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)