An interactive approach for the Multi-Criteria Portfolio Selection - - PowerPoint PPT Presentation

An interactive approach for the Multi-Criteria Portfolio Selection Problem

• N. Argyris, J.R. Figueira and A. Morton

LSE and IST

February 2010

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 1 / 11

Overview

The standard formulation of the MCPSP. New ‘integrated’ formulations. Enumerating e¢cient Portfolios. Incorporating preferences. Interactive procedures.

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 2 / 11

MOBO formulation of the MCPSP

’ max ’ (c1x, ..., cpx) s.t. aix bi 8i 2 I xj 2 f0, 1g 8j 2 J. J = f1, ..., ng : The set of n projects. cr, r 2 R = f1, ..., pg : Objective vectors, assumed non-negative. ai, i 2 I = f1, ..., mg : Resource utilisation vectors. bi, i 2 I : Resource levels.

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 3 / 11

The problem

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 4 / 11

The problem

Motivation: Can we identify supported e¢cient portfolios without selecting weights a-priori?

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 4 / 11

The problem

Motivation: Can we identify supported e¢cient portfolios without selecting weights a-priori? Integrate criterion weights with binary decision variables in a single

• ptimisation problem.

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 4 / 11

Integrated formulations

max ∑r wr(∑j cr

j xj)

s.t. aix bi 8i 2 I

∑r wr = 1

wr 0 8r 2 R xj 2 f0, 1g 8j 2 J.

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 5 / 11

Integrated formulations

max ∑r wr(∑j cr

j xj)

s.t. aix bi 8i 2 I

∑r wr = 1

wr 0 8r 2 R xj 2 f0, 1g 8j 2 J. (w , x) optimal ) x is a supported e¢cient portfolio.

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 5 / 11

Integrated formulations

max ∑r wr(∑j cr

j xj)

s.t. aix bi 8i 2 I

∑r wr = 1

wr 0 8r 2 R xj 2 f0, 1g 8j 2 J. (w , x) optimal ) x is a supported e¢cient portfolio. Linearised via the transformation wrxj = zr

j .

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 5 / 11

Integrated formulations

max ∑r wr(∑j cr

j xj)

s.t. aix bi 8i 2 I

∑r wr = 1

wr 0 8r 2 R xj 2 f0, 1g 8j 2 J. (w , x) optimal ) x is a supported e¢cient portfolio. Linearised via the transformation wrxj = zr

j .

max ∑r ∑j zr

j cr j

s.t. aix bi 8i 2 I

∑r wr = 1

wr 0 8r 2 R xj 2 f0, 1g 8j 2 J zr

j 0

zr

j wr

• 8(r, j) 2 R J

∑j zr

j xj, 8j 2 J.

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 5 / 11

Integrated formulations

max ∑r wr(∑j cr

j xj)

s.t. aix bi 8i 2 I

∑r wr = 1

wr 0 8r 2 R xj 2 f0, 1g 8j 2 J. (w , x) optimal ) x is a supported e¢cient portfolio. Linearised via the transformation wrxj = zr

j .

max ∑r ∑j zr

j cr j

s.t. aix bi 8i 2 I

∑r wr = 1

wr 0 8r 2 R xj 2 f0, 1g 8j 2 J zr

j 0

zr

j wr

• 8(r, j) 2 R J

∑j zr

j xj, 8j 2 J.

(z, w , x) optimal ) x is a supported e¢cient portfolio.

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 5 / 11

Enumerating supported e¢cient portfolios

Basic Idea: Identify a di¤erent portfolio through the introduction of two cutting planes.

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 6 / 11

Enumerating supported e¢cient portfolios

Basic Idea: Identify a di¤erent portfolio through the introduction of two cutting planes. Example: Suppose we have identi…ed xk and let Πk = fj 2 Jjxk

j = 1g

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 6 / 11

Enumerating supported e¢cient portfolios

Basic Idea: Identify a di¤erent portfolio through the introduction of two cutting planes. Example: Suppose we have identi…ed xk and let Πk = fj 2 Jjxk

j = 1g

1

∑

j2Πk

xj jΠkj 1

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 6 / 11

Enumerating supported e¢cient portfolios

Basic Idea: Identify a di¤erent portfolio through the introduction of two cutting planes. Example: Suppose we have identi…ed xk and let Πk = fj 2 Jjxk

j = 1g

1

∑

j2Πk

xj jΠkj 1

2 ∑

r

wr ∑

j2Πk

cr

j ∑r ∑j zr j cr j

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 6 / 11

Enumerating supported e¢cient portfolios

Basic Idea: Identify a di¤erent portfolio through the introduction of two cutting planes. Example: Suppose we have identi…ed xk and let Πk = fj 2 Jjxk

j = 1g

1

∑

j2Πk

xj jΠkj 1

2 ∑

r

wr ∑

j2Πk

cr

j ∑r ∑j zr j cr j

In this fashion we can enumerate the set of supported e¢cient portfolios.

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 6 / 11

Incorporating preferences

Preferential statements give rise to constraints on the weights.

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 7 / 11

Incorporating preferences

Preferential statements give rise to constraints on the weights. Example: xA xB

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 7 / 11

Incorporating preferences

Preferential statements give rise to constraints on the weights. Example: xA xB V (xA) V (xB) , ∑

r

wr(∑

j

cr

j xA j ) ∑ r

wr(∑

j

cr

j xB j ) , ∑ r

wrvr 0 (where vr = ∑

j

cr

j (xA j xB j ))

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 7 / 11

Incorporating preferences

Preferential statements give rise to constraints on the weights. Example: xA xB V (xA) V (xB) , ∑

r

wr(∑

j

cr

j xA j ) ∑ r

wr(∑

j

cr

j xB j ) , ∑ r

wrvr 0 (where vr = ∑

j

cr

j (xA j xB j ))

Overall, this restricts the space of weights to a polyhedral preference cone W : W = fw 2 Rp

j ∑ r

vr

f wr 0 8f 2 Pref g

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 7 / 11

Incorporating preferences

Preferential statements give rise to constraints on the weights. Example: xA xB V (xA) V (xB) , ∑

r

wr(∑

j

cr

j xA j ) ∑ r

wr(∑

j

cr

j xB j ) , ∑ r

wrvr 0 (where vr = ∑

j

cr

j (xA j xB j ))

Overall, this restricts the space of weights to a polyhedral preference cone W : W = fw 2 Rp

j ∑ r

vr

f wr 0 8f 2 Pref g

Preferences can be incorporated by appending W to our formulation.

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 7 / 11

A naive interactive procedure

Assume the existence of an implicit value function and consistency of the Decision Maker

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 8 / 11

A naive interactive procedure

Assume the existence of an implicit value function and consistency of the Decision Maker Let ¯ x be a current incumbent portfolio (best so far).

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 8 / 11

A naive interactive procedure

Assume the existence of an implicit value function and consistency of the Decision Maker Let ¯ x be a current incumbent portfolio (best so far). We compare ¯ x with x, identi…ed by solving the following:

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 8 / 11

A naive interactive procedure

Assume the existence of an implicit value function and consistency of the Decision Maker Let ¯ x be a current incumbent portfolio (best so far). We compare ¯ x with x, identi…ed by solving the following: max β = ∑

r ∑ j

wrxjcr

j ∑ r ∑ j

wr ¯ xjcr

j

s.t. ∑

r

wr = 1, w 2 W , x 2 X.

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 8 / 11

A naive interactive procedure

Assume the existence of an implicit value function and consistency of the Decision Maker Let ¯ x be a current incumbent portfolio (best so far). We compare ¯ x with x, identi…ed by solving the following: max β = ∑

r ∑ j

wrxjcr

j ∑ r ∑ j

wr ¯ xjcr

j

s.t. ∑

r

wr = 1, w 2 W , x 2 X. Solving iteratively, the incumbent solution converges to a preferred solution (when β = 0).

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 8 / 11

A general interactive scheme

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 9 / 11

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 10 / 11

Questions/Comments

Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 11 / 11