Predicting multi factor productivity--a comparative analysis between the neural network and multiple regression.
Gurunathan, M. ; Narayanan, S.
Introduction
The word 'Productivity' has become such a buzz word these
days that it is almost difficult not to find it used by any industry.
The presence of global competition, the shortage of critical resources
has compelled firms to focus on strategies for productivity
improvements. The overall performance of an industry is a complex
phenomenon. Among the other several factors namely, Contribution of the
industry to the society,
Its impact on imports/exports, employment, economic and
technological progress, productivity is preferred as a measure of
efficiency. One of the original authors of Productivity Measurement and
management 'Davis' has defined the productivity as '
change in product obtained for the resources expended'.
Productivity has several sub concepts. They are Partial, Total
Productivity and Multi Factor Productivity. Partial Productivity is the
ratio of gross or net output to a single factor input. This expression
is further classified based on the type of input Labor, Capital,
Material or Energy.
Total productivity is the ratio of total output to the sum of all
input factors. MFP is the ratio of 'net' output to the sum of
associated labor and capital inputs. Industry Multi Factor Productivity
(MFP) measures related output to the combined inputs of labor, capital
and intermediate purchases. Obviously enormous data are required to
compute MFP.The Bureau of Labor Statistics, U.S. Department of labor
(BLS) is publishing MFP statistics for various industries.
Forecasting Industry Level MFP
Measurement, data collection, collation and computing gross output
and total value of production to calculate MFP requires lot of time.
Hence before computation of actual MFP statistics, understanding the
trend and predicting the productivity is very much needed. Predicting
productivity serves the purposes:
--to understand the dynamic pace of the resources consumption
pattern
--to estimate the National Income, price changes, wages etc.,
--to align operational activities with strategic activities
--to ascertain the ability to meet the market, business and
competitive objectives of the industry
--to devise the action plans and policy changes proactively before
the damages could happen.
Earlier research studies used conventional statistical techniques
for predicting MFP. These correlations have been developed by both
theoretical and empirical methods. Many are not accurate enough and have
their own limitations. Multiple Regression (MR) Analysis has an proved
use of record for MFP forecasting. Recent studies make use of Artificial
Intelligence for forecasting and Artificial Neural Networks (ANN) find
Wide applications in forecasting Stock market indices movement,
pattern classification, recognition etc., .because of their ability to
understand the inter relationships among the casual factors. So far,
application of ANN in Productivity forecasting is only little.
Research Objectives
The main objective of this paper is to create models for predicting
MFP using the techniques MR and ANN and comparing their accuracy. In
this process, the following steps are involved,
--data collection--data issued by BLS--2004,for calculating MFP are
used
--applying ANN technique and using MATLAB version 6, predicting MFP
--applying MR method to forecast MFP
--comparing the performances through the results produced by the
two models.
The aim of this paper therefore, is not to dwell on the MFP
computing methodologies adopted by BLS, but rather concentrate on
comparing the performances of the two competitive yet complementary
techniques in forecasting.
Data Description and Methodology
The data issued by Bureau of Labor Statistics, U.S. Department of
labor (BLS), Feb. 10, 2004 for the Industry Classification - Total
manufacturing (SIC 20-39), of Table 6, Table 7 containing the value of
production and factor costs in billions of current dollars is used.
From the above table, the data on the value of production (Output)
is not used in the model, but the factors responsible for producing the
output are used. The industry Multi factor Productivity indices
calculate productivity growth by measuring changes in the relationship
between the quantity of an industry's output and the quantity of
inputs consumed in producing that output, where measured inputs include
Capital and intermediate purchases (including raw materials, purchased
services, and purchased energy) as well as labor input.
A Tornqvist index is used to calculate Multi factor Productivity
and the method followed by BLS to calculate MFP index is as follows:
Ln([A.sub.t]/ [A.sub.t-1]) = ln ([Q.sub.t]/[Q.sub.t-1]) - [[w.sub.k] (ln
[K.sub.t]/ [K.sub.t-1]) + [w.sub.1](ln [L.sub.t]/[L.sub.t-1]) +
[w.sub.ip] (ln [IP.sub.t]/[IP.sub.t-1])]
Where:
ln = the natural logarithm of the variable
A = multifactor productivity
Q = Output
K = capital Input
L = Labor Input
IP = intermediate purchases input
[W.sub.k],[w.sub.l],[w.sub.ip] = Cost share weights
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
The available data set was divided in to two sub sets. The training
sub set has thirty observations and the forecasting subset has ten
observations.
MR Model
La-Lun-Chou remarked, regression analysis attempts to establish the
nature of relationship between the variables--that is to study the
functional relationship between the variables 'X' and
'Y', and thereby provides a mechanism for prediction or
forecasting.
When the number of independent variables become more than one, in a
regression analysis model, it becomes Multiple regression model. In most
of the practical situations, the relationship between the variables is
not linear(Fig.1). In such cases it is required to quantify the
relationship through a mathematical function. When it is not possible to
define the non linear relationship exactly, the basic assumption to be
made is making variables linearly related. Then the model of multiple
linear regression is
Y = [B.sub.0] + [B.sub.1]X1 + [B.sub.2]X2+.... BiXi
Y is the dependent variable,
Xi, the [i.sup.th] independent variable (Predictor variable), i
=1,2,3 ... n. [B.sub.0] = constant, Bi = Co.efficients of the model, i =
1,2,3 ... n. Following the matrix method of solving the regression
model, co-efficients of the regression model are obtained using the
equation, b = [[X.sup.T] X].sup.-1] [[X.sup.T] Y] where X = vector
matrix of independent variables and Y = vector matrix of dependent
variables.
From Fig.1 it is evident that, the contributing factors have non
linear relationship with the dependent variable MFP. With the
approximated linearity between the variables, MR model for predicting
the productivity is expressed as:
67.1023(constant)+0.0513(capital)+0.1118(labor)-0.2495(energy)-0.0462
(material)-0.1582(purchased services).
ANN Model
ANNs are computing systems made up of a number of simple, highly
interconnected processing elements, which process information by their
dynamic state response to external inputs. ANNs are able to learn and
generalize from examples. Due to this ability, ANN models can predict by
abstracting essential characteristics from inputs in the pattern of
variable interconnection weights among the processing elements. In
essence, ANN mimics the human brain and are able to respond to new input
data to predict required output.
Among the various ANN architectures, the back propagation (BP)
neural net work is one of the simplest and most preferred networks that
can be well employed for prediction. Back propagation is a type of
supervised learning and requires a set of pre existing data patterns and
corresponding targets for training. In neural networks, the number of
input and output neurons is equal to the number of input and output
parameters. It has the ability to memorize complex nonlinear mappings
with increased reliability. A properly trained back propagation net work
can make reasonable predictions when presented with new inputs.
Developing the ANN model:
The following steps are followed to develop ANN model.
1. Selecting the architecture of ANN based on the number of
predictor variables,
2. Feeding the data and training the network,
3. Simulate the network to get the result
Microsoft Windows based ANN software, MATLAB version 6.1, was used
for the study on a P IV personal Computer. The flexibility of this
software allows the user to select the number of hidden layer, neurons,
learning functions, training function and iterations(epochs). In this
study, a multi layer feed forward back propagation network was created
with Levenberg-Marquardt's learning algorithm and sigmoidal
transfer function to predict the MFP. The network consists of an input
layer with five neurons, two hidden layer and one output layer with
single neuron.(Fig.3). Since the number of predictor variables are five
from the available data, the number of neurons in input layer are five.
The output layer has one response variable. The number of neurons
greatly influence the generalization characteristics of a neural
network. Since there are no specific rules to fix the number of neurons
in the hidden layer, a trial and error method was followed. Finally five
and ten hidden neurons in first and second hidden layers were used to
achieve the least mean squared error (mse) on the performance of the
network.
[FIGURE 3 OMITTED]
Comparison of Results
The predictive performances of MR model and ANN model are tabulated
in Table 1.
Results of MR model
Predicted
MFP MFP PE % APE
94 92.4657 1.632234043 1.5343
95.1 93.442 1.743427971 1.658
97.3 97.1314 0.17327852 0.16858
99.2 97.8689 1.341834677 1.331
100 95.7878 4.2122 4.2121
103.
1 99.4709 3.519980601 3.6291
105.
7 102.8486 2.697634816 2.8513
108.
7 103.1379 5.116927323 5.56209
111.
3 98.3664 11.62048518 12.9332
110.
3 93.7981 14.96092475 16.5018
Mean = 4.701892788 5.03815
Results of ANN model
Predicted
MFP MFP PE % APE
0.69
94 93.3012 0.743404255 88
0.20
95.1 95.3025 -0.21293375 25
1.13
97.3 98.4364 -1.16793422 64
0.08
99.2 99.1102 0.090524194 98
0.60
100 100.6012 -0.6012 12
0.48
103.1 102.6102 0.475072745 98
1.19
105.7 104.5058 1.129801325 42
3.59
108.7 105.1035 3.308647654 65
5.54
111.3 105.7501 4.986433064 99
5.87
110.3 104.428 5.323662738 2
1.94
Mean = 1.4075478 31
[FIGURE 4 OMITTED]
The comparative analysis is made using two prediction performance
measures namely:
(1) Mean Percentage Error (MPE) calculated by
MPE = [SIGMA] [PE.sub.i]/n, [PE.sub.i] =
([x.sub.i]-[p.sub.i]/[x.sub.i])100 %, i = 1 to n.
(2) Mean Absolute Percentage Error (MAPE) calculated by
MAPE = [SIGMA] AEi/n, AEi = [square root of ([(xi-pi).sup.2])]
Where PEi is the percentage error on prediction of productivity,
AEi is the absolute error on prediction of productivity, Pi is the
predicted MFP, xi is the Actual MFP, n is the number of observations.
From the table of results, the MPE and MAPE of MR model is 4.701% and
5.0381 respectively while that of ANN model is 1.407% and 1.9431
respectively. The scrutiny of MPE and MAPE of both the models reveal
that--both models tend to under predict the MFP but the ANN model is
able to produce more accurate forecasts for MFP growth.
Conclusion
The perpetual performance information in terms of Productivity
indices are essential for effective management of resources at three
levels -industry, firm and plant. An attempt was made through this study
to make use of AI information processing tool, the artificial Neural
Network to predict the Industry MFP growth and to compare the
model's performance with conventional statistical technique
Multiple Regression. The distinct advantage of using ANN for prediction
is, the contributing factors can be retained in their own units of
measure and there is no need for converting them to a common measure
like Rupees or Dollars, the monetary value. This study proves that the
ANNs can be used to predict MFP more accurately without the necessity to
capture the relationships between the variables irrespective of their
nature linear or non linear. In conclusion, the study has, therefore,
achieved its broad objective of demonstrating the accuracy and
versatility of ANN by its successful application in estimating MFP.
References
[1] Umit s. Bititci, 'Modeling of Performance Measurement
systems in Manufacturing Enterprises'. Intl. Jl of Production
Economics, Vol 42 (1995) p137-147.
[2] Andrea Rangone' An Analytical Hierarchy Process frame work
for comparing the overall performance of Manufacturing
Departments', Intl Jl. of Operations and Productions Management.
Vol. 16-no.8(1996) p104-119.
[3] Andy Neely, Mike Gregory, Ken Platts Performance Measurement
System Design',Intl. Jl. of Operations and Production Management,
Vol. 15-no.4 (1996)p80-116.
[4] David J. Edwards, Gary D. Holt and Frank C. Harris, A
Comparative Analysis between the Multilayer Perceptron Neural Network
and Multiple Regression analysis for Predicting construction plant
Maintenance costs', Jl. of Quality in Maintenance Engineering, Vol.
6 no1(2000) p45-60.
[5] U.S. Bititci, P. Suwignjo, A.S. Carrie, 'Strategy
Management through Quantitative Modeling of Performance Measurement
systems', Intl. Jl of Production Economics, Vol. 69 (2001) p15-22.
[6] Mika Hannula' 'Total Productivity Measurement based
on Partial Productivity ratios' Intl. Jl of Production Economics,
Vol. no.78 (2002)p 57-67.
[7] M. Munir ahmad and Nasreddin Dhafr., Establishing and improving
Manufacturing Performance measures' Robotics and Computer
Integrated Manufacturing, Vol. 18 (2002) p 171-176.
[8] M. Cabassud, N. Delgrange, L. Durand-bourlier, A Tool to
improve UF Plant Productivity' Desalination, Vol.
no145(2002)223-231.
[9] Srinivas Talluri and Joseph Sarkis, 'A Methodology for
monitoring System Performance' Intl. Jl. of Production Research,
Vol .no40,7(2002) p 1567-1582.
[10] Mustafa Yurdakul, Measuring a Manufacturing system's
Performance using Saaty's system with feed back approach,
Integrated Manufacturing Systems, Vol. no 13,1(2002) p 25-34.
[11] J. Ren, Y.Y. Yusuf, N.D. Burns, 'The effects of agile
attributes on Competitive priorities: a Neural Network approach'
Integrated Manufacturing Systems, Vol. 14 no6(2003) p489-497
[12] Joseph Sarkis, Quantitative Models for Performance Measurement
systems alternate considerations' Intl. Jl of Production Economics,
Vol. 86 (2003) p8190.
[13] Dr. Tarek el-fauly, Dr. Gamal M. Aly., Modeling of
Manufacturing Systems using Neural Networks', IEEE, 03.
[14] Mohan P. Rao and David M. Miller. 'Expert systems
applications fo Productivity analysis' Industrial Management &
Data System, Vol .104 no9 (2004) p 776-785.
[15] B.S. Sahay, Multi Factor Productivity Measurement Model for
Service Organization' Intl. Jl. of Productivity and Operations
Management, Vol. 54 no1(2005) p7-22.
[16] Liang-Hsuan chen, Shu-yi Liaw, Measuring Performance via
Production Management: a pattern analysis', Intl Jl. of
Productivity and Operations Management, Vol. 55 no1, 2006 pp 79-89
(1) M.Gurunathan and (2) S.Narayanan
(1) Counselor, Confederation of Indian Industry- Institute of
Logistics India, Chennai, E-mail: gurunathan_e@hotmail.com,
mgurunathan@ciionline.org (2) Prof & Dean, Vellore Institute of
Technology, India. * Author for correspondence
Data Used : Bureau Of Labor Statistics, U.S. Department Of Labor,
Feb. 10, 2004, Value Of Production And Factor Costs (Billions of
Current Dollars)
Value of
Year Prodn Capital Labor Energy
1962 239.337 43.952 112.06 5.184
1963 250.068 48.536 117.429 5.456
1964 265.54 51.477 125.759 5.776
1965 290.795 60.468 134.366 6.05
1966 321.917 65.129 149.628 6.47
1967 336.742 63.909 157.865 6.948
1968 361.551 69.616 171.649 7.293
1969 384.321 68.439 186.711 7.672
1970 378.435 61.763 187.706 8.192
1971 403.353 70.926 191.992 9.04
1972 454.358 80.501 212.257 10.068
1973 524.205 86.459 242.485 11.513
1974 599.108 85.206 267.174 15.563
1975 634.061 102.25 262.635 18.711
1976 721.974 120.545 300.568 22.695
1977 830.583 138.886 340.051 26.616
1978 933.633 151.617 384.54 30.671
1979 1052.865 155.224 432.569 36.151
1980 1153.33 159.315 461.541 41.88
1981 1263.23 184.172 500.83 48.497
1982 1248.414 182.765 502.688 51.174
1983 1309.412 202.073 523.039 54.539
1984 1467.964 240.424 577.429 58.546
1985 1500.569 234.467 608.259 55.897
1986 1482.966 230.765 629.268 51.552
1987 1567.223 269.917 653.792 51.342
1988 1691.69 311.642 688.924 51.848
1989 1771.109 319.286 713.278 52.215
1990 1847.681 329.521 727.302 52.835
1991 1822.467 318.029 730.234 52.01
1992 1917.995 329.503 776.195 54.726
1993 1997.157 345.573 806.324 58.097
1994 2125.288 399.184 848.327 58.938
1995 2256.868 445.569 875.537 57.425
1996 2350.65 471.011 885.211 61.328
1997 2467.672 516.324 933.594 63.278
1998 2493.475 511.826 981.147 72.247
1999 2598.556 534.965 1015.161 75.444
2000 2729.071 503.122 1072.951 90.916
2001 2592.192 445.517 1020.846 88.628
Year Material Pur.Services MFP
1962 59.927 18.214 71
1963 58.256 20.391 73.1
1964 60.356 22.172 75.2
1965 65.468 24.443 77.2
1966 72.331 28.359 77.5
1967 74.96 33.06 77.1
1968 78.968 34.025 79.4
1969 84.242 37.257 79.9
1970 83.026 37.748 78.8
1971 92.274 39.121 81
1972 107.548 43.984 84
1973 133.706 50.041 85.4
1974 173.757 57.408 80.8
1975 190.765 59.699 78.5
1976 212.032 66.133 81.3
1977 247.459 77.571 82.4
1978 278.36 88.445 83.1
1979 321.73 107.191 82.5
1980 378.348 112.246 81.2
1981 411.659 118.072 81.7
1982 400.764 111.023 83
1983 397.924 131.837 85.1
1984 443.556 148.01 87.7
1985 457.8 144.147 89.2
1986 418.738 152.643 90.7
1987 419.158 173.014 93.6
1988 442.538 196.739 95.3
1989 472.438 213.892 93.5
1990 509.663 228.36 93.3
1991 487.749 234.445 92.4
1992 492.707 264.864 94
1993 513.297 273.866 95.1
1994 526.37 292.469 97.3
1995 565.219 313.119 99.2
1996 611.375 321.726 100
1997 609.746 344.731 103.1
1998 577.438 350.817 105.7
1999 605.773 367.213 108.7
2000 680.384 381.698 111.3
2001 677.751 359.45 110.3
