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Processing Time Ambiguity and Port Competitiveness

SangHyun CheonDepartment of Urban Planning and Design, School of Engineering, Hongik University, Seoul 04066, Korea, [email protected]

Chung-Yee LeeDepartment of Industrial Engineering and Logistics Management, Hong Kong University of Science and Technology, Clear Water Bay,

Kowloon, Hong Kong, [email protected]

Yimin WangDepartment of Supply Chain Management, W. P. School of Business, Arizona State University, Tempe, Arizona 85287, USA,

[email protected]

C ontainer seaports play an important role in cross-border logistics as firms increasingly expand their global footprintin sourcing, manufacturing and distribution. Besides convenience of access to hinterland regions, a key metric for a

port’s attractiveness is its processing time, that is, its ability to clear goods within a consistent, predictable time frame.Due to differences in infrastructure, government regulations, and operating procedures, ports may exhibit differentdegrees of predictability in processing times: some are more predictable while others are more ambiguous. We study howambiguity in processing times affects a port’s attractiveness under various circumstances. We find that even if a portmaintains a consistent expected processing time, increased ambiguity can still affect its attractiveness to firms, althoughnot always negatively. The effect of ambiguity depends on its nature, whether the shipments are time-sensitive, attitudestoward ambiguity, and trade terms surrounding shipments.

Key words: processing time; ambiguity; port operationsHistory: Received: March 2015; Accepted: June 2017 by Eric Johnson, after 3 revisions.

1. Introduction

Managing transit lead times is a critical task in globalsupply chain management. This is especially true forperishable products, which include frozen foodsrequiring temperature-controlled containers, andproducts with short selling seasons or life cycles, suchas Christmas decorations and customized ASICs(application-specific integrated circuits). Because per-ishable products have a limited selling window,uncertainties around transit time can have seriousconsequences for profitability.Based on a survey of 60 international carriers,

Chung and Chiang (2011) report that only a handfulachieve a transit schedule reliability of 90% or greater.Uncertainties in transit time can be categorized intoseveral stages: hinterland to port (ocean, land, or air),port processing (export clearance), and port to desti-nation (which includes ocean/land/air shipping,import clearance, and transportation to the hinter-land). Many factors can influence transit time, someof which are external, such as weather, congestion,and road conditions, which affect all shipments, whileothers are operator-related, such as capacity, fleet net-work, and scheduling. The impact of these factors on

transit lead time, however, is often predictable: forexample, firms can check weather forecasts and exam-ine carrier schedules to make reasonable estimates.What is unpredictable and yet significantly influ-

ences transit time is processing efficiency in terms ofcustoms clearance. Grosso and Monteiro (2011) reportthat customs procedures and characteristics, such ashours and efficiency, are among the most importantfactors in port selection (pp. 152–154). Similarly, San-chez et al. (2011) and Sayareh and Alizmini (2014)note that time efficiency and delays are key contribut-ing factors to a port’s attractiveness (pp. 155; 89).Tiwari et al. (2003) find that congestion negativelyaffects port selection, while efficiency has a positiveimpact.1 Because of complex regulations and safetymandates, the time required for shipments to clearcustoms is highly unpredictable. Hitachi, for example,faces 4000 pages of regulations when shipping com-ponents to the United States, and “even if only onecomponent has been misclassified, the entire ship-ment is likely to be held up in customs until that itemis in compliance” (Gary 2005). Similar observationshave been made on exporting procedures, where gov-ernment regulations and port bureaucracy can signifi-cantly influence port processing times (Persson et al.

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Vol. 26, No. 12, December 2017, pp. 2187–2206 DOI 10.1111/poms.12752ISSN 1059-1478|EISSN 1937-5956|17|2612|2187 © 2017 Production and Operations Management Society

2009). Furthermore, such complex processing proce-dures can cause shipments to miss their vessels, lead-ing to further delays down the line (Veenstra 2014).Based on a dataset of 44,723 container shipments

from Asia to the United States for a large retailer in2011, we find an average time at sea of 17.07 days (co-efficient of variation 0.41) and average time at port(from ship arrival to out gate) of 7.50 days (coefficientof variation 0.71).2 This suggests that the average timespent at port can be both very significant (around 44%of the length of the sea journey itself) and highly vari-able (coefficient of variation 0.71). Besides physicallocation and access to hinterland (Cheon et al. 2017),therefore, a port’s attractiveness to a shipper can besignificantly influenced by its efficiency and pre-dictability in customs clearance and other proceduresrelated to the clearing of a shipment. In this study, westudy the impact of processing time predictabilityand physical location on a port’s attractiveness toshippers and forwarders. For the sake of clarity, wefocus our discussion on maritime ports of origin, butour analysis applies equally to land crossings or air-ports that handle either import or export processes.As an illustration, consider the ports of Shenzhen

and Hong Kong. Both are deep water ports capable ofhandling the largest container ships, but in terms ofphysical location, the Shenzhen port is closer to andthus offers more convenient access to the hinterlandof the Pearl River Delta (PRD) than the Hong Kongport does. However, the Hong Kong port is often con-sidered more efficient with more predictable process-ing times (Hong Kong Trade Development Council2013).3

Predictability refers to a port’s ability to consis-tently adhere to predesignated processing time stan-dards. As processing times are inherently uncertain,predictability does not imply the processing times isdeterministic but rather means that it has a consis-tently known distribution. In contrast, processingtimes are unpredictable when there is a set of possibledistributions. Such unpredictability can be caused bymany factors, such as opaque inspection procedures,paucity of information, and complex, caprice govern-ment regulations. Firms can combine these factorswith estimation techniques such as market research,expert opinions, and looks-like analysis to generate arange of possible distributions of a port’s processingtime.4 We define these possible distributions collec-tively as an ambiguity set. Because the ambiguity setcontains beliefs (estimates) about possible processingtime distributions, it will expand, that is, include awider range of possible distributions, with decreasedpredictability but will shrink with increased pre-dictability. Processing times do not exhibit ambiguity(but still exhibit uncertainty) when the ambiguity setcontains only one distribution.

Isolating ambiguity from processing time uncer-tainty (e.g., as captured by a known distribution)offers several advantages. Firstly, ambiguity setexplicitly captures a firm’s uncertainty about the rele-vance of a particular distribution as the “correct” one.Secondly, it allows multiple distributions to be incor-porated into a firm’s forecasting and decision-makingprocess. Thirdly, doing so allows the impact of ambi-guity to be contrasted with that of uncertainty over afirm’s preference for a port, particularly when the firmis ambiguity-averse. As such, although uncertaintyover processing times exists in both cases (predictableor ambiguous), the ramifications for the shipper canvary substantially. A key question we explore is howthe processing time ambiguity of a port affects itsattractiveness to shippers. It is of significant manage-rial interest to understand whether a reduction inprocessing time ambiguity can enhance a port’s com-petitiveness despite a locational disadvantage.5

A port’s attractiveness may also be influenced bythe International Commercial Terms (Incoterms), e.g.,whether the trade terms are free on board (FOB) orcost insurance and freight (CIF). According to Leeand Wong (2007), the shipper (supplier) typicallydecides on the port of export, and hence attractiveports to buyers may not necessarily appeal to suppli-ers. We therefore consider both suppliers’ and buyers’preferences and explore how ambiguity interacts withIncoterms to affect a port’s attractiveness.

2. Related Literature

Ambiguity in lead time is related to the uncertain sup-ply literature, which can be roughly classified intothree different but related streams: supply disruption,random yield, and stochastic lead time. This researchis most closely aligned with the third stream.The problem of stochastic lead time is typically

studied in multi-period inventory models as opposedto newsvendor models. Bagchi et al. (1986), Song(1994), and Chopra et al. (2008) study the effect oflead time variability on service levels as well as theexpected cost for a single-item inventory system.Humair et al. (2013) study an inventory model thatincorporates stochastic lead times with a guaranteedservice level. Chopra et al. (2008) point out that thecommonly-used normal approximation of lead timedistribution can lead to erroneous safety stock levels.They examine several individual distributions inde-pendently, and thus the issue of ambiguity is irrele-vant. Other related works, such as Chu et al. (1994),Fujiwara and Sedarage (1997) and Proth et al. (1997),investigate the effect of stochastic lead times onexpected cost in multi-item inventory models. Zipkin(2000) provides a more detailed treatment of stochas-tic lead times in multi-period inventory models.

Cheon, Lee, and Wang: Time Ambiguity and Port Competitiveness2188 Production and Operations Management 26(12), pp. 2187–2206, © 2017 Production and Operations Management Society

However, none of the above research examines theeffect of ambiguity on the attractiveness of supplysources such as ports.Specific to the ocean shipping industry, Vernimmen

et al. (2007) find significant uncertainty in ocean ship-ping transit times, which has serious cost implicationson shippers’ and consignees’ bottom lines. Djankovet al. (2010) further quantify the effect of ocean ship-ping delays on the volume of international trade.From a modeling perspective, Meng et al. (2013)broadly review the literature on container routingand scheduling which have significant implicationsfor ocean shipping lead times. While we focus on theocean shipping industry, our study is also applicable(albeit with non-trivial differences) to the air transportindustry, where transit time uncertainty also has asignificant impact on the attractiveness of competingairports and carriers (see, e.g., Hao and Hansen 2014,Koppelman et al. 2008, Koster et al. 2011).While the above stream of research treats stochastic

lead times as given, another stream of literature inves-tigates the active management or estimation ofstochastic lead times to improve supply chain perfor-mance. Notteboom (2006) points out the importanceof understanding the root cause of variability in oceantransit time and suggests methods of improving sched-ule stability. From a modeling perspective, Brookset al. (2016) propose a Bayesian hierarchical model thatallows a more refined estimation of ocean shippingtransit times via different ports in close proximity.Beyond the ocean shipping industry, Chaharsooghiand Heydari (2010) study the relative benefits of reduc-ing average lead time or variability in lead time, andfind through simulation that variability reduction hasa larger impact on supply chain performance.Our work is related to the stream of literature distin-

guishing ambiguity from uncertainty. Camerer andWeber (1992) summarize theoretical development inmodeling uncertainty and ambiguity in the economicsliterature, while Schrader et al. (1993) illustrate thatambiguity can influence choices of solution differentlyfrom uncertainty. Dequech (2000) distinguishes uncer-tainty from fundamental ambiguity, which refers tosituations where further information to increase thereliability of future estimates cannot be obtainedthrough waiting. In such scenarios, it is neither usefulnor relevant to investigate the commitment timing ofbuyers’ purchasing decisions. Recently, Klibanoffet al. (2005) explore decision-making under ambiguityand show that ambiguity and uncertainty can be sepa-rated such that a decision-maker may exhibit differentattitudes toward ambiguity and risk.Our work is also closely related to the literature on

factors that influence a port’s attractiveness. Malchowand Kanafani (2004) suggest that the most significantfactor influencing attractiveness is physical location. In

China, however, Tiwari et al. (2003) note that port con-gestion and fleet size of shipping lines also play impor-tant roles. From an organizational perspective, Cheonet al. (2010) show that a port’s ownership structureand institutional reforms can improve its efficiencyand total productivity. Our research complements theabove research in that ports cannot change their physi-cal location and other factors enhancing their attrac-tivenessmust therefore be considered.In closing, we note that while there is extensive

literature on stochastic lead times and inventory poli-cies, no paper to date has explicitly linked lead timeambiguity to firms’ preferences in the context of portliterature to the best of our knowledge. A key contri-bution of this research is to provide one such link.The rest of the study is organized as follows: wedescribe the model in section 3 and conduct a prelimi-nary analysis in section 4. The effect of ambiguity isstudied in section 5 and we complement this analysiswith a numerical study in section 6. We conclude thestudy in section 7. All proofs are provided in theonline supplement.

3. Model

Consider a buyer (consignee) that purchases a certainamount of a perishable product from a foreign sup-plier (shipper). The product has a short shelf life, suchas fresh/cold produce, or a short selling season, suchas seasonal or high-tech products, and is hence sensi-tive to transit lead times. The lead time is composedof several components: transportation from the sup-plier hinterland to an ocean or land port, port process-ing (e.g., customs inspection), ocean shipping andimport processing, and finally transportation to thebuyer’s local market.To focus on the effect of ambiguity in port process-

ing time, we single out the processing time of port i as~ei and aggregate all other components of the transitlead time from the hinterland to the buyer’s marketvia port i as li. The total transit lead time via port i istherefore li þ ~ei. To reveal the effect of ~ei on portattractiveness, we further assume that li is fairly pre-dictable and hence deterministic. Note that the case ofstochastic li can be approximated in our model bymaintaining the expected li and aggregating thestochastic portion into ~ei.Within a regular selling window from time t to T,

the product commands the full market price of pf butonce this window is exceeded the product can only besold at a discounted price pd < pf. For notational con-venience, we adopt the convention that the buyerplaces the order at time 0 and that the supplier’s pro-duction time is instantaneous, such that the orderarrives by time li þ ~ei when the product transitsthrough port i. If the shipment arrives before time t,

Cheon, Lee, and Wang: Time Ambiguity and Port CompetitivenessProduction and Operations Management 26(12), pp. 2187–2206, © 2017 Production and Operations Management Society 2189

the buyer incurs a unit holding cost of h per unit time.Figure 1 illustrates the relationship between trans-portation lead time (~ei þ li) and the regular sellingtime window (T � t).Market demand D is random and follows a general

distribution D � G(�). The buyer places an order attime 0 with an order size of Q to meet marketdemand. If the shipment arrives within the regularselling window, the system behaves similarly to anews vendor problem: unmet demand is lost and theleftover inventory is salvaged at price pd. On the otherhand, if the shipment misses the regular selling win-dow, then the product can only be sold at the dis-counted price pd.

3.1. Processing Time AmbiguityWe refer to port i as having a predictable processingtime if ~ei follows a known distribution, that is,~ei � Fið�Þ. In contrast, port i’s processing time isambiguous if there is a set of plausible distributionsF i ¼ fFi;1ð�Þ; Fi;2ð�Þ; . . .; Fi;Mi

ð�Þg, and the firm isunsure which distribution is ‘correct’ for ~ei. The firm’ssubjective assessment of the likelihood that ~ei isdrawn from Fi;kð�Þ, k 2 {1, . . ., M} is qi;k � 0, whereqi;k is the second-order probability over the first-orderdistribution Fi;k. The processing time at port i becomesmore ambiguous as the second-order probabilitybecomes more diffuse, that is, maxk 6¼l jqi;k � qi;lj ! 0,and becomes more predictable as maxk 6¼l jqi;k �qi;lj ! 1. As such, a predictable processing time is aspecial case of ambiguous processing time withqi;k ¼ 1 and qi,l6¼k = 0 for some k. Segal (1987), Kliban-off et al. (2005), and Saghafian and Tomlin (2016) pro-vide further examples, and discuss the properties andapplications of ambiguity represented by second-order probability.It is worth pointing out that if the firm is ambigu-

ity-neutral then an ambiguous processing time is

mathematically equivalent to a predictable processingtime defined by Fið�Þ � Rkqi;kFi;kð�Þ. Otherwise if thefirm is ambiguity-averse (or ambiguity-loving) thenthis equivalence breaks down (Proposition 1, Kliban-off et al. 2005). Recall that, even if the firm is ambigu-ity-neutral, separating predictable and ambiguousprocessing times is conceptually useful (see section 1).Suppose, for example, port 1 has a predictable pro-

cessing time with a known distribution F1(�), whereasport 2 has an ambiguous processing time with twoplausible distributions F2,1(�) and F2,2(�). If port 1 isselected, the firm knows that the shipment will clearthe port in x days with probability F1(x). In contrast, ifport 2 is selected, this probability is uncertain: it canbe either F2,1(x) or F2,2(x). In this case, the firm relieson a subjective assessment (i.e., second-order proba-bility) of whether F2,1 or F2,2 is more likely to be ‘cor-rect’ to estimate the probability that the shipment willclear the port in x days. Thus, in the former case theprobability of clearing in x days is deterministic,whereas in the latter case it is conceptually probabilis-tic. Modeling the latter case with second-order proba-bility is therefore conceptually more appealing if theprocessing time is influenced by several distinct ran-dom factors (e.g., congestion, inspection, demurrageand detention) and the firm is unsure which factorsare active.6

Increased ambiguity may not necessarily lead toincreased (overall) variance. Suppose port i’s process-ing time is governed by F i ¼ fFi;kg, k 2 {1, 2, 3}. Apredictable processing time can be constructed by set-ting the second-order probability qi,2 = 1 and qi,1 =qi,3 = 0. Reducing qi,2 and increasing qi,1 and qi,2 thusintroduces ambiguity, and maximum ambiguity isreached at qi,1 = qi,2 = qi,3 = 1/3. The two extremecases have identical overall variances if r2i;1 þ r2i;3 ¼8r2i;2, where ri,k is the standard deviation associatedwith Fi;k. In addition, ambiguity can also be increased

Stochastic port processing time

Fixed land and oceantransportation time

Regular selling window

Total transportation lead time

Shipment startsat time

Regular selling seasonstarts at time

Regular selling seasonends at time

Figure 1 Transportation Lead Time and the Time Window of the Regular Selling Season [Color figure can be viewed at wileyonlinelibrary.com]


or reduced without affecting the overall expected pro-cessing time, even if F i includes asymmetric distribu-tions (e.g., exponential) with different means.To consistently compare the efficiency of different

ports, we scale ~ei and ~ej such that E½~ei� ¼ E½~ej� for anygiven i 6¼ j. In other words, the expected port process-ing times are all identical, but have different degreesof ambiguity with some ports having better-definedprocessing time distributions than others do.

3.2. Buyer’s and Supplier’s ObjectivesThe buyer may place an order with the supplierunder two standard Incoterms: FOB or CIF. UnderFOB, the supplier is responsible for delivering thegoods to an origin port, while under CIF, the sup-plier must deliver the goods to a destination port(importing port) specified by the buyer. UnderFOB, the supplier absorbs transportation costs onlyup to the exporting port i, whereas under CIF thesupplier absorbs all of the transportation costs tothe destination port.To distinguish between these two arrangements, let

ai denote the fraction of the freight cost via port i. Thesupplier then incurs a freight cost of ai(/i + siQ)under FOB but incurs /i + siQ under CIF, where /i

and si are the fixed and variable (unit) freight costs,respectively. The supplier’s unit production cost is c,and it charges the buyer a unit selling price of w.Under FOB, the buyer’s objective is to determine an

order quantity Q that maximizes its expected profitwhen port i is selected.

pBðQÞ ¼ pfED minfQ;Dg½ � þ pdED ðQ�DÞþ� �� Probfli þ ~ei �Tg þ pdQProbfli þ ~ei [Tg� hQE~ei ðt� li � ~eiÞþ

� �� wQ

� ð1� aiÞð/i þ siQÞ; ð1Þ

where the first two terms are the expected revenueif the shipment arrives in time for the regular sellingseason or arrives late, respectively, and the last threeterms capture the expected holding (if the shipmentarrives early), purchasing, and transportation costsrespectively. The inventory carrying-cost along thevoyage is subsumed in the transportation cost si. IfCIF is used, the last term is dropped from the aboveexpression because the supplier absorbs all of thetransportation costs.The supplier’s corresponding objective function is

to determine the unit selling price w that maximizesits expected profit.

pSðwÞ ¼ ðw� cÞQ� aið/i þ siQÞ; ð2Þwhere the first term captures the supplier’s profitand the second term captures the transportation costunder FOB. If CIF is used, then the second term is

replaced by /i + siQ, that is, the supplier bears all ofthe transportation costs.In practice, the supplier can simply implement a

cost-plus pricing approach to determine its unit sell-ing price w. In such a case, the supplier can setw = c + aisi + c, where c is the supplier’s desired unitmargin. We will consider this alternative pricingscheme later.Several factors are not captured in the above

model but are worth discussing here. First, the sup-plier may be held liable for late shipments and thebuyer may assess a penalty against the supplier ifthe shipment arrives after time T. However, if theshipment delay is caused by port processing timeuncertainty, the supplier may claim force majeure,such as an interruption of the operations of a termi-nal accepting a tanker. A supplier that experiencesan unusually lengthy customs clearance, such as dueto a labor strike, can also claim force majeure, but formore usual situations the liabilities are industry-spe-cific. Second, the supplier may also be concernedabout cash flows involving accounts receivable and/or exchange rate risks that may influence the attrac-tiveness of FOB vs. CIF as well as the supplier’s sell-ing price w. Our model does not capture such issuesas our primary focus is on port processing time.Lastly, port delays may also generate additional portstorage-related costs for the supplier, but we ignoresuch issues here as well.

4. Basic Structure

4.1. Buyer’s Ordering DecisionIn the following, we examine the buyer’s objectivefunction for any given unit purchasing price w(quoted by the supplier). Simplifying Equation (1), wehave

pBðQÞ ¼Eqi;k

"ðpf � pdÞ

�Zx�Q

xdGðxÞ þQGðQÞ�

� Fi;kðT � liÞ � hQ

�ðt� liÞFi;kðt� liÞ

�Z~ei � t�li

~eidFi;kð~eiÞ�#

� �wþ ð1� aiÞ

si � pdÞQ� ð1� aiÞ/i: ð3Þ

The second term in the square brackets drops out ift ≤ li, in which case the shipment cannot arrivebefore the start of the selling season t. The followingproposition proves that the buyer’s order quantitydecision is well-behaved.

PROPOSITION 1. The buyer’s expected profit function isconcave in the order quantity Q. Furthermore, the


optimal (interior) order quantity Q� is given by thefollowing:

GðQ�Þ¼

hEqi;k ðt�liÞFi;kðt�liÞ�Z~ei�t�li

~eidFi;kð~eiÞ� þwþð1�aiÞsi�pd

ðpf�pdÞEqi;kFi;kðT�liÞ : ð4Þ

The following sensitivity results ensue:

COROLLARY 1. The buyer’s optimal order quantity is(a) increasing in unit selling price pf, discount price pd,the supplier’s share of transportation cost ai, and thelength of the selling season T; (b) decreasing in unitpurchasing cost w, unit holding cost h, unittransportation cost s, and the starting time of the sellingseason t.

The observation that the buyer orders largerquantities as T increases and smaller quantities as tincreases warrants further discussion. Firstly,because we have fixed the starting time, that is, theordering time, at time 0, an increase in T impliesthat the firm orders well in advance of the sellingseason and is hence more likely to receive the ship-ment in time. This reduced risk allows the buyer toorder more. However, as the buyer places its orderearly, its demand estimate becomes less accurate,which may dampen its preference for early orders.Wang and Tomlin (2009) treat this problem in moredetail.In contrast, when t increases, the selling window

T � t narrows and, although this also implies that thebuyer orders early with respect to the start of the sea-son, the reduced selling time span exerts dominateeffect on the buyer’s expected profit, impelling thebuyer to reduce its order to hedge against the higherrisk of a limited selling time.Analogously to Corollary 1, we can establish simi-

lar sensitivity results for the buyer’s optimal expectedprofit through the envelope theorem. Hence thebuyer’s expected profit is intimately related to itsorder quantity, and we can therefore study how sys-tem parameters (i.e., processing time ambiguity)influence the buyer’s expected profit by observing itsorder quantity decision.

4.2. Supplier’s Pricing DecisionWhen the supplier quotes unit selling price w, it takesinto consideration the buyer’s ordering decision asdescribed in section 4.1. As such, we use the notationQ�ðwÞ to stress that the buyer’s optimal order quantitydepends on the supplier’s price quote w. By Equation(2), we have

pSðwÞ ¼ ðw� c� aisiÞQ�ðwÞ � ai/i: ð5Þ

The following assumption enables us to character-ize the supplier’s optimal pricing decision.

ASSUMPTION 1. The demand distribution G(�) satisfiesan increasing failure rate (IFR), that is, gðxÞ=�GðxÞweakly increases in x for all G(x) < 1.

A wide range of distribution functions satisfy IFR,and Lariviere (2006) provides a detailed discussion ofthe IFR and increasing generalized failure rate(IGFR).7 The following proposition proves that undermild assumptions the supplier’s pricing decisionproblem is well behaved:

PROPOSITION 2. Suppose Assumption 1 holds. If pd � c ≤si, then the supplier’s objective function is concave in theunit selling price w quoted to the buyer. Furthermore, theoptimal unit selling price w� is given by

w� ¼ cþ aisi þ ðpf � pdÞgðQ�ÞQ�Eqi;k Fi;kðT � liÞ� �

; ð6Þ

where Q* is given by Proposition 1.

By Proposition 2, the supplier quotes a selling pricethat covers its unit production cost and its share ofvariable transportation costs, plus additional marginsthat depend on the buyer’s profitability as well as theselling time T.In practice, many suppliers still use the cost-plus

approach in quoting unit selling prices. In other words,the suppliers may simply add a fixed percentage of mar-gins to their cost (unit production and transportation) toarrive at their final price quotes. Although this practice istheoretically suboptimal, its ease of implementation leadsto its widespread usage. The cost-plus approach can beseen as a special case of the optimal pricing decisiondescribed in Equation (6), where w�

cp ¼ c þ aisi þ cand the profit margin c can be regarded as the supplier’sexpectations of Eqi;k ½ðpf � pdÞ Q�gðQ�ÞFi;kðT � liÞ�. Wewill highlight the impact of the cost-plus pricingapproach on the buyer’s and supplier’s preferences over(or tolerance of) processing time ambiguity.

4.3. Equilibrium BehaviorsGiven the buyer’s and supplier’s optimal orderingand pricing decisions, we can further characterizetheir equilibrium behaviors. Substituting Equation (6)into Equation (4), we have

GðQ�Þ¼Q�gðQ�Þ

þ

hEqi;k

�ðt� liÞFi;kðt� liÞ�

Z~ei�t�li

~eidFi;kð~eiÞ

þ cþ si�pdðpf �pdÞEqi;kFi;kðT� liÞ :

ð7Þ


It follows directly from Equation (7) that the buyer’soptimal order quantity does not depend on its shareof transportation costs ai. This suggests that regard-less of whether its transaction is based on FOB orCIF, all else being equal the buyer’s order quantitywill remain the same. This is consistent with casualobservations in practice that firms rarely adjust theirorder quantity in response to changes in Incoterms.In contrast, combining Equation (7) with Equation

(6), we find that the supplier’s pricing decision w doesdepend on its share of transportation costs. Neverthe-less, the supplier’s optimal selling price w� fullyabsorbs transportation costs, such that regardless ofwhether FOB or CIF is used, the supplier’s expectedprofit is also unaffected by Incoterms.The above observations are based on the assump-

tion that the supplier has full pricing power. In casethe supplier practices the cost-plus pricing schemedescribed at the end of section 4.2, such that the unitprice equals a fixed proportion of the production costplus transportation costs, that is, w = c + aisi + c, thenboth the supplier’s unit price and the buyer’s orderquantity are influenced by Incoterms. As a result,their expected profits are also affected by Incoterms.

5. Impact of Lead Time Ambiguity

Increased ambiguity in port processing time affectsthe port’s attractiveness to the buying firm by influ-encing the firm’s subjective assessment of the likeli-hood that the shipment will arrive in time for theselling season. The supplier’s preference is also indi-rectly affected by ambiguity levels through the buy-ing firm’s ordering behavior.If increased ambiguity is associated with a longer

expected mean processing time, the buying firmunderstandably finds the port less attractive and viceversa. In contrast, the buying firm’s preferences areless obvious when the expected mean processing timeremains constant as ambiguity levels vary. It is notimmediately clear a priori whether the buying firmalways prefers a port with less ambiguity, given thesame mean processing time. In the following, wefocus on the latter case where the expected mean pro-cessing time is kept constant while the ambiguitylevels are varied.To place our study of ambiguity into practical con-

text, let us first consider two different scenarios whereambiguity in port processing time may arise.

1. The port processing time is influenced by dis-crete, isolated events, but the firm is unsurewhich events will occur. For example, theshipment may or may not by delayed at cus-toms due to various operational factors. Ifthere is no delay, the shipment’s processing

time ~ei is characterized by Fi;1ð�Þ; if there is adelay, the processing time ~ei is characterizedby Fi;2ð�Þ. In this example, it is expected thatEFi;1ð�Þ½~ei�\EFi;2ð�Þ½~ei�, and ambiguity mainlyinfluences the firm’s assessment of theexpected port processing time.

2. The port processing time is influenced by fre-quent, random noise. For example, the hetero-geneity of the mix of shipments at the portvaries over time. These varying cargo mixesmay influence the variability of the port pro-cessing time but not necessarily the expectedprocessing time. In such cases ambiguitymainly influences the firm’s assessment of thevariability in the port processing time.

The above examples lead to two natural classifica-tions of processing time ambiguity based on whetherthe distributions (in the ambiguity set F i) satisfy first-order stochastic dominance (FOSD) or second-orderstochastic dominance (SOSD). Figures 2 and 3 illus-trate examples of these two types of processing timeambiguity.Figure 2 illustrates an ambiguity set F i within

which the three distributions satisfy the FOSD, that is,EFi;1 ½~ei�\EFi;2 ½~ei�\EFi;3 ½~ei�. The set of distributions inF i need not have identical variances. The anticipatedmean processing time for port i depends on qi;k, thefirm’s subjective assessment of the likelihood of ~eibeing drawn from Fi;k.In contrast, Figure 3 illustrates an ambiguity set

within which the three distributions satisfy the SOSD.The set of distributions in F i have identical mean butdifferent variances, and, as a result, the anticipatedmean processing time does not depend on qi;k but theanticipated variance of ~ei does. A mean-preservingspread of the set of distributions in F i reflects increasedprocessing time uncertainty but not ambiguity.Below, we first explore the ambiguity set with

FOSD and then that with SOSD, assuming that thebuyer is ambiguity neutral. We will consider

10 20 30 40Processing Time

0.05

0.10

0.15

0.20

Density

Figure 2 Illustration of the Ambiguity Set with Distributions SatisfyingFOSD [Color figure can be viewed at wileyonlinelibrary.com]


ambiguity aversion in section 5.3. For expositionalease, in the following we assume that Mi ¼ jF ij is anodd number, and define Mi ¼ ð1 þ MiÞ=2, that is, Mi

indexes the middle distribution in F i.

5.1. Lead Time Ambiguity Manifested throughFOSDBy the definition of FOSD, the distributions in theambiguity set F i satisfy Fi;kðxÞ [ Fi;lðxÞ for any1 � k\ l � Mi and any given x. To facilitate the fur-ther characterization of the buyer’s and supplier’soptimal behavior, we also need to consider the firm’ssubjective assessment of ~ei being drawn from Fi;k,that is, the structure of qi;1; qi;2; . . .; qi;Mi

. A naturalstructure that can represent values of qi;k is the dis-crete symmetric unimodal distribution qi(k, a), whereqi;k ¼ qiðk; aÞ and a 2 fnj1=Mi � n� 1g measures theheight (i.e., dispersion) of the qi(�, a) function. In par-ticular, a = 1 reduces the processing time to apredictable one because it implies that qi;Mi

¼ 1 (andqi;k 6¼Mi

¼ 0) due to the unimodal assumption. In con-trast, 1=Mi � a\ 1 corresponds to an ambiguous pro-cessing time distribution. In the special case ofa ¼ 1=Mi, the processing time ~ei is equally likely tobe drawn from any distribution from the ambiguityset F i.The qi(�, a) function behaves somewhat, but not

exactly, like a tent map function, where a larger valueof a corresponds to more concentrated and hencemore predictable, subjective assessments, whereas asmaller value of a corresponds to more dispersed andambiguous, subjective assessments. It follows fromthe above definition that for any given a and1 � k\Mi, we have qiðk; aÞ ¼ qiðMi � k þ 1; aÞ.Similarly, we have qi(k, a) < qi(l, a) for anyk\ l\Mi, and qi(k, a) > qi(l, a) for anyMi [ k [ l.Leveraging the above observations, we define a

useful metric to characterize the impact of ambiguity.For a given port i, an anticipated probability of thelead time being less than x under ambiguity level a isdefined as

Liðx; aÞ ¼XMi

k¼1

qiðk; aÞFi;kðxÞ

¼XMi�1

k¼1

qiðk; aÞ Fi;kðxÞ þ Fi;Mi�kþ1ðxÞ� �

þ qiðMi; aÞFi;MiðxÞ:

ð8Þ

By the definition of qi(�, a), we have @qiðMi; aÞ=@a [ 0 and @qi(k, a)/@a < 0 for some k\Mi. Noticethat

@Liðx; aÞ@a

¼XMi�1

k¼1

@qiðk; aÞ@a

Fi;kðxÞ þ Fi;Mi�kþ1ðxÞ� �

þ @qiðMi; aÞ@a

Fi;MiðxÞ: ð9Þ

Although the second term in Equation (9) is alwayspositive, the first term may be either positive or neg-ative. Hence it is not immediately obvious whetherLi(x, a) is monotone in a in general. With some regu-latory conditions, however, the following lemmacharacterizes the behavior of the Li(x, a) function.

LEMMA 1. For a given x > 0, if

Fi;kðxÞ þ Fi;Mi�kþ1ðxÞ� Fi;lðxÞ þ Fi;Mi�lþ1ðxÞ� Fi;MiðxÞð10Þ

for all 1 � k\ l\Mi, then @Li(x, a)/@a ≥ 0. Otherwiseif the inequality sign in Equation (10) is reversed then@Li(x, a)/@a < 0.

It is worth pointing out that the results in Lemma 1depend on the value of x, as different values of x ingeneral influence the inequality condition (10). Lever-aging Lemma 1, we can now investigate how leadtime ambiguity affects the buyer’s and supplier’spreference toward a particular port. Substitutingx = T � li into the Li(x, a) function, we haveLiðx; aÞ ¼ LiðT � li; aÞ ¼ Eqi;k ½Fi;kðT � liÞja�, where alarger value of a corresponds to a less ambiguous sub-jective assessment of the lead time distribution. Toillustrate how Eqi;k ½Fi;kðT � liÞja� changes with theambiguity level a (in relation to condition (10) inLemma 1), we consider two special cases where Fi;kfollows the uniform distribution and the exponentialdistribution.The uniform distribution is appropriate when the

firm is minimally informed about the processing timedistribution except plausible lower and upperbounds. The firm then relies on second-order proba-bility qi;k to assess the likelihood that a particular setof (lower and upper) bounds is correct.

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0.10

0.15

0.20

Density

Figure 3 Illustration of the Ambiguity Set with Distributions SatisfyingSOSD [Color figure can be viewed at wileyonlinelibrary.com]


In contrast, the exponential distribution is moreappropriate if the firm knows a set of plausible valuesof the mean processing time. The firm then relies onqi;k to assess the likelihood that a particular value forthe mean processing time is correct. The exponentialdistribution guarantees that the processing time ~ei ispositive, and because it has long tails, the exponentialdistribution is often an accurate approximation of ~eiwhen only the mean processing time is specified.

5.1.1. Ambiguity Set with Uniform Distribution.Suppose jF ij ¼ 3 and Fi;k � Uðli;k; ri;kÞ, whereli,1 < li,2 < li,3 and ri,1 = ri,2 = ri,3 = ri, such thatEFi;k ½~ei� increases with k but the variance of ~ei doesnot depend on k. This ensures that specifications ofFi;k indeed satisfy FOSD. For clarity, assume that thedistributions have positive support and do not over-lap, that is, li;k � ffiffiffi

3p

ri � li;k�1 þ ffiffiffi3

pri � 0. The sec-

ond-order probabilities are qið1; aÞ ¼ qið3; aÞ ¼12 ð1 � qið2; aÞÞ ¼ 1

2 ð1 � aÞ, where 13 � a � 1. We

have

Eqi;k

�Fi;kðT � liÞja

� ¼ LiðT � li; aÞ¼ qið1; aÞ

�Fi;1ðT � liÞ þ Fi;3ðT � liÞ

�þ qið2; aÞFi;2ðT � liÞ

¼ 1

2ð1� aÞ

(min

ðT � li � li;1 þffiffiffi3

priÞþ

2ffiffiffi3

pri

; 1

!

þminðT � li � li;3 þ

ffiffiffi3

priÞþ

2ffiffiffi3

pri

; 1

!)

þ aminðT � li � li;2 þ

ffiffiffi3

priÞþ

2ffiffiffi3

pri

; 1

!: ð11Þ

Notice that the above expression satisfies condition(10) in both ways for any given T � li, and hencethe sign of @Eqi;k ½Fi;kðT � liÞja�=@a cannot be solelydetermined by specifications of Fi;kð�Þ. As mentionedabove, the sign depends on the value of T � li. Sup-pose T � li is within the maximum range of the leadtime distributions, that is, li;1 � ffiffiffi

3p

ri � T � li �li;3 þ ffiffiffi

3p

ri.If li;2 � ffiffiffi

3p

ri � T � li � li;2, then by Equation (11)we have

@Eqi;k Fi;kðT� liÞja� �

@a¼�1

2þT� li�li;2þ

ffiffiffi3

pri

2ffiffiffi3

pri

�0: ð12Þ

It can be verified that @Eqi;k ½Fi;kðT � liÞja�=@a � 0 forany T � li ≤ li,2 and the sign is reversed for anyT � li > li,2.The above example illustrates that with uniform

distributions, increased ambiguity (i.e., more diffusesecond-order probability) leads to a shorter anticipatedlead time if T � li ≤ li,2 and vice versa. In the follow-ing, we use li (which equals li,2 in the above example)

to denote the midpoint processing time of port i. The fol-lowing proposition ensues.

PROPOSITION 3. Suppose Fi;k 2 F i follows a uniform dis-tribution, and t ≤ li or h 0. If T � li � li then as theprocessing time becomes more ambiguous,

(a) The buyer’s optimal order quantity Q� increases;(b) The supplier quotes a higher unit selling price w�;(c) Both the supplier’s and buyer’s optimal expected

profits increase;

The reverse is true if T � li [ li.

The buyer’s and supplier’s preference towards aparticular port therefore depends critically on howimportant it is for the shipment to meet the sellingtime window. For time-sensitive shipments, wherethe midpoint processing time is greater than the tran-sit buffer time, that is, li [ T � li, a port with anambiguous processing time is more appealingbecause there is a greater chance that the processingtime will turn out to be short so that the shipmentarrives in time for the selling season. Although thereis also an equally greater chance that the shipmentwill arrive late, the upside gain from its arriving intime dominates the downside loss from its arrivinglate.8 In contrast, with little ambiguity, the shipmentis almost guaranteed to be late for the selling season.Under practical settings where the shipment isplanned ahead with sufficient transit buffer time, thatis, li � T � li, the buying firm is likely to prefer aport with less ambiguity in processing time. Hence,buyers and suppliers that plan ahead will more likelychoose ports with less ambiguous processing times.Regardless of whether the shipment is time-sensi-

tive, that is, whether li [ T � li, the buyer andthe supplier are equally affected by ambiguity: theyeither both benefit from increased ambiguity or areboth hurt by it. Combining this observation withthe earlier result that both the buyer and the sup-plier are indifferent to Incoterms (see section 4.3),we can conclude that the buyer’s and supplier’spreferences toward ambiguity are not affectedbyIncoterms—as long as the supplier optimizes theunit wholesale price w.Such consistent preferences have significant man-

agerial implications. Because Incoterms have no per-ceivable impact on buyer and supplier preferences, aport’s attractiveness is significantly influenced byits overall transit buffer time and processing timeambiguity.For example, although the port of Shenzhen is clo-

ser to the hinterland of the PRD than that of HongKong, if the overall transit buffer time T � li is thesame their attractiveness to buyers or suppliersis largely influenced by whether the shipment is


time-sensitive and their relative operational efficiencyin processing the shipment. Suppose hypotheticallythat the Shenzhen port exhibits more ambiguity thanthe Hong Kong port. All else being equal both thebuyer and the supplier are likely to select the Shen-zhen port for time-sensitive shipments but pick theHong Kong port for most regular shipments. TheShenzhen port could therefore attract more regularshipments by reducing the processing time ambiguityand enhancing operational efficiency. In contrast, amore reasonable approach for the Hong Kong port toattract more shipments is to further reduce its mid-point processing time li while maintaining or reduc-ing the associated ambiguity levels.The above analysis thus far ignores inventory hold-

ing costs if a shipment arrives early, which will dam-pen the directional effect discussed above. Oursubsequent numerical study (section 6) suggests thatsuch a dampening effect does not qualitatively impactthe above observations.

5.1.2. Ambiguity Set with Exponential Distribu-tion. The exponential distribution is more realistic inpractice, as processing times often adhere to certainpublished standards but can be significantly longer(i.e., with long tails) when there are delays. SupposejF ij ¼ 3 and Fi;k � expðli;kÞ, where li,1 < li,2 < li,3such that Fi;k’s satisfy FOSD. The second-order proba-bilities are qð1; aÞ ¼ qð3; aÞ ¼ 1

2 ð1 � qð2; aÞÞ ¼ 12ð1 � aÞ, where 1

3 � a � 1. We have

Eqi;k Fi;kðT � liÞja� � ¼ LiðT � li; aÞ ¼ qið1; aÞ

�Fi;1ðT � liÞ

þ Fi;3ðT � liÞ�þ qið2; aÞFi;2ðT � liÞ

¼ 1

2ð1� aÞ

(1� exp �T � li

li;1

!

þ 1� exp �T � lili;3

!)

þ a 1� exp �T � lili;2

!( ): ð13Þ

Directly verifying the above expression with condi-tion (10) in Lemma 1 is analytically challenging. Thefollowing lemma, however, partially characterizesthe behavior of Eqi;k ½Fi;kðT � liÞja�.

LEMMA 2. Suppose values of li,j are symmetric suchthat they can be represented as li,1 = l � x, li,2 = l,and li,3 = l + x. Define hðk; xÞ ¼ ðe� k

lþx � e�klÞ=

ðe� klþx � e�

kl�xÞ. If h(T � li, x) ≥ 1/2, then @Eqi;k

½Fi;kðT � liÞja�=@a � 0, and the reverse is true ifh(T � li, x) < 1/2. Furthermore, for any given x, thereexists a �k such that h(k, x) ≥ 1/2 for any k � �k, andh(k, x) < 1/2 for any k\�k.

Similarly to the uniform distribution case,Eqi;k ½Fi;kðT � liÞja� can be increasing or decreasing in a,which depends partially on the shipping buffer timeT � li. Specifically, if the shipping buffer time exceedsa critical threshold then Eqi;k ½Fi;kðT � liÞja� alwaysincreases in a. Leveraging Lemma 2, the followingproposition describes the impact of ambiguity on portpreference when the ambiguity set F i consists ofexponential distributions.

PROPOSITION 4. Suppose Fi;k 2 F i is exponentially dis-tributed, and that t ≤ li or h 0. If the shipping buffertime T � li is sufficiently large (i.e., T � li � �k), then asthe processing time becomes more ambiguous,

(a) The buyer’s optimal order quantity Q� decreases;(b) The supplier quotes a lower unit selling price w�;(c) Both the supplier’s and buyer’s optimal expected

profits decrease.

The reverse is true if T � li \�k.

The supplier’s and buyer’s preferences thereforeexhibit similar traits regardless of whether the ambi-guity set consists of uniform or exponential distribu-tions. Ambiguity can influence the supplier’s andthe buyer’s choice of port in different ways. If ashipment is not time-sensitive both the supplier andthe buyer tend to prefer a port with less ambiguity.An important managerial insight is that a port canincrease its competitiveness by reducing the ambi-guity associated with its processing times for regu-lar shipments. This can even enable it to overcomea locational disadvantage, such as longer distancesto the hinterland or shorter transit buffer times(T � li), to gain business from both the supplier andthe buyer by improving its processing predictabilityfor regular shipments.On the other hand, if a shipment is time-sensitive,

their preferences reverse and they tend to prefer a portwith more ambiguity. Part of the intuition is that therealized processing time can equally be short or long,but the benefit from a short processing time (thusallowing the shipment to arrive in time for the sellingseason) offsets the cost of a long processing time. Animportant caveat is that, as we will see in our subse-quent numerical study (section 6), ambiguity can affectboth regular and time-sensitive shipments with asym-metrical impacts on the buyer and the supplier’sexpected profit, and this impact depends on the distri-butional assumption in the ambiguity set.The above observations are based on jF ij ¼ 3, that

is, the ambiguity set contains three plausible distribu-tions. While it has a natural interpretation in practice,that is, firms often rely on estimations for three pro-cessing time scenarios (optimistic, most likely, andpessimistic), it is unclear whether the above


observations continues to hold when the ambiguityset increases. We address this question in section 6.

5.2. Lead Time Ambiguity Manifested throughSOSDBy the definition of SOSD, distributions in ambiguityset F i possess different variances but the same meanprocessing time. In practice, this implies that the firmknows the mean processing time but is unsure of thecorrect variance.As noted in section 3, it is important to realize that

increased ambiguity may not necessarily correspondto increased variance. In particular, increased ambi-guity implies that distributions with both larger andsmaller variances are more likely to be true and hencethe overall variance may not necessarily increase.With SOSD, we stipulate that the set of distribu-

tions in F i have the same mean and the distributionscross only once, that is, they possess a single-crossingproperty. Such a set of distributions can be generatedthrough a mean-preserving spread approach. For thesecond-order probabilities we adopt a similar struc-ture and notation to those described in the FOSD case(see section 5.1).As in the FOSD case, for a given port i we define

the anticipated probability of the processing timesbeing less than x under ambiguity level a as

Liðx; aÞ ¼XMi�1

k¼1

qiðk; aÞ Fi;kðxÞ þ Fi;Mi�kþ1ðxÞ� �

þ qiðMi; aÞFi;MiðxÞ: ð14Þ

By the definition of qi(�, a), we have @qiðMi; aÞ=@a [ 0 and @qi(k, a)/@a < 0 for some k\Mi. Noticethat

@Liðx; aÞ@a

¼XMi�1

k¼1

@qiðk; aÞ@a

Fi;kðxÞ þ Fi;Mi�kþ1ðxÞ� �

þ @qiðMi; aÞ@a

Fi;MiðxÞ: ð15Þ

Similar to the FOSD case, although the second termin Equation (15) is always positive, the first termmay be either positive or negative. Hence it is notimmediately obvious whether the Li(x, a) function ismonotone in a in general. It turns out that Lemma 1continues to apply in the SOSD case, that is, the signof Equation (15) depends on the relative magnitudeof Fi;kðxÞ þ Fi;Mi�kþ1ðxÞ � Fi;lðxÞ þ Fi;Mi�lþ1ðxÞ �Fi;Mi

ðxÞ, as well as the value of x involved.Leveraging Lemma 1, we can investigate how

processing time ambiguity affects the buyer’s andsupplier’s preference toward a particular port. Substi-tuting x = T � li into the Li(x, a) function, we haveLiðx; aÞ ¼ LiðT � li; aÞ ¼ Eqi;k ½Fi;kðT � liÞja�, where a

higher a corresponds to a more predictable processingtime. To illustrate how Eqi;k ½Fi;kðT � liÞja� changeswith ambiguity level a (in relation to condition (10) inLemma 1), we consider the special case of an ambigu-ity set with jF ij ¼ 3 where Fi;k 2 F i follows a uniformdistribution. Note that the exponential distributiondoes not apply in the SOSD case because it is asym-metric.Suppose Fi;k � Uðli;k; ri;kÞ, where li,1 = li,2 = li,3 =

li and ri,1 > ri,2 > ri,3 such that Fi;k’s satisfy SOSD.The second-order probabilities are qð1; aÞ ¼ qð3; aÞ¼ 1

2 ð1 � qð2; aÞÞ ¼ 12 ð1 � aÞ, where 1

3 � a � 1. Wehave

Eqi;k

�Fi;kðT � liÞja

� ¼ LiðT � li; aÞ ¼ qið1; aÞ�Fi;1ðT � liÞ

þ Fi;3ðT � liÞ�þ qið2; aÞFi;2ðT � liÞ

¼ 1

2ð1� aÞ

(min

ðT � li � li þffiffiffi3

pri;1Þþ

2ffiffiffi3

pri;1

; 1

!

þminðT � li � li þ

ffiffiffi3

pri;3Þþ

2ffiffiffi3

pri;3

; 1

!)

þ aminðT � li � li þ

ffiffiffi3

pri;2Þþ

2ffiffiffi3

pri;2

; 1

!: ð16Þ

The above expression satisfies condition (10) whenT � li � li �

ffiffiffi3

pri;1 (or when T � li [ li þ

ffiffiffi3

pri;1),

and the sign is reversed otherwise. Hence, byLemma 1, we have @Eqi;k ½Fi;kðT � liÞja�=@a � 0 whenT � li is very small or moderately large; otherwisewhen T � li is moderately small or very large thesign is reversed. The following lemma formalizesthe above observation.

LEMMA 3. Suppose Fi;k 2 F i follows the uniform distri-bution satisfying SOSD. If x � li �

ffiffiffi3

pri;1ri;2= ð2ri;1 �

ri;2Þ or x [ li þffiffiffi3

pri;1ri;2=ð2ri;1 � ri;2Þ then @Li(x, a)/

@a ≤ 0; otherwise if li �ffiffiffi3

pri;1ri;2=ð2ri;1 �ri;2Þ\ x �

li þffiffiffi3

pri;1ri;2=ð2ri;1 � ri;2Þ the sign is reversed.

Lemma 3 suggests that the effect of SOSD differsfrom that of FOSD in the sense that the directionalchange of Li(x, a) cannot be unequivocally ranked byambiguity level, as the sign of @Li(x, a)/@a changesfrom region to region, depending on how time sensi-tive a shipment is relative to the transit buffer timeT � li.While it is straightforward to enumerate the direc-

tional effects of ambiguity on the firm’s preference(analogous to Proposition 3) by examining differentlevels of shipment time sensitivity, such levels con-tinue to increase as jF ij increases. This leads to a sys-tem in which the firm’s preference towards ambiguityoscillates as the ambiguity level increases. In practice,therefore, it is neither useful nor feasible to develop


managerial insights on how SOSD ambiguity influ-ences firms’ preferences. As such, we conduct anumerical study in section 6 to investigate the effectof ambiguity under more general settings.

5.3. Aversion to AmbiguityIn the preceding sections we have assumed that deci-sion-makers are ambiguity-neutral. In practice, deci-sion-makers can be ambiguity averse (Klibanoff et al.2005, Segal 1987). Following the smooth decision-making framework developed by Klibanoff et al.(2005), we can tailor (3) to incorporate the buyer’sambiguity attitude. For notational ease, define thebuyer’s utility function as

UFi;kðQÞ ¼ ðpf � pdÞZx�Q

xdGðxÞ þQGðQÞ� �

Fi;kðT � liÞ

� hQ ðt� liÞFi;kðt� liÞ �Z~ei � t�li

~eidFi;kð~eiÞ� �

:

ð17ÞThe buyer’s objective (under ambiguity aversion)can be expressed as

pBðQÞ ¼Eqi;k HðUFi;kðQÞÞ� �� wþ ð1� aiÞsi � pdð ÞQ� ð1� aiÞ/i;

ð18Þ

where Θ(x) is an increasing concave function. Analo-gously to Proposition 1, the following propositioncharacterizes the buyer’s objective under ambiguityaversion.

PROPOSITION 5. The buyer’s expected profit function(under ambiguity aversion) is concave in the orderquantity Q. Furthermore, the optimal (interior) orderquantity Q� satisfies the following:

Eqi;k H0ðUFi;kðQÞÞU0Fi;kðQÞ

h i¼ wþ ð1� aiÞsi � pd: ð19Þ

If Θ(�) is linear then H0(�) is constant, that is, theoptimal order quantity is not influenced by thebuyer’s attitude toward ambiguity. In particular, set-ting Θ(x) = x recovers our base model in which thefirm is ambiguity-neutral.The following corollary shows that increased ambi-

guity aversion reduces the buyer’s optimal orderquantity. For clarity (and concreteness), in the follow-ing, we assume that Θ(x) = �e�gx with g ≥ 0, which isconcave-increasing and has a constant coefficient ofambiguity g = �H00(x)/H0(x). A higher g correspondsto a higher degree of ambiguity aversion.

COROLLARY 2. If g ≥ 1 then @Q�/@g ≤ 0, that is, thebuyer’s optimal order quantity decreases as ambiguityaversion increases.

If the buyer is only slightly or moderately averse toambiguity, e.g., 0 ≤ g < 1, then its optimal order quan-tity can be higher than if it is ambiguity-neutral. As thebuyer becomes increasingly ambiguity-averse, how-ever, the optimal order quantity eventually declines.The supplier’s objective function is influenced by

ambiguity aversion through the buyer’s orderingdecisions. For notational convenience, let

zi;kðxÞ ¼dHðUFi;kðxÞÞ

dx¼ ge�gUFi;k

ðxÞU0Fi;kðxÞ: ð20Þ

The following proposition partially characterizes thesupplier’s optimal pricing decision under ambiguityaversion.

PROPOSITION 6. If z00i;kðxÞ � 0 then the supplier’s objectivefunction is concave in the unit selling price w quoted to thebuyer. Furthermore, the optimal unit selling price w� satisfies

w� ¼ cþ aisi �Q�Eqi;k z0i;kðQ�Þh i

; ð21Þ

where Q� is given by Proposition 5.

Note that z0i;kðQ�Þ ¼ H00ðUi;kðQ�ÞÞðU0i;kðQ�ÞÞ2 þ

H0ðUi;kðQ�ÞÞU00i;kðQ�Þ � 0. The condition in Proposi-

tion 6 is not straightforward to verify, but a sufficientcondition is

g2 U0i;kðxÞ

� �3�3gU0

i;kðxÞU00i;kðxÞ þU000

i;kðxÞ� 0; ð22Þ

in which the first two terms are both positive. Rec-ognizing that U000

i;kðxÞ ¼ �ðpf � pdÞg0ðxÞFi;kðT � liÞ, a(strong) sufficient but not necessary condition isg0(x) ≤ 0. The supplier’s optimal pricing decisioncan, however, still be well-behaved even if the con-dition is not satisfied.While it is straightforward to establish that, under

certain conditions, the optimal order quantity Q�

decreases in w, it is unclear whether the optimal w�

increases or decreases as the buyer’s ambiguity aver-sion increases. As a result, it is not immediately clearhow ambiguity aversion affects the buyer’s and sup-plier’s expected profits. We explore this questionthrough a numerical study in section 6.

6. Numerical Study

The purpose of the numerical study is threefold. First,we relax the assumption that the ambiguity set con-tains three distributions (jF ij ¼ 3) made in the ana-lytical work. We also incorporate positive inventoryholding costs with the possibility that the shipmentwill arrive on time. These relaxations allow us to test


the robustness of our analysis and examine the effectof a larger ambiguity set on firms’ preferences. Sec-ond, we examine the magnitude of the ambiguityeffect to gain a more realistic sense of the importanceof ambiguity to profitability. Third, we examine theeffect of ambiguity relative to other important param-eters, especially the selling time window T � t andthe transit buffer time T � li, which will allow us tofurther illustrate the effect of ambiguity in a morepractical setting.In our numerical study, we operationalize the sec-

ond-order probabilities (values of qi;k) through theBeta distribution with parameters a and b. We seta = b such that the weights follow a symmetric, uni-modal distribution. The reason we choose the Betadistribution is that this distribution is boundedbetween 0 and 1, which provides a natural domain forassigning second-order probabilities to Fi;k. Through-out the numerical study, we set jF ij ¼ 5, that is, theambiguity set contains five distributions. We imple-mented qi;k through the Beta distribution by varying aand b in lockstep from 1 to 3, 7, 15, and 31 (see Fig-ure 4). Hence we have five ambiguity levels: a = b = 1is associated with the most ambiguous processingtime because the values of qi;k are uniform, whereasa = b = 31 corresponds to the least ambiguous pro-cessing time because the values of qi;k are most con-centrated (high kurtosis).For each value of a (and b), we use the equidistant

set of discrete points x 2 (0.1, 0.3, 0.5, 0.7, 0.9) withinthe domain of the Beta distribution to obtain rawweights for qi;k ¼ BðxkÞ for k 2 (1, 2, . . ., 5). The val-ues of qi;k are then normalized to Rkqi;k ¼ 1.For market demand, we use the normal distribu-

tion and fix the parameters at ld = 100 and rd = 30.For the transit lead time parameters, we set T = 20and vary t from 13 to 17 with a step size of one suchthat the selling window varies from seven to three

periods. We also vary the total transit lead time (inthe absence of the port processing time) li from eightto 12 with a step size of one such that the transit buf-fer time (to allow port processing) varies from 12 toeight.For the financial parameters, we fix the selling price

pf = 10 and the discounted price pd = 3.0. We vary theunit production cost c from 1.0 to 3.0 with a step sizeof 0.5. We set the unit inventory holding costh = 5% 9 c and the unit freight cost si = 7% 9 c. Notethat the average transportation cost as a percentage ofsales is 7.87% (Ellis 2014).9 We set the fixed freightcost /i = 0, because its directional impact is straight-forward and it does not influence the firm’s attitudetoward processing time ambiguity. The parameter aifor the supplier’s share of shipping costs is set at 0.2and 0.8.In the FOSD case, we distinguish between uniform

and exponential distributions. For the uniform distri-bution, we allow the EFi;k ½~ei� to vary from one to fiveand then from eight to 12 for relatively short and rela-tively long mean processing times respectively. Thestep size is one for both cases. The coefficient of varia-tion for Fi;k is kept constant at 0.5 in both cases. Forthe exponential distribution, the mean processingtime is set similarly and the standard deviation is pro-portional to the mean by definition.In the SOSD case, we use uniform and normal dis-

tributions (because exponential distribution is asym-metric). For the uniform distribution, we set EFi;k ½~ei� atthree and 10 for relatively short and relatively longmean processing times respectively. The coefficient ofvariation for Fi;k is varied from 0.1 to 0.9 with a stepsize of 0.2 for both uniform and normal distributions.In total, we have 12,500 combinations in each case:

FOSD uniform, FOSD exponential, SOSD uniform,and SOSD normal.

6.1. FOSD AmbiguityTo examine the effect of ambiguity on the supplier’sand buyer’s profits (and hence preferences), we com-pute the percentage changes in their expected profit,with the expected profit at the lowest ambiguity levelas the base case. A negative/positive percentagechange therefore implies that the port becomes less/more attractive as ambiguity increases. Figures 5 illus-trates the percentage changes in the supplier’s andbuyer’s profits in the uniform case.10

Figure 5 corroborates our earlier analytical result(see Proposition 3) that both the supplier and thebuyer prefer less ambiguity with regular shipments(which have relatively long buffer time to selling sea-son) but both (in expectation) benefit from more ambi-guity with more time-sensitive shipments (withrelatively short buffer times). The impact of ambigu-ity is more significant for regular shipments than that0.2 0.4 0.6 0.8 1.0

1

2

3

4

5

6

Figure 4 Implementing qi;k through Beta Distribution (a and b Vary inLockstep from 1 to 3, 7, 15, and 31) [Color figure can beviewed at wileyonlinelibrary.com]


for time-sensitive shipments: Compared with theleast-ambiguity case, increased ambiguity reducesexpected profit by as much as 7% for regular ship-ments but improves them by only up to 1.5% for time-sensitive shipments. In the exponential distributioncase, we observe qualitatively similar results, that is,increased ambiguity reduces expected profit (by up to4.4%) for regular shipments but slightly improvesthem (by up to 0.3%) for time-sensitive shipments.While the above observations are related to Figure 3,

they also hold more generally: out of 12,500 combina-tions, increased ambiguity reduces expected profit byup to 7.1%/4.4% (uniform/exponential) for regularshipments but improves expected profit by only up to4.4%/1.8% (uniform/exponential) for time-sensitiveshipments. However, for time-sensitive shipments,increased ambiguity improves expected profit by anaverage of only 0.59% under the exponential case,much less than the 2.14% increase in the uniform case.This suggests that if the ambiguity set consists ofexponential distributions, both the supplier and thebuyer are more likely to benefit from reduced ambi-guity for regular shipments than from increasedambiguity for time-sensitive shipments.As the ambiguity set F i is enlarged, we observe that

the impact of increased ambiguity is dampenedregardless of whether the shipment is time-sensitive.For example, in the uniform case, while increasedambiguity reduces/improves expected profit by up to11.3%/5.9% for regular/time-sensitive shipmentswhen jF ij ¼ 3, these are dampened to 5.3%/3.4%when jF ij ¼ 9. Similarly, in the exponential case,increased ambiguity reduces/increases expectedprofit by up to 6.0%/2.3% for regular/time-sensitiveshipments when jF ij ¼ 3, and these are dampened to3.5%/1.4% when jF ij ¼ 9. Observe that the

magnitude of the ambiguity impact between uniformand exponential cases is also dampened as F i isenlarged. Because the distributional shape effect isweakened as jF ij increases, the effect of the shipmentbuffer time manifests itself more clearly and the firms’preferences under greater ambiguity is therefore pri-marily driven by whether a shipment is time-sensi-tive: for regular shipments, they tend to select a portwith less ambiguity (all else being equal), whereasambiguous processing times are more palatable fortime-sensitive shipments.Nevertheless, reasonably accurate estimations for

the three scenarios (e.g., optimistic, most likely, andpessimistic) enable firms to gain and utilize informa-tion on the shape of the distributions in F i to makeinformed decisions. In particular, increased ambigu-ity is likely to have a larger impact on expected profitwhen the ambiguity set consists of uniform instead ofexponential distributions, and reducing ambiguity ismore likely to be a robust strategy with an exponen-tial ambiguity set.

6.2. SOSD AmbiguityWe now turn our attention to the ambiguities mani-fested through SOSD. Similarly to the FOSD case, wecompute the percentage changes in the buyer’s andsupplier’s expected profits to deduce their prefer-ences. We observe that ambiguity has only a negligi-ble influence on firms’ expected profits. In theuniform case, for example, increasing processing timeambiguity from the lowest to the highest level reducesthe supplier’s/buyer’s expected profit by an averageof 0.07%/0.07% for regular shipments and 0.52%/0.58% for time-sensitive shipments. Similarly, in thenormal (distribution) case, the corresponding num-bers are 0.01%/0.01% for regular shipments and

0.0% -0.1% -0.4%

-2.1%

-7.0%

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-2.1%

-7.1%

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-1.0%

0.0%

)1,1()3,3()7,7()51,51()13,13(

Ambiguity parameters (α,β) (-> increasing ambiguity)

Percentage change in expected profit as port processing time becomes more ambiguous (Anticipated shipment buffer time 3-7 days)

SupplierBuyer

(a)

0.0%0.1%

0.3%

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1.5%

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0.3%

0.6%

1.3%

)1,1()3,3()7,7()51,51()13,13(


Percentage change in expected profit as port processing time becomes more ambiguous (Anticipated shipment buffer time 0-4 days)

SupplierBuyer

(b)

Figure 5 Illustration of the Effect of Port Processing Time Ambiguity (FOSD uniform) [Color figure can be viewed at wileyonlinelibrary.com]


1.10%/1.09% for time-sensitive shipments.11 This sug-gests that ambiguities characterized by the SOSDhave less of an impact on firms’ preferences. For regu-lar shipments with sufficient buffer times, theexpected profits are virtually the same at variousambiguity levels, and firms often forgo transactions ifshipment buffer times are excessively tight.Part of the intuition is related to the fact that

under an SOSD structure, elevated ambiguity levelsdo not necessarily correspond to a larger variance inthe anticipated processing times. Coupled with thefact that anticipated processing times are kept con-stant, the result is that firms’ expected profits arerelatively insensitive (although decreasing slightly)to ambiguities in processing times. As such, theanticipated mean processing time becomes muchmore important (than in FOSD), because longermean processing times can cause a transaction tobecome unprofitable to both the buyer and the sup-plier. An important managerial insight is that it ismore important to reduce anticipated mean process-ing times if ambiguities arise through the SOSDstructure.With either FOSD- or SOSD-manifested ambiguity,

the fraction of the supplier’s transportation costs aihas little impact on the effect of ambiguity. Weobserve no moderating effect of ai on ambiguitylevels. Part of the intuition is that the supplier fullyprices transportation costs into the unit wholesaleprice w, such that whether the supplier bears a largerportion of transportation costs does not affect its andthe buyer’s expected profits. Thus, whether theyadopt FOB or CIF Incoterms does not influence aport’s attractiveness. This implies that the buyer neednot insist on a particular port since its preference isaligned with that of the supplier.

On the other hand, being aware of the nature ofambiguity is important to recognizing its effect on aport’s attractiveness. When processing times areaffected by discrete, traceable events (FOSD), reduc-ing ambiguity enhances a port’s attractiveness for reg-ular shipments but not necessarily for time-sensitiveshipments, all else being equal. In contrast, when pro-cessing times are affected by frequent, random noises(SOSD), reducing the mean processing time is para-mount to enhancing a port’s attractiveness. Likewise,a port’s attractiveness is always enhanced with ashorter mean processing time, even under FOSD.

6.3. Freight CostThe freight cost for a port can be influenced by manyfactors, such as overall distance, commodity size andweight, port efficiency, and fleet network. Whenselecting a port, the buyer must often therefore strikea balance between freight costs and anticipated pro-cessing times. Below we explore how large of anincrease or decrease in the unit freight cost is requiredto compensate for increased processing time ambigu-ity. Specifically, we first calculate the buyer’s expectedprofit at the unit freight cost of 7% 9 c with the leastprocessing time ambiguity as a base case. We thenincrease processing time ambiguity and compute thecorresponding unit freight costs that will maintain thebuyer’s expected profit. We assume that the fixed costis absorbed in the unit freight cost, that is, /i = 0.12

Figure 6 illustrates the ensuing iso-freight costcurve.13

For regular shipments, a port with higher process-ing time ambiguity must compensate with a lowerunit freight cost rate, and the reduction of this ratecan be as much as �9% (6.38%/7% � 1) all else beingequal. This is intuitive as a port with a less predictable

7.00% 6.99% 6.96%

6.82%

6.38%

6.30%

6.40%

6.50%

6.60%

6.70%

6.80%

6.90%

7.00%

)1,1()3,3()7,7()51,51()13,13(


ISO-Freight cost as port processing time becomes more ambiguous

(Anticipated shipment buffer time 3-7 days)

(a)

7.00%7.04%

7.11%

7.26%

7.60%

7.00%

7.10%

7.20%

7.30%

7.40%

7.50%

7.60%

7.70%

)1,1()3,3()7,7()51,51()13,13(


ISO-Freight cost as port processing time becomes more ambiguous


(b)

Figure 6 Illustration of ISO-Freight Cost Based on Buyer’s Expected Profit (FOSD uniform) [Color figure can be viewed at wileyonlinelibrary.com]


processing time is not as attractive as one with a moreconsistent processing time, but reducing the unitfreight rate can compensate for such processing timeambiguity. It is worth pointing out that freight costsoften include both transport costs and handling costsat the port, many aspects of which are thereforebeyond the control of the port. Achieving a 9% reduc-tion requires either cost reductions by all parties or ahigher (more than 9%) cost reduction by the portitself. On the other hand, improving processing timepredictability enables a port to charge a higher freightrate and still maintain a competitive advantage.For time-sensitive shipments, a port with a higher

degree of processing time ambiguity can potentiallycharge a higher unit freight cost rate. This is surpris-ing, but consistent with our earlier results thatincreased ambiguity can potentially benefit the buyer:a higher level of ambiguity implies that the port canprocess the shipment more quickly than the antici-pated mean processing time with a non-negligibleprobability. This possibility is beneficial for time-sen-sitive shipments, especially when the unit productioncost is low and the profit margin is high. In practice,however, this observation may not hold as a normbecause the buyer can be ambiguity-averse.

6.4. Aversion to AmbiguityIf the buyer is ambiguity-averse, it is likely to increas-ingly prefer ports with less ambiguity—regardless ofwhether the shipment is time-sensitive. On the otherhand, it is not immediately clear how ambiguityaffects the supplier’s preference. We slightly adaptthe ambiguity aversion function developed in section5.3 by using Θ(x) = 1 � e�gx to calculate the buyer’sand supplier’s expected profits. We vary the coeffi-cient of ambiguity g = 0.005, 0.01, 0.05. Recall that a

larger g corresponds to higher ambiguity aversion.Figures 7 and 8 illustrate the effect of ambiguity aver-sion for regular and time-sensitive shipments respec-tively. All other parameters used are identical tothose in Figure 5. Note that with ambiguity aversionthe expected profit is transformed by Θ(x) = 1 � e�gx,and hence the percentage changes in expectedprofit cannot be directly compared with those inFigure 5.For regular shipments Figure 7a suggests that, con-

trary to our earlier intuition, an ambiguity-aversebuyer may not be worse off with increased processingtime ambiguity. In contrast, the supplier is alwaysworse off. This is partly because ambiguity aversionleads the buyer to reduce its order quantity (Corollary2), which in turn forces the supplier to quote lowerprices. The burden of ambiguous processing times istherefore partially transferred from the buyer to thesupplier (see Figure 7a and b at g = 0.05). The systemas a whole is worse off: the buyer places a reducedorder and the supplier earns less profit.For time-sensitive shipments, increased ambiguity

aversion curbs the buyer’s preference toward portswith more ambiguous processing times. Figure 8asuggests that as the buyer becomes more ambiguity-averse (g increases) the attractiveness of ports withambiguous processing times diminishes. Similar tothe regular shipment case, Figure 8b suggests that thesupplier is consistently worse off as the buyerbecomes increasingly ambiguity-averse.While the general observations in Figures 7 and 8

agree with our intuition that ambiguity aversiondampens the attractiveness of ports with ambiguousprocessing times for both regular and time-sensitiveshipments, the impacts on the buyer and supplier aremore nuanced. The buyer suffers less (or may even

0.0% -0.1% -0.3%

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)1,1()3,3()7,7()51,51()13,13(


Percentage change in buyer's expected profit as port processing time becomes more ambiguous (Anticipated shipment buffer time 3-7 days)

η=0.005η=0.01η=0.05

(a)

0.0% -0.1% -0.5%-2.5%

-8.1%

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-11.9%

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)1,1()3,3()7,7()51,51()13,13(


Percentage change in supplier's expected profit as port processing time becomes more ambiguous


η=0.005η=0.01η=0.05

(b)

Figure 7 Illustration of the Effect of Ambiguity in Port Processing Time When the Buyer is Ambiguity-Averse (FOSD uniform, regular shipments)[Color figure can be viewed at wileyonlinelibrary.com]


slightly benefit) from ambiguous processing times,which can transfer the burden of ambiguity to thesupplier, whereas the supplier is consistently worseoff. Because the system as a whole is worse off, how-ever, the supplier is significantly better off if the ambi-guity-averse buyer can select ports with lessambiguous processing times.

6.5. Fixed Wholesale PriceWhen the buyer is influential, the unit wholesale priceis often fixed on a cost-plus basis for the supplier. It istherefore important to determine whether our earlierresults continue to hold when the supplier quotes afixed, cost-plus wholesale price. This is admittedlysub-optimal for the supplier but it eliminates doublemarginalization, which is inherent in most two-playersupply chain models.We find that our previous observations on the effect

of ambiguity continue to hold with a fixed wholesaleprice. There are no qualitative differences in firms’preferences toward ambiguity with the exception oftime-sensitive shipments under ambiguity manifestedthrough an SOSD normal structure. Specifically, weobserve that at a fixed price the buyer’s expectedprofit increases with processing time ambiguity fortime-sensitive shipments. The supplier’s preference isnot influenced by ambiguity levels because thebuyer’s optimal order quantity is not influencedby ambiguity. The fixed wholesale price schemeeliminates the inherent inefficiency of doublemarginalization and hence makes it profitable for thebuyer to ship the product even if the shipment buffertime is very tight. We observe, however, that this isonly possible when the unit production cost is rela-tively low (i.e., c = 1.0); at higher costs the systembehaves as in the previous analysis with both firms

forgoing the transaction if the shipping buffer time isdeemed too tight.Thus under specific conditions, especially when the

unit production cost is relatively low and the ship-ment is time-sensitive, the buyer’s and supplier’spreferences are no longer strictly aligned: the supplieris indifferent between choosing a port with highambiguity or with low ambiguity whereas the buyerprefers a port with high ambiguity. In such cases, thebuyer is better off specifying the port itself in the con-tract, as opposed to delegating the decision to thesupplier.Another difference we observe at a fixed wholesale

price is that the supplier is consistently better off withhigher values of ai. This suggests that the supplierstrictly prefers a CIF contract to an FOB contract at afixed wholesale price. This is not the case when thesupplier optimizes the wholesale price and transfersthe transportation costs to the buyer. At a fixedwholesale price, the supplier earns an additional mar-gin on the transportation costs and hence prefers CIFcontracts, which incorporates these transportationcosts. This observation is consistent with real practicewhere suppliers usually charge a premium forarranging transportation all the way to the buyer’sdestination.As a result, if the total (origin to destination) trans-

portation cost is similar but the supplier is closer toport A than to port B, then all else being equal thesupplier is equally likely to choose either port at anoptimal wholesale price (either with FOB or CIF) butwill prefer port B at a fixed wholesale price (withCIF). Thus at a fixed wholesale price, the impact ofambiguity is moderated by a port’s proximity to thesupplier’s facility under a CIF contract. In particular,a fixed wholesale price reduces the alignment

0.0%0.2%

0.7%

1.5%

3.4%

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0.9%

1.9%

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0.5%

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4.5%

)1,1()3,3()7,7()51,51()13,13(


Percentage change in buyer's expected profit as port processing time becomes more ambiguous (Anticipated shipment buffer time 0-4 days)

η=0.005η=0.01η=0.05

(a)

0.0% -0.1% -0.3% -0.6%-1.3%

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-7.7%

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)1,1()3,3()7,7()51,51()13,13(


Percentage change in supplier's expected profit as port processing time becomes more ambiguous


η=0.005η=0.01η=0.05

(b)

Figure 8 Illustration of the Effect of Ambiguity in Port Processing Time When the Buyer is Ambiguity-Averse (FOSD uniform, time-sensitiveshipments) [Color figure can be viewed at wileyonlinelibrary.com]


between the preferences of the supplier and buyer.For example, for regular shipments both the supplierand the buyer prefer a port with less ambiguity (portC), but, because another port (port D) is farther fromthe supplier’s facility, the supplier may opt for port Ddespite port C’s less ambiguous processing times.

7. Conclusion

Processing time predictability is an important factorinfluencing a port’s competitiveness in global con-tainer transportation. Given that geographic loca-tions are fixed, it is both theoretically interestingand practically useful to understand whether andhow improved processing time predictability canenhance a port’s attractiveness to shippers. We con-sider a situation where a firm does not know thetrue distribution of a port’s processing time butrelies on second-order probabilities to obtain aplausible set of processing time distributions tooptimize its shipments.While it is always desirable for a port to reduce

average processing times, all else being equal, theassociated costs can be prohibitive. On the otherhand, improving predictability is often more cost-effective, as is the main thrust of this study. As an ini-tial step to explore ambiguity associated with portprocessing times, we find that the effect of ambiguityis not straightforward.We prove that an ambiguity-neutral firm should

select a port with less ambiguity for regular ship-ments but can be better off opting for a port withmore ambiguity for time-sensitive shipments. Such apreference is robust to differences in the firm’s beliefstructure (ambiguity set) about port processing times,which holds regardless of whether the firm forms anambiguity set through uniform or exponential distri-butions. The uniform distribution reflects situationswhere the firm can reasonably estimate the upper andlower bounds of the port processing times but is mini-mally informed about their distributional shapes,whereas an exponential distribution reflects situationswhere the firm knows the mean processing times butis unsure about their upper bounds. Despite these sig-nificant differences, the firm’s preference for less/more ambiguity for regular/time-sensitive shipmentscontinues to hold.An important caveat, however, is that the distribu-

tional shape in the ambiguity set does influence themagnitude of the expected benefit. In particular, whenport processing times are exponentially distributed(i.e., with long tails), a firm derives a greater benefit ifit selects a port with less ambiguity for regular ship-ments than if it selects a port with more ambiguity fortime-sensitive shipments. In contrast, with symmetric,bounded processing times (i.e., with a uniform

distribution), the firm can benefit equally from select-ing a port with more ambiguity for time-sensitiveshipments and from selecting a port with less ambi-guity for regular shipments. The firm thus benefitsmore significantly from selecting a port with lessambiguity for regular shipments when port process-ing times exhibit long tails.When the firm becomes less certain about the mean

or lower/upper bounds of processing times (i.e., anenlarged ambiguity set), the impact of ambiguity onthe expected profit dampens, as does the effect of theshape of the underlying distributions. The firm’s pref-erence is more saliently influenced by the time sensi-tivity of shipment: the firm prefers a port with lessambiguity for regular shipments but is better off withan ambiguous port for time-sensitive shipments. Aport with erratic operations may enable a shipment topass through more quickly than a port with more con-sistent operations given the same expected processingtime.If a firm is ambiguity-averse, however, a port with

less ambiguity is increasingly preferred, especially bythe supplier. Coupled with the fact that exponentialdistributions are often considered more realistic forestimating port processing times, it is safe to concludethat a port can indeed improve its competitiveness byimproving its operational predictability.Our aim is not to prescribe operational decision

support but rather to gain an improved understand-ing of how ambiguity can affect a port’s attractivenessin the global container industry. We hope that thisstudy will pave the way for future work to furtherinvestigate relevant factors that can improve a port’scompetitiveness.

Acknowledgments

The authors thank the department editor, the senior edi-tor, and an anonymous referee for their insightful com-ments and suggestions. This research is supported inpart by a grant from the Research Grants Council of theHKSAR, China (T32-620/11). The third author alsoacknowledges support of 2015 Dean’s Excellence Awardfrom W. P. Carey School of Business at Arizona StateUniversity, as well as sabbatical support by the depart-ment of IELM at Hong Kong University of Science andTechnology.

Notes

1Another factor that can influence port processing time isdemurrage and detention stipulations, which can alsoinfluence carrier/forwarder preferences.2The variability of average time at sea is primarily influ-enced by shipping routes.3We have also conducted informal interviews withseveral senior shipping executives, and learned that itusually takes a container about one day longer


(expected time) to pass through terminals in MainlandChina than through those in Hong Kong. The executivessuggested, however, that the predictability of processingtime and cost is their primary concern rather than expectedtime and is the main reason for preferring HongKong ports.4Looks-like analysis is also referred to as analogous fore-casting and a forecasting method for new products thatexhibit similar traits to old products (Mas-Machuca et al.2014).5Caldeirinha et al. (2011), for example, note that port loca-tion itself is not a critical factor determining port perfor-mance and efficiency (p. 64).6It is also methodologically more appealing when the sec-ond-order probability can be updated over time, such asthe updating process proposed by Saghafian and Tomlin(2016). The definition of ambiguity, however, does notintrinsically rest on the time dimension, that is, the sec-ond-order probability need not be updated over time (seethe verifiability issue discussed in Klibanoff et al. (2005, p.1856). We ignore the mechanics of the updating processfor the second-order probability to focus on the effect ofambiguity on port attractiveness. Nevertheless, our modelstill applies if there is an updating process—as long as theshipment decision needs to be made before the ambiguityis fully resolved.7The generalized failure rate is defined as xgðxÞ=�GðxÞ. AnIFR distribution always satisfies IGFR.8Note that if the buyer is risk averse (but not necessarilyambiguity averse) then all else being equal the port with amore ambiguous processing time is less attractive.9Here we use the unit production cost c as a baseline (asopposed to the unit selling price pf or pd) because freightcosts are more commonly considered as a percentage ofthe cost of goods sold. In reality, freight costs vary signifi-cantly between carriers as fleet networks and hence transittimes differ significantly. Lu et al. (2017) provides a moredetailed discussion on carrier selection based on differenttransit times and freight costs.10Figure 5a was obtained by setting t = 17, l = 12, c = 3,and ai = 0.2. The anticipated shipment buffer time isT � li � E½~ei� ¼ 20 � 12 � f5� 1g ¼ 3 � 7. Figure 5bwas obtained by setting t = 17, l = 8, c = 1, and ai = 0.2.The anticipated shipment buffer time is T � li � E½~ei� ¼20 � 8 � f12 � 8g ¼ 0 � 4.11With time-sensitive shipments, however, firms choose toforgo transactions in 48.8% of the 12,500 observations.Moreover, unlike the FOSD, increased ambiguity typicallyreduces both the buyer’s and supplier’s profits regardlessof whether the shipment is time-sensitive.12Most commercial quotes are based on costs per 20- or40-ft container (often with volume discounts), which sug-gests that fixed costs are somewhat absorbed into thequoted per-container cost.13Figure 6 is obtained using the same set of parameter val-ues as those in Figure 5.

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