Discrete Optimization An improved Benders decomposition algorithm for the tree of hubs location problem Elisangela Martins de Sá, Ricardo Saraiva de Camargo ⇑ , Gilberto de Miranda Department of Industrial Engineering, Federal University of Minas Gerais, Brazil article info Article history: Received 22 September 2011 Accepted 26 October 2012 Available online 17 November 2012 Keywords: Tree of hubs location problem Hub-and-spoke networks Benders decomposition method Benders cuts selection scheme abstract The tree of hubs location problem is a particularly hard variant of the so called hub location problems. When solving this problem by a Benders decomposition approach, it is necessary to deal with both opti- mality and feasibility cuts. While modern implementations of the Benders decomposition method rely on Pareto-optimal optimality cuts or on rendering feasibility cuts based on minimal infeasible subsystems, a new cut selection scheme is devised here under the guiding principle of extracting useful information even when facing infeasible subproblems. The proposed algorithm outperforms two other modern vari- ants of the method and it is capable of optimally solving instances five times larger than the ones previ- ously reported on the literature. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction In many-to-many distribution systems, in which several origin– destination pairs of nodes exchange flows, hub-and-spoke net- works have a great application appeal. In such networks, rather than directly connect each pair, hub facilities are used to consoli- date, route and distribute the traffic in order to take advantage of economies of scale on inter-hub connections. Flows from the same origin but addressed to different destinations are bundled at the hubs to other traffic that has different origins but the same desti- nation. The consolidation of flows at the hubs allows the exploita- tion of scale economies due to the use of more efficient and higher volume carriers on inter-hub connections (O’Kelly, 1998). Usually, in hub-and-spoke networks, it is assumed that there is a inter-hub connection between every hub pair; that no two non- hub nodes can be directly linked; that an origin–destination flow is routed through one or at most two hubs. Moreover, different assumptions may be contemplated including: Single (O’Kelly, 1987; Klincewicz, 1991; Skorin-Kapov et al., 1996; Aykin, 1995; Ernst and Krishnamoorthy, 1998b; Ebery, 2001) or multiple allocation (Campbell, 1994; Skorin-Kapov et al., 1996; Ernst and Krishnamoorthy, 1998a; Mayer and Wagner, 2002; Hamacher et al., 2004; Marı ´ n et al., 2006) of the non-hub nodes to the installed hubs, the number of hubs to be located may or may not be known beforehand, direct service between non-hub nodes may be enabled (Aykin, 1994, 1995), capacity constraints on the amount of traffic an installed hub can handle (Campbell, 1994; Ay- kin, 1994, 1995; Ernst and Krishnamoorthy, 1999; Ebery et al., 2000; Labbé et al., 2005; Costa et al., 2008; Contreras et al., 2009a), consideration of congestion effects at the installed hubs (Elhedhli and Hu, 2005; Camargo et al., 2009a; Elhedhli and Wu, 2010) and flow dependent economies of scale on inter-hub connec- tions (O’Kelly and Bryan, 1998; O’Kelly, 1998; Horner and OKelly, 2001; Klincewicz, 2002; Racunica and Wynter, 2005; Kimms, 2006; Camargo et al., 2009b) among other variants. A general review of different problems is presented on the exhaustive surveys of Campbell et al. (2002) and of Alumur and Kara (2008b). Recently, more flexible network policies of hub inter- connections have been proposed (Nickel et al., 2000; Labbé et al., 2004; Campbell et al., 2005a,b; Contreras et al., 2009b, 2010; Alumur et al., 2009; Calik et al., 2009) in order to broaden the applicability of hub-and-spoke systems to other areas. The idea is to disregard the assumption that every pair of hub has to be directly linked and to adapt the design of the network to the characteristics of the application being addressed. For instance, depending on the criterion used for establishing the inter-hub connections, many related applications can then be seen as special cases of hub-and-spoke systems: (i) Multi-modal urban transportation (Bruno et al., 1998; Nickel et al., 2000; cGelareh, 2008; Chen et al., 2008; Marı ´ n and Jaramillo, 2009), (ii) telecommunication networks (Hu, 1974; Kim and Tcha, 1992; Lee et al., 1994, 1996; Klincewicz, 1998), (iii) tree-shaped facilities location problems (Kim et al., 1996; Puerto and Tamir, 2005), (iv) ring-star network designs (Labbé et al., 2004; Laporte and Martı ´ n, 2007), and (v) incomplete hub networks (Campbell et al., 2005a,b; 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.10.051 ⇑ Corresponding author. Address: Department of Industrial Engineering, Federal University of Minas Gerais, Av. Antonio Carlos, 6627, 31270-901 Belo Horizonte, MG, Brazil. Tel.: +55 31 3409 4881; fax: +55 31 3409 4883. E-mail addresses: [email protected](E.M. de Sá), [email protected](R.S. de Camargo), [email protected](G. de Miranda). European Journal of Operational Research 226 (2013) 185–202 Contents lists available at SciVerse ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Transcript
European Journal of Operational Research 226 (2013) 185–202
Contents lists available at SciVerse ScienceDirect
European Journal of Operational Research
journal homepage: www.elsevier .com/locate /e jor
Discrete Optimization
An improved Benders decomposition algorithm for the tree of hubslocation problem
Elisangela Martins de Sá, Ricardo Saraiva de Camargo ⇑, Gilberto de MirandaDepartment of Industrial Engineering, Federal University of Minas Gerais, Brazil
a r t i c l e i n f o
Article history:Received 22 September 2011Accepted 26 October 2012Available online 17 November 2012
0377-2217/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.ejor.2012.10.051
⇑ Corresponding author. Address: Department of InUniversity of Minas Gerais, Av. Antonio Carlos, 6627MG, Brazil. Tel.: +55 31 3409 4881; fax: +55 31 3409
The tree of hubs location problem is a particularly hard variant of the so called hub location problems.When solving this problem by a Benders decomposition approach, it is necessary to deal with both opti-mality and feasibility cuts. While modern implementations of the Benders decomposition method rely onPareto-optimal optimality cuts or on rendering feasibility cuts based on minimal infeasible subsystems, anew cut selection scheme is devised here under the guiding principle of extracting useful informationeven when facing infeasible subproblems. The proposed algorithm outperforms two other modern vari-ants of the method and it is capable of optimally solving instances five times larger than the ones previ-ously reported on the literature.
� 2012 Elsevier B.V. All rights reserved.
1. Introduction
In many-to-many distribution systems, in which several origin–destination pairs of nodes exchange flows, hub-and-spoke net-works have a great application appeal. In such networks, ratherthan directly connect each pair, hub facilities are used to consoli-date, route and distribute the traffic in order to take advantage ofeconomies of scale on inter-hub connections. Flows from the sameorigin but addressed to different destinations are bundled at thehubs to other traffic that has different origins but the same desti-nation. The consolidation of flows at the hubs allows the exploita-tion of scale economies due to the use of more efficient and highervolume carriers on inter-hub connections (O’Kelly, 1998).
Usually, in hub-and-spoke networks, it is assumed that there isa inter-hub connection between every hub pair; that no two non-hub nodes can be directly linked; that an origin–destination flow isrouted through one or at most two hubs. Moreover, differentassumptions may be contemplated including: Single (O’Kelly,1987; Klincewicz, 1991; Skorin-Kapov et al., 1996; Aykin, 1995;Ernst and Krishnamoorthy, 1998b; Ebery, 2001) or multipleallocation (Campbell, 1994; Skorin-Kapov et al., 1996; Ernst andKrishnamoorthy, 1998a; Mayer and Wagner, 2002; Hamacheret al., 2004; Marı́n et al., 2006) of the non-hub nodes to theinstalled hubs, the number of hubs to be located may or may notbe known beforehand, direct service between non-hub nodes
may be enabled (Aykin, 1994, 1995), capacity constraints on theamount of traffic an installed hub can handle (Campbell, 1994; Ay-kin, 1994, 1995; Ernst and Krishnamoorthy, 1999; Ebery et al.,2000; Labbé et al., 2005; Costa et al., 2008; Contreras et al.,2009a), consideration of congestion effects at the installed hubs(Elhedhli and Hu, 2005; Camargo et al., 2009a; Elhedhli and Wu,2010) and flow dependent economies of scale on inter-hub connec-tions (O’Kelly and Bryan, 1998; O’Kelly, 1998; Horner and OKelly,2001; Klincewicz, 2002; Racunica and Wynter, 2005; Kimms,2006; Camargo et al., 2009b) among other variants. A generalreview of different problems is presented on the exhaustivesurveys of Campbell et al. (2002) and of Alumur and Kara (2008b).
Recently, more flexible network policies of hub inter-connections have been proposed (Nickel et al., 2000; Labbé et al.,2004; Campbell et al., 2005a,b; Contreras et al., 2009b, 2010;Alumur et al., 2009; Calik et al., 2009) in order to broaden theapplicability of hub-and-spoke systems to other areas. The idea isto disregard the assumption that every pair of hub has to bedirectly linked and to adapt the design of the network to thecharacteristics of the application being addressed.
For instance, depending on the criterion used for establishingthe inter-hub connections, many related applications can then beseen as special cases of hub-and-spoke systems: (i) Multi-modalurban transportation (Bruno et al., 1998; Nickel et al., 2000;cGelareh, 2008; Chen et al., 2008; Marı́n and Jaramillo, 2009), (ii)telecommunication networks (Hu, 1974; Kim and Tcha, 1992;Lee et al., 1994, 1996; Klincewicz, 1998), (iii) tree-shaped facilitieslocation problems (Kim et al., 1996; Puerto and Tamir, 2005), (iv)ring-star network designs (Labbé et al., 2004; Laporte and Martı́n,2007), and (v) incomplete hub networks (Campbell et al., 2005a,b;
186 E.M. de Sá et al. / European Journal of Operational Research 226 (2013) 185–202
Yoon and Current, 2008; Alumur and Kara, 2008a; Calik et al.,2009; Alumur et al., 2009).
In this work, the Tree of Hubs Location Problem (THLP), intro-duced by Contreras et al. (2009b, 2010), is addressed. The THLPis a single allocation hub location problem variant where hubsare located on a network and connected by means of a non-directed tree. Each non-hub node is single assigned to an installedhub. There are a per unit transportation cost associated with eachlink of the network and a fixed charge to install a hub in a node.The objective is to minimize the total cost that includes the trans-portation cost and can account or not for the installation costs ofthe hubs.
As remarked by Contreras et al. (2010), the design of the high-speed train network in Spain, which will place a train stationwithin 50 kilometer of every Spanish city with more the 10,000inhabitants and which it is expected to be finished by 2020, is aconcrete example of an application of the THLP. The design of rapidtransit systems for urban areas, in which citizens travel betweenorigin–destination pairs of stations, can also be seen as anothercase of the THLP.
The design of such networks has to be carefully done, since itinvolves great amounts of resources and has a major impact inthe operational costs and in the overall efficiency of the serviceafterwards. Hence efficient exact methods capable of solving morerealistic, large-scale instances in reasonable time are of the utmostimportance, despite posing as a difficult challenge.
Benders decomposition method (Benders, 1962) is one of theclassical exact methods suitable to address the THLP. Its prior suc-cessfulness in solving hub-and-spoke network problems (Camargoet al., 2008; Gelareh and Nickel, 2008; Camargo et al., 2009b;Camargo et al., 2009a; Contreras et al., 2011; Contreras et al.,2011b; Contreras et al., 2011a), as well as other problems (Costa,2005; Cordeau et al., 2006; Papadakos, 2009; Fortz and Poss,2009), has rekindled the interest of the research community forit and thus motivated the current work.
Benders decomposition is a method for solving mixed integerprogramming (MIP) problems that have special structure in theconstraint set, i.e. when fixing the complication variables (integervariables), the mathematical program reduces to an ordinary, easyto solve linear problem. The technique relies on projection andproblem manipulation, followed by solution strategies of dualiza-tion, outer linearization and relaxation (Lasdon, 1972; Minouxand Vajda, 1986).
In general terms, the complicating variables of the originalproblem are projected out, resulting into an equivalent model withfewer variables, but many more constraints. When attaining opti-mality, a large number of these constraints will not be binding,suggesting then a strategy of relaxation that ignores all but a fewof these constraints.
These constraints are then added on demand by observing twolevels of coordination. At a higher level, known as master problem(MP), a relaxed version of the original problem having the set of theinteger variables and its associated constraints is responsible forfixing the values of these integer variables and for providing a low-er bound (LB) for the problem. At a lower level, known as subprob-lem (SP), the dual of the original problem with the values of theinteger variables temporarily fixed by the MP is responsible forrendering a new cut or a Benders cut to be added to the MP andfor generating a upper bound (UB) for the problem. The algorithmiterates, solving the MP and the SP one at a time, until the UB andthe LB converge towards an optimal solution, if one exists.
Over the years, since its introduction by Benders (1962), differ-ent improvements have been proposed in order to speedup theperformance of the Benders decomposition algorithm. Basically,these enhancements can be classified into three categories: (i)Reformulations for tightening the linear programming (LP) relaxa-
tion; (ii) algorithmic add-ons; and (iii) Benders cut selectionschemes.
Geoffrion and Graves (1974) are the first authors to discuss howa tighter LP relaxation improves the overall efficiency of the algo-rithm, while McDaniel et al. (1977) show how the use of an initialset of Benders cuts, generated from the LP relaxation of the MPprior to starting the main iterations, also tightens the LP relaxationand betters the performance of the method.
Aiming at MIPs which depend on big-M coefficients to model log-ical implications, Codato and Fischetti (2006) propose a reformula-tion method for removing this dependency and, therefore, forimproving the LP relaxation of the formulation based on the additionof combinatorial inequalities to the MP instead of the traditionalBenders cuts. Their method appears to be more suitable for MIPswhose objective function depends only on the integer variables.
By applying local branching throughout the solution process,Rei et al. (2009) describe a Benders decomposition algorithm thatimproves both the LB and UB at each iteration.
Algorithmic add-ons are commonly used to ease the computa-tional effort of the technique. Cote and Laughton (1984), andPoojari and Beasley (2009) use heuristics for solving the MP andfor generating Benders cuts from the found integer feasiblesolutions, alleviating then the resolution of MP to optimality.
Working with two-stage stochastic problems, Birge andLouveaux (1988) show how to enrich the MP by disaggregatingthe Benders cuts into multiple cuts when the dual SPs are decom-posable. Fortz and Poss (2009) embed the Benders algorithm into abranch-and-cut framework, generating Benders cuts in the nodesof branch-and-bound tree of the MP. They obtain good results ona multi-layer network design problem.
Deploying a great variety of enhancements, Contreras et al.(2011) include in their algorithm: Generation of multiple Benderscuts based on the work of Birge and Louveaux (1988), reductiontests during the main iterations relying on the ideas of Mitchell(1997), and the use of a heuristic for generating optimality cutsprior to the start of the main iterations of the algorithm and forobtaining a good initial UB. The applied reduction tests are ableto eliminate a large number of the integer variables for theuncapacitated multiple allocation hub location problem.
Cut selection schemes have attracted the interest of manyresearchers since the work of Magnanti and Wong (1981). Magnan-ti andWong introduce the concepts of Pareto-optimal cuts, afternoticing that cuts of different strength can be assembled, whenthe Benders dual SPs are degenerate. They demonstrate how thestrongest possible one or the Pareto-optimal cut can be obtainedby solving an additional SP, given that a fixed core point or a pointbelonging to the relative interior of the MP convex hull is known.Although it requires the resolution of one more SP per Benderscycle, their scheme proves to be very efficient for solving networkproblems.
The scheme of Magnanti andWong for generating Pareto-optimal cuts has one major drawback: Numerical instability dueto a normalization constraint present in the additional SP. Address-ing this disadvantage, Papadakos (2008) shows how to disregardthis constraint when a new core-point is utilized at each Bendersiteration. Papadakos (2008) demonstrates that the convex combi-nation of the current MP solution for the integer variables andthe previous used core-point suffices for obtaining a new core-point at each cycle. Nevertheless, finding a starting core-point stillremains a challenge for some problems.
Working in the context of multi-commodity capacitated networkdesign problems, Costa et al. (2009) establish the relationshipsbetween three classes of inequalities: Benders, metric and cutset.They show that feasibility Benders cuts are always metric, andhow to strengthen the Benders cuts in order to attain metricinequalities.
E.M. de Sá et al. / European Journal of Operational Research 226 (2013) 185–202 187
An interesting and clever Benders cut selection scheme is pro-posed by Fischetti et al. (2010). Using the ideas of Fukuda et al.(1997), which state that obtaining a most-violated optimality cutis equivalent to finding an optimal vertex of a polyhedron with un-bounded rays, Fischetti et al. (2010) transformed the dual SP into apure feasibility problem, treating both optimality and feasibilitycuts in an unified framework. In practice, they show how to detecta ‘‘minimal source of infeasibility’’ of the SP, so as to detect a smallset of constraints that suffices to cut the master solution.
Another scheme for generating multiple strengthened cuts ispresented by Saharidis et al. (2010). Saharidis et al. (2010) devisea procedure that employs an auxiliary SP capable of examining fea-sibility cuts for variables not ‘‘covered’’ by the cut. The procedureiterates assembling a new cut for ‘‘covering’’ at least one of thesevariables until all of the variables of the MP are covered. Thescheme is more appropriate when the computational burden tosolve the MP is greater than the SP.
Directing their attention for cases in which more feasibility thanoptimality cuts are required for attaining optimality, Saharidis andIerapetritou (2010) propose a MIP based SP for finding the maxi-mum feasible subsystem (MFS). The solution of such SP generatesan extra cut that helps to restrict the value of objective function ofthe MP. These extra cuts may help reducing the total number ofiterations of the Benders algorithm. The weakness of this schemeis that the saved number of iterations may not be worth, sincethe time spend to solve the MFS SP is considerable.
More recently, Fortz and Poss (2009), Naoum-Sawaya andElhedhli (2010) integrate the Benders decomposition in a branch-and-cut algorithm where Pareto-optimal cuts are generated fromcore-points obtained by an analytic center cutting plane method(ACCPM). The found analytic centers are utilized as the core-points.Though not mentioned, the proposed dual SP is equivalent to theSP of Papadakos (2008). The effectiveness of the proposed methodon solving capacitated facility location and the multi-commoditycapacitated fixed charge network design problems relies on thequality of the attained core-points and on how iterations of theACCPM are saved by using the previous calculated analytic centeras a starting point. It is important to remark that, when facing fea-sibility cuts, they produce a traditional Benders cut based on a ran-dom selection of an extreme ray to cut off the infeasibility.
In this work, the main contribution is twofold: The developmentof a very effective specialized Benders decomposition algorithm forsolving the THLP, and the proposition of a cut selection scheme forthe method when facing both optimality and feasibility cuts. Theimplemented Benders algorithm takes advantage of the features ofthe THLP formulation of Contreras et al. (2009b), and of the genera-tion of Pareto-optimal cuts, even when facing infeasible solutions.
Furthermore, the proposed algorithm is further improved whenembedded in a branch-and-cut framework. In order to evaluateand assess its efficiency and limitations, extensive computationalexperiments are performed on instances from the classical Austra-lian Post (AP) data set.
This work is organized as follows. First, the definitions and thenotation used for the THLP are presented in Section 2. In Section 3,the two different Benders cut selection schemes – the one intro-duced by Fischetti et al. (2010), and the one proposed here – areexplained, while their computational assessed performances areshown in Section 4. The paper finishes with a section of final re-marks and possible future research lines.
2. Notation and definitions
The formulation introduced by Contreras et al. (2009b) for theTHLP requires the following definitions: Let N be the set of demandnode locations which exchange flows and let K be the set of node
candidates to become hubs. Usually K # N, but it is assumedhenceforth that all demand nodes are candidates to have an in-stalled hub, implying K � N. For all node pairs i and j (i, j 2 N:i – j), wij represents the flow demand from origin node i to destina-tion node j which is routed through at least one installed hub. Nor-mally wij – wji. Although, in the literature, there are some authors(Ernst and Krishnamoorthy, 1998b, 1999; Contreras et al., 2010)that consider demands originating and ending at the same node,i.e., the existence of wii (i 2 N), this will be disregarded here, sinceit is always possible to split nodes with this kind of demand intotwo without further complications. Let also Oi ¼
Pj2Nwij and
Di ¼P
j2Nwji be the total of demand that is originated from anddestined to node i 2 N, respectively. Further, let fk be the fixedinstallation cost of a hub at node k 2 N.
As the hubs are connected by means of a non-directed tree andas every non-hub node is single assigned to a hub, hence everypath connecting each pair of origin–destination i � j (i, j 2 N: i – j)is unique. Further, besides, possibly, the first and the last segmentsof this path, all of the links of it will be inter-hub connections.When only one hub is present on this path, there is no inter-hublinks. Hence, the cost per unit of flow of this path, between i � j(i < j), is the sum of the costs of each segment, where, on the in-ter-hub connections, it is applied a discount factor 0 6 a 6 1 repre-senting the scale economies. Each unit of product that traverses asegment from a non-hub node i 2 N to a hub k 2 N (likewise froma hub m 2 N to a non-hub node j 2 N) incurs in a cost cik P 0(cmj P 0), whereas when it passes through an inter-hub connec-tion, for example, linking hubs k and m (k,m 2 N), then the per unitcost associated is ackm. Moreover, the triangle inequality and thesymmetry of costs are not assumed.
The MIP formulation uses flow variables xijkm P 0 to representthe fraction of demand wij (i, j 2 N) that is routed through inter-hubconnection k �m (k,m 2 N); and, the integer variables zik 2 {0,1}and ykm 2 {0,1} to indicate if node i 2 N is allocated to hub k 2 N(zik = 1) or not (zik = 0) and if the inter-hub connection k �m(k < m 2 N) is utilized to link hubs k and m (ykm = 1) or not(ykm = 0), respectively. When a hub is located at node k 2 N, thenzkk = 1; otherwise zkk = 0.
In the THLP, as each non-hub node is allocated to a single in-stalled hub only, demands wij and wji are sent over the same pathand pass through the same inter-hub connections. Thus the num-ber of variables xijkm can be reduced to half. Furthermore, as theoutgoing and the incoming flows of a non-hub must go and arrivethrough the same hub, the cost component of local traffic can bewritten as
Pk–iðOi cik þ Di ckiÞ zik. In the remainder of this paper,
for the sake of simplicity in presentation, one must consider i, j,k, m 2 N and i < j. Hence the implied formulation introduced byContreras et al. (2009b) is given as:
minX
k
fkzkk þX
i
Xk:k–i
ðOi cik þ Di ckiÞzikþXi
Xj:j>i
Xk
Xm:m–k
ðckmwij þ cmkwjiÞ a xijkm ð1Þ
s:t: :zik 6 zkk 8 i – k ð2ÞXk
zik ¼ 1 8 i ð3Þ
ykm 6 zkk 8 k < m ð4Þykm 6 zmm 8 k < m ð5ÞX
k
Xm:m>k
ykm ¼X
k
zkk � 1 ð6ÞX
k:k–m
xijkm þ zim ¼X
r:r–m
xijmr þ zjm 8 i < j;m ð7Þ
xijkm þ xijmk 6 ykm 8i < j; k < m ð8ÞXm:m–k
xijkm 6 zkk 8 i < j; k ð9Þ
188 E.M. de Sá et al. / European Journal of Operational Research 226 (2013) 185–202
Xm:m–k
xijmk 6 zkk 8 i < j; k ð10Þ
xijkm P 0 8 i < j; k – m ð11Þ
ykm 2 f0;1g 8 k < m; ð12Þ
zik 2 f0;1g 8 i; k ð13Þ
The objective function (1) minimizes the total cost associated withthe installation of the hubs and the demand transportation costsfrom the non-hub nodes to the hubs and on the inter-hub connec-tions. Constraints (2) ensure that each node is assigned to a hub,whereas constraints (3) allow a node i to be allocated to hub k onlyif hub k is installed. Constraints (4) and (5) only admit inter-hubconnection k �m if hubs k and m are set, respectively.
Constraints (6) are required to guarantee the formation of a treeon the inter-hub connections, because the y variables alone are notsufficient to accomplish it. Constraints (7) are flow balancing con-straints for paths connecting each pair i � j (i < j). Constraints (8)guarantee that the flow of pair i � j can use the inter-hub connec-tion k �m only if this link is present. Constraints (9) and (10) as-sure that the flow of i � j uses inter-hub connection k �m only ifhubs k and m are opened, respectively. Constraints (11)–(13) arenon-negativity and integrality constraints, respectively.
The constraints 4 and 5 and 9 and 10 are not present on the ori-ginal formulation. Though they are redundant, they improve thequality of the LP relaxation. Moreover, when the number p of hubsto be installed is known a priori, the first term
Pkfkzkk
� �of the total
cost (1) can be disregarded and a further constraint is requiredPkzkk ¼ p
� �.
Although there is another formulation for the THLP (Contreraset al., 2010), the MIP (1)–(13) is chosen because it has a very tightLP bound and an interesting property: for fixed feasible vectors zand y, the resulting subproblem is decomposable into LPs, onefor each i � j pair. Therefore, in light of the aforementioned, adecomposition strategy that takes advantage of these featureshas an appealing motivation. Such a technique is the Bendersdecomposition method (Benders, 1962), subject of the nextsection.
3. Benders decomposition
As previously mentioned, Benders decomposition is a classicalpartitioning method for MIP problems that relies on a problemmanipulation using projection, followed by solution strategies ofdualization, outer linearization and relaxation. In this section, a ba-sic development of the method is presented for formulation (1)–(13). The original problem is divided into two smaller ones: a SPand a MP, subjects of Sections 3.1 and 3.2, respectively.
Since the dual SP may have multiple optimal solutions and sincethe strength of the Benders cuts is closely related to the quality ofoptimal dual solutions used to generate them, an effective cutselection scheme is proposed in Section 3.4. The procedure is capa-ble to choose the most appropriate dual solution for generatingPareto-optimal cuts for the MP, even when having to produceinfeasible cuts. In order to assess its efficiency, the proposed selec-tion scheme is compared with the one introduced by Fischetti et al.(2010), explained in Section 3.5.
3.1. Benders subproblem
For fixed values of the integer (complicating) variables, z ¼ �zand y ¼ �y, a primal linear SP is obtained. This primal SP is decom-posable in independent routing problems, one for each i � j (i < j),stated as:
minX
k
Xm:k–m
ðckmwij þ cmkwjiÞ a xijkm ð14Þ
s:t: :X
k:k–m
xijkm �X
r:r–m
xijmr ¼ �zjm � �zim 8 m ð15Þ
xijkm þ xijmk 6 �ykm 8 k < m ð16Þ
Xm:m–k
xijkm 6 �zkk 8 k ð17Þ
Xm:m–k
xijmk 6 �zkk 8 k ð18Þ
xijkm P 0 8 k – m ð19Þ
Whenever the integer variables induce a feasible allocation andspanning tree structure connecting the hubs, an UB for the originalproblem (1)–(13) is readily available if the optimal values of (14)are summed to the fixed quantities of
Pkfk�zkk andP
k–iðOi cik þ Di ckiÞ �zik. Further, the dual form of (14)–(19) can beattained after associating the dual variables uijm, eijkm, sijk and tijk
to constraints (15)–(18), respectively. Hence a dual SP, one for eachi � j pair (i < j), can be given as:
maxX
m
ð�zjm � �zimÞuijm �X
k
Xm:k<m
�ykmeijkm �X
k
�zkkðsijk þ tijkÞ ð20Þ
s:t: :uijm � uijk � eijkm � sijk � tijm 6 aðckmwij þ cmkwjiÞ 8k < m ð21Þuijm � uijk � eijmk � sijk � tijm 6 aðckmwij þ cmkwjiÞ 8k > m ð22Þuijm 2 R 8m ð23Þeijkm P 0 8k < m ð24Þtijm P 0 8m ð25Þsijm P 0 8m ð26Þ
Using the objective function (20), two types of Benders cuts can beproduced in order to enrich the MP at each iteration. When the dualSP (20)–(26) is unbounded, implying in an infeasible primal SP(14)–(19), a feasibility cut can be assembled:
Xi
Xj:j>i
Xm
ðzjm � zimÞ�uijm �X
k
Xm:k<m
ykm�eijkm �X
k
zkkð�sijk þ �tijkÞ" #
6 0
ð27Þ
where �uijm; �eijkm; �sijk and �tijk are the values of the extreme ray of theunbounded dual SP. Whereas, when dual SP (20)–(26) is bounded,implying in a feasible primal SP (14)–(19), an optimality cut canbe assembled with the help of an auxiliary variable g:
g PX
i
Xj:j>i
Xm
ðzjm � zimÞ�uijm �X
k
Xm:k<m
ykm�eijkm �X
k
zkkð�sijk þ �tijkÞ" #
ð28Þ
where �uijm; �eijkm, �sijk and �tijk are the optimal values of the dual vari-ables of the bounded dual SP of iteration h. The variable g P 0 is acontinuous variable that is used to underestimate the transporta-tion cost. As the transportation costs are non-negative, then the var-iable g is also non-negative.
3.2. Benders master problem
The original problem (1)–(13) is then equivalent to the follow-ing Benders MP:
E.M. de Sá et al. / European Journal of Operational Research 226 (2013) 185–202 189
minX
k
fkzkk þX
i
Xk:k–i
ðOi cik þ Di ckiÞ zik þ g
s:t: :ð2Þ � ð6Þ
g PX
i
Xj:j>i
Xm
ðzjm � zimÞuijm �X
k
Xm:k<m
ykmeijkm
"
�X
k
zkkðsijk þ tijkÞ#
8 ðu; e; s; tÞ 2 K
Xi
Xj:j>i
Xm
ðzjm � zimÞuijm �X
k
Xm:k<m
ykmeijkm
"
�X
k
zkkðsijk þ tijkÞ#6 0 8 ðu; e; s; tÞ 2 X
gij P 0 8 i < j
ykm 2 f0;1g 8 k < m
zik 2 f0;1g 8 i; k
where K and X are sets of extreme points and rays of the dual SP(20)–(26).
The Benders MP consists of the sets of the complicating(integer) variables and their respective constraints, the continuousvariable g, and the Benders cuts. The number of Benders cuts canbe exponential. Nevertheless, just some of them will be active inan optimal solution, suggesting thus a strategy of relaxation.
This stratagem solves the Benders MP iteratively, adding theBenders cuts when needed. As the Benders MP is now a relaxedreformulation of the original problem, it provides therefore a LB.This LB is improved by the addition of the Benders cuts (28) at eachiteration, if the dual SP is bounded. When the LB converges to theUB, an optimal solution of the original problem is obtained, if oneexists.
Furthermore, observing the Benders cuts (27) and (28), they canbe disassembled into different formats: One for each i � j pair (i < j)or n cuts for each i, where n = jNj. Remark that variable g has to beresized accordingly (Birge and Louveaux, 1988). The employed re-laxed MP can be stated as follows:
minX
k
fkzkk þX
i
Xk:k–i
ðOi cik þ Di ckiÞ zik þX
i
Xj>i
gij ð29Þ
s:t: :ð2Þ � ð6Þgij P
Xm
ðzjm � zimÞuhijm �
Xk
Xm:k<m
ykmehijkm
�X
k
zkk shijk þ th
ijk
� �8 i < j; h 2 H ð30Þ
Xm
ðzjm � zimÞuhijm �
Xk
Xm:k<m
ykmehijkm
�X
k
zkk shijk þ th
ijk
� �6 0 8 i < j; h 2 G ð31Þ
gij P 0 8 i < j ð32Þykm 2 f0;1g 8 k < m ð33Þzik 2 f0;1g 8 i; k ð34Þ
where H and G are the sets of iterations having the dual SP boundedand unbounded, respectively. The presence of constraints (31) arehere required, because, unfortunately, constraints (6) are not en-ough now to ensure that a spanning tree is going to be formed by
the integer variables z and y. And hence infeasible primal SPs areto be expected, implying that feasibility cuts (31) will be added tothe MP.
As the feasibility cuts do not help to improve the LB of the prob-lem, they are algorithmically undesirable. One way to prevent un-bounded dual SPs and consequently the use of constraints (31) is toincorporate a further set of constraints to the MP that guaranteethe formation of a spanning tree by the integer variables z and y.These additional constraints need to be carefully selected in orderto not overload the resolution of the MP (29)–(34).
The idea is to introduce constraints to the MP that can induce aspanning tree rooted at a fictitious node labeled 0, and that hasonly installed hub nodes and active inter-hub connections as itscomponents. This can be accomplished through the use of tradi-tional spanning tree formulations (Maculan et al., 2003). Usually,in these formulations, one node is chosen as the root node that actsas a supply node for the other demand nodes, and one unit of flowhas to arrive at each demand node by means of a connected tree.
Here, the number of installed hubs represents the flow quantitythat must leave the fictitious root node 0 through only one connec-tion to an installed hub. Each hub must then receive one unit ofthis flow and be connected by means of a tree. Using thus flowvariables fkm P 0, where k 2 N [ {0}, m 2 N and k – m, to representthe traffic leaving root node 0 and on the inter-hub connectionsk �m, and the integer variables qk 2 {0,1} that indicates if the rootnode link to hub k is active (qk = 1) or not (qk = 0), the followingconstraints can be added to the MP in order to induce a spanningtree on the integer variables z and y:
Xm2N
f0m ¼Xk2N
zkk ð35ÞX
m2N[f0g:m–k
fmk �Xm2Nm–k
fkm ¼ zkk 8 k 2 N ð36Þ
fkm 6 n ykm 8 k;m 2 N : k < m ð37Þfkm 6 n ymk 8 k;m 2 N : k > m ð38Þf0k 6 n qk 8 k 2 N ð39ÞXk2N
qk ¼ 1 ð40Þ
qk 6 zkk 8 k 2 N ð41Þfkm P 0 8 k 2 N [ f0g;m 2 N : k – m ð42Þqk 2 f0;1g 8 k 2 N ð43Þ
recalling that n = jNj. Constraint (35) assures that the flow quantitythat leaves the root node 0 is equal to the number of installed hubs.The flow balancing constraints (36) guarantee that every installedhub will be connected by means of a spanning tree. Constraints(37)–(39) ensure that the flow can use the connections only if theselinks are present. Constraint (40) allows the use of only one connec-tion leaving node 0, while constraints (41) state that these links areonly valid if there is an installed hub on its end. Constraints (42) and(43) are the non-negativity and integrality constraints, respectively.
Adding (35)–(43) to the MP induces the formation of a spanningtree on the integer variable z and y at the expense of an additionaln2 continuous and n integer variables, and of n2 + 2n + 2 con-straints. Hence Benders cuts (31) can now be avoided when theMP supplies integer values for the variables z and y. It is worthof remark here that, when generating Benders cuts at a branch-and-cut framework, i.e. when the values of z and y are fractional,constraints (35)–(43) do not guarantee the formation of a spanningtree. A basic Benders algorithm is illustrated in Algorithm 1, where/MP and /SP are the values of the objective functions of the MP andthe dual SP, respectively.
190 E.M. de Sá et al. / European Journal of Operational Research 226 (2013) 185–202
Algorithm 1. A basic Benders decomposition for the THLP.
UB 1, LB 0, h 0while UB – LB do
Solve MP (29)–(43)LB /MP(g,z,y)Solve SP (20)–(26)Add cut (30) to MPUpdate UB, if necessaryh h + 1
end while
3.3. Improving the efficiency of the method
The efficiency of Benders decomposition algorithm dependsmainly on the number of iterations required to attain globalconvergence. This number is close related to the strength of theBenders cuts generated, i.e. the values of the dual variables se-lected. Strong cuts usually mean fewer iterations. Recall that, fora given iteration, the dual SP usually may have alternative optimalsolutions, since it is usually degenerated. Hence, in order to havestrong cuts, the solution of the dual SP has to be judiciously doneat each iteration, because if the values of the dual variables arenot carefully chosen, then a poor behavior will be expected.
Magnanti and Wong (1981) propose a dual variable selectionscheme to tackle this issue and thus accelerate the Benders algo-rithm. They notice that, as the dual SP is highly degenerated inmost applications, multiple optimal solutions can assembledifferent Benders cuts. Rather than adding them all to the MP, theypropose a slightly different dual SP in order to find the strongestcut or a cut that dominates all the other cuts in the sense of Pareto-optimality. In the context of the THLP, the strength of a cutaccording to Magnanti and Wong can be stated as the followingdefinition:
Definition 1. A cut is Pareto-optimal if it is not dominated by anyother cut. For instance, let U = {(e,u,s, t):satisfying constraints(21)–(26)} be the set of feasible values for the dual variables. ABenders cut (28) corresponding to (u1,e1,s1, t1) 2 U dominates thatcorresponding to (u2,e2,s2, t2) 2 U, if:X
m
ðzjm � zimÞu1ijm �
Xk
Xm:k<m
ykme1ijkm �
Xk
zkk s1ijk þ t1
ijk
� �
PX
m
ðzjmzimÞu2ijm �
Xk
Xm:k<m
ykme2ijkm �
Xk
zkk s2ijk þ t2
ijk
� �
8 ðz; yÞ 2 Z
with strict inequality for at least one point (z,y) 2 Z, whereZ = {(z,y): constraints (2)–(6), (12) and (13) hold}. Similarly, it is saidthat (u1,e1,s1, t1) dominates (u2,e2,s2, t2) and it is called Pareto-optimal.
In order to build Pareto-optimal cuts, the notion of core pointsis required. For the THLP, a core point has the following definition:
Definition 2. A point (z0,y0) 2 Z is a core point if it belongs to therelative interior of the convex hull of Z or (z0,y0) 2 ri(Zc), where ri(�)and Zc are the relative interior and convex hull of set Z,respectively.
Originally, Magnanti andWong propose the solution of two dif-ferent SPs at each iteration: One dual SP associated to the currentMP solution and another related to an initial fixed given core point.For the THLP, for each i � j (i < j), zh and yh, and an initial core point(z0,y0), the SP of Magnanti andWong can be states as:
maxX
m
z0jm � z0
im
� �uijm �
Xk
Xm:k<m
y0kmeijkm �
Xk
z0kkðsijk þ tijkÞ ð44Þ
s:t: :ð21Þ � ð26Þ
Xm
zhjm � zh
im
� �uijm �
Xk
Xm:k<m
yhkmeijkm
�X
k
zhkkðsijk þ tijkÞ ¼ /SPij
ðu; e; s; tÞ ð45Þ
where /SPijis the optimal value of the objective function of the dual
SP associated with the solution (zh,yh) of the MP of iteration h. TheSP (44) and (45) generates Pareto-optimal cuts that speed up theconvergence of the Benders decomposition method. However thetime required for solving it and the number of Benders iterationsit saves have to be assessed prior to be effectively used. Observethat constraint (45) is very dense and prone to numerical instability.
However, Papadakos (2008) shows that it is possible to disre-gard constraint (45) by using a different core point (z0,y0) on theobjective function (44) at each iteration. This allows the SP ofMagnanti and Wong to be lighter, shortening thus the computa-tional effort for the generation of Pareto-optimal cuts. Further-more, Papadakos (2008) demonstrates that (z0,y0) does not evenneed to be a core point to give a Pareto-optimal cut. It can be apoint of Magnanti and Wong.
Definition 3. A point (z0,y0) is a point of Magnanti and Wong if theoptimal solution of the Magnanti and Wong (1981) SP (44) and(45) or the Magnanti and Wong (1981) SP (44), (21)–(26) gives aPareto-optimal cut.
An alternative definition for a point of Magnanti and Wong,proved by Papadakos (2008), is given by Definition 4. This defini-tion is going to be used for devising the new cut selection scheme,presented on Section 3.4.
Definition 4. If a point (z0,y0) is not one of the points that spansthe set of all possible projections into the variables of the MP; andwhen the space of feasible solutions of the dual SP is not empty,and the primal SP is feasible for (z0,y0), then (z0,y0) is a point ofMagnanti and Wong.
Although, Mercier et al. (2005) remark that there are not prac-tical methods to find good core points or points of Magnanti andWong, Papadakos (2008) shows that any convex combination ofa valid initial point of Magnanti and Wong (z0,y0) and (z,y) 2 Z isalso a valid point of Magnanti and Wong. Hence, in the contextof THLP, an approximation scheme can be successfully employed,such as:
z0hþ1 ¼ ð1� kÞz0
h þ kzh ð46Þ
y0hþ1 ¼ ð1� kÞy0
h þ kyh ð47Þ
where 0 < k < 1. Empirically k = 1/2 gives the best results (Papadakos,2008; Mercier et al., 2005). At each iteration h, the SP (44), (21)–(26)is solved after the updating scheme (46) and (47) is used. Notice thatconstraint (45) is not considered. Further, an valid initial point ofMagnanti and Wong (z0,y0) is required. For the THLP, a valid onecan be given by:
z0kk ¼ 1=2 8 k ð48Þ
z0ik ¼ 1=ð2 n� 2Þ 8 i – k ð49Þ
y0km ¼ ðn� 2Þ=ðn2 � nÞ 8 k < m ð50Þ
E.M. de Sá et al. / European Journal of Operational Research 226 (2013) 185–202 191
Proposition 1. The point (48)–(50) is a valid point of Magnanti andWong.
Proof of Proposition 1. The initial value of (z0,y0) can be con-structed by a convex combination of n + 1 different feasible solu-tions: n solutions having only a single hub, and one solutionwith all nodes as hubs, i.e. n installed hubs. By combining thesesolutions, it is possible to obtain a valid z0 vector by assigningproper values to the weights of the convex combination, and byobserving that the constraints of the MP and the definition ofconvex combination are respected. Hence a weight equals to1/2(n � 1) can be prescribed for the n first single solutions, whilea weight of (1/2 � 1/2(n � 1)) can be given for the solution withn installed hubs.
The y0 vector can be calculated in a similar manner, recallingthat the n first solutions with a single hub have a null y vector.Thus the y0 vector can be built by observing that it is possible toconvex combine n possible shortest path trees of the solutionhaving n hubs, each one having n hubs, using weights equal to(1/2 � 1/2(n � 1)). As each arc appears twice, then each compo-nent of the y vector assumes a 2/n value. Therefore, for eachk < m; y0
km ¼ ð1=2� 1=2ðn� 1ÞÞð2=nÞ, yielding y0km ¼ ðn� 2Þ=n
ðn� 1Þ. h
An outline of the Benders decomposition algorithm based onPapadakos (2008) is presented on Algorithm 2. Notice thatthe point of Magnanti andWong (z0,y0) is updated at everyiteration.
Algorithm 2. A Benders decomposition based on Papadakos(2008) for the THLP.
1: UB 1, LB 0, h 0, k 0.52: Find an initial point of Magnanti andWong (z0,y0)3: while UB – LB do4: Solve SP (44), (21)–(26)5: Add cut (30) to MP6: Solve MP (29)–(43)7: LB /MP(g,z,y)8: Solve SP (21)–(26)9: Add cut (30) to MP10: (z0,y0) (1 � k)(z0,y0) + k(zh,yh)11: Update UB, if necessary12: h h + 113: end while
McDaniel et al. (1977) demonstrate that solving the MP,while disregarding the integrality constraints at the initial cyclesof the algorithm, helps to speed it up. As the first MPs haslimited information to greatly improve the LB, it is more inter-esting to replace computational demanding, integer MPs forlighter, linear MPs. Adding Benders cuts produced from linearMPs at the beginning of the algorithm is known as a warm-startphase.
Furthermore, embedding the algorithm into a branch-and-cutframework has also shown to be very helpful in shortening thenumber of iterations and the computational effort (Fortz and Poss,2009; Naoum-Sawaya and Elhedhli, 2010; Contreras et al., 2011).In both aforementioned cases, Benders cuts are produced fromfractional values for the integer variables.
In order to incorporate these ideas into Algorithm 2, a differentapproach is needed, because the scheme of Papadakos (2008)requires that the solutions provided by the MPs induce feasibleprimal SPs. Otherwise traditional feasibility Benders cuts are gen-
erated, being the dual SP (44), (21)–(26) not solved and the pointof Magnanti and Wong not updated meanwhile, delaying thenthe convergence of the method.
Unfortunately, constraints (35)–(43) are not enough for guaran-teeing feasible primal SPs (14)–(19) when the values of the vari-ables z and y are fractional. Hence, infeasible primal SPs areexpected to happen during a warm-start phase or inside a nodeof one of the branch-and-bound trees of the MPs, implying thenthe generation of feasibility cuts. Therefore, a new cut selectionscheme that exploits solution vectors, which cause infeasible pri-mal SPs, by generating optimality cuts (30) from them has anappealing motivation, being the subject of the next Section 3.4.
3.4. A new cut selection scheme (Ok)
The idea of the new cut selection scheme is simple, but veryeffective. When the MP provides solutions (zh,yh) that result ininfeasible primal SPs, it is not possible to use a fixed value forparameter k to do the convex combination with the previous pointof Magnanti and Wong (line 10 of Algorithm 2). Because it canyield a point for which Definition 4 does not hold.
This situation is illustrated in Fig. 1a. The thin and bold linesrepresent the feasible space and the convex hull of set Z, whilethe bold triangle and square depict the previous point of Magnantiand Wong z0 and the solution zh provided by the MP at iteration h,respectively. The empty circle is the convex combination of z0 andzh. In Fig. 1a, the convex combination of z0 and zh using a fixedparameter k = 0.5, for instance, produces a point that is outside ofthe set Z, disrespecting thus Definition 4. However, when a valueof k is properly computed observing Definition 4, a new valid pointof Magnanti and Wong can be obtained, see Fig. 1b, and used togenerate Pareto-optimal cuts (30).
Hence, for those MP solutions that results in infeasible primalSPs, i.e. unbounded dual SPs (line 8 of Algorithm 2), it is importantto compute valid values for k prior to obtain new points ofMagnanti and Wong (line 10 of Algorithm 2). The value of k mustbe chosen such that the convex combination of the current MPsolution with the previous point of Magnanti and Wong resultsin a feasible primal SP. In other words, the parameter k can beobtained after solving the following subsystems, one for eachi � j (i < j):
maxkij ð51Þ
s:t: :X
k:k–m
xijkm �X
r:r–m
xijmr ¼ ð1� kijÞ z0jm � z0
im
� �
þ kij zhjm � zh
im
� �8 m ð52Þ
xijkm þ xijmk 6 ð1� kijÞy0km þ kijyh
km 8 k < m ð53ÞXm:m–k
xijkm 6 ð1� kijÞz0kk þ kijzh
kk 8 k ð54Þ
Xm:m–k
xijmk 6 ð1� kijÞz0kk þ kijzh
kk 8 k ð55Þ
xijkm P 0 8 k – m ð56Þ
0 6 kij 612
ð57Þ
where the value of k that satisfies all the subsystems (51)–(57) isthen given by min{kij}, " i, j: i < j, being entitled as optimal-k.
This methodology is worthwhile because it extends the updat-ing scheme of the point of Magnanti and Wong proposed byPapadakos (2008) to problems where the MP can generate infeasi-ble primal SPs. It is now possible to use convex combination in
z 0
z h
z 0
z h
(a) (b)Fig. 1. An invalid (a) and a valid (b) convex combination of a point of Magnanti and Wong z0 and a MP’s solution zh of itereation h, respectively.
192 E.M. de Sá et al. / European Journal of Operational Research 226 (2013) 185–202
order to produce a new point of Magnanti and Wong regardless thecurrent MP solution yields feasible or infeasible primal SPs. An out-line of the proposed Benders algorithm for the warm-start itera-tions, considering the optimal-k, is given in Algorithm 3, whereMaxIter is the maximum number of warm-start iterations. In thepresent work, this scheme is referred from now on as the Ok vari-ant of the Benders decomposition algorithm.
Algorithm 3. A Benders decomposition using k-optimal for thewarm-start phase.
LB 0, h 0Find an initial point of Magnanti and Wong (z0,y0)while h < MaxIter do
Solve SP (44), (21)–(26)Add cut (30) to MPSolve MP (29)–(43)LB /MP(g,z,y)Solve SP (20)–(26)if SP is unbounded then
Add cut (31) to MPFind the optimal-kk optimal-k
elseAdd cut (30) to MPk 0.5
end if(z0,y0) (1 � k)(z0,y0) + k(zh,yh)
h h + 1end while
In order to assess the performance of the proposed cut genera-tion strategy, two different cut selection schemes are used forcomparison and presented in the next sections: the minimuminfeasible systems proposed by Fischetti et al. (2010) and the gen-eration of feasibility cuts (31) when the point of Magnanti andWong is not valid.
3.5. Minimum Infeasible Subsystems (MISs)
Fischetti et al. (2010) introduce another cut selection criterion,specially addressed for problems where both feasibility and opti-mality cuts are present. In order to produce more clever cuts, theypropose an unified framework that generates a single type of cut.
This cut selection scheme is based on the fact that the primal SPcan always be reformulated as an infeasible problem, i.e., the dualSP can be transformed into an unbounded problem.
For the THLP, this is accomplished by transforming the objec-tive function (14) into a constraint, bounded by the value of thevariable �gij of the MP. The objective function is then replacedby a null one, yielding the following primal SP, one for eachi � j (i < j):
min 0 ð58Þ
s:t: :X
k:k–m
xijkm �X
r:r–m
xijmr ¼ �zjm � �zim 8 m ð59Þ
xijkm þ xijmk 6 �ykm 8 k < m ð60ÞXm:m–k
xijkm 6 �zkk 8 k ð61Þ
Xm:m–k
xijmk 6 �zkk 8 k ð62Þ
Xk
Xm:k–m
ðckmwij þ cmkwjiÞ a xijkm 6 �gij ð63Þ
xijkm P 0 8 k – m ð64Þ
The primal SP (58)–(64) is only feasible when the optimal solu-tion is found, implying then that constraint (63) is met with equal-ity. When this is not the case, the primal SP is infeasible resultingthus in an unbounded dual subproblem represented, for each i � j(i < j), as:
maxX
m
zhjm � zh
im
� �u1
ijm � u2ijm
� �� zh
mmðsijm þ tijmÞh i�X
k
Xm:k<m
yhkmeijkm � p0
ijghij ð65Þ
s:a : u1ijm � u2
ijm
� �� u1
ijk � u2ijk
� �� eijkm � sijk � tijm
6 p0ijaðckmwij þ cmkwjiÞ 8 k; m : k < m ð66Þ
u1ijm � u2
ijm
� �� u1
ijk � u2ijk
� �� eijmk � sijk � tijm
6 p0ijaðckmwij þ cmkwjiÞ 8 k; m : k > m ð67Þ
u1ijm;u
2ijm; sijm; tijm P 0 8 m ð68Þ
eijkm P 0 8 k; m : k < m ð69Þp0
ij P 0 ð70Þ
Table 1Ok with multiple cuts versus a single cut version.
Instance Ok with constraints (30) and (31) Ok with constraints (28) and (27)
Time (s) # Of iters. for Time (s) # Of iters. for GAP(%)
E.M. de Sá et al. / European Journal of Operational Research 226 (2013) 185–202 193
where p0ij P 0 is the dual variable associated to constraint (63).
From the extreme rays ð�u1; �u2; �e;�s;�t; �p0Þ of the set (66)–(70), validcuts can be produced for each i � j (i < j):
Xm
ðzjm � zimÞ �u1ijm � �u2
ijm
� �� zmmð�sijm þ �tijmÞ
h i
�X
k
Xm:k<m
ykm�eijkm � p0ijg
hij 6 0 ð71Þ
In order to compute the values of the dual variables for the ex-treme rays, Fischetti et al. (2010) propose the detection of smallcardinality set of cuts that suffices to cut the current MP solution,i.e. the identification of a ‘‘minimal source of infeasibility.’’ Asestablished by Gleeson and Ryan (1990), there is a correspondencebetween any row of any Minimal Infeasible Subsystem (MIS) andthe support of the vertices of the so called alternative polyhedron.In practical terms, this alternative polyhedron is assembled byadding the normalization constraint (72) to the dual SP (65)–(70).
Xm
u1ijm � u2
ijm þ sijm þ tijm
� �þX
k
Xm:m>k
eijkm þw0p0ij ¼ 1 ð72Þ
Its important to notice that there is a direct correspondenceamong the cuts (71), generated by the dual SP (65)–(72), and thefeasibility (27) and optimality (28) cuts. When the value ofp0 = 0, then a feasibility cut is attained, whereas when the valueof p0 > 0, then an optimality cut is produced. So the cuts obtainedfrom the SP (65)–(70), (72) remain being valid Benders cuts.
An outline of the Benders decomposition algorithm using cutsbased on MIS for the warm-start phase is presented in Algorithm4, where /MIS is sum of the values of the objective function ofthe dual SP (65)–(70), (72). Remark that, as the dual SP is maximi-zation problem, the algorithm stops when /MIS is equal to zero.This procedure is now on referred in the present text as the MISvariant.
Algorithm 4. A Benders decomposition using cut based on MIS forthe warm-start phase.
194 E.M. de Sá et al. / European Journal of Operational Research 226 (2013) 185–202
/MIS 1, LB 0, h 0while /MIS > 0 or h < MaxIter do
Solve MP (29)–(43)LB /MP(g,z,y)Solve MIS SP (65)–(70), (72)Add cut (71) to MPCompute /MIS(u1,u2,s, t,e,p0)h h + 1
end while
3.6. The Papadakos (2008)’ algorithm (CII)
Another cut selection scheme can also be implemented forassessing the performance of the optimal-k strategy. This schemehas already been proposed by Papadakos (2008) and it consistson updating the core-point only when the MP provides solutionsthat lead to a point of Magnanti and Wong, see Definition 4. Other-wise, random extreme rays are gathered in order to producefeasibility cuts.
An outline of the Benders Decomposition based on this strategyfor the warm-start phase is presented on Algorithm 5. The convexcombination is done by using the fixed value of k = 0.5. This algo-rithm is referred as the CII variant of the Benders decompositionmethod.
Algorithm 5. A Benders decomposition CII for the warm-startphase.
LB 0, h 0, k 0.5Find an initial point of Magnanti andWong (z0,y0)while h < MaxIter do
Solve MP Eqs. (29)–(43)LB /MP(g,z,y)Solve SP (20)–(26)if SP is unbounded then
Add cut (31) to MPelse
Add cut (30) to MP(z0,y0) (1 � k)(z0,y0) + k(zh,yh)Solve SP (44), (21)–(26)Add cut (30) to MP
end ifh h + 1
end while
3.7. Embedding the Benders cuts in a branch-and-cut framework
Usually, the presence of integrality constraints in the masterproblems makes their solution time to be greater than the solutiontime of the dual subproblems. Hence, whenever the number ofmaster problems solved and their solution time are shortened,the better is for the overall performance of the Benders decompo-sition algorithms.
Nowadays, one way of addressing both cases is by embeddingthe generation of Benders cuts into a branch-and-cut frameworkfor solving the master problem of each iteration (Fortz and Poss,2009; Naoum-Sawaya and Elhedhli, 2010), and by re-utilizingthese cuts from the branch-and-bound tree of the previous itera-tion in the next one by means of a pool P of cuts.
Embedding Benders cuts in a branch-and-cut framework allowsthe use of different strategies for producing these cuts. For in-
stance, the cuts can be generated in every feasible branch-and-bound node, or only in those nodes that have improved the lowerbound beyond a predefined threshold, or only when incumbentsolutions are found along the search process. One needs to care-fully assess which strategy is more appropriate depending on theproblem being address, because, if caution is not taken, too manyunnecessary cuts can be produced slowing down therefore thesolution of the master problem, instead of speeding it up.
In the present context, the number of Benders cuts (30) and (31)to be added each time can be considered large (O(n2)), hence a suit-able cut generation strategy is necessary. Since commercial solverscombine branch-and-bound and heuristics to search for integersolutions, these solutions can be found in any node of the searchtree. Whenever this happens, new cuts can be added to the cut poolP generated from both fractional or integer solutions. The formerhappens when the integer solutions are returned by the solverheuristics; while, in the latter, integer solutions come from integernodes. Recall that, in the integer nodes, feasibility cuts are neverproduced. The adopted strategy is akin to the one selected byFischetti et al. (2010) with the difference of having cuts generatedfrom fractional nodal solutions.
Such strategy is better illustrated in Algorithm 6, where thepool of cuts P is initialized with cuts coming from the warm startphase and adjoined by the cuts of each iteration and by the onesgenerated inside the branch-and-bound tree. This branch-and-cutframework is used to solve the master problems of the aforemen-tioned Benders decomposition algorithms.
Algorithm 6. A branch-and-cut framework (B&C) for Bendersbased algorithms.
Set tree T ¼ fog, where o has no branching constraints, buthas the cuts of pool P
while T is nonempty doSelect a node o0 2 TT T n fo0gSolve o0 and let (z⁄,y⁄) be an optimal solutionif (z⁄,y⁄) is integer or a new integer solution is found from(z⁄,y⁄) then
Add Benders cuts to PT T [ fo0g
elseBranch, resulting in nodes o00 and o000,T T [ fo00; o000g
end ifend while
4. Computational experiments
Three variants of the Benders decomposition method wereimplemented: The Ok, based on the new cut selection scheme(Section 3.4); the CII, based on feasibility cuts generated bythe Papadakos (2008) algorithm (Section 3.6); and the MIS, basedon the ideas of minimum infeasible subsystems (Section 3.5). Allvariants were coded in C++ using the Concert Technology (CPLEX12) set to execute only one thread to solve the MPs and the SPs.They all were devised to have a warm start phase and to generatecuts inside the branch-and-bound tree through the use of theCPLEX callback classes. Further, the integer phase of each versionis based on Algorithm 2.
All computational tests were carried out on an Intel Xeon with 8cores of 2.53 gehahertz and 24 gehabite of memory, running theLinux operating systems. Furthermore, instances generated from
Table 3Warm start phase.
Instance # Of iters. Ok CII MIS
Time (s) GAP (%) # Of cuts Time GAP (%) # Of cuts Time (s) GAP (%) # Of cuts
E.M. de Sá et al. / European Journal of Operational Research 226 (2013) 185–202 195
the Australian Post (AP) standard data set were utilized as the testbed. This data set, introduced by Ernst and Krishnamoorthy (1996,1998a), can produce instance sizes ranging from 10 up to 200nodes, and having discount factor a = {0.2,0.4,0.6,0.8}. However,as this set has fixed cost information for the first 50 nodes only,the fixed costs were then randomly generated for all of the nodesof all instances using a Gaussian distribution with average equalto fo and the coefficient of variance set to 40% in order to portrayhow the installation costs vary in real problems. The parameter fo
has been introduced by Ebery et al. (2000) and represents thescaled difference in objective value between a scenario in whichthere is a virtual hub located in the center of mass of the demandand scenario in which all nodes are hubs. For all the carried outtests, the wii components have been disregarded, since what makesthe THLP interesting and harder to solve than the previously ad-dressed variants in the literature is the design of the inter-hub con-nections, i.e. the back-bone. As the wii components do not flowthrough the inter-hub links, they do not affect the underlyingback-bone structure. Moreover, in applications in which it wouldbe necessary to connect the hubs by means of a tree, the wii com-ponents may not happen at all.
In order to assess the three implemented variants, the total timerequired to attain an optimal solution, up to a limit of 75 hours, thenumber of feasibility and optimality cuts, and the number of iter-ations were recorded for both the warm start phase and the maincycles, named integer phase, where the master problem is solvedto integer optimality.
Four sets of experiments were realized. On the first one, a com-parison between the Ok scheme with constraints (28) and (27)(single cut) versus a version having constraints (30) and (31) (mul-tiple cuts) is made in order to assess which one performs better.For both procedures, Benders cuts are generated both in the warmstart and branch-and-cut phases within a time limit of 2 hours(7200 seconds). As can be seen on Table 1, as the number of nodes
increases, the multiple cuts variant converges to an optimal solu-tion with less computational effort than the other one, being, onaverage, 2.72 times faster for instance sizes up to 30 nodes, andeven more fast for the instances with 40 nodes. Further, the singlecut version requires a greater number of integer cycles for conver-gence than the multiple cuts approach. Because of the aforemen-tioned and for sake of simplicity, for now on, only the varianthaving constraints (30) and (31) is considered.
The second experiment observes the isolated contributions ofthe warm start and the branch-and-cut phases on the Ok versionwith constraints (30) and (31). On this experiment, a time limitof 2 hours (7200 seconds) was imposed. As can be observed onTable 2, as the size of the instances increases, the combination ofwarm start and branch-and-cut phases presents the best behaviorwhen compared with the other variants having only one of thesefeatures. The version having only the branch-and-cut feature wasnot able to solve instances with 50 nodes under the prescribedtime limit, presenting large optimality gaps at the end.
On the third experiment, the three Benders variants (Ok, CII andMIS) are compared with one another on solving instances rangingfrom 10 up to 70 nodes. The results are reported on Tables 3 and 4,for the warm start and the integer phases, respectively. Further-more, the performance of the Ok scheme is assessed on addressinglarger instances, from 80 to 100 nodes, being the attained resultsshown on Table 5.
On Table 3, for each instance size, the same number of iterationsfor the warm start phase is set for all of the Benders decompositionvariants (see the second column of Table 3). The Ok takes longer tofinish this phase, in most cases, than the other variants. This addi-tional time can be explained by observing the number of SPs to besolved at each iteration by each algorithm. While the Ok versionsolves up to three SPs per iteration, the CII and the MIS variantshave only two and one SPs per iteration, respectively. Howeverthe gaps of the Ok algorithm are much smaller than the other
Table 4Computational results - integer phase.
Instances Ok CII MIS
# Of hubs # Of iters. Time (s) Total time (s) # Of cuts # Of iters. Time (s) Total time (s) # Of cuts CII/Ok total time # Of iters. Time (s) Total time (s) # Of cuts MIS/Ok total time
Opt. Feas. Opt. Feas. Opt. Feas. Total time
AP10.2 3 2 0.08 0.1 270 0 2 0.15 0.17 423 27 1.70 3 0.19 0.21 281 124 2.10AP10.4 2 2 0.14 0.16 270 0 1 0.07 0.09 201 24 0.56 2 0.1 0.12 159 21 0.75AP10.6 1 1 0.04 0.2 90 0 1 0.05 0.07 180 0 0.35 2 0.05 0.07 135 0 0.35AP10.8 1 1 0.03 0.06 90 0 1 0.04 0.06 180 0 1.00 2 0.05 0.07 135 0 1.17AP20.2 3 1 1.47 3.63 760 0 1 12.33 12.88 1540 550 3.55 3 3.96 4.25 1697 203 1.17AP20.4 2 1 0.55 2.57 760 0 2 17.21 17.77 1649 441 6.91 3 1.99 2.26 1333 187 0.88AP20.6 1 1 0.26 1.56 380 0 1 1.83 2.39 831 309 1.53 2 0.65 0.92 570 0 0.59AP20.8 1 1 0.62 1.67 760 0 1 0.46 1.06 760 0 0.63 2 0.41 0.68 570 0 0.41AP30.2 4 1 3.41 19 870 0 3 831.1 835.85 6931 1334 43.99 4 36.36 39.6 5097 993 2.08AP30.4 3 1 2.88 15.15 1740 0 1 511.04 516.48 4136 649 34.09 3 19.29 22.51 3442 473 1.49AP30.6 2 1 2.67 12.7 1740 0 2 119.62 125.3 4749 906 9.87 3 12.13 15.24 3034 446 1.20AP30.8 2 1 2.65 16.68 1740 0 2 104.11 109.41 3809 976 6.56 4 28.42 31.45 4115 1105 1.89AP40.2 5 1 76.59 155.11 6240 0 4 68307.61 68336.49 16727 5113 440.57 5 360.55 375.81 13829 2551 2.42AP40.4 3 2 260.25 344.67 6240 0 a a a a a a 5 835.22 851.13 15193 2747 2.47AP40.6 2 1 8.31 87.1 1560 0 3 79459.2 79495.36 12871 6629 912.69 4 508.67 523.73 10075 1625 6.01AP40.8 2 1 14.11 62.37 3120 0 3 13822.18 13902.11 9876 4164 222.90 4 389.02 404.02 10795 1685 6.48AP50.2 6 2 115.3 407.15 4900 0 a a a a a a 6 1569.92 1617.26 22727 6673 3.97AP50.4 6 2 1660.39 2032.92 17150 0 a a a a a a 6 4446.62 4490.97 26811 5039 2.21AP50.6 3 1 36.48 374.44 4900 0 a a a a a a 5 4127.08 4171.41 24301 3874 11.14AP50.8 2 1 26.63 264.01 4900 0 a a a a a a 4 828.91 872.98 14718 2432 3.31AP60.2 5 3 1071.13 1863.98 28320 0 a a a a a a 4 2025.38 2163.55 30210 6960 1.16AP60.4 4 3 3347.56 4193.64 28320 0 a a a a a a 5 9158.04 9272.65 33954 8526 2.21AP60.6 3 2 4142.83 4889.64 17700 0 a a a a a a 4 58003.71 58125.55 33047 7663 11.89AP60.8 2 2 2292.58 3001.27 10620 0 a a a a a a 3 67200.68 67327.11 25271 6589 22.43AP70.2 6 1 264.85 2244.73 4830 0 a a a a a a 4 8907.12 9255.2 39125 9175 4.12AP70.4 6 3 19905.07 22314.14 14490 0 a a a a a a 3 208622.35 208971.38 45407 14968 9.36AP70.6 4 1 3078.87 5244.30 4830 0 a a a a a a 4 150703.01 151160.14 48895 11480 28.82AP70.8 3 1 97.94 1939.72 4830 0 a a a a a a 3 57184.94 57517.06 35683 5372 29.65
a 75 hours Time limit exceeded
196E.M
.deSá
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Table 5Computational results - Ok for larger instances.
Instances # Of hubs Time W.S. (s) Total time (s) WS gap (%) # Of iters. # Of cuts
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competitive algorithms. Note that this effect is even more pro-nounced when larger instances are addressed. For instances great-er than 30 nodes, the optimality gap is smaller than 0.22% onaverage.
Table 4 completes the report of the first set of experiments,informing the results for the integer phase. It is possible to observethat the Ok version takes an equal or smaller number of iterations toreach optimality than the other variants. Only in one small instance– AP10.4 – this does not happen. The small number of iterations canbe explained by the strength of the optimality cuts generated by thenew cut selection scheme (Ok). As expected, the total time requiredto finish is also much smaller than the other algorithms, speciallyfor those instances having size greater than 30 nodes.
For instances greater than 30 nodes, the Ok algorithm is 83.9 and5.77 times, on average, faster than the CII and MIS variants, respec-tively. This is further illustrated when observing the plotted conver-gence of the UBs and LBs of the three variants on solving instanceAP30.8, see Fig. 2. While the Ok algorithm converges to the optimalsolution using only 15% of the total time required for the CII version,the MIS variant uses 30% of the time, twice as much.
Fig. 2. Convergence of the g
Furthermore, for the larger instances – AP60 and AP70 – the Okalgorithm is 9.42 and 17.99 times faster than the MIS version, onaverage. Please refer to columns CII/Ok and MIS/Ok total time forthe observed speedups. Unfortunately, the CII variant is unable tosolve instances greater than 50 nodes under the given time limit.
As can be inferred from Tables 3 and 4, the Ok algorithm is moreefficient than the other competing algorithms. This can be ex-plained by the additional information retrieved by the Ok proce-dure from the projected system. The method is able to extractuseful information even when infeasible SPs are faced. Recall thatan optimal solution is usually on the threshold of infeasibility,therefore several infeasible solutions may be topologically similarto an optimal solution, consisting then of good tentative approxi-mations for the optimal direction. The idea of generating Pareto-optimal optimality cuts from infeasible SPs is new, as far as theauthors know, and it has never been reported before.
Figs. 3 and 4 depict the number of accounted optimality andfeasibility cuts. Analysing these graphics separately, there is nogeneral trend favoring a specific algorithm. However, when observ-ing the total number of cuts, see Fig. 5, it is clear that the Okalgorithm produces an equal or smaller number of cuts than the
ap for instance AP30.8.
Fig. 4. Total number of feasibility cuts.
Fig. 3. Total number of optimality cuts.
198 E.M. de Sá et al. / European Journal of Operational Research 226 (2013) 185–202
other variants, implying also in shorter computational times, seeFig. 6.
When facing both optimality and feasibility cuts, the strategy ofrelying only on Pareto-optimal optimality cuts, while disregardingthe information extractable from the infeasible SPs, may result inpoor performance. Further, the alternative of improving the feasi-bility cuts at the cost of loosing the optimality cuts is not very
appealing, since it also yields a poor algorithmic behavior. Byaddressing the best of both worlds, the new cut selection schemehere proposed takes advantage of Pareto-optimal optimality cuts,while concurrently retrieves useful information available on theinfeasible SPs.
A scalability experiment was also carried out in order to showthe performance of the Ok algorithm on solving instances ranging
Fig. 6. Total time of the Ok and MIS variants.
Fig. 5. Total number of cuts.
E.M. de Sá et al. / European Journal of Operational Research 226 (2013) 185–202 199
from 80 up to 100 nodes to optimality. As fas as the authors know,it is the first time that such large size instances are optimallysolved in reasonable time.
With the purpose of providing a better assessment of the per-formance of Ok algorithm, the p-hub variant of the tree of hubslocation problem, introduced by Contreras et al. (2010), is ad-dressed on the fourth set of experiments. In this problem variant
any installation costs are disregarded and a fixed number (p) ofhubs to be installed is defined a priori. Although this is knownto be a harder variant, instances up to 100 nodes are also solvedto optimality within the stipulated time limit, as can be seen onTables 6 and 7. Please, recall that the literature reports optimalsolutions for instances up to 20 nodes only (Contreras et al.,2010).
Table 7Ok algorithm for the p-hub variant (larger instances).
Instances Installed hubs Time W.S. (s) Total time (s) # Of iters. # Of cuts
200 E.M. de Sá et al. / European Journal of Operational Research 226 (2013) 185–202
5. Conclusion and final remarks
This work proposes a new Benders cut selection scheme basedon the idea of retrieving cost information even when infeasibleSPs have to be addressed. This method extends the work of Pap-adakos (2008) by selecting a suitable value for the weight k render-ing a feasibility and also a Pareto-optimal optimality cut from theinfeasible SPs.
The proposed technique is powerful enough to solve to optimal-ity instances up to 100 nodes, while the other available methods onthe literature report optimal solutions only for instances up to 20nodes for the tree of hubs location problem. The new scheme alsooutperforms other modern variants of the Benders algorithm.
Moreover, it comprises a general technique for improving theBenders method convergence whenever both feasibility and opti-mality cuts are faced. Further research may assess the new schemeperformance in other well known hard problems.
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