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Comments on “Maintenance policies with two-dimensional warranty”
Reliability Engineering and System Safety 82 (2003) 105–109 www.elsevier.com/locate/ress
Discussion
Comments on “Maintenance policies with two-dimensional warranty” N. Jacka,*, D.N.P. Murthyb,c, B.P. Iskandard a
School of Computing and Advanced Technologies, University of Abertay Dundee, Dundee DD1 1HG, Scotland, UK b Department of Mechanical Engineering, The University of Queensland, St Lucia 4072, Australia c Norwegian University of Science and Technology, Trondheim, Norway d Departemen Teknik Industri, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia Received 18 March 2003; accepted 5 May 2003
Abstract Chen and Popova [Res. Engng Syst. Saf. 77 (2002) 61] discuss maintenance policies for items sold with a two-dimensional warranty. However, their paper fails to give a proper review of the literature and it also contains errors. In this note we first review the relevant literature and then comment on the errors in their analysis. q 2003 Elsevier Ltd. All rights reserved. Keywords: Two-dimensional warranty; Servicing strategies
1. Introduction A warranty is an integral component of most product sales. It requires the manufacturer to either rectify or compensate for any failures that occur within the warranty period. A variety of warranty policies have been studied, and a taxonomy of these can be found in Blischke and Murthy [1]. In a one-dimensional (1D) warranty policy the warranty period is usually a time interval, and in the two-dimensional (2D) case it is characterised by a 2D planar region with one coordinate normally representing time (or age) and the other representing usage. Offering any warranty results in additional costs to the manufacturer and these must be factored into the sale price. The cost analysis of many different one- and two-dimensional warranty policies is given in Ref. [1]. For repairable products sold with a FRW (free repair or replacement warranty), the manufacturer has the option of either repairing or replacing a failed item. The costs as well as the benefits derived from these two types of corrective maintenance action are different. Replacement by a new item, although costing more than a repair, reduces the chances of subsequent failures occurring under warranty. A manufacturer’s optimal servicing strategy involves choosing the appropriate corrective maintenance * Corresponding author. E-mail address: [email protected] (N. Jack). 0951-8320/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0951-8320(03)00119-4
actions on failure in order to minimise expected warranty servicing costs. Servicing strategies for 1D warranties have received a lot of attention in the literature, but considerably less attention has been paid to this problem for 2D warranties. Also, in the 2D case, two different approaches to modelling item failures have been proposed. Chen and Popova [2] consider servicing strategies for 2D warranties but do not give a proper review of the relevant literature, and their model analysis of item failures contains errors. The purpose of this note is to rectify these two deficiencies. The outline of the note is as follows. In Section 2 we discuss repair – replace decisions for items sold with a 1D FRW policy. Section 3 reviews the two different approaches to modelling failures in the analysis of 2D warranty policies. Section 4 reviews the literature dealing with optimal repair – replace decisions for a 2D FRW policy. In Section 5 we critically look at the analysis of Chen and Popova [2] and highlight the errors. We conclude with some topics for future research in Section 6.
2. Servicing strategies for one-dimensional FRW policies The simplest servicing strategies for a repairable item sold with a 1D FRW with period W are (i) to always replace on failure (Strategy 1), and (ii) to always repair (Strategy 2).
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Each replacement is understood to be with a new and identical item and all repair actions are ‘minimal’, so they restore a failed item to an operational state without affecting the hazard rate. The number of item failures under warranty, NðWÞ; is modelled by a renewal process for Strategy 1 and a non-homogeneous Poisson process (NHPP) for Strategy 2. The corresponding expected number of failures are then given by the renewal function MðWÞ and the cumulative intensity function LðWÞ; respectively, and expected servicing costs are found by multiplying these functions by the cost of a replacement item and the cost of a minimal repair. Nguyen [12] proposes two alternative servicing strategies, which we shall term Strategies 3 and 4. In each case, the warranty period W is divided into two sub-intervals ½0; W1 Þ and ½W1 ; WÞ: Under Strategy 3, if an item fails in ½0; W1 Þ then it is replaced and if it fails in ½W1 ; WÞ it is only minimally repaired. Under Strategy 4, the opposite behaviour occurs with minimal repair on failure in the first sub-interval and replacement in the second sub-interval. The decision variable is W1 ; which is selected optimally by minimising the expected warranty servicing costs. Jack and Van der Duyn Schouten [8] prove that Strategies 3 and 4 are not optimal. Using a dynamic programming model, they show that the optimal repair– replace decision is determined by comparing the item’s age on failure with a time-dependent control limit function hðtÞ: The item is replaced on failure at time t if and only if its age is greater than hðtÞ: A typical plot of this function is shown in Fig. 1. Under the optimal strategy, which we term Strategy 5, the warranty period is divided into three sub-intervals I ¼ ½0; W1 Þ; II ¼ ½W1 ; W2 Þ; and III ¼ ½W2 ; WÞ: The item is always minimally repaired on failure in sub-intervals I and III, with the possibility of replacement on failure in
sub-interval II. The actual shape of hðtÞ in this sub-interval determines the maximum number of replacements that will occur. In Fig. 1, hðtÞ lies above the indicated line L1 ; so at most one replacement will be carried out. In general, the shape of hðtÞ is determined by the reliability of the item and the ratio of repair costs to replacement costs. Jack and Murthy [7] suggest a sub-optimal servicing strategy, which we term Strategy 6, with the following structure. Every item failure in sub-intervals I and II is again minimally repaired but only the first failure in sub-interval II results in a replacement. In most cases, the expected cost of this sub-optimal strategy is very close to that of the optimal Strategy 5 and the strategy is far easier to implement.
3. Failure modelling for two-dimensional warranties A 2D warranty is characterised by a region V in a 2D plane with the horizontal axis signifying time and the vertical axis usage. We confine to the case where the FRW policy expires either at time K or when the total item usage exceeds a level L; so that V is a rectangle. See Refs. [10,13] for other possible shapes for V: Item failures are represented by points lying in the 2D ‘time-usage’ plane, and two different methods have been used to model these failures. In the first method, termed the ‘one-dimensional approach’, usage is assumed to be a function of age so that failures are effectively modelled using a 1D point process formulation (see Refs. [4,9]). The second method, termed the ‘two-dimensional approach’, uses a joint distribution function to model the age and usage of the item at its first failure and then subsequent failures are modelled using a 2D point process formulation (see Ref. [11]). We now give further details of these two techniques. 3.1. The one-dimensional approach In this approach, the usage of the item is modelled as a linear function of the item’s age with non-negative coefficient R: R represents the usage rate and is assumed to be a non-negative random variable to account for the varying usage across the consumer population. Conditional on R ¼ r; lðtlrÞdt is used to denote the probability that the item in use at time t will fail in the small interval ½t; t þ dtÞ and NðtlrÞ denotes the number of item failures in the interval ð0; t: The form of NðtlrÞ depends on the type of corrective maintenance actions taken at item failures. 3.2. The two-dimensional approach
Fig. 1. Optimal control limit function for repair– replace decision.
Let T and X denote the system’s age and usage at its first failure. ðT; XÞ is assumed to be a non-negative bivariate random variable with distribution function Fðt; xÞ ¼ P{T # xÞ ¼ Pr{T . t; X . x}; t; X # x}; survivor function Fðt; density function f ðt; xÞ ¼ ›2 Fðt; xÞ=›t ›x; and hazard xÞ: Nðt; xÞ denotes the number function rðt; xÞ ¼ f ðt; xÞ=Fðt;
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of system failures occurring in the rectangular region ½0; tÞ £ ½0; xÞ and, as in the 1D approach, the form of Nðt; xÞ depends on the type of corrective maintenance actions taken to restore failed items.
4. Servicing strategies for the two-dimensional FRW policy with rectangular region V We now discuss the 2D counterparts of Strategies 1 –6 from Section 2, and we refer to these as Strategies 1(2D) –6(2D), respectively. Under Strategy 1(2D), all item failures that occur under warranty are rectified through replacement, and under Strategy 2(2D) every failure is minimally repaired. In each case, we consider both the 1D and 2D failure modelling approaches. Under Strategy 1(2D), with the 1D approach, let r1 ¼ L=K: It follows that, given R ¼ r; the warranty will expire at time Kr ; where Kr ¼ K if r # r1 and Kr , K if r . r1 : Conditional on R ¼ r; NðKr lrÞ is a 1D renewal process, where the time to first Ðitem failure has distribution function FðtlrÞ ¼ 1 2 exp{ 2 t0 lðxlrÞdx}: The expected number of failures (replacements) over ½0; Kr Þ is given by MðKr lrÞ; where MðtlrÞ is the renewal function corresponding to FðtlrÞ: Finally, the conditioning in this expected failure count function is removed by multiplying by gðrÞ; the density function for R; and integrating over all possible values of r: Using the 2D modelling approach for Strategy 1(2D), we have a 2D renewal process for system failures. The expected number of failures (replacements) under warranty is then given by MðK; LÞ; where MðK; LÞ is the 2D renewal function. function is defined by MðK; LÞ ¼ P This ðnÞ E½NðK; LÞ ¼ 1 F ðK; LÞ; where F ðnÞ ðt; xÞ is the n-fold n¼1 bivariate convolution of Fðt; xÞ with itself. Alternatively, it be expressed as the solution of the 2D integral equation (see Ref. [3]) MðK; LÞ ¼ FðK; LÞ þ
ðK ðL 0
MðK 2 u; L 2 vÞf ðu; vÞdv du:
0
Under Strategy 2(2D), with the 1D approach, it follows that conditional on R ¼ r; NðKr lrÞ is a NHPP with intensity function lðtlrÞ; and the expected number of failures (minimal repairs) over ½0; KrÐÞ is given by the cumulative intensity function LðKr lrÞ ¼ K0 r lðtlrÞdt: The conditioning in this expression is then removed in the same way as in Strategy 1(2D). The first notion of minimal repair using the 2D failure modelling approach has recently been developed by Baik et al. [14]. They show that the counting process {Nðt; xÞ : t $ 0; x $ 0} is a 2D NHPP with intensity function lðt; xÞ ¼ rðt; xÞ; and so the expected number of failures (minimal repairs) under warranty for Strategy 2(2D) is Ð Ð then given by E½NðK; LÞ ¼ LðK; LÞ ¼ K0 L0 lðt; xÞdx dt: Note that, other notions of minimal repair can be
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formulated and these are currently under investigation by Baik et al. [14]. Expected servicing costs for Strategies 1(2D) and 2(2D) are again found by multiplying the expected number of failures by the cost of a replacement item and the cost of a minimal repair, respectively. Further details on Strategy 1(2D) and on the 1D approach to Strategy 2(2D) are given in Ref. [1]. Strategies 3(2D) and 4(2D) have been studied by Iskandar and Murthy [5]. In each case, the warranty region V ¼ ½0; KÞ £ ½0; LÞ is divided into two sub-regions—V1 and V2 ; where V1 ¼ ½0; K1 Þ £ ½0; L1 Þ; and V2 ¼ V\ V1 : Under Strategy 3(2D), all failures occurring in V1 are rectified by replacement and any failure in V2 is rectified by minimal repair. This is the strategy discussed by Chen and Popova [2]. Under Strategy 4(2D), all failures occurring in V1 are rectified by minimal repair and any failure in V2 is rectified by replacement. For both strategies, Iskandar and Murthy [5] use the 1D approach to obtain the expected number of replacements and minimal repairs under warranty. This again involves first conditioning on R ¼ r and then removing the conditioning by multiplying by gðrÞ; the density function for R; and integrating over all possible values of r: Expected servicing costs are a function of the two decision variables K1 and L1 : Iskandar et al. [6] propose Strategy 6(2D), where the warranty region is divided into three sub-regions V1 ; V2 ; and V3 (See Fig. 2). Under this strategy (i) all failures in V1 are minimally repaired, (ii) the first failure in V2 is rectified through replacement and subsequent failures in this region are rectified through minimal repair, and (iii) all failures in V3 are always minimally repaired. Again, the 1D failure modelling approach is used to obtain the expected number of replacements and minimal repairs under warranty, and hence the expected warranty servicing cost. This cost is now a function of the four decision variables K1 ; K2 ; L1 ; and L2 : The analysis of Strategy 6(2D) based on the 2D failure modelling approach is currently under investigation by
Fig. 2. Regions characterising Strategy 6(2D).
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the authors. This is also the case for the optimal servicing strategy, Strategy 5(2D), which involves complex repair– replace regions over the 2D age-usage plane.
5. Comments on the Chen and Popova [2] paper Chen and Popova [2] consider Strategy 3(2D) but seem to be unaware of the paper by Iskandar and Murthy [5] where analytical results for the expected number of repairs and replacements and expected servicing cost are found using the 1D modelling approach. However, these functions do not have closed form analytical solutions and computational schemes are required for their evaluation. Section 2.2 of the Chen and Popova paper contains a derivation of the expected servicing cost for Strategy 3(2D) but their results contain a number of fundamental flaws and many minor errors. They appear to be totally confused between the 1D and 2D approaches to modelling item failures. We shall concentrate only on identifying the fundamental flaws in their modelling but first we need to give a summary of some of the notation they use: {ð0; WT Þ £ ð0; WU Þ} the rectangular warranty region. A ¼ {ð0; Vt Þ £ ð0; Vu Þ} the replacement region. B ¼ {ð0; WT Þ £ ð0; WU Þ}=A the minimal repair region. ðt; uÞ a point [ {ð0; WT Þ £ ð0; WU Þ}: lðt; uÞ the ‘2D failure rate’ at the point ðt; uÞ: N1 ðt; uÞ the number of replacements in the region {ð0; tÞ £ ð0; uÞ}: N2 ðt; uÞ the number of minimal repairs in the region {ð0; tÞ £ ð0; uÞ}: SN1 ðt;uÞ the time of the last replacement prior to the point ðt; uÞ: r ¼ u=t the usage rate. Vt ; Vu the decision variables for the servicing strategy. At the beginning of Section 2.2, Chen and Popova state that the objective function to be minimised is the sum of the expected cost of replacements and repairs over the warranty region. Eq. (1) then gives a general expression for this function conditioned on the usage rate r: Thus, they seem to be using the 1D failure modelling approach and the un-conditioning should then be carried out with respect to this variable in order to obtain the unconditional expected cost. This is the method used by Iskandar and Murthy [5]. However, Chen and Popova then change to the 2D approach by introducing the variables ðt; uÞ; N1 ðt; uÞ; N2 ðt; uÞ; and SN1 ðt;uÞ : They re-define the objective function, with the conditioning now on ðt; uÞ and SN1 ðt;uÞ ; but the latter variable gives insufficient information regarding the position of the last replacement. Instead, the authors should have used the point (SN1 ðt;uÞ ; rSN1 ðt;uÞ ), and considered the two cases where the point lies in region A and region B, separately. The un-conditioning then needs to be done with
respect to r: Eq. (4) for the unconditional expected minimal repair cost is incorrect, because of this error. Eq. (5) is also incorrect. Firstly, minimal repairs are carried out over the region B. This result also uses a 1D modelling concept where, under minimal repair, the expected number of failures over a time interval is given by the integral of the failure intensity function with this function having the same form as the item’s hazard function for time to first failure. This same concept cannot be assumed without proof for the 2D case. Eq. (6) is incorrect. The right hand side is the probability that the first failure occurs outside region A (i.e. in or beyond region B) if Fðt; uÞ is the 2D failure distribution function of a new item. Note also that, if one starts with the failure rate function lðt; uÞ it is not possible to derive the failure distribution function Fðt; uÞ uniquely. The authors also state that Eq. (8) follows from the ‘key renewal theorem’. This is not the case and, once again, Eq. (8) seems to be an attempted extension of a well-known 1D modelling result. It should not be stated without proof or without a valid reference. Eqs. (4) –(8) deal with the expected number of minimal repairs in region B but few details are given on how to find the expected number of replacements in region A. There is a brief discussion on the 1D renewal function but item failures occur according to a 2D renewal process in region A. We discussed this type of process and the evaluation of the 2D renewal function in Section 4 of this note. Chen and Popova, however, attempt to reduce the 2D renewal function MðVt ; Vu Þ to 1D form by using the usage rate equation r ¼ Vu =Vt ; but then give no details on how to remove the conditioning on r: Chen and Popova state that ‘conventional means’ cannot be used to find the optimal values of the decision variables Vt and Vu from the results they give in Section 2.2. They then discuss an alternative simulation – optimization procedure in Section 3. This procedure still applies for a given value of r but no details are again provided about how to remove the dependence on r: It is important to note that the results of the simulation are only estimates whose accuracy depends on the number of independent replications carried out. The authors should have given confidence limits for these estimates to indicate their accuracy. Also, the simulation –optimization procedure must have been more time consuming then the exact solution method used by Iskandar and Murthy [5].
6. Conclusion The purpose of this note has been to highlight the errors in the paper by Chen and Popova [2]. Their review of the literature on servicing strategies for 2D warranties is seriously incomplete. In Sections 1 – 4 of this note, we have given a proper literature review of such strategies
N. Jack et al. / Reliability Engineering and System Safety 82 (2003) 105–109
for both 1D and 2D warranties and we have also described the 1D and 2D approaches to modelling item failures in the 2D warranty case. Chen and Popova’s paper concentrates on a particular servicing strategy that has already been discussed by Iskandar and Murthy [5]. They seem to be unaware of this paper and their stochastic model for the strategy contains fundamental flaws. In fact, they do not actually use this model for the optimization but instead resort to simulation. In Section 5 of this note, we identify the flaws in their stochastic model and suggest how they should be corrected. We also criticise the lack of detail they give in their description of the simulation procedure and we comment on the need for providing confidence limits that will indicate the accuracy of the simulation results. In our opinion, there is still scope for further research into servicing strategies for 2D warranties. We are currently investigating the 2D approach to modelling minimal repair in a warranty context. The techniques being developed can then be used to study the servicing strategies 3(2D)– 6(2D) discussed in Section 4 of this note using the 2D approach to failure modelling.
References [1] Blischke WR, Murthy DNP. Warranty cost analysis. New York: Marcel Dekker; 1994.
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[2] Chen T, Popova E. Maintenance policies with two-dimensional warranty. Rel Engng Syst Saf 2002;77:61–9. [3] Hunter JJ. Mathematical techniques for warranty analysis. In: Blischke WR, Murthy DNP, editors. Product warranty handbook. New York: Marcel Dekker; 1996. Chapter 7. [4] Iskandar BP. Modelling and analysis of two dimensional warranty policies. Unpublished Doctoral Thesis, The University of Queensland, Brisbane, Australia; 1993. [5] Iskandar BP, Murthy DNP. Repair replace strategies for twodimensional warranty. Proceedings of First Western Pacific and Third Australia–Japan Workshop on Stochastic Models in Engineering, Technology and Management, New Zealand; 1999. p. 206–13. [6] Iskandar BP, Murthy DNP, Jack N. A new repair–replace strategy for items sold with a two-dimensional warranty. Under Review; 2003. [7] Jack N, Murthy DNP. Servicing strategies for items sold under warranty. Jr Oper Res Soc 2001;52:1284 –8. [8] Jack N, Van der Duyn Schouten F. Optimal repair–replace strategies for a warranted product. Int J Prod Econ 2000;67:95 –100. [9] Moskowitz H, Chun YH. A Poisson regression model for twoattribute warranty policy. Naval Res Logist 1994;41:355 –76. [10] Murthy DNP, Wilson RJ. Modelling two-dimensional failure free warranties. Proceedings of the Fifth Symposium on Applied Stochastic Models and Data Analysis, held in Granada, Spain; 1991. [11] Murthy DNP, Iskandar BP, Wilson RJ. Two-dimensional failure free warranties: two-dimensional point process models. Oper Res 1995;43: 356–66. [12] Nguyen DG. Studies in warranty policies and product reliability. Unpublished PhD Thesis, The University of Queensland, Australia; 1984. [13] Singpurwalla ND, Wilson S. The warranty problem: its statistical and game theoretic aspects. SIAM Rev 1993;35:17 –42. [14] Baik J, Murthy DNP, Jack N. Two-dimensional failure modelling with minimal repair. Under Review; 2003.
Discussion
Comments on “Maintenance policies with two-dimensional warranty” N. Jacka,*, D.N.P. Murthyb,c, B.P. Iskandard a
School of Computing and Advanced Technologies, University of Abertay Dundee, Dundee DD1 1HG, Scotland, UK b Department of Mechanical Engineering, The University of Queensland, St Lucia 4072, Australia c Norwegian University of Science and Technology, Trondheim, Norway d Departemen Teknik Industri, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia Received 18 March 2003; accepted 5 May 2003
Abstract Chen and Popova [Res. Engng Syst. Saf. 77 (2002) 61] discuss maintenance policies for items sold with a two-dimensional warranty. However, their paper fails to give a proper review of the literature and it also contains errors. In this note we first review the relevant literature and then comment on the errors in their analysis. q 2003 Elsevier Ltd. All rights reserved. Keywords: Two-dimensional warranty; Servicing strategies
1. Introduction A warranty is an integral component of most product sales. It requires the manufacturer to either rectify or compensate for any failures that occur within the warranty period. A variety of warranty policies have been studied, and a taxonomy of these can be found in Blischke and Murthy [1]. In a one-dimensional (1D) warranty policy the warranty period is usually a time interval, and in the two-dimensional (2D) case it is characterised by a 2D planar region with one coordinate normally representing time (or age) and the other representing usage. Offering any warranty results in additional costs to the manufacturer and these must be factored into the sale price. The cost analysis of many different one- and two-dimensional warranty policies is given in Ref. [1]. For repairable products sold with a FRW (free repair or replacement warranty), the manufacturer has the option of either repairing or replacing a failed item. The costs as well as the benefits derived from these two types of corrective maintenance action are different. Replacement by a new item, although costing more than a repair, reduces the chances of subsequent failures occurring under warranty. A manufacturer’s optimal servicing strategy involves choosing the appropriate corrective maintenance * Corresponding author. E-mail address: [email protected] (N. Jack). 0951-8320/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0951-8320(03)00119-4
actions on failure in order to minimise expected warranty servicing costs. Servicing strategies for 1D warranties have received a lot of attention in the literature, but considerably less attention has been paid to this problem for 2D warranties. Also, in the 2D case, two different approaches to modelling item failures have been proposed. Chen and Popova [2] consider servicing strategies for 2D warranties but do not give a proper review of the relevant literature, and their model analysis of item failures contains errors. The purpose of this note is to rectify these two deficiencies. The outline of the note is as follows. In Section 2 we discuss repair – replace decisions for items sold with a 1D FRW policy. Section 3 reviews the two different approaches to modelling failures in the analysis of 2D warranty policies. Section 4 reviews the literature dealing with optimal repair – replace decisions for a 2D FRW policy. In Section 5 we critically look at the analysis of Chen and Popova [2] and highlight the errors. We conclude with some topics for future research in Section 6.
2. Servicing strategies for one-dimensional FRW policies The simplest servicing strategies for a repairable item sold with a 1D FRW with period W are (i) to always replace on failure (Strategy 1), and (ii) to always repair (Strategy 2).
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Each replacement is understood to be with a new and identical item and all repair actions are ‘minimal’, so they restore a failed item to an operational state without affecting the hazard rate. The number of item failures under warranty, NðWÞ; is modelled by a renewal process for Strategy 1 and a non-homogeneous Poisson process (NHPP) for Strategy 2. The corresponding expected number of failures are then given by the renewal function MðWÞ and the cumulative intensity function LðWÞ; respectively, and expected servicing costs are found by multiplying these functions by the cost of a replacement item and the cost of a minimal repair. Nguyen [12] proposes two alternative servicing strategies, which we shall term Strategies 3 and 4. In each case, the warranty period W is divided into two sub-intervals ½0; W1 Þ and ½W1 ; WÞ: Under Strategy 3, if an item fails in ½0; W1 Þ then it is replaced and if it fails in ½W1 ; WÞ it is only minimally repaired. Under Strategy 4, the opposite behaviour occurs with minimal repair on failure in the first sub-interval and replacement in the second sub-interval. The decision variable is W1 ; which is selected optimally by minimising the expected warranty servicing costs. Jack and Van der Duyn Schouten [8] prove that Strategies 3 and 4 are not optimal. Using a dynamic programming model, they show that the optimal repair– replace decision is determined by comparing the item’s age on failure with a time-dependent control limit function hðtÞ: The item is replaced on failure at time t if and only if its age is greater than hðtÞ: A typical plot of this function is shown in Fig. 1. Under the optimal strategy, which we term Strategy 5, the warranty period is divided into three sub-intervals I ¼ ½0; W1 Þ; II ¼ ½W1 ; W2 Þ; and III ¼ ½W2 ; WÞ: The item is always minimally repaired on failure in sub-intervals I and III, with the possibility of replacement on failure in
sub-interval II. The actual shape of hðtÞ in this sub-interval determines the maximum number of replacements that will occur. In Fig. 1, hðtÞ lies above the indicated line L1 ; so at most one replacement will be carried out. In general, the shape of hðtÞ is determined by the reliability of the item and the ratio of repair costs to replacement costs. Jack and Murthy [7] suggest a sub-optimal servicing strategy, which we term Strategy 6, with the following structure. Every item failure in sub-intervals I and II is again minimally repaired but only the first failure in sub-interval II results in a replacement. In most cases, the expected cost of this sub-optimal strategy is very close to that of the optimal Strategy 5 and the strategy is far easier to implement.
3. Failure modelling for two-dimensional warranties A 2D warranty is characterised by a region V in a 2D plane with the horizontal axis signifying time and the vertical axis usage. We confine to the case where the FRW policy expires either at time K or when the total item usage exceeds a level L; so that V is a rectangle. See Refs. [10,13] for other possible shapes for V: Item failures are represented by points lying in the 2D ‘time-usage’ plane, and two different methods have been used to model these failures. In the first method, termed the ‘one-dimensional approach’, usage is assumed to be a function of age so that failures are effectively modelled using a 1D point process formulation (see Refs. [4,9]). The second method, termed the ‘two-dimensional approach’, uses a joint distribution function to model the age and usage of the item at its first failure and then subsequent failures are modelled using a 2D point process formulation (see Ref. [11]). We now give further details of these two techniques. 3.1. The one-dimensional approach In this approach, the usage of the item is modelled as a linear function of the item’s age with non-negative coefficient R: R represents the usage rate and is assumed to be a non-negative random variable to account for the varying usage across the consumer population. Conditional on R ¼ r; lðtlrÞdt is used to denote the probability that the item in use at time t will fail in the small interval ½t; t þ dtÞ and NðtlrÞ denotes the number of item failures in the interval ð0; t: The form of NðtlrÞ depends on the type of corrective maintenance actions taken at item failures. 3.2. The two-dimensional approach
Fig. 1. Optimal control limit function for repair– replace decision.
Let T and X denote the system’s age and usage at its first failure. ðT; XÞ is assumed to be a non-negative bivariate random variable with distribution function Fðt; xÞ ¼ P{T # xÞ ¼ Pr{T . t; X . x}; t; X # x}; survivor function Fðt; density function f ðt; xÞ ¼ ›2 Fðt; xÞ=›t ›x; and hazard xÞ: Nðt; xÞ denotes the number function rðt; xÞ ¼ f ðt; xÞ=Fðt;
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of system failures occurring in the rectangular region ½0; tÞ £ ½0; xÞ and, as in the 1D approach, the form of Nðt; xÞ depends on the type of corrective maintenance actions taken to restore failed items.
4. Servicing strategies for the two-dimensional FRW policy with rectangular region V We now discuss the 2D counterparts of Strategies 1 –6 from Section 2, and we refer to these as Strategies 1(2D) –6(2D), respectively. Under Strategy 1(2D), all item failures that occur under warranty are rectified through replacement, and under Strategy 2(2D) every failure is minimally repaired. In each case, we consider both the 1D and 2D failure modelling approaches. Under Strategy 1(2D), with the 1D approach, let r1 ¼ L=K: It follows that, given R ¼ r; the warranty will expire at time Kr ; where Kr ¼ K if r # r1 and Kr , K if r . r1 : Conditional on R ¼ r; NðKr lrÞ is a 1D renewal process, where the time to first Ðitem failure has distribution function FðtlrÞ ¼ 1 2 exp{ 2 t0 lðxlrÞdx}: The expected number of failures (replacements) over ½0; Kr Þ is given by MðKr lrÞ; where MðtlrÞ is the renewal function corresponding to FðtlrÞ: Finally, the conditioning in this expected failure count function is removed by multiplying by gðrÞ; the density function for R; and integrating over all possible values of r: Using the 2D modelling approach for Strategy 1(2D), we have a 2D renewal process for system failures. The expected number of failures (replacements) under warranty is then given by MðK; LÞ; where MðK; LÞ is the 2D renewal function. function is defined by MðK; LÞ ¼ P This ðnÞ E½NðK; LÞ ¼ 1 F ðK; LÞ; where F ðnÞ ðt; xÞ is the n-fold n¼1 bivariate convolution of Fðt; xÞ with itself. Alternatively, it be expressed as the solution of the 2D integral equation (see Ref. [3]) MðK; LÞ ¼ FðK; LÞ þ
ðK ðL 0
MðK 2 u; L 2 vÞf ðu; vÞdv du:
0
Under Strategy 2(2D), with the 1D approach, it follows that conditional on R ¼ r; NðKr lrÞ is a NHPP with intensity function lðtlrÞ; and the expected number of failures (minimal repairs) over ½0; KrÐÞ is given by the cumulative intensity function LðKr lrÞ ¼ K0 r lðtlrÞdt: The conditioning in this expression is then removed in the same way as in Strategy 1(2D). The first notion of minimal repair using the 2D failure modelling approach has recently been developed by Baik et al. [14]. They show that the counting process {Nðt; xÞ : t $ 0; x $ 0} is a 2D NHPP with intensity function lðt; xÞ ¼ rðt; xÞ; and so the expected number of failures (minimal repairs) under warranty for Strategy 2(2D) is Ð Ð then given by E½NðK; LÞ ¼ LðK; LÞ ¼ K0 L0 lðt; xÞdx dt: Note that, other notions of minimal repair can be
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formulated and these are currently under investigation by Baik et al. [14]. Expected servicing costs for Strategies 1(2D) and 2(2D) are again found by multiplying the expected number of failures by the cost of a replacement item and the cost of a minimal repair, respectively. Further details on Strategy 1(2D) and on the 1D approach to Strategy 2(2D) are given in Ref. [1]. Strategies 3(2D) and 4(2D) have been studied by Iskandar and Murthy [5]. In each case, the warranty region V ¼ ½0; KÞ £ ½0; LÞ is divided into two sub-regions—V1 and V2 ; where V1 ¼ ½0; K1 Þ £ ½0; L1 Þ; and V2 ¼ V\ V1 : Under Strategy 3(2D), all failures occurring in V1 are rectified by replacement and any failure in V2 is rectified by minimal repair. This is the strategy discussed by Chen and Popova [2]. Under Strategy 4(2D), all failures occurring in V1 are rectified by minimal repair and any failure in V2 is rectified by replacement. For both strategies, Iskandar and Murthy [5] use the 1D approach to obtain the expected number of replacements and minimal repairs under warranty. This again involves first conditioning on R ¼ r and then removing the conditioning by multiplying by gðrÞ; the density function for R; and integrating over all possible values of r: Expected servicing costs are a function of the two decision variables K1 and L1 : Iskandar et al. [6] propose Strategy 6(2D), where the warranty region is divided into three sub-regions V1 ; V2 ; and V3 (See Fig. 2). Under this strategy (i) all failures in V1 are minimally repaired, (ii) the first failure in V2 is rectified through replacement and subsequent failures in this region are rectified through minimal repair, and (iii) all failures in V3 are always minimally repaired. Again, the 1D failure modelling approach is used to obtain the expected number of replacements and minimal repairs under warranty, and hence the expected warranty servicing cost. This cost is now a function of the four decision variables K1 ; K2 ; L1 ; and L2 : The analysis of Strategy 6(2D) based on the 2D failure modelling approach is currently under investigation by
Fig. 2. Regions characterising Strategy 6(2D).
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the authors. This is also the case for the optimal servicing strategy, Strategy 5(2D), which involves complex repair– replace regions over the 2D age-usage plane.
5. Comments on the Chen and Popova [2] paper Chen and Popova [2] consider Strategy 3(2D) but seem to be unaware of the paper by Iskandar and Murthy [5] where analytical results for the expected number of repairs and replacements and expected servicing cost are found using the 1D modelling approach. However, these functions do not have closed form analytical solutions and computational schemes are required for their evaluation. Section 2.2 of the Chen and Popova paper contains a derivation of the expected servicing cost for Strategy 3(2D) but their results contain a number of fundamental flaws and many minor errors. They appear to be totally confused between the 1D and 2D approaches to modelling item failures. We shall concentrate only on identifying the fundamental flaws in their modelling but first we need to give a summary of some of the notation they use: {ð0; WT Þ £ ð0; WU Þ} the rectangular warranty region. A ¼ {ð0; Vt Þ £ ð0; Vu Þ} the replacement region. B ¼ {ð0; WT Þ £ ð0; WU Þ}=A the minimal repair region. ðt; uÞ a point [ {ð0; WT Þ £ ð0; WU Þ}: lðt; uÞ the ‘2D failure rate’ at the point ðt; uÞ: N1 ðt; uÞ the number of replacements in the region {ð0; tÞ £ ð0; uÞ}: N2 ðt; uÞ the number of minimal repairs in the region {ð0; tÞ £ ð0; uÞ}: SN1 ðt;uÞ the time of the last replacement prior to the point ðt; uÞ: r ¼ u=t the usage rate. Vt ; Vu the decision variables for the servicing strategy. At the beginning of Section 2.2, Chen and Popova state that the objective function to be minimised is the sum of the expected cost of replacements and repairs over the warranty region. Eq. (1) then gives a general expression for this function conditioned on the usage rate r: Thus, they seem to be using the 1D failure modelling approach and the un-conditioning should then be carried out with respect to this variable in order to obtain the unconditional expected cost. This is the method used by Iskandar and Murthy [5]. However, Chen and Popova then change to the 2D approach by introducing the variables ðt; uÞ; N1 ðt; uÞ; N2 ðt; uÞ; and SN1 ðt;uÞ : They re-define the objective function, with the conditioning now on ðt; uÞ and SN1 ðt;uÞ ; but the latter variable gives insufficient information regarding the position of the last replacement. Instead, the authors should have used the point (SN1 ðt;uÞ ; rSN1 ðt;uÞ ), and considered the two cases where the point lies in region A and region B, separately. The un-conditioning then needs to be done with
respect to r: Eq. (4) for the unconditional expected minimal repair cost is incorrect, because of this error. Eq. (5) is also incorrect. Firstly, minimal repairs are carried out over the region B. This result also uses a 1D modelling concept where, under minimal repair, the expected number of failures over a time interval is given by the integral of the failure intensity function with this function having the same form as the item’s hazard function for time to first failure. This same concept cannot be assumed without proof for the 2D case. Eq. (6) is incorrect. The right hand side is the probability that the first failure occurs outside region A (i.e. in or beyond region B) if Fðt; uÞ is the 2D failure distribution function of a new item. Note also that, if one starts with the failure rate function lðt; uÞ it is not possible to derive the failure distribution function Fðt; uÞ uniquely. The authors also state that Eq. (8) follows from the ‘key renewal theorem’. This is not the case and, once again, Eq. (8) seems to be an attempted extension of a well-known 1D modelling result. It should not be stated without proof or without a valid reference. Eqs. (4) –(8) deal with the expected number of minimal repairs in region B but few details are given on how to find the expected number of replacements in region A. There is a brief discussion on the 1D renewal function but item failures occur according to a 2D renewal process in region A. We discussed this type of process and the evaluation of the 2D renewal function in Section 4 of this note. Chen and Popova, however, attempt to reduce the 2D renewal function MðVt ; Vu Þ to 1D form by using the usage rate equation r ¼ Vu =Vt ; but then give no details on how to remove the conditioning on r: Chen and Popova state that ‘conventional means’ cannot be used to find the optimal values of the decision variables Vt and Vu from the results they give in Section 2.2. They then discuss an alternative simulation – optimization procedure in Section 3. This procedure still applies for a given value of r but no details are again provided about how to remove the dependence on r: It is important to note that the results of the simulation are only estimates whose accuracy depends on the number of independent replications carried out. The authors should have given confidence limits for these estimates to indicate their accuracy. Also, the simulation –optimization procedure must have been more time consuming then the exact solution method used by Iskandar and Murthy [5].
6. Conclusion The purpose of this note has been to highlight the errors in the paper by Chen and Popova [2]. Their review of the literature on servicing strategies for 2D warranties is seriously incomplete. In Sections 1 – 4 of this note, we have given a proper literature review of such strategies
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for both 1D and 2D warranties and we have also described the 1D and 2D approaches to modelling item failures in the 2D warranty case. Chen and Popova’s paper concentrates on a particular servicing strategy that has already been discussed by Iskandar and Murthy [5]. They seem to be unaware of this paper and their stochastic model for the strategy contains fundamental flaws. In fact, they do not actually use this model for the optimization but instead resort to simulation. In Section 5 of this note, we identify the flaws in their stochastic model and suggest how they should be corrected. We also criticise the lack of detail they give in their description of the simulation procedure and we comment on the need for providing confidence limits that will indicate the accuracy of the simulation results. In our opinion, there is still scope for further research into servicing strategies for 2D warranties. We are currently investigating the 2D approach to modelling minimal repair in a warranty context. The techniques being developed can then be used to study the servicing strategies 3(2D)– 6(2D) discussed in Section 4 of this note using the 2D approach to failure modelling.
References [1] Blischke WR, Murthy DNP. Warranty cost analysis. New York: Marcel Dekker; 1994.
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[2] Chen T, Popova E. Maintenance policies with two-dimensional warranty. Rel Engng Syst Saf 2002;77:61–9. [3] Hunter JJ. Mathematical techniques for warranty analysis. In: Blischke WR, Murthy DNP, editors. Product warranty handbook. New York: Marcel Dekker; 1996. Chapter 7. [4] Iskandar BP. Modelling and analysis of two dimensional warranty policies. Unpublished Doctoral Thesis, The University of Queensland, Brisbane, Australia; 1993. [5] Iskandar BP, Murthy DNP. Repair replace strategies for twodimensional warranty. Proceedings of First Western Pacific and Third Australia–Japan Workshop on Stochastic Models in Engineering, Technology and Management, New Zealand; 1999. p. 206–13. [6] Iskandar BP, Murthy DNP, Jack N. A new repair–replace strategy for items sold with a two-dimensional warranty. Under Review; 2003. [7] Jack N, Murthy DNP. Servicing strategies for items sold under warranty. Jr Oper Res Soc 2001;52:1284 –8. [8] Jack N, Van der Duyn Schouten F. Optimal repair–replace strategies for a warranted product. Int J Prod Econ 2000;67:95 –100. [9] Moskowitz H, Chun YH. A Poisson regression model for twoattribute warranty policy. Naval Res Logist 1994;41:355 –76. [10] Murthy DNP, Wilson RJ. Modelling two-dimensional failure free warranties. Proceedings of the Fifth Symposium on Applied Stochastic Models and Data Analysis, held in Granada, Spain; 1991. [11] Murthy DNP, Iskandar BP, Wilson RJ. Two-dimensional failure free warranties: two-dimensional point process models. Oper Res 1995;43: 356–66. [12] Nguyen DG. Studies in warranty policies and product reliability. Unpublished PhD Thesis, The University of Queensland, Australia; 1984. [13] Singpurwalla ND, Wilson S. The warranty problem: its statistical and game theoretic aspects. SIAM Rev 1993;35:17 –42. [14] Baik J, Murthy DNP, Jack N. Two-dimensional failure modelling with minimal repair. Under Review; 2003.