Have selected an entering basic variable and a

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Unformatted text preview: nd a leaving basic variable. These can now be swapped so that we can move to the next basic feasible solution (feasible cornerpoint). The calculations so far are summarized in Tableau 4.2. basic variable Z s1 s2 s3 eqn. no. 0 1 2 3 Z x1 x2 s1 s2 s3 RHS MRT 1 0 0 0 -15 1 0 1 -10 0 1 1 0 1 0 0 0 0 1 0 0 0 0 1 0 2 3 4 never 2/1 = 2 no limit 4/1 = 4 Tableau 4.2: Updated ABC at the origin. Step 2.4: Update the tableau Now that the entering and leaving basic variables are known, we can move to the new and better basic feasible solution. This is easily done by putting the tableau back into proper form. It is not in proper form now because the exchange of the entering and leaving basic variables has not yet been done, so for example, s1 still shows as a basic variable when it should now be nonbasic, and the column for x1, the new basic variable does not have zeros everywhere except in its own row. Here are the steps for putting the tableau back into proper form: Step 2.4.1: In the column entitled basic variable, replace the leaving basic variable listed for the pivot row by the entering basic variable. In Tableau 4.2 the basic variable listed for equation 1 is s1, but in Tableau 4.3 it is x1. Practical Optimization: a Gentle Introduction http://www.sce.carleton.ca/faculty/chinneck/po.html ©John W. Chinneck, 2000 4 Step 2.4.2: The tableau element where the pivot row and the pivot column intersect is known as the pivot element. This will be the coefficient associated with the new basic variable for this row, so if it is not already +1, then we must divide all of the elements in the pivot row by the pivot element to obtain a +1 in the pivot element position. Luckily, we already have a +1 in the pivot element position in Tableau 4.2. Step 2.4.3: Now we need to eliminate all of the coefficients in the pivot column except the pivot element. This is done by simple Gaussian operations. To remove the pivot column element in some row k for example, proceed as follows: (new row k) = (row k) – (pivot column coefficient in row k)×(pivot row) Remember that the pivot row has been updated by now so that the pivot element is +1. For example, to clean out the -15 that appears in the objective function row, • new Z coefficient= 1 - (-15)×0 = 1 • new x1 coefficient = -15 - (-15)×1 = 0 • new x2 coefficient = -10 - (-15)×0 = -10 • new s1 coefficient = 0 - (-15)×1 = 15 • new s2 coefficient = 0 - (-15)×0 = 0 • new s2 coefficient = 0 - (-15)×0 = 0 • new RHS = 0 - (-15)×2 = 30 Always follow the procedure given above to make sure that there is never any change in the Z column, because Z will always be the basic variable for the objective function, so you must always maintain a +1 in the equation 0 row under the Z column. Tableau 4.3 shows the tableau after it has been put into proper form by eliminating all of the elements in the pivot row except the +1 for the pivot element. This now summarizes the second basic feasible solution (or cornerpoint, if thinking graphic...
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