Review Managing quality variance in the postharvest food chain Maarten L.A.T.M. Hertog a, * , Jeroen Lammertyn a , Bart De Ketelaere b , Nico Scheerlinck a and Bart M. Nicola € ı a,c a BIOSYST-MeBioS, Katholieke Universiteit Leuven, W. de Croylaan 42, B-3001 Leuven, Belgium (Tel.: D32 16 322376; fax: D32 16 322955; e-mail: [email protected]) b BIOSYST-MeBioS, Katholieke Universiteit Leuven, Kasteelpark Arenberg 30, B-3001 Leuven, Belgium c Flanders Centre of Postharvest Technology, W. de Croylaan 42, B-3001 Leuven, Belgium Data generated in postharvest research are characterised by an inherent large amount of biological variance. This variance generally obscures the behaviour of interest, complicating both the statistical and conceptual interpretation of the data. To be able to manage the postharvest chain, clear insight is required in the propagation of biological variance during postharvest. Recently, biological variance has gained interest within the postharvest community. To analyse experimental data taking into account biological variance several (statistical) modelling techniques have been applied by different authors. These vary from models that can be solved analytically to an- alyse the effect of experimental design parameters (e.g., mixed effects models), via hybrid techniques which combine analyt- ical and numerical techniques, to pure numerical models for dynamic simulations based on ordinary differential equations (like the variance propagation algorithm) and/or partial differ- ential equations (e.g., the FokkerePlanck equation). Each of these techniques has its own possibilities and limitations in terms of the type of variance that can be accounted for, their technical complexity, and their applicability to practical situations. Using tomato colour as an example, the different tech- niques are applied and compared to highlight their strong and weak points in interpreting biological variance. The added value of these techniques is discussed within the framework of managing biological variance in the postharvest chain. Introduction What differentiates the postharvest chain from most other handling chains are (1) the fact that postharvest is dealing with living tissue that inherently is slowly changing with time, and (2) the large sources of biological variance. Postharvest product behaviour is thus inherently affected by the omnipresent biological variance. Most of the time, post- harvest management aims at controlling the average batch behaviour and limiting biological variance as much as pos- sible by sorting and grading the product at the different stages in the postharvest chain. From a marketing point of view, one mostly deals with such batches and not with the individual product items (or objects). Currently, the hor- ticultural industry is not capable of differentiating their postharvest treatments to such an extent that they can actu- ally take into account batch specific handling requirements. For this, much more insight is required into the dynamics of quality change in general and the propagation of biological variance through the postharvest chain. Identification and quantification of the different sources of variance (often referred to as error) is of utmost importance to compare batches and to predict how an initial variance in quality attributes propagates during time. Generally, postharvest scientists tend to ignore the be- haviour of the individual objects and focus on the averaged batch behaviour based on large sample sizes. However, treating biological variance as a nuisance instead of a central part of the modelling effort, can lead to inefficient estimation of means and even to misleading conclusions (Carroll, 2003; Tijskens, Konopacki, & Simcı ` c, 2003). Only recently, new impulses have been given to take into account the underlying behaviour of the individual objects to interpret postharvest batch behaviour. One of the driving forces for this is the increased availability of non-destructive measuring techniques that allow monitoring of individual objects during time (De Baerdemaeker, Hertog, Nicola € ı, * Corresponding author. 0924-2244/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved.
et al., 2004, Hertog, Lammertyn et al., 2007) and Belgian
endive (Hertog, Scheerlinck et al., 2007).
Monte Carlo simulationsMonte Carlo simulation is a numerical stochastic tech-
nique used to solve mathematical problems. Credit for
X
Y
Y = f(X)
A
B
Fig. 1. Given the model equation Y¼ f(X ) defining how the model re-sponse (Y ) depends on the value of a single stochastic parameter (X ),the probability density distribution of the model response (B) is ob-tained by transforming (as indicated by the arrows) the original distri-bution of the stochastic model parameter (A) using the model
equation.
inventing the Monte Carlo method often goes to Stanislaw
Ulam (Eckhardt, 2005), a Polish born mathematician who
worked during World War II on the United States’ Manhat-
tan Project to build the first atomic bomb. The Monte Carlo
method, as it is understood today, encompasses any tech-
nique of statistical sampling employed to approximate solu-
tions to quantitative problems.
The term ‘Monte Carlo’ comes from the name of a city
in Monaco. The city’s main attractions are casinos, which
run games such as roulette wheels, dice and slot machines.
These games provide entertainment by exploiting the ran-
dom behaviour of each game. Similarly, Monte Carlo
methods randomly select values to create scenarios of
a problem. These values are taken from within a fixed range
and selected to fit a given probability distribution (Metro-
polis & Ulam, 1949). So, a Monte Carlo simulation is based
on some model system (either functions or differential
equations) that is defined as a function of random model
parameters characterised by their probability distribution
functions. Using the Monte Carlo method, the model
system is simulated multiple times after random sampling
parameter values from these probability distribution func-
tions. Based on the repeated simulations, the probability
density function of the response variable is constructed as
a function of time. Monte Carlo methods have been used
since the late 1940s, but only since the availability of large
computational power has the technique gained the status of
a numerical method capable of addressing large complex
applications. The technique is useful to obtain numerical
solutions to problems which are too complicated to solve
analytically. Generally, this technique is used for simula-
tion, not for statistical data analysis.
This technique is already extensively applied in the field
of food science, mainly related to food safety and risk anal-
Nicola€ı, Scheerlinck, Verboven, & De Baerdemaeker,
2000; Nicola€ı, Verboven, Scheerlinck, & De Baerdemaeker,
1998; Scheerlinck et al., 2000, 2001) or during cold storage
(Nicola€ı et al., 1999) and to model variability in fruit qual-
ity (Scheerlinck et al., 2004, 2005, 2006).
Selecting a techniqueThe various stochastic modelling techniques discussed
in the previous section all have their own strong and
weak points and can be suitable to different extents depend-
ing on the problem in question. At the same time, none of
these techniques are exclusive and can often be combined
to tackle the different aspects of interest. To give some
guiding in when the different techniques can be applied
a decision tree was developed (Fig. 2).
The first step in selecting a technique (step 1, Fig. 2)
would be to consider the number of random factors in-
volved. When the number of random effects increases, all
of the techniques become numerically quite demanding in
terms of computing time. Because of its simple concept
and equally long computing time, Monte Carlo simulations
would now be in favour. The Monte Carlo results will
mainly give information on the random model effects
showing the propagation of biological variance, allowing
for the reconstruction of 95% confidence intervals for the
stochastic dependent response variable.
When the number of random factors is limited one
should subsequently check (step 2, Fig. 2) whether there
is an analytic model description available or whether the
model only exists in the form of differential equations.
When the analytic solution is available one should consider
the main purpose of the modelling exercise (step 3a,
Fig. 2). When the aim is to statistically analyse the data es-
tablishing proper estimates and confidence bands for the
fixed effects, mixed effects models would be the first
choice. When the aim is to simulate the stochastic behav-
iour of the dependent response variable focussing on the
random effects of the model, the stochastic kinetic ap-
proach would be most suitable.
If there is no analytic solution available one should con-
sider whether the distributions of the random factors are
known (step 3b, Fig. 2). If so, the FokkerePlanck equation
allows to reconstruct the full stochastic behaviour including
the 95% confidence intervals of the dependent stochastic
response variable. If the distributions are not known, the
final option is to apply the variance propagation algorithm
to calculate the transient evolution of the mean and variance
of the modelled response variable.
Application areasThe techniques outlined above, all deal with variance in
complex systems but are not explicitly designed for or lim-
ited to a specific application area. Any application requiring
reliable predictions of complex phenomena has to deal with
sources of variance and requires techniques as outlined
above. The outlined techniques can be applied to perform
proper risk analysis in the nuclear weapon industry, the
oil industry, the financial world, for clinical trails, to eval-
uate building construction and also while safeguarding
food quality and food safety. Depending on the application
area, higher stakes might be involved to warrant a proper
full stochastic analysis of the complex system of interest.
When focussing on the wider food area, potential appli-
cations are manifold given the large sources of variance and
the batch oriented character of many food processes. From
an economical point of view, applications are more likely to
be worthwhile for products with a large added value, such
as functional foods. From a general food quality point of
view, proper insight is required in the propagation of vari-
ance when dealing with any food process that starts from
a highly variable raw product and tries to turn this into
a food product of a reproducible constant quality. Inher-
ently, batch oriented or semi-continuous processes, as gen-
erally applied in the food industry, all have to deal with
these constantly varying input streams.
Potential applications in the wider food area could focus
on production, processing or health related issues where
microbial food safety becomes important. Using a recent
volume of the current journal, Trends in Food Science &
Technology, as a guideline, topics were identified that could
Number of random
factors involved?
Analytic solution
available?
MC
Main purpose?Distribution
known?
MM VPSK FP
≤ > 33
NoYes
NoYesStatistical
inferenceSimulation
1
3a
2
3b
Fig. 2. Decision tree showing the steps of selecting the appropriatestochastic modelling techniques. (MM: Mixed effects models; SK:stochastic kinetic models; MC: Monte Carlo simulations; FP:
potentially benefit from the use of stochastic modelling
techniques (Table 1). This list from Table 1 is far from
complete but just tries to raise the consciousness of the
huge potential of stochastic modelling techniques when
analysing time dependent data prone to variance.
Case studyPostharvest research produces experimental time depen-
dent data characterised by large sources of biological vari-
ance. Just to stay within our own comfort zone the case
study discussed in this section focuses on a typical posthar-
vest example on the ripening of tomatoes as indicated by
their colour.
General model behaviourUsing tomato colour as an example, the different tech-
niques are applied and compared to highlight their strong
and weak points in interpreting biological variance. All
techniques use the same colour change model as developed
by Hertog et al. (2004) to enhance comparison between the
techniques. The general behaviour of colour as a function
of time is depicted in Fig. 3 showing the meaning of the dif-
ferent model parameters. In addition, the rate constant is as-
sumed temperature dependent. The model can be either
applied in the form of its differential equations or using
the corresponding analytic solution (see Hertog et al.,
2004, for full model details).
The model describing hue colour as a function of time is
deterministic in the sense that time evolution of colour is
entirely predetermined by an exact knowledge of the tem-
perature conditions, colour (or biological age) at harvest,
the rate of colour change and the colour limits. Due to
biological variance and variability in the physiological state
some of the model parameters are inevitably subjected to
random variance introducing variance in the dependent
response variable hue.
Experimental dataThe experimental data used for this case study consists
of two parts. One part (Fig. 4) considers storage data
from three tomato cultivars (‘Quest’, ‘Style’ and ‘Tradiro’)
stored for 21 days at three different constant temperatures
(12 �C, 15 �C or 18 �C) using 20 fruit per condition.
The second part (Fig. 5) considers one large batch of
‘Tradiro’ tomatoes (300 fruit) stored for 11 days at varying
temperature conditions. Fore more detailed experimental
information see Hertog et al. (2004), Hertog, Lammertyn
et al. (2007). As shown by the data, one encounters a con-
siderable amount of colour variability between tomatoes at
harvest, the magnitude of which is mainly determined by
harvest criteria, and how accurately these are applied in
practice. This variability typically decreases with time be-
cause all tomatoes of a batch asymptotically evolve to
a similar red colour after storage. This clearly illustrates
the aspect of heteroscedasticity in hue colour with time.
Data analysisUpon completion of an experiment one is confronted with
an amount of experimental variance. This experimental var-
iance is partly induced by experimental factors (e.g.,
Table 1. Potential application areas for stochastic modelling tech-niques identified based on manuscripts published in 2005 inTrends in Food Science & Technology (volume 16)
Production and breedinge (Micro)nutrient enhancement in plant crops(e.g. Storozhenko et al., 2005)
Processinge Microbial growth in relation to fermentation technology(e.g. Hammes et al., 2005)e High pressure processing(e.g. Estrada-Giron, Swanson, & Barbosa-Canovas, 2005)e Drying and rehydration of foods(e.g. Sam Saguy, Marabi, & Wallach, 2005)e Increasing natural food (micro)nutrients through bioprocessing(e.g. Jagerstad et al., 2005)e Formation of carcinogens during food processing(e.g. Claeys, De Vleeschouwer, & Hendrickx, 2005)
Storage and transporte Antimicrobial effect of postharvest UV treatment(e.g. Shama & Alderson, 2005)e Quality changes during processing and storage(e.g. Wrolstad, Durst, & Lee, 2005)e Generation of aroma compounds(e.g. Hansen & Schieberle, 2005)e Food chain design(e.g. Apaiah, Hendrix, Meerdink, & Linnemann, 2005)
Health and safetye Microbial growth in relation to food safety(e.g. De Vuyst & Neysens, 2005)e Reactive oxygen species in food(e.g. Wiseman, 2005)e (Micro)nutrient bioavailability, intake and metabolism(e.g. Gregory III, Quinlivan, & Davis, 2005)
k
H+ ∞
H− ∞
H0
Preharvest Postharvest
time
Hue
colo
ur
tage
Href
harvest
Fig. 3. Tomato colour can be measured as hue values (H in �) changingfrom immature green (H�N in �) to a fully ripe red (HþN in �) colourfollowing a sigmoid pattern. Tomatoes change colour following a cer-tain rate constant (k in d�1) that depends on temperature. The tomatoesare harvested at some initial colour (H0 in
�). The biological age of anindividual tomato (tage in d ) is defined as the time required to changecolour from a certain given reference point (Href in
�) to the initial col-our observed at harvest (redrawn from Hertog et al., 2007).
temperature and cultivar effect) while remaining variability
is considered as residual (unexplained) variance. The classi-
cal statistical approach to analyse the hue colour data from
different cultivars stored at different temperatures would be
to perform a nonlinear regression analyses of the data fitting
the model per cultivar. For the constant temperatures the an-
alytic solution can be used while for dynamic temperature
data the ordinary differential equation formulation has to
be used. This model approach describes the averaged batch
behaviour perfectly well (Fig. 4) but tries not to explain the
remaining variance as all of it is regarded as residual error.
When comparing the parameter estimates between cultivars,
conclusions will be hampered by the inaccurate standard
errors of the estimates as they are biased by the incomplete
error analysis. Using the stochastic techniques outlined
above the unexplained residual error from the classical ap-
proach will be dissected further and assigned to the various
sources.
Identification of sources of varianceTo identify the main source(s) of variance one can start
with a classical analysis of the multiple individual objects
to build distributions for the various model parameters esti-
mated. This provides information on the type of distribution
(e.g., normal, Poisson) and the distribution parameters (e.g.,
mean, standard deviation) of the stochastic model parame-
ters including the possible correlation structure between
these stochastic model parameters. This information is cru-
cial for any of the stochastic techniques applied, except for
the variance propagation algorithm which is a distribution
free approach.
Subsequently one of the available stochastic techniques
can be selected to implement models with one or more sto-
chastic model parameters. The goodness of fit of these nested
models can be compared based on the various criteria avail-
able (e.g., root mean square error, likelihood ratio, Akaike in-
formation criterion, Bayesian information criterion; see
Forster and Sober, 1994, for a non-technical introduction to
these criteria). By doing so the model structure is restricted
to include only the most important stochastic variables.
The first source of variance is often most dominant and cor-
responds to inevitable natural biological variability between
the observed objects. Accounting for this source of variance
greatly reduces the unexplained variance term. Using the
final model structure with the main sources of variance
selected one can focus on analysing the data in more detail.
Statistically comparing groupsIn a mixed model context one has the tools (e.g., intro-
duction of random effects) to subdivide the variance com-
ponent into different sources of variance, such as variance
0 5 10 15 20
40
50
60
70
Quest
time (d)
40
50
60
70
Style
Hu
e (
°)
40
50
60
70
Tradiro12 °C
15 °C
18 °C
Fig. 4. Change in hue colour (H in �) of three tomato cultivars (‘Tra-diro’, ‘Style’ and ‘Quest’) stored at three constant temperatures(12 �C, 15 �C and 18 �C) as a function of storage time. The symbols in-dicate the averaged colour of 20 tomatoes while the bars enclose the95% confidence interval based on the standard errors of the measure-ments. The lines represent the general model describing the average
batch behaviour (redrawn from Hertog et al., 2004).
0 2 4 6 8 10
40
50
60
70
80
90
Hu
e (
°)
time (d)
12
14
16
18
T (
°C)
Fig. 5. Change in hue colour (H in �) as a function of storage timeof 300 ‘Tradiro’ tomatoes stored at a varying temperature regime asindicated in the top graph. The lines represent the colour change of
the individual tomatoes.
between objects, variance within objects, measurement var-
iance and unexplained variance. Furthermore, mixed
models offer a framework to test which of the sources of
(biological) variance are significantly present. This allows
accurate accounting for the typical varianceecovariance
structure of the experimental data set which is a prerequisite
for building accurate predictive models, and for appropriate
testing of the significance of the experimental factors (e.g.,
cultivar effects). Therefore, the mixed model framework is
perfectly suited to statistically compare groups of observa-
tions. In the present case study it is of interest to test
whether there are significant differences in colour profile
between the different batches (e.g., tomato cultivars). The
average colour profile per cultivar is characterised by two
fixed parameters: a colour change rate (k) and the asymp-
totic end colour (HþN). The test to identify significant dif-
ferences is based on the varianceecovariance structure of
these fixed effects. The latter can only be estimated cor-
rectly by accounting for the different sources of variance.
The two main sources of variance identified are one random
effect related to initial colour (H0) and one random effect
related to the asymptotic end colour (HþN). The remaining
variance is assumed to be random error.
Fig. 6 shows the 95% joint confidence ellipses for the
colour change rate and the asymptotic colour for the three
cultivars. The cultivar ‘Tradiro’ clearly has a higher colour
change rate and a lower asymptotic hue value compared to
the other two cultivars. The cultivars ‘Quest’ and ‘Style’
have partly overlapping confidence ellipses referring to
similar average colour profiles. An F-test indicates no
significant difference in colour behaviour between both cul-
tivars at the 5% significance level.
Propagation of biological varianceThe mixed effects model approach results in, among
others, the distribution parameters of the stochastic model
parameters including their covariance structure but, by
itself, does not provide information on the propagation of
biological variance with time. A quick way to get some
idea of the stochastic behaviour of hue colour is obtained
by the variance propagation algorithm that evaluates first
order approximations of the mean and variance of hue
with time without providing information on the shape of
the distribution of hue with time (Fig. 7B, C).
More detailed information on the model response is ob-
tained using the other techniques. Using the distributions of
40 41 42 43 44 45
1.2
1.3
1.4
1.5
1.6
1.7
k (
10
-3·
d-1
)
H+∞ (°)
Quest
Style
Tradiro
Fig. 6. Ninety-five percent joint confidence ellipses for the colourchange rate (k) and the asymptotic colour (HþN) for the three cultivars
stored at constant temperatures.
0 5 10 15 20
2
3
4
5
C
time (d)
40
45
50
55
60
B
μH (
°)
0
20
40
60
80
100A
% f
ruit
H (
°)
12 °C
15 °C
18 °C
Fig. 7. A. Batch ripening expressed as the cumulative percentage offruit that reached a critical hue value of 50�. B. Propagation of the av-eraged hue colour (m) as predicted by the variance propagation algo-rithm. C. Propagation of the standard deviation of hue colour (s) aspredicted by the variance propagation algorithm. All graphs were
made for the cultivar ‘Quest’ stored at 12 �C, 15 �C or 18 �C.
the stochastic model parameters as identified by the mixed
effects model, Monte Carlo simulations can be performed
to generate an approximation of the full stochastic behav-
iour of the model showing the propagation of biological
variance in hue with time (Fig. 8). These results describe
in full detail the stochastic behaviour of the modelled re-
sponse variable with the accuracy depending on the number
of runs. To get smooth surfaces one typically has to do five
to ten thousand simulation runs. Using the stochastic
kinetic model approach or the FokkerePlanck equation,
the same response surface is obtained in a single run,
with the stochastic kinetic model approach being the less
computational intensive one of the three approaches.
When using the FokkerePlanck equation one would also
start from the distributions of the stochastic model para-
meters as derived from a classical analysis on the multiple
individual objects or as identified by the mixed effects
model. In the case of hue colour, biological variance can
be explicitly incorporated through the introduction of three
correlated random variables, i.e., initial hue colour (H0), the
rate of colour change (k) and the asymptotic colour value
(HþN). The random variables can be represented by means
of a three dimensional joint probability density function us-
ing a trivariate Gaussian distribution. The FokkerePlanck
equation describes how this density function propagates
through the three dimensional space as a function of
time. After integrating the three dimensional density distri-
butions the final result is obtained, which is comparable to
that of an approximate solution generated by a Monte Carlo
simulation (Fig. 8).
In case of the stochastic kinetic approach one can either
start from already established distributions of the stochastic
model parameters or one could actually fit the model
response surface on the observed data thus establishing
the distribution parameters for the stochastic model param-
eters while applying this stochastic kinetic technique. How-
ever, the standard errors of these estimates would not be as
statistically correct as when using mixed effects models.
The final result on the simulated bath behaviour is however
identical to that established through either Monte Carlo
simulations or by using the FokkerePlanck equation
(Fig. 8).
Batch acceptanceOnce the full stochastic behaviour of hue as a function
of time has been established this information can be used
to answer questions like which part of the batch is likely
to have a quality below or beyond a certain critical quality
limit. Even though the averaged batch quality might still be
acceptable some part of the batch might have crossed these
limits (Fig. 7A). This can be simply done by integrating the
area under the probability distribution function of hue as
a function of time. This kind of information basically
reduces the complex information from the probabilistic
response surface into a manageable form for the industry.
As can be seen from Fig. 7A, the rate at which batch
acceptance changes is directly related to temperature, as
the rate at which biological variance propagates is directly
related to the rate of colour change of the individual tomatoes
in the batch. For dynamic temperature conditions (Fig. 5),
temperature, and thus the rate of propagation changes by
the hour. However, using the stochastic kinetic model
approach, these dynamic temperature scenarios can be ana-
lysed as well (Fig. 9A). It can be seen how, during periods
of high temperature (Fig. 9A, going from day 1 to day 4),
the variation in hue colour propagates faster, than during
periods of lower temperature (Fig. 9A, going from day 0 to
day 1). This is directly reflected in the percentage of fruit
reaching the critical quality limit of 50�, increasing more or
less steeply depending on temperature (Fig. 9B).
Comparative overviewHaving come this far in reading this review on stochastic
modelling techniques one might have noticed that, although
the various techniques could all be applied to the same case
study, they do have their own specific possibilities and
limitations, both regarding technical and practical issues.
In this section we will explicitly summarise the main issues.
Model formulationGenerally two types of model formulations are encoun-
tered, the ordinary differential equations or the analytic so-
lution derived from them (an algebra€ıc equation). Mixed
effects models and stochastic kinetic models can only
deal with analytic model formulations while the variance
propagation algorithm and the FokkerePlanck equation
are based on the differential equation model formulation.
Monte Carlo simulations can be applied to either type of
model formulation. Still there is some flexibility as the
40 45 50 5560
6570 75
0
5
10
15
20
0.0
0.1
0.2
0.3
18 °C15 °C
12 °C
freq
uen
cy
tim
e (d)
Hue (°)
Fig. 8. Propagation of the relative frequency distribution of hue colour(H in �) of ‘Quest’ tomatoes stored at 12 �C, 15 �C or 18 �C. Startingfrom one and the same batch with a certain initial distribution ofhue (the front black curve) the frequency distributions propagate fasteror slower towards lower hue values depending on storage temperature
(the three ridges).
two types of model formulations can often be converted to
each other. The differential equations can be integrated to
obtain an analytic solution while an algebra€ıc equation
can be differentiated to obtain differential equations.
Prime purposeWhen using mixed effects models, emphasis is on analy-
sing data to establish proper estimators of the fixed and ran-
dom effects, thus allowing proper statistical inference. The
numerical techniques (Monte Carlo simulation, Fokkere
Planck equation and variance propagation algorithm) are
focussing on prediction of the complex system behaviour.
Theoretically one could combine this with parameter esti-
mation as well but generally this becomes rapidly unman-
ageable from a computational point of view. However,
modern computational algorithms and parallel computing
facilities allow to manage computational analysis of such
complex systems. The stochastic kinetic model approach
takes an intermediate position. As it works with an alge-
bra€ıc equation it is still feasible to use it for parameter
estimation while at the same time it can be used to predict
the stochastic system behaviour.
Prior knowledgeIn all cases one needs the model structure (either as dif-
ferential equations or as an algebra€ıc equation). For all
techniques but the variance propagation algorithm, one
has to decide on the type of distribution function underlying
the random effects. As the variance propagation algorithm
is a distribution free technique there is no need to identify
the type of distribution when using this approach. For the
numerical techniques one also needs prior knowledge on
all model parameter values (fixed effects) including the ini-
tial distribution parameters (e.g. average and standard devi-
ation) of the random effects responsible for the stochastic
model behaviour.
Stochastic dimensionalityThe more stochastic variables are included the more re-
alistic system behaviour can be obtained, but the more
computational intensive it becomes. In all of the mentioned
techniques every parameter could theoretically be consid-
ered a stochastic variable with an additional residual error
term containing that part of the observed variance that re-
mains unexplained by the stochastic model. When looking
at the food related literature the number of stochastic sour-
ces is generally limited to one to three.
System boundary conditionsWhen using an algebra€ıc model formulation, system
boundary conditions like temperature and gas conditions
have to be kept constant with time as the system behaviour
is generally affected by changing these boundary condi-
tions. When the model has to be able to cope with a system
responding to dynamic boundary conditions, the use of the
differential equation formulation is imperative. Conse-
quently, the numerical modelling techniques discussed
can be applied to both constant and dynamic boundary con-
ditions while the mixed effects models can only be applied
in the case of constant boundary conditions. The stochastic
kinetic model approach forms again an exception as in
some case one can work around the problem of dynamic
boundary conditions as was done for instance by introduc-
ing the physiological time concept. Consequently, also the
stochastic kinetic model approach can be applied to both
constant and dynamic boundary conditions.
Prime outcomeThe prime outcome of a mixed effect model approach is
the set of parameter values and their confidence intervals
describing the fixed and random effects. The prime out-
come of the stochastic kinetic model approach is the
4050
60
70
80 0
2
4
6
810
0.00
0.05
0.10
0.15
0.20
0.25
rela
tive f
req
uen
cy
Hue (°)tim
e (d
)
A
0 2 4 6 8 10
40
60
80
100
time (d)
% f
ruit
B
Fig. 9. A. Propagation of the relative frequency distribution of hue col-our (H in �) of ‘Tradiro’ tomatoes stored at dynamic temperature con-ditions (Fig. 5) showing the observed frequencies (symbols) ascompared to the modelled frequency distributions (curves; redrawnfrom Hertog, Lammertyn et al., 2007). B. Batch ripening of the samebatch of fruit expressed as the cumulative percentage of fruit reaching
a critical hue value of 50�.
density distribution of the response variable as a function of
time combined with the parameter estimates. In the case of
Monte Carlo simulations the prime outcome is a collection
of multiple individual simulations of the complex system
which can afterwards be collated into the full density distri-
bution of the response variable as a function of time. The
variance propagation algorithm provides as its prime out-
come the transient evolution of the mean and variance of
the modelled response variable. Finally, the FokkerePlanck
equation provides as its prime outcome the density distribu-
tion of the response variable as a function of time. So
depending on the research question one should choose
one or more of the proposed techniques.
ImplementationThe main statistical software packages, nowadays, have
standard procedures implemented to run mixed effects
models. All other techniques are either only supported by
highly dedicated software packages or have to be imple-
mented in engineering software taking into account the spe-
cific requirements from the particular model studied. From
these, Monte Carlo simulation is the most straightforward
technique to implement while the remaining three tech-
niques require a larger conceptual effort.
Computational requirementsWith increasing number of stochastic dimensions all of
the proposed techniques will exhibit a computational time
that is exponentially increasing. In these cases the only fea-
sible technique remaining is the Monte Carlo approach as
the others become too complex to manage. When using
up to three stochastic parameters all of the techniques are
feasible within acceptable computational time scales (in
the range of minutes) with the exact timing depending
on, for instance, the choice of starting values when estimat-
ing parameter values, and the numerical fine-tune strategies
applied for all of the numerical techniques.
Practical applicationThe stochastic modelling techniques advocated in this re-
view can be applied either within a research or a practical
commercial context. Within the research context emphasis
would be on the aspect of analysing experimental data to in-
crease insight in and understanding of the behaviour of the
food product of interest. These techniques provide, for in-
stance, a solid base to statistically screen different groups
(whether these are cultivars, growers, years, or growing con-
ditions) for fixed effects taking into account the full range of
variance observed. Within a practical commercial context
emphasis would be on the simulation aspect to take into ac-
count the full impact of the various sources of variance
when handling a certain batch of food products. These mod-
elling approaches can be either applied at the level of a single
process or at that of a whole chain. The stochastic modelling
techniques can be incorporated in a model predictive control
environment to optimise and control (parts of) the food
processing chain.With model predictive control, the control-
ler relies on an empirical model of the process which predicts
the behaviour of dependent variables of a dynamical system
based on independent variables to compute a costminimizing
control. Although model predictive control originates from
the chemical industries it has found its way to the agro-
food industry as well (Verdijck, Hertog, Weiss, & Preisig,
1999). The introduction of stochastic modelling techniques
in model predictive control would further enhance its poten-
tial value as it allows the controller to take into account batch
variance.
The added advantage of including stochastic information
in descriptive and predictive chain models will be obvious
when dealing with the real life situation of a complex logis-
tic food chain. Generally, when modelling a logistic chain,
Operation Research models are applied (Slats, Bhola,
Evers, & Dijkhuizen, 1995) that focus on the operational
side of the logistic chain. The number of management
models that do include product models (Broekmeulen,
2001) are relatively limited or often use fully empirical
models (Bogataj, Bogataj, & Vodopivec, 2005). The sto-
chastic model approaches discussed can potentially contrib-
ute to existing logistic chain models by allowing them to
optimise the operational ‘quality’ of the chain taking into
account the actual quality of the food products including
the complete range of variance presence.
AcknowledgementsThe authors gratefully acknowledge the financial support
from the Institute for the Promotion of Innovation by Science
and Technology in Flanders (Project IWT 030807 and IWT
040726). Nico Scheerlinck is postdoctoral fellow with the
Flemish Fund for Scientific Research (FWO-Vlaanderen)
and Bart De Ketelaere is postdoctoral fellow with the Indus-
trial Research Fund (IOF) of the K.U. Leuven.
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