Condition-based maintenance under performance-based contracting

Accepted Manuscript Condition-based Maintenance under Performance-based Contracting Yisha Xiang, Zhicheng Zhu, David W. Coit, Qianmei Feng PII: DOI: Reference:

S0360-8352(17)30336-4 http://dx.doi.org/10.1016/j.cie.2017.07.035 CAIE 4842

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

28 November 2016 4 May 2017 25 July 2017

Please cite this article as: Xiang, Y., Zhu, Z., Coit, D.W., Feng, Q., Condition-based Maintenance under Performance-based Contracting, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/10.1016/j.cie. 2017.07.035

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Condition-based Maintenance under Performance-based Contracting Yisha Xiang Department of Industrial Engineering, Lamar University, Beaumont, TX 77710, USA 001-409-880-7045 [email protected] Zhicheng Zhu Department of Industrial Engineering, Lamar University, Beaumont, TX 77710, USA [email protected] David W. Coit Industrial and Systems Engineering Department Rutgers University, Piscataway NJ 08854, USA 001-848-445-2033 [email protected] Qianmei Feng Department of Industrial Engineering University of Houston, Houston, TX 77204, USA 001-713-743-2870 [email protected]

Abstract In this paper, a new cost optimization model has been developed and demonstrated to determine cost-effective maintenance plans specifically for performance-based contracting. Maintenance, repair and overhaul (MRO) has become more critical in many industries as the global economy continues to be more service-based. Traditionally, the MRO services are performed under material-based contracts. A new support contracting approach, referred to as performance-based contracts, has emerged and received much attention in recent years. In this paper, we examine the impact of the performance-based contracts on maintenance decisions for repairable systems. We consider a stochastic degradation process that has substantial unit-specific heterogeneity. Periodic inspection/repair is implemented during field operations. Rewards for maintenance service providers are directly dependent on the average availability, and the goal of the maintenance providers is to maximize profit rates. Two benchmark models are developed for

comparison purposes, namely cost-minimization and availability-maximization. Numerical examples and sensitivity analysis are provided to examine the effectiveness of the adopted profitcentric approach. Key words: condition-based maintenance, performance-based contracting, random effects, imperfect maintenance

1. Introduction The role of support and maintenance has become more important as the global economy continues to be more service-based. For example, maintenance, repair and overhaul (MRO) business represents 8-10% of the gross domestic product in the U.S. The annual operation and sustainment cost for U.S. military equipment alone is $63 billion, and the maintenance is supported by 678,000 DoD personnel along with hundreds of private contractors [1]. According to a 2003 study by Accenture (see Dennis and Kambil [2]), after sales services and parts contribute only 25% of revenues across all manufacturing companies, but are responsible for 4050% of profits. Many manufacturing firms have begun to recognize the importance of the after sales services. In particular, some commercial companies have a heavy reliance on MRO as a key component of their business (e.g., Southwest Airlines, JetBlue, Jaguar, Rolls Royce, and Pratt Whitney), since the availability of their products directly depend on the success of MRO. As products become more complex and more reliable with long life cycles, we expect the MRO services to continue its growth, generate more profits and become more important. Therefore, more rigorous planning models are required. Traditionally, the MRO services are performed under material-based contracts (MBC). Under MBC, service providers are compensated each time product maintenance is needed. Three types of MBC are commonly used in practice. One is the fixed-price contract under which the MRO service providers are paid with a fixed fee for labor and materials consumed in MRO. Another MBC contracting approach is cost plus fixed fee under which customers pay for all resources consumed with a negotiated fixed fee added. The third MBC approach is cost plus incentive under which customers are willing to provide some gain sharing. MRO service providers are often not well motivated to improve the reliability and availability of the products under MBC, and instead they usually focus on how to minimize the MRO costs to increase the profits when they are compensated with fixed fees. While some industries are more sensitive to operational costs, achieved performance outcomes are top priorities in many other industries. For example, in capital-intensive industries, such as aerospace and defense, key performance outcomes including system readiness, system reliability and mission success rate are of great importance [3, 4]. Customers of the capitalintensive goods are usually more concerned about the performance outcomes and while cost is always important, they are relatively less sensitive to the costs. In these performance-oriented 2

industries, a new support contracting approach, performance-based contracts (PBC), has emerged and received much attention in recent years. PBC is a key element to the success of the Performance-based Logistics (PBL). PBL first gained popularity in the U.S. Department of Defense (DoD). One of the earliest PBL implementation dates back to 1998 when Lockheed Martin offered a system to DoD for supporting F-117 Nighthawk, which tied its compensation to the fighter’s performance outcome [5]. During the past few years, performance-based contracting has been popularized in the after-sales contracts for servicing capital-intensive goods in both commercial and military fields (e.g., General Electric, Lockheed Martin, Rolls-Royce and Boeing). There are various PBL definitions. According to the official definition from the Defense Acquisition Guidebook (2004) (see Berkowitz, et al. [6]), “Performance-Based Logistics is the purchase of support as an integrated, affordable, performance package designed to optimize system readiness and meet performance goals for a weapon system through long-term support arrangements with clear lines of authority and responsibility. (p. 53)” PBL contracts, referred to as PBC, need defined performance goals and appropriately structured incentives. Under PBC, service providers are paid for a level of performance, and the actual quantity of material delivered may not be a metric. Contractor performance is measured using a performance metric or metrics, and incentives, penalties and adjustments are then based on this metric [7]. One of the pricing methodologies under PBC is power-by-the-hour. With this contracting strategy, service providers are paid a set price for logistics support for each hour that a system functions properly. For example, payment for an aircraft engine provided could be a fixed amount for each engine-operating hour regardless of which MRO policies are implemented during field operations. There are often additional incentives associated with achieving better performance. Under PBC, MRO service providers are often better motivated to improve reliability and availability of products. In this paper, we focus on the impact of PBC on maintenance policies and seek optimal maintenance policies for service providers under PBC. We consider a repairable system subject to gamma degradation with random effects. An imperfect condition-based maintenance (CBM) policy is implemented during field service. The service provider is compensated based on the average availability during operations. In other words, the rewards of the maintenance service are directly related to the desired performance metric, i.e., availability. The main differences 3

between the proposed approach and the previous ones are: 1) investigate the impacts of a PBC approach on maintenance planning, and provide insights on how MRO service providers should adjust their policies under PBC; 2) allow imperfect condition-based maintenance which is closer to reality; and 3) consider unit-specific heterogeneity in degradation processes. The rest of the paper is organized as follows. Section 2 provides a brief review of the related literature. Section 3 develops the imperfect CBM model under the profit-centric approach. Section 4 provides numerical examples and sensitivity analysis. Finally, concluding remarks are given in Section 5.

2. Literature Review Our model represents an enriched condition-based maintenance model under the PBC approach. We first examine the existing background on PBL, and then a brief overview of the condition-based maintenance is provided. 2.1. Background of the PBL Performance-based logistic has witnessed increasing popularity in recent years.

It was

mandated in 2001 in the Quadrennial Defense Review (QDR) [8]. It has since become the preferred strategy for weapon system logistics support in the U.S. DoD and many government sectors. It has also gained significant recognition in some private industries, such as aerospace. Unlike the traditional fixed-price or cost-plus contracts that are based on buying spare parts, repair action, or equipment, PBC focuses on buying the realized performance outcome. Berkowitz, et al. [6] conduct research aimed at developing a working definition of PBL, the drivers for its use, and the infrastructure changes needed for its successful deployment. Their studies reveal numerous reasons for the adoption of PBL, and among the top drivers is that weapons systems are expensive to maintain, difficult to upgrade with new technology, and take a long time to be implemented in the field, and this is also true for the repair and maintenance of fielded systems.

Devries [7] conducts a survey to identify the most common barriers and

enablers as the DoD services go forward with PBL implementation, and evaluates the success of PBL implementation. Their findings show that some common barriers are funding, culture, and lack of incentives, and that some key enablers are PBC, performance metrics, and total life cycle system management. Kim, et al. [3] analyze the implications of performance-based relationships. In their model the customer faces a product availability requirement for the “uptime” of the end 4

product. They compare and analyze three contracting forms, i.e., a fixed payment, a cost-sharing incentive, and a performance incentive, and they demonstrate how the allocation of performance requirements and contractual terms change under various environmental assumptions. PBL has been credited with reducing the total life cycle costs and improving product performance. Randall, et al. [9] conduct a survey to examine the effectiveness of PBL. Their results indicate that for-profit systems should expect stronger competence and greater improved performance outcomes from their PBL efforts than not-for-profit systems. Guajardo, et al. [10] empirically investigate how product reliability is impacted by the use of two different types of maintenance support contracts: time and material contracts (T&MC) and PBC, using a proprietary data set provided by a major manufacturer of aircraft engines. Their results show that product reliability is higher by 25-40% under PBC compared to T&MC. Most of the existing literature has focused on the conceptual framework and the implementation of PBL. Only a few existing models are concerned with determination of the optimal policies given a PBC. Smith [11] propose a process for planning and estimating the cost of a reliability improvement program under a PBL contract. Jin and Wang [12] propose an analytical model for planning and contracting PBL with the consideration of reliability and uncertain system usage. More recently, Jin, et al. [1] present a novel modeling approach that links the operational availability and spare parts availability under PBC. As the contracting approach shifts from the traditional fixed-price contract to this new performance-based approach, it is expected that models that facilitate the planning of after-sales support and service in this new paradigm become more relevant and applicable.

2.2. Overview of the CBM CBM has been extensively studied in the past decades. CBM policies predict the remaining useful life based on the currently observed system state, and suggests system inspection and maintenance action. Jardine [13] provides a review of common strategies for implementing effective condition-monitoring approaches such as trend analysis and expert systems. The available accurate sensor technologies that can continuously provide performance indicators at low cost make CBM even more appealing. Sensor data on system condition (health), especially if it is collected in real time, can be analyzed so that maintenance technicians can intervene just prior to system failure (see examples in Feng and Kapur [14], Jensen and Peterson [15], and Kuo 5

and Kuo [16]). Also, given the abundant historical degradation observations, many stochastic process models can be developed to model the system deterioration, and efficient CBM policies are then proposed for such systems. There are three main types of inspection schedules in CBM: continuous monitoring [17-21], periodic [22, 23], and non-periodic inspection [24, 25]. The advantage of continuous monitoring lies in the possibility of preventively maintaining the system only when necessary, thus, eliminating wasted inspection and maintenance activities. Continuous monitoring is often implemented in systems such as nuclear power plants, and offshore installations. However, continuous monitoring often incurs high inspection costs, and there are also some systems where continuous monitoring is not applicable or possible. Periodic or nonperiodic inspections with lower costs can be an effective alternative. The disadvantage of nonperiodic inspection schedules is that more documentation and rescheduling work is needed, and the risk of human errors significantly increases with additional rescheduling. Suzan and Xiang [26] provides a review of CBM literature with emphasis on mathematical modeling and optimization approaches. The degradation processes considered in the above-mentioned CBM models assume that all separate units are independent and identically distributed. However, when units are observed over time it is often apparent that they degrade at different rates, even though no differences in treatment or environment are present. This substantial heterogeneity between the degradation paths of different individuals or units is often beyond what can be accounted for by conditioning on explanatory variables. Therefore, it is necessary to allow for such unexplained differences [27]. Unit-specific random effects are commonly used to model such variability. Another factor that has been largely ignored in the existing CBM literature is the effect of the maintenance actions. The majority of the CBM models assume that maintenance is perfect and restores the system to an as-good-as-new state, since the CBM models are usually mathematically complex by themselves. In practice, many maintenance actions are imperfect, and restore the system to somewhere between as-good-as-new and as-bad-as-old. In this paper, we develop an imperfect CBM model for repairable systems under the PBC approach, and allow for unit-to-unit heterogeneity in degradation paths. To illustrate the impact of PBC, we compare the optimal policies and costs/profits under the new profit-centric approach and two traditional ones, namely cost-minimization and availability-maximization. Our main contribution lies in the adoption of the novel PBC for the maintenance service for a repairable 6

system. We also enrich existing CBM literature by incorporating unit-specific random effects and allowing CBM to be imperfect. More specifically, we consider a (p, q) rule. Each preventive maintenance action restores the system to an as-good-as-new state with probability of p, and the system remains in its pre-maintenance condition with probability of q (q=1- p). The rule itself is simple, but the modeling and the computation involved are demanding, and serve as a useful approximation for several types of imperfect maintenance.

2.3. Notation X(t)

Random wear accumulated in time interval [0,t]

Xj(t)

Individual degradaton level of the jth system at time t

F(x; x0, t)

Cumulative distribution function (cdf) of X(t), Pr{ X(t)≤x, X(0)=x0}

f(x; x0, t)

Probability density function (pdf) of X(t)

T(x)

Random time for the wear to reach level x

G(t; x0, s)

Cumulative distribution function of time-to-failure, Pr{ T(s)≤t, X(0)=x0}

g(t; x0, s)

Probability density function of T(s)

R(t; x0, s)

Survival function

s

Failure threshold

δ

Inspection interval

ξ

Maintenance degradation threshold

cPM

Preventive repair cost

cCM

Corrective repair cost

TPM

Time required for preventive maintenance

TCM

Time required for corrective maintenance

Tinsp

Inspection time

μPM

Mean time required for preventive maintenance

μCM

Mean time required for corrective maintenance

μinsp

Mean inspection time

Tup

Uptime during a renewal cycle

Tdown

Downtime during a renewal cycle

π

Expected profit rate

7

ρ

Expected reward rate

Α

Expected average availability

C Expected cost rate

3. Description of the system 3.1. System degradation with random effects There have been abundant discussions of different degradation models and their uses in the literature. In practice, a degradation increment is usually strictly positive (e.g., wear), indicating that a degradation process is irreversible. Assuming a geometric Brownian motion for degradation ensures a degradation process is always positive. For some systems, a degradation process is not only always positive but also strictly increasing. The geometric Brownian motion is always positive but not strictly increasing. A gamma process is more appropriate to model such a degradation process. In many degradation problems there is substantial heterogeneity between the degradation paths of different individuals or units, beyond what can be accounted for by conditioning on explanatory variables, and it is common to introduce unit-specific random effects to model such variability [27]. In this paper, we consider a system that deteriorates over time, and the state of the system can be described by a gamma process with random effects. The random effect model considered in this study is different from the heterogeneous models in Ye, et al. [28] and Xiang, et al. [29]. The unit-specific random effects model the substantial heterogeneity within the same population, whereas the models in Ye, et al. (2012) and Xiang, et al. [29] consider population that consist of several distinct homogeneous sub-populations. Consider a gamma process {X(t), t ≥ 0} with random effect z that controls heterogeneity across units. Given z, the process {X(t)} has independent increments: for 0  t1  t2 , X(t2) - X(t1) is independent of X(t1) and has a gamma distribution Ga(α(t2- t1), z) with α0=0. The conditional density function of X(t2) - X(t1) , given z, is f  x z  , where the gamma density function can be expressed in the form

f  x z      t2  t1   z  t2 t1  x t2 t1 1e x z , for x  0 . 1

8

(1)

（ ,  1） as the distribution of z-1, so the It is mathematically convenient to assume Ga degradation process has a closed-form for pdf and cdf. The marginal density of X(t) is as follows

f  x   B( t , ) 1

  x t 1 , ( x   ) t 

(2)

where B(αt, θ)=Γ(αt) Γ(θ)/ Γ(αt+ θ). We note that θX(t)/γαt has an F distribution, F2 t ,2 . Therefore, the cdf of the degradation X(t) is

F  x; x0 , t  

2 t , 2

 x   .   t 

(3)

In this study, we let γ=θ so that z-1 has a mean 1 and variance θ-1. Therefore, a large value of θ indicates a small variance and less heterogeneity due to the random effects. In Figure 1 (blue lines represent degradation paths without random effects, and red lines represent degradation paths with random effects), we illustrate the effect of random effects with different θ values.

=15

=25 200

180

180

180

160

160

160

140

140

140

120

120

120

100

X(t)

200

X(t)

X(t)

=10 200

100

100

80

80

80

60

60

60

40

40

40

20

20

20

0

0

0

20

40

60 t

80

100

120

0

20

40

60 t

80

100

120

0

0

20

40

60 t

80

100

120

Figure 1 Degradation Paths with Random Effects Theorem 1 A system exhibits an increasing failure rate if its underlying degradation process is governed by a gamma process with gamma random effects (Proof is in Appendix). The increasing failure rate implies that the preventive maintenance is potentially worth implementing to improve the reliability/availability of the system.

9

3.2. Imperfect condition-based maintenance It is assumed that the degradation of the system cannot be directly and instantaneously observed during field use, and thus, the system is monitored through periodic inspections which reveal the exact state of the system without error. In practice, the system can be declared “failed” as soon as an important defect or deterioration is present (even though the system is still operating, but in a degraded or failed state) prior to the occurrence of a complete and abrupt failure, i.e., a pending failure is considered an actual failure of the system [30]. In this study, we consider a similar non-obvious failure. When the cumulative deterioration of the system exceeds a pre-specified failure threshold s, the system is considered as failed, but a “failed” state can only be detected through inspections, so the system continues to operate. At each inspection, if the current deterioration level is below the maintenance threshold, the system is left unchanged. If the deterioration level is between the maintenance threshold and failure threshold, the system is preventively repaired. Therefore, in the presence of a nonobvious failure, the system is not known to be failed until it is revealed by inspections. The preventive repair is assumed to be imperfect. More specifically, it is assumed that the system is returned to an as-good-as-new state (perfect PM) with probability p and remains in the same state just prior to the PM action with probability q = 1 - p. This method of modeling imperfect maintenance is referred to as (p, q) rule, that is, after maintenance (preventive) a system becomes as-good-as-new with probability p and as-bad-as-old with probability (1–p) [31-33]. If p = 1 then PM is perfect, so perfect PM is a special case of imperfect PM. Corrective repair is assumed to be perfect, and it always restores the system to the as-good-as-new state at extra time and cost. The times required for inspection, PM, and CM actions are random variables. Availability (sometimes called operation readiness) is the most commonly used key performance criterion in maintenance support services. In this study, we use availability to measure the realized performance outcomes, and assume that the reward is directly linked to the availability. We are interested in the average availability defined as follows: expected total up time per cycle cycle time expected total up time per cycle = . expected total uptime per cycle+expected downtime per cycle

Availability =

10

4. Maintenance Optimization Model PBC has recently become more popular and received increasing attention in many capitalintensive industries. The profit-centric approach commonly observed with PBC allows us to integrate the performance, cost and reward into one unified objective. More specifically, each policy in our model results in a different cost and average availability, and the reward directly depends on the average availability.

Therefore, the ultimate profit which is the difference

between the reward and the cost takes both the cost and the performance outcome into consideration. A policy that minimizes the cost but has a poor performance might not provide the optimal profit. On the other hand, a policy that gives the best performance outcome but requires a high maintenance cost might not yield the best profit either. We are seeking an optimal policy that best balances the trade-off between the cost and the performance, to maximize the profit from the service provider perspective. There are various forms of reward functions. A reward function can be linear, exponential or step-wise. Several typical examples of reward functions are provided in Figure 2.

Figure 2 Rewards as functions of availability

11

We denote the expected profit rate with π(·), and the reward rate with ρ(·). Let A denote the expected average availability and C(·) denote the expected cost rate. The optimization model with the objective of maximizing profit is given by Model P1: max   ,      A  ,     C  ,  

 *,  *  arg max   ,     0,   0 For comparison purposes, we propose two benchmark models with the objectives of minimizing cost and maximizing availability, respectively. Model P2:

min C  ,  

 *,  *  arg min C  ,     0,   0 and Model P3:

max

A  ,  

 *,  *  arg min  A  ,     0,   0

4.1. Availability The average availability and the associated costs can now be derived from the proposed CBM policy. We define the “renewal cycle” as the period of time between two consecutive perfect maintenance actions (e.g., perfect preventive repair, or corrective repair). A cycle can be the interval between two corrective repairs, two perfect preventive repairs, or a perfect preventive repair and a corrective repair. Figures 3 - 5 illustrate the four renewal scenarios. If a preventive repair does not return the system to an as-good-as new state, it is not considered as a renewal point. Downtime may include the time of inspection, and preventive/corrective repair. 12

State X(t)

tPM

tPM

delay of failure detection

s as bad as old after PM ζ

as good as new after PM δ

δ tinsp

tCM

δ tinsp tinsp renewal cycle

δ tinsp

Time t

Figure 3 Renewal cycle between a CM and a perfect PM

State X(t)

delay of failure detection

tPM

delay of failure detection

s as bad as old after PM

ζ

δ tCM

δ tinsp

δ

δ tinsp

tinsp

tinsp tCM

renewal cycle

Figure 4 Renewal cycle between two CMs 13

Time t

tPM

tPM

State X(t)

tPM

tPM

s

ζ as good as new after PM

as bad as old after PM

δ

δ tinsp

δ tinsp

tinsp

as good as new after PM

δ

renewal cycle

δ tinsp

Time tinsp

t

Figure 5 Renewal cycle between two perfect PMs 4.1.1. Derivation of uptime We first derive the expected total uptime, 

E Tup      Pr  device preventively repaired perfectly at the  th inspection   1



    Pr  device correctively repaired at the  th inspections   1

We now derive the probability of preventive repair perfectly performed at the λth inspection, and denote this probability by pλ (λ=1, 2, 3 …). Upon the first inspection, if the cumulative deterioration level is greater than the maintenance threshold ξ but lower than the failure threshold s, the preventive repair is performed perfectly with probability p. The probability of having a perfect PM at the first inspection is

p1  Pr{  X ( )  s} p . Perfect PM at the second inspection (λ = 2) includes two scenarios. One is the case where the effect of PM at the first inspection is as-bad-as-old, and the system did not fail during the second inspection interval, and it is preventively replaced perfectly at the second inspection. Another possibility is that PM is needed and performed perfectly at the second inspection but not needed 14

at the first inspection. Thus, we have the probability of perfect PM at the second inspection as follows

p2  Pr{  X  2   s &   X    s} 1  p  p  Pr{  X  2   s & X     } p . Perfect PM at the third inspection (λ = 3) has three possibilities. One possibility is that the system needs PM at both first and second inspections but neither PM is perfect, and the system still needs preventive repair at the third inspection (shown in Figure 6(a)). The second one is that PM is not needed until the second inspection, but the PM at the second inspection is not done perfectly, and requires PM at the third inspection (shown in Figure 6(b)). The last one is that PM is not needed until the third inspection (shown in Figure 6(c)). The probability of perfect PM at the third inspection is given by adding those three probabilities together

p3  Pr{  X (3 )  s &   X ( )  s} (1  p) 2 p  Pr{  X (3 )  s &   X (2 )  s & X ( )   } (1  p) p  Pr{  X (3 )  s & X (2 )   }  p State X(t)

tPM

tPM

tPM s

tPM

State X(t)

tPM

tPM

State X(t)

s s

ζ

ζ

ζ as good as new after PM

as good as new after PM

δ

δ

δ

tinsp

tinsp

(a)

tinsp

Time t

δ

δ tinsp

as good as new after PM

δ tinsp

tinsp

Time t

(b)

δ

δ tinsp

δ tinsp

tinsp t

Time

(c)

Figure 6 Perfect PM at the 3rd Inspection

Now we derive a general probability of a PM performed perfectly at the λth inspection as follows:

15

Pr{ X    s}  p,  Pr{  X  2   s &   X    s}  1  p  p  Pr{  X  2   s & X     }  p,  p    1  n  Pr{  X     s &   X  n   s & X   n  1     }  1  p  p  n 1   Pr{ X     s & X     1     }  p, 

for   1 for   2

for  >2

where

Pr{  X     s &   X  n   s & X   n  1     }   f  u;0,  n  1    f  ; u,  F  s  ;0,    n    d du 

s

0



and

Pr{  X     s & X     1     }   f  u;0,    1    F  s  u;0,    F   u;0,    du 

0

Next, we derive the probability of corrective repair at the λth inspection, and denote this probability by qλ (λ =1, 2, 3 …). The probability of performing a CM at the first inspection is

q1  Pr{ X ( )  s} . The necessity of CM at the second inspection (λ = 2) also includes two possibilities. One is the case where the effect of PM at the first inspection is as-bad-as-old, and the cumulative deterioration level at the second inspection is above the failure threshold. The other case is where PM is not needed at the first inspection, but the system failed between the first and the second inspections. The probability that CM is needed at the second inspection is thus given by

q2  Pr{X (2 )  s &   X ( )  s} (1  p)  Pr{ X (2 )  s & X ( )  } . There are three scenarios where CM is needed at the third inspection (λ = 3). One is that the system is not perfectly repaired preventively at either the first or the second inspection, and deteriorates to a “failed” state between the second and the third inspections (shown in Figure 7(a)). The second one is that the first time the system requires a PM is at the second inspection, but the PM at the second inspection is not done perfectly, and the system fails between the second and the third inspections (shown in Figure 7(b)). The last one is that PM is not needed at 16

either the first or the second inspection, and the system fails between the second and the third inspections (shown in Figure 7(c)). Therefore, we have the probability of CM being performed at the third inspection as follows:

q3   Pr{ X (3 )  s &   X ( )  s}  Pr{ X (2 )  s &   X ( )  s}  (1  p) 2  Pr{ X (3 )  s &   X (2 )  s & X ( )   }  (1  p)  Pr{ X (3 )  s & X (2 )   }

State X(t)

δ

tPM

tPM

δ

δ tinsp

tPM

State X(t)

tinsp tCM

tinsp

State X(t)

s

s

ζ

ζ

Time t

δ

δ tinsp

δ tinsp

Time t tinsptCM

s

ζ

δ

δ

δ tinsp

tinsp tCM

Time t

tinsp

(a)

(b)

(c)

Figure 7 CM at the 3rd Inspection The probability of CM at the λth inspection can be generalized as follows:

 =1 Pr{ X    s},   =2 Pr{ X  2   s &   X    s}(1  p)  Pr{ X  2   s & X     },  2  [Pr{ X     s &   X  n   s & X   n  1     }  q   n 1   Pr{ X     1    s &   X  n   s & X   n  1     }](1  p)   n    Pr{ X     s &   X     1    s & X     2      }(1  p)    Pr{ X     s & X     1     },  >2 where

Pr{ X ( )  s}  



s

f ( x;0,  )dx 17

Pr{ X     s &   X  n   s & X   n  1     }   f  u;0,  n  1   f ( ; u,  ) 1  F  s  ;0, (  n)   d du 

s

0



Pr{ X     s &   X     1    s & X     2      }   f  u;0, (  2)   f  ; u,   1  F  s  ;0,    du. 

s

0



4.1.2. Derivation of downtime In this section, we derive the expected downtime. The expected total downtime is as follows:

E Tdown   E  inspection time  +E  preventive repair time  +E  corrective repair time  Let E1,λ denote the expected downtime when PM is performed perfectly at the λth inspection, and E2,λ denote the expected downtime when CM is performed at the λth inspection. The expected downtime is as follows: 2



E Tdown    Ei , Tdown  . i 1  1

The expected downtime when PM is performed perfectly at the λth inspection is given by   uinsp  uPM   Pr{ X ( )  s}  p, for   1   2uinsp  2uPM   Pr{  X  2   s &   X    s}  1  p  p   E1, Tdown      2uinsp  uPM   Pr{  X  2   s & X     } p, for   2   1   u     n  1 u   Pr{  X     s &   X  n   s & X   n  1     } insp PM  n 1   n   (1  p) p   uinsp  uPM   Pr{ X ( )  s & X  (  1)    }  p, for  >2

18

The expected downtime when CM is performed at the λth inspection is given by

 uinsp  uCM   Pr{ X ( )  s},  =1   2uinsp  uCM  uPM   Pr{ X (2 )  s &   X ( )  s}  (1  p)     2uinsp  uCM   Pr{ X (2 )  s & X ( )   } ,  =2   2   uinsp  uCM     n  uPM   E2, Tdown    n 1  [Pr{ X     s &   X  n   s & X   n  1     }  Pr{ X     1    s   &   X  n   s & X   n  1     }](1  p )   n   u  u  u  insp CM PM    Pr{ X     s &   X     1    s & X     2      }(1  p)     u  u   Pr{ X ( )  s & X  (  1)    },  >2 insp CM  4.2.Cost rate The derivation of the maintenance cost during the field operation is similar to that of the downtime. For example, the inspection time is substituted with the inspection cost, and the PM/CM downtime is substituted with the corresponding PM/CM costs. The expected maintenance cost when PM is performed perfectly at the λth inspection is as follows:   cinsp  cPM   Pr{ X ( )  s}  p, for   1   2cinsp  2cPM   Pr{  X  2   s &   X    s}  1  p  p   E1,  Cost      2cinsp  cPM   Pr{  X  2   s & X     }  p, for   2   1    c     n  1 c   Pr{  X     s &   X  n   s & X   n  1     } insp PM  n 1   n   (1  p) p    cinsp  cPM   Pr{ X ( )  s & X  (  1)    }  p, for  >2 Similarly, we have the expected maintenance cost when a renewal is caused by a CM performed

at the λth inspection as follows

19

 cinsp  cCM   Pr{ X ( )  s},  =1   2cinsp  cCM  cPM   Pr{ X (2 )  s &   X ( )  s}  (1  p)     2cinsp  cCM   Pr{ X (2 )  s & X ( )   } ,  =2   2    cinsp  cCM     n  cPM  E2,  Cost    n 1   [Pr{ X     s &   X  n   s & X   n  1     }    Pr{ X     1    s &   X  n   s & X   n  1     }](1  p )   n      c  c  c   Pr{ X     s &   X     1    s & X     2      } insp CM PM    (1  p)    c  c   Pr{ X ( )  s & X  (  1)    },  >2 insp CM 

Therefore, we have the expected cost rate as follows: 2

C



E   Cost    i 1

1

i,

E Tdown   E Tup 

Note that if cCM / cinsp = uCM / uinsp and cPM / cinsp = uPM / uinsp, the policy that minimizes the cost rate or maximizes the average availability is also the optimal solution that maximizes the profit

 

rate. Let A  ,    1  A  ,    E Tdown  E Tup  E Tdown  . Then, maximization of the availability A is equivalent to the minimization of A . Let ω be the ratio of A to the cost rate given

the

same 2



  A  ,   C  ,     Ei , Tdown  i 1  1

maintenance 2

policy,

defined

as



E   Cost  . If cCM / cinsp = uCM / uinsp and cPM /   i 1

1

i,

cinsp = uPM / uinsp, the ratio is a constant, which implies that Models P2 and P3 are equivalent and the problem of maximizing the profit rate has the same optimal solution as that of minimizing the cost rate or maximizing the average availability. We expect the optimal solutions to be different when these constant parameters are not linearly related. In the next section, we investigate the impact of the profit-centric approach and other model parameters. 5. Numerical Example and Sensitivity Analysis

20

In this section, we investigate how the profit-centric approach affects the optimal policies. We compare cost rate, availability and profit rate between the profit-centric approach and the two benchmark ones. Consider a population where the degradation process can be represented by gamma processes with random effects. Suppose s=50, x0=0, α=1, and the random effect variable z controls heterogeneity across units. z-1 has a gamma distribution Ga(5, 1/5). We fix the inspection time and inspection cost (μinspt = 0.2 and cinspt = 5), and different levels of preventive/corrective repair times (cost) are selected for the purpose of sensitivity analysis. We assume that cPM< cCM, and μPM< μCM, otherwise it would not be necessary to perform preventive maintenance. We assume that the relationship between rewards and availability is step-wise linear, and the reward function is given by

if Availability  0.3 0, 5    Availability - 0.3 , otherwise

 

If the average availability is below 30%, the reward is zero and the service provider is penalized for poor performance. Three levels of rewards are considered, low (κ=2), medium (κ=30), and high (κ=50). We are interested in exploring how optimal policies change under different reward levels. Rosenbrock’s method is used to find the optimal inspection intervals and maintenance threshold [34]. The method of Rosenbrock does not employ line searches but rather takes discrete steps along the search directions. At each iteration, the new directions established by the Rosenbrock procedure are linearly independent and orthogonal. The optimization model and algorithm are implemented in Matlab. Results of the numerical examples are summarized in Tables 1 - 3. Table 1. Min Cost vs. Max Availability vs. Max Profit (κ =2) Min Cost

Max Availability

Max Profit

cPM

cCM

μPM

μCM

δ

ξ

C

A

P

δ

ξ

C

A

P

δ

ξ

C

A

P

50

1000

2

24

3.01

32.38

3.73

0.85

2.36

3.74

36.00

3.94

0.87

2.19

3.01

32.38

3.73

0.85

2.36

2

12

3.23

32.83

3.77

0.86

2.33

6.68

36.97

5.39

0.88

0.78

3.00

32.35

3.77

0.85

2.34

2

6

3.00

32.35

3.77

0.86

2.34

11.12

39.27

8.61

0.91

2.39

11.12

39.27

8.61

0.91

2.39

2

24

3.98

37.24

3.24

0.85

2.86

3.74

36.00

3.94

0.87

2.86

3.74

36.00

3.28

0.87

2.86

2

12

4.28

38.14

3.35

0.87

2.79

6.68

36.97

3.64

0.88

2.53

4.50

37.73

3.36

0.88

2.81

2

6

3.89

35.32

3.35

0.89

2.83

11.12

39.27

4.84

0.91

1.38

3.89

35.32

3.35

0.89

2.83

2

24

9.08

39.37

2.55

0.82

3.49

3.74

36.00

3.28

0.87

3.20

6.83

39.14

2.55

0.84

3.53

50

50

500

250

21

2

12

7.02

39.53

2.67

0.87

3.48

6.68

36.97

2.76

0.88

3.41

7.45

38.94

2.68

0.88

3.49

2

6

7.46

38.91

2.76

0.90

3.43

11.12

39.27

2.96

0.91

3.26

6.73

38.56

2.76

0.91

3.46

* C denotes the cost rate, A denotes the availability, and P denotes the profit rate. Table 2. Min Cost vs. Max Availability vs. Max Profit (κ =30) Min Cost

Max Availability

Max Profit

cPM

cCM

μPM

μCM

δ

ξ

C

A

P

δ

ξ

C

A

P

δ

ξ

C

A

P

50

1000

2

24

3.01

32.38

3.73

0.85

18.23

3.74

36.00

3.94

0.87

18.05

3.31

34.46

3.81

0.87

18.40

2

12

3.23

32.83

3.77

0.86

17.90

6.68

36.97

5.39

0.88

17.11

3.35

35.05

3.89

0.88

18.40

2

6

3.23

32.83

3.79

0.86

17.99

11.12

39.27

8.61

0.91

14.74

3.74

36.00

4.03

0.89

18.58

2

24

3.98

37.24

3.24

0.85

18.31

3.74

36.00

3.94

0.87

18.71

3.98

36.07

3.26

0.87

18.71

2

12

4.28

38.14

3.35

0.87

18.75

6.68

36.97

3.64

0.88

18.86

4.48

37.61

3.35

0.88

19.18

2

6

3.89

35.32

3.35

0.89

19.25

11.12

39.27

4.84

0.91

18.51

5.36

38.93

3.57

0.90

19.51

2

24

9.08

39.37

2.55

0.82

18.06

3.74

36.00

3.94

0.87

19.06

4.17

37.44

2.80

0.86

19.13

2

12

7.02

39.53

2.67

0.87

19.56

6.68

36.97

2.76

0.88

19.74

5.63

40.82

2.70

0.88

19.83

2

6

7.46

38.91

2.76

0.90

20.20

11.12

39.27

2.96

0.91

20.39

9.07

39.40

2.84

0.91

20.50

50

50

500

250

Table 3. Min Cost vs. Max Availability vs. Max Profit (κ =50) Min Cost

Max Availability

Max Profit

cPM

cCM

μPM

μCM

δ

ξ

C

A

P

δ

ξ

C

A

P

δ

ξ

C

A

P

50

1000

2

24

3.01

32.38

3.73

0.85

29.32

3.74

36.00

3.94

0.87

29.38

3.43

35.28

3.83

0.87

29.49

2

12

3.23

32.83

3.77

0.86

29.00

6.68

36.97

5.39

0.88

28.78

3.74

36.00

4.00

0.88

30.01

2

6

3.23

32.83

3.79

0.86

29.17

11.12

39.27

8.61

0.91

26.97

4.63

33.32

4.25

0.88

29.97

2

24

3.98

37.24

3.24

0.85

29.34

3.74

36.00

3.94

0.87

30.04

3.74

36.00

3.28

0.87

30.04

2

12

4.28

38.14

3.35

0.87

30.15

6.68

36.97

3.64

0.88

30.52

5.30

36.73

3.44

0.88

30.75

2

6

3.89

35.32

3.35

0.89

30.99

11.12

39.27

4.84

0.91

30.74

5.66

38.85

3.62

0.90

31.56

2

24

9.08

39.37

2.55

0.82

28.46

3.74

36.00

3.94

0.87

30.38

4.21

37.72

2.78

0.86

30.40

2

12

7.02

39.53

2.67

0.87

31.04

6.68

36.97

2.76

0.88

31.41

5.66

38.85

2.71

0.89

31.58

2

6

7.46

38.91

2.76

0.90

32.16

11.12

39.27

2.96

0.91

32.62

9.07

39.40

2.84

0.91

32.73

50

50

500

250

From Tables 1-3, we can see that the profit rates from the profit-centric approach are better than those from the two benchmark approaches even when κ is small (κ =2), which illustrates that simply minimizing cost rates or maximizing availabilities would not lead to best profit rates. Also note that the improvement in profit rates are per time unit, and the total gains period would become larger over time. We also observe that when incentives for better performance outcomes 22

are low, maintenance policies under cost-minimization are close to policies under the profitcentric approach. As this incentive increases, policies under the availability-maximization approach are closer to the ones under the profit-centric approach. Tables 2 and 3 both indicate that the performance outcome, i.e., availability, has a dominant role in determining the profit rate when there is more incentive for improvement in the average availability. In the majority of examples examined in Tables 2 and 3, the results under the profitcentric approach are close to those under the max-availability approach, if not exactly the same. Since cost parameters are in general much harder to obtain, and the repair times are often more available and accurate, we can implement the optimal policies from the max-availability model and still get satisfactory profits when the reward is performance-based and the incentive to improve the performance is high. Based on the above observations, we can conclude that the traditional cost-minimization approach in MRO services might not lead to optimal profits in many cases when reward is determined by the performance, and that service providers are better motivated to improve the service levels under the profit-centric approach. Another important finding is that policies that maximize availability provide reasonably good profits in most numerical examples examined. If cost parameters are not available, the optimal policies under the max-availability approach can be used as an appropriate substitute to maximize the profit.

6. Conclusion In this research, we examine the impact of PBC or profit-centric contracting on the maintenance decisions for repairable systems. We consider a more general stochastic degradation process

which

models

the

unit-specific

heterogeneity in

the

population.

Periodic

inspection/repair is implemented during the field operations, and preventive repair is imperfect. Rewards for the maintenance providers are directly dependent on the average availability, and the goal of the maintenance providers is to maximize the profit rates. Two benchmark models are developed for comparison purpose, cost-minimization and availability-maximization. Numerical examples and sensitivity analysis are provided to examine the effectiveness of the adopted profitcentric approach. Our results show that MRO service providers are better motived by the incentive provided under the profit-centric approach. We also show that the policies that minimize cost (or maximize profit rates) are not alike unless the conditions in Theorem 1 are met. 23

If cost data is not easy to obtain, optimal policies from the max-availability approach can provide reasonably good solutions for PBC. The present model assumes that the system is repairable and maintained through repair actions with no spare parts involved. If spare parts do not incur high procurement and holding cost, it might be economic to further improve the average availability with spare parts, in addition to maintenance actions. Therefore, a useful extension is to consider spare parts inventory and investigate how the profit-centric approach affects the maintenance and inventory decisions jointly. The PBC considered in this research is a straightforward profit-centric approach. Other more complicated performance-based contracts are worth consideration, such as the fixed price with performance adjustment or renewing contracts for extra years. Regarding maintenance policies, non-periodic inspections are worth consideration. Regarding the forms of reward functions, it is worthwhile to examine the impacts of different forms of reward functions (e.g., exponential) on maintenance policies. Also other types of imperfect maintenance can also be

considered. Moreover, the model developed in this paper mainly provide decision support for MRO service providers under performance-based contracting approach. To obtain more collective benefits from customers and MRO service providers, it will be appealing to develop models that reflects interests of both customers and service providers in future.

Reference [1]

[2] [3] [4]

[5]

T. Jin, Y. Xiang, and R. Cassady, "Understanding operational availability in performance-based logistics and maintenance services," in Reliability and Maintainability Symposium (RAMS), 2013 Proceedings-Annual, 2013, pp. 1-6. M. J. Dennis and A. Kambil, "SERVICE MANAGEMENT: BUILDING PROFITS AFTER THE SALE," SUPPLY CHAIN MANAGEMENT REVIEW, V. 7, NO. 3 (JAN./FEB. 2003), P. 42-48: ILL, 2003. S.-H. Kim, M. A. Cohen, and S. Netessine, "Performance contracting in after-sales service supply chains," Management Science, vol. 53, pp. 1843-1858, 2007. D. Nowicki, U. Kumar, H. Steudel, and D. Verma, "Spares provisioning under performance-based logistics contract: profit-centric approach," Journal of the Operational Research Society, vol. 59, pp. 342-352, 2006. H. Mirzahosseinian and R. Piplani, "A study of repairable parts inventory system operating under performance-based contract," European Journal of Operational Research, vol. 214, pp. 256-261, 10/16/ 2011.

24

[6] [7] [8] [9]

[10]

[11] [12] [13] [14]

[15] [16] [17]

[18] [19] [20]

[21]

[22]

[23]

[24]

[25]

D. Berkowitz, J. N. Gupta, J. T. Simpson, and J. B. McWilliams, "Defining and implementing performance-based logistics in government," DTIC Document2005. H. J. Devries, "Performance-based logistics-barriers and enablers to effective implementation," DTIC Document2005. D. Rumsfeld, "Report of the Quadrennial Defense Review," Washington DC, Department of Defense, 2001. W. S. Randall, D. R. Nowicki, and T. G. Hawkins, "Explaining the effectiveness of performancebased logistics: a quantitative examination," International Journal of Logistics Management, The, vol. 22, pp. 324-348, 2011. J. A. Guajardo, M. A. Cohen, S.-H. Kim, and S. Netessine, "Impact of performance-based contracting on product reliability: an empirical analysis," Management Science, vol. 58, pp. 961979, 2012. T. C. Smith, "Reliability growth planning under performance based logistics," in Reliability and Maintainability, 2004 Annual Symposium-RAMS, 2004, pp. 418-423. T. Jin and P. Wang, "Planning performance based contracts considering reliability and uncertain system usage," Journal of the Operational Research Society, vol. 63, pp. 1467-1478, 2012. A. K. S. Jardine, "Optimizing condition based maintenance decisions," in Reliability and Maintainability Symposium, 2002. Proceedings. Annual, 2002, pp. 90-97. Q. Feng and K. Kapur, "Selection of optimal precision levels and specifications considering measurement error," The International Journal of Advanced Manufacturing Technology, vol. 40, pp. 960-970, 2009. F. Jensen and N. E. Peterson, Burn-in: John Wiley & Sons Ltd., 1982. W. Kuo and Y. Kuo, "Facing the headaches of early failures: A state-of-the-art review of burn-in decisions," Proceedings of the IEEE, vol. 71, pp. 1257-1266, 1983. J. Barata, C. G. Soares, M. Marseguerra, and E. Zio, "Simulation modelling of repairable multicomponent deteriorating systems for ‘on condition’maintenance optimisation," Reliability Engineering & System Safety, vol. 76, pp. 255-264, 2002. F. De Carlo and M. A. Arleo, "Maintenance cost optimization in condition based maintenance: a case study for critical facilities," Int. J. Eng. Technol, vol. 5, pp. 4296-4302, 2013. H. Liao, E. A. Elsayed, and L.-Y. Chan, "Maintenance of continuously monitored degrading systems," European Journal of Operational Research, vol. 175, pp. 821-835, 2006. X. Liu, J. Li, K. N. Al-Khalifa, A. S. Hamouda, D. W. Coit, and E. A. Elsayed, "Condition-based maintenance for continuously monitored degrading systems with multiple failure modes," IIE Transactions, vol. 45, pp. 422-435, 2013. X. Zhou, L. Xi, and J. Lee, "Reliability-centered predictive maintenance scheduling for a continuously monitored system subject to degradation," Reliability Engineering & System Safety, vol. 92, pp. 530-534, 4// 2007. K. T. Huynh, A. Barros, C. Berenguer, and I. T. Castro, "A periodic inspection and replacement policy for systems subject to competing failure modes due to degradation and traumatic events," Reliability Engineering & System Safety, vol. 96, pp. 497-508, 2011. R. J. Ferreira, A. T. de Almeida, and C. A. Cavalcante, "A multi-criteria decision model to determine inspection intervals of condition monitoring based on delay time analysis," Reliability Engineering & System Safety, vol. 94, pp. 905-912, 2009. X. Zhao, M. Fouladirad, C. Bérenguer, and L. Bordes, "Condition-based inspection/replacement policies for non-monotone deteriorating systems with environmental covariates," Reliability Engineering & System Safety, vol. 95, pp. 921-934, 2010. R. Jiang, "Optimization of alarm threshold and sequential inspection scheme," Reliability Engineering & System Safety, vol. 95, pp. 208-215, 3// 2010. 25

[26]

S. Alaswad and Y. Xiang, "A review on condition-based maintenance optimization models for stochastically deteriorating system," Reliability Engineering & System Safety, vol. 157, pp. 54-63, 2017. J. F. Lawless and M. J. Crowder, "Covariates and Random Effects in a Gamma Process Model with Application to Degradation and Failure," Lifetime Data Analysis, vol. 10, pp. 213-227, 2004. Z.-S. Ye, Y. Shen, and M. Xie, "Degradation-based burn-in with preventive maintenance," European Journal of Operational Research, vol. 221, pp. 360-367, 2012. Y. Xiang, D. W. Coit, and Q. Feng, "Accelerated Burn-in and Condition-based Maintenance for nsubpopulations subject to Stochastic Degradation," IIE Transactions, 2014. M. Fouladirad, A. Grall, and L. Dieulle, "On the use of on-line detection for maintenance of gradually deteriorating systems," Reliability Engineering & System Safety, vol. 93, pp. 1814-1820, 2008. H. Pham and H. Wang, "Imperfect maintenance," European Journal of Operational Research, vol. 94, pp. 425-438, 1996. T. Nakagawa, "Imperfect preventive-maintenance," Reliability, IEEE Transactions on, vol. 28, pp. 402-402, 1979. T. Nakagawa, "Optimum policies when preventive maintenance is imperfect," Reliability, IEEE Transactions on, vol. 28, pp. 331-332, 1979. M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear programming: theory and algorithms: Wiley-interscience, 2006.

[27] [28] [29] [30]

[31] [32] [33] [34]

Appendix Proof of Theorem 1

Proof. Consider a gamma process Ga(αt, z) and z is the random effect that controls heterogeneity across units. Random effect z-1 has a gamma distribution Ga(θ, γ-1). From Equation (3), the cdf of degradation Xt is

F  x; x0 , t  

2 t , 2

 x      t 

The cdf of an F-distribution with parameters d1 and d2 is given by

Fd1 ,d2 ( x)  I

where I

d1 x d1 x  d 2

I x ( a, b) 

(

d1 x d1 x  d 2

(

d1 d 2 , ) 2 2

d1 d 2 , ) is the regularized incomplete beta function, defined as 2 2

x B( x; a, b) , and B( x; a, b)   t a 1 (1  t )b1 dt . 0 B ( a, b)

26

To show that the failure rate is increasing in time, we first compute the failure rate λ(t),

 (t ) 

f (t ) 1 f (t )t 1 Pr( X (t )  s, X (t  t )  s)  lim  lim Pr( X (t )  s) t 0 t F (t ) t 0 F (t ) t

Let X  X (t  t )  X (t ) be the increment of the process during the interval [t , t  t ] and f(y; 0, Δt) be its pdf. Then,

 (t )  lim

t 0

1 Pr( s  X  X (t )  s ) t Pr( X (t )  s )

 1 0  Pr( X (t )  s )  Pr( X (t )  s  y )  f ( y;0, t )dy   f ( y;0, t ) dy s t 0 t Pr( X (t )  s ) s Pr( X (t )  s  y ) 1 = lim (1   f ( y;0, t ) dy) 0 t 0 t Pr( X (t )  s ) s

= lim

Now we need show that

Pr( X (t )  s  y ) is a decreasing function of t. Let Pr( X (t )  s ) x y

x y x  y   t 1 u (1  u ) 1 du B( ; t ,  t , ) x  Pr( X (t )  x  y ) x  y   ( x, y , t )    1  x  x x Pr( X (t )  x)  t 1  1 B( ; t ,  t , ) 0x v (1  v) dv x 

Taking partial derivative to Ψ(x, y, t) with respect to t yields   x, y, t   t

x

x y x  y  x x 

Since G( x, y, t ) 

(1  u ) 1 [ln(u )  ln(v)] x  v t 1 (1  v) 1 dvu t 1 0

x x  0

(

v

 t 1

 1

(1  v)

dv)

du  0

2

Pr( X (t )  x  y ) Pr( X (t )  s  y ) is increasing with respect to t, we have is Pr( X (t )  x) Pr( X (t )  s )

decreasing in t. Therefore, λ(t) is increasing in t.

27

Highlights     

Develop a CBM model under the novel performance-based contracting approach Determine the optimal CBM policies given a performance-based contract Allow CBM to be imperfect Consider unit-specific random effects Provide indicators for choosing the adequate policies according to different system characteristics

28