Linear Programming: The Simplex Method in Software Encode pdf417 2d barcode in Software Linear Programming: The Simplex Method

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10.10 Linear Programming: The Simplex Method using software topaint pdf-417 2d barcode with web,windows application Java Platform is almost alway barcode pdf417 for None s no larger than of order m or n, whichever is larger. An interesting mathematical sidelight is that this second property, although known empirically ever since the simplex method was devised, was not proved to be true until the 1982 work of Stephen Smale. (For a contemporary account, see [3].

). 10.10.2 Writing the General Problem in Standard Form There is a stan PDF-417 2d barcode for None dard form for linear programming problems, and we have to learn how to write a general problem like (10.10.1) (10.

10.4) in this standard form. For de niteness, consider the problem Minimize with the x s nonnegative and also with 2x1 C x2 70 x1 C x2 40 x1 C 3x2 D 90 (10.

10.6) (10.10.

7) (10.10.8) D 40x1 60x2 (10.

10.5). First, we rewri te the inequalities as equalities. We do this by adding to the problem so-called slack variables xnC1 ; xnC2 ; : : : In our example, equations (10.10.

6) and (10.10.7) become 2x1 C x2 C x3 D 70 x1 x2 C x4 D 40 (10.

10.9) (10.10.

10). (A slack variab Software barcode pdf417 le like x4 for a inequality is sometimes called a surplus variable.) Requiring the slack variables to be nonnegative makes these equalities equivalent to the original inequalities. Once they are introduced, you treat slack variables on an equal footing with the original variables xi ; then, at the very end, you just ignore them.

The simplex solution for each slack variable is simply the amount by which the original inequality is satis ed. The key idea in the simplex method is to start with a feasible basic vector and make a sequence of exchanges between basic and nonbasic variables. At each step the vector stays feasible (satis es the constraints), and the objective function decreases (or at least does not increase).

How do we nd an initial feasible basic vector to start the procedure Suppose that our example were changed so that equations (10.10.7) and (10.

10.8) were both inequalities, like (10.10.

6). Then, after introducing slack variables, we would have 2x1 C x2 C x3 D 70 x1 C x2 C x4 D 40 x1 C 3x2 C x5 D 90 (10.10.

11) (10.10.12) (10.

10.13). In this case it is easy to write down a feasible basic vector: Set the original variables x1 and x2 to zero and take .x3 ; x4 ; x5 / D .70; 40; 90/.

Here n D 2 of the constraints, namely x1 0, x2 0, are satis ed as equalities, while m D 3 components of the feasible basic vector are nonzero. The variables .x3 ; x4 ; x5 / are called basic variables, while the variables that are zero, .

x1 ; x2 /, are called nonbasic variables. Note. 10. Minimization or Maximization of Functions that if we writ e equations (10.10.11) (10.

10.13) as a 3 5 matrix equation, then the last three columns of the matrix, corresponding to the slack variables .x3 ; x4 ; x5 /, form a 3 3 unit matrix.

So constraints are easy. But how do we handle constraints like equations (10.10.

7) and (10.10.8) The trick is again to invent new variables called arti cial variables.

We rewrite equation (10.10.8) as x1 C 3x2 C x5 D 90 (10.

10.14). Now equations ( Software pdf417 10.10.9), (10.

10.10), and (10.10.

14) are almost in the form to give us an easy initial feasible basic vector by setting x1 D x2 D 0. The obstacle is equation (10.10.

10), which would give a negative value for x4 . We have to precede the actual simplex procedure by a preliminary procedure, called phase one of the simplex method, to nd an initial feasible vector. (The actual optimization is called phase two.

) In phase one, we replace our objective function (10.10.5) by a so-called auxiliary objective function, 0 x4 (10.

10.15) We now perform the simplex method on the auxiliary objective function (10.10.

15) with the constraints (10.10.9), (10.

10.10), and (10.10.

14), starting with the basis given by x1 D x2 D 0. The variable x4 starts off negative (at 40). Minimizing the function (10.

10.15) drives x4 toward satisfying x4 0, the condition for feasibility. In fact, we don t even have to solve phase one all the way to the exact minimum.

As we do the exchanges between variables during this phase, we continually rede ne the auxiliary objective function at each iteration to be minus the sum of all negative basic variables. As soon as all the basic variables are nonnegative, we are done with phase one. And what if the rst phase doesn t drive the auxiliary objective function to a negative value (i.

e., all basic variables nonnegative) That signals that there is no initial feasible basic vector, i.e.

, that the constraints given to us are inconsistent among themselves. Report that fact, and you are done. An arti cial variable in an equality constraint is an example of a zero variable, a variable that must vanish in the optimal solution.

Typically the way a zero variable gets to be zero is by being nonbasic in the optimal solution. So we can precede phase one with a phase zero in which we exchange each zero variable out of the basis. One last piece of jargon: Slack and arti cial variables are often called logical variables, to distinguish them from the original independent variables, which are sometimes called structural variables.

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