# A Bayesian Semiparametric Analysis of the Reliability and Maintenance of Machine Tools

TL;DR: A Bayesian semiparametric proportional hazards model is presented to describe the failure behavior of machine tools, and development of optimal replacement strategies are discussed.

Abstract: A Bayesian semiparametric proportional hazards model is presented to describe the failure behavior of machine tools. The semiparametric setup is introduced using a mixture of Dirichlet processes prior. A Bayesian analysis is performed on real machine tool failure data using the semiparametric setup, and development of optimal replacement strategies are discussed. The results of the semiparametric analysis and the replacement policies are compared with those under a parametric model.

## Summary (4 min read)

### 1. INTRODUCTION

- The useful life of a machine tool is the duration of time that the tool maintains an acceptable quality of performance.
- Because of the variation in individual characteristics of machine tools and the possible omission of relevant variables (describing the operational environment) from the model, a fully parametric PHM for tool life may not be adequate.
- Neither group considered the effect of covariate information, however.
- Neither a nonparametric nor semiparametric approach nor the use of covariates has been considered from a Bayesian point of view for developing optimal replacement strategies.

### 2. A BAYESIAN SEMIPARAMETRIC PROPORTIONAL HAZARDS MODEL

- This section discusses the use of a PHM to analyze machine tool failures under varying operating conditions.
- Unlike the parametric analysis of Mazzuchi and Soyer (1989), the authors develop a semiparametric inference using an MDP approach, with inference furnished using the ef cient algorithms of MacEachern (1994).
- This development is, therefore, a nontrivial combination of previously published techniques that allows a full analysis in this particular application.
- This section outlines the necessary methodology for this combination of techniques.

### 2.1 A Proportional Hazards Model for Machine Tool Failure

- The model has been widely applied in survival and reliability analysis.
- The parameters of the model are suppressed in the notation ‹i4t3Zi5.
- An alternative analysis of the parametric model was developed by Dellaportas and Smith (1993) using Markov chain Monte Carlo techniques, although an analysis of the machine tool failure data using these methods has not yet been published.
- Approaches for modeling the baseline cumulative failure rate function include the gamma process proposed by Kalb eisch (1978) and criticized by Hjort (1990), the extended gamma process presented by Laud, Damien, and Smith (1996), and the beta process presented by Hjort (1990), with a computational model developed by Laud et al. (1998).
- (For a full review of other semiparametric approaches to inference on regression models, see Gelfand 1999.).

### 2.2 A Mixture of Dirichlet Processes Prior for the Proportional Hazards Model

- Under the MDP approach, the baseline failure rate is assumed to be some continuous function ‹04t3 ˆi5, where ˆi is the vector of unknown parameters speci c to the ith machine tool.
- One way to model this uncertainty is to follow the development of MacEachern (1994) and West, Muller, and Escobar (1994) and describe uncertainty about G by a Dirichlet process prior denoted by G ¹ DP4G01M51 where G0 is the baseline prior and M is the strength of belief parameter.
- (See Ferguson 1973 for a discussion of Dirichlet process priors.).
- Speci cation of the semiparametric PHM is completed by specifying a parametric prior for the covariate effects ‚, denoted by 4‚5, which is independent of the ˆi’s.
- In addition to its exibility and ability to capture individual characteristics of the machine tools, the proposed semiparametric PHM also provides an assessment of the completeness of the set of covariates included in the analysis.

### 2.3 Posterior Inference and Prediction

- Instead of attempting to perform inference on the mixing distribution G directly, one can perform simple inference using the Markov chain Monte Carlo methods in algorithm 1 of Escobar and West (1995) to obtain a sample from the posterior distribution of ä and ‚ given the data D.
- For their problem, the attractive feature of this approach is that computation of 4ä1‚—D5 based on the Gibbs sampler can be achieved without sampling from the posterior distribution of 4G—‚1 D5, thus reducing the problem to n dimensions.
- Samples from this distribution can be obtained using the methods discussed by Dellaportas and Smith (1993) for the parametric model, because given ä, a conditionally parametric model is speci ed by (1).
- The authors approach follows that of Escobar and West (1995), assuming a priori that M follows an arbitrary prior 4M5.

### 3. ANALYSIS OF MACHINE TOOL FAILURE DATA USING PARAMETRIC AND SEMIPARAMETRIC INFERENCE

- The data used in this analysis, given in Table 1, were rst presented by Taraman (1974).
- Each experimental run used a workpiece material of SAE 1018 cold-rolled steel, 4 inches in diameter and 2 feet long.
- The 24 machine tools used for the cutting were tungsten carbide disposable inserts mounted in a tool holder.
- The cutting operations were performed without using cutting uids.
- The semiparametric inference follows the methods developed in Section 2.

### 3.1 Comparison of the Posterior Distributions of the Model Parameters

- For comparison of the semiparametric inference method to the parametric method proposed by Mazzuchi and Soyer (1989), the conditional baseline failure rate in (2) is assumed to be a Weibull density with scale parameter i and shape parameter ƒ.
- Figure 1 shows marked differences between the posterior distributions of the scale parameters.
- (See Kass and Raftery 1995 for further discussion of Bayes factors and their approximation.).
- Comparison of the posterior distributions of the covariate effect parameters, ‚1, ‚2, and ‚3, and the shape parameter, ƒ, is shown in Figures 3–6 graphically and given in Table 2 numerically.
- Under the semiparametric model, using Jeffreys’s scale there is still decisive evidence that cutting velocity has an effect, but there is also substantial evidence that feed rate and depth of cut affect the lifetime of machine tools.

### 3.2 Comparison of the Predicted Reliability

- The distribution of the predictive reliability of a given machine tool can be found using the sample approximations given in (6).
- The variance in the predicted mission time reliabilities is greater under the semiparametric model.
- Figure 9 shows the medians of the posterior predictive distributions of the reliabilities of the same machine tool at different mission times, with the dotted line indicating the parametric model and the solid line indicating the semiparametric model.
- The question now becomes which of these two predictions is better.

### 3.3 Comparison of the Predictive Ability

- The parametric and semiparametric models can be compared using posterior predictive densities, as discussed by Gelfand (1996), and the DIC, as described by Spiegelhalter et al. (2002).
- Thus in their analysis 100 partitions were selected at random, and the posterior predictive densities were calculated under each model.
- For the semiparametric model, the effective number of parameters is about 12, although the model includes 3 covariate effect parameters, a shape parameter, and potentially 24 scale parameters.
- In conclusion, each predictive comparison criterion shows strong evidence of the superior predictive ability of the semiparametric model over the parametric model.

### 4. SEMIPARAMETRIC OPTIMAL REPLACEMENT STRATEGIES FOR MACHINE TOOLS

- As in the original article by Taraman (1974) and the later work by Balakrishnan and DeVries (1985), the aim of machine tool life modeling is to aid decisions concerning the operation of machine tools.
- In this section the authors demonstrate how the MDP setup can lead to markedly different recommendations when compared with the parametric model.
- Under the age-replacement protocol, a planned replacement is made at age tA if the item survives until then, or an in-service replacement is made whenever the item fails.
- This protocol can be applied to individual machine tools.
- The machine tools are nonrepairable and thus operate under goodas-new (GN) replacement.

### 4.1 Replacing Individual Machine Tools

- The rst term represents the cost per unit time of in-service failures, and the second term represents the cost per unit time of planned replacements.
- D5 d‚ dä1 where D represents the observed failure and covariate data.
- This shows that some individual variation not explained by the three covariates can result in higher predicted machine tool reliability under the semiparametric model.
- 1 per unit time of the age-replacement protocol is predicted to be lower under the semiparametric model compared with the parametric model, again because the machine tool is predicted to be more reliable.
- For other covariate combinations this situation may be reversed.

### 4.2 Replacing a Group of Machine Tools

- Whereas the age-replacement policies are adequate for maintenance of a single machine tool, when several machine tools are to be maintained it may be more cost-effective to replace them as a group rather than individually.
- An alternative approach is to simulate the renewal process and approximate the conditional renewal function Mi4tB—ˆ0i1‚1 Zi5 (see, e.g., Ross 1989).
- The covariate values of the rst four machine tools listed in Table 1 are chosen to illustrate block replacement.
- The three tools have signi cantly lower scale parameter values than the other tools (see Fig. 1).
- The predicted optimal replacement intervals are now 27 for the parametric model and 33 for the semiparametric model—a larger difference, with the expected cost again lower for the semiparametric model due to the lower scale parameter values.

### 5. CONCLUSIONS

- In this article the authors have presented a semiparametric model developed for the analysis of machine tool failure data.
- The difference in the two models was apparent, with the semiparametric results favored because of its superior predictive ability.
- Such an analysis would require the use of time-dependent covariates or dynamic environment modeling techniques different from those used here.

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...Cox (1972) proposed that the distribution of Ti could be made dependent on Zi via the failure rate by assuming that the failure rate of the ith item is a product of a common base failure rate function and a function of the covariates, explicitly ‹i4t3Zi5 D ‹04t5e‚ T Zi 1 where ‚ is a vector of p…...

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...Instead of attempting to perform inference on the mixing distribution G directly, one can perform simple inference using the Markov chain Monte Carlo methods in algorithm 1 of Escobar and West (1995) to obtain a sample from the posterior distribution of ä and ‚ given the data D....

[...]

...Whereas the foregoing set-up leads to a complex analysis, inference is developed using the framework of Escobar and West (1995) and the ef cient sampling algorithms of MacEachern (1994)....

[...]

...Our approach follows that of Escobar and West (1995), assuming a priori that M follows an arbitrary prior 4M5....

[...]

...Instead of attempting to perform inference on the mixing distribution G directly, one can perform simple inference using the Markov chain Monte Carlo methods in algorithm 1 of Escobar and West (1995) to obtain a sample from the posterior distribution of ä and ‚ given the data D. For our problem, the attractive feature of this approach is that computation of 4ä1‚—D5 based on the Gibbs sampler can be achieved without sampling from the posterior distribution of 4G—‚1 D5, thus reducing the problem to n dimensions. Following the derivations of Escobar and West (1995), it can be shown that the full conditional for each ˆi is...

[...]

...Our approach follows that of Escobar and West (1995), assuming a priori that M follows an arbitrary prior 4M5....

[...]