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July 2011

Master of Computer Application (MCA) Semester 4

MC0079 Computer Based Optimization Methods 4 Credits

(Book ID: B0902) Assignment Set 1

1. Use graphical method and solve the L.P.P.Maximize Z = 5x1 + 3x2

Subject to 3x1 + 5x2 155x1 + 2x2 10

x1, x2 0

Ans: A LPP with 2 decision variables x1 and x2 can be solved easily by graphical method. We consider the

x1 x2 plane where we plot the solution space, which is the space enclosed by the constraints.

Usually the solution space is a convex set which is bounded by a polygon; since a linear function

attains extreme (maximum or minimum) values only on boundary of the region, it is sufficient to

consider the vertices of the polygon and find the value of the objective function in these vertices. By

comparing the vertices of the objective function at these vertices, we obtain the optimal solution of

the problem.

The method of solving a LPP on the basis of the above analysis is known as the graphical method.2.4.1

An Algorithm for solving a linear programming problem by Graphical Method:

(This algorithm can be applied only for problems with two variables).

Step I: Formulate the linear programming problem with two variables (if the given problem has

more than two variables, then we cannot solve it by graphical method).

Step II: Consider a given inequality. Suppose it is in the form

a1x1 + a2x2 clip_image047 b (or a1x1 + a2x2 clip_image015 [8] b). Then consider the relation a1x1 +

a2x2 = b. Find two distinct points (k, l), (c, d) that lie on the straight line a1x1 + a2x2 = b. This can be

found easily: If x1 = 0, then x2 = clip_image049. If x2 = 0, then x1 = clip_image051. Therefore

(k, l) = (0, clip_image049 [1]) and (c, d) = (clip_image051 [1],0) are two points on the straight line

a1x1 +a2x2 = b.

Step III: Represent these two points (k, l), (c, d) on the graph which denotes XY-axis plane. Join

these two points and extend this line to get the straight line which represents a1x1 + a2x2 = b.

Step IV: a1x1 + a2x2 = b divides the whole plane into two half planes, which are a1x1+ a2x2

clip_image047 [1] b (one side) and a1x1 + a2x2 clip_image055 b (another side).Find the half plane

that is related to the given inequality.

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Step V: Do step-II to step-IV for all the inequalities given in the problem. The intersection of the half-

planes related to all the inequalities and x1 clip_image055 [1] 0,x2 clip_image055[2] 0 , is called the

feasible region ( or feasible solution space). Now find this feasible region.

Step VI: The feasible region is a multisided figure with corner points A, B, C, (say).Find the co-

ordinates for all these corner points. These corner points are called as extreme points.

Step VII: Find the values of the objective function at all these corner/ extreme points.

Step VIII: If the problem is a maximization (minimization) problem, then the maximum (minimum)

value of z among the values of z at the corner/extreme points of the feasible region is the optimal

value of z. If the optimal value exists at the corner/extreme point, say A(u, v), then we say that the

solution x1 = u and x2 = v is an optimal feasible solution.

Step IX: Write the conclusion (that include the optimum value of z, and the co-ordinates of the

corner point at which the optimum value of z exists).

2. Explain the algorithm for solving a linear programming problem by graphical method.Ans: Linear programming (LP) is a mathematical method for determining a way to achieve the best

outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of

requirements represented as linear equations.

More formally, linear programming is a technique for the optimization of a linear objective function,

subject to linear equality and linear inequality constraints. Given a polytope and a real-valued affine

function defined on this polytope, a linear programming method will find a point on the polytope

where this function has the smallest (or largest) value if such point exists, by searching through the

polytope vertices.

An Algorithm for solving a linear programming problem by Graphical Method: (This algorithm can be

applied only for problems with two variables).

Step I: Formulate the linear programming problem with two variables (if the given problem has

more than two variables, then we cannot solve it by graphical method).

Step II: Consider a given inequality. Suppose it is in the forma1x1 + a2x2 = b).

Then consider the relation a1x1+ a2x2=b. Find two distinct points (k, l), (c, d) that lie on the straight

line a1x1+ a2x2= b. This can be found easily: If x1= 0, then x2 = b / a2.If x2=0, then x1 = b / a1.

Therefore (k, l) = (0, b / a2) and (c, d) = (b / a1, 0) are two points on the straight line a1x1+a2x2= b.

Step III: Represent these two points (k, l), (c, d) on the graph which denotes XY-axis plane. Join

these two points and extend this line to get the straight line which represents a1x1+ a2x2= b.

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Step IV: a1x1 + a2x2= b divides the whole plane into two half planes, which area1x1+ a2x2 = b (another side). Find the half plane that is related to the given inequality.

Step V: Do step-II to step-IV for all the inequalities given in the problem. The intersection of the half-

planes related to all the inequalities and x1 >= 0,x2 >= 0 , is called the feasible region (or feasible

solution space). Now find this feasible region.

Step VI: The feasible region is a multisided figure with corner points A, B,C, (say). Find the co-

ordinates for all these corner points. These corner points are called as extreme points.

Step VII: Find the values of the objective function at all these corner/extreme points.

Step VIII: If the problem is a maximization (minimization) problem, then the maximum (minimum)

value of z among the values of z at the corner/extreme points of the feasible region is the optimal

value of z. If the optimal value exists at the corner/extreme point, say A (u, v), then we say that the

solution x1= u andx2= v is an optimal feasible solution.

Step IX: Write the conclusion (that include the optimum value of z, and the co-ordinates of the

corner point at which the optimum value of z exists).

3. A manufacturing firm has discontinued production of a certain unprofitable product line. Thiscreated considerable excess production capacity. Management is considering to devote this excess

capacity to one or more of three products: call them product 1, 2 and 3. The available capacity on

the machines which might limit output are given below:

Machine Type Available Time (in machine hours per week)

Milling Machine 250Lathe 150

Grinder 50

The number of machine-hours required for each unit of the respective product is given below:

Productivity (in Machine hours/Unit)

Machine Type Product 1 Product 2 Product 3

Milling Machine 8 2 3

Lathe 4 3 0

Grinder 2 1

The unit profit would be Rs. 20, Rs. 6 and Rs. 8 for products 1, 2 and 3. Find how much of eachproduct the firm should produce in order to maximize profit?

Ans: Let x1, x2, x3 units of products 1, 2 and 3 are produced in a week.

Then total profit from these units is

Z = 20 x1 + 6 x2 + 8 x3

To produce these units the management requires

8x1 + 2x2 + 3x3 machine hours of Milling Machine

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4x1 + 3x2 + 0 x3 machine hou

and 2x1 + x3 machine hours o

Since time available for these

8x1 +2x2 + 3x3 250

4x1 + 3x2 150

2x1 + x3 50.

Obviously x1, x2, x3 0

Thus the problem is to

Maximize Z = 20x1 + 6x2 + 8x

Subject to 8x1 + 2x2 + 3x3 2

4x1 + 3x2 150

2x1 + x3 50,

x1, x2, x3 0.

Rewriting in the standard for

Maximize Z = 20x1 + 6x2 + 8x

Subject to 8x1 + 2x2 + 3x3 + S

4x1 + 3x2 + S2 = 150

2x1 + x3 + S3 = 50,

x1, x2, x3, S1, S2, S 3 0.

The initial basic solution is

X0= =

The initial simplex table is giv

rs of Lathe

Grinder

three machines are 250, 150 and 50 hours respecti

50

+ 0S1 + 0S 2 + 0S 3

1 = 250

n by

ely, we have

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x1 enters the basic set of v

table:

x2 enters the basic set of var

table:

x3 enters the basic set of va

table:

Since all zj cj 0 in the last

which is achieved by produci

riables replacing the variable S3. The first iterati

iables replacing the variable s2. The second iterati

iables replacing the variable x1. The third iteratio

row, the optimum solution is 700 .i.e., the maxim

g 50 units of product 2 and 50 units of product 3

ngives the following

n gives the following

yields the following

um profit is Rs. 700/-

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4. Determine optimal solutionWrite down the differences

Ans: Various Methods for finding i

1. North west corner meth2. Minimum Matrix Method3. Vogels Approximation MSince the aggregate supply is

dummy market, M5, for an

demand), with all cost eleme

obtained by VAM is degener

cells occupied and not 8 (=

P3M1, P3M2, P3M4, P3M5, a

For removing degeneracy pla

This is done in the following t

This solution is found to be n

The improved solution is:

Therefore the total cost is:

Rs. (630) + (110) + (320

= 700.

CPM was developed by Duproject and its overall compl

completion times by spendin

project?)

Definition: In CPM activities

node network construction

Single estimate of activity Deterministic activity tim

to the problem given below. Obtain the initi

etween PERT and CPM.

itial solution to a transportation problem

od

(MMM)

ethod (VAM)

220 units and the aggregate demand is 200 units,

mount equal to 20 (the difference between the

ts equal to zero. The solution is given in a one tab

ate, since it contains only seven basic variables (s

+ 5 -1) required for non-degeneracy).Here emp

nd P4M3 are independent while others are not.

e in the cell P3M5and then test it for optimality.

able.

n-optimal.

) + (050) + (760) + (130)

ont and the emphasis was on the trade-off bettion time (e.g. for certain activities it may be poss

g more money - how does this affect the overall c

are shown as a network of precedence relations

time

s

al solution by VAM.

we shall introduce a

ggregate supply and

le. The initial solution

ince there are only 7

y cells P1M3, P2M3,

een the cost of theible to decrease their

mpletion time of the

ips using activity-on-

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USED IN: Production management - for the jobs of repetitive in nature where the activity time

estimates can be predicted with considerable certainty due to the existence of past experience.

PERT was developed by the US Navy for the planning and control of the Polaris missile program and

the emphasis was on completing the program in the shortest possible time. In addition PERT had the

ability to cope with uncertain activity completion times (e.g. for a particular activity the most likely

completion time is 4 weeks but it could be anywhere between 3 weeks and 8 weeks).

Basic difference between PERT and CPM: Though there are no essential differences between PERT

and CPM as both of them share in common the determination of a critical path and are based on the

network representation of activities and their scheduling that determines the most critical activities to

be controlled so as to meet the completion date of the project.

PERT:

1. Since PERT was developed in connection with an R and D work, therefore it had to cope with theuncertainties which are associated with R and D activities. In PERT, total project duration is

regarded as a random variable and therefore associated probabilities are calculated so as to

characterize it.

2. It is an event-oriented network because in the analysis of network emphasis is given importantstages of completion of task rather than the activities required to be performed to reach to a

particular event or task.

3. PERT is normally used for projects involving activities of non-repetitive nature in which timeestimates are uncertain.

4. It helps in pinpointing critical areas in a project so that necessary adjustment can be made tomeet the scheduled completion date of the project.

CPM:

1. Since CPM was developed in connection with a construction project which consisted of routinetasks whose resources requirement and duration was known with certainty, therefore it is

basically deterministic.

2. CPM is suitable for establishing a trade-off for optimum balancing between schedule time andcost of the project.

3. CPM is used for projects involving activities of repetitive nature.Project scheduling by PERT-CPM: It consists of three basic phases: planning, scheduling and

controlling.

1. Project Planning.

2. Scheduling.

3. Project Control.

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5. Explain Project ManagementAns: PERT: Complex projects requ

and others that can be perfor

tasks can be modeled as a

network model for project

estimate for each activity. W

variations that can have a gre

The Program Evaluation and

in activity completion times.

having thousands of contrac

complete a project. The Netw

In a project, an activity is a

completion of one or more acbe completed. Project netwo

originally was an activity on

milestones on the nodes. Ov

For this discussion, we will us

The PERT chart may have mu

of a PERT diagram:

The milestones generally are

than the beginning node. Incr

modifying the numbering of

letters along with the expect

Steps in the PERT Planning P

(PERT).

ire a series of activities, some of which must be p

med in parallel with other activities. This collection

network .In 1957the Critical Path Method (CPM)

management. CPM is a deterministic method th

hile CPM is easy to understand and use, it does

at impact on the completion time of a complex proj

eview Technique (PERT) is a network model that a

ERT was developed in the late 1950's for the U.S.

ors. It has the potential to reduce both the time

ork Diagram

task that must be performed and an event is a

tivities. Before an activity can begin, all of its prederk models represent activities and milestones by

arc network, in which the activities are represen

r time, some people began to use PERT as an acti

e the original form of activity on arc.

ltiple pages with many sub-tasks. The following is

PERT Chart

numbered so that the ending node of an activity

ementing the numbers by 10allows for new ones t

he entire diagram. The activities in the above dia

d time required to complete the activity.

ocess

rformed sequentially

of series and parallel

was developed as a

at uses a fixed time

ot consider the time

ect.

llows for randomness

Navy's Polaris project

and cost required to

ilestone marking the

cessor activities mustrcs and nodes. PERT

ted on the lines and

ity on node network.

very simple example

has a higher number

o be inserted without

ram are labeled with

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PERT planning involves the following steps:

Identify the specific activities and milestones.

Determine the proper sequence of the activities.

Construct a network diagram.

Estimate the time required for each activity.

Determine the critical path.

Update the PERT chart as the project progresses