## Loyola College M.Sc. Mathematics April 2012 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – APRIL 2012

# MT 4811 – OPERATIONS RESEARCH

Date : 18-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

All questions carry equal marks

I a) Explain sensitivity analysis. Is it really useful to a company?

(or)

1. b) Explain branch and bound method.                             (5)

1. c) Find the optimum integer solution to the following LPP.

(15)

(or)

1. Solve the problem

Discuss the effect of changing the requirement vector from  on the optimum solution.

II a) What is goal programming? How is it useful for a manufacturing company?

(or)

1. b) Mention the differences between LP and GP approach.                               (5)

1. c) A firm produces two products A and B. Each product must be processed through two departments. Department I has 30 hours of production capacity per day, and department II has 60 hours. Each unit of Product A requires 2 hours in department I and 6 hours in department II. Each unit of product B requires 3 hours in department I and 4 hours in department II. Management has rank ordered the following goals it would like to achieve in determining the daily product mix.

P1 : Minimize the underachievement of joint total production of 10 units.

P2 : Minimize the underachievement of producing 7 units of product B.

P3 : Minimize the underachievement of producing 8 units of product A. Formulate

this problem as a GP problem and illustrate with graph.                                (15)

(or)

1. d) A factory can manufacture two products A and B. The profit on a unit of A is `.80 and of B is `.40. The maximum demand of A is 6 units per day and of B is 8 units. The manufacturer has set up a goal of achieving a profit of `.640 per day. Formulate the problem as goal programming and solve it. (15)

III a) Explain the following terms in inventory: setup cost, holding cost, lead time,  optimal cost and stock out cost.

(or)

1. b) Explain the term Price Break. Is it advisable to accept it always? (5)

1. c) Group the items given below into an ABC classification.

 Item No. Units Unit cost in Rs. 1 7000 5 2 2400 3 3 1500 10 4 600 22 5 3800 1.50 6 40000 0.50 7 60000 0.20 8 3000 3.50 9 300 8.00 10 29000 0.40 11 11500 7.10 12 4100 6.20

Explain by graphical representation.

(or)

1. d) (i) A company operating 50 weeks in a year is concerned about its stock of

copper cables. One meter costs `.240 and there is a demand for 8000 meters per

week. The setup cost is `.2,700 and the holding cost is  25 % of  one meter cost.

Assuming no shortages are allowed, find the optimal inventory policy. Also find

the number of orders and total inventory cost.

(ii) Plastic drums are produced at the rate of 50 items per day. The demand

occurs at the rate of 25 items per day. If the setup cost is `.1000 per setup and

holding cost is `.1.00 per unit of item per day find the economic lot size for one

run, assuming that the shortages are not allowed. Also find the time of cycle

and the minimum total cost of one production run.                                   (8+7)

1. a) Explain optimistic time and pessimistic time in network model.

(or)

1. b) Explain Kendall’s notation for representing queuing models.       (5)

1. c) Use Branch and Bound technique to solve the following:

(15)

(or)

1. d) With usual notation show that the probability distribution of queue length is

given by  where .

V a) Write Kuhn-Tucker conditions for a quadratic programming problem.

(or)

1. b) State Wolfe’s algorithm. (5)

1. c) Using Kuhn-Tucker conditions

(15)

(or)

1. d) State the special features of dynamic programming technique. Find the shortest

route for traveling from city 1 to 10 using dynamic programming technique.

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# LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

M.A. DEGREE EXAMINATION – ECONOMICS

SECOND SEMESTER – APRIL 2003

## ST 2900 / s 872  –  OPERATIONS RESEARCH

30.04.2003

1.00 – 4.00                                                                                                       Max : 100 Marks

### PART – A                                         (5´ 4=20 marks)

1. Define Operations Research and highlight any four significant features.
2. How will you define basic solution to a system of ‘m’ simultaneous linear equations in ‘n’ unknowns (m < n)?
3. State the transportation problem mathematically as a linear programming problem.
4. Define : (i) Game  (ii)  Fair game (iii)  Strictly determinable game  (iv)  Saddle point.
5. Using dominance method solve the game:

Player B

Player A

1. Write a note on (i) CPM  (ii)
2. Define : (i) Setup cost  (ii)  Holding cost (iii)  Shortage cost  (iv)  Lead time.

### PART – B                                       (4´ 10=40 marks)

1. Find all the basic feasible solutions of the equations:.
2. Use graphical method to solve the following:

Maximize

Subject to the constraints :

1. Write the simplex algorithm used in finding an optimum solution to a linear programming problem.
2. Explain the ABC Inventory system.

1. Four professors are each capable of teaching any one of four different courses. Class preparation time in hours for different topics varies from professor to professor and is given in the table below. Each professor is assigned only one course. Determine an assignment schedule so as to minimize the total course preparation time for all courses:

 Professor Linear Programmes Queueing Theory Dynamic Programme Regression Analysis A 2 10 9 7 B 15 4 14 8 C 13 14 16 11 D 4 15 13 9
1. A small maintenance project consists of the following ten jobs whose precedence relations are identified by their node number:
 Job (i, j) (1, 2) (2, 3) (2, 4) (3, 5) (3, 6) Duration (days) 2 3 5 4 1 Job (i, j) (4, 6) (4, 7) (5, 8) (6, 8) (7,8) Duration (days) 6 2 8 7 4
• Draw an arrow diagram.
• Find the critical path.
1. Explain multiple item static model with storage limitation.

### PART – C                                         (2´20=40 marks)

1. Use Big M method to

Minimize

Subject to the constraints :

1. Consider the following transportation table showing production and transportation costs along with the supply and demand positions of factories/distribution centres:

 M1 M2 M3 M4 Supply F1 4 6 8 13 500 F2 13 11 10 8 700 F3 14 4 10 13 300 F4 9 11 13 3 500 Demand 250 350 1050 200

Find an initial basic feasible solution by using VAM.

• Find an optimal solution for the above problem.
1. Solve the game graphically :

Player B

Player  A

1. Explain single item static model with one price break with the necessary diagram in detail.

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## Loyola College B.Sc. Statistics April 2006 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

 AC 23

SIXTH SEMESTER – APRIL 2006

# ST 6601 – OPERATIONS RESEARCH

(Also equivalent to STA 601)

Date & Time : 21-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

SECTION A  (10 x 2 =20)

Answer ALL the questions.  Each carries two Marks

1. Define saddle point of a pay-off matrix
2. How many basic cells will be there in a starting solution where there are 4 destinations and 3 origins in a transportation problem ?
3. Distinguish between “pure” and “mixed” strategies
4. What is a minimal spanning tree ?
5. How do you identify the critical activities of a project ?
6. Give the formula for free float and total float corresponding to an activity defined by arcs i and j
7. How will you conclude the presence of more than one optima for an LPP ?
8. What is an unbalanced transportation problem ?
9. Give the fundamental difference between PERT and CPM
10. In what way Floyd’s algorithm is superior to Dijkstra’s algorithm ?

SECTION B (5 x 8 =40)

Answer any FIVE of the following. Each carries EIGHT marks

1. Diamond is an aspiring young student of Loyola College. He realizes that “all work and no play makes him a dull boy”. As a result, Diamond wants to apportion his available time of about 10 hours a day between work and play. He estimates that play is twice as much fun as work. He also wants to study at least as much as he plays. However, Diamond realizes that if he is going to get all his homework assignments done, he can not play more than 4 hours a day. How should Diamond allocate his time to maximize his pleasure from both work and play ?

1. Comment on the solution of the following LPP

Maximize

Subject to

1. Specify the range for value of the game in the following case assuming that the payoff is for player A

B1 B2       B3       B4

A1       1          9          6          0

A2       2          3          8          4

A3       -5         -2         10        -3

A4       7          4          -2         -5

1. Use the Hungarian method to solve the following Assignment Problem

\$3  \$9        \$2        \$3        \$7

\$6  \$1        \$5        \$6        \$6

\$9  \$4        \$7        \$10      \$3

\$2  \$5        \$4        \$2        \$1

\$9  \$6        \$2        \$4        \$5

1. Write the following problem in standard form :

Maximize

Subject to

unrestricted

and also form the initial simplex table (No need to solve the problem)

1. Obtain the minimal spanning tree corresponding to the following network

3

1

6

4

9

5

10

7             5          8

3

1. Draw the time schedule for the following network and identify red flagged activities

 25
 5

B                                                             F

1. Express a transportation problem as Linear programming problem.

SECTION C    (2x 20=40)

Answer any TWO of the following. Each carries TWENTY marks.

1. Solve the following problem using big-M method

Maximize

subject to

1. Solve the following network problem using Floyd’s algorithm

5

3

6                      4

10

15

1. Solve the following 2 x 4 game graphically

B1       B2       B3       B4

A1       2          2          3          -1

A2       4          3          2          6

1. Find the critical path corresponding to the following project network and draw the time schedule

`               A  5                                         J 7                           K 3

C 8

B 10                       F 10                                        L 8

D 9                   E 4

I 1                                                  H 5                       M 4

G 3

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## Loyola College B.Sc. Statistics April 2007 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

 AC 23

SIXTH SEMESTER – APRIL 2007

# ST 6601 – OPERATIONS RESEARCH

Date & Time: 18/04/2007 / 9:00 – 12:00 Dept. No.                                             Max. : 100 Marks

SECTION A

1. Define: Basic Feasible Solution.
2. Mention any two limitations in solving a LPP by graphical method.
3. Express the following LPP in its standard form:

Maximize Z = 3x1 + 2x2

subject to x1 + x2 ≤ 2; 5x1 + 4x2 ≥4; x1,x2 ≥ 0;

1. What is the difference between Big-M and Two Phase method?
2. What is an unbalanced Transportation Problem?
3. Comment on the following statement:

“Assignment problem is a particular case of Transportation problem”.

1. Define: Activity and Node with reference to network analysis.
2. Draw the network, given the following precedence relationships:

Event           :   1          2,3       4          5          6          7

Preceded by:   —          1          2,3       3          4,5       5,6

1. Verify whether saddle point exists for the following game:

Player B

Player A          B1        B2        B3

A1        5          8          1

A2        10        16        13

A3        25        22        20

1. Give the Laplace criteria for making decisions under uncertainty.

SECTION B

1. Show graphically that the maximum or minimum values of the objective function for the following problem are same:

Maximize (Minimize) Z = 5x1 + 3x2

Subject to:           x1 + x2 ≤ 6;

2x1 + 3x2 ≥ 3;

x1 ≥ 3;

x2 ≥ 3;

x1,x2 ≥ 0;

1. Use Big M- method to maximize Z = 2x1 + 3x2 subject to the constraints:

x1 + 2x2 ≤ 4;       x1 + x2 =3;     x1,x2 ≥ 0

1. Explain the MODI algorithm to obtain an optimum solution for a transportation problem.

1. Solve the following assignment problem.

 1 2 3 4 5 A 3 8 2 10 3 B 8 7 2 9 7 C 6 4 2 7 5 D 8 4 2 3 5 E 9 10 6 9 10

1. Explain the Floyd’s method of finding the shortest route between any two given nodes.
2. Draw the network based on the following information and determine the critical path.

Activity           Duration (in days)                   Activity           Duration (in days)

(1,2)                             3                                  (3,4)                             3

(1,3)                             1                                  (3,7)                             10

(1,4)                             15                                (4,5)                             10

(1,6)                             7                                  (4,7)                             22

(2,3)                             8                                  (5,6)                             5

(2,5)                             10                                (5,7)                             12

(6,7)                             7

1. A farmer wants to decide which of the three crops he should plant on his 100 acre farm. The profit from each is dependent on the rainfall during the growing season. The farmer has categorized the amount of rainfall as high, medium and low. His estimated profit for each crop is shown in the table below:

Rainfall           Crop A                        Crop B                        Crop C

High                8000                3500                5000

Medium           4500                4500                5000

Low                 2000                5000                4000

If the farmer wishes to plant only one crop, decide which should be his ‘best

Crop’ using Maximin, Savage and Hurwicz (α = 0.5) rule

1. Use simplex method to find the inverse of the following matrix

A =

SECTION C

1. a.) Explain the mathematical formulation of a Linear Programming Problem.

b.) Use two phase method to verify that there does not exist a feasible solution

to the following LPP:

Maximize z = 2x1 + 3x2 + 5x3 subject to

3x1 + 10x2 +5x3 ≤ 15;

33x1 – 10x2 + 9x3 ≤ 33;

x1 + 2x2 + x3 ≥ 4;

x1 , x2 , x3 ≥ 0.

1. Obtain an optimum solution to the following transportation problem where the entries denote the unit cost of transporting commodities from source to destination:
 From↓To→ I II III IV Supply A 2 3 11 7 6 B 1 0 6 1 1 C 5 8 15 9 10 Demand 7 5 3 2

Find initial solution by least cost method and Vogle’s method and use the best

among them as the starting solution.

1. a.) Solve the following traveling salesman problem so as to minimize the total

distance traveled:

To

From           A         B         C         D         E

A         —        20        4          10        —

B         20        —        5          —        10

C         4          5          —        6          6

D         10        —        6          —        20

E          —        10        6          20        —

• Explain the calculations in PERT to identify the critical path of a project.

1. ) Consider the following data that gives the distance between pairs of cities

1, 2…,6:

Route: (1,2)     (1,3)     (2,3)     (2,4)     (2,5)     (3,4)     (3,5)     (4,5)

Distance:   5          1           1          5          2          2          1          3                                           Route:   (3,6)        (4,6)     (5,6)

Distance:   4           4          3

Use Dijkstra’s algorithm to find the shortest route between the cities

i.) 1 and 4    ii.) 1 and 6.

b.) Solve graphically the 2 x 4 game whose pay-off matrix is given below

Player B

Player A          1          3          11        7

8          5          2          5

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## Loyola College B.Sc. Statistics April 2008 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

# NO 29

SIXTH SEMESTER – APRIL 2008

# ST 6601 – OPERATIONS RESEARCH

Date : 21/04/2008                Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION – A

Answer ALL questions:                                                                                (10 x 2 = 20)

1. Define basic feasible solution and optimal solution in a linear programming problem.
2. Rewrite into standard form:
1. What is the role of artificial variables in linear programming?
2. Write the dual of :
1. Explain a transshipment problem
2. What is an unbalanced transportation problem?
3. Distinguish between CPM and PERT in network analysis.
4. Define activity and event in network analysis.
5. Define decision under uncertainity
6. Give an example of a 3 x 3 game without a saddle point.

SECTION – B

Answer any FIVE  questions.                                                                      (5 x 8 = 40)

1. The standard weight of special type of brick is 5kg and it contains two basic ingredients B1 and B2. B1 costs Rs. 5/kg and B2 costs Rs. 8/kg. The strength considerations dictate that the brick contains not more than 4 kgs of B1 and a minimum of 2 kgs of B2. Find graphically the minimum cost of the brick satisfying the above conditions.
2. Show that the following linear programming problem has an unbounded solution.
1. Explain how you will solve a travelling salesman problem.
2. Draw the network diagram given the following activities and find the critical path.

Job                               :           A         B         C         D         E          F          G        H        I             J         K

Time (days)     :           13        8          10        9          11        10          8        6         7          14       18

Immediate                   –           A         B         C         B         E          D,F      E         H         G,I         J

Predecessor     :

1. Explain the maximal flow problem.

1. Solve the following problem for the minimum time:
 I II III IV V A 16 13 17 19 20 B 14 12 13 16 17 C 14 11 12 17 18 D 5 5 8 8 11 E 3 3 8 8 10

1. A decision problem is expressed in the following pay-off table (Rs.’ 000s). What is the maximax and maximin payoff action?

Strategy

 State I II III A – 1880 1840 1800 B – 1620 1600 1640 C – 1400 1420 1720

1. How will you solve a game problem using linear programming?

SECTION – C

Answer any TWO   questions.                                                                      (2 x 20 = 40)

1. a) Discuss in brief duality in linear programming (8)
2. b) Solve using dual simplex method:

1. a) Explain Vogel’s method of finding an initial solution to a transportation problem.      (8)
1. b) Use least cost method to find initial basic feasible solution and then find the minimum transportation cost for the following problem.

Destinations

 D1 D2 D3 D4 Supply 01 1 2 1 4 30 Origins 02 3 3 2 1 50 03 4 2 5 9 20 Demand 20 40 30 10 100
1. a) Explain  i) total float ii) free float and

iii)independent float in network analysis (8)

1. b) The time estimates in weeks for the activities of a PERT network are given below:

 Activity: 1-2 1-3 1-4 2-5 3-5 4-6 5-6 Opt.time: 1 1 2 1 2 2 3 Most likely time 1 4 2 1 5 5 6 Pessimistic time: 7 7 8 1 14 8 5
1. Draw the network diagram and find the critical path
2. Find the expected length and variance of the critical path.
• What is the probability that the project is completed in 13 weeks? (12)
1.  a) Explain Laplace and Hurintz criteria in decision theory
1. b) Solve the following game graphically.

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## Loyola College B.Sc. Statistics April 2009 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

 YB 29

SIXTH SEMESTER – April 2009

# ST 6601 – OPERATIONS RESEARCH

Date & Time: 21/04/2009 / 9:00 – 12:00    Dept. No.                                                      Max. : 100 Marks

PART A

Answer ALL questions:                                                                                 (10 x 2 = 20)

1. Define Operations Research.
2. How will you identify an infeasible solution in linear programming?
3. Distinguish between Big M and Two phase methods.
4. Write the dual of

Max z = x1 + x2

subject to

x1, x2 ≥ 0

1. What is a traveling salesman problem?
2. Define a transhipment problem.
3. Define a critical path.
4. What are the time estimates used in PERT?
5. What is meant by decision under uncertainty?
6. Define a two-person zero sum game

PART B

Answer any FIVE questions:                                                                                     (5 x 8 = 40)

1. A person requires 10, 12 and 12 units of chemicals A, B and C respectively for his garden. The liquid product contains 5, 2 and 1 units of A, B and C respectively per jar. A dry product contains 1, 2 and 4 units of A, B and C per carton. If the liquid product sells for Rs 3 per jar and dry product sells for Rs 2 per carton, how many of each should he purchase in order to minimize the cost using graphical method?
2. How will you identify unbounded and alternate solutions in a linear programming problem?
3. Explain the dual simplex method of solving a linear programming problem.
4. Four professors are each capable of teaching any one of the following courses. The table below shows the class preparation time in hours by the professors on the topics. Find the assignment of the courses to the professors to minimize the preparation time.
 Professor Linear Programming Queuing Theory Dynamic Programming Inventory Control A 2 10 9 7 B 15 4 14 8 C 13 14 16 11 D 4 15 13 9

1. Distinguish between PERT and CPM.

1. For the following network determine the maximum flow and optimum flow in each link:

1. Find the optimum strategy and value of the game for which there is no saddle point given the payoff matrix of  A:

1. Explain minimax and savage criteria in decision making with suitable examples.

PART – C

Answer any TWO questions:                                                                         (2 x 20 = 40)

1. (a) Explain primal-dual relationship in linear programming.

(b) Use penalty method to

Max z = 2x1 + x2 + 3x3

subject to

x1 + x2 + 2x3 ≤ 5

2x1 + 3x2 + 4x3 = 12

x1 , x2 , x3 ≥ 0

1. (a) Solve the following transportation problem with cost coefficients, demand and supply as

given below

(b) Solve using simplex method

Max  Z = 10x1 +  x2 + 2x3

subject to

x1 + x2 – 2x3  £10

4x1 + x2 + x3  £ 20

x1, x2, x3 ≥ 0

1. (a) State the rules for drawing the network diagram.

(b) The following table lists the jobs in a network along with the time estimates.

 Job 1 – 2 1 – 6 2 – 3 2 – 4 3 – 5 4 – 5 6 – 7 5 – 8 7 – 8 Optimistic time (days) 3 2 6 2 5 3 3 1 4 Most likely time (days) 6 5 12 5 11 6 9 4 19 Pessimistic time (days) 15 14 30 8 17 15 27 7 28

(i) Draw the network diagram

(ii) Calculate the length and variance of the critical path

(iii)Find the probability of completing the project before 31 days.

1. (a) A business man has three alternatives each of which can be followed by any one of the four possible events. The conditional payoff in rupees for each action – event combination are as given below:
 Alternative Payoff A B C D X 8 0 10 6 Y -4 12 18 -2 Z 14 6 0 8

Determine which alternative should the business man choose if he adopts (i) Hurwicz Criterion, the degree of optimism being 0.7 (ii) Laplace Criterion

(b) Solve following game graphically:

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## Loyola College B.Sc. Statistics April 2011 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B. Ss. DEGREE EXAMINATION – STATISTICS

SIXTH SEMESTER – April 2011

ST6604 / ST6601 – OPERATIONS RESEARCH

Time: 3 hrs                                                                                                    Max.:100 Marks

PART A

Answer ALL questions:                                                                                                             (10 x 2  = 20)

1. Define Basic feasible solution.
2. What do you mean by Slack variable?
3. When do we go for artificial variables?
4. What are the two rules for selecting the entering and leaving variables?
5. Write the Standard form for the given Linear programming problem.
1. What do you understand by transportation problem?
2. What is a traveling salesman problem?
3. Define a Two- person zero sum game.
4. Distinguish between PERT and CPM.
5. What do you mean by Minimax criterion?

PART B

Answer any FIVE questions:                                                                                                       (5 x 8  = 40)

1. Let us assume that you have inherited Rs. 1,00, 000 from your father – in – law that can be inversted in a combination of only two stock portfolios, with the maximum investment allowed in either portfolio set at Rs. 75, 000. The first portfolio has an average of 10%, whereas the second has 20%. In terms of risk factors associated with these portfolios, the first has a risk rating of 4 (on a scale of 0 to 10) and the second has a risk rating of 9. since you wish to maximize yours returns, you will not accept an average rate of return below 12% or a risk factor above 6. Hence, you face the important question. How much should you invest in each portfolio?

Formulate this as a Linear Programming Problem.

1. Explain the Two Phase method of solving an LPP.

1. Find the initial basic feasible solution for the following transportation problem using Vogel’s Approximation method.
 From To 9 12 9 6 9 10 5 7 3 7 7 5 5 6 6 5 9 11 3 11 2 6 8 11 2 2 10 9 4 4 6 2 4 2
1. Solve the following linear programming problem by graphical method:

Maximize z = x1  + 2x2

Subject to

x1   + 2x2 £ 15

2x1  +   x2 £ 20

x1  + x2 ³ 1

x1,  x2  ³ 0

1. Consider a firm having two factories. The firm is to ship its products from the factories to three retail stores. The number of units available at factories X and Y are 200 and 300 respectively, while those demanded at retail stores A, B and C are 100, 150 and 250 respectively. Rather than shipping directly from factories to retail stores, it is added to investigate the possibility of transhipment. The transportation cost (in rupees) per unit is given below:
 Factory Retail Store X Y A B C Factory X 0 8 7 8 9 Y 6 0 5 4 3 Retail Store A 7 2 0 5 1 B 1 5 1 0 4 C 8 9 7 8 0

Find the optimal shipping schedule.

1. Solve the following game graphically:

Player B

Player A

1. Determine the total, free and independent floats and identify the critical path for the given project network.

1. Differentiate between Simplex method and Dual Simplex method.

PART C

Answer any TWO questions:                                                                        (2 x 20  = 40)

1. a) Give the Dual for the following Primal.
 Maximize z = 5x1 + 2x2 Subject to 6x1 +   x2  ≥ 6 4x1 + 3x2 ≥ 12 x1 + 2x2 ≥ 4 x1,     x2  ≥ 0

19 b) Solve the given problem using Big M method.

Minimize z = x1 + 2x2 + 3x3 –  x4

Subject to

x1 + 2x2 + 3x3 = 15

2x1 +  x2 + 5x3 = 10

x1 + 2x2 + x3 + x= 10

x1, x2, x3, x4 ≥ 0                                                                  (5 + 15)

1. a)  Reduce a  m x n game to a linear programming problem.:

b) Solve the following assignment problem to find the minimum total expected cost.

 Men I II III IV Job A 42 35 28 21 B 30 25 20 15 C 30 25 20 15 D 24 20 16 12

21. a) State the rules for drawing the network diagram.

21. b) A project consists of eight activities with the following relevant information:

 Activity Immediate Predecessor Estimated duration (days) Optimistic Most likely Pessimistic A – 1 1 7 B – 1 4 7 C – 2 2 8 D A 1 1 1 E B 2 5 14 F C 2 5 8 G D, E 3 6 15 H F, G 1 2 3

(i)                            Draw the PERT network and find out the expected project completion time and variance

(ii)                          What duration will have 95% confidence for project completion?

\$\$\$\$\$\$\$\$

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## Loyola College B.Sc. Statistics April 2012 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B. Ss. DEGREE EXAMINATION – STATISTICS

SIXTH SEMESTER – April 2012

ST6604 / ST6601 – OPERATIONS RESEARCH

Date: 18-04-2012                   Dept. No.                                                            Max.:100 Marks

Time: 1.00 – 4.00

PART – A

Answer ALL questions:                                                                       (10 x 2  = 20)

1. What are the different phases of Operations Research?
2. What is a degenerate solution?
3. Give the dual for the following primal:
1. When can dual simplex method be applied to a LPP?
2. Define Two- person zero sum game.
3. Mention the two phases in the two-phase method.
4. When is there a necessity for artificial variables?
5. What are the methods used to solve a mixed strategy game?
6. What is CPM?
7. What is Minimax criterion?

PART – B

Answer any FIVE questions:                                                               (5 x 8  = 40)

1. Explain simplex method for solving an LPP.
2. The manager of an oil refinery must decide on the optimum mix of two possible blending processes of which the input and output production runs are as follows:
 Process Input Output Crude A Crude B Gasoline X Gasoline Y 1 6 4 6 9 2 5 6 5 5

The maximum amounts available of crudes A & B are 250 units and 200 units respectively. Market demand shows that at least 150 units of gasoline X and 130 units of gasoline Y must be produced. The profits per production run from process 1 and process 2 are Rs. 4 and Rs. 5 respectively. Formulate the problem for maximizing the profit.

1. What are the steps involved in solving an unbalanced transportation problem to get the optimum solution?

1. A departmental head has four subordinates and four tasks to be performed. The subordinates differ in efficiency and the tasks differ in their intrinsic difficulty. His estimate of the time each man would take to perform each task, is given in the matrix below:
 Tasks Men E F G H A 18 26 17 11 B 13 28 14 26 C 38 19 18 15 D 19 26 24 10

How should the tasks be allocated, one to a man, so as to minimize the total man-hours?

1. Solve the following graphically:

Player B

Player A

1. Differentiate between Simplex method and Dual Simplex method.
2. Explain Laplace and Savage criterion in detail.
3. Explain PERT algorithm in detail.

PART – C

Answer any TWO Questions:                                                           ( 2 x 20 = 40 marks)

1. a) Explain Primal – Dual Relationship.                                                          (5)

1. b) Use Duel Simplex method to solve the following LPP.                            (15)

Minimize z = 6x1 + 7x2 + 3x3 + 5x4

Subject to

5x1 + 6x2 – 3x3 + 4x4 ≥ 12

x2 – 5x3 –  6x≥ 10

2x1 + 5x2 + x3 +   x4   ≥  8

x1, x2, x3, x4 ≥ 0

1. a) Show that for any zero-sum two-person game where there is no saddle point and for

which A’s payoff matrix is

(15)

the optimal strategies (x1, x2) and (y1, y2) for A and B respectively are determined by

What is the value of the game?

1.             20. b) What is the principle of dominance in game theory?                                          (5)

1. Find the initial basic feasible solution for the given transportation problem using:

(i)  North West Carner rule        (ii) Vogels method          (iii) Least Cost method.

 From To Available A B C I 50 30 220 1 II 90 45 170 3 III 250 200 50 4 Requirement 4 2 2

And obtain the optimal solution.

1. a) What are the rules for constructing a network diagram?                            (5)

22.  b) A project consists of eight activities with the following relevant information:

 Activity Immediate Predecessor Estimated duration (days) Optimistic Most likely Pessimistic A – 1 1 7 B – 1 4 7 C – 2 2 8 D A 1 1 1 E B 2 5 14 F C 2 5 8 G D, E 3 6 15 H F, G 1 2 3

(i)  Draw the PERT network and find out the expected project completion time.

(ii) What duration will have 95% confidence for project completion?

(iii) If the average duration for activity F increases to 14 days, what will be its effect on the expected project completion time, which will have 95% confidence?

(15)

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## Loyola College B.Sc. Mathematics April 2007 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 CV 16

B.Sc.  DEGREE EXAMINATION –MATHEMATICS

FIFTH SEMESTER – APRIL 2007

MT 5504 – OPERATIONS RESEARCH

Date & Time: 03/05/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

SECTION –A

## Answer All:                                                                                     2 x 10 = 20

1. Define Operations Research.
2. What are the three methods to find Initial Basic Feasible solution in Transportation problem ?
3. Solve the Transportation problem by Least Cost Method.
 A1 A2 A3 Supply B1 4 6 2 5 B2 3 1 5 15 B3 4 5 3 15 Demand 10 10 10

1. Solve the game:
 2 1 4 1 4 3 2 2 6

1. Define Unbalanced situation in Transportation problem.
2. Define Feasible Solution.
3. Solve the Assignment problem
 3 7 5 4 7 2 5 4 6

1. What is Dummy activity in Network problem ?
2. Define Optimistic Time Estimate.
3. Define Economic Order Quantity.

SECTION –B

## Answer any five:                                                                              5x 8 = 40

1. Find the Initial Basic Feasible solution in Transportation problem using i) North West Corner Rule ii) Least Cost Method.
 A1 A2 A3 A4 Supply B1 4 2 1 3 20 B2 8 4 2 4 20 B3 1 2 3 4 30 B4 5 2 4 6 20 Demand 10 30 10 40

1. Using Graphical method solve the Linear Programming Problem

Max z = 2x1+4x2    subject to the constraints

2x1+4x≤ 5 ,       2x1+4x2 ≤ 4,         x1, x2 ≥ 0.

1. Solve the Assignment problem

 M1 M2 M3 M4 M5 J1 9 22 58 11 19 J2 43 78 72 50 63 J3 41 28 91 37 45 J4 74 42 27 49 39 J5 36 11 57 22 25

1. Solve using Matrix Oddment method

 -1 2 1 1 -2 2 3 4 -3

1. Define critical path and draw the Network diagram for

Activity:   A    B   C    D    E   F    G    H     I      J       K

Immediate predecessor:    –     –    –     A    B   B    C    D    E    H,I    F,G

1. Solve using Dominance property
 1 7 3 4 5 6 4 5 7 2 0 3

1. Solve the Transportation problem
 A1 A2 A3 A4 Supply B1 1 2 1 4 30 B2 3 3 2 1 50 B3 4 2 5 9 20 Demand 20 40 30 10

1. The probability distribution of monthly sales of certain item is as follows:

Number of items:    0         1          2        3         4          5          6

P(d)    :   0.02    0.05    0.30    0.27    0.20     0.10     0.06

The cost of carrying inventory  is Rs 10 per unit per month . Find the

shortage cost for one item for one unit of time.

(P.T.O)

SECTION –C

## Answer any two:                                                                              2x 20 = 40

1. Solve the following Linear Programming Problem using Simplex method

Max z = 3x1+2x2    subject to the constraints

x1+2x≤ 6 ,       2x1+x2 ≤ 8,       -x1+x2 ≤ 1,         x2 ≤ 2,        x1, x2 ≥ 0.         (20)

20 a) Solve the Transportation problem to maximize the profit

 A1 A2 A3 A4 Supply B1 40 25 22 33 100 B2 44 35 30 30 30 B3 38 38 38 30 70 Demand 40 20 60 30

1. b) Solve the following traveling sales man problem

 A B C D E A – 3 6 2 3 B 3 – 5 2 3 C 6 5 – 6 4 D 2 2 6 – 6 E 3 3 4 6 –

(10+10)

21 a) Solve the game graphically

 1 0 4 -1 -1 1 2 5

1. b) The annual demand for an item is 3200 units, the unit cost is Rs 6 and inventory

Carrying charges 25% per annum. If the cost of one procurement is Rs 150. Find

1. i) Economic Order Quantity ii) Time between two consecutive orders

iii) Number of orders per year             iv) The optimal total cost                      (10+10)

22 a) Draw the Network diagram ,the Critical path ,the project duration and the total float

for the following activities

Activity:    1-2       2-3      3-4        3-7       4-5        4-7       5-6        6-7

Duration:     3          4          4           4           2          2           3           2

1. b) What is the probability that the project will be completed in 27 days? Draw the

network diagram also .

Activity:    1-2       1-3      1-4        2-5      2-6         3-6        4-7       5-7        6-7

T0          :      3          2          6           2          5            3           3           1           2

Tm         :      6          5         12          5         11           6           9           4           5

Tp          :      15       14        30          8         17          15         27          7           8

(10+10)

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## Loyola College B.Sc. Mathematics Nov 2008 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

# AB 21

FIFTH SEMESTER – November 2008

# MT 5504 – OPERATIONS RESEARCH

Date : 07-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION A

Answer all the questions. (10 X 2 = 20)

1. Define feasible solution of a general Linear Programming Problem.
2. Give the symmetric form of a LPP.
3. State the necessary and sufficient condition for a Transportation problem to have a feasible solution.
4. How can you convert a maximization assignment problem to a minimization problem?
5. What is a two person zero sum game.
6. Define EOQ.
7. Define total float of an activity.
8. What is payoff matrix?
9. What is a project? List the 3 main phases of a project.
10. What are the different types of inventory?

SECTION B

Answer any five questions.(5 X 8 = 40)

1. Solve the following L.P.P by the Graphical method:

Max Z = 3x1 + 2x2 subject to

-2x1 + x2 ≤ 1,

x1≤ 2,

x1 + x2 ≤ 3 and x1, x2 ≥ 0.

1. Find the initial basic feasible solution for the following transportation problem by least cost method.

To                          Supply

From          1          2          1          4          30

3          3          2          1          50

4          2          5          9          20

Demand          20        40        30        10

1. Solve the following 2 X 2 game:

B

A         5          1

4          0

1. Construct the network for the project whose activities are given below and compute the total, free and independent float of each activity and hence determine the critical path and the project duration:

Activity: 0-1    1-2       1-3       2-4       2-5       3-4       3-6       4-7       5-7       6-7

Duration:3       8          12        6          3          3          8          5          3          8

1. For an item, the production is instantaneous. The storage cost of one item is Re. one per month and the set up cost is 25 per run. If the demand is 200 units per month, find the optimum quantity to be produced per set-up and hence determine the total cost of storage and set-up per month.

1. A commodity is to be supplied at a constant rate of 200 units per day. Supplies for any amounts can be had at any required time, but each ordering costs Rs. 50. Cost of holding the commodity in inventory is Rs. 2.00 per unit per day while the delay in the supply of the items induces a penalty of Rs.10 per unit per delay of one day. Formulate the average cost function of this situation and find the optimal policy (q,t) where t is the reorder cycle period and q is the inventory level after reorder. What should be the best policy if the penalty cost becomes infinite?

1. The processing time in hours for the jobs when allocated to the different machines is indicated below. Assign the machines for the jobs so that the total processing time is minimum.
 Machines Jobs M1 M2 M3 M4 M5 J1 9 22 58 11 19 J2 43 78 72 50 63 J3 41 28 91 37 45 J4 74 42 27 49 39 J5 36 11 57 22 25

1. Solve the following:

Maximize

Z=15x1 + 6x2+ 9x3 + 2x4

Subject to 2x1 + x2+ 5x3+6x4 ≤ 20

3x1+x2+3x3+25x4 ≤ 24

7x1 + x4 ≤70

x1, x2, x3, x4 ≥ 0.

SECTION C

Answer any two questions: (2 X 20 = 40)

19a) Use Penalty method to solve Z=2x1 + x2+ x3

Subject to 4x1 + 6x2+ 3x3 ≤ 8

3x1– 6x2– 4x3 ≤ 1

2x1 + 3x2 – 5x3  ≥ 4

x1, x2, x3 ≥ 0.

19b) A person wants to decide the constituents of a diet which will fulfill his daily requirements of essential nutrition at the minimum cost. The choice is to be made from four different types of foods. The yields per unit of these foods are given in the following table:

 Food type Yield/unit Cost / unit (Rs) Proteins Fats Carbohydrates 1 3 2 6 45 2 4 2 4 40 3 8 7 7 85 4 6 5 4 65 Minimum requirement 800 200 700

Formulate the linear programming model for the [problem.

20a) Solve the following Transportation problem to minimize the total cost of transportation:

 Destination origin D1 D2 D3 D4 supply O1 14 56 48 27 70 O2 82 35 21 81 47 O3 99 31 71 63 93 Demand 70 35 45 60 210

20b) Solve the following Travelling salesman problem:

 A B C D From A – 46 16 40 B 41 – 50 40 C 82 32 – 60 D 40 40 36 –

21a) Three time estimates (in months) of all activities of a project are given below:

 Time in months Activity a m b 1-2 0.8 1.0 1.2 2-3 3.7 5.6 9.9 2-4 6.2 6.6 15.4 3-4 2.1 2.7 6.1 4-5 0.8 3.4 3.6 5-6 0.9 1.0 1.1

Find the expected duration and standard deviation of each activity,

Construct the project network,

Determine the critical path, expected project length and expected variance of the project length.

21b) For the pay-off matrix given below, decide optimum strategies for A and B

B

• 2

A         1          200      80

2          110      170

22a) Explain the purchase inventory model with n-price breaks.

22b) Find the optimal order quantity for a product for which the price break is as follows:

Quantity                                  Unit cost

0 ≤ Q1 < 50                             Rs. 10

50 ≤Q2<100                            Rs. 9

100 ≤Q3                                   Rs. 8

The monthly demand for the product is 200 units, the cost of the storage is 25% of the unit cost and ordering cost is Rs. 20 per order.

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## Loyola College B.Sc. Mathematics April 2009 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

 ZA 31

FIFTH SEMESTER – April 2009

# MT 5507 / 5504 – OPERATIONS RESEARCH

Date & Time: 17/04/2009 / 9:00 – 12:00    Dept. No.                                                       Max. : 100 Marks

SECTION A

Answer  ALL  Questions                                                                               (10 x 2 = 20)

1. Write down any two uses of Operations Research.
2. Define Slack variables.
3. Which is the necessary and sufficient condition for the transportation problem to have

a feasible solution?

1. Describe a traveling salesman problem.
2. What is meant by mixed strategy?
3. What is the value of the game?
4. Define Total float
5. Explain: Minimal Spanning tree problem?
6. Give any two reasons for maintaining inventory
7. Define Recorder level.

SECTION B

Answer ANY FIVE Questions.                                                                     (5 x 8 = 40)

1. Solve the following Linear Programming Problem by graphical method

Maximize  Z = 5x1 + 8x2

Subject to:

15x1 + 10x≤  180

10x1 + 20x≤  200

15x1 + 20x≤  210

and x1, x≥ 0

1. Use simplex method to solve the following LPP

Maximize  Z = 4x1 + 10x2

Subject to:

2x1 + x≤  50

2x1 + 5x≤  100

2x1 + 3x≤  90

and x1, x≥ 0

1. Find the initial basic feasible solution for the following transportation problem
 D1 D2 D3 D4 Supply S1 21 16 25 13 11 S2 17 18 14 23 13 S3 32 27 18 41 19 Demand 6 10 12 15

1. Write the algorithm for solving Assignment problem.

1. Solve the following game using dominance property.

 I II III Row mini. I 1 7 2 1 II 6 2 7 2 III 6 1 6 1 Column max. 6 7 7

1. Draw the network for the project whose activities with their predecessor relationships

are given below:

A, C, D can start simultaneously; E >B, C;  F, G >D;  H, I > E ,F ; J >I, G ;

K > H; B > A.

1. The annual demand of a product is 10,000 units, each unit cost Rs.100 if orders

placed in quantities below 200 units but for orders of 200 or above, the price is Rs.95,

the annual inventory holding cost is 10% of the value of the item, and the ordering

cost is Rs.5 per order. Find the economic lot size.

1. The demand for an item in a company is 18,000 units per year and the company can

produce the item at a rate of 3000 per month. The cost of one set up is Rs. 500 and

the holding cost of one unit per month is 15 paise. The shortage cost of one unit is

Rs. 20 per month. Determine the optimum manufacturing quantity and the number

of shortage. Also determine the manufacturing time and time between setups.

SECTION C

Answer ANY TWO Questions.                                                                      (2 x 20 = 40)

1. Solve the following Linear Programming Problem by Dual Simplex method

Minimize  Z = x1 + x2

Subject to:

2x1 + x≥ 2

-x1 – x2 ≥ 1

and x1, x≥ 0

1. (a) Solve the following transportation problem using Least Cost Method to find the

Initial Basic Feasible solution.

 A1 A2 A3 A4 A5 Supply B1 4 1 2 6 9 100 B2 1 4 7 3 8 120 B3 7 2 4 7 7 120 Demand 40 20 70 90 90

.

(b)Solve the following traveling sales man problem

 M1 M2 M3 M4 M5 J1 9 22 58 11 19 J2 43 78 72 50 63 J3 41 28 91 37 45 J4 74 42 27 49 39 J5 36 11 57 22 25

(10 + 10)

1. (a) Solve the following game graphically
 B1 B2 B3 B4 A1 1 0 4 -1 A2 -1 1 2 5

(b) In a game of matching points with 2 players suppose A wins one unit value when

there are 2 heads, wins nothing when there are 2 tails, and loses  ½ unit value when

there are 1 head and 1 tail. Determine the pay-off matrix, the best strategy for each

player and the value of the game.                                                        (10+10)

22    (a) Draw the network, determine the critical path , project duration and the total float for the

following activities .

 Activity 1-2 2-3 3-4 3-7 4-5 4-7 5-6 6-7 Duration 3 4 4 4 2 2 3 2

(b) ABC manufacturing company purchases 9,000 parts of a machine for its annual

requirement, ordering one month’s usage at a time. Each part costs Rs.20.

The ordering cost per order is Rs.15, and the carrying charges are 15% of the

average inventory per year.

You have been asked to suggest a more economical purchasing policy for the

company. What advice would you offer and how much would it save the company

per year?                                                                                  (10+10)

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## Loyola College B.Sc. Mathematics April 2012 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2012

# MT 5507/MT 5504 – OPERATIONS RESEARCH

Date : 30-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART  – A

1. A firm plans to purchase at least 200 quintals of scrap containing high quality metal X and low quality metal Y. It decides that the scrap to be purchased must contain at least 100 quintals of X and not more than 35 quintals of Y. The firm can purchase the scrap from two suppliers (A and B) in unlimited quantities. The percentage of X and Y metals in terms of weight in the scrap supplied by A and B is given below:
 Metals Supplier A Supplier B X 25% 75% Y 10% 20%

The price of A’s scrap is Rs.200 per quintal and that of B is Rs. 400 per quintal. The firm wants to determine the quantities that it should buy from the two suppliers so that total cost is minimized. Construct a LP model.

1. Give at least four different contrasting properties of primal and dual of general LP problems
2. What is meant by unbalanced transportation problem?
3. Name four different methods to solve an assignment problem.
4. Consider the game with the following payoff table:
 Player B Player A B1 B2 A1 2 6 A2 -2 λ

Show that the game is strictly determinable, whatever λ may be.

1. When no saddle point is found in a payoff matrix of a game, how do we find the value of the game?
2. Give an example of weighted graph which is not a tree. Find two different minimal spanning trees of the weighted graph you constructed.
3. Why is in PERT network each activity time assume a Beta-distribution?
4. What are the basic information required for an efficient control of inventory?
5. What is Economic Order Quantity (EOQ) concept?

PART  – B

Answer any FIVE questions:                                                                             (5×8=40 marks)

1. Using Simplex Method, solve the following Problem and give your comments on the solution: Maximize Z= 6x1+4x2, Subject to: x1+x2£  5, x2≥ 8, x1, x2≥ 0.
2. Prove that dual of the dual is the primal.

 Tasks Clerks A B C D 1 4 7 5 6 2 – 8 7 4 3 3 – 5 3 4 6 6 4 2
1. Consider a problem of assigning four clerks to four tasks. The time (hours) required to complete the task is given below:

Clerk 2 cannot be assigned to task A and clerk 3 cannot be assigned to task B. Find all the optimum assignment schedules.

1. A company has three production facilities S1, S2 and S3 with production capacity of 7, 9 and 18 units (in 100s) per week of a product, respectively. These units are to be shipped to four warehouses D1, D2, D3 and D4 with requirement of 5, 6, 7 and 14 units (in 100s) per week, respectively. The transportation costs (in rupees) pre unit between factories to warehouses are given in the table below:
 D1 D2 D3 D4 Capacity S1 19 30 50 10 7 S2 70 30 40 60 9 S3 40 8 70 20 18 Demand 5 8 7 14 34

Use North-West Corner Method to find an initial basic feasible solution to the transportation problem.

1. A soft drink company calculated the market share of two products against its major competitor having three products and found out the impact of additional advertisement in any one of its products against the other
 Company B Company A B1 B2 B3 A1 6 7 15 A2 20 12 10

What is the best strategy for the company as well as the competitor? What is the payoff obtained by the company and the competitor in the long run? Use graphical method to obtain the solution.

1. Solve the following game after reducing it to a 2×2 game
 Player B Player A B1 B2 B3 A1 1 7 2 A2 6 2 7 A3 5 1 6
1. An assembly is to be made from two parts X and Y. Both parts must be turned on a lathe and Y must be polished whereas X need not be polished. The sequence of activities together with their predecessors is given below.
 Activity Description Predecessor  Activity A Open work order – B Get material for X A C Get material for Y A D Turn X on lathe B E Turn Y on lathe B, C F Polish Y E G Assemble X and Y D, F H Pack G

Draw a network diagram of activities for the project.

1. The production department for a company requires 3600 kg of raw material for manufacturing a particular item per year. It has been estimated that the cost of placing an order is Rs. 36 and the cost of carrying inventory is 25% of the investment in the inventories. The price is Rs. 10 per kg. Calculate the optimal lot size, optimal order cycle time, minimum yearly variable inventory cost and minimum yearly total inventory cost.

PART  – C

Answer any TWO questions:                                                                          (2×20=40 marks)

1. (a) Use graphical method and solve the LPP: Maximize Z= 15x1+10x2, Subject to: 4x1+6x2£  360,

3x1£  180, x2≥ 8, 5x2£  200, x1, x2≥ 0.                                                                         (7 marks)

(b) Use simplex method to solve the LPP: Maximize Z= 3x1+5x2+4x3, Subject to:  2x1+3x2£  8,

2x2+5x3£  10, 3x1+2x2+4x3£  15, x1, x2, x3≥ 0.                                                           (13 marks)

1. (a) ABC limited has three production shops supplying a product to five warehouses. The cost of        production varies from shop to shop and cost of transportation form one shop to a warehouse also varies. Each shop has a specific production capacity and each warehouse has certain amount of requirement. The costs of transportation are given below:
 Warehouse Shop I II III IV V Supply A 6 4 4 7 5 100 B 5 6 7 4 8 125 C 3 4 6 3 4 175 Demand 60 80 85 105 70 400

The cost of manufacturing the product at different production shops is given by:

 Shop Variable Cost CCost Fixed Cost A 14 7000 B 16 4000 C 15 5000

Find the optimal quantity to be supplied from each shop to different warehouses at minimum total cost.                                                                                                                           (10 marks)

(b) Two firms A and B make smart phones and tablets. Firm A can make either 150 smart phones in a

week or an equal number of tablets, and make a profit of Rs. 400 per smart phone and Rs. 300 per

tablet. Firm B can, on the other hand, make either 300 smart phones, or 150 smart phones and 150

tablets, or 300 tablets per week. It also has the same profit margin on the two products as A. Each

week there is a market of 150 smart phones and 300 tablets and the manufacturers would share

market in the proportion in which they manufacture a particular product. Write the payoff matrix

of A per week. Obtain graphically A’s and B’s optimal strategies and value of the game.

(10 marks)

1. A small project is composed 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 of the activities in the project (5 marks)
• Find the expected duration and variance for each activity. (5 marks)
• What are the expected project length and critical path? (5 marks)
• Calculate the standard deviation of the project length.                         (5 marks)

1. (a) The annual demand of a product is 10,000 units. Each unit costs Rs.100 if orders are placed in

quantities below 200 units but for orders of 200 or above the price is Rs. 95. The annual inventory

holding costs is 10% of the value of the item and the ordering cost is Rs. 5 per order. Find the

economic lot size.                                                                                                       (8 marks)

(b) A dealer supplies you the following information with regard to a product dealt in by her: Annual

demand=10000 units; Ordering Cost=Rs 10/order; Price=Rs.20/unit; Inventory carrying cost=20%

of the value of the inventory per year.  The dealer is considering the possibility of allowing some

backorder to occur. She has estimated that the annual cost of backordering will be 25% of the

value of inventory.

• What should be the optimum number of units of the product she should buy in one lot?
• What quantity of the product should be allowed to be backordered, if any?
• What would be the maximum quantity of inventory at any time of the year?
• Would you recommend her to allow backordering? If so, what would be the annual cost saving by adopting the policy of backordering?                                                      (12 marks)

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## Loyola College B.Sc. Mathematics Nov 2012 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – NOVEMBER 2012

# MT 5507/MT 5504 – OPERATIONS RESEARCH

Date : 06/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer any ALL questions:                                                                                    (10 x 2 = 20 Marks)

1. Define the following: (i) Basic solution (ii) Basic feasible solution
2. Express the following linear programming problem into standard form:

Maximize

Subject to

3

1. What is a transportation problem?
2. Give the mathematical formulation of an assignment problem.
3. Define a pure strategy in game theory.
5. Define a spanning tree in a network.
6. Define a critical path in a network.
7. What is the Economic order quantity?
8. Differentiate the deterministic and the probabilistic demand inventory models.

PART – B

Answer any FIVE questions:                                                                                               (5 x 8 = 40 Marks)

1. Use the graphical method to solve the following linear programming problem.

Minimize

Subject to

1. Solve the following LPP by dual simplex method.

Maximize

Subject to

1. Determine an initial basic feasible solution to the following transportation problem by Vogel Approximation Method.

Available

Requirement     6      10    12    15

1. Solve the following assignment problem.

Jobs

I         II       III

Men

1. Solve the following game graphically

1. A project consists of a series of tables labeled A, B, …, H, I with the following relationships (W < X, Y means X&Y cannot start until W is completed; X, Y < W means W cannot start until both X&Y are completed). With this notations construct the network diagram having the following constraints: A < D, E; B, D < F; C < G; B < H; F, G < I

1. Determine the critical path of the following network.

1. A particular item has a demand of quantity 9000 units/year. The cost of the one procurement is Rs.100 and the holding cost per unit is Rs.2.40 per year. The replacement is instantaneous and no shortages are allowed. Determine
• the economic lot size
• the number of orders per year
• the time between orders

PART – C

Answer any TWO questions:                                                                             (2 x 20 = 40 Marks)

1. a) Find the optimal solution for the following transportation problem using MODI method.

D1              D2               D3               D4       Supply

 S1 19 30 50 10 7 S2 70 30 40 60 9 S3 40 8 70 20 18 Demand 5 8 7 14 34

1. Use the penalty (Big-M) method to solve the following LP problem.  (10)

Minimize

Subject to

1. a) Define the Total float, free float and Independent float.                               (6)

1. b) The following indicates the details of the activities of a project.

The durations are in days.                                                                                                            (14)

 Activities TO TM TP 1 – 2 4 5 6 1 – 3 8 9 11 1 – 4 6 8 12 2 – 4 2 4 6 2 – 5 3 4 6 3 – 4 2 3 4 4 – 5 3 5 8
• Draw the network
• Find the critical path
• Find the mean and standard deviation of the project completion time

1. a) Reduce the following game to game and hence find the optimum strategies and the value

of the game.                                                                                                                  (12)

Player B

 I II III IV I 3 2 4 0 II 3 4 2 4 III 4 2 4 0 IV 0 4 0 8

Player A

1. b) Solve the following unbalanced assignment problem of minimizing total time for doing all the jobs.                                        (8)
 Jobs Operators 1 2 3 4 5 1 6 2 5 2 6 2 2 5 8 7 7 3 7 8 6 9 8 4 6 2 3 4 5 5 9 3 8 9 7 6 4 7 4 6 8

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## Loyola College B.Sc. Computer Science April 2011 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – COMPUTER SCIENCE

FIFTH SEMESTER – APRIL 2011

# CS 5402 – OPERATIONS RESEARCH

Date : 12-04-2011              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART-A

1. Write the steps involved in L.P model formulation.
2. What is pivot element?
3. What is the difference between regular simplex method and the dual simplex method?
4.  Define an assignment problem.
5. List out the methods of solving Transportation problem
1. What is a sequencing problem?
2. Define optimistic & pessimistic time.
1. What is dummy activity?
2. What is Holding Cost?
3. Write the formula to calculate EOQ for the system with constant demand rate.

PART-B

11 a) A company makes two kinds of leather belts. Belt A is a high quality belt, and belt

B is of lower quality. The respective profits are Rs.4 and Rs. 3 per belt. Each belt

of type A requires twice as much time as a belt of type B, and if all belts were of

type B, the company could make 1000 per day. The supply of leathers is sufficient

for only 800 belts per day (Both A and B combined). Belt A requires a fancy buckle

and only 400 per day are available. There are only 700 buckles a day available for

belt B. Formulate the mathematical model to determine the optimal product mix .

(OR)

11 b) Solve the following L.P.P graphically.

Max  Z = 10 x1 +15x2

Subject to            2 x1 +x2     ≤ 26

2 x1 +4x2     ≤ 56

– x1 +x2     ≤ 5

x1 ,x2 ≥0

12 a) Construct  the dual to the primal problem

Max  Z = x1 +2x2 +3  x3

Subject to            3 x1 +x+  x3   ≤ 12

x1 +2x2 +4  x3    ≤ 20

2 x1 +5x2   x3    ≤ 18

x1 ,x2,  x3 ≥0

(OR)

12 b) Find an initial allocation by Vogel’s approximation method for the following transportation problem

 D E F G Availability A 11 13 17 14 250 B 16 18 14 10 300 C 21 24 13 10 400 Demand 200 225 275 250

13a) Five men are available to do five different jobs. From past records, the time (in hours) that each man take to do each job is known and given in the table

 Jobs Men I II III IV V A 2 9 2 7 1 B 6 8 7 6 1 C 4 6 5 3 1 D 4 2 7 3 1 E 5 3 9 5 1

(OR)

13 b) Find the sequence that minimizes the total elapsed time required to complete the following tasks on two machines

Task           : A       B         C         D         E          F          G         H         I

Machine I  : 2        5          4          9          6          8          7          5          4

Machine II            : 6        8          7          4          3          9          3          8          11

14a) A project consists of a series of activities called A,B,..,I with the following

relationship<X,Y means X and Y cannot start until W is completed with this

notation construct a network diagram having the following constraints.

A<D,  A<E;    B<F;    D<F;    C<G; C<H;F<I;G<I;

Job:     A          B         C         D         E          F          G         H         I

Activity:8        10        8          10        16        17        18        14        9

(days)

(OR)

14 b) (i) Write down the difference between PERT & CPM.   (4)

(ii) Define the following terms:

a)dummy activity         b) Total float                        (4)

15a) Manufacture has to supply 600 units of his product/year. Shortages are not allowed

and storage cost amounts to Rs.0.60/unit/year. The set up cost/run is Rs.80.Find the

optimum run size and the minimum average yearly cost.

(OR)

15b) A commodity is to be supplied at the rate of 200 units per day. Ordering cost is Rs.50 and the holding cost is Rs.2 per day. The delay in supply induces penalty of Rs10 per unit per delay of one day. Find the optimal policy and reorder cycle period.

PART-C

16 a) Use Simplex method to solve the following L.P.P

Max  Z = 5 x1 +3x2

Subject to        x1 +x2     ≤ 2

5 x1 +2x2     ≤ 10

3x1 +8x2     ≤ 12

x1 ,x2 ≥0

1. b) Determine an initial basic feasible solution to the following transportation problem

by using (a) North west corner rule (b) Least cost method

 Destination Source D1 D2 D3 D4 Supply S1 21 16 15 3 11 S2 17 18 14 23 13 S3 32 27 18 41 19 Demand 6 10 12 15

17 a) A marketing manager has 5 salesmen and 5 sales districts. Considering the capabilities of the salesman and the nature of districts, the marketing manager estimates that sales per month (in hundred rupees) for each salesman in each district would be as follows:

 Salesman Sales District A B C D E 1 32 38 40 28 40 2 40 24 28 21 36 3 41 27 33 30 37 4 22 38 41 36 36 5 29 33 40 35 39

What is the maximum sale that may be expected if an optimum assignment is made?

17 b)Find the sequence that minimizes the total time required in performing the following

job on three machines in order ABC .A processing time (in hours) are given in the

following table.

Jobs     :1         2          3          4          5

Machine A      :8         10        6          7          11

Machine B       :5         6          2          3          4

Machine C       :4         9          8          6          5

18a) A stockiest has to supply 12,000 units of a product per year to his customer. The

demand is fixed and known and the shortage cost is assumed is to be infinite. The

inventory holding cost is Re.0.20 per unit per month and the ordering cost per order

is Rs.350. Determine the following

(i) The optimum lot size q0

(ii) Optimum scheduling period t0

(iii) Minimum total variable yearly cost

18b) A company uses annually 24,000 units of raw material which costs Rs1.25/unit

placing each order cost Rs.22.50 and the carrying cost is 5.4%/year of the average

inventory. Find the total cost including the cost of material.

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## Loyola College B.Sc. Computer Science April 2012 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – COMPUTER SCIENCE

FIFTH SEMESTER – APRIL 2012

# CS 5402/5503 – OPERATIONS RESEARCH

Date : 27-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART-A

Answer ALL questions:                                                                                                               10 X 2=20

1. Write a General form of LPP.
2. What is optimal solution?
3. Why we should prefer dual problem to solve LPP?
4. Define traveling salesman problem.
5. List out the methods of solving Transportation problem.
6. Define Activity & Node
7. What is a sequencing problem?
8. What is setup cost?
9. What is reordering level?

PART-B

Answer All questions:                                                                                                                   5 X 8=40

11 a) ) A Company produces refrigerators in unit I and heaters in unit II. The two

products are produced and sold in a weekly basis. Weekly production cannot

exceed 25 in unit I and 36 in unit II. Formulate this problem as an LP model

(OR)

11 b Solve the following l.p.p  graphically.

Max  Z = 10 x1 +15x2

Subject to            2 x1 +x2     ≤ 26

2 x1 +4x2     ≤ 56

– x1 +x2     ≤ 5

x1 ,x2 ≥0

12 a) Construct  the dual to the primal problem

Minimum Z =  x1 +x2 +x3

Subject to         x1 –3x+4 x3      = 5

x1 -2x2       ≤   3

2 x2 – x3    ≥  4

x1 ,x2 ≥0 x3  unrestricted in sign.

(OR)

12 b) Obtain the initial solution of the following transportation problem by the north-west corner rule and matrix minima given that (i) the requirements are 40, 90 and 100 units and (ii) the supply are 90, 70  and 70.

 Source Destination S1 S2 S3 D1 15 28 27 D2 24 24 25 D3 22 25 20

13 a) A department has five employees with  five jobs to be performed . From past records, the time (in hours) that each man take to do each job is known and given in the table

 Employee Jobs I II III IV V A 10 5 13 15 16 B 3 9 18 13 6 C 10 7 2 2 2 D 7 11 9 7 12 E 7 9 10 4 12

How should the jobs be allotted on per employee, so as to minimize the total number of hours

(OR)

13 b) Find the sequence that minimizes the total elapsed time required to complete the following tasks on two machines

Task           : A       B         C         D         E          F          G         H         I

Machine I  : 2        5          4          9          6          8          7          5          4

Machine II            : 6        8          7          4          3          9          3          8          11

14 a) A project consists of a series of activities called A,B,..,I with the following relationship<X,Y means X and Y cannot start until W is completed with this notation construct a network diagram having the following constraints.

A<D,E;  B,D<F;   C<G;   B<H;   F,G<I;

Time:     A       B         C         D         E          F          G         H         I

Activity:23      8          20        16        24        18        19        4          10

.

(OR)

14 b )  (i) Write down the difference between PERT & CPM.   (4)

(ii) Define the following terms:

a)dummy activity         b) total float                        .

15a)   Manufacture has to supply 600 units of his product/year. Shortages are not allowed and storage cost amounts to Rs.0.60/unit/year.The set up cost/run is Rs.80.Determine(i) optimum run size (ii) the minimum average yearly cost.

(OR)

15 b) The annual demand for an item is 3200 units. The unit cost is Rs. 6/- and

inventory carrying charges 25% per annum. If the cost of one procurement is

Rs. 150/- determine (i) Economic order quality (ii) time between two

consecutive orders   (iii) number of order per year (iv) the optimal total cost

PART-C

Answer any TWO                                                                             2 X 20=40

16 a) Use simplex method to solve the following L.P.P

Maximize  Z =  x1 +2x2+x3

Subject to        2 x1 +x2 – x3    ≤ 2

-2 x1 +x2 -5x3     ≥-6

4 x1 +x2 +x3    ≤ 6

x1 ,x2 ,x3≥0

1. b) Determine an initial basic feasible solution to the following transportation problem

by using (a) North west corner rule (b) Least cost method(c)Vogel’s

approximation.

 Destination Source D1 D2 D3 D4 Supply S1 21 16 15 3 11 S2 17 18 14 23 13 S3 32 27 18 41 19 Demand 6 10 12 15

17 a) Find the sequence that minimizes the total time required in performing the following    job on three machines in order ABC .A processing time (in hours) are given in the      following table.

Jobs     :1         2          3          4          5

Machine A      :8         10        6          7          11

Machine B       :5         6          2          3          4

Machine C       :4         9          8          6          5

1. b) The project has the following time schedules.
 Activity 1-2 1-6 2-3 2-4 3-5 4-5 6-7 5-8 7-8 t0 3 2 6 2 5 3 3 1 4 tm 6 5 12 5 11 6 9 4 19 tp 15 14 30 8 17 15 27 7 28
1. Draw the Project Network
2. Find the critical path.

.

18 a) A stockiest has to supply 12,000 units of a product per year to his customer. The

demand is fixed and known and the shortage cost is assumed is to be infinite. The

inventory holding cost is Re.0.20 per unit per month and the ordering cost per order

is Rs.350. Determine the following

(i) The optimum lot size q0

(ii) Optimum scheduling period t0

(iii) Minimum total variable yearly cost.

1. b) )  Given the following data:
 Job 1 2 3 4 5 6 Machine A 12 10 9 14 7 9 Machine B 7 6 6 5 4 4 Machine C 6 5 6 4 2 4

Order of Processing : ACB

Determine the optimal sequence & the total elapsed time associated with it.

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