Allocation strategies in hub networks

In this paper, we study allocation strategies and their effects on total routing costs in hub networks. Given a set of nodes with pairwise traffic demands, the p-hub median problem is the problem of choosing p nodes as hub locations and routing traffic through these hubs at minimum cost. This problem has two versions; in single allocation problems, each node can send and receive traffic through a single hub, whereas in multiple allocation problems, there is no such restriction and a node may send and receive its traffic through all p hubs. This results in high fixed costs and complicated networks. In this study, we introduce the r-allocation p-hub median problem, where each node can be connected to at most r hubs. This new problem generalizes the two versions of the p-hub median problem. We derive mixed-integer programming formulations for this problem and perform a computational study using well-known datasets. For these datasets, we conclude that single allocation solutions are considerably more expensive than multiple allocation solutions, but significant savings can be achieved by allowing nodes to be allocated to two or three hubs rather than one. We also present models for variations of this problem with service quality considerations, flow thresholds, and non-stop service.     

Meta-heuristic approaches with memory and evolution for a multi-product production/distribution system design problem

We consider a multi-product two-stage production/distribution system design problem (PDSD) where a fixed number of capacitated distribution centers are to be located with respect to capacitated suppliers (plants) and retail locations (customers) while minimizing the total costs in the system. We present a mixed-integer problem formulation that facilitates the development of efficient heuristic procedures. We provide meta-heuristic procedures, including a population-based scatter search with path relinking and trajectory-based local and tabu search, for the solution of the problem. We also develop efficient construction heuristics and transshipment heuristics that are incorporated into the heuristic procedures for the solution of subproblems. We present extensive computational results that show the high performance of the solution approaches. We obtain smaller than 1.0% average optimality gaps with acceptable runtimes, even for relatively large problems. The computational results also demonstrate the effectiveness of the construction and transshipment heuristics that impact the solution quality and overall runtimes.     

Capacitated facility location: Separation algorithms and computational experience

We consider the polyhedral approach to solving the capacitated facility location problem. The valid inequalities considered are the knapsack cover, flow cover, effective capacity, single depot, and combinatorial inequalities. The flow cover, effective capacity and single depot inequalities form subfamilies of the general family of submodular inequalities. The separation problem based on the family of submodular inequalities is NP-hard in general. For the well known subclass of flow cover inequalities, however, we show that if the client set is fixed, and if all capacities are equal, then the separation problem can be solved in polynomial time. For the flow cover inequalities based on an arbitrary client set and general capacities, and for the effective capacity and single depot inequalities we develop separation heuristics. An important part of these heuristics is based on the result that two specific conditions are necessary for the effective cover inequalities to be facet defining. The way these results are stated indicates precisely how structures that violate the two conditions can be modified to produce stronger inequalities. The family of combinatorial inequalities was originally developed for the uncapacitated facility location problem, but is also valid for the capacitated problem. No computational experience using the combinatorial inequalities has been reported so far. Here we suggest how partial output from the heuristic identifying violated submodular inequalities can be used as input to a heuristic identifying violated combinatorial inequalities. We report on computational results from solving 60 medium size problems. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Two modifications of the least cost per period heuristic for dynamic lot-sizing

This paper proposes two constructive heuristics for the well-known single-level uncapacitated dynamic lot-sizing problem. The proposed heuristics, called net least period cost (nLPC) and nLPC(i), are developed by modifying the average period cost concept from Silver and Meal's heuristic, commonly known as least period cost (LPC). An improved tie-breaking stopping rule and a locally optimal decision rule are proposed in the second heuristic to enhance performance. We test the effectiveness of the proposed heuristics by using 20 benchmarking test problems frequently used in the literature. Furthermore, we perform a large-scale simulation study involving three factors, 50 experimental conditions, and 100?000 randomly generated problems to evaluate the proposed heuristics against LPC and six other well-known constructive heuristics in the literature. The simulation results show that both nLPC and nLPC(i) produce average holding and setup costs lower than or equal to those of LPC in every one of the 50 experimental conditions. The proposed heuristics also outperform each of the six other heuristics evaluated in all experimental conditions, without an increase in computational requirements. Lastly, considering that both nLPC and nLPC(i) are fairly simple for practitioners to understand and that lot-sizing heuristics have been commonly used in practice, there should be a very good chance for practical applications of the proposed heuristics.     

A column generation approach for location-routing problems with pickup and delivery

In this paper we formulate an integer programming model for the Location and Routing Problem with Pickup and Delivery. We propose a column generation scheme and implement, for the subproblem, a label-setting algorithm for the shortest path with pickup and delivery and time windows problem. We also propose a set of heuristics to speed up this process. To validate the model, we implement the column generation scheme and test it on different instances developed in this paper. We also provide an analysis of how the costs of opening depots and the fixed cost of routes affect the optimal solution.     

Solving a continuous local access network design problem with a stabilized central column generation approach

In this paper, we focus on a variant of the multi-source Weber problem. In the multi-source Weber problem, the location of a fixed number of concentrators, and the allocation of terminals to them, must be chosen to minimize the total cost of links between terminals and concentrators. In our variant, we have a third hierarchical level, two categories of link costs, and the number of concentrators is unknown. To solve this difficult problem, we propose several heuristics, and use a new stabilized column generation approach, based on a central cutting plane method, to provide lower bounds.     

Sequential location-allocation problems on chains and trees with probabilistic link demands

In this paper, we consider multiperiod minisum location problems on networks in which demands can occur continuously on links according to a uniform probability density function. In addition, demands may change dynamically over time periods and at most one facility can be located per time period. Two types of networks are considered in conjunction with three behavioral strategies. The first type of network discussed is a chain graph. A myopic strategy and long-range strategy for locatingp-facilities are considered, as is a discounted present worth strategy for locating two facilities. Although these problems are generally nonconvex, effective methods are developed to readily identify all local and global minima. This analysis forms the basis for similar problems on tree graphs. In particular, we construct algorithms for the 3-facility myopic problem and the 2-facility long-range and discounted cost problems on a tree graph. Extensions and suggestions for further research on problems involving more general networks are provided.     

A branch-and-cut algorithm for a generalization of the Uncapacitated Facility Location Problem

Summary We introduce a generalization of the well-know Uncapacitated Facility Location Problem, in which clients can be served not only by single facilities but also by sets of facilitities. The problem, calledGaneralized Uncapacitated Facility Lacition Problem (GUFLP), was inspired by the Index Selection Problem in physical database design. We for mulate GUFLP as a Set Packing Problem, showing that our model contains all the clique inequalities (in polynomial number). Moreover, we describe and exact separation procedure for odd-hole inequalities, based on the particular structure of the problem. These results are used within a branch-and-cut algorithm for the exact solution of GUFLP. Computational results on two different classes of test problems are given.     

Simulated N-Body: New Particle Physics-Based Heuristics for a Euclidean Location-Allocation Problem

The general facility location problem and its variants, including most location-allocation and P-median problems, are known to be NP-hard combinatorial optimization problems. Consequently, there is now a substantial body of literature on heuristic algorithms for a variety of location problems, among which can be found several versions of the well-known simulated annealing algorithm. This paper presents an optimization paradigm that, like simulated annealing, is based on a particle physics analogy but is markedly different from simulated annealing. Two heuristics based on this paradigm are presented and compared to simulated annealing for a capacitated facility location problem on Euclidean graphs. Experimental results based on randomly generated graphs suggest that one of the heuristics outperforms simulated annealing both in cost minimization as well as execution time. The particular version of location problem considered here, a location-allocation problem, involves determining locations and associated regions for a fixed number of facilities when the region sizes are given. Intended applications of this work include location problems with congestion costs as well as graph and network partitioning problems.     

New models for locating a moving service facility

In this paper we analyze a new location problem which is a generalization of the well-known single facility location model. This extension consists of introducing a general objective function and replacing fixed locations by trajectories. We prove that the problem is well-stated and solvable. A Weiszfeld type algorithm is proposed to solve this generalized dynamic single facility location problem on L ^p spaces of functions, with p ∈(1,2]. We prove global convergence of our algorithm once we have assumed that the set of demand functions and the initial step function belong to a subspace of L ^p called Sobolev space. Finally, examples are included illustrating the application of the model to generalized regression analysis and the convergence of the proposed algorithm. The examples also show that the pointwise extension of the algorithm does not have to converge to an optimal solution of the considered problem while the proposed algorithm does.     

On the convergence of the Weiszfeld algorithm for continuous single facility location–allocation problems

A general family of single facility continuous location–allocation problems is introduced, which includes the decreasingly weighted ordered median problem, the single facility Weber problem with supply surplus, and Weber problems with alternative fast transportation network. We show in this paper that the extension of the well known Weiszfeld iterative decrease method for solving the corresponding location problems with fixed allocation yields an always convergent scheme for the location allocation problems. In a generic way, from each starting point, the limit point will be a locally minimal solution, whereas for each possible exceptional situation, a possible solution is indicated. Some computational results are presented, comparing this method with an alternating location–allocation approach. The research of the second author was partially supported by the grant of the Algerian Ministry of High Education 001BIS/PNE/ENSEIGNANTS/BELGIQUE.     

Two new heuristics for the location set covering problem

Summary In this paper we present two new greedy-type heuristics for solving the location set covering problem. We compare our new pair of algorithms with the pair GH1 and GH2 [Vasko and Wilson (1986)] and show that they perform better for a selected set of test problems.     

A cost operator approach to multistage location-allocation

The multistage factory-warehouse location-allocation problem is to decide on locations of warehouses and shipping amounts from the factories through the warehouses to meet customer demands in such a way that the total fixed plus variable costs are minimized. Capacity constraints at factories and warehouses are also imposed.We present a cost operator algorithm for solving this problem. The algorithm takes into account the network structure and the submodularity of the objective function. Computational results with problems taken from the literature as well as new problems are provided.     

Multiple level production planning in rolling horizon assembly environments

In this paper, nine multiple level planning heuristics are evaluated to characterize how rolling horizon results relate to fixed horizon results in a deterministic demand environment. The weighted order cycle (WOC) is introduced as a single expression of cost structure within a multiple item bill of materials. When planning horizons are restated in terms of WOC (versus time buckets), it becomes apparent that the cost performance of the majority of the heuristics is essentially the same for fixed horizon and rolling horizon conditions when the planning horizon is at least two WOC in length. The horizon sensitive logic of the best two heuristics in cost performance also exhibited less nervousness then several horizon myopic rules, a counter intuitive result. An established multiple level cost modification technique was found to reduce the nervousness of single item rules, in addition to its original goal of schedule cost reduction. To gauge cost performance, Lagrangian relaxation of a binary formulation of the problem was used to find bounds within an average of 1% of the optimal solution cost of each simulation.     

A Hybrid Genetic Algorithm for the Single Machine Scheduling Problem

A hybrid genetic algorithm (HGA) is proposed for the single machine, single stage, scheduling problem in a sequence dependent setup time environment within a fixed planning horizon (SSSDP). It incorporates the elitist ranking method, genetic operators, and a hill-climbing technique in each searching area. To improve the performance and efficiency, hill climbing is performed by uniting the Wagner-Whitin Algorithm with the problem-specific knowledge. The objective of the HGA is to minimize the sum of setup cost, inventory cost, and backlog cost. The HGA is able to obtain a superior solution, if it is not optimal, in a reasonable time. The computational results of this algorithm on real life SSSDP problems are promising. In our test cases, the HGA performed up to 50% better than the Just-In-Time heuristics and 30% better than the complete batching heuristics.     

A Tabu search-based approach for scheduling job-shop type flexible manufacturing systems

The problem of scheduling parts in a job-shop type flexible manufacturing system (FMS) is investigated when each part can have alternative process plans and each operation required of a part can be performed on alternative machines. The mixed-(binary) integer programming model developed for the problem is proven strongly NP-hard. A higher-level heuristic solution algorithm based on a concept known as ‘tabu search’ is developed to determine the best (near-optimal) solution for problems of industrial merit. A comparison of six different versions of tabu search-based heuristics (TSH 1-TSH 6) is performed to investigate the impact of using long-term memory and the use of fixed versus variable tabu-list sizes. A carefully constructed statistical experiment, based on randomized complete-block design, is used to test the performance on four problem structures ranging from 4–14 parts. The results show that, as the problem size increases, TSH 3 with fixed tabu-list size and long-term memory is preferred over the other heuristics. Further, the branch-and-bound technique, by failing to identify as good a solution as that determined by the heuristics (TSH 1-TSH 6), let alone an optimal solution, for a small problem reinforces the need for developing efficient heuristics for solving real problems encountered in industry practice.     

Probabilistic local search algorithms for concave cost transportation network problems

In practice concave cost transportation problems are characterized as NP-hard, therefore cost functions are usually simplified as linear in order to facilitate problem solving. However, linear cost functions may not reflect actual operations, which generally results in decreased operational performance. This research employs the techniques of simulated annealing and threshold accepting to develop several heuristics that would efficiently solve these concave cost transportation network problems. A network generator has also been designed to generate many instances on an HP workstation to test the heuristics. The preliminary results show that these heuristics are potentially useful.     

The variable radius covering problem

In this paper we propose a covering problem where the covering radius of a facility is controlled by the decision-maker; the cost of achieving a certain covering distance is assumed to be a monotonically increasing function of the distance (i.e., it costs more to establish a facility with a greater covering radius). The problem is to cover all demand points at a minimum cost by finding optimal number, locations and coverage radii for the facilities. Both, the planar and discrete versions of the model are considered. Heuristic approaches are suggested for solving large problems in the plane. These methods were tested on a set of planar problems. Mathematical programming formulations are proposed for the discrete problem, and a solution approach is suggested and tested.     

Multi-dimensional vector assignment problems

We consider a special class of axial multi-dimensional assignment problems called multi-dimensional vector assignment (MVA) problems. An instance of the MVA problem is defined by $m$ disjoint sets, each of which contains the same number $n$ of $p$-dimensional vectors with nonnegative integral components, and a cost function defined on vectors. The cost of an $m$-tuple of vectors is defined as the cost of their component-wise maximum. The problem is now to partition the $m$ sets of vectors into $n$   $m$-tuples so that no two vectors from the same set are in the same $m$-tuple and so that the sum of the costs of the $m$-tuples is minimized. The main motivation comes from a yield optimization problem in semi-conductor manufacturing. We consider a particular class of polynomial-time heuristics for MVA, namely the sequential heuristics, and we study their approximation ratio. In particular, we show that when the cost function is monotone and subadditive, sequential heuristics have a finite approximation ratio for every fixed $m$. Moreover, we establish smaller approximation ratios when the cost function is submodular and, for a specific sequential heuristic, when the cost function is additive. We provide examples to illustrate the tightness of our analysis. Furthermore, we show that the MVA problem is APX-hard even for the case $m=3$ and for binary input vectors. Finally, we show that the problem can be solved in polynomial time in the special case of binary vectors with fixed dimension $p$.     

A genetic algorithm for the problem of configuring a hub-and-spoke network for a LTL trucking company in Brazil

A heuristic based on genetic algorithms is proposed to the problem of configuring hub-and-spoke networks for trucking companies that operate less-than-truckload (LTL) services in Brazil. The problem consists of determining the number of consolidation terminals (also known as hubs), their locations and the assignment of the spokes to the hubs, aiming to minimize the total cost, which is composed of fixed and variable costs. The proposed formulation differs from similar formulations found in the literature in the sense that it allows variable scale-reduction factors for the transportation costs according to the total amount of freight between hub terminals, as occurs to less-than-truckload (LTL) freight carriers in Brazil. Our genetic algorithm approach incorporates an efficient local improvement procedure that is applied to each generated individual of the population. Computational results for benchmark problems are presented. A practical application to a real world problem involving one of the top-ten trucking companies in Brazil is also described.