Dynamic joint pricing and production policy for perishable products

• Author: Wansheng Tang,Jianxiong Zhang,Lin Feng
• Date: 01 November 2018
• DOI: http://doi.org/10.1111/itor.12239
• Published date: 01 November 2018

Intl. Trans. in Op. Res. 25 (2018) 2031–2051

DOI: 10.1111/itor.12239

INTERNATIONAL

TRANSACTIONS

IN OPERATIONAL

RESEARCH

Dynamic joint pricing and production policy

for perishable products

Lin Fenga, Jianxiong Zhangband Wansheng Tangb

aSchool of Economics and Management, Southwest Jiaotong University,Chengdu 610031, China

bInstitute of Systems Engineering, Tianjin University, Tianjin 300072, China

E-mail: linfeng@swjtu.edu.cn [Feng]; jxzhang@tju.edu.cn [Zhang]; tang@tju.edu.cn [Tang]

Received 2 September 2014; receivedin revised form 18 August 2015; accepted 6 November 2015

Abstract

A joint dynamic pricing and production problem for perishable products without shortages is considered.

The demand rate is price-dependent and time-varying. This paper constructs an optimal control model to

maximize the total profit under a general nonlinear production cost function. The featureof the optimal joint

dynamic pricing and production policy is analyzed by solving the corresponding optimal control problem

on the basis of improved Pontryagin’s maximum principle. Then, an effective algorithm is designed to obtain

the optimal joint policy. The case of the joint static optimal policy is also investigated and compared with

the dynamic one. Finally, numerical examples are presented to illustrate the effectiveness of the proposed

methods, and some managerial implications are provided for the management of perishable items.

Keywords:perishable products; dynamic pricing; production; optimal control

1. Introduction

Advances in information technologies and the corresponding evolution of the Internet and

e-commerce enable managers to collect market data, learn about customer behavior, and change

pricing decisions dynamically much easier so as to better matchsupply with demand. Whether from

a financial or an operational standpoint, price is one of the most effective variables that managers

can manipulate to encourage or discourage demand in the short term (Bitran and Caldentey, 2003).

McKinsey’s study (Marn et al., 2003) indicated that a 1% improvement in pricing can lead to an

8% improvement in profits for a typical S&P 1500 company (a stock market index of the US stocks

made by Standard & Poor’s). In recent years, dynamic pricing strategies and their further devel-

opment have been adopted in several industries, such as airlines, fashion retailers, hotels, electric

utilities, sporting events, and health care (Elmaghraby and Keskinocak, 2003; Pang, 2011). Most

of these industries face the problem of selling a fixed stock of items over a finite selling horizon,

for example, airlines selling tickets before planes depart, hotels renting rooms before midnight, and

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2032 L. Feng et al. / Intl. Trans.in Op. Res. 25 (2018) 2031–2051

retailers selling seasonal or fashionable goods such as fashion apparel before the end of the season,

etc. Dynamic pricing practices are significantly useful to enhance sales and avoid the loss caused by

deterioration of these products (Caroand Gallien, 2012). Of course, the prices of products influence

not only customer choice, but also how the demand will be allocated among the products, thereby

driving the production decisions (Aydin and Porteus, 2008). Hence, a dynamic pricing strategy by

itself is often insufficient and production scheduling should be incorporated to manage sales. The

profitability of the manufacturing systems can be significantly improved by adjusting the sale price

and production schedule simultaneously.

The literature on joint pricing and replenishment policy has been substantially growing since

Whitin (1955) first studied the newsvendor problem with price-dependent demand. In the past

decades, dynamic and static strategies, continuous and discrete strategies, single and multiple prod-

uct strategies were proposed to maximize revenues. One can refer to Bitran and Caldentey (2003),

Elmaghraby and Keskinocak (2003), and Ramasesh (2010) for overviews of recent developments

and current practices. Pekelman (1974), Adida and Perakis (2007), and Jørgensen et al. (1999)

proposed a continuous time optimal control model to solve the joint pricing and production strat-

egy problem by applying Pontryagin’s maximum principle. Their models were developed under the

following assumptions: no backorders and strictly convex increasing production cost. In particular,

the third study considered demand-learning effects and the more recent paper considered the case

in which multiple products are optimized with a limited production capacity. The continuous time

approach they used has an advantage of providing an exact solution of the system without any

approximations. Bakal et al. (2008) studied a profit-maximization problem and analyzed simultane-

ous market selection, pricing, and order quantity decisions. The benefits of dynamic pricing on the

single product economic order decision was analyzed by Transchel and Minner (2009), they proved

that the overall benefit could be increased from a discrete number of price changes. Kabirian (2012)

used Newton’s method to find the economic production quantity and sale price,where several linear

and nonlinear functions of demand and variable cost of production were considered. Zhang et al.

(2012) and Mardaneh and Caccetta (2013) considered a discrete-time, finite-horizon coordinated

decision model to obtain pricing and the capacitated production policies with the assumption that

the unsatisfied demand was backlogged. They developed a solution procedure based on dynamic

programming methodology. Under supply chain environment, Arcelus et al. (2006) studied the

effects of the retailer’s risk preferences on the pricing and ordering policies. Guajardo et al. (2013)

presented both single-period and multi-period models for joint optimization of internal pricing,

production, inventory, and distribution decisions in a divergent supply chain. The optimal polices

in their model were essentially static.

Considering the perishability of products, many firms and researchers have paid considerable

attention to their decision-making process owing to the rapidly changing technology and fierce

competition. Chare and Schrader (1963) first developed a deteriorating inventory model to study

the inventory control problem. An excellent reviewof the literature for perishable products has been

reported by Goyal and Giri (2001).

Perishables have significant impact on inventory control and pricing decisions. One of the impor-

tant research efforts in this field is attributed to Abad (2001). He considered a pricing and lot-sizing

problem for a product with variable rate of deterioration, allowing shortages and partial backlog-

ging. This work was amended by Dye (2007), who added both the backorder cost and the cost

of lost sales into the total profit. Sana (2011) considered a quadratic demand function of price to

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2015 The Authors.

International Transactionsin Operational Research C

2015 International Federation of OperationalResearch Societies