Computers and Electronics in Agriculture 79 (2011) 116–124

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Computers and Electronics in Agriculture

journal homepage: www.elsevier .com/locate /compag

Interval analysis on non-linear monotonic systems as an efficient toolto optimise fresh food packaging

Sebastien Destercke a,b,⇑, Valerie Guillard c

a INRA/CIRAD, UMR1208, 2 Place P. Viala, F-34060 Montpellier cedex 1, Franceb LIRMM, UMR 5506, 161 rue Ada, 34095 Montpellier, Francec Université Montpellier 2, UMR1208, 2 Place P. Viala, F-34060 Montpellier cedex 1, France

a r t i c l e i n f o

Article history:Received 20 October 2010Received in revised form 27 August 2011Accepted 29 August 2011

Keywords:Interval analysisRobust designFuzzy setsSensitivity analysisUncertainty analysis

0168-1699/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.compag.2011.08.014

⇑ Corresponding author at: INRA/CIRAD, UMR120Montpellier cedex 1, France.

E-mail addresses: sebastien.destercke@cirad.fr (Smontp2.fr (V. Guillard).

a b s t r a c t

When few data or information are available, the validity of studies performing uncertainty analysis orrobust design optimisation (i.e., parameter optimisation under uncertainty) with a probabilistic approachis questionable. This is particularly true in some agronomical fields, where parameter and variable uncer-tainties are often quantified by a handful of measurements or by expert opinions. In this paper, we pro-pose a simple alternative approach based on interval analysis, which avoids the pitfalls of a classicalprobabilistic approach. We propose simple methods to achieve uncertainty propagation, parameter opti-misation and sensitivity analysis in cases where the model satisfies some monotonic properties. As a real-world case study, we interest ourselves to the application developed in our laboratory that has motivatedthe present work, that is the design of sustainable food packaging preserving fresh fruits and vegetablesas long as possible.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

There are many sources of uncertainties in life science and inagronomy, the main reasons for it being the high variability ofliving organisms and the error of measurement devices. Also, thenumber of available samples for a given experiment may belimited (sometimes even reduced to one sample), due to cost orpractical limitations. In such situations, it can be hard to determinea meaningful probabilistic model of the parameters, let alone ajoint probabilistic model over all parameters. In other situations,uncertainty around a parameter or a constant can be describedby expert opinions, and whether these opinions can be faithfullytranslated by single probabilities is questionable (see, e.g., (Walley,1991, Section 4) or (Sandri et al., 1995)).

In such situations, it may be better to use interval modellingand interval analysis to perform uncertainty studies, simplybecause determining intervals requires less data and knowledge.Also, using interval analysis amounts to make no assumptionsabout parameter dependencies.

In this paper, we consider dynamical non-linear modelsdescribing the evolution of variables, with the aim to optimisesome of the parameter values w.r.t. some given objective. In such

ll rights reserved.

8, 2 Place P. Viala, F-34060

. Destercke), guillard@univ-

systems, the values of initial conditions, of non-modifiableparameter values or even of the objectives may be ill-known. Itis then desirable to perform some uncertainty analysis to achieverobust design. Performing such analysis with classical probabilisticmethods (Hertog et al., 2007; Bedford and Cooke, 2001) usuallyrequires to:

� specify the distribution of each input variable,� specify the dependency structures between input variables,� perform (costly) numerical analysis to evaluate the output

uncertainty.

Meeting such requirements necessitates an important amount ofinformation and data. It also involves the use of techniques havinga high computational cost. In practice, when not enough informa-tion is available, distribution shapes (e.g., normality) and depen-dence assumptions (e.g., independence between all variables) areoften chosen accordingly to some practical criterion rather thanto available information. However, the validity of such choices,not confirmed by experiments or available knowledge, may bequestioned, as well as the validity of subsequent analysis results(Ferson and Ginzburg, 1996). They then provide overly preciseand misleading conclusions, which may in turn lead to unwar-ranted and non-robust design choices.

When few data are available, an alternative is to use intervalanalysis (Jaulin et al., 2001) to perform the uncertainty analysis,that is to consider that only the bounds in which each parameter

S. Destercke, V. Guillard / Computers and Electronics in Agriculture 79 (2011) 116–124 117

may vary are known (an information that is often available). Suchan analysis comes down to consider that:

� variable and parameter distributions are unknown (up to theirbounds),� dependence structure between variables and parameters is

unknown.

Compared to probabilistic analysis, interval analysis can thereforebe seen as a conservative analysis, in the sense that it does notmake any additional hypothesis with regard to the available infor-mation, and possibly ignores some of the available information.However, in scientific analysis as well as in robust design, it is saferto use such conservative methods than to make unsupportedassumptions. Also note that, when function f has some monotonicproperties (Singer and Barton, 2006; Fortin et al., 2008) (a commoncase in life sciences and other domains (Ramdani et al., 2010;Baudrit et al., 2009), where simple models are often encountered),performing interval analysis may require very few computationscompared to, say, probabilistic Monte–Carlo analysis. When thesemonotonic properties are not satisfied, performing interval analy-sis requires more complex techniques (Ramdani et al., 2009) withincreased computational complexity (the computational cost thenbecoming comparable to the one of probabilistic methods). How-ever, whether the monotonic properties are satisfied have noimpact on the results conservativeness of interval analysis.

In this paper, we introduce a set of methods to achieve uncer-tainty propagation, parameter optimisation and sensitivity analy-sis on monotonic dynamical systems when uncertainty isdescribed by intervals. Notations and general problem formula-tions are introduced in Section 2. Section 3 then provides detailsabout the method itself.

Finally, we illustrate the method on the real-world case studythat have motivated the present work and that is currently treatedin our laboratory (Charles et al., 2005) (see website http://www.tailorpack.com/). It concerns the design of sustainable freshfood packaging, with the objective to preserve food from decayas long as possible. We use our method to, first, perform uncer-tainty analysis of a model describing gas exchanges between thepackaging atmosphere and exterior atmosphere and, second, opti-mise oxygen and carbon dioxide permeances of packaging materi-als for a given fruit or vegetable, here chicory. The whole case studyis described in Section 4.

2. Problem setting and notations

Vectors of model variables and parameters will be denoted bybold letters (x,p, . . .), while specific values of these vectors willde denoted by non-bold letters (x,p, . . .). As the models consideredin this paper are multivariate, indexed letters xj will denote the jthelement of vector x. Sets will be denoted by calligraphic letters(X;P; . . .). The real line will be denoted by R.

We consider a dynamical non-linear model _x ¼ f ðx;pE;pD; tÞwith f : Rmþnþ1 ! Rm a time-dependent function that describesthe evolution of m state variables x 2 Rm, i.e. x = (x1, . . .,xm). Theevolution of these variables depends on nE environmental parame-ters pE (whose values cannot be controlled) such as temperature orexternal pressure and nD design parameters pD (which values canbe modified) such as mechanical or chemical properties of syn-thetic compounds. They form a vector p = (pD,pE) of n parametersp ¼ ðp1; . . . ; pnÞ 2 Rn.

Given some values p 2 Rn of the parameters and some initialconditions xð0Þ 2 Rm of the state variables, the solution of the sys-tem represented by f at time t is x(t) = (x1(t), . . .,xm(t)) where xi(t)describes the state of the ith state variable xi at time t.

When initial conditions x(0) and environmental parameter val-ues pE are known, a classical design problem (Aster et al., 2005)consists in identifying the values bpD 2 RnD of the design parameterspD so that the solutions x(t) are as close as possible to a givenobjective bxðtÞ on the state variables. However, both the exact val-ues of initial conditions x(0), of parameters p or of the objectivebxðtÞ to reach are seldom known with certainty. In the next section,we detail how the problem can be treated when these values be-come interval and when the model satisfies some monotonicproperties.

3. Interval analysis and design optimisation

In this section, we start by giving some refreshers on classicalinterval analysis and interval analysis on dynamical monotonicmodels, illustrating them on a simple example. We then detailour proposed optimisation and sensitivity analysis methods forsuch models.

3.1. Basics of interval analysis

In the computational literature, interval analysis was firstdeveloped to take account of numerical errors (Moore, 1979).However, interval analysis is now mostly used to perform robust-ness analysis in applications (robotics (Jaulin et al., 2002), chemi-cal, biological, . . .) where variable values are imprecisely known(Jaulin et al., 2001).

A real interval [x] :¼ [x�,x+] of a variable X is a connected andclosed subset of R. The set of real intervals is usually denoted IR.An interval vector [x] over Rn (also called box) is the Cartesianproduct of n intervals. The classical problem of interval analysisconsists in replacing, in a given function y = f(x1, . . .,xn) from Rn toRm, the point values x = (x1, . . .,xn) of variables x = (x1, . . .,xn) byintervals [x] = ([x1], . . ., [xn]) and to compute the range

f ð½x�Þ ¼ ff ðxÞjx 2 ½x�g: ð1Þ

Usually, f([x]) is not a box over Rm, but a complicated subset of it(i.e., it cannot be expressed as a Cartesian product of intervals).Rather than computing the exact propagation f([x]), approximationtechniques can be used to compute a box [f]([x]) over Rm that willbe an inclusion function, i.e.,

f ð½x�Þ# ½f �ð½x�Þ: ð2Þ

This comes down to compute an outer approximating box of f. Thenotion is illustrated in Fig. 1. Among possible techniques to com-pute such inclusion functions is interval arithmetic (Moore, 1979),where classical arithmetic operations {+,�, /,�} are replaced bytheir interval equivalent. Note that [f]([x]) is not unique, anddepending on the technique and on the characteristic of the modelf, [f]([x]) will be more or less close to f([x]). A challenge is then tofind techniques to compute easily an inclusion function [f]([x]) thatis as close as possible to f([x]). In the next section, we recall tech-niques that gives best possible inclusion function and that can beapplied to the particular dynamical models we are interested in.

3.2. Interval analysis to propagate uncertainties in dynamical systems

Let us consider the more complex problem of evaluatingthe solution of the dynamical system _x ¼ f ðx;pE;pD; tÞ withf : Rmþnþ1 ! Rm. In this model, the evolution of each state variablexi, i = 1, . . .,m is described by an ordinary differential equation(ODE) such that _xi ¼ fiðx;pE;pD; tÞ. At a given time t, thesolution of the system are m values xi(t), i = 1, . . .,m of the statevariables. Here, values xi(t) are computed for a finite number oftime steps. We denote by T ¼ ½0; t� the time domain of the model

Fig. 1. Illustration of interval analysis with inclusion function (Eq. (2)).

118 S. Destercke, V. Guillard / Computers and Electronics in Agriculture 79 (2011) 116–124

and assume that each xi(t) is computed for T different times valuestk, k = 1, . . .,T.

For such models, the classical problem of interval analysis isformulated as follows: given m initial conditions xi(0) 2 [xi](0) :¼[xi�,xi

+](0), i = 1, . . .,m and n parameters intervals pj 2 [pj],j = 1, . . .,n, find the multivariate sets XðtkÞ such that

XðtkÞ ¼ fxðtkÞ 2 Rmj _x¼ f ðx;pE;pD; tÞ;p 2 �nj¼1½pj�;xð0Þ 2 �m

i¼1½xi�ð0Þgð3Þ

for each time set tk, k = 1, . . .,T. As in the non-dynamical case (seeEq. (2) and Fig. 1), sets XðtkÞ can have a very complex shape andan easier problem is to find some inclusion functions or boxes½x�ðtkÞ ¼ ½x�; xþ�ðtkÞ � XðtkÞ. That is, determine the lower (x�i ðtkÞ)and upper (xþi ðtkÞ) envelopes of every state variable xi for each timestep t = tk, k = 1, . . .,T.

Let us call configuration an element of the Cartesian productC :¼ �m

i¼1½xi�ð0Þ�nj¼1½pj�. Finding the best inclusion function then

consists in finding among configurations in C those for which solu-tions of _x ¼ f ðx;pE;pD; tÞ reach the bounds [xi

�,xi+](tk) for i = 1, . . .,m

and for any time step tk, k = 1, . . .,T (note that configurations reach-ing bounds for two different time steps and for two different statevariables may be different). This is the problem we will considerhere.

Among the set C of configurations, denote byE ¼ �n

i¼1fx�i ; xþi g�mj¼1fp�j ; pþj g the set of extreme ones, that is the

set of configurations whose elements are vertices of the boxes½x�ð0Þ :¼ �m

i¼1½xi�ð0Þ and ½p� :¼ �nj¼1½pj�. The number of such vertices,

i.e. of elements of E is 2(n+m), while C contains an infinity of them.To illustrate the approach used in this paper, we will use the fol-lowing simple example:

Example 1. Consider the very simple system from R! R

_x1 ¼ f1ðp1Þ ¼ p1;

where p1 2 [p1] and x1(0) 2 [x1](0) are positive numbers, and the(analytical) solution of the system is x1(t) = p1 � t + x1(0). Here, bothx = (x1) and p = (p1) are reduced to a single element each. Ourknowledge about p1 and initial condition is given by the intervals

Fig. 2. Example 1 set of (ex

p1 2 [0.5,1] and x1(0) 2 [1,3]. The sets of all possible combinationsof [p1] and [x1](0), together with their extreme combinations areillustrated in Fig. 2. In this example, the set C ¼ ½1;3� � ½0:5;1�,while the set E ¼ fð1;0:5Þ; ð1;1Þ; ð3;0:5Þ; ð3;1Þg is reduced to fourpoints.

As for interval analysis on classical functions (Fortin et al.,2008), there exist specific cases (Delanoue, 2009; Ramdani et al.,2010; Singer and Barton, 2006) where the bounds of [xi](t) aresolutions of the system _x ¼ f ðx;pE;pD; tÞ computed for specific ele-ments of the set H of extreme configurations, thus reducing thenumber of necessary computations to find these bounds. Beforerecalling what are these cases, let us first define the notion ofmonotonicity for dynamical system _x ¼ f ðx;pE;pD; tÞ.

Definition 1. fi is said to be dynamically increasing w.r.t. variablexj, j – i or parameter pk if

8p 2 ½p�; xð0Þ 2 ½x�ð0Þ; t; @fiðx;pE;pD; tÞ@xj

P 0 ð4Þ

or

8p 2 ½p�; xð0Þ 2 ½x�ð0Þ; t; @fiðx;pE;pD; tÞ@pk

P 0 ð5Þ

and it is said to be dynamically decreasing w.r.t. variable xj, j – i orparameter pk if

8p 2 ½p�; xð0Þ 2 ½x�ð0Þ; t; @fiðx;pE;pD; tÞ@xj

6 0 ð6Þ

or

8p 2 ½p�; xð0Þ 2 ½x�ð0Þ; t; @fiðx;pE;pD; tÞ@pk

6 0 ð7Þ

For a given function fi (or, equivalently, for a given variable Xi), let usdenote by p%,i,x%,i and p&,i,x&,i the vectors of parameters and vari-ables for which fi is dynamically increasing and decreasing, respec-tively. Note that they are disjoint, i.e. p%,i and p&,i do not containcommon elements, as well as x%,i and x&,i. We also assume fromnow on that these vectors form partitions of the parameters and

treme) configurations.

Fig. 3. Interval analysis with Example 2 model.

S. Destercke, V. Guillard / Computers and Electronics in Agriculture 79 (2011) 116–124 119

variables, in the sense that for a given i, (p%,i,p&,i) = p and(x%,i,x&,i) = x. When a function fi is dynamically increasing ordecreasing in each variable and parameter, the following proposi-tion (Ramdani et al., 2010) tells us how the bounds of [xi](t) canbe obtained for any time t.

Proposition 1. For any t, the bounds of [xi] (t) can be computed asfollows:

� xþi ðtÞ is the solution of the system _x ¼ f ðEi; tÞwith the configurationEi 2 E such that

Ei ¼

p ¼ pþ for all p 2 p%;i

p ¼ p� for all p 2 p&;i

xð0Þ ¼ xþð0Þ for all x 2 x%;i

xð0Þ ¼ x�ð0Þ for all x 2 x&;i

8>>><>>>: ð8Þ

� x�i ðtÞ is the solution of the system _x ¼ f ðEi; tÞwith the configurationEi 2 E such that

Ei ¼

p ¼ pþ for all p 2 p&;i

p ¼ p� for all p 2 p%;i

xð0Þ ¼ xþð0Þ forall x 2 x&;i

xð0Þ ¼ x�ð0Þ for all x 2 x%;i

8>>><>>>: ð9Þ

This means that if the monotonic properties of fi are known,

then one can solve the system with configuration Ei to get xþi ðtÞ,and with Ei to get x�i ðtÞ. This means that to retrieve the boundsof [xi](t) for any variable xi, i = 1, . . .,m, the system _x ¼f ðx;pE;pD; tÞ has to be solved twice with usual techniques not

(A)

(C)Fig. 4. Illustration of solution

involving intervals. For such systems, propagating uncertaintiesmodelled by intervals then becomes much easier than propagatingprobabilistic uncertainties (it comes down to solve at most 2 �mdifferent systems). Let us continue our example.

Example 2. Consider the system of Example 1. We have (assuming

a positive p1) that @f1@p1

P 0, hence f1 is dynamically increasing w.r.t.

p1 and x1. Hence p%,i = (p1), p&,i = (), x%,i = (x1) and x&,i = (). Hencethe upper envelope xþ1 ðtÞ is given by the configuration

E1 ¼ ðpþ1 ; xþ1 ð0ÞÞ, and the lower one by x�1 ðtÞ by E1 ¼ ðp�1 ; x�1 ð0ÞÞ.

Fig. 3 represents the two envelopes when [p1] = [0.5,1] and[x1](0) = [1,3] for t 2 [0,10].

3.3. Parameters optimisation

We now consider the problem of searching optimal valuesbpD 2 RnD of design parameters pD when initial conditions and envi-ronmental parameters are interval-valued, and when objectives(constraints) bxiðtÞ on the state variables xi, i = 1, . . .,m are inter-val-valued as well. We consider that, for each variable xi, the objec-tives bxi : cT ! IR can be given over some subset cT #T of thewhole time domain (for instance, constraints on values may onlybe specified for the steady state only, that is after a time t⁄ suchthat cT ¼ ½t�; t�; . . .). Note that objectives can be intervals, as theirexact values can themselves be uncertainly known (or there maybe many values that appear optimal in the context). This way offormulating an optimisation problem is not usual, even in inter-val-analysis literature (Raissi et al., 2004), where the objectivefunction is usually precisely valued (e.g., corresponds to preciseoutputs). Again, such a problem may be in general difficult to solve.We propose a general way of formulating the problem, beforeproposing an easy-to-apply algorithm in the case of monotonicsystems (i.e., systems satisfying Definition 1).

Consider some pre-defined objectives ½bxi�ðtÞ as well as someinitial conditions ½x�ð0Þ 2 IRm, and some interval-valued uncer-tainty ½p� 2 IRnE of environmental parameters pE. Consider some(precise) value cpD for design parameters pD. For a time t, letXbpDðtÞ be the set

X bPDðtÞ¼ fxðtÞ 2Rmj _x¼ f ðx;pE;pD;tÞ;pE¼ p2 ½p�;xð0Þ 2 ½x�ð0Þ;pD¼cPDg:

ð10Þ

Design parameter values cpD are said to form a guaranteed (resp. pos-sible) solution if X bPD

ðtÞ#�mi¼1½bxiðtÞ� (resp. if XbPD

ðtÞ \ �mi¼1½bxi�ðtÞ – ;)

(B)

s with Example 2 model

120 S. Destercke, V. Guillard / Computers and Electronics in Agriculture 79 (2011) 116–124

for every t 2 cT. A solution is guaranteed if despite of uncertaintieson initial conditions and environmental parameters, we are certainto remain within optimal bounds. It is possible if given the uncer-tainties on initial conditions and environmental parameters, wecould be within optimal bounds. We denote by GD # RnD the set ofguaranteed and by SD # RnD the set of possible solutions. Both thesesets may be empty, but we have the inclusion relationship GD #SD.Solutions that are totally outside the objective bounds are said non-admissible.

Example 3. Consider again the model of Example 2, except thatthis time parameter p1 is considered as a design parameter thatcan be tuned through some control process (e.g., a speed ofreaction controlled by some catalyser). Objective bx1ðtÞ on thevariable x1 is specified for bT ¼ f10g and is such that½bx1�ð10Þ ¼ ½8;11�. Fig. 4 illustrates the notions of guaranteed,possible and non-admissible solutions for this case. For example,when p1 = 0.75, we have [x1](10) = [x1

�,x1+](10) = [8.5,10.5] and

½x1�ð10Þ# ½bx1�ð10Þ.

The exact bounds of sets GD and SD are, in practice, hard to find.However one may search, for each element p of pD, intervals½Gp� ¼ ½g�p ; gþp � and ½Sp� ¼ ½s�p ; sþp � such that the constructed boxapproximates GD and SD. Those values are the solutions of the fol-lowing optimisation problems (provided such solutions exist):

g�p ¼min p; gþp ¼max p;

under the constraints

xð0Þ 2 ½x�ð0Þ; i 2 f1; . . . ;mg; pE 2 ½p�8t 2 cT; xþi ðtÞ 6 bxþi ðtÞ and x�i ðtÞP bx�i ðtÞ;and

s�p ¼min p; sþp ¼max p;

under the constraints

xð0Þ 2 ½x�ð0Þ; pE 2 ½p�8t 2 cT; i 2 f1; . . . ;mg; x�i ðtÞ 6 bxþi ðtÞ or xþi ðtÞP bx�i ðtÞ:Again, solving exactly such problems is in general difficult. How-ever, when the model1 _x ¼ f ðx;pE;pD; tÞ is dynamically monotonic(either increasing or decreasing) in each variable and each environ-mental parameter, we propose a simple heuristic method to identify[Gp] and [Sp]. Algorithm 1 suggests some means to find them wheneach component of _x ¼ f ðx;pE;pD; tÞ is dynamically monotonic w.r.t.a given design parameter p. In this algorithm, Ei

pE ;xand Ei

pE ;xare the

configurations of Proposition 1 reduced to environmental parame-ters and initial conditions, the values of design parameters being leftunspecified.

The algorithm simply uses the known monotonic properties ofthe model to compute boundary values. For instance, considerLines 2–3 and the case p being an element of p%,i. Considering a va-lue pD of pD, the system solution with configuration Ei

pE ;x;pDprovides

the upper bound xþi ðtÞ. Hence, any solution xi(t) obtained withother values of initial conditions and environmental parametersand with pD = pD would be such that xiðtÞ 6 xþi ðtÞ. Therefore, ifxþi ðtÞ 6 bxþi ðtÞ, we satisfy one of the inequalities to get guaranteedsolutions, and pD is a good candidate for the upper bound gþp , sincep is an element of p%,i. Therefore, running an identification algo-rithm on pD with configuration ðEi

pE ;x;pDprovides us with an optimal

candidate for gþp . The same reasoning is done for the other bounds.We still have [SP] # [GP]. Note that Algorithm 1 works only when fsatisfies Definition 1. In more complex cases, approximating sets

1 Note that design parameters pD are no longer uncertain.

GD and SD can be done by sampling different values of designparameters and then performing interval analysis (with propaga-tion methods adapted to more general models Ramdani et al.,2009) to check whether the sampled values are (guaranteed orpossible) solutions.

Example 4. Consider again the model of Example 2, with theconstraints of Example 3. Running Algorithm 1 to identify ½Gp1

� and½Sp1� gives the following solutions (note that here, initial conditions

and objectives ½bx1�ð10Þ provides each time two points, hencetotally determining p1 in the equation x1(t) = p1 � t + x1(0)):

Algorithm 1.Approximations of [Gp] and [Sp] of a design parameter p

_ bþ b�

Input model x ¼ f ðx;pE;pD; tÞ, objectives xi ðtÞ and xi ðtÞon cT,Initial condition uncertainty [x](0) and environmentalparameter uncertainty [p], Algid: identification algorithmreturning design parameter valuesOutput Intervals [Gp] and [Sp]1 For i = 1, . . .,m do2 Run Algid on bxþi ðtÞ with configuration Ei

pE ;x, get opti-

mal value bp of p;3 If p element of p%,i, set gþp;i ¼ bp, else set g�p;i ¼ bp;4 Run Algid on bx�i ðtÞ with configuration Ei

pE ;x, get opti-

mal value bp of p;5 If p element of p%,i, set g�p;i ¼ bp, else set gþp;i ¼ bp;6 Run Algid on bxþi ðtÞ with configuration Ei

pE ;x, get opti-

mal value bp of p;7 If p element of p%,i, set sþp;i ¼ bp, else set s�p;i ¼ bp;8 Run Algid on bx�i ðtÞ with configuration Ei

pE ;x, get opti-

mal value bp of p;9 If p element of p%,i, set s�p;i ¼ bp, else set sþp;i ¼ bp;10 Gp ¼ ½maxi¼1;...;mg�p;i;mini¼1;...;mgþp;i�with Gp = ; if max-

i=1,. . .,mgp, i- P mini=1,. . .,m gp,i

+;11 Sp ¼ ½maxi¼1;...;ms�p;i;mini¼1;...;msþp;i� with Sp = ; if max-

i=1,. . .,msp, i- P mini=1,. . .,msp,i

+;

� Line 2 of Algorithm 1, i = 1: Identification with bxþ1 ð10Þ ¼ 11 and

Ei

pE ;x¼ ðx1ð0Þ ¼ 3Þ, giving bp1 ¼ 0:8 as solution and gþp1 ;1

¼ 0:8(since p1 2 p%,1).� Line 4 of Algorithm 1, i = 1: Identification with bx�1 ð10Þ ¼ 8 and

EipE ;x¼ ðx1ð0Þ ¼ 1Þ, giving bp1 ¼ 0:7 as solution and g�p1 ;1

¼ 0:7� Line 6 of Algorithm 1, i = 1: Identification with bxþ1 ð10Þ ¼ 11 and

EipE ;x¼ ðx1ð0Þ ¼ 1Þ, giving bp1 ¼ 1 as solution and sþp1 ;1

¼ 1� Line 8 of Algorithm 1, i = 1: Identification with bx�1 ð10Þ ¼ 8 and

EipE ;x¼ fx1ð0Þ ¼ 3g, giving bp1 ¼ 0:5 as solution and s�p1 ;1

¼ 0:5

Finally (Lines 10 and 11 of Algorithm 1), we obtain intervals½Gp1� ¼ ½0:7;0:8� and ½Sp1

� ¼ ½0:5;1� providing approximate valuesof p1 that give guaranteed and possible solutions, respectively.

The obtained sets [Sp] and [Gp] for each design parameter p canthen be transformed into fuzzy sets (Zadeh, 1975). Indeed, whileguaranteed solutions all provide the same satisfaction to the de-signer or the decision maker (they all ensure that the true solutionis within the objective boundaries), possible solutions can be seenas having gradual satisfaction degrees, as some of them will have amore significant overlap with the objective bounds than others.

First recall that a fuzzy set l : V! ½0;1� is a mapping from aspace V (here the real line) to the unit interval, where l(v) isthe membership degree of element v. A trapezoidal fuzzy numberl~a : ½a1; a4� ! ½0;1� is defined by a tuple ~a ¼ fa1; a2; a3; a4g of fournumbers and is such that

Fig. 5. Fuzzy set of Example 4.

S. Destercke, V. Guillard / Computers and Electronics in Agriculture 79 (2011) 116–124 121

l~aðxÞ ¼

x�a1a2�a1

if a1 6 x 6 a2

1 if a2 6 x 6 a3x�a4

a3�a4if a3 6 x 6 a4

8><>: ð11Þ

In our case, the fuzzy set degree expresses some satisfaction degree(Dubois and Prade, 1997) provided by a parameter value w.r.t. aninterval-valued objective. We propose, for a design parameter p,to build the trapezoidal fuzzy number l~p such that ~p ¼ fs�p ;g�p ; g

þp ; s

þp g if Gp – ; and such that ~p ¼ fs�p ; ðs�p þ sþp Þ=2; ðs�p þ sþp Þ=2;

sþp g otherwise. Fig. 5 illustrates the fuzzy number obtained inExample 4.

3.4. Sensitivity analysis

When performing an uncertainty analysis, it is usual to per-form it with a sensitivity analysis (Helton et al., 2006a,b; Saltelliet al., 2008). Sensitivity analysis consists in searching what in-put parameters or variables most contribute to the outputuncertainty. Its results indicate the parameter or variable onwhich experimental efforts should concentrate in order toreduce the output uncertainty. Note that there are only veryfew works dealing with sensitivity analysis of interval ap-proaches (Moens and Vandepitte, 2007), contrarily to probabilis-tic approaches. In other cases, sensitivity analysis can includeother components such as information related to economicalcost (Helton et al., 1997). Ideas presented here could beextended to such considerations, but in our case we focus ourattention on uncertainty.

In this paper, we propose a very simple method to perform thissensitivity analysis for a given function fi. Let [xi](t) be the interval-valued output resulting from initial interval uncertainty. Then, ifwe denote by LðxiðtjÞÞ :¼ ðxþi ðtjÞ � x�i ðtjÞÞ the length of [xi](tj) forany time-step tj, we can define the overall imprecision I(xi(t)) ofxi(t) as

IðxiðtÞÞ :¼XT

j¼1

LðxiðtjÞÞ; ð12Þ

that is, the sum of interval lengths obtained at each time step. Notethat this measure of imprecision depends on the choice of the timesteps and of their numbers, that should remain constant andnumerous enough between two consecutive computations to havea reliable analysis. Now, to quantify the impact of each parameterand variable uncertainty on the output imprecision, we proposethe following procedure (similar to some existing propositions inimprecise probability literature (Ferson and Tucker, 2006)): reduce,for each parameter and variable, its uncertainty by a given fractionr 2 [0,1], such a reduction coming down to transform an interval[a,b] into an interval [a0,b0] such that

a0 ¼ Mð½a; b�Þ � Lð½a; b�Þ2

ð1� rÞ� �

;

b0 ¼ Mð½a; b�Þ þ Lð½a; b�Þ2

ð1� rÞ� �

; ð13Þ

with M([a,b]),L([a,b]) the middle and length of the interval,respectively.

Such a reduction gives a new interval-valued solution [x0i](t) in-cluded in the previous one, and with an overall imprecisionI(x0i(t)) 6 I(xi(t)). If [x0i](t) is the output obtained after reducingthe imprecision of parameter p (or variable x) by r, the gain inthe output precision generated by this reduction can then be de-fined as

Gðp; rÞ :¼IðxiðtÞÞ � Iðx0iðtÞ� �

IðxiðtÞÞ: ð14Þ

Let us illustrate this notion in our example.

Example 5. Let us consider again the model of Example 1, withinitial uncertainty given by p1 2 [0.5,1] and x1(0) 2 [1,3], theresulting x1(t) being pictured in Fig. 3. We have I(xi(t)) = 70 (inthis case, the area between the two lines of Fig. 3 can be computedanalytically).

If we reduce the uncertainty of P1 and X1(0) by 50 % (r = 0.5), weget p01 2 [0.625,0.875] and x01(0) 2 [1.5,2.5]. We then have

Gðp1;0:5Þ ¼ 70—45=70 ’ 0:36;Gðx1;0:5Þ ¼ 70—60=70 ’ 0:14:

From these results, it appears that reducing the uncertainty of p1 by50 % has a more important impact on the output than reducingx1(0) uncertainty by the same amount. Hence, further experimentsshould focus on reducing the uncertainty around p1.

Finally, note that a reduction of 100 % (r = 1) of a parameter orvariable comes down to consider the middle point of the interval itbelongs to.

4. Application to food packaging

In this section, we apply the above framework to the design andoptimisation of food packaging material, with the aim to maximiseshelf life of packaged food (here, fresh fruits and vegetables). Notethat this model will be integrated to an online decision supportsystem (http://www.tailorpack.com/), hence quick computa-tions and approximate (but reliable) answers should be privilegedover more accurate but computationally more demanding answers.

4.1. Problem presentation and model analysis

Preserving fresh food after harvest and on the shop shelves is animportant issue in the food industry. Providing food with opti-mised packaging is a sure way to avoid premature decay and pro-long food edibility. Such optimisation means that the packagingmust be permeable enough to oxygen and carbon dioxide so asto allow food respiration, but not too permeable to these two gases,so that maturation and decaying process are slowed down.

In modified atmosphere packaging, oxygen and carbon dioxidepartial pressures in packaging head-space are modified and settleto steady values after a transient phase. This modification in theinternal gas partial pressures is achieved due to the mass balancebetween oxygen and carbon dioxide flux through the packagingmaterial and O2 and CO2 consumption/production due to the prod-uct respiration. For a given fruit or vegetable, it is possible toexperimentally determine (up to some uncertainties) oxygen andcarbon dioxide partial pressures that will result in an optimal pres-ervation. From these optimal partial pressure values, it is thenpossible to determine optimal packaging permeances, by using a

Table 1Parameters names and units. � = non-relevant.

Parameter Name Units

PeO2 O2 permeance mol m�1 s�1 Pa�1

PeCO2 CO2permeance mol m�1 s�1 Pa�1

S pack. surface m2

e pack. thickness mpi

jpartial press. of j in i %

RRO2 O2 respiratory rate mmol kg�1 h�1

RRO2 max max. O2 respiratory rate mmol kg�1 h�1

KmappO2Michalis-Menten constant kPa

KiCO2CO2 inhibition constant kPa

m food mass kgQR respiration coefficient �

Table 3Parameters uncertainty for propagation.

Parameter Uncertainty Unit

PeO2 [878,1278] � 10�18 mol m�1 s�1 Pa�1

PeCO2 [3614,4634] � 10�18 mol m�1 s�1 Pa�1

S [12,16] � 10�2 m2

e [4,6] � 10�5 mm [0.45,0.55] kgRRO2max [1.3,1.5] mmol kg�1 h�1

KmappO2[8.26,10.26] 103 kPa

KiCO2[1025,2025] 103 kPa

QR [0.67,0.81]

122 S. Destercke, V. Guillard / Computers and Electronics in Agriculture 79 (2011) 116–124

mathematical model predicting the dynamic evolution of internalgases partial pressures. Several environmental parameters mustbe specified in this model such as respiration rate or respiratoryquotient, packaging geometry, or environment variables such astemperature. The mass balance between gas transfer and respira-tion can be written as:

p _pkgO2¼ PeO2 � S

eðpext

O2� ppkg

O2Þ � RRO2 �m ¼ f1 ð15Þ

p _pkgCO2¼ PeCO2 � S

eðpext

CO2� ppkg

CO2Þ þ RRO2 �m � QR ¼ f2 ð16Þ

with

RRO2 ¼RRO2max � ppkg

O2

ðKmappO2þ ppkg

O2Þ � 1þ

ppkgCO2

KiCO2

� � ð17Þ

where the first part of the right-hand side describes gas flux pertime unit through the packaging material, while the second part de-scribes gas consumption (and emission) by the vegetable or fruit(modelled using a Michaëlis-Menten-type equation, see (17)). Thereare two variables x ¼ ðppkg

O2;ppkg

CO2Þ. Table 1 summarises the parame-

ters with their names and units, while Table 2 summarises whetherthey are variable, environmental or design parameters, and theirmonotonicity w.r.t. each function f1 and f2. Note that ðpext

O2� ppkg

O2Þ

is always positive, while ðpextCO2� ppkg

CO2Þ is always negative, that pext

i

are constant and that all parameters are positive (otherwise modelwould not be monotonic w.r.t. every parameter).

4.2. Case study: chicory

Note that in our case study, the shape of the packaging (S and e)and the mass (m) of food inside it have already been fixed, so thatthe only parameters that can be adjusted to optimise the packagingare the material permeability properties (PeO2 and PeCO2 ). Note that,if a probabilistic approach was used, no less than 9 probability dis-

Table 2Parameters type and monotonic w.r.t. f1,f2. � = non-relevant.

Parameter pD,pE, x f1 f2

PeO2 pD p%,1 -PeCO2 pD – p&,2

S pE p%,1 p&,2

e pE p&,1 p%,2

pij

x x&,1 x&,2

RRO2 pE p&,1 p%,2

RRO2 max pE p&,1 p%,2

KmappO2pE p%,1 p&,2

KiCO2pE p&,1 p%,2

m pE p&,1 p%,2

QR pE – p%,2

tributions would have to be specified, and independence betweenvariables would have to be assumed, with some of the parameterssometimes measured only 3 times. Also, estimating boundaries byinterval analysis for ppkg

O2and ppkg

CO2here requires only 4 simulations

(far less than for a Monte–Carlo simulation (Helton et al., 2006a)).The considered food is chicory, which has been previously stud-

ied, but without any proper uncertainty analysis (Charles et al.,2005).

4.2.1. Uncertainty propagation stepKnowing the permeability of the packaging material classically

used to pack chicory, the mathematical model described by Eqs.(15)–(17) can be used to simulate the evolution with time of theinternal O2 and CO2 partial pressures. Parameters with their inter-val uncertainty are summarised in Table 3. Maximal shelf life isestimated to be 200 h, therefore the simulation time domain isT ¼ ½0;200� h and the domain is discretised so that each time stepis one minute.

The results of interval analysis are displayed in Fig. 6, and wecan see that, at the steady states, ppkg

O22 ½1:6;8:1�% and

ppkgCO22 ½1:9;5:3�%. Using Proposition 1, only four simulations were

needed (two for each gas) to estimate the bounds.Also note that, in the current case, initial conditions ppkg

O2ð0Þ and

ppkgCO2ð0Þ for the two variables are perfectly known and only param-

eters are uncertain, as initial partial pressures are the same as infree atmosphere (21% for oxygen and 0% for carbon dioxide).Therefore, only uncertainty pertaining to environmental and de-sign parameters PE and PD have to be taken into account here (con-trary to Example 1, where x1(0) value is also uncertain).

Fig. 6. Uncertainty propagation.

Fig. 7. Optimal O2 permeance.

Fig. 9. Uncertainty propagation with optimal permeances.

Table 4Values G(P,0.5) of sensitivity analysis on the parameters P after a reduction r = 0.5, forthe two variables ppkg

O2ðtÞ and ppkg

CO2ðtÞ.

P PeO2 PeCO2 e S m

ppkgO2ðtÞ 0.099 0 0.124 0.077 0.088

ppkgCO2ðtÞ 0.095 0.12 0.077 0.058 0.024

P QR RRO2maxKmappO2

KiCO2All

ppkgO2ðtÞ 0 0.057 0.045 0 0.501

ppkgCO2ðtÞ 0.094 0.02 0.019 0 0.505

S. Destercke, V. Guillard / Computers and Electronics in Agriculture 79 (2011) 116–124 123

4.2.2. Optimisation stepIn the case where best oxygen and carbon dioxide permeabili-

ties suiting a particular fruit or vegetable are not known a priori,it is possible to use Eqs. (15)–(17) to perform a reverse engineeringtask. In this case, optimal oxygen and carbon dioxide concentra-tions in the packaging modified atmosphere have to be specified.Once this is done, Algorithm 1 can be run to find optimal perme-abilities that allow reaching the specified goal.

In this study, Algorithm 1was run with the same uncertainty onenvironmental parameters as the one of Table 3 (i.e., all parametersexcept packaging material permeabilities to O2 and CO2, which arethe design parameters). Given that T ¼ ½0;200� h and the fact thatthere exists a transient phase, we have chosen cT ¼ ½150;200�,dppkg

O2ðtÞ ¼ ½4;10�% and dppkg

CO2ðtÞ ¼ ½2;5�% for every t 2 cT, that is we

want oxygen partial pressure to be between 4% and 10% andcarbon dioxide partial pressure to be between 2% and 5% at thesteady state. Fuzzy sets obtained from Algorithm 1 and from theintervals ½GPeO2

� ¼ ½1:59;1:92�10�15, ½GPeCO2� ¼ ½4:94;7:50�10�15and

½SPeO2� ¼ ½4:34;6:51�10�15, ½SPeCO2

� ¼ ½1:78;19:96�10�15 (values areexpressed in mol.m�1.s�1.Pa�1) are shown in Figs. 7 and 8 (Leven-berg-Marquardt Algorithm (Nocedal and Wright, 1999) was used

Fig. 8. Optimal CO2 permeance.

to identify parameters). Fig. 9 shows the result of the interval anal-ysis done with optimal parameters belonging to ½GPeO2

� � ½GPeCO2�.

We see that the resulting imprecise oxygen and carbon dioxidepartial pressures well lie in the objective bounds at the steadystate.

4.2.3. Sensitivity analysisTable 4 summarises the sensitivity analysis performed accord-

ing to the method presented in Section 3.4. For each variableppkg

O2ðtÞ and ppkg

CO2ðtÞ, we have evaluated the precision gain after an

uncertainty reduction of r = 0.5 of each (environmental and design)parameter. The last column (All) indicates the gain in the outputprecision when all parameters uncertainty is reduced by r = 0.5.

The results indicate, among other things, that while O2 perme-ability uncertainty has an important impact on both O2 and CO2

internal partial pressures uncertainty (the reduction resulting ina gain of about 0.1 for each), CO2 permeability uncertainty only im-pact the CO2 internal pressures, and have almost none effects on O2

internal pressure. As could be expected, the respiration rate QRuncertainty only impacts on the CO2 internal pressure, whileKiCO2 uncertainty, due to the large value of KiCO2 , has (almost) noimpact on the resulting uncertainty for both variables.

Finally, to reduce the most efficiently the uncertainty on O2 andCO2 internal partial pressures, one should focus on a better charac-terisation of O2 permeability and thickness, and on O2/CO2 perme-ability, respectively.

5. Conclusion

Interval analysis is a useful alternative to probabilistic analysisthat requires far less information to be applicable, thus avoiding

124 S. Destercke, V. Guillard / Computers and Electronics in Agriculture 79 (2011) 116–124

the need to introduce hypothesis unsupported by available infor-mation. In some situations where the models satisfy some mono-tonic properties, using intervals is also more computationallyefficient, as sampling methods are not needed to achieve computa-tions. In this paper, we have used interval analysis for two pur-poses: classical uncertainty propagation for monotonic dynamicalmodels and robust design under uncertainties. We have also pro-posed an easy method to perform some first sensitivity analysis.

The proposed robust design optimisation method, althoughapproximate, quickly produces optimal values for design parame-ters. It takes account of interval uncertainty and can cope withimprecisely specified goals, distinguishing between possible solu-tions (potentially satisfying the goal) and guaranteed solutions(certainly satisfying the goal). The difference between the twokinds of solutions is represented by the means of fuzzy setsdescribing optimal solutions.

Methodological perspectives to this work include the develop-ment of more precise methods, eventually ending up with betterand multi-dimensional approximations of the sets GD and SD. Also,it may be desirable to extend the current approach to hybriduncertainty models mixing interval and probabilistic uncertainty(Baudrit et al., 2009; Baudrit et al., 2006; Fuchs and Neumaier,2009; Nassreddine et al., 2010), as information concerning differ-ent parameters may vary in quantity and quality. However, thesetwo perspectives would involve more computationally demandingprocedures, thus reducing the number of models one could workwith.

More applied perspectives include the combination of the opti-misation system with a decision support system where a user cansearch a database for optimal packaging (Destercke et al., 2011)(fuzzy sets describing sets of optimal solutions will then be usedas user preferences) and the application of the system to other foodproducts (such as mushrooms).

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