Learning effects on maintenance outsourcing

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1 Available online at European Journal of Operational Researh 192 (2009) Stohastis and Statistis Learning effets on aintenane outsouring Hakan Taraki a, *, Kwei Tang b, Sunantha Teyarahakul a Melbourne Business Shool, University of Melbourne, Carlton, Vitoria 3053, Australia b Krannert Shool of Manageent, Purdue University, West Lafayette, IN 47907, USA Esse Business Shool, Cergy Pontoise Cedex, Frane Reeived 13 February 2007; aepted 7 Septeber 2007 Available online 15 Septeber Abstrat The objetive of this paper is to study effets on aintenane outsouring. We onsider a situation in whih a anufaturer offers a short-ter outsouring ontrat to an external ontrator who is responsible for sheduling and perforing preventive aintenane and arrying out inial repairs when the proess fails. The anufaturer s payent to the ontrator onsists of a fixed aount along with ost subsidization for eah aintenane operation perfored. We assue ours when the ontrator perfors preventive aintenane that redues both tie and ost. Two types of are onsidered: natural and by ostly efforts. We deonstrate that a well-designed payent shee an indue the ontrator to adopt the aintenane shedule that axiizes the anufaturer s profit. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Maintenane outsouring; Learning; Cost subsidization 1. Introdution Maintenane outsouring has beoe a prevalent pratie in any industries, suh as edial equipent anufaturing and aviation. Tiean (2002) estiates that annual edial equipent aintenane outsouring is worth $26 billion, and Jensen (2006) reports in a reent survey that 84% of the surveyed opanies eploy aintenane ontrators. Many reports have suggested that anufaturers an benefit fro a partnership relationship with external aintenane ontrators. Jensen (2006) also reports that 64% of the opanies in the sae survey believe ontrators to be of high value and that ontrat aintenane has beoe a growing industry. In aadei researh, aintenane is a very well-established area. Extensive surveys and reviews have been given by Pierskalla and Voelker (1976), Sherif and Sith (1981), Valdez-Flores and Feldan (1989), and, ost reently, Wang (2002). Most existing studies in the literature use a oon assuption that aintenane operations are ontrolled and perfored by the anufaturer and, onsequently, that all related deisions are also ade by the anufaturer. The results derived under this assuption ay not be appliable if aintenane operations are outsoured beause ontrators ay ontrol ertain deisions. To the best of our knowledge, only five aadei researh papers address aintenane outsouring in the literature. The first paper was written by Murthy and Asgharizadeh (1999), who designed a aintenane ontrat between a anufaturer and a ontrator that axiized the ontrator s profit. The seond paper, their follow-up, onsidered ultiple * Corresponding author. Tel.: ; fax: E-ail addresses: h.taraki@bs.edu (H. Taraki), ktang@exhange.purdue.edu (K. Tang), teyarah@esse.fr (S. Teyarahakul) /$ - see front atter Ó 2007 Elsevier B.V. All rights reserved. doi: /j.ejor

2 hoogeneous anufaturers that opete for a ontrator s servies (Asgharizadeh and Murthy, 2000). The fous of their paper was also on axiizing the ontrator s profit. The third paper, by Plabek and Zenios (2000), odeled aintenane outsouring as a ulti-period prinipal-agent proble. They axiized the disounted profit of the prinipal (a single anufaturer) by taking into aount the reservation profit and inentives of the agent (a single ontrator). The fourth paper, by Taraki et al. (2006a), studied oordination issues between a anufaturer and a aintenane ontrator. Their ai was not to axiize a single party s profit but rather the total profit of the syste. They proposed several inentive shees for ahieving the goal. In the fifth paper, they extended the analysis to a ore opliated situation, in whih the anufaturer has ultiple proesses and onsiders ultiple ontrators for perforing aintenane for these proesses (Taraki et al., 2006b). Note that both of their studies are built on a long-ter ontratual relationship between two parties. In our paper, we onsider effets in aintenane outsouring. Learning in prodution is a phenoenon that unit prodution tie or ost dereases as additional units are produed. The effets of have been analyzed extensively in the operations anageent area. Wright (1936) was the first to reognize and odel experiene-based, based on his observations in the airraft industry. Additional evidene for the phenoenon was later douented in suh studies as Preston and Keahie (1964) and Rapping (1965). Yelle (1979) provided a thorough review of the urve. The topi ontinues to reeive researh attention. Spene (1981) odeled the opetitive interation in the presene of a urve. Fine (1986) studied the effet of and quality iproveent on prodution systes. Mazzola and MCardle (1996) and Jovanovi and Nyarko (1996) investigated stohasti urves using a Bayesian approah. Argote et al. (1990) onduted an epirial study on the persistene and transfer of organizational. Dorroh et al. (1994) studied the reation of knowledge (or ) through investent in organizational resoures suh as siene and tehnology, software developent, and worker training. Learning in their study was not experiene-based, but was ahieved through ostly investent. In the aintenane literature, Wang and Lee (2001) estiated the urve for total produtive aintenane operations. In this paper, we study effets on aintenane outsouring. We onsider two types of aintenane operations used to iprove the reliability of the anufaturing proess: preventive aintenane (PM) and inial repair (MR). Preventive aintenane, whih involves an overhaul of the whole syste, is perfored periodially to avoid ostly repairs and disruptions of prodution aused by proess breakdowns. Minial repair is perfored iediately after the proess breaks down. The anufaturer plans to outsoure these aintenane funtions to a ontrator for a fixed tie horizon (ontrat period) and delegates aintenane deisions to the ontrator in the ontrat period. In other words, under this relationship, the ontrator selets the nuber and frequeny of PM operations and perfors neessary MR whenever the proess breaks down. The anufaturer s payent to the ontrator has two oponents: a fixed payent and a ost subsidization (CS) shee for every PM and MR operation that the ontrator perfors. This subsidization (inentive) shee has been proven effetive in induing the ontrator to adopt a aintenane strategy that axiizes the total profit of the syste. We assue both experiene-based (natural) and ostly through investent in PM operations. The objetive of the anufaturer is to find the payent shee that axiizes profit when effets are present in the ontrator s PM operations. The organization of the paper is as follows. In Setion 2, we introdue the basi short-ter ontrat with no. In Setion 3, we analyze effets in preventive aintenane. In Setions 3.1 and 3.2, we fous on experiene-based or natural and fro ostly effort or investent, respetively. Finally, in Setion 4, we disuss onlusions and further researh possibilities. All proofs of analytial results are given in the Appendix. 2. Basi short-ter ontrat (no ) H. Taraki et al. / European Journal of Operational Researh 192 (2009) Suppose the anufaturer has a ontinuous proess with a failure rate funtion kðtþ in tie t suh that kð0þ ¼0and k 0 ðtþ > 0. This iplies that kðtþ is an inreasing failure rate (IFR) funtion. Many oon probability distributions, inluding Weibull, Noral, and Gaa distributions, are IFR funtions. We use T p and C p to denote the average tie and ost, respetively, required for the ontrator to perfor a preventive aintenane operation. It is assued that preventive aintenane overhauls the proess and restores the failure rate to the original (best) state; i.e., kð0þ ¼0(Barlow and Hunter, 1960). This assuption is oonly used in aintenane/reliability studies, and the results of the paper reain unhanged as long as the failure rate is restored to a positive onstant after every PM. Furtherore, we use T r and C r to denote the average tie and ost, respetively, required for the ontrator to perfor a inial repair operation. Let the ontrat period be Y tie units. We assue that the ontrator starts by perforing a PM at the beginning of the ontrat period. Let N, a deision variable, represent the nuber of PM operations perfored during the ontrat

3 140 H. Taraki et al. / European Journal of Operational Researh 192 (2009) period and T i, another deision variable, represent the length of the ith PM interval, i.e., the tie between the end of the ith and the beginning of the (i + 1)th PM operations. We also let T N be reaining tie in the ontrat period after the Nth PM. Sine we assue that the ontrator akes all the aintenane related deisions, inluding the nuber and length of PM intervals, we use N and T i to denote the ontrator s seletion of N and T i. We also use T N as the reaining tie in the ontrat period after the Nth PM under the ontrator s seletion of N and T i. Let P P 0 be the guaranteed payent, deided by the anufaturer, to the ontrator over the ontrat period and 0 6 s p 6 1 and 0 6 s r 6 1 be the ratio of C p and C r that the anufaturer subsidizes the ontrator, respetively, for eah of the PM and MR operations. Therefore, the ontrator s net PM and MR osts beoe ð1 s p ÞC p and ð1 s r ÞC r, respetively. Note that s p and s r are deision variables. This inentive ehanis an be used by the anufaturer to indue the ontrator to selet a aintenane poliy that axiizes the anufaturer s expeted profit. We assue that the ontrator has a pre-deterined reservation (iniu) expeted profit over the prodution horizon Y, denoted by P 0, for partiipating in the ontratual relationship. In other words, the ontrator will aept the ontrat only if his or her expeted profit is greater than or equal to P 0. Let the net revenue of the anufaturer, after taking into aount the prodution-related osts, be R per unit of tie that the proess is in operation. We assue both the anufaturer and the ontrator are risk-neutral. We assue that inial repair erely restores the proess bak to operation, and the deterioration of the proess reains during inial repair. Under this assuption, the proess is restored to the deterioration state that it would have been in without breakdown and inial repair. This assuption is used to siplify our analysis and is valid when T r is relatively sall or a inial repair involves, for exaple, replaing inor oponents, while the rest of the proess is not stopped (Barlow and Hunter, 1960; Jardine, 1973; Elsayed, 1996; Taraki et al., 2006). The ontrator s expeted profit over the tie horizon depends on the payent/ost subsidization it reeives fro the anufaturer, on the nuber of PM operations it deides to perfor, and on the expeted nuber of failures in eah PM interval. We define MðT i Þ as the expeted nuber of failures in tie interval ½0; T i Š. Using MðT i Þ¼ R T i kðxþ dx (Srinivasan 0 and Lee, 1996), the ontrator s deision an be forulated as the following optiization proble: Max P ¼ P N ½ð1 s p ÞC p Š ½ð1 s r ÞC r Š XN N ;T 1 ;T 2 ;...;T N s:t: N T p þ XN T i ¼ Y : MðT i Þ The anufaturer s profit is deterined by the revenue fro the proess uptie inus the expeted total payent to the ontrator, whih inludes the fixed payent and the subsidy for the ontrator s PM and MR operations. The anufaturer s deision an be forulated as the following proble: Max P ¼ R½Y N T p Š N s p C p N ½s r C r þ RT r Š XN P;s p;s r s:t: P N ½ð1 s p ÞC p Š ½ð1 s r ÞC r Š XN MðT i Þ P P 0 : MðT i Þ P Expression (4) is used to ensure that the ontrator s expeted profit given by (1) is P 0 or higher. The anufaturer s deision is to selet P, s p, and s r to axiize his or her expeted profit. ð1þ ð2þ ð3þ ð4þ 3. Learning effets in preventive aintenane In this setion, we assue that the ontrator, through repetition and preparation, will learn how to perfor the preventive aintenane operations ore effiiently. We assue ours in both ost and tie of preventive aintenane. In Setion 3.1, we will assue happens naturally; i.e. the ontrator an perfor PM operations ore effiiently over tie without exerting any additional ost. In Setion 3.2, we assue in PM will take plae only through a ostly investent by the ontrator Natural We assue that the ontrator perfors the first PM at a ost of C p and tie of T p. The subsequent PM operations ost less and take less tie beause of. Speifially, we assue that for the ith PM, the ost is C p i a, and the duration is T p i a, where a P 0 represents the slope of the urve. A large a value orresponds to a high level. These

4 funtions have been oonly used in the literature (Keahie and Fontana, 1966; Adler and Nanda, 1974; Karwan et al., 1988; Rahaadugu and Tan, 1997). Note that a ¼ 0 represents no-. Therefore, the optiization probles in Setion 2 are speial ases of the probles presented in this subsetion with a ¼ 0. We assue the sae level of for tie and ost to siplify the analysis, but it is not diffiult to odify our odel to allow different levels of for tie and ost. With the effets, the ontrator s deision odel is given by: Max P 1 ðlþ ¼P ½ð1 s p ÞC p Š XN N ;T 1 ;T 2 ;...;T N s:t: X N i a T p þ XN T i ¼ Y : i a ½ð1 s r ÞC r Š XN MðT i Þ Please note that in, (5), s p still represents the subsidy ratio that the ontrator reeives for every PM. In other words, for the first PM, the ontrator reeives a subsidy aount of s p C p ; for the seond PM, this aount dereases to s p C p 2 a, and for the ith PM, it beoes s p C p i a. Lea 1. For any value of N, the optial PM intervals for the ontrator that axiize P 1 ðlþ are equal. That is, T 1 ¼ T 2 ¼¼T N T. In the prodution anageent literature, researhers have found results siilar to those of Lea 1. For exaple, for the basi eonoi order quantity (EOQ) odel involving onstant and deterinisti deand, Chand (1989) proved that P using equal lot sizes is optial when effets in the setup ost is present. Using Lea 1 and the onstraint that N i a T p þ P N T i ¼ Y, we rewrite the ontrator s expeted profit as: P 2 ðlþ ¼P ½ð1 s p ÞC p Š XN i a N ½ð1 s r ÞC r ŠM Y N T p X N ð5þ ð6þ i a 1 N!: ð7þ To siplify our analysis, we use the approxiation, P N i a R N i a di, in(7). It is known that this approxiation 0 works very well for values of a less than or equal to 0.6, whih overs virtually all pratial ases (Keahie and Fontana, 1966). However, this approxiation has a tendeny to slightly overstate the total ost (Ca et al., 1987), and the perentage of inauray diinishes as N inreases. Lea 2 (i) Define hðnþ ¼ Y k at p YN T a pn N a YN T pna M. Then, there is a unique N N 1 a 1 a 1 a 1 a, denoted by N ðlþ and not neessarily an integer, that axiizes P 2 ðlþ, satisfying the following relationship: hðn ðlþþ ¼ ð1 s pþc p : ð8þ ð1 s r ÞC r (ii) The integral N value, denoted by N ðlþ, that axiizes P 2ðLÞ an be found by rounding N ðlþ up or down. In other words, N ðlþ ¼ argax P 2 ðlþ: ð9þ b ðlþ ; j N ðlþj N2 N H. Taraki et al. / European Journal of Operational Researh 192 (2009) Lea 2 indiates that the ontrator s optial PM poliy depends on the siple ratio of the PM and MR osts. Using Leas 1 and 2, we forulate the anufaturer s deision odel: Max P 1 ðlþ ¼R Y T pn 1 a N 1 a s p C p P;s p;s r N½s rc r þ RT r ŠM Y N T pn a P ð10þ s:t: P ½ð1 s p ÞC p Š N 1 a N½ð1 s rþc r ŠM Y N T pn a P P 0 : ð11þ Note that onstraint (11) should be binding when P 1 ðlþ is axiized. Substituting for P in the objetive funtion, the anufaturer s expeted profit beoes a funtion of N as follows: P 2 ðlþ ¼R Y T pn 1 a N 1 a C p N½C r þ RT r ŠM Y N T pn a P 0 : ð12þ Let N denote the nuber of PM operations that would axiize the anufaturer s profit.

5 142 H. Taraki et al. / European Journal of Operational Researh 192 (2009) Lea 3 (i) There is a unique N, denoted by N M ðlþ and not neessarily an integer, that axiizes P 2ðLÞ, satisfying the following relationship: hðn ðlþþ ¼ RT p þ C p : ð13þ RT r þ C r (ii) The integral N value, denoted by N ðlþ, that axiizes P 2ðLÞ an be found by rounding N ðlþ up or down. In other words, N ðlþ ¼ argax P 2 ðlþ: ð14þ N2 b N ðlþ ; j N ðlþj (iii) N ðlþ inreases in a. The results in Lea 3 provide several iportant insights. First, the results indiate that the ritial easure for the anufaturer is the ratio of the total osts (revenue loss and atual aintenane ost) of PM and MR operations. The results also suggest that, if there is an iproveent in the PM operation beause of, the anufaturer s optial (not neessarily integral) nuber of PM operations inreases. The PM ost obviously affets the ontrator s profit; however, it influenes the anufaturer s profit as well through the ost subsidization shee. Therefore, the derease in C p akes PM ore appealing to the anufaturer. In addition, a shorter T p akes PM even ore attrative to the anufaturer. Consequently, the frequeny of PM operation inreases as a result of effets. The anufaturer, with full knowledge of his or her own and the ontrator s profit and ost strutures, an apply the following three steps to find the optial solution: 1. Find N ðlþ fro Lea Set s p and s r values suh that hðn ðlþþ ¼ ð1 s pþc p : ð1 s r ÞC r 3. Set P ¼½ð1 s p ÞC p Š N ðlþ1 a þ N ðlþ½ð1 s Y rþc r ŠM N ðlþ T pn ðlþ a þ P 0 : ð15þ ð16þ In Step 1, we find the PM frequeny that axiizes the anufaturer s expeted profit, P 2 ðlþ. In Step 2, we selet an inentive in ters of s p and s r to indue the ontrator to selet a PM frequeny that axiizes the anufaturer s expeted profit. Note that the optial values of s p and s r are not unique; there are an infinite nuber of obinations that satisfy (15). Finally, in Step 3, (16) is used to ensure that the ontrator s expeted profit is equal to the reservation profit and thus the ontrator is willing to partiipate in the outsouring relationship. Exaple 1. A anufaturer has a proess with a Weibull failure rate funtion given by kðtþ ¼kwt w 1, where k ¼ 0:05 and w ¼ 3 are sale and shape paraeters, respetively. Using this failure rate funtion, the expeted nuber of failures in tie interval ½0; T Š is MðT Þ¼kT w. The anufaturer s profit is R ¼ $1000 per day when the proess is operational, the anufaturing horizon is Y ¼ 300 days, and the ontrator s reservation profit is P 0 ¼ $20; 000. The ties and osts for aintenane operations for the ontrator are T p ¼ 1 day, T r ¼ 0:2 days, C p ¼ $500, and C r ¼ $70. We onsider the no- ase first. Using (13) with a ¼ 0, we find that the optial nuber of PM operations for the anufaturer is N ¼ 68:6. Using (14), we find that the integral optial nuber, N, is equal to 69, whih iplies that the optial PM interval for the anufaturer is days. To indue the ontrator to selet this aintenane shedule, the anufaturer has an infinite nuber of s p and s r obinations to selet. We give two of those obinations Option 1 gives the highest guaranteed payent of P, whereas Option 2 finds a uh lower value for P. The first option has s p ¼ 0:239 and s r ¼ 0, whih result in P ¼ $55; 305:01, and the seond option has s p ¼ 0:962 and s r ¼ 0:950, whih lead to P ¼ $21; 765:25. Under both options, the anufaturer s expeted profit is $141, over the ontrat period, and, as expeted, the ontrator s expeted profit is $20,000. Note that the ontrator s profits follow very different distributions under the two options, although the expeted profits are idential. This is showed in part 1a of Fig. 1. Note that the ontrator s profit distributions are derived fro the well-known result that the nuber of failures is a non-hoogeneous Poisson proess (Yu et al., 2007). It is evident that a risk-averse ontrator would prefer Option 2.

6 H. Taraki et al. / European Journal of Operational Researh 192 (2009) Next, we assue that ours in perforing PM operations with the slope of the urve, a ¼ 0:1. Using Lea 3, we find that the optial nuber of PM operations in the ontrat period, N ðlþ, is Please note that, as part (iii) of Lea 3 suggests, N ðlþ ¼80:7 is larger than that in no- situation. This result is intuitively expeted, beause when ours, PM beoes ore attrative, and, therefore, PM operations should be perfored ore frequently. Using (13), we obtain N ðlþ ¼81, whih iplies the optial PM interval for the anufaturer is days. To indue the ontrator to selet this aintenane shedule, the anufaturer an selet fro an infinite nuber of s p and s r obinations. Again, we give two obinations, whih result in the highest and lowest total payents to the ontrator: Option 1 with s p ¼ 0:233, s r ¼ 0, and P ¼ $49; 798:38; Option 2 with s p ¼ 0:962, s r ¼ 0:950, and P ¼ $21; 489:92. In both ases, the expeted profit of the anufaturer is $163, (an inrease of ore than $22,000 opared with the no ase), and the ontrator reeives an expeted profit of $20,000. In part 1b of Fig. 1, we opare the ontrator s profit distributions under these two options. Siilar to the no- ase, Option 1 with low subsidization ratios displays a uh larger variation. We use Fig. 2 to deonstrate the effets of the slope of the urve, a, on the optial solution. We onsider two values for P: $25,000 and $35,000. For eah value of P, we study the effet of a, ranging fro 0 (no ) to 0.4, on the ost subsidization ratios, s p and s r. As expeted, for any given P, the ost subsidization ratios are dereasing in a, beause, as the level inreases, the total aintenane ost dereases, and thus, the ontrator is willing to aept a lower a 0.05 Probability Option 2 Option Profit b 0.05 Probability Option 2 Option Profit Fig. 1. Profit distributions under different options. 1.a. Without 1.b. With.

7 144 H. Taraki et al. / European Journal of Operational Researh 192 (2009) ost subsidy. We also observe that, for a fixed level, the ost subsidization ratios inrease as P dereases. Again, this is expeted beause using a lower guaranteed payent, P, requires higher ost subsidies to ensure the ontrator s expeted reservation profit is et. Based on this exaple, we arry out additional analysis to deonstrate the robustness of the optial CS payent shee. For oparison, we define a foring ontrat under whih the anufaturer sets the PM shedule for the ontrator to follow. Suppose a CS ontrat and a foring ontrat are both designed for a level, a, of 0.1. The paraeters of the CS ontrat are hosen suh that ð1 spþcp ð1 s rþc r ¼ RT pþcp RT rþc r. There are an infinite nuber of ðs p ; s r Þ pairs to ahieve this. For exaple, we set s p ¼ 0:378, s r ¼ 0:2, and P ¼ $44; 091:69. The foring ontrat, on the other hand, would require that the ontrator perfor PM 81 ties (in equal intervals of days) during the ontrat period and be paid the reservation profit of $20,000 plus expeted (and unsubsidized) aintenane osts of $34, for a total payent of $54, Under both ontrats, the expeted profit of the anufaturer is $163, and that of the ontrator is $20,000. Now, suppose the atual level turns out to be 0.2 rather than 0.1, due to, for exaple, tehnologial advanes. In this ase, with the CS ontrat, the ontrator would deide to perfor 96 PM operations (whih would also be the new optial nuber of PM operations for the anufaturer). The resulting profits are $180, for the anufaturer and $24, for the ontrator. On the other hand, with the foring ontrat, the profits would be $172, and $28,636.70, respetively. With the CS ontrat, the anufaturer akes 4.1% ore profit opared with the foring ontrat. If the atual rate turns out to be 0.5, the differene is uh ore profound for the anufaturer: $217, with the CS ontrat versus $197, with the foring ontrat, iplying a shortfall of 18% for the foring ontrat. We further illustrate the advantage of the CS ontrat over the foring ontrat in Table 1. In Table 1, we used the sae ost and tie paraeter values fro Exaple 1. In addition, we looked at three levels of expeted : no ða ¼ 0Þ, low rate ða ¼ 0:2Þ, and oderate rate ða ¼ 0:4Þ. For eah of these assued levels, a CS ontrat and a foring ontrat are obtained to axiize the anufaturer s profit. We then opare the perforanes of these two ontrats if the ontrator s rate is atually greater than the assued value. The CS ontrat generally outperfors the foring ontrat. For exaple, if the anufaturer assues no in designing the ontrats while the true rate is 0.3, the anufaturer s profit is 20.76% higher if the CS ontrat is used. The results also show, as the differene between the assued and the atual levels inreases, so does the differene in profits between the CS and the foring ontrats. We also note that this differene between profits is uh greater when the assued levels are low Learning through effort In this setion, we assue that takes plae through ostly efforts. Speifially, we assue the ontrator inurs a ost of KðaÞ in an effort to ahieve the level of a. This ost is an initial investent and is inurred only one through the ontratual horizon. We assue that Kð0Þ ¼0, K0ðaÞ > 0, and K00ðaÞ > 0, whih iplies that the higher the level, the ore ostly to yield an iproveent in the PM ost and tie. We still assue that for the ith PM, the ost is C p i a, and the duration is T p i a. 1 sp / sr s p s r s p s r P = 25,000 P = 35, a Fig. 2. Effets of level and guaranteed payent on ost subsidization ratios.

8 H. Taraki et al. / European Journal of Operational Researh 192 (2009) Table 1 Effets of level on the anufaturer s expeted profit Assued level = 0 Assued level = 0.2 Assued level = 0.4 Manufaturer s profit under foring ontrat Manufaturer s profit under CS ontrat % Profit gain of CS ontrat over foring ontrat Atual level = 0.1 Atual level = 0.3 Atual level = 0.5 Atual level = 0.3 Atual level = 0.5 Atual level = 0.7 Atual level = 0.5 Atual level = 0.7 Atual level = 0.9 $150,375 $157,452 $159,321 $193,273 $202,692 $206,401 $225,554 $232,941 $231,569 $159,295 $190,144 $212,969 $198,838 $221,589 $235,662 $228,623 $242,716 $244, % 20.76% 33.67% 2.88% 9.32% 14.18% 1.36% 4.20% 5.45% Note that in the ase of natural in Setion 3.1, ours naturally through auulated experiene. In other words, the ontrator does not ontrol the level of ; i.e., the value of a in the odels. However, in this Subsetion, the ontrator an deterine the level of, although the deision has a ost ipliation. Consequently, one iportant deision for the anufaturer is to deterine the optial level of a for his or her expeted profit and design the ontrat to indue the ontrator to use the sae level in designing the aintenane poliy. The onepts of knowledge () reation through investing resoures were addressed by Dorroh et al. (1994); that is, the output rate is deterined by the investent in knowledge and the uulative stok of knowledge. We have applied siilar onepts to this paper. Inluding Kða Þ in the ontrator s profit funtion given in (7), the ontrator s axiu profit an be obtained by: Max P 1 ðl; eþ ¼P ½ð1 s p ÞC p Š XN N ;T 1 ;T 2 ;...;T N ;a s:t: X N i a T p þ XN T i ¼ Y : i a ½ð1 s r ÞC r Š XN MðT i Þ Kða Þ ð17þ ð18þ Lea 4. For any given value of N and a, the optial PM intervals for the ontrator that axiize P 1 ðl; eþ are equal. That is, T 1 ¼ T 2 ¼¼T N ¼ T. As in natural and no, with ostly effort also results in equal tie intervals. Using this result, the ontrator s profit funtion an be rewritten as:! P 2 ðl; eþ ¼P ½ð1 s p ÞC p Š XN i a N ½ð1 s r ÞC r ŠM Y X N 1 T p Kða Þ: ð19þ N Again, using the approxiation P N i a R N i a di, we an obtain the ontrator s optial PM poliy and effort level. 0 Lea 5 N (i) Define Y hðnðaþþ ¼ N at p k YN 1 a T pn a N a M YN T pn a : ð20þ Then, there is a unique N ða Þ, denoted by N ða Þ and not neessarily an integer, that axiizes P 2 ðl; eþ for any given value of a, the effort level. N ða Þ satisfies the following relationship: hðn ða ÞÞ ¼ ð1 s pþc p : ð1 s r ÞC r (ii) Let ðaðnþþ ¼ N 1 a ½ðÞlnðNÞ 1Š ð1 s p ÞC p ð1 s r ÞC r T p k Y N T pn a ð1 aþ 2 K 0 ðaþ: ð21þ ð22þ

9 146 H. Taraki et al. / European Journal of Operational Researh 192 (2009) Then, there is a unique a, denoted by a ðn Þ that axiizes P 2 ðl; eþ for any given value of N, the nuber of PM operations. a ðn Þ satisfies the following relationship: ða ðn ÞÞ ¼ 0: ð23þ Lea 5 states the properties of the best response funtion of N given any value of a and vie versa. The following iterative algorith an be used to find the optial N and a pair that axiizes P 2 ðl; eþ: 1. Set a ¼ Find N ða Þ, using (21). 3. Find a ðn ða ÞÞ, using (23). 4. Return to Step 2, and find the new N value. If the differene between the previous N value and the new one is within a sall range, say e, then go to Step 5. Otherwise, go to Step Let N denote the optial but not neessarily integral N value. The optial N ; a pair that axiizes P 2 ðl; eþ, denoted by N ðl; eþ and a ðl; eþ, will be either N and a or N and a. N N Siilar to the approah in Setion 3.1, we forulate the anufaturer s deision odel as: Max P 1 ðl; eþ ¼R Y T pn 1 a N 1 a s p C p P;s p;s r ðn½s rc r þ RT r ŠM Y N T pn a P s:t: P ½ð1 s p ÞC p Š N 1 a N½ð1 s rþc r Š M Y N T pn a KðaÞ ¼P 0 : ð24þ ð25þ Substituting for P in the objetive funtion, the anufaturer s expeted profit beoes a funtion of N and a as follows: Lea 6 P 2 ðl; eþ ¼R Y T pn 1 a N 1 a C p N½C r þ RT r ŠM Y N T pn a P 0 : (i) Using the definition of hðnðaþþ fro Lea 5, there is a unique N ða Þ, denoted by N ða Þ and not neessarily an integer, that axiizes P 2 ðl; eþ for any given value of a, the effort level that the anufaturer would like the ontrator to hoose. Then, N ða Þ satisfies the following relationship: ð26þ hðn ða ÞÞ ¼ RT p þ C p : RT r þ C r (ii) Let ðaðnþþ ¼ N 1 a ½ðÞlnðNÞ 1Š RT p þ C p ðrt r þ C r ÞT p k Y N T pn a ð1 aþ 2 K 0 ðaþ: ð27þ ð28þ Then, there is a unique a, denoted by a ðn Þ that axiizes P 2 ðl; eþ for any given value of N, the nuber of PM operations. a ðn Þ satisfies the following relationship: ða ðn ÞÞ ¼ 0: ð29þ Lea 6 desribes the properties of the optial effort level and optial nuber of PM operations for the anufaturer. To find this unique optial N and a pair that axiizes P 2 ðl; eþ, denoted by N ðl; eþ and a ðl; eþ, we an utilize an iterative algorith very siilar to the one used to find N ðl; eþ and a ðl; eþ. The details, therefore, are oitted. Sine the ðn ðl; eþ; a ðl; eþþ and ðn ðl; eþ; a ; ðl; eþþ pairs are unique, it follows that the anufaturer an axiize its profit only if N ðl; eþ ¼N ðl; eþ and also a ðl; eþ ¼a ðl; eþ. The following Lea gives the properties of s p and s r whih an be used to ahieve this goal. Lea 7. The ontrator selets the PM frequeny and the effort level pair that axiizes the anufaturer s profit only if s p ¼ RT p C p and s r ¼ RT r C r.

10 H. Taraki et al. / European Journal of Operational Researh 192 (2009) This Lea has four iportant and interesting ipliations. First, if we liit ourselves to s p and s r both being nonnegative, then the anufaturer annot ahieve its first-best solution via the traditional ost subsidization ethod. Instead, it needs to use the reverse ost subsidization ethod, where the ontrator pays the anufaturer every tie it arries out any type of aintenane ativity. This ethod ay see ipratial but its effets an be negated by setting a very high P value and by sheduling this payent to oinide with PM operations. Therefore, the ontrator ould reeive a positive ash flow during eah PM and in the end, ahieve its reservation profit of P 0. For this reason, we assue that the anufaturer would be able to install this reverse ost subsidization shee. The seond ipliation of Lea 7 is that the optial values for s p and s r are unique. Therefore, when requires a ostly effort on the ontrator s behalf, this ethod an ahieve the first-best solution for the anufaturer only with one set of paraeters. Third, when we look at the iplied ost of eah PM for the ontrator, we see that the 1st PM osts C p þ RT p, the seond PM osts C p 2 a þ RT p 2 a, the 3rd one osts C p 3 a þ RT p 3 a, et. The first ter for eah ost value represents the atual ost of arrying out PM, whih is usually inurred by the ontrator. The seond ter represents the ost of revenue loss due to downtie, usually a ost assoiated with the anufaturer. However, this reverse ost subsidization atually enables the anufaturer to pass this ost along to the ontrator. This allows the anufaturer to align its goals with the ontrator and, thus, the anufaturer an ahieve its first-best solution. Finally, the optial s p and s r values are independent of the effort level or its ost funtion. This eans that there is no need to inur any adinistrative osts to atually observe and enfore a ertain effort level. The effort needs not be even observable; the anufaturer only needs to know the general ost funtion of the effort. The following steps enable the anufaturer to ahieve its first-best solution that is, N ðl; eþ; a ðl; eþ assuing the reverse ost subsidization an be ipleented: 1. Find N ðl; eþ; a ðl; eþ via Lea Set s p ¼ RT p C p and s r ¼ RT C r r. 3. Set P ¼½RT p þ C p Š N 1 a þ N½RT r þ C r ŠM Y N T pn a where N and a stand for N ðl; eþ and a ðl; eþ, respetively. þ P 0 þ KðaÞ; ð30þ It an be shown that, if this four-step algorith is suessfully ipleented, then the anufaturer s profit is siply equal to RY P. Of ourse, when the CS ontrat is designed, P will depend on the ost of effort paraeters but will be fixed. Therefore, if the atual ost-of-effort paraeters turn out to be different than the expetations, the anufaturer s profit will not be affeted. Exaple 2. Consider the exaple fro Setion 3.1. We assue the following funtional for for the ost of effort of : KðaÞ ¼ Ca ; ð31þ where C > 0 is a onstant. This funtional for ensures that the first two derivatives of KðaÞ are indeed positive and that Kð0Þ ¼0. We also note that the ost of effort inreases slowly at first; however, as the effort, a, approahes the theoretially axiu value of 1, the ost approahes infinity and beoes prohibitively large. We assue C ¼ $100; 000. For this funtional for, we find N ðl; eþ ¼106 and a ðl; eþ ¼0:256, using Lea 6. Therefore, the anufaturer s optial PM frequeny is every days. To indue the ontrator to selet this PM frequeny, the anufaturer designs the CS ontrat with the unique set of paraeters as follows: s p ¼ 2, s r ¼ 2:857, and P ¼ $139; 527:13. Therefore, the ontrator reeives a very large fixed payent, P, but needs to pay the anufaturer every tie the ontrator oes in for either a PM or a MR. The anufaturer ould ensure that the ontrator does not renege on the ontrat by onditioning the payent of P on the arrival of the ontrator for a aintenane ativity. With this arrangeent, the anufaturer s expeted profit is $160, and the ontrator s expeted profit is still at $20, Conlusions We onsider a anufaturer that has a proess with an inreasing failure rate. To iprove the reliability of the proess, the anufaturer offers a short-ter outsouring ontrat to an external ontrator, under whih the ontrator is responsible for sheduling and perforing preventive aintenane and arrying out inial repairs when the proess fails.

11 148 H. Taraki et al. / European Journal of Operational Researh 192 (2009) The ontrator reeives a fixed payent fro the anufaturer, along with additional ost subsidization for perforing eah aintenane operation. For oparisons, we first develop the basi odel for to introdue a short-ter ontrat assuing no ours in aintenane operations. We then inorporate effets in preventive aintenane (PM) operations into the basi odel. Two types of effets are onsidered: experiene-based and investent-based. Under experiene-based, we find that the anufaturer an utilize the ost subsidization shee to axiize its profit. The results show that both parties profits are axiized when equal preventive aintenane intervals are used. There are an infinite nuber of obinations of fixed payents and ost subsidizations whih enable the ontrator to eet its expeted reservation profit We found that by designing the ontrat arefully, the anufaturer an ensure that the ontrator will still hoose the new PM shedule that axiizes the anufaturer s profit even if the rate iproves due to, for exaple, tehnologial advanes fro that used in designing the ontrat. Therefore, this robustness property akes the ost subsidization shee very attrative when we onsider effets in designing an outsouring ontrat. We also deonstrate that the ontrat with the ost subsidization shee outperfors a foring ontrat, in whih the anufaturer speifies a PM shedule, when the anufaturer ay not have aurate inforation about the ontrator s rate. Under investent-based, we find that there is a tradeoff between the ost of effort and the savings in the PM ost and tie. Both parties profits are still axiized with equal PM intervals. Further, we find that using the ost subsidization shee guarantees axiizing the anufaturer s profit with only one set of ost-ratio paraeters. These paraeters atually iply a negative ost subsidization; i.e. the ontrator, rather than the anufaturer, akes a payent when a aintenane ativity is done. This ay not see very plausible; however, we are onfident it an be taken are of by aking the fixed payent fro the anufaturer to the ontrator oinide with or be ontingent on the aintenane ativity. The ost subsidization shee has an additional benefit under investent-based : The anufaturer does not need to observe or verify the effort level of the ontrator; the ontrat guarantees that the ontrator s own optial effort level will also axiize the anufaturer s profit. This feature an be espeially useful if the effort level is unobservable or unenforeable beause of legal issues. As shown by Taraki et al., (2006), the ost subsidization shee is effetive in oordinating a servie hain. In this paper, we find that it also works very well in axiizing the anufaturer s profit. In addition, it has a very attrative feature that guarantees the anufaturer the axiu profit even when the true level is not available to the anufaturer. There are several extensions of this paper. We assue, in this paper, that the ontrator s reservation profit is alulated using the total expeted repair ost. Sine the nuber of failures is rando, the ontrator s atual profit ay deviate signifiantly fro the reservation profit. It is possible for the ontrator to require reeiving a iniu profit with a ertain level of likelihood (assurane). We use the odel without effets given in Setion 2 to illustrate this requireent. Let Z i be the nuber of failures in the ith PM interval. As desribed in Setion 3, it is known that, for given T i, Z i is a non-hoogenous Poisson proess P and follows a Poisson distribution with ean kt w i. Then, the total nuber of failures in the ontrat period, N Z i, follows a Poisson distribution with ean N kt w i. The ontrator ay require his or her profit to be larger than P 0 with a probability of d. This requireent an be expressed as: ProbfP N ½ð1 s p ÞC p Š ½ð1 s r ÞC r Š XN Z i P P 0 g P d: For illustration, we onsider the solution given in Exaple 1. We an find the total payent to the ontrator required to satisfy the above onstraint by a nuerial proess. Using P 0 ¼ $20; 000, s p ¼ 0:239, s r ¼ 0, and d ¼ 0:95, the new payent, P, is $56,601.98, whih is a 2.3% inrease to the original value. It is evident that this adjustent to the solution is not optial under the new onstraint. To obtain the true solution, whih requires a very different approah, is very hallenging ainly beause Z i follows a disrete distribution with a paraeter as a funtion of a deision variable. We an also integrate forgetting into the odel as has been done in the prodution and operations anageent literature (Teyarahakul et al., 2006). For instane, between PM operations, the ontrator ould forget soe of the experiene learned. Presuably, longer tie intervals would result in ore forgetting. Another diretion would be to analyze different payent shees and opare the with the ost subsidization shee. Appendix Proof of Lea 1. The Lagrangean funtion of P 1 ðlþ ; for any given N,is LðT i ; bþ ¼P ½ð1 s p ÞC p Š XN i a ½ð1 s r ÞC r Š XN MðT i Þþb XN i a T p þ XN! T i Y : ð32þ

12 The first-order onditions for optiality are: ol ¼ ½ð1 s r ÞC r ŠkðT i Þþb ¼ 0 for all i; ot i " # ol ob ¼ XN i a T p þ XN T i Y ¼ 0; and b P 0: These equations, obined with the 2nd derivative ondition, iply that kðt 1 Þ¼kðT 2 Þ¼¼kðT NÞ. Then, it follows that T 1 ¼ T 2 ¼¼T N sine kðt Þ is a stritly inreasing, one-to-one funtion. h Proof of Lea 2 H. Taraki et al. / European Journal of Operational Researh 192 (2009) (i) Taking the 2nd derivative of P 2 ðlþ with respet to N, we obtain o 2 p 2 ðlþ ½ð1 s r ÞC r Šk 0 Y T pna Y 2 N 1 a ¼ < 0: on 2 N 3 Thus, we have a unique solution, N ðlþ, sine the profit funtion is stritly onave. Taking the first derivative of P 2 ðlþ with respet to N and setting it equal to zero, we find that hðn ð1 spþcp ðlþþ ¼ ð1 s r ÞC r. (ii) This siply follows fro the onavity property of P 2 ðlþ. h Proof of Lea 3. (i) and (ii). The proofs for these two parts are siilar to the proofs for Lea 2: P 2 ðlþ, a onave funtion in N, has the unique solution, N ðlþ. (iii) Using the Ipliit Funtion Theore and the fat that hðn Þ is dereasing in both N and a, we have ðlþ=oa > 0. h on Proof of Lea 4. Siilar to the proof of Lea 1, by applying the Kuhn Tuker onditions to the Lagrangean funtion, we find that T 1 ¼ T 2 ¼¼T N ¼ T. h Proof of Lea 5 (i) For any given a, it an be shown that P 2 ðl; eþ is onave in N ða Þ. Therefore, the first-order onditions lead to the best response funtion of N ða Þ and N ða Þ, for the a given level of a. (ii) Siilar to part (i), it an be shown that P 2 ðl; eþ is onave in a for any given N. Solving the first-order onditions leads to a ðn Þ, the best response funtion of a. h Proof of Lea 6. Siilar to the proof of Lea 5, we an find the best response funtions, N ða Þ and a ðn Þ, that axiize P 2 ðl; eþ by applying the first-order onditions. h Proof of Lea 7. Sine N ðl; eþ ¼N ðl; eþ and a ðl; eþ ¼a ðl; eþ, using Leas 5 and 6, we have ða ðl; eþðn ðl; eþþþ ¼ ða ðl; eþðn ðl; eþþþ ¼ 0. Substituting in appropriate values, we have RT p þ C p ¼ð1 s p ÞC p and RT r þ C r ¼ð1 s r ÞC r. Solving for s p and s r, we obtain the result of Lea 7. h Referenes Adler, G.L., Nanda, R., The effets of on optial lot size deterination single produt ase. AIIE Transations 6, Argote, L., Bekan, S.L., Epple, D., The persistene and transfer of in industrial settings. Manageent Siene 36, Asgharizadeh, E., Murthy, D.N.P., Servie ontrats: A stohasti odel. Matheatial and Coputer Modelling 31, Barlow, R., Hunter, L., Optiu preventive aintenane poliies. Operations Researh 8, Ca, J.D., Evans, J.R., Woer, N.K., The unit -urve approxiation of total-ost. Coputers and Industrial Engineering, Chand, S., Lot sizes and setup frequeny with in setups and proess quality. European Journal of Operational Researh 42, Dorroh, J.R., Gulledge, T.R., Woer, N.K., Investent in knowledge: A generalization of by experiene. Manageent Siene 40, Elsayed, E.A., Reliability Engineering. Addison Wesley Longan, Reading, PA. Fine, C., Quality iproveent and in produtive systes. Manageent Siene 32, Jardine, A.K.S., Maintenane Replaeent and Reliability. John Wiley & Sons, New York. Jensen, D., Contrat worker servies: An inside view. Aviation Manageent July 1. Jovanovi, B., Nyarko, Y., Learning by doing and the hoie of tehnology. Eonoetria 64,

13 150 H. Taraki et al. / European Journal of Operational Researh 192 (2009) Karwan, K.R., Mazzola, J.B., Morey, R.C., Produt lot sizing under setup and worker. Naval Researh Logistis 35, Keahie, E.C., Fontana, R.J., Effet of on optial lot-size. Manageent Siene 13, Mazzola, J.B., MCardle, K.F., A Bayesian approah to anaging -urve unertainty. Manageent Siene 42, Murthy, D.N.P., Asgharizadeh, E., Optial deision aking in a aintenane servie operation. European Journal of Operational Researh 116, Pierskalla, W.P., Voelker, JA., A survey of aintenane odels: The ontrol and surveillane of deteriorating systes. Naval Researh Logistis 23, Plabek, E.L., Zenios, S.A., Perforane-based inentives in a dynai prinipal-agent odel. Manufaturing and Servie Operations Manageent 2, Preston, L.E., Keahie, E.C., Cost funtions and progress funtions: An integration. Aerian Eonoi Review 54, Rahaadugu, R., Tan, C.L., Poliies for lot sizing with setup. International Journal of Prodution Eonois 48, Rapping, L., Learning and World War II prodution funtions. The Review of Eonois and Statistis 47, Sherif, V.S., Sith, M.L., Optiu aintenane odels for systes subjet to failure: A review. Naval Researh Logistis 28, Spene, M.A., The urve and opetition. The Bell Journal of Eonois 12, Srinivasan, M.M., Lee, H.S., Prodution-inventory systes with preventive aintenane. IIE Transations 28, Taraki, H., Tang, K., Moskowitz, H., Plante, R., 2006a. Inentive aintenane outsouring ontrats for hannel oordination and iproveent. IIE Transations 38, Taraki, H., Tang, K., Moskowitz, H., Plante, R., 2006b. Maintenane outsouring of a ulti-proess anufaturing syste with ultiple ontrators. IIE Transations 38, Teyarahakul, S., Chand, S., Ward, J., Effet of and forgetting on bath sizes. Working Paper, Krannert Graduate Shool of Manageent, Purdue University, West Lafayette, Indiana. Tiean, J., In need of repairs. Modern Healthare 32, Valdez-Flores, C., Feldan, R.M., A survey of preventive aintenane odels for stohastially deteriorating single unit systes. Naval Researh Logistis 36, Wang, H., A survey of aintenane poliies of deteriorating systes. European Journal of Operational Researh 139, Wang, K.F., Lee, W., Learning urve analysis in total produtive aintenane. The International Journal of Manageent Siene 29, Wright, T.P., Fators affeting the ost of airplanes. Journal of Aeronautial Sienes 3, Yelle, L.E., The urve: Historial review and oprehensive survey. Deision Sienes 10, Yu, J.W., Tian, G.L., Tang, M.L., Preditive analysis for nonhoogeneous Poisson proesses with power law using Bayesian approah. Coputational Statistis and Data Analysis 51,