optimal spare ordering policy under a rebate warranty

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European Journal of Operational Research 186 (2008) 708–719

www.elsevier.com/locate/ejor

Stochastics and Statistics

Optimal spare ordering policy under a rebate warranty

Yu-Hung Chien a,*, Jih-An Chen b

a Department of Applied Statistics, National Taichung Institute of Technology, 129, Sec. 3, San-min Road,

Taichung City, Taichung 404, Taiwanb Department of Business Administration, Kao-Yuan University, 1821, Chung-Shan Road, Lu-Chu Hsiang,

Kaohsiung County 821, Taiwan

Received 18 February 2006; accepted 3 February 2007Available online 23 March 2007

Abstract

This paper develops, from the customer’s perspective, the optimal spare ordering policy for a non-repairable productwith a limited-duration lifetime and under a rebate warranty. The spare unit for replacement is available only by order andthe lead time for delivery follows a specified probability distribution. Through evaluation of gains due to the rebate and thecosts due to ordering, shortage, and holding, we derive the expected cost per unit time and cost effectiveness in the long runand examine the optimal ordering time by minimizing or maximizing these cost expressions. We show that there exists aunique optimum solution under mild assumptions. We provide a numerical example and illustrate sensitivity analysis.� 2007 Elsevier B.V. All rights reserved.

Keywords: Spare; Replacement; Ordering policy; Rebate warranty; Cost per unit time; Cost effectiveness

1. Introduction

In this paper, we consider a general spare ordering policy for a product under a rebate warranty. A rebatewarranty is one of the most common types of warranty policies. Under a rebate policy, the manufacturerrefunds a customer some proportion of the sales price if the product fails during the warranty period. Com-mon examples of products sold under rebate policies include batteries and tires. In contrast, a failure-free war-ranty, another common type of policy, obligates the manufacturer to maintain the product free of chargeduring the warranty period. Products sold under failure-free warranties might include electronics or householdappliances. Rebate policies also take two common forms: lump sum and pro rata rebates; failure-free policiesusually fall into two categories: renewing and non-renewing. Background and discussions of warranty policiesand related issues can be found in [15,2–4,16,14].

Recent research on warranty policies has focused on preventive maintenance (PM). For example, Yeh andLo [23] considered pre-specified warranty periods and jointly determined the optimal number of PM actions,

0377-2217/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2007.02.026

* Corresponding author. Tel.: +886 4 22196660; fax: +886 4 22196331.E-mail address: [email protected] (Y.-H. Chien).

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Y.-H. Chien, J.-A. Chen / European Journal of Operational Research 186 (2008) 708–719 709

the corresponding degrees of PM, and the maintenance schedule. Jung and Park [13] found optimal periodicPM policies following the expiration of failure-free renewing and non-renewing warranty periods. Yeh et al.[24] investigated the effects of a free-replacement renewing warranty on the age-replacement policy for a non-repairable product. Chien [6] determined the optimal out-of-warranty replacement age from the buyer’sperspective under a failure-free renewing warranty. Chen and Chien [5] also developed a framework to studycontinuous PM actions for products sold with a renewing warranty, and determined the optimal PM strategiesfrom buyer’s and the manufacturer’s perspective, respectively. Most of these papers assumed that whenever aproduct fails during the warranty period or after the warranty expires, a new one is immediately available forreplacement. However, this might not be true in many situations. As a simple example, the distributor may runout of stock and need to order a replacement; clearly, this can incur substantial costs for customers if the pro-duct is necessary for ongoing business operations.

Jhang [12] was the first to consider the lead time for product replacement under warranty. He derived theoptimal use period of a repairable product after its warranty expires assuming replacements are ordered anddelivered as need. However, he assumed that the lead time for delivery is fixed. Park and Park [18], Sheu andLiou [21], Sheu [20], Sheu and Chien [22], and Chien [7] studied questions related to spare ordering policieswith random lead times but did not consider warranties.

In this paper, we consider a non-repairable product sold under a rebate warranty and develop an optimalspare ordering policy from the customer’s perspective. We assume that the lead time for spare delivery followsa known probability distribution. By evaluating the benefits due to the rebate warranty and the costs due toordering, shortage, and holding, we derive optimality criteria based on expected long-run per unit time costsand cost effectiveness. We then obtain the optimum ordering times by minimizing per unit time costs and bymaximizing cost effectiveness. We show that there is a unique solution under mild conditions. In addition, wefind that the optimal spare ordering time that minimizes per unit time costs depends directly on the warrantyperiod.

The remainder of this paper is organized as follows. Section 2 describes our notations and model assump-tions. Sections 3 and 4 develop measures of expected per unit time cost and cost effectiveness, respectively.Based on the cost models, we derive the optimal spare ordering times that minimize per unit time costsand maximize cost effectiveness and summarize their structural properties in Theorems 1 and 2, respectively.In Section 5, sensitivity analysis is carried out in a numerical example. We present conclusions in Section 6.

2. Model description and assumptions

Consider a non-repairable product sold under a rebate warranty. Under this policy, the customer isrefunded a proportion of the sales price Cp if the product fails during the warranty period [0,W]. The refundamount, R(x), is a function of the failure time x. In this paper it is assumed that R(x) is a linear function of x

[17]; that is,

RðxÞ ¼kCp 1� ax

W

� �; for 0 6 x 6 W ;

0; for x > W ;

(ð1Þ

where 0 < k 6 1 and 0 6 a 6 1. Two special cases of this general form of R(x) are the lump sum rebate policy(when a ¼ 0) and the pro rata rebate policy (when a ¼ 1 and k ¼ 1).

We assume that the product is inherently unreliable in the sense that it will eventually fail. A product failurecan occur early due to manufacturing defects or late due to degradation (i.e., wear-and-tear). We also assumethat the product has a finite useful lifetime T at which point it is discarded without repair. If the product failsbefore time T, the spare unit for replacement is provided only by an order from the supplier, and the deliveringtime L is a random variable which follow a distribution function G(y). Specifically, assuming the original prod-uct is purchased at time 0 and fails before a specified time t0 ð0 6 t0 6 T Þ, an expedited order are executedimmediately at the failure time instant, and the failed product is replaced by the new one as soon as it is deliv-ered. On the other hand, if the product has not failed by t0, a spare for replacement is regular ordered in antic-ipation of failure at time t0, and the original product is replaced when it fails. After failure and replacement,this process repeats. We consider the time between successive replacements one cycle.

710 Y.-H. Chien, J.-A. Chen / European Journal of Operational Research 186 (2008) 708–719

The spare ordering time t0 is a decision variable. By accounting for the gains due to the rebate warranty andthe costs due to ordering, shortage, and holding, we derive cost models and examine costs per unit time andcost effectiveness as criteria for optimal choice of t0. Costs per unit time represent expected costs over an infi-nite time horizon, and cost effectiveness is defined as

s-availability

s-expected out of pocket cost rate;

(where ‘‘s-’’ implies statistical meaning) and balances system effectiveness and money invested.We use the following notational conventions.

X lifetime of a productW warranty expiration dateT useful lifetime limit of a product ð0 < T <1Þf ð�Þ, F ð�Þ pdf and Cdf of X

F ð�Þ survival function of X ðF ð�Þ ¼ 1� F ð�ÞÞrð�Þ failure rate function of Xðrð�Þ ¼ f ð�Þ=F ð�ÞÞL random lead time for delivering a sparegð�Þ, Gð�Þ pdf and Cdf of L

Gð�Þ survival function of L ðGð�Þ ¼ 1� Gð�ÞÞlL mean lead time ðE½L� ¼

R10 GðyÞdyÞ

Coe cost for an expedited orderCor cost for a regular orderCp product sales price per unitCh holding cost per unit timeCs shortage cost per unit time resulting from a failed product ðCs ¼ Cs1 þ Cs2, where Cs1 and Cs2 repre-

sent the out of pocket and opportunity costs, respectively)t0 ordering time for a spare unitCRðt0Þ the expected cost per unit time over an infinite time horizonCEðt0Þ cost effectiveness over an infinite time horizon

3. The cost per unit time minimization model

Let X �i denote the length of the ith replacement cycle and R�i the operational cost over the renewal intervalX �i for i ¼ 1; 2; 3; . . .. Also, let X 1;X 2;X 3; . . . be independent copies of X, the lifetime of a product. From therenewal reward theorem (see Ross [19, p. 52]), the expected cost per unit time over an infinite time horizon isthe expected cost per cycle divided by the expected cycle length: E½R�1�=E½X �1�. Since the product is under arebate warranty, the expected cost per cycle is the sum of the ordering, shortage, and holding costs less thegains due to the rebate if the product fails during the warranty period [0,W]. We derive the expressions forE½X �1� and E½R�1� separately for the cases t0 < W and t0 P W .

Case 1: t0 < W . In this case, the first replacement cycle X �1 and its corresponding operational cost R�1 can beexpressed as

X �1 ¼

X 1 þ L; if X 1 < t0;

t0 þ L; if t0 6 X 1 < W and X 1 < t0 þ L;

X 1; if t0 6 X 1 < W and t0 þ L < X 1 ði:e:; t0 þ L < X 1 < W Þ;X 1; if W 6 X 1 < T and t0 þ L < X 1;

t0 þ L; if W 6 X 1 < T and X 1 < t0 þ L;

T ; if t0 þ L < T 6 X 1;

t0 þ L; if T 6 X 1 and T 6 t0 þ L;

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð2Þ


and

R�1¼

ðCoeþCpÞþðCs1þCs2Þ �L�kCp 1� aX 1

W

� �; if X 1 < t0;

ðCorþCpÞþðCs1þCs2Þ � ðt0þL�X 1Þ�kCp 1� aX 1

W

� �; if t06X 1 <W and X 1 < t0þL;

ðCorþCpÞþCh � ðX 1� t0�LÞ�kCp 1� aX 1

W

� �; if t06X 1 <W and t0þL<X 1 ði:e:; t0þL<X 1 <W Þ;

ðCorþCpÞþCh � ðX 1� t0�LÞ; if W 6X 1 < T and t0þL<X 1;

ðCorþCpÞþðCs1þCs2Þ � ðt0þL�X 1Þ; if W 6X 1 < T and X 1 < t0þL;ðCorþCpÞþCh � ðT � t0�LÞ; if t0þL< T 6X 1;

ðCorþCpÞþðCs1þCs2Þ � ðt0þL�T Þ; if T 6X 1 and T 6 t0þL:

8>>>>>>>>>><>>>>>>>>>>:

ð3Þ

Thus, the expected cycle length is

E½X �1� ¼Z 1

0

Z t0

0

ðxþ yÞdF ðxÞdGðyÞ þZ W

t0

Z 1

x�t0

ðt0 þ yÞdGðyÞdF ðxÞ þZ W

t0

Z x�t0

0

xdGðyÞdF ðxÞ

þZ T

W

Z x�t0

0

xdGðyÞdF ðxÞ þZ T

W

Z 1

x�t0

ðt0 þ yÞdGðyÞdF ðxÞ þZ 1

T

Z T�t0

0

T dGðyÞdF ðxÞ

þZ 1

T

Z 1

T�t0

ðt0 þ yÞdGðyÞdF ðxÞ

¼ lL þZ t0

0

F ðxÞdxþZ T

t0

F ðxÞGðx� t0Þdx ¼ lL þZ t0

0

F ðxÞdxþZ T�t0

0

F ðt0 þ xÞGðxÞdx; ð4Þ

and the expected cost per cycle is

E½R�1� ¼Z t0

0

ðCoe þ CpÞ þ ðCs1 þ Cs2Þ � lL � kCp 1� axW

� �h idF ðxÞ

þZ W

t0

Z 1

x�t0

ðCor þ CpÞ þ ðCs1 þ Cs2Þ � ðt0 þ y � xÞ � kCp 1� axW

� �h idGðyÞdF ðxÞ

þZ W

t0

Z x�t0

0

ðCor þ CpÞ þ Ch � ðx� t0 � yÞ � kCp 1� axW

� �h idGðyÞdF ðxÞ

þZ T

W

Z x�t0

0

½ðCor þ CpÞ þ Ch � ðx� t0 � yÞ�dGðyÞdF ðxÞ þZ T

W

Z 1

x�t0

½ðCor þ CpÞ

þ ðCs1 þ Cs2Þ � ðt0 þ y � xÞ�dGðyÞdF ðxÞ þZ 1

T

Z T�t0

0

½ðCor þ CpÞ þ Ch � ðT � t0 � yÞ�dGðyÞdF ðxÞ

þZ 1

T

Z 1

T�t0

½ðCor þ CpÞ þ ðCs1 þ Cs2Þ � ðt0 þ y � T Þ�dGðyÞdF ðxÞ

¼ Cp 1� k 1� ð1� aÞF ðW Þ � aW

Z W

0

F ðxÞdx� ��

þ Coe � F ðt0Þ þ Cor � F ðt0Þ

þ ðCs1 þ Cs2Þ � lL �Z T

t0

F ðxÞGðx� t0Þdx� �

þ Ch �Z T

t0

F ðxÞGðx� t0Þdx: ð5Þ

Case 2: W 6 t0 6 T . Here, X �1 and R�1 can be expressed as

X �1 ¼

X 1 þ L; if X 1 < W ;

X 1 þ L; if W 6 X 1 < t0;

t0 þ L; if t0 6 X 1 < T and X 1 < t0 þ L;

X 1; if t0 6 X 1 < T and t0 þ L < X 1 ði:e:; t0 þ L < X 1 < T Þ;T ; if T 6 X 1 and t0 þ L < T ;

t0 þ L; if T 6 X 1 and T 6 t0 þ L;

8>>>>>>>><>>>>>>>>:

ð6Þ


and

R�1 ¼

ðCoeþCpÞþ ðCs1þCs2Þ � L� kCp 1� aX 1

W

� �; if X 1 <W ;

ðCoeþCpÞþ ðCs1þCs2Þ � L; if W 6 X 1 < t0;

ðCorþCpÞþ ðCs1þCs2Þ � ðt0þ L�X 1Þ; if t0 6 X 1 < T and X 1 < t0þ L;

ðCorþCpÞþCh � ðX 1� t0� LÞ; if t0 6 X 1 < T and t0þ L< X 1 ði:e:; t0þ L< X 1 < T Þ;ðCorþCpÞþCh � ðT � t0� LÞ; if T 6 X 1 and t0þ L< T ;

ðCorþCpÞþ ðCs1þCs2Þ � ðt0þ L� T Þ; if T 6 X 1 and T 6 t0þ L:

8>>>>>>>><>>>>>>>>:

ð7Þ

We can derive E½X �1� and E½R�1� using (6) and (7), and they are exactly given by (4) and (5), respectively. That is,it does not matter whether the ordering time t0 is within the warranty period or after the warranty expires; theexpressions for E½X �1� and E½R�1� are the same. Hence, the expected cost per unit time is

CRðt0Þ ¼E½R�1�E½X �1�

; ð8Þ

where 0 6 t0 6 T . Next, we discuss the optimal solution which minimizes CR(t0). Let Dðt0Þ ¼ E½X �1� andNðt0Þ ¼ E½R�1�, that is

Dðt0Þ ¼ lL þZ t0

0

F ðxÞdxþZ T�t0

0

F ðt0 þ xÞGðxÞdx ð9Þ

and

Nðt0Þ ¼ Cp 1� k 1� ð1� aÞF ðW Þ � aW

Z W

0

F ðxÞdx� ��

þ Coe � F ðt0Þ þ Cor � F ðt0Þ

þ ðCs1 þ Cs2Þ � lL �Z T

t0


þ Ch �Z T

t0


Then we have

D0ðt0Þ ¼ F ðT ÞGðT � t0Þ þZ T

t0

Gðx� t0ÞdF ðxÞ > 0 ¼ F ðt0Þ½1�Z T�t0

0

F ðxjt0ÞgðxÞdx�

¼ F ðt0Þ � D1ðt0Þ; ð11Þ

and

N 0ðt0Þ ¼ ðCoe � CorÞrðt0ÞF ðt0Þ þ ðCs1 þ Cs2Þ F ðt0Þ þZ T�t0

0

F ðt0 þ xÞdGðxÞ� �

� Ch �Z T�t0

0

F ðt0 þ xÞdGðxÞ

¼ F ðt0Þ � ðCoe � CorÞrðt0Þ þ ðCs1 þ Cs2Þ 1�Z T�t0

0

F ðxjt0ÞgðxÞdx� �

� Ch �Z T�t0

0


¼ F ðt0Þ � N 1ðt0Þ;

where

D1ðt0Þ ¼ 1�Z T�t0

0

F ðxjt0ÞgðxÞdx; ð12Þ

N 1ðt0Þ ¼ ðCoe � CorÞrðt0Þ þ ðCs1 þ Cs2Þ 1�Z T�t0

0


� Ch �Z T�t0

0

F ðxjt0ÞgðxÞdx; ð13Þ

and F ðxjt0Þ ¼ 1� F ðxjt0Þ ¼ F ðt0 þ xÞ=F ðt0Þ.Eq. (11) implies that the expected cycle length Dðt0Þ ¼ E½X �1� is strictly increasing with the spare ordering

time t0 and D1ðt0Þ > 0. We see that dCRðt0Þ=dt0 ¼ 0 if and only if


fðt0Þ ¼ Dðt0ÞN 1ðt0Þ � Nðt0ÞD1ðt0Þ ¼ lL þZ t0

0

F ðxÞdxþZ T�t0

0

F ðt0 þ xÞGðxÞdx�

� ðCoe � CorÞrðt0Þf

þðCs1 þ Cs2Þ 1�Z T�t0

0


� Ch �Z T�t0

0

F ðxjt0ÞgðxÞdx

� Cp 1� k 1� ð1� aÞF ðW Þ � aW

Z W

0

F ðxÞdx� ��

þCoe � F ðt0Þ þ Cor � F ðt0Þ þ ðCs1 þ Cs2Þ � lL �Z T

t0


þ Ch �Z T

t0

F ðxÞGðx� t0Þdx

� 1�Z T�t0

0


¼ 0: ð14Þ

Moreover, f(t0) can be rewritten as

fðt0Þ ¼ Dðt0ÞD1ðt0ÞN 1ðt0ÞD1ðt0Þ

� Nðt0ÞDðt0Þ

� �¼ Dðt0ÞD1ðt0Þ½/ðt0Þ � CRðt0Þ�;

where /ðt0Þ ¼ N 1ðt0Þ=D1ðt0Þ ¼ N 0ðt0Þ=D0ðt0Þ is the marginal cost function of this spare ordering policy. Themain results of the optimal ordering time t�01, which satisfies CRðt�01Þ ¼Min06t06T CRðt0Þ, are summarized below.

Theorem 1. Suppose rð�Þ is strictly increasing, Coe > Cor, and the condition

Qðt0Þ ¼ ðCs1 þ Cs2Þ �Z T

0

F ðxÞdx� Cp 1� k 1� ð1� aÞF ðW Þ � aW

Z W

0

F ðxÞdx� ��

� Coe � F ðt0Þ � Cor � F ðt0ÞP 0 ð15Þ

holds. Then,

(i) If fð0ÞP 0 (i.e., /ð0ÞP CRð0ÞÞ, then the optimal ordering time t�01 is 0; i.e., the customer should order a

spare when the original product is purchased.

(ii) If fð0Þ < 0 and fðW Þ > 0 (i.e., /ð0Þ < CRð0Þ and /ðW Þ > CRðW ÞÞ, then there exists an unique optimal

ordering time t�01 ð0 < t�01 < W Þ that satisfies fðt�01Þ ¼ 0; i.e., the customer should order a spare during

the warranty period.

(iii) If fðW Þ < 0 and fðT Þ > 0 (i.e., /ðW Þ < CRðW Þ and /ðT Þ > CRðT ÞÞ, then there exists an unique optimalordering time t�01 ðW < t�01 < T Þ that satisfies fðt�01Þ ¼ 0; i.e., the customer should order a spare after the

warranty expires.

(iv) If fðT Þ 6 0 (i.e., /ðT Þ 6 CRðT ÞÞ, then the optimal ordering time t�01 is T; i.e., the customer should order a

spare when the original product ends its useful life.

For proof of Theorem 1, see Appendix A. h

Remark 1. The Q(t0) in Eq. (15) can be divided into two parts. The first one is ðCs1 þ Cs2Þ �R T

0 F ðxÞdx, which is

the cost of shortage all the lifetime. The second one is Cp 1� k 1� ð1� aÞF ðW Þ � aW

RW0 F ðxÞdx

h in oþ

Coe � F ðt0Þ þ Cor � F ðt0Þ, which is the expected cost for ordering a spare at time t0 under the rebate warrantyperiod with W. Notice that if Qðt0Þ < 0, there should not be any ordering policy, since it implies that incurringthe cost of shortage all the lifetime is cheaper than the ordering cost. Thus, in order to warrant the order for aspare, Q(t0) must be greater than or equal to zero, and the existence of t�0 in the theorem follows trivially.

Remark 2. Theorem 1 mainly illustrated the possible range of optimal spare ordering time t�0 as well as thecorresponding strategy for the user under various situations. Especially, it depends on the relationshipbetween the marginal cost /(t) and operation cost per unit time CR(t) of this spare ordering policy at any timet. It is worth to point out that the case (iv) of Theorem 1 is equivalent to violating the condition (15) since that

Qðt0Þ 6 �ðCoe � CorÞrðT Þ lL þR T

0F ðxÞdx

h i< 0 is obtained by algebraic manipulation of condition fðT Þ 6 0.


Thus, the optimal spare ordering policy is that the customer should order a spare (planned) when the originalproduct ends its useful life.

4. The cost effectiveness maximization model

In general, the policy maximizing profit rate does not discriminate among large and small investments.Thus, the policy discussed in the preceding section might have this property. In this section, we adopt costeffectiveness as an alternative criterion, which is defined as (s-availability)/(s-expected out of pocket cost rate),reflecting efficiency per dollar spent. As described in [18], this criterion is useful for the effective use of availablemoney. Furthermore, this criterion is useful when the benefits obtained from investment are difficult to quan-tify. As an example, national security may benefit from weapon systems (see [11]), but expressing the gains inmonetary terms is quite challenging. Although the product (which can be regard as a system or an equipmentfor business production) under rebate warranty maybe just in commercial industrial, however, the cost effec-tiveness maximization model still provides a useful performance measure alternative to the cost-rate minimi-zation model. Park and Park [18] also applied such a cost effectiveness model as a criterion of optimality whenthey considered the problem of spare ordering policies.

Since in our formulation of the spare ordering and replacement process, each replacement is a regenerationpoint, we can rewrite the cost effectiveness

s-availability

s-expected out of pocket cost rate¼ expected uptime in a cycle

expected out of pocket cost per cycle:

The expected uptime per cycle is the expected cycle length minus the expected downtime per cycle. Accordingto (2) or (6), we can write the expected downtime in a cycle as

lL �Z T

t0


Thus, from (4) and (16), expected uptime per cycle is

lL þZ t0

0

F ðxÞdxþZ T

t0


� lL �Z T

t0


¼Z T

0

F ðxÞdx: ð17Þ

We obtain the expected out of pocket cost per cycle by setting Cs2 ¼ 0 in (5). Hence, the cost effectiveness isCEðt0Þ ¼
R T
0F ðxÞdx

Cpf1� k½1�ð1�aÞF ðW Þ� aW

R W0

F ðxÞdx�gþCoeF ðt0ÞþCorF ðt0ÞþCs1½lL�R T

t0F ðxÞGðx� t0Þdx�þCh

R Tt0

F ðxÞGðx� t0Þdx:

ð18Þ

Notice that since the numerator in (18) reflects the benefits obtained from business operations, the inclusion ofopportunity costs in the denominator results in double counting of shortages in both the numerator and thedenominator. Thus, the inclusion of opportunity costs in the cost effectiveness is incorrect [18].

Since the numerator in (18) is independent of the decision variable t0, maximizing CE(t0) is equivalent tominimizing p(t0), i.e., Maxt02½0;T �CEðt0Þ ¼Mint02½0;T �pðt0Þ, where p(t0) is given by

CoeF ðt0Þ þ CorF ðt0Þ þ Cs1 lL �Z T

t0


þ Ch

Z T

t0


We see that dpðt0Þ=dt0 ¼ 0 if and only if #ðt0Þ ¼ 0, where

#ðt0Þ ¼ ðCoe � CorÞrðt0Þ þ Cs1 1�Z T�t0

0


� Ch �Z T�t0

0

F ðxjt0ÞgðxÞdx: ð20Þ

We summarize the main results concerning the optimal ordering time t�02 satisfying CEðt�02Þ ¼Max06t06T CEðt0Þbelow.


Theorem 2. Suppose rð�Þ is strictly increasing, Coe > Cor. Then:

(i) If rð0ÞPCh�R T

0F ðxÞgðxÞdx�Cs1�½1�

R T

0F ðxÞgðxÞdx�

ðCoe�CorÞ , then the optimal ordering time t�02 is 0; i.e., the customer should

order a spare when the original product is purchased.

(ii) If rð0Þ <Ch�R T

0F ðxÞgðxÞdx�Cs1�½1�

R T

0F ðxÞgðxÞdx�

ðCoe�CorÞ and rðW ÞPCh �R T�W

0F ðxjW ÞgðxÞdx�Cs1�½1�

R T�W

0F ðxjW ÞgðxÞdx�

ðCoe�CorÞ , then there exists

an unique optimal ordering time t�02 ð0 < t�02 < W Þ satisfying #ðt�02Þ ¼ 0; i.e., the customer should order a

spare during the warranty period.

(iii) If rðW Þ <Ch�R T�W

0F ðxjW ÞgðxÞdx�Cs1�½1�

R T�W

0F ðxjW ÞgðxÞdx�

ðCoe�CorÞ , then there exists an unique optimal ordering time t�02

ðW < t�02 < T Þ satisfying #ðt�02Þ ¼ 0; i.e., the customer should order a spare after the warranty expires

but before the product reaches the end of its useful life.

For proof of Theorem 2, see Appendix B. h

Remark 3. Theorem 2 implies that, under the cost effectiveness criterion, the optimal spare ordering time t�02

always precedes the useful life limit (i.e., t�02 < T ).

Remark 4. The assumption Coe > Cor is a necessary condition both for Theorems 1 and 2. It is reasonablesince that an expedited ordering is more urgent as well as unexpected than a regular one. This means thatthe expedited ordering cost includes the extra cost due to dealing with such an urgent event.

5. Numerical example and sensitivity analysis

As an illustration, consider a product carrying a pro rata rebate policy with period W. Assume the lifetimeof the product has a Weibull distribution with shape parameter b and scale parameter h; that is,

F ðxÞ ¼ 1� e�xhð Þ

b

:

Let b ¼ 3, h ¼ 2, and T ¼ 5, so that the mean time to first failure is 1.8 years. Assume further that the leadtime has a normal distribution with mean lL ¼ 0:1 and standard deviation rL ¼ 0:01. Fix the cost parametersat Cp ¼ $1000, Cor ¼ $5, and Coe ¼ $100; we vary the remaining parameters to examine their influence on theoptimal solution. We calculate the optimal spare ordering time by minimizing the cost rate CR(t0) or maxi-mizing the cost effectiveness CE(t0), the results are summarized in Tables 1 and 2.

From Table 1 (the cost per unit time criterion), we make the following observations.

1. The optimal spare ordering time t�01 decreases as shortage costs Cs increase and increases as holding costs Ch

increase. This is reasonable because a customer should order a spare earlier for a product with greatershortage costs. Conversely, customers should wait as long as possible before ordering a spare for a productwith greater holding costs.

2. As the warranty expiration date W increases, the optimal t�01 decreases. The reason is that when the war-ranty expiration date W is longer, the expected rebate is greater, offsetting holding costs. We show analyt-ically that t�01 decreases with W in Appendix C.

3. The optimal cost per unit time CR� ¼ CRðt�01Þ increases as the shortage costs Cs and holding costs Ch

increase, and decreases as the rebate warranty expiration date W increases. This is also to be expected.

From Table 2 (the cost effectiveness criterion), we observe the following.

1. As under the cost per unit time optimum, the optimal spare ordering time t�02 decreases as the out of pocketcosts Cs1 increase and increases as the holding costs Ch increase.

2. The optimal spare ordering time t�02 is independent of the warranty expiration W since Maxt02½0;T �CEðt0Þ ¼Mint02½0;T �pðt0Þ, where p(t0) is given in (19).

Table 1Optimal spare ordering time t�01 and cost rate CR� ¼ CRðt�01Þ with T ¼ 5, Cp ¼ $1000, Cor ¼ $5, Coe ¼ $100, lL ¼ 0:1 and rL ¼ 0:01

t�01CR� Cs ¼ 1800 Cs ¼ 3600 Cs ¼ 7200

Ch ¼ 180 Ch ¼ 360 Ch ¼ 720 Ch ¼ 180 Ch ¼ 360 Ch ¼ 720 Ch ¼ 180 Ch ¼ 360 Ch ¼ 720

W ¼ 0:50 1.44 2.00 2.71 1.06 1.50 2.09 0.75 1.08 1.54639.5681 663.7193 674.8988 662.3096 712.4758 753.9180 681.9683 761.6213 857.7289

W ¼ 0:75 1.43 2.00 2.71 1.06 1.50 2.09 0.75 1.08 1.54634.5678 658.8062 670.0559 657.2597 707.4854 749.0173 676.8916 756.5736 852.7447

W ¼ 1:00 1.43 2.00 2.70 1.05 1.50 2.09 0.75 1.08 1.53625.1256 649.5302 660.9120 647.7252 698.0634 739.7647 667.3065 747.0434 843.3329

W ¼ 1:25 1.43 1.99 2.70 1.05 1.49 2.09 0.75 1.08 1.53610.3708 635.0289 646.6223 632.8231 683.3391 725.3062 652.3284 732.1510 828.6235

W ¼ 1:50 1.42 1.98 2.69 1.05 1.49 2.08 0.75 1.08 1.53590.0818 615.0908 626.9755 612.3364 663.0921 705.4248 631.7373 711.6777 808.4016

W ¼ 1:75 1.41 1.97 2.67 1.05 1.49 2.07 0.75 1.07 1.53564.8193 590.2635 602.5140 586.8326 637.8866 680.6731 606.1035 686.1896 783.2275

W ¼ 2:00 1.40 1.96 2.66 1.04 1.48 2.07 0.75 1.07 1.53535.8681 561.8094 574.4833 557.6085 609.0004 652.3069 576.7341 656.9818 754.3848

Table 2Optimal spare ordering time t�02 and cost effectiveness CE� ¼ CEðt�02Þ with T ¼ 5, Cp ¼ $1000, Cor ¼ $5, Coe ¼ $100, lL ¼ 0:1 andrL ¼ 0:01

t�02CE� Cs ¼ 1800 Cs ¼ 3600 Cs ¼ 7200

Ch ¼ 180 Ch ¼ 360 Ch ¼ 720 Ch ¼ 180 Ch ¼ 360 Ch ¼ 720 Ch ¼ 180 Ch ¼ 360 Ch ¼ 720

W ¼ 0:50 1.53 2.11 2.83 1.26 1.76 2.41 0.97 1.38 1.930.001576 0.0015276 0.0015082 0.0015388 0.0014586 0.0014115 0.0014981 0.0013775 0.0012790

W ¼ 0:75 1.53 2.11 2.83 1.26 1.76 2.41 0.97 1.38 1.930.0015893 0.0015395 0.0015198 0.0015510 0.0014696 0.0014217 0.0015096 0.0013873 0.0012874

W ¼ 1:00 1.53 2.11 2.83 1.26 1.76 2.41 0.97 1.38 1.930.0016140 0.0015627 0.0015424 0.0015745 0.0014906 0.0014415 0.0015319 0.0014060 0.0013035

W ¼ 1:25 1.53 2.11 2.83 1.26 1.76 2.41 0.97 1.38 1.930.0016542 0.0016003 0.0015790 0.0016126 0.0015248 0.0014734 0.0015680 0.0014364 0.0013296

W ¼ 1:50 1.53 2.11 2.83 1.26 1.76 2.41 0.97 1.38 1.930.0017127 0.0016550 0.0016322 0.0016682 0.0015744 0.0015196 0.0016205 0.0014803 0.0013671

W ¼ 1:75 1.53 2.11 2.83 1.26 1.76 2.41 0.97 1.38 1.930.0017916 0.0017286 0.0017038 0.0017430 0.0016409 0.0015815 0.0016909 0.0015389 0.0014170

W ¼ 2:00 1.53 2.11 2.83 1.26 1.76 2.41 0.97 1.38 1.930.0018915 0.0018214 0.0017939 0.0018374 0.0017242 0.0016588 0.0017796 0.0016120 0.0014787


3. The optimal cost effectiveness CE� ¼ CEðt�02Þ decreases as the out of pocket costs Cs1 and holding costs Ch

increase and increases as the expiration date W increases.

Moreover, all the numerical results confirms to the main theorems. For example, under the cost per unittime criterion, as Cs ¼ 3600, W ¼ 1:25 and Ch ¼ 180, then fð0Þ ¼ �320:52 < 0 and fðW Þ ¼ 24:75 > 0, whichyields that 0 < t�0 ¼ 1:05 < W . This result is consistent with the condition (ii) of Theorem 1. If Cs ¼ 3600, W ¼1:25 and Ch ¼ 360, then fðW Þ ¼ �186:91 < 0 and fðT Þ ¼ 7066:14 > 0, which yields that t�0 ¼ 1:49 > W . Thisresult is corresponds to the condition (iii) of Theorem 1.

The behavior about t�0 that confirms to Theorem 2 can also be verified under the cost effectiveness criterion.

6. Conclusion

In this paper, we consider a non-repairable product with a finite useful lifetime under the rebate warranty.We discuss a generalized spare ordering policy assuming the time necessary for delivery of a spare follows a


known probability distribution and develop cost models to derive optimality under two measures: expectedcost per unit time over an infinite time horizon and cost effectiveness. Theorems 1 and 2 present the structuralproperties of the optimal spare ordering time that minimize the former and maximize the latter. A numericalexample illustrates the behavior of the optimal solution when selected cost parameters are varied. The resultsare intuitive and match our expectations.

It is worth pointing out that, if the product is inexpensive, quantity purchases might be practical, and it isnatural to consider a stocking policy to determine how many spares to purchase with each order as Falkner[10] does rather than our simplified ordering policy. Indeed, products warranting an ordering policy are usu-ally components in a high-cost complex system. Nonetheless, a product warranty is an important factor inderiving an optimal spare ordering policy, and practitioners need to be able to weigh their value when plan-ning and scheduling operations. Further, it is reasonable to consider that the high-cost complex product/sys-tem is repairable. However, the reason why the repairable case did not considered in this study is that themodel proposed in this paper is fundamental and original, further extension of this problem is worth to dobase on this model in future. It is not so hard to extend this ordering model to a more general repairablecase. The authors have a lot of experience in dealing with such an issue, for example, see Sheu and Chien[22], Chien [6], Chien and Sheu [8], Chien et al. [9] and Chen and Chien [5]. We believe there remain a greatvariety of interesting and useful research directions to pursue in order to provide comprehensive models formulti-product mixed-warranty-type systems and to incorporate maintenance in the decision-makingframework.

Acknowledgements

The authors thank the referees for their valuable comments and suggestions which greatly enhanced theclarity of the paper. All of their suggestions were incorporated directly in the text. This research was supportedby National Science Council of Taiwan, under Grant No. NSC 94-2213-E-025-005 and NSC 95-2221-E-025-006-MY2.

Appendix A. Proof of Theorem 1

Differentiating CR(t0) in Eq. (8) with respective to t0 yields

dCRðt0Þdt0

¼ F ðt0Þ � fðt0ÞD2ðt0Þ

;

where fðt0Þ ¼ Dðt0ÞN 1ðt0Þ � Nðt0ÞD1ðt0Þ. The explicit forms of D1ðt0Þ, N 1ðt0Þ, and f(t0) are given in (12)–(14),respectively. Moreover, we can write f0ðt0Þ ¼ Dðt0ÞN 01ðt0Þ � Nðt0ÞD01ðt0Þ, where

D01ðt0Þ ¼F ðT ÞF ðt0Þ

gðT � t0Þ �Z T�t0

0

d

dt0

F ðxjt0Þ� �

gðxÞdx;

and

N 01ðt0Þ ¼ ðCoe � CorÞr0ðt0Þ þ ðCs1 þ Cs2 þ ChÞ � D01ðt0Þ:
Note that F ðxjtÞ and r(t) are inversely related; that is, F ðxjtÞ is decreasing (increasing) in t if and only if r(t) isincreasing (decreasing) in t (see [1, p. 23]). Therefore, we have D01ðt0Þ > 0. Next, observe that we can rewritef0ðt0Þ as
f0ðt0Þ ¼ r0ðt0ÞðCoe � CorÞDðt0Þ þ D01ðt0Þ � ½ðCs1 þ Cs2 þ ChÞ � Dðt0Þ � Nðt0Þ�

¼ r0ðt0ÞðCoe � CorÞDðt0Þ þ D01ðt0Þ � Qðt0Þ þ Ch � lL þZ t0

0

F ðxÞdx ��

> 0:

Hence, if the conditions of Theorem 1 are satisfied, then f0ðt0Þ > 0; in other words, f(t0) is strictly increasing.The four statements in the theorem follow immediately.


Appendix B. Proof of Theorem 2

Differentiating #ðt0Þ in (20) yields

#0ðt0Þ ¼ ðCoe � CorÞr0ðt0Þ þ ðCs1 þ ChÞ � D01ðt0Þ > 0;

that is, #ðt0Þ is strictly increasing in t0. We also have

#ð0Þ ¼ ðCoe � CorÞrð0Þ þ Cs1 � 1�Z T

0

F ðxÞgðxÞdx� �

� Ch �Z T

0

F ðxÞgðxÞdx;

and

#ðT Þ ¼ ðCoe � CorÞrðT Þ þ Cs1 > 0:

This directly implies the three statements in Theorem 2.

Appendix C. Proof that t�0 decreases with W

Since the optimal t�0 satisfies fðt�0Þ ¼ 0 and t�0 is a function of W, we can find ot�0=oW implicitly using theequation Dðt�0ÞN 1ðt�0Þ ¼ Nðt�0ÞD1ðt�0Þ. We have

D0ðt�0Þot�0oW

�� N 1ðt�0Þ þ Dðt�0Þ � N 01ðt�0Þ

ot�0oW

�

¼ N 0ðt�0Þot�0oW

�þ Cp �k 0� ð1� aÞð�rðW ÞÞF ðW Þ þ a

W 2

Z W

0

F ðxÞdx� aW

F ðW Þ� ��

� D1ðt�0Þ þ Nðt�0Þ � D01ðt�0Þot�0oW

�;

which simplifies to

ot�0oW

�� ½Dðt�0ÞN 01ðt�0Þ � Nðt�0ÞD01ðt�0Þ� ¼ kCp �ð1� aÞrðW ÞF ðW Þ � a

W 2

Z W

0

F ðxÞdxþ aW

F ðW Þ� �

� D1ðt�0Þ

Since t�0 satisfies Mint02ð0;T ÞCRðt0Þ ¼ CRðt�0Þ, we have by Appendix A that

f0ðt�0Þ ¼ Dðt�0ÞN 01ðt�0Þ � Nðt�0ÞD01ðt�0Þ > 0:

Moreover, D1ðt�0Þ > 0 and the term

�ð1� aÞrðW ÞF ðW Þ � a

W 2

Z W

0

F ðxÞdxþ aW

F ðW Þ < �ð1� aÞrðW ÞF ðW Þ � a

W 2

Z W

0

F ðW Þdxþ aW

F ðW Þ

¼ �ð1� aÞrðW ÞF ðW Þ < 0:

Hence, we have ot�0=oW < 0. That is, the optimal spare ordering time t�0, which minimizes the cost per unittime, is decreasing with the warranty expiration date W.

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optimal spare ordering policy under a rebate warranty

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