A neural network model for solving the lot-sizing problem

A neural network model for solving the lot-sizing problem

Lot® K. Gaafar*, M. Hisham Choueiki

Mechanical and Industrial Engineering Department, Kuwait University, P.O. Box 5969, Safat, 13060, Kuwait

Received 1 March 1998; accepted 1 June 1999

Abstract

Arti®cial neural network models have been used successfully to solve demand forecasting and productionscheduling problems; the two steps that typically precede and succeed Material Requirements Planning (MRP). Inthis paper, a neural network model is applied to the MRP problem of lot-sizing. The model's performance is

evaluated under di�erent scenarios and is compared to common heuristics that address the same problem. Resultsshow that the developed arti®cial neural network model is capable of solving the lot-sizing problem with notableconsistency and reasonable accuracy. # 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Neural network models; Lot-sizing; Heuristics; Design of experiments

1. Introduction

This paper investigates the applicability of arti®cial

neural networks (ANNs) to the problem of lot-sizing

in Materials Requirement Planning (MRP) for the case

of a deterministic time-varying demand pattern over a

®xed planning horizon. Speci®cally, we are interested

in the ANN's ability to generate the optimum order

pattern as compared to other commonly used heuris-

tics. Although an algorithm to obtain the optimum sol-

ution to this problem has been developed by Wagner

and Whitin [1], our motivation in developing a neural-

network-based solution is due to the fact that ANNs

are successfully being used in demand forecasting and

production scheduling; the two steps that usually pre-

cede and succeed MRP, respectively [2±9]. Thus,

extending the use of ANNs to the MRP lot-sizing pro-

blem will permit the integration of the production in-

formation system, a key requirement for its success.

With ANNs supporting all major functions in pro-

duction planning, historical data and other inputs may

be directly converted into planned order releases or

production schedules in a transparent way. Some ben-

e®ts of such integration have been documented in the

literature [10].

The ANN solution developed in this paper is com-

pared to the optimum solution, as well as to solutions

developed using other heuristics including Periodic

Order Quantity (POQ), Silver-Meal (SM), and MINS

[11]. POQ and SM are based on the EOQ approach,

which minimizes the total inventory cost per unit, in

di�erent ways. The POQ determines the average num-

ber of periods covered by the EOQ and then orders

the exact quantity to cover the demand for those

periods. The SM ®nds the number of periods for

which the total inventory costs per period is minimized

and then orders the exact quantity to cover the

demand for those periods. Details of these two heuris-

tics may be obtained from standard production plan-

ning textbooks (e.g., [12,13]). On the other hand,

MINS is a mnemonic that indicates the method's re-

petitive selection of the period with minimum demand

Omega. 28 (2000) 175±184

0305-0483/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.

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* Corresponding author. Tel.: +965-4811188 ext. 5805; fax:

+965-4847893.

E-mail address: [email protected] (L.K. Gaa-

far).

to explore the bene®t of accelerating the delivery of itsrequirements.

The chosen heuristics are simple, and on average,provide close to optimal solutions [11]. Therefore,comparing an ANN-based approach to these heuristics

provides a good indication of its performance. Forcomparison purposes, POQ was chosen to provide abottom line of acceptable performance as it is known

to be the least accurate of the three methods, but themost simple at the same time. SM was chosen becauseit is one of the most commonly used methods with

reasonable accuracy. Finally, MINS was chosen as oneof the most recently developed methods with reason-able simplicity and notable accuracy. These choiceswere made to evaluate the ANN model against com-

mon algorithms to assure acceptable performance insupport of the main objective of extending the use ofANNs to the lot-sizing problem for production plan-

ning integration. Zhiwei et al. [11] provide a simplicity/accuracy scale comparing various approaches to thelot-sizing problem that justi®es our choices.

2. Literature review

2.1. Lot-sizing

This paper addresses the problem of determining the

optimum quantities (lot sizes) to order in discrete timeperiods of a single item over N periods to satisfy a cer-tain demand pattern D while minimizing the sum ofordering and carrying costs. It is assumed that demand

of variable quantities occurs in each of the N consecu-tive periods. The demand that occurs in a given periodmay be satis®ed by an order during that period or

during an earlier period, as inventory is carried for-ward in time. This problem has received signi®cantinterest in the literature with hundreds of papers

addressing various aspects of the problem [14]. The op-timum solution developed by Wagner and Whitin [1]has been criticized as being hard to understand andcomputationally complex [15,16]. Simpler algorithms

that provide reasonable solutions have been developedover the past three decades. Two of the most commonalgorithms are the PPB algorithm developed by

DeMatteis [17] and the SM algorithm developed bySilver and Meal [18]. Both algorithms have receivedmany modi®cations (e.g., [15, 19±22]). Other algor-

ithms include TOPS [23] and MINS [11]. Comparativestudies have shown that many of these algorithms aregood alternatives to the Wagner±Whitin algorithm,

and may even outperform it in certain cases [24,25].

2.2. Arti®cial neural networks

Arti®cial neural networks (ANNs) are alternative

computational tools that consist of a large number of

simple processing elements, all interconnected, and inparallel. The design of these networks was inspired bythe biological neural network (the mammalian cerebral

cortex). ANNs have a number of features that havecaused them to be successfully implemented, with sig-ni®cant economic bene®ts. First, they are massively

parallel and interconnected, and thus, hardware andsoftware implementations are much easier [26]. Second,

ANNs have a high tolerance to noise [27]. Knowledgeand information are generally stored throughout thenetwork in the form of weights rather than in a com-

plex-structured central processing unit. It is for thisreason that ANNs typically exhibit graceful degra-dation to noise. Third, ANNs provide a means for

modeling arbitrarily complex non-linear functions. Acollection of processing elements with weighted inter-

connections realizes a powerful modeling capability. Infact, some ANNs are capable of modeling arbitrarilycomplex functions to an arbitrarily degree of accuracy

[28]. These are often called universal approximators.Lastly, a fourth feature of ANNs is the existence ofgeneral purpose learning rules [29]. These are rules

that enable a neural network to adjust to new changesin the behavior of the system that is being modeled.

A production planning problem, as mentioned ear-lier, spans three sub-problems; the demand forecastingproblem, the material requirements planning/lot-sizing

problem, and the production scheduling problem. Theliterature is saturated with successful applications of

neural network models in time-series demand forecast-ing [2±5].Applying neural network models to solve job-sche-

duling problems has been recently gaining somemomentum. Ntuen [8] used neural networks to mapproduction elements such as lead-time, time between

orders, service rate, and order release time to thedemand patterns. Laarhoven et al. [7] used simulated

annealing to minimize the makespan in a job shop.Yih et al. [9] trained a backpropagation feed-forwardneural network to act as a decision making tool in job

scheduling. Kim et al. [6] trained a neural network topredict several parameters that were then used in com-puting priority indices for jobs to be sequenced.

Empirical work on using ANNs in lot-sizing pro-blems is sparse. Ezziane et al. [30] used a neural net-

work approach in deciding whether or not to place anorder for raw material in an inventory control manage-ment system. Zweitering et al. [31] demonstrated that a

properly trained neural network could outperform thetraditional algorithms in solving the lot-sizing problem

for speci®c ordering and carrying costs. Their workwas later extended by Stehouwer et al. [32] to includethe overtime cost. Our research is yet another exten-

sion to the latter work in that the developed neuralnetwork model is based on thousands of lot-sizing pro-

L.K. Gaafar, M.H. Choueiki / Omega. 28 (2000) 175±184176

blems, each with a di�erent demand pattern, orderingcost, and carrying cost. Additionally, we solve pro-

blems with two alternative planning horizons; an 8-period and a 12-period horizon.

3. The simulation experiment

In this section we describe the experiment that wasused to evaluate the performance of the neural-net-work model (NNM) when compared to other heuris-tics in the literature for solving the lot-sizing problem.

A 2-replicate of a 22 factorial design is used in this ex-periment (i.e., an experiment involving two factorseach investigated at two levels, [33]). The ®rst factor

investigated is the planning horizon (N ). This factorwas set at two levels: 8 and 12 periods. The second fac-tor is the demand pattern (D ). Two demand patterns

were investigated: constant and seasonal. Factor levelswere chosen to span a large part from the domain oflot-sizing problems. We were ultimately interested in

comparing the NNM, POQ, SM, and MINS perform-ances based on the percentage of cases for which eachapproach generated the optimum order pattern (O ).Optimum order patterns were obtained using the

Wagner±Whitin algorithm. Additionally, the averagepercentage deviation from optimum cost (C ) was usedas a second response to provide another dimension for

comparison because of its common use in the litera-ture.

3.1. Data generation

Various models were used to generate the input datafor training the NNM. These data comprise the

demand pattern and the ordering and carrying costs.The constant demand pattern was generated using thefollowing model:

dt � a� et, 1EtEN

where, dt is the demand in period t, a is a constantgenerated from an exponential distribution with a

mean of 100, and et is a normally independently dis-tributed error component with a mean of 0 and a con-stant variance of s 2(s=0.1a ).

The seasonal demand pattern was generated usingthe following model:

dt � a1 � a2 sin2p�t�m�

N� et, 1EtEN

where, a1 is a constant generated from an exponentialdistribution with a mean of 100, a2 is the amplitude of

the sinusoidal curve (a2=0.5 a1), m is a constant gener-ated from a discrete uniform distribution rangingbetween 0 and N ÿ 1 to randomly vary the starting

point of the demand pattern, and et is a normally inde-pendently distributed error component with a mean of

0 and a constant variance of s 2(s=0.1a2).Ordering costs were generated randomly from a uni-

form distribution between the limits $50 and $400 per

order, while the carrying costs were generated ran-domly from a uniform distribution between the limits$0.5 and $4.0 per unit per period.

3.2. The neural network model

The most commonly used type of ANN is the multi-layered feedforward backpropagation trained neuralnetwork [34]. Networks of this type are very general,

can approximate accurately complex mappings [35,36],and possess the statistical property of consistency(learnability) for unknown regression functions [37].

This topology was used in this research due to itsdemonstrated success in a variety of function approxi-mation and classi®cation problems (quality control,

forecasting, speech and image recognition, control,robot systems, sonar classi®cation, etc.); especially,problems that are characteristically sparse and noisywith highly complex nonlinear relationships. It is im-

portant, however, to mention that there are otherarchitctures that are variations on the standard back-propagation neural network model such as the Jordan

network, the Elman network, probabilistic networks,and functional-link networks. These neural networkarchitectures o�er slightly di�erent processing that

may outperform the standard feedforward backpropa-gation network. Nevertheless, feedforward backpropa-gation architecture was chosen because of its

commonality in the literature.Multi-layered feedforward neural networks belong

to a class of nonparametric estimators of unknownmappings. They are nonparametric due to the fact that

they do not require any assumptions regarding the dis-tribution of the errors nor regarding the underlyingstructure of the true mapping. They are estimators

because they can successfully estimate a givenunknown analytical function F that describes a true re-lationship between an m-tuple input vector x and an n-

tuple output vector y, i.e., F:Rm4Rn.In building a feedforward neural network model Nw,

it is generally required to have a selected topology inmind and a sample training set w={(xe, ye ), e = 1,

2, . . . ,p } that is generated by F; i.e., F(xe )=ye, e = 1,2, . . . ,p. Given the latter, one can generally use thebackpropagation rule (also called the Generalized

Delta rule) for training the network. The backpropaga-tion rule essentially performs a gradient-descent optim-ization over the space of the interconnection weights

such that the network output error function E=P�x;y�2w k I�x� ÿNw k is minimized (the function k.k is

interpreted as the L2 norm).

L.K. Gaafar, M.H. Choueiki / Omega. 28 (2000) 175±184 177

In general, this network's performance is a functionof several design parameters; such as the number of

hidden layers, the number of nodes per hidden layer,the type of transfer function, and the training factors.Identifying a set of parameter combinations that can

lead to optimum neural-network performance for aspeci®c problem has been mostly done using the``change one factor at a time'' method of experimen-

tation, or by trial and error. Recently, using factorialand fractional factorial designs, a new experimental al-gorithm was developed for building a ``quasi optimal''

neural network to solve a given problem [38]. Thisapproach starts by identifying the set of architecturalfactors and their associated levels that in¯uence theperformance of the neural network model in solving a

given problem; such as the number of hidden layers,transfer function in hidden and output layers, stoppingrule, training algorithm, the presence of recurrence, the

presence of noise during training, etc. It then uses theDesign of Experiment theory to design a neural-net-work model. The results of applying this experimental

algorithm to solve the lot-sizing problem are summar-ized in Table 1.

3.2.1. Training ruleTraining is an important feature of neural networks.

The objective of the training process, as mentionedearlier, is to minimize the squared error between thenetwork output and the desired output. This is doneby adjusting the connection weights across the net-

work. The error is computed by making a forward cal-culation through the hidden and output layers of thenetwork. For weight adjustment, the network errors

are then propagated backward through the network bythe gradient descent algorithm in order to determinethe new direction of movement in weight space. In

other words, the new direction vector for movement isdetermined as the linear combination of the currentgradient vector multiplied by a given learning rate

(search step size) and the previous direction vectormultiplied by the momentum rate. The momentumrate is used to ``e�ectively ®lter out high frequencyvariations of the error surface in weight space'' [34].

Determining the correct values for the learning andmomentum rates are problem dependent. For thisresearch, we followed the recommendations of Kuan

and Hornik [40], and conducted preliminary exper-iments in order to identify the speci®c training sche-dules (Table 2) that proved e�ective during training.

3.2.2. Stopping rule

The stopping rule is a very important factor duringthe training process of a neural network. The questionof when do we stop training has had considerable dis- T

able

1

Neuralnetwork

model

de®nition

Architecture

.Feedforw

ard

neuralnetwork

.Inputlayer:10or14nodes

.Hidden

layer:9or13(Thenumber

wasdetermined

usingtheCascadeCorrelationAlgorithm

[39])

.Outputlayer:8or12

.Sigmoid

transfer

functionin

hidden

andoutputlayers

.Allinputdata

norm

alizedbetween0and1

Computations/Termination

.Training:Cumulativebackpropagationrule;updatingtheweights

iterativelyafter

acomplete

pass

throughthetraining

data

set

.Randomness:Weights

initializedto

smallnumbersfrom

uniform

(ÿ0.2,+0.2),withrandom

presentationoftraining

examplesduringtraining

.Trainingterm

ination:Stoptrainingwhen

therootmeansquarederroronanindependentdata

setisminim

ized


cussions in the literature. The most popular answersare as follows:

1. Stop training when the magnitude of the gradient is

signi®cantly small.2. Stop training when an a priori number of training

iterations has been performed (one training iteration

involves presenting the entire training data set tothe network).

3. Stop training when the root mean squared error onthe training data set is no longer decreasing.

4. Stop training when the root mean squared error onan independent data set (often called validation dataset) that comes from the same population as the

training data set is minimized.

The fourth stopping rule, often called in the literaturethe Concurrent Descent Algorithm, appears to be the

most successful due to the fact that it helps avoid ®t-ting the noise in the training data set, and thus, helpsproduce a trained network that is able to generalize

[5]. As a result, the fourth stopping rule was used forterminating training in this research.

3.2.3. Input calibrationWhile the design parameters mentioned above do

a�ect the topology of a neural network, they do notsu�ciently de®ne it. An important step in building aneural network is the selection and coding of the input

variables. There are no general rules to follow in thisprocess. It depends largely on experience, professionaljudgment, and preliminary experimentation. In this

regard, it is a known fact that a lot-sizing decisionover a future period is always a function of thedemand forecasts, the inventory carrying cost per unitper period, and the ordering cost per order. Therefore,

in the case of the problem at hand, it was clear whattypes of input and output data were needed, but notso clear was how, and in what form, to present the

data to the network. We conducted preliminary exper-imentation to investigate the methods for coding theinput and output data. This is analogous to what

scientists in other ®elds do to calibrate the research ap-paratus.Two types of variables were chosen as inputs to the

neural network; these were cost input (2 nodes) anddemand forecast input (8 or 12 nodes depending onthe scenario that was being run). Thus, nodes 1 and 2

presented to the network a particular ordering costand inventory holding cost, respectively. Nodes 3±10(or 3±14) presented the associated demand forecastpattern. The output layer had a total of 8 or 12 nodes

(again depending on the scenario that was being run).The output nodes produced values of zeros or ones;output node k produced a zero if the decision was to

not place an order in period k, and a one if the de-cision was to place an order in period k. The questionfor how many future periods to order is explicitly

answered by the zero-one coding where a value of onein any node indicates that an order should be placedin the period corresponding to that node for its

demand and the demand in all subsequent nodes untila value of one is encountered again. For example, foran 8-period demand pattern, the output vector(10001001) represents a lot-sizing decision in which

three orders are placed during the next 8 periods; the®rst order in period 1 should satisfy the demands forperiods 1, 2, 3, and 4, the second order in period 5

should satisfy the demands for periods 5, 6, and 7, and®nally, the third order in period 8 should satisfy thedemand for period 8. This output coding scheme was

used to ®lter out noise and to assure that the orderquantity matched the forecasted demand over the Nperiods. Fig. 1 displays the general topology of theNNM.

3.2.4. Training and testing dataAt each of the four design points, 3000 input/output

examples were generated. 1600 randomly chosenexamples were used for training. 400 randomly chosenexamples were used for cross validating while training.

The remaining 1000 examples were randomly split intotwo test sets, each of size 500, to form two replicatesof the design. The test sets were used to evaluate the

NNM against the optimal solution as determined bythe Wagner±Whitin algorithm, and to compare its per-formance to that of the POQ, SM, and MINS algor-

Table 2

Training schedules

Iteration (an iteration is a presentation of 1600 examples)

1±10 11±20 21±40 41±100 > 100

Learning rate (hidden layer) 0.60 0.40 0.20 0.10 0.05

Learning rate (output layer) 0.30 0.20 0.15 0.05 0.025

Momentum rate (both layers) 0.40 0.20 0.05 0.025 0.01


ithms. The next section describes the results and the re-lated analyses.

4. Results and analyses

Table 3 shows two observations on the percentageof times each method obtained the optimum order pat-

tern (O ) for each design point for the various methods,with the average of the two observations shown in par-enthesis. As stated earlier, each observation is based

on 500 randomly chosen test cases. The overall averageO for each method is shown in the last raw of Table 3.Table 4 presents similar information for the average

percentage deviation from optimum cost (C ). As can

be noticed from Table 3, the NNM seems to do very

well in generating the optimum order pattern. Theoverall average percentages for generating the opti-

mum pattern are 65.2, 46.5, 50.6, and 58.9 for NNM,

POQ, MINS, and SM, respectively. The overallaverages for the percentage increase in cost from opti-

mum are 2.83, 3.01, 1.47, and 0.73 for NNM, POQ,

MINS, and SM respectively. Clearly, the NNM does

not perform as well with the cost criterion. In light ofthe good classi®cation performance, the poor cost per-

formance implies that when the NNM is missing the

optimum pattern, it does so in a signi®cant way. Thisresult, however, is expected since the NNM was

trained to obtain the optimum order pattern only. In

other words, the deviation from the optimum cost was

not directly used in training the NNM. Nevertheless,

Fig. 1. The general topology of the neural network model.

Table 3

Percentage of times obtaining the optimum order pattern (O )

NNM POQ MINS SM

Constant Seasonal Constant Seasonal Constant Seasonal Constant Seasonal

8 70.6, 68.2 62.4, 65.8 53.4, 52.0 28.2, 28.4 58.8, 60.0 50.8, 45.8 69.6, 66.2 40.8, 40.2

N (69.4) (64.1) (52.7) (28.3) (59.4) (48.3) (67.9) (40.5)

12 62.0, 63.8 64.2, 64.8 50.6, 52.2 55.0, 51.8 44.6, 46.8 48.0, 49.6 61.6, 63.6 64.6, 64.8

(62.9) (64.5) (51.4) (53.4) (45.7) (48.8) (62.6) (64.7)

Overall average 65.2 46.5 50.6 58.9


an average cost increase of less than 3% may still beacceptable for many applications considering the po-

tential integration bene®ts from using ANNs through-out the production planning system.Table 5 shows analysis of variance (ANOVA)

results for the optimum pattern response (O ) for all

methods, while Table 6 shows ANOVA results forthe average percentage deviation from optimum cost(C ). Figs. 2 and 3 display the average responses at

each design point for both performance measures.These plots are helpful in presenting results and ininterpreting interactions.

Based on Tables 3±6 and Figs. 2 and 3, the follow-ing remarks are in order (ANOVA remarks are basedon a 0.01 statistical level of signi®cance):

1. The NNM shows a robust performance relative toO (Table 5a) as it is not a�ected by either of the

investigated factors (N or D ) or their interaction(N�D ). This result is consistent with the documen-ted robustness of ANNs in the literature [28]. Allother methods (Tables 5b±d) appear to be sensitive

to both factors and their interaction when consider-ing the percentage of times obtaining the optimumorder pattern. Notice that since the N�D interaction

is statistically signi®cant in Table 5c, it is reasonableto assume that both N and D are also signi®cant, asinteractions tend to mask out the signi®cance of

main e�ects [33, p. 199]. The sharp decline in theperformance of the POQ and SM algorithms in theseasonal pattern case with a horizon of 8 periods is

Table 4

Average percentage deviation from optimum cost (C )

NNM POQ MINS SM

Constant Seasonal Constant Seasonal Constant Seasonal Constant Seasonal

8 1.68, 1.84 3.87, 3.96 3.14, 2.88 4.42, 3.84 1.27, 1.13 2.08, 2.00 0.71, 0.75 1.87, 1.97

N (1.76) (3.91) (3.01) (4.13) (1.20) (2.04) (0.73) (1.92)

12 2.8, 2.5 3.38, 2.74 3.2, 3.0 2.3, 3.06 1.38, 1.46 1.20, 1.22 0.6, 0.7 0.61, 0.55

(2.6) (3.06) (3.1) (2.68) (1.42) (1.21) (0.6) (0.58)

Overall average 2.83 3.01 1.47 0.73

Table 5

ANOVA results for the percentage of times obtaining the op-

timum order pattern (O )

Source DFa SS MS F P-value

(a) NNM

N 1 18.605 18.605 7.11 0.0560

D 1 6.845 6.845 2.62 0.1810

N�D 1 23.805 23.805 9.10 0.0393

Error 4 10.46 2.615

(b) POQ

N 1 283.22 283.22 153.09 0.0002

D 1 250.88 250.88 135.61 0.0003

N�D 1 348.48 348.48 188.37 0.0002

Error 4 7.4 1.85

(c) MINS

N 1 87.12 87.12 20.60 0.0105

D 1 32.00 32.00 7.57 0.0513

N�D 1 100.82 100.82 23.83 0.0081

Error 4 16.92 4.23

(d) SM

N 1 178.605 178.605 89.53 0.0007

D 1 320.045 320.045 160.42 0.0002

N�D 1 435.125 435.125 218.11 0.0001

Error 4 7.980 1.995

a DF: Degrees of freedom; SS: Sum of squares; MS: Mean

square.

Table 6

ANOVA results for the average percentage deviation from

optimum cost (C )

Source DFa SS MS F P-value

(a) NNM

N 1 0.0006 0.0006 0.009 0.9282

D 1 3.289 3.289 49.31 0.0022

N�D 1 1.522 1.522 22.82 0.0088

Error 4 0.2667 0.0667

(b) POQ

N 1 0.9248 0.9248 7.24 0.0546

D 1 0.2450 0.2450 1.92 0.2383

N�D 1 1.1858 1.1858 9.29 0.0381

Error 4 0.5108 0.1277

(c) MINS

N 1 0.18605 0.18605 45.38 0.0025

D 1 0.19845 0.19845 48.40 0.0022

N�D 1 0.55125 0.55125 134.45 0.0003

Error 4 0.0164 0.0041

(d) SM

N 1 1.0082 1.0082 320.06 0.0001

D 1 0.6272 0.6272 199.11 0.0001

N�D 1 0.7938 0.7938 252.00 0.0001

Error 4 0.0126 0.00315

a DF: Degrees of freedom; SS: Sum of squares; MS: Mean

square.


due to the fact that this pattern causes a rapid dropin the demand, a phenomenon that is known todegrade their performance [12]. MINS seems to be

less a�ected by the seasonality factor. Nevertheless,the sensitivity of the POQ, MINS, and SM methodsto demand variation is well documented in the lit-erature [25,11]. On the other hand, the consistency

of the NNM gives it an edge over the other methodswhen the demand exhibits high variability.

2. The NNM is sensitive to both investigated factors

and to their interaction relative to C. Again, sincethe N�D interaction is statistically signi®cant (Table6a), we may assume that N is also signi®cant. The

MINS and SM methods exhibit similar sensitivity.The POQ method, on the other hand, seems to berobust to both factors. However, Fig. 2b shows that

an interaction is present. Examining the results inTable 6b, it seems that the signi®cance of the inter-action has been overshadowed by the large error

term (especially when compared with other error-

terms in Table 6).3. The POQ method shows the worst performafor

both criteria. This result may be explained by the

fact that the POQ method is based on the averagedemand and will only perform well when thedemand exhibits minor variability. Nevertheless, anaverage increase in cost of about 3% may still be

acceptable considering potential gains due to thesimplicity and computing e�ciency of this method.

4. MINS performed relatively poorly compared to the

SM. However, MINS's performance is consistentwith that reported by Zhiwei et al. [11]. Their resultsshow that MINS generated 33.8% to 95.7% of the

optimum demand patterns depending on the order-ing to holding cost ratio. They used a lumpydemand pattern that was generated from a uniform

Fig. 3. Plots of the average percentage deviation from opti-

mum cost (C ).

Fig. 2. Plots of the percentage of times obtaining the optimum

order pattern (O ).


distribution with a minimum of 1 and a maximumof 1000 with 20% of the periods being randomly

overlaid with zeros. They also used ordering toholding cost ratio of 200 and 800. Their results weresimulated over a horizon of 50 periods. The di�er-

ence between MINS and the SM may be explainedby the fact that both are a�ected by demand patternvariations with the pattern investigated in this paper

favoring the SM. MINS seems to bene®t more froma lumpy demand pattern and a prolonged planninghorizon. In general, these results emphasize point 1

above in that the performance of these algorithmsdepends on the demand pattern and, as such, con-clusions about their performance should not be gen-eralized.

5. On average, the di�erences among all methods maybe considered practically small; thus, permitting theuse of any method based on a di�erent criterion. In

such cases, the NNM may be preferred when arobust performance is sought.

5. Conclusions

In this paper, a fully developed neural network

model for solving the lot-sizing problem for the case ofa deterministic time-varying demand pattern was pre-sented. The NNM was trained using optimum order

patterns obtained from the Wagner±Whitin algorithm.A special output coding scheme was used to ®lter outnoise and to assure that the order quantity matched

the forecasted demand over the planning horizon. TheNNM displayed a robust performance in all test caseswith an average percentage of optimum order patternsexceeding that of the POQ, MINS, and SM. We have,

thus, demonstrated that a properly developed neuralnetwork model provides a valid alternative for solvingthe lot-sizing problem.

The paper also provided an investigation of thePOQ, MINS, and SM lot-sizing algorithms, withrespect to changes in demand patterns and planning

horizons. For the cases investigated, the performanceof the three algorithms was a�ected by the two selectedfactors, with both the MINS and the SM algorithmsoutperforming the POQ. The superiority of the MINS

and the SM algorithms over the POQ, however, maybe practically insigni®cant, and thus, allowing the sim-plest approach to be used. When combining results

presented in this paper with other results reported inthe literature, it is clear that a general accuracy scalefor all methods may be misleading.

Although the research results demonstrate the suc-cess of the NNM in generating the optimum order pat-tern, they also indicate that a better cost performance

may be obtained by training a NNM using optimumcost information. This may be achieved by adding a

computational layer (with no learning taking place)after the output layer where the cost is computed as afunction of the lot sizing decision described in the out-

put layer. Then, the cost error (the di�erence betweenthe optimum cost and the estimated cost from theNNM output) is computed and back propagated.

Future research will focus on developing such a NNM.Another extension of this research is to develop anintegrated NNM that supports the three major func-

tions of a production planning system: forecasting,MRP, and scheduling.

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A neural network model for solving the lot-sizing problem