Multi-commodity Network Flow Model Research Articles

The operational flight and multi-crew scheduling problem

This paper introduces a new kind of operational multi-crew scheduling problem which consists in simultaneously modifying, as necessary, the existing flight departure times and planned individual work days (duties) for the set of crew members, while respecting predefined aircraft itineraries. The splitting of a planned crew is allowed during a day of operations, where it is more important to cover a flight than to keep planned crew members together. The objective is to cover a maximum number of flights from a day of operations while minimizing changes in both the flight schedule and the next-day planned duties for the considered crew members. A new type of the same flight departure time constraints is introduced. They ensure that a flight which belongs to several personalized duties, where the number of duties is equal to the number of crew members assigned to the flight, will have the same departure time in each of these duties. Two variants of the problem are considered. The first variant allows covering of flights by less than the planned number of crew members, while the second one requires covering of flights by a complete crew. The problem is mathematically formulated as an integer nonlinear multi-commodity network flow model with time windows and supplementary constraints. The optimal solution approach is based on Dantzig-Wolfe decomposition/column generation embedded into a branch-and-bound scheme. The resulting computational times on commercial-size problems are very good. Our new simultaneous approach produces solutions whose quality is far better than that of the traditional sequential approach where the flight schedule has been changed first and then input as a fixed data to the crew scheduling problem.

A Multicommodity Network-Flow Problem with Side Constraints on Paths Solved by Column Generation

The multicommodity network-flow model concerns routing of a number of commodities through a capacitated network at minimal cost. In the basic model, it is assumed that for each commodity, the flow can be routed on any path connecting its origin and its destination. In telecommunication applications, where a commodity represents a communication pair, there are often additional time-delay or reliability requirements on paths that are used for routing. These requirements may vary by communication pair, represented by different priority classes. In this paper, we extend the basic multicommodity network-flow model to include such side constraints on paths. The extended problem is NP-hard with the constrained shortest-path problem as a special case. To solve the extended model, we use a column-generation approach, in which the solution is built up successively by path generation. The side constraints are efficiently handled in the path-generation subproblem. We further discuss various enhancements of this approach. Computational results show that the column-generation approach provides an efficient way for solving the extended model, even for fairly large networks.

Simplex and Interior Point Specialized Algorithms for Solving Nonoriented Multicommodity Flow Problems

Multicommodity network flow models arise in a wide variety of contexts, typical among which is the dimensioning of telecommunication networks. In this paper, we present various approaches based on specialization of the simplex algorithm and interior-point methods to solve nonoriented multicommodity flowproblems. Algorithms are tested with data from the France-Telecom Paris district transmission network. First, we focus on a specialization for the node-arc formulation of the problem. A Primal simplex and Dual Affine Scaling algorithms exploiting the particular structure of the constraint matrix are presented and compared. Numerical results are provided for problems up to about 800,000 constraints and 6,000,000 variables. However, much more powerful approaches based on specialized decomposition methods can be implemented for solving the problem. A Dantzig-Wolfe decomposition method is designed and compared with a specialized implementation of the Analytic Center Cutting Plane Method (ACCPM). Partitioning techniques are used to exploit the structure of the master programs involved in those methods.

An Optimization Model for the Simultaneous Operational Flight and Pilot Scheduling Problem

This paper describes and solves the operational pilot scheduling problem for one day of operations. The problem consists in simultaneously modifying, as necessary, the existing flight departure schedules and planned individual work days (duties) while keeping planned aircraft itineraries unchanged. It requires the covering of all flights from one day of operations with available pilots while minimizing changes in both the flight schedule and the next day's planned duties. The newly constructed personalized duties must not exceed the maximum duty duration. Flight precedence constraints, coming from existing fixed aircraft itineraries, must be respected as well. The problem is mathematically formulated as an integer nonlinear multicommodity network flow model with time windows and additional constraints. To solve the problem, a Dantzig-Wolfe decomposition combined with a branch-and-bound method has been used. The master problem comprises the flight-covering constraints and a new set of flight precedence constraints. Subproblems consisting of time-constrained shortest-path problems with linear time costs are solved by a specialized dynamic-programming algorithm. The proposed optimization approach has been tested on several input data sets. All of them have been successfully solved in very short computational time.

Modeling and analysis of the product assignment problem in single stage electronic assembly systems

Consider the production of an evolving family of similar products, each having a well-defined life cycle. The fundamental production resources are inherently flexible, i.e., reconfigurable and reprogrammable. Two distinct strategies can be followed in configuring production facilities: (1) focused facilities, where a facility is dedicated to one product at a time, but may be reassigned; and (2) nonfocused facilities, where setup operations permit a variety of products to be produced during a given planning period. When focused facilities are used, which is a common strategy in some electronics companies, products must be assigned to specific facilities. If facilities are not identical, and capacity is limited, then changing production requirements may force reassignment of products from one facility to another. Thus, the product assignment/reassignment decision may have a significant impact on the production capacity required. This paper concentrates on the product assignment/reassignment decision when a pure focused facility strategy is used. This problem is analyzed and a number of insights are developed. Based on this analysis, the problem is reformulated and an optimal solution procedure based on a multi-commodity network flow model is presented and tested for the product assignment/reassignment decision.

Modeling and analysis of the product assignment problem in single stage electronic assembly systems

Consider the production of an evolving family of similar products, each having a well-defined life cycle. The fundamental production resources are inherently flexible, i.e., reconfigurable and reprogrammable. Two distinct strategies can be followed in configuring production facilities: (1) focused facilities, where a facility is dedicated to one product at a time, but may be reassigned; and (2) nonfocused facilities, where setup operations permit a variety of products to be produced during a given planning period. When focused facilities are used, which is a common strategy in some electronics companies, products must be assigned to specific facilities. If facilities are not identical, and capacity is limited, then changing production requirements may force reassignment of products from one facility to another. Thus, the product assignment/reassignment decision may have a significant impact on the production capacity required. This paper concentrates on the product assignment/reassignment decision when a pure focused facility strategy is used. This problem is analyzed and a number of insights are developed. Based on this analysis, the problem is reformulated and an optimal solution procedure based on a multi-commodity network flow model is presented and tested for the product assignment/reassignment decision

Multi-depot vehicle scheduling problems with time windows and waiting costs

The multi-depot vehicle scheduling problem with time windows (MDVSPTW) consists of scheduling a fleet of vehicles to cover a set of tasks at minimum cost. Each task is restricted to begin within a prescribed time interval and vehicles are supplied by different depots. The problem is formulated as an integer nonlinear multi-commodity network flow model with time variables and is solved using a column generation approach embedded in a branch-and-bound framework. This paper breaks new ground by considering costs on exact waiting times between two consecutive tasks instead of minimal waiting times. This new and more realistic cost structure gives rise to a nonlinear objective function in the model. Optimal and heuristic versions of the algorithm have been extensively tested on randomly generated urban bus scheduling problem (UBSP) and freight transport scheduling problem (FTSP). The results show that such a general solution methodology outperforms specialized algorithms when minimal waiting costs are used, and can efficiently treat the case with exact waiting costs.

Improved Empty Freight Car Distribution

In this paper we consider the problem of distributing empty freight cars in a railway company. We describe and analyze the current planning process, identify the shortcomings of the process, and stress the importance of a reliable distribution process for satisfying customer demand and reducing capital costs. We show how the process can be improved using an optimization model which includes capacity constraints on the trains and adheres explicitly to the arrival and departure times of the trains. The optimization model can be characterized as a multicommodity network flow model with integer requirements. Computational tests show that the model can be solved in acceptable time for real size problems, and indicate that the model generates distribution plans that can improve the quality of the planning process.

A knowledge-based operation support system for network traffic management

The operation of a telephone network is subject to unexpected contingencies such as facilities impairments, trunk congestion, and focused overload. When contingencies happen in the network, the proper control actions must be taken promptly within a short period of time (e.g. 5 minutes); otherwise, the serious congestion will soon spread to the other parts of network. Until now, there has been a lack of an automated approach to generate a contigency routing plan to prevent network degradation. In this paper, a knowledge-based traffic control support system (TRACOSS) is proposed for network traffic management. TRACOSS not only contains four knowledge bases to accumulate the domain expertise used for contingencies detection, diagnosis, affected traffic estimation and control commands generation, but also contains a multicommodity network flow model to formulate the operation of telephone network. Based on this model, a heuristic algorithm is developed to design the contingency routing plan. TRACOSS has the following advantages: (1) The acceptable contingency plan can be obtained by the heuristic algorithm within the fixed time constraint. (2) Four knowledge bases can be used to accumulate the domain expertise to automate the network traffic management. (3) When new network components are installed, knowledge bases can be easily upgraded to adapt new technologies.

Multicommodity network flow model for long-term hydro-generation optimization

A model for long-term hydrogeneration optimization based on multicommodity network flows is presented. The problem of optimizing hydrogeneration with partially dependent water inflows in the reservoir systems is also addressed. The results obtained are consistent with the operating experience for the reservoir systems and meet expectations. The optimal results obtained for volume and discharge for one of the reservoirs is shown graphically. The same type of result can be obtained for any of the reservoirs. These graphs show that water use policies are specific for each commodity; thus it is worth optimizing simultaneously the different types of water use with common constraints in a single program as the model proposes. >

The multiperiod assignment problem: A multicommodity network flow model and specialized branch and bound algorithm

The multiperiod assignment problem, a specialization of the three dimensional assignment problem, is an optimization model that describes the situation of assigning people to activities, or jobs over time. We consider the most general case which has both a cost of assigning a person to an activity in each time period, and a cost of transferring the person from one activity in each period to another activity in the next period. In general, the number of time periods need not equal the number of persons and activities. We present a new formulation of the multiperiod assignment problem, that of an integer, multicommodity network flow model. The special structure of the model allows us to find a good feasible solution relatively quickly by a shortest path heuristic algorithm. We discuss a new branch and bound algorithm for solving this problem to optimality. The subproblems of the branch and bound tree are evaluated by solving a set of special-structured, shortest path problems all of which can be solved in order n 2 T time, where n is the number of persons and activities, and T is the number of time periods. We present computational tests of the algorithm on moderately sized problems.

The multicommodity network flow model revised to include vehicle per time period and node constraints

AbstractThe minimum‐cost formulation of the problem of determining multicommodity flows over a capacitated network subject to resource constraints has been treated in previobs papers. In those treatments only capacitated arcs were assumed and a uniform unit of measure like short tons was used for all commodities. This paper treats the effect of constraints on the nodes of the network, allows the commodities to be measured in their “natural” units and allows the network capacities to be expressed in vehicles per time period‐in some cases giving a more accurate representation of the capacities of the network. This paper describes the solution procedure which uses the column generation technique; it also discusses computational experience.

Computational Experiments With An EmptyFreight Car Distribution Model

In this paper we make computational experiments with an empty freight car distribution model. The model takes the limited transportation capacities of the trains into account, and it can be characterized as a multicommodity network flow model with integer requirements. The tests indicate that the model is solvable and that the model can be used for operational planning and strategic analyses.