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A Lagrangian relaxation approach to combinatorial exchange in freight logistics
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A Lagrangian relaxation approach to A Lagrangian relaxation approach to A Lagrangian relaxation approach to A Lagrangian relaxation approach to combinatorial exchange in freight logistics combinatorial exchange in freight logistics combinatorial exchange in freight logistics combinatorial exchange∗ in freight logistics ∗ ∗
Yuan Yuan Li Li ∗ Haoxun Haoxun Chen Chen ∗ Christian Christian Prins Prins ∗ Yuan Li ∗∗ Haoxun Chen ∗∗ Christian Prins ∗∗ Yuan Li Haoxun Chen Christian Prins ∗ ∗ ICD-LOSI, UMR CNRS 6281, Universit´ ICD-LOSI, UMR CNRS 6281, Universit´ee de de Technologie Technologie de de Troyes Troyes ∗ UMRCurie, CNRSCS 6281, Universit´ e Troyes de Technologie de Troyes 12 rue Marie 42060, 10004 Cedex, France ∗ ICD-LOSI, 12 rue Marie 42060, 10004e Troyes Cedex, France ICD-LOSI, UMRCurie, CNRSCS 6281, Universit´ de Technologie de Troyes 12 rue Marie Curie,[email protected], CS 42060, 10004 Troyes Cedex, France (e-mail: [email protected], [email protected]) (e-mail: [email protected], [email protected]) 12 rue Marie Curie,[email protected], CS 42060, 10004 Troyes Cedex, France (e-mail: [email protected], [email protected], [email protected]) (e-mail: [email protected], [email protected], [email protected]) Abstract: Abstract: This This paper paper addresses addresses a a Combinatorial Combinatorial Exchange Exchange (CE) (CE) problem problem raised raised in in CollabCollabAbstract: This paper addresses a Combinatorial Exchange (CE) among problemless-than-truckload raised in Collaborative (CL). The new mechanism orative Logistics Logistics (CL). addresses The CE CE is isa a aCombinatorial new transaction transaction mechanism Abstract: This paper Exchange (CE) among problemless-than-truckload raised in Collaborative LogisticsWith (CL). The CE is a new transaction mechanism among less-than-truckload (LTL) this mechanism, several carriers an and exchange (LTL) carriers. carriers. this mechanism, several carriers form form an alliance alliance andless-than-truckload exchange some some of of orative LogisticsWith (CL). The CE is a new transaction mechanism among (LTL) carriers. Withrequests this mechanism, several of carriers form an profits. allianceInand exchange some of their transportation for the purpose gaining more the CE, each carrier their transportation for the purpose gaining more theexchange CE, eachsome carrier (LTL) carriers. Withrequests this mechanism, several of carriers form an profits. allianceInand of their transportation requests for the purpose of gaining more profits. Inadvantages the CE, each carrier plays a buyer seller. This brings some compared plays transportation a double double role role of ofrequests buyer and and seller. This characteristic characteristic brings some compared their for the purpose of gaining more profits. Inadvantages the CE, each carrier plays a double role of buyer and seller. This characteristic bringsAsome advantages compared with traditional Combinatorial Auctions (CA) in freight logistics. 0-1 linear programming is with traditional Combinatorial Auctions (CA)characteristic in freight logistics. 0-1 linear programming is plays a double role of buyer and seller. This bringsAsome advantages compared with traditional Combinatorial Auctions (CA) in freight logistics.is Athen 0-1 developed linear programming is formulated for this problem. A Lagrangian relaxation approach to solve the formulated for this problem. A Auctions Lagrangian relaxation approach to solve the with traditional Combinatorial (CA) in freight logistics.is Athen 0-1 developed linear programming is formulated for thisrelaxed problem. A Lagrangian relaxation approach is then developed to solve the CE problem is into maximum weight problem CE problem. problem.forThe The problem is transformed transformed into aaapproach maximumisvertex vertex weight clique clique problem formulated thisrelaxed problem. A Lagrangian relaxation then developed to solve the CE problem. The relaxed problem is transformed into a maximum vertex weight clique problem (MVWCP) which is solved solvedproblem either by by atransformed multi-neighborhood tabu search search or weight by aa commercial commercial MIP (MVWCP) is either multi-neighborhood tabu or by MIP CE problem.which The relaxed is a into a maximum vertex clique problem (MVWCP) whichheuristic is solvediseither by a to multi-neighborhood tabu searchcaused or by aby commercial MIP solver. proposed cope the solver. A A repair repair proposed cope with with any any infeasibilities infeasibilities the constraint constraint (MVWCP) whichheuristic is solvediseither by a to multi-neighborhood tabu searchcaused or by aby commercial MIP solver. A repair heuristic is proposed to cope on with any infeasibilities caused by show the constraint relaxation. Extensive numerical experiments randomly generated instances that relaxation. Extensive numerical experiments randomly generated caused instances that the the solver. A repair heuristic is proposed to cope on with any infeasibilities by show the constraint relaxation. Extensive numerical experiments on randomly generated instances show that the Lagrangian relaxation approach provide quality Lagrangian Extensive relaxation numerical approach can can provide high high quality solutions. solutions. relaxation. experiments on randomly generated instances show that the Lagrangian relaxation approach can provide high quality solutions. Lagrangian approach canofprovide high qualityHosting solutions. © 2016, IFACrelaxation (International Federation Automatic Control) by Elsevier Ltd. All rights reserved. Keywords: Keywords: combinatorial combinatorial exchange, exchange, carrier carrier collaboration, collaboration, Lagrangian Lagrangian relaxation, relaxation, maximum maximum Keywords: combinatorial exchange, carrier collaboration, Lagrangian relaxation, maximum vertex weight clique problem, tabu search vertex weight clique problem, tabu search Keywords: combinatorial exchange, carrier collaboration, Lagrangian relaxation, maximum vertex weight clique problem, tabu search vertex weight clique problem, tabu search 1. an 1. INTRODUCTION INTRODUCTION an indeed indeed vigorous vigorous way way to to decrease decrease driving driving distances, distances, and and 1. INTRODUCTION an indeed vigorous way to decrease driving distances, and to especially avoid deadheads, i.e., parts of tours to especially avoid way deadheads, i.e., driving parts ofdistances, tours driven driven 1. INTRODUCTION an indeed vigorous to decrease and to especially avoid deadheads, i.e., parts of tours driven without any load load (Ackermann eti.e., al.,parts 2011).of tours driven Freight without any al., 2011). especially avoid(Ackermann deadheads,et Freight logistics logistics specializes specializes in in the the movement movement (or (or ”for”for- to without any load (Ackermann et al., 2011). Freight logistics specializes in the movement (or ”forwarding”) of or from one anany load (Ackermann et al., 2011). sub-problems, warding”) of freight, freight, or cargo, cargo, from one place place(orto to”foran- without Freight logistics specializes in the movement Generally, Generally, CL CL must must deal deal with with two two successive successive sub-problems, warding”) of freight, or In cargo, from onealong placewith to another (Wikipedia, 2015). recent years, the CL must deal with two successive other (Wikipedia, 2015). recent years, the Generally, warding”) of freight, or In cargo, from onealong placewith to ani.e., requestCL allocation/exchange and profit sub-problems, sharing (Kra(Krarequest allocation/exchange profit sharing Generally, must deal with two and successive sub-problems, other (Wikipedia, 2015). In recent years, along with the i.e., flourishment of E-commerce and economic globalization, request allocation/exchange andand profit sharing (Kraflourishment of E-commerce and economic globalization, other (Wikipedia, 2015). In recent years, along with the i.e., jewska and Kopfer, 2006; Berger Bierwirth, 2010; jewska and Kopfer, 2006; Berger and Bierwirth, 2010; i.e., request allocation/exchange and profit sharing (Kraflourishment of E-commerce and economic globalization, freight is nowadays playing playing an indispensable indispensable role jewska and Kopfer, 2006; Berger and Bierwirth, 2010; freight business business nowadays an role flourishment of is E-commerce and economic globalization, Robu et al., 2011; Dai and Chen, 2011, 2012; Dai et al., Robu et al., 2011; Dai and Chen, 2011, 2012; Dai et al., jewska and Kopfer, 2006; Berger and Bierwirth, 2010; freight business is nowadays playing an indispensable role in activities of society. the et al., and 2011; Dai and Chen, 2011, 2012; DaiDai et and al., in economic economic activities of human human society. However, the rapid rapid freight business is nowadays playing anHowever, indispensable role Robu 2014; Wang Kopfer, 2014; Wang et al., 2014; 2014; Wang Kopfer, 2014; Wang et al., 2014; et al., and 2011; Dai and Chen, 2011, 2012; DaiDai et and al., in economic activities oflogistics human society. However, the rapid Robu development of freight brings a fierce competition Wang and Kopfer, 2014; Wang et al., 2014; Dai and development of freightoflogistics brings aHowever, fierce competition in economic activities human society. the rapid 2014; Chen, 2015). Chen, Wang 2015).and Kopfer, 2014; Wang et al., 2014; Dai and development of freight logistics brings a fierce competition 2014; among freight carriers or forwarders. The introduction 2015). among freight carrierslogistics or forwarders. The introduction development of freight brings a fierce competition Chen, Chen, 2015). we focus on the first subproblem: a deamong freight carriers or life forwarders. Theincreasing introduction of with shorter cycles, fuel In this this study, study, we focus on the first subproblem: a deof products products with shorter cycles, the the fuel In among freight carriers or life forwarders. Theincreasing introduction In this study, we focus on the (CE) first mechanism subproblem:to aswap deof products with shorter life cycles, the increasing fuel sign of Exchange costs and labor labor prices, thelife growing transport legislation signthis of Combinatorial Combinatorial Exchange study, we focus on the (CE) first mechanism subproblem:to aswap decosts and the growing legislation of products withprices, shorter cycles,transport the increasing fuel In sign of Combinatorial Exchange (CE) mechanism to swap costs andheightened labor prices, the growing transport legislation requests among carriers. To the best of our knowledge, and the expectations of customers, all these requests among carriers. To the(CE) bestmechanism of our knowledge, of Combinatorial Exchange to swap and the expectations of transport customers,legislation all these sign costs andheightened labor prices, the growing among carriers. To the best upon of ourinknowledge, and thehave heightened expectations of customers, all these requests is conceptually touched the factors, shrunk margins of and carriers this topic topicamong is only only carriers. conceptually touched the literaliterarequests To the best upon of ourinknowledge, factors, shrunk profit profit margins of of shippers shippers andall carriers and thehave heightened expectations customers, these this this topic is only conceptually touched upon in the literafactors, have shrunk profit margins of shippers and carriers ture(Ackermann et al., 2011; Bloos and Kopfer, 2009) but (Cruijssen et al., 2007). Especially for small or medium ture(Ackermann al., 2011; Bloos and upon Kopfer, 2009) but topic is only et conceptually touched in the litera(Cruijssen etshrunk al., 2007). small and or medium factors, have profitEspecially margins offor shippers carriers this ture(Ackermann et al., 2011; Bloos and Kopfer, 2009) but (Cruijssen et al., 2007). Especially for small or medium never be studied in depth. sized enterprises (SMEs), how to survive under such an never be studied in depth. ture(Ackermann et al., 2011; Bloos and Kopfer, 2009) but sized enterprises (SMEs), how to survive under such an (Cruijssen et al., 2007). Especially for small or medium never be studied in depth. sized enterprises (SMEs), how toenvironment survive under such an unprecedented harsh competition with global be studied in depth. unprecedented harsh competition with global sized enterprises (SMEs), how toenvironment survive under such an never The reminder reminder of of this this paper paper is is organized organized as as follows. follows. In In unprecedented harsh competition environment with global The pressure real challenge. pressure poses poses a aharsh real competition challenge. environment with global section The reminder of this paper is organized as framework. follows. In unprecedented 2, the advantages of section 2, we we discuss discuss thepaper advantages of our our CE CE pressure poses a real challenge. The reminder of this is organized as framework. follows. In section 2, we discuss thea advantages of our CE formulation framework. pressure posesan a real challenge. 3, present formal According In section section 3, we we present formal mathematical mathematical section 2, we discuss thea advantages of our CE formulation framework. According to to an authorized authorized report report provided provided by by Eurostat Eurostat In In section 3, we present a formal approach mathematical formulation According to an authorized report provided by Eurostat of CE. A Lagrangian relaxation to CE is in 2013 (Business economy size class analysis), SMEs of CE. A Lagrangian relaxation to CEformulation is presentpresentsection 3, we present a formal approach mathematical in 2013 (Business economy report - size class analysis), SMEs In According to an authorized provided by Eurostat of CE. A Lagrangian relaxation approach to CE isprovided presentin 201354% (Business economy - size class transportation analysis), SMEs ed in section 4. Computational experiments are gained EU market share in total & ed in section 4. Computational experiments are provided of CE. A Lagrangian relaxation approach to CE is presentgained EU market share in total transportation & in 201354% (Business economy size class analysis), SMEs 1 ed section in section 4. Computational experiments are provided gained 54% EU share in total transportation & in 5. Finally, section 6 concludes this paper with 1 . market Therefore, a more effective operation/ storage sector in section 5. Finally, section 6 concludes thisare paper with in section 4. Computational experiments provided Therefore, a in more effective operation/ storage 54% sector gained EU share total transportation & ed 1 . market in section 5. for Finally, section 6 concludes this paper with . Therefore, a more effective operation/ storage sectorpattern perspectives future research. management to enhance the competitiveness and 1 perspectives futuresection research. section 5. for Finally, 6 concludes this paper with management to enhance the competitiveness and in . Therefore, a more effective operation/ storage sectorpattern management patternfortoSMEs enhance the competitiveness and perspectives for future research. extract in transport/logistics extract more more profits profits in freight freight transport/logistics management patternfortoSMEs enhance the competitiveness and perspectives for future research. extract more profits for SMEs in freight transport/logistics sector quite demanded over last decade. sector is ismore quite demanded over the lasttransport/logistics decade. CollaboCollaboextract profits for SMEs in the freight 2. 2. COMBINATORIAL COMBINATORIAL EXCHANGE EXCHANGE VERSUS VERSUS sector Logistics is quite demanded over theAslast decade. Collaborative (CL) focus. an effective way 2. COMBINATORIAL EXCHANGE VERSUS rative Logistics (CL) is is a a over focus. an decade. effectiveCollaboway to to sector is quite demanded theAslast COMBINATORIAL AUCTION COMBINATORIAL AUCTION 2. COMBINATORIAL EXCHANGE VERSUS rative Logistics (CL) is a to focus. As anvehicle effective way to cut empty backhauls and increase utilization COMBINATORIAL AUCTION cut empty backhauls anda to increase utilization rative Logistics (CL) is focus. As anvehicle effective way to COMBINATORIAL AUCTION cut empty backhauls and to increase vehicle utilization rate, CL has attracted a growing interest from industrial rate,empty CL hasbackhauls attractedand a growing interest from utilization industrial We briefly recall the mathematical formulation of CA for cut to increase vehicle rate, CL has and attracted a growing interest from industrial briefly recall the mathematical formulation of CA for practitioners academic research (Dai Chen, 2009). practitioners academic researchinterest (Dai and and Chen, 2009). We rate, CL has and attracted a growing from industrial We briefly recall the and mathematical formulation of CA for carrier collaboration its practitioners and academic research (Dai and Chen, 2009). carrier collaboration its bid bid construction: construction: briefly recall the and mathematical formulation of CA for From both economic and ecological point of view, CL From both economic and ecological point of Chen, view, 2009). CL is is We practitioners and academic research (Dai and carrier collaboration and its bid construction: From both economic and ecological point of view, CL is Winner carrier collaboration and its bid construction: determination problem problem (WDP) (WDP) of of CA: CA: From both economic and ecological point of view, CL is Winner determination 1 1 http://ec.europa.eu/eurostat/statistics-explained/index. 1 http://ec.europa.eu/eurostat/statistics-explained/index. http://ec.europa.eu/eurostat/statistics-explained/index. php/Business_economy_-_size_class_analysis 1 php/Business_economy_-_size_class_analysis http://ec.europa.eu/eurostat/statistics-explained/index.
Winner determination problem (WDP) of CA: Winner determination problem (WDP) of CA:
php/Business_economy_-_size_class_analysis php/Business_economy_-_size_class_analysis Copyright © 2016 1650 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 IFAC IFAC 1650Hosting by Elsevier Ltd. All rights reserved. Copyright ©under 2016 responsibility IFAC 1650Control. Peer review of International Federation of Automatic Copyright © 2016 IFAC 1650 10.1016/j.ifacol.2016.07.817
IFAC MIM 2016 June 28-30, 2016. Troyes, France
max
j∈B
Subject to:
j∈B
Yuan Li et al. / IFAC-PapersOnLine 49-12 (2016) 1650–1655
pj · x j
(1)
ejk · xj ≤ 1
∀k ∈ M
(2)
xj ∈ {0, 1}
∀j ∈ B
(3)
Where M represents a set of requests and B denotes a set of bids. Each request j, ∀j ∈ B, is associated with a price pj . ejk is a 0 − 1 variable where ejk equals 1 if bid j covers request k. The objective function (1) aims to maximize the total price of winning bids. Constraint (2) ensures each request is assigned to at most one bid. If there is no free disposal, an equation is used in place of the inequality in constraint (2) (Andersson et al., 2000; Sandholm et al., 2002). Clearly, to achieve a CA among carriers, all carriers should firstly offer their outsourcing requests to a virtual auctioneer and then bid for profitable requests in common request pool. Bids are constructed in form of bundle, e.g., {(r1 , r2 , r3 ) : 50} represents an asking bid to serve request r1 , r2 , r3 at a price of 50. Differing from the bids of traditional CA, Each bid of CE consists of two parts: selling-part and buying-part, e.g., one bid: {(+r1 , +r2 , −r3 , −r4 , −r5 ) : 100} represents a bundle of requests with 100 as price, in which, requests r1 , r2 are bought in while requests r3 , r4 , r5 are sold out (’+’: symbol of buying in; ’-’: symbol of selling out, same hereafter). All requests in one bid are bundled together, that means one bid can only be accepted or rejected as a whole. The primary reason to devise CE mechanism as an alternative of traditional CA for carrier collaboration is to explore more synergies among requests in freight transportation. In traditional CA, all carriers must submit outsourcing requests to a virtual auctioneer (e.g., an auction platform) in an offering phase and bid for profitable requests in a bidding phase. In such scenarios, carriers can only bid for which requests to serve in the request pool. Whereas in CE, each carrier submits bids bundling simultaneously selling requests and buying requests, which is a complete exchange portfolio. So CA is more likely to be considered as a request allocation process while our CE is viewed as a request exchange process. From the point of view of carrier collaboration, CE may exploit more ”complementary potential” compared with traditional CA. A simple example is given below to illustrate this point: In CA, we assume one carrier hosts a request r3 which is not profitable in its current routing planning. Meanwhile, two requests r1 , r2 in the request pool are quite profitable for this carrier. However, due to some side constraints such as vehicle capacity, time-windows, request r1 , r2 , r3 are not possible be served by the carrier simultaneously. In such a circumstance, one dilemma of Static Games of Incomplete Information (SGII) occurs. One case for example, bid (+r1 , +r2 ) is not a feasible one. Because whether r3 can be sold out to another carrier is still uncertain till the WDP is solved by the auctioneer. Imaging once r3 is not assigned to any other carrier, logically, the service
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of r3 will still be in charge of the original carrier. So the winning bid of (+r1 , +r2 ) and a return of r3 will be an unpredictable outcome contradicting the real service capacity of the carrier. On the other hand, the missing of bid (+r1 , +r2 ) leads to uncompleted exploration of ”complementary potential” of this carrier in case r3 is eventually taken by one of the other carriers in carrier coalition. Contrarily, in CE, to prevent the dilemma mentioned above from happening, the carrier can submit a bid (+r1 , +r2 , −r3 ) instead.
Main contributions of this study can be concluded as: i) We introduce a new request allocation/exchange mechanism for carrier collaboration. ii) A formal mathematical formulation is provided for the proposed framework. iii) As a solution method of CE, an efficient algorithm based on Lagrangian relaxation is proven to produce promising outcomes. 3. PROBLEM DESCRIPTION AND MATHEMATICAL MODEL Our CE can be viewed as a scheme of profit optimization under the assumption that all carriers concertedly consent to maximize of total profit as the primary task, whereas fair post-profit allocation among the carriers is considered as a separate task. Actually, as long as the total profit of the carrier coalition increases, any carrier’s individual profit will definitely not be lowered by some reasonable profit sharing mechanism (Wang and Kopfer, 2014). But the second issue is not discussed here since this topic goes beyond the scope of this research.
Let N = {1, · · · , n} be the set of carriers in coalition and let M = {1, · · · , m} be the set of all requests. Each carrier i hosts a set of requests owned by itself, denoted as Mi . It is assumed that each carrier can bid for any requests except the ones already owned, thus, the set is denoted as M \ Mi . Let us denote Bi as the set of bids of carrier i, ∀i ∈ N . Then B = ∪i∈N Bi represents the set of all bids. Each bid j is associated with a price pj , ∀j ∈ B. To simplify the mathematical formulation of CE, three matrix of coefficients and a set of binary variables are defined as follows: Matrix of Coefficients:
1 request k is in bid j 0 otherwise 1 request k is in Mi aik = 0 otherwise 1 request k is in M \ Mi bik = 0 otherwise ejk =
Binary variables:
xj =
1 0
bid j is a winning bid otherwise
Winning determination problem of CE:
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max
Yuan Li et al. / IFAC-PapersOnLine 49-12 (2016) 1650–1655
j∈B
Subject to:
i∈N j∈Bi
i∈N j∈Bi
i∈N j∈Bi
aik · ejk · xj =
pj · x j
(4)
aik · ejk · xj ≤ 1
∀k ∈ M
(5)
bik · ejk · xj ≤ 1
∀k ∈ M
(6)
bik · ejk · xj
∀k ∈ M
(7)
xj ∈ {0, 1}
∀j ∈ B
(8)
i∈N j∈Bi
The objective function (4) represents the total profit of all winning bids’ price. Constraint (5) imposes each request can only be sold out at most once by its owned carrier. Constraint (6) ensures each request can only be bought in at most once by one carrier. Constraint (7) is the sell-buy matching constraint, which indicates once any request is sold out, there must be one other carrier in coalition buy it in. 4. A LAGRANGIAN RELAXATION APPROACH Lagrangian relaxation transforms a difficult optimization problem into a simpler one, by relaxing some hard constraints and moving them into the objective function. The relaxed constraints are penalized in case of violation by associating weights (Lagrangian multipliers) with them in the objective function. This dualization process yields a Lagrangian relaxed problem which is easier to solve. The optimum of this problem for given multipliers provides an upper bound to the optimum of the original maximization problem. The Lagrangian dual problem consists in determining the multipliers that minimize this bound.
best Lagrangian multiplier and the corresponding lowest upper bound, i.e., min ZD (λ), λ ∈ R.
The general structure of our Lagrangian relaxation approach is sketched in Algorithm 1 and its components are detailed in the following subsections. The main loop performances nitersLR iterations to subsequently solve the Lagrangian relaxed problem Z(λ) and the Lagrangian dual problem. The Lagrangian relaxed problem is transformed into the MVWCP which is solved by the multi-neighborhood tabu search (MNTS): M N T S (section 4.2.2) for the previous nitersLR − 1 iterations while the GUROBI solver is called to solve the last iteration. Subgradient method (SG) (section 4.3) is used to solve the Lagrangian dual problem, i.e., improve the Lagrangian multiplier λ and update the Lagrangian relaxed problem Z(λ) for the following next iteration. A repair heuristic (RH) is described in section 4.4 which aims to refine any infeasible solutions to obtain at least one feasible solution as lower bound. The main loop stops when nitersLR is reached or the call of GUROBI and returns relative duality gap Gap. Some definitions to make the following algorithms more readable: B denotes the set of all bids, S represents a solution and f (S) is the corresponding total price. U BM N T S and LBM N T S designate upper bound and lower bound by solving M N T S(Z(λ)), respectively. U BGU ROBI corresponds to upper bound found by the GUROBI solver. Algorithm 1 – Pseudo code of our Lagrangian relaxation approach– LR(B) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16:
4.1 Framework of the Lagrangian relaxation Consider the mathematical formulation of CE (Eq.(4) to Eq.(8)). Let χ = (xj )j∈B be the vector of tendered bids while λ = (λk )k∈M be the Lagrangian multiplier associated with constraint (7). Then the Lagrangian relaxed problem can be formulated as: Z(λ) = max χ
(
i∈N j∈Bi
i∈N j∈Bi
aik · ejk · xj −
pj · x j +
i∈N j∈Bi
k∈M
i∈N j∈Bi
i∈N j∈Bi
Gap =
U BM N T S −max(LBM N T S ,LBRH ) U BM N T S
else U BGU ROBI ← GU ROBI(Z(λ)) Gap =
U BGU ROBI −max(LBM N T S ,LBRH ) U BGU ROBI
end if return Gap
4.2 Resolution of the Lagrangian relaxed problem λk ·
bik · ejk · xj )
(9)
Subject to:
iter ← 0 λ←0 Gap ← ∞ Update λ by SG if iter < nitersLR & Gap ≥ 0.01 then S, LBM N T S ← M N T S(Z(λ)) U BM N T S ← f (S) Update λ by SG LBRH ← RH(S) iter ← iter + 1
aik · ejk · xj ≤ 1
∀k ∈ M
(10)
bik · ejk · xj ≤ 1
∀k ∈ M
(11)
xj ∈ {0, 1}
∀j ∈ B
(12)
Z(λ) is also referred to the Lagrangian dual problem. For each value of the Lagrangian multiplier which will provide an upper bound on the optimal value of the original problem. The Lagrangian dual problem aims to find out the
The WDP is the core issue in CA which can be reduced to the MVWCP (Ausiello et al., 1980). Wu and Hao (2015) solve the WDP via the MVWCP and their MNTS heuristic (Wu et al., 2012) report so far the best results on a large range of 530 benchmark instances in terms of computing time and solution quality. Our computational results reveal the MNTS solving the Lagrangian relaxed problem of CE shows excellent performance not only on providing a quite tight upper bound on the original CE but also producing a lower bound (feasible solution) with high quality. However, the optimum of the Lagrangian relaxed problem can not be guaranteed since the MNTS is a heuristic algorithm. Therefore, the GUROBI solver is called to solve the last iteration of Lagrangian relaxed problem so that an accurate upper bound is surely obtained.
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Transformation into the maximum vertex weight clique problem (MVWCP): We turn first to recall the basic definition of the MVWCP to illustrate how to transform the Lagrangian relaxed problem (Eq.(9) to Eq.(12)) of CE into the MVWCP. Given an undirected graph G = (V, E), where V is the set of vertices and E set of edges, E ⊂ V × V . A clique C of G is a subset of V such that all vertices in C are pairwise adjacent, i.e., ∀υ, ϕ ∈ C, {υ, ϕ} ∈ E. A positive weight ωυ is associated with each vertex υ ∈ V . The weight W (C) of a clique C is defined as W (C) = υ∈C ωυ . The MVWCP aims to find out a clique C of G which maximizes W (C).
It is noteworthy that the weight ωυ is not necessary to be positive in our study since the price of a CE bid is likely to be a negative value. For a given CE instance B, where B = {B1 , B2 , B3 , ..., Bn } denotes the bids. Each bid: Bi = {(+Ri , −Ri ) : pi } (1 ≤ i ≤ n) , where +Ri is the buying requests set and −Ri is the selling requests set, pi is the price of Bi . Now we construct an undirected graph G = (V, E) as follows:
1
Obviously, a clique C with maximum weight W (C) of the undirected graph G = (V, E) built by a bids set B corresponds to the optimal solution for the Lagrangian relaxed problem of CE, in other words, any solution method for the MVWCP can also be applied to solve the Lagrangian relaxed problem. To demonstrate the transformation, we consider a CE instance with 2 carriers and 3 requests (carrier 1 hosts request 1 and request 3 while carrier 2 hosts request 2), i.e., M = {1, 2}, N = {1, 2, 3}, M1 = {r1, r3}, M2 = {r2}. Table.1 lists the corresponding bids information. The transformation into the MVWCP is illustrated in Fig.1 and Fig.2. Fig.1 gives the optimal solution of the Lagrangian relaxed problem with λ = 0, whereas Fig.2 shows the optimal solution of the original problem. We can clearly see that the optimal solution {Bid1, Bid3, Bid7, Bid8}
3
2
3
9
4 8
4 8
5
7
5
7
6
Fig. 1. Solution of Z(0)
6
Fig. 2. Solution of CE
with total profit 219 of the relaxation problem is not feasible for CE since no carrier sells request r1, i.e., the matching constraint is violated. The genuine optimal solution of this instance is a winning bids combination {Bid1, Bid3, Bid6, Bid8} with total profit 210 in Fig.2. Table 1. Bids information of the example Number of Bid 1 2 3 4 5 6 7 8 9
• Edges set E: for any two bids Bi , Bj ∈ B: if +Ri ∩ +Rj ∩ −Ri ∩ −Rj = ∅, then one edge {i, j} ∈ E is created, ∀i, j ∈ V .
On one hand, any two vertices i and j are tied together by an edge if and only if the two corresponding bids Bi = {(+Ri , −Ri ) : pi } and Bj = {(+Rj , −Rj ) : pj } share no common request which is bought in or sold out, i.e., edges of constructed MVWCP can be directly used to detect any pairs of bids violating the non-overlapping constraint. On the other hand, the matching constraint will not necessarily be hold with a solution of the MVWCP since we can not guarantee each request bought in must match the same request sold out in any clique C, vice versa.
1
2
9
• Vertices set V : for each bid Bi ∈ B, a vertex i is generated with a weight ωi which equals to pi , i.e., V = {1, 2, 3, ..., n}, ∀i ∈ V , ωi = pi .
In CE model, constraint (5) and constraint (6) impose each request must be sold out and bought in at most once, thus we label it non-overlapping constraints. Constraint (7) ensures sell-buy balance of each request, namely matching constraint.
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Bidder Carrier Carrier Carrier Carrier Carrier Carrier Carrier Carrier Carrier
1 1 1 1 1 2 2 2 2
Requests to sell
Requests to buy
Price
r3 r1,r3 ∅ r1 r1,r3 r2 r2 ∅ r2
∅ ∅ r2 r2 r2 ∅ r1 r3 r1,r3
65 47 63 -18 79 13 22 69 -10
Multi-neighborhood tabu search (MNTS): We directly implement the MNTS algorithm in Wu et al. (2012), thence only the main ingredients are briefly introduced here. We refer reader to Wu et al. (2012) for more details. Before each call of M N T S to solve the Lagrangian relaxed problem, Algorithm 2 transforms Z(λ) into an undirected weighted graph G = (V, E) (section 4.2.1) as input. The program returns a clique CGlobalBest with maximum weight W (CGlobalBest ) as output. Initial solution construction: A seeding vertex i, i ∈ V is randomly selected to be the current clique C. Then we randomly insert another vertex j to C where j ∈ V , j ∈ / C and j is connected to all other vertices of C. The process stops until no more such a vertex j can be found. This initial clique construction procedure is fast, easyimplemented and diversified. Neighborhoods and selection criteria: Three basic neighborhood moves are designed to jointly improve a current clique C, denoted as Nswap , Nadd and Ndrop . Nswap works on the basis of Setswap , where Setswap is composed of the vertices that are excluded from the current clique C and connected to all but one vertex of C: Setswap = {υ : υ ∈ V \ C, |A (υ) ∩ C| = |C| − 1} where A (υ) = {j : j ∈ V, {j, υ} ∈ E}. Nswap defines a move to swap two vertices i, j satisfying i ∈ Setswap , j ∈ C, {i, j} ∈ / E.
Setadd contains the vertices that are excluded from the current clique C and connected to all the vertices of C: Setadd = {υ : υ ∈ V \ C, {v, i} ∈ E, ∀i ∈ C}. Nadd defines a move to add a vertex i to the clique C, i ∈ Setadd . Ndrop simply removes a vertex i from the clique C, i ∈ C.
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Tabu list and tabu tenure length: A prohibition rule is adopted in Algorithm 2: once a vertex leaves the current clique C, it is banned to reenter C for the following T T Lswap and T T Ldrop iterations where T T Lswap = |C| + LengthT abu, T T Ldrop = LengthT abu and LengthT abu a tabu tenure parameter. Multi-start mechanism and stop criteria: A multistart strategy may produce a more fruitful outcome because of diversification. We also apply such a mechanism in the MNTS to reinforce the algorithm’s ability of exploring more areas in the solution space. A restart is triggered when a number of consecutive iterations without improvement is reached, denoted as DepthSearch. Each new round of restart will construct a new initial solution and three neighborhoods Nswap , Nadd , Ndrop search the solution space with tabu prohibition rule. A maximum iteration number nitersM N T S of M N T S is set before the run of algorithm.
4.4 Construction of feasible solutions Algorithm 4 sketches the repair heuristic used to derive a feasible solution to the original CE problem (SRH ) from the solution obtained by the MNTS (SM N T S ). A solution of the Lagrangian relaxed problem is infeasible to the original CE problem because some requests in the solution do not meet the matching constraints. Such a request is called no-balanced iff it is included in a winning bid of the relaxed problem while its corresponding matching constraint is violated. So, the idea to repair an infeasible solution is to bring more bids to the final solution which makes all no-balanced requests meet the matching constraints. The repair algorithm returns a feasible solution SRH with its cost LBRH of the original CE problem. Algorithm 4 – Repair heuristic – RH(S) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16:
Algorithm 2 – Multi-neighborhood tabu search – M N T S(Z(λ)) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24:
iterT S = 0 CGlobalBest = ∅ while iterT S < nitersM N T S do Initialize C Initialize TabuList iterRestart = 0 CLocalBest = C while iterRestart < DepthSearch do Construct three neighborhoods Nswap , Nadd and Ndrop from C
Execute a max gain neighborhood C ∈ Nswap ∪ Nadd ∪ Ndrop
C =C iterRestart = iterRestart + 1 iterT S = iterT S + 1 Update TabuList if W (C) > W (CLocalBest ) then iterRestart = 0 CLocalBest = C end if end while if W (CLocalBest ) > W (CGlobalBest ) then CGlobalBest = CLocalBest end if end while
4.3 Subgradient method for the Lagrangian dual problem Algorithm 3 shows how the Lagrangian multipliers λ = (λk )k∈M are updated in line 16 of Algorithm 1. The notation δ denotes a given step size, δ ≤ 2. Recall that m is the number of requests.
i∈N
j∈Bi
for k ← 1 to m do
4:
end for
λk ← λ k + t ·
k∈M
i∈N
xj ·ejk ·(aik −bik )
j∈Bi
k∈M
xj · ejk · (aik − bik )
record B repair in a set L end if end for sort all elements in set L in non-increasing order of their total profit best denote the first element in the sorted set L as Brepair best SRH ← S ∪ Brepair
return SRH with cost LBRH
5.1 Generation of instances
Worthmentioning, a high quality lower bound (feasible solution) is often generated during the search of M N T S, on other words, the matching constraint is occasionally satisfied even it is already dualized.
2: 3:
if B repair = Rmatching then
5. COMPUTATIONAL RESULTS
return Clique (CGlobalBest )
Algorithm 3 – Update of Lagrangian multipliers δ·(U BM N T S −LBRH ) 1: t ← 2
generate Rnopair from the infeasible solution S generate Rmatching from Rnopair for all bids brepair ∈ B do if brepair ∈ / S & brepair ∈ Rmatching then include bid brepair in the set of bids Brepair end if end for for all possible subsets B repair ⊆ Brepair do
Instances are randomly generated from an input of a number of carriers and a number of requests. Each carrier randomly owns some requests from the request pool. Each request i is set a sell price pisell and a buy price pibuy . Each bid j is labeled a price pj where pj = i i i∈+Rj pbuy + i∈−Rj psell · σ, σ ∈ [0.5, 1.5] is a synergy factor. Each instance is named with format N umberOf Carrier− N umberOf Request − N umberOf Bid − SerialN umber. Consider a instance 4 − 8 − 1020 − 001 as an example. There are 4 carriers, 8 requests and 1020 bids in total. Serial Number of this instance is 001. 5.2 Parameter setting All experiments were conducted on a desktop equipped with Intel(R) Core (TM) i7-2600 3.40GHz processor and 8 GB RAM. The operating system of this PC is 64bit Window 7 service pack 1. The Lagrangian relaxation algorithm was coded in Python (Interpreter: Cpython 3.4.3, IDE: JetBrains PyCharm 4.5.4). The binary integer programming of the relaxed problem was built and solved by GUROBI with Python interface (GUROBI version 6.0.5).
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Table 2. Parameter tuning of Lagrangian relaxation approach Symbol
Role
Value
nitersLR nitersM N T S δ LengthT abu DepthSearch
Iteration number of LR Iteration number of M N T S Scalar step size Length of tabu tenure Trigger of restart M N T S
100 10000 1.2 7 30
5.3 Test results on the relative duality gap
Table 3. Computational results on the relative duality gap (Gap) Instances
U BLR
LBLR
Gap(%)
CP ULR (s)
2-4-30-001 2-4-30-002 3-8-765-001 3-8-765-002 4-8-1020-001 4-8-1020-002 4-10-4092-001 4-10-4092-002 5-12-20475-001 5-12-20475-002
188.25 201.8 288.4 408.3 184.2 352.7 1127.9 506.1 429.5 591.23
188 197 287 408 183 352 1125 506 427 585
0.13 2.4 0.48 0.07 0.65 0.2 0.26 0.02 0.58 1.06
0.109 0.322 3.421 1.784 15.3 17.077 38.367 340.124 132.924 672.367
6. PERSPECTIVES AND CONCLUSION We proposed a new CE framework for carrier collaboration. A Lagrangian relaxation approach is developed to solve CE. 10 randomly generated instances are evaluated on the relative duality gap and the results reveal the performance is promising. Future research may focus on the bid generation problem (BGP) and bid price optimization problem (BPOP) to improve the effectiveness of CE. Combining our CE with routing problem is another challenging issue. Moreover, a fair post-profit allocation mechanism should also be established. REFERENCES Ackermann, H., Ewe, H., Kopfer, H., and K¨ ufer, K.H. (2011). Combinatorial auctions in freight logistics. In Computational logistics, 1–17. Springer. Andersson, A., Tenhunen, M., and Ygge, F. (2000). Integer programming for combinatorial auction winner determination. In MultiAgent Systems, 2000. Proceedings. Fourth International Conference on, 39–46. IEEE. Ausiello, G., D’Atri, A., and Protasi, M. (1980). Structure preserving reductions among convex optimization problems. Journal of Computer and System Sciences, 21(1), 136–153. Berger, S. and Bierwirth, C. (2010). Solutions to the request reassignment problem in collaborative carrier networks. Transportation Research Part E: Logistics and Transportation Review, 46(5), 627–638. Bloos, M. and Kopfer, H. (2009). Efficiency of transport collaboration mechanisms. Communications of SIWN, 6, 23–28.
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A Lagrangian relaxation approach to A Lagrangian relaxation approach to A Lagrangian relaxation approach to A Lagrangian relaxation approach to combinatorial exchange in freight logistics combinatorial exchange in freight logistics combinatorial exchange in freight logistics combinatorial exchange∗ in freight logistics ∗ ∗
Yuan Yuan Li Li ∗ Haoxun Haoxun Chen Chen ∗ Christian Christian Prins Prins ∗ Yuan Li ∗∗ Haoxun Chen ∗∗ Christian Prins ∗∗ Yuan Li Haoxun Chen Christian Prins ∗ ∗ ICD-LOSI, UMR CNRS 6281, Universit´ ICD-LOSI, UMR CNRS 6281, Universit´ee de de Technologie Technologie de de Troyes Troyes ∗ UMRCurie, CNRSCS 6281, Universit´ e Troyes de Technologie de Troyes 12 rue Marie 42060, 10004 Cedex, France ∗ ICD-LOSI, 12 rue Marie 42060, 10004e Troyes Cedex, France ICD-LOSI, UMRCurie, CNRSCS 6281, Universit´ de Technologie de Troyes 12 rue Marie Curie,[email protected], CS 42060, 10004 Troyes Cedex, France (e-mail: [email protected], [email protected]) (e-mail: [email protected], [email protected]) 12 rue Marie Curie,[email protected], CS 42060, 10004 Troyes Cedex, France (e-mail: [email protected], [email protected], [email protected]) (e-mail: [email protected], [email protected], [email protected]) Abstract: Abstract: This This paper paper addresses addresses a a Combinatorial Combinatorial Exchange Exchange (CE) (CE) problem problem raised raised in in CollabCollabAbstract: This paper addresses a Combinatorial Exchange (CE) among problemless-than-truckload raised in Collaborative (CL). The new mechanism orative Logistics Logistics (CL). addresses The CE CE is isa a aCombinatorial new transaction transaction mechanism Abstract: This paper Exchange (CE) among problemless-than-truckload raised in Collaborative LogisticsWith (CL). The CE is a new transaction mechanism among less-than-truckload (LTL) this mechanism, several carriers an and exchange (LTL) carriers. carriers. this mechanism, several carriers form form an alliance alliance andless-than-truckload exchange some some of of orative LogisticsWith (CL). The CE is a new transaction mechanism among (LTL) carriers. Withrequests this mechanism, several of carriers form an profits. allianceInand exchange some of their transportation for the purpose gaining more the CE, each carrier their transportation for the purpose gaining more theexchange CE, eachsome carrier (LTL) carriers. Withrequests this mechanism, several of carriers form an profits. allianceInand of their transportation requests for the purpose of gaining more profits. Inadvantages the CE, each carrier plays a buyer seller. This brings some compared plays transportation a double double role role of ofrequests buyer and and seller. This characteristic characteristic brings some compared their for the purpose of gaining more profits. Inadvantages the CE, each carrier plays a double role of buyer and seller. This characteristic bringsAsome advantages compared with traditional Combinatorial Auctions (CA) in freight logistics. 0-1 linear programming is with traditional Combinatorial Auctions (CA)characteristic in freight logistics. 0-1 linear programming is plays a double role of buyer and seller. This bringsAsome advantages compared with traditional Combinatorial Auctions (CA) in freight logistics.is Athen 0-1 developed linear programming is formulated for this problem. A Lagrangian relaxation approach to solve the formulated for this problem. A Auctions Lagrangian relaxation approach to solve the with traditional Combinatorial (CA) in freight logistics.is Athen 0-1 developed linear programming is formulated for thisrelaxed problem. A Lagrangian relaxation approach is then developed to solve the CE problem is into maximum weight problem CE problem. problem.forThe The problem is transformed transformed into aaapproach maximumisvertex vertex weight clique clique problem formulated thisrelaxed problem. A Lagrangian relaxation then developed to solve the CE problem. The relaxed problem is transformed into a maximum vertex weight clique problem (MVWCP) which is solved solvedproblem either by by atransformed multi-neighborhood tabu search search or weight by aa commercial commercial MIP (MVWCP) is either multi-neighborhood tabu or by MIP CE problem.which The relaxed is a into a maximum vertex clique problem (MVWCP) whichheuristic is solvediseither by a to multi-neighborhood tabu searchcaused or by aby commercial MIP solver. proposed cope the solver. A A repair repair proposed cope with with any any infeasibilities infeasibilities the constraint constraint (MVWCP) whichheuristic is solvediseither by a to multi-neighborhood tabu searchcaused or by aby commercial MIP solver. A repair heuristic is proposed to cope on with any infeasibilities caused by show the constraint relaxation. Extensive numerical experiments randomly generated instances that relaxation. Extensive numerical experiments randomly generated caused instances that the the solver. A repair heuristic is proposed to cope on with any infeasibilities by show the constraint relaxation. Extensive numerical experiments on randomly generated instances show that the Lagrangian relaxation approach provide quality Lagrangian Extensive relaxation numerical approach can can provide high high quality solutions. solutions. relaxation. experiments on randomly generated instances show that the Lagrangian relaxation approach can provide high quality solutions. Lagrangian approach canofprovide high qualityHosting solutions. © 2016, IFACrelaxation (International Federation Automatic Control) by Elsevier Ltd. All rights reserved. Keywords: Keywords: combinatorial combinatorial exchange, exchange, carrier carrier collaboration, collaboration, Lagrangian Lagrangian relaxation, relaxation, maximum maximum Keywords: combinatorial exchange, carrier collaboration, Lagrangian relaxation, maximum vertex weight clique problem, tabu search vertex weight clique problem, tabu search Keywords: combinatorial exchange, carrier collaboration, Lagrangian relaxation, maximum vertex weight clique problem, tabu search vertex weight clique problem, tabu search 1. an 1. INTRODUCTION INTRODUCTION an indeed indeed vigorous vigorous way way to to decrease decrease driving driving distances, distances, and and 1. INTRODUCTION an indeed vigorous way to decrease driving distances, and to especially avoid deadheads, i.e., parts of tours to especially avoid way deadheads, i.e., driving parts ofdistances, tours driven driven 1. INTRODUCTION an indeed vigorous to decrease and to especially avoid deadheads, i.e., parts of tours driven without any load load (Ackermann eti.e., al.,parts 2011).of tours driven Freight without any al., 2011). especially avoid(Ackermann deadheads,et Freight logistics logistics specializes specializes in in the the movement movement (or (or ”for”for- to without any load (Ackermann et al., 2011). Freight logistics specializes in the movement (or ”forwarding”) of or from one anany load (Ackermann et al., 2011). sub-problems, warding”) of freight, freight, or cargo, cargo, from one place place(orto to”foran- without Freight logistics specializes in the movement Generally, Generally, CL CL must must deal deal with with two two successive successive sub-problems, warding”) of freight, or In cargo, from onealong placewith to another (Wikipedia, 2015). recent years, the CL must deal with two successive other (Wikipedia, 2015). recent years, the Generally, warding”) of freight, or In cargo, from onealong placewith to ani.e., requestCL allocation/exchange and profit sub-problems, sharing (Kra(Krarequest allocation/exchange profit sharing Generally, must deal with two and successive sub-problems, other (Wikipedia, 2015). In recent years, along with the i.e., flourishment of E-commerce and economic globalization, request allocation/exchange andand profit sharing (Kraflourishment of E-commerce and economic globalization, other (Wikipedia, 2015). In recent years, along with the i.e., jewska and Kopfer, 2006; Berger Bierwirth, 2010; jewska and Kopfer, 2006; Berger and Bierwirth, 2010; i.e., request allocation/exchange and profit sharing (Kraflourishment of E-commerce and economic globalization, freight is nowadays playing playing an indispensable indispensable role jewska and Kopfer, 2006; Berger and Bierwirth, 2010; freight business business nowadays an role flourishment of is E-commerce and economic globalization, Robu et al., 2011; Dai and Chen, 2011, 2012; Dai et al., Robu et al., 2011; Dai and Chen, 2011, 2012; Dai et al., jewska and Kopfer, 2006; Berger and Bierwirth, 2010; freight business is nowadays playing an indispensable role in activities of society. the et al., and 2011; Dai and Chen, 2011, 2012; DaiDai et and al., in economic economic activities of human human society. However, the rapid rapid freight business is nowadays playing anHowever, indispensable role Robu 2014; Wang Kopfer, 2014; Wang et al., 2014; 2014; Wang Kopfer, 2014; Wang et al., 2014; et al., and 2011; Dai and Chen, 2011, 2012; DaiDai et and al., in economic activities oflogistics human society. However, the rapid Robu development of freight brings a fierce competition Wang and Kopfer, 2014; Wang et al., 2014; Dai and development of freightoflogistics brings aHowever, fierce competition in economic activities human society. the rapid 2014; Chen, 2015). Chen, Wang 2015).and Kopfer, 2014; Wang et al., 2014; Dai and development of freight logistics brings a fierce competition 2014; among freight carriers or forwarders. The introduction 2015). among freight carrierslogistics or forwarders. The introduction development of freight brings a fierce competition Chen, Chen, 2015). we focus on the first subproblem: a deamong freight carriers or life forwarders. Theincreasing introduction of with shorter cycles, fuel In this this study, study, we focus on the first subproblem: a deof products products with shorter cycles, the the fuel In among freight carriers or life forwarders. Theincreasing introduction In this study, we focus on the (CE) first mechanism subproblem:to aswap deof products with shorter life cycles, the increasing fuel sign of Exchange costs and labor labor prices, thelife growing transport legislation signthis of Combinatorial Combinatorial Exchange study, we focus on the (CE) first mechanism subproblem:to aswap decosts and the growing legislation of products withprices, shorter cycles,transport the increasing fuel In sign of Combinatorial Exchange (CE) mechanism to swap costs andheightened labor prices, the growing transport legislation requests among carriers. To the best of our knowledge, and the expectations of customers, all these requests among carriers. To the(CE) bestmechanism of our knowledge, of Combinatorial Exchange to swap and the expectations of transport customers,legislation all these sign costs andheightened labor prices, the growing among carriers. To the best upon of ourinknowledge, and thehave heightened expectations of customers, all these requests is conceptually touched the factors, shrunk margins of and carriers this topic topicamong is only only carriers. conceptually touched the literaliterarequests To the best upon of ourinknowledge, factors, shrunk profit profit margins of of shippers shippers andall carriers and thehave heightened expectations customers, these this this topic is only conceptually touched upon in the literafactors, have shrunk profit margins of shippers and carriers ture(Ackermann et al., 2011; Bloos and Kopfer, 2009) but (Cruijssen et al., 2007). Especially for small or medium ture(Ackermann al., 2011; Bloos and upon Kopfer, 2009) but topic is only et conceptually touched in the litera(Cruijssen etshrunk al., 2007). small and or medium factors, have profitEspecially margins offor shippers carriers this ture(Ackermann et al., 2011; Bloos and Kopfer, 2009) but (Cruijssen et al., 2007). Especially for small or medium never be studied in depth. sized enterprises (SMEs), how to survive under such an never be studied in depth. ture(Ackermann et al., 2011; Bloos and Kopfer, 2009) but sized enterprises (SMEs), how to survive under such an (Cruijssen et al., 2007). Especially for small or medium never be studied in depth. sized enterprises (SMEs), how toenvironment survive under such an unprecedented harsh competition with global be studied in depth. unprecedented harsh competition with global sized enterprises (SMEs), how toenvironment survive under such an never The reminder reminder of of this this paper paper is is organized organized as as follows. follows. In In unprecedented harsh competition environment with global The pressure real challenge. pressure poses poses a aharsh real competition challenge. environment with global section The reminder of this paper is organized as framework. follows. In unprecedented 2, the advantages of section 2, we we discuss discuss thepaper advantages of our our CE CE pressure poses a real challenge. The reminder of this is organized as framework. follows. In section 2, we discuss thea advantages of our CE formulation framework. pressure posesan a real challenge. 3, present formal According In section section 3, we we present formal mathematical mathematical section 2, we discuss thea advantages of our CE formulation framework. According to to an authorized authorized report report provided provided by by Eurostat Eurostat In In section 3, we present a formal approach mathematical formulation According to an authorized report provided by Eurostat of CE. A Lagrangian relaxation to CE is in 2013 (Business economy size class analysis), SMEs of CE. A Lagrangian relaxation to CEformulation is presentpresentsection 3, we present a formal approach mathematical in 2013 (Business economy report - size class analysis), SMEs In According to an authorized provided by Eurostat of CE. A Lagrangian relaxation approach to CE isprovided presentin 201354% (Business economy - size class transportation analysis), SMEs ed in section 4. Computational experiments are gained EU market share in total & ed in section 4. Computational experiments are provided of CE. A Lagrangian relaxation approach to CE is presentgained EU market share in total transportation & in 201354% (Business economy size class analysis), SMEs 1 ed section in section 4. Computational experiments are provided gained 54% EU share in total transportation & in 5. Finally, section 6 concludes this paper with 1 . market Therefore, a more effective operation/ storage sector in section 5. Finally, section 6 concludes thisare paper with in section 4. Computational experiments provided Therefore, a in more effective operation/ storage 54% sector gained EU share total transportation & ed 1 . market in section 5. for Finally, section 6 concludes this paper with . Therefore, a more effective operation/ storage sectorpattern perspectives future research. management to enhance the competitiveness and 1 perspectives futuresection research. section 5. for Finally, 6 concludes this paper with management to enhance the competitiveness and in . Therefore, a more effective operation/ storage sectorpattern management patternfortoSMEs enhance the competitiveness and perspectives for future research. extract in transport/logistics extract more more profits profits in freight freight transport/logistics management patternfortoSMEs enhance the competitiveness and perspectives for future research. extract more profits for SMEs in freight transport/logistics sector quite demanded over last decade. sector is ismore quite demanded over the lasttransport/logistics decade. CollaboCollaboextract profits for SMEs in the freight 2. 2. COMBINATORIAL COMBINATORIAL EXCHANGE EXCHANGE VERSUS VERSUS sector Logistics is quite demanded over theAslast decade. Collaborative (CL) focus. an effective way 2. COMBINATORIAL EXCHANGE VERSUS rative Logistics (CL) is is a a over focus. an decade. effectiveCollaboway to to sector is quite demanded theAslast COMBINATORIAL AUCTION COMBINATORIAL AUCTION 2. COMBINATORIAL EXCHANGE VERSUS rative Logistics (CL) is a to focus. As anvehicle effective way to cut empty backhauls and increase utilization COMBINATORIAL AUCTION cut empty backhauls anda to increase utilization rative Logistics (CL) is focus. As anvehicle effective way to COMBINATORIAL AUCTION cut empty backhauls and to increase vehicle utilization rate, CL has attracted a growing interest from industrial rate,empty CL hasbackhauls attractedand a growing interest from utilization industrial We briefly recall the mathematical formulation of CA for cut to increase vehicle rate, CL has and attracted a growing interest from industrial briefly recall the mathematical formulation of CA for practitioners academic research (Dai Chen, 2009). practitioners academic researchinterest (Dai and and Chen, 2009). We rate, CL has and attracted a growing from industrial We briefly recall the and mathematical formulation of CA for carrier collaboration its practitioners and academic research (Dai and Chen, 2009). carrier collaboration its bid bid construction: construction: briefly recall the and mathematical formulation of CA for From both economic and ecological point of view, CL From both economic and ecological point of Chen, view, 2009). CL is is We practitioners and academic research (Dai and carrier collaboration and its bid construction: From both economic and ecological point of view, CL is Winner carrier collaboration and its bid construction: determination problem problem (WDP) (WDP) of of CA: CA: From both economic and ecological point of view, CL is Winner determination 1 1 http://ec.europa.eu/eurostat/statistics-explained/index. 1 http://ec.europa.eu/eurostat/statistics-explained/index. http://ec.europa.eu/eurostat/statistics-explained/index. php/Business_economy_-_size_class_analysis 1 php/Business_economy_-_size_class_analysis http://ec.europa.eu/eurostat/statistics-explained/index.
Winner determination problem (WDP) of CA: Winner determination problem (WDP) of CA:
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j∈B
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j∈B
Yuan Li et al. / IFAC-PapersOnLine 49-12 (2016) 1650–1655
pj · x j
(1)
ejk · xj ≤ 1
∀k ∈ M
(2)
xj ∈ {0, 1}
∀j ∈ B
(3)
Where M represents a set of requests and B denotes a set of bids. Each request j, ∀j ∈ B, is associated with a price pj . ejk is a 0 − 1 variable where ejk equals 1 if bid j covers request k. The objective function (1) aims to maximize the total price of winning bids. Constraint (2) ensures each request is assigned to at most one bid. If there is no free disposal, an equation is used in place of the inequality in constraint (2) (Andersson et al., 2000; Sandholm et al., 2002). Clearly, to achieve a CA among carriers, all carriers should firstly offer their outsourcing requests to a virtual auctioneer and then bid for profitable requests in common request pool. Bids are constructed in form of bundle, e.g., {(r1 , r2 , r3 ) : 50} represents an asking bid to serve request r1 , r2 , r3 at a price of 50. Differing from the bids of traditional CA, Each bid of CE consists of two parts: selling-part and buying-part, e.g., one bid: {(+r1 , +r2 , −r3 , −r4 , −r5 ) : 100} represents a bundle of requests with 100 as price, in which, requests r1 , r2 are bought in while requests r3 , r4 , r5 are sold out (’+’: symbol of buying in; ’-’: symbol of selling out, same hereafter). All requests in one bid are bundled together, that means one bid can only be accepted or rejected as a whole. The primary reason to devise CE mechanism as an alternative of traditional CA for carrier collaboration is to explore more synergies among requests in freight transportation. In traditional CA, all carriers must submit outsourcing requests to a virtual auctioneer (e.g., an auction platform) in an offering phase and bid for profitable requests in a bidding phase. In such scenarios, carriers can only bid for which requests to serve in the request pool. Whereas in CE, each carrier submits bids bundling simultaneously selling requests and buying requests, which is a complete exchange portfolio. So CA is more likely to be considered as a request allocation process while our CE is viewed as a request exchange process. From the point of view of carrier collaboration, CE may exploit more ”complementary potential” compared with traditional CA. A simple example is given below to illustrate this point: In CA, we assume one carrier hosts a request r3 which is not profitable in its current routing planning. Meanwhile, two requests r1 , r2 in the request pool are quite profitable for this carrier. However, due to some side constraints such as vehicle capacity, time-windows, request r1 , r2 , r3 are not possible be served by the carrier simultaneously. In such a circumstance, one dilemma of Static Games of Incomplete Information (SGII) occurs. One case for example, bid (+r1 , +r2 ) is not a feasible one. Because whether r3 can be sold out to another carrier is still uncertain till the WDP is solved by the auctioneer. Imaging once r3 is not assigned to any other carrier, logically, the service
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of r3 will still be in charge of the original carrier. So the winning bid of (+r1 , +r2 ) and a return of r3 will be an unpredictable outcome contradicting the real service capacity of the carrier. On the other hand, the missing of bid (+r1 , +r2 ) leads to uncompleted exploration of ”complementary potential” of this carrier in case r3 is eventually taken by one of the other carriers in carrier coalition. Contrarily, in CE, to prevent the dilemma mentioned above from happening, the carrier can submit a bid (+r1 , +r2 , −r3 ) instead.
Main contributions of this study can be concluded as: i) We introduce a new request allocation/exchange mechanism for carrier collaboration. ii) A formal mathematical formulation is provided for the proposed framework. iii) As a solution method of CE, an efficient algorithm based on Lagrangian relaxation is proven to produce promising outcomes. 3. PROBLEM DESCRIPTION AND MATHEMATICAL MODEL Our CE can be viewed as a scheme of profit optimization under the assumption that all carriers concertedly consent to maximize of total profit as the primary task, whereas fair post-profit allocation among the carriers is considered as a separate task. Actually, as long as the total profit of the carrier coalition increases, any carrier’s individual profit will definitely not be lowered by some reasonable profit sharing mechanism (Wang and Kopfer, 2014). But the second issue is not discussed here since this topic goes beyond the scope of this research.
Let N = {1, · · · , n} be the set of carriers in coalition and let M = {1, · · · , m} be the set of all requests. Each carrier i hosts a set of requests owned by itself, denoted as Mi . It is assumed that each carrier can bid for any requests except the ones already owned, thus, the set is denoted as M \ Mi . Let us denote Bi as the set of bids of carrier i, ∀i ∈ N . Then B = ∪i∈N Bi represents the set of all bids. Each bid j is associated with a price pj , ∀j ∈ B. To simplify the mathematical formulation of CE, three matrix of coefficients and a set of binary variables are defined as follows: Matrix of Coefficients:
1 request k is in bid j 0 otherwise 1 request k is in Mi aik = 0 otherwise 1 request k is in M \ Mi bik = 0 otherwise ejk =
Binary variables:
xj =
1 0
bid j is a winning bid otherwise
Winning determination problem of CE:
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max
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j∈B
Subject to:
i∈N j∈Bi
i∈N j∈Bi
i∈N j∈Bi
aik · ejk · xj =
pj · x j
(4)
aik · ejk · xj ≤ 1
∀k ∈ M
(5)
bik · ejk · xj ≤ 1
∀k ∈ M
(6)
bik · ejk · xj
∀k ∈ M
(7)
xj ∈ {0, 1}
∀j ∈ B
(8)
i∈N j∈Bi
The objective function (4) represents the total profit of all winning bids’ price. Constraint (5) imposes each request can only be sold out at most once by its owned carrier. Constraint (6) ensures each request can only be bought in at most once by one carrier. Constraint (7) is the sell-buy matching constraint, which indicates once any request is sold out, there must be one other carrier in coalition buy it in. 4. A LAGRANGIAN RELAXATION APPROACH Lagrangian relaxation transforms a difficult optimization problem into a simpler one, by relaxing some hard constraints and moving them into the objective function. The relaxed constraints are penalized in case of violation by associating weights (Lagrangian multipliers) with them in the objective function. This dualization process yields a Lagrangian relaxed problem which is easier to solve. The optimum of this problem for given multipliers provides an upper bound to the optimum of the original maximization problem. The Lagrangian dual problem consists in determining the multipliers that minimize this bound.
best Lagrangian multiplier and the corresponding lowest upper bound, i.e., min ZD (λ), λ ∈ R.
The general structure of our Lagrangian relaxation approach is sketched in Algorithm 1 and its components are detailed in the following subsections. The main loop performances nitersLR iterations to subsequently solve the Lagrangian relaxed problem Z(λ) and the Lagrangian dual problem. The Lagrangian relaxed problem is transformed into the MVWCP which is solved by the multi-neighborhood tabu search (MNTS): M N T S (section 4.2.2) for the previous nitersLR − 1 iterations while the GUROBI solver is called to solve the last iteration. Subgradient method (SG) (section 4.3) is used to solve the Lagrangian dual problem, i.e., improve the Lagrangian multiplier λ and update the Lagrangian relaxed problem Z(λ) for the following next iteration. A repair heuristic (RH) is described in section 4.4 which aims to refine any infeasible solutions to obtain at least one feasible solution as lower bound. The main loop stops when nitersLR is reached or the call of GUROBI and returns relative duality gap Gap. Some definitions to make the following algorithms more readable: B denotes the set of all bids, S represents a solution and f (S) is the corresponding total price. U BM N T S and LBM N T S designate upper bound and lower bound by solving M N T S(Z(λ)), respectively. U BGU ROBI corresponds to upper bound found by the GUROBI solver. Algorithm 1 – Pseudo code of our Lagrangian relaxation approach– LR(B) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16:
4.1 Framework of the Lagrangian relaxation Consider the mathematical formulation of CE (Eq.(4) to Eq.(8)). Let χ = (xj )j∈B be the vector of tendered bids while λ = (λk )k∈M be the Lagrangian multiplier associated with constraint (7). Then the Lagrangian relaxed problem can be formulated as: Z(λ) = max χ
(
i∈N j∈Bi
i∈N j∈Bi
aik · ejk · xj −
pj · x j +
i∈N j∈Bi
k∈M
i∈N j∈Bi
i∈N j∈Bi
Gap =
U BM N T S −max(LBM N T S ,LBRH ) U BM N T S
else U BGU ROBI ← GU ROBI(Z(λ)) Gap =
U BGU ROBI −max(LBM N T S ,LBRH ) U BGU ROBI
end if return Gap
4.2 Resolution of the Lagrangian relaxed problem λk ·
bik · ejk · xj )
(9)
Subject to:
iter ← 0 λ←0 Gap ← ∞ Update λ by SG if iter < nitersLR & Gap ≥ 0.01 then S, LBM N T S ← M N T S(Z(λ)) U BM N T S ← f (S) Update λ by SG LBRH ← RH(S) iter ← iter + 1
aik · ejk · xj ≤ 1
∀k ∈ M
(10)
bik · ejk · xj ≤ 1
∀k ∈ M
(11)
xj ∈ {0, 1}
∀j ∈ B
(12)
Z(λ) is also referred to the Lagrangian dual problem. For each value of the Lagrangian multiplier which will provide an upper bound on the optimal value of the original problem. The Lagrangian dual problem aims to find out the
The WDP is the core issue in CA which can be reduced to the MVWCP (Ausiello et al., 1980). Wu and Hao (2015) solve the WDP via the MVWCP and their MNTS heuristic (Wu et al., 2012) report so far the best results on a large range of 530 benchmark instances in terms of computing time and solution quality. Our computational results reveal the MNTS solving the Lagrangian relaxed problem of CE shows excellent performance not only on providing a quite tight upper bound on the original CE but also producing a lower bound (feasible solution) with high quality. However, the optimum of the Lagrangian relaxed problem can not be guaranteed since the MNTS is a heuristic algorithm. Therefore, the GUROBI solver is called to solve the last iteration of Lagrangian relaxed problem so that an accurate upper bound is surely obtained.
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Transformation into the maximum vertex weight clique problem (MVWCP): We turn first to recall the basic definition of the MVWCP to illustrate how to transform the Lagrangian relaxed problem (Eq.(9) to Eq.(12)) of CE into the MVWCP. Given an undirected graph G = (V, E), where V is the set of vertices and E set of edges, E ⊂ V × V . A clique C of G is a subset of V such that all vertices in C are pairwise adjacent, i.e., ∀υ, ϕ ∈ C, {υ, ϕ} ∈ E. A positive weight ωυ is associated with each vertex υ ∈ V . The weight W (C) of a clique C is defined as W (C) = υ∈C ωυ . The MVWCP aims to find out a clique C of G which maximizes W (C).
It is noteworthy that the weight ωυ is not necessary to be positive in our study since the price of a CE bid is likely to be a negative value. For a given CE instance B, where B = {B1 , B2 , B3 , ..., Bn } denotes the bids. Each bid: Bi = {(+Ri , −Ri ) : pi } (1 ≤ i ≤ n) , where +Ri is the buying requests set and −Ri is the selling requests set, pi is the price of Bi . Now we construct an undirected graph G = (V, E) as follows:
1
Obviously, a clique C with maximum weight W (C) of the undirected graph G = (V, E) built by a bids set B corresponds to the optimal solution for the Lagrangian relaxed problem of CE, in other words, any solution method for the MVWCP can also be applied to solve the Lagrangian relaxed problem. To demonstrate the transformation, we consider a CE instance with 2 carriers and 3 requests (carrier 1 hosts request 1 and request 3 while carrier 2 hosts request 2), i.e., M = {1, 2}, N = {1, 2, 3}, M1 = {r1, r3}, M2 = {r2}. Table.1 lists the corresponding bids information. The transformation into the MVWCP is illustrated in Fig.1 and Fig.2. Fig.1 gives the optimal solution of the Lagrangian relaxed problem with λ = 0, whereas Fig.2 shows the optimal solution of the original problem. We can clearly see that the optimal solution {Bid1, Bid3, Bid7, Bid8}
3
2
3
9
4 8
4 8
5
7
5
7
6
Fig. 1. Solution of Z(0)
6
Fig. 2. Solution of CE
with total profit 219 of the relaxation problem is not feasible for CE since no carrier sells request r1, i.e., the matching constraint is violated. The genuine optimal solution of this instance is a winning bids combination {Bid1, Bid3, Bid6, Bid8} with total profit 210 in Fig.2. Table 1. Bids information of the example Number of Bid 1 2 3 4 5 6 7 8 9
• Edges set E: for any two bids Bi , Bj ∈ B: if +Ri ∩ +Rj ∩ −Ri ∩ −Rj = ∅, then one edge {i, j} ∈ E is created, ∀i, j ∈ V .
On one hand, any two vertices i and j are tied together by an edge if and only if the two corresponding bids Bi = {(+Ri , −Ri ) : pi } and Bj = {(+Rj , −Rj ) : pj } share no common request which is bought in or sold out, i.e., edges of constructed MVWCP can be directly used to detect any pairs of bids violating the non-overlapping constraint. On the other hand, the matching constraint will not necessarily be hold with a solution of the MVWCP since we can not guarantee each request bought in must match the same request sold out in any clique C, vice versa.
1
2
9
• Vertices set V : for each bid Bi ∈ B, a vertex i is generated with a weight ωi which equals to pi , i.e., V = {1, 2, 3, ..., n}, ∀i ∈ V , ωi = pi .
In CE model, constraint (5) and constraint (6) impose each request must be sold out and bought in at most once, thus we label it non-overlapping constraints. Constraint (7) ensures sell-buy balance of each request, namely matching constraint.
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Bidder Carrier Carrier Carrier Carrier Carrier Carrier Carrier Carrier Carrier
1 1 1 1 1 2 2 2 2
Requests to sell
Requests to buy
Price
r3 r1,r3 ∅ r1 r1,r3 r2 r2 ∅ r2
∅ ∅ r2 r2 r2 ∅ r1 r3 r1,r3
65 47 63 -18 79 13 22 69 -10
Multi-neighborhood tabu search (MNTS): We directly implement the MNTS algorithm in Wu et al. (2012), thence only the main ingredients are briefly introduced here. We refer reader to Wu et al. (2012) for more details. Before each call of M N T S to solve the Lagrangian relaxed problem, Algorithm 2 transforms Z(λ) into an undirected weighted graph G = (V, E) (section 4.2.1) as input. The program returns a clique CGlobalBest with maximum weight W (CGlobalBest ) as output. Initial solution construction: A seeding vertex i, i ∈ V is randomly selected to be the current clique C. Then we randomly insert another vertex j to C where j ∈ V , j ∈ / C and j is connected to all other vertices of C. The process stops until no more such a vertex j can be found. This initial clique construction procedure is fast, easyimplemented and diversified. Neighborhoods and selection criteria: Three basic neighborhood moves are designed to jointly improve a current clique C, denoted as Nswap , Nadd and Ndrop . Nswap works on the basis of Setswap , where Setswap is composed of the vertices that are excluded from the current clique C and connected to all but one vertex of C: Setswap = {υ : υ ∈ V \ C, |A (υ) ∩ C| = |C| − 1} where A (υ) = {j : j ∈ V, {j, υ} ∈ E}. Nswap defines a move to swap two vertices i, j satisfying i ∈ Setswap , j ∈ C, {i, j} ∈ / E.
Setadd contains the vertices that are excluded from the current clique C and connected to all the vertices of C: Setadd = {υ : υ ∈ V \ C, {v, i} ∈ E, ∀i ∈ C}. Nadd defines a move to add a vertex i to the clique C, i ∈ Setadd . Ndrop simply removes a vertex i from the clique C, i ∈ C.
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Tabu list and tabu tenure length: A prohibition rule is adopted in Algorithm 2: once a vertex leaves the current clique C, it is banned to reenter C for the following T T Lswap and T T Ldrop iterations where T T Lswap = |C| + LengthT abu, T T Ldrop = LengthT abu and LengthT abu a tabu tenure parameter. Multi-start mechanism and stop criteria: A multistart strategy may produce a more fruitful outcome because of diversification. We also apply such a mechanism in the MNTS to reinforce the algorithm’s ability of exploring more areas in the solution space. A restart is triggered when a number of consecutive iterations without improvement is reached, denoted as DepthSearch. Each new round of restart will construct a new initial solution and three neighborhoods Nswap , Nadd , Ndrop search the solution space with tabu prohibition rule. A maximum iteration number nitersM N T S of M N T S is set before the run of algorithm.
4.4 Construction of feasible solutions Algorithm 4 sketches the repair heuristic used to derive a feasible solution to the original CE problem (SRH ) from the solution obtained by the MNTS (SM N T S ). A solution of the Lagrangian relaxed problem is infeasible to the original CE problem because some requests in the solution do not meet the matching constraints. Such a request is called no-balanced iff it is included in a winning bid of the relaxed problem while its corresponding matching constraint is violated. So, the idea to repair an infeasible solution is to bring more bids to the final solution which makes all no-balanced requests meet the matching constraints. The repair algorithm returns a feasible solution SRH with its cost LBRH of the original CE problem. Algorithm 4 – Repair heuristic – RH(S) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16:
Algorithm 2 – Multi-neighborhood tabu search – M N T S(Z(λ)) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24:
iterT S = 0 CGlobalBest = ∅ while iterT S < nitersM N T S do Initialize C Initialize TabuList iterRestart = 0 CLocalBest = C while iterRestart < DepthSearch do Construct three neighborhoods Nswap , Nadd and Ndrop from C
Execute a max gain neighborhood C ∈ Nswap ∪ Nadd ∪ Ndrop
C =C iterRestart = iterRestart + 1 iterT S = iterT S + 1 Update TabuList if W (C) > W (CLocalBest ) then iterRestart = 0 CLocalBest = C end if end while if W (CLocalBest ) > W (CGlobalBest ) then CGlobalBest = CLocalBest end if end while
4.3 Subgradient method for the Lagrangian dual problem Algorithm 3 shows how the Lagrangian multipliers λ = (λk )k∈M are updated in line 16 of Algorithm 1. The notation δ denotes a given step size, δ ≤ 2. Recall that m is the number of requests.
i∈N
j∈Bi
for k ← 1 to m do
4:
end for
λk ← λ k + t ·
k∈M
i∈N
xj ·ejk ·(aik −bik )
j∈Bi
k∈M
xj · ejk · (aik − bik )
record B repair in a set L end if end for sort all elements in set L in non-increasing order of their total profit best denote the first element in the sorted set L as Brepair best SRH ← S ∪ Brepair
return SRH with cost LBRH
5.1 Generation of instances
Worthmentioning, a high quality lower bound (feasible solution) is often generated during the search of M N T S, on other words, the matching constraint is occasionally satisfied even it is already dualized.
2: 3:
if B repair = Rmatching then
5. COMPUTATIONAL RESULTS
return Clique (CGlobalBest )
Algorithm 3 – Update of Lagrangian multipliers δ·(U BM N T S −LBRH ) 1: t ← 2
generate Rnopair from the infeasible solution S generate Rmatching from Rnopair for all bids brepair ∈ B do if brepair ∈ / S & brepair ∈ Rmatching then include bid brepair in the set of bids Brepair end if end for for all possible subsets B repair ⊆ Brepair do
Instances are randomly generated from an input of a number of carriers and a number of requests. Each carrier randomly owns some requests from the request pool. Each request i is set a sell price pisell and a buy price pibuy . Each bid j is labeled a price pj where pj = i i i∈+Rj pbuy + i∈−Rj psell · σ, σ ∈ [0.5, 1.5] is a synergy factor. Each instance is named with format N umberOf Carrier− N umberOf Request − N umberOf Bid − SerialN umber. Consider a instance 4 − 8 − 1020 − 001 as an example. There are 4 carriers, 8 requests and 1020 bids in total. Serial Number of this instance is 001. 5.2 Parameter setting All experiments were conducted on a desktop equipped with Intel(R) Core (TM) i7-2600 3.40GHz processor and 8 GB RAM. The operating system of this PC is 64bit Window 7 service pack 1. The Lagrangian relaxation algorithm was coded in Python (Interpreter: Cpython 3.4.3, IDE: JetBrains PyCharm 4.5.4). The binary integer programming of the relaxed problem was built and solved by GUROBI with Python interface (GUROBI version 6.0.5).
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Table 2. Parameter tuning of Lagrangian relaxation approach Symbol
Role
Value
nitersLR nitersM N T S δ LengthT abu DepthSearch
Iteration number of LR Iteration number of M N T S Scalar step size Length of tabu tenure Trigger of restart M N T S
100 10000 1.2 7 30
5.3 Test results on the relative duality gap
Table 3. Computational results on the relative duality gap (Gap) Instances
U BLR
LBLR
Gap(%)
CP ULR (s)
2-4-30-001 2-4-30-002 3-8-765-001 3-8-765-002 4-8-1020-001 4-8-1020-002 4-10-4092-001 4-10-4092-002 5-12-20475-001 5-12-20475-002
188.25 201.8 288.4 408.3 184.2 352.7 1127.9 506.1 429.5 591.23
188 197 287 408 183 352 1125 506 427 585
0.13 2.4 0.48 0.07 0.65 0.2 0.26 0.02 0.58 1.06
0.109 0.322 3.421 1.784 15.3 17.077 38.367 340.124 132.924 672.367
6. PERSPECTIVES AND CONCLUSION We proposed a new CE framework for carrier collaboration. A Lagrangian relaxation approach is developed to solve CE. 10 randomly generated instances are evaluated on the relative duality gap and the results reveal the performance is promising. Future research may focus on the bid generation problem (BGP) and bid price optimization problem (BPOP) to improve the effectiveness of CE. Combining our CE with routing problem is another challenging issue. Moreover, a fair post-profit allocation mechanism should also be established. REFERENCES Ackermann, H., Ewe, H., Kopfer, H., and K¨ ufer, K.H. (2011). Combinatorial auctions in freight logistics. In Computational logistics, 1–17. Springer. Andersson, A., Tenhunen, M., and Ygge, F. (2000). Integer programming for combinatorial auction winner determination. In MultiAgent Systems, 2000. Proceedings. Fourth International Conference on, 39–46. IEEE. Ausiello, G., D’Atri, A., and Protasi, M. (1980). Structure preserving reductions among convex optimization problems. Journal of Computer and System Sciences, 21(1), 136–153. Berger, S. and Bierwirth, C. (2010). Solutions to the request reassignment problem in collaborative carrier networks. Transportation Research Part E: Logistics and Transportation Review, 46(5), 627–638. Bloos, M. and Kopfer, H. (2009). Efficiency of transport collaboration mechanisms. Communications of SIWN, 6, 23–28.
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