Traditional Approach . . . Traditional Approach . . . Fuzzy Target . . . Semi-Heuristic Target-Based Fuzzy Target . . . Fuzzy Target . . . Fuzzy Decision Procedures: Analyzing the Problem Our Main Idea Towards a New Interval How To Estimate . . . Justification Home Page Title Page Christian Servin 1 , Van-Nam Huynh 2 , ◭◭ ◮◮ and Yoshiteru Nakamori 2 ◭ ◮ 1 Computational Science Program, University of Texas at El Paso El Paso, TX 79968, christians@miners.utep.edu Page 1 of 12 2 Japan Advanced Institute of Science and Technology (JAIST) Go Back 1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan { huynh, nakamori } @jaist.ac.jp Full Screen Close Quit
Traditional Approach . . . 1. Traditional Approach to Decision Making: Re- Traditional Approach . . . minder Fuzzy Target . . . Fuzzy Target . . . • The quality of each possible alternative is characterized Fuzzy Target . . . by the values of several quantities. Analyzing the Problem • For example, when we buy a car, we are interested in Our Main Idea its cost, its energy efficiency, its power, size, etc. How To Estimate . . . Home Page • For each of these quantities, we usually have some de- sirable range of values. Title Page • Often, there are several different alternatives all of which ◭◭ ◮◮ satisfy all these requirements. ◭ ◮ • The traditional approach assumes that there is an ob- Page 2 of 12 jective function that describes the user’s preferences. Go Back • We then select an alternative with the largest possible Full Screen value of this objective function. Close Quit
Traditional Approach . . . 2. Traditional Approach to Decision Making: Lim- Traditional Approach . . . itations Fuzzy Target . . . Fuzzy Target . . . • The traditional approach to decision making assumes: Fuzzy Target . . . – that the user knows exactly what he or she wants Analyzing the Problem — i.e., knows the objective function – and Our Main Idea – that the user also knows exactly what he or she will How To Estimate . . . get as a result of each possible decision. Home Page Title Page • In practice, the user is often uncertain: ◭◭ ◮◮ – the user is often uncertain about his or her own preferences, and ◭ ◮ – the user is often uncertain about possible conse- Page 3 of 12 quences of different decisions. Go Back • It is therefore desirable to take this uncertainty into Full Screen account when we describe decision making. Close Quit
Traditional Approach . . . 3. Fuzzy Target Approach (Huynh-Nakamori) Traditional Approach . . . Fuzzy Target . . . • For each numerical characteristic of a possible decision, Fuzzy Target . . . we form two fuzzy sets: Fuzzy Target . . . – µ i ( x ) describing the users’ ideal value; Analyzing the Problem – µ a ( x ) describing the users’ impression of the actual Our Main Idea value. How To Estimate . . . Home Page • For example, a person wants a well done steak, and the steak comes out as medium well done. Title Page • In this case, µ i ( x ) corresponds to “well done”, and ◭◭ ◮◮ µ a ( x ) to “medium well done”. ◭ ◮ • The simplest “and”-operation (t-norm) is min( a, b ); so, Page 4 of 12 the degree to which x is both actual and desired is Go Back min( µ a ( x ) , µ i ( x )) . Full Screen • The degree to which there exists x which is both pos- Close sible and desired is d = max min( µ a ( x ) , µ i ( x )) . x Quit
Traditional Approach . . . 4. Detailed Derivation of the d -Formula Traditional Approach . . . Fuzzy Target . . . • We know: Fuzzy Target . . . – a fuzzy set µ i ( x ) describing the users’ ideal value; Fuzzy Target . . . – the fuzzy set µ a ( x ) describing the users’ impression Analyzing the Problem of the actual value. Our Main Idea • For crisp sets, the solution is possibly satisfactory if How To Estimate . . . Home Page some of the possibly actual values is also desired. Title Page • In the fuzzy case, we can only talk about the degree to which the proposed solution can be desired. ◭◭ ◮◮ • A possible decision is satisfactory if either: ◭ ◮ – the actual value is x 1 , and this value is desired, Page 5 of 12 – or the actual value is x 2 , and this value is desired, Go Back – . . . Full Screen • Here x 1 , x 2 , . . . , go over all possible values of the de- Close sired quantity. Quit
Traditional Approach . . . 5. Derivation of the d -Formula (cont-d) Traditional Approach . . . Fuzzy Target . . . • For each value x k , we know: Fuzzy Target . . . – the degree µ a ( x k ) with which this value is actual, Fuzzy Target . . . and Analyzing the Problem – the degree µ i ( x k ) to which this value is desired. Our Main Idea • Let us use min( a, b ) to describe “and” – the simplest How To Estimate . . . Home Page possible choice of an “and”-operation. Title Page • Then we can estimate the degree to which the value x k is both actual and desired as ◭◭ ◮◮ min( µ a ( x k ) , µ i ( x k )) . ◭ ◮ • Let us use max( a, b ) to describe “or” – the simplest Page 6 of 12 possible choice of an “or”-operation. Go Back • Then, we can estimate the degree d to which the two Full Screen fuzzy sets match as Close d = max min( µ a ( x ) , µ i ( x )) . x Quit
Traditional Approach . . . 6. Fuzzy Target Approach: How Are Membership Traditional Approach . . . Functions Elicited? Fuzzy Target . . . Fuzzy Target . . . • In many applications (e.g., in fuzzy control), the shape Fuzzy Target . . . of a membership function does not affect the result. Analyzing the Problem • Thus, it is reasonable to use the simplest possible mem- Our Main Idea bership functions – symmetric triangular ones. How To Estimate . . . Home Page • To describe a symmetric triangular function, it is suf- ficient to know the support [ x, x ] of this function. Title Page • So, e.g., to get the membership function µ i ( x ) describ- ◭◭ ◮◮ ing the desired situation: ◭ ◮ – we ask the user for all the values a 1 , . . . , a n which, Page 7 of 12 in their opinion, satisfy the requirement; Go Back – we then take the smallest of these values as a and the largest of these values as a ; Full Screen – finally, we form symmetric triangular µ i ( x ) whose Close support is [ a, a ]. Quit
Traditional Approach . . . 7. Fuzzy Target Approach: Successes and Remain- Traditional Approach . . . ing Problems Fuzzy Target . . . Fuzzy Target . . . • The above approach works well in many applications. Fuzzy Target . . . • Example: it predicts how customers select a hand- Analyzing the Problem crafted souvenir when their ideal ones is not available. Our Main Idea How To Estimate . . . • Problem: this approach is heuristic, it is based on se- Home Page lecting: Title Page – the simplest possible membership function and ◭◭ ◮◮ – the simplest possible “and”- and “or”-operations. ◭ ◮ • Interestingly, we get better predictions than with more complex membership functions and “and”-operations. Page 8 of 12 • In this paper, we provide a justification for the above Go Back semi-heuristic target-based fuzzy decision procedure. Full Screen Close Quit
Traditional Approach . . . 8. Analyzing the Problem Traditional Approach . . . Fuzzy Target . . . • Reminder: all we elicit from the experts is two inter- Fuzzy Target . . . vals: Fuzzy Target . . . – an interval [ a, a ] = [ � a − ∆ a , � a + ∆ a ] describing the Analyzing the Problem set of all desired values, and Our Main Idea – an interval [ b, b ] = [ � b − ∆ b , � b + ∆ b ] describing the How To Estimate . . . set of all the values which are possible . Home Page Title Page • Based on these intervals, we build triangular member- a and � ship functions µ i ( x ) and µ a ( x ) centered in � b . ◭◭ ◮◮ • For these membership functions, ◭ ◮ min( µ a ( x ) , µ i ( x )) = 1 − | � b − � a | Page 9 of 12 d = max . ∆ a + ∆ b x Go Back • This is the formula that we need to justify. Full Screen Close Quit
Traditional Approach . . . 9. Our Main Idea Traditional Approach . . . Fuzzy Target . . . • If we knew the exact values of a and b , then we would Fuzzy Target . . . conclude a = b , a < b , or b < a . Fuzzy Target . . . • In reality, we know the values a and b with uncertainty. Analyzing the Problem Our Main Idea • Even if the actual values a and b are the same, we may How To Estimate . . . get approximate values which are different. Home Page • It is reasonable to assume that if the actual values are Title Page the same, then Prob( a > b ) = Prob( b > a ), i.e., ◭◭ ◮◮ Prob( a > b ) = 1 / 2 . ◭ ◮ • If the probabilities that a > b and that a < b differ, Page 10 of 12 this is an indication that the actual value differ. Go Back • Thus, it’s reasonable to use | Prob( a > b ) − Prob( b > a ) | Full Screen as the degree to which a and b may be different. Close Quit
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