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Introduction D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 3 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.
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Fuzzy Optimization
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L1
Advanced Topics
Objectives
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L12
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
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L13
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
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L14
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…
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L15
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 ,,
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L16
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
Fuzzy Set Theory and Membership Functions…
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L17
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 Theory and Membership Functions…
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L18
Fuzzy set operations Union of fuzzy sets A and B
(5)
Intersection of fuzzy sets A and B
(6)
Complement of fuzzy sets A
Fuzzy Set Theory and Membership Functions…
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
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L19
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…
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L110
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
Fuzzy Optimization…
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L111
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.
Fuzzy Optimization…
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L112
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
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L113
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
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L114
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
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L115
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…
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L116
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
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L117
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…
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L118
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
Example: Symmetric Fuzzy LP…
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L119
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
Example: Symmetric Fuzzy LP…
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L120
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
D Nagesh Kumar, IIScWater Resources Planning and Management: M9L1
Thank You