Chapter4

# Any negative coefficients in the objective function

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Unformatted text preview: bjective function row. There are two potential entering basic variables in the tableau at this point, so we are not optimal yet: the simplex iterations must continue. Of course, we are only selecting from among currently nonbasic variables at this point: these are the only ones that could enter the basis. A nice feature of proper form is that only nonbasic variables can have a nonzero value in the objective function row, so you never have to worry about accidentally selecting a basic variable (which is already in the basic set) as the entering basic variable. Proper form enforces this by requiring that each basic variable have all zeroes in its column, except for the row in which it is the basic variable. Step 2.2: Select the entering basic variable As described above, when using the simplex tableau, the entering basic variable is the nonbasic variable in the objective function row (equation 0) that has the most negative coefficient. In Tableau 4.1, this is x1 whose objective function coefficient of –15 is more negative than the objective function coefficient of –10 for x2. The column for the entering basic variable is called the pivot column. The pivot column is highlighted for ease of reference in Tableau 4.2. Recall that the entering basic variable, which is currently nonbasic and therefore set to zero, will now be allowed to take on a positive value. Step 2.3: Select the leaving basic variable The minimum ratio test is used to determine the leaving basic variable. Recall that this test determines which constraint most limits the increase in the value of the entering basic variable. The most limiting constraint is the one whose basic variable is driven to zero first as the entering basic variable increases in value. Practical Optimization: a Gentle Introduction http://www.sce.carleton.ca/faculty/chinneck/po.html ©John W. Chinneck, 2000 3 For the minimum ratio test, look only at the entries in the pivot column (the column for the entering basic variable) for the constraint rows, and calculate: (RHS)/(coefficient of entering basic variable) There are two special cases: • if the coefficient of the entering basic variable is zero: enter no limit in the minimum ratio test column • if the coefficient of the entering basic variable is negative: enter no limit in the minimum ratio test column. The minimum ratio test is never applied to the objective function row. Why? The objective function row is not a constraint, so it can never limit the increase in the value of the entering basic variable. The objective function always goes along for the ride, just measuring the value of the objective at any point that it is given. As usual in the minimum ratio test, the leaving basic variable is associated with the row that has the minimum value of the ratio test. This row is called the pivot row and is highlighted for ease of reference in Tableau 4.2. At this point, we have determined that we are not optimal, and have selected an entering basic variable a...
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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.

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