Download presentation

Presentation is loading. Please wait.

Published byBrionna Colie Modified over 2 years ago

1
1 K Convexity and The Optimality of the (s, S) Policy

2
2 Outline optimal inventory policies for multi-period problems (s, S) policy K convexity

3
3 General Idea of Solving a Two-Period Base-Stock Problem D i : the random demand of period i; i.i.d. x ( ) : inventory on hand at period ( ) before ordering y ( ) : inventory on hand at period ( ) after ordering x ( ), y ( ) : real numbers; X ( ), Y ( ) : random variables D1D1 x1x1 D2D2 X 2 = y 1 D 1 y1y1 Y2Y2 discounted factor , if applicable

4
4 General Idea of Solving a Two-Period Base-Stock Problem problem: to solve need to calculate need to have the solution of for every real number x 2 D2D2 D1D1 x1x1 y1y1 X 2 = y 1 D 1 Y2Y2

5
5 General Idea of Solving a Two-Period Base-Stock Problem convexity optimality of base-stock policy convexity convex convexity convex in y 1 D2D2 D1D1 x1x1 y1y1 X 2 = y 1 D 1 Y2Y2

6
6 General Approach FP: functional property of cost-to-go function f n of period n SP: structural property of inventory policy S n of period n what FP of f n leads to the optimality of the (s, S) policy? How does the structural property of the (s, S) policy preserve the FP of f n ? period N period N-1 period N-2 period 2period 1 … FP of f N SP of S N FP of f N-1 SP of S N-1 FP of f N-2 SP of S N-2 FP of f 2 SP of S 2 FP of f 1 SP of S 1 … attainment preservation

7
7 Optimality of Base-Stock Policy period N period N-1 period N-2 period 2period 1 … convex f N optimality of BSP convex f N-1 optimality of BSP convex f N-2 optimality of BSP convex f 2 optimality of BSP convex f 1 optimality of BSP … attainment preservation

8
8 Functional Properties of G for the Optimality of the (s, S) Policy

9
9 A Single-Period Problem with Fixed-Cost convex G(y) function: optimality of (s, S) policy G 0 (x) = actual expected cost of the period, including fixed and variable ordering costs G 0 (x) not necessarily convex even if G(y) being so convex f n insufficient to ensure optimal (s, S) in all periods what should the sufficient conditions be? G(y)G(y) b y e a s S K x G0(x)G0(x) s S

10
10 Another Example on the Insufficiency of Convexity in Multiple Periods convex G t (y) c = $1.5, K = $6 (s, S) policy with s = 8, S = 10 no longer convex neither f t (x) (10, 30) (8, 36) (0, 60) (20, 60) Gt(y)Gt(y) y y (10, 30) (8, 36) (0, 36) (20, 60) y (10, 15) (8, 24) (0, 36) (20, 30)

11
11 s S Feeling for the Functional Property for the Optimality of (s, S) Policy Is the (s, S) policy optimal for this G? Yes K K y G(y)G(y)

12
12 ld e b a Feeling for the Functional Property for the Optimality of (s, S) Policy Are the (s, S) policies optimal for these G? y K K G(y)G(y) No b d a l e y K K G(y)G(y)

13
13 a s S Feeling for the Functional Property for the Optimality of (s, S) Policy key factors: the relative positions and magnitudes of the minima Is the (s, S) policy optimal for this G? y K G(y)G(y)

14
14 Sufficient Conditions for the Optimality of (s, S) Policy set S to be the global minimum of G(y) set s = min{u: G(u) = K+G(S)} sufficient conditions (***) to hold simultaneously (1) for s y S: G(y) K+G(S); (2) for any local minimum a of G such that S < a, for S y a: G(y) K+G(a) no condition on y < s (though by construction G(y) K+G(S)) properties of these conditions sufficient for a single period not preserving by itself functions with additional properties

15
15 f n satisfying condition *** What is needed? optimality of (s, S) policy in period n f n satisfying condition *** plus an additional property optimality of (s, S) policy in period n f n-1 with all the desirable properties additional property: K- convexity

16
16 K Convexity and K Convex Functions

17
17 Definitions of K-Convex Functions (Definition 8.2.1.) for any 0 < < 1, x y, f( x + (1- )y) f(x) + (1- )(f(y) + K) (Definition 8.2.2.) for any 0 < a and 0 < b, or, for any a b c, (differentiable function) for any x y, f(x) + f '(x)(y-x) f(y) + K Interpretation: x y, function f lies below f(x) and f(y)+K for all points on (x, y)

18
18 Properties of K-Convex Functions possibly discontinuous no positive jump, nor too big a negative jump satisfying sufficient conditions *** (a) K (b) K (c) K (a): A K-convex function; (b) and (c) non-K-convex functions

19
19 Properties of K-Convex Functions (a). A convex function is 0-convex. (b). If K 1 K 2, a K 1 -convex function is K 2 -convex. (c). If f is K-convex and c > 0, then cf is k-convex for all k cK. (d). If f is K 1 -convex and g is K 2 -convex, then f+g is (K 1 +K 2 )-convex. (e). If f is K-convex and c is a constant, then f+c is K-convex (f). If f is K-convex and c is a constant, then h where h(x) = f(x+c) is K- convex. (g). If f is K-convex and D is random, then h where h(x) = E[f(x-D)] is K-convex. (h). If f is K-convex, x < y, and f(x) = f(y) + K, then for any z [x, y], f(z) f(y)+K. f crosses f(y) + K only once (from above) in (- , y)

20
20 K-Convexity Being Sufficient, not Necessary, for the Optimality of (s, S) non K-convex functions with optimal (s, S) policy K K y G(y)G(y) K K y G(y)G(y)

21
21 Technical Proof

22
22 Results and Proofs assumption: h+ 0 and v T is K-convex conclusion: optimal (s, S) policy for all periods (possible with different (s, S)-values) dynamics of DP: G t (y) = cy + hE(y D) + + E(D y) + + E[f t+1 (y D)] approach f t+1 K-convex G t (y) K-convex (Lemma 8.3.1) G t K-convex an (s, S) policy optimal (Lemma 8.3.2) G t K-convex K-convex (Lemma 8.3.3) K-convex f t K-convex desirable result (Theorem 8.3.4)

Similar presentations

Presentation is loading. Please wait....

OK

25 seconds left…...

25 seconds left…...

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on asp dot net Ppt on op amp circuits examples Ppt on causes of 1857 revolt pictures Ppt on forward rate agreement quote Ppt on fibonacci numbers stocks Ppt on 21st century skills curriculum Free download ppt on the road not taken Ppt on history of badminton sport Ppt on basic geometrical ideas Ppt on 60 years of indian parliament structure