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 ’ ( c 1 x , ..., c p x ) s.t. a i x � b i 8 i 2 I x j 2 f 0 , 1 g 8 j 2 J . J = f 1 , ..., n g : The set of n projects. c r , r 2 R = f 1 , ..., p g : Objective vectors, assumed non-negative. a i , i 2 I = f 1 , ..., m g : Resource utilisation vectors. b i , 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 optimisation problem. Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 4 / 11

Integrated formulations max ∑ r w r ( ∑ j c r j x j ) s.t. a i x � b i 8 i 2 I ∑ r w r = 1 w r � 0 8 r 2 R x j 2 f 0 , 1 g 8 j 2 J . Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 5 / 11

Integrated formulations max ∑ r w r ( ∑ j c r j x j ) s.t. a i x � b i 8 i 2 I ∑ r w r = 1 w r � 0 8 r 2 R x j 2 f 0 , 1 g 8 j 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 w r ( ∑ j c r j x j ) s.t. a i x � b i 8 i 2 I ∑ r w r = 1 w r � 0 8 r 2 R x j 2 f 0 , 1 g 8 j 2 J . ( w � , x � ) optimal ) x � is a supported e¢cient portfolio. Linearised via the transformation w r x j = z r j . Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 5 / 11

Integrated formulations max ∑ r ∑ j z r j c r j max ∑ r w r ( ∑ j c r j x j ) s.t. a i x � b i 8 i 2 I s.t. a i x � b i 8 i 2 I ∑ r w r = 1 ∑ r w r = 1 w r � 0 8 r 2 R w r � 0 8 r 2 R x j 2 f 0 , 1 g 8 j 2 J x j 2 f 0 , 1 g 8 j 2 J . � z r j � 0 8 ( r , j ) 2 R � J z r j � w r ( w � , x � ) optimal ) x � is ∑ j z r j � x j , 8 j 2 J . a supported e¢cient portfolio. Linearised via the transformation w r x j = z r j . Argyris, Figueira, Morton (LSE / IST) A new approach for the MCPSP 02/10 5 / 11

Integrated formulations max ∑ r ∑ j z r j c r j max ∑ r w r ( ∑ j c r j x j ) s.t. a i x � b i 8 i 2 I s.t. a i x � b i 8 i 2 I ∑ r w r = 1 ∑ r w r = 1 w r � 0 8 r 2 R w r � 0 8 r 2 R x j 2 f 0 , 1 g 8 j 2 J x j 2 f 0 , 1 g 8 j 2 J . � z r j � 0 8 ( r , j ) 2 R � J z r j � w r ( w � , x � ) optimal ) x � is ∑ j z r j � x j , 8 j 2 J . a supported e¢cient portfolio. ( z � , w � , x � ) optimal ) Linearised via the x � is a supported e¢cient transformation w r x j = z r j . 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 x k and let Π k = f j 2 J j x k j = 1 g 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 x k and let Π k = f j 2 J j x k j = 1 g x j � j Π k j � 1 ∑ 1 j 2 Π k 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 x k and let Π k = f j 2 J j x k j = 1 g x j � j Π k j � 1 ∑ 1 j 2 Π k w r ∑ 2 ∑ j � ∑ r ∑ j z r c r j c r j r j 2 Π k 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 x k and let Π k = f j 2 J j x k j = 1 g x j � j Π k j � 1 ∑ 1 j 2 Π k w r ∑ 2 ∑ j � ∑ r ∑ j z r c r j c r j r j 2 Π k 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: x A � x B 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: x A � x B V ( x A ) � V ( x B ) , ∑ j ) � ∑ j ) , ∑ w r v r � 0 w r ( ∑ c r j x A w r ( ∑ c r j x B r j r j r (where v r = ∑ c r j ( x A j � x B j ) ) 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: x A � x B V ( x A ) � V ( x B ) , ∑ j ) � ∑ j ) , ∑ w r v r � 0 w r ( ∑ c r j x A w r ( ∑ c r j x B r j r j r (where v r = ∑ c r j ( x A j � x B j ) ) j Overall, this restricts the space of weights to a polyhedral preference cone W : f w r � 0 8 f 2 Pref g � j ∑ W = f w 2 R p v r r 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: x A � x B V ( x A ) � V ( x B ) , ∑ j ) � ∑ j ) , ∑ w r v r � 0 w r ( ∑ c r j x A w r ( ∑ c r j x B r j r j r (where v r = ∑ c r j ( x A j � x B j ) ) j Overall, this restricts the space of weights to a polyhedral preference cone W : f w r � 0 8 f 2 Pref g � j ∑ W = f w 2 R p v r r 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). x with x � , identi…ed by solving the following: We compare ¯ 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). x with x � , identi…ed by solving the following: We compare ¯ max β = ∑ j � ∑ w r x j c r w r ¯ x j c r r ∑ r ∑ j j j s.t. ∑ w r = 1 , w 2 W , x 2 X . r 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). x with x � , identi…ed by solving the following: We compare ¯ max β = ∑ j � ∑ w r x j c r w r ¯ x j c r r ∑ r ∑ j j j s.t. ∑ w r = 1 , w 2 W , x 2 X . r 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