Contracting for vendor‐managed inventory with a time‐dependent stockout penalty

• DOI: http://doi.org/10.1111/itor.12488
• Published date: 01 May 2020
• Author: Gordon Johnson,Jun‐Yeon Lee
• Date: 01 May 2020

Intl. Trans. in Op. Res. 27 (2020) 1573–1599

DOI: 10.1111/itor.12488

INTERNATIONAL

TRANSACTIONS

IN OPERATIONAL

RESEARCH

Contracting for vendor-managed inventory

with a time-dependent stockout penalty

Jun-Yeon Lee and Gordon Johnson

Systems and Operations Management, California State University,Northridge, CA, USA

E-mail: junyeon.lee@csun.edu [Lee]; gordon.johnson@csun.edu[Johnson]

Received 24 October 2016; receivedin revised form 8 August 2017; accepted 29 October 2017

Abstract

We examine vendor-managed inventory contracts in a (Q,r) inventory system between a supplier and a

retailer, in which a stockout penalty is charged to the supplier based on the length of the time period during

which stockouts occur at the retailer. Linear and quadratic forms of the time-dependent stockout penalty are

considered. For the deterministic demand case, we find that the quadratic form of time-dependent stockout

penalty is equivalent to a proportional stockout penalty per unit short per unit time. For the stochastic

demand case, we provide the exact cost expressions for the supplier and the retailer with a linear time-

dependent stockout penalty. We also discuss how the stochastic model can be extended to the case with a

quadratic time-dependent stockoutpenalty when there is at most one outstanding replenishment order atany

point of time. We provide several interesting computational results.

Keywords:vendor-managed inventory; contracting; time-dependent stockout penalty; (Q,r) inventory system

1. Introduction

Vendor-managed inventory (VMI) is a supply chain practice between a supplier and a retailer

(or customer) in which the supplier manages the inventory at the retailer’s premises and makes

replenishment decisions. Since VMI was successfully implemented in the 1980s between Wal-Mart

and Proctor & Gamble (Buzzell and Ortmeyer, 1995; Waller et al., 1999), it has been adopted

by many supply chains such as Dell, Campbell Soup, Barilla SpA, and Intel (Bookbinder et al.,

2010; Katariya et al., 2014). VMI has also been extensively studied in academia over a short

period of time (for literature reviews on VMI, see Govindan 2013; Lee et al. 2015; Marques et al.

2010). Academic researches on VMI mainly focus on (i) investigating the benefits of VMI, (ii)

operational decisions in VMI programs, and (iii) designing contracts for VMI programs (Guan and

Zhao, 2010).

In particular,to our knowledge, Fry et al. (2001) was the first theoretical paper thatexamines a (z,

Z) VMI contract, which specifies maximum and minimum inventory levels and their corresponding

C

2017 The Authors.

International Transactionsin Operational Research C

2017 International Federation ofOperational Research Societies

Published by John Wiley & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main St, Malden, MA02148,

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1574 J.-Y. Lee and G. Johnson / Intl. Trans. in Op. Res. 27 (2020) 1573–1599

Tabl e 1

Analytical papers on VMI contract

Paper VMI contract

Fry et al. (2001), Shah and Goh (2006) Maximum and minimum inventory levels, over- and

under-stocking penalties per unit per period

Darwish and Odah (2010), Darwish and Goyal (2011),

Hariga et al. (2013, 2014), and Chakraborty et al.

(2015)

Maximum inventory level,over-stocking penalty per unit

per unit time

Bichescu and Fry (2009) Stockout penalty per unit per unit time

Lee and Ren (2011), Lee et al. (2016) Stockout penalty per unit per unit time, consignment

stock

Lee and Cho (2014) Stockout penalty per unit per unit time, consignment

stock, fixed stockout penalty per stockoutoccasion

Cai et al. (2016, 2017) Option contract

Our paper Stockout penalty per unit time, consignment stock

over- and under-stocking penalties per unit per period. Since then, many papers have examined

(z, Z)-type VMI contracts. For example, Shah and Goh (2006) examine the supplier’s problem

under a (z, Z) VMI contract in a deterministic setting. Darwish and Odah (2010), Darwish and

Goyal (2011), Hariga et al. (2013, 2014), and Chakraborty et al. (2015) consider VMI contracts

that specify a maximum inventory level and an over-stocking penalty per unit per unit time. Cai

et al. (2016, 2017) examine an option contract for VMI supply chains, where the retailer pays the

supplier an option price ofor all units delivered and an exercise price efor units sold. Note that

consignment stock may be viewed as a special case of option contract where o=0andeis the

wholesale price. On the other hand,several papers have examined VMI contracts with stockout-cost

sharing, which can be viewed as a special case of (z, Z) VMI contract with z =0andZ =∞.

Bichescu and Fry (2009) examine a VMI model where stockout costs are split between the retailer

and the supplier. Lee and Ren (2011) consider a VMI model where the retailer charges the supplier

a stockout penalty that is equal to her backorder cost. Lee and Cho (2014) examine a VMI contract

where the supplier is charged fixed and proportional penalties when stockouts occur at the retailer.

Lee et al. (2016) examine supply chain coordination in VMI systems with stockout-cost sharing

under limited storage capacity. See Table 1 for the summary of these analytical papers on VMI

contract.

Although the (z, Z) VMI contract is widely used in academia and practice (Fry et al., 2001), it

requires that the inventorylevel at the retailer be observable and verifiable. However, while on-hand

inventory is relatively easy to observe, stockout quantity might be hard to observe. Hence, one

may argue that the retailer might charge the supplier according to an unreal stockout quantity

(Lee and Cho, 2014). To deal with this potential drawback of the (z, Z) VMI contract, Lee and

Cho (2014) propose a VMI contract where the supplier is charged a fixed penalty as well as a

proportional penalty per unit short per unit time when stockoutsoccur at the retailer. They compare

the optimal contract (with both types of penalties) with the optimal fixed-stockout-penalty-only

contract to assess the value of information sharing on the retailer’s stockout quantity in VMI

contracting.

C

2017 The Authors.

International Transactionsin Operational Research C

2017 International Federation ofOperational Research Societies