Multi-Period Problems: (s, S)

KConvexity
and
The Optimality of the (s, S) Policy
1
Outline
 optimal
inventory policies for multi-period
problems
 (s,
K
S) policy
convexity
2
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
discounted factor
, if applicable
f1 ( x1 , y1 )  c ( y1  x1 )  hE ( y1  D1 )    E ( D1  y1 )   E [ f 2* ( X 2 )]
y1
x1
D1
X2 = y1 D1
Y2
f1* ( x1 )  min  f1 ( x1 , y1 )
y1  x1
D2
f 2 ( x 2 , y 2 )  c ( y 2  x 2 )  hE ( y 2  D 2 )    E ( D 2  y 2 ) 
f 2* ( x 2 )  min  f 2 ( x 2 , y 2 )
y 2  x2
3
General Idea of Solving
a Two-Period Base-Stock Problem
y1
x1
D1
X2 = y1 D1
Y2
D2

problem: to solve f1* ( x1 )

need to calculate E [ f 2* ( X 2 )]

*
need to have the solution of f 2 ( x2 )
for every real number x2
f1 ( x1 , y1 )  c ( y1  x1 )  hE ( y1  D1 )    E ( D1  y1 )   E [ f 2* ( X 2 )]
f1* ( x1 )  min  f1 ( x1 , y1 )
y1  x1
f 2 ( x 2 , y 2 )  c ( y 2  x 2 )  hE ( y 2  D 2 )    E ( D 2  y 2 ) 
f 2* ( x 2 )  min  f 2 ( x 2 , y 2 )
y 2  x2
4
General Idea of Solving
a Two-Period Base-Stock Problem
y1
x1
D1
X2 = y1 D1
Y2
D2

convexity  optimality of base-stock policy

convexity  f 2* ( x2 ) convex

convexity  E [ f 2* ( y1  D1 )] convex in y1

convexity 
convex in y1
cy1  hE ( y1  D1 )    E ( D1  y1 )   E [ f 2* ( y1  D1 )]
f1 ( x1 , y1 )  c ( y1  x1 )  hE ( y1  D1 )    E ( D1  y1 )   E [ f 2* ( X 2 )]
f1* ( x1 )  min  f1 ( x1 , y1 )
y1  x1
f 2 ( x 2 , y 2 )  c ( y 2  x 2 )  hE ( y 2  D 2 )    E ( D 2  y 2 ) 
f 2* ( x 2 )  min  f 2 ( x 2 , y 2 )
y 2  x2
5
General Approach
period 1
FP of f1
…
period 2
FP of f2
period N-2
period N-1
period N
FP of fN-2
FP of fN-1
FP of fN
SP of SN-2
SP of SN-1
SP of SN
…
SP of S1
SP of S2
attainment




preservation
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?
6
Optimality of Base-Stock Policy
period 1
convex f1
…
period 2
convex f2
period N-2
period N-1
period N
convex fN-2
convex fN-1
convex fN
optimality
of BSP
optimality
of BSP
optimality
of BSP
…
optimality
of BSP
optimality
of BSP
attainment
preservation
7
Functional Properties of G
for the Optimality of the (s, S) Policy
8
A Single-Period Problem
with 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?
K
G(y)
a s
b
S
y
G0(x)
e
s
x
S
9
Another Example
on the Insufficiency of Convexity in Multiple Periods





convex Gt(y)
c = $1.5, K = $6
(8, 36)
min{G ( x ),
min[ K  G ( y )]}
(s, S) policy with s = 8, S = 10
(0, 36)
(10, 30)
min{G t ( x ), min[ K  G t ( y )]} no longer convex
y x
neither ft(x)
(20, 60)
t
y x
t
y
(0, 60)
(20, 60)
(8, 36)
Gt(y)
ft ( x )  cx
 min{Gt ( x ),
(10, 30)
(0, 36)
(8, 24)
(20, 30)
min[ K  Gt ( y )]}
y x
(10, 15)
y
y
10
Feeling for the Functional Property
for the Optimality of (s, S) Policy
 Is
the (s, S) policy optimal for this G?
K
G(y)
K
s
Yes
S
y
11
Feeling for the Functional Property
for the Optimality of (s, S) Policy
 Are
the (s, S) policies optimal for these G?
No
G(y)
K
a b
No
K
d e l
y
G(y)
K
K
e
a b d
l
y
12
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?
G(y)
K
s
y
S
a
13
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
additional
property: Kconvexity
What is needed?
fn satisfying
condition ***
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
15
KConvexity
and
KConvex Functions
16
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,
f ( x )  ba ( f ( x )  f ( x  b ))  f ( x  a )  K

or, for any a  b  c,
f (b ) f ( a )
ba


f (c ) f (b ) K
c b
(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
Properties
of K-Convex Functions
 possibly
 no
discontinuous
positive jump, nor too big a negative jump
 satisfying
sufficient conditions ***
K
K
(a)
K
(b)
(a): A K-convex function;
(b) and (c) non-K-convex functions
(c)
18
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 Kconvex.

(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
K-Convexity Being Sufficient,
not Necessary, for the Optimality of (s, S)

non K-convex functions with optimal (s, S) policy
K
G(y)
G(y)
K
y
K
K
y
20
Technical Proof
21
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)


f
(
x
)


cx

min
G
(
x
),
min
[
K

G
(
y
)]
 t

t
t
dynamics of DP:
y x





Gt(y) = cy + hE(yD)+ + E(Dy)+ + E[ft+1(yD)]
approach
 ft+1

  K  Gt ( S ), for x  s ,
Gt* ( x )  min Gt ( x ), min K  Gt ( y )   
y x
o.w.

  Gt ( x ),
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  G t* ( x ) K-convex (Lemma 8.3.3)

Gt* ( x ) K-convex  ft K-convex  desirable result (Theorem 8.3.4)
22

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