Talk:Operations Research/The Simplex Method

This page. It needs more latex done right.

See the following links:
 * http://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style_(mathematics)
 * http://en.wikipedia.org/wiki/Help:Formula

--78.22.53.64 (talk) 08:25, 16 August 2008 (UTC)

Standard form?
The first section gives a linear program in "standard form"; however, it'd be best if the term "standard" was avoided as there may be no "standard". For example, the "standard form" of Introduction to Algorithms (Cormen, Leiserson, Rivest, Stein), a widely used textbook, looks like this:

$$6x_1 + 4x_2 \leq 24$$

$$x_1 + 2x_2 \leq 6$$

$$-x_1 + x_2 \leq 1$$

$$x_2 \leq 2$$

--Adevish (talk) 21:54, 18 February 2009 (UTC)
 * What you want would have been more appropriate if the page was on wikipedia because a certain amount of latitude in notation is available on wikibooks. Most OR books include some definition of standard form, and this book is only obeying the established standard by including one. Nevertheless, if you still think it might confuse readers, perhaps we should use some other terminology, like canonical.--Shahab (talk) 08:34, 19 February 2009 (UTC)


 * The module could be edited to make it clear that it was defining "standard form" as such-and-such; the current phrasing doesn't make that clear. --Adevish (talk) 03:02, 21 February 2009 (UTC)
 * OK. Go ahead and fix it.--Shahab (talk) 04:31, 21 February 2009 (UTC)

The 2-Phase example is wrong, algorithm not clearly laid out
The example beginning:

Minimize: z=4x1 + x2

Subject to: 3x1 + x2 = 3

4x1 + 3x2 &gt;= 6

x1 + 2x2 &lt;= 4

x1, x2 &gt;= 0

has the unique solution: x1 = 2/5

x2 =9/5

z= 17/5

This highlights several problems: I am familiar with linear algebra and matrix manipulations, yet I am unable to determine the error, because the presentation of gaussian elimination and the process of "optimization" and pivot selection are not presented. It is unclear what the simplex method is. For example, how do I go about the minimization process? How is it different from maximization? I know what the answer is according to the present text, but only because I was able to replicate its matrix result, which is wrong. There are unnecessary steps, which I am happy to discuss in detail if anyone is currently managing this project.

Overall, I think the expectations on the reader are erratic. In some places, we are clearly invoking additional terminology and ad-hoc explanations to appeal to someone who is not interested in understanding the linear algebra behind it. But in other places, we get "the obvious matrix operation is", and yet this will be incomprehensible to anyone not already familiar with matrix manipulation.

I think that simple facts should be stated and demonstrated. There need to be examples. The reader should be told explicitly (and not as a throw-away to those "already familiar") that two rows can be added together without altering the solution. Then this row operation is avaiable to the user, and s/he can decide how to go about it. I, for one, do not find the "pivot" explanation to be at all clear, and it is ONLY because I already know the elementary row operations and the principle behind the "pivot" that it made any sense.

The presentation of graphical solutions is poor. The explanation for how to "tell" that a line bounds a region by plugging in numbers is again, useless without a straightforward explanation. For example, given the constraint

6x1 + 4x2 &lt;= 24

make it clear that we can draw the line x2 &lt;= -(3/2)x1 + 6

and then shade the region below the line because "x1 is equal to or less than this line (bound)". NOW the reader can double-check him/herself by plugging in values. Take a sentence or two to remind the reader that s/he can add quantities to both sides without affecting the inequality, just as with =, that positive scalar multiplication of both sides holds, and that multiplying by -1 changes the direction of the inequality.

I find that upper division pure math textbooks have no qualms taking multiple pages to remind readers of the basic algebraic operations, any time new algebraic ideas are introduced. And inequalities are puzzling for anyone who doesn't work with them regularly. This text depends upon using them variously, it can stand the addition of clear, basic procedures with examples.

Because the algorithm is broken, and it is never presented.

I found the answer easily using Gaussian elimination in a Jordan matrix, but that's because it's a trivial solution. I am not clear what the ramifications of sign are to the max/min procedure or to Phase1/Phase2, because at no point is the procedure unambiguously stated. "We are going to do THIS in EXACTLY THE FOLLOWING WAY, until we have achieved the following unambiguous result." The simplex method, I presume, is an algorithm. Then give it. Every step not in the text is a step that is missing. What use is an instructional book if, in the end, I have never been given the algorithm?

I eliminated sx6 in the 2 Phase example because Z + (1/5)sx6 = 18/5

wasn't optimized. But the positive sign in the Z row is cited explicitly as the reason that we are done. And yet these rules about signs are never given, the procedure by which the author reduces the Phase 1 matrix is never given (all that is said is that "the simplex method is now applied", and yet it is evident to anyone following along that it cannot be applied in EXACTLY THE SAME WAY as it was in the previous example), and the author is mistaken about them, whatever they actually are.

Finally, I should like a better example. This two-phase example is a waste of time. The first constraint is 3x1 + x2 = 3.

That's an equal sign. We're done.

x2 = 3 - 3x1

...and substitute into the other constraints. We find right away that x1 &lt;= 3/5

x1 &gt;= 2/5

are the two strictest constraints. We are minimizing Z = x1 + 3, and the answer is x1 = 2/5.

Normally, I would consider this to be a minor oversight, but at every step the author fails to stop and identify the criteria to be taken into account, and to then do so, as a straightforward and unhurried matter of demonstration. When doing multivariable optimization, it is ESSENTIAL to recognize, as a matter of procedure, simple ways to reduce the number of variables before entering time-consuming analysis. I understand that the author used a computer program. This screen shot is useless, because the answer is incorrect, and it is to me indicative of the single, crippling flaw of a text that I would very much like to see rewritten. Good self-instructing math texts are hard to come by and I would like to see this text take responsibility for introducing the ideas that it claims it does not take for granted, then takes for granted. It could be an excellent free resource, the overall layout plan is good. Structurally, I am all for it. Algorithms are unambiguous procedures. Name each step, and the text will correct itself, and will be accessible.

I'm not super into business, and my interests are largely elsewhere. I'd rather not be the one to try to rework this text. However, I would be happy to assist whomever is currently working on it, if these criticisms prove troublesome to rectify. My intention is not to be overly criticial. It's just math, that's all. Errors need to be corrected. When those errors raise questions about what the author intends, it is worth mentioning. But I never state objections as rhetorical questions. I mean the words literally, so if I say it's unclear to me what the author is doing, that does NOT mean "I don't like you and you're bad at this." That means, "I don't know what you are doing or why." And I am unable to assist or offer fully useful suggestions if I don't. Solutions require knowing what went wrong, and with the implementation of the rote mathematics of this text, I am not sure what is going wrong.

Ryan Slumberland (talk) 03:56, 19 October 2010 (UTC)

How we can find zj-cj
Please tels us how we can find zj in problem.