Solving Fuzzy Optimization Problems: Flexible Approaches using Simulated Annealing

In: Proceedings of the World Automation Congress, WAC´96, Vol 5, TSI Press, May 1996

Solving Fuzzy Optimization Problems:

Flexible Approaches using Simulated Annealing

Fernando Moura Pires João Moura Pires Rita Almeida Ribeiro

Universidade Nova Lisboa FCT, Dept. Informatica 2825 Monte Caparica

Portugal e-mail: [email protected]





Abstract

This paper discusses flexible approaches for fuzzy optimization problems. Specifically, it presents a new solving

method for fuzzy optimization problems (FOP) when the variables coefficients and/or the constraints limits

parameters are imprecise. The new method uses the simulated annealing algorithm as a resolution procedure

because it is a well-suited algorithm for solving all forms of fuzziness found in fuzzy optimization problems. An

example is given to illustrate the method.

Keywords: fuzzy optimization, simulated annealing, flexible approaches, fuzzy multiple objective decision

making.

1. Introduction

Negoita (Negoita, 1981) distinguishes two types of imprecision in fuzzy optimization problems: (a) flexible

programming for problems with fuzzy equations and objectives (vague nature) and (b) robust programming for

fuzzy coefficients or parameters (of an ambiguous nature). This paper extends the concept of flexibility to

include fuzzy coefficients of variables and fuzzy objective function costs. This method is specially well-suited

for problems where the decisor is willing to consider a degree of flexibility either in order to obtain better results

or in presence of infeasible solutions. To illustrate the method a crisp infeasible solution problem is discussed.

The most important approaches proposed in the literature to solve fuzzy coefficients in optimization problems

are given by Tanaka and Asai (Tanaka & Asai, 1984a; Tanaka & Asai, 1984b), Carlsson and Korhonen

(Carlsson & Korhonen, 1986), Lai and Hwang (Lai & Hwang, 1994) and Negoita and Ralescu (Zimmermann,

1986). The first three approaches may be included in the realm of possibility distributions. The later authors deal

with tolerance analysis and solutions using fuzzy intervals and strict inclusion. Comprehensive overviews of this

subject are given by Fredrizzi et al. (Fredizzi,Kacprzyk, & Verdegay, 1991) Dubois (Dubois, 1987) and Lai and

Hwang (Lai & Hwang, 1994).

This paper follows a different perspective form the above in considering that the fuzzy coefficients could be


flexible rather than considered imprecise. For example, in the case of an infeasible solution the decision maker

might consider that instead of estimating the time to produce an item as 3 hours this can become flexible at

“around 3”.

Simulated annealing is a stochastic algorithm with the physical analogy of melting a solid using high

temperatures and then slowly lowering the temperature until the system crystallizes/freezes and no more changes

occur (Kirkpatrick,Gelatt, & Vecchi, 1983). The melting process is a similar process to the constraints

flexibilisation in a fuzzy optimization problem and the crystallization is a similar process to the search for the

“best” solution. Simulated annealing is considered a good tool for solving crisp optimization problems that are

NP-complete (Kirkpatrick et al., 1983) and fuzzy optimization problems (Ribeiro & Moura-Pires, 1995a;

Ribeiro & Moura-Pires, 1995b).

The paper is organized in six sections. The first one is this introduction, the second section presents the basics of

the fuzzy optimization problem. The third section discusses the theoretical aspects of all possible fexibilisation

procedures for fuzzy optimization problems. The fourth section introduces a new solution method using the

simulated annealing algorithm for fuzzy optimization problems. The fifth discusses an example of flexibility in

an infeasible crisp linear problem. The sixth section presents the conclusions.

2. Fuzzy Optimization

In general, fuzzy optimization problems are concerned with the maximization or minimization of a single or

multiple objectives while satisfying the problem constraints, which represent the model limited resources. Fuzzy

optimization’s main aim is to find the most satisfying solution (decision alternative) within a fuzzy environment.

Bellman and Zadeh addressed fuzzy optimization problems in their seminal paper on “Decision making in a

fuzzy environment” (Bellman & Zadeh, 1970) where they consider that in a fuzzy environment there is no

distinction between objectives and constraints. Following this symmetric fuzzy decision concept Zimmermann

(Zimmermann, 1978) developed the first approach for solving fuzzy linear programming problems. This

approach, denoted maxmin, assumes that the decision maker stipulates a goal for each of the objective functions

and hence, both goals and constraints may be considered fuzzy constraints. Further, Zimmermann proposes a

crisp equivalent model for maximizing the satisfaction of the intersection of the constraints, which can be solved

by traditional methods such as the simplex algorithm. Zimmermann, as did Bellman and Zadeh, points out that

using flexible constraints blurs the difference between single and multiple objective problems.

The fuzzification of the single or multiple objective linear programming model usually includes four forms of

imprecision (Fredizzi et al., 1991) (more subtle distinctions are made in (Lai & Hwang, 1994)): 1. Problems

with fuzzy constraints. E.g. “the total distance time between warehouses should be considerably fewer than 4

hours ”; 2. Problems with fuzzy objectives (goals imposed on the objective functions). E.g. “the total cost for the

project should be kept well below 100.000 ECUs”; 3. Problems with fuzzy coefficients on the variables. E.g.


“the transportation cost per item (x) is about 10 ECUs”; 4. Combinations of the above.

This paper concentrates on the third and forth cases, hence completing previous research on the solution of the

main forms of vagueness found in fuzzy optimization using the simulated annealing algorithm (Ribeiro &

Moura-Pires, 1995a; Ribeiro & Moura-Pires, 1995b). The new method deals specifically with fuzzy coefficients

as vague concepts, which can be stretched in a controllable fashion as deviations from the ideal (crisp) values.

In summary, the fuzzy optimization problem with fuzzy coefficients and fuzzy parameters can be

mathematically expressed by:

max / min Z = ˜ C x

˜ A x !,",={ } ˜ B x ! 0

where ˜ C is the fuzzy costs vector, ˜ A is the technological matrix containing the fuzzy coefficients of the

objective(s) and of the constraints and ˜ B is the corresponding vector of goals and resource limits (parameters).

3. Fuzzy Flexible Approaches

This paper presents the fuzzification of a crisp optimization problem using flexible approaches. To facilitate the

theoretical discussion a case of maximization with constraints of ≤ signal is provided. Consider the following

classical crisp linear maximization problem:

maxx

ck

k =1

K

! xk

s.t. aikx

k" b

i

k=1

K

! i = 1,.. ., N

When either the problem has an infeasible solution, or the decision maker (DM) is willing to accept some

compromise on the variable coefficients or on the objective function costs, or the resources can be stretched,

then the problem can be relaxed.

In general, the flexibilisation approaches to be considered are: (1) fuzzification of the constraints; (2)

fuzzification of the objective function costs. Of course combinations of the two is allowed.

In the first case (1), constraints fuzzification, the flexibility process replaces crisp constraints by fuzzy

constraints, thus becoming a problem of constraint satisfaction with a matter of degree. Three possible flexible

types could be considered:

1.1. Fuzzifying bi. A fuzzyfied constraint is a

ikx

k

k=1

K

! = ˜ b i where ˜ b

i is a fuzzy set as depicted in the next

figure. Obviously the membership function used can be any other function that represents a relaxation of

≤ bi.


bi

1

µbi

(b)

The satisfaction level of the constraint i given a vector value x is:

µ i (x) = µ ˜ b i( a ikxk

k =1

K

! ) (1)

1.2. Fuzzifying aik

. A fuzzyfied constraint is ˜ a ik

xk

k=1

K

! " bi where ˜ a

ik is a fuzzy set that indicates how

acceptable are the values around aik

. The satisfaction level of the constraint i given a vector x is:

µ i (x) =yi1 ,K,y iK

y ikx k ! bik=1

K

"

sup min(µ ˜ a i1(yi1),K,µ˜ a iK

(yiK )){ } (2)

1.3. Fuzzifying both aik

and bi. A fuzzyfied constraint is now ˜ a

ikx

k

k=1

K

! = ˜ b i. The satisfaction level of

the constraint i given a vector x is:

µ i (x) =yi1 ,K,y iK

sup min(µ ˜ a i1(y i1),K, µ ˜ a iK

(y iK), µ ˜ b i( y ikxk

k =1

K

! ))" # $

% & '

(3)

The constraints global satisfaction level is given by the minimum of all constraints satisfaction levels:

µ(x) = mini=1,..., N

µ i (x) (4)

Let Xα be the set of x vector values that satisfy the constraints at level greater than α:

X ! = x:µ(x) > !{ } (5)

where µ(x) is calculated for the specified α using (4). Notice that for β < α X! " X# . If αmax is the maximal

α for which Xα is non-empty,

0 < !1<K< !

max< 1

X0" X!

1

"K" X!max

(6)

Then, the optimization problem can now be formulated as:

maxx

ck

k =1

K

! xk

s.t. x "X#

(7)

where the vector x* is a solution for the problem if there is no other x of Xα such that:

ck

k=1

K

! xk > ckk=1

K

! xk*


and this solution satisfies the constraints with level α (4) and is contained in the interval:

! < µ(x*) " !max

Let Zα denote the optimal value for a given Xα, then:

0 < !1<K< !

max" 1

Z0# Z!1

#K# Z! max

µ(x0

*) " µ(x!1

*) "K" µ(x!max

*) = !

max

(8)

The second case (2) handles the fuzzification of the objective function costs. In this case the (7) is rewritten as

maxx

˜ c k

k =1

K

! xk

s.t. x "X#

(9)

where ˜ c k is a fuzzy set that indicates how acceptable are the values around ck. Let w = (w1, w2, ..., wK) be a set

of objective function costs such that µ ˜ c k(wk ) > 0 . In fact, for each different combination of values wk. a

different function to maximize over x is obtained. Each set has a satisfaction level defined by:

µ ˜ C (w) =

kmin µ˜ c k

(w k )

and

W! = w:µ ˜ c (w) > !{ } (10)

Then the maximization problem can be sated as:

maxx,w

wk

k =1

K

! xk

s.t. x "X#

w "W#

(11)

The global satisfaction level is the minimum between the existing constraints and the cost satisfaction

constraints levels.

For a linear objective function, the maximization over w can be easily obtained by considering:

• the greatest wk such that µ ˜ c k(wk ) ! " , if c has a positive signal in the objective;

• the lowest wk such that µ ˜ c k(wk ) ! " , if c has a negative signal in the objective ;

Let wk

* be the selected set obtained, then (11) is simplified to:

maxx

wk

*x

k

k =1

K

!

s.t. x "X#

(11)

It should be noted that if the DM wants to have a better constraint satisfaction level this may cause a decrease in

the optimal value. Moreover, the DM has no way to predict what will be the increment or decrement of Z for an


arbitrary α. Indeed, the more the constraints are relaxed the more likely it is that a better optimization value for

Z will be found. In summary, modifications in the α-level have to be carefully controlled and justified because

the effects on the optimal value are unpredictable.

4. New Solving Method

This paper discusses an extension to include fuzzy coefficients and fuzzy costs of the objective function. The

whole process is called a flexible approach to fuzzy optimization problems and a new solving method is

proposed.

Using the most general equation for flexible constraints (3) and (11), the formalization proposed is as follows:

maxx

Z = ˜ c k.x

k

k=1

K

!

˜ a i,k

.xk

= ˜ b i

k =1

K

! i = 1,…, N

"

#

$ $

%

$ $

which are converted into the non-linear equation system:

maxx

Z = wk

.xk

k =1

K!

maxy, w

µ x( )

yi, k

.xk

= ˜ b i

k = 1

K!

yi, k

= ˜ a i, k

w k = ˜ c k k = 1,…, K e i = 1,…, N

"

#

$ $ $ $

%

$ $ $ $

"

#

$ $ $ $ $

%

$ $ $ $ $

In this process of flexing the fuzzy ck and aik become variables of the process. It may even be that one of these

coefficients is dependent on other ck or aik . These dependencies can easily be incorporated in the model by

expressing the existing relationships. Note that with this new method only the independent variables are

variables of the optimization problem.

The solving procedure for case 1, fuzzy constraints (1), using the simulating annealing algorithm, were first

proposed by Ribeiro and Moura Pires (Ribeiro & Moura-Pires, 1995a). That proposal includes solutions for non-

linear equations.

Based on equation (7) the solving method implemented in the simulated annealing algorithm is developed in two

stages: first step, to find the value of x,y,w (independent variables) which maximize the aggregated µ of the

constraints; second step, find the optimal value of Z that satisfies all constraints with α-cut = αmax.


5. Flexible Optimization Problem: An example

Consider the following crisp linear optimization problem:

max 40x1 + 30x2

s.t. (1) 0.4x1 + 0.5x2 ! 20

(2) 0.2x2 ! 5

(3) 0.6x1 + 0.3x2 ! 21

(4) x1 " 30

(5) x2 " 15

which has no solution, i. e. there are no values (x1, x2) that satisfy all the constraints, as shown in the next

figure. The next figure represents the region where the constraints 1, 2, 3 and 5 are satisfied. However, in

conjunction with constraint 4 (x1 ≥ 30) the space of satisfying solutions becomes an empty set.

10

15

20

25

30

35

40

10 15 20 25 30 35 40

(1)

(3)(4)

(2)

(5)

satisfying (1) (2) (3) and (5)

x1

x2

Since this problem has an infeasible solution it can be relaxed using the approaches described in section 3. To

simplify the example only the constraints are relaxed (case 1 of the flexibility process). The fuzzification is

performed with triangular fuzzy sets for the aik coefficients, where the semi-base of the triangle is 10% of the

value aik . For the bi parameters the same fuzzy sets are assumed. The next figure depicts the triangular fuzzy

sets used in this example.

bi

1

µbi

(1+ 0.1)bi bi

1

µbi

(1- 0.1)bi aik

1

(1+0.1)aik

µ aik

The problem formalization using the possible types of constraints’ flexibility are:


Case 1.1. Case 1.2. Case 1.3.

max 40x1 + 30x2

s. t. 0.4x1 + 0.5x2 ! 20

~

0.2x2 !˜ 5

0.6x1 + 0.3x2 ! 21~

x1 " 30~

x2 "15

~

max 40x1 + 30x2

s.t. y1x1 + y2 x2 ! 20

y3x2 ! 5

y4x1 + y5x2 ! 21

x1 " 30, x2 "15

y1 = 0.4~

, y2 = 0.5~

y3 = 0.2~

, y4 = 0.6~

y5 = 0.3~

max 40x1 + 30x2

s.t. y1x1 + y2 x2 ! 20~

y3x2 ! 5~

y4x1 + y5x2 ! 21~

x1 " 30, x2 "15~~

y1 = 0.4~

, y2 = 0.5~

y3 = 0.2~

, y4 = 0.6~

,

y5 = 0.3~

The results obtained for all possible cases of constraint fuzzification (flexibility), using the proposed approach

(section 4) and the simulated annealing algorithm, are:

max Z over Xαmax

x1 = 28.97 µC = 0.655

Ax = ˜ b x2 = 14.48 Z = 1593

x1 = 30 µC = 0.33

˜ A x ! b x2 = 15.02 Z = 1651

x1 = 29.31 µC = 0.77

˜ A x = ˜ b x2 = 14.67 Z = 1612

where the µc represents the maximum of the minimum membership value of all constraints.

As can be observed the results depict variations depending on the type of flexibility used. The factors which

most influence the results are flexibility of the variable coefficients providing the best value for Z at the cost of

only satisfying the constraints with level µc=0.33. However, considering the complete flexibilisation (the last

case) for a not so good value of Z a much better satisfaction level for the µc of 77% is obtained. As expected, the

last case shows that the µc is necessarily bigger or equal to the memberships of the other two cases and that

there is no relation between the two first cases. Further, it is clear that any change of a scale of 10% for the

coefficients and the parameter flexibility will also cause changes in results obtained. In summary, the

fuzzification process enhances the possible trade-offs between flexibility, constraint satisfaction and optimal

value that the decision maker has at his/her disposal.


Most of the literature on fuzzy optimization is concerned with fuzziness at the modeling level, from fuzzy goals

and constraints to fuzzy coefficients. Most authors dealing with fuzzy coefficients (Carlsson & Korhonen, 1986;

Lai & Hwang, 1994; Tanaka & Asai, 1984b) perform mathematical transformations in the initial fuzzy model in

order to solve a crisp linear or non-linear problem. Here the fuzzy problem is solved with the simulated

annealing algorithm which can solve any type of fuzzy optimization problem (Ribeiro & Moura-Pires, 1995a).

The benefits of using the simulated annealing algorithm to solve traditional optimization problems have already

been discussed in the literature (Connoly, 1992; Eglese, 1990).

6. Conclusions

This paper discussed types of flexible approaches for fuzzy optimization problem. Then, it develops a new

solution method which uses the simulated annealing algorithm. The proposed method is illustrated with an

example which shows how flexible and adaptable the method is for solving fuzzy optimization problems.

As illustrated, the main advantage of using the proposed method is the freedom for the decision making to

choose any model which s/he finds appropriate for a specific problem. The proposed method can handle any

type of fuzzy linear and non-linear optimization problems. Another advantage is the possibility of solving

unfeasible crisp problems, which is given by working in a fuzzy environment and the inclusion of imprecision or

vagueness when the coefficients or parameters are not known precisely.

In conclusion, the fuzzification method proposed here enhances the possible trade-offs between flexibility,

constraints satisfaction and optimal value that the decision maker has at his/her disposal.

Acknowledgement

The authors would like to thank Dr. Philip Powell for his assistance with this paper.

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Solving Fuzzy Optimization Problems: Flexible Approaches using Simulated Annealing

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