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Advanced Technique for Manufacturing System Management
Copyright © IFAC Manufacturing Systems: Modelling, Management and Control, Vienna, Austria, 1997
ADVANCED TECHNIQUE FOR MANUFACTURING SYSTEM MANAGEMENT Andrea D'Angelo, Massimo Gastaldi and Nathan Levialdi Universita degli Studi di Roma "Tor Vergata" Dipartimento di lnjormatica, Sistemi e Produzione Via delta Ricerca Scientifica 00133 Roma ltalia
Abstract: Artificial Neural Networks are able to analyse the data distribution obtained by experimental simulations. In this paper Neural Network is utilised as black box tool to provide a low cost control procedure able to examine system events related to a Printed Circuit Board Assembly (PCBA) plant. The obtained results are compared with those arising from a traditional statistical method. Keywords: Production Systems, Performance functions, Simulation, Neural Networks, Statistical Inference.
learning phase, the network has reconstructed the mapping function as a black box. To check the effectiveness of this function, a second set of data is submitted for a test phase in which the network response is compared with the one of a statistical model. This paper is organised as follows. Section 2 describes the reference system for the manufacturing of printed circuit boards. Section 3 covers the implementation of the model built for the mathematical simulation. In section 4 an advanced backpropagation algorithm is applied to train a neural network on the data obtained by simulation experiments. The results arising from the test phase are compared with the outcomes of statistical regressions, in section 5. Finally, section 6 concludes the study.
1. INTRODUCTION Our study focuses on the typical Job Shop plant configuration for semiautomatic manufacturing of parts produced in limited unitary quantities. The treatment consists of the quantitative evaluation of the technological performance of a manufacturing system with particular reference to the Printed Circuit Board Assembly (PCBA) sector (Crama et aI., 1990; McGinnes et al., 1992; D'Angelo et aI., 1996a and 1996b). We present a method for monitoring two types of plant performance: Lead Time and Work In Process. This method is based on the capability of Neural Networks to map an m-dimensional input vector to an n-dimensional output vector (HechtNielsen 1987, Caudill 1987), and is applied to the reality of a leading industry in the production of PCBA and electronic modules for telecommunications. System events are investigated through a numerical simulation model. In fact, the complexity due both to the large number of variables and to the non linearity of the correlation involves the inconvenience of an analytical approach. Generally, the data so obtained, are processed through statistical methods in order to highlight the correlation among such variables and system performances. The inference can be alternatively proposed by a Neural Network approach in order to compare the results obtained through the two methods. The simulation experiments provide data for the Network training which learns the implicit correlations between input and output factors. In the presented analysis, a backpropagation algorithm (Werbos 1988, Rumelhart et a1. 1986) is utilised. So, after the
2. PHYSICAL SYSTEM DESCRIPTION This study is dealt with in the context of a PCBA system . The process of assembling electronic components is by its nature a flexible one, as the manufacturing process involves a great number (some one thousand) of items (part types), being produced in limited unitary quantities. It follows that the potential influence of events which are both endogenous and exogenous to the system is of a clearly stochastic nature. For example, in the short term there may be variations in the size, quantity and frequency of the lots to the system, either separately or jointly, in relation to the exogenous fluctuations in demand. This condition translates into an extreme segmentation of the manufacturing process, which,
151
in our case, is composed of nine types of processing, performed at an equal number of Work Centres (WC).
SMT C/S
Component SetUp
1st Solder
PIH Assembly
1st Inspection
SMT S/S
undergoing the same sequencing represents a family of products. The layout in Figure 2, highlights: the processing path for each family (the number in the box is Work Centre defining) the number of machines present at each node (the left hand-side number of each box) the average processing time in minutes at each Work Centre for each family (the right hand-side number of each box). Within each family the part types undergo, at each Work Centre, operations which require different amounts of time for each part type. That is to say, the times spent at each Work Centre are normally distributed around an average value which varies from family to family. The paths identified represent an outline sequencing as, actually, not all part types belonging to the family undergo all the operations envisaged by the sequencing itself. The overall output of the system is divided up among the various families in accordance with a set mix of specific percentage values. Specifically the seven different families are processed with part mix ratios 39:27 :24:5 :2:2: 1.
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Fig. 2. System technological sequencing Fig. 1. System description
The existence of a non-constant distribution of the average hourly productivity values among the various Work Centres leads to a non-homogeneous exploitation of system's resources. Indeed, the global productive capacity depends on the minima values of this distribution which characterises the Work Centres defined as the "bottlenecks". Thus, these Work Centres guide the dynamics of the production flow by determining a partial underutilization of other resources. On the other hand, the values of this distribution depend on the variables which define the average hourly productivity. For the reference manufacturing system, these variables can have values in a very wide domain. The result is that there can be a fluctuation of the bottleneck within the system. Moreover, such a fluctuation also depends on other variables which have a significant impact on the dynamics of the system, in particular, the lots size of the individual families.
The complexity caused by the large number of part types makes it impossible to dynamically follow the system events due to the lot relating to the single part type. Hence, we need to use a clustering criteria. That is to say, the part types are grouped together on the basis of the component assembly sequence. This may be of the standard Pin In Hole (PIH) type or of the Surface Mounted Technology (SMT Component Side C/S or Solder Side S/S) type. In this way we can identify seven assembly sequences (families) for the part types and it is with reference to these that we conduct a study of system performances. Let us consider the manufacturing process depicted in Figure 1, where the boxes represent Work Centres and the arrows represent the possible processing paths (sequencing) of the products b~ing manufactured. The starting point of the process IS an entrance gate ("IN" store, for the storing of the boards to be assembled), while the lots finally exit the system through an exit gate ("OUT" store, for storing of finished products). The set of part types
152
3. THE SIMULATION MODEL
processing and self-learning. For project purposes, a backpropagation model (Rumehart et al. 1986), is here selected to rebuild the implicit relation between the output factors Y 1, Y 2 (lead time and work in process) and input variables (Xl, X2 , X3 , X4). In particular we choice a four perception network consisting in an input layer, two hidden layers and an output layer. The input layer is formed by four nodes, one for each input variable whereas the output layer consists of two nodes used to represent lead time and work in process levels. Notice that in this paper two neural networks are considered; the first one, described in Figure 3, contains six nodes on the first hidden layer and four on the second hidden layer whereas the second network has respectively four and two nodes (see Figure 4).
In order to quantify the criticality dynamics it is necessary to observe the behaviour of the system in terms of parameters variation. So, first of all we have to select the significant variables impacts on system performances. The mathematical simulation of events which can significantly disturb system dynamics enables us to quantify in precise terms the variation in the following performance indicators: average Work In Process average Lead Time. The implementation of the reference system in a mathematical simulation model has been conducted through WITNESS (1994), an appropriate software for simulation of industrial processes. The presented analysis is referred to the system steady state. Each simulation is run for 960 hours; the first 100 hours of each run is truncated to eliminate initialisation bias. We consider constant interarrival times of lots to the system (for each product family), according to the original part mix ratio. Besides we assume: Machine breakdowns (exponentially distributed) Total system load (50.000 pieces in 960 hours) • Zero Transportation time of parts between WCs Set Up times for part type changing (20% of mean processing time) WCs' Input Buffers capacity (200). Pilot runs highlight the presence of bottlenecks in WC1, WC6 and WC8 and hence individuate three variables, the number of machines in each work centre, which exert a systematic influence on the performance indicators ; a forth one is identified in lot size. Once these variables have been identified, a discrete interval of variation is defined for each of them, as shown in Table 1.
XI
Fig. 3. The "6-4" neural network
YI
Y2
Table I . Significant variables levels
-1 Levels
0 1
Xl 15 25 35
Variables X3 X2 6 6 8 8 10 10
X4 3 4 5
Hidden layer I
Input layer XI
From the combination of the possible levels for each variable we can define a series of 34 =81 system configurations. For each configuration a simulation experiment is run to calculate the value of the two performance indicators.
X2
X3
X4
Fig. 4. The "4-2" neural network The training phase consists on 64 data series (10 relative to replications on central point and 54 = 2/3 of total system configurations). During the learning phase, the feed-forward output state calculation was combined with the backward error propagation and weight adjustment calculations. In the test phase the remaining 27 data series are used. The aim of this choice is to verify the learning effectiveness of the network about the relation between the input and output factors . At this purpose pilot experiments show that the evaluation is better performed by "6-4" network than by the "4-2" one.
4. NEURAL NETWORK ANALYSIS Neural networks are a form of computing inspired by the structure and learning ability of the human brain. They consists of networks of a number of simple, highly-interconnected processing elements, which process information in a vaguely analogous way to biological neurones. Neural networks work quite differently then conventional computing and consist of many processing units (nodes) that are interconnected with the capability of parallel 153
5. RESULTS ANALYSIS In this section results of the performed analysis are discussed. In particular data obtained by the test phase of neural networks are compared with those obtained by the application of statistical model. In Tables 2-3 the outcome of the statistical regression on 64 data series (the same used in the neural network training) is shown.
27 VC nn = -'--_-=...:..._--
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Table 2. Lead time second order re!ll"ession model
tat
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where 27
Lead Time Number of Observations - 64 Variable Coeff. Std error T Cl. -190.262 545.315 -0.349 0.729 Xl 14979.994 3315.243 4.519 0.000 X2 -3956.096 3202.926 -1.235 0.223 X3 )4 -14519.812 6630.485 -2.190 0.033 9.639 7.155 1.347 0.184 Xl*Xl -376.792 203 .694 -1.850 0.070 X2*X2 880.994 178.423 4.938 0.000 X3*X3 2769.110 814.774 3.399 0.001 X4*)4 -21.325 27.504 -0.775 0.442 Xl*X2 -15.493 26.903 -0.576 0.567 Xl*X3 3.594 55.008 0.065 0.948 Xl*)4 -859.340 131.900 -6.515 0.000 X2*X3 -43.376 264.661 -0.164 0.870 X2*)4 263.801 -4.715 0.000 X3*)4 -1243.846 F= 62.836
Y
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_=i=:;..,.l__ m27
and Yfim represents the i-th valul! of the simulated performance functions (lead time or work in process). yr n and y~tat are the i-th test values relative to neural network and statistical model respectively. In aggregated terms the variation coefficients are shown in Table 4. Table 4. VC results comparison
Neural Network Statist. model
lead time 0.658 0.505
work in process 0.363 0.353
R2= 0.932
Such table shows that the two approaches are practically equivalent for work in process level prediction whereas the performance of statistical inference is higher when lead time factor is considered.
Table 3. Work in process first order re!ll"ession model Work in process Number of Observations - 64 Variable Coeff. Std error T Cl. Xl 35.849 23.962 1.496 0.140 X2 896.997 100.166 8.955 0.000 X3 -71.052 109.505 -0.649 0.519 )4 -899.494 200.332 -4.490 0.000
6. CONCLUSIONS !his paper is focused on monitoring two types of a Job shop plant performance: lead time and work in process. In order to capture the relation between input and output factor, two alternative approaches have been applied. The first one is based on the application of a classical statistical regression model whereas the second one is built on the advanced neural network technique. The former provides the existence of an explicit polynomial relation between variables and the latter, by means of a backpropagation algorithm, maps as a black box the m-dimensional input vector to the n-dimensional one. The data utilised on both inferences are obtained by a simulation model reproducing dynamics of the system under study. The obtained results show that neural network is a low-cost procedure to understand and forecast system events with an acceptable performance level. Such results seem to be related to the particular system complexity.
F= 126.442
Notice that, for goodness of fit point of view, based on the 27 data series used for neural network testing, a first order model has been selected to represent work in process function whereas a second order model has been used for lead time. In Figure 5 and 6 the 27 test values for the observed performance functions are presented with respect to the simulation data (values are normalised by absolute maximum value). Even if lines seem to be not so close, they comply with an evident periodicity. Hence, neural network approach, as well as statistical model, understands the relations between factors in relative terms but not in absolute ones. A quantitative measure VC (Variation Coefficient), in order to compare distance between lines, can be defined as:
154
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Since the number of total possible configurations exponentially grows with the number of considered input variables, the authors are currently working on the performance evaluation of neural network approach for systems with an higher input vector size. Preliminary analysis have shown that neural networks results seem to be more significant when the training data set is reduced with respect to the
155
REFERENCES Caudill, M., (1987). Neural Networks primer Parts 1-8. AI Expert, 2, 12. Crama, Y., A.WJ. Kolen, A.G. Oelermans and F.C.R. Spieksma, (1990). Throughput rate optimization in the automated assembly of printed circuit board. Annals of Operations Research, 26, 455-480. D'Angelo A., M. Gastaldi and N. Levialdi, (1996a). Dynamic analysis of the performance of a Flexible Manufacturing Systems: a real case application . Computer Integrated Manufacturing Systems, 9, 2, 101-110. D'Angel0 A. , M. Gastaldi and N. Levialdi, (1996b). Performance analysis of a Flexible Manufacturing Systems: a statistical approach. In press: International Journal of Production Economics. Hecht-Nielsen R., (1987) . Kolmogorov's mapping neural network existence theorem. Proceedings of the First IEEE International Conference on Neural Networks , (M. Caudill and C. Butler Eds), IEEE Service Center, Piscataway, New Jersey, 11-14. McGinnis, L.F., J.c. Ammons , M. Carlyle, L. Cranmer, G.W. Depuy, K.P. Ellis, C.A. Tovey and H. Xu, (1992). Automated process planning for printed circuit card assembly. IEEE Transaction on Industrial engineering R&D, 24, 4, 18-29. Rumelhart, D.E., G.E. Hinton and R.J. Williams, (1986) . Learning internal representation by error propagation . Parallel Distributed Processing, 1, (David E. Rumelhart and James L. McClelland Eds.), MIT Press, Cambridge, Massachusetts, 318-362. Werbos, P.J., (1988). Generalisation of back propagation with application to a recurrent gas market method. Neural Networks, 1, 4, 339356.
156
ADVANCED TECHNIQUE FOR MANUFACTURING SYSTEM MANAGEMENT Andrea D'Angelo, Massimo Gastaldi and Nathan Levialdi Universita degli Studi di Roma "Tor Vergata" Dipartimento di lnjormatica, Sistemi e Produzione Via delta Ricerca Scientifica 00133 Roma ltalia
Abstract: Artificial Neural Networks are able to analyse the data distribution obtained by experimental simulations. In this paper Neural Network is utilised as black box tool to provide a low cost control procedure able to examine system events related to a Printed Circuit Board Assembly (PCBA) plant. The obtained results are compared with those arising from a traditional statistical method. Keywords: Production Systems, Performance functions, Simulation, Neural Networks, Statistical Inference.
learning phase, the network has reconstructed the mapping function as a black box. To check the effectiveness of this function, a second set of data is submitted for a test phase in which the network response is compared with the one of a statistical model. This paper is organised as follows. Section 2 describes the reference system for the manufacturing of printed circuit boards. Section 3 covers the implementation of the model built for the mathematical simulation. In section 4 an advanced backpropagation algorithm is applied to train a neural network on the data obtained by simulation experiments. The results arising from the test phase are compared with the outcomes of statistical regressions, in section 5. Finally, section 6 concludes the study.
1. INTRODUCTION Our study focuses on the typical Job Shop plant configuration for semiautomatic manufacturing of parts produced in limited unitary quantities. The treatment consists of the quantitative evaluation of the technological performance of a manufacturing system with particular reference to the Printed Circuit Board Assembly (PCBA) sector (Crama et aI., 1990; McGinnes et al., 1992; D'Angelo et aI., 1996a and 1996b). We present a method for monitoring two types of plant performance: Lead Time and Work In Process. This method is based on the capability of Neural Networks to map an m-dimensional input vector to an n-dimensional output vector (HechtNielsen 1987, Caudill 1987), and is applied to the reality of a leading industry in the production of PCBA and electronic modules for telecommunications. System events are investigated through a numerical simulation model. In fact, the complexity due both to the large number of variables and to the non linearity of the correlation involves the inconvenience of an analytical approach. Generally, the data so obtained, are processed through statistical methods in order to highlight the correlation among such variables and system performances. The inference can be alternatively proposed by a Neural Network approach in order to compare the results obtained through the two methods. The simulation experiments provide data for the Network training which learns the implicit correlations between input and output factors. In the presented analysis, a backpropagation algorithm (Werbos 1988, Rumelhart et a1. 1986) is utilised. So, after the
2. PHYSICAL SYSTEM DESCRIPTION This study is dealt with in the context of a PCBA system . The process of assembling electronic components is by its nature a flexible one, as the manufacturing process involves a great number (some one thousand) of items (part types), being produced in limited unitary quantities. It follows that the potential influence of events which are both endogenous and exogenous to the system is of a clearly stochastic nature. For example, in the short term there may be variations in the size, quantity and frequency of the lots to the system, either separately or jointly, in relation to the exogenous fluctuations in demand. This condition translates into an extreme segmentation of the manufacturing process, which,
151
in our case, is composed of nine types of processing, performed at an equal number of Work Centres (WC).
SMT C/S
Component SetUp
1st Solder
PIH Assembly
1st Inspection
SMT S/S
undergoing the same sequencing represents a family of products. The layout in Figure 2, highlights: the processing path for each family (the number in the box is Work Centre defining) the number of machines present at each node (the left hand-side number of each box) the average processing time in minutes at each Work Centre for each family (the right hand-side number of each box). Within each family the part types undergo, at each Work Centre, operations which require different amounts of time for each part type. That is to say, the times spent at each Work Centre are normally distributed around an average value which varies from family to family. The paths identified represent an outline sequencing as, actually, not all part types belonging to the family undergo all the operations envisaged by the sequencing itself. The overall output of the system is divided up among the various families in accordance with a set mix of specific percentage values. Specifically the seven different families are processed with part mix ratios 39:27 :24:5 :2:2: 1.
Manual Completion
Fam.. I
"I"
4 WO 4.6
Test
10 WCt 12
In Circuit
4~O4
IO~13
Final Inspection
4~~2 6~16 2~1
Fam. :!
F;un. S
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hm. 3
"$" '$" "$" wc:!
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{~O.l I~OI 4~L~
'1" I WC4 0.5
4 WCS 3.1
10 WCl II
4
wc,
6.2
6~'.1 2 WC9 I
Fig. 2. System technological sequencing Fig. 1. System description
The existence of a non-constant distribution of the average hourly productivity values among the various Work Centres leads to a non-homogeneous exploitation of system's resources. Indeed, the global productive capacity depends on the minima values of this distribution which characterises the Work Centres defined as the "bottlenecks". Thus, these Work Centres guide the dynamics of the production flow by determining a partial underutilization of other resources. On the other hand, the values of this distribution depend on the variables which define the average hourly productivity. For the reference manufacturing system, these variables can have values in a very wide domain. The result is that there can be a fluctuation of the bottleneck within the system. Moreover, such a fluctuation also depends on other variables which have a significant impact on the dynamics of the system, in particular, the lots size of the individual families.
The complexity caused by the large number of part types makes it impossible to dynamically follow the system events due to the lot relating to the single part type. Hence, we need to use a clustering criteria. That is to say, the part types are grouped together on the basis of the component assembly sequence. This may be of the standard Pin In Hole (PIH) type or of the Surface Mounted Technology (SMT Component Side C/S or Solder Side S/S) type. In this way we can identify seven assembly sequences (families) for the part types and it is with reference to these that we conduct a study of system performances. Let us consider the manufacturing process depicted in Figure 1, where the boxes represent Work Centres and the arrows represent the possible processing paths (sequencing) of the products b~ing manufactured. The starting point of the process IS an entrance gate ("IN" store, for the storing of the boards to be assembled), while the lots finally exit the system through an exit gate ("OUT" store, for storing of finished products). The set of part types
152
3. THE SIMULATION MODEL
processing and self-learning. For project purposes, a backpropagation model (Rumehart et al. 1986), is here selected to rebuild the implicit relation between the output factors Y 1, Y 2 (lead time and work in process) and input variables (Xl, X2 , X3 , X4). In particular we choice a four perception network consisting in an input layer, two hidden layers and an output layer. The input layer is formed by four nodes, one for each input variable whereas the output layer consists of two nodes used to represent lead time and work in process levels. Notice that in this paper two neural networks are considered; the first one, described in Figure 3, contains six nodes on the first hidden layer and four on the second hidden layer whereas the second network has respectively four and two nodes (see Figure 4).
In order to quantify the criticality dynamics it is necessary to observe the behaviour of the system in terms of parameters variation. So, first of all we have to select the significant variables impacts on system performances. The mathematical simulation of events which can significantly disturb system dynamics enables us to quantify in precise terms the variation in the following performance indicators: average Work In Process average Lead Time. The implementation of the reference system in a mathematical simulation model has been conducted through WITNESS (1994), an appropriate software for simulation of industrial processes. The presented analysis is referred to the system steady state. Each simulation is run for 960 hours; the first 100 hours of each run is truncated to eliminate initialisation bias. We consider constant interarrival times of lots to the system (for each product family), according to the original part mix ratio. Besides we assume: Machine breakdowns (exponentially distributed) Total system load (50.000 pieces in 960 hours) • Zero Transportation time of parts between WCs Set Up times for part type changing (20% of mean processing time) WCs' Input Buffers capacity (200). Pilot runs highlight the presence of bottlenecks in WC1, WC6 and WC8 and hence individuate three variables, the number of machines in each work centre, which exert a systematic influence on the performance indicators ; a forth one is identified in lot size. Once these variables have been identified, a discrete interval of variation is defined for each of them, as shown in Table 1.
XI
Fig. 3. The "6-4" neural network
YI
Y2
Table I . Significant variables levels
-1 Levels
0 1
Xl 15 25 35
Variables X3 X2 6 6 8 8 10 10
X4 3 4 5
Hidden layer I
Input layer XI
From the combination of the possible levels for each variable we can define a series of 34 =81 system configurations. For each configuration a simulation experiment is run to calculate the value of the two performance indicators.
X2
X3
X4
Fig. 4. The "4-2" neural network The training phase consists on 64 data series (10 relative to replications on central point and 54 = 2/3 of total system configurations). During the learning phase, the feed-forward output state calculation was combined with the backward error propagation and weight adjustment calculations. In the test phase the remaining 27 data series are used. The aim of this choice is to verify the learning effectiveness of the network about the relation between the input and output factors . At this purpose pilot experiments show that the evaluation is better performed by "6-4" network than by the "4-2" one.
4. NEURAL NETWORK ANALYSIS Neural networks are a form of computing inspired by the structure and learning ability of the human brain. They consists of networks of a number of simple, highly-interconnected processing elements, which process information in a vaguely analogous way to biological neurones. Neural networks work quite differently then conventional computing and consist of many processing units (nodes) that are interconnected with the capability of parallel 153
5. RESULTS ANALYSIS In this section results of the performed analysis are discussed. In particular data obtained by the test phase of neural networks are compared with those obtained by the application of statistical model. In Tables 2-3 the outcome of the statistical regression on 64 data series (the same used in the neural network training) is shown.
27 VC nn = -'--_-=...:..._--
Ym
VCS
Table 2. Lead time second order re!ll"ession model
tat
= -'-_ _-=2:. :. 7_ __
Ym
where 27
Lead Time Number of Observations - 64 Variable Coeff. Std error T Cl. -190.262 545.315 -0.349 0.729 Xl 14979.994 3315.243 4.519 0.000 X2 -3956.096 3202.926 -1.235 0.223 X3 )4 -14519.812 6630.485 -2.190 0.033 9.639 7.155 1.347 0.184 Xl*Xl -376.792 203 .694 -1.850 0.070 X2*X2 880.994 178.423 4.938 0.000 X3*X3 2769.110 814.774 3.399 0.001 X4*)4 -21.325 27.504 -0.775 0.442 Xl*X2 -15.493 26.903 -0.576 0.567 Xl*X3 3.594 55.008 0.065 0.948 Xl*)4 -859.340 131.900 -6.515 0.000 X2*X3 -43.376 264.661 -0.164 0.870 X2*)4 263.801 -4.715 0.000 X3*)4 -1243.846 F= 62.836
Y
I. y~im I
_=i=:;..,.l__ m27
and Yfim represents the i-th valul! of the simulated performance functions (lead time or work in process). yr n and y~tat are the i-th test values relative to neural network and statistical model respectively. In aggregated terms the variation coefficients are shown in Table 4. Table 4. VC results comparison
Neural Network Statist. model
lead time 0.658 0.505
work in process 0.363 0.353
R2= 0.932
Such table shows that the two approaches are practically equivalent for work in process level prediction whereas the performance of statistical inference is higher when lead time factor is considered.
Table 3. Work in process first order re!ll"ession model Work in process Number of Observations - 64 Variable Coeff. Std error T Cl. Xl 35.849 23.962 1.496 0.140 X2 896.997 100.166 8.955 0.000 X3 -71.052 109.505 -0.649 0.519 )4 -899.494 200.332 -4.490 0.000
6. CONCLUSIONS !his paper is focused on monitoring two types of a Job shop plant performance: lead time and work in process. In order to capture the relation between input and output factor, two alternative approaches have been applied. The first one is based on the application of a classical statistical regression model whereas the second one is built on the advanced neural network technique. The former provides the existence of an explicit polynomial relation between variables and the latter, by means of a backpropagation algorithm, maps as a black box the m-dimensional input vector to the n-dimensional one. The data utilised on both inferences are obtained by a simulation model reproducing dynamics of the system under study. The obtained results show that neural network is a low-cost procedure to understand and forecast system events with an acceptable performance level. Such results seem to be related to the particular system complexity.
F= 126.442
Notice that, for goodness of fit point of view, based on the 27 data series used for neural network testing, a first order model has been selected to represent work in process function whereas a second order model has been used for lead time. In Figure 5 and 6 the 27 test values for the observed performance functions are presented with respect to the simulation data (values are normalised by absolute maximum value). Even if lines seem to be not so close, they comply with an evident periodicity. Hence, neural network approach, as well as statistical model, understands the relations between factors in relative terms but not in absolute ones. A quantitative measure VC (Variation Coefficient), in order to compare distance between lines, can be defined as:
154
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- - - - - - - - Neur. net.
- - - - - Statist.
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Since the number of total possible configurations exponentially grows with the number of considered input variables, the authors are currently working on the performance evaluation of neural network approach for systems with an higher input vector size. Preliminary analysis have shown that neural networks results seem to be more significant when the training data set is reduced with respect to the
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