Or Question Paper

1

Seat No.: _________ Enrolment No._______________

GUJARAT TECHNOLOGICAL UNIVERSITY BE SEM-VII Examination-Nov/Dec.-2011

Subject code: 171901 Date: 19/11/2011

Subject Name: Operation Research

Time: 10.30 am-01.00 pm Total marks: 70

Instructions:

1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.

Q.1 (a) Discuss the Various phases in solving an Operation Research model 07

(b) Use the graphical method to solve the following LP problems

Maximize Z=2X1+X2 Subject to the constraints:

X1 + 2X2

2

(b) State the general rules for formulating a dual LP problem from its primal.

Write the dual to the following LP problem.

Maximize Z = X1-X2+3X3 Subject to Constraints

X1 + X2 + X3

3

(b) A firm is considering replacement of a machine, whose cost price is Rs.

12,200 and the scrap value Rs. 200. The running costs are found from

experience to be as follows.

Year 1 2 3 4 5 6 7 8

Running

Cost Rs.

200 500 800 1,200 1,800 2,500 3,200 4,000

When should the machine be replaced?

07

OR

Q.4 (a) What is degeneracy in transportation problems? Explain how to resolve

degeneracy in a transportation problem

07

(b) A company has factories at F1, F2 and F3 which supply to warehouses at

W1, W2, W3. Weekly factory capacities are 200 ,160 and 90 units,

respectively. Weekly warehouses requirement are 180,120 and 150 units,

respectively. Unit shipping costs(in Rs.) are as follows. Determine the

optimal distribution to minimize total transportation cost

Warehouse Factory

W1 W2 W3 Supply

F1 16 20 12 200

F2 14 8 18 160

F3 26 24 16 90

Demand 180 120 150 450

07

Q.5 (a) Listed in the table are the activities and sequencing necessary for a

maintenance job on the heat exchangers in a refinery. Draw a network

diagram for the project.

Activity Description Predecessor

Activity

A Dismantle pipe connections -

B Dismantle heater ,closure, and

floating front

A

C Remove tube bundle B

D Clean bolts B

E Clean heater and floating head

front

B

F Clean tube bundle C

G Clean shell C

H Replace tube bundle F, G

I Prepare shell pressure test D,E,H

J Prepare tube pressure test and

reassemble

I

07

4

(b) A company management and the labour union are negotiating a new three

year settlement. Each of these has 4 strategies:

I: Hard and aggressive bargaining

II: Reasoning and logical approach

III:Legalistic strategy

IV: Conciliatory approach

The cost to the company are given in the fo;;owing table for every pair of

strategy choice. What strategy will the two side adopt ? Also determine the

value of the game.

Company Strategies Union

Strategies I II III IV

I 20 15 12 35

II 25 14 8 10

III 40 2 10 5

IV -5 4 11 0

07

OR

Q.5 (a) Define the following dynamic programming terms:

(i) Stage (ii) State Variable (iii) Decision variable (iv) Immediate

return(v)Optimal return(vi) State transformation function

07

(b) What are the advantages and limitations of simulation models 07 _______________

1

Seat No.: ________ Enrolment No.______________ GUJARAT TECHNOLOGICAL UNIVERSITY BE- VIIth SEMESTEREXAMINATION MAY/JUNE- 2012

Subject code: 171901 Date: 24/05/2012 Subject Name: Operation Research Time: 02:30 pm 05:00 pm Total Marks: 70 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.

07

(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 xsubject to

x x x x x x

x x

= +

+ + +

07

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.

Customers 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.

07

2

(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 2 102 4 4 ; 2 5 3 3 ; , , 0

Minimize Z x x xsubject to

x x x x x x x x x

x x x x x x x x x

= +

+ + + + + + =

03

04

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?

07

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

07

(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 Position 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?

07

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.

07

3

(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.

07

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.

07

(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?

07

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 0.07 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.

07

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?

03

04

4

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.

07

(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?

07

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

07

(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?

07

*************

Seat No.: ________ Enrolment No._________

GUJARAT TECHNOLOGICAL UNIVERSITY B. E. - SEMESTER VII EXAMINATION WINTER 2012

Subject code: 171901 Date: 26/12/2012 Subject Name: Operation Research Time: 10.30 am - 01.00 pm Total Marks: 70 Instructions:

1. Attempt any five questions. 2. Make suitable assumptions wherever necessary. 3. Figures to the right indicate full marks.

Q.1 (a) Do as directed:

(i) What is the degeneracy in simplex method? How it can be resolved? (ii) Illustrate graphically; (a) No-feasible solution (b) Unbounded solution. (iii) Classify the Assignment problems in detail. (iv) Explain the following term in the context of game theory (a) Saddle point (b) Two persons Zero-sum game

08

(b)

(i) Find the maximum value of following LPP using graphical approach

1 2

1 2

1 2

1 2

1 2

Z x 2xS / t, x 3x 10

x x 6x x 2x and x 0

= + +

+

(ii) Write down the dual of above LPP

04 02

Q.2 (a) Write note on scope of Operation research in the various sector. 04

(b) Solve following LPP using Penalty method 1 2

1 2

1 2

2

1 2

Maximize Z 3x xS / t, 2x x 2

x 3x 3x 4

x and x 0

= + +

10

OR (b) A firm manufactures two product A & B on which the profit earned per

unit are Rs. 3 and Rs.4, respectively. Each product is processed on two machines M1 and M2. Product A requires one minute of processing time on M1 and two minutes on M2, while product B requires one minute of processing time on M1 and one minute on M2. Machine M1 is available for not more than 7 hrs and 30 minutes, while machine M2 is available for 10hrs during any working day. Find

(i) Formulate the problem as LPP (ii) Solve the above LPP using Simplex method

04 06

Page 1 of 3

of 3

Q.3 (a) State the name of method which divides inventories into three classes and describe it in detail.

05

(b) Find the optimal solution of the following Transportation problem using MODI method. Use VAM to find IBFS.

M1 M2 M3 M4 Supply F1 3 2 4 1 20 F2 2 4 5 3 15 F3 3 5 2 6 25 F4 4 3 1 4 40

Demand 30 20 25 25

09

OR Q.3 (a) A machine cost Rs 500. Operation and maintenance cost are zero for

the first year and increases by Rs. 100 every year. If money is worth 5% every year, determine the best age at which the machine should be replaced. The resale value of the machine is negligibly small. What is the weighted average cost of owning and operating the machine?

07

(b)

The following information is provided for an item: Annual usage = 1200, Ordering cost = Rs 60 per order, Carrying costs 10%, Unit cost of item = Rs 10, and lead time 10 days. Find (i) EOQ (ii) Number of order per years (iii) Average usage if there are 300 working days per year (iv) Safety stock if highest usage rate is 70 units per day (v) R. O. L (vi) Average inventory (vii) Inventory carrying cost per year.

07

Q.4 (a) A chemical company distributes its products by trucks loaded at its only

loading station and loading station is working 24 hours, continuously. Both companys trucks and contractors trucks are used for this purpose. It was find out that on an average 10 minutes one truck arrived and average loading time was 6 minute. If 50% trucks are contractors trucks find (i) Traffic intensity factor, (ii) Weighting time of trucks in system, (iii) The expected weighting time of contractors trucks per day.

06

(b) The captain of cricket team has to allot five middle batting positions to five batsmen. The average runs scored by each batsman at these positions are as follows: Find the assignments of batsmen to positions which would give the maximum number of runs.

Batsmen Batting position I II III IV V

P 40 40 35 25 50 Q 42 30 16 25 27 R 50 48 40 60 50 S 20 19 20 18 25 T 58 60 59 55 53

08

OR

of 3

Q.4 (a) Solve the following game by using Dominance method

Player B B1 B2 B3

A1 4 5 8 A2 6 4 6 Player A A3 4 2 4

07

(b) Explain the various elements of queuing system 07

Q.5 (a) The details of activity in project management are given below.

Normal Crash Activity Time (Days) Cost (Rs.) Time (Days) Cost (Rs.) 1-2 3 300 2 400 2-3 6 480 4 520 2-4 7 2100 5 2500 2-5 8 400 6 600 3-4 4 320 3 360 4-5 5 500 4 520

Find (i) Critical path & Project duration (ii) Optimum project cost with considering indirect cost is 100 Rs.er day.

03 05

(b)

What is the simulation? Classify the simulation model? Explain the general Simulation methodology.

06

OR Q.5 (a) (I) The details of activity and duration are shown below.

Activity Immediate predecessor activity Duration (Days)

A - 10 B A 5 C A 4 D A 7 E B, C 6 F C, D 4 G E, F 7

Find (i) Draw a network (ii) Find the critical path (iii) Project duration

(II) What is the meaning float in project management? State the various types float.

02 02 01 02

(b)

Do as directed with reference to dynamic programming (i) Explain the terms: State, Stage and Policy (ii) Explain Bellmans Principle of Optimality (iii) Explain: Transformation function and Return function

03 02 02

*************

1

Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTERVII EXAMINATION WINTER 2013

Subject Code: 171901 Date: 26-11-2013 Subject Name: Operation Research Time: 10.30 am - 01.00 pm Total Marks: 70 Instructions:


Q.1 (a) What are phases of OR Project, explain in detail 07 (b) A coffee company mixes Brazilian, Columbian and African coffee to make two

brands of coffee plains A and B. The characteristics used in blending the coffee include strength, acidity and caffine. The test result of the available supply of Brazilian, Columbian and African coffee.

The requirement for A and B coffee are given as below.

Assume that 35000 kg of plains A and 25000 kg of plain B are to be sold formulate LPP.

Price/kg Strength Acidity %coffine Supply available

Brazilian 60 6 4 2 50000 Columbian 70 8 3 2.5 30000 African 65 5 3.5 1.5 25000

Plain coffee

Price/kg Min strength

Max acididty

Max % coffine

Quantity Demanded

A 75 6.5 3.8 2.2 65000 B 85 6.0 3.5 2 55000

07

Q.2 (a) Minimize z = -3x1 + x2 2x3 Subject to x1 + 3x2 + x3 5 2x1 x2 + x3 2 4x1 + 3x2 2x3 = 5 x1, x2, x3 0

07

(b) Consider the transportation problem shown in table below Find the initial basic feasible solution using Northwest corner method

1 2 3 4 5 supply PLANT 1 20 4 32 28 20 3000

2 12 36 24 26 32 5000 3 16 8 28 24 20 8250 4 28 44 40 16 36 3750

Demand 3500 4000 2500 1500 4000

07

OR (b) Solve above method by Least cost cell method 07

Q.3 (a) Consider the assignment problem shown in table below. In the problem 5 different jobs are to be assigned to 5 different operators such that the toal processing time is minimized. The matrix entries represent processing times in hours. Develop a zero-one programming model.

07

2

1 2 3 4 5 1 20 24 30 24 16 2 14 32 28 28 22 3 26 28 14 18 18 4 24 20 22 26 20 5 16 26 30 22 30

(b) Solve above problem with Hungerian method 07 OR

Q.3 (a) Solve the below game theory problem with the concept of dominance method

PLAYER B PLAYER

A I II III IV I 3 5 4 2 II 5 6 2 4 III 2 1 4 0 IV 3 3 5 2

07

(b) Determine the solution of game for the pay-off matrix given below PLAYER B PLAYER A I II III

I -3 -1 7 II 4 1 -2

07

Q.4 (a) The initial cost of a machine is Rs 71000 and scrape value is Rs 1000. The maintenance costs found from experience are as below. Find when should the machine be replaced?

Year 1 2 3 4 5 6 7 8 Maintenance 2000 3500 5000 7000 10000 13000 17000 21000

07

(b) Customers arrive at a one window drive according to the poisons distribution with the mean of 10 minutes and service time per customer is exponential with mean of 6 minutes. The space in front of the window can accommodate only three vehicles including the serviced one. Other vehicles have to wait outside the space. Calculate - Probability that an arriving customer can drive directly to the space in

front of the window - Probability that an arriving customer will have to wait outside the directed

space - How long an arriving customer is expected to wait before getting the

service?

07

OR

Q.4 (a) Inventory control manager of a firm has collected the following data on one item

- Minimum total cost per annum = Rs. 16000 - Inventory holding cost per unit per year = Rs. 4 - No of order per year = 10 - Price per unit = Rs. 25

Calculate annual demand of the item, procurement cost per order, inventory carrying cost as a percentage of average inventory investment and economic order quantity (EOQ)

07

(b) In a firm, the demand for a certain item is random. It has been established that the monthly demand of an item has a normal distribution with a mean of 1000 and a standard deviation of 150 units. The unit price of an item is Rs 20/-. The ordering cost is Rs 40/-, the inventory carrying cost is estimated to be 15% per

07

3

annum respectively. The procurement lead time is constant and is two months. Find the most economic ordering policy and the expected cost of controlling inventory given that the service level is 95%.

Q.5 (a) A small project is compsed of 7 activities whose time estimates are listed in the table below. Activities are identified by their beginning (i) and ending (j) node numbers

Activity Estimated duration (weeks) (i-j) Optimistic Most likely pessimistic 1-2 1 1 7 1-3 1 4 7 1-4 2 2 8 2-5 1 1 1 3-5 2 5 14 4-6 2 5 8 5-6 3 6 15

- Draw the network diagram of activities in the project - Find the expected duration and variance of each activity. What is the

expected project length? - Calculate the variance and standard deviation of the project length.

07

(b) What do you mean by linear programming? Define following terms : linear function, objective function, decision variable, constraints, feasible solution, optimal solution.

07

OR Q.5 (a) What is CPM and PERT. Discuss significance of using CPM and PERT.

07

(b) Define following terms with respect to CPM/PERT : event, merge event, burst event, activity, processor activity, successor activity, dummy activity

07

*************

1


GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTERVII EXAMINATION SUMMER 2014

Subject Code: 171901 Date: 22-05-2014


Time: 02:30 pm - 05:00 pm Total Marks: 70 Instructions:


Q.1 (a) State the phases of Operation Research. Discuss in brief the areas of application

of Operation Research

07

(b) Solve the following LPP by simple method :

Maximize Z = 3x1 + 2x2 subject to 2x1 + x2 < 5, x1 + x2 < 3 and x1and x2 > 0

07

Q.2 (a) Company wants to find out the minimum time require to complete four tasks by

available four workers with him so that he can take another from the order

party. Following table gives the time in hours for each workers for each job

A B C D

1 24 10 21 11

2 14 22 10 15

3 15 17 20 19

4 11 19 14 13

07

(b) Explain Monte Carlo simulation procedure. Also discuss its applications with

suitable example 07

OR

(b) A school wants to pick up students from five different areas. Cost in rupees of

going from one area to another is shown in table. Find the optimal route to the

bus drive such that no repetition of the area comes before picking up students

from all areas.

I II III IV V

I 0 30 60 80 20

II 70 0 40 90 30

III 90 80 0 50 80

IV 130 50 70 0 60

V 20 40 30 90 0

07

Q.3 (a) Reduce following matrix by rule of dominance

Player

Player

B1 B2 B3 B4

A1 6 4 8 0

A2 6 8 4 8

A3 8 4 8 0

A4 0 8 0 16

07

(b) Two companies are thinking on selecting the advertising media. There are three

medias available along with the pay of as shown in the pay of matrix

Player A

TV Radio Internet

TV 150 200 -400

Radio 0 75 -100

Internet 450 100 250

Value is in gain sales in (1000 rupees) suggest optimal strategy for the

marketing and find out the value of the game

07

2

OR

Q.3 (a) A person is planning to purchase a car. A new car is costing rupees 3 lacs. The

resale value of the car at the end of the year is 85 % of the previous year.

Maintenance and repair cost during the first year are rupees 10000 and they

increase by 15 % every year. The minimum resale value of the car can be

rupees 75000. (a) When should the car be replaced to minimize average annual

cost? (b) If interest rate of 12 % is assumed, calculate the average cost at the

end of 10 years

07

(b) A copy maker has one copy making machine and he operates as the order

comes. The order arrival is poison distribution having interval time of 0.5 min.

The average time to serve a copy is distributed with mean of 0.3 min.

Determine the following: (1) Utilization factor of the machine (2) Idle time for

machine in a day having working hours of 10 hours (3) No of persons waiting

in the system (4) No of persons waiting in the queue (5) Average waiting time

in the queue

07

Q.4 (a) From the following given data find out shortage cost for the item. C1 = Rs.900/-

and critical probability = 0.70

Units Stocked 50 58 65 70 75

Probability 0.2 0.25 0.14 0.34 0.07

07

(b) Consider the following given data and based on that find out critical path for the

given project.

Activity 1-2 1-3 2-4 3-4 3-5 3-6 4-6 5-6

Time in

days

6 9 3 4 8 12 7 1

07

OR

Q.4 (a) A utensil manufacturing company manufactures around 140 units of utensils.

Depending upon the availability of row material and other conditions the daily

production has been varying from 136 units of utensils to 144 units of utensils

whose probability is as given below.

Production

per day

136 137 138 139 140 141 142 143 144

Probability 0.03 0.06 0.14 0.13 0.22 0.16 0.12 0.08 0.06

The finished units of utensils are transported in a specially designed rickshaw

that can accommodate only 140 units of utensils. Using the following given

random numbers simulate the process to find out (1) What will be the average

number of utensils waiting in the factory (2) What will be the number of empty spaces in rickshaw

Random Numbers: 84, 72, 28, 52, 38, 65, 13, 79, 27, 54, 01

07

(b) Solve the cargo loading problem with following data and maximum weight

capacity is five.

Item (n) Weight

(Wn)

Return

(Rn)

1 1 3

2 2 7

3 3 10

07

Q.5 (a) Explain the following terms in connection with inventory management. (1) Re-

order point (2) Safety stock (3) Lead time (4) Economic lot size (5) Carrying

cost

07

(b) Explain the term crashing of network. Why it is required? 07

OR

Q.5 (a) The annual demand of a product is 15,000 units. Each unit cost Rs.50/- if the

orders are placed in quantity below 150 units. For order of 200 and above the

unit prize is Rs.44/-. Assume inventory holding cost is 12% of the value of item

and ordering cost is Rs.2/- per order find the economic lot size

07

3

(b) A company is presently buying an item of worth Rs.90,000/- from a supplier at

an optimal purchasing policy at a discount of 1%. Presently the ordering cost is

Rs.100/- per order and 20% as inventory handling cost of the average inventory

level. The company receives another two offers from the other suppliers. First

supplier offers 5% discount if the order is placed twice a year and second

supplier offers 3% discount if the order is placed quarterly a year. Which offer

the company should accept?

07

*************


GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTERVII EXAMINATION WINTER 2014

Subject Code: 171901 Date: 25-11-2014


Time: 10:30 am - 01:00 pm Total Marks: 70 Instructions:


Q:-1 (a) Write the dual of the following linear programming problem. Minimize, Z = 20 X1+ 23 X2

Subjected to,

- 4X1 - X2 -8 5X1 - 3X2 = - 4 X1, X2 0

Solve the Dual problem using simplex method and predict the value of variables X1,

X2 from the solution of dual linear programming problem.

07

(b) Optimum simplex table of the following linear programming problem has been given in the table:-1.

Maximize, Z = 60 X1+ 20 X2 +80 X3

Subjected to,

6X1+3X2+5X3 750 3X1+4X2+5X3 600 X1, X2, X3 0

Basic

Variable X1 X2 X3 W1 W2

60 X1 1 -1/3 0 1/3 -1/3 50

80 X3 0 1 1 -1/5 2/5 90

Table:-1

(i) If the RHS of the constraints changes to [750, 900]T, does it affect the optimum solution? If yes, then find the optimum solution using sensitivity

analysis approach.

(ii) If coefficient of X2 in the constraints change to [1, 1]T, does it affect the optimum solution? If yes, obtain the optimum solution using sensitivity

analysis approach.

(iii) If new constraint X1+X2+X3 90 is added to the LP Problem, does it affect the optimum solution? If yes, obtain the optimum solution using

sensitivity analysis approach.

02

02

03

Q:-2 (a) A Manufacturer wants to ship 8 loads of his product as shown in following matrix.

The matrix gives the mileage from origins, O to the destinations, D. The shipping cost

is Rs. 10 per load per mile. What will be the optimal schedule and optimal cost? Use

Vogels approximation method to find initial basis feasible solution and MODI method to obtain optimal solution.

D1 D2 D3 Supply

O1 50 30 220 1

O2 90 45 170 3

O2 250 200 50 4

Demand 4 2 2

07

(b) (i) Using dynamic programming solve the following L.P.P.,

Maximize, Z = X1+ 9 X2

Subjected to,

X2 11 2X1+X2 25 X1, X2 0

(ii) In brief, explain characteristics of operation research.

04

03

OR

(b) (i) A student of engineering wants to appear in the three competitive exam and he has

three days left before examination. He wants to revise the whole syllabus of the

subjects before examination by devoting a single day, two days or not a single day to

any subject based on given estimate of expected grade points as shown in matrix. How

he should plan his study?

Subjects

Days I II III

0 0 1 0

1 1 1 1

2 1 3 4

3 3 4 3

(ii) Write the definition of operation research given by author Churchman, Ackoff and

Arnoff.

06

01

Q:-3 (a) The owner of a small machine shop has four machinists available. To assign jobs for

the days. Five jobs are offered with the expected profit in rupees for each machinist on

each job has been shown in matrix below. Find the assignment of machinists to jobs

that will result in a maximum profit. Which job should be declined?

Jobs

A B C D E

Mac

hin

ist 1 6.20 7.80 5.00 10.10 8.20

2 7.10 8.40 6.10 7.30 5.90

3 8.70 9.20 11.10 7.10 8.10

4 4.80 6.40 8.70 7.70 8.00

07

(b) Determine the approximate solution of following game problem (do minimum 10

iteration)

B

1 2 3 4

A

1 3 2 4 0

2 3 4 2 4

3 4 2 4 0

4 0 4 0 8

07

OR

Q:-3 (a) A company is facing the problem of assigning six different machines to five different

jobs. The estimated costs are given in matrix as below.

Jobs

1 2 3 4 5

Mac

hin

e

1 2.5 5 1 6 1

2 2 5 1.5 7 3

3 3 6.5 2 8 3

4 3.5 7 2 9 4.5

5 4 7 3 9 6

6 6 9 5 10 6

Solve the problem assuming that the objective function is to minimize total cost. Is

there any alternate optimal solution exist? If yes, find the possible alternate solution.

07

(b) For a game shown below, if X1 : X2 = (1/2) : (2/3) and Y1 : Y 2 = (3/4) : (1/4). Find expected pay off. Are these strategies optimal for player I and II? Why?

II

I 1 4

3 2

07

Q:-4 (a) The activities A to H of a new project having relationships and timings shown in table

below.

Duration (in days) Relationship between

activities

Activity t0 tm tp A < C, D B < E

C < F

D < F

E, F < H

A 2 2 8

B 2 5 8

C 3 6 15

D 2 5 14

E 1 1 7

F 2 2 8

G 2 2 8

H 2 5 14

(1) Draw the network.

07

(2) Find the critical path and expected time of completion of the project. (3) What will be the standard deviation of the project completion duration? (4) What will be the probability of completing the project in expected time of

completion?

(b) (i) Explain in brief Monte carlo simulation.

(ii) Automobile arrives at a petrol pump having one petrol unit in poisson fashion

with an average of 10 units per hour. The service time is distributed exponentially

with a mean of 3 min. Find following:-

a. Average number of unit in system b. Average waiting time for customer in queue. c. Probability that number of units in system is 2. d. Probability that waiting time exceeds 30 min.

03

04

OR

Q:-4 (a) Following table shows jobs, normal and crash time, normal and crash cost of a

project. Indirect cost for the project is 300 Rs./day.

Jobs

i - j

Normal

Time

(Days)

Normal

Cost

(Rs.)

Crash

Time

(Days)

Crash

Cost

(Rs.) 1 - 2 6 1400 4 1900

1 3 8 2000 5 2800

2 3 4 1100 2 1500

2 4 3 800 2 1400

3 4 --- --- --- ---

2 5 6 900 3 1600

4 6 10 2500 6 3500

5 - 6 3 500 2 800

(i) Draw the network and find the critical path. (ii) What is normal duration and cost of project. (iii) Find optimal cost and duration.

07

(b)(i) Explain Kendalls notation for representing Queuing model.

(ii) Generate random numbers using (1) Mixed congruence method and (2) Additive

congruence method for the data: r0 = 2, a = 14, b = 12 and m = 32.

03

04

Q:-5 (a)

(i) Explain importance of replacement in brief.

(ii) The value of the money is 10 % per year. Machine-1 is to be replaced every 3

years and Machine -2 is to be replaced for every 6 years with yearly expenditure as

given below. Which machine costs less?

Expenditure ( in rupees)

Year Machine:-1 Machine:-2

1 2000 3400

2 400 200

3 800 400

4 --- 600

5 --- 800

6 --- 1000

02

05

(b) Derive the expression of optimal production lot size and optimum level of shortage

for the inventory model with gradual supply and shortage is allowed.

07

OR

Q:-5 (a) (i) Explain in brief the reason for replacement.

(ii) As new automobile vehicle costs of Rs. 10000 and it can be sold at the end of any

year with the selling price as shown below. The operating and maintenance cost are

given year wise in following table. Find when the automobile vehicle needs to be

replaced because of wear and tear.

Expenditure ( in rupees)

Year Selling

Price (Rs.)

Operating and

maintenance

cost (Rs.)

1 7000 1000

2 5000 1600

3 3000 1800

4 2000 2500

5 1000 3000

6 500 3500

03

04

(b) Explain ABC analysis. 07

*************

1


GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTERVII EXAMINATION SUMMER 2015

Subject Code: 171901 Date: 01/05/2015 Subject Name: Operations Research Time: 02.30pm-05.00pm Total Marks: 70 Instructions:


Q.1 (a) State the general rules for formulating a dual LP problem fro its primal.

(b) Nachiketa corporation manufactures two products A1&A2.The profit per unit of the two products is Rs.50& Rs.60 respectively. Both the products require processing in three machines. Below table indicates the available machine hours per week & time require on each machine for one unit of A1 &A2.Formulate as linear programming problem.

07 07

Q.2 (a) Explain significance of any two assumptions of LPP. 07 (b) Use graphical method to solve the following LPP .

Maximize Z=17 X1 +15 X2 Subject to: 15X1 +25X2 375 24X1+11X2 265 All X1 , X2 0

07

OR (b) Solve the following game whose payoff matrix is given below.

Player B Player A

1 8 6 2

07

Q.3 (a) Distinguish between transportation & transshipment problems in detail. 07 (b) Company has factories A1, A2 & A3 which supply to warehouses at W1 ,W2 &

W3.Weekly factory capacities are 240,200&130 units respectively. Weekly warehouses requirements are 190,150&110 units respectively. Unit

transportation in costs Rs. As follows:- Find I.B.S. BY VAM method & Optimum solution BY MODI method.

W1 W2 W3 SUPPLY

A1 16 20 12 240 A2 14 8 18 200 A3 26 24 16 130 DEMAND 190 150 110 450

07

OR

2

Q.3 (a) Give different practical applications of transportation problem. 07 (b) Obtain an I.B.F.S. to the following transportation problem using N-W Corner

method.& Optimum solution BY STEPPING STONE method.

Q1 Q2 Q3 Q4 SUPPLY P1 1 3 2 4 8 P2 5 4 2 0 10 P3 0 3 3 1 12 DEMAND 4 5 8 5

07

Q.4 (a) The production department for a company requires 3500kg.of row material for manufacturing a particular item per year. It has been estimated that the cost of placing an order is Rs.35& the cost of carrying inventory is 25 percentage of the investment in the inventories. The price is Rs.10 per kg.The purchase manager wishes to determine an ordering policy for raw material. Calculate (1)The optimal lot size(2)The minimum yearly variable inventory cost(3)The optimal order cycle time(4)The minimum yearly total inventory cost.

07

(b) Solve the following assignment problem by minimization method.

I II III IV V M1 12 5 9 18 11 M2 13 7 6 12 14 M3 3 2 3 4 5 M4 18 9 12 16 15 M5 12 6 14 19 10

07

OR Q.4 (a) A Project is represented by the Network shown below & has the following data.

Determine(1)Expected Time &Variance(2)Earliest & Latest times to reach each event.(3)The critical path Task A B C D E F G H I Optimistic time

5 18 26 16 15 6 7 7 3

Pessimistic time.

10 22 40 20 25 12 12 9 5

Most likely time.

8 20 33 18 20 9 10 8 4

07

(b) On an average 95 patients per 24 hrs.day require the service of an emergency clinic. Also on the average, a patient requires 12 minutes of an active attention. Assume that the facility can handle only one emergency at a time.Suppoce that it cost the clinic Rs.100 per patient treated to obtain an average servicing of 10 minutes & that minute of decreasing in this average time would cost Rs. 10 per patient treated. How much would have to be budgeted by the clinic to decrease the average size of the queue from one to one third patients to half a patient.

07

Q.5 (a) What is dynamic programming? Discuss the similarities between dynamic & linear programming. How it differs from linear programming?

07

3

(b) A firm is considering replacement of a machine whose cost price is Rs.12200& the scrap value Rs.200.The running costs are found from experience to be as follows. When should the machine be replaced?

Year 1 2 3 4 5 6 7 8 Running costRs.

200 600 700 1000 1200 1800 2500 4000

07

OR Q.5 (a) Explain steps in Monte Carlo simulation process. 07

(b) Explain definition & scope of operation research. 07

*************

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