Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 789306, 8 pages

http://dx.doi.org/10.1155/2015/789306

## Dynamic Pricing and Production Control of an Inventory System with Remanufacturing

^{1}Department of Management Science, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, China^{2}School of Business Administration, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, China

Received 14 September 2015; Revised 14 November 2015; Accepted 24 November 2015

Academic Editor: Young Hae Lee

Copyright © 2015 Chunyan Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider optimal pricing and manufacturing control of a continuous-review inventory system with remanufacturing. Customer demand and product return follow independent Poisson processes. Customer demand is filled by serviceable product, which can be either manufactured or remanufactured from the returned product. The lead times for both manufacturing and remanufacturing are exponentially distributed. The objective is to maximize the expected total discounted profit over an infinite planning horizon. We characterize the structural properties of the optimal policy through the optimality equation. Specifically, the optimal manufacturing policy is a base-stock policy with the base-stock level nonincreasing in the return inventory level. The optimal pricing policy is also a threshold policy, where the threshold level is nonincreasing in the return inventory level.

#### 1. Introduction

With increasing awareness of sustainable development, and more stringent government regulations, companies are gradually switching focus from profits to triple bottom lines (profit, people, and planet) in their business operations. Remanufacturing, which converts used products into like-new ones, has gained momentum recently as it not only reduces waste and consumption of natural resources but also produces companies economic benefit by reducing production costs and increasing service levels. Meanwhile, governments in Europe and North America have established legislation to urge manufacturers to accept used products and carry out remanufacturing. Examples of remanufacturing programs include those of Caterpillar, IBM, Kodak, and Fuji-Xerox.

One operational challenge in remanufacturing is production planning and pricing management because of uncertain returned product numbers and customer demand volatility. In this paper, we consider optimal pricing and production control of a continuous-review, infinite-horizon inventory system with remanufacturing. Serviceable products that fill customer demand can be either manufactured or remanufactured from the returned products (also called “core”). The system keeps both serviceable product inventory and core inventory. Demand that cannot be satisfied by available serviceable inventory is backlogged. Lead times for both manufacturing and remanufacturing are exponentially distributed. The objective is to maximize the expected total discounted profit over an infinite planning horizon. We characterize the optimal policy through the optimality equation. Specifically, the optimal manufacturing policy follows a base-stock policy with the base-stock level nonincreasing in the core inventory level. The optimal pricing policy is also a threshold policy with threshold level nonincreasing in the return inventory level.

Growth of the remanufacturing industry has stimulated extensive research in the field. Ferrer and Whybark [1], Guide [2], and Larson et al. [3] show that remanufacturing is environmentally friendly and economical. McConocha and Speh [4] note that firms benefit from remanufacturing through reduced production cost, reduced pollution, shortened production lead time, new market, and improved reputation. Our paper is closely related to prior studies on production planning and inventory control of remanufacturing systems. The related work can be divided into two streams: the periodic-review systems and the continuous-review systems. Regarding periodic-review systems, Simpson [5] considers a system with zero remanufacturing and manufacturing lead times. The optimal control policy is shown to be characterized by constant thresholds. Inderfurth [6] subsequently extends Simpson’s [5] work to models with positive lead times and shows that with equal and deterministic remanufacturing and manufacturing processing times, the optimal policy is the same as the zero lead time model. Zhou et al. [7] study a similar remanufacturing system but with multiple types of remanufacturable products. They derive that the optimal remanufacturing-manufacturing-disposal policy is characterized by a sequence of constant parameters. Tao et al. [8] further consider random yield in remanufacturing inventory systems. DeCroix and Zipkin [9] and DeCroix [10] study multiechelon inventory systems with returns. Our paper differs from such works in that we consider a continuous-review system rather than a periodic-review system. Furthermore, the literature on the periodic-review systems almost assumed that the lead times of the remanufacturing process and the manufacturing process are equal and deterministic. But our paper assumes that the remanufacturing process and the manufacturing process are stochastic and could not necessarily be equal.

Papers on continuous-review models mainly focus on performance evaluation. Heyman [11] and Muckstadt and Isacc [12] formulate the remanufacturing systems as queueing systems. Using the analysis of quasi-birth-and-death process, they obtain the stationary distributions of the systems. Van der Laan and Salomon [13] and Van der Laan et al. [14] consider the push and pull systems. DeCroix et al. [15] study an assemble-to-order system with return; they provide an algorithm to calculate the near-optimal base-stock level. Our paper differs from the previous literature in that we focus on exploring the structural properties of the optimal control policy rather than the performance evaluation. Kim et al. [16] consider a remanufacturing system with all the processes exponentially distributed. The final product is sold at a fixed price. They obtain the structural properties of the optimal policy, including optimal manufacturing, remanufacturing, and disposal policies. Our paper differs from their work in that we assume that the price of the final product is controllable. It could be sold at either high or low price based on the serviceable product inventory. We focus on exploring the optimal pricing and manufacturing strategy.

The remainder of this paper is organized as follows. We specify the model details in Section 2. In Section 3, we examine the structural properties of the value function and characterize the optimal control policies, concluding with a discussion on possible extensions in Section 4.

#### 2. The Model

Consider a firm managing a hybrid inventory system with remanufacturing. The serviceable product that fills customer demand is either manufactured from new parts or remanufactured from returned products (also called core). We assume that customers feel indifferent whether a serviceable product is manufactured or remanufactured. This happens in remanufacturing of refillable containers, such as printer cartridges and single-use cameras. Return arrives randomly and follows a Poisson process with rate . We assume that return process is independent of the demand arriving process. This assumption is widely used in the previous literature; see Kim et al. [16] and references therein. The returns enter the system and are stocked at the core inventory waiting for remanufacturing. A cost is incurred for holding one unit of core per unit time. The remanufacturing and manufacturing lead times are exponentially distributed with rates and , respectively. Let and denote the unit remanufacturing and manufacturing cost with , which means that the remanufacturing is more cost-efficient. A completed product from remanufacturing or manufacturing process is stocked at the serviceable inventory and incurs a holding cost per unit per unit time. The demand arrives according to a nonhomogeneous Poisson process with price-dependent arrival rate at that time. There are two sale prices: high price and low price , with corresponding arrival rates and . To make low price economically reasonable, we assume that and Each customer requires only one unit of the serviceable product. The demand that cannot be satisfied immediately from the on-hand serviceable inventory is backlogged and incurs a backorder cost per unit per unit time. Let which is defined as the marginal profit by changing the price from high to low. Furthermore, we require that , which means that the marginal loss through raising price is smaller than the net value of backlogging an order, and thus it has incentive to rase the price.

The firm controls both the manufacturing process and price with an aim to minimize the expected total discounted cost over an infinite planning horizon. Specifically, the manufacturing control characterizes whether to manufacture; the price decision controls the sale price of the serviceable product by observing the current inventory level. The model is illustrated in Figure 1.