TWO DIMENSIONAL WARRANTY COST MODELS

TWO DIMENSIONAL WARRANTY COST MODELS (CASE STUDY : PT. INDOMOBIL PRIMA NIAGA SIDOARJO) Nuke Eva Novita (1214 100 029), Authors : Adhella Dwi Nur Saidah Pratiwi (1212 100 078) Prof. Dr. Basuki Widodo M.Sc., Ph.D. ABSTRACT A warranty is a contractual agreement between manufacturer and buyer, which requires the manufacturer to rectify all item failures either through repair or replacement should failure occur within the period specified in the warranty. Selling a product with warranty, means providing additional cost to manufactures, namely warranty cost. A warranty cost include into cost of product sales. In this Final project, warranty cost models using Weibull distribution for representing failure product. Two dimensional warranty models using two variabels i.e age and usage. This model has been applied to predict warranty cost to be borne by the manufacturer. Results in this Final Project are formation of expected two dimensional warranty cost models, expected number of failures is 5,40881 and expected warranty cost is Rp. 2.704.405, and shows that the estimation the number of failures product and the warranty cost increase along with the increase number of age and usage of the products. Keywords : Warranty cost, Weibull distribution, Products failures, Repair

I.

Introduction

In the era of globalisation, each company must have a strategy to get customers through the interesting offer and maintain its consumers through a satisfaction guarantee. One of the strategies that can be taken by the company to achieve that goal is to provide the after-sales service (after sales) on consumers that one of the forms is an offer to guarantee (warranty). Warranty is contractually obligated to agreement between the manufacturer and consumers, where the manufacturer is willing to do the repairs or replacement of the product against damage during the warranty period that has been determined. Warranty for the benefit to the manufacturer and consumers. From the point of view of consumers, warranty guaranteed protection against defects that occur during the warranty period. For the manufacturer, warranty provide benefits to protect from consumer claims that cause the manufacturer losses. According to the dimensions warranty policy differentiated are grouped into two policy of a onedimensional warranty policy warranty policy and two-dimensional [1]. One-dimensional dikarakteristikkan warranty policy by one the attribute product age (for example: age) or wearing (for example: distance). While the policy guarantee two-dimensional dikarakteristikan by two, where one-dimensional attribute represents the time and the other dimension represents the usage. To perform repairs (retrifikasi), producers have two options action rektifikasi, namely improvement (repair and replacement (replace)[1]. Repair actions provide the cost is smaller than the replacement, although the cost of replacement is greater than the cost of the

improvement but rektifikasi with changes have probability smaller damage for the remainder of the warranty period. The failure of a product is usually due to an error procedure in the making process or system error against marketed product. The cost of the guarantee is part of the cost of product sales. The greatness of the estimation of the cost of the high warranty will make the selling price of a product to high so that the product is not competitive price. Studies of the cost of the guarantee has been done by earlier researchers. Previous research conducted by C. S. Kim, I. Djamaludin, D. N. P. Murthy [2] discuss about the cost of warranty on a one-dimensional model. The results of the research on the optimal warranty cost ekspetasi on warranty one-dimensional model. On the research considering only one variable only the usage of the product. Then the research done by N. Jack, B. P. Iskandar, D. N. P. Murthy [3] discuss two-dimensional model that is the age of the product and the usage of the product. The results of the research that determine the strategy improvement in determining the cost of warranty. The research. The research developed by Yukun Wang, Zixian Liu, Yiliu liu [4] to discuss about the maintenance strategy for products that can be repaired in two-dimensional model. The results from the research namely ekspetasi optimal maintenance costs. Further research done by J. Good, D. N. P. Murthy, N. The Jack [5] to discuss about the warranty twodimensional model and product penggantia when damage. The results from the research namely ekspetasi the cost of warranty replacement products when damage during the warranty period specified by the manufacturer.

II.

Problem Statement This paper discuss about the modeling warranty cost in the case of two-dimensional model with repairs when damaged

III.

Mathematics Modelling N. Raphson method used to complete the non-linier equation system. Assume known non-linear equation system with n similarities and n variables which are written in the following form :

Where :

IV.

Analytic / Numeric Solution

According to the data from the damage to the Appendix A which contains data about the engine damage on the truck types of 300 series on PT Indomobil Prima Niaga Sidoarjo, namely the offer warranty for 3 years or maximum usage 100,000 km on the Appendix A obtained 23 claims data stated valid. 

Estimation of parameters

for mengestimasi 4.3.1 one method of parameter that is used by the method of Maximum Likelihood Estimation (MLE). Known solid function opportunity to Weibull distribution is as follows : 𝑓(𝑥, 𝑦) =

𝛽1 𝛽2 𝜃1 𝛽1 𝜃2 𝛽2

𝑥 𝛽1 𝑦 𝛽2 −( ) −( ) 𝜃 𝜃 2 𝑒 1 𝑥 𝛽1 −1 𝑦 𝛽2−1

Then the likelihood function is : L(𝜃̂) = ∏𝑛𝑖=1 𝑓𝑖 (𝑥, 𝑦) , 𝑤𝑖𝑡ℎ 𝜃̂ = 𝜃1 , 𝜃2 , 𝛽1 , 𝛽2 𝑛

=∏ 𝑖=1

=(

𝛽1 𝛽2 𝜃1 𝛽1 𝜃2 𝛽2

𝛽1 𝛽2 𝜃1 𝛽1 𝜃2 𝛽2

𝑥 𝛽1 𝑦 𝛽2 −( ) −( ) 𝜃 𝜃 1 2 𝑒 𝑥 𝛽1 −1 𝑦 𝛽2−1

𝑛

) 𝑒

𝑥 𝛽1 𝑦 𝛽2 ∑𝑛 𝑖=1(−(𝜃1 ) −(𝜃2 ) )

𝑛

∏ 𝑥 𝛽1 −1 𝑦 𝛽2 −1 𝑖=1

The function of the log-likelihood is : LnL(𝜃̂) = ln [(

𝛽1 𝛽2 𝜃1 𝛽1 𝜃2

𝑛

𝛽2 ) 𝑒

∑𝑛 𝑖=1(−(

𝑥 𝛽1 𝑦 𝛽2 ) −( ) ) 𝜃1 𝜃2

𝑥

1) ∑𝑛𝑖=1 ln𝑥𝑖 + (𝛽2 − 1) ∑𝑛𝑖=1 ln𝑦𝑖 − ∑𝑛𝑖=1 ( 𝑖 )

∏𝑛𝑖=1 𝑥 𝛽1 −1 𝑦 𝛽2 −1 ] = 𝑛(ln𝛽1 + ln𝛽2 − 𝛽1 ln𝜃1 − 𝛽2 ln𝜃2 ) + (𝛽1 − 𝛽1

𝑦

− ∑𝑛𝑖=1 ( 𝑖 )

𝜃1

𝛽2

𝜃2

(1)

To get the parameters of assessment 𝛽1 , 𝛽2 , 𝜃1 , 𝜃2 Then the equation (1) revealed then equals zero, so 𝒏

∂L(𝜃̂) n𝛽1 𝛽1 𝑥𝑖 𝛽1 =− + ∑( ) = 0 ∂θ𝟏 θ𝟏 θ𝟏 θ𝟏

(𝟐)

𝒊=𝟏 𝒏

∂L(𝜃̂) n𝛽2 𝛽2 𝑦𝑖 𝛽2 =− + ∑( ) = 0 ∂θ𝟐 θ𝟐 θ𝟐 θ𝟐

(𝟑)

𝒊=𝟏

𝑛

𝑛

𝑖=1

𝑖=1

𝑛

𝑛

∂L(𝜃̂) 𝑛 𝑥𝑖 𝛽1 𝑥𝑖 = − 𝑛 lnθ1 + ∑ ln 𝑥𝑖 − ∑ [( ) ln ( )] = 0 ∂𝛽1 𝛽1 θ𝟏 θ𝟏 ∂L(𝜃̂) 𝑛 𝑦𝑖 𝛽2 𝑦𝑖 = − 𝑛 lnθ2 + ∑ ln 𝑦𝑖 − ∑ [( ) ln ( )] = 0 ∂𝛽2 𝛽2 θ𝟐 θ𝟐 𝑖=1

𝑖=1

To get the parameter values using the newton raphson method. ̂

̂

̂

̂

1

𝟐

1

2

∂L(𝜃) ∂L(𝜃) ∂L(𝜃) ∂L(𝜃) 𝑔1 (𝜃̂) = ∂𝜃 , 𝑔2 (𝜃̂) = ∂θ , 𝑔3 (𝜃̂) = ∂𝛽 , 𝑔4 (𝜃̂) = ∂𝛽 then

θ1

(𝑖)

= θ1

(𝑖−1)

+ 𝑒1 (𝑖−1)

θ2

(𝑖)

= θ2

(𝑖−1)

+ 𝑒2 (𝑖−1)

β1

(𝑖)

= β1

(𝑖−1)

+ 𝑒3 (𝑖−1)

β2

(𝑖)

= β2

(𝑖−1)

+ 𝑒4 (𝑖−1)

Where,

(4)

(5)

With the value of each element in the matrix errornya as follows :

Using the software Matlab obtained the parameters namely : 𝜃1 = 0.8, 𝜃2 = 0.4, 𝛽1 = 0.1, 𝛽2 = 0.1

Conclusion From the analysis and the discussion that has been presented in the previous chapter, it can be concluded that : 1. The formation of the model of two-dimensional ekspetasi warranty costs are as follows : E (P)= 𝐶𝑠 𝑀 (𝑢, 𝑤) 𝐶𝑠 : The average cost that must be issued by the manufacturer to repair a product (per item) who experience failure. 𝑤 𝑢 𝛽1 ( ) 𝑀(𝑢, 𝑤) = ( ) (ln |𝑒 𝜃2 𝜃1

𝛽2

(

𝑤 𝛽2 ( ) 𝑒 𝜃2

+ 𝑑𝑖𝑙𝑜𝑔 (

1−𝑒 + ln |𝑒

(

𝑢 ) 𝜃1

(

𝛽1

+𝑒

𝑢 𝛽1 ) 𝜃1

1

)

𝛽2

(

𝑤 𝛽2 ) 𝜃2

1 𝑢 ( ) − 𝑒 𝜃1

𝛽1

𝑤 𝛽2 ( ) + 𝑒 𝜃2

− 1| ln |

1−𝑒 𝑒

(

(

𝑤 𝛽2 ) 𝜃2

𝑢 𝛽1 ) 𝜃1

|

)

𝑒

− 1| ln |

(

𝑤 𝛽2 ) 𝜃2

1−𝑒 − 𝑑𝑖𝑙𝑜𝑔 (

𝑤

𝑢 𝑤 𝛽2 1 − 𝑒 𝜃2 ( ) − 1 − |( ) ) − 𝑑𝑖𝑙𝑜𝑔 ( ) − ln |𝑒 𝜃1 𝛽1 𝑢 𝜃2 ( ) 𝑒 𝜃1

) − ln |𝑒 𝛽1

(

(

𝑢 ) 𝜃1

𝛽1

𝑢 𝛽1 ) 𝜃1 | ln |

| − 𝑑𝑖𝑙𝑜𝑔 (𝑒

1 1−𝑒

(

𝑢 𝛽1 ) 𝜃1

(

𝑤 𝛽2 ) 𝜃2 )

2

1 𝑤 𝛽2 1 − (( ) ) + 𝜋 2 2 𝜃2 6

|

2. The cost of the guarantee will increase along with the increase in the amount of usage and the age of the related product.

Reference Dwi, Adhella. (2016). “Pemodelan Biaya Garansi Dua Dimensi”. Tugas Akhir S1 Jurusan Matematika ITS Surabaya Widodo, Basuki. (2012). “pemodelan Matematika”. ITS Press : Surabaya.