Minimum Cost Network Flow Research Articles

A network simplex based algorithm for the minimum cost proportional flow problem with disconnected subnetworks

In this study, we present a variant of the minimum cost network flow problem where the associated graph contains several disconnected subgraphs and it is required that the flows on arcs belonging to same arc subsets to be proportional. This type of network is mostly observed in large supply chains of assemble-to-order products. It is shown that any feasible solution of a reformulation of this problem has a special characteristic. By taking into account this fact, a network simplex based primal simplex algorithm is developed and its details are provided.

Risk informed optimization of a hazardous material multi-periodic transportation model

This paper presents a systematic framework toward the development of a Transportation Model for Hazardous Materials (HazMat). In practice, the proposed modeling framework is realized through an appropriate generalization of the traditional transportation network problem in the presence of safety constraints that need to be satisfied. The objective is to minimize transportation cost while reducing risks at the desired levels. In particular, the present research study identifies and evaluates different risk factors that influence the HazMat transportation network. Next, the transportation model is depicted graphically using nodes and arcs and optimal conditions are identified by solving the associated minimum cost flow network problem. The results show safety levels that help making informed decisions on choosing the optimal transportation configuration for hazardous material shipments. Within the proposed methodological context, appropriately parameterized simulation studies elucidate the effects of occurrence probabilities of the different risk events on transportation cost. Furthermore, as the appropriate management decisions must consider the effect of actions in one time period on future periods, the proposed model is structured as a multi-periodic model. Finally, the proposed methodological approach is employed to demonstrate the utility of proper analytical tools in decision making and particularly in ensuring that scientifically informed safety procedures are in place while transporting goods that can be potentially proven dangerous to the public and the surroundings.

A deterministic annealing algorithm for the minimum concave cost network flow problem

The existing algorithms for the minimum concave cost network flow problems mainly focus on the single-source problems. To handle both the single-source and the multiple-source problem in the same way, especially the problems with dense arcs, a deterministic annealing algorithm is proposed in this paper. The algorithm is derived from an application of the Lagrange and Hopfield-type barrier function. It consists of two major steps: one is to find a feasible descent direction by updating Lagrange multipliers with a globally convergent iterative procedure, which forms the major contribution of this paper, and the other is to generate a point in the feasible descent direction, which always automatically satisfies lower and upper bound constraints on variables provided that the step size is a number between zero and one. The algorithm is applicable to both the single-source and the multiple-source capacitated problem and is especially effective and efficient for the problems with dense arcs. Numerical results on 48 test problems show that the algorithm is effective and efficient.

A least-squares minimum-cost network flow algorithm

Node-arc incidence matrices in network flow problems exhibit several special least-squares properties. We show how these properties can be leveraged in a least-squares primal-dual algorithm for solving minimum-cost network flow problems quickly. Computational results show that the performance of an upper-bounded version of the least-squares minimum-cost network flow algorithm with a special dual update operation is comparable to CPLEX Network and Dual Optimizers for solving a wide range of minimum-cost network flow problems.

A direct barter model for course add/drop process

Even though course timetabling and student scheduling problems have been studied extensively, not much has been done for the optimization of student add/drop requests after the initial registration period. Add/drop registrations are usually processed with a first come first served policy. This, however, can introduce inefficiencies and dead-locks resulting in add/drop requests that are not satisfied even though they can, in fact, be satisfied. We model the course add/drop process as a direct bartering problem in which add/drop requests appear as bids. We formulate the resulting problem as an integer linear program. We show that our problem can be solved polynomially as a minimum cost flow network problem. In our model, we also introduce a two-level weighting system that enables students to express priorities among their requests. We demonstrate improvement in the satisfaction of students over the currently used model and also the fast performance of our algorithms on various test cases based on real-life registration data of our university.

A heuristic approach to project staffing

Abstract In this paper we consider a company with a matrix organization where employees are members of departments and project teams at the same time, i.e. the employees have to accomplish operational work within their department apart from project work. The employees have different skills while projects require some of these skills. It is the aim to minimize the number of assignments of employees to projects. For this staffing problem we present a minimum cost network flow model. We prove that the problem is NP-hard and provide a heuristic solution approach.

A particle swarm optimization-based hybrid algorithm for minimum concave cost network flow problems

Traditionally, minimum cost transshipment problems have been simplified as linear cost problems, which are not practical in real applications. Some advanced local search algorithms have been developed to solve concave cost bipartite network problems. These have been found to be more effective than the traditional linear approximation methods and local search methods. Recently, a genetic algorithm and an ant colony system algorithm were employed to develop two global search algorithms for solving concave cost transshipment problems. These two global search algorithms were found to be more effective than the advanced local search algorithms for solving concave cost transshipment problems. Although the particle swarm optimization algorithm has been used to obtain good results in many applications, to the best of our knowledge, it has not yet been applied in minimum concave cost network flow problems. Thus, in this study, we employ an arc-based particle swarm optimization algorithm, coupled with some genetic algorithm and threshold accepting method techniques, as well as concave cost network heuristics, to develop a hybrid global search algorithm for efficiently solving minimum cost network flow problems with concave arc costs. The proposed algorithm is evaluated by solving several randomly generated network flow problems. The results indicate that the proposed algorithm is more effective than several other recently designed methods, such as local search algorithms, genetic algorithms and ant colony system algorithms, for solving minimum cost network flow problems with concave arc costs.

Vehicle dispatching algorithms for container transshipment hubs

This paper seeks to address the dispatching problem for vehicles (or prime movers) in a transshipment hub by considering the quay cranes and yard cranes capacity. The objective of this paper is to minimize the makespan time at the quay side. This issue is particularly important for a port which uses information technology in making real time decision because the port can exploit information technology to make full use of the data in making good decision. A mixed integer programming (MIP) model is developed to formulate the problem. As the existing solver cannot solve the MIP model in reasonable time, we develop two heuristics to tackle the problem. The first method is based on the neighborhood search, while the second method is based on genetic algorithm (GA) and minimum cost flow (MCF) network model. Unlike the typical GA which usually represents the chromosome using job sequence, we use the ready time for jobs as the representation of the chromosome, and MCF model is then used to decode the chromosome to determine the job sequence for prime movers. The experiment results indicate the superiority of the GA–MCF-based algorithm over the neighborhood search algorithm.

Edgelist phase unwrapping algorithm for time series InSAR analysis

We present here a new integer programming formulation for phase unwrapping of multidimensional data. Phase unwrapping is a key problem in many coherent imaging systems, including time series synthetic aperture radar interferometry (InSAR), with two spatial and one temporal data dimensions. The minimum cost flow (MCF) [IEEE Trans. Geosci. Remote Sens. 36, 813 (1998)] phase unwrapping algorithm describes a global cost minimization problem involving flow between phase residues computed over closed loops. Here we replace closed loops by reliable edges as the basic construct, thus leading to the name "edgelist." Our algorithm has several advantages over current methods-it simplifies the representation of multidimensional phase unwrapping, it incorporates data from external sources, such as GPS, where available to better constrain the unwrapped solution, and it treats regularly sampled or sparsely sampled data alike. It thus is particularly applicable to time series InSAR, where data are often irregularly spaced in time and individual interferograms can be corrupted with large decorrelated regions. We show that, similar to the MCF network problem, the edgelist formulation also exhibits total unimodularity, which enables us to solve the integer program problem by using efficient linear programming tools. We apply our method to a persistent scatterer-InSAR data set from the creeping section of the Central San Andreas Fault and find that the average creep rate of 22 mm/Yr is constant within 3 mm/Yr over 1992-2004 but varies systematically with ground location, with a slightly higher rate in 1992-1998 than in 1999-2003.

Updating Network Flows Given Multiple, Heterogeneous Arc Attribute Changes

A dual ascent reoptimization technique is proposed for updating optimal flows for the minimum cost network flow problem (MCNFP) given any number of simultaneous, heterogeneous changes to the network attributes (i.e. supply at nodes, arc costs and arc capacities) and the optimal solutions to the prior primal and dual problems. Significant savings in computation time can be achieved through the use of reoptimization in place of solving a new MCNFP from scratch as each new problem instance (i.e. set of network attribute updates) arises. The proposed technique can be implemented with polynomial worst-case computational complexity. Extensive numerical experiments were designed and conducted to assess the computational benefits of employing the proposed reoptimization technique as compared with solution from scratch using comparable classic implementations of the original algorithms. This work was motivated by the need for the real-time provision of evacuation instructions to people seeking quick egress from a large sensor-equipped building that has come under attack by natural or terrorist forces, but has broad applicability.

Predictive/reactive scheduling with controllable processing times and earliness-tardiness penalties

In this study, a machine scheduling problem with controllable processing times in a parallel-machine environment is considered. The objectives are the minimization of manufacturing cost, which is a convex function of processing time, and total weighted earliness and tardiness. It is assumed that parts have job-dependent earliness and tardiness penalties and distinct due dates, and idle time is allowed. The problem is formulated as a time-indexed integer programming model with discrete processing time alternatives for each part. A linear-relaxation-based algorithm is used to assign the parts to the machines and to find a sequence on each machine. A non-linear programming model is proposed to find the optimal starting and processing times of the parts for a given sequence. The proposed non-linear programming model is converted to a minimum-cost network flow model by piecewise linearization of the convex manufacturing cost in the objective function. The proposed method is used to find initial schedules in predictive scheduling. The proposed models are revised to incorporate a stability measure for reacting to unexpected disruptions such as machine breakdown, arrival of a new job, delay in the arrival or the shortage of materials in reactive scheduling.

A multi-unit tender award process: The case of Transantiago

The Transantiago public transit services were grouped into five trunk service units and 10 feeder units, and were awarded to private operators through a simultaneous tender. To avoid problems of ownership concentration, limits were placed on the maximum number of units per operator. This paper describes the issues involved in such simultaneous multi-unit tenders where various actors bid for different business units to be awarded in a single tender process. The general problem is formulated as a minimum cost network flow problem (MCNFP). Tie-breaking criteria are discussed and modeled. The problem is then solved for the Transantiago case. There follows a discussion of the utility of handling this tender using the tool developed here.

Infield logistics planning for crop-harvesting operations

Crop-harvesting operations are typically carried out with combine harvesters. The harvested product is transferred to one or more tractors every time the combine harvester's storage capacity is reached. The efficiency of the process can be significantly improved by computing optimal routes and interactions for the harvest vehicles in the field. Furthermore, an automated method for generating itineraries for the harvest vehicles facilitates the planning for autonomous agricultural vehicles. The infield logistics problem is formulated as an integer linear programming vehicle routing problem with additional turn penalty constraints, but, because of the high number of decision variables, it is not possible to solve cases of realistic field size. The solution time of the infield logistics problem is considerably reduced by reformulating it as a modified minimum-cost network flow problem. This specific structure allows the exact solution of intermediate-size planning problems in a much shorter time period. The result of solving the infield logistics problem with the proposed modelling approaches is a set of itineraries (‘tours’), covering the entire field. Each ‘tour’ is characterized by the combine harvester's start and end points and the positions where the combine harvester needs to be unloaded. The planning models minimize non-productivity (i.e. the time when a combine harvester travels in a field without harvesting). The results indicate that coordination between combine harvesters and tractors is also improved.

A dual exterior point simplex type algorithm for the minimum cost network flow problem

A new dual simplex type algorithm for the Minimum Cost Network Flow Problem (MCNFP) is presented. The proposed algorithm belongs to a special 'exterior- point simplex type' category. Similarly to the classical network dual simplex algorithm (NDSA), this algorithm starts with a dual feasible tree-solution and reduces the primal infeasibility, iteration by iteration. However, contrary to the NDSA, the new algorithm does not always maintain a dual feasible solution. Instead, the new algorithm might reach a basic point (tree-solution) outside the dual feasible area (exterior point - dual infeasible tree).

A network simplex algorithm for solving the minimum distribution cost problem

To model the distillation or decomposition of products in some manufacturingprocesses, a minimum distribution cost problem (MDCP) for a specializedmanufacturing network flow model has been investigated. In an MDCP, aspecialized node called a D-node is used to model a distillation process thatconnects with a single incoming arc and several outgoing arcs. The flowentering a D-node has to be distributed according to a pre-specified ratioassociated with each of its outgoing arcs. This proportional relationshipbetween arc flows associated with each D-node complicates the problem andmakes the MDCP more difficult to solve than a conventional minimum costnetwork flow problem. A network simplex algorithm for an uncapacitated MDCPhas been outlined in the literature. However, its detailed graphicalprocedures including the operations to obtain an initial basic feasiblesolution, to calculate or update the dual variables, and to pivot flows havenever been reported. In this paper, we resolve these issues and propose amodified network simplex algorithm including detailed graphical operations ineach elementary procedure. Our method not only deals with a capacitated MDCP,but also offers more theoretical insights into the mathematical properties ofan MDCP.

A primal–dual simplex algorithm for bi-objective network flow problems

In this paper we develop a primal–dual simplex algorithm for the bi-objective linear minimum cost network flow problem. This algorithm improves the general primal–dual simplex algorithm for multi-objective linear programs by Ehrgott et al. (J Optim Theory Appl 134:483–497, 2007). We illustrate the algorithm with an example and provide numerical results.

An exterior simplex type algorithm for the Minimum Cost Network Flow Problem

In this paper a new Network Exterior Point Simplex Algorithm (NEPSA) for the Minimum Cost Network Flow Problem (MCNFP) is analytically presented. NEPSA belongs to a special simplex type category and is a modification of the classical network simplex algorithm. The main idea of the algorithm is to compute two flows. One flow is basic but not always feasible and the other is feasible but not always basic. A complete proof of correctness for the proposed algorithm is also presented. Moreover, the computational behavior of NEPSA is shown by an empirical study carried out for randomly generated sparse instances created by the well-known GRIDGEN network problem generator.

Incorporating lateral transfers of vehicles and inventory into an integrated replenishment and routing plan for a three-echelon supply chain

This paper focuses on developing an integrated replenishment and routing plan that takes into account lateral transfers of both vehicles and inventory for a three-echelon supply chain system including a single plant, multiple distribution centers and multiple retailers. A mixed integer programming model to the overall system is formulated first, and then an optimization-based heuristic consisting of three major components is proposed. The purpose of the first component is to assign retailers to distribution centers, and determine routing schedules for each distribution center. And the remaining two components are corresponding to two smaller optimization models – a variant of the classical transportation problem modeled for determining vehicle transfer between distribution centers, and a variant of the conventional minimum cost network flow problem modeled for determining inventory replenishment and transfer. Experimental results reveal that the proposed algorithm is rather computational effectiveness, and the pooling strategy that considers both vehicles and inventory transfers is a worthy option in designing supply chain operations.

Visualization software of the network exterior primal simplex algorithm for the minimum cost network flow problem

The Minimum Cost Network Flow Problem (MCNFP) constitutes perhaps the most important of the research area of Network Optimization. Recently a new category of algorithms for the MCNFP have been developed. These algorithms belong to a special “exterior type” simplex category and they feature significant tree modifications. This paper presents a new didactic tool for the teaching of this type of algorithms. The proposed educational software can be used in courses like “Graph Theory” or “Combinatorial Optimization”. This tool has been implemented using the Java Programming language and it is platform independed. It constitutes a friendly application, for the instructor, as also for the novice student. Furthermore, its use is analytically shown through an illustrative example. Benefits and drawbacks are thoroughly described in order to support the significance of this visualization tool in computer-aided education and also possible future work is discussed.

Heuristic solutions for general concave minimum cost network flow problems

AbstractWe address the single‐source uncapacitated minimum cost network flow problem with general concave cost functions. Exact methods to solve this class of problems in their full generality are only able to address small to medium size instances, since this class of problems is known to be NP‐Hard. Therefore, approximate methods are more suitable. In this work, we present a hybrid approach combining a genetic algorithm with a local search. Randomly generated test problems have been used to test the computational performance of the algorithm. The results obtained for these test problems are compared to optimal solutions obtained by a dynamic programming method for the smaller problem instances and to upper bounds obtained by a local search method for the larger problem instances. From the results reported it can be shown that the hybrid methodology improves upon previous approaches in terms of efficiency and also on the pure genetic algorithm, i.e., without using the local search procedure. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 67–76 2007