An integrated inventory model with variable transportation cost, two-stage inspection, and defective items

Biswajit Sarkar; Bijoy Kumar Shaw; Taebok Kim; Mitali Sarkar; Dongmin Shin

doi:10.3934/jimo.2017027

Article Contents

2017, Volume 13, Issue 4: 1975-1990. Doi: 10.3934/jimo.2017027

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An integrated inventory model with variable transportation cost, two-stage inspection, and defective items

1.
Department of Industrial & Management Engineering, Hanyang University, Ansan Gyeonggi-do, 15588, South Korea
2.
Department of Mathematics & Statistics, Banasthali Vidyapith, Banasthali, Rajasthan, 304 022, India
3.
Department of Industrial Engineering, Hanyang University 220 Wangsimni-ro, Seongdong-gu, Seoul, 04763, Korea

* Corresponding author: mitalisarkar.ms@gmail.com(Mitali Sarkar), Phone Number-010-7490-1981, Fax No +82-31-436-8146.
* Corresponding author: mitalisarkar.ms@gmail.com(Mitali Sarkar), Phone Number-010-7490-1981, Fax No +82-31-436-8146.

Received: December 31, 2015

Revised: May 31, 2016

Published: November 2017

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Abstract

The paper deals with an integrated inventory model with make-to-order policy from buyer to vendor. A variable transportation cost is used as a power function of the delivery quantity for either tapering or considering proportional rate data to maintain single-setup multi-delivery (SSMD) policy with reduced transportation cost. A two-stage inspection process is introduced by the vendor to ensure the perfect quality of product even though the first inspection process indicates a constant defective rate of imperfect production is present during the long-run production system and all defective items are reworked with some fixed cost. The aim is to minimize the total cost of the integrated inventory model by using classical optimization technique. Two numerical examples, sensitivity analysis, and graphical representations are given to illustrate the model.

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Mathematics Subject Classification: Primary: 90B05, 90B50; Secondary: 90B30.

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Figure 1. Integrated vendor-buyer production-inventory model.

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Figure 2. Graphical illustration of the unit time cost with respect to joint decision(n, q) of Example 1.

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Figure 3. The curve of total cost function TRC when $a$ varies as on Example 1.

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Figure 4. Graphical illustration of the unit time cost with respect to joint decision(n, q) of Example 2.

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Figure 5. The curve of total cost function TRC when $a$ varies as on Example 2.

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Table 1. A summary of production-delivery lot sizing models.

Author(s)	MTO/MTS strategy	Production quantity	Delivery quantity	Delivery cost	Inspection policy
C$\acute{a}$rdenas$-$Barr$\acute{o}$n and Sana [4,5]	MTO	Variable	$-$	$-$	$-$
Taleizadeh et al. [42]	MTO	Variable	$-$	$-$	$-$
Benerjee [1] and Goyal [11]	MTO	Variable	Costant	$-$	$-$
Goyal [10]	MTO	Costant	$-$	$-$	$-$
Sarker and Parija [28,29]	MTO	Variable	Costant	$-$	$-$
Sarkar and Majumder[24]	MTO	Costant	Variable	Considered	$-$
Sarkar et al.[32]	MTO	Costant	Variable	$-$	$-$
Golhar and Sarker [9]	MTO	Variable	Costant	$-$	$-$
Lu [17]	MTS	Variable	Costant	$-$	$-$
Hill [14,15]	MTS	Variable	Variable	$-$	$-$
Chan and Kingsman [6]	MTS	Variable	Variable	$-$	$-$
Ben$-$Daya et al. [2]	MTS	Variable	Variable	$-$	$-$
Glock [8]	MTS	Variable	Variable	$-$	$-$
Ertogral et al. [7]	MTS	Variable	Variable	Considered	$-$
Wee and Chung[47]	MTO	Costant	Costant	$-$	Included
Lee and Fu [16]	MTO	Variable	Variable	Considered	$-$
This study	MTO	Variable	Variable	Considered	Included
"$-$" indicates non-availability of the research contribution, $"MTO"$ denotes make$-$to$-$order, and $"MTS"$ denotes make$-$to$-$stock.

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Table 2. Cost sensitiveness based on the parameter $a$ from Example 1

$a$	$TRC$
$0$	15722.9
$0.1$	15816.6
$0.2$	15935.4
$0.3$	16085.9
$0.4$	16276.8
$0.5$	16518.7
$0.6$	16825.3
$0.7$	17214.0
$0.8$	17706.6
$0.9$	18331.1
$1$	19122.8

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Table 3. Cost sensitiveness based on the parameter $a$ from Example 2

$a$	$TRC$
$0$	12126.6
$0.1$	12193.3
$0.2$	12275.0
$0.3$	12375.0
$0.4$	12497.4
$0.5$	12647.3
$0.6$	12830.7
$0.7$	13055.3
$0.8$	13330.3
$0.9$	13666.9
$1$	14078.9

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Table 4. Sensitivity analysis of key parameters

Parameters	Changes(in %)	$TRC$
	-50%	-12.45
	-25%	-5.04
$A_1$	+25%	6.23
	+50%	12.45
	-50%	-2.31
	-25%	-1.15
$A_2$	+25%	1.15
	+50%	2.31
	-50%	-12.70
	-25%	-6.35
$h_1$	+25%	6.35
	+50%	12.70
	-50%	-8.32
	-25%	-4.16
$h_2$	+25%	4.16
	+50%	8.32

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