Adaptive Limits In Inventory Control

Adaptive Limits in Inventory Control Author(s): Samuel Eilon and Joseph Elmaleh Source: Management Science, Vol. 16, No. 8, Application Series (Apr., 1970), pp. B533-B548 Published by: INFORMS Stable URL: http://www.jstor.org/stable/2628659 Accessed: 06/03/2009 21:32 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=informs. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected].

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MANAGEMENT SCIENCE Vol. 16, No. 8, April, 1970 Printed in U.S.A.

ADAPTIVE LIMITS IN INVENTORY CONTROL* SAMUEL EILON

AND

JOSEPH ELMALEH

Imperial Collegeof Science and Technology In conventional reorder policies for inventory control, stocks are allowed to fluctuate between two prescribedlimits: a lower limit correspondingto a safety stock to guard against runouts, and an upper limit to which the stock is replenished from time to time. Methods have been suggested for computing optimal upper and lower limits for the case of stationary demand patterns. The adoption of two rigid limits for controlling inventories for nonstationary demand patterns, which include trends and seasonal fluctuations, is clearly not satisfactory. Published papers in inventory control rarely devote enough attention to the need for a link between forecasting and a reorder policy. A method is suggested in this paper for computing adaptive control limits based on a forecasting procedure that takes account of seasonal fluctuations and trends. The method has been simulated for inventory models with varying parameters. Invariably it has yielded results which were equal to and usually better than those obtained by the optimal "fixed-limits" method. For example, in the case of a normal demand distribution, and a given level of satisfying demand, the reduction in costs resulting from the adaptive control procedure was found to be well over 30%. Introduction

The essence of inventory control is to determine when to order and how much to order new stock. Most of the literature is concerned with the following alternative policies: s,S-also known as the two-bin system, where an order is placed when the stock level declines to s, the order quantity being Q = S - s; the stock level fluctuates between the two limits s and S, achieving the upper limit when replacement is instantaneous. S, T-also known as the cyclical review system, where replenishment is effected every interval T to bring the stock level to an upper limit S. s,S, T-this is essentially an s,S system, except that review of the stock is not carried out continuously but at every time interval T; if the stock limit is then below s, an order for replenishment is made for Q S - s. There has been a considerable amount of research into optimization of inventory policies. Soine of the main conclusions may be briefly summarized as follows: 1. It is impossible to draw a universal comparison between various inventory policies, since the behaviour of several depends on the initial conditions of the system in question. 2. Given a set of initial conditions, it can be shown that at least one of the optimal policies must be of the s,$ type. 3. In the case of a stationary demand pattern, an infinite horizon, a discounting factor (a < 1) and when a backlog of unsatisfied demand is allowed, an optimal s,S policy can be determined, irrespective of initial conditions. 4. When demand patterns change with time, when lead-times are variable, and when unsatisfied demand is lost, the mathematical treatment for formulating an optimal inventory policy becomes extremely complicated, very often insoluble. Wagner et al [4], who investigated 800 different inventory cases, demonstrated the computational difficulties involved in determining optimal control limits and suggested approximate empirical solutions. * Received January, 1968; revised May 1969.

B-534

SAMUEL

EILON

AND

JOSEPH

ELMALEH

TABLE 1 Case a Cycle time = 2

Mean inventory of finished goods Standard deviation of finished goods Mean quantity of work in progress Standard deviation of work in progress Mean inventory of raw materials Standard deviation of raw materials Mean total inventories and quantity of work in progress Standarderrorof forecast Percentage of demand of finished goods satisfied Percentage of demand of raw materials satisfied Ordering costs Run-out costs Total costs

Cycle time = 4

Cycle time = 6

Adaptive limits

Fixed limits 4= 0

Fixed limits k =1

Fixed limits =2

Adaptive limits

Fixed limits k=0

Fixed limits k=

Fixed limits 4k = 2

Adaptive limits

Fixed limits k=0

Fixed limits k=1

138

128

179

235

188

176

217

270

229

225

278

319

64

62

55

13

92

86

86

77

110

109

100

103

173

171

184

198

182

187

197

202

191

199

199

203

85

85

94

79

118

113

117

100

152

153

151

150

170

149

171

210

212

202

226

266

275

267

305

343

114

111

117

109

160

147

159

149

198

195

197

203

738

595

691

782

865

481

17

448

-

643

534

-

582

24

-

640

565

-

-

30

-

-

-

Fixed lirmits k= 2

-

89

86

94

98

93 -

94

96

100

96

96

99

99

88

88

95

99

94

95

97

1O3

97

96

100

100

374 3407

356 3709

373 1709

416 450

217 2034

225 1697

203 1049

158 1137

153 1193

158 0

154 0

12680

12030

14140

14530

12370

13520

15000

221 130 17240

17030

16600

18800

21350

Demand: Normal distribution-Mean 50; Standard deviation 5. Trend: None. Seasonality: None. Lead time: Normal distribution-Mean 4; Standard deviation 1.

In another study [1] of stock control for a few raw materials in a firm belonging to the food industry, we investigated through simulation the performance of the above mentioned policies and several others, namely: Q, T-similar to S, T, except that replenishment is effected by a fixed reorder quantity Q (instead of bringing up the stock level to an upper limit S), the value of Q is determined by the classical square root formula for minimum total costs per unit. T,s, S-this is a combination of the s,S and the S, T policies, namely replenishment is provided every interval T to bring the stock level up to the upper limit S, but if in between review periods the stock declines to s, an order for a replenishment of S - s is made. T, s, Q-this is a combination of the s, S and Q, T policies and therefore similar to T, s, S except that the replenishment takes the form of a fixed quantity Q. Of these various methods the T, s, S was found to be most promising and it was then decided to explore the advantages of introducing an adaptive element into the policy by allowing the main parameters s and S to be redetermined periodically. The Model The inventory system studied here was assumed to consist of three stages in series: -Inventory of raw materials -Work in progress -Inventory of finished goods

ADAPTIVE

LIMITS

IN INVENTORY

B-535

CONTROL

TABLE 2 Case b Cycle time - 2

Cycle time = 4

Cycle time - 6

Adaptive limits

Fixed limits k= 0

Fixed limits k= 1

Fixed limits k= 2

Adaptive limits

Fixed limits k= 0

Fixed limits k =1

Fixed limits k= 2

AdaPtive limits

Fixed limits k=0

Fixed limits k-I

Fixed limits k= 2

Mean inventory of 384 finished goods Standard deviation 261 of finished goods Mean quantity of 513 work in progress Standard deviation 326 of work in progress Mean inventory of 451 raw materials Standard deviation 380 of raw materials AMean total inven1348 tories and quantity of work in progress Standard error of 15 forecast Percentage of de86 mand of finished goods satisfied 87 Percentage of demand of raw materials satisfied 388 Ordering Costs 12840 Run-out Costs 36380 Total Costs

326

440

578

500

480

505

689

649

639

759

894

199

230

225

336

296

313

334

449

363

379

384

418

462

511

567

450

497

556

573

493

528

564

229

258

279

429

347

335

350

552

437

471

473

417

500

607

553

542

652

757

746

760

851

939

290

1328

341

491

429

450

462

699

540

576

599

1161

1402

1696

1620

1472

1744

2052

1968

1892

2138

2397

-

-

-

-

-

-

-

-

-

23

28

70

77

85

92

78

84

89

94

83

88

93

71

77

86

93

78

84

90

97

85

89

94

322 26750 48520

334 20580 48250

373 13040 47870

232 7511 39810

187 19910 51480

200 14410 52970

222 10240 55670

164 4996 48570

146 15310 59370

161 10620 61660

158 6241 64230

Demand: Normal distribution with trend superimposed-Mean 150, Standard deviation 58. Trend: Linear, rate of 5% a year. Seasonality: None. Lead time for finished goods: Normal distribution-Mean 4, Standard deviation 1.

The investigation took the form of a simulation over 500 periods for some 200 cases which were covered in the course of this research. The varying parameters included new demand distributions with several coefficients of variation, truncated normal' and gamma lead-time distributions, seasonal demand factors, cycle times of different lengths and several safety factors. The results reported in this paper relate to the following three cases: (a) Normal demand distribution, no trend, no seasonal variations, normally distributed lead-times. (b) Normal demand distribution with exponential trend superimposed, seasonal factors based on a particular industrial case study, normally distrituted lead-times. (c) Normal demand distribution with linear trend superimposed, no seasonal variations, normally distributed lead-times. Details of the parameters for the demand distributions and the trends for the three cases are given in Tables 1-3. Each of the three cases was investigated for the three values of the cycle time T = 2, 4, 6 (i.e. shorter, equal and longer than the mean lead-time L respectively). Two alternative T , s,S policies were defined: Fixed limits-where s and S were determined at the beginning of the simulation run, and kept fixed throughout. s was taken as the expected demand D during the I To preventcaseswherethe orderarrivesbeforeit is placed.

B-536

SAMUEL

EILON

ELMALEH

AND JOSEPH TABLE 3 Ca8e c

Cycle time = 2

Mean inventory of finished goods Standard deviation of finished goods Mean quantity of work in progress Standard deviation of work in progress Mean inventory of raw materials Standard deviation of raw materials Mean total inventories and quantity of work in progress Standard

error of

forecast Percentage Vf demand of finlshed goods satisfied Percentage of demand of raw materials satisfied Ordering Costs Run-out Costs Total Costs

Cycle time = 6

Cycle time = 4

Adaptive time

Fixed limits k =0

Fixed limits k=

Fixed limits k=2

Adaptive limits

Fixed limits k-0

Fixed limits k= I

Fixed limits k -- 2

Adaptive limits

Fixed limits k=0

Fixed limits k =1

Fixed limits k=2

210

247

324

413

274

339

401

480

334

411

478

558

147

105

117

108

197

159

167

182

260

186

196

206

258

247

255

269

288

252

280

287

317

280

288

296

212

129

142

155

223

177

173

197

323

225

230

233

286

267

323

401

286

353

419

491

396

453

507

569

273

161

170

184

273

237

225

256

357

286

290

294

17

902

761

754

-

-

-

944

848

1083

-

-

1100

1258

1047

1144

1273

1423

-

-

_

-

-

-

83

77

81

86

89

80

86

88

91

86

89

91

84

77

82

87

n92

80

88

89

96

88

89

93

429 8251 21250

350 11060 26650

355 9221 28920

372 6859 31870

269 5231 22090

186 9593 31290

224 6655 32390

209 5659 36640

175 4289 27230

150 6809 33950

156 5381 36440

161 4447 40320

Demand: Normal distribution-Mean 80; Standard deviation 51 (with trend superimposed). Trend: Exponential seasonality, Factors taken from an industrial product. Lead time: Normal distribution-Mean 4; Standard deviation 1.

lead-time plus some safety factor described by the coefficient k in the expression s = D + kd where d = mean demand per lead time. Three values for k were used (k = 0,1, 2). S was taken as s plus the expected demand dulringthe cycle time T (the stock review being performed every cycle) or during the lead-time L, whichever was the shorter. Adaptive limits-the method for determining s and S was similar to that used in fixed limits policy, except that the demand D and d, as well as the demand during the cycle time T, were based on forecasts made each period (instead of using expected values), and the upper and lower limits S and s were modified accordingly; s = 0 was used in computing s. To account for the possiblie seasonal pattern in demand a somewhat modified rule of that suggested by Winters [5] was employed for the adaptive policy (see Appendix 1). The modifications consisted of: 1. Introduction of a monitoring subroutine for the periodical testing and reassessment of the smoothing constants (This is a safer procedure than any of the tracking signals tested. In terms of computer time, it takes 0.8 minutes on an IBM 7090 computer to test 125 combinations of smoothing constants.) 2. Special treatment for cases where negative demand values are forecast. In this case there are two possibilities: (i) Equate the forecast to 0.

ADAPTIVE

LIMITS

IN INVENTORY

CONTROL

B-537

(ii) For practical purposes, treat the forecast as 0, but leave the negative value in the forecast equation. The choice as to which method to adopt depends on the relative forecasting error incurred and on the possible effect on production levels. For the three cases reported in this paper no negative forecasts were recorded. Results Figures 1-3 show some results using the adaptive policy in the three cases (a), (b) and (c). To economize in space the diagrams cover some 240 periods, starting at period 60 in the simulation run. The top two diagrams in each figure show the adaptive control limits for raw materials and for finished goods respectively and the curve in between shows the fluctuations in available stock (this curve is based on stock figures at the end of each period and is a reflection of the last event to occur in that period; the curve describes the level of available stock, which includes stock on hand and stock on order;in the case of raw materials, the available stock level is recorded after orders have been placed and this is why it sometimes appears to hug the upper control limit, whereas the curve for available stock for finished goods records the level before the orders are placed, resulting in the curve hugging the lower limit). As can be seen from Figure 1, in the case of a stationary demand pattern, the adaptive limits have little to offer in comparison with well-chosen fixed limits, since they themselves converge to almost horizontal parallel lines. Figure 2 shows how the adaptive limits cope with the linear trend and Figure 3 clearly demonstrates their use for controlling a seasonal product. In the lower diagram of Figures 1-3 three graphs are given for the raw materials stock on hand, for the finished goods stock on hand, and for the market demand respectively. It can be seen that there is a demand for finished goods generated by the market each period. The frequency of demand for raw materials, however, is not that regular, since it depends on orders which have already been placed and on their relative size. Figures 4(a), 4(b) and 4(c) are a cost comparison between the fixed and adaptive control limits. The following cost parameters, which were chosen arbitrarily for the purpose of an illustration, were taken: 1. Cost of unit of finished goods: ?1.2. 2. Storage costs: ?0.08 per unit per period. 3. Interest costs on capital tied up: 6 % a year. 4. Orderinigcosts: ?1.0 per order. 5. Runout costs: ?1.2 per unit. The first observation to be made after examining these graphs is that there is a point of minimum costs in the fixed-limits method. This minimum is a function of the safety factor k, so that any deviation from an optimal safety stock will cause an increase in the total costs. Computations for the three safety factors k = 0,1,2 provide us with adequate results from which the point of minimum costs can be determninedor extrapolated, and this point may then be compared to the costs achieved by the adaptive method. In Figuire4(a), where the case of a stationary demand is illustrated, the two methods seem to exhibit a similar performance. The fixed-limit method yields slightly lower costs, but the difference is too small to be significant. Figures 4(b) and 4(c), where a trend and seasonal variations are included, show distinct advantages of the adaptive method over the fixed-limits procedure. As expected, the adaptive method in

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B-539

-

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JOSEPH

4~~~~~~~~~~0~~~~~~~ -

Pe rcenttge Satisfied

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13,000

Cycle t;m.2

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120

W

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400

240

220

200

180

160

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K=2

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ELMALEIH

Costs

K saofty factor 2 / 14R

Costs 1 14,000

13,000 '

AND

EILON

SAMUEL

700

600

500

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16,000 Cycle btme=4

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15,000

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15,000

93 %

L^/,

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14,000

K 0

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2'40

2,20

R ~~ ~~~~~~~~~F

41

600

500

B00 8

700

%

- 20,000

20,000 K=1

R

F

W

240

1220

16,000

30

290

2c0

finished

700

600

340

320 Mean

F=

~~~~~~~~~~~~

KA

A-10 16,000

100 % 10,000

Cycle time= 6 18,000

800

900

Mean

stocks

goods, W=work in progress, R=raw materials,

X

* adaptive,

0

*

of F,W and R

total

fixed

FIGURE 4(a). Tott,l costs against mean stock of finished goods, mean quantity of work in progress, mean stock of raw materials and mean total of stock

Costs

(i)

tKsotety O

500

F

LF

_=

_

F

Percentoge

Costs

toctor

50 0oo

41

Sotist

Bty__r;~~~~~~~~~~~~~K2 . F R

70ed

85%

WV

--45,000

45,000 Cycle

time=

2 40,000

40,000 F

r -

55,000

Cycle

timenk

400

350

r

500

450

9*

50,000

X_-___-_-__

R

1100

K="1"""

2

6

_%K=

F

F

6_% 600

600

550

R

1500

1600

89 %

84'!

-45,000

_

60

1400

-50,000

_ _

_ 450

_ _ _ 500

_ _ 550

Q W _ 600

_

_ _ 650

_ 700

_ _

_ 750

_ 1400

41=2

C%000

Cycle

liO0

55

45,000

,._ _

1200

1500

166

1700

1800

1900

05,000

0-

60,000

,00o6

, 2000

2 '!.,0g*I

93

83t

_

timQ=6 55,000

55.000

F w R __ __ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ ___---_-__------__-___ ___-_-__

50,000

1

550 F = finished

6C0 goods,

650 W =work

700 in

750 progress,

8o0

050

mean stocks R=raw materials,

94','.

--

|

1600 X - adaptive,

1900 0

2000

2100

22G0

2300 Mean

total

2400

2500

of F,W and R

fixed.

FIGURE 4(b). Total costs against mean stock of finished goods, mean quantity of work in progress, mean stock of raw materials and mean total of stock

ADAPTIVE

(i)

costs

Cyl

iR

IN INVENTORY

K=sotety factor

Percentage Sat isf ied

35,000

---~~~~~~~~~~~~~

2

_t-

_

V>30,000

2 5,0 00

86 *

-_ 0- 010

861.

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-R

F-W

m t

R

K_L I-

F

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-

250

r

350

300

So--oN g g00

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~~~~~~~~~~~~~~~K-2

W /-_

35,000

B-541

CONTROL

Fosts

.

315,000

Cycle t;me= 2

LIMITS

900

1100

1000

a 8A

-.

35,000

Cycle timee 4 30,000

F

25 000

a /.

-0

-30,000

-2S,000

F R W A4

-A

1

L

I

350

300

400

IV

4tpoc ,

K=2 Oooo 0

L/

-

-

-

L-

1100

I0&)

,-j90*

1200

-

--

:1

_<.1

I

900

--e/

-

r3S0006-1 K

Cycle time=6 35,000

30,000

30,000 3;

_

F= finished

~~X

4

-

__

300

400

goods,

W-work in progress,

5001U Mean stocksWatoaofWanR R=raw mnterrtals, X adaPtive,

--

-

9 3%*

12 Mean total of F,w andR

0ftixed

FIGURE4(c). Total costs against mean stock of finished goods, mean quantity of work in progress, mean stock of raw materials and mean total of stock X - Adoptive limits O0- Fixed limits Percentage Sisfed

100 Finisht GoodsX

50

Ol- Fixed limits

K-o

Satisfied

02-Fixed

K:2

limits

100 1

90

-

80

$0

120

140

100

160

200

220

240

100 Raw Materials

K-O

Percentage

20

40

60

t0

100

120

140

160

120

140

160

0

100 1

90 -

90

80

80

120

140

160

180 Maoons

200

220

240

20

40

60 S'andard

t0

100

Oeviations

FIGURE 5(a). Percentage of demand satisfied against means and standard deviations (cycle

time

2) (data in Table 1)

both cases resultsin lower costs, but it is interestingto note that it also provides a higherdegreeof demandsatisfaction. The stock of work in progressis only slightly affected by the increase in safety factor. This is Inot surprisingwhen we recall that the purposeof the safety factor k is to raise the level of the bufferstock, and this is achievedby an occasionalincrease in the level of productioil.

B-542

SAMUEL

?

AND JOSEPH

EILON

ELMALEE

X- Adaptive limits Percentage Satisfied

Percentage Satisfied

0-Fixed limits ? 0- Fixed limits

K-O

limits

K(2

02-Fixed 100

100

9?

90

80

80

Finished Goods

400

500

300

50

400

100

100 Raw Materials

x

F

600

K:1

90

90?

xX

x

22 80

so0

COO

500 Means

200

600

400 300 Standard Deviations

FIGURE 5(b). Percentage of demand satisfied against means and standard deviations (cycle time - 2) (data in Table 2) X - Adaptive

limits

00- Fixed limits Percentage Satisfied

Percentage Satisfied

K_1

limits

K-2

02-F;xad

100 _

'100 Finished Goods

90 _

90 80

2

x

r

20.0

300

80x

C0O

100

200

300

200 Deviations

300

100

100 Raw Materials

K- 0

-Fixed limits

900 900 a a x

200

300 Means

400

100 Standard

FIGURE 5(c). Percentage of demand satisfied against means and standard deviations (cycle

time

=

2) (data in Table 3)

Figures 5(a), 5(b) and 5(c) demonstrate the relationship between demand satisfaction on the one hand and mean stock levels and their standard deviation on the other. These results are given for the three cases (a), (b) and (c), when the cycle time is T = 2. The relationship between the percentage of demand satisfaction and the mean stock level should theoretically assume an asymptotic form reminiscent of the shape of a cumulative gamma function. In a finite simulation experiment, however, it is not difficult to achieve 100% demand satisfaction, provided the safety stocks are high enough, and consequently the curves do not follow this asymptotic shape. A comparison between the adaptive and the fixed limits methods in Figure 5 again

ADAI'TIVE

IN INVENTORY

LIMITS

B-543

CONTROLi X- Adaptive

Parcaenacge Sal,isled

Percentage Sat isfied

limts

*0-FFxxd

limits

*I-Fixed

limits

*2F,xed

limits

K :0 X: I Kal

2 01

.ni3ihed Goods

0

90

.X

0

609

60

240

______ so_ +Z60

200

220

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160 140 120 1ev~~~~~~~~~~~~~~~~~~~

260

300

I100 -

20

40

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40

60

60

80

I100

Raw 90

Materials

80

80

ISO

000

240

622

260

260

300

6x

120

100

W0

160

140

Deviotions

Standard

Means

FIGURE 6(a). Percentage of demanfndsatisfied against meanrsand standard deviationis (cycle tiune = 4) (data in Table 1)

<>

b

Pe,nrlvto

Pel
5Isf ged

X-

Adaptlve

8o-

Fixed

limils

01-

F.xed

limits,

Ka

hnlls

K

130

-Fixed

2

1

.00

X

X 90

40~~~~~~~~~~~~~~~~6

/o2 200

600

Soo

4

30

100

100

x

X

R.1w Materials

KzO

Satisttd *

FinishedJ Goods

I-mits

9 0 _

90 _

6 0-

60

t

_

400

Soo MIeons

i00

200

400

300 Stondard

Devi'atons

FIGURE6(b). Perceintage of demanidsatisfied against means and stanlard deviationis (cycle timne= 4) (data in Table 2)

shows the advantages of the former, except for the case (a) of stationary demand, where the adaptive limits converge to fixed limits and thereby achieve equal results. Also, for a giveni meaanstock level the adaptive method achieves a higher degree of dlemai(l satisfaction; alternatively, for a given level of demand satisfaction a lower mean stock is require(l. However, we found that the standard deviation of stock levels for the adaptive method wlas always higher than the one obtained by the fixed limits method. Figures 6 and 7 summarize simuilation results when the cycle times are 4 and 6 respectively and verify the coniclusionisderived from Figure 5.

B-544

SAMUEL

EILON

AND JOSEPIH ELNIAI2UH X - Adaptive limits limits

K 0

,-Fixed

limits

K:

*2-Fixed

limits

K 2

4-Fixed

O

Percentage Satisfied

Percentage Satisfied .

100

100 Finished

:

X2

90

80

B

300

.00

O

200

500

300

400

300 Deviations

400

1 00

1 00

Materisa

c2

80 so

803

-

1,,1,

300

so1

-

500

400 Means

200 Standard

FIGURE6(C). Percentage of demand satisfied against means and standhrd deviatiolis (cycle time = 4) (data in Table 3) X- Adaptive

Percentoge Satisfied

Percentage Satisfied

Finished Goods

100

100 -

90

90

90

80

200

220

240

260

280

300

320

120

140

120

140 160 Standard

160

K-0

Q1- Fixed limits

K- 1

O2- Fixed limits

11:2

200

220

240

190 200 Deviations

220

240

1I0

1Kt2 to2

1 00 Raw Materials

tOO

limits

00- Fixed limits

gO

90

80

90

200

220

240

260 Means

280

300

320

r t0oo

FIGURE 7(a). Percerntageof demand satisfied against meanis and stanidard deviations (cycle time = 6) (data in Table 1)

Figure 8 is an attempt to relate the mean stock level to its variance to check for any possible exponential relationship. The graphs were plotted on logarithmic paper, but no such relationship could be found. Again, one can see from these graphs that, for a given mean stock level, tihevarianicein the case of the adaptive method is always higher than that for the fixed limits. An attempt was made to compare an adaptive s,S policy with an adaptive T',sa policy and an example of the results is shown in Table 4. These results suggest t;hat a more frequent periodical review of stocks, such as the one achieved by the T,sS policy, tends to lower the average inventory, but this advantage iiust of course be

B-545

CONTROL

LlIMITS IN INVENTORY

ADAPTIVE

Percentage

Percenta-g ;Sf;,ed $b45G

~atisfied

X-

Adaptive

0-

Fixed

Q-

FiXed

0-Fxed

K= O

limits

Ka I

limits

K=2

IOCo

100

2

F;nished Goods

9?

a80

0

80 v_

|

A_____

FOOd

_

.~~~1__.__ '>d0

500

4dOt

dxOo

6000

GL

soo

t 100

tOO

2

|

Raw

_4

limits

limits

0

t.r_.X1

80-

.

t

__

60

tO0

70td

o

I _1_t*I-__ __

40

T

600

503

Standard

Means

-

Deviations

FIuGRU. 7(b). Percenatageof demiandsatisfied againist means and standard deviations (cycle time -= 6) ((lata in Table 2) X-

O

Percentage Satistied

Percentage Satistied

Finished

_

90

_

100

K=:

t1-Fiwed

limits

K -t

limits

K:2

X

80

80

1

_

2Fixed

_

2 vo0-2

x

limits

limits

0

100

Adaptive

40- Fixed

300

400

500o

K

0

/

100

300

200

0

x

400

42

90

Ramw Materials

so

80

-

[

300

400 veans

500

1V

200 Standard

300 Deviations

400

FIGURE 7(c). Percentagc of demanidsatisfied agaitist means and standaird (leviations (cycle

time

6) (data in Table 3)

weighed against the increased costs of frequent reviews. A more detailed comparison between VarliOuis invenitorycontrol policies is given elsewhere [1]. Conclusion With tile exception of a hint by Lewis [2], who investigated inventory control policies with the aid of ani analogue computer, and a recent paper by Trigg [6], who suggested a method for adapting forecasting parameters, there appears to be little mention in the literatuire of atdaptive policies for the control of inventories. Most of

VGrGonce

(i

Variance

1~~~~~~~~0

Cycle

2

l;aT

Cycle

Va riance

062 0V

-

R

--_

52

W

F

-F

1 0 0 FX~~~~~~~~~~~~~~~~~

2-0h10

------F~~~~~~~F

F

-

0 W 2-

-6

I

1

= 6

~~~~01

r? 1 0

time

Cycle

4

t;me

lO

0

0

O - K:O 1 -

Ka 1

2 - K.2

X - Adaptive

- I-

_

I- I

''',I

I

8(a). Variance against means (F

I1 I I I I I

'lo? Mean

Ilean

Mean

FIGURE

II1?h I

I

0o

10C

finishe(dgoods, W = work in] progress, R

=

raw

materials) -67-

Variance

Variance

Variaknce

Cycle

2

time

Cycle

time=

4

Cycle

6

time

XR

2~~~~~~~~~~~~~

x

R-

t1~~

12~~

6

-

PF--

10

F-

-

r

2F-

0 W

--

-

2

Fo

lo4

1010

0

-

Kx 0

I -CKat 2

-CKe-2

X- Adaptive

Meon

Mean

FIGURE

8(b). Variance against means (F

=

finished goods, W

mnaterials)

B-546

Mean

w work in progress, R - raw

ADAPTIVE

Vaocnce

Q

B-547

LAIMITS IN INVENTTORY CONTItOL

vrvr.'nce

Cycir time :2

vor:onor

t

Cycle

e;

4

6

time

Cycle

Ox

10

1; Ox

~~~~~~WX

10

1US

_-

2

X

r

VX~~~~~~~~~~~~~~~F

0~~~~~~~~~~~~~~~~

F~~~~~~~~

0

c

~

-F~~~~~~~~~~~~~~~ 0! i,4?

-

--

J-bD

2

X

FIGURET8(C).

Variance again.st means (F

=

finished goods, W

=

K-0-

-Adaphi

01

work in progress, R

raw

materials) TABLE 4 Comparisonof Adaptive s, S and T, s, S Policie8 S 9,

Mean iniventory on hand Starndarddeviation of inventory on hand Pcrcentage of demand satisfied Ordering costs Itun-out costs Total costs

T, s, S

232 104 98 168

150 59 96 488

201 7489

490 5767

Demand: Normal distribution-Mean 50; Standard deviation 5. Trend: None. Seasonality: None. Lead time: Normal distribution-Mean: 6; Standard deviation 2. Cycle time: 2.

the discussion in papers on inventory control is devoted to models involving statioinary demand distributions and mutchof the analysis centres on a "once and for all" determination of fixed control parameters based on historical data. This is only a natural development of inventory theory and understandable in the light of the cost and availability of computers in ilhe past. Now that computers are beingnused more extenisively in iiidustry, there is less justification to confine such investigations to sta-

B-548

SAMUEL

EILON

AND

JOSEPH

ELMALEH

tionary models and adaptive control procedures in real time terms can be developed, even "on line" where appropriate. The studies reported in this paper clearly demonstrate the advantages to be gained from using adaptive control limits (wNhichaie periodically adjuste(dto take account 6f recent information) in cases where the demand is subject to seasonal variations and/or to trends. While the studies were mainly concerned with a T,s,S inventory policy, the same procedure for determining adaptive cointrol limits can be employed to explore and compare the performance of other policies for conitrolling inventory and for controlling production-iinventory systems. One such comparison reported in this paper iiidicates the advantage of the T,s,S policy compared with an s,S policy. Appendix 1 Winters forecasting rule [5]: If we denote expected demand by y, actual demand by x time by t and a smoothing factor by a, the formula for forecasting by simple exponential smoothing is yt = ax, + (1 - a)yt1i . (1) If the observations include seasonal variations, the above formula will liold only if the quantity in the observation, due to seasoiiality is removed. Tlherefore, taking ft as seasonal factor and L as the length of the season anid usinig an asterisk for denlotilig seasonal data, then y t = t*lft = axe*/ft-L + (2) (1 - a)yt--i And to convert the, deseasonalized forecast to one which takes account of the seasons: yt* = y t*ft-L+1 (3) The seasonal factors are computed using the formula: ft = Xt3x/y t + (1 (4) )fft-L where : is a smoothing constant. If the data contains a linear trend T, the formula then takes a more complex form

(5)

Yt =

fxtf*/ftL

+

(1 -

a)(yt-1

+ Tt-1).

For calcuilating the trend at every step: + (1 -y)Tt (6) Tt= -(yty) where y is a smoothing constant. To convert the unexpected demand figure from its value without trend or seasolnal variations into its actual value Yt(with trend) = (yt + TAt)ft_ L References 1. EILON, S. AND ELMALEH, J., "An Evaluation of Alternative IInvenltory Conltrol Policies,"

International Journal of Production Research, Vol. 7, No. 1 (1968), pp. 3-14. 2. LEWIS,C. D., "Generating a Continuous Trend Corrected, Exponentially Weighted Average on an Analogue Computer," OperationalResearchQuarterly,Vol. 17 (1966), p. 77. 3. VEINOTT, JR., A. F. AND WAGNER, H. M., "Computing Optimal (s, S) Iniveiitory Policies," ManagementScience, Vol. 11, No. 5 (March 1965), p. 525. 4. WAGNER, H. M., O'HAGAN, M. AND LUNDH, B. "An Empirical Study of Exactly and Approximately Optimal Inventory Policies," Management Science, Vol. 11, No. 7 (May 1965), p. 690. 5. WINTERS, P. R., "Forecasting Sales by Exponentially Weighted Moving Averages," IJanagementScience, Vol. 6, No. 3 (April 1960), p. 324. 6. TRImG, D. W. AND LEACH, A. G., "Exponential smoothing with an anlaptive response rate," OperationalResearchQuarterly,Vol. 18 (1967), p. 1.