Continuous Dynamic Programming

Continuous DP Suppose we have a hiring problem where the employment level needed for each season is given by cost of excess employment = $2,000 per season cost to hire/fire = $200 x (# hired/fired)2

Continuous DP Let sn = employment level at start of season n xn = employment level decision Ex: Suppose at start of 3 (winter) we had 250 employees & we decide to keep 220 250 220 s3 s4

Continuous DP Let sn = employment level at start of season n xn = employment level decision sn xn n n+1

Continuous DP Let rn = min level required for season n (xn - sn) = # hired or fired (xn - rn) = excess over min required

Continuous DP Cost at stage (season) n cn = 200(xn - sn)2 + 2,000(xn - rn) fn(sn, xn) = cn + fn+1*(xn)

Continuous DP Feasible range for season n = # required in current season n to # required in previous season n-1

Continuous DP Feasible range for season n = # required in current season n to # required in previous season n-1

Continuous DP Season 4 At n=4 (spring), we must have 255 since this min requirement is the max min for all seasons. Then xn = s4 cn = 200(255 - s4)2 + 2,000(255-255)

Continuous DP Season 4 At n=4 (spring), we must have 255 since this min requirement is the max min for all seasons. Then xn = s4 cn = 200(255 - s4)2 + 2,000(255-255)

Continuous DP Stage 3 (winter) start at s3 = employment decision at stage 2 240 < x3 - s3 < 255 f s x n 3 2 4 200 000 255 ( , ) * = - +

Continuous DP Stage 3 (winter) f s x 200 000 255 ( , ) = - + 4 200 000 255 ( , ) * = - + But, s4 = x3 f s x n 3 2 200 000 255 ( , ) = - +

Continuous DP Now, min f3(sn,xn) f s x 200 000 255 ( , ) = - + ¶ = - + 400 2 000 255 ( , )

Continuous DP x s 250 2 = + Now, min f3(sn,xn) f s x 200 000 255 ( , ) - + Now, min f3(sn,xn) ¶ = - + f s x n 3 400 2 000 255 ( , ) x s 3 250 2 = +

Continuous DP x s 250 2 = + f s x 50 250 1 000 150 260 ( , ) = - + 3 *

Continuous DP Stage 2 (autumn) s2 = x1 220 < s2 , 255 f s x 200 000 3 200 000 240 ( , ) * = - + = - + 200 2 000 240 50 250 260 3 ( ) , x s + - 1 000 150 3 , ( ) s

Continuous DP Stage 2 (autumn) s2 = x1 220 < s2 , 255 f s x 200 000 3 200 000 240 ( , ) * = - + = - + 200 2 000 240 50 250 260 3 ( ) , x s + - 1 000 150 3 , ( ) s but s2 = x1

Continuous DP f s x 2 200 000 240 50 250 260 1 150 ( , ) = - +

Continuous DP ¶ = - + f x s 400 000 100 250 260 1 ( ) , f s x 200 000 240 50 250 260 1 150 ( , ) = - + ¶ = - + f x s 2 400 000 100 250 260 1 ( ) ,

Continuous DP x s 240 3 = + ¶ = - + f x s 400 000 100 250 260 1 ( ) , 400 000 100 250 260 1 ( ) , x s 2 240 3 = +