Uncertain reasoning using the Dempster-Shafer method: an application in forecasting and marketing...

Article Uncertain reasoning using the Dempster-Shafer method: an application in forecasting and marketing management

E. Cortes-Rello and F. Golshani Department of Computer Science, Arizona State University, Tempe, AZ 85287, USA.

Abstract:The intendedpurpose of this article is twofold: to study techniques for uncertainty managemeH in expert systems, particularly the Dempster-Shafer theory of belief; and to use this method in the construction of an expert system for the$eld of forecasting and marketing management. Compared with the probabilistic approach, which can only deal with singleton possibilities, the Dempster-Shafer approach proves to be superior because it provides the ability to deal with sets ofpossibilities. Since market analysis and forecasting always have a strong element of uncertainty associated with them, and since, in general, we consider a combination of several different forecasting techniques for planning our marketing strategies, the Dempster-Shafer approach is particularly suitable. Here we present a short introduction to this theory, briefly discuss the domain of application (selection of forecasting techniques for marketing planning), discuss the interesting components of our expert system, and analyze our experiences in applying the theory to this domain.

1. Introduction The problem of incomplete, inexact or uncertain information comes up in most information processing systems such as databases and decision support systems. It is particularly troublesome in the context of automated reasoning and expert systems. Obviously, when knowledge in an expert system is not complete (or certain), the answer to any query is only an approximation to the true result. The aim is to get the best approximation.

Expert Systems, February 1990, Vol. 7, No. 1

Uncertainty in rule-based systems, which is the concern of this article, can be represented by associating a certainty factor (CF) with each rule that reflects an uncertain relationship. A,n example is:

CF(E,H,) E _ _ _ _ _ _ _ _ _ _ > H

where E is the evidence (better known as condition or situation), His the hypothesis (also referred to as the conclusion), and CF(E,H) is the certainty factor attached to the rule. Two major problems must be resolved:

- how to measure the degree of inexactness and calculate certainty factors for inexact situations,

- and how to propagate uncertainty in making inferences and arrive at best conclusions in spite of some rules not being definite.

A number of methods of management of uncertainty have been proposed by researchers from different fields with different points of view. Among them are the fuzzy logic approach of Zadeh (1979), the Bayesian approach (Lee et a1 1987), and the theory of belief of Dempster and Shafer (1976). These methods have been employed in the field of AI, each one with a different degree of success. For a comparative study of these methods see Bhatnagar and Kana1 (1986) and Zadeh (1986), Lee et a1 (1987).

Dempster-Shafer theory is considered to be a natural extension of traditional probability theory. The main difference comes from the fact that in classical probability theory we are interested in probabilities of the points in our universe of dis- course (called the frame of discernment) whereas Dempster- Shafer’s approach attempts to find probabilities for a collec- tion of points (i.e. subsets) in the frame of discernment. A significant advantage of the Dempster-Shafer approach is that it provides an effective method for combining (sometimes contradictory) information obtained from multiple knowledge sources.

While we will present a more detailed comparison later, we should note that, among other advantages, the Dempster- Shafer methodology is surprisingly flexible and it can be ap- plied to many different domains. In this study, we are applying this theory to the problem of selecting appropriate forecasting methods.

Forecasting is concerned with predicting the future. We need forecasts to make decisions that will affect the future of the business. Unfortunately, only one thing is certain about forecasts: they will always be partially in error. Fortunately, however, there exist many forecasting techniques, each one with some weaknesses and some strengths.

Given a forecasting situation, some techniques may work better than others. By a forecasting situation we mean the intentions and purpose of the forecast and the constraints such as time-span, resources available, data available, etc. The aim is to balance the advantages and disadvantages of the forecasting methods with regards to a forecasting situation’s limitations and requirements. The result will be a method, or a

combination of methods, suitable for the forecasting situation.

It is important to note that a combination of forecasting methods can offset any technique’s limitations with the advantages of another, while retaining the strengths of the first. This happens because various techniques can incorporate different underlying assumptions. This is a very difficult but important management task. We must realize, however, that forecasting has nc value on its own - it is only meaningful as it relates to and supports decision making.

In the next section, we will discuss the domain of our expert system. We will then present an introduction to the Dempster- Shafer approach, and, finally, we will discuss the interesting parts of our particular implementation.

2. The domain of the expert system The system aims at selecting a forecasting technique (or a combination of several techniques) based on the characteristics of a forecasting situation given by the user.

Marketing managers face new market realities and uncer- tainties, and must deal with environments that change rapidly. In such environments, marketing planning plays a vital role. Marketing planning requires quantitative estimates of spe- cific demand by product, territory and type of customer (Kotler 1980). Furthermore, we must estimate the future course of market demand. Demand estimates are essential in carrying out the analysis of market opportunities, the planning of the marketing activities and the control of the marketing performance.

In the estimation of future demand (called demand forecasting) several forecasting techniques can be used, but the selection of the best forecasting technique (or the best combination of forecasting techniques) is a complex process. Each forecasting technique has strengths and weaknesses, and every

forecasting situation is limited by constraints such as availability of data, expected cost of the forecast, expected accuracy, urgency, etc., as explained thoroughly in Chambers (1971) and Georgoff and Murdick (1986). Baiancing the strengths and weaknesses of the techniques with regard to a forecasting situation’s constraints and requirements is a very important task in the marketing environment.

To illustrate the decision process we will present two cases:

Case I. A manufacturer of capital goods (industrial boilers) will try a new approach to forecast the demand of boilers for electricity generation for the next ten years. This item is the main product, and the survival of the company depends on it. The sources of input data to the forecast model are: the chief executive officer’s files and reports, the chief financial officer’s files and reports, internal historic data, clients’estimations, macro- economic data, etc. This will be a long term forecast, and the company would like annual updates of the forecast. The users of the forecast will be sophisticated analysts, with high quantitative skills. The expected cost of the forecast is high, and the developers of the forecast have access to all of the company’s computer resources. Available as input to the forecast model are fifty years of real past demand, and ten years of past forecasts. The demand is cyclic, and the forecast model must consider possible environmental changes. The forecasting model must detect and reflect turning points and should be accurate.

Case 11. The same company needs a monthly demand forecast of parts and supplies (a minor product line). The users of the forecast are the sales managers and the sales director. This is a short term forecast, with monthly updates. The sources of input data are: the sales managers, the sales persons, clients (main-

L

naive extrapolation sales force composite jury o f executive opinion scenario methods

istorical analogy

industrial market survey consumer market survey

judgement methods

ethods

moving averages exponential smoothing time series decomposition box- jenkins

correlation methods regression models econometric models input-output models

time-series methods

Forecasting Methods

associa tion

Figure 1: Taxonomy of forecasting methods

10 Expert Systems, February 1990, Vol. 7, No. 1

tenance schedules etc.), and past data on parts and supplies demand (five years). The expected cost of forecasting is low, and the developers of the forecast have access to a micro computer. There is a need for a detailed forecast.

As we can see, these cases, even though within the same company, have very different forecasting requirements and constraints, and will demand different forecasting techniques. Therefore, in this context, an expert system can be a very powerful assistant in the marketing planning process.

2.1 . Requirements and constraints in forecasting The following issues must be considered by the system in dealing with a forecasting situation:

(1) Time element: the span of the forecast (short, medium, long), the urgency of the forecast, and the frequency of the forecast. Company resources: the company resources committed to the forecast situation such as the characteristics of the users and developers of the forecast, the expected cost, and the computer resources available. Inputs to forecast: characteristics of the inputs to the forecast model such as availability of past data, fluctu- ation of the primary series, and management or environmental changes expected. Outputs of forecast: characteristics of the outputs of the forecast model such as level of detail of the output, degree of accuracy, need of interval forecasting (probabilistic), need of reflection and detection of turning points.

(2)

(3)

(4)

2.2. Forecasting techniques We consider the following forecasting techniques in the process of building a knowledge base for the expert system (see Figure 1). While most of these methods are obvious by their name, the reader is referred to Kotler (1980) and Georgoff and Murdick (1986) for details of each technique.

Judging Methods: which include naive extrapolation, sales force composite, jury of executive opinion, scenario methods, Delphi technique and others such as historical analogy.

Counting Methods: such as market testing, consumer market survey and industrial market survey.

Time Series Methods: which include moving averages, exponential smoothing, time series decomposition and Box- Jenkins.

Association Methods: from which correlation methods, regression model, econometric models, and input-output models are better known.

Forecasting is a vital task. The survival of a business can depend heavily on the quality of demand forecasting. Our system advises the marketing manager on the selection of the best set of forecasting techniques given the characteristics of a forecasting situation.

3. The Dempster-Shafer theory of evidence

This theory can be viewed as a natural extension of the classical probability theory. Essentially, in probability theory, we assign probability values to the points in the universe. The extension in the Dempster-Shafer theory is that we can talk about probability values for subsets of the universe. By pro-

viding an effective method for combining these values, Dempster and Shafer propose an interesting technique for dealing with uncertainty.

The basic underlying idea of the Dempster-Shafer theory can be best understood through a concrete example. We use an example first presented in Zadeh (1984). Suppose a country, say Alpha, suspects that a hostile country, say Beta, has sent a submarine to the temtorial waters of Alpha. A natural course of action is that the government of Alpha calls upon a group of experts to determine the validity of this suspicion. Assume experts E l , E2, ... ,En are asked, to suggest the possible locations for the submarine. We will use the symbol Li to denote the locations proposed by expert Ej. Note that for i=l, ..., n, Li is a subset of territorial waters. Let us assume that for some m a , Lm= ... =Ln= 0 . (Some experts may believe that there is no submarine.) Suppose there is a particularly sensitive area within the territorial waters, say zone A. The immediate ques- tion is: ‘Is the submarine in A?’ Considering the opinions ex- pressed by the experts, we would be interested in two cases:

I.

11.

those who predicted that the location is definitely in A,. that is Li c A, and those who suggested a possibility for the location being is A, that is Li n Ais not empty.

Suppose k experts fall in the category I, and j experts fall in the second category. (Since I implies II, it should be obvious that j>k.)

One possible way of dealing with different opinions is aver- aging. By using this simple method, the average certainty is k -, and the average possibility is l. By disregarding the opi- n n nion of those who believed that there was no submarine, the

average certainty and possibility would be - and- m-1 (m-1)

respectively. This process is referred to as normalization, and many claim that it leads to counterintuitive results.

We can add another factor to the problem, namely ranks for the experts and the value (or weight) associated with each expert’s opinion. (We will see later that these weights corre- spond to the basic probability assignments.) Suppose that Wi is the weight associated the opinion of Ei, where

k

OSWiSl

and

Wl+W2+ ... +W,=I.

By using a concept similar to these weights (called basic probability assignments), Dempster and Shafer define belief and plausibility functions, denoted Bel(A) and PI(A) respectively, as follows:

(0 Bel(A) = (Cwj ) whenever Li i A i

1 6 (ii) Pl(A) = - (Cwj ) whenever Li” A 0 a .

i

Expert Systems, February 1990, Vol. 7, No. 1 11

where 6 is given by the following equation:

6 = 1 - C W i whenever Li n A= 0 . I

We can define a belief and a plausibility value for any subset of the domain of interest. However, there is an important difference with the Bayesian (probabilistic) approach. In the classical probability theory, if two two sets A and B are disjoint (that is A n B = 0 ), then we have:

Prob(A) + Prob(B) = Prob (A u B),

whereas a similar equation would not hold for belief functions - 1.e.

Bel(A) + Be1 (B) # Bel(A u B)

even if A and B are disjoint. Also, in contrast to probability theory, we have:

Bel(A) + Bel(T A) # 1.

Intuitively, this nonequality makes sense, since a person’s degree of belief in A does not give us any information on his belief in A.

There are a number of interesting observations to be made. The first, which follows immediately from the equations i and i i above is that:

Bel(A) 5; Pl(A).

Another important property is the following equation:

Bel(A) + P1( 1 A) = 1.

There exists an interesting relationship between ordinary probability, belief and plausibility. Given Prob(A) to be probability of A, we have

Bel(A) I Prob(A) 5 Pl(A).

Thus, one can view the degree of belief and the degree of plausibility on a hypothesis as the lower bound and upper bound on probability of that hypothesis.

This completes our informal overview of the theory. In the following section, we define more rigorously some relevant terminologies along with examples from the domain of interest, namely, the forecasting domain. Details of defini- tions can be found in Gordon (1985).

3.1. Definition of some basic concepts Frume of Discernment: The frame of discernment, denoted 0, is the set of all singleton hypotheses. These singleton hypotheses are assumed to be mutually exclusive and exhaustive.

Example: Aframe of discernment containing some forecasting methods can be:

0 = { Delphi-Technique, Market-test, Naive-extr, Market- survey, Movinkavg ]

Basic Probability Assignment: The basic probability assignment, denoted m, assigns a nun- ber in the interval [O,l] to every subset in @, such that:

the quantity m(a) is called a’s basic probability number. A basic probability assignment is also known as an opinion.

Example: Given a forecasting situation, an expert can give the following basic probability assignment:

m(De1phiTechnique )) = 0.1

m( [ Market-test, Market-survey } ) = 0.2

m( [ Market-test )) = 0.3

m(O) = 0.4

Note that m(O) denotes the proportion of uncommitted belief. Belief Functions: A belief function determines the measure of belief in each of the subsets of the frame of discernment. In general, for any a&, a belief function returns a measure of the total amount of belief in a, and not only the amount exactly committed to a. That is:

Bel( a ) = &(P).. pca

Example:

Bel( { Market-test , Market-survey ])= m( { Market-test , Market-survey ) ) + m( [ Market-test ) ) + m( { Market-survey 1)

= 0.2 + 0.3 + 0.0 = 0.5

Plausibility: Denoted Pl(a ), where a r O , establishes a ‘maximum likeli- hood’ for the hypothesis a. It is given by:

P l (a )= 1 - Bel( la)

where al is the complement with respect o 0 of a.

Example:

Bel( 1 { Market-test , Market-survey )) = Bel((De1- phi-Technique , Naive-extr, Moving-avg )) =0.1

and if

a = { Market-test, Market-survey )

then

P l (a )= 1 -Bel(+x ) = 1-0.1 =0.9


Belief Intervals.: The belief interval for hypothesis a, where ale, is given by:

[ Bel( a >, PI( a ) 1 Note that the value

Pl(a) - Bel( a )

represents the amount of uncertainty with respect to a.

Example: Suppose

a = { Market-test , Market-survey } . As computed in the previous examples, the interval of belief

for hypothesis a is:

a : [ 0.5 ,0.9 ]

The interpretation is : the evidence provides strong support

The amount of uncertainty of this hypothesis is 0.4. Note that if the interval of belief for a hypothesis is [ 1 , 11, it

means we have complete certainty on that hypothesis. On the other hand, the interval [0 ,1] represents total uncertainty.

One of the advantages of the D-S approach is that belief in a concept evolves as we accumulate evidence. This property is particularly important in such domains as legal analysishea- soning in which when several paths of reasoning lead to the same conclusion strengthen the argument.

for hypothesis a, and weak support for la.

4. The uncertainty management system The objective of our expert system is to assist a marketing manager in choosing an appropriate forecasting method (or a

combination of methods) by balancing the strengths and weaknesses of the forecasting techniques with regard to a forecasting situation constraints and requirements.

4.1. The components of the system.

In addition to the knowledge base, the expert system has three main components (see Figure 2).

The first one is the Knowledge Acquisition module which is used by the knowledge engineer to build a knowledge base (consisting of a rule base and a structure containing the frames of discernment). The knowledge base is built prior to the consultation phase and reflects the relationships between the forecasting method’s advantages and disadvantages and the forecasting situation’s characteristics.

The consultation phase uses the next two modules. The second module is the Dempster-Shafer module that will be used during consultation time to maintain a ‘belief base’ consisting of the current beliefs for each one of the frames of discernment. The third is the Inference Engine module that uses a backward chaining strategy and that calls the Dempster- Shafer module to update the ‘belief base’ each time a rule is‘ fired. 4.2. Knowledge representation The relation between forecasting methods and the characteristics of a forecasting situation is represented using rules with the following structure. A rule consists of a ‘rule identi- fier’, a ‘condition part’and an ‘action part’. The condition part is a conjunction of conditions, where each condition is an Object-Attribute-Value triplet. The action part consists of an Object-Attribute-Value triplet and an opinion associated with it. The opinion consists of a value (between 0 and 1) and a label that distinguishes the value as either ‘belief’ (confirm) or ‘disbelief’ in the action (disconfirm). For example, the rule:

EXPERT

I

Demspter-Shafer

module

Engine Inference 1- USER

Figure 2: Components of the system


(1 ( (forecast span (long) 1

(input past-data (not-available)) ) ( (forecast method (market-testing

consumer-market-surve y industrial-market-survey) )

( 0.6 confirm )) )

means:

Rule #1: If

the forecast span is long AND input past data is not available

the forecasting method is in the set { market-testing consumer-market-survey industrial-market-survey } with a belief 0.6.

Then

Similarly, the rule:

(20 ( (forecast need (immediate)

(forecast expected-cost(1ow) ))

( (forecast method (market-testing)) (0.9 disconfirm) ) )

means:

Rule 20 If

The forecast is needed immediately AND the expected cost of forecasting is low

The forecasting method should not be market testing with a belief of 0.9

Then

The action part of rule 20 can also be read as ‘the forecasting method is in the commplement of the set (market-testing}, with a belief 0.9’. Note that the complement is constructed with respect to the universal set (which is the frame of discernment) corresponding to forecasting methods.

In the knowledge acquisition phase we build a structure containing a frame of discernment for each and every one of the Object-Attribute pairs that appear in the rules. For example, in the case of rules number 1 and number 20 we have the pairs (forecast need), (forecast expected-cost), (forecast span), (input past-data) and (forecast method) and we need a frames of discernment for each one of them. A frame of discernment of the pair (forecast span) could be the set ( short medium long } . See Figure 3 for examples of actual rules from the rule base.

4.3. Propagation of belief values

The inference engine uses a backward chaining strategy. The whole consultation process starts trying to solve the goal (say, what forecast method). New sub-goals are created for the conditions of the rules whose action part matches the goal until no rule can be matched. When no rule can be matched with the goal, the inference engine prompts the user for an ‘opinion’

r u l e 85

input chanqes_in-manaqemcnt-declsio~s IS (expected)

disconfirm: forecast method is (sales_force_composlte)

if:

then:

with a belleC of 0.7

_________^^-_-__---------------~----~------------------

rule 16

forecast cxDected-coat is Ilow)

conflrm: forecast method is Isales_force_com~osite moving-averages)

i f :

then:

with a b e l i e f or 0.9 ....................................................... rule t 7

forecast expected-co8t is [low)

disconfirm: forecast method is [market-testing)

i f :

then:

with a b e l i e f of 1.0

Figure 3: Some rules from the rule base


regarding the goal: the frame of discernment of the object-attribute part of the goal is presented to the user, and a belief is entered for one or several of the elements of the frame discernment. That opinion is added to the ‘belief base’ (see Fig- ure 4)

The ‘belief base’ is also modified each time a rule fires. For example, let us suppose that rules #1 and #20 mentioned above fire during a consultation. The ‘belief base’ will be modified for the object-attribute pair (forecast method).

What we have in terms of the Dempster-Shafer theory is: Opinion 1 (rule#l):

m 1 ( { market-testing consumer-market-surve y industrial-market-survey ) ) = 0.6

ml ( O ) = 0 . 4

Opinion 2 (rule#20):

m2( 1 { market-testing ] ) = 0.9 m2( 0 ) = 0.1

Dempster rule combines the opinions in the manner presented in Figure 5. It works in the following way:

We have to calculate a new basic probability number for each one of the combinations among basic probability numbers in ml and m2. For the first combination. we have:

and

m 1 ( { market-testing consumer-market-survey Industrial-market-survey 1) = 0.6

m2( 1 [ market-testing 1) = 0.9

The calculation is done in two steps:

The intersection of the sets: (i)

{ consumer-market-survey industrial-market-survey market-testing ]

and

1 { market-testing ]

is:

[ consumer-market-survey industrial-market-survey ]

(ii) The new basic probability number for the intersection is:

0.6 * 0.9 = 0.54

The result is then:

m l 0 m2 ( { consumer-market-survey industrial-market-surve y ) ) = 0.54

We follow this same procedure for each possible combination among basic probability numbers in ml and m2.

As a result, we have a combination of opinions (a combination of belief of two rules having a common object-attribute pair in the action part) that is updated as the ‘current opinion’ corresponding to the object-attribute pair (forecast method) into the ‘belief base’. In this case:

what is your opinion about forecast sDdn 7

belief confirm or disconfirm 1. more than five years 0 C 2 . more-than-one ?ear 0 C 3. one-year_Gr-lSss 0 C 4. one-month-or-less 0 C

- - > 4 - belief (between 0 and 1 1 - -> 0.8

confirm or disconfirm (c or d ) --> c ---

- what is your opinion about forecast span 7

belief confirm or disconfirm 1. more than five years 0 C 2 . more-than-one $ear 0 C . 3. one vear or less 0 C 4. one~montii-o?-less 0 . 8 C

--> exit ----

Figure 4: A typical dialogue. User’s entries are shown underlined


rnl( { market-testing, consumermarketsurvey , indust r ia lmarketzurvey ))

0.6

e

’ 0.4*0.1= 0.04 0.4

mZ(~(market-testing))

0.9

m2( 8 1

0.1

(consumermarket-survey indust r ia lmarketzurvey

0.6*0.9= 0.54

-(market,testing)

0.4*0.9= 0.36

{ c o n s u m e r m a r k e t r u r v e y industrial-market-survey market-testing)

0.6 *O. 1 ~ 0 . 0 6

Figure 5: Dempster’s Rule of Cornhination

m( (market-testing} ) = 0.36 m( [ consumer-market-survey industrial-market-survey 1) = 0.54 m( (market-testing consumer-market-survey industrial-market-survey }) = 0.06 m( 0) =0.04

will be updated as the current opinion for (forecast method). At the end of the consultation the expert system can give a

recommendation based on the contents of the ‘belief base’ corresponding to the goal (forecast method) (see Figure 6). Let us suppose that rules #I and #20 were the only rules fired whose action part matched with the main goal (forecast method). In that case the expert system can calculate the intervals of belief for several hypotheses and select the best. For example if we compare the belief intervals of the hypothesis ‘(forecast method) is { market-testing } ’ against the hypothesis ‘(forecast method) is { consumer-market-survey industrial-market-survey ] ’we will have (based on the example above):

bel( ( market-testing 1) = m( (market-testing 1) = 0 bel( (market-testing]) = m( [ market-testing } ) +

m( [ consumer-market-survey industrial-market-survey ] ) = 0.36 + 0.54 = 0.90

and the belief interval for { market-testing} is:

(market-testing) : [O, 0.11

For the second hypothesis we have:

16

bel( [ consumer-market-survey industrial-market-survey } )= 0.54 bel(7 [ consumer-market-survey industrial-market-survey 1) = 0.0

and the belief interval is:

( consumer-market-survey industrial-market-survey} : [ 0.54,1]

The interpretation of the above belief intervals are:

- we have a strong evidence against market testing and no evidence in favor of it,

- and we have no evidence against the second hypothesis and some evidence (0.54) in favor of it.

5. Conclusions Good forecasting is the basis for good marketing planning. In particular, sales forecasting is the basis for all the other budgeting and planning activities in a company. To select the ‘best’ forecasting method given a situation is not a trivial task, and it is a very important and frequent decision that marketing managers must make. Furthermore, the ‘best’ solution is not a single method, but a combination of several methods.

Our expert system can advise a marketing analyst on the selection of the combination of forecasting techniques that best matches a situation. After an interactive session, the user gets the best set of forecasting techniques such that their weaknesses and strengths are balanced. We envision our system embedded into a decision support system (DSS) for mar- ke ti ng manage men t.

Expert Systems, February 1990, Vol. 7, No. 1

The intervals of belief are:

1 market-testinq , sales-force-composite : 10.6129 0.8961

I market-testing moving-averages : (0.4621 0.8492

I market-testing I : 10.4621 0.6757

{ sales-force_composite 1 : [0.1607 0.8061

I moving-averaqes I : [O.OOOO 0 . 2 4 0 9 1

Figure 6: Results of a consultation. Intervals of belief represent lower and upper bound on the probability values.

The knowledge that we have on how to select a set of forecasting methods is complex, incomplete and subjective. For these reasons, the system must be able to manage uncertainty. While there exists several methods for management of uncertainty in expert systems, the Dempster-Shafer approach proves superior for a number of reasons. Compared to ad hoc techniques, this technique is more desirable because of its rigorous mathematical underpinning. Compared with other probability-based methods such as Bayesian, the Dempster- Shafer theory is more powerful since it can work with probability of sets of points instead of probability ofjust individual points. In addition it can handle contradictory evidence in a satisfactory manner.

These properties of the Dempster-Shafer method are crucial for the type of situation we were modeling because:

- the solution is not a single method, but a set of methods. - the problem is complex, and the solution is based on

subjective (and possibly contradictory) opinions of expert in forecasting techniques. we can build for different levels of abstraction (for example, rules referring to an individual method, or rules referring to a ‘class’ of methods.

- the concept of methods having strengths and weaknesses can be modeled using rules with confirming and ‘decon- firming’ beliefs over sets of hypotheses.

-

A prototype of our ideas was implemented in Common Lisp on a VAX-VMS machine. A microcomputer version of the system is under way.

6. Acknowledgements The authors gratefully acknowledge generous financial support from Bull HN Worldwide Information Systems.

7. References BHATNAGAR, R.K. and L.N. KANAL (1986), ‘Handling Un-

certain Information: A Review of Numeric and Non- numeric Methods’, in Uncertainty in Artificial Intel- ligence, Elsevier Science Publishers B.V. (North Hol- land).

CHAMBERS, J.C. (1971), ‘How to Choose the Right Forecast- ing Technique’, Harvard Business Review, 49, pp. 45-74.

GEORGOFF, D.M. and R.G. MURDICK (1986), ‘Manager’s Guide to Forecasting’, Harvard Business Review, Jan-Feb 1986, pp. 191-202.

GORDON J. and E. SHORTLIFFE, (1985), ‘AMethod for Mana- ging Evidential Reasoning in a Hierarchical Hypothesis Space’, Artificial Intelligence, 26, pp. 323-327.

KOTLER, P. ( 1980), Marketing Management: Analysis, Plan- ning and Control, Prentice-Hall.

LEE, N.S., Y.L. GRIZE and K. DEHNAD (1987), ‘Quantitative Models for Reasoning under Uncertainty in Knowledge- based Expert Systems’, International Journal of Intelli- gent Systems, 11, pp. 15-38.

SHAFER, G. (1976), ‘A Mathematical Theory of Evidence’, Princeton University Press.

ZADEH, L.A. (1979), ‘A Theory of Approximate Reasoning’, in Hayes, Michie and Mikulich (eds), Machine Intel- Eigence, 9 , Ellis Horwood.

ZADEH, L.A. (1984), ‘Review of Shafer’s “A Mathematical Theory of Evidence” ’, The AZ Magazine, Fall 1984, pp.

ZADEH, L.A. (1986), ‘A simple view of the Dempster-Shafer Theory of evidence and its implication for the rule of combination’, The Al Magazine, Summer 1986, pp. 85-90.

81-83.


The authors Enrique Cortes-Rello

Enrique Cortes-Rello is a PhD student at Arizona State University. He has had theoretical and practical training in marketing and the development of information systems. Mr Cortes-Rello worked as Marketing Manager and was heavily involved in demand forecasting and the development of marketing information systems. His current research inter- ests are: reasoning under uncertainty, decision support systems and deductive databases.

Forouzan Golshani

Dr Forouzan Golshani is on the faculty of Computer Science at Arizona State University. Previously he was with the Department of Computing at Imperial College, University of London, England. He has over twelve years of experience in programming languages, artificial intelligence and database systems. He has been on the program committees of several international conferences including the 1988 IFIP Conference on ‘The Role of A1 in Databases and Information Processing’, the AAAI Workshop on Data- bases in Large A1 Systems and the IEEE International Con- ference on Computers and Communications.

Research and Development in Expert Systems V Edited by B. KELLY and A. RECTOR The papers from Expert Systems ‘88 cover many important aspects of current work - in particular they address theoretical topics, practical techniques and real applications of expert systems. The applications cover a wide spectrum of commercial and industrial interest - the theme of the conference was ’Integrating with Mainstream Software Development’. E32.50 net 0 521 37324 7 359 pp. 1989 British Computer Society Workshop Series

For further information write to Susan Chadwick at the address below. Ring 0223 325970 at any time to order any Cambridge book on your credit card.

Research and Development in Expert Systems VI Edited by N. SHADBOLT This volume contains the refereed and invited papers from Expert Systems ’89, held in September 1989 with the theme of ’Applications and Methods in Knowledge Based Systems’. The papers describe practical applications in finance, engineering, manufacturing, power utilities and the food industry. Other papers report important theoretical work in areas of current research, such as inference, knowledge representation, distributed problem solving, knowledge base maintenance, HCI and knowledge acquisition. €30.00 net 0 521 38477 X 320 pp. 1989 British Computer Society Workshop Series

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18 Expert Systems, Febrdary 1990, Vol. 7, No. 1

Uncertain reasoning using the Dempster-Shafer method: an application in forecasting and marketing...

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