the max-min delphi method and fuzzy delphi method via fuzzy integration

Fuzzy Sets and Systems 55 (1993) 241-253 241 North-Holland

The Max-Min Delphi method and fuzzy Delphi method via fuzzy integration*

Akira Ishikawa Aoyama Gakuin University, International University of Japan, Tokyo, Japan and The University of Texas at Austin, TX, USA

Michio Amagasa Daitou Bunka University, Tokyo, Japan

Tetsuo Shiga Hakuhoudou Inc., Tokyo, Japan

Giichi Tomizawa Science University of Tokyo, Japan

Rumi Tatsuta Dokkyo University, Saitama, Japan

Hiroshi Mieno Science University of Tokyo, Japan

to 'the attainable period with a high degree' and 'the unattainable period with a high degree'. Next, through the implementation of the Max-Min Fuzzy Delphi Method and the New Delphi Method via Fuzzy Integration, we have developed algorithms which enable forecasting attainable periods. Third, we have applied such algorithms to two concrete questions, compared the result with one obtained from the Delphi method, and ascertained the feasible outcome. While more examination needs to be undertaken, the new methods look valid and applicable to further analyses of other questions and items on questionnaires. While both methods can forecast attainable periods, using these methods simultaneously as well as the traditional Delphi method, may prove a really effective result.

Keywords: Fuzzy Delphi method; Fuzzy integration; membership functions; technological forecasting; extent of expertise; Max-Min normativism; triangular membership function.

Received November 1991 Revised February 1992

Abstract." The traditional Delphi method is one of the effective methods which enables forecasting by converging a possibility value through the feedback mechanism of the results of questionnaires, based on experts' judgments. Some points needing revision are: (1) By pinpointing the intuition of the first response on the part of experts, feasible inference values need to be extracted so that the quality-oriented and semantic structure of the responses may be analyzed. (2) By removing the effect caused by feedback in the Delphi method, natural and non-converged results need to be acquired; Moreover, two and more repetitive surveys are likely to cause a decline in the response rate, which may produce negative effects in the ensuing analyses. (3) In general, as it is repeated, the survey becomes more costly and time-consuming. In order to resolve these issues, we have identified two kinds of membership functions in regard

Correspondence to: Prof. Akira Ishikawa, School of International Politics, Economics and Business, Aoyoma Gakuin University, Shibuya, Tokyo 150, Japan.

* This paper is a revised version of the paper presented at the l l t h European Congress on Operations Research, held in Aachen, Germany, July 16-19, 1991.

1. Introduction

As one of the long-term forecasting methods, the Delphi method developed by Helmer and his associates has been widely used to date. One of the weaknesses of this method is that it requires repetitive surveys of the experts-ordinarily more than twice - to allow the forecast values to converge. The more we repeat surveys, the more costly they become. In addition, the response rate becomes lower, particularly so for a complicated survey.

Alternatively, the fuzzy Delphi method that might process fuzziness in forecasting has been proposed. This method applies a kind of 3-point estimation method to forecast values. On the basis of the values obtained by the method, triangular membership functions can be constructed. Then, all of the distances between the expected values of the forecast values and those provided by each forecaster are computed. If a distance that satisfies a given convergence criterion is found, this process has been

0165-0114/93/$06.00 © 1993--Elsevier Science Publishers B.V. All rights reserved

242 A. Ishikawa et al. / The Max-Min Delphi method and fuzzy Delphi method

completed and the corresponding expected value becomes a forecast value. Unfortunately, this method requires a multiple number of surveys. Thus, it cannot be said to be an optimal solution approach.

Accordingly, this paper discusses a method which can suffice for the weaknesses of the above and Delphi method and attempts to propose a new forecast method on the basis of fuzzy theory.

This method, at the stage of questionnaire surveys, embraces specialization in each forecast item, stipulates it as a criteria of data classification and fuzzy measure of the object, and seeks a forecast value through the Max-Min criterion and fuzzy integration. Each of these methods is identified herein as the Max-Min Delphi Method (MMFDM) and the Fuzzy Delphi Method via Fuzzy Integration (FDMFI), and the two together are identified as the New Fuzzy Delphi Method (NFDM).

The NFDM has the following merits: (1) Fuzziness is inescapably incorporated in

the findings. (2) It enables reduction in the number of

surveys. (3) The semantic structure of forecast items is

clarified. (4) Individual attributes of the expert (fore-

caster) are elucidated.

2. Development of the New Fuzzy Delphi Method

The developmental objectives and the process of the fuzzy Delphi method will be shown below:

2.1. Developmental objectives

They are summarized as follows: (1) To develop the format design of the

questionnaire so as to process, fuzziness in relation to the information contents of the respondents (fuzziness of the respondents).

(2) To limit the additional surveys to only one, in order to reduce the time and costs required (convergence).

(3) On the basis of the data obtained from the respondents, forecast period (year) is to be

computed as the forecast value via the Max-Min criterion and fuzzy integration. These results will be compared with the result obtained from the traditional Delphi approach (reliability).

2.2. Developmental process

The developmental process of the new fuzzy Delphi method is shown in Figure 1.

In Stage A, an image on the achievable period by the object to the forecast items was extracted by the questionnaire surveys, while in Stage B, membership functions were established on the basis of the result of the surveys. In Stage C, forecast values were obtained through the new fuzzy Delphi methods via the Max-Min criterion and fuzzy integration. In Stage D, further analyses were made on the basis of Stage C. And at the Stage E, based on the analyses and result, a final report was prepared. Following is a summary of Stages A, B, and C.

3. Questionnaire

In the new fuzzy Delphi method, the question items that forecast the period of realization consist of:

(1) the period where realization is absolutely impossible, and

(2) the period where realization is certainly possible.

In Figure 2, for example, the responses by one respondent to a question item says that realization is certainly possible from 1988, while realization is absolutely impossible till 1995. This means that the period between 1995 and 1998 is the uncertain period and therefore we define the period as a gray zone. In our daily judgments, we are very often faced with such gray zones and are likely to discard them because of their fuzziness and ambiguity.

Considering the semantics of these zones, however, we notice the significance and importance of them in that possibility and impossibility coexist and that the judgment is left to the decision maker as to whether or not a selective judgment is made. Thus, we, now look into such gray zones in more depth.

A portion of the questionnaire is shown in Table 1. The Table shows the degree of

A. lshikawa et al. / The Max-Min Delphi method and fuzzy Delphi method 243

Object ] J

m a o e o

as to realization (A)

period

(1)Max-Min

Normativism

(2)Fuzzy integration

Establishment of

fuzzy membership

function (B)

1 Computation of

forecast period

(c)

J Analysis~ (D)

Doeumen

tation

Fig. 1. Developmental process of the fuzzy Delphi method.

A

Impossible

Zone

1990 1995

G Fay

Z o n e

B

1

Possible Zone

2000 2005

Fig. 2. Forecast period A & B.

2010

importance, period of realization, and the extent of expertise, according to the following instructions:

(1) The degree of importance: To be recorded from zero to ten points. From the viewpoint of realization, should the respondent judge that realization is extremely important, then ten points are given. If, on the other hand, realization is not at all important, then zero points are given.

(2) The period of realization: For periods when realization is never possible, "impossible up until 199xx" is recorded, while for periods when realization is absolutely possible, "possible from 19xx".

(3) The extent of expertise: When no expertise exists, zero points are inserted, whereas for extremely high expertise, ten points.

Upon summing up the data on the basis of the above taxonomy, in-depth analyses were made.

4. How to determine the membership function

Based upon the data collected from subjects (n) by the extent of expertise, the fuzzy membership function has been determined for each item, as shown in Figure 3.

Two algorithms have been developed in connection with the membership function and determination of forecast values:

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A. Ishikawa et aL / The Max-Min Delphi method and fuzzy Delphi method 2 4 5

100

Value

\ / . 1 - - I

. / " I , Pc(x) , 7 " A 1 The permd

f P ~ (x) 1 wh . . . . . lization ca . . . . . . be achieved

. . . . . . . . . . . . 2 . . . . . ~/_ . . . . . . . . . . -I- . . . . . . . . . . ==---- - --=-T h~--P25~2a- . . . . . . . . . 50 | ,'~ I when realization can be s~rely achieved

m / l ~ . c . . . . po in t ' [

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.// \ ' f~(x)

0 ~ {Year)

1990 1995 2000 2005 2010 2015 2020

m e m l

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a2 a~ x~ b~ b, t

Gray Zone

F i g . 3. H o w t o c o n s t r u c t m e m b e r s h i p f u n c t i o n a n d M a x - M i n f o r e c a s t v a l u e . Example: R e a l i z a t i o n o f c o m p u t e r s w h i c h a r e v e r s e d

in common sense and inferences on the basis of daily life.

4.1. Algorithm via Max-Min normativism

In this algorithm, the following steps are included:

Step 1. Construct a table of cumulative frequency distribution, with F~(x): a function that denotes the period of realization with an extremely high degree of possibility, and F2(x): a function that denotes the period of non- realization with an extremely high degree of possibility. Both Fl(X) and Fz(x) denote cumulative frequency distributions.

Step 2. Both upper and lower quantiles of Fl(x) and F2(x) are obtained as shown at (CI, DO and (C2, D2), respectively. Furthermore, medians corresponding to F~(x) and F2(x) are designated as ml, and m2, respectively. The membership functions denoting 'realizable period' and 'unrealizable period' are Pl(X) that links C1, ml, DD and P2(x) that combines C2, m2, /92, respectively.

In this arrangement, the region where realization is achieved becomes the defined domain [al, bl] of P~(x) and the forecast period for realization X1 (E[a, b]) denotes the period when realization is most possible. In a like manner, the domain of non-realization is defined as [a2, b2] within P2(x) and becomes the forecast period of unrealization. X2 (e[a2, bad denotes the highest membership value out of the non-realization with an extremely high degree.

Step 3. The Max-Min forecast value X* is to be obtained by computing

MaxMin(Pl(x), P2(x)).

This is the value of the forecast period on the basis of two contrastive periods. The line connecting cl, m, with /92 becomes the membership function which synthesizes both pl(x) and p2(x). We call m a 'cross point', and the defined zone of the membership function a 'gray zone'. Thus, the Max-Min forecast value belongs to the gray zone in which both the

246 A. lshikawa et al. / The Max-Min Delphi method and fuzzy Delphi method

realizable and unrealizable periods show the same value of the membership function. The Max-Min forecast value is obtained by

MaxMin(fffx), fz(x)). X*

In order to verify the effectiveness and utility of the Max-Min Fuzzy Delphi Method, 'the period of realization of computers which are versed in common sense and inferences on the basis of daily life' as a forecast item will be picked up and both the traditional Delphi method and the Max-Min Fuzzy Delphi Method applied. The objects are members of the Japan Society for Fuzzy Theory and Systems and the methods of the questionnaires are based on both the traditional Delphi and the new questionnaire (refer to Table 1) approaches.

4.2. Result obtained by the Delphi method

Table 2 is the result of the first and second surveys by means of the traditional Delphi method. Figure 4 shows the identification of both quartiles and the median to the result.

As shown in Figure 4, the most likely period (year) falls in 1998, while the lower and upper quantiles fall in 1995 and 2000, respectively.

Table 2. Result by the Delphi method. "Realization period of computers which are well versed in common sense and

inferences on the basis of daily life"

Exp. 1 (1989) Exp. 2 (1990)

At present 2 1 After 2-3 years 10 12 After 5 years 29 32 After 10 years 25 27 Unknown 28 28

(N = 94) (N = 100)

4.3. Result obtained by the Max-Min Fuzzy Delphi Method

Table 3 indicates the respondence rate and a table of cumulative frequency distribution on the basis of 'the period where realization is absolutely impossible' and 'the period where realization is certainly possible' due to the new procedure and questionnaires shown in Table 1.

Using Table 3, we have determined the membership functions, fffx) and f2(x). Through application of the Max-Min criterion, the cross Point, m, is obtained. The corresponding forecast value and gray zone are shown in Figure 5. Here in this case, the gray zone ranges from 1996 to 2000, whereas the forecast value turns out to be 1998.

As may been clear by now, even for the same subject such as 'realization of computers which

Table 3. Respondence distribution to the questionnaire

(a) Impossible (b) Possible Period (N = 74) Period (N = 69)

R.R. C.P. R.R. C.P.

1990 % % 1990 1.4% 1.4% 1991 2.6 100.0 1993 1.4 2.8 1992 6.8 97.3 1995 16.7 19.4 1993 12.2 90.5 1996 2.8 22.2 1994 1.4 78.4 1998 11.1 33.3 1995 33.8 77.0 1999 9.7 43.1 1996 4.1 43.2 2000 19.4 62.5 1997 1.4 39.2 2001 4.2 66.7 1998 5.4 37.8 2002 1.4 68.1 1999 5.4 32.4 2005 6.9 75.0 2000 21.6 27.0 2010 11.1 86.1 2001 1.4 5.4 2020 4.2 90.3 2010 2.7 4.1 2021~ 9.8 100.0 2021- 1.4 1.4

Note: R.R. Response Rate, C.P.: Cumulative Percentage.

1990 1995 • 2000 19,98

(1 /4 ) (2/4) (3/4)

200 I~ ---- 5 YEAR

Fig. 4. Result from the Delphi method.

A. Ishikawa et al. / The Max-Min Delphi method and fuzzy Delphi method 247

Possible --

Impossible-- f 1

I I , 1990 1995 1996 • 2000(1/4) 2005

(3/4) (1/4)11998 (3/4)

Fig. 5. Result by Max-Min Fuzzy Delphi Method.

are versed in common sense and inferences on the basis of daily life', the image and result of the response are subject to the pursuit of the issue presented, as shown in Figure 5. In this analysis, the result of the traditional Delphi method indicates a smaller range of quantiles. Although it is likely that the influence on measure presentation for easy response might be reflected, further convergence is needed to narrow down the range to the point where rational agreement is created. This is inevitable, because the traditional Delphi method, in essense, attempts to eliminate ambiguity in a statistical and procedural manner.

On the other hand, the Max-Min Fuzzy Delphi Method pursues the accuracy of forecast from both possibility and impossibility stand- point. Furthermore, the gap between these two extremities can be preserved and utilized. Since human judgments, in general, are considered variable and movable within a certain range, rather than converging into a single point, it is worth exploring the period where neither impossible nor possible judgments are made. In other words, a real clue might be hidden in such a fuzzy domain.

In Figure 5, impossibility is higher from 1995 to 1996, whereas possibility is higher from 2000 to 2005. Possibility and impossibility are mixed from 1996 to 2000, and thus this range is regarded as a gray zone in this case. It is our basic conviction that the fuzzy period embraces the trigger of technological breakthrough, while clear zones can be discarded.

Although which forecast method is more appropriate cannot be judged until the realization of forecast items, following differences have been uncovered:

(1) While the range of quantiles in the traditional Delphi method is from 1995 to 2000 with a median of 1998, the gray zone in the

Max-Min Fuzzy Delphi Method is from 1996 to 2000 with a forecast value of 1998. Thus, both methods turn out the same forecast value.

(2) However, both methods are essentially different. While the former adopts the approach where fuzziness is gradually eliminated in order to enhance the accuracy and precision of the forecast, the latter adopts the approach where fuzziness is preserved by discarding clarified judgments.

(3) While the former places more emphasis on the median (the period one-half of the respondents acknowledge), the latter on the point where the synthesized membership functions of possibility and impossibility cross each other, i.e., the corresponding values are equal to the Max-Min forecast value. Thus, on the basis of the latter method, our survey enabled the afore-mentioned analyses from which the similar result was obtained.

4.4. Algorithm for obtaining forecast values via fuzzy integration (FDMFI)

This method, on the basis of the data obtained from the object through the questionnaires as shown in Table 1, constructs membership functions to the forecast items and computes the forecast value via fuzzy integration, where the extent of specialization becomes fuzzy measure. The following steps are to be taken to acquire the overall assessment value, i.e., the subjec- tively forecasted value:

Step 1. Establish a membership function for each subject with respect to the forecast period. For instance, in the case where the membership function for the i-th subject is to be determined, those years being 'predicted impossible for attaining such a technological breakthrough until approximately year Xu' and 'predicted possible

248 A. Ishikawa et al. / The Max-Min Delphi method and fuzzy Delphi method

M.F.

d(x)

ht(x)

hi(x)

hu(x)

Xu X * Xt ~Forecast Per iod (Year)

Fig. 6. The membership function of the i-th subject.

for attaining such an achievement after year Xt' are identified. Each identified membership function is shown as hu(x) and ht(x), respectively, as shown in Figure 6.

Thus, their synthetic function

hi (x ) = min(hu(x), ht(x))

becomes the membership function for the i-th subject, as shown in Figure 3. The membership function X* that gives hi = m a x h i ( x ) is the attainable forecast period (year) by subject.

As shown above, all membership functions appropriate to each subject, hi(x) , i = 1 , 2 , . . . . n, are identified.

Step 2. Using hi(x), i = 1, 2 , . . . , n, obtained in Step 1 and the extent of experitse for the corresponding forecast item, gi, i = 1, 2 . . . . . n, fuzzy integration f-hi(x)og~ is undertaken by each forecast period (year). The forecast values via fuzzy integration are obtained as shown below:

(1) hi(x) is rearranged in descending order. (2) H i = gi + Hi-1 q- Agi" H i - l , 111 = gl is to be

computed. (3) hi^Hi , the maximum value for 1 ~ i ~<n

will be obtained and designated as an assessment value for a given year.

Through repetition of the above procedure, it is possible to obtain the attainable value appropriate to each year. The maximum value from the values acquired denotes the year most attainable for the respective item under consideration.

Thus, the overall assessment value of each subject for each forecast year, taking the extent of expertise into consideration, can be decided.

In order to verify the effectiveness of the FDMFI algorithm, it is applied to two forecast items, 'realization of a retrieval system for cases similar to a given case from the databases, from the viewpoint of the retriever' and 'Realization of computers which are versed in common sense and inferences on the basis of daily life' and forecast values are obtained in two ways:

Forecast values via f u z zy integration (1). By applying our method to a particular forecast item, 'from the viewpoint of the retriever, this data retrieval system can retrieve cases similar to a particular case from the databases', the feasibility of this method will be examined.

Since we have had 39 responses on the relevant items, we have constructed triangular membership functions, as shown in Figure 7. While 20 triangular membership functions are shown in this figure, the remaining 19 are a duplication of these 20 functions.

When we have applied these membership functions to fuzzy integration, as shown in Figure 7, for example, in 1997, the following membership values were extracted (refer to Table 4).

By using the data in Table 4, the attainable forecast values via fuzzy integration are shown in Figure 9. Figure 8 indicates that the highest assessment value of the forecasts for each year

M. F.

1995 1997

1.0

/ 1990


2000 2005 2010

Fig. 7. Triangular membership functions of 39 data.

Table 4. 39 Membership functions and the Extent of Expertise (E.E.) in 1997 extracted from triangular

membership function

MF MF MF MF value E.E. value E.E. value E.E. value E.E.

0.26 1.00 0.50 0.40 0.00 0.30 0.00 0.20 0.80 0.50 0.75 0.10 0.00 0.40 0.00 0.60 0.67 0.30 0.67 0.10 0.00 0.50 0.00 0.20 0.80 0.20 0.67 0.50 0.00 0.90 0.00 0.80 0.80 0.10 0.80 0.70 0.00 0.30 0.00 0.60 1.00 0.10 0.80 0.70 0.00 0.50 0.00 0.80 0.80 0.30 0.40 0.70 0.00 0.10 0.00 0.60 0.50 0.50 0.33 0.70 0.00 0.20 0.00 0.70 0.54 0.10 0.84 0.70 0.00 0.40 0.00 0.60 1.00 0.10 0.00 0.70 0.00 0.20

from 1993 to 2000 is 0.4 in 1996; therefore, 1996 is considered as the realizable period for this item.

It is worth noting that this result is approximately the same as results obtained from the Delphi method and the Max-Min normativism (Fuzzy Delphi) Method.

We have acquired responses from 40 objects. As previously, triangular membership functions were contructed and MF values obtained from 1993 to 2000. As an illustration, the MF values in 1996 are shown in Table 5. By using the data in Table 5, achievable forecast values were obtained by fuzzy integration, where the extent of expertise was used as a fuzzy measure, as is shown in Figure 10.

This outcome implies that the corresponding item will be realized in about 1996. The result by the Delphi method is in 1997, and our method indicates realization one year earlier. However, taking the method of MF values computation and inherent errors on expertise into consideration, it may be said that both approaches show approximately the same result.

Through the aforementioned analyses for verification, the proposed Max-Min Fuzzy Delphi Method and the Fuzzy Delphi Method via Fuzzy Integration have been validated as prospective methods for long-term forecasting.

Forecast values' via fuzzy integration (2). In order to verify the effectiveness of the FDMFI, this method is also applied to 'the realization period of computers which are versed in common sense and inferences on the basis of daily life'.

5. The relationship between the extent of expertise and the degree of importance

The relationship between expertise and importance in the sample questionnaire is shown

250 A. lshikawa et al. / The Max-Min Delphi method and fuzzy Delphi method

M.F.

1.0

M.F.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Answer

No. of Data

Fig. 8. The result of 1996 via fuzzy integration.

0.4

0. 247 0. 097

1993 1994 1995 1996 1997 1998 1999 2000 Forecast Year

Fig. 9. Transfer of forecast values via fuzzy integration.

in Table 6 and Figure 11 as three-dimensional data.

Table 6 denotes the relationship between the extent of expertise and the degree of importance. The extent of expertise is divided into four groups, H (6-10 points), M (3-5 points), L (1-2 points), and None (0 points).

The degree of importance is also divided into four groups, i.e., 'very important ' , 'considerably important ' , 'not so important ' , and 'unimpor- tant'. In addition, the relations between the extent of expertise and the degree of importance are denoted in three dimensions and in-depth analyses are made by devising easier visualiza-

tion. In this instance, as shown in Tables 7 & 8 and Figures 12 & 13, as one's expertise becomes higher, the variances become smaller, whereby cognition on the degree of importance tends to be congruent and consistent.

Moreover , in the Max-Min Fuzzy Delphi Method, as expertise becomes higher, the range of the gray zone where possibility and impossibility are mixed tends to be narrower. This trend is due to the higher convergence of responses, compared to other layers. To put into another way, it indicates that the judgments of forecast are sharper in the layer of higher expertise. As has been delineated, the proposed


Table 5. 4 0 M F values and the extent of expertise in 1996 Table 6. The relationship between the extent of expertise extracted from triangular membership functions and the degree of importance

MF value E.E. MF value E.E.

0.13 1.00 0.40 0.50 0.33 0.50 1.00 0.30 0.40 0.20 0.40 0.10 0.50 0.10 0.40 0,30 1.00 0,40 1.00 0.50 0.50 0.30 0.36 0.10 0.50 0.10 1.00 0.10 1.00 0.10 0.88 0.50 1.00 0.70 0.40 0.70 0.40 0.70 0.20 0.70

20. Automatic translation system that deals with sentences of Japanese with Chinese characters and other languages

0.50 0.70 0.67 0.70 1 2 3 4 0.84 0.70 No. of Very Pretty Not so Not at all 0.00 0.70 samples impor- impor- impor- impor-

tant tant tant tant 0.00 0.80

0.00 0.40 Total 93 69.9 22.6 3.2 1.1 0.00 0.20 0.00 0.90 E.E. 0.00 0.30 H (0-10 pts) 8 87.5 12.5 - - 0.00 0.50 M (3-5 pts) 21 76.2 23.8 -

L (1-2 pts) 21 57.1 28.6 9.5 - 0.00 0.10 None (0 pts) 42 69.0 21.4 2.4 2.4 0.00 0.20 0.00 0.40 0.00 0.20 0.00 0.60

0.00 0.60 effectiveness by applying them to concrete 0.00 0.80 technological forecasting issues. 0.00 0.60 o.oo o.6o As a result, these two methods, compared with

methods not only uncover subtle relations between the extent of expertise and the degree of importance, but also enable more sophisti- cated analyses than those in the traditional method.

6. Concluding remarks

This paper has proposed two new fuzzy Delphi methods, the Max-Min Fuzzy Delphi Method and the Fuzzy Delphi Method via Fuzzy Integration, and verified their feasibility and

the traditional Delphi method, have revealed the following merits:

(1) Reduction of the number of repetitions of the survey.

(2) Processing the fuzziness of each forecast item by each forecaster more rationally and in a more desirable manner.

(3) Better clarification of the individual properties of each forecaster in relation to each forecast item.

(4) More economy in both time and costs, because of fewer repetitions.

In order to use these methods, it is desirable to employ them by keeping in mind the following characteristics:

(1) The Max-Min Delphi Method clarifies the

0.5

0.4

0.3

0.2

0.1

0

0.379

0.280

0.389 0.400

0.351

19'93 19'94 19'95 1996 1997

0.343

~ 0 . 1 0 6

19'98 1969 2~00 (Year)

Forecast Year

Fig. 10. Transfer of forecast values via fuzzy integration.

%

I

f

]

r / / ' " 2 3 4

i j

i 1 I "lfl ,/i II

1 I

~<- H ( 6 ~ 1 0 0 t s . )

3 -- 5 p t s . )

1

L ( 1 - - 2 p t s . )

3 4

N O N E ( 0 p t s . )

Fig. 11. The r e l a t i ons b e t w e e n expe r t i s e and impor t ance . Example: R e a l i z a t i o n pe r iod of a u t o m a t i c t r ans l a t i on cons i s t ing of

J a p a n e s e s en t ences in C h i n e s e cha rac te r s and in o t h e r ones.

T a b l e 7. The r e l a t i onsh ip b e t w e e n e x p e r t i s e and impor t ance .

Example: F u t u r e c o m p u t e r s which are e q u i p p e d wi th

c o m m o n sense

T a b l e 8. The r e l a t i onsh ip b e t w e e n expe r t i s e and impor t ance .

Example: F u t u r e r e t r i eva l sys tems

4. Future retrieval systems which can retrieve similar cases from the 3. Future computers which are equipped with common sense retriever's viewpoints

1 2 3 4 1 2 3 4 No. of Very Pretty Not so Not at all No. of Very Pretty Not so Not at all samples impor- impor- impor- impor- samples impor- impor- impor- impor-

tant tant tant tant tant tant tant tant

Total 93 36.6 51.6 7.6 1.1 Total 93 36.6 50.6 10.8 -

E.E. E.E. H (6-10 pts) 17 52.9 47.0 - - H (6 ~ 10 pts) 18 66.6 33.4 - - M (3-5 pts) 25 44.0 52.0 4.0 - M (3 - 5 pts) 19 36.9 47.4 15.8 - L (1-2 pts) 20 15.0 75.0 5.0 - L (1 - 2 pts) 24 33.4 58.4 8.3 - None (0 pts) 31 35.5 38.8 16.2 3.2 None (0 pts) 32 21.9 56.3 15.6 -

I I

/ / / " 1 /

/ / / /

L

/ -

2

/ / / / /

/ /

I [ ~ M ( 3 - - 5 p t s . ) . . . . . . 1 2 3 ~ . . . . . . . .

L (1 ~ 2pts.) 3 ~ . . . . . . . . . . . . . . . . . . . . . .

H (6 ~lOpts.)

. . . . . . . . . . . . . . . ~ £ E ~ 2 e t z 2 . . . . 3 4

Fig. 12. The r e l a t i onsh ip b e t w e e n expe r t i s e and impor t ance . Example: F u t u r e c o m p u t e r s which are e q u i p p e d wi th c o m m o n

sense.


~ / ~ 1 2 3 i / /

¢ /

I / / H ( 6 ~lOpts.)

/ /J" . . . . 1 2 ~--~'- . . . . . . . . . . . . . . . .

I J--'-'l--J~ t i i ~ M (3-Spts., . . . . . , 2 3 . . . . . . . . . . . . . . . . . . . . .

z I I I - -BJ L ~T--i 2 3 ~ .................... I ' NONE (Opts.) 4

Fig. 13. The relationship between expertise and importance. Example: Future retrieval systems which can retrieve similar cases from the retriever's viewpoints.

data of each forecaster by expertise, regards a gray zone as an interval estimate, and identifies the cross point as the most attainable period.

(2) The Fuzzy Delphi Method via Fuzzy Integration employs the extent of expertise by each forecaster as fuzzy measure and identifies a point estimate as the most attainable period by fuzzy-integrating each membership function on forecasting.

Consequently, while each method, including the traditional Delphi and the fuzzy Delphi method, can obtain a forecast value that is an attainable period, the best way is to use these methods in parallel, or simultaneously, so that we may gain more confidence, on such difficult forecasts as technological forecasting, with constant regards to the merits and demerits of

each method.

References

[1] M. Sugeno, Fuzzy measure and fuzzy integral, Trans. Soe. Instr. Control Engrs. 8 (2) (1972) 218-226.

[2] A. Ishikawa et al., Needs Study of Fuzzy Systems (LIFE, 1991).

[3] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353.

[4] J. Pill, The Delphi method: Substance, context, a critique and an annotated bibliography, Socio-Econ. Plan. Sci. 5 (1971) 55-71.

[5] O.H. Helmer, The Delphi Method for Systematizing Judgments about the Future (University of California, Institute of Government and Public Affairs, 1966).

[6] A. Kaufmann and M.M. Gupta, Fuzzy Mathematical Models in Engineering and Management Science (Elsevier Sciences Publishers, Amsterdam, 1988) Chapter 13.

the max-min delphi method and fuzzy delphi method via fuzzy integration

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