The Theory of the Simplex Method

The Theory of the Simplex Method

The Essence • Simplex method is an algebraic procedure • However, its underlying concepts are geometric • Understanding these geometric concepts helps before going into their algebraic equivalents • With two decision variables, the geometric concepts are easy to visualize • Should understand the generalization of these concepts to higher dimensions

Standard (Canonical) Form of an LP Model Maximize Z = c1x1 + c2x2 + … + cnxn subject to a11x1 + a12x2 + … + a1nxn≤ b1 a21x1 + a22x2 + … + a2nxn ≤ b2 … am1x1 + am2x2 + … + amnxn ≤ bm x1≥ 0, x2≥ 0, …, xn≥ 0

Extended Terminology max cx n variables s.to Ax  b m inequalities x  0 n inequalities max cx n+m variables s.to Ax + Ixs = b m equations x, xs  0 n+m inequalities • Constraint boundary equation • For any constraint, obtain by replacing its , =,  by = • It defines a “flat” geometric shape: hyperplane • Corner-point solution • Simultaneous solution of n constraint boundary equations ( )

Extended Terminology max cx + 0xsn+m variables s.to Ax + Ixs = b m equations x, xs 0 n+m inequalities • Indicating variables (in the augmented form)

Indicating variables, overall

Properties of CPF (BF) Solutions Property 1: • If there is exactly one optimal solution, then it must be a corner-point feasible solution • If there are multiple optimal solutions (and a bounded feasible region), then at least two must be adjacent corner-point feasible solutions Proof of (a) by contradiction

Properties of CPF (BF) Solutions Property 2: There are only a finite number of corner-point feasible solutions max cx n variables s.to Ax  b m inequalities x  0 n inequalities CP solution is defined as the intersection of n constraint boundary equations (n+m) choose n: Upper bound on # of CPF solutions

Properties of CPF (BF) Solutions Property 3: • If a corner-point feasible solution has no adjacent CPF solution that are better, then there are no better CPF solutions elsewhere • Hence, such a CPF solution is an optimal solution (assuming LP is feasible and bounded)

Simplex in Matrix (Product) Form max cx n variables s.to Ax  b m inequalities x  0 n inequalities max cx m+n variables s.to Ax + Ixs = b m equations x, xs  0 m+n inequalities Initial Tableau: ( ) XB=

Simplex in Matrix (Product) Form Intermediate iterations: • Let B denote the square matrix that contains the columns from [ A | I ] that correspond to the current set of basic variables, xB(in order) • Similarly, let cB be the vector of elements in c that correspond to xB • Then at any intermediate step, the simplex tableau is given by

Revised Simplex Method • InitializationFind an initial BFS (if one not immediately available, do Phase 1. If Phase 1 terminates with z1>0, LP is infeasible) • Optimality testCalculate cBB-1A-c for basic,and cBB-1for nonbasic variables. If all ≥ 0, stop with optimality • Iterative step • Determine the entering variable as before (steepest ascent) • Let xk be the entering variable • Determine the leaving variable as before (minimum ratio test), but now only need to calculate the coefficients of the pivot column (from B-1A or B-1) and the updated rhs (B-1b) • Let xl : the leaving variable r : the equation number of the leaving variable a’rk : coefficient of xk in rth equation (the pivot element) • Determine B-1new=E B-1old where E is I expect for rth column replaced with • Can calculate the new BFS using xB=B-1b and z=cBB-1b • Back to optimality test

Original data: Revised Simplex Example Maximize z = 2x1+ 3x2 subject to x1 + 2x2≤ 10 3x1+ x2≤ 15 x2≤ 4 x1,x2 ≥0 Maximize z = 2x1+ 3x2 subject to x1 + 2x2+s1= 10 3x1+ x2+s2 = 15 x2+s3= 4 x1,x2, s1, s2, s3 ≥ 0

Iteration 0

Iteration 0 • Calculate row-0 coefficients: cBB-1A-c = cBB-1 = Optimal? • Entering variable is • Calculate coefficients in pivot column using B-1A or B-1 • Calculate updated rhs using B-1b, do the ratio test • Leaving variable is • Update B-1

Iteration 1

Iteration 2

Iteration 3

The Theory of the Simplex Method