Reverse Multistar Inequalities and Vehicle Routing

Problems with lower bound capacities

Luis Gouveia, Jorge Riera and Juan-José Salazar-González

CIO − Working Paper 13/2009

Reverse Multistar Inequalities and

Vehicle Routing Problems

with lower bound capacities

Luis Gouveia ∗ Jorge Riera† Juan-Jose Salazar-Gonzalez†

December 28, 2009

Abstract

Different variations of vehicle routing problems can be formulatedby so-called Single-Commodity Flow (SCF) formulations. The SCFmodel is an example of an extended model, in contrast to a naturalmodel that involves only the design binary variables. A natural modelwith a linear programming (LP) relaxation bound equal to the LPrelaxation bound of the SCF model can be obtained by using a theoremby Hoffman.

In this paper we show that the set of projected inequalities given bythe theorem can be divided into two sets which exhibit different mod-eling properties. For many vehicle routing problems, one set containsredundant inequalities while the other contains inequalities that areknown to be facet defining under mild conditions. This classificationraises the question of knowing whether the “redundant” set is redun-dant “for all interesting vehicle routing problems”. We show that theseapparently redundant constraints become fundamental for a less stud-ied variation of the problem, namely the case where arc lower boundcapacities are involved.

Furthermore, we also show that other related and interesting in-equalities can be generated by manipulating the new projected in-equalities by coefficient division and rounding operations. We haveimplemented a branch-and-cut algorithm which uses the new inequal-ities and which has been tested with reasonable success on instanceswith up to 50 nodes.

∗Faculdade de Ciencias da Universidade de Lisboa, DEIO–CIO Bloco C/2 - CampoGrande, 1749-016 Lisbon, Portugal legouveia@fc.ul.pt†DEIOC, Universidad de La Laguna, 38271 Tenerife, Spain jriera@ull.es ,

jjsalaza@ull.es

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We also discuss a variation of the classic (and more known) Capac-itated Vehicle Routing Problem where lower bound capacity does notmake explicit part of the problem specification, but it is implicit in theproblem definition. We show that the new inequalities might be usefulfor this variation.

1 Introduction

Different variations of vehicle routing problems can be formulated as Single-Commodity Flow (SCF) formulations (see, for instance, Gavish and Graves[5], Letchford and Salazar-Gonzalez [10] and Toth and Vigo [11]).

The SCF model is an example of an extended model, in contrast to anatural model that involves only the design binary variables. Given a SCF-like model, we can use a tool of projection to create a natural model witha linear programming (LP) relaxation bound equal to the LP relaxationbound of the original SCF model. In this paper the tool of projection is atheorem by Hoffman [8].

The main message of this paper can be summarized in five key items:

(i) The set of projected inequalities given by the theorem can be dividedinto two sets which exhibit different modeling properties. For manyvehicle routing problems, including the standard Capacitated Vehi-cle Routing Problem (CVRP), one set contains redundant inequalitieswhile the other contains inequalities that are known to be facet defin-ing under mild conditions. This is the topic of Section 2. This clas-sification raises the question of knowing whether the redundant set isredundant “for all vehicle routing problems”.

(ii) The set of apparently non-interesting constraints become interestingfor a less studied variation of the problem, namely the case where arclower bound capacities are involved. This is the topic of Section 3,where we introduce the so-called Balanced Vehicle Routing Problem(BVRP).

(iii) We also show that in contrast with the other set of projected inequal-ities, the apparently non-interesting inequalities can be lifted, leadingto a stronger set. This is also discussed in Section 3.

(iv) Following what is known for other inequalities, we shall also showthat another related, relevant and quite intuitive set of inequalitiescan be obtained by division and rounding of coefficients. In Section

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4 we introduce these rounded inequalities and, by presenting a simpleexample, show that the new inequalities might be useful to solve BVRPinstances in a cutting plane fashion. Computational results presentedin Section 6 give empirical evidence of the usefullness of these newinequalities.

(v) Finally, we point out that a variation of the standard CVRP imposesan implicit lower bound capacity on the vehicles, and thus inequali-ties proposed in (ii), (iii) and (iv) might be of interest to solve thisvariation. This is the topic of Section 5.

2 Multistar and Reverse Multistar Inequalities

Routing problems are usually modeled through a directed graph G = (V,A).A special node in V = {1, 2, . . . , n}, node 1, represents a depot. Nodes inV \ {1} represent clients. Each node i has a demand di such that∑

i∈Vdi = 0.

An arc is represented by one index a, or by two indices ij when its head j andits tail i are convenient for the notation. Each arc a ∈ A is associated with alower capacity q

aand an upper capacity qa such that q

a≤ qa, meaning that

if arc a is in the solution (that is, used by a vehicle) then the vehicle loadwhen traversing this arc cannot be greater than qa and not smaller than q

a.

Also, each arc a is associated with a value ca representing the cost of usingthe arc by a vehicle. A generic SCF model uses two variables for each arc a:

(i) a design binary variable xa indicating whether arc a is used by a vehicleand

(ii) a continuous variable fa representing the load (flow) of a vehicle travers-ing the arc.

To simplify notation, if S, T ⊂ V then we write x(S : T ) instead of∑(i,j)∈A,i∈S,j∈T xij . Given S ⊆ V \ {1}, we denote V \ (S ∪ {1}) by S′.

For brevity of notation, we also write i instead of {i} for any i ∈ V . Inaddition, δ+(S) stands for {(i, j) ∈ A : i ∈ S, j 6∈ S} and δ−(S) stands for{(i, j) ∈ A : i 6∈ S, j ∈ S}.

Then, the model minimizes a cost function

min∑a∈A

caxa

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subject to the following two sets of constraints.One set involves only the design binary variables, and imposes the in-

degree and out-degree constraints on each client node:

x(i : V \ {i}) = x(V \ {i} : i) = 1 for all i ∈ V \ {1} (1)xa ∈ {0, 1} for all a ∈ A. (2)

The other set of constraints imposes the connectivity and the capacityconstraints, and involves the use of the continuous variables:∑

a∈δ−(i)

fa −∑

a∈δ+(i)

fa = di for all i ∈ V (3)

qaxa ≤ fa ≤ qaxa for all a ∈ A. (4)

As one specific example, a model for the unit-demand CVRP with vehiclecapacity Q ≥ 2 can be defined by setting

di :=

{1 if i ∈ V \ {1}1− |V | if i = 1

and

qa

= qa = 0 for all a ∈ δ−(1)

qa

= 1 and qa = Q for all a ∈ δ+(1)

qa

= 1 and qa = Q− 1 for all a 6∈ δ+(1) ∪ δ−(1).

Different variations of vehicle routing problems can be formulated bychanging some of the parameters given in the previous generic SCF formu-lation or setting new ones (see, e.g., Toth and Vigo [11]). The SCF modelis an example of compact model since it involves a number of variables andconstraints that is bounded by a polynomial function defined on the size(number of nodes and number of arcs) of the input instance. We can createa natural model involving only the xa variables and with a LP relaxationbound equal to the LP relaxation bound of the SCF model, by using thetool of projection. The procedure to project out the continuous variablesfrom the LP relaxation of the SCF model is based on the following result:

Theorem 2.1 (Hoffman 1960 [8]) Given xa values, there is a solution ofthe linear system (3)–(4) on the fa variables if and only if∑

a∈δ−(S)

qaxa ≥∑

a∈δ+(S)

qaxa +

∑i∈S

di for all S ⊂ V. (5)

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We can give some intuition on how to generate these inequalities fromthe SCF model. Suppose we add the flow conservation constraints (3) forall nodes i in a set S and cancel equal terms, leading to:∑

a∈δ−(S)

fa =∑

a∈δ+(S)

fa +∑i∈S

di.

Then, by using the upper bounding part in constraints (4) on the term in theleft-hand side of the previous equality and by using the lower bounding partof constraints (4) on the right-hand side, we obtain (5). In fact, by reversingthe bounding procedure just suggested (i.e., using the lower bounding partin (4) on the term in the left-hand side, and using the upper bounding partin (4) on the term in the right-hand side) we obtain the following alternativenecessary-and-sufficient condition for the theorem∑

a∈δ−(S)

qaxa ≤

∑a∈δ+(S)

qaxa +∑i∈S

di for all S ⊂ V. (6)

One can easily see that the inequality (6) associated with S coincideswith the inequality (5) associated with the set V \ S. This follows fromthe fact that δ+(S) = δ−(V \ S), δ−(S) = δ+(V \ S) and that

∑i∈S di +∑

i∈V \S di = 0. Thus, the two families of inequalities (5) and (6) are equiv-alent.

However, the two families of inequalities (5) and (6) (together with theway we suggested above for generating these inequalities) permit us to castHoffman’s theorem in an alternative form. We can divide each family ofinequalities in two groups: one group is associated with the sets S containingnode 1 and the other group with the sets S not containing node 1. Thenwe use one group in both families of inequalities to generate a completeprojection. These groups are the sets of inequalities (5) and (6) defined bysets S not containing node 1.

The reason for this option is that then it becomes easier to enhance thedifferent modeling properties of the inequalities from each group. First, formost of the standard vehicle routing problems, the first group of projectedinequalities (given by (5) for sets S not containing node 1) is quite interestingwhile the second group (given by (6) for sets S not containing node 1) iseasily seen to be redundant. Second, expression (6) permits us to detectquite easily less standard variations of vehicle routing problems where theseinequalities might be useful.

We use the name Multistar (MS) inequalities for constraints (5) with1 6∈ S, while the name Reverse Multistar (RMS) inequalities is used for

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constraints (6) with 1 6∈ S. An intuition for the designation “reverse” willbe given later on.

As noted before, it is well known (see, for instance, Gouveia [6], Letch-ford, Eglese and Lysgaard [9] and Letchford and Salazar-Gonzalez [10]) thatfor “capacitated” routing problems, that is, routing problems with an upperbound on the number of clients on each route (or more generally, an upperbound on the sum of the demands of the clients on each route), the result-ing MS inequalities turn out to be rather interesting inequalities. However,for these problems, the RMS inequalities are not of interest since they areimplied by other inequalities in the model.

To exemplify this, consider again the unit-demand CVRP. The resultingMS inequalities are as follows

Qx(1 : S) + (Q− 1)x(S′ : S) ≥ x(S : S′) + |S| for all S ⊆ V \ {1}. (7)

They are the directed version of known inequalities that define facets of theassociated undirected polytope (see Araque et al. [1]).

The resulting RMS inequalities for the unit-demand CVRP are given asfollows

x(1 : S) + x(S′ : S) ≤ (Q− 1)x(S : S′) + |S| for all S ⊆ V \ {1}.

It is easy to see that for a given set S the corresponding RMS inequality isimplied by the equality obtained by adding the in-degree constraints (1) fornodes i ∈ S. Thus, for the CVRP, the RMS inequalities are not of interest.The next section presents and discusses a variant of the CVRP where theRMS inequalities are not redundant, and they have a specific and intuitiveinterpretation.

3 Vehicle routing with lower capacities

As we have noted before, expression (6) permits us to “guess” situationswhere the RMS inequalities might be of interest. The first situation thatcomes to mind is one where the demand summation on the left-hand side of(6) is negative. This may happen, for instance, in situations where some ofthe client demands are negative. These situations arise in pick-up and de-livery variations of the CVRP (see, e.g., [4]) where typically the clients withpositive demands correspond to locations which receive some commodityfrom the depot and where clients with negative demands send some com-modity to the depot. It is not difficult to see that there are interesting

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MS inequalities as well as RMS inequalities for these pickup and deliveryproblems (see, e.g., Hernandez and Salazar-Gonzalez [7]). However, a RMSinequality corresponds to a MS inequality in the symmetrized version of theproblem (which is obtained by exchanging the sign of all demand numbers)and from a structural point of view, a RMS inequality is similar to a MSinequality. For this reason, we do not further explore these pickup-and-delivery variations on this paper.

The second situation, which is also motivated by a simple analysis ofexpression (6), is to consider variations of the CVRP where lower boundvalues on the arc flows are bigger than one. To illustrate such a situation,consider the so-called Balanced Vehicle Routing Problem (BVRP), wherea minimum number of clients Q and a maximum number of clients Q arerequired to each route in a feasible solution. In general, the designation“balanced” applies to a variant of the problem when Q − Q is small. Herewe relax this designation since our aim is to consider situations with a lowerbound Q (> 1) and we may even allow examples where Q = |V |− 1 (i.e., noupper capacity on the vehicles). The lower limited capacity requirement canbe easily modeled through a SCF model by setting the lower bound valueqa

on the arcs leaving the depot as being equal to Q. More precisely, a SCFmodel for the BVRP can be obtained by setting the parameters as follows:

qa

= qa = 0 for all a ∈ δ−(1)

qa

= Q and qa = Q for all a ∈ δ+(1)

qa

= 1 and qa = Q− 1 for all a 6∈ δ+(1) ∪ δ−(1).

Thus, it is quite easy to include lower bound information in a SCF formu-lation. On the other hand, finding from scratch inequalities involving onlythe xa variables to guarantee the minimum required number of clients ineach route seems to be far from easy. Fortunately, the tool of projectionon the SCF formulation permits us to obtain such set of inequalities. TheMS inequalities are exactly the ones given in (7). Note that they do not de-pend on the lower bound value Q. The RMS inequalities, instead, take intoaccount the upper bound capacity as well as the new lower bound capacity:

Qx(1 : S) + x(S′ : S) ≤ (Q− 1)x(S : S′) + |S| for all S ⊆ V \ {1}. (8)

These constraints guarantee that if a (partial) route does not have theminimum required number of clients, then it cannot be closed by connectingdirectly to the depot the two extremes of the route. This can easily be seenif we use the in-degree and out-degree equations (1) on the client nodes to

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rewrite the RMS inequality (8) associated with a set S as follows:

(Q− 1)|S| ≤ (Q− 1)x(S : S′) + (Q− 1)x(S′ : S) +Qx(S : S). (9)

Suppose now that we have a set S where all the nodes are in the same routeand connected, and that |S| < Q. Then, we have that x(S : S) = |S| − 1and the RMS inequality becomes

(Q− 1)x(S : S′) + (Q− 1)x(S′ : S) ≥ Q− |S|.

It states that the set S must be connected to at least one node in the setS′. A more elaborate, but similar, interpretation could be found for a set Ssuch that |S| ≥ Q.

The interpretation just given for the RMS inequality (8) is exactly theopposite interpretation given for the MS inequalities (7) (see, for instance,Araque et al. [1]) and that is why we have named these inequalities asreverse multistars.

The MS inequalities are known to define facets, under mild conditions,of the undirected polytope associated to the CVRP. Although no similarstudy has been done for the BVRP, we have no reason to suspect thatthe MS inequalities can be strengthened when non-trivial lower bounds areimposed on the vehicle capacity. In contrast, the RMS inequalities can bestrengthened by decreasing the coefficient of the variables in the right hand-side term leading to inequalities that we denote by Enhanced RMS (ERMS)inequalities and that only involve the lower bound capacity:

Qx(1 : S)+x(S′ : S) ≤ (Q−1)x(S : S′)+ |S| for all S ⊆ V \{1}. (10)

Proposition 3.1 The ERMS inequalities (10) are valid for the BVRP poly-tope.

Proof.Consider a feasible solution x′ and a set S. Let k := x′(1 : S) and p :=

x′(S′ : S), and consider the set S partitioned into the subsets S1, . . . , Sk+pwhere each subset Si corresponds to a connected component of the route x′

in S (i.e., a path). Trivially |S| ≥ k + p and the in-degree and out-degreeequations on each node guarantee that x′(S : 1) + x′(S : S′) = k + p.

If x′(S : S′) ≥ k then the inequality (10) for S is valid since Qk + p ≤k + p + Q(k − 1). Let us assume now that k1 := x′(S : S′) < k. Then atleast k2 = k − k1 subsets Si correspond to complete routes, i.e. are directly

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connected to the depot from both sides. Since each route has at least Qclients, we must have |S| ≥ k2Q + k1 + p. Then, Qk + p ≤ k2Q + k1 + p +(Q− 1)k1 and the inequality (10) for S is valid. �

It is easy to find solutions that satisfy all MS inequalities and RMSinequalities and violate some of the ERMS inequalities. This suggests thatthe ERMS inequalities might be useful in a cutting plane approach to solvethe BVRP.

The ERMS inequalities (10) were inspired by the fact that, in inequalities(9), the coefficients of the variables associated with arcs a ∈ δ+(S)∩ δ−(S′)are different from the coefficients of the variables associated with the “re-verse” arcs a ∈ δ+(S′) ∩ δ−(S). One may suspect that a similar set ofinequalities might also be derived for the undirected version of the problem(where undirected edge variables are used instead of directed arc variables).Note that the ERMS inequalities are equivalent to (9) where the coefficientof Q − 1 associated to the variables in the cut δ+(S) ∩ δ−(S′) is decreasedto Q−1, thus permitting us to write an undirected inequality. The strongerinequalities suggest the following question. Similarly to what happens tothe weaker RMS inequalities, could we find a compact model which impliesthe ERMS inequalities (10)? Or in a more broad way, can they be separatedin polynomial time? For the moment we do not have an answer to these tworelated questions. At first glance, it appears that we could obtain such acompact model by decreasing the coefficients qa in (4) from Q−1 to Q−1 forthe arcs a 6∈ δ+(1) ∪ δ−(1) and use the projection tool as before. However,these modified upper bound constraints are not valid since they also forceeach route to have at most Q clients.

The ERMS inequalities (10) apparently make the RMS inequalities (8)useless. However, Hoffman’s result guarantees that we have a polynomialformulation that implies all the RMS inequalities. Due to the fact that cur-rent available ILP solvers (e.g., CPLEX and XPRESS) are very efficient insolving compact models, using such a compact SCF model is worth consid-ering. Furthermore, and although using the ERMS inequalities (instead ofthe original RMS inequalities) should provide better LP based lower bounds,it is still unknown whether the ERMS inequalities (10) can be separated inpolynomial time or whether a compact model model implying them exists.We also emphasize that we would not have been able to derive the ERMSinequalities without knowing about the RMS inequalities.

We end this section noting that for the special case of the BVRP whereQ = Q the ERMS inequalities (10) do not provide new information sincethey can be shown to be equivalent to the MS inequalities (7). We next

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show that the MS inequality (7) for a given set S is equivalent to the ERMSinequality (10) for the complement set S′. For the proof we use the factthat, when Q = Q, all feasible solutions have a fixed number of vehiclesgiven by (|V | − 1)/Q, i.e. x(1 : V \ {1}) = (|V | − 1)/Q.

Proposition 3.2 When Q = Q, the ERMS inequality (10) for set S isequivalent to the MS inequality (7) for the set S′, and vice-versa.

Proof. Consider the ERMS inequality (10) for a given set S:

Qx(1 : S) + x(S′ : S) ≤ (Q− 1)x(S : S′) + |S|

Using the degree constraint for the depot, x(1 : V \ {1}) = (|V | − 1)/Q, weobtain

|V | − 1 + x(S′ : S) ≤ (Q− 1)x(S : S′) +Qx(1 : S′) + |S|

Since |V | − 1− |S| = |S′| we obtain the MS inequality (7) for the set S′. �

Clearly, this equivalence no longer holds for the more general BVRPwhere Q < Q since we have shown that the ERMS inequalities (10) (andeven the RMS inequalities (8)) are fundamental for modeling the BVRP.

4 Rounded Inequalities

It is well known that “projected” inequalities can be used to produce (byadequate division and rounding) other interesting sets of inequalities. As anexample, consider the MS inequalities (7) for the unit-demand CVRP. Di-viding by Q these inequalities and rounding we obtain the following roundedMS inequalities

x(1 : S) +⌈Q− 1Q

⌉x(S′ : S) ≥

⌊1Q

⌋x(S : S′) +

⌈|S|Q

⌉that correspond to the well-known generalized cut constraints

x(V \ S : S) ≥⌈|S|Q

⌉(11)

for all S ⊆ V \ {1}. These inequalities are the directed version of facet-defining inequalities for the undirected CVRP polytope (see, e.g. Camposet al. [2], Cornuejols and Harche [3] and Araque et al. [1]) and are by far

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x12 = 1 x13 = 1/3 x15 = 1/2 x17 = 1/2 x23 = 2/3 x24 = 1/6x26 = 1/6 x31 = 2/3 x34 = 1/3 x46 = 5/6 x48 = 1/6 x57 = 1/6x58 = 5/6 x61 = 2/3 x67 = 1/3 x74 = 1/2 x75 = 1/2 x81 = 1

Table 1: Fractional solution violating a rounded ERMS inequality

the most relevant inequalities in cutting-plane approaches for solving theCVRP and related problems.

We show next that, by performing a similar procedure starting from theERMS inequalities (10), we obtain a new set of inequalities that may berelevant for problems with lower bound capacities. We note that a similarprocedure can be applied to the weaker RMS inequalities (8). However, theobtained inequalities will be weaker than the ones obtained from the ERMSinequalities (10). For this reason we only focus on applying the derivationprocedure to (10).

As it has been done with the original RMS inequalities, an ERMS in-equality can be rewritten as follows:

(Q− 1)|S| ≤ (Q− 1)x(S : S′) + (Q− 1)x(S′ : S) +Qx(S : S).

Then, if we divide this inequality by Q, and then apply rounding as before,we obtain the new rounded ERMS inequality :

|S| −⌊|S|Q

⌋≤⌈Q− 1Q

⌉x(S : S′) +

⌈Q− 1Q

⌉x(S′ : S) + x(S : S) (12)

which is the same as

|S| −⌊|S|Q

⌋≤ x(S : S′) + x(S′ : S) + x(S : S).

It is not difficult to see that there is no dominance relationship betweenthe two sets of inequalities, the ERMS inequalities (10) and the roundedERMS inequalities (12). Clearly, one expects the rounding to perform betterwhen |S|/Q is not integer.

Table 1 presents a fractional solution for an instance with n = 8, Q = 4and Q = 3. This solution satisfies all MS inequalities (7), RMS inequalities(8) and rounded MS inequalities (11). However, it is easy to see that thesolution violates the rounded ERMS inequalities (12) defined by the setsS1 = {2, 3} and S2 = {2, 3, 4, 6}. Thus, the rounded ERMS inequalities (12)might also be useful in a cutting plane algorithm for solving instances of theBVRP. We give next some intuition on why this may happen.

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First, we note that one can exhibit, quite easily, a simple case where therounded ERMS inequality (12) is at least as strong as the corresponding(for the same set S) ERMS inequality (10). Consider S = V \ {1}. Then,both constraints give an upper bound on the number of vehicles leaving thedepot, but the upper bound given by the rounded ERMS inequality (12) isstronger when (|V | − 1)/Q is not integer.

More generally, consider the following situation with a set S consisting ofthree nodes linked together in a feasible solution for the problem and wherethe depot (node 1) is linked to one node in S. Assume that Q = 5 andQ = 6. The ERMS inequality (10) for these parameters becomes 5 + 0 ≤3 + 4x(S : S′), which is equivalent to 2/4 ≤ x(S : S′). The inequalitycombined with the in-degree and out-degree constraints states that the lastnode in this sequence must be connected to a client (because the route hasonly visited three nodes). From the LP relaxation point of view we wouldprefer to obtain an inequality in the same situation where the left-hand sidewould be equal to 1. However, this is given precisely by the rounded ERMSinequality (12).

We can use the equality constraints of the model to obtain another usefulway of expressing the rounded ERMS inequalities (12). Indeed, by using thein-degree and the out-degree constraints for every node in set S we obtain

2x(S : S) + x(S : 1) + x(1 : S) + x(S : S′) + x(S′ : S) = 2|S|,

and thus the rounded ERMS inequality associated with S can also be writtenas

x(S ∪ {1} : S ∪ {1}) ≤ |S|+⌊|S|Q

⌋. (13)

Constraint (13) is a normal subtour elimination constraint involving the setS∪{1} when S ⊆ V \{1} such that |S| < Q. This is quite intuitive since thelower bound constraints can also be viewed as imposing that small subtoursinvolving the depot are not allowed.

From a practical point of view, our computational experiments showthat these constraints, for sets S with cardinality smaller than Q, are quiterelevant for improving LP bounds. On the other hand, for the sets S withcardinality greater or equal than Q they are not (for |S| = Q, the corre-sponding ERMS inequality dominates the rounded version). However, froma theoretical point of view, we can produce (fractional) solutions that sat-isfy all MS inequalities, rounded MS inequalities, ERMS inequalities androunded ERMS inequalities for sets S with cardinality smaller than Q, andwhich violate a rounded ERMS inequality for a set S with cardinality largerthan Q.

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We end this section providing a similar result to the one given at theend of the previous section. For the special case of the BVRP where Q = Q,the rounded ERMS inequalities (12) do not provide new information sincethey are equivalent to the rounded MS inequalities (11). Following the sameassumptions given at the end of the previous section, we can state and provethe next result.

Proposition 4.1 When Q = Q, the rounded ERMS inequality (12) for setS is equivalent to the rounded MS inequality (11) for the set S′.

Proof. Consider the rounded ERMS inequality (12) for a given set S. Usingthe in-degree equations for the nodes in set S we obtain

|S| −⌊|S|Q

⌋≤ x(S : S′) + |S| − x(1 : S).

Using the fact that x(1 : V \ {1}) = (|V | − 1)/Q and rearranging we get

|V | − 1Q

−⌊|S|Q

⌋≤ x(S : S′) + x(1 : S′).

Since, the left-hand side of the resulting inequality is equal to d|S′|/Qe, weachieve the desired result. �

As shown by the fractional solution in Table 1, the above-cited equiva-lence no longer holds for the more general BVRP where Q < Q. However,when Q = Q + 1 we may obtain some interesting relations. For simplicityof notation we now assume Q = Q− 1. First note that in this situation wehave ⌈

|V | − 1Q

⌉≤ x(1 : V \ {1}) ≤

⌊|V | − 1Q− 1

⌋.

Consider the rounded ERMS inequality (12) for a set S

|S| −⌊|S|Q− 1

⌋≤ x(S : S′) + x(S′ : S) + x(S : S).

Using the same reasoning as before, we obtain

x(1 : S) + |S| −⌊|S|Q− 1

⌋≤ x(S : S′) + |S|.

Using d(|V | − 1)/Qe ≤ x(1 : V \ {1}) and cancelling equal terms we obtainthe following cut-like inequality for the set S′:⌈

|V | − 1Q

⌉−⌊|S|Q− 1

⌋≤ x(S : S′) + x(1 : S′), (14)

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which is quite similar to the rounded MS inequality (11) but with a differentright-hand side. For the situations where⌈

|V | − 1Q

⌉= x(1 : V \ {1}) =

⌊|V | − 1Q− 1

⌋the two sets of constraints (14) and (12) are equivalent, and then it is inter-esting to compare (14) and (11). We discuss next two different cases:

Case 1. n = 8, Q = 4, Q = 3: We obtain x(1 : V \{1}) = 2. In this case one caneasily see that the inequalities (14) either are equivalent to (11) or areeven weaker. More precisely, (14) is weaker than (11) if |S′| = 1, andthe two inequalities are equivalent if |S′| > 1. This analysis does notinvalidate the previous example since in that case the solution satisfiesd(|V | − 1)/Qe < 2.333 = x(1 : V \ {1}). However, if we had previouslyincluded the degree constraint x(1 : V \{1}) ≤ b(|V |−1)/(Q−1)c = 2(which is equivalent to a rounded ERMS (12) for S = V \ {1}) wewould be in the situation of applying the previous analysis.

Case 2. n = 10, Q = 4, Q = 3: Here we have x(1 : V \ {1}) = 3. It is inter-esting to point that for this case, in three out of the nine possibilities,inequality (14) is stronger than the corresponding (for the same set)rounded MS inequality (11), and they are equivalent for the remainingpossibilities. Thus, for this case, the rounded ERMS inequalities (12)make the rounded MS inequalities (11) redundant. For instance, when|S′| = 4 the rounded MS inequality (11) has a right-hand side equalto 1, while inequality (14) for S with |S| = 5 has the right-hand sideequal to 2, thus it is stronger. There is a quite intuitive interpretationto this: consider a set S′ such that |S′| = 4; the rounded MS inequal-ity (11) states that at least one arc incoming into S′ has positive xa;however if there is only one arc in this situation then all the nodes inS′ must be in one vehicle, which is full; but then it is impossible to putall the remaining 5 nodes into feasible routes. In other words, at leasttwo arcs must enter S′ with positive xa, as stated by the correspondingrounded ERMS inequality (12). A similar analysis holds for sets S′

with |S′| = 7 and 8.

5 CVRP with a fixed number of vehicles

In the previous section, we have studied a problem where the lower boundon the number of clients per route was part of the problem specification.

14

x13 = 1/3 x14 = 1/3 x15 = 1/3 x16 = 1/3 x17 = 1/3 x18 = 1/3 x21 = 5/9x23 = 4/9 x31 = 4/9 x32 = 2/9 x34 = 1/3 x41 = 1 x54 = 1/3 x56 = 2/3x62 = 4/9 x68 = 5/9 x73 = 2/9 x75 = 2/3 x78 = 1/9 x82 = 1/3 x87 = 2/3

Table 2: Fractional solution violating a RMS inequality

The main reason for the proposed study was to motivate a problem whereRMS inequalities are important for modeling the problem. In this sectionwe consider a standard variant of the CVRP where the problem specificationdoes not explicitly include lower bound information. However, sometimesthis lower bound is implicit on the problem specification, and thus it mightbe used to generate new valid inequalities for the problem.

Consider the CVRP with unit demands, an upper bound Q on the num-ber of clients per route, and a fixed number m of vehicles given explicitlythe equation

x(1 : V \ {1}) = x(V \ {1} : 1) = m. (15)

Most of the exposition given in this section also applies to the more realisticsetting where an upper bound on the number of vehicles is given. In orderto show how a lower bound capacity can be useful for such a problem, letus consider an instance with n = 8, Q = 4 and m = 2. It is clear thatin any feasible solution for the problem, each vehicle cannot visit less thanthree clients. Thus, we can reformulate the problem and add this lowerbound information on the capacity of the vehicles. More generally, thisCVRP variant is a BVRP with Q = (|V | − 1)− 1− (m− 1)Q and a degreeconstraint on the depot.

The question now is to know whether, without being strictly necessaryto write a valid formulation for the problem, the lower bound informationis good for developing inequalities that will tighten the LP relaxations ofnatural formulations for the problem with a fixed number of vehicles (or anupper bound on the number of vehicles).

Table 2 shows a fractional solution for an instance with n = 8, Q = 4 andm = 2. As noted before, Q = 3. This solution satisfies the MS inequalities(7), rounded MS inequalities (11) and the degree constraint (15) stating thatm vehicles must be used. However, it violates an ERMS inequality (10) (andeven the weaker RMS inequality (8)) for the set S = {2, 3, 4}.

This solution tells us that the ERMS inequalities (10) (and even theRMS inequalities (8)) might be useful in the context of a pure cutting planemethod using only the xa variables. We are more precise in the next twopropositions and give a sufficient and necessary condition for a fractional

15

solution (that satisfies the degree equations, the MS constraints and therounded MS constraints) to violate either an ERMS or a RMS inequality.

Proposition 5.1 For the CVRP with a fixed number of vehicles m, thecondition x(V \S : S) ≤ m− 1 holds if and only if the MS inequality (7) forset S implies the ERMS inequality (10) for set S′.

Proof. Consider the MS inequality (7) for set S. Using the degree constraintfor the depot we obtain

mQ− |S|+ (Q− 1)x(S′ : S) ≥ x(S : S′) +Qx(1 : S′). (16)

If we now add the inequality

|S′| − (mQ− |S|) + (Q−Q)x(S : S′) ≥ (Q−Q)x(1 : S′) (17)

to (16) we obtain the ERMS inequality (10) for the set S′. Thus, whenever(17) is valid, we have that the MS inequality (7) for set S implies the ERMSinequality (10) for set S′.

Using the fact that Q = (n− 1)− (m− 1)Q and that |S′|+ |S| = n− 1,and multiplying by −1, the inequality (17) can be rewritten as

mQ− (n− 1) + (mQ− (n− 1))x(S′ : S) ≤ (mQ− (n− 1))x(1 : S′),

which is equivalent to 1 + x(S′ : S) ≤ x(1 : S′), and also equivalent tox(1 : S) + x(S′ : S) ≤ m− 1. Clearly the reversed argument also holds andwe obtain the desired result. �

Thus, whenever x(V \S : S) ≤ m−1 we know that the ERMS inequality(10) for set S′ is not necessary since it is implied by (or equivalent to whenx(V \ S : S) = m − 1) the MS inequality (7) for set S. However, whenx(V \ S : S) ≥ m, the ERMS inequality for the corresponding set S′ is notredundant in a formulation containing the degree constraints as well as theMS inequalities. The solution mentioned at the beginning of this section(see Table 2) is in these conditions. We can state a similar but weaker resultfor the case of the original RMS inequalities (8). For simplicity we omit itsproof since it is similar to the proof of Proposition 5.1.

Proposition 5.2 For the CVRP with a fixed number of vehicles m, thecondition x(1 : S) ≤ m− 1 holds if and only if the MS inequality (7) for setS implies the RMS inequality (8) for set S′.

16

Thus only when x(1 : S) = m the RMS inequality (8) is of interest ina formulation containing the degree constraints (1) and (15) as well as theMS inequalities (7). The equivalent results given in Section 3 permit us tostate that in the context of a method that uses a SCF-like model, betterlower bounds can be obtained by changing the lower bound parameter q

afrom 1 to Q for arcs a leaving the depot. Furthermore, by performingthis change, we can guarantee that all RMS inequalities (8) are implicitlysatisfied by the corresponding LP relaxation solution. We note, however,that this equivalence result does not diminish the interest in performinga similar study with respect to the ERMS inequalities (more precisely, toknow whether they can be separated in polynomial time) since the ERMSinequalities are stronger than the RMS inequalities. Furthermore, as shownin Propositions 5.1 and 5.2, given a fractional solution satisfying the degreeconstraints and the MS inequalities, it is more likely to find a violated ERMSinequality (10) rather than a violated RMS inequality (8).

The previous analysis motivates a similar study with respect to therounded ERMS inequalities (12). Unfortunately, the analysis for these in-equalities lead to quite negative results as the following result shows.

Proposition 5.3 For the CVRP with a fixed number of vehicles m, therounded ERMS inequality for set S′ is implied by the rounded MS inequalityfor set S.

Proof. For the CVRP with a fixed number of vehicles m, we have Q =(n − 1) − (m − 1)Q. Let us assume that Q = Q − q for a certain integernumber q. Since Qm = n− 1 + q, the number (n− 1 + q)/Q is integer.

Using arguments shown before, the rounded ERMS inequalities for setS can be rewritten as:

n− 1 + q

Q+⌊|S|Q− q

⌋≤ x(S : S′) + x(1 : S′),

which is quite similar to the rounded MS inequality for the set S′:⌈|S′|Q

⌉≤ x(S : S′) + x(1 : S′).

Thus we only need to compare the left-hand sides of the two inequalities.Without loss of generality, assume that |S| = kQ+ p and |S′| = k′Q+ p′

with 0 ≤ p ≤ Q − 1 and 0 ≤ p′ ≤ Q − 1. Note that k + k′ = m − 1 andp+ p′ = Q.

17

We first consider that case where p′ > 0. Under this assumption,d|S′|/Qe = k′+1 and b|S|/(Q− q)c ≥ k′ since |S| = k(Q− q)+p+kq. Thus⌈

|S′|Q

⌉= k′ + 1 = m− k ≥ n− 1 + q

Q−⌊|S|Q− q

⌋and therefore the rounded MS inequality for set S′ dominates the roundedERMS inequality for set S.

Consider now the case where p′ = 0. Then |S| = kQ + Q − q and|S′| = k′Q. Under this assumption, d|S′|/Qe = k′ and b|S|/(Q−q)c ≥ k′+1since |S| = k(Q−q)+(Q−q)+kq. We have obtained the desired inequality.�

This result is a bit surprising considering the examples for the case Q =Q − 1 given in the previous section. The reason for these disappointingresults is that for this special version of the CVRP, the values of Q and midentify univocally one value for Q. An example of this dominance is Case1 given at the end of Section 4. On the other hand, in the BVRP we mayhave situations with fixed values of Q and Q, and several feasible values form.

The fact that the ERMS inequalities (10) and even the RMS inequali-ties (8) may be of interest for this special case of the CVRP, raises severalpossibilities, namely that a more through study of the BVRP polytope maylead to other inequalities of interest for this variant of the CVRP. A relatedquestion is to know whether the ERMS and RMS inequalities are really newfor the CVRP with a degree constraint on the depot. That is, could they beshown to be equivalent to other inequalities which are already known fromthe literature? This is a difficult question to answer since there are manyclasses of inequalities for the CVRP. The best we can say is that from thislarge class of inequalities, and as far as we know, only the degree constraintsuse information on the number of vehicles, and thus it is highly unlikely thatthe ERMS and RMS inequalities are equivalent to other inequalities.

6 Computational results

In this section we present computational results in order to evaluate ourcontributions for solving BVRP instances. We want to show that using thenew inequalities one can solve larger instances. To this end we compare twobranch-and-cut implementations. The first implementation is the basic one,where we approach the BVRP by using standard inequalities for the capaci-tated VRP as well as the RMS constraints (8). The second implementation

18

also uses the standard inequalities for the capacitated VRP, but uses theERMS (10) and the rounded ERMS (12) instead of the RMS constraints.In both implementations the standard inequalities for the capacitated VRPare the classical subtour elimination constraints, the MS inequalities (7) andthe rounded MS inequalities (11).

More precisely, our cutting-plane generation approach proceeds as fol-lows.

Step 1: Find (if any exists) a violated subtour elimination constraints byapplying a min-cut algorithm. We repeat this step while we are findingviolated constraints. The number of inequalities found at the root nodeis denoted by f1 and the total number of inequalities found during thewhole branch-and-bound search is denoted by g1.

Step 2: This is a heuristic search for violated rounded ERMS inequalities(13). It consists of finding a most violated (if any exists) inequality ofthe type

|S| − |S|Q≤⌈Q− 1Q

⌉x(S : S′) +

⌈Q− 1Q

⌉x(S′ : S) + x(S : S)

which is a weaker version of (13) and which can also be rewritten as

x(S ∪ {1} : S′) +|S|Q≥ x(1 : V \ {1}).

These inequalities can be exactly separated by using a min-cut algo-rithm. Again, this step is repeated while it generates violated inequal-ities. Let f2 be the number of violated rounded ERMS inequalitiesfound within this step at the root node, and g2 the total numberduring the whole search.

Step 3: This is another heuristic approach to find violated rounded ERMSinequalities. We explicitly enumerate and check for violation (13) forall S such that 1 6∈ S and |S| < Q. The step is repeated while violatedinequalities are found. The number of inequalities found at the rootnode is denoted by f3 and the total number found during the wholesearch by g3.

Step 4: Here we use another heuristic approach, this time to separaterounded MS inequalities (11). To this end, we apply a min-cut al-gorithm to find a most violated inequality of type x(A(S)) ≥ d(S)/Q,

19

which is a weaker version of the rounded MS inequality. The roundedMS inequality associated to an optimal min-cut set S is added to themodel. The number of inequalities found at the root node is denotedby f4 and the total number during the whole search by g4.

Step 5: Solve the linear system (3)–(4) with qa

and qa given in Section 3.If this system is infeasible, the dual extreme ray divides the node setV \ {1} in two sets, S and S′. Then the MS inequalities (7) associatedwith S and S′ are checked for potential violation. If one is violatedthen it is added to the model and the step is repeated. The number ofMS inequalities found at the root node is denoted by f5 and the totalnumber during all the search by g5. If no MS inequality is violatedthen we check for violation the ERMS inequalities (10) associated tothe same sets S and S′. If one such inequality is violated then it isadded to the model and the step is repeated. The number of theseERMS inequalities found at the root node is denoted by f6 and thetotal number during the whole search by g6. If no ERMS inequalityis violated then we check for violation the rounded ERMS inequalities(12) associated to the same sets S and S′. If one such inequality isviolated then it is added to the model and the step is repeated. Thenumber of these rounded ERMS inequalities found at the root nodeis denoted by f7 and the total number during the whole search by g7.Alternatively, instead of checking ERMS and rounded ERMS, one cancheck the RMS inequalities (8) associated with S and S′. The numberof RMS inequalities found at the root node is denoted by f8 and thetotal number during all the search by g8.

The first implementation applies Steps 1, 4, and 5 to find violated MSand RMS inequalities. The second implementation applies Steps 1–4, and 5to find violated MS, ERMS and rounded ERMS inequalities.

In our computational experiments we have used three sets of instances.The first set contains Euclidean instances taken from the VRP-library. Thesecond set consists of randomly generated instances, still using the Euclideandistance. The third set consists of asymmetric instances also taken from theVRP-library. All the instances have up to 50 clients with unit demand tobetter understand the impact of Q. Also for that reason, we have createddifferent instances by considering different values for Q. The algorithm wascoded in C++ on a Linux platform running in a Intel(R) Core(TM)2 CPU6700 @ 2.66GHz desktop computer with 2GB RAM. We have made use of theCHIPPS framework (see [12]) included in the COmputational INfrastructure

20

for Operations Research COIN-OR (open source for the Operations Researchcommunity) using their simplex solver CLP.

Tables 3 and 4 show the results obtained with the second implementa-tion, i.e., the one that uses all the new inequalities proposed in this paper.Table 3 refers to features at the root node, while Table 4 refers to featuresat the end of the branch-decision search tree. They show the name of theinstance, the number n of clients, the vehicle capacity limits Q and Q, theobjective value r-LB at the end of the root node, the computing time in sec-onds r-time at the end of the root node, the objective value opt at the end ofthe search, the total computing time time in seconds, the number of cutting-plane iterations LP iter, the number of branch-and-bound nodes nodes, thenumber m of routes in the obtained optimal solution, the maximum numbermax of clients in a route in the optimal solution, and the minimum numbermin of clients in a route in the optimal solution. In addition, they alsodisplay the number of inequalities found by the separation steps. Columnsf1,f2,f3,f4,f5,f6,f7 in Table 3 and Columns g1,g2,g3,g4,g5,g6,g7 in Table 4gives the numbers of inequalities as described above.

The results indicate that, in general, the Eilon instances are easier tosolve than the random instances. Also, the asymmetric instances are muchharder to solve. In general, the results indicate that the instances becomeslightly easier to solve when the Q value increases and the other input pa-rameters remain fixed.

Tables 5 and 6 show the performance of the first implementation, i.e.the basic algorithm that, besides the some CVRP inequalities, only uses theRMS inequalities. Here, for brevity and clarity, we only present some Eiloninstances with 30 clients, random instances with 39 clients and asymmetricinstances with 45 clients (we have run the first implementation to solveall the instances and similar conclusions can be obtained for the remaininginstances). The meaning of the headings in these tables are the same as inthe previous tables.

Comparing tables 3 and 5 one immediately concludes that much largerCPU times are needed to solve the same instances with the first implemen-tation. These results show that the new inequalities are worth having ina cutting plane method for solving the BVRP. An interesting observationregarding the LP bounds comes from the comparison of tables ?? and 6.The results show that the inclusion of the RMS inequalities in the first im-plementation do not alter by much the LP bound given by the “true uppercapacity case”, that is the instance with Q = 1. In fact, the LP bound in-creases very slowly (when it increases) with the value of Q, for a fixed value

21

of Q. This can be explained by analyzing the solutions of the LP relaxationof the SCF model. In most of the cases, the LP solution contains largefractional routes, implying that the upper bound inequalities on fa in (4)are either satisfied at equality or close to being satisfied at equality. Thismeans that adding the lower bound inequalities on fa in (4) may not haveany effect on the given LP bound if the given Q is not close to the givenQ. The LP bounds improve when the new inequalities are added. We notehowever that, in some cases, large gaps are still obtained.

7 Conclusions

In this paper we have examined a subset of the projected inequalities from astandard single flow formulation for the CVRP. Although for the CVRP it isknown that this subset is useless, we have pointed out that for other variantsof the CVRP this is may not be the case. This subset of inequalities arethe RMS inequalities. We have also described two new sets of inequalities,called ERMS and rounded ERMS inequalities. Computationally we havenoted that the new inequalities are of interest to solve CVRP variationswhere arc lower bounds on the vehicle loads are either explicitly or implicitlyconsidered.

References

[1] Araque, J.R., Hall, L., Magnanti, T.L.: Capacitated trees, capacitatedrouting and associated polyedra. Discussion Paper 90-61, CORE, Uni-versity of Louvin La Neuve, Belgium, 1990.

[2] Campos, V., Corberan, A., Mota, E.: Polyhedral resuls for a vehiclerouting problem. European Journal of Operational Research 52 (1991)75–85.

[3] Cornuejols, G., Harche, F.: Polyhedral study of the capacitated vehiclerouting problem. Mathematical Programming 60 (1993) 21–52.

[4] Desaulniers, G., Desrosiers, J., Erdmann, A., Solomon, M.M., Soumis,F.: VRP with pickup and delivery. In Toth, P., Vigo, D. (Eds.): TheVehicle Routing Problem. SIAM, Society for Industrial and AppliedMathematics, Philadelphia, Pennsylvania, 2000.

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[5] Gavish, B., Graves, S.: The traveling salesman problem and relatedproblems. Working Paper, Graduate School of Management, Universityof Rochester, New York (1979).

[6] Gouveia, L.: A result on projection for the vehicle routing problem.European Journal of Operational Research 85 (1995) 610–624.

[7] Hernandez-Perez, H., Salazar-Gonzalez, J.J.: The one-commoditypickup-and-delivery traveling salesman problem: Inequalities and al-gorithms. Networks 50 (2007) 258–272.

[8] Hoffman, A.J.: Some recent applications of the theory of linear inequal-ities to extremal combinatorial analysis. In Bellman, R., Hall, Jr., M.(Eds.): Proc. Symp. in Applied Mathematics. Amer. Math. Soc. 10(1960) 113–127.

[9] Letchford, A.N., Eglese, R.W., Lysgaard, J.: Multistars, partial multi-stars and the capacitated vehicle routing problem. Mathematical Pro-gramming 94 (2002) 21–40.

[10] Letchford, A., Salazar-Gonzalez, J.J.: Projection of Flow Variables forVehicle Routing. Mathematical Programming 105 (2006) 251–274.

[11] Toth, P., Vigo, D.: An overview of vehicle routing problems. In Toth,P., Vigo, D. (Eds.): The Vehicle Routing Problem. SIAM, Society forIndustrial and Applied Mathematics, Philadelphia, Pennsylvania, 2000.

[12] Xu, U., Ralphs, T., Ladanyi, L., Saltzman, M.: The COIN-OR High-Performance Parallel Search Framework (CHIPPS) project. The Com-putational Infrastructure for Operations Research (COIN-OR). 2007.http://projects.coin-or.org/CHIPPS

name n Q Q f1 f2 f3 f4 f5 f6 f7 r-LB r-time

eil22a 21 5 1 50 0 0 31 8 0 0 403.32 0.09eil22b 21 5 2 51 15 0 18 11 1 1 391.03 0.08eil22c 21 5 3 44 46 2 5 5 2 2 412.75 0.10eil22d 21 5 4 50 41 21 1 6 22 15 429.35 0.21eil22e 21 6 1 34 0 0 22 8 0 0 353.72 0.06eil22f 21 6 3 29 17 0 6 2 0 0 362.50 0.04eil22g 21 6 4 47 17 9 9 5 1 1 371.49 0.09eil22h 21 6 5 47 26 13 1 3 6 5 372.20 0.13eil23a 22 4 1 55 0 0 42 19 0 0 775.01 0.13eil23b 22 4 2 43 22 0 15 9 0 0 774.24 0.08eil23c 22 4 3 57 46 2 5 10 2 1 794.00 0.00eil23d 22 7 1 28 0 0 15 1 0 0 601.00 0.04eil23e 22 7 2 34 12 0 8 0 0 0 631.50 0.04eil23f 22 7 3 29 11 0 4 0 0 0 633.00 0.03eil23g 22 7 5 35 17 3 1 0 3 2 637.47 0.06eil31a 30 7 1 32 0 0 14 8 0 0 294.50 0.08eil31b 30 7 2 31 9 0 7 0 0 0 301.00 0.04eil31c 30 7 3 32 14 0 5 0 2 2 301.00 0.06eil31d 30 7 4 43 15 3 1 3 1 1 309.00 0.10eil31e 30 7 5 44 22 1 2 0 1 1 309.50 0.10eil31f 30 7 6 44 25 7 1 0 9 9 310.86 0.18eil33a 32 5 1 212 0 0 161 25 0 0 1248.72 0.99eil33b 32 5 3 267 134 7 55 40 0 0 1268.98 1.16eil33c 32 5 4 231 186 20 22 42 10 7 1289.40 1.49eil33d 32 4 1 260 0 0 258 63 0 0 1471.25 1.77Rand30a 29 7 1 6 0 0 10 7 0 0 1218.76 0.04Rand30b 29 7 2 5 1 0 7 2 0 0 1244.75 0.03Rand30c 29 7 3 5 6 0 4 2 0 0 1273.40 0.03Rand30d 29 7 4 5 6 2 4 2 0 0 1283.13 0.05Rand30e 29 7 5 5 9 1 1 0 1 1 1285.50 0.04Rand40a 39 7 1 5 0 0 7 6 0 0 1210.34 0.06Rand40b 39 7 4 4 4 0 3 4 0 0 1222.09 0.05Rand40c 39 7 5 5 7 0 3 4 0 0 1222.09 0.07Rand40d 39 7 6 6 9 1 1 3 3 3 1222.23 0.10Rand50a 49 7 1 11 0 0 22 10 0 0 1272.99 0.18Rand50b 49 7 6 12 19 1 7 8 0 0 1270.39 0.24Rand50c 49 9 1 14 0 0 14 6 0 0 1127.58 0.14Rand50d 49 8 1 14 0 0 17 3 0 0 1204.61 0.13Rand50e 49 6 1 17 0 0 35 21 0 0 1533.84 0.38Rand50f 49 5 1 15 0 0 27 13 0 0 1815.55 0.27Rand50g 49 9 6 11 11 1 2 0 0 0 1135.50 0.14Rand50h 49 9 7 11 12 1 2 0 0 0 1135.50 0.14Rand50i 49 9 8 10 9 1 1 0 2 2 1137.75 0.14Rand50j 49 8 4 12 10 0 5 1 0 0 1261.80 0.13Rand50k 49 8 5 13 14 2 2 0 0 0 1265.50 0.18Rand50l 49 8 6 12 17 0 1 0 1 1 1269.20 0.14Rand50m 49 8 7 10 19 1 1 0 1 0 1270.25 0.17A034-02fa 33 8 1 35 0 0 16 5 0 0 1788.44 0.08A034-02fb 33 8 2 34 6 0 25 13 0 0 1806.43 0.17A034-02fc 33 8 3 46 30 2 15 9 0 0 1839.51 0.20A034-02fd 33 8 4 47 22 17 10 11 1 1 1844.07 0.36A034-02fe 33 8 5 45 33 3 5 7 1 1 1855.70 0.21A034-02ff 33 8 6 48 41 5 1 1 0 0 1879.12 0.24A034-02fg 33 7 1 51 0 0 51 13 0 0 1884.02 0.26A034-02fh 33 7 4 54 33 2 23 13 0 0 1896.60 0.28A034-02fi 33 7 5 50 49 1 11 11 0 0 1896.31 0.25A034-02fj 33 7 6 49 45 6 2 19 4 4 1891.30 0.34A034-02fk 33 6 1 58 0 0 39 15 0 0 2033.65 0.24A034-02fl 33 6 4 56 74 8 20 10 0 0 2089.62 0.41A034-02fm 33 6 5 62 107 4 5 6 3 3 2108.12 0.45A036-03fa 35 8 1 44 0 0 29 10 0 0 1903.81 0.19A036-03fb 35 8 4 40 28 6 16 22 1 1 1932.42 0.33A036-03fc 35 8 5 40 58 5 9 15 0 0 1931.96 0.33A036-03fd 35 8 6 41 53 3 2 8 3 2 1936.74 0.28A036-03fe 35 8 7 41 45 4 1 0 8 6 1979.77 0.30A036-03ff 35 7 6 46 63 1 13 23 0 0 1988.32 0.40A036-03fg 35 6 1 48 0 0 35 20 0 0 2156.46 0.27A036-03fh 35 6 5 58 106 1 23 14 1 1 2187.21 0.54A039-03fa 38 8 1 43 0 0 42 28 0 0 1983.30 0.34A039-03fb 38 8 6 42 52 4 9 13 2 2 1993.33 0.36A039-03fc 38 8 7 48 44 2 2 18 2 2 1992.49 0.36A039-03fd 38 7 1 59 0 0 73 27 0 0 2157.03 0.51A039-03fe 38 7 5 49 96 1 16 3 0 0 2180.10 0.46A039-03ff 38 7 6 51 81 1 2 18 5 4 2195.23 0.52A039-03fg 38 6 1 75 0 0 90 38 0 0 2336.10 0.67A039-03fh 38 6 3 66 112 3 51 29 2 2 2362.48 0.81A039-03fi 38 6 4 60 108 13 26 36 5 3 2391.79 0.96A039-03fj 38 6 5 72 143 10 6 16 8 6 2419.03 0.94A045-03fa 44 8 1 51 0 0 98 18 0 0 2195.19 0.61A045-03fb 44 8 4 55 99 7 41 20 1 1 2208.87 0.95A045-03fc 44 8 5 59 99 5 32 28 0 0 2223.67 0.91A045-03fd 44 8 6 59 72 7 16 22 0 0 2221.15 0.81A045-03fe 44 8 7 73 100 3 2 13 5 2 2246.79 0.80A048-03ff 47 7 2 51 23 0 81 28 1 1 2563.66 0.81A048-03fg 47 7 3 73 65 1 69 32 0 0 2578.84 1.04

Table 3: Second implementation: the root node.

name g1 g2 g3 g4 g5 g6 g7 opt time LP iter nodes m max mineil22a 203 0 0 165 510 0 0 415 3.00 1065 463 5 5 1eil22b 3132 542 0 3127 5992 313 260 421 45.27 17549 9733 5 5 2eil22c 243 191 546 209 995 189 140 426 7.81 2894 1097 5 5 3eil22d 76 61 275 1 54 121 78 437 2.30 630 135 5 5 4eil22e 348 0 0 328 490 0 0 369 3.55 1319 581 4 6 3eil22f 69 21 11 61 60 4 2 369 0.76 266 127 4 6 3eil22g 53 29 41 12 32 4 4 375 0.51 168 59 4 6 4eil22h 47 33 22 1 4 10 9 377 0.26 92 15 4 6 5eil23a 55 0 0 42 19 0 0 776 0.13 87 3 6 4 2eil23b 44 22 0 15 10 0 0 776 0.10 72 9 6 4 2eil23c 57 46 2 5 10 2 1 794 0.13 91 1 6 4 3eil23d 380 0 0 565 464 0 0 634 5.13 1867 1069 4 7 1eil23e 84 15 0 64 30 0 0 645 0.72 277 219 4 7 2eil23f 60 15 23 41 16 8 8 642 0.44 198 119 4 7 4eil23g 37 21 3 1 0 3 2 642 0.08 42 5 4 6 5eil31a 18338 0 0 14364 31700 0 0 309 281.52 70479 18561 5 7 3eil31b 637 86 0 909 1422 46 46 309 18.61 4075 2019 5 7 3eil31c 632 197 233 785 1111 126 107 309 17.94 4020 2015 5 7 3eil31d 161 79 70 262 84 14 12 313 4.82 891 525 5 7 4eil31e 408 594 350 337 133 82 63 314 14.56 2542 1379 5 7 5eil31f 3031 5861 2624 1 0 3977 2651 320 111.75 18502 6413 5 6 6eil33a 11222 0 0 5011 26654 0 0 1260 271.73 49486 18001 7 5 2eil33b 2098 288 932 689 4747 282 254 1280 51.85 9547 1839 7 5 3eil33c 468 234 479 63 636 114 90 1294 14.67 2028 339 7 5 4eil33d 3611 0 0 2282 8058 0 0 1477 78.38 14420 2395 8 4 4Rand30a 46 0 0 112 225 0 0 1284 1.81 457 149 5 7 1Rand30b 11 1 0 30 24 0 0 1287 0.32 82 29 5 7 2Rand30c 9 8 7 13 15 0 0 1300 0.16 59 19 5 7 3Rand30d 7 6 9 9 11 0 0 1303 0.26 51 23 5 7 5Rand30e 5 12 1 1 0 1 1 1303 0.07 21 7 5 7 5Rand40a 701 0 0 3787 6532 0 0 1310 76.42 13035 3781 6 7 4Rand40b 406 212 847 1481 3632 72 72 1310 67.54 8036 2331 6 7 4Rand40c 220 690 404 466 1384 79 53 1320 33.28 4066 1597 6 7 5Rand40d 273 1300 474 176 1088 535 408 1322 40.01 4680 1625 6 7 6Rand50a 19379 0 0 93637 242114 0 0 1380 4495.68 399126 63937 7 7 7Rand50b 13810 30286 26787 38138 187129 1020 1007 1380 4255.03 331250 62825 7 7 7Rand50c 45 0 0 194 237 0 0 1153 5.22 547 123 6 9 6Rand50d 1187 0 0 3603 9566 0 0 1275 123.73 16315 3781 7 8 4Rand50e 28692 0 0 105527 651534 0 0 1659 8512.45 853443 99531 9 6 5Rand50f 325 0 0 1004 10524 0 0 1888 137.05 12672 1141 10 5 4Rand50g 17 44 22 9 46 5 5 1153 3.55 183 57 6 9 6Rand50h 18 94 38 7 37 8 7 1163 4.62 254 103 6 9 7Rand50i 559 3168 739 1 405 1154 928 1196 123.08 7640 3119 6 9 8Rand50j 13 10 4 13 12 0 0 1275 0.57 53 11 7 8 4Rand50k 15 19 4 2 0 1 1 1279 0.47 40 9 7 8 5Rand50l 21 67 28 4 18 4 2 1289 2.62 161 41 7 8 6Rand50m 9852 71054 24883 1 0 41349 28075 1380 3127.82 176811 55151 7 7 7A034-02fa 663 0 0 881 1365 0 0 1859 15.92 3173 773 5 8 1A034-02fb 1214 85 0 1296 2168 76 64 1876 21.21 5024 993 5 8 2A034-02fc 2148 450 800 2293 4282 211 178 1911 47.33 10923 2137 5 8 3A034-02fd 1491 594 1883 1168 2576 212 180 1914 55.02 8423 1455 5 8 4A034-02fe 356 587 325 176 491 64 39 1914 11.70 2121 349 5 8 5A034-02ff 8269 21413 9251 296 6210 3822 2670 1991 325.56 53888 11599 5 8 6A034-02fg 12806 0 0 16212 35679 0 0 1980 331.68 71147 16567 5 7 5A034-02fh 4716 1090 4602 4758 12870 402 401 1980 199.11 32171 8997 5 7 5A034-02fi 1856 1321 1449 1449 4694 108 95 1980 65.26 11420 1769 5 7 5A034-02fj 17991 28951 19625 4395 44404 9855 7317 2024 827.48 136635 24753 5 7 6A034-02fk 8804 0 0 11409 24446 0 0 2154 225.05 48340 9393 6 6 4A034-02fl 485 304 790 538 1428 56 48 2154 22.20 3785 363 6 6 4A034-02fm 395 412 399 40 610 192 134 2166 12.12 2110 293 6 6 5A036-03fa 3855 0 0 4341 10855 0 0 1993 101.67 20729 4873 5 8 4A036-03fb 632 201 796 608 1666 63 53 1993 31.45 4304 779 5 8 4A036-03fc 833 641 746 485 1577 94 89 1995 33.41 4683 799 5 8 5A036-03fd 1367 2438 2516 299 2897 640 483 2012 81.17 11188 2199 5 8 6A036-03fe 9832 26431 9507 1 0 17888 13009 2094 498.05 69411 14325 5 7 7A036-03ff 10581 7521 8623 10232 51800 774 772 2094 584.80 94729 13567 5 7 7A036-03fg 857 0 0 997 3492 0 0 2224 29.44 5516 389 6 6 5A036-03fh 122 198 66 101 293 23 21 2224 3.89 803 59 6 6 5A039-03fa 7912 0 0 9885 22322 0 0 2071 221.52 41830 6683 5 8 6A039-03fb 4079 2904 3252 2523 12162 509 493 2071 202.66 26460 2949 5 8 6A039-03fc 4051 5465 3063 634 10854 1532 1278 2084 191.32 26742 3575 5 8 7A039-03fd 104407 0 0 120426 379425 0 0 2264 3798.47 641326 74087 6 7 4A039-03fe 2277 2234 2627 1446 7386 567 493 2264 136.09 17480 2079 6 7 5A039-03ff 2740 3496 3161 103 5251 2739 1943 2272 155.32 18116 1379 6 7 6A039-03fg 10068 0 0 9285 37915 0 0 2415 424.09 60401 7773 7 6 3A039-03fh 704 153 265 392 2089 56 54 2415 22.01 3687 247 7 6 3A039-03fi 180 156 290 112 516 36 30 2427 10.46 1289 63 7 6 4A039-03fj 105 203 63 16 109 47 28 2436 3.34 527 21 7 6 5A045-03fa 114222 0 0 144324 479545 0 0 2297 6362.39 783092 138481 6 8 4A045-03fb 51546 14053 63300 52261 233793 6445 6162 2297 4441.31 437376 54409 6 8 4A045-03fc 46368 27835 54412 34897 190801 6997 6200 2305 4028.67 377220 49705 6 8 5A045-03fd 58883 87839 71484 29800 244871 15025 12231 2314 5376.30 535620 63589 6 8 6A045-03fe 27736 65326 24516 1118 62734 30074 22816 2342 2232.28 222863 33349 6 8 7A048-03ff 258868 17021 0 408741 1335825 16834 15493 2690 24966.50 2152014 258665 7 7 5A048-03fg 151251 16535 59979 188512 734279 11968 11832 2690 13362.31 1230159 163615 7 7 5

Table 4: Second implementation: the whole branch-and-bound search.

name n Q Q f1 f4 f5 f8 r-LB r-timeeil31a 30 7 1 32 14 8 0 294.50 0.07eil31b 30 7 2 32 14 9 3 294.50 0.08eil31c 30 7 3 32 14 7 3 294.50 0.08eil31d 30 7 4 35 16 6 2 295.13 0.08eil31e 30 7 5 32 15 6 5 294.97 0.08eil31f 30 7 6 31 17 0 7 296.59 0.07Rand40a 39 7 1 5 7 6 0 1210.34 0.06Rand40b 39 7 4 5 7 6 0 1210.34 0.05Rand40c 39 7 5 6 7 8 1 1210.40 0.07Rand40d 39 7 6 5 7 3 3 1213.36 0.05A045-03fa 44 8 1 45 92 21 0 2184.06 0.55A045-03fb 44 8 4 47 93 16 2 2184.21 0.53

Table 5: First implementation: the root node.

name g1 g4 g5 g8 opt time LP iter nodes m max mineil31a 14762 11721 27051 0 309 242.54 58657 15105 5 7 3eil31b 18430 14508 32446 4206 309 268.01 75392 18613 5 7 3eil31c 10563 8429 21061 3687 309 183.01 48251 12463 5 7 3eil31d 25121 24362 50076 15550 313 518.40 138728 53195 5 7 4eil31e 21062 13403 35220 21365 314 427.61 108903 41083 5 7 5eil31f 102717 28713 0 365222 320 3175.39 613780 250061 5 6 6Rand40a 1234 5898 11562 0 1310 116.62 22298 4243 6 7 4Rand40b 654 3190 5659 285 1310 65.78 11584 3467 6 7 4Rand40c 1271 5560 11991 1813 1320 131.92 24233 6913 6 7 5Rand40d 1928 4762 10203 6836 1322 161.99 28394 8457 6 7 6A045-03fa 158100 192911 663461 0 2297 8636.57 1069546 177193 6 8 4A045-03fb 267358 387505 1168899 56178 2297 16176.83 1982542 228229 6 8 4

Table 6: First implementation: the whole branch-and-bound search.