I l l 0 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 25, NO. 7, JULY 1995

Reasoning in Qualitatively Defined Systems Using Interval-Based Difference Equations

Melody Y. Kiang, Member, ZEEE, Aimo Hinkkanen, and Andrew B. Whinston

Abstract-Quantitative research and statistical techniques have long been regarded as superior ways of analyzing knowledge in social sciences. To deal with incomplete or imprecise knowl- edge while modeling systems, traditional approaches in social sciences (i.e., management science, operations research) have attempted to measure social facts by making approximations of the problem under analysis. Recent Artificial Intelligence (AI) research on qualitative reasoning which focuses on using qual- itative knowledge to reason about the everyday physical world, suggests an opportunity to extend the capability of current logico- mathematical instruments used by social scientists. This paper proposes an interval propagation difference equation method, a type of qualitative-quantitative simulation method, to model dynamic systems by abstracting from the underlying true model. The proposed difference equation method can be used to model problems requiring discrete-time analysis, such as applications involving time-lag relationships. Moreover, the method does not require the exact functional form of the problem under analysis to be known with certainty. The incomplete or imprecise knowledge available about the functional form of the true model, and the values of its variables, are represented with bounding functions and interval values respectively.

I. INTRODUCTION

PPLICATIONS in social sciences have a long tradition A of using management science approaches like mathemat- ical programming and statistical analysis to model problems. “Commonsense methods for acquiring knowledge have been widely regarded as unsystematic, irreplicable, and invalid, and have been displaced in social science by natural science meth- ods, especially statistics. In this manner, statistical procedures and quantitative methods held an almost monopolistic grip on social science until recent years [34].” Both commonsense and expert knowledge are always incomplete. No one understands, down to the minutest detail, how any real system actually works. Given that our knowledge about the real world is incomplete, we have two choices for dealing with incomplete- ness: approximation (the quantitative analysis approaches) and abstraction (the qualitativekausal reasoning approaches).

When precise numerical information is available, quanti- tative analysis is still considered the most appropriate and

Manuscript received May 29, 1993; revised February 27, 1994, and August 28, 1994.

M. Y. Kiang is with the Department of Decision and Information Systems, College of Business, Arizona State University, Tempe, AZ 85287-4206 USA.

A. Hinkkanen is with the Department of Mathematics, The University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA.

A. B. Whinston is with the Department of Management Science and Information Systems, The University of Texas, Austin, TX 78712-1 175 USA.

IEEE Log Number 9410813.

efficient reasoning method to apply. However, most of the time, complete numerical information about the problem under study is not available at the time of analysis. The possible reasons for not having accurate numerical information are: 1) There is not enough information available to formulate a quantitative model (i.e., incomplete knowledge), 2 ) the exact form of that function is not known with certainty (the underlying theory is not precise enough to support accurate quantitative models (i.e., imprecise knowledge)), and 3) the information about the exact form of that function is not needed for the type of analysis to be carried out (e.g., the costhenefit analysis of computational complexity versus model accu- racy [35]). Business and economics problems, because of their intrinsic complexity, generally lack complete quantitative models and thus offer particularly suitable application areas for qualitativekausal modeling.

Research development in qualitative/causal reasoning, while focusing on predicting the dynamic behavior of piece wise continuous physical systems, offers an opportunity to extend the capability of current reasoning systems in business problem domains. Qualitativekausal reasoning, which can capture the structure, behaviors, and causality underlying the system, is a valuable method for reasoning about partially known systems. The use of causal knowledge by humans while performing comprehension, explanation, prediction, and control tasks has been extensively studied in the psychology literature [ 11, [28]. Findings of empirical research in understanding human diagnostic reasoning processes [6] also suggest that experts make decisions using qualitative and causal reasoning. A system built on the qualitativekausal modeling approach can not only be able to render a final answer to the user but also provide an explanatory path, a cause-and-effect picture of the organization’s problems.

In this paper, an interval-based difference equations method, a type of qualitative-quantitative modeling approach, is em- ployed to model systems that can genuinely cope with discrete- time dynamic problems. Constraint propagation has been a popular tool in AI research that deals with inference about quantities. Interval analysis through constraint propagation is a particular kind of constraint propagation method, where quantities are labeled with intervals and the intervals are prop- agated through the constraint network that defines the causal relationship between variables [8], [22], [ 191. We develop a solution procedure that carries out constraint propagation for a dynamic system. The method is aimed at modeling prob- lems of dynamic nature when only imprecise (or incomplete)

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KIANG et al.: REASONING IN QUALITATIVELY DEFINED SYSTEMS 1 1 1 1

knowledge is available for the problem solving process. Time is modeled quantitatively in our method, and the simulation result includes the actual time elapsed during the process.

The balance of the paper is organized as follows: In Section 11, limitations of current modeling approaches in social sciences are presented. In Section 111, we discuss the motivation, as well as the problems, for using current qualitativekausal reasoning approaches. Section IV is devoted to the development of the mathematical foundation of the interval-based difference equations method. An informal sketch of the reasoning process is presented to give the readers an overview of the mechanism. This is followed by a formal discussion of the mathematical basis approach. An economic model, the cobweb model, and tested in Section V. The paper is concluded in VI with some suggestions for future research.

11. LIMITATIONS OF CURRENT MODELING APPROACHES IN SOCIAL SCIENCES

A model is a simplified description of a complex Because the real world is too complicated to be fully under- stood, a model is constructed with only the most important attributes of the system, omitting much of the detail. A certain amount of measurement imprecision is commonly accepted in reasoning about social sciences. Statistical analysis tends to approximate a non-deterministic mechanism in terms of probability distribution functions (pdfs). The tractability of a model is maintained by assuming at least i.i.d. (identical independent distribution) random variables with a constant probability distribution. However, deviations from the nor- mality assumption, at least in economics and finance, appear more likely to be the rule than the exception [14]. Moreover, statistical approaches usually require a substantial amount of information (and data), and need highly trained specialists in order to develop a well-specified model and interpret its results.

Another common limitation found in the statistical ap- proaches is known as the model specification problem. Gener- ally speaking, a statistical model requires a priori specification of a functional model which includes both the pertinent variables and the forms of functional relations. Any mis- specification occurring during the model-building process can lead to a significant deterioration of the performance of the resulting model. Due to incomplete knowledge or human mistakes, the problem of mis-specification in statistical anal- ysis is not rare. On the other hand, qualitative models using qualitative descriptions to represent the relationship between variables are considered more robust, though less precise, than quantitative equations. Each equatiodfunction in a qualitative model actually corresponds to a class of quantitative equations, and the model mis-specification problem can be alleviated.

of our is built Section

reality .

111. MOTIVATION FOR USING QUALITATIVE AND CAUSAL REASONING APPROACH

When choosing the appropriate level of detail and precision in model specification, the issue of tradeoffs arises 141. In our case, the fundamental trade-off is in the use of quantitative

detail versus the effort required to specify the qualitative description of a domain. The answer depends on many factors, including the nature of the decision itself. As scholars in many disciplines are interested in knowing how to best present information to decision makers, the cost-benefit issues of hu- man decision making process have been extensively examined 1331. Previous works in consumer decision-making processes also suggest that the selection among decision strategies is a tradeoff between the strategies’ effort and accuracy [21], [291. Information representation studies [20] attempt to determine which form of information presentation is most “effective” and “efficient” for the human decision-making process. For example, data with only ordinal properties may be easier to process under certain circumstances than numerical tables or figures. The qualitative abstraction mechanism adopted by qualitative/causal reasoning approaches matches well with the simplifying decision rules and information format proposed by many researchers. The qualitative abstraction mechanism, which filters and reformats the information presented to the decision makers, works like a screening tool that will sub- stantially reduce the effort in complicated decision making processes.

Qualitative reasoning relieves the modeler from the highly technical task of formulating a precise model by using ab- straction instead of approximation. The advantage of such an approach would be a more natural kind of model specification (natural in a sense of being closer to the incomplete and qualitative knowledge being dealt with) and a more natural interpretation of the results. The intent of the abstractions is to retain the core causal model of the system, but reduce it to its essence by removing all aspects of the model not essential to the decision under analysis. Thus, unimportant distinctions contained in the complete model are removed if they did not effect the decision. The resulting models generally conform to what might be called the naive, but often correct, predictions of common-sense. Expert decision makers appear to work within the framework of a strong causal model of their environment, but without the details explicit in a quantitative model of that environment. That is, they work in a qualitativekausal model setting. Qualitative reasoning has been demonstrated to be an effective means for representing and reasoning with incomplete and qualitative (imprecise) knowledge in a business setting with highly simplified examples 121, 131, however, the predictions do not have the numeric precision of the results derived from a quantitative simulation.

A. Limitations of Current Qualitative/Causal Modeling Approaches

Qualitative reasoning is a relatively new field of study originated from AI research in qualitative physics [9], [ l l ] , [ 151, [23], [36], and focuses on using commonsense or incom- plete knowledge to reason about the everyday physical world. The preliminary results of applying qualitative simulation techniques to diagnosis [IO], [ 131, [ 181 and design problems [16], tutoring systems [15], medicine and cell biology [17], and auditing [2] have been encouraging. However, as mentioned above, current qualitative reasoning approaches work well only

1112 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 25, NO. 1, JULY 1995

in a highly simplified problem domain. When the size andlor the complexity of the problem increases, the system usually encounters the intractable branching problem [7] or renders uninteresting results (i.e., the results are too general for the decision maker to draw any useful conclusion).

One current approach to solve the above mentioned problem is to combine qualitative and quantitative methods to allow stronger predictions than using qualitative reasoning alone while avoiding the assumptions required for a numerical simulation. Several quantitative methods, including proba- bilistic [26], interval [5], discrete time simulator [32], and fuzzy approaches [30], have been suggested by researchers to be coupled with qualitative reasoning methods to alleviate the problems of pure qualitative reasoning methods. Each of the suggested quantitative methods has its own strengths and weaknesses that are inherent in their underlying algo- rithms. Most of the research efforts in integrating qualitative and quantitative reasoning techniques mainly concentrate on solving physics and engineering problems that are usually continuous in nature. This paper demonstrates an altemative approach for combining qualitative and quantitative methods that is especially aimed at solving business problems (Le., problems with time lag relationships and involving discrete time analysis.) For instance, in economics (and in business and social sciences, in general) time is typically modeled using discrete time intervals. When modeling a business problem, time is not really part of the knowledge but rather has a given structure that might be imposed by law (as in accounting), or by practical considerations (such as time points of particular interest, e.g., the end of months, quarters, or fiscal years.) Sim- ilar type of problems can be found in engineering areas as well. For example, when modeling medical or chemical reaction systems, the actual time elapsed for a given perturbed system to reach equilibrium could provide valuable information for analyzing the possible outcomes of the system.

B. The Modeling of Time in Interval-Based Difference Equations Approach

Time is usually a critical piece of information that needs to be known explicitly in the analysis of manzgement problems. Unfortunately, numerical values of moments of time cannot be provided by the previously developed qualitativelcausal reasoning techniques (i.e., Q2 [24]). The information about how long (one year or ten years) it takes for a certain product to start generating net profit will substantially affect the final decision about whether the company should invest money in that product. Although sometimes more than enough information is given to reason about the value of time in the model, the information will never be considered during the reasoning process due to the limitation of previous approaches.

In macroeconomics theory, dynamic analysis examines sys- tems or models involving relationships which hold over time, that is the value of a variable depends not only on the values of other variables at the same time instant but also depends on the values at previous time point(s). These are the systems involving time lags, habituation, or cumulants and are commonly found in economics (and business in general). Our

method takes advantage of the difference equation approach and is especially suitable for modeling such dynamic systems. In our approach, “time” is modeled explicitly in the entire process. From the result of the simulation, we can not only determine the value of a certain variable at a certain time point, but also ascertain how long it takes for the system to approach equilibrium.

Kuipers and Berleant have developed 4 2 (for “Qualitative + Quantitative”) [24] and Q3 algorithms [SI, extensions of QSIM (a qualitative simulation package), to integrate incomplete quantitative knowledge into qualitative simulation. Since we feel that Q2 is the closest qualitative-quantitative reasoning algorithm to our approach, we use Q2 as the basis for comparison in this section. To compare our method with Q2, we use the water tank example in their research. The water tank has an open drain and a constant positive inflow. Rough estimates of the relationships between inflowloutflow and initial values of parameters are represented by bounding functions and interval values, respectively. The simulation results of 4 2 showed that the amount of water in the tank E [5.2, 56.71 when the equilibrium is reached and the process takes [0.644, m ] units of time. In Q2, the possible time range is inferred using the mean value theorem which is often too wide to provide useful information for decision-making purposes. (An asymptotic function will always give cc as the upper bound or -cc as the lower bound.) The problem has been lessened in Q3 by using more and denser time points, but has not been completely eliminated.

Due to the limitation of the finite precision machine used, when modeling using difference equations, the precision level of the results will depend on the precision level used for calculation. That is, the results will have only finite precision. However, in business problem domains, the type of variables modeled usually have discrete values, (e.g., cash, inventory, cost of goods sold, profit, etc.) and a finite precision for the results (say, with two decimal places) is generally more appropriate. The result derived by our approach tells us how long it takes for the system to reach equilibrium (or approach close enough for practical purposes). If the system modeled can reach equilibrium in a finite time range, then the results derived from both Q2 (and Q3) and our approach will be the same. The results will be different only when the function is asymptotic. Depending on the nature of the problem domain, most of the time the precision level of the model can be determined without loss of pertinent information.

The water tank example has analogy to a cash management system in a company. The inflowloutflow functions represent a company’s policies and operation constraints that govern the relationships among cash inflow, cash outflow, and the amount of cash on hand. The simulation result can help predict, based on certain cash management policies, how the company’s cash level fluctuates over time and whether it will approach equilibrium (i.e., stabilize) in a reasonable time period. For business related problems, knowing the change in values of variables during the process is usually as important as knowing the final result. In the cash management example, the result of the simulation may show that, at some point, the amount of cash on hand may go below 0 and later stabilize at a

KIANG et a/.: REASONING IN QUALITATIVELY DEFINED SYSTEMS 1113

positive value. This means that based on the company’s current policies, it’s possible that the company may not maintain enough cash on hand to support daily operations. Although, the result shows that the system will finally reach equilibrium, the company may never reach that point (it will go bankrupt before stabilizing.)

Based on the same amount of information about the water tank, our approach can not only predict similar behavior as Q2 does, i.e., amount lies in the interval [5.15, 56.641, but also provide additional information about the range of elapsed time that the water tank takes to approach equilibrium, i.e., time lies in the interval 111, 791. (Our method shows that after 11 units of time, 0 is a possible value for netflow, and after 79 units of time, netflow must be 0.) The above result is calculated with a precision of two decimal places for the amount of water. If less precision is required, the time range will be narrower. For example, if the desired precision level to express the quantity is one decimal place, then the time lies in the interval 14, 511; if the quantity is an integer, then the time lies in the interval [ 1 , 241. Our approach gives decision makers the flexibility of deciding the precision level of the model to trade for more pertinent information (the time taken to approach equilibrium).

I v . MATHEMATICAL FOUNDATION OF THE INTERVAL-BASED DIFFERENCE EQUATIONS MODELING METHOD

The qualitative-quantitative method proposed in this paper is essentially an interval-based version of difference equa- tions. The problem under analysis is represented by a set of functions:

Aj(t),tcT = ( to , t l , . . . , t,}andjtJ = {1,2, . . . ,n}

Although the system outputs described by the Aj(t ) have specific values, these values are not known. It is only known that the value of A j at time t; lies in a set denoted by Ezj and Eij contains all possible values of Aj at time ti. Generally speaking, the more imprecise our knowledge is, the larger the set Eij will be. For the special case in which all functional forms in the model are known with certain (complete) information, the results derived from our approach should be the same as those using quantitative analysis. However, most of the time the exact forms of the functions are not known with certainty. We assume that the values of the Aj lie in the real axis R. Thus the E;j are subsets of R.

We build a conceptual model using qualitative difference equations (an abstraction of the quantitative difference equa- tions) based on the modelers knowledge and understanding of the problem situation as the basis of our qualitative- quantitative analysis. The functions, represented by rules or by constraints, can be of any form: linear or nonlinear, monotonic or non-monotonic. They are only implicitly represented by bounding functions. Each qualitative function actually cor- responds to a class of quantitative functions confined in the bounding functions. In many practical problems, the bounding functions may be formulated as linear functions, although our solver can handle nonlinear functions as well. Parameters are defined by interval values. The intervals of our initial model are progressively refined through constraint propagation

by taking the intersections of those intervals. The algorithm terminates for two reasons: a) There is always a reasonable (finite) default value for the maximum number of iterations, and b) even in an ill-defined problem, the finite machine precision will eventually cause the E,s to converge in some iteration. A detailed discussion of the convergence properties is presented in the Appendix.

In the rest of this section, the formal mathematical basis of our approach to reasoning with imprecise knowledge is introduced. After some basic definitions, a simple example is given to demonstrate how our algorithm works.

A. The Problem Solving Processor

At the beginning of the process, the user inputs all the information he or she has about the problem under analysis, which should include the initial values of all Aj at time t o (i.e., sets EQ for 1 5 j 5 n) and all other available information about Eij (represented by intervals, rules, or constraints). The sets Eij are usually denoted by closed intervals [a , b], which means that at time ti , a 5 A, 5 b. An interval may, of course, reduce to a single point (i.e., [a,a]) . For simplicity, we assume that any finite endpoint is contained in the interval. When no information is available about the sets EQ, the interval (-co, co) is assigned. If it is known that Aj takes only nonnegative values, we set E,, to [0, 00). The basic idea of our method is to formulate all the available information in the form of rules and constraints and perform calculations (simulation) so that if all the sets E,j are known for 0 5 p 5 i - 1 and 1 5 j 5 n, then we can determine the sets Eij for 1 5 j 5 n. That is, we determine the sets E,j as time ti varies. In other words, since the sets Eoj for 1 5 j 5 n are known, we can perform the calculations to find sets Elj for 1 5 j 5 n. Then, further calculations can be performed to find the sets Ezj for 1 5 j 5 n. A more detailed discussion of the computational process appears in Section 1V.C.

B. Rules and Constraints The qualitative functions used to model the system can be

classified into three categories: Past rules, present rules and constraints.

A past rule states that a certain Aj, at a certain time ti, is a function of some or all of the A, at time t , where 1 5 p 5 n and 0 5 y < i . For example, in the cobweb model we present in Section V, the function that describes a relationship such as “the quantity of a certain crop harvested this year depends on the quantity planted last year,” is a past rule. Later, we shall discuss the kind of functions that might be allowed to model this rule. A present rule states the same fact as the past rule except that now 0 5 q 5 i, that is, time t , could also be equal to the present time ti. It is possible to apply such a rule to denote something about A, at time ti only if we have already determined Eik for all the functions Ak that appear in this rule. For example, a relationship such as “the price of a certain crop in a certain year is dependent on the quantity harvested that year,” can be modeled by a present rule.

1 I14 IEEE TRANSACTIONS ON SYSTEMS, MAN. AND CYBERNETICS, VOL. 25, NO. I , JULY 1995

3) A constraint is an equality or inequality stating that an expression depending on the AI, at time t, for 1 5 IC 5 n and 0 5 p 5 1: must be equal to zero, nonnegative, or positive; that is, a constraint is of the

> O or >O. Using the model described in Section V, the price of a certain crop at any time must be greater than 0 is an example of a constraint.

A complete description of the cobweb model [I21 is pre- sented in Section V of this paper. The reason of distinguishing between past rules, present rules, and constraints is mainly to help the reader to understand the different types of func- tions that can be modeled in the program. The program automatically decides the sequence of applying them during the simulation process. All the rules and constraints have to be satisfied at each time point. The purpose of the iteration process performed at each time point is to make sure that all the rules and constraints are satisfied and the smallest possible set of E,, is derived. In the next section, we describe the determination of E,, .

form f(Al(t~),...,A,(t~),Al(tl),...,A,,(t,)) = 0 or

C. The Iteration Process-Finding The Set E,, Computation is performed to determine the smallest set

interval that contains all the possible values of A, at time t,. This set will be assigned to I C t J . There must be at least one past rule that allows us to determine E,, for at least one J at any time t, using our knowledge of the A, at previous times. The number of past rules needed depends on the problem situation. The rules must be such that it is possible to determine all the A, at time t , by using, first, the past rules and, then, the present rules.

Whenever we use a past rule, we determine what the set E,, can be, on the basis of our knowledge of the sets Epy for times strictly before t , , and on the basis of the given function for A, at time t,. The set E,, so obtained may be made smaller later. At this point it is merely one set that contains all possible values of A, at time t,. In this way we get a set E,, for all .I at time 2. The application of present rules is demonstrated using the following example:

Let a present rule be i l l = A2 + 5. If now i = 1 and, say, A1 = Ell = [lo, 121 and

A2 = E12 = [6,90], we can replace these sets by:

and = [6, AI - 5 ) n [6,901+ ~5~71 n [61901+ [6,71),

= [11; 121((~2 + 5 ) n [lo, 121 3 [I]; 951 n [IO, 121 =+ [ll, 121).

We treat all the present rules in this manner. If there are constraints, we apply them as they might reduce the sets E,,. We repeat the process of applying present rules and constraints until no changes can be made to the sets Et j .

The user can specify the number of iterations required as an input to the program. The iteration process will stop when:

1) the user-specified number of iterations is reached, or 2) the system reaches equilibrium, Le., for all j , E,, = EL-11.7 or

3 ) the system enters a cyclic process, i.e., for all j , for any p with p 5 i - l,E,, = E,,, or

4) any contradiction in the model description is revealed, i.e., E,, = 4 .

D. Admissible Functional Forms of Rules and Constraints

The next step is to determine the kind of functions to be used in the rules. A variety of functions may be used, such as, algebraic operations, exponential functions and logarithms, functions that are defined by monotonic upper and lower bounds, and combinations of these functions.

Example of basic (admissible) operations include: Addition, multiplication, subtraction, division, multiplication by a real constant (say forming 5 A or 7rA as opposed to starting with two variables A and B and forming their product AB), applying the exponential function or the natural logarithm to a single (variable) quantity (a combination of such operations allows us to extract roots), and the application of an arbitrary function. In each case, the operation is a function of a suitable number of variables. For example, for the operation of addition, we may assume that there are two variables. (Any finite number of quantities can then be added by repeating this operation.)

A real valued function f defined on a subset E of the real axis is said to be positive monotonic if f ( . x ) 5 f ( y ) whenever x and y lie in E and x < y. A negative monotonic function is defined similarly by requiring that f ( x ) 2 f ( y ) when z < y. For simplicity, we assume that there is only one variable in a monotonic function. We shall take the domain of definition of such a function to be an interval, usually the set of all nonnegative or the set of all positive real numbers.

Sometimes the exact functional form of a monotonic func- tion is not known but we do have some knowledge about the possible ranges of f . In such a case, explicit upper or lower bound functions may be specified for f . If f ~ ( : c ) 5 f(z) 5 fc(:r;) for all n: in the domain of fl then f~ is called a lower bound for f and fu is called an upper bound for f . We assume that both f~ and f r r are monotonic in the same direction as f . If f is increasing, u < b. and f ~ ( a ) 5 f ( u ) . then f~(u) 5 f ( b ) . If no better lower bound for f ( b ) is known, we may use f~(a) as a lower bound for f ( b ) and thus assume that f ~ ( b ) 2 f L ( a ) , equality being quite possible. Similarly, if f ( b ) 5 f c r ( h ) then f ( u ) 5 f r ; ( b ) . Thus, we may use f ~ ~ ( b ) as an upper bound for f ( u ) and thus assume that f c ( a ) 5 f u ( b ) . A condition concerning the range of f may amount to an upper or lower bound. For example, if all the values of f are nonnegative then f ( x ) 2 0 so that the constant function f(z) = 0 is the lower bound. An upper or lower bound might be given by an explicit formula, or by a collection of formulas, each for a particular range of values of ic (e.g., by a step function which is constant on certain intervals).

E. The Application of Rules and Constraints

As discussed in the previous section, a rule is defined by an operatiodfunction and applied in several stages. The operation itself must be from the list of admissible operations, and all the variables of the operation must come from quantities that

KIANG et al.: REASONING IN QUALITATIVELY DEFINED SYSTEMS 1115

are already known. Thus, they must be among those A, at time t , where q < z , or among those A, at time t, that have been already calculated (this means that a rule giving such an A, at time t , has been already applied.) When the operation is performed, the new quantity found may be used in the following stages, or may represent the final result. That is, this particular A, at time t , is considered the last stage of the application of this rule. The following simple example illustrates the meaning of the above discussion. Consider the past rule that gives A2 at time t 2 as:

Az(t2) = f (Al ( t1) + Az(t1)).

And, as a result of previous calculations, we know that Ell = [7,8] and E12 = [12,20]. Let f be an increasing function defined as 2 / 3 5 f ( x ) 5 22 + 10 for all z 2 0. The first stage in the application of the rule corresponds to the addition of Al(t1) and Az(t1). We deduce that this quantity, denoted byB,l ies in[19,28] (EII+E12 = [7,8]+[12,20] = [19,28].) The second stage corresponds to applying f . If B lies in 119, 281 then clearly f ( B ) lies in 11913, 661 ([19,28]/3 5 f ( B ) 5 2[19,28] + 10 =$ [19/3,28/3] 5 f ( B ) 5 [48,66] + 19/3 5 f ( B ) 5 661.) Thus A2(t2) lies in [19/3, 661 so that we obtain the initial choice E22 = [19/3,66]. This is all that can be said from this particular application of the rule. However, it should be understood that this need not be the final set E22. At this point we certainly know that A2(t2) must lie in [ 19/3, 661 but later certain other rules and constraints will be applied and may help us make the set E22 smaller.

To be careful, we should not use the notation E22 for sets other than the final choice for E22. The sets considered before such a final choice is made are merely candidates for E 2 2 . All the other sets to which A, at time t , belongs are only auxiliary objects used in the calculations and inferences performed in order to find the eventual Ea,. Having made this theoretical point clear, we avoid more complicated notation and refer to these candidates for E,, as Ea,, with the understanding that these are sets in the process of change.

F. The Order of Applying Rules and Constraints We note that rules are applied in stages, and that the

development towards finding the sets Eij goes through the same stages. There are different ways of decomposing a given rule into stages. As a simple example, note that if A is to be B + C + D , we could also say that A is to be (B - C) + (2C + D) . Here each of B, C, and D might represent complicated expressions. They might arise from a given problem in such a way that it is natural to first consider the difference B - C, and then the sum 2C + D. The two expressions so obtained are then added. It might only be observed later on, and even then it might not be self-evident, that the same quantity C appears in both expressions and can be canceled out so as to yield B+C+D. The quantity C itself, in the two expressions, might have different representations that do not appear similar on the surface but whose equality can be proved by applying a difficult mathematical theorem.

Now, the way that a rule is decomposed into stages may significantly affect the set Eij that emerges when the rule is

applied. For example, if we subtract C and then add C we are certainly wasting information, this phenomenon is known in the interval mathematics literature as “excess width” problem 125, 311. For example, suppose that B lies in [ I , 21, C lies in [3,5], and D in [ I , 41. Then B+C+D lies in 15, I l l . However, the best we can say about B - C is that it belongs to 1-4, -11. Since 2 C + D is in 17, 141, we deduce that ( B - C ) + ( 2 C + D ) lies in 13, 131, which is a wider set than the set [5, 1 I] obtained by considering B + C + D. The set Eij that we eventually obtain is the smallest set to which Aj(t j ) is known to belong in view of our deductions but what we get may depend greatly on the method and order of reasoning we use and how the definitions of the functions used in each rule or constraint have been formulated.

v. THE COBWEB MODEL: AN APPLICATION OF QUALITATIVE-QUANTITATIVE MODELING

Many practical econometrics models are formulated as difference equation systems. Therefore, an economics model can serve as an illustrative example of this important class of problems. Consider the so-called “cobweb models” from economics: Commodity producers face a supply and demand curve where some delay exists between the decision to produce and the sales of the product. For example, farmers might plant winter wheat based on 1994 prices, but sell their wheat at 1995 prices. Similarly a class of engineering students might decide their majors based partially on 1992 salaries but not be hired until 1995. Intuitively, if the price is initially high, the suppliers will plant more. At harvest time, the excess supply will drive the price down, causing the suppliers to plant less. This in turn drives the price up, and so on. The result is an oscillatory behavior.

To model this problem, let Qtl Ht and Pt denote the quantity planted, quantity harvested, and the price of a crop, respectively, at time t where t takes only integer values. We relate these quantities by the equations:

where S(z) is a positive increasing function for which positive increasing lower and upper bounds are known, and D ( x ) is a positive decreasing function for which positive decreasing lower and upper bounds are known, both defined for 5 2 0.

Assuming that the delivery cost plus overhead of a certain commodity is known to be $2/unit, we can determine that the minimum price of that commodity is $2. By the same token, if the import price of that commodity is $80/unit, based on the assumption that there is no limit on the import supply, we know that the maximal price of that commodity is $80. We assume that some additional information about functions S and D is known from which it is possible to derive the upper and lower bounds of these functions. For the example, arbitrary functions were chosen to represent the upper and lower bounds of the function S and D. The following expressions show the

1116 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 25, NO. 7, JULY 1995

definition and the initial state of the model.

Past Rules: Ht = Qt-1

Present Rules: S L ( P ~ ) = 0.2*Ht - 0.015*H," + 80 Scr(F't) = 0.205*Ht - 0.0145*

H," + 80 D L ( Q ~ ) = O.0145"P: - 0.205Pt - 2 Du(Qt) = 0.015*P: - O.2Pt - 2

Constraints: Qtt [O, m)

Ptt [2,80] Htt [O, m)

Initial State: Hot [45.50]

We set the program to run at most 100 iterations. The system reached equilibrium and stopped after the twelfth iteration with Q12 = [0, 781, H12 = [O, 781, and P12 = [4.34, 801. The results show that the behavior of the model tends to diverge. The equilibrium state is reached because the artificial boundaries we defined for the price was between 2 and 80, therefore, the value of price will never exceed 80 or go below 2. If we had not specified the boundaries for price, the system would have kept diverging and would never have stabilized. A more detailed discussion of the cobweb model is presented in the following sections. Fig. 1 shows how the range of Eij changed during the iteration process for each variable in the model.

A. Some Variations of The Cobweb Model

The price elasticity of supply is a measure of the change of quantity supplied (Q) in response to a change in price (P) . It is defined as the percentage change in quantity supplied divided by the percentage change in price, Le., ( A Q / Q ) / ( A P / P ) . It can also be represented as the derivative of the logarithm of the quantity with respect to the logarithm of the price, Le., ( d log Q ) / ( d log P ) . The higher the price elasticity, the larger the effect of a price change on quantity is. Price elasticity theory says that different types of commodities may have different price elasticities and the supply and demand curves of commodities with different price elasticities behave differently as the price of the commodity changes. Based on the above theory, we can construct three different types of models in accordance with the three categories which are (1) price elasticity > 1, (2) price elasticity = 1, and (3) price elasticity < 1.

B. Price Elasticity Greater Than 1

Assume that all the information we have about the model in the previous example applies to all of the following three examples, therefore the past rules, constraints, and initial state of the model are unchanged. The only changes are in the lower and upper bounds of the functions defining the relation between P (Price) and H (Quantity Harvested) and between Q (Quantity Planted) and P (Price). Again, arbitrary functions, with slope > 1, were chosen to represent the lower and upper bound functions. The definition of the new model is:

3 t 3:

8 7 7 6 6 5

4 4 3 + + u b 3 4 Ib

2

1

I 2 3 4 5 6 7 8 9 1 0 1 1 1 2 Time

9 ub L Ib

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 T i e

0 ub + Ib

e l . . . . . . . . . . . I . . . I . . . . ' i 1 2 3 4 5 6 7 8 9 10 11 12

Time

Fig. 1. The simulation of the cobweb model

Past Rules: Ht = Qt-1

Present Rules: S L ( P ~ ) =80 - 1.015*Ht Su(Pt) = 80 - l.Ol*Ht

DL(Qt) = l .O l*P t - 2 Du(Qt) = 1.015*Pt - 2

Constraints: Qt t [O; co)

Pt& [2,801

Htf [O, Initial State: Hot [45; 501

We set the program to run at most 200 iterations. The program terminated at the 130th iteration when the system reached the equilibrium state. The final values of quantity planted, quantity harvested, and prices are: QIX, = [0.02, 79.1791, HISO = [0.02, 79.1791, and P130 = [2, 79.981. Similar to the previous model, the results clearly show the diverging behavior of the system. If we look carefully at the results, we can see that not only do the ranges of Eij get larger and larger after each iteration, but also the wave which represents the movement of the Eij becomes stronger and stronger. Reasoning intuitively, the reaction of suppliers to the high price is to produce more for next year. The diverging behavior shows that the suppliers are overreacting

KIANG et al.: REASONING IN QUALITATIVELY DEFINED SYSTEMS 1117

m

1 E o u b

8 -4- Ib .$ - 0 - l h

+ b

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 6 11 16 21 26 31 36 41 Time T i

U 2

6 Q ub -C. Ib

j --lh .C Ib

Time 1 6 11 16 21 26 31 36 41

Time

- u b - u b i -C Ib + l b

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1

T i 1 6 11 16 21 26 31 36 41 T i

The simulation of the cobweb model with price elasticity > 1. Fig. 3. The simulation of the cobweb model with price elasticity = 1.

to the changes in price: A slightly higher price results in big increases in production, and a slightly lower price is always followed by big cuts in production. In Fig. 2 we only show the simulation results between time tl and t42.

C. Price Elasticity Equal To 1 The same information as in the previous model is applied.

Arbitrary functions, with slope = 1, are chosen to represent the lower and upper bound functions. This time we set the lower and upper bounds to be the same for functions S and D , and Ho. This represents the situation that the user has precise knowledge about the functional form of the relations. The result derived from our approach would be the same as the results derived from a pure quantitative analysis if the initial state is known with certainty (i.e., Hoc [45, 451 in the following example). The new model is defined as follows:

Past Rules: Ht = Qt-1

Present Rules: S L ( P ~ ) = 80 - Ht S u ( P t ) = 80 - Ht

D L ( Q ~ ) =Pt - 2

D u ( Q t ) =Pt - 2 Constraints: Qtt [O, co)

Ptt [a, 801

Htf [O, 0) Initial State: Hot [45,50]

We set the program to run at most 100 iterations. The system entered a cyclic loop and stopped at the fourth iteration with Q4 = [45, 501, H4 = [28, 331, P 4 = [30, 351. In Fig. 3 we show the behaviors of each of the variables in the model over 20 iterations in order to give the reader a better idea about the oscillatory behavior of the model.

D. Price Elasticity Less Than I The same information as in the previous model is applied.

The functions, in this case, have slope < 1. The new model is defined as follows:

Past Rules: Ht = Qt-1

Present Rules: SL(P,) = 80 - 0.79*Ht Su(Pt) = 80 - 0.785*Ht

D L ( Q ~ ) = 0.985*Pt - 2 Dv(Qt ) = 0.99*Pt - 2

Constraints: Qtt [0, m)

Ptf [2,801 HtE [O, ..)

Initial State: Hoc [lo, 701

We again set the program to run at most 100 iterations. The system reached a steady state and terminated at the

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 25, NO. 7, JULY 1995

7

6 5 5 4 4

3 2 2

1

1 6 11 16 21 26 31 36 41 Tune

1 6 11 16 21 26 31 36 41 T i

8 7 7

6 5 5

4 3

2 2 1

1 6 11 16 21 26 31 36 41 nm

VI. CONCLUSION AND FURTHER EXTENSIONS

This paper discusses the potential of applying the interval- based difference equations reasoning method to solve business applications that require reasoning with imprecise knowledge and involve time-lag problems. The method explicitly deals with qualitative difference equations and offers a more com- prehensible modeling methodology than a statistical model (i.e., a linearized statistical model), that is, model specification and interpretation of the results are easier to communicate to management personnel. Moreover, in the statistical ap- proaches, the linearization and the introduction of a pdf for the error term require simplifying assumptions which are not needed in our approach. In practice, the information available at the time we need to make a decision is usually incomplete, and therefore the interval-based difference equations reasoning method may be more suitable than traditional quantitative methods to build a decision-making model. More research needs to be done in order to develop decision-making tools that can be used for analyzing large-scale (realistic) prob- lems.

The likely explosion of final interval widths is a common problem with interval-based methods. This is similar to the explosion of possible behaviors problem in purely qualitative methods like QSIM. While the problem of inefficient solution methods can be alleviated by further research and new devel- opments in high speed computing, the degree of information explosion simply corresponds to the state of the knowledge

4 ub -*- ib

4 ub -C Ib

* -C Ib

used for building the model. The excessively wide range of final result derived from the first cobweb model (Fig. 1) does not mean that our reasoning scheme is not a useful decision

Fig. 4. The simulation of the cobweb model with price elasticity < 1.

45th iteration with Q45 = [42.318, 44.3131, H45 = [42.318, 44.3131, P45 = [44.993, 46.7801. The results show that after the sixteenth iteration, the ranges of E,, converge very slowly. The steady state of the system is reached at the 45th iteration because of the finite precision (i.e. 3 deci- mal places) used in our computation. If higher precision is used, it would have taken more iterations for the system to approach equilibrium or approach a narrow cycle around a stable state. However, the final values of EZJ would still be somewhere within the range we derived at this point. Also reasoning intuitively, the behavior shows that the re- action of suppliers to high prices is to produce more, but not too much to drive prices away from equilibrium. The reaction to low prices is also a moderate reduction in total production, therefore the behavior of the model tends to con- verge. Fig. 4 shows the behavior of E,, during the simulation process.

The simulation of this cobweb model results in three dif- ferent patterns of behaviors, which correspond well with the three known possible behaviors of cobweb model: diverging, cyclic, and converging behaviors. Readers should be aware that the results generated by our approach (and all other interval- based reasoning systems) only give us bounds of each variable at every time point during the simulation. Infinite number of combinations of possible behaviors can be inferred from the above results.

support tool but reflects the implications of weaWimprecise knowledge about a particular model specification. This, again, relates to the tradeoff issues between information cost and model accuracy.

One advantage of using difference equations is that piece- wise monotonic functions can be easily incorporated into the model. In a piece-wise monotonic function, the functional form of two variables can change with time, and does not have to be continuous. Since the time factor is modeled explicitly in our method, it is possible to define the relationship between two variables to have one set of boundaries at or before a certain time point and another set of boundaries after that time point. However, there is no straight-forward method of modeling the same situation using differential equations, which are what most qualitativekausal reasoning methods are built upon. If the relationship between two variables in the model can only be represented by a single function, some information about the relationships between variables may be lost. For example, suppose we know that the lower and upper bound functions for f ( : c ) at time t i , % 5 p . are f ~ ( z ) = 32 + 3 and f ~ ; ( z ) = 4z + 5,z 2 0, respectively. We also know that the functional form of f ( x ) after time tp will change to f l ( z ) , and the new bounds for fl(.) are: f~1(z ) = 5 + 2 and f u l ( z ) = 2z + 2, respectively. To model the function using only one functional form, we need to find

KIANG et al.: REASONING IN QUALITATIVELY DEFINED SYSTEMS 1119

the bounds for f ( x ) and fl(x) that are valid for all time i . These are: f~z(x) = x + 2 and fuz(x) = 42 + 5, which are much looser (wider) bounds than the original functions f(.) and f l ( X ) .

APPENDIX CONVERGENCE PROPERTIES

The procedures used in our approach always have the simple convergence property described below. To simplify notation, E is used to represent Eij. When applying the rules and constraints, several candidates for the set E , say E’b for n 2 1 are obtained. Since the iteration process is to incorporate additional information to reduce the set E”, it is clear that the set En+’ should be always smaller than or equal to the set E”, that is, c E” for all n 2 0. The worst case is that no new information is obtained so that En+’ = E”. In the limit, the set S nc==, E” is derived where N is the number of applications of rules and constraints. If infinitely many applications are made then N = 00. (If N is finite then S = E N . ) In the set theoretic sense the sets E” obviously converge to their intersection S even if N = ca. Since all the E” are subsets of E’, a fixed set of finite width, it follows from the monotonic convergence theorem that the measures of the E” tend to converge to the measure of S as n --f ca. Hence if each E” is to be the union of at most k intervals [ani, b,,,], 1 5 i 5 k , then for every E > 0 there is a no such that if n 2 no then the measure of En\S is less than E . If m > n > no the measure of E”\E”, is equal to:

k

En\Em is also less than E so that for each i with 1 5 i 5 k , we have b,i - b,i < E and ani - ami < E when m > n 2 no. Hence the sequences ani and b,i converge as n + co, for each i . Note that if we are to have E7’+l c for all n 2 1 then the number of intervals used can never be decreased. Thus, we can use q instead of k in the above summation to avoid the situation where some ani and b,; are defined but the alrLi and blni are not, or where the number of terms to be added is not I C .

If the sets E” are unbounded (have an infinite linear measure) then the above convergence still takes place in the set theoretic sense, but it is no longer clear if there is convergence in measure. It may be that every E” will have an infinite measure. However, even if one of the E” (and therefore all those after it) has a finite measure (i.e., is bounded), there may not be any clear way of predicting whether this will happen, and if it does happen, after how many applications (72. =?) will it happen. Thus, the decision as to when one should stop the calculations and accept the set E“, needs to be based on other factors.

It may be noted that if the number of intervals whose union is E”, is bounded by a fixed number, independent of n, then the measure of E” converges to that of S, as described below. Either all these measures are infinite, or from some point

onwards the measure of E” is finite and from that point the monotonic convergence theorem referred to above applies. If the number of intervals of E” is allowed to increase to infinity, then it may be that each E” has an infinite measure while the measure of S is equal to zero. For example, let E“ be the union of [n,, 001 and b , j+ l /n] for 1 5 j 5 71-1 . Then E”+’ c E” and E” has an infinite measure while S = { 1 , 2 , 3 , . . .} is the set of positive integers which, being a countable set, has a zero measure. Interested readers should refer to H. L. Royden [27] for a comprehensive discussion on the monotonic convergence theorem.

ACKNOWLEDGMENT

The authors would like to thank Dan Berleant, Reiner Lang, and the anonymous referees for their valuable comments on an earlier draft of this paper.

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I120 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 25, NO. 7 , JULY 1995

Melody Yihwa Kiang (M’93) was born in Taipei, [ 191 E Hyvonen, “Constraint reasoning based on interval anthmetic. The tolerance propagation approach,” Art$ Intell., vol. 58, nos. 1-3, pp Taiwan, R.0 C. on June 29, 1962 She received the

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Management Science, vol 3 1, no. 4, pp 3 9 5 4 14, Apr 1985 ment information systems from the University of [21] J Klayman “Simulation of six decision strategies’ Comparisons of Wisconsin, Madison, in 1987, and the Ph D degree

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Deriving interests include the development and applications of artificial intelligence behavior from structure,” Arr(ticla1 Intel[ I VOl 24, PP 169-204, 1984 techniques to a vanety of management problems She is an associate editor

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D~ K~~~~ a member of TIMS

1311

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New York: Erlbaum, 1980, pp. 33-58.

Aim0 Hinkkanen received the Ph.D. degree in mathematics from the University of Helsinki, Fin- land, in 1980.

He has held faculty positions at the University of Michigan, Ann Arbor, and the University of Texas, Austin. Currently, he is Professor of Mathematics at the University of Illinois, Urbana-Champaign. He is also currently an Alfred P. Sloan Research Fellow. Dr. Hinkkanen’s principal research interests are in complex analysis, particularly complex dynamical systems and quasiconformal analysis.

Andrew B. Whinston is the Hugh Roy Cullen Chaired Professor at the University of Texas, Austin, and the Jon Newton Fellow at the IC’ Institute. His research focuses on organization information systems, economics of information systems and decision support systems. He has published over 200 papers and 16 books and is the Editor-in-Chief of Decision Support Systems and the Journal of Organizational Computing.