Abstract
This paper deals with finding most vital links of a network which carries flows over time (also called ”dynamic flows”). Given a network and a time horizon T, Single Most Vital Link Over Time (SMVLOT) problem looks for a link whose removal results in greatest decrease in the value of maximum flow over time (dynamic maximum flow) up to time horizon T between two terminal nodes. SMVLOT problem is formulated as a mixed binary linear programming problem. This formulation is extended to a general case called k-Most Vital Links Over Time (KMVLOT) problem, in which we look for finding those k links whose removal makes greatest decrease in the value of maximum flow over time. A Benders decomposition algorithm is proposed for solving SMVLOT and KMVLOT problems. For the case of SMVLOT problem, the proposed algorithm is improved to a fully combinatorial algorithm by adopting an iterative method for solving existing integer programming problem. However, our experimental results showed the superiority of proposed methods.
Highlights
Static most vital link problem seeks for a link whose removal from network causes maximum decrease in static maximum flow between source node and sink node
The traditional most vital link problem and what we study in this paper have completely different nature
We denote by ea an |A|−tuple vector with 1 as the element corresponding to link a and 0 otherwise. By this introduction we develop a mathematical model for Single Most Vital Link Over Time (SMVLOT) problem
Summary
Static (traditional) most vital link problem seeks for a link whose removal from network causes maximum decrease in static (not over time) maximum flow between source node and sink node. To find SMVLOT in this simple network, note that for a given time horizon T , in a maximum flow over time pattern total flow which arrives to node t up to time T from a1 and a2 are respectively (max{0, T − τa1})ua and (max{0, −τa2})ua; we must distinguish between following five cases: a) 0 ≤ T ≤ 10. In this case no flow enters terminal node up to time T since both links traverse times is greater than T , both links can be selected SMVLOT.
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