New cornerpoint technically figure 44 a tie for the

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Unformatted text preview: ch will 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 ©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.

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