dynamic lot sizing techniques: survey and comparison

JOURNAL OF OPERATIONS MANAGEMENT Vol. 7, No. 4, December 1988

DYNAMIC LOT SIZING TECHNIQUES: SURVEY AND COMPARISON

Klaus Zoller*

Andreas Robrade*

EXECUTIVE SUMMARY

Numerous heuristics have been proposed in the past two decades for the dynamic lot sizing problem, many of them in APICS journals. Their relative performance is explored in extensive numerical tests measuring ezpected costs, risks of higher than expected costs and computer time

consumed. The results indicate that users of pertinent standard software systems could benefit substantially from an incorporation of more recently proposed methods, specifically Groff’s (1979) stop rule and a fathoming algorithm expanding it to a look-ahead heuristic.

INTRODUCTION

Despite the invaluable contribution of JIT and Kanban concepts to manufacturing economy (cf. Mortimer) [12], batch production will remain a competitive mode of operation for many companies, as will procurement in discrete order quantities, and distribution to field warehouses in economic shipments. EOQ planning, particularly under conditions of time-varying demand (“dynamic lot sizing”), has lost none of its relevance, and little of its challenge to business practice.

* German Armed Forces University Hamburg, Federal Republic of Germany

JOURNAL OF OPERATIONS MANAGEMENT 125

A sample survey of pertinent standard software packa es shows that most systems do feature dynamic lot sizing techniques f Table 1). These represent the state of 1968, however, while theoretical work of the past two decades (Table 2) suggests that a reappraisal of the methodology implemented in commercial software might be indicated. Results of an extensive comparative study by Robrade and Zoller support this view [13].

Following a brief discussion of the notion of optimality in this context and its consequences with respect to experimental design (“Dynamic Lot Sizing and Optimality”), we discuss first myopic heuristics of the stop rule type (“Stop Rules”), then look-ahead algorithms which should produce better results at the expense of more computer time (“Algorithms”). In our numerical experiments we recorded as tentative measures of performance eqected CO&, rislc of higher than expected costs, and computer time consumed. These criteria are combined in the section “Numerical Results,” to facilitate a comparative assessment of the methods included in the tests. An appendix gives details of the fathoming algorithm proposed by the authors.

Not all claims made in this summary report can be substantiated explicitly in numbers and tables. The authors would be pleased, however, to make available the fully documented original report [13] to interested parties.

DYNAMIC LOT SIZING AND OPTIMALITY

Dynamic EOQ planning, in procurement, manufacture/assembly and distribution, is characterized by two facets.

l Time-varyin demand: P

demands are defined for T periods into the future dt, t = 1,2,...,T), representing either schedules (deterministic) or forecasts (stochastic requirements); they may vary considerably from period to period;

l Successive commitment: definite action will be taken one at a time, with respect to imminent replenishments only.

Clearly, optimality or otherwise of an imminent decision cannot be established without considering subsequent replenishments and, ultimately, a sequence of decisions covering the entire plannin horizon. Thus, the dynamic algorithm due to Wagner/Whitin [18 : f Denote by R the fixed cost of each replenishment, and by H the holding cost per unit and period; then, assuming zero stock at the end of the current (t = 0) periodi, and assuming that at least dt units must be in stock at the beginning of each period t, the instruction2

126 Vol. 7, No. 4

TABLE 1 LOT SIZING TECHNIQUES IN STANDARD SOFTWARE SYSTEMS

Supplier Product(s)')

Bull

CTM

Digital

Hewlett

IBM

Nixdorf

Philips

SAP

Siemens

Sperry

IMS-TD/MIACS-TD

PLANOS-MB

VAX-ProFi

HP-MPN-MM3000/PM3000

COPICS

COMET-TOP FEROS/LAWI

Fertigungsorganis. 4000

System RM

IS-BS PBV

UNIS

Technique

Harris') LUC3) PPR3) ---

X X

X

X

X X

X X X

X

X X

X X

X X

X X

1) trade marks marketed in West Germany in 19E6

2) Harris (1915)

3) cf. Table 2 -


TA

BL

E 2

LO

T SIZ

ING

TE

CH

NIQ

UE

S F

OR

DE

TE

RM

INIST

IC,

TIM

E-V

AR

YIN

G

DE

MA

ND

/RE

QU

IRE

ME

NT

S

Optimizing Technique

WW-T: Wagner/Whitin (1958)

Stop Rules

LUC: Least Unit Cost

(-? -

)

PPR: Part Period Rule

(De~~tteis/~lendoza 1968)

SMR: Silver/Meal Rule (1973)

GRR: Groff Rule (1979)

IOQ: Incremental Order Quantity

(Boe/Yilmax 1983)

Algorithm

s

0 Incremental Order Algorithms

IOA-TR: Trux

(1972)

IOA-GA: Gaither (1981)

0 Part Period Algorithms

PPA-FB: Forward/Backward (OeMatteis/Mendoza 1968)

PPA-BM: Blackburn/Millen (1979)

PPA-MG: Maximum Gain

(Karni 1981)

0 Silver/Meal Algorithm

SMA-AM: Absolute Minimum (Blackburn/Millen 1980)

0 Fathoming Algorithms (Robrade/ZoJJer 1987)

F-GRrs: based on GRR

F-SMrs: based on SMR

Ct,j = CT-1 + R + H*hit(h-t)*dh

where

j=t ,...,T t=l,...,T

Cg-i = min {Ch,t_i} (CI = 0) 15 h<t-1

WW-T

will eventually produce the minimum total cost of an optimal sequence of decisions covering the entire planning horizon?

c; = mi n i’,,,) (1) l<h<T

*

In practice, however, only the first of these, XT = ;;’ ‘dh, will be acted h=l

on. All subsequent decisions x;, &, . . . . although they may have affected XT, remain hypothetical: When x7 has been consumed (at the end of ry), a new xr will be computed following the same procedure but using an updated forecast/schedule which covers, not periods 1 to T, but r;+l to T+7-: (and possibly contains revised figures for the overlapping span from rr+l to T).

This results in a series of reorder quantities x; which are parts of optimal solutions to continually revised problems and as such not necessarily optimal themselves (Table 3): Prejudicial effects of xl, iz, . . . on xr can be avoided only if T is large relative to 7;; this, however, enhances the problem of uncertainty (likelihood of revisions) with respect to both forecasts and schedules.

Thus, under normal operating conditions4, optimality needs to be defined in terms of “actual” performance, i.e. with a view to the controllable costs of a (preferably large) number of effective decisions: Assume that in each instance i = 1,2,...,m, a perfect forecast d& t = 1,2,...,T is available; then the controllable cost is

ri C(Ti) = R + HhfI (h-1)-d:, i = 1,2,...,m

-

m and the series of decisions 71, 72,..., r,,, will cover T, = E ri periods at a total cost of i=l

Cbl ,-.*,T~) = F C(7i) i=l

(2)

129 JOURNAL OF OPERATIONS MANAGEMENT

TABLE 3 E(K) FOR WW-T, T = 7 / 10 / 13 / 16-ERRATIC DEMAND1

ww-07

ww- 10

ww- 13

ww- 16

1,000

0.000

0.006

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

2,500 5,000 7,500 10,000 15,000

Demand

Variability

0.685 0.908 0.776 3.380 3.695 20%

0.363 1.095 1.266 4.116 5.361 40%

0.321 1.082 1.737 3.492 6.001 60%

0.141 1.077 1.718 4.195 6.991 80%

0.143 0.812 1.991 3.662 7.426 100%

0.126 0.381 1.097 1.018 0.808 20%

0.099 0.310 0.885 1.112 1.077 40%

0.050 0.192 0.865 0.964 1.187 60%

0.047 0.192 0.456 0.858 1.321 80%

0.010 0.170 0.396 0.647 1.548 100%

0.067 0.159 0.640 0.225 1.425 20%

0.030 0.141 0.380 0.313 1.281 40%

0.021 0.059 0.249 0.502 0.826 60%

0.006 0.065 0.227 0.340 0.801 80%

0.007 0.047 0.102 0.363 0.551 100%

0.014 0.098 0.150 0.512 0.144 20%

0.003 0.077 0.113 0.380 0.299 40:

0.003 0.041 0.060 0.164 0.281 60%

0.000 0.014 0.085 0.159 0.259 80%

0.000 0.005 0.052 0.108 0.257 100%

Part-Periods

1) uniformly distributed demand, expected value 1,000 units This table summarizes expected incremental costs (per cent of optimum) incurred by WW-T as a result of limited (restrictive) planning horizons T; see (3).

130 Vol. 7, No. 4

An optimal series of decisions covering the same span T, can now be computed using WW-T,, and (1) together with (2) provides a comparative measure of performance:

Heuristics exhibiting, in repeated tests, low expected values E(K) and variances V( KC) together with reasonable computing times could be called operationally efficient, provided that a representative range of operating conditions is reflected in the case histories. These conditions concern (a) the statistical nature of demand and (b) the cycle length considered in numerical experiments.

(a) Since sub-optimalities occurring as a result of forecast errors cannot be separated from systematic deficiencies of a heuristic decision rule, the distinction between stochastic and deterministic demand is of little use in this contexts. Hence, we assume perfect forecasts and distinguish between systematic and erratic demands: 0 systematic demands arise from forecasting techniques which fit

(relatively smooth) mathematical functions to observed data, e.g. linear and nonlinear trends, harmonic seasonal effects, and additive or multiplicative superpositions of both;

l erratic demands exhibiting much higher rates of change (and less aut ocorrelation) are generated by numerical forecasting techniques, e.g. Winters’ seasonal coefficients [20], and by MRP calculations. In our experiments erratic demands are simulated as uniformly

distributed random variables having constant means of 1,000 units each, and ranges of *200/o, %40%, &60%, GO%, and l lOO% respectively. Systematic demands are represented by (altogether 11) mathematical functions as indicated.

(b) The range of part-periods covered in the experiments comprises

R : H = 1,000 / 2,500 / 5,000 / 7,500 / 10,000 / 15,000

Together with an average demand of 1,000 units per period, this amounts to average cycle lengths T ranging from slightly more than unity to just under six: 1 < 7 < 6.

A single observation K is generated by m cycles covering at least 40 periods: Tm_1 < 40 < Tm; thus 8 3 m 5 40. A test involving systematic demand comprises 10 observations, each pertaining to slight (coefficient) modifications of a given function and a given quotient R : H. A test involving erratic demand comprises 80 observations, each pertaining to a given range of variability and a


given quotient R : H. There are altogether 11 x 6 = 66 tests involving systematic and 5 x 6 = 30 tests involving erratic demand. Results will be presented following brief descriptions of the heuristics included in the experiments.

STOP RULES

Several of the heuristics proposed for dynamic lot sizing may be classified as stop rules. They increase the cycle length r until some transform of the controllable cost

C(r) = R + H .hil(h-l).dh -

(4)

indicates that r* has been reached. Although there are differences in degree, they qualify as myopic rules.

Least unit cost (LUC)

Perhans the earliest heuristics, and apparently very popular still, LUC tra&s the cost per unit

c(r) = C(r)/h$dh =

and stops at r*=r as soon as

AC(~) = c(r+l) -c(r) 1 0 (5)

Part Period Rule (PPRj

Introduced by DeMatteis [4] and Mendoza [ll], and occasionally referred to as Least Total Cost Method [7], PPR will increase r as long as total holding costs do not exceed the fixed cost of a replenishment,

132

H - i (h-l).dh 5.R or7 : (h-l).dh 5 R/H h=l h=l

Vol. 7, No. 4

so that T* = 7 as soon as

r+l AC@) = H. E (h-l).dh -R > 0

h=l (6)

Silver/Meal Rule (SMR)

Silver/Meal [14] track the cost per period

k(T) = C(T)/T

and stop at r* = 7 as soon as

Ak(T) = k(T+l) -k(T) > 0 (7)

GrofPs Rule (GRRI

Groff [8] balances the marginal reduction of fixed cost per period owing to the increase from 7 to (~+l),

R/7-- R/(7+1) = R/(r++l)),

against the marginal increase of holding cost thus induced,

Hence, T*= 7 as soon as

A(T) = ;.H.dr+l - R/(++l)) > 0 (8)

Incremental Order Buantitv (10621

Boe/Yilmax [3] suggest that 7 be increased as long as the incremental holding costs H. 7.D7+l does not exceed R. This implies

r* = 7 as soon as

H+d7+l-R>O (9)


A few transformations using (4) and the cut-off criteria listed above lead to the following summary comparison of structures,

r* 5 H.r*.E d,

h=l

7-* 7-*+l ’ < HhzI (h-l). h + H .S (h-l) .dh

h=l

C(r*) ( < H. (T*)s.dr*+I

\ 5 H.;* (h-l). h +$-Ha r*+*+l).dr*+l h=l

r*+l <H. z (h-l).dh

h=l

LUC (5’)

PPR (6’)

SMR (7’)

GRR (8’)

IOQ (9’)

which allow some preliminary conclusions: l As apparent from (5’), LUC is strictly backward oriented and in

that sense blind rather than myopic. Poor performance under conditions of erratic demand must be expected.

l The right hand side of (9’) is seen to grow less rapidly than that of any other criterion under most conditions, indicating that IOQ will tend to exaggerate cycle lengths and consequently poor performance.

l All stop rules excepting IOQ will approximate the static optimum [9] under conditions of constant demand.

ALGORITHMS

Whereas stop rules terminate after r* or (r*+l) iterations, algorithms seek to improve the decision by looking further ahead and comparing alternative solutions. In contrast to the former, they are sensitive, in their computing time requirements, to the numerical structure of the demand data.

Our classification (Table 2) reflects similarities in logic rather than precedence. Presentation is restricted to essential character- istics; for more detail, refer to the literature given, or to Robrade/ Zoller [13].

134 Vol. 7, No. 4

TABLE 4 FtANKS OF HEURISTICS

(SCORES OUT OF 66)-SYSTEMATIC DEMAND

Heuristic Rank

12 3 4 5 6 7 8 9 10 11 12 13 14 15

LUC 1

PPR 29 17

SMR 3

GRR 24 13

IOQ 13

IOA-TR

IOA-GA

PPA-FB

PPA-BM

PPA-MG

WA-AM

F-SM21

F-SM22

F-GR21

F-GR22

5 6

a 17

1

1

2

1

2 25 6

3 12 1

3 8 35 11

21 4 2

3 11 1

21 3 2

11

5 5 11 25

12 5

11

6 3 8

4 3 2 15 2 6

13

111 3

11

2 7 4 40

15 11 17 7 1

5 4 40 6

11 2 5 6

19 22 5 8 12

8 6 2 1 12 12 21 1

12111742

8 10 28 9 3

5 8 11 24 9 5 2

28 8 9 2

1 3 4 10 25 710 3 2


IOQ AInorithms [IOA-)

Two algorithms using the IOQ logic have been proposed.

IOA-TX: Trux [17] uses (9) to determine a safe maximum for T*, then checks exhaustive1 profitably split in two I

whether the corresponding lot can be holding cost reduction for the second lot

exceeding the fixed cost incurred for additional replenishment).

IOA-GA*: Gaither [6] uses (9) to determine two subsequent cycle lengths 71 and r;, then checks whether shifting d7i to the second lot is profitable (eliminated holding cost for dTi exceeding the increased holding cost for the second lot).

PPR Alporithms (PPA-)

In attempts to improve its performance under conditions of erratic demand, several authors have suggested lnodifications of PPR:

WA-FB: DeMatteis [4, pp.32ffl proposed that T* as determined by (7) should be subjected to a forward/backward scan, reducing the

cycle length to (7*-l) if dr* 5 i . dr_I, else increasing it to the

largest value of r for which still (r--l).dT 5 d7+1.

WA-BM Blackburn/Millen [2 (7) should be increased to (r*+l if a closer balance of replenishment 1

suggested that 7-* as determined by

and holding costs could thus be secured.

PPA-iUG: Karni [lo] uses the part period criterion to guide an iterative procedure; starting from xt = dt, t = 1,2,...,T, it combines that pair of successive lots into a new single replenishment which promises maximum gain in terms of net cost reduction.

SMR Algorithm &MA-)

SMA-AM Following the observation by Silver/Meal [14, p. 721 that the cost per period is not necessarily convex and may hence possess several local minima, of which SMR detects only the first, Blackburn/ Millen [2] suggested that the absolute minimum be ascertained by way of exhaustive enumeration of C(t)/t over t = 1,2,...,T.

Fathominp Algorithms (F-)

It is intuitively appealing to base the decision on r* on the outcome of an experiment which (a) uses an efficient stop rule to calculate ~12 successive cycle lengths rr,..., ri, (b) records the total

136 Vol. 7, No. 4

TABLE 5 E(n) FOR STOP RULES--ERRATIC DEMAND1

1,000 2,500 5,000 7,500

LUC 3.745 0.920

7.593 4.377

12.252 7.308

19.098 12.796

30.150 19.829

0.737 0.500 1.153 0.915 20%

3.204 2.405 2.668 2.439 40%

5.752 5.260 4.640 4.025 60%

9.626 8.632 8.646 7.326 80%

13.817 13.486 11.678 9.582 100%

PPR 0.387 0.717 0.682

0.806 1.541 I. 787

1.407 1.904 2.834

2.193 2.700 3.793

2.125 3.849 4.796

0.500 0.380 0.346 20%

I.854 1.181 1 f 1921 40%

2.896 2.606 2.023 60%

3.332 3.381 3.543 80%

4.639 4.417 4.319 100%

SEIR 0.387 0.460 0.465 0.576

0.606 0.924 0.900 0.641

1.364 0.976 1.210 0.967

1.276 1.255 1.445 1.399

1.157 1.670 1.601 1.961

GRR 0.387 0.473 0.530 0.500

0.806 0.750 0.889 0.800

1.407 0.988 1.247 1.073

1.525 1.149 1.791 1.259

1.178 1.528 1.367 2.013

IOQ 0.387

0.806

1.341

1.285

1.045

6.390 18.753 30.326 41.984 59.414 20%

6.431 16.598 25.577 36.354 50.101 40%

5.527 13.828 22.789 30.800 43,560 60%

4.283 12.258 17.827 24.951 37.547 80%

3.667 10.239 16.390 21.098 30.749 100%

Part-Periods

10.000 15,000

Demand

Variability

0.225 0.251 20%

0.537 0.695 40%

1.277 0.943 60%

1.808 I.639 80%

2.335 2.620 100%

0.199 0.237 20%

0.581 0.537 40%

1.205 0.899 60%

1.333 1.499 80%

2.181 I.954 100%

1) UnirOnlllY distributed demand, expected value 1.000 units ; see also footnote in Table 3


controllable cost implied by these, and (c) attempts to reduce it by means of revisions 7’ = +i, i = 1,2,...,r which are again supplemented by appropriately adapted successors, cf. (a). The result,

r*=r:+j*, -r<j*<r,

will be one which minimizes medium- rather than short-term costs. Discriminating on the basis of “cost per period,” Robrade/Zoller [13] examine two types of algorithms:

F-B&l: a fathoming algorithm which relies on SMR for calculation of all required cycle lengths, and

F-GRsr: an analogous device using GRR for that purpose.

Of course, F-SMlO (F-GRlO) duplicates SMR (GRR). Further details are given in Appendix 2.

NUMERICAL RESULTS

The results obtained from tests involving systematic demand may be summarized as follows.

GRR, PPR, SMR, and LUC-in that order-perform well in terms of mean incremental costs and maximum observations: rC< 0.1% and K max < 0.9% throughout 66 tests. Computer times (FORTRAN V code on UNIVAC 1100-80) were measured in msec of CPU time per period in order to neutralize the effects of varying cycle lengths (see “Dynamic Lot Sizing and Optimality”); they range from 0.015 (PPR) to 0.045 (SMR) msec/period. At that speed, these rules are faster (by factor lo), though not better than most look-ahead algorithms, and interestingly, both faster (by factor 100) and better in terms of K than WW-T for T = 7 / 10 / 13 / 16. The F+rderings and the K .,x-orderings of the heuristics are highly correlated in all 66 test@ in fact, in 58 cases we observe Spearman’s p 1 0.98 (cf. Spiegel, p. 246) [16]. Thus, Ic,ax is redundant in the sense that it does not add pertinent (rank) information to K Assuming a cost of. CPU time of not more than 2,090 $/h, which

amounts to 0.0005 $/msec, we feel justified in (a) ranking all heuristics by their rC, and (b) breaking ties via computer times, for each test. The ranks scored by the various heuristics are exhibited in Table 4. Clearly, GRR and PPR with altogether 62 (61) scores in the first four ranks lead SMR, LUC, and IOQ. The scores of the latter three heuristics are misleading by vice of the numerical triviality of some of the test problems (which brings to bear speed as the

138 Vol. 7, No. 4

IOA-GA

IOA-TR

PPA-BM

PPA-MG

PPA-FB

WA-AM

TABLE 6 E(K) FOR ALGORITHMtiERRATIC DEMAND1

1,000 2,500 5,000 7,500 10,000 15,000

Demand

Variability

0.085 2.222 13.284

0.260 3.185 11.295

0.472 3.197 10.661

0.753 2.527 a.786

0.605 2.217 7.173

25.128 36.378 53.571

20.400 30.929 45.206

la.833 25.900 40.056

14.932 21.229 33.167

12.101 17.155 26.620

20%

40%

60%

80%

100%

0.368 4.904 1.430 0.313 1.146 4.870 20%

0.762 2.765 2.049 1.151 1.062 2.923 40%

0.885 1.940 2.494 1.640 1.548 2.216 60%

0.472 1.222 2.393 2.216 1.896 1.262 80%

0.312 1.280 2.318 2.835 2.239 1.593 100%

3.769 5.545 3.296 0.759 1.097 0.894 20%

8.054 6.384 4.064 2.630 2.475 2.320 40%

12.427 7.644 4.825 4.338 3.996 3.569 60%

17.335 8.400 6.770 6.848 5.975 4.625 80%

22.938 10.078 9.337 8.008 8.109 7.270 100%

0.037 0.837 2.228 2.677 2.454 3.008 20%

0.093 0.919 2.051 2.255 2.171 2.541 40%

0.106 0.793 1.644 2.220 2.241 2.179 60%

0.109 0.726 1.425 1.614 1.977 2.598 80%

0.060 0.381 1.111 1.643 1.649 2.263 100%

0.875 0.652 0.689 0.455 0.381 0.313

I.837 1.466 1.758 1.715 1.096 1.164

2.995 1.364 2.217 2.187 2.221 1.825

2.903 1.075 2.549 2.549 2.594 2.822

4.167 1.429 2.630 3.168 3.176 3.333

20%

40%

60%

80%

100%

0.368 0.451 0.459 0.566 0.203 0.237 20%

0.762 0.966 0.885 0.859 0.636 0.583 40%

1.663 1.236 1.541 1.231 1.280 1.059 60%

3.169 2.471 2.325 2.315 1.991 1.349 80%

4.855 4.389 3.600 2.564 2.644 2.155 100%

Part-Periods

1) uniformly distributed demand, expected value 1,000 units ; see also footnote in Table 3


TABLE 7 E(c) FOR FATHOMING ALGORITHMS---ERRATIC DEMAND1

F-SMZl

F-SIC'2

F-GRPl

F-GRZZ

1,000 2.500 5,000 7,500 10,000 15,000

Demand

Variability

0.015 0.166 0.216 0.318 0.154 0.117 20%

0.013 0.338 0.408 0.366 0.396 0.458 40%

0.182 0.541 0.673 0.615 0.735 0.781 60%

0.532 0.722 0.680 0.843 1.173 1.257 80%

0.376 0.729 0.843 1.395 1.395 2.076 100%

0.015 0.166 0.207 0.243 0.176 0.123 20%

0.013 0.349 0.304 0.352 0.430 0.407 40%

0.092 0.493 0.468 0.411 0.466 0.635 60%

0.195 0.628 0.328 0.609 0.627 0.649 80%

0.128 0.455 0.464 0.702 0.719 1.032 100%

0.015 0.180 0.182 0.239 0.137 0.107 20%

0.013 0.289 0.397 0.423 0.337 0.385 40%

0.047 0.509 0.576 0.517 0.631 0.623 60%

0.212 0.567 0.610 0.792 0.840 0.967 80%

0.164 0.570 0.748 1.183 1.278 1.462 100%

0.015 0.180 0.180 0.239 0.145 0.120 20%

0.013 0.300 0.320 0.344 0.372 0.335 40%

0.047 0.492 0.378 0.386 0.489 0.538 60%

0.061 0.508 0.375 0.488 0.612 0.447 80%

0.060 0.430 0.538 0.667 0.913 0.795 100%

Part-Periods -~

1) UnifOrmlY distributed demand, expected value 1,000 units; see also footnote in Table 3

140 Vol. 7, No. 4

TABLE 8 RANKS OF HEURISTICS

(SCORES OUT OF 30)---ERRATIC DEMAND

Heuristic Rank

1234567 8 9 10 11 12 13 14 15

LUC 11 3 3 11 6 5

PPR 12 5 416 2

SMR 4 9 6 3 5 12

GRR 9 7 7 2 5

IOQ 2 3 1 2 1 21

IOA-TR 2 3 15 7 2 2 5 3

IOA-GA 3 2 114 2 17

PPA-FB 2 1 5 10 7 2 3

PPA-BM 3 13 4 6 4

PPA-MG 2 1 1811314332

WA-AM 3 4 7 6 3 3 13

F-SMLl 11 616 3 3

F-SM22 510 9 6

F-GR21 8 4 14 4

F-GR22 14 14 1 1

predominant criterion, IOQ being the fastest, LUC a distant second, and SMR a close third-in precise reversal of their K-performances).

The experimental results obtained for erratic demand, probably more interesting theoretically and in practice, may be evaluated in a similar fashion: l Expected values (Tables 5 to 7) and variances10 of 80 observations

K each -recorded in 30 tests-exhibit partly dramatic differences in K-performance among the heuristics.

a Rank correlations of V(n), 50%, 67%, 75%, and 95% quantiles with E(K) are very high, suggesting that, again, rankings by risk essentially duplicate those by expected performance.


0 At a computing cost of 0.0001 $/period, look-ahead algorithms are ten times more expensive than stop rules. This amounts to only $1 per 10,000 items and eriod, however.

l At a consistent E(K) < 1 0, the best algorithm (F-GR22) 5+ outperforms the best stop rule (GRR) by roughly two-thirds in terms of expected (incremental) cost, and by a comparable margin in terms of rislc. Ranking the heuristics again as in Table 4, we obtain for erratic

demand the scores shown in Table 8. The slightly inferior performance of F-SMsr is due to the fact that they are only occasionally better in K but consistently slower (cf. SMR) than comparable GRR-based fathoming algorithms. Finally, F-GR32 (not shown here) does not seem worth the additional computational effort.

By way of conclusion, a direct comparison (Table 9) of the heuristics included in our sample survey with the best stop rule and the highest performing look-ahead algorithm suggests that profitable use can be made in dynamic lot sizing, too, of the immense increase in computing power since the late 1960s.

TABLE 9

RANKS OF FOUR HEURISTICS-ERRATIC DEMAND

Heuristic w

1 2 3 4

LUC 30

PPR 1 29 GRR 29 1 F-GR22 30

ENDNOTES

1. 2.

Without loss of generality: dt, t=1,2...,T, represent net requirements.

Therein Ct_1 represents the minimum total cost of an optimum policy

covering periods 1 through (t-l) while the remainder describes the additional costs of replenishing in period t, and of carrying stock for periods t, t+l ,..., j<T. For an illustrative example, see Appendix 1.

The sequence XT, x$, . . . implicit in CI is reconstructed recursively starting from (l), cf. Wagner/Whitin [13]. WW-T guarantees optimality if the forecast is perfect (no revisions) and complete (dT+l= dT+2= . . . = 0).

Nonetheless, some techniques may be more sensitive to actual forecast errors than others.

142 Vol. 7, No. 4

6. We could not ascertain its exact origin. 7. This form is responsible for the designation “part period” which relates to

the dimension [parts x periods] on both sides of the inequality. 8. Presumably in response to comments from Silver [15] and Wemmerlov

[19], Gaither [6] p resented a modified version of IOA-GA which, however, we were unable to reconstruct from his paper.

9. For details, see Robrade/Zoller [13]. 10. Not reported here; cf. Robrade/Zoller [13].

ACKNOWLEDGEMENT: The authors gratefully acknowledge the co-operation of various suppliers of pertinent standard software in this study.

REFERENCES

1.

2.

6.

7.

8.

9.

10.

11.

Blackburn, J.D., and R.A. Millen, “Selecting a Lot-Sizing Technique for a Single Level Assembly Process: Part I-Analytical Results,” Production and Inventory Management, Vol. 20, No. 3 (3rd Qtr. 1979), pp. 4247. Blackburn, J.D., and R.A. Millen, “Heuristic Lot-Sizing Performance in a Rolling Schedule Environment,” Decision Science, Vol. 11, (1980), pp. 691-701. Boe, W.J., and C. Yilmax, “The Incremental Order Quantity,” Production and Inventory Management, Vol. 24, No. 2 (2nd Qtr. 1983), pp. 94-100. DeMatteis, J.J., “An Economic Lot-Sizing Technique. I: The Part- Period Algorithm,” IBM System Journal, Vol. 7 (1968), pp. 30-38. Gaither, N.G., “A Near-Optimal Lot-Sizing Model for Material Requirements Planning Systems,” Production and Inventory Management, Vol 22, No. 4 (4th Qtr. 1981), pp. 75-89. Gaither, N.G., “An Improved Lot-Sizing Model for MRP Systems,” Production and Inventory Management, Vol. 24, No. 3 (3rd Qtr. 1983), pp. 10-20. Gorham, T., “Dynamic Order Quantities,” Production and Inventory Management, Vol. 9, No. 1 (1st Qtr. 1968), pp. 75-79. Groff, G.K., “A Lot Sizing Rule for Time-phased Component Demand,” Production and Inventory Management, Vol. 20, No. 1 (1st Qtr. 1979), pp. 47-53. Harris, F., “Operations and Costs,” Factory Management Series, Shaw Co., Chicago, 1915, pp. 48-52. Karni, R. “Maximum Part-Period Gain (MPG)-A Lot Sizing Procedure for Unconstrained and Constrained Requirements Planning Systems,” Production and Inventory Management, Vol. 22, No. 2 (2nd Qtr. 1981), pp. 91-98. Mendoza, A.G. “An Economic Lot-Sizing Technique. II: Mathematical Analysis of the Part-Period Algorithm,” IBM Systems Journal, Vol. 7 (1968), pp. 39-46.


12.

13.

14.

15.

16.

17.

18.

19.

20.

Mortimer, J. (ed.), Just-in--Time. An Ezecutive Briefing, Berlin:

Springer, 1986. Robrade, A., and K. Zoller, Dynamische Bestellmengen- und Losgrb;pen-

planung. Research Report #87/l. Institut fiir Betriebliche Logistik und Organisation, Universitlt der Bundeswehr Hamburg, 1987. Silver, E.A., and H. Meal, “A Heuristic for Selecting Lot-Size Quantities for the Case of a Deterministic Time-Varying Demand Rate and Discrete Opportunities for Replenishment,” Production and Inventory Management,

Vol. 14, No. 2 (2nd Qtr. 1973), pp. 64-74. Silver, E.A., “Comments on ‘A Near--Optimal Lot-Sizing Model’,” Production artd Inventory Management, Vol. 24, No. 3 (3rd Qtr. 1983), pp. 115-116. Spiegel, M.R., Theory and Problems of Statistics. New York: Schaum, 1961. Trux, W., Einkauf und Lagerdisposition mit Datenverarbeitung. 2. Aufl., Munchen: Moderne Industrie, 1972. Wagner, H.M., and T.M. Whitin, “Dynamic Version of the Economic Lot Size Model,” Management Science, Vol. 5 (1958), pp. 89-96. WemmerlSv, U., “Comments on ‘A Near--Optimal Lot-Sizing Model for Material Requirements Planning Systems’,” Production and Inventory

Management, Vol. 24, No. 3 (3rd Qtr. 1983), pp. 117-121. Winters, P.R., “Forecasting Sales by Exponentially Weighted Moving Averages,” Management Science, Vol. 6 (1960), pp. 324-342.

APPENDIX 1 DYNAMIC PROGRAMMING ALGORITHM

(WAGNER/WHITIN 1958) ILLUSTRATIVE EXAMPLE

The first section of this paper required a presentation of WW-T in the somewhat awkward notation of dynamic programming. Readers not already

familiar with that algorithm may wish to consider the following example. Assume a fixed replenishment cost R = 60, a variable holding cost per unit

and period H = .50, and future net requirements as shown in Table 10. If we were to order (at the beginning of period t = 1) for periods j =

1 2 6 we should incur costs 3 ,‘*a, >

CIj=R+H.i (h-l).dh, j = 1,2,...,6 > h=l

as indicated in row 3 of Table 10. Considering period t = 2 now, and assuming that dl will have been supplied earlier (no backorders), an alternative way of meeting future requirements would be to place a second order covering periods j = 2,3,...,6. Total cost in this instance would be

144 Vol. 7, No. 4

C .=CII+R+H-~ (h-l)*dh, j 2>J I

= 2,3,...,6 h=2

as indicated in row 4. Similarly, if an order were to be placed in t = 3 to cover periods j = 3,...,6, the total cost would amount to

C 3,J

. = min (Cl 2 , ;C2,2j+R+H*& (h-l)*dh,j = 3,...,6 h=3

(see row 5), etc. Generally, Ct,j is the total cost of a policy which is optimal (has minimum

total cost) for periods 1,2,... ,t-1 and places an order for (j-t+l) periods in period t, j = t, t+l,..., T:

c t,j = min {‘h,t-1

Ishit-1 ) + R + Wl; (h-t).dh

h=t

=C;_l+R+& (h-t).dh h=t

where

c; 1= min - (C, t_l) , t =I 2,3 ,..., T l<h<t-1 ’

and, in order to start up, Ci = 0.

Note that

* CT = min

l<h<T {Ch,T} = Ch*,T = C6,6 = 2g5

gives the minimum total cost of a policy covering all T periods.

WW-T


DYNAMIC PROGRAMMING

net requirements:

1

20

TABLE 10 ALGORITHM-AN EXAMPLE (WW-6)

periotis j row no.

2 3 4 5 6

80 60 90 50 120

total costs C . t,J

t=1 60’ 100’ 160 295 395 695

t=2 - 120 150’ 240 315 555

t=3 _ _ 160 205’ 255 435

t=4 _ _ 210 235’ 355

t=5 _ - - 265 325

t=6 _ _ 2g5&‘%

A column minimum - indicates minimum total cost of covering periods 1 through j

% minimum total cost for T= 6 periods - implies that

- last order is for d6 only: x:=120

- before-last order is for x2= 90+50=140 (xE= 0)

- third-last order is for x;= 8Ot60= 140 (XT= 0)

- fourth-last (and initial) order is for dl only: x7= 20

hence, for this example, -rT=l

146 Vol. 7, No. 4

APPENDIX 2 STRUGTURE OF FATHOMING ALGORITHMS

The principal idea is to base the imminent decision (r*), not on the short-term cost of a single cycle, nor on the total cost of an extrinsic (predetermined) planning horizon, but on the medium-term cost of g successive cycles, g 2 2.

To this end, a reference solution

* * * 71, r2,...,rs

is generated, applying an efficient stop rule (preferably GRR). At a total controllable cost--see (4) & (2)-f

,...,T) = ii C(TL) , h=l

this schedule covers an intrinsic horizon of

TO = ; r; h=l

periods. With a view to the superior performance of “least cost per period” (SMR) relative to “least unit cost” (LUC), the former is used to standardize

cO = C(r;,...,r:)/To

This reference solution will be challenged by alternatives created in an iterative procedure:

For i = -r,...,- l,+l,..., +r (r>l and such that 7; > l),

define: 7’ = 7* +i 1 1

compute: 5

= ri(rh_l), h=2,3,...,si, where rL(rl’_l) designates

the GRR-optimal successor to ~;1_~ and si is defined by

8. -1

T Si-1

= ;: Si

h=l rh < To 5 c

h=l 7’ -T

h- si (ST)


evaluate: , where,

si - 1 if TO - T s.-_12 IT0-Ts.1 n. = 1 1

1 5. else

1

ensures compatibility of ci, i # 0, and c 0 .

Then

c.* - 1-

min {ci} * * y * 7 =7 +i -r < i<+r --

identifies the schedule i*E {-r,...,- l,O,+l,...,+r} which near-minimizes the

medium-term controllable cost and hence should be implied when taking the imminent decision r*.

In terms of the evaluation criteria delineated in the fourth section, “Numerical Results,” the preferred configuration is F-GR22. For a detailed account on this and other versions, see Robrade/Zoller [13].

148 Vol. 7, No. 4

dynamic lot sizing techniques: survey and comparison

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