Network Simplex Algorithm Research Articles

Addressing the urban-scale vehicle assignment and rebalancing problems in shared autonomous vehicle system while simultaneously considering immediate, reservation, shareable, and unshareable requests

Recently, the autonomous driving technology has developed rapidly with the shared autonomous vehicle system extensively investigated. However, little attention has been given to the diversity of the travel demands. In view of this, the paper proposes a dispatching strategy to solve the vehicle assignment and rebalancing problems with simultaneous consideration of immediate, reservation, shareable and unshareable requests. The centralised optimisation and the decentralised heuristic methods are combined to handle the different requests efficiently, which enables the dispatching strategy to address the urban scale problem. The centralised optimisation method assigns the short-term requests and plans the empty vehicle rebalance by the network simplex algorithm, while the decentralised heuristic method assigns the long-term requests and fully uses the previous step results in each solution step of the rolling horizon. The simulation of Manhattan, New York, with 4,000 vehicles and 1,823,152 requests, confirms the high operational performance and the efficiency of the proposed dispatching strategy compared to the existing approach and the benchmark strategy. It was also found that more reservation requests with the earlier reservation time did not generate significant improvement in operational performance, and the high sharable request ratio facilitated the reduction of the travel distance and empty distance.

A linear input dependence model for interdependent networks

We consider a linear relaxation of a generalized minimum-cost network flow problem with binary input dependencies. In this model the flows through certain arcs are bounded by linear (or more generally, piecewise linear concave) functions of the flows through other arcs. This formulation can be used to model interrelated systems in which the components of one system require the delivery of material from another system in order to function (for example, components of a subway system may require delivery of electrical power from a separate system). We propose and study randomized rounding schemes for how this model can be used to approximate solutions to a related mixed integer linear program for modeling binary input dependencies. The introduction of side constraints prevents this problem from being solved using the well-known network simplex algorithm, however by characterizing its basis structure we develop a generalization of network simplex algorithm that can be used for its computationally efficient solution.

Development of a heuristic to solve the general transportation problem

The transportation problem is well known and has very important applications. For this well-researched model, there are very efficient approaches for solving it that are available. These approaches include formulating the transportation problem as a linear program and then using the efficient methods such as the simplex method or interior point algorithms. The Hungarian method is another efficient method for solving both the assignment model and the general transportation model. An assignment problem is a special case of the transportation model in which all supply and demand points are 1. Every transportation problem can be converted into an assignment problem since rows and columns can be split so that each supply and each demand point is 1. The transportation simplex method is another method that is also used to solve the general transportation problem. This method is also called the modified distribution method (MODI). To use this approach, a starting solution is required and the closer the starting solution to the optimal solution, the fewer the iterations that are required to reach optimality. The fourth method for transportation models is the network simplex method, which is the fastest so far. Unfortunately, all these approaches for transportation models are serial in nature and are very difficult to parallelize, which makes it difficult to efficiently use the available massively parallel technology. There is a need for an efficient approach for the transportation problem, which is easily parallelizable. This paper presents a See-Saw approach for solving the general transportation problem. This is an extension of the See-Saw approach for solving the assignment problem. The See-Saw moves can be done independently, which makes the approach proposed in this paper more promising than the available methods for transportation models

Parallel network simplex algorithm for the minimum cost flow problem

AbstractIn this work, we contribute a parallel implementation of the network simplex algorithm that is used for the solution of minimum cost flow problem. In the network simplex algorithm, finding an entering arc requires searching through many arcs to decide which one should be included in the spanning tree solution on the next iteration. We propose finding the entering arc in parallel as it often takes the majority of the execution time. A usual strategy is to pick the arc violating the optimality the most out of all possible candidates. Scanning all arcs can take quite some time, so it is common to consider only a fixed number of arcs which is referred as the block search pivoting rule. Arc scans can easily be done in parallel to find the best candidate as the calculations are independent of each other. We used shared memory parallelism using OpenMP along with vectorization using AVX instructions. We also tried adjusting block sizes to increase the parallel portion of the algorithm. Our dataset consists of various natural and synthetic graphs with sizes up to a billion arc. Our experiments show speedups up to four are possible, though they are typically lower.

Multi-granularity hybrid parallel network simplex algorithm for minimum-cost flow problems

Minimum-cost flow problems widely exist in graph theory, computer science, information science, and transportation science. The network simplex algorithm is a fast and frequently used method for solving minimum-cost flow problems. However, the conventional sequential algorithms cannot satisfy the requirement of high-computational efficiency for large-scale networks. Parallel computing has resulted in numerous significant advances in science and technology over the past decades and is potential to develop an effective means to solve the computational bottleneck problem of large-scale networks. This paper first analyzes the parallelizability of network simplex algorithm and then presents a multi-granularity parallel network simplex algorithm (MPNSA) with fine- and coarse-granularity parallel strategies, which are suitable for shared- and distributed-memory parallel applications, respectively. MPNSA is achieved by message-passing interface, open multiprocessing, and compute unified device architecture, so that it can be compatible with different high-performance computing platforms. Experimental results demonstrated that MPNSA has very great accelerating effects and the maximum speedup reaches 18.7.

Smoothed Analysis of the Minimum-Mean Cycle Canceling Algorithm and the Network Simplex Algorithm

The minimum-cost flow (MCF) problem is a fundamental optimization problem with many applications and seems to be well understood. Over the last half century many algorithms have been developed to solve the MCF problem, and these algorithms have varying worst-case bounds on their running time. However, these worst-case bounds are not always a good indication of the algorithms' performance in practice. The Network Simplex (NS) algorithm needs an exponential number of iterations for some instances, but it is among the best algorithms in practice and performs very well in experimental studies. On the other hand, the Minimum-Mean Cycle Canceling (MMCC) algorithm is strongly polynomial, but performs badly in experimental studies. To explain these differences in performance in practice we apply the framework of smoothed analysis to NS and MMCC. We show an upper bound of $O(mn^2\log(n)\log(\phi))$ for the number of iterations of the MMCC algorithm. Here, $n$ is the number of nodes, $m$ is the number of edges, and $\phi$ is a parameter limiting the degree to which the edge costs are perturbed. We also show a lower bound of $\Omega(m\log(\phi))$ for the number of iterations of the MMCC algorithm, which can be strengthened to $\Omega(mn)$ when $\phi=\Theta(n^2)$. For the number of iterations of the NS algorithm we show a smoothed lower bound of $\Omega(m \cdot \min \{ n, \phi \} \cdot \phi)$.

Social and economic flows across multimodal transportation networks in the Greater Tokyo Area

We model the flow of human capital and resources across multimodal transportation networks throughout the Greater Tokyo Area. Our transportation networks include trains, buses, and roads integrated with a walking network among a geographically grounded hexagonal grid and connecting nodes of different modes. The hexagonal grid holds data on both the working population and number of jobs from which we built probability distributions for the origins and destinations of commuting trips. Using both the network simplex method and stochastically generated origin-destination trips we estimate the population flows necessary to satisfy this demand. Rather than micro-simulations of actual commuting patterns, congestion, or route planning, our approach aims to uncover patterns in the aggregate flow of human resources to and from economic opportunities. We describe the details of the socioeconomic data, network generation, and the results of our exploratory analysis, then discuss the implications of these findings for transportation usage and future work.

A concurrent approach to the periodic event scheduling problem

We introduce a concurrent solver for the periodic event scheduling problem (PESP). It combines mixed integer programming techniques, the modulo network simplex method, satisfiability approaches, and a new heuristic based on maximum cuts. Running these components in parallel speeds up the overall solution process. This enables us to significantly improve the current upper and lower bounds for all benchmark instances of the library PESPlib.

Simulation and Evaluation of Network Simplex Algorithm and its Extensions for Vehicle Scheduling Problems in Ports

Simulation and Evaluation of Network Simplex Algorithm and its Extensions for Vehicle Scheduling Problems in Ports

The Simplex Algorithm Is NP-Mighty

We show that the Simplex Method, the Network Simplex Method—both with Dantzig’s original pivot rule—and the Successive Shortest Path Algorithm are NP-mighty . That is, each of these algorithms can be used to solve, with polynomial overhead, any problem in NP implicitly during the algorithm’s execution. This result casts a more favorable light on these algorithms’ exponential worst-case running times. Furthermore, as a consequence of our approach, we obtain several novel hardness results. For example, for a given input to the Simplex Algorithm, deciding whether a given variable ever enters the basis during the algorithm’s execution and determining the number of iterations needed are both NP-hard problems. Finally, we close a long-standing open problem in the area of network flows over time by showing that earliest arrival flows are NP-hard to obtain.

Inverse optimization in minimum cost flow problems on countably infinite networks

AbstractGiven the costs and a feasible solution for a minimum cost flow problem on a countably infinite network, inverse optimization involves finding new costs that are close to the original ones and that make the given solution optimal. We study this problem using the weighted absolute sum metric to quantify closeness of cost vectors. We provide sufficient conditions under which known results from inverse optimization in minimum cost flow problems on finite networks extend to the countably infinite case. These conditions ensure that recent duality results on countably infinite linear programs can be applied to our setting. Specifically, they enable us to prove that the inverse optimization problem can be reformulated as a capacitated, minimum cost circulation problem on a countably infinite network. Finite‐dimensional truncations of this problem can be solved in polynomial time when the weights equal one, which yields an efficient solution method. The circulation problem can also be solved via the shadow simplex method, where each finite‐dimensional truncation is tackled using the usual network Simplex algorithm that is empirically known to be computationally efficient. We illustrate these results on an infinite horizon shortest path problem.

NETWORK SIMPLEX ALGORITHM ASSOCIATED WITH THE MAXIMUM FLOW PROBLEM

In the present paper, we apply the network simplex algorithm for solving the minimum cost flow problem, to the maximum flow problem. Then we prove that the cycling phenomenon which causes the infinite loop in the algorithm, does not occur in the network simplex algorithm associated with the maximum flow problem.

A 2-Layers Virtual Network Mapping Algorithm Based on Node Attribute and Network Simplex

Network virtualization is one of the most important techniques to overcome Internet ossification, which can enable multiple networks to run on a common infrastructure. Virtual network mapping is a key process of network virtualization to efficiently map virtual network onto the substrate network within constraints of node and link resource. However, previous VNM heuristic algorithms completely separate the node mapping stage from link mapping stage and ignore the effect of link mapping results on physical node ranking, which may lead to higher mapping cost and distributed distribution of virtual nodes on the substrate network. The shortest path algorithm is used at most to resolve virtual link mapping which results in longer mapping time. To address these issues, a two-layer virtual network mapping algorithm named simplex-VNM is proposed based on node attribute and network simplex in this paper. In node mapping layer, a new node ranking approach which considers topology attributes, global network resources, and mapped physical node information provides candidate nodes and node mapping cost for link mapping layer. Then, in link mapping layer, network simplex algorithm which has smaller time complexity compared to traditional shortest path algorithms has been introduced to find the optimal link mapping solution under the result of node mapping layer. Finally, according to the results of two layers, the virtual node and relevant links will be mapped with minimal total mapping cost. Extensive simulation results have demonstrated that our proposed method behaves better than four other algorithms that consider node attribute and use shortest path algorithm in terms of total mapping cost and mapping time.

Ship type decision considering empty container repositioning and foldable containers

This paper addresses a problem of ship type decision considering empty container repositioning and foldable containers, which determines the capacity of ships deployed in a given shipping route at a tactical level and empty container repositioning between ports at an operational level. It considers the use of foldable containers and aims to find under what conditions, a shipping liner needs to use the foldable containers. To solve the problem, we formulate a network flow model with a revised network simplex algorithm, based on which an exact solution approach is designed to determine the optimal ship type.

Hybrid cost and time path planning for multiple autonomous guided vehicles

In this paper, simultaneous scheduling and routing problem for autonomous guided vehicles (AGVs) is investigated. At the beginning of the planning horizon list of orders is processed in the manufacturing system. The produced or semi-produced products are carried among stations using AGVs according to the process plan and the earliest delivery time rule. Thus, a network of stations and AGV paths is configured. The guide path is bi-direction and AGVs can only stop at the end of a node. Two kinds of collisions exist namely: AGVs move directly to a same node and AGVs are on a same path. Delay is defined as an order is carried after the earliest delivery time. Therefore, the problem is defined to consider some AGVs and material handling orders available and assign orders to AGVs so that collision free paths as cost attribute and minimal waiting time as time attribute, are obtained. Solving this problem leads to determine: the number of required AGVs for orders fulfillment assign orders to AGVs schedule delivery and material handling and route different AGVs. The problem is formulated as a network mathematical model and optimized using a modified network simplex algorithm. The proposed mathematical formulation is first adapted to a minimum cost flow (MCF) model and then optimized using a modified network simplex algorithm (NSA). Numerical illustrations verify and validate the proposed modelling and optimization. Also, comparative studies guarantee superiority of the proposed MCF-NSA solution approach.

The diameters of network-flow polytopes satisfy the Hirsch conjecture

We solve a problem in the combinatorics of polyhedra motivated by the network simplex method. We show that the Hirsch conjecture holds for the diameter of the graphs of all network-flow polytopes, in particular the diameter of a network-flow polytope for a network with n nodes and m arcs is never more than $$m+n-1$$ . A key step to prove this is to show the same result for classical transportation polytopes.

A strongly polynomial Contraction-Expansion algorithm for network flow problems

This paper addresses the solution of the capacitated minimum cost flow problem on a network containing n nodes and m arcs. Satisfying necessary and sufficient optimality conditions can be done on the residual network although it can be quite time consuming as testified by the minimum mean cycle-canceling algorithm (MMCC). We introduce a contracted network which exploits these conditions on a much smaller network. Since the construction of this contracted network is very flexible, we study its properties depending on the construction choice. A generic contraction algorithm is then produced around the contracted network. Interestingly enough, it turns out it encapsulates both the MMCC and primal network simplex algorithms as extreme cases. By guiding the solution using a particular expansion scheme, we are able to recuperate theoretical results from MMCC. As such, we obtain a strongly polynomial Contraction-Expansion algorithm which runs in O(m3n2) time. There is thus no improvement of the runtime complexity, yet the expansion scheme sticks to very practical observations of MMCC’s behavior, namely that of phases and jumps on the optimality parameter. The solution time is ultimately significantly reduced, even more so as the size of the instance increases.

Finding extreme supported solutions of biobjective network flow problems: An enhanced parametric programming approach

We address the problem of determining a complete set of extreme supported efficient solutions of biobjective minimum cost flow (BMCF) problems. A novel method improving the classical parametric method for this biobjective problem is proposed. The algorithm runs in O(Nn(m +nlogn)) time determining all extreme supported non-dominated points in the outcome space and one extreme supported efficient solution associated with each one of them. Here n is the number of nodes, m is the number of arcs and N is the number of extreme supported non-dominated points in outcome space for the BMCF problem. The memory space required by the algorithm is O(n +m) when the extreme supported efficient solutions are not required to be stored in RAM. Otherwise, the algorithm requires O(N +m) space. Extensive computational experiments comparing the performance of the proposed method and a standard parametric network simplex method are presented.

Optimization of Constrained Liquidity Management

A consistent framework for optimal liquidity management is presented. This framework optimizes the cost of covering expected cashflow gaps without violating regulatory and business constraints. Anticipated economic value loss, cashflow loss, and adverse market impact are the major drivers of cost. The notion of a deployable liquidity resource, which is subsequently extend to the notion of a dated liquidity strategy, is introduced. A formalization of LCR as a typical regulatory constraints is presented and included in the formulation. The formulation includes a general arbitrage free market impact function. A decoupling between liquidity risk management and that of market and credit risks is assumed. Both linear and quadratic programming approaches for solving the resulting optimization problem are derived. This is followed by the introduction of a novel mapping algorithm which transforms the linear program to a network flow problem that is more efficiently solvable via the network simplex algorithm. Next an algorithm for generating a plausible starting point for the iterative optimization problem, is given. This is shown to be already optimal under the risk neutral measure. Finally, heuristics that can help speed up the satisfaction of regulatory constraints are discussed. Throughout the presentation attention is given to algorithmic complexity issues.

A network simplex method for the budget-constrained minimum cost flow problem

We present a specialized network simplex algorithm for the budget-constrained minimum cost flow problem, which is an extension of the traditional minimum cost flow problem by a second kind of costs associated with each edge, whose total value in a feasible flow is constrained by a given budget B. We present a fully combinatorial description of the algorithm that is based on a novel incorporation of two kinds of integral node potentials and three kinds of reduced costs. We prove optimality criteria and combine two methods that are commonly used to avoid cycling in traditional network simplex algorithms into new techniques that are applicable to our problem. With these techniques and our definition of the reduced costs, we are able to prove a pseudo-polynomial running time of the overall procedure, which can be further improved by incorporating Dantzig’s pivoting rule. Moreover, we present computational results that compare our procedure with Gurobi (2016).