AllocatingCosttoFreightCarriersinHorizontalLogistic...

Research ArticleAllocating Cost to Freight Carriers in Horizontal LogisticCollaborative Transportation Planning on LeadingCompany Perspective

Ning Liu 12 and Yaorong Cheng1

1School of Traffic and Transportation Engineering Central South University Changsha 410083 Hunan China2Hunan Modern Logistics College Changsha 410131 Hunan China

Correspondence should be addressed to Ning Liu nick_liu1982126com

Received 1 May 2020 Accepted 9 July 2020 Published 26 July 2020

Academic Editor Haipeng Peng

Copyright copy 2020 Ning Liu and Yaorong Cheng is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

Interest has been raised by the recent identification of cooperation cost through collaborative planning in horizontal logisticsoperations Even though cooperation cost can be realized one key question exists how should cooperation cost be divided amonga group of collaborating companies In this article the question is studied in a centralized framework context We divide theparticipants into two groups leading companies (LC) and nonleading companies (NLC) and propose five fair distribution rulesfrom the perspective of leading companies According to these distribution principles we developed an allocation method calledLeading-idealism Cost Allocation Model (LiCAM) and compared it with three existing classic allocation mechanisms whichviolate some of these criteria are discussed Computational results show that our method has acceptable calculation time stabilityconsistency andmonotony Our model can fully reflect the value as a leading company which is consistent with the actual practicerequirements We also illustrate the value and operability of our model by discussing the number of leading companies and thesize of the alliance

1 Introduction

With the rapid development of the world economy and theadvancement of science and technology the modern lo-gistics industry has become an important part of thebooming modern economy worldwide [1] Due to theglobalization and network characteristics of the logisticsindustry it is difficult for any logistics company to cover allareas and to efficiently undertake a variety of transportationmodes at the same time so the cooperation between logisticscompanies is continuously deepening ere are generallytwo forms of cooperation namely horizontal cooperationand vertical cooperation e European Union [2] defineshorizontal cooperation as ldquoan agreement or concertedpractice (y) between companies operating at the same level(s) in themarketrdquo Cooperating companies can compete witheach other or not But they should perform the same type of

activities andor services rather than complementary ac-tivities andor services which is related to vertical coop-eration A definition of horizontal cooperation based onlogistics was proposed by Cruijssen et al [3] He concludedthat horizontal logistics are two or more firms performing acomparable logistics function at the same level of the supplychain on the landside

Horizontal logistic is an effective way to improve logisticoperations ere are a number of papers reporting onhorizontal logistic transportation studies within specificcontexts such cooperative planning in express carrier net-works [4] collaborative transportation planning of less-than-truckload freight [5] specific possibility for horizontalcooperation by planning linked deliveries [6] forest fueltransportation collaboration in Sweden [7] and requestselection and exchange approach for carrier collaborationbased on auction of a single request [8]

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 4504086 10 pageshttpsdoiorg10115520204504086

In order to realize cooperation in practice two methodscan be used to solve the problem of cooperative logisticscentralized method and decentralized method In the cen-tralized method multiple carriers reach an agreement andform a cooperative alliance with the coordinator responsiblefor making cooperative transportation plans for them ecoordinator redistributed the alliancersquos transportation re-quest to all carriers as a way to maximize the total profit ofthe alliance under the restriction that each request is allo-cated to at most one carrier As for decentralized methodsthe framework involves two types of participants namelyauctioneers and bidders (carriers) Among them the auc-tioneerrsquos function is to determine and update the price ofeach service request and to maximize the total profit Eachbidder chooses its best requirements according to the priceproposed by the auctioneer to maximize its personal profite centralized framework can usually bring more profits tothe alliance than the total profits generated by the decen-tralized framework [9] while the decentralized methodsoffer carrier more autonomy

e cooperative gaming method as the mainstreammethod can be used to solve the problem of cost allocationIn recent years this issue has caused widespread concernCooperative game theory has some well-known solutionsconcepts such as the core theory the Shapley value theoryproposed by Shapley [10] and the nucleolus theory putforward by Schmeidler [11] e core theory requires therationality of individuals groups and alliances and theseattributes are the basis for cost allocation in most literatureHowever two drawbacks are visible because core is notunique and it will probably be empty e Shapley valuemethod is used in many literatures to conduct cost allocationor profit sharing ([12 13]) e nucleolus cost-allocatingmethod to solve routing problems also was studied in manyarticles eg Yin et al [14] e scholars discuss some newmethods based on the classic solution concept Frisk et al[15] studied the issue of forest transportation cooperativeplanning and proposed the EPM (equal profit method)Audy et al [16] proposed four business models to study oncoalition formation and costsavings allocation Naber et al[17] developed four emission allocation methods Sun et al[18] considered a contribution constrained packing modelfor cost allocation Liu and Cheng [19] developed LCAM(location cost allocation model) which take geographicallocation factor into account

2 Problem Descriptions

In practice common scenarios for horizontal logistics col-laborative transportation planning can be described asfollows e Figure 1 shows a typical non-cooperativescenario where each freight carrier designs its own set ofroutes to deliver its own customers In contrast the Figure 2shows the same routing problem in a cooperative scenariowhere each distribution unit is reasonably assigned to itsadjacent logistics facility It also realizes to improve actualutilization of vehicles during a roundtrip and reduce thecrisscross transportation phenomenon in complex trans-portation networksWe can find practical examples from the

article Krajewska et al [20] Frisk et al [15] Perez-Bernabeuet al [21] and so on Obviously prior to horizontal co-operation each carrier had an independent cost Afterhorizontal cooperation all carriers formed an alliance tocomplete the task together which resulted in lower totalcost

In this article it leads to a lower cost for the alliance as awhole using the centralized collaboration framework Underthis framework there are two roles in the alliance one isleading company (s) (LC) and the other is participant (NLC)e responsibility of the leading company is to developparticipants in order to form an alliance collect informationabout supply and demand formulate a feasible operationplan and make each participating company put the coop-eration plan into practice Two major problems can besummarized to be solved One problem is the collaborativeplanning e problem to achieve the lowest overall cost forthe carriers has been studied in many literatures includingbut not limited to vehicle routing optimization or inventoryrouting problems Another problem lies in the equitableallocation of the total cost amount to each carrier concernedleading to a lower cost for each carrier as a result of thecooperation Cruijssen et al [22] and Leitner et al [23] bothpoint to fair cost allocation as one of the most importantobstacles to horizontal logistics e paper emphaticallyprobes into the second problem More details on the allo-cation rule are provided in Section 4 Our concept of modelallows allocating cost to freight carriers on leading companyperspective according to the specific context of such acollaborative organization

3 Allocation Methods

31CoreAllocation e concept of core has some of themostfrequently used properties A participantrsquos subset is denotedwith coalition S and all participants are denoted by grandcoalition N Assume that every participant has the chance togenerate and cooperate in coalitions In the case of cooperationof coalition S we can generate the common cost C(S)According to the cooperative game theory such function ofcost is referred to as characteristic cost function and everyparticipant is known as a playererefore the problem of costallocation can be considered as a cooperation game

A cost allocation method is applied to the separation oftotal cost ie C(N) Among the participants j isin N is said tobe efficient that is 1113936jisinNyj C(N) in which yj denotes thecost which is allocated to the participant j A cost allocation canbe considered as individual rational if there is no participantwhich makes a payment higher than the ldquostand alone costrdquo(that is the cost of the participant) when no coalition isgenerated If we put it mathematically we can express theproperty as yj leC( j1113864 1113865) e core of the game is identified asthose who meet the requirements of the cost allocation It isconsidered that the cost allocation at the core is stable

yj leC( j1113864 1113865) (1)

1113944jisinN

yj C(N) (2)

2 Mathematical Problems in Engineering

32 ampe Shapley Value e Shapley value is a concept ofsolution with which we can have one and only solution tothe problem of cost allocation According to the formulabelow for calculation it is assumed that only one participantj is accessible to the grand coalition each time As the co-alition is entered by every participant the marginal cost isallocated which suggests that through such entry the totalcoalition cost is increased e amount received by partic-ipants in this program depends on the order in whichparticipants joined e cost assigned to participant j isequal

yj 1113944S sub NjisinS

(|S| minus 1)(|N| minus |S|)

|N|1113890c(S) minus c(S minus j1113864 1113865)1113891

(3)

where | middot | shows the number of participants in the coalitionconsidered e sum of this formula is equal to that over allcoalitions S that include the participant je value of c(S) minus

c(S minus j1113864 1113865) refers to the increase of the cost of the alliancewith the increased the participant j which is expressed bythe marginal cost of participant j regarding the coalition

(a) (b)

Figure 1 Non-cooperative scenario each provider delivers its own customers

(a) (b)

Figure 2 Cooperative scenario each customer is delivered by the closest provider

Mathematical Problems in Engineering 3

S minus j1113864 1113865 With Shapley value we can obtain one and only costallocation Nevertheless it cannot be guaranteed that it isstable For example the individual rationality is not satisfiednecessarily

33 ampe Nucleolus e nucleolus refers to the allocation xand it is individually rational and efficient for this reason inall allocations f (x) is a lexicographic minimum As reportedby Schmeidler [11] a coalition excess shows the ldquoattituderdquothat the coalition has towards such allocation while thenucleolus could be considered as the allocation which ismost accepted e nucleolus is located in the core and it isunique if the core is not empty

To identify the nucleolus a method adopted by Engevallet al [24] is used here With the method different problemsof linear programming (LP) are solved one by one First anallocation is identified by having the smallest excessesmaximized It should be noted that such excesses will benegative in the case of empty core For the allocation ob-tained if it is not a unique allocation then have the obtainedexcesses fixed for every coalition whose dual variables arepositive Based on the fixed excess of the coalitions for therest of the coalitions the smallest excesses are maximizede procedure is repeated until we find one and only so-lution to the problem of LP e one and only allocation isthe so-called nucleolus

34 Nash Bargaining Solution e solution of Nash bar-gaining is a very powerful instrument when the negotiatorinteractions are modeled and for bargaining games it is oneand only solution that meets the conditions of scale inde-pendence symmetry Pareto optimality and independentirrelevant alternatives According to past researches [25] theset including all individuals are denoted as N 1 2 n a vector of payoff refers to an factor of the payoff space RNand a feasible set S refers to a subset of the payoff space abreakdown point b refers to an factor of the payoff spaceerefore the problem of bargaining problem can bespecified as the triple (N S b) e solution is based on afunction which is related to every bargaining problem(N S b) and it can be expressed as F(N S b) where such afeasible set is convex and compact it includes several vectorsof payoff and then every payoff of individual is higher thanthe breakdown payoff of the individual in the case thesolution ensures that the four properties of Nash is uniqueand it can be worked out by solving the problem of max-imization as follows

Max1113945n

i1ui minus bi( 1113857 (4)

where ui refers to the payment vector of individual which isassigned by the cooperation and bi refers to the independentpayoff of individuals Make sure that the income distributedis equal to the general income

4 Leading-Idealism Cost AllocationModel (LiCAM)

41 Allocation Rule of Model ere is no single and all-purpose method to achieve cost allocation e cooperativegame theory provides a set of ideal attributes for the costallocation method among a group of participants When itcomes to choose an existing method or developing a newmethod we will look for a method that satisfies certainproperties that are considered essential in our context In thecontext of our cooperative organization the allocation rulesof the allocation model must satisfy five of these propertiesFirst the property of efficiency requires that the commoncost of a coalition must be entirely split among its playersSecond the property of coalition rationality is the strongeststability condition and implies individual rationality andgroup rationality e third property is a reinforced indi-vidual rationality the ratio of cost allocated to NLC tocoalition that include the NLC and LC should be less thanthe ratio of NLCrsquos independent cost to the NLC and LC totalindependent cost Fourth the monotonic property requiresthat the cost of a collaborating player does not increase witha new player to join in the alliance We also propose a fifthproperty called it leader allocation advantage which is LCshould be allocated less cost Because under the centralizedframework the LC acts as a coordinator and it has to bearmore work and invests a lot of alliance establishment andmanagement costs So in every coordinated transportationtask it should bear less operating costs Some authors havestudied the establishment cost and management cost in theliterature on alliance formation such as the connecting costis used to study establishment cost of alliance by Slikker et al[26] and Galeotti et al [27] While some other authors (egAudy et al [28]) use fixed unit costs to study the man-agement cost

42 AllocationModel e formulation of LiCAM is in fact amultistage model Let us take a look at the first stage Webuild the following linear programming model to achievexlowasti e model is as follows

P1 max 1113944iisinNLC

xi

c( i )minus 1113944

iisinLC

xi

c( i ) (5)

st 1113944iisinN

xi c(N) (6)

1113944iisinS

xi le c(S) (7)

xi le c(M) timesc( i )

1113936jisinMc( j1113864 1113865) i isin NLC (8)

xi ge 0 (9)Objective function (5) guarantees that LC is allocated

less cost and it corresponds to the leader allocation ad-vantage property Constraint (6) corresponds to the efficiency


condition which states total cost should be completely al-located Constraint (7) corresponds to the coalition ratio-nality conditions Constraint (8) demonstrates cost allocatedto NLC is related to coalition M and it corresponds to thereinforced individual rationality Constraint (9) demon-strates that being a collaborating player in a coalition mustbear a positive cost By solving model P1 we can get thevalue of xi denoted as xlowasti

e next step is the second stage At this time the al-liance has new participants to join in A new constraint mustbe added that is the cost (yi) of a collaborating player doesnot increase with a new player to join in the alliancee newconstraint (10) is as follows which corresponds to themonotonic property

yi lexlowasti i isin N (10)

is condition is too strict many times alliance leadingcompany will be willing to sacrifice some benefits in ex-change for other potential benefits (eg faster delivery timeto protect the companyrsquos market share and to broaden theirservices at the same time) and accept new members to joinConstraint (10) can be modified to the following constraint

yi le xlowasti i isin NLC (11)

ese formulas can be generalized in the following wayat iteration t+ 1 solve the following linear program

Model

max 1113944iisinNLCt+1

xt+1i

c( i )minus 1113944

iisinLCt+1

xt+1i

c( i ) (12)

st 1113944iisinNt

xt+1i c Nt+1( 1113857 (13)

1113944iisinSt+1

xt+1i le c St+1( 1113857 (14)

xt+1i le c Μt+1( 1113857 times

c( i )

1113936jisinMt+1c( j1113864 1113865)

i isin NLCt (15)

xt+1i lex

tlowast

i i isin NLCt (16)

xt+1i ge 0 (17)

If constraint (16) is xt+1i le xtlowast

i i isin Nt the model is calledthe strict model Model LiCAM is a linear programmingmodel which can be solved by many classical algorithmssuch as simplex algorithm ellipsoid method and Karmar-karrsquos algorithm e specific algorithm to be selected isdetermined by the software Relative indices sets param-eters and decision variables are illustrated in Table 1

5 Computational Results

51 Case Description e case study in this article is basedon horizontal cooperation among freight carriers A com-plete description of the case is found in Krajewska et al [20]ey have generated three instances for five carriers Each

carrier possesses one depot e vehicle fleet is unlimitedand homogenous for all carriers e instances are differentwith regard to the number of requests that each carrier has tofulfill For each instance all 31 possible coalitions consistingof one two three four and five carriers are assumedCoalition S the minimum total cost is cited in Table 2

In this section we present the results of applying theallocation methods to the cases We can evaluate the effect ofeach method and gain insight into the performance ofmethod in practice All results are calculated by Lingo11using an Intel (R) Core (TM) i5-5200220GHz with 8GBof RAM

By comparison with classic methods such as the Shapleynucleolus and Nash bargaining solution the aim is todemonstrate through simple and easily customized alloca-tion rules how the Leading-idealism Cost Allocation Modelcan affect cost allocation among the collaborating players

From Table 2 we further explain what the independentcost is and what the total independent cost is e inde-pendent cost is the cost of participant to complete tasksindependently e total independent cost includes NLCrsquosindependent cost and LCrsquos completion cost For example56037 is the independent cost of participant A 41569 is theindependent cost of participant B 45984 is the independentcost of participantC IfA is the LC and B is the NLC then thetotal independent cost is (56037 + 41569) for the allianceA B If A is the LC B is the LC and C is the NLC then thetotal independent cost is (85439 + 45984) for the allianceA B C

52 Allocation Result Analysis e allocation results ofdifferent coalition are calculated according to the methods ofthe Shapley nucleolus Nash and LiCAM proposed byarticle as shown in Tables 3ndash5 Here we assume that A is theleader of the alliance

521 Computation Time Among the four methods theNash method takes the longest time to calculate and themain reason is that it is a nonlinear programminge otherthree methods are very close e average time of eachmethod in the three cases is shown in Table 6

522 Stability Stable allocation is a concept from coop-erative game theory For such allocations no subset ofcarriers has an incentive to withdraw from the collaborativeplanning on the basis of the allocation As can be seen fromthe definition of LiCAM their solutions satisfy efficiencycondition corresponds to constraint (6) and coalition ra-tionality condition corresponds to constraint (7)e core ofgame is the set of all vectors satisfying constraints (6) and (7)An allocation that belongs to the core is said to be stableerefore the stability of LiCAM model is fully guaranteedin theory In the same way Nucleolus also guarantees itsstability by definition that is their solutions satisfy indi-vidual rationality and coalition rationality But the Shapleyvalue and Nash methods are not constructed by implicitlyusing any of the core criteria erefore from a stability


point of view the Shapley value and Nash methods performsworst However for each instance in Tables 3ndash5 all fourmechanisms develop a cost allocation that stays at the core

523 Consistency with the Leader Allocation AdvantageProperty In order to determine the consistency with theprinciple that the leading companyrsquos cost allocation is as small

as possible the calculation results are analyzed In cases 1 and2 as the number of alliance members continues to increase itcan be seen that the cost of leading company A calculated byLiCAMmethod is allocated the smallest cost in each phase ofthe alliance In case 3 only in the case of AB coalition thecost allocated to leading company A calculated by LiCAMmethod is a little more than that of the other methods

We use the following method to calculate averageconsistency e consistency is divided into 4 levels whichare scored 1 2 3 and 4 respectively e lowest cost al-location method for leading company A is 4 points thesecond lowest is 3 points and the highest is 1 point ecalculation formula is as follows

Average Consistency 1113936 1113936 pmethod

|C| times Sc

11138681113868111386811138681113868111386811138681113868

(18)

Table 1 Indices sets parameters and decision variables in the LiCAM

Indicest A stage in the coalition formation with t 1 2 |NLC|

SetsNt e set of all players at stage t Nt LCtcupNLCt

LCt e set of all leading player(s) at stage tNLCt e set of all nonleading players at stage t

ParametersC(Nt) Total cost is calculated by various multidepot optimization algorithms at stage tc( i ) e stand-alone cost of player i

c(St) e cost of coalition St

c(Mt) e cost of coalition Mt

St e coalition that forms between players at stage tMt e coalition that forms at stage t where Mt LCtcup it1113864 1113865 i isin NLCt

xtlowast

i e calculated cost value of player i by model P1Decision variable

xti Cost of player i allocated by alliance at stage t

Table 2 Coalition S the minimum total cost quoted from articleKrajewska et al [20]

Carriers in coalition S T1 T2 T3A 56037 74685 56037B 41569 63907 58906C 45984 72759 72759D 54069 75187 54069E 52369 72029 64922A B 85439 116388 93466A C 84338 128257 112239A D 98366 142367 98366A E 94903 130314 103831B C 7780 117857 111441B D 89431 128841 105371B E 83624 12053 101388C D 84884 130503 110128C E 90045 137717 130661D E 100282 146491 112176A B C 115884 169103 147745A B D 123946 178478 131799A B E 123381 170347 139713A C D 124854 188079 149843A C E 126716 185439 164463A D E 138274 197639 151815B C D 112811 169898 146287B C E 118797 176818 163352B D E 132056 184892 151256C D E 129101 194502 167688A B C D 152384 227316 187753A B C E 152628 227387 197554A B D E 165873 232983 182765A B D E 165732 245974 203378B C D E 154278 228309 197746A B C D E 189271 280053 234435

Table 3 Allocation results of T1

Coalition Shapley Nucleolus Nash LiCAMGrand coalition N A B

A 49954 49954 49954 49052B 35485 35485 35485 36387

Grand coalition N A BC

A 45078 43483 46802 41483B 34575 36558 32334 36387C 36232 35844 36749 38014

Grand coalition N A BC D

A 43827 44659 44718 39573B 32411 32616 30250 36387C 33244 33524 34665 38014D 42903 41586 42750 38410

Grand coalition N A BC D E

A 41421 41755 43886 34993B 30521 30301 29418 36387C 31770 30160 33833 38014D 42825 43405 41918 36643E 42734 43649 40218 43234


where pmethod is the consistency score obtained by the samemethod in every coalition in the case |C| is the number ofcases and |Sc| is the number of coalitions generated in thecase According to the above method it can be seen that theaverage consistency of LiCAM is the highest In terms offrequency the consistency illustrates the probability that the

LiCAMmethod will allow leading companies to be allocatedless cost e specific results are shown in Table 7

524 ampe Value of Being a Leading Company From theperspective of individual participants the value of being analliance leader is obvious We do the following test In theABCDEAlliance of Case 1 company Bwill save up to 3531if company B as the leader which also reflects the protectionof the leaderrsquos interests by this methode allocation resultsare shown in Table 8

From a holistic perspective the value of being a leader canbe reflected in the LiCAMmethod In case 1 the Nash methodhas the highest cost allocation to leading company A and thenucleolus method has the lowest LiCAM saves 1123 onaverage compared with the Nash method It also saved anaverage of 821 compared with the nucleolus method In Case2 the nucleolus method allocated the highest cost to leadingcompany A and the Shapley method has the lowest LiCAMsaved 897 on average comparedwith the nucleolusmethod Italso saved an average of 736 compared with the Shapleymethod In case 3 the nucleolus method has the highest costallocation to the leading company A and the Nash method hasthe lowest LiCAM saves 929 on average compared with thenucleolus method and also saves 691 on average comparedwith the Nashmethod In terms of quantity the value of being aleading company reflects how much LiCAM can let leadingcompanies allocate cost value less than other methods Table 9illustrates the average of the various methods in the three cases

525 ampe Number of Leading Companies e above dis-cussion is all about the situation where there is only oneleader in the alliance What if there are two three and fourleaders in the alliance Choose A A B A B C and AB C D as the leaders in three cases and the calculationresults are shown in Tables 10ndash12

When the number of leaders increases in fact the costallocation of the original leader will be more than theoriginal level ere is a conflict of interest Why did theoriginal leader need a new leader to join in Although costreduction is usually by far the most important of the po-tential benefits a question still remains what payoff is re-quired to incite a player to collaborate In practice this issueis much more complex ere exist several other potentialbenefits (eg faster delivery time to protect the companyrsquosmarket share and to broaden their services at the same time)than cost reduction to perform horizontal cooperation intransportation and logistics Several opportunities and ad-vantages to horizontal cooperation are also detailed in theliterature review of Cruijssen et al [22] An added benefit ofhorizontal cooperation is that not only cost savings can beobtained but also CO2 emissions can be reduced (eg Ballotand Fontane [29])


Coalition Shapley Nucleolus Nash LiCAMGrand coalition N A BA 63583 63583 63583 62719B 52805 52805 52805 53668

Grand coalition N A BCA 59973 59345 60602 52144B 49384 48945 49824 53668C 59745 60814 58676 63291

Grand coalition N A BC DA 60827 63320 59880 57418B 47187 45139 49102 53668C 55593 54740 57954 58017D 63709 64116 60382 58213

Grand coalition N A BC D EA 57574 59691 58982 51744B 44358 42026 48204 53668C 55237 56028 57056 57998D 62536 61624 59484 52666E 60348 60684 56326 63977

Table 5 Allocation results of T3Coalition Shapley Nucleolus Nash LiCAMGrand coalition N A BA 45299 45299 45299 45567B 48168 48168 48168 47899




Table 6 Computation timeMethod Shapley Nucleolus Nash LiCAMAverage (s) lt005 lt005 15 lt005

Table 7 Consistency with the leader allocation advantage propertyMethod Shapley Nucleolus Nash LiCAMAverage consistency () 6458 5208 5833 9792


526 Size of Grand Coalition e interesting result is thatadding more partners to the collaboration will reduce therevenue [6] e Shapley method and Nucleolus methodprove this When the number of participants increases itmay cause the cost of the existing alliance members to beallocated lower than the cost of the new alliance is will

cause alliance members to resist new members joining Infact this goes against the monotonic property discussed inSection 41 But both the Nash method and the LiCAMmethod ensure that the cost allocated tomembers by the newlarger alliance is still equal to or less than the cost allocated tothe original alliance as the members increase e originalmembers of the alliance will not resist the joining of newmembers e analysis results are shown in Table 13

If we use the strict model the results are shown in Ta-ble 14 Although the LiCAM method has increased the costof leaders as new members joined it must be seen that theinitial cost is much lower than other methods Even if thecost of the leader increases it is still much lower than the costallocated by other methods

From the calculation results although the later memberscan be guaranteed to be less than their own independentcosts they will undoubtedly be allocated more costs by thealliance is is also in line with the actual situation Whenthe members can no longer obtain less than their own in-dependent costs from the alliance the alliance membersreach the upper limit

6 Conclusion

In this paper we consider the cost allocation in horizontallogistic transportation planning Five fairness criteria areintroduced in this specific context And based on thesecriteria we propose the Leading-idealism Cost AllocationModel In our case study the calculation time of LiCAM isas good as that of Shapley and Nucleolus and the Nashmethod performs worst In consistent with leader

Table 8 e value of being a leading company for company B inthe ABCDE alliance of T1Allocation A B C D ECompany A as leader 34993 36387 38014 36643 43234Company B as leader 36631 23539 40862 41620 46619

Table 11 Cost allocation results for multiple leading companies inthe alliance (T2)Allocation A B C D ECompany A as theleader 51744 53668 57998 52666 63977

Companies A and B asthe leaders 51744 34079 65049 64060 65121

Companies A B and Cas the leaders 59753 34079 48335 69963 67923

Company A B C and Das the leaders 55065 34079 48335 75187 67387




Companies A B C andD as the leaders 36689 33091 67539 36881 60235

Table 13 e number of companies in grand coalition

CoalitionMethod

Shapley Nucleolus Nash LiCAMT1 T2 T3 T1 T2 T3 T1 T2 T3 T1 T2 T3

Twoparticipants radic radic radic radic radic radic radic radic radic radic radic radic

reeparticipants radic radic radic times radic radic radic radic radic radic radic radic

Fourparticipants radic radic radic radic radic radic radic radic radic radic radic radic

Fiveparticipants radic radic times times times times radic radic radic radic radic radic

Table 14 e number of companies in grand coalition in the strictmodel

CoalitionMethod




Fourparticipants radic times times times times times radic radic radic radic times times


Table 9 Comparison of different methods of value as a leadingcompanyValue of being a player T1 () T2 () T3 ()Shapley 9718 9787 9791Nucleolus 9698 9948 9985Nash 10000 9834 9747LiCAM 8877 9051 9056






allocation advantage property LiCAM performs bestwhich is 3334ndash4584 higher than that of the othermethods e value of alliance leaders is fully respectedreflecting the balance of interests and responsibilitiesFurther we analyzed the number of leaders in the allianceCalculation shows that too many leaders in the alliance willweaken the distribution of leadersrsquo interests Finally weanalyzed the size of the alliance When the members can nolonger obtain less than their own independent costs fromthe alliance the alliance members reach the upper limitOur proposed method ensures that the alliance remainsstable as the size of the alliance increases But the Shapleyand Nucleolus methods are not

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] A Jaiswal C Samuel and G A Ganesh ldquoPollution opti-misation study of logistics in SMEsrdquo Management of Envi-ronmental Quality An International Journal vol 30 no 4pp 731ndash750 2019

[2] European Union Commission notice guidelines on the ap-plicability of article 81 of the EC treaty to horizontal coop-eration agreements 2001

[3] F Cruijssen W Dullaert and H Fleuren ldquoHorizontal co-operation in transport and logistics a literature reviewrdquoTransportation Journal vol 46 no 3 pp 22ndash39 2007

[4] S Dahl and D Ulrich ldquoCooperative planning in expresscarrier networksmdashan empirical study on the effectiveness of areal-time Decision Support Systemrdquo Decision Support Sys-tems vol 51 no 3 pp 620ndash626 2011

[5] X Wang and H Kopfer ldquoCollaborative transportationplanning of less-than-truckload freightrdquo OR Spectrumvol 36 no 2 pp 357ndash380 2014

[6] B Adenso-Dıaz S Lozano S Garcia-Carbajal and K Smith-Miles ldquoAssessing partnership savings in horizontal cooper-ation by planning linked deliveriesrdquo Transportation ResearchPart A Policy and Practice vol 66 pp 268ndash279 2014

[7] P Flisberg M Frisk M Ronnqvist and M Guajardo ldquoPo-tential savings and cost allocations for forest fuel trans-portation in Sweden a country-wide studyrdquo Energy vol 85pp 353ndash365 2015

[8] J Li G Rong and Y Feng ldquoRequest selection and exchangeapproach for carrier collaboration based on auction of a singlerequestrdquo Transportation Research Part E Logistics andTransportation Review vol 84 pp 23ndash39 2015

[9] S Berger and C Bierwirth ldquoSolutions to the request reas-signment problem in collaborative carrier networksrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 627ndash638 2010

[10] L S Shapley ldquoA value for n-person gamesrdquo Annals ofMathematics Studies Princeton University Press PrincetonNY USA pp 307ndash317 1953

[11] D Schmeidler ldquoe nucleolus of a characteristic functiongamerdquo SIAM Journal of Applied Mathematics vol 17 no 6pp 1163ndash1170 1969

[12] A Kimms and I Kozeletskyi ldquoShapley-value-based cost al-location in the cooperative traveling salesman problem underrolling horizon planningrdquo EURO Journal on Transportationand Logistics vol 5 no 4 pp 371ndash392 2016

[13] H D Sherali and B J Lunday ldquoEquitable apportionment ofrailcars within a pooling agreement for shipping automo-bilesrdquo Transportation Research Part E Logistics and Trans-portation Review vol 47 no 2 pp 263ndash283 2011

[14] C Yin F Wang H Zou D Mcwilliams and L Bu ldquoNucleoluscost allocation of vehicle routing problem with split deliveryrdquo inProceedings of the Transportation Research Board 95th AnnualMeeting pp 35ndash42 Washington DC USA January 2016

[15] M Frisk M Gothe-Lundgren K Jornsten andM RonnqvistldquoCost allocation in collaborative forest transportationrdquo Eu-ropean Journal of Operational Research vol 205 no 2pp 448ndash458 2010

[16] J-F Audy S DrsquoAmours and M Ronnqvist ldquoAn empiricalstudy on coalition formation and costsavings allocationrdquoInternational Journal of Production Economics vol 136 no 1pp 13ndash27 2012

[17] S K Naber D A D Ree R Spliet and W V D HeuvelldquoAllocating CO2 emission to customers on a distributionrouterdquo Omega vol 54 pp 191ndash199 2015

[18] L Sun A Rangarajan M H Karwan and J M PintoldquoTransportation cost allocation on a fixed routerdquo Computersamp Industrial Engineering vol 83 pp 61ndash73 2015

[19] N Liu and Y Cheng ldquoAllocating cost to shippers in pickupand delivery servicerdquo Mathematical Problems in Engineeringvol 2019 Article ID 1283539 10 pages 2019

[20] M A Krajewska H Kopfer G Laporte S Ropke andG Zaccour ldquoHorizontal cooperation of freight carriers re-quest allocation and profit sharingrdquo Journal of the OperationalResearch Society vol 59 no 11 pp 1483ndash1491 2008

[21] E Perez-Bernabeu A A Juan J Faulin and B B BarriosldquoHorizontal cooperation in road transportation a case il-lustrating savings in distances and greenhouse gas emissionsrdquoInternational Transactions in Operational Research vol 22no 3 pp 585ndash606 2015

[22] F Cruijssen M Cools and W Dullaert ldquoHorizontal coop-eration in logistics opportunities and impedimentsrdquo Trans-portation Research Part E Logistics and TransportationReview vol 43 no 2 pp 129ndash142 2007

[23] R Leitner F Meizer M Prochazka and W Sihn ldquoStructuralconcepts for horizontal cooperation to increase efficiency inlogisticsrdquo CIRP Journal of Manufacturing Science and Tech-nology vol 4 no 3 pp 332ndash337 2011

[24] S Engevall M Gothe-Lundgren and P Varbrand ldquoetravelling salesman game an application of cost allocation in agas and oil companyrdquo Annals of Operations Research vol 82pp 203ndash218 1998

[25] J F Nash ldquoe bargaining problemrdquo Econometrica vol 18no 2 pp 155ndash162 1950

[26] M Slikker B Dutta A V D Nouweland and S Tijs ldquoPo-tential maximizers and network formationrdquo MathematicalSocial Sciences vol 39 no 1 pp 55ndash70 2000

[27] A Galeotti S Goyal and J Kamphorst ldquoNetwork formationwith heterogeneous playersrdquo Games and Economic Behaviorvol 54 no 2 pp 353ndash372 2006

[28] J-F Audy S DrsquoAmours and L-M Rousseau ldquoCost alloca-tion in the establishment of a collaborative transportationagreementmdashan application in the furniture industryrdquo Journal


of the Operational Research Society vol 62 no 6 pp 960ndash9702011

[29] E Ballot and F Fontane ldquoReducing transportation CO2emissions through pooling of supply networks perspectivesfrom a case study in French retail chainsrdquo ProductionPlanning amp Control vol 21 no 6 pp 640ndash650 2010


In order to realize cooperation in practice two methodscan be used to solve the problem of cooperative logisticscentralized method and decentralized method In the cen-tralized method multiple carriers reach an agreement andform a cooperative alliance with the coordinator responsiblefor making cooperative transportation plans for them ecoordinator redistributed the alliancersquos transportation re-quest to all carriers as a way to maximize the total profit ofthe alliance under the restriction that each request is allo-cated to at most one carrier As for decentralized methodsthe framework involves two types of participants namelyauctioneers and bidders (carriers) Among them the auc-tioneerrsquos function is to determine and update the price ofeach service request and to maximize the total profit Eachbidder chooses its best requirements according to the priceproposed by the auctioneer to maximize its personal profite centralized framework can usually bring more profits tothe alliance than the total profits generated by the decen-tralized framework [9] while the decentralized methodsoffer carrier more autonomy

e cooperative gaming method as the mainstreammethod can be used to solve the problem of cost allocationIn recent years this issue has caused widespread concernCooperative game theory has some well-known solutionsconcepts such as the core theory the Shapley value theoryproposed by Shapley [10] and the nucleolus theory putforward by Schmeidler [11] e core theory requires therationality of individuals groups and alliances and theseattributes are the basis for cost allocation in most literatureHowever two drawbacks are visible because core is notunique and it will probably be empty e Shapley valuemethod is used in many literatures to conduct cost allocationor profit sharing ([12 13]) e nucleolus cost-allocatingmethod to solve routing problems also was studied in manyarticles eg Yin et al [14] e scholars discuss some newmethods based on the classic solution concept Frisk et al[15] studied the issue of forest transportation cooperativeplanning and proposed the EPM (equal profit method)Audy et al [16] proposed four business models to study oncoalition formation and costsavings allocation Naber et al[17] developed four emission allocation methods Sun et al[18] considered a contribution constrained packing modelfor cost allocation Liu and Cheng [19] developed LCAM(location cost allocation model) which take geographicallocation factor into account

2 Problem Descriptions

In practice common scenarios for horizontal logistics col-laborative transportation planning can be described asfollows e Figure 1 shows a typical non-cooperativescenario where each freight carrier designs its own set ofroutes to deliver its own customers In contrast the Figure 2shows the same routing problem in a cooperative scenariowhere each distribution unit is reasonably assigned to itsadjacent logistics facility It also realizes to improve actualutilization of vehicles during a roundtrip and reduce thecrisscross transportation phenomenon in complex trans-portation networksWe can find practical examples from the

article Krajewska et al [20] Frisk et al [15] Perez-Bernabeuet al [21] and so on Obviously prior to horizontal co-operation each carrier had an independent cost Afterhorizontal cooperation all carriers formed an alliance tocomplete the task together which resulted in lower totalcost

In this article it leads to a lower cost for the alliance as awhole using the centralized collaboration framework Underthis framework there are two roles in the alliance one isleading company (s) (LC) and the other is participant (NLC)e responsibility of the leading company is to developparticipants in order to form an alliance collect informationabout supply and demand formulate a feasible operationplan and make each participating company put the coop-eration plan into practice Two major problems can besummarized to be solved One problem is the collaborativeplanning e problem to achieve the lowest overall cost forthe carriers has been studied in many literatures includingbut not limited to vehicle routing optimization or inventoryrouting problems Another problem lies in the equitableallocation of the total cost amount to each carrier concernedleading to a lower cost for each carrier as a result of thecooperation Cruijssen et al [22] and Leitner et al [23] bothpoint to fair cost allocation as one of the most importantobstacles to horizontal logistics e paper emphaticallyprobes into the second problem More details on the allo-cation rule are provided in Section 4 Our concept of modelallows allocating cost to freight carriers on leading companyperspective according to the specific context of such acollaborative organization

3 Allocation Methods

31CoreAllocation e concept of core has some of themostfrequently used properties A participantrsquos subset is denotedwith coalition S and all participants are denoted by grandcoalition N Assume that every participant has the chance togenerate and cooperate in coalitions In the case of cooperationof coalition S we can generate the common cost C(S)According to the cooperative game theory such function ofcost is referred to as characteristic cost function and everyparticipant is known as a playererefore the problem of costallocation can be considered as a cooperation game

A cost allocation method is applied to the separation oftotal cost ie C(N) Among the participants j isin N is said tobe efficient that is 1113936jisinNyj C(N) in which yj denotes thecost which is allocated to the participant j A cost allocation canbe considered as individual rational if there is no participantwhich makes a payment higher than the ldquostand alone costrdquo(that is the cost of the participant) when no coalition isgenerated If we put it mathematically we can express theproperty as yj leC( j1113864 1113865) e core of the game is identified asthose who meet the requirements of the cost allocation It isconsidered that the cost allocation at the core is stable

yj leC( j1113864 1113865) (1)

1113944jisinN

yj C(N) (2)






(3)



(a) (b)


(a) (b)







Max1113945n

i1ui minus bi( 1113857 (4)






xi

c( i )minus 1113944

iisinLC

xi

c( i ) (5)

st 1113944iisinN

xi c(N) (6)

1113944iisinS

xi le c(S) (7)












Model


xt+1i

c( i )minus 1113944

iisinLCt+1

xt+1i

c( i ) (12)

st 1113944iisinNt

xt+1i c Nt+1( 1113857 (13)

1113944iisinSt+1

xt+1i le c St+1( 1113857 (14)


c( i )

1113936jisinMt+1c( j1113864 1113865)

i isin NLCt (15)

xt+1i lex

tlowast

i i isin NLCt (16)

xt+1i ge 0 (17)


















|C| times Sc

11138681113868111386811138681113868111386811138681113868

(18)









xtlowast







A 49954 49954 49954 49052B 35485 35485 35485 36387


A 45078 43483 46802 41483B 34575 36558 32334 36387C 36232 35844 36749 38014


A 43827 44659 44718 39573B 32411 32616 30250 36387C 33244 33524 34665 38014D 42903 41586 42750 38410


A 41421 41755 43886 34993B 30521 30301 29418 36387C 31770 30160 33833 38014D 42825 43405 41918 36643E 42734 43649 40218 43234
























6 Conclusion












CoalitionMethod







CoalitionMethod













Data Availability




References





































(3)



(a) (b)


(a) (b)







Max1113945n

i1ui minus bi( 1113857 (4)






xi

c( i )minus 1113944

iisinLC

xi

c( i ) (5)

st 1113944iisinN

xi c(N) (6)

1113944iisinS

xi le c(S) (7)












Model


xt+1i

c( i )minus 1113944

iisinLCt+1

xt+1i

c( i ) (12)

st 1113944iisinNt

xt+1i c Nt+1( 1113857 (13)

1113944iisinSt+1

xt+1i le c St+1( 1113857 (14)


c( i )

1113936jisinMt+1c( j1113864 1113865)

i isin NLCt (15)

xt+1i lex

tlowast

i i isin NLCt (16)

xt+1i ge 0 (17)


















|C| times Sc

11138681113868111386811138681113868111386811138681113868

(18)









xtlowast







A 49954 49954 49954 49052B 35485 35485 35485 36387


A 45078 43483 46802 41483B 34575 36558 32334 36387C 36232 35844 36749 38014


A 43827 44659 44718 39573B 32411 32616 30250 36387C 33244 33524 34665 38014D 42903 41586 42750 38410


A 41421 41755 43886 34993B 30521 30301 29418 36387C 31770 30160 33833 38014D 42825 43405 41918 36643E 42734 43649 40218 43234
























6 Conclusion












CoalitionMethod







CoalitionMethod













Data Availability




References





































Max1113945n

i1ui minus bi( 1113857 (4)






xi

c( i )minus 1113944

iisinLC

xi

c( i ) (5)

st 1113944iisinN

xi c(N) (6)

1113944iisinS

xi le c(S) (7)












Model


xt+1i

c( i )minus 1113944

iisinLCt+1

xt+1i

c( i ) (12)

st 1113944iisinNt

xt+1i c Nt+1( 1113857 (13)

1113944iisinSt+1

xt+1i le c St+1( 1113857 (14)


c( i )

1113936jisinMt+1c( j1113864 1113865)

i isin NLCt (15)

xt+1i lex

tlowast

i i isin NLCt (16)

xt+1i ge 0 (17)


















|C| times Sc

11138681113868111386811138681113868111386811138681113868

(18)









xtlowast







A 49954 49954 49954 49052B 35485 35485 35485 36387


A 45078 43483 46802 41483B 34575 36558 32334 36387C 36232 35844 36749 38014


A 43827 44659 44718 39573B 32411 32616 30250 36387C 33244 33524 34665 38014D 42903 41586 42750 38410


A 41421 41755 43886 34993B 30521 30301 29418 36387C 31770 30160 33833 38014D 42825 43405 41918 36643E 42734 43649 40218 43234
























6 Conclusion












CoalitionMethod







CoalitionMethod













Data Availability




References







































Model


xt+1i

c( i )minus 1113944

iisinLCt+1

xt+1i

c( i ) (12)

st 1113944iisinNt

xt+1i c Nt+1( 1113857 (13)

1113944iisinSt+1

xt+1i le c St+1( 1113857 (14)


c( i )

1113936jisinMt+1c( j1113864 1113865)

i isin NLCt (15)

xt+1i lex

tlowast

i i isin NLCt (16)

xt+1i ge 0 (17)


















|C| times Sc

11138681113868111386811138681113868111386811138681113868

(18)









xtlowast







A 49954 49954 49954 49052B 35485 35485 35485 36387


A 45078 43483 46802 41483B 34575 36558 32334 36387C 36232 35844 36749 38014


A 43827 44659 44718 39573B 32411 32616 30250 36387C 33244 33524 34665 38014D 42903 41586 42750 38410


A 41421 41755 43886 34993B 30521 30301 29418 36387C 31770 30160 33833 38014D 42825 43405 41918 36643E 42734 43649 40218 43234
























6 Conclusion












CoalitionMethod







CoalitionMethod













Data Availability




References






































|C| times Sc

11138681113868111386811138681113868111386811138681113868

(18)









xtlowast







A 49954 49954 49954 49052B 35485 35485 35485 36387


A 45078 43483 46802 41483B 34575 36558 32334 36387C 36232 35844 36749 38014


A 43827 44659 44718 39573B 32411 32616 30250 36387C 33244 33524 34665 38014D 42903 41586 42750 38410


A 41421 41755 43886 34993B 30521 30301 29418 36387C 31770 30160 33833 38014D 42825 43405 41918 36643E 42734 43649 40218 43234
























6 Conclusion












CoalitionMethod







CoalitionMethod













Data Availability




References























































6 Conclusion












CoalitionMethod







CoalitionMethod













Data Availability




References


































Data Availability




References

































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