the optimal inventory policy for batch ordering

The Optimal Inventory Policy for Batch OrderingAuthor(s): Arthur F. Veinott Jr.Source: Operations Research, Vol. 13, No. 3 (May - Jun., 1965), pp. 424-432Published by: INFORMSStable URL: http://www.jstor.org/stable/167806 .

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TrHE OPTIMAL INVENTORY POLICY FOR

BATCH ORDERINGt

Arthur F. Veinott, Jr.

Stanford University, Stanford, California

(Received August 20, 1964)

We consider a single product dynamic inventory problem in which the demands in each period are independent and identically distributed random variables. There is a constant lead time, a discount factor 0 <a <1, a unit ordering cost c and an expected holding and penalty cost function L (.) for which -[c(1-a)y+L(y)] is unimodal, and total backlogging of unfilled demand. In addition each order for stock must be in some nonnegative integral multiple of Q, a fixed positive constant. It is shown that the (k, Q) policy is optimal for the finite and infinite period models. With the (k, Q) policy if the initial inventory on hand and on order in a period is less than k, an order is placed for the smallest multiple of Q that will bring the inventory on hand and on order to at least k; otherwise, no order is placed. The optimal value of k is easy to compute and is the same for the finite and infinite period models. The results are generalized to the case where the demand distributions and cost functions vary over time. The (k, Q) policy is not optimal in general for the case where there is also a fixed charge for placing an order.

V'1T E CONSIDER a dynamic inventory model in which the demands V VD1, D2, for a single product in periods 1, 2, are independent,

nonnegative random variables with common known distribution function

MODEL FORMULATION

At the beginning of each period the system is reviewed. An order may then be placed for any nonnegative integral multiple of Q, a fixed positive number. Thus orders must be placed in multiples of some standard batch size, e.g., a case, a barrel, or a truck load. An order placed at the beginning of period i( = 1, 2, ---) is delivered at the beginning of period i+X where X is a known nonnegative integer. When the demand exceeds the inventory on hand after receipt of incoming orders in a period, the excess demand is backlogged until it is subsequently filled by a delivery.

Let xi denote the inventory on hand and on order prior to placing any order in period i. Let yi denote the inventory on hand and on order after ordering in period i. It is possible for xi and yi to be negative indi-

t This work was supported by the National Science- Foundation under grant number GP-1625.

424



Optimal Inventory Policy 425

eating the existence of a backlog. In view of the backlogging assumption we have xi=yi_-Dii(i=2, 3, *. ).

At the beginning of period i the manager is assumed to have observed the vector

Hi=(xi, ... xi, yi, I ,yi-1Di, .. Di,_) I

representing the history of the process up to the beginning of period i. He bases his ordering decision in period i upon Hi.

An ordering policy is a sequence Y= (Y}, Y2, *. ) of real valued Borel functions to be used as follows. At the beginning of period i after having observed the past history Hi, the manager orders YF(Hi) -xi. This quantity is, of course, some nonnegative integral multiple of Q.

Three types of costs are considered: ordering, holding, and penalty. Assume that the cost of ordering z( >0) units is c z with the cost being incurred at the time of delivery of the order. Let g(y, t) denote the holding and penalty cost in a period when the stock on hand (not on order) after receipt of orders is y and the demand during the period is t. We assume that q( *, * ) is a Borel function of its arguments. Finally, let a (O ?< a ) be the one period discount factor.

Let Vl( -) denote the n-fold convolution of b(.). b?(D) is the degenerate distribution whose entire mass is concentrated at the origin. We assume that

L(y)- I (y-u, v) d'(u) dJ(b )

exists and is finite for each y. Assume also that L(y) is continuous in y. L(y) may be interpreted as the conditional expected holding and penalty cost in period i+ X given that the amount of stock on hand and on order after ordering in period i is y. Let

G(y) = (1-a)cy+L(y).

We assume that G(y)-* co as ly c co and that -G(y) is unimodal. The assumption that - G(y) is unimodal can be realized in at least two

ways. First, if g(y, t) is convex in y for each fixed t, then G(y) is convex and -G(y) is unimodal. Alternatively if g(y, t)=g(y-t), if D has a density so that is a Polya frequency function,[101 and if (1- a)c+g'(z) is piecewise continuous and changes sign once from - to + as z traverses the real line from - oo to + oc, then -G(y) is unimodal. To prove this one uses the fact that P+' is a Polya distribution because the convolution of two Polya distributions is also a Polya distribution. [1] Next one uses certain sign variation diminishing properties of Polya distributions. See references 4 and 5 for details of this approach and numerous examples.

It is convenient now to obtain a simple expression for the (uncon-



426 Arthur F. Veinott, Jr.

ditional) expected holding and penalty cost incurred in period i+X. By virtue of the backlogging assumption, the amount of stock on hand after ordering at the beginning of period i+X is yz-Dig- . -Di+xi,. Since the Dj are independent random variables and since yi is a function only of Di, .. Di-,, we see that yi and Di, -**, D+?x are stochastically independent. Thus the expected holding and penalty cost in period i+X is

E[g(yi-Di- -Di+x-~, Di+x)] =E{E[(yi-Di-* -Di+x1, D+x)JyJi]} =EL(yi).

Denote byf (xjI Y) the expected discounted cost over periods X+ 1, ,

X+n all discounted to the beginning of period X+ 1 when xi is the amount of stock on hand and on order at the beginning of period 1 and Y is the ordering policy followed. We have

fn(x1IY) E== ai'-[cE(yi-xi)+EL(yi)] -a &cE[n +1- Z=Zf'+1 Di] = Z &'-EG(yi) + [-cxi?l.uc 0 i-Z4 a&+a'cPXI],

where we assume ,u= fSt d3(t) < o. The second equality follows by using the fact that xi = yi- -Di and regrouping terms.

Note that we have assumed that stock left over at the end of period X+n can be salvaged with a return of the initial purchase cost. Similarly, any backlogged demand remaining at the end of period X+n can be satisfied by a purchase at this same cost.

Since the bracketed term in the last expression is not affected by the choice of Y, we find it convenient to redefine fn by setting

fn(x1jY) = i.1 at'EG(yt) (1)

Observe that this revised formula is that which would be obtained if the original model were changed by setting c equal to zero and replacing L(y) by G(y).

We may now formulate the n-period problem as that of finding an ordering policy Y* such that

fn (xi(Y) = minyf (xi IY).

If such a policy exists, it is termed optimal. For the infinite period model, we distinguish two cases. First, if

O<<1, we may set AX1 I Y) =liMn---fn (XI I Y) m

which exists (with + oo allowed as a value) since G is bounded below. Then we seek a policy Y*, termed optimal, such that

f(xi IY*) =minyf(xi IY).




If a =1, then f(x I Y) as defined above is usually infinite for every feasible Y. Hence we consider instead,

a (xi I Y) = libs( 1/n)f,, (xlI Y),8

which exists with + oc allowed as a value. We say that Y* is optimal if

a,(xi IY*) = minya(xi IY).

OPTIMALITY OF THE (k, Q) POLICY

OUR OBJECTIVE in this section is to prove that the (k, Q) ordering policy is optimal for the n-period and infinite period models with 0?_ a <1 3,6,81

The (k, Q) policy may be described as follows. If at the beginning of a

G(y)

r R2 - @ _ = = 0

k-2Q k-Q k Y k+Q y

Figure 1

period the stock on hand and on order is less than k, order the smallest integral multiple of Q that will bring the on-hand plus on-order level to at least k (and probably higher); otherwise, do not order. The same pa- ramneter k is used in each period.

The parameter k is chosen as follows. Let g denote an absolute minimum of G(y). Then k is anynumber forwhich k?!7?k+Q and G(k) = G(k+Q). These definitions are illustrated in Fig. 1. Note that if the initial inventory xi on hand and on order in period i lies in the region Rn (n=O, 1, ***), then the (k, Q) policy calls for ordering nQ units in period i. THEOREM 1. The (k, Q) policy is optimal for the n-period and infinite period models.

Proof. We first establish the result for the n-period model. Let Y be any feasible policy other than (k, Q). Let Y1, Y2, * - - be the inventory levels associated with Y and Y1', Y2', * - - the inventory levels associated with (k, Q) when the demands are Di, D2,

Since each policy calls for orders in multiples of Q and since unsatisfied demand is backlogged, yi'- yi is an integral multiple of Q for each i. Thus




if k-<yi'<k+Q, then either yi=yi' or lyi-yi'l _ Q. In the latter event G(yi)>G(k)_G(yi'). On the other hand, if k+Q<yi', it follows that no orders were placed with (k, Q) in the first i periods. Thus, k+Q<yi'-<yi and G(yi')! <G(yi). Finally, it is impossible to have yi'<k when following (k, Q). Thus, in all cases

G(yi) _~ G(yi') * (i= 1, 2, . . .

Hence, fn (xl| Y) fn (xlk, Q) . (2 )

This proves the theorem for the n-period model. The theorem follows for the infinite period model where 0? a <1 by letting n-o oc in (2), and where a= 1 by multiplying (2) by 1/n and taking Lim on both sides of the inequality.

A review of the proof of Theorem 1 reveals that we can weaken the hypothesis of unimodality of - G(y). We really require only that G(y) ? G(k) for k < y < k+ Q, that G(y) > G(k) for y < k, and that G(y) be non- decreasing in y for y_ k+Q.

We remark that if unsatisfied demand is not backlogged, e.g., if it is lost, our proof will not work. This is because (in the notation of the above proof) we can not be sure that yi'-yi is an integral multiple of Q.

We emphasize that Q is fixed and is not subject to the control of the inventory manager in Theorem 1. If to the contrary Q(>O) can be controlled, one may ask what is the optimal choice of Q. As the model stands, there does not ordinarily exist an optimal choice of Q. To see this, suppose for simplicity that xl< M, a=1, and G(y)>G(y) for y7=~j. Then

G (y) ' ( 1/n)fn (xi Ik, Q) _ G (k).

Thus since k Z 7 as Q\O, we have

limQ \,o( 1/n)f,,(xi k Q) =G (g) < ( I /n)fn,(xi I k, Q) .

This situation can be understood when one realizes that when Q(>O) is chosen very small, one is in effect approximating the situation in which orders for any nonnegative quantity of stock are permitted. And if orders in any (nonnegative) amounts are permitted, it is well known [25,71

that the optimal policy is a base stock policy, i.e., if xi < g at the beginning of period i, order g- x; otherwise, do not order in period i. Indeed obvious modifications of the proof of Theorem 1 give a simple proof of this fact.

If there is a fixed set-up cost K( >0) incurred every time an order for a positive multiple of the fixed batch size Q is placed, the (k, Q) policy is not optimal in general. The reader may convince himself of this fact for the single period model by considering the graph of G(y) in Fig. 1 with a large value of K. If K>0 and G(y) is assumed to be convex, then for the




one period model the optimal policy is characterized by two parameters, s and k, as follows: if xi <s, order nQ where n is the smallest nonnegative integer for which xi+nQ ~k; otherwise, do not order. The parameter k should be chosen as before; s( ?k) should be chosen so that G(s) =

K+G{ s+ [(k - s) /Q]Q} where [z] is here the smallest integer not less than z. It remains to be seen whether a policy of this type is optimal for the multi- period model.

Suppose we modify the above problem by permitting orders to be placed in any nonnegative amounts. We retain all other assumptions including the fixed positive set-up cost and the convexity of G(.). For such a model it is pointed out in references 3, 6, and 8 that it would be reasonable to restrict one's attention to the class of all (k, Q) policies and attempt to optimize over this class. In this optimization k and Q would be free to vary and k would not need to be chosen so that G(k) ==G(k+Q). However, for the model under consideration it is known[9] that the optimal policy is of the (s, S) type: i.e., if xi<s, order S-xi in period i; otherwise do not order in period i. Thus, even though some (k, Q) policy might perform quite well, there is at least one (s, S) policy that performs even better. Therefore, it would seem to be preferable to use the best (s, S) policy rather than the best (k, Q) policy.

The above remarks persuade us that the principal importance of the (k, Q) policy lies in situations in which Q is a natural fixed batch size. In cases where the order quantity need not be in multiples of some batch size, other ordering policies will ordinarily be preferred to the (k, Q) policy.

THE NONSTATIONARY CASE

IN THIS section we generalize our results to the case where the demand distributions and cost functions may change over time. In particular let (i( ( ) denote the demand distribution in period i and let (D *) denote the distribution of the sum of the demands in periods i, i+ 1, ., j. (Di,i-1 ) is the degenerate distribution with entire mass concentrated at the origin. Let ai be the supremum of those numbers a for which Di(a) = 0. Clearly, ai>O, i=1,2,)

Let gi(y, t) be the holding and penalty cost function, ci be the ordering cost, and ai the discount factor for period i, (i= 1, 2, * * * ). Assume that

0 0

Gi (y)=_(ci-a ci+1)y+ I gi+A(y-u, v) day i~+;<_(u) d~lg+A(v)

exists, is finite, and is continuous in y for all y and i. We also suppose that Gi(y)- x as IyI c? and that -Gi(y) is unimodal for each i.

Let i=- 1 and fitI^ - laj, i> 1. Let deno-te an absolute mlimuimu of Gi (y). We assume that j- l I3i Gi ( ji) I < X and Oi _ O. i > 1.




Except for the above changes we shall suppose that the model structure is as given in the opening section. Following the argument there we then see that an appropriate objective function for the infinite period model is

f(xiIy) =z 3i- = iEGi (yi).

There is no need to consider the n-period model for the present case since it is the special case of the infinite period model in which G (y)- 0, i>n.

A natural generalization of the (k, Q) policy for the present model is to choose numbers ki such that ki Pi ki+Q and Gi(ki) =Gi(ki+Q) for each i, and then to use (ki, Q) in period i( = 1, 2, ** ). We call this the generalized (k, Q) policy. THEOREM 2. The generalized (k, Q) policy is optimal if

ki - ai<ki+,. (i-In) 2, ... ) (3)

Proof. The proof is identical to that for Theorem 1 where we replace k by ki and G by Gi. The condition (3) is required in the proof to ensure that (in the notation of the proof of Theorem 1) if ki+Q<ysi, then no orders were placed with the generalized (k, Q) policy in the first i periods.

We now give some conditions under which (3) holds. LEMMA 1. If { nt} is a sequence of numbers for which

Gi'(y) 1G'+l(y-Hi) for all i and y, (4)

then ki- i k ki + for all i. Proof. Clearly Gi(z+Q)-Gi(z) is nonpositive for z<ki and is non-

negative for z> ki. But by (4) and the definition of ki, for z <ki. RZ+Q Z+Q

O_!! Gi(z+Q) -Gi(z) = Gil(t) dt>' Gc+1(t -i) dt

= Gili(z +Q - 7) - Gi+i (z-a)

Thus, k-ikic+ as claimed. This lemma reduces the problem of establishing (3) to that of es-

tablishing (4) provided that ri < ai for all i. We now give some conditions under which (4) holds.

It is convenient to introduce the notion of stochastic ordering between two demand distributions, 4) and T say. We say that 4) is stochastically smaller than T and write PcCT if cb(t) ?'4'(t) for all t. This means that every percentile of ib is less than or equal to the corresponding percentile of T. We say that the sequence {4ah} of demand distributions is stochastically increasing if 4I CJ2 C03 C - * ' . We say that the sequence {c(i} is stochastically X-increasing-in-translation if Qi*C4y+x+l for all i where Di*(t) fX(t-ai) for all t and i.




Let w(y, t) be a real valued function of two variables. Let Diw(y, t) denote the partial derivative of w(, ) with respect to the ith variable. Let Dijw(y, t) =Dj[Djw(y, t)].

The following two sets of sufficient conditions for (4) to hold will be established in a subsequent paper where several examples will also be given.

1. gi(Y, t) is convex in y for each fixed t and i; ci-aici+l+DI gi(y, t) is non- increasing in i for each fixed y and t; D21 gi(y, t) 0, ylt; (1CI2CfD3C (For this case qi=0 for all i.)

2. gi(y, t) =gi(y-t) with gi(z) being convex in z for each i; -a i ci++git(z) is noninereasing in i for each fixed z; fDcAi?i+x+ for all i. (For this case Ad =ai for all i.)

Both of the above sets of conditions imply that Gi(y) is convex in y. There is one special case, however, in which we can relax this hypothesis. Specifically, suppose that the It are translates of a common distribution functiont n, i.e., b(t)=b(t-vi) for all t and some i O (i=1,2, .) .

Suppose also that gi(y, t)=g(y-t), ai=a, and ci=c. Let G(y) denote the (composite) expected holding and penalty cost function where D is the demand distribution and the cost factors are as given above. We suppose that -G(y) is unim-odal and denote by (k, Q) the policy defined for G(y) in the preceding section.

Under these assumptions, it is easy to see that

Gi'(y) = G'( --qi - ,, *- 7+x) . (i = I,~ 2, ...*

Hence - G (y) is unimodal. Also by an argument analogous to that used in establishing Lemma 1, we have the useful relation

ki =k+ ni+ +,qi+x. (i- 1, 2, ..

In this case (3) holds since

Thus the generalized (k, Q) policy is optimal.

REFERENCES

1. ARROW, K., S. KARLIN, AND H. SCARF (eds.), Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, Stanford, California, 1958.

2. BELLMAN, R., I. GLICKSBERG, AND 0. GROSS, "On the Optimal Inventory Equation," Management Sci. 2, 83-104 (1955).

3. HADLEY, G., AND T. M. WHITIN, "A Family of Inventory Models," Management Sci. 7, 351-371 (1961).

4. KARLIN, S., "One Stage Models with Uncertainty," Chap. 8 in reference 1, pp. 114-118.




5. , "Optimal Inventory Policy for the Arrow-Harris-Marschak Dynamic Model," Chap. 9 in reference 1, pp. 139-142.

6. AND A. FABENS, "A Stationary Inventory Model with Markovian Demand," Chap. 11 in Mathematical Methods in the Social Sciences, 1959, K. ARROW, S. KARLIN, AND P. SUPPES (eds.), Stanford University Press, Stanford, California, 1960.

7. AND H. SCARF, "Inventory Models of the Arrow-Harris-Marschak Type with Time Lag," Chap. 10 in reference 1.

8. MORSE, P., "Solutions of a Class of Discrete-Time Inventory ProbleM-s," Opns. Res. 7, 67-78 (1959).

9. SCARF, H., "The Optimality of (S, s) Policies in the Dynamic Inventory Problem," Chap. 13 in K. ARROW, S. KARLIN, AND P. SUPPES (eds.), Jlathe- matical Methods in the Social Sciences, 1959, Stanford University Press, Stanford, California, 1960.

10. SCHOENBERG, I., "On Polya Frequency Function I. The Totally Positive Functions and Their Laplace Transforms," J. D'Analyse llMathematiqiie 1, 331-374 (1951).



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