Structural Properties of Optimal Maintenance Policies for k-out-of-n Systems with Interdependence Between Internal Deterioration and External Shocks †
Abstract
:1. Introduction
- We propose an optimal maintenance policy for k-out-of-n systems that accounts for variability in shock impacts across units and addresses the interdependence between unit deterioration and external shocks, making it applicable to real-world systems for efficient operation.
- We derive a structural property of the optimal policy, offering key insights for developing algorithms that enable efficient searching for the optimal solution.
System Design | Interdependence Among Units | Interdependence Between External Shocks and Internal Deterioration | Optimization Target | Optimization Framework | Structural Property | |
---|---|---|---|---|---|---|
Derman [3] | single-unit | - | - | maintenance action | MDP | ✓ |
Ohnishi et al. [4] | single-unit | - | - | maintenance action | partially observable MDP | ✓ |
Lovejoy et al. [5] | single-unit | - | - | maintenance action | partially observable MDP | ✓ |
Elwany et al. [6] | single-unit | - | - | maintenance action | MDP | ✓ |
Chen et al. [7] | single-unit | - | - | maintenance action | MDP | ✓ |
Zhu and Xiang [11] | n-unit | × | - | maintenance action | stochastic programming | ✓ |
Ashizawa and Jin [13] | 2-unit | ✓ | - | maintenance action | MDP | ✓ |
Liu et al. [14] | series | ✓ | - | maintenance action | MDP | ✓ |
Xu et al. [16] | k-out-of-n | ✓ | - | maintenance action | MDP | - |
Oakley et al. [17] | series–parallel | ✓ | - | maintenance action | Bayesian sequential decision | ✓ |
Sun et al. [18] | k-out-of-n | × | - | maintenance action | MDP | ✓ |
Zhang et al. [19] | k-out-of-n | ✓ | - | maintenance action | MDP | ✓ |
Rafiee et al. [24] | single-unit | - | × | inspection interval | cost rate | - |
Dui et al. [25] | n-unit | × | age→internal deterioration age→shocks | maintenance action | integer programming | - |
Yousefi et al. [26] | series parallel | × | × | maintenance action | MDP | - |
Yousefi et al. [27] | series | × | internal deterioration ↔ external shocks | inspection interval | cost rate | - |
Tajiani [28] | single-unit | - | internal deterioration ← external shocks | maintenance threshold | cost rate | - |
Kurt and Maillart [29] | single-unit | - | internal deterioration ← external shocks | maintenance action | MDP | ✓ |
Wang et al. [30] | single-unit | - | internal deterioration ↔ external shocks | maintenance action | MDP | ✓ |
Qi and Huang et al. [31] | single-unit | - | internal deterioration ↔ external shocks | inspection interval | cost rate | - |
Lorvand and Kelkinnama. [33] | k-out-of-n | × | age→external shocks | maintenance interval | cost rate | - |
Cao et al. [39] | single-unit | - | internal deterioration ← external shocks age→external shocks | inspection interval | multi objective programming | - |
Shafiee et al. [40] | series | × | - | maintenance threshold maintenance interval | cost rate | - |
This research | k-out-of-n | ✓ | internal deterioration ↔ external shocks | maintenance action | MDP | ✓ |
2. Model
2.1. System Deterioration
2.2. Actions and Costs in Condition-Based Maintenance
2.3. Optimal Maintenance Policy
3. Structural Properties of the Optimal Maintenance Policy
- 1.
- Base Case : By Equation (1) and Assumption 1, holds. Since each unit’s deterioration is independent of other unit deterioration states, it follows from Lemma 2 that .
- 2.
- Inductive Step: Assume that holds for all when .
4. Case Study: Three-Bladed Rotor System of Offshore Wind Turbines
4.1. System and Parameter Specifications
4.2. Optimal Maintenance Policy for a Three-Bladed Rotor System of Offshore Wind Turbines
Algorithm 1 The value iteration algorithm utilized to compute . |
|
4.3. Performance of the Proposed Policy
Algorithm 2 Calculate average total discounted cost and availability for each policy. |
|
4.4. The 2-out-of-3 System
5. Sensitivity Analysis
5.1. Preventive Replacement Cost ()
5.2. Corrective Replacement Cost ()
5.3. Downtime Cost ()
5.4. Setup Cost ()
5.5. Coefficient in Scale Parameter Model (b)
5.6. Shock Likelihood ()
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Nomenclature
Fixed inspection interval | |
Deterioration state of unit i | |
Deterioration state of system | |
Shock-induced cumulative deterioration of unit i | |
Shock-induced cumulative deterioration of system | |
L | Failure threshold of units |
Elapsed time from last inspection to system failure | |
Shape parameter of units | |
Scale parameter of unit i given | |
Expected down time cost given and | |
Constant preventive replacement cost | |
Constant corrective replacement cost | |
Constant setup cost | |
Constant downtime cost | |
Discount factor | |
Shock type | |
Set of units whose deterioration state increases due to shocks of type m | |
Transition probability density function from to given after time t | |
Probability density function that type m shock occurs at time t | |
Probability density function that the increment of unit deterioration is when a shock of type m occurs in unit i |
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Policy | Cost (EUR) | CRR (%) | Availability (%) | Frequency of Preventive RP (times/) | Frequency of Corrective RP (times/) |
---|---|---|---|---|---|
Proposed Policy | 3,226,835 | - | 96.7% | 0.282 | 0.048 |
(i) Deterioration Shock Policy (Conventional) | 3,579,910 | 9.9 | 95.4% | 0.236 | 0.066 |
(ii) Shock → Deterioration Policy | 3,566,466 | 9.5 | 95.4% | 0.236 | 0.065 |
(iii) Deterioration → Shock Policy | 3,269,933 | 1.3 | 96.8% | 0.290 | 0.046 |
(iv) Individual Optimal Policy [30] | 3,556,059 | 9.3 | 95.5% | 0.237 | 0.065 |
b | ||||||
---|---|---|---|---|---|---|
Low | 4 | 12 | 4 | 1 | ||
Base | 7 | 16 | 8 | 2 | ||
High | 10 | 20 | 12 | 3 |
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Kasuya, M.; Jin, L. Structural Properties of Optimal Maintenance Policies for k-out-of-n Systems with Interdependence Between Internal Deterioration and External Shocks. Mathematics 2025, 13, 716. https://doi.org/10.3390/math13050716
Kasuya M, Jin L. Structural Properties of Optimal Maintenance Policies for k-out-of-n Systems with Interdependence Between Internal Deterioration and External Shocks. Mathematics. 2025; 13(5):716. https://doi.org/10.3390/math13050716
Chicago/Turabian StyleKasuya, Mizuki, and Lu Jin. 2025. "Structural Properties of Optimal Maintenance Policies for k-out-of-n Systems with Interdependence Between Internal Deterioration and External Shocks" Mathematics 13, no. 5: 716. https://doi.org/10.3390/math13050716
APA StyleKasuya, M., & Jin, L. (2025). Structural Properties of Optimal Maintenance Policies for k-out-of-n Systems with Interdependence Between Internal Deterioration and External Shocks. Mathematics, 13(5), 716. https://doi.org/10.3390/math13050716