A Branch-and-Cut Algorithm for the Periodic Rural Postman Problem with Irregular Services

Transcription

1 A Branch-and-Cut Algorithm for the Periodic Rural Postman Problem with Irregular Services Enrique Benavent Ángel Corberán Demetrio Laganà Francesca Vocaturo DEIO, Universitat de València, Spain DIMEG, Università della Calabria, Italy DESF, Università della Calabria, Italy Conference on Computational Management Science University of Bergamo CMS 2017 May-June 2017, Bergamo 1 / 24

2 Preliminary Issues The aim of the Routing Problems is to design a number of vehicle routes that visit a given set of customers in order to service them. In the Periodic Routing Problems: the service occurs in some periods (days) of a given time horizon. Moreover, the customers requirements can be different; usually the customers are identified as nodes of a graph representing the streets network (Periodic Vehicle Routing Problems); when the requirement of visit can be identified with the arcs or edges of the graph, we talk about Periodic Arc Routing Problems. CMS 2017 May-June 2017, Bergamo 2 / 24

3 Scientific Literature: Some Pertinent Works Periodic Vehicle Routing Problem: It has been studied extensively! A.M. Campbell and J.H. Wilson, Forty years of periodic vehicle routing, Networks 63 (2014), Periodic Arc Routing Problem: The literature is still scarce! G. Ghiani, R. Musmanno, G. Paletta, and C. Triki, A heuristic for the periodic rural postman problem, Computers & Operations Research 32 (2005) F. Chu, N. Labadi, and C. Prins, A Scatter Search for the periodic capacitated arc routing problem, European Journal of Operational Research 169 (2006), P. Lacomme, C. Prins, and W. Ramdane-Chérif, Evolutionary algorithms for periodic arc routing problems, European Journal of Operational Research 165 (2005), I.M. Monroy, C.A. Amaya, and A. Langevin, The periodic capacitated arc routing problem with irregular services, Discrete Applied Mathematics 161 (2013), The Periodic Rural Postman Problem with Irregular Services (PRPP-IS) belongs to the class of Periodic Arc Routing Problems. To the best of our knowledge, it has not yet been tackled. CMS 2017 May-June 2017, Bergamo 3 / 24

4 Problem Description PRPP-IS: There is a single vehicle (no capacity restriction). A route has to be designed for each day of the time horizon in order to meet the service requirements. The required elements correspond to links of a graph G = (V, E, A). In particular, set E R E includes required edges, set A R A includes required arcs. L R = A R E R denotes the set of required links. Vertex 1 V represents the depot. Every required link has an own service plan that identifies the number of visits needed in sub periods of the time-horizon. This number is called frequency. The service requirements can be irregular (different sizes of the subperiods, different frequencies, etc.). The goal is to minimize the total cost over the time horizon. CMS 2017 May-June 2017, Bergamo 4 / 24

5 Problem Description PRPP-IS: There is a single vehicle (no capacity restriction). A route has to be designed for each day of the time horizon in order to meet the service requirements. The required elements correspond to links of a graph G = (V, E, A). In particular, set E R E includes required edges, set A R A includes required arcs. L R = A R E R denotes the set of required links. Vertex 1 V represents the depot. Every required link has an own service plan that identifies the number of visits needed in sub periods of the time-horizon. This number is called frequency. The service requirements can be irregular (different sizes of the subperiods, different frequencies, etc.). The goal is to minimize the total cost over the time horizon. CMS 2017 May-June 2017, Bergamo 4 / 24

6 Problem Description PRPP-IS: There is a single vehicle (no capacity restriction). A route has to be designed for each day of the time horizon in order to meet the service requirements. The required elements correspond to links of a graph G = (V, E, A). In particular, set E R E includes required edges, set A R A includes required arcs. L R = A R E R denotes the set of required links. Vertex 1 V represents the depot. Every required link has an own service plan that identifies the number of visits needed in sub periods of the time-horizon. This number is called frequency. The service requirements can be irregular (different sizes of the subperiods, different frequencies, etc.). The goal is to minimize the total cost over the time horizon. CMS 2017 May-June 2017, Bergamo 4 / 24

7 Problem Description PRPP-IS: There is a single vehicle (no capacity restriction). A route has to be designed for each day of the time horizon in order to meet the service requirements. The required elements correspond to links of a graph G = (V, E, A). In particular, set E R E includes required edges, set A R A includes required arcs. L R = A R E R denotes the set of required links. Vertex 1 V represents the depot. Every required link has an own service plan that identifies the number of visits needed in sub periods of the time-horizon. This number is called frequency. The service requirements can be irregular (different sizes of the subperiods, different frequencies, etc.). The goal is to minimize the total cost over the time horizon. CMS 2017 May-June 2017, Bergamo 4 / 24

8 Simple Example The time horizon includes 7 days (a week). All links are required (L R = A E). The costs of traversing/servicing the links are not shown. There are 5 sub periods (i.e., subsets of the time horizon): T 1 = {Mon, T ue}, T 2 = {W ed, T hu}, T 3 = {F ri, Sat, Sun}, T 4 = {Mon, T ue, W ed, T hu}, T 5 = {Mon, T ue, W ed, T hu, F ri, Sat, Sun}. T 3, T 4 1 T 5 4 T 3, T 4 6 T 5 T 1, T 2, T 3 T 5 5 T 3, T T 5 T 3, T 4 CMS 2017 May-June 2017, Bergamo 5 / 24

9 Simple Example T 3, T 4 1 T 5 4 T 3, T 4 6 T 5 T 1, T 2, T 3 T 5 5 T 3, T T 5 T 3, T 4 If we assume that the frequencies for each sub period and for each link are equal to 1, then: edge (2,4) must be serviced once over the first sub period (i.e., Monday and Tuesday), once over the second sub-period (i.e., Wednesday and Thursday), once over the third sub period (i.e., Friday, Saturday, and Sunday), and so on for the other links. CMS 2017 May-June 2017, Bergamo 6 / 24

10 Applications The PRPP-IS is a natural extension of the rural postman problem when a single route must be planned not just for a day, but for a set of days. Typically, the required elements correspond to streets and do not require to be serviced every day. There are various applications: waste collection, inspection of electric power lines, mowing vegetation on roads, etc. The assumption of irregularity makes the problem more general! Specific Example: Monitoring Activities Monroy, Amaya, and Langevin (2013) Road network monitoring is carried out periodically. Each road has different surveillance requirements (number of passages) during sub periods over the time horizon. CMS 2017 May-June 2017, Bergamo 7 / 24

11 Notation Parameters H: finite and discrete time horizon; it is a set of days; P l : set of disjoint sub periods for required link l (a sub period is a subset of H); f l T : frequency, i.e. number of visits needed in sub period T P l by link l (f l T T at most one visit per day); c l = c ij: traversal cost of link l = (i, j); c s l = c s ij: service cost of required link l = (i, j), with c s ij c ij; K: set of all route indices (since a route index corresponds to a day, and vice versa, a sub period T can also denote a subset of K). Decision Variables x k l = x k ij: number of times that link l = (i, j) is traversed in route k from i to j; y k l : binary variable that takes value 1 if link l is serviced in route k, and value 0 otherwise. CMS 2017 May-June 2017, Bergamo 8 / 24

12 Notation Additional Notation A (S : S ): set of arcs from S V to S V ; E (S : S ): set of edges between S V and S V ; (S : S ) = E (S : S ) A (S : S ) A (S : S); A + (S) = A (S : V \S); A (S) = A (V \S : S); A(S) = A + (S) A (S); E(S) = E (S : V \S); δ(s) = E(S) A(S) (cutset); γ(s): set of links with both endpoints in S V. These sets are defined in a similar way with respect only to the required links (subscript R): A + R (S), δr(s), etc. In addition, S = {i} is represented by simply i. CMS 2017 May-June 2017, Bergamo 9 / 24

13 Mathematical Formulation Min ) (c s l c l )yl k + c ij (x k ij + x k ji + c ijx k ij (1) k K l L R k K (i,j) E k K (i,j) A x k ij + x k ij = x k ji + x k ji i V, k K, (2) (i,j) A + (i) x k ij + j:(i,j) E(i) (i,j) A (S) (i,j) E:i V \S,j S (j,i) A (i) j:(i,j) E(i) x k ij y k l S V \{1}, l γ R(S), k K (3) x k ij yij k (i, j) A R, k K (4) x k ij + x k ji yij k (i, j) E R, k K (5) yl k = ft l l L R, T P l (6) k T yl k {0, 1} l L R, k K (7) x k ij 0 and integer (i, j) A, k K (8) x k ij, x k ji 0 and integer (i, j) E, k K. (9) CMS 2017 May-June 2017, Bergamo 10 / 24

14 Connectivity Constraints (3) For k = 1; S = {6, 7}; l = (6, 7) γ R (S): x 1 ij + x 1 ij yl 1 (i,j) A (S) (i,j) E:i V \S,j S V \S S 5 If the route concerning day 1 (i.e., Mon) does not service edge (6,7), then y 1 67 = 0 and the inequality is trivially satisfied; otherwise: x x CMS 2017 May-June 2017, Bergamo 11 / 24

15 Valid Inequalities We propose several valid inequalities for the PRPP-IS. They are used in the algorithm in order to make it more effective. Let S V be a vertex subset and δ(s) a link cutset. We consider: x k (δ(s)) = (i,j) δ(s) A xk ij + (i,j) δ(s) E (xk ij + xk ji ). Sub Period Aggregate Parity Inequalities Let T be a sub-period and δ R (S) a required link cutset such that f(δ R (S), T ) = l δ R (S):T P l ft l is odd: x k (δ(s)) f(δ R (S), T ) + 1. (10) k T CMS 2017 May-June 2017, Bergamo 12 / 24

16 Valid Inequalities Sub Period Aggregate Parity Inequalities For T = T 4 = {Mon, T ue, W ed, T hu}; S = {6, 7}; (3, 6) δ R (S) and f (3,6) T 4 = 3 odd: x x x x x x x x x x x x x x x x V \S S 5 CMS 2017 May-June 2017, Bergamo 13 / 24

17 Valid Inequalities Disaggregate Parity Inequalities Let F be a subset of δ R (S) such that F is odd: where y k (F ) = l F yk l. P -Aggregate Parity Inequalities x k (δ(s)) 2y k (F ) F + 1, (11) In addition, let K = {k 1, k 2,..., k P } be a subset of P < K routes such that f(f, K) = l F T P l min(ft l, T K ) is odd: y k (F ) f(f, K) + 1, (12) k K x k (δ(s)) 2 k K For K = {Mon, T ue, Sat, Sun} and F = {(3, 6)} with P (3,6) = {T 4 = {Mon, T ue, W ed, T hu} ; T 3 = {F ri, Sat, Sun}}, f (3,6) T 4 = 3 and f (3,6) T 3 = 1: f(f, K) = min(f (3,6) T 4, T 4 K ) + min(f (3,6) T 3, T 3 K ) = CMS 2017 May-June 2017, Bergamo 14 / 24

18 Valid Inequalities K-C Inequalities They refer to a partition of V into {M 0, M 1,..., M Q} such that each subset of vertices associated with the connected components induced by the required links (R-set) is contained in one of the sets M 0 M Q, M 1,..., M Q 1, each M i contains at least one R-set, the induced subgraphs G(M i) are connected, and (M 0 :M Q) contains a positive and even number of required links. 1 M 1 M M 1 0 Q 2 M Q 1 M Q 1 1 M i 1 M Q Disaggregate K-C Inequalities and Sub Period Aggregate K-C Inequalities. CMS 2017 May-June 2017, Bergamo 15 / 24

19 A Branch-and-Cut Algorithm: Simplified Outline Step 0. Let LB = 0 be the lower bound and UB = + the upper bound. (Note: the value of LB is opportunely updated during the search) Step 1. Define a relaxed problem by eliminating connectivity constraints (3) and integer conditions and insert it into a list Θ (first problem in the list). Step 2. If Θ is empty or LB = UB, then STOP. Otherwise, extract a problem from Θ. Step 3. Solve the current problem. Let OBJ be the solution value. If OBJ UB then go to Step 2. Step 4. Identify inequalities (3) and other valid inequalities for the PRPP-IS by using adequate separation procedures. Step 5. If some violated inequalities have been identified, add these inequalities to the problem and go back to Step 3. Otherwise, if the current solution is feasible, set UB = OBJ and go back to Step 2. Step 6. If the current solution is not integer, generate two subproblems by branching on a fractional variable. Insert the subproblems into Θ and go back to Step 2. CMS 2017 May-June 2017, Bergamo 16 / 24

20 Separation Procedures Some inequalities are separated at each node of the search tree. Other valid inequalities are separated at the root node solely. Connectivity Constraints: exact and heuristic procedures described by Benavent, Corberán, Plana, and Sanchis (2009) for the Min-Max K- vehicles Windy Rural Postman Problem. Sub period aggregate parity inequalities: heuristic procedure commonly used in the literature for the separation of the aggregate parity inequalities. Disaggregate and P -Aggregate Parity Inequalities: heuristic procedure described by Ghiani and Laporte (2000) for the Undirected Rural Postman Problem. Disaggregate and Sub Period Aggregate K-C Inequalities: three-phase heuristic procedure inspired by the one introduced by Corberán, Letchford, and Sanchis (2001) for the General Routing Problem. CMS 2017 May-June 2017, Bergamo 17 / 24

21 Computational Experiments Instances Instances derived from mval datasets designed for the Mixed Capacitated Arc Routing Problem by Belenguer, Benavent, Lacomme and Prins (2006). pcp-mval (periodic Chinese postman): the nature of the mval instances has been kept, i.e., all links of the graph are stayed required. For each required link l, a set P l of sub-periods has been generated. The number of times that required link l must be serviced in each sub period T P l was uniformly generated in [1,..., T /2 ]. spcp-mval (simplified periodic Chinese postman): pcp-mval instances have been simplified by switching off some sub periods (i.e., by setting to zero the frequencies concerning these sub periods). prp-mval (periodic rural postman): pcp-mval instances have been also modified by reducing set L R. Specifically, each link in the pcp-mval instance has been kept required with probability 0.6 (with the same sub periods and frequencies). sprp-mval (simplified periodic rural postman): prp-mval instances have been simplified by switching off some sub periods. CMS 2017 May-June 2017, Bergamo 18 / 24

22 Computational Experiments Instances Instances derived from mval datasets designed for the Mixed Capacitated Arc Routing Problem by Belenguer, Benavent, Lacomme and Prins (2006). pcp-mval (periodic Chinese postman): the nature of the mval instances has been kept, i.e., all links of the graph are stayed required. For each required link l, a set P l of sub-periods has been generated. The number of times that required link l must be serviced in each sub period T P l was uniformly generated in [1,..., T /2 ]. spcp-mval (simplified periodic Chinese postman): pcp-mval instances have been simplified by switching off some sub periods (i.e., by setting to zero the frequencies concerning these sub periods). prp-mval (periodic rural postman): pcp-mval instances have been also modified by reducing set L R. Specifically, each link in the pcp-mval instance has been kept required with probability 0.6 (with the same sub periods and frequencies). sprp-mval (simplified periodic rural postman): prp-mval instances have been simplified by switching off some sub periods. CMS 2017 May-June 2017, Bergamo 18 / 24

23 Computational Experiments Instances Instances derived from mval datasets designed for the Mixed Capacitated Arc Routing Problem by Belenguer, Benavent, Lacomme and Prins (2006). pcp-mval (periodic Chinese postman): the nature of the mval instances has been kept, i.e., all links of the graph are stayed required. For each required link l, a set P l of sub-periods has been generated. The number of times that required link l must be serviced in each sub period T P l was uniformly generated in [1,..., T /2 ]. spcp-mval (simplified periodic Chinese postman): pcp-mval instances have been simplified by switching off some sub periods (i.e., by setting to zero the frequencies concerning these sub periods). prp-mval (periodic rural postman): pcp-mval instances have been also modified by reducing set L R. Specifically, each link in the pcp-mval instance has been kept required with probability 0.6 (with the same sub periods and frequencies). sprp-mval (simplified periodic rural postman): prp-mval instances have been simplified by switching off some sub periods. CMS 2017 May-June 2017, Bergamo 18 / 24

24 Computational Experiments Instances Instances derived from mval datasets designed for the Mixed Capacitated Arc Routing Problem by Belenguer, Benavent, Lacomme and Prins (2006). pcp-mval (periodic Chinese postman): the nature of the mval instances has been kept, i.e., all links of the graph are stayed required. For each required link l, a set P l of sub-periods has been generated. The number of times that required link l must be serviced in each sub period T P l was uniformly generated in [1,..., T /2 ]. spcp-mval (simplified periodic Chinese postman): pcp-mval instances have been simplified by switching off some sub periods (i.e., by setting to zero the frequencies concerning these sub periods). prp-mval (periodic rural postman): pcp-mval instances have been also modified by reducing set L R. Specifically, each link in the pcp-mval instance has been kept required with probability 0.6 (with the same sub periods and frequencies). sprp-mval (simplified periodic rural postman): prp-mval instances have been simplified by switching off some sub periods. CMS 2017 May-June 2017, Bergamo 18 / 24

25 Computational Experiments Workstation and Software Intel(R)Core(TM) i7-3630qm CPU running at 2.40 GHz; 32 GB of memory. C++ for Linux and ILOG CPLEX Library Column Headings FILE instance name V, A, E number of vertices, arcs and edges A R, E R number of required arcs and edges LB, UB final lower bound and upper bound GAP percentage gap (from CPLEX) SEC computation time in seconds (TL: time limit 6 hours) CON number of connectivity inequalities DIP,SAP,PAP number of disaggregate, sub period aggregate and P -aggregate parity inequalities DKC,SKC number of disaggregate and sub period aggregate K-C inequalities NOD number of nodes processed in the search (apart from the root) CMS 2017 May-June 2017, Bergamo 19 / 24

26 Numerical Results: pcp-mval dataset Instance Features Main Results Other Results FILE V A E A R E R LB UB GAP SEC CON DIP SAP PAP DKC SKC NOD pcp-mval1a TL pcp-mval1b pcp-mval1c TL pcp-mval2a pcp-mval2b pcp-mval2c TL pcp-mval3a pcp-mval3b pcp-mval3c TL pcp-mval4a TL pcp-mval4b TL pcp-mval4c TL pcp-mval4d TL pcp-mval5a TL pcp-mval5b TL pcp-mval5c TL pcp-mval5d TL pcp-mval6a pcp-mval6b TL pcp-mval6c pcp-mval7a TL pcp-mval7b TL pcp-mval7c TL pcp-mval8a TL pcp-mval8b TL pcp-mval8c TL pcp-mval9a TL pcp-mval9b TL pcp-mval9c TL pcp-mval9d TL pcp-mval10a TL pcp-mval10b TL pcp-mval10c TL pcp-mval10d TL average gap % 1.10 maximum gap % 3.69 # optima 7 CMS 2017 May-June 2017, Bergamo 20 / 24

27 Numerical Results: spcp-mval dataset Instance Features Main Results Other Results FILE V A E A R E R LB UB GAP SEC CON DIP SAP PAP DKC SKC NOD spcp-mval1a TL spcp-mval1b spcp-mval1c spcp-mval2a spcp-mval2b spcp-mval2c spcp-mval3a spcp-mval3b spcp-mval3c spcp-mval4a TL spcp-mval4b TL spcp-mval4c TL spcp-mval4d TL spcp-mval5a TL spcp-mval5b TL spcp-mval5c TL spcp-mval5d TL spcp-mval6a TL spcp-mval6b spcp-mval6c spcp-mval7a TL spcp-mval7b spcp-mval7c TL spcp-mval8a TL spcp-mval8b TL spcp-mval8c TL spcp-mval9a spcp-mval9b TL spcp-mval9c TL spcp-mval9d TL spcp-mval10a TL spcp-mval10b TL spcp-mval10c TL spcp-mval10d TL average gap % 0.81 maximum gap % 3.15 # optima 12 CMS 2017 May-June 2017, Bergamo 21 / 24

28 Numerical Results: prp-mval dataset Instance Features Main Results Other Results FILE V A E A R E R LB UB GAP SEC CON DIP SAP PAP DKC SKC NOD prp-mval1a prp-mval1b prp-mval1c TL prp-mval2a prp-mval2b prp-mval2c prp-mval3a prp-mval3b prp-mval3c prp-mval4a TL prp-mval4b prp-mval4c prp-mval4d prp-mval5a TL prp-mval5b TL prp-mval5c prp-mval5d TL prp-mval6a TL prp-mval6b prp-mval6c prp-mval7a prp-mval7b prp-mval7c TL prp-mval8a prp-mval8b TL prp-mval8c TL prp-mval9a TL prp-mval9b TL prp-mval9c TL prp-mval9d TL prp-mval10a TL prp-mval10b TL prp-mval10c TL prp-mval10d TL average gap % 0.72 maximum gap % 5.28 # optima 17 CMS 2017 May-June 2017, Bergamo 22 / 24

29 Numerical Results: sprp-mval dataset Instance Features Main Results Other Results FILE V A E A R E R LB UB GAP SEC CON DIP SAP PAP DKC SKC NOD sprp-mval1a sprp-mval1b sprp-mval1c sprp-mval2a sprp-mval2b sprp-mval2c sprp-mval3a sprp-mval3b sprp-mval3c sprp-mval4a sprp-mval4b sprp-mval4c TL sprp-mval4d TL sprp-mval5a TL sprp-mval5b TL sprp-mval5c sprp-mval5d sprp-mval6a sprp-mval6b sprp-mval6c sprp-mval7a TL sprp-mval7b sprp-mval7c sprp-mval8a sprp-mval8b TL sprp-mval8c sprp-mval9a TL sprp-mval9b TL sprp-mval9c TL sprp-mval9d TL sprp-mval10a TL sprp-mval10b TL sprp-mval10c sprp-mval10d TL average gap % 0.49 maximum gap % 2.71 # optima 21 CMS 2017 May-June 2017, Bergamo 23 / 24

30 Final Remarks The PRPP-IS has been introduced and studied. Computational results confirm that our branch-and-cut algorithm is an effective solution approach to the problem. Details on procedures, applications and results will be provided in a paper next to submission E. Benavent, Á. Corberán, D. Laganà, F. Vocaturo (2017). Exact Solution of the Periodic Rural Postman Problem with Irregular Services on Mixed Graphs. THANK YOU! CMS 2017 May-June 2017, Bergamo 24 / 24