Sales Using ANN n Fuzzy

7/29/2019 Sales Using ANN n Fuzzy

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.Computers in Industry 37 1998 115

An intelligent sales forecasting system through integration ofartificial neural network and fuzzy neural network

R.J. Kuo a,), K.C. Xue b

aDepartment of Marketing and Distribution Management, National Institute of Technology at Kaohsiung, Kaohsiung County, 824, Taiwan

bGraduate School of Management Science, I-Shou Uniersity, Kaohsiung County, 80048, Taiwan

Received 3 May 1997; accepted 16 February 1998

Abstract

This paper attempts to develop an intelligent sales forecasting system which can consider the quantitative factors as well . .as the non-quantitative factors. The proposed forecasting system consists of four parts: 1 data collection, 2 general pattern

. .model, 3 special pattern model, and 4 decision integration. In the general pattern model, a feedforward neural network .with error-backpropagation EBP learning algorithm is employed to learn the time series data, or quantitative factors.

However, unique circumstances, e.g., promotion, may cause a sudden change in the sales pattern. To this end, this paper .utilizes fuzzy logic which is capable of learning fuzzy neural network, FNN to learn the experts knowledge regarding the

effect of promotion on the sales. Finally, the outputs from the above two mentioned models are integrated with time effect

through a feedforward neural network with EBP learning algorithm. Evaluation of the model results indicates that the

proposed system performs more accurately than the conventional statistical method and single artificial neural network

.ANN .q

1998 Elsevier Science B.V. All rights reserved.

Keywords: Sales forecasting; Artificial neural networks; Fuzzy neural networks

1. Introduction

To enhance the commercial competitive advan-

tage in a constantly fluctuating environment, an or-

ganizations management must make the right deci-

sion in time depending on the information at hand.

The decision lead time ranges from several years toseveral hours based on the types of business. Thus,

making an accurate decision plays a prominent role.

)

Corresponding author. Tel.: q886-7-6011000 ext. 4213;e-mail: [email protected]

Intuitively, forecasting models can provide rea-

sonable estimates by using historical data. Therefore,

if the marketing department can estimate the sales

quantity for the next period, the materials department

can then effectively control the inventory to achieve .just-in-time JIT . In addition, the production depart-

ment can make the scheduling and arrange the facil-ity utilization. Such an action may cause the produc-

tion cost to decrease. Therefore, obtaining an accu-

rate forecast appears to be critical. Statistical meth-

ods, such as regression models and ARMA, have

long been candidates for decision makers. However,

these methods are only efficient for data which are

seasonal or cyclical. If the data are influenced by

0166-3615r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. .PII S 0 1 6 6 - 3 6 1 5 9 8 0 0 0 6 6 - 9


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( )R.J. Kuo, K.C. Xuer Computers in Industry 37 1998 1152

special cases, like promotion, they are inaccurate. .Artificial neural networks ANNs that are better

w xthan the conventional statistical methods 1,2 havebeen recently employed, but even then the problem

still arises. Thus, this paper first attempts to develop

a forecasting system capable of handling such a

circumstance, i.e., promotion.

Fuzzy logic has been applied in the area of con-

trol and has shown highly promising results. Fuzzy

logic attempts to capture vague experts knowledge.

However, the setup of experts knowledge resulting

from the membership function setup of fuzzy sets is

quite subjective. Therefore, this paper also aims to

develop an learning algorithm for fuzzy logic, called .fuzzy neural network FNN .

Thus, this paper develops a sales forecasting sys- .tem which consists of four parts: 1 data collection,

. . .2 general pattern model ANN , 3 special pattern

. . .model FNN , and 4 decision integration ANN .To evaluate the proposed system, the actual data

provided by a chain supermarket company are used.

According to these results, the proposed system per-

forms more accurately than the conventional statisti-

cal method and single ANN, particularly when the

promotion is conducted.

The rest of this paper is organized as follows.

Section 2 provides some necessary background in-

formation while the proposed system is discussed in

Section 3. Section 4 summarizes the evaluation re-

sults. Discussion and concluding remarks are finallymade in Sections 5 and 6, respectively.

2. Background

This section briefly reviews the sales forecasting

methods and the applications of artificial neural net-

works in this area. In addition, fuzzy modeling and

fuzzy neural networks are also discussed.

2.1. Sales forecasting

Sales forecasting always plays a prominent role in

a decision support system. Obtaining effective sales

forecasting in advance can help the decision maker

calculate production and materials costs, even deter-w xmine the sale price 3 . This will result in a lower

inventory level and achieve the objective of just-in-

time. Regarding conventional sales forecasting meth-w xods 46 , most of them used either factors or time

series data to determine the forecast. However, the

relationship between the factors or the past time .series data independent variables and the sales data

.dependent variable is always quite complicated.Obtaining results through the above mentioned ap-

proaches is quite difficult. Therefore, various deci-

sion makers prefer using their own intuition, insteadof model-based approaches i.e., time series or re-

.gression models . However, there is a model-freeapproach, ANN, that can be applied in the area of

forecasting owing to its adequate performance in

control and pattern recognition. Thus, ANN is re-

viewed in Section 2.2.

2.2. Artificial neural networks in sales forecasting

.Artificial neural network ANN is a system de-rived through models of neurophysiology. In general,

it consists of a collection of simple nonlinear com-

puting elements whose inputs and outputs are tied

together to form a network. The learning algorithms

of ANNs can generally be divided into three differ-

ent types: supervised, unsupervised, and hybrid

learning.

Many studies have attempted to apply ANN to

time-series forecasting. However, their conclusions

are often contradictory. Some studies conclude that

w xANNs are better than conventional methods 7 , whilew xothers reach an opposite conclusion 8 . Weigen etw xal. 7 introduced the eight-elimination backpropa-

gation learning procedure to effectively deal with the

overfitting problem, and applied it to sunspots and anw xexchange rate time series. Tang et al. 8 compared

the ANN and BoxJenkins models, using interna-

tional airline passenger traffic, domestic car sales

and foreign car sales in the USA. They concluded

that the BoxJenkins models outperformed the ANN

models in short-term forecasting. On the other hand,

the ANN models outperformed the BoxJenkins inlong term forecasting.w xChakraborty et al. 1 presents an ANN approach

to multivariate time-series analysis. They accurately

predicted the flour prices in three cities in USA.

According to their results, the ANN approach is a

leading contender among statistical modeling ap-

proaches.


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( )R.J. Kuo, K.C. Xuer Computers in Industry 37 1998 115 3

w xLachtermacher and Fuller 2 developed a cali-brated ANN model. The model used BoxJenkins

methods to identify the lag components of the data,

that should be used as input variables. Also, it

employed a heuristic to suggest the number of hid-

den units needed in structuring the model. In examin-

ing the stationary series, they observed that the

calibrated ANN models have only a slightly better

overall performance than the conventional time-series

methods used in the benchmark. In the case of a

non-stationary series, the calibrated ANN models

outperformed the ARMA model for three of the four

series, and performed almost as well as the ARMA

in fourth series. The above survey indicates that

ANN is more appropriate for the time series data.

However, only considering the time series data may

result in a bad forecast. Including both the time

series data and factors in the forecasting model

seems to be preferable. Fuzzy logic has not onlybeen successful in engineering application, but also

capable of replacing these factors by fuzzy logic as

introduced in Section 2.3.

2.3. Fuzzy modeling

The theory of fuzzy sets was first developed byw xLotfi Zadeh 9 , primarily in the context of his

interest in the analysis of complex systems. How-

ever, some of the key ideas of the theory were

envisioned by Max Black, a philosopher, almost 30w xyears prior to Zadehs seminal paper 10 . Basically,the concept of the fuzzy set is a generalization of the

classical or crisp set.

The crisp set is defined in such a way as to

dichotomize the individuals in some given universeof discourse into two groups: members those that

. certainly belong in the set and nonmembers those.that certainly do not . A sharp, unambiguous distinc-

tion exists between the members and nonmembers of

the class or category represented by the crisp set.

However, many of the collections and categories do

not have this kind of characteristic. Instead, their

boundaries seem vague, and the transition from

member to nonmember appears gradual rather than

abrupt. Thus, the fuzzy set introduces vagueness by

eliminating the sharp boundary dividing members of

the class from nonmembers. The logic based on

fuzzy set theory is called fuzzy logic. A simple

example is used to explained the general concept of

the fuzzy modeling. For instance, two arbitrary rules

are illustrated as follows. . .Rule 1: IF X is Small A and Y is Small B1 1

THEN Z is f1 . .Rule 2: IF X is Large A and Y is Large B2 2

THEN Z is f2The membership functions for X and Y are shown

in Figs. 2 and 3. Both X and Y are divided into four . .linguistic variables, say very small VS , small S ,

. .large L , and very large VL . Each rule has apremise, or IF part, which contains several precondi-

tions and a consequent, or THEN part which de-

scribes the value of one or more output actions. Now

suppose there are two preconditions, x and y, for

fuzzy variables X and Y, respectively. Then, for rule

1, the membership function values are represented as . .m x and m y for A and B , respectively. Simi-1 1 1 1

. .larly for rule 2, we have m x and m y as the2 2membership function values. Hence, the firing

. .strength of rule 1 is obtained as w sm x m y ,1 1 1 . .while w s m x m y is the firing strength of rule2 2 2

2.

The overall output O is determine by using cen-

troid defuzzification where

w f i i x =y =f qx =y =f1 2 1 2 1 2iO s 1 .

x =y qx =yw 1 2 2 1 ii

and f is the consequence, or control action, value ofiw xrule i 11 .

During the past several years, fuzzy modeling has

been applied in many practical areas, ranging from

finance to engineering. In particular, fuzzy control

has emerged as one of the most promising areas for

research in the application of fuzzy modeling. In

many applications, such as control, fuzzy-based sys-

tems have proved to be superior in performance to

conventional systems.

2.4. Fuzzy neural network

An ANN, which is employed for recognition pur-

poses, generally lacks the ability to be developed for

a given task within a reasonable time. On the otherw xhand, fuzzy modeling 9,12 , which is applied to fuse


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the decisions from the different variables, requires anapproach to learn from experience i.e., data col-

.lected in advance . Therefore, how to successfullycombine these two approaches, ANNs and fuzzy

modeling, has become the a relevant concern of

further studies.

Recently, ANNs learning algorithm has been ap-

plied to enhance the performance of a fuzzy system

and has been demonstrated to be an innovative ap-

proach. Also, fuzzy ifthen rules were generated and

adjusted by learning methods using numerical data.w xLin and Lee 13,14 proposed the so-called Neural-

Network-Based Fuzzy Logic Control System NN-.FLCS . They introduced the low-level learning power

of neural networks in the fuzzy logic system and

provided high-level human-understandable meaning

to the normal connectionist architecture. Also, Kuow x15 introduced a feedforward ANN into fuzzy infer-

ence represented by Takagis fuzzy modeling.The above mentioned FNNs are only appropriate

for numerical data. However, the experts knowledge

is always of a fuzzy nature. Thus, some researchers

have attempted to address this problem. Ishibuchi etw xal. 16,17 proposed learning methods of neural net-

works to utilize not only numerical data but also

expert knowledge represented by fuzzy if then rules.w xLin and Lee 18 also presented a FNN, capable of

handling both the fuzzy inputs and outputs. Mean-w xwhile, Buckley and Hayashi 19 surveyed recent

results on learning algorithms and applications forFNNs.

3. Methodology

Section 2 has emphasized the relevance of sales

forecasting as well as some necessary background

information. The proposed forecasting system is dis-

cussed in more detail in this section.

The proposed intelligent forecasting system con- . .sists of 1 data collection, 2 general pattern model

. . . .ANN , 3 special pattern model FNN , and 4 .decision integration ANN . Fig. 1 shows the pro-

posed system architecture. Basically, the system de-

termines the forecasted product and the factors af-

fecting the sales first. Thereafter, the general pattern

of the sales is forecasted by an ANN while FNN

considers the effect on the sales if a promotion is

conducted. Finally, the decisions from these two

networks and time effect are integrated through the

other ANN. Each part is thoroughly discussed in the

following subsections.

3.1. Data collection

The first part of the proposed forecasting system, .data collection, concerns two data resources: 1

.quantitative factors and 2 non-quantitative factors.The quantitative factors can be accumulated from the

supermarket company while the questionnaire survey

can provide the non-quantitative data.

3.2. ANN architecture

.In this part, i.e., ANN , a feedforward ANN with .error backpropagation EBP learning algorithm is

employed. Fig. 2 shows the general structure of the

one-hidden-layer ANN. The ANNs input layer with

some neurons represents the previous sales data, say

period t-p to period t-1, which are connected to the

hidden layer. The hidden layer with some neurons is

connected with the output layer with one single

neuron which represents the sales for period t. Usu- . ally, the input neurons use y sf x sx no change

.in input and all the other neurons have the sig-

. ..y1

moidal function y sf x s 1q exp yx . Theobjective is to minimize a cost function defined as:

1 2Es T yO , 2 . . p p

2 p

where T and O are the desired and actual outputs,

respectively, and p is the sample number. The

derivation is based on the gradient descent technique

to converge toward improved weights and biases .thresholds . The updating rule for all the weightsand biases is

yEEW t syh q a W ty 1 , 3 . . .k h k h /EWk hwhere h is learning rate and a is the momentum.

Therefore, from the preceding setup and required

data, the ANN can learn the relationship between the

sales at period t and previous sales from periods t-1


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Fig. 1. The architecture of the proposed forecasting system.

to t-p. Thus, just input sales from periods from m-1

to m-p if we intend to forecast sales at period m.

3.3. FNN architecture

In the first part, the ANN can provide us sales

without promotion effects. However, if a promotion

is running, then ANNs forecast will be inaccurate.

Thus, this section discusses how to effectively han-

dle the circumstance of promotion by means of a

FNN. Since the FNN architecture is based on the

fuzzy logic which possesses both the precondition

and consequence, the precondition variables repre-

sent the effective factors while the sales represents

the consequence variable. First, the data and ifthen

rules are obtained through the fuzzy Delphi method.

After this procedure, the collected data can be ap-

plied to train the proposed FNN. The structure ofw xFNN presented in this study is similar to Ref. 17 .

The main difference is that the network employs the

asymmetrical bell shaped instead of triangular fuzzy

weights. In the following, the two components, FNN

and fuzzy Delphi, are discussed in more detail.


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Fig. 2. The ANN structure.

3.3.1. FNN with asymmetric bell-shaped fuzzy

weights

Most FNNs can only handle the actual real input

w xand output except 17,18 . Thus, this component

w xintends to modify Ishibuchis work 17 . InIshibuchis work, the input, weight, and output fuzzy

numbers are symmetric triangular. However, this

assumption is not similar to a human beings think-

ing. Thus, this paper replaces the triangular fuzzy

numbers with asymmetric Gaussian functions. The

inputoutput relation of the proposed FNN is dis-

cussed in the following. However, the operations of

fuzzy numbers are presented first.

3.3.1.1. Operations of fuzzy numbers. Before describ-

ing the FNN architecture, fuzzy numbers and fuzzynumber operations are defined by the extension prin-

ciple. In the proposed algorithm, real numbers and

fuzzy numbers are denoted by the lowercase letters .e.g., a, b, . . . and a bar placed over uppercase

.letters e.g., A, B, . . . , respectively.

Fig. 3. The FNN architecture.

Since input vectors, connection weights and out-

put vectors of multi-layer feedforward neural net-

works are fuzzified in the proposed FNN, the addi-

tion, multiplication and nonlinear mapping of fuzzy

numbers are necessary for defining the proposed

FNN. Thus, they are defined as follows:

Z z sX x qY y . . .


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For hidden layer,

L Uw x w x w xO a s O a ,O ap h p h p h

L Uw x w xs f Net a ,f Net a , 8 .p h p h / /

hs

1 , 2 , . . . ,n ,H

nI

L L Lw x w x w xNet a s W a PO ap h h i p iis1

Lw xW a G0hi

nI

L Uw x w xq W a PO a hi p iis1

Lw xW a -0hi

Lw xqQ a , 9 .hnI

U U Uw x w x w xNet a s W a PO ap h h i p iis1

Uw xW a G0hi

nI

U Lw x w xq W a PO a h i p iis1

Uw xW a -0hi

Uw xq Q a , 10 .h

For output layer,

L Uw x w x w xO a s O a ,O ap k p k p k

L Uw x w xs f Net a ,f Net a ,p k p k / /

s1 , 2 , . . . ,n , 11 .OnO

L L Lw x w x w xNet a s W a PO ap k k h p hks1

L

w xW a G0kh

nO

L Uw x w xq W a PO a k h p hks1

Lw xW a -0kh

Lw xqQ a , 12 .k

nO

U U Uw x w x w xNet a s W a PO ap k k h p hks1

Uw xW a G0kh

nO

U Lw x w xq W a PO a k h p hks1

Uw xW a -0kh

Uw xq Q a . 13 .kThe objective is to minimize the cost function

defined as:

nO

L UE s a E qE s E , 14 . . p k a . k a . p a .a aks1

where

nO

L U

E s a E qE 15 . .p a . k a . k a .ks11 2L LL w x w xE s T a yO a , 16 . /k a . p k p k 21 2U UU w x w xE s T a yO a . 17 . /k a . p k p k 2

Where EL and EU can be viewed as the squaredk a . k a .errors for the lower boundaries and the upper bound-

aries of the a-cut sets of a fuzzy weight. Other a-cut

sets of a fuzzy weight are independently modified to

reduce E . Otherwise, the fuzzy numbers afterp a .modifications are distorted. Therefore, each fuzzy

weight is updated similar to but different from thew xapproach of Ishibuchi 17 . That is, in the proposed

FNN, the membership functions are asymmetric .Gaussian functions i.e., a general shape which is

represented as:

1 x ymexp y , x-m

L / /2 s~ 1, x smA x s 18 . .

2

1 x ymexp y , otherwise U / /2 sThus, the asymmetric Gaussian fuzzy weights are

specified by their three parameters i.e., center right.width and left width . The gradient search method is

derived for each parameter similar to error-backprop-

agation learning algorithm.


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3.3.2. Fuzzy delphi

Section 3.3.1 has provided us a novel FNN to

learn the relationship between the fuzzy inputs and

fuzzy outputs. Thus, this subsection proposes a pilot

study, say fuzzy Delphi, for extracting the fuzzy

inputs and outputs.

Delphi method was first developed by Dalkey in

Rand Corporation. This approach has been widely

applied in many management areas, e.g., forecasting,

public policy analysis, or project planning. However,

the conventional Delphi method does not converge

very well. Besides, high survey frequencies alwaysw xresult in high costs. Thus, Ishikawa et al. 20 uti-

lized fuzzy sets theory in the Delphi method to

resolve the above shortcomings. This study modifiesw xthe method proposed by Ishikawa et al. 20 since the

Gaussian function instead of triangular function is

used. In addition, the other difference is that this

study puts three indexes pessimistic, optimistic and.average indexes in the questionnaire in order to

accelerate the convergence. Therefore, the proce-

dures of the modified fuzzy Delphi method for this

research are as follows.

a. Collect all the possible factors which may

affect the sales and make the sortition, grouping in

order to formulate the first questionnaire. In this

questionnaire, all the factors are separate and sur-

vey results are the FNN inputs.b. Select one event from each group or dimen-

.sion to formulate an ifthen rule in order to formthe second questionnaire which is a set of ifthen

rules.

c. Fuzzify the returned second questionnaires from

the senior managers and determine the pessimistic

index, optimistic index and average index. The for- . .mulations are as follows: 1 Pessimistic Minimum

index

l q l q . . . ql1 2 nl s 19 .

n

where l is the pessimistic index of the ith experti .and n is the number of the experts. 2 Optimistic .Maximum index

u qu q . . . qu1 2 nu s 20 .

n

.where u is the optimistic index of the ith expert. 3i .Average most appropriate index . For each interval

. .l , u , calculate the mid point, m s l qu r2i i i i iand then find

1rnms m =m =PPP=m 21 . .1 2 n

R L .Thereafter, the fuzzy number A s m, s , s ,which represents the mean, right width, and left

width, respectively, for an asymmetrically bell shaped

function, can be determined through the above in-

dexes:

l y mRs s 22 .

3u ym

Ls s 23 .3

d. Formulate the third questionnaire with the above

indexes and make the survey.

e. Repeat the procedure c.w xf. Employ dissemblance index rule 21 to exam-

ine the second and third questionnaire fuzzy num-

bers. Restated, the dissemblance index rule deter-

mines whether the membership functions have con- .verged or not Fig. 4 . If not, continue the next

questionnaire until converging. Otherwise, results of

the third questionnaire are the FNN output. The .distance between two fuzzy numbers A and B is

11w x w xd A ,B s d A a ,B a d as b y b . . .H 2 12as0

=

1 L L

w x w xA a yB aH /as0U Uw x w xq A a yB a d a 24 . /

where b and b are given any convenient values in1 2w x w x order to surround both A a s0 and B a s0 Fig.

. . w x3 . Basically, d A,B is in the interval 0,1 .

Fig. 4. The concept of dissemblance index of two fuzzy numbers.


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Fig. 5. The time series data.

3.4. Decision integration

From the above two parts, the ANN provides the

sales without any special promotion while the pro-

motion effect is forecasted by the FNN. To yield the

final result, for the sales which consider both the

general pattern and the promotional effect, only inte-

grating the results from the ANN and FNN is inade-

quate in that the promotional effect during the pro-

motion interval may differ. Thus, this part will em-

ploy the other ANN to combine the ANN result, the

FNN result, and the time effect.

4. Model evaluation results

Section 3 has presented the proposed forecasting

system. To verify the proposed systems feasibility, a

set of real data is collected. In addition, the proposed

system is also compared with the other methods,

such as single ANN and ARMA. Both the proce-

Table 1

The third questionnaire

Related products without promotion 15-day promotion length

Promotion methods Advertising media Reference index

Pessimistic Average Optimistic Affective interval

$10 discount 1. From 2000 to 2100 h on TV 7.2 7.8 8.6 _to _

2. At noon on TV 2 2.1 3.8 _to _

3. In the evening on TV 3.6 3.9 5.2 _to _

4. Radio 4 4.6 5.4 _to _

5. Newspaper 4.8 5.5 6.4 _to _

6. POP notice 6.4 7 8 _to _

7. Poster 6.2 6.5 7.4 _to _

$5 discount 1. From 2000 to 2100 h on TV 5.4 6 7.2_

to_

2. At noon on TV 1 1.1 2 _to _

3. In the evening on TV 3 3.3 4.4 _to _

4. Radio 3 3.4 4.4 _to _

5. Newspaper 3.6 4.2 5.2 _to _

6. POP notice 4.6 5 6.2 _to _

7. Poster 5 5.3 6.2 _to _

$3 discount 1. From 2000 to 2100 h on TV 1 1.3 2.3 _to _

2. At noon on TV 0.8 1 1.8 _to _

3. In the evening on TV 0.8 1 1.8 _to _

4. Radio 1.8 2 3.2 _to _

5. Newspaper 1 1.3 2.3 _to _

6. POP notice 1.8 1.9 3.3 _to _

7. Poster 2.8 2.9 4.2_

to_

Buy two get one free 1. From 2000 to 2100 h on TV 6.6 7.3 8.2 _to _

2. At noon on TV 3.2 3.2 4.6 _to _

3. In the evening on TV 3.6 3.7 5 _to _

4. Radio 4.6 5.2 6.2 _to _

5. Newspaper 4.8 5.4 6.4 _to _

6. POP notice 5.8 6.5 7.4 _to _

7. Poster 6 6.6 7.4 _to _


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dures and results are sequentially shown in the fol-

lowing subsections.

4.1. Data collection

The data are provided by a locally well-known

chain supermarket. Since the forecasting pattern is

divided into two categories, general pattern and spe-

cial pattern, the data collection is also comprised of

two parts, as follows.

4.1.1. Time series data

The company provides the daily sales of 500 cc

papaya milk. The total number of the data points is

379 as shown in Fig. 5. The sudden convex indicates

that a promotion is conducted. In total, there are five

promotions. The date lasts from January 1, 1995 to

January 14, 1996. For the purpose of testing, these

379 data points are further divided into trainingsamples and testing samples. The front one has 334

data points while the latter one has 45 data points.

4.1.2. Expert questionnaire

To survey all the possible factors of promotion

and their effects on the sales, this paper employs

fuzzy Delphi method. The questionnaires setup is

based on related references, such as professional

journals and senior managers knowledge. The pro-

Fig. 6. Four fuzzy numbers.

cedures based on the modified fuzzy Delphi method

presented in the above section are as follows.

4.1.2.1. Factors determination. A large number of

factors can generally affect the sales. However, a

different product has different characteristics. Afterdiscussing with the senior managers, all the factors

are divided into three dimensions. The first dimen-

sion represents the methods of promotion, while the

types of advertising media are presented in the sec-

ond dimension. The third dimension represents the

competitors actions.

4.1.2.2. First questionnaire formulation. After the

events have been determined in the above procedure,

Table 2

The fuzzy number of each event for the third questionnaire

R L . . . . .Factors Events Average m s s lym r 3 s s u ym r 3

Promotion methods $10 discount 0.85 0.0310 0.0338

$5 discount 0.48 0.0257 0.0378

$3 discount 0.21 0.0206 0.0454

Buy two get one free 0.70 0.0266 0.0369

Advertising media At night on TV 0.77 0.0345 0.0387

At noon on TV 0.32 0.0159 0.0523

In the evening on TV 0.36 0.0223 0.0524

Radio 0.44 0.0126 0.0534

Newspaper 0.43 0.0213 0.0492

POP notice 0.69 0.0255 0.0325Poster 0.65 0.0290 0.0365

Promotion length 15 days 0.54 0.0221 0.0432

20 days 0.58 0.0343 0.0477

30 days 0.68 0.0227 0.0413

45 days 0.62 0.0245 0.0442

Others Related products without promotion 0.73 0.0259 0.0348

Related products with promotion 0.48 0.0177 0.0343


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Table 3

The combinatorial fuzzy number for each linguistic term

R LLevels Mean m Right standard deviation s Left standard deviation s

Poor 0.234188 0.039563 0.066271

Medium 0.446196 0.046899 0.059435

Good 0.637987 0.047829 0.049004

Very good 0.778053 0.023351 0.029316

these events can be used to formulate the first ques-

tionnaire. Provide the questionnaires to the senior

managers for survey. Calculate the membership func-

tion for each event. Results obtained from the first

questionnaire can provide the degree of importance

for each event and are the FNN inputs.

4.1.2.3. Second questionnaire formulation. Since the

questionnaire attempts to provide training data forFNN, the events of all three dimensions can be used

to formulate the second questionnaire which is a set .of ifthen rules Table 1 . Totally, there are 56

.4=7=2 ifthen rules. For example, the first ruleof the questionnaire is

if

discount is $10 and advertising time is from

2000 to 2100 h on TV and no related product is

under promotion

then

the effect on sales ranges from _ to _.

After the survey of this questionnaire, all the

required information, i.e., pessimistic, optimistic,

and average indexes, is calculated on the basis of

Section 3.

4.1.2.4. Third questionnaire surey. Formulate the

third questionnaire which includes the pessimistic,optimistic and average indexes. Continue the next .third survey and calculate all the information again.Table 2 presents the fuzzy number of each event.

However, directly using these fuzzy numbers is

time-consuming and complicated. Therefore, all the

similar events should be combined together. Finally,

only four linguistic variables are used: bad, medium,

good and very good. Fig. 6 and Table 3 present the

fuzzy numbers of these four linguistic terms.

Table 4

Dissemblance index rule testing results for separate events

.Factors Events d A,B ds0.06 ds0.08

Promotion methods $10 discount 0.010842 Accept Accept

$5 discount 0.012683 Accept Accept

$3 discount 0.055033 Accept Accept

Buy two get one free 0.021496 Accept Accept

Advertising media at night on TV 0.008304 Accept Accept

at noon on TV 0.072394 Reject Accept

In the evening on TV 0.036122 Accept Accept

Radio 0.020922 Accept Accept

Newspaper 0.039173 Accept Accept

POP notice 0.017868 Accept AcceptPoster 0.011316 Accept Accept

Promotion length 15 days 0.032976 Accept Accept

20 days 0.015148 Accept Accept



Others Related products without promotion 0.031801 Accept Accept

Related products with promotion 0.02144 Accept Accept


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Table 5

The dissemblance index rule test results for combinatorial events

.Sales linguistic terms d A,B ds0.06 ds0.08

Poor 0.018939 Accept Accept

Medium 0.004207 Accept Accept

Good 0.013379 Accept Accept

Very good 0.005568 Accept Accept

4.1.2.5. Similarity testing. To determine the neces-

sity of the next survey, the dissemblance index rule

should be utilized. The single event results and

combinatorial events results are summarized in Ta-

bles 4 and 5, respectively. This finding suggests that

the membership function of each linguistic term has

converged. The next survey does not need to be

performed. Therefore, this knowledge base will be

applied to train the FNN and represents the FNN

outputs.

( )4.2. General pattern model ANN

Only 379 data points are used for training. How- .ever, in a general pattern model ANN , the data

points under promotion are not included in the train-

ing samples. Thus, after subtracting the number of

promotion data points from 379, the number of

training samples is 288. This study will test different

alternatives, say network structure. For instance, pat-

tern 1 in Table 6 suggests that the input nodes are . .t-1 th and t-2 th sales which are connected to twohidden nodes which are connected to one output

.node, t th sales. However, all of these data pointsw xshould be normalized in 0,1 . The training rate and

momentum are set to be 0.3 and 0.8, respectively.

The network will not stop learning until 50,000

Table 7 y3 .MSEs =10 for different FNN setup

Momentum Training rate

0.1 0.3 0.5

0.1 8.31 9.10 8.46

0.5 8.15 8.52 8.63

0.8 13.85 8.96 11.30

epochs. In addition, statistical method, ARMA, is .also used to model the above time series data. AR 3

has the MSE value 2.21=10y3, while the network

whose structure is 7=7=1 has the lowest MSE

value 2.20=10y3.

( )4.3. Special pattern model FNN

Data collection demonstrated that the question-

naire should have fifty six ifthen rules. After using

unsimilarity index rule to test the similarity, the .result indicates that both the event newspaper A

. .and event radio B are quite similar in that d A,Bis equal to 0.017966, i.e., being smaller than d, 0.08.

Thus, combining the two events results in the total .number of rules is 48 4=6=2 .

The setup of proposed FNN with asymmetric bell

shaped fuzzy weights is similar to the general ANN.

The network structure consists of three input nodes,

i.e., dimensions, which are connected to six hiddennodes, i.e., numbers which are connected to one

output node. The network will not stop learning until

30,000 epochs. The a-level sets are 0.2, 0.4, 0.6,

0.8, and 1.0. Different training rates and momentum

terms may yield different results. Testing results .Table 7 indicate that the lowest MSE value is

Table 6

The testing alternatives for general pattern model

y3 . . .Pattern number Network structure I=H=O MSE 10 Percentage increase %

.0 AR 3 2.21 ))

1 2=2=1 2.36 6.79

2 2=2=2=1 2.36 6.78

3 5=5=1 2.22 0.45

4 5=5=5=1 2.23 0.76

5 7=7=1 2.20 y0.456 7=7=7=1 2.20 y0.52


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Table 8

MSE values of training results for different models

Forecasting model MSE Percentagey3 .=10 increase of MSE

Integration ANN 3.37 )) .ARMA 2,5 4.88 46.13

.ANN1 5=5=1 4.95 48.35

.ANN2 5=5=5=1 5.50 49.80 .ANN3 10=10=1 4.76 44.77 .ANN4 10=10=10=1 4.77 43.10

8.15=10y3 as training rate and momentum are 0.1

and 0.5, respectively. According to these results, the

smallest MSE is equal to 8.15=10y3 as the training

rate and momentum are equal to 0.1 and 0.5, respec-

tively. Finally, this network becomes integrated with

general pattern model, ANN, in the next part, deci-

sion integration.

( )4.4. Decision integration model ANN

The above two parts can provide results obtainedfrom the general pattern and special pattern promo-

.tion models. This part attempts to integrate theabove two models. Basically, the inputs are originat-

ing from the following three sources:

a. General pattern model provides one input node

which is the general pattern trend;

b. A special pattern model provides five inputnodes which are from FNN output fuzzy number

as a is equal to 0.6, 0.8 and 1.0, respectively; and

c. The promotion length implies one input node.

Therefore, the total number of integration ANN

input nodes is seven. Consequently, the network

structure is 7=7=1. The training rate and momen-

tum are 0.3 and 0.8, respectively. The training limit

is 50,000 epochs. Table 8 presents the MSE value,

which is equal to 3.37=10y3. For the purpose of

Table 9The forecasting results

Forecasting model MSE Percentagey3 .=10 increase of MSE

Integration ANN 3.10 ))

Short-term memory ANN 6.61 113.63

Long-term memory ANN 6.78 118.99

Fig. 7. The integration ANN forecasting result.

comparison, this paper also uses the collected data to

formulate the ARMA model and ANN model. Table

8 summarized those results.

So far, the paper has presented the training

results. In the following, three models, i.e., integra-

.tion ANN, short term memory ANN 5=5=1 , and .long term memory ANN 10=10=1 , are used toforecast the 45 data points with one time of promo-

tion. The data points are the last forty five data

points shown in Fig. 5. The related results are listed

in Table 9 and Figs. 79.

Fig. 8. The short-term memory ANN forecasting result.

Fig. 9. The long-term memory ANN forecasting result.


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5. Discussion

Section 4 has presented evaluation results based

on data accumulated from a chain supermarket com-

pany. The data collection appears to be subjective

since the data are provided by either the senior

managers or experts. However, the number of ex-

perts is twenty, implying that the subjective factors

can be eliminated to a minimum. In particularly the

fuzzy Delphi method is employed; the above consid-

eration can also be reduced. Among all types of

promotion methods, the event $10 discount is the

most effective event while the worst event is $3

discount. Regarding the types of the advertising

media, the event advertising time from 2000 to 2100

h on TV affects sales the most. Undoubtedly, since

2000 to 2100 h is the so-called golden interval, the

promotion of the related product yields a negative

effect.Notably, the pessimistic, optimistic, and average

indexes are put in the third questionnaire. The reason

is that providing the others experts opinions may

accelerate the converge of the fuzzy numbers. In the

third questionnaire, 13 fuzzy numbers are combined

into four fuzzy numbers. The main reason is to

reduce the computational complexity. The forecasts

precision may not differ significantly. Moreover, the

third questionnaire also used the unsimilarity index

rule to determine the necessity for the next survey.

Table 5 indicates that either d is equal to 0.06 or0.08, all the pairing fuzzy numbers are similar. Even

by viewing the separate events result, only the event

advertising at noon on TV is not similar as d is .0.06. Therefore, using these data or fuzzy numbers

to train the FNN is obviously subjective. Basically,

this is a pilot study to indicate the sources of thew xfuzzy inputs and fuzzy outputs. Both Refs. 18,17 ,

i.e., two representative papers considering the fuzzy

inputs and outputs, did not provide a mean to find

the fuzzy inputs and outputs for training.

In the second part of the proposed system, six

alternatives, from short-term to long term memories,

are tested. The MSE values indicate that the network .with long-term memory seven outperforms the net-

.work with short-term memory two or five . How-ever, continuously increasing the length of memory

does not yield a better result. The reason is that the

unnecessary information may mislead the networks

memory. Regarding the number of hidden layers, the

results indicate either one or two hidden layers can

provide a similar forecast. Owing to computational

considerations, one hidden layer is more feasible.

Moreover, it is the reason to select 7=7=1 net-

work instead of 7=7=7=1 network to integrate

with FNN.

Table 8 indicates that integration ANN outper-forms all other forecasting methods, e.g., ARMA 2,

.5 and single ANN. The reason is that the integrationANN prioritizes the promotion effect on the sales

pattern. Even with the forecasting result, the integra-

tion ANN is also second to none. Interestingly, the

short and long term ANNs always have one day

delay. It is because this kind of ANN only depends

on the previous data. However, the integration ANN

poses the promotion effect in the network.

6. Conclusions

This paper has developed a forecasting system

based on FNN to solve the sales forecasting problem

under promotion. Though directly using a single

ANN to model the sales pattern has been shown to

be better than the conventional statistical methods, it

still need further improvement. Integrating the ANN

and the FNN can provide more reliable forecast. In

addition, the fuzzy Delphi method was applied to

collect the fuzzy inputs and outputs for the FNN. Toour knowledge, this is the first paper to consider this

existing but unresolved problem. In the future, the

authors would like to further improve the FNN, like

the pruning of the fuzzy weights and training speed

of the network. From a marketing perspective, more

factors, which may yield a more precise result, can

be included.

Acknowledgements

The authors would like to thank the National

Science Council, Republic of China for partially

supporting this manuscript under Contract No. NSC

86-2416-H-327-003-E10. Mr. L.C. Shie is also ap-

preciated for providing the daily sales data and his

valuable discussion regarding chain store promotion.

In addition, the authors thank the anonymous refer-


15/15


ees for reading the paper and offering many helpful

comments.

References

w x1 K. Chakraborty, K. Mehrotra, C.K. Mohan, Forecasting the

behavior of multivariate time series using neural networks, .Neural Networks 5 1992 961970.

w x2 G. Lachtermacher, J.D. Fuller, Backpropagation in Time- .series forecasting, J. Forecasting 14 1995 381393.

w x3 G.S. LeVee, The key to understanding the forecasting pro- . .cess, J. Business Forecasting 11 4 1992 1216.

w x4 G.G. Meyer, Marketing research and sales forecasting at . .Schlegel Corporation, J. Business Forecasting 12 2 1993

2223.w x5 C.W. Chase, Ways to improve sales forecasts, J. Business

. .Forecasting 12 3 1993 1517.w x6 M.M. Florance, M.S. Sawicz, Positioning sales forecasting

. .for better results, J. Business Forecasting 12 4 1993

2728.w x7 A.S. Weigen, D.E. Rumelhart, B.A. Huberman, Generaliza-tion by weight-elimination with application to forecasting,

.Adv. Neural Inf. Processing Syst. 3 1991 875882.w x8 Z. Tang, C. Almeida, P.A. Fishwick, Times series forecast-

ing using neural networks vs. BoxJenkins methodology,

Simulations, Simulations Councils, Nov., 1991, pp. 303310.w x9 L.A. Zadeh, The concept of a linguistic variable and its

application to approximate reasoning: Parts 13, Inf. Sci. 8 .1965 .

w x10 M. Black, Vagueness: An Exercise in Logical Analysis,Philosophy of Science 4, pp. 427 455, 1937.

w x11 T. Takagi, M. Sugeno, Derivation of Fuzzy Control Rulesfrom Human Operators Control Actions, Proc. of the IFAC

Symposium on Fuzzy Information, Knowledge Representa-tion and Decision Analysis, Marseilles, France, July 1983,

pp. 5560.w x12 C.C. Lee, Fuzzy logic in control systems: fuzzy logic con-

trollerParts I and II, IEEE Trans. Syst. Man, Cybernetics . .SMC 20 2 1990 404435.

w x13 C.T. Lin, C.S.G. Lee, Neural-network-based fuzzy logic .control and decision system, IEEE Trans. Comput. C 40 12

.1991 13201336.w x14 C.T. Lin, C.S.G. Lee, Reinforcement structurerparameter

learning for neural network based fuzzy logic control sys- . .tems, IEEE Trans. Fuzzy Syst. 2 1 1994 4663.

w x15 R.J. Kuo, P.H. Cohen, Manufacturing process control throughintegration of artificial neural networks and fuzzy model,

. .Fuzzy Sets and Systems 98 1 1998 1531.w x16 H. Ishibuchi, H. Okada, R. Fujioka, H. Tanaka, Neural

networks that learn from fuzzy ifthen rules, IEEE Trans. . .Fuzzy Syst. FS 1 2 1993 8597.

w x17 H. Ishibuchi, K. Kwon, H. Tanaka, A learning algorithm offuzzy neural networks with triangular fuzzy weights, Fuzzy

.Sets Syst. 71 1995 277293.w x18 C.T. Lin, C.S.G. Lee, Fuzzy adaptive learning control net-

work with on-line neural learning, Fuzzy Sets Syst. 71 .1995 2545.

w x19 J.J. Buckley, Y. Hayashi, Fuzzy neural networks: a survey, .Fuzzy Sets Syst. 66 1994 113.

w x20 A. Ishikawa, M. Amagasa, G. Tomiqawa, R. Tatsuta, H.Mieno, The mix-min Delphi method and fuzzy Delphi method

.via fuzzy integration, Fuzzy Sets Syst. 55 1993 241253.w x21 A. Kaufmann, M.M. Gupta, Introduction to Fuzzy Arith-

metic, North-Holland, Amsterdam, 1985.

R.J. Kuo received an MS degree in In-

dustrial and Manufacturing Systems En-gineering from the Iowa State Univer-

sity, Ames, IA, in 1990 and Ph.D. de-

gree in Industrial and Management Sys-

tems Engineering from the Pennsylvania

State University, University Park, PA, in

1994. Currently, he is an Associate Pro-

fessor in the Department of Marketing

and Distribution Management, National

Institute of Technology at Kaohsiung,

Taiwan. His research interests include

architecture issues of neural networks, fuzzy logic, and genetic

algorithm, and their applications in process control, forecasting

and marketing.

K.C. XUE earned his MS degree in

Management Science from the I-Shou

University, Taiwan, in 1996. His re-

search interests include neural networks,

fuzzy logic, and their applications in

marketing. Currently, he is in the mili-

tary service.

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Sales Using ANN n Fuzzy