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Linear Programming
- 1.
- 2. INTRODUCTION ▪ Linear programmingis a method to achieve the best outcome in a mathematical model. ▪ It is a special case of mathematical programming. ▪ Use of “programming here means “choosing a course of action”. ▪ Linear programming involves choosing a course of action when the mathematical model of the problem contains only linear functions.
- 3. Linear Programming (LP)Problem ▪ The maximization or minimization of some quantity is the objective in all linear programming problems. ▪ All LP problems have constraints that limit the degree to which the objective can be pursued. ▪ A feasible solution satisfies all the problem's constraints. ▪ An optimal solution is a feasible solution that results in the largest possible objective function value when maximizing (or smallest when minimizing).
- 4. Linear Programming (LP)Problem ▪ Linear functions are functions in which each variable appears in a separate term raised to the first power and is multiplied by a constant (which could be 0). ▪ Linear constraints are linear functions that are restricted to be "less than or equal to", "equal to", or "greater than or equal to" a constant. ▪ If both the objective function and the constraints are linear, the problem is referred to as a linear programming problem.
- 5. Problem Modelling ▪ Isthe process of translating a verbal statement of a problem into a mathematical statement. ▪ Every LP problems has some unique features, but most problems also have common features.
- 6. Guidelines for ModelFormulation ▪ Understand the problem thoroughly. ▪ Describe the objective. ▪ Describe each constraint. ▪ Define the decision variables. ▪ Write the objective in terms of the decision variables. ▪ Write the constraints in terms of the decision variables.
- 7. Graphical Solution Procedure ▪Prepare a graph of the feasible solutions for each of the constraints. ▪ Determine the feasible region that satisfies all the constraints simultaneously. ▪ Draw an objective function line. ▪ Move parallel objective function lines toward larger objective function values without entirely leaving the feasible region. ▪ Any feasible solution on the objective function line with the largest value is an optimal solution.
- 8. Slack And SurplusVariables ▪ A linear program in which all the variables are non-negative and all the constraints are equalities is said to be in standard form. ▪ Standard form is attained by adding slack variables to "less than or equal to" constraints, and by subtracting surplus variables from "greater than or equal to" constraints. ▪ Slack and surplus variables represent the difference between the left and right sides of the constraints. ▪ Slack and surplus variables have objective function coefficients equal to 0. ▪ A linear program in which all the variables are non-negative and all the constraints are equalities is said to be in standard form. ▪ Standard form is attained by adding slack variables to "less than or equal to" constraints, and by subtracting surplus variables from "greater than or equal to" constraints. ▪ Slack and surplus variables represent the difference between the left and right sides of the constraints. ▪ Slack and surplus variables have objective function coefficients equal to 0.
- 9. Maximization Problem Simple ExampleOf Maximization Problem Max 5x1 + 7x2 s.t. x1 < 6 2x1 + 3x2 < 19 x1 + x2 < 8 x1 > 0 and x2 > 0 Objective Function “Regular” Constraints Non-negativity Constraints
- 10. Graphical Solution ▪ FirstConstraint Graphed x2 x1 x1 = 6 (6, 0) 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 Shaded region contains all feasible points for this constraint
- 11. Graphical Solution ▪ SecondConstraint Graphed 2x1 + 3x2 = 19 x2 x1 (0, 6 1/3) (9 1/2, 0) 8 7 6 5 4 3 2 1 Shaded region contains all feasible points for this constraint 1 2 3 4 5 6 7 8 9 10
- 12. Graphical Solution ▪ ThirdConstraint Graphed x2 x1 x1 + x2 = 8 (0, 8) (8, 0) 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 Shaded region contains all feasible points for this constraint
- 13. Graphical Solution ▪ Combined-ConstraintGraph Showing Feasible Region x1 x2 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 2x1 + 3x2 = 19 x1 + x2 = 8 x1 = 6 Feasible Region
- 14. Graphical Solution ▪ ObjectiveFunction Line x1 x2 (7, 0) (0, 5) Objective Function 5x1 + 7x2 = 35 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
- 15. Graphical Solution ▪ SelectedObjective Function Lines x1 x2 5x1 + 7x2 = 35 8 7 6 5 4 3 2 1 5x1 + 7x2 = 42 5x1 + 7x2 = 39 1 2 3 4 5 6 7 8 9 10
- 16. Graphical Solution ▪ OptimalSolution x1 x2 Maximum Objective Function Line 5x1 + 7x2 = 46 Optimal Solution (x1 = 5, x2 = 3) 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
- 17. Extreme Points andthe Optimal Solution ▪ The corners or vertices of the feasible region are referred to as the extreme points. ▪ An optimal solution to an LP problem can be found at an extreme point of the feasible region. ▪ When looking for the optimal solution, you do not have to evaluate all feasible solution points. ▪ You have to consider only the extreme points of the feasible region.
- 18. Extreme Points Examle x1 Feasible Region 1 2 3 4 5 x2 8 7 6 5 4 3 2 1 12 3 4 5 6 7 8 9 10 (0, 6 1/3) (5, 3) (0, 0) (6, 2) (6, 0)
- 19. Computer Solution ▪ LPproblems involving 1000s of variables and 1000s of constraints are now routinely solved with computer packages. ▪ Linear programming solvers are now part of many spreadsheet packages, such as Microsoft Excel.
- 20. Application ▪ Business ▪ Industrial ▪Military ▪ Marketing
- 21. Advantages ▪ Imroves thequality of the decisions ▪ Provides better tools for meeting the changing conditions. ▪ Highlights the bottleneck in the production process.
- 22. Limitation ▪ Applicable toonly static situation. ▪ Deals with the problems with single objective. ▪ For large problems the computational difficulties are enormous. ▪ It may yield fractional value answer to decision variables.
- 23. References LINKS ▪ https://en.wikipedia.org/wiki/Linear_programming ▪ https://faculty.mu.edu.sa/public ▪https://www.slideshare.net/tiwarimanutiwari/linear-programing
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