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.
• 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
An equation of the form
4x1 + 5x2 = 1500
5x1 + 3x2 = 1575
x1 + 2x2 = 420
defines a straight line in the x1-x2 plane
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.