Mathematical Model and Solution Approach for Collaborative Logistics inLess Than Truckload (LTL) Transportation

Bo Dai, Haoxun Chen

Charles Delaunay Institute (lCD, FRE CNRS 2848), Industrial Systems Optimization LaboratoryUniversity of Technology of Troyes, 10010, France (bo.dai@utt.fr, haoxun.chen@utt.fr)

ABSTRACT

Collaborative logistics is achieved when two or more carriers or shippers form partnerships to optimize theirtransportation operations by sharing vehicle capacities and delivery tasks in order to cut empty back hauls and toincrease vehicle utilization rate. This paper studies collaborative logistics in less than truckload transportation anddevelops a general mathematical model and a Lagrangian relaxation approach to solve this problem. From the model'soptimal solution, a set of feasible vehicle tours corresponding to the transportation operations in collaborative logisticscan be constructed. The model is suitable for both shipper collaboration and carrier collaboration. Furthermore, tenrandomly generated examples are tested to demonstrate the validity of our proposed model and solution approach.

Keywords: collaborative logistics; collaborative transportation management; less than truckload; modeling;Lagrangian relaxation

1. INTRODUCTION

In recent years, due to increasing fuel costs whichforced transportation companies to operate theirvehicles more efficiently, collaborative logisticsattracted a growing interest of industrial practitionersand academic researchers. Collaborative logistics isachieved when two or more companies formpartnerships to optimize their transportation operationsby sharing vehicle capacities and delivery tasks in orderto cut empty back hauls and to increase vehicle fillingrate. In collaborative logistics, suppliers, consumers,and even competitors, can be potential collaborativepartners and the information connectivity of thesepartners can be provided by the Intemet'".

A topic strongly related to collaborative logistics isCollaborative Transportation Management (CTM). Inorder to manage the collaboration among differententerprises, Voluntary Inter-industry CommerceStandards Association (VICS) has established someguidelines for CTM. CTM includes two different typesof collaboration: shipper collaboration and carriercollaboration. Generally, they are consideredindependently due to the benefit difference betweenshippers and carriers.

The road transportation has two modes, namely,Truckload (TL) and Less Than Truckload (LTL)transportation. Both of them have respective applicationdomains and advantages. TL is often used in thetransportation of a single product to a client with largedemand, whereas LTL is usually used to transportmultiple products in small volumes to multiple clientssuch as in parcel delivery. The advantage of TL carriersis that the freight is never handled on route, whereas anLTL shipment will typically be transported on severaldifferent trailersr". The advantage of LTL is that a

978-1-4244-4136-5/09/$25.00 ©2009 IEEE

shipment may be performed with a fraction of the costof hiring an entire vehicle and trailer for an exclusiveshipment. In addition, a number of accessorial servicesare available for LTL, which are not typically offered byTL[2]. Furthermore, in order to satisfy customerdemands quickly, shippers (suppliers) tend to delivergoods to the customers more frequently andconsequently the quantity of each delivery becomessmaller. In this case, shipments must be consolidatedthrough LTL to increase vehicle utilization rate. That'swhy LTL became more popular in recent years.

The key of collaborative logistics is to identify andreduce hidden costs, such as that induced by assetrepositioning'!', Asset repositioning is expensive, butwhen a carrier performs shipments for different shippers,it often has to reposition its assets. Through reducingasset repositioning operations, we can reducetransportation costs.

Up to now, the studies on collaborative logistics inthe literature are focused on (Full) TruckLoadtransportation. They can be divided into two categoriesin terms of the two different types of collaboration.

Shipper collaboration considers only asingle-carrier and focuses on finding optimal routingdecisions under the collaboration among differentshippers. It was first introduced by Ergun, Kuyzu andSavelsberghl'r", They focused on the full truckloadsegment of the transportation industry, and translatedthe truckload shipper collaboration problem into theLane Covering Problem (LCP), which is to fmd aminimum cost set of simple cycles to cover a givensubset of arcs (lanes) in a directed Euclidean graph.They showed that LCP is polynomially solvable.Furthermore, they considered some variants of LCP,such as the cardinality constrained LCp[l], the lengthconstrained LCp[3], the LCP with dispatch windows'I'

767

and/or driver restrictions'", and showed that all thesevariant problems are NP-hard.

Differing from shipper collaboration, carriercollaboration considers how carriers can reduce costs bycollaboration, given the set of lanes that they have toserve. The study of carrier collaboration started just twoyears ago. Firstly, Ergun, Ozener and Savelsberghstudied the full truckload carrier collaboration problemand designed some simple lane exchange processes tofacilitate collaboration among carriers''". They alsoevaluated the effectiveness of their simple exchangemechanisms.

Besides, Nadarajah and Bookbinder studied theless than truckload carrier collaboration problem andgave some simple examples to illustrate the benefits ofimplementing such a collaboration''". They proposedthat carriers exchange freight at logistics platformslocated at the entry point to a city[7l . This is referred toas entry-point collaboration. For the collaborationproblem, they presented an integrated three-phasesolution method. The first phase solves thecorresponding vehicle routing problems with timewindows resulted from entry-point collaboration. Thesecond phase is to locate facilities. The third phase is tobuild collaborative routes. Computational tests weredone on randomly generated data sets. Overall resultsshow the effectiveness of their method. However, tobest of our knowledge, they didn't give a generalmathematical model for the less than truckload carriercollaboration problem.

In this paper, we will present a generalmathematical model for collaborative logistics in lessthan truckload transportation. The model is ageneralization of the lane covering model forcollaborative logistics in truckload transportation.Except for the variables representing the number oftimes that each arc is visited by vehicles, our model alsocontains the flow variables representing the quantity ofeach product transported through each arc. A method isthen developed for constructing from the model'soptimal solution a set of feasible vehicle tourscorresponding to the transportation operations incollaborative logistics..

Note that although we mentioned full truckloadand less than truckload, the "truck" in our study maymean van or other types of vehicles. So the two terms"truck" and "vehicle" are exchangeable hereafter.

The rest of this paper is organized as follows.Section 2 describes the problem definition and outlinesour solution methodology. Section 3 presents amathematical model with a modeling example. Section4 presents a method for constructing the set of feasiblevehicle tours for this model. Section 5 proposes aLagrangian relaxation approach to solve this model.Section 6 demonstrates the validity of our proposedmethods by 10 randomly generated examples. Section 7concludes the paper with some remarks.

2. PROBLEM DEFINITION

We consider the collaborative logistics problem inless than truckload transportation as follows.

Multi-shippers and/or multi-carriers are involvedin a transportation network. Each shipper has one ormultiple products to be delivered to its clients with agiven delivery quality for each product in each deliveryinterval. Before the collaboration, each shipper has itsown carrieres) (one or more) to execute its deliverytasks, and each carrier has its individual and optimaltransportation tours to realize its all deliveries. After thecollaboration, all involved shippers and carriers form acollaborative alliance which puts all their delivery tasksand all their vehicle capacities in common. Thetransportation mode in the alliance is less than truckload.The primary objective of this alliance is to find a globaloptimal solution represented by a set of feasible vehicletours so as to minimize the total transportation cost ofthe whole alliance while satisfying the delivery quantityof each product for all customers. The secondaryobjective is to design a fair cost and profit allocationmechanism for the enterprises in the alliance so that thecollaborative logistics among them is possible. Since thetwo objectives can be considered successively, in thispaper we only consider the primary objective with thesecondary objective to be addressed in the futureresearch. In order to satisfy the delivery quantities ofmultiple customers simultaneously in the same tour,carriers could pick-up and delivery products during theirtours both in collaboration case and in non-collaborationcase. For simplicity, we assume that the method forcalculating the transportation cost is the same for eachcarrier. The cost depends on the distance traveled byeach vehicle. In addition, we assume all carriers employvehicles of the same capacity.

From the above discussion, the collaborativelogistics problem with less than truckload transportationmode can be more formally stated as follows.

Given a transportation network represented by adirected graph D = (N, A) with node set N and arc set

A. Each arc is associated with a transportation costwhich linearly depends on the length of the arc. Given aset of transportation tasks represented by a set ofpick-up and delivery pairs. The objective is to find anoptimal vehicle tour set T that satisfies the pick-up anddelivery requirements of all shippers and customers soas to minimize the total transportation cost of the wholealliance, where each tour is a simple cycle composed ofa set of arcs in the network.

A simple example is first designed to help theunderstanding of the collaborative logistics problem andits mathematical model given in Section 3. Consider atransportation network with 8 nodes and 28 arcs. Fourenterprises are involved in the network. They are twoshippers, A and B; and two carriers, C and D. Before thecollaboration, A is served by C, B is served by D, and Cand D have their individual and optimal transportationtours (cycles). Node 1 is the depot of A, node 8 is thedepot of B, and other nodes are their customers. Thearcs represented by thin lines belong to thetransportation routes (tasks) of C, the arcs represented

768

(I)

(2)

(3)

(4)

K

Iq~ ~ C,xij,i,j = I, ...,N,k =l

N N

Z=MinL L <:»,i=l j= l,j~i

{ ,~"q;-,~:q;,=,tu;-,tu;"k -I,..., K ,l -I, ..., N,

Xij ~ O'Xij E Z ,i,j = I, , Nil « j , (5)

q~ ~ O, q~ ER,k=I, ,K,i,j=I, ...,N,i;;,t.j,(6)

The objective function (I) represents the totaltransportation cost. Constraints (2) ensure that thenumber of vehicles leaving from a node is equal to ofthe number of vehicles arriving at the node. Constraints(3) are the vehicle capacity constraints. Constraints (4)are the product flow conservation equations, assuringthe flow balance ofeach product at each node.

If all carriers transport only with full truckloadmode, that is, each vehicle carries only one product andtransport it from one node to another node directly, then

q~ = u~ for any k; i, j with i ;;,t. j. It follows that

constraints (4) always hold and can be removed frommodel P.

In addition, if we define L ~ A as a set of arcscontaining all the full truckload transportation lanes,then L can be described as following.

Subject to:N N

I X ij = I xji ,i =I, ...,N,j = I,}"#; j = I,}"#;

k = I,. . . , K product index, where K represent Kdifferent types of productsParametersC vehicle capacity

c ij shipping cost from node i to j for each vehicle

where cij = cj i

and the triangle inequality,

Cik + ckj ~ cij , holds for any i, i. k with

k ;;,t.i,k;;,t. j

u~ quantity of product k delivered from node i to node

j , the delivery may be realized directly and/or

indirectly via the other nodes. That is, u~ units of

product k are picked up at node i and are deliveredto node j by a single vehicle tour or multiplevehicle tours. The tour or the tours may traveldirectly from node i to node j or indirectly from theformer node to the latter node via other nodes.

Variables

q~ quantity of product k transported through arc (i, j)

x ij number of times that arc (i,j) is visited by vehicles

With the notation, a mathematical model, denotedby P, is given as (I).Model P:

B 8

- Carriere- Carrier O

6

Figure I Transportation network and tasksFor this example, the shipping cost between two

nodes is given by matrix MJ, and the quantity of eachproduct transported from one node to another directlyand indirectly (via the other nodes) is given by matrixM2• All the diagonal elements of the matrixes are set 0and an alphabet a, b, c or d is added to the subscript ofeach non-zero element of M2 to represent the producttype involved in the transportation. From figure I andmatrix M2, we know that before the collaboration carrierA delivers 5 units of product a from node I to node 2, 5units of product b from node 2 to node 3, ... ; carrier Bdelivers 5 units of product c from node 8 to node 5, 5units of product d from node 5 to node 3, .... It isassumed that the capacity ofeach vehicle is 10.

o 3 3 3 3 7 7.5 5

3 0 I 5 5.5 8.5 9.5 7.5

3 I 0 4.5 5 8 9 7

3 5 4.5 0 I 5 6 4.5M1 = 3 5.5 5 I 0 4.5 6 4

7 8.5 8 5 4.5 0 2 3

7.5 9.5 9 6 6 2 0 3

5 7.5 7 4.5 4 3 3 0

o 5" 0 5" 0 0 5" 0

o 0 5' 0 0 0 0 0

00000000

o 0 0 0 5' 0 0 0M =

005dOOOOO

o 0 0 0 0 0 5d 0

o 0 0 0 0 5' 0 0

o 0 0 0 5' 5' 0 0

In this section , a mathematical model is proposedto formulate the collaborative logistics problem definedin the last section.

3. MATHEMATICAL MODEL

The notation used in the model is given as follows.Indicesi, j = I,. . . , N node index, where N represent thenumber of nodes in the transportation network. Thenodes include the customers, the product depots of theshippers and the vehicle depots of the carriers.

3.1 Model Formulation

by thick lines belong to the transportation routes (tasks)of D. The number associated with each arc represents itstransportation cost. Both C and D transport two differenttypes of products in this example. Figure I gives therelevant information.

769

K

xij ~ L(Iu~ IC l),xij E Z, V(i,j) E L, (10)k=l

k =1, ...,K,i,j =1, ...,N,Consequently, the original problem can be

transformed into the problem defined as following:

Xij ~O,xij EZ,V(i,j)EA\L, (11)

This is Lane Covering Problem (LCP) [21. So LCPis a special case ofour model.

Subject to:N N

L xij = L xji,i=l, ...,N, (9)j=l,j"#i j=l,j"#i

Finally, we consider the collaboration of the fourenterprises with less than truckload transportation modeby applying our model P. The objective is also tominimize the transportation cost of the whole alliance.We solve it by using Cplex, the optimal transportationcost is 29.5 for the whole alliance. So the collaborationwith LTL can reduce the total transportation costsignificantly.

Table 1 summarizes the costs of each carrier andtheir total costs in different collaboration models for theexample. Because of page limitation, the optimalsolution of each model is not given here.

1: bl 1 t . b 11 b tive lozi tia e cos savings )y co a ora rve OgIS ICSWithout collaboration With collaboration

Carrier C I Carrier D FTL LTL25.5 I 22

47 29.547.5

(7)

(8)N N

Z = MinL L cij· xiji=l j=l,j"#i

L={(i,J)lu~>O},V(i,J)EA

3.2 Model comparison4. CONSTRUCTION OF FEASIBLE VEHICLE

TOURS

5.1 Relaxation framework

5. MODEL SOLUTION APPROACH

For mixed integer programming model P, the

constraints coupling integer variables xi) and real

To solve model P, we propose a Lagrangianrelaxation based solution method in this section.Lagrangian relaxation method has been applied to solvemany hard integer programming problems!'?',

are constraints (3), however, thevariables

By solving model P in Section 3, xi) and q~ are

obtained. They can be used to construct a feasiblesolution of the original collaborative logistics problem,namely, a set of feasible vehicle tours. The constructionis divided into two steps, i.e., cycle (tour) generationand freight allocation.

Cycle generation is to generate a set of cycles forvehicles, each cycle is composed of some arcs under thecondition that the number of times that each arc (i, j) is

included in all cycles must be equal to xi).

After cycle generation, we allocate the flow q~

of each arc (i, j) to each cycle while satisfying thevehicle capacity constraints and the flow balanceconstraints at each node.

Under an assumption that vehicles can exchangetheir freight at each node they commonly visit, the twostep construction of feasible vehicle tours is alwaysfeasible. Here "exchange of freight among vehicles"means that when multiple vehicles arrive at a node, eachvehicle can unload part of its freight and/or load part ofthe freight unloaded by other vehicles, under thecondition that the flow balance constraint and thevehicle capacity constraint are satisfied.

We use the example presented in Section 2 todemonstrate the cost savings that can be achieved bycollaborative logistics. Initially, without collaboration, Ctransports products a and b, and D transports products cand d; C and D transport products using their individualoptimal transportation plans (tours) with less thantruckload transportation mode. The related problems todetermine the two optimal transportation plansrespectively can be formulated by our proposed model P.They can then be solved by Cplex. The optimalobjective value of the model, namely, the transportationcost, is 25.5 for (the problem of) C and 22 for (theproblem of) D, and the total cost is 47.5.

Next, we consider the collaboration of the fourenterprises with full truckload transportation mode. Theobjective is to minimize the transportation cost of thewhole alliance. This is Lane Covering Problem whichhas been described by (7) to (11). We also solve thisproblem by Cplex, the optimal transportation cost is 47for the whole alliance.

To show the novelty of our model, we compare it(referred to as CLLTL, the collaborative logisticsproblem with LTL transportation) with VRPPD (vehiclerouting problem with pick-up and delivery) models[S,9J:

(1) In a VRPPD model, each vehicle leaves from adepot and returns to the depot after performingpick-up/delivery tasks. In our model, each vehicle canstart from any node and return to this node.

(2) In most VRPPD models, it is assumed thatthere is no split delivery because it makes the problemfar more complicated. But in our model, goods can betransported to each customer in multiple vehicle routeswith split delivery.

(3) A general VRPPD model doesn't consider theinterchange of goods among vehicles (vehicle routes)during their travels. In our model, the interchange ofgoods is allowed at some pickup/delivery points.

3.3 Cost savings by collaborative logistics

770

Subject to constraints (2), (5), and

-. ~rtqi~ Ielq~iSgiVen, i,j=l, ...,N,i*j, (22)

Subject to constraints (3), (4), (6), (14), (16).Subproblem P":

N N

Z" =Min"'" "'" A.. · X'.. (20)L,; L,; lJ lJi=l j=l,j"#i

relaxation of (3) by using Lagrange multipliers leads topoor results. For this reason, we first reformulate the

model by introducing integer variables x'i)' relaxing

integer variables xi) to real variables, and equivalently

replacing constraints (3) and (5) by six constraints (12),(13), (14), (15), (16) and (17).

(26)

2. The sum of the value of each integer variablei=l j=l,j"#i

Let Zm denotes the lower bound obtained in

iteration m. In equation (25), initially, () is evaluated

as 1; then In iteration m, if Zm > Z,~, '

() = Max (1, () -1), otherwise, () = () +1. Parameters

wand p aretakenas wE[0.1,1.0], pE[I.1,1.5].

5.2 Lagrange multipliers updating rules

The optimal value of Lagrange multipliers A canbe found by solving the corresponding Lagrangian dualproblem. Because the relaxed problem is onlyapproximately solved, the subgradient method forupdating Lagrange multipliers cannot be used in solvingthe dual problem here. Instead, we use the surrogatesubgradient (SSG) method[ll] to update the multipliers.

Given an initial value ,10

, the value of Lagrangianmultipliers A in the m-th iteration of the solution process

of Lagrangian relaxation, denoted by Am, is generatedby the equation (23).t.=Max{~; ": .(X'ij-xij)' o} (23)

Z,j -1, ...,N,z 7= j,

In equation (23), xi) and x'ij are obtained by

solving P' and P" at the m-th iteration; tm is a

positive scalar step size, it is generated by equation (24).

tm =p.(D* -Z:)/IIX'ij - Xij lr (24)

In equation (24), fJ is a scalar satisfying

o< fJ < 1, Z,~, is the best (surrogate) lower bound

prior to iteration m, D * is estimated by equation (25).

D* ={I+m/BP).Z: (25)

The integer programming subproblem P" can beregarded as a quasi-network-flow problem with networkflow constraints (15) and an additional constraint (13)with xi} given. Normally, it is easy to solve if its problemsize is not too large. However, when the size of thesubproblem is very large, its resolution may becometime consuming. So in order to accelerate the resolution,we need to add some valid constraints to thesubproblem to reduce its solution space.

1. Let N* denote the number of nodes whichhave a pick-up or delivery task. The sum of the values

of integer variables xi} must be equal to or more than

N* . This constraint can be added to subproblem P".

By solving problem P "', we can get a feasiblesolution and an upper bound to the original problem.

5.3 Valid constraints for the relaxed subproblems

(13)

(14)

(21)

(12)

(15)

i=l j=l,jot:-ii=l j=l,jot:-i

N N

I »; = I X'ji,i =1, ...,N,j=l,j*i j=l,j*i

N N

I Xij = I xji,i =1, ...,N,j=l,I*i j=l,I*i

N N

Z'" = MinL L cij ·xiji=l j=l,j"#i

»', :::; xi),i,} = 1, ...,N,i 7=}

N N N N

I I x'ij~I I Xij

Subject to constraint (13), (15), and (17).

Because of variables xi) appeared in constraint

(13), the two subproblems are not independent. Todecouple them, we solve the two subproblemssequentially (thus solve the relaxed problem

approximately) with xi) in P" is evaluated by xi)

in the solution of subproblem P'.

Thirdly, after solving P' and P", xi) , x'ij and

q~ are obtained, by substituting q~ into constraints (3)

of model P, we can get the following network flowproblem P"'.

Xi) ~O'Xi) ER,i,}=l, ....Nsi .«], (16)

, > 0 ' Z·' -1 N·· (17)Xij_,XijE ,l,j-, ... , ,l*j,

Secondly, we relax constraints (12) by introducing

Lagrange multipliers .,1= {Aij ~ O,i,} =1,...,N,}, then

the relaxed problem can be formulated as (18):

z* =Min{t j~#Cij ·xij +t j~#~j .(Xlij-Xij)} (18)Subject to:Constraints (3), (4), (6), (13), (14), (15), (16), (17).Consequently, it can be decomposed into a linear

programming subproblem P' and an integerprogramming subproblem P".

Subproblem P':N N

Z' =MinL L (cij -Aij ).xij (19)i=l j=l,j*i

771

K

4. If L q~ > 0 In the solution of subproblem

Xi) multiplied by the shipping cost cij must be equal to or

less than the best upper bound found prior to iteration m,

denoted by Z:;. This constraint can also be added to

subproblem P".

3. In a special case, when each arc is visited byvehicles at most once in the original problem, the

constraints that the value of each integer variable x.. isIj

equal to or less than 1 can be added to subproblems P'and P".

N N

L L < ,xij ~ Z:*i=l j=l,ji:-i

(27)

The collaborative logistics problem with less thantruckload transportation mode has been studied in thispaper. To solve this problem, we have proposed ageneral mathematical model, which is suitable for bothshipper collaboration and carrier collaboration. ALagrangian relaxation approach is developed to solvethis model. The solution of the model can be used toconstruct a set of feasible vehicle tours corresponding tothe transportation operations in collaborative logistics.Ten randomly generated examples are tested todemonstrate our proposed model and method. In ourfuture work, we will try to find more valid constraintsfor the model to reduce its solution space and toaccelerate its resolution.

k=!

6. COMPUTATIONAL EXPERIMENTS

7. CONCLUSIONS

[1] Ergun, 6., Kuyzu, G., Savelsbergh, M., "Shippercollaboration", Computers & Operations Research,Vol. 34, No.6, ppI551-1560, 2007

[2] Solistics solutions Inc. Irvine, California, USA."Freight Terms", www.solistics.com/Documents/Freight%20Terms.pdf

[3] Ergun, 6., Kuyzu, G., Savelsbergh, M., "The lanecovering problem", http.z/wwwz.isye.gatech.edu/vmwps/publications/ijoc030404.pdf, 2003

[4] Ergun, 6., Kuyzu, G., Savelsbergh, M., "Reducingtruckload transportation costs through collaboration",Transportation Science, Vol. 41, No.2, pp206-221 ,2007

[5] Ozener, 0.6., Ergun, 6., Savelsbergh, M.,"Collaboration for truckload carriers",Transportation Science, submitted

[6] Nadarajah, S., Bookbinder, J.H., "Enhancingtransportation efficiencies through carriercollaboration". The 2007 Business Process CouncilWorldConference Proceedings, Mumbai, India

[7] Nadarajah, S., "Collaborative Logistics in VehicleRouting", Master of Applied Science, University ofWaterloo, Waterloo, Canada, 2008

[8] Savelsbergh, M., Sol, M., "The General Pickup andDelivery Problem". Transportation Science, Vol. 29,No.2, ppI7-29, 1995

[9] Parragh, S.N., Doerner, K.F., Hartl, R.F., "A surveyon pickup and delivery problems Part I:Transportation between customers and depot".Journal fur Betriebswirtschaft, Vol. 58, No.2,pp21-5I,2008

[10] Fisher, M.L., "The Lagrangian Relaxation methodfor solving integer programming problems".Management Science, Vol. 50, No. 12, ppI861-187I,2004

[11] Yu, Y.G., Chen, H.X., Chu, F., "A new model andhybrid approach for large scale inventory routingproblems". European Journal of OperationalResearch, Vol. 189, No. 3, ppI022-I040, 2008

REFERENCES

a e esu so e exampes WI no esExample 1 2 3 4 5

FTL 3150.1 3261.9 2754.8 3288.5 3568.6Cplex 1313.2 1145.7 806.7 1401 1963.4

LR 717 705.5 650.1 731.6 828.7Time 3004.9 3086 3061 3482.3 3256.5

Example 6 7 8 9 10FTL 4513.5 3434.8 3870.1 4920.9 3898.3

Cplex 1358.8 833.2 988.8 1545.6 1294.5LR 865.5 813.8 845.6 983.5 844.1

Time 3167.2 3090.2 2969.5 3137.2 3090

Because it is very time consuming to find anoptimal solution for each example by using Cplex andthe lower bound found by our proposed method is notan actual lower of our model, we only compare theperformance of our method with that of Cplex run in alimited computation time of 3600 seconds (one hour).From the table, we can see that the upper bounds foundby our proposed method are better than the optimalcosts of FTL and those found by directly solving themodel P by using Cplex with the time limitation.

P' in the current iteration, the following constraintsare added to subproblem P' in the next iteration with

X'i) evaluated as the solution of subproblem P".

xij ~ x'ij,i,j =I, ...,N,i 7:- j, (28)

Note that constraints (28) are not strictly valid forthe original problem but effective in improving theperformance of the Lagrangian relaxation method.

In this section, we design some computationalexperiments to evaluate the performance of the modeland solution method proposed in this paper. Werandomly generate 10 examples with 50 nodes and thefreights delivered on each arc less than 3. ParametersfJ, OJ and p are taken as 0.5, 0.6 and 1.3,

respectively. The results are given in table 2.1: bI 2 R It f th 10 I'th 50 d

772