# Fuzzy programming

Author: Irina Baek

Steward: Dajun Yue and Fenqi You

## Contents |

# Introduction

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.

# 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]

# Methods

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:

We use ~ to identify the fuzzy (or flexible) parameters.

By making this relation fuzzy, the user of the program can set an approximate goal to minimize/maximize an objective function rather than a completely realistic value.

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

# Applications

# Examples

# Conclusion

# References

[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