71 research outputs found
Accelerated Benders Decomposition and Local Branching for Dynamic Maximum Covering Location Problems
The maximum covering location problem (MCLP) is a key problem in facility
location, with many applications and variants. One such variant is the dynamic
(or multi-period) MCLP, which considers the installation of facilities across
multiple time periods. To the best of our knowledge, no exact solution method
has been proposed to tackle large-scale instances of this problem. To that end,
in this work, we expand upon the current state-of-the-art
branch-and-Benders-cut solution method in the static case, by exploring several
acceleration techniques. Additionally, we propose a specialised local branching
scheme, that uses a novel distance metric in its definition of subproblems and
features a new method for efficient and exact solving of the subproblems. These
methods are then compared through extensive computational experiments,
highlighting the strengths of the proposed methodologies
Combinatorial optimization and vehicle fleet planning : perspectives and prospects
Bibliography : p.52-60.Research supported in part by the National Science Foundation under grant 79-26225-ECS.by Thomas L. Magnanti
Analysis of large scale linear programming problems with embedded network structures: Detection and solution algorithms
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Linear programming (LP) models that contain a (substantial) network structure frequently
arise in many real life applications. In this thesis, we investigate two main questions; i) how an embedded network structure can be detected, ii) how the network structure can be exploited to create improved sparse simplex solution algorithms. In order to extract an embedded pure network structure from a general LP problem we
develop two new heuristics. The first heuristic is an alternative multi-stage generalised upper bounds (GUB) based approach which finds as many GUB subsets as possible. In order to identify a GUB subset two different approaches are introduced; the first is based on the notion of Markowitz merit count and the second exploits an independent set in the corresponding graph. The second heuristic is based on the generalised signed graph of the coefficient matrix. This heuristic determines whether the given LP problem is an entirely pure network; this is in contrast to all previously known heuristics. Using generalised signed graphs, we prove that the problem of detecting the maximum size embedded network structure within an LP problem is NP-hard. The two detection
algorithms perform very well computationally and make positive contributions to the
known body of results for the embedded network detection. For computational solution
a decomposition based approach is presented which solves a network problem with side constraints. In this approach, the original coefficient matrix is partitioned into the network and the non-network parts. For the partitioned problem, we investigate two alternative decomposition techniques namely, Lagrangean relaxation and Benders decomposition. Active variables identified by these procedures are then used to create
an advanced basis for the original problem. The computational results of applying these techniques to a selection of Netlib models are encouraging. The development and computational investigation of this solution algorithm constitute further contribution
made by the research reported in this thesis.This study is funded by the Turkish Educational Council and Mugla University
On orbital allotments for geostationary satellites
The following satellite synthesis problem is addressed: communication satellites are to be allotted positions on the geostationary arc so that interference does not exceed a given acceptable level by enforcing conservative pairwise satellite separation. A desired location is specified for each satellite, and the objective is to minimize the sum of the deviations between the satellites' prescribed and desired locations. Two mixed integer programming models for the satellite synthesis problem are presented. Four solution strategies, branch-and-bound, Benders' decomposition, linear programming with restricted basis entry, and a switching heuristic, are used to find solutions to example synthesis problems. Computational results indicate the switching algorithm yields solutions of good quality in reasonable execution times when compared to the other solution methods. It is demonstrated that the switching algorithm can be applied to synthesis problems with the objective of minimizing the largest deviation between a prescribed location and the corresponding desired location. Furthermore, it is shown that the switching heuristic can use no conservative, location-dependent satellite separations in order to satisfy interference criteria
Multi-period sales districting problem
In the sales districting problem, we are given a set of customers and a set of salesmen in some area. The salesmen have to provide services at the customers' locations to satisfy their requirements. The task is to allocate each customer to one salesman, which partitions the set of customers into subsets, called districts. Each district is expected to have approximately equal workload and travel time for each salesman to promote fairness among them. Also, the districts should be geographically compact since they are more likely to reduce unnecessary travel time, which is desirable for economic reasons. Moreover, each customer can require recurring services with different visiting frequencies such as every week or two weeks during a planning horizon. This problem is called the `Multi-Period Sales Districting Problem (MPSDP)' and can be found typically in regular engineering maintenance and sales promotion.
In addition to determining the sales districts, we also want to get valid weekly visiting schedules for the salesmen corresponding to the customers' visiting requirements. The schedules should result in weekly districts with the following desirable characteristics: each weekly district should be balanced in weekly workload and geographically compact. The compactness in the schedules provides benefits when a salesman has to deal with short-term requests from customers or change a visiting plan during the week. Namely, the salesman can postpone a visit to another day if necessary, without increasing the travel time too much compared to the original schedule. This is beneficial when the salesman has to deal with unexpected situations, for example, road maintenance, traffic jams, or short notice of time windows from customers. Although the problem is very practical, it has been studied only recently. Since most of the previous literature on general scheduling problems did not consider compactness, a few recent studies have begun to focus on solving the scheduling part of the problem.
The purpose of this research is to develop a more sophisticated exact solution approach as well as an efficient high-quality heuristic to solve the scheduling part. Eventually, with an effective elaborate method to solve the scheduling part, we aim for a robust algorithm to solve the districting and scheduling part of the problem simultaneously.
This thesis contains three main parts. The first part introduces the problem and provides a mixed-integer linear programming formulation for only the scheduling part and formulation for the whole problem. The second part presents solution approaches, including an exact method and a heuristic, for only the scheduling part. The last part is dedicated to further development of a successful approach from the second part to solve the districting and scheduling part of the problem simultaneously.
For solving the scheduling part, Benders' decomposition is developed as a new exact solution method. The linear relaxation of the problem is strengthened by adding several Benders' cuts derived from fractional solutions at the beginning of the algorithm. Moreover, a good-quality integer solution derived from a location-allocation heuristic is used to generate cuts beforehand, which significantly improves the upper bound of the objective function value. Nondominated optimality cuts are implemented to guarantee the strongest Benders' cuts in each iteration. Also, instead of generating a Benders' cut per iteration, we exploit the decomposable structure of the problem formulation to generate multiple cuts per iteration, resulting in a noticeable improvement in the lower bound of the objective function value. In the classical Benders' decomposition, one of the main factors that slow down the algorithm is that one has to solve the integer programmes from scratch in each iteration. To alleviate this problem, a modern implementation creates only one branch-and-bound tree and adds Benders' cuts derived from a solution in each node in a solution cut pool. This method is called branch-and-Benders' cut. To assess the suitability of the algorithm, we compare its performance on small data instances that contain 3050 customers to the Benders' algorithm in CPLEX and show that our algorithm is highly competitive.
Since an exact solution method usually struggles to solve realistic large data instances, a meta-heuristic called tabu search is proposed. A high-quality initial solution to start the algorithm is derived from the location-allocation heuristic. Three different neighbourhoods based on information about week centres or customers' week patterns are created within which we search for the best solution. An infeasible solution is allowed in the search to expand the search space. During the search, the size of a whole neighbourhood can be excessively large, so we limit the search to promising areas of the solution space to save computational time. Also, a surrogate objective value is used to save on computational time in cases when computing the real objective value is too time-consuming. Although the tabu search defines a list of forbidden moves to avoid the cycle of solutions, the algorithm can still struggle to avoid being trapped around a local optimum. Therefore, a diversification scheme is proposed for such cases. The algorithm is also accelerated by combining all neighbourhoods and selecting the appropriate neighbourhood for each iteration by a roulette wheel selection. It shows impressive results in small data instances that contain 3050 customers. The comparison with built-in heuristics in CPLEX confirms the robustness of the tabu search algorithm. Finally, we combine the tabu search algorithm with our developed Benders' decomposition. Numerical results show that the tabu search method improves the upper bound of the Benders' decomposition algorithm. However, the overall performance is not satisfying so the combination of these two techniques still requires more proper development.
As the tabu search algorithm performs well on the scheduling part, it is extended to solve the whole problem, i.e., the districting and scheduling part at the same time. Computational results on large data instances, which contain between 100 and 300 customers, demonstrate its capacity to derive a high-quality solution within a reasonable amount of time, i.e., less than 17 minutes. At the same time, the Benders' decomposition algorithm in CPLEX, which is a benchmark in this case, and the built-in heuristics in CPLEX cannot even find any feasible integer solution for most of the instances within an hour. Importantly, there is a conflict between the districting part and the scheduling part so we recommend solving both parts simultaneously for tackling the MPSDP.
The multi-period sales districting problem is highly practical and challenging to solve. To the best of our knowledge, we are the first to propose a single integrated solution approach to solve the whole problem. Further studies including adding more realistic planning requirements into consideration and effective solution approaches to solve the problem are still required
Two-Stage Stochastic Mixed Integer Linear Optimization
The primary focus of this dissertation is on optimization problems that involve uncertainty unfolding over time. In many real-world decisions, the decision-maker has to decide in the face of uncertainty. After the outcome of the uncertainty is observed, she can correct her initial decision by taking some corrective actions at a later time stage. These problems are known as stochastic optimization problems with recourse. In the case that the number of time stages is limited to two, these problems are referred to as two-stage stochastic optimization problems. We focus on this class of optimization problems in this dissertation. The optimization problem that is solved before the realization of uncertainty is called the first-stage problem and the problem solved to make a corrective action on the initial decision is called the second-stage problem. The decisions made in the second- stage are affected by both the first-stage decisions and the realization of random variables. Consequently, the two-stage problem can be viewed as a parametric optimization problem which involves the so-called value function of the second-stage problem. The value function describes the change in optimal objective value as the right-hand side is varied and understanding it is crucial to developing solution methods for two-stage optimization problems.In the first part of this dissertation, we study the value function of a MILP. We review the structural properties of the value function and its construction methods. We con- tribute by proposing a discrete representation of the MILP value function. We show that the structure of the MILP value function arises from two other optimization problems that are constructed from its discrete and continuous components. We show that our representation can explain certain structural properties of the MILP value function such as the sets over which the value function is convex. We then provide a simplification of the Jeroslow Formula obtained by applying our results. Finally, we describe a cutting plane algorithm for its construction and determine the conditions under which the pro- posed algorithm is finite.Traditionally, the solution methods developed for two-stage optimization problems consider the problem where the second-stage problem involves only continuous variables. In the recent years, however, two-stage problems with integer variables in the second- stage have been visited in several studies. These problems are important in practice and arise in several applications in supply chain, finance, forestry and disaster management, among others. The second part of this dissertation concerns the development and implementation of a solution method for the two-stage optimization problem where both the first and second stage involve mixed integer variables. We describe a generalization of the classical Benders’ method for solving mixed integer two-stage stochastic linear optimization problems. We employ the strong dual functions encoded in the branch-and-bound trees resulting from solution of the second-stage problem. We show that these can be used effectively within a Benders’ framework and describe a method for obtaining all required dual functions from a single, continuously refined branch-and-bound tree that is used to warm start the solution procedure for each subproblem.Finally, we provide details on the implementation of our proposed algorithm. The implementation allows for construction of several approximations of the value function of the second-stage problem. We use different warm-starting strategies within our proposed algorithm to solve the second-stage problems, including solving all second-stage problems with a single tree. We provide computational results on applying these strategies to the stochastic server problems (SSLP) from the stochastic integer programming test problem library (SIPLIB)
Models and solution approaches for intermodal and less-than-truckload network design with load consolidations
Logistics and supply chain problems arising in the context of intermodal transportation and less-than-truckload (LTL) network design typically require commodities
to be consolidated and shipped via the most economical route to their destinations.
Traditionally, these problems have been modelled using network design or hub-and-
spoke approaches. In a network design problem, one is given the network and flow
requirements between the origin and destination pairs (commodities), and the objective is to route the flows over the network so as to minimize the sum of the fixed
charge incurred in using arcs and routing costs. However, there are possible benefits, due to economies-of-scale in transportation, that are not addressed in standard
network design models. On the other hand, hub location problems are motivated by
potential economies-of-scale in transportation costs when loads are consolidated and
shipped together over a completely connected hub network. However, in a hub location problem, the assignment of a node to a hub is independent of the commodities
originating at, or destined to, this node. Such an indiscriminate assignment may not
be suitable for all commodities originating at a particular node because of their different destinations. Problems arising in the area of LTL transportation, intermodal
transportation and package routing generally have characteristics such as economies-
of-scale in transportation costs in addition to the requirement of commodity-based
routing. Obviously, the existing network design and hub location-based models are not directly suitable for these applications. In this dissertation, we investigate the
development of models and solution algorithms for problems in the areas of LTL and
intermodal transportation as well as in the freight forwarders industry. We develop
models and solution methods to address strategic, tactical and operational level decision issues and show computational results. This research provides new insights
into these application areas and new solution methods therein. The solution algorithms developed here also contribute to the general area of discrete optimization,
particularly for problems with similar characteristics
Accelerating Benders decomposition for network design.
Thesis. 1978. Ph.D.--Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING.Vita.Bibliography: leaves 136-144.Ph.D
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