OMBD - QB Wth Answer - June 2012-1

VINAYAKA MISSIONS UNIVERSITY, SALEM

FACULTY OF MANAGEMENT STUDIES – VMKV ENGINEERING COLLEGE

MBA – II SEMESTER (MBA 2009 REGULATIONS)

47109203 - Optimization Models for Business

Decisions

UNIT-I: Linear Programming Problem (LPP)

Part – A (1 Mark)

1. Define Operations Research.

“Operations Research is the application of scientific methods to complex problems

arising from operations involving large systems of men, materials, and money in industry,

business, government and defence”.

- UK Operations Research Society

2. What are the different phases in OR?

(i). Judgment phase

(ii). Research phase

(iii). Action phase

3. What is a Linear Programming Problem?

Linear Programming is the mathematical technique which involves the allocation of

scarce or limited resources in an optimal manner on the basics of a given criterion of

optimality.

4. Define feasible region.

A region in which all the constraints are satisfied, it is called as feasible region. It is also

called as Solution Space.

5. What is alternative optimal solution in LPP?

If a LPP has more than one optimal solution, it is said to be an Alternative/Multiple optimal

solution.

6. Define basic feasible solution.

A feasible solution to an LPP which is also the basic solution is called the basic feasible

solution. i.e., all basic variables assume non – negative values.

7. Define unbounded solution.

When the value of decision variables in LPP is permitted to increase infinitely without

violating the feasibility condition, then the solution is said to be unbounded solution.

MBA – OMBD Question Bank – 2012 1

QUESTION BANK

8. When an LPP is said to be infeasible?

In LPP there are no points that satisfy all the constraints of the given problem, it is said to

be infeasible (or) no feasible solution.

9. What are the methods used to solve an LPP involving artificial variables?

a. Big M – Method (or) Method of Penalties (or) Charnes Penalty Method

b. Two - Phase simplex Method.

10. State the fundamental conditions in the basis of the simplex method.

The Feasibility condition: It ensures that if the starting solution is basic feasible, only

basic feasible solutions will be obtained during computations.

The Optimality condition: It guarantees that only better solutions (as compared to the

current solution) will be encountered.

11. What is key column and how is it selected?

Key column is the column which gives the entering variable column and is selected by most

negative value in the net evaluation (Zj – Cj). It is also called as Pivot column.

12. When does the simplex method indicate that the LPP has unbounded solution?

The indication of unbounded solution of LPP can be obtained if all the variables in the key

column are negative.

13. What is a slack variable?

Slack variable is a non – negative variable which is added to the left hand side of the

constraint equation to convert the inequality less than or equal (≤) in to equality equation.

Slack variables refer to the amount of unused resources like raw materials, labour and money.

14. What is a surplus variable?

Surplus variable is a non – negative variable which is subtracted from the left hand side of

the constraint to convert the inequality greater than or equal (≥) into an equation. Surplus

variable is the amount of resources by which the left hand side of the equation exceeds the

minimum limit.

15. What is an artificial variable?

Artificial variable is a non – negative variable introduced in the constraint greater than or

equal (≥) in order to get the initial basic feasible solution. Artificial variables are temporary

slack variables which are used for purposes of calculation, and are removed later.

16. Define Iso Profit Line.

The objective function of maximization is represented by assuming suitable outcome

(revenue) and is known as Iso Profit line.


17. Define Iso Cost Line.

The objective function of minimization problem is represented by assuming suitable cost, a

line, known as Iso Cost line.

18. List out the Model of OR.

Models by Function Descriptive model

Predictive model

Normative model

Models by Structure Iconic or Physical model

Analogue or Schematic models

Symbolic or Mathematical models

Models by nature of the Environment Deterministic models

Probabilistic models

Models by degree of abstraction Language Models

Case studies

Models by the extent of Generality General models

Specific models

19. State the advantages of OR Models.

OR Models provides a logical and systematic approach to the problem.

OR Models indicates the scope as well as limitations of a problem.

OR Models helps in finding avenues for new research and improvements in a system.

OR Models makes the overall structure of the problem more comprehensible and helps

in dealing with the problem in its entirety.

20. Write the standard form of LPP.

Standard Form of LPP

Objective function is

Maximize or Minimize, Z =

Subject to constraints,

, i = 1, 2, 3, ……….., n

x1, x2, x3,………xn ≥ 0


21. Define Redundant Constraint.

The constraint does not affect the solution space, it is said to be redundant constraint.

22. Define degeneracy of LPP.

If the minimum positive ratio for two or more rows are same, tie exist. If there is a tie,

selection of key row is very difficult. In such case the problem is said to be degenerate. It can

be solved by choosing the key row arbitrarily.

23. When does an LPP possess a Pseudo-optimal Solution?

An LPP possesses a pseudo-optimal solution if at least one artificial variable is in the basis

at positive level even though the optimality conditions are satisfied.

24. What is the indication of alternative optimal solution in simplex table?

In optimal simplex table, in the net evaluation (Zj – Cj) row if zero occurs for non-basic

variables it indicates that LPP has alternate solution.

25. List out the limitations of LPP?

Linear programming is applicable only to problems where the constraints and objective

function are linear i.e., where they can be expressed as equations which represent

straight lines. In real life situations, when constraints or objective functions are not

linear, this technique cannot be used.

Factors such as uncertainty, weather conditions etc. are not taken into consideration.

This method doesn’t take into account the effect of time.

When the constraint equations are more, the computational difficulties are enormous

even when computers are used.

Part – B (8 marks)

1. A firm manufactures two products A & B on which the profits earned per unit are Rs.3/- &

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 B requires one

minute on M1 and one minute on M2. Machine M1 is available for not more than 7 hrs.

30minutes while machineM2 is available for 10 hours during any working day. Find the

number of units of products A & B to be manufactured to get maximum profit.

2. Naga & Co has two bottling plants, one located at Gandhi Nagar and another at Nehru

Nagar. Each plant produces three drinks, whisky, beer and fruit juices named A, B and C

respectively. The numbers of bottles produced per day are as follows.


ProductsPlant at

Gandhi Nagar Nehru Nagar

Whisky (A) 1500 1500

Beer (B) 3000 1000

Fruit Juices (C) 2000 5000

A market survey indicates that during the month of April, there will be a demand of

20,000 bottles of whisky, 40,000 bottles of beer and 44,000 bottles of fruit juices. The

operating costs per day for plants at Gandhi Nagar and Nehru Nagar are 600 and 400

monetary units. For how many days each plant be run in April so as to minimize the

production cost, while still meeting the market demand?

3. (i) A farmer has 1000 acres of land on which he can grow corn, wheat or soyabeans. Each

acre of corn costs Rs.100 for preparation, requires 7 man-days of work and yields a profit

of Rs.30. an acre of wheat costs Rs.120 to prepare 10 man – days of work and yields a

profit of Rs.40. an acre of soybeans costs Rs.70 to prepare 8 man – days of work and

yields a profit of Rs.20. if the former has Rs1, 00,000 for preparation and can count on

8,000 man – days of work. Formulate this as an LPP. (4 Marks)

(ii) Solve the following LPP graphically

Min. Z = 4X1 + 5X2

Subject to the constraints

X1 + X2 ≥ 10

2X1 + 5X2 ≥ 35

& X1, X2 ≥ 0 (4 Marks)

4. (i) A person wants to decide the constituents of a diet which will fulfill his daily

requirements of proteins, fats, carbohydrates 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

below.

Food typeYield per unit Cost per unit

(Rs)Proteins Fats Carbohydrates

A 3 2 6 45

B 4 2 4 40

C 8 7 7 85

D 6 5 4 65

Minimum requirement 800 200 700


Formulate the linear programming model for the problem. (4 Marks)

(ii) Solve the following LPP graphically

Maz. Z = 3X1 + 2X2


X1 - X2 ≤ 1

X1 + X2 ≥ 3

& X1, X2 ≥ 0 (4 Marks)

5. Use Simplex method to solve the following LPP

Max. Z = 100X1 + 200 X2 + 50X3


5X1 + 5X2 + 10X3 1000

10X1 + 8X2 + 5X3 2000

10X1 + 5X2 500

and X1, X2, X3 0

6. Use the simplex method to solve the LPP

Min Z = -X1 + X2 -3X3


X1 + X2 + X3 10

-2X1 + X3 -2

2X1 -2X2 + 3X3 0

and X1, X2, X3 0

7. A firm manufactures two types of products P1 and P2 and sells them at a profit of Rs.2/- on

type P1 and Rs. 3/- on type P2. Each product is processed on two machines G and H. Type

P1 requires one minute of processing time on G and two minutes on H; type P2 requires one

minute on G and one minute on H. The Machine G is available for not more than 6 hours 40

minutes while Machine H is available for 10 hours during any working day. Formulate the

problem as a LPP and solve graphically.

8. Use two phase simplex method to solve the following LPP

Max.Z = 5X1 + 3X2


2X1 + X2 1


X1 + 4X2 6

and X1, X2 0

9. Show that there does not exists any feasible solution to the following LPP Max. Z = 2X1 + 3X2 + 5X3


3X1 +10 X2 + 5X3 15

33X1 -10 X2 + 9X3 33

X1 + 2X2 + X3 4

and X1, X2, X3 0

10. Show that the following LPP has an unbounded solution.

Max. Z = 4X1 + X2 + 3X3 + 5X4


4X1 - 6X2 - 5X3 - 4X4 - 20

3X1 -2X2 + 4X3 + X4 10

8X1 - 3X2 + 3X3 + 2X4 20

and X1, X2, X3,X4 0

Unit – II: Transportation and Assignment problems

Part - A (1 Mark)

1. Define a transportation problem.

TP is a special kind of LPP in which goods are transported from a set of sources to a set of

destinations subject to the supply and demand of the source and destination, respectively, such

that the total cost of transportation is minimized.

2. Write the mathematical form of a transportation problem.

Minimize Z

Subject to


3. Define basic feasible solution.

A feasible solution is called a basic feasible solution if the number of non – negative

allocations is equal to m + n – 1 where m is the number of rows, n is the number of columns in

a TP table.

4. Define optimal solution.

Optimal solution is a feasible solution (not necessarily basic) which minimizes the total

cost.

5. Define non – degeneracy in a transportation problem.

A basic feasible solution to a (m X n) TP that contains exactly m + n – 1 non – negative

allocations in independent position.

6. What do you understand by degeneracy in a transportation problem?

A basic feasible solution to a (m X n) TP that contains less than m + n – 1 non – negative

allocations is called degeneracy in a TP.

7. How to resolve the degeneracy in a transportation problem?

This degeneracy in a TP can be resolved by adding one (more) empty cell having the least

cost and is of independent position with a non – negative allocation ( >0).

8. Define balanced transportation problem.

A TP is said to be balanced, if the total supply from all the sources is equal to the total

demand in all the destinations. i.e.,

9. How do you convert an unbalanced transportation problem in to a balanced one?

The unbalanced TP is converted into a balanced one by adding a dummy row (source) or

dummy column (destination) whichever is necessary. The unit TP cost for the dummy

row/column elements are assigned zero. Then the problem is solved by the usual procedure.

10. List out the methods used to obtain an IBFS in a transportation problem.

a. North – West Corner Rule (NWC)

b. Row Minima Method (RMM)

c. Column Minima Method (CMM)

d. Least Cost Method (LCM)

e. Vogel’s Approximation Method (VAM)

11. List out the methods used to obtain an optimal solution in a transportation problem.

Stepping Stone Method

MODI method


12. State the necessary and sufficient condition for the existence of a feasible solution to a

TP.

The necessary and sufficient condition for the existence of a feasible solution is a solution

that satisfies all conditions of supply and demand.

13. When does a TP have a unique solution?

While doing optimality test, if the entire empty (non-basic) cell evaluation,

Δj = Cij – (ui + vj) are positive, then the problem is said to have an unique solution.

14. What is the indication for an alternate optimal solution in TP?

While performing optimality test, if some of empty (non-basic) cell evaluation,

Δj = Cij – (ui + vj) = 0, then the problem is said to have an alternate optimal solution.

15. Vogel’s Approximation Method results is the most economical initial basic solution?

Why?

Yes, it is true. In this method we are allocating the items only at the least cost and hence

this method reduces the number of iterations towards optimality. Also it gives near optimal

solution. Some time it may be optimal solution.

16. List the merits and demerits of using North-West Corner Rule.

Merits: Since the cost factor is not considered in each time, this method is easy to compute the

total transportation cost.

Demerits: Since the allocations have been made without considering the unit cost, the solution

obtained may not be the best solution. Therefore, to get an optimal solution it requires more

number of iterations.

17. What is an assignment problem?

The assignment problem is a special type of allocation problem in which a number of

operations are to be assigned to an equal number of operators, where each operator performs

only one operation at a minimum cost or maximum profit.

18. What is a traveling salesman problem?

In this model a salesman has to visit ‘n’ cities. He has to start from a particular city (home

city), visit each city once and then return to his starting point (home city),

19. State the objectives of traveling salesman problem.

The main objective of a salesman is to select the best sequence in which he visited all cities

in order to minimize the total distance traveled or minimize the total time.

20. When is there an indication of an alternate solution in an assignment problem?

If the final cost matrix contains more than the required number of zeros for assignment at

independent position, then it indicates that the problem has an alternate optimal solution.


21. State the difference between the TP and the AP.

The cost matrix in TP is not necessarily a square matrix, whereas in AP, it is a square

matrix.

Supply and demand at any source and at any destination may be positive quantity a i, bj

in TP, whereas in AP it will be 1 i.e., ai = bj = 1.

The allocations Xij in the case of TP can take any positive values satisfying the rim

requirements, whereas in AP, Xij will take only two possible values 1 or 0.

TP will not provide degeneracy always, whereas in AP will always provide degeneracy.

22. Write the mathematical form of AP.

Minimize Z

Subject to

23. How do you solve the maximization case of AP?

To solve the maximization problem, first we convert the given maximization matrix into an

equivalent minimization matrix form by subtracting all the elements in the given profit matrix

from the highest element in that matrix. Then the problem is a minimization and can be solved

by the usual assignment method (Hungarian Method).

24. Give any two applications of assignment problem in Business.

AP is used in production environment in which the number of jobs are assigned to

number of workers or machines in such a way that the total time (or cost) to complete

all the jobs will be minimum.

AP is used in traveling salesman problem.

25. Why can the TP technique or the simplex method not be used to solve an assignment

problem?

The TP technique or the simplex method not be used to solve an assignment problem

because of degeneracy.


Part – B (8 Marks)

1. A company has 3 plants A, B, C and three warehouses X, Y, Z. The number of units

available at the plants are 60, 70, & 80 and the demand at X, Y, & Z are 50, 80, & 80

respectively. The unit cost of transportation has been displayed in the table.

X Y Z

A 8 7 3

B 3 8 9

C 11 3 5

Formulate the above data as transportation modal and obtain an optimum solution.

2. A product is manufactured by 4 factories A, B, C, & D. Their unit production costs are

Rs. 2/-, Rs. 3/- Re.1/- and Rs. 5/- respectively. Production capacity of the factories are 50,

70, 40, and 50 units respectively. The product is supplied to four stores S1, S2, S3, & S4,

requirements of which are 25, 35, 105, and 20 units respectively. Unit transportation costs

are given below.

S1 S2 S3 S4

A 2 4 6 11

B 10 8 7 5

C 13 3 9 12

D 4 6 8 3

Find the transportation plan such that the total production and transportation cost is

minimum.

3. There are four jobs to be assigned to the machines. Only one job could be assigned to one

machine. The amount of time in hours required for the jobs in machines are given below.

JobMachine

A B C D E

1 4 3 6 2 7

2 10 12 11 14 16

3 4 3 2 1 5

4 8 7 6 9 6Find an optimum assignment of jobs to the machines to minimize the total processing

time. Find for which machine no job is assigned. What is the total processing time to

complete all the jobs?


4. Solve the following TP,

Warehouse

Factory W1 W2 W3 W4 Capacity

F1 19 30 50 10 7

F2 70 50 40 60 9

F3 40 8 70 20 18

Requirement 5 8 7 14

5. Solve the transportation problem with unit TP cost in rupees and units of demand and

supply as given below.

Destination

Source D1 D2 D3 D4 Capacity

S1 10 13 11 8 65

S2 9 12 12 10 44

S3 13 9 11 9 41

Requirement 60 40 55 45

6. A company produced two products B and C. These products are marketed in four cities.

The following table gives you the sales figures of the firm. Find the transportation plan to

maximize the sales.

Cities

Product Trichy Salem Madurai Chennai Supply

B 90 90 100 110 200

C 50 70 130 85 100

Demand 75 100 100 30

7. A Marketing student has to select one and only one elective in each semester and the same

elective should not be selected in different semesters. Due to various reasons, the expected

grades in each subject if selected in different semesters, vary and they are given below.


Semester Subject

Marketing Research

Services Marketing

Retail Management

Event Marketing

I F E D C

II E E C C

III C D C A

IV B A H H

The grade points are H = 10, A = 9, B = 8, C = 7, D = 6, E = 5, and F = 4. How will the

student select the electives in order to maximize the total expected points and what will be

his maximum expected total points?

8. The processing time in hours for the jobs when allocated to the different machines is given

below. Assign the machines for the jobs so that the total processing time is minimum.

Mechines

Jobs M1 M2 M3 M4

1 5 7 11 6

2 8 5 9 6

3 4 7 10 7

4 10 4 8 3

9. A department head has four tasks to be performed and three subordinates, the subordinates

differ in efficiency. The estimates of the time, each subordinate would take to perform, is

given below in the table. How should he allocate the tasks one to each man, so as to

minimize the total man hours?

Men

Task A B C

I 9 26 15

II 13 27 6

III 35 20 15

IV 18 30 20

10. A head of department in a college has the problem of assigning courses to teachers with a

view to maximizing educational quality in his department. He has available to him one

professor, two associate professors and one teaching assistant (TA). Four courses must be


offered and, after appropriate evaluation, he has arrived at the following relative ratings

(100 = best rating) regarding the ability of each instructor to teach each of the four courses.

TeachersCourse - I Course - II Course - III Course - IV

Professor -1 60 40 60 70

Asso. Prof - 1 20 60 50 70

Asso. Prof – 2 20 30 40 60

T.A 30 10 20 40

How should he assign his staff to the courses to realize his objective?

11. A salesman has to visit five cities A1, A2, A3,……., A5. The distances given in kilometers

between the five cities are as follows.

To

From A1 A2 A3 A4 A5

A1 - 4 10 14 2

A2 12 - 6 16 4

A3 16 14 - 8 14

A4 24 8 12 - 10

A5 2 6 4 16 - If the salesman starts from city A1 and has comeback to city A1, which route should he

select so that the total distance traveled by him is minimized.

Unit – III: Sequencing problem and Game theory

Part - A (1 Mark)

1. What is a sequencing problem?

The selection of an appropriate order for number different tasks to be performed on a finite

number of service facilities in order to make effective use of available facilities and achieve

greater output is called sequencing.

2. Define total elapsed time.

Total elapsed time is the time between the starting time of the first job and completion time

of the last job.

3. List out the types of sequencing problems.

Processing n jobs through 2 machines

Processing n jobs through 3 machines


Processing n jobs through m machines

Processing 2 jobs through m machines

4. Define the term idle time.

Idle time is the time for which a machine remains idle during the total elapsed time.

5. Define processing time.

Processing time is the time required by each job on each machine.

6. Write any two assumptions in sequencing problem.

No machine can process more than one operation at a time.

Each operation, once started, must be performed till completion.

7. Write the conditions for applying Johnson’s Algorithm for finding the optimal

sequencing order of ‘N’ jobs through 3 machines.

The following conditions (either one or both) must be satisfied.

Minimum processing time on Machine A ≥ Maximum Processing time on Machine B

Minimum processing time on Machine C ≥ Maximum Processing time on Machine B

If any one of the conditions satisfied, the given three machines problem can be converted

into two machines problems namely G & H. where G = A + B and H = B + C

8. Write the conditions for applying Johnson’s Algorithm for finding the optimal

sequencing order of ‘N’ jobs through ‘M’ machines.

The following conditions (either one or both) must be satisfied.

Min. processing time on M/C. M1 ≥ Max. Processing time on M/C [M2, M3, …, Mm-1]

Min. processing time on M/C. Mm ≥ Max. Processing time on M/C [M2, M3, …, Mm-1]

If any one of the conditions satisfied, the given M machines problem can be converted into

two machines problems namely G & H.

Where,

G = M1 + M2 +……. + Mm-1 and

H = M2 + M3 +……..+Mm

9. Define a game.

The competitive situation with a finite number of competitor’s with conflicting interests and

where the action of one depends upon the action taken by the other.

(or)

Game is defined as an activity between two or more persons involving activities by each

person according to a set of rules, at the end of which each person receives some befit or

satisfaction or suffers loss (negative benefit).


10. Distinguish between pure and mixed strategies.

If a player plays only one particular course of action all the time, he is said to be adopting

pure strategy, whereas player plays each of his course of action a certain part of the time (i.e.,

in fixed proportion), he is said to use mixed strategies.

11. Define a saddle point.

A saddle point is that position in the payoff matrix where the maximin value of the row

coincides with minimax value of the column.

12. What is a two persons zero sum game?

A game with two players, where the sum of the points won equals the sum of the points lost

(i.e., gain of one player equals the loss of the other) is known as Two-person zero sum game.

13. Define zero sum game. Give an Example.

A game wherein the gain of one player equals the loss of other is called zero sum game. For

example, A and B are the players, A’s gain is equal the loss of the player B. two person zero

sum game is always zero sum game.

14. What is meant by minimax principle?

Minimax principle is the principle minimizes the maximum losses. Minimax is the

maximum of each column and take the minimum among the maximum values (i.e., maximum

of row minima).

15. State the rules of dominance.

(i) If all the elements in a column (say K th column) are greater than or equal to he

corresponding elements in another column (Rth column), Kth column is dominated by Rth

column.

(ii) If all the elements in a row (say Ith row) are less than or equal to the corresponding

elements in another row (say Jth row), Ith row is dominated by Jth row.

16. What are the types of games that can be solved graphically?

The games in which the payoff matrix is of the form (m X 2) and (2 X n) are solved

graphically.

17. Define the term players.

Finite number of participants (competitors) in a game is called players. The players may be

any two companies competing for tenders, two persons bidding in a game etc…

18. When do you say a game is stable?

A game is sad to be stable, if there is a saddle point.


19. Define value of the game.

The value of the game is defined as the expected outcome of the game when players follow

their optimal strategies.

20. What kind of games are solved graphically?

The games in which the payoff matrix in the form of (m x 2) or (2 x n), they can b solved

by using graphical method.

21. Find the Dominating row and Dominating column in the 5 4 2 payoff Matrix -2 1 -1 -4 8 5

Row II is dominated by row I and Column II is dominated by Column III.

22. What is meant by fair game?

A game is fair, if the value of minimax and the value of maximin is equal to zero

i.e., Minimax = Maximin = 0.

23. When do you say a game is strictly determinable?

A game is strictly determinable, if the value of maximin and minimax is equal to the game

value. i.e., Maximin = V = Minimax.

24. Define Strategy.

Strategy is a set of rules (Programmes) that specifies which of the available courses of action

the player should make at each play.

(Or)

Strategy is defined as a complete set of plans of action specifying precisely what the player

will do under every possible future contingency that might occur during the play of the game.

25. When do players apply mixed strategies?

Players apply mixed strategies when there is no saddle point in the given payoff matrix.



1. There are five jobs each of which has to go through four machines M1, M2, M3, & M4. in the

order M1 - M2 - M3 - M4. Processing times are given below.

MachineJob

A B C D E

M1 11 13 9 16 17

M2 4 3 5 2 6

M3 6 7 5 8 4

M4 15 8 13 9 11Find the optimal sequence and the total minimum elapsed time. And also calculate

idle time of the machines.

2. Use graphic method to find the minimum elapsed total time sequence of 2 jobs and five

Machines, when we are given the following information.

Machines

Job-1 Sequence A B C D E

Time (in hrs) 2 3 4 6 2

Job-2 Sequence C A D E B

Time (in hrs) 4 5 3 2 6

3. (i) Find the sequence that minimizes the total elapsed time (in hrs) required to complete the

following jobs on 2 machines in the order M1 - M2.

Job : A B C D E F G

Machine M1 : 8 6 10 11 10 14 4

Machine M2 : 5 3 7 12 8 6 7

Also find out the idle time for the machines. (5 Marks)

(ii) Solve the following game (3 Marks)

Player B

I II

I 3 2

Player A II - 2 -3

III - 4 -5


4 (i) Consider the game G with the following pay - off

Player B

I II

I 2 6

Player A II - 2

a) Show that G is strictly determinable, whatever may be

b) Determine the value of G. (4 Marks)

(ii) Two players A & B match coins. If the coins show two heads, A wins Rs.2/-, if there are

two tails A wins nothing, and loses Re.1/-. When there are one head and one tail. Determine

the optimum strategies for the players and the value of the game. (4 Marks)

5. Find the sequence that minimizes the total elapsed time (in hrs) required to complete the

following jobs on 3 machines in the order M1 - M2 - M3.

Job : A B C D E F G

Machine M1 : 3 8 7 4 9 8 7

Machine M2 : 4 3 2 5 1 4 3

Machine M3 : 6 7 5 11 5 6 12

Also calculate the idle time for the machines.

6. Find an optimal sequence for the following problem. There are four jobs, each of which must

go through five machines in the prescribed order. (Passing is not allowed). Processing times

(in hours) are given below.

Machines

Jobs A B C D E

1 7 5 2 3 9

2 6 6 4 5 10

3 5 4 5 6 8

4 8 3 3 2 6

Also find the total minimum elapsed time and idle time.

7. Using dominance property to solve the game

Player B

I II III IV

I -5 3 1 20

Player A II 5 5 4 6

III -4 -2 0 -5


8. Using graphical method solve the rectangular game.

Player B

I II III IV V

I 2 -1 5 -2 6

Player A

II -2 4 -3 1 0

9. Using graphical method solve the rectangular game.

Player B

I II

I 6 5

Player A II 3 6

III 8 4

10. Solve the following rectangular game by using graphical method.

Player B

I II III IV

I 3 3 4 0

Player A

II 5 4 3 7

Unit – IV: Network Project Scheduling & Replacement theory

Part - A (1 Mark)

1. What do you mean by Project?

Project is a combination of interrelated activities all of which must be executed in a

certain order before the entire task can be completed to achieve a set goal.

2. What is a network?

Network is a graphical representation of a project’s operations and is composed of all

the events and activities in sequence, along with their inter relationships and inter dependence.

3. Define an activity.

An activity is any portion of a project which consumes time or resources and has a

definable beginning and ending. It is denoted by an arrow with description and time estimates

within parenthesis along the arrow.


4. What do you mean by concurrent activities?

Activities which can be accomplished concurrently or together are called concurrent activities.

5. Define the term total float.

Total float is the amount of time within which an activity can be delayed without affecting

the project completion time. It is the difference between the LST and EST of the same activity.

6. Define earliest start time.

Earliest start time (EST) for an activity represents the time at which an activity can begin at

the earliest.

7. Define latest finish time.

The latest finish time (LFT) of an activity represents the latest by which an activity must be

completed without delaying the completion of the project.

8. What is slack time in a network?

It is a free time in a network. It is equal to the difference between the latest finish time and

the earliest finish time (or) the difference between latest start time and the earliest start time of

the activity.

9. Define Optimistic time.

It is the smallest time taken to complete the activity if every thing goes on well or under

best possible conditions.

10. Define total float.

Total float is the amount of time by which the completion of an activity can be delayed

beyond the earliest expected completion time without delaying the overall project. It is the

difference between the late start and early start times of the same activity.

11. Define critical path.

The critical path in a network diagram is the longest continuous chain of activities (a path along

which it takes the longest possible time duration) through the network starting from first to the

last event.

12. State the relationship among the floats.

Independent Float ≤ Free Float ≤ Total Float

13. State any one difference between direct cost and indirect cost.

Direct costs are directly dependent upon the amount of resources involved in the execution

of all the activities of the project. Increase in direct cost will decrease in project duration.

Indirect cost is associated with general and administrative expenses, insurance cost, taxes etc….

Increase in indirect cost will increase in project duration.


14. Define the cost-time slope of an activity.

The cost slope represent the rate of increase in the cost of performing the activity per unit

decrease in time and is called cost-time slope (or) cost-time trade off. It varies from activity to

activity.

Cost slope =

15. Define Crashing.

The deliberate reduction of activity time by putting an extra effort is called crashing the

activity.

16. Define crash time.

Crash time is the absolute minimum time required to achieve the crash cost i.e., possible

and any attempts to further crash would only raise the activity cost without reducing the time.

17. Define Crash cost.

Crash cost is the least cost of completing an activity by employing all possible means like

overtime, additional resources etc… i.e., the activity cost corresponding to the crash time is

called the crash cost.

18. Define optimum duration of a project.

The total project cost is the sum of direct cost and indirect cost. It is the time duration of a

project at which the total project cost associated it is minimum.

(Or)

Optimum duration of a project is the duration, for which the total cost associated with it is

minimum.

19. Define the least duration of a project.

Least duration of a project is the minimum duration beyond which no more crashing the

activity is possible. Here, the total cost associated with it is not necessarily minimum.

20. What is resource scheduling?

Resource scheduling is the effective allocation of available resources to various activities in

such a manner that the allocation is considered as acceptable under the given situation.

21. What is resource leveling?

Resource leveling refers to the scheduling of activities within the limits of the available

floats in such a way that the variations in resource requirements are minimized.

(Or)


There are various activities in a project demanding varying levels of resources. The overall

resources of the organization are limited. The demand on certain specified resources should not

go beyond the specified level. This operation of resource allocation is called resource leveling

or Load leveling.

22. What is replacement Problem?

The replacement problems are concerned with the situations that arise when some items

such as men, machines, electric light bulbs, etc… need replacement due to their decreased

efficiency, failure of breakdown. Such decreased efficiency or complete breakdown may either

be gradual or all of a sudden.

(Or)

Efficiency of certain items decreases due to wear and tear and require expensive

maintenance cost. Certain items like bulb fail suddenly. In such cases the old items have to be

replaced by new ones to prevent any further increase in expenditure. This is called replacement.

23. Mention any two situations where the replacement of items is necessary?

The old item has become in worse condition and works badly or requires expensive

maintenance.

The old item has failed due to accident or otherwise and does not work at all, or the old

item is expected to fail shortly.

A better or more efficient design of machine or equipment has become available in the

market.

24. What is salvage value?

Salvage value is the actual worth or cost of machine at the given period of time. When the

age of the machine increases, the salvage value decreases. It is also known as resale value.

25. What is meant by running cost?

The cost which is required to maintain and run to operate the machine is called running

cost.

26. Define optimum replacement policy.

Replace the equipment at the end of n years, if the maintenance cost in the (n + 1) th year is

more than the average total cost in the nth year and the nth year’s maintenance cost is less than

the previous year’s average total cost.

27. What are the types of failures?

Gradual failure

Sudden failure

o Progressive failure


o Regressive failure

o Random failure

28. Define group replacement.

Under this policy we take decisions as to when all the items should be replaced, irrespective

of whether they have failed or not, with a provision that if any item fails before the replacement

time it may be replaced individually.

29. What are the types of failures in replacement theory?

(a) Replacement of items with gradual deterioration

Replacement of items that deteriorate with time, ignoring time value of money.

Replacement of items that deteriorate with time, considering time value of money.

(b) Replacement of items that fail suddenly

Individual Replacement Policy

Group Replacement Policy

30. What is discount factor?

Let r% be the rate at which money value decreases. The present worth factor of unit amount

to be spend after one year is given by V = (1 + r)-1 where, V is called the discount factor. It is

also called present worth factor.

(Or)

The present worth factor of unit amount to be spent after one year is given by V = (1 + r) -1,

where r is the interest rate, then V is called discount factor or discount rate.


1. Consider a project consisting of 7 jobs A, B, …….., G with the following precedence

relations and time estimates(days).

Job Predecessors a m b

A - 2 5 8

B A 6 9 12

C A 5 14 17

D B 5 8 11

E C, D 3 6 9

F - 3 12 21

G E, F 1 4 7

Draw the project network and find the probability that the project will be completed,


(i) 4 days earlier than expected.

(ii) Not more than 4 days later than expected

(iii) If the project due date is 30 days, find the probability of meeting the due date.

(iv) If the project due date is 35 days, find the probability of meeting the due date.

2. (i) Distinguish between PERT and CPM. (4 Marks)

(ii) Construct a network for each of the projects whose activities and their precedence

relationships are given below. (4 Marks)

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

Predecessor - - - A B B C D E H,I F,G

3. (i) Identify the critical path for the following network diagram. (4 Marks)

(ii) A firm is considering replacement of a machine whose cost price is Rs.17500 and the

scrap value is Rs.500. The maintenance costs (in Rs.) are found from experience to be as

follows.

Year : 1 2 3 4 5 6 7 8

Maintenance

Cost (Rs.) : 200 300 3500 1200 1800 2,400 3,300 4,500

When the machine should be replaced? (4 Marks)

4. (i) The cost of equipment is Rs.10, 000. The annual operation and maintenance cost; and

resale values are given below.

Year 1 2 3 4 5 6 7 8

O & M cost (Rs.) 1000 1200 1400 1700 2000 2500 3000 3500

Resale value (Rs.) 6000 4000 3200 2600 2500 2400 2000 1600Determine at what time it could be replaced? (4 Marks)

(ii) A machine cost is Rs.500/- . Operation and maintenance costs are zero for the first year

and increase by Rs.100/- every year. If money is worth 5% every year determine the best


3

5

4

1

3

8

5

4

1

2

2

1

2

3

4

6

7

5

8 9

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? (4 Marks)

5. A project consists of a series of labeled A, B, C, …….., H, I with the following

relationships [W< X, Y means X & Y can not start until W is completed; X, Y <W means

W can not start until both X & Y are completed] with this notation construct the network

diagram having the following constraints.

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

Find also the minimum time of completion of the project, when the time (in days) of

completion of each task is as follows.

Task : A B C D E F G H I

Time : 23 8 20 16 24 18 19 4 10

6. Calculate earliest start, earliest finish, latest start and latest finish of each activity of the

project given below and determine the critical path of the project, Total float, Independent

float and free float.

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

Duration (in weeks)

8 7 12 4 10 3 5 10 7 4

7. A project schedule has the following characteristics.

Activity 1- 2 2- 3 2- 4 3- 5 4- 5 4- 6 5- 7 6- 7 7- 8 7- 9 8- 10 9- 10

to1 1 1 3 2 3 4 6 2 4 1 3

tm2 2 3 4 3 5 5 7 4 6 2 5

tp3 3 5 5 4 7 6 8 6 8 3 7

a) Construct a PERT network diagram.

b) Identify the critical path and compute the expected project completion time.

c) What is the probability factor for completing the project in 30 weeks?

8. The following mortality rates have been observed for a certain electric bulb.

Week 1 2 3 4 5 6 7 8

Percentage of failure 5 13 25 43 68 88 96 100There are 1000 bulbs in a factory and its costs Rs. 4/- to replace individual bulb. If

all bulbs are replaced simultaneously then the cost would be only Re.1/- per bulb. Determine

the best replacement policy.


9. The following failure rate have been observed for certain items

Month End 1 2 3 4 5

Probability of failure to date 0.10 0.30 0.55 0.85 1.00The cost of replacing an individual item is Rs.1.25. The decision is made to replace all

items simultaneously at fixed intervals and also to replace an individual item as they fail. If

the cost of group replacement is 30 paise, what is the best interval of group replacement?

10. The yearly cost of two machines A & B when the money neglected is as follow.

Year 1 2 3

MA 2900 2600 2700

MB 3400 2100 2700

Find their cost patterns if money value is 10% worth and hence find which machine is

most economical.

Unit – V: Decision Theory

Part - A (1Mark)

1. What is decision making?

Decision making is an integral part of any business organization. It involves by choosing

the best among several alternative decisions through a proper evaluation to achieve desired

outcome.

2. List out the different environments in which decisions are made.

Decision - making under certainty

Decision - making under uncertainty

Decision - making under risk

Decision - making under conflict.

3. Define payoff.

It is the effectiveness associated with specified combination of course of action and state of

nature. It is also known as profits or conditional values.

4. Define opportunity loss table.

Opportunity loss table is a payoff table which represents the cost or loss incurred because of

failure to take the best possible action. It is the numerical difference between the optimal

outcome and the actual outcome for a given decision.

5. Define uncertainty.


Uncertainty refers to a completion when one cannot specify the relative probability

distribution of the outcomes of his decision.

(Or)

State of knowledge in which one or more alternatives result in a set of possible specific

outcomes, but where the probabilities of the outcomes are neither known nor meaningful.

6. Define certainty.

An environment in which the decision maker is in a position to know exactly what will

happen by his decision during the period in which such a decision is made.

7. Define risk.

Risk refers to a situation in which one can specify the relative probability distribution of the

outcomes of his decision.

8. Define the term “Act”.

The acts are the alternative courses of action of strategies that are available for decision

maker. The objective is to choose the best alternative from the available alternatives. Decision

alternatives are also called Action or Strategies or Courses of action.

9. Define the term “Event”.

Events are the occurrences which affect the achievement of the objectives. Events are also

called States of nature or outcomes for the decision problem. The events constitute a mutually

exclusive and exhaustive set of outcomes, which describe the possible behaviour of the

environment in which the decision is made.

10. Define decision maker.

Decision maker is a person who is responsible for making decision. He has to select one

from a set of possible courses of action.

11. What are the approaches that are used in decision environment with risk?

Expected Monetary Value Criterion (EMV)

Expected Opportunity Loss Criterion (EOL)

Expected Value of Perfect Information Criterion (EVPI)

12. What is Expected Monetary Value?

The expected monetary value for a given course of action is the expected value of the

conditional payoff for that action. The conditional payoffs are obtained for each action by

considering various act-event combinations.

13. What are the methods used for decision making under uncertainty?

Maximax (or) Minimin criterion - Criterion of Optimism

Maximin (or) Minimax criterion – Criterion of Pessimism


Laplace criterion – Equally likely decision

Criterion of regret (savage Criterion)

Hurwicz criterion – Criterion of realism

14. Define Laplace criterion.

The Laplace criterion uses of all information by assigning equal probabilities to the possible

payoff for each action and then selecting that alternatives which corresponds to the maximum

expected payoff. It is an equally likely decision criterion.

15. Define decision tree.

A decision tree is a graphical representation of various decision alternatives, states of

nature, probabilities attached to the states of nature and the conditional benefits and losses.

(Or)

Type of tree-diagram used in determining the optimum course of action, in situations

having several possible alternatives with uncertain outcomes. The resulting chart or diagram

(which looks like a cluster of tree branches) displays the structure of a particular decision, and

the interrelationships and interplay between different alternatives, decisions, and possible

outcomes.

16. Define states of nature.

States of nature are the occurrences which affect the achievement of the objectives. It is

also known as events. The events constitute a mutually exclusive and exhaustive set of

outcomes. A future event not under the control of decision maker.

17. What is maximin principle?

The maximin criterion is based upon the conservative approach to assume that the worst

possible is going to happen. The decision maker considers each strategy and locates the

minimum payoff for each; and then selects that alternative which maximizes the minimum

payoff.

18. State the relationship between decision and decision making.

Decision making is the systematic process of selecting the suitable course of action among

the several alternatives where as decision is the final outcome of decision making process.

19. What are the steps involved in decision making process?

Determine the various alternative course of action from which the final decision is to be

made.

Identify the possible outcomes called the states of nature which are available to the

decision maker.


Tabulate the payoff function for all the decision alternatives with respect to the states of

nature from different combinations.

Construct the regret or opportunity loss table.

Select the optimum decision criterion which results in largest payoff.

20. What is Expected Opportunity Loss Criterion?

The Expected Opportunity Loss (EOL) is the cost of not picking the best solution. In other

words, it's the amount lost by not picking the best alternative. The minimum expected

opportunity loss is found by constructing an opportunity loss table and computing EOL for

each alternative.

21. What is EVPI Criterion?

The expected profit with perfect information is the expected return, in the long run, if we

have perfect information before a decision is made. It may be defined as the maximum amount

one would be willing to pay, to acquire effect information as to which event would occur.

22. Define Hurwicz criterion.

Hurwicz criterion stipulates that a decision maker’s view may fall somewhere between the

extreme pessimism and the extreme optimism of the maximum criterion. It is made by

assigning weights with certain degrees of optimism and pessimism.

23. Distinguish between decision under certainty and decision under uncertainty.

If the decision maker knows with certainty the consequences of every alternative or

decision choice, the problem is a decision problem under certainty.

If the decision maker faces multiple states of nature but he has no means to arrive at

probability values to the likelihood of occurrence of these states of nature, the problem is a

decision problem under uncertainty.

24. What are the steps involved in Maximin criterion?

Following steps are involved in Maximin criterion

Determine the minimum assured payoff for each alternative.

Choose that alternative which corresponds to the maximum of the above minimum

payoffs.

25. What are the steps involved in EMV criterion?

Following steps are involved in EMV Criterion

List conditional profit for each act-event combinations, along with the corresponding

event probabilities.

For each act, determine the expected conditional profits.

Determine EMV for each act.


Choose the act at which corresponds to the optimal EMV.


1. Under an unemployment promotion program, it is proposed to allow sale of news papers on

the buses during the off peak hours. The vendor can purchase the news paper at a special

concessional rate of 25 paise per copy against the selling price of 40 paise. Any unsold

copies are, however a dead loss. A vendor has estimated the following probability

distribution for the number of copies demanded.

Number of copies 15 16 17 18 19 20

Probability 0.04 0.19 0.33 0.26 0.11 0.07

(a) How many copies should he order so that his expected profit will be maximum?

(b) Compute EPPI.

(c) The vendor is thinking of spending on a small market survey to obtain additional

information regarding the demand levels. How much should he be willing to spend on

such a survey?

2. (i) A decision problem has been expressed in the following payoff table.

Customer CategorySupplies level

A1 A2 A3 A4

E1 7 12 20 27

E2 10 9 10 25

E3 23 20 14 23

E4 32 24 21 17

Using Hurwicz principle to find optimal solution [ =0.5] (4 Marks)

(ii) The details for two competing alternatives are shown in table below.

Alternative - I Alternative - II

Annual Revenue Probability Annual Revenue Probability

2000 0 .15 4000 0. 3

3000 0.2 2500 0.4

5000 0.4 3500 0.2

4000 0.25 1200 0.1


Find the best alternative, when the interest rate is 0%. (4 Marks) 3. The research department of Hindustan lever has recommended to the marketing department

to launch a shampoo of three different types. The marketing manager has to decide one of

the types of shampoo to be launched under the following estimated payoffs for various

levels of sales.

Estimated Levels of Sale(units)

Types of shampoo 15,000 10,000 5,000

Egg Shampoo 30 10 10

Clinic Shampoo 40 15 5

Deluxe Shampoo 55 20 3What will be the marketing manager’s decision if (i) Maximin (ii) Minimax

(iii) Maximax (iv) Minimin (v) laplace and (vi) Hurwicz criterion [ =0.5] is applied?

4. A motor parts dealer finds that the cost of a particular item in stock for a week is Rs.20/-

and the cost of a unit shortage is Rs. 50/-. The probability distribution of weekly sales in

terms of hundreds is as follows.

Weekly sales (’00 Units) 0 1 2 3 4 5 6

Probability 0.1 0.1 0.2 0.2 0.2 0.15 0.05

How many units per week should the dealer order? Also find the expected value of

perfect information.

5. The parker flower shop promises its customers delivery within four hours on all flower

orders. All flowers are purchased on the previous day and delivered to parker by 8.00AM.

the next morning. Parker’s daily demand for roses is as follows

Demand 7 8 9 10

Probability 0.1 0.2 0.4 0.3

Parker purchases roses for Rs.10.00 per dozen and sells them for Rs.30.00. All unsold

roses are donated to a local hospital. How many dozens of roses should parker order each

evening to maximize its profits? What is the optimum expected profit?

6. An ice-cream retailer buys ice-cream at a cost of Rs.5 per cup and sells it for Rs.8 per cup;

any remaining unsold at the end of the day can be disposed of at a salvage price of Rs. 2 per

cup. Past sales have ranged between 15 and 18 cups per day; there is no reason to believe


that sales volume will take on any other magnitude in future. Find the EMV if the sale

history has the following probabilities.

Market size 15 16 17 18


7. A company is going to develop a new product in the market. Three alternative decisions are

available for the management.

A1: Advertising in a television. Advertising cost is Rs. 3000/- per day

A2: Appointing salesmen for marketing. Cost is Rs. 1200/- per day

A3: Conducting exhibition cost Rs. 900/- per day.

The unit price is fixed at Rs. 25/- product and the cost of units associated with the

respective decision alternatives are 9, 5 and 11. The expected demands for the item are as

follows.

Demand 200 300 400 500


The company has to decide the best alternative decisions among the three decisions.

8. Dhana & Co is currently working with a process which after paying for materials, labour,

etc… brings a profit of Rs. 12,000. The following alternatives are made available to the

company.

a) The company can conduct research (R1) which is expected to cost Rs. 10,000

having 90% chances of success, the company gets a gross income of Rs. 25,000

b) The company conduct research (R2) which is expected to cost Rs. 8,000 having a

probability of 60% success, the gross income will be Rs. 25,000.

c) The company can pay Rs. 6000 as royalty for a new process which will bring a

gross income of Rs. 20,000.

d) The company continues the current process.

Because of limited resources, it is assumed that only one of the two types of research

can be carried out at a time. Use decision tree analysis to locate the optimal strategy for the

company.


9. A department store with a bakery section is faced with the problem of how many cakes to

buy in order to meet the day’s demand. The departmental store prefers not to sell day old

cakes in competition, left over cakes are therefore, a complete loss. On the other hand, if a

customer desires a cake and all of them have been sold, the customer will buy elsewhere

and the sales will be lost. The store has therefore, collected information on the past sales

based on selected 100 day period as shown in table below.

Sales per day 15 16 17 18

No. of days 20 40 30 10


Construct the conditional profit and the opportunity loss tables. What is the optimal number

of cakes that should be bought each day? A cake costs Rs. 2/- and sells for Rs. 2.50.

10. The research department of consumer products division has recommended to the marketing

department to launch soap with three different perfumes. The marketing manager has to

decide the type of perfume to launch under the following estimated payoff for the various

levels of sales.


Types of Perfume 20,000 10,000 2,000

I 250 15 10

II 40 20 5

III 60 25 3Examine which type can be chosen under (a) Maximax; (b) Minimax; (c) Maximin;

(d) Laplace and (e) Hurwicz criterion [use =0.5]

-------------------


VINAYAKA MISSIONS UNIVERSITY, SALEM

FACULTY OF MANAGEMENT STUDIES

(MBA 2009 REGULATION)

II SEMESTER

Optimization Models for business decisions

Model Question Paper

Duration: 3 hrs Max Marks: 50

Answer all questions. All questions carry equal marks

Part – A (10 X 1 = 10 Marks)

1. Define Operations Research.

2. What is a slack variable?

3. Define non – degeneracy in a transportation problem.

4. State the objectives of travelling salesman problem.

5. Define the term idle time.

6. Define zero sum game. Give an Example.

7. Define an activity.

8. What are the types of failures in replacement theory?

9. List out the different environments in which decisions are made.

10. Define state of nature.

Part – B (5 X 8 = 40 Marks)

11. (a) Naga & Co has two bottling plants, one located at Gandhi Nagar and another at Nehru

Nagar. Each plant produces three drinks, whisky, beer and fruit juices named A, B and

C respectively. The numbers of bottles produced per day are as follows.

ProductsPlant at

Gandhi Nagar Nehru Nagar

Whisky (A) 1500 1500

Beer (B) 3000 1000

Fruit Juices (C) 2000 5000


A market survey indicates that during the month of April, there will be a demand of

20,000 bottles of whisky, 40,000 bottles of beer and 44,000 bottles of fruit juices. The

operating costs per day for plants at Gandhi Nagar and Nehru Nagar are 600 and 400

monetary units. For how many days each plant be run in April so as to minimize the

production cost, while still meeting the market demand?

(Or)

(b) Show that there does not exists any feasible solution to the following LPP Max. Z = 2X1 + 3X2 + 5X3


3X1 +10 X2 + 5X3 15

33X1 -10 X2 + 9X3 33

X1 + 2X2 + X3 4

and X1, X2, X3 0

12. (a) Solve the transportation problem with unit TP cost in rupees and units of demand and

supply as given below.

Destination

Source D1 D2 D3 D4 Capacity

S1 10 13 11 8 65

S2 9 12 12 10 44

S3 13 9 11 9 41

Requirement 60 40 55 45.

(Or)

(b) A department head has four tasks to be performed and three subordinates, the

subordinates differ in efficiency. The estimates of the time, each subordinate

would take to perform, is given below in the table. How should he allocate the

tasks one to each man, so as to minimize the total man hours?

Men

Task A B C

I 9 26 15

II 13 27 6

III 35 20 15


IV 18 30 2013. (a) Find the sequence that minimizes the total elapsed time (in hrs) required to complete the

following jobs on 3 machines in the order M1 - M2 - M3.

Job : A B C D E F G

Machine M1 : 3 8 7 4 9 8 7

Machine M2 : 4 3 2 5 1 4 3

Machine M3 : 6 7 5 11 5 6 12

Also calculate the idle time for the machines.

(Or)

(b) Using dominance property to solve the game

Player B

I II III IV

I -5 3 1 20

Player A II 5 5 4 6

III -4 -2 0 -5

14. (a) Consider a project consisting of 7 jobs A, B, …….., G with the following precedence

relations and time estimates(days).

Job Predecessors a m b

A - 2 5 8

B A 6 9 12

C A 5 14 17

D B 5 8 11

E C, D 3 6 9

F - 3 12 21

G E, F 1 4 7

Draw the project network and find the probability that the project will be completed,

(i) 4 days earlier than expected.

(ii) Not more than 4 days later than expected

(iii) If the project due date is 30 days, find the probability of meeting the due date.

(iv) If the project due date is 35 days, find the probability of meeting the due date.

(Or)


(b) The yearly cost of two machines A & B when the money neglected is as follow.

Year 1 2 3

MA 2900 2600 2700

MB 3400 2100 2700

Find their cost patterns if money value is 10% worth and hence find which machine is

most economical.

15. (a) The research department of Hindustan lever has recommended to the marketing

department to launch a shampoo of three different types. The marketing manager

has to decide one of the types of shampoo to be launched under the following

estimated payoffs for various levels of sales.


Types of shampoo 15,000 10,000 5,000

Egg Shampoo 30 10 10

Clinic Shampoo 40 15 5

Deluxe Shampoo 55 20 3What will be the marketing manager’s decision if (i) Maximin (ii) Minimax

(iii) Maximax (iv) Minimin (v) laplace and (vi) Hurwicz criterion [ =0.5] is applied?

(Or)

(b) The parker flower shop promises its customers delivery within four hours on all flower

orders. All flowers are purchased on the previous day and delivered to parker by

8.00AM. the next morning. Parker’s daily demand for roses is as follows

Demand 7 8 9 10


Parker purchases roses for Rs.10.00 per dozen and sells them for Rs.30.00. All unsold

roses are donated to a local hospital. How many dozens of roses should parker order each

evening to maximize its profits? What is the optimum expected profit?

--------------------



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