Difference between revisions of "Fuzzy programming"

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Continuing with the age analogy - if young ranges from ages 0 to 30, we can define 0 to 18 as being definitely young, so u = 1. However, as we increase the age from there young is not explicitly defined, and can be given lower u values, as these ages are defined as "less young".  
 
Continuing with the age analogy - if young ranges from ages 0 to 30, we can define 0 to 18 as being definitely young, so u = 1. However, as we increase the age from there young is not explicitly defined, and can be given lower u values, as these ages are defined as "less young".  
  
=Methods=
+
=Method: Flexible Programming=
There are several types of fuzzy programming that can deal with different situations. Flexible programming and possibilistic programming will be described here.
+
There are several types of fuzzy programming that can deal with different situations. Flexible programming will be described here. 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:
==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:
+
  
 
<math>
 
<math>
Line 75: Line 73:
 
</math>
 
</math>
  
A new variable λ is used to construct an LP that can be solved easily
+
A new variable λ is used to construct an LP that can be solved easily. All membership functions must be greater than or equal to λ
 
+
 
<math> \max  \lambda \;\;\;\; s.t.  \bar{A} x + \lambda \le \bar{b}, x \ge 0, 0 \le \lambda \le 1 </math>
 
<math> \max  \lambda \;\;\;\; s.t.  \bar{A} x + \lambda \le \bar{b}, x \ge 0, 0 \le \lambda \le 1 </math>
  
where <math> \bar{A} \; \text{and} \; \bar{b} \; \text{are} \; \bar{a}_{ij} = \hat{a}_{ij}/ \Delta b_i  \; \text{and} \; \bar{b}_i = 1 + (\hat(b_i) / \Delta b_i) </math>
+
where <math> \sum_{k=1}^n \bar{A} \; \text{and} \; \bar{b} \; \text{are} \; \bar{a}_{ij} = \hat{a}_{ij}/ \Delta b_i  \; \text{and} \; \bar{b}_i = 1 + (\hat(b_i) / \Delta b_i) </math>
  
==Possibilistic programming==
+
n= m+1, total number of constraints including objective function.
In possibilistic programming
+
  
 
=Applications=
 
=Applications=
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This simple example for Water Resources Management [3] shows how triangular fuzzy programs can be applied to real life situations. NO1 and NO2 provide water to other nodes, while A1 and A2 are water consumers. The goal of this problem is to calculate the amount of water that should be supplied to A1 and A2 in order to maximize profit while still meeting the given constraints.
 
This simple example for Water Resources Management [3] shows how triangular fuzzy programs can be applied to real life situations. NO1 and NO2 provide water to other nodes, while A1 and A2 are water consumers. The goal of this problem is to calculate the amount of water that should be supplied to A1 and A2 in order to maximize profit while still meeting the given constraints.
  
 +
[[File:Water.png]]
 +
 +
This is the linear problem:
 
<math>
 
<math>
 
\max x_1 + 1.5x_2 \;\; \text{Revenue from consumers: A1 (1/L), A2 (1.5/L)}
 
\max x_1 + 1.5x_2 \;\; \text{Revenue from consumers: A1 (1/L), A2 (1.5/L)}
 
</math>
 
</math>
 
 
<math>
 
<math>
 
\begin{array}{lcl}
 
\begin{array}{lcl}
x_1 &\le& 66 \;\; \text{amount of water into N1} \\
+
x_1 &\le& 66 \;\; \text{1 amount of water into N1} \\
x_2 &\le& 59 \;\; \text{amount of water into N2} \\
+
x_2 &\le& 59 \;\; \text{2 amount of water into N2} \\
x_1 &\le& 49 \;\; \text{maximum demand from A1} \\
+
x_1 &\le& 49 \;\; \text{3 maximum demand from A1} \\
x_2 &\le& 35 \;\; \text{maximum demand from A2} \\
+
x_2 &\le& 35 \;\; \text{4 maximum demand from A2} \\
66+59 - 0.7x_1 - x_2 &\ge& 47 \;\; \text{at least 47L into N4, 30 percent of A1 enters N3} \\
+
66+59 - 0.7x_1 - x_2 &\ge& 47 \;\; \text{5 at least 47L into N4, 30 percent of A1 enters N3} \\
59-x_2 &\ge& 8 \;\; \text{at least 8L into N3} \\
+
59-x_2 &\ge& 8 \;\; \text{6 at least 8L into N3} \\
 
x_1,x_2 &\ge& 0
 
x_1,x_2 &\ge& 0
 
\end{array}
 
\end{array}
 
</math>
 
</math>
  
[[File:Water.png]]
+
The fuzzy problem involves equations 3 and 4 being set as fuzzy triangular numbers, and the minimum amount of water into N3 and N4 become fuzzy values as well.
 
+
 
<math>
 
<math>
 
p_1 = \begin{pmatrix}
 
p_1 = \begin{pmatrix}
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<math>
 
<math>
 
QN_1 = \begin{pmatrix}
 
QN_1 = \begin{pmatrix}
49 \\
+
45 \\
52 \\
+
47 \\
55
+
49
 
\end{pmatrix}
 
\end{pmatrix}
 
</math>
 
</math>
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<math>
 
<math>
 
QN_1 = \begin{pmatrix}
 
QN_1 = \begin{pmatrix}
49 \\
+
5 \\
52 \\
+
8 \\
55
+
11
 
\end{pmatrix}
 
\end{pmatrix}
 
</math>
 
</math>
 +
 +
We proceed to set up our membership functions:
 +
<math>
 +
u_p1(x1) =
 +
\begin{cases}
 +
1  & {x_1 \le 49} \\
 +
1- \frac{x_1-49}{55-49} & 49 < x_1 \le 55 \\
 +
0 &  x1 \ge 55
 +
\end{cases}</math>
 +
 +
<math>
 +
u_p2(x1) =
 +
\begin{cases}
 +
1  & {x_1 \le 35} \\
 +
1- \frac{x_1-35}{39-35} & 35 < x_1 \le 39 \\
 +
0 &  x1 \ge 39
 +
\end{cases}</math>
 +
 +
Equation 5 from the LP becomes
 +
<math>
 +
66+59 - 0.7x_1 - x_2 &\ge& (45,49) \;\; \text{5 at least 47L into N4, 30 percent of A1 enters N3} \\
 +
0.7x_1 + x_2 \in (75,105)
 +
</math>
 +
and Eqn 6 becomes
 +
<math>
 +
x_2 \in (59-11,59-5)
 +
</math>
 +
Finally, the objective function becomes
 +
<math>
 +
x_1 + 1.5x_2 \in (75,105)
 +
</math>
 +
 +
Following the method described for flexible programming, we set up a new LP
 +
 +
<math>
 +
\max \lambda
 +
</math>
 +
<math>
 +
\begin{array}{lcl}
 +
x_1 + 1.5x_2 \ge 75 + 30\lambda\\
 +
x_1 &\le& 66 \;\; \text{does not change from LP} \\
 +
x_2 &\le& 59 \;\; \text{does not change from LP} \\
 +
x_1 &\le& 49 + (1-\lambda)6 \\
 +
x_2 &\le& 35 + (1-\lambda)4 \\
 +
0.7x_1 + x_2 \le 76 + (1-\lambda)5
 +
x_2 \le 48 + (1-\lambda)6
 +
x_1,x_2 &\ge& 0\\
 +
0 \le \lambda \le
 +
\end{array}
 +
</math>
 +
 +
  
 
=Conclusion=
 
=Conclusion=

Revision as of 22:46, 7 June 2015

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, or an exact value is not critical to the problem. 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 membership function (u), which describes the 'grey spectrum' where constraint violation is permitted [1]


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

Continuing with the age analogy - if young ranges from ages 0 to 30, we can define 0 to 18 as being definitely young, so u = 1. However, as we increase the age from there young is not explicitly defined, and can be given lower u values, as these ages are defined as "less young".

Method: Flexible Programming

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


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

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"


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

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


\begin{align}
Find \; x \\
s.t. \; & c^t x \tilde{\ge} z \\
& Ax \tilde{\le} b\\
& x \tilde{\ge} 0
\end{align}

The two constraints can be combined and the problem is further simplified to:


\begin{align}
Find \; x \\
s.t. \; & \hat{A}x \; \tilde{\le} \hat{b} \\
\end{align}

u_o is the membership function for the initial objective while u_i is the membership function for the constraints. These membership functions describe how closely the fuzzy inequalities are satisfied, we can describe the optimal decision to be:


\max \min {u_i(x)} = \max u_o(x)

An optimal solution to this problem can be found by solving


max \; min \;  1- \frac{\hat{A}_i x - b_i}{\Delta b_i}

A new variable λ is used to construct an LP that can be solved easily. All membership functions must be greater than or equal to λ  \max  \lambda \;\;\;\; s.t.  \bar{A} x + \lambda \le \bar{b}, x \ge 0, 0 \le \lambda \le 1

where  \sum_{k=1}^n \bar{A} \; \text{and} \; \bar{b} \; \text{are} \; \bar{a}_{ij} = \hat{a}_{ij}/ \Delta b_i  \; \text{and} \; \bar{b}_i = 1 + (\hat(b_i) / \Delta b_i)

n= m+1, total number of constraints including objective function.

Applications

Examples

This simple example for Water Resources Management [3] shows how triangular fuzzy programs can be applied to real life situations. NO1 and NO2 provide water to other nodes, while A1 and A2 are water consumers. The goal of this problem is to calculate the amount of water that should be supplied to A1 and A2 in order to maximize profit while still meeting the given constraints.

Water.png

This is the linear problem: 
\max x_1 + 1.5x_2 \;\; \text{Revenue from consumers: A1 (1/L), A2 (1.5/L)}

\begin{array}{lcl}
x_1 &\le& 66 \;\; \text{1 amount of water into N1} \\
x_2 &\le& 59 \;\; \text{2 amount of water into N2} \\
x_1 &\le& 49 \;\; \text{3 maximum demand from A1} \\
x_2 &\le& 35 \;\; \text{4 maximum demand from A2} \\
66+59 - 0.7x_1 - x_2 &\ge& 47 \;\; \text{5 at least 47L into N4, 30 percent of A1 enters N3} \\
59-x_2 &\ge& 8 \;\; \text{6 at least 8L into N3} \\
x_1,x_2 &\ge& 0
\end{array}

The fuzzy problem involves equations 3 and 4 being set as fuzzy triangular numbers, and the minimum amount of water into N3 and N4 become fuzzy values as well. 
p_1 = \begin{pmatrix}
49 \\
52 \\
55 
\end{pmatrix}


p_2 = \begin{pmatrix}
35 \\
37 \\
39 
\end{pmatrix}


QN_1 = \begin{pmatrix}
45 \\
47 \\
49 
\end{pmatrix}


QN_1 = \begin{pmatrix}
5 \\
8 \\
11 
\end{pmatrix}

We proceed to set up our membership functions: 
u_p1(x1) = 
\begin{cases}
1  & {x_1 \le 49} \\
1- \frac{x_1-49}{55-49} & 49 < x_1 \le 55 \\
0 &  x1 \ge 55
\end{cases}


u_p2(x1) = 
\begin{cases}
1  & {x_1 \le 35} \\
1- \frac{x_1-35}{39-35} & 35 < x_1 \le 39 \\
0 &  x1 \ge 39
\end{cases}

Equation 5 from the LP becomes Failed to parse(syntax error): 66+59 - 0.7x_1 - x_2 &\ge& (45,49) \;\; \text{5 at least 47L into N4, 30 percent of A1 enters N3} \\ 0.7x_1 + x_2 \in (75,105)

and Eqn 6 becomes 
x_2 \in (59-11,59-5)
Finally, the objective function becomes 
x_1 + 1.5x_2 \in (75,105)

Following the method described for flexible programming, we set up a new LP


\max \lambda

\begin{array}{lcl}
x_1 + 1.5x_2 \ge 75 + 30\lambda\\
x_1 &\le& 66 \;\; \text{does not change from LP} \\
x_2 &\le& 59 \;\; \text{does not change from LP} \\
x_1 &\le& 49 + (1-\lambda)6 \\
x_2 &\le& 35 + (1-\lambda)4 \\
0.7x_1 + x_2 \le 76 + (1-\lambda)5
x_2 \le 48 + (1-\lambda)6
x_1,x_2 &\ge& 0\\
0 \le \lambda \le
\end{array}


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 [3] http://www.ewra.net/ew/pdf/EW_2004_7-8_03.pdf