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Inventory control and investment policy
Int. J. Production Economics 81–82 (2003) 309–316
Inventory control and investment policy E.V. Bulinskaya Faculty of Mechanics and Mathematics, Moscow State University, 119992 Moscow GSP-2, Russia
Abstract The idea underlying this paper is to consider inventory control as part of a wider class of economic problems, namely financial risk management. Only discrete-time models are investigated here. In the framework of the Cox–Ross– Rubinstein model of a financial market a combined inventory replenishment and investment policy is obtained. Company solvency is studied as well. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Inventory control; Financial market; Cost approach; Reliability approach; Solvency
1. Introduction The aim of this paper is to draw attention of inventory researchers to new perspectives opened by treating inventory problems within the wider scope of financial risk management. In the year 2001 we celebrate the half century anniversary of the pioneering work of Arrow et al. (1951). Along with the seminal papers of Dvoretzky et al. (1952, 1953) it formed the foundation of modern inventory theory (or, more strictly, its cost approach). It is interesting to recall that the 1971 Nobel Prize winner, the Honorary President of ISIR Kenneth Arrow, has also greatly contributed to the development of financial economics. The classical (AHM) inventory model and its various modifications (see, e.g. Chika! n, 1986) usually took into account inventory replenishment and holding costs, as well as a shortage penalty. Although some authors (see, e.g., Rustenburg E-mail address: [email protected] (E.V. Bulinskaya).
et al., 1999) considered budget restrictions, for the most part the cash amount available for inventories was assumed to be unlimited and the decisions were aimed at the minimization of expected accumulated costs (since demand was supposed to be random). There was a lot of discussion about capital tied up in inventories (see, e.g., Pujawan and Kingsman, 1999). We mention in passing that, in an attempt to avoid the undesirable effects of inventory holding, a new just-in-time concept has been introduced in inventory theory and practice (see, e.g., Groenevelt, 1993). One could also say that implicitly the existence of a financial market was taken into account from the outset by discounting future expenses. Now the time has come to use explicitly and in full the powerful tools of financial mathematics by combining inventory control with investment policy. For simplicity we consider below only discretetime models. Section 2 contains some preliminary results. First we point out the effect of a budget constraint in the classical inventory model. Then
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we drop the nonnegativity restriction on inventory order, allowing for seasonal sales. The one-period demand is assumed to be a random variable with a finite mean (taking negative values as well in order to incorporate the possibility of customers returning bought product or remanufacturing). In Section 3 we introduce the Cox–Ross– Rubinstein model of a financial market (for more details see the fundamental book by Shiryaev (1999)). Assuming an inventory replenishment policy to be fixed we establish the optimal investment strategy. Using a reliability approach we treat the company solvency problem. In the framework of the cost approach we obtain the optimal inventory replenishment and investment policy. Almost all proofs are omitted due to lack of space. In Section 4 we draw conclusions and outline further research directions.
with
2. Preliminary results
Moreover, y% ¼ limn-N yn is the solution of the following equation: Z y jðsÞ ds ¼ ðp c1 þ ac1 Þðp þ hÞ1 :
To illustrate the influence of various constraints on optimal inventory control we consider the simplest case. The replenishment order, made at the beginning of a period, is delivered immediately and can be used to satisfy the arising demand. The inventory left is stored for future use and a penalty is paid for a stock-out. Let the demand process be described by a sequence of independent identically distributed random variables fxi giX1 : We suppose xi to be nonnegative and to have a density jðsÞ > 0; s > 0; as well as a finite mean. All the cost functions are assumed to be linear, their slopes being, respectively, c1 ; the ordering cost per unit, h; the holding cost per unit per period, and p; the shortage penalty per unit per period. Unsatisfied demand is backlogged and hence x can be negative. We also introduce the discount factor a; 0oao1: Denote, as usual, by fn ðxÞ the minimal expected n-period cost, x being the initial inventory level. Then we have the following dynamic programming equation for nX1: fn ðxÞ ¼ c1 x þ min Gn ðyÞ; yXx
ð1Þ
Gn ðyÞ ¼ c1 y þ LðyÞ þ a
Z
N
fn1 ðy sÞjðsÞ ds ð2Þ 0
and LðyÞ ¼ h
Z
y
0
þp
ðy sÞjðsÞ ds Z
N
ðs yÞjðsÞ ds:
ð3Þ
y
The y-value, say yn ðxÞ; minimizing the righthand side of (1), is called the optimal ordering level. It is well known (see, e.g., Bulinskaya, 1990) that the following result is valid. Theorem 1. If p > c1 ; then there exists an increasing sequence fyn gnX1 such that ( yn for xpyn ; yn ðxÞ ¼ x for x > yn :
0
As in Bulinskaya (1990) the following result can be proved. Theorem 2. The stationary inventory policy determined by a single critical level y% is asymptotically optimal. (An inventory policy is said to be asymptotically optimal if the long-run average cost per period, under this policy, is equal to limn-N n1 fn ðxÞ with fn ðxÞ defined by (1).) Now let u be the cash amount available each period for inventory replenishment. Then the corresponding minimal costs f%n ðxÞ satisfy, instead of (1), a similar relation with minimization carried out over the interval ½x; x þ uc1 1 : Theorem 3. Under the assumption of Theorem 1 the optimal ordering policy y% n ðxÞ is determined by the
E.V. Bulinskaya / Int. J. Production Economics 81–82 (2003) 309–316
increasing sequence namely 8 1 > < x þ uc1 y% n ðxÞ ¼ y% n > : x
fy% n gnX1 of critical levels, for
xoy% n uc1 1 ;
for for
y% n uc1 %n; 1 pxpy x > y% n :
ð4Þ
Furthermore y% 1 ¼ y1 and y% n > yn ; n > 1: The proof, being standard induction, is omitted. The limit behaviour of y% n ; as n-N; has been studied as well. Thus we see that the budget constraints compel us to build up inventories. Next we suppose that xi can take negative values and include in the model the sales at the beginning of any period, the price being c2 per unit. In this case we have, instead of (1) and (2), the following recurrent relation: fn ðxÞ ¼ min fc1 ðy xÞþ þ c2 ðy xÞ þ LðyÞ y Z N fn1 ðy sÞjðsÞ dsg; þa
ð5Þ
N
where f0 ðxÞ 0; aþ ¼ maxða; 0Þ; a ¼ minða; 0Þ and in the first term of (3) we replace the lower limit of integration 0 by N: Although in this case the optimal policy is determined by two critical levels, as in (4) or for an ðs; SÞ-policy, it has the different form given below. Theorem 4. Suppose c2 oc1 op; then there exist two sequences fyn;1 g and fyn;2 g; yn;1 pyn;2 ; nX1; such that 8 > < yn;1 for xpyn;1 ; for yn;1 oxoyn;2 ; yn ðxÞ ¼ x ð6Þ > : yn;2 for xXyn;2 : For other parameter combinations the optimal policy is more complicated than (6). It is interesting to mention that all the critical levels in Theorems 1–4 depend on a; being finite even for a ¼ 1: It will be clear from the next section that the introduction of a means that we take into account only investments in a risk-free asset. Moreover a ¼ ð1 þ rÞ1 where r is the asset’s one-period interest rate.
311
3. Main results It has already been mentioned in Section 1 that the cost minimization has been the main objective of inventory research. However (see, e.g., Pujawan and Kingsman, 1999), it is not an appropriate performance measure for supply chains. In a twostage lot sizing problem, even if demand is deterministic, ‘‘the decision which minimizes cost does not necessarily equate to that which maximizes profit. Hence, from the supplier’s point of view, the performance of the lot sizing decisions has to be measured in terms of profit instead of cost’’. Pujawan and Kingsman (1999) use profit per unit of item delivered by the supplier to the buyer for the whole simulation period. Profit is obtained by subtracting the total costs associated with purchase price, inventory holding and ordering from the revenue received. All of these components are measured in terms of present values. The present value approach was used in . (1999) as well. Grubbstrom Taking into account explicitly a securities market provides a more adequate approach for calculating present values of future random payments. Furthermore, given an initial capital u; it is possible to establish whether one will be able to meet (with probability 1) a specified payments stream g ¼ ðgn Þn¼1;y;N : If the answer is ‘‘yes’’ the next question arises: how to choose the trading strategy in order to maximize the expected profit. Note that some trading strategies might be inadmissible. If the answer is ‘‘no’’ then one would like to estimate the probability of insolvency (or ruin). Cost and reliability approaches could be combined, aiming at profit maximization, subject to the assumption that the insolvency probability does not exceed a given value p0 : In the framework of the Cox–Ross–Rubinstein (CRR) model of a securities market we give below solutions to some of the problems mentioned above. 1: Let the supplier use the stationary replenishment policy determined by a single critical level y% introduced in Section 2. If the initial level is equal to y% as well then the stream of future payments
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312
can be described by a sequence of independent identically distributed random variables þ Zn ¼ c1 xn þ hðy% xn Þþ þ pðxn yÞ % ;
where xn ; as previously, is the demand amount within the time interval ½n 1; nÞ: These payments are assumed to be independent of the securities market. Let the cash amount provided by an external source for inventory control be equal to v > 0: The capital Rn available at time n (after the payment due) can be invested in stocks and/or bonds according to some trading strategy with non-negative components ðbn ; gn ÞnX1 : Hence, we have the following relations: Rn1 ¼ bn Bn1 þ gn Sn1 ; ð7Þ
nX1;
ð11Þ
where m ¼ Ern : It is easily seen that the decision depends on the difference m r only. Thus if mor (stocks are less profitable than bonds) one takes gn ¼ 0: That is, all the available capital is invested in bonds. If m > r (stocks are more profitable than bonds) one invests all in stocks. If m ¼ r any investor’s portfolio provides the same expected profit. Remark 1. The same strategy is obtained if, instead of (11), we consider the unconditional expectation as our objective function. Since 2 VarðRn jFn1 Þ ¼ g2n Sn1 Varðrn Þ þ VarðZn Þ
with Bn ¼ ð1 þ rÞBn1 ;
Sn ¼ ð1 þ rn ÞSn1 ;
nX1: ð8Þ
Eq. (8) describe the prices’ dynamics for two assets. The first asset (a bank account or bond) is risk-free, its time-n value Bn being already known at time n 1; whereas the second one is risky, its time-n value Sn being only known when all the information up to time n is available. In the Cox–Ross–Rubinstein model it is assumed that a one-period interest rate r ¼ const and r ¼ ðrn ÞnX1 is a Bernoulli sequence of independent identically distributed random variables, namely Pðrn ¼ aÞ ¼ q; and
EðRn jFn1 Þ ¼ ð1 þ rÞRn1 þ ðm rÞgn Sn1 þ v EZn ;
n ¼ 1; y; N;
Rn ¼ bn Bn þ gn Sn þ v Zn ;
expectation
Pðrn ¼ bÞ ¼ 1 q
1oaorob:
ð9Þ
In this case the s-field Fn ¼ sðr1 ; y; rn Þ can be interpreted as information available to all the investors at time n: Random variables bn and gn are supposed to be Fn1 -measurable. Simple calculations taking into account (7) and (8) give Rn ¼ ð1 þ rÞRn1 þ ðrn rÞgn Sn1 þ v Zn :
ð10Þ
Wishing to maximize the next period profit, we choose as our objective function the conditional
is minimal for gn ¼ 0; we obtain the following result. Theorem 5. Let the objective function be given by (11). Then it is optimal to invest all the available capital either in stocks or in bonds (choosing the more profitable). A risk averse investor would prefer to invest in bonds. 2: Suppose additionally that we are interested only in those strategies which exclude insolvency with a given probability p0 : A supplier is said to be insolvent at time n if Rn o0: According to (10), given all the information available at time n 1; the decision is specified by fixing gn : Recalling (9) we derive the expression PðRn o0jFn1 Þ % þ rÞRn1 ðr aÞgn Sn1 þ vÞ ¼ qFðð1 % þ rÞRn1 þ ðb rÞgn Sn1 þ vÞ; þ ð1 qÞFðð1 ð12Þ % ¼ PðZn > tÞ is assumed to be continuous where FðtÞ in t: Our objective is to maximize (11) subject to condition that the right-hand side of (12) does not exceed p0 :
E.V. Bulinskaya / Int. J. Production Economics 81–82 (2003) 309–316
Let F ðtÞ be the distribution function of Zn and introduce the following notations: cðg; R; SÞ ¼ qF ðð1 þ rÞR ðr aÞgS þ vÞ þ ð1 qÞF ðð1 þ rÞR þ ðb rÞgS þ vÞ and AðR; SÞ ¼ fg: cðg; R; SÞX1 p0 g: If this set is nonempty put gn ðR; SÞ ¼ supfg: gA½0; R=S -AðR; SÞg; g * ðR; SÞ ¼ inffg: gA½0; R=S -AðR; SÞg; and denote by g% n the solution of the constrained maximization problem EðRn jFn1 Þ-max
313
there exists a unique positive solution g# to the equation cðg; R; SÞ ¼ 1 p0 : Corollary 1. If cðRn1 =Sn1 ; Rn1 ; Sn1 ÞX1 p0 ; then g% n ¼ Rn1 =Sn1 : One can investigate the dependence of g% n on market parameters and constraints. Proposition 1. g%n is (i) inversely proportional to Sn1 ; (ii) independent of Bn1 ; (iii) an increasing function of Rn1 ; (iv) increasing in a and in b; (v) increasing in r (as long as rom), (vi) increasing in v; (vii) decreasing in q; (viii) increasing in p0 : Case 2: mor:
subject to : PðRn o0jFn1 Þpp0 :
Lemma 4. If F ðð1 þ rÞR þ vÞX1 p0 ; then g% n ¼ 0:
It is not difficult to see that the following result is valid.
Lemma 5. If F is concave, then the maximum of cðg; R; SÞ; for gA½0; R=S ; is attained at g ¼ 0:
Lemma 1. Let Rn1 ; Sn1 be such that gn and g * exist for R ¼ Rn1 and S ¼ Sn1 : Then g% n ¼ gn ðRn1 ; Sn1 Þ if m > r and g% * ðRn1 ; Sn1 Þ if mor:
Corollary 2. Under the assumptions of Lemmas 4 and 5, investment in bonds maximizes profit, simultaneously minimizing the insolvency probability.
To provide more detailed information concerning g% n we treat the two cases separately, making additional assumptions about F :
Corollary 3. If F is concave and F ðð1 þ rÞR þ vÞ o1 p0 then, for any investor’s portfolio, the probability of insolvency is greater than p0 :
Case 1: m > r: Using the inequalities
Proposition 2. g% n is decreasing in Sn1 ; Rn1 ; a; b; r; v; p0 and increasing in q:
F ðð1 þ aÞR þ vÞpcðg; R; SÞpF ðð1 þ bÞR þ vÞ; obvious for gA½0; R=S ; one easily proves the following: Lemma 2. (i) If F ðð1 þ aÞR þ vÞX1 p0 ; then gn ðR; SÞ ¼ R=S: (ii) If F ðð1 þ bÞR þ vÞo1 p0 ; then gn ðR; SÞ is not defined.
3: We have established that under certain assumptions it is optimal to use only risk-free assets. Hence, it is interesting to investigate the probability of multistep solvency under such a strategy. Define, for any kX0 and R0 ¼ x; the probability of becoming insolvent not later than at time k Ck ðxÞ ¼ Pð(npk: Rn o0Þ;
In other words, if assumption (i) of Lemma 2 is valid for R ¼ Rn1 then g% n ¼ Rn1 =Sn1 (investments in stocks). If assumption (ii) is satisfied for R ¼ Rn1 ; for any investor’s portfolio, the probability of insolvency at time n is greater than p0 : Lemma 3. Let F be strictly concave and cð0; R; SÞ ¼ F ðð1 þ rÞR þ vÞX1 p0 X1 q; then
then C0 ðxÞ ¼ 0 for xX0; C0 ðxÞ ¼ 1 for xo0 and the following recurrent equation is valid for kX0: Ckþ1 ðxÞ % þ rÞx þ vÞ ¼ Fðð1 Z ð1þrÞxþv þ Ck ðð1 þ rÞx þ v tÞ dF ðtÞ: ð13Þ 0
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314
Using induction arguments we can establish an upper bound for Ck ðxÞ:
We omit the cumbersome expression for CN ðxÞ ¼ limk-N Ck ðxÞ:
Theorem 6. Let R > 0 satisfy inequality Z N eRt dF ðtÞpeRv ;
4: In the last model considered below the inventory replenishment policy is not specified. In the framework of the cost approach and the CRR-model of a securities market we obtain a combined inventory replenishment and investment policy maximizing the expected one-period profit. Now let x be the initial inventory level and u be the cash amount available at the beginning of the period for inventory replenishment and investment. Then
0
then for all x and kX1 Ck ðxÞpeRx : If bn ¼ 0 for all n (only a risky asset is used) then, instead of (13), one has the integral equation Ckþ1 ðxÞ % þ aÞx þ vÞ þ ð1 qÞFðð1 % þ bÞx þ vÞ ¼ qFðð1 Z ð1þaÞxþv þq Ck ðð1 þ aÞx þ v tÞ dF ðtÞ 0
þ ð1 qÞ
Z
u ¼ cz þ bB0 þ gS0 ;
ð1þbÞxþv
Ck ðð1 þ bÞx 0
þ v tÞ dF ðtÞ:
ð14Þ
Remark 2. Theorem 6 is valid for solutions of (14). One can obtain better bounds for exponential distributions. Theorem 7. If F ðxÞ ¼ 1 elx ; x > 0; and r > elv ; then solutions of (13) satisfy the following inequality: r elð1þrÞx : Ck ðxÞp lv re 1 Moreover in the exponential case one can establish an explicit form for Ck ðxÞ: lx
Theorem 8. If F ðxÞ ¼ 1 e ; x > 0; then k X m Ck ðxÞ ¼ elxð1þrÞ dmk ; kX1;
where z is a replenishment order, c is the ordering cost per unit and B0 and S0 are assets prices, b and g being their quantities. For simplicity we assume the order to be delivered by the end of the period, in time to satisfy arising demand. Then the inventory holding cost hx does not depend on the replenishment decision. The demand x is a random variable with a known distribution having density jðsÞ > 0; s > 0: If the demand exceeds the inventory available an emergency delivery is required, the price being p per unit. The cash inflow during the period is u þ brB0 þ grS0 (with r distributed as rn in (9)). Whereas the costs are cz þ hx þ pðx x zÞþ and the problem to solve is Gðz; b; gÞ-max subject to : cz þ bB0 þ gS0 ¼ u; zX0; bX0; gX0; where Gðz; b; gÞ ¼ u þ brB0 þ gmS0 cz Z N p ðs x zÞjðsÞ ds hx:
satisfy the recurrent relations k X elv d k þ elv ; ¼ ð1 þ rÞm 1 m m¼1
where d1kþ1
m
kþ1 dmþ1
elvð1þrÞ dk ; ¼ ð1 þ rÞm 1 m
m ¼ 1; k:
ð16Þ
xþz
m¼1
dmk
ð15Þ
Denote by z1 and z2 ; respectively, the solutions of the equations Z z1 jðsÞ ds ¼ 1 cð1 þ rÞp1 ; Z0 z2 jðsÞ ds ¼ 1 cð1 þ mÞp1 ; ð17Þ 0
E.V. Bulinskaya / Int. J. Production Economics 81–82 (2003) 309–316
and let zn ðxÞ; bn ðxÞ and gn ðxÞ be the optimal decision. Rewriting (15) in the form uð1 þ rÞ czð1 þ rÞ þ gðm rÞS0 Z N p ðs x zÞjðsÞ ds-max
a nondecreasing sequence of s-fields in a probability space ðO; F; PÞ and F0 ¼ fO; |g: Hence, we get instead of (11) EðRn jFn1 Þ ¼ ð1 þ rn ÞRn1 þ ðm# n rn Þgn Sn1 þ c EZn ;
xþz
subject to : cz þ gS0 pu zX0; gX0 enables us to prove the following: Theorem 9. (1) If mor then one invests only in bonds and gn ðxÞ 0: Moreover if ppcð1 þ rÞ the replenishment order zn ðxÞ ¼ 0 for all x; whereas for p > cð1 þ rÞ the optimal order has the form zn ðxÞ ¼ min½uc1 ; ðz1 xÞþ and bn ðxÞ ¼ ðu czn ðxÞÞB1 0 : (2) If m > r then it is optimal to invest only in stocks and bn ðxÞ 0: Moreover if ppcð1 þ mÞ then zn ðxÞ 0; whereas for p > cð1 þ mÞ the optimal replenishment order is zn ðxÞ ¼ min½uc1 ; ðz2 xÞþ and gn ðxÞ ¼ ðu czn ðxÞÞS01 : Remark 3. Turning to the classical model of Section 2, we replace the last two terms in expression (16) of Gðz; b; gÞ by Lðx þ zÞ with Lð Þ defined in (3). Then the analogue of Theorem 9 is valid, with obvious modifications to Eq. (17) defining z1 and z2 :
4. Conclusions and further research directions The results of the previous section justify the commonly used present value calculation of future cash-flow streams for risk averse investors (in the framework of the Cox–Ross–Rubinstein model). In a general ðB; SÞ-model the dynamics of assets prices is described by two discrete stochastic differential equations DBn ¼ rn Bn1 ;
DSn ¼ rn Sn1 ;
B0 > 0;
315
S0 > 0;
here we use the notation DAn ¼ An An1 : The corresponding one-period interest rates rn are Fn1 -measurable and rn are Fn -measurable for all nX1: In this case ðFn ÞnX0 is a filtration, that is
where m# n ¼ Eðrn jFn1 Þ: Therefore a result similar to Theorem 5 is valid if, for each n; m# n and rn are ordered with probability 1. The choice of an appropriate objective function in the general case is one of the standing problems. Comparing Theorem 3, for n ¼ 1; and the result of Remark 3, explicitly taking into account investment, one can see that in the latter case the optimal replenishment policy is still specified by critical levels. However these parameters depend on asset profitability. A multiperiod version of such models is treated in Bulinskaya (2000), which also incorporates investment in the models described by Eq. (5). Here we considered discrete-time models. Another interesting research direction is an investigation of continuous-time models. However, this involves intricate mathematical tools such as continuous-time stochastic processes, Ito’s calculus, Girsanov’s theorem and so on.
Acknowledgements The author would like to thank two anonymous referees for their helpful suggestions. The research was partially supported by RFBR grant 00-0100131.
References Arrow, K.J., Harris, T., Marschak, J., 1951. Optimal inventory policy. Econometrica 19, 250–272. Bulinskaya, E.V., 1990. Inventory control in case of unknown distribution. Engineering Costs and Production Economics 19, 301–306. Bulinskaya, E.V., 2000. On some aspects of modern inventory theory. Obozrenie Prikladnoj i Promyshlennoj Matematiki 7 (2), 481 (in Russian). Chik!an, A. (Ed.), 1986. Bibliography of inventory literature, Budapest, ISIR.
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Dvoretzky, A., Kiefer, J., Wolfowitz, J., 1952. The inventory problem. Econometrica 20, l87–222, 450–466. Dvoretzky, A., Kiefer, J., Wolfowitz, J., 1953. On the optimal character of the (s,S)-policy in inventory control. Econometrica 21, 586–596. Groenevelt, H., 1993. The just-in-time system. In: Graves, S.C., et al. (Ed.), Handbooks in Operations Research and Management Science, Vol. 4, Elsevier, Amsterdam, pp. 629–670. Grubbstr.om, R.W., 1999. A net present value approach to safety stocks in a multi-level MRP system. International Journal of Production Economics 59, 361–365.
Pujawan, J.N., Kingsman, B.G., 1999. Evaluating the performance of supplier operations in a two-stage lot sizing problem. In: Hill, R., Smith, D. (Eds.), Inventory Modelling. A Selection of Research Papers presented at the fourth ISIR Summer School, University of Exeter, pp. 147–157. Rustenburg, W.D., van Houtum, G.J., Zijm, W.H.M., 1999. Large budget restricted resupply of spare parts at the Royal Netherlands Navy. In: Hill, R., Smith, D. (Eds.), Inventory Modelling. A Selection of Research Papers presented at the fourth ISIR Summer School, University of Exeter, pp. 41–56. Shiryaev, A.N., 1999. Essentials of Stochastic Finance (Facts, Models, Theory). World Scientific, Singapore.
Inventory control and investment policy E.V. Bulinskaya Faculty of Mechanics and Mathematics, Moscow State University, 119992 Moscow GSP-2, Russia
Abstract The idea underlying this paper is to consider inventory control as part of a wider class of economic problems, namely financial risk management. Only discrete-time models are investigated here. In the framework of the Cox–Ross– Rubinstein model of a financial market a combined inventory replenishment and investment policy is obtained. Company solvency is studied as well. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Inventory control; Financial market; Cost approach; Reliability approach; Solvency
1. Introduction The aim of this paper is to draw attention of inventory researchers to new perspectives opened by treating inventory problems within the wider scope of financial risk management. In the year 2001 we celebrate the half century anniversary of the pioneering work of Arrow et al. (1951). Along with the seminal papers of Dvoretzky et al. (1952, 1953) it formed the foundation of modern inventory theory (or, more strictly, its cost approach). It is interesting to recall that the 1971 Nobel Prize winner, the Honorary President of ISIR Kenneth Arrow, has also greatly contributed to the development of financial economics. The classical (AHM) inventory model and its various modifications (see, e.g. Chika! n, 1986) usually took into account inventory replenishment and holding costs, as well as a shortage penalty. Although some authors (see, e.g., Rustenburg E-mail address: [email protected] (E.V. Bulinskaya).
et al., 1999) considered budget restrictions, for the most part the cash amount available for inventories was assumed to be unlimited and the decisions were aimed at the minimization of expected accumulated costs (since demand was supposed to be random). There was a lot of discussion about capital tied up in inventories (see, e.g., Pujawan and Kingsman, 1999). We mention in passing that, in an attempt to avoid the undesirable effects of inventory holding, a new just-in-time concept has been introduced in inventory theory and practice (see, e.g., Groenevelt, 1993). One could also say that implicitly the existence of a financial market was taken into account from the outset by discounting future expenses. Now the time has come to use explicitly and in full the powerful tools of financial mathematics by combining inventory control with investment policy. For simplicity we consider below only discretetime models. Section 2 contains some preliminary results. First we point out the effect of a budget constraint in the classical inventory model. Then
0925-5273/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 0 2 ) 0 0 3 5 5 - 9
E.V. Bulinskaya / Int. J. Production Economics 81–82 (2003) 309–316
310
we drop the nonnegativity restriction on inventory order, allowing for seasonal sales. The one-period demand is assumed to be a random variable with a finite mean (taking negative values as well in order to incorporate the possibility of customers returning bought product or remanufacturing). In Section 3 we introduce the Cox–Ross– Rubinstein model of a financial market (for more details see the fundamental book by Shiryaev (1999)). Assuming an inventory replenishment policy to be fixed we establish the optimal investment strategy. Using a reliability approach we treat the company solvency problem. In the framework of the cost approach we obtain the optimal inventory replenishment and investment policy. Almost all proofs are omitted due to lack of space. In Section 4 we draw conclusions and outline further research directions.
with
2. Preliminary results
Moreover, y% ¼ limn-N yn is the solution of the following equation: Z y jðsÞ ds ¼ ðp c1 þ ac1 Þðp þ hÞ1 :
To illustrate the influence of various constraints on optimal inventory control we consider the simplest case. The replenishment order, made at the beginning of a period, is delivered immediately and can be used to satisfy the arising demand. The inventory left is stored for future use and a penalty is paid for a stock-out. Let the demand process be described by a sequence of independent identically distributed random variables fxi giX1 : We suppose xi to be nonnegative and to have a density jðsÞ > 0; s > 0; as well as a finite mean. All the cost functions are assumed to be linear, their slopes being, respectively, c1 ; the ordering cost per unit, h; the holding cost per unit per period, and p; the shortage penalty per unit per period. Unsatisfied demand is backlogged and hence x can be negative. We also introduce the discount factor a; 0oao1: Denote, as usual, by fn ðxÞ the minimal expected n-period cost, x being the initial inventory level. Then we have the following dynamic programming equation for nX1: fn ðxÞ ¼ c1 x þ min Gn ðyÞ; yXx
ð1Þ
Gn ðyÞ ¼ c1 y þ LðyÞ þ a
Z
N
fn1 ðy sÞjðsÞ ds ð2Þ 0
and LðyÞ ¼ h
Z
y
0
þp
ðy sÞjðsÞ ds Z
N
ðs yÞjðsÞ ds:
ð3Þ
y
The y-value, say yn ðxÞ; minimizing the righthand side of (1), is called the optimal ordering level. It is well known (see, e.g., Bulinskaya, 1990) that the following result is valid. Theorem 1. If p > c1 ; then there exists an increasing sequence fyn gnX1 such that ( yn for xpyn ; yn ðxÞ ¼ x for x > yn :
0
As in Bulinskaya (1990) the following result can be proved. Theorem 2. The stationary inventory policy determined by a single critical level y% is asymptotically optimal. (An inventory policy is said to be asymptotically optimal if the long-run average cost per period, under this policy, is equal to limn-N n1 fn ðxÞ with fn ðxÞ defined by (1).) Now let u be the cash amount available each period for inventory replenishment. Then the corresponding minimal costs f%n ðxÞ satisfy, instead of (1), a similar relation with minimization carried out over the interval ½x; x þ uc1 1 : Theorem 3. Under the assumption of Theorem 1 the optimal ordering policy y% n ðxÞ is determined by the
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increasing sequence namely 8 1 > < x þ uc1 y% n ðxÞ ¼ y% n > : x
fy% n gnX1 of critical levels, for
xoy% n uc1 1 ;
for for
y% n uc1 %n; 1 pxpy x > y% n :
ð4Þ
Furthermore y% 1 ¼ y1 and y% n > yn ; n > 1: The proof, being standard induction, is omitted. The limit behaviour of y% n ; as n-N; has been studied as well. Thus we see that the budget constraints compel us to build up inventories. Next we suppose that xi can take negative values and include in the model the sales at the beginning of any period, the price being c2 per unit. In this case we have, instead of (1) and (2), the following recurrent relation: fn ðxÞ ¼ min fc1 ðy xÞþ þ c2 ðy xÞ þ LðyÞ y Z N fn1 ðy sÞjðsÞ dsg; þa
ð5Þ
N
where f0 ðxÞ 0; aþ ¼ maxða; 0Þ; a ¼ minða; 0Þ and in the first term of (3) we replace the lower limit of integration 0 by N: Although in this case the optimal policy is determined by two critical levels, as in (4) or for an ðs; SÞ-policy, it has the different form given below. Theorem 4. Suppose c2 oc1 op; then there exist two sequences fyn;1 g and fyn;2 g; yn;1 pyn;2 ; nX1; such that 8 > < yn;1 for xpyn;1 ; for yn;1 oxoyn;2 ; yn ðxÞ ¼ x ð6Þ > : yn;2 for xXyn;2 : For other parameter combinations the optimal policy is more complicated than (6). It is interesting to mention that all the critical levels in Theorems 1–4 depend on a; being finite even for a ¼ 1: It will be clear from the next section that the introduction of a means that we take into account only investments in a risk-free asset. Moreover a ¼ ð1 þ rÞ1 where r is the asset’s one-period interest rate.
311
3. Main results It has already been mentioned in Section 1 that the cost minimization has been the main objective of inventory research. However (see, e.g., Pujawan and Kingsman, 1999), it is not an appropriate performance measure for supply chains. In a twostage lot sizing problem, even if demand is deterministic, ‘‘the decision which minimizes cost does not necessarily equate to that which maximizes profit. Hence, from the supplier’s point of view, the performance of the lot sizing decisions has to be measured in terms of profit instead of cost’’. Pujawan and Kingsman (1999) use profit per unit of item delivered by the supplier to the buyer for the whole simulation period. Profit is obtained by subtracting the total costs associated with purchase price, inventory holding and ordering from the revenue received. All of these components are measured in terms of present values. The present value approach was used in . (1999) as well. Grubbstrom Taking into account explicitly a securities market provides a more adequate approach for calculating present values of future random payments. Furthermore, given an initial capital u; it is possible to establish whether one will be able to meet (with probability 1) a specified payments stream g ¼ ðgn Þn¼1;y;N : If the answer is ‘‘yes’’ the next question arises: how to choose the trading strategy in order to maximize the expected profit. Note that some trading strategies might be inadmissible. If the answer is ‘‘no’’ then one would like to estimate the probability of insolvency (or ruin). Cost and reliability approaches could be combined, aiming at profit maximization, subject to the assumption that the insolvency probability does not exceed a given value p0 : In the framework of the Cox–Ross–Rubinstein (CRR) model of a securities market we give below solutions to some of the problems mentioned above. 1: Let the supplier use the stationary replenishment policy determined by a single critical level y% introduced in Section 2. If the initial level is equal to y% as well then the stream of future payments
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312
can be described by a sequence of independent identically distributed random variables þ Zn ¼ c1 xn þ hðy% xn Þþ þ pðxn yÞ % ;
where xn ; as previously, is the demand amount within the time interval ½n 1; nÞ: These payments are assumed to be independent of the securities market. Let the cash amount provided by an external source for inventory control be equal to v > 0: The capital Rn available at time n (after the payment due) can be invested in stocks and/or bonds according to some trading strategy with non-negative components ðbn ; gn ÞnX1 : Hence, we have the following relations: Rn1 ¼ bn Bn1 þ gn Sn1 ; ð7Þ
nX1;
ð11Þ
where m ¼ Ern : It is easily seen that the decision depends on the difference m r only. Thus if mor (stocks are less profitable than bonds) one takes gn ¼ 0: That is, all the available capital is invested in bonds. If m > r (stocks are more profitable than bonds) one invests all in stocks. If m ¼ r any investor’s portfolio provides the same expected profit. Remark 1. The same strategy is obtained if, instead of (11), we consider the unconditional expectation as our objective function. Since 2 VarðRn jFn1 Þ ¼ g2n Sn1 Varðrn Þ þ VarðZn Þ
with Bn ¼ ð1 þ rÞBn1 ;
Sn ¼ ð1 þ rn ÞSn1 ;
nX1: ð8Þ
Eq. (8) describe the prices’ dynamics for two assets. The first asset (a bank account or bond) is risk-free, its time-n value Bn being already known at time n 1; whereas the second one is risky, its time-n value Sn being only known when all the information up to time n is available. In the Cox–Ross–Rubinstein model it is assumed that a one-period interest rate r ¼ const and r ¼ ðrn ÞnX1 is a Bernoulli sequence of independent identically distributed random variables, namely Pðrn ¼ aÞ ¼ q; and
EðRn jFn1 Þ ¼ ð1 þ rÞRn1 þ ðm rÞgn Sn1 þ v EZn ;
n ¼ 1; y; N;
Rn ¼ bn Bn þ gn Sn þ v Zn ;
expectation
Pðrn ¼ bÞ ¼ 1 q
1oaorob:
ð9Þ
In this case the s-field Fn ¼ sðr1 ; y; rn Þ can be interpreted as information available to all the investors at time n: Random variables bn and gn are supposed to be Fn1 -measurable. Simple calculations taking into account (7) and (8) give Rn ¼ ð1 þ rÞRn1 þ ðrn rÞgn Sn1 þ v Zn :
ð10Þ
Wishing to maximize the next period profit, we choose as our objective function the conditional
is minimal for gn ¼ 0; we obtain the following result. Theorem 5. Let the objective function be given by (11). Then it is optimal to invest all the available capital either in stocks or in bonds (choosing the more profitable). A risk averse investor would prefer to invest in bonds. 2: Suppose additionally that we are interested only in those strategies which exclude insolvency with a given probability p0 : A supplier is said to be insolvent at time n if Rn o0: According to (10), given all the information available at time n 1; the decision is specified by fixing gn : Recalling (9) we derive the expression PðRn o0jFn1 Þ % þ rÞRn1 ðr aÞgn Sn1 þ vÞ ¼ qFðð1 % þ rÞRn1 þ ðb rÞgn Sn1 þ vÞ; þ ð1 qÞFðð1 ð12Þ % ¼ PðZn > tÞ is assumed to be continuous where FðtÞ in t: Our objective is to maximize (11) subject to condition that the right-hand side of (12) does not exceed p0 :
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Let F ðtÞ be the distribution function of Zn and introduce the following notations: cðg; R; SÞ ¼ qF ðð1 þ rÞR ðr aÞgS þ vÞ þ ð1 qÞF ðð1 þ rÞR þ ðb rÞgS þ vÞ and AðR; SÞ ¼ fg: cðg; R; SÞX1 p0 g: If this set is nonempty put gn ðR; SÞ ¼ supfg: gA½0; R=S -AðR; SÞg; g * ðR; SÞ ¼ inffg: gA½0; R=S -AðR; SÞg; and denote by g% n the solution of the constrained maximization problem EðRn jFn1 Þ-max
313
there exists a unique positive solution g# to the equation cðg; R; SÞ ¼ 1 p0 : Corollary 1. If cðRn1 =Sn1 ; Rn1 ; Sn1 ÞX1 p0 ; then g% n ¼ Rn1 =Sn1 : One can investigate the dependence of g% n on market parameters and constraints. Proposition 1. g%n is (i) inversely proportional to Sn1 ; (ii) independent of Bn1 ; (iii) an increasing function of Rn1 ; (iv) increasing in a and in b; (v) increasing in r (as long as rom), (vi) increasing in v; (vii) decreasing in q; (viii) increasing in p0 : Case 2: mor:
subject to : PðRn o0jFn1 Þpp0 :
Lemma 4. If F ðð1 þ rÞR þ vÞX1 p0 ; then g% n ¼ 0:
It is not difficult to see that the following result is valid.
Lemma 5. If F is concave, then the maximum of cðg; R; SÞ; for gA½0; R=S ; is attained at g ¼ 0:
Lemma 1. Let Rn1 ; Sn1 be such that gn and g * exist for R ¼ Rn1 and S ¼ Sn1 : Then g% n ¼ gn ðRn1 ; Sn1 Þ if m > r and g% * ðRn1 ; Sn1 Þ if mor:
Corollary 2. Under the assumptions of Lemmas 4 and 5, investment in bonds maximizes profit, simultaneously minimizing the insolvency probability.
To provide more detailed information concerning g% n we treat the two cases separately, making additional assumptions about F :
Corollary 3. If F is concave and F ðð1 þ rÞR þ vÞ o1 p0 then, for any investor’s portfolio, the probability of insolvency is greater than p0 :
Case 1: m > r: Using the inequalities
Proposition 2. g% n is decreasing in Sn1 ; Rn1 ; a; b; r; v; p0 and increasing in q:
F ðð1 þ aÞR þ vÞpcðg; R; SÞpF ðð1 þ bÞR þ vÞ; obvious for gA½0; R=S ; one easily proves the following: Lemma 2. (i) If F ðð1 þ aÞR þ vÞX1 p0 ; then gn ðR; SÞ ¼ R=S: (ii) If F ðð1 þ bÞR þ vÞo1 p0 ; then gn ðR; SÞ is not defined.
3: We have established that under certain assumptions it is optimal to use only risk-free assets. Hence, it is interesting to investigate the probability of multistep solvency under such a strategy. Define, for any kX0 and R0 ¼ x; the probability of becoming insolvent not later than at time k Ck ðxÞ ¼ Pð(npk: Rn o0Þ;
In other words, if assumption (i) of Lemma 2 is valid for R ¼ Rn1 then g% n ¼ Rn1 =Sn1 (investments in stocks). If assumption (ii) is satisfied for R ¼ Rn1 ; for any investor’s portfolio, the probability of insolvency at time n is greater than p0 : Lemma 3. Let F be strictly concave and cð0; R; SÞ ¼ F ðð1 þ rÞR þ vÞX1 p0 X1 q; then
then C0 ðxÞ ¼ 0 for xX0; C0 ðxÞ ¼ 1 for xo0 and the following recurrent equation is valid for kX0: Ckþ1 ðxÞ % þ rÞx þ vÞ ¼ Fðð1 Z ð1þrÞxþv þ Ck ðð1 þ rÞx þ v tÞ dF ðtÞ: ð13Þ 0
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314
Using induction arguments we can establish an upper bound for Ck ðxÞ:
We omit the cumbersome expression for CN ðxÞ ¼ limk-N Ck ðxÞ:
Theorem 6. Let R > 0 satisfy inequality Z N eRt dF ðtÞpeRv ;
4: In the last model considered below the inventory replenishment policy is not specified. In the framework of the cost approach and the CRR-model of a securities market we obtain a combined inventory replenishment and investment policy maximizing the expected one-period profit. Now let x be the initial inventory level and u be the cash amount available at the beginning of the period for inventory replenishment and investment. Then
0
then for all x and kX1 Ck ðxÞpeRx : If bn ¼ 0 for all n (only a risky asset is used) then, instead of (13), one has the integral equation Ckþ1 ðxÞ % þ aÞx þ vÞ þ ð1 qÞFðð1 % þ bÞx þ vÞ ¼ qFðð1 Z ð1þaÞxþv þq Ck ðð1 þ aÞx þ v tÞ dF ðtÞ 0
þ ð1 qÞ
Z
u ¼ cz þ bB0 þ gS0 ;
ð1þbÞxþv
Ck ðð1 þ bÞx 0
þ v tÞ dF ðtÞ:
ð14Þ
Remark 2. Theorem 6 is valid for solutions of (14). One can obtain better bounds for exponential distributions. Theorem 7. If F ðxÞ ¼ 1 elx ; x > 0; and r > elv ; then solutions of (13) satisfy the following inequality: r elð1þrÞx : Ck ðxÞp lv re 1 Moreover in the exponential case one can establish an explicit form for Ck ðxÞ: lx
Theorem 8. If F ðxÞ ¼ 1 e ; x > 0; then k X m Ck ðxÞ ¼ elxð1þrÞ dmk ; kX1;
where z is a replenishment order, c is the ordering cost per unit and B0 and S0 are assets prices, b and g being their quantities. For simplicity we assume the order to be delivered by the end of the period, in time to satisfy arising demand. Then the inventory holding cost hx does not depend on the replenishment decision. The demand x is a random variable with a known distribution having density jðsÞ > 0; s > 0: If the demand exceeds the inventory available an emergency delivery is required, the price being p per unit. The cash inflow during the period is u þ brB0 þ grS0 (with r distributed as rn in (9)). Whereas the costs are cz þ hx þ pðx x zÞþ and the problem to solve is Gðz; b; gÞ-max subject to : cz þ bB0 þ gS0 ¼ u; zX0; bX0; gX0; where Gðz; b; gÞ ¼ u þ brB0 þ gmS0 cz Z N p ðs x zÞjðsÞ ds hx:
satisfy the recurrent relations k X elv d k þ elv ; ¼ ð1 þ rÞm 1 m m¼1
where d1kþ1
m
kþ1 dmþ1
elvð1þrÞ dk ; ¼ ð1 þ rÞm 1 m
m ¼ 1; k:
ð16Þ
xþz
m¼1
dmk
ð15Þ
Denote by z1 and z2 ; respectively, the solutions of the equations Z z1 jðsÞ ds ¼ 1 cð1 þ rÞp1 ; Z0 z2 jðsÞ ds ¼ 1 cð1 þ mÞp1 ; ð17Þ 0
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and let zn ðxÞ; bn ðxÞ and gn ðxÞ be the optimal decision. Rewriting (15) in the form uð1 þ rÞ czð1 þ rÞ þ gðm rÞS0 Z N p ðs x zÞjðsÞ ds-max
a nondecreasing sequence of s-fields in a probability space ðO; F; PÞ and F0 ¼ fO; |g: Hence, we get instead of (11) EðRn jFn1 Þ ¼ ð1 þ rn ÞRn1 þ ðm# n rn Þgn Sn1 þ c EZn ;
xþz
subject to : cz þ gS0 pu zX0; gX0 enables us to prove the following: Theorem 9. (1) If mor then one invests only in bonds and gn ðxÞ 0: Moreover if ppcð1 þ rÞ the replenishment order zn ðxÞ ¼ 0 for all x; whereas for p > cð1 þ rÞ the optimal order has the form zn ðxÞ ¼ min½uc1 ; ðz1 xÞþ and bn ðxÞ ¼ ðu czn ðxÞÞB1 0 : (2) If m > r then it is optimal to invest only in stocks and bn ðxÞ 0: Moreover if ppcð1 þ mÞ then zn ðxÞ 0; whereas for p > cð1 þ mÞ the optimal replenishment order is zn ðxÞ ¼ min½uc1 ; ðz2 xÞþ and gn ðxÞ ¼ ðu czn ðxÞÞS01 : Remark 3. Turning to the classical model of Section 2, we replace the last two terms in expression (16) of Gðz; b; gÞ by Lðx þ zÞ with Lð Þ defined in (3). Then the analogue of Theorem 9 is valid, with obvious modifications to Eq. (17) defining z1 and z2 :
4. Conclusions and further research directions The results of the previous section justify the commonly used present value calculation of future cash-flow streams for risk averse investors (in the framework of the Cox–Ross–Rubinstein model). In a general ðB; SÞ-model the dynamics of assets prices is described by two discrete stochastic differential equations DBn ¼ rn Bn1 ;
DSn ¼ rn Sn1 ;
B0 > 0;
315
S0 > 0;
here we use the notation DAn ¼ An An1 : The corresponding one-period interest rates rn are Fn1 -measurable and rn are Fn -measurable for all nX1: In this case ðFn ÞnX0 is a filtration, that is
where m# n ¼ Eðrn jFn1 Þ: Therefore a result similar to Theorem 5 is valid if, for each n; m# n and rn are ordered with probability 1. The choice of an appropriate objective function in the general case is one of the standing problems. Comparing Theorem 3, for n ¼ 1; and the result of Remark 3, explicitly taking into account investment, one can see that in the latter case the optimal replenishment policy is still specified by critical levels. However these parameters depend on asset profitability. A multiperiod version of such models is treated in Bulinskaya (2000), which also incorporates investment in the models described by Eq. (5). Here we considered discrete-time models. Another interesting research direction is an investigation of continuous-time models. However, this involves intricate mathematical tools such as continuous-time stochastic processes, Ito’s calculus, Girsanov’s theorem and so on.
Acknowledgements The author would like to thank two anonymous referees for their helpful suggestions. The research was partially supported by RFBR grant 00-0100131.
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