Robust ordinal regression for multiple criteria sorting problems within MAUT

(1)

Outline

Robust ordinal regression for multiple

criteria sorting problems within MAUT

Miłosz Kadzi ´nski

Pozna ´n University of Technology, Poland, milosz.kadzinski@cs.put.poznan.pl

(2)

1 _Introduction

2 _{Sorting with a set of value function}

3 _{The most representative value function}

Principle

Three stage-procedure

One-stage procedure

Illustrative example

4 _Conclusions

(3)

Outline

Outline

1 _Introduction

2 _{Sorting with a set of value function}

3 _{The most representative value function}

Principle

Three stage-procedure

One-stage procedure

Illustrative example

4 _Conclusions

(4)

1 _Introduction

2 _{Sorting with a set of value function}

3 _{The most representative value function}

Principle

Three stage-procedure

One-stage procedure

Illustrative example

4 _Conclusions

(5)

Outline

Outline

1 _Introduction

2 _{Sorting with a set of value function}

3 _{The most representative value function}

Principle

Three stage-procedure

One-stage procedure

Illustrative example

(6)

Characteristics

Actions

described by evaluation

vectors

to be assigned to classes

Pre-defined ordered

(or unsorted)

classes

Classes have

a semantic definition

Assignment to classes is grounded

on absolute evaluation of actions

No relative comparisons

is required,

6= choice, ranking

(7)

Introduction

Sorting with a set of value function The most representative value function Conclusions

Multiple Criteria Sorting Problems

Characteristics

Actions

described by evaluation

vectors

to be assigned to classes

Pre-defined ordered

(or unsorted)

classes

Classes have

a semantic definition

Assignment to classes is grounded

on absolute evaluation of actions

No relative comparisons

is required,

6= choice, ranking

(8)

Real-world sorting problems

Financial management and economics:

business failure

prediction, credit risk assessment for firms and consumers,

country risk assessment

Evironmental and energy management, ecology:

analysis

of different energy policies

Human resources management:

assigment to appropriate

occupation groups, incentive package groups

Marketing:

customer satisfaction measurement,

development of market penetration stategies

(9)

Introduction

Definitions and notation

Given data

A = {a

1 ,

a

2 , . . . ,

a

i

, . . . ,

a

m

} - a finite set of m

actions

g

1 ,

g

2 , . . . ,

g

j

, . . . ,

g

n

- n evaluation

criteria

, g

j

:

A → R for

all j ∈ G = {1, 2, . . . , n}

X

j

= {x

j

∈ R : g

j

(a

i

) =

x

j

,

a

i

∈ A} - the set of all different

evaluations

on g

_j

, j ∈ G

x

_j

0 ,

x

_j

1 , . . . ,

x

m

j

- the ordered values of X

j

,

x

_j

k

<

x

_j

k +1

,

k = 0, 1, . . . , m

j

− 1

C

1 ,

C

2 , . . . ,

C

p

- p predefined preference ordered

classes

,

where C

h+1

is preferred to C

h

, h = 1, . . . , p − 1, moreover,

(10)

Preference model

To represent DM’s preferences, we use an

additive value function

such that:

U(a) =

n

X

j=1

u

j

(g

j

(a)),

where the marginal value functions u

j

are such

that:

u

j

(x

jk

) ≤

u

j

(x

jk +1

),

k = 0, 1, . . . , m

j

− 1, j ∈ G

To normalize U so that U(a) ∈ [0, 1], ∀a ∈ A, we

set:

u

j

(x

j0

) =

0, ∀j ∈ G and

n

X

j=1

u

j

(x

mj j

) =

1

(11)

Introduction

Definitions and notations

Preference information

A

R

_{⊆ A - a set of}

_{reference actions}

_,

An

assignment example

is an action

a

∗

∈ A

R

_{for which the DM defined a}

desired assignment

a

∗

→ [C

_L

DM

_(a

∗

₎

,

C

_R

DM

_(a

∗

₎

],

i.e. to an interval of contiguous

classes

C

_L

DM

_(a

∗

₎

,

C

_L

DM

_(a

∗

₎₊₁

, ...,

C

_R

DM

_(a

∗

₎

An assignment example is said to be

precise

if L

DM

_(a

∗

_{) =}

_R

DM

_(a

∗

_{), and}

imprecise

, otherwise

(12)

Compatible preference information

Given a value function U, a set of assignment examples is

said to be

consistent with U

iff:

∀a

∗

,

b

∗

∈ A

R

,

U(a

∗

) ≥

U(b

∗

) ⇒

R

DM

(a

∗

) ≥

L

DM

(b

∗

),

which is equivalent to:

∀a

∗

,

b

∗

∈ A

R

,

L

DM

(a

∗

) >

R

DM

(b

∗

) ⇒

U(a

∗

) >

U(b

∗

)

We will suppose the DM provides a set of assignment

examples consistent with U

(13)

Introduction

Definitions and notation

Compatible preference information

Reference actions: A

R

= {a

1 ,

a

2 ,

a

3 }

Classes: C

1 <<

C

2 <<

C

3 <<

C

4 A set of

assignment examples

:

a

1 ⇒ [C

1 ,

C

2 ],

a

2 ⇒ [C

2 ,

C

3 ],

a

3 ⇒ [C

4 ,

C

4 ]

Resulting constraints

:

U(a

3 ) >

U(a

1 )

(14)

Definition

Value driven

sorting procedures aim to assign each action to

one class or a set of contiguous classes, using a value function

U in such a way that if

U(a) > U(b)

then a is assigned to a

class

not worse

than b

Sorting procedures

Given a single additive value function U,

two

different sorting

procedures can be considered:

threshold-based

value driven sorting procedure

(15)

Introduction

Value driven sorting procedures

Threshold-based sorting procedure

Action a ∈ A is assigned to class C

h

, denoted as

a → C

h

,

iff

U(a) ∈ [b

_h−1

U

,

b

h

)

Threshold b

_h−1

corresponds to the

minimum

value for an

action a to be assigned to class C

h

Threshold b

h

corresponds to the

supremum

value for an

action a to be assigned to class C

_h

We impose b

h−1

<

b

h

, ∀h ∈ H and we set b

0 =

0 and

(16)

(17)

Introduction

Value driven sorting procedures

Example-based sorting procedure

It is driven by a value function U and its associated

assignment examples A

R

⊆ A

It assigns an action a to an interval of classes:

[C

_L

U

_(a)

,

C

_R

U

_(a)

]:

L

U

(a) = Max

n

L

DM

(a

∗

) :

U(a

∗

) ≤

U(a), a

∗

∈ A

R

o

R

U

(a) = Min

n

R

DM

(a

∗

) :

U(a

∗

) ≥

U(a), a

∗

∈ A

R

o

(18)

Example-based sorting procedure

For each non-reference action a ∈ A \ A

R

_{the indices}

satisfy the following condition:

L

U

(a) ≤ R

U

(a)

Each reference action a

∗

∈ A

R

_{is assigned to an interval of}

classes, such that:

L

U

(a

∗

) ≥

L

DM

(a

∗

)

R

U

_(a

∗

_{) ≤}

_R

DM

_(a

∗

₎

(19)

Introduction

Threshold-based vs example-based sorting

Proposition

Consider the case where L

DM

_(a

∗

_{) =}

_R

DM

_(a

∗

_{), ∀a}

∗

_{∈ A}

R

Assuming the use of a single value function U in the

example-based sorting procedure, if we choose the threshold

b

U

_h

,

h = 1, . . . , p − 1 in the interval

]Max

a

∗

_→C

h

{U(a

∗

_)},

_Min

a

∗

_→C

h+1

{U(a

∗

_)}]

we obtain a

threshold-based sorting procedure

that restores the

assignment examples and assigns each non-reference action

a ∈ A \ A

R

_{to a single class in the interval [C}

L

U

(a), C

_R

U

(a)]

stemming from the

example-based sorting procedure

(20)

(21)

Introduction

Threshold-based vs example-based sorting

Consider b ∈ A

L

U

_{(b) = C}

(22)

Proposition

Consider the case where L

DM

_(a

∗

_{) ≤}

_R

DM

_(a

∗

_{), ∀a}

∗

_{∈ A}

R

Assuming the use of a single value function U in the

example-based sorting procedure, if we choose the threshold

b

U

h

,

h = 1, . . . , p − 1 in the interval

]Max

_a

∗

_:R

DM

_(a

∗

_)≤h

{U(a

∗

)},

Min

_a

∗

_:L

DM

_(a

∗

_)>h

{U(a

∗

)}[

with b

U

h

<

b

U

h+1

,

we obtain a

threshold-based sorting procedure

that restores the

assignment examples and assigns each non-reference action

a ∈ A \ A

R

_{to a single class in the interval [C}

L

U

(a), C

_R

U

(a)]

stemming from the

example-based sorting procedure

(23)

Introduction

(24)

(25)

Introduction

Threshold-based vs example-based sorting

Consider b ∈ A, with U(b) ∈]U(a

6 ),

U(a

7 )[

(26)

Consider b ∈ A, with U(b) ∈]U(a

6 ),

U(a

7 )[

(27)

Introduction

Robust ordinal regression

Ordinal regression paradigm

Comprehensive preferences on a subset of

reference actions

is known a priori

Consistent criteria aggregation model

is inferred from this

information to be applied on the set of all actions

Observations

The set of

all

preference models compatible with the stated

indirect preference information can be

quite large

Traditionally,

only one specific set is used

to give

a recommendation

The choice of a single preference model is either

arbitrary or left

to the DM

(28)

Comment

(29)

Introduction

Robust ordinal regression

Comment

(30)

Comment

(31)

Introduction

Robust ordinal regression

Comment

(32)

Comment

(33)

Introduction

Robust ordinal regression

Aim

Take into account

all the sets of preference models compatible

with the preference information given by the DM

Developed methods

UTA

GMS

and GRIP (choice and ranking problems)

UTADIS

GMS

(sorting problems)

Robust ordinal regression applied to Choquet integral

Robust ordinal regression applied to group decisions

Interactive and evolutionary multiobjective optimization

methodology

(34)

Questions

Is action x ∈ A sorted in the same way by

all

compatible

value functions?

Is there

at least one

compatible value function sorting

action x ∈ A to a given class?

(35)

Introduction

Sorting with a set of value function

The most representative value function Conclusions

The UTADIS-GMS method

Principles

The preference information is composed of assignment

examples on A

R

⊆ A

A value function is called

compatible

if it is able to restore

all assignment examples

In result, one obtains

two assignments

for any action

a ∈ A:

1

_the

_{possible assignment}

_C

P

(a):

C

P

(a) = {h ∈ H :

∃

U ∈ U

A

R

for which h ∈ [L

U

(a), R

U

(a)]}

2

_the

_{necessary assignment}

_C

N

(a):

C

N

(a) = {h ∈ H :

∀

U ∈ U

A

R

it holds h ∈ [L

U

(a), R

U

(a)]}

where

L

U

(a)

and

R

U

(a)

are, respectively, the

worst

and the

(36)

General additive compatible value function

An additive value function U(a) =

P

n

j=1

u

j

(g

j

(a)) satisfying the

following set of constraints:

compatibility

with the preference information:

∀a

∗

_,

_b

∗

_{∈ A}

∗

_,

_L

DM

_(a

∗

_{) >}

_R

DM

_(b

∗

_{) ⇒}

_U(a

∗

_{) >}

_U(b

∗

₎

monotonicity

of the family of criteria G:,

u

i

(g

i

(a

τi(j)

)) −

u

i

(g

i

(a

τi(j−1)

)) ≥

0, i = 1, . . . , n, j = 2, . . . , m.

where τ

i

is the permutation on the set of indices of actions from A

that reorders them according to the increasing evaluation on

criterion g

i

:

g

i

(a

τi(1)

) ≤

g

i

(a

τi(2)

) ≤ . . . ≤

g

i

(a

τi(m−1)

) ≤

g

i

(a

τi(m)

)

normalization

to the interval [0, 1]:

u

i

(α

i

) =

0, i = 1, . . . , n,

P

n

(37)

Introduction

Remarks about linear constraints

Transforming strict inequalities into weak inequalties

x > y ⇒ x ≥ y + ε

Set of functions satisfying the particular condition

ε

∗

=

max ε

x ≥ y + ε

constraints defining set of functions

)

if

ε

∗

>

0 then there exists

at least one

function satisfying

condition x > y in the defined set of functions,

if

ε

∗

≤ 0

then there is

no function

satisfying condition x > y in

the defined set of functions.

(38)

Verification whether the

set

of all compatible value functions U

A

R

is

not empty

U

_A

R

6= ∅ ⇔ ε

∗

>

0 where:

ε

∗

=

max ε

U(a

∗

) ≥

U(b

∗

) + ε ⇔

L

DM

_(a

∗

_{) >}

_R

DM

_(b

∗

_{) } ∀a}

∗

_,

_b

∗

_{∈ A}

R

u

i

(g

i

(a

τ

i

(j)

)) −

u

i

(g

i

(a

τ

i

(j−1)

)) ≥

0, i = 1, . . . , n, j = 2, . . . , m

u

i

(g

i

(a

τ

i

(1)

)) ≥

0, u

i

(g

i

(a

τ

i

(m)

)) ≤

u

i

(β

i

),

i = 1, . . . , n

u

i

(α

i

) =

0, i = 1, . . . , n

P

n

i=1

u

i

(β

i

) =

1 









(E

A

R

)

(39)

Introduction

Possible preference relation

Definition

a

P

b

means that U(a) ≥ U(b) for

at least one

compatible

value function

a

P

b ⇔ ε

∗

>

0 where:

ε

∗

=

max ε

U(a) ≥ U(b)

U(a

∗

) ≥

U(b

∗

) + ε ⇔

L

DM

_(a

∗

_{) >}

_R

DM

_(b

∗

_{) } ∀a}

∗

_,

_b

∗

_{∈ A}

R

u

i

(g

_i

j

) −

u

i

(g

_i

j−1

) ≥

0, i = 1, . . . , n, j = 1, . . . , ω + 1

u

i

(g

i

0 ) =

0, i = 1, . . . , n

P

n

i=1

u

i

(g

i

ω+1

) =

1 









(40)

Definition

a

N

_b

_{means that U(a) ≥ U(b) for}

_all

_{compatible value}

functions

a

N

b ⇔ ε

∗

≤ 0

where:

ε

∗

=

max ε

U(b) ≥ U(a) + ε

U(a

∗

) ≥

U(b

∗

) + ε ⇔

L

DM

(a

∗

) >

R

DM

(b

∗

) } ∀a

∗

,

b

∗

∈ A

R

u

_i

(g

_i

j

) −

u

_i

(g

_i

j−1

) ≥

0, i = 1, . . . , n, j = 1, . . . , ω + 1

u

_i

(g

0 i

) =

0, i = 1, . . . , n

P

n

i=1

u

i

(g

_i

ω+1

) =

1 









(41)

Introduction

Possible assignments

Minimum possible

class:

L

U

_P

(a)

=

Max

n

L

DM

(a

∗

) : ∀U ∈ U

A

R

,

U(a

∗

) ≤

U(a), a

∗

∈ A

R

o

=

Max

n

L

DM

(a

∗

) :

a

N

a

∗

,

a

∗

∈ A

R

o

Maximum possible

class:

R

_P

U

(a)

=

Min

n

R

DM

(a

∗

) : ∀U ∈ U

_A

R

,

U(a

∗

) ≥

U(a), a

∗

∈ A

R

o

=

Min

n

R

DM

(a

∗

) :

a

∗

N

a, a

∗

∈ A

R

o

Assign to each a ∈ A its

possible assignment

C

P

(a) = [L

U

P

(a), R

P

U

(a)]

(42)

Potentially

minimum necessary

class:

L

U

_N

(a)

=

Max

n

L

DM

(a

∗

) : ∃U ∈ U

A

R

for which U(a

∗

) ≤

U(a), a

∗

∈ A

R

o

=

Max

n

L

DM

(a

∗

) :

a

P

_a

∗

_,

_a

∗

_{∈ A}

R

o

Potentially

maximum necessary

class:

R

_N

U

(a)

=

Min

n

R

DM

(a

∗

) : ∃U ∈ U

A

R

for which U(a

∗

) ≥

U(a), a

∗

∈ A

R

o

=

Min

n

R

DM

(a

∗

) :

a

∗

P

_{a, a}

∗

_{∈ A}

R

o

Assign to each a ∈ A its

necessary assignment

which is

C

N

(a) = [L

U

N

(a), R

N

U

(a)] in case L

U

N

(a) ≤ R

N

U

(a) and C

N

(a) = ∅

otherwise

(43)

Introduction

The UTADIS − GMS method

Example

A = {a

1 ,

a

2 ,

a

3 ,

a

4 ,

a

5 ,

a

6 }

G = {g

1 ,

g

2 } - a consistent family G of 2 criteria with an

increasing direction of preference

C = {C

1 ,

C

2 ,

C

3 ,

C

4 ,

C

5 } where C

h+1

C

h

,

h = 1, 2, 3, 4

Evaluation table:

a

1 a

2 a

3 a

4 a

5 a

6 g

1

2

6

7

5

8

(44)

Example

Step1:

The DM provides

exemplary assignments

for actions in

the reference set A

R

= {a

1 ,

a

3 ,

a

5 }. (S)he feels confident that:

a

1 → [C

1 ,

C

1 ]

a

3 → [C

4 ,

C

4 ]

a

5 → [C

2 ,

C

3 ]











Step 2:

Provided preference information is

consistent

and,

consequently, a set of compatible value functions is not empty

(45)

Introduction

The UTADIS − GMS method

Example

Step1:

The DM provides

exemplary assignments

for actions in

the reference set A

R

= {a

1 ,

a

3 ,

a

5 }. (S)he feels confident that:

a

1 → [C

1 ,

C

1 ]

a

3 → [C

4 ,

C

4 ]

a

5 → [C

2 ,

C

3 ]











Step 2:

Provided preference information is

consistent

and,

consequently, a set of compatible value functions is not empty

(46)

Example

Step 3:

Matrix of relations

P

_and

N

P

N

a

1

a

2

a

3

a

4

a

5

a

6

a

1

a

2

a

3

a

4

a

5

a

6

a

1

T

F

-

T

-

F

T

F

-

F

-

F

a

2

T

-

T

-

T

F

-

T

-a

3

-

T

-

T

-

T

-

F

a

4

T

-

F

T

-

T

-

F

T

F

-a

5

-

T

-

T

F

-

F

-

T

F

a

6

T

-

T

-

T

-

T

-

T

Step 4:

Calculate boundary indices

L

U P

(a

2

) =

Max

L

DM

(a

1

) =

C

1

,L

DM

(a

5

) =

C

2

=

C

2

R

U P

(a

2

) =

Min

R

DM

(a

3

) =

C

4

=

C

4

L

U_N

(a

2

) =

Max

L

DM

(a

1

) =

C

1

,

L

DM

(a

3

) =

C

4

,

L

DM

(a

5

) =

C

2

=

C

4

R

U N

(a

2

) =

Min

R

DM

(a

3

) =

C

4

,

R

DM

(a

5

) =

C

3

=

C

3











(47)

Introduction

The UTADIS − GMS method

Example

Step 3:

Matrix of relations

P

_and

N

P

N

a

1

a

2

a

3

a

4

a

5

a

6

a

1

a

2

a

3

a

4

a

5

a

6

a

1

T

F

-

T

-

F

T

F

-

F

-

F

a

2

T

-

T

-

T

F

-

T

-a

3

-

T

-

T

-

T

-

F

a

4

T

-

F

T

-

T

-

F

T

F

-a

5

-

T

-

T

F

-

F

-

T

F

a

6

T

-

T

-

T

-

T

-

T

Step 4:

Calculate boundary indices

L

U P

(a

2

) =

Max

L

DM

(a

1

) =

C

1

,L

DM

(a

5

) =

C

2

=

C

2

R

U P

(a

2

) =

Min

R

DM

(a

3

) =

C

4

=

C

4

L

U_N

(a

2

) =

Max

L

DM

(a

1

) =

C

1

,

L

DM

(a

3

) =

C

4

,

L

DM

(a

5

) =

C

2

=

C

4

R

U N

(a

2

) =

Min

R

DM

(a

3

) =

C

4

,

R

DM

(a

5

) =

C

3

=

C

3











(48)

Example

Step 5, 6:

Matrix of possible and necessary assignments

L

P

R

P

L

N

R

N

a

1 C

1 a

2 C

4 a

3 C

4 C

4 a

4 C

1 C

3 a

5 C

2 C

3 C

2 C

3 a

6 C

4 C

5 C

4 C

4 L

U

P

(a) ≤ L

U

N

(a) and R

U

N

(a) ≤ R

U

P

(a).

(49)

Introduction Sorting with a set of value function

The most representative value function

Conclusions Principle Three stage-procedure One-stage procedure Illustrative example

Outline

1 _Introduction

2 _{Sorting with a set of value function}

3 _{The most representative value function}

Principle

Three stage-procedure

One-stage procedure

Illustrative example

4 _Conclusions

(50)

Why to search for the function?

To

see and know

the most representative value function

among all the compatible ones

To assess

relative importance

of the criteria

To

assign a score

(value) to the actions

To work out the

most representative assignments

To identify the most representative function

without loosing

the advantage of taking into account all compatible value

functions

To

exhibit

it explicitly along with the results of the

UTADIS

GMS

method

(51)

Conclusions

Principle

Three stage-procedure One-stage procedure Illustrative example

The most representative value function

Question

Which value function is the most representative one in the set of value

functions compatible with the preference infromation?

(52)

One for all, all for one

One for all:

one value function is representing all

compatible value functions

All for one:

all compatible value functions contribute to the

definition of the most representative value function

(53)

Conclusions

Principle

The most representative value function in sorting

First idea

1

_Maximize

_{the minimal difference between values of actions}

a, b ∈ A for which possible assignments are

disjoint

:

[L

U

_P

(a), R

_P

U

(a)] ∩ [L

U

_P

(b), R

U

_P

(b)] = ∅

2

_Minimize

_{the maximal difference between values of actions}

a, b ∈ A for which possible assignments are

not disjoint

:

[L

U

_P

(a), R

_P

U

(a)] ∩ [L

U

_P

(b), R

U

_P

(b)] 6= ∅

Comment

(54)

First idea

1

_Maximize

_{the minimal difference between values of actions}

a, b ∈ A for which possible assignments are

disjoint

:

[L

U

_P

(a), R

_P

U

(a)] ∩ [L

U

_P

(b), R

U

_P

(b)] = ∅

2

_Minimize

_{the maximal difference between values of actions}

a, b ∈ A for which possible assignments are

not disjoint

:

[L

U

_P

(a), R

_P

U

(a)] ∩ [L

U

_P

(b), R

U

_P

(b)] 6= ∅

Comment

(55)

Conclusions

Principle

(56)

Final idea

1

_Maximize

_{the minimal difference between values of actions}

a, b ∈ A for which

possible assignments are disjoint

2

_Maximize

_{the minimal difference between values of actions}

a, b ∈ A for which:

for

all

value function U a is assigned to a class

not worse

than

the class of b,

and

, for

at least one

compatible value function a is assigned to

a class which is

better

than the class of b

3

_Minimize

_{the maximal difference between values of actions}

a, b ∈ A

being in the

same class

for all compatible value functions U

or

, for which the order of classes is

not univocal

(57)

Outline

1 _Introduction

2 _{Sorting with a set of value function}

3 _{The most representative value function}

Principle

Three stage-procedure

One-stage procedure

Illustrative example

4 _Conclusions

(58)

Condition in the first stage

Possible assignments for actions a, b ∈ A are disjoint:

[L

U

_P

(a), R

_P

U

(a)] ∩ [L

U

_P

(b), R

U

_P

(b)] = ∅.

The difference between values of actions should be as large

as possible.

(59)

Three-stage procedure

First stage

1

_{For all pairs of actions (a, b), such that L}

U

P

(a) > R

U

P

(b), add

the following constraint to the linear programming

constraints of UTADIS

GMS

:

U(a) ≥ U(b) + ε.

2

_Maximize

_ε.

3

_{Add the constraint ε = ε}

∗

_{, with ε}

∗

₌

_{max ε from point 2), to}

(60)

Condition in the second stage

a

→

b ⇔ ∀U ∈ U

A

R

: (L

U

(a) ≥ R

U

(b))

and (∃U ∈ U

A

R

: L

U

(a) > R

U

(b));

The difference between values of actions should be as large

as possible.

(61)

Three-stage procedure

Second stage

1

_{For all pairs of actions (a, b), such that a}

→

_{b, add the}

following constraint to the linear programming constraints

of point 3) of first stage:

U(a) ≥ U(b) + γ.

2

_Maximize

_γ.

3

_{Add the constraint γ = γ}

∗

_{, with γ}

∗

₌

_{max γ from point 2), to}

(62)

Condition in the third stage

a ∼

→

b ⇔ (∀U ∈ U

_A

R

: L

U

(a) = L

U

(b) and R

U

(a) = R

U

(b))

or (∃U

1 ,

U

2 ∈ U

A

∗

: L

U

1

(a) > R

U

1

(b) and L

U

2

(b) > R

U

2

(a))

The difference between values of actions should be as small

as possible.

(63)

Three-stage procedure

Third stage

1

_{For all pairs of actions (a, b), such that a ∼}

→

_{b add the}

following constraints to the linear programming constraints

of point 3) of second stage:

U(a) − U(b) ≤ δ and U(b) − U(a) ≤ δ.

(64)

1 _Introduction

2 _{Sorting with a set of value function}

3 _{The most representative value function}

Principle

Three stage-procedure

One-stage procedure

Illustrative example

4 _Conclusions

(65)

One-stage procedure

1

_{For all pairs of actions (a, b), such that L}

U

P

(a) > R

P

U

(b) add:

U(a) ≥ U(b) + ε.

to the linear programming constraints of UTADIS

GMS

.

2

_{For all pairs of actions (a, b), such that a}

→

_{b add:}

U(a) ≥ U(b) + γ.

to the linear programming constraints from point 1).

3

_{For all pairs of actions (a, b), such that a ∼}

→

_{b add:}

U(a) − U(b) ≤ δ and U(b) − U(a) ≤ δ.

to the linear programming constraints from point 2).

4

_Maximize

_{Mε + Nγ − δ}

(66)

1 _Introduction

2 _{Sorting with a set of value function}

3 _{The most representative value function}

Principle

Three stage-procedure

One-stage procedure

Illustrative example

(67)

Conclusions

Principle

Three stage-procedure One-stage procedure

Illustrative example

Global MBA classes

Problem statement and given data

Actions: 30 MBA programs offered by diffrent universities

(originally from the Financial Times;

reconsidered by Koksalan et al., 2008)

Criteria: 20 factors grouped under three main criteria: alumni

career progress, diversity and idea generation

Classes: 5 preference-ordered

Evaluation table

Program

Alumni career

Diversity idea

Idea

name

progress

generation

London Business

68.78

62.03

59.87 Yale University

79.01

25.98

51.84 Carnegie Mellon

54.02

18.69

71.93 Duke University

64.05

27.25

64.68

(68)

Example

Step1:

Ask the DM for possibly imprecise sorting examples.

The DM provides

exemplary assignments

for 8 actions in the

reference set:

London Business School → 5

University of North Carolina → 3

University of Maryland → 1

...











Step 2:

Verify whether the set of compatible value functions is not

empty

(69)

Conclusions

Principle

The UTADIS − GMS method

Example

Step1:

Ask the DM for possibly imprecise sorting examples.

The DM provides

exemplary assignments

for 8 actions in the

reference set:

London Business School → 5

University of North Carolina → 3

University of Maryland → 1

...











Step 2:

Verify whether the set of compatible value functions is not

empty

(70)

Example

Step 3:

Determine the possible sorting C

P

(a) for each

considered action a ∈ A.

Step 4:

For all pairs of actions (a, b), such that L

U

_P

(a) > R

_P

U

(b),

add the following constraint to the linear programming

constraints of UTADIS

GMS

:

U(a) ≥ U(b) + ε.

(71)

Conclusions

Principle

The UTADIS − GMS method

Example

Step 3:

Determine the possible sorting C

P

(a) for each

considered action a ∈ A.

Step 4:

For all pairs of actions (a, b), such that L

U

_P

(a) > R

_P

U

(b),

add the following constraint to the linear programming

constraints of UTADIS

GMS

:

U(a) ≥ U(b) + ε.

(72)

Example

Step 5:

Determine the relation

→

for all pairs of actions (a,b)

with a, b ∈ A.

Step 6:

For all pairs of actions (a, b), such that a

→

b, add the

following constraint to the linear programming constraints from

step 4):

U(a) ≥ U(b) + γ.

(73)

Conclusions

Principle

The UTADIS − GMS method

Example

Step 5:

Determine the relation

→

for all pairs of actions (a,b)

with a, b ∈ A.

Step 6:

For all pairs of actions (a, b), such that a

→

b, add the

following constraint to the linear programming constraints from

step 4):

U(a) ≥ U(b) + γ.

(74)

Example

Step 7:

Determine the relation ∼

→

for all pairs of actions (a,b)

with a, b ∈ A.

Step 8:

For all pairs of actions (a, b), such that a ∼

→

b add the

following constraints to the linear programming constraints from

step 6):

U(a) − U(b) ≤ δ and U(b) − U(a) ≤ δ.

(75)

Conclusions

Principle

The UTADIS − GMS method

Example

Step 7:

Determine the relation ∼

→

for all pairs of actions (a,b)

with a, b ∈ A.

Step 8:

For all pairs of actions (a, b), such that a ∼

→

b add the

following constraints to the linear programming constraints from

step 6):

U(a) − U(b) ≤ δ and U(b) − U(a) ≤ δ.

(76)

Example

Step 9: Maximize Mε + Nγ − δ

U(a) ≥ U(b) + ε ⇔ L

U_P

(a) > R

_PU

(b) } ∀a, b ∈ A

U(a) ≥ U(b) + δ ⇔ a

→

b } ∀a, b ∈ A

U(a) − U(b) ≤ γ ⇔ a ∼

→

b } ∀a, b ∈ A

U(b) − U(a) ≤ γ ⇔ a ∼

→

_{b } ∀a, b ∈ A}

U(a

∗

_{) ≥}

_U(b

∗

_{) + ε ⇔}

_L

DM

_(a

∗

_{) >}

_R

DM

_(b

∗

_{) } ∀a}

∗

_,

_b

∗

_{∈ A}

R

u

i

(g

i

(a

τi(j)

)) −

u

i

(g

i

(a

τi(j−1)

)) ≥

0, i = 1, . . . , n, j = 2, . . . , m

u

i

(g

i

(a

τi(1)

)) ≥

0, u

i

(g

i

(a

τi(m)

)) ≤

u

i

(β

i

),

i = 1, . . . , n

u

i

(α

i

) =

0, i = 1, . . . , n

P

n i=1

u

i

(β

i

) =

1 









E

εδγ

where M and N are arbitrarily large positive constant such that

M >> N >> 1.

(77)

Conclusions

Principle

The most representative value function in sorting

Results of optimization

ε =

0.2,

e.g. {U(LBS) = 0.857, U(Yale) = 0.657},

{U(LBS) = 0.857, U(Maryland) = 0.057},

γ =

0.029,

e.g. {U(LBS) = 0.857, U(Pennsylv .) = 0.828},

{U(LBS) = 0.857, U(Maryland) = 0.057},

δ =

0.714,

e.g. {U(Pennsylv .) = 0.114, U(W .Ontario) = 0.114},

{U(Rotterdam) = 0.828, U(W .Ontario) = 0.114}.

(78)

Characteristics

The characteristic points correspond to the evaluation

values of the considered actions

The constructed functions are usually not strictly

monotonic, which results from the form of optimized

function Mε + Nγ − δ

Although in the figure connections between characteristic

points are linear, it would be sufficient if they reflected the

monotonic character

(79)

Conclusions

Principle

The UTADIS − GMS method

Example

Step 10:

Optionally, conduct an example-based sorting

procedure driven by the value function from point 9 and

assignment examples from point 1 in order to determine the

most representative assignment for each action in the

considered set

(80)

Aim of the platform

Decision Deck (D2) aims to develop a tool to support

decision makers in evaluating actions in a multiple criteria

and multiple experts context

(81)

Conclusions

Principle

Decision Deck Plugins

Technical aspects

Current implementation of plugin works on the

second

version of Decision Deck

platform (1.1)

The plugin is an

OSGI bundle

- dynamically loadable

collection of classes, resources, and configuration files

Data access is achieved through

Hibernate

To analyze potential inconsistency and verify the truth of

preference relations it uses

GLKP linear solver

To visualize relations and assignments of actions in form of

tables

it uses standard Java classes and for visualisation of

the most representative value function it uses

JChart

(82)

(83)

Introduction Sorting with a set of value function The most representative value function