1. Fuzzy Optimization D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 Advanced Topics

2. Objectives D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 2  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

3. 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.

4. Fuzzy Set Theory and Membership Functions D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 4 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

5. Fuzzy Set Theory and Membership Functions… D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 5 Fuzzy set A is characterized by the set of all pairs of points denoted as (2) where μ A (x) is the membership function of x in A Closer the value of μ A (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)}

6. D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 6 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 40 20 25 3035 4550 55 60 Triangular shaped membership function Fuzzy Set Theory and Membership Functions…

7. D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 7 If X is a finite set {x 1, x 2, x 3,…, x n } 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 μ A and μ B be their membership functions as shown 0 1 0 1 μAμA μBμB Membership functions of A and B Fuzzy Set Theory and Membership Functions…

8. D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 8 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… 0 1 0 1 0 1 μAμA μBμB Union Intersection Compliment

9. Fuzzy Optimization D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 9 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

10. Fuzzy Optimization… D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 10 Then,the decision can be described by the membership function, μ D (x) as (8) Membership function of objective, μ f (x) Membership function of constraint, μ g (x) Membership function of decision, μ D (x) Fuzzy decision

11. Fuzzy Optimization… D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 11 Formulation: Let the conventional optimization problem be Minimize f(X) Subject to m constraints ll j ≤ g j (X) ≤ ul j for j = 1,2,…,m where ll j is the lower bound and ul j is the upper bound of the jth constraint. The fuzzy optimization problem can be stated as Minimize f(X) Subject to m constraints g j (X) G j for j = 1,2,…,m where G j is the fuzzy interval the constraint g j (X) should belong.

12. Fuzzy Optimization… D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 12 Formulation: Feasible region of this fuzzy system is the intersection of all these G j ’s Defined by the membership function Optimum value is the maximum value of the intersection of objective function and feasible domain where

13. Fuzziness in LP Model D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 13 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.

14. Fuzzy LP D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 14 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: c T x z Ax b; x ≥ 0 (10) ≥ ~ ≤ ~

15. Symmetric Fuzzy LP D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 15 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 p i is tolerance interval (subjectively chosen).

16. Symmetric Fuzzy LP… D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 16 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 λp i + B i x ≤ d i + p i 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

17. Example: Symmetric Fuzzy LP D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 17 A farming company wanted to decide on the size and number of pumps required for lift irrigation. Four differently sized pumps (x 1 through x 4 ) 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 x 1 + 44,300 x 2 + 48,100 x 3 + 49,100 x 4 Subject to 0.84 x 1 + 1.44 x 2 + 2.16 x 3 + 2.4 x 4 ≥ 170 16x 1 + 16 x 2 + 16 x 3 + 16 x 4 ≥ 1300 x 1 ≥ 6 x 2, x 3, x 4 ≥ 0

18. Example: Symmetric Fuzzy LP… D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 18 The solution of this problem using classical LP is Min Cost = Rs. 38,64,975 x 1 = 6, x 2 = 16.29, x 3 = 0, x 4 = 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: d 1 = 37,00,000; d 2 = 170; d 3 = 1,300; d 4 = 6 Spreads: p 1 =5,00,000; p 2 =10; p 3 =100; p 4 =6 Objective function in the classical LP problem is transformed as a constraint 41,400 x 1 + 44,300 x 2 + 48,100 x 3 + 49,100 x 4 +  42,00,000

19. Example: Symmetric Fuzzy LP… D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 19 Optimization problem constraints are (acc. to eqn. 13) Maximize Subject to0.083 x 1 + 0.089 x 2 + 0.096 x 3 + 0.098 x 4 +  8.4 0.084 x 1 + 0.144 x 2 + 0.216 x 3 + 0.240 x 4 - ≥ 17 0.16 x 1 + 0.16 x 2 + 0.16 x 3 + 0.16 x 4 - ≥ 13 0.167 x 1 - ≥ 1, x 2, x 3, x 4 ≥ 0

20. Example: Symmetric Fuzzy LP… D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 20 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 LPFuzzy LP Z = 38,64,975Z = 39,88,250 x 1 = 6 ; x 2 = 16.29 ; x 4 = 59.96x 1 = 17.41 ; x 2 = 0 ; x 4 = 66.54 Constraints: 1. 1701. 174.33 2. 13002. 1343.328 3. 63. 17.414

21. D Nagesh Kumar, IISc Water Resources Planning and Management: M9L1 Thank You