1

OutlineOutline

multi-period stochastic demand base-stock policy

convexity

2

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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Two-Period Problem: Two-Period Problem:

Base Stock PolicyBase Stock Policy

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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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Multi-Period Problem: Multi-Period Problem:

Base Stock PolicyBase Stock Policy

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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