Showing papers on "Integer programming published in 2001"

Mixed integer programming for multi-vehicle path planning

Abstract: This paper presents a new approach to fuel-optimal path planning of multiple vehicles using a combination of linear and integer programming. The basic problem formulation is to have the vehicles move from an initial dynamic state to a final state without colliding with each other, while at the same time avoiding other stationary and moving obstacles. It is shown that this problem can be rewritten as a linear program with mixed integer/linear constraints that account for the collision avoidance. A key benefit of this approach is that the path optimization can be readily solved using the CPLEX optimization software with an AMPL/Matlab interface. An example is worked out to show that the framework of mixed integer/linear programming is well suited for path planning and collision avoidance problems. Implementation issues are also considered. In particular, we compare receding horizon strategies with fixed arrival time approaches.

Algorithms for Hybrid MILP/CP Models for a Class of Optimization Problems

Abstract: The goal of this paper is to develop models and methods that use complementary strengths of Mixed Integer Linear Programming (MILP) and Constraint Programming (CP) techniques to solve problems that are otherwise intractable if solved using either of the two methods. The class of problems considered in this paper have the characteristic that only a subset of the binary variables have non-zero objective function coefficients if modeled as an MILP. This class of problems is formulated as a hybrid MILP/CP model that involves some of the MILP constraints, a reduced set of the CP constraints, and equivalence relations between the MILP and the CP variables. An MILP/CP based decomposition method and an LP/CP-based branch-and-bound algorithm are proposed to solve these hybrid models. Both these algorithms rely on the same relaxed MILP and feasibility CP problems. An application example is considered in which the least-cost schedule has to be derived for processing a set of orders with release and due dates using a set of dissimilar parallel machines. It is shown that this problem can be modeled as an MILP, a CP, a combined MILP-CP OPL model (Van Hentenryck 1999), and a hybrid MILP/CP model. The computational performance of these models for several sets shows that the hybrid MILP/CP model can achieve two to three orders of magnitude reduction in CPU time.

A Mixed Integer Disjunctive Model for Transmission Network Expansion

Abstract: The classical nonlinear mixed integer formulation of the transmission network expansion problem cannot guarantee finding the optimal solution due to its nonconvex nature. We propose an alternative mixed integer linear disjunctive formulation, which has better conditioning properties than the standard disjunctive model. The mixed integer program is solved by a commercial branch and bound code, where an upper bound provided by a heuristic solution is used to reduce the tree search. The heuristic solution is obtained using a GRASP metaheuristic, capable of finding sub-optimal solutions with an affordable computing effort. Combining the upper bound given by the heuristic and the mixed integer disjunctive model, optimality can be proven for several hard problem instances.

Creating bus timetables with maximal synchronization

Abstract: This paper addresses the problem of generating a timetable for a given network of buses so as to maximize their synchronization. It attempts to maximize the number of simultaneous bus arrivals at the connection (transfer) nodes of the network. Transit schedulers, taking into account the satisfaction and convenience of the system's users, appreciate the importance of creating a timetable with maximal synchronization, which enables the transfer of passengers from one route to another with minimum waiting time at the transfer nodes. In this paper, the problem is formulated as a mixed integer linear programming problem, and a heuristic algorithm is developed to solve the problem in polynomial time. The efficiency of this algorithm, compared to optimal solutions, is illustrated through a series of examples.

Optimal Capacitor Placement in Radial Distribution Networks

Abstract: The capacitor placement (replacement) problem for radial distribution networks determines capacitor types, sizes, locations, and control schemes. Optimal capacitor placement is a hard combinatorial problem that can be formulated as a mixed integer nonlinear program. Since this is a nonpolynomial time (NP) complete problem, the solution approach uses a combinatorial search algorithm. The paper proposes a hybrid method drawn upon the Tabu search approach, extended with features taken from other combinatorial approaches such as genetic algorithms and simulated annealing, and from practical heuristic approaches. The proposed method has been tested in a range of networks available in the literature with superior results regarding both quality and cost of solutions.

Integrating SQP and Branch-and-Bound for Mixed Integer Nonlinear Programming

Abstract: This paper considers the solution of Mixed Integer Nonlinear Programming (MINLP) problems. Classical methods for the solution of MINLP problems decompose the problem by separating the nonlinear part from the integer part. This approach is largely due to the existence of packaged software for solving Nonlinear Programming (NLP) and Mixed Integer Linear Programming problems. In contrast, an integrated approach to solving MINLP problems is considered here. This new algorithm is based on branch-and-bound, but does not require the NLP problem at each node to be solved to optimality. Instead, branching is allowed after each iteration of the NLP solver. In this way, the nonlinear part of the MINLP problem is solved whilst searching the tree. The nonlinear solver that is considered in this paper is a Sequential Quadratic Programming solver. A numerical comparison of the new method with nonlinear branch-and-bound is presented and a factor of up to 3 improvement over branch-and-bound is observed.

Experiences with Mixed Integer Linear Programming-Based Approaches in Short-Term Hydro Scheduling

Abstract: This paper describes experiences with mixed integer linear programming (MILP) based approaches on the short-term hydro scheduling (STHS) function. The STHS is used to determine the optimal or near-optimal schedules for the dispatchable hydro units in a hydro-dominant system for a user-definable study period at each time step while respecting all system and hydraulic constraints. The problem can be modeled in detail for a hydro system that contains both conventional and pumped-storage units. Discrete and dynamic constraints such as unit startup/shutdown and minimum-up/minimum-down time limits are also included in the model for hydro unit commitment (HUC). The STHS problem is solved with a state-of-the-art package which includes an algebraic modeling language and a MILP solver. The usefulness of the proposed solution algorithm is illustrated by testing the problem with actual hydraulic system data. Numerical experiences show that the solution technique is computationally efficient, simple, and suitable for decision support of short-term hydro operations planning. In addition, the proposed approaches can be easily extended for scheduling applications in a deregulated environment.

Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme

Abstract: In the topology optimization of structures, compliant mechanisms or materials, a density-like func- tion is often used for material interpolation to overcome the computational difficulties encountered in the large "0-1" type integer programming problem. In this paper, we illustrate that a gradually formed continuous peak function can be used for material interpolation. One of the advantages of introducing the peak function is that multiple materials can easily be incorporated into the topology optimization without increasing the number of design variables. By using the peak function and the op- timality criteria method, we synthesize compliant mech- anisms with multiple materials with and without the ma- terial resource constraint. The numerical examples in- clude the two-phase, three-phase, and four-phase materi- als where void is treated as one material. This newdesign method enables us to optimally juxtapose stiff and flex- ible materials in compliant mechanisms, which can be built using modern manufacturing methods.

The Traveling Tournament Problem Description and Benchmarks

Abstract: The Traveling Tournament Problem is a sports timetabling problem that abstracts two issues in creating timetables: home/away pattern feasibility and team travel. Instances of this problem seem to be very difficult even for a very small number of teams, making it an interesting challenge for combinatorial optimization techniques such as integer programming and constraint programming. We introduce the problem, describe one way of modeling it, and give some interesting classes of instances with base computational results.

Cold-standby redundancy optimization for nonrepairable systems

Abstract: A solution methodology is described and demonstrated to determine optimal design configurations for nonrepairable series–parallel systems with cold-standby redundancy. This problem formulation considers non-constant component hazard functions and imperfect switching. The objective of the redundancy allocation problem is to select from available components and to determine an optimal design configuration to maximize system reliability. For cold-standby redundancy, other formulations have generally required exponential component time-to-failure and perfect switching assumptions. For this paper, there are multiple component choices available for each subsystem and component time-to-failure is distributed according to an Erlang distribution. Optimal solutions are determined based on an equivalent problem formulation and integer programming. Compared to other available algorithms, the methodology presented here more accurately models many engineering design problems with cold-standby redundancy. Previously, it has been difficult to determine optimal solutions for this class of problems or even to efficiently calculate system reliability. The methodology is successfully demonstrated on a large problem with 14 subsystems.

Aggregation and Mixed Integer Rounding to Solve MIPs

Abstract: In this paper, we discuss the use of mixed integer rounding (MIR) inequalities to solve mixed integer programs. MIR inequalities are essentially Gomory mixed integer cuts. However, as we wish to use problem structure, we insist that MIR inequalities be generated from constraints or simple aggregations of constraints of the original problem. This idea is motivated by the observation that several strong valid inequalities based on specific problem structure can be derived as MIR inequalities.Here we present and test a separation routine for such MIR inequalities that includes a heuristic row aggregation procedure to generate a single knapsack plus continuous variables constraint, complementation of variables, and finally the generation of an MIR inequality. Inserted in a branch-and-cut system, the results suggest that such a routine is a useful additional tool for tackling a variety of mixed integer programming problems.

On the use of linear programming for dynamic subchannel and bit allocation in multiuser OFDM

Abstract: Adaptive subcarrier allocation and adaptive modulation for multiuser orthogonal frequency division multiplexing (OFDM) is considered. The optimal subcarrier and bit allocation problems, that have been formulated in Wong et al., (1999), and Rhee et al., (2000), as nonlinear optimizations, are converted into linear ones and solved by integer programming (IP). A suboptimal approach that separately performs subcarrier allocation and bit loading is proposed. It is shown that subcarrier allocation in this approach can be optimized by the linear programming (LP) relaxation of the IP. Comparison through computer simulation indicates that performance of the suboptimal approach can be close to that of the optimal.

Solving the Asymmetric Travelling Salesman Problem with time windows by branch-and-cut

Abstract: Many optimization problems have several equivalent mathematical models. It is often not apparent which of these models is most suitable for practical computation, in particular, when a certain application with a specific range of instance sizes is in focus. Our paper addresses the Asymmetric Travelling Salesman Problem with time windows (ATSP-TW) from such a point of view. The real–world application we aim at is the control of a stacker crane in a warehouse.¶We have implemented codes based on three alternative integer programming formulations of the ATSP-TW and more than ten heuristics. Computational results for real-world instances with up to 233 nodes are reported, showing that a new model presented in a companion paper outperforms the other two models we considered – at least for our special application – and that the heuristics provide acceptable solutions.

Linear and integer programming; Theory and practice

Abstract: Linear optimisation basic concepts Dantzig's simplex method duality and optimality sensitivity analysis karmarkar's interior path method integer linear optimisation linear network models computational complexity issues model building, case studies, and advanced techniques solutions to selected exercises Appendices: linear algebra convexity graph theory optimisation theory computer package INTPM

Modelling Practical Lot-Sizing Problems as Mixed-Integer Programs

Abstract: In spite of the remarkable improvements in the quality of general purpose mixed-integer programming software, the effective solution of a variety of lot-sizing problems depends crucially on the development of tight formulations for the special problem features occurring in practice. After reviewing some of the basic preprocessing techniques for handling safety stocks and multilevel problems, we discuss a variety of aspects arising particularly in small and large bucket time period models such as start-ups, changeovers, minimum batch sizes, choice of one or two set-ups per period, etc. A set of applications is described that contains one or more of these special features, and some indicative computational results are presented. Finally, to show another technique that is useful, a slightly different supply chain application is presented, for which the a priori addition of some simple mixed-integer inequalities, based on aggregation, leads to important improvements in the results.

Transmission network expansion planning under a Tabu Search approach

Abstract: This paper describes an implementation of Tabu Search to cope with long-term transmission network expansion planning problems. Tabu Search is a metaheuristic proposed in 1989 to be applied to combinatorial problems. To assess the potential of our approach we test it with two cases of transmission network expansion planning. The results obtained by our approach let us to conclude that TS is a robust and promising technique to be applied in this problem.

Combined word-length optimization and high-level synthesis of digital signal processing systems

Abstract: Conventional approaches for fixed-point implementation of digital signal processing algorithms require the scaling and word-length (WL) optimization at the algorithm level and the high-level synthesis for functional unit sharing at the architecture level. However, the algorithm-level WL optimization has a few limitations because it can neither utilize the functional unit sharing information for signal grouping nor estimate the hardware cost for each operation accurately. In this study, we develop a combined WL optimization and high-level synthesis algorithm not only to minimize the hardware implementation cost, but also to reduce the optimization time significantly. This software initially finds the WL sensitivity or minimum WL of each signal throughout fixed-point simulations of a signal flow graph, performs the WL conscious high-level synthesis where signals having the similar WL sensitivity are assigned to the same functional unit, and then conducts the final WL optimization by iteratively modifying the WLs of the synthesized hardware model. A list-scheduling-based and an integer linear-programming-based algorithms are developed for the WL conscious high-level synthesis. The hardware cost function to minimize is generated by using a synthesized hardware model. Since fixed-point simulation is used to measure the performance, this method can be applied to general, including nonlinear and time-varying, digital signal processing systems. A fourth-order infinite-impulse response filter, a fifth-order elliptic filter, and a 12th-order adaptive least mean square filter are implemented using this software.

Approximating optimal spare capacity allocation by successive survivable routing

Abstract: Spare capacity allocation (SCA) is an important part of a fault tolerant network design. In the spare capacity allocation problem one seeks to determine where to place spare capacity in the network and how much spare capacity must be allocated to guarantee seamless communications services survivable to a set of failure scenarios (e.g., any single link failure). Formulated as a multi-commodity flow integer programming problem, SCA is known to be NP-hard. We provide a two-pronged attack to approximate the optimal SCA solution: unravel the SCA structure and find an effective algorithm. First, a literature review on the SCA problem and its algorithms is provided. Second, a integer programming model for SCA is provided. Third, a simulated annealing algorithm using the above INP model is introduced. Next, the structure of SCA is modeled by a matrix method. The per-flow based backup path information are aggregated into a square matrix, called the spare provision matrix (SPM). The size of the SPM is the number of links. Using the SPM as the state information, a new adaptive algorithm is then developed to approximate the optimal SCA solution termed successive survivable routing (SSR). SSR routes link-disjoint backup paths for each traffic flow one at a time. Each flow keeps updating its backup path according to the current network state as long as the backup path is not carrying any traffic. In this way, SSR can be implemented by shortest path algorithms using advertised state information with complexity of O( Link/sup 2/). The analysis also shows that SSR is using a necessary condition of the optimal solution. The numerical results show that SSR has near optimal spare capacity allocation with substantial advantages in computation speed.

Algorithm for traffic grooming in optical networks to minimize the number of transceivers

Abstract: We study the problem of traffic grooming to reduce the number of transceivers in optical networks. We show that this problem is equivalent to a certain traffic maximization problem. We give an intuitive interpretation of this equivalence and use this interpretation to derive a greedy algorithm for transceiver minimization. We discuss implementation issues and present computational results comparing the heuristic solutions with the optimal solutions for several small example networks. For larger networks, the heuristic solutions are compared with known bounds on the optimal solution obtained using integer programming tools.

Branch-and-Check: A Hybrid Framework Integrating Mixed Integer Programming and Constraint Logic Programming

Abstract: We present Branch-and-Check, a hybrid framework integrating Mixed Integer Programming and Constraint Logic Programming, which encapsulates the traditional Benders Decomposition and Branch-and-Bound as special cases. In particular we describe its relation to Benders and the use of nogoods and linear relaxations. We give two examples of how problems can be modelled and solved using Branch-and-Check and present computational results demonstrating more than order-of-magnitude speedup compared to previous approaches. We also mention important future research issues such as hierarchical, dynamic and adjustable linear relaxations.

Optimal spilling for CISC machines with few registers

Abstract: Many graph-coloring register-allocation algorithms don't work well for machines with few registers. Heuristics for live-range splitting are complex or suboptimal; heuristics for register assignment rarely factor the presence of fancy addressing modes; these problems are more severe the fewer registers there are to work with. We show how to optimally split live ranges and optimally use addressing modes, where the optimality condition measures dynamically weighted loads and stores but not register-register moves. Our algorithm uses integer linear programming but is much more efficient than previous ILP-based approaches to register allocation. We then show a variant of Park and Moon's optimistic coalescing algorithm that does a very good (though not provably optimal) job of removing the register-register moves. The result is Pentium code that is 9.5% faster than code generated by SSA-based splitting with iterated register coalescing.

The Batch Loading and Scheduling Problem

Abstract: This paper discusses the problem of batching and scheduling of certain kinds of batch processors. Examples of these processors include heat treatment facilities, particularly in the steel and ceramics industries, as well as a variety of operations in the manufacture of integrated circuits. In general, for our problem there is a set of jobs waiting to be processed. Each job is associated with a given family and has a weight or delay cost and a volume. The scheduler must organize jobs into batches in which each batch consists of jobs from a single family and in which the total volume of jobs in a batch does not exceed the capacity of the processor. The scheduler must then sequence all the batches. The processing time for a batch depends only on the family and not on the number or the volume of jobs in the batch. The objective is to minimize the mean weighted flow time.The paper presents an integer programming formulation for this problem, generates a lower bound from a partial LP relaxation, provides a polynomial algorithm to solve a special case, and tests a set of heuristics on the general problem. The ability to pack jobs into batches is the key to efficient solutions and is the basis of the different solution procedures in this paper. The heuristics include a greedy heuristic, a successive knapsack heuristic, and a generalized assignment heuristic. Optimal solutions are obtained by complete enumeration for small problems.The conclusions of the computational study show that the successive knapsack and generalized assignment heuristics perform better than the greedy. The generalized assignment heuristic does slightly better than the successive knapsack heuristic in some cases, but the latter is substantially faster and more robust. For problems with few jobs, the generalized assignment heuristic and the knapsack heuristic almost always provide optimal solutions. For problems with more jobs, we compare the heruistic solutions' values to lower bounds; the computational work suggests that the heuristics continue to provide solutions that are optimal or close to the optimal. The study also shows that the volume of the job relative to the capacity of the facility and the number of jobs in a family affect the performance of the heuristics, whereas the number of families does not. Finally, we give a worst-case analysis of the greedy heuristic.

An Improved RTN Continuous-Time Formulation for the Short-term Scheduling of Multipurpose Batch Plants

Abstract: This paper presents an improved general mathematical programming formulation for optimal scheduling of batch processes based on the resource−task network (RTN) representation. The formulation uses a continuous-time representation and results in a mixed integer linear programming problem. It is a relaxation of the problem presented by Schilling (Schilling, G. Optimal Scheduling of Multipurpose Plants. Ph.D. Thesis, University of London, London, U., K., 1997). By allowing, if possible, finite storage within the processing tasks resource equipments of the involved raw materials and/or products, the proposed approach leads to simpler and less degenerate mathematical models. These models can be solved in significantly less CPU time, when compared to other RTN continuous-time formulations. Three published example problems are presented to illustrate the effectiveness of the proposed formulation. Finally, we show that the STN-based continuous-time scheduling formulation of Ierapetritou and Floudas (Ierapetritou,...

101 optimal PDB structure alignments: a branch-and-cut algorithm for the maximum contact map overlap problem

Abstract: Structure comparison is a fundamental problem for structural genomics. A variety of structure comparison methods were proposed and several protein structure classification servers e.g., SCOP, DALI, CATH, were designed based on them, and are extensively used in practice. This area of research continues to be very active, being energized bi-annually by the CASP folding competitions, but despite the extraordinary international research effort devoted to it, progress is slow. A fundamental dimension of this bottleneck is the absence of rigorous algorithmic methods. A recent excellent survey on structure comparison by Taylor et.al. [23] records the state of the art of the area: In structure comparison, we do not even have an algorithm that guarantees an optimal answer for pairs of structures …In this paper we provide the first rigorous algorithm for structure comparison. Our method is based on developing an effective integer linear programming (IP) formulation of protein structure contact maps overlap (CMO), and a branch-and-cut strategy that employs lower-bounding heuristics at the branch nodes. Our algorithms identified a gallery of optimal and near-optimal structure alignments for pairs of proteins from the Protein Data Bank with up to 80 amino acids and about 150 contacts each — problems of instance size of about 300. Although these sizes also reflect our current limitations, these are the first provable optimal and near-optimal algorithms in the literature for a measure of structure similarity which sees extensive practical use. At the heart of our success in finding optimal alignments is a reduction of the CMO optimization to the maximum independent set (MIS) problem on special graphs. For CMO instances of size 300, the corresponding MIS graph instance contains about 10,000 nodes. While our algorithms are able to solve to optimality MIS problem of these sizes, the known optimal algorithms for the MIS on general graphs can at present only solve instances with up to a few hundred nodes. This is the first effective use of IP methods in protein structure comparison; the biomolecular structure literature contains only one other effective IP method devoted to RNA comparison, due to Lenhof et.al. [18].The hybrid heuristic approach that worked well for providing lower bounds in the branch and cut algorithm was tried on large proteins in a test set suggested by Jeffrey Skolnick. It involved 33 proteins classified into four families: Flavodoxin-like fold CheY-related, Plastocyanin, TIM Barrel, and Ferratin. Out of the set of all 528 pairwise structure alignments, we have validated the clustering with a 98.7% accuracy (1.3% false negatives and 0% false positives).

The airport gate assignment problem: mathematical model and a tabu search algorithm

Abstract: Considers an airport gate assignment problem that dynamically assigns airport gates to scheduled flights based on passengers' daily origin and destination flow data. The objective of the problem is to minimize the overall connection times during which passengers walk to catch their connection flights. We formulate this problem as a mixed 0-1 quadratic integer programming problem and then reformulate it as a mixed 0-1 integer problem with a linear objective function and constraints. We design a simple tabu search meta-heuristic to solve the problem. The algorithm exploits the special properties of different types of neighborhood moves, and create highly effective candidate list strategies. We also address issues of tabu short-term memory, dynamic tabu tenure, aspiration rules, and various intensification and diversification strategies. Preliminary computational experiments are conducted, and the results are presented and analyzed.