Irreducibility of joint inventory positions in an assemble-to-order system under (r, nQ) policies

Jiejian FENG, Liming LIU, Yat Wah WAN

Research output: Journal PublicationsJournal Article (refereed)

7 Citations (Scopus)

Abstract

In a typical assemble-to-order system, a customer order may request multiple items, and the order may not be filled if any of the requested items are out of stock. A key customer service measure when unfilled orders are backordered is the order-based backorder level. To evaluate this crucial performance measure, a fundamental question is whether the stationary joint inventory positions follow an independent and uniform distribution. In this context, this is equivalent to the irreducibility of the Markov chain formed by the joint inventory positions. This article presents a necessary and sufficient condition for the irreducibility of such a Markov chain through a set of simultaneous Diophantine equations. This result also leads to sufficient conditions that are more general than those in the published reports.
Original languageEnglish
Pages (from-to)18-25
Number of pages8
JournalNaval Research Logistics (NRL)
Volume59
Issue number1
DOIs
Publication statusPublished - Feb 2012

Fingerprint

Irreducibility
Markov processes
Markov chain
Customers
Backorder
Simultaneous equations
Sufficient Conditions
Diophantine equation
Uniform distribution
Performance Measures
Policy
Order systems
Assemble-to-order
Necessary Conditions
Evaluate
Customer service
Performance measures

Bibliographical note

This work is partially supported by grant PolyUG-YX82 and RGC grant PolyU6145/04E of Hong Kong, grant NSERC 371939-2009 of Saint Mary’s University, Canada, and grant NSC99-2410-H-259-046 of Taiwan.

Keywords

  • assemble-to-order
  • diophantine equations
  • irreducibility
  • necessary and sufficient condition
  • order-based backorders
  • unit-skeleton condition

Cite this

@article{780d3eb8ef00485ca8140af497f27c7f,
title = "Irreducibility of joint inventory positions in an assemble-to-order system under (r, nQ) policies",
abstract = "In a typical assemble-to-order system, a customer order may request multiple items, and the order may not be filled if any of the requested items are out of stock. A key customer service measure when unfilled orders are backordered is the order-based backorder level. To evaluate this crucial performance measure, a fundamental question is whether the stationary joint inventory positions follow an independent and uniform distribution. In this context, this is equivalent to the irreducibility of the Markov chain formed by the joint inventory positions. This article presents a necessary and sufficient condition for the irreducibility of such a Markov chain through a set of simultaneous Diophantine equations. This result also leads to sufficient conditions that are more general than those in the published reports.",
keywords = "assemble-to-order, diophantine equations, irreducibility, necessary and sufficient condition, order-based backorders, unit-skeleton condition",
author = "Jiejian FENG and Liming LIU and WAN, {Yat Wah}",
note = "This work is partially supported by grant PolyUG-YX82 and RGC grant PolyU6145/04E of Hong Kong, grant NSERC 371939-2009 of Saint Mary’s University, Canada, and grant NSC99-2410-H-259-046 of Taiwan.",
year = "2012",
month = "2",
doi = "10.1002/nav.20486",
language = "English",
volume = "59",
pages = "18--25",
journal = "Naval Research Logistics",
issn = "0894-069X",
publisher = "John Wiley and Sons Inc.",
number = "1",

}

Irreducibility of joint inventory positions in an assemble-to-order system under (r, nQ) policies. / FENG, Jiejian; LIU, Liming; WAN, Yat Wah.

In: Naval Research Logistics (NRL), Vol. 59, No. 1, 02.2012, p. 18-25.

Research output: Journal PublicationsJournal Article (refereed)

TY - JOUR

T1 - Irreducibility of joint inventory positions in an assemble-to-order system under (r, nQ) policies

AU - FENG, Jiejian

AU - LIU, Liming

AU - WAN, Yat Wah

N1 - This work is partially supported by grant PolyUG-YX82 and RGC grant PolyU6145/04E of Hong Kong, grant NSERC 371939-2009 of Saint Mary’s University, Canada, and grant NSC99-2410-H-259-046 of Taiwan.

PY - 2012/2

Y1 - 2012/2

N2 - In a typical assemble-to-order system, a customer order may request multiple items, and the order may not be filled if any of the requested items are out of stock. A key customer service measure when unfilled orders are backordered is the order-based backorder level. To evaluate this crucial performance measure, a fundamental question is whether the stationary joint inventory positions follow an independent and uniform distribution. In this context, this is equivalent to the irreducibility of the Markov chain formed by the joint inventory positions. This article presents a necessary and sufficient condition for the irreducibility of such a Markov chain through a set of simultaneous Diophantine equations. This result also leads to sufficient conditions that are more general than those in the published reports.

AB - In a typical assemble-to-order system, a customer order may request multiple items, and the order may not be filled if any of the requested items are out of stock. A key customer service measure when unfilled orders are backordered is the order-based backorder level. To evaluate this crucial performance measure, a fundamental question is whether the stationary joint inventory positions follow an independent and uniform distribution. In this context, this is equivalent to the irreducibility of the Markov chain formed by the joint inventory positions. This article presents a necessary and sufficient condition for the irreducibility of such a Markov chain through a set of simultaneous Diophantine equations. This result also leads to sufficient conditions that are more general than those in the published reports.

KW - assemble-to-order

KW - diophantine equations

KW - irreducibility

KW - necessary and sufficient condition

KW - order-based backorders

KW - unit-skeleton condition

UR - http://commons.ln.edu.hk/sw_master/2306

UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-84855642500&doi=10.1002%2fnav.20486&partnerID=40&md5=d0fc53dae3acf8081e5b5506965d1514

U2 - 10.1002/nav.20486

DO - 10.1002/nav.20486

M3 - Journal Article (refereed)

VL - 59

SP - 18

EP - 25

JO - Naval Research Logistics

JF - Naval Research Logistics

SN - 0894-069X

IS - 1

ER -