Process modelling and optimization aidFONTEIX ChristianProfessor of Chemical EngineeringPolytechnical National Institute of LorraineChemical Engineering Sciences Laboratory

Process modelling and optimization aid

MultiCriteria Decision AidFONTEIX ChristianProfessor of Chemical EngineeringPolytechnical National Institute of LorraineChemical Engineering Sciences Laboratory

Multicriteria decision aidGeneralitiesModelling of human preferencesOpposite objectivesSubjectivityContext influence (social, economic)LobbyingExpert system structure of the modelMathematical functionsDifficulty to determine the best structure of the model and its characteristics (rules) or parametersRobustness of a decision

Multicriteria decision aidGeneralitiesDifferent methodsSequential search by single pointThe decision maker indicate at each step the search direction. Disadvantages : no entire view of the problem, a priori expression of the preferences

Using a preferences modelling on an alternatives listTo list all the alternatives at first and choose a modelTo determine the prefered alternatives with the model Advantages : entire view and a posteriori expression of the preferences

Multicriteria decision aidPreferences modelling typesMCDA is just an aidIt is an information for the decision makerThe decision maker choose the final decision

Parameters determination of preference modellingPreferences extraction by cognitive determinationPreferences extraction by alternatives comparizon Preferences extraction by parametric identification from preferences measurementGroup consensus

Multicriteria decision aid Preferences modelling typesParametric identification from preferences measurementChoice of a representative set of alternativesThe decision maker score each alternative of the setThe decision maker rank the alternatives of the set

3 classifications for preferences modellingClassification by the result (ranking of classification of the totality of the alternatives, or classification) Compensative or non compensative modelsPartial or total aggregation

Multicriteria decision aid Preferences modelling typesClassification by the resultAlternatives comparizon 2 by 2Determination of a Preferences Relationship SystemDetection and delating of CyclesDetermination of several classificationChoice of one classification

Multicriteria decision aid Preferences modelling typesClassification by the compensation typeCompensative type : for each alternative a bad criterion value can be compensate by a good criterion value (2 criteria)Non compensative type : for each alternative a bad criterion value cannot be compensate by a good criterion value (2 criteria)

Classification by aggregation typeTotal aggregation : the score depends on one alternativePartial aggregation : the score of one alternative depends also on the others

Multicriteria decision aid Preferences modelling types

TypeTotal aggregationPartial aggregationCompensativeMAUT (AHP)Non compensativeOWAChoquet integralPROMETHEE ELECTRERough Sets

Multicriteria decision aid Preferences modelling typesMAUT : Multi Attribute Utility TheoryAHP : Analytical hierarchical Process (method for MAUT parameters determination)

OWA : Ordered Weighted AverageChoquet integral : complex model (criteria interactions)

ELECTRE : ELimination Et Choix Traduisant la REalitPROMETHEE : Preference Ranking Organisation METHod for Enrichment EvaluationsGAIA : Graphical Analysis for Interactive Assistance (for PROMETHEE)

Multicriteria decision aid Preferences modelling typesRough Sets : mathematical theory which extend the set theory (as fuzzy sets)Authors have used the rough sets theory to develop a simple fast multicriteria decision aid

In rough sets theory a set is defined by a maximum set and a minimum set

Multicriteria decision aid MAUTx : alternative of which we calculate the scorefi(x) : value of the criterion i for the alternative xui(fi(x)) : utility value corresponding to criterion iwi : weight associated to criterion is(x) : score of the alternative xu(s(x)) : total utility of the alternative xn : number of criteriau(), ui() : utility functions

Multicriteria decision aid MAUTThe weights define the relative importance of criteriaThe utility functions define the intrinsic exigency of the decision maker on each criterionThe utility ([0, 1]) must be maximizedFor a criterion to be minimized :

Multicriteria decision aid MAUTProblem on neutral caseExample (2 criteria to be maximized) :

Parabole with one minimumPrefered alternatives are f1 or f2 maximum w1No compromise

Multicriteria decision aid MAUTA limit case is given by :

If w10.5 the prefered alternative is f1 maximumIf w1=0.5 all the alternatives are equivalent

Multicriteria decision aid MAUTIn practice :

Multicriteria decision aid OWAx : alternative of which we calculate the scorefi(x) : value of the criterion i for the alternative xui(fi(x)) : performance value corresponding to criterion iwj : weight non associated to criterion i, associated to a performance level jExample : 0.3, 0.1, 0.8, 0.5 are ranked as 0.8, 0.5, 0.3, 0.1and (1)=3, (2)=4, (3)=1, (4)=2s(x) : score of the alternative xu(s(x)) : total performance of the alternative xn : number of criteriau(), ui() : performance (utility) functions

Multicriteria decision aid OWAThe weights define the relative importance of performance levelThe performance functions define the intrinsic exigency of the decision maker on each performance levelThe performance ([0, 1]) must be maximized

Multicriteria decision aid OWAExampleWeight associated to performance level 1, w1=0.5Weight associated to performance level 2, w2=0.2Weight associated to performance level 3, w3=0.2Weight associated to performance level 4, w4=0.1u1=0.3, u2=0.1, u3=0.8, u4=0.5u1 is the performance 3, the associated weight is w3=0.2 u2 is the performance 4, the associated weight is w4=0.1u3 is the performance 1, the associated weight is w1=0.5u4 is the performance 2, the associated weight is w2=0.2u=0.2*0.3+0.1*0.1+0.5*0.8+0.2*0.5=0.57

Multicriteria decision aid PROMETHEEThe score of one alternative depends also on the othersAnalogy with football :PROMETHEE is a championshipThe alternatives are the teamThe score of one alternative is the goal averageThe number of goals obtained by team i on team j is ijThe number of teams is NThe score of i is :

Multicriteria decision aid PROMETHEEThere is one match between teams i and jThe goals obtained by team i depends of its player quality (and the player quality of team j)Each player of team i play against the corresponding player of team jThe participation of a player to the score depends on the quality difference of the 2 corresponding players :

fk is the value of criterion k

Multicriteria decision aid PROMETHEEThe number of goals obtained by team i on team j is :

n is the number of criteriaCijk depends of 2 thresholds :The indiference threshold qk The preference threshold pk

Multicriteria decision aid PROMETHEE0pkCijk1ijkqk

Multicriteria decision aid ExampleEmulsion polymerization : latex productionPareto domainPareto frontier

Multicriteria decision aid ExampleEmulsion polymerization : latex productionPareto domainPareto frontier

Multicriteria decision aid ExampleEmulsion polymerization : latex productionPareto domainPareto frontier

Multicriteria decision aid ExampleDieProduction of cow food by extrusionBarrel temperaturePrefered zone

Multicriteria decision aid ExampleBest technicalrobustnessBest decisionnalrobustnessProduction of cow food by extrusion

Optimisation multicritre dun procd de mise en pte haut rendementJ. Thibault (Universit dOttawa)R. Lanouette (Universit du Qubec TR)C. Fonteix (LSGC, Nancy)L.N. Kiss (Universit Laval)

IntroductionModelling of the fermentation processMulticriteria OptimisationPareto domainRanking algorithm : Net Flow MethodResultsConclusionOutline

In complex processes with a large number of input and output variables, the determination of a suitable set of input variables that would provide an optimal set of outputs is a major chalenge.

Usually dealing with numerous conflicting objectives.

There is a need to develop new optimisation methods to capture the preferences of the decision-maker to lead to an acceptable compromise solution.Introduction

New techniques in multicriteria optimisation are now emerging in the field of engineering to resolve problems of conflicting criteria.

Two of these techniques have been used by the authors:Net Flow MethodRough Sets

They are very useful because they allow one to rank the full domain of non-dominated solutions (Pareto domain).Introduction

Schematic of the Process

Pulp and Paper ProcessModel InputsFirst stage:TemperaturePlate gap Second stage:Consistency Interstage Treatment:H2O2 chargeTemperatureRetention timeNaOH (Y or N)For each output, the model is a stackedneural network (10 levels) having 7 inputs

Model OutputsBrightness Index maximizeSpecific Refining Energy Extractive Content Breaking LengthmaximizeminimizeminimizePulp and Paper Process

Optimisation ProcessExperimental Design

OptimisationApproximation of the Pareto domain (with or without NaOH leading to two separate models)Output 1SimulationOne solutionOutput 2Output 3Output 4

Point 1 66.71 7.599 0.1251 4.229Point 2 65.76 8.603 0.1745 3.837Dominance and Pareto DomainProcedure to approximate the Pareto domain:Randomly select the six input variables and calculate the four criteria for a total of 6000 points.Compare each point and discard those points that are dominated for all three criteria by at least one other point.

Point 1 66.71 7.599 0.1251 4.229Point 3 65.76 8.603 0.1745 4.784A large number of points are generated and compared until 6000 non dominated points are identified.Dominance and Pareto Domain

OptimisationProcedure to approximate the Pareto domain:Randomly select the four input variables and calculate the three criteria for a total of 6000 points.Compare each point and discard those points that are dominated for all three criteria by at least one other point.Keep all non-dominated solutions and 30% of dominated points that are dominated fewest number of times.Generate new points (up to 6000 points) by random interpolation between two points.Redo the procedure until 6000 non-dominated points are obtained.

OptimisationThe Net Flow Method requires a human expert to give some appreciation as to the nature of each criterion. Four pieces of information must be provided:

The relative importance of each criterion (a relative weighting WK);

The indifference threshold (QK);

The preference threshold (PK);

The veto threshold (VK).0 Qk Pk Vk

OptimisationConcordance index:Fk being minimizedIndividualGlobal

OptimisationDiscordance index:Fk being minimized

ResultsY1 - ISO BrightnessY2 - Specific refining energyBest points are in red color

ResultsY3 - ExtractivesY4 - Rupture lengthBest points are in red color

ResultsX1 - Temperature of first stageX2 - Plate spacing200 best points are in dark

ResultsX3 - ConsistencyX4 - H2O2 Charge200 best points are in dark

ResultsX5 - Temperature of second stageX6 - Residence time200 best points are in dark

A multicriteria optimisation routine has been applied to optimise a pulping process.

The ranked points of the Pareto domain allow to visualize the zone of optimal operation and help in designing a control strategy.Conclusion

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