Reverse Multistar Inequalities and Vehicle Routing Problems with lower bound capacities. Luis Gouveia, Jorge Riera and Juan-José Salazar-González

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1 Reverse Multistar Inequalities and Vehicle Routing Problems with lower bound capacities Luis Gouveia, Jorge Riera and Juan-José Salazar-González CIO Working Paper 13/2009

2 Reverse Multistar Inequalities and Vehicle Routing Problems with lower bound capacities Luis Gouveia Jorge Riera Juan-José Salazar-González December 28, 2009 Abstract Different variations of vehicle routing problems can be formulated by so-called Single-Commodity Flow (SCF) formulations. The SCF model is an example of an extended model, in contrast to a natural model that involves only the design binary variables. A natural model with a linear programming (LP) relaxation bound equal to the LP relaxation bound of the SCF model can be obtained by using a theorem by Hoffman. In this paper we show that the set of projected inequalities given by the theorem can be divided into two sets which exhibit different modeling properties. For many vehicle routing problems, one set contains redundant inequalities while the other contains inequalities that are known to be facet defining under mild conditions. This classification raises the question of knowing whether the redundant set is redundant for all interesting vehicle routing problems. We show that these apparently redundant constraints become fundamental for a less studied variation of the problem, namely the case where arc lower bound capacities are involved. Furthermore, we also show that other related and interesting inequalities can be generated by manipulating the new projected inequalities by coefficient division and rounding operations. We have implemented a branch-and-cut algorithm which uses the new inequalities and which has been tested with reasonable success on instances with up to 50 nodes. Faculdade de Ciências da Universidade de Lisboa, DEIO CIO Bloco C/2 - Campo Grande, Lisbon, Portugal legouveia@fc.ul.pt DEIOC, Universidad de La Laguna, Tenerife, Spain jriera@ull.es, jjsalaza@ull.es 1

3 We also discuss a variation of the classic (and more known) Capacitated Vehicle Routing Problem where lower bound capacity does not make explicit part of the problem specification, but it is implicit in the problem definition. We show that the new inequalities might be useful for this variation. 1 Introduction Different variations of vehicle routing problems can be formulated as Single- Commodity Flow (SCF) formulations (see, for instance, Gavish and Graves [5], Letchford and Salazar-González [10] and Toth and Vigo [11]). The SCF model is an example of an extended model, in contrast to a natural model that involves only the design binary variables. Given a SCFlike model, we can use a tool of projection to create a natural model with a linear programming (LP) relaxation bound equal to the LP relaxation bound of the original SCF model. In this paper the tool of projection is a theorem by Hoffman [8]. The main message of this paper can be summarized in five key items: (i) The set of projected inequalities given by the theorem can be divided into two sets which exhibit different modeling properties. For many vehicle routing problems, including the standard Capacitated Vehicle Routing Problem (CVRP), one set contains redundant inequalities while the other contains inequalities that are known to be facet defining under mild conditions. This is the topic of Section 2. This classification raises the question of knowing whether the redundant set is redundant for all vehicle routing problems. (ii) The set of apparently non-interesting constraints become interesting for a less studied variation of the problem, namely the case where arc lower bound capacities are involved. This is the topic of Section 3, where we introduce the so-called Balanced Vehicle Routing Problem (BVRP). (iii) We also show that in contrast with the other set of projected inequalities, the apparently non-interesting inequalities can be lifted, leading to a stronger set. This is also discussed in Section 3. (iv) Following what is known for other inequalities, we shall also show that another related, relevant and quite intuitive set of inequalities can be obtained by division and rounding of coefficients. In Section 2

4 4 we introduce these rounded inequalities and, by presenting a simple example, show that the new inequalities might be useful to solve BVRP instances in a cutting plane fashion. Computational results presented in Section 6 give empirical evidence of the usefullness of these new inequalities. (v) Finally, we point out that a variation of the standard CVRP imposes an implicit lower bound capacity on the vehicles, and thus inequalities proposed in (ii), (iii) and (iv) might be of interest to solve this variation. This is the topic of Section 5. 2 Multistar and Reverse Multistar Inequalities Routing problems are usually modeled through a directed graph G = (V, A). A special node in V = {1, 2,..., n}, node 1, represents a depot. Nodes in V \ {1} represent clients. Each node i has a demand d i such that d i = 0. i V An arc is represented by one index a, or by two indices ij when its head j and its tail i are convenient for the notation. Each arc a A is associated with a lower capacity q a and an upper capacity q a such that q a q a, meaning that if arc a is in the solution (that is, used by a vehicle) then the vehicle load when traversing this arc cannot be greater than q a and not smaller than q a. Also, each arc a is associated with a value c a representing the cost of using the arc by a vehicle. A generic SCF model uses two variables for each arc a: (i) a design binary variable x a indicating whether arc a is used by a vehicle and (ii) a continuous variable f a representing the load (flow) of a vehicle traversing the arc. To simplify notation, if S, T V then we write x(s : T ) instead of (i,j) A,i S,j T x ij. Given S V \ {1}, we denote V \ (S {1}) by S. For brevity of notation, we also write i instead of {i} for any i V. In addition, δ + (S) stands for {(i, j) A : i S, j S} and δ (S) stands for {(i, j) A : i S, j S}. Then, the model minimizes a cost function min a A c a x a 3

5 subject to the following two sets of constraints. One set involves only the design binary variables, and imposes the indegree and out-degree constraints on each client node: x(i : V \ {i}) = x(v \ {i} : i) = 1 for all i V \ {1} (1) x a {0, 1} for all a A. (2) The other set of constraints imposes the connectivity and the capacity constraints, and involves the use of the continuous variables: f a f a = d i for all i V (3) a δ (i) a δ + (i) q a x a f a q a x a for all a A. (4) As one specific example, a model for the unit-demand CVRP with vehicle capacity 2 can be defined by setting { 1 if i V \ {1} d i := 1 V if i = 1 and q a = q a = 0 for all a δ (1) q a = 1 and q a = for all a δ + (1) q a = 1 and q a = 1 for all a δ + (1) δ (1). Different variations of vehicle routing problems can be formulated by changing some of the parameters given in the previous generic SCF formulation or setting new ones (see, e.g., Toth and Vigo [11]). The SCF model is an example of compact model since it involves a number of variables and constraints that is bounded by a polynomial function defined on the size (number of nodes and number of arcs) of the input instance. We can create a natural model involving only the x a variables and with a LP relaxation bound equal to the LP relaxation bound of the SCF model, by using the tool of projection. The procedure to project out the continuous variables from the LP relaxation of the SCF model is based on the following result: Theorem 2.1 (Hoffman 1960 [8]) Given x a values, there is a solution of the linear system (3) (4) on the f a variables if and only if q a x a d i for all S V. (5) a δ (S) a δ + (S) q a x a + i S 4

6 We can give some intuition on how to generate these inequalities from the SCF model. Suppose we add the flow conservation constraints (3) for all nodes i in a set S and cancel equal terms, leading to: f a = d i. a δ (S) a δ + (S) f a + i S Then, by using the upper bounding part in constraints (4) on the term in the left-hand side of the previous equality and by using the lower bounding part of constraints (4) on the right-hand side, we obtain (5). In fact, by reversing the bounding procedure just suggested (i.e., using the lower bounding part in (4) on the term in the left-hand side, and using the upper bounding part in (4) on the term in the right-hand side) we obtain the following alternative necessary-and-sufficient condition for the theorem q a x a d i for all S V. (6) a δ (S) a δ + (S) q a x a + i S One can easily see that the inequality (6) associated with S coincides with the inequality (5) associated with the set V \ S. This follows from the fact that δ + (S) = δ (V \ S), δ (S) = δ + (V \ S) and that i S d i + i V \S d i = 0. Thus, the two families of inequalities (5) and (6) are equivalent. However, the two families of inequalities (5) and (6) (together with the way we suggested above for generating these inequalities) permit us to cast Hoffman s theorem in an alternative form. We can divide each family of inequalities in two groups: one group is associated with the sets S containing node 1 and the other group with the sets S not containing node 1. Then we use one group in both families of inequalities to generate a complete projection. These groups are the sets of inequalities (5) and (6) defined by sets S not containing node 1. The reason for this option is that then it becomes easier to enhance the different modeling properties of the inequalities from each group. First, for most of the standard vehicle routing problems, the first group of projected inequalities (given by (5) for sets S not containing node 1) is quite interesting while the second group (given by (6) for sets S not containing node 1) is easily seen to be redundant. Second, expression (6) permits us to detect quite easily less standard variations of vehicle routing problems where these inequalities might be useful. We use the name Multistar (MS) inequalities for constraints (5) with 1 S, while the name Reverse Multistar (RMS) inequalities is used for 5

7 constraints (6) with 1 S. An intuition for the designation reverse will be given later on. As noted before, it is well known (see, for instance, Gouveia [6], Letchford, Eglese and Lysgaard [9] and Letchford and Salazar-González [10]) that for capacitated routing problems, that is, routing problems with an upper bound on the number of clients on each route (or more generally, an upper bound on the sum of the demands of the clients on each route), the resulting MS inequalities turn out to be rather interesting inequalities. However, for these problems, the RMS inequalities are not of interest since they are implied by other inequalities in the model. To exemplify this, consider again the unit-demand CVRP. The resulting MS inequalities are as follows x(1 : S) + ( 1)x(S : S) x(s : S ) + S for all S V \ {1}. (7) They are the directed version of known inequalities that define facets of the associated undirected polytope (see Araque et al. [1]). The resulting RMS inequalities for the unit-demand CVRP are given as follows x(1 : S) + x(s : S) ( 1)x(S : S ) + S for all S V \ {1}. It is easy to see that for a given set S the corresponding RMS inequality is implied by the equality obtained by adding the in-degree constraints (1) for nodes i S. Thus, for the CVRP, the RMS inequalities are not of interest. The next section presents and discusses a variant of the CVRP where the RMS inequalities are not redundant, and they have a specific and intuitive interpretation. 3 Vehicle routing with lower capacities As we have noted before, expression (6) permits us to guess situations where the RMS inequalities might be of interest. The first situation that comes to mind is one where the demand summation on the left-hand side of (6) is negative. This may happen, for instance, in situations where some of the client demands are negative. These situations arise in pick-up and delivery variations of the CVRP (see, e.g., [4]) where typically the clients with positive demands correspond to locations which receive some commodity from the depot and where clients with negative demands send some commodity to the depot. It is not difficult to see that there are interesting 6

8 MS inequalities as well as RMS inequalities for these pickup and delivery problems (see, e.g., Hernández and Salazar-González [7]). However, a RMS inequality corresponds to a MS inequality in the symmetrized version of the problem (which is obtained by exchanging the sign of all demand numbers) and from a structural point of view, a RMS inequality is similar to a MS inequality. For this reason, we do not further explore these pickup-anddelivery variations on this paper. The second situation, which is also motivated by a simple analysis of expression (6), is to consider variations of the CVRP where lower bound values on the arc flows are bigger than one. To illustrate such a situation, consider the so-called Balanced Vehicle Routing Problem (BVRP), where a minimum number of clients and a maximum number of clients are required to each route in a feasible solution. In general, the designation balanced applies to a variant of the problem when is small. Here we relax this designation since our aim is to consider situations with a lower bound (> 1) and we may even allow examples where = V 1 (i.e., no upper capacity on the vehicles). The lower limited capacity requirement can be easily modeled through a SCF model by setting the lower bound value q a on the arcs leaving the depot as being equal to. More precisely, a SCF model for the BVRP can be obtained by setting the parameters as follows: q a = q a = 0 for all a δ (1) q a = and q a = for all a δ + (1) q a = 1 and q a = 1 for all a δ + (1) δ (1). Thus, it is quite easy to include lower bound information in a SCF formulation. On the other hand, finding from scratch inequalities involving only the x a variables to guarantee the minimum required number of clients in each route seems to be far from easy. Fortunately, the tool of projection on the SCF formulation permits us to obtain such set of inequalities. The MS inequalities are exactly the ones given in (7). Note that they do not depend on the lower bound value. The RMS inequalities, instead, take into account the upper bound capacity as well as the new lower bound capacity: x(1 : S) + x(s : S) ( 1)x(S : S ) + S for all S V \ {1}. (8) These constraints guarantee that if a (partial) route does not have the minimum required number of clients, then it cannot be closed by connecting directly to the depot the two extremes of the route. This can easily be seen if we use the in-degree and out-degree equations (1) on the client nodes to 7

9 rewrite the RMS inequality (8) associated with a set S as follows: ( 1) S ( 1)x(S : S ) + ( 1)x(S : S) + x(s : S). (9) Suppose now that we have a set S where all the nodes are in the same route and connected, and that S <. Then, we have that x(s : S) = S 1 and the RMS inequality becomes ( 1)x(S : S ) + ( 1)x(S : S) S. It states that the set S must be connected to at least one node in the set S. A more elaborate, but similar, interpretation could be found for a set S such that S. The interpretation just given for the RMS inequality (8) is exactly the opposite interpretation given for the MS inequalities (7) (see, for instance, Araque et al. [1]) and that is why we have named these inequalities as reverse multistars. The MS inequalities are known to define facets, under mild conditions, of the undirected polytope associated to the CVRP. Although no similar study has been done for the BVRP, we have no reason to suspect that the MS inequalities can be strengthened when non-trivial lower bounds are imposed on the vehicle capacity. In contrast, the RMS inequalities can be strengthened by decreasing the coefficient of the variables in the right handside term leading to inequalities that we denote by Enhanced RMS (ERMS) inequalities and that only involve the lower bound capacity: x(1 : S) + x(s : S) ( 1)x(S : S ) + S for all S V \ {1}. (10) Proposition 3.1 The ERMS inequalities (10) are valid for the BVRP polytope. Proof. Consider a feasible solution x and a set S. Let k := x (1 : S) and p := x (S : S), and consider the set S partitioned into the subsets S 1,..., S k+p where each subset S i corresponds to a connected component of the route x in S (i.e., a path). Trivially S k + p and the in-degree and out-degree equations on each node guarantee that x (S : 1) + x (S : S ) = k + p. If x (S : S ) k then the inequality (10) for S is valid since k + p k + p + (k 1). Let us assume now that k 1 := x (S : S ) < k. Then at least k 2 = k k 1 subsets S i correspond to complete routes, i.e. are directly 8

10 connected to the depot from both sides. Since each route has at least clients, we must have S k 2 + k 1 + p. Then, k + p k 2 + k 1 + p + ( 1)k 1 and the inequality (10) for S is valid. It is easy to find solutions that satisfy all MS inequalities and RMS inequalities and violate some of the ERMS inequalities. This suggests that the ERMS inequalities might be useful in a cutting plane approach to solve the BVRP. The ERMS inequalities (10) were inspired by the fact that, in inequalities (9), the coefficients of the variables associated with arcs a δ + (S) δ (S ) are different from the coefficients of the variables associated with the reverse arcs a δ + (S ) δ (S). One may suspect that a similar set of inequalities might also be derived for the undirected version of the problem (where undirected edge variables are used instead of directed arc variables). Note that the ERMS inequalities are equivalent to (9) where the coefficient of 1 associated to the variables in the cut δ + (S) δ (S ) is decreased to 1, thus permitting us to write an undirected inequality. The stronger inequalities suggest the following question. Similarly to what happens to the weaker RMS inequalities, could we find a compact model which implies the ERMS inequalities (10)? Or in a more broad way, can they be separated in polynomial time? For the moment we do not have an answer to these two related questions. At first glance, it appears that we could obtain such a compact model by decreasing the coefficients q a in (4) from 1 to 1 for the arcs a δ + (1) δ (1) and use the projection tool as before. However, these modified upper bound constraints are not valid since they also force each route to have at most clients. The ERMS inequalities (10) apparently make the RMS inequalities (8) useless. However, Hoffman s result guarantees that we have a polynomial formulation that implies all the RMS inequalities. Due to the fact that current available ILP solvers (e.g., CPLEX and XPRESS) are very efficient in solving compact models, using such a compact SCF model is worth considering. Furthermore, and although using the ERMS inequalities (instead of the original RMS inequalities) should provide better LP based lower bounds, it is still unknown whether the ERMS inequalities (10) can be separated in polynomial time or whether a compact model model implying them exists. We also emphasize that we would not have been able to derive the ERMS inequalities without knowing about the RMS inequalities. We end this section noting that for the special case of the BVRP where = the ERMS inequalities (10) do not provide new information since they can be shown to be equivalent to the MS inequalities (7). We next 9

11 show that the MS inequality (7) for a given set S is equivalent to the ERMS inequality (10) for the complement set S. For the proof we use the fact that, when =, all feasible solutions have a fixed number of vehicles given by ( V 1)/, i.e. x(1 : V \ {1}) = ( V 1)/. Proposition 3.2 When =, the ERMS inequality (10) for set S is equivalent to the MS inequality (7) for the set S, and vice-versa. Proof. Consider the ERMS inequality (10) for a given set S: x(1 : S) + x(s : S) ( 1)x(S : S ) + S Using the degree constraint for the depot, x(1 : V \ {1}) = ( V 1)/, we obtain V 1 + x(s : S) ( 1)x(S : S ) + x(1 : S ) + S Since V 1 S = S we obtain the MS inequality (7) for the set S. Clearly, this equivalence no longer holds for the more general BVRP where < since we have shown that the ERMS inequalities (10) (and even the RMS inequalities (8)) are fundamental for modeling the BVRP. 4 Rounded Inequalities It is well known that projected inequalities can be used to produce (by adequate division and rounding) other interesting sets of inequalities. As an example, consider the MS inequalities (7) for the unit-demand CVRP. Dividing by these inequalities and rounding we obtain the following rounded MS inequalities 1 x(1 : S) + 1 x(s : S) x(s : S ) + S that correspond to the well-known generalized cut constraints S x(v \ S : S) (11) for all S V \ {1}. These inequalities are the directed version of facetdefining inequalities for the undirected CVRP polytope (see, e.g. Campos et al. [2], Cornuejols and Harche [3] and Araque et al. [1]) and are by far 10

12 x 12 = 1 x 13 = 1/3 x 15 = 1/2 x 17 = 1/2 x 23 = 2/3 x 24 = 1/6 x 26 = 1/6 x 31 = 2/3 x 34 = 1/3 x 46 = 5/6 x 48 = 1/6 x 57 = 1/6 x 58 = 5/6 x 61 = 2/3 x 67 = 1/3 x 74 = 1/2 x 75 = 1/2 x 81 = 1 Table 1: Fractional solution violating a rounded ERMS inequality the most relevant inequalities in cutting-plane approaches for solving the CVRP and related problems. We show next that, by performing a similar procedure starting from the ERMS inequalities (10), we obtain a new set of inequalities that may be relevant for problems with lower bound capacities. We note that a similar procedure can be applied to the weaker RMS inequalities (8). However, the obtained inequalities will be weaker than the ones obtained from the ERMS inequalities (10). For this reason we only focus on applying the derivation procedure to (10). As it has been done with the original RMS inequalities, an ERMS inequality can be rewritten as follows: ( 1) S ( 1)x(S : S ) + ( 1)x(S : S) + x(s : S). Then, if we divide this inequality by, and then apply rounding as before, we obtain the new rounded ERMS inequality: S 1 1 S x(s : S ) + x(s : S) + x(s : S) (12) which is the same as S S x(s : S ) + x(s : S) + x(s : S). It is not difficult to see that there is no dominance relationship between the two sets of inequalities, the ERMS inequalities (10) and the rounded ERMS inequalities (12). Clearly, one expects the rounding to perform better when S / is not integer. Table 1 presents a fractional solution for an instance with n = 8, = 4 and = 3. This solution satisfies all MS inequalities (7), RMS inequalities (8) and rounded MS inequalities (11). However, it is easy to see that the solution violates the rounded ERMS inequalities (12) defined by the sets S 1 = {2, 3} and S 2 = {2, 3, 4, 6}. Thus, the rounded ERMS inequalities (12) might also be useful in a cutting plane algorithm for solving instances of the BVRP. We give next some intuition on why this may happen. 11

13 First, we note that one can exhibit, quite easily, a simple case where the rounded ERMS inequality (12) is at least as strong as the corresponding (for the same set S) ERMS inequality (10). Consider S = V \ {1}. Then, both constraints give an upper bound on the number of vehicles leaving the depot, but the upper bound given by the rounded ERMS inequality (12) is stronger when ( V 1)/ is not integer. More generally, consider the following situation with a set S consisting of three nodes linked together in a feasible solution for the problem and where the depot (node 1) is linked to one node in S. Assume that = 5 and = 6. The ERMS inequality (10) for these parameters becomes x(S : S ), which is equivalent to 2/4 x(s : S ). The inequality combined with the in-degree and out-degree constraints states that the last node in this sequence must be connected to a client (because the route has only visited three nodes). From the LP relaxation point of view we would prefer to obtain an inequality in the same situation where the left-hand side would be equal to 1. However, this is given precisely by the rounded ERMS inequality (12). We can use the equality constraints of the model to obtain another useful way of expressing the rounded ERMS inequalities (12). Indeed, by using the in-degree and the out-degree constraints for every node in set S we obtain 2x(S : S) + x(s : 1) + x(1 : S) + x(s : S ) + x(s : S) = 2 S, and thus the rounded ERMS inequality associated with S can also be written as S x(s {1} : S {1}) S +. (13) Constraint (13) is a normal subtour elimination constraint involving the set S {1} when S V \{1} such that S <. This is quite intuitive since the lower bound constraints can also be viewed as imposing that small subtours involving the depot are not allowed. From a practical point of view, our computational experiments show that these constraints, for sets S with cardinality smaller than, are quite relevant for improving LP bounds. On the other hand, for the sets S with cardinality greater or equal than they are not (for S =, the corresponding ERMS inequality dominates the rounded version). However, from a theoretical point of view, we can produce (fractional) solutions that satisfy all MS inequalities, rounded MS inequalities, ERMS inequalities and rounded ERMS inequalities for sets S with cardinality smaller than, and which violate a rounded ERMS inequality for a set S with cardinality larger than. 12

14 We end this section providing a similar result to the one given at the end of the previous section. For the special case of the BVRP where =, the rounded ERMS inequalities (12) do not provide new information since they are equivalent to the rounded MS inequalities (11). Following the same assumptions given at the end of the previous section, we can state and prove the next result. Proposition 4.1 When =, the rounded ERMS inequality (12) for set S is equivalent to the rounded MS inequality (11) for the set S. Proof. Consider the rounded ERMS inequality (12) for a given set S. Using the in-degree equations for the nodes in set S we obtain S S x(s : S ) + S x(1 : S). Using the fact that x(1 : V \ {1}) = ( V 1)/ and rearranging we get V 1 S x(s : S ) + x(1 : S ). Since, the left-hand side of the resulting inequality is equal to S /, we achieve the desired result. As shown by the fractional solution in Table 1, the above-cited equivalence no longer holds for the more general BVRP where <. However, when = + 1 we may obtain some interesting relations. For simplicity of notation we now assume = 1. First note that in this situation we have V 1 x(1 : V \ {1}) V 1 1 Consider the rounded ERMS inequality (12) for a set S S S x(s : S ) + x(s : S) + x(s : S). 1 Using the same reasoning as before, we obtain S x(1 : S) + S x(s : S ) + S. 1 Using ( V 1)/ x(1 : V \ {1}) and cancelling equal terms we obtain the following cut-like inequality for the set S : V 1 S x(s : S ) + x(1 : S ), (14) 1 13.

15 which is quite similar to the rounded MS inequality (11) but with a different right-hand side. For the situations where V 1 V 1 = x(1 : V \ {1}) = 1 the two sets of constraints (14) and (12) are equivalent, and then it is interesting to compare (14) and (11). We discuss next two different cases: Case 1. n = 8, = 4, = 3: We obtain x(1 : V \{1}) = 2. In this case one can easily see that the inequalities (14) either are equivalent to (11) or are even weaker. More precisely, (14) is weaker than (11) if S = 1, and the two inequalities are equivalent if S > 1. This analysis does not invalidate the previous example since in that case the solution satisfies ( V 1)/ < = x(1 : V \ {1}). However, if we had previously included the degree constraint x(1 : V \{1}) ( V 1)/( 1) = 2 (which is equivalent to a rounded ERMS (12) for S = V \ {1}) we would be in the situation of applying the previous analysis. Case 2. n = 10, = 4, = 3: Here we have x(1 : V \ {1}) = 3. It is interesting to point that for this case, in three out of the nine possibilities, inequality (14) is stronger than the corresponding (for the same set) rounded MS inequality (11), and they are equivalent for the remaining possibilities. Thus, for this case, the rounded ERMS inequalities (12) make the rounded MS inequalities (11) redundant. For instance, when S = 4 the rounded MS inequality (11) has a right-hand side equal to 1, while inequality (14) for S with S = 5 has the right-hand side equal to 2, thus it is stronger. There is a quite intuitive interpretation to this: consider a set S such that S = 4; the rounded MS inequality (11) states that at least one arc incoming into S has positive x a ; however if there is only one arc in this situation then all the nodes in S must be in one vehicle, which is full; but then it is impossible to put all the remaining 5 nodes into feasible routes. In other words, at least two arcs must enter S with positive x a, as stated by the corresponding rounded ERMS inequality (12). A similar analysis holds for sets S with S = 7 and 8. 5 CVRP with a fixed number of vehicles In the previous section, we have studied a problem where the lower bound on the number of clients per route was part of the problem specification. 14

16 x 13 = 1/3 x 14 = 1/3 x 15 = 1/3 x 16 = 1/3 x 17 = 1/3 x 18 = 1/3 x 21 = 5/9 x 23 = 4/9 x 31 = 4/9 x 32 = 2/9 x 34 = 1/3 x 41 = 1 x 54 = 1/3 x 56 = 2/3 x 62 = 4/9 x 68 = 5/9 x 73 = 2/9 x 75 = 2/3 x 78 = 1/9 x 82 = 1/3 x 87 = 2/3 Table 2: Fractional solution violating a RMS inequality The main reason for the proposed study was to motivate a problem where RMS inequalities are important for modeling the problem. In this section we consider a standard variant of the CVRP where the problem specification does not explicitly include lower bound information. However, sometimes this lower bound is implicit on the problem specification, and thus it might be used to generate new valid inequalities for the problem. Consider the CVRP with unit demands, an upper bound on the number of clients per route, and a fixed number m of vehicles given explicitly the equation x(1 : V \ {1}) = x(v \ {1} : 1) = m. (15) Most of the exposition given in this section also applies to the more realistic setting where an upper bound on the number of vehicles is given. In order to show how a lower bound capacity can be useful for such a problem, let us consider an instance with n = 8, = 4 and m = 2. It is clear that in any feasible solution for the problem, each vehicle cannot visit less than three clients. Thus, we can reformulate the problem and add this lower bound information on the capacity of the vehicles. More generally, this CVRP variant is a BVRP with = ( V 1) 1 (m 1) and a degree constraint on the depot. The question now is to know whether, without being strictly necessary to write a valid formulation for the problem, the lower bound information is good for developing inequalities that will tighten the LP relaxations of natural formulations for the problem with a fixed number of vehicles (or an upper bound on the number of vehicles). Table 2 shows a fractional solution for an instance with n = 8, = 4 and m = 2. As noted before, = 3. This solution satisfies the MS inequalities (7), rounded MS inequalities (11) and the degree constraint (15) stating that m vehicles must be used. However, it violates an ERMS inequality (10) (and even the weaker RMS inequality (8)) for the set S = {2, 3, 4}. This solution tells us that the ERMS inequalities (10) (and even the RMS inequalities (8)) might be useful in the context of a pure cutting plane method using only the x a variables. We are more precise in the next two propositions and give a sufficient and necessary condition for a fractional 15

17 solution (that satisfies the degree equations, the MS constraints and the rounded MS constraints) to violate either an ERMS or a RMS inequality. Proposition 5.1 For the CVRP with a fixed number of vehicles m, the condition x(v \ S : S) m 1 holds if and only if the MS inequality (7) for set S implies the ERMS inequality (10) for set S. Proof. Consider the MS inequality (7) for set S. Using the degree constraint for the depot we obtain m S + ( 1)x(S : S) x(s : S ) + x(1 : S ). (16) If we now add the inequality S (m S ) + ( )x(s : S ) ( )x(1 : S ) (17) to (16) we obtain the ERMS inequality (10) for the set S. Thus, whenever (17) is valid, we have that the MS inequality (7) for set S implies the ERMS inequality (10) for set S. Using the fact that = (n 1) (m 1) and that S + S = n 1, and multiplying by 1, the inequality (17) can be rewritten as m (n 1) + (m (n 1))x(S : S) (m (n 1))x(1 : S ), which is equivalent to 1 + x(s : S) x(1 : S ), and also equivalent to x(1 : S) + x(s : S) m 1. Clearly the reversed argument also holds and we obtain the desired result. Thus, whenever x(v \S : S) m 1 we know that the ERMS inequality (10) for set S is not necessary since it is implied by (or equivalent to when x(v \ S : S) = m 1) the MS inequality (7) for set S. However, when x(v \ S : S) m, the ERMS inequality for the corresponding set S is not redundant in a formulation containing the degree constraints as well as the MS inequalities. The solution mentioned at the beginning of this section (see Table 2) is in these conditions. We can state a similar but weaker result for the case of the original RMS inequalities (8). For simplicity we omit its proof since it is similar to the proof of Proposition 5.1. Proposition 5.2 For the CVRP with a fixed number of vehicles m, the condition x(1 : S) m 1 holds if and only if the MS inequality (7) for set S implies the RMS inequality (8) for set S. 16

18 Thus only when x(1 : S) = m the RMS inequality (8) is of interest in a formulation containing the degree constraints (1) and (15) as well as the MS inequalities (7). The equivalent results given in Section 3 permit us to state that in the context of a method that uses a SCF-like model, better lower bounds can be obtained by changing the lower bound parameter q a from 1 to for arcs a leaving the depot. Furthermore, by performing this change, we can guarantee that all RMS inequalities (8) are implicitly satisfied by the corresponding LP relaxation solution. We note, however, that this equivalence result does not diminish the interest in performing a similar study with respect to the ERMS inequalities (more precisely, to know whether they can be separated in polynomial time) since the ERMS inequalities are stronger than the RMS inequalities. Furthermore, as shown in Propositions 5.1 and 5.2, given a fractional solution satisfying the degree constraints and the MS inequalities, it is more likely to find a violated ERMS inequality (10) rather than a violated RMS inequality (8). The previous analysis motivates a similar study with respect to the rounded ERMS inequalities (12). Unfortunately, the analysis for these inequalities lead to quite negative results as the following result shows. Proposition 5.3 For the CVRP with a fixed number of vehicles m, the rounded ERMS inequality for set S is implied by the rounded MS inequality for set S. Proof. For the CVRP with a fixed number of vehicles m, we have = (n 1) (m 1). Let us assume that = q for a certain integer number q. Since m = n 1 + q, the number (n 1 + q)/ is integer. Using arguments shown before, the rounded ERMS inequalities for set S can be rewritten as: n 1 + q S + x(s : S ) + x(1 : S ), q which is quite similar to the rounded MS inequality for the set S : S x(s : S ) + x(1 : S ). Thus we only need to compare the left-hand sides of the two inequalities. Without loss of generality, assume that S = k + p and S = k + p with 0 p 1 and 0 p 1. Note that k + k = m 1 and p + p =. 17

19 We first consider that case where p > 0. Under this assumption, S / = k + 1 and S /( q) k since S = k( q) + p + kq. Thus S = k + 1 = m k n 1 + q S q and therefore the rounded MS inequality for set S dominates the rounded ERMS inequality for set S. Consider now the case where p = 0. Then S = k + q and S = k. Under this assumption, S / = k and S /( q) k +1 since S = k( q)+( q)+kq. We have obtained the desired inequality. This result is a bit surprising considering the examples for the case = 1 given in the previous section. The reason for these disappointing results is that for this special version of the CVRP, the values of and m identify univocally one value for. An example of this dominance is Case 1 given at the end of Section 4. On the other hand, in the BVRP we may have situations with fixed values of and, and several feasible values for m. The fact that the ERMS inequalities (10) and even the RMS inequalities (8) may be of interest for this special case of the CVRP, raises several possibilities, namely that a more through study of the BVRP polytope may lead to other inequalities of interest for this variant of the CVRP. A related question is to know whether the ERMS and RMS inequalities are really new for the CVRP with a degree constraint on the depot. That is, could they be shown to be equivalent to other inequalities which are already known from the literature? This is a difficult question to answer since there are many classes of inequalities for the CVRP. The best we can say is that from this large class of inequalities, and as far as we know, only the degree constraints use information on the number of vehicles, and thus it is highly unlikely that the ERMS and RMS inequalities are equivalent to other inequalities. 6 Computational results In this section we present computational results in order to evaluate our contributions for solving BVRP instances. We want to show that using the new inequalities one can solve larger instances. To this end we compare two branch-and-cut implementations. The first implementation is the basic one, where we approach the BVRP by using standard inequalities for the capacitated VRP as well as the RMS constraints (8). The second implementation 18

20 also uses the standard inequalities for the capacitated VRP, but uses the ERMS (10) and the rounded ERMS (12) instead of the RMS constraints. In both implementations the standard inequalities for the capacitated VRP are the classical subtour elimination constraints, the MS inequalities (7) and the rounded MS inequalities (11). More precisely, our cutting-plane generation approach proceeds as follows. Step 1: Find (if any exists) a violated subtour elimination constraints by applying a min-cut algorithm. We repeat this step while we are finding violated constraints. The number of inequalities found at the root node is denoted by f1 and the total number of inequalities found during the whole branch-and-bound search is denoted by g1. Step 2: This is a heuristic search for violated rounded ERMS inequalities (13). It consists of finding a most violated (if any exists) inequality of the type S S 1 x(s : S ) + 1 x(s : S) + x(s : S) which is a weaker version of (13) and which can also be rewritten as x(s {1} : S ) + S x(1 : V \ {1}). These inequalities can be exactly separated by using a min-cut algorithm. Again, this step is repeated while it generates violated inequalities. Let f2 be the number of violated rounded ERMS inequalities found within this step at the root node, and g2 the total number during the whole search. Step 3: This is another heuristic approach to find violated rounded ERMS inequalities. We explicitly enumerate and check for violation (13) for all S such that 1 S and S <. The step is repeated while violated inequalities are found. The number of inequalities found at the root node is denoted by f3 and the total number found during the whole search by g3. Step 4: Here we use another heuristic approach, this time to separate rounded MS inequalities (11). To this end, we apply a min-cut algorithm to find a most violated inequality of type x(a(s)) d(s)/, 19

21 which is a weaker version of the rounded MS inequality. The rounded MS inequality associated to an optimal min-cut set S is added to the model. The number of inequalities found at the root node is denoted by f4 and the total number during the whole search by g4. Step 5: Solve the linear system (3) (4) with q a and q a given in Section 3. If this system is infeasible, the dual extreme ray divides the node set V \ {1} in two sets, S and S. Then the MS inequalities (7) associated with S and S are checked for potential violation. If one is violated then it is added to the model and the step is repeated. The number of MS inequalities found at the root node is denoted by f5 and the total number during all the search by g5. If no MS inequality is violated then we check for violation the ERMS inequalities (10) associated to the same sets S and S. If one such inequality is violated then it is added to the model and the step is repeated. The number of these ERMS inequalities found at the root node is denoted by f6 and the total number during the whole search by g6. If no ERMS inequality is violated then we check for violation the rounded ERMS inequalities (12) associated to the same sets S and S. If one such inequality is violated then it is added to the model and the step is repeated. The number of these rounded ERMS inequalities found at the root node is denoted by f7 and the total number during the whole search by g7. Alternatively, instead of checking ERMS and rounded ERMS, one can check the RMS inequalities (8) associated with S and S. The number of RMS inequalities found at the root node is denoted by f8 and the total number during all the search by g8. The first implementation applies Steps 1, 4, and 5 to find violated MS and RMS inequalities. The second implementation applies Steps 1 4, and 5 to find violated MS, ERMS and rounded ERMS inequalities. In our computational experiments we have used three sets of instances. The first set contains Euclidean instances taken from the VRP-library. The second set consists of randomly generated instances, still using the Euclidean distance. The third set consists of asymmetric instances also taken from the VRP-library. All the instances have up to 50 clients with unit demand to better understand the impact of. Also for that reason, we have created different instances by considering different values for. The algorithm was coded in C++ on a Linux platform running in a Intel(R) Core(TM)2 CPU 2.66GHz desktop computer with 2GB RAM. We have made use of the CHIPPS framework (see [12]) included in the COmputational INfrastructure 20

22 for Operations Research COIN-OR (open source for the Operations Research community) using their simplex solver CLP. Tables 3 and 4 show the results obtained with the second implementation, i.e., the one that uses all the new inequalities proposed in this paper. Table 3 refers to features at the root node, while Table 4 refers to features at the end of the branch-decision search tree. They show the name of the instance, the number n of clients, the vehicle capacity limits and, the objective value r-lb at the end of the root node, the computing time in seconds r-time at the end of the root node, the objective value opt at the end of the search, the total computing time time in seconds, the number of cuttingplane iterations LP iter, the number of branch-and-bound nodes nodes, the number m of routes in the obtained optimal solution, the maximum number max of clients in a route in the optimal solution, and the minimum number min of clients in a route in the optimal solution. In addition, they also display the number of inequalities found by the separation steps. Columns f1,f2,f3,f4,f5,f6,f7 in Table 3 and Columns g1,g2,g3,g4,g5,g6,g7 in Table 4 gives the numbers of inequalities as described above. The results indicate that, in general, the Eilon instances are easier to solve than the random instances. Also, the asymmetric instances are much harder to solve. In general, the results indicate that the instances become slightly easier to solve when the value increases and the other input parameters remain fixed. Tables 5 and 6 show the performance of the first implementation, i.e. the basic algorithm that, besides the some CVRP inequalities, only uses the RMS inequalities. Here, for brevity and clarity, we only present some Eilon instances with 30 clients, random instances with 39 clients and asymmetric instances with 45 clients (we have run the first implementation to solve all the instances and similar conclusions can be obtained for the remaining instances). The meaning of the headings in these tables are the same as in the previous tables. Comparing tables 3 and 5 one immediately concludes that much larger CPU times are needed to solve the same instances with the first implementation. These results show that the new inequalities are worth having in a cutting plane method for solving the BVRP. An interesting observation regarding the LP bounds comes from the comparison of tables?? and 6. The results show that the inclusion of the RMS inequalities in the first implementation do not alter by much the LP bound given by the true upper capacity case, that is the instance with = 1. In fact, the LP bound increases very slowly (when it increases) with the value of, for a fixed value 21

23 of. This can be explained by analyzing the solutions of the LP relaxation of the SCF model. In most of the cases, the LP solution contains large fractional routes, implying that the upper bound inequalities on f a in (4) are either satisfied at equality or close to being satisfied at equality. This means that adding the lower bound inequalities on f a in (4) may not have any effect on the given LP bound if the given is not close to the given. The LP bounds improve when the new inequalities are added. We note however that, in some cases, large gaps are still obtained. 7 Conclusions In this paper we have examined a subset of the projected inequalities from a standard single flow formulation for the CVRP. Although for the CVRP it is known that this subset is useless, we have pointed out that for other variants of the CVRP this is may not be the case. This subset of inequalities are the RMS inequalities. We have also described two new sets of inequalities, called ERMS and rounded ERMS inequalities. Computationally we have noted that the new inequalities are of interest to solve CVRP variations where arc lower bounds on the vehicle loads are either explicitly or implicitly considered. References [1] Araque, J.R., Hall, L., Magnanti, T.L.: Capacitated trees, capacitated routing and associated polyedra. Discussion Paper 90-61, CORE, University of Louvin La Neuve, Belgium, [2] Campos, V., Corberán, A., Mota, E.: Polyhedral resuls for a vehicle routing problem. European Journal of Operational Research 52 (1991) [3] Cornuejols, G., Harche, F.: Polyhedral study of the capacitated vehicle routing problem. Mathematical Programming 60 (1993) [4] Desaulniers, G., Desrosiers, J., Erdmann, A., Solomon, M.M., Soumis, F.: VRP with pickup and delivery. In Toth, P., Vigo, D. (Eds.): The Vehicle Routing Problem. SIAM, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania,

24 [5] Gavish, B., Graves, S.: The traveling salesman problem and related problems. Working Paper, Graduate School of Management, University of Rochester, New York (1979). [6] Gouveia, L.: A result on projection for the vehicle routing problem. European Journal of Operational Research 85 (1995) [7] Hernández-Pérez, H., Salazar-González, J.J.: The one-commodity pickup-and-delivery traveling salesman problem: Inequalities and algorithms. Networks 50 (2007) [8] Hoffman, A.J.: Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. In Bellman, R., Hall, Jr., M. (Eds.): Proc. Symp. in Applied Mathematics. Amer. Math. Soc. 10 (1960) [9] Letchford, A.N., Eglese, R.W., Lysgaard, J.: Multistars, partial multistars and the capacitated vehicle routing problem. Mathematical Programming 94 (2002) [10] Letchford, A., Salazar-González, J.J.: Projection of Flow Variables for Vehicle Routing. Mathematical Programming 105 (2006) [11] Toth, P., Vigo, D.: An overview of vehicle routing problems. In Toth, P., Vigo, D. (Eds.): The Vehicle Routing Problem. SIAM, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, [12] Xu, U., Ralphs, T., Ladányi, L., Saltzman, M.: The COIN-OR High- Performance Parallel Search Framework (CHIPPS) project. The Computational Infrastructure for Operations Research (COIN-OR)

25 name n f1 f2 f3 f4 f5 f6 f7 r-lb r-time eil22a eil22b eil22c eil22d eil22e eil22f eil22g eil22h eil23a eil23b eil23c eil23d eil23e eil23f eil23g eil31a eil31b eil31c eil31d eil31e eil31f eil33a eil33b eil33c eil33d Rand30a Rand30b Rand30c Rand30d Rand30e Rand40a Rand40b Rand40c Rand40d Rand50a Rand50b Rand50c Rand50d Rand50e Rand50f Rand50g Rand50h Rand50i Rand50j Rand50k Rand50l Rand50m A034-02fa A034-02fb A034-02fc A034-02fd A034-02fe A034-02ff A034-02fg A034-02fh A034-02fi A034-02fj A034-02fk A034-02fl A034-02fm A036-03fa A036-03fb A036-03fc A036-03fd A036-03fe A036-03ff A036-03fg A036-03fh A039-03fa A039-03fb A039-03fc A039-03fd A039-03fe A039-03ff A039-03fg A039-03fh A039-03fi A039-03fj A045-03fa A045-03fb A045-03fc A045-03fd A045-03fe A048-03ff A048-03fg Table 3: Second implementation: the root node.