6 research outputs found
Integrating facility location and production planning decisions
We consider a metric uncapacitated facility location problem where we must assign each customer to a facility and meet the demand of the customer in future time periods through production and inventory decisions at the facility. We show that the problem, in general, is as hard to approximate as the set cover problem. We therefore focus on developing approximation algorithms for special cases of the problem. These special cases come in two forms: (i) specialize the production and inventory cost structure and (ii) specialize the demand pattern of the customers. In the former, we offer reductions to variants of the metric uncapacitated facility location problem that have been previously studied. The latter gives rise to a class of metric uncapacitated facility location problems where the facility cost function is concave in the amount of demand assigned to the facility. We develop a modified greedy algorithm together with the idea of cost-scaling to provide an algorithm for this class of problems with an approximation guarantee of 1.52. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/64912/1/20315_ftp.pd
Separable Concave Optimization Approximately Equals Piecewise-Linear Optimization
We study the problem of minimizing a nonnegative separable concave function
over a compact feasible set. We approximate this problem to within a factor of
1+epsilon by a piecewise-linear minimization problem over the same feasible
set. Our main result is that when the feasible set is a polyhedron, the number
of resulting pieces is polynomial in the input size of the polyhedron and
linear in 1/epsilon. For many practical concave cost problems, the resulting
piecewise-linear cost problem can be formulated as a well-studied discrete
optimization problem. As a result, a variety of polynomial-time exact
algorithms, approximation algorithms, and polynomial-time heuristics for
discrete optimization problems immediately yield fully polynomial-time
approximation schemes, approximation algorithms, and polynomial-time heuristics
for the corresponding concave cost problems.
We illustrate our approach on two problems. For the concave cost
multicommodity flow problem, we devise a new heuristic and study its
performance using computational experiments. We are able to approximately solve
significantly larger test instances than previously possible, and obtain
solutions on average within 4.27% of optimality. For the concave cost facility
location problem, we obtain a new 1.4991+epsilon approximation algorithm.Comment: Full pape
Strongly Polynomial Primal-Dual Algorithms for Concave Cost Combinatorial Optimization Problems
We introduce an algorithm design technique for a class of combinatorial
optimization problems with concave costs. This technique yields a strongly
polynomial primal-dual algorithm for a concave cost problem whenever such an
algorithm exists for the fixed-charge counterpart of the problem. For many
practical concave cost problems, the fixed-charge counterpart is a well-studied
combinatorial optimization problem. Our technique preserves constant factor
approximation ratios, as well as ratios that depend only on certain problem
parameters, and exact algorithms yield exact algorithms.
Using our technique, we obtain a new 1.61-approximation algorithm for the
concave cost facility location problem. For inventory problems, we obtain a new
exact algorithm for the economic lot-sizing problem with general concave
ordering costs, and a 4-approximation algorithm for the joint replenishment
problem with general concave individual ordering costs
Integrated production-distribution systems : Trends and perspectives
During the last two decades, integrated production-distribution problems have attracted a great deal of attention in the operations research literature. Within a short period, a large number of papers have been published and the field has expanded dramatically. The purpose of this paper is to provide a comprehensive review of the existing literature by classifying the existing models into several different categories based on multiple characteristics. The paper also discusses some trends and list promising avenues for future research
Service System Design with Immobile Servers, Stochastic Demand and Economies of Scale
The service system design problem seeks to locate facilities, determine their capacity,
and assign customers to them in order to improve the service quality and the customers'
experience while minimizing the capacity acquisition cost, the customer access cost, and
the average waiting cost. While the centralization of facilities will lead to economies of
scale, decentralizing them will lead to faster response times. Traditionally, the capacity
acquisition costs were assumed linear with a xed setup cost. In this work, we explicitly
account for economies of scale by modeling the cost as a concave function of capacity.
In this thesis, we model and provide solution methodologies for the service system
design problem with immobile servers, stochastic demand and economies of scale. We
start by reformulating the problem, and then provide solution approaches based on piece-wise linearization, Second Order Cone Programming (SOCP), and Lagrangian Relaxation.Extensive numerical testing on a standard data set is provided and the results analyzed
Combinatorial optimization problems with concave costs
Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2009.Includes bibliographical references (p. 83-89).In the first part, we study the problem of minimizing a separable concave function over a polyhedron. We assume the concave functions are nonnegative nondecreasing on R+, and the polyhedron is in RI' (these assumptions can be relaxed further under suitable technical conditions). We show how to approximate this problem to 1+ E precision in optimal value by a piecewise linear minimization problem so that the number of resulting pieces is polynomial in the input size of the original problem and linear in 1/c. For several concave cost problems, the resulting piecewise linear problem can be reformulated as a classical combinatorial optimization problem. As a result of our bound, a variety of polynomial-time heuristics, approximation algorithms, and exact algorithms for classical combinatorial optimization problems immediately yield polynomial-time heuristics, approximation algorithms, and fully polynomial-time approximation schemes for the corresponding concave cost problems. For example, we obtain a new approximation algorithm for concave cost facility location, and a new heuristic for concave cost multi commodity flow. In the second part, we study several concave cost problems and the corresponding combinatorial optimization problems. We develop an algorithm design technique that yields a strongly polynomial primal-dual algorithm for a concave cost problem whenever such an algorithm exists for the corresponding combinatorial optimization problem.(cont.) Our technique preserves constant-factor approximation ratios as well as ratios that depend only on certain problem parameters, and exact algorithms yield exact algorithms. For example, we obtain new approximation algorithms for concave cost facility location and concave cost joint replenishment, and a new exact algorithm for concave cost lot-sizing. In the third part, we study a real-time optimization problem arising in the operations of a leading internet retailer. The problem involves the assignment of orders that arrive via the retailer's website to the retailer's warehouses. We model it as a concave cost facility location problem, and employ existing primal-dual algorithms and approximations of concave cost functions to solve it. On past data, we obtain solutions on average within 1.5% of optimality, with running times of less than 100ms per problem.by Dan Stratila.Ph.D