2. A Generic Formulation and the Single-Commodity Flow Formulation for the CVRP

Transcription

1 On Time-Dependent Models for Unit Demand Veicle Routing Problems Maria Teresa Godino, CIO-Dep. de Matemática, Instituto Politécnico de Beja, Luis Gouveia, CIO-DEIO, Faculdade de Ciências da Universidade de Lisboa, Tomas L. Magnanti, Dep. of Electrical Engineering and Computer Science and Sloan Scool of Management, MIT, Cambridge, Pierre Pesneau, Université de Bordeaux, Institut IMB, France, José Pires, CIO-ISCAL, Portugal, Keywords: Veicle Routing Problem, Time-dependent Formulations, Flow Based Formulations Introduction Te unit-demand Capacitated Veicle Routing Problem (CVRP) is defined on a given directed grap G = (V,A) wit node set V = {,,n}, 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\{} visited exactly once and eac route containing at most Q nodes (plus te depot). Te CVRP is closely related wit delivery type problems and appears in a large number of practical situations concerning te distribution of commodities. Te book by Tot and Vigo [2] provides surveys on te problem, including variants and solution tecniques. Papers by Lysgaard, Letcford and Eglese [0] and Fukasawa et al. [3] discuss te most successful algoritms for solving tis problem as well as general demand cases. Te paper by Letcford and Salazar- Gonzalez [9] 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 [5] presents and compares te linear programming relaxation of several socalled multicommodity flow time-dependent formulations. In tis paper we study te relationsip between te linear programming relaxation of a well-known single-commodity flow model due to Gavis and Graves [4] (presented in Section 2) and pure time-dependent formulation, presented in Section 3, tat is a modified version of te well-known Picard and Queyranne [] formulation for te TSP (see Section 3). In section 4 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. 2. A Generic Formulation and te Single-Commodity Flow Formulation for te CVRP Consider te following generic formulation for te CVRP:

2 minimize subject to ( i, j) A j V x = for all j V \{} x = for all i V \{} c x {( i, j) : x = does not contain routes wit more tan Q nodes} x {0,} for all ( i, j) A. Tere are several ways to model te implicit route constraints using inequalities involving only te x variables (see Letcford and Salazar-Gonzalez [9]). An alternative modelling approac is to use extra variables to express te implicit constraints. Probably, te most well-known suc formulation for te CVRP is te following single commodity flow formulation (SCF) due to Gavis and Graves [4] tat uses additional flow variables f indicating te amount of flow on arc (i,j) (assuming tat te depot, node, sends one unit of flow to every oter node): minimize ( i, j) A ji \{} f f = for all j V \{} j j V \{} = n f ( Q ) x for all ( i, j) A, i, j c x subject to x Assign f f j Q x j for all j V \{} f x for all ( i, j) A, j f 0 for all ( i, j) A. Te linking constraints f j Q x j 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. 3. Te Modified Picard and Queyranne Formulation Te well-known Picard and Queyranne formulation [] for te TSP can be easily modified for te CVRP. As in te original Picard and Queyranne formulation, we use an expanded layered grap. Two copies of node, te source and te destination, are on te leftmost position and rigtmost position. A node j (=,,Q-), indicates tat a copy of node j (j = 2,,n) is in layer. Te network contains: i) arcs from te source version of node to any node in te levels to Q (we will say tat an arc from te source node to any node in te level, as position ), 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.

3 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 = 2,,Q in order to allow veicle routes wit fewer tan Q nodes, and ii) allow more tan one pat linking te source and te destination (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 z = z for all j = 2,..., n 2 j ji \{} \{} z = z for all j = 2,..., n and = 2,..., Q Q Q+ j j j =,..., Q + ji z = z for all j = 2,..., n x = z for all j = = 2,.., Q Q+ j j { } 2,...., n x = z for all i, j = 2,..., n x = z for all j = 2,..., n z 0, for all (, j) A and =,..., Q or ( i, j) A, i, j and = or ( i,) A and = Q +. 2,..., Q In tis model, te variable z indicates tat arc (i,j) is in a pat tat contains Q-+ nodes after te arc (including node j). Te equality constraints imposed upon te z 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 version to te destination version 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 z 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 z variables permit us to rewrite te MPQ formulation wit only te z variables. Following Gouveia and Voβ [7] one can sow tat te linear programming relaxation of te MPQ formulation implies te linear programming relaxation of te SCF formulation. Some computational results (to be presented in te talk) taken from instances wit up to 80 nodes will sow tat, in general, MPQ L improves on SCF L. Tese results suggest tat it may be wort knowing wat are te inequalities implied by te linear programming relaxation of MPQ but are not redundant in te linear programming relaxation of SCF. Section 4 gives a partial answer to tis question. 4. Some Inequalities Implied by MPQ In tis section we describe some inequalities in te space of te variables x and f tat are implied by te linear programming relaxation of MPQ and tat are not dominated 2

4 by te linear programming relaxation of te SCF formulation. Consider te following constraints (wose validity is easy to establis) f ( Q 2) x + x for all ( i, j) A, i, j i f 2 x x for all ( i, j) A, j. j (4.a/b) We can add tem to te single commodity flow model to tigten te linear programming relaxation. Tere are two ways we would like to generalize te constraints (4.a/b): (i) bound te flow in arcs tat are more tan 2 arcs away from te depot, and (ii) consider constraints for arc sets instead of a single arc (i,j). Te constraints (4.a/b) bound te flow in arcs tat are at least two arcs away from te depot. To obtain analogous bounds for arcs tat are at least 3 arcs away from te depot, for simplicity, we consider only a generalization of te upper bound constraint, by writing it in te form f ( Q k) x + Exp( x, k) for all ( i, j) A, i, j, k =,..., Q 3 wit Exp(x,k) denotes 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. Suc constraints can be written by using te concept of jump-sets of arcs (see Dal [2] and Godino, Gouveia and Magnanti [5]). 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}. For any pair of node sets A and B let x( A, B) = x. Consider te inequalities i A, j B f ( Q k) x + ( q p ) x( S, S ) for all ( i, j) A, k =,..., Q - 2 p q p= 0,..., k- q= p+ 2,..., k+ and all partitions ( S,..., S ) of V wit S = {} and S = { i}. 0 k+ 0 k + (4.2a) Note tat wen k = we obtain te inequalities (4.a). Note, also, 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 ( q p ) x ( S, S ) 0,..., - 2,..., p= k q= p+ k + p q appearing in te two inequalities satisfies te conditions previously given for te generic term Exp(x,k). Next, we analyze constraints of te form f ( Q k) x +?? for S, S ' V \{}, S S ' = {}. (4.3a) tat are stronger tan te constraints obtained by adding (4.2a) for all te arcs in te cut [S,S]. For simplicity, we start wit te following generalization of (4.a). f ( Q 2) x + x i for S, S ' V \{}, S S ' = {}. (4.4a) i S ' 3

5 Te validity of tese constraints is easy to establis. Te key item in constraints (4.4a) is te coefficient on te rigtand term for every set S. Tat is, tese constraints are stronger tan te ones obtained by adding S * S corresponding constraints (4.a). It is not difficult to see ow to generalize te more general constraints (4.2a) in a similar way to p= 0,..., k- q= p+ 2,..., k+ f ( Q k) x + ( q p ) x( S p, Sq ) for all S, S ' V \{}, S S ' = {}, k =,..., Q 2 (4.5a) and all 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. Tus, for a given k and S, te generalized inequality is stronger tan te constraint obtained by adding S single node set constraints. In a similar way we produce te following set of lower bounding inequalities + p= 0,..., k- q= p+ 2,..., k+ f ( k ) x ( q p ) x( S p, Sq) for all S, S ' V \{}, S S ' = {}, k =,..., Q 2 (4.5b) and all partitions ( S0,..., Sk+ ) of V wit S0 = S and S k + = {}. Te following result (we omit te proof in tis Abstract) sows tat a special case of (4.5a/b) are implied by te linear programming relaxation of MPQ. It is still open weter all inequalities (4.5a/b) are implied by te linear programming relaxation of MPQ. Proposition 4. Te inequalities (4.5a/b) are implied by te linear programming relaxation of MPQ for all Q if one of te following olds: i) k = ii) k = 2,,Q-2 and S S S2 (for inequality 4.5a) or k = 2,,Q-2 and S Sk- Sk (for inequality 4.5b). We conclude tis section by pointing out tat inequalities in te space of te variables x tat are implied by te linear programming relaxation of te MPQ model can be obtained by combining te previous inequalities (and tus, tese inequalities are also implied by te linear programming relaxation of te SCF model augmented wit (4.5a/b) in te space of te x variables). To do tis we can follow te procedure used by Gouveia [6] (see also Letcford and Salazar-Gonzalez [9]) for generating te multistar inequalities (see Araque, Hall and Magnanti []). Details about new projected inequalities will be given at te presentation. 4

6 References [] Araque, J.R Hall, L. Magnanti, T.L. Capacitated trees, capacitated routing and associated polyedra, Discussion Paper 90-6, CORE, University of Louvin La Neuve, Belgium, 990. [2]Dal, G Notes on Polyedra Associated wit Hop-Constrained Pats, Operations Researc Letters, 25, 97-00, 999. [3]R. Fukasawa, H. Longo, J. Lysgaard, M. Poggi de Aragão, M. Reis, E. Ucoa and R. F. Werneck, Robust branc-and-cut-and-price for te capacitated veicle routing problem, Matematical Programming, 06 (3), 49-5, [4] Gavis, B., Graves, S. "Te Travelling Salesman Problem and Related Problems," Working Paper, Graduate Scool of Management, University of Rocester, 978. [5] M.T. Godino, L. Gouveia, T.L. Magnanti, Combined Route Capacity and Route Lengt Models for Unit Demand Veicle Routing Problems, Working Paper nº / 2005, Centro de Investigação Operacional. [6] L. Gouveia, A result on projection for te veicle routing problem, European Journal of Operational Researc, 85, , 995. [7] L. Gouveia, and S. Voβ, "A Classification of Formulations for te (Time-Dependent) Travelling Salesman Problem," European Journal of Operations Researc, Vol 83, pp 69-82, 995. [8] A. N. Letcford, R.W Eglese and J. Lysgaard, Multistars, Partial Multistars and te Capacitated Veicle Routing Problem, Matematical Programming, 94 (), 2-40, [9] A.N. Letcford and J.J. Salazar-Gonzalez, Projection Results for Veicle Routing Matematical Programming B (05), , [0] J. Lysgaard, A.N. Letcford and R.W. Eglese, A new branc-and-cut algoritm for te capacitated veicle routing problem, Matematical Programming, 00 (2), , [] J. C. Picard, J.C. and M. Queyranne, Te time-dependent travelling salesman problem and its application to te tardiness in one-macine sceduling, Operations Researc 26, 86 0, 978. [2] P. Tot, D. Vigo (eds.), Te Veicle Routing Problem, SIAM Monograps on Discrete Matematics and Applications, Piladelpia,