Allocating redundancy, maintenance and spare parts for minimizing system cost under decentralized repairs

doi:10.1007/s42524-024-0145-3

Front. Eng

2024, Vol. 11

Issue (3) : 377-395 https://doi.org/10.1007/s42524-024-0145-3

Industrial Engineering and Intelligent Manufacturing

Allocating redundancy, maintenance and spare parts for minimizing system cost under decentralized repairs

Tongdan JIN¹, Shubin SI², Wenjin ZHU²(

)

¹. Ingram School of Engineering, Texas State University, San Marcos, TX 78666, USA
². School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, China; Key Laboratory of Industrial Engineering and Intelligent Manufacturing (Ministry of Industry and Information Technology), Xi’an 710072, China

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Abstract

Reliability-redundancy allocation, preventive maintenance, and spare parts logistics are crucial for achieving system reliability and availability goal. Existing methods often concentrate on specific scopes of the system’s lifetime. This paper proposes a joint redundancy-maintenance-inventory allocation model that simultaneously optimizes redundant component, replacement time, spares stocking, and repair capacity. Under reliability and availability criteria, our objective is to minimize the system’s lifetime cost, including design, manufacturing, and operational phases. We develop a unified system availability model based on ten performance drivers, serving as the foundation for the establishment of the lifetime-based resource allocation model. Superimposed renewal theory is employed to estimate spare part demand from proactive and corrective replacements. A bisection algorithm, enhanced by neighborhood exploration, solves the complex mixed-integer, nonlinear optimization problem. The numerical experiments show that component redundancy is preferred and necessary if one of the following situations occurs: extremely high system availability is required, the fleet size is small, the system reliability is immature, the inventory holding is too costly, or the hands-on replacement time is prolonged. The joint allocation model also reveals that there exists no monotonic relation between spares stocking level and system availability.

Keywords system availability installed base decentralized repair redundancy-maintenance-inventory model superimposed renewal process

Corresponding Author(s): Wenjin ZHU

Just Accepted Date: 28 April 2024 Online First Date: 27 June 2024 Issue Date: 26 September 2024

Cite this article:

Tongdan JIN,Shubin SI,Wenjin ZHU. Allocating redundancy, maintenance and spare parts for minimizing system cost under decentralized repairs[J]. Front. Eng, 2024, 11(3): 377-395.

URL:

https://academic.hep.com.cn/fem/EN/10.1007/s42524-024-0145-3
https://academic.hep.com.cn/fem/EN/Y2024/V11/I3/377

Fig.1 A system comprised of N redundant subsystems in series.

Fig.2 Product-service integration with decentralized repair services.

Notation	Definition
$x i$	Redundancy level for part type $i$ , for $i = 1, 2, …, N$
$s i$	Base-stock level for part type $i$ , for $i = 1, 2, …, N$
$τ i$	Replacement age or interval for part type $i$ , for $i = 1, 2, …, N$
$p i$	Number of renewing servers for part type $i$ , for $i = 1, 2, …, N$
$q i$	Number of repair servers for part type $i$ , for $i = 1, 2, …, N$

Tab.1 Decision variables

Fig.3 The

M / M / q / ∞

queueing model for parts repair process.

Fig.4 A graphical illustration of the bisection search.

Notation	Benchmark	Sensitivity Analysis	Unit
$α$	0.2	[0.2, 1.5]	failure/year
$β$	3	[1.5, 5]	n/a
$k$	10	10	item
$n max$	13	13	item
$m$	50	[10, 200]	system
$t s$	8	[4, 48]	hour
$1 μ p$	6	[3, 18]	day
$1 μ q$	12	[6, 24]	day
$c L R U$	50,000	[25000, 200000]	$/item
$c u$	3,000	n/a	$/item
$c v$	4,500	n/a	$/item
$c h$	10,000	[5000, 50000]	$/item/year
$c p$	480,000	[0.5 $c p$ , 1.5 $c p$ ]	$/server
$c q$	640,000	[0.5 $c q$ , 1.5 $c q$ ]	$/server
$A min$	0.99	[0.9, 0.999]	n/a
$θ$	0.7	[0.5, 1.1]	n/a
$γ min, γ max$	0.5, 2	0.5, 2	n/a
$φ 1, φ 2$	0.1295, 0.2310	0.1295, 0.2310	n/a

Tab.2 Parameter values of ATE system (n/a=not applicable, item=LRU)

Fig.5 System cost and availability for different fleet sizes.

Fig.6 Optimal solutions for different fleet sizes.

Fig.7 Spare part stock level and parts availability for benchmark study.

Case	Parameter	$x$	$s$	$τ$	$p$	$q$	$A s y s$	$A p a r t$	$R (τ)$	Cost ($)
1	$A min = 0.999$	1	23	3.728	2	2	0.9992	0.89	0.661	137,348
1	$A min = 0.999$	0								No Solution
	$A min = 0.99$	0	15	3.710	2	2	0.9901	0.962	0.661	126,407
	$A min = 0.99$	1	19	3.751	2	2	0.9901	0.590	0.656	135,599
	$A min = 0.9$	0	9	3.773	2	2	0.9014	0.482	0.651	123,757
	$A min = 0.9$	1	0	3.840	2	2	0.9131	0	0.636	127,316
2	$θ = 0.5$	0	15	3.710	2	2	0.9901	0.962	0.661	126,407
	$θ = 0.5$	1	19	3.751	2	2	0.9901	0.590	0.656	135,599
	$θ = 0.7$	0	15	3.710	2	2	0.9901	0.962	0.661	126,407
	$θ = 0.7$	1	19	3.751	2	2	0.9901	0.590	0.656	135,599
	$θ = 1.1$	0	16	5.068	1	3	0.9902	0.963	0.353	129,623
	$θ = 1.1$	1	0	5.068	1	4	0.9902	0.001	0.322	142,776
3	$α = 0.2$	0	15	3.728	2	2	0.9901	0.962	0.661	126,388
3	$α = 0.2$	1	19	3.751	2	2	0.9901	0.590	0.656	135,599
	$α = 0.5$	0	24	1.331	6	4	0.9902	0.991	0.745	211,168
	$α = 0.5$	1	24	1.500	5	5	0.9906	0.678	0.656	222,028
	$α = 1.0$	1	33	0.746	10	10	0.9920	0.787	0.661	366,491
	$α = 1.0$	0								No Solution
4	$t s = 8$	0	15	3.728	2	2	0.9901	0.962	0.661	126,388
4	$t s = 8$	1	19	3.751	2	2	0.9901	0.590	0.656	135,599
	$t s = 24$	0	18	3.817	2	2	0.9903	0.991	0.641	127,592
	$t s = 24$	1	19	3.728	2	2	0.9917	0.633	0.661	135,624
	$t s = 48$	1	19	3.728	2	2	0.9907	0.633	0.661	135,624
	$t s = 48$	0								No Solution
5	$c h = 10, 000$	0	15	3.706	2	2	0.9901	0.962	0.666	126,412
5	$c h = 10, 000$	1	19	3.751	2	2	0.9901	0.590	0.656	135,599
	$c h = 20, 000$	0	15	3.728	2	2	0.9901	0.962	0.661	129,388
	$c h = 20, 000$	1	0	3.483	3	2	0.9904	0.001	0.713	137,365
	$c h = 50, 000$	1	0	3.483	3	2	0.9904	0.001	0.713	137,365
	$c h = 50, 000$	0	15	3.728	2	2	0.9901	0.962	0.661	138,388

Tab.3 Comparison between redundancy and spares inventory for Cases 1 to 5

Case	Parameter	$x$	$s$	$τ$	$p$	$q$	$A s y s$	$A p a r t$	$R (τ)$	Cost ($)
6	$β = 1.5$	0	11	6.771	1	4	0.9919	0.957	0.207	140,773
6	$β = 1.5$	1	7	6.048	1	4	0.9901	0.226	0.264	146,835
	$β = 3$ 3	0	15	3.728	2	2	0.9901	0.962	0.661	126,388
	$β = 3$ 3	1	19	3.751	2	2	0.9901	0.590	0.656	135,599
	$β = 4.5$	0	10	3.262	3	1	0.9905	0.946	0.864	120,742
	$β = 4.5$	1	6	3.308	3	1	0.9901	0.180	0.856	126,407
7	$1 μ p = 3$	0	12	2.500	2	1	0.9910	0.964	0.882	115,023
7	$1 μ p = 3$	1	9	2.433	2	1	0.9902	0.324	0.891	121,816
	$1 μ p = 6$	0	15	3.728	2	2	0.9901	0.962	0.661	126,388
	$1 μ p = 6$	1	19	3.751	2	2	0.9901	0.590	0.656	135,599
	$1 μ p = 12$	0	17	5.090	2	3	0.9908	0.968	0.348	139,453
	$1 μ p = 12$	1	9	5.961	1	4	0.9906	0.327	0.184	146,719
8	$1 μ q = 6$	0	7	5.202	1	2	0.9914	0.926	0.324	112,741
8	$1 μ q = 6$	1	0	5.202	1	2	0.9941	0.003	0.324	117,176
	$1 μ q = 12$	0	15	3.728	2	2	0.9901	0.962	0.661	126,388
	$1 μ q = 12$	1	19	3.751	2	2	0.9901	0.590	0.656	135,599
	$1 μ q = 24$	0	32	2.679	3	2	0.9907	0.987	0.857	145,399
	$1 μ q = 24$	1	12	2.478	4	2	0.9903	0.434	0.885	154,906
9	$c p = 240 K$	0	13	2.433	4	1	0.9902	0.963	0.891	115,728
9	$c p = 240 K$	1	11	2.456	4	1	0.9906	0.408	0.888	122,575
	$c p = 480 K$	0	15	3.728	2	2	0.9901	0.962	0.661	126,388
	$c p = 480 K$	1	19	3.751	2	2	0.9901	0.590	0.656	135,599
	$c p = 720 K$	0	16	5.068	1	3	0.9902	0.963	0.353	134,223
	$c p = 720 K$	1	21	4.934	1	3	0.9910	0.636	0.383	143,841
10	$c q = 320 K$	0	16	5.068	1	3	0.9902	0.963	0.353	110,223
10	$c q = 320 K$	1	0	5.202	1	4	0.9919	0.001	0.324	117,176
	$c q = 640 K$	0	15	3.728	2	2	0.9901	0.962	0.661	126,388
	$c q = 640 K$	1	19	3.751	2	2	0.9901	0.590	0.656	135,599
	$c q = 960 K$	0	32	2.679	3	1	0.9923	0.990	0.857	138,999
	$c q = 960 K$	1	11	2.456	4	1	0.9906	0.408	0.888	148,175

Tab.4 Comparison between redundancy and spares inventory for Cases 6 to 10

$m$	$s$	$p$	$q$	$τ$	$A s y s$	Cost	Cost Diff (%)	Algorithm
10	2	1	1	4.063	0.9900	191,147.79	0.000	BS
	2	2	2	5.210	0.9952	302,831.52	58.428	GA
	2	2	2	5.209	0.9952	302,831.52	36.880	PSO
20	7	1	1	3.795	0.9903	138,598.13	0.000	BS
	3	2	2	4.959	0.9900	189,764.39	36.917	GA
	3	2	2	4.957	0.9900	189,764.54	36.917	PSO
30	9	2	1	3.170	0.9911	135,742.37	0.000	BS
	5	2	2	4.386	0.9900	152,922.17	12.656	GA
	5	2	2	4.374	0.9902	152,927.02	12.660	PSO
40	21	1	2	4.510	0.9909	129,934.56	0.000	BS
	8	2	2	3.998	0.9900	135,191.86	4.046	GA
	8	2	2	3.974	0.9906	135,209.85	4.060	PSO
50	15	2	2	3.706	0.9901	126,407.38	0.000	BS
	16	2	2	3.829	0.9900	126,717.73	0.246	GA
	16	2	2	3.774	0.9926	126,770.23	0.287	PSO
60	15	3	2	3.349	0.9914	126,344.98	0.031	BS
	13	2	3	4.270	0.9900	127,384.86	0.854	GA
	15	3	2	3.372	0.9905	126,306.04	0.000	PSO
70	30	2	3	4.161	0.9903	125,154.35	0.000	BS
	13	3	3	3.865	0.9900	126,991.05	1.468	GA
	32	2	3	4.141	0.9948	125,780.55	0.500	PSO
80	22	3	3	3.706	0.9901	123,073.06	0.000	BS
	22	2	4	4.523	0.9900	124,541.17	1.193	GA
	23	3	3	3.775	0.9901	123,269.10	0.159	PSO
90	21	4	3	3.488	0.9906	123,109.26	0.001	BS
	21	4	3	3.488	0.9900	123,107.83	0.000	GA
	21	4	3	3.477	0.9909	123,123.63	0.013	PSO
100	38	3	4	4.018	0.9914	123,057.65	0.000	BS
	17	3	5	4.376	0.9900	124,731.38	1.360	GA
	39	3	4	3.980	0.9907	123,299.45	0.196	PSO
110	29	4	4	4.465	0.9903	121,513.23	0.001	BS
	29	4	4	3.745	0.9900	121,512.23	0.000	GA
	29	4	4	3.713	0.9917	121,545.98	0.028	PSO
120	26	5	4	3.527	0.9901	121,363.67	0.000	BS
	24	4	5	4.025	0.9900	121,838.51	0.391	GA
	26	5	4	3.492	0.9917	121,408.94	0.037	PSO
130	25	6	4	3.349	0.9903	121,614.60	0.002	BS
	25	6	4	3.349	0.9900	121,611.91	0.000	GA
	22	5	5	3.760	0.9907	121,811.97	0.165	PSO
140	35	5	5	3.743	0.9901	120,495.55	0.000	BS
	35	5	5	3.740	0.9900	120,496.74	0.001	GA
	37	5	5	3.744	0.9937	120,800.53	0.253	PSO
150	31	6	5	3.559	0.9903	120,307.38	0.001	BS
	31	6	5	3.558	0.9900	120,305.60	0.000	GA
	28	5	6	3.930	0.9902	120,556.70	0.209	PSO
160	29	7	5	3.393	0.9904	120,459.42	0.000	BS
	25	5	7	4.104	0.9900	121,176.29	0.595	GA
	26	6	6	3.716	0.9907	120,636.36	0.147	PSO
170	40	6	6	3.706	0.9915	119,746.86	0.000	BS
	29	8	5	3.288	0.9900	120,812.72	0.890	GA
	42	6	6	3.669	0.9915	120,039.51	0.244	PSO
180	36	7	6	3.572	0.9918	119,613.35	0.000	BS
	44	8	5	3.258	0.9900	120,155.51	0.453	GA
	39	7	6	3.502	0.9943	120,061.78	0.375	PSO
190	33	8	6	3.438	0.9900	119,650.11	0.000	BS
	30	7	7	3.732	0.9900	119,782.91	0.111	GA
	34	8	6	3.444	0.9916	119,752.30	0.085	PSO
200	42	6	8	4.024	0.9903	119,390.37	0.000	BS
	42	6	8	4.019	0.9900	119,391.56	0.001	GA
	29	8	7	3.600	0.9901	119,991.31	0.503	PSO

Tab.5 Comparisons of BS, GA and PSO results (underscores are the lowest)

Subsystem	$i = 1$	$i = 2$	$i = 3$	$i = 4$	Unit
$α$	0.25	0.3	0.35	0.4	failure/year
$β$	1.5	2.5	3	3.5	n/a
$k$	10	7	5	3	item
$n max$	13	10	7	5	item
$t s$	12	18	24	30	hour
$1 μ p$	6	9	12	15	day
$1 μ q$	12	14	18	21	day
$c L R U$	60,000	90,000	110,000	130,000	$/item
$c u$	3,000	5,000	6,000	7,500	$/item
$c v$	4,500	7,500	9,500	11,250	$/item
$c h$	12,000	18,000	22,000	26,000	$/item/year
$c p$	250,000	320,000	375,000	420,000	$/server
$c q$	350,000	384,000	494,000	630,000	$/server
$γ min, γ max$	0.5, 2	0.5, 2	0.5, 2	0.5, 2	n/a

Tab.6 Reliability and cost data for series-parallel system (n/a=not applicable)

$m$	10	20	30	40	50	60	70	80	90	100
Cost	776,664	701,735	666,538	645,222	634,393	627,222	627,411	619,270	612,960	610,007
$A s y s$	0.99001	0.99001	0.99001	0.99000	0.99001	0.99000	0.99000	0.99000	0.99001	0.99000
$x 1$	1	1	0	0	0	0	0	0	0	0
$x 2$	1	1	1	1	1	1	1	1	1	1
$x 3$	1	1	1	1	1	1	1	1	1	1
$x 4$	0	0	0	0	0	0	0	0	0	0
$s 1$	5	7	15	17	17	19	25	29	32	34
$s 2$	3	6	10	13	16	19	14	16	18	20
$s 3$	4	8	8	12	16	20	23	20	23	25
$s 4$	4	8	16	15	15	18	19	25	26	26
$τ 1$	3.033	4.044	5.416	5.416	5.416	5.416	5.416	5.416	5.416	5.416
$τ 2$	2.810	3.372	3.667	3.845	4.022	4.141	3.224	3.342	3.460	3.519
$τ 3$	2.398	2.781	2.679	2.322	2.526	2.653	2.730	2.424	2.526	2.602
$τ 4$	2.272	2.564	2.497	2.699	2.834	2.497	3.104	2.969	2.654	2.744
$p 1$	1	1	1	1	1	1	1	1	2	2
$p 2$	1	1	1	1	1	1	3	3	3	3
$p 3$	1	1	2	3	3	3	3	5	5	5
$p 4$	1	1	1	1	1	2	1	1	2	2
$q 1$	1	2	3	4	5	6	7	8	8	9
$q 2$	1	2	3	4	5	6	6	7	8	9
$q 3$	1	2	3	3	4	5	6	6	7	8
$q 4$	1	2	2	3	4	4	6	6	6	7

Tab.7 The solution for systems with four redundant subsystems

Notation	Definition
$α, β$	Weibull scale and shape parameters, respectively
$λ p$	Part demand rate of a single-item system in planned replacement
$λ q$	Part demand rate of a single-item system in failure replacement
$λ m$	Aggregate part demand rate of a fleet of single-item systems
$λ F, p$	Aggregate part demand rate of a fleet in planned replacement
$λ F, q$	Aggregate part demand rate of a fleet in failure replacement
$λ F$	Aggregate part demand rate of a fleet, and $λ F = λ F, p + λ F, q$
$ρ p, ρ q$	Part renewing and repair traffic intensity rate, respectively
$φ 1$	Capital recovery factor of system
$φ 2$	Capital recovery factor of spare part or LRU
$θ$	Percentage of mean time between failures of LRU
$μ$	Number of returned items during parts turn-around time
$μ p, μ q$	Part renewing rate and repair rate, respectively
$τ min, τ max$	The lower and upper limit of maintenance intervals
$γ min, γ max$	Minimum and maximum percentage of MTBF for LRU
$c L R U$	Unit cost for a spare part or LRU
$c h$	Unit holding cost per year
$c u, c v$	Cost for renewing and repairing a part, respectively
$c p, c q$	Cost for operating a renewing and repairing server, respectively
$k$	The minimum number of required working item in a system
$m$	System fleet size or installed base
$n max$	Maximum number of components a system can install
$t A T T$	Average part turn-around time
$t p$	Part renewing turn-around time
$t q$	Part repair turn-around time
$t s$	Hands-on time for replacing a part
$B (q)$	Probability for a part waiting in a repair shop
$C (p)$	Probability for a part waiting in a renewing shop
$A$	Availability of a single-item system
$A R$	Availability of a $k$ -out-of- $n$ redundant system
$A min$	Target or desired system availability
$A s y s$	Actual system availability
$A p a r t$	Actual parts availability
$f (t)$	Probability density functions of the LRU lifetime
$R (t)$	Reliability function of the LRU
$F (t)$	Cumulative distribution function of the LRU lifetime
$O$	Steady-state inventory on-order, a random variable
$T p$	Mean downtime in a planned replacement
$T q$	Mean downtime in a failure replacement
$T ¯$	Mean-time-between-failures of the LRU
$N$	Number of redundant subsystems in a system

Table A1 Model parameters

Fig.8 Weibull reliability for

τ =

0.5, 1 and 2MTBF

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[1]	Tongdan JIN. Bridging reliability and operations management for superior system availability: Challenges and opportunities[J]. Front. Eng, 2023, 10(3): 391-405.

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