CS6704 RESOURCE MANAGEMENT TECHNIQUESpit.ac.in/pitnotes/uploads/CS6704_QB_CSE.pdfregion of an LPP is...

CS6704 – RESOURCE MANAGEMENT TECHNIQUES

CS6704 RESOURCE MANAGEMENT TECHNIQUES L T P C

3 0 0 3

OBJECTIVES: The student should be made to:

Be familiar with resource management techniques.

Learn to solve problems in linear programming and Integer programming.

Be exposed to CPM and PERT.

UNIT I LINEAR PROGRAMMING 9

Principal components of decision problem – Modeling phases – LP Formulation and graphic

solution – Resource allocation problems – Simplex method – Sensitivity analysis.

UNIT II DUALITY AND NETWORKS 9

Definition of dual problem – Primal – Dual relationships – Dual simplex methods – Post

optimality analysis – Transportation and assignment model - Shortest route problem.

UNIT III INTEGER PROGRAMMING 9

Cutting plan algorithm – Branch and bound methods, Multistage (Dynamic) programming.

UNIT IV CLASSICAL OPTIMISATION THEORY 9

Unconstrained external problems, Newton – Ralphson method – Equality constraints – Jacobean

methods – Lagrangian method – Kuhn – Tucker conditions – Simple problems.

UNIT V OBJECT SCHEDULING 9

Network diagram representation – Critical path method – Time charts and resource leveling –

PERT.

TOTAL: 45

PERIODS OUTCOMES: Upon Completion of the course, the students should be able to:

Solve optimization problems using simplex method.

Apply integer programming and linear programming to solve real-life applications.

Use PERT and CPM for problems in project management

TEXT BOOK:

1. H.A. Taha, “Operation Research”, Prentice Hall of India, 2002.

REFERENCES:

1. Paneer Selvam, „Operations Research‟, Prentice Hall of India, 2002

2. Anderson „Quantitative Methods for Business‟, 8th Edition, Thomson Learning, 2002.

3. Winston „Operation Research‟, Thomson Learning, 2003.

4. Vohra, „Quantitative Techniques in Management‟, Tata Mc Graw Hill, 2002.

5. Anand Sarma, „Operation Research‟, Himalaya Publishing House, 2003.

Course Outcome

C102.1 Ability to solve Linear Programming problems.

C102.2 Having the knowledge about duality and Networks and ability to solve

problems Transportation and Assignment problems.

C102.3 Ability to solve Integer Programming problems.

C102.4 Having the knowledge about Classical Optimization Theory.

C102.5 Ability to use CPM and PERT for problems in project management.

UNIT I – LINEAR PROGRAMMING

PART – A

1. What is Operations Research?

Answer:

Operations research is a study of optimization techniques. It is applied decision theory.

OR is the application of scientific methods, techniques and tools to problems involving the

operations of systems so as to provide these in control of operations with optimum solutions

to the problem.

2. List some applications of OR.

Answer:

Optimal assignment of various jobs to different machines and different operators.

To find the waiting time and number of customers waiting in the queue and

system in queuing model.

To find the minimum transportation cost after allocating goods from different

origins to various destinations in transportation model.

Decision theory problems in marketing, finance and production planning and

control.

3. What are the various types of models in OR?

Answer:

Models by function

Descriptive model

Predictive model

Normative model.

Models by structure

Iconic model

Analogue model

Mathematical model

Models by nature of environment

a. Deterministic model

b. Probabilistic model

4. What are main characteristics of OR?

Answer:

Examination of functional relationship from a system overview.

Utilization of planned approach

Adaptation of planned approach

Uncovering of new problems for study

5. Name some characteristics of good model.

Answer:

The number of assumptions made should be as few as possible

It should be easy as possible to solve the problem

The number of variables used should be as few as possible.

It should be more flexible to update the changes over a period of time without

change in its framework.

6. What are the different phases of OR?

Answer:

Formulation of the problem.

A mathematical model has to be constructed, with an objective function to be

optimized and constraints in the form of inequalities or equalities.

The solution of the model can be obtained by analytic or iterative method

depending on the structure of the model.

7. List out the advantages of OR.

Answer:

Optimum use of managers production factors

Improved quality of decision

Preparation of future managers by improving their knowledge and skill

Modification of mathematical solution before its use.

8. What are the limitations of OR?

Answer:

Mathematical model do not take into account the intangible factors such as

human relations etc. cannot be quantified.

Mathematical models are applicable to only specific categories of problems.

Requires huge calculations. All these calculations cannot be handled manually

and require computers which bear heavy cost.

9. What is linear programming?

Answer:

Linear programming is a technique used for determining optimum utilization of limited

resources to meet out the given objectives. The objective is to maximize the profit or

minimize the resources (men, machine, materials and money)

10. What is the mathematical model of linear programming?

Answer:

The general linear programming problem with decision variables and constraints can be

stated in the following form

Optimize (Max. or Min.)

∑

Su ject to the linear constraints ∑ ( )

and

11. What are the characteristic of LPP?

Answer:

There must be a well defined objective function.

There must be alternative course of action to choose.

Both the objective functions and the constraints must be linear equation or

inequalities.

12. What are the characteristic of standard form of LPP?

Answer:

The objective function is of maximization type.

All the constraint equation must be of equal type by adding slack or surplus

variables.

RHS of the constraint equation must be positive type.

All the decision variables are of positive type.

13. What are the characteristics of canonical form of LPP?

Answer:

In canonical form, if the objective function is of maximization type, then all constraints

are of ≤ type. Similarly if the objective function is of minimization type, then all constraints

are of ≥ type. But non-negative constraints are ≥ type for both cases.

14. A firm manufactures two types of products A and B and sells them at profit of Rs 2 on

type A and Rs 3 on type B. Each product is processed on two machines M1 and

M2.Type A requires 1 minute of processing time on M1 and 2 minutes on M2 Type B

requires 1 minute of processing time on M1 and 1 minute on M2. Machine M1 is

available for not more than 6 hours 40 minutes while machine M2 is available for 10

hours during any working day. Formulate the problem as a LPP so as to maximize the

profit.

Solution:

Maximize Su ject to the constraints

and

15. A company sells two different products A and B, making a profit of Rs.40 and Rs. 30

per unit on them, respectively. They are produced in a common production process and

are sold in two different markets; the production process has a total capacity of 30,000

man-hours. It takes three hours to produce a unit of A and one hour to produce a unit

of B. The market has been surveyed and company official feel that the maximum

number of units of A that can be sold is 8,000 units and that of B is 12,000 units. Subject

to these limitations, products can be sold in any combination. Formulate the problem as

a LPP so as to maximize the profit.

Solution:


and

16. What is feasibility region? Is it necessary that it should always be a convex set?

Answer:

Each constraint of the LPP represents a region and the region common to all these

(intersection of the sets) gives the feasible region of the given LPP.

A region in which all the constraints are satisfied is called feasible region. The feasible

region of an LPP is always convex set.

17. Define feasible solution, basic feasible solution and optimal solution of LPP.

Answer:

In a LPP If a set of values of the variables satisfy all the constraints then it is called a

feasible solution.

If a basic solution is also a feasible solution then it is a basic feasible solution.

A basic feasible solution which optimizes the objective function is called optimal

solution.

18. State the applications of linear programming.

Answer:

Linear programming is the most widely used technique of decision making in business

and industry and in military, agricultural applications, production management, Financial

management, marketing management etc.

19. State the Limitations of LP.

Answer:

Linear programming treats all relationships among decision variables as linear. However,

generally, neither the objective function nor the constraints in real life situations concerning

business and industrial problems are linearly related to the variables.

Linear programming model does not take into consideration the effect of time and

uncertainty.

20. What do you understand by redundant constraints?

Answer:

In a given LPP any constraint does not affect the feasible region or solution space then

the constraint is said to be a redundant constraint.

21. Define unbounded solution and Multiple Optimal solutions?

Answer:

Unbounded solution: If a feasible region does not exist then the given LPP has no

solution and if the feasible region is not closed polygon then the problem has unbounded

solution.

Multiple Optimal solutions: If two or more set of values of decision variables give the

same optimum value then the LPP has multiple optimal solution.

22. What is slack variable?

Answer:

The variable which is used to convert the constraint inequality is of upper bound

type ( ) into equality is called slack variable.

23. What are surplus variables?

Answer:

The variable which is used to convert the constraint inequality is of lower bound

type ( ) into equality is called surplus variable.

24. Define non degenerate basic feasible solution and degenerate basic solution?

Answer:

A basic feasible solution is called degenerate if value of at least one basic variable is

zero.

A basic feasible solution is called non degenerate if values of all basic variables are

non zero and positive.

25. From the optimum simplex table how do you identify that the LPP has no solution?

Answer:

To find the leaving variables the ratio is computed. The ratio is then there is an

unbounded solution to the given LPP.

26. Define non basic variable and basic variable in linear programming.

Answer:

If the number of equations is less than the number of variables then the variables in

excess are called non basic variables and in the process of solving the equations they are

treated as constraints.

The variables taken for solution are called basic variables.

27. Define artificial variable.

Answer:

If a LPP has type inequality constraint, the surplus variable cannot become a starting

basic variable. In order to obtain an initial basic feasible solution we introduce another

variable called artificial variable.

28. What do you understand by degeneracy?

Answer:

The concept of obtaining a degenerate basic feasible solution in LPP is known as

degeneracy. This may occur in the initial stage when at least one basic variable is zero in the

initial basic feasible solution.

29. How do you identify that LPP has no solution in a two phase method?

Answer:

If all & then at least one artificial variable appear in the optimum basis at

non zero level the LPP does not possess any solution.

30. Define unrestricted variable.

Answer:

A variable is unrestricted if it is allowed to take on positive, negative or zero values.

31. What is pseudo optimal solution?

Answer: If atleast one artificial variable appears in the basis at non zero level(with positive value

in XB column) and the optimality condition is satisfied, then the original problem has no feasible

solution . The solution satisfies the constraints but does not optimize the objective function since

it contains a very large penalty M and is called pseudeo optimal solution

32. What is sensitivity analysis?

Answer: The investigation that deal with changes in the optimal solutions due to discrete variations in the

parameters , and cj are called sensitivity analysis.

PART – B

1. Use graphical method to solve the following LPP.


and

2. Solve LPP by graphical method.

Maximize Su ject to

and

3. Solve by graphically.


and

4. Solve LPP by graphical method.

Maximize Su ject to and

5. Use Two – Phase simplex method to solve the following LPP.


6. Use Big-M method to solve the following LPP.



Minimize Su ject to

and




Maximize Su ject to 3

and

10. Use artificial variable technique to solve the LPP.

Maximize Su ject to

and

11. Use Simplex method to solve the LPP.

Minimize Su ject to

12. In the LPP.


and

If is kept fixed at 45 determine how much can be changed without affecting the optimal

solution?

13. In the LPP

Maximize Su ject to

and

(i) What is the new optimal solution when is increased from 2 to 12?

(ii) Find the effect of changing the objective function to .

14. Solve the following LPP.


Discuss the change in the optimal solution when (i) a constraint is added (ii) a

constraint is added.

15. A manufacturer makes two components, T and A, in a factory that is divided in to two

shops. Shop I ,which performs the basic assembly operation, must work 5 man-days on

each component T but only 2 man- days on each component A. Shop II, which performs

finishing operation, must work 3 man-days for each of component T and A it produces.

Because of men and machine limitations, Shop I has 180 man-days per week available,

While Shop II has 135 man-days per week. If the manufacturer makes a profit of Rs.300 on

each component T and Rs. 200 on each component A, how many of each should be

produced to maximize his profit. Use simplex method.

16. Explain the types of models. Also explain the characteristics of a good model along with the

principles involved in modeling.

17. Elucidate the procedure for formulating a linear programming problems. Explain the

advantages and limitation of linear programming.

18. Discuss the steps involved in formulation of LP problem

19. Solve by graphical method Maximize Z= 0.75a+b

Subject to : a+b 0 ,-0.5a+b 1 where a,b

20. A factory manufactures two products A and B on three machines X,Y and Z. Product A

requires 10 hours of machine X and 5 hours of machine Y a one hour of machine Z. The

requirement of product B is 6 hours , 10 hours and 2 hours of machine X,Y and Z

respectively. The profit of the contribution of the products A and B are Rs.23 per unit and

Rs.32 per unit respectively. In the coming planning period the available capacity of

machines X,Y and Z are 2500 hours,2000 hours and 500 hours respectively. Find the

optimal product mix for maximizing the product.

21. An automobile manufacturer makes auto –mobile and trucks in a factory that is divided into

two shops. Shop A, which performs the basic assembly operation must work 5 man days on

each truck but only 2 man days on each automobile. Shop B, which performs finishing

operation must work 3 man days for each truck or automobile it produces. Because of men

and machine limitations shop A has 180 man days per week available while Shop B has

135 man days per week. If the manufacturer makes a profit Rs.800 on each Truck and

Rs.200 on each automobile , how many of each should he produce to maximize his profit.

22. Garden Ltd.has two product Rose and Lotus. To produce one unit of Rose, 2 unit of materil

X and 4 units of material Y are required. To produce one unit of Lotus, 3 unit of materiel X

and 2 units of material Y are required. Atleast 16 units of each materiel must be used in

order to meet the committed sales of Rose and Lotus cost per unit of material X and Y are

Rs.2.50 per unit and Rs.0.25 per unit respectively

i) To formulate the mathematical model

ii) To solve it for minimum cost(Graphically)

UNIT II – DUALITY AND NETWORKS

PART – A

1. What is dual?

Answer:

In the context of linear programming, duality implies that each linear programming

problem can be analyzed in two different ways but having equivalent solutions. Each LPP

stated in its original form has associated with another LPP called dual LPP or in short dual.

2. What is dual simplex method?

Answer:

The dual simplex method is used to solve problems which start with dual principal i.e

whose primal is optimal but infeasible. In this method the solutions starts with optimum but

infeasible and remains infeasible until the true optimum is reached at which the solution

becomes feasible.

3. State the fundamental theorem of duality.

Answer:

If either the primal or dual problem has a finite optimal solution, then the other one also

possess the same, and the optimal values of the objective function of the two problems are

equal.

4. Define transportation problem.

Answer:

It is a special type of linear programming model in which the goods are shipped from

various origins to different destinations. The objective is to find the best possible allocation

of goods from various origins to different destinations such that the total transportation cost

is minimum.

5. Define the following: feasible solution & basic feasible solution.

Answer:

Feasible solution: A set of non-negative decision values (

) satisfies the constraint equation is called a feasible solution.

Basic feasible solution: A basic feasible solution is said to be basic if the number of

positive allocations are ( ) ( -origin and -destination).If the number of

allocations are less than ( ) it is called degenerate basic feasible solution.

6. Define optimal solution in transportation problem.

Answer:

A feasible solution is said to be optimal, if it minimizes the total transportation cost.

7. What are the methods used in transportation problem to obtain the initial basic feasible

solution.

Answer:

North West corner method

Row minima method

Column minima method

Least cost method

Vogel‟s approximation method

8. Write down the basic steps involved in solving a transportation problem.

Answer:

Formulate the problem and set up in the matrix form

Obtain an initial basic feasible solution

Test the initial solution for optimality

Updating the solution

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

Answer:

When the number of positive allocation (Values of decision variables) at any stage of the

feasible solution is less than the required number (Rows+Columns-1), i.e. number of

independent constraint equations, the solution is said to be degenerate, otherwise non-

degenerate.

10. What is balanced transportation problem & unbalanced transportation problem?

Answer:

When the total supply equals total demand, the problem is called balanced transportation

problem, otherwise unbalanced transportation problem.

11. How do you convert an unbalanced transportation problem into a balanced one?

Answer:

The unbalanced transportation problem can be made balanced by adding a dummy supply

center (Row) or a dummy demand center (Column) as the need arises.

12. Explain how the profit maximization transportation problem can be converted to an

equivalent cost minimization transportation problem.

Answer:

If the objective is to maximize the profit or maximize the expected sales we have to

convert these problems by multiplying all cell entries by -1.Now the maximization problem

becomes a minimization and it can be solved by the usual algorithm

13. Define transshipment problem.

Answer:

A problem in which available commodity frequently moves from one source to another

source or destination before reaching its actual destination is called transshipment problems.

14. What is the difference between Transportation problem & Transshipment Problem?

Answer:

In a transportation problem there are no intermediate shipping points while in

transshipment problem there are intermediate shipping points

15. What is assignment problem?

Answer:

An assignment problem is a particular case of transportation problem where the objective is

to assign a number of resources to an equal number of activities so as to minimize total cost

or maximize total profit of allocation.

16. Explain the difference between transportation and assignment problems?

Answer:

S.No. Transportation Problems Assignment Problems

1. Supply at any source may be a

positive number.

Supply at any source will be 1.

2. Demand at any destination may be a

positive number.

Demand at any destination will be

1.

3. One or more source to any number of

destinations.

One source one destination.

17. Define unbounded assignment problem and describe the steps involved in solving it?

Answer:

If the no. of rows is not equal to the no. of column in the given cost matrix the problem is

said to be unbalanced. It is converted to a balanced one by adding dummy row or dummy

column with zero cost.

18. Explain how a maximization problem is solved using assignment model?

Answer:

Since ( ), we multiply all the elements of the table

by and then apply Hungarian method to find the solution.

19. What do you understand by restricted assignment? Explain how you should overcome

it?

Answer:

The assignment technique, it may not be possible to assign a particular task to a particular

facility due to technical difficulties or other restrictions. This can be overcome by assigning a

very high processing time or cost (it can be ∞) to the corresponding cell.

20. How do you identify alternative solution in assignment problem?

Answer:

Sometimes a final cost matrix contains more than required number of zeroes at the

independent position. This implies that there is more than one optimal solution with some

optimum assignment cost.

21. What is a traveling salesman problem?

Answer:

The travelling salesman problem is also an assignment problem in which going from a city to

itself is not allowed and the travel should be a cycle passing through all the cities and no city

visited twice.

22. Define route condition.

Answer:

The salesman starts from his headquarters and passes through each city exactly once.

23. Give the areas of operations of assignment problems.

Answer:

Assigning jobs to machines.

Allocating men to jobs/machines.

Route scheduling for a traveling salesman.

24. How do you convert the unbalanced assignment problem into a balanced one?

Answer:

If the number of jobs and the number of workers are not equal it is called unbalanced

assignment problem. We introduced a dummy job or a dummy worker in order to balance the

problem. We assign the cost zero for every cell in the newly introduced row or column.

PART – B

1. Apply dual simplex method to solve:

Minimize Su ject to 2

and

2. Use dual simplex method to solve:


3. Apply dual simplex method to solve:

Minimize Su ject to 2 and



and


Maximize Su ject to

and


Minimize Su ject to 3

nd

7. Find the initial basic feasible solution for the transportation problem by VAM

Availability

11 13 17 14 250

16 18 14 10 300

21 24 13 10 400

Requirement 200 225 275 250

8. Solve the following transportation problem

Availability

6 8 8 5 30

5 11 9 7 40

8 9 7 13 50

Demand 35 28 35 25 120

9. Solve the transportation problem

Availability

8 7 3 60

3 8 9 70

11 3 5 80

Demand 50 80 80 210

10. Find the IBFS of the following TP by VAM and hence find the optimum solutions

Supply

5 1 7 10

6 4 6 80

3 2 5 15

Demand 45 20 40

11. Solve the following transportation problem

Supply

4 2 3 2 6 30

5 4 5 2 1 40

6 5 4 7 3 50

Demand 35 28 35 25

12. Solve the following transportation problem to maximize the total profit

Market

Supply

15 51 42 33 23

Source 80 42 26 81 44

90 40 66 60 33

Demand 23 31 16 30 100

13. Solve the following assignment problem

Jobs

10 15 24 30

Workers 16 20 28 10

12 18 30 16

9 24 32 18

14. Solve the following AP

Operators

10 5 13 15

Machines 3 9 18 3

10 7 3 2

5 11 9 7

15. Solve the following AP

Machines

1 2 3 4 5

5 5 - 2 6

7 4 2 3 4

Operators 9 3 5 - 3

7 2 6 7 2

6 5 7 9 1

16. A company has 4 salesmen and . These salesmen are to be allotted 4 districts

and . The estimated profit per day for each salesman in each district is given in the

following table

2 3 4

16 10 14 11

14 11 15 15

15 15 13 12

13 12 14 15

What is the optimal assignment which will yield maximum profit?

17. There are five jobs and four machines. The expected profits on each job on each machine are

given below. Determine an optimal assignment of the machines to the jobs so that the total

profit is maximum.

Job

1 2 3 4 5

I 62 78 50 101 82

Machine II 71 84 61 73 59

III 87 92 111 71 81

IV 48 64 87 77 80

18. Solve the following travelling salesman problem

- 4 7 3

4 - 6 3

7 6 - 7

3 3 7 -

19. Solve the following travelling salesman problem

- 5 12 6 4 8

6 - 10 5 4 3

8 7 - 6 3 11

5 4 11 - 5 8

5 2 7 8 - 4

6 3 11 5 4 -

20. Four captain pilots (CP1,CP2, CP3,CP4) has evaluated four flight officers

(FO1,FO2,FO3,FO4) according to perfection, adaptation, morale motivation in a 1-20 scale

(`1: very good, 20: very bad). Evaluation grades are given in the table. Flight company wants

to assign each flight officers to a captain pilot according to evaluation. Determine possible

flight crews

FO1 FO2 FO3 FO4

CP1 2 4 6 10

CP2 2 12 6 5

CP3 7 8 3 9

CP4 14 5 8 7

21. Obtain and optimum basic feasible solution to the following transportation problem:

TO Available

From

7 3 2 2

2 1 3 3

3 4 6 5

Demand 4 1 5 10

22. Solve the following assignment problem for maximization given the profit matrix(profit

in rupees)

Machines

P Q R S

A 51 53 54 50

Job B 47 50 48 50

C 49 50 60 61

D 63 64 60 60

UNIT III – INTEGER PROGRAMMING

PART – A

1. What is Integer Programming Model?

Answer:

In a LPP if all the decision variables are required to assume non negative integer values is

called pure integer values, it is called integer programming model.

2. Explain the importance of Integer programming problem.

Answer:

In LPP the values for the variables are real in the optimal solution. However in certain

problems this assumption is unrealistic. For example if a problem has a solution of 81/2 cars

to be produced in a manufacturing company is meaningless. These types of problems require

integer values for the decision variables. Therefore IPP is necessary to round off the

fractional values.

3. List out some of the applications of IPP.

Answer:

IPP occur quite frequently in business and industry.

All transportation, assignment and traveling salesman problems are IPP, since the

decision variables are either Zero or one.

All sequencing and routing decisions are IPP as it requires the integer values of the

decision variables.

Capital budgeting and production scheduling problem are PP. In fact, any situation

involving decisions of the type either to do a job or not to do can be treated as an IPP.

All allocation problems involving the allocation of goods, men, machines, give rise to

IPP since such commodities can be assigned only integer and not fractional values.

4. List the various types of integer programming.

Answer:

Pure integer problem

Mixed integer problem

5. What is pure IPP?

Answer:

In a linear programming problem, if all the variables in the optimal solution are restricted

to assume non-negative integer values, then it is called the pure (all) IPP.

6. What is Mixed IPP?

Answer:

In a linear programming problem, if only some of the variables in the optimal solution are

restricted to assume non-negative integer values, while the remaining variables are free to

take any non-negative values, then it is called A Mixed IPP.

7. What is Zero-one problem?

Answer:

If all the variables in the optimum solution are allowed to take values either 0 or 1 as in

„do‟ or „not to do‟ type decisions, then the problem is called Zero-one problem or standard

discrete programming problem.

8. What is the difference between pure integer programming & mixed integer

programming?

Answer:

In an optimization problem, if all the decision variables are restricted to take integer

values, then it is referred as pure integer programming. If some of the variables are allowed

to take integer values, then it is referred as mixed integer programming.

9. Explain the importance of Integer Programming?

Answer:

In linear programming problem, all the decision variables allowed to take any non-

negative real values, as it is quite possible and appropriate to have fractional values in many

situations. However in many situations, especially in business and industry, these decision

variables make sense only if they have integer values in the optimal solution. Hence a new

procedure has been developed in this direction for the case of LPP subjected to additional

restriction that the decision variables must have integer values.

10. Why not round off the optimum values instead of resorting to IP?

Answer:

There is no guarantee that the integer valued solution (obtained by simplex method) will

satisfy the constraints. i.e. ., it may not satisfy one or more constraints and as such the new

solution may not feasible. So there is a need for developing a systematic and efficient

algorithm for obtaining the exact optimum integer solution to an IPP.

11. What are methods for IPP?

Answer:

Integer programming can be categorized as

i. Cutting methods.

ii. Search Methods.

12. What is cutting method?

Answer:

A systematic procedure for solving pure IPP was first developed by R. E. Gomory in

1958. Later on, he extended the procedure to solve mixed IPP, named as cutting plane

algorithm; the method consists in first solving the IPP as ordinary LPP. By ignoring the

integrity restriction and then introducing additional constraints one after the other to cut

certain part of the solution space until an integral solution is obtained.

13. What is search method?

Answer:

It is an enumeration method in which all feasible integer points are enumerated. The

widely used search method is the Branch and Bound Technique. It also starts with the

continuous optimum, but systematically partitions the solution space into sub problems that

eliminate parts that contain no feasible integer solution. It was originally developed by A.H.

Land and A.G. Doig.

14. What is Branch and Bound Technique?

Answer:

The widely used search method is the Branch and Bound Technique. It starts with the

continuous optimum, but systematically partitions the solution space into sub problems that

eliminate parts that contain no feasible integer solution. It was originally developed by A.H.

Land and A.G. Doig.

15. Give the general format of IPP?

Answer:

The general form of IPP is given by

and some or all varia les are integers

16. Write an algorithm for Gomory’s Fractional Cut algorithm?

Answer:

1) Convert the minimization IPP into an equivalent maximization IPP and all the

coefficients and constraints should be integers.

2) Find the optimum solution of the resulting maximization LPP by using simplex

method

3) Test the integrity of the optimum solution.

4) Rewrite each .

5) Express each of the negative fractions if any, in the row of the optimum simplex

table as the sum of a negative integer and a non-negative fraction.

6) Find the fractional cut constraint.

7) Add the fractional cut constraint at the bottom of optimum simplex table obtained in

step 2

8) Go to step 3 and repeat the procedure until an optimum integer solution is obtained.

9)

17. What is the purpose of Fractional cut constraints?

Answer:

In the cutting plane method, the fractional cut constraints cut the unuseful area of the

feasible region in the graphical solution of the problem. i.e. cut that area which has no

integer-valued feasible solution. Thus these constraints eliminate all the non-integral

solutions without losing any integer-valued solution.

18. A manufacturer of baby dolls makes two types of dolls, doll and doll . Processing of

these dolls is done on two machines and . Doll requires 2 hours on machine and

6 hours on Machine . Doll requires 5 hours on machine and 5 hours on

Machine . There are 16 hours of time per day available on machine and 30 hours on

machine . The profit is gained on both the dolls is same. Format this as IPP?

Answer:

Let the manufacturer decide to manufacture number of doll and number of doll

so as to maximize the profit. The complete formulation of the IPP is given by

Maximize Su ject to the constraints and and are integers

19. Explain Gomory’s Mixed Integer Method?

Answer:

The problem is first solved by continuous LPP by ignoring the integrity condition. If the

values of the integer constrained variables are integers, then the current solution is an optimal

solution to the given mixed IPP. Else select the source row which corresponds to the largest

fractional part among these basic variables which are constrained to be integers. Then

construct the Gomarian constraint from the source row. Add this secondary constraint at the

bottom of the optimum simplex table and use dual simplex method to obtain the new feasible

optimal solution. Repeat this procedure until the values of the integer restricted variables are

integers in the optimum solution obtained.

20. What is the geometrical meaning of portioned or branched original problem?

Answer:

Geometrically it means that the branching process eliminates portion of the feasible

region that contains no feasible-integer solution. Each of the sub-problems solved separately

as a LPP.

21. What is standard discrete programming problem?

Answer:

If all the variables in the optimum solution are allowed to take values either 0 or 1 as in

„do‟ or „not to do‟ type decisions, then the problem is called standard discrete programming

problem.

22. What is the disadvantage of branched or portioned method?

Answer:

It requires the optimum solution of each sub problem. In large problems this could be

very tedious job.

23. How can you improve the efficiency of portioned method?

Answer:

The computational efficiency of portioned method is increased by using the concept of

bounding. By this concept whenever the continuous optimum solution of a sub problem

yields a value of the objective function lower than that of the best available integer solution it

is useless to explore the problem any further consideration. Thus once a feasible integer

solution is obtained, its associative objective function can be taken as a lower bound to delete

inferior sub-problems. Hence efficiency of a branch and bound method depends upon how

soon the successive sub-problems are fathomed.

24. What are the essential characteristics of dynamic programming problems?

Answer:

a) The problem can be divided in to stages with a policy decision required at each stage.

b) Each stage has a number of stages associated with it. The states are various possible

conditions in which the system may find itself at that stage of the problem. The number

of states may be finite or infinite.

c) The effect of the policy decision at each stage is to transform the current state into a state

associated with the next stage.

The current situation of the system at a stage is described by a set of variables called state

variables. It is defined to reflect the status of the constraints that bind all stages together

25. What is Dynamic programming?

Answer:

Dynamic programming is a mathematical technique of optimization using multistage

decision process. That is, the process in which a sequence of interrelated decision has to

be made. It provides a systematic procedure for determining the combination of decision

which maximize the overall effectiveness.

PART – B

1. Solve the following Integer Programming Problems (IIP)


and and are integers

2. Solve the IIP


2 and and are integers

3. Solve the IIP y Gomary‟s cutting plane method



4. Solve the IIP y Gomary‟s algorithm



5. Solve the following mixed integer programming problem


(

) and is an integers

6. Solve the following mixed IPP


and is an integers

7. Solve the following mixed IPP y Gomory‟s cutting plane algorithm


and is an integers

8. Using branch and bound method solve the IPP


and and integer





Minimize Su ject to the constraints







and are non negative integers

13. Factorize a positive number into factors such that their sum is minimum.

14. Solve the following problem using dynamic programming.

Minimize

Su ject to the constraints

15. Solve using dynamic programming


and



and



and

18. Solve the following LPP using dynamic programming approach:


and

19. Use Branch and Bound method to solve the following


and and integers

20. Bring out the characteristics of Dynamic programming.

21. A vessel is to e loaded with stocks of 3 items. Each items “j” has a weight of wj and a

value of vj. The maximum cargo weight the vessel can take is 5 and the details of the three

items are as follows

j wj vj

1 1 30

2 3 80

3 2 65

Develop the recursive equation for the above case and find the most valuable cargo load

without exceeding the maximum cargo weight by using Dynamic programming

UNIT IV – CLASSICAL OPTIMISATION THEORY

PART – A

1. What are stationary or critical points?

Answer:

The domain limits or end points of a continuous function ( ) of single independent

variable are generally called stationary or critical points.

2. What are local or relative extreme points?

Answer:

Local extreme points represent the maximum or minimum values of the function in the

given range of values of the variable.

3. What is global or absolute maximum?

Answer:

The global maximum value of a function is the maximum value among all local

maximum values of the function in the domain.

4. What is the necessary condition for to be an extreme point of ( )?

Answer:

( )

5. State Lagrange’s necessary and sufficient conditions in non linear programming.

Answer:

The necessary conditions for an optimum (Maximum or Minimum) of ( ) or ( ) are

the ( ) equations to be solved for ( )

unknowns ( )

∑

These ( ) necessary conditions are also become sufficient conditions for maximum (or

minimum) of the objective function ( ), in case it is concave (or convex) and constraints

are equalities respectively.

6. State Kuhn – Tucker necessary and sufficient conditions in non linear programming.

(Nov/Dec 2018)

Answer:

Necessary conditions:

∑

( ) ( )

Sufficient conditions:

The Kuhn – Tucker necessary conditions for the NLP problem are also sufficient

conditions if ( ) is concave and all ( ) are convex functions of .

7. What is Newton Raphson method

Answer: Newton Raphson method is finding the root of the real function by successive

approximation xn+1 = xn − f(xn)/ f1 (xn)

PART – B

1. Find the stationary points of ( ) ( ) ( ) using Newton – Raphson method.

2. Use N – R method to solve ( ) .

3. Use Newton – Raphson method to solve ( ) . for the root in (1,2)

4. Solve ( )

( ) by Nweton – Raphson method.

5. Consider the problem

Maximize ( )

Su ject to the constraints ( )

( )

Apply the Jacobian method to find ( ) in the neighborhood of the feasible point ( ).

Assume that this neighborhood is specified by

6. Obtain the solution to the following problem by using the method of Lagrangian multiplier

Minimize


7. Obtain the solution to the following problem by using the method of Lagrangian

Minimize

Su ject to

8. Obtain the solution to the following problem by using the method of Lagrangian multiplier

Maximize


9. Obtain the solution to the following problem by using the method of Lagrangian

Minimize

Su ject to

10. Use the method of Lagrangian multipliers to solve the following NLP problem. Does the

solution maximize or minimize the objective function.

Optimize

S.C

11. Use Kuhn – Tucker conditions to solve the following NLP

Maximize

Su ject to 2


Maximize

Su ject to


Minimize

Su ject to g1=

14. Apply Kuhn – Tucker conditions to solve the following NLP

Maximize Su ject to 2


Maximize

Su ject to

16. Apply Kuhn – Tucker conditions to solve the following NLP

Minimize ( ) ( )

Su ject to 0≤

17. Illustrate Kuhn-Tucker conditions with an example

18. Illustrate Newton –Raphson method with suitable example

UNIT V – OBJECT SCHEDULING

PART – A

1. What do you mean by project?

Answer:

A project is defined as a combination on inter related activities with limited resources

namely men, machines materials, money and time all of which must be executed in a defined

order for its completion.

2. What are the three main phases of project?

Answer:

Project planning phase

Scheduling phase

Project control phase

3. What are the two basic planning and controlling techniques in a network analysis?

Answer:

Critical Path Method (CPM)

Programme Evaluation and Review Technique (PERT)

4. What are the advantages of CPM and PERT techniques?

Answer:

It encourages a logical discipline in planning, scheduling and control of projects.

It helps to effect considerable reduction of project times and the cost.

It helps better utilization of resources like men, machines, materials and money with

reference to time.

It measures the effect of delays on the project and procedural changes on the overall

schedule.

5. Differentiate between PERT and CPM?

Answer:

PERT

An event oriented network.

Probabilistic nature.

Three time estimation.

CPM

Network is built on the basis of activity.

Deterministic nature.

One time estimation.

6. What is network?

Answer:

Networks are diagrams representing a sequence of activities involved in the project work.

7. What is Event in a network diagram?

Answer:

Event in a network diagram represent project milestones, such as the start or completion

of an activity or activities, and occur at a particular instant of time at which some specific

part of the project has been or is to be activated.

8. Define activity.

Answer:

Activities in a network diagram represent project operations or tasks to be conducted.

9. Define non critical activities.

Answer:

In a Network diagram critical activities are those whose if consume more than estimated

time the project will be delayed.

10. Define non critical activities?

Answer:

Activities which have a provision such that the event if they consume a specified time

over and above the estimated time the project will not be delayed are termed as non critical

activities.

11. Define Dummy Activities.

Answer:

An activity which does not consume either any resource and/or time is known as dummy

activity.

12. Define duration.

Answer:

It is the estimated or the actual time required to complete a trade or an activity.

13. Define total project time & Critical path.

Answer:

Total project time: It is time taken to complete to complete a project and just found from

the sequence of critical activities. In other words it is the duration of the critical path.

Critical path: It is the sequence of activities which decides the total project duration. It is

formed by critical activities and consumes maximum resources and time.

14. Define float or slack.

Answer:

The float (also called slack) of an event is the difference between its latest occurrence

time ( ) and the earliest occurrence time ( ). That is .

15. Define total float, free float and Independent float?

Answer:

Total Float: It is the amount of time by which an activity can be delayed if all its

preceding activities take place at their earliest possible times and following activities can be

allowed to wait until three latest permissible times.

Free Float: The time by which the completion of an activity can be delayed without

causing any delay in its immediate succeeding activities.

Independent Float: Independent float is the amount of time available when preceding

activities take place at their latest permissible times and all the following activities can still

take place at their earliest possible times. ( ) ( )

{ }

16. Define Optimistic.

Answer:

This is the shortest possible time required to perform an activity, assuming that

everything goes well. It is denoted by

17. Define Pessimistic.

Answer:

This is the longest possible time required to perform an activity under extremely bad

conditions. It is denoted by

18. Define most likely time.

Answer:

This is the most likely time required to perform an activity. If the activity was repeated

many times, then it is the duration that would occur most frequently. It is denoted

by

19. What is a critical path?

Answer:

The critical path is the continuous chain of critical activities in a network diagram. It is

the longest path starting from first to the last event and is shown by a thick line or double

lines in the network diagram.

20. What is a parallel critical path?

Answer:

When critical activities are crashed and the duration is reduced other paths may also

become critical such critical paths are called parallel critical path.

PART – B

1. Explain the following i) Difference between PERT and CPM. ii) Lagrangian method and

Khun-Tucker condition.

2. What is Critical Path Method and further bring out the usefulness of it?

3. The following ta le indicates the details of a project. The duration are in days „a‟ refers to

optimistic time, „m‟ refers to most likely time and „ ‟ refers to pessimistic time duration.

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

a 2 3 4 8 6 2 2

m 4 4 5 9 8 3 5

b 5 6 6 11 12 4 7

i)Draw the network ii) Find the critical path. iii) Determine the expected standard

deviation of the completion time.

4. A project consists of a series of jobs such that

. The time of completion of each job is given below

Job Time

(Days) 23 8 20 16 24 18 19 4 10

5. The following table gives the activities and their durations in their order of a project. Draw

the network and find the minimum time required for completion of the project.

Activity Preceding

Activity

Duration

(In Days)

-

-

-

9

4

7

8

7

5

10

8

6

6. Find the critical path of the project with the following activities

Job ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Time

(Days) 4 1 1 1 6 5 4 8 1 2 5 7

7. Find the critical path of the project having the tasks as given below:

Job Time Job Time

( ) ( ) ( ) ( ) ( ) ( )

2

7

3

3

5

3

( ) ( ) ( ) ( ) ( ) ( )

5

8

4

4

1

7

8. A project has the following data:

Task Optimistic time 2 3 5 5 8 7 9 3 6

Most likely time 3 5 8 7 10 9 11 8 8

Pessimistic time 4 7 11 9 12 11 13 13 10

Determine (i) Expected task times, (ii) Variance of the tasks, (iii) Critical path.

9. A project consists of the following activities and times estimates.

Activity Estimated duration in weeks

Optimistic Most likely Pessimistic

( ) ( ) ( ) ( ) ( ) ( ) ( )

1

1

2

1

2

2

3

1

4

2

1

5

5

6

7

7

8

1

14

8

15

i. Draw the network.

ii. Find the expected time and variance of each activity.

iii. What is the probability that the project will be completed 4 weeks earlier than the

expected time?

10. The activities and their estimates of time of a project are given below:

3 2 6 2 5 3 3 1 4 1 2

6 5 12 5 11 6 9 4 19 3 4

12 14 30 8 17 15 27 7 28 8 12

Precedence relationships are

What is the probability that the project will be completed in 35 days?

11. Draw the network diagram exactly with two dummies

Activity Must be

proceeded by

A -

B -

C B

D A,C

E A

F E

G E

H G

I D,F

J G,I

K G,I

L H,K

12. A small project is composed of 7 activities whose time estimates are listed below.

Activities are being identified by their beginning( i)and ending (j) node numbers

Activities Time in weeks

i j

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

i) Draw the network

ii) Calculate the expected variance for each

iii) Find the expected project completed time.

iv) Calculate the probability that the project will be completed at least 3 weeks than

expected.

v) If the project due date is 18 weeks. What is the probability.

13. For the project represented by the following network and the table showing time and cost,

find the optimum duration and cost.

Activity Normal Crash

Time

(Days)

Cost

(Rs.)

Time

(Days)

Cost

(Rs.)

( ) ( ) ( ) ( ) ( ) ( )

8

4

2

10

5

3

100

150

50

100

100

80

6

2

1

5

1

1

200

350

90

400

200

100

2

2

1

5

4

2

100

200

40

300

100

20

50

100

40

60

25

10

Indirect cost per day.

14. Find the optimum cost schedule for the project with the following schedule.


Time

(Days)

Cost

(Rs.)

Time

(Days)

Cost

(Rs.)

( ) ( ) ( ) ( ) ( ) ( ) ( )

5

6

3

5

7

6

8

100

120

60

80

100

120

160

3

3

2

3

4

2

6

220

195

100

170

250

240

220

2

3

1

2

3

4

2

120

75

40

90

150

120

60

60

25

40

45

50

30

30

Indirect cost is zero.

8

4

5

3

2

10

15. The following table gives the cost particulars of a project


Time

(Days)

Cost

(Rs.)

Time

(Days)

Cost

(Rs.)

( ) ( ) ( ) ( ) ( ) ( )

8

4

2

10

5

3

100

150

50

100

100

80

6

2

1

5

1

1

200

350

90

100

200

100

2

2

1

5

4

2

100

200

40

300

100

20

50

100

40

60

25

10

If the indirect cost is per day, find the least cost schedule (optimum duration).

16. A project has the following activities and other characteristics:

Time estimates (in weeks)

Activity Preceding

Activity

Most Optimistic Most likely Most

Pessimistic

A - 4 7 16

B - 1 5 15

C A 6 12 30

D A 2 5 8

E C 5 11 17

F D 3 6 15

G B 3 9 27

H E,F 1 4 7

I G 4 19 28

Required:

i) Draw the PERT network diagram

ii) Identify the critical path

iii) Prepare the activity schedule for the project

iv) Determine the mean project completion time

v) Find the probability that the project is completed in 36 weeks.

17. Draw the network form the following activity and the find the critical path and total

duration of project.

Activity Immediate

predecessors

Duration (weeks)

A - 3

B - 8

C A 9

D B 6

E C 10

F C 14

G C,D 11

H F,G 10

I E 5

J I 4

K H 1

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