Table 1 - uploaded by David Applegate
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We use a branch-and-cut search to solve the Whizzkids'96 vehicle routing problem, demonstrating that the winning solution in the 1996 competition is in fact optimal. Our algorithmic framework combines the LP-based traveling salesman code of Applegate, Bixby, Chvátal, and Cook, with specialized cutting planes and a distributed search algorithm, per...
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... speed up the computation, we call Concorde's version of the Chained Lin-Kernighan TSP heuristic (Martin, Otto, and Felten (1992)) as a preliminary step in the TSP solution procedure; if the heuristic returns a tour of value B or less, then we do not need to run the branch-and-bound algorithm. In the first two entries of Table 1, we compare Hurkens' feasibility model and the layered optimization model, where we use a bound of 1,182 in the feasibility model, that is, B = 1182 in (3) (all distances are integer valued, and 1,182 is 1 better than the value of the best solution found in the Whizzkids'96 competition). The much larger graph used in the layered model translates into a larger running time. ...
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... sets S ∈ H B ; Karger's algorithm can again be used to sample candidate sets. The third entry in Table 1 gives the running time required for this combined approach. ...
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... results reported in Table 1 were obtained on a Compaq XP10000 workstation, with a 500 MHz Alpha EV6 processor. The Concorde TSP code was run using ILOG's CPLEX 6.5 LP solver. ...
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... feasibility constraints (3) force 0/1 solutions to the newspaper routing model to use at least 2 paths through the specified set of customers S, with the justification that any single path would have length greater than that allowed by the objective bound B. The effectiveness of these simple constraints in increasing the LP lower bound (see Table 1) suggests the use of more general inequalities ...
Citations
... concorde.html, Applegate et al. 2002). Int the 2nd step (see Step 2 from Figure 1), the toolpaths points were considered as nodes from the integrated in Concorde TSP Solver algorithms, in order to find an optimal route through them ( Figure 2a). ...
While investigating the variator transmission of vehicles, the relationship between the technological and service parameters of the working surfaces of conical disks treated by technological methods was established. The service properties are proposed to be enhanced by Regular MicroReliefs (RMRs) created on such surfaces. The optimal technological processing conditions were found, which allow retaining the greatest amount of lubricant. The causes of surface defects, formed on the working surfaces of conical disks of the Continuously Variable Transmission (CVT), are systematized and classified. The wear resistance of such surfaces is proposed to be enhanced by technological methods, in particular, by forming partially RMRs on them. Their application facilitates relaxation processes on the material near to the surface, reduces shear stresses and strains, thus preventing the formation of burrs and extending the life of the conical disks of the CVT. A novel approach for obtaining the toolpaths of the deforming element, based on the so-called “Commis–Voyageur problem” algorithms, is employed in order to research the possibilities for involving that methods in toolpath generation. Dependences between the partial RMR’s formation conditions (deforming forces and feedrate) and microgeometric quality parameters are established. The latter include surface roughness, with a partially RMR applied onto the face surfaces of the test specimen (rotary body). It is found that these microreliefs enhance the ability of oil retaining in plastically deformed traces, formed over the operational surfaces, in comparison with those, that are processed by traditional cutting methods, as turning for example.
... An exact algorithm for the min-max VRP, able to prove the optimality of a solution provided to the problem presented at the Whizzkids '96 competition, has been proposed by Applegate et al. in [6]. The algorithm is based on branch-andcut and requires a very lare computing time on a computer network. ...
This work is inspired by the problem of planning sequences of operations, as welding, in car manufacturing stations where multiple industrial robots cooperate. The goal is to minimize the station cycle time, \emph{i.e.} the time it takes for the last robot to finish its cycle. This is done by dispatching the tasks among the robots, and by routing and scheduling the robots in a collision-free way, such that they perform all predefined tasks. We propose an iterative and decoupled approach in order to cope with the high complexity of the problem. First, collisions among robots are neglected, leading to a min-max Multiple Generalized Traveling Salesman Problem (MGTSP). Then, when the sets of robot loads have been obtained and fixed, we sequence and schedule their tasks, with the aim to avoid conflicts. The first problem (min-max MGTSP) is solved by an exact branch and bound method, where different lower bounds are presented by combining the solutions of a min-max set partitioning problem and of a Generalized Traveling Salesman Problem (GTSP). The second problem is approached by assuming that robots move synchronously: a novel transformation of this synchronous problem into a GTSP is presented. Eventually, in order to provide complete robot solutions, we include path planning functionalities, allowing the robots to avoid collisions with the static environment and among themselves. These steps are iterated until a satisfying solution is obtained. Experimental results are shown for both problems and for their combination. We even show the results of the iterative method, applied to an industrial test case adapted from a stud welding station in a car manufacturing line.
... This study utilizes four TSP algorithms: an exact algorithm implemented in Concorde (Applegate et al., 2002) and three simple heuristic algorithms of nearest insertion, farthest insertion, and cheapest insertion. To create the remainder of the initial population, we randomly select one of the available TSP tours, modify it, and then apply the Split algorithm to obtain the individual. ...
This paper proposes a hybrid genetic algorithm for solving the Multiple Traveling Salesman Problem (mTSP) to minimize the length of the longest tour. The genetic algorithm utilizes a TSP sequence as the representation of each individual, and a dynamic programming algorithm is employed to evaluate the individual and find the optimal mTSP solution for the given sequence of cities. A novel crossover operator is designed to combine similar tours from two parents and offers great diversity for the population. For some of the generated offspring, we detect and remove intersections between tours to obtain a solution with no intersections. This is particularly useful for the min-max mTSP. The generated offspring are also improved by a self-adaptive random local search and a thorough neighborhood search. Our algorithm outperforms all existing algorithms on average, with similar cutoff time thresholds, when tested against multiple benchmark sets found in the literature. Additionally, we improve the best-known solutions for 21 out of 89 instances on four benchmark sets.
... Task balance between robots has been studied in several tasks, such as vehicle routing problem [20], area coverage [21], etc. For the area coverage, Kapoutsis et al. [22] improve the workload balance among robots with a balanced partition map. ...
Line coverage is to cover linear infrastructure modeled as 1D segments by robots, which received attention in recent years. With the increasing urbanization, the area of the city and the density of infrastructure continues to increase, which brings two issues: (1) Due to the energy constraint, it is hard for the homogeneous robot team to cover the large-scale linear infrastructure starting from one depot; (2) In the large urban scene, the imbalance of robots' path greatly extends the time cost of the multi-robot system, which is more serious than that in smaller-size scenes. To address these issues, we propose a heterogeneous multi-robot approach consisting of several teams, each of which contains one transportation robot (TRob) and several coverage robots (CRobs). Firstly, a balanced graph partitioning (BGP) algorithm is proposed to divide the road network into several similar-size sub-graphs, and then the TRob delivers a group of CRobs to the subgraph region quickly. Secondly, a balanced ulusoy partitioning (BUP) algorithm is proposed to extract similar-length tours for each CRob from the sub-graph. Abundant experiments are conducted on seven road networks ranging in scales that are collected in this paper. Our method achieves robot utilization of 90% and the best maximal tour length at the cost of a small increase in total tour length, which further minimizes the time cost of the whole system. The source code and the road networks are available at https://github.com/suhangsong/BLC-LargeScale.
... We are not aware of any attempts to extend these to unconventional layouts. Given a distance matrix is provided, however, TSP's can be optimized reasonably fast using OR-tools [19] or Concorde [2,3]. Concorde, for example, is almost guaranteed to find the shortest path of a TSP with 100 nodes in less than one second [3]. ...
Order Picking in warehouses is often optimized using a method known as Order Batching, which means that one vehicle can be assigned to pick a batch of several orders at a time. There exists a rich body of research on Order Batching Problem (OBP) optimization, but one area which demands more attention is computational efficiency, especially for optimization scenarios where warehouses have unconventional layouts and vehicle capacity configurations. Due to the NP-hard nature of the OBP, computational cost for optimally solving large instances is often prohibitive. In this paper, we compare the performance of two approximate optimizers designed for maximum computational efficiency. The first optimizer, Single Batch Iterated (SBI), is based on a Seed Algorithm, and the second, Metropolis Batch Sampling (MBS), is based on a Metropolis algorithm. Trade-offs in memory and CPU-usage and generalizability of both algorithms is analyzed and discussed. Existing benchmark datasets are used to evaluate the optimizers on various scenarios. On smaller instances, we find that both optimizers come within a few percentage points of optimality at minimal CPU-time. For larger instances, we find that solution improvement continues throughout the allotted time but at a rate which is difficult to justify in many operational scenarios. SBI generally outperforms MBS and this is mainly attributed to the large search space and the latter’s failure to efficiently cover it. The relevance of the results within Industry 4.0 era warehouse operations is discussed.
... The Metric RuralPostCover is a restricted case of RuralPostCover where all the edge lengths obey the triangle inequality. Typical applications of RuralPostCover (or ChinesePost-Cover) and its variants encompass newspaper delivery (Applegate et al. 2012), route design for security guard patrols (Willemse and Joubert 2012), urban snow plowing (Quirion-Blais et al. 2017), and so on. ...
In this paper, we devise improved approximation algorithms for the Min–Max Rural Postmen Cover Problem (RuralPostCover) and the Min–Max Chinese Postmen Cover Problem (ChinesePostCover), which are natural extensions of the classical Rural Postman Problem and the Chinese Postman Problem where multiple postmen are available. These results are based on some key observations, a new approach to derive closed walks from (open) walks and an efficient postmen allocation procedure in the literature. As an application of the algorithm for RuralPostCover, we give the first constant-factor approximation algorithms for the Min–Max Subtree Cover Problem (SubtreeCover) and its generalization, called the Min–Max Steiner Tree Cover Problem with Vertex Weights (SteinerTreeCover), using simple approximation preserving reductions. Moreover, we devise specialized algorithms for SteinerTreeCover (SubtreeCover) with better approximation ratios.
... The minimax problem is a typical non-smooth optimization problem, and widely used in fields of data fitting [1], vehicle routing [2], structural optimization [3], location [8], optimal- [9], resource-allocation [16], engineering design [18] and etc, see for example [4,5] and references therein. ...
In this paper, an active set strategy is presented to address the ill-conditioning of smoothing methods for solving finite linear minimax problems. Based on the first order optimality conditions, a concept of the strongly active set composed of a part of active indexes is introduced. In the active set strategy, a strongly active set is obtained by solving a linear system or a linear programming problem, then an optimal solution with its active set and Lagrange multipliers is computed by an iterative process. A hybrid algorithm combining a smoothing algorithm and the active set strategy is proposed for solving finite linear minimax problems, in which an approximate solution is obtained by the smoothing algorithm, then an optimal solution is computed by the active set strategy. The convergences of the active set strategy and the hybrid algorithm are established for general finite linear minimax problems. Preliminary numerical experiments show that the active set strategy and the hybrid algorithm are effective and robust, and the active set strategy can effectively address the ill-conditioning of smoothing methods for solving general finite linear minimax problems.
... While in the former the authors improved a Clarke and Wright savings heuristics algorithm using local search, in the latter a greedy randomised adaptive search procedure enhanced with heuristic concentration was used. The relevance of balancing workload in VRPs is evident in different other real-life contexts, such as delivery of newspapers or perishable goods (Applegate et al., 2002), planning of third-party logistics services to convenience stores (Liu et al., 2006), periodic metermaintenance routes (Mendoza et al., 2009), home healthcare logistics (Liu et al., 2013), soft drink distribution (Martínez-Salazar et al., 2014), and humanitarian relief (Golden et al., 2014). It is also relevant in waste collection management. ...
The focus of this paper is the Smart Waste Collection Routing Problem (SWCRP) with workload concerns, a variant of the well-known Vehicle Routing Problem (VRP). Specifically, we propose a solution methodology to address medium to large size problems consisting of two phases. In the first phase, a look-ahead heuristic is used to decide when to perform collection routes considering that real-time information is available through sensors located inside the waste bins. In the second phase, it defines the routes to perform using either an optimisation-based approach or a hybrid metaheuristic approach. A large-size real case study is used to test these approaches. Significant improvements on the key performance indicators characterising the waste collection operations demonstrate the benefits that can be achieved through the proposed solution methodology, within balanced collection routing plans are designed. Moreover, computational and managerial insights about adding workload concerns and performing multi-municipality versus single-municipality operations are provided for the case study.
... The complex nature of the problem frequently leads to solution approaches involving the combination of separate procedures including heuristics. The relevance of balancing workload in VRPs is evident in different other reallife contexts, such as delivery of newspapers or perishable goods (Applegate et al. 2002), planning of third-party logistics services to convenience stores (Liu et al. 2006), periodic meter-maintenance routes (Mendoza et al. 2009), home healthcare logistics (Liu et al. 2013), soft drink distribution (Martínez-Salazar et al. 2014), and humanitarian relief (Golden et al. 2014). It is also relevant in waste collection management. ...
This thesis focuses on the waste collection field, which has become an increasingly important concern in global economies. Due to a high uncertainty associated with waste accumulation, waste collection is usually performed in an inaccurate, inefficient, and expensive way. To reduce the associated uncertainty, volumetric sensors can be installed inside waste bins to transmit real-time information about the amount of waste inside them. The sensor information may be used to feed computer systems based on operational research techniques so that smart waste collection routes are designed, resulting in a more efficient operation with higher service levels.
In this thesis, optimization-based tools are developed to optimize waste collection by defining smart waste collection routes that take into account the fill levels of the bins and their locations in order to maximize the amount of waste collected while minimizing the total distance travelled, considering different planning horizons (short and medium term). Furthermore, a solution methodology is developed to address the decision on which bins, from the whole set of waste bins, should receive a sensor, considering both the expensive investment value of the sensor and the economic gain provided by the early knowledge of information about the bin fill levels. In this thesis, solution methods are developed to solve real large-scale smart waste collection routing problems, and the proposed approaches are validated using real case studies.
This thesis contributes to waste collection planning managers, who based on the results obtained by applying the developed optimization approaches to real case studies can understand the impact these policies may have on the design of waste collection routes. Also, this thesis contributes to waste collection practice, which through studying the problem of selecting a sample of bins to monitor, considering the expensive cost of investment, can be informed on the higher value of the information handled.
... (minsum mTSP) min F (ϕ) = m k=1 T SP (r k ) subject to ∪ m k=1 r k = V r k ∩ r k = {0}, k = k , 1 ≤ k, k ≤ m (1) where ϕ = {r 1 , r 2 , . . . , r m } is a feasible solution with r k (k ∈ {1, · · · , m}) representing the kth tour composed of the vertices visited by the kth salesman, and T SP (r k ) is the length of the tour r k . ...
... For example, a branch-and-cut algorithm [1] was presented to solve a minmax vehicle routing problem on instances up to 120 cities and 4 vehicles. Benders decomposition algorithms [5] were proposed to optimally solve the mTSP with load balancing on instances with up to 171 cities and 10 salesmen. ...
We present an effective hybrid algorithm with neighborhood reduction for solving the multiple traveling salesman problem (mTSP). This problem aims to optimize one of the two objectives: to minimize the total traveling distance (the minsum mTSP) or to minimize the longest tour (the minmax mTSP). The proposed algorithm hybridizes inter-tour optimization with an efficient neighborhood search based on tabu search and intra-tour optimization using the traveling salesman heuristic EAX. A dedicated neighborhood reduction strategy is introduced to avoid the examination of non-promising candidate solutions and thus speed up the neighborhood search. Results of extensive computational experiments are shown on 41 popular instances from several sources and 36 new large instances. Comparisons with five state-of-the-art methods in the literature demonstrate a high competitiveness of the proposed algorithm. Additional experiments on applying a classical TSP heuristic to the minsum mTSP instances show excellent results.
