Amarasinghe A.T.A.S(061005G)Chandrasekara S.A.A.U(061009X)

Dayarathna K.H.L.R(061012B)Jeewan D.M.L.C(061026V)

Mathematical Modeling Technique design to optimize the usage of limited recourses.most extensively used in business

and economics, but can also be utilized for some engineering problems

Economics-Determination of shadow prices

Business application- Maximization of profit

Engineering-Design of structures.

Facilitates decision-making process Keep focus on profit under any scenario provides the targets and operating

strategies Optimize utilization of assets. Optimize utilization of inventory.

Military Industry Textile industry Agriculture Transportation Energy Telecommunications

• Optimize utilization of the assets• Optimize inventory management• Optimize capacity utilization and shutdown planning •Minimize losses

1) Textile industry -

2) Petroleum -Refinery

• Optimize inventory management

• Optimize black oil generation and up gradation

• optimize overall product mix and dispatch

Graphical Method Tabular Method Simplex Method

Example:-

An equation of the form

4x1 + 5x2 = 1500

5x1 + 3x2 = 1575

x1 + 2x2 = 420

defines a straight line in the x1-x2 plane

The optimal solution will be:

X1 = 270

X2 = 75

Advantages.Easy to Analyze.

Disadvantages.Handle only up to 3 variables.Need to draw according scale.

Method:-All the constraints are converted into equal

sign by introducing Slack variable and calculate the solution for set of equation to find out the corner points of the feasible region.

Substitute all corner point of the feasible region in objective function and thereby determine the corresponding optimal solution.

Maximize Z =13x1 + 11x2

4x1 + 5x2 + x3 = 1500

5x1 + 3x2 + x4 = 1575

x1 + 2x2 + x5 = 420

Basic Variabl

e

Independent

Columns

Basic solution

Feasible

Extreme Point

Z value

(X1, X2, X3)

Yes (270,75,45,0,0) Yes (270,75) 4335

(X1, X2, X4)

Yes (300,60,0,-105,0) No

(X1, X2, X5)

Yes (260,92,0,0,-24) No

(X1, X3, X4)

Yes (420,0,-180,-525,0)

No

(X1, X3, X5)

Yes (315,0,240,0,105) Yes (315,0) 4095

(X1, X4, X5)

Yes (375,0,0,-300,45) No

(X2, X3, X4)

Yes (0,110,950,1245,0)

Yes (0,110) 1210

(X2, X3, X5)

Yes (0,525,-1125,0,-630)

No

(X2, X4, X5)

Yes (0,300,0,675,120) Yes (0,300) 3300

(X3, X4, X5)

Yes (0,0,1500,1575,420)

Yes (0,0) 0

Introducing these slack variables into the inequality constraints and rewriting the objective function such that all variables are on the left-hand side of the equation

Identify the variable that will be assigned a nonzero value in the next iteration so as to increase the value of the objective function. This variable is called the entering variable.Identify the variable, called the leaving variable, which will be changed from a nonzero to a zero value in the next solution.

Advantages Identifies geometric extreme points

algebraicallyAlways directed towards final objective

DisadvantagesThe simplex method is not used to examine

all the feasible solutions.