Some Probabilistic Multiperiod Inventory Problems

Copyright © IFAC Modelling and Control of National and Regional Economies, Queensland, Australia, 1995

Some Probabilistic Multiperiod Inventory Problems Masanori KODAMA Fa-c.

0/ Economics,

Kyu6hv. Uni1lemty 27, Hakouki, Higa.6hiJru, FukuoktJ 812, JAPAN

Abstract: The probabiIistic inventory models with various partial returns and additional orders are considered. There are the ordering costs, holding costs and penalty costs in the models. Specific values of the variables that minimize the total expected costs in multistage are found. Mathematical generalization of these problems is made and various properties are discussed. Keywords: Convex optimization, Dynamic programming, Inventory control, Optimization proDlems, Probabilistic models, Time lag

1. INTRODUCTION

We consider the function C(b, z) where b is any given value of random variable B and z is a decision variable. Suppose that C(b, z) divides into many different functions, depending on the value of b and z. We will calculate the expectation of C(B, z) and derive the method to minimize the value of E{ C( B, z)} . Finally we will apply this method to the probabilistic multiperiod inventory model under general demand. Different from models so far considered, our model has the following prominent features:

Cf(b, z),b < hHz) = ~J(b, z), h~ (z) $ b < ~(z)

C(b,z) =

am-l

e

$ b<

=

' 1 ( b,z ) ,h 1 - (z) nl nl 1

=

C~l+1(b,z),b~h~I(Z)

$ z

1( z) hnl

(1)

< am

C(b,z) = Cj(b,z),b
1) Mathematical generalization of the model with partial returns and additional orders is made, (See Section 1 and Section 2) 2) The multistage models with ba.cklogging of demand are treated. (See Section 3) 3) Demand occurs according to general demand pattern. (See Section 3) Our model is a generalization of Kabak (1984) and Sorai, et al. (1986), in a point that it treats the model with 1),2) and 3).

=

C:",(b,z),h~",_l(Z) $

b < h~",(z)

= C:",+I(b,z),b ~ h~",(z) (2) z ~ am , n = nm+1 + 1 C(b,z) = Cj+1(b,z),b
2. MATHEMATICAL FORMULATION OF THE PROBLEM

=

m +1 (z) < - b < hn",+1

1 c:n+ (b" z) b> hm +1 (z) n - n",+1

(3)

where hj(z) is the function of z, and dhj(z)/dz > O,j = 1,2, ... , ni; i = 1,2, ... , m + 1, ni is any given natural number such that ni ~ 1, and when ni = 1, let h~'+1 (z) = 00.

Consider now a random variable B with a continuous distribution, and let q,(b) and ~(b) denote the p.d.f. and the d.f., respectively, of B. Let C(b, z) denote the cost function, where b is any given value of a random variable B and z is a decision variable. Suppose that C(b, z) is given by the following relation (1)-(3):For any given values

Let E{C(B, z)} denote the expectation of C{B, z). Since B is a random variable, so we con217

And also h' (*) denotes the derivative of h~(*) with respect to *. Similarly 4>' (h~(z» denotes the derivative of 4>(h~(z» with respect to h~(z). The second derivative of Hi(Z) with respect to z is given by

sider to minimize E{C(B, z)} with respect to the decision variable z.Let us define for any given real value of z and constant c > O.

where

d · dz 4>(hk(z»

» q (hi:(z ,z ,f

am~z
•

(4)

: { Hm(z), Hm+1(Z),

am-1 am

~ ~

z < am z < 00

.'

(5)

=

+

100

+

+

lh~(Z) oC~(b, z) 4>(b)db

dzAq(hA,(z),z)

d·

.

Ib=h~(z)1

I 1

[oACi(b, z) h( (z) ob b=h~(z)

= 0, k = 1,2"", nij i

= 1,2,,, " m + 1

Hence we have

(6)

k=1

=

where

+

ACt(hL(z),z)

=Ct(h~(z), z) - ct+! (h~(z), z)

=

{Ct(h~(z), z) - ct+! (h~(z), z)}

ACt,' (h~(z), z)

.

L .!lC~(h~(z), z)4>(h~(z»h~ (z)

I

oCi(b, z) dh~(z) x-ob b=h~(z) dz

If the cost function Ct(b, z) is continuous at branching points of b, then

oz k=2 h'.-1 (z) oCn ,+! (b, z) 4>(b)db h~.(z) oz

n,

I

oCt(b, z) oz b=h~(z)

+

oz

-00

t

:z

dh~(z)

a:;-

4>' (hL(z»h( (z) dCt(h~(z), z) dz

= [OAC~(b, z)

jhi (z) oC i (b ) 1 ,z 4>(b)db oz

+

.

Aq (hl,(z),z)

Next, we shall find a real number z for which H(z) is a minimum. Suppose that Hi(Z) is twice differentiable (i = 1, ... , m + 1).

H;(z) =

dh~(z)'

=

H1(Z), H(z) =

d4>(h~(z»

= = =

(7) 218

hi(Z) oCi(b ) 1 ,z 4>(b)db j OZ

-00

t

[h~(z) oCt(b, z) 4>(b)db

k=2 Jh~_1 (z)

OZ

(9)

roo

+ H;' (z)

=

lh~,(z) hi(z)

11

8C~i+l(b,z)4>{b)db 2

8

+ a

I~'{x)

2

8 CL(b, z) 4>{b)db 8z 2

.

lh~ . (z)

+

El 1:=1

I; (x) =

8z 2

=

I . j4>(h'

[86.CL(b, z) (z»h, (z), 8z b=hl(z) i=I,2,···,m+l (11)

=

H' (z)

+ a 100 I~-l (z -

b)4>{b)db

= 0,

I~(·)

=0

(14)

=

H(z) + a

(i) Case when k = 1. It is clear that there exists a unique Xl that H(z) attains its absolute minimum at z = Xl where H' (Xl) = O. IT x < Xl, then Minz~.,{H(z)} = H(Xl) and x ~ Xl then

100 II:-l(Z - b)4>(b)db, = 0, k = 1,· .. , N

(13)

Minz~.,{H{z)l=

lim F~{z)

li~ F~(z) >

z~oo

Theorem 1 IT conditions of Assumption 1 are satisfied, then we have

-ex + H{xI),x < Xl

=

-ex+H{X),X~Xl

=

-ex + H(xl:)

+

a

=

k=2,···,N -ex + H{x)

+

a

= =

-c,x < Xl -c+H'(x),x ~Xl

100

!A:-l(XI: -

b)4>(b)db,x < XI:

100 11:-1 (x - b)4>(b)db,x ~

=

lim H' (z) - ac

z~-oo

lim H' (z)

%-+00

<0

>0

Since F~' (z) ~ 0 for all z with the possible exception of the points z = a2 and z = X2, where leftand right hand bounded nonnegative derivatives exist, F~ (z) is nondecreasing. Hence F~ (z) = 0 possesses a unique root X2 that F2 (z) attains its absolute minimum z = X2. From the same analysis as in the case when k = 1, we get the case when k = 2.

(ill) Assuming that the theorem 1 hold the integer 1, then we have

II(x)

XI:

lim I~(x) > 0 .,-00

=

=

%-+-00

H;(ai) = n;+1{ai).

=

H(x).

(ii) Case when k = 2. Then we have from (i)

Assumption 1 The function H (z) is convex in z, limz_-ooH'{z) < O,limz_ooH'(z) > c and each Hi{Z) is twice differentiable (i = 1,2,···,m+ 1),

I~{x)

= 1,···, m + 1)

(12)

We shall discuss under the following conditions

I~(x)

XI:

Proof (by induction)

lo{-}

11: (x)

100 1;-1 {x - b)4>{b)db,x >

z~.,

Let

hex)

k = 2,···,N H"(x)

where at x = ai(i = 1,2,···,m + 1) and x = xl:(k = 1,2, ··· ,N), each I~(x) has left-and right hand bounded derivatives and also XI: is a unique root of each equations

Min{-ex+H(z)}

100 II:-l(Z-b)4>{b)db},k=2,3,···,N

FI:(z)

O,X
(X:f: ai, i

11: (x) = Min{ -ex + H{z) z~., +a

XI:

O,X Xl

+ a

Next, we shall define II:{x)(k = 1,2,···, N) using H(z), constant c > 0, real value x and 0 < a < 1,

hex)

=

=

L..J lhi (z) 1:=2 '-I oo + B2C~i+l Cb, z) 4>{b)db

r

100 1~_l{X - b)4>(b)db,x ~

lim I~(x) > 0 "-00

.

CiC~, z) 4>{b)db 8z

-00

+ ~ (h~(z)

(10)

8z

I; (x)

-c,x < XI: -c+H'(x)

219

=

-ex + H(x)

+

a

=

-ex + H(x)

+

a

= =

-c,x < XI -c+H'(x)

100 !I-I (X - b)4>(b)db, x < XI 100 1,-l{X - b)4>(b)db,x ~ XI

00

et 10

+ lim

z-+oo

1;-1 (X - b)tP(b)db,x

et 100 I~ (Xl = -ete < 0

~ Xl

+

lax) > 0

1,"( ... ... ) J,

=

o,x < X,

=

H"(x)+et x

Hence we have Xl < X2. H x ~ Xl then I~(x) I~(x) H Xl < X ~ X2, then 12'( x)

=

100 1;~I(X-b)tP(b)db,

H'(x) ~-c

where Xl is a unique root of each equations

f

1;_I(X-b)tP(b)db=O

Xl> XI-I

I~(x) ~ 1~_l(X)

X

< Xl

And also

F;+1(x) = H' (x) +

=

x"

and

H x 5 XI then I; (x) = 1;+1 (x) = -c H Xl < x ~ Xl+1, then 1:+1 (x) = -c, I; (x) ~ -c Hence if x ~ X;+1, then 1:+1(x) ~ I;(x)

oo H' (z)

=

=

x",k ~ 2

~

I,,(x) - 1"-1 (x) I

I

I~(x) - I~(x)

= et =

1

~ et - h, "

=

Hence we have -Cc + h) < I;(x) - I~(x) ~ 0, x ~ X2 H x > X2, then we have

I

h[l - et"-l + (k - 1)(et - l)et"-2]}

=

~(x) ~-c

h(l- et"-l) I-et l
+

=

ii) Case when k = 2. From Theorem 1 we have Xl < X2 H x ~ Xl then I~(x) = I;(x) = -c H Xl < X ~ X2, then I;(x) -c, I~(x) -c +

(ill) -c ~ I~(x) < h,k = 1

(l~et)2{C(I-et)(1-et"-1)

=

1 , Since H" (x) ~ (ill) i) Case when k 0, limz _ oo H' (x) c + h, limz _ oo I~ (x) h it follows that H' (x) < c + hand -c ~ I~ (x) < h

(ii) 1~+1(x) ~ I~(x),

-

< Xl+1

Hence we get X/

> X"

x ~

I; (Xl - b)tP(b)db

=

Theorem 2 H conditions of Assumption 1 are satisfied, then we have

I

10

~ H' (Xl) + et 100 1;-1 (X 1- b)tP(b)db = 0

I,,(x).

And also for constant h > 0, if limz _ e + h, then we have

b)tP(b)db

00

F;+1 (Xl) = H' (Xl) + et

=

We have the following theorem about

et 100 I; (x -

is nondecreasing and F;+l (x/+1) = 0

Since F;~1 (z) ~ 0 for all z with the possible exception of the points z = a'+1 and z Xl+1, where left-and right hand bounded nonnegative derivatives exist, F;+ 1 ( z) is nondecreasing. Hence F;+1 (z) 0 possesses a unique root xHl that F'+1(Z) attains its absolute minimum z = Xl+1. From the same analysis as in the case when k 1, we get the case when k = I + 1.

Ic+

11'( x) = -e +

ii) Assuming that the case(i) and (ii) hold the integer I, then we have

Hence we have

(i) Xk+l

= -c = -e,

Hence if x ~ X2, then I~(x) ~ I~(x)

> Xl (x :F ai, i = 1,2,"·' m + 1)

F;(x)=H'(x)+et

b)tP(b )db

et

100 I~(x - b)tP(b)db

l

z1

z -

I~ (x -

b)tP(b )db

+ etl~zl 1~(x-b)tP(b)db

x> x",k ~ 2

~ eth J:-Zl tP(b)db -etC J:"Zl tP(b)db ~ eth ~ -etC JoZ-Zl tP(b )db { -etC JZ~%l tP(b)db = -etC

Proof (by induction) (i),(ii) i)Case when k = 1. F~(x) is nondecreasing and F~(X2) O. And also we have

=

Hence we have -etC ~ I~(x) -l~(x) ~ eth,x 220

> X2

-

h(1- ai-I)] 1_ a < fz (x) - 1,-1 (x) ~ 0, I

[C +

a {c(1-a)(1-a' - 1 ) (1- a)2 h[1- a ' - 1 + (1- 1)(a _ 1)al - 2]) I; (x) - 1;-1 (x) ~ a - 1h,x > XI

+ ~

F;

. '() F,"() x ~ 0, lim F, x z ..... oo lim z ..... oo

From the principle of optimality each Ik(X) satisfies the functional equation(12) and optimal policy is given by if x < Xk, order (Xk - x) if x ~ XI:, do not order where XI: is a unique finite root of (14)

=

l

= c + h(1-a 1- a )'

I; (x) = h(11-- a

l )

a

3. APPLICATIONS

it follows that

F{(x) < c +

h(1- a l ) 1_ a

I

-c -< f I (x) <

Application 1 '

Model and Notation

al)

h(l--'---"":'" 1- a

1) The multiperiod model with backlogging of demand will be investigated under general demand without setup cost. The stock relenishment occurs instantaneously. 2) Ordering takes at the beginning of each period and unit purchasing cost c is charged(each period length is t). Let x be the initial stock level and let z be the amount on hand in the initial period a.fter an order is received. 3) Let h and p be the holding and shortage costs per unit per period ( c < p).4) Demand B in the initial period is nonnegative random variable with known distribution 4i(b) and density 0(0 ~ x ~ 1). If b ~ s, then exists a unique positive Tit such that s = g(Tlt)b. Let it be designated by Tit = g-l(slb). For a given demand b, two typica.l situations arise in this system, depending on the relative values of z and b. Then we see that the average amount in inventory 11 (b, z), the avara.ge shortage 12 (b, z) for a given demand b and z ~ 0 are given by (i) b < z

Hence we have

h(1 _ a l )] 1- a < 11+l(x) - fz(x) ~ O,x ~ Xl+l I

- [c+

I

> XI+b then we have

If x

I;+l (x) - I; (x)

=

a

= a

10

00

Io z -

[1;(X-b)-I;_I(X-b)
z /[/;(X-b)-I;_I(X-b)]
l~Z/[/;(X-b)-I;-I(x-b)I
+

a

+ 0 . Jz~z/
~ -(I~:)dc(1- a)(1- ai-I) + h[1 - a ' - 1 + (1- 1)(a - 1)]al - 2]}x JoZ - Z/
>

Hence we have

-

+

I

I

Let H(z)-cx denote the expected one-period loss, given z is amount on hand a.fter an order is placed and let h,(x) denote minimum total expected loss for period 1,2, .. . , k, given x is the initial stock level.

'

From Theorem we have XI ~ XI+l If x ~ XI, then I; (x) = I;+l (x) = -c If XI < X ~ XI+l, then 1;+1 (x) = -c, I; (x) -c+ (x) ~-c Since

.

Now, we shall consider the probabilistic multiperiod inventory model with zero delivery 1a.g, bacldogging of demand and linear purchasing cost [c(y) = c· y].

I

X ~YI

_

I

1,+1 (x) - I, (x) < a h,x > XI+l The proof is complete. ~

ill) Assuming that the case (ill) hold the integer I, then we have

a {c(1-a)(1-a l ) (1- a)2 h[1- a ' + I(a - 1)al - 1 ])

Il(b,z)

= ~ lot

= 221

[z - 9 (~) b] dT

z- G(1)b

12(b, z)

=

(ii) b ~ z =

Il(b,

z)

=

12(b,

z)

0, G(y)

~

=

1"

1'1 [z -

zg-1

= ~

=

1:

g(t)dt

=

C(b,z) =

= tg- 1 (i)

(~) b - z] dT

(i))] b -Z(l- g-1 (i)) [G(l) - G(g-1

z~O

where

= cf (b, z), b < z

=

C1(b,z),O~b
=

C?(b,z),z-R1 ~b«Z-Rd/g(t:)

= = =

Ci(b, z), (z - Rt}/g(t:)

~ b < z/g(t:)

2 to to C4 (b, z), z/g( T) ~ b < (z + R2)/g( T)

to Cs2( b,z ) ,b~ (Z+R2)/g(T)

where we omit the value of CL(b, z). This model is the special case when z is defined for 0 ~ z < 00 and m = 2, al = R},a2 = oo,i = 1,2,k = 1,2,,·· ,nijnl = 2, n2 = 4, hlCz) = z/g(!t), ~(z) = (z + R2 )/g(!t)jhi(z) z - R 1,hi(z) (zRt}/g(!t),h~(z) = z/g(!t),h~(z) = (z +

Hence we have the average cost per period for

C(b,z)

t

z ~ RI

(i) - bG(g-1 (i)), [g

t

= CJ(b,z),b~ (Z+R2)/9(t;)

9 (~) b] dT

tl

.

Ci(b, z), z/g( :) ~ b < (z + R2)/g( :)

CJ(b,z),b ~ z

=

=

R 2)/g(!t )

Cf(b, z) = c(z - x) + h[z - G(I)b] CJ(b, z) = c(z - x)

We shall note that there are a lot of models to be considered as the special case of our problem and we omit the details.

(i) - bG (g-1 (i))] +p [[G(l) - G (g-1 (i)) jb -z (1-g- (i))]

+h [Zg-1

4. CONCLUSION

1

a.

We note that lot of models are considered as special cases of our model. The present author feels that it is interesting to obtain the simple OJr timal policy in the case when H(z) is not a convex function of z.

This model is the special case when z is defined for 0 ~ z < 00 and m = 1, al = 00, i = 1, nl = 1,hHz) = z.

Application 2 The assumptions 1),3),4),and 5) are the same as Application 1. 2) Regular ordering takes at the beginning of each period and purchasing cost c is charged (the period length is t) and the commodity is sold at unit price Tl (Tl > c). Let x be the initial stock level and let z be the amount on hand in initial period after an order is received. 6)The partial returns in the case of surplus at any given to in each period (0 < to ~ t), and the addtional orders in the case of shortage at any given to in each period (0 < to ~ t) are, respectively, allowed less than or equal to the maxima permitted. The maxima permitted are RI and R2, respectively. Let T2(0 ~ T2 < c) and C2(C ~ C2) be the cost per unit of the partial returns and the addtional orders, respectively. Let Il(b,z),12(b,z) and CL(b,z)(i = 1,2,k = 1,2"", nij nl = 2, n2 = 4) be as defined in AJr plication 1 and Section 2. Then we have (we omit the details) o ~ Z < RI to C(b,z) = Cl(b,z),O ~ b < z/g(T)

REFERENCES Kobak, I. W. (1984). Partial rturns in the single period inventory model. lE News 19(2), 1-3. Sorai, M., I.Arizono and H.Ohta (1986). A solution of single period inventory model with partial returns and additional orders. JlMA 37(2), 1O~105.

222