Showing papers on "Integer programming published in 2004"

The Lagrangian Relaxation Method for Solving Integer Programming Problems

Abstract: (This article originally appeared in Management Science, January 1981, Volume 27, Number 1, pp. 1-18, published by The Institute of Management Sciences.) One of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints. Dualizing the side constraints produces a Lagrangian problem that is easy to solve and whose optimal value is a lower bound (for minimization problems) on the optimal value of the original problem. The Lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm. This approach has led to dramatically improved algorithms for a number of important problems in the areas of routing, location, scheduling, assignment and set covering. This paper is a review of Lagrangian relaxation based on what has been learned in the last decade.

A new branch-and-cut algorithm for the capacitated vehicle routing problem

Abstract: We present a new branch-and-cut algorithm for the capacitated vehicle routing problem (CVRP). The algorithm uses a variety of cutting planes, including capacity, framed capacity, generalized capacity, strengthened comb, multistar, partial multistar, extended hypotour inequalities, and classical Gomory mixed-integer cuts. For each of these classes of inequalities we describe our separation algorithms in detail. Also we describe the other important ingredients of our branch-and-cut algorithm, such as the branching rules, the node selection strategy, and the cut pool management. Computational results, for a large number of instances, show that the new algorithm is competitive. In particular, we solve three instances (B-n50-k8, B-n66-k9 and B-n78-k10) of Augerat to optimality for the first time.

Global optimization of mixed-integer nonlinear programs: A theoretical and computational study

Abstract: This work addresses the development of an efficient solution strategy for obtaining global optima of continuous, integer, and mixed-integer nonlinear programs. Towards this end, we develop novel relaxation schemes, range reduction tests, and branching strategies which we incorporate into the prototypical branch-and-bound algorithm. In the theoretical/algorithmic part of the paper, we begin by developing novel strategies for constructing linear relaxations of mixed-integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties. We then use Lagrangian/linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming. This theory leads to new range reduction schemes, including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch-and-bound tree. Finally, we incorporate these relaxation and reduction strategies in a branch-and-bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems. In the computational part of the paper, we describe our implementation discussing, wherever appropriate, the use of suitable data structures and associated algorithms. We present computational experience with benchmark separable concave quadratic programs, fractional 0–1 programs, and mixed-integer nonlinear programs from applications in synthesis of chemical processes, engineering design, just-in-time manufacturing, and molecular design.

A Decomposition Approach for the Inventory-Routing Problem

Abstract: In this paper, we present a solution approach for the inventory-routing problem. The inventory-routing problem is a variation of the vehicle-routing problem that arises in situations where a vendor has the ability to make decisions about the timing and sizing of deliveries, as well as the routing, with the restriction that customers are not allowed to run out of product. We develop a two-phase approach based on decomposing the set of decisions: A delivery schedule is created first, followed by the construction of a set of delivery routes. The first phase utilizes integer programming, whereas the second phase employs routing and scheduling heuristics. Our focus is on creating a solution methodology appropriate for large-scale real-life instances. Computational experiments demonstrating the effectiveness of our approach are presented.

Impact-based search strategies for constraint programming

Abstract: A key feature of constraint programming is the ability to design specific search strategies to solve problems. On the contrary, integer programming solvers have used efficient general-purpose strategies since their earliest implementations. We present a new general purpose search strategy for constraint programming inspired from integer programming techniques and based on the concept of the impact of a variable. The impact measures the importance of a variable for the reduction of the search space. Impacts are learned from the observation of domain reduction during search and we show how restarting search can dramatically improve performance. Using impacts for solving multiknapsack, magic square, and Latin square completion problems shows that this new criteria for choosing variables and values can outperform classical general-purpose strategies.

An Exact Algorithm for the Capacitated Vehicle Routing Problem Based on a Two-Commodity Network Flow Formulation

Abstract: The capacitated vehicle routing problem (CVRP) is the problem in which a set of identical vehicles located at a central depot is to be optimally routed to supply customers with known demands subject to vehicle capacity constraints. In this paper, we describe a new integer programming formulation for the CVRP based on a two-commodity network flow approach. We present a lower bound derived from the linear programming (LP) relaxation of the new formulation which is improved by adding valid inequalities in a cutting-plane fashion. Moreover, we present a comparison between the new lower bound and lower bounds derived from the LP relaxations of different CVRP formulations proposed in the literature. A new branch-and-cut algorithm for the optimal solution of the CVRP is described. Computational results are reported for a set of test problems derived from the literature and for new randomly generated problems.

A finite branch-and-bound algorithm for two-stage stochastic integer programs

Abstract: This paper addresses a general class of two-stage stochastic programs with integer recourse and discrete distributions. We exploit the structure of the value function of the second-stage integer problem to develop a novel global optimization algorithm. The proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination. Computational experiments on standard test problems indicate superior performance of the proposed algorithm in comparison to those in the existing literature.

Multistage and coordinated planning of the expansion of transmission systems

Abstract: In this paper, an efficient genetic algorithm (GA) is presented to solve the problem of multistage and coordinated transmission expansion planning. This is a mixed integer nonlinear programming problem, difficult for systems of medium and large size and high complexity. The GA presented has a set of specialized genetic operators and an efficient form of generation of the initial population that finds high quality suboptimal topologies for large size and high complexity systems. In these systems, multistage and coordinated planning present a lower investment than static planning. Tests results are shown in one medium complexity system and one large size high complexity system.

An Exact Method for the Car Pooling Problem Based on Lagrangean Column Generation

Abstract: Car pooling is a transportation service organized by a large company which encourages its employees to pick up colleagues while driving to/from work to minimize the number of private cars travelling to/from the company site. The car pooling problem consists of defining the subsets of employees that will share each car and the paths the drivers should follow, so that sharing is maximized and the sum of the path costs is minimized. The special case of the car pooling problem where all cars are identical can be modeled as a Dial-a-Ride Problem. In this paper, we propose both an exact and a heuristic method for the car pooling problem, based on two integer programming formulations of the problem. The exact method is based on a bounding procedure that combines three lower bounds derived from different relaxations of the problem. A valid upper bound is obtained by the heuristic method, which transforms the solution of a Lagrangean lower bound into a feasible solution. The computational results show the effectiveness of the proposed methods.

Decentralized model predictive control of cooperating UAVs

Abstract: This paper implements robust decentralized model predictive control (DMPC) for a team of cooperating uninhabited aerial vehicles (UAVs). The problem involves vehicles with independent dynamics but with coupled constraints to capture required cooperative behavior. Using a recently-developed form of DMPC, each vehicle plans only for its own actions, but feasibility of the sub-problems and satisfaction of the coupling constraints are guaranteed throughout, despite the action of unknown but bounded disturbances. UAVs communicate relevant plan data to ensure that decisions are consistent across the team. Collision avoidance is used as an example of coupled constraints and the paper shows how the speed, turn rate and avoidance distance limits in the optimization should be modified in order to guarantee robust constraint satisfaction. Integer programming is used to solve the non-convex problem of path-planning subject to avoidance constraints. Numerical simulations compare computation time and target arrival time under decentralized and centralized control and investigate the impact of decentralization on team performance. The results show that the computation required for DMPC is significantly lower than for its centralized counterpart and scales better with the size of the team, at the expense of only a small increase in UAV flight times.

The optimal cordon-based network congestion pricing problem

Abstract: This paper investigates the cordon-based second-best congestion-pricing problems on road networks, including optimal selection of both toll levels and toll locations. A road network is viewed as a directed graph and the cutset concept in graph theory is used to describe the mathematical properties of a toll cordon by examining the incidence matrix of the network. Maximization of social welfare is sought subject to the elastic-demand traffic equilibrium constraint. A mathematical programming model with mixed (integer and continuous) variables is formulated and solved by a combined use of a binary genetic algorithm and a grid search method for simultaneous determination of the toll levels and cordon locations on the networks. The model and algorithm are demonstrated with a numerical example.

Receding horizon path planning with implicit safety guarantees

Abstract: This paper extends a recently developed approach to optimal path planning of autonomous vehicles, based on mixed integer linear programming (MILP), to account for safety. We consider the case of a single vehicle navigating through a cluttered environment which is only known within a certain detection radius around the vehicle. A receding horizon strategy is presented with hard terminal constraints that guarantee feasibility of the MILP problem at all future time steps. The trajectory computed at each iteration is constrained to end in a so called basis state, in which the vehicle can safely remain for an indefinite period of time. The principle is applied to the case of a UAV with limited turn rate and minimum speed requirements, for which safety conditions are derived in the form of loiter circles. The latter need not be known ahead of time and are implicitly computed online. An example scenario is presented that illustrates the necessity of these safety constraints when the knowledge of the environment is limited and/or hard real-time restrictions are given.

Decentralized Cooperative Trajectory Planning of Multiple Aircraft with Hard Safety Guarantees

Abstract: This paper presents a framework for provably safe decentralized trajectory planning of multiple (autonomous) aircraft. Each aircraft plans its trajectory individually using a receding horizon strategy based on mixed integer linear programming (MILP). A constrained, inertial, first-order linear model is used to capture the dynamics and kinematics of the vehicle. Safety is guaranteed by maintaining, at each time step, a dynamically feasible trajectory for each aircraft that terminates in a loiter pattern. Conflicts between multiple aircraft are resolved in a sequential, decentralized fashion, in which each aircraft takes into account the latest trajectory and loiter pattern of the other aircraft. Besides maintaining feasibility, if the problem is too complex to be solved within the time constraints of a realtime system, this approach also provides an a priori safe rescue solution consisting of the previous trajectories and individual loiter patterns. Several examples of conflict situations resolved by the proposed method are presented.

Semantic role labeling via integer linear programming inference

Abstract: We present a system for the semantic role labeling task. The system combines a machine learning technique with an inference procedure based on integer linear programming that supports the incorporation of linguistic and structural constraints into the decision process. The system is tested on the data provided in CoNLL-2004 shared task on semantic role labeling and achieves very competitive results.

Integration of AI and OR techniques in constraint programming for combinatorial optimization problems : first International Conference, CPAIOR 2004, Nice, France, April 20-22, 2004 : proceedings

Abstract: Invited Paper- Using MILP and CP for the Scheduling of Batch Chemical Processes- Technical Papers- SIMPL: A System for Integrating Optimization Techniques- A New Exact Solution Algorithm for the Job Shop Problem with Sequence-Dependent Setup Times- Simple Rules for Low-Knowledge Algorithm Selection- Filtering Algorithms for the Same Constraint- Cost Evaluation of Soft Global Constraints- SAT-Based Branch & Bound and Optimal Control of Hybrid Dynamical Systems- Solving the Petri Nets Reachability Problem Using the Logical Abstraction Technique and Mathematical Programming- Generating Benders Cuts for a General Class of Integer Programming Problems- A Constraint Programming Model for Tail Assignment- Super Solutions in Constraint Programming- Local Probing Applied to Network Routing- Dynamic Heaviest Paths in DAGs with Arbitrary Edge Weights- Filtering Methods for Symmetric Cardinality Constraint- Arc-Consistency Filtering Algorithms for Logical Combinations of Constraints- Combining Forces to Solve the Car Sequencing Problem- Travelling in the World of Local Searches in the Space of Partial Assignments- A Global Constraint for Nesting Problems- Models and Symmetry Breaking for 'Peaceable Armies of Queens'- A Global Constraint for Graph Isomorphism Problems- Echelon Stock Formulation of Arborescent Distribution Systems: An Application to the Wagner-Whitin Problem- Scheduling Abstractions for Local Search- O(nlog n) Filtering Algorithms for Unary Resource Constraint- Problem Decomposition for Traffic Diversions- Short Papers- LP Relaxations of Multiple all_different Predicates- Dispatching and Conflict-Free Routing of Automated Guided Vehicles: A Hybrid Approach Combining Constraint Programming and Mixed Integer Programming- Making Choices Using Structure at the Instance Level within a Case Based Reasoning Framework- The Challenge of Generating Spatially Balanced Scientific Experiment Designs- Building Models through Formal Specification- Stabilization Issues for Constraint Programming Based Column Generation- A Hybrid Branch-And-Cut Algorithm for the One-Machine Scheduling Problem

Safe bounds in linear and mixed-integer linear programming

Abstract: Current mixed-integer linear programming solvers are based on linear programming routines that use floating-point arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers. An example is given where many state-of-the-art MILP solvers fail. It is then shown how, using directed rounding and interval arithmetic, cheap pre- and postprocessing of the linear programs arising in a branch-and-cut framework can guarantee that no solution is lost, at least for mixed-integer programs in which all variables can be bounded rigorously by bounds of reasonable size.

Traffic grooming, routing, and wavelength assignment in optical WDM mesh networks

Abstract: In this paper, we consider the traffic grooming, routing, and wavelength assignment (GRWA) problem for optical mesh networks. In most previous studies on optical mesh networks, traffic demands are usually assumed to be wavelength demands, in which case no traffic grooming is needed. In practice, optical networks are typically required to carry a large number of lower rate (sub-wavelength) traffic demands. Hence, the issue of traffic grooming becomes very important since it can significantly impact the overall network cost. In our study, we consider traffic grooming in combination with traffic routing and wavelength assignment. Our objective is to minimize the total number of transponders required in the network. We first formulate the GRWA problem as an integer linear programming (ILP) problem. Unfortunately, for large networks it is computationally infeasible to solve the ILP problem. Therefore, we propose a decomposition method that divides the GRWA problem into two smaller problems: the traffic grooming and routing problem and the wavelength assignment problem, which can then be solved much more efficiently. In general, the decomposition method only produces an approximate solution for the GRWA problem. However, we also provide some sufficient condition under which the decomposition method gives an optimal solution. Finally, some numerical results are provided to demonstrate the efficiency of our method.

A Branch-and-Price Algorithm for Multistage Stochastic Integer Programming with Application to Stochastic Batch-Sizing Problems

Abstract: In this paper, we present a branch-and-price method to solve special structured multistage stochastic integer programming problems. We validate our method on two different versions of a multistage stochastic batch-sizing problem (SBSP). One version adopts a recourse formulation, and the other is based on probabilistic constraints. Our algorithmic approach is applicable to both formulations. Our computational results suggest that both classes of problems can be solved using relatively few nodes of a branch-and-price tree. The success of our approach calls for extensions in methodology as well as applications.

An enhanced 0-1 mixed-integer LP formulation for traffic signal control

Abstract: An enhanced 0-1 mixed-integer linear programming formulation based on the cell-transmission model is proposed for the traffic signal optimization problem. This formulation has several features that are currently unavailable in other existing models developed with a similar approach, including the components for handling the number of stops, fixed or dynamic cycle length and splits, and lost time. The problem of unintended vehicle holding, which is common in analytical models, is explicitly treated. The formulation can be utilized in developing strategies for adaptive traffic-control systems. It can also be used as a benchmark for examining the convergence behavior of heuristic algorithms based on the genetic algorithm, fuzzy logic, neural networks, or other approaches that are commonly used in this field. The discussion of extending the proposed model to capture traffic signal preemption in the presence of emergency vehicles is given. In terms of computational efficiency, the proposed formulation has the least number of binary integers as compared with other existing formulations that were developed with the same approach.

Survey, Categorization, and Comparison of Recent Tour Scheduling Literature

Abstract: The employee tour scheduling problem involves the determination of both work hours of the day and workdays of the week for each employee. This problem has proven difficult to solve optimally due to its large size and pure integer nature. During the last decade, numerous approaches for modeling and solving this problem have been proposed. In this paper, employee tour scheduling literature published since 1990 is reviewed and classified. Solution techniques are classified into ten categories: (1) manual solution, (2) integer programming, (3) implicit modeling, (4) decomposition, (5) goal programming, (6) working set generation, (7) LP-based solution, (8) construction and improvement, (9) metaheuristics, and (10) other methods. The objective is to identify broad classifications, present typical mathematical models, compare the different methods, and identify future research directions.

Ant direction hybrid differential evolution for solving large capacitor placement problems

Abstract: This paper presents an ant direction hybrid differential evolution (ADHDE) with integer programming which is effective and efficient for solving large capacitor placement problems in distribution systems. The use of proper mutation operator in hybrid differential evolution (HDE) can accelerate the search of a global solution. However, the selection of mutation operator depends on the problem. In this study, the ADHDE method utilizes the concept of ant colony search to search the proper mutation operator to accelerate searching out the global solution. Various-scale application systems are used to compare the performance of the proposed method with HDE, simulated annealing, and ant system. Numerical results show that the performance of the proposed ADHDE method is better than the other methods. Also, the ADHDE method is superior to some other methods in terms of solution power loss and costs.

Optimal scheduling of rehabilitation activities for multiple pavement facilities: exact and approximate solutions

Abstract: This paper presents a mathematical programming model for optimal highway pavement rehabilitation planning which minimizes the life-cycle cost for a finite horizon. It extends previous researches in this area by solving the problem of multiple rehabilitation activities on multiple facilities, with realistic empirical models of deterioration and rehabilitation effectiveness. The formulation is based on discrete control theory. A nonlinear pavement performance model and integer decision variables are incorporated into a mixed-integer nonlinear programming (MINLP). Two solution approaches, a branch-and-bound algorithm and a greedy heuristic, are proposed for this model. It is shown that the heuristic results provide a good approximation to the exact optima, but with much lower computational costs.

CalSim: Generalized Model for Reservoir System Analysis

Abstract: The California State Department of Water Resources and the United States Bureau of Reclamation Mid-Pacific Region have developed a general-purpose reservoir–river basin simulation model for the planning and management of the State Water Project and the federal Central Valley Project. The California Water Resources Simulation Model brings a fundamental change to modeling of these systems. Model users specify system objectives as input to the model. System description and operational constraints are specified using a new water resources engineering simulation language. A mixed integer linear programming solver efficiently routes water through the system network given the user-defined priorities or weights. Simulation cycles at different temporal scales allow for successive layering of constraints. The power and flexibility of the model is demonstrated by its ability to simulate the operation of complex new environmental water accounts.

A Branch-and-Cut Approach for Solving Railway Line-Planning Problems

Abstract: An important strategic phase in the planning process of a railway operator is the development of a line plan, i.e., a set of routes (paths) in a network of tracks, operated at a given hourly frequency. We consider a model formulation of the line-planning problem where total operating costs are to be minimized. This model is solved with a branch-and-cut approach, for which we develop a variety of valid inequalities and reduction methods. A computational study of five real-life instances based on examples from Netherlands Railways (NS) is included.

1001 optimal PDB structure alignments: integer programming methods for finding the maximum contact map overlap.

Abstract: Protein structure comparison is a fundamental problem for structural genomics, with applications to drug design, fold prediction, protein clustering, and evolutionary studies. Despite its importance, there are very few rigorous methods and widely accepted similarity measures known for this problem. In this paper we describe the last few years of developments on the study of an emerging measure, the contact map overlap (CMO), for protein structure comparison. A contact map is a list of pairs of residues which lie in three-dimensional proximity in the protein's native fold. Although this measure is in principle computationally hard to optimize, we show how it can in fact be computed with great accuracy for related proteins by integer linear programming techniques. These methods have the advantage of providing certificates of near-optimality by means of upper bounds to the optimal alignment value. We also illustrate effective heuristics, such as local search and genetic algorithms. We were able to obtain for...

School redistricting: embedding GIS tools with integer programming

Abstract: The paper deals with a school redistricting problem in which blocks of a city must be assigned to schools according to diverse criteria. Previous approaches are reviewed and some desired properties of a good school districting plan are established. An optimization model together with a geographic information system environment are then proposed for finding a solution that satisfies these properties. A prototype of the system is described, some implementation issues are discussed, and two real-life examples from the city of Philadelphia are studied, one corresponding to a relatively easy to solve problem, and the other to a much harder one. The trade-offs in the solutions are analysed and feasibility questions are discussed. The results of the study strongly suggest that ill-defined spatial problems, such as school redistricting, can be addressed effectively by an interaction between objective analysis and subjective judgement.