Existence of average optimal strategies in inventory problems

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Wijngaard, J. (1976). Existence of average optimal strategies in inventory problems. (Memorandum COSOR; Vol. 7607). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1976

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PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 76-07

Existence of average optimal strategies in inventory problems

by

J. Wijngaard

Eindhoven, March 1976 The Netherlands

Existence of average optimal strategies in inventoEY problems

by

J. Wijngaard

O. Introduction

In

[4J

conditions were derived for the existence of optimal strategies of

stationary Markovian decision problems. These conditions were given 1n very general terms and are therefore not so easy to verify in a concrete case. In this paper we consider a class of one point inventory problems. Using

the results of

[4J,

easy verifiable conditions for the existence of optimal

strategies are derived.

1. Preliminaries

for BEL, by (PBf) (u).

is denoted by (~P)(E).

denoted by (Pf)(u) and the integral

Throughout this paper we assume that (V,~) is a measurable space. For a cornr

plex valued function f and a complex measure V the integral

f

f(u)v(du), if

V

f and a Markov process on (V,~)

f

P(u,ds)f(s), is existing, is

V

J

P(u,ds)f(s),

B

For ~ a measure the integral

J

P(u,E)v(du), E E I,

V

The space B(V,L) is the Banach space of all complex valued, bounded,

measu-rable functions on

V

with norm II f II := sup

I

feu)

I.

The space

M(V,I)

is the

UEV

Banach space of all complex measures on L wi th norm II ~ II := v (V), where v (V)

~ V

is the total variation of ~ on

V.

Each Markov process P on

(V,I)

defines a

existing, is denoted by ~f. For a function

with transition probability P the integral

bounded linear operator in B(V,L) by Pf and a bounded linear operator in

M(V,I)

by ~P. These operators are almost adjoint to each other (see

[4J).

A stationary Markovian decision problem (SMD) on (V,L) is a set of pairs

{(P ,r )}, a E

A

where P is a Markov process on (V,L) for each a and r a

a a a a

nonnegative function on V, the costfunction of a. The elements of

A

are

called strategies and can be interpreted as the stationary policies 1n a

Markovian decision process. The average costs of a, starting in u, are equal

to n-1 lim.!. ') n ~ n-+<x> £=0 £ (P r ) eu) , a a

if this limit exists, and are denoted by ga(u). A strategy aa E A ~s called

optimaZ

if g (u) ~ g (u) for all a E

A,

u E V.

aa a

In [4J conditions for the existence of an optimal strategy are given. Since

of these conditions stated a certain recurrency to a subset A of V. To

for-mulate these conditions we introduce the concept (A,f)-recurrency.

Definition I. Let f be a nonnegative measurable real valued function on V

and let A be a measurable set. The Markov process P is said to be

(A,f)-recurrent if

i) P;f exists for all m E:IN, (B:= V\A),

00

ii) the sum

L

(P;f)(u) exists for all u E V,

m=O 00

iii) the convergence of

L

(P:f)(u) is uniform on A and

m=O

00

L

(P:f) (u) is

m=O

bounded on A.

A set of conditions sufficient for the existence of an optimal strategy is the following (see [4J).

a) There is a set A E L such that

i) for all a E A the Markov process P

a ~s (A,lv)-recurrent and (A,ra

)-recurrent.

ii) the functions

I

P:Bl_v and

I

n=O n=O

rence time and recurrence costs

00 00

n PaBr

a on A, with B := V\A, (the

recur-to A) are uniformly bounded on

A.

b) For all a E

A

the embedded Markov process Q of P on A is quasi-compact

a a

(see [4J) and has only one ergodic set.

c) There is a metric p on

A

such that

i)

A

is compact.

II Qk(Q - Q ) II "" 0 for all a

O E A and some k, where II· II

~s

a a a

O

k

Qa(Qa -

Q

_a ) as operator in B(A,E_A)·

o

lim p(a,a O)-){) the norm of ii)

iii) lim p(a,aO)+O

o

for all a O E A for all a

O

E

A

where T ex. co

:=

I

pn and TI is

o

aB a_O the stationary probability of aO'

Notice that condition b) guarantees that the average costs ga are constant on V. In [4J it is shown that under some weak conditions one may restrict the at-tention, without loss of generality, to such strategies.

• To verify the recurrence conditions a) for inventory problems we shall use

the spaces Band M •

w w

Definition 2. Let wbe a positive function on V. The space of all complex

valued functions f on

V

such that f

:=!

E

B(V,E)

is denoted by

B (V,E).

w w w

The space of all complex measures ~ such that the measure ~ definedby

w

~W(E)

:=

f

w(u)~(du),

E E L , E

is an element of

M(V,E),

is denoted by M

(V,E).

w

With the norms II f II := II f II and II ~ II := 11].l II, respectively, these spaces are

w w w w

Banach spaces. For w(u)

=

1, u E V the spaces are equal to the spaces

B(V,l:)

and

M(V,E).

It is easy to show that the integral ].If exists for ].l E

M

and

].l

fEB and that (].lP)f

=

].l(Pf) for ].l E M and fEB if P is a transition

pro-w w w

bability such that Pw E B •

w

2. Inventory problems

The inventory problems considered in this paper are one-point inventory

pro-blems with 1eadtime 0 and backlagging. The state of the system is given by

the inventory level. We assume therefore that V is the real line and E the

a-field Borel sets. An

inventory problem

is an SMD {(P ,r )}, ex E

A

on

(V,E)

a CI.

i) A is a subset of the set of all nonnegative measurable functions on V.

ii) There is a probability distribution function F with F(a) = 0, a <

a

and

F(O)

:f

1 such that b Pa(u,Ca,b» "" -

J

dF(u+a(u) - v) a u+a (u)-a

J

dF (x) u+a(u)-b iii) There are nonnegative measurable functions rl,r

Z such that

for all u E V, a EA.

The distribution function F is the distribution function of the demand per

period. The functions r _l and r

Z give the ordering and inventory costs. For

u E V, a(u) is the quantity to order under strategy a.

Throughout this section we consider an inventory problem {(P ,r )}, a EAwith a a

00

f

eXdF(x) < 00. To prove the existence of an optimal strategy we use the

0-spaces B

w and Mw with w(u)

:=

e 1ul , U E V. The functions r 1 and rZ are

as-sumed to be elem:nts of

B •

w

Subsequently the conditions ~, ~ and ~ of section 1 will be considered.

First the concept of an (m,M,R)-inventory problem is introduced.

Definition 3. Let m,M,R be real numbers such that m< 0 < M, R ~ M-m, and

00

•

J

e dF(x)x < e R• The inventory problem {(P ,r )}, a E A is an

(m,M,R)-O- ct a

problem i f for all a E A

a(u) ~ R for u ~ m

u + a(u) ~ M for u :::; M

a (u) =

a

for u > M.

2. I. The recurrence conditions (condition a»

In the following lemma and corollary it is shown that an (m.M.R)-inventory

Lemma 4. Let {(P ,r )}, a E A be an (m,M,R)-inventory problem. Define a a

A :=

[m,M] and B :=

V\A.

Then

P

B is a bounded operator in

B

for all a E

A

a . w

and there is an a > 0 and a p, 0 < p < I such that II pn

BII ~ apn for all

a w

a E

A,

n E IN.

Proof. First we have to proof that PaBf E

B

w

for all f E

B

w

and a E

A.

Let

a E A and fEB • w For u ~ m we have 00

II

Hence (PaBf)(u) = - 0 0 m

J

f(v)dF(u+a(u) -v) =

J

f(u+a(u) -x)dF(x) u+a(u)-m where

I

(PaBf) (u)

I

1

---:rur-

=

e-u • m ~ II f II w - 0 0 00

f

f (u+a (u) - x)dF(x)

I

~

u+a (u)-m 00

J

ex-a(u)dF(x), u+a (u)-m 00

•

So (1) m II f II w - 0 0 m I II f II._{w a u}() - 0 0 e

o

00

~ _~f

II w• e-R

J

eXdF(x) •

o

For m ~ u ~ M~.,e have

00

(PaBf)(u) =

f

f(u+a(u) - x)dF(x) • u+a (u)-m

I

(p Bf) (u)

I

00 m

J

e-u-a (u)+xdF(x) (2)

----;rur- -

a < II £11 ~ w -u. -<Xl _e u+a (u)-m 00 00 m

J

ex-a(u)dF(x) m

J

II £11 II f II x ~ ~ e dF(x)

.

w w -00 -00 0 0 For u > M we have (PaBf)(u) =

f

f(u-x)dF(x) + u-m Hence

f

u-Mf (u - x) dF(x) - 0 0 - 0 0 (3) m II f II w -00 00

I

x-u 00

!:..-.-

dF(x) +II f II • eU M W u-m

f

U - M u-x ~ dF(x) ~ e m ~ II f II • W _00

o

00

J

eXdF(x) +

~

Mf IIW•

o

00

The relations

(I), (2), (3)

imply that PaBf E

B

w for f E

B

w and for all

a E

A.

Now we have to consider P:Bf. Let

•

r := Then by (I)

o

00 x e dF (x) and q:=

o

00

J

e-xdF(x)

(4)

n rn • II f II w - 0 0 for u ~ m and by (2) (5 ) m n R r •e • II f Ii w -co for m ~ u ~ M • For u > M we have

I

I

(P:Bf) (u)

I

m 00 (6) R IIpn-I f II n-I el ul :s; _{r • e}

.

_{+ q •IIP B f II} :s; :s; aB w _{M a w} -00 m 00 n n-] n-I r)eR II f II n II f II :s; _{(r +qr +••. + q}

.

_{+ q} :s; w w -00 M n n-] n-I n R • II f II :s; _{(r + qr +•.. + q r +q )e}

₌

w n+ I n+ I = _r_ _-_S,L..'_ •eR •II f II r - q w

The relations (4), (5), (6) completes the proof.

It is a direct consequence of this lemma that an (m,M,R)-inventory problem satisfies the condition a).

Corollary 5. Let {(P ,r )}, a E

A

be an (m,M,R)-problem and let A := [m,MJ

- a a

and B := V\A. Then for all a E

A

the Markov process is (A,I)-recurrent and

(A, r )-recurrent, and the boundedness on A of ttl", functions a lJ 00

I

pnBr n=O a a 1.S uniform on

A.

and 00

all a E: A and that II r II

a w IV E:

B

w'

Further we have

Hence

•

Proof. By lemma 4 it is sufficient to prove that IV E

B ,

that r E B for

w I a w

is bounded on

A.

The boundedness of

w

implies that

ra(u)

=

rl(a(u» + rZ(u + a(u» _{and rl,r Z E Bw'}

r (u) $ II rIll .ea(u) + II rZ11 .e

l

u+a(u)

I •

a

w

w

Using that {(Pa,r

a)}, a E: A is an (m,M,R)-problem it 1.S easy to show that

r E B for a E:

A

and that II r II is bounded on

A.

[l

a w a w

We defined w(u) := e 1ul but it is possible of course to use the same methods

2.2. The conditions on Q (condition b» a

In this subsection we assume that {(P ,r )}, a E

A

is an (m,M,R)-inventory

a a

problem. By subsection 2. 1 the embedded Markov process Q of P on

a a

A

:= [m,MJ exists for all a E

A.

We have to state conditions which guarantee

the quasi-compactness and ergodicity of Q . a The Markov process Q

a on

(A,L

A

)

is quasi-compact if there is a compact

ope-rator K ~n B(A,l.A) and an integer n such that IIQ: - Kil < I. Quasi-compactness,

defined ~n this way, is equivalent with the well-known Doeblin condition

(see [2J).

Qn for some n ~

a ')

of Q- is given. a

Sufficient for the quasi-compactness of Q is of course the compactness of

a

1. In the next lemma, a condition sufficient for compactness

Lemma 6. I f F has a bounded density cp, Q2 is compact for a E

A.

a

Proof. Let cp (u):= cp(u + a(u) ~ v) for a E

A,

v E V, U E V. Then

c/,v Qa(u,E)

=

f

qa(u,v)dv , E where co q",(u., v) :=

I

(pn_BCP ) (u) • '" n=O a av

By lemma 4 the boundedness of cP implies the boundedness of qa

_C-,-)

on A x A

for all a E A.

a E A

[IJ, IV.9.2 and VI.4. 1 we infer that Q is weakly

2 a

compact for all a E

A.

This implies the compactness of Q , (see [IJ, VI.8. 13

C/,

Now let

A

be the Lebesgue measure on

A.

It is easy to show that for all

lim (~Q)(E) = 0 uniformly for all measures ).1 on l.A 'Nith 11).111:$ 1.

i-.(E)~ a

Using Dunford-Schwartz

and the remarks at the end of VI. 8). [I

It is clear that the existence of a bounded density of F is not necessary for

quasi-compactness of Q but weaker conditions are verj difficult to check if a

one makes no extra assumptions abollt A.

To guarantee tllat Qa, hal:> unly OlW l'q;odi(' la-! WI.' call !i1.IIII' lor illtll:UI!'I' lilill

2.3. The topology on

A

(condition c))

on P and r

a et

cHi). We assume

The continuity conditi.ons cii) and cHi) are given in terms of Q , T

lv'

a a

T r • These conditions are very difficult to check since the decision pro-a pro-a

blem is given in terms of P and r • In the following two lemma's conditions

a a

are given which are sufficient for the conditions cii) and

that {(P ,r )}, a E

A

is an (m,M,R)-inventory problem and

ex a

F has a bounded density.

•

Lemma 7. Let p be a metric on

A

and a

O an element of

A

such that for all

n

=

0,1,2, ... ( 1) (2)

o .

Then lim II Q (Q - Q ) II =

a .

( ) 0 a ex et

o

p ex,a O-+ Proof. We have Q f = a 00

I

P:BPaAf for all a E

A,

fEB. Let

n=O II filA := sup uEA If(u)1 for £ E

B

and II wII A := sup !w(u)

I •

UEA

By lemma 4, for each e: > 0 there is an integer N such that for all a E A

E:: and fEB 00 II

I

n=N E. n P BP Af11 ~ E::.11P AfII • a ex w a w Hence 00 for u E: A , and

00 00

But

Therefore i t is ·sufficient to prove for all n = 0,1,2, ••• (3) lim II Q (pnBP A - pn BP A) II = 0

( ) 0 a a a 0.0 CiO

P 0.,0.

0 +

It is easy to show that

•

n-!

k _{_ P )pn - 1-k p +pn (P -P}

n _{pn p}

I

A)

PaBPaA - _{aOA aDA} = PaB(PaB

k=O aOB aOB aOA aB aA aO

Hence, by the assumptions (I) and (2)

(4) lim II P A(pnBP A - pn BP A) II = 0 ( ) 0 a a a 0.0 aD p a,a O+ for n = 0, 1,2, • •• • 00 By lemma 4 II

I

P:

B

IL

is bounded on

A.

Let K be an upperbound. Then for all n=O fEB we get 00 :::; II

I

pnBII .11P AfII .11 wII A :::; K.II f 11.11 wIIA n=O a w a w

Together with

(4)

this implies

(3).

Lemma 8. Let CJ.

e

E A and let p be a metric on A such that for each ~ E M

w

which is continuous with respect to the Lebesgue measure A, the following properties hold

LJ

( I )

o

(2) lim

I

I

(P Bf ) (u) - (P Bf ) (u)

I

~

(du)

p(a,aO)+O V a 0.0

o

for all fEB w

Then condition ciii) is satisfied for a O'

Proof. First we shall show by induction that for all n

=

0,1,2, .•.

which are continuous with respect to

A.

For n

=

0 (3)

is a direct consequence of assumption

(I).

Now let it be true

for n = k. We have

f

J

V B By assumption (2) P (u,ds)

I

(pk_{Br ) (s) -} (pk Br ) (s) Ill(du) + a a a 0. 0 0.0 (4)

° .

Further

I

J

pa(u , ds)

I

(pkaBr )a (s) - (pk0. Br )(S)lll(du) = 0 0.0 V B =

f

(llP ) (ds)

I

(pkBr ) (s) - (Pk Br ) s)( 1I a a a 0. 0 0.0 . B

where llP is an element of M • llP is continuous with respect to

A.

By

(4)

a w a

and the induction assumption we see that (3) is true for n = k + 1. Hence

(3) is true for all n 0,1,2, . . . .

For each E: > 0, lemma 4 and the boundedness of 11r lion A, imply the

exis-a w

tenee of an integer N such that

E: Henee 00

L

n=N E: n p Br II a a w < E: for all a EO

A .

J

00 00

II

_{p Br}n

- I

pn Br

I

(U)11(du) ~ II ]..I" .ZE: for all 11 E M

N a a N aO aO w w

V E: E:

Using this and (3) we get for all A-continuous 11 E M

w

(5) lim

J

I

(T r )(u) - (T r ) (u)

III

(du) =

o .

p(a.aO)~O _V a a aO aO

Simi 1arly we can prove

(6 )

J

I

_{(Ta IV) (u) - (Tao lV) (u)}

I

_{11(du) = 0 •}

V

Let TI' for a E

A

be the measure on L defined by

al

EEL •

nlen TI~l E

M

_w' The existence of a bounded density of F implies the

A-conti-nuity of TI~I' Substitution of 11 := TI~I ~n (5) and (6) completes the proof.D

The next problem is the introduction of a metric p on

A

such that the

conti-nuity conditions of the lemma's 7 and 8 are satisfied and

A

is compact.

Since F has a bounded density it is not necessary to distinguish between strategies which are almost everywhere equal. One can interprete the

strate-gies a E

A

as classes of functions which are almost everywhere equal. Since

{(P .r )}. a E A is an (m.M.R)-inventory problem the integral

J

a«u» du

~s

a a V w u

finite for all a E

A.

The functions a

:=

a are elements of the space

w w

L1(V.L.A). where A is the Lebesgue measure. This induces a metric p on

A

de-fined by p (a 1.(2)

_J

laI(u) - az(u)1 du • := w(u) V

This metric is called the w-metric.

The space A with this metric is isometrically isomorphic with the space A

w

of all functions'::' with the L1-metric. Hence compactness of

A

implies

com-w w

Lemma 9. Let A be such that

lim

J

x+O

_v

ja(u + x) _ a(u)ldu =

w(u

+ x)

w(u)

0) uniform on A •

Then the closure of

A

w ~n L1 is compact.

Proof. Using that {(P .r )}) a E

A

IS an (m)M)R)-problem) we can see that

a(u) d a a

J

w(u) u is bounded on A. and

V +a lim { -00 -a

J

wa (u)(u) du + +00

J

a(u) du}w(u) = 0• uniform on A •

By [IJ. IV.8.20 this implies that the closure of

A

is compact.

w

If

A

~s closed we have here a sufficient condition for compactness of

A.

w

Now we have to consider the continuity conditions of the lemma's 7 and 8 with p equal to the w-metric.

o

Lemma 10. If the density ~ of F has a bounded derivative ~'. the conditions

(I) and (2) of lemma 7 and condition (2) of lemma 8 are satisfied for all

0.

0 E

A

and for p equal to the w-metric on

A.

Proof. First we shall prove that for all a ::; m. £ > 0 and for each n E :IN

there are n finite intervals B. := [a. ,MJ. i = It ••. ,n such th~t for all

1 ~

a E (1)

A

and all u E [a.M]

I

(P~Bw)(u) - (P P ••• P

aB w)(u)! ::; E •

"" aB_{I aB 2} n

For all a ~ m and all s > 0 it is possible to choose an a ~ m such that

£

for u E [a.M] and for all a E

A

a

£

J

~(u

+ a(u) - v)w(v)dv < s •

This proves (I) for n

=

1.

By lemma 4 P BW E

B

and II P BWII is bounded on A. Hence, the induction

as-a W a W

sumption implies for each a ~ m and each s >

a

the existence of finite in-tervals B. := [a.,MJ, 1. = l, ••• ,k, such that

1. 1.

I

(P"'Bw)(u)- (Pk+1 P ••• PBPBw)(u), <£foruE[a,MJ,aEA.

~ aB

I aB2 a k a Let the interval B

k+I := [~+I ,MJ be such that

for U E [~,MJ, a EA.

Then for all u E [a,MJ and a E

A

we get

I

(pk+ Iw) (u) - (p P _{••• PBPB W}) ( )_U I_{1<£+ M-a,M_}( ) € ₌ 2_£,

aB _{aB 1 aB2} a k a k+l K ~

which shows that (I) is true for 11 k + I and hence for all n E IN. We can use this result to show for each £ > a the existence of intervals

B

i := [ai,MJ, 1.

=

I, ...,n, such that for all a,aa E A

(2) _{IIP A}PnB (p A - P A) - P AP B P B ••• P B (P A - P A) II < £ ,

a a a aa a a I a 2 a n a aa

and the ens tence of intervals B. := [a. ,MJ, 1.

1. 1. 1, •••,n+ I, such that for

(3) _{\I P} ApnB(P B - P B) - P AP B ••• P B (P B - P", B ) II < £ •

a a a aa a a I a n a n+l ~O n+l

Now let a

Obe an arbitrary element of A, t:,a :=

I

a - aO

I

for a E A, II r.p' II 1.S

the supremum of ~'(u), and C := [a,bJ is an arbitrary finite interval. Then =

I (

Cp(u+a(u) -v) -cp(u+aa(u) -v}£(v)dv!

~

J

I(P C - P C)f (u)I a aa

c

~

l1

a(u).lIrp' II.

I

!f(v)ldV5: Da(u).llcp' 11.(b-a).lifll for all f E B .

c

Let C

Ip

A

P

C

...

P e (P e - P e)f(u)1 s a a I a n a a_O But (P C L'. )(u) Ct, a n

r

J

IJ. (v)

j

'

cp(u+a(u) -v)L~ (v)dvsllwll .llcpll.a

e

a()w v C n C n dv s where II cp II

s

II wile .11 cpII.p(a ,a_O) , n

sup cp (u). Hence

UEV

a .

Using the results (I) and (2), this proves the conditions (I) and (2) of

lemma 7 for all a

O E A.

Now we have to prove condition (2) of lemma 8 far all a_O E

A.

Using the

fact that {(P ,r )}, a E A is an (m,M,R)-problem, we see that

a a

f

I

(PaBf) (u) - (PaoBf) (u) If.l(du)

V

M

I

!CPaBf)(u) - (PaoBf)(u)If.l(dU).

_00

For each E: >

o

and each )l E M there ~s an a < m such that

w E:

a

E:

f

1(PaBf)(u) - (P Bf)(u)!)l(du) < E: for all a,a

O E

A .

a O

- 0 0

Hence, it ~s sufficient to prove for all f.l E

M

wand aO E

A

lim

f

!(PaBf)(u) - (PaoBf)(u)I)l(dU)

=

a

p(a,ao)-+O C

for all ,fini te intervals e. This can be done by approximating B by a finite interval, as in the first part of the proof.

2.4. Existence of an optimal strategy

Taking the results of the subsections 2. I, 2.2 and 2.3 together we get the

following theorem.

Theorem II. Let {(P ,r )}, a E

A

be an (m,M,R)~problem such that

a a

i) F has a bounded dens i ty q> such that cp(x) > 0 for x > 0 and qJ has a boun-ded derivative.

ii) lim

x+O

+00

I

,a(uiw(u ++ x) - a(u)!dux) w(u)

-00

0, uniform on A

iii)

A

w is closed ~n L1

(V,E,A);

i v) For each J.l E

M

which is continuous with respect to the Lebes gue

mea-w

sure A and for p the w-metric

lim

I

I

r (u) - r (U)!fl(du) =

a

for all a

O E A

.

p(a,aO)-+O _V a aO

In this case an optimal strategy exists.

Especially the conditions on F are very strong. These can be weakened if one

makes extra assumptions on the set

A

of order strategies; for instance is one

restricts A to the set of (s,S)-strategies (see [3J).

References

[I] Dunford, N. and Schwartz, J.T. (1958): Lineare Operators, part I.

Inter-science publishers, New York.

[2J Neveu, J. (1965): Mathematical foundations of the calculus of

probabili-ty. Holden-Day, San Francisco.

[3J Scarf, H. (1960): The optimality of (S,s)-policies in the dynamic

inven-tory problem. Mathematical Methods in the Social Sciences, Stanford University Press, Stanford.

[4J Wijngaard, J. (1975): Stationary Markovian decision problems I and II.

Memorandum-COSOR 75-14 and 75-15, Technological University Eindhoven, The Netherlands.