Fuzzy programming

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Author: Irina Baek
Steward: Dajun Yue and Fenqi You



Fuzzy programming is one of many optimization models that deal with optimization under uncertainty. This model can be applied when situations are not clearly defined and thus have uncertainty. For example, categorizing people into young, middle aged and old is not completely clear, so overlap of these categories may exist as can be seen in the image below.

Young, middle-aged, and old are not strictly defined categories, and may result in overlap.

Logical Reasoning

Unlike binary models, where an event is either black or white, fuzzy programming allows for a grey spectrum between the two extremes. As a result, it increases the possible applications since most situations are not bipolar, but consist of a scale of values. A linear function is often used to describe the 'grey spectrum' [1]

u(x) = 
1  & {ax \le b} \\
1- \frac{ax-b}{ \Delta b} & b < ax \le b+ \Delta b \\
0 &  b + \Delta b < ax


There are several types of fuzzy programming that can deal with different situations. Flexible programming and possibilistic programming will be described here.

Flexible programming

This type of programming can be applied when there is uncertainty in the coefficient values, and a certain amount of deviation is acceptable. Starting from a typical LP model defined as:

\max c^t x \\
s.t. \; & Ax \le b \\
& x \ge 0 

We use ~ to identify the fuzzy (or flexible) parameters. By making the inequalities fuzzy, the user of the program can set an approximate goal to minimize/maximize an objective function rather than a completely realistic value. Furthermore, this fuzzy relation can be interpreted as "essentially smaller than or equal" instead of "smaller than or equal"

\tilde{max} c^t x \\
s.t. \; & Ax \tilde{\le} b \\
& x \ge 0 

If the user has a certain objective value they would like to reach, this can be further simplified to:

Failed to parse(syntax error): Find x \\ s.t. c^t x tilde{\ge} z \\ \; Ax \tilde{\le} b\\ & x \tilde{ge} 0

Possibilistic programming

In possibilistic programming





[1] http://www.researchgate.net/profile/Nikolaos_Sahinidis/publication/222687527_Optimization_under_uncertainty_state-of-the-art_and_opportunities/links/5463babb0cf2c0c6aec4f7a8.pdf [2] http://www.worldacademicunion.com/journal/jus/jusVol01No2paper03.pdf