A vehicle routing problem with multi-trips and time windows for circular items

This article presents a vehicle routing problem with time windows, multiple trips, a limited number of vehicles and loading constraints for circular objects. This is a real problem experienced by a home delivery service company. A linear model is proposed to handle small problems and a two-step heuristic method to solve real size instances: the first step builds an initial solution through the modification of the Solomon I1 sequential insertion heuristic, and the second step improves the initial solution through the Tabu search algorithm proposed; in both steps, the problems related to circle packing with different sizes and bin packing are solved jointly with the use of heuristics. Finally, the computing results for two different sets of instances are presented.     

A library for continuous convex separable quadratic knapsack problems

The Continuous Convex Separable Quadratic Knapsack problem (CQKnP) is an easy but useful model that has very many different applications. Although the problem can be solved quickly, it must typically be solved very many times within approaches to (much) more difficult models; hence an efficient solution approach is required. We present and discuss a small open-source library for its solution that we have recently developed and distributed.     

A new regularized quasi-Newton method for unconstrained optimization

In this paper, we propose a new regularized quasi-Newton method for unconstrained optimization. At each iteration, a regularized quasi-Newton equation is solved to obtain the search direction. The step size is determined by a non-monotone Armijo backtracking line search. An adaptive regularized parameter, which is updated according to the step size of the line search, is employed to compute the next search direction. The presented method is proved to be globally convergent. Numerical experiments show that the proposed method is effective for unconstrained optimizations and outperforms the existing regularized Newton method.     

Enumeration and Search Procedures for a Hub Location Problem with Economies of Scale

Within a communications or transportation network, in which a number of locations exchange material or information, hubs can be used as intermediate switching points. In this way, traffic can be consolidated on inter-hub links and, thus, achieve economies of scale in transport costs. Recently, O'Kelly and Brian in 1998 proposed a model (termed the FLOWLOC model) that treats these economies of scale by means of piecewise-linear concave cost functions on the interhub arcs. We show that, for a fixed set of hubs, the FLOWLOC model can be solved using the classic Uncapacitated Facility Location Problem (UFLP). This observation then motivates an optimal enumeration procedure for the FLOWLOC model, as well as some search heuristics that are based upon tabu search and greedy random adaptive search procedures (GRASP). These search procedures would be especially applicable for large-sized problems. Some computational experience is described.     

Optimal search for one of many objects hidden in two boxes

A discrete search model for one of many objects hidden in two boxes is studied. The number of objects is assumed to be a random variable with a known prior distribution. When box i is searched, a cost c_i > 0 is paid and the conditional probability of finding a specific object given it was hidden there is α_i. We are interested in determining a search strategy which finds at least one object with minimum expected cost. Zones of the state space for which Blackwell's rule [3] is optimal are characterized. Based on these results an algorithm for constructing an optimal search sequence is suggested and demonstrated in the case where the number of hidden objects has a geometric distribution.     

A global optimization approach for solving three-dimensional open dimension rectangular packing problems

The three-dimensional open dimension rectangular packing problem (3D-ODRPP) aims to pack a set of given rectangular boxes into a large rectangular container of minimal volume. This problem is an important issue in the shipping and moving industries. All the boxes can be any rectangular stackable objects with different sizes and may be freely rotated. The 3D-ODRPP is usually formulated as a mixed-integer non-linear programming problem. Most existing packing optimization methods cannot guarantee to find a globally optimal solution or are computationally inefficient. Therefore, this paper proposes an efficient global optimization method that transforms a 3D-ODRPP as a mixed-integer linear program using fewer extra 0–1 variables and constraints compared to existing deterministic approaches. The reformulated model can be solved to obtain a global optimum. Experimental results demonstrate the computational efficiency of the proposed approach in globally solving 3D-ODRPPs drawn from the literature and the practical applications.     

A phase model for point spread function estimation in ground-based astronomy

In ground-based astronomy,images of objects in outer space are acquired via ground-based telescopes.However,the imaging system is generally interfered by atmospheric turbulence and hence images so acquired are blurred with unknown point spread function(PSF).To restore the observed images,aberration of the wavefront at the telescope’s aperture,i.e.,the phase,is utilized to derive the PSF.However,the phase is not readily available.Instead,its gradients can be collected by wavefront sensors.Thus the usual approach is to use regularization methods to reconstruct high-resolution phase gradients and then use them to recover the phase in high accuracy.Here,we develop a model that reconstructs the phase directly.The proposed model uses the tight frame regularization and it can be solved efciently by the Douglas-Rachford alternating direction method of multipliers whose convergence has been well established.Numerical results illustrate that our new model is efcient and gives more accurate estimation for the PSF.     

Algorithms for generalized halfspace range searching and other intersection searching problems

In a generalized intersection searching problem, a set S of colored geometric objects is to be preprocessed so that, given a query object q, the distinct colors of the objects of S that are intersected by q can be reported or counted efficiently. These problems generalize the well-studied standard intersection searching problems and have many applications. Unfortunately, the solutions known for the standard problems do not yield efficient solutions to the generalized problems. Recently, efficient solutions have been given for generalized problems where the input and query objects are iso-oriented (i.e., axes-parallel) or where the color classes satisfy additional properties (e.g., connectedness). In this paper, efficient algorithms are given for several generalized problems involving objects that are not necessarily iso-oriented. These problems include: generalized halfspace range searching in , for any fixed d ≥ 2, and segment intersection searching, triangle stabbing, and triangle range searching in for certain classes of line segments and triangles. The techniques used include: computing suitable sparse representations of the input, persistent data structures, and filtering search.     

A new algorithm for two-dimensional object inspection using string matching

In this paper, a string matching scheme is proposed to inspect two-dimensional objects for dimensional and shape verification in industrial environment. This approach consists of two stages. First, the procedures of determining the invariant starting point for boundary tracing and locating the cornerpoints of a curved object for polygon approximation are derived. To speed up the process, an optimization-based unconstrained line search method is used to locate the cornerpoints of the polygon image of a curved object. These cornerpoints are then recorded as feature string. At last, the feature string for each tested object are utilized to find the exact correspondence to one of several model objects. In contrast to conventional matching methods, which requires translation and rotation of the tested image before matching, the proposed method proves to be computationally efficient for real-time applications.     

A mathematical programming-based scheduling framework for multitasking environments

A new paradigm along with a mixed (binary) integer-linear programming model is developed for scheduling tasks in multitasking environments, for which the number of completed tasks is not a good measure. One special case falls into the realm of deteriorating jobs. Polynomial time optimal solution algorithms are presented for this and one other special case. As the complexity of the original problem is believed to be strongly NP-hard, an efficient solution algorithm, based on tabu search, is developed to solve the problem. Small, medium, and large size problems are solved, and the solution obtained from the algorithm is compared with that of the optimal solution or the upper bound found from using the Lagrangian relaxation. Where it was measurable, the search algorithm gave quantifiably good quality solutions, and in all cases it had a much better time efficiency than the branch-and-bound enumeration method. A detailed statistical experiment, based on the split-plot design, is developed to identify the characteristics of the tabu search algorithm, thus guaranteeing a solution that is significantly better in quality. A conjecturing technique is introduced for problems with very large planning horizons. This technique had remarkable time efficiency with no apparent loss of quality.     

Subgraph extraction and metaheuristics for the maximum clique problem

The maximum clique problem involves finding the largest set of pairwise adjacent vertices in a graph. The problem is classic but still attracts much attention because of its hardness and its prominent applications. Our work is based on the existence of an order of all the vertices whereby those belonging to a maximum clique stay close enough to each other. Such an order can be identified via the extraction of a particular subgraph from the original graph. The problem can consequently be seen as a permutation problem that can be addressed efficiently by metaheuristics. We first design a memetic algorithm (MA) for this purpose. Computational experiments conducted on the DIMACS benchmark instances clearly show that our MA not only outperforms existing genetic approaches, but it also compares very well to state-of-the-art algorithms regarding the maximal clique size found after different runs. Furthermore, we show that a hybridization of MA with an iterated local search (ILS) improves the stability of the algorithm. This hybridization (MA-ILS) permits to find two distinct maximal cliques of size 79 and one of size 80 for the C2000.9 instance of the DIMACS benchmark.     

Two-stage search algorithm for the inbound container unloading and stacking problem

This study focuses on the inbound container unloading and stacking problem at container terminals and achieves both a reasonable unloading sequence and the optimal yard stacking distribution. A formulation is proposed as the relational expression between the expected number of rehandles and the stacking height. Based on the formulation, an integer programming model is established to both find the optimal stacking distribution and unloading sequence and attempt to minimize the expected number of rehandles. The model can be solved by the commercial solver for small-scale instances. To solve for large-scale instances in the real world, a two-stage search algorithm is designed, therein incorporating an initial stage for generating the feasible solution and a neighborhood search stage for finding the optimal solution. The algorithm can find an optimal solution in polynomial time, which is proved by theoretical methods and evidenced by numerical experiments.     

A heuristic approach for the truck and trailer routing problem

In this paper, we propose an approach based on mathematical programming and local search to cope with the truck and trailer vehicle routing problem. The mathematical programming framework models two subproblems that are solved sequentially, that is, the customer-route assignment problem (CAP), with the objective of minimizing the fleet size used to service clients, and the route definition problem, with the objective of minimizing the total tour length given the set of clients assigned to each vehicle. Since the route assignment model can return infeasible solutions, the local search plays the role of possibly retrieving a feasible solution. The mathematical formulations and the local search work iteratively, embedded in a multiple restarting mechanism able to diversify solutions by (i) identifying additional constraints for the CAP formulation to be taken into account during the algorithm progress, (ii) using a tabu like customer-route matrix to avoid assignments already analysed in the previous iterations of the algorithm. Also a lower bound to assess the solution quality is given. Experiments and comparison with competing approaches suggest that the results of the proposed machinery are promising, producing, on average,a smaller total tour lengths on benchmarks.     

Parametric linear programming and cluster analysis

In the cluster analysis problem one seeks to partition a finite set of objects into disjoint groups (or clusters) such that each group contains relatively similar objects and, relatively dissimilar objects are placed in different groups. For certain classes of the problem or, under certain assumptions, the partitioning exercise can be formulated as a sequence of linear programs (LPs), each with a parametric objective function. Such LPs can be solved using the parametric linear programming procedure developed by Gass and Saaty [(Gass, S., Saaty, T. (1955), Naval Research Logistics Quarterly 2, 39–45)]. In this paper, a parametric linear programming model for solving cluster analysis problems is presented. We show how this model can be used to find optimal solutions for certain variations of the clustering problem or, in other cases, for an approximation of the general clustering problem.     

A new model and hybrid approach for large scale inventory routing problems

This paper studies an inventory routing problem (IRP) with split delivery and vehicle fleet size constraint. Due to the complexity of the IRP, it is very difficult to develop an exact algorithm that can solve large scale problems in a reasonable computation time. As an alternative, an approximate approach that can quickly and near-optimally solve the problem is developed based on an approximate model of the problem and Lagrangian relaxation. In the approach, the model is solved by using a Lagrangian relaxation method in which the relaxed problem is decomposed into an inventory problem and a routing problem that are solved by a linear programming algorithm and a minimum cost flow algorithm, respectively, and the dual problem is solved by using the surrogate subgradient method. The solution of the model obtained by the Lagrangian relaxation method is used to construct a near-optimal solution of the IRP by solving a series of assignment problems. Numerical experiments show that the proposed hybrid approach can find a high quality near-optimal solution for the IRP with up to 200 customers in a reasonable computation time.     

Heuristic and exact methods for the cutting sequencing problem

Cutting stock problems deal with the generation of a set of cutting patterns that minimizes waste. Sometimes it is also important to find the processing sequence of this set of patterns to minimize the maximum queue of partially cut orders. In such instances a cutting sequencing problem has to be solved. This paper presents a new mathematical model and a three-phase approach for the cutting sequencing problem. In the first phase, a greedy algorithm produces a good starting solution that is improved in the second phase by a tabu search, or a generalized local search procedure, while, in the last phase, the problem is optimally solved by an implicit enumeration procedure that uses the best solution previously found as an upper bound. Computing experience, based on 300 randomly generated problems, shows the good performance of the heuristic methods presented.     

Optimal search strategies using simultaneous generalized hill climbing algorithms

Optimal search strategies for conducting reconnaissance, surveillance or search and rescue operations with limited assets are of significant interest to military decision makers. Multiple search platforms with varying capabilities can be deployed individually or simultaneously for these operations (e.g., helicopters, fixed wing or satellite). Due to the timeliness required in these operations, efficient use of available search platforms is critical to the success of such missions. Designing optimal search strategies over multiple search platforms can be modeled and solved as a multiple traveling salesman problem (MTSP). This paper demonstrates how simultaneous generalized hill climbing algorithms (SGHC) can be used to determine optimal search strategies over multiple search platforms for the MTSP. Computational results with SGHC algorithms applied to the MTSP are reported. These results demonstrate that when limited computing budgets are available, optimal/near-optimal search strategies over multiple search platforms can be obtained more efficiently using SGHC algorithms compared to other generalized hill climbing algorithms. Applications and extensions of this research to other military applications are also discussed.     

On external-memory MST, SSSP and multi-way planar graph separation

Recently external memory graph problems have received considerable attention because massive graphs arise naturally in many applications involving massive data sets. Even though a large number of I/O-efficient graph algorithms have been developed, a number of fundamental problems still remain open.The results in this paper fall in two main classes. First we develop an improved algorithm for the problem of computing a minimum spanning tree (MST) of a general undirected graph. Second we show that on planar undirected graphs the problems of computing a multi-way graph separation and single source shortest paths (SSSP) can be reduced I/O-efficiently to planar breadth-first search (BFS). Since BFS can be trivially reduced to SSSP by assigning all edges weight one, it follows that in external memory planar BFS, SSSP, and multi-way separation are equivalent. That is, if any of these problems can be solved I/O-efficiently, then all of them can be solved I/O-efficiently in the same bound. Our planar graph results have subsequently been used to obtain I/O-efficient algorithms for all fundamental problems on planar undirected graphs.     

One-pass heuristics for large-scale unconstrained binary quadratic problems

Many significant advances have been made in recent years for solving unconstrained binary quadratic programs (UQP). As a result, the size of problem instances that can be efficiently solved has grown from a hundred or so variables a few years ago to 2000 or 3000 variables today. These advances have motivated new applications of the model which, in turn, have created the need to solve even larger problems. In response to this need, we introduce several new “one-pass” heuristics for solving very large versions of this problem. Our computational experience on problems of up to 9000 variables indicates that these methods are both efficient and effective for very large problems. The significance of problems of this size is that they not only open the door to solving a much wider array of real world problems, but also that the standard linear mixed integer formulations of the nonlinear models involve over 40,000,000 variables and three times that many constraints. Our approaches can be used as stand-alone solution methods, or they can serve as procedures for quickly generating high quality starting points for other, more sophisticated methods.     

Solution procedures for sizing of warehouses

In the past, researchers presented a linear programming formulation for the economic sizing of warehouses when demand is highly seasonal and public warehouse space is available on a monthly basis. The static model was extended for the dynamic sizing problem in which the warehouse size is allowed to change over time. By applying simplex routine, the optimal size of the warehouse to be constructed could be determined. In this paper, an alternative and simple method of arriving at an optimal solution for the static problem is given. Three extensions of the static model are given. These extensions involve costs varying over time, economies of scale in capital expenditure and/or operating cost and stochastic version. The dynamic warehouse sizing problem is shown to be a network flow problem which could be solved by using network flow algorithms. The structure of an optimal solution is also given. The concave cost version of the dynamic warehouse sizing problem is also discussed and it is shown that this problem can be solved efficiently using dynamic programming.