Bayesian decision making in determining optimal leased term and preventive maintenance scheme for leased facilities

Chih-Chiang Fang

doi:10.3934/jimo.2020127

Article Contents

2021, Volume 17, Issue 6: 3445-3473. Doi: 10.3934/jimo.2020127

This issue Previous Article Impact of cap-and-trade regulation on coordinating perishable products supply chain with cost learning Next Article Two penalized mixed–integer nonlinear programming approaches to tackle multicollinearity and outliers effects in linear regression models

Bayesian decision making in determining optimal leased term and preventive maintenance scheme for leased facilities

Chih-Chiang Fang^,

School of Computer Science and Software, Zhaoqing University, Guangdong 526061, China

^*Corresponding author: Chih-Chiang Fang
^*Corresponding author: Chih-Chiang Fang

Received: June 2019

Revised: June 2020

Early access: August 2020

Published: November 2021

Abstract / Introduction Full Text(HTML) Figure(13) / Table(5) Related Papers Cited by

Abstract

Under a business competitive environment, quite a few enterprises choose capital leasing to reduce tax payment and investment risk instead of buying facilities. Since the durability and service life of leased facilities will be longer, the breakdowns and deterioration of leased facilities are inevitable during lease period. Accordingly, in order to reduce the related costs and keep the facility's health during lease period, preventive maintenances are required to perform to reduce the cost of free-repair warranty and maintain customers' satisfaction. However, performing preventive maintenance is not easy to scheme due to the scarcity of historical failure data. Accordingly, the study integrates lease and maintenance decisions into a synthetic strategy, and it can be applied under the situation of only expert's evaluation and/or scare historical failure data by employing Bayesian analyses. In this study, the mathematical models and corresponding algorithms are developed to determine the best preventive maintenance scheme and the optimal term of contract for leased facilities to maximize the expected profit. Moreover, the computerized architecture is also proposed, and it can help the lessor to solve the issue in practice. Finally, numerical examples and the sensitive analyses are provided to illustrate the managerial strategies under different leased period and the preventive maintenance policies.

Keywords:

Mathematics Subject Classification: Primary: 62C10, 62C25; Secondary: 90B25.

Citation:

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Figure 1. Timeline of the PM Model

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Figure 2. Maintenance Scheme under Imperfect Recovery

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Figure 3. The Flow Chart of the Heuristic Process for Obtaining $ N^* $ and $ q^* $

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Figure 4. Flowchart for the Bayesian Solution Algorithm

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Figure 5. Computerized Implementation Architecture

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Figure 6. Average Profits per Unit and Year for Maintenance Plans 1, 2, 3 Estimated by Prior Analysis

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Figure 7. Average Profits per Unit and Year for Maintenance Plan 1 Estimated by Prior and Posterior Analyses

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Figure 8. The Impact of $ E $($ \alpha $), $ E $($ \beta $), $ \sigma(\alpha) $ and $ \sigma(\beta) $ on Average Profit

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Figure 9. The Impact of Minimal Repair Cost on Average Profit

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Figure 10. The Impact of Base Cost for a PM action on Average Profit

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Figure 11. The Impact of Increasing Rate of PM Cost on Average Profit

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Figure 12. The Impact of Time Discount Rate on Average Profit

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Figure 13. The Impact of Depreciation Rate on Average Profit

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Table 1. The detailed information of three maintenance plans

	Maintenance Plan 1	Maintenance Plan 2	Maintenance Plan 3
Parameters for the deterioration judged by experts	${u_\alpha}$= 1.60, ${u_\beta}$= 2.10, ${\sigma_\alpha}$= 1.10, ${\sigma_\beta}$= 0.80
Age reduction factors	${\delta ^{M_P^1}}$ = 0.7	${\delta ^{M_P^2}}$ = 0.8	${\delta ^{M_P^3}}$ = 0.9
Base cost for a PM action	$C_F^{M_P^1}$=600	$C_F^{M_P^2}$=750	$C_F^{M_P^3}$=900
Periodically increasing rates of PM cost	${\tau ^{M_P^1}}$=0.2	${\tau ^{M_P^2}}$=0.225	data ${\tau ^{M_P^3}}$=0.25
Depreciation rate	$\rho$=0.15
Interval of PM; Time segment	$x$=0.5 years; $T_S$=0.5 year
The minimal and maximal planned lease terms	$T_L^{Min}$=2 years; $T_L^{Max}$=12 years
Rental of per half-year	$R_0$=9800
Time discount rate	$\epsilon$=0.02
Production cost of an equipment	$V$=9800
Penalty cost for repair time over the time limit	$C_Penalty= 170$
Expectation of performing a minimal repair	$E(t_r)$= 9 hours
Standard deviation of performing a minimal repair	$\sigma(t_r)$= 5 hours
Tolerable waiting time limit for performing a minimal repair	$\varphi$=4.5 hours
Expected cost of performing a minimal repair	$C_mr$=350

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Table 2. Expected failures, repair costs, preventive costs, production cost, residual value and average profits per unit and year for maintenance plan 1, 2, 3 estimated by prior analysis

	$\mathop E\limits_{{\text{Prior}}} [\Phi ({T_L},x,{\delta ^{M_P^q}},\alpha ,\beta )]$			PM Cost			Repair Cost			$V$	${V_{residual}}$	$\mathop E\limits_{{\text{Prior}}} \left[ \pi \right]$
Time	Plan1	Plan2	Plan3	Plan1	Plan2	Plan3	Plan1	Plan2	Plan3			Plan1	Plan2	Plan3
2	3.06	2.74	2.44	2760	3495	4194	1500	1343	1194	98000	70805	3292	3003	2729
2.5	4.24	3.69	3.17	3600	4575	5490	2077	1807	1553	98000	65279	3472	3190	2926
3	5.61	4.75	3.96	4500	5738	6885	2746	2327	1938	98000	60184	3625	3352	3099
3.5	7.17	5.93	4.79	5460	6983	8379	3514	2904	2349	98000	55487	3752	3491	3251
4	8.95	7.23	5.69	6480	8310	9972	4387	3542	2786	98000	51157	3854	3608	3381
4.5	10.97	8.66	6.63	7560	9720	11664	5373	4243	3250	98000	47164	3932	3703	3492
5	13.22	10.23	7.64	8700	11213	13455	6480	5011	3741	98000	43483	3987	3779	3584
5.5	15.75	11.94	8.70	9900	12788	15345	7716	5848	4261	98000	40089	4021	3835	3659
*6	18.55	13.79	9.81	11160	14445	17334	9090	6758	4809	98000	36961	*4033	3874	3718
6.5	21.65	15.80	10.99	12480	16185	19422	10611	7744	5386	98000	34076	4025	3896	3761
*7	25.08	17.98	12.23	13860	18008	21609	12289	8810	5993	98000	31417	3998	*3902	3790
7.5	28.85	20.32	13.53	15300	19913	23895	14135	9959	6631	98000	28965	3951	3893	3805
*8	32.98	22.85	14.90	16800	21900	26280	16160	11194	7301	98000	26704	3885	3869	*3808
8.5	37.50	25.55	16.33	18360	23970	28764	18375	12521	8002	98000	24620	3802	3831	3798
9	42.43	28.46	17.83	19980	26123	31347	20793	13943	8736	98000	22698	3701	3780	3778
9.5	47.81	31.56	19.40	21660	28358	34029	23426	15464	9504	98000	20927	3583	3716	3746
10	53.65	34.88	21.03	23400	30675	36810	26289	17089	10306	98000	19294	3448	3640	3705
10.5	59.99	38.41	22.74	25200	33075	39690	29394	18822	11143	98000	17788	3296	3553	3654
11	66.85	42.18	24.52	27060	35558	42669	32758	20668	12016	98000	16400	3128	3455	3595
11.5	74.28	46.19	26.38	28980	38123	45747	36396	22632	12926	98000	15120	2944	3346	3527
12	82.29	50.45	28.31	30960	40770	48924	40324	24719	13874	98000	13940	2744	3227	3451

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Table 3. Expected failures, repair costs, preventive cost, production cost, residual value and average profits per unit and year for prior and posterior analyses

	$E[\Phi ({T_L},x,{\delta ^{M_P^q}},\alpha ,\beta )]$		Repair Cost		PM Cost	$V$	${V_{residual}}$	$E\left[ \pi \right]$
Time	Prior	Posterior	Prior	Posterior				Prior	Posterior
2	3.06	3.90	1500	1911	1500	98000	70805	3292	3087
2.5	4.24	5.82	2077	2856	2077	98000	65279	3472	3161
3	5.61	8.09	2746	3965	2746	98000	60184	3625	3219
3.5	7.17	10.68	3514	5233	3514	98000	55487	3752	3261
4	8.95	13.58	4387	6655	4387	98000	51157	3854	3287
*4.5	10.97	16.79	5373	8226	5373	98000	47164	3932	*3298
5	13.22	20.29	6480	9944	6480	98000	43483	3987	3295
5.5	15.75	24.09	7716	11805	7716	98000	40089	4021	3277
*6	18.55	28.18	9090	13807	9090	98000	36961	*4033	3247
6.5	21.65	32.54	10611	15946	10611	98000	34076	4025	3204
7	25.08	37.19	12289	18222	12289	98000	31417	3998	3150
7.5	28.85	42.10	14135	20631	14135	98000	28965	3951	3085
8	32.98	47.29	16160	23173	16160	98000	26704	3885	3009
8.5	37.50	52.74	18375	25845	18375	98000	24620	3802	2923
9	42.43	58.46	20793	28645	20793	98000	22698	3701	2829
9.5	47.81	64.43	23426	31573	23426	98000	20927	3583	2725
10	53.65	70.67	26289	34627	26289	98000	19294	3448	2614
10.5	59.99	77.15	29394	37805	29394	98000	17788	3296	2495
11	66.85	83.89	32758	41107	32758	98000	16400	3128	2369
11.5	74.28	90.88	36396	44532	36396	98000	15120	2944	2236
12	82.29	98.12	40324	48077	40324	98000	13940	2744	2097

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Table 4. The impact of $E(\alpha)$ and $\sigma(\alpha)$ on expected failure times and repair cost

$E(\alpha)$
	1.2	1.4	1.6	1.8	2.0	2.2	1.2	1.4	1.6	1.8	2.0
Time	Expected failure times					Expected repair cost
2	2.32	2.69	3.06	3.43	3.79	4.15	1136	1319	1500	1680	1858
3	4.08	4.84	5.61	6.38	7.16	7.95	2000	2371	2746	3125	3507
4	6.30	7.61	8.95	10.34	11.75	13.21	3085	3727	4387	5065	5759
5	9.00	11.07	13.22	15.47	17.81	20.25	4412	5423	6480	7581	8726
6	12.26	15.31	18.55	21.98	25.59	29.42	6005	7500	9090	10769	12541
7	16.10	20.42	25.08	30.07	35.41	41.13	7890	10005	12289	14736	17352
8	20.60	26.51	32.98	40.01	47.61	55.88	10096	12989	16160	19603	23331
9	25.82	33.69	42.43	52.06	62.60	74.20	12653	16508	20793	25508	30675
10	31.83	42.09	53.65	66.54	80.83	96.74	15596	20622	26289	32602	39606
11	38.69	51.83	66.85	83.80	102.81	124.23	18959	25398	32758	41060	50378
12	46.49	63.08	82.29	104.23	129.14	157.52	22781	30910	40324	51075	63280
$\sigma(\alpha)$
	0.9	1.0	1.1	1.2	1.3	1.4	0.9	1.0	1.1	1.2	1.3
Time	Expected failure times					Expected repair cost
2	3.04	3.05	3.06	3.07	3.08	3.09	1488	1494	1500	1505	1509
3	5.71	5.66	5.61	5.56	5.51	5.47	2800	2772	2746	2723	2701
4	9.36	9.14	8.95	8.78	8.63	8.48	4584	4480	4387	4304	4228
5	14.14	13.65	13.22	12.85	12.51	12.20	6928	6688	6480	6295	6128
6	20.27	19.34	18.55	17.85	17.24	16.68	9932	9478	9090	8749	8446
7	27.98	26.41	25.08	23.93	22.92	22.02	13709	12940	12289	11725	11230
8	37.53	35.05	32.98	31.20	29.66	28.30	18390	17175	16160	15290	14532
9	49.23	45.51	42.43	39.83	37.58	35.61	24124	22300	20793	19515	18412
10	63.43	58.05	53.65	49.96	46.80	44.07	31080	28442	26289	24480	22933
11	80.51	72.95	66.85	61.78	57.48	53.78	39447	35747	32758	30271	28163
12	100.91	90.56	82.29	75.48	69.75	64.87	49444	44376	40324	36986	34179

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Table 5. The impact of $E(\beta)$ and $\sigma(\beta)$ on expected failure times and repair cost

$E(\beta)$
	1.5	1.7	1.9	2.1	2.3	2.5	2.7	1.5	1.7	1.9	2.1	2.3	2.5	2.7
Time	Expected failure times							Expected repair cost
2	3.02	3.07	3.08	3.06	3.03	2.98	2.92	1481	1502	1507	1500	1482	1458	1428
3	4.88	5.14	5.38	5.61	5.81	6.01	6.20	2391	2517	2636	2746	2849	2946	3038
4	7.07	7.65	8.28	8.95	9.66	10.40	11.17	3463	3749	4059	4387	4732	5094	5476
5	9.64	10.66	11.87	13.22	14.73	16.40	18.26	4723	5226	5816	6480	7219	8038	8946
6	12.65	14.25	16.22	18.55	21.25	24.35	27.92	6199	6981	7950	9090	10410	11932	13681
7	16.17	18.47	21.45	25.08	29.42	34.59	40.70	7923	9050	10510	12289	14418	16947	19943
8	20.27	23.42	27.66	32.98	39.52	47.50	57.20	9932	11475	13552	16160	19366	23276	28026
9	25.04	29.19	34.97	42.43	51.82	63.53	78.08	12269	14301	17138	20793	25394	31132	38261
10	30.58	35.88	43.54	53.65	66.64	83.17	104.12	14982	17582	21336	26289	32653	40752	51016
11	37.00	43.63	53.52	66.85	84.31	106.94	136.13	18129	21377	26225	32758	41313	52400	66704
12	44.44	52.56	65.08	82.29	105.23	135.44	175.07	21777	25754	31890	40324	51561	66367	85783
$\sigma(\beta)$
	0.5	0.6	0.7	0.8	0.9	1.0	1.2	0.5	0.6	0.7	0.8	0.9	1.0	1.2
Time	Expected failure times							Expected repair cost
2	3.08	3.08	3.07	3.06	3.04	3.02	2.99	1507	1508	1505	1500	1491	1480	1466
3	5.78	5.72	5.66	5.61	5.56	5.52	5.50	2832	2802	2773	2746	2724	2706	2696
4	9.29	9.15	9.03	8.95	8.92	8.94	9.03	4551	4482	4426	4387	4369	4378	4424
5	13.63	13.42	13.27	13.22	13.29	13.51	13.94	6680	6574	6505	6480	6511	6618	6833
6	18.85	18.59	18.47	18.55	18.88	19.55	20.76	9239	9107	9052	9090	9249	9579	10172
7	24.99	24.71	24.72	25.08	25.91	27.46	30.19	12245	12110	12114	12289	12698	13455	14792
8	32.07	31.87	32.12	32.98	34.69	37.75	43.25	15716	15615	15740	16160	16997	18496	21192
9	40.15	40.11	40.79	42.43	45.53	51.06	61.40	19673	19652	19985	20793	22309	25019	30088
10	49.25	49.50	50.83	53.65	58.82	68.23	86.78	24134	24256	24905	26289	28824	33432	42524
11	59.42	60.12	62.36	66.85	75.04	90.32	122.54	29118	29458	30558	32758	36768	44259	60044
12	70.70	72.03	75.53	82.29	94.71	118.73	173.42	34644	35295	37010	40324	46407	58178	84977

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