The directed profitable rural postman problem with incompatibility constraints

ARTICLE IN PRESS

JID: EOR

[m5G;February 27, 2017;11:32]

European Journal of Operational Research 0 0 0 (2017) 1–14

Contents lists available at ScienceDirect

European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

The directed profitable rural postman problem with incompatibility constraints Marco Colombi a,∗, Ángel Corberán b, Renata Mansini a, Isaac Plana c, José M. Sanchis d a

Department of Information Engineering, University of Brescia, Brescia, Italy Department of Statistics and Operational Research, University of Valencia, Burjassot, Spain Department of Mathematics for Economics and Business, University of Valencia, Valencia, Spain d Department of Applied Mathematics, Polytechnic University of Valencia, Valencia, Spain b c

a r t i c l e

i n f o

Article history: Received 10 December 2015 Accepted 3 February 2017 Available online xxx Keywords: Routing Rural postman problem Incompatibility constraints Generalized independent set problem

a b s t r a c t In this paper, we study a variant of the directed rural postman problem (RPP) where profits are associated with arcs to be served, and incompatibility constraints may exist between nodes and profitable arcs leaving them. If convenient, some of the incompatibilities can be removed provided that penalties are paid. The problem looks for a tour starting and ending at the depot that maximizes the difference between collected profits and total cost as sum of traveling costs and paid penalties, while satisfying remaining incompatibilities. The problem finds application in the domain of road transportation service, and in particular in the context of horizontal collaboration among carriers and shippers. We call this problem the directed profitable rural postman problem with incompatibility constraints. We propose two problem formulations and introduce a matheuristic procedure exploiting the presence of a variant of the generalized independent set problem (GISP) and of the directed rural postman problem (DRPP) as subproblems. Computational results show how the matheuristic is effective outperforming in many cases the result obtained in one hour computing time by a straightforward branch-and-cut approach implemented with IBM CPLEX 12.6.2 on instances with up to 500 nodes, 1535 arcs, 1132 profitable arcs, and 10,743 incompatibilities. © 2017 Elsevier B.V. All rights reserved.

1. Introduction The directed profitable rural postman problem with incompatibility constraints (DPRPP-IC) can be defined as follows. Let G(V, A) be a strongly-connected directed graph, where V = {0, . . . , n} is the set of nodes (node 0 represents the depot), and A is the set of m directed arcs. A traveling cost cij is associated with each arc (i, j) ∈ A. Let R⊆A be a subset of arcs that require a service and that yield a positive profit in case of service (profitable arcs). The positive profit pij assigned to each arc (i, j) ∈ R represents the prize that can be collected if the corresponding arc is served in the solution. The profit is available only the first time the arc is traversed. We indicate as VI ⊂ V the set of nodes i, i ∈ V, with at least one arc (i, j) ∈ R, that is the set of the initial nodes of profitable arcs. Nodes belonging to VI may be related to each other by some incompatibility conditions (incompatibility constraints). The following three situations may occur between any pair of nodes i, j ∈ VI :

∗

Corresponding author. E-mail address: [email protected] (M. Colombi).

a) Nodes i and j are strongly incompatible: the profitable arcs starting at node i and the ones leaving from node j can never be jointly selected in a solution; b) Nodes i and j are weakly incompatible: the profitable arcs starting at node i and the ones leaving from node j can be jointly selected in a solution only if a finite positive cost (penalty) c¯i j is paid to remove incompatibility between i and j; c) Nodes i and j are compatible: the joint selection of any profitable arcs starting from nodes i and j can be done. The problem looks for a tour starting and ending at the depot that maximizes the difference between the total profit collected and the total cost as sum of the traveling cost of the tour and the penalty cost paid to eliminate weak incompatibilities, while satisfying the remaining incompatibilities. We represent incompatibility constraints among nodes in VI by means of an incompatibility graph G(V , E1 ∪ E2 ), where V ⊆ VI is the set of nodes i ∈ VI which are incompatible with at least another node j ∈ VI {i} and E1 ∪ E2 is the edge set. If two nodes i, j ∈ V are connected by an edge {i, j} ∈ E1 , then they are strongly incompatible (case a). If two nodes i, j ∈ V are connected by an edge {i, j} ∈ E2 , then they are weakly incompatible (case b). When no edge

http://dx.doi.org/10.1016/j.ejor.2017.02.002 0377-2217/© 2017 Elsevier B.V. All rights reserved.

Please cite this article as: M. Colombi et al., The directed profitable rural postman problem with incompatibility constraints, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.002

JID: EOR 2

ARTICLE IN PRESS

[m5G;February 27, 2017;11:32]

M. Colombi et al. / European Journal of Operational Research 000 (2017) 1–14

Fig. 1. Incompatibility graph for an instance of the DPRPP-IC with |VI | = 9, |V | = 7 , |E1 | = 2 and |E2 | = 7.

connects two nodes i, j ∈ V , then no incompatibility exists between them (case c). Nodes in VI \ V are not incompatible with any other node, and thus profitable arcs leaving them can be freely selected. In Fig. 1, an example is presented to clarify the problem. In Fig. 1(a), we show the original problem graph G, where arcs in dashed–dotted lines are profitable arcs. In Fig. 1(b), we draw the incompatibility graph G presenting the incompatibilities existing between nodes and thus among the profitable arcs leaving them. In graph G edges in E1 and E2 are represented as continuous and dashed lines, respectively. For instance, profitable arc (4,5) and profitable arcs (3,2),(3,7) cannot be served jointly in the same tour because they are leaving nodes 4 and 3 that are strongly incompatible in graph G (they are connected by an edge belonging to E1 ). Profitable arcs (6,10) and (11,7) can both be served provided that a penalty c6,11 is paid, since nodes 6 and 11 are weakly incompatible (connected by a dashed edge) in graph G. Arcs (14,10) and (14,15) can be jointly selected with (3,2) and (3,7) since nodes 14 and 3 are not connected by any edge. Finally, arcs (0,1) and (5,6) can be freely served in a tour, since nodes 0 and 5 belong to the set VI \ V and their arcs can be freely served with any other profitable arc. Note that, for sake of clarity, in Fig. 1(b), we also draw nodes 0 and 5 even if they are not incompatible with the remaining ones, and thus do not make part of the incompatibility graph. The described type of incompatibility constraints, where incompatibilities among profitable arcs are induced by incompatibilities among nodes, comes from a real case application in the domain of transportation services. The problem has been proposed by a local association of haulers and shippers in North Italy. In many European cities there exist associations among companies active in shipping, haulage, logistics, and in all the auxiliary activities related to the movement and storage of goods. These associations pursue different aims such as the protection and general representation of categories, the support for economic, technical, tax and union activities required to create the best working conditions and business development for all their member companies. One of the most important objectives of these associations is to promote the horizontal collaboration among members, mainly on road transportation. Indeed, collaboration is widely seen as one of the best ways to deal with increasingly complex business sectors in order to create an advantage, as pointed out in Fugate, Davis-Sramek, and Goldsby (2009), Stefansson (2006), and Cruijssen, Cools, and Dullaert (2007). Horizontal collaboration concerns companies at

the same level of the supply chain that decide to cooperate even if they are competing companies performing comparable logistics functions on the land side. Few papers with relevant models have been published and several problems on collaboration routing remain to be studied (see, for instance, Fernández, Fontana, & Speranza, 2016). Transportation companies that participate or form coalitions want to fulfill customer requests by sharing their services to possibly reduce their costs. In this context, the local association acts as a central third party, able to gather information in a truthful way, and to impose constraints that have to be satisfied by any company when providing a service for other association members. In this paper, we are interested in the problem faced by a potential hauler that may decide to use one of the vehicles of his/her fleet (consisting of medium and large size vehicles) to execute transportation services (so called lanes), possibly including services belonging to other haulers and shippers, while satisfying the constraints imposed by the association. A lane is defined as a transportation service in full truckload mode associated with a given customer that requires a transportation service from an origin (typically the hub of the customer) to a specified destination on a dedicated vehicle. For each executed service the hauler earns a predefined revenue. When deciding the lanes to serve and the tour serving them, the hauler has to take into account the following constraints imposed by the association to guarantee a fair service: •

•

Size of the lane. The transportation services are classified by the association according to the size of the quantity transported in small, medium and large size lanes. To avoid non-optimized vehicle loading, the association has imposed strong incompatibility constraints between small and large size lanes, and weak incompatibility constraints between large and medium size lanes, and between small and medium size ones. A penalty can be paid by the hauler to remove a weak incompatibility. Penalty is a cost assigned by the association for a non-optimized vehicle use and represents a measure of the environmental cost produced by a vehicle that pollutes by traveling with a partial load without completely exploiting its capacity. Type of goods. Incompatibility between nodes may be due to the type of good transported by lanes leaving them. Food and chemicals, although not jointly transported, are strongly incompatible and lanes transporting them cannot be served by the same vehicle in a tour. On the contrary detergents and clothing can be considered weakly incompatible and the penalty to pay

Please cite this article as: M. Colombi et al., The directed profitable rural postman problem with incompatibility constraints, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.002

JID: EOR

ARTICLE IN PRESS

[m5G;February 27, 2017;11:32]

M. Colombi et al. / European Journal of Operational Research 000 (2017) 1–14

•

is the cost of an additional insurance against possible contamination. Customer’s requirements. Some customers may require the exclusivity of the transportation service. This implies that they do not want that their lanes are jointly served with those of some other customers (this is common when customers are competitors) and the incompatibility is strong. In some cases, lanes from two competitors can be served jointly but the transportation company is charged with a penalty measuring the discount (or a similar incentive) applied to the customers to accept to be jointly served.

Lanes starting from the same node (central hub, depot or production center) are associated with a given customer and are characterized by the same type of transported goods, the same lane size and the same requirements. Thus, they are all compatible by definition. A customer may be associated with more than one starting point (node), and these nodes are not necessarily located at the same geographical position. In practice, this is always associated with lanes that differ in their size or the type of products transported. Thus, although associated with the same customer, such nodes may have a different type of incompatibilities with other nodes. For instance, a company has two different starting nodes for its lanes. In both cases lanes are associated with the same good but in one case lanes have a small size, whereas in the other they are of medium size. This implies that while they are weakly incompatible each other, they have different types of incompatibility with other large size lanes. The small ones are strongly incompatible while the medium size ones are only weakly incompatible. The resulting problem is the one we describe. Lanes represent profitable arcs and define the set R. The hauler wants to select the lanes to serve so as to maximize the difference between the total revenue earned and the cost of traveling and penalty costs, while satisfying all incompatibility constraints imposed by the association. We start studying the single vehicle case. The natural extension to the multi-vehicle case, possibly involving different haulers, requires the formalization of additional constraints and is left for future research. To the best of our knowledge the problem is new and provides the first tentative to deal with incompatibility constraints in arc routing problems. We propose two different mathematical formulations for the problem involving a different number of variables and constraints, and introduce a matheuristic exploiting the double nature of the DPRPP-IC, including a routing part (the profitable directed rural postman problem) and a combinatorial side related to the selection of compatible arcs and formulated as a variant of the generalized independent set problem (GISP). Incompatibilities associated with nodes are side-constraints that can be formulated as GISP constraints (see Colombi, Mansini, & Savelsbergh, 2016c). More precisely, when transferred to arcs, the incompatibility constraints give rise to a variant of the GISP where profitable arcs become nodes of the GISP and incompatibilities are introduced among clusters of nodes. This variant of the GISP, called clustered GISP, has never been studied before. To solve the clustered GISP, we exploit the quadratic nature of the problem by introducing a Greedy Randomized Adaptive Search Procedure where the improvement phase is a Tabu Search algorithm (GRASP-TS). Computational results on a large set of randomly generated instances have shown that the matheuristic approach is efficient and effective compared to an exact approach obtained by solving the best of the two problem formulations by means of a branch-andcut algorithm implemented with IBM CPLEX 12.6.2 that separates the connectivity inequalities heuristically. The paper is organized as follows. In Section 2, we first analyze the literature on arc routing problems with profits

3

(revenues, prizes). In Section 3, we provide two different mathematical formulations for the problem discussing simple properties. Section 4 is devoted to the presentation of the clustered GISP as a new variant of the GISP. We provide a mathematical formulation for the problem and two solution algorithms, the GRASP-TS and an exact approach applied to an approximation of the problem. Section 5 proposes the solution algorithms for the DPRPP-IC. Other than the branch-and-cut approach, we present the matheuristic that exploits the solutions of the clustered GISP and of the directed rural postman problem (DRPP) as subproblems. Finally, in Section 6, we present the computational results. We discuss the comparison of the two formulations and the performance of the matheuristic method implemented in four different configurations. The latter comes out to be quite effective, being able to get the optimal solution in many instances and in many others to find, in few minutes, a solution value better than the one found by the branch-and-cut method in one hour of computing time. Conclusions and future developments also concerning variants of the presented problem are drawn in Section 7. 2. Literature review The interest in arc routing problems with profits has strongly increased in the last years mainly due to a huge number of real applications from different domains that involve profits. The Chinese postman problem (CPP) is one of the most central problems in arc routing. The first historical contribution of an arc routing problem with profits concerns a variant of the CPP and it is due to Malandraki and Daskin (1993). The authors introduce the maximum benefit Chinese postman problem (MBCPP) on a directed graph where all arcs are profitable. The benefit associated with an arc can be collected only for a finite number of times. More precisely, they assume that the gross benefit for each arc is decreasing with the number of traversals (or alternatively the servicing cost is increasing with the number of traversals). Thus, the net benefit of a traversal is non-increasing. The MBCPP looks for the route starting and ending at the depot with maximum total net benefit. The problem is NP-hard as the rural postman problem (RPP) is a special case. Malandraki and Daskin propose a branchand-bound algorithm generating subtour elimination constraints when needed, while retaining the minimum cost flow formulation. The approach is used to solve an instance with only 25 vertices. In Corberán, Plana, Rodríguez-Chía, and Sanchis (2013), the authors study the undirected variant of the MBCPP. They propose a mathematical formulation for the problem with only two net benefits for each link, and provide a polyhedral study of the formulation. Two classes of valid inequalities are also proposed to strengthen the formulation, namely the K–C inequalities and the p-connectivity inequalities. The proposed branch-and-cut algorithm is able to solve instances with up to 10 0 0 vertices and 30 0 0 edges in one hour of computing time. Heuristic and approximation algorithms were proposed in Pearn and Wang (2003) and Pearn and Chiu, 2005). The first paper considers the undirected version of the problem, whereas the second paper the directed one. In both papers, it is assumed that the benefit is decreasing with the number of traversals. Pearn and Wang (2003) present a heuristic which first finds a minimum spanning tree on an expanded graph obtained by replacing each edge of the original graph with a set of edges with positive net benefit. Then, a minimum cost matching on odd-degree vertices is solved to obtain a route. Finally, negative net benefit cycles are removed. The algorithm is tested on an instance with only 15 vertices and 26 edges. Pearn and Chiu, 2005) propose three heuristic methods, namely a branch and scan algorithm, a connection algorithm and a directed tree expansion algorithm. Tests are performed on instances with up to 30 vertices and 783 arcs.

Please cite this article as: M. Colombi et al., The directed profitable rural postman problem with incompatibility constraints, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.002

JID: EOR 4

ARTICLE IN PRESS

[m5G;February 27, 2017;11:32]

M. Colombi et al. / European Journal of Operational Research 000 (2017) 1–14

Variants and extensions of the undirected RPP including profits have also been studied. Two of these are the prize-collecting RPP studied in Aráoz, Fernández, and Meza (2009b) and Aráoz, Fernández, and Zoltan (2006), and the profitable arc tour problem analyzed in Feillet, Dejax, and Gendreau (2005). The prizecollecting RPP, introduced in Aráoz et al. (2006), can be seen as a special case of the MBCPP where the profit of an edge can be collected at most once. The authors introduce different problem formulations one of which, based on the property that there always exists an optimal solution where each edge is traversed at most twice, has been strengthened with the introduction of inequalities derived from dominance rules in Aráoz et al. (2009b). In Aráoz et al. (2009b), the authors propose both an exact and some heuristic approaches tested on benchmark instances from the RPP duly modified. All instances have been solved to optimality within one hour of computing time. In Aráoz, Fernández, and Franquesa (2009a) the authors analyze a variant of the problem with customers clustered in groups and profits associated with groups (the clustered prize-collecting arc routing problem). Constraints impose that if a customer of a group is served, then all customers of the same group have to be served. They study the problem properties and provide a mathematical formulation developing a branch-and-cut algorithm to solve it. Tests are made on the instances proposed in Aráoz et al. (2009b) opportunely modified. All instances have been solved to optimality in almost all cases in less than one hour of computing time. The same problem is studied in Corberán, Fernández, Franquesa, and Sanchis (2011) with an extension to a windy graph. The authors propose a mathematical formulation and present polyhedral results including facet-defining and valid inequalities. They propose a cutting plane algorithm solving instances with up to 196 vertices and 316 edges. In Guastaroba, Mansini, and Speranza (2010) and afterwards in Archetti, Guastaroba, and Speranza (2014) and in Colombi and Mansini (2014), a variant of the directed version of the prizecollecting rural postman problem is addressed. The problem is called profitable because the goal is to find the route that maximizes the difference between the total collected profit and the traveling cost (equivalent to the minimization of the sum of the traveling cost and the penalties for unserved arcs). The shipper’s lane selection problem (SLSP), as referred to in Guastaroba et al. (2010), considers a shipper that has to decide which transportation requests (lanes) to undertake for a direct service with its own vehicle, and which to assign to external carriers paying an outsourcing cost (the penalty). The authors provide a mathematical formulation using binary instead of integer variables, thus allowing a single traversal for each arc. More recently, the SLSP has been reformulated by Archetti et al. (2014) who called it the directed profitable rural postman problem (DPRPP) and introduce an effective Tabu Search algorithm with the addition of an ex-post ILPrefinement. Computational results on 108 instances adapted from other arc routing problems show that the average gap with respect to the optimal solution, when available, is below 1%. They use a branch-and-cut to solve the problem to optimality. All but 22 instances were solved in one hour of computing time. Colombi and Mansini (2014) introduce a tighter problem formulation than the one proposed in Archetti et al. (2014) by adding different valid inequalities and use it to develop a branch-and-cut algorithm as well as an effective matheuristic exploiting information provided by a problem relaxation. The proposed exact method results to be highly efficient, always finding the optimal solution in less than one hour of computing time for all the benchmark instances proposed in Archetti et al. (2014) including the 22 open ones. Heuristics also behave very well outperforming state-of-theart algorithms in many sets of instances. Ávila, Corberán, Plana, and Sanchis (2016) propose a formulation for the profitable windy rural postman problem, which contains the DPRPP as a particu-

lar case, and study its associated polyhedron, designing a branchand-cut algorithm that is able to solve large-sized instances of up to 1500 vertices optimally in less than one hour of computing time. Black, Eglese, and Wøhlk (2013) study a variant of the directed version of the prize-collecting RPP with time dependent costs assuming that the cost of traversing each arc depends on the time at which the traversal is made. They propose a non-linear formulation and introduce two metaheuristic algorithms, a Variable Neighborhood Search and a Tabu Search. Tests on benchmark instances generated from real road networks with up to 50 vertices, 2500 arcs and 350 profitable arcs show that both algorithms find good solutions, though the VNS performs better on average. Finally, two additional problems with profits have been studied. The first one is the arc orienteering problem (AOP) introduced by Souffriau, Vansteenwegen, Berghe, and Van Oudheusden (2011). The problem is defined on a directed graph where all arcs are profitable and the objective is to find a route from vertex 0 to a destination vertex n with maximum profit and total traveling time not greater than a predefined value Tmax . The authors propose a Greedy Randomized Adaptive Search Procedure (GRASP) solving instances from a real network consisting of 989 nodes and 2963 arcs. The procedure provides good results in a limited computational time of only 1 second being devoted to solve on-line problems. The second problem is the one-period bus touring problem (BTP) studied by Deitch and Ladany (20 0 0) and concerning the problem of finding a route for a bus that has to visit tourist sites and scenic routes. Profits, measuring attractiveness, are associated with both arcs and nodes and the problem looks for a route with maximum attractiveness while satisfying side constraints on maximum touring time or cost. We refer to the recent book by Corberán and Laporte (2015), particularly the chapter by Archetti and Speranza (2014), for more details on arc routing problems with profits including single and multi-vehicle variants. Interesting enough, while different works have introduced side constraints (typically on the total cost or time associated with a route) none of them take incompatibility constraints into account. For instance, from the arc routing side, problems incorporate constraints about forbidden turns and turn penalties (see Corberán, Martí, Martínez, & Soler, 2002), hierarchies (see Colombi, Corberán, Mansini, Plana, & Sanchis, 2016b; 2017), and clusters (see Dror & Langevin, 1997). Contrarily, incompatibilities are somehow present on the side of vehicle routing problems, even though they are applied on products and not to the nodes of the graph. In this field, one can read the recent works by Manerba and Mansini (2015) and Gendreau, Manerba, and Mansini (2016) dealing with strong incompatibilities among the collected products in a multi-vehicle traveling purchaser problem with unitary demands. The introduction of weak and strong incompatibility constraints, as frequent components in practical problems, and of efficient methods to deal with them, opens up a new research stream. 3. Problem formulations We introduce two integer linear programing formulations for the DPRPP-IC. The first model uses the following sets of variables: • •

•

•

xij , (i, j) ∈ A: indicates the number of times arc (i, j) is traversed, yij , (i, j) ∈ R: takes value 1 if the profitable arc (i, j) is served and its profit collected, and 0 otherwise, zi , i ∈ V : takes value 1 if at least one profitable arc (i, j) ∈ R leaving vertex i is served, and 0 otherwise, uij , {i, j} ∈ E2 : takes value 1 if the penalty associated with weak incompatibility between nodes i and j is paid and the incompatibility can be ignored.

Please cite this article as: M. Colombi et al., The directed profitable rural postman problem with incompatibility constraints, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.002

ARTICLE IN PRESS

JID: EOR

[m5G;February 27, 2017;11:32]

M. Colombi et al. / European Journal of Operational Research 000 (2017) 1–14

As usual, each vertex set S⊆V defines an arc cutset δ + (S ), formed by all the arcs leaving S, and an arc cutset δ − (S ), formed by all the arcs entering S. If S consists of only one vertex i, we denote the corresponding arc cutsets as δ + (i ) and δ − (i ), respectively. Moreover, given S⊆V, profitable arcs (i, j) ∈ R with both endpoints in S define the set R(S). The problem can be formulated as follows:



(A ) w = max

pi j yi j −

(i, j )∈R

s.t. :



(i, j )∈A



x ji =

( j,i )∈δ + ( j )

ci j xi j −



j ∈V

xi j

(1)

Proof. Consider an optimal solution of the reformulated model (A). If at least one profitable arc leaving node i is selected (i.e. yis = 1, for some s) then zi = 1 due to constraints (4). Otherwise all yis = 0 for any s, and zi could take a positive value. In this case, we can set zi = 0 obtaining another optimal solution, since the objective value is the same. The only constraints affecting the value of uij are inequalities (6) and uij ≥ 0. Since uij has a negative coefficient in the objective function, we can assume that in any optimal solution ui j = max{0, zi + z j − 1}, which is a binary value (since as we have seen, zi and zj can be made binary). 

(2)

Model (A) can be reformulated without variables z. The new formulation (B) is as follows:

c¯i j ui j

{i, j}∈E2

(i, j ) ∈ R

xi j ≥ yi j





(i, j )∈δ − ( j )

xi j ≥ yks

S⊆V

\ {0}, (k, s ) ∈ R(S )

(3)

(i, j )∈δ + (S )

yi j ≤ zi

{i, j} ∈ E1

zi + z j ≤ 1

zi + z j − ui j ≤ 1

zi ∈ {0, 1} ui j ∈ {0, 1}

{i, j} ∈ E2 (i, j ) ∈ A

xi j ≥ 0 integer yi j ∈ {0, 1}

(B )

(i, j ) ∈ R i∈V

{i, j} ∈ E2 .



max

pi j yi j −

(i, j )∈R

s.t. : i ∈ V , (i, j ) ∈ R

5

 (i, j )∈A

ci j xi j −



c i j ui j

{i, j}∈E2

( 1 )– ( 3 )

(11)

(4) yis + y jt ≤ 1

{i, j} ∈ E1 , (i, s ) ∈ R, ( j, t ) ∈ R

(12)

(5) yis + y jt − ui j ≤ 1

{i, j} ∈ E2 , (i, s ) ∈ R, ( j, t ) ∈ R

(13)

(6) (7) (8) (9) (10)

The objective function maximizes the net profit, that is, the difference between the profit collected and the total cost, as sum of the traveling cost of the tour plus the penalties paid to ignore weak incompatibilities. Constraints (1) imply that arc (i, j) ∈ R can be served and its profit collected (yi j = 1) if and only if it has been traversed (xij ≥ 1). Constraints (2) are the so called symmetry constraints and impose the equivalence between the number of arcs entering and leaving each node j ∈ V. Constraints (3) are connectivity constraints. Inequalities (4) state that if arc (i, j) is served, then variable zi must take value 1. Inequalities (5) impose that two arcs leaving strongly incompatible nodes cannot be jointly served. Inequalities (6) impose that two arcs leaving weakly incompatible nodes i and j can be jointly served provided that the corresponding penalty ci j is paid, i.e. variable ui j = 1. Finally, constraints (7)–(10) define integer and binary conditions. Note that constraints (5) and (6) correspond to inequalities of the generalized independent set problem (GISP), a generalization of the independent set problem where a set of removable edges with associated removal costs are given, and the goal is to find an independent set that maximizes the net profit, i.e., the difference between the profits collected for the vertices in the independent set and the costs incurred for any removal of edges with both endpoints in the independent set. Interested readers are referred to Colombi et al. (2016c). Moreover, differently from a pure GISP, integrality conditions (9) and (10) are not necessary and can be removed, as stated in the following proposition. Proposition 1. Reformulate model (A) by substituting conditions (9) and (10) with 0 ≤ zi ≤ 1, i ∈ V , and 0 ≤ uij ≤ 1, {i, j} ∈ E2 . Then there always exists at least one optimal solution in which variables zi , i ∈ V and uij , {i, j} ∈ E2 , take value 0 or 1.

xi j ≥ 0 integer

(i, j ) ∈ A

(14)

yi j ∈ {0, 1}

(i, j ) ∈ R

(15)

ui j ∈ {0, 1}

{i, j} ∈ E2 .

(16)

Formulation (A) requires the introduction of |V | additional variables z and a number of constraints equal to |R| + |E1 | + |E2 | in order to model the incompatibilities. Formulation (B) does not need for the additional variables z, but requires a larger number of constraints, O((|E1 | + |E2 | ) ∗ |R|2 ), to model the incompatibilities. Although the two formulations (A) and (B) require a different number of variables and constraints, they provide the same bound value when solving their linear relaxations, LRA and LRB , as it is shown in the following proposition. Proposition 2. The linear relaxations LRA and LRB are equivalent. Proof. We will prove that for any feasible solution of LRA , there is a feasible solution of LRB with the same cost and viceversa, and therefore the optimal solutions of both relaxations coincide. Let (xAi j , yAi j , ziA , uAi j ) be a feasible solution of LRA with cost cA . We

define the vector (xBi j , yBi j , uBi j ) = (xAi j , yAi j , uAi j ), which obviously has the same cost cB = cA and satisfies constraints (1) to (3). In order to check that constraints (12) are satisfied, let {i, j} ∈ E1 , (i, s) ∈ R, and (j, t) ∈ R. From (4), yBis + yBjt ≤ ziA + zAj , and then from (5) we have yBis + yBjt ≤ 1. Similarly, from (6) it can be shown that constraints

(13) are satisfied. Therefore, (xBi j , yBi j , uBi j ) is a feasible solution of LRB with the same cost. Reciprocally, consider a feasible solution (xBi j , yBi j , uBi j ) of LRB with

cost cB . We now construct the vector (xAi j , yAi j , ziA , uAi j ) with xAi j = xBi j ,

yAi j = yBi j , uAi j = uBi j , and ziA = max{yAis : (i, s ) ∈ R}. This vector has the same cost cA = cB and satisfies constraints (1)–(3). Let us now see that (4)–(6) are satisfied. From the definition of ziA , constraints (4) hold. Given {i, j} ∈ E1 , ziA + zAj = max{yAis } + max{yAjt } = max{yAis + yAjt } ≤ 1, where the last inequality is implied by (12). A similar argument shows that constraints (13) are also satisfied. Finally, since ziA ≤ 1 holds, (xAi j , yAi j , ziA , uAi j ) is a feasible solution of LRA with the same cost. 

Please cite this article as: M. Colombi et al., The directed profitable rural postman problem with incompatibility constraints, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.002

ARTICLE IN PRESS

JID: EOR 6

M. Colombi et al. / European Journal of Operational Research 000 (2017) 1–14

4. The clustered GISP We have already emphasized the importance of the clustered GISP as a subproblem of the DPRPP-IC. In this section, we provide a mathematical formulation for this new variant of the GISP and two solution algorithms to tackle it. Constraints (5), (6), (9) and (10) are classical constraints of the generalized independent set problem (GISP). The problem, introduced by Hochbaum and Pathria (1997) in the context of forest management and harvesting, has been recently studied from a polyhedral point of view in Colombi et al. (2016c). In the GISP, a revenue is associated with each vertex of the incompatibility graph and edges are partitioned into removable edges, that may be eliminated provided that a positive cost is paid, and non-removable ones. The problem looks for an independent set, i.e. a set of vertices such that no two vertices in the set are adjacent, that maximizes the difference between the total revenue associated with the vertices in the set and the total cost associated with the removal of edges with both endpoints in the set. In the DPRPP-IC an incompatibility on two nodes determines an incompatibility among the profitable arcs leaving these two nodes, as shown in constraints (12) and (13) and (15) and (16) of formulation (B). This means that we can formulate the selection of profitable arcs to serve as a GISP variant where vertices are the profitable arcs with revenues equal to the arc profits, the profitable arcs leaving a same node become the vertices belonging to a same cluster, while strong and weakly incompatibility constraints are introduced among pairs of clusters. Paying the penalty associated with a removable edge (weak incompatibility) connecting a pair of clusters implies that any vertices of the two clusters can be selected in the independent set. On the contrary, if a non-removable edge (strong incompatibility) connects two clusters, then no pair of vertices, belonging each one to a different cluster, can be selected. We call this problem the clustered generalized independent set problem (clustered GISP). Let N be a set of vertices and let wh be the revenue associated with each vertex h ∈ N. Let C = {Ci : i = 1, . . . , r} be the set of vertex clusters into which set N is partitioned. Given a graph G˜ = (C, H1 ∪ H2 ), where H2 is the set of removable edges {i, j} connecting pairs of clusters with associated removal costs c¯i j , whereas H1 is the set of non-removable edges among pairs of clusters. The problem seeks to find an independent set that maximizes the net benefit, i.e., the difference between the revenues collected from the vertices in the independent set and the costs incurred for any removal of edges associated with pairs of clusters in H2 .

(Clustered GISP )

max

  wh vh − c i j si j h∈N

s.t. :

[m5G;February 27, 2017;11:32]

zi ≥ vh

zi + z j ≤ 1

h ∈ Ci , i = 1, . . . , r

{i, j} ∈ H1

zi + z j − si j ≤ 1

{i, j} ∈ H2

(17)

{i, j}∈H2

(18) (19) (20)

vh ∈ {0, 1}

h∈N

(21)

si j ∈ {0, 1}

{i, j} ∈ H2 .

(22)

A binary variable zi is associated with each cluster Ci , i = 1, . . . , r, and a binary variable vh with each vertex h ∈ N. A cluster Ci is selected (i.e. zi = 1) if at least one vertex vh belonging to the cluster enters the independent set (inequalities (18)). If {a, b} ∈ H1 , then no vertex in Ca can be selected with any vertex of Cb . On the contrary if {a, b} ∈ H2 , then any vertex in Ca can be selected

with any vertex in Cb only if the penalty cab is paid to free all the vertices of the two clusters. The clustered GISP is a new problem that, to the best of our knowledge, has never been studied before. In order to solve it, we propose two alternative procedures: 1. We approximate the clustered GISP instance with a GISP instance, and then solve it exactly by means of a branch-and-cut algorithm. 2. A GRASP-Tabu Search. In the first approach, a GISP instance is obtained by introducing a removable edge for each pair of vertices (h1 , h2 ) such that h1 ∈ Ca and h2 ∈ Cb and {a, b} ∈ H2 . In particular, if ci j is the cost associated with two weakly incompatible nodes in DPRPP-IC, then in the GISP instance this cost is repeated for each pair of vertices belonging to the weakly incompatible clusters, and is thus overestimated. We have also tried a variant in which the cost assigned to a removable edge in the GISP instance was equal to the original cost divided by the number of all the profitable arcs leaving the nodes in DPRPP-IC and selected in the tour representing the feasible solution. However, the resulting underestimation of the costs seems to provide worse results, and therefore this version was discarded. A polyhedral study of the GISP can be found in Colombi et al. (2016c). Authors propose several families of valid inequalities, which we have used within a branch-and-cut algorithm to solve this GISP instance. The implemented approach adds violated generalized clique inequalities (GCI) and generalized odd-cycle inequalities (GOCI) to cut fractional solutions. More precisely, when a fractional solution is found, heuristic separation procedures search for generalized cliques (containing removable and non-removable edges) and generalized odd-cycles using the random procedures described in Colombi et al. (2016c). All the violated GCI and GOCI found are added. The second procedure is a Greedy Randomized Adaptive Search Procedure that makes use of a Tabu Search algorithm as local search (GRASP-TS). The method is an adaptation to the clustered GISP of the probabilistic GRASP-TS proposed for unconstrained binary quadratic problems in Wang, Lü, Glover, and Hao (2013). To obtain a quadratic formulation for the GISP, removable edges between vertices in a generalized independent set can be represented by products of variables of their endpoints, considering non-removable edges as removable ones with huge penalty costs. GRASP is a multi-start algorithm characterized by two phases iteratively repeated until a stopping rule is satisfied. The first phase is a randomized greedy solution construction, whereas the second phase is a local improvement procedure. The randomized greedy construction procedure we used, is described in Algorithm 2, where the main while loop stops when the solution value does not further improve or all the vertices have been included in the independent set. The method adds vertices to the independent set by randomly selecting them. When the solution obtained by including a new vertex is worse than the previous one, the while loop is ended and the latter is retained as final solution (Steps 8 and 9). When Algorithm 2 terminates, a feasible solution is available. We then apply a local improvement phase corresponding to the Tabu Search procedure described in Algorithm 3. The Tabu Search is based on two simple neighborhoods represented by procedures ADD and RE MOV E _CL. In the first one, a move corresponds to the introduction of an external vertex in the solution I, whereas in the second one a move corresponds to the removal of a cluster from the current solution I. Removing a cluster implies the removal of all the vertices belonging to it. Each time we carry out a move, the reverse move is forbidden for the next tt (tabu tenure) iterations. The tabu tenure is computed as t t = const 1 + rand ([1, 10] ) for neighborhood ADD, and as t t = const 2 + rand ([1, 10] ) for neighborhood RE MOV E _CL, where const1 and const2 are instance depen-

Please cite this article as: M. Colombi et al., The directed profitable rural postman problem with incompatibility constraints, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.002

ARTICLE IN PRESS

JID: EOR

[m5G;February 27, 2017;11:32]

M. Colombi et al. / European Journal of Operational Research 000 (2017) 1–14

|N | r dent constants set to 100 and to 100 with r the number of clusters, respectively. In both formulas rand([1, 10]) is a random number in the interval [1,10]. In both Algorithms 2 and 3, to update the objective function in linear time and search the neighborhoods effectively we make use of four different vectors q1 , q2 , q3 and q4 . While the first one has a size equal to the number of vertices and is initialized with their profit values, the remaining three have a size equal to the number of clusters and are initialized with the null vector. The vector q1 represents, for each vertex, not in the independent set, its contribution to the objective function value when added to the generalized independent set. The vector is updated each time a vertex h is added to the independent set or a cluster is removed. In particular, when vertex h enters the solution, its q1 [h] value is set to zero (it is no longer convenient for it to enter the solution as it is already there), while when a cluster is removed then all its vertices belonging to the solution are also removed and their q1 values are updated setting them back to their initial profit value (they may be considered again for possible future selection). Similarly, vector q2 indicates the total profit associated with each cluster belonging to the solution. Every time a vertex is added to the current solution, as in the greedy randomized algorithm or when an ADD move is performed, the profit of the entering vertex is added to the total profit of the cluster to which it belongs. On the contrary, when a RE MOV E _CL move on a cluster Ck is applied, the cluster is removed from the solution, and the corresponding value of vector q2 [k] is set to zero. The main role is however performed by vector q3 . The following update logic for q3 is applied (with C¯ a large positive number): •

•

ADD move (vertex h of cluster Ck enters the solution I):  q3 [ k] ← − q3 [ k] ;  q3 [s]←q3 [s] − ck,s , if no vertex of cluster Cs belongs to I and {k, s} ∈ H2 ;  q3 [s]←q3 [s] − C¯, if no vertex of cluster Cs belongs to I and {k, s} ∈ H1 ;  q3 [s]←q3 [s] + ck,s , if at least one vertex of cluster Cs belongs to I and {k, s} ∈ H2 ;  q3 [s]←q3 [s] + C¯, if at least one vertex of cluster Cs belongs to I and {k, s} ∈ H1 . RE MOV E _CL move (cluster Ck is removed from the solution I):  q3 [ k] ← − q3 [ k] ;  q3 [s]←q3 [s] + ck,s , if no vertex of cluster Cs belongs to I and {k, s} ∈ H2 ;  q3 [s]←q3 [s] + C¯, if no vertex of cluster Cs belongs to I and {k, s} ∈ H1 ;  q3 [s]←q3 [s] − ck,s , if at least one vertex of cluster Cs belongs to I and {k, s} ∈ H2 ;  q3 [s]←q3 [s] − C¯, if at least one vertex of cluster Cs belongs to I and {k, s} ∈ H1 .

Note that when a vertex is added to the solution, the vector q3 is updated only if this move implies the entering of a new cluster. Finally, vector q4 is a binary vector indicating which clusters are currently in the solution. In the Tabu Search procedure, we allow ADD and RE MOV E _CL moves improving the best solution value found so far, even when they are tabu (an aspiration criterion). Routine ADD searches the neighborhood consisting of all the clustered generalized independent sets that contain one more vertex than the current one. The size of the neighborhood is bounded by O(|V|). When an ADD move is performed and a vertex belonging to cluster Ck is selected to enter the independent set I, before updating the values of vectors q1 , q2 , q3 and q4 , the objective function value f(I) is modified as follows:

f ( I ) ← f ( I ) + q 1 [h] + q 3 [k] ∗ ( 1 − q 4 [k] ).

7

Note that q3 [k] does not contribute when q4 [k] = 1, i.e. the cluster Ck is already in the solution. In the routine RE MOV E _CL the neighborhood contains all the clustered generalized independent sets that differ from the current one by the removal of one cluster. The size of the neighborhood is bounded by the number of clusters. When a RE MOV E _CL move is performed and cluster Ck is selected to leave the independent set I, before updating the values of vectors q1 , q2 , q3 and q4 , the objective function value f(I) is modified as follows:

f ( I ) ← f ( I ) + q3 [k] − q2 [k]. The Tabu Search ends when a maximum time tTS is reached, whereas the stopping rule of the whole method (Algorithm 1) is given by a maximum computational time tmax . Algorithm 1 GRASP-TS. Require: Incompatibility graph on vertex clusters, costs and revenues Ensure: A clustered generalized independent set I∗ with objective value f (I∗ ) 1: Initialize I∗ with empty solution 2: while (stopping rule is not satisfied) do Construct a greedy randomized solution I 3: 4: I ← TabuSearch(I) if f (I ) > f (I∗ ) then 5: I∗ ← I

6: f (I ∗ ) ← f (I ) 7: 8: end if 9: end while 10: return I∗ and f (I∗ )

Algorithm 2 Greedy randomized solution. Require: An empty solution Ensure: A feasible solution 1: Initialize vectors q1 , q2 , q3 , q4 2: while (stopping rule is not satisfied) do I ← I , f ( I ) ← f ( I ); 3: Randomly select a vertex v not in I (all vertices have equal 4: probability to be selected) Set I ← I ∪ {v}, and update f (I ) 5: Update vectors q1 , q2 , q3 , q4 6: 7: end while 8: if f (I ) > f (I ) then return I , f (I ) 9: 10: else 11: return I, f (I ) 12: end if

5. Solution algorithms In this section, we first describe the branch-and-cut approach used to exactly solve the two problem formulations, and then we introduce the matheuristic exploiting the presence of the clustered GISP and of the DRPP as subproblems once some of the decisions have already been taken. 5.1. Branch-and-cut algorithm To implement the branch-and-cut algorithm, the initial graph has been reduced by only maintaining nodes adjacent to profitable arcs, plus the depot if needed. Then we added all shortest directed paths between every pair of nodes and removed the ones for which ci j = cik + ck j .

Please cite this article as: M. Colombi et al., The directed profitable rural postman problem with incompatibility constraints, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.002

JID: EOR 8

ARTICLE IN PRESS

[m5G;February 27, 2017;11:32]

M. Colombi et al. / European Journal of Operational Research 000 (2017) 1–14

Algorithm 3 Tabu Search.

Algorithm 4 Sequential Simplified Exacts (SSE).

Require: Solution I with value f (I ) Ensure: Best solution I found so far and its value f (I ) 1: Initialize q1 , q2 , q3 , q4 2: I ← I, f (I ) ← f (I ) 3: while (stopping rule is not satisfied) do I1 ← ADD(I ) 4: I2 ← RE MOV E _CL(I ) 5: if f (I1 ) > f (I2 ) then 6: I

← I1 7: Let v be the vertex associated with the best add 8: Make the removal of v tabu with tenure t t 1 = const1 + 9: rand ([1, 10] ) else 10: I

← I2 11: 12: Let Ck be the cluster associated with the best removal Make the selection of vertices of Ck tabu with tenure t t 2 = 13: const2 + rand ([1, 10] ) end if 14: if f (I

) > f (I ) then 15: I ← I

16: f (I ) ← f (I

) 17: end if 18: I ← I

19: 20: Update q1 , q2 , q3 , q4 21: end while 22: return I and f (I )

Require: Graph G, profitable arcs set R, incompatibility graph G Ensure: Best solution Best found so far and its value w∗ 1: (Best, w∗ ) ← (0, 0 ) 2: (Sol _rel, wrel ) ← call RELAX 3: if Sol _rel is optimal for the DPRPP-IC then (Best, w∗ ) ← (Sol _rel, wrel ) 4: 5: return (Best, w∗ ) 6: else (Sol _dr pp, wdr pp ) ← call DRPP(Sol _rel) 7: if wdr pp > w∗ then (Best, w∗ ) ← (Sol _dr pp, wdr pp ) 8: (Sol _gisp, wgisp ) ← call ClusteredGISP(Sol _dr pp) 9: if wgisp > w∗ then (Best, w∗ ) ← (Sol _gisp, wgisp ) 10: (Sol _rem, wrem ) ← call LocalREMOVE(Best) 11: if wrem > w∗ then 12: (Best, w∗ ) ← (Sol _rem, wrem ) 13: (Sol _gisp, wgisp ) ← call ClusteredGISP(Sol _rem) 14: if wgisp > w∗ then (Best, w∗ ) ← (Sol _gisp, wgisp ) 15: end if 16: (Sol _add, wadd ) ← call LocalADD(Best) 17: if wadd > w∗ then 18: (Best, w∗ ) ← (Sol _add, wadd ) 19: (Sol _gisp, wgisp ) ← call ClusteredGISP(Sol _add) 20: if wgisp > w∗ then (Best, w∗ ) ← (Sol _gisp, wgisp ) 21: end if 22: 23: return (Best, w∗ ) 24: end if

Let us call RELA and RELB the problem relaxations obtained by removing connectivity constraints (3) and integrality conditions from formulations (A) and (B) respectively. Let (xˆ, yˆ, zˆ, uˆ ) be a fractional solution of problem RELA (similarly for RELB ). Consider the directed support graph Gˆ having an arc (i, j) for each variable xˆi j > 0, with an associated weight xˆi j . Connectivity inequalities (3) can be separated exactly in polynomial time by computing a maximum directed flow between every pair of vertices of graph Gˆ (similarly to what is done in Benavent, Corberán, & Sanchis, 20 0 0). Since the exact procedure is computationally expensive, we apply a heuristic separation algorithm based on the computation of connected components (see Benavent et al., 20 0 0 for more details). In particular, for each ε = 0, 0.25, 0.5, we compute the connected components of the graph Gˆ induced by the arcs with a weight xˆi j > ε . Note that, differently from Benavent et al. (20 0 0), the nodes of a connected component induced by profitable arcs cannot be shrunk into a single node, since not all the profitable arcs in the component are mandatorily served. Then, for each connected component on Gˆ with set of nodes S, we check if the corresponding inequality (3) is violated. The checking is performed for each profitable arc with both endpoints in S.

5.2. The matheuristic Sequential Simplified Exacts We have already emphasized the double nature of the problem including both the clustered GISP and the DRPP as subproblems. In fact, once the profitable arcs to serve are selected, the problem reduces to a DRPP. On the other hand, if we ignore the routing costs, in order to select the compatible arcs that maximize the profit, a clustered GISP can be solved. Both such components are used in the following matheuristic, the pseudo-code of which is shown in Algorithm 4. The method makes use of formulation (A) and since it sequentially solves different mixed integer linear programing subproblems, we call it Sequential Simplified Exacts (SSE).

The method receives as input the initial graph G, the set of profitable arcs R, and the incompatibility graph G. The incumbent solution Best is initialized with an empty solution where all variable values and the objective function value w∗ are set to zero (Step 1). This null solution corresponds to staying at the depot without serving any arc, and thus without collecting profits. The matheuristic starts by solving a problem relaxation of formulation (A) to collect information about which profitable arcs would be the most convenient to serve (Step 2). If the optimal solution found by the problem solved by procedure RELAX is feasible and thus optimal for the original problem, the matheuristic stops and the solution found is returned as the final one (Steps 3–5). Otherwise a feasible tour visiting the profitable arcs selected by the solution of the relaxation (Sol _rel) is identified by solving a DRPP (Step 7). Since the solution obtained by solving procedure DRPP is a tour that might traverse profitable arcs without serving them, we solve a clustered GISP in order to identify the best combination of profitable arcs that we can serve in the selected tour, while satisfying the incompatibility constraints of the problem (Step 9, procedure ClusteredGISP). Then, two local search procedures are sequentially applied. Procedure LocalREMOVE tries to improve the incumbent solution by removing connected components of profitable arcs from the solution (Step 11). If the improvement is successful, a clustered GISP is solved on the new solution to possibly further improve the solution by changing the profitable arcs served (Step 14). Procedure LocalADD receives as input the incumbent solution and tries to improve it by adding connected components of profitable arcs (Step 17). If the procedure succeeds in finding a better solution, a clustered GISP problem is again solved on it (Step 20). Each time a different routine is called, the main algorithm checks if the value of the new solution found is better than the incumbent one (Best) and updates it if this is the case (see Steps 8, 10, 15 and 21). Finally, in Step 23, the algorithm ends by returning the best solution found. In what follows, we describe the components of the algorithm in more detail.

Please cite this article as: M. Colombi et al., The directed profitable rural postman problem with incompatibility constraints, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.002

JID: EOR

ARTICLE IN PRESS

[m5G;February 27, 2017;11:32]

M. Colombi et al. / European Journal of Operational Research 000 (2017) 1–14

5.2.1. Procedure RELAX Procedure RELAX aims at identifying promising profitable arcs to serve. To this purpose, the procedure solves a relaxation problem corresponding to the mixed integer linear programing problem obtained by removing from formulation (A) the integrality conditions on the binary variables z and u (see Proposition 1) and the connectivity constraints (3). Such a relaxation problem is solved with a branch-and-cut algorithm where violated connectivity constraints are separated as follows: •

•

at the root node: on the integer solutions and, if computing time required for this task is lower than tlim1 , also on fractional solutions; at any node of the search tree: only on fractional solutions until a computing time equal to tlim1 is reached.

After tlim1 units of time, the search for violated connectivity constraints on fractional solutions is stopped. The idea is to enforce the formulation with useful connectivity inequalities only at the beginning of the search. The procedure terminates when a time limit tlim2 , where tlim2 > tlim1 , is reached, and the best solution found is then returned. Thus, between times tlim1 and tlim2 , no more connectivity inequalities are added and the focus of the method is on finding integer solutions. Note that if the algorithm ends at the root node before reaching time tlim2 , then the solution is optimal for the original problem since variables x and y are integer, variables z and u are integer or can be easily made integer (Proposition 1), and connectivity is ensured by the check for violated connectivity inequalities made on all integer solutions. When the algorithm stops due to the reaching of the time limit tlim2 (either at the root node or at any node of the branchand-bound tree), the feasible solution of the relaxation we obtain might have (and usually has) isolated subtours. However, this does not affect the validity of the solution in terms of the profitable arcs selected (identification of promising arcs). Finally, if no feasible solution is found within the time limit tlim2 , the algorithm keeps working until one is obtained. 5.2.2. Procedure DRPP Given an input solution Sol, this procedure labels the profitable arcs visited by the solution as required arcs for a DRPP instance. This DRPP instance is solved with the branch-and-cut procedure proposed in Ávila, Corberán, Plana, and Sanchis (2015). The obtained DRPP tour is a feasible solution for the DPRPP-IC. 5.2.3. Procedure ClusteredGISP Given a feasible solution of the DPRPP-IC representing a tour visiting some profitable arcs and possibly traversing some other profitable arcs without serving them, the problem of deciding which of all the traversed profitable arcs have to be served so as to maximize the net profit while satisfying incompatibility constraints, can be formulated as a clustered GISP. We have already described the solution algorithms introduced for solving the clustered GISP. Two variants of ClusteredGISP procedure have been implemented according to the algorithm chosen to solve the problem. 5.2.4. Procedure LocalREMOVE Given an incumbent solution Best for the DPRPP-IC, the procedure LocalREMOVE tries to improve it by removing profitable arcs. The procedure identifies the connected components of the graph induced by the profitable arcs served in the current solution Best. Then, a local search is started. The procedure moves from a solution to another solution by changing the subset of profitable arcs. The neighborhood consists of all the solutions that differ from the current one in the removal of the profitable arcs of a

9

given connected component. After removing some profitable arcs, the construction of a new solution requires the determination of a new tour by solving a DRPP on the remaining profitable arcs. The neighborhood is searched following a first-improvement strategy. As soon as a solution better than the current one is found, the current solution is updated and the local search restarts looking in the neighborhood of the new solution, until reaching a local optimum. 5.2.5. Procedure LocalADD Procedure LocalADD works in the opposite direction with respect to the LocalREMOVE one. Also in this case the starting point is the current best feasible solution Best. This time however, the procedure tries to improve the solution value with the introduction of new profitable arcs. The procedure identifies all the connected components of the graph induced by the profitable arcs not served in the current solution Best. The neighborhood of the local search consists of all the solutions that differ from the current one in the addition of the profitable arcs of an external connected component. After inserting the new component of profitable arcs, a new tour visiting the new included arcs is found by solving a DRPP instance having as required arcs the initial set of profitable arcs and those of the added component. The local search follows a first-improvement strategy, analogously to procedure LocalREMOVE. 6. Experimental analysis In what follows, we first describe the construction of the instances, then we compare the solutions obtained when solving the two integer formulations (A) and (B) of the problem and their relaxations RELA and RELB . Finally, we analyze the performance of different variants of the proposed matheuristic. All tests have been performed by using an Intel i7 processor with 2.93 gigahertz and 8 gigabytes of RAM in Windows Seven operating system. Algorithms have been coded in C++ and models implemented in ILOG Concert Technology 2.9 and solved with IBM CPLEX 12.6.2. We run experiments using a single core processor. 6.1. Instances Instances have been generated by using the same graphs proposed for the directed general routing problem (DGRP) in Ávila et al. (2015). They can be downloaded from the site http://www. uv.es/corberan/instancias.htm. More precisely, we considered 12 graphs with 500 nodes. Required arcs of the original DGRP instances have been used as profitable arcs of the DPRPP-IC with an integer profit pij , (i, j) ∈ R, randomly generated in the intervals [3 cij , 4 cij ] and [4 cij , 5 cij ] as suggested by the real case. The sets E1 and E2 of the incompatibility graph have been randomly generated. In particular, for each pair of nodes in VI not connected by a profitable arc, we randomly generate a value r1 in the interval [1, 10 0 0]. If r1 ≤ 10 0 0 α , with 0 < α < 1, then an edge is introduced between the two nodes, thus parameter α controls the size of the incompatibility graph. Among all the inserted edges, the ones belonging to set E2 are selected as follows. For each edge, a random number r2 in the interval [1, 100] is generated. If r2 ≤ 100 β , with 0 ≤ β ≤ 1, then the edge is put in E2 , otherwise it belongs to E1 . Thus, the larger the parameter β value, the higher the number of weak incompatibilities. Finally, to each edge {i, j} ∈ E2 , we assign a cost ci j = γ [val (i ) + val ( j )] , where 0 < γ < 1 and val (q ) is a fictitious value associated with each node  q in VI and equal to s∈(q,s )∈R ( pqs − cqs ). By setting α = 0.01, 0.05, β = 0, 0.5, 1 and γ = 0.01, 0.025, 0.05, a total number of 432 instances has been created. Table 1 summarizes the main characteristics of the generated instances.

Please cite this article as: M. Colombi et al., The directed profitable rural postman problem with incompatibility constraints, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.002

ARTICLE IN PRESS

JID: EOR 10

M. Colombi et al. / European Journal of Operational Research 000 (2017) 1–14

Table 1 Main characteristics of the generated instances.

α

β

|V|

|A|

|R|

|E 1 |

|E 2 |

0.01

0.00 0.5 1.00

500 500 500

1166–1571 1166–1571 1166–1571

218–1132 218–1132 218–1132

324–2214 155–1087 0

0 157–1114 347–2184

0.00 0.5 1.00

500 500 500

1166–1571 1166–1571 1166–1571

218–1132 218–1132 218–1132

1724–10,553 836–5334 0

0 836–5405 1699–10,743

0.05

Table 2 Continuous relaxations: average solution times and values.

α

β

RELA

RELB

Avg. time [seconds]

Avg. Obj.

Avg. time [seconds]

Avg. Obj.

0.01

0.0 0.5 1.0 All

0.68 0.83 0.61 0.71

26930.47 28260.06 45195.26 33461.93

1.05 1.55 0.65 1.08

26930.47 28260.06 45195.26 33461.93

0.05

0.0 0.5 1.0 All

1.10 1.36 3.43 2.08

26762.62 26768.74 31225.88 28252.41

2.14 4.25 19.75 8.72

26762.62 26768.74 31225.88 28252.41

1.39

30857.17

4.90

30857.17

Total

[m5G;February 27, 2017;11:32]

6.2. Formulations comparison Table 2 shows the average computing time expressed in seconds (“Avg. time [seconds]”) and the average value of the optimal solutions (“Avg. Obj.”) of the two problem relaxations RELA and RELB for different combinations of parameters α and β . For the sake of brevity, and because parameters α and β are among the most relevant ones to control the structure of the underlying incompatibility graph, we do not extend the same analysis to results averaged with respect to other parameters. Formulation (A) performs slightly better than formulation (B). The global average computing time required to solve the continuous relaxation more than triplicates when moving from formulation (A) (1.39 seconds) to formulation (B) (4.90 seconds). In order to decide which formulation to use in the final tests, we also run some additional experiments to compare the performance of the two models when looking for exact solutions in a limited amount of time. To this aim we selected a graph (DG552), and for each combination of parameters (α , β , γ and profit structure) we solved both integer problem formulations by using the branch-and-cut procedure described in Section 5, by setting a time limit of one hour. A total number of 36 instances were solved. Table 3 reports the values for the 11, out of 36, instances in which the branch-and-cut was able to obtain the optimal solution for at least one of the two formulations. In particular, the first four columns specify the parameters values (the profit structure, and the parameters α , β and γ ) to identify the solved instances. Column “Opt.” shows the optimal solution value, whereas in the remaining columns we report for each formulation the computing time, in seconds, to obtain the optimal solution. If one of the formulations was not able to find the optimal solution, we put “–” instead of the computing time. The last row of the table presents the average values computed by taking into account only the 8 instances for which the branch-and-cut finds the optimal solution for both formulations. Table 3 shows that the branch-and-cut was able to solve to optimality, in one hour of computing time, 11 out of the 36 considered instances. In particular, it has found 10 optimal solutions using formulation (A) and 9 with formulation (B). Moreover, in the instances where the exact approach has found the optimal solution

Table 3 Comparison between optimal integer solution values: instances generated on graph DG552. Problem

Formulation

pij

α

β

γ

Opt.

(A) Time [seconds]

(B) Time [seconds]

[3, 4] [3, 4] [3, 4] [3, 4] [3, 4] [4, 5] [4, 5] [4, 5] [4, 5] [4, 5] [4, 5] Average

0.01 0.01 0.01 0.01 0.05 0.01 0.01 0.01 0.01 0.01 0.05

0.5 1.0 1.0 1.0 1.0 0.5 0.5 1.0 1.0 1.0 1.0

0.025 0.010 0.025 0.050 0.010 0.025 0.050 0.010 0.025 0.050 0.010

8527 14,200 12,796 11,844 10,960 14,395 13,557 26,265 24,628 21,600 20,380

73.0 26.0 7.0 74.5 59.3 – 2817.2 11.9 9.2 16.7 43.3 31.0

– 15.4 33.0 137.0 187.9 2093.8 – 10.0 8.8 13.6 63.5 58.6

for both formulations, the average computing time almost doubles when moving from formulation (A) (31 seconds) to formulation (B) (59 seconds) and the maximum time increases from 74.55 to 187.91 seconds. For the sake of brevity, we do not report here the results obtained on the remaining instances. We only point out that in these instances, the algorithm when solving formulation (A) obtains a higher average feasible solution value (2892) and a stronger average upper bound (8621) than when solving formulation (B) (2693 and 8777, respectively). To conclude, after analyzing the results of these preliminary tests, we decided to exactly solve the whole set of 432 instances by using formulation (A). 6.3. Matheuristic results After some preliminary tests, we decided to set the following parameter values: tmax = 20 seconds and tT S = 4 seconds in the GRASP-TS, whereas tlim1 = 100 seconds and tlim2 = 200 seconds, respectively. We tested different variants of our SSE algorithm. The first variant, named SSE1(exact), solves the clustered GISP subproblems by using the exact approach on the problem approximation, whereas the second one, called SSE1(GRASP-TS), uses the developed GRASPTS metaheuristic. For both variants we have also considered a different parameters setting by testing their performance with the values for tlim1 and tlim2 increased to 30 0–60 0 seconds, respectively. We called SSE2(exact) and SSE2(GRASP-TS) these two additional configurations. Table 4 shows the average results of the four SSE variants for instances grouped according to the values of parameters α and β . Among all the possibilities, we finally decided to show this combination of parameters that reflects the “incompatibility structure” of the problem. Additional tables grouping the results according to other parameters can be found in the technical report (Colombi, Corberán, Mansini, Plana, & Sanchis, 2016a). The first two columns of the table show the parameters identifying the set of instances. Columns “Avg. gap” and “Max. gap” provide the percentage average and maximum gaps of the solutions found by each algorithm configuration (SSE1(exact), SSE2(exact), SSE1(GRASP-TS) and SSE2(GRASP-TS)) with respect to the optimal solution (if any) or a feasible one obtained by the branch-andcut when solving formulation (A) with a time limit of one hour. Column “Time [seconds]” reports the average computing time in seconds. Finally, the last column of the table indicates the average computing time in seconds of the exact algorithm (“Exact Time [seconds]”). In 129 out of 432 instances the exact algorithm could not find any solution better than the null one received as input (corresponding to remaining at the depot without servicing any

Please cite this article as: M. Colombi et al., The directed profitable rural postman problem with incompatibility constraints, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.002

ARTICLE IN PRESS

JID: EOR

[m5G;February 27, 2017;11:32]

M. Colombi et al. / European Journal of Operational Research 000 (2017) 1–14

11

Table 4 Results with respect to the values of parameters α and β . SSE1(GRASP-TS)

SSE2(GRASP-TS)

Exact

α

β

Avg. gap (%)

Max. gap (%)

Time [seconds]

Avg. gap (%)

Max. gap (%)

Time [seconds]

Time [seconds]

0.01

0.00 0.50 1.00 0.00 0.50 1.00

−36.37 −4.29 0.21 −111.54 −93.37 −129.96

43.68 23.79 3.19 79.75 62.42 31.48

348.2 339.4 108.7 309.3 321.3 339.8

−42.98 −8.60 0.19 −185.50 −146.50 −119.07

33.44 11.86 3.19 88.25 64.38 21.45

697.0 670.9 122.4 646.5 686.5 522.9

3601.9 3486.6 322.6 3510.4 3600.8 2552.6

α

β

Avg. gap (%)

Max. gap (%)

Time [seconds]

Avg. gap (%)

Max. gap (%)

Time [seconds]

Time [seconds]

0.01

0.00 0.50 1.00 0.00 0.50 1.00

−36.28 −5.37 0.22 −192.12 −276.62 −57.52

29.14 23.79 3.19 87.87 59.46 30.57

285.1 287.8 86.9 274.4 296.7 305.1

−51.51 −8.46 0.20 −172.01 −290.87 −137.35

22.81 9.55 3.19 92.59 31.78 22.41

643.0 608.0 98.6 637.6 643.8 491.7

3601.9 3476.2 322.6 3510.4 3600.8 2552.6

0.05

SSE1(exact)

0.05

SSE2(exact)

Exact

Table 5 Comparison of SS1(exact) and SS2(exact): number of optimal and best known solutions. Exact

SSE1(exact)

SSE2(exact)

α

β

#Opt.

#Feas.

#BF

#BK

#Opt.

#Feas.

#BF

#BK

#Opt.

#Feas.

#BF

#BK

0.01

0.00 0.50 1.00 0.00 0.50 1.00

0 5 71 1 0 17

51 59 1 29 32 37

15 23 0 11 7 17

15 28 71 12 7 34

0 1 36 1 0 7

72 71 36 71 72 65

15 13 1 10 9 6

25 17 37 36 28 14

0 1 37 1 0 8

72 71 35 71 72 64

24 26 0 9 16 17

35 32 37 43 43 42

0.05

value−SSE value , we profitable arc). Since gaps are computed as Exact Exact value excluded such instances from the computation of the results reported in the following tables. It is worth noticing that in all these 129 instances, the matheuristic (all its four variants) has always been able to find a feasible solution different from zero (with an average absolute profit above five thousands). Table 4 shows that, in terms of average gaps, all variants work quite well. A negative average gap means that the solution found by the heuristic is better than the best feasible solution found by the branch-and-cut algorithm in one hour. Maximum average gaps are in some cases quite high, but if we go into more detail analyzing the results it comes out that this worse performance is concentrated only on a bunch of instances, whereas in the remaining ones max gaps usually range between 5% and 10% and in many instances are lower than 2%. Interestingly enough, this bunch of instances shares a common set of underlying graphs (DG532, DG545, DG557, and DG567). This characteristic, rather than their incompatibility structure, seems to explain the complexity of these instances. Computing times for the two variants SSE1(exact) and SSE1(GRASP-TS) are comparable. The quality of the solutions provided by SSE1(GRASP-TS) is in general better than the one of SSE1(exact), when β = 1, i.e. when the incompatibility graph contains a large percentage of weak incompatibilities. In such cases, the problem approximation with an overestimation of penalty costs used to solve the clustered GISP in SSE1(exact) provides a worse performance than the use of GRASP-TS directly on the clustered GISP as in SSE1(GRASP-TS). The opposite is true in the remaining instances, which implies that, when the relevance of weak incompatibilities is limited, the clustered GISP is closer to a normal GISP and thus an exact approach on the latter performs definitely better than a heuristic approach on the former. In Table 5, we compare the variants SSE1 and SSE2 when the clustered GISP is solved by using the exact algorithm on the approximated problem. In Table 6 we make the same comparison but

using GRASP-TS to solve the clustered GISP. The tables present the results for any combination of parameters α and β with respect to the branch-and-cut algorithm. More precisely, for each algorithm, they report the number of optimal solutions (column “#Opt.”), the number of feasible solutions (not proved to be optimal) (column “#Feas.”), the number of times such feasible solutions represent the best known values among the variants compared at the corresponding table, excluding the instances where the branch-and-cut was not able to find a feasible solution different from the null one (column “#BF ”), and the total number of times (out of all the instances) the algorithm finds the best known value (column “#BK”). Note that for a heuristic algorithm we know if its solution is optimal only when we compare it with the optimal solution of the exact algorithm. Thus, it might be that the number of optimal solutions found by the heuristic variants is higher than the one reported in the tables. Moreover, it is worth recalling that all the heuristic variants always find a solution with positive value in the 129 instances where the branch-and-cut terminates with a null value (in Tables 5 and 6, the sum of columns “#Opt.” and “#Feas.” for the heuristics is always equal to 72, the number of instances of each set, whereas this is not the case for the exact algorithm). In Tables 7 and 8 we compare, in terms of number of best known solutions, all the four heuristic variants with the branchand-cut and a fictitious heuristic taking the best result among the ones provided by the four SSE variants (we call it SSE(Best)). Results are grouped with respect to parameter β , which is relevant for the incompatibility structure of the problem and the intervals for profit generation. Interestingly enough, the global number of best known values found by SSE(Best) is quite large and definitely higher than the number found by each heuristic, thus indicating that the heuristics behave in a complementary way. Table 7 still confirms that variants SSE1(exact) and SSE2(exact) globally outperform SSE1(GRASP-TS) and SSE2(GRASP-TS) except for instances with β = 1, where the large presence of removable edges in the clustered GISP justifies the application of an ad hoc method for its solution.

Please cite this article as: M. Colombi et al., The directed profitable rural postman problem with incompatibility constraints, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.002

ARTICLE IN PRESS

JID: EOR 12

[m5G;February 27, 2017;11:32]

M. Colombi et al. / European Journal of Operational Research 000 (2017) 1–14

Table 6 Comparison of SS1(GRASP-TS) and SS2(GRASP-TS): number of optimal and best known solutions. Exact

SSE1(GRASP-TS)

SSE2(GRASP-TS)

α

β

#Opt.

#Feas.

#BF

#BK

#Opt.

#Feas.

#BF

#BK

#Opt.

#Feas.

#BF

#BK

0.01

0.00 0.50 1.00 0.00 0.50 1.00

0 5 71 1 0 17

51 59 1 29 32 37

19 23 0 17 12 17

19 28 71 18 12 34

0 1 37 0 0 7

72 71 35 72 72 65

12 12 1 4 7 8

24 14 38 34 30 20

0 1 38 0 0 7

72 71 34 72 72 65

23 28 0 9 14 15

32 36 38 37 35 35

0.05

Table 7 Number of best known values out of all instances grouped with respect to parameter β .

β

Exact

SSE1 (exact)

SSE2 (exact)

SSE1 (GRASP-TS)

SSE2 (GRASP-TS)

SSE (Best)

0.0 0.5 1.0

23 33 105 161

55 42 47 144

67 62 58 187

32 15 56 103

37 31 70 138

122 112 86 320

Table 8 Number of best known values out of all instances grouped with respect to profit values. Profit

Exact

SSE1 (exact)

SSE2 (exact)

SSE1 (GRASP-TS)

SSE2 (GRASP-TS)

SSE (Best)

[3,4] [4,5]

81 80 161

69 75 144

94 93 187

48 55 103

66 72 138

153 167 320

Regarding the profit structure, it seems that the two intervals do not determine any relevant discrepancies in the performance of all methods. What is evident is that SSE2(exact) is definitely the best approach. In Table 9, we show the average gaps of the heuristics in the 94 instances for which the exact algorithm has found the optimal solution. Instances are always grouped with respect to the most relevant parameters α and β . The third column “#Opt.” provides the number of optimal solutions found by the exact algorithm, while the remaining columns have the same meaning as before. When no optimal solution has been found for a certain combination of parameters, a “–” is shown in all the columns. In these instances, where optimal solutions are known, SSE1(exact) has an average gap of 0.79%, which reduces to 0.57% for SSE2(exact) in an average computing time of about 100 seconds versus 360 seconds of the exact algorithm.

Table 10 Procedures contribution. Procedure

# Best. SSE1(exact)

# Best. SSE1(GRASP-TS)

RELAX DRPP ClusteredGISP-DRPP LocalREMOVE ClusteredGISP-LocalREMOVE LocalADD ClusteredGISP-LocalADD (a)

24/432 41/432 1/432 170/432 2/432 160/432 34/432

24/432 38/432 5/432 212/432 4/432 115/432 35/432

SSE1

SSE2

Procedure

Min.

Avg.

Max.

St. Dev.

Min.

Avg.

Max.

St. Dev.

LocalREMOVE LocalADD ClusteredGISP (b)

0.00 0.00 0.00

2.54 2.00 0.58

23.00 25.00 3.00

2.93 3.30 0.67

0.00 0.00 0.00

2.26 1.61 0.31

13.00 25.00 3.00

2.71 3.03 0.64

It seems clear that the difficulty of the instances depends on the underlying incompatibility structure. High average gaps can be observed for β = 0.5. Nevertheless, we can observe that such a gap for SSE1(exact) reduces from 8.05% to 4.07% with SSE2(exact). In the same row, the average computing time taken by the exact algorithm is very high in comparison with the ones in the other rows. Since such high average gaps are mainly due to a bunch of instances, we have just considered the five instances with worst average performance and run SSE1(exact) with a new configuration where parameters tlim1 and tlim2 have been set to 600 and 1200 seconds, respectively. Interesting enough, the new average gap obtained is 1.79% in an average computing time of 475 seconds. This proves that the higher the time available to the heuristics the better the results. Table 10 (a) and (b) analyze the contribution of the different procedures defining the matheuristic by comparing SSE1(exact) to SSE1(GRASP-TS). In particular, column “# Best.” in Table 10(a)

Table 9 Average results with respect to α and β on the optimally solved instances. SSE1(GRASP-TS)

SSE2(GRASP-TS)

Exact

α

β

#Opt.

Avg. gap (%)

Max. gap (%)

Time [seconds]

Avg. gap (%)

Max. gap (%)

Time [seconds]

Time [seconds]

0.01

0.00 0.50 1.00 0.00 0.50 1.00

0 5 71 1 0 17

– 7.98 0.27 10.15 – 0.71

– 18.87 3.19 10.15 – 5.49

– 210.4 106.2 224.0 – 194.3

– 4.37 0.25 10.15 – 0.70

– 9.55 3.19 10.15 – 5.49

– 347.2 115.8 344.0 – 243.1

– 1954.9 274.0 845.6 – 258.4

0.05

SSE1(exact)

α

β

#Opt.

0.01

0.00 0.50 1.00 0.00 0.50 1.00

0 5 71 1 0 17

0.05

Avg. gap (%) – 8.05 0.28 0.00 – 0.87

SSE2(exact) Max. gap (%)

Time [seconds]

– 20.16 3.19 0.00 – 5.71

– 159.0 84.4 204.2 – 166.9

Avg. gap (%) – 4.37 0.26 0 – 0.76

Exact Max. gap (%) – 9.55 3.19 0 – 5.71

Time [seconds]

Time [seconds]

– 309.4 92.2 603.8 – 209.9

– 1954.9 274.0 845.6 – 258.4

Please cite this article as: M. Colombi et al., The directed profitable rural postman problem with incompatibility constraints, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.002

ARTICLE IN PRESS

JID: EOR

[m5G;February 27, 2017;11:32]

M. Colombi et al. / European Journal of Operational Research 000 (2017) 1–14

13

Table 11 Comparison between the original instance on graph DG532 and the modified one. Exact

SSE2(GRASP-TS)

Instance

|V|

|R|

|AR|

|E 1 |

|E 2 |

Obj

UB

Gap [%]

Time [seconds]

Obj

Time [seconds]

Original Modified

500 528

218 218

953 1148

173 269

185 245

3046 0

3665.75 3154.78

0.00 –

3537 3600

3046 2034

72.6 668.9

reports the number of times each procedure finds the final solution of the matheuristic. In column “Procedure”, ClusteredGISPDRPP, ClusteredGISP-LocalREMOVE and ClusteredGISP-LocalADD show the effectiveness of calling the procedure ClusteredGISP after the methods DRPP, LocalREMOVE and LocalADD in Steps 9, 14 and 20, respectively. As can be seen in Table 10(a), each procedure contributes actively to the identification of the best final solutions. In particular, in terms of local search approaches, procedure LocalREMOVE is globally more effective than procedure LocalADD. It is evident how the solution of the clustered GISP instance instead of the approximated one is, in general, more effective. Table 10(b) shows further details for the procedures that are called more than once for each instance. Note that if we exclude the calls inside the local search methods, procedure GISP is called at most 3 times for each instance. Columns “Min.”, “Avg.” and “Max.” report the minimum, average and maximum number of times each procedure improves the solution, while “St. Dev.” gives the standard deviation. These results show that, although the procedures may not contribute for some instances, procedures LocalREMOVE and LocalADD improve the solution twice on each instance on average, with a maximum of 23 and 25 times, and of 13 and 25, respectively. Note that there are instances in which procedure ClusteredGISP provides an improvement each time it is called. In the real life problem described in the introduction, a customer may be associated with different nodes. However, such nodes are physically represented by separated hubs located at different geographical positions. This means that, although related to the same customer, these nodes are practically treated as different customers, each one with its own separated incompatibilities. Although not relevant for our application, it might however happen that incompatibilities can arise among arcs starting from the same node (same geographical location). Dealing with this situation requires the introduction of replicas of the node with null traveling costs among them. As an example for this specific case, we selected and modified one of the instances for which the branchand-cut algorithm was able to find an optimal solution (namely, the instance on graph DG532 with profit interval [3, 4], α =0.01, β =0.5 and γ =0.025). In particular, for each node with more than one profitable arc leaving it, we generated as many copies of that vertex as the number of profitable arcs leaving it minus one, and assumed that they are strongly incompatible with each other. Each copy inherits the same incompatibilities of the original node with respect to the remaining nodes of the graph. Traveling costs among all the copies and the original node are set to zero. Table 11 reports the results obtained by the exact algorithm and the matheuristic (variant SSE2(GRASP-TS)) on both the original and the new instance. Notice that, with respect to the original instance, the number of nodes and arcs in the modified instance is larger by about 6% and 21% respectively, whereas the number of strong incompatibilities increases by about 55%. The results suggest that finding the optimal solution seems to be more difficult in the modified instance. The branch-and-cut algorithm does not find the optimal solution now, and it is not even able to obtain a feasible solution in one hour of computing time. The matheuristic keeps performing well, and it is able to provide a feasible solution in a few minutes. Generating and testing more structured instances as the one we present above is surely an interesting direction for future research.

7. Conclusions In this paper, we analyze a variant of the directed profitable rural postman problem where incompatibility constraints are introduced between profitable arcs leaving weak/strong incompatible nodes. We call this problem the directed profitable rural postman problem with incompatibility constraints. A matheuristic that combines the solution of two problems, a DRPP and a variant of the GISP (called clustered GISP), is proposed. The latter is a new problem never studied before and for which we propose a GRASP Tabu Search procedure to solve it. We tested four different variants of the method, all of them providing good results in comparison to those obtained by an exact algorithm with a time limit of one hour. The proposed branch-and-cut algorithm is only based on the separation of connectivity constraints. We think that this algorithm could be further improved through a polyhedral analysis of the problem and the separation of valid inequalities obtained from both the DRPP and the GISP. This will be the subject of future work. Finally, it should be noted that the incompatibilities between nodes, and thus among arcs leaving them, presented here and motivated by a real case problem do not conclude the analysis of incompatibility constraints on arc routing problems. One can easily figure out situations where there may be direct incompatibilities between arcs (independently of the nodes where they originate). This will be considered for future research along with the possibilities to extend the problem to a multi-vehicle case. Acknowledgments The work by Ángel Corberán, Isaac Plana, and José M. Sanchis was supported by the Spanish Ministerio de Economía y Competitividad and Fondo Europeo de Desarrollo Regional (FEDER) through project MTM2015-68097-P (MINECO/FEDER) and by the Generalitat Valenciana (project GVPROMETEO2013-049). References Aráoz, J., Fernández, E., & Franquesa, C. (2009a). The clustered prize-collecting arc routing problem. Transportation Science, 43(3), 287–300. Aráoz, J., Fernández, E., & Meza, O. (2009b). Solving the prize-collecting rural postman problem. European Journal of Operational Research, 196(3), 886–896. Aráoz, J., Fernández, E., & Zoltan, C. (2006). Privatized rural postman problems. Computers & Operations Research, 33(12), 3432–3449. Archetti, C., Guastaroba, G., & Speranza, M. G. (2014). An ILP-refined tabu search for the directed profitable rural postman problem. Discrete Applied Mathematics, 163, 3–16. Archetti, C., & Speranza, M. G. (2014). Arc routing problems with profits. In A. Corberán, & G. Laporte (Eds.), Arc routing: Problems, methods, and applications (pp. 281–299). SIAM. Ávila, T., Corberán, Á., Plana, I., & Sanchis, J. M. (2015). The stacker crane problem and the directed general routing problem. Networks, 65, 43–55. Ávila, T., Corberán, Á., Plana, I., & Sanchis, J. M. (2016). A branch-and-cut algorithm for the profitable windy rural postman problem. European Journal of Operational Research, 249(3), 1092–1101. Benavent, E., Corberán, A., & Sanchis, J. M. (20 0 0). Linear programming based methods for solving arc routing problems. In M. Dror (Ed.), Arc routing: Theory, solutions and applications (pp. 231–275). Kluwer. Black, D., Eglese, R., & Wøhlk, S. (2013). The time-dependent prize-collecting arc routing problem. Computers & Operations Research, 40(2), 526–535. Colombi, M., Corberán, Á., Mansini, R., Plana, I., & Sanchis, J. M. (2016a). The directed profitable rural postman problem with incompatibility constraints. Department of Information Engineering, University of Brescia. Technical Report RO@DII 04.2016

Please cite this article as: M. Colombi et al., The directed profitable rural postman problem with incompatibility constraints, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.002

JID: EOR 14

ARTICLE IN PRESS

[m5G;February 27, 2017;11:32]

M. Colombi et al. / European Journal of Operational Research 000 (2017) 1–14

Colombi, M., Corberán, Á., Mansini, R., Plana, I., & Sanchis, J. M. (2016b). The hierarchical mixed rural postman problem. Transportation Science. http://dx.doi.org/ 10.1287/trsc.2016.0686 Colombi, M., Corberán, Á., Mansini, R., Plana, I., & Sanchis, J. M. (2017). The hierarchical mixed rural postman problem: Polyhedral analysis and a branch-and-cut algorithm. European Journal of Operational Research, 257(1), 1–12. Colombi, M., & Mansini, R. (2014). New results for the directed profitable rural postman problem. European Journal of Operational Research, 238(3), 760–773. Colombi, M., Mansini, R., & Savelsbergh, M. (2016c). The generalized independent set problem: Polyhedral analysis and solution approaches. European Journal of Operational Research.. http://dx.doi.org/10.1016/j.ejor.2016.11.050 Corberán, Á., Fernández, E., Franquesa, C., & Sanchis, J. M. (2011). The windy clustered prize-collecting arc-routing problem. Transportation Science, 45(3), 317–334. Corberán, Á., & Laporte, G. (2015). Arc routing: Problems, methods, and applications. MOS-SIAM Series 20. Corberán, Á., Martí, R., Martínez, E., & Soler, D. (2002). The rural postman problem on mixed graphs with turn penalties. Computers & Operations Research, 29(7), 887–903. Corberán, Á., Plana, I., Rodríguez-Chía, A. M., & Sanchis, J. M. (2013). A branch-and– cut algorithm for the maximum benefit chinese postman problem. Mathematical Programming, 141(1–2), 21–48. Cruijssen, F., Cools, M., & Dullaert, W. (2007). Horizontal cooperation in logistics: opportunities and impediments. Transportation Research Part E: Logistics and Transportation Review, 43(2), 129–142. Deitch, R., & Ladany, S. P. (20 0 0). The one-period bus touring problem: Solved by an effective heuristic for the orienteering tour problem and improvement algorithm. European Journal of Operational Research, 127(1), 69–77. Dror, M., & Langevin, A. (1997). A generalized traveling salesman problem approach to the directed clustered rural postman problem. Transportation Science, 31(2), 187–192. Feillet, D., Dejax, P., & Gendreau, M. (2005). The profitable arc tour problem: solution with a branch-and-price algorithm. Transportation Science, 39(4), 539–552.

Fernández, E., Fontana, D., & Speranza, M. G. (2016). On the collaboration uncapacitated arc routing problem. Computers & Operations Research, 67, 120–131. Fugate, B. S., Davis-Sramek, B., & Goldsby, T. J. (2009). Operational collaboration between shippers and carriers in the transportation industry. The International Journal of Logistics Management, 20(3), 425–447. Gendreau, M., Manerba, D., & Mansini, R. (2016). The multi-vehicle traveling purchaser problem with pairwise incompatibility constraints and unitary demands: A branch-and-price approach. European Journal of Operational Research, 248(1), 59–71. Guastaroba, G., Mansini, R., & Speranza, M. G. (2010). Modeling the pre auction stage the truckload case. In Innovations in distribution logistics. In Lecture Notes in Economics and Mathematical Systems: 619 (pp. 219–233). Springer. Hochbaum, D. S., & Pathria, A. (1997). Forest harvesting and minimum cuts: A new approach to handling spatial constraints. Forest Science, 43(4), 544–554. Malandraki, C., & Daskin, M. S. (1993). The maximum benefit Chinese postman problem and the maximum benefit traveling salesman problem. European Journal of Operational Research, 65(2), 218–234. Manerba, D., & Mansini, R. (2015). A branch-and-cut algorithm for the multi-vehicle traveling purchaser problem with pairwise incompatibility constraints. Networks, 65(2), 139–154. Pearn, W.-L., & Chiu, W. (2005). Approximate solutions for the maximum benefit chinese postman problem. International journal of systems science, 36(13), 815–822. Pearn, W.-L., & Wang, K. (2003). On the maximum benefit chinese postman problem. Omega, 31(4), 269–273. Souffriau, W., Vansteenwegen, P., Berghe, G. V., & Van Oudheusden, D. (2011). The planning of cycle trips in the province of east flanders. Omega, 39(2), 209–213. Stefansson, G. (2006). Collaborative logistics management and the role of third– party service providers. International journal of physical distribution & logistics management, 36(2), 76–92. Wang, Y., Lü, Z., Glover, F., & Hao, J.-K. (2013). Probabilistic grasp-tabu search algorithms for the UBQP problem. Computers & Operations Research, 40(12), 3100–3107.

Please cite this article as: M. Colombi et al., The directed profitable rural postman problem with incompatibility constraints, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.002