## Fuzzy Logic for Planning and Decision Making (1)

As I also pointed out in the comments, I’ve completed *And Then There Were None *(you can read my brief* *review of the book here), but* *it’s not easy to say much about that book without providing spoilers so instead I decided to blog the book mentioned in the post title.

It’s not easy to blog the other book either, so I won’t say very much about it. I’ve only read roughly the first three chapters (out of eight) but reading it requires absolute concentration and focus, which makes it feel suspiciously much like actual work. It’s quite technical, here’s the description of the book from goodreads:

“This book starts with the basic concepts of Fuzzy Logic: the membership function, the intersection and the union of fuzzy sets, fuzzy numbers, and the extension principle underlying the algorithmic operations. Several chapters are devoted to applications of Fuzzy Logic in Operations Research: PERT planning with uncertain activity durations, Multi-Criteria Decision Analysis (MCDA) with vague preferential statements, and Multi-Objective Optimization (MOO) with weighted degrees of satisfaction. New items are: Fuzzy PERT using activity durations with triangular membership functions, Fuzzy SMART with a sensitivity analysis based upon Fuzzy Logic, the Additive and the Multiplicative AHP with a similar feature, ELECTRE using the ideas of the AHP and SMART, and a comparative study of the ideal-point methods for MOO. Finally, earlier studies of colour perception illustrate the attempts to find a physiological basis for the set-theoretical and the algorithmic operations in Fuzzy Logic. The last chapter also discusses some key issues in linguistic categorization and the prospects of Fuzzy Logic as a multi-disciplinary research activity. Audience: Researchers and students working in applied mathematics, operations research, management science, business administration, econometrics, industrial engineering, information systems, artificial intelligence, mathematical psychology, and psycho-physics.”

A little bit of stuff from the book:

“The mathematical theory of fuzzy sets (Zadeh, 1965), alternatively referred to as fuzzy logic, is concerned with the degree of truth that the outcome belongs to a particular category, not with the degree of likelihood that the outcome will be observed. […] Fuzzy logic agrees that an element may with a positive degree of truth belong to a set and with another positive degree of truth to the complement of the set, whereby it violates the law of noncontradiction (a statement cannot be true and not-true at the same time). Fuzzy logic also violates the law of the excluded middle (a statement is either true or false, “tertium non datur “).”

(Fuzzy numbers are weird:)

“Let us consider the fuzzy number ã, that is, the fuzzy set of numbers which are roughly equal to the crisp number a. […] In general, a fuzzy number has a membership function which increases monotonically from o to 1 on the left-hand side; thereafter, there is a single top or a plateau at the level 1; and finally, the membership function decreases monotonically to 0 on the right-hand side. […] For practical purposes we shall further limit our attention to a convenient subclass of fuzzy numbers: to those with a triangular membership function. Such a fuzzy number ã is characterized by three parameters: the lower value a-1, the modal value a-m, and the upper value a-u. The interval (a-1, a-u) constitutes the basis of the triangle, and a-m is the position of the top. [slight change in notation due to wordpress formatting problems, US] […] we shall denote triangular fuzzy numbers as ordered triples, so that we simply write ã = (a-1, a-m, a-u) […]

In general, a triangular fuzzy number does not […] have a proper opposite number. […] In general, a triangular fuzzy number does not […] have a proper inverse. […] The maximum of two triangular fuzzy numbers is not necessarily triangular …”

…

Chapter 2 mostly deals with basic properties of fuzzy numbers and -sets, calculation rules, etc. In chapter 3 the ‘for planning and decision making’-part of the title enters the equation as well – this chapter deals with *Stochastic and fuzzy **PERT*. You can probably read and understand most of the first couple of chapters without having much math/stats background knowledge, but here I assume it may start to get a little more difficult if you don’t. I don’t want to blog equations (formatting problems is a pain in the neck) but you should be able to get an impression of which type of book this is from the comments below anyway:

“In stochastic optimization it is generally very difficult, if not impossible, to calculate the probability distribution of an optimal solution. Therefore, stochastic optimization problems are sometimes converted into deterministic problems with optimal solutions which have a particular risk to be unfeasible and/or suboptimal (see Kall and Wallace, 1994). This mode of operation has the advantage that it supplies a deterministic schedule for action. The simplest procedure is to replace every stochastic quantity in the problem by its expectation. The literature on stochastic optimization, however, does not always recommend the user to do so because it is such a crude simplification: the deterministic solution may have a high chance to be unfeasible.” […]

“A stochastic optimization problem can also be converted into a deterministic problem under constraints which have a controlled chance to be violated. The PERT problem, for instance, can accordingly be written as [a] so-called chance-constrained programming problem […] the chance-constrained version of PERT can be rewritten as [a] linear programming problem […]

In summary, the simple and the extended versions of PERT are particular formulations of a stochastic optimization problem. The manner in which these versions are employed is unusual in the field of stochastic optimization, however, because the risk·of constraint violations is high. And indeed, PERT has not been designed to make a plan that can be followed throughout the duration of the project. In practice, frequent recalculations are necessary in order to review the plan in the light of what happens during the execution of the activities. […]

When the activity durations are modelled as fuzzy numbers we have several pragmatic approaches of fuzzy optimization at our disposal to make a time schedule …”

…

The edition I’m sitting with is from before the year 2000, and I assume that when it comes to some of the aspects covered the updated version of the book (there’s also a 2011 edition I believe) is better; a lot of stuff has happened in CS and related fields in the meantime, and presumably some of the approaches described which were back then considered computationally too expensive to use no longer are.

Despite the low page count (a couple hundred pages) this book is hard work, and it’s not a book you can just sit down and read in an afternoon. At least not if you want to actually learn something.

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