Warranty Claim Cost Forecasting based on ARIMA Box-Jenkins Approach

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Warranty Claim Cost Forecasting based on ARIMA Box-Jenkins Approach

Nursafian Haris^1*, Ahmad Kadri Junoh², Safwati Ibrahim³, Amran Ahmed⁴, Wan Zuki Azman Wan Muhamad⁵

1 Faculty of Applied and Human Science, Universiti Malaysia Perlis, Arau, Malaysia

*Corresponding Author: [email protected]

Accepted: 15 June 2021 | Published: 1 July 2021

__________________________________________________________________________________________

Abstract: Warranty claims reported in recent months might carry more up-to-date information than those reported in earlier months. Depending on the technological development, forecasting of produced quantity rejection is main aspect of a manufacturing company to the plan of services; predict the approximate warranty cost and customer satisfactions. An attempt has been made to develop a forecasting model from the existing seasonal timely behaviour warranty claim cost using Box-Jenkins approach - Auto Regressive Integrated Moving Average (ARIMA) methodology for building forecasting model. This includes the observation and monitoring of warranty trend from the existing actual warranty data, by plotting warranty claim cost over Repair Month or Complaint Month. These trends have been deployed into statistical means of Box-Jenkins approach, to define the best Model Equation. The proposed model equation has a significant impact towards a reliable and convincing figure – another key factor in warranty budgeting and accrual task.

Keywords: ARIMA Box-Jenkins, Warranty Forecasting, Box-Cox Transformation

___________________________________________________________________________

1. Introduction

In today’s challenging world, any manufacturing organization strives for the highest financial efficiency, by making the highest profit with the most reduced cost. For automotive manufacturers and its vendors, it is needed to take into account, warranty claim as one of the costs as well. As most of the people underestimate warranty cost, this may lead to unexpected profit loss to the manufacturing organization.

As far as product quality is concerned, it is needed to take into consideration that deviations from the expected performance may occur when the vehicle is in use (Verband der Automobilindustrie, 2009). According to the 2009 General Motors annual report, the company had total revenue of US$104.2 billion and the future warranty cost on sold cars was estimated to be US$2.7 billion-about 2.6% of the annual sales turnover (Shafiee & Chukova, 2013). To assess warranty performance thoroughly, traditional methods are insufficient and improved assessment methods are needed to aid Vehicle Lighting management teams in decisions related to the anticipated incoming future cost. In order to keep up with the current assessment methods, making failure number forecasting is main aspect of a manufacturing company to the plan of services, resulting in prediction of approximate warranty cost and customer satisfactions (Ahammed & MesbahulAlam, 2012). To have a clear overview of warranty situation, considering the previous actual warranty cost data, an effective time series

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forecasting method needs to be derived, to represent warranty cost impact anticipated in the next coming years.

In this paper, the Time Series ARIMA or also called Box Jenkins method is proposed to solve the limitation of the previous traditional forecasting methods. It is the simplest and benchmarked methods, as far as time series forecasting is concerned (Singh & Mishra, 2015).

It is explained as a systematic method of identifying, fitting, checking, and using average (ARIMA) time series models.

2. Literature Review

Over time, time series trend can be explained by characterizing certain unique attributes (Nopiah et al., 2012). In trends of moving towards industry 4.0, time series data are also made into good use when monitoring real-time industrial processes or tracking corporate business metrics (Ismail et al., 2019). It is a innovative ground of analysis where almost all of the article found in literature are dedicated towards generic frameworks presentation or specific applications solution (Zurita et al., 2016). There are countless research methods available within forecasting patterns analysis, such as exponential smoothing models, ARIMA models and its variants including seasonal ARMAX models, vector ARIMA models using variable time series, and ARIMA models (Mishra & Jain, 2014). Nowadays, industrial time series modelling represents an area of growing interest for many researchers due to the convergence of industry and new information technologies.

Time Series ARIMA or also called Box Jenkins method is one of the simplest and benchmarked technique as far as time series forecasting is concerned (Singh & Mishra, 2015). It is described a process to identify, fit, check, and use average (ARIMA) time series models in an organized technique. The technique is suitable for medium to long length time series. As many statistical researchers and users are facing, various actual data sets are found in reality approximately not normal (Osborne, 2010), (Nwakuya & Nwabueze, 2018), (Chen & Takaishi, 2014), (Vélez et al., 2015). In the event where this is detected, usually in seasonal fluctuation set like in this paper, transforming seasonal non-normal dependent parameters into a normal form by means of Box Cox transformation is a way to make it happen.

ARIMA Box Jenkins method has been used extensively in almost across all applications and different purposes. For example, Li et al., (2013) suggested an ARIMA technique based on Hadoop framework, where an operative weather data evaluation and prediction system is managed to be executed. Eni & Adeyeye, (2015) have utilized hydrological data and manipulating it in a time series seasonal (ARIMA) models to forecast rainfall in Warri Town.

In terms of currency exchange rate, Mong & Ngan, (2016) explored ARIMA method with four steps to anticipate year 2016 exchange rate between Vietnam Dong and US Dollars, by manipulating factual foreign exchange rate data from January 2013 December 2015. On the other hand, Mondal et al., (2014) deployed ARIMA method on 56 Indian stocks prices forecasting, where each stocks are from various industries. Going to electricity load demand, Mohamed et al., (2010) succeeded implementing a double seasonal ARIMA prediction model using an actual 12 months half hourly Malaysia load demand data.

Rai & Singh, (2009) has considered ARIMA method for developing a monthly warranty spend forecasting model by identifying several important defect types, which are identified to have seasonality influences. For example, complaints like engine difficult to start usually have more tendency to occur during winter season compared to summer season, and more headlamp

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water-penetration complaints during rainy season. Other seasonal affected factor such as calendar month seasonality, dealers’ operating days per month, and increase of vehicle sales, are also incorporated in the ARIMA model.

The ARIMA unique crucial reference is Box and Jenkins (1970), and ARIMA models are occasionally named Box-Jenkins models. Should the variable violate the assumption of data stationary and no seasonality or any pattern according to scale or location, therefore the first thing to do is to make it stationary in preparation for Box-Jenkins model application (Ali et al., 2016). Stationarity concept of a stochastic method can be illustrated as a statistical equilibrium configuration (Hipel & McLeod, 1994).

Many statistical tests and intervals are based on the assumption and concept of normality (Sakia, 1992), (Gopal et al., 2017), (Ahmad et al., 2008). The Box Cox transformation is named after statisticians George Box and Sir David Roxbee Cox who collaborated on a 1964 paper and developed the technique (Box & Cox, 1964).

As data normality is an important assumption for many statistical techniques, if a data set isn’t normal, applying a Box-Cox provides the ability of running a broader number of tests. At the core of the Box Cox transformation is an exponent, lambda (λ), which varies from -5 to 5. All values of λ are considered and the optimal value for the data is selected. In Minitab statistical software, for Box-Cox transformation, the confidence interval of 95% was used to determine an appropriate transformation, as follows.

⚫ A λ value of 1 is equivalent to using the original data. Therefore, if the confidence interval for the optimal λ includes 1, then no transformation is necessary.

⚫ If the confidence interval for λ does not include 1, a transformation is appropriate.

Then, the transformation of Y has the form as below equation 1 and equation 2 (Sakia, 1992):

𝑦₁^𝜆 = {

𝑦₁^𝜆− 1

𝜆 , 𝑖𝑓 𝜆 ≠ 0 𝑙𝑜𝑔 𝑦_𝑖, 𝑖𝑓 𝜆 = 0

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This test only works for positive data. However, Box and Cox did propose a second formula that can be used for negative y-values.

𝑦₁^𝜆 = {

(𝑦₁+ 𝜆₂)^𝜆¹− 1

𝜆 , 𝑖𝑓 𝜆 ≠ 0

𝑙𝑜𝑔(𝑦_𝑖+ 𝜆₂) , 𝑖𝑓 𝜆 = 0

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Using plots of the data and outputs from ACF and PACF, a class of simple ARIMA models is selected, is denoted by ARIMA (p,d,q), where “p” stands for the order of the auto regressive process(AR), ‘d’ is the order of the data homogeneity and ‘q’ is the order of the moving average process(MA). The order of the homogeneity ‘d’, was determined by as a the number of times, series to be differenced to yield a stationary series until the ACF approached zero.

The ARIMA common formulation (p,d,q) is written as below (Judge et al., 1982).

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∆^𝑑𝑦_𝑡= 𝛿 + 𝜃₁∆^𝑑𝑦_𝑡−1 + 𝜃₂∆^𝑑𝑦_𝑡−2+ ⋯ +𝜃_𝑝𝑦_𝑡−𝑝+ 𝑒_𝑡−1𝛼𝑒_𝑡−1− 𝛼₂𝑒_𝑡−2𝛼_𝑞𝑒_𝑡−2 (3)

Where, ∆^𝑑 denotes differencing of order d, i.e., ∆𝑦_𝑡 = ∆𝑦_𝑡− ∆𝑦_𝑡−1, ∆²= ∆𝑦_𝑡− ∆𝑦_𝑡−1,

∆²𝑦_𝑡 = ∆𝑦_𝑡− ∆𝑦_𝑡−1, and so on. 𝑌_𝑡−1… 𝑌_𝑡−𝑝 are previous lags observations, δ,θ1 …… θp are factors (constant and coefficient) to be expected alike to regression coefficients of the Auto Regressive progression (AR) of order “p” represented by AR (p) and is formulated as:

𝑌 = 𝛿 + 𝜃₁𝑦₁+ 𝜃₂𝑦₂ + ⋯ + 𝜃_𝑡𝑦_𝑡−𝑝+ 𝑒_𝑡 (4) While, et is forecast error, assumed to be independently distributed across time with mean θ and variance θ2e, et-1, et-2……et-q are past forecast errors, α1,…… αq are moving average (MA) coefficient that needs to be estimated.

Where MA model of order q, MA (q) is formulated as below:

𝑌_𝑡= 𝑒_𝑡− 𝛼₁𝛼_𝑡−1− 𝛼₂𝛼_𝑡−2… 𝛼_𝑞𝛼_𝑡−𝑞 (5)

3. Methodology

All of the processes and steps are summarized as per below flow chart (Figure 1).

Figure 1: Flow Chart of Box Jenkins (ARIMA) method

Modeling time series by Box-Jenkins methodology has five main steps and procedures to go through as explained below (Mečiarová, 2007).

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Data Preparation

Data preparation is where the requirement on modeling and forecasting time series of having a set of stable data is illustrated in this study. This study selects monthly data for Warranty Claim Cost starting from January 2015 until December 2019. The data are collected from the Warranty Database.

Model Identification

The crucial task in ARIMA modeling technique is to choose the most appropriate values for the p, d, and q. This problem can be partially resolved by looking at the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) for the series (Nopiah et al., 2012), (Fattah et al., 2018), (Bakar & Rosbi, 2017). ACF is more crucial when examining stationarity and when selecting from various non-stationary models.

Model parameters estimation

This is done by using the suitable method in order to estimate numerical values of model parameters (Oliveira et al., 2017). After determining ‘d’, a stationary series Δ^dyt, its autocorrelation function and partial autocorrelation were examined to determined values of p and q, next step was to estimate the model using Minitab statistical software (Judge et al., 1988).

Diagnostic Verification of Model

To verify model in a qualitative techniques, it can be based on expert opinion and/or personal judgment (Alsharif et al., 2019). A time series plot of the model estimated parameters was a useful aid for judgment. In a well fitted model the residuals obtained are expected to have the property of white noise (Abdullah, 2012). This was done by checking the obtained results for inadequacies by considering the autocorrelations of the residual series (Dritsaki, 2015). Next, a time series plot of residuals was plotted using the obtained results. The applied model was declared as proper when the plot made a rectangular scatter around a zero horizontal level with no trend. On the other hand, identification of normality also functions as the third diagnostic verification. Model was declared normal and proper when model passed normality test. Then, it was plotted a histogram of the residuals. Finding out the fitness of good served as the final check. Residual against corresponding fitted values were plotted. When the plot showed no pattern, model was verified a good fit (Iqbal et al., 2005).

Forecast

As the main expected outcome of Box-Jenkins methodology, finally in the last phase, forecasting as the process of making predictions of the future with inputs of past and present data, by trend analyses done at step 1 - 4. This was where extrapolated prognosis was calculated of time series and also includes confidence lag of prognosis.

4. Result and Analysis

This section consists of three main parts, the first one is data validations for normality and seasonality. Second part is data transformation using Box Cox Transformation, and finally the third part is ARIMA implementation results of Warranty Cost forecasting.

This study adopted a quantitative research design in which real-time data was retrieved from the customer warranty database. Data consists of details of the complained vehicle including VIN number, service dealer location, complained part number, part names and types, repair

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costs, repair dates, vehicle assembly dates and other traceability-related information. For the purpose of this study, only repair dates and cost are useful and fully utilized.

4.1. Data Preparation

Warranty Claim Cost data have been validated and verified to pass Anderson Darling normality test (see figure 2 below), before further methodologies are proceeded.

700 600 500 400 300 200 100 0 99.9

99

95 90 80 7060 50 4030 20 10 5

1

0.1

Cost

Percent

Mean 383.6 StDev 97.53

N 60

AD 0.440

P-Value 0.282

Probability Plot of Warranty Claim Cost Normal

Figure 2: Probability Plot of Warranty Claim Cost

Box-Jenkins methodology on modeling and forecasting warranty claim cost (from January 2015 to December 2019). Plot of warranty claim cost is illustrated in figure 3.

Year Month

2019 2018

2017 2016

2015

Jul Jan Jul Jan Jul Jan Jul Jan Jul Jan 700

600

500

400

300

200

Cost

Time Series Plot of Warranty Claim Cost

Figure 3: Time Series Plot of Warranty Claim Cost.

4.2. Model Identification

Another critical effect on time series is the presence of seasonality (Rotela Junior et al., 2014).

As illustrated in figure 4, results of the first 3 lags of warranty cost, the autocorrelation coefficient is beyond bounds of statistical signification that is supported, which in this time series is seasonality. Slow decay is also detected, supporting seasonality data.

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15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

Lag

Partial Autocorrelation

Partial Autocorrelation Function for Warranty Claim Cost (with 5% significance limits for the partial autocorrelations)

Figure 4: Autocorrelation Function for Warranty Claim Cost

Figure 5: Partial Autocorrelation Function for Warranty Claim Cost

Warranty Claim Cost Data was then transformed using Box-Cox Transformation. 2 times transformation was required to achieve lambda (λ) rounded value of 1. See figure 6 for transformation 1 and figure 7 for the final transformation.

5.0 2.5 0.0 -2.5 -5.0 0.00030

0.00025

0.00020

0.00015

0.00010

Lambda

StDev

Lower CL Upper CL

Limit

Estimate 0.80 Lower CL -0.02 Upper CL 1.79 Rounded Value 1.00 (using 95.0% confidence)

Lambda (λ) Box-Cox Plot of Warranty Claim Cost T1

Figure 6: Box-Cox Transformation of Warranty Claim Cost

Figure 7: Box-Cox Transformation of Warranty Claim Cost T1

Then the stationary characteristics for first difference of Warranty Claim Cost is evaluated.

Figure 8 and 9 illustrates the ACF and PACF analysis for first difference of Warranty Claim Cost. ACF is still showing a significant spike on lag 1 and lag 12, which indicates that moving average (MA) can be represented by order of 2. PACF is also showing significant spike on lag 1, which indicates the autoregressive part may be represented by order of 1. Hence, the first difference of Warranty Claim Cost can be represented by ARIMA (2,1,1)

Figure 8: Autocorrelation Function for Warranty Claim Cost first difference.

Figure 9: Partial Autocorrelation Function for Warranty Claim Cost first difference.

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

Lag

Autocorrelation

Autocorrelation Function for Warranty Claim Cost (with 5% significance limits for the autocorrelations)

5.0 2.5 0.0 -2.5 -5.0 40

35

30

25

20

15

Lambda

StDev

Lower CL Upper CL

Limit

Estimate -0.80 Lower CL -1.79 Upper CL 0.02 Rounded Value -1.00 (using 95.0% confidence)

Lambda (λ) Box-Cox Plot of Warranty Claim Cost

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

Lag

Autocorrelation

Autocorrelation Function for Warranty Claim Cost first difference (with 5% significance limits for the autocorrelations)

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

Lag

Partial Autocorrelation Function for Warranty Claim Cost first difference (with 5% significance limits for the partial autocorrelations)

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4.3. Model Parameters Estimation

Now the estimation of model parameters for ARIMA (2,1,1) is presented. It is completed by using Minitab Statistical software, resulting as per below figure 10 and table 1. Blue line area is the estimated region, where center line is the estimation, upper and lower lines are 95%

confidence upper and lower limit.

Figure 10: Time Series Plot for Warranty Claim Cost ARIMA (2,1,1)

Table 1: Final estimates of Warranty Claim Costs

Type Coef. St. Dev. t-ratio p-value

AR1 -0.4320 0.4319 -1.00 0.323

AR2 0.2974 0.1542 1.93 0.061

SAR12 1.1451 0.7950 1.44 0.157

SAR24 -0.1558 0.7480 -0.21 0.836

MA1 -0.6075 0.4310 -1.41 0.166

SMA12 0.6069 0.8706 0.70 0.490

Differencing: 1 regular, 1 seasonal of order 12

Number of observations: Original series = 60, after differencing = 47

Residuals: SS = 291.893 (backforecasts excluded) MS = 7.119 DF = 41

Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 48

Chi-Square 7.4 17.3 19.6 * DF 6 18 30 * P-Value 0.288 0.503 0.927 *

4.4. Diagnostic Verification of Model

In figure 10 and table 1. In figure 10, two trend lines coloured in red are added as a guideline.

Using the result of this ARIMA model, it is compared with another ARIMA models F2(1,0,0), F3(1,0,1), F4(1,1,1), and F5(2,1,2). This is to tabulate and visualize appropriateness of ARIMA (2,1,1), also named F1, towards all these ARIMA models. From the observation, ARIMA F5(2,1,2) and ARIMA F4 (1,1,1) are out of the red lines and declared not a proper model.

84 78 72 66 60 54 48 42 36 30 24 18 12 6 1 700 600 500 400 300 200 100 0

Time

Cost

Time Series Plot for Warranty Claim Cost ARIMA(2,1,1) (with forecasts and their 95% confidence limits)

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Figure 11: ARIMA Model Parameter Estimations of ARIMA F1, F2, F3, F4 and F5.

Table 2: ARIMA Model Parameter Estimations of ARIMA F1, F2, F3, F4 and F5.

Repair

Month F1 (2,1,1) F2 (1,0,0) F3 (1,0,1) F4 (1,1,1) F5 (2,1,2)

Jan-20 293.52 296.82 294.36 293.70 295.06

Feb-20 257.20 257.25 257.79 255.32 257.10

Mar-20 231.27 227.58 230.84 227.12 228.84

Apr-20 202.00 194.94 200.96 196.03 197.67

May-20 198.00 187.04 195.16 190.00 191.47

Jun-20 195.91 184.09 191.36 185.21 185.62

Jul-20 183.90 175.20 181.66 173.87 173.45

Aug-20 186.61 178.18 181.09 172.11 170.62

Sep-20 190.94 183.15 185.31 175.91 174.42

Oct-20 196.47 189.10 188.13 177.75 175.59

Nov-20 206.63 202.97 200.80 189.87 186.79

Dec-20 217.72 217.83 214.41 203.12 199.03

Jan-21 229.67 236.46 230.22 217.72 215.57

Feb-21 197.05 197.33 197.47 179.32 178.32

Mar-21 175.05 167.98 173.63 151.27 150.44

Apr-21 149.51 135.70 147.04 120.41 119.76

May-21 149.39 127.90 142.79 114.67 114.01

Jun-21 148.56 124.99 140.38 109.55 107.81

Jul-21 134.00 116.21 132.45 97.78 94.98

Aug-21 136.57 119.18 132.98 95.20 91.38

Sep-21 140.11 124.11 137.94 98.78 95.01

Oct-21 145.27 130.01 141.55 99.91 95.53

Nov-21 151.63 143.75 154.30 111.70 106.15

Dec-21 158.89 158.47 167.90 124.65 117.80

Total 4475.831 4276.239 4420.514 3960.926 3922.373

Next, the property of white noise are inspected in the residuals for the first difference of Warranty Claim Cost ARIMA F1(2,1,1), F2(1,0,0), and F3(1,0,1) by performing ACF and PACF analysis. For the first difference of Warranty Claim Cost ARIMA F1(2,1,1),figure 12

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and figure 13 shows that residual is not significant. Therefore white noise properties exists in the residuals.[34]

Figure 12: Autocorrelation Function for Warranty Claim Cost F1(2,1,1) residuals.

Figure 13: Partial Autocorrelation Function for Warranty Claim Cost F1(2,1,1) residuals.

For the first difference of Warranty Claim Cost ARIMA F2(1,0,0), figure 14 and figure 15 shows that residual is significant as there are trends visible. Therefore residuals cannot be considered as white noise, and ARIMA F2(1,0,0) is declared not a proper model.

For the first difference of Warranty Claim Cost ARIMA F3(1,0,1), figure 16 and figure 17 shows that residual is not significant. Therefore white noise properties also exists in the residuals.

12 11 10 9 8 7 6 5 4 3 2 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

Lag

Autocorrelation

ACF of Residuals for Warranty Claim Cost ARIMA F1(2,1,1) (with 5% significance limits for the autocorrelations)

12 11 10 9 8 7 6 5 4 3 2 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

Lag

PACF of Residuals for Warranty Claim Cost ARIMA F1(2,1,1) (with 5% significance limits for the partial autocorrelations)

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

Lag

PACF of Residuals for Warranty Claim Cost ARIMA F2(1,0,0) (with 5% significance limits for the partial autocorrelations)

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

Lag

Autocorrelation

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

Lag

PACF of Residuals for Warranty Claim Cost ARIMA F3(1,0,1) (with 5% significance limits for the partial autocorrelations) 15

14 13 12 11 10 9 8 7 6 5 4 3 2 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

Lag

Autocorrelation

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Next, a time series plot of residuals was plotted using the obtained ARIMA F1(2,1,1) and ARIMA F2(1,0,1) results. Both figure 18 and figure 19 made a scatter around a zero horizontal level with no trend.

Figure 18: Time Series Plot for Warranty Claim Cost ARIMA F1(2,1,1) residuals.

Figure 19: Time Series Plot for Warranty Claim Cost ARIMA F2(1,0,1) residuals.

Function as the third diagnostic verification, figure 20 and figure 21 demonstrates identification of normality for both ARIMA model. Both passes normality test but only ARIMA F1(2,1,1) p- value has the best result of <0.005, ARIMA F2(1,0,1) scores 0.011.

Figure 20: Normality Test for Warranty Claim Cost ARIMA F1(2,1,1) residuals.

Figure 21: Normality for Warranty Claim Cost ARIMA F2(1,0,1) residuals.

Then, it was plotted a histogram of the residuals as in figure 22 and figure 23. both ARIMA models F1(2,1,1) and F2(1,0,1) are normal with mean value of -0.1189 and -0.5340 respectively. But ARIMA F1(2,1,1) has the better distribution compared to ARIMA F2(1,0,1).

Figure 22: Histogram for Warranty Claim Cost ARIMA F1(2,1,1) residuals.

Figure 23: Histogram for Warranty Claim Cost ARIMA F2(1,0,1) residuals.

60 54 48 42 36 30 24 18 12 6 1 10.0

7.5 5.0 2.5 0.0 -2.5 -5.0 -7.5 -10.0

Month

Residuals

Time Series Plot of Warranty Claim Cost ARIMA F1(2,1,1) Residuals

60 54 48 42 36 30 24 18 12 6 1 8 6 4 2 0 -2 -4 -6 -8

Month

Residuals

Time Series Plot of Warranty Claim Cost ARIMA F2(1,0,1) Residuals

10 5

0 -5

99

95 90 80 70 60 50 40 30 20 10 5

1

Residuals

Percent

Mean -0.1189 StDev 2.516

N 47

AD 3.106

P-Value <0.005

Probability Plot of Warranty Claim Cost F1(2,1,1) Residuals Normal

5 0

-5 -10

99.9

99 95 90 80 70 6050 40 30 20 10 5 1

0.1

Residuals

Percent

Mean -0.5340 StDev 2.369

N 60

AD 1.007

P-Value 0.011

Probability Plot of Warranty Claim Cost ARIMA F2(1,0,1) Residuals Normal

8 4

0 -4

-8 30

25

20 15

10

5

0

Residuals

Frequency

Mean -0.1189 StDev 2.516

N 47

Normal

Histogram for Warranty Claim Cost ARIMA F1(2,1,1) Residuals

6 3

0 -3

-6 20

15

10

5

0

Residuals

Frequency

Mean -0.5340 StDev 2.369

N 60

Normal

Histogram for Warranty Claim Cost ARIMA F1(1,0,1) Residuals

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Finally, fitness of good result is illustrated as in figure 24 and figure 25. Residual against corresponding fitted values plot for both ARIMA showed no pattern, therefore in this topic, both ARIMA model was verified a good fit.

Figure 24: Versus Fit for Warranty Claim Cost ARIMA F1(2,1,1) residuals.

Figure 25: Versus Fit for Warranty Claim Cost ARIMA F2(1,0,1) residuals.

4.5. Forecast

Now extrapolated prognosis is calculated of time series. Based on all the diagnostic verification and qualitative done, researcher decides to opt for ARIMA F1(2,1,1) as the optimum model to represent warranty claim cost forecast for the next 2 years from January 2020 to December 2021. Result is illustrated as in Table 2.

Table 2: Forecast Result using ARIMA F1(2,1,1) Repair Month F1 (2,1,1)

Jan-20 293.52

Feb-20 257.20

Mar-20 231.27

Apr-20 202.00

May-20 198.00

Jun-20 195.91

Jul-20 183.90

Aug-20 186.61

Sep-20 190.94

Oct-20 196.47

Nov-20 206.63

Dec-20 217.72

Jan-21 229.67

Feb-21 197.05

Mar-21 175.05

Apr-21 149.51

May-21 149.39

Jun-21 148.56

Jul-21 134.00

Aug-21 136.57

Sep-21 140.11

Oct-21 145.27

Nov-21 151.63

Dec-21 158.89

Total 4475.831

550 500 450 400 350 300 250 200 10

5

0

-5

Fitted Value

Residual

Versus Fits for Warranty Claim Cost ARIMA F1(2,1,1) (response is Cost)

700 600

500 400

300 200

7.5

5.0

2.5

0.0

-2.5

-5.0

Fitted Value

Residual

Versus Fits for Warranty Claim Cost ARIMA F2(1,0,1) (response is Cost)

(13)

Year Month

2021 2020

2019 2018

2017 2016

2015

Jan Jan

Jan 700 600 500 400 300 200 100 0

Cost (RM k)

0 Time Series Plot of Warranty Claim Cost (Actual & Forecasted)

Figure 26: Time Series Plot of Warranty Claim Cost (Actual & Forecasted)

5. Discussion and Conclusion

ARIMA model time series forecasting provides a good technique for any variables with fluctuated trends. The main advantage ARIMA models is that the method is suitable for any time series regardless of its pattern of trends, furthermore it does not require predictors or forecaster parameters to initiate calculation. But ARIMA models are only possible if it is a long time series data, involving more than 50 observations, which in this study involves 60 data. As other forecasting method, accurate and precise result is not assured by using this technique.

Nevertheless, this technique successfully implemented for preparing long time series data forecasting.

A clear overview of warranty situation is now visible to be presented, where ARIMA model for warranty claim cost has been successfully developed. As illustrated in figure 26, now the forecasted region is plotted over time (Repair Month). It demonstrated a match in terms of downtrending event, as what previous actual warranty claim cost data is demonstrating. An effective time series forecasting method successfully have been derived, to represent warranty cost impact anticipated in the next coming years. As we know, making defect cost forecasting, including warranty claim is main aspect of a manufacturing company to the plan of services, thus resulting in prediction of approximate warranty cost and increase customer satisfactions.

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