Subtour Elimination Research Articles

A note on the Miller-Tucker-Zemlin model for the asymmetric traveling salesman problem

Abstract An enhancement of the Miller-Tucker-Zemlin (MTZ) model for the asymmetric traveling salesman problem is presented by introducing additional constraints to the initial formulation. The constraints account for ordering of boundary nodes as well as all successive nodes in the salesman tour. The enhanced MTZ subtour elimination constraints are computationally compared with the basic MTZ constraints and the version of MTZ lifted by Desrochers and Laporte. The proposed enhancement shows improved performance on a number of asymmetric TSPLIB instances.

An Improved Integrality Gap for Asymmetric TSP Paths

The asymmetric traveling salesperson path problem (ATSPP) is one where, given an asymmetric metric space (V, d) with specified vertices s and t, the goal is to find an s-t path of minimum length that passes through all the vertices in V. This problem is closely related to the asymmetric TSP (ATSP), which seeks to find a tour (instead of an s-t path) visiting all the nodes: for ATSP, a ρ-approximation guarantee implies an O(ρ)-approximation for ATSPP. However, no such connection is known for the integrality gaps of the linear programming (LP) relaxations for these problems: the current-best approximation algorithm for ATSPP is O(ln n/ln ln n), whereas the best bound on the integrality gap of the natural LP relaxation (the subtour elimination LP) for ATSPP is O(ln n). In this paper, we close this gap, and improve the current best bound on the integrality gap from O(ln n) to O(ln n/ln ln n). The resulting algorithm uses the structure of narrow s-t cuts in the LP solution to construct a (random) tree spanning tree that can be cheaply augmented to contain an Eulerian s-t walk. We also build on a result of Oveis Gharan and Saberi and show a strong form of Goddyn’s conjecture about thin spanning trees implies the integrality gap of the subtour elimination LP relaxation for ATSPP is bounded by a constant. Finally, we give a simpler family of instances showing the integrality gap of this LP is at least 2.

Generating subtour elimination constraints for the TSP from pure integer solutions

The traveling salesman problem (TSP) is one of the most prominent combinatorial optimization problems. Given a complete graph G = (V, E) and non-negative distances d for every edge, the TSP asks for a shortest tour through all vertices with respect to the distances d. The method of choice for solving the TSP to optimality is a branch and cut approach. Usually the integrality constraints are relaxed first and all separation processes to identify violated inequalities are done on fractional solutions. In our approach we try to exploit the impressive performance of current ILP-solvers and work only with integer solutions without ever interfering with fractional solutions. We stick to a very simple ILP-model and relax the subtour elimination constraints only. The resulting problem is solved to integer optimality, violated constraints (which are trivial to find) are added and the process is repeated until a feasible solution is found. In order to speed up the algorithm we pursue several attempts to find as many relevant subtours as possible. These attempts are based on the clustering of vertices with additional insights gained from empirical observations and random graph theory. Computational results are performed on test instances taken from the TSPLIB95 and on random Euclidean graphs.

Simultaneous subtour elimination model for single-stage multiproduct parallel batch scheduling with sequence dependent changeovers

In this paper a mixed-integer linear programming (MILP) model is presented to minimize makespan of single-stage multiproduct parallel batch production with sequence dependent changeovers. The computational inefficiency and suboptimal problems are addressed by the tight and rigorous formulation of the proposed model. Subtours (subcycles) are eliminated simultaneously so that the optimal solution is obtained in one step. The proposed model is tested with two examples. The results show that the model obtains the global optimal solutions with significant improvement in solution time.

Optimal Patrol Planning Using STEN for Road Maintenance

The authors propose a method for road maintenance patrol planning using multiple vehicles, which is modeled as an integer-linear problem. Specifically, the authors consider a model for reducing the number of patrol vehicles and optimizing the patrol routes. Difficulties arise in solving such routing problems because of sub-tour elimination with multiple vehicles. To address this sub-tour problem, the authors use the Space-Time Extended Network (STEN). The authors investigate a virtual network using the proposed method, and describe the behavior of the proposed model. The time required for the calculations is short, so that the system is suitable for practical applications.

The traveling salesman problem with time-dependent service times

This paper introduces a version of the classical traveling salesman problem with time-dependent service times. In our setting, the duration required to provide service to any customer is not fixed but defined as a function of the time at which service starts at that location. The objective is to minimize the total route duration, which consists of the total travel time plus the total service time. The proposed model can handle several types of service time functions, e.g., linear and quadratic functions. We describe basic properties for certain classes of service time functions, followed by the computation of valid lower and upper bounds. We apply several classes of subtour elimination constraints and measure their effect on the performance of our model. Numerical results obtained by implementing different linear and quadratic service time functions on several test instances are presented.

Quadratic assignment problem mathematical modelling for process planning

Increased global competition and frequent unpredictable market changes are current challenges facing manufacturing enterprises. Part design and engineering specifications changes trigger frequent and costly changes in process plans, which often require changes in their manufacturing system. Process planning is a key support function that should be further developed to cope with these challenges. A sequential hybrid approach at the macro-level has been exploited. At the heart of the proposed method, a new mathematical model based on the popular Quadratic Assignment Problem (QAP) is solved, where sub-operations are assigned positions and clustered to represent operations in one-dimensional space. A linearisation of the quadratic model is performed and, hence, solved for optimality. The proposed model cures the conceptual flaws in the classical solutions of the Traveling Salesperson Problem (TSP). It also overcomes the complexity of the sub-tour elimination constraints and, for the first time, mathematically formulates precedence constraints, which is a cornerstone of the process planning problem. The developed methods, their limitations and merits are conceptually and computationally analysed, compared and validated against other models in the literature using detailed industrial case studies. General algebraic modeling system language, its SBB mixed-integer non-linear programming (MINLP) solver and CPLEX solvers are used. The presented innovative new concepts and novel formulations represent significant contributions to knowledge in the field of process planning. Their effectiveness and applicability are validated in different domains of applications, namely metal cutting, inspection and assembly.

A relax and cut approach using the multi-commodity flow formulation for the traveling salesman problem

<p class="ADYNAAbstrac"><span lang="EN-US">In this paper we explore the multi-commodity flow formulation for the Asymmetric Traveling Salesman Problem (ATSP) to obtain dual bounds. The procedure employed is a variant of a relax and cut procedure proposed in the literature that computes the Lagrangean multipliers associated to the subtour elimination constraints preserving the optimality of the multipliers associated to the assignment constraints. The results obtained by the computational study are encouraging and show that the proposed algorithm generated good dual bounds for the ATSP with a low execution time.</span></p>

A matheuristic algorithm for the three-dimensional loading capacitated vehicle routing problem (3L-CVRP)

This paper presents a hybrid algorithm for solving the Capacitated Vehicle Routing Problem with practical three-dimensional loading constraint. This problem is known as 3L-CVRP (Three-dimensional Loading Capacitated Vehicle Routing Problem). The proposed methodology consists of two phases. The first phase uses an optimization procedure based on cuts to obtain solutions for the well-known Capacitated Vehicle Routing Problem (CVRP). The second phase validates the results of the first phase of a GRASP algorithm (Greedy Randomized Adaptive Search Procedure). In particular, the GRASP approach evaluates the packing constraints for each performed route of the CVRP. The proposed hybrid algorithm uses a relaxation of the classical model of two sub-indices for the vehicle routing problem. Specifically different types of cuts are added: subtour elimination, capacity-cut constraints, and packing-cut constrains. The proposed algorithm is compared with the most efficient approaches for the 3L-CVRP on the set of benchmark instances considered in the literature. The computational results indicate that the proposed approach is able to obtain good solutions, improving some of the best-known solutions from the literature.

A generalization of extension complexity that captures P

In this paper we propose a generalization of the extension complexity of a polyhedron Q. On the one hand it is general enough so that all problems in P can be formulated as linear programs with polynomial size extension complexity. On the other hand it still allows non-polynomial lower bounds to be proved for NP-hard problems independently of whether or not P=NP. The generalization, called H-free extension complexity, allows for a set of valid inequalities H to be excluded in computing the extension complexity of Q. We give results on the H-free extension complexity of hard matching problems (when H are the odd-set inequalities) and the traveling salesman problem (when H are the subtour elimination constraints).

A Study of Different Subsequence Elimination Strategies for the Soft Drink Production Planning

The production of soft drinks involves two main stages: syrup preparation and bottling. To obtain the lots sequence inthe bottling stage, three approaches are studied. They are based on the sub-tour elimination constraints used inmathematical models for the Asymmetric Traveling Salesman Problem. Two of the mathematical models are from theliterature and use classical constraints. The third model includes multi-commodity flow constraints to eliminatedisconnected subsequences. The computational behavior of the three models is studied using instances generatedwith data from the literature. The numerical results show that there are considerable differences among the threemodels and indicates that the multi-commodity formulation provides good results but it requires far morecomputational effort when the instances are solved by a commercial software.

A Node-based Mathematical Model towards the Location Routing Problem with Intermediate Replenishment Facilities under Capacity Constraint

In this paper, we study the location routing problem with intermediate replenishment facilities (LRPIRF), an extension of the location routing problem (LRP), where the vehicles can replenish at some intermediate facilities. Vehicles leave the depot with load on-board, serve customers until out of load, may return to an intermediate facility to replenish, and finally return to the depot, completing their route. In this paper, we initiate a mathematical mixed integer programming model with new kind of subtour elimination constraints for this problem. Moreover, the facility location phase is considered besides vehicle routing phase in our problem. The objective of the problem is to find routes for vehicles to serve all customers at a minimal cost in terms of total travel cost and total facility location cost, without violating the capacity constraint of the vehicles. The solution to the LRPIRF is obtained through CPLEX solver in commercial software GAMS 23.5.1 ,Genetic Algorithm and Tabu Search algorithm. Computational results are obtained on a set of randomly generated instances and indicate the effectiveness of the proposed algorithms in terms of solution time and quality.

Modeling lotsizing and scheduling problems with sequence dependent setups

Several production environments require simultaneous planing of sizing and scheduling of sequences of production lots. Integration of sequencing decisions in lotsizing and scheduling problems has received an increased attention from the research community due to its inherent applicability to real world problems. A two-dimensional classification framework is proposed to survey and classify the main modeling approaches to integrate sequencing decisions in discrete time lotsizing and scheduling models. The Asymmetric Traveling Salesman Problem can be an important source of ideas to develop more efficient models and methods to this problem. Following this research line, we also present a new formulation for the problem using commodity flow based subtour elimination constraints. Computational experiments are conducted to assess the performance of the various models, in terms of running times and upper bounds, when solving real-word size instances.

Steiner Ring Star 문제를 해결하기 위한 새로운 Mixed-Integer Programming Modeling

In this paper, we deal with a Steiner Ring Star (SRS) problem arising from the design of survivable telecommunication networks. We develop two mixed integer programming formulations for the SRS problem by implementing MillerTucker-Zemlin (MTZ) and Sarin-Sherali-Bhootra (SSB) subtour elimination constraints, and then apply the reformulationlinearization technique (RLT) to enhance the lower bound obtained by the LP relaxation. By exploiting the ring-star structure of underlying network, we devise some valid inequalities that tighten the LP relaxation. Computational results demonstrate the effectiveness of the proposed solution procedure. Keyword:Steiner Ring Star Problem, Mixed Integer Programming, Reformulation-Linearization Technique 한국경영과학회지 제39권 제1호 2014년 3월

Robust FIPP p-cycles against dual link failures

We propose a new generic flow formulation for Failure-Independent Path-Protecting (FIPP) p-cycles subject to multiple failures. While our new model resembles the decomposition model formulation proposed by Orlowski and Pioro (Networks, 2011) in the case of classical shared path protection, its originality lies in its adaptation to FIPP p-cycles. When adapted to that last pre-configured pre-cross connected protection scheme, the bandwidth sharing constraints must be handled in a different way in order to take care of the sharing along the FIPP p-cycles. It follows that, instead of a polynomial-time solvable pricing problem as in the model of Orlowski and Pioro (Networks, 2011), we end up with a much more complex pricing problem, which has an exponential number of constraints due to some subtour elimination constraints. Consequently, in order to efficiently solve the pricing problem, we consider: (i) a hierarchical decomposition of the original pricing problem; (ii) heuristics in order to go around the large number of constraints in the pricing problem. Performance evaluation is made in the case of FIPP p-cycles subject to dual failures. For small to medium size networks, the proposed model remains fairly scalable for increasing percentages of dual failures, and requires much less bandwidth than p-cycle protection schemes (ratio varies from 2 to 4). For larger networks, heuristics are required in order to keep computing times reasonable. In the particular case of single link failures, it compares very favorably (5 to 10 % of bandwidth saving) to the previously proposed column generation ILP model of Rocha, Jaumard and Stidsen (Telecommun. Syst., 2012).

An extended approach for lifting clique tree inequalities

We present a new lifting approach for strengthening arbitrary clique tree inequalities that are known to be facet defining for the symmetric traveling salesman problem in order to get stronger valid inequalities for the symmetric quadratic traveling salesman problem (SQTSP). Applying this new approach to the subtour elimination constraints (SEC) leads to two new classes of facet defining inequalities of SQTSP. For the special case of the SEC with two nodes we derive all known conflicting edges inequalities for SQTSP. Furthermore we extend the presented approach to the asymmetric quadratic traveling salesman problem (AQTSP).

Requiem for the Miller–Tucker–Zemlin subtour elimination constraints?

The Miller–Tucker–Zemlin (MTZ) Subtour Elimination Constraints (SECs) and the improved version by Desrochers and Laporte (DL) have been and are still in regular use to model a variety of routing problems. This paper presents a systematic way of deriving inequalities that are more complicated than the MTZ and DL inequalities and that, in a certain way, “generalize” the underlying idea of the original inequalities. We present a polyhedral approach that studies and analyses the convex hull of feasible sets for small dimensions. This approach allows us to generate generalizations of the MTZ and DL inequalities, which are “good” in the sense that they define facets of these small polyhedra. It is well known that DL inequalities imply a subset of Dantzig–Fulkerson–Johnson (DFJ) SECs for two-node subsets. Through the approach presented, we describe a generalization of these inequalities which imply DFJ SECs for three-node subsets and show that generalizations for larger subsets are unlikely to exist. Our study presents a similar analysis with generalizations of MTZ inequalities and their relation with the lifted circuit inequalities for three node subsets.

A note on the separation of subtour elimination constraints in elementary shortest path problems

This note proposes an alternative procedure for identifying violated subtour elimination constraints (SECs) in branch-and-cut algorithms for elementary shortest path problems. The procedure is also applicable to other routing problems, such as variants of travelling salesman or shortest Hamiltonian path problems, on directed graphs. The proposed procedure is based on computing the strong components of the support graph. The procedure possesses a better worst-case time complexity than the standard way of separating SECs, which uses maximum flow algorithms, and is easier to implement.

The traveling salesman problem on cubic and subcubic graphs

We study the traveling salesman problem (TSP) on the metric completion of cubic and subcubic graphs, which is known to be NP-hard. The problem is of interest because of its relation to the famous 4/3-conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal value of a TSP instance and that of its linear programming relaxation (the subtour elimination relaxation), is 4/3. We present the first algorithm for cubic graphs with approximation ratio 4/3. The proof uses polyhedral techniques in a surprising way, which is of independent interest. In fact we prove constructively that for any cubic graph on TeX vertices a tour of length TeX exists, which also implies the 4/3-conjecture, as an upper bound, for this class of graph-TSP. Recently, Momke and Svensson presented an algorithm that gives a 1.461-approximation for graph-TSP on general graphs and as a side result a 4/3-approximation algorithm for this problem on subcubic graphs, also settling the 4/3-conjecture for this class of graph-TSP. The algorithm by Momke and Svensson is initially randomized but the authors remark that derandomization is trivial. We will present a different way to derandomize their algorithm which leads to a faster running time. All of the latter also works for multigraphs.

Refinery Production Planning: Multiperiod MINLP with Nonlinear CDU Model

In a previous paper, Alattas, Grossmann, and Palou-Rivera (2011) developed a single-period, nonlinear programing refinery planning model using the fractionation index (FI) for the crude distillation unit (CDU) equations. In this paper, the single period model is modified to a mixed-integer nonlinear programming (MINLP) model to determine the sequencing, changeovers, and processing times of crude oils over multiple time periods. The MINLP equations include traveling salesman problem constraints to generate the crude oil sequences that maximize profit. Moreover, the disjunction for the fractionation index (FI) is formulated with mixed integer constraints as opposed to the Heaviside function formulation of the previous work. The resulting model is shown to be robust and relatively fast. When subcycles arise, they are eliminated by adding appropriate subtour elimination constraints. Examples with up to 5 crude oils and 6 weeks time horizon are presented to illustrate the application of the proposed model.