Network Simplex Research Articles

Solving Multicommodity Flow Problems with a Primal Embedded Network Simplex Algorithm

This article describes the authors’ experience solving large multicommodity flow problems with an embedded network simplex algorithm augmented with a fast-start heuristic for choosing an initial basis. The heuristic makes successive capacity allocations in an attempt to find a feasible initial basis. Our implementation of the heuristic makes use of piece-wise linear convex costs. The efficacy of our heuristic, and of the embedded network simplex method, is demonstrated on large publicly available multicommodity flow problems. Comparisons with other published computational results are given.

Airline Scheduling for the Temporary Closure of Airports

The poor scheduling of flights or a fleet resulting from temporary closure of airports may cause a substantial loss of profit and decreased levels of service for airline carriers. This research develops a framework in order to help carriers handle schedule perturbations resulting from the temporary closure of airports. The framework is based on a basic model constructed as a time-space network from which several strategic network models are developed for scheduling. These network models are formulated as pure network flow problems or network flow problems with side constraints. The former are solved using the network simplex method whereas the latter are solved using a Lagrangian relaxation-based algorithm. To test how well the model may be applied in the real world, a case study regarding the international operations of a major Taiwan airline was performed. The results show that the framework could be useful in actual operations.

A Subtree-Partitioning Algorithm for Inducing Parallelism in Network Simplex Dual Updates

This paper reports on the development of a very efficient method for partitioning the network simplex basis subtree in which dual values must be updated during a pivot. The partitioning procedure may be concurrently executed by multiple processes. The resulting rapid decomposition of the subtree allows an arbitrary number of processes to be utilized in the actual dual update. This approach alleviates a primary limitation of the most efficient parallel network simplex implementation published to date. The new code performs at least as well as the previous implementation on medium-scale problems and reduces average solution time by over 34% on large-scale problems.

Simplex Memory Neural Networks

The biological neural network—simplex memory neural network—is proposed to describe the mechanisms of pattern memory in the brain. A mathematical model of the simplex memory neural network is constructed to memorize any binary pattern with content-addressable memory function. Under Hebbian learning rule, the new network has some important functions in accord with the learning and memory behaviors of the brain. Copyright © 1996 Elsevier Science Ltd.

The Water Resource Management System (WREMS): Linking data management and operational optimization

There have been many management science models proposed for optimizing various operational aspects of municipal water systems. However, our practical experience indicates that water managers view such modeling attempts with skepticism. Too much effort has been spent on developing elegant, complex models, instead of focusing on providing water managers with systems that support their total information needs. This paper presents the design and development of the Water Resource Management System (WREMS). The main distinction of the WREMS system is that the optimization model used to offer operational guidance is imbedded within an inventory data management information system, the "hook" that interests the city water managers in trusting and, ultimately, using the system. The optimization model incorporated within WREMS is a customized network simplex imple-mentation of a minimum cost network flow problem, designed to assist in making daily decisions such as which wells to produce and how to manage water levels in reservoirs. Details of the linkages between the data management and optimization components of the system are highlighted, illustrating how IS and MS/OR can be synergistically coupled.

A decision support framework for handling schedule perturbation

Perturbations of flight schedules may occur everyday. Poor scheduling of flights may result in a substantial loss of profit and decreased levels of service for air carriers. This research aims at developing a framework to help carriers in handling schedule perturbations caused by the breakdown of aircraft. The framework is based on a basic schedule perturbation model constructed as a dynamic network from which several perturbed network models are developed for scheduling following incidents. These network models are formulated as pure network flow problems or network flow problems with side constraints. The former are solved using the network simplex method while the latter are solved using Lagrangian relaxation with subgradient methods. To show how to apply the framework in the real world, a case study focusing on the operations of a major Taiwan air carrier in East Asia is presented.

A decision support framework for multi-fleet routing and multi-stop flight scheduling

Fleet routing and flight scheduling are important in airline carrier operations. Ineffective and inefficient fleet routing and flight scheduling will result in a substantial loss of profits. This research aims at developing a framework to help carriers adjust their draft timetables and fleet routes, when market demand conditions are expected to change in the near future. The framework is based on a basic model, constructed as a multi-fleet time-space network from which several strategic models are developed, to help carriers in fleet routing and flight scheduling. These models are formulated as multiple commodity network flow problems. The Lagrangian relaxation accompanied by the network simplex method, a Lagrangian heuristic and a modified sub-gradient method are suggested to solve the problems. A flow decomposition algorithm is also suggested in order to trace every aircraft route. To show how to apply this framework in the real world, a case study regarding the international operations of a major Taiwan airline was performed. The results show that, the framework would be useful for actual operations.

Efficient multicast routing in high speed networks

This paper presents two algorithms: for a minimal cost shared multicast tree construction (MCTC), and for a scalable multicast session call establishment, called multicast tree development (MCTD). Both algorithms are of original design based on a network simplex presentation, providing a new effective approach to the MCTC, and an object-oriented modelling framework providing the highest scalability for sparse groups with high dynamics of membership, different quality of service requirements and multiple senders. A new approach is proposed for the MCT construction-bidirectional growing of the MCT without pruning; instead the MCTD makes use of extra branches to improve reliability and advertise new receivers.

Performance driven spacing algorithms using attractive and repulsive constraints for submicron LSI's

This paper proposes one-dimensional spacing algorithms that minimize [maximize] the degree of separation [proximity] among the specified elements of a given integrated circuit layout. Number of problems on chip and MCM systems such as poor performance, higher crosstalk, lower yield etc. are related to the degree of separation [proximity]. The proposed algorithms utilize the attractive [repulsive] constraints to shrink [expand] the distances among the specified elements while keeping the layout free of any design rule errors. The spacing problem is reduced to a parametric linear programming problem and a network simplex algorithm is proposed for solving it. The proposed algorithms are implemented into a system called PERFECT for performance enhancement and crosstalk reduction. Given a routed design, PERFECT minimizes delay due to the coupled capacitance, which could contribute as much as 50-75% to the interconnect delay in near future. PERFECT utilizes the repulsive constraints between interconnect segments for the purpose. The experimental results show a significant delay improvement and the crosstalk reduction for submicron technologies. >

A comprehensive simplex-like algorithm for network optimization and perturbation analysis

The family of network optimization problems includes the following prototype models: assignment, critical path, max flow, shortest path, and transportation. Although it is long known that these problems can be modeled as linear programs (LP), this is generally not done. Due to the relative inefficiency and complexity of the simplex methods (primal, dual, and other variations) for network models, these problems are usually treated by one of over 100 specialized algorithms. This leads to several difficulties. The solution algorithms are not unified and each algorithm uses a different strategy to exploit the special structure of a specific problem. Furthermore, small variations in the problem, such as the introduction of side constraints, destroys the special structure and requires modifying andjor restarting the algorithm. Also, these algorithms obtain solution efficiency at the expense of managerial insight, as the final solutions from these algorithms do not have sufficient information to perform postoptimality analysis. Another approach is to adapt the simplex to network optimization problems through network simplex. This provides unification of the various problems but maintains all the inefficiencies of simplex, as well as, most of the network inflexibility to handle changes such as side constraints. Even ordinary sensitivity analysis (OSA), long available in the tabular simplex, has been only recently transferred to network simplex. This paper provides a single unified algorithm for all five network models. The proposed solution algorithm is a variant of the self-dual simplex with a warm start. This algorithm makes available the full power of LP perturbation analysis (PA) extended to handle optimal degeneracy. In contrast to OSA, the proposed PA provides ranges for which the current optimal strategy remains optimal, for simultaneous dependent or independent changes from the nominal values in costs, arc capacities, or suppliesJdemands. The proposed solution algorithm also facilitates incorporation of network structural changes and side constraints. It has the advantage of being computationally practical, easy for managers to understand and use, and provides useful PA information in all cases. Computer implementation issues are discussed and illustrative numerical examples are provided in the Appendix

A Method of Determining Target Configurations for Power System Restoration Using a Mixed Integer Programming Approach

AbstractWhen electricity supply is interrupted due to a fault, it is essential that the power system be restored promptly according to an adequately planned restoration procedure. A problem of obtaining an appropriate target system for restoration is referred to as a power system restoration problem. The authors have already proposed a very efficient method based on the network simplex method which is in the category of mathematical programming approaches. Although the method is effective for most system conditions, it is likely to give an undesirable result in the case of abnormal network conditions.This paper proposes an effective and computationally fast solution algorithm based on the mixed integer programming approach, which can resolve the shortcoming of the former method The solution speed is improved greatly by incorporating the, restoration strategies of system operators. The following assumptions are postulated in this approach: (1)0–1 variables are allocated to branches and loads to present their status; (2) continuous variables are taken to represent branch flows: and (3) costs are assigned to branches to represent their priorities during restorative operations and power flow limits. The proposed approach can be used to complement the former approach.The new approach has been applied to restoration problems of practical size, and simulation results demonstrate its advantage over the former method for intricate operating conditions.

Application of computer software in transmission expansion planning using variable load structure

The present paper proposes a computer-rated transmission expansion planning approach using the network simplex technique. A six-bus sample system is used for illustration. Assuming an increased load demand, an optimum network configuration is designed. Software is developed to obtain the optimum network configuration, which uses variable load structure and generator cost curves for economic generation. Curves are presented of the diversity factor versus mileage and number of circuits, assuming ten different load curves.

A generator and a simplex solver for network piecewise linear programs

This is a brief report on our recent work in network piecewise linear programming (NPLP), and it consists of two parts. In the first park, we describe a generator for NPLP problems which is derived from the classical network linear program generator NETGEN. The generator creates networks of the same topological structures as NETGEN, but each arc is associated with a convex piecewise linear cost. The purpose of this program is to provide a set of standard test problems which can be used to compare the performance of various algorithms for NPLP. In the second part, we introduce a network simplex method that directly solves a network piecewise linear program without reformulating it as a network linear program of higher dimension. Forty benchmark NPLP problems are solved by this method and a reformulation method. The computational results are in favor of the direct method and show that solving an NPLP problem is not much harder than solving a network linear program of the same dimension.

A practical anti-degeneracy row selection technique in network linear programming

A characteristic feature of the primal network simplex algorithm (NSA) is that it usually makes a large number of degenerate iterations. Though cycling and even stalling can be avoided by recently introduced pivot rules for NSA, the practical efficiency of these rules is not known yet. For the case when the simplex algorithm is used to solve the continuous linear programming (LP) problem there exists a practical anti-cycling procedure that proved to be efficient. It is based on an expanding relaxation of the individual bound on the variables. In this paper we discuss the adaptation of this method to NSA, taking advantage of the special integer nature of network problems. We also give an account of our experience with these ideas as they are experimentally implemented in the MINET network LP solver. Reductions of CPU time have been achieved on a smaller set of specially structured real-life problems.

Studies of lexicography in the generalized network simplex method

This paper introduces an analytical approach for studying lexicography in generalized network problems. The equations obtained can help us to understand and to extend the existing theory. First, it is verified that all nonzero elements have the same sign in each row vector of a basis inverse for a generalized network (GN) problem with positive multipliers. However, this property does not necessarily hold when there exist negative multipliers. Second, we developed a strategy to select the dropping arc in the GN simplex algorithm when addressing GN problems with positive andnegative multipliers. This strategy is also based on lexicography and requires performing some comparisons. However, the values to be compared are already known since they can be obtained as a by-product of the calculations necessary to compute the basis representation of the entering arc. Consequently, the computational effort per pivot step isO(n) in the worst case. This worst case effort is the same as that required by the strongly convergent rules for selecting the dropping arc in the method of strong convergence.

An Implementation of the Dual Affine Scaling Algorithm for Minimum-Cost Flow on Bipartite Uncapacitated Networks

This paper describes an implementation of the dual affine scaling algorithm for linear programming specialized to solve minimum-cost flow problems on bipartite uncapacitated networks. This implementation uses a preconditioned conjugate gradient algorithm to solve the system of linear equations that determines the search direction at each iteration of the interior point algorithm. Two preconditioners are considered: a diagonal preconditioner and a preconditioner based on the incidence matrix of an approximate maximum weighted spanning tree of the network. Under dual nondegeneracy this spanning tree allows for early identification of the optimal solution. By applying an $\epsilon $-perturbation to the cost vector, an optimal extreme-point primal solution is produced in the presence of dual degeneracy. The implementation is tested by solving several large instances of randomly generated assignment problems, comparing solution times with the network simplex code NETFLO and the relaxation algorithm code RELAX....

A Primal Partitioning Solution for the Arc-Chain Formulation of a Multicommodity Network Flow Problem

We present a new solution approach for the multicommodity network flow problem (MCNF) based upon both primal partitioning and decomposition techniques, which simplifies the computations required by the simplex method. The partitioning is performed on an arc-chain incidence matrix of the MCNF, similar within a change of variables to the constraint matrix of the master problem generated in a Dantzig-Wolfe decomposition, to isolate a very sparse, near-triangular working basis of greatly reduced dimension. The majority of the simplex operations performed on the partitioned basis are simply additive and network operations specialized for the nine possible pivot types identified. The columns of the arc-chain incidence matrix are generated by a dual network simplex method for updating shortest paths when link costs change.

Polynomial dual network simplex algorithms

We show how to use polynomial and strongly polynomial capacity scaling algorithms for the transshipment problem to design a polynomial dual network simplex pivot rule. Our best pivoting strategy le...

Proximal minimizations with D-functions and the massively parallel solution of linear network programs

This paper discusses the massively parallel solution of linear network programs. It integrates the general algorithmic framework of proximal minimization with D-functions (PMD) with primal-dual row-action algorithms. Three alternative algorithmic schemes are studied: quadratic proximal point, entropic proximal point, and least 2-norm perturbations. Each is solving a linear network problem by solving a sequence of nonlinear approximations. The nonlinear subproblems decompose for massively parallel computing. The three algorithms are implemented on a Connection Machine CM-2 with up to 32K processing elements, and problems with up to 16 million variables are solved. A comparison of the three algorithms establishes their relative efficiency. Numerical experiments also establish the best internal tactics which can be used when implementing proximal minimization algorithms. Finally, the new algorithms are compared with an implementation of the network simplex algorithm executing on a CRAY Y-MP vector supercomputer.

A technique for drawing directed graphs

A four-pass algorithm for drawing directed graphs is presented. The fist pass finds an optimal rank assignment using a network simplex algorithm. The seconds pass sets the vertex order within ranks by an iterative heuristic, incorporating a novel weight function and local transpositions to reduce crossings. The third pass finds optimal coordinates for nodes by constructing and ranking an auxiliary graph. The fourth pass makes splines to draw edges. The algorithm creates good drawings and is fast. >