Download Understanding Linear Programming: Simplex Method I and Basic Solutions and more Study notes Engineering in PDF only on Docsity! Lecture 3 Simplex Method I January 27, 2009 Solving Linear Programs The graphical method is only applicable for simple problems (e.g. problems with two variables). However, it provides some very important observations. • The feasible region has finite many vertices (corner points); • If the problem is bounded, then at least one optimal solution is also a vertex of the feasible region (the problem could have multiple optimal solutions, but at least one of them is a vertex of the feasible region); • If a vertex is not an optimal solution then there exists an adjacent vertex which is better than the current vertex in terms of the objective function value. These observations form the basis of the Simplex Method. 2 Linear Programs in the Standard Form Is each of the following linear programs in the standard form? max 3x1 + 3x2 + 4x3 x1 − x2 + x3 ≥ 4 x1 + 2x2 − x3 = 3 x1, x2, x3 ≥ 0 min 2x1 + 3x2 + x3 x1 + 2x2 + 3x3 = 4 x1 + 2x2 − 2x3 = 3 x1, x2 ≥ 0 min x1 + 3x2 + 4x3 x1 − x2 + x3 + 2 = 4 x1 + 2x2 − x3 − 1 = 3 x1, x2, x3 ≥ 0 min x1 + 3x2 + 4x3 x1 − x2 + x3 = 4 x1 + 2x2 − x3 = 3 x1, x2, x3 ≥ 0 Any linear program can be converted to the standard form. 5 Handling Inequality Constraints Introduce slack variables. • For each “≤” constraints, add a nonnegative slack variable to the left hand side. x1 + x2 + x3 ≤ 8 ⇒ x1 + x2 + x3 + s = 8. • For each “≥” constraints, subtract a nonnegative slack vari- able to the left hand side. x1 + x2 + x3 ≥ 8 ⇒ x1 + x2 + x3 − s = 8. • The slack variables have zero weight in the objective func- tion, or they don’t appear in the objective function 6 Handling Unrestricted Variables Unrestricted variables: no sign restriction on the variable xi. • Introduce a pair of nonnegative variables x−i ≥ 0 and x+i ≥ 0, substitute all xi’s with x + i −x−i (in both the objective function and constraints). • In any simplex solution, at most one of the pair of variables can be positive. In fact, if both of them are positive, we can always make one of them 0 without changing the objective value. e.g. x+i = 5, x − i = 3 can be changed to x + i = 2, x − i = 0. The objective value dose not change because x+i − x−i always appears in the objective function as a whole. 7 From Vertex to Basic Solutions Example 1: max 2x1 + 3x2 2x1 + x2 ≤ 4 x1 + 2x2 ≤ 5 x1, x2 ≥ 0, ⇒ max 2x1 + 3x2 2x1 + x2 + s1 = 4 x1 + 2x2 + s2 = 5 x1, x2, s1, s2 ≥ 0, 1 2 3 4 5 x1 x2 1 2 3 4 2x1+x2 = 4 x1+2x2 = 5 1 2 3 4 5 x1 x2 1 2 3 4 s1 = 0 s2 = 0 Observations: after adding slacks, the boundaries of the feasible region can be represented by one of the variables being 0, the vertices can be represented by two of the variables being 0. 10 Basic Solutions • After converting an LP to the standard form, we have a linear system with n nonnegative variables and m equations. Usually, n > m. • By setting n−m variables to 0 and solving for the rest of the variables, we get a basic solution of the linear system. The maximum number of basic solutions is given by Cnm = n! m!(n−m)! • Some of the basic solutions might not be feasible. Those which are feasible are called basic feasible solutions (BFS). • In a basic solution, the n − m variables being set to 0 are called nonbasic variables, the other variables are called ba- sic variables. The set of all basic variables is called the basis. Note: basic variables can also be 0, this situation is called degeneracy. 11 Back to Example 1 2x1 + x2 + s1 = 4 x1 + 2x2 + s2 = 5 x1, x2, s1, s2 ≥ 0, 1 2 3 4 5 x1 x2 1 2 3 4 s1 = 0 s2 = 0 A B C D E F set x1 and x2 to 0, we get s1 = 4, s2 = 5 which is point A. set s1 and s2 to 0, we get x1 = 1, x2 = 2 which is point C. set s1 and x1 to 0, we get x2 = 4, s2 = −3 which is point F (infeasible). 12
