Modified critical fractile approach for a class of partial postponement problems

$Page 1: Modified critical fractile approach for a class of partial postponement problems$
Int. J. Production Economics 136 (2012) 185–193

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Int. J. Production Economics

0925-52

doi:10.1

n Corr

E-m

bizteocp

journal homepage: www.elsevier.com/locate/ijpe

Modified critical fractile approach for a class of partialpostponement problems

Qi Fu a,n, Chung-Yee Lee a, Chung-Piaw Teo b

a Department of Industrial Engineering and Logistics Management, HKUST, Hong Kongb Department of Decision Sciences, NUS Business School, Singapore

a r t i c l e i n f o

Article history:

Received 30 July 2010

Accepted 10 October 2011Available online 12 November 2011

Keywords:

Partial postponement

Delayed customization

Critical fractile

Newsvendor problem

73/$ - see front matter & 2011 Elsevier B.V. A

016/j.ijpe.2011.11.003

esponding author.

ail addresses: [email protected] (Q. Fu), [email protected]

@nus.edu.sg (C.-P. Teo).

a b s t r a c t

This paper studies a single period partial postponement problem, which is motivated by an inventory

planning problem encountered in Reebok NFL replica jersey supply chain. We consider a group of

regular products each facing a random demand. There are two stocking options. One is to procure the

regular products. The other is to stock a common component which can be customized later to any one

of the regular products, after demand realization.

We obtain an insightful interpretation of the optimality conditions for this class of problems, and

use it to obtain rules of thumb that practitioners can incorporate into their inventory models to

determine the stocking levels to minimize the supply chain cost. Instead of proposing a numerical

procedure to obtain the optimal solution, we propose an adaptation of the classical critical fractile

approach for this class of partial postponement problem. The closed-form formula obtained are

surprisingly effective. Our numerical results suggest that this simple approach to inventory planning

often comes close to the performance of the optimal solution obtained from numerical method.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

Postponement is a product design, manufacturing, and deliv-ery strategy that retains the product in a neutral and non-committed status, and delays the point of consumer specificationas much as possible along the supply chain. There are manyadvantages of such strategy: companies can (i) reduce the amountof inventory investment due to the risk pooling effect of holdingcustomizable components and (ii) respond quickly to demandshifts among the family of products, leading to less stock outsituations. Moreover, (iii) the salvage value of non-customizedcomponent can be higher than the customized products.

Postponement is best suited for situations when the productsshare a common platform, and the demands are highly affectedby consumers’ perception and thus have large forecast variability.Many companies in a wide variety of industries have adopted andsuccessfully implemented various postponement strategies intheir manufacturing, assembly, packaging and delivery processes.Practical application of the concept can be traced back to the1920s and the first detailed empirical descriptions appeared inthe 1960s (Pagh and Cooper, 1998). The strategy is now used by

ll rights reserved.

k (C.-Y. Lee),

companies such as Burger King, General Motors, Honda, Embraer,Dade Behring, Reebok, McGraw-Hill, Imation, etc. Readers mayrefer to Rietze (1970) for a thorough review of postponement casestudies. In the academic literature, the concept was originallyproposed by Alderson (1950) and later expanded by Bucklin(1965). Afterwards, there has been a stream of literature addres-sing various types of postponement problems, e.g., Lee and Tang(1997), Li et al. (2007), Anupindi and Jiang (2008), Granot and Yin(2008), etc. Readers may refer to Aviv and Federgruen (1999),Cheng et al. (2010) and Swaminathan and Lee (2003) for anoverview of concepts and models of postponement problems.

A partial postponement problem arises in the situation that afamily of similar products share a common platform and differonly in some minor features (for example, color of apparels,taste of foods, flavor of drinks, etc.). By postponing the point ofdifferentiation, companies can maintain flexibility to cope withdemand uncertainties. However, this is usually accompaniedwith a higher production costs, due to the additional operations.Therefore, companies should trade-off between the benefit andcost of postponement. This makes a partial postponementstrategy optimal, where companies need to determine the bestcombination of finished-goods inventory and non-customizedproduct inventory. Parsons (2004) provided a thorough descrip-tion of the partial postponement strategy used in Reebok’s NFLreplica jersey supply chain: Blank jerseys were stored and laterprinted with appropriate design depending on the popularity of

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raw

materials

contract

manufacturers

retailersReebok DC

blank

jerseys

blank

jerseys

screen

printing

Q. Fu et al. / Int. J. Production Economics 136 (2012) 185–193186

the players in the season. Silver and Minner (2005) studieda similar inventory replenishment problem for a pizza store:demand for a particular type of pizza is first met with the finishedpizzas (end products), and then upon stock out of the finishedpizzas, met by appropriately adding toppings to the commonunfinished shells (customizable product). Note that under apartial postponement strategy, part of the stock is customizedonly after the realization of demands for the various end products.This problem can be viewed as an inventory substitution problem.Bassok et al. (1999) presented structural results for such problemwhen the products can be ordered and higher ranked productscan be used as substitute for lower ranked products. Tibben-Lembke and Bassok (2005) used these results to develop severalnew insights to this class of problems—when one generic com-ponent can be customized to substitute for any one of the m

regular products. They derived necessary and sufficient condi-tions for (i) not keeping inventory of the generic component and(ii) keeping only the generic component. Furthermore, based onthe optimality conditions, they developed an efficient routineto obtain the optimal procurement quantities for the genericcomponent and the m regular products. Graman (2010) investi-gated a partial postponement problem with two products anddecisions need to be made on both the finished-goods inventoryand postponement capacity, and examined the sensitivity of themodel.

In this paper, we investigate this type of partial postponementproblem for the case when the cost structure of the m regularproducts are identical, although the demand distribution could differ.This situation holds, for instance, in the Reebok supply chain, wherethe m regular products refer to the jerseys of m different players. Thisproblem is also similar to the inventory replenishment problem in asingle period one-warehouse-multi-retailer system, where we needto determine jointly the inventory levels at the warehouse and atthe retailers, assuming that the inventory at the warehouse can beused to replenish the demand of the retailers at an added cost(but identical for all retailers). The added cost component models theemergency shipping cost from warehouse to retailer, or the back-logging cost incurred while shipping goods from warehouse to theretailers. In this case, the inventory at the warehouse plays the role ofthe blank jersey in the Reebok’s supply chain. In a different vein, byre-interpreting the terms in the profit function, our problem alsomodels the portfolio option contract procurement problem for multi-ple products, where the problem is to determine the amount ofoption contract to purchase, given the different parameters forreservation and execution costs.

We show that the optimality conditions for the partial postpone-ment problem simplify dramatically in this case, into something thatresembles the critical fractile formula for the simple newsvendormodel. This gives rise to a closed-form solution to the case with asingle regular product (i.e., single warehouse single retailer problem,or in the case of single option contract procurement problem). Weexploit the new optimality conditions to develop bootstrap heuristicfor retailers and warehouse to obtain near optimal procurementquantities.

The rest of this paper is organized as follows. In Section 2, weprovide a detailed description of the problem studied in thispaper. Section 3 discusses the solution approach and the mainresults obtained. Section 4 presents the numerical results, usingthe Reebok NFL replica jersey case as the background for theexperiments.

dressed

jerseys

dressed

jerseys

2 -16

weeks

8

weeks

1

weeks

Fig. 1. Reebok supply chain.

2. Model specification

Consider the following partial postponement problem: afamily of regular products can be acquired either in the form of

finished goods from the manufacturer, or in the form of acommon component, which can be customized later to one ofthe regular products. This is a problem encountered, for instance,in the Reebok’s supply chain as shown in Fig. 1: the contractmanufacturers cut, sew and assemble the jerseys. Some jerseysare printed with player name and number (dressed jerseys), whileother jerseys are left blank, according to the orders from Reebok.Then both the blank jerseys (common component) and thedressed jerseys (regular products) are shipped to the distributioncenter (DC) of Reebok located in Indianapolis. The distributioncenter owns on-site screen printing facility, which can burnblank jerseys to meet customer orders on time after demand isobserved.

Our objective is to maximize the expected system profit whichconsists of three components, the expected revenue from sellingthe products, the cost of procurement and production, and thesalvage value of leftover inventory. The sequence of events canbe described as follows. The company faces random demandsDi,i¼ 1, . . . ,m for the m regular products. The company needs todetermine the procurement quantities Qi,i¼ 1, . . . ,m for thefinished products at a unit cost of c01, and the procurementquantity Qo for the common component at a unit cost of c02. Theselling price of each unit of the regular products is p. If the stockof one or more finished products is depleted, additional demandsare satisfied by customizing the common component based ona first-come-first-serve rule. The unit customization cost is r2.Keeping the common component in stock postpones the point ofcustomization and therefore is more flexible in meeting customerdemand but it also comes with a higher unit cost, so we assumec02oc01oc02þr2op. At the end of the season, leftover inventory ofregular products can be salvaged at a discount price of s1, andcommon component can be salvaged at a discount price of s2. It isreasonable to assume c024s2Zs1, as the uncustomized compo-nent normally has a higher salvage value than the finishedproduct.

We formulate the partial postponement problem as follows:

maxQi ,i ¼ 1,...,m,Qo

PðQ1, . . . ,Qm,QoÞ,

where

PðQ1, . . . ,Qm,QoÞ ¼ pXmi ¼ 1

E½minðQi,DiÞ�

(

þðp�r2ÞE min Qo,Xm

i ¼ 1

ðDi�QiÞþ

" # !�c02Qo

�c01Xm

i ¼ 1

Qiþs1

Xm

i ¼ 1

E½ðQi�DiÞþ�þs2E Qo�

Xmi ¼ 1

ðDi�QiÞþ

" #þ):

k ¼ 1

Q. Fu et al. / Int. J. Production Economics 136 (2012) 185–193 187

This is clearly a convex optimization problem (cf. Tibben-Lembkeand Bassok, 2005). Note that without the common component(i.e., Qo¼0), the above problem reduces to the classical newsvendorproblem, with Qi chosen so that

PðDi4QiÞ ¼c01�s1

p�s1:

Let Q ¼ ðQ1, . . . ,Qm,QoÞ denote an arbitrary procurement deci-sion. Since E½ðb�xÞþ � ¼ b�E½minðb,xÞ�, where b is a constant and x

is a random variable, by rearranging terms the above profitfunction can be rewritten as follows:

PðQ Þ ¼ pXm

i ¼ 1

EðDiÞ�pXm

i ¼ 1

ðDi�QiÞþ�Qo

" #þ

�ðc01�s1ÞXm

i ¼ 1

Qi�ðc02�s2ÞQo

�s1

Xm

i ¼ 1

E½minðQi,DiÞ��ðr2þs2ÞE min Qo,Xm

i ¼ 1

ðDi�QiÞþ

" # !:

In the above expression pPm

i ¼ 1 EðDiÞ is a constant independent ofthe ordering decisions. Let

CðQ Þ ¼ ðc01�s1ÞXmi ¼ 1

Qiþðc02�s2ÞQoþp

Xm

i ¼ 1

ðDi�QiÞþ�Qo

" #þ

þs1

Xm

i ¼ 1

E½minðQi,DiÞ�þðr2þs2ÞE min Qo,Xmi ¼ 1

ðDi�QiÞþ

" # !:

ð1Þ

The expression of CðQ Þ can be interpreted as the procurementcost of using mþ1 option contracts with: reservation costc1 ¼ c01�s1, and execution cost h1 ¼ s1 for each unit of the endproducts i¼ 1, . . . ,m; reservation cost c2 ¼ c02�s2 and executioncost h2 :¼ r2þs2 for each unit of the common component; and anyresidual demand is met by a spot market with cost p per unit. It iseasy to see that c14c2 and c1þh1oc2þh2op. That is, the optioncontract for the final products ðc1,h1Þ is cheaper than the optioncontract for the common component ðc2,h2Þ, but is less flexibledue to the higher reservation cost. The spot market has thehighest cost, but no reservation is required and therefore is mostflexible.

The decisions Qi’s and Qo are now merely the number of optioncontracts to purchase. Demand will be met by first exercising theoptions, and then buying from the spot market. The first twoterms in the expression (1) are the option reservation costs, andthe rest terms are the execution and spot purchase costs, based ondemand realization. Thus the two problems can be linked by thefollowing equation:

max PðQ Þ ¼ pXm

i ¼ 1

EðDiÞ�min CðQ Þ:

That is, maximizing the expected profit of a partial postponementproblem is equivalent to minimizing the expected cost of anoption procurement problem under the above transformation.Portfolio option procurement problem has been studied in theliterature normally for meeting a single random demand withmultiple supply sources (option contracts). Reader may refer toFu et al. (2010) for a detailed analysis of the portfolio optionprocurement problem. Our problem differs in that there aremultiple random demand classes sharing the same commoncomponent, which makes the problem rather complicated. Inthe following we will provide solution approaches based on theequivalent option procurement problem instead of the originalpartial postponement problem.

3. Solution approach

We consider the case when the demands for the m types ofproducts are independent of each other, as the exposition issimpler. The same approach can be extended to the general case,but the optimality conditions derived will not be as insightful.

Theorem 1. For a partial postponement problem with multiple

independent demand sources stated above, the optimality condition

on the optimal procurement quantity of the common component Qo is

PXm

i ¼ 1

ðDi�QiÞþ4Qo

!¼

c2

p�h2ð2Þ

and the optimality condition on the optimal procurement quantity for

the customized products Qi,i¼ 1, . . . ,m, is

PðDi4QiÞ ¼c1�c2þðp�h2ÞPð

Pka iðDk�QkÞ

þ4QoÞ

h2�h1þðp�h2ÞPðP

ka iðDk�QkÞþ4QoÞ

: ð3Þ

Proof. If the optimal solution Qo does not satisfy the optimalitycondition (2), say ‘‘o ’’ instead of ‘‘¼ ’’ holds, then we can perturbthe optimal solution, changing Qo by E ðE40Þ. The objective valuewill change by the following amount:

�
Reservation cost will change by c2E.P � In the event m
i ¼ 1ðDi�QiÞþoQo, the change does not affect

the execution cost and hence has no impact on the total cost.P
� When m
i ¼ 1ðDi�QiÞþ4Qo, since we have E more units of the

common component. The execution cost will change byEðh2�pÞ:

Then change of expected total cost can be expressed as

E c2�ðp�h2ÞPXmi ¼ 1

ðDi�QiÞþ4Qo

!( )o0:

Thus, the objective value can be reduced, contradicting theoptimality of the solution. On the other hand, if ‘‘4 ’’ instead of‘‘¼ ’’ holds, perturbing Qo by �E, we will arrive at the samecontradiction. Therefore Eq. (2) must hold at the optimality.

If the optimal solution Qi does not satisfy the optimality

condition (3), for example ‘‘o ’’ instead of ‘‘¼ ’’ holds, then we

can perturb the optimal solution, changing Qi by E ðE40Þ. The

objective value will change by the following amount:

�
Reservation cost will change by c1E. � In the event DirQi, no impact on the execution cost. � When Di4Qi, savings on execution cost can be realized since
we have E more units of product i. The cost will change by

Eðh1�h2Þ ifXm

k ¼ 1

ðDk�QkÞþoQo,

and

Eðh1�pÞ ifXm

k ¼ 1

ðDk�QkÞþ4Qo:

The change of expected total cost can be expressed asðnote p�h1 ¼ ðp�h2Þþðh2�h1ÞÞ

E c1�ðh2�h1ÞP Di4Qi,Xm

k ¼ 1

ðDk�QkÞþoQo

!"

�ðp�h1ÞP Di4Qi,Xmk ¼ 1

ðDk�QkÞþ4Qo

!#

¼ E c1�ðh2�h1ÞPðDi4QiÞ�ðp�h2ÞP Di4Qi,XmðDk�QkÞ

þ4Qo

!" #:


Since

P DioQi,Xm

k ¼ 1

ðDk�QkÞþ4Qo

!

¼ PXmk ¼ 1

ðDk�QkÞþ4Qo9DioQi

!PðDioQiÞ

¼ PXka i

ðDk�QkÞþ4Qo

!PðDioQiÞ,

where the second equality comes from the independency of thedemands, we have

P Di4Qi,Xm

k ¼ 1

ðDk�QkÞþ4Qo

!

¼ PXmk ¼ 1

ðDk�QkÞþ4Qo

!�P

Xka i

ðDk�QkÞþ4Qo

!PðDioQiÞ:

Substituting the above equation into the change of expected totalcost, the change of the expected cost can be expressed as

E c1�ðh2�h1ÞPðDi4QiÞ�ðp�h2ÞPXm

k ¼ 1

ðDk�QkÞþ4Qo

!"

þðp�h2ÞPXka i

ðDk�QkÞþ4Qo

!PðDioQiÞ

#:

By Eq. (2) the above expression can be rewritten as:

E"

c1�ðh2�h1ÞPðDi4QiÞ�c2

þðp�h2ÞPXka i

ðDk�QkÞþ4Qo

!PðDioQiÞ

#

¼ E c1�ðh2�h1ÞPðDi4QiÞ�c2

�þðp�h2ÞP

Xka i

ðDk�QkÞþ4Qo

!

�ðp�h2ÞPXka i

ðDk�QkÞþ4Qo

!PðDiZQiÞ

#

¼ E c1�c2þðp�h2ÞPXka i

ðDk�QkÞþ4Qo

!(

� ðh2�h1Þþðp�h2ÞPXka i

ðDk�QkÞþ4Qo

!" #PðDi4QiÞ

)o0:

Thus, the objective value can be reduced, contradicting theoptimality of the solution. The case of ‘‘4 ’’ can be analyzedsimilarly by perturbing Qi by �E. &

The left-hand sides of the optimality conditions (2) and (3) inTheorem 1 correspond to the optimal stock-out probabilities forthe common component and the customized products, respec-tively. However, the demand terms in the equations cannot bedecoupled, so the problem needs to be solved by iterativelysearching solutions to Qi and Qo, until the set of optimalityconditions are all met. Convergence of the solution is guaranteeddue to the convexity of the problem. This numerical proceduremay not be so straightforward to be used in practice. Therefore, inthe following, we discuss some rules of thumb and develop asimple heuristic that can help practitioners in making this kind ofdecisions.

In the case when m¼1, the optimality conditions simplify tothe following:

Corollary 1. For a partial postponement problem with a single type

of product, the optimal procurement quantity for the customized

product Q1 satisfies the following condition:

PðD14Q1Þ ¼c1�c2

h2�h1

and the optimal procurement quantity for the standard component

Qo can be determined by the following optimality condition:

PðD14Q1þQoÞ ¼c2

p�h2:

Remark. In the classical newsvendor model, with unit sellingprice p, cost c01, and salvage value s1, the optimal orderingquantity satisfies the well known critical fractile formula:

PðD1rQ1Þ ¼p�c01p�s1

,

or in stock-out probability : PðD14Q1Þ ¼c01�s1

p�s1:

However, if the product can be purchased in an intermediateform, at a cost of c02 and salvage value s2, and the final customiza-tion cost is r2, then the optimal ordering quantity for the final andintermediate product should be set at

PðD14Q1Þ ¼ðc01�s1Þ�ðc

02�s2Þ

ðr2þs2Þ�s1, PðD14Q1þQoÞ ¼

c02�s2

p�ðr2þs2Þ:

As an immediate corollary, since

PXmi ¼ 1

ðDi�QiÞþ4Qo

!ZP

Xia j

ðDi�QiÞþ4Qo

0@

1A

for all j¼ 1, . . . ,n,

we have

PðDi4QiÞ ¼c1�c2þðp�h2ÞP½

Pka iðDk�QkÞ

þ4Qo�

h2�h1þðp�h2ÞP½P

ka iðDk�QkÞþ4Qo�

rc1�c2þðp�h2ÞP½

Pmk ¼ 1ðDk�QkÞ

þ4Qo�

h2�h1þðp�h2ÞP½Pm

k ¼ 1ðDk�QkÞþ4Qo�

¼

c1�c2þðp�h2Þc2

p�h2

h2�h1þðp�h2Þc2

p�h2

¼c1

c2þh2�h1:

Similarly,

c2

p�h2¼ P

Xm

i ¼ 1

ðDi�QiÞþ4Qo

!

¼ PXm

i ¼ 1

ðDi�QiÞþ4Qo j DioQi

!PðDioQiÞ

þPXm

i ¼ 1

ðDi�QiÞþ4Qo j DiZQi

!PðDiZQiÞ

rPXka i

ðDk�QkÞþ4Qo

!PðDioQiÞþPðDi4QiÞ:

Using the optimality condition, we can rewrite

PXka i

ðDk�QkÞþ4Qo

!¼ðh2�h1ÞPðDi4QiÞ�ðc1�c2Þ

ðp�h2ÞPðDioQiÞ:

We can re-arrange terms to have

c2

p�h2rðh2�h1ÞPðDi4QiÞ�ðc1�c2Þ

p�h2þPðDi4QiÞ,

i.e.,

PðDi4QiÞZc1

p�h1:

1 We say that jersey i has a stock out if Di 4Qn

i , i.e., without consider the

possibility of converting blank jerseys into printed one.


It also follows directly from the optimality condition (3) that

PðDi4QiÞZc1�c2

h2�h1:

Thus we have the following corollary:

Corollary 2.

maxc1�c2

h2�h1,

c1

p�h1

� �rPðDi4QiÞr

c1

c2þh2�h1

for each i¼ 1, . . . ,m.

In a nutshell, the rule of thumb for the replenishment quantityof jersey i is thus given by

Rule of Thumb 1: Order enough printed jerseys so that the chancesof stock out is between maxððc1�c2Þ=ðh2�h1Þ,c1 =ðp�h1ÞÞ� 100%and c1=ðc2þh2�h1Þ � 100%.

In the Reebok’s case, the two ratios are close to each other if(i) c2þh2 � p, i.e., the sum of procurement and customization costc02þr2 of the blank jersey is close to the selling price p, or(ii) c2 ¼ c02�s2 � 0, i.e., the salvage value of the blank jersey isclose to the procurement cost.

In the rest of this section, we assume that the demands of theproducts are independent normally distributed with mean mi andstandard deviation si. In the classical newsvendor model, thecritical fractile depends only on the cost parameters, and allproducts with the same cost parameters will have the same stockout probability in the optimal solution. In the partial postpone-ment case, with the incorporation of the postponement option(blank jerseys), interestingly this may not be the case. We discussnext how the incorporation of si changes the stocking decisions.

Theorem 2. Let Qn

k ,k¼ 1, . . . ,m, denote the optimal ordering quan-

tity for product k. When the demands Dk,k¼1,y,m, follow normal

distribution ðmk,skÞ, if siZsj, then PðDi4Qn

i ÞrPðDj4Qn

j Þ. In parti-

cular, if the demands have identical standard deviation, then all

optimal stock out probabilities are identical.

Proof. Suppose PðDi4Qn

i Þ4PðDj4Qn

j Þ. Since demands are nor-mally distributed, we have

Qn

i �mi

sir

Qn

j�mj

sj:

Note that

PðDi4Qn

i Þ ¼c1�c2þðp�h2ÞP½

Pka i,jðDk�Qn

kÞþþðDj�Qn

j Þþ4Qn

o�


ka i,jðDk�Qn

kÞþþðDj�Qn

j Þþ4Qn

o�,

and

PðDj4Qn

j Þ ¼c1�c2þðp�h2ÞP½

Pka i,jðDk�Qn

kÞþþðDi�Qn

i Þþ4Qn

o�


ka i,jðDk�Qn

kÞþþðDi�Qn

i Þþ4Qn

o�:

Hence

PX

ka i,j

ðDk�Qn

kÞþþðDj�Qn

j Þþ4Qn

o

24

35

4PX

ka i,j

ðDk�Qn

kÞþþðDi�QiÞ

þ4Qn

o

24

35: ð4Þ

Note also that ðDk�mkÞ=sk ¼ z,k¼ 1, . . . ,m normalizes all Dk to

standard normal random variable z. For all xZ0,

PðDi4Qn

i þxÞ ¼ PDi�mi

si4

Qn

i þx�mi

si

� �¼ P

Dj�mj

sj4

Qn

i þx�mi

si

� �

ZPDj�mj

sj4

Qn

j�mj

sjþ

x

si

!ZP

Dj�mj

sj4

Qn

j�mj

sjþ

x

sj

!

¼ PðDj4Qn

j þxÞ:

Thus whenP

ka i,jðDk�Qn

kÞþrQn

o , let x¼ Qn

o�P

ka i,jðDk�Qn

kÞþ .

We have

PX

ka i,j

ðDk�Qn

kÞþþðDj�Qn

j Þþ4Qn

o

24

35

¼ P ðDj�Qn

j Þþ4Qn

o�X

ka i,j

ðDk�Qn

kÞþ

24

35

¼ P Dj�Qn

j 4Qn

o�X

ka i,j

ðDk�Qn

kÞþ

24

35

rP Di�Qn

i 4Qn

o�X

ka i,j

ðDk�Qn

kÞþ

24

35

¼ PX

ka i,j

ðDk�Qn

kÞþþðDi�Qn

i Þþ4Qn

o

24

35:

On the other hand, whenP

ka i,jðDk�Qn

kÞþZQn

o ,

PX

ka i,j

ðDk�Qn

kÞþþðDj�Qn

j Þþ4Qn

o

24

35

¼ PX

ka i,j

ðDk�Qn

kÞþþðDi�Qn

i Þþ4Qn

o

24

35:

This is a contradiction to (4). &

This can be translated to the following rule of thumb:

Rule of Thumb2: If printed jersey i has higher variability thanprinted jersey j, then in the optimal ordering solution, jersey i

has lower stock out1 probability than jersey j.

Next, we propose a simple heuristic to the partial postpone-ment problem. By optimality condition (2), optimality condition(3) can be rewritten as:

PðDi4QiÞ ¼

c1�c2þc2PðP

k a iðDk�QkÞ

þ 4QoÞ

PðP

kðDk�QkÞ

þ 4QoÞ

� �

h2�h1þc2PðP

k a iðDk�QkÞ

þ 4QoÞ

PðP

kðDk�QkÞ

þ 4QoÞ

� � :

To evaluate the optimal ordering quantity, we need to understandthe behavior of the ratio:

PXka i

ðDk�QkÞþ4Qo

!,PXm

k ¼ 1

ðDk�QkÞþ4Qo

!:

It is difficult to find a closed-form formula to the above expres-sion. By Markov’s inequality:

PXka i

ðDk�QkÞþ4Qo

!rE

Xka i

ðDk�QkÞþ

!,Qo,

PXm

k ¼ 1

ðDk�QkÞþ4Qo

!rE

Xk

ðDk�QkÞþ

!,Qo:


We compare the two expressions on the right hand side instead toobtain

PXka i

ðDk�QkÞþ4Qo

!,PXm

k ¼ 1

ðDk�QkÞþ4Qo

!

� EXka i

ðDk�QkÞþ

!,EX

k

ðDk�QkÞþ

!

¼Xka i

skLððQk�mkÞ=skÞ

, Xk

skLððQk�mkÞ=skÞ

!,

where

LðzkÞ ¼

Z 1zk

ðz�zkÞf ðzÞ dz

is the unit loss function for standard normal random variable z,and f ð�Þ is the pdf of the standard normal distribution.

We use this to propose the following rule-based heuristic:

1.

TabDem

D

N

B

LA

B

V

B

S

O

Initially, for each product i, order Qi such that the stock-outprobability satisfies the following condition:

PðDi4QiÞ ¼

c1�c2þc2

Pk a i

skPksk

h2�h1þc2

Pk a i

skPksk

: ð5Þ

2.
Let zi ¼ ðQi�miÞ=si. Calculate LðziÞ, where Lð�Þ corresponds tothe unit loss function of the standard normal random variable.Refine the ordering quantity to satisfy:
PðDi4QiÞ ¼

c1�c2þc2

Pk a i

skLðzkÞPkskLðzkÞ

h2�h1þc2

Pk a i

skLðzkÞPkskLðzkÞ

: ð6Þ

3.
Order Qo such that
PXm

i ¼ 1

ðDi�QiÞþ4Qo

!¼

c2

p�h2:

4. Numerical results

In this section, we use the Reebok’s case and data described inParsons (2004) to conduct our numerical experiments. Based onforecast, jerseys of the six key players, listed in Table 1, are themost popular in terms of jersey sales.

The demands for the key players are satisfied first by the stockof dressed (player-specific) jerseys and then by decorating theblank jerseys. The demands for other players are met solely bydecorating blank jerseys.

Parsons (2004) studied the above problem using the news-vendor model in his thesis. Two approaches are discussed

le 1and forecast of New England Patriots 2003.

escription Mean Std. dev.

EW ENG PATRIOT total 87 680 19 211

RADY,TOM #12 30 763 13 843

W,TY #24 10 569 4756

ROWN,TROY #80 8159 3671

INATIERI,ADAM #54 7270 4362

RUSCHI,TEDY #04 5526 3316

MITH,ANTOWAIN #32 2118 1271

ther players 23 275 10 474

and compared, assuming the demands are normally distributed.Approach 1 uses the newsvendor solution for each demand classindependently, where dressed jerseys are for the key players andblank jerseys are for the rest of the players. Approach 2 uses acomplicated modified newsvendor solution to determine theamount of dressed jerseys for each key player, by incorporatingthe information on the availability of blank jerseys. The data usedin his experiments are shown in Table 2.

Using Corollary 2 and the above parameters, the stock outprobabilities (i.e., PðDi4QiÞÞ for regular products are confined inthe tight range [0.741, 0.796] in the optimal solution. The critical

fractiles (i.e., the in-stock probabilities PðDirQiÞ ¼ 1�PðDi4QiÞ)are thus in the range [0.204, 0.259]. In the numerical study, wewill report the critical fractiles instead of the stock-out probabil-ities. Note that the demands of jerseys for other players are metsolely by the stock of blank jerseys. Let Dr denote this portion ofdemand. The portion of demand to be satisfied by blank jerseysisP

kðDk�QkÞþþDr . We could view Dr as demand for a product

with Qr fixed at 0. Thus we should modify the optimalityequations in Theorem 1 as follows. The optimal procurementquantity for dressed jerseys Qi, i¼ 1, . . . ,6 satisfies the followingcondition:

PðDi4QiÞ ¼c1�c2þðp�h2ÞP½

Pka iðDk�QkÞ

þþDr 4Qo�


ka iðDk�QkÞþþDr 4Qo�

, ð7Þ

and the optimal procurement quantity for blank jerseys Qo

satisfies the following condition

PX6

i ¼ 1

ðDi�QiÞþþDr 4Qo

!¼

c2

p�h2: ð8Þ

To search for the optimal solution, we need to know thedistribution of the following two terms:

Xka i

ðDk�QkÞþþDr and

X6

i ¼ 1

ðDi�QiÞþþDr :

Parsons approximates these distributions by first finding themean of each random term, then multiplying it by the originalcoefficient of variance to obtain its variance. We use the heuristicequations (5) and (6) to approximate the optimality condition (7)and find the order quantities for the printed jerseys, and then usemonte-carlo simulation (with Qi fixed by the heuristic, whileincorporating the demand due to Dr) to find the appropriate levelof the blank jerseys (Qo) needed to maintain the service levelstipulated in the optimality condition (8).

We compare the performance of the four solution approaches,i.e., Parsons’s approaches (denoted by P-1 and P-2), the heuristicapproach, and the optimal solution which can be solved byoptimality conditions (8) and (7). The results are presented inTable 3. For the heuristic approach, we tabulate the order quan-tities, the critical fractiles in step 1 ðCF1Þ, followed by the criticalfractiles obtained in step 2 ðCF2Þ. From the table, we can see thatthe performances of the several approaches are rather close. This is

Table 2Cost parameters.

Description Notation Value

Wholesale price p 24.00

Cost of a dressed jersey c01 10.90

Cost of a blank jersey c02 9.50

Cost of decorating a blank jersey to a dressed jersey r2 2.40

Salvage value of a dressed jersey s1 7.00

Salvage value of a blank jersey s2 8.46


due to the choice of cost and profit structure in the problemsetting: the cost of obtaining a dressed jersey by decorating a blankone is only slightly higher than the cost of procuring a dressedjersey. Moreover, the unit shortage cost (net margin) is at least$12.10¼$24–$9.5–$ 2.4 and the unit overage cost is at most$3.9¼$10.9–$7.0, so the risk of Reebok NFL replica jerseys is quitelow and therefore the differences among the several approachesare not significant. Furthermore, the critical fractiles obtained fromstep 1 are already extremely close to the ratios from step 2.

We also conduct another numerical study by altering the systemparameters. Specifically, we decrease the selling price and thesalvage values to increase the risk level. The results are shown inTable 4. Note that Corollary 2 pins the critical fractiles to the range[0.053, 0.147], i.e., stock out probabilities in the range [0.853, 0.947].We observe that for this case with a higher business risk, the ratio ofthe quantity of blank jerseys to the total quantity of dressed jerseyin the optimal solution increase significantly.

We conduct a more comprehensive comparison of the perfor-mance of five different approaches, i.e.,the previous fourapproaches, and an approach storing only blank jerseys (denotedby BL). The latter corresponds to the full postponement strategyand reduces to a simple newsvendor problem. In presenting theresults, we only report the critical fractiles from step 2 of ourheuristic. However, we found that the solutions obtained fromstep 1 perform also pretty well.

Figs. 2(a)–5(b) show the sensitivity analysis based on the bench-mark problem in Table 4. Interestingly, the proposed heuristicperforms quite well in all instances: its performance in terms ofexpected system profit is extremely close to the optimum. Specifi-cally, Fig. 2(a) shows that as the cost of a dressed jersey c01 increases,the expected system profit exhibits a downward trend except underthe pure blank jersey policy, which is not affected by c01. Moreover,with c01 increasing, the performance of BL improves and can even benear optimal, because the unit cost of blank jerseys plus the unit

Table 3Comparison of solutions.

Desc Solution

P-1 P-2

BRADY,TOM #12 Q1 41 018 24 8

LAW,TY #24 Q2 14 092 8532

BROWN,TROY #80 Q3 10 878 6587

VINATIERI,ADAM #54 Q4 10 501 5402

BRUSCHI,TEDY #04 Q5 7982 4106

SMITH,ANTOWAIN #32 Q6 3060 1574

Blanks Qo 38 052 59 8

Expected profit 9.9412�105 1.04

Table 4Comparison of solutions for the benchmark problem: p¼ 13,c01 ¼ 10:9,c02 ¼

Desc Solution

P-1 P-2

BRADY, TOM #12 Q1 18 657 17 2

LAW, TY #24 Q2 6410 594

BROWN, TROY #80 Q3 4949 458

VINATIERI, ADAM #54 Q4 3455 302

BRUSCHI, TEDY #04 Q5 2626 230

SMITH, ANTOWAIN #32 Q6 1006 882

Blanks Qo 14 848 45 7

Expected profit 6.8582�104 9.77

decorating cost is only slightly higher than the cost of dressed jerseys.Figs. 3(a)–5(a) show that the expected system profits are decreasingin c02 and r2 while increasing in s2, which are intuitive. The numericalresults show that the heuristic always outperforms P-1, P-2 andBL, and none of these three approaches always works well.Figs. 2(b)–5(b) depict the critical fractiles for dressed jerseys underthe four approaches, P-1, P-2, the optimal solution and the heuristic,except BL (the fractile is zero and therefore is not depicted). P-1 andP-2 use the same critical fractile for all dressed jerseys, whereas forthe optimal solution and the proposed heuristic, each type of dressedjerseys has a distinct critical fractile, so here we only plot themaximum and minimum values (according to Theorem 2, themaximum and minimum critical fractiles correspond to Brady andSmith). From these graphs, we can observe that P-1 and P-2 tend tooverestimate the critical fractile for dressed jerseys. For our heuristic,the critical fractiles are rather close to the optimal ones, especially forthose ratios with relatively smaller values.

The numerical study validates the effectiveness of our simpleheuristic. Its performance is almost as good as the optimal solution. Inaddition, we observe that most of the critical fractiles for dressedjerseys are close to the lower bound, except for the maximum criticalfractile, which is more sensitive to the change of system parameters.The reason can be found from the demand distribution information(cf. Table 1) that the largest standard deviation dominates the others.This observation is also consistent with our heuristic approach.

5. Conclusion

This paper investigates a partial postponement problem whichexists in many practical situations. We derive the optimalityconditions on the procurement quantities for the group ofcustomized products as well as the common component. Solvingfor the optimal solution requires an iterative searching procedure

Heuristic (CF1, CF2) Optimal (CF)

34 20 204 (0.2196, 0.2228) 21 687 (0.2560)

6739 (0.2091, 0.2103) 6960 (0.2240)

5185 (0.2080, 0.2089) 5317 (0.2194)

3750 (0.2087, 0.2098) 3932 (0.2221)

2834 (0.2076, 0.2084) 2945 (0.2181)

1074 (0.2054, 0.2057) 1091 (0.2096)

68 76 591 74 604

96�106 1.0546�106 1.0547�106

9:0,r2 ¼ 2:4,s1 ¼ 2,s2 ¼ 3.

Heuristic (CF1, CF2) Optimal (CF)

98 10 522 (0.0675, 0.0718) 11 505 (0.0821)

3 3122 (0.0574, 0.0587) 3167 (0.0598)

8 2369 (0.0564, 0.0574) 2397 (0.0583)

7 422 (0.0570, 0.0582) 462 (0.0593)

1 283 (0.0560, 0.0569) 306 (0.0577)

82 (0.0542, 0.0546) 85 (0.0548)

56 56 992 56 005

25�104 1.0467�105 1.0472�105

9 9.5 10 10.5 11 11.50.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 105

c’1

Avera

ge P

rofit

P−1P−2BLOptimalHeuristic

9 9.5 10 10.5 11 11.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

c’1

Critical F

ractile

s for

Dre

ssed J

ers

eys

P−1P−2Optimal (max)Optimal (min)Heuristic (max)Heuristic (min)

Fig. 2. (a) Performance of the solutions. (b) Critical fractiles for dressed jerseys.

8.8 9 9.2 9.4 9.6 9.8 10 10.2 10.40

2

4

6

8

10

12

x 104

c’2

Ave

rag

e P

rofit


8.8 9 9.2 9.4 9.6 9.8 10 10.2 10.40

0.05

0.1

0.15

0.2

0.25

c’2

Critica

l F

ractile

s f

or

Dre

sse

d J

ers

eys



2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.60

2

4

6

8

10

12

14x 104

r2

Ave

rag

e P

rofit


2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

0

0.05

0.1

0.15

0.2

0.25

r2

Critica

l F

ractile

s f

or

Dre

sse

d J

ers

eys




to meet all the optimality conditions. We thus discuss somerules of thumb and propose a simple bootstrap heuristic tothe problem, which resembles the critical fractile approach for

the classic newsvendor problem. We then apply our approach tothe inventory planning problem of Reebok NFL replica jerseys,and compare with Parsons’s (2004) two heuristic approaches.

2 3 4 5 6 7 8 9

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4x 105

s2

Ave

rag

e P

rofit


2 3 4 5 6 7 8 9

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

s2

Critica

l F

ractile

s f

or

Dre

sse

d J

ers

eys




The numerical results demonstrate that the performance of ourheuristic approach is near-optimal in all problem cases.

Acknowledgements

The authors thank the anonymous referee for the usefulcommons that have helped to improve the paper. The first authoris supported in part by Hong Kong Research Grants Council(RGC)-DAG11EG01.

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Modified critical fractile approach for a class of partial postponement problems