# Optimal single ordering policy with multiple delivery modes and Bayesian information updates

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Computers & Operations Research 31 (2004) 19651984www.elsevier.com/locate/dsw

Optimal single ordering policy with multiple delivery modesand Bayesian information updatesTsan-Ming Choi, Duan Li, Houmin Yan

Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong,Shatin, N.T., Hong Kong

Abstract

We investigate in this paper a retailers optimal single ordering policy with multiple delivery modes. Dueto the existence of di3erent delivery modes, the unit delivery cost (and hence the product cost) is formulatedas a decreasing function of the lead-time. Market information can be collected in the earlier stages and usedto update the demand forecast by using a Bayesian approach. The trade-o3 between ordering earlier or lateris evident. The former enjoys a lower product cost but su3ers a less accurate demand forecast. The latterpays a higher product cost, but bene7ts from a lower uncertainty in the demand forecast. In this paper, amulti-stage dynamic optimization problem is formulated and the optimal ordering policy is derived usingdynamic programming. The characteristics of the ordering policy are investigated and the variance of pro7tassociated with the ordering decision is discussed. Numerical analyses through simulation experiments arecarried out to gain managerial insights. Implementation tips are also proposed.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Inventory management; Optimal stopping problem; Bayesian normal conjugate family; Dynamicprogramming; Computer implementation

1. Introduction

In supply chain inventory management, one of the challenges is how to manage demand uncer-tainty. As an example, consider a problem faced by a fashion apparel retailer: The fashion productsshelf-life is short, the ordering lead-time is long and market demand is highly volatile. In fact,demand uncertainty is the root of evil which accounts for the large supply chain inventory cost. Inorder to reduce demand uncertainty, companies of fashion clothing have proposed the well-knownquick response policy. Quick response basically refers to the concept of reducing the ordering

Corresponding author. Tel.: +852-31634062; fax: +852-26035505.E-mail addresses: tmchoi@se.cuhk.edu.hk (T.-M. Choi), dli@se.cuhk.edu.hk (D. Li), yan@se.cuhk.edu.hk (H. Yan).

0305-0548/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0305-0548(03)00157-6

1966 T.-M. Choi et al. / Computers & Operations Research 31 (2004) 19651984

lead-time and adjusting the stocking decision quickly in response to market changes. In fact, weknow that the closer to the selling season, the more information we have about the uncertain de-mand and hence a smaller demand uncertainty results. Quick response is now widely applied invarious industries all around the world and its impact is substantial. Reducing demand uncertaintyby shortening the lead-time is well-recognized as an important achievement of supply chainmanagement.In the literature, many articles focus on the issues of postponing the stocking decision and using

information updates for forecast improvement. Among them, Iyer and Bergen [1] provide an indepthanalysis of quick response policy and information updating in a manufacturer-retailer channel. They7nd that by postponing the ordering decision under quick response, the retailer is always better-o3.They also suggest methods under which the manufacturer also bene7ts from quick response. A con-sideration of the use of early sales signals from the market for reducing cost and demand uncertaintyunder a quick response type of ordering system is studied in Fisher and Raman [2]. They modeland analyse the ordering decisions under quick response for a single selling season and propose amethod for estimating the demand distribution required in their model. A fashion skiwear 7rm ischosen as the analysis case and the use of information is shown to be very signi7cant. In additionto the work by Fisher and Raman [2], with the consideration of using advance demand informa-tion, Gilbert and Ballou [3], Gallego and Ozer [4] and Tang et al. [5] have formulated inventorymodels under which the early demand signals or commitments are used to improve the forecast ofthe product during the normal selling season. The impacts brought by advance demand informationhave also been discussed. On the other hand, with the idea of including Lexibility into the retailersstocking decisions, Lau and Lau [6], Gurnani and Tang [7], Yan et al. [8] and Choi et al. [9] havestudied the ordering model with two ordering opportunities: After placing the 7rst order, the retailercan make market observation and improve the prior forecast. The retailer can then decide how muchto order at the second ordering chance with the updated and more accurate forecast.Some papers are devoted to modeling the inventory decisions with multiple supply or delivery

modes. For example, in Zhang [10], ordering policies with three supply modes are developed, witha heuristic method proposed to solve the problem. Other related papers for information updates andmultiple production/delivery modes include Murray and Silver [11], Azoury and Miller [12], Azoury[13], Hausmann et al. [14], Brown and Lee [15], Choi et al. [16], Barnes-Schuster et al. [17], Huanget al. [18], Donohue [19], Chiang [20], Chung and Flynn [21], Sethi et al. [22], and Lau and Lau[23].In this paper, we investigate an inventory stocking problem with multiple delivery modes and

information updates. A retailer is considering placing an order for a (newsvendor type of) seasonalproduct with an uncertain demand. There are di3erent delivery modes available for him to chooseand the faster delivery mode is also the more expensive one. If he chooses a faster delivery mode,he can delay his stocking decision more and obtain more market observation (e.g. by observing thesales of other related pre-seasonal product). To be precise, the retailer can use market observationto revise and update his current knowledge about the demand distribution in a Bayesian fashion.This results in a better forecast and a lower demand uncertainty. Thus, a tradeo3 between deliverycost and demand uncertainty exists. In this paper, we are interested in studying an optimal singleordering policy. The retailer can only place one order. This situation arises when: (1) The 7xedordering setup cost is high. For instance, if the retailer places one big order, he can make use of afull truckload for delivery; if he makes two orders, he needs to use two which can cost a lot if the

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trucks capacity is expensive. (2) A quantity discount scheme o3ered by the supplier is attractiveenough to entice the retailer to make one big order. (3) There is a minimum order quantity imposedby the supplier and the retailers demand is not large enough to support him to order more thanonce.We formulate the ordering policy as an optimal stopping problem (see Bertsekas [24]) and use

dynamic programming to derive the optimal policy. Notice that this paper basically extends theordering system in Iyer and Bergen [1] with the incorporation of multiple delivery modes, multipleBayesian information updates and delivery cost di3erence. This enriches the ordering policy andaddresses the possibility of having multiple information updates. The optimal ordering decision ina multi-stage setting is notorious for a growth of complexity with respect to the number of stages.The lack of analytical solutions also negatively a3ects the models applicability. Thus, we proposemethods which enhance the implementation of the optimal policy by a computer program. We alsostudy the impact of observation on the variance of pro7t for the stocking problem.The organization of the rest of this paper is as follows. In Section 2, we de7ne the basic mathemat-

ical model and the uncertainty structure of the problem. In Section 3, we determine the optimal orderquantity and derive the optimal ordering policy using dynamic programming. Section 4 is devoted tothe numerical analyses with a real case example. The variance of pro7t associated with the optimalpolicy is discussed in Section 5. We conclude in Section 6 with a discussion of implementation.

2. Mathematical model

We derive in this section the mathematical model for the optimal ordering problem. First, we de7nesome notation: We denote probability density function (pdf) and cumulative distribution function(cdf) of the standard normal by () and (), respectively. The inverse of () is represented by1(). We de7ne the standard normal linear loss function, (), as follows:

(a) = a

(x a) d(x): (2.1)Notice that () can be written as

(a) = (a) a[1 (a)]: (2.2)In this paper, we have a multi-stage model in which a retailer can choose one delivery mode amongmany for shipping the ordered products. The faster the delivery mode, the more expensive it is.This is intuitive in real-life. For instance, using air-transport is much faster and requires a muchshorter lead-time than ship-cargo, but it is also much more expensive. Now, suppose there are N +1di3erent delivery modes available for the retailer to choose (N is any non-negative integer). Weorder the delivery modes in a decreasing sequence of their respective shipping times and the fastestdelivery mode is named Stage N and the slowest delivery mode is called Stage 0. As a result, fork = 0; 1; : : : ; N : Stage ks delivery time is longer than the delivery time of Stage k + 1. (Notice thatwe assume in this paper that the delivery lead-time of each delivery mode be deterministic and theorder placed with a particular delivery mode will arrive on time before the selling season starts. Inreal-life, however, a probable variation of delivery lead time does exist. As a result, the retailer shouldmake sure that the delivery mode is reliable by, for example, signing a supply warranty contractto ensure that the product will be delivered on time (or at least within a small time interval).

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This is essentially important because failing to guarantee the reliability of the delivery time, themodeling for the delivery modes becomes inaccurate, which then directly a3ects the optimality of thesolution.) When the retailer chooses to order later (e.g. at Stage 1 instead of Stage 0), he can makeobservations in the market and use this information to revise his forecast estimate. To be speci7c, fork =0; 1; : : : ; N 1, market information gathered by the retailer at Stage k can be used to update thedemand forecast at Stage k+1 by using a Bayesian approach. The observations can be visualized asthe market signals obtained from the sales of other related pre-seasonal product/product attributein the market. For example, let us consider the seasonal product as a yellow jacket which is goingto be sold in the up-coming selling season. Before the selling season of this yellow jacket starts,observations can be made from another clothing item which is currently selling in the market (e.g. ayellow t-shirt) and the sales of this related pre-seasonal product can help to improve the demandforecast for the yellow jacket. In fact, the color yellow is an important product attribute here whichis observed in the observation process. This type of example is well-studied in the literature andmore discussions can be found in Iyer and Bergen [1]. We denote the predicted demand of theseasonal product at Stage k by xk , where k = 0; 1; : : : ; N . Following the basic demand uncertaintystructure as that of Iyer and Bergen [1], we take the distribution of xk to be a normal distributionas follows:

xk N (k ; ); (2.3)where k is the mean of xk and is also normally distributed with a mean of k and a variance ofdk ,

k N (k; dk): (2.4)The variance in (2.3) is called the seasonal products inherent demand uncertainty. It reLects theuncertainty of xk for given k and also captures the uncertainty in using the sales of the related (butnot the same) pre-seasonal products in the market to estimate the seasonal products demand (forthe details, please refer to Iyer and Bergen [1, p. 561]).At Stage k + 1, we have an observation between Stage k and Stage k + 1 and we call it xk . By

Bayes Theorem, with xk , we can update the demand distribution with the following parameterschanges:

k+1 = [dk=(dk + )]xk + [=(dk + )]k; (2.5)

dk+1 = dk=(dk + ): (2.6)

Before obtaining the observation xk , the unconditional distribution for the predicted demand at Stagek + 1; xk+1, is

xk+1 N (k+1; dk+1 + ): (2.7)Using (2.5) and (2.6), we have the unconditional distribution of k+1,

k+1 N (k; 2k); (2.8)where 2k = d

2k =(dk + ).

T.-M. Choi et al. / Computers & Operations Research 31 (2004) 19651984 1969

As a remark, from (2.3) to (2.8), we assume that the uncertain demand is normally distributed.Since demand is non-negative, in order to have a valid model, the coeQcient of variation of xk ,de7ned by

dk + =k , should be suitably small (for example, less than 0.4). 1

Next, we discuss the cost-revenue structure for the seasonal product. We take the seasonal productas a newsvendor type of product. The retailer buys the product with a 7xed unit purchase cost calledpc. The holding cost and salvage price of each unit of product left over at the end of the sellingseason are hc and v, respectively. We represent hc v by h (the unit cost of the product left over atthe end of the selling season). Observe that h can be positive or negative depending on the valuesof hc and v. The revenue (i.e. selling price) per product sold is called r. We denote the unit deliverycost for using the delivery mode at Stage k as lk . In our multi-stage model, we have the followingstructure for the unit products delivery cost:

l06 l1 6 lN ; (2.9)Eq. (2.9) is a sound assumption because the faster delivery mode should be more expensive (orat least not cheaper) and the larger stage number implies a faster delivery mode in our modelformulation. To simplify the notation, we de7ne the unit ordering cost at Stage k as the summationof unit purchase cost and the unit delivery cost at Stage k:

ck = pc + lk for k = 0; 1; : : : ; N: (2.10)

Obviously, from (2.9) and (2.10), we have the following:

c06 c16 c2 6 cN16 cN : (2.11)To avoid trivial cases, we have r ck , for k = 0; 1; : : : ; N . Furthermore, we de7ne the service level(also called the 7ll-rate), which describes the probability of having no stockout during the sellingseason (see Nahmias [25, Chapter 5]), when the retailer orders at Stage k, by

sk = (r ck)=(r + h) for k = 0; 1; : : : ; N: (2.12)Fig. 1 shows the basic model structure of the problem. It also shows the trade-o3 between orderingcost and forecast accuracy.

Fig. 1. The basic model structure of the single ordering problem.

1 Thanks are given to an anonymous reviewer for reminding us of this important model validity condition.

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