View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Periodica Polytechnica (Budapest University of Technology and Economics) Ŕ periodica polytechnica Sigmoid functions in reliability based Social and Management Sciences management 15/2 (2007) 67–72 doi: 10.3311/pp.so.2007-2.04 Tamás Jónás web: http://www.pp.bme.hu/so c Periodica Polytechnica 2007 RESEARCH ARTICLE Received 2008-09-14 Abstract 1 Introduction This paper introduces the theoretical background of a few A sigmoid function is a mathematical function that produces possible applications of the so-called sigmoid-type functions in a curve having an "S" shape, and is defined by the manufacturing and service management. 1 σλ,x (x) (1) An extended concept of reliability, derived from fuzzy theory, 0 λ(x x0) = 1 e− − is discussed here to illustrate how reliability based management + formula. The sigmoid function is also called sigmoidal curve decisions can be made consistent, when handling of weakly de- [1] or logistic function. The interpretation of sigmoid-type func- fined concepts is needed. tions – I use here – is that any function that can be transformed I demonstrate how performance growth can be modelled us- into σλ,x (x) through substitutions and linear transformations ing an aggregate approach with support of sigmoid-type func- 0 is considered as a sigmoid-type one. There are several well- tions. known applications of sigmoid-type functions. A few examples Suitably chosen parameters of sigmoid-type functions allow are: threshold function in neural networks [2], approximation these to be used as failure probability distribution and survival of Gaussian probability distribution, logistic regression [3], or functions. If operation time of an item has a given sigmoid-type membership function in fuzzy theory [4]. failure probability distribution function, then its hazard function My objective is to conclude hypotheses on how this function is proportional to the failure probability distribution function. family is applicable in certain areas of reliability based manufac- Furthermore, this hazard function can be a model of the third turing and service management, along with brief interpretations part of the bathtub failure rate curve. and demonstrations of these possible applications. Besides the (1) generic form, different other forms of the sigmoid function Keywords such as P(x), µ(m R), Fλ,t (t) and Rλ,t (t) with different pa- Performance growth sigmoid function aggregate perfor- 0 0 · · rameters will be used here. These different forms are different mance performance growth extended concept of reliability · · · appearances of the same function, and the notations always fit to survival function hazard function · the notations that are commonly used in the fields of particular applications. 2 Modelling performance growth The manufacturing as well as the service processes can be characterized by various indicators and metrics, which are usu- ally functions of several process variables, parameters and con- stants. The overall performance of a process depends on its in- puts, and it is common that finally, there is one aggregated in- dicator or metric used to characterize the overall performance. These kind of aggregate indicators are commonly associated with some financial metrics, and so whenever a new process is being introduced its financial performance can be monitored through the chosen aggregate performance indicator. Certainly, Tamás Jónás the ultimate goal is to find the highest performance resulting in- Flextronics International Ltd.„ 1183 Budapest , Hangár u. 5-37, Hungary put set as quickly as possible. However, in reality, the manufac- e-mail: tamas.jonas@hu.flextronics.com turing and service processes are too complex, with a large num- Sigmoid functions in reliability based management 2007 15 2 67 2.1 The nature of performance growth Let x be such an independent input variable that is proportional with the per- formance development and improvement eorts. Furthermore, let P (x) note the aggregate performance as function of x, Pi the initial value and Pt the target value of P (x). With other words, we look at the growth of aggregate perfor- mance as function of x, providing that P (x) increases from Pi to Pt, and Pt represents the level of operational performance corresponding to the targeted nancial results. Numerous practical observations conrm that the same ¢x eort increase results dierent ¢P (x) performance increases, depending on the actual level of performance, at which the eorts were made. It may be assumed that when ¢P (x) is at a low level and is close to Pi the speed of growth is low, namely, the early eorts do not yield much of improvement. As the performance increases, the speed of its growth increases as well. It may be explained so that as the improvement and development eorts result higher and higher level of process specic knowledge and skills, the impact of the same sized every new eort results greater performance increase. This tendency, however, is only valid until a certain level of performance. Although the performance increases as the eorts increase, but after a certain time, the growth slows down as the new eorts result low-rate increase in process specic knowledge and skills. Finally, the performance gets close to an upper limit. If the Pt target value of ¢P (x) is set to this upper limit, then it can be said that growth speed of ¢P (x) decreases when P (x) is near Pt. ber of input factors, and commonly, with unknown dependen- cies and interactions among them. That is, searching for depen- dencies between the input set and the aggregate performance is cumbersome and works with difficulties in most of the practical cases. Instead of trying to handle many inputs and outputs, the approach shown here is using one independent variable that is proportional to and so represents all those efforts that contribute to the aggregate performance growth. For example, we may con- sider the time spent on process development and improvement as an aggregate input variable, and so we can look at the aggre- gate performance growth as function of the so-interpreted time variable. 2.1 The nature of performance growth Let x be such an independent input variable that is propor- tional with the performance development and improvement ef- forts. Furthermore, let P(x) note the aggregate performance FigureFig. 1: Growth1. Growth speed speed of performance of performance in its different in ranges its dierent ranges as function of x, Pi the initial value and Pt the target value of P(x). With other words, we look at the growth of aggregate per- where λ > 0 is a process specific proportionality coefficient. formance as function of x, providing that P(x) increases from ∗ Figure 1 illustratesTurning the into growth infinitesimal speed quantities of performance results the following in its dierent differ- ranges, and P to P , and P represents the level of operational performance i t t ential equation. corresponding to the targeted financial results. Numerous practical observations confirm that the same 1x dP(x) λ∗ P(x)3 Pi Pt P(x) (3) effort increase results different 1P(x) performance increases, dx = − − depending on the actual level of performance, at which the ef- Eq. 3 is known as logistic equation and is also used as a model forts were made. It may be assumed that when 1P(x) is at a of population growth [5]. Population models using the logis- low level and is close to Pi the speed of growth is low, namely, tic growth can be found in Murray’s book [9] and the book by the early efforts do not yield much of improvement. As the per- Clark [10] introduces its applications in economics. Solving this formance increases, the speed of its growth increases as well. It equation results the may be explained so that as the improvement and development λ(x a) Pt e Pi P x − (4) efforts result higher and higher level of process specific knowl- ( ) λ(x +a) = 1 e − edge and skills, the impact of the same sized every new effort + function, where λ λ (Pi Pt ) λ (Pt Pi ) is a positive results greater performance increase. This tendency, however, = − ∗ − = ∗ − number. If a x0, Pi 0, and Pt 1, then is only valid until a certain level of performance. Although the = = = performance increases as the efforts increase, but after a certain eλ(x x0) 1 P(x) − . (5) time, the growth slows down as the new efforts result low-rate = 1 eλ(x x0) = 1 e λ(x x0) + − + − − increase in process specific knowledge and skills. Finally, the It means that function (4) is a sigmoid-type function. performance gets close to an upper limit. If the Pt target value of 1P(x) is set to this upper limit, then it can be said that growth 2.2 Attributes of the performance growth function speed of 1P(x) decreases when P(x) is near Pt . The P(x) function derived above has four parameters: λ, a, Fig. 1 illustrates the growth speed of performance in its dif- Pi , and Pt . Interpretation of these parameters and the basic ana- ferent ranges, and shows how a small 1x effort increase results lytical properties of P(x) are introduced in this subsection. low-rate performance increase when P(x) is near Pi or Pt , as well as shows greater slope of P(x) when it is more distant both Derivative from Pi and Pt . This illustration assumes that the chosen 1x is Derivative of P(x) is small enough to assume linear relationship between x and P(x) λ(x a) dP(x) λe (Pt Pi ) in the 1x intervals. − − (6) dx = (1 eλ(x a))2 Thus, my assumption is that the nature of performance growth + − is so that its speed is proportional to the P(x) Pi and Pt P(x) − − Monotonicity and limits differences.