碎形理論在品質管理上之研究(II)

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(1)行政院國家科學委員會專題研究計畫成果報告碎形理論在品質管理上之研究(2/2) 研究成果報告(完整版). 計計執執. 畫畫行行. 類編期單. 別號間位. ：個別型： NSC 95-2221-E-011-076： 95 年 08 月 01 日至 96 年 07 月 31 日：國立臺灣科技大學工業管理系. 計畫主持人：周碩彥計畫參與人員：碩士級-專任助理：徐啟桓、林奕誠、詹佳喜. 報告附件：出席國際會議研究心得報告及發表論文. 處理方式：本計畫涉及專利或其他智慧財產權，2 年後可公開查詢. 中. 華民國 96 年 10 月 30 日.

(2) Contents 摘. 要 …………………………………………………………………………..I. Abstract ……………………………………………………………………….…II Contents …………………………………………………………………………III 1. Introduction …………………………………………………………………….1 2. Research Objective ……………………………………………………………..1 3. Literature Review ………………………………………………………………2 3.1 The Hurst phenomenon and Hurst law …………………………………....2 3.2 The Brownian Motion …………………………………………………..….4 3.3 The Fractional Brownian Motion ………………………………………….6 3.5 The Autoregressive T2 Control Chart ……………………………………..7 4. Research Methods ……………………………………………………….…...…8 4.1 Performance Comparison ………………………………………………….8 4.2 The Guideline for Monitoring fBm Processes ……………..…………….10 4.3 Illustrative Example ……………………………………………………….14 5. Conclusion ……………………………………………………….……..………17 References ………………………………………………………..………………..18 Plan achievement ………...……………..…………………………………………..19. I.

(3) 1. Introduction Control charts are effective tools in many industries to monitor processes with the objective of improving process quality and productivity. In recent years, statistical process control for autocorrelated processes has received a great deal of attention. This is due in part to the improvements in measurement and data collection technology that allow processes to be sampled at higher rate which often leads to autocorrelation. There are two types of autocorrelated processes: short-term dependent processes and long-term dependent processes. Fractional Brownian motion (fBm) is one of the latter that was put forward by Mandelbrot after Hurst’s findings, fractal theory. It has been successfully used to model the variate of many natural phenomena because it considers the dependence between each variate of natural phenomena. In addition to natural phenomena, it has been proved that fBm is the suitable model to describe many artificial processes, such as stock trend line of finance[1], the pressure signal in time at a certain height of industrial airlift reactor[16], hedging options in market models [3], and so forth. Recently, statistical process control (SPC) for short-term autocorrelated processes has received a great deal of attention. Approaches for dealing with such processes exist. But so far, no approaches are available for dealing with long-term autocorrelated processes such as fBm, which has been proved a suitable model to describe certain processes. Therefore, it is necessary to bring up an idea to help manager to monitor that. Since fBm process has some characteristic properties like self-similarity, increment stationarity, variance law and so on, this paper designs a guideline for monitoring by transforming the observations according to its properties.. 2. Research Objective The objective was to develop a guideline for monitoring fBm process. Generally, the most widely researched approaches of SPC for autocorrelated processes are residual-based control charts, which involve fitting some form of time series model to the data and the forecast errors or residuals from the model are used in the control charts. However, because of the significant and long-time autocorrelation, estimating the parameters of time series model like autoregressive moving average (ARMA) model will be difficult and inaccurate. The autoregressive T2 chart requires only the process autocorrelation function up to a pre-specified lag to form a multivariate vector for using. And this paper finds the relationship between the Hurst exponent of fBm and the autocorrelation function of the transformed data. Hence use the autoregressive 1.

(4) T2 chart to deal with the fBm data after transforming. In many situations, the autoregressive T2 control chart compares favorably. In addition, it possesses other advantages over the other control chart. A single autoregressive T2 chart design is often suitable for a wide range of mean shift sizes. This has practical significance, in that is generally difficult to select one mean shift size that is of primary interest. It is, therefore, desirable that a control chart performs well over a range of mean shift sizes and could match with fBm process. Besides, use R to simulate fBm process by the technique of Cholesky that has been chosen as one of the fundamental fBm synthesizers and deal with that according to the guidelines provided in this paper. In the reminder of this paper, we present the Hurst phenomenon and Hurst law, the characters of fractional Brownian motion, the autoregressive T2 control chart, the performance comparison with other control charts and giving the guidelines for monitoring fBm processes, illustrative example and conclusion.. 3. Literature Review 3.1 The Hurst phenomenon and Hurst law Hurst spent his lifetime studying the Nile to determine the storage capacity of an ideal reservoir based upon the given record of observed discharges from the Great lake of the Nile Basin[12]. The broad plan includes storage of water from good years for use in bad ones and designs reservoirs of sufficient capacity to meet the shortages that might occur during a century. Much research related to the capacity of such reservoirs was used to determine the size of the reservoirs required in the Great Lakes of the Nile Basin for what was called “century storage.” Hurst analyzed the data form such research and discovered that are long-term records of natural events which are shown to have some certain similarities, no matter they may be records of stream flow, river discharge, rainfall, temperature, annual growth rings of trees, or annual deposits of clay in lakes. These natural phenomena appear that they all have chaotic and irregular behaviors. According to the results of investigation, Hurst developed a new statistic theory, the rescaled range analysis, to explain the chaotic and irregular behaviors of several natural phenomena. The Rescaled Range Analysis In the case of Lake Albert (in the Belgian Congo and the Uganda Protectorate), the problem of what storage is required on a stream to give a certain minimum discharge was first investigated. It was to determine the size of a storage reservoir 2.

(5) required to balance the outflow over a number of years. The basic questions of designing a capacity reservoir are how its outflow can be regulated economically to meet the shortages of low years, and what the capacity required to guarantee a certain minimum discharge is in these years. The size of reservoir required to maintain the maximum possible steady discharge during the period can be determined if the past annual discharges from the lake are given. This is the mean discharge for the period n and capacity of the reservoir is obtained by adding, algebraically, the departures from the mean to form a series of accumulated totals. The difference between the highest and lowest of these accumulated totals—that is, the range R— is (a) the maximum accumulated deficit when there is never a deficit, (b) the maximum accumulated deficit when there is never any storage, or (c) their sum when there is both storage and deficit. This range R is the storage required to maintain the average discharge. According to this concept, Hurst supposed rescaled range analysis: There is a reservoir fed by the discharge of water from a lake. In any given year the reservoir receives an influx from the lake and a regulated volume of water is released. Hurst let x(t ) denote the inflow in year t then the average inflow over a period of T years is simply (1 T )[ x(1) + x(2) + ⋅⋅⋅ + x(T )] . Call this average inflow xT . Now define the departure of the inflow from this average over a t-year time horizon to be X (t , T ) = [( x(1) − x ) + ( x(2) − x ) + ⋅⋅⋅ + ( x(t ) − x )] . The difference between the minimum and the maximum accumulated inflow over a period of T years is what we call the range, denote by RT : RT = max X (t , T ) − min X (t , T ) T. T. T. and the sample standard deviation is[12] ST = {(1 T )[( x(1) − xT )2 + ( x(2) − xT )2 + ⋅⋅⋅ + ( x(T ) − xT )2 ]}. 1. 2. What Hurst discovered is that a large number of natural phenomena seem to be governed by a simple relation involving what’s now called the rescaled range, defined by the dimensionless ratio RT/ST. This quantity, in effect, scales the range by its standard deviation. Form a qualitative point of view it gives a standardized measure of the path length covered over a given time interval by the stochastic process. It is a statistical method used to study a wide range of phenomena. Hurst’s results showed that a lot of natural phenomena, ranging from flood levels on the Nile to trends in global temperature, changes seem to obey the relation RT ST = aT H , where H is a number called the Hurst exponent and this relation is called Hurst Law. 3.

(6) To determine H, Hurst set RT /ST equal to (T/2)H, that is, a equals to 1/2, and compute H for each time series. Therefore, H is given by [22]. H =. lo g (. RT. lo g ( T. ST. ). ) 2 It can be shown that the Hurst exponent H must differ from 0.5 for long period T when the process is dependent and has persistent/anti-persistent behavior. And many researches suggest that a lot of natural processes belong to that. It proves that there is some long-term memory present in processes like sunspot fluctuation, river discharges and rainfall levels, all of which have values of H differing from 0.5. To understand why these sorts of natural phenomena display values of H differing from 0.5, Hurst devised an experiment. The experiment suggested that these kinds of natural processes “remember” what happened earlier. For example, the discharge of a river depends not only on the current level of rainfall but also on earlier rainfall. Analysis shows that the Hurst exponent, H, provides double information on the underlying stochastic process. For H ∈ (0, 0.5) , there is an anti-persistent behavior. For instance, if for some time in the past we had an increase in rainfall levels, then we will see a decrease in the future—on the average. Consequently, an increasing trend in the past implies a decreasing trend in the future for all processes with H less than 0.5 and we call this situation Hurst Phenomenon. On the other hand, the process has a persistent or trend-reinforcing behavior for H ∈ (0.5,1) . In these situations an increase in the past implies an increase in the future.. 3.2 The Brownian Motion Randomness is inherent in all natural phenomena. Even the most perfect crystal has many impurities and other defects placed at random. The study of random functions has been overwhelmingly devoted to sequences of independent random variables, to Markov processes, and to other random functions having the property that sufficiently distant samples of these functions are independent, or nearly so. But according to the empirical findings of the design of water system [12] and Hurst’s law there was a common basic feature of many phenomena, namely “span of interdependence”. Mandelbrot and Van Ness proposed to designate a family of random functions, called fractional Brownian motion (fBm) [2] and believed that fBm do provide useful models for a host of natural time series and wish therefore to present their curious properties to scientists, engineers and statisticians[13]. The study of Brownian motion dates back to 1827, when Scots botanist Robert 4.

(7) Brown observed that small particles immersed in a liquid exhibit ceaseless irregular behavior. In 1905, Albert Einstein explained this motion in terms of perpetual 0.5 collisions with molecules of the surrounding medium. He set R = T Norbert Wiener, in 1923, gave the first satisfactory mathematical construction of Brownian motion sample path and proved their “almost sure” (in the language of probability theory) continuity. Definition: A one-variable Brownian motion or Wiener process on an interval [a,b] is a Gaussian process X(t) such that. 1. X(0) = 0 and X(t) is almost surely continuous. 2. Gaussian increment property: The increments ΔX = X (t 2 ) − X (t1 ). for t2＞t1 have Gaussian distributions with mean zero and variance σ2(t2 - t1), where σ is a positive constant. Equivalently, u2 Pr[ΔX < x] = ∫ exp(− 2σ 2 (t 2 − t1 ) )du 2πσ 2 (t 2 − t1 ) −∞ 1. x. Variance Law and Stationarity: Restating the variance part of property 2 gives us the variance law for increments of Brownian motion var[ X (t 2 ) − X (t1 )] = σ 2 t 2 − t1. for any t1 and t 2 in the interval [a,b]. Brownian motion is a “self-similar” process and satisfies “Hurst law.” Thus, we obtain Standard deviation of X(t+s) - X(t) = σ s for every t and s[4] Independent Increments Property: Two random variables X and Y are independent provided that for any real numbers x and y,. Pr[X < x and Y < y] = Pr[X < x] Pr[Y<y] Brownian motion has independent increments in the sense then the increments are independent random variables. Markov property: A Brownian motion, and indeed any independent increment process, is a Markov process. This implies that the conditional probability that X(t2),. reaches a certain value, given the value of X(t1), where t1 < t2 depends only on t1 and t2. It does not depend on the history of X(t) for t < t1. One can think of a random 5.

(8) walk process in which each step is taken anew with no recollection of how the process got to the present state. Statistical Self-similarity: The increments of a sample path of Brownian motion satisfy the statistical self-similarity property,. X (t + Δt ) − X (t )Δ. 1. r. ( X (t + rΔt ) − X (t )). for any r > 0. (The symbol. Δ. stands for the two random variables follow identical. distribution.). 3.3 The Fractional Brownian Motion Classical Brownian motion given the value at X(t1), where t1< t2, depends only on t1 and t2, not on the history of X(t). Clearly, there is a need for a version of the random process that has some memory of the past. Such a process was introduced and analyzed under the name fractional Brownian motion (fBm) by Mandelbrot and Van Ness in 1968[2]. A fractional Brownian motion is defined in terms of a parameter H with H ∈ (0,1) . For H = 1/2 the definition of a fractional Brownian motion reduces to that of classical Brownian motion. Falconer defines the fBm as flows. Definition: A real stochastic process X(t), with t ∈ [0, ∞) , is a fractional Brownian motion with index H ∈ (0,1) , called the Hurst exponent, if 1. X(0) = 0 with probability one, 2. X(t) is continuous almost everywhere for all t ∈ [0, ∞) Stationarity: The autocorrelation of fractional Brownian motion is. φB. (t , s ) = σ H 2. H BH. 2. (t. 2H. +s. 2H. − t−s. 2H. ). with. σ H2 =. ⎡0⎛ 1 ⎤ (H − 12 ) (H − 12 ) ⎞ − −τ 1−τ ⎟ dτ + ⎥ 2 ⎢ ∫⎜ ⎠ 2H ⎦ 1 ⎞ ⎣−∞⎝ ⎛ Γ⎜ H + ⎟ 2⎠ ⎝. σ2. Non-independence of Increments: By contrast with ordinary Brownian motion, which satisfies the independent increments property, a fractional Brownian motion with parameter H ≠ 1/2 does not satisfy this property. Non-Markovian Behavior: If H > 1/2, then X(t) - X(0) and X(t+h) - X(t) tend to have the same sign, and X(t) tends to be increasing in the future if it is increasing in the past.. If H < 1/2, then X(t) - X(0) and X(t+h) - X(t) tend to have opposite signs, and the trends tend to reverse. These observations combine to show that a fractional Brownian 6.

(9) motion is not a Markov process, except when H = 1/2. Statistical Self-similarity: The increments of a fractional Brownian motion satisfy. B H (t + Δt ) − BH (t )Δ. 1 ( BH (t + rΔt ) − BH (t )) , for any r > 0. rH. and it is obtained from filtering of B(t , w) as given by B (t , w) =. t 1 (t − s) H −1/ 2 dB( s, w) ∫ 0 Γ( H + 1/ 2). where ∞. Γ(α ) ≡ (α − 1)! ≡ ∫ e − t t α −1dt 0. The increment process of fractional Brownian motion which called fractional Gaussian noise, X BH (t , Δ ) , which. X BH (t , Δ ) = B H (t ) ……..…….(1) are normally distributed with mean zero and variance σ H2 Δ2 H for all t ∈ [0, ∞) and Δt ∈ [0, ∞) , where σ H2 is a positive constant and Δ is the sampling period. If H≠0.5, the increments of fBm are stationary but dependent random variables. Particularly, it is not short-term memory, but the long-term memory that is influenced the most by the latest increments. The autocorrelation of fractional Gaussian noise is. φB. (t , s, Δ ) = σ H (s − t + Δ 2 H + s − t − Δ 2 H − 2 s − t 2 H ) 2. H. BH. 2. …………………(2). 3.4The Autoregressive T2 Control Chart The basic idea of the T2 approach is to form a multivariate vector of a moving window of observations from a univariate autocorrelated process, and then apply multivariate control charts. The T2 approach supposes the process is Gaussian with autocovariance function γ k = E[( xt − ξ )( xt + k − ξ )] . Let ξ0 denotes the mean when the process is in-control, and assume ξ 0 is known. Consider the sequence of p-dimensional vectors X t = [ xt − p +1 xt − p + 2 … xt ]T formed from observations of the univariate process. The covariance matrix of X t is. 7.

(10) ⎛ γ0 ⎜ Σ = ⎜ γ1 ⎜γ ⎝ p −1. γ1. …. γ p −1. γ1. γ0. γ0 γ1 …. ⎞ ⎟ ⎟ …………………………………..(3) ⎟ ⎠. and when the process is in-control the mean vector is ξ0 = [ξ0 ξ0. ξ0 ]T . If ∑ is. known, the T2 statistic Tt 2 = [ X t − ξ 0 ]T Σ −1[ X t − ξ 0 ] ……………………………….(4) which follows a chi-square distribution with p degrees-of-freedom[11] when the process is in-control, could be used to monitor for departures from the in-control state. One can specify a false alarm probability α and compare Tt 2 to the 1 − α percentile of the chi-square distribution with p degrees-of-freedom, denoted χ 2 (1 − α , p) . According to the complicated properties of fBm, the data modeled with ARMA process may be with a large number of parameters, potentially obfuscating connections between the fBm process and its model. Furthermore, great caution must be used in applying the common σ / n error estimate as it can be a significant under-approximation.[13] Therefore, the residual-based control chart is not appropriate to detecting fBm process. Moreover, with significant and decaying slowly autocorrelation, the approach plotting original observations is not applicable for fBm process, either. The T2 approach using a multivariate vector of a moving window of observations and needs autocovariance and autocorrelation function only can be an acceptable method for monitoring fBm process.. 4. Research Methods 4.1 Performance Comparison In this section, the performance comparison with the autoregressive T2 control chart, residual-based CUSUM charts and Shewhart individual charts for a variety of fBm processes are shown in Table 1. There are 1,000 Monte Carlo trials in all simulations. These data were generated by fBm processes (with H=0.55, 0.7, 0.8, 0.9). We just discuss the value of Hurst exponent greater than 0.5. When Hurst exponent is smaller than 0.5, the fBm process has negative autocorrelation, a situation that is uncommon in industry. A range of mean shift is denoted μ1 to μ4. For each fBm process and each mean shift magnitude, there are four different autoregressive T2 charts (with p =2, 6, 10, and 20), four different CUSUM charts (with k=0.5 h=17.73, k=0.75 h=13.02, k=1 h=7.13, k=1.5 h=5.53, where k is reference value and h is decision threshold), and a Shewhart chart were compared. 8.

(11) TABLE1. Comparsion of Out-of-Control ARLs for various Mean Shifts and Various Hurst Exponents.(All charts have an in-control ARL of 500) T2 ARL. mean shift. CUSUM ARL Shewart ARL. H. μ1. μ2. p. Α. μ1. μ2. k. h. μ1. μ2. μ1. μ2. 0.55. 1. 2. 2. 0.0019. 195.3. 99.86. 0.5. 17.73. 47.13. 27.31. 70.85. 39.24. 0.55. 1. 2. 6. 0.0033. 138.7. 91.66 0.75 13.02. 30.85. 15.71. 0.55. 1. 2. 10. 0.0044. 141.4. 91.12. 1. 7.13. 32.12. 22.3. 0.55. 1. 2. 20. 0.0141. 121. 82.82. 1.5. 5.53. 40.39. 20.79. 0.7. 2. 3. 2. 0.0019. 110.5. 29.47. 0.5. 17.73. 77.62. 37.82. 73.65. 46.63. 0.7. 2. 3. 6. 0.0033. 108.9. 32.82 0.75 13.02. 87.26. 34.33. 0.7. 2. 3. 10. 0.0045. 115.6. 31.78. 1. 7.13. 58.22. 25.6. 0.7. 2. 3. 20. 0.0144. 82.99. 27.82. 1.5. 5.53. 100.7. 41.7. 0.8. 2. 3. 2. 0.0021. 189.4. 39.88. 0.5. 17.73. 160.8. 100.3. 147.5. 74.61. 0.8. 2. 3. 6. 0.0033. 157.1. 36.29 0.75 13.02. 151.6. 92.51. 0.8. 2. 3. 10. 0.0045. 203.7. 44.02. 1. 7.13. 216.3. 123.3. 0.8. 2. 3. 20. 0.0145. 206.8. 39.06. 1.5. 5.53. 215.1. 127.4. 0.9. 2. 3. 2. 0.0021. 277.6 123.22 0.5. 17.73. 258.7. 162.54. 0.9. 2. 3. 6. 0.0033. 296.7. 142.9 0.75 13.02. 314.2. 184.2. 0.9. 2. 3. 10. 0.0049. 237.6. 109.7. 7.13. 375.1. 188.3. 0.9. 2. 3. 20. 0.0162. 290.8 119.92 1.5. 5.53. 264.1. 167.5. 1. 297.8 167.54. The in-control average run-lengths (ARL) of all charts approximate 500. Results for other in-control ARLs exhibited the same general trends and are not shown. A following section of this paper discusses guidelines for selecting p and specifying α that is needed to achieve a desired in-control ARL for the autoregressive T2 control chart. The residual-based CUSUM chart signaled when the upper or lower CUSUM statistics exceeded the decision threshold h. The upper and lower CUSUM statistics are calculated recursively via St+ = max{0, St-1+ + et – k} and St– = max{0, St-1– + et – k} Where et is the residual for the observation at time t. The Shewhart chart signaled when the observation exceeds the upper control limit = μ + cσ or is below lower control limit = μ + cσ, where c is a constant we selected to make the in-control ARL approximate 500.In Table 1 we show the out-of-control ARLs of the different control charts with mean shift=1, 2, 3, 4, 5. In general, the autoregressive T2 chart outperforms the residual-based CUSUM charts and Shewhart charts when H or mean shift is large. For example, when H=0.9, the autoregressive T2 chart with p=10 has lower out-of-control ARL than residual-based 9.

(12) TABLE 1. Continued T2 ARL. mean shift H. μ3. μ4. p. 0.55. 3. 4. 2. 0.55. 3. 4. 0.55. 3. 0.55. α. μ3. CUSUM ARL Shewart ARL. μ4. k. h. μ3. μ4. μ3. μ4. 0.0019 25.17 4.17. 0.5. 17.73. 11.77. 5.21. 7.25. 3.18. 6. 0.0033 14.92 3.39. 0.75 13.02. 7.93. 2.01. 4. 10. 0.0044 14.01 2.83. 1. 7.13. 9.82. 2.93. 3. 4. 20. 0.0141 22.28 1.54. 1.5. 5.53. 9.73. 3.72. 0.7. 4. 5. 2. 0.0019 14.21 4.21. 0.5. 17.73. 50.18. 19.37. 11.53. 7.12. 0.7. 4. 5. 6. 0.0033 16.58. 0.75 13.02. 51.88. 18.53. 0.7. 4. 5. 10. 0.0045 17.81 5.78. 1. 7.13. 32.98. 12.17. 0.7. 4. 5. 20. 0.0144 15.74 3.75. 1.5. 5.53. 66.36. 33.61. 0.8. 4. 5. 2. 0.0021 28.24 9.87. 0.5. 17.73. 74.36. 27.18. 21.78. 10.35. 0.8. 4. 5. 6. 0.0033 24.66 6.54. 0.75 13.02. 70.11. 24.64. 0.8. 4. 5. 10. 0.0045 27.78 7.63. 1. 7.13. 89.12. 36.87. 0.8. 4. 5. 20. 0.0145 27.91 7.64. 1.5. 5.53. 80.78. 31.36. 0.9. 4. 5. 2. 0.0021 65.96 20.52. 0.5. 17.73. 100.3. 61.87. 89.81. 29.82. 0.9. 4. 5. 6. 0.0033 69.34 26.68 0.75 13.02. 113.2. 75.92. 0.9. 4. 5. 10. 0.0049 49.52 19.1. 1. 7.13. 117.9. 79.37. 0.9. 4. 5. 20. 0.0162 58.81 22.12. 1.5. 5.53. 92.83. 54.43. 4.11. CUSUM and Shewhart chart. But when H=0.7 and mean shift=2, 3, the ARL1 of autoregressive T2 are not lowest between the residual-based CUSUM chart and Shewhart chart. As Table 1 display we know the autoregressive T2 chart aims at the high Hurst exponent of fBm process. And when the mean shift is moderate to large, the autoregressive T2 chart also has the better performance than the residual-based CUSUM and Shewhart chart. To detect a mean shift effectively for the various fBm processes, we must to find the optimal value of p. The method for selecting optimal value of p for the autoregressive T2 control chart is mentioned in the subsequent section.. 4.2 The Guideline for Monitoring fBm Processes In order to monitor the fBm process by SPC method, we have to use fGn observations to remove the variation of its variance. According to the properties of fBm, we can use the Statistical Self-similarity to do that. BH(t) and the increment process X BH (t , Δ) (fGn; fractional Gaussian noise) are Statistical self-similar.. 10.

(13) Since. X (t + Δ t ) − X (t ) Δ. 1 ( X ( t + r Δ t ) − X ( t )) (for any r > 0) and the rH. fGn (increment process of fBm) X BH (t , Δ ) = B H (t + Δ ) − B H (t ) ~ N (0, σ H2 Δ2 H ) and it becomes the stationary autocorrelated process. We have to get the Hurst exponent to determine the variance of increment process. It has introduced the most widely used method by the previous section, R/S analysis:. H =. lo g (. RT. lo g ( T. ST 2. ). ). Next, we can find the autocorrelation of the fGn data and use T2 chart to monitor that. The autocorrelation is. ρk =. σ H2 2. (k +Δ. 2H. + k −Δ. 2H. −2k. 2H. ).. After that, we can design the T2 control chart: Let. [. Yt = X BH (t − p + 1, Δ), X BH (t − p + 2, Δ ),. , X BH (t , Δ). ]. T. , we can get Σ, the covariance matrix of Yt,. ⎛ γ0 ⎜ Σ = ⎜ γ1 ⎜γ ⎝ p −1. γ1. …. γ p −1. γ0 γ1 …. γ1. γ0. ⎞ ⎟ ⎟ ⎟ ⎠. when the process is in-control, the mean vector is μ0 = [μ0 μ0. μ0 ]T . And then. calculate the T 2 statistic from equation (4). Because it is fGn process, we can say μ0 of the autocorrelated process equal to 0.. ⇒ Tt. 2. = Yt T Σ −1Yt ………………………………………………………………...(5). which follows a chi-square distribution with p degrees-of-freedom when the process is in-control, could be used to monitor for departures from the in-control state. From [11], we know that to implement the autoregressive T2 control chart. The user must specify p and α. We first discuss guidelines for selecting p. As discussed previously, the simulation results indicate that the value of p that provides the lowest 11.

(14) out-of-control ARL does not depend strongly on the size of the mean shift. In other words, p can be selected based only on the autocovariance structure of the process. We specify the p value of T2 chart by [11]. We show that the Cholesky factorization of Σ is Σ = Β −1' DΒ −1 ,where. ⎛ 1 − β1,1 − β 2, 2 ⎜ − β1, 2 1 ⎜ ⎜ B= 1 ⎜ 0 ⎜ ⎜ ⎝. − β p −1, p −1 ⎞ ⎛ σ 02 ⎟ ⎜ − β p − 2, p −1 ⎟ σ 12 ⎟ , D=⎜ ⎜ ⎟ ⎜ − β1, p −1 ⎟ ⎜ ⎟ ⎝ 1 ⎠. ⎞ ⎟ ⎟ ⎟ ⎟ σ p2 −1 ⎟⎠. Let n* be the lag after which the magnitude of the parameters drops below some small value, say 0.1. In other words, n* is defined as the smallest integer such that. β j ,∞ < 0.1 for all j>n*. If p is set as n* + 1, then Σ will also capture the dynamics of the process. Setting p* = n* + 1 appears to be an effective rule-of-thumb for selecting the optimal value of p. After selecting p, one potential means of selecting α is to fix the false alarm probability. Clearly the false alarm probability will be α for any given isolated time if 2 2 the threshold for Tt ~ χ (1 − α , p) . However, Apley and Tsung [11] have proved with. a large value of p , the sensitivity of T2 control chart will decrease when the process is out-of-control. They observed a relationship between ARL when the process is in-control (ARL0) and α : log( ARL0 ) ≅ c0 − c1 log(α ) ………………………………………………(6). where c0 and c1 are constants that depend predominantly on p. Table 2 and 3 show the values of c0 and c1 respectively. For each fBm model we show the c0 and c1 with range of p from 2 to 20. The constants are generated by 1,000 Monte Carlo simulation. We can use the c0 and c1 and specify a desire in-control ARL to get α from equation (6).. 12.

(15) TABLE 2.Value for c0 for various p and H H. p. 2 4 6 8 10 12 14 16 18 20. 0.2. 0.3. 0.55. 0.7. 0.8. 0.9. 0.861 0.975 1.124 1.258 1.341 1.402 1.463 1.507 1.564 1.613. 0.987 1.084 1.189 1.352 1.392 1.441 1.493 1.521 1.589 1.668. 1.009 1.115 1.208 1.389 1.442 1.509 1.56 1.614 1.652 1.683. 1.049 1.189 1.296 1.408 1.461 1.513 1.573 1.619 1.678 1.732. 1.171 1.289 1.372 1.451 1.486 1.521 1.592 1.653 1.709 1.746. 1.237 1.352 1.458 1.589 1.652 1.709 1.762 1.813 1.844 1.877. TABLE 3.Value for c1 for various p and H H. p. 2 4 6 8 10 12 14 16 18 20. 0.2. 0.3. 0.55. 0.7. 0.8. 0.9. 0.854 0.879 0.909 0.931 0.945 0.958 0.981 1.021 1.066 1.082. 0.849 0.867 0.897 0.924 0.933 0.946 0.971 0.989 1.057 1.079. 0.834 0.853 0.876 0.899 0.908 0.936 0.961 0.977 1.042 1.063. 0.826 0.843 0.859 0.871 0.879 0.920 0.945 0.964 1.014 1.058. 0.815 0.832 0.847 0.851 0.869 0.882 0.924 0.934 0.981 1.056. 0.807 0.827 0.831 0.843 0.859 0.867 0.907 0.924 0.970 1.052. Finally, we can know the threshold for T2 and apply it to monitor the fBm process. The procedure is summarized as following: Step 1. Use R/S analysis to get Hurst exponent of time series Step 2. Transform the observations of fBm process to the fGn model by the statistical self-similarity properties of fBm. Step 3. Calculate the autocorrelation function of the fGn data to specify the optimal order of ARMA model that is used to fit the time series and the dimension of the vector using in T2 chart.. 13.

(16) Step 4. Find the autocovariance of the transformed observations to form the covariance matrix in the T2 control chart Step 5. Identify p Step 6. Identify α by equation (6) Step 7. Apply an autoregressive T2 control chart with p and α to the fGn process and monitor it. 4.3 Illustrative Example We have formulated a model and discussed the effect of the parameters. In order to display the difficulty in monitoring fBm model and implement the model developed by the guideline for monitoring fBm process, now we will simulate a fBm process and detect it by the approach provided in this paper respectively. There are a lot of researches about the techniques for synthesizing a fBm process. The midpoint displacement technique is not considered here based on previously reported inaccuracies[5], and spectral techniques were omitted due to the problems noted in [19]. We will adopt Cholesky algorithm for fBm synthesis [19] because of its good performance within much smaller sample sizes while the algorithm of Mandelbrot-van Ness performs significant better for 2048 points [19]. This paper implements it in R. Figure 3.1 is the fBm process of H=0.8. As the figure shows, An fBm process is with high autocorrelation and detected hardly.. Figure 1. fBm processes with H=0.8 Consider we want to monitor a fBm process. And assume the value of Hurst exponent is 0.8 and the mean of the process is zero when the process is in-control (that means μ0 = 0 ). We synthesize 100 observations, the before 80 of it are 14.

(17) in-control while the last 20 are out-of-control with the mean shifts is 3. These observations are plotted on the Figure 2.. Figure 2. Observation from fBm processes Suppose we know the H value of the process when it is in-control, set H = 0.8. And then we begin to detect this process: First, transform the observations by equation (1). Figure 3 is the observations after transformed and the original ones.. Figure 3. Observation after transforming Using equation (5) gives every Tt 2 .The Σ can be factored as Σ = Β −1' DΒ −1 , we. 15.

(18) get the matrix B and select the optimal p is 6. From Table 2 and 3, c0=1.372 and c1=0.847 with p=6. Applying equation (6) with those parameters gives α=0.00329 for approximate ARL0 of 500. And that leads to the threshold is 19.5803. Figure 2 is the Autoregressive T2 chart for this example.. Figure 4. Autoregressive T2 chart example As the above figure shows, the T2 chart signal at the 76 sample point. We will know the shift in process is at sample 75 or 76 because p=6 and 75+5=80 (75 sample point is the one before signal), 76+5=81 (every sample in T2 chart involves 6 observations). In this case, the T2 chart signaled on the first observation that the mean shift occurred.. 5. Conclusion In the paper, we have been presented the method for monitoring the fBm process. The concept we use is to combine the autoregressive T2 control chart with fractional Brownian motion. Most important, this paper transforms the fBm process to fGn according to the self-similarity and stationarity. And we find the relationship between Hurst exponent and the dimension of the multivariate vector using in T2 approach. From the performance comparison, we know the autoregressive T2 control chart is superior to the residual-based CUSUM chart and Shewhart individual chart when the high autocorrelation (Hurst exponent of fBm is high) or the mean shift is moderate to large. But for small mean shifts and the residual-based CUSUM chart always has better performance than the other two control charts. Therefore, the autoregressive T2 control chart is efficiency applied to the fBm process with high Hurst exponent or moderate to large mean shift. There are more and more applications of the SPC method. In addition to the industrial field, scientist applied it to finance field, even the efficiency of employees 16.

(19) has been measured by that. But the model adopted to fit is not certainly accuracy. It is intuitive that fBm can be a good model to fit those. The method for monitoring fBm process presented in this paper will be useful in these applications. Based on the method and procedure presented in this paper, a program for monitoring fBm process may be developed in the QC software. Besides, a comparison with the existing method for monitoring autocorrelated process may be made by Monte Carlo simulation. And then we will clearly know the advantages of every method. Moreover, the Hurst exponent may be used to estimate the standard deviation of the process while it is usually estimated by the range and variance of the samples. If we find the relationship between the two, maybe we can develop a control chart designed for fBm process based on original observations without transformed. And its advantage is easily understood. Since a lot of synthesis techniques of fractional Brownian motion exit, it is possible to find a way to synthesize an fBm process in terms of its parameters and properties to fit the one want to detect, and then computes the residuals. The residuals will be regarded as random variables if it is an unbiased estimation. Hence we can plot those residuals on the standard control chart and monitor them. Those subjects and directions may be able to be extended from this article in the future.. References [1] Andrew W. Lo, "Long-term memory in stock market prices," Econometrics, 59(1991), 1279—1313 [2] Benoit B. Mandelbrot and W.V. John, “Fractional Brownian motions, fractional noises and applications,” SIAM Review, 10(1968), 422—437 [3] B. Djehiche and M. Eddahbi, "Hedging options in market models modulated by the fractional Brownian motion," Stochastic Analysis and Applications, 19(2001), 753—770 [4] B. Mandelbrot, “Comment on computer rendering fractal stochastic models,” Communication of the ACM, 25(1982), 581—583. [5] Camarasa, E., Carvalho, E., Meleiro, L.A.C., Maciel Filho, R., Domingues, A., Wild, G., Poncin, S., Midoux, N., and Bouillard, J., “Development of a complete model for an air-lift reactor,” Chemical Engineering Science, 56(2000), 493—502. [6] Carl J.G. Evertsz, “Fractal geometry of finance time series,” Fractals, 3(1995), 609—616. [7] C.-W. Lu and Marion R. Reynolds, Jr., “Control charts for monitoring the mean and variance of autocorrelated processes,” Journal of Quality Technology, 31(1999), 259—274. 17.

(20) [8] C.-W. Lu, M.R. Reynolds, Jr., “Cusum charts for monitoring an autocorrelated process,” Journal of Quality Technology, 33(2001), 316—334. [9] C.C. Hsieh, T.W. Hu and H.K. Chang, “Discussing the meaning of Hurst exponent (in Chinese),” Construction and Building Activities and Water Conservancy, (1997), 7—18. [10] Dejian Lai, Barry R. Davis and Robert J. Hardy, "Fractional Brownian motion and clinical trials," Journal of Applied Statistics, 27(2000), 103— 108 [11] D.W. Apley and F. Tsung, “The autoregressive T2 chart for monitoring univariate autocorrelated processes,” Journal of Quality Technology, 34(2002), 80—96. [12] H.E. Hurst, “Long-term storage capacity of reservoirs,” American Society of Civil Engineers, 116(1956), 770—808. [13] J. Beran, “A test of location for data with slowly decaying serial correlations,” Biometrika, 76(1989), 261—269. [14] O. Magre’ and M. Guglielmi, “Modelling and analysis of fractional Brownian motions,” Chaos, Solitons & Fractals, 8(1997), 377—388. [15] P. Grau-Carles , "Tests of Long Memory: A Bootstrap Approach," Computational Economics, 25(2005), 103–113 [16] R. Scheffer and R.M. Filho, “The fractional Brownian motion as a model for an industrial airlift reactor,” Chemical Engineering Science, 56(2001), 707—711. [17] S. Dueck, “Synthesis of fractional Brownian motion: an evaluation of the simulation techniques of Mandelbrot-van Ness and Cholesky,” Physica S, 1(2003), 1—29. [18] Van Brackle, L.N. and Reynolds, M.R., Jr., “EWMA and CUSUM control charts in the presence of correlation,” Communications in Statistics—Simulation and Computation, 26(1997), 979—1008 [19] B.B. Mandelbrot and J.R.WALLS, “Computer experiments with fractional Gaussian noises part 3, mathematical appendix,” International Business Machines Research Center, Yorktown Heights, New York, 5(1969), No.2. [20] Douglas C., Montgomery, Introduction to Statistical Quality Control,” Wiley, 4th Edition (2003). [21] J. Holton Wilson – Barry Keating, John Galt Solutions, Inc., “Business Forecasting,” McGraw-Hill, 4th Edition (2004), 287—341. [22] J.R. Wallis, “Small sample properties of h and k-estimators of the Hurst coefficient h,” IBM Research Center, Yorktown Heights, New York.. 18.

(21) Plan achievement In the paper, we successfully develop a method for monitoring the fBm process. The concept we use is to combine the autoregressive T2 control chart with fractional Brownian motion. We make the statistical self-similarity of fractal as the foundation to construct the method similar to the SPC. And then find the time that the process is out-of-control and repair the process ahead of time. Among the literatures of the foreign and domestic, this plan makes fractal theory apply in the quality control field first of all. Our objective is to rely on the fBm process, to minimize the cost of engineering process, to increase the profit of industry, to raise the industry competition and there are innovative contributions to the field of Quality Control.. 19.

(22) Report for Attending CE 2006 By Shuo-Yan Chou. This conference had over 120 international scholars participating in the 91 talks and 9 keynote speeches. The countries of the participants include Taiwan, Hong Kong, China, Singapore, Korea, Japan, France, Italy, Spain, Greece, Netherlands, UK, Belgium, Switzerland, Germany, Poland, Sweden, Finland, Brazil, Canada, USA and so on. This conference is sponsored by International Society of Productivity Enhancement which was founded in US. The current president of ISPE is Professor Nel Wognum from the Netherlands. The conference is about productivity enhancement in particular in the area of design, that is, how to make the design process more productive. I had the opportunity to have a very extensive discussion with scholars from a lot of other countries. It was also decided by the board of ISPE that the conference of CE2009 will be held in Taiwan. I will be serving as the general chair for the conference. Also, I will be serving as the program chair for the conference held in Ireland in 2008. Professor Amy R. Trappey of National Tsing Hua University will be at least co-chair the program chair with me in 2008. In general, international scholars have great interests in Taiwan and coming to Taiwan for the conference. Hopefully, this conference will serve as a platform to attract awareness of the people in Taiwan to the area of concurrent engineering and subsequently enhance the capability of our designers and the quality of our designs..

(23) A Fuzzy QFD Approach for Designing Internal Processes for Air Cargo Transportation Services Shuo-Yan Choua,1, Yao-Hui Chang a,b and Chih-Hsien Chen c Dept. of Industrial Management, National Taiwan Univ. of Science & Technology b Department of Industrial Management, Lunghwa Univ. of Science and Technology c Dept. of Industrial Engineering and Management, Lee-Ming Institute of Technology a. Abstract. The objective of this study is to develop an integrated approach to process design for air cargo transportation. This work integrated the methodologies of fuzzy set theory (FST), balanced scorecard (BSC), and theory of constraints (TOC) to ensure that the internal process design meets the needs of employees, shareholders, and customers, concurrently. This integrated approach is constructed by three stages. In the first stage, the strategies, strategic objectives, and key performance indicators (KPIs) of the company are developed through the conceptual framework of BSC. In the second stage, the initiator’s actions are developed through the five logic tools of TOC. Furthermore, in the third stage, through the integration of BSC and TOC with fuzzy quality function deployment (QFD), KPIs and initiator’s actions were transformed into the design requirements and the technical requirements. Finally, the house of quality (HOQ) is built for designing the service process of air cargo transportation based on fuzzy QFD. Keywords. air cargo transportation, fuzzy quality function deployment, fuzzy set theory, balanced scorecard, theory of constraints.. Introduction Air cargo transportation industry has three characteristics: strict control, peak time influence, and internationalization. Because of these factors stated above, air cargo transportation industry is very distinctive from traditional transportation industry. Therefore, the process management is also a crucial to air cargo transportation industry according to the complexity of air cargo transportation logistics chain and the characteristics of air cargo transportation industry. As air cargo industry focuses on process management beyond organizational boundaries, there is a need to develop an integrated process design for air cargo transportation. The most logical and viable alternative under the present circumstances would be to become more efficient through better design of existing facilities, resources and process flow. In our study, we are concerned with process design of the air cargo transportation. Meanwhile, the purpose of this study is to propose an integrated process design based on BSC, TOC, QFD, and FST for air cargo transportation. 1. Literature Review Ahn [1] employed the BSC framework proposed by Kaplan and Norton [2]-[5] to integrate a company’s mission, values, vision, strategy into the four perspectives of 1. Corresponding Author: 43 Keelung Road, Section 4, Taipei, Taiwan; [email protected]..

(24) BSC, which subsequently evolve into the company’s performance targets and indicators. With respect to applications in the field of transportation, Poli and Scheraga [6] designed a BSC framework to elucidate customer satisfaction from five perspectives. The results revealed that transportation operators must find a balance among all quality perspectives and prioritize the needs of key customers. Rouse, Putterill, and Ryan [7] spent four years monitoring the performance and studying the control systems used by international airlines in maintenance. Goldratt proposed the TOC in 1982, the complexity of a factory is decided not only by the number of constraints, but also by the relativity of the constraints. Rahman [8] indicated that in order to address the policy constraints and effectively implement the process of on-going improvement, Goldratt developed the TP in 1990. Therefore, the TP of TOC is a systematic approach for investigating, analysis, and solving complex design problem [9]- [11]. Besides, Klein and Debruine [12] presented a TP for establishing management policies of American companies. QFD process may input various linguistic data, which are assessed with human perceptions and judgments. As such, QFD process typically involves the imprecision and vagueness inherent in linguistic assessment. This environment can be to imprecision with the help of FST [13]. As a result, it has been widely used in QFD recently [14]-[17]. Companies having implemented QFD have reported a variety of advantages and also shortcomings with method [18], [19]. Several approaches have been made to overcome the shortcomings in working the QFD process. Two of these trends are considered in this work: application of the FST and homo/heterogeneous group decision making to determine priorities of design requirements and technical requirements in QFD. 2. Derivation of Equations This work integrated the methodologies of BSC, TOC, and fuzzy QFD to ensure that the service process design meets the needs of employees, shareholders, and customers, concurrently. BSC seeks for the balance of short-term objectives and long-term objectives, financial perspective and non-financial perspective, lagging indicators and leading indicators, quantitative indicators and qualitative indicators, local performance and global performance, internal performance and external performance, and top-down analysis and bottom-up analysis. This paper, for satisfying the need of customers and the commitment of shareholders, applies BSC in service process design to develop KPIs for air cargo transportation. Mission statement transform into KPIs by the top-down procedure, whereas the four perspectives analysis are gained by the bottom-up procedure. Through HOQ of QFD, this study also transforms KPIs to design requirements. General speaking, there are two kinds of organization’s limitations which is physical constraints and policy constraints. There are three main physical constraints of an organization. Number one, vendors’ constraints, most of the companies complain about the vendors, but this normally won’t be a real constraint. For example, a company might face shortage of material supply, the root cause might be the ineffective of its procurement or inventory strategy, not caused by vendor – except for when vendor face a continuous lack of supply of material (normally it happens to the.

(25) whole industry). Resource constraints are the second constraint, and market constraints are the third constraint [10]. However, the policy constraints include behavior constraint, managerial constraint, and logistical constraint. Five logic trees have been developed by Goldratt’s TP. That helps to solve physical constraints and policy constraints. TP has probed the core problem deeply by the current reality tree (CRT). TP finds out conflict conquering win-win alternatives in order to enhance process quality and effectiveness by the evaporating cloud (EC) diagram and future reality tree (FRT). The final step, pre-requisite tree (PRT) and transition tree (TT) infer the effective alternatives and initiator’s actions in order to solve the core problem and achieve the desirable effects. QFD is a well-structured design tool; HOQ is the core of QFD. The conventional HOQ to quantify the relationships is accomplished using 3-point scale or 5-point scale to denote relationships between each WHATs and each HOWs [20]. However, in practice the relationships are usually vague and imprecise, and can be described in linguistic terms. In this work, the relationships are represented as linguistic terms, and conversion scales are applied to transform linguistic terms into fuzzy numbers. A scale of five for importance weight and relationship are used in this study. Table 1 presents relative importance of WHATs and relationship between each WHATs and each HOWs considered as linguistic variables. The trapezoidal fuzzy number is easy to use and interpret. For example, a very significant weight of a specific requirement can be measured by a trapezoidal fuzzy number and denoted by (7, 10, 10, 10). Table 1. Linguistic variables and fuzzy numbers for the relative importance or relationship Relative importance Very Low (VL) Low(L) Moderate(M) High(H) Very High (VH). Relationship Very Weak (VW) Weak (W) Moderate (M) Strong (S) Very Strong (VS). Fuzzy numbers (0, 0, 0, 3) (0, 3, 3, 5) (2, 5, 5, 8) (5, 7, 7, 10) (7, 10, 10, 10). After obtaining the necessary data from the participants in the company, the following computational procedure is performed to determine the normalized design requirement scores. The arithmetic operations on trapezoidal fuzzy numbers are employed. Step 1: Determine the degree of importance (or reliability) of the participants. If the degrees of importance (or reliability) of participants are equal, then the group of participants is called a homogeneous group. Otherwise, the group is called a heterogeneous (non-homogeneous) group [21]. Assume that there are k participants ( Dt ,t = 1, 2 , ...,k ) who are responsible for assessing the relationship between m HOWs ( Ai , i = 1, 2 , ..., m ) and n WHATs (C j , j = 1, 2, ..., n) as well as the importance of the WHATs. The degrees of importance (or reliability) of participants are I t , t = 1, 2,..., k , where I t ∈ [0,1] and the sum equal to 1. If the relative importance (or reliability) and weight of each participant is considered, then the most important participant is selected among participants and a weight of 10 is assigned, i.e., ω t = 10. Furthermore, the lth participant is compared with the most important participant and a relative weight for the lth.

(26) participant ω l , l = 1, 2, ..., k , where ω l ∈ I and ω l ∈ [0,10], is obtained. Finally, the degree of importance I t is defined as follows: k I t = ω t / ∑ ω t , t = 1, 2, ..., k . (1) t =1. If I1 = I 2 = ... = I k , the group of participants is called a homogeneous group; otherwise, the group of participants is called a heterogeneous (non-homogeneous) group.. Step 2: Introduce linguistic weighting variables (Table 1) for participants to assess the fuzzy relative importance of WHATs (customer needs), and then compute aggregated fuzzy relative importance of WHATs. Let W% jt = ( a jt , b jt , c jt , d jt ) , j = 1, 2,..., n ; t = 1, 2, ..., k , to be the linguistic weight given to WHATs C1, C2, …, Cn. The aggregated fuzzy relative importance, W% j = ( a j , b j , c j , d j ) , j = 1, 2 , ..., n , of WHATs C j assessed by k participants is defined as: W% j = ( I1 ⊗ W% j 1 ) ⊕ ( I 2 ⊗ W% j 2 ) ⊕ ⋅ ⋅⋅ ⊕ ( I k ⊗ W% jk ), (2). where a j = ∑ I a , b j = ∑ I b , c j = ∑ I c , d j = ∑ I d jt . k. k. t. t =1. jt. k. t. t =1. jt. k. jt. t. t =1. t. t =1. Step 3: Use linguistic relation variables (Table 3.1) for participants to assess the degree of relationship between each WHATs and each HOWs, and then pool them to obtain the aggregated fuzzy relationship between each WHATs and each HOWs. Let x%ijt = ( oijt , pijt , qijt , sijt ) , i = 1, 2, ..., m, j = 1, 2, ...,n,t = 1,2,…,k, be the linguistic suitability degree of relationship between each WHATs and each HOWs assigned to HOWs Ai for WHATs C j by participant Dt . Let us further define x%ij as the aggregated fuzzy relationship between HOWs Ai and WHATs C j , such that x%ij = ( I1 ⊗ x%ij 1 ) ⊕ ( I 2 ⊗ x%ij 2 ) ⊕ ... ⊕ ( I k ⊗ x%ijk ), (3) which can subsequently be represented and computed as x%ij = ( oij , pij , qij , sij ) , i = 1, 2, ..., m, j = 1, 2, ...,n, (4) where oij = ∑ I t oijt , pij = ∑ I t pijt , qij = ∑ I t qijt , sij = ∑ I t sijt . k. k. k. k. t =1. t =1. t =1. t =1. Step 4: Construct a fuzzy relationship matrix based on fuzzy relationships. ~ The fuzzy relationship matrix M can be concisely expressed in matrix format: M% = [ x%ij ] , (5) where x%ij , ∀i, j is the aggregated fuzzy relationship of HOWs Ai , i = 1, 2, ..., m with respect to WHATs C j . Step 5: Derive fuzzy HOWs (design requirement) scores are computed by multiplying the fuzzy relationship matrix with the corresponding weight vector W , i.e., T F% = M% ⊗ W% = [ f%i ]m1 (6) % where fi = ( ri , si , ti , ui ) , i = 1, 2 , ...,m. Step 6: Normalization ensures a more meaningful representation of the fuzzy HOWs scores. Hence, a linear scale of measurement to vary precisely in the [0,1] interval is employed to normalized the resulting HOWs scores as follows: N% = f% / u* (7) i. i. where N% i denotes the normalized HOWs scores andu* = maxi ui.. 3. Integrated Framework.

(27) The fuzzy QFD takes only customer requirements into consideration for the product design, whereas the design of service process for air cargo transportation have to include customers, employees as well as shareholders requirements into consideration, due to the fact that the process is more complicated and has many upstream and downstream interfaces. Through the application of BSC, consideration of customers, employees, and shareholders requirements, and TOC, analysis of bottleneck process, the authors apply fuzzy QFD to integrate BSC and TOC into HOQ. This article focuses the research on the internal process perspective and proposes an integrated design framework. First, we use the BSC methodology for a company to develop the framework for establishing its design requirements in the first stage of deployment of fuzzy QFD. In the second stage of deployment of fuzzy QFD, this study applies the TP of TOC to deploy the technical requirements for the internal process perspective. Finally, the HOQ is built for designing the internal process of air cargo transportation based on fuzzy QFD methodology (as shown in Fig. 1).

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(37) . Fig. 1 A design framework for integrating BSC and TOC into fuzzy QFD. Stage one: BSC to meet customer requirements and design requirements The BSC framework is used to develop the customer requirements and design requirements in this stage to meet in air cargo transportation design (as shown in Fig. 2). It ensures that all employees are in line and are striving toward a common mission. HOWs Design requirements. WHATs Financial perspective BSC. Customer perspective Internal process perspective Learning & growth perspective. Fuzzy Relative Importance. Fuzzy Relationship matrix. Fuzzy HOWs Score. Fig. 2 BSC to meet customer requirements and design requirements. Kaplan and Norton [4] stated that BSC has four characteristics, which help to link the long-term objectives and short-term actions together. First, build consensus through the discussion of company’ s mission and vision. Second, through cause-and-effect analysis,.

(38) strategic map and KPI, link the employee performance to company’ s overall strategy. Third, incorporation of operation plan and financial plan, to ensure the long-term plan can be implemented. Fourth, build up the feedback and learning systematic through the learning & growth prospective of BSC, so that the company is able to examine and discuss its strategy. Based on the application of BSC, the financial perspective remains an important tool for companies, since it reflects the results of any actions taken to improve the four perspectives. Customer acquiring, customer retention, and meeting customers’ expectations are typically key determinants of customer perspective. The internal process perspective should enhance the value-add of products or services used by customers. The indicators of employees’ learning & growth perspective are used mainly to ensure outstanding internal process, which will in turn be the basis of internal process, customer, and financial perspectives. Stage two: TP to meet technical requirements This research utilizes the following three steps of TP of TOC to deploy the technical requirements (as shown in Fig. 3). In the first step, this work applies the five groups of weak links, including behavior constraint, managerial constraint, capacity constraint, market constraint, and logistical constraint, to develop the CRT in the first step. The CRT is a diagram built on the cause-effect relationships between the UDEs and their immediate causes. The objective is to find the core problem—what to change? HOWs WHATs. TP. Design Fuzzy requirements Relative of internal Importprocess ance perspective. Technical requirements. Fuzzy Relationship matrix. Fuzzy HOWs Score. Fig. 3 TP to meet technical requirements. The second step in the TP deals with the search for a solution to the core problem—what to change to? This step consists of the EC diagram and the FRT. After developed the CRT formed by core problems and UDEs, through the EC diagram, second logic tool of TP, we define the desirable objective of core problem. We then find out the conflict that exists in their reality and work out the injection to achieve the desirable objective. The third logic tool of TP is FRT, it is used to model the changes created after defining the injection from the EC diagram and uncover the intermediate objectives. The objective of the third step in the TP is to find the initiator’ s actions—How to change? This step consists of the PRT and the TT. The fourth logic tool of TP is the PRT, this is to find and define clearly the intermediate obstacles arise while moving from CRT to FRT. We first define the intermediate objectives of DEs, and then find out the intermediate obstacles to achieve the intermediate objectives, link the intermediate objectives and obstacles to form the PRT. The last logic tool of TP is the TT; this is to help to develop the details of the initiator’ s actions, where it is developed through solution finding for the obstacles of PRT. All initiator’ s actions can help to achieve intermediate objectives by overcoming the obstacles, after all intermediate objectives is.

(39) achieved, the injection can be achieved, thus solve the core problem. Finally, the initiator’ s actions construct the technical requirements for the service process design. Stage three: The integration of BSC and TP with fuzzy QFD The core of fuzzy QFD is the matrix, called the HOQ. In the first deployment, the four perspectives of BSC constitute the “WHATs” matrix, which are the needs of all the stakeholders, of fuzzy QFD. The procedure of balanced business scorecard for formulating the KPIs can be applied to derive the “HOWs” matrix to meet the customers’ requirements associated with fuzzy QFD. In the second deployment, the KPIs of BSC constitute the “WHATs” matrix, which are the design requirements, of fuzzy QFD. The procedure of TP for formulating the initiator’ s actions constructs the “HOWs” matrix, which can be applied to derive the technical requirements associated with fuzzy QFD.. 4. Conclusions As the air cargo transportation industry incorporates a complex industrial logistics chain, the air cargo transportation industry should focus on process management beyond organizational boundaries. Thus, this study proposes an integrated approach to process design for air cargo transportation based on the methodologies of BSC, TP, and fuzzy QFD. This approach is constructed by three stages. First, we use the BSC methodology for a company to develop the framework for establishing its design requirements. In the second stage of deployment of fuzzy QFD, this study applies the TP of TOC to deploy the technical requirements. Finally, the HOQ is built for designing the process of air cargo transportation based on fuzzy QFD. The design and redesign of service process for air cargo transportation need huge capital expenditure, thus, it also needs the voice of shareholders and employees, not only the voice of customers. The BSC methodology involves the voice of shareholders, customers, and employees through the four perspectives—financial perspective, customer perspective, internal process perspective, and learning & growth perspective. TP of TOC can also be used to deploy the design requirements of four perspectives, and the technical requirements are built from the five logic tools. Fuzzy QFD can be used to predict problems in advance of operation or post sales service during the design stage. Through fuzzy QFD, the KPIs that developed by BSC can be transformed into the design requirements, and the initiator’ s actions, which developed, by TP can also be transformed into the technical requirements. This study only considers the framework and procedures of process design for air cargo transportation is one of the limitations of this research. Therefore, how to apply the proposal method systematically to a case study need to be included in future research. Besides, how to consider the quantitative factors into the proposal method in the development of the optimal model for service process design of air cargo transportation also needs to be included in future work.. References 1. H. Ahn, “Applying the balanced scorecard concept: an experience report,” Long Range Planning, vol. 34, pp. 441—461, 2001. 2. R. S. Kaplan and D. P. Norton, “The balanced scorecard—measures that drive performance,” Harvard Business Review, pp. 71—79, January-February, 1992..

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