Lecture6-Dr. V.H.Bajaj

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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
OBJECTIVES
1.
2.
3.
4.
5.
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.
WHAT IS FUZZINESS?
•
•
•
•
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.
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
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
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.
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 nonspecificity 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
FUZZY SETS


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.
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.
NORMAL SET AND FUZZY SET
ONE REPRESENTATION FOR THE FUZZY NUMBER "ABOUT 600".
Type of sets
REPRESENTING TRUTHFULNESS (CERTAINTY) OF EVENTS AS
FUZZY SETS OVER THE [0,1] DOMAIN.
MEMBERSHIP FUNCTIONS REPRESENTING THREE
FUZZY SETS FOR THE VARIABLE "HEIGHT".
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.
FUZZY SET OPERATORS

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))
OPERATIONS WITH TWO FUZZY SETS A AND B
APPROXIMATELY REPRESENTED IN A GRAPHICAL FORM
FUZZY RULES



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



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 EXPERT SYSTEMS



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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 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
FUZZINESS VERSUS PROBABILITY
PROBABILITY DENSITY FUNCTION FOR THROWING A DICE AND
THE MEMBERSHIP FUNCTIONS OF THE CONCEPTS "SMALL"
NUMBER, "MEDIUM", "BIG".
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)=
 Pi
1 - S(u)
- function:
 P(u)=S(u), u<=b; P(u)=Z(u), u>b.
MEMBERSHIP FUNCTIONS
STANDARD TYPES OF MEMBERSHIP FUNCTIONS: Z
FUNCTION;
¶ FUNCTION; S FUNCTION; TRAPEZOIDAL FUNCTION;
TRIANGULAR FUNCTION; SINGLETON.
MF PLOTS
LINGUISTIC VARIABLES
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

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 Imprecision
Unusual 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 are near the junction, start braking slowly.
LINGUISTIC VARIABLES

Fuzzy Logic Reflects Real-Life Scenario
Membership Assignment
Membership
1
0
16
24
Age of Operators at a Factory
40
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.
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.
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.
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.
FUZZY MADM METHOD FOR SELECTION OF SHOP
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, lowermiddle, 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 APPLIED FOR SELECTION OF
TELEVISION
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 APPROACH FOR SELECTION OF
DISTRIBUTION CENTER LOCATION IN LOGISTICS
SYSTEMS
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)
A FUZZY APPROACH TO SOLVE MULTIOBJECTIVE TRANSPORTATION PROBLEM
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.
FUZZY MULTI-OBJECTIVE MODEL FOR SUPPLIER
SELECTION WITH LONG-TERM RELATIONSHIP IN A
SUPPLY CHAIN
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.
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. Fuzzy Approach 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.
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. Fuzzy multi-objective multi-index transportation problem; Advances in
Information Mining, Vol.2, No.1, 2010,pp.01-07.
8. A Fuzzy Approach to Solve Multi-objective Transportation Problem,
International Journal of Agri. and Stat. Science, Vol.5, No.2, 2009, pp.443452.
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.
10. Additive fuzzy multiple 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 itemsdefuzzification 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.
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, International Journal 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, International Journal 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.
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.
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.
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.
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,pp 317-326.
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).
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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