Unformatted text preview: will be zero
because putting the tableau into proper form Figure 4.5: An unbounded linear
requires that those coefficients be zeroed. But program.
the coefficients of the nonbasic variables are
normally nonzero, and in fact are normally positive at the optimum (if any are negative,
you are not yet at the optimum). So what does it mean if a nonbasic variable has a
coefficient of zero in the objective function at the optimum? It means that choosing this
variable as the entering basic variable increases Z at the rate of zero, i.e. that choosing
this variable as the entering basic variable has no effect on Z!
But choosing such a variable as the entering basic variable means that you will pivot to a
different basic feasible solution. And the resulting new basic feasible solution will have
the same value of Z. Two different basic feasible solutions, same value of Z? That
means that there must be multiple optimum solutions. You will have the same value of the
objective function at both of the two basic feasible solutions and at any point between
those two solutions. Remember that there may in fact be more than two basic feasible
solutions with the same value of Z.
You may wish to find the other basic feasible solutions. Sometimes one solution is
preferred over another for nonquantifiable reasons, so you would like to see a selection
of solutions that give the optimum value. To see the other optima, choose one of the
nonbasic variables whose objective function coefficient is zero as the entering basic
variable, and pivot to another basic feasible solution as you normally would. Practical Optimization: a Gentle Introduction http://www.sce.carleton.ca/faculty/chinneck/po.html ©John W. Chinneck, 2000 9...
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 Fall '10
 SETHURAMAN
 Optimization, objective function, basic variable, W. Chinneck

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