# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020013

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

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

* Corresponding author: Tien-Yu Lin

Received  December 2018 Revised  August 2019 Published  January 2020

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.

Citation: Tien-Yu Lin. Effect of warranty and quantity discounts on a deteriorating production system with a Markovian production process and allowable shortages. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020013
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##### References:
The inventory level for imperfect manufacturing system with allowable shortages
The three-dimension graph of the expected total cost
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
 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
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 84514.92 130 1050 480 84572.06 0.26 100 883.1 448 84652.73 130 905.1 459.2 84719.84 7.8 0.2 100 1050 480 85848.04 130 1050 480 85905.19 0.26 100 879.8 446.4 85984.22 130 901.9 457.6 86051.58 650 6 0.2 100 1050 480 84800.63 130 1062.1 485.5 84857.59 0.26 100 1050 532.7 84958.68 130 1050 532.7 85015.83 7.8 0.2 100 1050 480 86133.76 130 1059 484.1 86190.8 0.26 100 1050 532.7 86291.81 130 1050 532.7 86348.95 0.13 500 6 0.2 100 1050 480 84518.1 130 1050 480 84575.24 0.26 100 884.3 448.7 84656.5 130 906.3 459.8 84723.52 7.8 0.2 100 1050 480 85853.41 130 1050 480 85910.55 0.26 100 881.9 447.4 85990.62 130 903.9 458.6 86057.8 650 6 0.2 100 1050 480 84803.81 130 1050 480 84860.95 0.26 100 1050 532.7 84961.86 130 1050 532.7 85019 7.8 0.2 100 1050 480 86139.12 130 1060.9 485 86196.11 0.26 100 1050 532.7 86297.17 130 1050 532.7 86354.32
 $p$ $K$ $c_w$ $I$ $R$ $L^{*}$ $S^{*}$ $ATC^{*}$ 0.1 500 6 0.2 100 1050 480 84514.92 130 1050 480 84572.06 0.26 100 883.1 448 84652.73 130 905.1 459.2 84719.84 7.8 0.2 100 1050 480 85848.04 130 1050 480 85905.19 0.26 100 879.8 446.4 85984.22 130 901.9 457.6 86051.58 650 6 0.2 100 1050 480 84800.63 130 1062.1 485.5 84857.59 0.26 100 1050 532.7 84958.68 130 1050 532.7 85015.83 7.8 0.2 100 1050 480 86133.76 130 1059 484.1 86190.8 0.26 100 1050 532.7 86291.81 130 1050 532.7 86348.95 0.13 500 6 0.2 100 1050 480 84518.1 130 1050 480 84575.24 0.26 100 884.3 448.7 84656.5 130 906.3 459.8 84723.52 7.8 0.2 100 1050 480 85853.41 130 1050 480 85910.55 0.26 100 881.9 447.4 85990.62 130 903.9 458.6 86057.8 650 6 0.2 100 1050 480 84803.81 130 1050 480 84860.95 0.26 100 1050 532.7 84961.86 130 1050 532.7 85019 7.8 0.2 100 1050 480 86139.12 130 1060.9 485 86196.11 0.26 100 1050 532.7 86297.17 130 1050 532.7 86354.32
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