Vol. 48, No. 16, 15 August 2010, 4725–4744
Dynamic performance measures for tools with multi-state wear processes
and their applications for tool design and selection
Ming-Hung Shua, Bi-Min Hsub and Kailash C. Kapurc*
a
Department of Industrial Engineering and Management, National Kaohsiung University of Applied Sciences, Kaohsiung 807, Taiwan;bDepartment of Industrial Engineering and Management, Cheng-Shiu University, Kaohsiung 833, Taiwan;cDepartment of Industrial and Systems Engineering, University of Washington, Box 352650, Seattle, WA 98195-2650, USA
(Received 16 December 2008; final version received 15 May 2009) Traditional models to evaluate the reliability and performance of tools are binary models, working (success) or failure, to classify the state of the tool. Most machine tools degrade with time and thus a multi-state discrete classification is more realistic for the continuous degradation of the tool. We propose a non-homogeneous continuous-time Markov process model for tool degradation, because the length of time the machine tool stays in a certain state depends not only on the current state, but also on how long the tool has been in the current state. The traditional reliability and life performance measures focus on the mean time between failure or the failure rate. The performance measures must capture the total experience of the manufacturer over the target life of the tool and the impact of its degradation on the quality of the products to the downstream customers. We propose several new measures for tool performance. These measures can be used to evaluate different tool designs or we can use them to select the best tool for a certain application based on the economic/disutility functions.
Keywords: quality engineering; process capability; fuzzy data analysis; process control
1. Introduction
With the development of modern manufacturing, machine tools have played a major role in most areas of industrial fabrication for the production of quality goods. They perform tasks such as turning, boring, milling, drilling and tapping in order to machine simple and complex components in many materials ranging from alloys to plastics (Jeang 1998). Since machine-tool-related costs make up a significant portion of the expense of producing a part, continuous advancement in automation and computer-aided manufacturing have all increased the importance of tool evaluation and selection in a machining process to assist the manufacturer in sustaining worldwide competition.
An effective performance measure for machine tools is required in order to determine the reliability of a machining process (Sun et al. 2005). For any machining process, one of
*Corresponding author. Email: [email protected]
ISSN 0020–7543 print/ISSN 1366–588X online ß 2010 Taylor & Francis
DOI: 10.1080/00207540903071385 http://www.informaworld.com
the most important pieces of information in the process is related to tool failure, such as tool wear degradation or tool breakage. The state of tool wear in the machining process is very useful information for the determination of the machining process reliability. For example, surface roughness is one of the most important requirements in the machining process. The surface roughness quality is related to tool wear. When tool wear increases, the surface roughness also increases (Ozcelik and Bayramoglu 2006).
However, most earlier work focused on the identification and analysis of two kinds of states of the tool, fresh and worn, or in terms of reliability—success and failure. Tool wear is a dynamic process. For the wear of a cutting tool, a sharp cutting edge is quickly worn down and finite wear is established at the flank face. The wear then progresses at a slightly increasing rate and lasts for a certain period. Finally, the wear rate increases and breakdown occurs. Therefore, a multi-state classification of the states of tool wear is recommended instead of the binary classification (Sun et al. 2004), and multi-state classification provides more useful information concerning the condition of the tool.
The multi-state classification of tool wear has been studied by several authors (Balakrishnan et al. 1988, Moriwaki and Tobito 1988, Li and Yuan 1998, Fish et al. 2003, Sun et al. 2004). Monitoring the state of the tool was based on vibration or acoustic emission (AE) signals, and is an on-going research problem (Dornfeld 1992, Pai and Rao 2002, Sun et al. 2004, Yesilyurt and Ozturk 2007). The typical power spectral density (PSD) of the machining signal over time was suggested for monitoring the wear state of the machining tool (Bhattacharyya et al. 2008). The magnitude of the spectrum peaks in the lower-frequency region is utilised as an index of various wear states. In general, tool wear can be classified into five states, i.e. the initial wear, normal wear, micro fracture, whole wear and tool breakage state. This behaviour of tool wear was also reported by Fang (1994) and Khrais and Lin (2007).
While much work has been carried out on the multi-state classification of tool wear using modern monitoring techniques, for tool wear with multi-state wear data there is a lack of published research on the development of the performance measures and selection of tools based on these new and useful performance measures. Tool wear is a gradual deterioration process. The machine tool normally starts in an initial wear state and remains in that state for a random period of time, before proceeding to severe wear states, and eventually breakage (see Figure 1). This progressive degradation is a stochastic process, and it has commonly been modeled as a Markov process by assuming that the
j 1 0 m m – 1 lm–1,1 lm,m–1 l1,0 lj,0 lj,1 lm,0 lm,1 lm, j lm–1, j lm–1,0
Figure 1. A multi-state machine tool with minor and major degradation.
next state of machine tool wear only depends on its current wear state (Ross 1996, Bre´maud 1998). A non-homogeneous continuous-time Markov process (NHCTMP) model is used in this paper for tool degradation, because the length of time the machine tool stays in a certain state depends not only on the current state, but also on how long the tool has been in the current state. The goal of this research is to develop performance measures that capture the total experience of the manufacturer over the target life of the tool. In addition, we integrate economic/disutility functions with these performance measures to evaluate different tool designs and select the best tool in real applications.
2. Machine tool deterioration based on general Markov processes
In traditional binary models for evaluating machine tool wear, the state of the machine
tool is a function of time t denoted by ~ðtÞ ¼1 and the state of failure at time t is denoted
by ðtÞ ¼~ 0. In practice, most machine tools degrade gradually and perform at
intermediate states between working perfectly and complete failure, thus the states of tool wear have more than two states of working efficiency (i.e. (t) 2 {0, 1, 2, . . . , m}, m 1) (Brunelle and Kapur 1998, Lisnianski and Levitin 2003).
The degradation of a machine tool from the perfect state, (0) ¼ m, to lower states (t) 2 {m 1, m 2, . . . , 0} is modeled using a general stochastic model known as a non-homogeneous continuous-time Markov process model (NHCTMP), which assumes that the next state of the machine tool and the length of time that the machine tool stays in a certain state depend not only on the current state, but also on how long the system has been in the current state. NHCTMP is a stochastic model with discrete states and continuous time (Ross 1996, Bre`maud 1998), and is a generalization of traditional Markov processes to capture the effect of age on the state change of the machine tool.
2.1 General Markov process model
A discrete-state continuous-time stochastic process (t) 2 {0, 1, 2, . . . , m} is called a
Markov chain if, for t15 t25 5 tn15 tn, its conditional probability mass function
satisfies the relation
PrfðtnÞ ¼knjðtn1Þ ¼kn1, . . . , ðt2Þ ¼k2, ðt1Þ ¼k1g
¼PrfðtnÞ ¼knjðtn1Þ ¼kn1g: ð1Þ
Introducing the notation tn1¼tand tn¼t þDt, expression (1) is simplified to
Prfðt þ DtÞ ¼ j j ðtÞ ¼ i g ¼ Pi, jðt, DtÞ, ð2Þ
for i, j 2 {0, 1, . . . , m}. These conditional probabilities are called the transition probabilities of NHCTMP (Ross 1996, Bre`maud 1998, Lisnianski and Levitin 2003). The transition probabilities satisfy
Pi, jðt, DtÞ 0 and
Xm
j¼0
Pi, jðt, DtÞ ¼ 1, ð3Þ
for t, Dt40. The Chapman–Kolmogorov equation, which follows from the Markov property and the rule for total probability, is given by (Høyland and Rausand 1994, Ross 1996)
Pi, jðt þDtÞ ¼
Xm
k¼0
Pi,kðtÞ Pk,jðt, DtÞ, ð4Þ
for t, Dt40. From Equation (4), we have
Pi, jðt þDtÞ Pi, jðtÞ ¼ Pi, jðtÞ Xm k¼0,k6¼j Pj,kðt, DtÞ þ Xm k¼0,k6¼j Pi,kðtÞ Pk,jðt, DtÞ: ð5Þ
After dividing by Dt and letting Dt ! 0, let
i, jðtÞ ¼ lim Dt!0 Prððt þ DtÞ ¼ j j ðtÞ ¼ iÞ Dt ¼ lim Dt!0 Pi, jðt, DtÞ Dt , ð6Þ and j,jðtÞ ¼ Xm k¼0,k6¼j j,kðtÞ, ð7Þ
where i, j(t) is the transient degradation rate from state i to state j and j,j(t) is the transient
degradation rate from state j to state j. Then, for the deterioration processes of machine tools followed by a NHCTMP, their state equations can be expressed as
Pi, j0 ðtÞ ¼ Pi, jðtÞ Xm k¼0,k6¼j j,kðtÞ þ Xm k¼0,k6¼j Pi,kðtÞ k,jðtÞ: ð8Þ
Let us now assume that (0) ¼ i ¼ m, namely that the machine tool is in state i ¼ m at time
t ¼0. Then, the notation of the state equations shown in Equation (8) can be simplified by
omitting the index i, that is
P0 jðtÞ ¼ PjðtÞ Xm k¼0,k6¼j j,kðtÞ þ Xm k¼0,k6¼j PkðtÞ k,jðtÞ, ð9Þ
where Pm(0) ¼ 1 and Pk(0) ¼ 0 for k 6¼ m and Pj(t) is the probability that the machine tool is
in state j at time t.
3. Multi-state tool wear with minor and major degradation
Definition 3.1: An (m þ 1) 3 multi-state deterioration process followed by a
non-homogeneous continuous-time Markov process (NHCTMP) is defined as a minor and major degradation process if the process is capable of degrading directly to any lower state from state i to state j for all j i 1 and i 2 {m, m 1, . . . , 1} during a transition.
Figure 1 shows an (m þ 1) 3 multi-state machine tool with minor and major degradation followed by a NHCTMP. Its state equations described as Equation (9) can be explicitly expressed as a matrix equation,
P0 mðtÞ P0 m1ðtÞ .. . P0 1ðtÞ P0 0ðtÞ 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 ¼ m, mðtÞ 0 0 0 0 m,m1ðtÞ m1,m1ðtÞ 0 0 0 m,m2ðtÞ m1,m2ðtÞ m2,m2ðtÞ 0 .. . .. . .. . .. . .. . . . . 0 0 m,1ðtÞ m1,1ðtÞ m2,1ðtÞ 1,1ðtÞ 0 m,0ðtÞ m1,0ðtÞ m2,0ðtÞ 1,0ðtÞ 0 2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 5 PmðtÞ Pm1ðtÞ .. . P1ðtÞ P0ðtÞ 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 :
Since state 0 is an absorbing state, all the transition rates from this state are equal to zero. Thus the elements of the column corresponding to the absorbing state are all equal to zero. Without losing any information concerning the probability that the machine tool is in each
state at time t, Pm(t), Pm1(t), . . . , P0(t), the matrix equation can be reduced to
P0 mðtÞ P0 m1ðtÞ .. . P0 2ðtÞ P0 1ðtÞ 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 ¼ m, mðtÞ 0 0 0 m, m1ðtÞ m1, m1ðtÞ 0 0 m,m2ðtÞ m1, m2ðtÞ m2, m2ðtÞ 0 .. . .. . .. . .. . . . . 0 m,1ðtÞ m1,1ðtÞ m2,1ðtÞ 1,1ðtÞ 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 PmðtÞ Pm1ðtÞ .. . P2ðtÞ P1ðtÞ 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 , ð10Þ where ðtÞ ¼ m, mðtÞ 0 0 0 m,m1ðtÞ m1,m1ðtÞ 0 0 m,m2ðtÞ m1,m2ðtÞ m2,m2ðtÞ 0 .. . .. . .. . .. . . . . 0 m,1ðtÞ m1,1ðtÞ m2,1ðtÞ 1,1ðtÞ 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5
is the transient degradation rate matrix. Suppose now (t) has a full rank, then P0(t), the
probability that the machine tool is in state 0 at time t, can be found from
P0(t) ¼ 1 P1(t) Pm(t).
We assume that the amount of time Ti, j that the tool spends in each state i before
proceeding to the next state j has a Weibull distribution with a scale parameter i, jand a
shape parameter , i.e. Ti, j Weibull(i, j, ), as shown in Equation (11). During the
degradation phase, the machine tool lifetime follows a Weibull distribution with 41 (Høyland and Rausand 1994)
fi, jðtÞ ¼ i, jt1eði, jtÞ
, ð11Þ
for t40, the transient degradation rate of which is given by
i, jðtÞ ¼ i, jt
1
: ð12Þ
Based on Equations (11) and (12), we are able to model the practical behaviour of transient degradation rates in the multi-state deterioration process of tool wear with minor and major degradation, that is:
. the transient degradation rates given by Equation (12) increase as the machine tool running time increases; and
. the deterioration process has less likelihood of degrading to a far state than degrading to a near state.
Example 3.2: Suppose that the machine tool wear with five different possible states
(t) 2 {0, 1, 2, 3, 4} (4 being the best, and 0 the worst) and its degradation process follow a
NHCTMP with the amount of time Ti, jspent in each state i before proceeding to the next
state j being Weibull distributed with scale parameter i, j¼1/(i 0.4j) and shape
parameter ¼ 3 i.e. Ti, jWeibull(1/(i 0.4j), 3). For this case, the Weibull distributions
for i ¼ 4 and j 2 {0, 1, 2, 3} are displayed in Figure 2(a) and their corresponding transient degradation rates are given by
i, jðtÞ ¼
3t2
ði 0:4j Þ3,
where i, j 2 {0, 1, 2, 3, 4}, i4j, and
j, jðtÞ ¼ X4 k¼0,k6¼j 3t2 ðj 0:4kÞ3: 0 1 2 3 4 5 6 7 8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 i = 4, j = 0 i = 4, j = 1 i = 4, j = 2 i = 4, j = 3 i = 4, j = 0 i = 4, j = 1 i = 4, j = 2 i = 4, j = 3 (a) (b)
Figure 2. (a) Weibull distributions with scale parameter ¼ 1/(i 0.4j) and shape parameter ¼ 3. (b) Transient degradation rates for i ¼ 4 and j 2 {0, 1, 2, 3}. (Available in colour online).
The transient degradation rate matrix I(t) in this case is IðtÞ ¼ 4,4ðtÞ 0 0 0 4,3ðtÞ 3,3ðtÞ 0 0 4,2ðtÞ 3,2ðtÞ 2,2ðtÞ 0 4,1ðtÞ 3,1ðtÞ 2,1ðtÞ 1,1ðtÞ 2 6 6 6 4 3 7 7 7 5 ¼ 0:3394t2 0 0 0 0:1367t2 0:5635t2 0 0 0:0916t2 0:2817t2 1:1074t2 0 0:0643t2 0:1707t2 0:7324t2 3t2 2 6 6 6 4 3 7 7 7 5: ð13Þ
Figure 2(b) shows the transient degradation rates for this multi-state tool wear.
4. State-by-state integration method
In order to determine the probability of the machine tool being in each state j at time t, Pj(t),
for j 2 {m, m 1, . . . , 0}, several differential equations described by Equation (10) have to be solved, which is somewhat difficult in practice. Traditionally, a truly multi-state deterio-ration process was reduced to a binary-state process for evaluation. However, by doing this, very significant instantaneous degradation information is lost and leads to suboptimal decisions concerning tool performance and its evaluation. In this section we present a state-by-state integration method based on Liu and Kapur (2008) with some modifications and extensions. By means of graphical aids and basic integration techniques, the problem of the multi-state deterioration process of tool wear becomes straightforward and easy to understand and solve. Figure 3 shows various state-by-state cases for the machine tool being in a particular state at time t. Using all possible cases, the probability that the machine tool is
in state j at time t, Pj(t), for j 2 {m, m 1, . . . , 1, 0}, can be determined.
. Case I: The probability that the machine tool will be in state m at time t is given by
PmðtÞ ¼e Rt 0m,mðÞd, ð14Þ where m, mðtÞ ¼ Xm k¼0,m6¼j m,kðtÞ:
. Case II: The probability that the machine tool will be in state m 1 at time t is given
by Pm1ðtÞ ¼ Z t 0 e R1 0 m, mðÞdm,m1ð1Þ e Rt 1m1,m1ðÞdd1: ð15Þ
. Case III: There are two possibilities for the probability that the machine tool will be
in state m 2 at time t:
. the tool degrades from m to m 2 directly, i.e. m ! m 2 (shown as the red line in Figure 3); or
. the tool degrades from m to m 1 and then from m 1 to m 2, i.e.
m ! m 1 ! m 2 (shown as the black line in Figure 3).
Similarly,
IELVIIð2:635Þ ¼ 935:5:
This result indicates that if the machining process was using Tool II between time 0 and time 2.635, the customer’s total expected loss value would be less (935.5 5 960.7), which means that Tool II is a better choice.
. Comparison of the total expected loss value for Tools II and III over the period (0, t* ¼ 1.175) (see Figure 7(d)): IELVIIð1:175Þ ¼ 100 Z 1:175 0 X4 ðtÞ¼0 ð4 ðtÞÞ2PðtÞðtÞdt ¼84:0: Similarly, IELVIIIð1:175Þ ¼ 45:0:
Clearly, Tool III gives the smaller total expected loss value to the customer over the period (0, 1.175), thus Tool III would be a better choice than Tool II. For all
three types of friction drill tools, since IELVI(1.175) ¼ 33.7 5 45.0 5 84.0, Tool I
is the best choice in this case.
6. Conclusions
Performance evaluation of the life characteristics of machine tools using traditional reliability methods use binary models for the state of the tool. Machine tools generally degrade over time and thus there are a range of states between perfect functioning and total failure. Recent research has also supported this idea for monitoring tool states and several states are used to classify the state of the tool. Binary models are too simple to evaluate the performance of a tool and its impact on downstream quality based on variation of the output due to tool wear. In addition, traditional measures based on success and failure models such as mean time between failures and failure rate do not capture the total impact on the quality of the product of tool wear over time. The performance measures developed in this paper capture the total experience of the manufacturer over the target life of the tool. The examples given in this paper illustrate how the proposed performance measures help with the evaluation process for tool design and selection from several alternative tools with different degradation processes.
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