplex method;
1 2 1 2 1 2
( ) 6 4
2 3 30 ; 3 2 24 ; 3
Maximize Z x x
x x x x x x
+ £ + £ + ³
Q.2 (a) A transport company has 5, 10, 7 and 3 trucks available at four different
sites A, B, C and D. Its customers have requirement of 5, 8 and 10 trucks
at three different destinations X, Y and Z respectively. The distance (in
kms.) from an origin to destination is summarized in following table.
Sites X Y Z
A 70 30 60
B 40 60 80
C 50 80 40
D 80 40 30
Formulate above problem as a transportation problem and determine
strategy for a company using VAM. Test the optimality of VAM solution
and determine optimum strategy for the transport company.
(b) i. What do you mean by Infeasibility and Unboundness in LPP? How
are the following issues identified from the simplex tableau?
ii. Construct the dual of following Primal Problem;
1 2 3
1 2 3 1 2 3 1 2 3
1 2 3 1 2 3 1 2 3
( ) 5 6 4
3 4 6 9 ; 3 2 5; 7 2 10
2 4 4 ; 2 5 3 3 ; , , 0
Minimize Z x x x
x x x x x x x x x
x x x x x x x x x
= − +
+ + ³ + + ³ − − £
− + ³ + − = ³
(b) The following tableau for a maximization type LPP is produced after few
iterations of simplex method;
8.5 10.5 0 0 0
Mix Qty (bi) X1 X2 S1 S2 S3
X2 300 0 1 3/5 -2/5 0
X1 300 1 0 -2/5 3/5 0
S3 400 0 0 -1/5 -1/5 1
Answer the following questions with brief reasons from above table;
i. Does the tableau represent an optimal solution? If not, carry out
necessary iterations and obtain an optimal solution.
ii. Is this solution degenerate?
iii. Are there more than single optimal solution to above problem?
iv. What are the shadow prices or dual values of resources?
v. What is optimum objective function value for the problem?
vi. If S1 represent the slack for production capacity constraint, how
much should company be willing to pay for each additional unit of
Q.3 (a) What do you understand by ‘zero-sum’ in the context of game theory?
Explain the meaning following terms used in game theory;
i. Saddle Point
ii. Pure Strategy
iii. Mixed Strategy
(b) The captain of a cricket team has to allot five middle order batting
positions to six batsmen available for selection. The average runs scored
by each batsmen at these positions are summarized in a table below
Batting Positio n
Batsman III IV V VI VII
A 40 40 35 25 50
B 42 30 16 25 27
C 50 48 40 60 50
D 20 19 20 18 25
E 58 60 59 55 53
F 45 52 38 50 49
Using Assignment model, determine the assignment of batsmen to
positions which would give maximum runs in favor of team. Which
batsmen will not qualify for selection based on the solution obtained?
Q.3 (a) What is ‘dominance rule’ in game theory? How can a ‘two person-zero
sum game’ problem be converted into LP problem? Illustrate with
(b) A solicitors’ firm employs typists on hourly piece-rate basis for daily
work. There are five typists available with hourly charges and speed
mentioned in table below.
Typist A B C D E
Rate per hour (Rs.) 5 6 3 4 4
No. pages typed/hour 12 14 8 10 11
There are five jobs available to the firm and it wishes to allocate one job
to one typist only. The typist is paid for full hour even if he works for
fraction of an hour. The details of job are given in table below.
Job P Q R S T
No. of Pages 199 175 145 298 178
Find least cost allocation for the firm using Assignment model.
Q.4 (a) Following failure rates have been observed for certain type of light bulbs;
Month 1 2 3 4 5
Percentage of items failing by end of month 10 25 50 80 100
There are total 1000 bulbs in use and it costs Rs. 10 to replace an
individual bulb which has fused out. If all bulbs are replaced
simultaneously, it would cost Rs. 4 per bulb. Two policies are being
considered for replacement of bulbs; First, replace all bulbs
simultaneously at fixed interval whether failed or not and do individual
replacement in intermediate periods. Secondly, individual replacement of
bulbs as and when it fails. Determine the optimum policy for replacement
of bulbs based on above failure data and costs.
(b) What is simulation? What are different phases of simulation process?
Differentiate between deterministic and stochastic simulation models.
What are the advantages and limitations of simulation?
Q.4 (a) An electronic item contains 10000 resistors. When any resistor fails, it is
replaced. The cost of replacing a resistor individually is Rs. 1 only. If all
resistors are replaced at the same time, the cost per resistor reduces to 35
paisa. The probability of failure of a resistor by the end of month is given
in table below.
Month 1 2 3 4 5 6
Prob. of items failing by
end of month
0.03 0. 0.2 0.4 0.15 0.15
Two policies are being considered for replacement of resistors; First,
replace all items simultaneously at fixed interval whether failed or not
and do individual replacement in intermediate periods. Secondly,
individual replacement of items as and when it fails. Determine optimum
policy for replacement of bulbs based on above failure data and costs.
Q.4 (b) i. Explain the meaning of following items in inventory management;
a. Re-order Level
b. Buffer Stock
ii. A purchase manager has decided to place an order for a minimum
quantity of 500 units of a particular item of inventory in order to get
discount of 10%. Past records reveal that 8 orders (each of 200 units)
were placed last year. Given ordering cost = Rs. 500 per year,
Inventory carrying cost = 40% of inventory value and price of item =
Rs. 400 per unit. What is the effect of this decision on company?
Q.5 (a) Arrival rate of telephone calls at a telephone booth follows Poisson
distribution with an average time of 9 minutes between two consecutive
calls. The length of telephone call is assumed to be exponentially
distributed with mean of 3 minutes.
i. Determine the probability that a person arriving at the telephone
booth have to wait.
ii. Find the average queue length that is formed from time to time.
iii. The telephone company will install a second booth when convinced
that an arrival would expect to have to wait at least four minutes for
the phone. Find increase in rate of arrival which will justify a second
(b) A small project is composed of 7 activities whose time estimates are
listed in the table below. Activities are identified by their beginning and
ending node numbers.
Activity 1-2 1-3 1-4 2-5 3-5 4-6 5-6
Optimistic 1 1 2 1 2 2 3
Most Likely 1 4 2 1 5 5 6
(weeks) Pessimistic 7 7 8 1 14 8 15
i. Draw the project network.
ii. Find the expected duration and variance for each activity.
iii. What is the expected project length and standard deviation?
iv. What is the probability that the project will be completed 3 weeks
later than the expected time?
Q.5 (a) How is dynamic programming problem different from LPP? Explain the
meaning of following terms used in dynamic programming;
iii. Principle of optimality
(b) The time estimates and precedence relationships of different activities
constituting a small construction project is given in following table;
Activity A B C D E F G H I
Predecessor – – B B A A F C, E, G F
3 8 6 5 13 4 2 6 2
i. Draw the project network.
ii. Determine project completion time.
iii. What is critical path?
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