Linear Programming Lecture Notes - Penn State Personal Web Server
Linear Programming Lecture Notes - Penn State Personal Web Server
Linear Programming Lecture Notes - Penn State Personal Web Server
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point in Y we know that at most m elements of the vector x are non-zero and therefore,<br />
every extreme point in Y corresponds to an extreme point of the original problem P . Since<br />
Problem P has a finite solution, it follows that the optimal solution to problem P occurs<br />
at some point in Y, by Theorem 5.1. Furthermore the value of the objective function for<br />
Problem P is precisely the same as the value of the objective function of Problem PM for<br />
each point in Y because xa = 0. Define:<br />
(6.22) z min<br />
P<br />
= min<br />
y∈Y {cT x : y = [x xa] T }<br />
At each extreme point in Ya, the value of the objective function of Problem PM is a<br />
function of the value of M, which we are free to choose. Therefore, choose M so that:<br />
(6.23) max{c<br />
y∈Ya<br />
T x − Me T xa : y = [x xa] T } < z min<br />
P<br />
Such a value exists for M since there are only a finite number of extreme points in Y. Our<br />
choice of M ensures that the optimal solution to PM occurs at an extreme point where xa = 0<br />
and the x component of y is the solution to Problem P . <br />
Remark 6.10. Another way to look at the proof of this theorem is to think of defining<br />
M in such a way so that at any extreme point where xa = 0, the objective function can<br />
always be made larger by moving to any extreme point that is feasible to Problem P . Thus<br />
the simplex algorithm will move among the extreme points seeking to leave those that are<br />
not feasible to Problem P because they are less desirable.<br />
Theorem 6.11. Suppose Problem P is infeasible. Then there is no value of M that will<br />
drive the all the artificial variables from the basis of Problem PM.<br />
Proof. If such an M existed, then xa = 0 and the resulting values of x represents<br />
a feasible solution to Problem P , which contradicts our assumption that Problem P was<br />
infeasible. <br />
Remark 6.12. The Big-M method is not particularly effective for solving real-world<br />
problems. The introduction of a set of variables with large coefficients (M) can lead to<br />
round-off errors in the execution of the simplex algorithm. (Remember, computers can only<br />
manipulate numbers in binary, which means that all floating point numbers are restricted<br />
in their precision to the machine precision of the underlying system OS. This is generally<br />
given in terms of the largest amount of memory that can be addressed in bits. This has<br />
led, in recent times, to operating system manufacturers selling their OS’s as “32 bit” or “64<br />
bit.” When solving real-world problems, these issue can become a real factor with which to<br />
contend.<br />
Another issue is we have no way of telling how large M should be without knowing that<br />
Problem P is feasible, which is precisely what we want the Big-M method to tell us! The<br />
general rule of thumb provided earlier will suffice.<br />
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