A result on projection for the vehicle routing ptoblem

' i

ELSEVIER European Journal of Operational Research 85 (1995) 610-624

EUROPEAN JOURNAL

OF OPERATIONAL RESEARCH

Theory and Methodology

A result on projection for the Vehicle Routing Problem

Luis Gouveia DEIO - CEAUL, Faculdade de Ci~ncias da Universidade de Lisboa, Bloco C / 2 - Campo Grande, Cidade Universitaria,

1700 Lisboa, Portugal

Received December 1992; revised September 1993

Abstract

In this paper we present a result on projection for the Vehicle Routing Problem (VRP). The VRP is closely related to delivery-type problems and appears in a large number of practical situations concerning the distribution of commodities. The present work focuses on a commodity flow formulation presented by Gavish and Graves. This formulation includes two sets of variables and, hence, it also must include coupling constraints between the two sets of variables. These coupling constraints can be defined in several ways. The main result of this work establishes that when the strongest form of the coupling constraints is used in the flow formulation, the equivalent formulation using only the Xq variables satisfies the so called multistar constraints which, for certain parameters, induce facets of the non-directed VRP polytope. Using an idea taken from Gouveia, we show how to derive a more compact representation, in the number of constraints, of the multistar constraints. Some consequences of our projection result are also discussed.

Keywords: Mathematical programming; Vehicle routing; Projection

1. Introduction

In the Vehicle Routing Problem (VRP) we are given one depot, a set of n customers with a positive integer valued demand function q defined on them, a fleet of an unlimited number of trucks of capacity Q and a traveling cost between all pairs of points. We want to find a minimal cost set of routes originating and terminating at the depot such that each one of the n customers is visited exactly once and the total weight of all customers on the same route is at most Q.

The VRP is closely related to delivery-type problems and appears in a large number of practical situations concerning the distribution of commodities such as the distribution of foods in a school system, salesman routing, maintenance inpection tours, collection of mail from mailboxes, etc. Two good surveys on the VRP considering both its real-world applications and solution methods can be found in [6] and [23].

A particular instance of the VRP is obtained when the weights associated with each node are equal to one. This is known as the unit-weight VRP and for this particular case, the capacity constraints can be

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L. Gouveia / European Journal of Operational Research 85 (1995) 610-624 611

restated as: each route contains no more than Q customers. It is remarkable that much of the research on lower bounding methods and on heuristics has focused mainly on the general case, i.e., the case with non-units weights on the customers. See, for instance [1,7-9,11,12,16,17,24,26]. Note that each method described for the general VRP can be simply adapted for the identical customer VRP. However, it is interesting to note that in many of the above works, computational results for the unit-weight case are simply ommitted.

On the other hand, it is precisely the unit-weight version of the problem which has been receiving, lately, a lot of attention from the polyhedral combinatorics point of view. This can be simply explained by the fact that the polyhedral structure associated to the unit-weight version of the problem is much simpler than the polyhedral structure of the general case. As far as we know, only the very recent reference [10] addresses the polyhedral structure of the non-unit weight VRP. Note also, !that before tackling the general case, one must have a proper understanding of the polyhedral structure of the simpler case.

It is well known that the convex hull of the set of feasible solutions of many combinatorial optimization problems (including the VRP) can be described by a set of linear inequalities. This means that, in theory, such problems can be solved by linear programming methods if such a set isl completely known. However, for the VRP and many other combinatorial problems, such a description! is far from being known. From a practical point of view, one may simply use a subset of the faCet-inducing inequalities involved in the description of the corresponding convex-hull. Then, nearly optimal solutions can be generated by using efficient linear programming techniques combined with cutting methods. This approach was successfully applied for the travelling salesman problem (TSP) in [29]. In fact, it was the impressive results reported in the same work which lead to a very active research on polyhedral descriptions of other combinatorial optimization problems.

A subset of facet inducing inequalities was already established for the unit-weight VRP. In fact, Araque [2] presents a list of facet-inducing inequalities for the unit-weight VRP. The list includes among others, generalized subtour elimination constraints, multistar constraints and partial multistar constraints. He reports solving some VRP instances with up to 48 nodes by branch-and-cut techniques. The facetial properties of the generalized subtour elimination constraints are also studied in [5]. The combined work of Araque with Hall and Magnanti [4] also presents a reasonable list of facet-inducing inequalities both for the unit-weight VRP and the Capacitated Minimal Spanning Tree Problem (CMSTP). As far as we know this is the first work that establishes a close relationship between the polyhedral structures of the VRP and the CMSTP. Laporte and Nobert [22] show how to generalize comb inequalities for the VRP. However, their inequalities are dominated by the set of inequalities given in [3].

In this work we also focus on the unit-weight VRP (for simplicity, VRP in the sequel). Our main result establishes that the set of inequalities which includes the directed version of every multistar constraint (which are facets of the VRP polytope when some adequate conditions are verified) has a polynomial representation, i.e., there exists a formulation involving a polynomial number of variables and a polynomial number of constraints which satisfy every multistar constraint.

Let us note that the search for polynomial representations of sets of inducing-facet inequalities for the VRP has not received much attention from the polyhedral combinatorics community. As will be discussed later, the existence of such a representation for a particular set of facets i of a given combinatorial problem, may have a strong effect on the performance of a branch-and-cut imethod for solving it. Additionally, such a representation may suggest that the associated separation problem is polynomially solvable. Note that the complexity of the separation problem associated with each of the non-trivial facets listed in the previous works, is not yet known. This includes the separation problem for the multistar constraints.

In the present work, we start by looking closer at the Gavish and Graves one-commodity flow

612 L. Gouveia / European Journal of Operational Research 85 (1995) 610-624

formulation for the directed VRP (see [14]). This formulation involves two types of variables: binary variables Xij that specify if an arc (i, j ) is in the solution and, non-negative variables Y/j that specify the flow in arc (i, j). Clearly, this formulation must include coupling constraints between the two types of variables. One way of assessing the 'strength' of its linear relaxation is by characterizing completely their associated set of feasible solutions using only the Xij variables. For this, we shall use the concept of projection of a space into a subspace. The original set of feasible solutions defined with variables Xij and Y~j will be projected into the subspace defined only by variables Xij. We will show that the equivalent formulation using only the Xij variables verifies the weakest form of the generalized subtour elimination constraints. This gives, in a sense, a weak 'status' to the Gavish and Graves formulation.

We show how to derive a stronger linear relaxation by applying a lifting procedure to the coupling constraints involved in the original Gavish and Graves formulation. The main result of this work establishes that when the lifted coupling constraints are used in the one-commodity flow formulation, the 'equivalent' formulation using only the Xii variables satisfies the so called multistar constraints which, for certain parameters, induce facets of the non-directed VRP polytope (see, for instance [4]). As far as we know, this work and reference [19] are the only works which present a multistar constraint as a stronger and lifted version of the weakest form of a generalized subtour elimination constraint.

However, the 'lifted' Gavish and Graves formulation involves O(n 2) variables and O(n 2) constraints which can lead to fairly large linear programming problems even for moderate values of n. Using a different set of variables introduced in [20], we also present an alternative way to produce the optimal value of that linear programming relaxation. This will yield a reformulation for the VRP involving n2Q variables and only 3n constraints which is more attractive to use than the original formulation.

The remainder of this work is organized as follows. In Section 2, we present a graph-theoretical definition of the VRP and discuss the Gavish and Graves formulation. We show how to lift its coupling constraints. In Section 3, we present the main result of this work. In Section 4 we discuss some implications of the projection result. In Section 5, we present a more compact representation of the multistar constraints. In Section 6, we use a hard instance taken from [2] in order to show how our main result can be used in the context of a branch-and-cut method for the VRP. In Section 7 we present some concluding remarks.

2. A flow based formulation for the VRP [14]

The VRP can be described as follows:

Consider a graph G = (V, A) where V= {0, 1 . . . . , n}, with costs cij for each edge (i, j) ~ A and a natural number Q. We want to find a minimal cost set of circuits such that every circuit begins and ends at node 0 (the depot), each node i, i = 1 . . . . . n, is included in exactly one circuit and the number of nodes (except the depot) in any circuit cannot be superior to Q.

The VRP is trivial for Q = 1. For Q = 2, it can be polynomially solved by matching techniques. When Q ___ n, it reduces to the TSP if the costs satisfy the triangular inequality. Unfortunately, for n > Q > 3 it is NP-complete [25].

Several (mixed) integer linear programming formulations were presented for the VRP (see, for instance [23]). In the present work we discuss a one-commodity flow formulation presented in [14]. Consider the binary variables Xij such that Xij = 1 if arc (i, j ) is in the solution and Xij = 0 otherwise. Consider also the non-negative variables Y~ which represent the amount of flow produced by the depot and going through arc (i, j). We have then the following formulation for the unit weight VRP:


(Formulation F)

min ~ ~cijXij (1) i = 0 j=O

s.t. ~ X i j = l , j = l . . . . . n, (2a) i = o

X j i = l , j = l . . . . . n, (2b) i=O

Y/y- Yyi = 1, j = 1 . . . . . n, (3) i = 0 i=O

Yij<_QXij, i, j = O , . . . , n , (4)

Y/i>0 and Xi j~{0 ,1} , i , j = O , . . . , n . (5)

To simplify the indexing, we have not considered variables Xii (i = 0 . . . . . n). Constraints (2a) an (2b) define the well known assignment relaxation for the VRP. Constraints (3) are the flow conservation constraints for each node and ensure that circuits involving the depot are not included in the solution. Constraints (4) guarantee that the maximum flow in any arc leaving the root is equal to Q. It is easy to see that the flow in arc (i, j) gives the number of clients that have yet to be visited in the corresponding circuit. Hence, constraints (4) guarantee that no circuit with more than Q nodes can be included in the solution.

Three different lower bound schemes based on the Lagrangean relaxation of some constraints in F are presented in [27]. The three approaches differ precisely on the set of constraints being relaxed. To the best of our knowledge, no computational results related to these schemes have been published; However, the relaxed problem associated with each relaxation verifies the integrality property (see [15]). This simply means that the theoretical best lower bound associated with each scheme cannot be better than the cost of the optimal solution of the linear relaxation of formulation F. Therefore, these loWer bound schemes depend strongly on the value of that linear bound.

Following one idea presented in [13] for the CMSTP, the linear relaxation of formulation F can be tightened by replacing the set of constraints (4) by the following tighter set of constraints:

d j X i : < Y i y < ( Q - d i ) X i j , i , j = O , 1 . . . . . n, (4')

where do = 0 and d i = 1 (i = 1 . . . . , n). Constraints (4') simply state that the maximum flow of any arc not leaving the depot is equal to Q - 1. Additionally, the flow in any arc not entering the depot is at least equal to 1.

In the sequel, let FF denote the formulation (1)-(4') and (5). In this paper, we will give a complete description of the set of feasible solutions of the linear relaxation of formulations F and FF, using only variables X~j. For this, we shall project the set of feasible solutions defined by variables Xiy and Y,.j into the subspace defined only by the X~j variables.

3. The main result

Some notation is needed before presenting the main result. Let E(S) be the set of arcs having both endpoints in S, i.e. E(S) = {(i, j): i, j ~ S}, and let E(S1, S 2) be the set of arcs directed from S 1 to $2, i.e., E(S1, $2) = {(i, j): i ~ S 1 and j ~ $2}. Finally, if A is a set of arcs, let

X ( A ) = E Xij and Y( A ) = E Y,'j. (i,j)~A (i,j)~A

614 L. Gouveia / European Journal of Operational Research 85 (I 995) 610-624

Using this notation, constraints (2a), (2b) and (3) may be rewritten as

X ( E ( V \ { j } , { j } ) ) = 1 , j = l . . . . . n,

X ( E ( { j } , V \ { j } ) ) = 1, j = 1 . . . . . n,

Y ( E ( V \ { j } , { j } ) ) - Y ( E ( { j } , V \ { j } ) ) = I , j = l . . . . . n,

(2a)

(2b)

(3)

where S \ {j} denotes the set obtained from S by removing element j. In a directed formulation for the VRP, the number of variables Xij is equal to n + n 2. The same happens with respect to variables Yq. Hence, let X and Y be vectors of dimension n + n 2 such that

S .~- ( S o l , . . . , S o n , S l 0 , X 1 2 . . . . , S i n , S n o , . . . , S n , n - 1 ) ,

Y = ( Yol . . . . ,Yon, Y10, Y,2 . . . . . Yln , Y.o . . . . . Yn,n-- , ) "

In the following, the pair of vectors (X, Y) represents a vector of dimension 2(n + n2). Let (5') denote the constraints Y/j > 0 and Xij >_ 0 (i, j = 0 . . . . . n; i Oj), let PL denote the linear

relaxation of a given formulation P and let ADM(P) denote its set of feasible solutions. Then, we have

ADM(FFL) = {(X, Y): (X , Y) satisfies (2a), (2b), (3), (4') and (5')}.

For a given set U c o4+~2(n+nb, let P R O J x ( U ) b e the set such that

P ROJ x( U ) = {X ~ ~n+,2. there exists Y ~ R n+"2 with (X , Y) ~ U} ~ + • + •

Note that the set PROJx(U) is the projection of the set U into the subspace defined by the variables Xij. We present next a complete description of the set PROJx(ADM(FFL)). For this, we introduce the concept of DEGREE(S) for each set of nodes S such that S _c V 0 --- V\{0}. Let

D E G R E E ( S ) = X ( E ( S, SC\{0})) + X ( E( SC\ {O}, S ) )

where S c = V \ S . Observe that DEGREE(S) is the cardinal of the set of arcs that have exactly one endpoint in S, excluding the arcs leaving or entering into the depot (node 0).

Consider the set of inequalities

X ( E ( S ) ) + D E G R E E ( S ) / Q < I SI - IS I / Q v s c_ v o. (6)

Each inequality in (6) is called a multistar constraint and they correspond to the directed case of the multistar constraints mentioned in [2] and [4]. For certain parameters, these inequalities induce facets of the non-directed VRP polytope (see, for instance [4]). If for a given S, we multiply each side of the multistar constraint by Q, we obtain the following alternative and usual description (see, for instance [2] and [4]) for that constraint:

Q X ( E ( S ) ) + D E G R E E ( S ) < (Q - 1) I SI VS c_ V0. (6')

As pointed out in [19] any multistar constraint can be seen as a generalized degree constraint which for any feasible solution gives an upper bound on the number of arcs (not linked to the root) that have exactly one endpoint in a given subset of nodes S. To see this, consider a set of customer nodes S which are spanned by a path. Note that such a path might be included in a feasible solution for the VRP. The assignment constraints (2a) and (2b) guarantee that the number of arcs with both endpoints in S is equal to I S I - 1 and we have Q X ( E ( S ) ) = Q( I S I - 1). Therefore, the multistar constraint for such a set S is equivalent to

D E G R E E ( S ) < Q - I SI (6")

which states that the number of arcs (not linked to the root) that have exactly one endpoint in S cannot be greater than Q - I SI. Now, let us consider the following four cases which are illustrated by using a


5-customer route which includes the arcs (0, 1) (1, 2), (2, 3), (3, 4), (4, 5) and (5, 0) (recall that node 0 corresponds to the depot):

Case 1: Q < I SI. In this case, (6") cannot be satisfied because the value of DEGREE(S) cannot be a negative number. However, this is consistent with the fact that such a case corresponds to an infeasible solution. In fact, the given path spans more than Q nodes and cannot be included in any feasible solution for the VRP with capacity Q. Notice that this case also illustrates how the multistar constraints eliminate subtours with more than Q customer nodes. As an example consider the 5 customer route Q = 3 and the path given by the arcs (1, 2), (2, 3), (3, 4) and (4, 5). Clearly, this path is not feasible for any instance with Q = 3. The multistar constraint corresponding to the set S = {1, 2, 3, 4, 5} is violated by this solution. Actually, the same happens with the multistar constraints associated to the sets S = {1, 2, 3, 4} and S = {2, 3, 4, 5}.

Case 2: Q = I S I. In this case, we have DEGREE(S) < 0. According to the definition of DEOREE(S), this implies that the only node in S c which can be directly connected with nodes in S is precisely the node 0 (the depot). Due to the assignment constraints, the depot have to be connected with the two endpoints of the path that spans S. Notice that this corresponds to the case where we have a path with exactly Q nodes. In order to mantain feasibility, the two endpoints of the path have to be directly linked with the depot. Considering again the same path given by the arcs (1, 2), (2, 3), (3, 4) a n d (4, 5) and Q = 5 we can easily check that to mantain feasibility, the two endpoints of that path, nodes 1 and 5, have to be directly linked to the depot.

Case 3: Q - 1 = I S I. In this case, we have DEGREE(S) < 1. According to the definition of DE- GREE(S), this implies that in order to mantain feasibility one of the endpoints of the path that spans S has to directly linked with the depot. Consider again Q = 5 but now the path (1, 2), (2, 3) and (3, 4). Only one of the nodes 1 and 4 have to be linked to the root. In this case, node 1 is linked to the root. The other node may be linked to another customer node, and in this particular route, it was linkedl to node 5.

Case 4: Q - 2 >__ IS I . In this case, we have D E G R E E ( S ) < 2. This corresponds to case where the multistar constraint is non-binding. Due to the assignment constraints the inequality DEGREE(S) < 2 is always true when the set S is spanned by a path. In this case, both endpoints of the path may be linked to other customer nodes because the capacity in the path is not yet filled. Consider again Q = 5 and the path which contains only the arc (2, 3) and (3, 4). The two endpoints, nodes 1 and 4, may be linked to customer nodes. In fact, node 2 is linked to node 1 and node 4 is linked to node 5.

The multistar constraints are needed to define PROJx(ADM(FFL)). Let

X iy~{0 ,1} , i = O , . . . , n , j = l . . . . . n, (7)

X i j ~ O , i = 0 . . . . . n, j = l , . . . , n . (7')

Consider, now, the main result:

Result 1. PROJx(ADM(FFL)) = {X ~ R~_2: X satisfies (2a), (2b), (6), (7').

Proof. We will show that the following holds: (i) PROJx(ADM(FFL)) ___ {X ~ R~2. • X satisfies (2a), (2b), (6), (7')}.

n 2 . . (ii) { X ~ R+: X satisfies (2a), (2b), (6), (7')} ___ PROJx(ADM(FFL)).

(i): Consider any feasible solution {Xij, Y/j} for FF L. We will show that the subsolution {Xij} satisfies the multistar constraints and so it will satisfy (2a), (2b), (6) and (7').

Consider any set S ___ V 0. Adding constraints (3) for j ~ S and removing equal terms in both sums, we obtain

Y ( E ( S c, S)) - Y ( E ( S , SC)) = ISl VSC_Vo,

616 L. Gouveia / European Journal of Operational Research 85 (1995) 610-624

which is equivalent to

Y(E({0} , S ) ) + Y( E( SC\ {O}, S)) = I S I + Y ( E ( S , SC\{0}) ) + Y ( e ( S , {0})). (8)

By (8) and (4'), it follows that

Q X( E( {O}, S) ) + ( Q - 1 ) X( E( SC\ {O}, S) ) > ]SI + X( E( S, S ~ \ { 0 } ) ) . (9)

Adding X(E(S~\{O}, S)) to each side of (9) we obtain

a X ( E ( S ~, S)) > I SI + D E G R E E ( S ) VS _c V 0. (10)

Next, we show that constraint (10) is simply, another equivalent description for a multistar constraint. Adding constraints (2a) for j ~ S ___ V 0 we have

X ( E ( V , S ) ) = ISI ¥SC_Vo.

Since

x ( e ( v , S)) = X ( E ( S ) ) + X ( E ( S c, S)) VSc_V o,

we obtain

X ( E ( S ) ) + S ( E ( S ~, S)) = l S I VS___ V0,

which is equivalent to

X ( E ( S ~, S) ) = I S I - X ( E ( S ) ) VSc_V o. (11)

Replacing X(E(S c, S)) in (10) by the right hand-side of (11), the multistar constraints (6') are obtained. Hence, we get PROJx(ADM(FFL))__. {X ~ R_]_~ X satisfies (2a), (2b), (6), (7')}.

(ii): Let {Xij} be a solution satisfying (2a), (2b), (6) and (7'). Consider any feasible solution {Xij, Y/j} for FF L such that the value of the variables Xij in FF L is equal to the value of the same variables in the original solution. The constraints (2a) and (2b) of FF L which only use variables Xij , are dearly satisfied by this solution. We must show next that, for X# defined in this way, it is possible to define a flow Y~j that satisfies the constraints (3), (4') and (5').

In this case, we only present an existence proof. Hence, we will simply show that a flow satisfying (3), (4') and (5') must necessarily exist. Searching for such a flow is equivalent to searching for a feasible flow in an adequate graph with capacities in the arcs. Let c(i, j) be the minimum capacity on arc (i, j ) and let C(i, j) be the maximum capacity on the same arc. A flow Y/j is feasible in a graph with capacities c(i, j) and C(i, j) if and only if (i) c(i, j) < Yij < C(i, j) for each arc (i, j);

(ii) Ej ~ vYij - Ej ~ vYji = 0 for each node i in the graph. Finding a flow that satifles (3), (4') and (5') is the same as finding a feasible flow in a graph with

capacities c(i, j) and C(i, j) (i, j = 0,.. . , n) defined as follows:

c ( i , j )=X i j , C ( i , j ) = ( Q - d ~ ) X i j , i=O, . . . ,n , j = l , . . . , n , (12)

c( j, O) = C( j, 0) = 1. (13)

It is easy to see that a flow is feasible in the above graph if and only if it satisfies every constraint (3), (4') and (5').

Following a result due to Hoffii~an (see [18]), we know that a feasible flow exists in a graph with capacities in the arcs if and only if for each set of nodes S ___ V the following relation holds:

C( E( S, S¢)) - c ( E( S ¢, S)) >_ O, (14)


where

C(E(S1, S2))= • E C(i , j ) and c(E(S 1,$2))= E E c ( i , J ) - i~S 1 j ~ S 2 i~Sl j ~ S 2

To see the validity of (14) for the graph with capacities given by (12) and (13), two different cases have to be considered. When the depot (note 0) is in S and when the depot is not in S.

If 0 ¢ S, we have by (12) and (13) that

c( e( s c, S)) =X( E( S c, S)), (15)

C(E(S, SO)) = ISl + ( Q - 1) X(E(S, SO\{0})) . (16)

By constraints (2a), the right-hand side of (15) cannot be greater than IS 1. Hence, the rigllt-hand side of (16) is greater than the right-hand side of (15) and the inequality (14) is verified for 0 ff S:

On the other hand, if 0 ~ S, by using (12) and (13) it follows that

C( E( S, Sc) ) = ( Q -1 ) X( E( S \ {O}, SO)) + Q X( E( {O}, S¢)), (17)

c( E(S ~, S)) = Iacl +X( E(S c, S \ { 0 } ) ) . (18)

Since {Xij} is a feasible solution for MS L, it must satisfy the constraint

a X(E(S, SC)) > I S°l +DEGREE(SO), (10)

which is another valid description for a multistar constraint (see part (i) of the proof). Considering the definition of DEGREE(SC), the above inequality is equivalent to

Q X(E(S, S~)) -X(E(S \ {O} , SO) >_ I ScI+X(E(S c, S \ { 0 } ) ) ,

which, in turn, is equivalent to

Q X( E( {O}, SO)) + ( Q -1 ) X( E( S \ {O}, S~) ) > Iacl +X( E( S c, S \ { 0 } ) ) .

By (17) and (18), we can see that the above expression is equivalent to

C( E( S, sO) >__ c( e( s ~, S) ).

Hence, the inequality (14) is also valid when 0 ~ S. This means that from any {Xij} solution satis~ing (2a), (2b), (6) and (7'), it is possible to find a feasible solution, {Xij, Yo}, to FF L. Hence {X ~ R~: X satisfies (2a), (2b), (6), (7')} c_ PROJx(ADM(FFL) ) and Result 1 is proved. []

In [19] it is shown that a similar result holds for the CMSTP. It is well known that for many network design problems which can be modelled by one commodity flow formulations, the corresponding linear programming relaxations give very poor results. The result given in [19] and Result 1 show that for some capacitated versions of well known combinatorial problems, the corresponding one commodity flow formulations have interesting facetal properties. This indicates that for some capacitated problems, this class of apparently weak formulations may lead to successful lower bounding schemes when used together with other sets of valid inequalities. In fact, such an approach has already been used with some success for the CMTSP (see [20]).

Consider, now, the following set of constraints:

X(E(S ) )<_IS I - IS I /Q VS___{1 . . . . . n} and I S I > 2 . (19)

Constraints (19) correspond to the weaker version of the generalized subtour elimination constraints and they can be obtained from the multistar constraints (6) (I S I>__ 2) simply by omitting the DEGREE(S)/Q term. Note that the usual (and strongest) form of these constraints is Obtained by replacing the I S[/Q term with [I S I/Q]. As shown in [2], [4] and [5] the generalized subtour elimination

618 L. Gouoeia / European Journal of Operational Research 85 (1995) 610-624

constraints define facets of the non-directed VRP polytope when some conditions on iS I are verified. Therefore the weakest form of a generalized subtour elimination constraint can be seen as a starting inequality for deriving two stronger inequalities: (i) its stronger version by rounding up the I S I /Q term of the right hand-side; (ii) a multistar by applying a lifting procedure. Notice also that no dominance relationship can be established between the set involving every generalized subtour elimination constraint and the set involving every multistar. This is a consequence of the fact that both sets include the directed version of different facet-inducing constraints for the VRP.

One interesting point of research is to know whether it is possible to derive a polynomial representation of the generalized subtour elimination constraints. As far as we know, no such representation exists. It is possible to derive polynomial multicommodity reformulations which verify the generalized subtour elimination constraints for I SI _< Q . However, the cases with I SI > Q are far from being verified by those type of formulations. We believe that either a polynomial representation of the complete set involving every generalized subtour elimination constraint will never be found, or that new concepts must be devised to obtain polynomial representations of that set for capacitated problems.

Result 1 suggests that the same does not happen with respect to the weaker version of the generalized subtour elimination constraints (19). In fact, the original Gavish and Graves formulation F gives a representation of the set (19). The next result states that the linear relaxation of the original Gavish and Graves formulation F verifies the set of constraints (19). Its proof is similar to the proof of Result 1 and so we will omit it.

Result 2. PROJx(ADM(FL)) + {X ~ R+~ X satisfies (2a), (2b), (19), (7')}.

Using Results 1 and 2 we can see that the effect of lifting the set of constraints (19) into constraints (6) corresponds precisely to the effect of lifting constraints (4) into constraints (4') in the extended flow based formulation. While it is relatively straighforward to apply the 'rounding-up' procedure used in transforming constraints (19) into their strongest form, we do not believe that it is easy to derive the multistar constraints (6) directly from constraints (19). Results 1 and 2 may provide an indirect 3-step scheme for deriving the multistar constraints. In the first step, Result 2 is used to derive an equivalent extended formulation which satisfies the set of constraints (19). The new formulation, F, involves a new set of variables. The information related to the new variables leads to a more compact formulation involving a smaller set of constraints. Additionally, the constraints in the new formulation may be easier to understand than the constraints in the original formulation. This means that it should be easier to lift constraints in the compact formulation than to lift constraints in the original formulation. Therefore, in step 2, constraints (4) are tightened into constraints (4'). This leads to formulation FF. In Step 3, the set of feasible solutions of the FF formulation is projected into the X o. space and we obtain the multistar constraints (6), which were lifted from the original constraints (19). This 3-step scheme is illustrated in Fig. 1.

4. Implications

Araque has been able to solve some VRP instances with up to 47 customer nodes to proven optimality by branch-and-cut techniques [2]. As pointed out in the Introduction, one relevant requirement for developing successfully a branch-and-cut algorithm for a specific combinatorial problem consists of having a good understanding of its polyhedral structure, which corresponds to knowing a set of facet-inducing inequalities for the problem. Such a list of facet-inducing inequalities for the unit-weight VRP can be found in [2] and [4].


(2a), (2b ) , ( 7 ' ) Extend L[ Formulat ion F I and w I

I X(E(S)) < ISI - ISl/O Yij < Q xi j

I I I L i f t

I 12a) , 12b), (7') and

X(E(S)) + OEOREE(S)tQ ~; ISI - ISl,Q

11 Lift

Fig. 1. The 3-step scheme 'Extend, Lift and Project' is equivalent to the lifting at the left.

Additionally, the success of a branch-and-cut technique also depends on solving the 'constraint identification' problem, i.e., given a linear solution that satisfies a given subset of constraints, :determine whether the solution is feasible for the original problem and if not, find a valid inequality violated by that solution. Currently, no polynomial-time separation algorithm is known for any of the non-trivial facets presented in the two above papers. This includes the multistar constraints. Notice that from a computational point of view, good separation heuristics may suffice. However, we do not know: yet of any work involving separation heuristics for solving the VRP.

Araque [2] has overcome this problem by visually identifying violated inequalities. He Considered multistar constraints as well as other facet inducing inequalities, including the well-known generalized subtour elimination constraints. As reported, the generalized subtour elimination constraints! proved to be the most effective constraints in the process of obtaining the optimal solution. The multistar constraints were not as effective as the generalized subtour elimination constraints but their inclusion in the linear model also produced a reasonable contribution for obtaining the optimal solution.

One consequence of the main result is that as the formulation FF only uses O(n 2) variables and O(n 2) constraints, its linear programming relaxation may be used as a starting model in a branCh-and-cut algorithm for the VRP. This simply means that the starting linear solution satisfies every multistar constraint. Let us point out that only a small number of these inequalities should be necessary in the process of obtaining the optimal VRP solution. However, by using such compact formulation, We save the effort of looking for such inequalities in the process of obtaining the optimal solution. Additionally, the lower bound given by such initial formulation may be much better than the bound given by using for instance, the assignment model defined by (1), (2a), (2b) and (7'). The branch-and-cut algorithm can be completed by subsequently adding any other facet-inducing inequality violated by the current solution. Considering the results given in [2], a set of 'good' inequalities to be added to the FF L model corresponds precisely to the set of the generalized subtour elimination constraints.

Notice also that any directed formulation for the VRP can be used for the non-directed version of the problem by considering symmetric costs. Additionally, notice that any directed formulation for the VRP can be 'symmetrized' by using the following relation:

Z i j = X i j - - ] - X j i , i = 1 . . . . . n - l , j = i + l . . . . . n ,

where Zi~ = 1 if edge (i, j) is in the solution and Ziy = 0 otherwise. It can be easily seen that if the directed formulation satisfies the directed version of the multistar constraints then the 'symmetrized' version also satisfies the non-directed version of the multistar constraints. This may also justi~ the use of the FF formulation in a branch-and-cut algorithm, even when attempting to solve non-directed VRP instances.

The main disadvantage of using the FF formulation as an initial model in the branch-and-cut process is given by the 2n 2 constraints (4') which can lead to fairly large linear programming problems even for

620 L. Gouveia ~European Journal of Operational Research 85 (1995) 610-624

moderate values of n. To overcome this deficiency we follow an idea given in [20] for the CMSTP and obtain a reformulation for the VRP involving a fewer number of constraints. Such a reformulation is presented in the next section.

5. A more compact representation of the multistar constraints

All the information provided by the (Xiy , Yi~) variables can be duplicated by the set of binary variables Zi j q (i = 0 . . . . . n; j = 0 . . . . . n; q = 0 . . . . . Q - d i ) such that Zij q = 1 if a flow of value q goes through arc (i, j ) and Zi j q = 0 otherwise. This type of variables were introduced in [20] and the two following relations show how the X~j and the Y,.j variables used in formulation FF can be written in terms of the new variables.

Q-d~

S i r = E Zijq, i , j = 0 . . . . . n , (20) q = 0

O -d i Yii = ~., qZ i j q, i , j = O . . . . . n . (21)

q=O

If we replace in (1)-(3), (4') and (5) the X i j and Y,.j variables by the corresponding right hand-sides of (20) and (21), the following formulation is obtained:

(Formulation FQ)

min n Q-di

E E cijZijq ( 2 2 ) i = 0 j=O q=O

n Q-di s.t. Y'. ~_, Z i j q = 1 j = 1 . . . . . n , (23)

i = 0 q = l

n Q - 1

~ , ~., Z i j q = l i = l , . . . , n , (24) j=O q=O

n Q-di ~ Q - 1 ~_, ~_, qZ i j q - Y'. qZ i j q = l , j = l , . . . , n ,

i = 0 q = l i = 0 q = 0

ZijqE{0,1}, i = 0 . . . . . n, j = O . . . . . n, q - -O . . . . . Q - d i.

(25)

(26)

To simplify the indexing, we have not considered variables Zuq (i = 1 . . . . . n). The same happens with variables Zio q (i = 1 , . . . , n; q = 1 . . . . . Q ) and with variables Zi/o (i = 0 . . . . . n; j = 1 , . . . , n). T h e main point of replacing in (1)-(5) the X i j and Y/j variables by the corresponding right hand-sides of (20) and (21) is that the set of constraints corresponding to the 'copy' of constraints (4') in the new variables are always true and therefore they can be omitted from FQ. In fact, this is the reason why this transformation leads to an O(n) constraint formulation for the VRP. Notice that the capacity constraints are guaranteed in FQ by the range of variation of the index q.

One simple consequence of the transformation based on (20) and (21) is that there is a one to one correspondence between the set of feasible solutions of FF and the set of feasible solutions of FQ. It is also clear that for any feasible solution of FF there exists a feasible solution to FQ with the same cost,


and vice versa. This shows that the optimal solutions in both formulations must have the same objective value. These two arguments indicate that FQ is a valid formulation for the directed VRP problem.

Actually, by 'borrowing' a result from [20] we can even show that the linear programming relaxation of both formulations produce optimal solutions with the same cost. The proof of the following result is similar to the proof given in [20] and we shall omit it.

Result 3. v(FQ L) = V(FFL).

The above result shows that the multistar constraints have a more compact representation. The number of constraints involved in FQ, indicates that this formulation should be prefered tO FF if one wants to obtain lower bounds for the CMST by solving the associated linear programming ~elaxations. Additionally, our sugestion of using a polynomial representation for the multistar constraints as an initial model in a branch-and-cut method might become more attractive if we use the FQ formulation.

6. A numerical result

To exemplify, let us use the 47-customer symmetric instance presented in [2] with capacity Q = 15. The optimal solution for this instance is also given in [2] and its cost is equal to 12 641. As reported in [2], the problem was hard to solve and required the creation of a search tree with 65 nodes. About 270 constraints (including 133 generalized subtour elimination constraints and 40 multistar constraints) were added to the original model.

Our results were produced in a PC DX486/66MHz. We used the LPS-867 package for solving the linear programming models. The lower bound given by the assignment relaxation is equal to 8300 and 4 seconds of CPU were needed for obtaining it. 12 minutes and 30 seconds were needed for producing the optimal FF L solution with cost equal to 10548.8 which is within 17% of the optimum. By using the more compact FQ L model, we obtain the same bound after only 2 minutes of CPU time.

It should be pointed out that the reported bound is still far from the optimum. However, this does not invalidate the fact that the FQ L model might be used as a starting model in a branch-and-cut method. In fact, by showing that the optimal FQ L solution for a 47 customer instance can be obtained after about 2 minutes of CPU, we showed the feasibility of using such an initial formulation in a branch-and-cut method. Additionally, we saved the effort of looking afterwards for multistar constraints (in particular, the 40 multistars which were needed by Araque for obtaining the optimal solution).

More computational results are needed for testing the potential and effectiveness of using I a compact representation for the multistar constraints as an initial model in a branch-and-cut method for the VRP. Our preliminary computational results indicate, however, that such a possibility should be considered for the VRP. Considering the results presented in [20] for the CMSTP we also believe that the contribution of the multistar constraints in a lower bounding scheme for the VRP becomes stronger in highly capacitated cases, i.e., cases with small values of Q.

Finally, we should point out that as an alternative to linear programming based methods, an approximation of the optimal VRP solution can be obtained by Lagrangean relaxation based schemes combined with implicit generation schemes for dualizing some of the generalized subtour elimination constraints. This approach has already been used with some success for the CMSTP (see [20]).

7. Conclusions

In this paper, we examined and discussed a stronger version of the Gavish and Graves [14] one-commodity flow formulation for the VRP. We showed that the directed version of a set of facets for

622 L. Gouveia /European Journal of Operational Research 85 (1995) 610-624

the VRP, the multistar constraints, can be obtained by projecting into the Xiy space, the set of feasible solutions of the linear relaxation of the stronger one-commodity flow formulation. This formulation involves only O(n 2) variables and O(n 2) constraints. By using an alternative set of variables we even showed how to obtain a O(n) constraint representation of the multistar constraints. This simply means that such formulations (in particular, the O(n) constraint formulation) may be used as starting model in a branch-and-cut algorithm for the VRP. In this manner, the first linear solution satisfies every multistar constraint.

We should point out that the set of variables involved in the FQ formulation might be used to derive new valid inequalities and sharper formulations for the VRP. In our opinion, a good place to look for such inequalities and formulations are references [20] and [21]. The last reference gives a fairly extensive catalog of formulations for the Time-Dependent Travelling Salesman Problem. These formulations use a similar set of variables and can be adapted for the VRP by simply restricting the range of variation of the third index. We believe that such formulations should be investigated in the contest of the VRP problem.

Our result also suggests the following question: Is there any relationship between the existence of a polynomial representation of a set of facet-inducing inequalities and the existence of a polynomial algorithm for solving the corresponding constraint identification problem? In fact, Martin shows how to derive polynomial representations of sets of facet-inducing inequalities from the corresponding constraint identification algorithms [28]. In the context of this paper we are concerned with the reciprocal implication because the complexity of identifying the multistar constraints is still open. It is not clear yet if the existence of such a representation implies that the multistar constraints can be polynomial identified. However, it may suggest that such constraint identification problem is polynomially solvable and it may induce the polyedral combinatorics community to look for such an algorithm for the multistar constraints.

Based on the ideas discussed in the last paragraphs, we believe that searching for polynomial representations of sets of facet-inducing inequalities may yet prove to be a relevant feature when studying the polyhedral structure of a given combinatorial optimization problem. However, apart from the multistar constraints and a subset of the generalized subtour elimination constraints (i.e., the subset including every constraint with IS I_< Q) we do not know of any 'polynomial representation' result concerning any other non-trivial set of facets for the VRP.

Finally, we show that the concept of 'projection' can be used as tool for generating valid inequalities for a combinatorial problem. Consider the non unit-weight version of the VRP. Some generalizations of the subtour elimination constraints are presented in [10] and [23] for this general version of the problem. Generalizations for the non-unit case of comb inequalities and path inequalities are presented in [10]. However, we have not yet seen any inequality which may correspond to a non-unit weighted version of a multistar constraint. One way of deriving such an inequality consists of using a projection result similar to Result 1 but using instead, a non-unit weighted and lifted version of the Gavish and Graves formulation. In fact, a non-unit weighted formulation for the VRP was presented in the original paper by Gavish and Graves [14]. Such a formulation can be derived by generalizing the flow conservation constraints (3) in the following way:

n n

E Y / j - E Y i i = q i , J = l . . . . . n, (3') i f f i0 i f f i0

where qj is the weight associated with node j (j = 1 . . . . . n). In the non-unit weight case, constraints (4) can be lifted in the following way:

qjdjXi j < Yij < ( Q - q i d i ) X i j , i, j = O , 1 . . . . . n, (4")

where d o = 0 and d i = 1 (i = 1 . . . . . n).


We state without proof (the proof will be similar to the proof of Result 1) that the linear relaxation of formulation (1), (2a), (2b), (3'), (4") and (5) is equivalent to the linear relaxation of the formulation (1), (2a), (2b), (27) and (6), where (27) is given by

E ( X ( S ) ) + ( ~s E q jX i j+ E E q i X i j ) < _ s - ( ~ s q i ) / / a . (27) i j~SC\{O} i~SC\{O} j~S i

The above constraints may be considered as a generalization of the multistar constraints for the non-unit weight case. The facetal properties of these inequalities are not studied here. By presenting them we only want to illustrate another outcome of a projection result, namely the generation of valid inequalities for a combinatorial optimization problem.

Acknowledgements

The author wishes to thank the referees for several suggestions which have greatly improved the paper.

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A result on projection for the vehicle routing ptoblem

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