Abstract
Graphs are excellent mathematical tools applied in many fields such as transportation, communication, informatics, economy,…. A network and a flow network is a useful device to solve many problems in many fields in reality. However, most of the network applications in traditional graphs have only considered the weights of edges and vertexes independently, in which the length of a path is the sum of weights of the edges and the vertexes on the path. However, in many practical problems, weights at a vertex are not the same for all paths passing the vertex, but depend on the edges coming to and leaving the vertex. For example, the transit time on the transport network depends on the direction of transportation: turn right, turn left or go straight, even some directions are forbidden. Furthermore, on a network, there are many types of commodities, each of which are at different costs. Types of commodities share the capacity of edges and vertexes. Therefore, it is necessary to study a network with multiple commodities at multiple costs. The article builds a model of extended multi-commodity multi-cost network in order to modelise practical problems more exactly and effectively. The maximal concurent multi-commodity multi-cost flow limited cost problems, that are modelized by implicit linear programming problems. On the basis of duality theory in linear programming, an effective polynomial approximation algorithm is developed.
Highlights
O(ω−2.(cemax/dmax).(χ+k).m.n3.log2(m+n+1)), where m is the number of edges, n is the number of vertices of the network, k = k1+...+kr, cemax = max{ce(e).ze(e) | e∈E }
The maximal concurent limited cost flow problems are modeled as implicit linear optimization problems
On the base of dual theory in linear optimization, an effective polynomial approximate algorithm is developed
Summary
Multi-commodity Multi-cost Network the set of edges incident the node v∈V. Node switch cost function i, i=1, 2,..., r, bvi:V×E×E→R*, where bvi(v,e,e’) is the cost of transferring a converted unit of commodity of type i from edge e∈Ev through v∈V to edge e’∈Ev. The set (V, E, ce, ze, cv, zv,{bei, bvi, qi|i=1..r}) is called an extended multi-cost multi-commodity network. The cost of passing a converted unit of commodity of kind i, i = 1, 2, ..., r, through the path p, is denoted by the symbol bi(p), and calculated by the following formula:. Let Pi,v denote the set of paths in Pi passing through the node v, ∀v∈V. is called a multi-commodity flow on the extended multi-cost multi-commodity network, if it satisfies the edge capacity constraints:. I =1 is called the flow value of the multi-commodity flow F
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