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Unformatted text preview: r example, equation 3 could be written out in full as:
0Z + 1x1 +1x2 + 0s1 + 0s2 + 1s3 = 4
or more simply as:
x1 + x2 + s3 = 4
which you will recognize as the equality version of the original constraint on the metal
finishing machine production limit.
The objective function is also rewritten to align with the constraints. The original version
was Z = 15x1 + 10x2, but we have moved all of the variables to the left hand side to create
this equivalent version: Z – 15x1 – 10x2 = 0.
The MRT column is for the minimum ratio test. We will fill this column in as we begin
the simplex calculations. The basic variable column shows the single basic variable that
occurs in each equation.
The most important thing about Tableau 4.1 is that it is in proper form (remember,
standard form is a type of LP, but proper form is the way that a tableau is written). A
tableau in proper form has these characteristics:
• exactly 1 basic variable per equation.
• the coefficient of the basic variable is always exactly +1, and the coefficients
above and below the basic variable in the same column are all 0.
• Z is treated as the basic variable for the objective function row (equation 0).
A big advantage of proper form is that you can always read the current solution directly
from the tableau. This is because there is exactly one basic variable per row, and the
coefficient of that variable is +1. All the other variables in the row are nonbasic (set to
zero, remember?), so the value of any variable is just given by the value shown in the
right hand side column. For example, in Tableau 4.1, the value of s3 is 4. What’s the
value of x1? Well, x1 does not appear in the list of basic variables, which means it must
be nonbasic, and nonbasic variables are always zero, hence x1 is zero in this basic feasible
solution. What’s the value of the objective function in this tableau? Because Z is the Practical Optimization: a Gentle Introduction http://www.sce.carleton.ca/faculty/chinneck/po.html ©John W. Chinneck, 2000 2 basic variable for equation 0, we can read it’s current value from the RHS column: its
value is 0.
Now we can use the tableau to keep track of our calculations as we go through the steps
of phase 2 of the simplex method as described in Figure 4.1. Step 2.1: Are we optimal yet?
We are optimal if no entering basic variable is available. Recall that the entering basic
variable chosen is the one that gives the fastest rate of increase in the objective function
value. In the original version of the objective function, Z = 15x1 + 10x2, the obvious
entering basic variable is x1. In the tableau, however, we have moved all of the variables
to the left hand side of the equation, to obtain the form Z – 15x1 – 10x2 = 0. Because of
this the rule is a little bit different: choose the variable in the objective function row that
has the most negative value as the entering basic variable.
In this step, then, just check to see if there are any negative coefficients in the o...
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This note was uploaded on 09/22/2013 for the course IEOR 4004 taught by Professor Sethuraman during the Fall '10 term at Columbia.
- Fall '10