[PPT] - Balanced portfolio selection Olivier Cailloux Vincent Mousseau Jun PowerPoint Presentation

SLIDE 1

. . . . . . .

Balanced portfolio selection

Olivier Cailloux Vincent Mousseau Jun Zheng

ILLC - Universiteit van Amsterdam

October 7, 2013

SLIDE 2

Introduction Problem Formulation Mathematical program Green labelling Conclusion

Outline

. .

1

Introduction . .

2

Problem Formulation . .

3

Mathematical program . .

4

Green labelling .

5

Conclusion

2 / 47

SLIDE 3

Introduction Problem Formulation Mathematical program Green labelling Conclusion Context Literature review

Outline

. .

1

Introduction . .

2

Problem Formulation . .

3

Mathematical program . .

4

Green labelling .

5

Conclusion

3 / 47

SLIDE 4

Introduction Problem Formulation Mathematical program Green labelling Conclusion Context Literature review

A few examples

.

Enroll students in a university

. . . . . Select the “best” 𝑦 students among the candidates Possibly select less than 𝑦 if not enough good students Also: obtain a good balance (e.g. gender balance) “I want those highly motivated and ≥ 8 in literature”? motivation math. literature … gender Student 1 good 9 7 … female Student 2 good 5 8 … male Student 3 average 6 4 … female Student 4 bad 7 … male Student 5 good 4 9 … male …

4 / 47

SLIDE 5

Introduction Problem Formulation Mathematical program Green labelling Conclusion Context Literature review

A few examples

.

Allocate budget to research proposals

. . . . . Fund the “best” proposals Evaluation on multiple criteria: redaction quality; scientific quality; experience of the team; … Only fund projects if they are good enough! with a good balance between risk, field… . . . redac . sci . exp . … . budget . field . . Prj 1 . 3 . . 2 . . 5 . . . . 10k . . OR . . Prj 2 . 5 . . 4 . . 2 . . . . 13k . . AI . . Prj 3 . . . 5 . . 5 . . . . 7k . . AI . . … . . . . . . . . . . . . . 𝐷: fund project . 𝐷: maybe . 𝐷: reject . ? . ? . ?

5 / 47

SLIDE 6

Introduction Problem Formulation Mathematical program Green labelling Conclusion Context Literature review

A few examples

.

Reward improvements (sports team)

. . . . . I want to reward some team members Accounting for their improvements on multiple criteria Those that improved a lot Only the best of those (at most 5%) Also reward those that improved moderately (at most 10% of them) → comparisons, as in a ranking → …however not exactly a ranking .

Category sizes

. . . . . Category sizes increase (5% best, 10% moderate, …)

6 / 47

SLIDE 7

Introduction Problem Formulation Mathematical program Green labelling Conclusion Context Literature review

A few examples

.

Partition in similar classes

. . . . . Split students in language classes Classes should be of approximately the same size Classes should be of homogeneous level

7 / 47

SLIDE 8

Introduction Problem Formulation Mathematical program Green labelling Conclusion Context Literature review

A few examples

.

Yes / No / I don’t know

. . . . . Sort old items in a shop (or in your attic) OK category: could still be sold (or kept) KO category: goes to trash Intermediary cases: ask the boss (or your wife or husband) Limited patience of the boss/wife/husband limit the number of such items

8 / 47

SLIDE 9

Introduction Problem Formulation Mathematical program Green labelling Conclusion Context Literature review

Old problem revisited

This is a variant of an old problem! Classical portfolio selection: select the best subset of items satisfying some constraints (knapsack) Multicriteria portfolio problem: value is given by multiple criteria Usual approaches use one value per portfolio (no flexibility!) Only two categories Only relative value considered No intuitive explanation of the selection . . . redac . sci . exp . … . budget . . field . . Prj 1 . . 3 . . 2 . . 5 . . . . 10k . . OR . . Prj 2 . . 5 . . 4 . . 2 . . . . 13k . . AI . . Prj 3 . . . . 5 . . 5 . . . . 7k . . AI . . … . . . . . . . . . . . . . 𝐷: select . 𝐷: reject . ? . ? . ?

9 / 47

SLIDE 10

Introduction Problem Formulation Mathematical program Green labelling Conclusion Context Literature review

Excluding with constraints

Selecting a Portfolio of Solar Energy Projects Using Multiattribute Preference Theory [Golabi et al., 1981] Goal is simply to choose one portfolio A portfolio has a value (using value theory) Screen-out insufficiently good alternatives Screen-out portfolio with unsatisfactory balance Choose best valued portfolio among remaining ones No explanation about why an alternative is selected

10 / 47

SLIDE 11

Introduction Problem Formulation Mathematical program Green labelling Conclusion Context Literature review

A balance model

Evaluating the whole portfolio: a balance model to measure the distribution of specific attributes [Farquhar and Rao, 1976] Choose the highest valued portfolio According to balance on attributes Attributes evaluated on the same scale (same distance measure) Preference model: weights of the attributes No absolute evaluation Resulting model does not compare to norms Multiple categories not supported

11 / 47

SLIDE 12

Introduction Problem Formulation Mathematical program Green labelling Conclusion Context Literature review

Robust portfolio selection

Combining preference programming with portfolio selection [Liesiö et al., 2007, Liesiö et al., 2008] Robust portfolio selection Screen-out portfolios based on constraints and robust decisions No absolute comparisons Selection not easily explained

12 / 47

SLIDE 13

Introduction Problem Formulation Mathematical program Green labelling Conclusion General description Électre Tri variant Preference elicitation Method summary

Outline

. .

1

Introduction . .

2

Problem Formulation . .

3

Mathematical program . .

4

Green labelling .

5

Conclusion

13 / 47

SLIDE 14

Introduction Problem Formulation Mathematical program Green labelling Conclusion General description Électre Tri variant Preference elicitation Method summary

Decision situation

.

Decision situation

. . . . . A set of alternatives To be sorted in ordered categories Evaluated using criteria and attributes (Possibly) A decision to be repeated . . Alt 1 . Alt 2 . Alt 3 . Alt 4 . 𝐷: “Good” . 𝐷: “Average” . 𝐷: “Bad”

14 / 47

SLIDE 15

Introduction Problem Formulation Mathematical program Green labelling Conclusion General description Électre Tri variant Preference elicitation Method summary

Objectives

.

Objective of our method

. . . . . Obtain a sorting function .

Desirable features of the sorting function

. . . . . Comparison of alternatives to norms Judge alternatives on the same ground Select a portfolio using:

the quality of individuals the overall portfolio quality (e.g. good balance?)

Easily explain assignments

15 / 47

SLIDE 16

Introduction Problem Formulation Mathematical program Green labelling Conclusion General description Électre Tri variant Preference elicitation Method summary

.

Principle of the sorting function

. . . . . Uses the alternatives performances on the criteria The sorting function reflects the Decision Maker (DM) preferences thanks to a preference model . . preference model . Alt 1 . Alt 2 . … . 𝐷: Good . 𝐷: Average . 𝐷: Bad .

Preference information considered at two levels

. . . . . .

1 Intrinsic alternatives evaluations: “is it good enough?”

. .

2 Portfolio evaluations: balance? category size? … 16 / 47

SLIDE 17

Introduction Problem Formulation Mathematical program Green labelling Conclusion General description Électre Tri variant Preference elicitation Method summary

Explanation of the method

. . preference model . Alt 1 . Alt 2 . … . 𝐷: Good . 𝐷: Average . 𝐷: Bad .

1 Assuming the preference model is known, explain how the sorting

function proceeds . .

2 Then: explain how to obtain the preference model 17 / 47

SLIDE 18

Introduction Problem Formulation Mathematical program Green labelling Conclusion General description Électre Tri variant Preference elicitation Method summary

Notations

.

Objective data

. . . . . Alternatives: 𝒝 (one alternative: 𝑏) Criteria: 𝒦 (one criterion: 𝑘) Categories: 𝒟 (one category: 𝐷, 1 ≤ ℎ ≤ 𝑙) Performance of 𝑏 on criterion 𝑘: 𝑕(𝑏), 𝑕 ∶ 𝒝 → 𝑌 ⪰, ≻ defined on 𝑌 (here: ≥, >) .

Preference parameters

. . . . . Lower limit of category 𝐷 on criterion 𝑘: 𝑚

∈ 𝑌

Weight of criterion 𝑘: 𝑥 ∈ ℝ Majority threshold: 𝜇 ∈ ℝ

18 / 47

SLIDE 19

Introduction Problem Formulation Mathematical program Green labelling Conclusion General description Électre Tri variant Preference elicitation Method summary

Sorting problem: a variant of Électre Tri

.

Preference model: 𝜕 = ⟨𝑀, 𝑋, 𝜇⟩

. . . . . Category limits 𝑀 = 𝑚, 𝐷 ∈ 𝒟 ∖ 𝐷: determine when the alternative is good enough on a criterion; weights 𝑋 = 𝑥, 𝑘 ∈ 𝒦, and a majority threshold 𝜇: determine when the alternative is globally good enough. .

Alternatives 𝒝

. . . 𝑘 𝑘 𝑘 𝑏 3 1 1 𝑏 5 3 3 𝑏 5 1 𝑏 2 2 .

Cat. limits 𝑀

. . 𝑘 𝑘 𝑘 𝐷: Good 𝑚 4 4 3 𝐷: Average 𝑚 3 3 2 𝐷: Bad .

Weights 𝑋, 𝜇

. . . . . 𝑘 𝑘 𝑘 𝑋 0.2 0.6 0.2 𝜇 = 0.8

19 / 47

SLIDE 20

Introduction Problem Formulation Mathematical program Green labelling Conclusion General description Électre Tri variant Preference elicitation Method summary

Sorting problem: a variant of Électre Tri

𝑘 in favor of 𝑏 reaching 𝐷 iff 𝑕(𝑏) ≥ 𝑚

.

𝑏 may reach at least 𝐷 iff ∑ in favor 𝑥 ≥ 𝜇 (𝐷 ≠ 𝐷). Thus, 𝑏 sorted into the best category s.t.

𝑥 ≥ 𝜇.

.

Alternatives 𝒝

. . . 𝑘 𝑘 𝑘 𝑏 3 1 1 𝑏 5 3 3 𝑏 5 1 𝑏 2 2 .

Cat. limits 𝑀

. . 𝑘 𝑘 𝑘 𝐷: Good 𝑚 4 4 3 𝐷: Average 𝑚 3 3 2 𝐷: Bad .

Weights 𝑋, 𝜇

. . . . . 𝑘 𝑘 𝑘 𝑋 0.2 0.6 0.2 𝜇 = 0.8

19 / 47

SLIDE 21

Introduction Problem Formulation Mathematical program Green labelling Conclusion General description Électre Tri variant Preference elicitation Method summary

Where are we?

We now know how the sorting function proceeds Now: how to obtain the preference model? .

(Reminder) Preference information considered at two levels

. . . . . .

1 Intrinsic alternatives evaluation: “is it good enough?”

. .

2 Portfolio evaluation: balance? category size? …

.

Parameters to be elicited

. . . . . Category limits 𝑚

∀𝑘 ∈ 𝒦, 𝐷 ∈ 𝒟

Weights 𝑥 ∀𝑘 ∈ 𝒦 Majority threshold 𝜇

20 / 47

SLIDE 22

Introduction Problem Formulation Mathematical program Green labelling Conclusion General description Électre Tri variant Preference elicitation Method summary

Preference elicitation

.

Intrinsic alternatives evaluation

. . . . .

DM gives holistic preference examples

The sorting function must match these examples These examples constrain the set of possible preference models Classical approach [Mousseau and Słowiński, 1998] Recently: implemented as a Mixed Integer Program (MIP) [Meyer et al., 2008] .

Example

. . . . . Alt “Student 1” → 𝐷=“Good”

21 / 47

SLIDE 23

Introduction Problem Formulation Mathematical program Green labelling Conclusion General description Électre Tri variant Preference elicitation Method summary

Preference elicitation about portfolio

.

Portfolio evaluation

. . . . .

DM gives general category size constraints

The sorting function must satisfy these constraints It constrains the set of possible preference models .

Examples

. . . . . Number of students in 𝐷 ≤ 25 Number of students “male” in 𝐷 ≈ number of students “female” in 𝐷

22 / 47

SLIDE 24

Introduction Problem Formulation Mathematical program Green labelling Conclusion General description Électre Tri variant Preference elicitation Method summary

Method summary

.

Proposed method

. . . . . Individual comparison constraints (e.g. examples) Portfolio quality constraints (category size) Obtain a preference model This preference model defines the sorting function .

Alternative use

. . . . . Individual comparison constraints Obtain a preference model Sort the alternatives Tweak the model to account for category size constraints

23 / 47

SLIDE 25

Introduction Problem Formulation Mathematical program Green labelling Conclusion Stating the Problem Assignment constraints Portfolio constraints MIP to be solved

Outline

. .

1

Introduction . .

2

Problem Formulation . .

3

Mathematical program . .

4

Green labelling .

5

Conclusion

24 / 47

SLIDE 26

Introduction Problem Formulation Mathematical program Green labelling Conclusion Stating the Problem Assignment constraints Portfolio constraints MIP to be solved

Problem variables

We want to find a preference model capturing: what DM considers a “good” / “average” / “bad” alternative; how balanced the categories should be. .

Preference variables

. . . . . Category limits 𝑚

∀𝑘 ∈ 𝒦, 𝐷 ∈ 𝒟

Weights 𝑥 ∀𝑘 ∈ 𝒦 Majority threshold 𝜇 Find values such that sorting 𝐵 into 𝒟 yields an appropriate partition.

25 / 47

SLIDE 27

Introduction Problem Formulation Mathematical program Green labelling Conclusion Stating the Problem Assignment constraints Portfolio constraints MIP to be solved

From constraints to inequalities

We want to find 𝑚

, 𝑥, 𝜇 satisfying constraints.

E.g. 𝑏 → 𝐷. 𝑏 having performances (𝑕), 𝑘 ∈ 𝒦 Hence: ∑

𝑥 ≥ 𝜇

∑

𝑥 < 𝜇

26 / 47

SLIDE 28

Introduction Problem Formulation Mathematical program Green labelling Conclusion Stating the Problem Assignment constraints Portfolio constraints MIP to be solved

Using mathematical programming

In order to keep decent performances, we want to use only linear inequalities: 1 ≤ 𝑚

≤ 3

𝜇 ≤ 𝑥 + 𝑥 NOT 𝑚

≤
We will however use some binary variables.

Mixed Integer Program (MIP) not usually solvable in polynomial time but solvable in most of the cases of interest to us.

27 / 47

SLIDE 29

Introduction Problem Formulation Mathematical program Green labelling Conclusion Stating the Problem Assignment constraints Portfolio constraints MIP to be solved

Using linear inequalities

We want to express this using linear constraints:

𝑥 ≥ 𝜇.

28 / 47

SLIDE 30

Introduction Problem Formulation Mathematical program Green labelling Conclusion Stating the Problem Assignment constraints Portfolio constraints MIP to be solved

Comparing alternatives to categories (criterion 𝑘)

Point of view of criterion 𝑘: Is an alternative 𝑏 good enough for a category 𝐷? .

Arguments in favor of 𝑏 ≽ 𝐷 (binary)

. . . . . 𝑐

= 1 ⇔ 𝑕(𝑏) ≥ 𝑚
𝑐
= 0 ⇔ 𝑕(𝑏) < 𝑚
.

Arguments in favor of 𝑏 ≽ 𝐷 (valued)

. . . . . 𝑤

= 𝑥 ⇔ 𝑕(𝑏) ≥ 𝑚
𝑤
= 0 ⇔ 𝑕(𝑏) < 𝑚
29 / 47

SLIDE 31

Introduction Problem Formulation Mathematical program Green labelling Conclusion Stating the Problem Assignment constraints Portfolio constraints MIP to be solved

Comparing alternatives to categories (global)

Is an alternative 𝑏 good enough for a category 𝐷? .

Sum of weights in favor of 𝑏 ≽ 𝐷

. . . . .

𝑥 =
𝑥 =
𝑐
𝑥 =
𝑤
.

Decision

. . . . . 𝑏 ≽ 𝐷 ⇔

𝑤
≥ 𝜇

30 / 47

SLIDE 32

Introduction Problem Formulation Mathematical program Green labelling Conclusion Stating the Problem Assignment constraints Portfolio constraints MIP to be solved

Computing the assignments: 𝑐𝑏,ℎ

𝑘

.

Constraints

. . . . . 𝑕(𝑏) − 𝑚

𝑁

< 𝑐

≤

𝑕(𝑏) − 𝑚

𝑁

+ 1 𝑁 an arbitrary big value ensuring −1 <

< 1

guarantee 𝑐

= 1 ⇔ 𝑕(𝑏) ≥ 𝑚

:

𝑕(𝑏) − 𝑚

≥ 0 ⇒
+ 1 ≥ 1 ⇒ 𝑐
≤
+ 1

𝑕(𝑏) − 𝑚

≥ 0 ⇒ 0 ≤
< 1 ⇒ 𝑐
= 1

𝑕(𝑏) − 𝑚

< 0 ⇒ −1 <
< 0 ⇒
< 𝑐
𝑕(𝑏) − 𝑚
< 0 ⇒ 0 <
+ 1 < 1 ⇒ 𝑐
= 0

31 / 47

SLIDE 33

Introduction Problem Formulation Mathematical program Green labelling Conclusion Stating the Problem Assignment constraints Portfolio constraints MIP to be solved

Computing the assignments: 𝑤𝑏,ℎ

𝑘

.

Constraints

. . . . . ⎧ ⎨ ⎩ 𝑥 + 𝑐

− 1 ≤ 𝑤
≤ 𝑐
𝑤
≤ 𝑥

(1) (2) guarantee 𝑤

= 0 ⇔ 𝑐
= 0 ∧ 𝑤
= 𝑥 ⇔ 𝑐
= 1:

𝑐

= 0 ⇒ 𝑥 + 𝑐
− 1 ≤ 0 ⇒ 𝑤
≤ 0 (per (1))

𝑐

= 1 ⇒ 𝑥 ≤ 𝑤
≤ 1 (per (1)), hence 𝑥 ≤ 𝑤
≤ 𝑥

(using (2))

32 / 47

SLIDE 34

Introduction Problem Formulation Mathematical program Green labelling Conclusion Stating the Problem Assignment constraints Portfolio constraints MIP to be solved

Assignment constraints

The DM wants: 𝑏 → 𝐷 Remember that: 𝑏 ≽ 𝐷 ⇔

𝑤
≥ 𝜇

.

Constraints to force 𝑏 → 𝐷

. . . . . ∑ 𝑤

≥ 𝜇 (hence, 𝑏 ≽ 𝐷)

∑ 𝑤

< 𝜇 (hence, 𝑏 ≺ 𝐷)

33 / 47

SLIDE 35

Introduction Problem Formulation Mathematical program Green labelling Conclusion Stating the Problem Assignment constraints Portfolio constraints MIP to be solved

Measuring the category size

.

Category size

. . . . . 𝑜 = 1 ⇔ 𝑏 → 𝐷 ∑ 𝑜: number of alternatives in 𝐷 .

Weighted category size

. . . . . 𝑄(𝑏) the weight of the alternative 𝑏 in the category size constraint Example: 𝑄(𝑏) = 1 ⇔ the student 𝑏 is male, 0 otherwise ∑ 𝑜𝑄(𝑏): number of alternatives in 𝐷, weighted Example 2: 𝑄(𝑏) = price of some alternative

34 / 47

SLIDE 36

Introduction Problem Formulation Mathematical program Green labelling Conclusion Stating the Problem Assignment constraints Portfolio constraints MIP to be solved

Constraining the category size

The DM wants, for given 𝐷, 𝑄, 𝑜, 𝑜: 𝑜 ≤

𝑜𝑄(𝑏) ≤ 𝑜

.

Define 𝑜

. . . . . 𝑜 ≤ 1 +

𝑤
− 𝜇

𝑜 < 1 + 𝜇 −

𝑤
∀𝑏 ∈ 𝐵 ∶
𝑜 = 1

35 / 47

SLIDE 37

Introduction Problem Formulation Mathematical program Green labelling Conclusion Stating the Problem Assignment constraints Portfolio constraints MIP to be solved

Define 𝑜 (justification)

.

Define 𝑜

. . . . . 𝑜 ≤ 1 +

𝑤
− 𝜇

(3) 𝑜 < 1 + 𝜇 −

𝑤
(4)

∀𝑏 ∈ 𝐵 ∶

𝑜 = 1

(5) We have 𝑜 = 1 ⇒ 0 ≤ ∑ 𝑤

− 𝜇 ⇒ 𝑏 ≽ 𝐷 (3)

We have 𝑜 = 1 ⇒ 0 < 𝜇 − ∑ 𝑤

⇒ 𝑏 ≺ 𝐷 (4)

We have 𝑜 = 0 ⇒ ∃ℎ ∣ 𝑜 = 1 (5), hence 𝑏 ↛ 𝐷

36 / 47

SLIDE 38

Introduction Problem Formulation Mathematical program Green labelling Conclusion Stating the Problem Assignment constraints Portfolio constraints MIP to be solved

max 𝑡 s.t. ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

𝑥 = 1

𝑚

≤ 𝑚
.

Variables

. . ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝑚

, ∀𝑘 ∈ 𝒦, 𝐷 ∈ 𝒟

𝑥, ∀𝑘 ∈ 𝒦 𝜇 𝑐

(binaries)

𝑤

𝑜 (binaries)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (𝑕(𝑏) − 𝑚

) + 𝜁

𝑁 ≤ 𝑐

≤

𝑕(𝑏) − 𝑚

𝑁

+ 1 𝑤

≤ 𝑥; 𝑐
+ 𝑥 − 1 ≤ 𝑤
≤ 𝑐
𝑤
≥ 𝜇 + 𝑡

∀𝑏 → 𝐷, ℎ ≥ 2

𝑤
+ 𝑡 ≤ 𝜇 − 𝜁

∀𝑏 → 𝐷, ℎ < 𝑙 𝑜 ≤ 1 +

𝑤
− 𝜇

𝑜 ≤ 1 + 𝜇 −

𝑤
− 𝜁
𝑜 = 1

𝑜 ≤

𝑜𝑄(𝑏) ≤ 𝑜

∀ 𝐷, 𝑄, 𝑜, 𝑜

37 / 47

SLIDE 39

Introduction Problem Formulation Mathematical program Green labelling Conclusion Stating the Problem Assignment constraints Portfolio constraints MIP to be solved

Objective

Our formulation does not impose an objective Possibly: find a satisfying solution (no objective) Possibly: maximize slack between sum of weights and majority threshold Other variants (minimize number of vetoes…) .

With slack

. . . . . max 𝑡 s.t. ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

𝑤
≥ 𝜇 + 𝑡

∀𝑏 → 𝐷, ℎ ≥ 2

𝑤
+ 𝑡 ≤ 𝜇 − 𝜁

∀𝑏 → 𝐷, ℎ < 𝑙

38 / 47

SLIDE 40

Introduction Problem Formulation Mathematical program Green labelling Conclusion Stating the Problem Assignment constraints Portfolio constraints MIP to be solved

In practice

.

Interactive use

. . . . . Possibly many satisfying portfolios Possibly infeasible Interactive use: constrain less or further Or search for minimal relaxation of the constraints .

Result

. . . . . Publish the preference model 𝑚

, 𝑥, 𝜇

Explains assignments of each alternative If stable distribution: can reuse it later

39 / 47

SLIDE 41

Introduction Problem Formulation Mathematical program Green labelling Conclusion The case Process

Outline

. .

1

Introduction . .

2

Problem Formulation . .

3

Mathematical program . .

4

Green labelling .

5

Conclusion

40 / 47

SLIDE 42

Introduction Problem Formulation Mathematical program Green labelling Conclusion The case Process

Context

.

Objective

. . . . . Evaluate ecological quality of consumer products Criteria: pollutant 1, pollutant 2, road distance, … In 5 categories: A+, A, B, C, D We want at most 5% products in A+, at most 10% in A, … .

Challenge

. . . . . Reasonable category limits and weights difficult to assess: not too easy, not impossible!

41 / 47

SLIDE 43

Introduction Problem Formulation Mathematical program Green labelling Conclusion The case Process

Objective

.

We want to

. . . . . fix norms by comparing products assign labels by absolute evaluation (comparison to fixed norms) permits industrials to plan achieving a given ecological quality avoids badly fixed norms (too easy to achieve, e.g.)

42 / 47

SLIDE 44

Introduction Problem Formulation Mathematical program Green labelling Conclusion The case Process

Process

.

Process

. . . . . Use a representative set of products Set category limits and weights using intervals Use suggested method to find adequate preference model .

Results

. . . . . Obtain a preference model Publish these norms Yields transparent procedure to assess for quality No need to compare each new product to a set of existing ones Reasonable norms (reaching category A+ is not easy, nor impossible)

43 / 47

SLIDE 45

Introduction Problem Formulation Mathematical program Green labelling Conclusion

Outline

. .

1

Introduction . .

2

Problem Formulation . .

3

Mathematical program . .

4

Green labelling .

5

Conclusion

44 / 47

SLIDE 46

Introduction Problem Formulation Mathematical program Green labelling Conclusion

A method for balanced portfolios

.

Method features

. . . . . Advantage of affirmative action: balanced categories …with fair treatment! Uses natural preference statements At both individual level and portfolio level Compare alternatives to norms …yet satisfy portfolio constraints Sorting decision easily explained

45 / 47

SLIDE 47

Introduction Problem Formulation Mathematical program Green labelling Conclusion

Future research

Robust recommendation as a result of incomplete preference information Compare formally to existing methods Model balance as objectives Decision model (even) easier to explain Efficient solving

46 / 47

SLIDE 48

Introduction Problem Formulation Mathematical program Green labelling Conclusion

Thank you for your attention!

Check next page for bibliography.

47 / 47

SLIDE 49

Bibliography Projects funding Students selection

Outline

. .

6

Bibliography . .

7

Projects funding . .

8

Students selection

1 / 24

SLIDE 50

Bibliography Projects funding Students selection

Bibliography I

Farquhar, P. H. and Rao, V. R. (1976). A balance model for evaluating subsets of multiattributed items. Management Science, 22(5):528–539. Golabi, K., Kirkwood, C. W., and Sicherman, A. (1981). Selecting a portfolio of solar energy projects using multiattribute preference theory. Management Science, 27(2):174–189. Le Cardinal, J., Mousseau, V., and Zheng, J. (2011). Multiple criteria sorting: An application to student selection. In Salo, A., Keisler, J., and Morton, A., editors, Portfolio Decision

Analysis. Springer-Verlag New York Inc.

2 / 24

SLIDE 51

Bibliography Projects funding Students selection

Bibliography II

Liesiö, J., Mild, P., and Salo, A. (2007). Preference programming for robust portfolio modeling and project selection. European Journal of Operational Research, 181(3):1488 – 1505. Liesiö, J., Mild, P., and Salo, A. (2008). Robust portfolio modeling with incomplete cost information and project interdependencies. European Journal of Operational Research, 190(3):679–695. Meyer, P., Marichal, J., and Bisdorff, R. (2008). Disaggregation of bipolar-valued outranking relations. In Le Thi, H. A., Bouvry, P., and Pham Dinh, T., editors, Proc.

f MCO’08 conference, pages 204–213, Metz, France. Springer.

3 / 24

SLIDE 52

Bibliography Projects funding Students selection

Bibliography III

Mousseau, V. and Słowiński, R. (1998). Inferring an ELECTRE TRI model from assignment examples. Journal of Global Optimization, 12(2):157–174.

4 / 24

SLIDE 53

Bibliography Projects funding Students selection The case First stage Second stage Third stage

Outline

. .

6

Bibliography . .

7

Projects funding . .

8

Students selection

5 / 24

SLIDE 54

Bibliography Projects funding Students selection The case First stage Second stage Third stage

An illustrative example

.

Context

. . . . . 100 research proposals: which to finance? Criteria: redaction quality; scientific quality; experience of the team; … Attributes: budget; field; … . . . . redac . sci . exp . … . budget . field . . Prj 1 . 3 . . 2 . . 5 . . . . 10k . . OR . . Prj 2 . 5 . . 4 . . 2 . . . . 13k . . AI . . Prj 3 . . . 5 . . 5 . . . . 7k . . AI . . … . . . . . . . . . . . . . . Prj 100 . 1 . . 2 . . 4 . . . . 14k . . St . 𝐷: fund project . 𝐷: maybe . 𝐷: reject . ? . ? . ? . ?

6 / 24

SLIDE 55

Bibliography Projects funding Students selection The case First stage Second stage Third stage

First stage

.

Assignment examples

. . . . .

DM gives 30 examples of past research proposals

NB: Only criteria matter, not attributes . . . redac . sci . exp . … . . Ex 1 . 5 . . 1 . . 2 . . . . Ex 2 . 5 . . 4 . . 3 . . . . Ex 3 . 4 . . 2 . . 1 . . . . … . . . . . . . . . . Ex 30 . . 1 . . 3 . . 5 . . . 𝐷: fund project . 𝐷: maybe . 𝐷: reject .

Iteration

. . . . . MIP is run, finds preference model matching the examples

7 / 24

SLIDE 56

Bibliography Projects funding Students selection The case First stage Second stage Third stage

First iteration with the preference model

.

Run with the first preference model

. . . . . We have found a preference model Now, assign the whole set (100 projects) with this model . . . . redac . sci . exp . … . budget . field . . Prj 1 . 3 . . 2 . . 5 . . . . 10k . . OR . . Prj 2 . 5 . . 4 . . 2 . . . . 13k . . AI . . Prj 3 . . . 5 . . 5 . . . . 7k . . AI . . … . . . . . . . . . . . . . . Prj 100 . 1 . . 2 . . 4 . . . . 14k . . St . preference model . 𝐷: fund project . 𝐷: maybe . 𝐷: reject

8 / 24

SLIDE 57

Bibliography Projects funding Students selection The case First stage Second stage Third stage

First iteration with the preference model: results

.

Results with the first preference model

. . . . . 22 projects in category 𝐷: “fund project” Unsatisfactory because the costs of these 22 projects (718 k$) exceed the available budget (400 k$) .

Next iteration

. . . . . We may add the supplementary constraint: sum of the budget of projects in 𝐷 ≤ 400𝑙$

9 / 24

SLIDE 58

Bibliography Projects funding Students selection The case First stage Second stage Third stage

Second stage

.

Second stage: the cost constraint at portfolio level

. . . . . MIP re-run with 30 examples and supplementary constraint Also assigns the set of 100 projects Different preference model found Result: 11 projects in 𝐷, leading to a total cost below 400 BUT unsatisfactory because of unbalanced domain: the AI domain has 7 projects; 1 project in the OR domain

10 / 24

SLIDE 59

Bibliography Projects funding Students selection The case First stage Second stage Third stage

Third stage

.

Third stage: the balance of domains at portfolio level

. . . . . Constraint added: the domain OR must have at least 2 projects in 𝐷 And so on…(e.g. obtain a better balance among the originating countries) At some point, possibly no solutions any more: problem too constrained Then inconsistency resolution techniques must be used (see other application!)

11 / 24

SLIDE 60

Bibliography Projects funding Students selection Context The preference model Portfolio constraints Conflicts

Outline

. .

6

Bibliography . .

7

Projects funding . .

8

Students selection

12 / 24

SLIDE 61

Bibliography Projects funding Students selection Context The preference model Portfolio constraints Conflicts

Context

A real case: student selection [Le Cardinal et al., 2011] Students from Ecole Centrale Paris, end of second year Students choose 1 major among 9

DM: dean of one of these major, IE (Industrial Engineering)

Dean wants at most 50 students in IE Each year, more than 50 applications Students who choose IE as major also choose 1 stream among 4 Students also choose 1 professional track among 6

13 / 24

SLIDE 62

Bibliography Projects funding Students selection Context The preference model Portfolio constraints Conflicts

Two concerns, two stages

.

First stage: Who is good enough?

. . . . . 6 criteria

grade point average on 1st and 2nd year motivation professional career plan maturity/personality general knowledge of Industrial Engineering and its career

pportunities

.

Second stage: Adequate balance

. . . . . Also, want adequate balance: gender, track, streams Course opens only if at least 10 students (cf. streams) Students should be well distributed among professional tracks

14 / 24

SLIDE 63

Bibliography Projects funding Students selection Context The preference model Portfolio constraints Conflicts

Problem definition

When |𝐷| + |𝐷| ≤ 50: may admit less than 50 students …or add those from 𝐷 Goal: select at most 50 students from 76 applications (this year) . . . . 1st y . 2nd y . moti . proj . pers . . kno . . Student 1 . 12 . . 13 . . 4 . . 4 . . 4 . . 4 . . Student 2 . 14 . . 14 . . 1 . . 1 . . 2 . . 1 . . Student 3 . 14 . . 14 . . 3 . . 2 . . 2 . . 2 . . … . . . . . . . . . . . . . . Student 76 . 11 . . 12 . . 3 . . 2 . . 1 . . 4 . 𝐷: accept . 𝐷: sufficient . 𝐷: insufficient . 𝐷: reject . ? . ? . ? . ?

15 / 24

SLIDE 64

Bibliography Projects funding Students selection Context The preference model Portfolio constraints Conflicts

.

Determining the preference model

. . . . . Direct assessments of category limits To determine weights: 5 examples “second-year grade not more important than motivation” NB: vetoes omitted here . . . . 1st y . 2nd y . moti . proj . pers . . kno . . Ex 1 . 13 . . 13 . . 3 . . 2 . . 3 . . 2 . . Ex 2 . 13 . . 13 . . 4 . . 3 . . 4 . . 3 . . Ex 3 . 12 . . 14 . . 5 . . 5 . . 5 . . 4 . . Ex 4 . 12 . . 13 . . 4 . . 4 . . 3 . . 3 . . Ex 5 . 12 . . 14 . . 2 . . 2 . . 3 . . 3 . 𝐷: accept . 𝐷: sufficient . 𝐷: insufficient . 𝐷: reject . preference model . category limits . 𝑥 ≤ 𝑥

16 / 24

SLIDE 65

Bibliography Projects funding Students selection Context The preference model Portfolio constraints Conflicts

Assignments

With the resulting preference model: assign the 76 students . . . . 1st y . 2nd y . moti . proj . pers . . kno . . Student 1 . 12 . . 13 . . 4 . . 4 . . 4 . . 4 . . Student 2 . 14 . . 14 . . 1 . . 1 . . 2 . . 1 . . Student 3 . 14 . . 14 . . 3 . . 2 . . 2 . . 2 . . … . . . . . . . . . . . . . . Student 76 . 11 . . 12 . . 3 . . 2 . . 1 . . 4 . preference model . 𝐷: 29 . 𝐷: 27 . 𝐷: 4 . 𝐷: 16

17 / 24

SLIDE 66

Bibliography Projects funding Students selection Context The preference model Portfolio constraints Conflicts

Assignments (analyze)

.

Analyze

. . . . . Assigns 29, 27, 4, 16 students to 𝐷, 𝐷, 𝐷, 𝐷 29+27 = 56 students in 𝐷 ∪ 𝐷: too many 18 girls among these 56 students: too few Distribution among the streams: ⟨19, 14, 12, 11⟩, OK (all > 10) Among the professional tracks: ⟨0, 13, 26, 0, 8, 9⟩, too unbalanced .

Proceed further

. . . . . Decision: keep the 29 admitted students (those in 𝐷) New task: select at most 21 students among 27 students in 𝐷 Choose the one giving the best portfolio quality!

18 / 24

SLIDE 67

Bibliography Projects funding Students selection Context The preference model Portfolio constraints Conflicts

Portfolio constraints

.

Students

. . . . . 27 students in 𝐷: 𝑦, … , 𝑦 binaries 𝑦 = 1 ⇔ the student 𝑗 is selected ∑ 𝑦 ≤ 21 (relax: 22, 23, etc) .

Equilibrate streams

. . . . . Nb students in each stream ≥ 10

Define 𝑄(𝑗), 𝑗 the student, 𝑡 the stream (1 to 4) 𝑄(𝑗) = 1 ⇔ student 𝑗 has chosen stream 𝑘 𝑜(𝐷) = nb students in 𝐷 who choose the stream 𝑘 ∀𝑡 ∶ ∑ 𝑦𝑄(𝑗) + 𝑜(𝐷) ≥ 10 (relax: 9, 8, etc)

Nb students in each professional track ≤ 20

∀𝑢 ∶ ∑ 𝑦𝑄(𝑗) + 𝑜(𝐷) ≤ 20 (relax: 21, 22, etc)

19 / 24

SLIDE 68

Bibliography Projects funding Students selection Context The preference model Portfolio constraints Conflicts

Portfolio constraints (2)

.

Gender balance

. . . . . 𝑄girl(𝑗) = 1 ⇔ the student 𝑗 is a girl nb girls between 20 and 30 (on the total of 50) 𝑜girls(𝐷) = nb girls in 𝐷 20 ≤ ∑ 𝑦𝑄girl(𝑗) + 𝑜girls(𝐷) ≤ 30 relax: 19 ≤ … ≤ 31, 18 ≤ … ≤ 32, …

20 / 24

SLIDE 69

Bibliography Projects funding Students selection Context The preference model Portfolio constraints Conflicts

All portfolio constraints

.

Variables

. . . . . 𝑦, … , 𝑦, binaries .

Constraints

. . . . . ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

𝑦 ≤ 21
𝑦𝑄(𝑗) + 𝑜(𝐷) ≥ 10

∀𝑡 ∈ streams

𝑦𝑄(𝑗) + 𝑜(𝐷) ≤ 20

∀𝑢 ∈ professional tracks 20 ≤

𝑦𝑄girl(𝑗) + 𝑜girls(𝐷) ≤ 30

No satisfying solution!

21 / 24

SLIDE 70

Bibliography Projects funding Students selection Context The preference model Portfolio constraints Conflicts

All portfolio constraints

.

Variables

. . . . . 𝑦, … , 𝑦, binaries .

Constraints

. . . . . ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

𝑦 ≤ 21
𝑦𝑄(𝑗) + 𝑜(𝐷) ≥ 10

∀𝑡 ∈ streams

𝑦𝑄(𝑗) + 𝑜(𝐷) ≤ 20

∀𝑢 ∈ professional tracks 20 ≤

𝑦𝑄girl(𝑗) + 𝑜girls(𝐷) ≤ 30

No satisfying solution!

21 / 24

SLIDE 71

Bibliography Projects funding Students selection Context The preference model Portfolio constraints Conflicts

Conflict resolution

Impossible to satisfy all the constraints Add relaxed constraints Search solutions with least possible number of disabled constraints Add 𝑒 variables, binaries, for each constraint 𝑑 When 𝑒 = 1, constraint is disabled Minimize ∑ 𝑒

22 / 24

SLIDE 72

Bibliography Projects funding Students selection Context The preference model Portfolio constraints Conflicts

min ∑ 𝑒 s.t. ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

𝑦 ≤ 21 + 𝑁𝑒
𝑦 ≤ 22 + 𝑁𝑒
𝑦 ≤ 23 + 𝑁𝑒

.

Variables

. . 𝑦, … , 𝑦 𝑒, ∀𝑑, binaries ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

𝑦𝑄(𝑗) + 𝑜(𝐷) + 𝑁𝑒 ≥ 10, ∀𝑡
𝑦𝑄(𝑗) + 𝑜(𝐷) + 𝑁𝑒 ≥ 9, ∀𝑡
𝑦𝑄(𝑗) + 𝑜(𝐷) + 𝑁𝑒 ≥ 8, ∀𝑡
𝑦𝑄(𝑗) + 𝑜(𝐷) ≤ 20 + 𝑁𝑒, ∀𝑢
𝑦𝑄(𝑗) + 𝑜(𝐷) ≤ 21 + 𝑁𝑒, ∀𝑢
𝑦𝑄(𝑗) + 𝑜(𝐷) ≤ 22 + 𝑁𝑒, ∀𝑢

20 − 𝑁𝑒 ≤

𝑦𝑄girl(𝑗) + 𝑜girls(𝐷) ≤ 30 + 𝑁𝑒

19 − 𝑁𝑒 ≤

𝑦𝑄girl(𝑗) + 𝑜girls(𝐷) ≤ 31 + 𝑁𝑒

…

23 / 24

SLIDE 73

Bibliography Projects funding Students selection Context The preference model Portfolio constraints Conflicts

Results

Four ex-æquo best possibilities Only 15 girls (wanted ≥ 20), all other constraints satisfied Only 16 girls, 1 prof. track has 21 students (wanted ≤ 20) Only 17 girls, 1 prof. track has 22 students Only 18 girls, 1 prof. track has 23 students .

Final choice

. . . . . The DM may observe the different ways to resolve the conflicts and choose its preferred one.

24 / 24