An inventory model with imperfect item, inspection errors, preventive maintenance and partial backlogging in uncertainty environment

Javad Taheri-Tolgari; Mohammad Mohammadi; Bahman Naderi; Alireza Arshadi-Khamseh; Abolfazl Mirzazadeh

doi:10.3934/jimo.2018097

Article Contents

2019, Volume 15, Issue 3: 1317-1344. Doi: 10.3934/jimo.2018097

This issue Previous Article Note on : Supply chain inventory model for deteriorating items with maximum lifetime and partial trade credit to credit risk customers Next Article Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy

An inventory model with imperfect item, inspection errors, preventive maintenance and partial backlogging in uncertainty environment

Department of Industrial Engineering, Faculty of Engineering Kharazmi University, 15719-14911 Tehran, Iran

* Corresponding author: Mohammad Mohammadi
* Corresponding author: Mohammad Mohammadi

Received: August 31, 2017

Revised: September 30, 2017

Early access: June 2018

Published: April 2019

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Abstract

In this paper investigate a production system with the defective quality process and preventive maintenance to establish the inspection policy and optimum inventory level for production items with considering uncertainty environment. The shortage occurs because of preventive maintenance and is considered as partial backlogging. Through the production process, at a random moment, the production of items from the state in-control turns into an out-of-control mode, so that parts of the defective product are manufactured an in-control state and outside of the control process mode. The online item inspection process begins after a time variable through the production period. The human inspection process has also been considered for the classify of defective goods. Uninspected products are accepted to the customer/buyer with minimal repair warranty and the defective items classify by the inspector at fixed cost before being shipped subject to salvaged items. Also, the inspection process of manufactured goods includes a human inspection error. Therefore, two types of classification errors (Type Ⅰ & Ⅱ) are considered to be more realistic than the proposed model. The input parameters of the model are considered as a triangular fuzzy environment, and the output parameters of the model are solved by the Zadeh's extension principle and nonlinear parametric programming. As a final point, a numerical example by graphical representations is obtainable to illustrate the proposed model.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Schematic representation of inventory system

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Figure 2. Flowchart the solution procedure

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Figure 3. Behavior of the target function at $\alpha = 0$

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Figure 4. Behavior of the target function at $\alpha = 0.1$

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Figure 5. Behavior of the target function at $\alpha = 0.2$

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Figure 6. Behavior of the target function at $\alpha = 0.3$

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Figure 7. Behavior of the target function at $\alpha = 0.4$

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Figure 8. Behavior of the target function at $\alpha = 0.5$

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Figure 9. Behavior of the target function at $\alpha = 0.6$

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Figure 10. Behavior of the target function at $\alpha = 0.7$

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Figure 11. Behavior of the target function at $\alpha = 0.8$

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Figure 12. Behavior of the target function at $\alpha = 0.9$

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Figure 13. Behavior of the target function at $\alpha = 1$

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Figure 14. Membership function of output parameters-optimal solution

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Figure 15. Membership function of output parameters-optimal solution

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Figure 16. Membership function of output parameters-optimal solution

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Figure 17. Membership function of output parameters-optimal solution

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Figure 18. Membership function of Hessian matrix

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Table 1. General data - input parameters

		Triangular Fuzzy Number: [a, b, c]
General Data	Symbol	a	b	c
Production rate	$P$	300	500	700
Annual demand rate	$D$	250	450	650
Setup cost	$k$	400	600	800
Holding cost	$c_h$	1.5	2.5	4
Variable cost	$c_m$	75	100	125
Inspection cost	$I_c$	0.75	1	1.25
Repair cost for warranty	$w$	35	50	65
Salvage cost	$s$	10	15	20
Cost of falsely accepted imperfect item	$c_a$	23	28	33
Cost of falsely rejected perfect item	$c_r$	7	10	13
Maintenance cost	$M_c $	75	100	125
Shortage cost	$s_c$	5.5	6.5	7.5
Fraction of product shortage that is backorder	$\gamma$	0.6	0.75	0.9
Percentage of defective products/in control	$\theta_1$	0.05	0.15	0.25
Percentage of defective products/out-of-control	$\theta_2$	0.05	0.30	0.60

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Table 2. General data - input parameters

General Data	Symbol	Probability Density Function
Type-Ⅰ error	$\alpha$	$f(\alpha)=0.04e^{0.04\alpha}$
Type-Ⅱ error	$\beta$	$f(\beta)=0.05e^{0.05\beta}$
Preventive maintenance time	$PM_T$	$f(PM_T)=0.25e^{0.25PM_T}$
Time after which the production process shifts in control to out-of-control state	$\sigma$	$f(\sigma)=0.33e^{0.33\sigma}$

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Table 3. General data - $\alpha$ cuts input parameters

		Lower and upper bound of $P$		Lower and upper bound of $D$		Lower and upper bound of $k$
		$P_{\alpha}^L$	$P_{\alpha}^U$	$D_{\alpha}^L$	$D_{\alpha}^U$	$k_{\alpha}^L$	$k_{\alpha}^U$
$\alpha~ level$	0	300	700	250	650	400	800
	0.1	320	680	270	630	420	780
	0.2	340	660	290	610	440	760
	0.3	360	640	310	590	460	740
	0.4	380	620	330	570	480	720
	0.5	400	600	350	550	500	700
	0.6	420	580	370	530	520	680
	0.7	440	560	390	510	540	660
	0.8	460	540	410	490	560	640
	0.9	480	520	430	470	580	620
	1	500	500	450	450	600	600

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Table 4. General data - $\alpha$ cuts input

		Lower and upper bound of $c_h$		Lower and upper bound of $c_m$		Lower and upper bound of $I_c$
		${c_h}_{\alpha}^L$	${c_h}_{\alpha}^U$	${c_m}_{\alpha}^L$	${c_m}_{\alpha}^U$	${I_c}_{\alpha}^L$	${I_c}_{\alpha}^U$
$\alpha~ level$	0	1.5	4	75	125	0.75	1.25
	0.1	1.6	3.85	77.5	122.5	0.775	1.225
	0.2	1.7	3.7	80	120	0.8	1.2
	0.3	1.8	3.55	82.5	117.5	0.825	1.175
	0.4	1.9	3.4	85	115	0.85	1.15
	0.5	2	3.25	87.5	112.5	0.875	1.125
	0.6	2.1	3.1	90	110	0.9	1.1
	0.7	2.2	2.95	92.5	107.5	0.925	1.075
	0.8	2.3	2.8	95	105	0.95	1.05
	0.9	2.4	2.65	97.5	102.5	0.975	1.025
	1	2.5	2.5	100	100	1	1

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Table 5. General data - $\alpha$ cuts input parameters

		Lower and upper bound of $w$		Lower and upper bound of $s$		Lower and upper bound of $c_a$
		$w_{\alpha}^L$	$w_{\alpha}^U$	$s_{\alpha}^L$	$s_{\alpha}^U$	${c_a}_{\alpha}^L$	${c_a}_{\alpha}^U$
$\alpha~ level$	0	35	65	10	20	23	33
	0.1	36.5	63.5	10.5	19.5	23.5	32.5
	0.2	38	62	11	19	24	32
	0.3	39.5	60.5	11.5	18.5	24.5	31.5
	0.4	41	59	12	18	25	31
	0.5	42.5	57.5	12.5	17.5	25.5	30.5
	0.6	44	56	13	17	26	30
	0.7	45.5	54.5	13.5	16.5	26.5	29.5
	0.8	47	53	14	16	27	29
	0.9	48.5	51.5	14.5	15.5	27.5	28.5
	1	50	50	15	15	28	28

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Table 6. General data - $\alpha$ cuts input parameters

		Lower and upper bound of $c_r$		Lower and upper bound of $M_c$		Lower and upper bound of $s_c$
		${c_r}_{\alpha}^L$	${c_r}_{\alpha}^U$	${M_c}_{\alpha}^L$	${M_c}_{\alpha}^U$	${s_c}_{\alpha}^L$	${s_c}_{\alpha}^U$
$\alpha~ level$	0	7	13	75	125	5.5	7.5
	0.1	7.3	12.7	77.5	122.5	5.6	7.4
	0.2	7.6	12.4	80	120	5.7	7.3
	0.3	7.9	12.1	82.5	117.5	5.8	7.2
	0.4	8.2	11.8	85	115	5.9	7.1
	0.5	8.5	11.5	87.5	112.5	6	7
	0.6	8.8	11.2	90	110	6.1	6.9
	0.7	9.1	10.9	92.5	107.5	6.2	6.8
	0.8	9.4	10.6	95	105	6.3	6.7
	0.9	9.7	10.3	97.5	102.5	6.4	6.6
	1	10	10	100	100	6.5	6.5

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Table 7. General data - $\alpha$ cuts input parameters

		Lower and upper bound of $\theta_1$		Lower and upper bound of $\theta_2$		Lower and upper bound of $\gamma$
		${\theta_1}_{\alpha}^L$	${\theta_1}_{\alpha}^U$	${\theta_2}_{\alpha}^L$	${\theta_2}_{\alpha}^U$	$\gamma_{\alpha}^L$	$\gamma_{\alpha}^U$
$\alpha~ level$	0	0.6	0.9	0.05	0.25	0.05	0.6
	0.1	0.615	0.885	0.06	0.24	0.075	0.57
	0.2	0.63	0.87	0.07	0.23	0.1	0.54
	0.3	0.645	0.855	0.08	0.22	0.125	0.51
	0.4	0.66	0.84	0.09	0.21	0.15	0.48
	0.5	0.675	0.825	0.1	0.2	0.175	0.45
	0.6	0.69	0.81	0.11	0.19	0.2	0.42
	0.7	0.705	0.795	0.12	0.18	0.225	0.39
	0.8	0.72	0.78	0.13	0.17	0.25	0.36
	0.9	0.735	0.765	0.14	0.16	0.275	0.33
	1	0.75	0.75	0.15	0.15	0.3	0.3

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Table 8. General data - $\alpha$ cuts input parameters

		Lower and upper bound of $E[TCU]$		Lower and upper bound of $I_m$		Lower and upper bound of $\delta$
		$E[TCU]_{\alpha}^L$	$E[TCU]_{\alpha}^U$	${I_m}_{\alpha}^L$	${I_m}_{\alpha}^U$	$\delta_{\alpha}^L$	$\delta_{\alpha}^U$
$\alpha~ level$	0	87.0133	145.1221	261.5060	334.5721	0	1
	0.1	89.9842	142.218	268.8547	332.0059	0	0.6745
	0.2	92.9146	139.3142	276.4437	329.4848	0	0.4356
	0.3	95.8269	136.4106	282.4803	327.0103	0	0.3054
	0.4	98.7300	133.5071	287.6358	324.5838	0	0.2221
	0.5	101.627	130.6038	292.2257	322.2072	0	0.1640
	0.6	104.522	127.7005	296.4265	319.8825	0	0.1211
	0.7	107.414	124.7973	300.3485	317.6118	0	0.0882
	0.8	110.305	121.8939	304.0651	315.3975	0	0.0622
	0.9	113.195	118.9902	307.6279	313.2330	0.0096	0.0412
	1	116.0856	116.0856	311.0745	311.0745	0.0239	0.0239

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Table 9. General data - $\alpha$ cuts Hessian matrix

		Lower and upper bound of $H_1$		Lower and upper bound of $H_2$
		$H_1^L$	$H_1^U$	$H_2^L$	$H_2^U$
$\alpha~ level$	0	0.176880867	5.9687669	9.23117E-06	0.000239
	0.1	0.366359442	5.691024185	1.82229E-05	0.000228
	0.2	0.652038831	5.417609624	3.09138E-05	0.000218
	0.3	0.947627216	5.148450421	4.33739E-05	0.000208
	0.4	1.25140767	4.883472592	5.57048E-05	0.000197
	0.5	1.562433439	4.622600569	6.79622E-05	0.000187
	0.6	1.880135101	4.365756781	8.01786E-05	0.000177
	0.7	2.204158168	4.112861206	9.23724E-05	0.000166
	0.8	2.534281476	3.863830732	0.000104554	0.000156
	0.9	2.870373498	3.546574322	0.000116726	0.000143
	1	3.212363938	3.212363938	0.000128889	0.000129

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Table 10. General data - optimal output parameters in crisp state

	Defuzzification method
Variable	Centroid	Signed distance
$E[TCU]$	116.07	116.08
$I_m$	302.38	304.56
$\delta$	0.3413	0.262
Hessian matrix $H_1$	3.1193	3.1426
Hessian matrix $H_2$	0.0001	0.0001

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