MS

Fuzzy Sets and Systems 51 (1992) 1-27 North-Holland

Invited Review

Fuzzy expert systems for IE/OR/MS I.B. Turksen* LIFE Chair of Fuzzy Theory, Department of Systems Science, Tokyo Institute of Technology, Yokohama, Japan Received April 1992 Revised April 1992

Abstract: Fuzzy expert systems can be developed for the effective use of management within the domains of concern associated with Industrial Engineering, IE, Operations Research, OR, and Management Science, MS. These models are designed with: (1) expressive powers of representation embedded in linguistic variables and their linguistic values in natural language expressions, and (2) improved methods of inference based on fuzzy logic which is a generalization of

multi-valued logic. The results of these fuzzy expert system models are either (1) approximately good in comparison with their classical counterparts, i.e., within a few percentage points of the optimal results of classical models, as for example, for the cases of production, and spare parts planning, or (2) much better than their counterparts, as for example, for the cases of job shop scheduling and consumer preference prediction in market share analysis. In order to establish a sound basis for the preceding claims, basic issues of representation and inference are first reviewed and then several prototype case studies are presented to support these claims. Keywords: Expert systems; industrial engineering; management science; measurement; representation; reasoning; prototypes.

Editorial note The IFSA Journal on Fuzzy Sets and Systems has been publishing review articles in the general area of fuzzy sets and systems with applications to the fields such as management science, systems and control, decision analysis, medical systems, and reliability analysis etc. These review articles are solicited from researchers and experts in the field. If you wish to write a review article, please contact Dr. Madan M. Gupta, the Review Editor, Intelligent Systems Research Laboratory, College of Engineering, University of Saskatchewan, Saskatoon, S7N 0W0, Canada, Telephone: (306) 966-5451, and Fax: (306) 966-8710. In this issue, we present an invited review on Fuzzy Expert Systems by Professor I. B. Turksen. I hope that general readers as well as experts in the field will enjoy reading this well-written and well-illustrated review article. Madan M. Gupta Editor, Special Invited Reviews

Correspondence to: Prof. I.B. Turksen, 4259 Nagatsuta, Midori-ku, Yokohama 227, Japan, * On leave from the Department of Industrial Engineering, University of Toronto, Toronto, Ontario, M5S 1A4, Canada. Research supported in part by the Natural Science and Engineering Research Council of Canada and in part by the Manufacturing Research Corporation of Ontario.

1. Introduction

Generally in Engineering and Sciences, and particularly in Industrial Engineering, IE, Operations Research OR, and Management Science, MS, our tradition of quest and respect

0165-0114/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved

LB. Turksen / Fuzzy expert systems for IE/OR/MS

for what is quantitative and precise, and disdain for what is qualitative and vague or imprecise, have let us concentrate on models based on linear, non-linear, and/or differential equations that can be essentially analyzed with addition and multiplication operations of algebra and mathematics. However, it is well known from the foundations of the measurement theory that these operations require data and information to be on ratio and absolute scales, respectively. We are aware that these precise and quantitative models of I E / O R / M S provide valuable insights into a system's behavior under specific assumptions and identify useful guidelines. However, they are often not implemented. There are several reasons for this: (1) either managers do not understand the basic assumptions of such models; (2) such models are too complex and are expressed in a mathematical language that does not translate easily into the natural language of management; or (3) the precise information required to determine the values of the parameters of such models cannot be obtained easily or is not available. Hence, such models are, by default, inadequate on their own to help managers cope with the natural behavior of many real-life systems. This, in turn, entails our need to resort to qualitative judgements of managers in the final analysis of many of these systems. With the emergence of fuzzy sets, initially, some valuable research efforts were directed into fuzzifying classical models of I E / O R / M S in the development of such models of fuzzy linear, non-linear programming [6, 43, 44], fuzzy dynamic programming [5], fuzzy games [ll, etc. However, these models also require, generally speaking, addition and multiplication operations. Thus the data and information for these models also have to be on the ratio and absolute scales as in the case of the classical precise I E / O R / M S models. Thus, our traditional models of IE/OR/MS, in the main, or their fuzzified versions, do not allow us to build and analyze models with data and information that are vague or imprecise to the degree that they could be on an ordinal scale. Exceptions are the minimax and maximin criteria of optimistic and pessimistic evaluations, respectively, of the payoff values that we are

introduced in the initial stages of a 'Decision Theory' course; where one generally reviews the scale strength attribute of data and information. Even the 'expected utility' approach in decision analysis require that the utilities be on an interval scale while probabilities by their very definition has to be on an absolute scale in order that computations of expected utilities be admissible under the requirements of the measurement theory. Such measurement scale requirements essentially render these precise models of IE/OR/MS inapplicable to real-life situations; and thus they are not implemented or if they are implemented at all, their costs are very high because precision is obtained with an associated cost. Naturally, if the cost of precision outweighs the marginal benefit so gained, it is not practicable to implement such models. Furthermore such models are very complex and are expressed in a mathematical language that cannot be translated easily into the natural language of management. In addition, most precise models are built to identify optimal solutions that are generally characterized as 'a single point solution' as opposed to 'ball park' (set) solutions. Of course sensitivity analyses are exceptions to these amongst others. Finally, in most cases, these models do not, generally, consider hierarchies associated with real-life situations. They are, generally, non-hierarchical models. Again there are exceptions such as decomposition models of Linear Programming in economic activity analyses, etc.

1.1. First generation expert systems Natural language needs of management together with hierarchies associated with various problem domains eventually lead to the development of expert systems within AI. Thus natural language expressions were formulated for the construction of rules in rule bases that allowed certain structural hierarchies to be built as meta-rules. Together with two-valued logic and constraint directed search techniques, what might be called 'the first generation expert systems' were built and implemented for consumption by engineers and management. In such first generation expert systems, if our

LB. Turksen / Fuzzy expert systems for 1E/OR/MS

knowledge base consists of the set of rules: i f X is A~, then Y is B1, if X is A2, then Y is Bz, •

(1)

if X is AN, then Y is BN, and we observe that X is Ak, where k = 1, 2 , . . . , that

(2) N, then we can conclude

Y is Bk,

(3)

In such first generation expert systems, Ag's and Bk'S, k = 1, 2 . . . . , N, are represented either by precise numerical values or by precise symbolic values which may be linguistic values of linguistic variables but without any allowance for the degree of belonging to borderline cases, i.e., #ak(Xi) and /*Bk(Yj)e {0, 1} due to the requirement of two valued logic upon which such models are built. Thus during the inference process, for any rule to be fired, we need to have an observed system state to be an exact match to one of the left-hand side antecedent terms, Ak, k e {1, 2 . . . . . N}. Such limitations on the representation of knowledge and hence the inference methods, naturally limit the domains of the first generation expert systems• It should also be noticed that the requirements that the rules be precise and an observed system state has to match exactly with the antecedents of a rule for it to be fired lead to the generation of a very large number of rules in realistic system models. In response to these limitations, fuzzy sets and logic were introduced to the design of expert systems within the last few years. Thus we have entered into a new era of expert systems. I. 2. Second generation expert systems

Fuzzy expert system should be called 'the second generation expert systems'. For at least two reasons: (1) from the perspective of expert systems, they are the next logical generation beyond the two valued 'first generation expert systems', and (2) from the perspective of

IE/OR/MS, also they are the next logical generation beyond the fuzzification of the classical I E / O R / M S models. There are two essential and unique advantages for the design of expert systems with fuzzy sets and logic: (1) fuzzy knowledge representation, and (2) fuzzy inference methods. First, knowledge representation is enhanced with the use of linguistic variables and their linguistic values that are defined by contextdependent fuzzy sets whose meanings are specified by graded membership functions. The values of these graded membership functions are generally subjective assessments (ratings) assigned by experts. They may be obtained through the well known measurement theoretical direct and/or reverse rating methods during knowledge acquisition [25]. Their measurement scale could be as weak as 'ordinal', i.e., minimally specific. Fuzzy sets provide a framework for knowledge representation, with such minimally specific information, to deal with (a) fuzzy predicates, (b) fuzzy truth values, (c) fuzzy quantifies, (d) fuzzy probabilities, and (e) fuzzy hedges [40,41,42]. Thus knowledge representation gains a far greater expressive power than traditional systems. Secondly, fuzzy logic provides a unique computational base for inference in knowledge based systems. Inference methods such as generalized modus ponens, tollens, etc., are based on fuzzy logic which form the bases of approximate reasoning with pattern matching scores of similarity and fuzzy subsethood measures instead of exact matching required by two valued logic. Unlike traditional logic systems, fuzzy logic is aimed at providing modes of reasoning which are approximate and analogical rather than exact. In this perspective, the importance of fuzzy logic derives from the fact that almost all of human reasoning, in particular common sense reasoning, is approximate and analogical in nature. With these unique, advantages, we are now able to build knowledge bases which consist of rules if X is Ak, then Y is Bk,

k = 1, 2 . . . . .

N,

(4)

where generally Ak'S and B~'s are linguistic

LB. Turksen / Fuzzy expert systemsfor IE/OR/MS values of linguistic variables which are specified by their degree of belonging in a meaning representation scheme through membership functions ~Ak(Xi) and /~m(YJ)e [0, 1], i = 1 , . . . , n ; j = 1 , . . . ,m. Furthermore, if we observe that

X is A',

(5)

where A' does not necessarily match exactly with any of the antecedents of the rules in the knowledge base, we can obtain via approximate reasoning the conclusion Y is B',

(6)

such that ItA'(Xi), I~B'(Yj), I~Ak(Xi) and /~t~(Yi) [0,1], i = l , . . . , n ; ] = l , . . . , m . Since A' and B' can represent many linguistic variation of linguistic values and since the approximate reasoning can cope with such variations, we need to identify only a set of basic anchor rules in the design and implementation of these 'second generation expert systems' thus limiting the rule explosion on the one hand and providing an added form of intelligence that can handle various linguistic variations.

i. 3. Fuzzy I E / O R / M S models With these unique advantages, we can design, develop and implement fuzzy expert systems that can bridge the communication's gap between IE/OR/MS model builders and the managers of system operations. In our approach, we were able to do this by utilizing the I E / O R / M S models and model builders as knowledge sources. In this sense, our approach may be classified as 'expert and model-based' design. Furthermore, in this approach, the fuzzy logic based approximate reasoning requires only ordinal scale data and information for the design and development of fuzzy expert systems for the use of operations management. It should be clear that fuzzy expert systems are, on the one hand, qualitative models because they use linguistic variables and their linguistic values. But on the other hand, with the inclusion of membership functions that represent linguistic values that are at least on the ordinal scale, such models are both qualitative and quantitative, however, quasi-precise, i.e., ordinal, and hence approximate.

In this new approach of qualitative and quantitative, i.e., quasi-precise modeling, it is important to be reminded from time to time for the benefit of 'novice researchers in fuzziness' that Zadeh's [40] max-max-pseudo-complement based fuzzy set theory essentially relaxes only two axioms of the classical (two-valued) set theory, i.e., the law of excluded middle, ALI A - - I , is relaxed to become A tOA c_ I and its dual, the law of contradiction, A fq A = 0 is relaxed to become AI'qA ~0. But the other eleven axioms, i.e., (1) commutativity, (2) associativity, (3) idempotency, (4) distributivity, (5) null-set, (6) identity, (7) absorption, (8) De Morgan's Laws, (9) involution, (10) equivalence, and (11) symmetric difference, still form the basis of Zadeh's max-min-pseudocomplement based fuzzy logic. While later research has shown that there are many other fuzzy set theories based on t-norms and t-conorms [22] that require only the four basic axioms, i.e., (1) Boundary:

T(O, O)= O,

T(a, 1) = T(1, a) = a.

(2) Monotonicity:

T(a, b) <<-T(c, d)

whenever a ~
(3) Symmetry:

T(a, b) = T(b, a). (4) Associativity:

TIT(a, b), c] = T[a, T(b, c)]. As it will be illustrated in Section 4, these relaxations together with the ordinal scale data and information requirements permit us to build models with least cost and most benefit. Unfortunately, there was and continues to be a grave misunderstanding in some parts of our scientific community in particular and our societies in general. Because 'fuzzy' has a pejorative connotation, it is assumed that under

I.B. Turksen / Fuzzy expert systems for IE/OR/MS the rubric of 'fuzzy', researchers relax into non-scientific principles. Whereas, the fuzzy logic researchers do in fact abide by most of the axiomatic foundations of the previous theories as pointed out above. But by relaxing certain axioms and allowing the use of data and information that are on the ordinal scale and by providing new forms of reasoning methods, the community of fuzzy researchers have advanced the reach of scientific achievement in harmony with the scientific tenets of quest and respect for precision when the only available information and knowledge is quasi-precise. Before proceeding with a discussion of second generation expert system's applications in the domains of I E / O R / M S , we will review two essential topics briefly: (1) foundations of measurement theory which is important for the acquisition of membership functions from experts in the representation of expert rules that dictate the behavior of systems, subject to management control, and (2) approximate reasoning methods which are essential for the derivation of control actions from quasi-precise rules extracted from human experts, in our case the I E / O R / M S model builders.

2. Foundations of measurement

Both the 'fundamental' and the 'conjoint' measurement issues should be closely scrutinized for the design and development of fuzzy expert systems. With the fundamental measurement, we are to assure ourselves that membership function values are at least on the ordinal scale. With the conjoint measurement, we are to assure ourselves that the combination of membership values preserve the weakmonotonicity of the alternatives in decisionaction space.

2.1. Fundamental measurement The fundamental measurement view of gradual set-membership function can be formalized as the construction of homomorphisms from an 'empirical relational membership structure' (O, ~>m) to a 'numerical relational membership structure' (~A(O),~), where 0 6 0 are the elements in the empirical domain of concern, ~>

is a subjective gradual membership relation over these elements with respect to a fuzzy set A, a linguistic value of a linguistic variable, and /~A either e[0,1] or • ( 0 , 1 ] , depending on the representation, is the corresponding numerical membership assignment with the corresponding numerical relation 1> which forms the vertical axis in the representation of fuzzy sets. There is also an intervening numerical relational structure (V(O), >I) on the basis of the suppression of which we allow the referential set X for the attribute V to form the horizontal axis of a planar view of the representations instead of O. That is, X stands for O in the representations even though we continue to say I~A(O) but we write/~a(X). 2.2. Ordinal scale The 'weak order' relation is an essential starting point of the establishment of an 'ordinal scale' in the fundamental measurement of membership functions [15]. Let O be a set and ~>A be a binary relation on O, i.e., ~>a is a subset of O ' = O × O . The empirical relational membership structure ( O , ~>a) has weak order if and only if for all Oi, Oj, Ok • O, the following axioms are satisfied: 1. Connectedness: Either Oi ~A Oj or 0j ~>a 0i. 2. Transitivity: If Oi>~a Oj and Oj>~AOk, then

Oi >~a Ok. An empirical relation membership structure (O, mA) is 'bounded' if there exist elements 0 . . . . 0min, such that 0max ~Z 0 and 0 ~A 0rain for all 0 ~ O. The relation 0i~>z 0j holds in the structure if an observer, an expert, judges that "Oi is at least as much a member of the fuzzy set A as Oi"; or " 0 / s degree of membership in A is at least as large as 0j's degree of membership in A,,. It can be shown as an ordinal scale theorem that for a finite non-empty set O, if an empirical gradual-set membership structure ( O , ~ Z ) is a weak order, then there exists a bounded real-valued membership function /~A ~ [0, 1] on O such that for all Oi, Oj • O, Oi ~Z Oj if and only if #z(O~)>I IJA(Oj). Moreover, if /~a is another real valued membership function for A on O, it has the same property if and only if there exists a transformation function f , monotonic on the interval [0, 1], such that for all 0 ~ O,

6

I.B. Turksen / Fuzzy expert systems for 1E/OR/MS

#~4(0) =f[#z(O)]. That is ~A is an ordinal scale unique up to transformation functions f , if and only if, these two conditions are satisfied [25]. A natural consequence of the existence of an ordinal scale property is that 'max-min' operations suggested by Zadeh [40, 41,42] are meaningful operations, in the measurement theoretic sense, within the context of fundamental measurement over different linguistic values, say, A and B, of the same linguistic variable V, e.g., 'low', 'high' of 'temperature', or 'low', 'high' of 'inventory'. That is, since A and B are on the same ordinal scale applicable to V, there exists a transformation function f , monotonic on [0, 1], such that f[l~AnB(X)] = min{f[ItA(X)], f (t~B(X)]} and

f[IIAuB(X)] = max{f[/~a(x)], f(/~n(x)]}. This means there is an 'ordinal scale invariance' over V. For example, suppose f(#A) = rain{l, /~Z + a~} where a~ is a constant, 0 < c~< 1. Then it is clear that min{1, I~znn(x) + Ol} = min{min{1, l~m(X) + 0:}, min{1,/tB(x) + o:}}. Since by definition

#AnB(X) = min{#z(X), /tB(x)}, we have

variable which we need to measure have no internal additive structure since no extensive concatenation is apparent [20]. However, there are entities 0 c O whose subjective attribute V which we wish to represent may be composed of two or more subjective components, V,., i = 1, 2 . . . . . each of which affects the attribute in question. An example is 'comfort' of various 'humidity' and 'temperature' combinations. Another example is 'production rate' dependent on various combinations of 'inventory level' and 'demand rate', etc. In these examples, 'comfort' is a composite linguistic variable of its component linguistic variables 'humidity' and 'temperature', and 'production rate' is a composite of 'inventory level' and 'demand rate', etc. In this section we are concerned with the construction of membership functions and their measurement scales for composite linguistic variables. Furthermore, we are interested to know in particular whether the composite linguistic variable would inherit the monotonic weak order property of its component linguistic variables under various t-norm and t-conorm operators. The following two results have been shown to hold in Turksen [25].

2. 3.1. T-normed conjoint measurement In general for all the t-norm family of operators [22], the following result is obtained: if ar ~ fq iff T(l~j(a), Irk(r)) >i T(#j(f), ~k(q)),

(7)

and fp >~br

min{1, min(/zA(X), /zB(x)) + a'} = min{min{1,/la(x) + oc}, min{1,/us(x) + 0~}}. This function f merely causes an upward shift in the membership function. A similar argument would be put forth for f ( . ) = max{0,f(.) - a'} which causes a downward shift in the membership function. Naturally, other transformation functions may be found when necessary.

2.3. Conjoint measurement Most if not all of the subjective evaluations represented by linguistic terms of a linguistic

iff T(l~j(f), I~k(P)) >1 T(t~i(b), I~k(r)),

(8)

then ap >~bq iff T(lzj(a), I~k(p)) >! T(t~j(b), #k(q)),

(9)

where

At(O ) =a, Ak(O)=p,

Aj(O') =f, Aj(O") = b, Ak(O')=r, Ak(O")=q

are the linguistic assessments of 0, 0', 0"~ O for the linguistic values Aj of Vj and A~ of V~, and #j(a), I~j(f), 12j(b), #k(P), I~k(r), l~k(q) are the corresponding numerical membership values, respectively. From the monotonicity of t-norms, the weak-order property of ordinal scale membership functions, and (7) and (8),

I.B. Turksen / Fuzzy expert systems for IE/OR/MS

By cancelling 0:#j(f) and (1 - a)/~k(r) from both side we obtain

whenever

#t(a) >i Itt(f),

7

uk(r) >! Uk(q),

/~t(f) >~/~t(b) ' and /~k(P) >~ #k(r),

aq~j(a) + (1 - O0#k(p ) >1 oq~j(b ) + (1 - oOl~k(q)

that is, when ~t(a) ~>/~t(f) ~> ~t(b) and /~k(P) >I/~k(r) >i/~k(q),

which leads to ap >~bq as desired. Thus convex linear combination also preserves the monotonic weak ordering of components in its composite. The method used above is known as the 'double-cancellation' [15]. A natural extension of this method leads to triple cancellation and, in general, to a new form of transitivity of monotonic weak order which allows us to extend the results discussed so far to more than two components, i.e., multicomponent composition. In summary, the results obtained for t-norms, and hence by duality for s-norms, and convex linear combinaton of linguistic values of linguistic variables hold in general in conjoint measurement. It is clear that the ordinal scale properties can be verified with a minimum cost and effort. However, in order to interpret these numerical assignments as probabilities or beliefs, one needs to verify that they are at least on a ratio or absolute scale, which require a lot more tests and additional costs. Such tests have to validate first the existence of a cardinal scale, and then the existence of an 'absolute zero' and an 'absolute unit', which are not generally justifiable in context dependent domains of subjective ratings [20].

then we obtain

T(ttt(a), t~k(p)):->- T(ltt(b), l~k(q)) and

ap >~bq. This is an important and significant result. It states that members of the t-norm family of operators preserve the monotonic weak ordering of component attributes A t and Ak in their composite attribute A. Thus if we are only interested in the relative ordering of alternatives of a composite linguistic term A, then we can apply any t-norm to the combination of component linguistic terms A j, Ak provided that we have grounds for the monotonic weak ordering of its components, A t and At.

2. 3. 2. Convex linear conjoint measurement At times, we are interested in convex linear combination of linguistic concepts such as where 0 ~< o: ~< 1, and j, k ~ {1, 2 . . . . }. It can be shown in a straightforward manner that if ar >, fq iff aq~t(a ) + (1 - OOl~k(r) >i ac/tt(f) + (1 - o:)u~(q),

(7')

and fp >~br iff oq~t(f) + (1 - aC)l~k(p) /> a~ut(b ) + (1 - ae)uk(r ),

(8')

then ap >. bq

iff ¢~/~t(a) + (1 - a ) / ~ ( p ) ~> aq~t(b ) + (1 - a0uk(q),

(9')

where again a , f , b e A j and p , r , q e A k as defined above. By adding side by side (7') and (8'), we obtain olut(a ) + (1 - OOIZk(r) + oq~t(f) + (1 - c 0 u , ( p )

1> aut(f) + (1 - o0uk(q) + o~uj(b) + (1 -- O¢)Uk(r).

3. Approximate reasoning There are many versions of approximate reasoning methods. We will review here only two of these methods which were implemented in our investigations: (1) Compositional Rule of Inference, CRI, also known as the Generalized Modus Ponens, proposed by Zadeh [41], and (2) Approximate Analogical Reasoning, proposed by Turksen and Zhong [38, 39]. Before we review these two methods, however, it should be mentioned that, amongst the other approximate reasoning methods, a notable class is the Japanese approach used in many applications of fuzzy control. These

I.B. Turksen / Fuzzy expert systemsfor I E / O R / M S

8

methods generally rely on a defuzzification of rules and on an interpolation technique either weighted or unweighted. A well known method amongst these is Takagi-Sugeno method [23].

3.1. Generalized modus ponens Amongst algorithms that infer the conclusion (6) from a single rule and the observed premise (5), the most commonly used formula is the celebrated Compositional Rule of Inference [17, 26, 41]: #a(Yj) = max (min(#A(x~), #A~B(X,, Yj))), x, f o r j = 1, 2 . . . . .

m;i=l,

2. . . . .

(10)

n.

More generally, the algorithm is

#n,(Y~) = X/ (T(#m'(Xi), ~a--~B(Xi, Yj))), xi

(11)

or via

#R(Xi, yj) = #a,-,a,(Xi, yj) ~ " " " 0 ~AAAr~BN(Xi, yj), (14) and #a'(Yj) = V (T(~A,(X,), #~(x,, Y3)),

(15)

xi

for i = 1 , 2 . . . . .

n;j=l,

2. . . . .

m.

While the Compositional Rule of Inference has been used extensively in database applications, and the choice of representations makes it flexible for utilization in different inference processing applications, there are some conceptual problems associated with its use [26]. Another commonly used routine for inferring a conclusion of the form (6) from the set of rules (4) and an observation (5) is given, in a generalized form, by the Interpolation Rule [42]:

for j = l, 2, . . . , m ; i = l, 2 . . . . n. Different T-norms have been used in different applications for computing the conjunction of A ' and {A---~B} [17,19]. It is also possible to generalize (11) by using interval valued representation for AND and OR [26, 27, 28]. When there are several rules in the knowledge base, as is the case in (4), then one can use the Compositional Rule of Inference for obtaining the conclusion that can be inferred from each rule with the observed premise (5). But then the inferred conclusions should be combined in order to obtain the final conclusion of the form (6). Alternatively one can first combine the rules in the knowledge base (4) and use the resulting joint membership function with the observed premise (5) to infer the conclusion (6) via CRI [3]. In general, if we denote the combination operator by ~ , ~) e {v, A} or more generally e {S, T}, the derivation of the final conclusion from the set of rules (4) and observation (5) can either be carried out via

#a,k(Yj) = V (T(#A,(x,), #Ak~a~(X,, Yj))), xi

for i = 1, 2 . . . . . n ; j = l , k = l , 2 . . . . . N,

2. . . . .

(12)

m;

for j = 1, 2 . . . . .

m;

k

(16) where

for k = l ,

2. . . . .

N;i=l,

2. . . . .

N.

The interpolation formula, it is claimed, approximates the Compositional Rule of Inference [17]. Indeed, if we have the precise observation A' = {xj}, i.e., /&,(xi) = 1, for i = 1, 2 . . . . . n, (18) then we conclude from (17) that rk = V UA~(X/), xi

(19)

and so we have

#a'(Yi) = V V T(~Ak(Xi), #n~(Yi))" k

xi

(20)

On the other hand, the use of Compositional Rule of Inference (12) with disjunctive combination of consequents in (13) for the same case of the precise observation (18) yields k

~ #B'N(Yj), (13)

(17)

r k = V ( T ( l ' l a k ( X i ) , [AA'(Xi))), xi

#a,(Yj) = V V l&~ak(x~, Yj).

and

i~B.(yi) = IAB,l(yj) {t) #n'2(Yj) ~ ' ' "

I~B'(Yj)=V(T(Yk, #Bk(Yj))), f o r j = 1, 2 . . . . . m,

xi

(21)

It is seen that (20) and (21) will coincide, for example, with the choice of the minimum operator for the T-norm and Mamdani representation for the implication (or generally for a

I.B. Turksen / Fuzzy expert systems for IE/OR/MS

class of Mamdani type representations for the implication) [17]. However, there is less correspondence between the two procedures when asymmetrical representations are used for the implication, as well as when the observation is not point valued or is fuzzy [31].

3. 2. Approximate analogical reasoning The approximate reasoning approach proposed in our studies draws upon an approximate analogical reasoning routine that has recently been elaborated [31, 38, 39]. Briefly, the algorithm used by the Approximate Analogical Reasoning, AAR, for inferring a conclusion of the form (6) from the knowledge base (4) and the observation (5) is as follows: Step 1. for k := 1, 2 . . . . . N compute a pattern matching measure Pk between A' and Ak; Step 2. determine the set R* of the rules that are to be evoked, i.e., if k e R* then the rule (if X is Ak, then Y is Bk} is to be evoked; Step 3. for all k e R* modify the consequents Bk (call the modified consequents B~) using the pattern matching measure Pk; the pattern vector for B;, is a function of the pattern vector for B~ and Pk; Step 4. combine the modified consequents by using the combination formula (13) to obtain the desired conclusion B'. It can be seen that the algorithm is very general and has a high degree of flexibility. One can use many different similarity measures as well as the fuzzy subsethood measure in Step 1 [16, 45]. Even for interval valued fuzzy sets, one can choose different pattern matching measures by defining an appropriate function for measuring analogy [39]. There are also several possibilities in the second step, including: (a) evoking only the rule corresponding to the maximum pattern matching measure Pk (with an arbitrary tie breaking formula when it is not unique); (b) evoking rules corresponding to a set of the highest pattern matching measures (again with some appropriate tie-breaker); (c) evoking those rules corresponding to pattern matching measures above a predetermined threshold with a routine for reporting the case when there is no such rule and an option for

repeating the step with a lower threshold; (d) Evoking all the rules. There is further flexibility in the third step where one can choose different modification functions or no modification at all. Two modification functions suggested in [38, 39] are the 'membership function expansion' form given by #B'k(Yj) = min(1, #Bk(Yj)/Pk),

(22)

and the 'membership function reduction' form

12B,(yj) = y I~n~(Yj) "Pk.

(23)

It can be seen that the modification formula (23), which reduces the membership function in accordance with a pattern matching measure PK, approximates the operation of the Interpolation algorithm and the Compositional Rule of Inference more closely. One can generalize this modification formula via /~B'(Yj) = V T(~B,(yi), Pk) k

(24)

which is similar to the operation of the T-norm in (16). Recently, a new set of modification functions have been introduced based on an analysis of bounds on multiple antecedent fuzzy Simplication and reasoning that establishes a link between CRI and AAR [32]. This latter approach is characterized as 'gradual shifts in Interval-valued reasoning' [34]. Note that the Analogical Reasoning approach, unlike the Compositional Rule of Inference, will always infer the conclusion Y is Bk if the observation X is Ak. It is very intuitive and has a high degree of flexibility.

4. Applications In an attempt to display the power and reach of fuzzy sets and logic, we have build IE/OR/MS expert system prototypes in cooperation with our colleagues in this area. In these prototype studies, a cooperative effort is exerted on the problem domain in the following manner: (1) first, we take a classical problem, such as production, inventory or spare-parts problem or activity analysis or market share analysis; (2) we

10

I.B. Turksen / Fuzzy expert systems for IE/OR/MS

investigate these problem domains with the classical methods of deterministic or stochastic methodologies to learn about the behavior of the system under various restricted assumptions: (3) next, we build a fuzzy set and logic based representation of the same problem domain with the specification of linguistic variables, i.e., their linguistic values together with their membership functions and the specification of linguistic rules that govern the behavior of the system; (4) then we simulate various observed system behavior patterns with the original deterministic or stochastic model and the fuzzy expert system model; (5) finally we compare the results of simulated system behavior that is produced by (a) fuzzy, i.e., second generation, expert system and (b) by the original, deterministic or stochastic model. Our approach relies on the proposition that IE/OR/MS model builders are often experts who can interpret the results of their models and can express their insightful expert knowledge about such systems behavior in a natural language that usually contains vague, ambiguous, uncertain, fuzzy linguistic terms. Such linguistic terms provide (1) a flexible expression of the system behavior subject to various uncertainties and imprecisions and at the same time (2) help model builders communicate their complex results to management in an appropriate natural language context. Hence closing the communications' gap between I E / O R / M S model builders and managers of operations in integrated manufacturing environments. Such concerns lead us to propose the development of expert system prototypes via an approximate reasoning framework based on fuzzy logic but relying on insights obtained from I E / O R / M S models with the model builders acting as the experts. In this regard, so far, we have conducted (i) aggregate production planning [27], (ii) inventory capacity planning [29], (iii) spare-parts planning [33], (iv) scheduling [30, 35,37], and (v) market share analysis studies [36]. In all these studies we have been able to show that ordinal scale data can be used to obtain good 'ball park' results from fuzzy expert systems that produce results within a few percentage points of the so called optimal results of analytical models that require absolute scale data. It should also be

pointed out that in one real life case of aggregate production planning, one real life case study of job shop scheduling, and in an experimental case study of market share analysis fuzzy expert system models produced much better results than the original I E / O R / M S models as will be pointed out later. In the rest of this section we will review briefly four of these real-life data-based case studies of our fuzzy expert systems models and their comparison with the corresponding analytical I E / O R / M S models.

4.1. Aggregate production planning prototype (APPP) Our first investigation with the proposed approach was for the aggregate production planning. For this study we used the paint factory data and compared it with the classic Holt et al. HMMS analysis [11, 12, 13]. The reasons for choosing the aggregate production planning and the associated paint factory data are two fold: (1) the case uses real-world data and is sufficiently well documented to be the basis of such a comparative analysis; and (2) the HMMS paint factory linear decision rule (LDR) solutions have become the standard by which other aggregate planning models compared [10, 14, 18, 24]. (Some of these models are also considered in our comparisons.) Details of this prototype may be found in Turksen [26, 27]. Here we briefly summarize the content of this prototype.

4.1.1. Knowledge base The knowledge base for the aggregate production planning prototype contains the following: (1) a rule base which consists of management decision rules; and, (2) a database, which consists of all other information needed to support the inference system in order to infer an expert system advice.

4.1.2. Rule base In cooperation with the experts in building and analyzing APPP models, it was decided that management decision rules for aggregate production planning be expressed in the meta

LB. Turksen / Fuzzy expert systems for IE/OR/MS

language representation as: IF Xj is A1 AND X 3

AND X 2

is A

is

11

Table 2. S u m m a r y of the total cost (in $100)

A2

3

THEN Y is (should be) B, where X1 is the base variable of sales forecast St, X2 is the inventory level /,-1, and X3 is the work force level Wt_I. Furthermore, the Ai's are the linguistic values describing these independent base variables, i.e., linguistic variables, where linguistic values are represented by their membership functions.

Name

1949-1953

1949-1950

LDR PPP Rink's PVFS(40) T u k s e n ' s IVFS-CRI(40) T u k s e n ' s IVFS(27) C o m p a n y ' s actual

2058 2055 2156 2109 2163 a 2552

734 740 -683 721 b 742

a A A R , b CRI.

Abbreviations:

L D R , linear decision rules; IVFS, interval-valued fuzzy sets; PVFS, point-valued fuzzy; PPP, parametric production planning.

4.1.3. Database The database consists of three basic components: (1) weights for the sales forecast: (2) membership values for the linguistic values allowed to be used in the system; (3) descriptions of observed system characteristics. In general, components 1 and 2 would be permanent components of the database, which may require adjustments from time to time to accommodate environmental changes, whereas component 3 would be temporary within the span of a particular system diagnosis or system planning for the period of concern.

4.1.4. Membership values The linguistic values high (H), low (L), average (A), etc., are specified with standard membership functions for the production variable Pt, the sales forecase variable St, the inventory level variable lt-l, the work force W,_I, and the change in the work force AW,. The domains of applicability for these functions are shown in Table 1 for each of these base variables.

4.1.5. Comparison of models In Table 2, the first row, for the year 1949 to 1953, shows a comparison of costs associated Table 1. Bounds on the base variables of 'paint factory' Variables

Lower b o u n d

Upper bound

Wt- 1 AW, P, 1, I

60 - 10 25O 150

115 10 750 490

s,

250

750

with LDR [11, 12, 13], parametric production planning (PPP) [14], Rinks' [21] point-valued fuzzy sets (PVFS) and our interval-valued fuzzy sets (IVFS) approaches based on a knowledge base formed out of 40 rules, and our AAR approach based on 27 rules, together with the company's actual total cost. First of all, it is clear that every model solution results in a cost that is much better than the company's actual cost. Secondly, it can be observed that Rinks' model produced a cost 4.7% more than LDR, whereas our IVFS approach produced a result 2-4% more than LDR. It needs to be noted that both Rink's model and one of ours are based on CRI and 40 rules. Finally our AAR approach produced a result 5% more than LDR, but this was based on 27 rules in knowledge base. In the second row of Table 2, for the years 1949 to 1950, which avoids the disturbance due to Korean War conditions, we observe that the PPP approach results in a cost slightly larger than the LDR approach. Our IVFS-CRI approach with 27 rules produces a result 3.8% smaller than the LDR approach, whereas the IVFS-CRI approach with 40 rules produces a result 7.3% less than LDR approach. In Figure 1, a graphic comparison is shown between an AAR result and the actual performance of the company.

4. 2. Hierarchical job-shop scheduling ISIS is a hierarchical job-shop scheduling model developed by [4, 7, 8, 9] using Artificial Intelligence technology. In the design of our approximate reasoning-based scheduling model,

I.B. Turksen / Fuzzy expert systemsfor IE/OR/MS

12

MONTHLY C O S T $ 8O

$70

-

ACTUAL

AAR

$ 60 -

09

o o

$ 50

\

l

'i ! II

!

/" 1 \ l "1

t

I

$ 40 I

/ I I D~ $ 30 -

I\

$ 20 1949

1950

1951

1952

1953

MONTH

Fig. 1. A graphic Comparison of the A A R result and the actual company performance.

the structure of ISIS is taken as a framework to provide a structural basis for the comparison of the two models' performance. ISIS represents the scheduling environment through constraints. Constraints are interrelated and linked to the components of the scheduling model using a variety of relationships. The method of constraint-directed search is used to generate schedules. The scheduling environment of our model is also represented through constraints which are in the form of (fuzzy) linguistic values of linguistic variables. The representation of the constraints using (fuzzy) linguistic values of linguistic variables results in 'elastic constraints' and has a smoothing effect on the system behaviour and thus its performance. Although our frame work of modeling the scheduling activity is similar to ISIS, there are some conceptual differences in the designs of the two model. Our models [30, 35, 37] are

based on fuzzy set theory. Therefore, the representation and utilization of expertise is expressed in a qualitative manner with the representation of linguistic variables via membership values extracted from experts. The imprecise expertise regarding the scheduling environment is incorporated into the decisionmaking process through the use of fuzzy logic.

4. 2.1. Details of the second investigation The details of our first investigation could be found in [35]. Here we will review our second investigation [30] which is based on the job shop studied by Chiang, Fox, and Ow [4]. In this investigation, there are 120 jobs, produced in batches (without job splitting). Requested start dates (release dates), due dates, processing times and setup times of the jobs are known prior to scheduling. Each job has a priority class assigned to it. There are three different classes of jobs, i.e., products, with different routings as shown in

I.B. Turksen

Fuzzy expert systems for IE/OR/MS

RB1 2&5

1&4

13

3&6

I i i i

i

I1

I11

II

q

\ \

/

/

11

I

I l !

6 Work Stations

~ WA8

WA10

f

/

I

/

RB2

I

j

i iiii

~; t / / / •1 ¢i /

I

I

,,," 3 Work Station /

2~i~/" 4 Work Stations

" WAll

Fig. 2. Routings of different products.

Figure 2. Part of the data for different parts are shown in Table 3(a)-(d). We considered two performance measures. They are average tardiness cost ( A T C ) and average flow time (AFT) and are computed as follows: ATe = ~

ti • Pi, i=1

1

AFT =

6

-~ i~=lfi ° Pi,

where ti is the total tardiness for batch i, ti = max{ci - d i , 0},

p/is its penalty, ci is its job completion date, di is its due date, f,- is its flow time, f~ = c / - r/, and r~ is its release date, i.e., requested start data. 4.2.1.1. R u l e base. As shown in Figure 2, there are two rule bases, RB1 and RB2. The second rule base is needed to rearrange the job sequences based on their remaining processing time due to merges from different routines. The first rulebase, RB1, takes into account the following factors: PRIORITY, SLACK TIME, REQUESTED START DATE~

I.B. Turksen / Fuzzy expert systemsforlE/OR/MS

14

Table 3. An example of parts of a job data op.

m/c

setup

(a) Job data for Parts 1 and 4 opl WA1 25200 op2a WA2 49500 op2b WA3 59400 op3 WA8 900 op4 WA9 5400 op5 WA10 10800 op6 WAll 14400

piece

1238 471 759 1123 162 860 658

(b) Job data for Parts 2 and 5 opl WA1 25200 op2a WA4 55800 op2b WA5 55800 op3 WA7 16019 op4 WA9 5400 op5 WA10 10800 op6 WAll 14400

957 2995 2995 396 205 619 468

(c) Job data for Parts 3 and 6 opl WA1 18000 op2a WA5 54000 op2b WA6 54000 op3 WAll 25200

356 2818 2818 345

(d) Tardiness and flow time cost

di is its due date, t is the current time and rpt i is the remaining processing time multiplied by batch quantity. The rules in RB1 and RB2 control the flow of all jobs. An example of the rules in the first rulebase, RB1 is: IF PRIORITY IS 'LOW' AND SLACK TIME IS ~SOMEWHAT LOW' AND REQUESTED START DATE IS tHIGH' THEN SELECTIBILITY IS ~MEDIUM'

An example of the rules in the second rulebase is: IF PRIORITY IS ~LOW' AND SLACK TIME IS ~SOMEWHAT LOW' AND REMAINING PROCESSING TIME IS ~HIGH' THEN SELECTIBILITY IS ~MEDIUM'

where the linguistic variable PRIORITY can take on the following linguistic values:

VH (very high), H (high), SH (somewhat high), M (medium), SL (somewhat low), and L (low). The linguistic variables, SLACKTIME, REQUESTED START DATE a n d REMAINING PROCESSING TIME c a n

Priority class

Penalty for tardiness and flow time ($/batch/day)

1 2 3 4 5 6

1 4 8 12 16 20

where the slack time for batch i is given by Si =

d i -

ei -

max{t,

ri} ,

di is the due date for batch i, ei is its expected remaining processing time multiplied by its batch quantity, re is its requested start time and t is the current time. The second rulebase, RB2, takes into account the following factors: PRIORITY, SLACK TIME, REMAINING PROCESSING TIME~

where the slack time for batch i is given by s~ = di - rpt~ - t,

take on the following linguistic values: L (low), SL (somewhat low), M (medium), SH (somewhat high), and H (high). If should be noted that there are 6 . 5 * 5 = 150 rules in each rulebase. Thus there are 300 rules in total. 4.2.1.2. S e q u e n c i n g . A job assignment in the scheduling sequence is determined by a selectibility factor that is the right hand side of each rule. The selectibility factor, SF, is determined with importance li~(i) and weighting W,. factors where /~w)'s are the importance factors within the variables, i.e., within the linguistic values of a linguistic variable and W/s are the weights assigned to each linguistic variable. In this study, the following importance factors are assigned to each of the linguistic values of the linguistic variables. The linguistic values of PRIORIaW,VH, H, SH, M, SL, L, are assigned the importance factors of 6, 5, 4, 3, 2, 1, respectively; and the linguistic values o f REQUESTED START DATE, REMAINING PROCESSING TIME, and SLACK TIME, L, SL, M, SH,

I.B. Turksen / Fuzzy expert systems for IE/OR/MS H, are assigned the importance factors of 5 , 4 , 3 , 2 , 1, respectively. Thus, it should be noted that here we are defuzzifying the linguistic values to obtain a simple solution. A basic question is "How to combine Iq's and W~'s to find SF?", e.g., if the rule is " L AND SL AND H", then what should be the selectibility factor, SF, of this rule on the right hand side, i.e.,

WI ~ I1,L (~ W2 ~ I2,sL (~ W3 ~ I3,H ----> 9. The combination operators, ® and ~ , in this study are chosen to be max-min (v, ^) based on the arguments presented in the measurement section; in other words the subjective assessments of the experts, i.e., the importance factors lq and the weights W~, are on the ordinal scale for the fundamental measurement and that conjoint measurement preserves the ordinality. That is, SF = V A [w,, i

I,j ol.

where i corresponds to PRIORITY, SLACKTIME and REQUESTED START DATE in RB1 and to PRIORITY, SLACK TIME and REMAININGPROCESSINGTIME in RB2 and j(i) corresponds to VH, H, SH, M, SL, L for PRIORITY and L, SL, M, SH, H for SLACK TIME, REQUESTED START DATE and REMAINING PROCESSING TIME in each rule within each rulebase. For example, if for a rule we have W, ^ 1,.L V W2^I2.sLV W3 A/3,H and the weights are chosen to be W1 = WE= W3= ½ then 31A-~V~^4V~A½---> 0.333. As will be discussed later in the comparison of the experiments, the algebraic plus and times (+, .) operators are also considered in the investigation, under the assumption when and if interval scale acquisition of those subjective weights would be obtained which on our view would not be cost effective. Based on such considerations, the rule bases are formulated. An example of a section of RB1 is shown in Table 4.

4. 2. 2. Computational experience Simulation experiments are conducted on 13 data sets of 120 jobs each that came from

15

Table 4. An example of a section of the first rulebase (RB1) Priority

SlackT.

StartT.

Select.

L L L L L L L

L L L L L SL SL

L SL M SH H L SL Z

0.722 0.656 0.589 0.522 0.456 0.656 0.589

SL SL SL SL SL SL SL

i~ L L L L SL SL

L SL M SH H L SL

0.778 0.711 0.644 0.578 0.511 0.711 0.644

Chiang, Fox and Ow investigation [4]. In Tables 5 and 6, we show a comparison of our results based on four different experiments: 1. ISIS; used Chiang, Fox and Ow's initial sequence, with a routing of first come first serve, FCFS, within the job shop. 2. Static Expert System Solution: where RB1 is called prior to the first work station WA1 and that sequence is maintained through all the routes in the job shop. 3. Dynamic Expert System SolutionCommon Weights: where both RB1 and RB2 is executed in the flow as shown in routing diagram in Figure 2, but the weights are common to both rule bases. 4. Dynamic Expert System SolutionDifferent Weights: where both RB1 and RB2 are executed in the flow as shown in routing diagram in Figure 2, but first the best weights for the RB1 are determined and then a different set of best weights for RB2 are determined. Thus the two rule bases are different with respect to weights. For the experiments 2, 3 and 4, there are 16 runs for each of 13 data sets. In Tables 5 and 6, the minimum average tardiness cost and average flow time are shown in darker print. Results of our experiment for this second investigation of the hierarchical job shop scheduling problem can be summarized as

I.B. Turksen / Fuzzy expert systems for IE/OR/MS

16

Table 5. Comparison of average tardiness cost Data file # 4 5 6 7 9 10 13 14 15 16 17 19 20

Stat. expert system

Common W~

Different V¢~

max-min

(+,-)

max-min (+,-)

max-min

80.57 45.48 86.05 7 0 . 1 5 223.17 1 2 4 . 1 6 72.36 43.83 193.46 162.55

57.32 113.05 126.69 80.26 126.57

38.36 51.90 89.10 38.61 151.85

162.79

116.99

35.35 53.41 89.10 33.77 149.32 93.34 85.42 74.25 57.48 23.78 139.59

ISIS

116.99

135.31 9 5 . 3 3 140.18 8 9 . 7 2 149.39 64.22 79.97 32.60 175.47 1 5 1 . 8 4 27.16 30.45 76.33 47.67

148.62 158.26 150.65 78.43 147.94 52.17 104.05

17.55

64.06

47.59 76.41 101.24 46.47 112.37 109.94 150.96 169.42 145.77 101.95 139.59 27.02 99.76

88.21 85.42 74.25

57.48 23.78 139.72 25.38 75.49

Table 6. Comparison of average flow time Data File # 4

5 6 7 9 10 13 14 15 16 17 19 20

Stat. expert system

Common W~

Different WI

ISIS

max-min

(+, .)

max-min (+, .)

max-min

244.67 195.03 386.48 208.36 307.44 333.81 247.08 255.15 329.97 211.50 293.94 155.04 223.24

146.54 154.39 183.01 147.34 278.12 206.77 146.71 150.07 198.20 132.25 251.72 130.63 165.23

155.64 166.24 214.71 174.25 228.82 206.77 192.73 190.19 172.80 150.67 242.62 154.52 200.70

136.40 145.22 173.77 138.69 273.64 202.71 144.62 146.62 184.59 126.32 245.32 127.44 158.87

135.80 141.04 258.07 136.30 345.60 273.65 168.21 180.15 173.93 126.32 278.78 125.47 165.7l

follows: 1. M a x - M i n vs. P l u s - T i m e s : M a x - M i n p e r f o r m e d (in g e n e r a l ) b e t t e r t h a n P l u s - T i m e s c o m b i n a t i o n in t h e s e l e c t i b i l i t y c a l c u l a t i o n o t h e r factors b e i n g e q u a l . 2. I S I S vs. Static M a x - M i n : Static M a x - M i n p e r f o r m e d ( a l m o s t a l w a y s ) b e t t e r t h a n I S I S in b o t h t a r d i n e s s a n d flow t i m e m e a s u r e s . 3. Static ( F C F S ) vs. D y n a m i c : R e s e q u e n c i n g j o b s within t h e j o b s h o p with t h e s e c o n d rule base RB2 outperformed (almost always) the FCFS job selection criteria. 4. C o m m o n vs. S e p a r a t e w e i g h t s in R u l e Bases: I n c o n c l u s i v e . 5. T h e r e is n o t a single rule b a s e ( i . e . , a weight v e c t o r ) t h a t a l w a y s gives b e t t e r result b u t

158.60 159.96 205.26 126.44 214.60 203.24 192.89 170.86 180.10 148.76 238.7 145.62 189.07

it is f o u n d t h a t WpRIORITV 1> 0.5 in all o f t h e b e s t solutions.

4.3. Spare-parts system A s p a r e - p a r t s service c e n t r e m o d e l is d e v e l o p e d b y B e r g a n d P o s n e r [2] in t h e t r a d i t i o n a l o p e r a t i o n a l r e s e a r c h a p p r o a c h using t h e s t o c h a s tic m o d e l i n g t e c h n i q u e s . W e h a v e a n a l y z e d this classical o p e r a t i o n a l r e s e a r c h m o d e l ; a n d with the e x p e r t k n o w l e d g e o f t h e s t o c h a s t i c m o d e l b u i l d e r s , we d e v e l o p e d a fuzzy e x p e r t s y s t e m for the s p a r e - p a r t s c e n t r e o p e r a t i o n s [33]. T h e service c e n t r e s y s t e m o p e r a t e s as follows: w h e n a c u s t o m e r a r r i v e s with a f a i l e d i t e m , t h e failed

I.B. Turksen / Fuzzy expert systems for IE/OR/MS

17

c SERVERS CUSTOMER QUEUE

n INITIAL SPARES J t ~

~

FAILED ITEMS

~

[

0000

\, /

SHELF

,,t CUSTOMERS ARRIVE


WORLD / J

Fig. 3. A repair service center.

item is sent to the item-repair shop for repair; and the customer proceeds to the shelf. If there is a spare on the shelf, the customer takes it and leaves the system. The delay of the customer in this case is 0. If however, there is no spare on the shelf the customer will have to wait until an item is repaired and put on the shelf. In reality customers may not arrive in person. They may ship the failed item to it, and similarly there may not be an actual physical customer queue but just a list of customers to whom the service centre owes spares at any given time. The system behaviour is described in Figure 3. With the stochastic system model developed in classic operational research approach, a solution is found for the mean customer delay E(W) as follows: cc-lp n

n0c!(l_p)2

forn>ic-1,

:ro(cp)~_ , (1 + (c - n - 1)(1 - p)) e(w) =

c! (1 - p)2 ~-"-'

+ z°(cP)c-' ~

j=l

](coy (n + j)l '

for n ~ < c - 1 . where c is the number of servers; n is the number of initial spares; p = Mcl~ is the system utilization; E(W) is the mean customer delay

and _

z°l

(cp)

c!(1-p)

c-1 ( c o y

=

i!

Based on this stochastic model, the system's behaviour is analyzed at length numerically and graphically. Table 7 and Figure 4, for example, Table 7. A subset of the exact results obtained from the analytical model n=2

n=8

n=12

c p

e(w)

p

e(w)

p

e(/v)

2 0.20 0.40 0.70 0.90

0.04167 0.19048 0.96078 4.26316

0.20 0.40 0.70 0.90

0.00000 0.00078 0.11304 2.26562

0.20 0.40 0.70 0.90

0.00000 0.00002 0.02714 1.48647

4 0.20 0.40 0.70 0.90

0.07485 0.23620 0.72900 2.43134

0.20 0.40 0.70 0.90

0.00000 0.00097 0.08577 1.29211

0.20 0.40 0.70 0.90

0.00000 0.00002 0.02059 0.84775

8 0.20 0.40 0.70 0.90

0.20431 0.44498 0.76020 1.59964

0.20 0.40 0.70 0.90

0.00004 0.00385 0.11275 0.87692

0.20 0.40 0.70 0.90

0.00000 0.00010 0.02707 0.57534

16 0.20 0.40 0.70 0.90

0.44124 0.68979 0.84846 1.23070

0.20 0.40 0.70 0.90

0.00254 0.06691 0.33470 0.81595

0.20 0.40 0.70 0.90

0.00001 0.00391 0.10981 0.56167

LB. Turksen / Fuzzy expert systemsforlE/OR/MS

18

MEAN DELAY, E(W) SPARES n - - 8 3

2

1.C=2 2. C = 6 3. C = 1 0 4. C = 1 4

3 4

I

0

/~

I

I

,

,

,

I

I

I

UTILIZATION,

p

0.1 0.2 0.3 0.4 0.5 06. 0.7 0.8 0.9 1.0 Fig. 4. Relation between variables in system model, i.e., {c, n, p}--* E(W).

show the behaviour patterns of the mean delay E(W) for various parameters of the system model. It should be noted that E(W) shows several regions of 'ball park' stability. Based on this analysis, several decision structures could be considered as follows in classical operations research approach: Pr(W ~ 1 - tr;

(25)

E(W) ~ E0;

(26)

(1 - X)Eo <~ W' <~ (1 + X)Eo.

(27)

Alternatively, these stochastic expressions may be expressed in natural language with linguistic values as follows: Probability that the customer delay W is SHORTis (should be) HIGH;

(28)

Expected Customer delay E(W) should be LOW;

(29)

Actual customer mean delay W' should be within a FEW percentage points of a desired value E0. (30) Clearly (25), (26) and (27) are abstract expressions of decision-making in operational research methodology, whereas (28), (29) and (30) are linguistic expressions of decision-making

akin to real-life situations. A decision model should relate, c, p and E(W) to the determination of spares n as (c, o = ~lc~, E ( W ) } - * n .

In order to develop a fuzzy expert system model of the spare parts problem, dependent only on ordinal scale database, we identify the linguistic variables and their linguistic values with the help of the OR model builder as follows: First, we treat all the system variables as linguistic variables: c,E(W), p, and n. Secondly we determine the linguistic values of these linguistic variables: For this study, the linguistic values of the linguistic variables are determined as follows: c:

VS (Very Small), S (Small), RS (Rather Small), M (Medium), RL (Rather Large), L (Large), VL (Very Large). E(W): VS (Very Short), S (Short), M (Medium). p: L (Low), M (Medium), H (High), VII (Very High). n: VS (Very Small), S (Small), RS (Rather Small), M (Medium), RL (Rather Large), L (Large), VL (Very Large). For an example, the definition of membership

19

I.B. Turk.sen / Fuzzy expert systems for IE/OR/MS

Table 8. Membership functions of c for spare parts VS (Very Small)

(1/0.1, 0.5/0.15, 0/0.2, 0/0.25, 0/0.325, 0/0.4, 0/0.45, 0/0.5, 0/0.55, 0/0.6, 0/0.65, 0/0.7, 0/0.75, 0/0.8, 0/0.9)

S (Small)

(0/0.1, 1/0.15, 0.75/0.2, 0.5/0.25, 0.1/0.325, 0/0.4, 0/0.45, 0/0.5, 0/0.55, 0/0.6, 0/0.65, 0/0.7, 0/0.75, 0/0.8, 0/0.9)

RS (Rather Small)

(0/0.l, 0/0.15, 0/0.2, 0.5/0.25, 1/0.325, 0.5/0.4, 0/0.45, 0/0.5, 0/0.55, 0/0.6, 0/0.65, 0/0.7, 0/0.75, 0/0.8, 0/0.9)

M (Medium)

(0/0.1, 0/0.15, 0/0.2, 0/0.25, 0.1/0.325, 0.5/0.4, 0.7/0.45, 1/0.5, 0.7/0.55, 0.5/0.6, 0.1/0.65, 0/0.7, 0/0.75, 0/0.8, 0/0.9)

RL (Rather Large)

(0/0.1, 0/0.15, 0/0.2, 0/0.25, 0/0.325, 0/0.4, 0/0.45, 0/0.5, 0.1/0.55, 0.5/0.6, 1/0.65, 0.5/0.7, 0.1/0.75, 0/0.8, 0/0.9)

L (Large)

(0/0.1, 0/0.15, 0/0.2, 0/0.2, 0/0.25, 0/0.325, 0/0.4, 0/0.45, 0/0.5, 0/0.55, 0/0.6, 0.1/0.65, 0.5/0.7, 0.7/0.75, 1/0.8, 1/0.9)

VL (Very Large)

(0/0.1, 0/0.15, 0/0.2, 0/0.25, 0/0.325, 0/0.4, 0/0.45, 0/0.5, 0/0.55, 0/0.6, 0/0.65, 0/0.7, 0.1/0.75, 0.5/0.8, 1/0.9)

In order to show the effectiveness of the fuzzy expert system, we have simulated both the analytical stochastic model and the fuzzy expert system. We executed 20 simulation experiments. We show seven of these simulation results in Table 10. We tested two null hypotheses that: "there is no significant difference between the analytical results and the fuzzy expert system results" when the fuzzy expert rules are fixed at the re-threshold of (i) rl = 0.9 similarity, and (ii) r2--0.5 similarity. For both cases, the A N O V A computations show that the hypotheses can not be rejected at Fo.os, i.e., F = 0.02 < Fo.o5(38, 1). It is clearly observed that the results of the fuzzy expert system are approximately similar to that of the analytical stochastic model. However, we should iterate the two advantages of the fuzzy expert system

functions for c are shown in Table 8 and in Figure 5. Based on extensive analysis of the stochastic model, the structure of the rules of this system is determined as: iv A = (A1, A2, A3) THEN B

=

B1

where A ~ = n u m b e r of servers, A 2 = d e s i r e d customer delay, A 3 = s y s t e m utilization, B~ = number of spares required (details may be found in [33]). It should be clear that there should be 84 rules in rule base. A subsection of this rulebase is shown in Table 9. For example: Rule # 2 6 is read as follows: IF number of servers is RATHER LARGE AND customer mean delay is MEDIUM AND the utilization is HIGH, THEN the number of spares is RATHERSMALL. MF VS

S

RS

M

RL

L

VL

0.5 (NORMALIZED) o

o.1 0.2 o,3 0.4 0.5 0.6 o.7 o.a 0.9 1.o

NUMBER C/C

OF

SERVERS

MAX

Fig. 5. Membership function of C/Cm~,: MF(L) = {0/0.1, 0/0.15, 0/0.25, 0/0.325, 0/0.4, 0/0.45, 0/0.5, 0/0.55, 0/0.6, 0.2/0.65, 0.5/0.7, 1/0.75, 1/0.8, 1/0.9}.

I.B. Turksen / Fuzzy expert systems for IE/OR/MS

20

Table 9. A subset of the rule base for spare parts Rule No.

C

E(W)

p

n

½4 25 26 27 28 29 30 31 32 33 34

L RL RL RL RL RL RL RL RL RL RL

VS M M M M S S S S VS VS

L VH H M L VH H M M VH H

S RL RS S VS WL M RS S VVL RL

model: (1) ordinal data is required as opposed to absolute scale data; and (2) rules are expressed with linguistic values of linguistic variables in a

manner very close to natural language expressions of management. Hence, fuzzy expert systems demand least cost information but provide effective user friendly interface closing much of the communication's gap between management and OR model builders.

4. 4. Conjoint analysis Conjoint analysis is widely used in Marketing, i.e., in new product planning and design, pricing, competitive analysis, market segmentation and advertising research. Existing crisp methods have at least the following disadvantages: (a) Estimation must be done across subjects and large numbers of attributes (8) and products (25) must be rated by subjects. (b) Prediction results vary little across models or estimation methods.

Table 10. A sample of experiment results for spare parts

Variable name

Crisp values of variables

Results based on analytical model

FES results* n (tnrf)

n (range of n)

'r I

~'2

1 c E o, [1 + 0.2]E o p

10 0.24, [0.19, 0.29] 0.37

3 (2-5)

3 (2)

4 (9)

2

c E o, [1 + 0.2]E o p

17 0.14, [0.11, 0.17] 0.45

6 (4-9)

5 (4)

5 (4)

c E o, [1 ± 0.2]E o p

10 0.50, [0.40, 0.60] 0.74

6 (4-9)

7 (2)

8 (9)

e Eo, [1 + 0.2]Eo p

16 0.13, [0.11, 0.16] 0.60

10 (8-12)

9 (4)

9 (4)

c E0, [1 -1-0.2]E0 p

10 0.13, [0.10, 0.16] 0.81

11 (8-13)

10 (4)

9 (18)

c E o, [1 + 0.2]E o p

16 0.26, [0.21, 0.31] 0.40

5 (3-6)

4 (2)

4 (2)

c

10 0.22, [0.18, 0.26] 0.60

5 (4-7)

5 (2)

5 (7)

3

4

5

6

7

eo, [1 + 0.2lEo p

n: number of spares. tnrf: total number of rules fired. rt: the threshold for similarity measure is set to 0.9. r2: the threshold for similarity measure is set to 0.5.

21

LB. Turksen / Fuzzy expert systems for IE/OR/MS (c) There are no representation techniques for linguistic variables in existing models. (d) Complex models are needed to respond to changing choice environments; however, the characteristics of subjective assessment data may not satisfy measurement requirements of cardinal scales that are essential for the existing crisp methods. Therefore, there is opportunity to address these problems using: (a) an appropriate representation of vague linguistic ratings, i.e., subjects' response information; (b) new individual-level models based on sound measurement principles that can be fully 'estimated' and adjusted within a typical interaction with the subject; (c) interactive experimental software that uses only salient attributes, appropriate descriptions and range~, of attribute values; and (d) more m ~ n g f u i prediction tests beyond just first choice. We have proposed fuzzy sets and logic based models to address these problems [36] by: (a) modifying an existing crisp model with fuzzy set representation, and (b) developing a fuzzy expert system model. In order to compare our results, we have also investigated the crisp conjoint model that we fuzzified. Thus, we have investigated and compared the results of three models: (1) a crisp conjoint model (2) a fuzzy conjoint model, and (3) a fuzzy expert system model.

with W~ a directly elicited crisp attribute weight for the i-th attribute over all the products; l~8,(yj, m) a computed membership degree estimation of a linguistic value assignment B' for a holdout product m over each of its domain values, yj = 1, 2 . . . . . k; l~a~(xj, m) is a membership degree of a subjects' linguistic rating A i e {Very Poor, Poor, Somewhat Poor, Neutral, Somewhat Good, Good, Very Good} for the i-th attribute of a holdout product m over each of its domain values xj = 1, 2 . . . . . k; and k is a parameter that defines the number of domain values to be used in an experiment for the definition or elicitation of membership functions, e.g., k = 7 or 15. Furthermore, the membership values IZa,(Xj, ") could be either standard, i.e., pre-defined to be selected by subjects for assignment to each holdout product m or they could be elicited directly from each subject. An example of a fuzzy conjoint calculation is given below in order to clarify these symbols and their combinations. Suppose the pre-defined fuzzy sets for the linguistic values are given for k = 7 as follows: Very Poor = {(1, 1), (2, 0.7), (3, 0.2), (4, 0.1), (5, 0), (6, 0), (7, 0)}, Poor = {(1, 0.6), (2, 1), (3, 0.6), (4, 0.3), (5, 0.1), (6, 0), (7, 0)}, Somewhat Poor = {(1,0.2), (2, 0.7), (3, 1),

4. 4. 1. Crisp conjoint model Here

(4, 0.7), (5, 0.2), (6, 0.1),

(7, 0)},

4

y(m) = ~ Wi" ei(m) + Wo,

(31)

i=1

with W/ the estimated weight for the i-th attribute, i = 1. . . . . 4; ei(m) the i-th attribute evaluation on a 1 to 7 crisp scale for a holdout product m; and y ( m ) is the estimated rating of a holdout product M. This is generally known as the vector-weight model where attribute weights W~ are estimated by ordinary least squares regression.

=

Wiwi~Ai(Xj,m),

~ 24=1

i=1

(4, 1), (5, 0.7), (6, 0.1), (7, 0)}, Somewhat Good = {(1, 0), (2, 0.1), (3, 0.2), (4, 0.7), (5, 1), (6, 0.7),

(7, 0.2)}, Good = {(1, 0), (2, 0), (3, 0.1), (4, 0.3), (5, 0.6), (6, 1), (7, 0.6)}, Very Good = {(1.0), (2, 0), (3, 0), (4, 0.1), (5, 0.2), (6, 0.7), (7, 1)}.

4. 4. 2. Fuzzy conjoint model Here Iza'(Yi, m)

Neutral = {(1, 0), (2, 0.1), (3, 0.7),

(32)

Furthermore, suppose that a subject's ratings for each attribute A1, A2, A3 and A4 of a holdout product m are 'Somewhat Poor', 'Somewhat

LB. Turksen / Fuzzy expert systems for IE/OR/MS

22

Good', 'Good', 'Somewhat Good', respectively. If the subject's overall product rating for the holdout product is 'Somewhat Good', and his directly elicited weights for the attributes Ai, i = 1, 2 . . . . ,4, for all products are given as 3, 5, 7, 6 respectively, then

similar fuzzy set is found to be 'Somewhat Good'. Finally, if the subject's overall evaluation if 'Somewhat Good', then this is a potential prediction.

Wl = ~1,

(33) with B* the computed fuzzy set; A* =A~' ANO A~ AND A~ AND A~ is the combined rating of a subjects' individual ratings for each of the four attributes for a holdout product m; A---)B is the rule whose left hand side A =A1 AND A2 AND A 3 AND A 4 is the combined rating of subjects' ratings for each of the four attributes using the estimation products; and B is the right-hand side of the rule determined during the estimation analysis. Naturally, for the computational purposes, the compositional rule of inference is re-written for the membership domain expressions either in the point-valued fuzzy sets representation as discussed in the previous section or in the interval-valued fuzzy sets representation [28]. In this study interval-valued fuzzy sets were implemented as in most of our studies considered earlier [with the exception of a point-valued defuzzification method used in the second scheduling project discussed above].

W2=~l,

W3=h,

W4 = ~1.

Here B* =A*o(A---) B )

Next, membership degrees of the assigned linguistic values for each of a subject's ratings, are re-scaled by these weights; i.e., AI: multiply with 3 each membership the fuzzy set 'Somewhat Poor', Az: multiply with ~ each membership the fuzzy set 'Somewhat Good', A3: multiply with 7 each membership the fuzzy set 'Good', A4: multiply with 6 each membership the fuzzy set 'Somewhat Good'.

4. 4. 3. Fuzzy expert system model

value of value of value of value of

Thus, we obtain: A 1 = {(1, 0.029), (2, 0.1), (3, 0.14), (4, 0.04), (5, 0.029), (6, 0.014)}, A2 = {(2, 0.024), (3, 0.48), (4, 0.167), (5, 0.238), (6, 0.167), (7, 0.048)}, A3 = {(3, 0.033), (4, 0.1), (5, 0.2), (6, 0.33), (7, 0.2)}, A4 = {(2, 0.029), (3, 0.057), (4, 0.2), (5, 0.286), (6, 0.2), (7, 0.057)}. Next, the membership degrees of each attribute are added separately for each value of the domain set xj to determine the membership degree for each value of the domain set of yj. Thus, we get: B ' = {(1, 0.029), (2, 0.153), (3, 0.278), (4, 0.507), (5, 0.753), (6, 0.711), (7, 0.305)}. In order to determine the linguistic value to be assigned to B' we compute similarity (1/(1 + dist.)) to each of the 7 pre-defined sets representing allowable linguistic values. For example, if the Euclidian distance is used, we find that the similarities of B' to the fuzzy set 'Very Poor' is 0.279, to 'Poor' is 0.311, to 'Somewhat Poor' is 0.379, to 'Neutral' is 0.528, to 'Somewhat Good' is 0.893, to 'Good' is 0.774, and to 'Very Good' is 0.486. Thus, the most

4. 4. 4. Data used in models

For a realistic comparison, data used in models has the following characteristics. (a) Crisp and Fuzzy models all used the identical data, subject attribute/overall evaluations on a 1 to 7 scale, with linguistic terms defined as responses. (b) Evaluation scale values are: Very Poor (1), Poor (2), Somewhat Poor (3), Neutral (4), Somewhat Good (5), Good (6), Very Good (7). A fuzzy set is defined for each linguistic value. (c) Identical scale is used for both attribute and overall evaluations, with identical set elements in domain variables X, Y and identical membership functions for a given set for all attribute and overall evaluations. (d) There were eighteen sample products that are used for estimation; and six holdout products to test the predictive power of each model. The computations are carried out in the following manner.

23

I.B. Turksen / Fuzzy expert systemsfor IE/OR/MS

4. 4. 5. Conjoint experiment results Among other things, we investigated the following two issues important from the perspective of fuzzy set theory and fuzzy expert system design and development: (1) Comparison of the three models: (a) crisp conjoint model, (b) fuzzy conjoint model, and (c) fuzzy expert systems model. (2) Effects of the fuzzy set definitions: (a) pre-defined sets, (b) elicited sets, and (c) set size effect. Our investigation was carried for the consumer preference in both the pizza selection and the compact car selection. The results are summarized in Table 11. It is observed that: (1) Fuzzy conjoint (FC) prediction is very

(1) Crisp Conjoint Model: A crisp number is computed for the overall evaluation based on the attribute evaluations (1-7) and derived weights (from regression on estimation products). (2) Fuzzy Conjoint Model: An overall evaluation set is computed as a convex linear combination of the sets defined for each attribute evaluation, e.g., GOOD, using the crisp weights to re-scale membership for each attribute set. (3) Fuzzy Expert System Model: (a) A rule base is formed from the response to the 18 estimation products, i.e., attribute evaluations imply overall evaluation. (b) Closest rule(s) to input state A* were found for the holdout product rated by a subject using the analogy of similarity. B* is computed from the rule(s) fired via GMP.

Table 11. Summary of prediction results (Overall results (Pizza+ Car);

n = 50) Prediction

Crisp Conjoint

Fuzzy Conjoint

Fuzzy Fuzzy Conj. El. Rules

Fuzzy Rule El.

1st choice (% subjects)

25 50%

41 82%

40 80%

31 62%

24 48%

1st & 2nd choice lst-3rd choices lst-4th choices lst-5th choices lst-6th choices

12 4 2 1 1

29 17 9 8 4

27 18 6 1

26 16 5 2 1

13 8 6 4 2

Sum of choices

45

108

96

81

57

Weighted sum

80

250

198

167

130

138%

107%

71%

24%

2.16

1.92

1.62

1.14

4.53 (0.0001)

3.69 (0.0006)

2.51 (0.015)

0.82

% Improvement Prediction mean T-value of mean (fuzzy-crisp) (p<).

0.90

Notes: Crisp: Crisp conjoint, vector-weight formula, with individually estimated attribute weights using 18 estimations products and OLS regression. Fuzzy Conjoint: Fuzzy conjoint method using pre-defined sets as per all previous studies, with cut-off determined on estimation products. Fuzzy Conj. El.: Above using elicited set based only on individual subject responses with direct set elicitation methods. Fuzzy Rules: GMP (Generalized Modus Pones) performed on individual fuzzy rule base composed only of estimation product ratings using identical pre-defined sets as per previous studies and fuzzy conjoint methods. Fuzzy Rules El.: GMP as above using elicited sets based only on individual subject response with direct set elicitation methods.

24

I.B. Turksen / Fuzzy expert systems for IE/OR/MS

Table 12. Set size effects for fuzzy conjoint analysis Study

1-elem. sets

3-elem. sets

7-elem. sets a

15-elem. setsa

1. Pizza 1 (1989) (n = 32) 1st Choice 14 Sum of Choices 23 Weighted Sum 44

20 44 85

24 67 152

24 68 155

2. Pizza 2 (1989) (n = 32) 1st Choice 8 Sum of Choices 19 Weighted Sum 38

14 32 65

17 50 117

17 49 115

3. Overall (Pizza + Car) (1991) (n = 50) 1st Choice 31 37 Sum of Choices 54 78 Weighted Sum 94 154

41 88 174

41 88 174

Mean Prediction:

1.692

1.692

2.43 (0.01) 0.655

2.41 (0.01) 0.652

1.04

1.50

T-value for difference in means (p<): 1-element sets to: 1.70 (0.03) 3-element sets to:

Example of 'Somewhat Good' (5), defined in 4 levels of detail over domain of possible scale responses/linguistic terms 1-7, and intermediate points: 1-element set; {(1,0), (2, 0), (3, 0), (4, 0), (5, 1.0), (6, 0), (7, 0)}. 3-element set: {(1,0), (2, 0), (3, 0), (4, 0.70), (5, 1.00), (6, 0.70), (7, 0)} 7-elementset: {(1,0), (2,0.10), (3,0.20), (4,0.70), (5,1.00), (6,0.70), (7, 0.20)} 15-elementset: {(0.5,0), (1,0), (1.5,0.05), (2,0.10), (2.5,0.15), (3,0.20), (3.5, 0.45), (4, 0.70), (4.5, 0.85), (5, 1.00), (5.5,0.85), (6, 0.70), (6.5,0.45), (7,0.20), (7.5, 0.15)}

m u c h b e t t e r than fuzzy e x p e r t system ( F E S ) prediction which is m u c h b e t t e r than crisp conjoint (CC) prediction. (2) T h e first choice of six h o l d o u t p r o d u c t s are 82%, 62% and 50% for FC, F E S and C C , respectively. (3) the overall i m p r o v e m e n t s of FC and F E S o v e r C C are 138% and 7 1 % , respectively. (4) G e n e r a l l y , pre-defined sets gave a higher rate of prediction with F C m o d e l in c o m p a r i s o n to elicited sets (except pizza, 88% with the elicited sets). In T a b l e 12, we s u m m a r i z e the effect of set size definition. It is found that the predictive p o w e r i m p r o v e d as we increased the n u m b e r of e l e m e n t s in the d o m a i n of fuzzy set definitions from 1 to 7 but no a p p r e c i a b l e i m p r o v e m e n t was o b t a i n e d when we increased the d o m a i n e l e m e n t s f r o m 7 to 15 points. E x a m p l e s of 7- and 15-element fuzzy set definitions are shown in Figures 6 and 7.

5. Conclusions Second g e n e r a t i o n of expert systems for I E / O R / M S could be designed with: (1) the expressive p o w e r s of fuzzy sets that i m p r o v e knowledge r e p r e s e n t a t i o n and close the c o m munications' gap b e t w e e n I E / O R / M S model builders and m a n a g e r s of o p e r a t i o n a l systems, and (2) the effective inference m e t h o d s of fuzzy logic which p r o v i d e ' g o o d results' with h u m a n like reasoning capability and intelligence, in the sense of, not requiring an exact m a t c h b e t w e e n the left-hand side of a rule and an o b s e r v e d system state which is a must for the first generation expert systems based on t w o - v a l u e d logic. F u r t h e r m o r e , as a result of o u r studies, we suggest that such fuzzy expert systems should be i m p l e m e n t e d either (1) w h e n the cost of precise information acquisition, in the sense of absolute scale i n f o r m a t i o n , outweighs the m a r g i n a l

I.B. Turksen / Fuzzy expert systemsfor IE/OR/MS D E G R E E O F M E M B E R S H I P

25

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1

2

3

4

5

15 elements ( d o m a i n = s c a l e

~-

VERY POOR I POOR S. GOOD O GOOD

6

7

responses)

+ S. POOR -13- NEUTRAL - ~ - VERY GOOD

(S.=SOMEWHAT) (4-6 non-zero elements) SET SIZE EFFECT (2 of 4)

Fig. 6. 7-element pre-defined fuzzy sets; one set per linguistic term (subject 1, category pizza). D E G R E E O F M E M B E R S H I P

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1

2

3

4

5

6

7

15 elements (domain - scale responses) VERY POOR-d--POOR -x-- S.GOOD - O - GOOD

~ ~-

S. POOR t3 VERY GOOD

NEUTRAL

(S.=SOMEWHAT) (9-13 non-zero elements) SET SIZE EFFECT (1 of 4)

Fig. 7. 15-element pre-detined fuzzy sets; one set per linguistic term (subject 1, category pizza).

benefits to be gained by restructuring a model based on ordinal scale information to that based on absolute scale information, or (2) when the information is inherently vague in the minds of its users, such as consumer information and supervisory scheduling control information. To illustrate these two categories of possibilities, we selected two examples for each category. The aggregate production and spareparts centre models demonstrate the effectiveness of fuzzy expert systems in the first category. While the job shop scheduling and conjoint analysis for market share determination

demonstrate the effectiveness of the second category. It should be once again emphasized that classical IE/OR/MS models are valuable in gaining insights into particular domains of behaviour under various restricted assumptions that usually require absolute scale data and information for the use of arithmetic operations of plus ( + ) and times (-). However, with knowledge obtained from such models together with their builders, we can develop and implement fuzzy expert systems that are natural language based and exhibit a human-like

26

LB. Turksen / Fuzzy expert systems for IE/OR/MS

reasoning capability while providing approximately 'good results' as we demonstrated above with fuzzy expert systems for the aggregate production planning and spare-parts centre planning. In the cases of job shop scheduling and consumer choice modeling, we have attempted to show that fuzzy logic base models produce much better results than classical I E / O R / M S and AI based models. Finally, we have suggested that the success of these last two models are due to inherent vagueness of their information sources.

Acknowledgements The design, development and analysis of fuzzy expert system models for I E / O R / M S took a period of six years and much effort contributed by some of my graduate students, including but not limited to, T. Bilgic, K. Demirli, Y. Modi, Y. Tian, D. Ulguray, Q. Wang, T. Yurtsever, Z. Zhong, and I. Willson, a recent Ph.D., and my colleague Dr. M. Berg. The typing of the manuscript and the preparation of Tables and Figures and their revision took many hours and was done by Mrs M. Mitake, Mrs L. Kung, Mr. H. Narazaki, and Mr. H. Sugawara. The author is indebted to all.

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