7. Fuzzy Soft-Constraint Portfolio Optimisation · optimisation methodology: overlaying a...

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7. Fuzzy Soft-Constraint Portfolio Optimisation

7.1 Introduction

A Mathematical Programming (MP) [Bazaraa & Shetty 1979; Solow 1984] framework for constrained

optimisation analysis generally refers to the process of performing a criterion-driven search over a

reduced decision space, whereby the problem’s objective function directs the search path and its

constraint relations define the boundaries of the feasible region. In practice, there is often a degree of

imprecision and/or flexibility with regard to the constraint specifications. A fuzzy set [Bellman & Zadeh

1970; Fuller & Zimmermann 1993] framework can address both the numerical imprecision of the

coefficients in the mathematical expressions as well as the decisional flexibility concerning the satisfaction

of each constraint relation. This latter feature enables us to better capture our optimisation/decision model.

The use of soft-constraint relations facilitates modelling system objectives which are best represented as

statements of goal-satisfactions as well as system constraints which can be relaxed, i.e. at some incurred

penalties. In particular, our work is motivated by a decision environment whereby a primary optimising

objective function is supplemented by secondary goal-satisfaction objectives and by soft-bounds on the

decision variables.

We propose to model these supplementary criteria as fuzzy soft-constraint relations [Dubois & Prade

1980] with hyperbolic membership characteristics [Dhingra et al. 1992]. The fuzzy soft-constraint

relations, in turn, are enforced via penalty-reward functions, linear in the membership values themselves.

Under a Multi-Criteria Decision Model (MCDM) [Chankong & Haimes 1983] framework, these

penalty-rewards are then combined together with the primary optimising criterion, resulting in a fuzzy

multi-criteria performance evaluation function P X: →ℜ , defined over the (vector) solution space X.

With this, P x( ) simultaneously reflects how well a particular solution x X∈ optimises the primary

optimisation objective as well as how much it satisfies, or fails to satisfy, the various soft-constraints

specified.

Our use of one optimising criterion amidst a multiplicity of soft-constraints is a legacy of our motivating

application’s Linear Programming (LP) [Dantzig 1966; Solow 1984] based formulation. In a general

case, our framework extends naturally to the case containing a multiplicity of optimising criteria as well.

7 Fuzzy Soft-Constraint Portfolio Optimisation

211

Our hyperbolic fuzzification of the soft-constraints now renders our P( ) a nonlinear function, whence

necessitating a Nonlinear Programming (NLP) [Bazaraa & Shetty 1979] type formulation.

Our general strategy is to construct P x( ) for a particular problem by examining its optimisation/decision

model and its soft-constraint modelling requirements, then maximise this non-linear function, subject only

to the problem’s remaining ‘hard’-constraints, as summarised by x F∈ .

It is hoped that our proposed fuzzy soft-constraint formulationwith shared commonalities with Fuzzy

Mathematical Programming (FMP) [Tanaka et al. 1974; Fuller & Zimmermann 1993] Goal

Programming (GP) [Charnes et al. 1955] and particularly Fuzzy Goal Programming (FGP)

[Zimmermann 1978; Narasimhan 1980; Hannan 1981; Ignizio 1982; Rao et al. 1992]achieves the soft-

constraint modelling realism of fuzzy formalism along with the multi-criteria modelling practicality of

goal-satisficing decision models. Moreover, we hope that it affords the optimisation/decision model a

greater degree of modelling flexibility, allowing optimising criteria to exist alongside goal-aspirations,

hard-bounds alongside soft-bounds, and crisp definitions with fuzzy entities, and so on.

In term of implementation, we first note thatP x( ) is mathematically equivalent to performing a series of

linear-algebraic operations followed by hyperbolic sigmoidal set-partitionings, i.e. the functionalities of a

feed-forward mapping Neural Network (NN), particularly the Multi-Layered Perceptron (MLP)

architecture [Rumelhart & McCelland 1986; Pao 1989; Haykin 1994]. As such, any readily available

Object-Oriented (OO) [Stroustrup 1991] implementation of NN function classes, e.g. [Masters 1993],

can be tailored to encode this solution-performance mapping P x( ) . The NN-encoded parameters can be

divided into two groups: those pertaining to the problem itself (i.e. the coefficients of the objective

function and the constraint relations), and those pertaining to the fuzzification/MCDM/optimisation model

(i.e. the sigmoidal temperature parameters which control the shapes of the hyperbolic membership

functions, the maximum penalty/reward points, etc.). Let us refer to the latter collectively as the fuzzy

utility parameters. Here, as with MCDM in general, a decision maker is distinguished by his/her set of

fuzzy utility parameters. Unlike in part one of the thesis, here the model parameters are not optimised

variables.

In theory, a learning NN, partially encoded with the problem data only, can be trained to ‘self-

parameterise’ the fuzzy utility parameters, i.e. based on how a particular decision maker ranks or rates

different solutions to the same problem. These parameters can then be extracted from a trained NN. Such

a modelling exercise brings to the fore two of the cornerstones in a neural information processing system:

a neural architecture’s ability to implement a nonlinear mapping, and a neuro-model’s ability to perform

experiential learning and induce an implicit mapping model. We propose such a learning

NN/connectionist framework as a basis for future direction in adaptive, case-based induction of the

decision maker’s implicit fuzzy utility function.

At any rate, the optimisation problem posed is of the form { }max ( )P x x F∈ , where P x( ) , NN-

Evolutionary Optimisation and Financial Model-Trading

212

encoded or otherwise, is nonlinear and our F is discrete. This leaves us two choices in terms of an

optimisation methodology: overlaying a branch-and-bound search over a succession of continuous-valued

NLP relaxations, or employing a stochastic-evolutionary search engine, i.e. an Evolutionary

Optimisation (EO) [Fogel 1994a] algorithm, to perform direct (numerical) optimisation [Schwefel 1995]

over the discretised space itself. We opted for the latter strategy, which addresses the more general case of

when P x( ) might not be differentiable and may contain multiple local maxima. In such a case, it is

inappropriate to employ an indirect (analytical) approach, as in classical NLP.

The chapter is organised as follows: section 7.2 examines a particular multi-criteria FX/Futures trading

portfolio optimisation problem, for which section 7.3 derives a fuzzy soft-constraint formulation. Section

7.4 describes the NN encoding and EO method for solving the formulated discrete nonlinear optimisation

problem and proposes how to generalise the hybrid fuzzy neuro-evolutionary framework for optimisation

analysis. In section 7.5, we examine an example portfolio solution and run through the model specification

steps in our prototype software, and section 7.6 concludes the discussion.

7.2 FX/Futures Trading Portfolio Model

The Quantitative Research and Trading (QRT) group at Chemical Bank (now Chase Manhattan), London,

had developed daily technical trading models (DTMs) for a number of liquid foreign exchange (FX)

rates in the spot market as well as for interest-rate and equity-index futures. Suppose that QRT has taken

on to model-trade n FX/Futures contracts, i.e. take positions according to n DTMs, each of which may

be reading a ‘Long’, ‘Square’, or ‘Short’ signal at a given time. The portfolio optimisation problem is to

determine the set of portfolio weights { }w i n wi ii≥ = =∑0 1 1, , , ,K each wi of which, when

multiplied to some fixed notional, gives the size of the ‘Long’ or ‘Short’ position to be taken or

maintained in the ith underlying spot/contract when the corresponding ith DTM does generate a ‘Long’ or

‘Short’ signal.

This clearly forces a break with the ‘invest-and-hold’ definition of the portfolio weights, as per the

traditional asset allocation problem. In essence, what is being allocated is the proprietary account’s

exposure to the performance of each DTM in anticipating price movements and generating appropriate

trading signals. For this and other considerations, this ‘portfolio of models’ was not formulated in the

classical Markowitz [Markowitz 1952; Markowitz 1959] risk-return framework. Moreover, because

FX/Futures transaction orders are carried out in even (US$-denominated) lot sizes, each portfolio weight

is discretised to a multiple of some percentage unit (e.g. 5%) corresponding to an appropriate minimum

transaction order size. Formally, { }w D N N N N Ni N n∈ =≥ 0 1 2, , , ,K (e.g. N = 20).


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Our portfolio model is based principally on maximising the weighted sum of the individual DTM Risk-

Adjusted Returns (RAR)1. This maximising criterion is supplemented by three goal-satisfaction criteria

based on technical trading performance statistics [Kaufman 1987; Kaufman 1995], namely that the

weighted sums of the Relative-Return (RR), Sharpe-Ratio (SR) and the Probability-of-Ruin (PR)

measures should be above (or below) their respective thresholds. The rationale given is that while the

supplementary criteria are crucial in capturing the profitability of the trading portfolio and the money

management viability of the portfolio trading, they are not as information-rich, nor as directly applicable

to portfolio models in general, as the risk-adjusted-return measure.2 Furthermore, we include lower and

upper soft-bound exposure limits on each weight variable to ensure intra-portfolio diversification.

In general, we denote fuzzy concepts of ‘at least’, ‘at most’, ‘equal’, and ‘not equal’, respectively, with

‘ ~> ’, ‘ ~< ’, ‘ ~= ’, and ‘~≠ ’. Letting x denote a vector of n portfolio allocation weights, the n -DTM

portfolio optimisation model is stated thus:

{ }

max~

~ ~

c x

Ax b

L x U

x W x D x

t

Nn

subject to >

< <

∈ = ∈ =1

1

(1)

where c

RAR

RARn

=

1

M , A

RR RR

SR SR

PR PR

n

n

n

=

− −

1

1

1

L

L

L

, and b

RR

SR

PR

thresh

thresh

thresh

=

−

, with RAR j , RR j , SR j , and PR j ,

j n= 1, ,K , denoting, respectively, the jth DTM’s individual risk-adjusted-return, relative-return, Sharpe-

ratio and probability-of-ruin measures, while the subscript thresh denotes setting the performance goal

thresholds, i.e. the goal-aspiration levels that the solution portfolio is to achieve. The lower and upper

bounds are uniformly applied: [ ] [ ]L U L U j nj j, , , , ,= = 1K . We consider two such intervals in our

experiments, [0.05,0.80] and [0.10,0.60].

7.3 Fuzzy Soft-Constraint Optimisation/Decision Model

First, we introduce some notations:

1 The correlations of returns (analogous to the off-diagonal elements in the Markowitz variance-covariance matrix) are negligible---a correlation filter having been applied in the portfolio selection process in order to ensure diversification---and no longer enters into the analysis. In contrast, the Markowitz approach places a closer scrutiny on the off-diagonal elements, whence its concept of portfolio risk differs markedly from a collection of individual return risks. In particular, it seeks to exploit the possibility of portfolio risk reduction via optimal diversification, which, once again, is more meaningful within the ‘portfolio of investments’ context. Perhaps a unified risk management framework bridging the underlying-price-volatility and model-trading-return concepts of risk is needed and warrants further investigations.

2 The relative-return does not include a risk measure. The Sharpe-ratio uses the underlying price movement volatility in place of the DTM volatility of returns. The probability-of-ruin criterion echos the belief in a deliberate integration between investment and money management strategies [Dunis & Feeny 1989].


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Sigmoidal Transfer Function:

[ ]℘ =℘ ≡+

≡− +

∈ ∈ℜ− −( ) ( )tanh(( ) / )

( , ) , ,,( )/v v

e

vvv

θ τθ τ

θ τ1

1

2 1

20 1

(2)

where τ > 0 is the shape (temperature) parameter controlling the steepness of this monotone function,

and θ is the threshold parameter. Note ℘ = ′℘ =( ) . , ( )θ θ τ05 1 4 .

Linear Re-Scaling Function:

[ ][ ] [ ] [ ]ζ ζ ζ( ) ( )

( )

( )( ) , , ( ) ,

,

,v v c

d c

b av a y a b v c d

a b a

c d= ≡ +

−−

− ∈ ⇒ ∈≠

(3)

7.3.1 Fuzzy Set Membership Functions

Let us focus initially on a fuzzy soft-constraint relation { }A x bi i• >~ , for which we define a slack

‘function’, s x A x bi i i( ) ≡ −• . Consider a hyperbolic fuzzy set membership function:

{ }

[ ]µ µ τθ τi A x b i ix x s x

i i

i i( ) ( ) ( ( )) ( , ) ,~,= ≡℘ ∈ >

• >0 1 0 (4)

This particular form of µi x( ) exhibits several desirable features from the optimisation/decision modelling

point of view. First and foremost, µi x( ) serves as a set-partition function, defining the boundary between

the constraint-satisfying subspace { }x A x bi i• ≥ and its complement; solutions in the former are to be

rewarded, those in the latter to be penalised. On the other hand, as a robust fuzzy set function, µi x( )

‘blurs’ this constraint-satisfied/constraint-violated set boundary. That is, µi x( ) is functionally smooth,

and makes for a relatively gradual transition from 0 05< <µi x( ) . for negative s xi ( ) , to µi x( ) .= 05 at

s xi ( ) = 0 and onto 05 1. ( )< <µi x for positive s xi ( ) , saturating to unity as s xi ( ) >> 0 and diminishing to

zero as s xi ( ) << 0 . Moreover, µi x( ) is also explicitly parameterised by θi which locates the set-partition

boundary, i.e. where µi x( ) .= 05 , and by τ i which controls the ‘sharpness’ of the set-partitioning. They

are the fuzzy parameters of our optimisation model. This hyperbolic membership function can also be said

summarily to exhibit a decreasing coefficient of membership satiation [Dhingra et al. 1992] m( )x ,

defined as the second derivative of the membership function:

m( ) ( )x x= ′′µ (5)

which is positive for x with negative slack ( ( ) )s xi < 0 and negative for x with positive slack

( ( ) )s xi > 0 .

7.3.2 Fuzzy Penalty-Reward Functions

Here we derive a fuzzy penalty-reward function, denoted Φi x( ) , which assigns some negative point

(penalty) to a solution identified with the fuzzy “{ }A x bi i• ≥ -violating”' concept and some positive point


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(reward) to one identified with “{ }A x bi i• ≥ -satisfying”. Each Φi x( ) is to be in direct proportionality, not

with the slack measure s xi ( ) , but with the membership function value µi x( ) itself, whence a linear re-

scaling of µi x( ) . Let [ ]P Ri i≤ >0 0, denote the maximum penalty and the minimum reward points, i.e.

assigned to solutions with µi x( ) approaching zero and one respectively. The fuzzy penalty-reward

function is symbolically represented by the composition ( )Φ( ) ( )= ℘ζ o , and is given by:

[ ][ ]

{ } ( )ΦiP R

A x b i ix x P R i mi i

i i( ) ( ) , , , , ,

,

,~=

∈ =

• >ζ µ

0 11K

(6)

where θi now locates the boundary where a solution is neither penalised nor rewarded (Φi x( ) exactly

equals zero). The fuzzy penalty-reward functions enforcing the soft-bound limits on the portfolio weights

are similarly constructed, except that the two-sided bound on each variable requires two sigmoidal

functions, two sets of fuzzification parameters, and is a scalar function of the individual elements

x x xj n1, , , ,K K :

[ ][ ]

{ }

[ ][ ]

{ }

Ψ Ψ Ψ

Ψ

Ψ

j j jL

j jU

j

jL

j

P R

x L j

jU

j

P R

x U j j j

x x x

x x

x x L U j n

jL

jL

j j

jU

jU

j j

( ) ( ) ( )

( ) ( ) ,

( ) ( ) , , , ,

,

,

~

,

,

~

= +

=

=

< =

>

<

ζ µ

ζ µ

0 1

0 11K

(7)

7.3.3 Fuzzy Multi-Criteria Objective Function

For the portfolio problem, the optimising objective function is optionally3 re-scaled as:

[ ][ ]

Λ( ) ( ) ,min max

max

,

,x c x

c c

o t=>

ζ0 0

(8)

where c cj n jmin , ,min { }= =1K and c cj n jmax , ,max { }= =1K are determined, respectively, by ‘solving’

{ }min c x x Wt ∈ and { }maxc x x Wt ∈ [Dhingra et al. 1992].

Let omax and the P ’s and the R ’s be our MCDM parameters, reflecting the relative importance among

each of the ( )1+ +m n criteria. Together with the fuzzy parameters (the τ ’s and the θ ’s) already

defined, they constitute the fuzzy utility parameters of our optimisation/decision model. The fuzzy multi-

criteria performance evaluation function P x( ) is simply the summation of a (re-scaled) objective

function and the fuzzy penalty-rewards:

3 Because the optimising objective may grow, in this case, linearly in x , without bound, while the maximum penalty points are finite, the problem is bounded only by the remaining ‘hard’ constraints. In this case, F W= suffices, i.e. constrains the solution space, but in general, the optimising objective function is to be normalised [Rao et al. 1992].


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}

{P x x x xii

m

jj

n

( ) ( ) ( ) ( )

Fuzzy Multi-CriteriaPerformance Evaluation

optimisingcriterion supplementary

performancecriteria

soft-boundconstraintson variables

= + += =∑ ∑Λ Φ Ψ

1 11 24 34 1 24 34

(9)

7.3.4 Solution-Parameter Optimisation Problem

Altogether, we formalise the following discrete non-linear optimisation problem:

max ( ) ( ) ( ) ( )P x x x x

x W

ii

m

jj

n

= + +

∈= =∑ ∑Λ Φ Ψ

1 1

subject to

(10)

This is solution-parameter optimisation problem in the sense that our solution is a set of scalar quantities.

Our fuzzy soft-constraint formulation retains a formal equivalence with the FGP model. To wit, let there

be no penalty assignment (maximum penalty points of zero), define w o0 = max, w R i mi i≡ =, , ,1K ,

and w R R j n wm j jL

jU

ll

m n+ =

+≡ + = =∑, , , ,1 1

0K , and artificially introduce a piecewise-linear

membership function [Bellman & Zadeh 1970; Dhingra et al. 1992] on the objective function:

[ ][ ] ( )µ ζ00 1

0 1( ) max ,min ,min max,

,x c x

c ct≡

. As µ0( )x remains completely linear within the range

c c x ctmin max≤ ≤ , the optimising criterion component of P x( ) is not affected (the effect of the linear

rescaling can be reversed with a new set of MCDM weightings).

Clearly, this modified form ′ ==

+∑P x w xll

m nl( ) ( )

0µ is a weighted additive fuzzy achievement function, as

per FGP formulation [Rao et al. 1992]. In other word, FGP derives P x( ) directly from the fuzzy

membership functions defined on both the goal-satisficing as well as on the optimising criterion. Our

handling the fuzzy soft-constraints through a penalty-reward scheme, however, affords what we believe to

be a more precise as well as intuitive control of the fuzzy-constraint set-partitioning. Moreover, with

penalty assignments it is relatively straightforward to benchmark the fuzzy soft-constraint reformulation

against the hard-bound version, i.e. a LP, which can be thought of as maximising P x( ) with very large

penalty assignments and the sigmoidal shape parameters approaching zero.

As an illustration, a hypothetical two-variable problem is depicted graphically in Figure 7-1. It resembles

a LP problem, except within the ‘true’ feasible region, the F ≡ ×[ , ] [ , ]0 1 0 1 square, the shaded ‘pentagon’

represents the sub-region bounded by the soft-constraints. The thick arrow represents the objective

coefficient vector. For some given set of fuzzy utility parameters4, P x x( , )1 2 yields an evaluation surface

4 As a fuzzy-based analysis is generally robust w.r.t. the fuzzy parameters/functions themselves, in terms of actual modelling exercise, the analyst may assume a few ‘sensible’ sets of fuzzy utility parameters and interactively discuss the solution alternatives with the decision maker.


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such as the one depicted in Figure 7-2. Notice the rectangular depression corresponding the fuzzy penalty-

reward functions enforcing the [ . , . ]0 05 080 soft-bounds on both variables as well as the 5-sided ‘plateau’

directly above the said soft-constrained sub-region, whose ‘plane’ can be seen to tilt upward in the

direction corresponding to the linear improvement in the underlying LP’s objective function. The

remaining optimisation task is to locate the highest point of this surface over [ , ] [ , ]0 1 0 1× :

x2

x1

Figure 7-1: Linear Programming Backdrop of a Fuzzy Soft-Constraint Optimisation Problem

Figure 7-2: Fuzzy Multi-Criteria Performance Evaluation Function/Surface

Our portfolio problem is essentially a higher dimensional extension of this example, but with introduced

granularity or discretisation W D Nn⊆ as well as the portfolio weight definition { }W x⊆ =

11 .


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7.4 Neuro-Evolutionary Methodology & Extensions

7.4.1 Mapping NN Architecture & Evolutionary Optimisation

Consider the feed-forward neural architecture depicted in Figure 7-3. The first set of connections forms a

dot product c xt for a given input (solution) vector x ∈ℜ4 , and this value is then re-scaled into Λ( )x .

The 3-by-5 weight matrix implements the Ax multiplication, yielding the inputs to the three hidden-layer

neurons fuzzifying the goal-satisfaction soft-constraints. Each portfolio weight variable is also input to a

pair of input-layer neurons whose sigmoidal non-linearities fuzzify the soft-bound exposure limits. The

connections to the output nodes linearly re-scales the neurons’ outputs (fuzzy membership values) into

penalty/reward assignments. For a given set of NN-encoded fuzzy utility parameters, the NN-mapped

output, NN( )x , is functionally equivalent to the performance evaluation function P x( ) :

c xt

Ax

P x( )

x2x1 x3 x4

x1

x2

x3

x4

Figure 7-3: Neural Architecture Encoding Fuzzy Performance Function

It remains for an EO engine is to search over W to maximise the NN-encoded P x( ) , whence the latter

constitutes the EO algorithm's fitness evaluation function. To stress the practical benefits over a MP

approach, we note how a structural constraint such as the discretisation of the portfolio weight space

x D Nn∈ can be implemented simply and elegantly by designing a genotype-phenotype map, as in the

classical Genetic Algorithm (GA) [Holland 1975], which actually decodes onto a structurally feasible


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solution set, i.e. only generates solutions from W itself. In contrast, one only has to imagine an equivalent

branch-and-bound formulation over the n -dimensional Cartesian grid, whereby each node corresponds to

a complete NLP problem!

In our proposed framework, the particular EO algorithm employed, Simulated Annealing Genetic

Evolution (SAGE), is based on introducing genetic search (crossover) operator into Evolutionary

Simulated Annealing (ESA) [Yip & Pao 1993b], itself an approach based on direct parameter-encoding

Evolutionary Programming (EP) [Fogel et al. 1966], but one which additionally utilises Simulated

Annealing (SA) [Kirkpatrick et al. 1983; Cerny 1985] as a local mechanism for escaping local optima and

as a stochastic method of fitness evaluation.

Note that the portfolio optimisation problem considered here is intrinsically linear; the non-linearities in

P x( ) are due to fuzzification. As such, it lends itself neatly to a ‘standard’ feed-forward NN architecture

with ‘dot-producted’ connection weights. Nonetheless, it is a straight-forward matter to generalise our

neural representation of P x( ) to problems containing non-linear expressions, e.g. a Quadratic

Programming (QP) [Bazaraa & Shetty 1979] formulation of a fund manager's Markowitz [Markowitz

1959] portfolio. For higher-order functionalities, we refer to a Functional Link Net (FLN) architecture

[Pao 1989], which has similarly well-defined mapping and learning behaviours.

So far, we have only partially utilised the neural information processing capability, namely a neural

architecture’s ability to implement a nonlinear mapping. Next, we explore the more definitive part of

neural information processing, namely a neuro-model’s ability to perform experiential learning and induce

an implicit mapping model.

7.4.2 Learning NN & Fuzzy Multi-Criteria Utility Induction

The above mathematical/conceptual equivalence between the mathematical P x( ) and the neural P x( ) is

essentially a matter of implementation convenience, an OO exercise in labour saving. Operating strictly in

‘recall mode’, the encoded-NN is employed solely for its mapping functionality.

Now we propose to exploit its learning functionality. Suppose a decision maker is asked, in a MCDM

setting, to rank and/or rate U solutions, presumably on basis of how well each solution x k pk , , ,= 1K ,

optimises the primary objective function as well as how much it satisfies, or fails to satisfy, the various

soft-constraints specified by the problem.5 Clearly, we now have p examples, or patterns, each consisting

of an ‘input’ solution vector x u and the corresponding ‘output’ numerical evaluation, Pu ∈[ , ]0 1 . Let us

denote this training set by ( ) ( ) ( ){ }l K K= x P x P x Pu u U U1 1, , , , , , , .

5 To maintain compatibility with FGP proper, further stipulate that such a ranking/rating is bounded to within [0,1].


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Suppose that we specify a particular feed-forward NN architecture to satisfy the functional requirements

necessary for implementing P x( ) , but then only partially encode the NN parameters, i.e. only those

which pertain to the problem data themselves, i.e. in our case these correspond to the RAR, RR, SR, and

PR numbers and the lower and upper bounds. We would then let the NN learn, in an l -supervised

manner, allowing only adjustments to those NN parameters which correspond to the fuzzy utility

parameters. As usual, the NN training process corresponds to the following mapping error minimisation:

( )min , ,1

1

1

1 2pu u d

u

U d

P P x d−

=

=∑

(11)

In this way, the decision maker’s inconsistency does actually destroy the consistency of the multi-criteria

utility model. Here, the decision maker’s inconsistent rankings will show up as data fitting noise. This

constitutes an example/case-based induction of the decision maker's implicit fuzzy utility parameters,

which are then extracted from a trained NN. It is hoped that future work exploring this possibility will tie

in with theoretical and practical research on fuzzy utility theory [Yoneda et al. 1993; Nishizaki & Seo

1994; Billot 1995].

7.4.3 Evolutionary NN Learning & TMP Formulation

The true power of NN learning is in its ability to create a universal approximation model for any

continuous map. Our situation does not require the full generality of the universal approximation theorem

[Hornik et al. 1989]. In fact, not only is our NN architecture dictated by the number of optimising criteria

and/or soft-constraints, some NN parameters may actually need to be hard-coded by various problem-

defined constants. For example, in our case, some connection weights correspond to the DTMs’ various

performance statistics, and are hard-coded accordingly.

But what might be required within our generalised fuzzy multi-criteria utility induction framework above

and beyond tradition neuro-modelling exercises, especially those based on the MLP architecture, is the

ability to combinatorially mix and match different classes of neuronal transfer functions.

In our application, the soft-constraints correspond to the fuzzy notion of goal-satisfaction, as embodied by

the ‘~> ’ and ‘ ~< ’ notations. As such, the appropriate neuronal transfer is the hyperbolic/logistic sigmoid,

which is a monotonic, amodal function. Suppose, on the other hand, that in some other applications, some

soft-constraints correspond to the fuzzy notion of equality, embodied by the ‘~= ’ notation. In such a case,

a unimodal, non-monotonic function, i.e. a Gaussian radial basis, would be needed.

Now, in some applications, it may be possible to a priori decide which soft-constraint is of what form,

sigmoidal, piecewise linear, piecewise unimodal, Gaussian, etc. However, the truly general framework

should be able to induce this type of information from the training set l itself. Gradient-based MLP

learning [Werbos 1974; Werbos 1994], however, is generally about parameterising the NN map, given the

architecture. The issue of NN architecture is often a matter of specifying the number of neurons, the

number of layers, and the extent of inter-neuronal connectivity, generally given the types of neuronal


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transfer functions. In our case, the number of neurons correspond to the number of objective criteria and

soft-constraints; there is essentially one ‘hidden layer’; and the NN connectivity is determined by the

problem itself.

The open question which remains concerns what combination of neuronal transfer function classes is

appropriate. Suppose we had M > 1 multi-criteria, each of whose fuzzification is to be achieved via a

parameterised neuron from one of the K = 6 classes available: k Increas ingPiecewiseLinear,

k Decreas ingPiecewiseLinear, k Increas ingLogisticSigmoid, k Decreas ingLogisticSigmoid, k GaussianRadialBasis, and

k InvertedGaussianBas in. These classes correspond, respectively, to ~> , ~< , ~> , ~< , ~= , and ~≠ fuzzy concepts.

Considered as a single specialised functionality, all K = 6 classes are referred to by a single class

template group K NeuroFuzzySetPartition. The two-stage problem of specifying the class configuration as

well as the various neurons’s object-specific parameterisation can thus be viewed as a modular-type Thai

Menu Problem [TMP]. In this case, each criterion m m M, , ,= 1K , within this M-criteria analysis

corresponds to the same class template group, whence K K K1= = =L

M NeuroFuzzySetPartition.

Note that unlike most NN treatments, here we see k IncreasingLogisticSigmoid as being fundamentally different

from k DecreasingLogisticSigmoid. In fact, k IncreasingLogisticSigmoid is closer to k IncreasingPiecewiseLinear in that both

epitomise the ‘~< ’ fuzzy concepts. In terms of implementation consideration, we could even cite the OO

inheritance relation k kIncreasingLogisticSigmoid IncreasingPiecewiseLinear: 6, where the ‘virtual’ mapping object

method f IncreasingPiecewiseLinear:ℜ→ℜ (defined to implement the axon activation part of a [McCulloch

& Pitts 1943] neuron) gets overridden by f IncreasingLogisticSigmoid:ℜ→ℜ via the automatic ‘operator

name overloading mechanism’ in C++ [Stroustrup 1991].

Had we been able to a priori specify which criterion is of the ‘~> ’ sort, which is of the ~< sort, etc., then

we could define different class template groups:

{ }{ }{ }{ }

K k k

K k k

K k

K k

IncreasingFuzzyNeuron IncreasingPiecewiseLinear IncreasingLogisticSigmoid

DecreasingFuzzyNeuron DecreasingPiecewiseLinear DecreasingLogisticSigmoid

EqualityFuzzifyingNeuron GaussianRadialBasis

InequalityFuzzifyingNeuro InvertedGaussianBasis

≡

≡

≡

≡

,

,

(12)

In particular, for our portfolio model, we have a total of M m n n= + + +1 multi-criteria, i.e. 1 for RAR,

m = 3 for RR, SR, and PR, n for lower bounds, and n for upper bounds, so that there are M modular

components, each corresponding to a modularised functionality of representing a fuzzy function.

6 Here we are making a historical reference to the fact that an artificial neuron was modelled and implemented with a piecewise-

linear activation profile [McCulloch & Pitts 1943] long before the use of a smooth sigmoid, so that k IncreasingPiecewiseLinear

could have been defined years before k IncreasingLogisticSigmoid is derived.


222

The first neuron (fuzzifying the optimising RAR criteron) is a priori specified to the class:

k k K1 11= ∈ ⇒ =IncreasingPiecewiseLinear IncreasingFuzzyNeuron κ

(13)

The next three (fuzzifying the goal-satisficing RR, SR, and PR criteria) are specified in terms of the class

template groups:

k k K

k k K

k k K

2

3

4

= ∈

= ∈

= ∈

IncreasingFuzzyNeuron IncreasingFuzzyNeuron

IncreasingFuzzyNeuron IncreasingFuzzyNeuron

DecreasingFuzzyNeuron DecreasingFuzzyNeuron

,

,

(14)

where, for instance, we specified:

k k

k k

k k

2 2

3 3

4 4

2

2

2

= ⇒ =

= ⇒ =

= ⇒ =

IncreasingLogisticSigmoid

IncreasingLogisticSigmoid

DecreasingLogisticSigmoid

κκκ

(15)

For example, κ 42= means the neuron fuzzifying the 4th criterion (the PR criterion) is from the

k Decreas ingLogisticSigmoid class, which is the 2nd class listed in the class template group:

{ }K K k k4= ≡

DecreasingFuzzyNeuron DecreasingPiecewiseLinear DecreasingLogisticSigmoid, (16)

Altogether, the class configuration vector is given by:

κ

κκκκκ

κ

κ

κ

=

=

+

+

+ +

+ +

1

2

3

4

4 1

4

4 1

4

1

2

2

22

22

2

M

M

M

M

n

n

n n

n

n

n

n

(17)

With κ specified as a problem-defined constant, the NN-learning is purely parametric. But in a general

case, κ would be variable, and subjected to optimisation, whence the overall fuzzy multi-criteria utility

induction problem becomes correspondingly combinatorial-parametric, and formulated as a modular-type

TMP:

max

, , ,

Φi

Mi

i i i M m n

=⊕

∈ = = + +1

1 2 1

o

k Ksubject to K

(18)


223

7.5 FX/Futures Trading Portfolio Solutions

7.5.1 Experimental Futures Trading Portfolio Solutions

We present an example portfolio of n = 6 DTMs, designated [a][a][a][a],...,[f][f][f][f], three of which model-trade

government bond futures while the others model-trade equity market index futures. Table 7-1 lists the

portfolio data, i.e. the past RAR, RR, SR and PR performance statistics recorded for each DTM. From

inspection, it is clear that a sort of dominance relationship emerges among them. That is, [b][b][b][b] not only

achieves the highest RAR figure, but also outperforms most other DTM’s with respect to the three

supplementary performance criteria. So one expects to find the optimal portfolio assignment to be

essentially driven by the need to satisfy the minimum exposure bounds, L , and the desire to assign as

much weighting as possible to [b][b][b][b]. This is due to the fact that the four performance criteria are, to an

extent, correlated measures (a DTM which is doing well with respect to one is likely to do well with

respect to the others) and to the fact that [b][b][b][b] performed exceptionally well during the trading period

evaluated, bearing in mind that the performance multi-criteria pertained to individual DTMs. Indeed, a

sensible heuristics would be to make the assignment: x x x x x L[ ] [ ] [ ] [ ] [ ]a c d e f= = = = = , x L[ ] .b = −10 5 .In

contrast, [d][d][d][d] is dominated by all other DTMs with respect to every criterion:

Model-Trading Performance Measures [a][a][a][a] [b][b][b][b] [c][c][c][c] [d][d][d][d] [e][e][e][e] [f][f][f][f]

Risk-Adjusted Return (RAR) 12.00 17.54 5.25 1.88 10.45 7.74 Relative Return (RR) 128.4 123.9 188.4 65.4 88.8 104.8 Sharpe Ratio (SR) 1.558 2.095 0.997 0.197 1.333 0.920 Probability of Ruin (PR) 0.083 0.096 0.518 0.835 0.367 0.384

Table 7-1: Financial Futures Model-Trading Performance Measures

Next, we specify the supplementary performance goal thresholds, i.e. the bi ’s (or the ‘right-hand-side’

values), as well as the soft-bound exposure limits, i.e. the interval [ , ]L U . While it is possible to specify

bi ’s in an a priori manner, we choose instead to determine each bi as a function of the ‘left-hand-side’

numbers ( )A Ai i,[ ] ,[ ], ,a fK . We consider two methods of calculations: ‘straight-average’,

( )b A Ai i i= + +,[ ] ,[ ]a fK 6 , and ‘α -median’, ( )b A Ai ibest

iworst= + − < <• •α α α1 0 1, . We specify α = 0 75. .

In this particular data set, the ‘75%-median’ numbers are higher, and therefore make for tougher goal-

satisfaction thresholds, than the ‘straight-average’ numbers, e.g. 0.75 * 188.4 + 0.25 * 65.4 = 158 >

(128.4 + 123.9 + 188.4 + 65.4 + 88.8 + 104.8)/6 = 117. With regard to the upper and lower soft-bound

exposure limits, we try two different intervals: [ , ]L U = [5%,80%],[10%,60%]. Thus altogether we have a

total of four problem variations.

For each of the four problems considered, we further specify four sets of fuzzy utility parameters. For

simplicity and uniformity, we let Φi x( ) [ , ]∈ −11 , i m= =1 3, , ;K Ψ j jx( ) [ , ]∈ −11 , j n= =1 6, ,K . In

other word, each supplementary criterion and each soft-bound exposure limit is worth a maximum of -1


224

penalty point and +1 reward. With regard to omax , which signifies the relative importance between the

primary optimising criterion (RAR) and the m+n soft-constraints (m = 3 secondary performance criteria

and n = 6soft-bound exposure limits), we specify o k m n kmax ( ) , ,= + = 1 2. In terms of interpretation,

k = 1 implies a multi-criteria trade-off profile whereby a hypothetical solution which achieves cmax , but

which also earns the maximum total of − +( )m n penalty points for violating all the soft-constraints, is

considered to be worth exactly zero. On the other hand, k = 2 states that such a solution would be worth

as much as another hypothetical solution which earns the maximum total of + +( )m n reward points, but

only manages cmin with respect to the primary optimising criterion.

Furthermore, we also specify ‘high’ and ‘low’ values for the τ ’s. The higher the value, the ‘more fuzzy’

the distinction or set-partitioning between soft-constraint satisfaction and violation, while the lower the τ

value, the sharper, more ‘hard-constraint’ like, the fuzzy soft-constraint relation becomes.

The four problem variations (1,2,3,4) together with the four combinations of the fuzzy utility parameters

(i,ii,iii,iv) yield a total of sixteen different cases of the optimisation problem { }max ( )P x x W∈ , which

are solved respectively using our EO algorithm. The results are listed in below:

Prob. Problem Specification Fuzzy Utility Parameter Portfolio Solution Weights (in %)

(Vers.) bi [L,U] τ omax

[a][a][a][a] [b][b][b][b] [c][c][c][c] [d][d][d][d] [e][e][e][e] [f][f][f][f]

1(i) [5%,80%] ‘high’ 2(m+n) 5 75 5 5 5 5 1(ii) (m+n) 10 65 10 5 5 5 1(iii) ‘low’ 2(m+n) 5 75 5 5 5 5 1(iv) ( )A Ai i,[ ] ,[ ]a f+ +K 6

(m+n) 5 75 5 5 5 5 2(i) [10%,60%] ‘high’ 2(m+n) 15 55 10 0 10 10 2(ii) (m+n) 15 40 15 10 10 10 2(iii) ‘low’ 2(m+n) 10 50 10 10 10 10 2(iv) (m+n) 10 50 10 10 10 10 3(i) [5%,80%] ‘high’ 2(m+n) 5 75 5 5 5 5 3(ii) (m+n) 5 75 5 5 5 5 3(iii) ‘low’ 2(m+n) 5 75 5 5 5 5 3(iv) 0 75 0 25. .A Ai

bestiworst

• •+ (m+n) 5 75 5 5 5 5 4(i) [10%,60%] ‘high’ 2(m+n) 15 55 10 0 10 10 4(ii) (m+n) 15 55 10 0 10 10 4(iii) ‘low’ 2(m+n) 10 60 10 0 10 10 4(iv) (m+n) 10 60 10 0 10 10

Table 7-2: Fuzzy Soft-Constraint Optimisation Problems, Model Parameters, and Solutions

Let us observe some of the patterns which emerge among the sixteen solutions, their differences having

been magnified here by the relatively ‘crude’ discretisation (in steps of 5%). On the whole, one sees that

all solutions weight heavily toward [b][b][b][b], as already anticipated. This is more pronounced in problems 1

and 3, where a `wider' interval of soft-bound exposure limit ([5%,80%] vs. [10%,60%]) was specified. To

a lesser extent, this is also noticeable when comparing solutions to problems with

b A Ai ibest

iworst= +• •0 75 0 25. . (problems 3 and 4) and those with ( )b A Ai i i= + +,[ ] ,[ ]a fK 6 (problems 1 and

2), where the former, all else being equal, are more demanding with respect to the supplementary


225

performance criteria and therefore exhibit greater bias toward [b][b][b][b], which, once again, tends to dominate.

Only with respect to the RR criterion is [b][b][b][b] outperformed by both [a][a][a][a] and [c][c][c][c], and this explains the

difference between case 1(i) and case 1(ii), the latter giving less bias toward the primary optimising

criterion (RAR). Meanwhile, [d][d][d][d] is so comprehensively outperformed by all other DTMs that it is only

ever included in the portfolio to satisfy the lower soft-bound exposure limit. In fact, as a tighter

[10%,60%] limit reigns in x[ ]b , the algorithm is ‘willing’ to suffer penalty (for violating x[ ] .d > 010) in

order to ‘free up’ additional assignable weight for the better DTMs. Moreover, note how high τ values

tend to push the weight variables ‘inward’, because the fuzzy boundaries are less sharp and the variables

have to move ‘deeper’ inside the bounds to converge to higher reward points. On the other hand, higher τ

values also mean that a given extent of soft-bound violation is less penalised, and therefore more likely to

be tolerated (see 2(i) vs. 2(iii)). Lastly, notice how [a][a][a][a] is generally favoured over [c][c][c][c], the former

outperforming the latter with respect to all but the RR criterion.

We find it more useful to enumerate the varying combinations of problem/model parameters in this way,

rather than to try to enumerate the decision model prior to the optimisation analysis, as in traditional utility

theory/MCDM framework. This provides insights into the interactions among the modelled multi-criteria.

7.5.2 FX Trading Portfolio Run on Prototype Software

Here we go thorough the data/model specification steps in our prototype software, which implements

logistic, as opposed to hyperbolic, sigmoid. This is in keeping with the classical FGP formulation where

the lowest goal-satisfaction value equates to the fuzzy number value of zero, and no penalty term is

applied.

Data Input

First, the FX model-trading performance measures:

Model-Trading Performance Measures DEMJPYDEMJPYDEMJPYDEMJPY DEMITLDEMITLDEMITLDEMITL GBPJPYGBPJPYGBPJPYGBPJPY USDCHFUSDCHFUSDCHFUSDCHF USDCADUSDCADUSDCADUSDCAD

Risk-Adjusted Return (RAR) 7.2604 4.3729 14.1036 3.1489 1.8053 Relative Return (RR) 77.2115 141.8040 117.7551 71.8000 99.0200 Sharpe Ratio (SR) 0.9686 1.3365 1.7041 1.8053 0.6219 Probability of Ruin (PR) 11.7777 14.3421 9.6910 52.3960 5.9739

Table 7-3: FX Model-Trading Performance Measures

is loaded from file and/or manually entered/edited:


226

Figure 7-4: Manual Data Entry/Edit Dialogue

Model Specification

Fuzzy Function Classes

Here the decision maker specifies the choices of goal-aspiration and/or optimising function for each of the

multi-criteria. In this example:

Figure 7-5: Specifying Combination of Optimising Criteria/Fuzzy Goal Functions

the RAR is considered to be the only optimising criterion. The SR is to be represented as a logistic

sigmoid fuzzy function, as is the PR criterion and upper bound soft-constraint. The RR criterion and the

lower bound soft-constraint are represented by a piecewise linear function.

Fuzzy Function Parameters

For each of these function, the criterion-specific parameters have to be specified. For example, the RAR is

simply scaled so that the portfolio solution assigning all the weightings to the trading model with the


227

highest RAR number, i.e. RAR xj jj=∑ DEMJPY

USDCAD = { }max

jjRAR = RARGBPJPY = 14.1036, is assigned the

value one, while the portfolio solution assigning all the weightings to the trading model the lowest RAR

number, i.e. RAR xj jj=∑ DEMJPY

USDCAD = { }min

jjRAR = RARUSDCAD = 1.8053, is assigned the value zero:

Figure 7-6: Parameterising an Optimising Criterion

On the other hand, the SR criterion is fuzzified so that a portfolio solution whose weighed sum of the SR

numbers is about half a standard deviation below the mean SR number, i.e. 85 ((101.5 - 29.01/2), is

accorded a fuzzy value of 0.1, while a portfolio solution whose weighted sum of the SR numbers is about

half a standard deviation above the mean SR number, i.e. 115 ((101.5 + 29.01/2), is accorded a fuzzy

value of 0.9:

Figure 7-7: Parameterising a Fuzzy Goal Function

Note how within this context, any criterion, not just RAR, can be considered an optimising criterion. In

fact, RAR can also be captured as a fuzzy goal as well.


228

Multi-Criteria Utility Weighting

Finally, the decision maker specifies the MCDM weighting amongst the multi-criteria:

Figure 7-8: Specifying the Multi-Criteria Utility Weightings

In this example, the RAR criterion is given a relative importance of 8/(8+6+2+4+2+10) = 0.25. On the

other hand, the lower bound soft-constraint is weight very little at 2/(8+6+2+4+2+10) = 0.625. With this,

an FX trading model which is comprehensively dominated by all others would never be assigned any

weighting.

Portfolio Solution Quantisation

Finally, the quantisation D Nn is specified, for example, to D NN

n= ⇒ =50 1 2%:

Figure 7-9: Specifying the Portfolio Solution Quantisation

Evolutionary Optimisation Run

In Figure 7-10, the evolving solutions are depicted as points on the fuzzy goal functions. Underneath each


229

fuzzy function, the length of each thick bar indicates the relative MCDM importance weighting:

Figure 7-10: Top 10 Model-Trading Portfolio Solutions, 10th Generation

In Figure 7-11, we see that by 500 generations, the population converges onto a single solution:

Figure 7-11: Converged Model-Trading Portfolio Solutions, 500th Generation


230

The detailed result of the optimised portfolio solution is shown below:

DEMJPYDEMJPYDEMJPYDEMJPY DEMITLDEMITLDEMITLDEMITL GBPJPYGBPJPYGBPJPYGBPJPY USDCHFUSDCHFUSDCHFUSDCHF USDCADUSDCADUSDCADUSDCAD

x j 0.08 0.32 0.6 0.0 0.0 x j∑ 1.0

RAR xj j 0.580832 1.399328 8.46216 0.0 0.0 RAR xj j∑ 10.44232

SR xj j 6.17692 45.37728 70.65306 0.0 0.0 SR xj j∑ 122.20726

RR xj j 0.077488 0.42768 1.02246 0.0 0.0 RR xj j∑ 1.527628

PR xj j 0.942216 4.589472 5.8146 0.0 0.0 PR xj j∑ 11.346288

Table 7-4: Converged Portfolio Solution and Model-Trading Performance Multi-Criteria

The evolutionarily optimised solution assigns 8%, 32%, and 60% portfolio weightings, respectively, to the

DEMJPYDEMJPYDEMJPYDEMJPY, DEMITLDEMITLDEMITLDEMITL, and GBPJPYGBPJPYGBPJPYGBPJPY DTMs, while the USDCHFUSDCHFUSDCHFUSDCHF and USDCADUSDCADUSDCADUSDCAD DTMs are completely unweighted

in this optimised portfolio, reflecting the abysmal performance of the latter two, the tolerable performance

of the DEMJPYDEMJPYDEMJPYDEMJPY DTM (still completely dominated by the GBPJPYGBPJPYGBPJPYGBPJPY DTM), the strengths of the DEMITLDEMITLDEMITLDEMITL and

GBPJPYGBPJPYGBPJPYGBPJPY DTMs, and our heavy leaning (MCDM utility weighting-wise) toward the RAR figure, which is

highest for the GBPJPYGBPJPYGBPJPYGBPJPY DTM.

7.6 Concluding Remarks

We have introduced a fuzzy set framework within the context of portfolio theory. Whilst our particular

portfolio model does not resemble a Markowitz [Markowitz 1959] formulation, the latter can be captured

via a non-linear extension of our framework. We have also introduced explicit penalty-reward concepts

within a FGP type framework, whilst retaining a formal equivalence to the classical FGP formulation.

In term of solution methodology, we forgo MP-type approach in favour of a general search strategy based

on an EO algorithm, citing practical and modelling benefits. The fuzzy soft-constraint

optimisation/decision model is illustrated on a portfolio of daily trading models which traded equity-index

and government bond futures contracts. The patterns of weight assignments which emerge are consistent

with the problem data and the variously specified fuzzy utility parameters. We proposed NN, firstly as a

mapping architecture to encode our fuzzy multi-criteria performance evaluation function, and, secondly as

a learning machine which in theory is capable of performing a case-based induction of a decision maker's

implicit fuzzy utility function. We put forward this fuzzy, multi-criteria optimisation/decision model

together with the neuro-evolutionary methodology as a flexible, analytical framework for managing a

portfolio of FX/Futures model-trading activities in particular, and for tackling soft-constraint optimisation

problems in a general MCDM setting.


231

7. Fuzzy Soft-Constraint Portfolio Optimisation __________________210 7.1 Introduction ____________________________________________________ 210 7.2 FX/Futures Trading Portfolio Model_________________________________ 212 7.3 Fuzzy Soft-Constraint Optimisation/Decision Model ____________________ 213

7.3.1 Fuzzy Set Membership Functions ______________________________________ 214 7.3.2 Fuzzy Penalty-Reward Functions ______________________________________ 214 7.3.3 Fuzzy Multi-Criteria Objective Function ________________________________ 215 7.3.4 Solution-Parameter Optimisation Problem _______________________________ 216

7.4 Neuro-Evolutionary Methodology & Extensions _______________________ 218 7.4.1 Mapping NN Architecture & Evolutionary Optimisation____________________ 218 7.4.2 Learning NN & Fuzzy Multi-Criteria Utility Induction _____________________ 219 7.4.3 Evolutionary NN Learning & TMP Formulation __________________________ 220

7.5 FX/Futures Trading Portfolio Solutions ______________________________ 223 7.5.1 Experimental Futures Trading Portfolio Solutions _________________________ 223 7.5.2 FX Trading Portfolio Run on Prototype Software _________________________ 225

7.6 Concluding Remarks _____________________________________________ 230

Figure 7-1: Linear Programming Backdrop of a Fuzzy Soft-Constraint Optimisation Problem______ 217 Figure 7-2: Fuzzy Multi-Criteria Performance Evaluation Function/Surface ___________________ 217 Figure 7-3: Neural Architecture Encoding Fuzzy Performance Function_______________________ 218 Figure 7-4: Manual Data Entry/Edit Dialogue ___________________________________________ 226 Figure 7-5: Specifying Combination of Optimising Criteria/Fuzzy Goal Functions _______________ 226 Figure 7-6: Parameterising an Optimising Criterion ______________________________________ 227 Figure 7-7: Parameterising a Fuzzy Goal Function _______________________________________ 227 Figure 7-8: Specifying the Multi-Criteria Utility Weightings ________________________________ 228 Figure 7-9: Specifying the Portfolio Solution Quantisation _________________________________ 228 Figure 7-10: Top 10 Model-Trading Portfolio Solutions, 10th Generation ______________________ 229 Figure 7-11: Converged Model-Trading Portfolio Solutions, 500th Generation __________________ 229 Table 7-1: Financial Futures Model-Trading Performance Measures _________________________ 223 Table 7-2: Fuzzy Soft-Constraint Optimisation Problems, Model Parameters, and Solutions _______ 224 Table 7-3: FX Model-Trading Performance Measures _____________________________________ 225 Table 7-4: Converged Portfolio Solution and Model-Trading Performance Multi-Criteria _________ 230


232

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