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Constrained Multicriteria Sorting Method Applied to Portfolio Selection

Constrained Multicriteria Sorting Method Applied to Portfolio Selection

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Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection Conclusion<strong>Constrained</strong> <strong>Multicriteria</strong> <strong>Sorting</strong> <strong>Method</strong> <strong>Applied</strong><strong>to</strong> <strong>Portfolio</strong> <strong>Selection</strong>Jun Zheng Olivier Cailloux Vincent MousseauLabora<strong>to</strong>ire Génie Industriel, École Centrale Paris, FranceLamsade1 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection Conclusion1 Introduction2 Problem Formulation3 Mathematical program4 Projects funding5 Green labelling6 Students selection7 Conclusion2 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContextLiterature review1 Introduction2 Problem Formulation3 Mathematical program4 Projects funding5 Green labelling6 Students selection7 Conclusion3 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContextLiterature reviewA few examplesAllocate budget <strong>to</strong> research proposalsFund the “best” proposalsEvaluation on multiple criteriaOnly fund projects if they are good enough!. . . with a good balance between risk, subjects. . .Common pointsAbsolute evaluation: good enough?Relative evaluation: among the best?Relative evaluation: portfolio balance?5 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContextLiterature reviewA few examplesReward improvements (sports team)I want <strong>to</strong> reward some team membersAccounting for their improvements on multiple criteriaThose that improved a lotOnly the best of those (at most 5%)Also reward those that improved moderately (at most 10% ofthem)→ comparisons, as in a ranking→ . . . however not exactly a rankingCategory sizesCategory sizes increase (5% best, 10% moderate, . . . )6 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContextLiterature reviewA few examplesPartition in similar classesSplit students in language classesClasses should be of approximately the same sizeClasses should be of homogeneous level7 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContextLiterature reviewA few examplesYes / No / I don’t knowSort old items in a shop (or in your attic)OK category: could still be sold (or kept)KO category: goes <strong>to</strong> trashIntermediary cases: ask the boss (or your wife)Limited availability of the boss limit the number of such items8 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContextLiterature reviewA balance modelEvaluating the whole portfolio: a balance model <strong>to</strong> measure thedistribution of specific attributes [Farquhar and Rao, 1976]Choose the highest valued portfolioAccording <strong>to</strong> balance on attributesAttributes evaluated on the same scale (same distance measure)Preference model: weights of the attributesNo absolute evaluationResulting model does not compare <strong>to</strong> normsMultiple categories not supported9 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContextLiterature reviewExcluding with constraintsSelecting a <strong>Portfolio</strong> of Solar Energy Projects Using MultiattributePreference Theory [Golabi et al., 1981]Goal is simply <strong>to</strong> choose one portfolioA portfolio has a value (using value theory)Screen-out insufficiently good alternativesScreen-out portfolio with unsatisfac<strong>to</strong>ry balanceChoose best valued portfolio among remaining onesNo explanation about why an alternative is selected10 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContextLiterature reviewRobust portfolio selectionCombining preference programming with portfolio selection [Liesiöet al., 2007, 2008]Robust portfolio selectionScreen-out portfolios based on constraints and robust decisionsNo absolute comparisons<strong>Selection</strong> not easily explained11 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionGeneral description Electre Tri variant Preference elicitation <strong>Method</strong> summary1 Introduction2 Problem Formulation3 Mathematical program4 Projects funding5 Green labelling6 Students selection7 Conclusion12 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionGeneral description Electre Tri variant Preference elicitation <strong>Method</strong> summaryDecision situationDecision situationA set of alternativesTo be sorted in ordered categoriesQuantitative and qualitative criteria(Possibly) A decision <strong>to</strong> be repeatedAlt 1Alt 2Alt 3Alt 4C 3 : “Good”C 2 : “Average”C 1 : “Bad”13 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionGeneral description Electre Tri variant Preference elicitation <strong>Method</strong> summaryObjectivesObjective of our methodObtain a sorting functionDesirable features of the sorting functionComparison of alternatives <strong>to</strong> normsJudge alternatives on the same groundSelect a portfolio using:the quality of individualsthe overall portfolio quality (e.g. good balance?)Easily explain assignments14 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionGeneral description Electre Tri variant Preference elicitation <strong>Method</strong> summaryPrinciple of the sorting functionUses the alternatives performances on the criteriaThe sorting function reflects the DM preferences thanks <strong>to</strong> apreference modelpreference modelAlt 1Alt 2. . .C 3 : “Good”C 2 : “Average”C 1 : “Bad”Preference information considered at two levels1 Intrinsic alternatives evaluations: “is it good enough?”2 <strong>Portfolio</strong> evaluations: balance? category size? . . .15 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionGeneral description Electre Tri variant Preference elicitation <strong>Method</strong> summaryExplanation of the methodpreference modelAlt 1Alt 2. . .C 3 : “Good”C 2 : “Average”C 1 : “Bad”1 Assuming the preference model is known, explain how thesorting function proceeds2 Then: explain how <strong>to</strong> obtain the preference model16 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionGeneral description Electre Tri variant Preference elicitation <strong>Method</strong> summary<strong>Sorting</strong> method: a variant of Electre TRIPreference parametersA set of category limits: determine when the alternative is goodenough on a criterionWeights, for each criteria, and a majority threshold: determinewhen the alternative is globally good enoughExample: alternativesg1 g2 g3Alt 1 3 1 1Alt 2 5 3 3Alt 3 0 5 1Alt 4 2 0 2Example: cat. limitsg 1 g 2 g 3C 3 :Goodl C 34 4 3C 2 :Averagel C 23 3 2C 1 :BadExample: weightsg 1 g 2 g 3W 0.2 0.4 0.4λ = 0.617 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionGeneral description Electre Tri variant Preference elicitation <strong>Method</strong> summaryNotationsObjective dataAlternatives: A (one alternative: a)Criteria: J (one criterion: j)Categories:C (one category: C h , 1≤h≤ k)Performance of a on criterion j: g j (a)≥,> defined on the image of g j (a)PreferencesLower limit of category C h on criterion j: l C hjWeight of criterion j: w jMajority threshold:λ18 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionGeneral description Electre Tri variant Preference elicitation <strong>Method</strong> summaryComparing alternatives <strong>to</strong> categories (criterion j)Point of view of criterion j: Is an alternative a good enough for acategory C h ?Arguments in favor of a C h (binary)b a,C hjb a,C hj= 1⇔g j (a)≥l C hj= 0⇔g j (a)


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionGeneral description Electre Tri variant Preference elicitation <strong>Method</strong> summaryComparing alternatives <strong>to</strong> categories (global)Is an alternative a good enough for a category C h ?Sum of weights in favor of a C h∑ ∑ ∑w j = w j = b a,C hw j j =j∈J|g j (a)≥l C hjj∈J|b a,C hj=1j∈J∑j∈Jv a,C hjDecision∑a C h ⇔j∈Jv a,C hj≥λ20 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionGeneral description Electre Tri variant Preference elicitation <strong>Method</strong> summaryWhere are we?We now know how the sorting function proceedsNow: how <strong>to</strong> obtain the preference model?(Reminder) Preference information considered at two levels1 Intrinsic alternatives evaluation: “is it good enough?”2 <strong>Portfolio</strong> evaluation: balance? category size? . . .Parameters <strong>to</strong> be elicitedCategory limits l C hjWeights w jMajority thresholdλ∀j∈ J, C h ∈C∀j∈ J21 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionGeneral description Electre Tri variant Preference elicitation <strong>Method</strong> summaryPreference elicitationIntrinsic alternatives evaluationDM gives holistic preference examplesThe sorting function must match these examplesThese examples constrain the set of possible preference modelsClassical approach (e.g. Mousseau and Słowiński [1998])Recently: implemented as a Mixed Integer Program (MIP) [Meyeret al., 2008]ExampleAlt “Student 1”→C 3 =“Good”!22 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionGeneral description Electre Tri variant Preference elicitation <strong>Method</strong> summaryPreference elicitation about portfolio<strong>Portfolio</strong> evaluationDM gives general category size constraintsThe sorting function must satisfy these constraintsIt constrains the set of possible preference modelsExampleNumber of students “male” in C 3 ≈ number of students “female” in C 323 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionGeneral description Electre Tri variant Preference elicitation <strong>Method</strong> summary<strong>Method</strong> summaryProposed methodIndividual comparison constraints (e.g. examples)<strong>Portfolio</strong> quality constraints (category size)Obtain a preference modelThis preference model defines the sorting functionAlternative useIndividual comparison constraintsObtain a preference modelSort the alternativesTweak the model <strong>to</strong> account for category size constraints24 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionStating the Problem Assignment constraints <strong>Portfolio</strong> constraints MIP <strong>to</strong> be solved1 Introduction2 Problem Formulation3 Mathematical program4 Projects funding5 Green labelling6 Students selection7 Conclusion25 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionStating the Problem Assignment constraints <strong>Portfolio</strong> constraints MIP <strong>to</strong> be solvedProblem variablesPreference variablesCategory limits l C hjWeights w jMajority thresholdλ∀j∈ J, C h ∈C∀j∈ J26 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionStating the Problem Assignment constraints <strong>Portfolio</strong> constraints MIP <strong>to</strong> be solvedComputing the assignmentsDataDM gives a set of example alternatives A ∗ ⊆ A,and assignment examples: a→ C h , a∈ A ∗ , C h ∈C.Sum of weights in favor of a C h∑ ∑ ∑w j = w j = b a,C hw j j =j∈J|g j (a)≥l C hjTechnical variablesj∈J|b a,C hj =1j∈J∑j∈Jv a,C hjb a,C hj(binaries) ∀j∈ J, a∈ A, C h ∈Cv a,C hj(continuous) ∀j∈ J, a∈ A, C h ∈C27 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionStating the Problem Assignment constraints <strong>Portfolio</strong> constraints MIP <strong>to</strong> be solvedComputing the assignments: b a,C hjConstraintsg j (a)−l C hjM< b a,C hj≤ g j(a)−l C hjM+ 1M an arbitrary big value ensuring−1< g j(a)−l C hjguarantee b a,C hjg j (a)−l C hjg j (a)−l C hjg j (a)−l C hjg j (a)−l C hj= 1⇔g j (a)≥l C hj?≥ 0⇒ g j(a)−l C hjM≥ 0⇒0≤ g j(a)−l C hjM< 0⇒−1< g j(a)−l C hjM< 0⇒0< g j(a)−l C hjM+ 1≥1⇒b a,C hjM< 1< 1⇒b a,C hj= 1< 0⇒ g j(a)−l C hjM≤ g j(a)−l C hjM+ 1< b a,C hj+ 1


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionStating the Problem Assignment constraints <strong>Portfolio</strong> constraints MIP <strong>to</strong> be solvedComputing the assignments: v a,C hjConstraints⎧⎪⎨ w j + b a,C hj⎪⎩− 1≤v a,C hj≤ b a,C hj(1)j≤ w j (2)v a,C hguarantee v a,C hjb a,C hjb a,C hj= 0⇔b a,C hj= 0∧v a,C hj= w j ⇔ b a,C hj= 1?= 0⇒w j + b a,C h− 1≤0⇒v a,C h≤ 0 (per (1))jj= 1⇒w j ≤ v a,C hj(using (2))≤ 1 (per (1)), hence w j ≤ v a,C hj≤ w j29 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionStating the Problem Assignment constraints <strong>Portfolio</strong> constraints MIP <strong>to</strong> be solvedAssignment constraintsThe DM wants: a→ C hRemember that:∑a C h ⇔j∈Jv a,C hj≥λConstraints <strong>to</strong> force a→ C h∑j∈J v a,C hj≥λ(hence, a C h )∑j∈J v a,C h+1j


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionStating the Problem Assignment constraints <strong>Portfolio</strong> constraints MIP <strong>to</strong> be solvedMeasuring the category sizeCategory sizen(a, C h ) = 1⇔a→ C h∑a∈A n(a, C h ): number of alternatives in C hWeighted category sizeP(a) the weight of the alternative a in the category sizeconstraintExample: P(a) = 1⇔the student a is male, 0 otherwise∑a∈A n(a, C h )P(a): number of alternatives in C h , weightedExample 2: P(a) = price of some alternative31 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionStating the Problem Assignment constraints <strong>Portfolio</strong> constraints MIP <strong>to</strong> be solvedConstraining the category sizeThe DM wants, for given C h , P, n h , n h :∑n h ≤ n(a, C h )P(a)≤n ha∈ADefine n∑n(a, C h )≤1+ v a,h −λjj∈J∑n(a, C h )


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionStating the Problem Assignment constraints <strong>Portfolio</strong> constraints MIP <strong>to</strong> be solvedDefine n (justification)Define n∑n(a, C h )≤1+ v a,h −λ (3)jj∈J∑n(a, C h )


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionStating the Problem Assignment constraints <strong>Portfolio</strong> constraints MIP <strong>to</strong> be solvedmax s s.t.Variables⎧∑w j = 1⎪⎨j∈J⎪⎩ l C h≤ l C h+1j j⎧l C h,∀j∈ J, Cjh ∈Cw j ,∀j∈ J⎪⎨ λ(binaries)b a,C hjv a,C hj⎪⎩ n(a, C h ) (binaries)⎧(g j (a)−l C h)+εj≤ b a,hjM≤ gj(a)−l ChjM+ 1v a,C h≤ wj j ; b a,C h+ wj j − 1≤v a,C h≤ b a,C hj j∑v a,C h≥λ+s ∀a→ Cj h , h≥ 2j∈J∑v a,C h+1+ s≤λ−ε ∀a→ Cjh , h< kj∈J ⎪⎨ ∑n(a, C h )≤1+ v a,C h−λjj∈J∑n(a, C h )≤1+λ− v a,C h+1−εjj∈J∑n(a, C h ) = 11≤h≤k∑n h ≤ n(a, C h )P(a)≤n h ∀ 〈 〉C h , P, n h , n h⎪⎩a∈A34 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionStating the Problem Assignment constraints <strong>Portfolio</strong> constraints MIP <strong>to</strong> be solvedAll constraints (smaller version)max s s.t.⎧∑w j = 1⎪⎨j∈J⎪⎩ l C h≤ l C h+1j jVariables⎧w j ,∀j∈ J;λ⎪⎨⎪⎩l C h,∀j∈ J, Cjh ∈Cb a,C hj⎧⎪⎨⎪⎩(binaries); v a,C hj;(g j (a)−l C h)+εj≤ b a,hjMv a,C hj∀j∈ J, C h ∈Cn(a, C h ) (binaries),∀j∈ J, a∈ A, C h ∈C≤ gj(a)−l ChjM+ 1 ∀j∈ J, a∈ A, h≥ 2≤ w j ; b a,C h+ wjj − 1≤v a,C h≤ b a,C h∀j∈ J, a∈ A, h≥ 2j j⎧∑v a,C h≥λ+s ∀a→ Cj h , h≥ 2⎪⎨ j∈J∑v a,C h+1+ s≤λ−ε ∀a→ Cjh , h< k⎪⎩j∈J⎧ ∑n(a, C h )≤1+ v a,C h−λ ∀a∈ A, h≥ 2jj∈J∑n(a, C h )≤1+λ− v a,C h+1−ε ∀a∈ A, h≤ k− 1j⎪⎨j∈J∑n(a, C h ) = 1 ∀a∈ A1≤h≤k∑n⎪⎩ h ≤ n(a, C h )P(a)≤n h ∀ 〈 〉C h , P, n h , n ha∈A35 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionStating the Problem Assignment constraints <strong>Portfolio</strong> constraints MIP <strong>to</strong> be solvedObjectiveOur formulation does not impose an objectivePossibly: find a satisfying solution (no objective)Possibly: maximize slack between sum of weights and majoritythresholdOther variants (minimize number of ve<strong>to</strong>es. . . )With slackmax s s.t.⎧∑⎪⎨ j∈J∑⎪⎩j∈Jv a,C hj≥λ + s ∀a→ C h , h≥ 2v a,C h+1j+ s ≤λ−ε ∀a→ C h , h< k36 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionStating the Problem Assignment constraints <strong>Portfolio</strong> constraints MIP <strong>to</strong> be solvedRemarksPossibly many satisfying portfoliosPossibly infeasibleInteractive use: constrain less or furtherOr search for minimum disagreements37 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionThe case First stage Second stage Third stage1 Introduction2 Problem Formulation3 Mathematical program4 Projects funding5 Green labelling6 Students selection7 Conclusion38 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionThe case First stage Second stage Third stageAn illustrative exampleContext100 research proposals: which <strong>to</strong> finance?Criteria: redaction quality; scientific quality; experience of theteam; . . .Attributes: budget; field; . . .redac sci exp . . . budget fieldPrj 1 3 2 5 10k ORPrj 2 5 4 2 13k AIPrj 3 0 5 5 7k AI. . .Prj 100 1 2 4 14k St????C 3 : fund projectC 2 : maybeC 1 : reject39 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionThe case First stage Second stage Third stageFirst stageAssignment examplesDM gives 30 examples of past research proposalsNB: Only criteria matter, not attributesredac sci exp . . .Ex 1 5 1 2Ex 2 5 4 3Ex 3 4 2 1. . .Ex 30 1 3 5C 3 : fund projectC 2 : maybeC 1 : rejectIterationMIP is run, finds preference model matching the examples40 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionThe case First stage Second stage Third stageFirst iteration with the preference modelRun with the first preference modelWe have found a preference modelNow, assign the whole set (100 projects) with this modelpreference modelredac sci exp . . . budget fieldPrj 1 3 2 5 10k ORPrj 2 5 4 2 13k AIPrj 3 0 5 5 7k AI. . .Prj 100 1 2 4 14k StC 3 : fund projectC 2 : maybeC 1 : reject41 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionThe case First stage Second stage Third stageSecond stageSecond stage: the cost constraint at portfolio levelMIP re-run with 30 examples and supplementary constraintAlso assigns the set of 100 projectsDifferent preference model foundResult: 11 projects in C 3 , leading <strong>to</strong> a <strong>to</strong>tal cost below 400BUT unsatisfac<strong>to</strong>ry because of unbalanced domain: the AIdomain has 7 projects; 1 project in the OR domain43 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionThe case First stage Second stage Third stageThird stageThird stage: the balance of domains at portfolio levelConstraint added: the domain OR must have at least 2 projectsin C 3And so on. . . (e.g. obtain a better balance among the originatingcountries)At some point, possibly no solutions any more: problem <strong>to</strong>o muchconstrainedThen inconsistency resolution techniques must be used (seeother application!)44 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionThe caseProcess1 Introduction2 Problem Formulation3 Mathematical program4 Projects funding5 Green labelling6 Students selection7 Conclusion45 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionThe caseProcessContextObjectiveEvaluate ecological quality of consumer productsCriteria: pollutant 1, pollutant 2, road distance, . . .In 5 categories: A+, A, B, C, DWe want at most 5% products in A+, at most 10% in A, . . .Experts assess minimum required <strong>to</strong> reach given quality?ChallengeReasonable category limits and weights difficult <strong>to</strong> assess: not <strong>to</strong>oeasy, not impossible!46 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionThe caseProcessObjectiveWe want bothFix norms by comparing productsassign labels by absolute evaluation (comparison <strong>to</strong> fixed norms)Permits industrials <strong>to</strong> plan achieving a given ecological qualityAvoids badly fixed norms (<strong>to</strong>o easy <strong>to</strong> achieve, e.g.)47 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionThe caseProcessProcessProcessResultsUse a representative set of productsSet category limits and weights using intervalsUse suggested method <strong>to</strong> find adequate preference modelObtain a preference modelPublish these normsYields transparent procedure <strong>to</strong> assess for qualityNo need <strong>to</strong> compare each new product <strong>to</strong> a set of existing onesReasonable norms (reaching category A+ is not easy, norimpossible)48 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContext The preference model <strong>Portfolio</strong> constraints Conflicts1 Introduction2 Problem Formulation3 Mathematical program4 Projects funding5 Green labelling6 Students selection7 Conclusion49 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContext The preference model <strong>Portfolio</strong> constraints ConflictsContextA real case: student selection [Le Cardinal et al., 2011]Students from Ecole Centrale Paris, end of second yearStudents choose 1 major among 9DM: dean of one of these major, IE (Industrial Engineering)Dean wants at most 50 students in IEEach year, more than 50 applicationsStudents who choose IE as major also choose 1 stream among 4Students also choose 1 professional track among 650 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContext The preference model <strong>Portfolio</strong> constraints ConflictsTwo concerns, two stagesFirst stage: Who is good enough?6 criteriagrade point average on 1 st and 2 nd yearmotivationprofessional career planmaturity/personalityGeneral knowledge of Industrial Engineering and its careeropportunitiesSecond stage: Adequate balanceAlso, want adequate balance: gender, track, streamsCourse opens only if at least 10 students (cf. streams)Students should be well distributed among professional tracks51 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContext The preference model <strong>Portfolio</strong> constraints ConflictsProblem definitionWhen C 3 + C 4 ≤ 50: may admit less than 50 students. . . or add those from C 2Goal: select at most 50 students from 76 applications (this year)1 st y 2 nd y moti proj pers knoStudent 1 12 13 4 4 4 4Student 2 14 14 1 1 2 1Student 3 14 14 3 2 2 2. . .Student 76 11 12 3 2 1 4????C 4 : acceptC 3 : sufficientC 2 : insufficientC 1 : reject52 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContext The preference model <strong>Portfolio</strong> constraints ConflictsDetermining the preference modelDirect assessments of category limitsTo determine weights: 5 examples“second-year grade not more important than motivation”NB: ve<strong>to</strong>es omitted here1 st y 2 nd y moti proj pers knoEx 1 13 13 3 2 3 2Ex 2 13 13 4 3 4 3Ex 3 12 14 5 5 5 4Ex 4 12 13 4 4 3 3Ex 5 12 14 2 2 3 3C 4 : acceptC 3 : sufficientC 2 : insufficientC 1 : rejectcategory limitspreference modelw 2 ≤ w 453 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContext The preference model <strong>Portfolio</strong> constraints ConflictsAssignmentsWith the resulting preference model: assign the 76 studentspreference model1 st y 2 nd y moti proj pers knoStudent 1 12 13 4 4 4 4Student 2 14 14 1 1 2 1Student 3 14 14 3 2 2 2. . .Student 76 11 12 3 2 1 4C 4 : 29C 3 : 27C 2 : 4C 1 : 1654 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContext The preference model <strong>Portfolio</strong> constraints ConflictsAssignments (analyze)AnalyzeAssigns 29, 27, 4, 16 students <strong>to</strong> C 4 , C 3 , C 2 , C 129+27 = 56 students in C 4 ∪ C 3 : <strong>to</strong>o many18 girls among these 56 students: <strong>to</strong>o fewDistribution among the streams:〈19, 14, 12, 11〉, OK (all > 10)Among the professional tracks:〈0, 13, 26, 0, 8, 9〉, <strong>to</strong>o unbalancedProceed furtherDecision: keep the 29 admitted students (those in C 4 )New task: select at most 21 students among 27 students in C 3Choose the one giving the best portfolio quality!55 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContext The preference model <strong>Portfolio</strong> constraints Conflicts<strong>Portfolio</strong> constraintsStudents27 students in C 3 : x 1 ,...,x 27 binariesx i = 1⇔the student i is selected∑i x i ≤ 21 (relax: 22, 23, etc)Equilibrate streamsNb students in each stream≥10Define P s (i), i the student, s the stream (1 <strong>to</strong> 4)P s (i) = 1⇔student i has chosen stream jn s (C 4 ) = nb students in C 4 who choose the stream j∀s : ∑ i x i P s (i)+n s (C 4 )≥10 (relax: 9, 8, etc)Nb students in each professional track≤20∀t : ∑ i x i P t (i)+n t (C 4 )≤20 (relax: 21, 22, etc)56 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContext The preference model <strong>Portfolio</strong> constraints Conflicts<strong>Portfolio</strong> constraints (2)Gender balanceP girl (i) = 1⇔the student i is a girlnb girls between 20 and 30 (on the <strong>to</strong>tal of 50)n girls (C 4 ) = nb girls in C 420≤ ∑ i x i P girl (i)+n girls (C 4 )≤30relax: 19≤...≤31, 18≤...≤32,...57 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContext The preference model <strong>Portfolio</strong> constraints ConflictsAll portfolio constraintsVariablesx 1 ,...,x 27 , binariesConstraints⎧∑x i ≤ 21i∑x i P s (i)+n s (C 4 )≥10⎪⎨ i∑x i P t (i)+n t (C 4 )≤20i∑20≤ x ⎪⎩i P girl (i)+n girls (C 4 )≤30i∀s∈ streams∀t∈ professional tracks58 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContext The preference model <strong>Portfolio</strong> constraints ConflictsAll portfolio constraintsVariablesx 1 ,...,x 27 , binariesConstraints⎧∑x i ≤ 21i∑x i P s (i)+n s (C 4 )≥10⎪⎨ i∑x i P t (i)+n t (C 4 )≤20i∑20≤ x ⎪⎩i P girl (i)+n girls (C 4 )≤30i∀s∈ streams∀t∈ professional tracksNo satisfying solution!58 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContext The preference model <strong>Portfolio</strong> constraints ConflictsConflict resolutionImpossible <strong>to</strong> satisfy all the constraintsAdd relaxed constraintsSearch solutions with least possible number of disabledconstraintsAdd d c variables, binaries, for each constraint cWhen d c = 1, constraint is disabledMinimize ∑ c d c59 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContext The preference model <strong>Portfolio</strong> constraints Conflictsmin ∑ c d c s.t.⎧∑x i ≤ 21+Md 1i∑⎪⎨ x i ≤ 22+Md 2i∑x i ≤ 23+Md 3⎪⎩iVariables⎧⎪⎨ x 1 ,..., x 27⎪⎩ d c ,∀c,binaries⎧∑x i P s (i)+n s (C 4 )+Md 4 ≥ 10,∀si∑x i P s (i)+n s (C 4 )+Md 5 ≥ 9,∀si∑x i P s (i)+n s (C 4 )+Md 6 ≥ 8,∀si∑x i P t (i)+n t (C 4 )≤20+Md 7 ,∀ti⎪⎨ ∑x i P t (i)+n t (C 4 )≤21+Md 8 ,∀ti∑x i P t (i)+n t (C 4 )≤22+Md 9 ,∀ti∑20−Md 10 ≤ x i P girl (i)+n girls (C 4 )≤30+Md 10i∑19−Md 11 ≤ x i P girl (i)+n girls (C 4 )≤31+Md 11i⎪⎩...60 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionContext The preference model <strong>Portfolio</strong> constraints ConflictsResultsFour ex-æquo best possibilitiesOnly 15 girls (wanted≥20), all other constraints satisfiedOnly 16 girls, 1 prof. track has 21 students (wanted≤20)Only 17 girls, 1 prof. track has 22 studentsOnly 18 girls, 1 prof. track has 23 studentsFinal choiceThe DM may observe the different ways <strong>to</strong> resolve the conflicts andchoose its preferred one.61 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection Conclusion1 Introduction2 Problem Formulation3 Mathematical program4 Projects funding5 Green labelling6 Students selection7 Conclusion62 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionOur contribution<strong>Method</strong> featuresUses natural preference statementsAt both individual level and portfolio levelCompare alternatives <strong>to</strong> norms. . . yet satisfy portfolio constraints<strong>Sorting</strong> decision easily explained63 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionFuture researchRobust recommendation as a result of incomplete preferenceinformationModel balance as objectivesDecision model (even) easier <strong>to</strong> explainMore applications64 / 65


Introduction Problem Formulation Mathematical program Projects funding Green labelling Students selection ConclusionThank you for your attention!65 / 65


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