Effect of warranty and quantity discounts on a deteriorating production system with a Markovian production process and allowable shortages

Tien-Yu Lin

doi:10.3934/jimo.2020013

Article Contents

2021, Volume 17, Issue 3: 1101-1118. Doi: 10.3934/jimo.2020013

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Effect of warranty and quantity discounts on a deteriorating production system with a Markovian production process and allowable shortages

Tien-Yu Lin^,

Economics and Management College, Zhaoqing University, Zhaoqing City 526061, Guangdong Province, China

^* Corresponding author: Tien-Yu Lin
^* Corresponding author: Tien-Yu Lin

Received: December 2018

Revised: August 2019

Early access: January 2020

Published: May 2021

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Abstract

This paper explores the retailer's optimal lot sizing and quantity backordering for a deteriorating production system with a two-state Markov production process in which quantity discounts are provided by the supplier. The products are sold with the policy of free reasonable repair warranty employing the fraction of nonconforming items in a lot size. Unlike the traditional economic production quantity (EPQ) model with warranty policy based on the elapsed time of the system in the control state follows an exponential distribution, this paper not only constructs an alternative mathematical model for EPQ model based on the fraction of nonconforming items in a lot size for an imperfect production system but also extends the topics of optimal quantity and shortage to a wider scope of academic research and further finds that some results are different from the traditional EPQ models. We seek to minimize the expected total relevant cost through optimal lot sizing and quantity backordering. We also demonstrate that the optimal lot size is bounded in a finite interval. An efficient algorithm is developed to determine the optimal solution. Moreover, a numerical example is given and sensitivity analysis is conducted to highlight management insights.

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Mathematics Subject Classification: Primary: 90B05, 90B30; Secondary: 91B06.

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Figure 1. The inventory level for imperfect manufacturing system with allowable shortages

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Figure 2. The three-dimension graph of the expected total cost

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Table 1. The values of the parameters for numerical example

Description and parameters	Value	Unit
Production rate ($M$)	10, 000	units/year
Demand rate ($D$)	2, 000	units/year
Setup cost ($K$)	500	＄/cycle
Holding cost rate for a unit (a fraction of dollar value) ($I$)	0.26	＄/unit/year
Backordering cost ($b$)	6	＄/unit/year
Repair cost/warranty cost ($c_w$)	5	＄/unit
Restoration cost ($R$)	100	＄/cycle
Probability that the system from controlled state shifts to uncontrolled state ($p$)	0.1	N/A
Percentage of nonconforming items when the process is controlled state ($\lambda_1$)	0.1	N/A
Percentage of nonconforming items when the process is in uncontrolled ($\lambda_2$)	0.75	N/A

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Table 2. The values of $L^{}, S^{*}$, and $ATC^{*}$ corresponding to 32 combinations of $p, K, c_w, I, R$

$p$	$K$	$c_w$	$I$	$R$	$L^{*}$	$S^{*}$	$ATC^{*}$
0.1	500	6	0.2	100	1050	480	84514.92
			0.2	130	1050	480	84572.06
			0.26	100	883.1	448	84652.73
			0.26	130	905.1	459.2	84719.84
		7.8	0.2	100	1050	480	85848.04
			0.2	130	1050	480	85905.19
			0.26	100	879.8	446.4	85984.22
			0.26	130	901.9	457.6	86051.58
	650	6	0.2	100	1050	480	84800.63
			0.2	130	1062.1	485.5	84857.59
			0.26	100	1050	532.7	84958.68
			0.26	130	1050	532.7	85015.83
		7.8	0.2	100	1050	480	86133.76
			0.2	130	1059	484.1	86190.8
			0.26	100	1050	532.7	86291.81
			0.26	130	1050	532.7	86348.95
0.13	500	6	0.2	100	1050	480	84518.1
			0.2	130	1050	480	84575.24
			0.26	100	884.3	448.7	84656.5
			0.26	130	906.3	459.8	84723.52
		7.8	0.2	100	1050	480	85853.41
			0.2	130	1050	480	85910.55
			0.26	100	881.9	447.4	85990.62
			0.26	130	903.9	458.6	86057.8
	650	6	0.2	100	1050	480	84803.81
			0.2	130	1050	480	84860.95
			0.26	100	1050	532.7	84961.86
			0.26	130	1050	532.7	85019
		7.8	0.2	100	1050	480	86139.12
			0.2	130	1060.9	485	86196.11
			0.26	100	1050	532.7	86297.17
			0.26	130	1050	532.7	86354.32

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