By Veldre V Ya, Damburg R Ya, Peterkop R K. (eds.)

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5). According to the maximum principle, the Hamiltonian is maximized at every point of time with respect to admissible controls u(t). 17) is non-negative, then it is optimal. e. u(t)=O . 3 Step 1: If the average demand as a fuction of time, ~ f t d(r)dr is greater than 0 c- -4 x (2 T- t) for all t e [0, T], then set the optimal sol uti on as cu u(t) = 2_ (T- t) over the entire planning horizon, STOP; 2cu otherwise go to the next step. Step 2: Define the following function of t 1 If the smallest positive root of this function is such that and for all t e (t 1 , T] , then set the optimal solution as c- u(t) = _x_(T- t) for t 1 ~ t ~ T; 2cu Otherwise, go to the next step.

K-l as m -:t. K m -:t. K m =K m=K then do Subroutine 1, otherwise then do Subroutine 2, otherwise then do Subroutine 3, otherwise then do Subroutine 4. If all pairs of nodes have been checked in this step, go to Step 3, otherwise, link a pair of nodes which has not yet been checked. Step 3: Find the shortest feasible path from node T 1 to node -r K in graph G by the following sequence-dependent, shortest path procedure 52 (i) Define the solved and unsolved sets of nodes, S = 0 and S = E , respectively.

Step 7: If i < n , then i Step 8: =i+ 1 and go to Step 6. Otherwise, go to the next step. Remove j from F. If k < n -I, then assign k=k+ 1 and go to Step 2. Otherwise, STOP. 3 initializes a working set, F, of nodes whose shortest paths are to be detennined. This set first contains all nodes. Steps 2-7 determine a shortest path to a node from the working set. At Step 8 tllis node is removed from the working set by assigning a negative value to the path length to this node from the initial one. Step 6 ensures the path is updated each time when a shorter arc is selected.