PII: 0888-613X(91)90010-J An Expert System Prototype for Inventory Capacity Planning: An Approximate Reasoning Approach I. B. Turksen Department of lndustrial Engineering, University of Toronto, Toronto, Ontario M. Berg Department of Statistics, Haifa University, Haifa, Israel A B S T R A C T A n approximate reasoning f r a m e w o r k is suggested f o r the d e v e l o p m e n t o f an expert system p r o t o t y p e to aid m a n a g e m e n t in planning inventory capacities. The development is considered to be a stage that comes after the analysis o f a stochastic m o d e l Such a m o d e l would p r o v i d e the requisite insight and knowledge about the inventory s y s t e m under specific assumptions. A s a consequence, the model builder(s) would act as expert(s). The restructuring process f r o m the stochastic m o d e l into the approximate reasoning f r a m e w o r k is described in a case s t u d y analysis f o r a M a r k o v i a n p r o d u c t i o n model. The stochastic m o d e l considers a relatively simplified p r o d u c t i o n process: one machine, constant production rate, a c o m p o u n d Poisson d e m a n d process f o r the p r o d u c t together with the reliability f e a t u r e comprising the machine f a i l u r e process and the ensuing repair action. In this context, the authors p r o p o s e an approximate reasoning f r a m e w o r k and describe (1) the identification o f the managerial decision-making rules, which usually contain uncertain (vague, ambiguous, f u z z y ) linguistic terms; and (2) the specification o f m e m b e r s h i p f u n c t i o n s that represent the meaning o f such linguistic terms within context-dependent domains o f concern. They then define a new Address correspondence to L B. Turksen, Department o f Industrial Engineering, University o f Toronto, Toronto, Ontario, Canada M5S 1A4. International Journal of Approximate Reasoning 1991; 5:223-250 © 1991 Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas, New York, NY 10010 0888-613X/91/$3.50 2 2 3 224 I.B. Turksen and M. Berg universal logic incorporating these rules and functions and apply it to inventory capacity planning. Two case examples and a simulation experiment consisting o f 21 cases are summarized with a discussion o f results. KEYWORDS: e x p e r t s y s t e m , k n o w l e d g e acquisition, a p p r o x i m a t e reason- ing, i n v e n t o r y capacity p l a n n i n g , s i m u l a t i o n e x p e r i m e n t s 1. I N T R O D U C T I O N The purpose o f this paper is to suggest a possible approach to bridging the communication gap between stochastic model builders and the managers o f inventory capacity planners. We use an approximate reasoning framework based on fuzzy logic for the development o f an expert system prototype that can serve as an aid to managerial decision making in manufacturing. It is generally known that although stochastic models provide valuable insights into a system's behavior under specific assumptions and identify useful guidelines, more often than not they are not implemented, either because managers do not understand the basic assumptions o f the model, or because they are too complex, or because the precise information required to determine the values o f the model parameters cannot be obtained or are not available. Hence, such models are, b y default, inadequate to help managers to cope with the natural behavior o f many real-life systems. However, stochastic model builders are often experts who can interpret the results o f their models and can express their insightful expert knowledge about such system behavior in a natural language that usually contains vague, ambiguous, uncertain, fuzzy linguistic terms. Such linguistic terms provide (1) a flexible expression o f 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. Such concerns lead us to suggest the development o f expert system proto- types via an approximate reasoning framework based on fuzzy logic but relying on insights obtained from stochastic models with the model builders acting as the experts. In order to explain in detail how such a development would and could take place, we first summarize briefly both the approximate reasoning framework and the stochastic model under consideration in this paper. Approximate Reasoning Informally, approximate reasoning is the process or processes by which a possible imprecise conclusion is deduced from a collection o f imprecise premises (Zadeh [1]). A number o f alternative approaches are available for reasoning in the design o f knowledge-based systems (Turksen [2], Turksen and Inventory Capacity Planning 225 Zhong [3]). Before we discuss the details o f an alternative approach, let us first review the structure of a possible knowledge-based system that may be used in prototype studies. (Clearly, the following is just a skeleton; other relevant aspects of the development of such a system are omitted in this paper.) 1. Factual knowledge (observed system state) ("facts"): X t is A* AND X 2 is A T A N D . . - AND X n is A* (1) for short: A* = A* AND A* A N D . . - AND A* The user o f an expert system inputs the factual knowledge, choosing the appropriate linguistic terms A T, A* . . . . A* that describe the observed states o f the system. Users may be given the option to state the meaning o f these linguistic terms by specifying membership values a n d / o r functions. Alternatively, they may be given the option to rely on the definitions provided by the domain experts. 2. Rule base (expert rules) ("rules"): I F X 1 is A l A N D X 2 is A 2 A N D . . . A N D X n is A n, T H E N Y is B (2) for short: A ~ B , where A = A l A N D A 2 A N D . . . A N D A n Domain experts (for this discussion, stochastic model builders) pro- vide such rules during the knowledge acquisition phase, where A~, A 2 . . . . . A n and B are the appropriate set o f linguistic terms specified by the domain experts. The domain experts must also provide the meaning o f these linguistic terms by specifying their membership values a n d / o r functions over the domain o f discourse. Rules are then encoded into the knowledge base by a knowledge engineer during the design and development phase. 3. Expert system advice ("response"): Y is (should be) B* (3) The expert system " r e s p o n s e " is provided to a user after the inference subsystem operates on the " r u l e s " and the given " f a c t s " in accordance with an inference scheme. In the current literature o f fuzzy sets, it may be observed that the meanings o f linguistic terms and their logical combinations are generally represented by " p o i n t - v a l u e d " membership assignments. However, it has been observed that 226 I.B. Turksen and M. Berg membership values obtained from domain experts and their logical combina- tions usually turn out to be " i n t e r v a l - v a l u e d " in most measurement experi- ments (Turksen [ 4 - 6 ] , Chameau and Santamarina [7]). In this paper we start out with a knowledge representation scheme based on point-valued membership functions obtained from experts (the stochastic model builders). That is, A 1, A 2 . . . . . A n, B; A T, A* . . . . . A* will all be repre- sented by point-valued membership functions. However, the representation o f the logical combination o f AND and the logical implication I F . . . T H E N will be computed with the disjunctive and conjunctive normal forms, DNF and CNF, respectively. These representations and computations will be further discussed in appropriate sections in the remainder o f the paper. FRAMEWORK The basic approximate reasoning framework used in this paper consists essentially of the following: (i) An interval-valued fuzzy set representation o f " r u l e s " and " f a c t s " (ii) A search for the " r u l e s " closest to the " f a c t s " based on a similarity measure (iii) An inference based on the implication o f the " r u l e s " found in (ii) and the " f a c t s " (iv) An advice based on the linguistic approximation o f the result(s) found in (iii) The details o f approximate reasoning methodology and its framework may be found in Turksen [2, 4 - 6 , 8, 9]. S t o c h a s t i c M o d e l Production/inventory/reliability models have been at the focus o f produc- tion modeling for a long time now. The rule o f inventory is to accommodate fluctuations from the demand side, and the fundamental problem is to relate the production process to the demand process so that, on the one hand, shortages are kept at a desirable low level but, on the other hand, no excessive inventory is built up. An important but frequently neglected element in such systems is the imperfection o f production systems. Posner and Berg [10] studied a model that incorporates reliability (or, indeed, unreliability) factors into the analysis and obtained closed-form analytical results under certain assumptions on the nature of the randomness in the production inventory system and the reliability factors. MODEL ASSUMPTIONS We consider a Markovian production model in which a single machine produces items at a constant production rate 1 (without loss o f generality) up to a level N , the inventory capacity. The production is halted whenever the inventory level reaches N and is resumed at the next demand epoch. Inventory Capacity Planning 22"/ The demand process is compound Poisson: Demands arrive according to a Poisson process at rate X, and order sizes are i.i.d, random variables exponen- tially distributed with a mean /x -~. There is no backlogging o f demand, so excess demand beyond the available inventory is lost. The operating time o f the machine is exponentially distributed with parameter 0. Thus 0 is the failure rate of the operating machine. The repair time o f the failed machine is exponentially distributed with mean a -~. ANALYTICAL RESULTS Let X ( t ) be the inventory level at time t, and set W ( t ) -= N - X ( t ) to be the slack storage capacity at time t. Clearly, W ( t ) [O, N ] . We introduce the variable W ( t ) because to deal with X ( t ) directly is less convenient than to deal with W ( t ) , but at the same time these two variables are equivalent to our understanding o f the behavior o f this system. The limiting density and distribution function of W = l i m t ~ o W ( t ) are f ( . ) and F ( - ) , respectively. Since the knowledge of W alone does not indicate whether or not the machine is in repair, a supplementary variable approach is implemented through the generalized technique o f " s y s t e m p o i n t " in level-crossing analysis (Brill and Posner [11, 12]). W should be partitioned into two parts, W o and W 1, where W 0 indicates the portion o f W while the machine is not under repair, and W~ is the portion of W while the machine is under repair. Correspondingly, both f ( . ) and F ( . ) should be partitioned into two parts, denoted by f o ( ' ) , f l ( ' ) and F0(.), F I ( . ) , respectively. Naturally we have the following two equations: f ( ' ) = f o ( ' ) + f , ( ' ) F ( ' ) = Fo(- ) -t- F1(" ) It is to be observed that aside from the partial densities with respect to the state of the machine, there is also a probability mass f o - Pr(W0 = 0) associated with a full inventory accompanied by production shutoff, and f ~ = P r ( W 1 = N ) associated with an empty inventory and no active produc- tion. The result of the mathematical analysis due to Posner and Berg [10] is summarized in the Appendix. The inventory capacity level N is the decision variable through which a desirable level of " t h e fraction o f satisfied customer d e m a n d " p can be achieved. For computational convenience we define a surrogate decision variable. N - N o v - (4) No 228 I . B . Turksen and M. Berg where N O is the existing inventory capacity. V is thus the fraction o f increase in the existing inventory capacity needed to achieve P0, the desirable level o f p. F o r specificity, we shall set Po = 0.95 and let N O -- 1.5. Setting N O --- 1.5 means an inventory capacity o f 1.5 " l o t s , " where a lot corresponds to a prespecified number o f items. This unit is also used in the definition o f # - t, the mean size o f demand. Another basic unit here is the unit o f time that is used in the definition o f the parameters X, a ~, and 0. The production rate is based on both units and their standardized form, which corresponds to produc- ing 1 lot per unit o f time. K N O W L E D G E A C Q U I S I T I O N The analytical result provides us with a thorough understanding o f this production/inventory/reliability system. It is, however, not very practical from the point o f view o f management. The exactness o f the system parameters is unnecessary to the managers, and furthermore it does not provide managers with insights into the operation o f this system, which is o f vital importance to the effective management o f such a complex system. Many analytical models bear the same language barriers. Even though these models clearly depict the characteristics o f a system for the well-trained model builders and their results could be effectively interpreted b y their builders, they are usually inappropriate for managerial implementation and use. An important, and at times critical, step in the design and development o f an expert system prototype is knowledge acquisition in the f o r m o f rules and their components. Preliminary Considerations In accordance with the analytical model, we have decided to use the fraction o f satisfied customer demand p as our performance criterion and the inventory capacity N as the decision variable, with other parameters tacitly assumed to be outside our control. Furthermore, in agreement with the approximate reasoning rule f o r m (2), we opt for the following rule structure: I F X i s A I A N D / ~ - 1 i s A2 A N D 0 i s A3 A N D a - i is Z 4 A N D p is high T H E N take action B (with respect to N or, equivalently, V ) The Ai, i = 1 . . . . . 4, and B stand for sets o f relevant linguistic descrip- tors. A basic set o f descriptors for each o f the A i in this case is low, Inventory Capacity Planning 229 medium, high (Turksen [2]). (If necessary, this categorization can be made finer by adding linguistic descriptors such as very low o r moderately high, as shown in the simulation experiments.) Thus, an e x a m p l e o f the rules w e aim to construct is: I f the d e m a n d arrival rate )x is low, and the m e a n d e m a n d size /x i is medium, and the failure rate o f the m a c h i n e 0 is low and its repair rate a - ~ is high, then in order to have a high fraction o f satisfied d e m a n d p, increase N moderately (or, equivalently, set a moderate value f o r V). The linguistic descriptors such as low, medium, and high are imprecise, but these are terms that a m a n a g e r not only understands but is also generally willing to give a m e a n i n g representation to by specifying a m e m b e r s h i p function ( Z y s n o [13], T u r k s e n [ 4 - 6 ] , N o r w i c h and T u r k s e n [14], Z i m m e r m a n and Z y s n o [15]). I n contrast, to use the exact model (i.e., the stochastic model) as is, the m a n a g e r w o u l d be required to substitute exact figures f o r the parameters - - a responsibility he o r she m a y be reluctant to assume because o f the ever-acute shortage o f data and the often encountered statistical inference difficulties, and so on. W e thus sacrifice some o f the exactness o f the original model in order to turn it into an implementable d e c i s i o n - m a k i n g tool w h e n e v e r w e are c o n f r o n t e d with either a lack o f data o r a limited a m o u n t o f data. Indeed, the exact analysis can be v i e w e d in this f r a m e w o r k as a building b l o c k in the construc- tion o f this m a n a g e r a l aid (the expert system) w h e r e the other building blocks relate to the qualitative/quantitative interpretations o f the linguistic descriptors as represented b y the c o r r e s p o n d i n g m e m b e r s h i p functions and inferencing methods, and so on. T h e detailed d e v e l o p m e n t o f this p r o c e d u r e f o r the p r o d u c t i o n / i n v e n t o r y / r e l i a b i l i t y case u n d e r consideration is described in the next section. INTERPRETATION B e f o r e w e turn to the actual construction o f the rules, it is useful to review s o m e o f o u r interpretations regarding the regions o f their applicability. T h e p a r a m e t e r N , the i n v e n t o r y capacity, is the sole decision variable b y means o f w h i c h w e want to achieve a desirable level Po for the fraction o f satisfied c u s t o m e r d e m a n d . While p = p(N) is an increasing function o f N , it is not at all clear, o r indeed true, that any desirable P0 can be achieved by m e r e l y increasing N o r V [because w e m a y very well have p(oo) < Po]- This is likely to h a p p e n w h e n the effective production rate is small c o m p a r e d to the d e m a n d rate )~/x-l, b e c a u s e in this case i n v e n t o r y does not build up fast e n o u g h to allow c h a n g e s on N relevant to our goal o f reaching Po- (By effective p r o d u c t i o n rate w e mean the theoretical production rate 1 minus the lost p r o d u c t i o n rate due to machine unreliability.) At the other extreme, w e have the cases w h e r e even a m u c h smaller level o f 230 I . B . Turksen and M. Berg N than the present N O is good enough to achieve P0. Generally speaking, a very low demand rate relative to the effective production rate makes a relatively small N sufficient to generate a high enough p. When the demand rate is close to the production rate 1, the objective V and the action on N are very sensitive to the failure rate, particularly i f the mean repair time o - ~ is high, because a greater or smaller failure rate can make all the difference between adequate supply and the occurrence o f a shortage. Note that this statement refers not only to the expectation o f the shortage but also to its variability. Thus, in the Poisson process, which in our theoretical model characterizes the failure process, the mean number o f failure events per unit o f time is proportional to the variance o f this random quantity, and therefore a high failure rate can significantly increase the range o f required changes on N , thereby detrimentally affecting the stability o f the expert s y s t e m ' s responses. This undesirable effect can be avoided b y limiting the set o f rules to low failure rates. Such a restriction is justified b y the fact that machines with medium or high failure rates are, with a high possibility, not likely to be retained by the management. Consequently, rules involving medium or high failure rates will be eliminated f r o m the system development. Linguistic Descriptors In order to design and develop an expert system that operates with the principles o f approximate reasoning and p e r f o r m s a task heuristically equiva- lent to the analytical model considered earlier, we require that the model builders not only provide us with the linguistic descriptors that identify the aggregate patterns o f behavior but also provide us with " m e a n i n g representa- t i o n " for these linguistic descriptors. As suggested in the previous section these linguistic terms were identified as L , low; M , medium; H , high for each o f the system (independent) parameters; that is, Demand rate k A 1 Demand size # - 1 A 2 Failure rate 0 A 3 Repair rate o - l A 4 Turning to the set o f actions B on N (or, alternatively, on V) that will yield the desirable performance level p, we consider the following four possible Inventory Capacity Planning 231 actions: Low (L) Increase N a bit (which includes the case o f no increase at all) or, equivalently, set a low value for V. Medium (M) Increase N moderately o r , equivalently, set a moderate value for V. High (H) Increase N a lot or, equivalently, set a high value for V. Very high ( V H ) Increase N quite a lot or, equivalently, set a very high value for N . (The term " i n c r e a s e " is interpreted in these actions in percentage terms.) In practice, the meaning representations o f these membership functions are specified either by an expert o f the production system under consideration or by its manager, in accordance with universally accepted forms and on the basis o f relevant considerations (as delineated above for the failure rate). For this exercise, we have determined the membership functions in a cooperative effort between the operations research (OR) specialist and the knowledge engineer. Due to the context-dependent nature o f the curves, these functions need to be adjusted and modified for a given production system. For all parameters, we have kept the convention o f scaling the p a r a m e t e r ' s ranges into the [0, 1] interval by normalizing each base variable with its m a x i m u m in the following manner: Normalized demand rate: D R = k / k m a x , (5a) Normalized demand size: DS = # - l//Zmalx, (5b) Normalized failure rate: FR = 0/0ma x (5c) Normalized repair rate: RR -- a - J/Om-alx (5d) Normalized decision variable: DV = V~ Vma x ( 5 e ) where the m a x i m u m values are set to )kma x 1.0 - l = , ~tma x = 1.0, 0ma x = 0 . 0 5 , Un~alx --- 5 . 0 , Vma x = 7 . 0 MEMBERSHIP FUNCTIONS The notion o f crossover point (Zadeh [16]) with the membership value o f 1/2 plays an important role in determining the membership functions. F o r each variable, we must first identify the points that are significant enough to divide the interval o f the universal set [0, 1] into corresponding linguistic subintervals o f low (L), medium (M), and high (H) or very high (VH), varying according to different system states, in order to 232 I.B. Turksen and M. Berg T a b l e 1. Subintervals of Linguistic Terms Variable Low Medium High Very high DR [0, 0.3] (0.3, 0.7] (0.7, 1.0] DS [0, 0.3] (0.3, 0.7] (0.7, 1.01 RR [0, 0.3] (0.3, 0.8] (0.8, 1.0] FR [0, 0.1] (0.1, 0.6] (0.6, 1.0] DV [0,0.15] (0.15,0.35] (0.35,0.54] (0.54, 1.0] acquire meaningful membership functions for every linguistic term. Again these subintervals are identified by experts as shown in Table 1. It should be noted that the separation point between regions is the crossover point o f the two curves representing the membershp functions o f the two corresponding linguistic terms, and the membership grade (MG) o f this point has the largest uncertainty (Kosko [17]) for both o f these fuzzy sets. For example, consider DR. The point separating the low and medium subintervals is 0.3: MGLow(0.3 ) = MGMedium(0.3 ) = 0.5 An element o f a fuzzy set with a membership grade greater than 0.5 is more likely to belong to this set than not, while a membership grade less than 0.5 indicates that an element is less likely to belong to this set. For example, the interval [0, 0.3] o f DR is intended to be regarded as a low-level demand rate rather than a medium-level demand rate by experts. Therefore 0.3 is chosen as the point separating the low-level and medium-level intervals o f DR, and accordingly a membership grade o f 0.5 is assigned for the fuzzy sets o f both low DR and medium DR. The membership curves are shown in Figures 1 -5 . A different case is the fuzzy set o f Very High for the service level, as shown in Figure 5. Even though the VH curve does not intersect the H curve except at the extreme right side o f the figure, it still has a membership grade o f 0.5 at MG l L M H 1 0.0 ' - ~ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 1. Four membership functions of VL, L, M, H for DR. Inventory Capacity Planning 233 MG L M H 0.5 0.1 0.2 0.3 0.4 0.5 0,6 0.7 0.8 0.9 1.0 Figure 2. Four membership functions of L, M, AAM, H for DS. MG ' L M H 1 0.0 c Vc~.~ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 9 1.0 Figure 3. Four membership functions of L, M, LBH, H for RR. MG L M H 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 4. Three membership functions of L, M, H for FR. point 0.54, which separates intervals o f H and VH, following the same philosophy as we discussed above. Each o f these meaning representations o f the membership curves needs to be further justified by the expert in an appropriate manner. For example, note the steep slope near FR = 0. This is due to the sensitivity o f the notion o f ' " l o w failure rate" to even moderate (absolute) increases; thus, whereas a 0.05 failure rate may look to the manager to be low with a degree o f determination 234 I . B . Turksen and M. Berg MG M H 1 0 . 5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 J' V/Vm~ Figure 5. Four membership functions of L, M, H, VH for DV. 0.95, a 0.1 failure rate is assigned a degree o f determination 0.5, indicating the highest level o f entropy in the assessment (Kosko [17]). However, with a parameter like the demand rate DR, such sensitivity is unlikely, because demand volatility is anticipated anyway because o f its dependence on human choices (as opposed to machine performance, for which a relatively high degree o f precision is expected). Similar considerations help determine the shapes o f all the membership curves (Figures 1 - 5 ) . Some figures include the membership curves for additional linguistic terms that will be needed for the case study examples discussed later (after we present the inference procedure). Structure o f t h e R u l e B a s e Let us now proceed to the actual construction o f the basic set o f rules o f the expert system. In principle, we should have 81 basic rules, since each o f the A i is assigned three linguistic terms. However, as mentioned earlier, rules involving " a b o v e - l o w " failure rates are impractical because the machine is not likely to be used. Hence, fixing the failure rate input state to " l o w " reduces the number o f input state vectors to the manageable size o f 27. Suppose a manager assesses all parameter values to be low. This will correspond to an input state vector I = ( L , L , L , L ) where the first, second, third, and fourth components o f I correspond to the normalized parameters DR, DS, FR, and RR, respectively. The question is what (minimal) action on N , or equivalently on V - - L , M, H, or V H - - i s required to achieve the desired performance level Po (0.95 in the illustration here). To answer this question we again require expertise. The expertise needed now is o f a m o r e general nature. Such expertise should be valid for an entire class o f production/inventory systems (with machine imperfection incorpo- Inventory Capacity Planning 235 T a b l e 2. Midpoints o f Linguistic Subintervals (Normalized) Variable Low Medium High DR 0.15 0.5 0.85 DS 0.15 0.5 0.85 RR 0.15 0.55 0.9 FR 0.05 rated), and it could be obtained, for example, f r o m production/inventory/reli- ability system managers. In our case, however, the source o f this expertise is again the O R experts who derived the mathematical results and interpreted them with the following simulated analysis. For each independent variable DR, DS, RR, FR, choose the midpoint o f the three linguistic subintervals (L, M, H) as a representation o f this linguistic term. These points are shown in Table 2. It should be noted that a decision to consider only the Low failure rates means that FR is set at 0.05 for L. First, the expert chooses these midpoints as a representation o f the linguistic terms in deciding the formation o f the left-hand side o f the rules. Next, the corresponding values o f these midpoints are determined by (5a)-(5d) and substituted into the analytical solution procedure that is shown in the Appendix to compute the corresponding inventory level and hence by Eq. (4) the service level. The service level in turn is converted to the corresponding subinterval by (5e), which identifies the value o f the base variable in Figure 5 and determines the linguistic term for the response. Thus, the linguistic term is determined for the right-hand side o f the rule. All 27 rules are determined in this manner; the results are presented in Table 3. R E P R E S E N T A T I O N A N D I N F E R E N C E Observe from Table 3 that each rule is a fuzzy relation, with the left-hand side o f the rule being an A N D combination among three variables, connected with the right-hand side o f the rule by an I F . . . T H E N (implication) relation. In order to carry out the operation o f A N D combination and I F . . . T H E N relation, a number o f finite support points for each linguistic term must be identified in Figures 1 - 5 . The values o f the support points for each term are a compromise between the efficiency o f computation and the accuracy o f the inference results. In this study we choose seven points for each t e r m as shown in Table 4. Recall f r o m the Introduction that in this paper the linguistic combinations are 236 I . B . Turksen and M. Berg T a b l e 3. T h e Rule Base for T h r e e I n d e p e n d e n t Variables a Rule Variable Independent Variables Decision No. DR DS RR DV 1 L L L L 2 L L M L 3 L L H L 4 L M L M 5 L M M M 6 L M H M 7 L H L M 8 L H M M 9 L H H M 10 M L L L 11 M L M L 12 M L H L 13 M M L M 14 M M M M 15 M M H M 16 M H L H 17 M H M H 18 M H H H 19 H L L L 20 H L M L 21 H L H L 22 H M L M 23 H M M M 24 H M H M 25 H H L VH 26 H H M VH 27 H H H VH aRecall that the failure rate FR is set to L only. based on the d i s j u n c t i v e a n d c o n j u n c t i v e n o r m a l forms, D N F a n d C N F , respectively, which are e x t e n s i o n s o f the c a n o n i c a l forms in Boolean logic. A N D C o m p o s i t i o n T h e D N F a n d C N F for A N D c o m b i n a t i o n are O N F ( A A N D B ) = A I"1 B (6) C N F ( A A N D B ) = ( A U B ) CI ( A U B c) ¢3 (A ~UB) (7) Inventory Capacity Planning 237 Table 4. Typical Linguistic Descriptors and Their Membership Values for Each Fuzzy Set DR: VL = (1.0/0.0, 0.4/0.25, 0.32/0.3, 0.0/0.5, 0.0/0.7, 0.0/0.8, 0.0/1.0) SL = ~1.0/0.0, 0.57/0.25, 0.25/0.3, 0.0O64/0.5, 0 . 0 / 0 . 7 , 0.0/0.8, 0.0/1.0) LBL = ~1.0/0.0, 0.87/0.25, 0.71/0.3, 0.28/0.5, 0 . 0 / 0 . 7 , 0 . 0 / 0 . 8 , 0.0/1.0) L = (1.0/0.0, 0.76/0.25, 0.5/0.3, 0.08/0.5, 0.0/0.7, 0.0/0.8, 0.0/1.0) M = (0.0/0.0, 0.24/0.25, 0.5/0.3, 1.0/0.5, 0.5/0.7, 0.2/0.8, 0.0/1.0) QBM = (0.0/0.0, 0.057/0.25, 0.25/0.3, 1.0/0.5, 0.25/0.7, 0.04/0.8, 0.0/1.0) H = ~0.0/0.0, 0 . 0 / 0 . 2 5 , 0 . 0 / 0 . 3 , 0.08/0.5, 0 . 5 / 0 . 7 , 0 . 8 / 0 . 8 , 1.0/1.0) DS: L = (1.0/0.0, 0.76/0.25, 0.5/0.3, 0.08/0.5, 0.0/0.7, 0./0.8, 0.0/1.0) M = (0.0/0.0, 0.24/0.25, 0.5/0.3, 1.0/0.5, 0.5/0.7, 0.2/0.8, 0.0/1.0) QBM = (0.0/0.0, 0.1/0.25, 0.24/0.3, 1.0/0.5, 0.3/0.7, 0.15/0.8, 0.0/1.0) H = (0.0/0.0, 0.0/0.25, 0.0/0.3, 0.08/0.5, 0.5/0.7, 0.8/0.8, 1.0/1.0) VH = (0.0/0.0, 0.0/0.25, 0.0/0.3, 0.0/0.5, 0.25/0.7, 0.64/0.8, 1.0/1.0) RR: VL = (1.0/0.0, 0.64/0.25, 0.25/0.3, 0.0/0.55, 0.0/0.8, 0.0/0.86, 0.0/1.0) L = (1.0/0.0, 0.8/0.25, 0.5/0.3, 0.01/0.55, 0.0/0.8, 0.0/0.86, 0.0/1.0) NM = (1.0/0.0, 0.73/0.25, 0.5/0.3, 0.0/0.55, 0.5/0.8, 0.74/0.86, 1.0/1.0) M = (0.0/0.0, 0.27/0.25, 0.5/0.3, 1.0/0.55, 0.5/0.8, 0.26/0.86, 0.0/1.0) H = (0.0/0.0, 0.0/0.25, 0.0/0.3, 0.01/0.55, 0.5/0.8, 0.8/0.86, 1.0/1.0) LBH = (0.0/0.0, 0.0/0.25, 0.0/0.3, 0.17/0.55, 0.8/0.8, 1.0/0.86, 1.0/1.0) VH = (0.0/0.0, 0.0/0.25, 0.0/0.3, 0.0/0.55, 0.25/0.8, 0.64/0.86, 1.0/1.0) DV: L = (1.0/0.0, 0.5/0.15, 0.2/0.25, 0.09/0.35, 0.0/0.5, 0.0/0.54, 0.0/1.0) M = (0.0/0.0, 0.5/0.15, 1.0/0.25, 0.5/0.35, 0.13/0.5, 0.04/0.54, 0.0/1.0) H = (0.0/0.0, 0.06/0.15, 0.18/0.25, 0.5/0.35, 0.95/0.5, 1.0/0.54, 1.0/1.0) VH = (0.0/0.0, 0.0/0.15, 0.02/0.25, 0.1/0.35, 0.32/0.5, 0.5/0.54, 1.0/1.0) where f3, U , and superscript c are used in the set notation to correspond to the intersection, union, and complementation operators, respectively. All the fuzzy sets we have defined so far are point-valued. However, by applying Eqs. (6) and (7) for AND combination, interval-valued fuzzy sets are generated, with CNF(.) and DNF(.) being the upper and lower bounds, respectively. This is certainly not a surprise, as it is argued in Turksen [4-6] that most of the linguistic combinations would be interval-valued when impre- cise knowledge is extracted from experts. That means that our four indepen- dent variables and one decision variable could most probably have been interval-valued fuzzy sets. For ease of computation we adopt a point-valued approach and choose the midpoint of an interval for fuzzy relations whenever such an interval is given or is generated by DNF and CNF operations. Let us choose rule 6 from Table 3 as an example to explain these calcula- 238 I . B . Turksen and M. Berg tions. The left-hand side o f this rule is RLM H = D R L A N D DS M A N D RR H (8) A finite support membership value is chosen f r o m each o f these three fuzzy sets, say a i ~ D R L , b j E DS M , c k E R R H, 1 _< i, j , k _< 7. Let us also choose Max and Min operations, denoted by v and A, to correspond to the U and f) in the set notation o f C N F and D N F in Eqs. (6) and (7) and throughout this paper. This choice is needed because there are a large collection o f operators that correspond to U and O (Turksen [ 4 - 6 ] ) . Hence the membership grades (MGs) for the first two variables are com- puted as M G ( D N F ( D R L A N D DSM) ) = a i A b j (9) M G ( C N F ( D R L A N D D S M ) ) = ( a i v b j ) A ( a i V b f ) A ( a C v b j ) (10) where the complementation is chosen to be the pseudocomplement: c = 1 - a i, b f = 1 - b j , c ~ = 1 - cg a i The midpoint G j o f the interval-valued fuzzy set R L M -~ D R L A N D DS M is computed as rii = ( 1 / 2 ) [ M G ( D N F ( . ) ) + M G ( C N F ( . ) ) ] (11) Next rij is used to compute the C N F and D N F o f A N D with c k similarly determining rijk, which is the midpoint o f the interval-valued fuzzy set o f RLM H in Eq. (8). RLM is a 7 × 7 matrix, and RLM n is a 7 × 7 × 7 matrix. Altogether we have 27 such matrices, since we have 27 rules in our rule base. An c~-cut for each matrix or each rule needs to be computed by a = V ( r i j k A r / ~ , ) , 1 _< i, j , k _< 7, (12) to satisfy the sufficiency condition required for the generalized modus ponens (Turksen [2]). Thus, we have 27 ot's in all, that is, a , , 1 _< n < 27, which are then used to truncate the decision space as discussed next. I F . . . THEN Composition The 27 three-dimensional matrices are then combined with the right-hand side o f each rule, the decision space, b y I F . . . T H E N composition to construct the fuzzy relation o f each rule in Table 3. But we must first truncate the decision space b y c~-cuts, where the o~'s are given by Eq. (12). For example, Inventory Capacity Planning 2 3 9 suppose we have t~ 6 = 0.5. Rule 6 shows that DV should be M, with the following finite support: DV M : ( 0 . 5 , 1 , 0 . 5 , 0 . 1 3 , 0 . 0 4 ) After truncation, we get D v(T) ( 0 . 5 , 1 , 0 . 5 ) v M Those elements that are less than c¢ 6 are discarded, because they are insignifi- cant in carrying the information through this inference procedure (Turksen [2-61). Now we are ready to do DNF and CNF evaluation o f the I F . . . T H E N relation between R and DV. The DNF and CNF expressions o f I F . . . T H E N composition are D N F ( A ~ B ) = ( A A B ) U ( A C A B ) tO ( A c A B ¢) (13) C N F ( A --} B ) = A c O B (14) In the membership domain, again with the application o f Max and Min operators, we get rij~, = (1/2)[MG(DNF(RLM~I ~ D V ~ ) ) ) + MG(CNF(RLM n ~ D V ~ )))] (15) where riik t is the midpoint o f the interval-valued fuzzy sets in matrix RLMHM , that is, R LMHM = R LMH ~ ~--Ml')~ur ( T ) Equation (15) will reduce interval-valued membership grades to point-valued ones. The 27 matrices generated in this manner are represented in four-dimen- sional matrices with membership values. So far we have constructed a set o f rules and the database that is the set o f these matrices, which are required for the inferences in approximate reasoning. I n f e r e n c e P r o c e d u r e A number of alternative procedures are available for the approximate reasoning approach (Turksen [2, 9], Turksen and Zhong [3]). In this paper, we discuss Zadeh's composition rule o f inference [16]. In addition, from among the many approximate reasoning modes, we choose the modus ponens known as the generalized modus ponens (GMP) in our discussion. For GMP, compositional inference is written as R* o ( R ~ DV) = DV* (16) where o is the compositional rule o f inference; R* is an AND combination of I. B. Turksen and M. Berg observed states o f the system, DR*, DS*, RR*; and R ~ DV is a rule to be selected from the rule base with a suitable distance or similarity measure. DV* is the expert system advice (response) to be provided to a user for the combined observed system state R*. Such expert system responses are deter- mined in the following manner. Suppose a user inputs DR*, DS*, and RR* (the observed states); then the inference engine first computes R'LOWER = D N F ( D R * AND DS* AND RR*) R'UPPER ~--- C N F ( D R * AND DS* AND RR*) M G ( R * ) = (1/2)[MG(R~.oWER) + MG(R~ppER) ] Next the closeness o f R* is checked against R , the left-hand side o f each o f the 27 rules. Among the many distance measures (Zwick et al. [18]), we choose the Hamming distance as the closeness measure for this paper, which is d H ( R - R*) = Z rijk-- r*'kl (17) ijk The rule whose left-hand side R has the minimum distance among the 27 rules to the observed system state R* is chosen for G M P computations. The compositional rule o f inference is applied for the rule so chosen. For example, suppose the rule so chosen has left-hand side and right-hand side denoted by R * and DVd', respectively; then R* o ( R ~ --} DV*) = DV* (18) By substituting DNF and CNF operations o f R* ~ DV* into Eq. (18), we can obtain the DNF and CNF o f DV* and the midpoint o f this interval as M G ( D V * ) = ( 1 / 2 ) [ M G ( D N F ( D V * ) ) + MG(CNF(DV*)I (19) Since DVg' in Eq. (18) is already truncated in the implication composition by a0 given by Eq. (12), DV* has only those elements corresponding to the truncated DV*. Finally, DV* is compared to all the linguistic terms in the decision space, that is, D e { L , M , H , VH} to find the closest linguistic term based on the distance measure, in this study the Hamming distance d n given by Eq. (17), that is, d H ( D V * , D V ) = ~ I M G ( D V * ) - M G ( D V ) I Thus, the closest linguistic term is taken as the linguistic approximation to D V * . The output o f the fuzzy inference engine is this linguistic term so chosen. Inventory Capacity Planning 241 Please note that when we make the distance comparison between DV* and the linguistic terms in decision space, D V d' is truncated by the a0-cut [Eq. (12)]. Thus, we should truncate the linguistic terms in the decision space accordingly in order to calculate the H a m m i n g distance properly. To indicate the closeness between D V * and D V we m a y choose another measure, besides the distance measure, which is somewhat more general in the sense that it is in the interval [0, l] and is independent o f any particular situation. This measure is called the s i m i l a r i t y m e a s u r e (Turksen and Zhong [3]) and is defined as 1 S - l + d where d is a distance measure. In our c a s e , d H is the H a m m i n g distance; therefore, 1 S - (20) 1 + d n When d n = O, S = 1, indicating that the two linguistic terms are exactly the same; and when d H = oo, S = O, meaning that the two terms are very far apart. It should be noted that d n and S are equivalent measures o f closeness. S i m u l a t i o n E x p e r i m e n t The proposed approximate reasoning approach was p r o g r a m m e d in LISP and was run on an Apollo workstation in a U N I X environment for two simulation experiments with 21 hypothetical case data shown in Table 6. The membership values o f the linguistic variables are shown in Table 4. The first experiment was run with the application o f max-min operators, the second with the application o f bold union/intersection operators. The results o f these experiments are shown in Tables 7 and 8. Before we discuss the outcome o f these experiments, let us illustrate the inference procedure described in the previous subsection with two examples. EXAMPLE 1 I f the state o f the system we have observed matches exactly to one o f the left-hand sides o f the 27 r u l e s - - f o r example; DR* is L, DS* is M, and RR* is M, that is, rule 5 in Table 3 a n d / o r case 11 in Table 6 - - t h e n R * = R L M M and d H ( R * , R L M M ) ----- 0. I f we input R * into the fuzzy inference engine, we get DV* = DVd', which is M. Thus, M is the linguistic t e r m we should choose according to this observed state o f the system. This process is illustrated in Table 5. The output is M with a similarity measure o f 1. The result is what we should have expected, because the observed state is exactly the left-hand side o f rule 5. The 242 I . B . Turksen and M. Berg T a b l e 5. I n f e r e n c e P r o c e d u r e Observed states DR* = (1.0, 0.76, 0.50, 0.08, 0.0, 0.0, 0.0) DS* = (0.0, 0.24, 0.50, 1.0, 0.50, 0.20, 0.0) RR* = (0.0, 0.27, 0.50, 1.0, 0.50, 0.26, 0.0) R* = DR* AND DS* AND RR* min[ R* - R I = 0 identifies R~ o f rule 5, with a 5 = 0.5. Hence, DV~' is M. DV* = R* o(R~ --* DV~) DV* = (0.5, 1., 0.5) and dH(DV ~, DV*) = 0 1 S = - - = I 1 + d H fuzzy inference engine does certainly p r o d u c e , as it should, an output that is the right-hand side o f rule 5. This e x a m p l e illustrates the fact that an expert system based o n o u r inference p r o c e d u r e p r o d u c e s the expected result w h e n there is an exact match between the o b s e r v e d s y s t e m state and the left-hand side o f a rule in the rule base. H o w e v e r , the real p o w e r o f o u r approximate reasoning has a bit m o r e intelligence built into it in the following sense. Suppose there is no exact match between the o b s e r v e d s y s t e m state and the left-hand side o f a n y rule in the rule base. T h e n the question is, W h a t can o u r a p p r o x i m a t e reasoning p r o c e d u r e p r o v i d e f o r the user? W e s h o w in the next e x a m p l e that even w h e n there is not an exact m a t c h , o u r approximate reasoning has the p o w e r to generate appropri- ate advice f o r the users. EXAMPLE 2 I f the state o f the s y s t e m w e have o b s e r v e d is an arbitrary o n e - - s a y , D R * is very low, DS* is quite a bit medium, and R R * is a little bit high--then the inference p r o c e d u r e is as follows. I f they are not available in the s y s t e m already, the expert (the model builder) should give us m e a n i n g representations o f all the allowable linguistic terms Inventory Capacity Planning 243 T a b l e 6. Simulation Inputs: O b s e r v e d S y s t e m States Case Linguistic Descriptors of Input Parameters No. DR* DS* RR* 1 L M H 2 VL QBM LBH 3 VL M VH 4 VL QBM VL 5 SL QBM VL 6 SL QBM VH 7 VL M NM 8 L L L 9 M M M 10 H H H 11 L M M 12 VL L VH 13 VL L VL 14 H L VH 15 H L LBH 16 LBL L H 17 M H VL 18 M H LBH 19 M VH VL 20 QBM H VH 21 QBM H VL such as v e r y low (VL), quite a bit m e d i u m ( Q B M ) , a little bit high (LBH), specifying their m e m b e r s h i p functions. ( F o r the case study, o u r expert identi- fied these m e m b e r s h i p functions as s h o w n in Figures 1 - 3 and as defined in Table 4.) Thus, D R * is V L , D S * is Q B M , R R * is L B H , that is, case 2 in Table 6. O b s e r v e that this does not m a t c h the left hand-side o f any o f the rules in Table 3. First R * = D R * A N D D S * A N D R R * is c o m p u t e d b y D N F and C N F expressions o f A N D combination. T h e n the nearest R , denoted b y R * , is c h o s e n f r o m a m o n g the 27 left-hand sides o f the rules in the rule base. I n this case, rule 6 is selected; hence, D V d' is M. W i t h the G M P and an c~-cut o f 0.5, w e get D V * = R * o ( R * --* DV~') T h e result is D V * = ( 0 . 5 / 0 . 1 5 , 1 . 0 / 0 . 2 5 , 0 . 5 / 0 . 3 5 ) . T h e c o m p a r i s o n o f distance b e t w e e n D V * and other linguistic terms in 244 I . B . Turksen and M. Berg d e c i s i o n s p a c e D c a n b e s u m m a r i z e d as f o l l o w s : D e c i s i o n S p a c e D H a m m i n g D i s t a n c e M e a s u r e [DVd' - D V * [ S i m i l a r i t y L 1.21 0 . 4 5 M 0 . 0 1.0 H 1.26 0 . 4 4 V H 3 . 1 4 0 . 2 4 T h e r e f o r e , the l i n g u i s t i c t e r m M is the r e s p o n s e o f the e x p e r t s y s t e m for the o b s e r v e d s y s t e m state. T h a t is, w h e n the d e m a n d rate is v e r y l o w , the d e m a n d s i z e is q u i t e a b i t m e d i u m , a n d the r e p a i r r a t e is a little bit h i g h , w e s h o u l d e x p e c t the s e r v i c e l e v e l to b e m e d i u m . SIMULATION EXPERIMENTS L e t us n o w l o o k at the t w o s i m u l a t i o n e x p e r i - T a b l e 7. S i m u l a t i o n E x p e r i m e n t s R e s u l t s - - ( a ) S i m u l a t i o n w i t h M a x - M i n O p e r a t o r s ; (b) S i m u l a t i o n w i t h B o l d U n i o n / I n t e r s e c t i o n O p e r a t o r s Selected Rule Linguistic Descriptor Number of System Response Case No. (a) Co) (a) (b) 1 6 6 M M 2 6 6 M M 3 6 6 M M 4 4 4 M M 5 4 4 M M 6 6 6 M M 7 5 6 M M 8 1 1 L L 9 14 14 M M 10 27 27 VH VH 11 5 5 M M 12 3 3 L L 13 1 1 L L 14 12 12 L L 15 12 12 L L 16 3 3 L L 17 16 16 H H 18 18 18 H H 19 16 16 H H 20 18 18 H H 21 16 16 H H Inventory Capacity Planning 245 T a b l e 8. Simulation E x p e r i m e n t s - - C o m p a r i s o n o f DV* for Max-Min and Bold Operator with or-Cut o f 0.5 Case No. DV*, Max-Min DV*, Bold 2 (0.50, 1.00, 0.50) (0.65, 1.00, 0.65) 3 (0.50, 1.00, 0.50) (0.50, 1.00, 0.50) 4 (0.50, 1.00, 0.50) (0.506, 1.00, 0.506) 5 (0.50, 1.00, 0.50) (0.646, 1.00, 0.646) 6 (0.50, 1.00, 0.50) (0.646, 1.00, 0.646) 7 (0.75, 1.00, 0.75) (1.00, 1.00, 1.00) 12 (1.00, 0.50, 0.00) (1.00, 1.00, 0.00) 13 (1.00, 0.50, 0.00) (1.00, 1.00, 0.00) 14 (1.00, 0.75, 0.00) (1.00, 1.00, 0.00) 15 (1.00, 0.75, 0.00) (1.00, 1.00, 0.00) 16 (1.00, 0.50, 0.00) (1.00, 1.00, 0.00) 17 (0.95, 1.00, 1.00) (1.00, 1.00, 1.00) 18 (0.95, 1.00, 1.00) (1.00, 1.00, 1.00) 19 (0.95, 1.00, 1.00) (1.00, 1.00, 1.00) 20 (0.95, 1.00, 1.00) (1.00, 1.00, 1.00) 21 (0.95, 1.00, 1.00) (1.00, 1.00, 1.00) ments based on the 21 cases shown in Table 6. Both o f these experiments were run using the H a m m i n g distance measure and the associated similarity measure as defined by Eq. (20) in identifying the rule to be selected and in identifying the linguistic descriptor to be displayed as the system response. It is observed in Table 7 that, with the exception o f case 7, there were no differences in the rule selection. In particular, for case 7, we observe that max-min operators selected rule 5 but bold union/intersection selected rule 6. Furthermore, there were no differences in the system response in terms o f the linguistic descriptors on the surface. H o w e v e r , if we look at the internal value o f DV* vectors before it is approximated to a linguistic descriptor via the use o f H a m m i n g distance and similarity measures, we observe some differences (see Table 8). Hence, we realize that the similarity measures have a smoothing effect on the system response in terms o f the linguistic descriptors. Whether such a smooth- ing effect is desirable or not m a y be context- and domain-dependent. There- fore, a system designer in cooperation with the users can decide whether to output the system response in terms o f just the linguistic descriptors or just DV* or both. It should be noted that in Table 8 we list only cases 2 - 7 and 1 2 - 2 1 , where the observed system states do not match the left-hand side o f any rule in the rule base. Since cases 1, 8, 9, 10, and 11 have an exact match, we have no reason to list those cases. As explained in Example 1, the system response corresponds to the right-hand side o f the rule as expected. 246 I.B. Turksen and M. Berg Let us now reconsider cases 2 - 7 and 12-21. They were selected randomly, and case 2 is explained in detail in Example 2. It is rather interesting to note that cases 2 - 7 describe an observed system behavior around " m e d i u m " and the system response is " m e d i u m , " corresponding to our expectations. Simi- larly, when the system behavior is around " l o w , " as in cases 12-16, the system response is " l o w , " and when it is around " h i g h " as in cases 17-21, the system response is " h i g h , " again corresponding to expectations. This is an indication o f robust system behavior. Clearly these results support the hypothe- sis that approximate-reasoning-based expert systems would better serve the operations managers o f such robust systems. Finally, these simulation experiments appear to suggest that once the rule is identified with the help o f a similarity measure we could directly fire the rule, that is, give the right-hand side o f the rule as a response linguistic variable. This needs to be validated theoretically and experimentally in the future. I f this finding is true, then there would be no need for a compositional rule o f inference. This would be analogous to modus ponens in two-valued logic. Could this be true for the case of robust systems? C O N C L U S I O N S In this paper, we have described our approximate reasoning approach for an expert system design and development as an aid to management confronted with a production/inventory capacity problem. We have attempted to show how operations research and approximate reasoning can be synthesized for the solution o f real-life problems in the era of knowledge-based systems. Opera- tions research methodologies can generate valuable insights into the under- standing o f problem domains. Approximate reasoning provides a framework where the insight gained from OR models could be restructured for real-life problems either where there is insufficient information to identify parameters o f the system at hand or where such data are not available owing to various factors and uncertain environmental conditions. Hence, the best we can do is rely on OR experts' assessments and the interpretation o f real system behavior via model analysis. Since such assessments can best be expressed in a natural language setting with linguistic terms providing flexibility o f expression in human judgments, and since approximate reasoning based on fuzzy logic can handle such linguistic uncertainties and imprecision, it appears that the mar- riage o f operations research and approximate reasoning is inevitable with the advance o f expert systems. Approximate reasoning with linguistic variables and their terms provides aggregation o f human knowledge as well as user friendliness. Thus, approxi- mate reasoning is a reasonably good analog to human reasoning. This was illustrated with the second example in the last section. Furthermore, the Inventory Capacity Planning 247 simulated cases 2 - 7 and 12-21 discussed in the previous section indicate and support the commonsense reasoning that managers in charge o f robust systems would get the appropriate support without detailed precision with the aid o f approximate-reasoning-based expert systems. The approximate reasoning shows an advantage o v e r analytical models in that it allows a certain degree o f freedom f r o m accuracy in the information and knowledge acquisition. This freedom and flexibility combined with the p o w e r o f the aggregation due to linguistic variables allows expert system designers and developers to summa- rize the available knowledge in terms o f far fewer rules than if one were to design expert systems that require precise information leading to rule explo- sion. On the other hand, the extra intelligence provided in the inference procedure gives us a way to cope with situations not explicitly included in the rule base and hence with unforeseen future conditions in the observed system behavior. This means that the use o f an expert system is not confined to the set o f rules provided in the knowledge base. There are certain issues that need m o r e elaboration. Some o f these are: 1. Throughout our discussion, point-valued fuzzy sets are employed instead o f interval-valued ones, with the exception o f C N F and D N F expres- sions, which creates interval-valued results for the logical combinations. H o w e v e r , by taking the average o f the upper and lower bounds, respec- tively, we reduce the intervals to points. H o w e v e r , several experimental results obtained so far suggest that interval-valued fuzzy sets represent experts' assessment m o r e naturally. Even though we have shown that interval-valued inference is quite possible, computations required for interval-to-interval inference are a lot more complex and costly at this point in our research (Turksen [9], Turksen and Zhong [3]). 2. In the inference procedure, we have used G M P and the H a m m i n g distance and a similarity measure based on this distance measure. This choice needs justification. It is known that there are other distance measures and other similarity measures (Zwick et al. [18], Turksen and Zhong [3]). H o w e v e r , the choice o f a distance and a similarity measure is still an open question. Until we identify the context-dependent effects o f these distance and similarity measures, we cannot know how to choose an appropriate measure. These issues are left for future research. A C K N O W L E D G M E N T S This investigation was supported in part b y the Natural Science and Engi- neering Council o f Canada and in part by the Manufacturing Research Corpo- ration o f Ontario. The software developments and experiments were carried 248 I . B . Turksen and M. Berg out by graduate students H. Zhao, I. Wilson, and P. Grant. We are grateful for all the support we have received. A P P E N D I X f o ( W ) = A ( e - ~ W - re-~2~,) A[ r ] A ( ~ o ) -- --ff ( . - x - ~ 1 ) ~ - ~ , ~ - - ( . - x - ~ ) e - ~ x A ( 1 - r) f o - X and f N = --~[ # - ~ k - /3' ( e - u N e - # ' N ) - r ( # - ~ k - /32) ( e - " N - - I a /32 -- I z where X A - I = ~1-1 1 - /32 ~'l /31] [/32-/3,_ + ( . - x - / 3 , ) ( . ~ , ~ ; ~ / 3 2 ) e " N e - 8 1 N e - B 2 N + - - - ~ ~ - / 3 2 B = - A r and ( ~ - X - / 3 , ) / ( ~ - / 3 , ) r = (# - x - / 3 2 ) / ( ~ , - / 3 2 ) where A , B , /3~, /3, and r are intermediate parameters used in the analytical solution procedure. 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