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

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

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

3. discounted factor , if applicable y1 Y2 X2 = y1 D1 D1 D2 General Idea of Solving a Two-Period Base-Stock Problem • Di: 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 x1 3

4. y1 Y2 x1 X2 = y1 D1 D1 D2 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 x2 4

5. y1 Y2 x1 X2 = y1 D1 D1 D2 General Idea of Solving a Two-Period Base-Stock Problem • convexity  optimality of base-stock policy • convexity  convex • convexity  convex in y1 • convexity  convex in y1 5

6. … period 1 period 2 period N-1 period N-2 period N attainment preservation General Approach • FP: functional property of cost-to-go function fn of period n • SP: structural property of inventory policy Sn of period n • what FP of fn leads to the optimality of the (s, S) policy? • How does the structural property of the (s, S) policy preserve the FP of fn? FP of fN FP of f2 FP of fN-1 FP of f1 FP of fN-2 … SP of SN SP of S2 SP of SN-1 SP of S1 SP of SN-2 6

7. … period 1 period 2 period N-1 period N-2 period N attainment preservation Optimality of Base-Stock Policy convex fN convex f2 convex fN-1 convex f1 convex fN-2 … optimality of BSP optimality of BSP optimality of BSP optimality of BSP optimality of BSP 7

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

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

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

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

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

13. G(y) K a s S y 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? 13

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 14

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

16. KConvexity and KConvex Functions 16

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) 17

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

19. Properties of K-Convex Functions • (a). A convex function is 0-convex. • (b). If K1K2, a K1-convex function is K2-convex. • (c). If f is K-convex and c > 0, then cf is k-convex for all kcK. • (d). If f is K1-convex and g is K2-convex, then f+g is (K1+K2)-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) 19

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

21. Technical Proof 21

22. Results and Proofs • assumption: h+ 0 and vT is K-convex • conclusion: optimal (s, S) policy for all periods (possible with different (s, S)-values) • dynamics of DP: • Gt(y) = cy + hE(yD)+ + E(Dy)+ + E[ft+1(yD)] • approach • ft+1K-convex Gt(y) K-convex (Lemma 8.3.1) • GtK-convex  an (s, S) policy optimal (Lemma 8.3.2) • GtK-convex  K-convex (Lemma 8.3.3) • K-convex ftK-convex  desirable result (Theorem 8.3.4) 22