Abstract
This paper presents an algorithm for solving Bi-criteria Minimum Cost Dynamic Flow (BiCMCDF) problem with continuous flow variables. The approach is to transform a bi-criteria problem into a parametric one by building a single parametric linear cost out of the two initial cost functions. The algorithm consecutively finds efficient extreme points in the decision space by solving a series of minimum parametric cost flow problems with different objective functions. On each of the iterations, the flow is augmented along a cheapest path from the source node to the sink node in the time-space network avoiding the explicit time expansion of the network.
Highlights
This paper presents an algorithm for solving Bi-criteria Minimum Cost Dynamic Flow (BiCMCDF) problem with continuous flow variables
Classical network flow models have been well known as valuable tools for many applications [1] and efficient algorithms have been developed
In transportation problems or in network flows problems, the criteria that can be considered are the minimization of the cost for selected routes, the minimization of arrival time at the destinations, the minimization of the deterioration of goods, the minimization of the load capacity that would not be used in the selected vehicles, the maximization of safety, reliability, etc
Summary
Classical (static) network flow models have been well known as valuable tools for many applications [1] and efficient algorithms have been developed. They fail to capture the dynamic property of many real-life problems, such as traffic planning, production and distribution systems, communication systems, and evacuation planning. In transportation problems or in network flows problems, the criteria that can be considered are the minimization of the cost for selected routes, the minimization of arrival time at the destinations, the minimization of the deterioration of goods, the minimization of the load capacity that would not be used in the selected vehicles, the maximization of safety, reliability, etc. In the presentation to follow, some familiarity with flow algorithms is assumed and many details are omitted, since they are straightforward modifications of known results
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