Maintenance Scheduling Optimization based on Reliability and Prognostics Information Zhaojun Li, PhD, CRE, Western New England University, Springfield, MA, USA Jian Guo, PhD student, Western New England University, Springfield, MA, USA Ruolin Zhou, PhD, Western New England University, Springfield, MA, USA Key Words: maintenance, aircraft, preventive maintenance, prognostic, remaining flying hours SUMMARY & CONCLUSIONS Maintaining a large fleet of aircraft can be very challenging due to the variety of aircraft models, ages, and operating environments. Usually, the maintenance activities are implemented in a few centered facilities with different levels of maintenance capabilities. For instance, the Air Force's Air Logistics Complexes (ALC) are responsible for maintaining the Air Force's aging aircraft fleets. Current static methods for maintenance planning are based on the long-established Programmed Depot Maintenance (PDM) interval method, which can result in schedule delays, low throughput, and aircraft availability shortfalls. The situations are further complicated due to the increasing age of aircraft, high operational demands, and overall lack of fiscal resources. This paper is to propose innovative condition-based maintenance scheduling methodologies by integrating complex data processing, feature extraction, prognostic algorithm, and maintenance scheduling optimization. The proposed framework of prognostic-based maintenance scheduling is able to provide tradeoff analysis in terms of key performance metrics such as command possession rate, cost, and capacity expansion. The optimized maintenance schedule based on fleet health status will lead to higher aircraft availability, lower unscheduled maintenance cost, and meeting the continuous improvement initiatives such as the transitioning from PDMbased maintenance to High Velocity Maintenance (HVM) paradigm. The research outcomes will lead to more predictable and efficient maintenance scheduling capability. A numerical example shows how the aircraft reliability and health information can be integrated into the maintenance scheduling and planning optimization. NOTATION AND ACRONYM
I J
|I| |J| T lj
Set of all aircraft i Î I Set of sortie types j Î J Number of aircraft available Number of unique sortie types Length of time horizon Length of sortie type, j Î J
s tj
Minimum number of sorties of type j Î J
hmax
xijt
Maximum number of remaining phase hours an aircraft can have before being entered into maintenance The average time period required to complete maintenance Flying hours remaining on aircraft i Î I at the beginning of the horizon Maximum number of aircraft that can be in maintenance any given time period t without penalty Whether aircraft i Î I flies sorties type j Î J
mit
Whether aircraft i Î I enters maintenance in period t
k
bi
M
h
Life remaining of aircraft i Î I at period t
Z
Objective function value
t i
1 INTRODUCTION Higher Reliability, Availability, Maintainability, and Safety (RAMS) standard is the goal for many military applications such as weapon and surveillance systems. However, achieving such goals becomes challenging due to factors such as materials degradation, technology gaps, and inadequate operating and service support. For example, maintaining a required command possession (CP) rate or certain level of operating availability for Air Force's fleets of aircraft can be very challenging due to the increasing age of aircraft and limitations of manpower, facility, and other resources. One key element in realizing the RAMS goal is to enable optimal predictive maintenance scheduling and rescheduling. With built-in prognostic capability for maintenance scheduling, maintenance delays and interruptions will be greatly reduced. Meanwhile, more reliable and safer aircraft will be served with higher availability and operation readiness level. To implement prognostic-based maintenance scheduling, a systematic approach must be followed using a variety of the state-of-the-art technologies and methodologies including an efficient data processing and management system, advanced data mining and computing techniques, sophisticated prognostic algorithms, as well as optimization algorithms for solving the maintenance scheduling problem of complex and large-scale optimization formulations. We summarize the major
challenges for prognostic-based maintenance into the following elements. Current aircraft maintenance scheduling process mainly includes two steps. The first one is to determine flying schedules and sortie requirements. Flying Hour Program (FHP), which is assigned by MAJCOM, flying schedules and sortie requirements. Here we can consider this as an input. In other words, we can consider sortie requirements of each day as demand of aircraft. In order to meet the sortie requirements, a collective plan is needed in which all aircraft in the fleet are considered comprehensively. The remainder of the paper is organized as follows. Section 2 presents the existing literature in maintenance plans and maintenance scheduling. In Section 3, the proposed prognosticbased maintenance scheduling model is formulated. Section 4 provides a numerical example to apply the proposed model into a real application. Section 5 presents the conclusion and future research work.
the importance qualitatively and quantitatively. It is common for mathematical maintenance models to take static reliability and failure rate information, such as remaining flying hours, into account. The development of aircraft maintenance schedules is a complicated task involving the synthesis of a range of economic, political, legal and technical factors. Sriram et.al [11] proposed a heuristic approach to solve the maintenance scheduling problem with the objective function of minimizing the maintenance cost Cho [12] proposed a mixedinteger programming to solve the maintenance scheduling of fighter aircraft. However, very few maintenance scheduling models based on prognostics have been applied in aircraft operations, especially for fighter aircraft maintenance. Therefore, the main objective of this paper is to propose innovative prognostic-based maintenance scheduling methodologies by integrating complex data processing, feature extraction, prognostic algorithm, and maintenance scheduling optimization in the aircraft maintenance area.
2 LITERATURE REVIEW
3 PROGNOSTIC-BASED MAINTENANCE SCHEDULING
Maintenance problems have been studied extensively. Lyonnet [1] summarized maintenance policies, maintenance planning, and mathematics for maintenance optimization. Dekker [2] reviewed the maintenance optimization models and discussed potential research opportunities. Ben-Daya et al. [3] provided the latest development in the area of maintenance planning and scheduling in one volume of literature. Optimization models aim to help in determining effective and efficient schedules and plans, taking all kinds of constraints into account. A newer maintenance technique, condition-based maintenance (CBM), which is conducted in a situation where the performance indices are periodically or continuously monitored [4, 5], is needed. CBM is designed to balance the maintenance cost during corrective maintenance (CM) and preventive maintenance (PM), and it has attracted a lot of attention from both researchers and industry. Advances in sensing and information collection enable maintenance planners to collect and analyze the real-time status of target systems. Not only the current conditions, but also the future failure prediction based on prognostics can be applied in maintenance optimization. Prognostics provide information for maintenance planners and operators to make proactive maintenance decisions. Prognostic methods also have been studied extensively [6]. Researchers started to study maintenance scheduling based on reliability and prognostics. Grall et al. [7] proposed a decision model to optimize maintenance planning for systems with a Gamma deteriorating process. Yang et al. [8] proposed a method for scheduling of maintenance operations in a manufacturing system using the continuous assessment and prediction of the level of performance degradation of manufacturing equipment, as well as the complex interaction between the production process and maintenance operations. Camci [9] extended the idea of Yang et al. into non-production system by considering the system operations and maintenance constraints. In the area of aircraft operations, maintenance is important for the sake of safety and effectiveness. Kiyak[10] explained
With developed prognostic capability and prognostic outcomes, such as the downtime, the remaining useful life estimation, and other prognostic results can be incorporated for optimal maintenance scheduling. Our research formulates optimal maintenance scheduling by incorporating stochastic and uncertain inputs such as maintenance duration and the number of incoming airplanes at given time period. The research objective is to maximize maintenance performance including throughput, cost, and the mission capable rate of alternative maintenance schedules. 3.1 Maintenance practice and activity scheduling Maintenance includes calendar-based maintenance and usage-based maintenance. Phase maintenance is the most important one in usage-based maintenance, which is actually the condition-based maintenance. In phase maintenance, staff checks each part/system of an aircraft following a predetermined standard operation process. Other types of maintenance are PDM and time compliance technical orders (TCTOs) which are usually implemented at depot level and require an aircraft to be flown to a MDS-specific location where highly specialized maintenance personnel conduct a complete overhaul of the aircraft. Personnel conduct any necessary upgrades and can perform extensive maintenance that is beyond the capabilities of the maintenance personnel at the base level. Since depot level maintenance is carried out at a single consolidated location for a given mission, design and series (MDS), operations and maintenance schedulers at the base level have no input as to when an aircraft is scheduled for depot maintenance. Base level schedulers are provided with specific dates on which a given aircraft must be delivered to depot maintenance. TCTOs could be at base or depot level because they are special orders for a specific MDS. Given TCTOs need to be completed by a certain date, date and downtime of these TCTO maintenance activities can be considered constraints in the maintenance scheduling formulation.
mit + å j xijt + y £ 1, "t Î [1, T - k + 1], y Î [0, k - 1], i Î I (10)
3.2 Maintenance scheduling Maintenance scheduling can be formulated at both base and depot operation levels, and in this section it is shown how the predicted aircraft health metrics and the amount of workload of each maintenance activity as well as prediction uncertainties will be input to a prognostic-based maintenance scheduling optimizer. Aircraft maintenance scheduling usually includes two steps. The first one is to determine flying schedules and sorties requirements. Flying schedules and sortie requirements are decided by Flying Hour Program (FHP) which determines the total number of flying hours that must be flown each year [12]. The requirement can consider sortie requirements of each day as the demand of an aircraft. Flying schedules and sorties are all planned in half days. So maintenance schedules should also be planned in the same time unit. After the flying schedule and sortie requirements have been generated according to FHP requirements, the next step is to assign specific aircraft to each sortie requirement while also scheduling downtime for required preventative maintenance. In order to meet the sortie requirements, a collective plan in which all aircraft in the fleet should be considered comprehensively. 3.3 Mathematical formulation for prognostic maintenance scheduling A preliminary prognostic-based maintenance scheduling model is formulated as follows. Given the characteristics of Air Force maintenance, the constraint of maintenance cost is set up as a threshold capability, M , in terms of the number of aircraft, assessing at which number a “penalty” would happen. Assume that one aircraft only flies one sortie in one period (half day) and only one sortie is required for different types of sorties in half day. Decision variables:
{ m = {1 if aircraft i Î I enters maintenance period t 0 otherwise
xijt = 1 if aircraft i Î I flies sorties type j Î J in period t 0 otherwise t i
Objective function:
minimize Z = max(å i
t
( mt - M )), t Î [k + 1, T ] å t i
=t - k
(1)
Subject to:
hi1 = bi , "i Î I
t +1 i
h
(2)
£ h - å j x l + hmax m , "t , i Î I t i
t ij j
t i
hit +1 £ hit - å j xijt l j , "t , i Î I t +1 i
hmax m £ h t i
å
j
£ hmax , "t , i Î I
x £ 1, "t , i Î I
åx
t ij
t i ij
£ 1, "t , j Î J
hmax - h ³ hmax m , "t , i Î I
å
t i
j
(4) (5) (6) (7)
t i
(8)
t +1 ij
(9)
x ³ å j x , "odd t , i Î I t ij
(3)
mit + mit + y £ 1, "t Î [1, T - k ], y Î [1, k ], i Î I , t Î T h ³ 0, "i Î I , t x Î{0,1}, "i Î I , j Î J , t t i
t ij
mit Î {0,1}, "i Î I , t
(11) (12) (13) (14)
(1) is the objective function where the number of aircraft in maintenance in any average maintenance duration period does not excess the maintenance capacity. (2)-(14) are the constraints. (2) initializes the remaining flying hours of the fleet of interest. To incorporate the stochastic process of the remaining flying hours, the input used is a distribution. (3), (4), and (5) ensure the remaining flying hours on each aircraft are tracked depending on whether an aircraft flies or enters maintenance. (6) and (7) enforce all of the operational sortie requirements and the aircraft availability. (8) models the condition-based maintenance policy where only when the remaining hours reaches the threshold does the aircraft enter into maintenance. (9) ensures that the number of aircraft flown in the afternoon is less than the morning period. (10) and (11) keep the downtime long enough to complete the maintenance. (12), (13), and (14) define the decision variables. 4 NUMERICAL EXAMPLE This section will illustrate the application of maintenance optimization for aircraft operations. Effective maintenance schedules can help keep an appropriate availability level of fighter aircraft and an acceptable level of maintenance cost. 4.1 Model inputs In the numerical example, Air Force base with a fleet of twenty fighter aircraft is considered. The proposed model can help obtain the optimal maintenance schedule. FHP requires 1050 flying hours for two types of sorties in one month. The characteristics of fighter aircraft uses half day as one scheduling period, i.e. scheduling the morning and afternoon flying missions. Since usually flying and maintenance are not implemented during the weekend, the total scheduling horizon is set to be 50 periods which is about one month. The initial remaining flying hours of each aircraft in one schedule horizon and maintenance duration are inputs based on the prognostic process with uncertainties quantified using certain probability distributions. The initial remaining flying hours can be derived from the results of last schedule horizon and the usage of each fighter aircraft in a fleet should be opportunistically equal. The initial value of available remaining flying hours is set up as 200 hours. For simplicity, the distribution describing the uncertain initial remaining flying time is assumed to be a uniform distribution, U (0, 200) . The maintenance duration of each aircraft varies from the part(s) that needs inspection and maintenance activities. Based on the analysis in the work of Hurst [13], we assume that the maintenance duration is a normal distribution N (3,1) , i.e. the maintenance duration is normally distributed around 3 periods with 1 period standard deviation. The completed model inputs
are shown in Table I.
32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
Table I Model input Input
I
J
|I| |J| T
Value 1, 2, …, 20 1,2 20 2 50 3, 2, j = 1, 2.
lj k
bi
N (3,1) U (0, 200)
M
0
In this example, the maximum maintenance capacity of M is assigned as zero to realize the objective of high availability of the fighter aircraft. That is, the capacity of the maintenance station of this base is zero and any fighter aircraft entering into maintenance would cost a penalty. In general, the objective function in the maintenance scheduling optimization is to minimize the maximum difference between the number of aircraft under maintenance and the maintenance capacity at any given period of planning horizon.
S1 S1 S1 S2 S2 S1 S1 S1 S1 S2 S2 S2 S2 S1 S2 S1 S2 S1 S2
S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S2 S1 S1 S1 S1 S1 S1 S2
S1 S1 S1 S2 S2 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S2
USM USM USM USM USM USM USM USM USM USM USM USM USM USM USM USM USM USM USM
S1 S1 S1 S2 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1
S2 S2 S2 S2 S2 S2 S2 S2 S2 S1 S2 S2 S2 S2 S1 S1 S2 S2 S2
S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S2 S1 S2 S2 S2 S1 S1 S2
S1 S1 S1 S1 S1 USM USM USM USM USM USM USM USM S1 S1 S1 S1 USM USM
S2 S2 S2 S2 S2 S2 S2 S2 S2 S1 S2 S1 S1 S1 S1 S2 S1 S1 S1
S2 S2 S2 S1 S1 S2 S2 S2 S2 S2 S2 S2 S2 S2 S2 S2 S2 S2 S1
The program for solving the optimization problem is written in AMPL and solved by CPLEX. Table II presents part of the maintenance scheduling results, i.e., the flying and maintenance schedule of aircraft 1-10 in the schedule horizon (50 periods), where “S1”, “S2”, “M”, and “USM” respectively stand for flying sortie type 1, flying sortie type 2, under maintenance, and unscheduled maintenance, i.e. neither entering maintenance or flying. Fig. 1 shows the remaining flying hours change profile of aircraft 1-10 over time.
4.2 Results and analysis Table II the maintenance scheduling result of aircraft 1-10 A/C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
1 S1 S1 S1 S1 S1 S1 S1 S2 S1 S1 S2 S2 S2 S1 S1 S1 S1 S1 S1 S1 S1 S1 S2 S1 S2 S2 S1 S1 S2 S1 S1
2 S2 S1 S1 S2 S2 S2 S2 S1 S1 S2 S1 S2 S1 S1 S2 S2 S2 S1 S2 S2 S1 S2 S2 S2 S2 S1 S1 S2 S1 S1 S1
3 S2 S1 M USM USM USM USM USM S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1
4 USM USM USM USM USM USM USM USM S1 S1 USM USM USM USM USM USM USM USM USM USM USM USM USM USM USM USM USM USM USM USM USM
5 S1 S1 S2 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S2 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1 S1
6 S1 M USM USM USM USM S2 S1 S1 S1 S2 S2 S2 S1 S2 S2 S1 S1 S1 S2 S2 S2 S1 S1 S1 S1 S2 S2 S1 S2 S1
7 S1 S1 S1 S1 S2 S1 S2 S1 S1 S1 S2 S1 S2 S1 S1 S1 S2 S2 S1 S1 S2 S2 S2 S2 S1 S1 S1 S1 S1 S1 S1
8 USM USM S1 S1 S1 S1 S1 S1 USM USM USM USM USM USM S1 S1 S1 S1 S2 S1 USM USM S1 S1 S1 S1 S1 S1 S1 S1 S1
9 S1 S2 S2 S2 S2 S2 S2 S2 S2 S2 S2 S2 S2 S2 S1 S2 S2 S2 S1 S2 S2 S2 S2 S2 S2 S2 S2 S2 S2 S2 S2
10 S2 S2 S2 S2 S2 S2 S2 S2 S2 S2 S2 S1 S2 S2 S2 S1 S1 S2 S2 S1 S2 S1 S2 S2 S2 S2 S2 S2 S2 S2 S2
Fig. 1 Remaining flying hours over 50 periods The optimized schedule indicates that aircraft 3 and 6 should enter into maintenance starting at the second period for three periods, while aircraft 4 basically will stay in unscheduled maintenance except flying for sortie type 1 in period 9 and 10. Likewise, the maintenance schedule of other aircraft can be seen from Table II. 5 DISCUSSION AND CONCLUSION This paper aims to investigate the fighter aircraft phase maintenance schedule problem based on prognostic information. A MIP model is formulated to maximize maintenance resource utilization given flying hours and aircraft
availability requirements. The uncertainties of initial remaining flying hours and maintenance durations are incorporated into the optimization model using probability distributions. In the future, we will investigate how aircraft fleet availability and the optimal maintenance schedules change over different levels of maintenance capacity. REFERENCES 1. 2.
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Zhaojun Li Department of Industrial Engineering and Engineering Management
Western New England University 1215 Wilbraham Road Springfield, MA, 01119, USA e-mail:
[email protected] Jian Guo Department of Industrial Engineering and Engineering Management Western New England University 1215 Wilbraham Road Springfield, MA, 01119, USA e-mail:
[email protected] Roulin Zhou Department of Electrical and Computer Engineering Western New England University 1215 Wilbraham Road Springfield, MA, 01119, USA e-mail:
[email protected] 2
