The Theory of the Simplex Method Asst. Prof. Dr. Nergiz Kasımbeyli ISE 203 OR I Chapter 5 The Theory of the Simplex Method Asst. Prof. Dr. Nergiz Kasımbeyli
x1=0 and x1=4 x2=0 and 2x2=12
Fig. 5.3 Not a convex set!
Whenever a constraint boundary equation is one of the defining equations for a CP solution, its indicating variable has a value of zero in the augmented form of the problem. Each such indicating variable is called a nonbasic variable for the corresponding basic solution.
Degenerate solution A BF solution is a basic solution where all m basic variables are nonnegative (≥ 0). A BF solution is said to be degenerate if any of these m variables equals zero. Thus, it is possible for a variable to be zero and still be a basic variable for the current BF solution (Another constraint boundary equation is satisfied in addition to its n defining equations).
The Matrix Form of Simplex
The Matrix Form of Simplex
Fundamental Insight or Z rhs Decision variables Slack variables rhs All you need to know is B-1 and cbB-1. You can calculate these from the initial tableau. or You can read them directly off the final tableau.
Fundamental Insight We replace cBB-1 with y*; and B-1 with S* Z rhs Decision variables Slack variables rhs We replace cBB-1 with y*; and B-1 with S*
Fundamental Insight Z Decision variables Slack variables rhs y* plays a very special role. These are shadow prices. We will often write the final tableau like this. We can use the fundamental insight for sensitivity analysis.
Apply Fundamental Insight Here is part of the final Tableau for Wyndor Use the fundamental insight to find the values of the decision variables and the profit.
cBB-1b cBB-1 B-1b B-1
b
Apply Fundamental Insight Here is part of the final Tableau for Wyndor Use the fundamental insight to find the values of the decision variables and the profit.
Revised Simplex Method Z Decision variables Slack variables rhs Find entering basic variable Calc top row Find leaving basic variable Calc rhs and column under entering basic variable New Solution Calc new B-1
A more efficient Simplex Method Revised Simplex A more efficient Simplex Method Instead of computing B-1 in every iteration, it derives B-1new from B-1old .
Revised Simplex