# Price planning for time-definite less-than-truckload freight services

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Logistics 2009 Elsevier Ltd. All rights reserved.ervicrphywidto totgrouexpanits prots. Price planning has yet to be well studied and incorporated into the overall marketing strategy by the time-deniteLTL freight delivery industry. In practice, prices are simply set at a constant percentage above costs across the market. Thisensures that the carrier will earn at least that percentage of prot.Studies of pricing strategies and capacity allocation in revenue management have been carried out for perishableasset industries, such as airline and liners (Feng and Xiao, 2006; McGill and van Ryzin, 1999; Subramanian et al., 1999;1366-5545/$ - see front matter 2009 Elsevier Ltd. All rights reserved.* Corresponding author. Tel.: +886 6 275 7575x53240; fax: +886 275 3882.E-mail address: cclin@mail.ncku.edu.tw (C.-C. Lin).Transportation Research Part E 45 (2009) 525537Contents lists available at ScienceDirectTransportation Research Part Edoi:10.1016/j.tre.2008.12.004nancial services. With this expansion in services, to sort, load and transfer all products according to their contractualand temporal needs becomes both a key competitive advantage and an inescapable requirement for shippers. In response,the 3rd-party LSPs provide various service levels, all with guaranteed delivery times.The time-denite less-than-truckload (LTL) freight delivery common carriers, one of the 3rd-party LSP, publish tariffs anddeliver small shipments door-to-door with various service levels all with guaranteed delivery times for shippers. To besuccessful, cost minimization is a basic and effective strategy. For this reason, most research has focused on the design ofa cost-effective operations plan. However, cost consciousness is but one of several successful factors. In fact, carriers muststrategically combine prices with a cost-effective delivery network and optimized operations plan in order to maximizeHub-and-spoke networkLagrangian RelaxationImplicit enumeration1. IntroductionThe mission of 3rd-party logistic scustomized services to shippers (Mu3rd-party LSPs. To meet the one-stopdirections, (1) from a single functionhave chosen to expand their singlereplenish management, and also toe providers (LSPs) is to establish a long-term relationship with and provide broaderand Poist, 2000). Transportation and warehousing are two common services of thee-range integrated logistics services, their services recently have expanded in twoal solutions, (2) from domestic to global services. To be a full service provider, theynd to multiple modal transportation services, from warehousing to automaticd physical distribution to integrate their operations with e-commerce as well asPrice planning for time-denite less-than-truckload freight servicesCheng-Chang Lin *, Dung-Ying Lin, Melanie M. YoungDepartment of Transportation and Communication Management Science, National Cheng Kung University, 1 University Road, 701 Tainan, Taiwan, ROCa r t i c l e i n f oArticle history:Received 3 October 2006Received in revised form 29 September2008Accepted 15 December 2008Keywords:Pricinga b s t r a c tPrice planning simultaneous determines the service demand (with associated prices) andan operational plan to maximize a carriers prot. We modeled this integral-constrainedconcave program in the link formulation and proposed an implicit enumeration embeddedwith Lagrangian Relaxation upper bounds to determine the optimal prices. Computationson Taiwans time-denite less-than-truckload freight market showed that the carrier needsto simultaneously re-evaluate its network capacity while determining prices. The commonpractice of distance-based pricing that sets price by a base rate over direct shipment dis-tance underestimates operating cost, specically operating losses for short distanceshipments.journal homepage: www.elsevier .com/locate / t reWeatherford and Bodily, 1992). These studies determine discriminating prices and simultaneously allocate capacity to max-imize their prots. The pricing scheme is implemented through booking as well as limited overbooking. Such an approachcannot be applied to common carriers, who must provide service indiscriminately for anyone who pays the published rates.This constraint motivates us to study the pricing planning for time-denite LTL freight delivery common carriers. In this re-search, we make the following assumptions: (1) The demand is a continuous and invertible function of price (which is ver-ied in the computational results in Section 6); (2) the revenue function is a concave continuous function; (3) the capacity inthe hub-and-spoke network is xed. The pricing planning is dened so as to simultaneously determine the demand (withassociated prices) for service and develop an operational plan in which the prot is maximized while meeting the servicecommitment, capacity and other operational restrictions.The structure of this paper is as follows. In Section 2, we give a brief overview of carriers line-haul operations in a purehub-and-spoke network and review the research on operational planning. In Section 3, we represent the carrier pricing plan-ning problem in a capacitated directed operations network for mathematical formulation and algorithmic design. The math-ematical model in the link formulation is formulated in Section 4, resulting in an integer concave program. In Section 5, wepropose an exact algorithm, an implicit enumeration on paths with embedded concave programming subproblem to deter-mine the optimal pricing for carrier. The subproblem is solved by FrankWolfe algorithm. The Lagrangian Relaxation (LR)upper bounds, by relaxing the capacity constraints, are implemented to improve the computational efciency. Conceptually,the algorithmic scheme needs to maintain feasibility while searching for optimality. In Section 6, we select a small pure hub-and-spoke network of one of the three top time-denite LTL freight delivery carriers in Taiwan to provide a basis for numer-ical testing. The computational results are presented, analyzed and discussed. We conclude our research in the eld of526 C.-C. Lin et al. / Transportation Research Part E 45 (2009) 525537pricing in the last section.2. Line-haul operations in a pure hub-and-spoke networkThe line-haul operations in a pure hub-and-spoke network consist of facilities, centers and hubs, and long-haul feeders thatare carrying equipment feeding freight between facilities. All the feeders must either depart or end at hubs (Fig. 1). As a re-sult, no center-to-center direct feeds are allowed with the result that all the freight requires at least one handle operation athub facilities. Each center serves an exclusive geographic area for delivering shipments to consignees and, subsequently,picking up new shipments from shippers, using a eet of package cars. Pickups are typically completed at twilight, at whencenter runs a local sort operation. New shipments are unloaded from the package cars, sorted and reloaded onto long-haulfeeders. Feeders are subsequently dispatched to the hubs, which are points of consolidation for partial loads. All the hubs willoperate at least a night sort, unload inbound freight, rehandle and reload unto outbound feeders. Providing sufcient staffrotation times in between, hubs may operate additional sorts to increase the total handling volume to lower unit overheadcost, if necessary. At dawn, centers receive daily delivery freight, when they run a preload sort to unload freight from thefeeders, rehandle and reload unto package cars for local deliveries. Thus, local sorts and preload sorts are freight originsand destination, together is called OD pair for each shipment in the pure hub-and-spoke network.The pure hub-and-spoke network may substantially reduce center-to-center partial loads, resulting in a lower total oper-ating cost. Carriers develop the most cost-effective line-haul operations plan to guide daily operations. The plan consists offreight routing planning (Lin, 2001), trailer assignment and balancing planning (Lamar and Shef, 1987), and feeder schedulingplanning (Lin and Lin, 2001) that, respectively, determine freight paths, a balanced feeder network and feeder schedules tophysically move the loads/empties. To account for mutually interactive effects, the load planning simultaneously determinesthe freight routes and a balanced trailer network (Leung et al., 1990). Lin and Wu (2001) extended the single-frequency tocenter 2center 1hub-Ahub-BLine-haul operationsLocal services Local servicesshippers/congineesfeeders package carscenter 3Fig. 1. Time-denite delivery operations network illustration.multiple-frequency load planning problem, while Lin (2004) extended the deterministic to stochastic in demand. Lastly, theload planning with aircraft scheduling problem not only determines the air cargo routes but also designs a balanced aircraftnetwork with schedules. Barnhart and Schneur (1996) developed a branch-and-bound algorithm for the set of feasibleschedules with high reduced cost values to study a single air hub case for air express time-denite services. Overall, the pric-ing plan to optimize the carriers prot is yet to be fully studied.3. The pricing planning networkThe line-haul operations hub-and-spoke network can be represented as a capacitated directed pricing planning network(N,A) of a set N of i nodes and a set A of ij links. Each center has two nodes associated with local and preload sorts, whileeach hub has as many nodes as its respective sorts. A network conguration for two hubs (with night sorts) and three centersis shown in Fig. 2. The given attributes for a node i e N are ci [cd ], ti [td ] and U^i, respectively, denoted as the unit handling cost(of a center preload), the consolidation duration in hours (of a center preload) and the capacity. The unit handling cost is anaggregated cost for a unit of freight that sums the unloading, handling, reloading and facility-related costs. A center localnode o 2 O together with a center preload node d 2 D forms an origin-destination (OD) pair od 2 OD. The carrier servesthe set OD of od origin-destination pairs with the endogenously determined demand of qod (with associated price) and theexogenously published service level of T^od. The service level is the number of days (for example, 2 days) after the day of pickup(Monday) that the consignees will receive the shipment (Wednesday). Thus, it is the elapsed time in hours from start to endbetween local (origin) and preload (destination) sorts.There are directed links connecting from all local (origin) nodes to all hub-sort nodes; all hub-sort nodes to all preload(destiin thetationC.-C. Lin et al. / Transportation Research Part E 45 (2009) 525537 527ity U^ij. There is no associated unit cost or capacity limitation on any of all holdover/repositioning links. The unittransportation cost sums carrying equipment and transport costs. The elapsed time on links is the interval from end to startof the handling operation between directly connecting nodes. It must be sufciently large for traveling, otherwise, an incre-ment of 24 h will be added for next day arrival.4. The mathematical modelWe assume the capacity of the carrier to be xed, that is, we assume there is no resize or reschedule of the feeder eet,and that there is no expansion of the facilities during the planning period. However, freight paths may be altered, which re-sults in changing the contents of the hubs and feeders to realize a higher prot. Denote podqod as the inverse demand functionfor OD pair od. The pricing planning of a carrier is to unilaterally determine the demand for each OD pair (with associatedprice) and develop an operational plan that maximizes prots while meeting the level of service and operational restrictions.This results an OD-based pricing system which means shipments are charged based on their origin and destination locations.The model in the link formulation with the inverse demand function ismaxWq; x; z Xodpodqodqod XodXijci cijxodij cd( )qod 1nightnightpreloadpreloadlocallocalcenter 3center 2Hub-BHub-Atransportation linkspreloadlocalcenter 1holdover linksrepositioning linksFig. 2. Pricing planning network illustration.nation) nodes; and all hub-sort nodes to other hub-sort nodes with no intermediate nodes. Links connecting two sortssame building are holdover (in terms of freight) or repositioning (in terms of feeders) links; while others are transpor-links. The given attributes of each link ij 2 A include per unit transportation cost cij, elapsed time tij, and carrying capac-pair. Cregardeithermusttogethand feform agether Center-1 local? Hub-B night? Center-2 preload and center-3 local? Hub-B night? Hub-A night? Center-2 preloadare noall bin5. ThThlems,guraall ODthe pr(along5.1. TTowhileof an528 C.-C. Lin et al. / Transportation Research Part E 45 (2009) 525537t. Lastly, the demand for its service is continuous; while the freight path and outbound node decision variables areary as stated in (8). We denote the feasible set as X.e algorithmic designe pricing planning problem yields to an integer concave program. Structurally, the problem integrates two subprob-(a) a binary program that determines a time-feasible path for each of all OD pairs while meeting a directed in-tree con-tion for each of all destinations, and (b) a concave program that determines the prices for a determined path of each ofpairs. Based on this observation, in this research we developed an exact algorithm, the implicit enumeration to solveoblem. We implicitly enumerate path and directed in-tree binary variables. When feasible, we determine demandswith the resulting prices) by FrankWolfe algorithm, an approach for concave programs.he implicit enumeration algorithmguide the search procedure, we visually construct a two-dimensional search tree. OD pairs are on the vertical axis,the time-feasible paths (TFP) of each OD pair are on the horizontal axis (see Fig. 3). Each tree node represents a TFPOD pair.load and Center-3 local? Hub-B night? Center-2 preload is a directed in-tree to the destination Center-2 preload; but to-onstraint (6) states that at any node, carrier will ow the freight of a destination to at most one outbound node dis-ing their origins. As an example, all center-2 preload freight at Hub-A night can only assign at most one outbound node,Hub-B night or center-2 preload but not both. Constraint (7) requires that at any node, all the freight of a destinationow together to the assigned outbound node. That is, at Hub-B night, every piece of center-2 delivery freight is shippeder to the center-2 preload, even though part of them are center-1 origin and others are center-3 origin but rehandled atd by Hub-B night. Constraints (6) and (7) require that freight paths headed to a single destination from all origins mustdirected in-tree rooted at that destination. Thus, together Center-1 local? Hub-A night? Hub-B night? Center-2 pre-subject to :Xijti tijxodij td T^od 8od 2 OD 2XodXjqodxodij U^i 8i 2 N 3Xodqodxodij U^ij 8ij 2 A 4Xjxodji Xjxodij 1; if i o;1; if i d;0;otherwise;8>: 8i 2 N; od 2 OD 5Xjzdij 1 8d 2 D; i 2 N 6Xoxodij Bzdij 8ij 2 A;d 2 D 7qod 2 R; zdij; xodij 2 f0;1g 8ij 2 A; od 2 OD 8with parameter: B as is a huge number; and decision variables:qod : the demand on OD pairod; 8od 2 ODxodij :1; if link ij is chosen for OD pairod;0; otherwise;8ij 2 A; od 2 ODzdij :1; if j is the outbound node for destination d at node i;0; otherwise;8ij 2 A; d 2 DIt is an integer concave program. The objective function is the carriers prot. The total operating cost, the second term is thesum of handling costs at nodes and transportation cost between all pairs of nodes, whilePijci cijxodij cd is the unit pathcost for an OD pair od.Constraint (2) is the service levels, the maximum elapsed time on OD pairs. The path time is the sum of duration time atnodes and elapsed time on links. These are time-feasible paths if they meet the desired level of service. Constraints (3) and (4)separately state that the ows on a node or link cannot exceed its respective handling or carrying capacity. Constraint (5) isthe ow conservation constraint. It requires all of the pickups to be shipped out of the origins; all of the delivery volumesarrive at the destinations; and no freight is staged in the hubs. As a result, there is one and only one freight path for each OD pC.-C. Lin et al. / Transportation Research Part E 45 (2009) 525537 529The search tree structure is constructed as follows. Let Rod be a set of r time-feasible paths for OD pair od. Wheneverassigning the path r for an OD pair od, its associated path variables, fxodij ; ij 2 Ag are determined, that is, xodij 1 if ij on pathr, otherwise, xodij 0. Thus, the unit operating cost for path r of OD pair od is the sum of the handling and transportation costson that path, codr Pijci cijxodij cd. How the TFP are ordered may impact on the computational efciency. Thus, we orderTFP paths of each OD pair by their potential prots, Eodr ; r 2 Rod. This is the maximum prot that the capacity on the path rmaysustain excluding any demands from all other OD pairs. Computationally, the optimal demand on path r, q^odr minf~qodr ; qodr g isdetermined by the smaller of the following numbers: (1) the quantity at which the marginal revenue equals to the unit pathod od od od od P od od ^ od od ^|M|thNnew current tree nodeNCNcurrent tree nodesuperior upper boundedge caseInfeasible or inferior upper boundODCCN edge caseFig. 3. The search tree structure and branching scheme.cost, ~qThereprotThIt sis, r 2pair fIn(1)(2)(3)Time feasible pathsairs1st2nd1st 2nd |(P|-1)th |P|thr : MR~qr cr , or (2) the maximum capacity on the path, qr maxqr : f jxij qr Ui; xij qr Uij; 8i 2 N; ij 2 Ag.fore, the potential prot is: Eodr fpodq^odr codr gq^odr for path r of OD pair od. Upon completion, the maximum potentialpath r 2 Rod becomes the rst tree node of its respective row, Eodr maxfEodr ; r 2 Rodg.e complete procedure of the implicit enumeration is shown in Fig. 4.tarts at the rst tree node and proceeds. An iteration represents one TFP for each of a subset of OD pairs is selected, thatRod for OD f. . . ; od; . . .g#OD. Collectively, fr 2 Rod; od 2 ODg is dened as the current set, while the TFP of the last ODr 2 RjODjg is denoted as the current tree node and its associated OD pair is the current OD pair, jODj on the search tree.each iteration, the computational procedure is as follows:(Directed in-tree). To maintain the feasibility, it requires that there is a directed in-tree conguration for each of alldestinations, that is, fxodij ;zdij; 8j 2 N; ij 2 A; d 2 Dg X for the current set fr 2 Rod; od 2 ODg. When this fails, it is aninfeasible solution. Go to branching at (4).(Upper bounding). When solution is yet to be feasible ODOD, that is, the current set does not contain one TFP for eachof all OD pairs, we determine its upper bound that will be described in Section 5.2. Upon completion, go branching at(4).(A feasible solution). If the current set consists of one TFP for each of all OD pairs, OD OD, the pricing problemreduces to a concave program. That is, given fxodij ;zdij; 8j 2 N; ij 2 A; od 2 ODg X, that satises constraints (2) and(5)(7), the pricing planning is simplied as a concave program:MaxXodfpodqodr codr gqodr 9XodXjqodr xodij U^i 8i 2 N 10Xodqodr xodij 6 U^ij 8ij 2 A 11qodr 2 530 C.-C. Lin et al. / Transportation Research Part E 45 (2009) 525537Contain all OD 0.StartStopYes1. Directed in-trees ?YesNo3. Determine the 4. Move In this research, the FrankWolfe algorithm was used to solve this concave program. The procedure is as follows.FW_1. (Linearization). At each LR iteration t, we linearize the concave revenue function resulting in the following LP:MaxPodfMRqodr t codr gqodr , subject toPodPjqodr xodij U^i andPodqodr xodij U^ij. We solve and obtain its primalqodr ; 8od 2 OD .FW_2. (Line search). Determine a step size 0 6 x 6 1, such that qodr t1 xqodr t 1xqodr is the optimum tox : MaxPodfpodxqodr t 1xqodr codr gxqodr t 1xqodr .FW_3. (Convergent test). If jqodr t1 qodr t j=qodr t e. The demands fqodr gt1 and its associated protPodfpodqodr t1 codr gqodr t1 become a feasible solution, the FrankWolfe algorithm terminates. Otherwise, go to stepFW_1. If the solution determined is better than the incumbent solution, we update the incumbent,fqodr qodr t1; Podqodr ; r 2 Rod; od 2 ODg. Go branching at (4).(4) (Branching) The result of an iteration can be classied into three scenarios, (a) infeasible, (b) upper bound is lowerthan the incumbent, Wq; x; z; x Podfpodqodr codr gqodr or a feasible solution is determined whether or not theincumbent is updated, (c) upper bound is greater than the incumbent. For the scenario (c) the procedure moves(downward) to the rst tree node (TFP) of the next of current OD pair in which it becomes a current tree node.2. Determine the upper boundpairs ?4. Available?NoYes4. Backward and determine a non-edge current setNoNoNoYesNoYesoptimal prices for the current setBetter solution ?4. Edge tree node?Update incumbentYesWorse than the incumbent ?4. Move downward rightward Fig. 4. The implicit enumeration algorithmic procedure.5.2. UAtprobleq x q x U 8i 2 N 15We destrain(15) aC.-C. Lin et al. / Transportation Research Part E 45 (2009) 525537 531XijbijXod2ODqodxodij Xod2ODnODqodxodij U^ij0@1A Max Wq; y; z;a;bXod2ODpodqod Xijci ai cij bijxodij cd ad" #( )qodXod2ODnODpodqod Xijci ai cij bijxodij cd ad" #( )qod 22subject to: (14), (17)(21)XiaiXod2ODXjqodxodij Xod2ODnODXjqodxodij U^i0@1Aod2ODnODp q ijci cijxij cd qod2OD od2ODnODXjxodji Xjxodij 1; if i 2 o;1; if i 2 d;0;otherwise;8>: 8i 2 N; od 2 OD n OD 17Xjzdij zdij 1 8d 2 D; i 2 N 18Xo:od2ODxodij Xo:od2ODnODxodij Bzdij 8ij 2 A;d 2 D 19zdij; xodij 2 f0;1g 8d 2 D; ij 2 A; od 2 OD n OD 20qod 2 R 8od 2 OD 21velop a Lagrangian Relaxation approach to determine its upper bound by relaxing both the node and link capacity con-ts. As a result we determine the prot in an uncapacitated network. When we relax the capacity constraints on nodesnd links (16), the relaxed problem becomesMaxWq; x; z;a;b; x Xod2ODpodqod Xijci cijxodij cd" #( )qodXod odXod" #( )odod2OD jijod2ODnOD jij iXqodxodij Xqodxodij U^ij 8ij 2 A 16(rightward) to the next tree node (TFP) of the current OD pair in which it becomes the current tree node. However,if it is an edge node, the procedure moves backward till one OD pair contains a non-edge next tree node. Whenevera new current tree node is identied, the current set is updated and a new iteration starts. The branching procedure isshown in Fig. 3. Of course, when no new current tree node exists, the program terminates. Then the incumbent solu-tion is the optimal solution to the pricing planning.pper boundseach iteration OD pairs are classied into two sets, one with selected TFP, OD#ODwhile the other does not. The pricingm becomesMaxWq; x; z; x Xod2ODpodqod Xijci cijxodij cd" #( )qod Xod2ODnODpodqod Xijci cijxodij cd" #( )qod 13subject to :to Xijti tijxodij T^od 8od 2 OD n OD 14X XododX Xod od ^However, for the rst two scenarios (a) and (b), if the current tree node is not an edge node, the procedure moveswith 1% of the tolerance rate of convergence in the line search step. It was performed on a Pentium-IV with a CPU speed of532 C.-C. Lin et al. / Transportation Research Part E 45 (2009) 5255373.06 GHz under the Windows OS environment. We ran a maximum of 100 iterations for Lagrangian Relaxation upperbounds. The initial step size was set at 2 and halved when dual solutions failed to improve by more than 1% in 50 iterations.The initial volume for any OD pair was initialized at 0.In addition to experiments in the base operations network, we also performed two sensitivity analyses on carrying capac-ity and the transportation costing structure. The total derived demand did not exhaust the available carrying capacities in thebase operations network. Thus, we labeled the base operations network as an uncapacitated network. In order to realize theimpact of carrying capacity on prices and prots, we performed the sensitivity analysis on carrying capacity. With the sameThe relaxed problem may decompose into (1) the determination of the directed in-tree constrained shortest path and (2)the prot optimization two subproblems for each OD pair. That is for each OD pair, the rst subproblem determines the leastcost path while meeting the service level and directed in-tree (Lin, 2001). Upon determination, the second subproblem deter-mines the demand for that OD pair at where its marginal revenue equals to its cost (of that least cost path). As a result, thedual is the upper bound with respect to this iteration. We update the Lagrangian multipliers for nodes and links by the Sub-gradient method (Held et al., 1974). The number of iterations increments and the procedure repeats.6. Computational resultsWe used one of the three largest time-denite less-than-truck load (LTL) freight delivery common carriers in Taiwan forthe numerical test. The carrier currently has 49 closely located centers along with three major hubs located in the vicinity ofthree of top four most populated cities, Taipei, Taichung, and Tainan. They are, respectively, the central cities of Taiwans topthree major metropolitan areas in the Northern, Central and Southern regions. The carrier currently owns a heterogeneoustrailer eet. However, 85% of the eet is 20 tons in weight with 14 tons in carrying capacity. But, the current load factor is of70% (10 tons) because of the freight irregularity.In this research, we selected a subset of current hub-and-feeder operations network to construct a base operations networkfor the numerical test. The network consists of all 3 hubs and 10 remote centers evenly scattered in Taiwan. Only a singlesort, night, for each hub starting at 11pm and lasting for 3 h is operated. On the other hand, the start times for local and pre-load sorts are 5 pm and 6 am and both last for 2 h. The base networks operational capacities are the current facility handlingand feeder carrying capacities. The level of service is identical to the carriers current commitments, which takes a day todeliver. We used the carriers operating cost statements to calculate the handling and transportation unit costs. The facilityxed and handling variable costs per unit across the facilities are, respectively, $350 and $50 per ton; while the carryingequipment xed and transport variable costs per unit are, respectively, $2.5, and $0.75 per ton-km. All the currencies arein New Taiwanese Dollar (NTD).Currently, the LTL industry in Taiwan implements a highly regulated distance-based pricing system. The tariff includesloading/unloading unto carrying equipment and transport two charge items. The central government reviews their base ratesonce a year and makes adjustment if the economic conditions change, prior to ofcially publishing to the general public.Their base rates are, respectively, $147 per ton and $9.324 per ton-km. Using the base rates, an OD shipments loading/unloading charge is determined by its weight, while both weight and origin-destination direct distance determines thetransport charge. The initial charge for transport is the fee for the distance of 12 km. The charge increases by linear incre-ments from the base rate for distances that exceed 12 km. However, there are discounts for the long distance services over100 km. The portions of 100200 km, and 200 km and above will, respectively, be discounted by 15% and 30%. An additionalsurcharge is levied for empty carrying equipment repositioning which is 70% of the initial charge, if the distance is no morethan 12 km.The carriers daily average demand by distance is shown in Fig. 5. Demands are relatively at over service area, except forfour peaks at distances of 150, 250, 290 and 340 km. There are, respectively, the distances from/to the top four most pop-ulated cities in Taiwan, which are Taipei, Taichung, Tainan and Kaoshiung. We used the daily average demand with the pub-lished tariff to calibrate the inverse demand function. We assumed a continuous nonlinear inverse demand function ofp lqh:Therefore, the price elasticity is g 1h. The result is l = 317.03 and h = 0.372 (p 317:03q0:372), with R2 = 0.999. How-ever, we may expect a relative inelastic demand (a smaller absolute value of price elasticity) for long distance services, sincethey have relatively fewer alternative means to transport shipments than short distance services. Thus, in addition to assum-ing the same price elasticity for all OD pairs, namely a single-segmentmarket (Oum et al., 1992), we calibrated the respectivedemand functions for dual-segment, short (no more than 180 km, approximately a half of the Taiwans longest delivery) andlong (180 km and above) distance markets (Beuthe et al., 2001). The short and long distance demand functions, respectively,are l = 109.29 and h = 0104 (p 109:29q0:104) with R2 = 0.998, and l = 317.35 and h = 0.266 (p 317:35q0:266) withR2 = 0.999. Overall, the CobbDouglas functional form of demand function ts well for the Taiwans LTL industry. Price elas-ticities are all negative values. Since all the absolute values of elasticity are greater than 1, they showed that the demand isprice elastic. Furthermore, the 0 < h < 1 of both market segmentations ensures that the revenue functions are concave, whilethe marginal revenue functions are strictly monotonic decreasing function. They together with high R2s verify the assump-tion and facilitate a numerical example.The program was coded in C with embedded Cplex callable function to compute the linearized FrankWolfe subproblem0501001502002503003504004505000 100 200 300 400 500Volume (in hundred kg)C.-C. Lin et al. / Transportation Research Part E 45 (2009) 525537 533network structure (nodes and links), we created two capacitated networks. All the links in the same network have the samecarrying capacity. But, one (10 hundred kilograms) has 1/10 of carrying capacity of the other (100 hundred kilograms) net-work. Furthermore, hauling long distance freight may increase the on-road utilization of carrying equipment. As a result, thecarrying equipment and transport costs per unit may be reduced. Thus, we experimented with a 10% decrease (increase) ofper unit carrying equipment and transport costs for the long (short) distance transport, a 2-tier cost structure. The cut offdistance is 120 km, an approximate maximum distance to reach the nearest hub.The computational result, together with the sensitivity analyses on the carrying capacity and transportation costing struc-ture, is organized in Table 1. The overall performance of the algorithm is promising. Analytically, there were several conclu-sions on the issues of operational capacity, transportation costing structure, preferable shipments, pricing schemes andmarket segmentation under the price elastic freight market.(1) Operational capacity: the higher the insufcient operational capacity, the lower are the revenue and prot. Observe thepricing and demand curves for single-segment market in Figs. 6 and 7 and dual-segment market in Figs. 8 and 9.For freight with price elastic when an ample of capacity (in the uncapacitated network) is available, the carriermay charge a lower price but may stimulate a higher demand. The volume increase outweighs the price loss withthe result of higher total revenue and prot. However, with limited capacity (in the capacitated networks), the carriermay be able to increase prices, but this results in a lower in demand. The revenue and prot are both lower. Thus,Distance (Km)Demand AverageFig. 5. Carriers current daily demand by distance.under the price elastic freight demand market, the revenue and the prot decrease as the insufciency of operatingcapacity increases.(2) Transportation costing structure: a slight increase on returns, assuming a better carry equipment utilization over long haultransports. The computational results showed that the price (Figs. 6 and 8) and demand curves (Figs. 7 and 9) for sin-gle- and 2-tier cost structures are, respectively, nearly overlapped, and prots (Table 1) are within a small margin in allTable 1Computational results.Capacity CoststructureMarketsegmentIteration Revenue(NTD$)Cost (NTD$) Prot(NTD$)CPU (s)Handling Transport TotalCurrent (uncap) Same based Single 1747 24,373 9259 6046 15,305 9068 36.38Dual 1442 33,141 17,974 9567 27,541 5600 32.392-Tier based Single 3218 24,098 8988 6143 15,131 8967 89.81Dual 1176 31,520 16,357 9522 25,879 5641 33.61Cap (100 hundredkilograms)Same based Single 1747 24,373 6047 9259 15,306 9067 36.22Dual 2380 30,165 15,465 9158 24,623 5542 142.832-Tier based Single 3218 24,098 8989 6143 15,131 8966 89.20Dual 2107 29,498 14,685 9215 23,900 5598 151.11Cap (10 hundred kilograms) Same based Single 2526 17,486 5228 4015 9244 8243 989.84Dual 1785 13,776 5017 4594 9611 4165 1147.162-Tier based Single 2868 17,751 5335 4181 9516 8235 1025.19Dual 5026 14,125 4927 4920 9847 4278 2864.41534 C.-C. Lin et al. / Transportation Research Part E 45 (2009) 52553750100150200250300Price/Cost(NTLeast costTariffUncap/Cap100Uncap/Cap100 2-tier costCap10Cap10 2-tier costthe market segments. However, with the 2-tier cost structure, the demands are marginally lower (higher), while theprices are marginally higher (lower) for short (long) distance shipments. On an average, the better utilization of car-rying equipment for long haul transports resulted in a lower long haul transportation cost that increases the prots by0.36%. The reason for a marginal return is because the handling-related costs share a major portion of the total oper-ating cost when compared with the transportation-related costs. Since, all the shipments require at least one hubrehandle, the lower in transportation costs may not directly translate into a signicant return.(3) Types of shipments: the carrier will favor short over long distance shipments, if they require the same number of hub han-dlings. The hub consolidation reduces the center-to-center partial loads resulting in a lower operating cost. The totaloperating cost of a shipment for LTL operations is the sum of handling and transportation costs which means, the dis-tance alone is not a sole cost factor. This implies that short distance shipments may not necessarily have a lower oper-ating cost than long distance shipments. However, if shipments with different distances in the same market segment(implies the same demand function) require the same number of hub rehandlings, the shorter the distance, the loweris its operating cost. Suppose a carrier charges the same prices, it will generate the same amount of demand quantitywith a result of shorter the distance, the higher is the unit and total prots. Thus, the carrier will favor short distance00 50 100 150 200 250 300 350 400Distance (km)D)Fig. 6. Price and cost by distance in the single-segment market.0510152025300 50 100 150 200 250 300 350Distance (km)Demand (in hundred kg)Uncap/Cap100Uncap/Cap100 2-tier costCap10Cap10 2-tier costFig. 7. Demand by distance in the single-segment market.C.-C. Lin et al. / Transportation Research Part E 45 (2009) 525537 535100150200250300Price/Cost(Least costTariffUncapUncap 2-tier costCap100Cap100 2-tier costshipments for as long as their demands prevail. Otherwise, the carrier will try to exhaust its overall operating capacityto generate higher revenue by lling up the residual capacity with long distance shipments. Our computational resultsshowed that all the OD pairs require one hub rehandle. Thus, higher demands result for short distance services in thesingle-segment (Fig. 7) or dual-segment market (Fig. 9). This observation also applies to the 2-tier cost structure, sincethe handling-related cost is the dominated portion of the operating cost as discussed in (2).(4) OD-based pricing: it deviates from the distance-based pricing, a constant base rate over shipments direct distances. In thecurrent practice, the carrier determines a shipment tariff based on its direct distance and weight. That is, it applies apublished transport base rate over its direct distance with an additional loading/unloading unto carrying equipmentcharge. The transport charge may be adjusted for additional surcharge or discounts for respective short or long dis-tances, if applicable. The current distance-based pricing shows a price ratio of 8.5 between long (350 km) and short(2 km) distance freight. However, the LTL industry requires hub rehandling to consolidate partial loads which arenot linearly proportionate to shipments distance. Furthermore, the indirect routes for hub consolidation imply thatthe short distance shipments will haul longer than their direct distances. Thus, the current pricing does not properly0500 50 100 150 200 250 300 350 400Distance (km)NTD)Cap10Cap10 2-tier costFig. 8. Price and cost by distance in the dual-segment market.0204060801001201401600 50 100 150 200 250 300 350Distance (km)Demand (in hundred kg)UncapUncap 2-tier costCap100Cap100 2-tier costCap10Cap10 2-tier costFig. 9. Demand by distance in the dual-segment market.536 C.-C. Lin et al. / Transportation Research Part E 45 (2009) 525537reect its operating cost with a result of operating losses for short distance shipments as shown in Figs. 6 and 8. TheOD-based pricing simultaneously plans operational routes and prices. Costs of operational routes are used to deter-mine their associated prices. As a result, there were no operating losses. The computational result showed a highershort distance price with a ratio of 2.1 (1.6) for the same pair of distances in the uncapacitated (capacitated) network.(5) Market segmentation: single-segment with the smallest (absolute) price elasticity has higher prices and prots than dual-segment market. The CobbDouglas function form ts well for the Taiwans LTL industry. However, the single or dualmarket segmentation resulted in different price elasticities. The single-segment market has the smallest (absolutevalue of) price elasticity. Thus, the prices are higher for all the OD pairs than their counterparts in the dual-segmentmarket. The higher the prices the lower are the demands (Figs. 6 and 8). As a result, the prots were almost doubledfor single-segment than dual-segment market.7. ConclusionsPricing is one of the most important instruments to generate higher revenues with an increase in prots. Simultaneouslydetermining a price structure and operating plan will help time-denite common carriers, a 3rd-party logistics service pro-vider, to achieve higher prots in addition to improving operations. The pricing planning with inverse demand function for acarrier is to simultaneously determine the demand (with associated prices) for each OD pair and develop an operational planto ll the available carrying capacity in a pure hub-and-spoke network so that its prot is maximized. It is an OD-based pric-ing system which means the shipments are charged based on their origin and destination locations. We modeled this in thelink formulation as an integral-constrained concave programming problem. We implemented an implicit enumeration withLagrangian Relaxation upper bounds to determine the optimal prices. We tested the algorithm using one of the three largesttime-denite common carriers in Taiwan. The CobbDouglas inverse demand function form ts well for the Taiwans LTLindustry. The computational results are encouraging and promising.The economic implications under the price elastic demand freight market are as follows. (1) The higher the shortfall inoperational capacity, the lower is the revenue and prot for price elastic freight demand. (2) Since the handling-related costdominates the total operating cost, high carry equipment utilization over long haul transports may only contribute a mar-ginal increase on returns. (3) In the same market segment, the carrier will favor short over long distance shipments, if theyrequire the same number of hub rehandles. (4) The current distance-based pricing, that is, a base rate over shipments directdistance with an additional loading/unloading unto carrying equipment charge, does not properly reect its operating costwith a result of operating losses for short distance shipments. Using operating costs of planned routes, the OD-based pricingdeviates from the current practice with higher (lower) prices for short (long) distance shipments. (5) Prices and carriersprots are sensitive to the price elasticity. The smallest (absolute value of) price elasticity will result in highest pricesand carriers prot, which is the case in the single-segment market.Thus, there are the following managerial implications. First, the hub rehandling and indirect routes to hubs are two oper-ational characteristics of the LTL industry. Therefore, the operating cost is not linearly proportion to shipments direct dis-tance. The distance-based pricing method in practice that sets the prices by a base rate over direct distance may result inlosses for short distance shipments, which may not serve as an optimal pricing scheme for the LTL industry. Second, theoperational capacity impacts the carriers prot. While determining prices, the carrier has to simultaneously re-evaluateits network capacity. Third, different ways to segment a market may result in different price elasticity for demand. Theymay impact the optimal prices and prots in any conguration of uncapacitated and capacitated networks.AcknowledgmentsThe authors would like to thank two anonymous reviewers for their helpful comments on earlier version of this paper.This research was partially supported by Grant NSC 94-2416-H-006-003 from the National Science Council, Taiwan, ROC.ReferencesBarnhart, C., Schneur, R.S., 1996. Air network design for express shipment service. Operations Research 44 (6), 852863.Beuthe, M., Jourquin, B., Geerts, J.-F., Ha, C., 2001. Freight transportation demand elasticities: a geographic multimodal transportation network analysis.Transportation Research Part E 37 (4), 253266.Feng, Y., Xiao, B., 2006. Integration of pricing and capacity allocation for perishable products. 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Lin et al. / Transportation Research Part E 45 (2009) 525537 537Price planning for time-definite less-than-truckload freight servicesIntroductionLine-haul operations in a pure hub-and-spoke networkThe pricing planning networkThe mathematical modelThe algorithmic designThe implicit enumeration algorithmUpper boundsComputational resultsConclusionsAcknowledgmentsReferences