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define a different partitioning of the variables into basic and nonbasic sets.
So how do we choose between the two possibilities for the leaving basic variable?
Again, simply choose arbitrarily. The variable that is not chosen as the leaving basic
variable will remain basic, but will have a calculated value of zero. In contrast, the
variable chosen as the leaving basic variable will of course be forced to zero by simplex.
Both variables must be zero simultaneously because both constraints are active at that
point; it’s just that simplex only needs one of them to define the basic feasible solution.
When there is a tie for the leaving basic variable as we have described, the basic feasible
solution defines what is known as a degenerate solution. Degenerate solutions can lead
to an infinite loop of solutions that traps the simplex solution method; this is known as
cycling. This does not happen in 2-dimensional problems, so it is difficult to show a
simple diagram. But here is the general idea of how cycling happens, if it could happen
in two dimensions (refer to Figure 4.4): first define the basic feasible solution using
constraints A and C, then pivot to a new basic feasible solution defined by constraints B
and C, then pivot back to the basic solution defined by constraints A and C, then continue
around this loop infinitely.
Cycling is actually fairly rare in real problems, and most good-quality commercial linear
programming solvers have special code to detect and exit from cycling behavior. One of
the most often used anti-cycling routines for network LPs was in fact invented by former
Carleton University professor William Cunningham.
Now suppose that all of the minimum ratio tests are tied at “no limit”: what does this
mean? The interpretation is straightforward: no constraint puts a limit on the increase in
the value of the entering basic variable! Because of this, there is then no limit on the
increase in the value of the objective function. Your result might then show that you can
make an infinite amount of profit for the Acme Bicycle Company, for example. Sounds
great, but it usually means that you forgot a constraint. You may have assumed, for
example, that you can sell whatever quantity of widgets you manufacture, or that a key
ingredient is available in infinite quantities. Problems of this type are called unbounded,
and have an unbounded solution. Figure 4.5 gives an example of an unbounded problem. Practical Optimization: a Gentle Introduction http://www.sce.carleton.ca/faculty/chinneck/po.html ©John W. Chinneck, 2000 8 An unbounded problem is recognized while
pivoting when all of the entries in the pivot
column (the entering basic variable column)
are zero or negative, and hence place no limit
on the increase in the value of the entering
basic variable. At the Optimum, the Coefficients of
Some Nonbasic Variables are Zero
in the Objective Function Row
You expect that the objective function
coefficients of the basic variables...
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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