What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
What is a Decision Problem?
Alberto Colorni1
1 Design
2 LAMSADE
Alexis Tsoukiàs2
Dept., Politecnico di Milano,
- CNRS, Université Paris-Dauphine
Labex MS2T, 13/10/2015
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Outline
1
What is the problem?
2
What is a decision problem?
3
What does really matter?
4
Why this is important?
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Problems
Patients triage in emergency room;
Identification of classes of similar DNA sequences;
Star ratings of hotels;
Waste collection vehicle routing;
Vendor rating and bids assessment;
Optimal mix of sausages;
Chip-set lay out;
Airplanes priority landing;
Tennis tournament scheduling ...
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
What is important?
Characterisation
Is there a way to characterise problems which does not depend
from the methods used?
What does really matter?
In designing, choosing, applying, implementing, understanding,
explaining, justifying, a method?
What are the primitives?
And what is the derived information and the expected
outcomes?
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
What is important?
Characterisation
Is there a way to characterise problems which does not depend
from the methods used?
What does really matter?
In designing, choosing, applying, implementing, understanding,
explaining, justifying, a method?
What are the primitives?
And what is the derived information and the expected
outcomes?
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
What is important?
Characterisation
Is there a way to characterise problems which does not depend
from the methods used?
What does really matter?
In designing, choosing, applying, implementing, understanding,
explaining, justifying, a method?
What are the primitives?
And what is the derived information and the expected
outcomes?
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
What is my perspective?
A general theory for scholars and practitioners such that:
it is easy to teach;
it is easy to use “practically”;
it is easy to implement (if necessary ...).
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
What is a decision problem?
Consider a set A established as any among the following:
an enumeration of objects;
a set of combinations of binary variables;
a set of profiles within a multi-attribute space;
a vector space in Rn .
Without loss of generality:
Q
A ⊆ j Xj where ∀jXj = {x1j , · · · xnj }, Xj being separable and
ordered;
Technically:
A Decision Problem is a partitioning of A under some desired
properties.
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
What is a decision problem?
Consider a set A established as any among the following:
an enumeration of objects;
a set of combinations of binary variables;
a set of profiles within a multi-attribute space;
a vector space in Rn .
Without loss of generality:
Q
A ⊆ j Xj where ∀jXj = {x1j , · · · xnj }, Xj being separable and
ordered;
Technically:
A Decision Problem is a partitioning of A under some desired
properties.
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
What is a decision problem?
Consider a set A established as any among the following:
an enumeration of objects;
a set of combinations of binary variables;
a set of profiles within a multi-attribute space;
a vector space in Rn .
Without loss of generality:
Q
A ⊆ j Xj where ∀jXj = {x1j , · · · xnj }, Xj being separable and
ordered;
Technically:
A Decision Problem is a partitioning of A under some desired
properties.
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Problem Statements
Practically we partition A in n classes. These can be:
Pre-defined wrt
some external norm
Defined only through
relative comparisons
Ordered
Not Ordered
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Problem Statements
Practically we partition A in n classes. These can be:
Ordered
Not Ordered
Pre-defined wrt
some external norm
Rating
Assigning
Alberto Colorni and Alexis Tsoukiàs
Defined only through
relative comparisons
Ranking
Clustering
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Problem Statements
Practically we partition A in n classes. These can be:
Ordered
Not Ordered
Pre-defined wrt
some external norm
Rating
Assigning
Defined only through
relative comparisons
Ranking
Clustering
Two special cases:
- there are only two classes (thus complementary);
- the size (cardinality) of the classes is also predefined.
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Claims 1
Recall: A ⊆
Q
j
Xj
A decision problem exists iff
- ∃Xj such that Xj is known and
- ∀Xj such that Xj is unknown these are not separable.
Claim 1
Establishing the set A is on its turn a decision problem. We get
a fix point A0 using the known attributes and we establish a
recursion where each step of is a decision problem.
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Claims 1
Recall: A ⊆
Q
j
Xj
A decision problem exists iff
- ∃Xj such that Xj is known and
- ∀Xj such that Xj is unknown these are not separable.
Claim 1
Establishing the set A is on its turn a decision problem. We get
a fix point A0 using the known attributes and we establish a
recursion where each step of is a decision problem.
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Claims 2
Claim 2
All the previously mentioned problems boil down in aggregating
some ordering relations applied on the set A.
Claim 3
From an algorithmic point of view a decision problem boils
down to an optimisation algorithm.
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Claims 2
Claim 2
All the previously mentioned problems boil down in aggregating
some ordering relations applied on the set A.
Claim 3
From an algorithmic point of view a decision problem boils
down to an optimisation algorithm.
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Claims 2
Claim 2
All the previously mentioned problems boil down in aggregating
some ordering relations applied on the set A.
Claim 3
From an algorithmic point of view a decision problem boils
down to an optimisation algorithm.
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
What is a ranking problem?
Primitive
The input is a binary relation on A: ⊆ A × A to be read
“at least as good as”.
Result
The result is a partitioning of A in [A1 ], · · · [An ] such that:
[Aj ] ≥ [Ai ] ⇔ j ≥ i and
∀x ∈ [Aj ], y ∈ [Ai ] : x 0 y
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Discussion 1
What is a choice problem?
We partition A in two classes [A1 ] ≥ [A2 ]. Thus [A1 ] = supA (0 ).
What is an optimisation problem?
A choice problem for which:
- =0
- x y ⇔ f (x) ≥ f (y ).
- Thus [A1 ] = maxA f (x)
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Discussion 1
What is a choice problem?
We partition A in two classes [A1 ] ≥ [A2 ]. Thus [A1 ] = supA (0 ).
What is an optimisation problem?
A choice problem for which:
- =0
- x y ⇔ f (x) ≥ f (y ).
- Thus [A1 ] = maxA f (x)
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Discussion 2
Why is 0 different from ?
Generally speaking is not an ordering relation since
preferences can be partial and or inconsistent. If we have to
proceed with some operational procedure we need to transform
to an ordering relation 0 .
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Discussion 2
Why is 0 different from ?
Generally speaking is not an ordering relation since
preferences can be partial and or inconsistent. If we have to
proceed with some operational procedure we need to transform
to an ordering relation 0 .
How do we learn ?
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Discussion 2
Why is 0 different from ?
Generally speaking is not an ordering relation since
preferences can be partial and or inconsistent. If we have to
proceed with some operational procedure we need to transform
to an ordering relation 0 .
How do we learn ?
What properties should 0 fulfill?
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
What is a clustering problem?
Primitive
The input is a set of binary relations on A: ≈l ⊆ A × A
to be read “similar to”.
Result
The result is a partitioning of A in [A1 ], · · · [An ] such that:
∃ ≈l : ∀x, y ∈ [Aj ] x ≈ y and
∀x ∈ [Aj ], y ∈ [Ai ] : ¬(x ≈ y )
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Discussion 1
Indiscernibility.
In case ≈l are equivalence relations then the partitioning of A
results in constructing the indiscernibility relation on A.
However, this is not generally the case and [Aj ] = supA (≈l ).
In other terms we try to maximise similarity within classes
(clusters) and minimise similarity among classes (clusters).
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Discussion 1
Indiscernibility.
In case ≈l are equivalence relations then the partitioning of A
results in constructing the indiscernibility relation on A.
However, this is not generally the case and [Aj ] = supA (≈l ).
In other terms we try to maximise similarity within classes
(clusters) and minimise similarity among classes (clusters).
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Discussion 2
Distances.
If ≈l are nested similarity relations with nice properties then we
can establish a metric:
- s(x, y ): how similar is x to y ?
- d(x, y ): how distant is x from y ?
Then [Ay ] = {x| maxA F (s(x, y ))},
F being a measure of the overall similarity of the elements of
[Ay ] with respect to y .
What properties should F and the metrics fulfill?
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
What is a rating problem?
Primitive
The input is a binary relation on A: ⊆ A × P ∪ P × A
to be read “at least as good as”.
P being the set of external “norms” characterising the ordered
classes C1 B · · · B Cn
Result
The result is to assign each element of A in a Cj such that:
x ∈ Cj ⇔ x 0 pj , pj+1 , · · · pn and p1 · · · pj−1 0 x
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Discussion 1
Constraint Satisfaction
If ∀x, y ∈ A ∪ P x y ⇔ f (x) ≥ f (y ).
Then x ∈ Cj ⇔ f (pj−1 ) ≥ f (x) ≥ f (pj ).
This is a Constraint Satisfaction Problem.
Why is 0 different from ?
Generally speaking is not an ordering relation since
preferences can be partial and or inconsistent. If we have to
proceed with some operational procedure we need to transform
to an ordering relation 0 .
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Discussion 1
Constraint Satisfaction
If ∀x, y ∈ A ∪ P x y ⇔ f (x) ≥ f (y ).
Then x ∈ Cj ⇔ f (pj−1 ) ≥ f (x) ≥ f (pj ).
This is a Constraint Satisfaction Problem.
Why is 0 different from ?
Generally speaking is not an ordering relation since
preferences can be partial and or inconsistent. If we have to
proceed with some operational procedure we need to transform
to an ordering relation 0 .
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
What is an assigning problem?
Primitive
The input is a set of binary relations on A: ≈l ⊆ A × P ∪ P × A
to be read “similar to”.
P being the set of external “norms” characterising the classes
C1 · · · Cn
Result
The result is to assign each element of A in a Cj such that:
x ∈ Cj ⇔ ∃ ≈l : x ≈l pj
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Discussion 1
Constraint Satisfaction
If ∀x, y ∈ A ∪ P x ≈l y ⇔ f (x) = f (y ).
This is once again a Constraint Satisfaction Problem.
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Why optimisation is important?
Any unsupervised decision problem is an optimisation
problem.
Any supervised decision problem is a constraint
satisfaction problem.
Since any constraint satisfaction problem can be seen
as an optimisation problem,
we can definitely focus only to the later ones
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Why is not straightforward?
multiple opinions
multiple values
multiple scenarios
+ algorithmic aspects
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Where primitives come from?
6multiple criteria
•
•
•
•
•
•
multiple scenarios
and
epistemic states
-
•
multiple stakeholders
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Where primitives come from?
6multiple criteria
•
•
•
'$
•
•
&%
•
Optimisation
multiple scenarios
and
epistemic states
-
•
multiple stakeholders
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Where primitives come from?
6multiple criteria
'$
•
•
Compromise
&%
•
•
•
•
multiple scenarios
and
epistemic states
-
•
multiple stakeholders
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Where primitives come from?
6multiple criteria
•
•
•
•
•
'$
•
Agreement
multiple scenarios
and
epistemic states
-
•
&%
multiple stakeholders
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Where primitives come from?
6multiple criteria
Robust
•
•
•
•
Optimisation
'$
•
multiple scenarios
and
epistemic states
-
&%
•
•
multiple stakeholders
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Where primitives come from?
6multiple criteria
Robust
•
•
•
•
•
'$
•
Agreement
multiple scenarios
and
epistemic states
-
•
&%
multiple stakeholders
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Where primitives come from?
6multiple criteria
'$
Robust
•
•
Compromise
&%
•
•
•
•
multiple scenarios
and
epistemic states
-
•
multiple stakeholders
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Where primitives come from?
6multiple criteria
Agreed
•
•
Compromise
'$
•
•
•
&%
•
multiple scenarios
and
epistemic states
-
•
multiple stakeholders
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Where primitives come from?
6multiple criteria
•
•
'$
•
•
•
&%
•
MESS
multiple scenarios
and
epistemic states
-
•
multiple stakeholders
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Is that all?
Behind a criterion other criteria may be considered in a
hierarchy of criteria (objectives);
Behind a stakeholder other actors may have to be
considered, that precise stakeholder being a speaker for a
community;
Behind a state of the nature other uncertainties may have
to be considered;
Any combination of the above may in reality occur as
complex as possible.
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Hierarchy
A decision problem can be represented as a sequence of
preference aggregations along an hierarchy of actors, criteria
and states of the nature, combined arbitrarily.
d
H
A H
d
d Ad HHd
J
B
BJ
d d d BdJd
@
B
B@
d Bd @d
d
d
@
B
B@
d Bd @d
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Given a set A
Consider a binary relation , separable in one or more
asymmetric parts () and one or more symmetric ones (∼). We
can have ⊆ A × A, ⊆ 2A × 2A , ⊆ A × P ∧ P × A:
single or multi-attribute preferences
single item or extended preferences
relative or absolute preferences
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Given a set A
Consider a binary relation , separable in one or more
asymmetric parts () and one or more symmetric ones (∼). We
can have ⊆ A × A, ⊆ 2A × 2A , ⊆ A × P ∧ P × A:
single or multi-attribute preferences
single item or extended preferences
relative or absolute preferences
These are first order preferences: the primitives
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Second Order Preferences
Consider a set H of criteria, stakeholders or scenarios such
that:
∀i ∈ H∃ i ⊆ A × A or of any other type
Suppose we have a relation <⊆ 2H × 2H . Then
- < represents the relative importance of each subset of criteria;
- < represents the power of each coalition of stakeholders;
- < represents the likelihood of each subset of scenarios.
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Second Order Preferences
Consider a set H of criteria, stakeholders or scenarios such
that:
∀i ∈ H∃ i ⊆ A × A or of any other type
Suppose we have a relation <⊆ 2H × 2H . Then
- < represents the relative importance of each subset of criteria;
- < represents the power of each coalition of stakeholders;
- < represents the likelihood of each subset of scenarios.
These are second order preferences: derivable from the
primitives
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Four fundamental questions
1
How do we consider differences of preferences along
a single criterion/dimension?
2
How do we consider differences of preferences among
several different criteria/dimensions?
3
How do we consider preferential dependencies?
4
How do we consider negative preference statements?
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Four fundamental questions
1
How do we consider differences of preferences along
a single criterion/dimension?
2
How do we consider differences of preferences among
several different criteria/dimensions?
3
How do we consider preferential dependencies?
4
How do we consider negative preference statements?
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Four fundamental questions
1
How do we consider differences of preferences along
a single criterion/dimension?
2
How do we consider differences of preferences among
several different criteria/dimensions?
3
How do we consider preferential dependencies?
4
How do we consider negative preference statements?
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Four fundamental questions
1
How do we consider differences of preferences along
a single criterion/dimension?
2
How do we consider differences of preferences among
several different criteria/dimensions?
3
How do we consider preferential dependencies?
4
How do we consider negative preference statements?
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
The Air Quality index
pollutant
t1
t2
t3
CO2
3
1
7
Alberto Colorni and Alexis Tsoukiàs
SO2
3
3
7
O3
8
8
7
dust
8
2
7
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
The Air Quality index
pollutant
t1
t2
t3
CO2
3
1
7
SO2
3
3
7
O3
8
8
7
dust
8
2
7
For the ATMO index t3 is better than t2 . Is this legitimating (and
legitimated)?
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Axiomatic Considerations
Health danger assessment
may require imposing a non compensation axiom.
Air pollution reduction policy effectiveness
may require imposing a compensation axiom
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Horse Races Betting
rain
not rain
horse
jockey
v r (t) = 0.5v r (h) + 0.5v r (j)
v n (t) = 0.5v n (h) + 0.5v n (j)
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Horse Races Betting
rain
not rain
team
v (b) = 0.5v r (t) + 0.5v n (t)
v (b) = 0.25v r (h) + 0.25v r (j) + 0.25v n (h) + 0.25v n (j)
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Horse Races Betting
rain
not rain
horse
jockey
What should we do now?
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Pragmatic Considerations
Solving the horse betting problem
may require reformulating the problem refining the alternatives
and the preference statements.
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
The Knapsack problem
max
X
vj xj
j
st
X
pj xj ≤ P
j
xj ∈ {0, 1}
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Depending objects
What if objects in the knapsack are not independent? Such as
in a committee selection or in portfolio management.
.
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Depending objects
What if objects in the knapsack are not independent? Such as
in a committee selection or in portfolio management.
If we consider each xj as a “criterion”, then a Choquet integral
or a k-additive utility could replace the objective function. This
will be an (almost) linear function.
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Depending objects
What if objects in the knapsack are not independent? Such as
in a committee selection or in portfolio management.
If we consider each xj as a “criterion”, then a Choquet integral
or a k-additive utility could replace the objective function. This
will be an (almost) linear function.
The complexity only moves around: we need to know the
k-additive utilities ...
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Algorithmic Considerations
Reformulating a problem
through equivalent problem statements allows to obtain
“solvable” problem formulations.
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?
What is the problem?
What is a decision problem?
What does really matter?
Why this is important?
Conclusions
We have a general theory
We have decision problems archetypes
We know what characterises such archetypes
We can make equivalent reformulations
Alberto Colorni and Alexis Tsoukiàs
What is a Decision Problem?