Effects of discounting and demand rate variability on the EOQ

intemationel journal of

production economics

ELSEVIER Int. J. Production Economics 54 (1998) 173-192

Effects of discounting and demand rate variability on the EOQ

Wiilem K. Klein Haneveld, Ruud H. Teunter*

Department of Econometrics. University of Groningen, P.0. 50.x 800, 9700 A V Groningen, Netherlands

Received 6 November 1995; accepted 14 October 1997

Abstract

The Economic Order Quantity (EOQ) formula is probably the most well-known formula in inventory theory. It determines the optimal value of the ordering quantity by minimizing the average cost in an ordering cycle. It is based on the assumptions that there is a fixed lead time and a constant demand rate. These assumptions are usually not fulfilled in practice. Especially when dealing with slow-moving items, demand seems to fluctuate. Furthermore, since the time between two successive orderings is often large for slow moving items, discounted costs instead of average costs should be minimized. In this paper we determine the optimal ordering quantity if demand is modelled by a Poisson process and the expected discounted costs are minimized. We also analyze the case where demand is uniform and/or the average cost criterion is used. We give graphical presentations of the optimal ordering quantities. Besides easily determining the optimal values, these allow us to show what the effects of changing the objective function and of changing the demand distribution are. We derive a simple (discounted cost) ordering quantity formula, denoted by EOQ,, for slow moving items. We show that for items with a small demand rate and high fixed ordering costs the EOQ, is much closer to the optimal (discounted cost) ordering quantity than the traditional EOQ-formula. (0 1998 Elsevier Science B.V. All rights reserved.

Keywords: Inventory control; Economic order quantity; Discounted cost

1. Introduction

Consider an inventory system in which we focus on a specific item. The demand for this item is low and hard to predict. We seek an optimal ordering strategy for this item. A well-known approximate solution for such a problem is the Economic Order- ing Quantity or shortly EOQ (see for instance Peterson and Silver [I]). However, underlying the

* E-mail: [email protected].

EOQ are some rigorous assumptions that are sel- dom fulfilled in practice. We shortly list them. There is a uniform demand. That is, at most one item is demanded at each time and the time between two successive demands is fixed. The lead time is constant. Shortages are not allowed. The holding cost is proportional to the number of items being held in stock and the time of carrying stock. Yet, despite of these restrictive assumptions, the EOQ formula is the most cited and used model for an approximate solution even in computer pack- ages of inventory control, see works of Chik6n [2]

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174 W.K. Klein Haneveld, R.H. Teunterjlnt. J. Production Economics 54 (1998) 173-l 92

and Oakland and Sohal [3]. The EOQ formula was NPV of profits as a function of the investment in first published by Harris [4]. See also the works of inventory. However, this result only holds when Erlenkotter [S], where early literature on the applying a first degree approximation (see Ding EOQ-model and specially Harris’ publications on [I 11). Schroeder and Krishnan [12] discuss the this topic are discussed. Since then, a lot of general- Return On Investment (ROI) as an alternative deci- izations have appeared. An overview of the results sion criterion. However, their analysis is question- is given by Chikan [2]. In most papers, the result- able, since the average instead of the discounted ing ordering quantities minimize the average cost return on investment is maximized (see Gurnani in an ordering cycle. Hadley [6] was the first who [13]). Gurnani claims that there is a substantial considered determining ordering quantities that difference between the NPV and the ACC criterion. minimize the discounted cost, rather than the tradi- But, like in Trippi and Lewin [9] opportunity costs tional criterion Average Cycle Cost (ACC). That is, are not included in the ACC model. Furthermore, he was the first who applied the Net Present Vulue Kim and Chung [14] show that they made some (NPV) criterion to the problem of finding optimal serious derivational errors. Kim et al. [7] consider, ordering quantities. For the most important advant- among other things, the more general EOQ-setting age of the NPV-criterion, we quote Kim et al. [7]. where shortages are allowed.

The net present value criterion permits correct use of opportunity costs and out of pocket costs by recognizing separately their financial implications in applying an inventory system.

Hadley [6] concludes that in most cases there is only a small difference between the optimal ordering quantities with and without discounting. However, he also gives some examples with large differences, for instance:

Example 1.1 (Hadley [6]). Let the annual discount rate be 15%, the setup cost 10, the variable cost 0.10 and the average annual demand 0.5. Then the optimal ordering quantities with and without discounting are respectively 10 and 20.

In the last two decades, a number of papers discussing the effect of inflation on the EOQ have appeared. Although inflation is not discussed in this paper, a uniform inflation rate could easily be incorporated in the NPV-approach by substracting the inflation rate from the discounting factor. The analysis becomes more complicated when one dis- tinguishes between an inflation rate for the general economy (the so-called external inflation rate) and an inflation rate within the company (the so-called internal inflation rate). See for instance Chandra and Bahner [ 151. Overviews of the results for infla- tionary models are given by Wirth [16] and Bose et al. [17].

Since then, a number of papers dealing with the effect of changing the traditional average cost criterion to the NPV-criterion have appeared. We give a brief overview of the results. Beranek [S] shows that carrying costs are underestimated in traditional models, causing the traditional ACC model to overestimate the optimal ordering quantity. Trippi and Lewin [9] compare the NPV criterion to the ACC criterion and conclude that the NPV is less sensitive to errors in the ordering quantity. However, they do not include opportunity or interest costs in the ACC model. Therefore, they make an unfair comparison. Thompson [lo] notes that the well known square root formula from inventory theory results from maximizing the

Robert W. Grubbstrom [18-201 wrote a number of papers on determining the correct capital costs of work in process and inventory. He gave five basic axioms for comparing series of payments over time. He showed that the only objective criterion that satisfies these properties is the Net Present Value (or a monotonic transformation of this value). He also showed that the traditional ACC criterion clearly underestimates the holding cost of inventory. This holds even if the opportunity or interest rate is included in the holding cost parameter. The reason for this is intuitively clear. The traditional accounting disregards interest payment on the holding cost, interest payment on interest, and so on. Thorstenson [21] discusses the net present value approach, and applies it on models that are more general than the traditional EOQ model, namely models with varying deterministic demand

W.K. Klein Haneveld, R.H. Teunterllnt. J. Production Economics 54 (1998) 173-192 175

(deterministic lot-size models) and models with stochastic demand. Thorstenson also reviews the literature on EOQ models with a disounted cost approach.

It appears from this short review, that there is still considerable disagreement concerning the effect of discounting on the optimal order quantity. In this paper we show on which characteristics the difference between the traditional ACC and the NPV-approach depends. We also show under which circumstances the difference is large. Al- though we only derive the discounted expected cost in case stockouts are not allowed and if there is an infinite replenishment rate, the analysis could easily be extended to cases where these assumptions are dropped. Just as all models in the papers mentioned above, we consider uniform demands. We conclude that especially for expensive slow-moving items the NPV-approach is better, and that the traditional model gives nearly optimal solutions for fast moving items.

Especially for slow-moving items, which often constitute a large part of the stock (see Mitchell [22]), the constant demand rate assumption is hardly ever fulfilled. The demand pattern of slow moving items can often be well described by a Pois- son process, see for instance Peterson and Silver [l] or Sherbrooke [23]. Therefore, we also consider a model comparable to the one mentioned so far, but with random demands, generated by a Poisson process. The results are quite comparable to those of the deterministic models. Moreover, since these items are often stocked for long periods, minimizing the discounted cost seems more appropriate then minimizing the average cost.

The organization of the paper is as follows. In Section 2 we describe the ordering quantity models that are compared in this paper. These models differ in the objective function (either average cost or discounted cost) and in the demand process (either uniform or Poisson). The uniform demand models are discussed in Section 3. The Poisson demand models in Section 4. For each model, the optimal solution is represented in a graph. Further- more, necessary and sufficient conditions for optimality of one-for-one replenishment are given. In Section 5 we compare the optimal solutions of the models. For all four models, the optimal ordering

quantity is shown to depend on the same two characteristics. We also discuss the cost-robustness of the traditional EOQ-formula. In Section 6, we give a number of examples that show for which combinations of parameters the traditional EOQ- formula performs well, and for which it does not. In the last section we derive an improved but still simple EOQ-formula for slow-moving items and compare it to the traditional one.

2. Model

The goal is to find ordering quantities that minimize the total cost. We make the following modelling assumptions: l Time is modelled as a continuous variable

t E [0, cc ). That is, we assume that the planning horizon is infinite.

l Items are held in stock. In order to increase stock, additional items may be ordered at any time t E [0, co ). It is assumed that the lead time is negligible compared to the mean time between two demands. It is also assumed that shortages are not allowed. As a consequence, a new order will be placed as soon as the stock drops to zero, so that at all times the stock is at least one.

l The following costs are incurred: Purchase cost: If Q items are ordered, then the purchase costs are K + Q. That is, there is a fixed ordering cost K and the price of an item is used as a cost unit, i.e. it is equal to 1. Holding cost: If S(t) items are held in stock at time t, then the holding cost in period [t, t + dt] is hS(t)dt, where the holding cost rate h is positive. That is, the holding cost is proportional to the number of items being held in stock and the time of carrying stock.

References about how to set h and CI in the model are works of Aucamp [24], Christensen and Fry- man [25], Singhal and Raturi [26,27] and Corbey and Jansen [28].

In this paper, four different ordering quantity models, schematically shown in Table 1, are compared.

When demand over time is uniform, at most one item is demanded at each time and the time

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176 W.K. Klein Haneveld, R.H. Teunterllnt. J. Production Economics 54 (1998) 173-192

Table 1 Four different models

Objective function

Uniform demand Poisson demand

average cost

Section 3.1 Section 4.1

net present value

Section 3.2 Section 4.2

between two successive demands is fixed. When demand over time is modelled as a Poisson process with intensity 1, at any moment of time at most one item is demanded and the time between two successive demands is exponentially distributed with mean value 2-i.

When we apply the net present value criterion, all costs are discounted with a fixed continuous discounting parameter SI. That is, ordering Q items at time t contributes e-“‘(K + Q) to the total discounted expected cost. And holding S(t) items in stock at time t, t E [0, CC ), contributes hl;e-“‘S(t)dt.

3. Uniform demand

In this section we discuss the uniform demand models. First, in Section 3.1 we assume that demand is uniform with fixed interarrival times l/A and that the average cost criterion is used. The optimal solution for this model is the well-known Economic Order Quantity, denoted by Q& (* for optimal, a for average cost, u for uniform demand). The Economic Order Quantity can be approximated by the well known square root formula, denoted by EOQa. Second, in Section 3.2, we assume uniform demand and apply the discounted cost criterion. The optimal order quantity in this case is denoted by Qz,, (* for optimal, d for discounted cost, u for uniform demand).

3.1. Un[form demand and average cost criterion

(EOQ-formula)

The economic ordering quantity, denoted by

Q* a,", is the ordering quantity that minimizes the average cost in an ordering cycle, assuming that the

demand is uniform with fixed inter arrival times l/L That is, at most one item is demanded at each time and the time between two successive demands is l/L Since the lead time is fixed and known, a new order arrives at any moment at which the stock is 0 and an item is demanded. Note that, since demand arrival times are fixed and known, an inventory level of 0 is allowed. The average cost in a cycle, when demand is uniform and the ordering quantity is Q is given in most books dealing with inventory theory. It is

1

F,,,(Q) = K^ + jb + cr+h

Q +Q - 1).

Note that in the holding cost we include an interest component. That is, the holding cost is taken to be sum of the out-of-pocket expenses h and the interest/opportunity cost ~1. In fact, the interest component is equal to the discounting parameter multiplied by the price of the item, which has been set 1, as we recall, already at the beginning.

Define AF,,,(Q), Q = 1, 2, . . . , as the difference between F,,,(Q + 1) and F,,,(Q), i.e.

@a,,(Q) = Fa,dQ + 1) - F,,,(Q)> Q = 1, 2, ...

Since F,,,(Q) is a convex function of Q, the optimal Q can be characterized in terms of the differences

@w(Q) = f'w(Q + 1) - Fa,u(Q)

+-+&)

a+h Kl =----

2 Q(Q + 1)'

via

Q:u =

i

1 if and only if AF,,,(l) > 0,

Q 3 2 if and only if AF,,,(Q - 1) GO <AF,JQ).

Hence, Q& = Q 3 1 if and only if

Q(Q - 1) < z ,< Q(Q + 1).

W.K. Klein Haneveld, R.H. Teunterjlnt. J. Production Economics 54 (1998) 173-192 17-l

In particular, one-for-one replenishment is optimal y = %/(a + h). In Fig. 1, the boundary line if between Qz,, = Q and Q& = Q + 1 is determined

K < (a + II)/&

whereas both Q and Q + 1 are optimal if

by

)! = Q(Q + 1) 2x ’

Q(Q + 1) = = = 2; 5. C.?+h

(3.1) Q=l,2 ,...,_ x=,l/a,y=Ka/(a+h).

This representation is useful (see sections to A graphical representation of Q&, based on come) in comparing the optimal ordering quantit- Eq. (3.1) is given in Fig. 1. Let x = A/a and let ies of the different models.

25

20

X0: a+h 15

10

5

0 0 2 4 6 8 10

x a

Fig. I. Boundary lines between Q,*,,. = Q and Q$,. = Q + 1, Q = 1, 2, , 23. The numbers in the figure indicate the value of Q& in the area between the boundary lines to the left and to the tight of that number

178 W.K. Klein Haneveld, R.H. Teunterllnt. J. Production Economics 54 (1998) I73-I92

Finally we note that Q& can be approximated by the well-known square root formula, rounded to the nearest strictly positive integer

EOQa = 2K1 u-l iZa .

3.2. Uniform demand and discounted cost criterion

The effect of replacing the average cost criterion by the discounted cost criterion is studied by Hadley [6]. There, an expression for the total discounted cost in period [0, cc) is derived if demand is uniform with fixed interarrival times l/1 and the ordering quantity is Q. This expression is

In Appendix B it is shown that Fd,JQ) is convex in Q. In previous studies concerning discounted cost [6,7,10,13] similar expressions for the discounted cost where derived for EOQ-type models with uniform demand. However, in none of these studies the cost function was shown to be convex. In fact, the convexity proof given in Appendix B is also valid in the discounted cost case treated in the next section. Rachamadugu proves that the discounted cost is convex if demand is continuous with a fixed rate 1 (and shortages are not allowed). In Ding [11] a sufficient condition, satisfied by a class of models including the ones dealt with in this paper, for strict quasi-convexity of the total discounted expected cost in EOQ-type models with an infinite planning horizon is given.

We now will discuss the characteristics of the optimal order quantity Q&. Since Fd,“(Q) is convex in Q, the optimal Q can be characterized in terms of the differences

A~.&?) = Fd.u@ + 1) - F&Q)

via

QL =

i

1 if and only if AF,,,(l) > 0,

Q > 2 if and only if AF+(Q - 1) ,<O Q AF&Q).

Calculations show that

where

C(Q) = (1 + (h/a))(l - eCnin)

(,@A _ I)(1 _ e-*(Q+l)iz)

is positive for all Q = 1, 2, . . . . Hence, the condition AF,,,(l) > 0 is equivalent to

e a/k -1 1 _ ,-aia _l>Sh.

which can be rewritten as

Therefore,

a + h Q& = 1 if and only if K Q - e cx (

a/i - 1).

Moreover, the condition for which both Q and Q + 1 are optimal is AF,,,(Q) = 0, that is

(3.3)

A graphical representation of Q&, based on Eq. (3.3), is given in Fig. 2. Let x = A/a and let y = Ka/(a + h). In Fig. 2, the boundary line between Q& = Q and Q& = Q + 1 is determined by

eQ/x - 1 Y= 1 _ e-l’” - Q>

Q = 1, 2, . . . . x = Ala, y = Ka/(a + h).

In the next section, we compare Qa*,” and Q& to the optimal ordering quantities if demand is no longer uniform, but Poisson distributed.

W.K. Klein Haneveld, R.H. Teunter/Int. J. Production Economics 54 (1998) 173-192 179

30

25

20

3 15

10

5

0 I -

0 2 4 G 8 10 A *

Fig. 2. Boundary lines between QJ,, = Q and Q ;I;. = Q + 1, Q = 1, 2, . . . , 16. The numbers in the figure indicate the value of Q$,. in the area between the boundary lines to the left and to the right of that number.

4. Poisson demand

In this section we show what the effect of intro- ducing variability in the demand is. To that end, we compare the optimal ordering quantity of the previous section to the optimal quantity in comparable models, with demand modelled by a Poisson process with intensity A. In Section 4.1 we derive the ordering quantity that minimizes the average cost. The optimal ordering quantity Q& (* for optimal, a for average cost, p for Poisson demand)

is shown to be equal to Q.&. In Section 4.2, we consider the discounted cost criterion. The optimal order quantity in this case is denoted by Q$,,, (* for optimal, d for discounted cost, p for Poisson demand).

4.1. Poisson demand and average cost criterion

The goal is to minimize the average expected cost over the planning horizon [IO, cc). If the ordering


quantity is Q, the ordering process can be seen as a renewal process where a renewal occurs every Q failures. To derive the average expected cost of this renewal process we apply the renewal reward theorem, which states that the average long-run cost of a renewal process is the expected cost during a renewal cycle divided by the average length of a renewal cycle.

Definition 4.1. Let 0 6 t < t’ d cc. We define F!;;‘(Q) as the total discounted cost in period [t, t’), given that the size of an order is always Q E (1, 2, . . . } and the first order is placed at time 0.

The goal is to find the Q that minimizes E(F&m’(Q)). We first determine E(F!y$(Q)) by conditioning on the time at which the first ordering cycle ends, i.e. the time at which the Qth demand occurs. Let Mi, i = 1, 2, . . , be the time at which the ith demand occurs. Since demand is represented as a Poisson process with intensity A, Mi is Gamma- distributed with parameters i and 1 (density gi,k).

We first determine the expected cost in a ordering cycle by conditioning on M,. Since demand is modelled by a Poisson process and shortages are not allowed, an order is placed if a demand occurs and the stock at that moment is 1.

=K+Q+(h+a)E [

f Mi(M,=t i=l 1

=K+Q+(h++Q-l$+r)

=K+Q+ (h + d(Q + ‘It.

2

Since M, is Gamma distributed with parameters Q and 1 (mean Q/1), the average long-run cost of applying ordering quantity Q is

Fan(Q) = EF:(‘i”~‘( Q)

EM,

=K + Q + 0 + 4(Q + l)P)Q/i Q/A

=K,,+A+ (h + 4(Q + 1) Q 2 .

Comparison with Section 3.1 shows that

L,(Q) = L(Q) + (h + 4

That is, under the average cost criterion Poisson demand is equivalent to uniform demand. Conse- quently, both models have the same optimal ordering quantity

Q: := Q:p = QZ,,.

Remark 1. Apparently, the randomness in the moments of demand is irrelevant for the average expected costs. The constant difference, h + ~1, has an obvious explanation. Since shortages are not allowed, in the Poisson case a new order is placed as soon as the stock drops to 0, so that at all times the stock is at least one. On the other hand, in the uniform case the moments of demand are deterministic, so that a new order is placed as soon as the stock drops to - 1. As a consequence, the average stock in case of Poisson demand is precisely one more than in case of uniform demand.

4.2. Poisson demand and discounted cost criterion

The goal is to minimize the total discounted expected cost over the planning horizon [0, co), given that at time 0 the stock is empty and hence an order is placed. We discount all costs to time 0.

Definition 4.2. Let 0 < t < t’ < co. Analogous to Section 4.1, we define F;;:“(Q) as the total discounted cost in period [t, t’), given that the size of an order is always Q E { 1, 2,. . . } and the first order is placed at time 0.

We first determine E(F&“,“‘(Q)) by conditioning on the time at which the first ordering cycle ends, i.e. the time at which the Qth demand occurs. Let Mi, i = 1, 2, .., , be the time at which the ith demand occurs. By definition we have

E[Fiy;P”‘(Q) 1 MQ = t] = E[Fb:g’(Q) ( MQ = t]

+ E[Ff*“‘(Q) 1 MQ = t]. .P

Since the expected discounted cost in [MQ, co)

given MQ = t is equal to the cost in [0, co), except


for a discounting factor, it follows that

E[F;“$(Q) ( M, = t] = E[F[d9;‘(Q) I M, = t]

Therefore,

W!&"(QWfp = tl

+ e-“‘E[Ff’;P”‘(Q)].

Since E,Je -I”p) = (L/(2 + CX))~ (see Eq. (A.l) in _ e-l{, _ emar)). at

Appendix-A), we have Although at deriving this formula we implicitly

W[dtk'(QN = &&W?;P"'(Q)I MQ]) assumed that Q > 2, the result is valid for Q = 1 too. Using Eqs. (A.l) and (A.2) in Appendix A it

1 m

1 -W/v. + 41Q I ~~&FC%;'(Q~/MQ = tldt. follows that

0

(4.1) Fd,p(Q) = W%:;;"'(Q))

Hence, in order to get a formula for E(F$y;m’(Q)) we only have to calculate the conditional expected cost = i - (n/(/X + Cr))Q 1 (K+Q+$-&)~

in [0, MQ) given MQ = t. This is done as follows

-W!?;'(Q)I M, = tl

+g--)"'-;))

e?dsjM, = t 1 = K + Q + hE F !(I --e-aM’)(Mp = t i=l a 1 =K+Q+; E

(II ‘&-e-““‘)lMp=t i=l 1

+ (1 - emat) )

.

Since under the condition MQ = t the distributions of M Ir . . . , Mp 1 can be described as the result of ordering Q - 1 independent random variables U1, . . . , uQ_ 1 that are uniformly distributed on

CO, t), we get

Q-l

1 [ Q-1

i;l (l-e-“Mz)IMQ=t =E 1 (l-e-““J) j=l 1

= (Q - l)E[l - eP”I]

=(Q - 1) i

di(l -e-““)du

= 7 - (/l/(1 + a))Q ’ (K+Q+~(Q+(~F

- l)(&JQ -i))

= 1 - (n/(n + ct))Q ' (~+Q+z(j

1

= i - (L/(1 + L7))Q K+Q+;Q

hA = -3+

a l-(n!fn+.))Q(K+(l+~)Q)

hA h =--- acc+ l+- ( >

(KaAa + h) + Q) (4.2)

a 1 - (A/(2 + a))Q ’

In Appendix B we show that Fd,JQ) is convex with respect to Q.


We now will discuss the characteristics of the optimal order quantity Q&,. Since F&Q) is convex with respect to Q, the optimal Q is characterized as follows

Q = 1 is optimal if and only if AF,,,(l) > 0, (4.3)

Q > 2 is optimal if and only if AF&Q - 1) < 0

Q AFd,,(Q).

Moreover, if AFd,p(Q) is precisely 0, then both Q and Q + 1 are optimal. We can rewrite

In general, it is not possible to derive a closed-form formula for Q& However, we can give a simple condition for optimahty of Q:,, = 1. This is an important condition, since many inventory models for slow-moving items are based on the assumption

K”: 15 a+h

234 5

I

I

234

AF,+(Q) 3 0 as (see Appendix D)

1 - (L/(2 + a))Q K KC!

(/?/(A + ct))Q(a/(l + a)) - Q 3 1 + (h/a) = a+

0 2 4 6 8 10 x c*

Fig. 3. Boundary lines between Q& = Q and Qz,, = Q + 1, Q = 1, 2,. . , 17. The numbers in the figure indicate the value of Q$, in the

area between the boundary lines to the left and to the right of that number.


that one-for-one replenishment is optimal. Using Eq. (4.3) we can rewrite AF,,,(l) > 0 as

1 - (A/(/? + a)) KG!

(A/(1 + M))(M/(i + c()) - l a CI + h’

which is equivalent to

a+h KG---

/I .

Therefore, if K is at most (a + h)/i, one-for-one replenishment is optimal. If this condition is not satisfied then the optimal ordering quantity is at least 2.

Note that Q&, only depends on two characteristics, namely k/a and Ka/(a + h). We can therefore graphically represent Qt,. Let x = i/a and y = Ka/(a + h). In Fig. 3, the boundary line between Q& = Q and Qz,, = Q + 1 is determined by

1 - (X/(X + 1))Q

y = (x/(x + l))Q(l/(x + 1)) - QI (4.4)

Q = 1, 2, . . , x = A/a, y = Ka/(a + h).

With Fig. 3, Qz,p can be determined for any set of parameter values.

All four models have now been discussed. In the next section the optimal ordering quantities will be compared.

5. Comparison of the models

In the previous sections we analyzed four different models and determined the optimal ordering quantity for each one. These models are: 1. uniform demand, average cost criterion (Q& =

Qz, approximated by EOQJ, 2. uniform demand, discounted cost criterion (Q&), 3. Poisson demand, average cost criterion (Q& =

Qa*), 4. Poisson demand, discounted cost criterion (Q&J.

In this section we compare these four models. In Section 5.1 it is shown that the optimality of Q only depends on the same two characteristics for all models. In Sections 5.2 and 5.3 we discuss the influence of the variability of the demand and the effect of discounting on the ordering quantity.

5.1. C’haracteristics of the parameter values

It is interesting to observe that in all models the optimality of Q only depends on the same two characteristics, one measuring the demand rate i, and the other measuring the fixed cost K, relative to price 1 and holding rate h (both adjusted to the discounting parameter a), namely

x = Ala and y = Ka/(a + h).

For this reason, not only the optimal ordering quantities can be determined quite easily for each of the models separately, from Figs. 1-3, but also it is easy to compare the optimal solutions. In all three models the boundary line for which both Q and Q + 1 are optimal, is the graph of a convex function y = he(x), decreasing from + 00 at x = 0 to 0 at x = c;cj . These functions are, for Q = 1, 2, . . . ,

hap(x): = 1 - (x/(x + 1))”

(x/(x + l))Q(l/(x + 1)) - Q, x = O7

4(x): = Q(Q + l), 2x

x = 0

h;“(x): eQ/x _

= 1 _ e_ I,X 1 - Q, x = 0,

respectively. In Appendix C it is shown that in all cases the optimal ordering quantity is nondecreas- ing in both characteristics x and y, as is to be expected from their interpretations. The graphs suggest that for all x > 0 and all Q = 1, 2, . . .

hQ(x) < hsP(x) < ha”(x),

implying that for all values of the characteristics x and y,

Q * < d,u ’ > QZp 6 Q: = EOQa,

where the results for the average cost model seem to approach those for the discounted Poisson demand model better than those for the discounted uniform model. In the next subsection we will go to more details. The differences between the results of the three models can and will be analyzed in terms of the two characteristics x and y. Special attention is paid to slow-moving items.

184 W.K. Klein Haneveld, R.H. Teunterjlnt. J. Production Economics 54 (1998) 173-192

5.2. The influence of the variability of the demand

When the average cost criterion is used, the optimal ordering quantities for Poisson demand Qa*,, and uniform demand Qa*,” are equal. Hence, the variability of the demand seems to have no influence on the optimal ordering quantity if the average cost criterion is applied.

In this section we compare the optimal ordering quantities of the model with Poisson demands (Sec- tion 4.2) and of the model with uniform demand (Section 3.2) both with the discounted cost criterion. The first model has a high variability in the moments that demands occur, whereas the second one has fixed interarrival times. All parameters are supposed to be equal, hence also both characteristics x = 1/a and y = &~/(a + h) are equal, so the differences between the models can be attributed to the influence of demand variability. Lemma 5.1 states that uniform demand with interarrival times l/X = (l/cr)ln(l + E/A) is ‘equivalent’ to Poisson demand with rate 1. By F,,,(Q,n) we denote the expected discounted cost if demand is uniform with interarrival times l/A and all other parameters are fixed. By F&QJ) we denote the expected discounted cost if demand is Poisson distributed with rate 1 and all other parameters are fixed.

Lemma 5.1. Under the discounted cost criterion, Poisson demand with arrival rate A’ 2 IE is equivalent to uniform demand with rate 1, i.e.

F&Q,l) = Fa,JQ,l’) + h/cr for all Q = 1, 2, . . . .

Remark 2. As in the average cost case, there is an extra holding cost term h/cc, since (see Remark 1) on average the stock in case of Poisson demand is one larger than in case of uniform demand. Contrary to the case of average cost, where Poisson demand is equivalent to uniform demand with the same arrival rate, in case of discounted cost Poisson demand is equivalent to uniform demand with a larger rate.

Proof. Since (rewriting Eq. (4.2))

hIa h - 1 - (A/@ + a)) + c(

and (see Eq. (3.2))

hIa 1 _ e-“/l’

F,,,(QJ) = F&Q,,?‘) + h/a is true if

A -a/i’ G=e ,

that is

a F=


q

As a consequence of Lemma 5.1 it holds that

Qd*p>Qd*u, since Q& is increasing in 1 (see Lemma C.1 in Appendix C). Also, the condition for Q&, = 1 is more restrictive than the condition for Q& = 1. This can also be seen by comparing Figs. 2 and 3. The exact conditions for optimality of one-for-one replenishment in the models with Poisson and uniform demand are

Q& = 1 if and only if y 6 e”” - 1,

Q;,, = 1 if and only if y 6 l/x,

respectively. Especially for small values of x, i.e. for slow-moving items, their is a considerable difference between these conditions.

Comparison of Figs. 2 and 3 also reveals that for all O<x<lO and 0~~~30 it holds that Qz,, d Q& + 1. Hence, the variability of the demand has only very little effect on the optimal ordering quantity. There are extreme cases, where the difference between Qt, and Q& is more than one (see Example 5.2).

Example 5.2. Let n/a = 1 and Ka/(a + h) = 200. Then Q&, = 7 and Q&, = 5.

5.3. The efSect of discounting

In this section we compare the optimal ordering quantities of the model with the average cost

W.K. Klein Haneveld, R.H. Teunterjlnt. J. Production Economics 54 (1998) 173-192 185

criterion and of the model with the discounted cost criterion. In Section 5.3.1 we compare the two models with uniform demand. In Section 5.3.2 we compare the Poisson demand models. We assume in both subsections that the goal is to minimize the total discounted expected cost, and hence the NPV-approach is interpreted as the ‘right’ approach, whereas the results of the corresponding ACC-models are seen as approximations.

Table 2

Relative deviation of Q$ compared to Q&, i.e.

C(Q: - QXJ/Qd*..l x 100%

Kz!(r + h) i,Jr

2 4 6 8 10

5.3.1. Uniform demand We compare Qz and Q& using Figs. 2 and 3.

First of all, note that Qz is larger than Q&. This was to be expected, since in determining Q$ holding costs are underestimated (see Introduction). One of the consequences is that the condition for Qz = 1 is more restrictive than the condition for Q& = 1. These conditions are

30 83% 50% 46% 41% 41%

25 61% 56% 42% 43% 38%

20 80% 63% 36% 38% 33%

15 60% 38% 30% 25% 31%

10 50% 22% 38% 30% 21%

5 33% 20% 33% 13% 11%

Table 3

Relative cost increase of Q: compared to Q&. i.e. [(F&Q:)

- Fd.,(Q~.u))/Fd,“(Qd*,u)l x 100%

Q,* = 1 if and only if y < l/x,

Q&=1 ifandonlyifyde”“-1,

Ka/(cc -t h) h/cc I&

2 4 6 8 10

respectively. In general, the (relative) gap between the solutions seems to be decreasing in J/cc and increasing in Kc~/(GL + h). This is illustrated in Table 2. Note that there are some exceptions to these rules. This is explained by the fact that we are dealing with discrete quantities. In interpreting Table 2, one has to bear in mind that all examples concern slow-moving items. For instance, 1,‘~ = 10 could correspond to an annual interest rate of 20% and an annual demand of 2 items.

30 0 8.7% 5.8% 5.3% 4.5% 3.3%

30 0.5 8.9% 6.0% 5.6% 4.8% 3.5%

30 2 9.1% 6.2% 5.9% 5.1% 3.1% 25 0 8.0% 5.8% 4.3% 3.9% 2.9% 25 0.5 8.2% 6.1% 4.5% 4.1% 3.1%

25 2 8.4% 6.3% 4.8% 4.4% 3.3% 20 0 1.1% 6.0% 3.3% 3.4% 2.1% 20 0.5 8.0% 6.3% 3.5% 3.6% 2.9% 20 2 8.2% 6.6% 3.8% 3.9% 3.2% 15 0 1.5% 4.4% 2.6% 2.0% 1.1% 15 0.5 1.8% 4.7% 2.8% 2.1% 1.9% 15 2 8.2% 5.0% 3.0% 2.3% 2.1% 10 0 4.0% 3.2% 2.3% 2.2% 1.2% 10 0.5 4.2% 3.5% 2.5% 2.4% 1.3% 10 2 4.5% 3.8% 2.8% 2.1% 1.5%

5 0 1.1% 1.0% 1.4% 0.8% 0.6% 5 0.5 1.2% 1.2% 1.1% 0.9% 0.1% 5 2 1.3% 1.3% 1.9% 1 1% 0.8%

It appears that, especially for slow-moving items with large fixed ordering cost, approximation Qz is much larger than the optimal value Q&. Now the question arises what the effect of applying the wrong order quantity on the discounted cost is. We showed that the difference between Qz and

Qd*,u only depends on the two characteristics n/cc and &/(a + h). Using Eq. (3.2) it can easily be Since the stock level is 0 at the start, at least one checked that the discounted cost increase depends item, at price K + 1 has to be ordered at t = 0. All on the same two characteristics and on the value of additional costs are influenced by the choice of the h/a. Table 3 shows the discounted cost increase in ordering strategy. Hence, we define the influential a number of cases. discounted cost as

The performance of any cost minimizing strategy should be based only on the effect of that strategy on the costs it can influence. In this case, a part of the costs are independent of the ordering strategy.

F::’ = F,,, - K - 1.

Table 4 shows what the effect of applying the wrong order quantity on the influential discounted

186 W.K. Klein Haneveld, R.H. TeunterJInt. J. Production Economics 54 (1998) 173-192

Table 4 Relative influential cost increase of Q: compared to Qt.., i.e. [(Ftt(Q.*) - F$t’(Qz”))/F~t’(Q,*.“)] x 100% (h is set 0)

KaJ(a + h) A/a

2 4 6 8 10

30 47.7% 19.9% 14.8% 10.8% 7.1% 25 39.5% 18.4% 10.9% 8.6% 5.9% 20 33.5% 17.1% 7.7% 6.9% 5.0% 15 28.3% 11.1% 5.4% 3.6% 2.4% 10 12.5% 6.9% 4.1% 3.6% 1.8% 5 2.6% 1.8% 2.2% 1.1% 0.8%

cost is. We set h = 0, since h has only a minor influence on the cost increase (see Table 3).

Using Tables 24 we analyze the effect of discounting on the ordering quantity the uniform demand case. Table 2 shows that there can be a considerable difference between Q& and Qz, especially when I/or is small and Ka/(a + h) is large. As a consequence, in these cases there is a large influential cost increase (see Table 3) when Qz is applied whereas Qtu is optimal. Hence, the performance of Qz is very poor for slow-moving items with large fixed ordering costs. On the other hand, the discounted cost increase is less than ten percent in all cases. In fact, only in cases where n/cc is small and Kcr/(cc + h) is large the cost increase is more than five percent. Hence, in general, applying the traditional EOQ-analysis gives good results. The cost-robustness of Qz can be attributed for a large part to the fact that, in cases with a large cost increase, a large part of the cost is independent of the ordering strategy.

We end this section with a comparison to results of previous studies concerning the effect of discounting on the ordering quantity. Rachamadugu [29] considers an EOQ-model with a fixed continuous demand rate instead of uniform demand. It is concluded that the optimal ordering quantity for the discounted model is at most the optimal solution for the average cost model. This is consistent with our findings. Also, bounds for the increase in cost are derived. Rachamadugu concludes that

As the ratio of noncapital related holding costs (such as material handling and warehousing) to

the total holding charges increases, the relative error for the total discounted cost increases.

In other words, the relative error is increasing in h/(h + a), and hence increasing in h/a. However, this result only holds if the values of L/CX and Ka/(a + h) are fixed. In fact, if the value of h/a is varied, while all other model parameters are fixed, the effect is just opposite. This can easily be checked using Table 3. Assume for instance that a = 0.2, /I = 1.2 and K = 30. If h = 0, then 3L/c( = 6, &~/(a + h) = 30 and h/a = 0, so that the cost increase is 5.3%. If h = 0.4, then J/a = 6, Ka/(ct + h) = 10 and h/cr = 2, so that the cost increase is 2.8%.

5.3.2. Poisson demand Since in most cases Qz,” 6 Qz,,p $ Q& + 1 and

since Qf = Q& = Q& the effect of discounting on the ordering quantity if demand is Poisson is almost equal to the effect if demand is uniform. The results for the average cost model approach those of the discounted Poisson demand model some- what more closely than those of the discounted uniform demand model. In fact, the conditions for optimality of one-for-one replenishment in the average cost model and the discounted Poisson demand model are equal. This condition is

Qz = Q$,, = 1 if and only if K d (a + h)/l.

We end the comparison of Qz and Q.& here, and refer to Figs. 1 and 2 and the results of the previous section for more details.

6. EOQ-formula for slow-moving items

In the previous section we showed that, if the NPV-approach is interpreted as the ‘right’ approach, for items with a small demand rate and large fixed ordering costs the EOQ formula does not give nearly optimal solutions. As an alternative, we presented graphs with which the optimal quantity can be determined, both in the case of uniform demand (Fig. 2) and in the case of Poisson demand (Fig. 3). In this section we derive a simple order quantity formula, denoted by EOQd, that performs better for slow-moving items than EOQ,.

W.K. Klein Haneveld, R.H. Teunterjlnt. J. Production Economics 54 (1998) 173-192 187

We first comment on an approximation for Q&which is derived in Bartmann and Beckmann [30]. They propose as an ordering quantity

2K3.

J-- ai+

This approximation however is not very good. For small demand rates (i < l), it is even larger than EOQa. For large demand rates it is much too small.

In determining EOQd we assume that there is a uniform demand. We expect EOQd to perform well for a large number of demand distributions, since there is hardly any difference between

QLI and Q&. We therefore expect the demand distribution to have only little effect on the optimal ordering quantity. The total discounted cost, when demand is uniform, is (see Eq. (3.2))

F&Q) = l_d-~~,i(K+Q+~(Q-l~~~~:))

= Const. + &++(~+~)Q)~

We use a Maclaurin expansion to approximate F,+ In Grubbstrom et al. [18,19] these expansions were shown to be useful and accurate in determining the total discounted cost.

F,,"(Q) = Const. + h+l+!Q_ C13Q3

aQ 2 122 ??$

+ + + (1 +;)Q).

In Table 5 the accuracy of the 2-term and the 3-term approximation are shown for different values of olQ/L.

The reader can easily check using Fig. 2 that in most cases clQ&/A is less than 5, and hence in these cases the 3-term approximation (near Q&J is within 10% of the true value. Considering only the first two terms of the Maclaurin expansion, minimizing F&Q) leads to the well-known square root formula EOQ, (this is left to the reader). See also Grubbstrijm and Thorstenson [18], where it is shown that applying a 2-term approximation, when there is a fixed continuous demand rate and a constant production rate, leads to the well-known

Table 5

Maclaurin approximation error

(aQ)/i Maclaurin approximation error

2-term (%) 3-term (%)

0.1 - 0.1 0.0

0.5 - 1.6 0.0

1 - 5.2 0.1

2 - 13.5 0.9

5 - 30.5 10.9

10 - 40.0 43.3

finite production rate square root formula. How- ever, when we include another term, we get

F&Q) z Const + ?- + A+ 2 LXQ 2 122 >

++(I+;),)

= Const’ + Fi + TQ+~$Q

a+h +=Q2 = G,(Q).

We minimize F&(Q) by solving i3/aQ(F,“,JQ)) = 0. This gives

Ki 1 a+h c&C or+h ---z+-

ci Q 2!X +i%+ 6A -Q=O,


6Ki2 -_- a(a + h)

+zQ”+ &Q2 + Q3 = 0. (6.1) ct

Solving this equation does not give an easy formula for Q. We can however determine a simple upper bound for the optimal Q by solving

6KA2 ” -___ cr(a + h)

+?Q’+ a

&Q2 =a (6.2)

Recall, that there is a large difference between EOQ, and Q&, when A/a is small and Kcr/(c.t + h) is large. But, for large values of Ka/(a + h), the solutions of Eqs. (6.1) and (6.2) will not differ much.

188

Table 6

W.K. Klein Haneveld, R.H. TeunterJInt. J. Production Economics 54 (1998) 173-192

Comparison of Qz, EOQ,, Q$,., Q& and EOQd

n/a K&Au + h) QZ EOQa QZ,u Qb, EOQd Cost increase”

Total

EOQa EOQd

Influential

EOQa EOQd

10 5 10 10 9 9 10 0.6% 0.6% 0.8% 0.8%

2 30 11 11 6 7 6 8.7% 0.0% 47.7% 0.0%

3.3 60 21 20 10 11 10 10.1% 0.0% 59.1% 0.0%

4 200 40 40 16 17 13 9.2% 0.7% 105.0% 8.2%

“discounted cost increase compated to Q& if demand is uniform.

Solving Eq. (6.2) and rounding to the nearest strictly positive integer gives

EOQd := 2K1/(a + h)

1 + (Kc?/6/2(~1 + h)) 1 EOQa

E ’ 1 + (&‘/64x + h)) 1 In Table 6 EOQd is compared to Q&, Q&,

Qz and EOQa. Note that EOQd, like Qz,,, Q&, Qz and EOQ, only depends on the two characteristics ~/CI and Ka/(ct + h).

The performance of EOQd seems good. Indeed, for cases with small demand rates and large fixed ordering costs, EOQd is much closer to Q& (or Q&J than EOQa. Note that, although EOQd is an upper bound for the solution of Eq. (6.1), in the last example of Table 6 we have EOQd < Q&. This can occur, since we used an (Maclaurin) approximation of the expected discounted cost.

7. Conclusion

We analyzed four different models and determined the optimal ordering quantity for each one. These models are:

uniform demand, average cost criterion (Q& = Qz, approximated by EOQJ, uniform demand, discounted cost criterion (Q&), Poisson demand, average cost criterion (Q& =

Q3> Poisson demand, discounted cost criterion (Q&j.

The optimality of Q in all four models mentioned above only depends on two characteristics, namely n/a and Ka/(a + h). We graphically represented and compared the optimal solutions of all three models, and gave conditions for optimality of one- for-one replenishment. First, we analyzed the influence of the variability of the demand on the optimal ordering quantity. We showed that Q& is always at least Q&, and in almost all cases at most Q& + 1, and that Q& = Qz,,, = Qz. Hence, the variability of demand seems to have only a minor influence on the optimal ordering quantity. Second, we analyzed the effect of discounting on the optimal ordering quantity. For fast moving items there are only small differences between Qz, Q&, and Q&. For slow-moving items with large fixed ordering costs Qz can substantially deviate from Q& z Q$,,. In fact, we saw that applying Qz in a discounted cost model can cause large increases in the influential discounted cost increase, especially when items are slow moving and have large fixed ordering costs. However, in such cases, a large part of the expected costs cannot be influenced by the ordering strategy. Hence, even in cases where there is a large difference between Qz and Q&, the discounted cost increase is small, confirming the robustness of Qz and EOQ,. Finally, we derived an alternative but still simple EOQ-model for the discounted cost model, that is much closer to Q& than

EOQa. Other studies have shown (see for instance

Chand and Tang [31]) that under slightly different modelling assumptions the difference between the ordering quantities resulting from the ACC and the


NPV-approach is larger. It would be interesting to see whether the characteristics A/a and Ka/(a + h) also play an important role in these models, and can be used as an analyzing tool.

We show that F(Q) is strictly convex by showing that g1 and g2 are strictly convex.

Lemma B.l. Functions g1 and g2 are strictly convex.

Appendix A. Expectations of functions of the Gamma distribution

Proof. Since

gI(x) = f, eekx k=O

Let M be Gamma distributed with parameter n and II, n = 1, A > 0. Its density gn,i is denoted by

is a sum of convex functions, g1 is convex in x for all x > 0.

g&t) = “tnp ‘epnr, r(n)

t 2 0.

Then

The first derivative of g2 is

d &S2(4 =

1 _ eeX _ xeex

(1 -e-“)* ’

and

The second derivative of g2 is n>l, a> -i, 64.1) d2

&-$I*(4 =

So that

E(n-l)s=;(l -($--)‘-‘),

- 2e-“(1 - eeX - xeeX)) -x

= (1 “,_,,),(x + xeFX - 2 + 2eF”)

u2e-m

= (, _ ,-,$U +e-“) (; - G)

P-’

(B.l)

>o %h(r)

n 3 2, I* > - c1. (A.2) We next show that h(x) is positive for all x > 0.

Appendix B. Convexity of the discounted cost-function

Claim 1. The following holds for all y > 0:

Y ’ l- e-2y 1 + ep2”

Both in the uniform demand case and in the case where demand is modelled by a Poisson process, the total discounted cost in [0, co) is of the form

bQ + c F(Q) = Const. + 1 _e_aQ, Q = 1, 2 ,...,

where a > 0, b > land c > 0. We define functions

g1 and g2 as

91(x) = L 1 _ e-x’ XE(O, cc) and

It is easy to see that the claim holds for y > 1. Let 0 < y < 1. We can rewrite the claim as

I-Y - e-2Y’1 +Y

for all 0 < y < 1,

which holds if and only if

e2y + 1 1 __- - 2 <l-y

for all 0 < y < 1.

Using Taylor expansions this can be written as

g*(x) = -?- 1 _ eex’ x E (0, co ). f PnY" < f Y" for all 0 < y < 1,

n=O n=O

190 W.K. Klein Haneveld, R.H. TeunterJlnt. J. Production Economics 54 (1998) 173-192

where p. = 1 and

1 2” P”=j; foralln>l,

Since p. = p1 = 1 and pn = (2/n)p, _ I for all n > 2, we have that P,, < 1 for all II 2 0 with strict inequal- ity for n 2 2. Hence, Claim 1 holds.

From Eq. (B.l) and Claim 1 it follows that h(x) > 0 for all x > 0. Hence g2(x) is strictly convex. 0

Appendix C. Derivatives of functions /Z&X)

Let x = A/CI and y = Ka/(cr + h). In Section 5 it was shown that the boundary line for which both Q and Q + 1 are optimal is the graph of a function y = he(x), Q = 1, 2, . . . , where these functions h,(x) are

hap(X): = 1 - (x/(x + 1))Q

(x/(x + l))Q(l/(x + 1)) - QY x = O,

&(x): = Q(Q + 11, 2x

x = 0 2

eQ/x _ 1

h;“(x): = 1 _ e- I,x - Q, x = 0,

respectively. In this Appendix we prove that in all cases the optimal ordering quantity is nondecreas- ing in both characteristics x and y. For this, it is sufficient to proof that the functions h,(x) are increasing in Q and decreasing in x. It is easy to see that h;(x) has these properties.

Lemma C.1. Function ha”(x) is increasing in Q = 1, 2, . . and decreasing in x > 0.

Proof. We first show that ha”(x) is decreasing in x for all x > 0. This is equivalent to showing that @“(l/s) is increasing in s for all s > 0. Differenti- ating @“(l/s) with respect to s gives

(1 _ e-s)2$h$” 0 f s

= (1 - e-“)Qe@ - (eQs - l)e-’

= c@(Q - (Q + l)e-’ + e-cQ+l)y.

Note that for all Q > 1 the last term on the r.h.s. is 0 if s = 0. Moreover, since

$Q - (Q + l)e-’ + e-(Q+‘)s)

= (Q + l)ems - (Q + l)e-(Q+l)s > 0

for all s > 0 and Q > 1, it holds that

> 0 for all s > 0 and Q = 1, 2, . . .

Next, we show that ha”(x) is increasing in Q = 1, 2, . . . for all s > 0. This is equivalent to showing that @“(l/s) is increasing in Q for all s > 0. Differentiating @“(l/s) with respect to Q gives

Using a first-order Taylor expansion of ems we have

dhd,U J. 0 seSQ >--l>O.

dQQs s

Hence, ha”(x) is increasing in Q for all s > 0. 0

Lemma C.2. Function hsP(x) is increasing in Q = 1, 2 ) . . . and decreasing in x > 0.

Proof. We rewrite hap(x) as

hap(x) = 1 - (x/(x + 1))Q

(x/(x + l))Q(lMx + 1)) - Q 1 _ ,-QlW+W)

= ,-QW +W(l _ ,-Ml +(W) - Q.

Note that ln(1 + (l/x)) is positive and increasing in l/x for all x > 0. Applying Lemma C. 1 completes the proof. 0

Appendix D. Rewriting Fd,,(Q) > 0

In Section 4.2 it is shown that

1 Fd,p(Q) = - 2 + 1 _ (n/(n + a))Q

x[:+[l +->e> h

CC .

WK. Klein Haneveld, R.H. Teunterllnt. J. Production Economics 54 (1998) 173-192 191

Hence, AF&Q) > 0 can be rewritten as

hi -3+

@I 1 - (n/(2 ' (K+(l+$Q+i,) + M))Q+ l

h2

a - 2 + 1 - (I_/@ + cc))Q ' (K+(l+$Q),


b +(I +a)e)(1 -ii,(~+r))O'l

1 ) 1+1 -l-(A/(A+cc))Q +l-(n/(n+a))Q+‘aO~

This can be written as

@+(l+k)Q)((&JQ+l-(A)Q)

+(I +~)(1-(g-)Q)20,

Or

(I+:)* -(K+(l +;)Q)aO.

Therefore, AF&Q) 2 0 can be rewritten as

1 - @/(A + a))Q K Kc!

(A/(/l + ct))Q(cc/(l + 2)) - Q 3 1 + (A/Lx) = - a + h’

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Effects of discounting and demand rate variability on the EOQ

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