1

KKConvexity Convexity

and and

The Optimality of the (The Optimality of the (ss, , SS) Policy ) Policy

2

OutlineOutline

optimal inventory policies for multi-period problems (s, S) policy

K convexity

3

General Idea of Solving General Idea of Solving a Two-Period Base-Stock Problema 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

D1

x1

D2

X2 = y1 D1

y1 Y2

*1 1 1 1 1 1 1 1 1 2 2( , ) ( ) ( ) ( ) [ ( )]f x y c y x hE y D E D y E f X

2 2

*2 2 2 2 2( ) min ( , )

y xf x f x y

2 2 2 2 2 2 2 2 2( , ) ( ) ( ) ( )f x y c y x hE y D E D y

1 1

*1 1 1 1 1( ) min ( , )

y xf x f x y

discounted factor , if applicable

4

General Idea of Solving General Idea of Solving a Two-Period Base-Stock Problema Two-Period Base-Stock Problem

problem: to solve need to calculate need to have the solution of

for every real number x2

D2D1

x1

y1

X2 = y1 D1

Y2

2 2

*2 2 2 2 2( ) min ( , )

y xf x f x y

*1 1 1 1 1 1 1 1 1 2 2( , ) ( ) ( ) ( ) [ ( )]f x y c y x hE y D E D y E f X

2 2 2 2 2 2 2 2 2( , ) ( ) ( ) ( )f x y c y x hE y D E D y

1 1

*1 1 1 1 1( ) min ( , )

y xf x f x y

*1 1( )f x

*2 2[ ( )]E f X

*2 2( )f x

5

General Idea of Solving General Idea of Solving a Two-Period Base-Stock Problema Two-Period Base-Stock Problem

convexity optimality of base-stock policy

convexity convex

convexity convex in y1

convexity convex in y1

D2D1

x1

y1

X2 = y1 D1

Y2

2 2

*2 2 2 2 2( ) min ( , )

y xf x f x y

*1 1 1 1 1 1 1 1 1 2 2( , ) ( ) ( ) ( ) [ ( )]f x y c y x hE y D E D y E f X

2 2 2 2 2 2 2 2 2( , ) ( ) ( ) ( )f x y c y x hE y D E D y

1 1

*1 1 1 1 1( ) min ( , )

y xf x f x y

*2 1 1[ ( )]E f y D

*2 2( )f x

*1 1 1 1 1 2 1 1( ) ( ) [ ( )]cy hE y D E D y E f y D

6

General ApproachGeneral 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?

period Nperiod N-1period N-2period 2period 1 …

FP of fN

SP of SN

FP of fN-1

SP of SN-1

FP of fN-2

SP of SN-2

FP of f2

SP of S2

FP of f1

SP of S1

…

attainment preservation

7

Optimality of Base-Stock PolicyOptimality of Base-Stock Policy

period Nperiod N-1period N-2period 2period 1 …

convex fN

optimality of BSP

convex fN-1

optimality of BSP

convex fN-2

optimality of BSP

convex f2

optimality of BSP

convex f1

optimality of BSP

…

attainment preservation

8

Functional Properties of G

for the Optimality of the (s, S) Policy

9

A Single-Period Problem A Single-Period Problem with Fixed-Costwith Fixed-Cost

convex G(y) function: optimality of (s, S) policy G0(x) = actual expected cost of the period, including fixed 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?

G(y)

by

ea s S

K

x

G0(x)

s S

10

Another Example Another Example on the Insufficiency of Convexity in Multiple Periodson 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, 30)

(8, 36)

(0, 60) (20, 60)

Gt(y)

y

y

min{ ( ),

min[ ( )]}t

ty x

G x

K G y

(10, 30)

(8, 36)

(0, 36)

(20, 60)

( )

min{ ( ),

min[ ( )]}

t

t

ty x

f x cx

G x

K G y

y(10, 15)

(8, 24)(0, 36) (20, 30)

min{ ( ), min[ ( )]}t ty x

G x K G y

11

s S

Feeling for the Functional Property Feeling for the Functional Property for the Optimality of (for the Optimality of (ss, , SS) Policy) Policy

Is the (s, S) policy optimal for this G?

YesK

K

y

G(y)

12

ld eba

Feeling for the Functional Property Feeling for the Functional Property for the Optimality of (for the Optimality of (ss, , SS) Policy) Policy

Are the (s, S) policies optimal for these G?

y

KKG(y)

No No

b da le

y

KK

G(y)

13as S

Feeling for the Functional Property Feeling for the Functional Property for the Optimality of (for the Optimality of (ss, , SS) Policy) Policy

key factors: the relative positions and magnitudes of the minima

Is the (s, S) policy optimal for this G?

y

K

G(y)

14

Sufficient Conditions Sufficient Conditions for the Optimality of (for the Optimality of (ss, , SS) Policy) 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

fn satisfying condition ***

What is needed?What is needed?

optimality of (s, S) policy in period n

fn satisfying condition

*** plus an additional property

optimality of (s, S) policy in

period n

fn-1 with all the desirable properties

additional property: K-

convexity

16

KConvexity

and

KConvex Functions

17

Definitions Definitions of of KK-Convex Functions-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

Kaxfbxfxfxfba )())()(()(

bcKbfcf

abafbf

)()()()(

Interpretation: x y, function f lies

below f(x) and f(y)+K for all points

on (x, y)

18

Properties Properties of of KK-Convex Functions-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

Properties Properties of of KK-Convex Functions-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)

20

KK-Convexity Being Sufficient, -Convexity Being Sufficient, not Necessary, for the Optimality of (not Necessary, for the Optimality of (ss, ,

SS)) non K-convex functions with optimal (s, S) policy

K

K

y

G(y)

K

K

y

G(y)

21

Technical Proof

22

Results and ProofsResults 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(yD)+ + E(Dy)+ + E[ft+1(yD)]

approach ft+1 K-convex Gt(y) K-convex (Lemma 8.3.1)

Gt K-convex an (s, S) policy optimal (Lemma 8.3.2)

Gt K-convex K-convex (Lemma 8.3.3)

K-convex ft K-convex desirable result (Theorem 8.3.4)

..

,for

),(

),()(min),(min)(*

wo

sx

xG

SGKyGKxGxG

t

tt

xytt

)]([min),(min)( yGKxGcxxf t

xytt

)(* xGt

)(* xGt