Optimization using soft computing

Optimization using Soft Computing (Fuzzy Sets) (07/12/2017) STTP from 4/12/2017 to 9 /12/2017 on “Recent Trends in Optimization Techniques & Applications in Science & Engineering” Dr. Purnima Pandit Assistant Professor (purnima.pandit-appmath@msubaroda.ac.in) Department of Applied Mathematics Faculty of Technology and Engineering The Maharaja Sayajirao University of Baroda

What is soft computing ? Techniques used in soft computing?

What is Soft Computing ? (adapted from L.A. Zadeh) • Soft computing differs from conventional (hard) computing in that, unlike hard computing, it is tolerant of imprecision, uncertainty, partial truth, and approximation. In effect, the role model for soft computing is the human mind.

What is Hard Computing? • Hard computing, i.e., conventional computing, requires a precisely stated analytical model and often a lot of computation time. • Many analytical models are valid for ideal cases. • Real world problems exist in a non- ideal environment.

Many contemporary problems do not lend themselves to precise solutions such as – Recognition problems (handwriting, speech, objects, images) – Mobile robot coordination, forecasting, combinatorial problems etc.

Guiding Principles of Soft Computing • The guiding principle of soft computing is: – Exploit the tolerance for imprecision, uncertainty, partial truth, and approximation to achieve tractability, robustness and low solution cost.

Techniques of Soft Computing • The principal constituents, i.e., tools, techniques, of Soft Computing (SC) are – Fuzzy Logic (FL), Neural Networks (NN), Support Vector Machines (SVM), Evolutionary Computation (EC), and – Machine Learning (ML) and Probabilistic Reasoning (PR)

Properties of Soft computing • Learning from experimental data • Soft computing techniques derive their power of generalization from approximating or interpolating to produce outputs from previously unseen inputs by using outputs from previous learned inputs • Generalization is usually done in a high dimensional space.

Applications using Soft Computing • Handwriting recognition • Automotive systems and manufacturing • Image processing and data compression • Decision-support systems • Power systems • Intelligent systems • Adaptive control

Future of Soft Computing (adapted from L.A. Zadeh) • Soft computing is likely to play an especially important role in science and engineering, but eventually its influence may extend much farther. • Soft computing represents a significant paradigm shift in the aims of computing – a shift which reflects the fact that the human mind, unlike present day computers, possesses a remarkable ability to store and process information which is pervasively imprecise, uncertain and lacking in categorization.

Introduction • In 1965* Zadeh published his seminal work "Fuzzy Sets" which described the mathematics of Fuzzy Set Theory. • FST has numbers of applications in various fields- artificial intelligence, automata theory, computer science, control theory, decision making, finance etc. • It is being applied on a major scale in industries for machine-building (cars, engines, ships, etc.) through intelligent robots and controls. *L. A. ZADEH, Fuzzy Sets, Information Control, 1965, 8, 338-353.

This approach provides a way to translate a linguistic model of the human thinking process into a mathematical framework for developing the computer algorithms for computerized decision-making processes. crisp fuzzy very cold

In general, fuzziness describes objects or processes that are not acquiescent to precise definition or precise measurement. Thus, fuzzy processes can be defined as processes that are vaguely defined and have some uncertainty in their description. The data arising from fuzzy systems are in general, soft, with no precise boundaries.

Fuzziness in Everyday Life John is tall; Temperature is hot; The girl next door is pretty; The sun is getting relatively hot; The people living close to Vadodara; My car is slow.

Characteristic Function in the Case of Crisp Sets and Fuzzy Sets P: X  {0,1} P(x) = A : X  [0,1] A = {X, A(x)} if x  X A Fuzzy Set is a generalized set to which objects can belongs with various degrees (grades) of memberships over the interval [0,1].      Xx Xx if0 if1

Difference between (a) crisp set and (b) fuzzy set

• Crisp set: This is defined in such a way as to dichotomize the individuals in some given universe of discourse into the two groups- members and nonmembers. Full membership and full non-membership in the fuzzy set can still be indicated by the values 1 and 0, respectively. • Fuzzy set: Mathematically, if U is the universe discourse, the fuzzy set is defined as a pair given as For each , the value is called a grade of membership of x in . Here is called membership function of fuzzy set A. We can consider the concept of a crisp set to be a restricted case of the more general concept of a fuzzy set.  ,A  ]1,0[: AA Ux )(x  ,A 

Properties of Crisp sets ( ) ( ) ( Involution Commutativity and Associativity and Distrubutivity and Idem ) ( ) potence and Absor ( ) ( ) ( ) ( ) ( i ) p ( ) t A A A B B A A B B A A B C A B C A B C A B C A B C A B A C A B C A B A C A A A A A A                                         on and( ) ( )A A B A A A B A     

Absorption by X and Ø and and Identity Law of contradiction Law of excluded middle Demorgan’s l ( ) ( ) Out of above aw listed laws 'Fu an zz d A X X A A A A X A A A A A X A B A B A B A B                                  y Set' does not satisfy "Laws of contradiction" and "Laws of excluded middle".

Fuzzy Union : The union of fuzzy sets that is fuzzy union can be defined in many ways. The most commonly used method for fuzzy union is to take the maximum. For two fuzzy sets A and B with membership functions and Fuzzy Intersection: Fuzzy complement: A A(x)= 1- (x)  BA ))(),(max()()( xxx BABA   ))(),(min()()( xxx BABA   Operations on Fuzzy sets

Convex fuzzy set: A fuzzy set is fuzzy convex set  -cut is convex for all  1,0 . A fuzzy set is normal if    1, xx  . Let U is set of real number. A fuzzy number is a convex normal fuzzy set  , ~ RA  whose membership function is at least segmentally continuums. If    babaxx  ,,1, they also fuzzy number call fuzzy interval.

           If we have any set then, / 0 / 1 / If height of is 1 then is . Otherwise it is . of set = { / } A Support of A x A x Core of A x A x Height of A x max of A x A A normal subnormal cut A A x A x                  , 0,1 .  Some definitions

Some Concepts 1 0 a b X = [a,b] core(A) supp(A) height(A) = 1 (normal fuzzy set) Membership function has a trapezoidal form

Fuzzy Number       •A fuzzy set , whose membership function defined on possess at least following three properties: i must be a normal fuzzy set. ii must be a closed interval for every 0,1 i ( ) ii The support A R A A   0 Then, elements of that se of , must be bounde ts is known as Fuzzy n d. umber. A A

𝐴1 𝑥 = 0, 𝑥 < 50 𝑥 − 50 5 , 50 < 𝑥 ≤ 55 60 − 𝑥 5 , 55 < 𝑥 ≤ 60 𝛼 𝐴1 = 5𝛼 + 50, 60 − 5𝛼 𝐴2 𝑥 = 0, 𝑥 < 50 𝑥 − 50 2.5 , 50 < 𝑥 ≤ 52.5 1, 52.5 ≤ 𝑥 < 57.5 60 − 𝑥 2.5 , 60 < 𝑥 𝛼 𝐴2 = 2.5𝛼 + 50, 60 − 2.5𝛼

Fuzzy number in parametric form Parametric representation has advantage of allowing flexible and easy to control shapes of the fuzzy number. A fuzzy number in parametric form is an order pair of the form u = (𝑢 α , 𝑢(α))where 0 ≤ α ≤ 1 satisfying the following conditions:  𝑢 α is a bounded left continuous increasing function in the interval [0, 1]  𝑢 α is a bounded right continuous decreasing function in the interval [0, 1]  𝑢 α ≤ 𝑢(α), α ∈ 0,1 . If for each α, 𝑢 α = 𝑢 α then 𝑢 is a crisp number.

Fuzzy Arithmetic: Let 𝐴 and 𝐵 are two fuzzy numbers, then on taking alpha-cut we get, 𝐴 𝛼 = 𝐴, 𝐴 and 𝐵 𝛼 = 𝐵, 𝐵 Then, 𝑖 𝐴 𝛼 + 𝐵 𝛼 = [𝐴 + 𝐵, 𝐴 + 𝐵] 𝑖𝑖 𝐴 𝛼 − 𝐵 𝛼 = [𝐴 − 𝐵, 𝐴 − 𝐵] 𝑖𝑖𝑖 𝐴 𝛼. 𝐵 𝛼 = [min( 𝐴 . 𝐵, 𝐴 . 𝐵, 𝐴 . 𝐵, 𝐴 . 𝐵), max ( 𝐴. 𝐵, 𝐴 . 𝐵, 𝐴 . 𝐵, 𝐴. 𝐵)] 𝑖𝑣 𝐴 𝛼/ 𝐵 𝛼 = 𝐴 𝛼. 1/𝐵 𝛼

Fuzzy Variables: • Several fuzzy sets representing linguistic concepts such as low, medium, high, and so one are often employed to define states of a variable. Such a variable is usually called a fuzzy variable. • For example:

Fuzzy Variables: • Consider three fuzzy sets that represent the concepts of a young, middle-aged, and old person. The membership functions are defined on the interval [0,80] as follows: Find line passing through (x,y) and (20,1): 1/(35-20) = y/(35-x)

Fuzzy Variables: • For example:

Fuzzy Variables: • For example: consider the discrete approximation D2 of fuzzy set A2

Linear programming Problem (LPP) The typical linear programming problem is, 𝑀𝑎𝑥 𝑍 = 𝐶𝑋 s.t. 𝐴𝑋 ≤ 𝐵 𝑋 ≥ 0 where, 𝐶 = (𝑐1, 𝑐2, 𝑐3,…, 𝑐 𝑛), 𝐶 is called cost coefficients 𝑋 = (𝑥1, 𝑥2, 𝑥3, … , 𝑥 𝑛) 𝑇 is a vector of variables and 𝐵 = (𝑏1, 𝑏2, 𝑏3, … , 𝑏 𝑚) 𝑇 is called the resource vector. The vector 𝑋 satisfy all given constraints, is called feasible solution.

In many practical and real life situation, it is may not be possible to obtain precise value of the coefficients in the objective functions, coefficients in the LHS of the constraints or/and the resources. In such case, we model and solve the problem as Fuzzy Linear Programming Problem (FLPP). Fuzzy Linear programming Problem (FLPP)

Fuzzy LPP The crisp LPP form changes to following most general FLPP, max 𝑍 = 𝑗=1 𝑛 𝐶𝑗 𝑋𝑗 𝑗=1 𝑛 𝐴𝑖𝑗 𝑋𝑗 ≤ 𝐵𝑖 𝑋𝑗 ≥ 0, 𝑖, 𝑗 ∈ 𝑁 (1)

Fuzzy LPP (Case I) In this case, we consider 𝑨𝒊𝒋 and 𝑩𝒊 are fuzzy triangular number. 𝐴𝑖𝑗 = (𝑙𝑖𝑗, 𝑚𝑖𝑗, 𝑟𝑖𝑗) And, 𝐵𝑖 = (𝐿𝑖, 𝑀𝑖, 𝑅𝑖) The FLPP is of the form max 𝑍 = 𝑗=1 𝑛 𝐶𝑗 𝑋𝑗 𝑗=1 𝑛 𝐴𝑖𝑗 𝑋𝑗 ≤ 𝐵𝑖 𝑋𝑗 ≥ 0, 𝑖, 𝑗 ∈ 𝑁 (2)

Fuzzy LPP (Case I) The FLPP becomes max 𝑍 = 𝑗=1 𝑛 𝐶𝑗 𝑋𝑗 𝑗=1 𝑛 (𝑙𝑖𝑗, 𝑚𝑖𝑗, 𝑟𝑖𝑗) 𝑋𝑗 ≤ (𝐿𝑖, 𝑀𝑖, 𝑅𝑖) 𝑖 ∈ 𝑁 𝑚 𝑋𝑗 ≥ 0, 𝑗 ∈ 𝑁 𝑛,

Fuzzy LPP (Case I) The FLPP (2) becomes max 𝑍 = 𝑗=1 𝑛 𝐶𝑗 𝑋𝑗 𝑗=1 𝑛 𝑙𝑖𝑗 𝑋𝑗 ≤ 𝐿𝑖 𝑗=1 𝑛 𝑚𝑖𝑗 𝑋𝑗 ≤ 𝑀𝑖 𝑗=1 𝑛 𝑟𝑖𝑗 𝑋𝑗 ≤ 𝑅𝑖 𝑋𝑗 ≥ 0, 𝑗 ∈ 𝑁 𝑛, (3)

Example-1: Consider the following fuzzy LPP, max 𝑍 = 5𝑥1 + 4 𝑥2 (2,4,5) 𝑥1+(2,5,6) 𝑥2 ≤ (19, 24,32) (3,4,6) 𝑥1+(0.5, 1,2) 𝑥2 ≤ (6,12,15) 𝑥1, 𝑥2 ≥ 0

By using equation (3), we can rewrite equations, max 𝑍 = 5𝑥1 + 4𝑥2 s.t. 4𝑥1 + 5𝑥2 ≤ 24 4𝑥1 + 𝑥2 ≤ 12 2𝑥1 + 2𝑥2 ≤ 19 3𝑥1 + 0.5𝑥2 ≤ 6 5𝑥1 + 6𝑥2 ≤ 32 6𝑥1 + 2𝑥2 ≤ 15 𝑥1, 𝑥2 ≥ 0

Solving graphically: Max z is obtained at X* = (1.2,3.8) and z *= 21.2

When RHS is fuzzy (case-2) In problem (3), when only 𝐵𝑖 is fuzzy we get: 𝑚𝑎𝑥 𝑗=1 𝑛 𝐶𝑗 𝑋𝑗 𝑗=1 𝑛 𝐴𝑖𝑗 𝑋𝑗 ≤ 𝐵𝑖 𝑋𝑗≥ 0, 𝑖, 𝑗 ∈ 𝑁 (4)

In this case 𝐵𝑖 is typically have form given below, 𝐵𝑖 𝑥 = 1 𝑤ℎ𝑒𝑛 𝑥 ≤ 𝑏𝑖 𝑏𝑖 + 𝑝𝑖 − 𝑥 𝑝𝑖 𝑤ℎ𝑒𝑛 𝑏𝑖 < 𝑥 < 𝑏𝑖 + 𝑝𝑖 0 𝑤ℎ𝑒𝑛𝑏𝑖 < 𝑥 where 𝑥 ∈ 𝑅, for each vector 𝑥 = (𝑥1, 𝑥2, … , 𝑥 𝑛) We first calculate the degree 𝐷𝑖 𝑥 to which x satisfies the ith constraint by this formula 𝐷𝑖 𝑥 = 𝐵𝑖( 𝑗=1 𝑛 𝑎𝑖𝑗 𝑥𝑗). These degrees are fuzzy sets on 𝑅 𝑛 and their integration, 𝑖=1 𝑚 𝐷𝑖 is a fuzzy feasible set. 𝑏𝑖 𝑏𝑖 + 𝑝𝑖 0 1

Next, we determine fuzzy set of optimal values, this is done by calculating the lower and upper bounds of the optimal values first. The lower bound of the optimal values 𝑍𝑙, is obtained by solving the standard LPP max 𝑍 = 𝑗=1 𝑛 𝐶𝑗 𝑋𝑗 𝑗=1 𝑛 𝑎𝑖𝑗 𝑋𝑗 ≤ 𝑏𝑖, 𝑗 ∈ 𝑁 𝑚 𝑋𝑗≥ 0, 𝑖, 𝑗 ∈ 𝑁 𝑛 (5) and upper bound 𝑍 𝑢 is obtained by replacing, 𝑏𝑖 to 𝑏𝑖 + 𝑝𝑖, 𝑎𝑠 max 𝑍 = 𝑗=1 𝑛 𝐶𝑗 𝑋𝑗 𝑗=1 𝑛 𝑎𝑖𝑗 𝑋𝑗 ≤ 𝑏𝑖 + 𝑝𝑖, 𝑗 ∈ 𝑁 𝑚 𝑋𝑗 ≥ 0, 𝑖, 𝑗 ∈ 𝑁𝑛 (6)

Now the fuzzy set of optimal values, G , which is fuzzy subset of 𝑅 𝑛 is defined by, G 𝑥 = 1 𝐶𝑋−𝑍 𝑙 𝑍 𝑢−𝑍 𝑙 0 𝑤ℎ𝑒𝑛 𝑍 𝑢 ≤ 𝐶𝑋 𝑤ℎ𝑒𝑛 𝑍𝑙 ≤ 𝐶𝑋 ≤ 𝑍 𝑢 𝑤ℎ𝑒𝑛 𝐶𝑋 < 𝑍𝑙 Now the problem (4) becomes, max 𝜆 𝜆(𝑍 𝑢- 𝑍𝑙)-𝐶𝑋 ≤ − 𝑍𝑙 𝜆 𝑝𝑖 + 𝑗=1 𝑛 𝑎𝑖𝑗 𝑋𝑗 ≤ 𝑏𝑖 + 𝑝𝑖 𝜆 , 𝑋𝑗 ≥ 0, 𝑖, 𝑗 ∈ 𝑁 (7)

Example-2: Assume that company makes two products, 𝑃1 & 𝑃2. Product 𝑃1 has $0.40 profit and 𝑃2 has $0.30 profit. Each unit of product 𝑃1 requires twice as many labor hours of product 𝑃2. The total available labor hours are at least 500 hours per day and may possibly be extended to 600 hours per day, due to arrangement of overtime work. The supply of material is at least sufficient for 400 units for products 𝑃1 & 𝑃2 per day and may possibly be extended to 500 units per day according to previous experience. The problem is how many units of products 𝑃1 & 𝑃2 should be made per day to maximize profit?

The given problem can be formulated as following fuzzy LPP, max 𝑍 = 0.4𝑥1 + 0.3 𝑥2 𝑥1+𝑥2 ≤ 𝐵1 2𝑥1+ 𝑥2 ≤ 𝐵2 𝑥1, 𝑥2 ≥ 0 where, 𝐵1 and 𝐵2 are fuzzy and are is defined as,

𝐵1 𝑥 = 1 𝑤ℎ𝑒𝑛 𝑥 ≤ 400 500 − 𝑥 100 𝑤ℎ𝑒𝑛 400 < 𝑥 ≤ 500 0 𝑤ℎ𝑒𝑛 500 < 𝑥 𝐵2 𝑥 = 1 𝑤ℎ𝑒𝑛 𝑥 ≤ 400 600 − 𝑥 100 𝑤ℎ𝑒𝑛 500 < 𝑥 ≤ 600 0 𝑤ℎ𝑒𝑛 600 < 𝑥

By using equation (5) and (6), we can calculate 𝑍𝑙 and 𝑍 𝑢 for the example 2 by solving max 𝑍 = 0.4𝑥1 + 0.3 𝑥2 𝑥1+𝑥2 ≤ 400 2𝑥1+ 𝑥2 ≤ 500 𝑥1, 𝑥2 ≥ 0 (8) And max 𝑍 = 0.4𝑥1 + 0.3 𝑥2 𝑥1+𝑥2 ≤ 500 2𝑥1+ 𝑥2 ≤ 600 𝑥1, 𝑥2 ≥ 0 (9) giving us, 𝑍𝑙 = 130 and 𝑍 𝑢 = 160. Now the example (2) becomes,

Then the solution of fuzzy LPP can now be obtained by solving the following LPP: max 𝜆 30𝜆-(0.4𝑥1 + 0.3𝑥2)≤ −130 100𝜆 +𝑥1+𝑥2 ≤ 500 100 𝜆+2𝑥1+ 𝑥2 ≤ 600 𝜆 , 𝑥1, 𝑥2 ≥ 0 Using Simplex Algorithm we get: 𝑥1 = 100, 𝑥2 = 350, 𝜆=0.5 and 𝑍 = 145.

The fully fuzzy linear programming problems are given as: Maximize Z = 𝑗=1 𝑛 𝑐𝑗 𝑥 𝑗 s. t. 𝑗=1 𝑛 𝑎𝑖𝑗 𝑥𝑗 ≤ 𝑏𝑖, 𝑖 = 1, … , 𝑚 𝑥𝑗 ≥ 0, 𝑗 = 1, … , 𝑛 (10) where, 𝑐𝑗, 𝑎𝑖𝑗 and 𝑏𝑖 are fuzzy numbers and 𝑥𝑗 are fuzzy variables whose states are fuzzy numbers. Fully fuzzy LPP (case-3)

Considering the fuzzy parameters as triangular fuzzy numbers the problem (10) can be given as Maximize Z where, Z = 𝑗=1 𝑛 (𝑐𝑙, 𝑐𝑚, 𝑐𝑟) 𝑗 𝑥 𝑗 , s. t. 𝑗=1 𝑛 (𝑎𝑙, 𝑎𝑚, 𝑎𝑟)𝑖𝑗 𝑥𝑗 ≤ (𝑏𝑙, 𝑏𝑚, 𝑏𝑟)𝑖, 𝑖 = 1, … , 𝑚 𝑥𝑗 ≥ 0, 𝑗 = 1, … , 𝑛 where, (𝑐𝑙, 𝑐𝑚, 𝑐𝑟) 𝑗 is the jth fuzzy coefficient in the objective function, (𝑎𝑙, 𝑎𝑚, 𝑎𝑟)𝑖𝑗 is the fuzzy coefficient of jth variable in the ith constraint, (𝑏𝑙, 𝑏𝑚, 𝑏𝑟)𝑖 is the ith fuzzy resource.

Thakre et al. solved such problem by converting (10) into to equivalent crisp multi-objective linear problem as given below )Maximize (Z1, Z2, Z3 where, Z1 = 𝑗=1 𝑛 𝑐𝑙𝑗 𝑥 𝑗 , Z2 = 𝑗=1 𝑛 𝑐𝑚𝑗 𝑥 𝑗 , Z3 = 𝑗=1 𝑛 𝑐𝑟𝑗 𝑥 𝑗 s. t. 𝑗=1 𝑛 𝑎𝑙𝑖𝑗 𝑥𝑗 ≤ 𝑏𝑙𝑖, 𝑖 = 1, … , 𝑚 𝑗=1 𝑛 𝑎𝑚𝑖𝑗 𝑥 𝑗 ≤ 𝑏𝑚𝑖 𝑖 = 1, … , 𝑚 𝑗=1 𝑛 𝑎𝑟𝑖𝑗 𝑥 𝑗 ≤ 𝑏𝑟𝑖 𝑖 = 1, … , 𝑚 𝑥𝑗 ≥ 0, 𝑗 = 1, … , 𝑛 (11)

Maximize (𝑍1 1 , 𝑍2 1 , 𝑍3 1 , 𝑍1 2 , … , 𝑍3 𝑘 𝑍1 𝑘 = 𝑗=1 𝑛 𝑐𝑙𝑗 𝑘 𝑥𝑗 ; 𝑍2 𝑘 = 𝑗=1 𝑛 𝑐𝑚𝑗 𝑘 𝑥𝑗; 𝑍3 𝑘 = 𝑗=1 𝑛 𝑐𝑟𝑗 𝑘 𝑥𝑗 𝑘 = 1, … , 𝐾 s. t. 𝑗=1 𝑛 𝑎𝑙𝑖𝑗 𝑥𝑗 ≤ 𝑏𝑠𝑖, 𝑖 = 1, … , 𝑚 𝑗=1 𝑛 𝑎𝑚𝑖𝑗 𝑥𝑗 ≤ 𝑏𝑚𝑖, 𝑖 = 1, … , 𝑚 𝑗=1 𝑛 𝑎𝑙𝑖𝑗 𝑥𝑗 ≤ 𝑏𝑙𝑖, 𝑖 = 1, … , 𝑚 𝑥𝑗 ≥ 0, 𝑗 = 1, … , 𝑛 (12) Multi-objective Fully fuzzy LPP (case-4)

Consider the Multi objective fuzzy Linear Problem 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑍1 , 𝑍2 where, 𝑍1 = (7,10,14)𝑥1 + (20,25,35)𝑥2 𝑍2 = (10,14,25)𝑥1 + (25,35,40)𝑥2 subject to the constraints 3,2,1 𝑥1 + 6,4,1 𝑥2 ≤ (13,5,2) 4,1,2 𝑥1 + 6,5,4 𝑥2 ≤ (7,4,2) (13) Example-3:

It is equivalent to solving the MOLPP 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 7𝑥1 + 20𝑥2, 10𝑥1 + 25𝑥2, 14𝑥1 + 35𝑥2, 25𝑥1 + 40𝑥2 subject to the constraints 3𝑥1 + 6𝑥2 ≤ 13 𝑥1 + 2𝑥2 ≤ 8 4𝑥1 + 7𝑥2 ≤ 15 4𝑥1 + 6𝑥2 ≤ 7 3𝑥1 + 𝑥2 ≤ 3 6𝑥1 + 10𝑥2 ≤ 9 𝑥1, 𝑥2 ≥ 0 (12)

That is, solving 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑤1 7𝑥1 + 20𝑥2 + 𝑤2 10𝑥1 + 25𝑥2 +𝑤3 14𝑥1 + 35𝑥2 + 𝑤4(25𝑥1 + 40𝑥2) subject to the constraints 3𝑥1 + 6𝑥2 ≤ 13 𝑥1 + 2𝑥2 ≤ 8 4𝑥1 + 7𝑥2 ≤ 15 4𝑥1 + 6𝑥2 ≤ 7 3𝑥1 + 𝑥2 ≤ 3 6𝑥1 + 10𝑥2 ≤ 9 𝑥1, 𝑥2 ≥ 0 such that wi = S = 2

Sr. No 𝒘 𝟏 𝒘 𝟐 𝒘 𝟑 𝒘 𝟒 𝒙 𝟏 ∗ , 𝒙 𝟐 ∗ 1 0 1 1 0 (0, 0.9) 2 0 1 0.5 0 (0, 0.9) 3 0.2 0.4 0.5 0.2 (0, 0.9) 4 0.1 0.2 0.3 0.4 (0, 0.9) 5 0 0.3 0 0.4 (0, 0.9) 6 0.2 0.4 0.6 0.8 (0, 0.9) 7 0.5 0 0.5 0 (0, 0.9) 8 0 1 1 0 (0, 0.9) 9 0 0 0 0.5 (0, 0.9) 10 0.3 0.1 1 1 (0, 0.9) 11 0.5 0.5 0.5 0.5 (0, 0.9) 12 0 0 0.5 0.5 (0, 0.9) 13 0.2 0.5 0.5 0.5 (0, 0.9) 14 0.1 0.2 0.3 0.4 (0, 0.9) 15 0 0.2 0 0.2 (0, 0.9)

1. Klir G, Yuan B, Fuzzy Sets and Fuzzy Logic Theory and Applications, Prentice Hall, (1997). 2. Pandit P, Multi-objective Linear Programming Problems involving Fuzzy Parameters, International Journal of Soft Computing and Engineering (IJSCE) ISSN: 2231-2307, Volume-3, Issue-2, May (2013). 3. Tanaka H, Ichihashi H, Asai K, Formulation of fuzzy linear programming problem based on comparison of fuzzy numbers, Control Cybernetics 3 (3): 185-194. (1991). 4. Thakre P A, Shelar D S, Thakre S P, Solving Fuzzy Linear Programming Problem as Multi Objective Linear Programming Problem, Proceedings of the World Congress on Engineering, Vol II WCE, July 1 - 3, (2009). 5. Zadeh LA, Fuzzy sets as a basic for theory of possibility, FSSI, 3-28,(1978) . References:

Optimization using soft computing

Optimization using soft computing

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Optimization using soft computing