**GUJARAT TECHNOLOGICAL UNIVERSITY**

**BE- VII****th ****SEMESTER–EXAMINATION – MAY/JUNE- 2012**

**Subject code: 171901**

**Subject Name: Operation Research**

**Mechanical Engineering**

** **

**Instructions:**

**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

Problem (LPP).

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 1 2

1 2

( ) 6 4

2 3 30 ; 3 2 24 ; 3

, 0

*Maximize Z x x*

*subject to*

*x x x x x x*

*x x*

= +

+ £ + £ + ³

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.

Customer s

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*

*subject to*

*x x x x x x x x x*

*x x x x x x x x x*

= − +

+ + ³ + + ³ − − £

− + ³ + − = ³

** **

**OR**

**(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

production capacity?

** **

**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?

** **

**OR**

**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

example.

** **

** (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?

** **

**OR**

**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

booth.

** **

**(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

Time

Estimates

(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?

** **

**OR**

**Q.5 (a) **How is dynamic programming problem different from LPP? Explain the

meaning of following terms used in dynamic programming;

i. Stages

ii. States

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

Duration

(days)

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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