By Stuart E. Dreyfus
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9) and (1. 1 with the additional stipulation that each direction change costs 3. P ( x , y, z ) = U means that up is the optimal second decision if we start at ( x , y ) and move first in the direction indicated by z and a similar definition holds for P ( x , y, z ) = D. P(4, 2, 1) = D ; 8. SOLUTION OF THE EXAMPLE 15 [ ] ” + min 3 +S(59 = 7, S(5, 1, 1) 3 + S(5, - 1,O) S(4, 0, 1) = 8 + rnin = 12, S(4, 0, 0 ) = 2 S(4, - 2, 0 ) = 4 i - 1, 1) S(5, - 1,O) S(5, + min 3 + S(5, - 1, S(4, - 2, 1) = 00. , x = 3), S(3, 3, 0 ) = 00; S(3, 3, 1) =3 + min [ 3;m] S(3, 1,O) = 3 + min [ O0 + = 10, ] = 13, + min S(3, - 1,O) = 2 + min ~ ( 3, 1, I ) = 2 + min[ 3 2 5 1 = 10, ~ ( 3, 3 , 0 ) = 2 + min[ 21 = 7, S(3, 1, 1) P ( 3 , 3, 1) =D; ~ ( 3 1,0) , =D; =4 ~ ( 3 -, 1 , 1) = U; ~ ( 3 -, 3 , 0 ) = U ; S(3, - 3, 1) = 00.
Give the backward dynamic-programming solution of the problem and determine the approximate number of additions and comparisons. 7. , that costs in year k differ from those in year j for j # k. , u5(4,2) = cost of trading a 4-year-old machine for a 2-year-old one at the start of year 5, etc. Give the backward dynamic-programming solution. 5. More Complex Equipment-Replacement Models We now complicate the problem in a more fundamental way, thereby requiring a new definition of the optimal value function.
Solve for h = 5 . 5? 7. Justlflcatlon of the Procedure In this section we shall justify the procedure that we have outlined above. 1. If (xi(A),yi(A))is an optimal solution to p