Showing papers on "Integer programming published in 1999"

A Computational Study of Search Strategies for Mixed Integer Programming

Abstract: The branch-and-bound procedure for solving mixed integer programming (MIP) problems using linear programming relaxations has been used with great success for decades. Over the years, a variety of researchers have studied ways of making the basic algorithm more effective. Breakthroughs in the fields of computer hardware, computer software, and mathematics have led to increasing success at solving larger and larger MIP instances. The goal of this article is to survey many of the results regarding branch-and-bound search strategies and evaluate them again in light of the other advances that have taken place over the years. In addition, novel search strategies are presented and shown to often perform better than those currently used in practice.

A scalable solution to the multi-resource QoS problem

Abstract: The problem of maximizing system utility by allocating a single finite resource to satisfy discrete Quality of Service (QoS) requirements of multiple applications along multiple QoS dimensions was studied previously. In this paper we consider the more complex problem of apportioning multiple finite resources to satisfy the QoS needs of multiple applications along multiple QoS dimensions. In other words, each application, such as video-conferencing, needs multiple resources to satisfy its QoS requirements. We evaluate and compare three strategies to solve this provably NP-hard problem. We show that dynamic programming and mixed integer programming compute optimal solutions to this problem but exhibit very long running times. We then adapt the mixed integer programming problem to yield near-optimal results with smaller running times. Finally, we present an approximation algorithm based on a local search technique that is less than 5% away from the optimal solution but which is more than two orders of magnitude faster. Perhaps more significantly, the local search technique turns out to be very scalable and robust as the number of resources required by each application increases.

Stochastic integer programming:General models and algorithms

Abstract: We survey structural properties of and algorithms for stochastic integer programmingmodels, mainly considering linear two‐stage models with mixed‐integer recourse (and theirmulti‐stage extensions).

Computational study of a column generation algorithm for bin packing and cutting stock problems

Abstract: This paper reports on our attempt to design an efficient exact algorithm based on column generation for the cutting stock problem. The main focus of the research is to study the extend to which standard branch-and-bound enhancement features such as variable fixing, the tightening of the formulation with cutting planes, early branching, and rounding heuristics can be usefully incorporated in a branch-and-price algorithm. We review and compare lower bounds for the cutting stock problem. We propose a pseudo-polynomial heuristic. We discuss the implementation of the important features of the integer programming column generation algorithm and, in particular, the implementation of the branching scheme. Our computational results demonstrate the efficiency of the resulting algorithm for various classes of bin packing and cutting stock problems.

An Optimal Routing Algorithm for a Transfer Crane in Port Container Terminals

Abstract: This paper focuses on how to optimally route transfer cranes in a container yard during loading operations of export containers at port terminals. Decision variables are the number of containers that a transfer crane picks up at each yard-bay and the sequence of yard-bays that atransfer crane visits during a loading operation. This routing problem is formulated as a mixed integer program. The objective function of the formulation is to minimize the total container handling time of a transfer crane, which includes setup time at each yard-bay and travel time between yard-bays. Based on the mixed integer program, an optimizing algorithm is developed.

Solving Parallel Machine Scheduling Problems by Column Generation

Abstract: We consider a class of problems of scheduling n jobs on m identical, uniform, or unrelated parallel machines with an objective of minimizing an additive criterion. We propose a decomposition approach for solving these problems exactly. The decomposition approach first formulates these problems as an integer program, and then reformulates the integer program, using Dantzig-Wolfe decomposition, as a set partitioning problem. Based on this set partitioning formulation, branch-and-bound exact solution algorithms can be designed for these problems. In such a branch-and-bound tree, each node is the linear relaxation problem of a set partitioning problem. This linear relaxation problem is solved by a column generation approach where each column represents a schedule on one machine and is generated by solving a single machine subproblem. Branching is conducted on variables in the original integer programming formulation instead of variables in the set partitioning formulation such that single machine subproblems ...

Surrogate gradient algorithm for Lagrangian relaxation

Abstract: The subgradient method is used frequently to optimize dual functions in Lagrangian relaxation for separable integer programming problems. In the method, all subproblems must be solved optimally to obtain a subgradient direction. In this paper, the surrogate subgradient method is developed, where a proper direction can be obtained without solving optimally all the subproblems. In fact, only an approximate optimization of one subproblem is needed to get a proper surrogate subgradient direction, and the directions are smooth for problems of large size. The convergence of the algorithm is proved. Compared with methods that take effort to find better directions, this method can obtain good directions with much less effort and provides a new approach that is especially powerful for problems of very large size.

Algorithmic Aspects of the Core of Combinatorial Optimization Games

Abstract: We discuss an integer programming formulation for a class of cooperative games. We focus on algorithmic aspects of the core, one of the most important solution concepts in cooperative game theory. Central to our study is a simple but very useful observation that the core for this class is nonempty if and only if an associated linear program has an integer optimal solution. Based on this, we study the computational complexity and algorithms to answer important questions about the cores of various games on graphs, such as maximum flow, connectivity, maximum matching, minimum vertex cover, minimum edge cover, maximum independent set, and minimum coloring.

Optimal capacitor placement using deterministic and genetic algorithms

Abstract: A procedure for solving the power capacitor placement problem is presented. The objective is to determine the minimum investment required to satisfy suitable reactive constraints. Due to the discrete nature of reactive compensation devices, optimal capacitor placement leads to a nonlinear programming problem with mixed (discrete and continuous) variables. It is solved with an iterative algorithm based on successive linearizations of the original nonlinear model. The mixed integer linear programming problem to be solved at each iteration of the procedure is tackled by applying both a deterministic method (branch and bound) and genetic algorithm techniques. A hybrid procedure, aiming to exploit the best features of both algorithms is also considered. The proposed procedures are tested and compared with reference to a small CIGRE system and two actual networks derived from the Italian transmission and distribution system.

Supplier Selection and Order Quantity Allocation: A Comprehensive Model

Abstract: SUMMARY In today's competitive environment, it is extremely difficult to successfully produce high quality, low cost products without considering a satisfactory set of suppliers. A useful approach to ensure the reliability of a manufacturer's supply stream is to follow a multiple sourcing policy. Mathematical programming techniques lend themselves nicely by providing optimal solutions to several instances of the problem. However, a recent review of supply selection methods (Weber et al. 1991) identified very few articles that have proposed mathematical programming techniques to analyze supplier selection decisions. Given the economic importance and inherent complexity associated with the supplier selection problem, this article proposes a mixed integer programming approach to solve the supplier selection program. The modelSUPPSEL (Supplier Selection) simultaneously decides the set of suppliers and order quantity allocations among them.

Lifted Cover Inequalities for 0-1 Integer Programs: Complexity

Abstract: We investigate several complexity issues related to branchand-cut algorithms for 0-1 integer programming based on lifted cover inequalities (LCIs). We show that given a fractional point, determining a violated LCI over all minimal covers is NP-hard. The main result is that there exists a class of 0-1 knapsack instances for which any branch-and-cut algorithm based on LCIs has to evaluate an exponential number of nodes to prove optimality.

Primal-Dual RNC Approximation Algorithms for Set Cover and Covering Integer Programs

Abstract: We build on the classical greedy sequential set cover algorithm, in the spirit of the primal-dual schema, to obtain simple parallel approximation algorithms for the set cover problem and its generalizations. Our algorithms use randomization, and our randomized voting lemmas may be of independent interest. Fast parallel approximation algorithms were known before for set cover, though not for the generalizations considered in this paper.

Improved Approximation Guarantees for Packing and Covering Integer Programs

Abstract: Several important NP-hard combinatorial optimization problems can be posed as packing/covering integer programs; the randomized rounding technique of Raghavan and Thompson is a powerful tool with which to approximate them well. We present one elementary unifying property of all these integer linear programs and use the FKG correlation inequality to derive an improved analysis of randomized rounding on them. This yields a pessimistic estimator, thus presenting deterministic polynomial-time algorithms for them with approximation guarantees that are significantly better than those known.

A branch-and-cut algorithm for capacitated network design problems

Abstract: We present a branch-and-cut algorithm to solve capacitated network design problems. Given a capacitated network and point-to-point traffic demands, the objective is to install more capacity on the edges of the network and route traffic simultaneously, so that the overall cost is minimized. We study a mixed-integer programming formulation of the problem and identify some new facet defining inequalities. These inequalities, together with other known combinatorial and mixed-integer rounding inequalities, are used as cutting planes. To choose the branching variable, we use a new rule called “knapsack branching”. We also report on our computational experience using real-life data.

Multimodal Express Package Delivery: a Service Network Design Application

Abstract: The focus of this research is to model and solve a large-scale service network design problem involving express package delivery. The objective is to find the cost minimizing movement of packages from their origins to their destinations, given very tight service windows, limited package sort capacity, and a finite number of ground vehicles and aircraft. We have developed a model for large scale transportation service network design problems with time windows. With the use of route-based decision variables, we capture complex cost structures and operating regulations and policies. The poor linear programming bounds limit our ability to solve the problem, so we strengthen our linear programming relaxation by adding valid inequalities. By exploiting problem structure using a specialized network representation and applying a series of novel problem reduction methods, we achieve dramatic decreases in problem size without compromising optimality of the model. Our solution optimization approach synthesizes column and row generation optimization techniques and heuristics to generate solutions to an express package delivery application containing hundreds of thousands of constraints and billions of variables, using only a small fraction of the constraint matrix. The results are potential savings in annual operating costs of tens of millions of dollars, reductions in the fleet size required, dramatic decreases in the time required to develop operating plans, and scenario analysis capabilities for planners and analysts. Through this and additional computational experiments, we conclude that, although state-of-the-art integer programming methods can work well for relatively small, uncongested service network design problems, they must be used in concert with heuristics to be effective for large-scale, congested problems encountered in practice.

Linear programming with fuzzy coefficients in constraints

Abstract: This paper presents a new method for solving linear programming problems with fuzzy coefficients in constraints. It is shown that such problems can be reduced to a linear semi-infinite programming problem. The relations between optimal solutions and extreme points of the linear semi-infinite program are established. A cutting plane algorithm is introduced with a convergence proof, and a numerical example is included to illustrate the solution procedure.

An Exact Method for the Vehicle Routing Problem with Backhauls

Abstract: We consider the problem in which a fleet of vehicles located at a central depot is to be optimallyused to serve a set of customers partitioned into two subsets of linehaul and backhaul customers. Each route starts and ends at the depot and the backhaul customers must be visited afterthe linehaul customers. A new (0-1) integer programming formulation of this problem is presented. We describe a procedure that computes a valid lower bound to the optimal solution cost by combining different heuristic methods for solving the dual of the LP-relaxation of the exact formulation. An algorithm for the exact solution of the problem is presented. Computational tests on problems proposed in the literature show the effectiveness of the proposed algorithms in solving problems up to 100 customers.

Formulations and hardness of multiple sorting by reversals

Abstract: We consider two generalizations of signed Sorting Bg Reversals (SBR), both aimed at formalizing the problem of reconstructing the evolutionary history of a set of species. In particular, we address Multiple SBR, calling for a signed permutation at minimum reversal distance from a given set of signed permutations, and Dee SBR, calling for a tree with the minimum number of edges spanning a given set of nodes in the complete graph where each node corresponds to a signed permutation and there is an edge between each pair of signed permutations one reversal away from each other. We describe a graph-theoretic relaxation of MSBR, which is the counterpart of the so-called alternating-cycle decomposition relaxation for SBR., illustrating a convenient mathematical formulation for this relaxation. Moreover, we use this relaxation to show that, even if the number of given permutations equals 3, MSBR is NP-hard, and hence so is nee SBR. In fact, we show that the two problems are APX-hard, i.e. they do not have a polynomial-time approximation scheme unless P=NP. Finally, we mention known Zapproximation algorithms for two general problems which generalize MSBR and Tree SBR, respectively. To our knowledge, this work is the f?rst one discussing the complexity of MBSR (and Tree SBR), as well as potential solution approaches to the problem based on the use of a tight relaxation.

On the use of integer programming models in AI planning

Abstract: Recent research has shown the promise of using propositional reasoning and search to solve AI planning problems In this paper, we further explore this area by applying Integer Programming to solve AI planning problems The application of Integer Programming to AI planning has a potentially significant advantage, as it allows quite naturally for the incorporation of numerical constraints and objectives into the planning domain Moreover, the application of Integer Programming to AI planning addresses one of the challenges in propositional reasoning posed by Kautz and Selman, who conjectured that the principal technique used to solve Integer Programs--the linear programming (LP) relaxation--is not useful when applied to propositional search We discuss various IP formulations for the class of planning problems based on STRIPS-style planning operators Our main objective is to show that a carefully chosen IP formulation significantly improves the "strength" of the LP relaxation, and that the resultant LPs are useful in solving the IP and the associated planning problems Our results clearly show the importance of choosing the "right" representation, and more generally the promise of using Integer Programming techniques in the AI planning domain

A new approach to integrating mixed integer programming and constraint logicprogramming

Abstract: This paper represents an integration of Mixed Integer Programming (MIP) and ConstraintLogic Programming (CLP) which, like MIP, tightens bounds rather than adding constraintsduring search. The integrated system combines components of the CLP system ECLiPSe[7] and the MIP system CPLEX [5], in which constraints can be handled by either one orboth components. Our approach is introduced in three stages. Firstly, we present an automatictransformation which maps CLP programs onto such CLP programs that any disjunction iseliminated in favour of auxiliary binary variables. Secondly, we present improvements ofthis mapping by using a committed choice operator and translations of pre‐defined non‐linearconstraints. Thirdly, we introduce a new hybrid algorithm which reduces the solutionspace of the problem progressively by calling finite domain propagation of ECLiPSe aswell as dual simplex of CPLEX. The advantages of this integration are illustrated by efficientlysolving difficult optimisation problems like the Hoist Scheduling Problem [23]and the Progressive Party Problem [27].