# Towards a holistic approach for multi-objective optimization of food processes: a critical review

Martial Madoumier, Gilles Trystram, Patrick Sébastian, Antoine Collignan

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Martial Madoumier, Gilles Trystram, Patrick Sébastian, Antoine Collignan. Towards a holistic approach for multi-objective optimization of food processes: a critical review. Trends in Food Science and Technology, 2019, 86, pp.1-15. 10.1016/j.tifs.2019.02.002. hal-02050477

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Towards a holistic approach for multi-objective optimization
of food processes: a critical review

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Martial Madoumier <sup>a</sup>, Gilles Trystram <sup>b</sup>, Patrick Sébastian <sup>c</sup>, Antoine Collignan <sup>a</sup>

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a QualiSud, Montpellier SupAgro, CIRAD, Univ d'Avignon, Univ Montpellier, Univ de La Réunion, Montpellier, France

b Université Paris-Saclay, Agroparistech, Inra, UMR 1145, F-75005 Paris, France

c Université de Bordeaux, I2M, UMR CNRS 5295, Esplanade des Arts et Métiers, 33405 Talence cedex, France

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Abstract

While Multi-objective Optimization (MOO) has provided many methods and tools for solving design problems, food processes have benefitted little from them. MOO encompasses the identification of performance indicators, process modelling, preference integration, trade-off assessment, and finding the best trade-offs. In this review, the use of these five elements in the design of food processes through MOO is analysed. A number of studies dealing with food processes MOO have been identified. Even though these studies improve the design process, they often approach MOO in a simplified and insufficiently rationalized way. Based on this review, several research issues are identified, related to the improvement of the different models and methods, and to the development of more holistic MOO methods for food processes.

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Key words

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Multi-objective optimization; food process design; multi-criteria decision aids

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1. Introduction

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In food process engineering, most design problems are aimed at several objectives, which can often be contradictory. Thus, maximizing food product quality (texture, nutrients concentration, flavour...) is often in conflict with process performance objectives, such as minimizing energy consumption, maximizing profit, or ensuring safety in the case of heat treatments. For the last two decades, solving multi-objective design problems has been a major concern as sustainable development practices also need to be integrated in the design process. Many kinds of objectives can be defined by the decision-maker, all with potential antagonistic effects, e.g. maximizing one has the effect of minimizing one or several others.

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To solve multi-objective design problems, different kinds of methods have been
34 developed, with the earliest being gradient-based methods and experiments-based
35 methods. Gradient-based methods, such as the method of Lagrange multipliers, are
36 based on the resolution of differentiable equation systems, and although they yield fast
37 computation times, they converge toward local optima only, which may not be global
38 optima. Experiments-based methods, and more specifically Response Surface
39 Methodology, were and remain a common optimization approach in the food processing
40 industry (Banga et al., 2008). Since then, new optimization methods for multi-objective
41 problems have been developed, which are able to efficiently identify global optima. They
42 have been grouped under the term "multi-objective optimization (MOO) methods".
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44 MOO is a general methodology aimed at identifying the best trade-off(s) between
45 several conflicting objectives. Numerous applications in engineering can be found, from
46 the design of a single mechanical part (Collignan et al., 2012) to the optimization of a
47 worldwide supply chain (Wang et al., 2011). MOO consists in a) a multi-objective
48 processing method, to transform the original multi-objective problem into a solvable
49 problem, and b) an optimization algorithm, to search for trade-off solutions to the
50 multi-objective problem (Collette and Siarry, 2013).
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52 A multi-objective processing method requires the following elements, in the food
53 processes framework:

54 1) Optimization objectives and associated indicators. The decision maker defines
55 objectives, i.e. changes that the decision-maker(s) wish(es) to cause in the
56 process (profit increase, productivity increase, environmental impact decrease...),
57 and these changes are quantified or described by suitable performance
58 indicators (margin, yield, carbon dioxide emissions...) (Church and Rogers,
59 2006). Indicators are also called by the term "criteria", which can itself be used as
60 an equivalent to "objectives" (Craheix et al., 2015). In this work, the terminologies
61 "objectives" and "indicators" will be used.

62 2) A **predictive food process model**: the effect of different values of the design
63 variables (input variables, i.e. operating conditions, equipment size, process
64 structure...) on the indicators is predicted by a process model. Thus, the different
65 design solutions available can be evaluated. The predictive model should provide
66 a satisfying level of prediction accuracy, while optimizing efficiently for reasonable
67 computation times.

68 3) A preference model, where the decision-maker **preferences and expert
69 knowledge** are integrated. Preferences may be specified at two different levels
70 (figure 1): i) objectives may be weighted according to their relative significance for
71 the decision-maker and/or qualified experts; ii) desirability functions may be used
72 to integrate satisfaction levels of experts according to indicator values. The
73 decision-maker may have sufficient knowledge to specify preferences at both
74 levels. However, it is considered in this work that the experts have more
75 qualifications to specify preferences on indicator values, based on a good
76 scientific and/or technical knowledge of the process and the installation context.
<figure><img src="image_2.png" /><figcaption>Figure 1: Relationships between objectives, indicators, and preference integration</figcaption></figure>

4) A **selection method** to choose the “best trade-off” by sorting, ranking or scoring the design solutions available. The selection method generally consists in aggregating preferences and indicators to build an objective function for optimization, but may also in consist in different approaches.

Regarding the optimization algorithm, it integrates these four elements to search for trade-offs among possible design solutions.

Numerous methods and algorithms can be used to build a multi-objective processing method to be combined with an optimization algorithm. Detailed taxonomies and information on these methods can be found in reference books such as (Chen and Hwang, 1992; Collette and Siarry, 2013; Ehrgott, 2005; Miettinen, 1998). It is also noteworthy that predictive food process models and preferences models are used in single-objective (mono-objective) optimization, in order to obtain a single performance indicator. These elements are not specific to MOO, and a detailed comparison of single- and multi-objective optimization can be found in Rangaiah et al. (2015).

In this context, the application of MOO to food processing was studied, that is the transformation of biological raw materials by one or several unit operations to produce edible food products. The investigation field of this review was restricted to MOO for food process design, which excludes:

* process control (or closed loop optimal control, as defined in Banga et al. (2008) – see for example Trelea et al. (1997));

* product formulation (or mixture design - see for example Chen et al. (Chen et al., 2004));

* model parameter optimization.

The design problems included were:

* selection of fixed or variable operating conditions (i.e. open loop optimal control – Banga et al. (2008));
108 • equipment sizing;

109 • number and structure of unit operations in the process.

110 A number of articles have been reviewed to discuss the methods used by the authors to perform MOO. From these studies it was established that despite the advanced development of MOO as a generic design methodology, the tools and methods of MOO have not yet fully reached the area of food process design:

114 • MOO is infrequent in the design of food processes compared to chemical processes: around 40 articles on MOO application in food processing had been published in scientific journals before 2009 (Abakarov et al., 2009), whereas around 360 papers regarding MOO in chemical engineering applications had been published until mid-2012 (Rangaiah and Bonilla-Petriciolet, 2013). Several authors (Banga et al., 2003; Trystram, 2012) have identified two major hindrances: i) physical properties, and consequently quality parameters of food materials, are difficult to predict because of the complexity of food materials; ii) many food process models are unsuitable for optimization purposes, since they have been developed to understand the behaviour of food materials as biological reactors (with reaction kinetics and transfers), rather than predict its behaviour as a function of process control variables and size.

126 • Most studies focus on the optimization of operating conditions for design or process control; many of them concern heat treatment processes. In contrast only a few MOO studies concentrate on the integrated design of food processes, where both unit operations structure and equipment sizing are optimized (see for example Nishitani and Kunugita (1979)).

131 • Most MOO design studies published are limited to the production of the Pareto front, i.e. the set of trade-off solutions for the process design (Kiranoudis and Markatos, 2000; Kopsidas, 1995; Nishitani and Kunugita, 1979, 1983; Stefanis et al., 1997; Yuen et al., 2000 ...). Multi-criteria decision making (MCDM) methods, which help to select the best trade-off amongst Pareto-efficient solutions, are seldom applied in these studies. MCDM methods can help include the preferences of the decision-maker in the design process, and rank the possible solutions to identify one (or a small set of) "best" trade-off(s) for process design.

139 • Very few design approaches are systemic: most optimization objectives are evaluated with "raw" indicators of process performance (nutrient retention, energy consumption, processing time...) and do not involve the interactions of the process with its environment (environmental impact based on LCA, overall economic profit, nutritional interest...).

144 Thus, the potential for developing more advanced MOO methods and associated tools for the design of food processes is high: most studies only partially use the constituent elements of MOO, while a variety of methods and tools are available to perform MOO. Hence it seemed relevant to study and review these methods and tools along with their use for food process design.

149 In this paper, a critical review of multi-objective optimization methods which have been used in food process design studies is developed. The main purpose is to demonstrate how design methods engineering can solve design problems in food processing, which however requires a choice among existing MOO methods.

153 The different sections of this review match the aforementioned elements which
154 constitute a MOO method:

155 • Section 2 is a critical analysis of **indicators** which describe design objectives;

156 • Section 3 briefly reviews **process models** used for MOO of food processes;

157 • Section 4 deals with the integration of **preferences** in decision-making;

158 • Section 5 handles the methods used in the literature to **select the best solutions**;
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160 • Section 6 explores **optimization algorithms** for MOO from the perspective of methods engineering;
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162 • Section 7 describes some holistic MOO methods, which include all elements for MOO (process indicators and model, preference model, ranking method, optimization algorithm) and discusses research issues.
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## 2. Design indicators

### 2.1. Raw and integrative indicators

167 The indicators required for optimization are produced by a set of more or less complex models, based on knowledge of the process. The indicators are mostly quantitative, but may possibly be qualitative (one product more appreciated than another, soft or hard texture, sanitary risk present or absent, etc.); the latter case is not considered in this work. A quantitative indicator may be an integer variable, but is more generally a real variable in the field of food processing. Process sizing parameters, such as a number of effects of an evaporator or number of cleaning cycles, may be represented by an variable integer, which is common in the field of chemical processing (see for example Morandin et al. (2011) and Rangaiah and Bonilla-Petriciolet (2013)). Quantitative indicators can be constructed simply using the physical variables (or chemical, biochemical, biological variables) of the process, or be generated by an economic model or environmental impact model, making it possible to quantify global objectives (especially sustainability). In this work, the indicators are categorized in two families:

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• Raw indicators, i.e. variables of physical, chemical, biochemical or even biological origin, calculated using the process model, such as the product treatment temperature, its sensorial qualities (texture, colour...), steam consumption, retention rate of a compound of interest, etc.
• Integrative indicators, which combine raw indicators referring to different (but linked) phenomena into a unique variable, according to scientific and/or technical or statistical principles, or even rule-like principles. They are constructed from economic, environmental, social, or even product quality models. They correspond to the definition of the composite indicators given by Von Shirdning (2002).

191 In the case of raw indicators, interpretation i.e. the relationship between indicators and objectives, is left to the decision-maker, which assumes a degree of expertise in the process under study. A decision-maker not specializing in the process will be less capable of analysing the solutions proposed, since the raw indicators may not be explicit in terms of the objectives sought. Thus, the exergy proposed by Nishitani and Kunugita
196 (1983) requires an ability to understand this concept in terms of environmental impact;
197 the "head kernel" yield for optimizing rice drying by Olmos et al. (2002) cannot quantify
198 the economic implications of this indicator. Similarly, selecting a raw indicator could
199 partly conceal, or even bias, the information required for evaluating the objectives. Thus
200 Stefanis et al. (1997) opted to characterize the environmental impact in wastewater by
201 BOD (Biological Oxygen Demand), which represents a highly partial view of the
202 environmental impact that a process may have. In Nishitani and Kunugita (1979), the
203 exchange surface contributes only partially to the cost of the evaporator, and so appears
204 to be an incomplete indicator in terms of the defined economic objective.

205 Conversely, raw indicators can be tailored to specific contexts, where the process
206 objectives can be expressed directly by physical variables derived from the process
207 model: in Yuen et al. (2000), the objective is to remove alcohol from beer while
208 minimizing loss of chemicals associated with taste, which is explicitly expressed by an
209 "alcohol removal" indicator and an "extract removal" indicator. In particular in the case of
210 explicitly known product quality objectives, they can be expressed by selecting certain
211 nutritional compounds, such as in Tarafdar et al. (2017), where the indicators are
212 contents of nutritional compounds of interest.

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214 On ther other hand, integrative indicators can link the process physical variables to
215 variables of interest/which are meaningful for the decision-maker: a return on investment
216 time for example will be easier to interpret for an investor than an investment cost and
217 an operating cost taken separately. Sebastian et al. (2010) defined a total cost of
218 ownership, bringing together the operating cost (electricity and fluids consumption) and
219 an investment cost (purchasing and manufacturing costs), which can be used to quantify
220 what the equipment costs over a planned service life of twenty years.

221 However, due to the construction of the associated functions, integrative indicators entail
222 a risk of bias in the interpretation. Firstly, the models used may be subject to debate; in
223 the case of impact scores based on Life Cycle Analysis (LCA) for example, modelling of
224 the environmental impacts varies according to the impact calculation methodologies, and
225 there is not always an established consensus on these models (Hauschild et al., 2008).
226 Then, the weighting of different kinds of indicators (greenhouse effect and
227 eutrophication, texture and colour...) for the purpose of aggregating them in an
228 integrative indicator may also entail a bias. Finally, constructing integrative indicators
229 assumes use of data which is sometimes uncertain; thus it is not always possible, at the
230 scale of a process situated in a larger system (e.g. factory), to predict its profitability or
231 maintenance cost.

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233 The indicators encountered in the various articles studied in this work are rarely
234 integrative indicators. While aggregating raw indicators can produce an indicator which
235 is meaningful for the decision-maker, the way in which they are grouped induces a risk
236 of information loss. Thus, raw indicators of major significance in design choices may find
237 themselves concealed by the integrative indicator, as in the case of the SAIN-LIM
238 indicator which conceals the effect of certain nutrients on the overall score (Achir et al.,
239 2010). So the development of relevant indicators means finding a balance between an
240 excessive number of raw indicators, which is difficult to interpret and discuss, and an
241 integrative indicator, which would cause major information loss through aggregation.

242 **2.2. Relevance of indicators**

243 Besides the advantages and shortcomings of raw and integrative indicators, the
244 question of choice of indicators is an issue of interest, firstly in terms of the meaning
245 given to the indicators. There are numerous approaches for constructing more or less
246 integrative indicators which are meaningful for the decision-maker in view of their
247 objectives. An overview of some of these approaches is proposed here, via the four
248 dimensions of sustainability of food engineering processes: economic sustainability and
249 product quality, which are the most frequently encountered dimensions, plus
250 environmental and social sustainability.
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252 Economic evaluation of processes makes it possible to establish the cost that they
253 represent, and/or their profitability in the shorter or longer term. In the context of
254 optimization, it must be possible to predict their operating cost and the investment they
255 represent; there are correlations for predicting investment as a function of sizing
256 choices, the best known of which is from Guthrie (1969). Benchmark works provide
257 values for the parameters of this correlation (Maroulis and Saravacos, 2007; Turton et
258 al., 2008). Based on the economic and financial information on the process and the
259 company, it becomes possible to construct integrative economic indicators, the best
260 known of which are the internal profitability rate, return on investment time, discounted
261 income, net present value and net cumulative cash flow (Chauvel et al., 2001; Turton et
262 al., 2008). Other approaches are being developed, such as thermo-economics, which
263 associates a cost with exergy (a measure of energy quality to determine energy
264 degradation in the system), to evaluate economic feasibility and profitability (Rosen,
265 2008). In keeping with the "life cycle" approach, Life Cycle Costing (LCC), where the
266 financial, environmental and social costs are factored into the life cycle as a whole
267 (Norris, 2001), is another approach under development. Like thermo-economics, it still
268 requires construction of databases large enough for the economic indicators proposed to
269 evaluate the food engineering processes.
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270 Food quality needs to be described through a holistic perspective which covers all
271 consumer requirements. Among several possible approaches, an attempt was made by
272 Windhab (2009) to provide such holistic perspective, known by the acronym PAN:
273 Preference (organoleptic and usage properties), Acceptance (religious, cultural,
274 GMO...), Need (health, nutrition...). However, the indicators used in the literature
275 primarily relate to the P and N dimensions. Only fragmentary elements of food quality
276 are dealt with, which were classified in three categories:
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278 • Nutritional indicators are generally nutritional or anti-nutritional compound
279 degradation kinetics (Abakarov et al., 2009; Garcia-Moreno et al., 2014 ...) and
280 are thus raw indicators, although there are some integrative indicators in the form
281 of algebraic equations. Hence, among other approaches, the SAIN-LIM indicators
282 were developed in an attempt to classify foods by their nutritional value, by
283 quantifying their favourability or unfavourability for human health (Darmon et al.,
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2007). However, they are ill-suited to optimization, as they are insufficiently sensitive to the process control parameters (Achir et al., 2010; Bassama et al., 2015).

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* Organoleptic quality is described either by denaturing kinetics (or conversely development kinetics) of compounds relating to organoleptic appraisal of a product (Gergely et al., 2003; Kahyaoglu, 2008; Yuen et al., 2000 ...), or by sensory scores. These scores directly express the appraisal of product quality by the consumer, but they are based on a posteriori evaluation (Abakarov et al., 2013; Singh et al., 2010 ...). Sensory scores can be aggregated to produce integrative indicators of overall appraisal, provided they have been evaluated on a common scale (e.g. 1 to 9 from worse to best).

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* Finally, sanitary quality, which is generally a feasibility constraint rather than an indicator for optimization, is described by microorganism mortality kinetics, or development kinetics of compounds hazardous to humans (Arias-Mendez et al., 2013; Garcia-Moreno et al., 2014).

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Quality indicators represent a particularly topical problem, with growing market demands in terms of health, and consequently a research issue for modelling the links between process, nutrition and health.

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The issue of environmental impact indicators is particularly topical. While there are numerous environmental impact approaches, they are all debatable in terms of relevance regarding the process studied, and of over- or under-estimating the impact. Three of the best known environmental impact evaluation methods are listed below:

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* Life Cycle Analysis (LCA) is the most commonly used method, and most comprehensive for evaluating the environmental impacts of a system (Azapagic et al., 2011; Jacquemin et al., 2012; Manfredi et al., 2015). The indicators produced are calculated based on the inventory of emissions and resources consumed throughout the life cycle of the product in question, LCI (Life Cycle Inventory). An LCI analysis methodology is employed to convert the emissions surveyed from the entire system in question into environmental impact scores, using characterization factors specific to the method used. LCA is a widely described and analysed method (Jolliet et al., 2010), but rarely used in optimizing food engineering processes: it was partially used in the study by Romdhana et al. (Romdhana et al., 2016), where only the Global Warming Potential (GWP) indicator, relating to climate change, was used, and in the works of Stefanis et al. (1997), which defined several indicators comprising air pollution, water pollution, solid wastes, photochemical oxidation, and stratospheric ozone depletion. Although standardized and comprehensive, LCA contains possible biases caused by the choice of inventory analysis method, functional unit, system and impact allocation.

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* Thermodynamic methods, based on the second law of thermodynamics, quantify changes of thermodynamic state in the system under study, making it possible to identify "degradations" caused by the process and thereby quantify the impact. For example, the exergetic analysis, which quantifies quality loss of the energy entering the system, i.e. destruction of exergy; this makes it possible to determine
the "available energy" in outgoing currents in the form of "exergetic efficiency", 330
which is used as an environmental impact indicator (Ouattara et al., 2012). Used 331
in Nishitani and Kunugita (1983), this seems to be the most developed 332
thermodynamic method, though there are still insufficient thermodynamic data to 333
be able to generalize its application. 334

• The Sustainable Process Index (SPI) is an indicator measuring the environmental 335
impact in terms of surface of the planet used to provide goods or services 336
(Steffens et al., 1999). Assuming that the sole external input into the system is 337
solar energy, any process occupies a more or less large fraction of the Earth's 338
surface for its workings "from cradle to grave" (raw materials, energy, personnel, 339
environmental emissions...). Thus, a low SPI will indicate an efficient process. 340
This approach provides a sole indicator, independent of modelling environmental 341
damage, but it lacks data for the area attributed to each substance or process, 342
and there are inconsistencies when the use of fossil or mineral resources is 343
analysed (Hertwich et al., 1997). 344

Mention may be made of other methods, such as the WAR (Waste Reduction) algorithm, 345
and the IChemE indicators, which are both (like LCA) based on using impact factors, 346
and the AIChE metrics developed for petrochemical processes, though these cannot be 347
used to evaluate the potential damage. 348

Finally, the social dimension of sustainability is not represented in the literature studied 349
for this work, since it is hard to quantify at the process design stage. The concept of 350
social LCA is relatively recent (early 2000s), suffers among other things from a lack of 351
data (Norris, 2014), and has hitherto been applied to fields such as industrial 352
management and product development (Jørgensen et al., 2008); only a few works 353
(Schmidt et al., 2004) mention consideration of social objectives in the field of processes 354
for comparative studies. True, indicators such as job creation, safety and nuisance 355
generation have been proposed (Azapagic et al., 2011), but they often relate to the 356
operational phase, and are difficult to associate with indicators in the preliminary design 357
phase. Employment could be a relevant indicator for processes, for example via the 358
number of total local jobs (You et al., 2012), depending for example on the quantity of 359
labour required by each piece of equipment included in the process. 360

Thus, the choice from among all these indicators affects the meaning given to the 362
optimization, but also the results. Indeed, the results derived from the optimization of the 363
same process are dependent on the decision-maker's objectives, and more generally on 364
the specific context of the optimization study. By way of example, the mass of the 365
equipment, used to quantify its transportability in Sebastian et al. (2010), would not be a 366
relevant indicator for a fixed process in a factory. Thus it is clear that the ranking of a 367
solution is closely linked to the mathematical construction of the indicators, hence the 368
usefulness of considering their relevance. Achir et al. (2010) for example showed that 369
the number of nutrients factored into the SAIN-LIM indicator affects the ranking of a 370
product by this indicator. Yet to our knowledge, no study has taken an in-depth look into 371
the subject, as the indicators are pre-selected, and not questioned thereafter. So 372
evaluation of the relevance of the indicators for optimization is a relevant research 373
question, but a difficult task. 374
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# 3. Models for multi-objective optimization

377 Although there are optimization approaches without models (especially sequential 378 experimental strategies such as the simplex method), the exploration of various 379 scenarios, and the need to rank them to identify the best (especially if the question is to 380 find a compromise between several objectives), it would seem that optimization 381 definitely requires numerical models.

382 For food and biological processes, there are numerous long-standing modelling 383 approaches. Table 1 presents and categorizes the approaches listed in the literature.

384 These process model construction and validation methods present various 385 characteristics, and may be classified in three categories (Banga et al., 2003; Perrot et 386 al., 2011; Roupas, 2008): knowledge-driven models (“white box” type), which are derived 387 from the physical laws governing the behaviour of the process; data-driven models 388 (“black box” type), which are solely based on empirical data; and hybrid models (“grey 389 box” type), which are a combination of the two.
<table><caption>Table 1: Process model types</caption><thead><tr><th>Model type</th><th>Categories</th><th>References</th><th>Comments</th></tr></thead><tbody><tr><td rowspan="3">Data-driven models ("Black box")</td><td>Response surface methodology (RSM)</td><td>Abakarov et al. (2013); Annor et al. (2010); Collignan and Raoult-Wack (1994); Corzo and Gomez (2004); Eren and Kaymak-Ertekin (2007); Garcia-Moreno et al. (2014); Gergely et al. (2003); Kahyaoglu (2008); Karimi et al. (2012); Kowalski and Ganjya (2018); Lespinard et al. (2015); Noshad et al. (2012); Singh et al. (2010); Tarafdar et al. (2017); Themelin et al. (1997); Yuan et al. (2018)</td><td>Provides insight into process behaviour without knowledge of intrinsic mechanisms; minimizes the number of experiments to model the process; it is risky to use the process models out of their validity domain.</td></tr><tr><td>Artificial neural networks (ANN)</td><td>Asgari et al. (2017); Chen and Ramaswamy (2002); Izadifar and Jahromi (2007); Karimi et al. (2012); Taheri-Garavand et al. (2018); Winiczenko et al., (2018a-c)</td><td>Require no a priori knowledge of relationships between input and output variables; however a high amount of experimental data is needed.</td></tr><tr><td>Gene expression programming</td><td>Kahyaoglu (2008)</td><td></td></tr><tr><td rowspan="4">Knowledge-driven models ("White box")</td><td>Batch thermal treatment</td><td>Abakarov et al. (2009); Erdoğdu and Balaban (2003); Sendín et al. (2010)</td><td></td></tr><tr><td>Fermentation</td><td>Rodman and Gerogiorgis (2017)</td><td>Mainly mechanistic approach; provide robust and resilient models which account for involved mechanisms; however the models may take into account a limited number of variables; precision in prediction may also be low in a given domain compared to data-driven models.</td></tr><tr><td>Continuous heat treatment</td><td>De Jong (1996); Sidaway and Kok, (1982)</td><td></td></tr><tr><td>Evaporation</td><td>Nishitani and Kunugita (1979, 1983); Sebastian et al. (2010); Sharma et al. (2012)</td><td></td></tr><tr><td>Filtration</td><td>Yuen et al. (2000)</td><td></td></tr><tr><td>Hybrid models ("Grey box")</td><td>-</td><td>Arias-Mendez et al. (2013); Ferrández et al. (2018a,b); Goñi and Salvadori (2012); Hadiyanto et al. (2009); Kiranoudis and Markatos (2000); Olmos et al. (2002); Romdhana et al. (2016); Sicard et al. (2012)</td><td>Combine mechanistic models (heat & mass transfer, chemical reactions) with empirical equations, and even expert knowledge; trade-off between precision in prediction and range of validity domain.</td></tr></tbody></table>
392 White box models, also known as “mechanistic” models, are now capable of addressing
393 various scales (from molecular to macroscopic), making it possible to produce just as
394 great a variety of indicators. Purely mechanistic models are uncommon, since the links
395 between molecular and macroscopic scales are still difficult to establish. Various
396 approaches have been proposed which take into account prediction of phase changes in
397 food matrices with the SAFES methodology (Systematic Approach for Food Engineering
398 Systems; Fito et al. (2007)), but in which information requirements on the systems to be
399 modelled go beyond current knowledge (Trystram, 2012). The models considered in this
400 work as knowledge-driven use well-known laws, within specific domains:

401 • Some white box models relate to heat treatment in a container. The classic
402 equations of diffusion and convection of mass and heat, as well as the
403 degradation kinetics of compounds of interest, are used to describe the
404 phenomena occurring in the container.

405 • The works of Rodman and Gerogiorgis (2017) consider only part of the chemical
406 reactions occurring during fermentation, which makes it possible to use generic
407 kinetic parameters.

408 • Modelling of heat exchangers, with or without phase change, is abundantly
409 covered in the literature, especially in chemical engineering. This means that it
410 can be applied to food engineering processes, but using empirical correlations for
411 the exchange coefficients specific to the food products. Thus Sharma et al. (2012)
412 designed an evaporator treating milk, while Sidaway and Kok (1982) developed a
413 heat exchanger sizing program for heat treatment.

414 • Yuen et al. (2000) modelled the performance of a beer dialysis module, including
415 the molecular scale in the solute transfer rate calculation. Although simplified, this
416 is the closest model to a purely mechanistic model.

417 White box models are often characterized by long calculation times, inherent in the
418 partial derivative equations which have to be solved. Although computing power aids
419 simulation, the most complex models are not necessarily the most appropriate for multi-
420 objective optimization. That is why model reduction techniques are proposed to create
421 quick tools, containing all the degrees of freedom with optimization at the core, and
422 which are sometimes broken down into hybrid (grey box) models - quick, efficient and
423 simple to employ.

425 Black box models are based on experimental or compiled data, and require approaches
426 which employ model parameter identification algorithms to be determined once the
427 mathematical structure has been chosen. There are countless examples of modelling
428 approaches in the literature; Response Surface Methodology (RSM) is the most
429 common in food processing, particularly for modelling osmotic dehydration (Abakarov et
430 al., 2013; Arballo et al., 2012; Corzo and Gomez, 2004; Eren and Kaymak-Ertekin, 2007;
431 Singh et al., 2010; Themelin et al., 1997; Yuan et al., 2018), in which the complex
432 mechanisms involved (transfer through vegetable cell membranes) are well-suited to the
433 black box approach. The field of possible modelling approaches is wide, also
434 encompassing Artificial Neural Networks (Asgari et al., 2017; Chen and Ramaswamy,
435 2002; Izadifar and Jahromi, 2007; Karimi et al., 2012), gene expression programming
436 (Kahyaoglu, 2008), fuzzy logic, pure algorithms, etc. The main advantage of these black
437
box models is probably the calculation speed, which enables use of a wide variety of optimization algorithms. Nonetheless, these modelling approaches are often very data-hungry (and demanding in terms of data quality), especially when a random dimension is present in at least one of the indicators. In addition, black box models are limited by their ability to cover all the influencing variables, and if one of the variables is not taken into consideration, the whole work needs to be redone. Finally, due to the fact that the modelling is based on incomplete or non-existing prior knowledge, extrapolation is impossible or hazardous, and in this case confidence in the results obtained is generally low.

Improvements to black box models are designed and applied when knowledge based on expert opinion or experimental results is used. This knowledge makes it possible to describe a priori a black box models structure, which entails at least some degree of robustness after identification of the parameters. Numerous graph-based models enable such approaches to be used (e.g. Bayesian graphs, dynamic or not, fuzzy graphs); see for example Baudrit et al. (2010) and the review of Perrot et al. (2011). The modelling approach used in Sicard et al. (2012) combines a mechanistic model with expert knowledge to model the system dynamic. Thus in many cases, a compromise between a first principle (white box) model based on explicit knowledge, and coupled black box models is available, resulting in the creation of hybrid (grey box) models. For example in Olmos et al. (2002), a mathematical model for transfer into a rice grain was combined with empirical models of transfer coefficient and of quality deterioration. One of the advantages of these models is their applicability on various scales, or ability to contribute to multi-scale modelling, which is a major challenge for food engineering processes.

There is a great variety of modelling approaches, which is why it is important to be able to evaluate the model quality in terms of optimization, yet there are practically no analysis methods that have been developed to this end. Vernat et al. (2010) proposed rating the quality of a model by four aspects, united under the acronym “PEPS”:

* Parsimony: a model must be as simple as possible, which is quantified by the number of variables and mathematical relationships. This aspect could be supplemented by an execution time indicator for compatibility with optimization;

* Exactitude (accuracy): the distance between the results derived from the model and the experimental measurements/observations must be as low as possible. This aspect touches on the concept of physical (or chemical, biological) robustness, which means that whatever the simplification employed, the physical laws and the consequent behaviour of the model are still conserved;

* Precision: the uncertainty over the results derived from the model must be as low as possible;

* Specialization: the restriction of the model's field of application must be minimal.

Two additional aspects could be added to the PEPS framework:

* The identifiability of unknown model parameter values (transfer coefficient, activation energy of a reaction...) is validated.

* Sensitivity is established (and quantified) between the degrees of freedom for optimization and the key variables.
484 Model quality analyses are often limited to exactitude (accuracy), by comparison with
485 experimental results, and to the sensitivity of the model's responses to the operating or
486 sizing parameters. Hence the process models used are often developed specifically for
487 a unit operation or a process (Diefes et al., 2000), which means a high degree of
488 specialization. The development of more generic food engineering process models,
489 using IT tools able to easily evaluate model performances, would make it possible to
490 establish a logic of model quality compliance for optimization.

491
492 Once the indicators have been defined (section 2) and the process model is operational
493 (section 3), a method for selecting the best compromise must be chosen. This method
494 must be able to integrate the preferences of the decision-maker and/or experts in
495 evaluating the solutions. Multi-criteria analysis, which employs multiple criteria decision
496 analysis (MCDA) methods (also known as multi-criterion decision making – MCDM – or
497 multiple attribute decision making - MADM), refers to methods able to address this
498 issue. The following sections propose a review of methods of integrating preferences
499 and methods of identifying the best-performing solutions, used in the food engineering
500 literature.

501 4. Integrating preferences

502 Preferences apply to the indicator values and to the comparative significance of the
503 objectives. These preferences may be integrated before or after the optimization
504 process, or indeed during the process, i.e. interactively. Hence there are methods to
505 integrate these preferences in order to make the decision-making process more rational.
506 The articles reviewed in which the preferences are integrated via specific methods have
507 been classified in table 2, depending on whether the preferences are on the indicators,
508 the significance of the objectives, or whether they are integrated interactively.

509
<table><caption>Table 2: Preference integration methods</caption><thead><tr><th>Preference level</th><th>Methods</th><th>References [Product & process type]</th></tr></thead><tbody><tr><td rowspan="5">Preferences on indicator values</td><td>Desirability function: Harrington</td><td>Sebastian et al. (2010) [Wine evaporation]</td></tr><tr><td>Desirability function: Derringer</td><td>Arballo et al. (2012) [Pumpkin, kiwi, pear osmotic dehydration]; Corzo and Gomez (2004) [Sweet melon osmotic dehydration]; Eren and Kaymak-Ertekin (2007) [Potato osmotic dehydration]; Kowalski and Ganjya (2018) [Pea extrusion]; Lespinard et al. (2015) [Pumpkin pasteurization]; Karimi et al. (2012) [Wormwood leaves drying]; Taheri-Garavand et al. (2018) [Banana convective drying]</td></tr><tr><td>Sigmoid desirability function</td><td>Raffray et al. (2015) [Fish hot-smoking]</td></tr><tr><td>Desirability function not given (use of software)</td><td>Alam et al. (2010) [Indian gooseberry osmotic dehydration]; Azarpazhooh et al. (2012) [Apple osmotic dehydration]; Kahyaoglu (2008) [Pistachio nut roasting]; Noshad et al. (2012) [Quince dehydration process]; Vieira et al. (2012) [Guava osmotic dehydration]; Yadav et al. (2012) [Peach osmotic dehydration]</td></tr><tr><td>AHP weighting</td><td>Abakarov et al. (2013) [Carrot osmotic dehydration]</td></tr><tr><td>Preferences on objectives</td><td>Ordinal ranking: lexicographic ordering</td><td>Erdoğdu and Balaban (2003) [Food product thermal processing]</td></tr><tr><td>Interactive methods</td><td>NIMBUS</td><td>Hakanen et al. (2007) [Glucose-fructose separation]</td></tr></tbody></table>

511

512 The preferences may relate to the values adopted by the indicators. They originate from expert knowledge, functional analysis of the process to be designed, data mining, market studies... Their usefulness is based on:

515 • Reducing the search space for possible solutions, by proposing desired valves (upper and lower) associated with each indicator. This may prove particularly useful in the case of raw indicators, for which context-specific limitations may be integrated; thus for example in Sebastian et al. (2010), the maximum acceptable mass of the equipment makes it possible to evaluate the transportability objective of the equipment.

521 • Favouring certain indicator values over others, by means of desirability functions. 522 Desirability functions convert the value of an indicator into a dimensionless variable of 523 between 0 and 1, known as "satisfaction index", which quantifies the satisfaction of the 524 decision-maker on the performance of the indicator. They require the determination of a
high value for the indicator (associated with an upper or lower desirability value) and a low value (associated with a lower or upper desirability value, respectively), in order to demarcate the desirability domain. The most commonly used functions in the literature (Arballo et al., 2012; Corzo and Gomez, 2004; Eren and Kaymak-Ertekin, 2007; Lespinard et al., 2015) are those from Derringer (1980), one able to express increasing or decreasing desirability (one-sided), and the other to express maximum desirability in one domain, and decreasing when an indicator moves away from this domain (two-sided). There are other forms of desirability function, in various mathematical forms, such as from Harrington (1965), used in Sebastian et al. (2010), and the sigmoid function (Raffray et al., 2015). All these functions lead to normalized indicators (expressed on a common scale), which can facilitate ranking the solutions by aggregating the scores. The choice between the different existing functions depends on how the desirability values are seen, as a function of the values of the indicator under study. Thus for example, the functions from Derringer (1980) strictly demarcate the indicator's domain of variation, while the sigmoid function from Raffray et al. (2015) remains discriminant in the vicinity of the domain under study.

The decision-maker may also formulate preferences over the relative significance of their objectives, i.e. on the comparative significance of the indicators. This may involve weighting the objectives, or ranking them in order of significance. If the decision-maker is faced with a multitude of objectives, it may be difficult to rationally and consistently attribute the weights. That is why there are methods to help the decision-maker to prioritize the objectives: the AHP method (Analytic Hierarchy process – Saaty (1990)) for example, which designates a method even capable of ranking the solutions, includes a step of defining the weights by comparing the objectives (or indicators) in pairs, is used in Abakarov et al. (2013). A score of between 1 and 9 is attributed to each objective depending on its significance compared to every other objective, and the results are aggregated using a given formula to provide a numerical value for the weight of each objective. Other methods use pairwise comparison, a non-exhaustive list of which is given in Siskos and Tsotsolas (2015). Ranking the objectives in order of significance does not require priorization methods. It has been used by Erdoğan and Balaban (2003) and named "lexicographic ordering". This approach seems uncommon, since most decision-making aid and optimization methods require quantification of the significance of the objectives for calculating the objective functions. Otherwise, lexicographic ordering of the indicators must be implemented in the optimization algorithm, as is the case in Erdoğan and Balaban (2003). Another possibility is to use a lexicographic approach to produce a weighting (Sebastian et al., 2010): the objectives are ranked by significance, and a mathematical function attributes a weight to each objective according to its level of significance. This approach is similar to the SMARTER method (Edwards and Barron, 1994), and to other hybrid approaches of this type, such as: the Simos method (Figueira and Roy, 2002; Simos, 1990a, 1990b), where cards are used to order the objectives and quantify their relative significance, and the SWING method, in which the objectives are ranked based on solutions with the best possible value for one indicator, and the worst possible value in all the others. Interested readers can find a detailed review of weighting methods in Wang et al. (2009).

Finally, there are optimization methods in which the decision-maker formulates their
572 preferences through an iterative design process, in which solutions are presented to
573 them. These so-called interactive methods generally proceed in three phases (Coello,
574 2000):

575 1. Calculate a Pareto-efficient solution;

576 2. Put together the decision-maker's preferences on this solution, and its possible
577 improvements;

578 3. Repeat steps 1 and 2 until the decision-maker is satisfied.

579 The advantages of this type of method lie mainly in the low requirement for calculations
580 (few solutions calculated in each iteration), the absence of need for an overall
581 preferences diagram, and the possibility for the decision-maker to correct their
582 preferences and therefore learn through the optimization process (Taras and
583 Woinaroschy, 2012). Conversely, it is assumed that the decision-maker has the
584 necessary time and capacities to take part in the decision-making process, and that the
585 information supplied to the decision-maker is comprehensible and relevant (Miettinen,
586 1998). Although a substantial number of interactive optimization methods are available
587 (Collette and Siarry, 2013; Miettinen, 1998; Miettinen and Hakanen, 2009...), only
588 Hakanen et al. (2007) have used them, with the NIMBUS method (Miettinen and
589 Mäkelä, 1995, 2006). In NIMBUS, when a solution is presented to the decision-maker,
590 the latter specifies for each indicator how they would like it to evolve - for example if an
591 indicator needs to be improved, is satisfactory, or may be downgraded - and these
592 preferences are used to converge toward the most satisfactory possible solution for the
593 decision-maker.

594

595 If the optimization problem encountered has not been solved by an interactive method,
596 the preferences integration methods (desirability functions, weighting methods and
597 ranking methods for objective) prove useful in providing a framework for formulating the
598 preferences. To this end, the desirability function best suited to the objectives to be
599 optimized must be chosen, in particular preventing an indicator from adopting
600 undesirable values. The choice of weighting method meanwhile will depend primarily on
601 the user's affinity with one method or the other, and the ease with which they can
602 formulate their preferences.
603 5. Selection methods

604 The quantified preferences of the decision-maker may then be used to select the most
605 acceptable solution for the decision-maker. So in the case of an optimization problem,
606 this involves constructing a function or a mathematical criterion able to evaluate the
607 performances of the solutions generated by the process model. Yet it is also possible
608 that the decision-maker will be unable to formulate preferences, or that they are not
609 provided, in the absence of a decision-making context for example. That is why the
610 reviewed articles are classified in two major categories:

611 • "No information" (Table 3): in the absence of information from the decision-maker,
612 it is possible to calculate a relevant set of solutions ("Sorting / Filtering"), which
613 can then be compared in a decision-making context, or to select a solution
614 anyway without reference to the decision-makers formulated preferences
615 ("Ranking with weight elicitation");

616 • "Preferences expressed" (Table 4): the decision-maker's preferences are
617 expressed, so a solution acceptable under the decision-maker's criteria can be
618 selected.

619
<table><caption>Table 3: Selection methods – no information from the decision-maker</caption><thead><tr><th>Optimization problem</th><th>Methods</th><th>References [Product & process type]</th></tr></thead><tbody><tr><td>Filtering</td><td>Pareto front / Dominance</td><td>Abakarov et al. (2009) [Pork puree sterilization]; Arias-Mendez et al. (2013) [Potato frying]; Chen and Ramaswamy (2002) [Food conduction heating]; Ferrández et al. (2018a,b) [High-pressure thermal treatment]; Garcia-Moreno et al. (2014) [Fish oil extraction]; Goñi and Salvadori (2012) [Beef roasting]; Kawajiri and Biegler (2006) [Glucose-fructose separation]; Kiranoudis and Markatos (2000) [Potato drying]; Kopsidas (1995) [Table olive preparation]; Kurup et al. (2005) [Glucose-fructose separation]; Massebeuf et al. (1999) [Food extrusion / granulation]; Nishitani and Kunugita (1979, 1983) [Milk evaporation]; Sendín et al. (2010) [Canned fish sterilization]; Sharma et al. (2012) [Milk evaporation]; Subramani et al. (2003) [Glucose-fructose separation]; Stefanis et al. (1997) [Cheese manufacturing]; Winiczenko et al., (2018a-c) [Apple convective drying]; Yuen et al. (2000) [Beer dialysis]; Zhang et al. (2004) [Fructose syrup production]</td></tr><tr><td rowspan="3">Ranking with weight elicitation</td><td>Objective sum (weighted sum with equal weights)</td><td>Erdoğdu and Balaban (2003) [Food product thermal processing]; Rodman and Gerogiorgis (2017) [Beer fermentation]</td></tr><tr><td>Geometric average</td><td>Corzo and Gomez (2004) [Sweet melon osmotic dehydration]; Kahyaoglu (2008) [Pistachio nut roasting]; Karimi et al. (2012) [Wormwood leaves drying]; Kowalski and Ganjya (2018) [Pea extrusion]; Taheri-Garavand et al. (2018) [Banana convective drying]; Vieira et al. (2012) [Guavas osmotic dehydration]</td></tr><tr><td>TOPSIS</td><td>Madoumier (2016) [Milk evaporation]</td></tr><tr><td></td><td>MaxiMin</td><td>Raffray et al. (2015) [Fish hot-smoking]</td></tr></tbody></table>
<table><caption>Table 4: Selection methods – preferences of the decision-maker(s) are expressed</caption><thead><tr><th>Optimization problem</th><th>Methods</th><th>References [Product & process type]</th></tr></thead><tbody><tr><td rowspan="2">Filtering</td><td>Superimposition of contour plots</td><td>Annor et al. (2010) [Tempeh preparation process]; Collignan and Raoult-Wack (1994) [Fish dewatering and salting]; Ozdemir et al. (2008) [Pepper osmotic dehydration]; Singh et al. (2010) [Carrot osmotic dehydration]</td></tr><tr><td>Tabular method</td><td>Abakarov et al. (2013) [Carrot osmotic dehydration]; Winiczenko et al., (2018b) [Apple convective drying]</td></tr><tr><td rowspan="6">Ranking</td><td>Weighted sum</td><td>Asgari et al. (2017) [Olive oil bleaching]; De Jong (1996) [Milk sterilization]; Hadiyanto et al. (2008a; 2008b) [Bakery]; Izadifar and Jahromi (2007) [Oil hydrogenation]; Sidaway and Kok (1982) [Food continuous sterilization]</td></tr><tr><td>Weighted geometric mean / Derringer's desirability function</td><td>Arballo et al. (2012) [Pumpkin, kiwi, pear osmotic dehydration]; Eren and Kaymak-Ertekin (2007) [Potato osmotic dehydration]; Lespinard et al. (2015) [Pumpkin pasteurization]; Sebastian et al. (2010) [Wine evaporation]</td></tr><tr><td>AHP</td><td>Abakarov et al. (2013) [Carrot osmotic dehydration]</td></tr><tr><td>Loss-minimization function</td><td>Gergely et al. (2003) [Wine membrane filtration]</td></tr><tr><td>Custom partial aggregation method based on ELECTRE and PROMETHEE</td><td>Massebeuf et al. (1999) [Food extrusion / granulation]</td></tr></tbody></table>
## 626 5.1. No information

627 If the decision-maker's preferences cannot be formulated, the approach most commonly 628 used in the literature is obtaining the Pareto front, i.e. a larger or smaller set of non-629 dominated solutions. The concept of Pareto efficiency or dominance is illustrated in 630 figure 2, where the Pareto front covers all the solutions which are not inferior to any 631 solution at any point (i.e. for each indicator). Thus many authors have opted for this 632 approach (Abakarov et al., 2009; Kiranoudis and Markatos, 2000; Kopsidas, 1995; 633 Massebeuf et al., 1999; Nishitani and Kunugita, 1979...) for the purpose of providing 634 Pareto efficient design solutions uncoupled from any context, on which a decision-maker 635 can formulate their preferences. So there is no bias, hence it is possible to optimize 636 without a priori knowledge of the decision-maker's preferences (Massebeuf et al., 1999), 637 since an initial sort is carried out by eliminating the dominated solutions. However, this 638 method entails the risk of generating a large number of Pareto-efficient solutions 639 (Raffray et al., 2015), or even absurd solutions for the decision-maker, due to the low 640 solution filtering capacity (Scott and Antonsson, 1998). Indeed, a solution which is 641 extremely poor under one of the indicators may be among the non-dominated solutions, 642 but could be useless as a design solution. In addition, as identified by several authors 643 (Hadiyanto et al., 2009; Hakanen et al., 2007; Subramani et al., 2003), Pareto efficient 644 solutions may be presented to the decision-maker in graphic form for two or three 645 indicators, but interpretation becomes difficult after three. 646

<figure><img src="image_3.png" /><figcaption>Figure 2: Graphic representation of a Pareto front for two indicators (yᵢ and yⱼ). The solutions are designated by the symbols S¹ (Collignan, 2011)</figcaption></figure>
Another possibility, making it possible to go beyond the Pareto front while maintaining as neutral an approach as possible, is to use an aggregation function which eliminates weighting of the objectives ("weight elicitation" - Wang et al. (2009)). Thus it is possible to calculate the weighted sum, definitely one of the simplest and most commonly used aggregation functions, with the normalized indicators, assuming equal weight for each indicator. In Erdoğdu and Balaban (2003), the weighted sum became a simple "objective sum" (Marler and Arora, 2004). Another neutral function, the geometric mean, is used in several works (Corzo and Gomez, 2004; Kahyaoglu, 2008; Vieira et al., 2012). Product aggregation functions, like the geometric mean, are said to be more "aggressive" than sum functions (Quirante, 2012), since a low value for one indicator will have a big impact on the total score, and consequently better discrimination of the compromise solutions. Another possible approach is calculating the distance (Euclidian distance, with two or more dimensions) from "utopian" or "ideal" solutions; in the TOPSIS method ("Technique for Order Preference by Similarity to Ideal Solution") used in Madoumier (2016), the solutions are ranked by a function which aggregates the distance of a given solution from the "ideal" solution (comprising the best values for each indicator) and from the "anti-ideal" solution (comprising the worst values for each indicator), with the best solutions evidently being the closest to the former and the furthest from the latter. A shortcoming of these aggregation functions is their compensatory logic, i.e. a high value for one indicator may counterbalance a low value for another indicator (Collignan, 2011). To offset this shortcoming, there are so-called "conservative" aggregation functions (Otto and Antonsson, 1991), such as minimum aggregation (Raffray et al., 2015): the score of a solution is represented by the lowest value among its indicators. So maximizing this score comes down to selecting the "least worst" of all the solutions. According to the same logic, maximum aggregation gives the score of a solution as being the best value among its indicators, but this logic is not suited to a design context (Scott and Antonsson, 1998).

## 5.2. Preferences expressed

If the decision-maker's preferences are expressed, they can be used to more finely filter a set of solutions. Within the framework of RSM modelling (Response Surface Methodology), a graphic method of filtering the response surfaces was developed by Lind et al. (1960): the overlaid contour plots method comprises overlaying the contour plots for the various indicators, the value of which is determined according to the decision-maker's preferences, in order to isolate a zone in which the indicator values are most satisfactory. Used with success by several authors (Annor et al., 2010; Collignan and Raoult-Wack, 1994; Ozdemir et al., 2008; Singh et al., 2010), this graphic method does however lose some efficiency when the number of design variables is greater than two (Khuri and Mukhopadhyay, 2010), and when the optimization requirements are more complex (Myers et al., 2016). Some authors (Alam et al., 2010; Arballo et al., 2012) use in addition an MADM based on desirability functions to select the best solution from those filtered. Another method, this time based on using tables (known as the "Tabular method"), is used in Abakarov et al. (2013). Its principle is to rank the values adopted by each indicator according to whether they must be maximized or minimized. Hence each row in the table no longer corresponds to one solution. This then enables the decision-maker's preferences to be applied to the indicators to eliminate the undesirable values.
694 If the remaining values correspond to proposed solutions, these are adopted. The risk 695 with this type of approach is that if there is no solution corresponding to the preferences 696 on the indicators, it forces the decision-maker to revise their requirements downward. 697

698 To obtain a ranking of solutions or select the best compromise, the decision-maker's 699 preferences may be integrated into the aforementioned aggregation functions, in the 700 form of weighting. The most classic are the weighted sum, used in Asgari et al. (2017), 701 Hadiyanto et al. (2008a; 2008b) and Sidaway and Kok (1982), and the weighted 702 geometric mean (or weighted product) used in four studies (Arballo et al., 2012; Eren 703 and Kaymak-Ertekin, 2007; Lespinard et al., 2015; Sebastian et al., 2010). Proposed by 704 Derringer (1994), the geometric mean is used in the four studies to aggregate 705 normalized indicators by means of desirability functions. A potential shortcoming of the 706 weighted geometric mean is that the meaning given to the weights is less intuitive than 707 in a weighted sum, since the indicators between them have an exponential relative 708 significance instead of a proportional relative significance (Collignan, 2011). However, it 709 makes it possible to eliminate solutions where an indicator adopts a very low value or 710 zero, under an "aggressive" strategy as mentioned above. Besides these "primary" 711 aggregation functions (as per Marler and Arora (2004)), it is possible to adopt more 712 complex aggregation strategies, at least two of which have been identified within this 713 work:

714 • Integration strategy within a more complex decision-making aid framework, such 715 as the AHP method employed in Abakarov et al. (2013): the steps for determining 716 the weights, set out in section 4, lead to a weighted sum aggregation. 717

718 • "Mathematical" strategy, aimed at increasing the complexity of the aggregation 719 functions. An example is the function derived from an optimization method known 720 as "loss-minimization method" (Equation 1), corresponding to the weighted sum 721 (weight $w_i$) of variables defined as the relative difference between an indicator 722 (Q$_i$) and its optimal value (Q$_i^*$) (Gergely et al., 2003). This function requires prior 723 single-objective optimization of the indicators, to obtain their optimal value. 724

725 Equation 1 726

727 The aggregation functions mentioned above belong to full aggregation approaches, 728 characterized by the synthesis of several indicators into a single score, which can be 729 distinguished from so-called partial aggregation approaches or outranking approaches 730 (Brans and Vincke, 1985). The latter are based on construction of binary relationships 731 between solutions, based on the decision-maker's preferences (Wang et al., 2009). 732 Hence it is possible to do without an overall aggregation function, but it is also 733 necessary to be able to compare the solutions in twos. This means that partial 734 aggregation methods are applicable only when a sufficiently small set of solutions has 735 been generated. Thus in Massebeuf et al. (1999), the best solution is selected after 736 obtaining Pareto efficient solutions. The partial aggregation method employed in 737 Massebeuf et al. (1999) is constructed from methods such as ELECTRE (Elimination 738 and choice translating reality) and PROMETHEE (Preference ranking organization 739 method for enrichment evaluation); these two terms represent method families suited to 740
741 various types of issue (sorting for Electre-Tri, choice for ELECTRE I and PROMETHEE
742 I, ranking for ELECTRE III and PROMETHEE II, ...), the general principles of which are
743 in brief:

744 • ELECTRE (Roy, 1968) provides a ranking or preference relationships between
745 solutions, without calculating a cardinal score, based on concordance &
746 discordance indices, and threshold values (Wang et al., 2009). The relationships
747 between solutions are obtained based on pair comparison under each of the
748 decision-maker's objectives.
749

750 • PROMETHEE (Brans and Vincke, 1985) is based on quantified comparison of
751 solutions, i.e. the relationship between solutions under a given indicator will be
752 described by a preference function evaluating the intensity of this preference.
753 This information is used to calculate the "incoming" and "outgoing" flows of a
754 solution, i.e. the quantitative measurement of confidence and regret, respectively,
755 relating to a solution (Wang et al., 2009).
756

755 For selection or ranking issues, methods such as PROMETHEE are deemed easier to
756 use than methods such as ELECTRE (Velasquez and Hester, 2013), and were indeed
757 designed as an improvement on the latter (Brans and Vincke, 1985).
758

758

759 **5.3. Normalization**

760 A question rarely addressed, concerning the application (or possible development) of
761 selection methods, is that of normalization. Most of these methods require normalization
762 of the indicators to enable their comparison on a common scale; if normalization is not
763 applied by a desirability function, simple mathematical operators are applied, such as
764 division by an optimum or a reference value. Yet the works of Pavličić (2001) indicate
765 that the selection results may depend on the mathematical operation applied, and that
766 use of vector type normalization (Equation 2 – xij is the value of the jth indicator for
767 solution i, and ij is the corresponding normalized indicator) should be reconsidered. So it
768 seems that the normalization operator must be wisely chosen for the applying the
769 selection methods.
770

770
771

$$
r_{ij}^{V} = x_{ij} / \sqrt{\sum_{i=1}^{m} x_{ij}^{2}}
$$

Equation 2

772 **5.4. Difficulty of choosing a selection method**

773 In view of these technical considerations and noting the diversity of approaches, the
774 question of choosing a selection method is potentially complex. Indeed, the diversity of
775 solution selection methods, and more generally MADMs, is accentuated by possible
776 combinations between methods. For example in Abakarov et al. (2013), the AHP method
777 and tabular method are combined, and in Massebeuf et al. (1999), the Pareto front is
778 filtered by a partial aggregation method, constructed with the elements of two well-
779 known methods. While none of the methods appears to be the best, their respective
780 advantages and shortcomings may make them incompatible with certain applications
781 (Velasquez and Hester, 2013). Thus partial aggregation methods may be incompatible
782 with a decision-maker pressed for time, and so must be replaced by a full aggregation
783 method requiring little interaction with the decision-maker. As with weighting methods
784 (section 4), the choice will also be partly subjective, since it depends on the affinity with
785 one method or another. Moreover, they can lead to different results, as observed by
786 Wang and Rangaiah (2017): they compared 10 selection methods, obtained different
787 results for the same problem, and discussed the relevance of these methods for dealing
788 with an optimization problem. Considering a number of criteria (amount of user inputs,
789 simplicity, and applicability), they even recommend 3 methods: TOPSIS, GRA (Gray
790 relational Analysis) and SAW (Simple Additive Weighting).

791 6. Multi-objective optimization algorithms

792 An optimization algorithm refers to a more or less automated process, employed to seek
793 combinations of design variables leading to the best solutions, according to the MADM
794 used. Since the size of the space of the possible solutions varies exponentially with the
795 number of design variables and the number of values that these variables may adopt, an
796 "optimization engine" (as per Marler and Arora (2004)) is necessary to efficiently identify
797 the best solutions (in a given context). There are many possible optimization strategies,
798 and in the present work they have been grouped into five categories:

799 • Exhaustive search;

800 • Graphic optimization;

801 • Deterministic indirect search and direct search methods;

802 • Methods using stochastic metaheuristics;

803 • Interactive methods.

804 Exhaustive search and graphic optimization are considered as approaches without
805 optimization engine, while the other three categories are considered as approaches
806 using optimization engines. Thus, the reviewed articles are given in Tables 5 and 6,
807 which correspond to the two groups respectively.

808

809 It should be noted that in the literature, a considerable number of authors do not
810 explicitly give the algorithm employed for optimization purposes, if at all. Thus it appears
811 that the question of the optimization algorithm is often neglected, due to a probable lack
812 of knowledge and command of the subject (highly mathematical approach). his leads to
813 papers which are not reproducible by other researchers. One easy solution consists in
814 using software equipped with dedicated optimization functions.
<table><caption>Table 5: Optimization approaches and methods without optimization engines</caption><thead><tr><td>Optimization approaches</td><td>Optimization methods</td><td>References [Product &amp; process type]</td></tr></thead><tbody><tr><td>Exhaustive search</td><td>-</td><td>Lespinard et al. (2015) [Pumpkin pasteurization]; Nishitani and Kunugita (1979) [Milk evaporation]; Sidaway and Kok (1982) [Food continuous sterilization]; Taheri-Garavand et al. (2018) [Banana convective drying]</td></tr><tr><td>Graphic</td><td>Overlaid contour plots</td><td>Alam et al. (2010) [Indian gooseberry osmotic dehydration]; Annor et al. (2010) [Tempeh preparation]; Arballo et al. (2012) [Pumpkin, kiwi, pear osmotic dehydration]; Collignan and Raoult-Wack (1994) [Fish dewatering and salting]; Ozdemir et al. (2008) [Pepper osmotic dehydration]; Singh et al. (2010) [Carrot osmotic dehydration]</td></tr></tbody></table>
<table><caption>Table 6: Optimization approaches and methods with optimization engines</caption><thead><tr><th>Optimization approaches</th><th>Optimization algorithms</th><th>References [Product & process type]</th></tr></thead><tbody><tr><td rowspan="6">Deterministic – indirect search</td><td>Custom gradient-based method</td><td>Hadiyanto et al. (2008a; 2009) [Bakery]</td></tr><tr><td>Max-sensitive method</td><td>Nishitani and Kunugita (1983) [Milk evaporation]</td></tr><tr><td>Newton-based method IPOPT</td><td>Kawajiri and Biegler (2006) [Glucose-fructose separation]</td></tr><tr><td>Sequential quadratic programming (SQP)</td><td>Goñi and Salvadori (2012) [Beef roasting]; Manivannan and Rajasimman (2008) [Beetroot osmotic dehydration]; Olmos et al. (2002) [Rice drying]; Rodman and Gerogiorgis (2017) [Beer fermentation]</td></tr><tr><td>Complex method</td><td>Erdoğdu and Balaban (2003) [Food product thermal processing]</td></tr><tr><td>Control vector parameterization</td><td>Hadiyanto et al. (2008a; 2008b) [Bakery]</td></tr><tr><td rowspan="2">Deterministic – direct search</td><td>Direct method in Design-Expert software</td><td>Azarpazhooh and Ramaswamy (2012) [Apple osmotic dehydration]; Corzo and Gomez (2004) [Sweet melon osmotic dehydration]; Eren and Kaymak-Ertekin (2007) [Potato osmotic dehydration]; Kahyaoglu (2008) [Pistachio nut roasting]; Noshad et al. (2012) [Quince dehydration process]; Vieira et al. (2012) [Guava osmotic dehydration]; Yadav et al. (2012) [Peach osmotic dehydration]</td></tr><tr><td>Hooke-Jeeves method</td><td>De Jong (1996) [Milk sterilization]; Gergely et al. (2003) [Wine membrane filtration]</td></tr><tr><td rowspan="6">Stochastic metaheuristics</td><td>Differential evolution</td><td>Sendín et al. (2010) [Canned fish sterilization]</td></tr><tr><td>Genetic algorithms</td><td>Asgari et al. (2017) [Olive oil bleaching]; Chen and Ramaswamy (2002) [Food conduction heating]; Ferrández et al. (2018a,b) [High-pressure thermal treatment]; Izadifar and Jahromi (2007) [Oil hydrogenation]; Kowalski and Ganjya (2018) [Pea extrusion]; Kurup et al. (2005) [Glucose-fructose separation]; Massebeuf et al. (1999) [Food extrusion / granulation]; Raffray et al. (2015) [Fish hot-smoking]; Romdhana et al. (2016) [Alfalfa and beet pulp dehydration]; Sebastian et al. (2010) [Wine evaporation]; Sharma et al. (2012) [Milk evaporation]; Subramani et al. (2003) [Glucose-fructose separation]; Winiczenko et al., (2018a-c) [Apple convective drying]; Yuen et al. (2000) [Beer dialysis]; Zhang et al. (2004) [Fructose syrup production]</td></tr><tr><td>Particle swarm</td><td>Romdhana et al. (2016) [Alfalfa and beet pulp dehydration]</td></tr><tr><td>Scatter search</td><td>Arias-Mendez et al. (2013) [Potato frying]</td></tr><tr><td>Adaptive random search</td><td>Abakarov et al. (2009) [Pork puree sterilization]; Abakarov et al. (2013) [Carrot osmotic dehydration]</td></tr></tbody></table>
818 **6.1. Exhaustive search and graphic methods**

819 It is possible to do without optimization engines, generally when the problem is simple,
820 i.e. when it comprises a small number of design variables and objective functions, and
821 when the calculation time of a solution is sufficiently short. In this case, all possible
822 solutions are generated, and an MADM can be applied to rank them and/or select the
823 best. In Lespinard et al. (2015), the total desirability is calculated over the entire
824 feasibility domain based on polynomial regression models. In Nishitani and Kunugita
825 (1979), the number of possible solutions is limited to 6 in the first optimization study (6
826 possible flow patterns), and in the second optimization study, the 6 solutions are
827 recalculated for various temperature levels of the incoming product. Finally, in Sidaway
828 and Kok (1982), the indicators are evaluated for all temperature and holding time
829 combinations which comply with a given sterility constraint.
830

831 Under RSM, it is also possible to do without an optimization engine, using graphic
832 optimization methods, such as overlaid contour plots, as mentioned above (section 5).
833 So this method has both a filtering role according to the preferences, and search role in
834 the solutions feasibility space. Indeed, overlaying certain surface response contours of
835 various indicators makes it possible to reduce the search space without calculations,
836 and thereby provide a smaller set of acceptable solutions. Use of this method has been
837 found in six studies (Alam et al., 2010; Annor et al., 2010; Arballo et al., 2012; Collignan
838 and Raoult-Wack, 1994; Ozdemir et al., 2008; Singh et al., 2010) as an optimization
839 engine, and although its usefulness is recognized, it does have a number of
840 shortcomings, mentioned in section 5.
841

841 **6.2. Optimization engines**

842 Conversely, when there are a high number of possible solutions and indicators, it
843 becomes necessary to employ an optimization procedure enabling automated searches
844 for the best solutions. The search methods may be divided into two categories:
845 deterministic methods and stochastic methods.
846

846
847 Deterministic optimization methods guarantee that a solution representing an optimum
848 will be obtained (Miri et al., 2008). However, depending on the method employed, the
849 optimum found may only be local, i.e. the adopted solution is Pareto efficient in only a
850 portion of the search space, but may be dominated by other solutions situated in other
851 portions of the search space. There are two types of deterministic method, namely
852 indirect search methods and direct search methods (Romdhana et al., 2016). Indirect, or
853 gradient-based, search methods require derivable objective functions, which is seldom
854 the case, since many variables are discrete and/or discontinuous (Pailhès et al., 2011).
855 So gradient-based methods are applicable only to certain types of problem. Thus the
856 algorithms implementing these methods converge toward an optimum, which achieves
857 rapid convergence in the case of a single extremum. However, most multi-criteria
858 problems involving several extrema ("multi-extremal" or "multimodal" problems),
859 gradient-based methods are unable to converge unfailingly toward an overall optimum
860 (Banga et al., 2003). When these methods are used to generate a Pareto front, they are
861 coupled with aggregation functions. Thus, Goñi and Salvadori (2012), Kawajiri and
Biegler (2006), Nishitani and Kunugita (1983) and Olmos et al. (2002) use the $ε$-constraint method, the formulation of which is to optimize one indicator at a time, considering the other indicators as constraints limited by a given value $ε$ (Seng and Rangaiah, 2008). So the gradient-based method makes it possible to solve every single-objective problem, which entails solving several optimization problems, and requires the values $ε$ to be defined, which may prove difficult without a priori knowledge of the possible optimal value of the indicators. Furthermore, varying only one design variable in the search space (with the others fixed) does not necessarily make it possible to converge toward an optimum due to failure to factor in interactions between variables (Myers et al., 2016). Another example is use of the weighted sum with variable weights to generate the Pareto front by Hadiyanto et al. (2009). In order to identify overall optima and not only local ones, some deterministic methods have been developed. These methods, known as direct search methods, evaluate the objective functions without calculating their derivative. For example, the Hooke-Jeeves method used by De Jong (1996) and Gergely et al. (2003), consists in evaluating the objective function around a start point, and shifting the search zone in the direction that improves the objective function until a stop criterion is obtained (Benasla et al., 2008). Certain authors (Azarpazhooh and Ramaswamy, 2012; Corzo and Gomez, 2004; Eren and Kaymak-Ertekin, 2007; Kahyaoglu, 2008; Noshad et al., 2012; Vieira et al., 2012; Yadav et al., 2012) have used a direct search method implemented in the Design-expert software, dedicated to RSM (Myers et al., 2016). Direct search methods have the major shortcoming of converging less and less quickly as the size of the problem to solve increases (number of design variables and of objective functions) (Banga et al., 2003). In addition, just like indirect search methods, they require an objective function in algebraic form.

Due to the fact that deterministic optimization methods are not always suitable, methods based on random draws and iterative procedures have been developed. These methods, known as stochastic methods, or using stochastic metaheuristics, are generally inspired by natural phenomena or everyday life, and have been developed in order to solve problems for which conventional deterministic methods proved ineffective (Collette and Siarry, 2013). They make it possible to couple the optimization algorithm to the problem without having to formulate it in an algebraic form, by directly optimizing based on indicator values returned by the process model. In addition, stochastic methods are able to converge more quickly than deterministic methods in the case of complex problems, but with no guarantee of obtaining an overall optimum (Banga et al., 2003). Recently, a literature review of the application of metaheuristics in food engineering (in the broad sense, including formulation of foods and production of pharmaceutical products) was conducted by Wari and Zhu (2016); it emerged that stochastic methods, despite their complexity, are seeing increasing use with the development of computer calculation capacities. The three major common points of stochastic methods are: i) seeking an overall optimum for the entire feasibility domain of the design solutions; ii) the stochastic (random) nature of the calculation of new solutions in each iteration; iii) they authorize downgrading of the indicators to explore the search space more widely. In the literature, the genetic algorithms are by far the most popular, with no fewer than 12 studies (Table 5). Genetic algorithms imitate the process of genetic evolution: an initial “population” of solutions undergoes “genetic modifications”
909 by crossover, mutation, and selection of elements of its DNA (its "genes" correspond to
910 the values adopted by the indicators) according to the performances of each initial
911 individual (i.e. solution) to form new individuals, i.e. potentially better-performing
912 solutions for the multi-objective problem (Hugget et al., 1999; Wari and Zhu, 2016).
913 Several optimization methods employ genetic algorithms, such as MOGA (Multiple
914 Objective Genetic Algorithms) or NSGA (Non-dominated Sorting Genetic Algorithms),
915 the differences between which reside in the Pareto efficiency calculation for the
916 individuals. It is possible to couple together stochastic methods, as in Romdhana et al.
917 (2016), where a genetic algorithm was coupled to a particle swarm algorithm. Wari and
918 Zhu (2016) provided some guidelines on selecting a metaheuristic tailored to the design
919 problem encountered.
920

921 **6.3. Interactive methods**

922 Interactive methods, presented in section 4, are not optimization engines in themselves,
923 but rather "interaction principles" (Collette and Siarry, 2013). They may require use of an
924 optimization engine to generate a small number of solutions to present to the decision-
925 maker in each iteration. In the NIMBUS method for example, used in Hakanen et al.
926 (2007), several single-objective sub-problems are defined according to the decision-
927 maker's preferences, and a gradient-based method is used to optimize each of the sub-
928 problems. It should be noted that there are a host of heuristics, including the one known
929 as "simplex", which do not require an optimization engine to progress in the interactive
930 search process.

931 **7. Toward holistic design approaches**

932 Hitherto, optimization frameworks have been constructed primarily either with a view to
933 generating the Pareto front, or by partially employing MADMs. Various functions coupled
934 to optimization algorithms have been used to generate a Pareto front, leaving
935 expression of decision-maker preferences outside of the field of study. Use of partial
936 aggregation methods and interactive methods, which provide a framework for
937 preference integration, has been encountered once for each of these types of method
938 (Massebeuf et al. (1999) and Hakanen et al. (2007) respectively). Highly diverse solution
939 selection methods have been employed, such as weighted sum, which is among the
940 best known, or weighted geometric mean recommended by Derringer (1994).
941 Conversely, certain aspects of MOO are often neglected; for example, use of methods
942 able to help the decision-maker weigh the objectives remains restricted.
943

944 Yet the diversity of decision-making aid methods makes it possible to construct holistic
945 design frameworks, i.e. frameworks which handle all aspects of MOO in a structured
946 way. The constituent elements of the MOO associated with decision-making aid and
947 optimization (preference integration methods, selection methods and optimization
948 methods) may be defined under various approaches, the overview of which given in this
949 work is far from exhaustive. Thus there are holistic approaches for handling a multi-
950 objective problem, such as OIA design methodology (Gero and Kannengiesser, 2007).
This approach, used in Sebastian et al. (2010) and Raffray et al. (2015), combines
process modelling to link design variables and indicators (Observation), integrating
preferences on the indicators using desirability functions (Interpretation), and
constructing an objective function with an aggregation function (Aggregation). A
stochastic optimization algorithm may then be integrated into the design framework to
obtain the best solutions. This is a general methodology, which must be tailored to the
specific context of the study, i.e. the type of problem and field of application (Miettinen,
1998).

Once the study requirements have been identified, the methods to be employed need to
be considered. This is a complex subject, into which the advantages and shortcomings
of each method have to be factored, as well as the cognitive aspect of the decision-
making process; thus, it is important to reduce the "cognitive load" on the decision-
maker to facilitate the decision-making process (Hakanen et al., 2007). An initial avenue
of consideration resides in the moment, in the design process, when the preferences of
the decision-maker and/or expert are articulated. Thus optimization methods are often
classified according to whether the preferences are articulated a priori, interactively (or
progressively), or a posteriori (Collette and Siarry, 2013; Marler and Arora, 2004;
Miettinen, 1998), or even in a fourth category with no preferences articulated ("no-
preference") (Andersson, 2000; Erdoğdu, 2008; Miettinen and Hakanen, 2009). Thus, if
the preferences are formulated a priori, use of desirability functions, a weighting method
and an aggregation function will make it possible to construct an objective function
which will be incorporated into an optimization algorithm in order to generate the best
solution. In the case of a posteriori formulation, obtaining a Pareto front is relevant, but it
is easier for the decision-maker to select the best solution from a small set. Marler and
Arora (2004) also postulated that it is important to define, prior to choosing a method,
the type of preferences provided, as well as the quantity of information. Collette and
Siarry (2013) provided clues to helping choose multi-objective optimization methods,
based in particular on analysing the complexity of the problem and analysing the
objective functions. Miettinen (1998) proposed an organization chart for the choice of
multi-objective optimization methods, but it proved relatively complex to use. If no
method is entirely suitable, various methods may be combined to combine the
advantages and compensate for the shortcomings. Examples of these hybrid solving
approaches can be found in Abakarov et al. (2013), which combined two solution
selection methods, and in Romdhana et al. (2016), which combined two stochastic
optimization methods.

The authors would like to emphasize that holistic approaches do not provide a
guarantee of obtaining the best solution in terms of the decision-maker. Like any design
approach, MOO approaches are part of an iterative decision-making process, which is
only facilitated by using decision-making aid methods. Thus it is unlikely that a
satisfactory solution will be found in the first iteration; that is why it is important to
develop high-performance decision-making tools, which facilitate the optimization
procedure.

In this regard, some research questions have been identified:

* Evaluation of the relevance of the indicators has been identified as a difficult
search question. Thus the definition of the indicators can have a crucial impact on the optimization results, and few authors have looked into this subject. The question of relevance of the sustainability indicators is particularly topical, especially with regard to the social dimension, which remains hard to integrate into the process. The challenge is the integration of all the sustainability dimensions using appropriate indicators.

* The development of food engineering process simulators suitable for design purposes. To develop these simulators, it should be defined what is expected of a model tailored to optimization, especially in terms of compromise between calculation speed and accuracy of results. The development of evaluation frameworks based for example on the PEPS framework (Vernat et al., 2010) would make it possible to select the models best suited to a given context.

* Although there are some weight definition aid tools (AHP, SMARTER, Simos, SWING...), decision-maker weighting of the objectives remains a difficult task, for example if it involves weighting economic objectives against environmental objectives. Knowledge derived from psychology and social sciences could develop weighting methods best meeting the decision-maker's preferences, while avoiding as far as possible cognitive biases, and integrating data such as opinion surveys.

* A huge amount of reasearch has been carried out in stochastic metaheuristics since the 90's. However, they find limited applications, in particular in food process engineering, which brings many opportunities to researchers in metaheuristics. We identified two search fronts in particular: simplifying the search space to limit the number of calculations, and integrating robustness criteria in the selection of the best solutions, in order to directly eliminate the solutions most sensitive to small variations in the design variables. Thus, collaborations between researchers in metaheuristics and researchers in food process design could produce better-suited algorithms for food process design.

* Finally, in the more general framework of food systems sustainability, how can processing design be integrated into a more global system? In fact, design of sustainable industries actually entails factoring in food production as well as processing. Consequently, the performances of processing must be evaluable on a larger scale, and models must be tailored to optimization of the industry as a whole.

## 8. Conclusion

This review has identified various MOO methods employed in the design of food engineering processes, which are mostly fragmentary. Indeed, the possibilities offered by design engineering and decision-making aid are still under-exploited in the food engineering field. The associated methods facilitate the design process, by clearly defining the preferences of the experts and/or decision-maker, and by optimizing the process depending on the type of problem encountered.

The lack of development of holistic methods can be explained by several factors. First, the difficulty in selecting suitable indicators, which can give meaning to the decision-
1042 making variables, in particular for certain sustainability dimensions. Then, while it is
1043 sometimes possible to find the desired balance between calculation time and accuracy
1044 of the results, on the other hand gaps in the modelling of food engineering processes
1045 have been observed, as well as a need for simulators tailored to optimization. Also,
1046 weighting of the objectives is difficult, and the optimization algorithms require further
1047 improvements to be able to rapidly converge toward high-performance solutions.
1048

1049 Hence it would seem pertinent to develop an overall methodological framework which
1050 could guide the designer step-by-step in handling a design problem. This framework
1051 would greatly facilitate integration of the various decision-making aid and optimization
1052 tools. In this way it could rationalize the methodological choices and simplifications
1053 necessary for optimization.

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