1 Outline multi-period stochastic demand base-stock policy convexity.
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Transcript of 1 Outline multi-period stochastic demand base-stock policy convexity.
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Properties of Convex FunctionsProperties of Convex Functions
let f and fi be convex functions cf: convex for c 0 and concave for c 0
linear function: both convex and concave f+c and fc: convex sum of convex functions: convex f1(x) convex in x and f2(y) convex in y: f(x, y) = f1(x) + f2(y) convex in
(x, y)
a random variable D: E[f(x+D)] convex
f convex, g increasing convex: the composite function gf convex
f convex: sup f convex
g(x, y) convex in its domain C = {(x, y)| x X, y Y(x)}, a convex set, for a convex set X; Y(x) an non-empty set; f(x) > -∞: f(x) = inf{yY(x)}g(x, y) a convex function
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Illustration of the Last PropertyIllustration of the Last Property
Conditions: g(x, y) convex in its domain C
C = {(x, y)| x X, y Y(x)}, a convex set
X a convex set
Y(x) an non-empty set
f(x) > -∞
Then f(x) = inf{yY(x)}g(x, y) a convex function
Try: g(x, y) = x2+y2 for -5 x, y 5. What is f(x)?
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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
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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
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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 of f2 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
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Problem SettingProblem Setting
N-period problem with backlogs for unsatisfied demands and inventory carrying over for excess inventory
cost terms no fixed cost, K = 0
cost of an item: c per unit
inventory holding cost: h per unit
inventory backlogging cost: per unit
assumption: > (1)c and h+(1)c > 0 (which imply h+ 0)
terminal cost vT(x) for inventory level x at the end of period N
: discount factor
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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
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
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Necessary and Sufficient Condition Necessary and Sufficient Condition for the Optimality of the Base Stock Policy for the Optimality of the Base Stock Policy
in a Single-Period Problemin a Single-Period Problem
H(y): expected total cost for the period for ordering y units the necessary and sufficient condition for the optimality of the
base stock policy: the global minimum y* of H(y) being the right most minimum
y
H(y) H(y)
y y
H(y)
problem with the right-most-global-minimum property: attaining (i.e., implying optimal base stock policy) but not preserving (i.e., fn being right-most-global-minimum does not necessarily lead to fn-1 having the same
property)
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fn with right most global minimum
What is Needed?What is Needed?
optimality of base- stock policy in
period n
fn with right most global minimum plus an additional
property
optimality of base-stock
policy in period n
fn-1 with all the desirable properties
additional property: convexity
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Properties of Convex FunctionsProperties of Convex Functions
let f and fi be convex functions cf: convex for c 0 and concave for c 0
linear function: both convex and concave f+c and fc: convex sum of convex functions: convex f1(x) convex in x and f2(y) convex in y: f(x, y) = f1(x) + f2(y) convex in
(x, y)
a random variable D: E[f(x+D)] convex
f convex, g increasing convex: the composite function gf convex
f convex: sup f convex
g(x, y) convex in its domain C = {(x, y)| x ∈ X, y Y(x)}, a convex set, for a convex set X; Y(x) an non-empty set; f(x) > -∞: f(x) = inf{yY(x)}g(x, y) a convex function
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Illustration of the Last PropertyIllustration of the Last Property
Conditions: g(x, y) convex in its domain C
C = {(x, y)| x X, y Y(x)}, a convex set
X a convex set
Y(x) an non-empty set
f(x) > -∞
Then f(x) = inf{yY(x)}g(x, y) a convex function
Try: g(x, y) = x2+y2 for -5 x, y 5. What is f(x)?
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Period Period NN
GN(y): a convex function in y if vT being convex
minimum inventory on hand y* found, e.g., by differentiating GN(y)
if x < y*, order (y*x); otherwise order nothing
( ) min ( ) ( ) ( ) [ ( )]N Ty x
f x c y x hE y D E D y E v y D
( ) min ( )N Ny x
f x G y cx
( ) ( ) ( ) [ ( )]N TG y cy hE y D E D y E v y D
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Period Period NN-1-1
fN(x): a convex function of x
fN-1(x): in the given form
GN-1(y): a convex function of y
implication: base stock policy for period N-1
( ) min ( ) ( ) ( ) [ ( )]N Ty x
f x c y x hE y D E D y E v y D
1 ( ) min ( ) ( ) ( ) [ ( )]N Ny x
f x c y x hE y D E D y E f y D
1 1( ) ( ) ( ) [ ( )]N NG y cy hE y D E D y E f y D
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Example 7.3.3
two-period problem backlog system with vT(x) = 0
cost terms unit purchasing cost, c = $1
unit inventory holding cost, h = $3/unit
unit shortage cost, = $2/unit
demands of the periods, Di ~ i.i.d. uniform[0, 100]
initial inventory on hand = 10 units
how to order to minimize the expected total cost
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A Special Case A Special Case with Explicit Base Stock Level with Explicit Base Stock Level
single period with vT(x) = cx
objective function:
c(yx) + hE(yD)+ + E(Dy)+ + E(vT(yD))
c(1)y + hE(yD)+ + E(Dy)+ + c cx
optimal: (1 )
( )c
Sh
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A Special Case A Special Case with Explicit Base Stock Levelwith Explicit Base Stock Level
ft+1: convex and with derivative c
Gt(x)=cx+hE(xD)++E(Dx)++E(ft+1(xD))
same optimal as before:(1 )
( )c
Sh
problem: derivative of fN c for all x
( ) , if ,( )
( ) , . .N
NN
G S cx x Sf x
G x cx o w
fortunately good enough to have derivative c for x S, i.e., if vT(x) = cx, all order-up-to-level are the same