Delayed payment policy in multi-product single-machine economic production quantity model with repair failure and partial backordering

Ata Allah Taleizadeh; Biswajit Sarkar; Mohammad Hasani

doi:10.3934/jimo.2019002

Article Contents

2020, Volume 16, Issue 3: 1273-1296. Doi: 10.3934/jimo.2019002

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Delayed payment policy in multi-product single-machine economic production quantity model with repair failure and partial backordering

1.
Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran
2.
Department of Industrial Engineering, Yonsei University, 50 Yonsei-ro, Sinchon-dong, Seodaemun-gu, Seoul, 03722, South Korea
3.
Islamic Azad University, Tehran South branch, School of Industrial Engineering, Tehran, Iran

^* Corresponding author: bsbiswajitsarkar@gmail.com (Biswajit Sarkar), Phone Number-+82-10-7498-1981, Office Phone: +82-31-400-5259, Fax: +82-31-436-8146
^* Corresponding author: bsbiswajitsarkar@gmail.com (Biswajit Sarkar), Phone Number-+82-10-7498-1981, Office Phone: +82-31-400-5259, Fax: +82-31-436-8146

Received: April 30, 2017

Revised: March 31, 2018

Early access: March 2019

Published: March 2020

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Abstract

This study develops a single-machine manufacturing system for multi-product with defective items and delayed payment policy. Contradictory to the literature limited production capacity and partial backlogging are considered for more realistic result. The objective of this research is to obtain the optimal cycle length, optimal production quantity, and optimal backorder quantity of each product such that the expected total cost is minimum. The model is solved analytically. Three efficient lemmas are developed to obtain the global optimum solution of the model. An improved algorithm is designed to obtain the numerical solution of the model. An illustrative numerical example and sensitivity analysis are provided to show the practical usage of proposed method.

Keywords:

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

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Figure 1. Graphical representation of inventory system

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Figure 2. Graphical representation of interest earned and interest charged for $ M<t_a $

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Figure 3. Graphical representation of interest earned and interest charged for $ t_d \leq M < t_a $

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Figure 4. Graphical representation of interest earned and interest charged for $ M \geq t_d $

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Table 1. Author(s) contribution Table

Author(s)	EPQ	Imperfect Production	Multi-product Single machine	Delay-in-payment	Backorder	Repair
C$ \acute{a} $rdenas-Barr$ \acute{o} $net al. [2]	$ \surd $					$ \surd $
Chiu et al. [4]	$ \surd $				$ \surd $	$ \surd $
Goyal and C$ \acute{a} $rdenas-Barr$ \acute{o} $n [8]	$ \surd $	$ \surd $
Huang [9]	$ \surd $
Taleizadeh [38]	$ \surd $		$ \surd $
Sana et al.[21]	$ \surd $	$ \surd $			$ \surd $
Li et al.[12]	$ \surd $				$ \surd $
Taleizadh et al. [48]		$ \surd $	$ \surd $		$ \surd $	$ \surd $
Sarkar et al. [25]	$ \surd $	$ \surd $			$ \surd $	$ \surd $
Kang et al. [10]	$ \surd $	$ \surd $			$ \surd $
Ouyang et al. [16]					$ \surd $
This Model	$ \surd $	$ \surd $	$ \surd $	$ \surd $	$ \surd $	$ \surd $

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Table 2. The values of the parameters

$ P $	$ P_i $	$ P_i^1 $	$ \lambda_i $	$ K_i $	$ C_i $	$ C_R^i $	$ b_i $	$ h_i $
1	10000	600	2000	750	2	0.5	0.25	0.2
2	10500	650	1500	700	1.5	0.6	0.5	0.15
3	11000	750	1000	650	1	0.7	0.75	0.1

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Table 3. The parametric values

$P$	$M_j$	$I_c$	$I_e$	$S_i$	$v_i$	$SE_i$	$X_i$
1	0.04	0.09	0.05	4	1.5	0.003	0.05
2	0.04	0.09	0.05	3	1	0.004	0.075
3	0.04	0.09	0.05	2	0.5	0.005	0.1

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Table 4. Optimal solutions table

$P$	$T_{Min}$	$T$	$T^*=T_1$	$Q_i$	$B_i$	$Z$
1				5158	2331.9
2	0.128	2.579	2.579	3868.5	1061	9046.93
3				2579	375.7

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Table 5. Optimal solutions for different values of $ M $

$ M $	$ T_{Min} $	$ T $	$ T^* $	$ Q_i $	$ B_i $	$ Z^*=Z_1 $
				5159	2332
0.03	0.128	2.579	2.579	3869	1061	9051.666
				2579	376
				5158	2331.9
0.04	0.128	2.579	2.579	3868.5	1061	9046.93
				2579	375.7
				5157	2331.5
0.05	0.128	2.578	2.578	3868	1061	9042.125
				2578.6	375.5
				51555	2330
0.06	0.128	2.577	2.577	3866	1060	9037.253
				2577	375

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Table 6. Optimal solutions for different values of $ I_c $

$ I_c $	$ T_{Min} $	$ T $	$ T^* $	$ Q_i $	$ B_i $	$ Z^*=Z_1 $
				5354.9	2323.3
0.07	0.128	2.67	2.67	4016.2	1037.7	8990.8
				2677.5	367.2
				5158	2331.9
0.09	0.128	2.579	2.579	3868.5	1061	9046.93
				2579	375.7
				4987	2336
0.11	0.128	2.49	2.49	3740	1082	9098.927
				2493	383
				4838	2339
0.13	0.128	2.41	2.41	3628	1101	9147.354
				2149	392

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Table 7. Optimal solutions for different values of $ I_e $

$ I_e $	$ T_{Min} $	$ T $	$ T^* $	$ Q_i $	$ B_i $	$ Z^*=Z_1 $
				5159	2332
0.03	0.128	2.579	2.579	3869	1061	9047.469
				2579	375
				5158	2331.9
0.05	0.128	2.579	2.579	3868.5	1061	9046.93
				2579	375.7
				5156	2330
0.05	0.128	2.57	2.57	3867	1059	9046.39
				2578.6	374
				5154	2329
0.06	0.128	2.56	2.56	3865	1058	9045.851
				2577	372

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