An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts

Tien-Yu Lin; Bhaba R. Sarker; Chien-Jui Lin

doi:10.3934/jimo.2020043

Article Contents

2021, Volume 17, Issue 1: 467-484. Doi: 10.3934/jimo.2020043

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An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts

1.
School of Economics and Management, Sanming University, Sanming, Fujian 365004, China
2.
Department of Mechanical and Industrial Engineering, Louisiana State University, Baton Rouge, LA 70803, USA
3.
Department of International Business, National Chengchi University, Taipei 11605, Taiwan (R.O.C)

^* Corresponding author: Chien-Jui Lin (j698102@gmail.com)
^* Corresponding author: Chien-Jui Lin (j698102@gmail.com)

Received: June 30, 2018

Revised: July 31, 2019

Early access: March 2020

Published: January 2021

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Abstract

The purpose of this paper concentrates on an economic production quantity model with the factors of imperfect quality and quantity discounts, in which the inspection action occurs during the production stage. There is specific consideration of there being a finite production rate, and the quantity discounts offered by the supplier serves the purpose of stimulating buying greater quantities. This is in contrast to EPQ models that do not take these added factors into consideration. The objective of this paper is to determine the setup cost reduction, which is a function of capital investment, and inventory lot size. An alternative solution procedure was developed that does not employ the Hessian Matrix concavity in the expected total profit. We develop an algorithm to determine the optimal solution for this model. Theoretical results are discussed and a numerical example is proposed. Managerial insights are also examined.

Keywords:

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

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Figure 1. The behavior of the inventory level per cycle

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Figure 2. The expected total profit

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Table 1. Procurement cost structure for the manufacture

$r$	$Q_{r-1} \sim Q_{r}$	$c_{r}$
1	$0 < Q < 150$	$c_{1} = 20.05$
2	$150 \leq Q < 400$	$c_{2} = 20.04$
3	$400 \leq Q < 800$	$c_{3} = 20.03$
4	$800 \leq Q < 1250$	$c_{4} = 20.02$
5	$Q \geq 800$	$c_{5} = 20.01$

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Table 2. The values of $Q^{*}, S^{*}$ and $E T P U^{*}$ corresponding to 32 combinations of $\sigma, f_{g} M, i, U(d)$

$\sigma$	$f_{g}$	$M$	$i$	$U(d)$	$Q^{*}$	$S^{*}$	$E T P U^{*}$
9600	0.05	192000	0.2	0.04	858.9	175.3	274724.7
			0.2	0.056	866.3	175.4	275046.5
			0.28	0.04	910.9	200	274036.9
			0.28	0.056	918.8	200	274358.8
		268800	0.2	0.04	846.1	163.3	274694.8
			0.2	0.056	853.5	172.8	275016.6
			0.28	0.04	897.4	200	274004.6
			0.28	0.056	905.2	200	274326.4
	0.07	192000	0.2	0.04	800	163.3	274570.9
			0.2	0.056	800	162	274892.8
			0.28	0.04	858.9	200	273906.7
			0.28	0.056	866.4	200	274228.7
		268800	0.2	0.04	800	163.3	374599.6
			0.2	0.056	800	162	274862.7
			0.28	0.04	846	200	273872.3
			0.28	0.056	853.5	200	274194.3
13440	0.05	192000	0.2	0.04	877.4	128	385689.5
			0.2	0.056	885	128.1	386139.7
			0.28	0.04	930.5	190	384839
			0.28	0.056	938.6	190.1	385289.5
		268800	0.2	0.04	858.9	125.2	385646.9
			0.2	0.056	866.3	125.3	386097
			0.28	0.04	910.9	186	384779.3
			0.28	0.056	918.8	186.1	385229.8
	0.07	192000	0.2	0.04	812.5	118.4	385535.8
			0.2	0.056	819.6	118.5	385986.1
			0.28	0.04	877.4	179.1	384674.4
			0.28	0.056	885.1	179.2	385125.1
		268800	0.2	0.04	800	116.7	385493.1
			0.2	0.056	800	115.7	385943.3
			0.28	0.04	858.9	175.3	384614.7
			0.28	0.056	866.4	175.4	385065.3

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