Inequalities Implied by a Time-Dependent Model for the Unit Demand Vehicle Routing Problem

Transcription

1 Inequalities Implied by a Time-Dependent Model for te Unit Demand Veicle Routing Problem Maria Teresa Godino, Luis Gouveia, Tomas L. Magnanti, Pierre Pesneau and José Pires CIO Working Paper 0/009

2 Inequalities Implied by a Time-Dependent Model for te Unit Demand Veicle Routing Problem Maria Teresa Godino a, Luis Gouveia a,c, Tomas L. Magnanti 3b,c, Pierre Pesneau 4, José Pires 5a Departamento de Matemática, Escola Superior de Tecnologia e Gestão Instituto Politécnico de Beja, Portugal mtgodino@estig.ipbeja.pt Faculdade de Ciências da Universidade de Lisboa, DEIO - CIO Bloco C/ - Campo Grande, CIDADE UNIVERSITARIA, Lisboa, Portugal legouveia@fc.ul.pt 3 Department of Electrical Engineering and Computer Science and Sloan Scool of Management MIT, Cambridge, MA USA, magnanti@mit.edu 4 Université de Bordeaux, Institut de Matématique de Bordeaux 35, Cours de la Libération, Talence Cedex, FRANCE Pierre.Pesneau@mat.u-bordeaux.fr 5 ISCAL CIO Av. Miguel Bombarda, 0, Lisboa, Portugal jopires@sapo.pt May 009 Abstract We study, in te context of te Unit-Demand Veicle Routing Problem, te relationsip between te linear programming relaxation of a single-commodity flow model and te linear programming relaxation of a pure time-dependent formulation tat is closely related to a formulation of te travelling salesman problem. We sow tat te time-dependent formulation implies a new large class of upper bounding and lower bounding flow constraints tat are not implied by te linear programming of te single commodity flow model. Using te timedependent formulation we also sow ow to generate two new sets of inequalities in te space of te design variables: one set contains inequalities tat are related to te well-known multistar constraints wile te oter contains inequalities tat are related to double jump inequalities. Computational results sow tat te time dependent formulation can be used to easily solve instances wit up to 80 nodes, wen te veicle capacity is reasonably small. Keywords: Veicle Routing Problem, Time-dependent Formulations. Introduction Te unit-demand Capacitated Veicle Routing Problem (CVRP) is defined on a given directed grap G = (V,A) wit a node set V = {,,n} and an arc set A wit an integer weigt (cost) c a associated wit eac arc a of A, as well as a given natural number Q. Te problem seeks a minimum cost set of routes originating and terminating at te depot (we assume tat node is te depot) wit eac node in V\{} a Supported by FCT project POCTI-ISFL--5 b Supported in part by te Singapore-MIT Alliance (SMA) c Supported in part by te MIT-Portugal Program

3 visited exactly once and eac route containing at most Q nodes (plus te depot). Te CVRP is closely related to delivery type problems and appears in a large number of practical situations concerning te distribution of commodities. Te book by Tot and Vigo (00) provides surveys on te problem, including variants and solution tecniques. Papers by Araque et al. (994), Lysgaard, Letcford and Eglese (004) and Fukasawa et al. (006) discuss te most successful algoritms for solving tis problem as well as general demand cases. Te Araque et al metod is a cutting plane metod tat views te CVRP as an equivalent pat partitioning problem (PPP) and uses routines for separating several facet defining inequalities of te PPP polytope, suc as generalied subtour elimination constraints, and large, intermediate and small multistars tat were identified and introduced in te paper by Araque, Hall and Magnanti (990). Te Lysgaard, Letcford and Eglese metod is a cutting plane metod using rater efficient routines for separating well-known inequalities for te CVRP suc as generalied subtour elimination constraints, multistars, knapsack multistars, framed capacity inequalities, combs inequalities, and ypotours inequalities. Te metod described in Fukusawa et al. goes a step furter by efficiently combining cutting planes wit column generation. Several autors ave studied te polyedral structure of te CVRP, among tem, Laporte and Nobert (984), Campos, Corberan and Mota (99), Araque, Hall and Magnanti (990), Araque (990), Augerat (995) and Letcford, Eglese and Lysgaard (00). Te paper by Letcford and Salaar-Gonale (006) provides a recent comparison of te linear programming relaxation of several formulations presented in te literature and te recent paper by Godino, Gouveia and Magnanti (007) presents and compares te linear programming relaxation of several so-called multicommodity flow time-dependent formulations. In tis paper we study te relationsip between te linear programming relaxation of a pure timedependent formulation (tat can be seen as modified version of te well-known Picard and Queyranne (978) formulation for te TSP) and te linear programming relaxation of a well known single-commodity flow model due to Gavis and Graves (978). In particular, we sow tat te time-dependent formulation implies a new large class of upper bounding and lower bounding flow constraints tat are not implied by te linear programming of te single commodity flow model. By using some well-known tecniques for generating inequalities in te space of te design variables from te linear programming feasible set of single commodity flow models (see, for instance, Gouveia (995) and, Letcford and Salaar-Gonale (006)) we sow ow to generate new inequalities tat are related to te well-known multistar constraints (see Araque, Hall and Magnanti (990)). We also sow ow to generate a conditional version of te double jump inequalities proposed in Godino, Gouveia and Magnanti (008). Finally, we present some computational results sowing tat an enanced version of te time dependent formulation can be used to easily solve instances wit up to 80 nodes, wen Q is reasonably small. In Section we review a well-known single commodity flow formulation for te CVRP and in Section 3 we present te modified Picard and Queyranne formulation togeter wit a set of valid inequalities in te space of te time-dependent variables tat improve te linear programming relaxation of te modified

4 formulation. In Section 4 we state a result relating te linear programming relaxations of tese two formulations. Section 5 describes some new upper bound inequalities and new lower bound flow inequalities in te space of te design and flow variables and Section 6 describes new conditional doublejump inequalities and multistar-like inequalities tat are defined in te space of te design variables. Section 7 presents computational results comparing te linear programming bounds of te formulations discussed in te paper.. A Generic Formulation and te Single-Commodity Formulation for te CVRP Trougout tis paper, for any formulation P, F(P) denotes its set of feasible solutions, v(p) te value of an optimal solution, and P L its linear programming relaxation. If g is any quantity defined on te arcs (i,j), we let g( AB, ) denote. Wen A = B, we will simply use te notation ( ) i A g j B g A and wen A (or B) is a single node set, for instance A = {i}, we will use g(, ib. ) In particular, we will use tis notation wen referring to binary arc variables x associated wit arc inclusion. Consider te following generic formulation for te CVRP: minimie c x (, i j) A subject to x Assign ( i, j) A {( i, j) : x = does not contain routes wit more tan Q nodes} wit Assign denoting te feasible set of te well-known assignment relaxation arising in formulations for te problem: xv (, j) = j V \ {} xiv (, ) = i V \ {} x {0,} ( i, j) A. Tere are several ways to model te implicit route constraints. We obtain probably te most wellknown formulation for te CVRP by modelling tem as te standard generalied subtour elimination constraints x( S) S S / Q for all S V\{} (see, for instance, Araque, Hall and Magnanti (990)) tat limit te sie of eac route by guaranteeing a sufficient number of arcs incoming into eac subset of nodes and so ensuring tat te nodes in te subset can be split into feasible routes. Letcford and Salaar- Gonale (006) summarie and discuss several variants of tese inequalities from te literature. An alternative modelling approac is to use extra variables to express te implicit constraints (see, for instance, Letcford and Salaar-Gonale (006) and Godino, Gouveia and Magnanti (007)). Probably, te most well-known suc formulation for te CVRP is te following single commodity flow formulation

5 (see Gavis and Graves (978)) tat uses additional flow variables (i,j) (assuming tat te depot, node, sends one unit of flow to every oter node): f indicating te amount of flow on arc minimie (, i j) A f( V, j) f( j, V \{}) = j V \{} f(, V \{} = n f ( Q ) x ( i, j) A, i, j c x subject to x Assign f \{} j Qx j j V f x ( i, j) A, j f 0 ( i, j) A. We assume te flow on eac arc entering or leaving te depot node is equal to ero (tat is f j = 0 for all j V\{}). Te linking constraints fj Qxj guarantee tat te flow on eac arc leaving te root does not exceed Q. Tis restriction, togeter wit te flow conservation constraints and te remaining linking constraints, guarantees tat eac route cannot contain more tan Q client nodes. Gouveia (995) as sown tat te projection of te feasible set of te linear programming relaxation of te single commodity formulation is completely described by te equalities in Assign, plus nonnegativity constraints imposed upon te x variables and te so-called multistar constraints (see Section 6) tat are known to be facet defining for te CVRP polytope (see Araque, Hall and Magnanti (990)). Araque, Hall and Magnanti (990) and Letcford (00) ave also proposed variations of tese constraints. Letcford and Salaar- Gonale (005) discuss similar classes of inequalities tat result from a similar projection result for several oter variants of te CVRP. In Section 6 we describe variations of tese constraints wic, as far as we know, appear to be new. 3. Te Modified Picard and Queyranne Formulation Te well-known Picard and Queyranne formulation for te TSP (see Picard and Queyranne (978)) is easily modified for te CVRP (as far as we know, no one as previously given explicit reference to tis formulation for te CVRP). As in te Picard and Queyranne formulation for te TSP, we use an expanded layered grap. Two copies of node serve as te source and te destination nodes. A node j (=,,Q), represents a copy of node j (j =,,n) in layer. Te network contains: i) arcs from te source version of node to any node in te levels to Q, ii) arcs from nodes in level to nodes in level + ( =,,Q-), but we do not allow arcs linking copies of te same original node and, iii) arcs from nodes in level Q to te destination version of node. We will also say tat an arc directed into a node at level, as position ). 3

6 As an illustration, Figure 3. sows te layered grap representation of te feasible CVRP solution depicted also in Figure 3., an instance wit n=7 and Q=3 (labels associated wit te arcs correspond to te value of te flow in te arc, tat is te number of nodes still to be visited in te route) Layers: Layer Layer Layer 3 Figure 3. A feasible solution for a CVRP instance wit n = 7 and Q = 3 and te corresponding layered grap. Te main difference between tis network and te layered grap construct for te TSP is tat now we i) allow arcs leaving te source node to nodes in layers =,,Q (to allow for veicle routes wit fewer tan Q nodes), and ii) allow more tan one pat linking te source and destination versions of node (to represent several veicle routes). We obtain te Modified Picard and Queyranne formulation, MPQ, by replacing te implicit part of te generic formulation wit te following system: 4

7 = j =,..., n j ji i V\{} i V i V = j =,..., n and =,..., Q + ji i V\{} Q = j =,..., n Q+ j x = j =,..., n j j =,..., Q x = =,.., Q Q+ j j { } i, j =,..., n x = j =,..., n 0, (, j) A and =,..., Q or ( i, j) A, i, j and =,..., Q or ( i,) Aand= Q+. In tis model, te variable indicates tat arc (i,j) is in a pat tat contains Q-+ nodes after te arc (including node j, but not node ). Te equality constraints imposed upon te variables simply define a network flow system in tis layered grap wose solution (in integer variables) are pats (corresponding to routes in te original grap) from te source to te destination versions of node. Tese constraints permit eac pat to visit several copies of te same original node. However, in te overall problem, te constraints linking te wit te x variables and te assignment constraints in te x variables rule out tat situation. Note tat te equality constraints relating te x and formulation wit only te variables. variables permit us to rewrite te MPQ Te existence of tese two models, suggest tree questions: Q. Wat is te relationsip between te linear programming relaxation of te two formulations, SCF and MPQ? In Section 4 we will sow tat te linear programming relaxation of MPQ is at least as good as te linear programming relaxation of SCF. Tis result suggests two oter questions: Q. Wat are te improvements, if any, of te linear programming bounds of MPQ over te linear programming bounds of SCF? Q3. Wat inequalities are implied by te linear programming relaxation of MPQ tat are not redundant in te linear programming relaxation of SCF? Section 9 gives some computational results sowing wat we can gain by using te MPQ model instead of te SCF model wen solving CVRP instances and Sections 5 and 6 give a partial answer to question Q3. 5

8 Before closing tis section we introduce one set of inequalities in te space of te variables tat considerably improves te linear programming bound of te MPQ formulation. Te new inequalities state tat if arc (j,i) is in position +, ten some arc oter tan (i,j) is in position + kj ji for all i, j =,..., n and =,..., Q. (3.) k V; k i As far as we know, tese inequalities were first proposed in Gouveia (999) for tree problems and were later on used as valid inequalities to enance op-indexed formulations for te capacitated minimum spanning tree problem (see Gouveia and Martins (999,000). Recently, Cordeau, Costa and Laporte (009) ave used use a formulation involving inequalities (3.) for a variation of te Steiner Tree Problem wit revenues, budget and op constraints. Note tat by adding to eac side of tis inequality, by using te constraints from MPQ and canceling equal terms, we can rewrite tese constraints in symmetrical form (as compared to (3.)) as follows: + jk for all i, j =,..., n and =,..., Q. k V\{}; k i We let MPQ* denote te MPQ model augmented wit inequalities (3.) and we ave Proposition 3.: v(mpq* L ) v(mpq L ). As noted before, it is quite easy to find instances for wic te previous inequality is strict. Clearly, questions similar to questions Q and Q3 also apply for te MPQ* model in relation wit te models MPQ and SCF. 4. Relating te linear programming relaxations of MPQ and SCF For te next result, we need to add to MPQ te following definitional constraints tat link te variables wit te flow variables f of te SCF model. Tis addition does not alter its linear programming bound (for simplicity, we will use te same designation MPQ to refer to tis augmented model) f = ( Q + ) j =,..., n (4.) j j =,..., Q f = ( Q + ) ( i, j) A; i, j (4.) =,.., Q f = 0 j =,..., n. (4.3) Q+ j j 6

9 Te following result and proof are a simple adaptation of a similar result and proof given in Gouveia and Voss (995) for te Traveling Salesman Problem (for simplicity we skip te proof and refer te reader to Godino et al. (007)). Proposition 4.: Te projection of F(MPQ L ) into te space of te variables x and contained in F(SCF L ). f is Corollary: v(mpq L ) v(scf L ). 5. Some Inequalities Implied by MPQ (and MPQ*) in te x and f Space As noted in Section 4 te linear programming relaxation of te MPQ formulation is at least as strong as te linear programming relaxation of te SCF formulation. Some computational results in Section 9 sow tat, in general, te linear programming relaxation of MPQ strictly dominates te linear programming relaxation of SCF (see also Figure 5. to follow). In tis section we describe several inequalities in te space of te variables x and f tat are implied by te linear programming relaxation of MPQ and tat are not implied by te linear programming relaxation of te SCF formulation. As motivating one suc set of constraints, we note tat we could ave defined a valid version of te SCF model wit a weaker linear programming relaxation by using te weaker linking constraints: f Qx ( i, j) A, f 0 ( i, j) A. Te stronger set included earlier in te SCF formulation models te fact tat te flow in any internal arc wit x = cannot exceed Q- (since te first node in any route absorbs one unit of flow) and cannot be less tan (since only te arc returning to te depot would ave flow equal to ero). We can generalie tis concept by introducing a variation of te stronger constraints tat reflects te maximum or minimum flow in arcs in positions two away (after or before) node. Te validity of te following constraints is easy to establis: f ( Q ) x + x ( i, j) A, i, j i f x x ( i, j) A, j. j (5.a/b) Note tat tese constraints do not imply te previous set. However, we can add tem to te single commodity flow model to tigten te linear programming relaxation. Figure 5. depicts a feasible solution for te linear programming relaxation of SCF for an instance wit n = 6 and Q = 3. On te left, te labels associated wit te arcs correspond to values (tose greater tan ero) of te x variables and on te rigt 7

10 te labels associated wit te arcs correspond to values (again tose greater tan ero) of te flow variables. 5/6 5/6 5/3 /6 /6 /6 / /6 / /6 /6 5/ / / / 3/ / Figure 5. A feasible solution for te SCF L model. It is easy to see tat te solution sown in Figure 5. violates te constraint f6 x6 x6. Note tat increasing te flow in tis arc would violate te upper bound constraint f6 x6 + x. Tere are two ways we would like to generalie te constraints (5.a/b): (i) bound te flow in arcs tat are more tan arcs away from te depot, and (ii) consider constraints for arc sets instead of a single arc (i,j). 5. Generaliing Inequalities (5.a/b) for Arcs Tat Are More Tan Arcs Away From te Depot In any feasible solution to te problem, we will say tat an arc is at least k arcs away from te depot if it is te (k+) st or later arc on a pat (route) from te depot. Te constraints (5.a/b) bound te flow on arcs tat are at least two arcs away from te depot. To obtain analogous bounds for arcs tat are at least k arcs away from te depot, for simplicity, we consider only a generaliation of te upper bound constraint (5.a). We begin by writing a constraint of te form f ( Q k) x + Exp( x, k) ( i, j) A, i, j, k =,..., Q 3 8

11 wit Exp( x, k ) denoting a linear term in te x variables tat depends on k and sould equal p (p < k) if te pat from node to node i contains k-p+ arcs. To motivate tis expression, we consider a set of inequalities for k =, but involving quadratic terms. f ( Q 3) x + x + ( x x ) ( i, j) A, i, j. i k ki k V\{, i} To sow tat tis inequality is valid, we note te following: i) if te pat from node to node i contains tree or more arcs, ten te flow in arc (i,j) is at most Q-3, te two terms on te rigt are ero, and te inequality is valid; ii) if arc (,i) is in te solution, ten te middle term is equal to, te rigt most term is equal to ero and again, te inequality is valid; iii) if te pat from node to node i contains two arcs, ten one of te terms x k x ki equals, te middle term equals ero, and since te flow on arc (i,j) is at most Q-, te inequality is valid. Using te fact tat x k xki x k and xk xki xki for eac k in V\{,i}, we can generate te following set of linear inequalities from te previous nonlinear one f ( Q 3) x + x + x(, P) + x( V \ ( P {}), i) ( i, j) A, i, j, P V \{, i}. (5.a) i Notice tat tis inequality captures condition iii) from te previous argument since for any node set P, te node in te middle of te -pat joining node to node i eiter is in te set P (and ten, te second term from te rigt is equal to ) or is not in P (and ten, te first term from te rigt equals ). It is easy to see ow to strengten te inequalities (5.a) by excluding node j from P. Examples can be found sowing tat tese inequalities are not implied by te linear programming relaxation of MPQ. Later, we sow tat te linear programming relaxation of MPQ* does imply tese inequalities. Te previous inequalities suggest te following more general class of inequalities tat are based on te concept of jump-sets of arcs (see Dal (999) and Godino, Gouveia and Magnanti (007)). Let S 0, S,...,S k+ be node-disjoint nonempty sets defining a partition of te node set V wit S 0 = {} and S k+ = {i}. Consider te inequalities k- k+ f ( Q k) x + ( q p ) x( S, S ) ( i, j) A, k =,..., Q- (5.3a) p q p= 0 q= p+ and ( S,..., S ) of V wit S = {} and S = { i}. 0 k+ 0 k+ Note tat wen k = and, we obtain te inequalities (5.a) and (5.a) described previously. Note tat if a variable corresponds to an arc jumping over t intermediate sets S j (i.e., (q - p ) = t), ten its coefficient equals t. It is not difficult to ceck tat te term p= 0,..., k- q= p+,..., k+ ( q p ) x( S, S ) p q 9

12 appearing in te inequalities satisfies te conditions previously given for te generic term Exp( x, k ). In te next section, we sow ow to generalie inequalities (5.3a) for arc sets and we will also discuss similar lower bounding inequalities 5. Generaliing constraints (5.3a) for Arc Sets In tis subsection we will analye constraints of te form f( S', S) ( Q k) x( S', S) +?? S, S' V \{}, S S' = φ (5.4a) tat are stronger tan te constraints obtained by adding (5.3a) for all te arcs in a cut [S,S]. For simplicity, we start wit te simplest case, te following generaliation of inequality (5.a). f( S', S) ( Q ) x( S', S) + x(, S') S, S' V \{}, S S' = φ. (5.5a) Te validity of tese constraints is easy to establis. Te key item in constraints (5.5a) is te coefficient of te rigtand term for every pair of sets S and S. Tese constraints are stronger tan te ones obtained by adding S * S corresponding constraints (5.a) and tis fact will prove to be crucial wen, in te next subsection, we use projection tecniques to generate new inequalities in te space of te variables x. It is not difficult to see ow to generalie any of te constraints (5.3a) in a similar way to k- k+ f( S', S) ( Q k) x( S', S) + ( q p ) x( Sp, Sq) S, S' V \{}, S S' = φ, (5.6a) p= 0 q= p+ k =,..., Q and partitions ( S,..., S ) of V wit S = {} and S = S '. 0 k+ 0 k+ Again, we note tat te coefficients of te variables witin te rigtand side summation are independent of S and S. Tus, for a given k, S and S, te generalied inequality is stronger tan te constraint obtained by adding S * S single node set constraints (5.3a). In a similar way we write te following set of lower bounding inequalities: k k+ f( S, S') ( k+ ) x( S, S') ( q p ) x( S, S ) S, S' V \{}, S S' = φ, p= 0 q= p+ p q k =,..., Q and partitions ( S,..., S ) of V wit S = S ' and S = {}. 0 k+ 0 k+ (5.6b) Te following result (wit a proof given in te Appendix) sows tat a special case of (5.6a/b) is implied by te linear programming relaxation of MPQ. It is still open weter all inequalities (5.6a/b) are implied by te linear programming relaxation of MPQ. 0

13 Proposition 5. Te inequalities (5.6a) (resp. (5.6b)) are implied by te linear programming relaxation of MPQ for all values Q if one of te following conditions are satisfied: i) k = ii) k =,,Q- and S S S (resp. S S k- S k ). To conclude tis section, we sow tat if we make some assumptions on te partitions involved in te different inequalities, we can obtain inequalities tat are stronger versions of inequalities tat result from adding eiter several inequalities (5.6a) or (5.6b). Again, tese inequalities migt prove to be relevant wen using projection tecniques suc as te ones described in Section 6 for obtaining inequalities defined in te space of te x variables tat are implied by te linear programming relaxation of MPQ. To motivate tis set of constraints consider, te following tree constraints (5.6a) for k = 4, 3 and, respectively, and for a specific family of pairwise disjoint node sets S, S, S3, S4 and S: 4 f ( S4, S) ( Q 4) x( S4, S) + ( q p ) x( A, A ) p= 0q= p+ 3 f ( S3, S) ( Q 3) x( S3, S) + ( q p ) x( B, B ) p= 0q= p+ f( S, S) ( Q ) x( S, S) + x(, S). p p q q In te first constraint, te node sets in te partition are A 0 = {}, A = S S, A = S, A 3 = S3 and A 4 = S4. In te second constraint, te node sets in te partition are B 0 = {}, B = S S, B = S and B 3 = S3. Note tat te sets S, S, S3 and S4 appear in increasing order in tese tree partitions. In suc a case, it is possible to derive te inequality f ( S4, S) + f( S3, S) + f( S, S) ( Q 4) x( S4, S) + ( Q 3) x( S3, S) + ( Q ) x( S, S) 4 + ( q p ) x( A, A ). p= 0q= p+ p q Tis inequality is stronger tan adding te previous tree inequalities since it does not include te rigtmost terms of te second and tird inequalities. Tis inequality is easily generalied yielding te following inequalities: k+ k+ k- k+ f( S, S) ( Q u) x( S, S) + ( q p ) x( A, A ) k =,..., Q u u p q u= u= p= 0 q= p+ and partitions ( S,..., S, S) of V \{} suc tat 0 k + A = {}, A = S S and A = S ( u =,..., k+ ). u u (5.7a) In a similar manner we can write lower bounding inequalities as follows:

14 k+ k+ k- k+ f( S, S) u x( S, S) ( q p ) x( A, A ) k =,..., Q u u p q u= u= p= 0 q= p+ and partitions ( S,..., S, S) of V suc tat k + A = S, A = S ( u =,..., k ), A = S S, and A = {}. 0 k+ u k u k k+ (5.7b) Te following result (wit a proof given in te Appendix) sows tat tese inequalities are implied by te linear programming relaxation of MPQ. Proposition 5. Te inequalities (5.7a/b) are implied by te linear programming relaxation of MPQ. 5.3 Some Inequalities Implied by MPQ* in te x and f Space Some (and peraps) all te inequalities derived in te previous subsections are implied by te linear programming relaxation of te stronger model MPQ. Finding inequalities tat in te space of te variables x and f tat are implied by te linear programming relaxation of MPQ* but are not implied by te linear programming relaxation of MPQ is also of interest but does not seem to be an easy task. Here we produce one suc set of inequalities. In Subsection 5. we ave mentioned te following sligtly stronger form of inequalities (5.a): f ( Q 3) x + x + x(, P) + x( V \ ( P {, j}), i) ( i, j) A, i, j, P V \{, i, j}. (5.8a) i As noted before, tese inequalities differ from (5.a) by considering more sligtly restricted index sets in te last term and are not implied by te linear programming relaxation of MPQ. We can also write a similar and sligtly stronger form of te corresponding lower bounding inequalities (5.b) as follows: f 3 x x x( P,) x( j, V \( P {, i})) ( i, j) A, i, j, P V \{, i, j}. (5.8b) j Proposition 5.3: Te inequalities (5.8a/b) are implied by te linear programming relaxation of te MPQ* model. It does not seem easy to generalie (5.8a) in te same way tat we generalied constraints (5.a) for sets of nodes (see (5.6a) for k = ). It is also not clear ow to use te constraints (3.) to obtain stronger versions of te upper and lower bounding constraints for arcs tat are more tan 3 arcs away from te depot. 6. Some Inequalities implied by MPQ in te x space In tis section we describe several inequalities in te space of te variables x tat are implied by te linear programming relaxation of te MPQ model. Altoug à priori we do not know te structure of suc

15 inequalities, fortunately, we can use te generalied flow bounding constraints of te single commodity flow model derived in te previous section to suggest tese inequalities. Tese new inequalities will be obtained eiter in a trivial way (see Subsection 6.) or by using projection tecniques similar to te ones described in Gouveia (995) and Letcford and Salaar-Gonale (006) wic will be described in Subsection 6.. Note tat te way tese inequalities are obtained sows tat tey are also implied by te linear programming relaxation of te SCF model augmented wit (5.6a/b) and/or (5.7a/b) in te space of te x variables. We will also sow tat wit exception to a few cases, we can obtain stronger inequalities by finding direct derivations from MPQ. We note, owever, tat te new flow bounding inequalities togeter wit te known projection tecniques ave suggested te nature of some of te inequalities implied by te linear programming relaxation of MPQ. 6. Conditional Double-Jump Inequalities Te first set of inequalities is quite simple. Adding te upper bound constraint (5.6a) and te lower bound constraint (5.6b) for k = and any arc cut set [S,S] and canceling equal terms, we obtain te constraints 0 ( Q 4) x( S', S) + x(, S')+ x( S,) S, S' V \{}, S S' = φ (6.) wic apparently are uninteresting since for Q > 3 tey are redundant. However, wen Q = 3 we obtain te intuitive inequality x( S', S) x(, S') + x( S,). Note tat if we set S = {} and S = {6}, te solution sown in Figure 5. violates te resulting inequality (6.). We can obtain more complicated inequalities by combining oter general lower and upper bounding inequalities from te entire class (5.6a/b). However, we believe tat, in general, tose inequalities are easily dominated by oter inequalities implied by te linear programming relaxation of MPQ. As an example, if we combine (5.6a) for k = wit (5.6b) for k = to eliminate te term f ( S', S ), we obtain, after rearranging, 0 ( Q 5) x( S', S) + x(, S')+ x(, P) + x( V \ ( P {}), S')+ x( S,) SS, ' V\{}, S S' = {}, P V\{ S'}. (6.) Wen Q = 4, te inequality (6.) becomes x( S', S) x(, S')+ x(, P) + x( V \( P {}), S')+ x( S,) SS, ' V\{}, S S' = {}, P V\{ S'}. (6.3) However, it is easy to see ow to strengten tis inequality by replacing te coefficient of of te term x(, S ') wit a. Tese resulting inequalities belong to a more general class of inequalities tat are implied by te linear programming of MPQ but, besides a few special cases (for example, te inequalities 3

16 x( S', S) x(, S') + x( S,) for Q = 3), do not appear to be implied by te linear programming relaxation of SCF augmented by inequalities (5.6a/b) and/or (5.7a/b). Tese inequalities are similar to te double-jump inequalities introduced in Godino, Gouveia and Magnanti (008). To motivate te new inequalities, assume tat an arc (i,j) (wit i,j ) is in te solution in one of te routes. Ten, for any position p ( < p < Q), eiter te pat from node to i as lengt less or equal tan p-, or te pat from node j to node as lengt less or equal tan Q-p. Tis observation suggests te following conditional version of a double jump inequality. Let S 0, S,...,S p define a partition of te node set V wit S 0 = {}, S p = {i}. Let [S j, S j+ ] = {(u,v) A: u S i, v S j } and define J = J(S 0, S,..., S p ) = [i+<j] [S i, S j ]. Let ' S 0, and S = {} and define J = J( S, ' Q-p+ ' 0 ' S,..., ' S,..., S ' Q-p+ define anoter partition of te node set V wit ' S 0 = {j} ' S Q-p+ ) = [i+<j] [S i, S j ]. For a given arc (i,j) and position p ( < p < Q), we consider a conditional double jump set of arcs defined as CDJ = J(S 0, S,..., S p ) ' ' ' J( S,S, K,S 0 Q-p+ ). We refer to suc a set CDJ as a ((i,j),q,p)-conditional double jump and we let CDJ ((i,j),q,p) denote te set of all ((i,j),q,p)-conditional double jumps. To explain te constraints to be presented next, assume tat arc (i,j) is in te solution and consider a fixed position p. Ten, any veicle pat from te depot to node i tat contains an arc from every set [S j, S j+ ] taken from J(S 0, S,..., S p ) must contain at least p arcs and any veicle pat from node j to te depot tat contains an arc from every set [S j, ' ' ' S j+ ] taken from J( S,S, K,S 0 Q-p+ ) takes at least Q-p+ arcs. Tese two pats togeter wit te arc (i,j) cannot constitute a feasible route. Tus, eiter te first pat contains at most p- arcs (and at least one arc from a jump set [S i, S j ] wit i+ < j taken from J(S 0, S,..., S p ) ) or te second pat contains at most Q-p arcs (and at least one arc from a jump set [S i, S j ] wit i+ < j taken from J(S 0, S,..., S Q-p+ )). Tese arguments sow tat te following inequalities are valid, x( CDJ ) x for all (i,j) A, i,j and CDJ CDJ ((i,j),q,p). (6.4) Notice tat wen Q = 3 and S = S = V\{,j}, tis inequality becomes te constraint x x i + xj and wen Q = 4 and op = 3 we obtain te stronger version of te inequality (6.3). A simple generaliation of te previous argument sows tat constraints (6.4) can also be generalied for arc sets. To define tese more general constraints we need to generalie a little bit, te concept of conditional double jump. For a given arc set (A,B) wit A B = φ and A,B V\{}, and position ( < < Q), we consider, now, a ' ' ' conditional double jump set of arcs defined as CDJ = J(S 0, S,..., S p ) J( S,S, K,S tis time, setting S p = A and 0 Q-p+ ) as before but, ' S 0 = B. We refer to suc a CDJ set as a ((A,B),Q,)-conditional double jump and we let CDJ ((A,B),Q,p) denote te set of all ((A,B),Q,p)-conditional double jump. Te inequalities are: 4

17 x( CDJ ) x( A, B) for all A,B V\{}, A B =φ and CDJ CDJ ((A,B),Q,p). (6.4 ) Note tat constraints (6.4 ) contain constraints (6.4) as special cases. Te next result, proved in te Appendix, states tat some of tese inequalities are implied by te linear programming relaxation of MPQ. It is still open weter all of te inequalities (6.4 ) are implied by te linear programming relaxation of MPQ. Proposition 6. Te inequalities (6.4 ) are implied by te linear programming relaxation of MPQ for all Q if one of te following conditions old: iii) p < 3 iv) p > 4, B S S and A S Q-p- S Q-p. We note from te conditions in te statement of te proposition tat te result is valid wen Q = 3 and 4 independent of te value of p. 6. Flow Based Projected Inequalities We can obtain a different class of inequalities by using projection tecniques applied to te SCF model augmented wit te flow bounding inequalities derived in Section 5. We start by reproducing te procedure used by Gouveia (995) (see also Letcford and Salaar-Gonale (006)) for generating te multistar inequalities. By adding te flow conservation constraints for nodes j in a set S V\{} (for simplicity, let S denote te set V\(S )) we obtain: f (, S) + f( S', S) = S + f( S, S') for all S V\{}. (6.5) Using te flow upper bound constraints f j Qx j and f ( Q ) x on te leftand side, te flow lower bound constraints f x on te rigtand side, and te assignment constraints, we obtain te multistar constraints Qx( S) + x( δ ( S)) S ( Q ). for all S V\{}. (6.6) In tis expression δ ( S) denotes te set of arcs wit only one endpoint in S, excluding arcs directed into or out of te depot node. As noted by Letcford and Salaar-Gonale (006), we can use te same procedure to obtain projected inequalities for several versions of te CVRP problem, including variations wit pick-up and delivery and variations wit distance constraints. In our context we could obtain many different projected inequalities for te CVRP by using te same procedure and bounding te flow sums in equation (6.5) wit alternate expressions (5.6a/b)) (note tat since te first summation of expression (6.5) is always bounded by te same expression (Q x(, S ) ), we will not 5

18 consider options for it in tis discussion). For te moment, we will consider only te following four upper bound and lower bound constraints for arcs not entering or leaving node. Constraints () and (4) correspond to te inequalities (5.5a/b) for k =. () f( A, B) ( Q ) x( A, B) A, B V \{,}, A B = φ () f( A, B) ( Q ) x( A, B) + x(, A) A, B V \{,}, A B = φ (3) f( A, B) x( A, B) A, B V \{,}, A B = φ (4) f( A, B) x( A, B) - x( B,) A, B V \{,}, A B = φ. However, a straigtforward and naïve use of tis procedure will lead to new, but uninteresting inequalities in te sense tat (see Godino (009)) all te inequalities obtained by using te same upper bound expression () and () for all te arcs in te cuts [S,S] and te same lower bound expression (3) and (4) for te arcs in te cut [S,S ] are implied by te multistar constraints (6.6). Fortunately, we can generalie tis procedure by partitioning te cut sets into two subsets and using different upper (or lower) bounds for eac partition. Suppose we partition te set S into two sets S and S and for te arcs in te cut [S,S] we bound te flow by using te original upper bound inequality () and for te arcs in te cut [S,S] we bound te flow using te inequalities (). We can perform a similar manipulation wit te summation on te rigtand side by partitioning te cut [S,S ] into two cuts [S,S] and [S,S]. Table 6. sows ow to obtain, after some manipulation using te assignment constraints, symmetric inequalities and nonsymmetric inequalities using tis procedure. Option for Bounding f ( S, S) + f( S, S) Option for Bounding f ( SS, ) + f( SS, ) Resulting Inequality () + () (3) + (4) Qx( S) + x( S) + x( δ ( S)) + x( δ ( S)) + x( SS, ) + xs (, S) S ( Q ) + S (6.7) () + () (4) + (3) Qx( S) + x( S) + x( S) + x( δ ( S)) + + x( S, S)) S ( Q ) + S + S (6.8) Table 6.. Inequalities generated from bounding aggregated flow constraints. We note tat wen eiter S or S is empty, we obtain dominated inequalities (since tese inequalities are obtained by using te same expression for all te arcs in te cuts [S,S] and [S,S ]). Note also tat wen S is empty, te inequalities (6.7) become te multistar constraints (6.6). 6

19 To see ow effective te constraints (6.7) and (6.8) migt be, we tested tem on small problem instances wit n = 5, 6 and 7, Q = 3, 4 and 5 and almost all coices of S, S and S (we conducted te test by solving te linear programming relaxation of te SCF model wit an objective function equal to te leftand side of te constraint (6.7) or (6.8)). For tese small networks inequalities (6.7) were implied by te multistar constraints (6.6) for Q = 3. In addition for Q = 3, we can formally sow tat te inequality (6.7) is redundant since it is implied by te multistar constraint (6.6) for te set (S S). However, for Q > 3, we can find solutions tat are feasible for te linear programming relaxation of SCF but are violated by (6.7). In contrast, for many of tese small instances, te asymmetric instances (6.8) were not dominated. As an example, te solution depicted in Figure 5. violates te asymmetric constraint (6.8) for S={,6}, S ={5} and S = {3,4}. Tese inequalities demonstrate someting we ave mentioned in te introduction of tis section, namely we can tigten some inequalities obtained by manipulating inequalities from SCF augmented wit te constraints (5.6a/b) or (5.7a/b) by using direct derivations from te model MPQ. In fact, for te specific case of Q = 3, we can tigten inequalities (6.7) and (6.8). As we ave noted before te inequality (6.7) is dominated by a multistar constraint and inequality (6.8) is dominated by te following inequality 3( x S) + x() S + x( S)+ (( x δ S)) + (, x S S) + x( S, S) S + S + S (6.9) wic is implied by te linear programming relaxation of MPQ. We refer to Godino (009) for a proof tat tis inequality is implied by te linear programming relaxation of MPQ. Altoug, tese inequalities are implied by te linear programming relaxation of MPQ, tey can be used to generate inequalities tat are not implied by te linear programming relaxation of MPQ. As an illustration, recall te following well-known fact. If, for a given set S, we divide te corresponding multistar constraint by Q and apply integer rounding, we obtain te generalied subtour elimination constraint (see Section ) for te set S. Applying te same procedure to constraints (6.7) and (6.8) for a given set S leads to te same inequality. However, if for a given set S, we divide inequality (6.8) by and use integer rounding we obtain te following inequality Q/ x( S) + x( S, S)) + x( δ ( S)) ( S ( Q ) + S + S ) /. (6.0) A few experiments wit small instances for Q = 3, 4 and 5 ave indicated tat te inequalities (6.0) are not implied by te linear programming relaxation of MPQ. In Figure 6. we indicate one suc solution for a situation wen Q = 3. It is easy to see tat te solution sown in te figure violates constraint (6.0) for S = {} 7, S = { 56, } and S { 34,, } subtour elimination constraints. =. Note also tat tis solution does not violate any of te generalied 7

20 7 7 7 / / / 6 6 /4 6 / 5 5 /4 5 /4 /4 / 4 3 / /4 4 3 / /4 3/ /4 / /4 /4 /4 3 6 /4 / / / /4 / 7 / 5 3/4 /4 4 3/4 Figure 6. A feasible solution for te MPQ L model tat violates inequality (6.0). Te topmost figure sows te values of te variables and te bottommost te values of te x variables. We conclude tis section by generaliing te previous procedure in te following ways (te reader is referred to Godino (009) for details): i) We can partition te cut [S,S] into two cuts [S,S] and [S,S] and te cut [S,S ] into two cuts [S,S3] and [S,S4] (S3 does not necessarily equal S or S). Te inequalities (6.7) and (6.8) are particular cases of tese more general inequalities, obtained wen S = S3 and S = S4 or S = S4 and S = S3. ii) We ave to tis point partitioned te set S in te expression (6.5). Instead we can partition te set S into two sets S and S. We note (but do not include te proof ere again, see Godino (009)) tat i) inequalities obtained by partitioning te cut [S,S] into two cuts [S,S] and [S,S] and partitioning te cut [S,S ] into two cuts [S3,S ] and [S4,S ] and ii) inequalities obtained by partitioning eiter te UB term or te LB term (but not bot), also lead only to inequalities tat are implied by te multistar constraints. 8

21 iii) To tis point, in our effort to identify inequalities implied by te model MPQ we ave partitioned te set S in te flow expression (6.5) into two sets and used te simple upper and lower bounds () - (4). To uncover oter, more complicated, inequalities we migt use partitions wit more sets and use te more complicated jump bounds (5.7a/b). Tere are many suc possibilities. Godino (009) describes inequalities tat arise wen we partition te set S in te cuts [S,S] and [S,S ] into tree subsets S, S and S3. Once we set te (arbitrary) order S, S, S3 for te upper bounding, we ave six different options for ordering te sets in te lower bounding. More general inequalities can be obtained by using te general upper bound and lower bound constraints (5.7a) (5.7b). Furtermore, an inequality similar to inequality (6.9) wit a lead coefficient of 4 applies wen Q=4. Oter related inequalities wit a lead coefficient of Q are also possible. 7. Computational Results In tis section we empirically evaluate te quality of te lower bounds given by te two time-dependent models presented in te previous sections. For comparing te models, we use data for complete graps wit 40 and 80 nodes plus te depot. We ave considered tree test sets. For te first two sets, te nodes ave been randomly selected in a square grid wit Euclidean arc lengts. Tese tests are divided according to te location of te depot. Te designation TEM refers to Euclidean cost instances wit te depot located in te middle of te grid and te designation TEC refers to Euclidean cost instances wit te depot located in te corner. We ave also considered two sets of random costs instances, one set wit symmetric edge costs (denoted by TRS) and anoter set wit asymmetric cost instances (denoted by TRA). Te costs of te edges of te TRS instances were generated from an uniform distribution in te interval [,00] wile te costs of te arcs of te TRA instances were generated from an uniform distribution in te same interval. Eac group includes 5 complete grap instances. Te models tat we are comparing are te models SCF, MPQ, MPQ* and SCF+ (wic is SCF augmented wit inequalities (5.a/b) of Section 5). Table 7. presents te average linear programming gaps (computed as [(Optimal Value v(p L ))/Optimal value]*00 for eac group. We solved te linear programming relaxations using te CPLEX 9.0 software package on a Pentium IV, 3.4GH computer wit Gb of RAM, and obtained te optimal solutions by running te model MPQ* wit te branc-and-cut algoritm of CPLEX. Table 7. presents te CPU times for solving te linear programming relaxations of eac model and for finding te integer solution. n Q SCF SCF+ MPQ MPQ* TEM , 3, 3,,09 5 8,59 7,66 4,94,95 3,0,0,0 0,5 5 5,4 3,6,39 0,94 9

22 TEC 3,96,96,96 0, ,40 3,79,87,0 3,48,48,48 0, ,90,84, 0,8 TRS 3,45,45,45, ,79 4,30 4,6 0,68 3,6,6,6, ,9,36,35 0,9 TRA 3 5,57 3,30 3,03, ,4 5,37,56,08 3 3,30,3,6, ,88 3,8,3,3 Table 7. Average gaps of te linear programming relaxation bounds of several models. N Q SCF SCF+ MPQ MPQ* TEM TEC TRS TRA ,7 4,7 0,05 0,4 6,86 5, 4,4 0,9,78 755,47 3,4 37,59 0,3,35 90,40 5,03 8,57,34 4, ,9 3,9 6,3 0,05 0,84 3,8 5,5 5,64 0,5,3 684,80 3 4,39 5,5 0,3,47 598,8 5,78 40,06,0 3, ,57 3,74 6,3 0,04 0,7 0,6 5,68 4,3 0,35,79 3, 3 5,0 38, 0,8,67 5, 5,54 4,94 4, 4,57 49,0 3,70 6,07 0,05 0,4 0,6 5,55 3,8 0,3,78 3,4 3 7,48 03,74 0,3,7 6, 5,94 70,0 5,05,3 34,8 Table 7. Average CPU times in seconds of te linear programming relaxation bounds of several models. Te rigtmost column specifies te CPU times for obtaining te optimal integer solutions wit model MPQ*. 0

23 Te single commodity models contain *n variables and te time-dependent models contains Q*n variables. Te MPQ model contains Q*n constraints and te MPQ* contains Q*(n + n ) constraints. Terefore, te largest model being solved (MPQ*) contains more tan 3,805 variables and 3,400 constraints. Te results indicate tat te linear programming relaxation of te model MPQ produces reasonable improvements over te linear programming of SCF for te Q = 5 instances. For Q = 3 te reported gaps for all te symmetric instances (instances TEM, TEC and TRS) are te same for te two models. However, we obtain reasonable improvements for te TRA instances. Te discrepancy between asymmetric and symmetric results reported for Q = 3 conform to our analysis (see Section 6) of te inequalities implied from te linear programming relaxation of MPQ. Te results also indicate tat for te instances wit Q = 5 (were te linear programming relaxation of MPQ improves upon te linear programming relaxation of SCF), te gap reduction from model SCF obtained by te linear programming relaxation of te model SCF+ is about 60 percent of te gap reduction obtained by te linear programming relaxation of te model MPQ (wit a range of 5 to 99 percent). For Q = 3 and te asymmetric instances, it appears tat te simple flow bounding inequalities (5.a/b) are very effectives since te gaps produced by SCF+ are quite close to te ones obtained by MPQ. Te computations also sow a difference between te gaps provided by te two op indexed models MPQ and MPQ* for te symmetric and asymmetric instances (recall tat MPQ* is obtained by adding inequalities (3.) to te MPQ model). For symmetric instances te addition of tese inequalities leads to a reduction of about 50 per cent from te linear programming relaxation gaps of te original MPQ model. Te model MPQ* seems to be a good option for solving small and medium sied symmetric instances (up to 8 nodes) wenever Q is reasonably small. In tese instances, te model is very attractive to use witin an ILP package because it is small and it provides reasonable lower bounding values. To obtain te optimal integer solutions, te average CPU times do not sow te ig variability of te individual problem instances as compared wit te variability from one group of problem instances to anoter. For instance, for te EM instances wit n = 8 and Q = 3, one instance is solved witin 00 seconds wile anoter is solved after 40,000 seconds. We migt note tat te computational results we ave obtained are not competitive wit oter, more intricate metods (as for instance, te metods described in Lysgaard, Letcford and Eglese (004) and Fukasawa et al. (006)). However, it is wort aving a model tat essentially anyone can implement and tat can solve instances witin certain parameters (i.e., small values of Q).. Peraps, one way of explaining wy adding constraints (3.) to MPQ is not tat effective for te asymmetric cases, is to see wy it is effective for te symmetric cases. Note tat for a symmetric instance, an attractive edge {i,j} (one wit a small cost) is very likely used in te two directions, allowing bot of

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