Development and assessment of the SHARPand RandSHARP algorithms for the arc

routing problem

Sergio Gonzáleza, Alejandra Pérez-Bonillaa, Angel A.Juana, Daniel Rieraa

{sgonzalezmarti, aperezbon, ajuanp, drierat, jcastellav}@uoc.edu

aDepartment of Computer Science, Multimedia and Telecommunication, IN3 – Open University of Catalonia, Barcelona, Spain.

http://dpcs.uoc.edu

Cyted-Harosa Int. WorkshopValparaiso, Chile. November 12-13, 2012

1. The Capacitated ARP (deterministic)

The Capacitated Arc Routing Problem (CARP) is a well-known NP-hard problem Golden et al. (1981):

The set of edges constitute an undirected, non-complete graph or network.

A set of edges’ demands must be supplied by a fleet of (homogeneous) vehicles.

Resources are available from a depot.

Moving a vehicle from one node i to another j has associated (symmetric) costs c(i, j) > 0.

An edge might be traversed several times by different vehicles, but it is served by just one vehicle.

Additional constraints must be considered: maximum load capacity per vehicle, service times, etc.

Depot(resources)

Edges with demands

Nodes

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Goal: to obtain an ‘optimal’ solution, i.e. a set of routes satisfying all constraints with minimum costs

2. Variants of the CARP

Different ARP variants have been proposed:

Directed CARP (DCARP): A version of the CARP where the Arcs can be traversed in a single direction (Maniezzo & Roffilli, 2008).

Mixed CARP (MCARP): A version of the CARP with mixed graphs (Belenguer, Benavent, Lacomme, & Prins, 2006).

Min-Max k-CPP: CARP-like problem excluding capacity constraints on vehicles (Ulusoy, 1985).

Periodic CARP (PCARP): CARP extended to the planning of P days (Lacomme, Prins, Ramdane-Chérif, 2002).

CARP with Stochastic demands (ARPSD): CARP where the demand to be served is not known beforehand (Fleury, Lacomme, & Prins, 2005).

Etcetera.

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3. Practical Applications

Some ARP practical applications are:

Refuse collection (Almeida & Mourão, 2000).

Snow removal (Eglese, 1994).

Inspection of distributed systems (Lee, 1989).

Routing of electric meter readers (Stern & Dror, 1979).

School bus routing (Ferland & Gudnette, 1990).

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4. Mathematical Model

The first ILP formulation was probably the one presented by Golden and Wong in 1981. (B. L. Golden and R. T. Wong 1981).

A formulation using undirected variables is presented by Belenguer and Benavent. In his PhD

dissertation, Letchford gives several ILP formulations of the CARP, and derives additional valid inequalities and separation algorithms for the problem.

Other mathematical formulations are due to Welz [22] and Eglese [23], all of them for the undirected case.

5. Solving Approaches (1/2)

Exact Methods (optimal solutions for small-scale problems):

Branch and Bound techniques (Kiuchi, Hirabayashi, Saruwatari, & Shinano, 1995).

Tour construction algorithms (Hirabayashi, Nishida, & Saruwatari, 1992).

Subtour elimination algorithms (Saruwatari, Hirabayashi, & Nishida, 1992).

ILP formulations (Belenguer & Benavent, 1992).

Heuristics (fast, ‘good’ solutions) :

Path Scanning algorithm, Augment-Merge algorithm, Construct-Strike algorithm (Golden, DeArmon, & Baker, 1983).

Node duplication heuristic (Wøhlk, 2005).

The Cycle Assignment algorithm (Benavent, Campos, Corberan, & Mota, 1990).

Parallel insert algorithm (Chapleau, Ferland, Lapalme, & Rousseau,1984).

6. Solving Approaches (2/2)

Meta-heuristics (pseudo-optimal solutions):

Tabu Search algorithms (Hertz, Laporte, & Mittaz, 2000; Greistorfer, 2003).

Simulated Annealing (Eglese, 1994).

Genetic Algorithms (Lacomme, Prins, & Ramdane-Chérif, 2001).

Ant Colony Optimization (Lacomme, Prins, & Tanguy, 2004).

Memetic Algorithms (Lacomme, Prins, & Ramdane-Chérif, 2004).

7. Our SHARP procedure for the CARP Based on the savings heuristic for the CVRP

(Clarke & Wright, 1964).

Designed to provide a ‘good’ starting point for CARP metaheuristics.

SHARP main steps:

Use the Floyd-Warshall algorithm to compute the shortest paths for all pairs of nodes in the network. This allows to treat the graph as if it was a complete one.

Compute the savings associated with demanding edges (real or virtual) connecting any two nodes and create a sorted savings list.

Create a dummy (initial) solution by assigning a route to each demanding arc.

Iterate over the list of savings and look at each node in the selected arc to see which routes (if any) have that node as an exterior node, attempting to merge these routes if possible.

Reconstruct the final solution by computing the shortest path between the edges in the route.

González, S.; Juan, A.; Riera, D.; Castellà, Q.; Muñoz, R.; Pérez, A. (2012): ”Development and Assessment of the SHARP and RandSHARP Algorithms for the Arc Routing Problem”. AI Communications, Volume 25, pp. 173-189 (indexed in ISI SCI, 2011 IF = 0.500, Q4). ISSN: 0921-7126.

González, S.; Juan, A.; Riera, D.; Castellà, Q.; Muñoz, R.; Pérez, A. (2012): ”Development and Assessment of the SHARP and RandSHARP Algorithms for the Arc Routing Problem”. AI Communications, Volume 25, pp. 173-189 (indexed in ISI SCI, 2011 IF = 0.500, Q4). ISSN: 0921-7126.

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SHARP the first edge (the one with the most savings) is the one selected.

RandSHARP introduces randomness in this process by using a quasi-geometric statistical distribution edges with more savings will be more likely to be selected at each step, but all edges in the list are potentially eligible.

Notice: Each time RandSHARP is run, a random feasible solution is obtained. By construction, chances are that this solution outperforms the SHARP one hundreds of ‘good’ solutions can be obtained after some seconds/minutes.

Good results with0.10 < α < 0.20

Good results with0.10 < α < 0.20

8. RandSHARP: a Multi-Start approach

Juan, A.; Faulin, J.; Jorba, J.; Riera, D.; Masip; D.; Barrios, B. (2011): “On the Use of Monte Carlo Simulation, Cache and Splitting Techniques to Improve the Clarke and Wright Savings Heuristics”. Journal of the Operational Research Society, Vol. 62, pp. 1085-1097.

Juan, A.; Faulin, J.; Jorba, J.; Riera, D.; Masip; D.; Barrios, B. (2011): “On the Use of Monte Carlo Simulation, Cache and Splitting Techniques to Improve the Clarke and Wright Savings Heuristics”. Journal of the Operational Research Society, Vol. 62, pp. 1085-1097.

Average Gap w.r.t. BKS in MaxTime = 180s

Averages Heuristics BEST10 AVG10

Set Instances Nodes Arcs Density PSH SHARP RandPSH RandSHARP RandPSH RandSHARP

egl 24 109 144 2,65% 24,61% 6,20% 13,34% 1,32% 13,76% 1,75%

gdb 23 12 29 53,77% 7,09% 7,56% 0,46% 0,26% 0,72% 0,53%

kshs 6 8 15 55,95% 9,87% 9,87% 0,98% 0,62% 0,98% 0,62%

val 34 36 63 10,59% 14,98% 11,93% 2,74% 1,60% 4,00% 2,85%

Totals 87

Averages 47,79 73 22,94% 15,20% 9,05% 4,94% 1,10% 5,62% 1,78%

(*) BKS values obtained from (2010)

(**) Java SE 1.6.0 - Intel Core Quad Q9400 @2.5GHz 8GB RAM

9. Computational Experiments

RandSHARP >> RandPSH >> SHARP >> PSH

RandSHARP >> RandPSH >> SHARP >> PSH

RandSHARP >> RandPSH >> SHARP >> PSH

RandSHARP >> RandPSH >> SHARP >> PSH

9. Computational Experiments

We have presented the SHARP procedure solving the CARP. This procedure is based on the savings heuristic for the CVRP.

We have introduced the RandSHARP algorithm for the CARP. This algorithm uses biased randomization to improve the SHARP procedure.

We have also proposed an approach for solving the ARPSD. This approach combines MCS with parallel computing and the RandSHARP algorithm.

The basic idea is to consider a vehicle capacity lower than the actual VMC when constructing CARP solutions. This way, this capacity surplus or safety stocks can be used when necessary to significantly reduce the expected costs due to expensive route failures.

Our approach provides the decision-maker with a set of alternative solutions with different properties (number of routes, fixed and expected variable costs, reliability indices, etc.)

It offers flexibility since it does not assume any particular behavior of the customers’ stochastic demands. Therefore, the probabilistic distributions which describe demands can be generic.

The randomized algorithm is easily parallelizable, which allows to generate pseudo-optimal solutions in ‘reasonable’ clock times.

18. Conclusions

http://dpcs.uoc.edu | http://ajuanp.wordpress.com

Development and assessment of the SHARPand RandSHARP algorithms for the arc

routing problem

Sergio Gonzáleza, Alejandra Pérez-Bonillaa, Angel A.Juana, Daniel Rieraa

{sgonzalezmarti, aperezbon, ajuanp, drierat, jcastellav}@uoc.edu

aDepartment of Computer Science, Multimedia and Telecommunication, IN3 – Open University of Catalonia, Barcelona, Spain.

Cyted-Harosa Int. WorkshopValparaiso, Chile. November 12-13, 2012

Thanks for y

our atte

ntion!

Thanks for y

our atte

ntion!