Fuzzy reasoning for solving fuzzy multiple objective linear programs ∗ Robert Full´ er†
[email protected]
Christer Carlsson
[email protected]
Abstract We interpret fuzzy multiple objective linear programming (FMOLP) problems with fuzzy coefficients and fuzzy inequality relations as multiple fuzzy reasoning schemes (MFR), where the antecedents of the scheme correspond to the constraints of the FMOLP problem and the facts of the scheme are the objectives of the FMOLP problem. Then the solution process consists of two steps: first, for every decision variable x ∈ Rn and objective s ∈ {1, . . . k}, we compute the effective attainment of the s-th objective function, Es (x), via sup-min convolution of the antecedents/constraints and the s-th fact/objective, then a solution to FMOLP problem is any point which is a good compromise solution to the crisp multiple objective problem max {H1 (x), . . . Hk (x)},
x∈Rn
where Hs is an application function (measuring the degree of fulfillment of the decision maker’s requirements about the s-th objective function) s ∈ {1, . . . k}.
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Statement of FMOLP problems with fuzzy coefficients
We consider MOLP problems, in which all of the coefficients are fuzzy quantities (i.e. fuzzy sets of the real line R), of the form maximize (˜ c1 x, . . . , c˜k x) < ˜ bi , i = 1, . . . , m, subject to a ˜i x ∼
(1)
where x ∈ Rn is the vector of decision variables, a˜i , ˜bi and c˜s are vectors of fuzzy quantities, the operations addition and multiplication by a real number of fuzzy quan< for tities are defined by Zadeh’s extension principle [13] and the inequality relation ∼ ∗
The final version of this paper appeared in: R.Trappl ed., Cybernetics and Systems ’94, Proceedings of the Twelfth European Meeting on Cybernetics and Systems Research, World Scientific Publisher, London, 1994, vol.1, 295-301. † Partially supported by the Hungarian Research Foundation for Scientific Research (OTKA) under contract I/3-2152.
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constraints is given by a certain fuzzy relation. We can now state (1) as follows: find an x∗ ∈ Rn such that the grades of effective attainments of all the objective functions at x∗ fulfill the decision maker’s requirements (achievement of goals, nearness to an ideal point, satisfaction, etc.) as far as possible.
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Preliminaries
In this section we set up the notations and recall some fuzzy inference rules needed for the proposed solution principle. A fuzzy set a ˜ of the real line R is called a fuzzy quantity. We denote the family of fuzzy quantities by F. A fuzzy number a ˜ is a fuzzy quantity with a continuous, finite-supported, fuzzy-convex and normalized membership function a ˜: R → [0, 1]. The family of all fuzzy numbers will be denoted by F(R). An α-level set of a fuzzy quantity a ˜ is a non-fuzzy set denoted by [˜ a]α and is defined by [˜ a]α = {t ∈ R|˜ a(t) ≥ α} for α ∈ (0, 1] and [˜ a]α = cl(supp a ˜) for α = 0. We metricize F(R) by the metrics, Z
Dp (˜ a, ˜b) = (
0
1
d([˜ a]α , [˜b]α )p )1/p
for 1 ≤ p ≤ ∞, especially, for p = ∞ we get D∞ (˜ a, ˜b) = sup d([˜ a]α , [˜b]α ), α∈[0,1]
where d denotes the classical Hausdorff metric in the family of compact subsets of R2 . Let X be a non-empty set. A binary fuzzy relation W in X is a fuzzy subset of the Certesian product X × X and defined by its membership function µW . If the < in (1) is modeled by a fuzzy implication operator I then for all inequality relation ∼ n x ∈ R we have µa˜i x≤˜bi (u, v) = I(µa˜i x (u), µ˜bi (v)), < is given by the G¨ e.g. if ∼ odel implication operator then we get
µa˜i x≤˜bi (u, v) =
(
1 if µa˜i x (u) ≤ µ˜bi (v), µ˜bi (v) otherwise.
(2)
We shall use the compositional rule of inference scheme with several relations (called Multiple Fuzzy Reasoning Scheme) [13] which has the general form Fact Relation 1: ... Relation m: Consequence:
X is P X and Y are in relation W1 ... X and Y are in relation Wm Y is Q
where X and Y are linguistic variables taking their values from fuzzy sets in classical sets U and V , respectively, P and Q are unary fuzzy predicates in U and V , respectively, Wi is a binary fuzzy relation in U × V , i=1,. . . ,m. 2
The consequence Q is determined by [13] Q=P◦
m \
Wi
i=1
or in detail, µQ (y) = sup min{µP (x), µW1 (x, y), . . . , µW1 (x, y)}. x∈U
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Multiply fuzzy reasoning for solving FMOLP problems
Generalizing the results of [3, 5, 9] we consider FMOLP problems with fuzzy coefficients and fuzzy inequality relations as MFR schemes, where the antecedents of the scheme correspond to the constraints of the FMOLP problem and the facts of the scheme are the objectives of the FMOLP problem. Then the solution process consists of two steps: first, for every decision variable x ∈ Rn and objective s ∈ {1, . . . k}, we compute the effective attainment of the s-th objective function, Es (x), via sup-min convolution of the antecedents/constraints and the s-th fact/objective, then a solution to FMOLP problem is any point which is a good compromise solution to the crisp multiple objective problem max {H1 (x), . . . Hk (x)},
x∈Rn
where Hs is an application function (measuring the degree of fulfillment of the decision maker’s requirements about the s-th objective function) s ∈ {1, . . . k}. For every decision variable x ∈ Rn and objective s ∈ {1, . . . k} we interpret the FMOLP problem (1) as MFR schemes of the form Antecedent 1 ... Antecedent m
< ˜ Constraint1 (x) := a ˜11 x1 + · · · + a ˜1n xn ∼ b1 ... < ˜ Constraintm (x) := a ˜m1 x1 + · · · + a ˜mn xn ∼ bm
Fact Consequence
Goals (x) := c˜s1 x1 + · · · + c˜sn xn Es (x)
where the consequence (i.e. the effective attainment of the s-th objective function subject to constraints at x) Es (x) is computed as follows Es (x) = Goals (x) ◦
m \
Constrainti (x),
i=1
i.e. µEs (x) (v) = sup min{µGoals (x) (u), µConstraint1 (x) (u, v), . . . , µConstraintm (u, v)}. u
Then (1) turns into the following unconstrained fuzzy decision problem max{(E1 (x), . . . Ek (x)) | x ∈ Rn }. 3
(3)
Following [5] suppose that we have two reference points from F(R), denoted by m ˜s ˜ s , which represent undesired and desired levels for each objective function Es . and M We can now state (1) as follows: find an x∗ ∈ X such that Es (x∗ ) is as close as possible to the desired point Es and as far as possible from the undisered point Es , for each s = 1, . . . , k. Supposing now that Es (x) ∈ F(R), ∀x ∈ Rn we can use the following family of application functions [5] ½
Hs (x) = min 1 −
1 1 , ˜ s , Es (x)) 1 + D(m ˜ s , Es (x)) 1 + D(M
¾
or, more generally, µ
Hs (x) = T 1 −
1 1 , ˜ s , Es (x)) 1 + D(m ˜ s , Es (x)) 1 + D(M
¶
(4)
where T is a t-norm, D is a metric (e.g Dp ) in F(R). It is clear that the bigger the value of Hs (x) the closer the value of the s-th objective function to the desired level or/and further from the undesired level, and vica versa the smaller the value of Hs (x) the closer its value to the undesired level or/and further from the desired level. In (4) the t-norm T measures the degree of satisfaction of two (conflicting) goals ”to be far from the undesired point and to be close to the desired point”. The particular t-norm T should be chosen very carefully, because it can occur that Hs (x) attends its maximal value at a point which is very far from the undesired point, but not close enough to the desired point. For example, if T is the weak t-norm (T (x, y) = min{x, y} if max{x, y} = 1 and T (x, y) = 0 otherwise) then Hs (x) positive ˜ s , i.e. we have managed to reach completely the desired point, if and only if Es (x) = M ˜ s is not necessarily in the range of Es . Another which is rarely the case, because M ˜ s) ˜ s, M crucial point is the relative setting of the desired and undesired points. If D(m is small then it is impossible to find an x∗ ∈ X satisfying the condition ”Es (x) is close ˜ s and is far from m to M ˜ s ”. Thus, similarly to the crisp case, the fuzzy decision problem (3) turns into the singleobjective problem max T (H1 (x), . . . , Hk (x)). (5) x∈X
It is clear that the bigger the value of the objective function of problem (5) the closer the fuzzy functions to their desired levels. Remark 3.1 Apart from the deterministic MOLP, max{(c1 x, . . . , ck x) | Ax ≤ b}, where we simply compute the value of s-th objective function as cs1 y1 + · · · + csn yn at any feasible point y ∈ Rn and do not care about non-feasible points, in FMOLP problem (1) we have to consider the whole decision space, because every point y from Rn has a (fuzzy) degree of feasibility (given by the fuzzy relations Constrainti (y), i = 1, . . . , m). We have right to compute the value of the s-th objective function of (1) at y ∈ Rn as c˜s1 y1 + · · · + c˜sn xn if there are no constraints at all (if there are no rules in a fuzzy reasoning scheme then the consequence takes the value of the observation automatically).
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4
Concluding remarks
We have interpreted FMOLP problems with fuzzy coefficients and fuzzy inequality relations as MFR schemes and shown a method for finding an optimal compromise solution. In the general case the computerized implementation of the proposed solution principle is not easy (see [1, 3, 4, 10, 11, 12, 14]). To compute Es (x) we have to solve a generally non-convex and non-differentiable mathematical programming problem. However, the stability property of the consequence in MFR schemes under small changes of the membership function of the antecedents [8] guarantees that small rounding errors of digital computation and small errors of measurement in membership functions of the coefficients of the FMOLP problem can cause only a small deviation in the membership function of the consequence, Es (x), i.e. every successive approximation method can be applied to the computation of the linguistic approximation of the exact Es (x).
References [1] J.J.Buckley , Multiobjective possibilistic linear programming, Fuzzy Sets and Systems, 35(1990) 23-28. [2] C.Carlsson, Tackling an MCDM-problem with the help of some results from fuzzy sets theory, European Journal of Operational research, 3(1982) 270-281. [3] C.Carlsson, Approximate reasoning for solving fuzzy MCDM problems, Cybernetics and Systems: An International Journal, 18(1987) 35-48. [4] C.Carlsson, Approximate reasoning through fuzzy MCDM-models, Operational Research’87, North-Holland, 1988 817-828. [5] C.Carlsson and R.Full´er, Interdependence in fuzzy multiple objective programming, (submitted to Fuzzy Sets and Systems). [6] C.Carlsson and R.Full´er, Multiple Criteria Decision Making: The Case for Interdependence, Computers & Operations Research (submitted). [7] C.Carlsson, On optimization with interdependent multiple criteria, in: R.Lowen and M.Roubens eds. Fuzzy Logic: State of the Art, Kluwer, Dordrecht, 1993, 287-300. [8] R.Full´er and H.-J.Zimmermann, On Zadeh’s compositional rule of inference, in: R.Lowen and M.Roubens eds., Proc. of Fourth IFSA Congress, Vol. Artifical Intelligence, Brussels, 1991 41-44. [9] R.Full´er and H.-J.Zimmermann, Fuzzy reasoning for solving fuzzy mathematical programming problems, Fuzzy Sets and Systems, 63(1993) (to appear [10] M.K.Luhandjula and M.Sakawa, Multiple objective linear programming problems in the presence of fuzzy coefficients, in: R.Slowinski and J.Teghem eds., Stochastic versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty, Kluwer Academic Publishers, Dordrecht, 1990 301-320. 5
[11] H.Rommelfanger, Entscheiden bei Unsch¨arfe, Springer-Verlag, Berlin, 1989. [12] E. Stanley Lee and R.J.Li, Fuzzy multiobjective programming and compromise programming with Pareto optimum, Fuzzy Sets and Systems, 53(1993) 275-288. [13] L.A.Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. on Systems, Man and Cybernetics, Vol.SMC-3, No.1, 1973 28-44. [14] H.-J.Zimmermann, Fuzzy Sets Theory and Mathematical Programming, in: A.Jones et al. (eds.), Fuzzy Sets Theory and Applications, D.Reidel Publishing Company, 1986 99-114.
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