A Composite Algorithm for the Concave Cost Ltl Consolidation Problem
A Composite Algorithm for the Concave Cost Ltl Consolidation Problem
Anantaram Balakrishnan
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For our work, we set v. As the length of the shortest path from R R R 0(k) to i on graph G with arc length C + (^jj/'^^jj)- "here R is p the number of cost segments on arc (i. J) and M . = E dj^ . This ■^ k initial choice of multipliers results in a solution to [SP(v)] in which the flow on all arcs is zero, i. E. f . . = ij Vi, j, k, where f '^ . = E x'^': . The optimal value for [SP(v)] for this ij ^ iJ choice of v is then 35 where v . Is the length of the shortest path from 0(k) to D(k) D ( k... ) using the arc lengths given above. To improve this initial choice of multipliers, we can use an §§fI§Ql_Br2cedure, the intent of which is to change iteratively the multipliers so that z(v) increases monoton i ca 1 1 y . We can write z ( V ) as z(v) E Z . . (V ' I ^k^D(k) (i, J) where Z|j(v) is the optimal value to [SPij(v)l To increase z(v) we employ an iterative strategy where we adjust v so that z j j ( v ) remains unchanged for all (i, j) and d v, . Increases for some k. To keep z^j(v) unchanged, we note that [SPjj(v)] depends on k k k the difference v.
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