Linear Programming ١ C H A P T E R 2

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1 Linear Programming ١ C H A P T E R 2

2 Problem Formulation Problem formulation or modeling is the process of translating a verbal statement of a problem into a mathematical statement. The Guidelines of formulation are 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 ٢

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 (feasible solution region F.R.). An optimal solution is a feasible solution that results in the largest possible objective function value when maximizing (or smallest when minimizing). A graphical solution method can be used to solve a linear program with two variables. ٣

4 Example: Graphical Solution-Example-1 ٤ 1. Objective function(o.f) Max 5x 1 + 7x 2 2. Constraints (subject to( s.t.)) x 1 < 6 2x 1 + 3x 2 < 19 x 1 + x 2 < 8 Non-negativity x 1, x 2 > 0

5 Graphical Solution-Example-1 ٥ Combined-Constraint Graph x 2 8 x 1 + x 2 < x 1 < x 1 + 3x 2 < x 1

6 Graphical Solution-Example-1 x 2 Feasible Region: satisfy all the constraints ٦ Feasible Region x 1

7 Extreme Points and the 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. You have to consider only one value of the extreme points of the feasible region. Max. value of o.f. (profit) or Min. value of o.f. (cost). ٧

8 Extreme points Graphical Solution-Example-1 x 2 8 ٨ Feasible Region x 1 3

9 Graphical Solution-example 1 ٩ Max x Objective Function 5*5+ 7*3= 46 Optimal Solution (x 1 = 5, x 2 = 3) x 1

10 Graphical solution- Example-2 ١٠ Solve Min 5x 1 + 2x 2 s.t. 2x 1 + 5x 2 > 10 4x 1 - x 2 > 12 x 1 + x 2 > 4 x 1, x 2 > 0

11 Graphical Solution-Example-2 ١١ Sol x 2 Min z = 5x 1 + 2x 2 4x 1 - x 2 > 12 x 1 + x 2 > x 1 + 5x 2 > Optimal: x 1 = 16/5 x 2 = 4/5 x o.f=

12 the Graphical Solution Procedures Prepare a graph of the feasible solutions for each of the constraints. Determine the feasible region that satisfies all the constraints simultaneously. Find the value of objective function at corner points (extreme points) of feasible region. Any solution with the largest value (Max.) or smallest value (Min.) of the objective function is an optimal solution. ١٢

13 Special Cases Alternative Optimal Solutions If there are more than extreme points give the same optimal solution. There are alternate optimal solutions, with all points on this line segment being optimal. Infeasibility A linear program which is overconstrained so that no point satisfies all the constraints is said to be infeasible. Unboundedness open area ١٣

14 Special cases:infeasible Problem ١٤ Solve graphically for the optimal solution: Max 2x 1 + 6x 2 s.t. 4x 1 + 3x 2 < 12 2x 1 + x 2 > 8 x 1, x 2 > 0

15 Special cases: Infeasible Problem There are no points that satisfy both constraints, hence this problem x 2 has no feasible region, and no optimal solution. ١٥ 8 2x 1 + x 2 > 8 4 4x 1 + 3x 2 < x 1

16 Special cases: Unbounded Problem Solve graphically for the optimal solution: ١٦ Max 3x 1 + 4x 2 s.t. x 1 + x 2 > 5 3x 1 + x 2 > 8 x 1, x 2 > 0

17 Special cases: Unbounded Problem T*he feasible region is unbounded and the objective function can be increased infinitely. x 2 ١٧ 3x 1 + x 2 > x 1 + x 2 > 5 x 1

18 Slack and Surplus Variables A linear program in which all the variables are nonnegative 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. ١٨