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Transportation Research Part E 45 (2009) 525–537

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Transportation Research Part E

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

Cheng-Chang Lin *, Dung-Ying Lin, Melanie M. YoungDepartment of Transportation and Communication Management Science, National Cheng Kung University, 1 University Road, 701 Tainan, Taiwan, ROC

a r t i c l e i n f o

Article history:Received 3 October 2006Received in revised form 29 September2008Accepted 15 December 2008

Keywords:PricingLogisticsHub-and-spoke networkLagrangian RelaxationImplicit enumeration

1366-5545/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.tre.2008.12.004

* Corresponding author. Tel.: +886 6 275 7575x5E-mail address: [email protected] (C.-C. Lin

a b s t r a c t

Price planning simultaneous determines the service demand (with associated prices) andan operational plan to maximize a carrier’s profit. 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 Taiwan’s time-definite 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, specifically operating losses for short distanceshipments.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The mission of 3rd-party logistic service providers (LSPs) is to establish a long-term relationship with and provide broadercustomized services to shippers (Murphy and Poist, 2000). Transportation and warehousing are two common services of the3rd-party LSPs. To meet the one-stop wide-range integrated logistics services, their services recently have expanded in twodirections, (1) from a single function to total solutions, (2) from domestic to global services. To be a full service provider, theyhave chosen to expand their single ground to multiple modal transportation services, from warehousing to automaticreplenish management, and also to expand physical distribution to integrate their operations with e-commerce as well asfinancial 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-definite 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 maximizeits profits. Price planning has yet to be well studied and incorporated into the overall marketing strategy by the time-definiteLTL 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 profit.

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;

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526 C.-C. Lin et al. / Transportation Research Part E 45 (2009) 525–537

Weatherford and Bodily, 1992). These studies determine discriminating prices and simultaneously allocate capacity to max-imize their profits. 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-definite 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-ified 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 fixed. The pricing planning is defined so as to simultaneously determine the demand (withassociated prices) for service and develop an operational plan in which the profit 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 carrier’s 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 Frank–Wolfe algorithm. The Lagrangian Relaxation (LR)upper bounds, by relaxing the capacity constraints, are implemented to improve the computational efficiency. 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-definite 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 field ofpricing in the last section.

2. Line-haul operations in a pure hub-and-spoke network

The 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 fleet 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 sufficient 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 Sheffi, 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 to

center 2

center 1hub-A

hub-B

Line-haul operations

Local services L

ocal

ser

vice

s

shippers/congineesfeeders package cars

center 3

Fig. 1. Time-definite delivery operations network illustration.

C.-C. Lin et al. / Transportation Research Part E 45 (2009) 525–537 527

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-definite services. Overall, the pric-ing plan to optimize the carrier’s profit is yet to be fully studied.

3. The pricing planning network

The 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 configuration 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 Ui, 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 Tod. 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(destination) nodes; and all hub-sort nodes to other hub-sort nodes with no intermediate nodes. Links connecting two sortsin the same building are holdover (in terms of freight) or repositioning (in terms of feeders) links; while others are transpor-tation links. The given attributes of each link ij 2 A include per unit transportation cost cij, elapsed time tij, and carrying capac-ity Uij. 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 sufficiently large for traveling, otherwise, an incre-ment of 24 h will be added for next day arrival.

4. The mathematical model

We assume the capacity of the carrier to be fixed, that is, we assume there is no resize or reschedule of the feeder fleet,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 profit. Denote podðqodÞ 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 profits 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 is

max Wðq; x; zÞ ¼X

od

podðqodÞqod �X

od

Xij

ðci þ cijÞxodij þ cd

( )qod ð1Þ

night

night

preload

preload

local

local

center 3

center 2Hub-B

Hub-A

transportation links

preload

local

center 1

holdover linksrepositioning links

Fig. 2. Pricing planning network illustration.


subject to :Xij

ðti þ tijÞxodij þ td � Tod 8od 2 OD ð2Þ

Xod

Xj

qodxodij � Ui 8i 2 N ð3Þ

Xod

qodxodij � Uij 8ij 2 A ð4Þ

Xj

xodji �

Xj

xodij ¼

�1; if i ¼ o;

1; if i ¼ d;

0;otherwise;

8><>: 8i 2 N; od 2 OD ð5Þ

Xj

zdij � 1 8d 2 D; i 2 N ð6Þ

Xo

xodij � Bzd

ij 8ij 2 A;d 2 D ð7Þ

qod 2 Rþ; zdij; x

odij 2 f0;1g 8ij 2 A; od 2 OD ð8Þ

with parameter: B as is a huge number; and decision variables:

qod : the demand on OD pairod; 8od 2 OD

xodij :

1; if link ij is chosen for OD pairod;

0; otherwise;

�8ij 2 A; od 2 OD

zdij :

1; if j is the outbound node for destination d at node i;

0; otherwise;

�8ij 2 A; d 2 D

It is an integer concave program. The objective function is the carrier’s profit. The total operating cost, the second term is thesum of handling costs at nodes and transportation cost between all pairs of nodes, while

Pijðci þ cijÞxod

ij þ 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 flows on a node or link cannot exceed its respective handling or carrying capacity. Constraint (5) isthe flow 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 ODpair. Constraint (6) states that at any node, carrier will flow the freight of a destination to at most one outbound node dis-regarding their origins. As an example, all center-2 preload freight at Hub-A night can only assign at most one outbound node,either Hub-B night or center-2 preload but not both. Constraint (7) requires that at any node, all the freight of a destinationmust flow together to the assigned outbound node. That is, at Hub-B night, every piece of center-2 delivery freight is shippedtogether to the center-2 preload, even though part of them are center-1 origin and others are center-3 origin but rehandled atand fed by Hub-B night. Constraints (6) and (7) require that freight paths headed to a single destination from all origins mustform a directed in-tree rooted at that destination. Thus, together Center-1 local ? Hub-A night ? Hub-B night ? Center-2 pre-load and Center-3 local ? Hub-B night ? Center-2 preload is a directed in-tree to the destination Center-2 preload; but to-gether Center-1 local ? Hub-B night ? Center-2 preload and center-3 local ? Hub-B night ? Hub-A night ? Center-2 preloadare not. Lastly, the demand for its service is continuous; while the freight path and outbound node decision variables areall binary as stated in (8). We denote the feasible set as X.

5. The algorithmic design

The pricing planning problem yields to an integer concave program. Structurally, the problem integrates two subprob-lems, (a) a binary program that determines a time-feasible path for each of all OD pairs while meeting a directed in-tree con-figuration for each of all destinations, and (b) a concave program that determines the prices for a determined path of each ofall OD pairs. Based on this observation, in this research we developed an exact algorithm, the implicit enumeration to solvethe problem. We implicitly enumerate path and directed in-tree binary variables. When feasible, we determine demands(along with the resulting prices) by Frank–Wolfe algorithm, an approach for concave programs.

5.1. The implicit enumeration algorithm

To guide the search procedure, we visually construct a two-dimensional search tree. OD pairs are on the vertical axis,while the time-feasible paths (TFP) of each OD pair are on the horizontal axis (see Fig. 3). Each tree node represents a TFPof an OD pair.

|M|th

N

new current tree node

N

C

N

current tree nodesuperior upper bound

edge case

Infeasible or inferior upper bound

Time feasible paths

OD

pai

rs

C

CN ed

ge c

ase

1st

2nd

1st 2nd |(P|-1)th |P|th

Fig. 3. The search tree structure and branching scheme.


The 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, fxod

ij ; ij 2 Ag are determined, that is, �xodij ¼ 1 if ij on path

r, otherwise, �xodij ¼ 0. Thus, the unit operating cost for path r of OD pair od is the sum of the handling and transportation costs

on that path, codr ¼

Pijðci þ cijÞ�xod

ij þ cd. How the TFP are ordered may impact on the computational efficiency. Thus, we orderTFP paths of each OD pair by their potential profits, Eod

r ; r 2 Rod. This is the maximum profit that the capacity on the path r maysustain excluding any demands from all other OD pairs. Computationally, the optimal demand on path r, qod

r ¼minf~qodr ; �q

odr g is

determined by the smaller of the following numbers: (1) the quantity at which the marginal revenue equals to the unit pathcost, ~qod

r : MRð~qodr Þ ¼ cod

r , or (2) the maximum capacity on the path, �qodr ¼max qod

r : fP

j�xod

ij qodr � Ui; �xod

ij qodr � Uij; 8i 2 N; ij 2 Ag.

Therefore, the potential profit is: Eodr ¼ fpodðqod

r Þ � codr gqod

r for path r of OD pair od. Upon completion, the maximum potentialprofit path r� 2 Rod becomes the first tree node of its respective row, Eod

r� ¼maxfEodr ; r 2 Rodg.

The complete procedure of the implicit enumeration is shown in Fig. 4.It starts at the first tree node and proceeds. An iteration represents one TFP for each of a subset of OD pairs is selected, that

is, �r 2 Rod for OD ¼ f. . . ; od; . . .g# OD. Collectively, f�r 2 Rod; od 2 ODg is defined as the current set, while the TFP of the last ODpair f�r 2 RjODjg is denoted as the current tree node and its associated OD pair is the current OD pair, jODj on the search tree.

In each iteration, the computational procedure is as follows:

(1) (Directed in-tree). To maintain the feasibility, it requires that there is a directed in-tree configuration for each of alldestinations, that is, f�xod

ij ;�zdij; 8j 2 N; ij 2 A; d 2 Dg � X for the current set f�r 2 Rod; od 2 ODg. When this fails, it is an

infeasible solution. Go to branching at (4).(2) (Upper bounding). When solution is yet to be feasible OD–OD, that is, the current set does not contain one TFP for each

of all OD pairs, we determine its upper bound that will be described in Section 5.2. Upon completion, go branching at(4).

(3) (A feasible solution). If the current set consists of one TFP for each of all OD pairs, OD ¼ OD, the pricing problemreduces to a concave program. That is, given f�xod

ij ;�zdij; 8j 2 N; ij 2 A; od 2 ODg � X, that satisfies constraints (2) and

(5)–(7), the pricing planning is simplified as a concave program:

MaxX

od

fpodðqod�r Þ � cod

�r gqod�r ð9Þ

Xod

Xj

qod�r

�xodij � Ui 8i 2 N ð10Þ

Xod

qod�r �xod

ij 6 Uij 8ij 2 A ð11Þ

qod�r 2 <þ 8od 2 OD ð12Þ

2. Determine the upper bound

Contain all OD pairs ?

0.Start

Stop

4. Available?

Yes

No

Yes

4. Backward and determine a non-edge current set

No

No

No

Yes

No

Yes

1. Directed in-trees ?

Yes

No

3. Determine the optimal prices for

the current set

Better solution ?

4. Edge tree node?

Update incumbent

Yes

Worse than the incumbent ?

4. Move downward

4. Move rightward

Fig. 4. The implicit enumeration algorithmic procedure.


In this research, the Frank–Wolfe 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:Max

PodfMRð�qod

�r Þt � cod

�r gqod�r , subject to

Pod

Pjq

od�r �xod

ij � Ui andP

odqod�r �xod

ij � Uij. We solve and obtain its primal�qod

�r ; 8od 2 OD .FW_2. (Line search). Determine a step size 0 6 x 6 1, such that ð�qod

�r Þtþ1 ¼ xð�qod

�r Þt þ ð1�xÞð�qod

�r Þ is the optimum tox : Max

Podfpodð½xð�qod


�r Þ�Þ � cod�r g½xð�qod


�r Þ�.FW_3. (Convergent test). If jð�qod

�r Þtþ1 � ð�qod

�r Þt j=ð�qod

�r Þt � e. The demands f�qod

�r gtþ1 and its associated profitP

odfpodð�qod�r Þ

tþ1 � cod�r gð�qod

�r Þtþ1 become a feasible solution, the Frank–Wolfe algorithm terminates. Otherwise, go to step

FW_1. If the solution determined is better than the incumbent solution, we update the incumbent,fqod�

�r ¼ ð�qod�r Þ

tþ1; Podðqod��r Þ; �r 2 Rod; od 2 ODg. Go branching at (4).

(4) (Branching) The result of an iteration can be classified into three scenarios, (a) infeasible, (b) upper bound is lowerthan the incumbent, Wðq; x; z; �xÞ �

Podfpodð�qod�

�r Þ � cod��r g�qod�

�r 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 first tree node (TFP) of the next of current OD pair in which it becomes a current tree node.


However, for the first two scenarios (a) and (b), if the current tree node is not an edge node, the procedure moves(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 identified, 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.

5.2. Upper bounds

At each iteration OD pairs are classified into two sets, one with selected TFP, OD # OD while the other does not. The pricingproblem becomes

MaxWðq; x; z; �xÞ ¼X

od2OD

podðqodÞ �X

ij

ðci þ cijÞ�xodij þ cd

" #( )qod þ

Xod2ODnOD

podðqodÞ �X

ij


" #( )qod ð13Þ

subject to :

to þX

ij

ðti þ tijÞxodij � Tod 8od 2 OD n OD ð14Þ

Xod2OD

Xj

qod�xodij þ

Xod2ODnOD

Xj

qodxodij � Ui 8i 2 N ð15Þ

Xod2OD

qod�xodij þ

Xod2ODnOD

qodxodij � Uij 8ij 2 A ð16Þ

Xj

xodji �

Xj

xodij ¼

�1; if i 2 o;

1; if i 2 d;

0;otherwise;

8><>: 8i 2 N; od 2 OD n OD ð17Þ

Xj

ð�zdij þ zd

ijÞ � 1 8d 2 D; i 2 N ð18ÞX

o:od2OD

�xodij þ

Xo:od2ODnOD

xodij � Bzd

ij 8ij 2 A;d 2 D ð19Þ

zdij; x

odij 2 �f0;1g � 8d 2 D; ij 2 A; od 2 OD n OD ð20Þ

qod 2 Rþ 8od 2 OD ð21Þ
We develop a Lagrangian Relaxation approach to determine its upper bound by relaxing both the node and link capacity con-straints. As a result we determine the profit in an uncapacitated network. When we relax the capacity constraints on nodes(15) and links (16), the relaxed problem becomes
MaxWðq; x; z;a;b; �xÞ ¼X

od2OD

podðqodÞ �X

ij

ðci þ cijÞ�xodij þ cd

" #( )qod

þX

od2ODnOD

podðqodÞ �X

ij


" #( )qod

�X

i

ai

Xod2OD

Xj

qod�xodij þ

Xod2ODnOD

Xj

qodxodij � Ui

0@

1A

�X

ij

bij

Xod2OD

qod�xodij þ

Xod2ODnOD

qodxodij � Uij

0@

1A

¼Max Wðq; y; z;a;bÞ

¼X

od2OD

podðqodÞ �X

ij

ðci þ ai þ cij þ bijÞ�xodij þ ðcd þ adÞ

" #( )qod

þX

od2ODnOD

podðqodÞ �X

ij

ðci þ ai þ cij þ bijÞxodij þ ðcd þ adÞ

" #( )qod ð22Þ

subject to: (14), (17)–(21)


The relaxed problem may decompose into (1) the determination of the directed in-tree constrained shortest path and (2)the profit optimization two subproblems for each OD pair. That is for each OD pair, the first 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 results

We used one of the three largest time-definite 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 Taiwan’s topthree major metropolitan areas in the Northern, Central and Southern regions. The carrier currently owns a heterogeneoustrailer fleet. However, 85% of the fleet 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 network’s operational capacities are the current facility handlingand feeder carrying capacities. The level of service is identical to the carrier’s current commitments, which takes a day todeliver. We used the carrier’s operating cost statements to calculate the handling and transportation unit costs. The facilityfixed and handling variable costs per unit across the facilities are, respectively, $350 and $50 per ton; while the carryingequipment fixed 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 officially 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 shipment’s 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 100–200 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 carrier’s daily average demand by distance is shown in Fig. 5. Demands are relatively flat 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 ¼ lq�h:

Therefore, the price elasticity is g ¼ � 1h. The result is l = 317.03 and h = 0.372 (p ¼ 317:03q�0:372), 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-segment market (Oum et al., 1992), we calibrated the respectivedemand functions for dual-segment, short (no more than 180 km, approximately a half of the Taiwan’s 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 ¼ 109:29q�0:104) with R2 = 0.998, and l = 317.35 and h = 0.266 (p ¼ 317:35q�0:266) withR2 = 0.999. Overall, the Cobb–Douglas functional form of demand function fits well for the Taiwan’s 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 R2’s verify the assump-tion and facilitate a numerical example.

The program was coded in C with embedded Cplex callable function to compute the linearized Frank–Wolfe subproblemwith 1% of the tolerance rate of convergence in the line search step. It was performed on a Pentium-IV with a CPU speed of3.06 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 profits, we performed the sensitivity analysis on carrying capacity. With the same

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Fig. 5. Carrier’s current daily demand by distance.


network 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 insufficient operational capacity, the lower are the revenue and profit. 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 profit. 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 profit are both lower. Thus,under the price elastic freight demand market, the revenue and the profit decrease as the insufficiency 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 profits (Table 1) are within a small margin in all

Table 1Computational results.

Capacity Coststructure

Marketsegment

Iteration Revenue(NTD$)

Cost (NTD$) Profit(NTD$)

CPU (s)

Handling Transport Total

Current (uncap) Same based Single 1747 24,373 9259 6046 15,305 9068 36.38Dual 1442 33,141 17,974 9567 27,541 5600 32.39

2-Tier based Single 3218 24,098 8988 6143 15,131 8967 89.81Dual 1176 31,520 16,357 9522 25,879 5641 33.61

Cap (100 hundredkilograms)

Same based Single 1747 24,373 6047 9259 15,306 9067 36.22Dual 2380 30,165 15,465 9158 24,623 5542 142.83

2-Tier based Single 3218 24,098 8989 6143 15,131 8966 89.20Dual 2107 29,498 14,685 9215 23,900 5598 151.11

Cap (10 hundred kilograms) Same based Single 2526 17,486 5228 4015 9244 8243 989.84Dual 1785 13,776 5017 4594 9611 4165 1147.16

2-Tier based Single 2868 17,751 5335 4181 9516 8235 1025.19Dual 5026 14,125 4927 4920 9847 4278 2864.41

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Fig. 6. Price and cost by distance in the single-segment market.

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Fig. 7. Demand by distance in the single-segment market.


the 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 profits 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 significant 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 profits. Thus, the carrier will favor short distance

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Fig. 8. Price and cost by distance in the dual-segment market.

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Fig. 9. Demand by distance in the dual-segment market.


shipments for as long as their demands prevail. Otherwise, the carrier will try to exhaust its overall operating capacityto generate higher revenue by filling 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 shipment’s 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 shipment’s 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 properly


reflect 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 profits than dual-segment market. The Cobb–Douglas function form fits well for the Taiwan’s 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 profits were almost doubledfor single-segment than dual-segment market.

7. Conclusions

Pricing is one of the most important instruments to generate higher revenues with an increase in profits. Simultaneouslydetermining a price structure and operating plan will help time-definite common carriers, a 3rd-party logistics service pro-vider, to achieve higher profits 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 fill the available carrying capacity in a pure hub-and-spoke network so that its profit 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-definite common carriers in Taiwan. The Cobb–Douglas inverse demand function form fits well for the Taiwan’s 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 profit 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 shipment’s directdistance with an additional loading/unloading unto carrying equipment charge, does not properly reflect 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 carrier’sprofits are sensitive to the price elasticity. The smallest (absolute value of) price elasticity will result in highest pricesand carrier’s profit, 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 shipment’s 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 carrier’s profit. 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 profits in any configuration of uncapacitated and capacitated networks.

Acknowledgments

The 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.

References

Barnhart, C., Schneur, R.S., 1996. Air network design for express shipment service. Operations Research 44 (6), 852–863.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), 253–266.Feng, Y., Xiao, B., 2006. Integration of pricing and capacity allocation for perishable products. European Journal of Operational Research 168, 17–34.Held, M., Wolfe, P., Crowder, H.P., 1974. Validation of subgradient optimization. Mathematical Programming 6, 62–88.Lamar, B.W., Sheffi, Y., 1987. An implicit enumeration method for LTL network design. Transportation Research Record 1120, 1–11.Leung, J.M., Magnanti, T.L., Singhal, V., 1990. Routing in point-to-point delivery systems: formulations and solution heuristics. Transportation Science 24 (4),

245–260.Lin, C.-C., 2001. The freight routing problem of time-definite freight delivery common carriers. Transportation Research Part B 35 (6), 525–547.Lin, C.-C., 2004. The load planning with uncertain demands for time-definite freight common carriers. Transportation Research Record 1873, 17–24.Lin, C.-C., Lin, D.-Y., 2001. The feeder scheduling problem for time-definite ground delivery common carriers. Journal of Eastern Asia Society for

Transportation Studies 4 (4), 259–369.Lin, C.-C., Wu, Y.-C., 2001. The multiple frequency delivery operations in the time-definite freight delivery industry. Journal of the Operational Research

Society 52 (11), 1215–1224.McGill, J.I., van Ryzin, G.J., 1999. Revenue management: research overview and prospects. Transportation Science 33, 233–256.


Murphy, P.R., Poist, R.F., 2000. Third-party logistics: some user versus provider perspectives. Journal of Business Logistics 21 (1), 121–133.Oum, T., Waters, W., Yong, J.S., 1992. Concepts of price elasticities of transport demand and recent empirical estimates: an interpretative survey. Journal of

Transport Economics and Policy 26 (2), 139–154.Subramanian, J., Stidham, S., Lautenbacker, A.C., 1999. Airline yield management with overbooking, cancellations and no-shows. Transportation Science 33,

147–167.Weatherford, L.R., Bodily, S.E., 1992. A taxonomy and research overview of perishable-asset revenue management: yield management, overbooking and

pricing. Operations Research 40, 831–844.

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