The effect of discounting on inventory lot sizing models

Engineering

Costs and Production

Economics,

Elsevier Science Publishers B.V., Amsterdam

35

16 ( 1989) 35-48 - Printed in The Netherlands

THE EFFECT OF DISCOUNTING ON INVENTORY SIZING MODELS

LOT

Chan S. Park Department

of Industrial

Engineering,

Auburn

University,

AL 36849

(U.S.A.)

and Young K. Son Departmenr

of Management,

Baruch College, The City University

of New York, New York, NY 700 IO (U.S.A.)

ABSTRACT

This paper examines the conditions under which discounting effects should be considered in modeling inventory systems. Four cIassica1 i~?ventor~modems were considered: basic EOQ, finite product.ion rate, backorder, and general.

The results indicate that the significance of the discounting effects depends on the parametric values in the first two models; whereas the discounting effects are sign~~cant regardless of the parametric values in the last two models.

INTRODUCTION

However, Kim and Chung [ 5 ] pointed out that Gurnani made several fundamental errors. Accordingly, the signi~~ance of the dismounting effects on other models could not be addressed in perspective. Other authors also addressed the combining effects of inflation and the time value of money on the basic EOQ [ 1] and the finite production rate model [2 1. Inflation is not discussed in this paper. The brief literature review indicates that most conclusions on the discounting effects on EOQ were drawn from a typical set of numerical examples. Most authors failed to recognize some parametric relationships in the EOQ models, and these relationships determine under which condition the discounting effect can be significant. Therefore, some of the conclusions drawn from a typical numerical example may not be valid for other situations. Inventory investment decisions are based on the to-

Recently, several authors addressed the discounting effects on classical inventory economic lot sizing models [ l-7 1. Different views were held, however, in assessing the discounting effects on economic order quantities (EOQ ). Hadley [ 41, Misra and Wortham [ 7 ] developed discounted cost expressions of the basic EOQ model. They compared EOQ between the discounted cost approach and the conventional average annual cost approach, concluding that the discounting effects on EOQ were negiigib~e. Iladley further suggested that the same conclusion would be observed in other EOQ models. More recently, Gurnani [ 3 ] expanded the development of discounted cost expressions to other EOQ models and claimed that the discounting effects on EOQ were substantial. 0167-188X/89/$03.50

0 1989 Elsevier Science Publishers B.V.

36 tal present equivalent cost savings. Hence, the critical issue to resolve is how well the conventional EOQ will perform when discounting is present. This paper examines the various conditions under which the discounting effects on EOQ and on total costs can be critical. The objective is to minimize the total costs per unit time. (Kim et al. [ 5 ] performed a study similar to [ 3 ] but based their analysis on profit maximization rather than on cost minimization.) A brief review of assumptions on the classic inventory models is followed by their discounted cost models, and by a sensitivity analysis of some critical inventory parameters. THE TRADITIONAL ANALYSIS

INVENTORY

In classic inventory models, inventory carrying costs are not distinguished analytically into two component costs: opportunity costs (capital cost ) and out-of-pocket costs. They are commonly classified into four inventory systems or models (see Fig. 1) - the basic (BA), the finite production rate (FP), the backorder (BO) and the general (GE) - with the following operating conditions: BA model - infinite replenishment rate without stockout, FP model - finite replenishment rate without stockout, BO model - intinite replenishment rate with stockout backordered, and GE model - finite replenishment with stockout backordered. Both the BO and GE models assume that backorders are satisfied before a new demand is received. The BA and BO models assume that order quantities are delivered in a single lot at the same time which the FP and GE models assume items are delivered continuously during a production period. For each model we assume cost minimization to be the objective function in seeking the optimal lot

;

Fig. 1. Classical inventory systems: Q=economic lot size. f(t)=demand function, T= inventory cycle, I, = maximum inventory level, b=backorder position.

size. Further, for each model we assume the following: . no consideration of time value of money and inflation, deterministic and known demand rate and lead time, constant unit cost, 0 sufficient storage space for inventory, no item being obsolete during inventory. Under these assumptions, the GE model’s average annual total cost (K), the optimal lot size ( Qa) and the maximum backorder level l

l

l

(b,) are K=AD/Q+CD+hC{Q(l-D/P)-bj2/{2Q(1 -D/P)}+{2abD+irb2/(1-D/P)}/(2Q),

(1)

31

(hC)]‘{(hC+ir)/fc}~ b,=(hCQ,-nD)(l-D/P)/(hC+ri),

(2) (3)

D = demand rate year, per rate per year, A = ordering (or setup) cost per cycle, C=purchase (or production) cost per unit, h=inventory holding charge rate including capital cost, rc=tixed backorder cost per unit, and Tz= variable backorder cost per unit per year. The other three models (BA, FP and BO) can be special versions of the GE model, respectively. To obtain the EOQ for the BA model, we let P+CC and n=72=0 in eqn. (2). To obtain the result for the FP’s Qa, we let r=ti=Oineqn. (2).LettingP+c0ineqns. (2) and (3) yields Q, and b, for the BO model. where

P= production

parts: cost of capital r and the remainder I. Under these assumptions and definitions, we can derive the discounted cost equation for each Instead of Iinding the inventory model. expression of the net present value (NPV) of the inventory costs over the planning horizon, we use the annual equivalent inventory cost AE(r) for computational simplicity. When a cycle of cash flows repeats, such as periodic patterns in an inventory system, the AE(r) is preferred to the NPV criterion since AE(r) is independent of a planning horizon. We will show this by first developing the AE(r) expression for the BA model. Consider Fig. 2 (a) for the basic cost components for the BA model. Since the order quantity Q and the demand rate D are constant (see Fig. 1 (a) ), the inventory level of the first cycle is f;(t)=Q-Dt,

DEVELOPMENT COST MODELS

OF THE DISCOUNTED

To develop a series of discounted cost models, we assume the following: ordering and purchase costs occur at the time of order delivery, inventory carrying and variable backorder costs occur continuously, fixed backorder cost occurs at the end of each cycle, all costs are discounted continuously over time, demand occurs continuously over time, present time (time 0) indicates an arbitrary delivery point in the business process, not the establishment point of a new business, the purchase cost is viewed as a cost for which there is no return flow (the inventory investment is not viewed as working capital). This view may not be realistic in practice but it is made to compare various models on a common basis. For discounting purpose, the annual inventory carrying charge rate h is divided into two

O
(4a)

Since f, (t) =O at t= T, we have T= Q/D. Let the discounted ordering cost for a cycle be PW( r) 1;the discounted purchase cost PW( r)2; and the discounted inventory carrying cost PW( r) 3. Assuming that the ordering and purchase costs occur at the beginning of each cycle, we obtain PW(r),

=A

(4b)

PW(r),=CQ

(4c)

PW(r),=/ZC(Q-Dt)exp(-rt)dt 0

(4d)

Then, the discounted cycle is PW(r),,,=PW(r),

total

cost for the first

+PW(r),+PW(r),

(4e)

Iem PWrhol =pWrLyc ,go (-r$Y 1 The total inventory

cost over N cycles is

N-

(4f) Finally, we obtain the annual equivalent

cost

38 ~~(r)=~Wr),otl(e’-

1I/

[ 1 -exp(

-m)]},

where n=NQ/D

Note that the A?(r) in eqn. (4h) is independent of planning horizon n. We can obtain the annual equivalent cost expressions for other models in a similar fashion using Figs. 2 (b )(d). For the FP model, we have

(4g)

=PW(r),,,(e’-l)/[l-exp(-rT)],whereT=Q/D or,

[ 1 -exp(

-rQ/D)]

-ZCD/r’}

I

A/.

(5)

AE(r)=(e’-l)[{A+[C(I+r)P/r’]x

AE(r)=(e’-l){[A+C(I+r)Q/r]/

(a)

.

(4h)

cost

orderh

[l-exp(-rQ/P)]}/[l-exp(-rQ/D)]-ZCD/r*]

I

Al .

.

L

I

L

1

0

T

0

T

(b) Furchase

rmr

Cost

.

.

.

.

.

(a)

mrr!

.

.

.

.

.

set-up co!st

(b) ProductionCost .

.

a

BA Model

I

A/ .

.

I

I

0

T

(a)

OrderingCost

(b) Rvchase

Al.

.

0

T

I

Cost

.

..I..

.

I

1

I

.

I

I

T

2T

(d)

Fixed

cost

.

.

.

./

I

. I

BOModel

Frdudion cost

2T

T

Tl

T3

*

d

GE Model

in the classical inventory models (adapted from Ref. [ 31).

(d) Fixti

cost

:

,

Fig. 2. Cost components

(bl .

T

T2

Backorder

0

set-up CO&

!

0

.

.

(a)

2T

Backorder

39 For the BO Model, AE(r)={(A-IC/r’)+C(I+r)Q/r-ICb/r + (nbr-rib-irD/r)exp(

-rQ/D)/r

(6)

+(ZC+f)Dexp[-r(Q-b)/D]/r’}

or

ew[(r/D)Qdl

- (r/D)Qd

(f’-1)/[1-exp(-rQlD)l =[(r/D)/(~/r+1)l(A/C)+I

For the GE Model, AE(r)={A-Ec(P-D)/r’+(CP+fb)/r

(7)

+(ZC+f)(P-D)exp[-rb/(P-D)]/r’ -CP(I+r)exp(

-rQ/P)/r’

+ (rrbr-tib-fD/r)exp(

-rQ/D)/r

+(IC+ir)Dexp(-r(Q-b)/D]/r’}(e’-l)/

[I-ew-rQ/D)l Note that to obtain the BA model from the BOmodelwelet6=Oand7r=fi=Oineqn. (6). Similarly, we can obtain the FP model from the GE model by substituting b = 0 and n = 72= 0 in eqn. (7). DISCOUNTING MODEL

EFFECTS ON BASIC

In this section, we examine the discounting effects on both Q and AE( Y) for the BA model. To examine the effects on Q, we obtain numerically the optimal Qd that minimizes AE( r), then compare this Qd with Q, in eqn. ( 1). To examine the effects on AE( r), we compare the AE( Y) value obtained using Qd with that value obtained using Q,. 1. Discounting

effects on EOQ

The discounting effect on Q depends on the A/C value. To show this, we use eqns. (2 ) and (4).BylettingP+coandb=Oineqn. (2) and by rearranging the terms Qi=j2D/(Z+r)}(A/C).

(8)

Equating the first derivative of eqn. (4) to zero and rearranging the result yields

(9b)

Note that, as shown in eqns. (8) and (9>, Qa and Qd are functions of A/C. Table 1 which shows the numerical comparison of Qd and Q, for various parametric combinations, supports our critical observation. For cases 5-7 and 9- 12 in the table where A/C ( = 100-1200) is relatively large, the Q differences are significant; however, in the remaining cases where A/C ( = 0.1-7.5 ) is relatively small, they are insignificant. Cases l-7 are, in fact, Hadley’s examples [ 41; case 8, Misra and Wortham’s example [ 7 1, and case 9, Gurnani’s example [ 3 1, respectively. Note that these authors’ total present equivalent cost equations are consistent to our annual equivalent cost of eqn. (4) that should generate the same optimal Q under the present value formulation. With a cycle period T ( = Q/D) considered as the major factor for determining effects on Q, Hadley concluded that the discounting effects are negligible (cases 1-5) except for an unrealistic T value such as T> 10 years (cases 6 and 7 ). However, cases lo-12 indicate that this is not true: the Q difference is significant with some realistic T values (less than 10 years ) . Cases 6 and 7 may be unrealistic not because T is large, but because D-c 1. Misra and Wortham also made a conclusion similar to that of Hadley’s by just observing the result from case 8. Conversely, Gurnani concluded that the discounting effects are substantial, reasoning these to be due to the discount rate Y.Cases 1-4, 8 and 13- 15, however, indicate that r is not the major factor that causes the Q differential, if any. Analysis of Table 1 also indicates that the

40 TABLE

1

Optimal

Qd and Q, for the basic model

Ref.

Case

141 141

2

141

3

141

4

141

5

141

6

141

7

171

8

I31

9 10 11 12 13 14 15

by varying

r values

Q

Parameter

A=10 D= 100 A=O.SO D=O.SO A=25 D= 100 A=O.SO D= 1000 A=5 D= 100 A=10 D=O.SO A=50 D=O.SO A=150 D=508 A = 12000 D= 8000 A/C= 100 D= 1000 A/C=200 D=200 A/C= 500 D= 100 d/C=2 D= 100 AfC=2 D= 100 A/C=O.l D=20

c=5 IzO.10 C=O.lO I=O.lO c=5 I=O.lO c-=0.10 I=O.lO C=O.lO I=O.lO CEO.10 I=O.lO C=O.lO I=O.lO c=20 I=O.lO C=lO 1=0.03 1=0.25 1=0.05

I=O.Ol I= 0.20 1~0.30 I=O.lO

discounting effect on Q is insensitive to the other parameters ifA/Cis relatively small. For example, cases 2-4 show negligible Q differences for various D values; cases 1, 13 and 14, for various I values; and cases l-4, 8 and 1315, for various Yvalues. Note that A/C is less than 7.5 in all cases examined. If A/C is relatively large, however, the discounting effects are sensitive to these parameters. The following sensitivity analyses will support these findings: &ficts of A, C and r on Q differences. Figure 3 (a) shows that as A/C and/or Yincrease, the Q differences become large. When A/C= 500, the Q difference is 12 to 25% even for realistic rvalues (0.1 Grd0.3). Effects of D and r on Q differences. Figure

r 0.10

0.20

0.30

0.40

0.50

44 45 4 5 70 71 223 224 216 224 13 22 20 50 194 195 11853 12153 744 756 688 730 823 953 36 37 31 32 4 4

36 37 3 4 57 58 181 183 172 183 9 18 12 41 158 159 8802 9137 653 667 517 566 562 690 31 32 28 28 3 3

31 32 3 4 49 50 157 158 147 158 7 16 9 35 136 138 7281 7628 585 603 427 478 443 568 28 28 25 26 3 3

28 28 2 3 43 45 140 141 129 141 5 14 7 32 121 123 6330 6682 535 555 370 422 373 494 25 26 24 24 3 3

25 26 2 3 39 41 128 129 117 129 5 13 6 29 111 113 5664 6019 495 516 329 381 325 443 24 24 22 22 3 2

3 (b) shows that the Q difference increases as D decreases and/or r increases. When D= 10, the difference is 42 to 80% for 0.1~ rd 0.3. As D increases, Qd also increases (case 2-4). Hence, 1Qa-- Qd 1/Qd can decrease when (Qa-Q,) increases. For example, if Qa= 1,050,OOO and Qd= l,OOO,OOO, then the 1Qa-- Qd 1/Qd value is just 0.05 (or 5%) but the ( Qa- Qd) value is as much as 50,000. This example implies that a small-percentage Q difference can be significant with a large EOQ. Effects of I and r on Q differences. Figure 3 (c) shows that the Q differences increase as I/r decreases. Note that Misra and Wortham’s analysis was based on the I/r value. Observing that the Q difference is less than 7% for many examples, they concluded that the discounting

41

Fig. 3. Sensitivity analysis: (a) Effects of A, C and r on Q difference, and r on Q difference (Q+_JQd).

(b) effects of D and

r on Q

difference, and (c) effects of J

Fig. 4. Sensitivity analysis - Discounting effects on total cost difference for the three situations considered in Fig. 3 (C=AE( Y),_,J AE(r),).

43 effects are minimal. However, Fig. 3 (c) indicates that the Q difference is as much as 14 to 27% for 1=0.05 and O.ldrd0.3 (1/6<1/ rd l/2 ). The small-percentage Q difference (less than 7%) can be significant when the EOQ becomes large, as shown above. Therefore, Misra and Wortham’s conclusion is misleading. Gurnani’s example (case 9 in Table 1) showed a large Q difference since A/C ( = 1200 ) is unrealistically large and I ( = 0.03 ) is relatively small. He claimed that the discounting effects on Q are substantial. The ( Qa-- Qd) values are indeed large, but the 1Q, - Qd 1/Qd ratios are less than 7%. Because a Q difference less than 7% was considered insignificant according to Misra and Wortham, the discounting effects on Q, though noticeable, are not substantial in this example. In summary, the discounting effects on Q depend on the A/C ratio. As this ratio increases, the discounting effects become pronounced; otherwise, they are negligible. The effects become more pronounced when I/r and D are small. Previous authors failed to mention the role of this A/C factor and based conclusions on isolated numerical examples. Therefore, their conclusions on the discounting effects were inconsistent and, to some extent, only partially valid. 2. Discounting

effects

on total cost

In this section, we examine the discounting effects on the actual cost of operating the inventory system under the BA model. To facilitate our discussion, we define Q;,-d=Qn-Qd

DISCOUNTING MODELS

EFFECTS

ON OTHER

In this section, we examine the discounting effects on other classical inventory models.

AE(r),~,=AE(r),-AE(r),.

Then, the difference is

When both A and C increase k times respectively, Q, and Qd values are unchanged since A/C is constant in eqns. (8) and (9), increasing k!?(r),_, value k times in eqn. (10). Therefore, if the setup and production costs increase ten times, then the annual cost differential is increased by a factor of ten. Figure 4 depicts 1,4E(r),_, 1/Al?(r), values for the three situations considered in Fig. 3. For example, when we compare the situation in Fig. 3 (a) with that in Fig. 4 (a), the (2 difference ranges between 0 and 33.3% while the AE( Y) differences are within 2.9%. When the situation in Fig. 3 (b) is compared with that in Fig. 4(b), the Q difference ranges between 0 and 104.5% while the AE( v) differences are only within 9.2%. This lower sensitivity of AE( r) is due to the trade-offs among AE( Y)‘S components. The ordering cost is a monotonically decreasing function of Q, but the purchase and inventory carrying costs are monotonically increasing functions of Q. The net result is that the k!?(r) function is rather flat as (2 deviates from the optimum in the positive direction and much flatter about the optimal Q. For all BA model examples considered thus far, we have Qa > Qd although this relationship is not always true. Hence, AE( r) differences are rather small even for considerable (2 differences. The small-percentage savings in total cost by using the discounting approach can be significant, however. For a $100 million inventory system, only a 1% saving results in $1 million.

in annual equivalent

AE(r),_,=C(&l)(Z/r+l){Qa/

[1-exp(--Q,l~)l-Q,/[1--exp(--Q,lD)l}.

cost (10)

1. FP model

Discounting effects on EOQ. Recall eqns. (2 ) and (5 ). Letting h=O in eqn. (2) and rearranging the terms, we obtain

44 Q6=i2D/(l+~))lP/(P_D)](.~/C)

(11)

Equating the first derivative of eqn. (5 ) to zero and rearranging the terms, we have (I/v+

1); (D/r)

[exp(rQ,lD)

- 1 lexp(

-rQ,If’)

-(P/r)[l-exp(-rQ‘,/P)]i=.-1/C

(12)

Equations ( 11) and ( 12 ) show that Qa and Qd are again the functions of A/C. Table 2 shows the numerical comparison of Qd and Qa for various parametric combinations. We confirm that ifii/Cis relatively small

(0.S5), the Q differences are insignificant (cases l-9 ) , but if r-1/C is relatively large ( 501200), they are significant (cases 10 and 11, 13-l 5 and 17). Unlike the BA model, however, the Q differences can be negligible despite a relatively large LI/ Cvalue ( 200 or 1200 ), as seen in cases 12 and 16. We also observe in Table 2 that the discounting effects on Q are insensitive to other parameters if A/C is relatively small. For example, cases l-3 show negligible Q differences for various P values; cases 4-6, for various Z values: cases 7-9, for various D values; and cases

T.ABLE 2 Optimal Case

Q,, and Q, for the finite production

model with varying

Q

Parameter

r values

I 0.10

I

.d/C‘=O.5 P=llO

2

.-1/C=O.5 P=200

3

.-I/C=O.5 P= 700

4

.-l/C’=2 P= 300

5

.1/C‘=2 P= 300

6

:1/C=2 P= 300

7

.1/c’=5 P= I200 .-1/C= 5 P= 1200

8 9

IO II

.l/c‘=5 P= I200 :l/C=SO P=llO :I /C’= 200 P= 1600

I?

.J/C=200 P= 3000

I3

.-1/c‘=200 P= 30000

I4

A/(‘= 500 P=250

15

.1/c‘= 1200 P=8800

16

A/C’= 1200 P= 30000

17

A/C’= 1200 P= 30000

I=O.lO D= 100 I=O.lO D= 100

d

I=O.lO D=lOO I=O.lO D= 100 1=0.20 D= 100 1=0.30 D= 100 I=O.lO D= 100 I=O.lO 0~500 f=O.lO D= 1000 I=O.lO D= 100 l=O.lO D= 1500 I=O.lO D=l500

:

/=O.lO D= 1500 1=0.05 D=200 /=0.03 D= 8000 1=0.03 D= 15000 1zo.03 D=8000

:

:

: Fl a d : : : Ti : :

: : : : a

75 74 32 32 24 24 55 55 45 45 39 39 73 74 207 207 551 548 823 742 7423 6928 2449 2449 I746 1777 2918 2582 44403 40307 23528 23534 13995 14191

0.20 62 61 76 26 20 20 44 45 39 39 35 35 59 60 169 169 452 447 719 606 6342 5657 1999 2000 1410 1451 2391 2000 35625 30303 17683 17693 10447 10669

0.30

0.40

0.50

54 52 22 22 I7 17 38 39 34 35 31 32 51 52 146 146 392 387 656 524 5691 4899 1730 1732 1211 1257 2091 1690 31500 25298 14758 14771 8674 8907

48 47 20 20 I5 15 34 35 31 32 29 29 46 47 I31 131 352 346 612 469 5246 4382 I546 I549 1075 II24 1886 1491 29167 22163 12924 12940 7565 7803

44 43 18 18 14 14 31 32 29 29 27 27 41 43 II9 120 322 316 579 428 4918 4000 1411 1414 976 1026 1733 1348 27653 19962 II637 11655 6787 7028

45 1-9, for various

Yvalues. Note that A/C is less than 5 in all cases examined. If A/C is relatively large, however, the discounting effect in sensitive to these parameters in most cases considered. To examine the conditions under which the discounting effects are most pronounced, we performed sensitivity analyses and observed a trend similar to that of the BA model. For example, the discounting effects on Q are significant if ( 1) A/C becomes large, (2 ) P/D deviates from 2, or ( 3 ) I/r becomes small. In case 17 of Table 2 that meets the three conditions above, Gurnani claimed that the Q difference was substantial. But we can easily show that the Q differences are just 1 to 3% for 0.1 drdO.3. If P were 8,800 (P/D= 1.1) instead of 30,000 in his example, the Q difference would be more noticeable (9 to 20% for O.l~r~0.3) as shown in case 15. If D were 15,000 with P/D=2 instead of 8,000, the difference would be negligible (less than 0.1% for 0.1 d rd 0.3 ) as shown in case 16. More recently, Kim et al. [ 5 ] concluded that the discounting effects on Q were negligible based on poorly chosen sets of examples. The parameter values used in the examples are rather unlikely r=0.0005 with D=l, and P= 5, 0.0003 <16 0.04. Discomting effects on total cost. The total cost difference is given by AE(r),,_,=C(p’-l)[(Z+r)P/r’]{[l-exp(-rQ,/P)]j

(13)

Once again, as with the BA model, if both A and C increase k times, the AE ( r ) a_ d value will also increase k times from eqn. ( 13 ) , while Qa and Qd values remain constant as seen from eqns. ( 1 1) and ( 12). Because of trade-offs among AE(r)‘s components seen in the BA model, the AE( r) differences are much smaller than the Q differences. However, even small AE( r) differences can be again critical in economic analysis of inventory systems.

2. BO Model Table 3 shows the two types of comparisons for various parametric combinations: ( 1) between Qd and Qa, and (2) between AE( r)d and AE( r),. The Q differences are substantial in all cases, ranging from 80% (case 10) to 6 14% (case 8) for r=0.2. Because of the substantial Q differences, the AE(r) differences are noticeable in most cases regardless of the tradeoffs among the cost components. The cost savings by using the discounting approach range from 1% (case 7) to 64% (case 1) for r=O.2. Cases 8-10 in Table 3 show that a small-percentage AE( r) difference results in significant savings. Cases 5-7 indicate that the total cost savings increase as the variable backorder cost 7i decreases. 3. GE Model Table 4 shows the comparisons between Qd and Q,, and between AE(r), and AE(r), for various parametric combinations. The Q differences are again substantial in all cases, ranging from 76% (case 10) to 583% (case 8) for r=0.2. In particular, cases 6 and 9 indicate that the Q difference is still significant even with P/D=2. [Note that the condition P/D=2 indicates the Q difference to be insignificant in the FP model. ] Because of the substantial Q differences, the AE( Y) differences are noticeable in most cases regardless of the trade-offs among the cost components, ranging from 1% (case 7) to 49% (case 1) for r=0.2. Cases 810 show that even a small-percentage AE( Y) difference results in significant savings. Cases 5-7 indicate that the total cost savings by using the discounting approach increase as P/D increases. Note that this was not the situation in the FP model. In summary, we have shown that, for the BO and GE models, the discounting effects on both Q and AE( r) can be significant regardless of the parametric combinations. The major reasons are that the b differences are substantial

C=8000 A= LOO I=O.O5 C=lO ri=l.l I=O.O3

D=8000

l=O.lO

D=SO

c’=lO A=O.lO I=O.O5 C=lOO Ts=l.O I=O.lO C=lOO fi=5 /=O.lO (‘=I00 ir=lO I=O.lO C=2500 ir=lO

‘1-400 n=O.lO D= 20 .A=12000 n=o

1=0.10

DdOO

.,i z 300 7rZO.07 D3200 ‘4 = 50 7rZO.S D=200 A=50 nz0.5 D=200 A z so n=O.5 D=200 A=250 x= 1.0

C=l irzo.02 I=O.20 c’=l ri=O.Ol I=O.O5 C-=4 ii-o.1

a

17951

16063

13 8938

13 12175

a d

51 7

54 10

a d 50 3

68 28

70 36

a d

50 4

142 24

143 29

143 38

a d

a d

111 25

1117 30

1131 42

a d

15255

13 7360

so 3

50 6

67 24

1023 189

1014 268

1031 231

1020 325

1033 443

a d

3225 166

I046 325

3240 205

3266 291

a d

765

0.30

a d

932

1281

d

.-l=lOO ir=O D=lOO .,1=50 n=O D= 100 A= 100 i7zO.l

Q 0.20

Optimal 0.10

r

Parameter

14805

13 6384

50 2

49 5

66 21

142 21

1107 22

1019 163

1011 233

3215 142

663

0.40

14518

13 5703

50 2

48 4

66 19

142 19

1105 19

1016 145

1009 208

3207 126

592

0.50

9723

12 2169

49 3

33 9

51 13

131 19

1052 29

874 254

968 264

3062 248

1171

0.10

Optimal

Qd. Q,. b,, h,,. :lE( y),< and .-lI:‘(I’), with varying I’ values for the hackordcr

Optimal

Case

3

TABLE

10866

12 2091

49 2

36 7

55 Y

135 13

1069 I8

913 177

981 172

3086 176

854

0.20

A

model

11441

13 1757

49 2

38 6

5-J 7

136 IO

1076 14

933 142

986 128

3101 143

702

0.30

11789

13 1550

50 2

39 5

58 6

137 8

1080 II

947 121

989 101

3111 123

609

0.40

12023

13 1405

50 2

40 4

59 5

138 6

1083 9

955 107

991 82

3118 110

545

0.50

100894

173188 98743

134338 171100

21310 132576

21720 21192

21889 21427

2774 21526

2395 2465

170 2321

1283 139

1209

0.10

115271

186124 109771

143991 181264

22649 140022

23417 22369

23736 22696

3547 22831

2716 2751

242 2518

1543 163

1337

0.20

Clptimal AE(r)

130338

200285 120688

154257 191911

24012 147896

25258 23616

25752 24026

4416 24193

3067 3030

320 2716

1839 186

1463

0.30

146565

215594 131930

165151 203142

25584 156217

27237 24945

27926 25436

5365 25635

3452 3317

399 2922

2174 209

1593

0.40

164209

232150 143714

176710 215161

27196 165233

29385 26365

30311 26938

6382 27170

3873 3617

478 3139

2552 233

172Y

0.50

10

9

8

7

6

5

4

3

2

1

Case

1=0.20 A=50 n=O P= 500 1=0.05 A= 100 7r=O.l P= 1500 I=O.lO A=300 x=0.07 P=900 1=0.05 A=50 7r=o.5 P= 800 I=O.lO A=50 n=0.5 P=400 I=O.lO A=50 7l=o.5 P=220 I=O.lO A=250 n=l P=60 1=0.10 A=400 7c=o.10 PC40 1=0.05 A= 12000 n=O P= 30000 I=O.O3

a=0 P= 8000

A= 100

C=IO ri= 1.1 D=8000

C= 8000 ir= 100 D=20

C=2500 ir=lO D=50

c= 100 ir=l.O D=200

c= 100 ir= 1.0 D=200

c= 100 ti=l.O D=200

c= 10 k=O.lO D=200

c=4 EE=O.l D= 500

C=l ir=O.Ol D= 100

C=l ti=O.O2 D= 1000

Qd, Q., bd. b,, AE(r),

Parameter

Optimal

TABLE 4

5

10601 18758

20962

a

18

14363

19

a d

6

123

124

a d

18

25

17814

8764

18

4

123

17289

7624

18

3

123

12

473

74

201

31

164

25

1256

188

1250

293

113.0

163

3436

714

0.40

r values

14

475

476

480

a d

85

105

202

202

143

204

a

36

43

d

60

165

165

166

a d

29

1260

35

1266

217

49

1282

a

266

1256

336

1134

190

3448

824

0.30

d

372

1267

1287

a d

406

551

1140

1155

a d

234

329

3464

3491

a d

1001

0.20

Q

1373

0.10

Optimal

with varying

d

r

andAE(r),

16953

6830

18

8327

2329

9

2

20

123 3

4

41

473 11

11

95

68

201

23

115

164 28

27

928

227

726

232

866

223

2864

1097

0.10

22

1253

168

1246

262

1128

145

3429

639

0.50

9305

1743

9

2

20

3

42

8

96

15

117

17

943

159

754

157

877

159

2887

801

0.20

b

model

Optimal

for the general

9798

1455

9

2

20

2

42

6

97

12

119

13

950

128

768

121

882

130

2901

661

0.30

21641

22032

21729

21659

10096

1276

10295

1150

103681

101171

119878

113281

190229

9 175064 9

153289 182938

138723

142421

23172

137088

125205

206900

194159

169388

150980

24720

24357

24940

23292 22975

24349

26162

24336

22964

24013

1 172259

20

2 134313

42

5

98 21759

8

120

22947

4972

3829 2885 21613

3124

3282

2852 2823

2807

394

276 2592

193

1906

1580 169

1484

0.30

1353

0.20

2514

2469

2376

181

143

1300

1220

0.10

AE( r)

I

20

2

42

5

98

10

119

8

955

953 10

97

784

778 110

83

886

100

2917

513

0.50

99

884

112

2910

573

0.40

Optimal

155929

137445

225038

205833

187188

160016

26372

25820

26709

25814

28529

25805

6325

3432

3769

3031

534

219

2283

1618

0.40

176780

150252

244883

218276

206850

169671

28145

27379

28639

27374

31123

27367

7902

3753

4318

3266

694

245

2715

1759

0.50

P -4

48 and that b, and b, are inversely related to each other as r increases. See Tables 3 and 4. Note that Q is a function of b. The b, value decreases as the discount rate r increases. However, the b, value increases as we increase the inventory carrying cost consisting of C, I and r. For example, with r=0.2, the b differences range from 262% (case 3) to 650% (case 5) in the BO model, and from 260% (case 3) to 588% (case 5) in the GE model.

CONCLUSIONS

The objective of this paper was to examine the conditions under which the discounting effects on optimal lot sizing should be considered for modeling realistic inventory systems. For this, we developed discounted cost expressions of classical inventory models such as BA, FP, BO and GE, and compared them with classical average annual inventory cost models. We have shown that the discounting effects on Q depend on the parametric value of A/C in the BA and FP models. For the BO and GE models, however, they were significant regardless of the parametric values. The same sorts of analyses were used to study how the actual cost would vary from the optimum cost derived under the discounted cash flow approach if we were to estimate the Q from the classical EOQ models. As expected, the discounting effects on the total annual equivalent cost can be significant and the magnitude in dollars is a function of the Q differential. Therefore, the parametric combinations that affect the Q differential dictate the significance of the actual cost. The fact that some of the published models under discounting did not view the inventory investment as working capital does not mean we accept their views. We simply adopted their views to compare their models on a common basis. We believe that a profit maximization formulation considering the timing of revenues is needed.

NOTATION replenishment - infinite rate without stockout rate withFP model - finite replenishment out stockout replenishment BO model - infinite rate without stockout backordered with stockGE model - finite replenishment out backordered D = demand rate per year P = production rate per year A = ordering (acquisition or setup) cost per cycle C =purchase (or production ) cost per unit h =inventory holding charge rate including capital cost = fixed backorder cost per unit n 72 = variable backorder cost per unit per year K = average annual total cost Qa = the optimal lot size under no discounting Qd = the optimal lot size under discounting b, = the maximum backorder level under no discounting b, =the maximum backorder level under discounting r = cost of capital I =h-r BA model

REFERENCES Bierman, H. and Thomas, J., 1977. Inventory decisions under inflationary conditions. Decision Sci., 10: 15 l-l 55. Chandra, M.J. and Bahner, M.L., 1985. The effects of inflation and the time value of money on some inventory systems. Int. J. Prod. Res., 23(4): 723-730. Gurnani, C., 1983. Economic analysis of inventory systems. Int. J. Prod. Res., 21(2): 261-277. Hadley, G., 1964. A comparison of order quantities computed using the average annual cost and the discounted cost. Manage. Sci., 10(3): 472-476. Kim. Y.H. and Chung, K.H., 1985. Economic analysis of inventory systems: A clarifying analysis. Int. J. Prod. Res., 23: 76 l-767. Kim, Y.H., Philippatos. G.C. and Chung, K.H., 1986. Evaluating investment in an inventory policy: A net present value framework. Eng. Economist, 3 1(2 ): 119- 136. Misra, R.B. and Wortham, A.W., 1977. The EOQ model with continuous compounding. OMEGA - Int. J. Manage. Sci., 5( 1 ): 98-99. (Received

April

10, 1987;

accepted

February

2,

1988)