## Abstract

Accurate prediction of product failures and the need for repair services become critical for various reasons, including understanding the warranty performance of manufacturers, defining cost-efficient repair strategies, and compliance with safety standards. The purpose of this study is to use machine learning tools to analyze several parameters crucial for achieving a robust repair service system, including the number of repairs, the time of the next repair ticket or product failure, and the time to repair. A large data set of over 530,000 repairs and maintenance of medical devices has been investigated by employing the Support Vector Machine (SVM) tool. SVM with four kernel functions is used to forecast the timing of the next failure or repair request in the system for two different products and two different failure types, namely, random failure and physical damage. Frequency analysis is also conducted to explore the product quality level based on product failure and the time to repair it. Besides, the best probability distributions are fitted for the failure count, the time between failures, and the time to repair. The results reveal the value of data analytics and machine learning tools in analyzing post-market product performance and the cost of repair and maintenance operations.

## 1 Introduction

Most products experience some sort of failure during their lifespan. Several reasons for product failures include insufficient reliability, poor quality, wrong design, uncertain usage, and production conditions [1]. Product failure can cause adverse effects such as financial cost, consumer dissatisfaction, and ruining manufacturers’ market image [2]. Product repair is often part of the product lifecycle events. For example, once the product reaches the end-of-use or the end-of-life, customers may decide to upgrade their product due to technological advances or repair, refurbish, or remanufacturer to extend the product lifespan [3].

The concept of end-of-use product recovery and extending the product lifecycle is nothing new in the literature. Previous studies have already discussed product recovery challenges that are facing different industries, such as building material stocks [4], automobiles [5], and household electric and electronic equipment [6]. Mathematical models have been developed to overcome some of those challenges (e.g., uncertain quality and quantity of waste stream) and help remanufacturers decide what to do with used devices [7,8].

Among the six common recovery strategies (e.g., reuse, repair, recondition/refurbishment, remanufacturing, cannibalization, and material recovery), repair has received less attention in the literature. However, predicting product failure has always been of interest to both academia and industry. Several studies have discussed the forecasting of product failure. To name a few, Yoon and Sohn built several random-effects regression models to forecast the mean time between failure (MTBF) [9]. Al-Garni et al. used the Weibull family to build a relationship with aircraft air-conditioning failure [10]. Wang and Yuan completed the failure rate prediction based on the AR model [11]. Wang and Yin adopted Weibull distribution to failure rate prediction [12]. Sexton et al. used different rules such as single-tag and rules-based to extract keywords to estimate median time to failure [13]. Brundage et al. also studied technical language processing by considering human-in-the-loop, natural language processing tools, and text analysis to analyze documents that contain the equipment maintenance data [14]. While the previous studies have discussed product failure and maintenance, the number of studies that have worked with real industry data sets is very limited. The practical insights derived from analyzing big industry data sets help us define proper repair and maintenance strategies for corporations toward cost reduction and sustainability goals.

This study aims to show how machine learning tools can help businesses analyze the overwhelming data set of product post-market performance and identify the depth and significance of the demand for repair and maintenance services. The Support Vector Machine (SVM) regression model simulates the product failure scenario and predicts the timing of the next repair service. The forecasting accuracy of SVM in regression is already confirmed by several studies [1517]. In this study, the data set is collected from 2004 to 2018 under different management policies. Although some bias and data set noise influence the decision to repair, the SVM model can still forecast failure time.

Besides, this study runs a frequency analysis to analyze the product quality based on the probability distributions of failure count, TBF, and time to repair (TTR). The frequency analysis provides information about each product maintenance condition.

This study provides an overview of products maintenance conditions by frequency analysis and offers a way to forecast product failure and the timing of the subsequent repair service request. Figure 1 shows the overall structure of this study.

Fig. 1
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## 2 Overview of Data Set

### 2.1 Elements of the Data Set.

In the current study, we investigate 536,597 records of repair and maintenance of medical devices, 90,278 individual products, and 9,351 product types from 2004 to 2018. The management policies may affect the collection and documentation of the repair data set and might have imposed some biases in how the data are reported. The data come from one of the largest health care providers in the United States. The data set includes 9351 different products with different functions and purposes, such as infusion pumps, pulse detection, and oximeters. The average number of repair orders is 35,773 per year or 98 repair orders per day. Therefore, the need for a maintenance plan is significant for the health care provider.

Table 1 shows an overview of the available fields of data for each repair and maintenance record. First Asset Number is the unique code to identify an individual product. First Asset Description is the information of product function, and First Asset Model Number is the name of each type of product. Date Created and Completed Date fields are the start time and the completion time on the repair. Based on the Date Created and Completed Date, we can identify TBF and TTR. The TBF is the time between an individual product’s failures, and the TTR is the repair start time to repair completion time. TBF and TTR are good indicators of product reliability and cost of repair. The long TBF and short TTR mean that a product has high reliability and may be easier to be maintained.

Table 1

The description of data set fields

Data attributeDescription
1. Work order numberThe work order of repair
2. Type codeProduct type
3. First asset numberProperty unique number
4. First asset descriptionFunction of product
5. First asset manufacturer nameManufacturer
6. First asset model numberProduct name
7. First asset serial numberUnique serial number
8. Date createdRepair start time
9. Completed dateRepair completion time
10. Repair attributeThe reason for repair
Data attributeDescription
1. Work order numberThe work order of repair
2. Type codeProduct type
3. First asset numberProperty unique number
4. First asset descriptionFunction of product
5. First asset manufacturer nameManufacturer
6. First asset model numberProduct name
7. First asset serial numberUnique serial number
8. Date createdRepair start time
9. Completed dateRepair completion time
10. Repair attributeThe reason for repair

Figure 2 shows the distribution of 35 repair reasons among the top 10 product categories. We have only relabeled the equipment names, including CA type 1, CA type 2, M8 type 1, PA type 1, PC type 1, MF type 1, CSE type 1, VE type 1, R type 1, and M8 type 2. We have not relabeled the repair codes, such as failure reasons and product type, to train the models. As shown in Fig. 2, 35 repair reasons are listed for the relabeled products. The repair attributes field in the data set provides specific details on why an individual product needs repair. For example, random failure (RF) is a physical failure with excessive stress of the device and can happen at any time [18,19].

Fig. 2
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Figures 3 and 4 show the number of RF and physical damage (PD) for the top 10 categories of products, respectively. The top 10 categories of products have the highest records of data among all 9351 product types. Among the top 10 categories, the CA type 1 and CA type 2 are selected for the forecasting part of the study due to enough sample size. The number of RF and PD is the highest among other failure reasons.

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According to Table 2, some product categories serve the same functions. For example, there are many products with infusion pumps and multitherapy functions. Four product categories, including CA type 1, M8 type 1, PA type 1, and M8 type 2, have the same function.

Table 2

The summary of function for the top 10 product categories

ProductFirst asset description
CA type 1Infusion Pumps, Multitherapy
CA type 2Controllers, Infusion Pump Systems, Programmable
M8 type 1Infusion Pumps, Multitherapy
PA type 1Infusion Pumps, Multitherapy
PC type 1Hospital Communication Systems, Nurse Call
MF type 1Infusion Pumps, Multitherapy, Syringe
CSE type 1Circulatory Assist Units, Peripheral Compression, Sequential
VE type 1Circulatory Assist Units, Peripheral Compression, Sequential
R type 1Oximeters, Pulse
M8 type 2Infusion Pumps, Multitherapy
ProductFirst asset description
CA type 1Infusion Pumps, Multitherapy
CA type 2Controllers, Infusion Pump Systems, Programmable
M8 type 1Infusion Pumps, Multitherapy
PA type 1Infusion Pumps, Multitherapy
PC type 1Hospital Communication Systems, Nurse Call
MF type 1Infusion Pumps, Multitherapy, Syringe
CSE type 1Circulatory Assist Units, Peripheral Compression, Sequential
VE type 1Circulatory Assist Units, Peripheral Compression, Sequential
R type 1Oximeters, Pulse
M8 type 2Infusion Pumps, Multitherapy

### 2.2 Data Frequency Analysis.

This section describes the frequency analysis of the failure count, TBF, and TTR. To analyze the reliability based on the above indices, we need to fit the best probability distribution for each product type. In the previous studies, Patil applied Normal, Lognormal, and Weibull distributions to reliability and maintainability analysis on hardware and software failures [20]. Lampreia et al. used the Weibull probability density function to analyze MTBF on the reliability analysis [21]. Sukhwani et al. applied Gamma distribution to analyze the reliability of NASA space flight software [22]. Tronskar et al. analyzed corrosion damage for pipelines and pressure vessels by using Gumbel distribution [23]. Four probability distributions, including Gumbel, Gamma, Normal, and Weibull distributions, have been applied to this study. The frequency analysis is also conducted for five categories of products: CA type 1, CA type 2, M8 type 1, PA type 1, and M8 type 2.

The best-fitted probability distributions for the failure count, TBF, and TTR are determined by the Akaike information criterion (AIC) and Bayesian information criterion (BIC). The AIC test that measures the discrepancy between the true model and simulated model is proposed by Akaike (1973) [24], and the BIC test is based on the Bayesian framework [25]. Several studies had applied the AIC or BIC to compute the best-fitted probability distributions and showed the AIC and BIC are good indices for this purpose [2629]. The probability distribution with the minimum AIC and BIC, among other distributions, is the best-fitted distribution. The AIC is constituted by the principle of maximum entropy and is expressed as
$AIC=2n−2ln(h^)$
(1)
where $h^$ indicates the maximum value of the likelihood function, and n is the number of probability distribution parameters. The BIC is constructed by the Bayesian framework and is given as
$BIC=ln(s)n−2ln(h^)$
(2)
where s expresses the number of samples.

Table 3 shows the best-fitted probability distributions of RF and PD. As seen, the best-fitted probability distributions are either Weibull or Gamma. It is reasonable as the histograms of failure count, TBF, and TTR are similar to exponential shape. The example shown in Figs. 5 and 6 illustrates the probability density function (PDF) and the cumulative distribution function (CDF) of failure count for RF in CA type 2.

Fig. 5
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Fig. 6
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Table 3

The best-fitted probability distribution of RF and PD

ProductFailure countTBF (year)TTR (10 days)
CA type 1Wei, WeiGam, GamWei, Wei
AIC: 4922, 737
BIC: 2947, 763
AIC: 926, 585
BIC: 944, 604
AIC: 3653, 5411
BIC: 3679, 5437
CA type 2Wei, WeiWei, GamWei, Wei
AIC: 1819, 583
BIC: 1842, 907
AIC: 604, 606
BIC: 627, 622
AIC: 6981, 4670
BIC: 7005, 4688
M8 type 1Wei, WeiWei, GamWei, Wei
AIC: 2109, 373
BIC: 2130, 387
AIC: 570, 342
BIC: 595, 346
AIC: 36948, 485
BIC: 36975, 465
PA type 1Wei, WeiGam, GamWei, Wei
AIC: 398, 410
BIC: 415, 427
AIC: 542, 461
BIC: 553, 471
AIC: 664, 999
BIC: 681, 1015
M8 type 2Wei, WeiGam, GamWei, Wei
AIC: 1388, 530
BIC: 1401, 540
AIC: 491, 445
BIC: 508, 446
AIC: 5208, 13214
BIC: 5225, 1325
ProductFailure countTBF (year)TTR (10 days)
CA type 1Wei, WeiGam, GamWei, Wei
AIC: 4922, 737
BIC: 2947, 763
AIC: 926, 585
BIC: 944, 604
AIC: 3653, 5411
BIC: 3679, 5437
CA type 2Wei, WeiWei, GamWei, Wei
AIC: 1819, 583
BIC: 1842, 907
AIC: 604, 606
BIC: 627, 622
AIC: 6981, 4670
BIC: 7005, 4688
M8 type 1Wei, WeiWei, GamWei, Wei
AIC: 2109, 373
BIC: 2130, 387
AIC: 570, 342
BIC: 595, 346
AIC: 36948, 485
BIC: 36975, 465
PA type 1Wei, WeiGam, GamWei, Wei
AIC: 398, 410
BIC: 415, 427
AIC: 542, 461
BIC: 553, 471
AIC: 664, 999
BIC: 681, 1015
M8 type 2Wei, WeiGam, GamWei, Wei
AIC: 1388, 530
BIC: 1401, 540
AIC: 491, 445
BIC: 508, 446
AIC: 5208, 13214
BIC: 5225, 1325

Note: wei, weibull; gam, gamma.

Table 4 represents the frequency analysis results for RF and PD with the best-fitted PDF. The P(X ≥ 1) for failure count means the probability that at least one failure happens. For TBF, it means the probability that an individual product has a normal operation of at least one year before it is broken. For TTR, it means the probability that an individual product needs at least ten days to be repaired.

Table 4

The probability (0 ∼ 100%) of RF and PD considering the best-fitted pdf: P(X ≥ 1) to P(X ≥ 5)

ProductRandom failure, physical damage
P(X ≥ 1)P(X ≥ 2)P(X ≥ 3)P(X ≥ 4)P(X ≥ 5)
Failure count
CA type 1100, 10040, 3232, 1328, 525, 2
CA type 2100, 10033, 3014, 126, 53, 2
M8 type 1100, 10043, 1426, 917, 612, 5
PA type 199, 10029, 2226, 724, 223, 1
M8 type 294, 10077, 0755, 237, 024, 0
TBF (year)
CA type 151, 5329, 2716, 139, 65, 3
CA type 244, 5523, 2712, 137, 64, 3
M8 type 139, 5819, 3610, 225, 143, 9
PA type 150, 4528, 2416, 139, 76, 4
M8 type 238, 6817, 428, 233, 122, 6
TTR (10 days)
CA type 140, 3133, 2229, 1726, 1424, 12
CA type 225, 2219, 1515, 1112, 0911, 7
M8 type 131, 4619, 2713, 179, 117, 7
PA type 120, 210, 0.36, 04, 03, 0
M8 type 239, 6630, 5026, 3923, 3121, 26
ProductRandom failure, physical damage
P(X ≥ 1)P(X ≥ 2)P(X ≥ 3)P(X ≥ 4)P(X ≥ 5)
Failure count
CA type 1100, 10040, 3232, 1328, 525, 2
CA type 2100, 10033, 3014, 126, 53, 2
M8 type 1100, 10043, 1426, 917, 612, 5
PA type 199, 10029, 2226, 724, 223, 1
M8 type 294, 10077, 0755, 237, 024, 0
TBF (year)
CA type 151, 5329, 2716, 139, 65, 3
CA type 244, 5523, 2712, 137, 64, 3
M8 type 139, 5819, 3610, 225, 143, 9
PA type 150, 4528, 2416, 139, 76, 4
M8 type 238, 6817, 428, 233, 122, 6
TTR (10 days)
CA type 140, 3133, 2229, 1726, 1424, 12
CA type 225, 2219, 1515, 1112, 0911, 7
M8 type 131, 4619, 2713, 179, 117, 7
PA type 120, 210, 0.36, 04, 03, 0
M8 type 239, 6630, 5026, 3923, 3121, 26

Comparing the failure count of RF and PD, the overall probability of RF is larger than PD. For example, the P(X ≥ 1) to P(X ≥ 5) of CA type 1 is from 100% to 25% on RF but is from 100% to 2% on PD. It reveals that random failure is the most repair reason for these products. The P(X ≥ 1) and P(X ≥ 2) of CA type 1 to CA type 2 on RF are almost similar, but the P(X ≥ 3) to P(X ≥ 5) of CA type 2 is smaller than that of CA type 1. It means that CA type 2 is more stable than CA type 1. Also, comparing CA type 1, M8 type 1, PA type 1, and M8 type 2 under the same functional purposes (Infusion Pumps, Multitherapy) on RF, the PA type 1 is more reliable among these products. Although P(X ≥ 4) and P(X ≥ 5) of PA type 1 are slightly higher than those of M8 type 1, the overall P(X ≥ 2) to P(X ≥ 3) are smaller among the same functional products. M8 type 2 is not reliable among products with the same function since it has a high-value P(X ≥ 2). Almost 77% of M8 type 2 products fail at least twice.

The overall P(X ≥ 1) to P(X ≥ 5) for TBF among RF and PD are very similar. The TBF reveals that products can normally operate at least 1 year to 3 years before they fail. CA type 1 and PA type 1 have higher P(X ≥ 1) on RF among all products. It means that CA type 1 and PA type 1 have higher probabilities of operating for more than 1 year, among others. M8 type 2 can endure more PD because it has a higher probability of P(X ≥ 1) to P(X ≥ 3) than others with the same functional purpose.

The TTR results reveal that CA type 1 and M8 type 2 are not easy to repair among all products. These two products have higher P(X ≥ 1) to P(X ≥ 5) than others, especially in P(X ≥ 5). Once a random failure occurs, it takes up to 50 days to repair the device. Also, CA type 1, M8 type 2, and M8 type 1 take a longer time to repair in the case of PD. Therefore, the cost of repair is likely higher for CA type 1 and M8 type 2. In contrast, PA type 1 is easier to be repaired among other products. Therefore, after analyzing Table 4, the PA type 1 category is more cost-effective for Infusion Pumps and Multitherapy functions.

## 3 Method: Support Vector Machine

Vapnik proposed support vector networks in 1995 to solve the classification problem and later developed it for regression analysis [30]. SVM has confirmed its forecasting accuracy through previous work [1517]. It has been applied to different fields for various predictions, such as dewpoint temperature prediction [31], stock prediction [32], and flood forecasting [16,33]. Further details on SVM can be found in Ref. [34].

According to Ref. [16], it is given the training set of [(x1, y1), (x2, y2), (x3, y3),…, (xn, yn)] with the input vector x and the target data y. The regression function can be expressed as
$y^=f(x)=wTΦ(x)+b$
(3)
where $y^$ is the output, w is the weight, Φ(x) is a nonlinear function, and b is the bias. According to the structural risk minimization (SRM) principle, the w and b can be derived by minimizing the structural risk function as follows
$MinR(w,b,ξ,ξ*)=12‖w‖2+c*∑i=1nLε(ξ+ξ*)$
(4)
$Subjectto{yi−y^i=y^i−(wTΦ(x)+b)≤ε+ξiyi−y^i=(wTΦ(x)+b)−y^i≤ε+ξi′ξi≥0ξi′≥0i=1,2,…,n$
(5)
where $c*$ is a penalty parameter for making a tradeoff between model complexity, and ɛ is the error tolerance as a range between target data y and output data $y^$. ξ and $ξ*$ are slack variables. The Lagrange multipliers with $aanda*$ can be used to solve the above-mentioned optimization problem
$f(x)=∑i=1v(ai−ai*)K(xk,x)+b$
(6)
where v is the number of support vector, xk is the support vector, and K(xk, x) is a kernel function. SVM has four kernel functions as follows:
Linear function:
$K(xi,xj)=xiT⋅xj$
(7)
Polynomial function:
$K(xi,xj)=(r⋅xiT⋅xj+c)d$
(8)
$K(xi,xj)=exp(−r‖xi−xj‖2)$
(9)
Sigmoid function:
$K(xi,xj)=tanh(r⋅xiT⋅xj+c)$
(10)
where r is a scaling factor, c is the bias term, and d is the degree term. This study uses the grid search method [35] to find parameters (r, $c*$, d, and ɛ). Figure 7 shows an overview of the SVM structure.
Fig. 7
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## 4 Forecasting the Time of Next Random Failure or Repair Service Ticket

### 4.1 The Input of Support Vector Machine.

Four different kernel functions, including Linear (LN), Polynomial (PL), Radial basis (RBF), and Sigmoid (SIG) functions, are selected to see which one has better forecasting performance. The input variable to the SVM model is the cumulative seconds, as illustrated in Table 5. The definition of cumulative seconds is the difference between the current failure time and the subsequent failure time. We should note that the current and upcoming failures may not be for the same individual product. The output of SVM is the forecasted time of the next failure. When the failure time is t + 1, t + 2, or t + 3, the SVM model will predict the next first, second, and third failure times in advance. The relationship of the failure time to the input factor is shown in Table 6. The T(t), T(t−1), and T(t−2) denote current, previous, and the second-to-last product failure time as an input factor, respectively. This study selects T(t−6), T(t−5), T(t−4), T(t−3), T(t−2), T(t−1), T(t) as input factors to forecast the time of the next failure since these input factors have a high correlation coefficient (CC) with output data. Also, when the time of failure increases, the number of input factors is reduced. The reason for reducing the input factor is to maintain the high correlation coefficient between input and output data.

Table 5

Example of calculating cumulative seconds (e.g., time of the first failure and the second failure)

OrderFailure timeCumulative seconds
12008-07-23 11:00:000 s (Time 1–Time1)
22008-07-24 08:00:0075600 s (Time 2–Time 1)
32008-07-28 09:00:00424800 s (Time 3–Time 1)
42008-08-08 09:00:001375200 s (Time 4–Time 1)
52008-08-08 09:00:001375200 s (Time 5–Time 1)
62008-09-04 09:00:003708000 s (Time 6–Time 1)
OrderFailure timeCumulative seconds
12008-07-23 11:00:000 s (Time 1–Time1)
22008-07-24 08:00:0075600 s (Time 2–Time 1)
32008-07-28 09:00:00424800 s (Time 3–Time 1)
42008-08-08 09:00:001375200 s (Time 4–Time 1)
52008-08-08 09:00:001375200 s (Time 5–Time 1)
62008-09-04 09:00:003708000 s (Time 6–Time 1)
Table 6

Basic information about CA type 1 and CA type 2

InformationCA type 1CA type 2
Training RF data set51413256
Testing RF data set22031395
Data start time2008-7-23 11:00:002011-6-25 14:00:00
Data end time2018-12-31 8:14:232018-12-27 16:16:40
Data set length (years)10.57.51
Failure time (order)Input factor
t + 1T(t−6), T(t−5), T(t−4), T(t−3,) T(t−2), T(t−1), T(t)
t + 2T(t−5), T(t−4), T(t−3,) T(t−2), T(t−1), T(t)
t + 3T(t−4), T(t−3,) T(t−2), T(t−1), T(t)
InformationCA type 1CA type 2
Training RF data set51413256
Testing RF data set22031395
Data start time2008-7-23 11:00:002011-6-25 14:00:00
Data end time2018-12-31 8:14:232018-12-27 16:16:40
Data set length (years)10.57.51
Failure time (order)Input factor
t + 1T(t−6), T(t−5), T(t−4), T(t−3,) T(t−2), T(t−1), T(t)
t + 2T(t−5), T(t−4), T(t−3,) T(t−2), T(t−1), T(t)
t + 3T(t−4), T(t−3,) T(t−2), T(t−1), T(t)

We have selected two products, including CA type 1 and CA type 2, for prediction since these two products have enough sample sizes and are commonly used in medical applications. Table 6 shows the basic information for CA type 1 and CA type 2.

The proportion of training and test sample sizes is 7 and 3. The available data on the start time and the end time for CA type 1 and CA type 2 are from July 23, 2008, 11:00:00 to December 31, 2018, 8:14:23 and from June 25, 2011, 14:00:00 to December 27, 2018, 16:16:40, respectively. The length of the data set is at least 7.5 years for both products. The forecasting results will be discussed in Sec. 4.2.

### 4.2 The Forecasting Results for Random Failure.

The training and testing results for SVM-LN, SVM-PL, SVM-RBF, and SVM-SIG are shown in Tables 7 and 8. According to the tables, the SVM-PL has better performance on training and testing, among other kernel functions. When the time is t + 1, t + 2, and t + 3, both SVM-LN and SVM-PL can forecast the time of the next random failure for CA type 1 and CA type 2 shown in Figs. 811. The codes SVM-LN, SVM-PL, SVM-RBF, and SVM-SIG mean the kernel of SVM, including linear, polynomial, radial basis, and sigmoid function, respectively. As time increases, it is obvious that the forecasting performance is decreased. Figures 9 and 11 provide specific details on the above-mentioned discussion. Based on Tables 7 and 8 and Figs. 811, the performance of SVM models with different kernel functions can forecast the next random failure without overfitting. For the indices such as root-mean-square-error (RMSE), mean absolute error (MAE), and CC, the test results have a good forecasting performance, as shown in each Figure. Each SVM model based on different kernel functions catches the trend of observation. Also, when the time is increasing, the SVM-PL will give overestimated values for both products. Although the SVM-PL will overestimate values as time increases, the model does not lose its forecasting performance with higher failure time. In Fig. 9 on orders 5184 to 5189 and Fig. 11 on orders 3277 to 3,283, the SVM-PL with time t + 3 provides a forecasting value near to the observation. The actual failure time and the forecasted failure time for these orders are shown in Table 9.

Fig. 8
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Fig. 9
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Fig. 10
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Fig. 11
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Table 7

The training and testing results for random failure in CA type 1

Failure time (order)Train, test
RMSE (s)MAE (s)CC
SVM-LN
t + 1400871, 49757116881, 405831.0, 1.0
t + 2518767, 61001137235, 466011.0, 1.0
t + 3674727, 80082156101, 553031.0, 1.0
SVM-PL
t + 1316582, 41586104599, 258921.0, 1.0
t + 2479382, 59246126297, 436091.0, 1.0
t + 3587345, 72843149849, 541311.0, 1.0
SVM-RBF
t + 1657926, 180843148978, 1454771.0, 1.0
t + 2767078, 186720166230, 1532021.0, 1.0
t + 3869516, 210683190072, 1740581.0, 1.0
SVM-SIG
t + 1561663, 68182132766, 474321.0, 1.0
t + 2728044, 81871160764, 573911.0, 1.0
t + 3820536, 89412192585, 631751.0, 1.0
Failure time (order)Train, test
RMSE (s)MAE (s)CC
SVM-LN
t + 1400871, 49757116881, 405831.0, 1.0
t + 2518767, 61001137235, 466011.0, 1.0
t + 3674727, 80082156101, 553031.0, 1.0
SVM-PL
t + 1316582, 41586104599, 258921.0, 1.0
t + 2479382, 59246126297, 436091.0, 1.0
t + 3587345, 72843149849, 541311.0, 1.0
SVM-RBF
t + 1657926, 180843148978, 1454771.0, 1.0
t + 2767078, 186720166230, 1532021.0, 1.0
t + 3869516, 210683190072, 1740581.0, 1.0
SVM-SIG
t + 1561663, 68182132766, 474321.0, 1.0
t + 2728044, 81871160764, 573911.0, 1.0
t + 3820536, 89412192585, 631751.0, 1.0
Table 8

The training and testing results for random failure in CA type 2

Failure time (order)Train, test
RMSE (s)MAE (s)CC
SVM-LN
t + 1107236, 9858378451, 673291.0, 1.0
t + 2156948, 132817126061, 1055251.0, 1.0
t + 3181359, 162836134489, 1201741.0, 1.0
SVM-PL
t + 195388, 8742771958, 569681.0, 1.0
t + 2133387, 12472491799, 845601.0, 1.0
t + 3168557, 149977132054, 1195831.0, 1.0
SVM-RBF
t + 1175961, 146894118780, 1132431.0, 1.0
t + 2201158, 167534135506, 1285701.0, 1.0
t + 3221640, 182546150537, 1431881.0, 1.0
SVM-SIG
t + 1128303, 12991688436, 889811.0, 1.0
t + 2171729, 146100126134, 1167601.0, 1.0
t + 3200389, 179164142448, 1324391.0, 1.0
Failure time (order)Train, test
RMSE (s)MAE (s)CC
SVM-LN
t + 1107236, 9858378451, 673291.0, 1.0
t + 2156948, 132817126061, 1055251.0, 1.0
t + 3181359, 162836134489, 1201741.0, 1.0
SVM-PL
t + 195388, 8742771958, 569681.0, 1.0
t + 2133387, 12472491799, 845601.0, 1.0
t + 3168557, 149977132054, 1195831.0, 1.0
SVM-RBF
t + 1175961, 146894118780, 1132431.0, 1.0
t + 2201158, 167534135506, 1285701.0, 1.0
t + 3221640, 182546150537, 1431881.0, 1.0
SVM-SIG
t + 1128303, 12991688436, 889811.0, 1.0
t + 2171729, 146100126134, 1167601.0, 1.0
t + 3200389, 179164142448, 1324391.0, 1.0
Table 9

The actual failure time and forecasting failure time with SVM-PL for orders 5184 to 5189 of CA type 1 and orders 3277 to 3283 of CA type 2

Actual failure timeForecasting failure time
t + 1t + 2t + 3
CA type 1
2017-05-02 09:20:162017-05-02 11:29:202017-05-02 14:07:122017-05-02 06:36:32
2017-05-02 12:07:122017-05-02 19:00:002017-05-02 17:07:282017-05-02 20:07:44
2017-05-02 23:59:442017-05-02 22:45:042017-05-03 02:53:042017-05-02 21:14:56
2017-05-02 23:59:442017-05-03 09:38:242017-05-03 04:22:402017-05-03 05:08:00
2017-05-03 15:22:562017-05-03 10:00:482017-05-03 17:30:562017-05-03 09:15:28
2017-05-03 15:24:002017-05-04 01:01:362017-05-03 19:46:242017-05-03 20:53:36
CA type 2
2016-09-01 10:28:002016-09-01 22:02:402016-09-01 06:14:082016-09-01 14:28:16
2016-09-02 08:08:162016-09-02 02:16:322016-09-02 06:30:082016-09-02 05:23:28
2016-09-02 15:09:522016-09-02 21:25:202016-09-02 08:03:442016-09-03 06:19:44
2016-09-02 15:10:402016-09-03 06:33:042016-09-03 06:19:442016-09-03 08:06:24
2016-09-03 21:29:522016-09-03 08:06:242016-09-03 13:40:322016-09-04 07:02:24
2016-09-04 13:56:162016-09-04 09:29:202016-09-03 13:27:122016-09-04 13:29:52
2016-09-04 14:02:242016-09-05 03:31:282016-09-04 19:30:402016-09-04 12:36:16
Actual failure timeForecasting failure time
t + 1t + 2t + 3
CA type 1
2017-05-02 09:20:162017-05-02 11:29:202017-05-02 14:07:122017-05-02 06:36:32
2017-05-02 12:07:122017-05-02 19:00:002017-05-02 17:07:282017-05-02 20:07:44
2017-05-02 23:59:442017-05-02 22:45:042017-05-03 02:53:042017-05-02 21:14:56
2017-05-02 23:59:442017-05-03 09:38:242017-05-03 04:22:402017-05-03 05:08:00
2017-05-03 15:22:562017-05-03 10:00:482017-05-03 17:30:562017-05-03 09:15:28
2017-05-03 15:24:002017-05-04 01:01:362017-05-03 19:46:242017-05-03 20:53:36
CA type 2
2016-09-01 10:28:002016-09-01 22:02:402016-09-01 06:14:082016-09-01 14:28:16
2016-09-02 08:08:162016-09-02 02:16:322016-09-02 06:30:082016-09-02 05:23:28
2016-09-02 15:09:522016-09-02 21:25:202016-09-02 08:03:442016-09-03 06:19:44
2016-09-02 15:10:402016-09-03 06:33:042016-09-03 06:19:442016-09-03 08:06:24
2016-09-03 21:29:522016-09-03 08:06:242016-09-03 13:40:322016-09-04 07:02:24
2016-09-04 13:56:162016-09-04 09:29:202016-09-03 13:27:122016-09-04 13:29:52
2016-09-04 14:02:242016-09-05 03:31:282016-09-04 19:30:402016-09-04 12:36:16

Since the observed trend is close to linear or polynomial function, SVM-LN and SVM-PL have a better performance than the other two models. SVM-PL is more suitable than SVM-LN since the trend of observation is like a polynomial function.

All models have a value of 1.0 for CC. They perform with similar trends, as shown in Figs. 811. Although all models have a similar trend, they do not guarantee small residual errors based on RMSE and MAE listed in Tables 7 and 8.

Figures 12 and 13 show the error between forecasting t + 1 and observation for CA type 1 and CA type 2. SVM-PL has a better performance than others. In addition, the trend of error has a cycle period in both Figures. Figure 12 shows the models overestimate the actual day, but sometimes the models can catch the actual failure time. Figure 13 also shows a similar trend, such as error with the cycle period. Based on the current observation, future work suggests training an error model that can forecast the next error between forecasting and actual failure time to reduce the error between forecasting and observation.

Fig. 12
Fig. 12
Close modal
Fig. 13
Fig. 13
Close modal

The outcomes provide helpful insights into enhancing the sustainability of medical devices. For example, the forecasting results provide information on product maintenance needs and the degree of repairability. Also, predicting time between failures gives repair service providers and technicians sufficient advance notice for predicting the resources needed for repair and proper resource management for repair services. Moreover, the prediction outcomes can be used for forecasting the amount of waste generated if the malfunctioning medical devices would not be repaired. Forecasting waste provides further information on how to define strategies for the reduction of greenhouse gas (GHG) emissions.

## 5 Conclusion

This study aims to forecast the arrival time of the next repair order or the failure time by applying the SVM model into a medical device data set with over 530,000 records of repair and maintenance activities. Four SVM kernel functions, including SVM-LN, SVM-PL, SVM-RBF, and SVM-SIG, have been compared for forecasting the next failure time of two categories of products labeled as CA type 1 and CA type 2. The results reveal that the SVM-LN and SVM-PL perform well in forecasting the time of the next repair request or failure time. Among all functions, SVM-PL has the best forecasting performance based on the RMSE and MAE values for testing.

In addition, frequency analysis has been run to analyze the data further and identifies the best-fitted distributions to the failure count, the time between failure, and the time to repair. Four probability distributions, including Gumbel, Gamma, Normal, and Weibull distribution, have been analyzed. Analyzing the failure count reveals that PA type 1 is more reliable among products with the same function. CA type 1 and PA type 1 have higher P(X ≥ 1) of TBF among all products in terms of random failure. M8 type 2 has a higher P(X ≥ 1) of TBF in terms of PD among other products with the same function. Finally, analyzing TTR reveals that CA type 1 and M8 type 2 are not easy to repair among all products.

The study shows the importance of machine learning tools such as SVM models in predicting the time of repair orders. Forecasting the time to failure and time to repair provides guidelines to designers for improving product repairability. The SVM model can further be used in forecasting the waste generation rate.

The study can be extended in several ways. First, other categories of products and types of failures can be compared together to identify the most vulnerable products and the type of repairs needed in each category. Second, the study outcome, which includes after-market repair and maintenance requests, can be linked to the product design features to identify design improvement directions. Third, the data analytics results can be fed to cost modeling techniques to quantify the business outcomes of repair services and identify the best repair strategies companies need to adopt. Decisions such as whether to handle repair services by in-house teams or outsource them to original equipment manufacturers or third-party repair service providers, depending on the demand and repair cost, can be informed by the current data analyses. Fourth, the study outcomes can be fed to resource allocation and scheduling models to help enterprises manage workforce requirements for repair services. Fifth, other machine learning tools can be employed to facilitate data analytics efforts. Finally, training an error model that can forecast the following error between forecasting and actual failure time will be further analyzed.

## Acknowledgment

This material is based upon work supported by the National Science Foundation–USA under Grant Nos. CMMI-2017968 and CBET-2017971. Any opinions, findings, and conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

• b =

bias

•
• d =

degree term

•
• n =

number of the probability distribution parameters

•
• r =

scaling factor

•
• s =

the number of samples

•
• x =

input vector

•
• v =

number of support vector

•
• w =

weight of support vector machine

•
• R =

structural risk function

•
• $c*$ =

penalty parameter

•
• $h^$ =

the maximum value of the likelihood function

•
• $y^$ =

target data

•
• xk =

support vector

•
• $a,a*$ =

Lagrange multipliers

•
• K(x, xk) =

Kernel function

•
• $Lε(y^)$ =

Vapnik’s ɛ-insensitive loss function

•
• ɛ =

error tolerance

•
• $ξ,ξ*$ =

slack variables

•
• Φ(x) =

nonlinear function

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