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

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

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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

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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

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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

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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

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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

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

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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

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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)

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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)

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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)

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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)

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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

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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

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16 K Convexity and K Convex Functions

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17 Definitions of K-Convex Functions (Definition ) for any 0 < < 1, x y, f( x + (1- )y) f(x) + (1- )(f(y) + K) (Definition ) 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)

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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

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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)

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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)

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21 Technical Proof

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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)

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