Inventory model with two stocks

production economics ELSEVIER

Int. J. Production Economics 45 (1996) 361-368

Inventory model with two stocks S. Bylka*, J. Komar Institute of Computer Science, Polish Academy of Sciences, Ordona 21, 01-237 Warsaw, Poland

Abstract The model we present generalizes in a very simplified mode a problem that could arise in such different places as a steel factory and a hydro-electric power plant. We deal with a two stock model. The first one of a constant level (we ought to define it), the other one (limited or unlimited) changes according to the random demand. We formulate some necessary conditions for the volume of the first stock and optimal policies which guarantee the minimal expected average cost. An

algorithm to determine the optimum decision rule is also presented. Keywords:

Inventory

model; Deterioration;

(s, S) policy; (Q, S) policy

1. Introduction Most of the classical discrete time inventory models discussed in the literature deal with the situation of a single stock, see [l, 21. Because of capacity limitations a single stock would not always be sufficient. Additional stock is necessary to store excess items. The effect of deterioration is so vital in many inventory systems that it cannot be neglected while analyzing the system, see [3]. In most production systems the objective is to strike a balance between too much inventory, which causes excessive holding costs and too small inventory, which leads to stockouts and poor customer service. In many models of single-item inventory systems with special cost functions, it is well known that an optimal policy exists within the class of (s, S) policies. Tijms [4] discussed the role of (Q, S)

* Corresponding author Copyright 0 SSDI 0925-5273(96)00006-O

0925-5273/96/$15.00

policies in an inventory control system without cost functions but with a restricted order size. The goal was to find the order level to satisfy the following service level constraint: The fraction of a demand satisfied directly from stock on hand is not less than a, where c( is a pre-specified value. We consider a model with two stocks and a constant deterioration rate d in the first stock. The model we present generalizes in a very simplified mode a problem that could arise in a steel factory, see also [S]. In the manufacture of steel large quantities of oxygen are required and it is necessary to install an oxygen plant. This plant supplies most of the requirements but when demand exceeds the capacity supplementary supply is utilized from the cylinders of oxygen at a greater cost per unit than the cost of gas taken from the plant. When the demand rate is less than the production rate of the plant the excess oxygen is lost to the atmosphere, the deterioration rate d = 1. Given the fluctuating nature of the demand we ought to determine the

1996 Elsevier Science B.V. All rights reserved

362

S. Bylka, J KomarJInt.

J. Production Economics 45 (1996) 361-368

Let I, denote the level of M2 in the end of the tth period (also in the beginning of the (t + 1)th period). Knowing I,_ I we decide to increase it to a,. Then comes the demand j > 0 (which is a realization of 5,). The basic system is illustrated in Fig. 1. We assume that all 5, have the same distribution and have the positive mean

size of the plant so that the total cost of supply both from the plant and from cylinders shall be a minimum. Moreover we adapt an algorithm given by Bylka [6]. Expressing demands as multiples of a standard unit, we assume without loss of generality, that random demands take only integer values.

E(L) =

2. Model description

P >

0.

Many variations of this system can be defined. For the purpose of this study we shall restrict our attention to the case described by the following assumptions: Let us denote the probability that 5, takes the real value j by

The demands for a single product in the successive periods t = 1,2, . . . are independent random variables {,. We assume that the inventory position in the stock M2 is received every period and the lead time of any replenishment order is negligible. There is a limitation A on the size of the replenishment order. The demand in the beginning of each period is fulfilled either with the first stock Ml or if the demand exceeds the volume of Ml, then the difference is obtained from the second stock MZ. If the demand is less than the volume of M, then a fraction 1 = 1 - d of the difference is used to replenish the stock M2. The surplus over M, is neglected. We have to determine the volume of the first stock so as the overall cost will be a minimum. You could guess the solution right away but mathematical explanations are a little bit harder.

P{&=j}=cp(j), l

t=l,2

,....

Let Q be the amount of production that comes to the stock Ml. Assume, I,_ i is the inventory level of M2 from t - 1 to t and a,, where 0 6 a, < A + I, _ i, is our decision. If j is a realization of the demand 4, then the level of M2, in the beginning of the next period, will be I, = max{a,-(j-Q), a, + 4Q -3

-b}

ifj>Q, if j d Q and a, + n(Q -j)

B

Fig. 1.

otherwise,

Q B,

363

S. Bylka. J. KomarJInt. J. Production Economics 45 (1996) 361-368

where b > 0 is the limitation of the size of backlog, B is the limitation of M, and A = 00, b = co, B = ocj in unlimited cases. If - b < I, < 0 then we pay for a backlog and this demand will be realized in future. If I, < - b then we lost - b - I, of demands in excess of the stock on hand. So this is a model with restricted backlogging in one of the next periods. We assume the following costs: K, 3 0 the cost of the decision that the volume of stock Ml will be Q. m(Q) 2 0 the one-period cost of production of Q. L(z) b 0 the one-period expected cost of storage in MZ, including backlog penalty cost, if z is the stock on hand before the realization of the demand. K 2 0 the setup cost of a positive replenishment order in M2. c > 0 the price per unit of the ordered product. If in the beginning of the tth period the volume of M2 is I,_ 1 and our decision is a, the cost will be

‘XQ, a,) = m(Q) + KG,

- I,- 11+ +t - 1,-d

+ -wJ, where 6(i) =

1

if i > 0,

i 0

if i = 0.

And our problem is to find such a policy (Q*, f *) called optimal (or T-optimal) which minimize C(Q, f) (or CT(Q, f) respectively). Remark 1. If Q = 0 and A = CC we have the case of the (s, S)-model which was considered (among others) by Scarf [7] and Zheng and Federgruen [S] for the case of infinite time horizon. If i = 0 then we have the situation of deterioration rate as in steel factor [S]. If 0 < 2 < 1 then we have the situation of a hydro-lectric plant. This time our aim is to define the volume of the water storage Ml. We give another interpretation of this case ,4 = 1 in the last paragraph.

3. Optimal policies If b or B is finite, we define L(i) = + co for i < - b or i > B. In this section we make the following assumptions on L L(x) = + 0. Al lim,.,,, A2 There exist u and U such that L(x) is nonincreasing for x < u and non-decreasing for x > u. Let j*(f) be a random variable which is the sum of all replenishments of the stock M2 given policyf, i.e.,

Definition 1. By a policy we mean the pair (Q,f) where Q is the volume of Ml andf= (fi,f2, . . . ) is a sequence such that 0
c’(QJl=&j+WQ)+E

~~&CU-I)-L-d i t=1

+ 4f,v-1)

- It-11 + W(~,-l)

Thus the average expected a policy (Q,f) will be C(Q,f)

= lim inf T-CC

cost connected

C'(Q,f1 T

.

Lemma 1. For every policy (Q,f)

10 + Bdf) + AQt- “gl L 6 1, G 10 +

A(f) + Qt - A

c i;,,

v=l

I

.

for every t = 1,2, . . . .

with

Proof. Let

{vG t It, d Q}, = {vd t I<, > Q).

J(t) = JIM

we have

S. Bylka, J. KomarJInt.

364

J. Production Economics 45 (1996) 361-368

Z, = Zo + Bt(f) + 1. 1

(Q - 5,) -

1

+I

(5, - Q),

AQ I J(t) I + Q I JI (4 I

Take E = (A.Q - ,n)/2 then numbers we have that

5, - (1 - A) 1 4, - (1 - n)lJ(t) I Q, VGJ,

Z, 6 Zo + Bt(f) +

v’6 > 03tvt

t

from

the law of large

Because AQ > p there exists T 3 t* such that for any t > T

lim EL(Z,) = + co. *+*

Thus

Proof. We can assume lo = 0. Let a, = EL(Z,) so for any number M

us denote

V’6 > 03TVt

> T

3 M}.

If L(i) CM),

u1 = max{iI L(i) < M}, and

then

iv) -“F,

We

AQ-:>,(+E.

Y, = Z%(f) +

3

> t*

Lemma 2. Let (Q,f) be a policy such that 1Q > p. For the one-period holding costs in M2 we have

u. = min{il

t

Qt - A i t, 0=l

and this ends the proof of the right inequality. 0 prove the left one in the same way.

c(, 3 MP{L(Z,)

u1

>p\

1, = 10 + Bt + Q I J(t) I + Q I JI (~1I

0=l

t

,

I

- A c c’” - & L t.EJ(I) L

- 2 i

f _--

]vQ+P(f)

otJ,(t)

veJ(t) 1, = 10 + Bt +

f

I

Then we have

JbQt- i tv L.=l

from this ~1,> SO P{y, 2 ul} > 1 - 6 and ends the proof. 0 This M(1 - 6) for 6 > 0. In the sequel we shall use the following lemmas. Lemma 3. Zflim,,,ai= = +cx2.

simple

+co, then lim,+,(CT= lailn)

Lemma 4. If ,I > 0 and L jiilfills Al, A2 then there existsf’ such that C(O,f”)< + co. For each policy (QJ) such that C(Q,f) < + cc we have Q;1 < p. Proof. From Lemmas 2 and 3 it follows that policy (Q, f) where AQ > p we obtain infinite age cost. When Q = 0 and f O is a stationary such that h
using averpolicy H are

365

S. Bylka. J. Komar/Int. J. Production Economics 45 (1996) 361-368

arbitrary

Remark 4. Given Q Theorem 1 characterizes policy f and strengthens Lemma 4 because given L “good” values for Q fulfill the inequality

real numbers.

E(K&f’(I,-

1) - I,- 1) + c(f”(lt-1)

+ L(fOUP

- I,- 1)

1)))

,LL- 2.Q 3 fis

< K + max{I,,

H} +

max

where fif 3 0.

L(x).

h
But this means that costs in all periods are bounded uniformly and this guarantees that C(0, f”) < + ccj. 0 Remark 2. In Lemma 2 we were using only the fact that L is integrable and that its limits in infinity are + cc. In Lemma 4 both assumptions are essential. Remark 3. We have assumed if ,l = 0 it suffices to consider able 5” =

;_Q

that A > 0. However a new random vari-

;; : 2;’

i and use the model with one stock. Lemma 4 gives a necessary condition for Q. Now we give also a necessary condition for a policyf: Let

4. The algorithm In this section we assume that 2 = 1, A = + cx and that the demand has a discrete distribution i.e. P{t,=i}=cp(i),

Theorem 2. Let [, be as in Theorem 1, let it additionally have a discrete distribution, and let 1 = 1. Moreover let there be no restrictions on replenishments. Then there exist optimal policies with decision functions satisfying h(i) =

Proof. We can assume I, = 0. It follows from our assumptions and the law of large numbers that the sequence E(I,) is bounded: +-Q+i

i

$++Q+

i v=l

lJ=l

From

this

S, for i i

~+~Q+p

Proof. We can take u = b, U = B if either backlogs or M2 are limited. The quantity V,(i) which expresses the total expected cost of an optimal Tperiod policy with the initial level i of the second stock M2 satisfies V,(i) = 0,

;.

I&)

= L(i) + f

Because E(I,) is bounded

i-m

1

t

which ends the proof.

cp(j)vr-,(i

-j

+

Q),

j=O

~Fjy+, CKW - 4 + &+41. . .

From our assumptions Al and A2 about the function L we can conclude that there are two integers u < U such that

t

EMf &U - Q < lim -

i < ut,

fori>Ut,

for some u, < U,.

V,(i) =

E(IJ --Q++<

,....

To obtain an optimal policy, we formulate an algorithm, which consists of solving problems of increasing lengths, using standard dynamic equations. As a solution of a finite T-period problem with a given Q we mean a sequence (Q; fi, . . . ,ft) such thatf,, . . ,fi are used as decision functions in 1, , T period, respectively.

B/ = ii”, (EP,- l(f)lr). Theorem 1. Assume 5, is a sequence of independent random variables, having the same distribution and El, = p > 0 and L fulfills Al and A2. If C(Q,f) < co then 1.~ - Q < /3f < n - AQ.

i=O,1,2


L(i) > min L(j) + K

for i < u,

j20

L(i) > min L(j) j>O

+ 2K

for i > U,

366

S. Byika, J, Komarllnt.

J. Production Economics 45 (1996) 361-368

and L(i) is monotonic for i < u and i > U. It is easy to check that for every t the function L,(i) is monotonic for i < u and we have L,(i) > min L,(j) + K j >0

for i > U.

That proves the theorem.

3. Construct Z, as the set of decision functions f such that q*(i) i=u,

= .st+ . . ..U

r,(f(i))

+ KG(f(i)

-

i)

for

and

0

f(i)

=

In the case of the limited M2 or

f(u) 1

QO} we can adapt an algorithm given in [6]. It is enough to know two sequences of functions defined for each i such that u 6 i < U:

fori < 4 for U < id B.

In general, we may ask if there exists a sequence (gJ such that

and V,(i) - gT converges as T + co. Note that for our algorithm

r,(i) = L,(i) - L,(u), qdi) = v,(i) - v&4

qT(i) = v,(i) - gT

For the co-period problem let us define

where gT

as the minimal expected average cost. If these sequences Y,, qt converge to Y, and qm respectively, then we can also find an optimal stationary policy for the infinite period problem with average cost as a criterion. For connections between stationary optimal polices for infinite horizon time and optimal long time horizon policies, see [9]. The Algorithm

Step 0. To start computation one has to calculate r(i) = L(i) - L(u) for i = u, . . . , U and has to set qo(i) = 0 and go to Step 1. Step t. For each t > 0 it is performed as follows: 1. Compute r,(i) for i = u, . . . , U:

=

TL(u) -

f

&,

t=1

and SC0= L(u) - s, where E,

=

.

Remark 5. In the unlimited case u and U are determined by the cost functions, see assumptions Al and A2. If b and B are finite numbers then no assumptions on L is needed. It is enough to take u=b and U=B. If b<+co, then there is no possibility to assume A = + cc and Theorem 2 is true.

cP(jk- 16 -j + Q).

r,(i) = r(i) + f j=O

5. Numerical

2. Compute E, and qt(i) for i 6 U: E,=

-K’-

min r,(j) u$jGU

where K’ = K if K is finite and K’ is an arbitrary number otherwise. We set qt(i) = 0 for i < u and for 1 = 24,. . . ) U q*(i) = E, + min r,(i),

min r,(j) + K i
. I

examples and some conclusions

By way of illustration, we apply the algorithm to the following simple example. We set b = + cc, B = 5, ~(1) = (p(4) = 3, K = 10, L(i) = - 6i for i < - 2 and we take Table 1 as valid. It is enough to set u = - 2 and U = 5. For Q = 0 we compute Table 2. For every t the set Z, is a set of decision functions of (s, S)-type. gm = L(u) - E, = 15 - 6 = 9.

S. Bylka, J. KomarJInt.

J. Production

g,=15-7$$=7&. The stationary policy (1; ( - 1,0)) is optimal for the infinite-time-horizon problem.

model Table

A control rule of (s, S)-type assumes that there is no limitation on the size of the replacement order. In practice this assumption is not always satisfied. In such a case Tijms [4] discussed the role of the following (Q, S) policies: at each review the inventory position is ordered up to the level S provided that the order size does not exceed Q; otherwise, the amount Q is ordered. Here Q is a given number, where it is assumed that Q > p. This prevents the

6. Let us notice that when ,l = 1 our with two stocks could be replaced by

1

i L(i)

-1

-2 15

9

0 1

1 3

2 6

3 9

4 12

361

45 (1996) 361-368

a model with one stock and the random demand 5, - Q similary as in Remark 3. But in this case it can take negative values (but is still bounded from below). This small change unables us to use standard ways of proofs concerning the existence of optimal policies.

For Q = 1 and Q = 2 we have, respectively, Tables 3 and 4. IfQ=ltheng,=15-9=6andifQ=2then

Remark

Economics

5 21

Table 2 i

-2

f.1 41 r2 42 r3 q3 r4 q4 r5 45 r6 q6

0 0 0 0 0 0 0 0 0 0 0 0

qm

0

-1 -6 -2 -6

0

0 - 14 - 10 - 15 -8 - 14 -8 - 14 -8 - 14 -8 - 14 -8

1 - 12 -8 - 17 - 10 - 16 - 10 - 16 - 10 - 16 - 10 - 16 - 10

2 -9 -5 - 13 -6 - 14 -8 - 14 -8 - 14 -8 - 14 -8

0

-8

- 10

-8

0 -6 0 -6 0 -6 0 -6

3 -6 -2 - 94 - 2: -9 -3 - 10 -4 - 10 -4 - 10 -4 -4

4 -3 1 -9 -2 - s+ - 24 - 84 - 2f -9 -3 -9 -3 -3

5 6 10 - 8: - l’i 0 6 -k

Cl =4 ZI = {( - LO)) Eg= I 22 = ((0, 1)) tzj = 6 z, = {( - 1, lb Ed= 6

5i

24 = {( - 1, lb (0, 1)) es = 6

5s

z5 = it - 1, lb (0, 1)) Ed= 6

5:

z, = i( - 1, lb (0, 1)) E, = 6

5f

z,

-a -t

= it - 1, lb (O,l))

Table 3 i rl 41 rz q2 r3 q3 r4 q4 r5 q5 r6 q6

qm

- -2 0 0 0 0 0 0 0 0 0 0 0 0 0

-1 -6 -2 -7 0 -6

-

0 14 10 19 10 19

1 - 12 -8 - 16 -1 -154

0 -- 10 19

-6

0

-

0

- 10

0 -6 0 -6

10 19 10 19 10

- -6: 1st - 6+ - 154 -4 - 15& -6h -6

2 -9 -5 - 124 - 3: -lo$ 1 1; -3 -9i% -A -9& -3zi 0

3 -6 -2 - 12 -3 - 124 - 3: - 124 - 33 - 124 - 34 - 12% - 3E -4

4 -3 + -

1 64 3: 42 44 4Q

f4& -3+i + 5& -3% + 5% 6

(0, 1))

5 6 10 - 84 - 1: 5 14 - 12+ 214 16if 254 17% 26% 30

E, =4 z1 = i( - LO)) E2= 9 22 = NO,0,) Ej = 9 -G = ((0,O)j E‘$= 9 24 = {(O, 0,) Es = 9 z, = ((0,0,) Eg= 9 zs = {no)) E, =9 zcc = {CO,0))

368

S. Bylka, J. Komar/Int.

J. Production

Economics 45 (1996) 361-368

Table 4 i rl

-2

-1 -6

41 r= 42

0 - 18 0

0

- 3# - 11 -31945

0 ~ 14

1 - 12

-9

2

- 10 - 17s - 10

-7& - 14% -7&

-2& -9% -2&

inventory position drifting to minus infinity. In the case with limited backlog, see Remark 5, we propose to use our Algorithm to determine “the fraction of demand satisfied directly from stock on hand” is not less than CC,where CIis a pre-specified value (e.g. a = 0.95). It is enough to set u = 6, U = S, K = ccj and the function L as follows:

3 -6

4 -3

8% l& 8%

2% 13% 21%

5 6 35% 28g 35%

E, = 78 2, = (( - 1,0,;

the machine is subject to failures and times between failures are stochastic. As a good policy they used a (Q, S) control, in which the machine is turned on when the inventory in M, falls below Q and turned off when it rises above S.

References L(i) = f

q(j)G(min{i

-j

+ Q, S -j})

.i=O

where G(1) = 1 for negative i and G(i) = 0 otherwise. Additionaly we set K’ = 1 and we redefine Y, in the following way: r(i) = r(i) + f

q(j)

j=O

xq,_i(max{

-b,min{i-j+Q,S}})

or (closely to Tijms’s case) r,(i) = r(i) + F

q(j)

j=O x

qt_i(max{

- b, min{i -j

+ Q, S -j}>).

In this case the expected average cost of an optimal policy is exactly the frequency of demand nonsatisfied directly from stock on hand. Other more complicated policies could be devised. Let us consider such a problem where Ml can be controlled by turning a production process on and off. In such case a content of M2 can be seen as a buffer inventory. Hopp et al. [lo] investigated a production system consisting of a machine (Ml) and a tank M2. They assumed that

Cl1 Wagner,

H.M., 1969. Principles of Operations Research. Prentice Hall, New Jersey. PI Bensoussan, A., Crouhy, J. and Proth, J.M., 1983. Mathematical Theory of Production Planning, North-Holland, Amsterdam. c31 Pakkala, T.P.M. and Achary K.K., 1992. Discrete time inventory model for deteriorating items with two warehouses. Opsearch, 29(2). ideas are often M Tijms, H.C., 1989. Good probabilistic simple, in: 25 Years of Operations Research in The Netherlands, CWI Tract 70. S., 1963. Mathematical Techniques of Operac51 Goddard, tional Research. Pergamon, Oxford. in dynamic inC61 Bylka, S., 1980. Policies of (multi-S)-type ventory problem, Proc. 1st Internat. Symp. on Inventories, Budapest. Publishing House of the Hungarian Academy of Sciences, pp. 3 19-329. c71 Scarf, H., 1960. The optimality of generalized (s, S) policies in the dynamic inventory problem, in: Mathematical Methods in Social Sciences, Chapter B. Stanford Uni. Press, Stanford. PI Zheng, Y.S. and Federgruen, A.. 1991. Finding optimal (s, S) policies is about as simple as evaluating a single policy. Oper. Res., 39: 6544665. results c91 Hordijk, A. and Tijms, H.C., 1974. Convergence and approximations for optimal (s, S) policies. Mgmt. Sci., 1432-1438. Cl01 Hopp, W.J., Pati, N. and Jones, Ph.C., 1989. Optimal inventory control in a production flow system with failures. Int. J. Prod. Res.. 27: 136771384.