fuzzy optimization d nagesh kumar, iisc water resources planning and management: m9l1 advanced...

Fuzzy Optimization

D Nagesh Kumar, IIScWater Resources Planning and Management: M9L1

Advanced Topics

Objectives


To briefly discuss the fuzzy set theory and membership

functions

To incorporate fuzziness in optimization problems

To discuss fuzzy linear programming and its applications in

water resources

Introduction


Models discussed so far in this lecture are crisp and precise in nature

Crisp: Dichotonomous i.e., yes-or-no type (or true or false) and not more-or-less type

This indicates that the model is unequivocal or it contains no ambiguities

Most of the real situations are not crisp; but are vague

Fuzziness: Vagueness in the events, phenomena or statements

(For eg. “tall men”, “beautiful flower”, “profitable deal” etc.)

In planning, fuzziness can be expressed as plan A is better than plan B or plan A is

more acceptable to some and less acceptable to others.

Fuzzy Set Theory and Membership Functions


Let X be a crisp set of integers, whose elements are denoted by x Consider a set of integer numbers ranging from 20 – 30

Set A = [20, 30]. In classical or crisp set theory, any number, say x either exists in A or not, i.e., set A

is crisp Hence membership in a classical subset A of X can be expressed as a characteristic

function

(1) Set [0, 1] is called the valuation set Suppose when it is not certain about the existence of x in A, then set A is fuzzy The degree of truth attached to that statement is defined by a membership function

AxifAxif

xA 01

Fuzzy Set Theory and Membership Functions…


Fuzzy set A is characterized by the set of all pairs of points denoted as

(2)

where μA(x) is the membership function of x in A

Closer the value of μA(x) is to 1, the more x belongs to A

For example, let the possible releases X from a reservoir be

X = {25 30 35 40 45 50}

and the irrigation demand be 40.

Then the fuzzy set A of “satisfiable releases without causing crop damage” may be

A = {(25,0.25), (30,0.5), (35,0.75), (40,1), (45,0.75), (50,0.5)}

XxxxA A ,,


Membership function is normally represented by a geometric shape which maps

each point x to a membership value between 0 and 1 Membership function ranges from 0 (completely false) to 1 (completely true). Commonly used membership function shapes are triangular, trapezoidal and bell

shape (gaussian). For the above example, the membership function assumed is a triangular one Release X = {25 30 35 40 45 50}

Demand = 40

Fuzzy set A

A = {(25,0.25), (30,0.5), (35,0.75), (40,1), (45,0.75), (50,0.5)}

Releases

0

1

4020 25 30 35 45 50 55 60

Triangular shaped membership function



If X is a finite set {x1, x2, x3,…, xn} then the fuzzy set can be expressed as

(3)

If X is infinite

(4)

Fuzzy set operations Basic set theory operations: Union, Intersection and Compliment

Let A and B be two fuzzy sets and μA and μB be their membership functions as shown

n

ixiAxnAxAxA

inxxxxA

121

21 ...

x

xA xA

0

1

0

1

μAμB

Membership functions of A and B



Fuzzy set operations Union of fuzzy sets A and B

(5)

Intersection of fuzzy sets A and B

(6)

Complement of fuzzy sets A


BAB

BAA

BABA

ifxifx

xxx

,max

BAB

BAA

BABA

ifxifx

xxx

,min

xx AA 1

0

1

0

1

0

1

μA μB

Union

Intersection

Compliment

Fuzzy Optimization


Conventional optimization models find the optimum value of design variables which

optimizes the objective function subject to the stated constraints If the system is fuzzy, then this optimization problem needs to be revised Fuzzy system: Objective and constraints are expressed by the membership functions Decision: Intersection of the fuzzy objective and constraint functions Consider the water allocation problem in which the objective function is

“The water allocated for irrigation should be substantially greater than 10”. Membership function for objective function f is

(7) Let the constraint be

“The amount of water allocated should be around 11.5” Membership for this constraint is

12101

100

x

xifxf

135111

.xxg

Fuzzy Optimization…


Then ,the decision can be described by the membership function, μD(x) as

(8)

105111101

1001312 xifxx

xif

xxx gfD

.,min

Membership function of objective, μf (x)Membership function

of constraint, μg (x)

Membership function of decision, μD (x)

Fuzzy decision



Formulation: Let the conventional optimization problem be

Minimize f(X)

Subject to m constraints

llj ≤ gj (X) ≤ ulj for j = 1,2,…,m

where llj is the lower bound and ulj is the upper bound of the jth constraint.

The fuzzy optimization problem can be stated as

Minimize f(X)

Subject to m constraints

gj (X) ϵ Gj for j = 1,2,…,m

where Gj is the fuzzy interval the constraint gj (X) should belong.



Formulation:

Feasible region of this fuzzy system is the intersection of all these Gj’s

Defined by the membership function

Optimum value is the maximum value of the intersection of objective function and

feasible domain

where

XgX jGmjS j

,...,,min

21

XX DD max*

XgXX jGmjfD j

,...,,min,min

21

Fuzziness in LP Model


In an LP model, the coefficients of the vectors b or c or of the matrix A itself can

have a fuzzy character.

This can happen either because they are fuzzy in nature or because perception of

them is fuzzy

In classical LP, the violation of any single constraint by any amount renders the

solution infeasible

In real situation, the decision maker might accept small violations of constraints

May also attach different (crisp or fuzzy) degrees of importance to violations of

different constraints

Fuzzy LP offers a number of ways to allow for all those types of vagueness.

Fuzzy LP


Goal and constraints are represented by fuzzy sets Then aggregate them in order to derive a maximizing decision (Bellman-Zadeh's

approach) In contrast to classical LP, FLP is NOT a uniquely defined type of model but many

variations are possible, depending on the assumptions or features of the real situation to be modeled

Symmetric Fuzzy LP Decision maker can establish an aspiration level, z, for the value of the objective

function Each of the constraints is modeled as a fuzzy set Fuzzy LP can then be formulated as:

cTx z Ax b; x ≥ 0 (10)

≥~≤~

Symmetric Fuzzy LP


Objective function is converted to a fuzzy goal Fuzzified version of

≥ has the linguistic interpretation “essentially greater than or equal” ≤ has the linguistic interpretation “essentially smaller than or equal”

Each constraint and Objective function will be represented by a Fuzzy set with a

membership function i(x)

Membership function i(x) increases monotonously from 0 to 1 with a value 0 if the constraints (including objective function) are strongly violated and a value 1 if they are very well satisfied (i.e., satisfied in the crisp sense)

Membership function can expressed as

(11)

where pi is tolerance interval (subjectively chosen).

iii

iiii

ii

i

pdxBifmipdxBdif

dxBifx

01,...2,11,0

1

Symmetric Fuzzy LP…


Assuming a linear increase over the tolerance interval i(x) will be

(12)

Hence, fuzzy LP model can be defined as

Maximize λ

Subject to λpi + Bix ≤ di + pi i = 1,2,…,m+1 (13)

x ≥ 0

where is one new variable. Optimal solution is the vector (, x*) Hence in fuzzy LP model maximizing solution can be obtained by solving one standard

(crisp) LP with only one more variable and one more constraint than the original

crisp LP model

iii

iiiii

ii

ii

i

pdxBif

mipdxBdifp

dxBdxBif

x

0

1,...2,11

1

Example: Symmetric Fuzzy LP


A farming company wanted to decide on the size and number of pumps required for lift irrigation. Four differently sized pumps (x1 through x4) were considered. The objective

was to minimize cost and the constraints were to supply water to all fields (who have a strong seasonally fluctuating demand). That meant certain quantities had to be supplied (quantity constraint) and a minimum number of fields per day had to be supplied (routing constraint). For other reasons, it was required that at least 6 of the smallest pumps should be included. The management wanted to use quantitative analysis and agreed to the following suggested linear programming approach. The available budget is Rs. 42 lakhs. The optimization problem is

Minimize 41,400 x1 + 44,300 x2 + 48,100 x3 + 49,100 x4

Subject to 0.84 x1 + 1.44 x2 + 2.16 x3 + 2.4 x4 ≥ 170

16x1 + 16 x2 + 16 x3 + 16 x4 ≥ 1300 x1

≥ 6 x2, x3, x4 ≥ 0

Example: Symmetric Fuzzy LP…


The solution of this problem using classical LP is

Min Cost = Rs. 38,64,975

x1= 6, x2 = 16.29, x3= 0, x4 = 58.96.

Fuzzy LP

As the demand forecasts had been used to formulate the constraints, there was a danger of

not being able to meet higher demands

It is safe to stay below the available budget of Rs. 42 lakhs.

Therefore, bounds and spread of the tolerance interval are fixed as follows

Bounds: d1 = 37,00,000; d2 = 170; d3 = 1,300; d4 = 6

Spreads: p1=5,00,000; p2=10; p3=100; p4=6

Objective function in the classical LP problem is transformed as a constraint

41,400 x1 + 44,300 x2 + 48,100 x3 + 49,100 x4 + 42,00,000



Optimization problem constraints are (acc. to eqn. 13)

Maximize

Subject to 0.083 x1 + 0.089 x2 + 0.096 x3 + 0.098 x4 + 8.4

0.084 x1 + 0.144 x2 + 0.216 x3 + 0.240 x4 - ≥ 17

0.16 x1 + 0.16 x2 + 0.16 x3 + 0.16 x4 - ≥ 13

0.167 x1 - ≥ 1

, x2, x3, x4 ≥ 0



Solutions obtained using classical and fuzzy LP

Through Fuzzy LP, a "leeway" has been provided with respect to all constraints and at

additional cost of 3.2% Decision maker is not forced into a precise formulation because of mathematical reasons even

though he/she might only be able or willing to describe his/her problem in fuzzy terms

Classical LP Fuzzy LP

Z = 38,64,975 Z = 39,88,250

x1 = 6 ; x2 = 16.29 ; x4 = 59.96 x1 = 17.41 ; x2 = 0 ; x4 = 66.54

Constraints:

1. 170 1. 174.33

2. 1300 2. 1343.328

3. 6 3. 17.414


Thank You

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