Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry

Semidefinite programming (SDP) bounds for the quadratic assignment problem (QAP) were introduced in Zhao et?al. (J Comb Optim 2:71–109, 1998). Empirically, these bounds are often quite good in practice, but computationally demanding, even for relatively small instances. For QAP instances where the data matrices have large automorphism groups, these bounds can be computed more efficiently, as was shown in Klerk and Sotirov (Math Program A, 122(2), 225–246, 2010). Continuing in the same vein, we show how one may obtain stronger bounds for QAP instances where one of the data matrices has a transitive automorphism group. To illustrate our approach, we compute improved lower bounds for several instances from the QAP library QAPLIB.     

Lagrangean bounds for the optimum communication spanning tree problem

This paper considers the Optimum Communication Spanning Tree Problem. An integer programming formulation that yields tight LP bounds is proposed. Given that the computational effort required to obtain the LP bounds considerably increases with the size of the instances when using commercial solvers, we propose a Lagrangean relaxation that exploits the structure of the formulation. Since feasible solutions to the Lagrangean function are spanning trees, upper bounds are also obtained. These bounds are later improved with a simple local search. Computational experiments have been run on several benchmark instances from the literature. The results confirm the interest of the proposal since tight lower and upper bounds are obtained, for instances up to 100 nodes, in competitive computational times.     

The variable neighborhood search for the two machine flow shop problem with a passive prefetch

We consider the two machine flow shop scheduling problem with passive loading of the buffer on the second machine. To compute lower bounds for the global optimum, we present four integer linear programming formulations of the problem. Three local search methods with variable neighborhoods are developed for obtaining upper bounds. Some new large neighborhood is designed. Our methods use this neighborhood along with some other well-known neighborhoods. For computational experiments, we present a new class of test instances with known global optima. Computational results indicate a high efficiency of the proposed approach for the new class of instances as well as for other classes of instances of the problem.     

Relaxations and discretizations for the pooling problem

The pooling problem is a folklore NP-hard global optimization problem that finds applications in industries such as petrochemical refining, wastewater treatment and mining. This paper assimilates the vast literature on this problem that is dispersed over different areas and gives new insights on prevalent techniques. We also present new ideas for computing dual bounds on the global optimum by solving high-dimensional linear programs. Finally, we propose discretization methods for inner approximating the feasible region and obtaining good primal bounds. Valid inequalities are derived for the discretized models, which are formulated as mixed integer linear programs. The strength of our relaxations and usefulness of our discretizations is empirically validated on random test instances. We report best known primal bounds on some of the large-scale instances.     

An algorithm and upper bounds for the weighted maximal planar graph problem

In this paper, we investigate the weighted maximal planar graph (WMPG) problem. Given a complete, edge-weighted, simple graph, the WMPG problem involves finding a subgraph with the highest sum of edge weights that is maximal planar, namely, it can be embedded in the plane without any of its edges intersecting, and no additional edge can be added to the subgraph without violating its planarity. We present a new integer linear programming (ILP) model for this problem. We then develop a cutting-plane algorithm to solve the WMPG problem based on the proposed ILP model. This algorithm enables the problem to be solved more efficiently than previously reported algorithms. New upper bounds are also provided, which are useful in evaluating the quality of heuristic solutions or in generating initial solutions for meta-heuristics. Computational results are reported for a set of 417 test instances of size varying from 6 to 100 nodes including 105 instances from the literature and 312 randomly generated instances. The computational results indicate that instances with up to 24 nodes can be solved optimally in reasonable computational time and the new upper bounds for larger instances significantly improve existing upper bounds.     

Maximum-entropy sampling and the Boolean quadric polytope

We consider a bound for the maximum-entropy sampling problem (MESP) that is based on solving a max-det problem over a relaxation of the Boolean quadric polytope (BQP). This approach to MESP was first suggested by Christoph Helmberg over 15 years ago, but has apparently never been further elaborated or computationally investigated. We find that the use of a relaxation of BQP that imposes semidefiniteness and a small number of equality constraints gives excellent bounds on many benchmark instances. These bounds can be further tightened by imposing additional inequality constraints that are valid for the BQP. Duality information associated with the BQP-based bounds can be used to fix variables to 0/1 values, and also as the basis for the implementation of a “strong branching” strategy. A branch-and-bound algorithm using the BQP-based bounds solves some benchmark instances of MESP to optimality for the first time.     

Improved lower bounds for the Split Delivery Vehicle Routing Problem

This paper presents an algorithm to obtain lower bounds for the Split Delivery Vehicle Routing Problem. An extended formulation over a large set of variables is provided and valid inequalities are identified. The algorithm combined column and cut generation and improved the best known lower bounds for all instances from the literature. Some reasonably sized instances are solved to optimality for the first time.     

The crane scheduling problem: models and solution approaches

In this paper, we study the crane scheduling problem for a vessel after the vessel is moored on a terminal and develop both exact and heuristic solution approaches for the problem. For small-sized instances, we develop a time-space network flow formulation with non-crossing constraints for the problem and apply an exact solution approach to obtain an optimal solution. For medium-sized instances, we develop a Lagrangian relaxation approach that allows us to obtain tight lower bounds and near-optimal solutions. For large-sized instances, we develop two heuristics and show that the error bounds of our heuristics are no more than 100%. Finally, we perform computational studies to show the effectiveness of our proposed solution approaches.     

On the Applicability of Lower Bounds for Solving Rectilinear Quadratic Assignment Problems in Parallel

The quadratic assignment problem (QAP) belongs to the hard core of NP-hard optimization problems. After almost forty years of research only relatively small instances can be solved to optimality. The reason is that the quality of the lower bounds available for exact methods is not sufficient. Recently, lower bounds based on decomposition were proposed for the so called rectilinear QAP that proved to be the strongest for a large class of problem instances. We investigate the strength of these bounds when applied not only at the root node of a search tree but as the bound function used in a Branch-and-Bound code solving large scale QAPs.     

An exact algorithm for large multiple knapsack problems

The Multiple Knapsack Problem (MKP) is the problem of assigning a subset of n items to m distinct knapsacks, such that the total profit sum of the selected items is maximized, without exceeding the capacity of each of the knapsacks. The problem has several applications in naval as well as financial management. A new exact algorithm for the MKP is presented, which is specially designed for solving large problem instances. The recursive branch-and-bound algorithm applies surrogate relaxation for deriving upper bounds, while lower bounds are obtained by splitting the surrogate solution into the m knapsacks by solving a series of Subset-sum Problems. A new separable dynamic programming algorithm is presented for the solution of Subset-sum Problems, and we also use this algorithm for tightening the capacity constraints in order to obtain better upper bounds. The developed algorithm is compared to the mtm algorithm by Martello and Toth, showing the benefits of the new approach. A surprising result is that large instances with n=100 000 items may be solved in less than a second, and the algorithm has a stable performance even for instances with coefficients in a moderately large range.     

Improved lower bounds for the early/tardy scheduling problem with no idle time

In this paper, we consider the single machine earliness/tardiness scheduling problem with no idle time. Two of the lower bounds previously developed for this problem are based on Lagrangean relaxation and the multiplier adjustment method, and require an initial sequence. We investigate the sensitivity of the lower bounds to the initial sequence, and experiment with different dispatch rules and some dominance conditions. The computational results show that it is possible to obtain improved lower bounds by using a better initial sequence. The lower bounds are also incorporated in a branch-and-bound algorithm, and the computational tests show that one of the new lower bounds has the best performance for larger instances.     

Branch-and-cut algorithms for the split delivery vehicle routing problem

In this paper we present two exact branch-and-cut algorithms for the Split Delivery Vehicle Routing Problem (SDVRP) based on two relaxed formulations that provide lower bounds to the optimum. Procedures to obtain feasible solutions to the SDVRP from a feasible solution to the relaxed formulations are presented. Computational results are presented for 4 classes of benchmark instances. The new approach is able to prove the optimality of 17 new instances. In particular, the branch-and-cut algorithm based on the first relaxed formulation is able to solve most of the instances with up to 50 customers and two instances with 75 and 100 customers.     

On multi-item economic lot-sizing with remanufacturing and uncapacitated production

In this study, we consider the multi-item economic lot-sizing problem with remanufacturing and uncapacitated production. By observing that the problem comprises several independent single-item problems, we show how very high quality feasible solutions and bounds can be obtained by solving each item separately using an effective recently proposed approach. Computational experiments demonstrate that our approach improves the best known feasible solutions and lower bounds for all the available instances. In addition, we show that 86 instances can be solved to optimality and the remaining open gap is below 0.5% for almost all the unsolved instances.     

Improved state space relaxation for constrained two-dimensional guillotine cutting problems

Christofides and Hadjiconstantinou (1995) introduced a dynamic programming state space relaxation for obtaining upper bounds for the Constrained Two-dimensional Guillotine Cutting Problem. The quality of those bounds depend on the chosen item weights, they are adjusted using a subgradient-like algorithm. This paper proposes Algorithm X, a new weight adjusting algorithm based on integer programming that provably obtains the optimal weights. In order to obtain even better upper bounds, that algorithm is generalized into Algorithm X2 for obtaining optimal two-dimensional item weights. We also present a full hybrid method, called Algorithm X2D, that computes those strong upper bounds but also provides feasible solutions obtained by: (1) exploring the suboptimal solutions hidden in the dynamic programming matrices; (2) performing a number of iterations of a GRASP based primal heuristic; and (3) executing X2H, an adaptation of Algorithm X2 to transform it into a primal heuristic. Extensive experiments with instances from the literature and on newly proposed instances, for both variants with and without item rotation, show that X2D can consistently deliver high-quality solutions and sharp upper bounds. In many cases the provided solutions are certified to be optimal.     

Survivable network design with shared-protection routing

In this paper we study the problem of designing a survivable telecommunication network with shared-protection routing. We develop a heuristic algorithm to solve this problem. Recent results in the area of global re-routing have been used to obtain very tight lower bounds for the problem. Our results indicate that in a majority of problem instances, the average gap between the heuristic solutions and the lower bounds is within 5%. Computational experience is reported on randomly generated problem instances with up to 35 nodes, 80 edges and 595 demand pairs and also on the instances available in SNDlib database.     

A Boolean satisfiability approach to the resource-constrained project scheduling problem

We formulate the resource-constrained project scheduling problem as a satisfiability problem and adapt a satisfiability solver for the specific domain of the problem. Our solver is lightweight and shows good performance both in finding feasible solutions and in proving lower bounds. Our numerical tests allowed us to close several benchmark instances of the RCPSP that have never been closed before by proving tighter lower bounds and by finding better feasible solutions. Using our method we solve optimally more instances of medium and large size from the benchmark library PSPLIB and do it faster compared to any other existing solver.     

Improved bounds for the traveling umpire problem: A stronger formulation and a relax-and-fix heuristic

Given a double round-robin tournament, the traveling umpire problem (TUP) consists of determining which games will be handled by each one of several umpire crews during the tournament. The objective is to minimize the total distance traveled by the umpires, while respecting constraints that include visiting every team at home, and not seeing a team or venue too often. We strengthen a known integer programming formulation for the TUP and use it to implement a relax-and-fix heuristic that improves the quality of 24 out of 25 best-known feasible solutions to instances in the TUP benchmark. We also improve all best-known lower bounds for those instances and, for the first time, provide lower bounds for instances with more than 16 teams.     

A loose Benders decomposition algorithm for approximating two-stage mixed-integer recourse models

We propose a new class of convex approximations for two-stage mixed-integer recourse models, the so-called generalized alpha-approximations. The advantage of these convex approximations over existing ones is that they are more suitable for efficient computations. Indeed, we construct a loose Benders decomposition algorithm that solves large problem instances in reasonable time. To guarantee the performance of the resulting solution, we derive corresponding error bounds that depend on the total variations of the probability density functions of the random variables in the model. The error bounds converge to zero if these total variations converge to zero. We empirically assess our solution method on several test instances, including the SIZES and SSLP instances from SIPLIB. We show that our method finds near-optimal solutions if the variability of the random parameters in the model is large. Moreover, our method outperforms existing methods in terms of computation time, especially for large problem instances.

     

Lower bounds for the ITC-2007 curriculum-based course timetabling problem

This paper describes an approach for generating lower bounds for the curriculum-based course timetabling problem, which was presented at the International Timetabling Competition (ITC-2007, Track 3). So far, several methods based on integer linear programming have been proposed for computing lower bounds of this minimization problem. We present a new partition-based approach that is based on the “divide and conquer” principle. The proposed approach uses iterative tabu search to partition the initial problem into sub-problems which are solved with an ILP solver. Computational outcomes show that this approach is able to improve on the current best lower bounds for 12 out of the 21 benchmark instances, and to prove optimality for 6 of them. These new lower bounds are useful to estimate the quality of the upper bounds obtained with various heuristic approaches.     

A hybrid imperialist competitive algorithm for minimizing makespan in a multi-processor open shop

In this paper a new mixed-integer linear programming (MILP) model is proposed for the multi-processor open shop scheduling (MPOS) problems to minimize the makespan with considering independent setup time and sequence dependent removal time. A hybrid imperialist competitive algorithm (ICA) with genetic algorithm (GA) is presented to solve this problem. The parameters of the proposed algorithm are tuned by response surface methodology (RSM). The performance of the algorithm to solve small, medium and large sized instances of the problem is evaluated by introducing two performance metrics. The quality of obtained solutions is compared with that of the optimal solutions for small sized instances and with the lower bounds for medium sized instances. Also some computational results are presented for large sized instances.