The problem of defining the tool path in a 3DP process is a generalization of the Rural Postman Problem (RPP), that is a particular version of the more general class of Arc Routing Problems (see, e.g., Corber´an and Laporte [7]). The RPP is the problem of finding the minimum cost eulerian cycle that visits a set of arcs at least once, by using also a second set of arcs, if needed, where each arc is associated with a non-negative cost. The RPP represents a generalization of the Chinese Postman Problem (CPP), that is the problem of finding the minimum cost path that visits at least once all the arcs. The CPP can be solved in polynomial time, while the RPP (directed or undirected) is an NP-hard problem if the set of required arcs is not strongly connected (Laporte [27]).
The RRP arises in many real-world situations such as the snow plowing (see, e.g., Haslam and Wright [24]), the garbage collection (see, e.g., Beltrami and Bodin [4]), the mail delivery (see, e.g, Levy [29]), the school bus routing (see, e.g, Angel et al. [2]), the meter reading (see, e.g., Stern and Dror [32]), and the street sweeping (see, e.g., Bodin and Kursh [5]). The interested reader can find other work related to these applications in Eiselt et al. [16]. More recent application of the RRP are referred to the control of plotting and drilling machines (see, e.g. Groetschel [22]) and to the optimization of laser-plotter beam movements (see, e.g. Ghiani and Improta [19]).
Many polyhedral studies and exact algorithms, mostly cutting plane and branch-and-cut algorithm, have been proposed in the last years for the RRP and its generalizations. See, e.g., Corber´an and Sanchis [12, 13], Laprte and Ghiani [21] (where they use a formulation with just 1/0 variables), Reinelt and Theis [31], and Fern´andez et al. [17] (where they use a completely dif- ferent approach and formulation from the previous works). According to the survey redacted by Corber´an and Prins [11], the largest RPP instances that have been solved to optimality are characterized by up to 1000 vertices, 3080 edges and 204 required components and are solved in about one hour in average on a standard PC by using the branch-and-cut described in Cor- ber´an et al. [10] for the Windy General Routing Problem (which contains the RPP).
Probably, the most famous heuristic algorithm for the RPP is the one proposed by Frederikson [18], which is built on the well known heuristic for the TSP by Christofides, and, similarly, presents a worst case bound of 3/2. Hertz et al. [25] propose some improving techniques to be applied to the Frederickson algorithm and to a simply newly developed heuristic. Corber´an
et al. [9] present two heuristics approaches to solve the Mixed RPP where the problem is defined on a graph with edges and arcs and required components can be both edges or arcs. Groves and Vuuren [23] present an effective local search framework for the URRP based on local searches for the TSP. They solve heuristically very large instances with up to about 5000 vertices and 30000 edges. Ghiani et al. [20] propose a constructive heuristic for the Undirected RPP (URPP) with a local post-optimization. The procedure is competitive with the Frederickson one. Holmberg [26] proposes some heuristics for the RPP built on the Frederikson heuristic by changing the order of the components. The author includes also two post-processing heuristics to improve the solution.
We report now a set of variants of the RPP that are interesting for 3DP. Dror and Langevin [15] introduced the Directed Clustered RPP and solved it by transforming the problem into a Generalized TSP. They separate the arcs in clusters that must be finished before stepping to another cluster. Letchford and Eglese [28] present the RPP with deadline classes, a RPP with time windows where the time windows have only a maximum time to be respected. They propose a formulation and a cutting plane algorithm. Corber´an et al. [8] consider the mixed RPP with turn penalties. They associate a penalty to every turn and take into account also the existence of forbidden turns. The problem is to find the minimal tour avoiding forbidden tours and paying for turn penalties. The authors transform the problem into an ATSP and solve it by using exact and heuristic algorithms for the ATSP. Some relevant applications of the RPP are listed in the following. Ghi- ani and Improta [19] consider the problem of drawing and decorating metal surfaces by means of a laser plotter, that works as a 2D printer, minimiz- ing the total length of the non-drawing moves. The so-called Laser-Plotter Beam Routing Problem is modeled as a RPP with additional constraints and solved transforming it into a RPP. By using the branch-and-cut pre- sented in Ghiani and Laporte [21] istances with up to 225 vertices can be solved. Moreira et al. [30] describe the problem that arises when manufac- turing high-precision tools. They intend to determine the shortest path for the cutting of given pieces. The problem is represented as a particular RPP where non-cutting movements are allowed only after the complete cut of a piece. Hence the problem is called Dynamic RPP and solved by means of a heuristic algorithm.
7.3
Problem Description
We call 3D printing Routing Problem (3DRP) the generalization of the URPP that is related to the tool path definition in a 3DP process. In particular, the nozzle must start from its starting position and return to the same position when the print is terminated. In addition, the edges are set
into clusters to be visited in a sequence, that are represented by the layers of the print. On each layer the set of compulsory edges represents the set of edges on which the nozzle is required to depose material (the first time they are used), while the set of the optional edges represents the other edges. Because of the printed material, no layer can be started before terminating the previous one, and the nozzle cannot move back to previous layers. Then one and only one optional edge can be used to move from one layer to the next one and no edge between two non-subsequents layers is present. Lastly, only optional edges are present between two layers. We will formalize the 3DRP in Section 7.4. The 3DRP represents the basic problem of optimizing the tool path of a 3DP process. In the following we are focusing on the basic problem, but some additional characteristics can arise across the 3DP process: we will list them in Section 7.7.