Solving the traveling repairman problem on a line with general processing times and deadlines

doi:10.1016/j.ejor.2015.02.009

European Journal of Operational Research

Volume 244, Issue 3, 1 August 2015, Pages 690-703

https://doi.org/10.1016/j.ejor.2015.02.009 Get rights and content

Highlights

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Resolving the complexity status of the Line-TRP with processing times and deadlines.
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Proof of strong NP-completeness by a reduction from 3-PARTITION.
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A new practically applicable best-first Branch & Bound procedure is proposed.
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Development of various lower bounds and dominance rules.
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Efficiency of the new approach is validated by a computational study.

Abstract

This paper resolves the complexity status of the well-known Traveling Repairman Problem on a line (Line-TRP) with general processing times at the request locations and deadline restrictions. It has long remained an open research question whether an exact solution procedure with pseudo-polynomial running time can be developed for this version of the Traveling Repairman Problem that was known to be at least binary $N P$ -hard. The presented proof of strong $N P$ -completeness of the problem is provided by a reduction from 3-PARTITION. Since recent literature provides significant new results for further variants of the Line-TRP and the Line-TSP, a brief updated overview of the complexity status of the different variants is given. Another major contribution is that a practically applicable exact best-first search Branch&Bound approach that optimally solves instances of real-world size in reasonable time is proposed. By applying sophisticated dominance rules as well as lower bounds, the number of enumerated partial solutions is kept limited. The efficiency of the new approach and the applied instruments is validated by a computational study.

Section snippets

Introduction and literature review

In this paper, the Line-Traveling Repairman Problem (Line-TRP) is considered. In the literature this problem is also known as the Delivery Man Problem or the Minimum Latency Problem (Luo, Qin, Lim, 2014, Sitters, 2004). In this problem the pure delivery or pickup tour of a single vehicle has to be planned in order to minimize the total customer waiting time. If, however, the objective is to minimize the tour length or tour duration the Line-TRP becomes the Line-Traveling Salesman Problem

Problem formulation

In this section, the considered Line-Traveling Repairman Problem with general processing or handling times and deadlines is mathematically defined.

Complexity analysis

In what follows, we prove that the Line-TRP with general processing times and request deadlines is strongly $N P$ -hard by using a reduction from 3-PARTITION (which is known to be strongly $N P$ -hard (Garey and Johnson, 1979)). By proving that this problem is $N P$ -hard, the existence of a solution procedure that guarantees an optimal solution in pseudo-polynomial time is ruled out, unless it holds that $P = N P$ .

An instance of 3-PARTITION is defined by an integer $B \in Z^{+}$ and a set A with $| A | = 3 q$ . Set A contains

A best-first Branch&Bound approach

This section introduces a new best-first Branch&Bound procedure. It solves the Line-TRP with deadlines and general processing times to optimality. According to the mathematical definition of the problem in Section 2, (partial) schedulesα that are explored by this procedure are stored as partial sequences of requests that are assigned to their final position.

In what follows, a formal definition of a (partial) schedule is provided. We assume that all requests are sorted according to

Computational results

In what follows, the efficiency of the introduced Branch&Bound approach is validated. Specifically, the effectiveness of the proposed dominance rules as well as of the lower bounds is analyzed in detail. For this purpose, this section provides a brief introduction of the utilized test environment, the generated test instances, and the applied versions of the Branch&Bound approach.

Conclusions

This paper resolves the complexity status of the well-known Traveling Repairman Problem on a line with general processing times at the request locations and deadlines. This had remained an open research question for a long time. In the paper, the problem variant is found to be strongly $N P$ -hard. Therefore, there is no exact solution procedure with strongly polynomial or pseudo-polynomial running time possible unless it holds that $N P = P$ . Apart from this significant result, the paper proposes the

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View full text

Discrete OptimizationSolving the traveling repairman problem on a line with general processing times and deadlines

Highlights

Abstract

Section snippets

Introduction and literature review

Problem formulation

Complexity analysis

A best-first Branch&Bound approach

Computational results

Conclusions

European Journal of Operational Research

European Journal of Operational Research

Annals of Discrete Mathematics

European Journal of Operational Research

The complexity of the traveling repairman problem.

Theoretical Informatics and Applications

Network flows

Minimizing sequence-dependent setup costs in feeding batch processes under due date restriction

Journal of Scheduling

A constant-factor approximation algorithm for the k-MST problem

Proceedings of the twenty-eighth annual ACM symposium on theory of computing (STOC 1996)

Computer-aided complexity classification of dial-a-ride problems.

INFORMS Journal on Computing

Optimally loading tow trains for just-in-time supply of mixed-model assembly lines.

IIE Transactions

Two-processor scheduling with start times and deadlines.

SIAM Journal on Computing

Discrete Optimization
Solving the traveling repairman problem on a line with general processing times and deadlines