210 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, Px () 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.
24
Embed
7. Fuzzy Soft-Constraint Portfolio Optimisation · optimisation methodology: overlaying a branch-and-bound search over a succession of continuous-valued NLP relaxations, or employing
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
210
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
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].
Evolutionary Optimisation and Financial Model-Trading
214
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
7 Fuzzy Soft-Constraint Portfolio Optimisation
215
(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].
Evolutionary Optimisation and Financial Model-Trading
216
}
{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.
7 Fuzzy Soft-Constraint Portfolio Optimisation
217
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
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
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
Evolutionary Optimisation and Financial Model-Trading
232
Bazaraa, M. S. and C. M. Shetty (1979). Nonlinear Programming: Theory and Algorithms. New York, John Wiley & Sons. Bellman, R. E. and L. A. Zadeh (1970). “Decision Making in a Fuzzy Environment.” Management Science 17(2 B): 141-164. Billot, A. (1995). “An Existence Theorem for Fuzzy Utility Functions, a New Elementary Proof.” Fuzzy Sets and Systems 74(2): 271-276. Cerny, V. (1985). “Thermodynamical Approach to the Travelling Salesman Problem: an Efficient Simulation Algorithm.” Journal of Optimisation Theory and Applications 45: 41-51. Chankong, V. and Y. Haimes (1983). Multiobjective Decision Making: Theory and Methodology. New York, North-Holland. Charnes, A., W. W. Cooper and R. O. Ferguson (1955). “Optimal Estimation of Executive Compensation by Goal Programming.” Management Science 1(2): 138-151. Dantzig, G. B. (1966). Lineare Programmierung und Erweiterungen (Linear Programming and Extension). Berlin, Springer. Dhingra, A. K., S. S. Rao and V. Kumar (1992). “Nonlinear Membership Functions in Multiobjective Fuzzy Optimisation of Mechanical and Structural Systems.” AIAA Journal 30(1): 251-260. Dubois, D. and H. Prade (1980). “System of Linear Fuzzy Constraints.” Fuzzy Sets and Systems 3(37-48). Dunis, C. and M. Feeny, Eds. (1989). Exchange Rate Forecasting, Woodhead-Faulkner. Fogel, D. B. (1994a). “An Introduction to Simulated Evolutionary Optimisation.” IEEE Transactions on Neural Networks 5(1): 3-14. Fogel, L. J., A. J. Owens and M. J. Walsh (1966). Artificial Intelligence through Simulated Evolution. New York, John Wiley. Fuller, R. and H.-J. Zimmermann (1993). “Fuzzy Reasoning for Solving Fuzzy Mathematical Programming Problems.” Fuzzy Sets and Systems 60: 121-133. Hannan, E. L. (1981). “On Fuzzy Goal Programming.” Decision Sciences 12: 522-531. Haykin, S. (1994). Neural Networks: A Comprehensive Foundation. New York, McMillan College Publishing Company. Holland, J. H. (1975). Adaptation in Natural and Artificial Systems. Ann Arbor, MI, The University of Michigan Press. Hornik, K., M. Stinchcombe and H. White (1989). “ Multilayer Feedforward Networks are Universal Approximators.” Neural Networks 2(5): 359-366. Ignizio, J. P. (1982). “On the (Re)discovery of Fuzzy Goal Programming.” Decision Sciences 13: 331-336. Kaufman, P. (1987). The New Commodity Trading Systems and Methods. New York, John Wiley & Sons. Kaufman, P. (1995). Smarter Trading: Improving Performance in Changing Markets. New York, McGraw-Hill. Kirkpatrick, S., C. D. Gelatt and M. P. Vecchi (1983). Optimisation by Simulated Annealing. Science: 671-680.
7 Fuzzy Soft-Constraint Portfolio Optimisation
233
Markowitz, H. M. (1952). “Portfolio Selection.” Journal of Finance 7(March): 77-91. Markowitz, H. M. (1959). Portfolio Selection: Efficient Diversification of Investments. New York, John Wiley & Sons. Masters, T. (1993). Practical Neural Network Recipes in C++, Academic Press. McCulloch, W. S. and W. Pitts (1943). “A logical calculus of the ideas of immanent in nervous activity.” Bulletin of Mathematical Biophysics 5: 115-133. Narasimhan, R. (1980). “Goal Programming in a Fuzzy Environment.” Decision Sciences 11: 325-336. Nishizaki, I. and F. Seo (1994). “Interactive Support for Fuzzy Trade-off Evaluation in Group Decision-Making.” Fuzzy Sets and Systems 68(3): 309-325. Pao, Y.-H. (1989). Adaptive Pattern Recognition and Neural Networks. Reading, MA, Addison-Wesley. Rao, S. S., K. Sundararaju, B. G. Prakash and C. Balakrishna (1992). “Fuzzy Goal Programming Approach for Structural Optimisation.” AIAA Journal 30(5): 1425-1432. Rumelhart, D. E. and J. L. McCelland, Eds. (1986). Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Cambridge, MA, MIT Press. Schwefel, H.-P. (1995). Evolution and Optimum Seeking. New York, John Wiley & Sons. Solow, D. (1984). Linear Programming: An Introduction to Finite Improvement Algorithms, North-Holland. Stroustrup, B. (1991). The C++ Programming Language. Reading, MA, Addison-Wesley. Tanaka, H., T. Okuda and K. Asai (1974). “On Fuzzy Mathematical Programming.” Journal of Cybernetics(3): 37-46. Werbos, P. J. (1974). Beyond Regression: New Tools for Prediction and Analysis in the Behavioural Sciences. Cambridge, MA, Harvard University. Werbos, P. J. (1994). The Roots of Backpropagation: from Ordered Derivatives to Neural Networks and Political Forecasting. NewYork, John Wiley & Sons. Yip, P. and Y.-H. Pao (1993b). A New Evolutionary Computational Technique for Optimisation: An Evolutionary Simulated Annealing Approach. The Fifth International Conference on Genetic Algorithm. Yoneda, M., S. Fukami and M. Grabisch (1993). “Interactive Determination of Utility Function Represented as a Fuzzy Integral.” Information Sciences 71(1-2): 43-64. Zimmermann, H.-J. (1978). “Fuzzy Programming and Linear Programming with Several Objective Functions.” Fuzzy Sets and Systems 1: 45-55.