Heuristics for the container loading problem

The knapsack container loading problem is the problem of loading a subset of rectangular boxes into a rectangular container of fixed dimensions such that the volume of the packed boxes is maximized. A new heuristic based on the wall-building approach is proposed, which decomposes the problem into a number of layers which again are split into a number of strips. The packing of a strip may be formulated and solved optimally as a Knapsack Problem with capacity equal to the width or height of the container. The depth of a layer as well as the thickness of each strip is decided through a branch-and-bound approach where at each node only a subset of branches is explored.Several ranking rules for the selection of the most promising layer depths and strip widths are presented and the performance of the corresponding algorithms is experimentally compared for homogeneous and heterogeneous instances. The best ranking rule is then used in a comprehensive computational study involving large-sized instances. These computational results show that instances with a total box volume up to 90% easily may be solved to optimality, and that average fillings of the container volume exceeding 95% may be obtained for large-sized instances.     

Heuristics for scheduling in a flow shop with multiple processors

This study investigates the performance of scheduling heuristics in a flow shop with multiple processors. We investigated five better performing flow shop heuristics for their performances of makespan and mean flow time criteria in a flow shop with multiple processors. The study examined the effects of problem characteristics (number of jobs, number of machine stages and number of parallel processors at each stage) and the performance of heuristics using regression analysis. We found that although structural characteristics explain most of the variation in performance, heuristics also had an effect. The experimental results showed that flow shop heuristics developed by Nawaz, Enscore, and Ham and that of Ho were comparable in performance in a flow shop with multiple processors. However, the former was slightly more consistent in results for both criteria.     

Heuristics for multimode scheduling problems with dedicated resources

We study scheduling problems with multiple modes and dedicated resources arising in production and project management, which constitute a special class of the general multimode resource-constrained project scheduling problem. A task may require simultaneously a set of discrete, renewable resources to be processed and the processing can be performed in different modes, that is with different resource sets, processing times, or costs. Precedence constraints can exist among tasks. The total budget that can be allocated to the project can be limited. The problem consists of identifying a mode for each task and a starting time for its processing respecting precedence, resource, and budget constraints. A graph model and an iterative solution scheme are presented. Specific heuristic algorithms for the cases with and without budget constraints are given and computational results are discussed.     

An ellipsoidal distance-based search strategy of ants for nonlinear single and multiple response optimization problems

Various continuous ant colony optimization (CACO) strategies are proposed by researchers to resolve continuous single response optimization problems. However, no such work is reported which also verifies suitability of CACO in case of both single and multiple response situations. In addition, as per literature survey, no variant of CACO can balance simultaneously all the three important aspects of an efficient search strategy, viz. escaping local optima, balancing between intensification and diversification scheme, and handling correlated variable search space structure. In this paper, a variant of CACO, so-called ‘CACO-MDS’ is proposed, which attempts to address all these three aspects. CACO-MDS strategy is based on a Mahalanobis distance-based diversification, and Nelder–Mead simplex-based intensification search scheme. Mahalanobis distance-based diversification search ensures exact measure of multivariate distance for correlated structured search space. The proposed CACO-MDS strategy is verified using fourteen single and multiple response multimodal function optimization test problems. A comparative analysis of CACO-MDS, with three different metaheuristic strategies, viz. ant colony optimization in real space (ACO_R), a variant of local-best particle swarm optimization (SPSO) and simplex-simulated annealing (SIMPSA), also indicates its superiority in most of the test situations.     

Heuristics for unequal weight delivery problems with a fixed error guarantee

This paper presents heuristics that are based on optimal partitioning of a travelling salesman tour, for solving the unequal weight delivery problem. The worst case error performance is given as a bound on the worst case ratio of the cost of the heuristic solution to the cost of the optimal solution. A fully polynomial procedure which consists of applying the optimal partitioning to a travelling salesman tour generated by Christofides' heuristic has a worst case error bound of 3.5−3/Q where Q is the capacity limit of the vehicles. When optimal partitioning is applied to an optimal travelling salesman tour, the worst case error bound becomes 3−2/Q.     

Towards a new strategy for solving fuzzy optimization problems

Fuzzy Optimization models and methods has been one of the most and well studied topics inside the broad area of Soft Computing. Particularly relevant is the field of fuzzy linear programming (FLP). Its applications as well as practical realizations can be found in all the real world areas. As FLP problems constitute the basis for solving fuzzy optimization problems, in this paper a basic introduction to the main models and methods in FLP is presented and, as a whole, Linear Programming problems with fuzzy costs, fuzzy constraints and fuzzy coefficients in the technological matrix are analyzed. But fuzzy sets and systems based optimization methods do not end with FLP, and hence in order to solve more complex optimization problems, fuzzy sets based Meta-heuristics are considered, and two main operative approaches described. Provided that these techniques obtain efficient and/or effective solutions, we present a fuzzy rule based methodology for coordinating Meta-heuristics and in addition, to provide intelligence, we propose a process of extraction of the knowledge to conduct the coordination of the system.     

A prototype column generation strategy for the multiple container loading problem

The multiple container loading cost minimization problem (MCLCMP) is a practical and useful problem in the transportation industry, where products of various dimensions are to be loaded into containers of various sizes so as to minimize the total shipping cost. The MCLCMP can be naturally formulated as a set cover problem and solved using column generation techniques, which is a popular method for handling huge numbers of variables. However, the direct application of column generation is not effective because feasible solutions to the pricing subproblem is required, which for the MCLCMP is NP-hard. We show that efficiency can be greatly improved by generating prototypes that approximate feasible solutions to the pricing problem rather than actual columns. For many hard combinatorial problems, the subproblem in column generation based algorithms is NP-hard; if suitable prototypes can be quickly generated that approximate feasible solutions, then our strategy can also be applied to speed up these algorithms.     

Replenishment routing problems between a single supplier and multiple retailers with direct delivery

We consider the replenishment routing problems of one supplier who can replenish only one of multiple retailers per period, while different retailers need different periodical replenishment. For simple cases satisfying certain conditions, we obtain the simple routing by which the supplier can replenish each retailer periodically so that shortage will not occur. For complicated cases, using number theory, especially the Chinese remainder theorem, we present an algorithm to calculate a feasible routing so that the supplier can replenish the selected retailers on the selected periods without shortages.     

Heuristics for sourcing from multiple suppliers with alternative quantity discounts

In this paper, we analyze the impact of supplier pricing schemes and supplier capacity limitations on the optimal sourcing policy for a single firm. We consider the situation where the total quantity to be procured for a single period is known by the firm and communicated to the supplier set. In response to this communication, each supplier quotes a price and a capacity limit in terms of a maximum quantity that can be supplied to the buyer. Based on this information, the buyer makes a quantity allocation decision among the suppliers and corresponding to this decision is the choice of a subset of suppliers who will receive an order. Based on industry observations, a variety of supplier pricing schemes from the constituent group of suppliers are analyzed, including linear discounts, incremental units discounts, and all units discounts. Given the complexity of the optimization problem for certain types of pricing schemes, heuristic solution methodologies are developed to identify a quantity allocation decision for the firm. Through an extensive computational comparison, we find that these heuristics generate near-optimal solutions very quickly. Data from a major office products retailer is used to illustrate the resulting sourcing strategies given different pricing schemes and capacity limitations of suppliers in this industry. We find for the case of capacity constrained suppliers, the optimal quantity allocations for two complex pricing schemes (linear discount, and incremental units discount) are such that at most one selected supplier will receive an order quantity that is less than its capacity.     

Heuristics and memetic algorithm for the two-dimensional loading capacitated vehicle routing problem with time windows

This work deals with a new combinatorial optimization problem, the two-dimensional loading capacitated vehicle routing problem with time windows which is a realistic extension of the well known vehicle routing problem. The studied problem consists in determining vehicle trips to deliver rectangular objects to a set of customers with known time windows, using a homogeneous fleet of vehicles, while ensuring a feasible loading of each vehicle used. Since it includes NP-hard routing and packing sub-problems, six heuristics are firstly designed to quickly compute good solutions for realistic instances. They are obtained by combining algorithms for the vehicle routing problem with time windows with heuristics for packing rectangles. Then, a Memetic algorithm is developed to improve the heuristic solutions. The quality and the efficiency of the proposed heuristics and metaheuristic are evaluated by adding time windows to a set of 144 instances with 15–255 customers and 15–786 items, designed by Iori et al. (Transport Sci 41:253–264, 2007) for the case without time windows.     

Heuristics for multiple facility location in a network using the variance criterion

In cyclic networks the p-variance location problem is NP-hard, and therefore it is suitable to use heuristic methods to find approximate solutions to the problem. To this end, two strategies are explored, the first using combinatorial search procedures over the vertex set, whereas the second searches for the solution over the entire network. The initial vertex set solutions are generated by using tabu search, variable neighbourhood search and interchange procedures. The heuristics have been tested on instances of up to 30 vertices and 70 edges, and their performances compared.     

A cluster insertion heuristic for single and multiple depot vehicle routing problems with backhauling

We investigate an extension to the classical insertion-based heuristic for the vehicle routing problem with backhauling (VRPB). It is based on the idea of inserting more than one backhaul at a time. This method is tested on data sets with single and multiple depots with encouraging results at no additional computational burden. This approach can also be useful in generating good starting solutions for the more computer-intensive meta-heuristics.     

A goal-driven prototype column generation strategy for the multiple container loading cost minimization problem

In the multiple container loading cost minimization problem (MCLCMP), rectangular boxes of various dimensions are loaded into rectangular containers of various sizes so as to minimize the total shipping cost. The MCLCMP can be naturally modeled as a set cover problem. We generalize the set cover formulation by introducing a new parameter to model the gross volume utilization of containers in a solution. The state-of-the-art algorithm tackles the MCLCMP using the prototype column generation (PCG) technique. PCG is an effective technique for speeding up the column generation technique for extremely hard optimization problems where their corresponding pricing subproblems are NP-hard. We propose a new approach to the MCLCMP that combines the PCG technique with a goal-driven search. Our goal-driven prototype column generation (GD-PCG) algorithm improves the original PCG approach in three respects. Computational experiments suggest that all three enhancements are effective. Our GD-PCG algorithm produces significantly better solutions for the 350 existing benchmark instances than all other approaches in the literature using less computation time. We also generate two new set instances based on industrial data and the classical single container loading instances.     

Linear programming with multiple choice constraints for single chain undiscounted Markov decision problems

This paper develops an efficient LP algorithm for solving single chain undiscounted Markov decision problems. The algorithm imposes, in the framework of the simplex method, the multiple choice constraints that exactly one basic variable be chosen from each Markov state. It is proved that the algorithm converges to an optimal solution in a finite number of steps.     

Heuristics for solving a capacitated multiple allocation hub location problem with application in German wagonload traffic

In this paper, we present a capacitated multiple allocation hub location problem, which arose from a network design problem in German wagonload traffic. We develop heuristic solution approaches based on local improvements. We solve the problem with the heuristics and CPLEX on test data sets provided by our partner Deutsche Bahn AG. The computational results are presented and compared.     

Heuristics for multi-attribute vehicle routing problems: A survey and synthesis

The attributes of vehicle routing problems are additional characteristics or constraints that aim to better take into account the specificities of real applications. The variants thus formed are supported by a well-developed literature, including a large variety of heuristics. This article first reviews the main classes of attributes, providing a survey of heuristics and meta-heuristics for Multi-Attribute Vehicle Routing Problems (MAVRP). It then takes a closer look at the concepts of 64 remarkable meta-heuristics, selected objectively for their outstanding performance on 15 classic MAVRP with different attributes. This cross-analysis leads to the identification of “winning strategies” in designing effective heuristics for MAVRP. This is an important step in the development of general and efficient solution methods for dealing with the large range of vehicle routing variants.     

A new approach to variational problems with multiple scales

We introduce a new concept, the Young measure on micropatterns, to study singularly perturbed variational problems that lead to multiple small scales depending on a small parameter ε. This allows one to extract, in the limit ε → 0, the relevant information at the macroscopic scale as well as the coarsest microscopic scale (say ε^α) and to eliminate all finer scales. To achieve this we consider rescaled functions Rx (t) := x (s + ε^αt) viewed as maps of the macroscopic variable s ∈ Ω with values in a suitable function space. The limiting problem can then be formulated as a variational problem on the Young measures generated by R^εx. As an illustration, we study a one‐dimensional model that describes the competition between formation of microstructure and highest gradient regularization. We show that the unique minimizer of the limit problem is a Young measure supported on sawtooth functions with a given period. © 2001 John Wiley & Sons, Inc.     

A new branching strategy for time constrained routing problems with application to backhauling

In this paper, we explore a new branching strategy for branch-and-bound approaches based on column generation for the vehicle routing problems with time windows. This strategy involves branching on resource variables (time or capacity) rather than on network flow variables. We also examine criteria for selecting network nodes for branching. To test the effectiveness of the branching strategy, we conduct computational experiments on time window constrained vehicle routing problems where backhauling is permitted only after all the shipments to clients have been made. The branching method proved very effective. In cases where time was the more binding constraint, time-based branching succeeded in decreasing the number of nodes explored by two thirds and the total computation time by more than half when compared to flow-based branching. The computational results also show that the overall algorithm was successful in optimally solving problems with up to 100 customers. It produced an average cost decrease of almost 7% when backhauling was permitted as compared to the cost involved when the client and the distributor routes were distinct.     

Optimal selling strategy with a large block of stock

In this paper, we develop an optimal stock selling strategy with the stochastic upper bound of selling rate over an infinite time horizon. Moreover, the temporary and permanent price impact are considered. We treat the problem by using a fluid model. In the model that the number of shares is treated as fluid (continuous) and the overall liquidation is dictated by the rates of selling over time. The goal is to maximize the overall return under state constraints. The corresponding value function with the selling strategies is shown to be continuous and the unique viscosity solution to the associated HJB equation. Finally, a numerical example is given to illustrate the result.