- Email: [email protected]
Feature-based cost estimation for packaging products using neural networks
ELSEVIER
Computers
Feature-based
in Industry 32 (1996) 95-113
cost estimation
for packaging networks
products using neural
Y.F. Zhang *, J.Y.H. Fuh, W.T. Chan Department
of Mecka.~ical
and Production
Engineering,
National Uniuersity Singapore
Received 25 April 1995; revised 29 February
of Singapore,
IO Kent Ridge Crescent,
Singapore
119260,
1996; accepted 26 June 1996
Abstract Cost estimation plays an important role in the product development cycle. For instance, a proper cost estimation can help designers make good trade-off decisions regarding product structures, material, and manufacturing processes. In this paper, a feature-based product cost estimation using back-propagation neural networks is proposed. A system using this approach has been successfully developed for estimating the cost of packaging products. The cost-related features in both design and manufacturing aspects were extracted and quantified according to their cost drivers. The correlation between the cost-related features and the estimated costs of the product was obtained by training and validating a back-propagation neural network based on 60 existing products with their designs, process routings, and actual cost data. To illustrate, the testing results of the trained neural network based on 20 actual products are presented. The performances of the neural network are compared to those of the company’s method and a linear regression model. The results show that the neural network model outperformed both the other methods in respect to performance measures such as average relative deviation and maximum relative deviation. Keywords:
Back-propagation
neural networks;
Cost estimation:
Cost-related
1. Introduction
Cost estimation has traditionally been the province of manufacturing engineers. Typically, the product cost is derived frolm the summation of the various cost components such as material, machine hours, direct labour, administration, and engineering costs. However, during the early stage of product engineering, it is always difficult to obtain these cost elements accurately. Moreover, to allow the designer to
S Corresponding
author. Email: [email protected]
features; Packaging
products
make a good trade-off decision [1,2], product cost must be identified early, i.e., during the designing stage where a large proportion of the manufacturing cost is determined. The work reported in this paper is a result of an industrial project with a local packaging company. A new cost estimation method for packaging products before the actual production run has been established to replace the company’s current method which uses a linear function of a limited number of factors. Many cost estimation techniques have been developed for different products based on various cost drivers. The explicit cost drivers, such as material cost, can be obtained directly; while the implicit
0166.3615/96/$15.00 ‘Copyright 0 1996 Elsevier Science B.V. All rights reserved. PII SO166-3615(96)00059-O
96
Y.F. Zhmg et al./Computers
ones, such as design complexity-related cost, have to be derived through analysis of historical cost data. Therefore, the key to product cost estimation is how to use the historical cost data to understand the implicit cost drivers. This paper discusses how the neural network approach can be applied to product cost estimation. A new approach, feature-based cost estimation, is proposed for packaging products based on the assumption that all the design and process requirements of a product contribute to the final product cost. Since it is difficult to obtain an explicit relationship between these requirements and the final cost, neural networks are selected for approximating this relationship as they can be trained to perform tasks by presenting them with examples rather than specifying the procedures [3-51. The approach involves (1) cost-related feature extraction and quantification, and (2) the identification of the cost estimation function using historical cost data. The cost-related features were classified based on material, printing plates, printing, and production processes. A back-propagation neural network was then constructed and trained using historical product cost data to approximate the relationships between the cost-related features and the product cost. For the packaging products in the company, 32 cost-related features have been identified and quantified. The network was trained and validated using 60 samples, The testing results on another 20 samples are very promising. In this paper, some related work is reviewed in Section 2. The back-propagation neural network and its application in cost estimation are introduced in Section 3. The cost-related feature classification and quantification of packaging products are described in Section 4. The neural network architecture selection and the training procedures are presented in Section 5 and Section 6. Testing results and discussions are given in Section 7, and conclusions are presented in Section 8.
2. Previous work on product cost estimation Much literature has been published on product cost estimation. Most approaches can be broadly classified into the following categories: (1) traditional detailed-breakdown cost estimation, (2) sim-
in Industry 3.2 (1996) 95-113
plified breakdown cost estimation, (3) group technology (GT)-based cost estimation, (4) regression-based cost estimation, and (5) activity-based cost estimation. In the traditional detailed-breakdown approach, the total product cost is typically the summation of all the cost incurred during the production cycle of the product, such as material cost, manufacturing cost, labour cost, and overheads. The calculation, however, requires detailed information of product design, process routing, machining time, etc. The accuracy of the cost estimation depends on the availability of the required information and the variance of the actual activities. A comprehensive description of this method is given by Ostwald [6]. The simplified breakdown methods are mainly developed for cost estimation in the early design stage when typically a process plan is not available. The method is based on the assumption that the manufacturing process will eventually take place under ideal processing conditions. A shortcoming of this approach is that it is difficult to update the cost estimate based on actual cost information. The work of Dewhurst and Boothroyd [7] on early cost estimating models for machining and injection moulding components, of Boothroyd and Reynolds [8] on a cost model for typical turned parts, of Dewhurst and Blum [9] on a cost model for die-casting, and of Boothroyd and Radovanovic [ 101 on a cost model for machined components are the representative examples of this category. The GT-based approach is based on the similarity principle. It typically uses a basic cost value while taking into account the effects of variable cost factors such as size and complexity. Linear relationships between the final cost and the variable cost factors are assumed. Among the examples of this category, Hundal [2] described a cost model for predicting costs of products made in different size ranges, and Poli et al. [l 11 developed a tool to estimate injection-moulding tool costs based on a six-digit code of a part design. Look-up tables were used to extract the cost drivers based on the code. The regression approach tries to find the dependence relationships between costs and product characteristics such as size and material. The coefficients and exponents are derived through the regression analysis of historical cost data. The problem with
Y.F. Zhang et al./Computers in Industry 32 (1996) 95-113
this approach is that the nature of the non-linearity relationship between the cost and the product’s characteristics is not known and hence has to be assumed. The work of Pscyna et al. [12] on a cost model for grey iron castings is an example of this category. Activity-based costing (ABC) is a method for accumulating product costs by determining all cost drivers associated with the activities required to produce the product. Based on historical, observed, or estimated data, the cost per unit of the activity’s output is calculated. The estimated cost for a new product can be obt,ained according to the product’s consumption of these activities. However, it is difficult to obtain the cost per unit of the activity’s output. The work of Ong 1131 on cost tables to support wire harness design in the assembly of electrical connections is, an example of this category.
Little research has been done on product cost estimation using neural networks. Shtub and Zimerman [ 141 applied back-propagation neural networks to estimate the cost of assembly systems. The emphasis was given on the comparison of performance between the neural network model and the regression model. Our research contributes to the neural network application in product cost estimation of another domain. Moreover, the concept of cost-related feature based on design and manufacturing of products is introduced which could be used for cost estimation of other product domains. In this paper, we demonstrate the potential of neural networks in cost estimation by using a backpropagation neural network to obtain the relationship between the product cost and cost-related features of packaging products through supervised learning based on historical cost data. The performance of the
Product coding r-
-
+Neurons in each layer I
Fhxduct
,
4 neural network specification & supervised training
“Knowledge” base
definition & quantification of cost-related features
(a) Preparatory stage
+Neurcno in each layer
“Knowledge” base
(b) Production stage Fig. 1. Product cost estimation:
97
A back-propagation
neural network approach.
98
Y. F. Zhang et al. / Computers in Industry 32 (1996) 95-113
network is compared with both the performance of the company’s method and with a linear regression model.
3. Back-propagation neural network approach for cost estimation Artificial neural networks (ANN) are an information processing technique that simulate biological neurons using computers. In engineering fields, one of the most important application of ANN is modelling a system with an unknown input-output relation. Usually, the accurate information of the system is not available and we can only utilise a number of examples observed from the actual system, called training set. Given a fixed architecture of the networks, learning is carried out through modifying the parameters by the stochastic gradient decent method which eventually minimises a certain loss function. Among various ANN architectures, back-propagation neural network [3,15-171 is a common method for learning of multilayered perceptrons with sigmoidal functions. Since an explicit function for product cost estimation without detailed information on machining time is difficult to obtain, it provides a potential ground for back-propagation neural network applications. To utilise a neural network for product cost estimation, a framework which outlines the procedures is created. As shown in Fig. 1, the proposed neural network approach for product cost estimation has two operational stages: a preparatory stage and a production stage. The preparatory stage is required to construct and train a neural network by using existing products and their actual cost data. During this stage, the product domain is defined, and all the cost-related features of the product are identified. A cost-related feature quantification scheme is established in which different states of a feature are quantified according to their influence on the product cost. Next, existing products are coded based on the cost-related feature definition and quantification scheme to form cost-related feature vectors. A backpropagation neural network architecture is then selected which is often done by trail and error. The training is started by assigning random weights for each interconnection and using the cost-related fea-
ture vectors as inputs and the products’ actual cost as the target output. The training stops when the sumsquared error between the actual outputs and the target outputs of all the training samples reaches an error goal given by the trainer. The network is then tested using samples which have not been used in the training process. If the testing results are not acceptable, testing samples with large deviations are added to the training samples and the network is re-trained. Otherwise, training is completed. The final weights for all the interconnections is then stored in a weight matrix and the neural network is ready for cost estimation of new products. The production stage occurs when the neural network is in use for cost estimation of a new product. During this stage, an incoming product is first coded to obtain its cost-related feature vector according to its design specifications and manufacturing requirement. The vector is then input to the network. By using the existing weight matrix obtained from training, an estimated cost of this product can be obtained. It is worth noting that the preparatory stage can be repeated if the estimated cost for a new product differs too much from the actual cost obtained after production. This can be achieved by adding the new product to the existing training samples and re-training the network accordingly.
4. Cost-related features in the packaging product A cost-related feature of a product can be generally defined as a characteristic associated with the product, which makes a certain contribution to the total cost of the product. It can be either a design specification or a processing requirement. For example, the material type is a cost-related feature in the design domain, while a particular process required for the product is a cost-related feature in the process domain. Also, cost-related features should be product-type dependent. It can be broadly classified into categories, such as machined products, casting products, injection-moulded products, packaging products. In the following sections, the packaging products and processes of the company are introduced and the cost-related features and their quantification are described.
Y. F. Zhang ef al. / Computers in Industry 32 11996) 95-l I3 Fluting
ABC -W? -
ABC -rY9-
I
I
I
First End
Second Front
Fig. 2. The layout of an unfolded RX
4.1. Packaging prohcts
Single
lace
wall
Double
Triple
wall
Wdl
Fig. 3. Types of corrugated
design.
and processes
and quantification
Single
Second End
The type of packaging products studied is the regular slotted cari’on (RSC). An example of an unfolded RSC design is given in Fig. 2. The raw material for the packaging products is paper which has to be converted into corrugated boards before it can be used for manufacturing the carton boxes. Most of the cartons consist of some printing which may be wording or iogos. The RSC products require all or some of the following manufacturing processes: 1. Creating artwork!; and negatives. 2. Making printing plates. 3. Making prototypes. 4. Creating product master cards. (A master card contains the specification of the specification of a product.) 5. Producing corrugated boards. 6. Printing. 7. Stitching/gluing. 8. Packing. 4.2. Classification features
Liner
rw ‘=x%0=
rnxsao~
First Front
99
board construction.
cost-related features were extracted and classified under the aspects of material, printing plates, printings, and manufacturing processes. Since these features have to be used as input to a neural network and a neural network can only perform arithmetic operations, each cost-related feature of a product had to be assigned with a value between 0 and 1. This value is called the cost-related feature attribute (CFA). This process is called the quantification of cost-related features. The magnitude of a CFA of a feature is an indication of the relative cost influence among the various possible states of that feature. A state which contributes to a higher cost will be assigned a higher value. The details of the cost-related features and their quantification are described in the following sections. 4.2.1. Based on material (1) Type of corrugated board construction (CFAI) There are four types of corrugated board construction: single face, single wall, double wall, and triple wall (see Fig. 3):
of cost-related
0.3
CFAI = Based on our study of the RSC product design and manufacturing processes in the company, 32
I ;.;
0:s
single face single wall
(1)
double wall triple wall.
Table 1 Quantification
of liner quality (x = 2, 3, 4, 5)
Type
K4
K5
K6
K7
K8
K9
w5
W6
WI
W8
CFA x
0.52
0.5
0.42
0.44
0.46
0.48
0.54
0.56
0.58
0.6
Type
T4
T5
T6
T7
Ml
M2
M3
M4
M5
M6
CFA x
0.34
0.36
0.38
0.4
0.18
0.2
0.22
0.24
0.26
0.28
Y.F. Zhang et al. / Computers in Industry 32 (1996) 95-I 13
100
r-l
ABC
Fluting’s height
reversecut
normal cut
Fig. 4. The fluting’s height.
Fig. 5. Types of cut for rubber plate.
(2) Quality of raw material (CFA2, CFA3, CFA4, CFAS, CFA6, CFA7, CFA8) The quality of the material is defined as the weight per unit area of the paper. Based on the type of corrugated board construction, CFA2, CFA3, CFA4, CFAS refer to the quality of the first liner, the second liner, the third liner, and the fourth liner, respectively. The quantification of liner quality is given in Table 1. Likewise, CFA6, CFA7, CFA8 refer to the quality for the first fluting, the second fluting, and the third fluting, respectively. The detailed quantification of fluting quality is given in Table 2. (3) Height of fluting (CFA9, CFAlO, CFAll) The height of the fluting is defined as the distance between two liners as shown in Fig. 4. There are 3 fluting heights, i.e., 0.1,0.15, and 0.2. CFA9, CFAlO, CFAll refer to the height of the first fluting, the second fluting, and the third fluting, respectively. (4) Area of the carton (CFA12)
(2) Material of printing plates (CFA14) The types of material used for printing plates are rubber and photopolymer:
area of carton (m* ) CFA12 =
(2)
’
20
CFA14 =
(3)
both.
(3) Types of cut for the rubber plate (CFA15) The cut for the rubber plate can be either nomal or reverse (see Fig. 5). The normal type has wording or logos printed with colours. A reverse cut on the other hand prints the surroundings with colours leaving the wording or logos uncoloured. In general, reverse cutting is more difficult and hence costs more.
CFA15 =
0.2 0.3 i 0.25
normal reverse both.
(4)
(4) Size of the printing plates (CFA16, CFA17) The size of the printing plates is defined as the area of the rubber or photopolymer required: rubber area (m2)
where 20 m2 is the largest area of a product which can be produced by the company.
CFA16 =
4.2.2. Based on printing plates (1) Availability of printings (CFA13) Products with printings cost more than the ones without printing features. If there are printings CFA13 equals 0.5.
CFA17 =
Table 2 Quantification
rubber photopolymer
0.2 0.6 i 0.4
(5)
’
2.5 photopolymer
area (m* ) t
0.7
(6)
where 2.5 m2 and 0.7 m2 are the largest printing plates size which can be produced for rubber and photopolymer, respectively.
of fluting quality (x = 6, 7, 8)
Type
F5
F6
Ml
M2
M3
M4
M5
M6
T4
T5
T6
n
CFAx
0.3
0.32
0.18
0.2
0.22
0.24
0.26
0.28
0.34
0.36
0.38
0.4
Y.F. Zhang et al./Computers
4.2.3. Based on printings ( 1) Number of C:O~OU~S (CFA 18) The larger the number of colours, the more set-up time is required:
\ 0.7
No No No No
of of of of
colours colours colours colours
= = = >
1 2 3 3.
P/A
< 30%
30% I P/A 50 I P/A \0.8
P/A
< 50 < 70%
(8)
2 70%.
(3) Effects of colour trapping (CFA20, CFA21, CFA22) Colour trapping refers to a situation where the printing consists of two or more different colours that are printed in contact with one another (see Fig. 6). The problem of colour trapping arises from the misalignment of the printing plates. As a result, more samples are usually required during the process of aligning the printing plates. CFA20 is 0.5 if there is colour trapping. CFA21 is based on the number of printing plates (NPP) for different colour trapping and CFA22 is based on the type of colour trapping. 0.3 CFA21 = 0.4 i 0.5
NPP=2 NPP=3 NPP=4.
0.3 CFA22 = 0.6 i 0.5
between dark and light between light only
101
32 (1996) 95-113 colour 2
colour 1
ur3
(7)
(2) The printed area (CFA19) The printed area determines the amount of paint required directly. Moreover, it affects the quality of the print as well. Quantification is based on the printed area percentage out of the total area (P/A): (0.2
in Mushy
Fig. 6. An example of colour trapping.
4.2.4. Based on production processes (1) Number of processes (CFA24) In general, the maximum number required for a product is less than 10. number of processes CFA24 =
10
(11)
.
(2) Requirement for reverse corrugated boards (CFA25) The use of reverse corrugated boards requires additional processes and hence additional cost occurs. CFA25 equals 0.5 if reverse corrugated board is required. (3) Requirement of printing machine (CFA26) If the product needs a printing machine, CFA26 equals 0.5. (4) Stitching (CFA27, CFA28) If stitching is required, CFA27 equals 0.5. Since the distance between each stitch is fixed at 50 mm, the number of stitching required depends on the working height (see Fig. 7). working height (mm) CFA28 =
(9)
( 10)
of processes
The maximum (5) Gluing Gluing can a machine or
50 x 25
(12)
’
number of stitches is 25. (CFA29, CFA30) be carried out either automatically on manually. In general, manual gluing
both.
(4) Availability of printing inks (CFA23) If the required ink is not available, it has to be purchased and often there will be some left over after the printing is done. Extra cost is therefore incurred. CFA23 equals 0.2 if required printing inks are available; otherwise CFA23 equals 0.4.
I Fig. 7. The working height.
working height
0.52
0
0.2
0
0.1
0
0.089
0.5
0.2
0.2
CFA4
CFA6
CFA7
CFA9
CFAIO
CFAIZ
CFAl3
CFAl4
CFAI
0.15
0.1
0.2
0.2
0.5
I.812
1.80,
0.2
1.1801.102
1.1651.104
0.4
0
0
0
0.2
0.2
0.5
0
0
0.5
0.1
0.2
I.563
1.579
I .O% - I .3% 0.2%
CFA19
CFAZO
CFAZI
CFA22
CFA23
CFA24
CFA26
CFA27
CFA28
CFA29
CFA31
CFA32
Cost(T)
Cost(E)
RDa
0.5
0.5
0
0
0.5
0.1
0.2
0.28
0
0.5
0.4
0.2
0
0.2
0
0
0
0.4
0.2
0.2
0
0
0
0.4
0.2
- 0.6%
0.2
0.1
0.5
0
0
0.5
0.2
0
0
0
0
0
0
0
0.2
0
CFAID
0
0
0
0
0
0
0.084
0.15
0.1
0.2
0.2
0.52
-7%
6
0.15
0.1
0.2
0.2
0.42
0.18
0.42
0.6
0.2
0.2
0.5
3.8%
0.2%
1.920
0.424.418
0.2
0
0
0
0
0
0. I
0
0
0
0
0
0
I.917
0.2
0
0.28
0.5
0.5
0.3
0.2
0.2
0.2
0
0
0
0
0
0.454.366
0.2
0.10
0.5
0
0
0.5
0.2
0.2
00
00
00
0.2
0.2
00
0.022X074
0.2
0.2
0.5
0. IO0
0.15
0. I
0.2
0.2
0.52
0.18
0.52
0.6
1
Es of training
0.02Xl.057
0
0.1
0
0.2
0
0.5
0.5
0.52
0.18
0.4
CFA17
0.1260.145
0.25
0.2
0.5
costs
0.6
0.126
0.2
0.2
0.5
0.0650.042
0
0.1
0
0.2
0
0.18
0.5
0.6
CFA16
5
0.5
0.5
CFA3
0.52
0.4
0.4
CFAI CFA2
4
5
3
1
2
actual costs Ts, and estimated
The CFAs,
Table 3
0.795
0.413 2.8%
0.773
02
0.1
0.5
0
0
0.5
0.2
0.4
0.3
0.3
0.5
0.8
0.3
0
0.284
0.25
0.2
0.5
0.439
0.2
0.1
0.5
0
0
0.5
0.2
0
0
0
0
0
0
0
0
0
0
0
0.035
0.026
0. I
0
0.2
0
0.52
0.4 0.54
0
9
0
0.1
0
0.2
0
0.52
0.4 0.52
-5.9%
8
samples 10
11 12
-1.28-128
- 2.6%
0.952
0.367
0.41 I
0.2
0.1
0.5
0
0
0.5
0.3
0.4
0.3
0.3
0.5
0.2
0.4
0.288
0
0
0.6
0.5
0.042
0
0.1
0
0.2
0
0.42
0.4 0.54
0.4160.977
0 2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.6
0.2
0.44
0
0
0.6
0.5
0.023
0
0.1
0
0.2
0
0.5
0.4 0.5
0.416
0.2
0.1
0.5
0
0
0.5
0.2
0.4
0
0
0
0.6
0.2
0.44
0
0
0.6
0.5
0.023
0
0.1
0
0.2
0
0.5
0.4 0.5
0.2
0.4
0.5
- 2.7%
7.7%
0.7900.586
0.596
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.2
0.8120.544
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.3
0.612
0.2
0.1
0.5
0
0
0.5
0.2
0
0
0
0.1
0
0.2
0
0.0720.016 0.6
0
0.5 0.5
0.1060.007
0.25
0.4
0.5
0 0
15 0.4
0.0410.030
0
0.15
0
0.2
0
0.42
0.42
0.4
0.2
14
0
0
0
0
0
0
0.028
0.15
0.1
0.2
0.2
0.5
0.18
0.5
0.6
- 2.6%
13
16 17
0
0.15
0
0.2
0
0.42
0.42
0.4
0.2
0.2
0.5
I.086
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.6
0.2
0.1%
1.8%
1.4621.106
I.461
0.2
0
0
0.24
0.5
0.5
0.3
0.2
0.3
0.3
0.5
0.4
0.3
0.2120
0.0790.216
0.25
0.4
0.5
0.0490.055
0.15
0.1
0.2
0.2
0.42
0.18
0.42
0 6
I. 120
I.174
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.6
0.2
0.08 I
0.123
0.25
0.4
0.5
0.041
0.15
0.1
0.2
0.2
0.44
0.18
0.44
0.6
-4.6%
18
0.409
0.429
0.2
0.1
0.5
0
0
0.5
0.2
0.4
0
0
0
0.6
0.2
0.12
0
0
0.6
0.5
0.019
0
0.1
0
0.2
0
0.42
0.54
0.4
-4.7%
19
20
0.5%
1.658
I.649
0.2
0
0
0.28
0.5
0.5
0.3
0.4
0.3
0.4
0.5
0.6
0.4
0
0.167
0.2
0.2
0.5
0.052
0.15
0.1
0.2
0.2
0.42
0.18
0.54
0.6
0.469
0.9580.505
0
0.3
0.5
0.5
0.12
0
0
0.2
1.438
I.432
CFA23
CFA24
CFA26
CFA27
CFA28
CFA29
CFA3l
CFA32
Cost(T)
Cost(E)
* RD:
Relative
- 0.4%
0.9750.487
0
CFA22
RD=
0.447
0.2
0
0
0
0.5
0.1
0
0
0.5
0.1
deviation.
CFA5
I .7% 3.7%
0.2
0.5
0.2
0.2
0.5
0.2
0
0
0
0.2
0
0
0
0.2
0.2
= 0, CFA8
4.9%
0 0.2
0.1
0.5
0
0.5
0.2
0.2
0
0
0
0.6
0.2
0
= 0. CPA
-3.3%11.2%
1.5700.981
1.6230.882
0
0.1
0.20
0.5
0.5
0.3
0.2
0
0
0.5
0
0
0.5
0.3
0.2
0
0
0
0
0.2
0.4 0
0.3 0.6
0.2
0.6
0.4
0.2
0
0
0
0
CFAZI
0
CFAl7
0.2
0.2
0.5
0.0490.1
0.25
0.067
0.2
0.2
0.5
0.0270.047
0.2
0.2
CFAZO
0
CFA16
0.25
0.2
0.5
0
0
CFAIS
0.2
0.5
0
0
CFA14
0.5
0.15
0.1
0.2
0.2
0.5
0.18
0.6 0.5
26
0.0600.038
0.15
0.019
0
0.2
0.2
0.42
0.18
0.6 0.42
0.1
25
0.1
0
0.2
0
0.5
0.4 0.5
24
CFAl8
0
CFA13
0
0.1
0
0.2
0
0.5
0.0430.028
0.15
0.1
0.2
23
0.4 0.5
CFAl9
0.15
0.1
CFA9
0.058
0.2
CFA7
CFAIZ
0.2
0.2
CFA6
CFAIO
05
0.42
CFA4
0.18
0.18
CFA3
0.6 0.5
22
0.6 0.42
21
CFAI CFAZ
2.1%
1.625
I.591
0.2
0.1
0.5
0
0
0.5
0.2
0.4
0
0
0
0.2
0.2
0
0.059
0.2 0.059
0 0
0.971 - 1.5%
2.696 0.4%
-6%
= 0, and CFA30
0.986
0.2
0.1
2.685
0.2
0
0.5
0
0.36 0
0
0.5
0.5 0.5
0.2
0.3
0
0 0.4
0
0 0
0
0
0.2 0.2
0
0
0
0.2
0
0
0.5
0
0.15 0
0.1
0.1 0.054
0
0.2
0.119
0.2
0.2
0.741 5.4%
0.632 5.2%
0.703
0.2
0.1
0.5
0.7%0.5%
1.0640.826
1.0570.822
0.2
0
0.2
0
0.1
0.2
0.5
0.5
0.3
0
0
0
0
0
0
0
0
0
0.5
0
0
0
0.2
0.2
0
0
0
0.6
0.3
0. I280
0.2630
0.25
0.4
0.5
0.5
0
0
0.1
0
0.2
0
0.52
0.52
0.4
35
0.0510.043
0
0.15
0
0.5
0.601
0.2
0.1
0.5
0
0
0.5
0.2
0
0.2 0.2
0
0
0
0
0
0 0
0.2 0.2 0
0
0
0
0
0
0.041
0
0.1
0
0.2
0
0.2
0.42
0
0.42
0.5 0.5
0.4
34
0.4
33
0
0.037
0.2
0.2
0.5
0.031
0
0.1
0
0.2
0
0.42
0.42
0.4
32
= 0 for all samples
0.9%
0.916
0.908
1.323 I.244
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.2
0.2
0
0.062
0.2
0.2
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.4
0.2
0
0.003
0.2
0.2
0.5
0.040
0.5
0.15
0.1
0.2
0.2
0.52
0.18
0.52
0.6
0.057
31
0.15
0.1
0.2
0.2
0.5
0.18
0.5
0.18 0
0.5
0.5
0.5
0.6
30
0.4
29
0.6 0.5
28
I I = 0, CFA25
I I
0.2
0.2
0.5
0.070
0.15
0. I
0.2
0.2
0.5
0.18
0.6 0.5
27
0.15
0.1
0.2
0.2
0.5
0.18
0.42
0.6
37
0.2
0.1
0.5
0
0
0.5
0.2
0
0
0
0
0
0
0
0
0
0
~4.4%
1.8%
1.5581.439
1.6301.414
0.2
0
0
0.2
0.5
0.5
0.3
0.2
0
0
0
0.2
0.3
0
0.0720
0.25
0.2
0.5
0.0600.061
0.15
0.1
0.2
0.2
0.42
0.18
0.42
0.6
36
-
0.339 -2.0%
1.594
0.2
0.1
11%
0.729
0.656
0.2
0
0
0.5
0.5
0.3
0.2
0
0
0
0.4
0.2
0
0.619
0.25
0.2
0.5
0.026
0.15
0.1
0.2
0.2
0.4
0.18
0.5
0.346
1.1%
0.6 0.5
0.2
40
0
0
0.5
0.3
0.2
0
0
0
0.2
I.612
0.2
0.1
0.s
0
0
0.5
0.3
0.2
0
0
0
0.4
0.2
0
0 0.3
0.004
0.2
0.2
0.5
0.015
0
0.1
0
0.2
0
0.52
0.52
0.4
39
0.069
0.25
0.2
0.5
0.080
0
0.1
0
0.2
0
0.5
0.5
0.4
38
4
0.5
0.2
0.2
0.1
0.15
0.043
0.5
0.2
0.2
0.108
0
0.2
0.4
0.52
0.2
0.2
0.1
0.15
0.057
0.5
0.2
0.2
0.126
0
0.2
0.4
0
0
CPA4
CFA6
CFA7
CFA9
CFAIO
CFAIZ
CFAI3
CFAI4
CFAI.5
CFA16
CFAI7
CFAI8
CFA19
CFAZO
CFA2
0.5
0.24
0
0
0.2
0.5
0.24
0
0
0.2
I.471
1.372
CFA27
CFA28
CFA29
CFA31
CFA32
Cast(T)
Cost(E)
RD:
Relative
-6.7%
1.142
0.5
0.5
CFA26
RD’
I.146
0.3
0.3
CFA5
6.9%
1.066
0.2
0.1
0.5
0
0
0.5
0.2
=
-0.3%
0.2
0.1
0.5
0
0
0.5
0.2
0
0
CFA24
0.2
0
0.2
0
0.2
0
CFAB
0.843
0.2
0.1
0.5
0
0
0.5
0.3
0.4
0.3
0.3
0.5
0
0
0
0.2
0
0.4
0.288
0
0
0
0
0.6
0.5
=
0.9%
0.5% 1
0.773
0.769
0.2
0.1
0.5
0
0
0.5
0.2
0.4
0
0
0
0.2
0.2
0
0.059
0.2
0.2
0.5
CFAI
2.422
0.2
0.1
0.5
0
0
0.5
0.2
0.4
0
0
0
0.2
0.2
0
0.059
0.2
0.2
0.5
0.041
0
0.1
0
0.2
0
0.5
0.5
0.4
1
-1.1%
0.737
0.4
samples
9
0, CFA25
0.616
0.2
0.1
0.5
0
0
0.5
0.2
0
0
0
0
0
0
0
0
0
0
0
0.036
0
0. I
0
0.2
0
0.5
0.5
0.4
a
Es of vahdation
0.108
0.15
0.1
0.2
0.2
0.5
0.18
0.5
0.6
6
costs
0.035
0
0.1
0
0.2
0
0.42
0.54
0.4
5
0
0
0
0
0.052
0.15
0.1
0.2
0.2
0.5
0.18
0.5
0.6
CFA23
0
0
0.2
0.3
0
0.039
0.2
0.2
0.5
0.056
0
0.1
0
0.2
0
0.5
0.5
4
CFA22
I
0.18
0.18
CFA3
0
0.5
0.52
CFA2
0.4
3
costs Ts, and estimated
2
0.6
actual
0.6
1
CFA I
The CFAs,
Table
0. and
-6.6%
0.607
0.650
0.4
10
= 0
0.914
0.4
11
-4.3%
0.598
0.625
0
0.5
0.4
13
all samples.
-2.5%
0.895
0.6
12
- 1.6%
0.839
0.5
0.5
0.479
0.4
15
0.4
14
0.980
0.900
0.42
0.42
0.6
16
-9.1%
I.159
0.42
0.4
17
-0.2%
1.203
0.5
0.4
18
I.147
I.058
0.52
0.4
19
0.805
0.1
0.5
0
0.5
0.2
0
0
0.2
0.221
0
0.6
0.5
0
0.15
0.2
0
0.42
0.4
- 12%
20
0.52 0
0.2
0.2
0.1
0.15 0
0.0630.080
0.5
0.2
0.2
0.0750.059
0
0.2
0.2
0.2
0.2
0
0.5
0.1
0.2
0
0.2
0
0.15
0
0.036
0.5
0.6
0
0
0.096
0.2
0.2
0
0
0
0.2
0.2
0.5
0
0
0.5
0.1
0.2
0.701
0.648
CFA4
CFA6
CFAl
CFA9
CFAIO
CFAl2
CFA13
CFAl4
CFAIS
CFAl6
CFAI7
CFAIS
CFA19
CFAZO
CFAZI
CFA22
CFA23
CFA24
CFA26
CFA27
CFA28
CFA29
CFA3l
CFA32
Cost(T)
Cost(E)
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
- 8.0%
- 7.5% - 3.2% 7.0%
1.475
I.489 1.740
1.750
0.2
0
0
0.32
0.5
0.5
0.3
0.4
0.3
0.4
0.5
0.4
0.4
0.466
0
0
0.6
0.5
0.057
0.15
0.1
0.2
0.2
0.42
0.18
0.54
0.6
3.3% -0.6%
I.523
0.2
0.2
I .370
0
6
0.1
0.1
0.641
0.566
1.4%
0.573
0.565
0.2
0.5
0.5 0.2
0
0
0.5
0.2
0.2
0
0
0
0
0.5
0.2
0.2
0
0
0
0.6
0.6 0
0.2
0.2
0.062
0.25
0.2
0.5
0.03 I
0
0.1
0
0.2
0
0.5
0.5
0.4
0
9
0
0.121
0.25
0.2
0.5
0.031
0
0.1
0
0.2
0
0.5
0.5
0.4
8
3.8% 13.3%
I.055
I.016
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.6
0.3
0.1
0.166
0.25
0.4
0.5
0.051
0
0.15
0
0.2
0
0.42
0.42
0.4
7
0.804
0.955
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.2
0.2
0
O.OSi
0.2
0.2
0.5
0.044
0
0.15
0
0.2
0
0.44
0.44
0.4
- 16%
10
1.283
1.327
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.2
0.2
1.3% -3.3%
1.037
1.024
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.4
0.2
0
0.003
0.077 0
0.25
0.2
0.5
0.057
0.15
0.1
0.2
0.2
0.42
0.18
0.42
0.6
12
0.2
0.2
0.5
0.055
0
0.15
0
0.2
0
0.42
0.42
0.4
11
and cost relative deviatmns of testing samples
0
0
0.24
0.28
0
0.5
0.5
0.4
0.4 0.5
0.3
0.3
0.5
0.4
0.4
0.3
0.5
0.5
0.3
0.4
0.4
0
0.191
0.2
0.2
0.5
0.047
0.15
0.1
0.2
0.2
0.42
0.18
0.54
0.6
0.4
0.4
0.188
0.276
0.25
0.4
0.5
0.048
0.15
0.1
0.2
0.2
0.42
0.18
I .423 I.522
1.4701.422
0
0.5
0
0
0
0.4
0.3
0
0.25
0.2
0.5
0.1
0
0.2
0.5
0.54
0.6
5
a RD: Relative deviation. CFAS = 0. CFAS = 0, CFAI I = 0, CFA2S = 0, and CFA30 = 0 for all samples
RDa
0.18
0.42
0.5
CFA3
0.52
0.42
0.4
*
estimated costs Es,
CFA2
0.6
3
0.4
2
CFA I
1
The CFAs. actual costs Ts,
Table 5
14
0.15
0.1
0.2
0.2
0.5
0.18
0.44
0.6
0.2
0.2
0.5
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.2
0.2
0
-8.6%
1.6%
I.360 I .054
I.488 I.037
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.2
0.2
0
0.0620.061
0.2
0.2
0.5
0.0560.042
0.15
0.1
0.2
0.2
0.5
0.18
0.44
0.6
13
16
0.5
0.5
0.4
0
0.1
0
0.2
0.2
0.1
0.5
0.5
0.2
0.4
0.2
0.2
0.049
0.2
0.2
0.5
-2.7q4.41
0.9840.917
1.0110.878
0.2
0.1
0.5
00
00
0.5
0.2
0
00
00
00
0
0
00
0
0
0
0
0.0460.051
0.15
0.1
0.2
0.2
0.52 0
0.18
0.52
0.6
15 17
2.0%
2.422
2.374
0.2
0.1
0.5
0
0
0.5
0.2
0.4
0
0
0
0.2
0.2
0
0.059
0.2
0.2
0.5
0.108
0.15
0.1
0.2
0.2
0.5
0.18
0.5
0.6
18
4.6%
1.447
1.384
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.4
0.2
0.235
0.098
0.2
0.4
0.5
0.060
0.15
0.1
0.2
0.2
0.5
0.18
0.5
0.6
-11%
19
0.544
0.608
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.2
0.2
0.2
0
0
0.6
0.5
0.026
0
0.2
0
0.26
0
0.44
0.44
0.4
- 1.3%
I.544
I .564
0.2
0
0
0.28
0.5
0.5
0.3
0.2
0
0
0
0.4
0.2
0
0.104
0.2
0.2
0.5
0.083
0
0.1
0
0.2
0
0.5
0.5
0.4
20
Y.F. Zhang
106
et al./Compurers in Industry 32 (1996) 95-113
incurs a higher cost. If gluing is required, equals 0.5. If manual gluing is applied, equals 0.5. (6) Number of glued joints (CFA31) 0.1 CFA31 = 0.4 i 0.6
No of joints = 1 2 I No of joints < 4
(13)
No of joints = 4.
(7) Method of packaging 0.2 CFA32 = 0.25 i 0.4
CFA29 CFA30
(CFA32)
bundling shrink - wrapped
(14)
plastic strips.
If a feature is not available in a product or the a feature is not applicable to a product, the respective CFA is set to 0. For example, if a product does not use reverse corrugated board, CFA25 = 0. It is worth noting that the values assigned to various states of a cost-related feature are arbitrary since they are mainly based on experience and the study of the product cost. 4.3. Preparing samples
input and output
data of training
The training and testing samples were collected from the existing products of the company. They must be representative enough to cover most cost-related features of the whole product spectrum. By doing so the neural network can be trained and tested comprehensively by exposing to various situations. As a result, 80 existing products were selected. These samples include RSCs with or without printings as well RSCs with or without colour trapping. According to the cost-related feature classification and quantification scheme, their CFAs were then obtained based on their design specifications and processes. Both the CFAs and actual cost data (Ts) provided by the company are listed in Table 3, Tables 4 and 5. One may find that among the training samples, all the CFAl 1, CFA25, and CFA30 are zero. This is caused by the absence of products with these features in the present product range. These CFAs can be activated as soon as the company produces products with these features in the future.
5. The neural network architecture
A feed-forward back-propagation neural network is implemented with a program written using the MATLAB Neural Network Toolbox [4] platform. The first step is to select a good neural network architecture, i.e., number of layers and number of neurons in each layer. The number of neurons in the input layer is the same as the dimension of the cost-related feature attribute vector of CFA, i.e., 32 in this application. The output layer has only one neuron which represents the cost. The determination of the number of hidden layers and neurons in each hidden layer is by trial and error based on the training and testing results of the network. Two types of total 12 neural networks were studied: 1. Six neural networks with one hidden layer. The number of hidden neurons in the hidden layer was 3, 5, 7, 9, 12, and 15. 2. Six neural networks with two hidden layers. The number of neurons in the first and second hidden layers, respectively, was (3, 5), (5, 4), (6, 4), (4, S), (4, lo), and (12, 6). Training of these networks was carried out using a set of 40 randomly selected samples. All the networks were trained for 8000 times. The trained networks were then tested on another set of 20 samples. Four criteria were used as the measure of performance: (1) The average deviation of training samples
2 ICostT, “=
i=l
CostEil
(15)
’
N
where N: costq: Cost E,:
number of samples, target cost of the ith sample, estimated cost of the ith sample.
(2) The maximum samples
relative
deviation
of training
ICost7” - Cost Eil C2=
Max i=l,...,N
CostT,
.
(16)
Networks 3 0.024 1 28.1% 0.0254 29.2%
Network architecture
No of neurons in hidden layer(s)
Cl c2 c3 c4
Table 6 Criteria of the tested network architectures
0.0219 24.2% 0.023 1 24.8%
5
_ ..
c.
0.0219 25.5% 0.029 1 29.2%
7
with one hidden layer
.,,
..-
-
0.0241 21.4% 0.0259 27.6%
9
,,
.
,
rn
-
0.0234 27.4% 0.027 1 30.8%
12
i
-
-
0.023 1 27.8% 0.0273 30.3%
15
-
^
-
0.0222 29.3% 0.0268 30.6%
3, 5
_
_”
-
0.0257 28.5% 0.029 1 30.0%
5.4
_
-
0.0241 21.2% 0.0266 36.4%
6,4
_1
Networks with two hidden layers
.I
0.0217 26.8% 0.0307 35.8%
4, 8
_
_
i
0.0259 28.1% 0.0287 30.1%
4, 10
“.
0.0240 28.3% 0.0299 28.9%
12,6
_
108
Y.F. Zhang et al./Computers
in Industry 32 (1996) 95-113 oumut laver
hidden layer
input layer
CFAl
CFA2
CFA3
AZ=purelin(WZ’Al
CFA32
+B2)
Al =tansig(Wl ‘CFA+Bl)
Where, CFA=
[CFA~
CFAZ
Bl = [Bi(l)
M(2)
Al = [Al(l)
Al(2)
Wl=
...
M(3) Al(3)
CFA32]’ Bl(4)
E1(5)]T
Al(4)
A1(5)lT
82 = [B2(1)] A2 = [A2(1)]
Wl(1,l)
W1(1,2)
...
W1(1,32)
W2(1)
Wl(2,l)
W1(2,2)
...
W1(2,32)
W2(2)
Wl(3,l)
W1(3,2)
...
W1(3,32)
Wl(4,l)
W1(4,2)
.,.
W1(4,32)
~4)
Wl(5,l)
Wl(52)
...
W1(5,32)
W2(5)
w2=
T
w2(3)
Fig. 8. The back-propagation neural network architecture for cost estimation
(3) The average deviation
of testing samples
N ICostT, - cost E, I c3=
c i= 1
N
(17)
’
(4) The maximum samples
relative
deviation
of testing
(CostT, - CostEJ C4 =
Max
i=l,...,N
costq
(18)
The criteria for each type of architecture trained and tested using the training and testing samples are summarised in Table 6. Based on the results, the network with one hidden layer of five neurons was selected. It is worth mentioning that the selected architecture is by no means the best. This is because the number of neural network architectures is infinite. The selected neural network architecture is shown in Fig. 8. CFA is the input vector, Wl and WZ are
the weight matrices for the interconnection between the input layer and hidden layer and that between hidden layer and output layer, respectively. Bl and B2 are the bias vectors for the neurons in the hidden layer and output layer, respectively. Al represents the output from the hidden layer, while A2 represents the output from the output layer, which represents the product cost. The transfer function TanSigmoid tansig is used for the hidden layer, while linear function purelin is used for the output layer in order that the final cost value can reach any value.
6. Training of the back-propagation
network
The training of the selected neural network was carried out automatically by running the program. It includes the following steps:
0.4196 0.8758 - 0.5202 - 0.6382 - 0.3649
0.9543 0.1914 - 0.3235 0.4734 - 0.5642
w1,,26-w15.26
Wl
,.25-w15.25
-0.2721 0.4840 0.3830 0.1376 0.1033
0.5707 - 0.0179 0.1775 0.9967 0.7806
-0.1878 - 0.6366 - 0.4378 - 0.2922 0.4474
w1,.27-w15.27
-
w1,,,9-w1,,,9
WI
Wl 0.1593 0.0288 0.0235 0.0980 0.2956
- 0.6670 - 0.0270 0.7953 0.8184 - 0.8789
0.3119 0.2515 -0.6021 - 0.4807 - 0.4447
-0.8919 0.1466 0.7629 - 0.3964 - 0.2256
,,,8-w15.,8
WI,.,, -WI,.,,
Wl ,.,0-W1~,,,
w1,,9-w15,9
,,17-WlS,l7
0.5603 - 0.3532 - 0.3858 -0.1246 - 0.9883
- 0.3673 0.0723 0.5707 -0.8149 - 1.3620
W~l,3--W’5,3
Wl,,z-Wl,,,
0.8764 - 0.5200 0.4387 0.3899 0.2717
-
Wl,,,-Wl,,,
- 0.3567 0.3545 0.4357 - 0.7577 0.763 1
w1,.28-w15,28
0.1458 -0.6533 - 0.5099 - 0.8737 - 0.7072
w1,,2,-w15.20
0.5880 4.3772 - 1.9846 0.3573 0.8331
Wl,.,*-Wl,,,,
- 1.1402 0.3568 0.1118 0.8102 0.1451
WI,.,-Wls,4
Table 7 Weights for interconnection between input neurons and hidden neurons
- 0.0389 -0.1931 0.2385 0.0815 -0.1151
w1,.Z9-w15,29
0.1612 0.3138 - 0.9853 -0.0681 - 0.0148
w1,,2,-w1S,2,
0.3176 - 0.2329 0.0359 0.7853 - 0.9688
w11,,3-w15.13
0.0539 -0.8161 0.3078 -0.1680 0.4024
Wl,,,-w15.5
- 0.0719 0.9222 - 0.7479 - 0.6005 -0.3615
w1,,30-w15,30
0.9534 0.1431 0.0586 0.6432 0.2410
w1,,2,-w1,,22
0.5451 - 0.4040 - 0.6457 0.6419 0.0125
w11,14-W’5.14
0.8109 0.5200 - 0.3828 - 0.8691 0.2821
w~l,6-w~s.6
0.2872 - 0.6420 0.3243 0.3587 0.5877
w11,31-w15,31
0.3271 - 0.2723 - 0.5400 0.3798 - 0.8032
w1,.23-w15,23
- 0.0072 0.0990 0.2374 0.4544 0.4773
w~,.Is-w15.,5
-0.6383 0.6601 0.4093 0.9413 - 0.4870
W~,,,-W~5,7
- 0.4322 - 0.0922 -0.0710 - 0.5492 - 1.0776
w11,3Z-w’5,32
- 0.6984 - 0.2542 - 1.2134 -0.1419 - 1.1484
w11.24-w15,24
0.5124 - 0.7305 - 1.8707 0.7668 0.7011
w~l,l6-w~5.16
- 0.5059 0.965 1 0.4453 0.5067 0.3030
w1l,8-w15,8
110
KF. Zhang et al./Computers
(1) Initialising the training session Before the training starts, the parameters of the network for training must be pre-set. The initial weights and biases were generated with values between - 1 and + 1 by the random functions provided by the MATLAB Neural Network Toolbox [4]. (2) Training procedures (a) The training data CFAs and Ts are presented to the network in the form of vectors [CFA’, CFA2, . . . . . . ) CFA”]*, and [T’, T2, . . . . .., Tm]*, where m is the number of training samples. (b) The input vector of each sample is pre-multiplied by a set of weights, and summed up with the biases through the transfer functions to generate an estimated cost vector [ A2’, A22, . . . . . . , A2”]*. (c) The sum-squared error E is then obtained by E=
f i=
(T’-A2’)‘.
(19)
I
(d) Generally, the training will stop when either (i) the sum-squared error is smaller than the error goal specified by the user, or (ii) the maximum number of training epochs is reached. However, the target error goal was unknown. We also observed that the sum-squared error tended to converge to a certain value as the number of training epochs increased. Therefore, we used the relative changing rate (RCR) of the sum-squared error to determine when the training has reached a satisfying result. The RCR was calculated at any point where every 1000 training epochs has been completed:
qi-
Table 9 Weights for interconnection between neuron, and bias for output neuron
1)X1000
’
and output
W20)
W2(2)
W2(3)
W2(4)
W2(5)
B2(1)
3.9244
- 2.2210
0.3380
-0.4153
0.1255
The target error goal was given a very small value, 0.001, that cannot be reached. The maximum number of epochs is given a very large value, 500000. This was to ensure that it will go to step (e) to continue training in between the checking points. At the checking point, if 0.99 < RCR, < 1.01, the training stops; otherwise, go to step (e). (e) The derivatives of the error for each neuron layer is calculated and back-propagated to its proceeding layers to alter the weights and biases - go back to step (b). (3) Validation The trained network is then validated using a set of samples that have not been used in training to check its generalisation ability. The estimated cost of each validating sample was obtained by inputting its CFA to the trained network. The estimated cost relative deviation for each sample was calculated as: Cost Ej - Costq RD;=
CostT,
x loo%,
(21)
where CostT =
(20)
hidden neurons
0.8591
1000
EiX
RCR, =
in Industry 32 (1996) 95-113
cost Ei =
actual cost of sample i, estimated cost of sample 1.
where EiXIOOO
E(i-
=
1)X 1000 =
i=
sum-squared error at (i X 1000) epochs, sum-squared error at ((i - 1) X 1000) epochs, 1) 2, 3, . . . . . .
Table 8 Biases for the hidden neurons El(l)
Bl(2)
Bl(3)
Bl(4)
Bl(5)
0.7545
-0.1690
- 0.2547
1.0750
- 1.1295
If the relative deviations of some validating samples are out of +20%, there must be important information in these validating samples that the network has not learnt. These samples are then added to the training set. The same number of new samples is added to the validation set. The network is re-trained using this augmented training set and the validating process is repeated. Initially, 20 samples were used as training set and 20 other samples were used as validating set. After a number of rounds of training and validating, the size of training set was increased to 40 when the trained network performs well on the validation set of 20.
Y.F. Bang
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et al. / Computers in Industry 32 (1996) 95-1 I3
The cost estimates and their relative deviations from the actual costs of the training and validating samples are shown in Tables 3 and 4. The final sumsquared error of the training samples is 0.0504. The weights and biases of the trained neural network are shown in Table 7, Table 8, and Table 9, respectively.
examples) were used in a regression analysis. The regression analysis was performed using the SPSS for Windows Release 6.0 [18] on a PC. The following linear equation was used: Estimated cost = (Y+ E ( pi X CFA( i)). i=
1
(22)
20 testing samples were then used to test the performance of the linear regression model obtained. The values of the criteria for measuring the performance of both the neural network and the linear regression equation on the training, validating, and testing samples, i.e., the average relative deviation and the maximum relative deviation, are summarised in Table 10. It can be seen that for both the 60 training and validating samples and the 20 testing samples, the neural network outperformed the linear regression model. Neither the company’s current method nor the neural network model can effectively deal with products with cost-related features beyond the training data. These products can only be added to the training dam after their production and the neural network is then re-trained to accommodate the products with such features in the future.
The
7. Testing results and discussions In order to test the capability of the trained neural network, the remai.ning 20 samples were used as the testing set which lhad not been used in the training and validation process. The cost estimates and relative deviations of the testing samples are shown in Table 5. The company’s current method utilises a linear function. The factors considered include only 16 out of the 32 cost-related features incorporated by the neural network m’odel, i.e., the area of the carton, quality of raw material, type of corrugated board construction, height of fluting, size of printing plate, number of colours., the printing area, and the number of processes. The average relative deviation of cost estimates is about 25%. The performance of the neural network model is based on the 20 testing samples. From Table 5, it can be observed that the relative cost deviations of 17 samples (or 85% of the total samples) are within i 10%. Cost estimation on 13 samples (or 65% of the total samples) were even within the relative deviation of f 5%. The maximum cost relative deviation from the actual cost is about 16% which is still considered acceptable by the company. Moreover, the results obtained by using the trained neural network for the training, validation, and testing samples are consistent (see Table 3, Table 4, and Table 5). To further evaluate the performance of the neural network, the training and validating samples (60
8. Conclusions The major drawbacks of the traditional approaches for product cost estimation include the requirement of detailed cost information from production, poor function approximation, and inability to update estimation algorithms by using actual cost data. Although the regression analysis approach can utilise new cost data for updating, the relationship between cost and input parameters may not be the same as the approximated function assumes. An
Table 10
Measures
of performance
for the neural network and the linear regression
Average relative deviat ton Maximum relative deviation
60 training and validating
samples
Neural network model
Linear regression
3.7% 12.0%
8.8% 29.0%
model 20 testing samples model
Neural network model
Linear regression
5.2% 16.0%
13.2% 36.4%
model
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Y.F. Zhang et al. / Computers in Industry 32 (19%) 95-113
algorithm using the back-propagation neural networks has been presented with a system developed for cost estimation of packaging products, which shows promise of reducing, if not eliminating these drawbacks. Using cost-related features as input. the neural network model is trained using historical cost data. The testing results suggest that the neural network approach can support good product cost estimation. A performance comparison between the neural network and a linear regression analysis based on the training, validating, and testing samples was carried out. The results showed that the neural network model outperformed the linear regression model. The advantages and limitation of this approach are given below. Advantages 1. Detailed information on processing time is not necessary. 2. There is no need to know the actual cost function. With the actual cost data, the neural network is capable of learning knowledge and performing function approximation. 3. The neural network can be updated by re-training the neural network using new cost data from time to time to cover new cost-related features and/or adjust itself to the new manufacturing environment and practices. 4. The neural network helps to improve conceptual design by evaluating the costs from different designs and process alternatives. Limitations 1. Determining the number of hidden layers and neurons in each hidden layer is a trial and error process. There is no guarantee that the selected neural network structure is the best. 2. Quantification of cost-related features is subjective thus needs extensive study on the product cost aspects. 3. Sensitivity study of the cost-related features on the product cost is difficult. Therefore, the current neural network model does not provide the means which can guide the designer towards better solutions. Further work is to incorporate more cost-related features (e.g., lot size) into the network for improving the performance of cost estimation. Sensitivity analysis of the cost-related features is another important area to be further investigated.
Acknowledgements
The authors would like to thank Mr. S.M. Loh and Mr. K. Liang from the company in providing the product information for this study and the anonymous referees for their helpful comments and suggestions on the earlier version of this paper.
References [tl G. Boothroyd,
“Product design for manufacture and assembly”, Computer-Aided Design, Vol. 26, No. 7, 1994. pp. 505-520. El MS. Hundal, “Design to cost”, in: H.R. Parsaei and W.G. Sullivan (eds.), Concurrent Engineering-Contemporary Issues and Modem Design Tools, Chapman and Hall, 1993, pp. 330-35 1. [31 R. Beale and T. Jeckson, Neural Computing: An Introduction, Adam Hilger, 1990. [41 The MathWorks Inc., Neural Network Toolbox for Use with MATLAB, 1993. bl B. Wu, “An introduction to neural networks and their appliJournal of Intelligent Manufaccations in manufacturing”, turing, Vol. 2, 1992, pp. 391-403. 161P.F. Ostwald, Engineering Cost Estimating (3rd ed.), Prentice Hall, 1992. [71 P. Dewhurst and G. Boothroyd, “Early cost estimating in product design”, Journal of Manufacturing Systems, Vol. 7, No. 3, 1988, pp. 183-191. k31G. Boothroyd and C. Reynolds, “Approximate cost estimates for typical turned products”, Journal of Manufacfuring Systems, Vol. 8, No. 3, 1989, pp. 185-193. [9] P. Dewhurst and C. Blum, “Supporting analysis for the economic assessment of diecasting in product design”, Annals of CIRP, Vol. 38, No. 1, 1989, pp. 161-164. [lo] G. Boothroyd and P. Radovanovic, “Estimating the cost of machined components during the conceptual design of a product”, Annals of CIRP, Vol. 38, No. 1, 1989, pp. 157160. [Ill C. Poli, J. Escudero and R. Fermandez, “How part design affects injection-moulding tool costs”, Machine Design, November 24, 1988, pp. 101-104. [12] H. Pacyna, A. Hillebrand and A. Rutz, Early cost estimation for castings, VDI Berichte Nr. 457, Designers Lower Manufacturing Costs, Ddsseldorf, VDI Verlag, 1982. [13] N.S. Ong, “Activity-based cost tables to support wire harInternational Journal of Production Econess design”, nomics, Vol. 29, No. 3, 1993, pp. 271-289. [14] A. Shtub and Y. Zimerman, “A neural-network-based approach for estimating the cost of assembly”, International Journal of Production Economics, Vol. 32, No. 2, 1993, pp. 189-207.
Y.F. Zhang et al./Computers [15] D.E. Rumelhart, GE. Hinton and R.J. Williams, Learning representations by back-propagation errors”, Nature, No. 323, 1986, pp. 533-536. [16] T. Khanna, Foundztions of Neural Nehvorks, Addison-Wesley, 1990. [17] R. Hecht-Nielsen, “Theory of the backpropagation neural networks”, Proceedings of the 3rd IEEE Annual International Conference on Neural Networks, Washington DC., 1989, pp. 1.593-1.626. [18] SPSS Inc., SPSSfor Windows - Release 6.0, 1989-1993. Y.F. Zhang obtained his B.Eng. (Mech. Eng.) from Shanghai Jiao Tong University in 1985 and his Ph.D. (Mech. Eng.) from the University of Bath in 1991. He joined the National University of Singapore in 1992 as a Research Scientist, and is currently a Lecturer at the Department of Mechanical and Production Engineering. His research interests are CAPP, DPMA, computer-aided injection mould design, and artificial neural network applications. He is a senior member of SME.
in Industry 32 (19%) 95-113
113
Jerry Y.H. Fuh received his Ph.D. (Mech. Eng.) from the University of California at Los Angeles (UCLA) in 1992. Before he joined the National University of Singapore (NUS) in 1993, he was an Engineering System Engineer at the Unigraphics Division of Electronics Data Systems (EDS) in the US. Currently, he is a Senior Lecturer at the Department of Mechanical and Production Engineering. His research interests are rapid prototype manufacturing, plastic injection mould design, and fixture planning analysis. He is a member of ASME and SME, a Certified Manufacturing Engineer (CMfgE) from CASA/SME and a registered Professional Engineer in manufacturing engineering from California, USA. W.T. Chan obtained his B.Eng. (Mech. Eng.) from the National University of Singapore in 1995. He is a Project Engineer at the Singapore Defence Materials Organisation.
Computers
Feature-based
in Industry 32 (1996) 95-113
cost estimation
for packaging networks
products using neural
Y.F. Zhang *, J.Y.H. Fuh, W.T. Chan Department
of Mecka.~ical
and Production
Engineering,
National Uniuersity Singapore
Received 25 April 1995; revised 29 February
of Singapore,
IO Kent Ridge Crescent,
Singapore
119260,
1996; accepted 26 June 1996
Abstract Cost estimation plays an important role in the product development cycle. For instance, a proper cost estimation can help designers make good trade-off decisions regarding product structures, material, and manufacturing processes. In this paper, a feature-based product cost estimation using back-propagation neural networks is proposed. A system using this approach has been successfully developed for estimating the cost of packaging products. The cost-related features in both design and manufacturing aspects were extracted and quantified according to their cost drivers. The correlation between the cost-related features and the estimated costs of the product was obtained by training and validating a back-propagation neural network based on 60 existing products with their designs, process routings, and actual cost data. To illustrate, the testing results of the trained neural network based on 20 actual products are presented. The performances of the neural network are compared to those of the company’s method and a linear regression model. The results show that the neural network model outperformed both the other methods in respect to performance measures such as average relative deviation and maximum relative deviation. Keywords:
Back-propagation
neural networks;
Cost estimation:
Cost-related
1. Introduction
Cost estimation has traditionally been the province of manufacturing engineers. Typically, the product cost is derived frolm the summation of the various cost components such as material, machine hours, direct labour, administration, and engineering costs. However, during the early stage of product engineering, it is always difficult to obtain these cost elements accurately. Moreover, to allow the designer to
S Corresponding
author. Email: [email protected]
features; Packaging
products
make a good trade-off decision [1,2], product cost must be identified early, i.e., during the designing stage where a large proportion of the manufacturing cost is determined. The work reported in this paper is a result of an industrial project with a local packaging company. A new cost estimation method for packaging products before the actual production run has been established to replace the company’s current method which uses a linear function of a limited number of factors. Many cost estimation techniques have been developed for different products based on various cost drivers. The explicit cost drivers, such as material cost, can be obtained directly; while the implicit
0166.3615/96/$15.00 ‘Copyright 0 1996 Elsevier Science B.V. All rights reserved. PII SO166-3615(96)00059-O
96
Y.F. Zhmg et al./Computers
ones, such as design complexity-related cost, have to be derived through analysis of historical cost data. Therefore, the key to product cost estimation is how to use the historical cost data to understand the implicit cost drivers. This paper discusses how the neural network approach can be applied to product cost estimation. A new approach, feature-based cost estimation, is proposed for packaging products based on the assumption that all the design and process requirements of a product contribute to the final product cost. Since it is difficult to obtain an explicit relationship between these requirements and the final cost, neural networks are selected for approximating this relationship as they can be trained to perform tasks by presenting them with examples rather than specifying the procedures [3-51. The approach involves (1) cost-related feature extraction and quantification, and (2) the identification of the cost estimation function using historical cost data. The cost-related features were classified based on material, printing plates, printing, and production processes. A back-propagation neural network was then constructed and trained using historical product cost data to approximate the relationships between the cost-related features and the product cost. For the packaging products in the company, 32 cost-related features have been identified and quantified. The network was trained and validated using 60 samples, The testing results on another 20 samples are very promising. In this paper, some related work is reviewed in Section 2. The back-propagation neural network and its application in cost estimation are introduced in Section 3. The cost-related feature classification and quantification of packaging products are described in Section 4. The neural network architecture selection and the training procedures are presented in Section 5 and Section 6. Testing results and discussions are given in Section 7, and conclusions are presented in Section 8.
2. Previous work on product cost estimation Much literature has been published on product cost estimation. Most approaches can be broadly classified into the following categories: (1) traditional detailed-breakdown cost estimation, (2) sim-
in Industry 3.2 (1996) 95-113
plified breakdown cost estimation, (3) group technology (GT)-based cost estimation, (4) regression-based cost estimation, and (5) activity-based cost estimation. In the traditional detailed-breakdown approach, the total product cost is typically the summation of all the cost incurred during the production cycle of the product, such as material cost, manufacturing cost, labour cost, and overheads. The calculation, however, requires detailed information of product design, process routing, machining time, etc. The accuracy of the cost estimation depends on the availability of the required information and the variance of the actual activities. A comprehensive description of this method is given by Ostwald [6]. The simplified breakdown methods are mainly developed for cost estimation in the early design stage when typically a process plan is not available. The method is based on the assumption that the manufacturing process will eventually take place under ideal processing conditions. A shortcoming of this approach is that it is difficult to update the cost estimate based on actual cost information. The work of Dewhurst and Boothroyd [7] on early cost estimating models for machining and injection moulding components, of Boothroyd and Reynolds [8] on a cost model for typical turned parts, of Dewhurst and Blum [9] on a cost model for die-casting, and of Boothroyd and Radovanovic [ 101 on a cost model for machined components are the representative examples of this category. The GT-based approach is based on the similarity principle. It typically uses a basic cost value while taking into account the effects of variable cost factors such as size and complexity. Linear relationships between the final cost and the variable cost factors are assumed. Among the examples of this category, Hundal [2] described a cost model for predicting costs of products made in different size ranges, and Poli et al. [l 11 developed a tool to estimate injection-moulding tool costs based on a six-digit code of a part design. Look-up tables were used to extract the cost drivers based on the code. The regression approach tries to find the dependence relationships between costs and product characteristics such as size and material. The coefficients and exponents are derived through the regression analysis of historical cost data. The problem with
Y.F. Zhang et al./Computers in Industry 32 (1996) 95-113
this approach is that the nature of the non-linearity relationship between the cost and the product’s characteristics is not known and hence has to be assumed. The work of Pscyna et al. [12] on a cost model for grey iron castings is an example of this category. Activity-based costing (ABC) is a method for accumulating product costs by determining all cost drivers associated with the activities required to produce the product. Based on historical, observed, or estimated data, the cost per unit of the activity’s output is calculated. The estimated cost for a new product can be obt,ained according to the product’s consumption of these activities. However, it is difficult to obtain the cost per unit of the activity’s output. The work of Ong 1131 on cost tables to support wire harness design in the assembly of electrical connections is, an example of this category.
Little research has been done on product cost estimation using neural networks. Shtub and Zimerman [ 141 applied back-propagation neural networks to estimate the cost of assembly systems. The emphasis was given on the comparison of performance between the neural network model and the regression model. Our research contributes to the neural network application in product cost estimation of another domain. Moreover, the concept of cost-related feature based on design and manufacturing of products is introduced which could be used for cost estimation of other product domains. In this paper, we demonstrate the potential of neural networks in cost estimation by using a backpropagation neural network to obtain the relationship between the product cost and cost-related features of packaging products through supervised learning based on historical cost data. The performance of the
Product coding r-
-
+Neurons in each layer I
Fhxduct
,
4 neural network specification & supervised training
“Knowledge” base
definition & quantification of cost-related features
(a) Preparatory stage
+Neurcno in each layer
“Knowledge” base
(b) Production stage Fig. 1. Product cost estimation:
97
A back-propagation
neural network approach.
98
Y. F. Zhang et al. / Computers in Industry 32 (1996) 95-113
network is compared with both the performance of the company’s method and with a linear regression model.
3. Back-propagation neural network approach for cost estimation Artificial neural networks (ANN) are an information processing technique that simulate biological neurons using computers. In engineering fields, one of the most important application of ANN is modelling a system with an unknown input-output relation. Usually, the accurate information of the system is not available and we can only utilise a number of examples observed from the actual system, called training set. Given a fixed architecture of the networks, learning is carried out through modifying the parameters by the stochastic gradient decent method which eventually minimises a certain loss function. Among various ANN architectures, back-propagation neural network [3,15-171 is a common method for learning of multilayered perceptrons with sigmoidal functions. Since an explicit function for product cost estimation without detailed information on machining time is difficult to obtain, it provides a potential ground for back-propagation neural network applications. To utilise a neural network for product cost estimation, a framework which outlines the procedures is created. As shown in Fig. 1, the proposed neural network approach for product cost estimation has two operational stages: a preparatory stage and a production stage. The preparatory stage is required to construct and train a neural network by using existing products and their actual cost data. During this stage, the product domain is defined, and all the cost-related features of the product are identified. A cost-related feature quantification scheme is established in which different states of a feature are quantified according to their influence on the product cost. Next, existing products are coded based on the cost-related feature definition and quantification scheme to form cost-related feature vectors. A backpropagation neural network architecture is then selected which is often done by trail and error. The training is started by assigning random weights for each interconnection and using the cost-related fea-
ture vectors as inputs and the products’ actual cost as the target output. The training stops when the sumsquared error between the actual outputs and the target outputs of all the training samples reaches an error goal given by the trainer. The network is then tested using samples which have not been used in the training process. If the testing results are not acceptable, testing samples with large deviations are added to the training samples and the network is re-trained. Otherwise, training is completed. The final weights for all the interconnections is then stored in a weight matrix and the neural network is ready for cost estimation of new products. The production stage occurs when the neural network is in use for cost estimation of a new product. During this stage, an incoming product is first coded to obtain its cost-related feature vector according to its design specifications and manufacturing requirement. The vector is then input to the network. By using the existing weight matrix obtained from training, an estimated cost of this product can be obtained. It is worth noting that the preparatory stage can be repeated if the estimated cost for a new product differs too much from the actual cost obtained after production. This can be achieved by adding the new product to the existing training samples and re-training the network accordingly.
4. Cost-related features in the packaging product A cost-related feature of a product can be generally defined as a characteristic associated with the product, which makes a certain contribution to the total cost of the product. It can be either a design specification or a processing requirement. For example, the material type is a cost-related feature in the design domain, while a particular process required for the product is a cost-related feature in the process domain. Also, cost-related features should be product-type dependent. It can be broadly classified into categories, such as machined products, casting products, injection-moulded products, packaging products. In the following sections, the packaging products and processes of the company are introduced and the cost-related features and their quantification are described.
Y. F. Zhang ef al. / Computers in Industry 32 11996) 95-l I3 Fluting
ABC -W? -
ABC -rY9-
I
I
I
First End
Second Front
Fig. 2. The layout of an unfolded RX
4.1. Packaging prohcts
Single
lace
wall
Double
Triple
wall
Wdl
Fig. 3. Types of corrugated
design.
and processes
and quantification
Single
Second End
The type of packaging products studied is the regular slotted cari’on (RSC). An example of an unfolded RSC design is given in Fig. 2. The raw material for the packaging products is paper which has to be converted into corrugated boards before it can be used for manufacturing the carton boxes. Most of the cartons consist of some printing which may be wording or iogos. The RSC products require all or some of the following manufacturing processes: 1. Creating artwork!; and negatives. 2. Making printing plates. 3. Making prototypes. 4. Creating product master cards. (A master card contains the specification of the specification of a product.) 5. Producing corrugated boards. 6. Printing. 7. Stitching/gluing. 8. Packing. 4.2. Classification features
Liner
rw ‘=x%0=
rnxsao~
First Front
99
board construction.
cost-related features were extracted and classified under the aspects of material, printing plates, printings, and manufacturing processes. Since these features have to be used as input to a neural network and a neural network can only perform arithmetic operations, each cost-related feature of a product had to be assigned with a value between 0 and 1. This value is called the cost-related feature attribute (CFA). This process is called the quantification of cost-related features. The magnitude of a CFA of a feature is an indication of the relative cost influence among the various possible states of that feature. A state which contributes to a higher cost will be assigned a higher value. The details of the cost-related features and their quantification are described in the following sections. 4.2.1. Based on material (1) Type of corrugated board construction (CFAI) There are four types of corrugated board construction: single face, single wall, double wall, and triple wall (see Fig. 3):
of cost-related
0.3
CFAI = Based on our study of the RSC product design and manufacturing processes in the company, 32
I ;.;
0:s
single face single wall
(1)
double wall triple wall.
Table 1 Quantification
of liner quality (x = 2, 3, 4, 5)
Type
K4
K5
K6
K7
K8
K9
w5
W6
WI
W8
CFA x
0.52
0.5
0.42
0.44
0.46
0.48
0.54
0.56
0.58
0.6
Type
T4
T5
T6
T7
Ml
M2
M3
M4
M5
M6
CFA x
0.34
0.36
0.38
0.4
0.18
0.2
0.22
0.24
0.26
0.28
Y.F. Zhang et al. / Computers in Industry 32 (1996) 95-I 13
100
r-l
ABC
Fluting’s height
reversecut
normal cut
Fig. 4. The fluting’s height.
Fig. 5. Types of cut for rubber plate.
(2) Quality of raw material (CFA2, CFA3, CFA4, CFAS, CFA6, CFA7, CFA8) The quality of the material is defined as the weight per unit area of the paper. Based on the type of corrugated board construction, CFA2, CFA3, CFA4, CFAS refer to the quality of the first liner, the second liner, the third liner, and the fourth liner, respectively. The quantification of liner quality is given in Table 1. Likewise, CFA6, CFA7, CFA8 refer to the quality for the first fluting, the second fluting, and the third fluting, respectively. The detailed quantification of fluting quality is given in Table 2. (3) Height of fluting (CFA9, CFAlO, CFAll) The height of the fluting is defined as the distance between two liners as shown in Fig. 4. There are 3 fluting heights, i.e., 0.1,0.15, and 0.2. CFA9, CFAlO, CFAll refer to the height of the first fluting, the second fluting, and the third fluting, respectively. (4) Area of the carton (CFA12)
(2) Material of printing plates (CFA14) The types of material used for printing plates are rubber and photopolymer:
area of carton (m* ) CFA12 =
(2)
’
20
CFA14 =
(3)
both.
(3) Types of cut for the rubber plate (CFA15) The cut for the rubber plate can be either nomal or reverse (see Fig. 5). The normal type has wording or logos printed with colours. A reverse cut on the other hand prints the surroundings with colours leaving the wording or logos uncoloured. In general, reverse cutting is more difficult and hence costs more.
CFA15 =
0.2 0.3 i 0.25
normal reverse both.
(4)
(4) Size of the printing plates (CFA16, CFA17) The size of the printing plates is defined as the area of the rubber or photopolymer required: rubber area (m2)
where 20 m2 is the largest area of a product which can be produced by the company.
CFA16 =
4.2.2. Based on printing plates (1) Availability of printings (CFA13) Products with printings cost more than the ones without printing features. If there are printings CFA13 equals 0.5.
CFA17 =
Table 2 Quantification
rubber photopolymer
0.2 0.6 i 0.4
(5)
’
2.5 photopolymer
area (m* ) t
0.7
(6)
where 2.5 m2 and 0.7 m2 are the largest printing plates size which can be produced for rubber and photopolymer, respectively.
of fluting quality (x = 6, 7, 8)
Type
F5
F6
Ml
M2
M3
M4
M5
M6
T4
T5
T6
n
CFAx
0.3
0.32
0.18
0.2
0.22
0.24
0.26
0.28
0.34
0.36
0.38
0.4
Y.F. Zhang et al./Computers
4.2.3. Based on printings ( 1) Number of C:O~OU~S (CFA 18) The larger the number of colours, the more set-up time is required:
\ 0.7
No No No No
of of of of
colours colours colours colours
= = = >
1 2 3 3.
P/A
< 30%
30% I P/A 50 I P/A \0.8
P/A
< 50 < 70%
(8)
2 70%.
(3) Effects of colour trapping (CFA20, CFA21, CFA22) Colour trapping refers to a situation where the printing consists of two or more different colours that are printed in contact with one another (see Fig. 6). The problem of colour trapping arises from the misalignment of the printing plates. As a result, more samples are usually required during the process of aligning the printing plates. CFA20 is 0.5 if there is colour trapping. CFA21 is based on the number of printing plates (NPP) for different colour trapping and CFA22 is based on the type of colour trapping. 0.3 CFA21 = 0.4 i 0.5
NPP=2 NPP=3 NPP=4.
0.3 CFA22 = 0.6 i 0.5
between dark and light between light only
101
32 (1996) 95-113 colour 2
colour 1
ur3
(7)
(2) The printed area (CFA19) The printed area determines the amount of paint required directly. Moreover, it affects the quality of the print as well. Quantification is based on the printed area percentage out of the total area (P/A): (0.2
in Mushy
Fig. 6. An example of colour trapping.
4.2.4. Based on production processes (1) Number of processes (CFA24) In general, the maximum number required for a product is less than 10. number of processes CFA24 =
10
(11)
.
(2) Requirement for reverse corrugated boards (CFA25) The use of reverse corrugated boards requires additional processes and hence additional cost occurs. CFA25 equals 0.5 if reverse corrugated board is required. (3) Requirement of printing machine (CFA26) If the product needs a printing machine, CFA26 equals 0.5. (4) Stitching (CFA27, CFA28) If stitching is required, CFA27 equals 0.5. Since the distance between each stitch is fixed at 50 mm, the number of stitching required depends on the working height (see Fig. 7). working height (mm) CFA28 =
(9)
( 10)
of processes
The maximum (5) Gluing Gluing can a machine or
50 x 25
(12)
’
number of stitches is 25. (CFA29, CFA30) be carried out either automatically on manually. In general, manual gluing
both.
(4) Availability of printing inks (CFA23) If the required ink is not available, it has to be purchased and often there will be some left over after the printing is done. Extra cost is therefore incurred. CFA23 equals 0.2 if required printing inks are available; otherwise CFA23 equals 0.4.
I Fig. 7. The working height.
working height
0.52
0
0.2
0
0.1
0
0.089
0.5
0.2
0.2
CFA4
CFA6
CFA7
CFA9
CFAIO
CFAIZ
CFAl3
CFAl4
CFAI
0.15
0.1
0.2
0.2
0.5
I.812
1.80,
0.2
1.1801.102
1.1651.104
0.4
0
0
0
0.2
0.2
0.5
0
0
0.5
0.1
0.2
I.563
1.579
I .O% - I .3% 0.2%
CFA19
CFAZO
CFAZI
CFA22
CFA23
CFA24
CFA26
CFA27
CFA28
CFA29
CFA31
CFA32
Cost(T)
Cost(E)
RDa
0.5
0.5
0
0
0.5
0.1
0.2
0.28
0
0.5
0.4
0.2
0
0.2
0
0
0
0.4
0.2
0.2
0
0
0
0.4
0.2
- 0.6%
0.2
0.1
0.5
0
0
0.5
0.2
0
0
0
0
0
0
0
0.2
0
CFAID
0
0
0
0
0
0
0.084
0.15
0.1
0.2
0.2
0.52
-7%
6
0.15
0.1
0.2
0.2
0.42
0.18
0.42
0.6
0.2
0.2
0.5
3.8%
0.2%
1.920
0.424.418
0.2
0
0
0
0
0
0. I
0
0
0
0
0
0
I.917
0.2
0
0.28
0.5
0.5
0.3
0.2
0.2
0.2
0
0
0
0
0
0.454.366
0.2
0.10
0.5
0
0
0.5
0.2
0.2
00
00
00
0.2
0.2
00
0.022X074
0.2
0.2
0.5
0. IO0
0.15
0. I
0.2
0.2
0.52
0.18
0.52
0.6
1
Es of training
0.02Xl.057
0
0.1
0
0.2
0
0.5
0.5
0.52
0.18
0.4
CFA17
0.1260.145
0.25
0.2
0.5
costs
0.6
0.126
0.2
0.2
0.5
0.0650.042
0
0.1
0
0.2
0
0.18
0.5
0.6
CFA16
5
0.5
0.5
CFA3
0.52
0.4
0.4
CFAI CFA2
4
5
3
1
2
actual costs Ts, and estimated
The CFAs,
Table 3
0.795
0.413 2.8%
0.773
02
0.1
0.5
0
0
0.5
0.2
0.4
0.3
0.3
0.5
0.8
0.3
0
0.284
0.25
0.2
0.5
0.439
0.2
0.1
0.5
0
0
0.5
0.2
0
0
0
0
0
0
0
0
0
0
0
0.035
0.026
0. I
0
0.2
0
0.52
0.4 0.54
0
9
0
0.1
0
0.2
0
0.52
0.4 0.52
-5.9%
8
samples 10
11 12
-1.28-128
- 2.6%
0.952
0.367
0.41 I
0.2
0.1
0.5
0
0
0.5
0.3
0.4
0.3
0.3
0.5
0.2
0.4
0.288
0
0
0.6
0.5
0.042
0
0.1
0
0.2
0
0.42
0.4 0.54
0.4160.977
0 2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.6
0.2
0.44
0
0
0.6
0.5
0.023
0
0.1
0
0.2
0
0.5
0.4 0.5
0.416
0.2
0.1
0.5
0
0
0.5
0.2
0.4
0
0
0
0.6
0.2
0.44
0
0
0.6
0.5
0.023
0
0.1
0
0.2
0
0.5
0.4 0.5
0.2
0.4
0.5
- 2.7%
7.7%
0.7900.586
0.596
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.2
0.8120.544
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.3
0.612
0.2
0.1
0.5
0
0
0.5
0.2
0
0
0
0.1
0
0.2
0
0.0720.016 0.6
0
0.5 0.5
0.1060.007
0.25
0.4
0.5
0 0
15 0.4
0.0410.030
0
0.15
0
0.2
0
0.42
0.42
0.4
0.2
14
0
0
0
0
0
0
0.028
0.15
0.1
0.2
0.2
0.5
0.18
0.5
0.6
- 2.6%
13
16 17
0
0.15
0
0.2
0
0.42
0.42
0.4
0.2
0.2
0.5
I.086
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.6
0.2
0.1%
1.8%
1.4621.106
I.461
0.2
0
0
0.24
0.5
0.5
0.3
0.2
0.3
0.3
0.5
0.4
0.3
0.2120
0.0790.216
0.25
0.4
0.5
0.0490.055
0.15
0.1
0.2
0.2
0.42
0.18
0.42
0 6
I. 120
I.174
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.6
0.2
0.08 I
0.123
0.25
0.4
0.5
0.041
0.15
0.1
0.2
0.2
0.44
0.18
0.44
0.6
-4.6%
18
0.409
0.429
0.2
0.1
0.5
0
0
0.5
0.2
0.4
0
0
0
0.6
0.2
0.12
0
0
0.6
0.5
0.019
0
0.1
0
0.2
0
0.42
0.54
0.4
-4.7%
19
20
0.5%
1.658
I.649
0.2
0
0
0.28
0.5
0.5
0.3
0.4
0.3
0.4
0.5
0.6
0.4
0
0.167
0.2
0.2
0.5
0.052
0.15
0.1
0.2
0.2
0.42
0.18
0.54
0.6
0.469
0.9580.505
0
0.3
0.5
0.5
0.12
0
0
0.2
1.438
I.432
CFA23
CFA24
CFA26
CFA27
CFA28
CFA29
CFA3l
CFA32
Cost(T)
Cost(E)
* RD:
Relative
- 0.4%
0.9750.487
0
CFA22
RD=
0.447
0.2
0
0
0
0.5
0.1
0
0
0.5
0.1
deviation.
CFA5
I .7% 3.7%
0.2
0.5
0.2
0.2
0.5
0.2
0
0
0
0.2
0
0
0
0.2
0.2
= 0, CFA8
4.9%
0 0.2
0.1
0.5
0
0.5
0.2
0.2
0
0
0
0.6
0.2
0
= 0. CPA
-3.3%11.2%
1.5700.981
1.6230.882
0
0.1
0.20
0.5
0.5
0.3
0.2
0
0
0.5
0
0
0.5
0.3
0.2
0
0
0
0
0.2
0.4 0
0.3 0.6
0.2
0.6
0.4
0.2
0
0
0
0
CFAZI
0
CFAl7
0.2
0.2
0.5
0.0490.1
0.25
0.067
0.2
0.2
0.5
0.0270.047
0.2
0.2
CFAZO
0
CFA16
0.25
0.2
0.5
0
0
CFAIS
0.2
0.5
0
0
CFA14
0.5
0.15
0.1
0.2
0.2
0.5
0.18
0.6 0.5
26
0.0600.038
0.15
0.019
0
0.2
0.2
0.42
0.18
0.6 0.42
0.1
25
0.1
0
0.2
0
0.5
0.4 0.5
24
CFAl8
0
CFA13
0
0.1
0
0.2
0
0.5
0.0430.028
0.15
0.1
0.2
23
0.4 0.5
CFAl9
0.15
0.1
CFA9
0.058
0.2
CFA7
CFAIZ
0.2
0.2
CFA6
CFAIO
05
0.42
CFA4
0.18
0.18
CFA3
0.6 0.5
22
0.6 0.42
21
CFAI CFAZ
2.1%
1.625
I.591
0.2
0.1
0.5
0
0
0.5
0.2
0.4
0
0
0
0.2
0.2
0
0.059
0.2 0.059
0 0
0.971 - 1.5%
2.696 0.4%
-6%
= 0, and CFA30
0.986
0.2
0.1
2.685
0.2
0
0.5
0
0.36 0
0
0.5
0.5 0.5
0.2
0.3
0
0 0.4
0
0 0
0
0
0.2 0.2
0
0
0
0.2
0
0
0.5
0
0.15 0
0.1
0.1 0.054
0
0.2
0.119
0.2
0.2
0.741 5.4%
0.632 5.2%
0.703
0.2
0.1
0.5
0.7%0.5%
1.0640.826
1.0570.822
0.2
0
0.2
0
0.1
0.2
0.5
0.5
0.3
0
0
0
0
0
0
0
0
0
0.5
0
0
0
0.2
0.2
0
0
0
0.6
0.3
0. I280
0.2630
0.25
0.4
0.5
0.5
0
0
0.1
0
0.2
0
0.52
0.52
0.4
35
0.0510.043
0
0.15
0
0.5
0.601
0.2
0.1
0.5
0
0
0.5
0.2
0
0.2 0.2
0
0
0
0
0
0 0
0.2 0.2 0
0
0
0
0
0
0.041
0
0.1
0
0.2
0
0.2
0.42
0
0.42
0.5 0.5
0.4
34
0.4
33
0
0.037
0.2
0.2
0.5
0.031
0
0.1
0
0.2
0
0.42
0.42
0.4
32
= 0 for all samples
0.9%
0.916
0.908
1.323 I.244
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.2
0.2
0
0.062
0.2
0.2
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.4
0.2
0
0.003
0.2
0.2
0.5
0.040
0.5
0.15
0.1
0.2
0.2
0.52
0.18
0.52
0.6
0.057
31
0.15
0.1
0.2
0.2
0.5
0.18
0.5
0.18 0
0.5
0.5
0.5
0.6
30
0.4
29
0.6 0.5
28
I I = 0, CFA25
I I
0.2
0.2
0.5
0.070
0.15
0. I
0.2
0.2
0.5
0.18
0.6 0.5
27
0.15
0.1
0.2
0.2
0.5
0.18
0.42
0.6
37
0.2
0.1
0.5
0
0
0.5
0.2
0
0
0
0
0
0
0
0
0
0
~4.4%
1.8%
1.5581.439
1.6301.414
0.2
0
0
0.2
0.5
0.5
0.3
0.2
0
0
0
0.2
0.3
0
0.0720
0.25
0.2
0.5
0.0600.061
0.15
0.1
0.2
0.2
0.42
0.18
0.42
0.6
36
-
0.339 -2.0%
1.594
0.2
0.1
11%
0.729
0.656
0.2
0
0
0.5
0.5
0.3
0.2
0
0
0
0.4
0.2
0
0.619
0.25
0.2
0.5
0.026
0.15
0.1
0.2
0.2
0.4
0.18
0.5
0.346
1.1%
0.6 0.5
0.2
40
0
0
0.5
0.3
0.2
0
0
0
0.2
I.612
0.2
0.1
0.s
0
0
0.5
0.3
0.2
0
0
0
0.4
0.2
0
0 0.3
0.004
0.2
0.2
0.5
0.015
0
0.1
0
0.2
0
0.52
0.52
0.4
39
0.069
0.25
0.2
0.5
0.080
0
0.1
0
0.2
0
0.5
0.5
0.4
38
4
0.5
0.2
0.2
0.1
0.15
0.043
0.5
0.2
0.2
0.108
0
0.2
0.4
0.52
0.2
0.2
0.1
0.15
0.057
0.5
0.2
0.2
0.126
0
0.2
0.4
0
0
CPA4
CFA6
CFA7
CFA9
CFAIO
CFAIZ
CFAI3
CFAI4
CFAI.5
CFA16
CFAI7
CFAI8
CFA19
CFAZO
CFA2
0.5
0.24
0
0
0.2
0.5
0.24
0
0
0.2
I.471
1.372
CFA27
CFA28
CFA29
CFA31
CFA32
Cast(T)
Cost(E)
RD:
Relative
-6.7%
1.142
0.5
0.5
CFA26
RD’
I.146
0.3
0.3
CFA5
6.9%
1.066
0.2
0.1
0.5
0
0
0.5
0.2
=
-0.3%
0.2
0.1
0.5
0
0
0.5
0.2
0
0
CFA24
0.2
0
0.2
0
0.2
0
CFAB
0.843
0.2
0.1
0.5
0
0
0.5
0.3
0.4
0.3
0.3
0.5
0
0
0
0.2
0
0.4
0.288
0
0
0
0
0.6
0.5
=
0.9%
0.5% 1
0.773
0.769
0.2
0.1
0.5
0
0
0.5
0.2
0.4
0
0
0
0.2
0.2
0
0.059
0.2
0.2
0.5
CFAI
2.422
0.2
0.1
0.5
0
0
0.5
0.2
0.4
0
0
0
0.2
0.2
0
0.059
0.2
0.2
0.5
0.041
0
0.1
0
0.2
0
0.5
0.5
0.4
1
-1.1%
0.737
0.4
samples
9
0, CFA25
0.616
0.2
0.1
0.5
0
0
0.5
0.2
0
0
0
0
0
0
0
0
0
0
0
0.036
0
0. I
0
0.2
0
0.5
0.5
0.4
a
Es of vahdation
0.108
0.15
0.1
0.2
0.2
0.5
0.18
0.5
0.6
6
costs
0.035
0
0.1
0
0.2
0
0.42
0.54
0.4
5
0
0
0
0
0.052
0.15
0.1
0.2
0.2
0.5
0.18
0.5
0.6
CFA23
0
0
0.2
0.3
0
0.039
0.2
0.2
0.5
0.056
0
0.1
0
0.2
0
0.5
0.5
4
CFA22
I
0.18
0.18
CFA3
0
0.5
0.52
CFA2
0.4
3
costs Ts, and estimated
2
0.6
actual
0.6
1
CFA I
The CFAs,
Table
0. and
-6.6%
0.607
0.650
0.4
10
= 0
0.914
0.4
11
-4.3%
0.598
0.625
0
0.5
0.4
13
all samples.
-2.5%
0.895
0.6
12
- 1.6%
0.839
0.5
0.5
0.479
0.4
15
0.4
14
0.980
0.900
0.42
0.42
0.6
16
-9.1%
I.159
0.42
0.4
17
-0.2%
1.203
0.5
0.4
18
I.147
I.058
0.52
0.4
19
0.805
0.1
0.5
0
0.5
0.2
0
0
0.2
0.221
0
0.6
0.5
0
0.15
0.2
0
0.42
0.4
- 12%
20
0.52 0
0.2
0.2
0.1
0.15 0
0.0630.080
0.5
0.2
0.2
0.0750.059
0
0.2
0.2
0.2
0.2
0
0.5
0.1
0.2
0
0.2
0
0.15
0
0.036
0.5
0.6
0
0
0.096
0.2
0.2
0
0
0
0.2
0.2
0.5
0
0
0.5
0.1
0.2
0.701
0.648
CFA4
CFA6
CFAl
CFA9
CFAIO
CFAl2
CFA13
CFAl4
CFAIS
CFAl6
CFAI7
CFAIS
CFA19
CFAZO
CFAZI
CFA22
CFA23
CFA24
CFA26
CFA27
CFA28
CFA29
CFA3l
CFA32
Cost(T)
Cost(E)
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
- 8.0%
- 7.5% - 3.2% 7.0%
1.475
I.489 1.740
1.750
0.2
0
0
0.32
0.5
0.5
0.3
0.4
0.3
0.4
0.5
0.4
0.4
0.466
0
0
0.6
0.5
0.057
0.15
0.1
0.2
0.2
0.42
0.18
0.54
0.6
3.3% -0.6%
I.523
0.2
0.2
I .370
0
6
0.1
0.1
0.641
0.566
1.4%
0.573
0.565
0.2
0.5
0.5 0.2
0
0
0.5
0.2
0.2
0
0
0
0
0.5
0.2
0.2
0
0
0
0.6
0.6 0
0.2
0.2
0.062
0.25
0.2
0.5
0.03 I
0
0.1
0
0.2
0
0.5
0.5
0.4
0
9
0
0.121
0.25
0.2
0.5
0.031
0
0.1
0
0.2
0
0.5
0.5
0.4
8
3.8% 13.3%
I.055
I.016
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.6
0.3
0.1
0.166
0.25
0.4
0.5
0.051
0
0.15
0
0.2
0
0.42
0.42
0.4
7
0.804
0.955
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.2
0.2
0
O.OSi
0.2
0.2
0.5
0.044
0
0.15
0
0.2
0
0.44
0.44
0.4
- 16%
10
1.283
1.327
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.2
0.2
1.3% -3.3%
1.037
1.024
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.4
0.2
0
0.003
0.077 0
0.25
0.2
0.5
0.057
0.15
0.1
0.2
0.2
0.42
0.18
0.42
0.6
12
0.2
0.2
0.5
0.055
0
0.15
0
0.2
0
0.42
0.42
0.4
11
and cost relative deviatmns of testing samples
0
0
0.24
0.28
0
0.5
0.5
0.4
0.4 0.5
0.3
0.3
0.5
0.4
0.4
0.3
0.5
0.5
0.3
0.4
0.4
0
0.191
0.2
0.2
0.5
0.047
0.15
0.1
0.2
0.2
0.42
0.18
0.54
0.6
0.4
0.4
0.188
0.276
0.25
0.4
0.5
0.048
0.15
0.1
0.2
0.2
0.42
0.18
I .423 I.522
1.4701.422
0
0.5
0
0
0
0.4
0.3
0
0.25
0.2
0.5
0.1
0
0.2
0.5
0.54
0.6
5
a RD: Relative deviation. CFAS = 0. CFAS = 0, CFAI I = 0, CFA2S = 0, and CFA30 = 0 for all samples
RDa
0.18
0.42
0.5
CFA3
0.52
0.42
0.4
*
estimated costs Es,
CFA2
0.6
3
0.4
2
CFA I
1
The CFAs. actual costs Ts,
Table 5
14
0.15
0.1
0.2
0.2
0.5
0.18
0.44
0.6
0.2
0.2
0.5
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.2
0.2
0
-8.6%
1.6%
I.360 I .054
I.488 I.037
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.2
0.2
0
0.0620.061
0.2
0.2
0.5
0.0560.042
0.15
0.1
0.2
0.2
0.5
0.18
0.44
0.6
13
16
0.5
0.5
0.4
0
0.1
0
0.2
0.2
0.1
0.5
0.5
0.2
0.4
0.2
0.2
0.049
0.2
0.2
0.5
-2.7q4.41
0.9840.917
1.0110.878
0.2
0.1
0.5
00
00
0.5
0.2
0
00
00
00
0
0
00
0
0
0
0
0.0460.051
0.15
0.1
0.2
0.2
0.52 0
0.18
0.52
0.6
15 17
2.0%
2.422
2.374
0.2
0.1
0.5
0
0
0.5
0.2
0.4
0
0
0
0.2
0.2
0
0.059
0.2
0.2
0.5
0.108
0.15
0.1
0.2
0.2
0.5
0.18
0.5
0.6
18
4.6%
1.447
1.384
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.4
0.2
0.235
0.098
0.2
0.4
0.5
0.060
0.15
0.1
0.2
0.2
0.5
0.18
0.5
0.6
-11%
19
0.544
0.608
0.2
0.1
0.5
0
0
0.5
0.2
0.2
0
0
0
0.2
0.2
0.2
0
0
0.6
0.5
0.026
0
0.2
0
0.26
0
0.44
0.44
0.4
- 1.3%
I.544
I .564
0.2
0
0
0.28
0.5
0.5
0.3
0.2
0
0
0
0.4
0.2
0
0.104
0.2
0.2
0.5
0.083
0
0.1
0
0.2
0
0.5
0.5
0.4
20
Y.F. Zhang
106
et al./Compurers in Industry 32 (1996) 95-113
incurs a higher cost. If gluing is required, equals 0.5. If manual gluing is applied, equals 0.5. (6) Number of glued joints (CFA31) 0.1 CFA31 = 0.4 i 0.6
No of joints = 1 2 I No of joints < 4
(13)
No of joints = 4.
(7) Method of packaging 0.2 CFA32 = 0.25 i 0.4
CFA29 CFA30
(CFA32)
bundling shrink - wrapped
(14)
plastic strips.
If a feature is not available in a product or the a feature is not applicable to a product, the respective CFA is set to 0. For example, if a product does not use reverse corrugated board, CFA25 = 0. It is worth noting that the values assigned to various states of a cost-related feature are arbitrary since they are mainly based on experience and the study of the product cost. 4.3. Preparing samples
input and output
data of training
The training and testing samples were collected from the existing products of the company. They must be representative enough to cover most cost-related features of the whole product spectrum. By doing so the neural network can be trained and tested comprehensively by exposing to various situations. As a result, 80 existing products were selected. These samples include RSCs with or without printings as well RSCs with or without colour trapping. According to the cost-related feature classification and quantification scheme, their CFAs were then obtained based on their design specifications and processes. Both the CFAs and actual cost data (Ts) provided by the company are listed in Table 3, Tables 4 and 5. One may find that among the training samples, all the CFAl 1, CFA25, and CFA30 are zero. This is caused by the absence of products with these features in the present product range. These CFAs can be activated as soon as the company produces products with these features in the future.
5. The neural network architecture
A feed-forward back-propagation neural network is implemented with a program written using the MATLAB Neural Network Toolbox [4] platform. The first step is to select a good neural network architecture, i.e., number of layers and number of neurons in each layer. The number of neurons in the input layer is the same as the dimension of the cost-related feature attribute vector of CFA, i.e., 32 in this application. The output layer has only one neuron which represents the cost. The determination of the number of hidden layers and neurons in each hidden layer is by trial and error based on the training and testing results of the network. Two types of total 12 neural networks were studied: 1. Six neural networks with one hidden layer. The number of hidden neurons in the hidden layer was 3, 5, 7, 9, 12, and 15. 2. Six neural networks with two hidden layers. The number of neurons in the first and second hidden layers, respectively, was (3, 5), (5, 4), (6, 4), (4, S), (4, lo), and (12, 6). Training of these networks was carried out using a set of 40 randomly selected samples. All the networks were trained for 8000 times. The trained networks were then tested on another set of 20 samples. Four criteria were used as the measure of performance: (1) The average deviation of training samples
2 ICostT, “=
i=l
CostEil
(15)
’
N
where N: costq: Cost E,:
number of samples, target cost of the ith sample, estimated cost of the ith sample.
(2) The maximum samples
relative
deviation
of training
ICost7” - Cost Eil C2=
Max i=l,...,N
CostT,
.
(16)
Networks 3 0.024 1 28.1% 0.0254 29.2%
Network architecture
No of neurons in hidden layer(s)
Cl c2 c3 c4
Table 6 Criteria of the tested network architectures
0.0219 24.2% 0.023 1 24.8%
5
_ ..
c.
0.0219 25.5% 0.029 1 29.2%
7
with one hidden layer
.,,
..-
-
0.0241 21.4% 0.0259 27.6%
9
,,
.
,
rn
-
0.0234 27.4% 0.027 1 30.8%
12
i
-
-
0.023 1 27.8% 0.0273 30.3%
15
-
^
-
0.0222 29.3% 0.0268 30.6%
3, 5
_
_”
-
0.0257 28.5% 0.029 1 30.0%
5.4
_
-
0.0241 21.2% 0.0266 36.4%
6,4
_1
Networks with two hidden layers
.I
0.0217 26.8% 0.0307 35.8%
4, 8
_
_
i
0.0259 28.1% 0.0287 30.1%
4, 10
“.
0.0240 28.3% 0.0299 28.9%
12,6
_
108
Y.F. Zhang et al./Computers
in Industry 32 (1996) 95-113 oumut laver
hidden layer
input layer
CFAl
CFA2
CFA3
AZ=purelin(WZ’Al
CFA32
+B2)
Al =tansig(Wl ‘CFA+Bl)
Where, CFA=
[CFA~
CFAZ
Bl = [Bi(l)
M(2)
Al = [Al(l)
Al(2)
Wl=
...
M(3) Al(3)
CFA32]’ Bl(4)
E1(5)]T
Al(4)
A1(5)lT
82 = [B2(1)] A2 = [A2(1)]
Wl(1,l)
W1(1,2)
...
W1(1,32)
W2(1)
Wl(2,l)
W1(2,2)
...
W1(2,32)
W2(2)
Wl(3,l)
W1(3,2)
...
W1(3,32)
Wl(4,l)
W1(4,2)
.,.
W1(4,32)
~4)
Wl(5,l)
Wl(52)
...
W1(5,32)
W2(5)
w2=
T
w2(3)
Fig. 8. The back-propagation neural network architecture for cost estimation
(3) The average deviation
of testing samples
N ICostT, - cost E, I c3=
c i= 1
N
(17)
’
(4) The maximum samples
relative
deviation
of testing
(CostT, - CostEJ C4 =
Max
i=l,...,N
costq
(18)
The criteria for each type of architecture trained and tested using the training and testing samples are summarised in Table 6. Based on the results, the network with one hidden layer of five neurons was selected. It is worth mentioning that the selected architecture is by no means the best. This is because the number of neural network architectures is infinite. The selected neural network architecture is shown in Fig. 8. CFA is the input vector, Wl and WZ are
the weight matrices for the interconnection between the input layer and hidden layer and that between hidden layer and output layer, respectively. Bl and B2 are the bias vectors for the neurons in the hidden layer and output layer, respectively. Al represents the output from the hidden layer, while A2 represents the output from the output layer, which represents the product cost. The transfer function TanSigmoid tansig is used for the hidden layer, while linear function purelin is used for the output layer in order that the final cost value can reach any value.
6. Training of the back-propagation
network
The training of the selected neural network was carried out automatically by running the program. It includes the following steps:
0.4196 0.8758 - 0.5202 - 0.6382 - 0.3649
0.9543 0.1914 - 0.3235 0.4734 - 0.5642
w1,,26-w15.26
Wl
,.25-w15.25
-0.2721 0.4840 0.3830 0.1376 0.1033
0.5707 - 0.0179 0.1775 0.9967 0.7806
-0.1878 - 0.6366 - 0.4378 - 0.2922 0.4474
w1,.27-w15.27
-
w1,,,9-w1,,,9
WI
Wl 0.1593 0.0288 0.0235 0.0980 0.2956
- 0.6670 - 0.0270 0.7953 0.8184 - 0.8789
0.3119 0.2515 -0.6021 - 0.4807 - 0.4447
-0.8919 0.1466 0.7629 - 0.3964 - 0.2256
,,,8-w15.,8
WI,.,, -WI,.,,
Wl ,.,0-W1~,,,
w1,,9-w15,9
,,17-WlS,l7
0.5603 - 0.3532 - 0.3858 -0.1246 - 0.9883
- 0.3673 0.0723 0.5707 -0.8149 - 1.3620
W~l,3--W’5,3
Wl,,z-Wl,,,
0.8764 - 0.5200 0.4387 0.3899 0.2717
-
Wl,,,-Wl,,,
- 0.3567 0.3545 0.4357 - 0.7577 0.763 1
w1,.28-w15,28
0.1458 -0.6533 - 0.5099 - 0.8737 - 0.7072
w1,,2,-w15.20
0.5880 4.3772 - 1.9846 0.3573 0.8331
Wl,.,*-Wl,,,,
- 1.1402 0.3568 0.1118 0.8102 0.1451
WI,.,-Wls,4
Table 7 Weights for interconnection between input neurons and hidden neurons
- 0.0389 -0.1931 0.2385 0.0815 -0.1151
w1,.Z9-w15,29
0.1612 0.3138 - 0.9853 -0.0681 - 0.0148
w1,,2,-w1S,2,
0.3176 - 0.2329 0.0359 0.7853 - 0.9688
w11,,3-w15.13
0.0539 -0.8161 0.3078 -0.1680 0.4024
Wl,,,-w15.5
- 0.0719 0.9222 - 0.7479 - 0.6005 -0.3615
w1,,30-w15,30
0.9534 0.1431 0.0586 0.6432 0.2410
w1,,2,-w1,,22
0.5451 - 0.4040 - 0.6457 0.6419 0.0125
w11,14-W’5.14
0.8109 0.5200 - 0.3828 - 0.8691 0.2821
w~l,6-w~s.6
0.2872 - 0.6420 0.3243 0.3587 0.5877
w11,31-w15,31
0.3271 - 0.2723 - 0.5400 0.3798 - 0.8032
w1,.23-w15,23
- 0.0072 0.0990 0.2374 0.4544 0.4773
w~,.Is-w15.,5
-0.6383 0.6601 0.4093 0.9413 - 0.4870
W~,,,-W~5,7
- 0.4322 - 0.0922 -0.0710 - 0.5492 - 1.0776
w11,3Z-w’5,32
- 0.6984 - 0.2542 - 1.2134 -0.1419 - 1.1484
w11.24-w15,24
0.5124 - 0.7305 - 1.8707 0.7668 0.7011
w~l,l6-w~5.16
- 0.5059 0.965 1 0.4453 0.5067 0.3030
w1l,8-w15,8
110
KF. Zhang et al./Computers
(1) Initialising the training session Before the training starts, the parameters of the network for training must be pre-set. The initial weights and biases were generated with values between - 1 and + 1 by the random functions provided by the MATLAB Neural Network Toolbox [4]. (2) Training procedures (a) The training data CFAs and Ts are presented to the network in the form of vectors [CFA’, CFA2, . . . . . . ) CFA”]*, and [T’, T2, . . . . .., Tm]*, where m is the number of training samples. (b) The input vector of each sample is pre-multiplied by a set of weights, and summed up with the biases through the transfer functions to generate an estimated cost vector [ A2’, A22, . . . . . . , A2”]*. (c) The sum-squared error E is then obtained by E=
f i=
(T’-A2’)‘.
(19)
I
(d) Generally, the training will stop when either (i) the sum-squared error is smaller than the error goal specified by the user, or (ii) the maximum number of training epochs is reached. However, the target error goal was unknown. We also observed that the sum-squared error tended to converge to a certain value as the number of training epochs increased. Therefore, we used the relative changing rate (RCR) of the sum-squared error to determine when the training has reached a satisfying result. The RCR was calculated at any point where every 1000 training epochs has been completed:
qi-
Table 9 Weights for interconnection between neuron, and bias for output neuron
1)X1000
’
and output
W20)
W2(2)
W2(3)
W2(4)
W2(5)
B2(1)
3.9244
- 2.2210
0.3380
-0.4153
0.1255
The target error goal was given a very small value, 0.001, that cannot be reached. The maximum number of epochs is given a very large value, 500000. This was to ensure that it will go to step (e) to continue training in between the checking points. At the checking point, if 0.99 < RCR, < 1.01, the training stops; otherwise, go to step (e). (e) The derivatives of the error for each neuron layer is calculated and back-propagated to its proceeding layers to alter the weights and biases - go back to step (b). (3) Validation The trained network is then validated using a set of samples that have not been used in training to check its generalisation ability. The estimated cost of each validating sample was obtained by inputting its CFA to the trained network. The estimated cost relative deviation for each sample was calculated as: Cost Ej - Costq RD;=
CostT,
x loo%,
(21)
where CostT =
(20)
hidden neurons
0.8591
1000
EiX
RCR, =
in Industry 32 (1996) 95-113
cost Ei =
actual cost of sample i, estimated cost of sample 1.
where EiXIOOO
E(i-
=
1)X 1000 =
i=
sum-squared error at (i X 1000) epochs, sum-squared error at ((i - 1) X 1000) epochs, 1) 2, 3, . . . . . .
Table 8 Biases for the hidden neurons El(l)
Bl(2)
Bl(3)
Bl(4)
Bl(5)
0.7545
-0.1690
- 0.2547
1.0750
- 1.1295
If the relative deviations of some validating samples are out of +20%, there must be important information in these validating samples that the network has not learnt. These samples are then added to the training set. The same number of new samples is added to the validation set. The network is re-trained using this augmented training set and the validating process is repeated. Initially, 20 samples were used as training set and 20 other samples were used as validating set. After a number of rounds of training and validating, the size of training set was increased to 40 when the trained network performs well on the validation set of 20.
Y.F. Bang
111
et al. / Computers in Industry 32 (1996) 95-1 I3
The cost estimates and their relative deviations from the actual costs of the training and validating samples are shown in Tables 3 and 4. The final sumsquared error of the training samples is 0.0504. The weights and biases of the trained neural network are shown in Table 7, Table 8, and Table 9, respectively.
examples) were used in a regression analysis. The regression analysis was performed using the SPSS for Windows Release 6.0 [18] on a PC. The following linear equation was used: Estimated cost = (Y+ E ( pi X CFA( i)). i=
1
(22)
20 testing samples were then used to test the performance of the linear regression model obtained. The values of the criteria for measuring the performance of both the neural network and the linear regression equation on the training, validating, and testing samples, i.e., the average relative deviation and the maximum relative deviation, are summarised in Table 10. It can be seen that for both the 60 training and validating samples and the 20 testing samples, the neural network outperformed the linear regression model. Neither the company’s current method nor the neural network model can effectively deal with products with cost-related features beyond the training data. These products can only be added to the training dam after their production and the neural network is then re-trained to accommodate the products with such features in the future.
The
7. Testing results and discussions In order to test the capability of the trained neural network, the remai.ning 20 samples were used as the testing set which lhad not been used in the training and validation process. The cost estimates and relative deviations of the testing samples are shown in Table 5. The company’s current method utilises a linear function. The factors considered include only 16 out of the 32 cost-related features incorporated by the neural network m’odel, i.e., the area of the carton, quality of raw material, type of corrugated board construction, height of fluting, size of printing plate, number of colours., the printing area, and the number of processes. The average relative deviation of cost estimates is about 25%. The performance of the neural network model is based on the 20 testing samples. From Table 5, it can be observed that the relative cost deviations of 17 samples (or 85% of the total samples) are within i 10%. Cost estimation on 13 samples (or 65% of the total samples) were even within the relative deviation of f 5%. The maximum cost relative deviation from the actual cost is about 16% which is still considered acceptable by the company. Moreover, the results obtained by using the trained neural network for the training, validation, and testing samples are consistent (see Table 3, Table 4, and Table 5). To further evaluate the performance of the neural network, the training and validating samples (60
8. Conclusions The major drawbacks of the traditional approaches for product cost estimation include the requirement of detailed cost information from production, poor function approximation, and inability to update estimation algorithms by using actual cost data. Although the regression analysis approach can utilise new cost data for updating, the relationship between cost and input parameters may not be the same as the approximated function assumes. An
Table 10
Measures
of performance
for the neural network and the linear regression
Average relative deviat ton Maximum relative deviation
60 training and validating
samples
Neural network model
Linear regression
3.7% 12.0%
8.8% 29.0%
model 20 testing samples model
Neural network model
Linear regression
5.2% 16.0%
13.2% 36.4%
model
112
Y.F. Zhang et al. / Computers in Industry 32 (19%) 95-113
algorithm using the back-propagation neural networks has been presented with a system developed for cost estimation of packaging products, which shows promise of reducing, if not eliminating these drawbacks. Using cost-related features as input. the neural network model is trained using historical cost data. The testing results suggest that the neural network approach can support good product cost estimation. A performance comparison between the neural network and a linear regression analysis based on the training, validating, and testing samples was carried out. The results showed that the neural network model outperformed the linear regression model. The advantages and limitation of this approach are given below. Advantages 1. Detailed information on processing time is not necessary. 2. There is no need to know the actual cost function. With the actual cost data, the neural network is capable of learning knowledge and performing function approximation. 3. The neural network can be updated by re-training the neural network using new cost data from time to time to cover new cost-related features and/or adjust itself to the new manufacturing environment and practices. 4. The neural network helps to improve conceptual design by evaluating the costs from different designs and process alternatives. Limitations 1. Determining the number of hidden layers and neurons in each hidden layer is a trial and error process. There is no guarantee that the selected neural network structure is the best. 2. Quantification of cost-related features is subjective thus needs extensive study on the product cost aspects. 3. Sensitivity study of the cost-related features on the product cost is difficult. Therefore, the current neural network model does not provide the means which can guide the designer towards better solutions. Further work is to incorporate more cost-related features (e.g., lot size) into the network for improving the performance of cost estimation. Sensitivity analysis of the cost-related features is another important area to be further investigated.
Acknowledgements
The authors would like to thank Mr. S.M. Loh and Mr. K. Liang from the company in providing the product information for this study and the anonymous referees for their helpful comments and suggestions on the earlier version of this paper.
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Y.F. Zhang et al./Computers [15] D.E. Rumelhart, GE. Hinton and R.J. Williams, Learning representations by back-propagation errors”, Nature, No. 323, 1986, pp. 533-536. [16] T. Khanna, Foundztions of Neural Nehvorks, Addison-Wesley, 1990. [17] R. Hecht-Nielsen, “Theory of the backpropagation neural networks”, Proceedings of the 3rd IEEE Annual International Conference on Neural Networks, Washington DC., 1989, pp. 1.593-1.626. [18] SPSS Inc., SPSSfor Windows - Release 6.0, 1989-1993. Y.F. Zhang obtained his B.Eng. (Mech. Eng.) from Shanghai Jiao Tong University in 1985 and his Ph.D. (Mech. Eng.) from the University of Bath in 1991. He joined the National University of Singapore in 1992 as a Research Scientist, and is currently a Lecturer at the Department of Mechanical and Production Engineering. His research interests are CAPP, DPMA, computer-aided injection mould design, and artificial neural network applications. He is a senior member of SME.
in Industry 32 (19%) 95-113
113
Jerry Y.H. Fuh received his Ph.D. (Mech. Eng.) from the University of California at Los Angeles (UCLA) in 1992. Before he joined the National University of Singapore (NUS) in 1993, he was an Engineering System Engineer at the Unigraphics Division of Electronics Data Systems (EDS) in the US. Currently, he is a Senior Lecturer at the Department of Mechanical and Production Engineering. His research interests are rapid prototype manufacturing, plastic injection mould design, and fixture planning analysis. He is a member of ASME and SME, a Certified Manufacturing Engineer (CMfgE) from CASA/SME and a registered Professional Engineer in manufacturing engineering from California, USA. W.T. Chan obtained his B.Eng. (Mech. Eng.) from the National University of Singapore in 1995. He is a Project Engineer at the Singapore Defence Materials Organisation.