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Optimal solutions of multivariate coupling problems

100%
Rüschendorf L.
Applicationes Mathematicae
|
1995-1996
|
tom 23
|
nr 3
325-338
EN
Some necessary and some sufficient conditions are established for the explicit construction and characterization of optimal solutions of multivariate transportation (coupling) problems. The proofs are based on ideas from duality theory and nonconvex optimization theory. Applications are given to multivariate optimal coupling problems w.r.t. minimal $l_p$-type metrics, where fairly explicit and complete characterizations of optimal transportation plans (couplings) are obtained. The results are of interest even in the one-dimensional case. For the first time an explicit criterion is given for the construction of optimal multivariate couplings for the Kantorovich metric $l_1$.
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A relation of dominance for the bicriterion bus routing problem

75%
Widuch J.
International Journal of Applied Mathematics and Computer Science
|
2017
|
tom 27
|
nr 1
133-155
EN
A bicriterion bus routing (BBR) problem is described and analysed. The objective is to find a route from the start stop to the final stop minimizing the time and the cost of travel simultaneously. Additionally, the time of starting travel at the start stop is given. The BBR problem can be resolved using methods of graph theory. It comes down to resolving a bicriterion shortest path (BSP) problem in a multigraph with variable weights. In the paper, differences between the problem with constant weights and that with variable weights are described and analysed, with particular emphasis on properties satisfied only for the problem with variable weights and the description of the influence of dominated partial solutions on non-dominated final solutions. This paper proposes methods of estimation a dominated partial solution for the possibility of obtaining a non-dominated final solution from it. An algorithm for solving the BBR problem implementing these estimation methods is proposed and the results of experimental tests are presented.
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