Application of Fuzzy Set Theory in Optimization Techniques Dr. V. H. Bajaj Professor and Head,

1. Application of Fuzzy Set Theory in Optimization Techniques Dr. V. H. Bajaj Professor and Head, Department of Statistics, Dr. B. A. M. University, Aurangabad, India. E-mail : vhbajaj@gmail.com

2. OBJECTIVES • To introduce fuzzy sets and how they are used. • To define some types of uncertainty and study. what methods are used to with each of the types. • To define fuzzy numbers, fuzzy Rules, fuzzy logic, fuzzy membership function and how they are used • To study methods of how fuzzy sets can be constructed. • To see how fuzzy set theory is used and applied in Inventory Models, Assignment Problems, Transportation Problem, Replacement Theory and Decision Making Problem, supply chain management.

3. According to the Oxford English Dictionary….. The word fuzzy means blurred, fluffy, frayed or indistinct. • Fuzziness is deterministic uncertainty. • Fuzziness is connected with the degree to which events occur rather than the likelihood of their occurrence (probability). • For example, the degree to which a person is young/ smart is a fuzzy event rather than a random event. What is Fuzziness?

4. A. Why fuzzy sets? - Modeling with uncertainty requires more than probability theory - There are problems where boundaries are gradual Examples: What is the boundary of INDIA? Is the boundary a mathematical curve? What is the area of INDIA? Is the area a real number? Where does a tumor begin in the transition? What is the habitat of rabbits in 20km radius from here? What is the depth of the ocean 30 km east of Juhu Beach? 1. Data reduction – driving a car, computing with language 2. Control and fuzzy logic a. Appliances, automatic gear shifting in a car b. Air conditioner control panel c. Washing machine

5. Temperature control in NASA space shuttles IF x AND y THEN z is A IF x IS Y THEN z is A … etc. If the temperature is hot and increasing very fast then air conditioner fan is set to very fast and air conditioner temperature is coldest. 3. Pattern recognition, Cluster analysis - A bank that issues credit cards wants to discover whether or not it is stolen or being illegally used prior to a customer reporting it missing - Given a cat-scan determine the organs and their position; given a satellite imagine, classify the land/cover use - Given a mobile telephone, send the signal to/from a particular receiver to/from the telephone

6. 4. Decision making - Locate mobile telephone receptors/transmitters to optimally cover a given area - Position a satellite to cover the most number of mobile phone users - Deliver sufficient radiation to a tumor to kill the cancerous cells while at that same time sparing healthy cells (maximize dosage up to a limit at the tumor while minimizing dosage at all other cells) - Design a product in the following way: I want the product to be very light, very strong, last a very long time and the cost of production is the cheapest.

8. AMBIGUITY: A one to many relationship; for example, he is tall, he is handsome. There are a variety of alternatives 1. Non-specificity: Suppose one has a heart blockage and is prescribed a treatment. In this case “treatment” is a non-specificity in that it can be an angioplasty, medication, surgery (to name three alternatives) 2. Dissonance/contradiction: One physician says to operate and another says start yoga and pranayam. VAGUENESS – lack of sharp distinction or boundaries, our ability to discriminate between different states of an event, undecidability (is a glass half full/empty) For example: Distinction between Baby, Child, Teenager, Adult (Age wise) Height-Short, Medium, Tall

9. A crisp set is a set for which each value either is or is not contained in the set. • Examples: Set of integers – a real number is an integer or not You are either in an airplane or not Your bank account is x dollars and y cents Fuzzy set theory was introduced by Dr Lotfi A Zadeh (1965) For a fuzzy set, every value has a membership value, and so is a member to some extent. • The membership value defines the extent to which a variable is a member of a fuzzy set. • The membership value is from 0 (not at all a member of the set) to 1. Fuzzy Sets

10. Basic Definitions and Terminology If X is space of objects and x be a generic element of X. Then Fuzzy set A in X is defined as a set of of ordered pairs: A={(x, µA (x)) |x ε X}, Where µA(x) is called the Membership Function (MF) the fuzzy set A. Two parameters must be defined for the quantization procedure: the number of the fuzzy labels; the form of the membership functions for each of the fuzzy labels.

11. Normal Set and Fuzzy Set

12. One representation for the fuzzy number "about 600".

13. Type of sets

14. Representing truthfulness (certainty) of events as fuzzy sets over the [0,1] domain.

15. Membership functions representing three fuzzy sets for the variable "height".

16. Crisp Set Operators • Not A – the complement of A, which contains the elements which are not contained in A. • A  B – the intersection of A and B, which contains those elements which are contained in both A and B. • A  B – the union of A and B which contains all the elements of A and all the elements of B. • Fuzzy sets use the same operators, but the operators have different meanings.

17. Fuzzy set operators can be defined by their membership functions • M¬A(x) = 1 - MA(x) • MA  B (x) = MIN (MA (x), MB (x)) • MA  B (x) = MAX (MA (x), MB (x)) • We can also define containment (subset operator): • B  A iff x (MB (x)  MA (x)) Fuzzy Set Operators

18. Operations with two fuzzy sets A and B approximately represented in a graphical form

19. A fuzzy rule takes the following form: IF A op x then B = y • op is an operator such as >, < or =. • For example: IF temperature > 50 then fan speed = fast IF height = tall then trouser length = long IF study time = short then grades = poor Fuzzy Rules

20. Fuzzy rules may be expressed in terms such as ``If the room gets hotter, spin the fan blades faster'' The temperature of the room and speed of the fan's blades are both imprecisely (fuzzily) defined quantities, and ``hotter'' and ``faster'' are both fuzzy terms. Fuzzy logic, with fuzzy rules, has the potential to add human-like subjective reasoning capabilities to machine intelligences - usually based on bivalent boolean logic. Fuzzy Rules

21. A fuzzy expert system is built by creating a set of fuzzy rules, and applying fuzzy inference. • In many ways this is more appropriate than standard expert systems since expert knowledge is not usually black and white but has elements of grey. • The first stage in building a fuzzy expert system is choosing suitable linguistic variables. • Rules are then generated based on the expert’s knowledge, using the linguistic variables. Fuzzy Expert Systems

22. Fuzzy Vs Probability • PA(x):S→[0,1] determines the probability that an element x belongs to the set A. μ A(x) ≠ pA(x) • though both map x to a value in [0,1]. PA(x) measures our knowledge or ignorance of the truth of the event that x belongs to the set A. However the belongingness of x to A is not of degree but crisp. • μA(x) measures the degree of belongingness of x to set A and there is no interest regarding the uncertainty behind the outcome of the event x. Event x has occurred and we are interested in only making observations regarding the degree to which x belongs to A

23. Fuzziness versus probability Probability density function for throwing a dice and the membership functions of the concepts "Small" number, "Medium", "Big".

24. Standard membership functions • single-valued, or singleton • triangular • trapezoidal • S-function (sigmoid function): • S(u) = 0, u<=a • S(u) = 2((u-a)/(c-a))2 , a <u <= b • S(u) = 1 - 2((u-a)/(c -a))2 , b <u <= c • S(u) = 1, u > c. • Z function: • Z(u)= 1 - S(u) • Pi - function: • P(u)=S(u), u<=b; P(u)=Z(u), u>b.

25. Membership Functions

26. Standard types of membership functions: Z function;¶ function; S function; trapezoidal function; triangular function; singleton.

27. MF Plots

28. Variables used in fuzzy systems to express qualities such as height, which can take values such as “tall”, “short” or “very tall”. • Example of Linguistic ImprecisionUnusual and Real-Life Quotes • How was the weather like yesterday? • Oh! It was rainy with 98% humidity and hot with temperature of 35.5 deg C • Oh! It was very humid and really hot. • Fuzzy logic can handle such linguistic imprecision where other techniques have difficulty in handling • When you are at 10 metres from the junction start braking at 50% pedal level. • When you arenear the junction, start braking slowly. Linguistic Variables

29. Fuzzy Logic Reflects Real-Life ScenarioMembership Assignment Linguistic Variables 1 Membership 0 40 16 24 Age of Operators at a Factory

30. Application of Fuzzy Set Theory in Inventory Models In crisp inventory models, all the parameters in the total inventory cost are known and have definite values without ambiguity. But in reality, it is not so certain. Hence there is a need to consider the fuzzy inventory models. So far very little research has been done for the solution of Multi-objective Fuzzy non-linear problems in production inventory. An EOQ model for deteriorating items using two warehouses is developed in fuzzy sense. A rented warehouse is used to store the excess units over the fixed capacity of the own warehouse. The capacity of rented warehouse is unlimited. Deterioration rates of two warehouses are considered to be different due to change in environment. The parameters such as holding cost and deteriorating cost for two warehouses are considered as fuzzy number. Triangular and trapezoidal both types of fuzzy numbers are considered to represent the fuzzy parameters. Total inventory costs as well as optimum order quantity is obtained in fuzzy sense. Signed distance and Function principle methods are used for defuzzification.

31. Fuzzy approach for solving multi-objective assignment problem In Assignment problem, time and cost are taken as fixed numbers. But in realistic situation time/cost is imprecise, vague and flexible in nature i.e. the elements of the effective matrix should be imprecise number instead of fixed real numbers as because time/cost for doing a job by a facility (machine/ person) might vary due to different reasons and their values are varied within some ranges (intervals). We use a special type of linear and non-linear membership functions to solve the multi-objective assignment problem. It gives an optimal compromise solution. The result obtained by using a linear membership function has been compared with the solution obtained by using non-linear membership functions. If we use the hyperbolic membership function, then the crisp model becomes linear. The optimal compromise solution does not change significantly if we compare with the solution obtained by the linear membership function.

32. Use of Fuzzy Multiple Criteria Decision Making Method In Replacement problem The question of vagueness in replacement problem is usually ignored. The concept of fuzzy multiple criteria decision making (FMCDM) theory may provide a suitable tool to tackle this problem.Fuzzy multiple criteria decision making evaluation methodology is used to solve possible replacement, reducing failure and higher utilization of component life. ere, triangular membership function is used to identify the most efficient maintenance approach. Simple additive weighting method and multiplicative exponential weighting method are used to verify the efficiency of present method. a numerical example is taken to illustrate and verify effectiveness of the proposed methodology.

33. Numerical Example In a Sugar Mill, we can identify the failure causes (criteria) of sugar cane crushing bearings. The Weight =“WEIGHT” estimated using the following possible ratings : {Very Low (VL), Low (L), More or Less Low (MLL), Medium (M), More or Less High (MLH), High (H), Very High (VH)}. we propose a framework based on the fuzzy MCDM approach for predicting most efficient maintenance policy/strategy for a replacement problem. The fuzzy linguistic variable is used to carry out the ranking of the strategies with respect to the decided criteria. Two different methods have been established to select most efficient/optimal maintenance strategy under fuzzy decision criteria for given machine. These methods are used for minimizing the number of failures and planned replacement. By using the proposed fuzzy evaluation methodologies we are able to identify and select, in advance, the optimal maintenance approach for the replacement purpose. Consequently we get higher product quality, improve efficiency and higher productivity and hence better economy and profitability.

34. In real life, decision-making is the process of finding best option from all the feasible alternatives. The use of Multiple Attribute Decision Making (MADM) approach is studied. We propose MADM method in solving a shop selection problem with four alternatives: Monthly rent, Size of the shop, Distance from house and Locality (Area) . For linguistic attribute locality (rich, upper-middle, lower-middle, middle, poor) the decision maker makes use of scales/ weights. The selection of alternative and the weight of each criterion are described. A Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is proposed and tried to obtain ideal solution. Fuzzy MADM Method For Selection of Shop

35. A new approach to product configuration by applying the theory of fuzzy multiple attribute decision making (FMADM), which focus on uncertain and fuzzy requirements the customer submits to the product supplier. The proposed method can be used in e-commerce websites, with which it is easy for customers to get his preferred product according to the utility value with respect to all attributes. The main concern of this paper is customer’s requirements about configuration of television in his mind is vague. Further verify the validity and the feasibility of the proposed method compared with Weighted Product Method (WPM). Finally, Selection of television is taken as an example with four attributes as no. of Speakers, Watt output, Price, Warranty period. Fuzzy MADM applied For selection of Television

36. Multiple Attribute Decision Making (MADM) problem under fuzzy environment, in which the preference values take the form of triangular fuzzy numbers is considered. The important weights of evaluation criteria are obtained by utilizing entropy analysis. Therefore, an approach to deal with attribute weights which are completely unknown are developed by using entropy weights of fuzzy variables. Furthermore, in order to make a decision or choose the optimum alternative, an expected value method is presented under the assumption that attribute weights are known fully. We consider a numerical example to illustrate the proposed method for the MADM problem of selecting distribution center location in logistics systems. Logistic company considers four attributes (Service level, Social Benefit, Natural environment, Public Infrastructure and Cost) Fuzzy MADM Approach for Selection of Distribution Center Location in Logistics Systems

37. In Transportation problem, time, cost, demand (destination) and supply (origin) are taken as fixed numbers. But in realistic situation time/cost/ demand/ supply is imprecise, vague and flexible in nature i.e. the elements of the effective matrix should be imprecise number instead of fixed real numbers as because transportation time/cost for transportation of goods might vary due to different reasons and their values are varied within some ranges (intervals). Multi-objective transportation problem refers to a special class of vector minimum linear programming problem in which the constraints are equality type and all the objective are conflict with each other. All the methods so far developed either generate a set of non-dominated solutions or find a compromise solution. We use a special type of linear and non-linear membership functions to solve the multi-objective transportation problem. It gives optimal compromise solution. A FUZZY APPROACH TO SOLVE MULTI-OBJECTIVE TRANSPORTATION PROBLEM

38. This paper presents a fuzzy decision –making approach to deal with Multi-objective supplier selection in a supply chain. During recent years, how to determine suitable suppliers in the supply chain has become difficult. The ultimate success of a firm will depend on its managerial ability to integrate and coordinate the intricate network of business relationships among supply chain partners. There are several objectives, such as price, quality, on time delivery and long term relationship. Also there are several constraints, such as demand from buyers, capacity of suppliers and total budget of suppliers. Now a day’s suppliers play a key role in achieving corporate competition. In practice, vagueness and imprecision of goals, constraints and parameters in this problem make it difficult. So, we use fuzzy multiple criteria decision making (MCDM). In this paper we study fuzzy programming for obtaining solution of this Multi-objective supplier selection problem. Fuzzy Multi-objective Model for Supplier Selection With Long-Term Relationship in a Supply Chain

39. LIST PUBLICATIONS ON APPLICATION OF FUZZY CONCEPT IN OPTIMIZATION TECHNIQUES DURING 2007-2012 (By Prof. Bajaj & Research Scholars) 1. Multi-item fuzzy EOQ model for deteriorating items; International Jr. of Agri. and Statistical Sciences, 2007, Vol. 3, No. 2, pp. 597-608. 2. FuzzyApproach with linear and non-linear membership functions for solving multi-objective assignment problems; Advances in Computational Research, 2009, Vol.1, No. 2, pp.14-17. 3. An order level inventory model for item with finite rate of replenishment dependent on inventory level and time dependent Deterioration; International Jr. of Statistical Sciences, 2007, No. 2, pp. 533-540. 4. Reliability analysis for components using Fuzzy membership function; Advances in Computational Research, 2009, Vol.1, No. 2, pp. 30-33.

40. 5.An application of Fuzzy Multiple Attribute Decision Making method, International Journal of Agri. and Stat. Science, Vol.5, No.2, 2009, pp. 331-334. 6. Multi-objective fuzzy Inventory model of deteriorating with fuzzy Lead Time; Journal of Statistical Sciences, Vol. 1,No. 2, 2009,pp.149-161. 7.Fuzzymulti-objective multi-index transportation problem; Advances in Information Mining, Vol.2, No.1, 2010,pp.01-07. 8. A Fuzzy Approach to SolveMulti-objective Transportation Problem, International Journal of Agri. and Stat. Science, Vol.5, No.2, 2009, pp.443-452. 9. Fuzzy Multiple Attribute Decision Making by Utilizing Entropy-Based Approach, International Journal of Agri. and Stat. Science, Vol.5, No.2, 2009, pp.613-621.

41. 10. Additive fuzzymultiple goal programming model for unbalanced multi-objective transportation problem; International Journal of Machine Intelligence, Vol.2, No.1, 2010, pp.29-34. 11. Fuzzy inventory models for deteriorating items- defuzzification by centroid and signed distance method; Int. Jr. of Agri. & Statistical Sciences, 2007, Vol. 3, No. 2, pp. 351-359. 12.Fuzzy Inventory model with shortages; Sankhya Vignan, Vol. 4, No.1, June -2008, pp.01-08. 13.A New Approach to Solve Fuzzy Multi-objective Unbalanced Assignment Problem, Int. Jr. of Agri. and Stat. Science, Vol.6, No.1, 2010, pp.31-40. 14. A Comparative approach used for solving Fuzzy MADM Problem;Int. Jr. of Math. Sci. & Engg. Applications, Vol.4 No. 2, June 2010, pp. 21-40.

42. 15. Equipment Replacement Analysis using Fuzzy Linguistic Theory, Int. Jr. of Agri. and Stat. Science, Vol.6, No.1, 2010, pp.299-306. 16.A Fuzzy Method for Solving Unbalanced Assignment Problems with Interval Valued Coefficients, Int. Jr. of Commerce & Business Mgmt, Vol.3, No.1, 2010, pp.82-87. 17.Fuzzy Present Worth Approximation of cash flow in Replacement analysis, InternationalJournal of Commerce and Business Management, Vol. 3, No. 1, 2010, pp.155-160. 18.Use of Fuzzy Multiple Criteria Decision making method In Replacement problem, InternationalJournal of Statistics and System,Vol. 5, No. 2, 2010, pp.155-163. 19. Multi-item, multi-objective fuzzy model of deteriorating items under two constraints with hyperbolic and linear membership; Journal of Statistical Sciences,Vol. 1,No. 2, 2009,pp.163-171.

43. 20. Fuzzy Multi-objective Transportation Problem with Interval Cost, International Journal of Agri. and Stat.Science, Vol.6, No.1, 2010, pp.187-196. 21.MADM Method Used in Investment Company: A Fuzzy Approach ;International Journal of Math. Sci. & Engg. Applications, Vol.4 No.3, August 2010, pp. 21-32. 22. A comparative FMADM method used to solve real life problem; International Journal of Machine Intelligence, Vol.2, No.1, 2010, pp.35-39. 23. A new fuzzy MADM approach used for finite selection; Advances in Information Mining, Vol.2, No.1, 2010,pp.08-12. 24. Application of fuzzy multiple attribute decision making method solving by interval numbers;Advances in Computational Research, Vol.2, No.1, 2010,pp.01-05.

44. 21.MADM Method Used in Investment Company: A Fuzzy Approach ;International Journal of Math. Sci. & Engg. Applications, Vol.4 No.3, August 2010, pp. 21-32. 22. A comparative FMADM method used to solve real life problem; International Journal of Machine Intelligence, Vol.2, No.1, 2010, pp.35-39. 23. A new fuzzy MADM approach used for finite selection; Advances in Information Mining, Vol.2, No.1, 2010,pp.08-12. 24. Application of fuzzy multiple attribute decision making method solving by interval numbers;Advances in Computational Research, Vol.2, No.1, 2010,pp.01-05. 25. Fuzzy method for Solving Multi-objective Assignment Problem with interval cost; Journal of Statistics and Mathematics, Vol. 1, No. 1, 2010, pp.01-09.

45. 26. Solving Multi-objective Assignment Problem by using Additive Fuzzy Programming Techniques, International Journal of Mathematics Research, Vol. 2, No. 1, pp. 175- 184. 27.Fuzzy programming technique to solve multi-objective Solid transportation problem with some non-linear membership functions ; Advances in Computational Research, Volume 2, Issue 1, 2010, pp-15-20. 28 . Fuzzy programming technique to solve bi-objective transportation problem; International Journal of Machine Intelligence, Vol. 2, Issue 1, 2010, pp - 46-52. 29. Fuzzy approach to solve multi-objective capacitated Transportation problem; Int. Jr. of Bioinformatics Research, Vol. 2, Issue 1, 2010, pp-10-14.

46. 30. Fuzzy Programming for Multi-objective Transportation and Inventory Management Problem with Retailer Storage. Int. Jr. of Agri. & Statistical Sciences,Vol.7,No.1, 2011,pp317-326. 31.Application of Fuzzy TOPSIS Method for Solving Job-Shop Scheduling Problem, Int. Jr. of Operation Research &Optimization , Vol. 2, No.2, July Dec-2011, pp. 333-342. 32. Single Machine Scheduling Problem Using Fuzzy Processing Time and Fuzzy Due Dates. Int. Jr. of Computer Engg. Sciences, Vol. 2, No.5 Issue, May-2012, pp.12-18. 33. Selection of Job Shop Scheduling Problem Using Fuzzy Linguistic variables; Int. Jr. of Statistika & Mathematika, Vol.3, No-3, 2012, pp-1-5, (ISSN NO. 2277- 2790 E-ISSN: 2249-8605).

47. 34. Flow-Shop Scheduling Problem Using Fuzzy Approach; Int. Jr. of Agri. & Statistical Sciences, Vol.8, No.2, Dec-2012. 35.An MILP Model for Oil Vessel at Refinery, Int. Jr. of Statistika & Mathematika, Vol.3, No-3, 2012, pp -12-15, (ISSN NO.2277- 2790 E-ISSN: 2249-8605).

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