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Optimal Price/Lead-Time Menus for Queues withCustomer Choice: Segmentation, Pooling, and StrategicDelayPhilipp Afèche, J. Michael Pavlin

To cite this article:Philipp Afèche, J. Michael Pavlin (2016) Optimal Price/Lead-Time Menus for Queues with Customer Choice: Segmentation,Pooling, and Strategic Delay. Management Science 62(8):2412-2436. http://dx.doi.org/10.1287/mnsc.2015.2236

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MANAGEMENT SCIENCEVol. 62, No. 8, August 2016, pp. 2412–2436ISSN 0025-1909 (print) � ISSN 1526-5501 (online) http://dx.doi.org/10.1287/mnsc.2015.2236

© 2016 INFORMS

Optimal Price/Lead-Time Menus forQueues with Customer Choice:

Segmentation, Pooling, and Strategic DelayPhilipp Afèche

Rotman School of Management, University of Toronto, Toronto, Ontario M5S 3E6, Canada, afeche@rotman.utoronto.ca

J. Michael PavlinSchool of Business and Economics, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, Canada, mpavlin@wlu.ca

How should a firm design a price/lead-time menu and scheduling policy to maximize revenues fromheterogeneous time-sensitive customers with private information about their preferences? We consider a

queueing system with multiple customer types that differ in their valuations for instant delivery and their delaycosts. The distinctive feature of our model is that the ranking of customer preferences depends on lead times:patient customers are willing to pay more than impatient customers for long lead times, and vice versa forspeedier service. We provide necessary and sufficient conditions, in terms of the capacity, the market size, and theproperties of the valuation-delay cost distribution, for three features of the optimal menu and segmentation: pricingout the middle of the delay cost spectrum while serving both ends, pooling types with different delay costs into asingle class, and strategic delay to deliberately inflate lead times.

Keywords : congestion; delay; incentives; lead times; mechanism design; pooling; pricing; priorities;quality of service; queueing systems; revenue management; scheduling; segmentation; servicedifferentiation; strategic delay

History : Received September 14, 2011; accepted February 1, 2015, by Assaf Zeevi, stochastic models and simulation.Published online in Articles in Advance February 16, 2016.

1. IntroductionFirms such as Amazon, Dell, or Federal Express, servetime-sensitive customers whose willingness to pay fora product or service also depends on the lead timebetween order placement and delivery. To exploit het-erogeneous customer preferences—some value speedyservice more than others—firms may offer a menuof differentiated price/lead-time options (same day,two day, etc.) as a revenue management tool, givingimpatient customers the option to pay more for fasterdelivery while charging less for longer lead times.This paper studies the joint problem of designing therevenue-maximizing price/lead-time menu and the cor-responding scheduling policy for a monopoly providerwho cannot tell apart individual customers but only hasaggregate information on their preferences, e.g., basedon market research. We study this problem withina queueing model and consider customers who areheterogeneous in their valuations for instant deliv-ery and their delay costs. This problem has recentlyreceived some attention, but as detailed below, signif-icant gaps remain in understanding its solution forthe case with multiple delay cost types considered here.This paper contributes to closing these gaps. We focuson the case where the valuations of types are strictlyincreasing and affine in their delay costs, and the

valuation-to-delay cost ratio is decreasing in impatience.This yields the simplest model with multiple delaycosts that gives rise to a lead-time-dependent rankingof types. That is, patient customers are willing to paymore or less than impatient customers for a givenlead time, depending on whether it is above or belowsome indifference threshold. This is an important novelfeature of our model and is critical for our results. Itallows us to capture, in a one-dimensional type model,a key property of preferences over two attributes:one or the other dominates depending on the serviceoption. To our knowledge, this paper presents the firstresults that specify the optimal menu for multiple typeswith lead-time-dependent ranking. We address threequestions:

1. Customer segmentation. Which customer types—most, least, or moderately time sensitive—should beserved?

2. Priorities, pooling, and strategic delay. Should themenu target a distinct price/lead-time class to eachcustomer type based on her delay cost, and prioritizetypes accordingly? Or, is it optimal to offer less thana full range of classes and target some types withdifferent delay costs to be pooled in the same price/lead-time class? Should the scheduling policy be workconserving, or involve strategic delay to deliberately

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inflate some lead times above operationally feasiblelevels?

3. Impact of capacity and demand attributes. How dothe optimal segmentation and menu depend on thecapacity, the market size, and customer preferences?

We identify low, medium, and high capacity regimes.At low capacity it is optimal to differentiate lead timesbased on delay costs. At medium capacity it is optimalto pool intermediate types but sell differentiated leadtimes to more and less patient types. If valuations arebelow some threshold, then at medium capacity itis also optimal to price out some intermediate types.These medium capacity results hinge on the lead-time-dependent ranking of types. Unlike typical poolingresults in screening models, our pooling result holdseven if the type distribution has a monotone hazardrate, as our analysis assumes. At high capacity it isoptimal to differentiate lead times either for all types, oronly for sufficiently impatient ones while strategicallydelaying more patient ones.

1.1. Literature and PositioningThis paper bridges research streams on queueing sys-tems and mechanism design. See Stidham (2002) for asurvey of research on the analysis, design, and controlof queueing systems in settings where the systemmanager is fully informed and determines all job flows.Our analysis builds on the achievable-region approach,pioneered by Coffman and Mitrani (1980).

Mechanism design tools have been applied to manyresource allocation problems under private information.Rochet and Stole (2003) survey screening studies inthe economics literature. Among these, papers on thedesign of price-quality menus are closest to ours. Intheir seminal paper Mussa and Rosen (1978) consider amodel with one-dimensional types. Rochet and Choné(1998) study its multidimensional version. Althoughquality pooling is known to be potentially optimalin these “standard” screening models, they rule outoperational interdependencies among quality levels. Withthis simplifying assumption these models can focus oninterdependencies due to customer self-selection, andtheir results are invariant to capacity and market size.These models are therefore not designed to generatemeaningful prescriptions in our setup where opera-tional interdependencies among lead times play animportant role in addition to customer self-selection.The capacity constraint and queueing effects implyexternalities among service classes, and the providercontrols these externalities through the price/lead-time menu and the scheduling policy. These featuresconsiderably complicate the problem as explainedin §2.2.

Several papers that study variations of the classicprice-quality design problem also ignore queueingeffects. Dana and Yahalom (2008) introduce a capac-ity constraint. Bansal and Maglaras (2009) consider a

model with multiple consumer types that are satisfic-ing as opposed to utility maximizing, and rely on adeterministic analysis of the optimal menu, so thatthe queueing effects reduce to a capacity constraint.Neither study reports the phenomena we identify.Quality degradation similar to strategic delay has beenstudied in the damaged goods literature (Deneckereand McAfee 1996, McAfee 2007). Anderson and Dana(2009) unify the results of the damaged goods literatureand other well-known pricing results, by identifyinga necessary condition for price discrimination to beprofitable in models with a quality constraint but amplecapacity. We discuss these connections to our strategicdelay results in §6.3.

This paper is part of a research stream on pricingand operational decisions for queueing systems withself-interested and time-sensitive customers. See Hassinand Haviv (2003) and Stidham (2009) for surveys. Weconsider static price/lead-time menus, unlike paperson dynamic price and/or lead-time quotation (e.g.,Plambeck 2004, Çelik and Maglaras 2008, Ata andOlsen 2013).

Three problem features jointly distinguish this frommost papers on static price/lead-time optimization:(1) Revenue maximization. The objective is to maximizethe provider’s revenue, not the total system benefit(see Mendelson and Whang 1990, Van Mieghem 2000).(2) Customer choice over menu options. Types have privateinformation on their preferences and can choose theirclass, unlike in models that restrict each type to asingle class (see Boyaci and Ray 2003, Maglaras andZeevi 2005, Zhao et al. 2012). (3) Scheduling optimization.The provider chooses the scheduling policy, unlike inpapers that fix the policy (see Naor 1969, Mendelson1985, Rao and Petersen 1998, Afèche and Mendelson2004). In studies that lack feature (1) or (2) neitherpooling nor strategic delay can be optimal.

Only a few papers on static price/lead-time menuscapture all three attributes. Afèche (2004, 2013), hence-forth AF, is the first in the queueing literature toidentify strategic delay and characterize its optimality.His model considers only two delay cost types, butallows heterogeneous valuations for each delay costlevel, unlike our model. Katta and Sethuraman (2005),henceforth KS, is the first and only other queueingpaper that considers optimal pooling. Like our model,theirs also considers multiple types with perfectlycorrelated valuations and delay costs. However, con-trary to this paper, they restrict the valuation-to-delaycost ratio to be decreasing in impatience. This superfi-cially trivial distinction is important: it implies that theranking of types is lead-time-invariant, which rules outthe key features in our findings. We further discussthe relationship to KS in §6.2. Maglaras et al. (2016)consider the menu-design problem for an unrestrictedvaluation-delay cost distribution, that is, without the

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perfect correlation assumed in this paper. However,unlike this paper, they provide no analytical results thatidentify which solution features arise in terms of thedemand and capacity parameters. Rather, they focus onoptimal decisions in large-scale systems, that is, withlarge capacity and market size. They propose a deter-ministic relaxation (DR) of the problem that ignoresqueueing effects but captures the essential behaviorof these large-scale systems. The DR is typically notsolvable in closed form in multitype settings, althoughit is computationally more tractable than the originalstochastic problem. They show how to derive fromthe DR solution a policy that is asymptotically nearoptimal in large-scale systems.

In brief, our contributions are as follows:1. Customer type model. Our model is distinctive in

that it jointly considers multiple delay costs, unlike AF,and a lead-time-dependent ranking of types, unlike KS. Ityields novel results and offers a unifying frameworkfor explaining the presence or absence of these resultsin related models. We return to these connections in §6.

2. Optimal customer segmentation and price/lead-timemenu. We provide necessary and sufficient conditionson the demand and capacity parameters, for threestriking solution features: (i) Pricing out the middle of thedelay cost spectrum while serving both ends; this featuredoes not arise in KS. (ii) Pooling types with differentdelay costs; this feature does not arise in the two-typemodel of AF. Our pooling results are more general andmore informative than those in KS. Most importantly,unlike in KS, in our model pooling arises even fordelay cost distributions with a monotone hazard rate.(iii) Strategic delay. Our pooling results complementthose of AF. We identify optimal strategic delay ina setting with multiple, not just two, types, but weignore the effect of valuation heterogeneity.

2. Model, Problem Formulation, andAnalysis Roadmap

We model a service or make-to-order manufacturingprovider as an M/M/1 queueing system. Potentialcustomers with unit demand arrive according to anexogenous Poisson process with rate or market size å.The system has i.i.d. exponential service times withmean 1/�, where � is the capacity. The capacity isnot a decision variable, but our results specify howthe optimal menu varies with �. We normalize themarginal cost of service to zero.

Preferences. Customers differ in their valuations forimmediate delivery of the product or service, and intheir linear delay costs. We use the term “lead time”to refer to the entire time between order placementand delivery. We consider a continuum of customertypes indexed by c, which denotes the customer’sdelay cost per unit of lead time. Types c are i.i.d.draws from a continuous distribution F with p.d.f. f ,

which is assumed strictly positive and continuouslydifferentiable on the interval £¬ 6cmin1 cmax7⊂ 601�5.Let F̄ = 1 − F . The service time distribution and thedelay cost distribution are mutually independent andindependent of the arrival process.

Valuations and delay costs are perfectly correlated.A type c customer has positive valuation V 4c5 forimmediate delivery, where V 2 £→�+ is a monotoneand continuous function. The analysis focuses on V 4c5=

v+ c · d, where v and d are constants. The base value vis a scale parameter for valuations. As discussed below,the slope of the valuation-delay cost relationship d canalso be viewed as a threshold lead time that determinesthe ranking of customers’ willingness to pay for agiven service class. The paper focuses on the case v > 0and d > 0 since it gives rise to novel results. It alsocovers v ≤ 0 < d and v > 0 ≥ d, in which case our modelspecializes to related models. Section 6.4 outlines howour results generalize if V 4c5 is not affine.

The case of perfectly positively correlated valuationsand delay costs (d > 0) is well suited for settings wheredelays deflate values. A variety of important phenom-ena lead to delay-driven value losses (see Afèche andMendelson 2004), including physical decay of perish-able goods during transportation delays, technologicalor market obsolescence of short life-cycle productssuch as computer chips or fashion items, and delayedinformation in industrial and financial markets.

A type c customer’s net value or willingness to payfor an expected lead time w is N4c1w5¬ V 4c5− c ·w =

v+c ·4d−w5, and her utility at price p is v+c ·4d−w5−p.This net value function has two standard properties.First, it decreases in the lead time, i.e., the partialderivative Nw4c1w5 < 0. This captures the notion ofvertical product differentiation in that service classescan be objectively ranked from fastest to slowest, orfrom highest to lowest “quality.” Second, the moretime sensitive a type, the sharper her net value declineas the lead time increases: Ncw4c1w5 < 0. This is thesingle-crossing condition. The distinctive feature ofour model is that the ranking of types’ net values islead-time-dependent. Specifically, Nc4c1w5= d−w, so thatnet values for lead times w < d increase, whereas thosefor lead times w > d decrease, in the time-sensitivity c.In this sense, the parameter d represents a thresholdlead time. This feature describes situations whereimpatient customers are willing to pay more thanpatient customers for speedy service (e.g., overnightdelivery) but less than patient customers for slowservice (e.g., delivery in several business days).

To rule out the case where no type has a positiveexpected net value from service even in the absence ofwaiting, we assume that �−1 <d+ v/cmin if v > 0, and�−1 <d+ v/cmax if v ≤ 0.

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Information. The provider knows the arrival process,the delay cost distribution f , the function V 4c5, and theservice time distribution. However, a customer’s type cis her private information. Service time realizationsbecome known only once jobs are processed to comple-tion. Only the provider observes the system queues.Customers lack this information.

Decisions. The provider announces a static price/lead-time menu and chooses a scheduling policy tomaximize her long-run average revenue rate. We outlinethe set of admissible scheduling policies below. Themenu specifies for each class two attributes, the priceand the expected lead time. We simply say “lead time”for the expected lead time of a class. Because weconsider a continuum of types with respect to theirdelay preferences, the provider will optimize over acontinuum of service classes. (“Class” refers to theattributes of a service option, “type” refers to those of acustomer.) Because customers have private information,they can choose among all classes, and the providermust consider their choice behavior.

Formally, the provider selects a menu of lead-time/price options 84w1P4w552 w ∈ ·9, where ·denotes the set of offered lead times and the functionP2 ·→� assigns prices to lead times.

Customer arrival times are exogenous. However,customers are strategic in their purchase decisions.We assume they are risk neutral with respect to lead-time uncertainty. Upon arrival at the facility, cus-tomers decide, based on the posted menu, whichservice class to purchase, if any. Specifically, a cus-tomer of type c determines the lead-time/price pair4w1P4w55 that maximizes her expected utility fromservice, U4w1P4w53 c5¬ v+c · 4d−w5−P4w5. Let w4c5¬arg maxw∈·8U4w1P4w53 c59 and p4c5¬ P4w4c55 denotethe preferred lead-time/price pair for type c, and letU4c5¬U4w4c51 p4c53 c5= v+c4d−w4c55−p4c5 denote thecorresponding expected utility from service. Customerswho do not purchase balk and receive zero utility, sothey only purchase if their expected utility from serviceis nonnegative. We assume no retrials and no reneging.We keep track of purchase decisions with the acceptancefunction a2 £→ 80119, where a4c5= 1 if type c buysservice, choosing the lead-time/price pair 4w4c51 p4c55,and a4c5= 0 otherwise. Let £a ¬ 8c ∈ £2 a4c5= 19 denotethe set of types that buy service and °£a = £\£a itscomplement. Let �¬å

∫

x∈£af 4x5 dx denote the resulting

arrival rate. Best responses to a menu satisfy c ∈£a ifU4c5 > 0 and c ∈ °£a if U4c5 < 0. Types with zero expectedutility may or may not purchase as discussed in §3.

Lead Times and Admissible Scheduling Policies. Weassume that customers base their decisions on theannounced expected lead times. However, we requirethat the announced expected lead times equal theaverage steady-state lead times that are realized giventhe capacity �, the scheduling policy, and the customers’

purchase decisions that are induced by the menu.This consistency requirement reflects the notion thatreputation effects and third-party auditors instill in theprovider the commitment to perform in line with herannouncements (see Afèche 2013).

We do not assume a specific scheduling policy butrather let the provider optimize over the followingset of admissible scheduling policies: (1) We focus onnonanticipative and regenerative policies. This appearsto be the most general, easily described restrictionunder which the existence of long-run lead-time aver-ages may be verified. (2) We do not restrict attention towork conserving policies. Specifically, we allow theinsertion of strategic delay whereby the provider artifi-cially increases the lead times for a subset of serviceclasses above the levels that are operationally achiev-able. See Afèche (2004, 2013) for a detailed discussionof strategic delay. (3) We allow preemption, which doesnot affect the results but simplifies the analysis. Inparticular, under priority scheduling, with preemptionthe lead time of a given class does not depend on thearrival rates to lower-priority classes.

We build on the achievable region approach tomulticlass scheduling problems; see Coffman andMitrani (1980). Problem 1 specifies the achievable regionfor the set of admissible policies defined above.

2.1. Mechanism Design FormulationWe formalize the provider’s problem as a mechanismdesign problem. Based on the revelation principle(e.g., Myerson 1979), we restrict attention, withoutloss of generality, to direct mechanisms in which cus-tomers have the incentive to truthfully report theirtype. The procedure described above, whereby cus-tomers make decisions based on a menu, is strictlyspeaking an indirect mechanism, but it is more descrip-tive of how services are sold, and it is equivalent toa direct revelation mechanism in which customerstruthfully reveal their type. This requires that the func-tions 4a1w1p5 satisfy the individual rationality (IR) andincentive-compatibility (IC) constraints. IR requires thatthe expected utility from service be nonnegative fortypes who are targeted for service and nonpositive forall others: U4c5≥ 0 for c ∈ £a and U4c5≤ 0 for ∈ °£a. ICrequires that each type c maximizes her expected utilityif it truthfully reports its type: U4c5≥U4w4c′51 p4c′53 c5for c 6= c′.

Problem 1.

maxa2£→801191 w2£→�1 p2£→�

å∫ cmax

cmin

a4x5f 4x5p4x5 dx (1)

subject to

�>å∫

x∈£a

f 4x5dx1 (2)

å

�

∫

x∈sf 4x5w4x5dx≥

4å/�5∫

x∈sf 4x5dx

�−å∫

x∈sf 4x5dx

1

∀s⊂£a1 (3)

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U4c5=v+c4d−w4c55−p4c5≥01 ∀c∈£a1 (4)

U4c5=v+c4d−w4c55−p4c5≤01 ∀c∈ °£a1 (5)

w4c5·c+p4c5≤w4c′5·c+p4c′51 ∀c 6=c′0 (6)

Constraint (2) ensures that the system is stable. Con-straints (3) ensure that the lead times 8w4c52 c ∈£a9 areoperationally achievable. The right-hand side (RHS) of (3)is the long-run average work in the system under awork conserving policy that gives all admitted cus-tomers in the set s strict preemptive priority over allothers. It equals the average work in a first-in-first-out(FIFO) M/M/1 system with arrival rate å

∫

x∈sf 4x5dx

and capacity �. A scheduling policy is work conservingif (3) is binding for s = £a. Constraints (4)–(5) capture IRand (6) capture IC. The menu corresponding to a feasi-ble 4a1 p1w5 satisfies ·= 8w4c52 c ∈ £9 and P4w4c55= p4c5for w4c5 ∈·.

First-Best Benchmark2 Observable Types. The first-bestproblem, in which the provider observes the types,yields a considerably simpler version of Problem 1as the IC constraints (6) are dropped. The providercan charge each type the full amount of her net value;that is, type c pays p4c5= v+ c4d−w4c55. In this case,a standard work conserving strict priority policy isoptimal. It prioritizes admitted types by their delaycosts. Hence the menu offers all lead times withinan interval and each admitted type buys a differentlead time.

2.2. Analysis RoadmapWe develop the solution of Problem 1 using the follow-ing three-step approach.

Step 1. Incentive-compatible segmentation and lead times,optimal prices (§3). We translate the IR and IC con-straints (4)–(6) into equivalent properties that anyfeasible and revenue-maximizing triple 4a1 p1w5 mustsatisfy. These properties yield up to three segments ofcustomer types, and also imply the optimal prices forgiven segmentation and lead times, reducing Problem 1to one of choosing the arrival rates and lead times forthese segments.

Step 2. Optimal segmentation and lead times for fixedarrival rate (§4). We characterize the optimal segmenta-tion and lead-time menu depending on �1å1�, d1 andthe distribution f .

Step 3. Optimal arrival rate, segmentation, and lead times(§5). We characterize the solution at the optimal �, forfixed capacity �, and as a function of � for givendemand parameters.

Step 1 is based on standard mechanism design meth-ods, but steps 2 and 3 are not. The capacity constraintand queueing delays introduce operational interdepen-dencies among lead times, which significantly compli-cate the analysis. Following the seminal work of Mussaand Rosen (1978), price-quality menu design problems

in the economics literature rule out operational interde-pendencies among quality levels, which simplifies theanalysis. To be specific, if one removes the queueing-related operational constraints (3) in Problem 1 andintroduces instead a quality cost function in the objec-tive function,1 then under regularity conditions on thedistribution f the problem is quickly solved pointwisefor each type c, and the solution is invariant to the mar-ket size. This pointwise approach fails in the presenceof queueing effects. Steps 2 and 3 consider these effectsby building on the achievable region approach, butour problem calls for three important modifications:we account for the IC and IR constraints, we optimizeover arrival rates, and we allow strategic delay.

3. Incentive-Compatible Segmentationand Lead Times, Optimal Prices

Given a triple 4a1w1p5 we partition the set of admittedtypes £a into the following three segments:

Cl ¬ 8c∈£a2 w4c5>d91

Cm ¬ 8c∈£a2 w4c5=d91 (7)

Ch ¬ 8c∈£a2 w4c5<d90

For simplicity we suppress the dependence of Cl, Cm,and Ch on a. We call classes with w>d low lead-timequality or l classes, those with w < d high lead-timequality or h classes, and the class with the thresholdlead time w = d the medium lead time or m class.

Proposition 1. Fix a triple 4a1w1p5. Define the mar-ginal types cl and ch as follows:

cl ¬{

supCl if Cl 6= �1

cmin otherwise3

ch ¬{

infCh if Ch 6= �1

cmax otherwise0

Suppose that 4a1w1p5 maximizes the revenue rate. Then4a1w1p5 satisfies the IR and IC constraints (4)–(6) if andonly if the following properties hold.

1. The lead times w4c5 are nonincreasing, the prices p4c5are nondecreasing, and cl ≤ ch.

2. If there is a segment of types who buy low lead-timequalities (Cl 6= �) then (a) it is an interval that includes cmin,i.e., c < cl ⇒ c ∈ Cl; (b) the prices and expected utilitiesfrom service satisfy

p4c5= v+c ·4d−w4c55−∫ cl

c4w4x5−d5dx1 ∀c∈ 6cmin1cl71

where p4c5<v1 for c<cl1 (8)

1 The model of Mussa and Rosen (1978) is also simpler than thisquality-cost version of our model, because they consider types witha quality-independent ranking, whereas in our model the ranking oftypes is lead-time-dependent.

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U4c5=

∫ cl

c4w4x5−d5dx1 ∀c∈ 6cmin1cl71

where U4c5>0=U4cl51 for c<cl0 (9)

3. If there is a segment of types who buy high lead-timequalities (Ch 6= �) then (a) it is an interval that includes cmax,i.e., c > ch ⇒ c ∈Ch; (b) the prices and expected utilitiesfrom service satisfy

p4c5 = v+c ·4d−w4c55−∫ c

ch

4d−w4x55dx1 ∀c∈ 6ch1cmax71

where p4c5>v1 for c>ch1 (10)

U4c5 =

∫ c

ch

4d−w4x55dx1 ∀c∈ 6ch1cmax71

where U4c5>0=U4ch51 for c>ch0 (11)

4. If there is a segment of types who buy the mediumlead time (Cm 6= �) then (a) Cm ⊂ 6cl1 ch7; (b) the pricesand expected utilities from service satisfy p4c5 = v andU4c5= 01 for c ∈Cm.

5. Types in 4cl1 ch5 buy the medium lead time or do notbuy at all, i.e., 4cl1 ch5⊂Cm ∪ °£a, and

U4c5 = U4cl5−∫ c

cl

4w4x5−d5dx

= U4ch5−∫ ch

c4d−w4x55dx≤01 ∀c∈ 6cl1ch71 (12)

where U4c5 = 0 for all c ∈ 6cl1 ch7 if some types buy themedium lead time (Cm 6= �).

All proofs are in the appendix. Consider the netvalue as a function of the type c and the lead time w,that is, N4c1w5= V 4c5− cw. Part 1 follows becauseN4c1w5 decreases in lead time.

Parts 2–5 follow because the net value N4c1w5 fora fixed lead time w changes at the rate V ′4c5−w =

d−w as the customer type increases, where d and wcapture the rate of increase in valuation and delaycost, respectively. As a result, net values for lead timesw> d are decreasing, whereas net values for leadtimes w < d are increasing, in customer impatience.Therefore, in part 2 of Proposition 1, if a type c choosesto buy the lead time w4c5 > d, then more patienttypes c′ < c have a higher net value for this leadtime than c. Since IR holds for type c, more patienttypes must get strictly positive utility for IC to hold,i.e., they must be served. Therefore, the set Cl is aninterval that includes the most patient type cmin, andby (9) the expected utilities of customers who buylow lead-time qualities decrease in their impatience.By (8) the price paid by a type c < cl decreases in thelead times of more impatient types in Cl, because thelonger these lead times the more type c values themin comparison to more impatient types. Similarly, inpart 3 of Proposition 1, if a type c buys a lead timew4c5 < d, then more impatient types c′ > c must beserved with strictly positive utility. Therefore, the set Ch

is an interval that includes the most impatient type cmax,

and by (11) the expected utilities of customers who buyhigh lead-time qualities increase in their impatience. Incontrast to the prices (8) that decrease in lead times,by (10) the price paid by a type c > ch increases in thelead times of more patient types in Ch. This holdsbecause the longer these lead times the less type cvalues them in comparison to more patient types. Byparts 4 and 5, the set of customers Cm who purchasethe threshold lead time d is a subset of 6cl1 ch7 thatneed not be an interval. If Cm 6= �, then the lead time dis offered at a price of v, every type has zero expectedutility from this option, but only types in 6cl1 ch7 haveno better option available and are indifferent betweenbuying and not doing so.

By Proposition 1 it may be optimal to price the most,the least, or only moderately impatient types out of themarket. Furthermore, choosing Cl, Cm, and Ch reducesto choosing the corresponding arrival rates, which weuse to replace the acceptance function a. Let �l, �m,and �h be the rates for l, m, and h classes, respectively,where � = �l + �m + �h. Note that �l ¬ åF 4cl5 and�h ¬åF̄ 4ch5 uniquely determine the correspondingfunction a for types c < cl and c > ch, respectively,whereas �m only determines the mass of types c ∈ 6cl1 ch7who buy the medium lead time. Any feasible triples4a1w1p5 and 4a′1w1p5 that only differ in the set, butnot the mass, of types who buy the medium leadtime are revenue equivalent. If �m > 0 then a positivemass of different types are pooled at the mediumlead time. If in addition �m <å−�l −�h then thereare two positive masses of types in 6cl1 ch7 with zeroexpected utility, those who buy the medium lead timeand those who do not buy any service. This featurearises because V 4c5 is affine, but as outlined in §6.4,our main structural results remain valid for a broaderclass of V 4c5 functions.

4. Optimal Segmentation andLead Times for Fixed Arrival Rate

We turn to step 2 of the solution approach outlined in§2.2 and characterize the optimal segmentation andlead times for fixed arrival rate �. In §4.1 we define thevirtual delay costs. In §4.2 we preview how they playa key role for the solution structure. We then discussthe solution in detail, in §4.3 for fixed �, and in §4.4 asa function of � and å.

4.1. Virtual Delay CostsWrite ç4�l1�h1�1w5 for the revenue as a function ofthe arrival rate �, the segmentation characterized by �l

and �h, and the lead-time function w. Substituting theprices (8) and (10) into (1) yields

ç4�l1�h1�1w5 ¬ �v+å∫ cl4�l5

cmin

f 4c5fl4c54d−w4c55dc

+å∫ cmax

ch4�h5f 4c5fh4c54d−w4c55dc1 (13)

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where cl4�l5= F −14�l/å5, ch4�h5= F̄ −14�h/å5 and thefunctions fl and fh are defined as follows:

fl4c5¬ c+F 4c5

f 4c51 for c ∈ 6cmin1 cl71 (14)

fh4c5¬ c−F̄ 4c5

f 4c51 for c ∈ 6ch1 cmax70 (15)

We call fl and fh the virtual delay cost functions. Theymeasure the marginal revenue effect from a reductionin the lead time of a given type. The virtual delaycost of a type c depends on its lead time: it is fl4c5for low quality (w4c5 > d) and fh4c5 for high quality(w4c5 < d). The virtual delay cost consists of an ownprice effect and an external price effect. The own priceeffect is the delay cost c in (14) and (15). It simplymeasures how a lead-time reduction for type c allowsan increase in its own price. The external price effect isthe second summand in (14) and (15). It measures howa lead-time reduction for type c changes the prices ofclasses targeted to other types, in order to maintainIC. By (14) the external price effect is positive for lowlead-time qualities, so fl4c5 > c (for c > cmin). That is,reducing the lead time w4c5 > d of a type c in Cl allowsprice increases for classes targeted to more patienttypes c′ < c while maintaining IC. (By (8) the pricepaid by a type in Cl decreases in the lead times ofmore impatient types in Cl.) As explained in §3, thisfollows because a customer’s net value for a lead timew > d is higher the more patient that customer. Incontrast, by (15) the external price effect is negativefor high lead-time qualities, so fh4c5 < c (for c < cmax).That is, reducing the lead time w4c5 < d of a type cin Ch requires price decreases for classes targeted tomore impatient types c′ > c in order to maintain IC.(By (10) the price paid by a type in Ch increases in thelead times of more patient types in Ch). As explainedin §3, this follows because a customer’s net valuefor a lead time w < d is higher the more impatientthat customer. Furthermore, note that fh4c5 < 0 for atype c if its negative external price effect dominates itsown price effect. In this case increasing the lead timew4c5 < d of this type increases revenues.

4.2. Solution PreviewMaximizing the revenue (13) calls for lead times thatare strictly decreasing in virtual delay costs, whereas ICrequires that the lead times be appropriately ranked rel-ative to the threshold d and nonincreasing in delay costs.This gives rise to three striking solution features.

1. Pricing out the middle of the delay cost spectrum. Itmay be optimal to price out intermediate types. This isthe only one of the three features that also arises underthe first-best menu.

2. Pooling. It may be optimal to target a commonclass with a single lead time to multiple types with

different delay costs. If it is optimal to pool sometypes into the same class, then virtual delay costsmust be decreasing over a subset of these types. Thisnecessary condition has three variations. If pooling is(strictly) optimal at some lead time w>d, it must bethat f ′

l 4c5 < 0 for some pooled type. Similarly, f ′

h4c5 < 0for some pooled type if pooling is optimal at somelead time w< d. If pooling is optimal at the thresholdlead time d, then fl4c15 > fh4c25 for some pooled typesc1 < c2.

Note that optimal pooling at the lead-time thresh-old d can arise for every type distribution. Example 1in §4.3 explains why. In contrast, pooling at a leadtime w 6= d is not optimal if f ′

l 1 f′

h > 0, which holds formany common probability distributions, includingthose with log-concave density function (see Bagnoliand Bergstrom 2005). Examples include the uniform,normal, logistic, Laplace, and power function distribu-tions, and the gamma and Weibull distributions withshape parameter ≥ 1. However, delay cost distribu-tions that are mixtures of unimodal distributions easilyyield nonmonotone virtual delay cost functions. Suchdistributions might describe markets with multiple seg-ments where across-segment delay cost differences arelarge relative to those within segments. We henceforthassume f ′

l 1 f′

h > 0. We further discuss this assumptionin §6.1.

3. Strategic delay. It may be optimal to intentionallyinflate the lead times of some types above operationallyfeasible levels, which is not work conserving. In ourmodel, doing so can only be optimal at the thresholdlead time d. In this case all types c with fh4c5 < 0 buythe lead time d. Hence, strategic delay also impliespooling in our model, but the converse does not hold.

In contrast to the first-best solution, a menu thatinvolves pooling, with or without strategic delay, hasone or more “gaps” between the offered lead times. Thisimplies less lead-time differentiation among pooledtypes and more differentiation relative to neighboringtypes buying different classes.

4.3. Optimal Segmentation and Lead Timesfor Fixed Arrival Rate

We now discuss step 2 of the solution approach outlinedin §2.2. Let �∗

l 4�5, �∗m4�5 and �∗

h4�5 denote the optimalcustomer segmentation as a function of the arrivalrate �, where �∗

l 4�5+�∗m4�5+�∗

h4�5= �. Write c∗

l 4�5¬F −14�∗

l 4�5/å5 and c∗

h4�5¬ F̄ −14�∗

h4�5/å5 for the corre-sponding marginal types. Let C∗

l 4�5, C∗m4�5 and C∗

h4�5denote the sets of types buying low, medium, and highlead-time qualities, respectively. Finally, let w∗4c3�5denote the optimal lead time function.

Lemma 1. Fix � ∈ 401å7∩ 401�5. Assume strictly in-creasing virtual delay cost functions fl, fh.

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1. The optimal lead-time menu and the correspondingscheduling policy have the following properties:

(a) The lead times satisfy

w∗4c3�5=

�

4�−åF̄ 4c552<d1 if c∈C∗

h4�51

d1 if c∈C∗m4�51

�

4�−6�−åF 4c5752>d1 if c∈C∗

l 4�50

(16)

(b) If low lead-time qualities are sold (Cl 6= �) thenthe optimal policy is work conserving.

2. Under the optimal customer segmentation, the virtualdelay costs are positive and increasing over types with lowor high lead-time quality: (a) fh4c∗

h4�55≥ 0. (b) If �∗

l 4�5 > 0and �∗

h4�5 > 0 then fl4c∗

l 4�55 ≤ fh4c∗

h4�55; if in addition�∗m4�5 > 0 then fl4c

∗

l 4�55= fh4c∗

h4�55.

Lemma 1 provides a set of simple rules to determinethe optimal lead-time menu for a particular arrivalrate. First, Lemma 1.1(a) limits pooling to types thatbuy the medium lead time d, i.e., types in C∗

m4�5.Customers in C∗

h4�5 receive strict priority over typesin C∗

m4�5, which receive strict priority over types inC∗

l 4�5. Different types in C∗

l 4�5 and C∗

h4�5 buy differentlead times and are strictly prioritized in the orderof their delay cost. Different types in C∗

m4�5 buy thesame lead time and are pooled into a single FIFOservice class. Second, Lemma 1.1(b) limits strategicdelay to the case where no customers are served in thelow lead-time quality segment. Third, Lemma 1.2(a)rules out serving customers with negative virtual delaycosts in the high-quality segment; if it is operationallyfeasible to serve such customers with lead time lowerthan d, they should instead be offered the lead time d,which yields strategic delay. Finally, Lemma 1.2(b)requires monotone virtual delay costs across admittedcustomers who are not pooled at lead time d.

Figure 1 (Color online) Example 1: Optimal Menu with Intermediate Types Priced Out or Pooled

1.0 1.5 2.00

1

2

cl cl′ ch

′ ch

(a) Lead times w(c)

1.0 1.5 2.0cl cl

′ ch′ ch

0

1

2

3

(b) Prices p(c)

1.0 1.5 2.0

cl cl′ ch

′ ch

0

1

2

3

4

V(c)

(c) Full prices p(c) + cw(c)

c c c

Notes. Optimal menu properties represented by solid lines, first-best menu properties by dashed lines. Parameters: �= 3, å= 2, �= 1075, v = 1, d = 1, and f 4c5

uniform with 6cmin1 cmax7= 61127.

Substituting the lead-time function (16) in (13) yieldsthe revenue function

ç4�l1�h1�5

¬�v−å∫ cl4�l5

cmin

f 4x5fl4x5

(

�

4�−6�−åF 4x5752−d

)

dx

+å∫ cmax

ch4�h5f 4x5fh4x5

(

d−�

4�−åF̄ 4x552

)

dx0 (17)

Using Proposition 1 and Lemma 1, we reformulateProblem 1 to the following program.

Problem 2.

max�l≥01 �h≥01 �

ç4�l1�h1�5 (18)

subject to

�l +�h ≤ �≤å1 (19)

�<�1�

4�− 6�−�l754�−�h5≤ d1 if �−�l > 01 (20)

�

4�− 6�−�l752

≥ d1 if �l > 00 (21)

Constraint (20) ensures that the lead times in thehigh- and medium-quality classes do not exceed d,and (21) ensures that the policy is work conservingif there are customers that are served with low lead-time qualities. Constraint (20) implies the low-capacitythreshold �= 1/d: at lower capacities, only low lead-time qualities can be offered (that is, �−�l = 0), andneither pooling nor strategic delay are optimal byLemma 1.

We illustrate Lemma 1 with two examples. In bothcases we compare the optimal menu to the first-best(FB) benchmark, which strictly prioritizes all typesthat are served in the order of their delay costs andoperates a work conserving policy.

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Afèche and Pavlin: Segmentation, Pooling, and Strategic Delay2420 Management Science 62(8), pp. 2412–2436, © 2016 INFORMS

Figure 2 (Color online) Example 2: Optimal Menu with Low End of Delay Cost Spectrum Strategically Delayed

0.25 1.250

1

2

0.75 0.25 1.250.75 0.25 1.250.75ch = co

c c c

ch = co ch = co

0

1

2

0

1

2

3

V(c)

(a) Lead times w(c) (b) Prices p(c) (c) Full prices p(c) + cw(c)

Notes. Optimal menu properties represented by solid lines, first-best menu properties by dashed lines. Parameters: �= 3, å= 1, �= 1, v = 1, d = 1, and f 4c5

uniform with 6cmin1 cmax7= 60025110257 and fh4c05= 0 for c0 = 00625.

Example 1. Pricing out or pooling intermediatetypes. In this example, shown in Figure 1, it is notfeasible to serve all customers with a lead time shorterthan d under strict priorities, because d = 1 and thearrival rate �= 1075 <å= 2 is relatively high given thecapacity �= 3.

Consider the FB lead times, shown by the dashedline in Figure 1(a). The type c′

h receives w4c5= d understrict priorities, and it is the lower bound on thetypes that are served in the high-quality segment(w< d). The remaining types that are served must beadmitted into the low-quality segment, that is, theirlead times exceed d. Because the net value for such leadtimes decreases in impatience, the optimal low-qualitysegment is an interval that includes the most patienttype, i.e., 6cmin1 c

′

l7. Because � < å the types in theremaining intermediate interval 4c′

l1 c′

h5 are priced out.Next consider how and why the optimal lead times

involve pooling. The marginal types c′

l and c′

h underthe FB lead times are close enough so that their virtualdelay costs are strictly decreasing. That is, fl4c′

l5 > fh4c′

h5and c′

l < c′

h, which violates the necessary conditionfor optimality in Lemma 1.2(b). Recall from §4.1 thatfl4c5 and fh4c5 have identical own price effects (equalto c), and that the external price effect of fl4c5 ispositive, whereas that of fh4c5 is negative. Becausethe types c′

l and c′

h are close enough, their externalprice effects dominate their own price effects, so thatfl4c

′

l5 > fh4c′

h5 and, unlike in the FB menu, it is notoptimal to serve c′

h with a shorter lead time than c′

l:speeding up c′

l at the expense of c′

h increases revenues.2

Starting with the FB lead times, the optimal lead timesare obtained by pooling a set of customers 4cl1 c

′

l5 fromthe low-quality segment with a set of customers 4c′

h1 ch5

2 If the marginal types of the low- and high-quality segments aresufficiently far apart, their virtual delay costs are nondecreasing andthe FB lead times are optimal. The discussion of Figure 3 includesthat case.

from the high-quality segment into a common classsuch that fl4cl5= fh4ch5, as required by Lemma 1.2(b).As shown in Figure 1(a), compared to the FB leadtimes, the optimal lead times are lower for the pooledtypes in 4cl1 c

′

l7, higher for the pooled types in 6c′

h1 ch5,and unchanged for the remaining strictly prioritizedcustomers.

Figure 1(b) shows the prices. Compared to the FBprices, the optimal prices are lower only for the pooledtypes in 6c′

h1 ch5, because their lead times are longer thanin the FB menu. However, pooling increases revenueas the price reductions for these types are more thanoffset by price increases for all other types: comparedto the FB prices the provider can increase prices forthe pooled types in 4cl1 c

′

l7 because their lead timesare shorter, for the strictly prioritized patient typesin 6cmin1 cl7 because of the shorter lead times of themore impatient types in 4cl1 c

′

l7, and for the strictlyprioritized impatient types in 6ch1 cmax7 because of thelonger lead times of the more patient types in 6c′

h1 ch5.Figure 1(c) shows for both menus the value function

V 4c5 = v + cd and the full price p4c5+ cw4c5. Theirdifference is the customer’s utility. The full prices ofthe optimal menu strictly exceed those of the FB menu,resulting in lower customer utility and higher providerrevenue.

Example 2. Strategic delay for the most patienttypes. This example, shown in Figure 2, differs in threeimportant respects from Example 1. First, the marketsize å= 1 is smaller, so that under the FB menu allcustomers can be served with lead time lower thand = 1; refer to Figure 2(a). Second, all customers areserved, i.e., �=å. Third, the types at the low end ofthe delay cost spectrum are so patient that their virtualdelay costs are negative, i.e., fh4c5 < 0 for c < c0 = 00625and fh4c05= 0. For these types the external price effectof their virtual delay cost dominates their own priceeffect. That is, increasing the lead time for a type c

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Afèche and Pavlin: Segmentation, Pooling, and Strategic DelayManagement Science 62(8), pp. 2412–2436, © 2016 INFORMS 2421

with fh4c5 < 0 improves the revenue on more impatienttypes by more than it reduces the type-c price.

By Lemma 1.2(a), optimality requires that all typesin the high-quality segment have a nonnegative virtualdelay cost, i.e., fh4ch5 ≥ 0. Therefore, as shown inFigure 2(a), compared to the FB lead times, it is optimalto increase the lead times of the types in 6cmin1 c07 upto d = 1 by inserting strategic delay, and to leave thelead times of the types in 6c01 cmax7 unchanged.

Figure 2(b) shows that, compared to the FB prices,the optimal prices are lower for the types in 6cmin1 c05,but this revenue loss is more than offset by higherprices for the types in 6c01 cmax7.

Figure 2(c) shows again the loss in customer surplusunder the optimal menu relative to the FB menu.

4.4. Optimal Segmentation and Lead TimesDepending on the Arrival Rate and Market Size

Proposition 2 (see the appendix) specifies the solution ofProblem 2 as a function of � and å. Figure 3 illustratesthese results for the case where high lead-time qualitiescan be offered (�> 1/d). It shows for each 4å1�5 whichlead-time classes are sold in the optimal menu, high (h),medium with strategic delay (msd) or without (m), andlow (l). For instance, for Example 1 where å= 2 and� = 1075, Figure 3(a) confirms that all three qualityclasses 4h1m1 l5 are sold. (Proposition 2 characterizesthe market size thresholds å1, å2, å3, å4, åsd, and å̄sd

shown in Figure 3.)Pricing Out the Middle. Figure 3 illustrates the condi-

tions for optimally pricing out some intermediate typeswhile selling low and high lead-time qualities to themost patient and the most impatient types, respectively.Such a menu is optimal if the market size å is suffi-ciently large, and the arrival rate is smaller than themarket size (�<å) but too large to serve all types inthe high- and/or medium-quality segments. In Figure 3this holds for �<å in the regions labeled 4h1 l5 and4h1m1 l5.

Figure 3 Illustration of Optimal Segmentation as a Function of � and å

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 1 2 3 4 5 6 7

�P

�

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

�P

�

�1�2 �3 �4� �

(h)

(h,m, l )

(h,m)

(h, l )

(h)

(h,m, l)

(h, l)

(a) Strictly positive virtual delay costs ( fh(c) > 0)

0 1 2 3 4 5 6 7

(b) Segment of negative virtual delay costs ( fh(cmin) < 0)

�sd– �sd–

(h,m)

(h,msd)

Note. Parameters: �= 3, v = 11 d = 1, and f 4c5 uniform with 6cmin1 cmax7= 61127 in panel (a) and 6cmin1 cmax7= 60025110257 in panel (b).

Pooling at w = d. Figure 3(a) illustrates the conditionsfor optimal pooling in the case of positive virtual delaycosts fh (which rules out optimal strategic delay). Atlow arrival rates �≤ �P =�−

√

�/d = 1027, pooling isnot optimal because capacity is sufficient to admit allcustomers into the high lead-time quality segment 4h5.For higher arrival rates, �> �P , there is a thresholdon the market size å (which depends on �), such thatfor å below this threshold some customers are pooled(the regions labeled 4h1m5 and 4h1m1 l5), whereas for åabove this threshold all customers are strictly priori-tized (the region labeled 4h1 l5). This structure followsbecause for fixed �, increasing å increases the mass ofeach type so that the difference between the types inthe high-quality versus low-quality segments growslarger. Therefore, for sufficiently large market size, themarginal types of the low- and high-quality segmentsunder the FB menu are sufficiently far apart that theirvirtual delay costs are nondecreasing (i.e., fl4c′

l5≤ fh4c′

h5),and so pooling is suboptimal by Lemma 1.

Strategic Delay. Figure 3(b) illustrates the conditionsfor optimal strategic delay in the case where the mostpatient types have negative virtual delay costs fh.Specifically, fh4c5 < 0 for the types c ∈ 6cmin = 00251 c0 =

006257, and serving them with lead time w < d issuboptimal as shown in Lemma 1 and in Example 2.Strategic delay is optimal in the region of Figure 3(b)labeled 4h1msd5. This region reflects two conditionsfor optimal strategic delay. First, the arrival rate �must be sufficiently large relative to the market size å,so some types with negative virtual delay costs areserved. Second, � must be sufficiently small relative tothe capacity �, so these types’ “natural delays” underwork conserving scheduling (FIFO, but with lowerpriority than types with positive virtual delay costs)are insufficient to increase their lead times up to themedium lead time d.

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Afèche and Pavlin: Segmentation, Pooling, and Strategic Delay2422 Management Science 62(8), pp. 2412–2436, © 2016 INFORMS

5. Optimal Arrival Rate, Segmentation,and Lead Times

We turn to step 3 of the solution approach outlined in§2.2 and discuss the optimal segmentation and lead-time menu at the optimal �, in §5.1 for fixed capacity,and in §5.2 as a function of capacity.

5.1. Fixed CapacityThe solution hinges on Proposition 2 and the followingresult. (We write gx and gxy for the first- and second-order partial derivatives of a bivariate function g4x1y5.)

Lemma 2. Fix the market size å. Write ç∗4�1�5 for therevenue function under the optimal segmentation and leadtimes.

1. (a) ç∗�4�1�5 ≥ v for every 4�1�5 if the optimal

segmentation is 4h5 and ç∗�4�1�5 = v if it is 4h1msd5.

(b) ç∗��4�1�5≤ 0 ≤ç∗

��4�1�5 for every 4�1�5 if the opti-mal segmentation is 4h5 or 4h1msd5, and ç∗

��4�1�5 < 0 <ç∗

��4�1�5 if it is 4l51 4h1m51 4h1 l51 4h1m1 l5 or 4m1 l5.2. (a) ç∗

�4�1�5 is continuous in 4�1�5. (b) The partialderivatives ç∗

��4�1�5, ç∗��4�1�5 are continuous in 4�1�5

under each optimal segmentation; they are also continuousat every 4�1�5 where a transition takes place between twoof the optimal segmentations 4l51 4h1m51 4h1 l51 4h1m1 l5,and 4m1 l5.

For fixed � the optimal segmentation and lead-timemenu do not depend on the base value v. The basevalue v scales the valuations V 4c5= v+ c · d, so theprofitability of all types increases in v. Write �∗4v5 forthe optimal arrival rate as a function of v. Lemma 2implies that �∗4v5 increases in v with �∗4v5→ min4�1å5as v → �. Furthermore, if v > 0 then selling h classeseither exclusively or together with the strategicallydelayed medium lead-time class (the segmentations4h5 and 4h1msd5, respectively) can only be optimalif the entire market is served, i.e., �∗4v5 = å: underthese segmentations each additional customer increasesrevenue by at least v. These properties imply thefollowing result on optimal pooling and strategic delayas a function of market size.

Theorem 1. Fix a capacity �> 0 and assume that f ′

l > 0and f ′

h > 0.1. If d ≤�−1 or v ≤ 0, then pricing out the middle of the

delay cost spectrum, pooling, and strategic delay are notoptimal.

2. If d > �−1 and v > 0, then there are market sizethresholds å1 < å2 < å3 < å4 and åsd < å3 such thatpooling and strategic delay are optimal as shown in Table 1.

In the case of nonnegative virtual delay costs(fh4cmin5≥ 0), pooling is optimal only if the market sizeis in some intermediate range 4å11å45: at smaller mar-ket sizes all customers are served with strict prioritiesin h classes, at larger market sizes it is optimal to sell

Table 1 Fixed �: Optimality of Pooling, Strategic Delay

fh4cmin5≥ 0 fh4cmin5 < 0

å≤ å1 No pooling, onlyh classes

å<åsd Strategic delay withpooling iff v > 0

å ∈ 4å11å35 Pooling iff v > 0 å ∈ 6åsd 1å35 Pooling iff v > 0å ∈ 4å31å45 Pooling iff v

sufficiently largeå ∈ 4å31å45 Pooling iff v

sufficiently largeå≥ å4 No pooling, h and l

classeså≥ å4 No pooling, h and l

classes

only h and l classes and price out the middle, becausethere are enough profitable patient and impatient typeswith sufficiently different virtual delay costs. In thepresence of patient customers with negative virtualdelay costs (fh4cmin5 < 0), the results are the same formarket sizes larger than åsd, but strategic delay withpooling is optimal if the market size is smaller than åsd:in this case there is more than enough capacity to serveall customers with lead times w ≤ d, such that typeswith positive virtual delay costs are strictly prioritizedand all others are pooled with w = d.

5.2. Impact of CapacityTheorem 2 specifies how pricing out the middle, pool-ing, and strategic delay, depend on the capacity.

Theorem 2. Let �∗4�5 be the optimal arrival rate as afunction of capacity. If f ′

l 1 f′

h > 0, d > 0, and v > 0, theoptimal segmentation and lead times are as follows. Definethe capacity thresholds

�min¬1

d+v/cmin<

1d<�H¬å+

1+√

1+4då2d

0 (22)

If fh4cmin5 < 0, let the strategic delay threshold �SD be theunique solution in � ∈ 4å+ d−11�H 5 of

�−�/d

�−å=åF̄ 4f −1

h 40550 (23)

1. Pricing out the middle of the delay cost spectrum isoptimal iff � ∈ 4d−11�A5, where �A is the market coveragethreshold. The optimal arrival rate �∗4�5 is strictly increasingon 6�min1�A7, where �A >�min and �∗4�5=å⇔�≥�A.If fh4cmin5 ≥ 0 then �A < �H , and if fh4cmin5 < 0 then�A <�SD <�H .

2. Pooling and strategic delay. There is a unique capacitythreshold �P such that

(a) if fh4cmin5 ≥ 0 then �P ∈ 6d−11�H 5, pooling isoptimal iff � ∈ 4�P 1�H 5, and strategic delay is suboptimal;

(b) if fh4cmin5 < 0 then �P ∈ 6d−11�SD5, pooling isoptimal iff �>�P , and strategic delay is optimal iff �>�SD.

3. Effect of base value v. There are thresholds vP <vA,such that pooling is optimal for all � ∈ 4d−11�H 5 iff v ≥ vP ,and serving the entire market is optimal for all �≥ d−1 iffv ≥ vA.

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Pricing out the middle can be optimal only for capacityin the range 4d−11�H 5. In this range the providercan sell high lead-time qualities to some but not allcustomers. For fixed capacity, increasing the arrival rateby selling service to more intermediate types generatesadditional revenue, but it also slows down the servicethat can be provided to more patient customers, whichreduces the revenue from these customers. The gainfrom additional intermediate types increases in theirvaluations, the loss on the more patient customersdecreases in the capacity level, giving rise to two cases(parts 1 and 3 of Theorem 2). If valuations are below athreshold (v < vA) then pricing out the middle of thedelay cost spectrum is optimal in the capacity range4d−11�A5, where �A <�H : under these conditions therevenue gain from additional intermediate types isinsufficient to offset the loss on more patient customers,so market coverage is not optimal, but capacity is stillsufficient to profitably sell both low- and high-lead-time qualities. In contrast, if valuations are sufficientlyhigh (v ≥ vA), then at every capacity in the range4d−11�H 5 the revenue from additional intermediatetypes dominates the congestion-driven revenue losseson more patient types, so market coverage is optimal(and �A ≤ d−1).

Pooling at w = d is optimal in the intermediate capac-ity range 4�P 1�H 5. In this range, capacity is insufficientto serve all customers with high quality, but stillsufficient to profitably serve such large low- and high-quality segments that their marginal types cl and chare similar enough to be served in a single class (seeExample 1).

Strategic delay is optimal for sufficiently large capac-ities (�>�SD) if and only if the most patient typeshave negative virtual delay costs (fh4cmin5 < 0).

6. Connections and Extensions6.1. Pooling with Increasing vs. Nonmonotone

Virtual Delay CostsWe call a delay cost distribution f irregular if it yieldsnonmonotone fh and/or fl. In this case the conditions ofTheorem 2 for pricing out the middle, and for poolingand strategic delay at the threshold lead time d, remainstructurally the same. However, pooling may also beoptimal within the high- and low-quality segments,and the thresholds �P and �SD may change.

Conditions for Pooling at w 6= d. Table 2 lists necessaryand sufficient conditions for pooling at lead times w 6= d.To summarize, for pooling with lead time smaller than d

Table 2 Nonmonotone Virtual Delay Costs: Conditions for Optimal Pooling at Lead Times w < d or w > d

Lead time Necessary conditions Sufficient conditions

w < d v >−dcmax1 d > 0 f ′

h≯ 0 �> max41/d11/4d + v/cmax55 � > �H 1 v > 0, ∃c s.t. f ′

h4c5 < 0 < fh4c5

w > d v > 01 d >−v/cmin f ′

l ≯ 0 � ∈ 4�min1 �H 5 if d > 0, or �> �min if d ≤ 0 �>å, v large, d ≤ 0, f ′

l ≯ 0

(greater than d), a type c where f ′

h4c5 < 0 (f ′

l 4c5 < 0) mustbe admitted into the high-quality (low-quality) segment.Therefore, pooling at w < d is optimal if there is enoughcapacity to serve everyone with lead time shorter thand (d > 0 and �>�H ), all types are profitable (v > 0),and some types c should be pooled without strategicdelay (fh4c5 > 0 > f ′

h4c5). Similarly, pooling at w>d isoptimal if there is enough capacity to serve everyone(�>å), only lead times longer than d can be offered(d ≤ 0), all types are profitable (v large), and some typesc should be pooled (f ′

l 4c5 < 0).Contrasting Pooling at w = d with Pooling at w 6= d.

Pooling at w = d arises only if it is profitable to offersome lead times shorter than and some longer than d.This requires v > 01 d > 0, and intermediate capacitylevels—neither too low, nor too high (Theorem 2).In this case, two disjoint type intervals are served,with different virtual delay cost functions: for eachtype c in the low-quality (high-quality) segment, theexternal price effect of its virtual delay cost, fl4c5(fh4c5), is positive (negative); that is, reducing its leadtime increases (decreases) the prices for more patient(impatient) types. Therefore, fl4c5 > fh4c5 holds for everydelay cost distribution, and as discussed in §§4–5.1,pooling at the threshold lead time is optimal if thelow- and high-quality segments are sufficiently close.Pooling at w 6= d differs in two ways from pooling atw = d: first, it can also arise if it is profitable to onlyserve a single interval of types, all of them with leadtimes either shorter or longer than d. This only requiresv > 0 or d > 0, but not both. Second, because all typeswithin a segment share the same virtual delay costfunction (fl or fh), pooling can only be optimal if therelevant virtual delay cost function is nonmonotone.

To summarize, pooling at w = d can occur for everydelay cost distribution but only at intermediate capacity.Pooling at w 6= d can only occur for an irregular delaycost distribution but merely requires enough capacityso that some types with nonmonotone virtual delaycost are profitable.

6.2. Special Cases with Lead-Time-IndependentRanking of Types

Net values satisfy N4c1w5= V 4c5− cw = v+ c4d−w5. Ifv > 0 and d > 0 then both lead times w< d and w> dmay be profitable, which implies a lead-time-dependentranking of types: their net values increase in c for w < dand decrease in c for w>d. This property gives riseto pricing out the middle of the delay cost spectrum,work conserving pooling at w = d, and strategic delay.

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Afèche and Pavlin: Segmentation, Pooling, and Strategic Delay2424 Management Science 62(8), pp. 2412–2436, © 2016 INFORMS

We discuss three parameter regimes that give rise to alead-time-independent ranking of types.

Increasing V 4c5/c Ratio: v ≤ 0. If v ≤ 0 <d then onlylead times shorter than d are profitable. To contrast thiscase with Theorem 2 we state the following intuitiveLemma without proof.

Lemma 3. If v ≤ 0 < d then the following holds. (1) Pric-ing out the middle of the delay cost spectrum, pooling atw = d, and strategic delay are not profitable. (2) The optimalarrival rate �∗4�5 increases in �, but lim�→� �∗4�5=å ifand only if fh4cmin5≥ �v�/d.

Part 1 of Lemma 3 follows because only lead timesw < d are profitable for v ≤ 0. The ranking of customertypes is invariant for such lead times, i.e., their netvalues are increasing in impatience because Nc4c1w5=

d−w > 0. Therefore, the set of types served is contigu-ous and includes the most impatient ones; this alsoprecludes pooling at the threshold lead time. Strategicdelay is not profitable because types c with negativevirtual delay costs fh4c5 < 0 are willing to pay at mostv ≤ 0, so the provider gains (does not lose) by notserving them. Therefore, with v ≤ 0 <d, the only non-standard solution feature is pooling at lead times w < d,which is optimal only if f ′

h ≯ 0. Part 2 of Lemma 3 fol-lows by noting that a type c with positive virtual delaycosts contributes v+ fh4c54d−w4c55 to total revenues;as capacity gets large, lead times go to zero, so thecondition in the lemma is required for the most patienttype to have a positive revenue contribution.

With v ≤ 0 < d our model specializes to that forpriority auctions in Afèche and Mendelson (2004),3

and it is essentially equivalent to the model of KS.Nazerzadeh and Randhawa (2015) consider essentiallythe same demand model as Afèche and Mendelson(2004), but they focus on the asymptotic performanceof menus that offer only two classes, for systemswith large potential demand and capacity. Afèche andMendelson (2004) restrict attention to work conserv-ing strict priority policies. Our analysis shows this iswithout loss of optimality in their model. Strategicdelay is not optimal since v ≤ 0. Pooling at lead timesw < d is not optimal because they assume f ′

h > 0 (whichis equivalent to their assumption that the function�ê̄−14�/å5 is strictly concave in �, where ê is thec.d.f. of valuations). KS mainly analyze a discrete-typeversion of the model of Afèche and Mendelson (2004),but do not restrict the scheduling policy. However,they restrict attention to the case where the valuation-delay cost ratio increases in the delay cost, that is,

3 Afèche and Mendelson (2004) fix �= 10 They consider i.i.d. val-uations x with continuous c.d.f. ê4x5 for x ∈ 6v1 v̄7 and perfectlycorrelated delay costs, given by xd+ c for x ∈ 6v1 v̄7, where d ∈ 401�5and c ≥ 0 are constants. See Assumptions 1, 4, and 5. A simple changeof variable yields v = −c/d ≤ 0, d = 1/d > 1/�, and F 4c5=ê4v+ cd5for c ∈ 6cmin1 cmax7, where cmin = vd+ c and cmax = v̄d+ c. To avoidconfusion we use here c, d, and x for their c1d, and v, respectively.

vi+1/ci+1 <vi/ci, where ci+1 < ci. Although KS do notassume an affine 4vi1 ci5-relationship, their model is inessence equivalent to ours with v < 0. In our model, theratio V 4c5/c = v/c+d increases in c if and only if v ≤ 0.Models with increasing V 4c5/c ratio but nonaffine V 4c5,such as the one of KS, yield the same fundamental prop-erties as ours with v ≤ 0: because V 4c5/c is increasingif and only if V ′4c5 > V 4c5/c, and N4c1w5 > 0 ⇔

V 4c5/c >w, a lead time w is profitable for type c onlyif w <V ′4c5. The ranking of types is invariant for suchlead times in that Nc4c1w5= V ′4c5−w > 0. In particular,Lemma 3 applies, and pooling can be optimal onlyat lead times w <V ′4c5 and if f ′

h ≯ 0. KS analyze asegmentation algorithm that computes the optimal leadtimes for their discrete-type model and yields poolingonly if the discrete version of our virtual delay cost fhis not monotone increasing.

Identical Valuations or Negative Value-Delay Cost Corre-lation: d ≤ 0. If v > 0 ≥ d then net values decrease in cfor all feasible lead times. Therefore, only lead timesw> d are offered, the set of types served is contiguousand includes the most patient ones, and there is nopooling at the threshold lead time. Strategic delay isnot profitable because reducing the lead times of typeswith w>d increases the prices of more patient types(fl > 0 because the external price revenue effect in thelow-quality segment is positive). Therefore, pooling isoptimal only at lead times w> d and if f ′

l ≯ 0.Identical Delay Costs: v → −�, d → �. In models

where types have the same delay cost c but hetero-geneous valuations, a single class is optimal, e.g.,Mendelson (1985). Such a model emerges as the limit-ing case of ours if one lets 6cmin1 cmax7 get small, so thatv → −� and d → �.

6.3. Strategic Delay: Log Supermodularity,Damaged Goods, Queueing Effects

Anderson and Dana (2009), henceforth AD, consider amaximum quality constraint in the standard monopolyprice discrimination model with quality-independenttype ranking, that is, for every quality the customer sur-plus is increasing in the type. AD ignore capacity andqueueing effects. They show that price discrimination,i.e., offering at least two quality levels, is optimal if(i) the surplus function is log supermodular, and (ii) itis not profitable to serve the lowest types at any quality.Together these conditions imply that it is optimal todegrade quality for the lowest profitable type.

In our model, log supermodularity of the net valuefunction is equivalent4 to the condition v > 0 (or V 4c5−cV ′4c5 > 0 in general). This condition, together withd > 0 (or V ′4c5 > 0 in general) is necessary but notsufficient for optimal strategic delay. In our model, log

4 Let the lead times be a function w4q5 of quality q ∈ 6q1 q̄7, with w4q̄5=

0 and w′4q5 < 0. Then the net value N4c1w4q55 is log supermodular in4c1 q5 iff v > 00

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supermodularity implies that all types are profitable ifserved with lead time w = d. For strategic delay to beoptimal, we also require fh4cmin5 < 0, i.e., the existenceof types whose quality can be lowered at a net benefit;and �>�SD, i.e., enough capacity so that their leadtimes must be increased artificially rather than throughqueueing delays.

To highlight the connection to AD, consider amplecapacity (� = �). The revenue contribution of thelowest type is v + fh4cmin54d −w5. Given v > 0 andd > 0, condition (ii) in AD implies v+ fh4cmin5d < 0 forw = 0, which implies fh4cmin5 < 0. Therefore, condition(ii) in AD is stronger, but more generally applicablethan fh4cmin5 < 0 in our model. The model and resultsof AD apply to the damaged goods literature (seeDeneckere and McAfee 1996; McAfee 2007, §4.3 in AD).These papers assume a nonincreasing quality cost,as in our model, but ignore capacity constraints. Atample capacity, the solution in our model resembles thedamaged goods solution for a model with zero costs: ahigh-quality segment 6c01 cmax7 pays a high price V 4c05for the lead time w4c5= 0, and a low-quality segment6cmin1 c07 pays v for the “damaged lead time” d > 0.

However, queueing effects play a significant rolefor strategic delay: queueing increases the minimumcapacity threshold for optimal strategic delay. In ourmodel this threshold exceeds the market size, that is,�SD >å by (23). In a damaged goods model withoutqueueing, this threshold is smaller than the market size.To see this, drop the work conservation constraints (3)in Problem 1 to eliminate queueing. Let �0 be the rate ofcustomers with nonnegative virtual delay cost fh, where�0 <å if fh4cmin5 < 0. Then for v > 0, d > 0, strategicdelay is optimal if and only if �>�0, where �0 <�SD.

6.4. Nonaffine Value-Delay Cost RelationshipThe affine value-delay cost relationship V 4c5= v+ c · dyields the simplest type model with multiple delaycosts and lead-time-dependent ranking. It has therestriction that all types value the threshold lead-time dequally. However, our key results hold for a broaderclass of V 4c5 functions, and our analysis sheds lighton the underlying demand and capacity conditions.We discuss these conditions for strictly convex andstrictly concave V 4c5. In both cases, types differ in theirnet value for all lead times. The discussion draws ontwo properties of any IC menu (see Proposition 1): theutility changes at the rate U ′4c5= V ′4c5−w4c5 and leadtimes decrease in impatience (w′4c5≤ 0).

For convex V 4c5 it may be optimal to price out and/orpool intermediate types, as in the affine case, in bothcases because U4c5 is convex. Specifically, for increas-ing V 4c5 and intermediate capacity levels, intermediatetypes may be the least profitable customers. In thiscase U4c5 is decreasing-increasing, so two disjoint inter-vals of types are served with positive utility, includingthe most and the least patient types. Different virtual

delay cost functions, fl and fh, apply to these disjointsegments, hence pooling improves revenue if they areclose enough, i.e., fl4cl5 > fh4ch5. However, the thresholdlead time now depends on V 4c5, f 4c5, and �.

For concave V 4c5 the segmentation may resemble orreverse the structure of the affine case, but our poolingresult does not hold. The segmentation depends onthe concavity of V 4c5. For sufficiently low levels ofconcavity, U4c5 may be convex, with intermediatetypes priced out as in the affine case. For sufficientlyconcave V 4c5, the intermediate types may be the mostprofitable and U4c5 will be concave. Contrary to theaffine model, in this case it may be optimal to serveonly intermediate types and price out both the moreand the less patient types. For concave V 4c5 pooling issuboptimal with monotone virtual delay costs, becauseit results in concave U4c5 for the pooled types. That is,unlike for (strictly) convex V 4c5, for concave V 4c5 thepooled types have positive utility and form a singleinterval (with the same virtual delay cost function).Therefore, the provider can extract more surplus fromthese types by differentiating their lead times.

As outlined in §6.3, strategic delay can be optimalfor every V 4c5 function that is increasing and logsupermodular, provided that fh4cmin5 < 0 and there isenough capacity.

Finally, the menu-design problem for two-dimen-sional types with an unrestricted valuation-delay costdistribution remains open. The difficulty arises because(1) the relationship between segmentation structureand model parameters is even more intricate thanunder perfect correlation, and (2) the number of ICconstraints is quadratic in the number of types, andlocal IC between neighboring types in the same segmentdoes not ensure global IC across segments. Mechanismdesign problems with unrestricted multidimensionaltype distributions are notoriously cumbersome, even inthe absence of queueing constraints; see Rochet andChoné (1998).

Supplemental MaterialSupplemental material to this paper is available at http://dx.doi.org/10.1287/mnsc.2015.2236.

AcknowledgmentsThe authors thank the department editor, the associate editor,and the referees for their many helpful suggestions.

Appendix. ProofsTable A.1 summarizes the notation and indicates as applicablewhere it is introduced.

Proof of Proposition 1The constraints 445–465 imply parts 1–5:

Write U4c′3 c5 for the expected utility of a type c thatreports type c′, and U4c5¬U4c3 c5, where

U4c′3 c5 ¬ U4w4c′51 p4c′53 c5= v+ c4d−w4c′55− p4c′5

= U4c′5+ 4c− c′54d−w4c′550 (24)

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Afèche and Pavlin: Segmentation, Pooling, and Strategic Delay2426 Management Science 62(8), pp. 2412–2436, © 2016 INFORMS

Table A.1 Notation

å, � Market size (maximum arrival rate), capacityc, v , d Delay cost (customer type), base value, threshold lead timecmin, cmax Minimum and maximum delay costF 4c5, F̄ 4c5, f 4c5 Delay cost CDF, its complement and PDFa4c5 Acceptance function indicating whether type c buys service or not£a, °£a Sets of customers who, respectively, buy and do not buy servicep4c5, w4c5, U4c5 Price, expected lead time, and expected utility of type c

ç Revenue rate� Arrival rate of customers to all classes�l , �m , �h Arrival rates to l, m, h classesfl 4c5, fh4c5 Virtual delay cost functions of customers in Cl and Ch

Proposition 1Cl , Cm , Ch Sets of customers who buy low (l), medium (m), high (h) classescl , ch Marginal customer types of segments buying l and h classesU4c′3 c5 Expected utility of type c who reports type c′

Um Expected utility of a type c who buys the m class

Lemma 1D4�l 1 �h1w5 Virtual delay cost rate�∗

l , �∗

m , �∗

h , w∗ Optimal arrival rates to l, m, h classes, and optimal lead times, for fixed �

c∗

l , c∗

h Marginal types under optimal segmentation for fixed �

C∗

l , C∗

h Sets of customers who buy l and h classes under optimal segmentation for fixed �

Proposition 2l, m, msd , h Indicator of positive arrival rate to classes l, m, m with strategic delay, and h

å11å21å31å41åml 1 å̄ml Market size thresholds for poolingåsd , å̄sd Market size thresholds for strategic delay�mh, �∗

mh4�5 Total arrival rate to m and h classes, and optimal �mh as function of �ç4�mh1 �h5 Revenue rate as function of arrival rates �mh and �h�F , �P Upper bounds, on arrival rate �mh under FIFO, and on �h under strict priorities�̄h4�mh5, �h4�mh5 Maximum feasible �h, and optimal �h, as function of �mh

c0 = f−1h 4051 �0 = åF̄ 4c05 Type with zero virtual delay cost, arrival rate of higher types to h classes

g4�1 �mh5 Difference between virtual delay costs of marginal types cl and ch if arrival rate to h classes is �̄h4�mh5

�1, �2, �3, �sd Arrival rate thresholds (Lemmas 5, 6, and 8)

Lemma 2ç∗4�1�5 Optimal revenue rate as function of arrival rate � and capacity �

Theorem 2�∗4�5 Optimal arrival rate as function of capacity �

�min Lower capacity threshold for profitable operation�A, �P , �SD , �H Lower capacity thresholds for market coverage, pooling, strategic delay, and serving all customers with h classesvA, vP Lower base value thresholds for market coverage, pooling over entire applicable capacity range

The IC constraints (6) require that the expected utilities fromservice satisfy

U4c5 = v+ c4d−w4c55− p4c5≥U4c′3 c5

⇔ c ·w4c5+ p4c5≤ c ·w4c′5+ p4c′51

for ∀ c 6= c′0 (25)

Similarly, we must have U4c′5≥U4c3 c′5, so from (24) the ICconstraints (6) are equivalent to

4c− c′54d−w4c55≥U4c5−U4c′5≥ 4c− c′54d−w4c′551

for ∀ c 6= c′0 (26)

It follows from (26) that the expected utility from serviceU4c5 is continuous in the type.

Part 1. It follows from (25) and (26) that w4c5 is nonincreas-ing and p4c5 is nondecreasing in c. If c < c′ (26) impliesw4c′5≤w4c5, and (25) implies p4c′5−p4c5≥ c4w4c5−w4c′55≥ 0.

Because w is nonincreasing in c, it follows that cl ≤ ch, andthat w is Riemann integrable. Therefore, (26) implies

U4c′′5−U4c′5=

∫ c′′

c′4d−w4x55dx1 for all c′ < c′′0 (27)

Part 2. We first show (a). The case Cl = 8cmin9 is trivial.Suppose that Cl 6= � and cl > cmin. Fix c ∈ 6cmin1 cl5. We showthat c ∈Cl. Apply (27) with c = c′ and cl = c′′ to get

U4c5=U4cl5+∫ cl

c4w4x5− d5dx >U4cl5≥ 00 (28)

The first (strict) inequality follows since w4x5 > d for x < cl:otherwise, if w4x5≤ d for some x < cl then w4x′5≤w4x5≤ dfor x′ > x since w4c5 is nonincreasing, contradicting thatcl = supCl. That U4cl5≥ 0 follows since U4c5 is continuousin c: if U4cl5 < 0 then by continuity it must be that U4c5 < 0for all c in some interval 6cs1 cl7 and the IR constraint (4) forc ∈ 6cs1 cl7 can only hold if c yCl. But this contradicts thatcl = supCl, so U4cl5≥ 0. Therefore, U4c5 > 0 for c < cl; the

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IR constraint (5) for c only holds if c ∈ £a, and since w4c5 > dfor c < cl it follows that c ∈ Cl. To prove 4b5 it remains toshow that Cl 6= � ⇒U4cl5= 0 (which we prove with part 5)for then (28) reduces to (9) and the price equation (8) followssince U4c5= v+ c4d−w4c55− p4c5.

Part 3. Follows from the same line of argument as in theproof of part 2, by applying (27) with ch = c′ < c = c′′ andshowing that Ch 6= � implies U4ch5= 0, which we prove withpart 5.

Part 4. Suppose that Cm 6= �. Parts 2 and 3 imply Cm ⊂

6cl1 ch7. Fix c ∈Cm. Then

U4c′5≥U4c3 c′5=U4c5= v− p4c5≥ 01 for ∀ c′6= c0 (29)

The first inequality follows from the IC constraints (6), theequalities hold since w4c5= d, and the IR constraint (4) impliesthe second inequality. It follows from (29) that U4c′5=U4c5for all c′ ∈Cm. Define Um ¬U4c5= v−p4c5 as the commonutility for all c′ ∈Cm.

It remains to show that Um = 0, which we prove withpart 5.

Part 5. That 4cl1 ch5⊂Cm ∪ °£a is immediate from the def-initions of cl and ch. The expression (12) for U4c5 followsfrom (27). We prove that U4c5≤ 0 for c ∈ 6cl1 ch7 for threeexhaustive cases.

(i) Not all types are served ( °£a 6= �) but some types buythe medium lead time (Cm 6= �). This implies that cl < ch. Then(5) and (29) imply 0 ≥U4c′5≥U4c5=Um ≥ 0 for any typesc ∈ Cm and c′ ∈ °£a; therefore we have U4c5= 0 for all c ∈ 4cl1 ch5.Since U4c5 is continuous it follows that U4cl5= 0 =U4ch5.Since Cm ⊂ 6cl1 ch7 it follows that U4c5=Um = 0 for c ∈Cm.

(ii) Not all types are served ( °£a 6= �) and no types buy themedium lead time (Cm = �). It follows that 4cl1 ch5⊂ °£a. TheIR constraints (5) and the continuity of U4c5 imply U4c5≤ 0for c ∈ 6cl1 ch7. If Cl 6= � then (28) implies U4cl5≥ 0 and soU4cl5= 0. Similarly, U4ch5= 0 if Ch 6= �.

(iii) All types are served ( °£a = �). Let Umin = minc U4c5.The IR constraints (4) require Umin ≥ 0, and revenue-maximization requires Umin = 0. The proof is complete ifU4c5=Umin for c ∈ 6cl1 ch7. First note that U4c5=U4cl5=U4ch5for c ∈ 6cl1 ch7. If cl = ch this is trivial. If cl < ch this holdssince then 4cl1 ch5⊂Cm, part 4 implies U4c5=Um for c ∈Cm,and by continuity Um =U4cl5=U4ch5. By parts 2 and 3 wehave U4c5 >U4cl5=U4ch5 for c y 6cl1 ch7, which proves thatU4c5=Umin for c ∈ 6cl1 ch7.

Parts 1–5 imply 445–4652 parts 2–5 imply the IR con-straints (4)–(5). The IC constraints (6) are equivalent to (26).Substituting for U4c5 from parts 2–5, (26) is equivalent to

4c′′−c′54d−w4c′′55≥U4c′′5−U4c′5

=

∫ c′′

c′4d−w4x55dx≥ 4c′′

−c′54d−w4c′551 for ∀c′<c′′0 (30)

By part 1, w4c′5≥w4x5≥w4c′′5 for x ∈ 6c′1 c′′7, which estab-lishes both inequalities. �

Proof of Lemma 1From Problem 1, Proposition 1, and the revenue rate (13), itfollows that for fixed �, the revenue-maximization problemis equivalent to minimizing the virtual delay cost rate

D4�l1�h1w5 ¬ å∫ cl4�l5

cmin

f 4x5fl4x54w4x5− d5dx

+å∫ cmax

ch4�h5f 4x5fh4x54w4x5− d5dx

over �l1�h, and the lead-time function w2 £→�, subjectto �m = �−�l −�h ≥ 0, increasing w4c5, w4c5 > d >w4c′5 forc < cl4�l5 and c′ > ch4�h5, and

å∫ cmax

cf 4x5w4x5dx≥

åF̄ 4c5

�−åF̄ 4c51 ∀c∈ 6ch4�h51cmax71 (31)

å∫

x∈6c1cl7∪6ch1cmax7f 4x5w4x5dx+�m ·d≥

�−åF 4c5

�−6�−åF 4c571

∀c∈ 6cmin1cl4�l570 (32)

Recall that fl and fh satisfy (14) and (15), f ′l 1 f

′h > 0, and that

cl4�l5= F −14�l/å5 and ch4�h5= F̄ −14�h/å5 are the marginaltypes corresponding to �l and �h, respectively. Supposethe scalars �∗

l 1�∗h and the function w∗ are a solution of this

problem and write c∗l = cl4�

∗l 5 and c∗

h = ch4�∗h5.

The proof hinges on three necessary optimality conditions.

C1. If c ∈ 4c∗h1 cmax7 then fh4c5 > 0.

Proof of C1. We argue by contradiction. If fh4c05=0 forsome c0 ∈ 4c∗

h1 cmax5, where fh4cmax5= cmax > 0, then fh4c5 < 0for c ∈ 6c∗

h1 c05 since fh is strictly increasing, and w∗4c5 < d forc ∈ 4c∗

h1 c07. By inspection it is clear that we can reduce thevirtual delay cost rate by perturbing the lead time functionfrom w∗ to wo , where wo agrees with w∗ except that wo4c5= dfor c ∈ 6c∗

h1 c07. Then wo is feasible and D4�∗l 1�

∗h1w

o5 <D4�∗

l 1�∗h1w

∗5. Under the menu wo the marginal type c∗h moves

to c′h = c0 > c∗

h and fh4c5 > 0 holds for c ∈ 4c′h1 cmax7.

C2. If the set of types with high lead-time qualities is nonempty(C∗

h 6= �) then the constraints 4315 are binding for c ∈ 6c∗h1 cmax7,

and w∗4cmax5= 1/�.

Proof of C2. This is trivial if c∗h = cmax ∈ C∗

h . Supposethat c∗

h < cmax. If the property is not satisfied, there exists afeasible perturbation wo of w∗, which reduces the virtualdelay cost rate D4�l1�h1w5 by lowering the lead times fortypes 4c21 c2 + �27⊂ 6c∗

h1 cmax7 and by increasing the lead timesfor lower types 4c11 c1 + �17⊂ 6c∗

h1 cmax7, where �11 �2 > 0 andc1 + �1 ≤ c2. This holds since fh4c5 > 0 for c > c∗

h by C1, andbecause f ′

h > 0.

C3. If C∗l is nonempty, then the constraints 4325 bind for

c ∈ 6cmin1 c∗l 7, and w∗4cmin5=�/4�−�52.

Proof of C3. This follows from a similar argument asfor C2, because fl > 0 and f ′

l > 0.Part 1(a). Properties C2 and C3 imply optimality of the

lead times shown in (16). This is immediate for c∗h = cmax ∈

C∗h and/or cmin = c∗

l ∈ C∗l . If c∗

h < cmax then by C2 the con-straints (31) are binding for c ∈ 6c∗

h1 cmax7. Solving the result-ing integral equation in w∗ yields w∗4c5=�/4�−åF̄ 4c552.If cl > cmin the RHS of (32) satisfies

�−åF 4c5

�− 6�−åF 4c57

=6�l −åF 4c57�

4�− 6�−åF 4c5754�−�m −�h5

+�m�

4�−�m −�h54�−�h5+

�h

�−�h

1 c ∈ 6cmin1 cl70

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By C2 and C3 for an optimal solution the constraints (32)therefore simplify to

å∫ c∗l

cf 4x5w∗4x5dx+�∗

m · d

=å6F 4c∗

l 5− F 4c57�

4�− 6�−åF 4c5754�−�∗m −�∗

h5

+�∗m�

4�−�∗m −�∗

h54�−�∗h51 c ∈ 6cmin1 c

∗

l 70 (33)

Solving this integral equation in w∗ yields

w∗4c5=�

4�− 6�−åF 4c57520

Part 1(b). This claim follows directly from C2 and C3. IfC∗l 6= � then (32) is binding for c = cmin, which implies that

the policy is work conserving.Part 2(a). Follows from C1 since fh is continuous.Part 2(b). Follows by substituting w∗4c5 from (16) in the rev-

enue function (13) and analyzing its partial derivatives withrespect to �l and �h. Refer to the proof of Proposition 2. �

Optimal Segmentation and Lead Times Dependingon � and åProposition 2 specifies, for fixed capacity �, how the optimalset of lead-time classes changes as the arrival rate � increases,and how these transitions depend on the market size å.This result yields the optimal revenue as a function of �1å,and �, which is key for characterizing the optimal menu atthe optimal arrival rate, depending on å (Theorem 1) and �(Theorem 2).

Proposition 2 uses the following notation to describe thesestructural properties. The optimal set of lead-time classesfor a given arrival rate is denoted by a subset of the lettersh1m1msd, and l, shown in parentheses. For instance 4h1 l5indicates that h and l classes have a strictly positive arrivalrate, but the m and msd class have a zero arrival rate, in theoptimal menu. The notation 4x5→ 4y5 indicates the existenceof a threshold arrival rate such that the optimal set of classeschanges from 4x5 to 4y5 as the arrival rate � crosses thethreshold from below.

Proposition 2. Fix a capacity �> 0 and assume that f ′l > 0

and f ′h > 0. The optimal customer segmentation and lead-time

menu depend as follows on the market size å and the arrivalrate �.

1. For d ≤�−1 the segmentation (l) is optimal for all � and å.2. For d >�−1 denote by �P ¬�−

√

�/d and �F ¬�− 1/dthe arrival rates at which the maximum lead time equals d underwork conserving priority and FIFO service, respectively.

(a) If fh4cmin5≥ 0 and �−1 ≤ F 4f −1l 4cmax55 ·d then there are

unique thresholds å1 <å2 <å3 <å4, where å1 = �P <å2 <�F

and �<å4, which yield the following structure:

Classes with positive rate as �Market size increases on 601å7∩ 401�5

å ∈ 401å17 4h5å ∈ 4å11å27 4h5→ 4h1m5å ∈ 4å21å35 4h5→ 4h1m5→ 4h1m1 l5å ∈ 4å31å45 4h5→ 4h1 l5→ 4h1m1 l5å ∈ 4å41�5 4h5→ 4h1 l5

(34)

The optimal policy is work conserving.

(b) Selling only medium- and low-quality classes, 4m1 l5, isoptimal for some 4�1å5 iff F 4f −1

l 4cmax55 · d <�−1 <d. If fh4cmin5≥ 0 and F 4f −1

l 4cmax55 · d < �−1 < d then (34) is modified byadditional thresholds åml < å̄ml, where �F <åml <�< å̄ml <å4:

Classes with positive rate as �Market size increases on 601å7∩ 401�5

å ∈ 4åml1 å̄ml5 4h5→ 4h1m5→ 4h1m1 l5→ 4m1 l5,if å<å3

4h5→ 4h1 l5→ 4h1m1 l5→ 4m1 l5,if å>å3

(35)

For åy 4åml1 å̄ml5 the structure of (34) applies.(c) Strategic delay is optimal for some 4�1å5 iff fh4cmin5

< 0 and d > �−1. If fh4cmin5 < 0 and �−1 ≤ F 4f −1l 4cmax55 · d

then (34) changes: two thresholds åsd < å̄sd replace å1 and yieldthe following structure, where �P <åsd ≤å2 and åsd < å̄sd ≤

å3 <å4:

Classes with positive rate as �Market size increases on 601å7∩401�5

å∈ 401åsd5 4h5→ 4h1msd5å∈ 4åsd1å̄sd5 4h5→ 4h1msd5→ 4h1m51 if å≤å2

4h5→ 4h1msd5→ 4h1m5→ 4h1m1l5,if å>å2

(36)

where msd indicates that the lead time d involves strategic delay.For å≥ å̄sd the optimal scheduling policy is work conserving

and (34) applies.

Proof.

Preliminaries. The optimal segmentation and lead-timemenu described in Proposition 2 are obtained by solvingProblem 2 for fixed �. Refer to (17)–(21).

Part 1 is trivial: in this case all lead times exceed d, so 4205requires �l = �.

For part 2 we reformulate Problem 2. Define the aggregaterate for the m class and the h classes,

�mh ¬ �m +�h0

For fixed � write the revenue (17) as

ç4�mh1�h5

¬�v−å∫ cl4�−�mh5

cmin

f 4x5fl4x5

(

�

4�−6�−åF 4x5752−d

)

dx

+å∫ cmax

ch4�h5f 4x5fh4x5

(

d−�

4�−åF̄ 4x552

)

dx0 (37)

We define two threshold arrival rates. Define the maximumfeasible rate for h classes,

�P ¬�−√

�/d0 (38)

This is the maximum arrival rate of customers underwork conserving priority service (optimal for h classes byLemma 1) so that their lead times are shorter than d. Definethe maximum aggregate rate for m and h classes

�F ¬�− 1/d0 (39)

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At this rate the lead time is d under work conserving FIFOservice. Note that �d > 1 implies 0 <�P <�F .

Let �̄h4�mh5 be the maximum feasible rate of h classes asa function of the total rate �mh to m and h classes, so thatthe medium lead time d is achievable, i.e., (20) holds. Thenby (38)–(39):

�̄h4�mh5 ¬ min(

�mh1�−�/d

�−�mh

)

=

�mh1 if �mh ∈ 601�P 71

�−�/d

�−�mh

≤ �P 1 if �mh ∈ 6�P 1�F 70(40)

For �mh ≤ �P the entire �mh can be allocated to h classes, so�̄h4�mh5= �mh, which increases on 601�P 7 with �̄h4�P 5= �P .For �mh >�P only a portion �̄h4�mh5 < �mh can be allocated toh classes, and �̄h4�mh5 decreases on 6�P 1�F 7, with �̄h4�F 5= 0.Having �mh >�F violates (20).

Constraint (20) in Problem 2 holds if and only if �mh ≤

�F and �h ≤ �̄h4�mh5. Constraint (21) holds if and only if�P ≤ �mh <� or �mh = �≤ �P . Problem 2 for fixed � is thusequivalent to

max�mh1�h

ç4�mh1�h5 (41)

s.t. min4�1�P 5≤ �mh ≤ min4�1�F 51 (42)

0 ≤ �h ≤ �̄h4�mh50 (43)

The lower bound in (42) ensures work conservation if �l > 0: if�mh < min4�1�P 5, strategic delay is required for l classes (d >�/4�−�mh5

2 from (38)), which is suboptimal by Lemma 1.1(b).

Proof Steps. We prove part 2 by characterizing the solutionof (41)–(43) in four steps:

1. For fixed �mh, we characterize the optimal rate �h4�mh5.2. For fixed �, we derive the optimality conditions for

the optimal segmentation, specifically, for the optimal rates�∗mh4�5 and �∗

h4�5= �h4�∗mh4�55. These conditions, summarized

in (46) and Table A.2, are stated in terms of the virtual delaycosts of appropriately chosen customer types.

3. We translate the virtual delay cost conditions identifiedin step 2 into conditions on � and å. We organize this analysisinto the technical Lemmas 4–8.

4. We prove parts 2(a)–2(c) of Proposition 2 by combin-ing (46), Table A.2, and Lemmas 4–8.

Step 1. Optimal �h for Fixed �mh. For fixed �mh ∈ 401min4�1�F 57, the optimal �h satisfies

�h4�mh5¬ arg max�h

{

ç4�mh1�h5 s.t. 0 ≤ �h ≤ �̄h4�mh5}

0

From (37) we have

¡ç4�mh1�h5

¡�h

=fh4ch4�h55

(

d−�

4�−�h52

)

1 for �h ≤ �̄h4�mh51

where åf 4ch4x55c′h4x5= −1. The multiplier of fh4ch4�h55 is

nonnegative by (38) and since �̄h4�mh5≤ �P by (40). Sincefh4ch4055= cmax > 0 and f ′

hc′h < 0, the maximizer �h4�mh5 is

unique:

�h4�mh5

=

{

�̄h4�mh51 if fh4ch4�̄h4�mh555≥01

�0¬åF̄ 4f −1h 4055<�̄h4�mh51 if fh4ch4�̄h4�mh555<00

(44)

If fh4ch4�̄h4�mh555≥ 0 then it is optimal to sell the maximumpossible rate �̄h4�mh5 to h classes and �mh − �̄h4�mh5≥ 0 to themedium lead-time class. This policy is work conserving.

If fh4ch4�̄h4�mh555 < 0 then it is optimal to sell �0 to hclasses, less than the maximum possible rate �̄h4�mh5, and�mh −�0 > 0 to the medium lead time d. At �0 defined in (44)the virtual delay cost of the corresponding marginal type iszero: fh4ch4�055= fh4F̄

−14�0/å55= 0. This policy is not workconserving: the lead time d involves strategic delay since�0 < �̄h4�mh5:

�

4�−�mh54�−�05<

�

4�−�mh54�− �̄h4�mh55≤ d0 (45)

Step 2. Optimal Segmentation for Fixed �: Virtual Delay CostConditions. This step derives optimality conditions for theoptimal arrival rates of h, m, and l classes, denoted by �∗

h4�5,�∗m4�5, and �∗

l 4�5. To this end, we characterize the optimaltotal arrival rate to h and m classes,

�∗

mh4�5 ¬ arg max�mh

{

ç4�mh1�h4�mh55

s.t. min4�1�P 5≤ �mh ≤ min4�1�F 5}

0

Then �∗h4�5 follows from (44) with �mh = �∗

mh4�5, namely,�∗h4�5= �h4�

∗mh4�55. Furthermore, �∗

m4�5= �∗mh4�5−�∗

h4�5 and�∗l 4�5= �−�∗

mh4�5.Optimal segmentation for �≤ �P . For �≤ �P it is not optimal

to sell l classes: (42) requires �mh = �, so the maximizersatisfies �∗

mh4�5 = �. By (40) the entire � can be sold toh classes, so �̄h4�

∗mh4�55= �̄h4�5= �. Therefore, (44) yields the

following optimal segmentations, where msd denotes that themedium lead time involves strategic delay.

Optimal segmentations for �> d−1 and �≤ �P

Segments Virtual delay cost condition �∗

mh4�5 �∗

h4�5 �∗m4�5 �∗

l 4�5

4h5 fh4ch4�̄h4�555= fh4ch4�55≥ 0 � � 0 04h1msd5 fh4ch4�̄h4�555= fh4ch4�55 < 0 � �0 �−�0 > 0 0

(46)

Optimal segmentation for � > �P . In this case �mh ∈

6�P 1min4�1�F 57 by (42). Using (44) to substitute for �h

into (37), the total derivative of the revenue with respectto �mh satisfies

dç4�mh1�h4�mh55

d�mh

=¡ç4�mh1�h4�mh55

¡�mh

+¡ç4�mh1�h4�mh55

¡�h

·�′

h4�mh5

=fl4cl4�−�mh55

(

�

4�−�mh52−d

)

+fh4ch4�h4�mh555

·

(

d−�

4�−�h4�mh552

)

�′

h4�mh51 (47)

where åf 4cl4x55c′l4x5= 1 and −åf 4ch4x55c

′h4x5= 1. By (44), we

have two cases: if fh4ch4�̄h4�mh555 < 0 then �h4�mh5= �0 and�′h4�mh5= 0. If fh4ch4�̄h4�mh555≥ 0 then �h4�mh5= �̄h4�mh5=

�−�/d4�−�mh5, where the second equality holds by (40), so�′h4�mh5= −�/d4�−�mh5

2.

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Table A.2 Optimal Segmentations for �> d−1 and � > �P

Conditions (other than � > �P ) Arrival rates for each segment

Segments � Virtual delay costs �∗

mh4�5 �∗

h4�5 �∗

m4�5 �∗

l 4�5

4h1msd 5 � < �F fh4ch4�̄h4�555 < 0 � �0 �− �0 04h1m5 � < �F fh4ch4�̄h4�555≥ 01 g4�1 �5≥ 0 � �̄h4�5 �− �̄h4�5 04h1 l5 — g4�1 �P 5≤ 0 �P �P 0 �− �P4h1m1 l5 — g4�1 �P 5 > 0 > g4�1min4�1 �F 55 �∗

mh4�5 �̄h4�∗

mh5 > 0 �− �∗

mh

4m1 l5 � > �F g4�1 �F 5≥ 0 �F 0 �F �− �F

Substituting for �h4�mh5 and �′h4�mh5 into (47) yields

dç4�mh1�h4�mh55

d�mh

=

fl4cl4�−�mh55

(

�

4�−�mh52−d

)

1

if fh4ch4�̄h4�mh555<01

g4�1�mh5

(

�

4�−�mh52−d

)

1

if fh4ch4�̄h4�mh555≥01

(48)

for �mh ∈ 6�P 1min4�1�F 57 (the feasible range by (42)), wherethe function

g4�1�mh5¬ fl(

cl4�−�mh5)

− fh(

ch4�̄h4�mh55)

0 (49)

It measures the difference between the virtual delay costsof the marginal types cl and ch as a function of � and �mh,when allocating the corresponding maximum feasible rate�h = �̄h4�mh5 to h classes and �l = �−�mh to l classes. Thesign of g4�1�mh5 is important for the optimal segmentation:g4�1�mh5 > 0 indicates more pooling increases profits.

The sign of the revenue derivative in (48) only depends onfh4ch4�̄h4�mh555 and g4�1�mh5, because fl > 0 and the commonfactor in both cases of (48) is zero at �mh = �P and positive for�mh >�P . The maximizer �∗

mh4�5 is unique since the followingproperties hold for �mh ∈ 6�P 1min4�1�F 57:

(i) fh4ch4�̄h4�mh555 increases in �mh since �̄′h4�mh5 < 0 by (40)

and f ′hc

′h < 0.

(ii) g4�1�mh5 decreases in �mh since f ′l c

′l > 0, and by (i).

Properties (i) and (ii) reflect that, as �mh increases, boththe maximum feasible h rate �̄h4�mh5 and the l rate �l =

�−�mh decrease, so the corresponding marginal types andtheir virtual delay costs (because f ′

l 1 f′h > 0) move apart:

ch4�̄h4�mh55 and fh4ch4�̄h4�mh555 increase while cl4�−�mh5 andfl4cl4�−�mh55 decrease in �mh.

Properties (i) and (ii) and (48) determine the maxi-mizer �∗

mh4�5, and (44) determines �∗h4�5= �h4�

∗mh4�55. The

optimal segmentations and the corresponding conditions aresummarized in Table A.2. The optimal segmentation sets �mh

and �h to minimize the virtual delay cost of types served,subject to the constraints (42) and (43). A segmentation isoptimal if and only if it minimizes �mh ∈ 6�P 1min4�F 1�57, andmaximizes �h ≤ �̄h4�mh5, subject to two conditions.

(1) The marginal h type’s virtual delay cost is nonnegative:fh4ch4�h55≥ 0.

(2) If both h and l are offered (�h > 01�−�mh > 0) then thevirtual delay cost of the marginal h type (with the shorterlead time) is higher: fh4ch4�h55≥ fl4ch4�−�mh55. We discussthese conditions for the solutions summarized in Table A.2,in the order 4h1 l5–4h1m1 l5–4m1 l5–4h1m5–4h1msd5.

Segmentation 4h1 l5: If g4�1�P 5≤ 0, conditions (1)–(2) holdat �mh = �P , which yields maximum allocations to h and l,using the entire �.

Segmentations 4h1m1 l5: If g4�1�P 5 > 0 >g4�1min4�1�F 55,then �mh = �P violates 425 whereas �mh = min4�1�F 5 satisfies425 but does not minimize �mh. In this case increase �mh tothe point where the marginal types’ virtual delay costs areequal: g4�1�mh5= 0 for �mh = �∗

mh4�5. This reduces the l andh rates and increases the rate �m.

Segmentations 4m1 l5, 4h1m5 and 4h1msd5: In these casesg4�1�mh5 > 0 for every �mh < min4�1�F 5, so condition 425is violated for every segmentation that sells both h and lclasses; note that fh4ch4�̄h4�555 < 0, the condition for 4h1msd5,implies g4�1�5 > 0. In each case condition (2) is met bynot offering both l and h classes: set �∗

mh4�5 = min4�1�F 5.If � > �F , no h classes are sold: allocate �F to the m classand the rest to l classes. If �< �F , no l classes are sold: set�∗mh4�5= �, then maximize �h ≤ �̄h4�5 subject to condition 415,

by applying (44) with �mh = �. If all of �̄h4�5 can be sold totypes with nonnegative fh this yields 4h1m5. If not, 4h1msd51with strategic delay is optimal: sell h classes to �0 < �̄h4�5,where fh4ch4�055= 0, and the m class (lead time = d) to theremaining �−�0 consisting of types with negative fh.

Step 3. Virtual Delay Cost Conditions as Functions of � and å.The optimality conditions of step 2, (46) for �≤ �P and theconditions presented in Table A.2 for �>�P , are stated interms of the signs of the virtual delay cost fh4ch4�̄h4�555 andof the virtual delay cost difference g4�1�mh5 for �mh = �P

and �mh = min4�1�F 5. Next, we translate these conditionsinto conditions on � and å. For this purpose, we state thetechnical Lemmas 4–8 (proofs are in the online appendix,available as supplemental material at http://dx.doi.org/10.1287/mnsc.2015.2236).

Lemmas 4–6 characterize the signs of g4�1�P 5 andg4�1min4�1�F 55 depending on � and å. These properties areimportant to determine whether pooling is optimal.

Lemmas 7–8 characterize, for the case fh4cmin5 < 0, the signof fh4ch4�̄h4�555 depending on � and å. These properties areimportant to determine whether strategic delay is optimal.

Lemma 4. Fix �>d−1 and å>�P . Consider g4�1�mh5 for�mh = �P and �mh = min4�1�F 5.

1. The virtual delay cost difference g4�1�P 5, where �h = �P ,�m = 0, increases in �≥ �P , where

g4�1�P 5= fl4cl4�−�P 55− fh4ch4�P 550 (50)

2. The virtual delay cost difference g4�1min4�1�F 55 varies asfollows with �.

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Figure A.1 Virtual Delay Cost Differences g4�1 �P 5 and g4�1min4�1 �F 55 as Functions of �

Market size � < �F g(�,�P)

�P �F

�

g(�,�)

�

g(�P,�P)

g(�,�P)g(�,�)cmin – cmax < 0

0

Market size � > �Fg(min(�,�),�P)

�P �F

�

min(�,�)

g(min(�,�),�F)

g(�,�P)g(�,min(�,�F))cmin – cmax < 0

g(�P,�P)0

(a) It decreases in � ∈ 6�P 1min4�F 1å57, where �h = �̄h4�5,�m = �− �̄h4�5 > 0, and

g4�1min4�1�F 55= g4�1�5 = cmin − fh4ch4�̄h4�5551

� ∈ 6�P 1min4�F 1å570 (51)

(b) It increases in � ∈ 6�F 1min4å1�57, where �h = 0, �m =

�F , g4�F 1�F 5= cmin − cmax, and

g4�1min4�1�F 55= g4�1�F 5 = fl4cl4�−�F 55− cmax1

� ∈ 6�F 1min4å1�570 (52)

Figure A.1 illustrates Lemma 4. In both panels g4�1�P 5increases in � (part 1). At left å<�F and g4�1min4�1�F 55decreases in �. At right å>�F and g4�1min4�1�F 55 is min-imized and equal to cmin − cmax < 0 at � = �F , where alltypes are served FIFO (part 2). In both panels g4�1�P 5 >g4�1min4�1�F 55 for fixed �: as explained in step 2, g4�1�mh5decreases in �mh for fixed �.

From the two panels in Figure A.1, note that increasing ådecreases g4�1�P 5 and g4�1min4�1�F 55. This holds because forfixed 4�1�mh5 (where �mh = �P or �mh = min4�1�F 5), increas-ing å lowers the marginal type cl4�−�mh5= F −144�−�mh5/å5,increases ch4�̄h4�mh55= F̄ −14�̄h4�mh5/å5, and changes theirvirtual delay costs accordingly. To make the dependenceon å explicit we write

g4�1�mh3å5= fl4cl4�−�mh3å55− fh4ch4�̄h4�mh53å551

for � ∈ 6�P 1min4�1å57 and �mh ∈ 6�P 1min4�1�F 57.Lemma 5 describes how the sign of g4�1min4�1�F 53å5

depends on the arrival rate � and the market size å. Thisdetermines whether it is optimal to pool maximally: Ifg4�1min4�1�F 53å5≥ 0, it is profitable to pool the maximumfeasible set of customers.

Lemma 5. Fix �> d−10 Consider g4�1min4�1�F 53å5 as afunction of �> �P and å, where

g4�1min4�1�F 53å5

=

g4�1�3å5=cmin −fh

(

F̄ −1

(

�̄h4�5

å

))

1

�∈ 6�P 1min4å1�F 571

g4�1�F 3å5=fl

(

F −1

(

�−�F

å

))

−cmax1

�∈ 6�F 1min4å1�570

(53)

1. For � ∈ 6�P 1min4å1�F 57 the sign of g4�1min4�1�F 53å5=

g4�1�3å5 depends on the thresholds

å2 ¬�

2

(

1

F̄ 4f −1h 4cmin55

+ 1)

−

√

�2

4

(

1

F̄ 4f −1h 4cmin55

− 1)2

+�

dF̄ 4f −1h 4cmin55

1 (54)

å3 ¬�P

F̄ 4f −1h 4cmin55

1 (55)

where å2 ∈ 4�P 1�F 5 satisfies g4å21å23å25 = 0 and å3 >å2satisfies g4�P 1�P 3å35= 0.

(a) If å≤å2 then g4�1�3å5≥ 0 for � ∈ 6�P 1å7, whereå2 <�F .

(b) If å ∈ 4å21å35, then there is an arrival rate threshold�1 ∈ 4�P 1�F 5, given by

�1 ¬�−�/d

�−åF̄ 4f −1h 4cmin55

1 such that (56)

g4�1�3å5

> 01 if � ∈ 6�P 1�151

= 01 if �= �11

< 01 if � ∈ 4�11min4�F 1å570

(57)

(c) If å>å3, then g4�1�3å5 < 0 for � ∈ 6�P 1min4�F 1å57.2. For � ∈ 6�F 1min4å1�57 the sign of g4�1min4�1�F 53å5=

g4�1�F 3å5 depends on the thresholds

åml ¬�F

F̄ 4f −1l 4cmax55

> �F 1 (58)

å̄ml ¬1

dF 4f −1l 4cmax55

1 (59)

where g4åml1�F 3åml5= 0 and g4�1�F 3 å̄ml5= 0.(a) If F 4f −1

l 4cmax55 · d < �−1 < d then åml <�< å̄ml. Ifå ∈ 6åml1 å̄ml5 then there is a threshold

�3 ¬ �F +åF 4f −1l 4cmax551 such that (60)

g4�1�F 3å5

< 01 if � ∈ 4�F 1�351

= 01 if �= �31

> 01 if �> �30

(61)

If åy 6åml1 å̄ml5 then g4�1�F 3å5 < 0 for all feasible �> �F .

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Afèche and Pavlin: Segmentation, Pooling, and Strategic Delay2432 Management Science 62(8), pp. 2412–2436, © 2016 INFORMS

(b) If �−1 ≤ F 4f −1l 4cmax55 · d then g4�1�F 3å5 < 0 for all

feasible �> �F .

Lemma 6 describes how the sign of g4�1�P 3å5 dependson the arrival rate � and the market size å. This determineswhether any pooling is optimal: if g4�1�P 3å5≤ 0 then poolingis not optimal.

Lemma 6. Fix �>d−10 The sign of g4�1�P3å5 depends asfollows on �> �P and å. Let

å4 ¬{

å≥ �P 3å=

√

�/d

F 4f −1l 4fh4ch4�P 3å5555

}

1 (62)

where å4 satisfies g4�1�P3å45= 0. Moreover, å4 > max4å31å̄ml1�5.

1. If å<å3 then g4�1�P 3å5 > 0 for �> �P .2. If å ∈ 4å31å45 then there is an arrival rate threshold �2 >�P

where

�2 ¬ �P +åF 4f −1l 4fh4ch4�P 3å55551 such that (63)

g4�1�P 3å5

< 01 if � ∈ 6�P 1�251

= 01 if �= �21

> 01 if �> �20

(64)

3. If å≥å4 then g4�1�P 3å5≤ 0 for �> �P .

Lemma 7 describes how an increase in � changes thelowest virtual delay cost of types buying h classes, given weallocate the maximum feasible rate, i.e., �̄h4�5, to h classes.

Lemma 7. Fix �> d−1 and å. Let fh4cmin5 < 0. Consider thevirtual delay cost fh4ch4�̄h4�555.

1. It decreases in � ∈ 601min4�P 1å57, where �̄h4�5= �1 themaximum fh4ch4055= cmax > 0, and

fh4ch4�55��=min4�P 1å5 =

{

fh4cmin5 < 01 å≤ �P 1

fh4F̄−14�P/å551 å> �P 0

(65)

2. If å > �P , it increases in � ∈ 6�P 1min4�F 1å57, where�̄h4�5=�−�/d4�−�5 and

fh4ch4�̄h4�555��=min4�F 1å5

=

{

fh4F̄−14�̄h4å5/å551 å ∈ 4�P 1�F 51

cmax > 01 å≥ �F 0(66)

Write fh4ch4�̄h4�53å55 to make the dependence on å explicit.For fixed � the marginal type ch4�̄h4�55 = F̄ −14�̄h4�5/å5increases in å, hence fh4ch4�̄h4�53å55 increases in å.

Lemma 8 describes the sign of fh4ch4�̄h4�53å55 dependingon � and å. This determines whether strategic delay isoptimal: if fh4ch4�̄h4�53å55 < 0 then allocating the maximumfeasible rate of customers to h classes would result in negativevirtual delay costs for the marginal type.

Lemma 8. Fix �> d−10 Let fh4cmin5 < 0. Let

åsd ¬�

2

(

1

F̄ 4f −1h 4055

+ 1)

−

√

�2

4

(

1

F̄ 4f −1h 4055

− 1)2

+�

dF̄ 4f −1h 4055

1 (67)

å̄sd ¬�P

F̄ 4f −1h 4055

1 (68)

where fh4ch4�̄h4åsd53åsd55= 0 and fh4ch4�P 3 å̄sd55= 0. Moreover�P < åsd ≤ å2 and åsd < å̄sd ≤ å3. Define the arrival ratethresholds

�0 ¬åF̄ 4f −1h 40551 (69)

�sd ¬�−�/d

�−�00 (70)

The sign of fh4ch4�̄h4�53å55 depends as follows on � and å.1. If å<åsd then fh4ch4�̄h4�53å55 < 0 if and only if �> �0,

where �0 <�P .2. If å∈ 6åsd1 å̄sd5 then fh4ch4�̄h4�53å55 < 0 if and only if

� ∈ 4�01�sd5, where �0 <�P <�sd <�F .3. If å≥ å̄sd then fh4ch4�̄h4�53å55≥ 0 for all feasible �.

Step 4. Proofs of parts 2(a)–2(c) of Proposition 2. We completethe proof by combining the optimality conditions (46) andthose listed in Table A.2 from step 2 with Lemmas 4–8 fromstep 3.

Part 2(a). Suppose that fh4cmin5≥ 0 and 1/�≤ F 4f −1l 4cmax55d.

The optimality conditions Table A.2 for �≤ �P and thosefrom Table A.2 for �> �P , combined with Lemmas 5–6, implythe results. Since fh4cmin5≥ 0, segmentation 4h1msd5 cannotbe optimal: the optimality condition is fh4ch4�̄h4�555 < 0by (46) and by Table A.2, which cannot hold. Since 1/�≤

F 4f −1l 4cmax55 ·d, segmentation 4m1 l5 cannot be optimal: the

condition is g4�1�F 3å5≥ 01 see Table A.2, and Lemma 5.2(b)rules it out.

The conditions (46) and Table A.2, combined with Lem-mas 5–6, imply the transitions among 4h5, 4h1m5, 4h1 l5and 4h1m1 l5 listed in Table (34). We illustrate how for å ∈

4å21å35. By (34) the optimal segmentation transitions as4h5→ 4h1m5→ 4h1m1 l5 as � increases. For �≤ �P segmenta-tion 4h5 is optimal by Table A.2. For �> �P Lemma 5 specifiesthe sign of g4�1min4�1�F 53å5; Lemma 5.1(b) applies sinceå ∈ 4å21å35, and Lemma 5.2(b) since 1/�≤ F 4f −1

l 4cmax55 ·d.Together they specify that g4�1min4�1�F 53å5≥ 0 for �≤ �1

and g4�1min4�1�F 53å5 < 0 otherwise. Lemma 6 specifies thesign of g4�1�P 3å5, where Lemma 6.1 applies since å<å3:it specifies that g4�1�P3å5 > 0 for �> �P . It follows fromTable A.2 that 4h1m5 is optimal for �≤ �1, and 4h1m1 l5 isoptimal for �> �1.

Part 2(b). Suppose that fh4cmin5≥ 0 and F 4f −1l 4cmax55 · d <

�−1 <d. The proof follows the same logic as explained forpart 2(a). However, by Lemma 5.2(a) segmentation 4m1 l5 isoptimal if and only if å ∈ 6åml1 å̄ml5 and �≥ �3: in this caseg4�1�F 3å5≥ 0, which is the optimality condition for 4m1 l5 byTable A.2. The optimal segmentation for å ∈ 6åml1 å̄ml5 and�< �3 follows from Lemmas 5–6 and Table A.2 in exactly thesame way as in part 2(a). This yields Table (35).

Part 2(c). Suppose that fh4cmin5 < 0 and 1/�≤ F 4f −1l 4cmax55 ·d.

The segmentation 4h1msd5 is optimal if and only iffh4ch4�̄h4�555 < 0, where �̄h4�5 is defined in (40). Refer to (46)for �≤ �P and Table A.2 for �> �P . This condition, combinedwith Lemmas 5, 6, and 8, imply Table (36).

The case å<åsd in Table (36) is immediate from (46),Table A.2, and Lemma 8.1. For å ∈ 6åsd1 å̄sd5 Lemma 8.2.implies the transition 4h5→ 4h1msd5 for �< �sd . For �≥ �sd

Lemmas 5–6 and Table A.2 imply 4h1m5 if å≤å2 or 4h1m5→

4h1m1 l5 if å>å2. �

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Afèche and Pavlin: Segmentation, Pooling, and Strategic DelayManagement Science 62(8), pp. 2412–2436, © 2016 INFORMS 2433

Table A.3 Characteristics of the Arrival Rates �∗

mh and �∗

h as Functions of � and �

Segments � < � �−1 �∗

mh �∗

h ¡�∗

mh/¡� ¡�∗

mh/¡� ¡�∗

h/¡� ¡�∗

h/¡�

4l5 any � ≥d 0 0 0 0 0 04h5 ≤�P <d � � 1 0 1 04h1msd 5 ∈ 4�P 1 �F 5 <d � �0 = åF̄ 4f−1

h 4055 < �P 1 0 0 04h1m5 ∈ 4�P 1 �F 5 <d � �∗

h = �̄h4�5= �− 4�/d5/4�− �5 < �P 1 0 <0 >04h1 l5 >�P <d �P �P 0 >0 0 >0

4h1m1 l5 >�P <d(i) �∗

h = �− 4�/d5/4�− �∗

mh5 < �P < �∗

mh ∈ 40115 ∈ 40115 <0 >0(ii) fl 4cl 4�− �∗

mh55= fh4ch4�∗

h55

4m1 l5 >�F <d �F 0 0 >0 0 0

Proof of Lemma 2This Lemma characterizes the partial derivatives of themaximum revenue, that is, the revenue under the optimalsegmentation and menu, with respect to � and �.

The maximum revenue satisfies ç∗4�1�5=ç4�1�∗mh1�

∗h1�5,

where

ç4�1�∗

mh1�∗

h1�5

¬ �v−å∫ cl4�−�∗

mh5

cmin

f 4x5fl4x5

(

�

4�− 6�−åF 4x5752− d

)

dx

+å∫ cmax

ch4�∗h5f 4x5fh4x5

(

d−�

4�−åF̄ 4x552

)

dx1

and �∗mh1�

∗h depend on 4�1�5 as tabulated in Table A.3. The

entries for �1�∗mh1�

∗h and � are from the proof of Proposition 2;

refer in particular to (46) and Table A.2 and their discussion.They directly imply the partial derivatives of �∗

mh and �∗h,

except for 4h1m1 l5, where we derive them below. The proofrefers to these properties of �∗

mh and �∗h and derives those of

ç below. We first review some important facts. Recall fromProposition 2 and its proof: �P =�−

√

�/d and �F =�− 1/dare well defined if �−1 < d, so �P < �F ; �̄h4�5 is definedin (40); and �0 is defined in (44).

We suppress the arguments of ç∗, ç1 �∗mh and �∗

h.Recall that cl4x5= F −14x/å5, so åf 4cl4x55c

′l4x5= 1, c′

l > 0;ch4x5= F̄ −14x/å5, so åf 4ch4x55c

′h4x5= −1, c′

h < 0; fl1f ′l 1 f

′h > 0

and fh4ch4�∗h55≥ 0. Therefore,

ç∗

� =¡ç

¡�+

¡ç

¡�∗mh

¡�∗mh

¡�+

¡ç

¡�∗h

¡�∗h

¡�1 where (71)

¡ç

¡�∗mh

= fl4cl4�−�∗

mh55

(

�

4�−�∗mh5

2− d

)

≥ 01

¡2ç

¡�¡�∗mh

≥ 01 and¡2ç

¡�¡�∗mh

< 01 (72)

¡ç

¡�= v−å

∫ cl4�−�∗mh5

cmin

f 4x5fl4x52�4�−�+åF 4x553

dx−¡ç

¡�∗mh

1

¡2ç

¡�2≤ 01 and

¡2ç

¡�¡�> 01 (73)

¡ç

¡�∗h

= fh4ch4�∗

h55

(

d−�

4�−�∗h5

2

)

≥ 01

¡2ç

¡�∗2h

≤ 01 and¡2ç

¡�¡�∗h

≥ 00 (74)

Part 1(a). It follows from Table A.3 and (71)–(74) thatç∗

� = v for 4h1msd5 and ç∗� ≥ v for 4h5.

Part 1(b). We show that ç∗�� < 0 <ç∗

�� for 4h1m1 l5 andç∗

�� ≤ 0 ≤ç∗�� for any segmentation other than 4h1m1 l5. We

omit the remaining straightforward checks that ç∗�� < 0 <ç∗

��

for 4l51 4h1m5, 4h1 l5, and 4m1 l5.Proof that ç∗

�� < 0 < ç∗�� for 4h1m1 l5. Table A.3 claims

0 < ¡�∗mh/¡�1¡�

∗mh/¡� < 1, which is implied by equations

(i)–(ii) in the table and the fact that f ′l c

′l > 0 > f ′

hc′h:

sign(

1 −¡�∗

mh

¡�

)

= − sign(

¡�∗h

¡�

)

= sign(

¡�∗mh

¡�

)

1

so 0 <¡�∗

mh

¡�< 10 (75)

sign(

¡�∗mh

¡�

)

= sign(

¡�∗h

¡�

)

and

1 − ¡�∗mh/¡�

�−�∗mh

+1 − ¡�∗

h/¡�

�−�∗h

=1�1 so 0 <

¡�∗mh

¡�< 10 (76)

The first equations in (75) and (76) each follow from (ii) andf ′l c

′l > 0 > f ′

hc′h, the second equations each follow from (i). For

fixed � the total derivative dç/d�mh = 0 at �mh = �∗mh: see (48)

in the proof of Proposition 2, where g4�1�∗mh5= 0 for 4h1m1 l5.

It follows that

ç∗

� =¡ç

¡�+

¡�∗mh

¡�

[

¡ç

¡�∗mh

+¡ç

¡�∗h

¡�∗h

¡�∗mh

]

=¡ç

¡�1

for all 4�1�5 with 4h1m1 l50 (77)

We show that the following holds:

ç∗

�� =¡2ç

¡�2+

¡2ç

¡�¡�∗mh

¡�∗mh

¡�

≤ −fl4cl4�−�∗

mh552�4�−�∗

mh53

−¡2ç

¡�∗mh¡�

(

1 −¡�∗

mh

¡�

)

< 01 (78)

ç∗

�� =¡2ç

¡�¡�+

¡2ç

¡�¡�∗mh

¡�∗mh

¡�> 00 (79)

The equations for ç∗��, ç∗

�� hold by (77) and because¡2ç/¡�¡�∗

h = 0 by (74). The first inequality in (78) holds by(72)–(73); the second by (72) and (75). The inequality in (79)holds by (72)–(73) and (76).

Proof that ç∗�� ≤ 0 ≤ ç∗

�� for segmentations other than4h1m1 l5. First consider ç∗

��. Since ¡2ç/¡�¡�∗h = 0 by (74) and

¡2�∗mh/¡�

2 = 0 follows from Table A.3, we have

ç∗

�� =¡2ç

¡�2+ 2

¡2ç

¡�¡�∗mh

¡�∗mh

¡�+

¡2ç

¡�∗2mh

(

¡�∗mh

¡�

)2

+

[

¡2ç

¡�∗2h

(

¡�∗h

¡�

)2

+¡ç

¡�∗h

¡2�∗h

¡�2

]

0

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Afèche and Pavlin: Segmentation, Pooling, and Strategic Delay2434 Management Science 62(8), pp. 2412–2436, © 2016 INFORMS

The terms in brackets are nonpositive by (74) and ¡2�∗h/¡�

2 ≤ 0from Table A.3. The other terms satisfy

¡2ç

¡�2+2

¡2ç

¡�¡�∗mh

¡�∗mh

¡�+

¡2ç

¡�∗2mh

(

¡�∗mh

¡�

)2

≤

((

¡�∗mh

¡�

)2

−1)

fl4cl4�−�mh552�4�−�mh5

3−

(

1−¡�∗

mh

¡�

)2 ¡2ç

¡�mh¡�

by (72)–(73). The RHS is nonpositive since ¡2ç/¡�mh¡�≥ 0 by(72) and ¡�∗

mh/¡�≤ 1 as shown in Table A.3.Next consider ç∗

��. As shown in Table A.3 we have4¡�∗

mh/¡�5 · 4¡�∗mh/¡�5= ¡2�∗

mh/¡�¡�= 0, which implies

ç∗

�� =

(

¡2ç

¡�¡�+

¡2ç

¡�∗mh¡�

¡�∗mh

¡�

)

+

(

¡2ç

¡�¡�∗mh

¡�∗mh

¡�

)

+

(

d

d�

[

¡ç

¡�∗h

¡�∗h

¡�

])

0 (80)

We show that each bracket is nonnegative. For the first,Table A.3 and (72)–(73) imply

¡2ç

¡�¡�+

¡2ç

¡�∗mh¡�

¡�∗mh

¡�≥ −

¡2ç

¡�∗mh¡�

(

1 −¡�∗

mh

¡�

)

≥ 00

The second bracket of (80) is nonnegative by Table A.3and (72). The third bracket of (80) is also nonnegative.For segmentations other than 4h1m5 and 4h1m1 l5 we have¡�∗

h/¡�= 0 or = 1 by Table A.3 and ¡2ç/¡�∗h¡�≥ 0 by (74).

For 4h1m5 substitute for �∗h =�−�/d4�−�5−1 from Table A.3

to get

¡ç

¡�∗h

¡�∗h

¡�= fh4ch4�

∗

h55

(

d−�

4�−�∗h5

2

)

¡�∗h

¡�

= v+ fh4ch4�∗

h55

(

d−�

4�−�52

)

0 (81)

This expression increases in �: the virtual delay costfh4ch4�

∗h55≥ 0 decreases in � since f ′

hc′h < 0 and ¡�∗

h/¡�> 0by Table A.3, and its multiplier is negative (� > �P ) andincreases in �.

Part 2. The function ç∗4�1�5=ç4�1�∗mh1�

∗h1�5 is defined

piecewise: �∗mh4�1�5 and �∗

h4�1�5 depend on the optimalsegmentations, which vary with 4�1�5 as specified by Propo-sition 2. We establish continuity within each piece and at thetransition points.

Table A.4 Possible Transitions Between Optimal Segmentations

From To At 4�1�5 �∗

mh Virtual delay cost Lemma

Transitions in optimal segmentation involving 4h5 or 4h1msd 5

4h5 4h1msd 5 �= �0, �> d−1 � fh4ch4�055= 0 8.14h1msd 5 4h1m5 �= �sd , �> d−1 � fh4ch4�̄h4�sd 555= 0 8.24h5 4h1 l5 �= �P , �> d−1 �

4h5 4h1m5 �= �P , �> d−1 �

Transitions in optimal segmentation involving neither 4h5 nor 4h1msd 5

4l5 4h1 l5 � < �= d−1 04l5 4m1 l5 � < �= d−1 04h1m5 4h1m1 l5 �= �1, �> d−1 � fh4ch4�̄h4�1555= fl 4cl 4�− �∗

mh55= cmin 5.1(b)4h1m1 l5 4m1 l5 �= �3, �> d−1 �F fl 4cl 4�3 − �F 55= cmax 5.2(a)4h1 l5 4h1m1 l5 �= �2, �> d−1 �P fh4ch4�P 55= fl 4cl 4�2 − �P 55 6.2

By the definition of ç and by Table A.3, all first- andsecond-order derivatives of ç, �∗

mh and �∗h with respect to

4�1�5 are continuous for each segmentation. Therefore so arethe functions ç∗

�4�1�5, ç∗��4�1�5 and ç∗

��4�1�5. It remains toshow the stated properties at each 4�1�5 with a transitionbetween two optimal segmentations. By Proposition 2, thepossible transitions are as shown in Table A.4.

The following facts establish (a) and (b). At 4�1�5 where thesegmentation transitions from 4h5 or 4h1msd5 we have ç∗

� = v.At 4�1�5 with a transition involving neither 4h5 nor 4h1msd5the two expressions for ç∗

�� (one for each segmentation)agree, and ditto for ç∗

�u. We omit these checks; they arestraightforward using Table A.4 and the formulae for ç∗

�,ç∗

�� and ç∗�� derived above. �

Proof of Theorem 1The results follow from Proposition 2 and Lemma 2. �

Proof of Theorem 2As a preliminary, we prove two key properties of the thresh-olds �H , defined by (22), and �SD, defined by (23), whichwill simplify the proofs of parts 1–3. The optimal arrival rate�∗4�5 satisfies

�∗4�5= arg max�

{

ç∗4�1�5 s.t. � ∈ 601å7∩ 601�5}

0 (82)

P1. Suppose that fh4cmin5≥ 0. Then (i) the optimal rate �∗4�5=

å for �≥�H , and (ii) the optimal segmentation is 4h5 if and onlyif �≥�H .

Proof of P1. Recall that �P = � −√

�/d for � > d−1 asdefined in Proposition 2. We write �P 4�5 to make its depen-dence on � explicit. For fixed � the following facts imply thatP1 holds if the condition “�≥�H” is replaced by “å≤ �P 4�5.”First, segmentation 4h5 is optimal for fixed � if and only if�> d−1 and �≤ �P 4�5; this holds by Proposition 2.2 and itsproof. Second, ç∗

�4�1�5≥ v > 0 for all 4�1�5 where segmenta-tion 4h5 is optimal, and ç∗

�4�1�5 is continuous in 4�1�5; seeLemma 2. The proof of P1 is complete if �≥�H ⇔å≤ �P 4�5.This holds since å= �P 4�H 5 by the definition (22), �P 4d

−15= 0,and �′

P 4�5= 1 − 1/42√

�d5 > 0 for �≥ d−1.

P2. Suppose that fh4cmin5 < 0. Then (i) �∗4�5=å for �≥�SD,and (ii) the optimal segmentation is 4h1msd5 if and only if �>�SD.

Proof of P2. Recall the threshold åsd from Proposition 2.2(c)and its proof; it is defined in (67) of Lemma 8. We writeåsd4�5 to make its dependence on � explicit. For fixed �the following facts imply that P2 holds if the conditions

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Afèche and Pavlin: Segmentation, Pooling, and Strategic DelayManagement Science 62(8), pp. 2412–2436, © 2016 INFORMS 2435

“�≥�SD” and “�>�SD” are replaced by “å≤åsd4�5” and“å<åsd4�5,” respectively. First, the optimal segmentation is4h1msd5 at the largest feasible � if and only if å<åsd4�5;this holds by Proposition 2.2(c); for details see Lemma 8.Second, the revenue ç∗ satisfies ç∗

�4�1�5 ≥ v > 0 for all4�1�5 where segmentation 4h1msd5 is optimal, ç∗

��4�1�5≥

0 for all 4�1�5, and ç∗�4�1�5 is continuous in 4�1�5; see

Lemma 2. We complete the proof of P2 by showing that�=�SD4å5⇔å=åsd4�5 and �>�SD4å5⇔å<åsd4�5. Wewrite �SD4å5 to emphasize that �SD depends on å throughits defining Equation (23). Fix � and solve (23) for å toget å =åsd4�5 as defined in (67) of Lemma 8. Note that�′

SD4å5 > 0 since the LHS of (23) increases in � and decreasesin å. For fixed å the fact that å+ d−1 <�SD <�H followssince the LHS of (23) is 0 for �=å+ d−1, and 1 for �=�H

since �P 4�H 5=�H −√

�H/d =å.Part 1. For � ≤ �min, �∗4�5 = 0 since no type buys at

a positive price: w4c5 ≥ 1/�min = d + v/cmin implies v +

c · 4d−w4c55 ≤ v41 − c/cmin5 ≤ 0. For � = �min we haveç∗

�401�5= 0. For � ∈ 4�min1d−15 segmentation 4l5 is optimal

for all � by Proposition 2.1. By Lemma 2, ç∗��4�1�5 > 0>

ç∗��4�1�5 under segmentation 4l5; therefore �∗4�5 > 0 for

� > �min. Since ç∗�4�1�5 is continuous in 4�1�5 we have

�∗4�5 <å for � ∈ 4�min1�min + �5 and small �> 0.We next show that there exists a unique threshold �A >

�min such that �∗4�5=å if and only if �≥�A, where �A <�H

if fh4cmin5 ≥ 0 and �A < �SD if fh4cmin5 < 0. By Lemma 2,ç∗4�1�5 is concave in � for fixed �, and ç∗

��4�1�5≥ 0 for all4�1�5. It follows that �∗4�5=å⇔ç∗

�4å1�5≥ 0 for any �,and if �∗4�5=å for some � then �∗4�′5=å for all �′ >�. Itremains to show that there exists � that satisfies �∗4�5=åand either �<�H if fh4cmin5≥ 0, or �A <�SD if fh4cmin5 < 0.This holds since P1 implies that ç∗

�4å1�H 5= v > 0, P2 impliesthat ç∗

�4å1�SD5= v > 0 if fh4cmin5 < 0, and because ç∗��4�1�5

is continuous in 4�1�5 by Lemma 2.2.It remains to show that �∗4�5 is strictly increasing on

6�min1�A7. Lemma 2.1 implies that for fixed �, ç∗4�1�5 hasa unique maximizer �∗4�5, and that if �∗4�5 <å then theoptimal segmentation is 4l51 4h1 l51 4h1m51 4m1 l1 5 or 4h1m1 l5.Under each of these segmentations, ç∗

��4�1�5 < 0 <ç∗��4�1�5

(Lemma 2.1) and ç∗��4�1�5 and ç∗

��4�1�5 are continu-ous in 4�1�5 (Lemma 2.2). We have ç∗

�4�∗4�51�5 = 0 for

� ∈ 6�min1�A7. By the implicit function theorem �∗4�5 isdifferentiable and �∗′4�5 = −ç∗

��4�∗4�51�5/ç∗

��4�∗4�51�5

> 0 for � ∈ 6�min1�A7.Part 2. We first prove two key properties.

P3. If � ∈ 4d−11�H 5 and �∗4�5 = å, then pooling must beoptimal.

Proof of P3. For � ∈ 4d−11�H 5, by P1–P2 and Proposition 2,only one of 4h1 l51 4h1m51 4m1 l1 51 4h1m1 l5 or 4h1msd5 canbe optimal. Only 4h1 l5 has no pooling, but it cannot beoptimal if �∗4�5=å, for this rules out the necessary opti-mality condition fl4c

∗l 4å55≤ fh4c

∗h4å55 (Lemma 1.2). Recall

that c∗l 4�5= F −14�∗

l 4�5/å5 and c∗h4�5= F̄ −14�∗

h4�5/å5, where�∗l 4�5 and �∗

h4�5 are, respectively, the optimal arrival rates to land h classes for fixed �. Under 4h1 l5 with �∗4�5=å, theysatisfy �∗

h4å5=�−√

�/d and �∗l 4å5=å−�∗

h by Table A.3. Itfollows that c∗

l 4å5= c∗h4å5. The proof is complete since by

definition fl4c5 > fh4c5 for all c.

P4. Suppose pooling is not optimal for a fixed �<�H . Thenpooling is not optimal for �′ <�.

Proof of P4. If �≤ d−1 this follows since segmentation 4l5without pooling is optimal for all �′ <� by Proposition 2.1.Suppose that � ∈ 4d−11�H 50 By P3 segmentation 4h1 l5 mustbe optimal for �, and �∗4�5 <å. Let �∗

h4�5=�−√

�/d and�∗l 4�5= �∗4�5−�∗

h4�5 be the corresponding optimal rates.Optimality requires fl4F

−14�∗l 4�5/å55≤ fh4F̄

−14�∗h4�5/å55 by

Lemma 1.2, and

0 =ç∗

�4�∗4�51�5

⇔ v =å∫ F −14�∗

l 4�5/å5

cmin

f 4x5fl4x52�4�−�∗4�5+åF 4x553

dx

=å∫ F −14�∗

l 4�5/å5

cmin

f 4x5fl4x52�

4√

�/d−�∗l 4�5+åF 4x553

dx1 (83)

where the second equation holds since �−�∗4�5=√

�/d−

�∗l 4�5. We have �∗′

l 4�5 > 0, since the RHS of this equa-tion strictly decreases in � for fixed �∗

l 4�5 and strictlyincreases in �∗

l 4�5 for fixed �. Noting that �∗′h 4�5 = 1 −

1/42√

�d5 > 0 implies that fl4F −14�∗l 4�5/å55−fh4F̄

−14�∗h4�5/å55

strictly increases in �. Therefore, the optimality conditionsfor 4h1 l5 hold for every �′ ∈ 4d−11�5.

Part 1 and P3–P4 imply that there is a unique �P ∈ 6d−11�H 5such that pooling is not optimal for � ≤ �P and optimalfor � ∈ 4�P 1�H 5. Together with P1–P2 shown above, thisproves 2(a)–2(b).

Part 3. Threshold vA. By part 1 we need �A ≤ d−1, whichholds iff ç∗

�4å1�5≥ 0 for �= d−1. Segmentation 4l5 is optimalfor �= d−1. Substituting v = vA, �∗

l 4�5=å, �∗h4�5= 0 in (83)

yields

ç∗

�4å1�5��=d−1 = vA −å∫ cmax

cmin

f 4x5fl4x52d2

41 − dåF̄ 4x553dx = 00 (84)

Threshold vP . By part 2, we need �P = d−1. If v ≥ vA then�∗4�5=å for �≥ d−1, and it follows from P3 that poolingis optimal for � ∈ 4d−11�H 5. If v < vA then �∗4�5 < å for� = d−1, so ç∗

�4�∗4�51�5 = 0 for � = d−1. By P4 pooling

is optimal for � ∈ 4d−11�H 5 if and only if segmentation4h1 l5 is not optimal at any such �. This in turn holds ifffl4F

−14�∗l 4�5/å55− fh4F̄

−14�∗h4�5/å55≥ 0 for �= d−1, because

this virtual delay cost difference strictly increases in � asshown in proving P4. Noting that �∗

h4�5= 0 for �= d−1, thiscondition is equivalent to �∗

l 4�5≥åF 4f −1l 4cmax55. Substituting

in (83) the capacity �= d−1, the rate for l classes �∗l 4�5=

åF 4f −1l 4cmax55 and v = vP yields ç∗

�4�∗4�51�5= 00 The proof

is complete since �∗l 4�5 strictly increases in v for �= d−1. �

ReferencesAfèche P (2004) Incentive-compatible revenue management in queue-

ing systems: Optimal strategic delay and other delay tactics.Working paper, University of Toronto, Toronto, ON.

Afèche P (2013) Incentive-compatible revenue management in queue-ing systems: Optimal strategic delay. Manufacturing Service Oper.Management 15(3):423–443.

Afèche P, Mendelson H (2004) Pricing and priority auctions inqueueing systems with a generalized delay cost structure.Management Sci. 50(7):869–882.

Anderson ET, Dana JD (2009) When is price discrimination profitable?Management Sci. 55(6):980–989.

Ata B, Olsen M (2013) Congestion-based leadtime quotation andpricing for revenue maximization with heterogeneous customers.Queueing Systems 73(1):35–78.

Dow

nloa

ded

from

info

rms.

org

by [

142.

1.11

.215

] on

02

Aug

ust 2

016,

at 1

3:24

. Fo

r pe

rson

al u

se o

nly,

all

righ

ts r

eser

ved.

Afèche and Pavlin: Segmentation, Pooling, and Strategic Delay2436 Management Science 62(8), pp. 2412–2436, © 2016 INFORMS

Bagnoli M, Bergstrom T (2005) Log-concave probability and itsapplications. Econom. Theory 26(2):445–469.

Bansal M, Maglaras C (2009) Product design in a market withsatisficing customers. Netessine S, Tang CS, eds. Consumer-DrivenDemand and Operations Models (Springer, New York), 37–62.

Boyaci T, Ray S (2003) Product differentiation and capacity costinteraction in time and price sensitive markets. ManufacturingService Oper. Management 5(1):18–36.

Çelik S, Maglaras C (2008) Dynamic pricing and lead-time quotationfor a multiclass make-to-order queue. Management Sci. 54(6):1132–1146.

Coffman EG Jr, Mitrani I (1980) A characterization of waitingtime performance realizable by single-server queues. Oper. Res.28(3):810–821.

Dana JD, Yahalom T (2008) Price discrimination with a resourceconstraint. Econom. Lett. 100(3):330–332.

Deneckere RJ, McAfee RP, Damaged goods (1996) J. Econom. Manage-ment Strategy 5(2):149–174.

Hassin R, Haviv M (2003) To Queue or Not to Queue (Kluwer, Boston).Katta A, Sethuraman J (2005) Pricing strategies and service differenti-

ation in queues—A profit maximization perspective. Workingpaper, Columbia University, New York.

Maglaras C, Zeevi A (2005) Pricing and design of differentiatedservices: Approximate analysis and structural insights. Oper. Res.53(2):242–262.

Maglaras C, Yao J, Zeevi A (2016) Optimal price and delay differenti-ation in large-scale queueing systems. Working paper, ColumbiaUniversity, New York.

McAfee RP (2007) Pricing damaged goods. Econom.: The Open-AccessOpen-Assessment E-J. 1(2007-1):1–19.

Mendelson H (1985) Pricing computer services: Queueing effects.Comm. ACM 28(3):312–321.

Mendelson H, Whang S (1990) Optimal incentive-compatible prioritypricing for the M/M/1 queue. Oper. Res. 38(5):870–883.

Mussa M, Rosen S (1978) Monopoly and product quality. J. Econom.Theory 18(2):301–317.

Myerson RB (1979) Incentive compatibility and the bargainingproblem. Econometrica 47(1):61–74.

Naor P (1969) On the regulation of queue size by levying tolls.Econometrica 37(1):15–24.

Nazerzadeh H, Randhawa R (2015) Near-optimality of coarse servicegrades for customer differentiation in queueing systems. Workingpaper, University of Southern California, Los Angeles.

Plambeck E (2004) Optimal leadtime differentiation via diffusionapproximations. Oper. Res. 52(2):213–228.

Rao S, Petersen ER (1998) Optimal pricing of priority services. Oper.Res. 46(1):46–56.

Rochet J, Choné P (1998) Ironing, sweeping, and multidimensionalscreening. Econometrica 66(4):783–826.

Rochet J, Stole L (2003) The economics of multidimensional screening.Dewatripont M, Hansen LP, Turnovsky SJ, eds. Advances inEconomics and Econometrics: Theory and Applications (CambridgeUniversity Press, Cambridge, UK), 150–198.

Stidham S Jr (2002) Analysis, design and control of queueing systems.Oper. Res. 50(1):197–216.

Stidham S Jr. (2009) Optimal Design of Queueing Systems (CRC, BocaRaton, FL).

Van Mieghem JA (2000) Price and service discrimination in queuingsystems: Incentive compatibility of Gc� scheduling. ManagementSci. 46(9):1249–1267.

Zhao X, Stecke KE, Prasad A (2012) Lead time and price quotationmode selection: Uniform or differentiated? Production Oper.Management 21(1):177–193.

Dow

nloa

ded

from

info

rms.

org

by [

142.

1.11

.215

] on

02

Aug

ust 2

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. Fo

r pe

rson

al u

se o

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righ

ts r

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1

Online Appendix: Proofs of Technical Lemmas 4-8

Proof of Lemma 4. By (49) we have g (λ,λmh) = fl (cl (λ−λmh))− fh(ch(λh (λmh)

)). Set λmh =

λP and note that λh (λP ) = λP to get (50). Set λmh = λ and note that cl (0) = cmin to get (51). Setλmh = λF and note that λh (λF ) = 0 and ch (0) = cmax to get (52). Parts 1. and 2(b) follow sincef ′l c′l > 0 and the rate of l classes λ− λmh increases in λ, while λh is fixed. Part 2(a) follows since

f ′hc′h < 0 and λ

′h (λ)< 0 for λ∈ [λP , λF ] by (40), while λl = 0. �

Proof of Lemma 5. From Lemma 4.2, for fixed Λ the function g (λ,min(λ,λF ) ;Λ) is decreasingin λ ∈ [λP ,min(Λ, λF )], negative at λ = λF and increasing in λ ∈ [λF ,min(Λ, µ)]. Hence it has atmost two roots in λ, one smaller and the other larger than λF . These roots are determined by thesigns of g (λ,min(λ,λF ) ;Λ) for λ= λP , for λ= Λ when Λ ∈ (λP , λF ), and for λ= min(Λ, µ) whenΛ>λF . The proof identifies thresholds on Λ and λ that determine these signs.Part 1. The threshold Λ3 determines the sign of g (λ,min(λ,λF ) ;Λ) for λ= λP . It is the unique

solution of g (λP , λP ;Λ) = cmin−fh(F−1

(λP/Λ)) = 0 in Λ≥ λP . This follows because g (λP , λP ;λP ) =

cmin −fh (cmin)> 0 and since g (λP , λP ;Λ) decreases in Λ with limΛ→∞

g (λP , λP ;Λ) = cmin − cmax < 0.Therefore g (λP , λP ;Λ)> 0 if Λ<Λ3 and conversely for Λ>Λ3. Solving for Λ3 yields (55).The threshold Λ2 determines the sign of g (λ,min(λ,λF ) ;Λ) for λ= Λ. It is the unique solution

of g (Λ,Λ;Λ) = cmin − fh(F−1 (

λh (Λ)/Λ)) = 0 in Λ ∈ (λP , λF ). This follows since g (λP , λP ;λP ) =

cmin − fh (cmin) > 0 > g (λF , λF ;λF ) = cmin − cmax, and g (Λ,Λ;Λ) decreases in Λ ∈ [λP , λF ] as thefraction λh (Λ)/Λ allocated to h classes decreases in Λ. It follows that g (Λ,Λ;Λ)≥ 0 if Λ≤Λ2 andg (min (Λ, λF ) ,min(Λ, λF ) ;Λ)< 0 if Λ>Λ2. Solving for Λ2 yields (54).To show that Λ2 < Λ3, first note that g (λP , λP ;Λ2) > g (Λ2,Λ2;Λ2) = 0 by Lemma 4.2(a) since

λP <Λ2 <λF . Since g (λP , λP ,Λ) decreases in Λ it follows that Λ2 <Λ3.Parts 1(a)-(c). The above analysis implies for λ ∈ [λP ,min(Λ, λF )] the function g (λ,λ;Λ) is

non-negative if Λ ≤ Λ2, as in 1(a), and strictly negative if Λ > Λ3, as in 1(c). For 1(b), ifΛ ∈ (Λ2,Λ3) then g (λ,λ;Λ) has an unique root λ1 ∈ [λP ,min(Λ, λF )] since g (λP , λP ;Λ) > 0 >

g (min (Λ, λF ) ,min(Λ, λF ) ;Λ). Solving g (λ1, λ1;Λ) = cmin−fh(ch(λh (λ1) ;Λ

))= 0 yields (56)-(57).

Part 2. The thresholds Λml and Λml determine the sign of g (λ,min(λ,λF ) ;Λ) for λ= min(Λ, µ)

when Λ>λF . When Λ>λF and λ= min(Λ, µ), the maximum feasible total rate of h and m classesis λF : λmh = min(λF , λ) = min (λF ,min(Λ, µ)) = λF . By (53) the virtual delay cost difference is

g (min(Λ, µ), λF ;Λ) =

{g (Λ, λF ;Λ) = fl

(F−1

(Λ−λF

Λ

))− cmax, Λ∈ [λF , µ],

g (µ,λF ;Λ) = fl(F−1

(1dΛ

))− cmax, Λ≥ µ.

(85)

The function g (Λ, λF ;Λ) increases in Λ ∈ [λF , µ] as the fraction 1 − λF/Λ in l classes increases,g (µ,λF ;Λ) decreases in Λ≥ µ as the fraction 1/ (dΛ) in l classes decreases, and lim

Λ→∞g (µ,λF ;Λ) =

cmin−cmax<0. Hence g (min(Λ, µ), λF ;Λ) has an unique maximum for Λ≥ λF at Λ = µ where

g (µ,λF , µ) = fl

(F

(1

dµ

))− cmax > 0⇔ F (f−1

l (cmax)) · d< µ−1. (86)

Part 2(a). It follows that if F (f−1l (cmax)) · d < µ−1 < d then Λml ∈ (λF , µ) is the unique solution

of g (Λ, λF ;Λ) = 0 and Λml >µ is the unique solution of g (µ,λF ;Λ) = 0, where Λml and Λml satisfy(58)-(59). Furthermore, g (Λ, λF ;Λ)> 0 for Λ∈ (Λml, µ) and g (µ,λF ;Λ)> 0 for Λ∈ [µ,Λml). Becauseby Lemma 4.2(b) the function g (λ,λF ;Λ) satisfies g (λF , λF ;Λ)< 0 and increases in λ≥ λF for fixedλF <Λ, it has an unique root λ3 if Λ∈ [Λml,Λml). Solving g (λ3, λF ;Λ) = 0 yields (60)-(61).

2

If Λ /∈ [Λml,Λml), the above discussion and Lemma 4.2(b) imply that g (λ,λF ;Λ)< 0 for λ≥ λF .Part 2(b). If µ−1 ≤ F (f−1

l (cmax)) · d then the above analysis of g (min(Λ, µ), λF ;Λ) and (86),together with Lemma 4.2(b) again imply that g (λ,λF ;Λ)< 0 for all feasible λ≥ λF . �

Proof of Lemma 6. By Lemma 4.1, for fixed Λ>λP the virtual delay cost difference g (λ,λP ;Λ)

increases in λ. Hence for fixed Λ the function g (λ,λP ;Λ) has at most one root in λ∈ [λP ,min(Λ, µ)]

which the proof characterizes.The threshold Λ3 >λP is defined in Lemma 5 and determines the sign of g (λ,λP ;Λ) for λ= λP ,

where g (λP , λP ;Λ3) = 0 and g (λP , λP ;Λ)> 0 (< 0) if Λ< (>)Λ3.The threshold Λ4 defined in (62) determines the sign of g (λ,λP ;Λ) for λ= min(Λ, µ) where

g (min(Λ, µ), λP ;Λ) =

g (Λ, λP ;Λ) = fl(F−1

(Λ−λP

Λ

))− fh

(F−1 (λP

Λ

)), Λ∈ [λP , µ],

g (µ,λP ;Λ) = fl(F−1

(µ−λP

Λ

))− fh

(F−1 (λP

Λ

)), Λ≥ µ.

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For Λ ≤ µ and λ = min(Λ, µ) we have g (Λ, λP ;Λ) > 0: when all types are served the segmentswith high and low lead time qualities have the same marginal type cl (Λ−λP ;Λ) = ch (λP ;Λ) =

F−1

(λP/Λ), and (14)-(15) imply that fl(c) > fh(c) for c ∈ [cmin, cmax]. The threshold Λ4 is theunique solution of g (µ,λP ;Λ) = 0 in Λ ∈ [µ,∞), because g (µ,λP ;Λ) strictly decreases in Λ withlim

Λ→∞g (µ,λP ;Λ)< 0. Noting that µ−λP =

√µ/d and solving for Λ4 yields (62).

To summarize, for λ= min(Λ, µ) we have g (λ,λP ;Λ)> 0 if Λ<Λ4, and g (λ,λP ;Λ)≤ 0 if Λ≥Λ4.To prove Λ3 <Λ4 we show g (λ,λP ;Λ3)> 0 for λ= min(Λ3, µ). This follows since g (λP , λP ;Λ3) = 0

and Λ3 >λP as noted above, and because g (λ,λP ;Λ) increases in λ≥ λP by Lemma 4.1.To prove Λml <Λ4, since Λ4 >µ we show for the nontrivial case Λml >µ that g

(µ,λP ;Λml

)> 0.

Recall that g(µ,λF ;Λml

)= 0 as defined in Lemma 5.2, and that for fixed λ and Λ the virtual delay

cost difference g (λ,λmh;Λ) decreases in the rate λmh allocated to h and m classes. Setting λ= µ

and Λ = Λml and noting that λP <λF implies that g(µ,λP ;Λml

)> g

(µ,λF ;Λml

)= 0.

Parts 1-3. The claims follow from the established properties of Λ3 and Λ4, combined with the factof Lemma 4.1 that g (λ,λP ;Λ) increases in λ for fixed Λ>λP . �

Proof of Lemma 7. The claims on the slope of fh(ch(λh (λ)

))hold since f ′hc′h < 0, and λh (λ)

increases in λ < λP and decreases in λ ∈ (λP , λF ) by (40). In Part 1 the values of fh (ch (λ)) forλ= 0 and λ= min(λP ,Λ) follow as ch (λ) = F

−1(λ/Λ) and fh (cmax) = cmax. In Part 2 the fact that

fh(ch(λh (λ)

))= cmax for λ= λF ≤Λ holds since λh (λF ) = 0 by (40) and ch (0) = cmax = fh (cmax).

�

Proof of Lemma 8. Parts 1-3 follow by Lemma 7 and since fh(ch(λh (λ) ;Λ

))increases in Λ.

Part 3. For fixed Λ, if fh(ch(λh (λ) ;Λ

))≥ 0 at λ = min(λP ,Λ), then Lemma 7 implies that

fh(ch(λh (λ) ;Λ

))≥ 0 throughout. This condition holds if and only if Λ≥ Λsd: By (65) in Lemma

7, for λ = min(λP ,Λ) this virtual delay cost equals fh (cmin) < 0 if Λ ≤ λP , otherwise it equalsfh(F

−1(λP/Λ)) and increases in Λ>λP , and limΛ→∞ fh (ch (λP ;Λ)) = fh (cmax) = cmax > 0. Solving

for Λ sucht that fh(F−1

(λP/Λ)) = 0 yields Λsd >λP as in (68).Part 1. For fixed Λ, if fh

(ch(λh (λ) ;Λ

))< 0 at λ = min(λF ,Λ), then Lemma 7 implies that

fh(ch(λh (λ) ;Λ

))has an unique root λ0, where λ0 < λP and fh

(ch(λh (λ) ;Λ

))is nonnegative iff

λ≤ λ0. The condition holds if and only if Λ<Λsd: we have

fh(ch(λh (λ) ;Λ

))∣∣λ=min(λF ,Λ)

=

fh (ch (Λ;Λ)) = fh (cmin)< 0, Λ≤ λP ,fh(ch(λh (Λ) ;Λ

))= fh(F

−1(λh (Λ)/Λ)), Λ∈ (λP , λF ) ,

fh(ch(λh (λF ) ;Λ

))= cmax > 0, Λ≥ λF ,

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3

where fh(ch(λh (Λ) ;Λ

))= fh(F

−1(λh (Λ)/Λ)) increases in Λ ∈ (λP , λF ) since λh (Λ)/Λ decreases

in Λ. Hence Λsd ∈ (λP , λF ) is the unique solution of fh(ch(λh (Λ) ;Λ

))= 0 and satisfies (67). The

fact that Λsd <Λsd follows by the properties of Λsd.Part 2. If Λ∈ [Λsd,Λsd) then Parts 1 and 3 together with Lemma 7 imply the stated properties.It remains to rank Λsd and Λsd relative to Λ2 and Λ3, which are defined in Lemma 5. To see that

Λsd ≤Λ2, recall that g (Λ2,Λ2;Λ2) = 0 which is equivalent to fh(ch(λh (Λ2) ;Λ2

))= cmin. The rank-

ing follows since fh(ch(λh (Λsd) ,Λsd

))= 0≤ cmin and fh

(ch(λh (Λ) ;Λ

))increases in Λ ∈ [λP , λF ].

To see that Λsd ≤Λ3 recall that g (λP , λP ;Λ3) = 0 which is equivalent to fh (ch (λP ;Λ3)) = cmin. Theranking follows since fh

(ch(λP ,Λsd

))= 0≤ cmin and fh (ch (λP ;Λ)) increases in Λ. �