New tighter polynomial length formulations for the asymmetric traveling salesman problem with and without precedence constraints

We propose a new formulation for the asymmetric traveling salesman problem, with and without precedence relationships, which employs a polynomial number of subtour elimination constraints that imply an exponential subset of certain relaxed Dantzig-Fulkerson-Johnson subtour constraints. Promising computational results are presented, particularly in the presence of precedence constraints.     

An exact algorithm for solving a capacitated location-routing problem

In location-routing problems, the objective is to locate one or many depots within a set of sites (representing customer locations or cities) and to construct delivery routes from the selected depot or depots to the remaining sites at least system cost. The objective function is the sum of depot operating costs, vehicle acquisition costs and routing costs. This paper considers one such problem in which a weight is assigned to each site and where sites are to be visited by vehicles having a given capacity. The solution must be such that the sum of the weights of sites visited on any given route does not exceed the capacity of the visiting vehicle. The formulation of an integer linear program for this problem involves degree constraints, generalized subtour elimination constraints, and chain barring constraints. An exact algorithm, using initial relaxation of most of the problem constraints, is presented which is capable of solving problems with up to twenty sites within a reasonable number of iterations.     

The sector design and assignment problem for snow disposal operations

Winter road maintenance operations involve a host of decision-making problems at the strategic, tactical, operational, and real-time levels. Those operations include spreading of chemicals and abrasives, snow plowing, loading snow into trucks, and hauling snow to disposal sites. In this paper, we present a model and two heuristic solution approaches based on mathematical optimization for the problem of partitioning a road network into sectors and allocating sectors to snow disposal sites for snow disposal operations. Given a road network and a set of planned disposal sites, the problem is to determine a set of non-overlapping subnetworks, called sectors, according to several criteria related to the operational effectiveness and the geographical layout, and to assign each sector to a single snow disposal site so as to respect the capacities of the disposal sites, while minimizing relevant variable and fixed costs. Our approach uses single street segments as the units of analysis and we consider sector contiguity, sector balance and sector shape constraints, hourly and annual disposal site capacities, as well as single assignment requirements. The resulting model is based on a multi-commodity network flow structure to impose the contiguity constraints in a linear form. The two solution approaches were tested on data from the city of Montreal in Canada.     

Integer linear programming formulations of multiple salesman problems and its variations

In this paper, we extend the classical multiple traveling salesman problem (mTSP) by imposing a minimal number of nodes that a traveler must visit as a side condition. We consider single and multidepot cases and propose integer linear programming formulations for both, with new bounding and subtour elimination constraints. We show that several variations of the multiple salesman problem can be modeled in a similar manner. Computational analysis shows that the solution of the multidepot mTSP with the proposed formulation is significantly superior to previous approaches.     

Subtour elimination constraints imply a matrix-tree theorem SDP constraint for the TSP

De Klerk et al., (2008) give a semidefinite programming constraint for the Traveling Salesman Problem (TSP) based on the matrix-tree theorem. This constraint says that the aggregate weight of all spanning trees in a solution to a TSP relaxation is at least that of a cycle graph. In this note, we show that the semidefinite constraint holds for any weighted 2-edge-connected graph and, in particular, is implied by the subtour elimination constraints.     

Optimizing over the subtour polytope of the travelling salesman problem

A commonly studied relaxation of the travelling salesman problem is obtained by adding subtour elimination constraints to the constraints of a 2-factor problem and removing the integrality requirement. We investigate the problem of solving this relaxation for a special type of objective function. We also discuss some ways in which this relates to the concept of rank introduced by Chvátal.     

An Exact Reformulation Algorithm for Large Nonconvex NLPs Involving Bilinear Terms

Many nonconvex nonlinear programming (NLP) problems of practical interest involve bilinear terms and linear constraints, as well as, potentially, other convex and nonconvex terms and constraints. In such cases, it may be possible to augment the formulation with additional linear constraints (a subset of Reformulation-Linearization Technique constraints) which do not affect the feasible region of the original NLP but tighten that of its convex relaxation to the extent that some bilinear terms may be dropped from the problem formulation. We present an efficient graph-theoretical algorithm for effecting such exact reformulations of large, sparse NLPs. The global solution of the reformulated problem using spatial Branch-and Bound algorithms is usually significantly faster than that of the original NLP. We illustrate this point by applying our algorithm to a set of pooling and blending global optimization problems.     

A stabilized column generation scheme for the traveling salesman subtour problem

Given an undirected graph with edge costs and both revenues and weights on the vertices, the traveling salesman subtour problem is to find a subtour that includes a depot vertex, satisfies a knapsack constraint on the vertex weights, and that minimizes edge costs minus vertex revenues along the subtour.We propose a decomposition scheme for this problem. It is inspired by the classic side-constrained 1-tree formulation of the traveling salesman problem, and uses stabilized column generation for the solution of the linear programming relaxation. Further, this decomposition procedure is combined with the addition of variable upper bound (VUB) constraints, which improves the linear programming bound. Furthermore, we present a heuristic procedure for finding feasible subtours from solutions to the column generation problems. An extensive experimental analysis of the behavior of the computational scheme is presented.     

The asymmetric travelling salesman problem and a reformulation of the Miller–Tucker–Zemlin constraints

In this paper we compare the linear programming (LP) relaxations of several old and new formulations for the asymmetric travelling salesman problem (ATSP). The main result of this paper is the derivation of a compact formulation whose LP relaxation is characterized by a set of circuit inequalities given by Grotschel and Padberg (In: Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A., Shmoys, D.B. (Eds.), The Travelling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, New York, 1985). The new compact model is an improved and disaggregated version of a well-known model for the ATSP based on the subtour elimination constraints (Miller et al., Journal of ACM 7 (1960) 326–329). The circuit inequalities are weaker than the subtour elimination constraints given by Dantzig et al. However, each one of these circuit inequalities can be lifted into several different facet defining inequalities which are not dominated by the subtour elimination inequalities. We show that some of the inequalities involved in the previously mentioned compact formulation can be lifted in such a way that, by projection, we obtain a small subset of the so-called D_k^← and D_k^→ inequalities. This shows that the LP relaxation of our strongest model is not dominated by the LP relaxation of the model presented by Dantzig et al. (Operations Research 2 (1954) 393–410). The new models motivate a new classification of formulations for the ATSP.     

Partial linear characterizations of the asymmetric travelling salesman polytope

We consider the linear programming formulation of the asymmetric travelling salesman problem. Several new inequalities are stated which yield a sharper characterization in terms of linear inequalities of the travelling salesman polytope, i.e., the convex hull of tours. In fact, some of the new inequalities as well as some of the well-known subtour elimination constraints are indeed facets of the travelling salesman polytope, i.e., belong to the class of inequalities that uniquely characterize the convex hull of tours to an-city problem.     

A class of lifted path and flow-based formulations for the asymmetric traveling salesman problem with and without precedence constraints

In this paper, we present a new class of polynomial length formulations for the asymmetric traveling salesman problem (ATSP) by lifting an ordered path-based model using logical restrictions in concert with the Reformulation–Linearization Technique (RLT). We show that a relaxed version of this formulation is equivalent to a flow-based ATSP model, which in turn is tighter than the formulation based on the exponential number of Dantzig–Fulkerson–Johnson (DFJ) subtour elimination constraints. The proposed lifting idea is applied to derive a variety of new formulations for the ATSP, and we explore several dominance relationships among these. We also extend these formulations to include precedence constraints in order to enforce a partial order on the sequence of cities to be visited in a tour. Computational results are presented to exhibit the relative tightness of our formulations and the efficacy of the proposed lifting process.     

Finding low cost TSP and 2-matching solutions using certain half-integer subtour vertices

Consider the traveling salesman problem (TSP) defined on the complete graph, where the edge costs satisfy the triangle inequality. Let TOUR denote the optimal solution value for the TSP. Two well-known relaxations of the TSP are the subtour elimination problem and the 2-matching problem. If we let SUBT and 2M represent the optimal solution values for these two relaxations, then it has been conjectured that TOUR/SUBT ≤4/3, and that 2M/SUBT ≤10/9.In this paper, we exploit the structure of certain 1/2-integer solutions for the subtour elimination problem in order to obtain low cost TSP and 2-matching solutions. In particular, we show that for cost functions for which the optimal subtour elimination solution found falls into our classes, the above two conjectures are true. Our proofs are constructive and could be implemented in polynomial time, and thus, for such cost functions, provide a 4/3 (or better) approximation algorithm for the TSP.     

Formulations and Valid Inequalities for the Heterogeneous Vehicle Routing Problem

We consider the vehicle routing problem where one can choose among vehicles with different costs and capacities to serve the trips. We develop six different formulations: the first four based on Miller-Tucker-Zemlin constraints and the last two based on flows. We compare the linear programming bounds of these formulations. We derive valid inequalities and lift some of the constraints to improve the lower bounds. We generalize and strengthen subtour elimination and generalized large multistar inequalities.     

New formulations of the Hop-Constrained Minimum Spanning Tree problem via Miller-Tucker-Zemlin constraints

Given an undirected network with positive edge costs and a natural number p, the Hop-Constrained Minimum Spanning Tree problem (HMST) is the problem of finding a spanning tree with minimum total cost such that each path starting from a specified root node has no more than p hops (edges). In this paper, we develop new formulations for HMST. The formulations are based on Miller-Tucker-Zemlin (MTZ) subtour elimination constraints, MTZ-based liftings in the literature offered for HMST, and a new set of topology-enforcing constraints. We also compare the proposed models with the MTZ-based models in the literature with respect to linear programming relaxation bounds and solution times. The results indicate that the new models give considerably better bounds and solution times than their counterparts in the literature and that the new set of constraints is competitive with liftings to MTZ constraints, some of which are based on well-known, strong liftings of Desrochers and Laporte (1991).     

Value-Estimation Function Method for Constrained Global Optimization

A novel value-estimation function method for global optimization problems with inequality constraints is proposed in this paper. The value-estimation function formulation is an auxiliary unconstrained optimization problem with a univariate parameter that represents an estimated optimal value of the objective function of the original optimization problem. A solution is optimal to the original problem if and only if it is also optimal to the auxiliary unconstrained optimization with the parameter set at the optimal objective value of the original problem, which turns out to be the unique root of a basic value-estimation function. A logarithmic-exponential value-estimation function formulation is further developed to acquire computational tractability and efficiency. The optimal objective value of the original problem as well as the optimal solution are sought iteratively by applying either a generalized Newton method or a bisection method to the logarithmic-exponential value-estimation function formulation. The convergence properties of the solution algorithms guarantee the identification of an approximate optimal solution of the original problem, up to any predetermined degree of accuracy, within a finite number of iterations.     

A simple LP relaxation for the asymmetric traveling salesman problem

It is a long-standing open question in combinatorial optimization whether the integrality gap of the subtour linear program relaxation (subtour LP) for the asymmetric traveling salesman problem (ATSP) is a constant. The study on the structure of this linear program is important and extensive. In this paper, we give a new and simpler LP relaxation for the ATSP. Our linear program consists of a single type of constraints that combine both the subtour elimination and the degree constraints in the traditional subtour LP. As a result, we obtain a much simpler relaxation. In particular, it is shown that the extreme solutions of our program have at most 2n ? 2 non-zero variables, improving the bound 3n ? 2, proved by Vempala and Yannakakis, for the ones obtained by the subtour LP. Nevertheless, the integrality gap of the new linear program is larger than that of the traditional subtour LP by at most a constant factor.     

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.     

Dual formulations and subgradient optimization strategies for linear programming relaxations of mixed-integer programs

This paper examines algorithmic strategies relating to the formulation of Lagrangian duals, and their solution via subgradient optimization, in the context of solving linear programming relaxations of mixed-integer programs. As in the current trend in integer programming, several researchers have found it beneficial to generate and add additional constraints to the model, prior to its solution, in order to tighten its linear programming relaxation. It becomes necessary, therefore, to be able to efficiently derive strong surrogate constraints and bounds based on such reformulations. This paper addresses the design and testing of the most viable technique available for exploiting such tight reformulations, namely, using Lagrangian relaxation in coordination with subgradient optimization. A computational study is conducted to test the efficiency of various Lagrangian dual formulations, and to investigate in the context of using subgradient optimization the effects of step size choices, problem scaling, conducting pattern moves, and projecting dual solutions onto subsets of violated dual constraints. Based on this study, certain recommendations are made regarding the manner in which one should implement such an approach.     

A note on the lifted Miller–Tucker–Zemlin subtour elimination constraints for routing problems with time windows

We propose lifted versions of the Miller–Tucker–Zemlin subtour elimination constraints for routing problems with time windows (TW). The constraints are valid for problems such as the travelling salesman problem with TW, the vehicle routing problem with TW, the generalized travelling salesman problem with TW, and the general vehicle routing problem with TW. They are corrected versions of the constraints proposed by Desrochers and Laporte (1991).     

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.