Air Freight Service Network Design for Stochastic Demand

Ta-Hui Yang and Chen-Cheng Chen

Abstract-This paper introduced a two-stage stochasticmodel to address the strategic air freight network designproblems under stochastic demand. Most of existing approachesfor the airline network design problems are restricted todeterministic environment. However, the demand in airlinemarket usually varies seasonally. The first stage of the proposedmodel is to determine the number and the location of hubs. Thesecond stage consists of deciding the flight routes to transportflows from origins to destinations in an optimal fashion basedupon the hub location and realized uncertain scenario. Finally,the data from air passenger market in Taiwan and China is usedto test the proposed model.

I. INTRODUCTION

A~rline networ~ d.esign is a strategic level of planning andInvolves deciding the configuration of the service

network. The strategic configuration ofservice network has along lasting impact on the airline company. However, the airfreight market is fraught with uncertainty. Seasonal demandvariation is one of the major sources of uncertainty. Most ofexisting approaches for the airline service network designproblems are restricted to deterministic environment. That is,the parameters and variables of the models are alldeterministic. Most commonly used method is to apply theaverage values to the parameters or variables, which can onlyrepresent a single situation. However, the demand in airfreight market usually varies seasonally. The result based onthe deterministic model cannot response to the demandfluctuations. This study explicitly considers the uncertainty ofdemand and formulates the airline network design problem asa two-stage stochastic program. The proposed two-stageframework matches the decision process in real-worldpractice. The choice ofhub location is a long-term investmentand would not change according to seasonal demandvariation. Therefore, the hub location would not beinfluenced by the randomness of demand and belongs to thefirst stage decision of the proposed model. However, thechoice of delivery paths and allocation of flows does vary inresponse to the change of demand. Hence, they are affectedby stochastic demand and belong to the second stage of theproposed model.

Manuscript received March 1,2009.Ta-!Iui Y~ng is ~ith the Department of Logistics Management, National

Kaohs~ung FIrst University of Science and Technology, 2 Jhuoyue Rd.,Kaohsiung 81164, Taiwan (phone: +886-7-601-1000 Ext. 3223, fax:+886-7-601-1040, e-mail: yang@ccms.nkfust.edu.tw).

~hen-Cheng.Chen is is with the Department of Logistics Management,National K~ohslung First University of Science and Technology, 2 JhuoyueRd., Kaohsiung 81164, Taiwan (e-mail: jjen@ccms.nkfust.edu.tw).

978-1-4244-3541-8/09/$25.00 ©2009 IEEE

Since the network design is a key issue in airline operations,many studies have addressed this problem. There are twomost commonly used network systems in airline operations:point-to-point and hub-and-spoke. The point-to-pointnetwork is relatively simple to the hub-and-spoke network.Therefore, most of studies focused the design ofhub-and-spoke network [4]. first formulated the discrete hublocation problem as a quadratic integer program. Severalresearches were studies based on this quadratic integerformulation [5]-[7]. Campbell proposed a new method todefine the variables for discrete hub location problems [8], [9].The method reduced the hub location problems to linearinteger problems, but the benefits come at the cost ofadditional variables and constraints. After that, many modelsor solution techniques were proposed based on Campbell'smethod. The model was discussed and enhanced in differentdirections, including incorporating with concave costfunctions [10], [11], considering different hubbing policy[12], [13], and modifying Campbell's formulation [14], [15].

The significance of uncertainty has prompted a number ofresearchers to address stochastic parameters in networkdesign problems. Mirchandani and Odoni considereduncertainty on link travel times and addressed stochasticnetwork median problems [16]. Louveaux studied stochasticfacility location problems for different applications in bothprivate and public sector [17]. The uncertainty comes fromdemands, production costs, transportation costs, and prices.Louveaux and Thisse used a two-stage stochastic model toformulate a location-production problem with uncertaindemands [18]. Barbarosoglu and Arda applied two-stagestochastic programming model in transportation networkplanning for disaster response [19]. Santoso et al. addressedstochastic strategic supply chain planning problems wherethe uncertainty comes from processing costs, transportationcosts, demands, supplies, and capacities [20]. Listes andDekker employed stochastic programming approach to aproduct recovery network design problem where theuncertainty happens on demands and/or supplies [21].

The idea of network design under uncertainty has beenapplied in different areas. However, in our review of relatedliterature we did not find any that addressed the particularissues of airline network design with seasonal demandvariation. This study employed a two-stage stochasticprogramming model to formulate the airline network designproblem with demand uncertainty. The reminder ofthis paperis organized as follows: In the next section the notation andvariables are first defined and a stochastic airline networkdesign model is then introduced. In Section 3, a case study

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based on real air passenger market in Taiwan and China isused to illustrate the proposed model. Finally, someconcluding remarks are discussed in Section 4.

II. MODEL DEVELOPMENT

2.1 Problem Description

In a real-world air freight market, the demand usually hasobvious seasonal variation. Airline network design is astrategic level of planning which is a long-term planning andis impossible to have precise information on seasonal demandvariation. To deal with the stochastic nature of demand, thispaper assumed that the demand has a discrete distributionwith finite number of possible realizations, or calledscenarios. The purpose of this study is to determine hublocations and design a service network to transport thedemands to the destinations at the lowest operation cost whiletaking seasonal demand variation into consideration.

2.2 Notation and Variable Definition

The notation and variables used in the model are defined asfollows:

transportation costs.

f3(OJ) : the discount factor for the transportation costsbetween a non-hub origin and a hub, or betweena hub and a non-hub destination for scenario OJ •

f3(OJ) is used to approximate the economies of

scale for the transportation costs between huband non-hub. In generally, 0 S a(OJ) S I3(OJ) S 1.

cij : the unit transportation cost for the nonstopservice between i and j

cilaj (OJ) : the unit transportation cost for hub-connectedservice from i to j and transshipped at hubs k

and t . In order to represent the effect ofeconomies of scale on hub-connected flight, Ciktj

is calculated by13(OJ) . Cik +a (OJ) .Ckt +13(OJ) . Ctj •

fk : the fixed cost to set up hub k.

V : a very large positive number.

~(OJ) : a random data vector consisting ofthe uncertainparameters and quantities.

(2)

(1)

The stochastic airline network design problem can bewritten as follows:

Problem (P):Stage-One (P.1)

Min LfkXk + E[Q(x,~(OJ))]x keN

The variables X k > Yij (OJ), and Yiktj (OJ) are decision

variables. In practice, setting hub is a long-term planning andinvestment which cannot be easily changed in short time. Thehub location would not alter due to demand variation. The

hub-location variable x k is, therefore, not affected by

randomness of demand. However, the path used to transportthe demand may change due to different demand levels.

Hence, variables Yij (OJ) and Yiktj (OJ) would be different

due to different realized demands. This study uses a two-stagestochastic program to formulate the airline network designproblem under seasonal demand variation. The first stageconsists of the determining the number and the location ofhubs which is a strategic level of decision and usually doesnot change in short term. The second stage decision is toconstruct a service network, which includes determination ofdelivery routes and allocation of flows, in an optimal fashionbased upon the hub location and realized uncertain scenario.

Weare now ready to state a two-stage stochastic model forthe stochastic airline network design problem.

2.3 Two-Stage Stochastic Model

Subject to

Xi E {O,l}

the set of nodes. In this study, a node isrepresented as an airport.a realized uncertain scenario

the set of all possible scenarios and, OJ E Q.

a realized demand level from node i to node j

for scenario OJ • In this study, the demand levelis treated as discrete random variable.the probability for scenario OJ .

the decision variable for hub location. If xk =1,

node k is set to be a hub; otherwise, xk =0 .

if Y ij (OJ) =1 , the demand is transported

through the non-stop path i - j for scenario

OJ ; otherwise Y ij (OJ) =o. By definition, both

nodes i and j are not hubs. The path that

transports demand may change due to differentdemand levels. Therefore, the decisionvariables Yij (OJ) has stochastic feature.

if Yilaj (OJ) =1, the demand is transported from

i to j and transshipped at hubs k and t for

scenario OJ ; otherwise Yilaj(OJ) =0 . The path

that transports demand may change due todifferent demand levels. Therefore, thedecision variables Yilaj (OJ) has stochastic

feature.the discount factor for inter-hub transportationcosts for scenario OJ . a(OJ) is used to

approximate the economies of scale for inter-hub

p(OJ) :

Q:

D(OJ) :

a(OJ) :

N:

OJ:

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Subject to

establish a service network through the determination offlight paths and allocation of flows for each path (Le. Yij(OJ)

and Yilaj (OJ)) after the values of the random parameters

become known and first stage decision, xk ' has been taken.

Clearly once the value of xk is chosen in the first stage, the

second stage decisions Yij(OJ) and Yiktj (OJ) are likely to

change under different realizations of OJ. In other words,when the scenario OJ is realized, the second-stage unknownquantities Dij (OJ), p(OJ), and Ciktj (OJ) become known. Then

the second-stage decision Yij(OJ) and Yiktj (OJ) can be taken.

Constraint (2) requires the variable x, to be either 0 or 1.

Constraint (4) ensures that all demand on each origin has totransport to its destination. Constraint (5) enforces that thenonstop flight should represent as Xkkki ' Xikkk ' Xiikk ' or Xkkii '

not Xik or xki ,ifnode k is a hub. Constraint (6) requires that

only nonstop and one-hub-stop services are allowed if originor destination is a hub. Constraint (7) requires that onlynonstop service is allowed when both origin and destinationare hubs. Constraint (8) requires that no hub-connectedservices are allowed transshipping at node k ifnode k is nota hub. Constraint (9) states that at least one ofhub-stop flightsmust transship at node k , ifnode k is a hub. Constraints (10)and (11) enforce the flow fractions being within the rangebetween 0 and 1.

\:;fi,k,OJ: i *-k (6)

Miny

LYik(m) + LYki(m)::; V(1-Xk)ieN ieN

LYkkti(aJ) + LYitkk(aJ) 2:: 2xkteN teN

(3)LLLp( OJ)Dij (OJ)eijY ij (OJ) + LLLLLp( OJ)Dij (OJ )Ciklj (OJ )Yiklj (OJ)~~N~ ~~~~~

Yij(m) + LLYilqj(m)=1keN teN

where Q(x,~) is the optimal value ofthe following second

stage problem:

Stage-Two (P.2)

III. AN ILLUSTRATIVE EXAMPLE

The random data vector ~(OJ) consists of the coefficients

of every term in formulas (1) to (11). E(·) denotes

mathematical expectation operation. (P.l) and (P.2) combineas a two-stage stochastic program. The objective function (1)

contains a deterministic term L fk Xk ' which is the hubkeN

setup costs, and the expectation of the second-stage objectiveQ(x,~) taken over all realizations ofrandom event OJ , which

is the expected value of the transportation costs (objectivefunction 3). The first stage problem corresponds to theinvestments that must be made for opening hubs (xk ) prior to

knowing the actual realizations of the random parameters~(OJ) . The second stage decision (P.2) corresponds to 10The number of nodes (airports)

TABLE I DATA SETTINGS FOR TEST NETWORK

3.1 Data Collection

The data from air freight market in Taiwan and China wereused to test the proposed model. The top 10 cities ranked bythe economic exchange activities between Taiwan and Chinawere selected. The selected cities accounted for more than88% ofsurveyed economic exchange market between Taiwanand China in 1999 [1], [2]. Currently direct flights betweenTaiwan and China are banned due to political issues. We try tosimulate the air freight service network after Taiwan andChina relax the restrictions on direct flights.

Based on the historical data the demand can be roughlyseparated to 3 levels, middle (100%), high (about 120% ofmiddle level), and low (about 75% of middle level). Theircorresponding probabilities are 0.58 (middle), 0.25 (high),and 0.17 (low) respectively. Previous studies suggest that areasonable value of a is between 0.6 to 0.8 and a reasonablevalue of f3 is between 0.7 to 0.9 [4], [14], [13]. The discount

factors for different demand levels and the market share weregiven by reasonable assumption. The other testing parameterswere collected or estimated based on the data from a leadingairline company in Taiwan. The data were all converted toannual basis and the detailed testing data are listed in Table I.

(10)

(11)\:;fi,k,t,j,OJ:

i*- j

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In order to facilitate solution procedure, the two-stagemodel (P) is rewritten to an extensive form. The extensiveform explicitly describes the second stage problem for everyscenario and combines with first stage problem [3]. Thestochastic model (P), therefore, becomes an MIP problem.The MIP problem is then implemented by the commercialsoftware GAMS and the MIP solver GAMS/OSL is employedto solve the problem. All tests were executed in a PersonalComputer equipped with Pentium 4, CPU 3.2 GHz, and 1 GBRAM in Microsoft Windows XP operation system.

3.2 Test Results

As discussed earlier, the first stage of Model (S) seeks thebest hub location over planning horizon before knowingexact demands. The second stage decision is to construct theoptimal service network based on the hub location afterobserving the realized demands. The results showed that thestochastic model chooses Hangzhou as hub for all demandlevels. However, the model provides different servicenetwork (delivery paths and flow allocation) in response todifferent demand levels. As the demand increases, the use ofhub-stop flights increases. This is because higher demandwould increase the saving on transportation costs due to theeffect of scale economies.

IV. SUMMARY AND CONCLUSIONS

The strategic airline network design problem is a key factorfor operational activities. Therefore, it has a long lastinginfluence to airline companies. The demand level is one ofthefundamental input parameters to airline network designproblem. Traditional deterministic model is to assume asingle demand level as a given input and then design theoptimal service network based on the given demand. Averagedemand is the most commonly used value. However, thedemand has obvious seasonal variation in air market. A singledemand level is not sufficient to represent the demandvariation. This study explicitly considers the demandvariation into network design process and proposed atwo-stage stochastic network design model. The decisions areseparated to two stages. The first stage seeks the best hublocation under different demand levels across the planninghorizon. The second stage decision is to determine thedelivery paths and allocate the path flows in response to thechange of demand. The two stage decisions match thedecision procedure of airline companies in practice: thechoice of hub location belongs to the long-term investment

The setup costs for a hub (New Taiwan Dollar)

The unit transportation costs (New TaiwanDollar)

Demand levels (high / middle / low)

Probability sets (high / middle / low)

Discount factors a(w), f3(W) (high /

middle / low)

420,000,000

7.74 NTDlkm·ton

120% /100% / 75%

0.25 /0.58/0.17

(0.5, 0.7) / (0.6, 0.8) /(0.7,0.9)

and will not change due to different demand levels, while thedelivery paths and their flows are usually adjusted in responseto the change ofdemand. The proposed model was tested by areal network data collected from the air passenger market inTaiwan and China. The preliminary tests showed that themodel is able to effectively respond to demand variation.

Since the stochastic airline network design model takespossible demands into consideration, the problem size ingeneral will be increased. Developing an efficient solutionalgorithm to solve the proposed model could be a directionfor future research.

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