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FUZZY sets and systems Fuzzy Sets and Systems 88 (1997) 69 79 ELSEVIER Temporal interdependence in fuzzy MCDM problems Ralf Ostermark Department of Business Administration, Abo Akademi University, Henriksgatan 7, 20500 Abo, Finland Received September 1995; revised January 1996 Abstract Contemporaneous MCDM methodology is based on the simplifying assumption of independent objectives. This restriction was partly relaxed through the concept of (static) interdependent objectives introduced by C. Carlsson and R. Fuller. Most real world managerial decision problems involve interdependent objectives, yet in a temporal setting. In the paper we generalize the static concept to temporal fuzzy multiobjective programming problem. We introduce the concepts temporal support and temporal conflict in the objective set within both infinite and finite planning horizons. We also formulate dynamic versions of membership functions for interdependent objectives in crisp and fuzzy multiobjective programming problems. The new concepts are used to describe and model temporal goal conflicts in numerical illustrations. © 1997 Elsevier Science B.V. Keywords: Temporal interdependence; Fuzzy multiobjective programming; Dynamic membership functions 1. Introduction The concept of interdependence in multiple criteria decision making (MCDM) was introduced by Carlsson and Fuller [1, 2]. The authors showed that fuzzy set theory can be successfully applied to resolve multiple criteria problems with interdependent objectives [13]. While the authors focused on static M C D M problems, most decision models become both more realistic and complicated when time is explicitly recognized [10]. In the present paper we consider the interdependence concept in a dynamic setting. We show that the approach can be naturally extended to temporal cases. We then apply the temporal concept to describe goal conflicts in a multiperiod firm model in which the concept of static interdependence would fail. Finally, the static membership function by Carlsson and Fuller [1] is generalized to a dynamic membership function for both multiobjective programming (MOP) and fuzzy multiobjective programming (FMOP) problems. 2. Temporal interdependence Consider the following static M C D M problem: m a x {fl (x), fE(x) . . . . , fk(X)} 0165-0114/97/$17.00 © 1997 ElsevierScienceB.V. All rights reserved PII S0165-01 14(96)00046-2 (2.1) R. Ostermark / Fuzzy Sets and Systems 88 (1997) 69 79 70 wheref~: N" ~ N1 are objective functions, x e 0~" is the vector of decision variables and X is a subset of N". Numerous ways to solve (2.1) exist if the problem is mathematically well-formulated [6, 15, 16]. To describe the relationship between the objectives in (2.1) we consider the following. Definition 2.1. (Static interdependence, Carlsson and Fuller [1]). Let f and fj (i-¢j) be two objective functions of (2.1). We say that (i) f~ supportsfj (denoted byJ~ T fj), i f f (x') ~>f~(x) implies fj (x') ~>fj (x), Vx', x ~ x . (ii) f/is in conflict withfj (denoted byJ~ J, fj), if f/(x') >~f/(x) implies Jj (x') ~<fj (x), Vx',x ~ X. (iii) Otherwise, f and f~ are independent. IfX := N", thenf~ supports or is in conflict withfj globally. In the static case the support (T) and conflict ($) operators possess the following properties: (i) symmetry: iff~ Tfj then f~ i"f~ Off Sf~ thenf~ ~f~), (ii) associativity: iff~ Tfi andfj Tft then J~ Tfz (iff~ J, fj andf~ lJ~ then f~ SJ~), (iii) reflexivity: fl ~ fi, V i, (iv) iff~ Tf~ andfj Sft then f~ ~ ft (if f~ Sf~ andfj Tfz then f~ ]"f~). Carlsson and Fuller [1] defined the grade of interdependency of a function f with respect to a certain system of objective functions, denoted by A (f,-), as follows: A(f~)= ~ 1-- ~ 1, i = l . . . . . k. fi'~fj iv~j (2.2) fl.tfj i#j If A (f~) is positive and large, thenf~ supports the majority of the objectives; if A (J~) is negative and large thenf~ is in conflict with the majority of objectives; if A (f~) is positive and small then j] supports more objectives than it hinders and if A(f~) is negative and small then J~ hinders more objectives than it supports. Finally, if A (f~) = 0, thenf~ is said to be independent of the other objectives or supports the same number of objectives as it hinders. The concept was explicitly used in Carlsson and Fuller [2] to solve M C D M problems with conflicting criteria (Fig. 1). They introduced a nonlinear membership function of the following form [16]: (1) Supportive objectives (A (f/) > 0) if f/(x)/> Mi, G~(x) = 1 ~ ~ ~ d 0 if ml < f (x) < Mi, (2.3a) if f/(x) ~< mi. (2) Conflicting objectives (A (f~) <0) Gi(x) = i 1 0 if J~(x) 1> MI, M i __ mi J if mi < f (x) < Mi, if J~ (x) ~< mi. (2.3b) R. Ostermark / Fuzzy Sets and Systems 88 (1997) 69-79 71 y Fig. 1. A typical example of conflict in ~2. If 3 i, A (f/) = 0, then the linear shape for that objective is maintained. If, however, A (f/) > 0, then the linear membership function for the ith objective is raised to the power 1/(A (f/) + 1). On the other hand, if A (f/) < 0, then the corresponding membership function is raised to the power ([A(f/)[ + 1). The rationale for these perturbations is that supportive objectives are made relatively more attractive by concave membership functions, whereas conflicting criteria (A(f/)< 0) are made less important through convex membership functions in the optimization problem. The notion of interdependency becomes somewhat different in the dynamic case. Consider the following discrete-time dynamic problem, where, for simplicity, the solution space X is assumed to be fixed and the function dependent on time only through the decision variables: DMOP: max {fl(xt),f2(xt), ... ,fk(X,)}. xt~X (2.4) The multiperiod firm planning problem considered in the numerical experiment below is an example of such a decision problem. The characteristic property of the problem is that the parameters (e.g., sales prices Pt, t = 1. . . . . h) are assumed to be known and fixed a priori. The case with unknown parameter values, e.g., future sales prices, interest rates etc. is not considered in the present study. Compared to the static concept of Carlsson and Fuller [2], we have a set of k objective function trajectories defined over the planning horizon, not merely k (static) objective function values. The conflict vs. support between objectives now also has to be considered at different time points. For example, whereas in the static situation an objective f/never is in conflict with itself, in the temporal situation the values off/(t) at time points tl and tz may well be conflicting. In the following the concept of interdependence is extended to the dynamic situation. In order to simplify the judgement of the trajectories, we assume below that the objectives have been discounted by the risk-adjusted weighted average cost of capital of the firm [3]. The impact of discounting on the dynamic membership function is considered explicitly in Section 4. Definition 2.2 (Temporal interdependence). Let f(x,) :=f/(t) and fi(xt):=f~ (t) be two objective functions of (2.4) and assume that, prior to conflict resolution, the objectives are properly discounted, i.e., the present value off/(t) is compared to the present value offj (t). Assume also that the conditions stated below are valid for some compact subset (D) of the feasible set, i.e., D _ X. We can then distinguish between interdependence situations under the assumption of an infinite vs. finite planning horizon as follows: (i) Infinite planning horizon, f~(t) strongly supports J) (t), i # j , iff/(t) Tf~(t + k) is true for any k >/0, i.e., fi(x't) ~f/(xt)~f/(X't+k)>>-f/(Xt+k), k >>-0 (the case with strong conflict, fi(t)+fj(t + k), is obtained by reversing the inequalities). R. Ostermark / Fuzzy Sets and Systems 88 (1997) 69 79 72 (ii) Finite planning horizon. Let H = { 1,2 . . . . . h} denote the set of time points in the planning horizon of length h. (iia) Strong temporal support fi(t) T fj(s) holds for t, s ~ H, Vi, j. (2.5) All relevant objectives are mutually supportive. It is important to note that if the trivial case (t = s) A (i = j) is disregarded then i ~ j is not essential: a good value on the ith objective today may well support a good value on it tomorrow. Strong conflict is obtained by reversing the arrow in (2.5): (iib) Strong temporal conflict fl (t) J, J) (s), t, s e H holds for t va s, Vi, j and for t = s, i ~a j. (2.6) The trivial case t = s, i = j is excluded due to impossibility of m o m e n t a r y self-conflict. (iic) Weak temporal support. Let Sl={(i,j,t,s)[fi(t)Tfj(s);t,s~H}, S2={(i,j,t,s) lfi(t)~fj(s);t,s~H}. (2.7) Assume that all elements o f f T (t) = (fl (t), . . . , fk (t)) are equally important over H. Then the objectives are weakly supportive in the planning horizon iffthe n u m b e r of instances (4-dimensional parameter vectors) in $1, N(SI), is not less than that in $2. (IfN(S~) < N(S2) t h e n f ( t ) is weakly conflicting, ifS~ = $2 = 0 thenf~, Vi are independent). As will be shown later, under the stated simplification the static concept of interdependent objectives and the T - n o r m solutions are directly extendable to discrete time finite horizon problems. We note that even the single objective problem turns multiobjective in the temporal case. The assumption of equal importance is not very realistic, however. The number of objectives is an insufficient indicator of support vs. conflict. In the temporal case the time difference between realizations should be properly recognized. (Affecting the possibility to gain on fj t o m o r r o w through variations on f~ is another critical aspect. This will not be addressed in the present study, however). The temporal aspect is recognized in the following definition. (iid) Qualified weak temporal support f ( t ) ~ fj(t + k), 3k >~O, if f(x(t)) >f~(x'(t)) ~ fj(x(t + k)) > f j ( x ' ( t + k)), Vi, j (2.8) for all trajectories x(t) and x'(t) over It, t + k] (analogously for qualified conflict and independence). This definition recognizes the evolution of the function(al)f(t) between t and t + k. U n d e r the qualified definition we get the following properties: (1) Temporal symmetry iff~(t) Tfj(t + k) t h e n f j ( t + k) T f~(t) (ifJ~(t) +fj(t + k) t h e n f j ( t + k) Sf~(t)) (2.9) (2a) Temporal associativity [f~(t) Tf~(t + k)] A [ f j ( t + k) l"f~(t + m)] ~ Eft(t) Tf~(t + m)] (2.10) for all i, j, l and all k, m such that 0 ~< k ~< m. (Note: i = j = l is meaningful here, e.g., if k ¢ 0). Complete analogy holds for conflict (J,). (2b) Temporal conflict [fi(t) ~ f j ( t + k)] A [ f j ( t + k) Sj~(t + m)] ~ Efi(t) +ft(t + m)] (2.11) for all (i,j,l) A 0 <~k < m (example: k = 0, m = 1, i = j = I) and for all (0 ~< k ~< m) A (j # l) (example: k=m=O,i=j, jCl). R. Ostermark / Fuzzy Sets and Systems 88 (1997) 69-79 73 In the next section we discuss the extended interdependence concept within the framework of a multiperiod optimization model for strategic firm planning. 3. Interdependence in multiperiod firm planning 3.1. Model specification A multiperiod optimization system for strategic firm planning was designed in PC-Windows-environment by S6derlund and Ostermark [14]. The key features of the model are presented below. The mathematical programming formulation contains seven actual decision variables for each planning period (Turnover, New debt, Loan repayment, Investment, New issues, Dividends, and Depreciations) and five deviational variables for possible violations of the constraint set. For each period, ten constraints are specified, seven of which describe the interrelations of the firm and its environment, and three of which determine a set of nonnegativity requirements. The constraints are presented below. The time index is suppressed with the understanding that the constraints are defined separately for each period. RI: R2: R3: R4: R5: R6: R7: Turnover ~< Production capacity, defined as a multiple of fixed assets, w * Fixed assets (w > 0), Loan repayment = q , Debt from the previous period (q > 0), New issues ~< s , Initial equity (s > 0). A target debt ratio is justified in the presence of taxes [4, 7], Cash dividend ~< Free (unrestricted) equity, Cash dividend ~> m , Equity (0 < m < 1), Depreciation = d , Fixed assets (0 < d < 1), Equity ~<f , Debt ( f > 0). The last three constraints limit the feasible set to the nonnegative orthant: R8: Cash ~> 0, R9: Debt >~ 0 (redundant if q < 1 in R2), R10: Fixed assets ~> 0 (redundant if selling of fixed assets is not allowed). Besides the parameters in the above constraints, the following parameters are to be specified: c = operating costs per turnover, t = tax rate, i = interest rate, h = sales receivables/turnover, l = current liabilities/operating costs, a = miscellaneous financial income and expenses/financial assets, r = discount factor. From a managerial viewpoint it is noteworthy that the planning system requires very few input parameters for generating full-fledged firm strategies. The imprecision inherent in the parameters (e.g., prices, tax rates, interest rates, etc.) may be recognized in Monte Carlo runs [-5]. By proper tuning of the parameters, envelope strategies encompassing the extreme outcomes are swiftly produced. 3.2. Numerical experiment The model has been tested on data from some Finnish listed companies, with reasonable strategies generated in all cases. A hypothetical firm is studied below. The parameter vector (see Section 3.1) was set to R. Ostermark / Fuzzy Sets and Systems 88 (1997) 69-79 74 (w = 0.5, q = 0.08, s = 0.05, m = 0.04, d = 0 . 1 5 , f = 1, c = 0.6, t = 0.25, i = 0.09, h = 0.33, 1 = 0.2, a = 0.67, r = 0.2). The system was optimized separately for three linear objectives over a five-year planning horizon (periods 5-9). All objectives are to be maximized: fl: cumulative discounted dividend payments, f2: cumulative discounted net income, f3: cumulative discounted cash flows. The optimal trajectories are presented in Figs. 2 and 3 (Complete numerical results may be obtained from the author on request). The maximal cash flow program coincides with the p r o g r a m for maximal net income. In the sample firm dividend maximization is coupled with smaller investments and growth than maximization of net income [-14]. This observation is consistent with economic theory, which predicts that retained earnings are reinvested up to the point where marginal income from new investment equals marginal cost I-3, 8]. With the optimal programs at hand, we m a y operationalize temporal interdependence in the context of optimal firm planning. The optimization results show that improved current net income (f2) induces improved future cash flows (f3) and vice versa. In fact, for the sample firm, maximization of cash flows is equivalent to maximization of net income. Hence, f2 (t) T f3 (t + k) and f3 (t) T f2 (t + k), t = 5 and 0 ~< k -%<5, since any action improving the value of total discounted net income for the sample firm will automatically improve its cash flows and vice versa. We also see that dividend maximization (fO is temporally in conflict with maximal net income, i.e., [fz(t) Tf3(t+k)]A[f3(t+k)+fl(t+m)]=c,[fz(t)$fl(t+m)], t=5, O<~k<m<~5. 350.00 Net income I Dividends --x--Operating cash tow 300.00 250.0( ×~ x~x 4 5 200.00 150.00 100.00 50.00 0.0O 2 3 6 -50.00 time Fig. 2. Goal trajectories for maximal dividends. 7 I I 8 9 (3.1) 75 R. Ostermark / Fuzzy Sets and Systems 88 (1997) 69 79 900.00 --13--Net income ~Dividends --x--Operating cash fto~ 800.00 700.00 600.00 500.00. 400.00 300.00, 200.00 x~ x 100.00 I 0.00 2 3 , . 4 5 6 I i E 7 8 9 -100.00 time Fig. 3. Goal trajectories for maximal net income. In other words, increasing future dividends from the minimum level at any time point will perturb the maximal net income (or cash flow) trajectory: dividend increments are, ceteris paribus, possible only through a less intensive investment program involving smaller net income. 4. A dynamic membership function The concept of temporal interdependence was applied above to describe the relationship between the objectives in a crisp multiperiod firm planning model. In order to model temporal interdependence in a decision support system we need a dynamic version of the membership function (2.3) for both D M O P and D F M O P situations. 4.1. An application function for dynamic MOP As a first attempt to capture interdependence over time we suggest a direct generalization of the static membership function Gi(x) defined in (2.3). We will here focus on the case with a finite planning horizon in a discrete setting, using the definitions for strong temporal support (2.5) and conflict (2.6) and the assumption of equally important objectives. An extension to unequally important objectives as defined in (2.8) (2.11) is left for future research. The temporal grade of interdependency is defined as follows (cf. (2.2)): A(f/(t))= ~, I ~ s= 1 I f'(t)Tfi(s) ~ . i c j i f t=s 1-- ~, 11, f~(tLf j(s) i , a j i f t=s J i = 1 . . . . . k. (4.1) 76 R. Ostermark / Fuzzy Sets and Systems 88 (1997) 69-79 In (4.1), self-support and self-conflict have been excluded, i.e., i = j =~ s 4: t and s = t ~ i [email protected] The dynamic membership function is now defined for t ~ H, i -- 1 . . . . . k as follows: (1) If A ( f ( t ) ) > 0, then i Gi(t) = 1 if f . ( t ) ~ Mi(t), M i ( t ) - - f i ( t ) :/~A~Z(0)+1) mi(t) -- mi(t) if mi(t) <f/(t) < m i (t), 1 0 (4.2) if f/(t) ~< m i (t). (2) If A(fi(t)) < O, then if fi(t) >~M~(t), 1 oi(t) 1 = Mi(t)--f/(t)] IAtf'(t))l+l if M~ (t) - mi ( t ) / 0 mi(t) <f/(t) < Mi(t), (4.3) if fi (t) ~< m~(t). In order to formulate the auxiliary problem for D M O P (2.4) [16], we have to perform a two-dimensional aggregation of the membership functions, once over time and once over the objectives: max xtE X r 1 {Z 2 {a 1(1) . . . . . G 1 (h)}, ..., T2 {G~(1). . . . . Gk (h)}}, (4.4) where the T2-norm aggregates the membership functions for each individual objective over time and T1 aggregates the objectives themselves. 4.2. An application function for dynamic FMOP Thus far we have assumed that the optimization problem (DMOP) consists of crisp numbers. In fuzzy multiple objective programming (DFMOP) the optimization problem is 'defined by fuzzy numbers. The dynamic F M O P is stated as follows [16]: DFMOP: max {fl(t), ... ,~(t)}. (4.5) x~X The parameters of the problem are LR-type fuzzy numbers ci = (a, 2, fl) represented by their modal value (a) and left (:0 and right (fl) spreads respectively (cf., e.g., [,-12-]). Using the notation Gi(t) = 9i(fi(t)), Gi(t) may be considered as the degree of membership ofxt in the fuzzy set "good solutions" for the ith fuzzy objective at time t. Carlsson and Fuller [-2-] discuss some alternative application functions 9~ : ~ (N) --* [0, 1]. They suggest the use of a family of static application functions, which we extend to the dynamic case as follows: ( ) Gi(t) = T3 1 -- 1 + Dv(rni(t),~(t))' 1 + Dp(f/ii(t),fi(t)) ' 46, where T3 is a t-norm (for example, the connective AND), and Dp is a metric in ~(N), 0 ~< p ~< oo. Below we will use the metric p = oo represented by D~(2,5) = sup D([2] ~, [5]~), ~[0,11 (4.7) R. Ostermark / Fuzzy Sets and Systems 88 (1997) 69-79 77 where [2]" = {y e R 12(y) >~ e} is a nonfuzzy 0~-levelset [2]. In order to recognize the presence of support vs. conflict in the objectives we change the shape of Gi(t) as in the crisp case by the value of A(~(t)) under the assumption of equally important objectives as follows: If A(~(t)) > 0 then Gi(t, A~(t)) := Gi(t) 1/~AG'~°)+1). (4.8) If A(~(t)) < 0 then Gi(t,A ~(t)) := Gi(t) IdG'~°)l+ l G~(t, A (j~(t))) measures simultaneously the distance from the value of the objective functionj~(t) at xt to the undesirable fuzzy level rfiz(t) and to the desirable fuzzy level _~ri(t). Therefore, Gi(t) provides a measure of overall fitness of the solution as being far from the undesirable level and close to the desirable level. The D F M O P (4.5) can be solved as in the crisp dynamic case by the two-dimensional aggregation procedure defined in (4.4), using the fuzzy membership function (4.8) in place of the crisp counterpart defined in (4.2) (4.3) and measuring the grade of interdependency (4.1) with fuzzy objectives. 4.3. Example We illustrate the measurement of temporal interdependence with fuzzy objectives (cf. (4.1)), the dynamic membership function (4.8), and the two-dimensional aggregation operations in (4.4) by a two-period, bi-objective D F M O P . The example is a dynamic version of the one presented by Carlsson and Fuller [2]. Consider the following discrete-time problem, where the objective functions are fuzzy numbers with the same widths: DFMOP: max {fl(t),L(t)}, (4.9) O~<x~<l t~[1,2] where il(t) = + r)t, ~,~ , f2(t) = + r)t' 2' ' r = the risk-adjusted discount rate (opportunity cost) of the firm. The discount rate incorporates the risk faced by the firm [9]. We assume the following reference numbers (desired (Mi(t)) vs. undesired (n~i(t)) points) for the firm: 1~1 (1) = M1(2) = (1, ½, ½), rill (1) = rhl (2) = (0, ½, ½), ]~z (1) = ~r2 (2) = (0, ½, ½), rfi2 (1) = rfi2(2) = (1, ½, ½). (4.10) We measure the distance between the fuzzy objective values and the reference points by the Doo-metric (4.7) and the following t-norm aggregates in (4.6) and (4.4): T1 = product(), T 2 = addition(), T3 = product(). Then we obtain the following distances (note that we assume xl D(rfia (1)'j71 (1)) - 1 + r' D()~t 1 (1), j71 (1)) - 1 - xl 1+ r ' D(rfia (2)'j7~ (2)) D()VIa _ x 2 r) 2' (1 + 1 -- X 2 (2),j71 (2)) -- (1~ '+ r) ffti(t ) ~ ( t ) (4.11) ~ ~Ii(t), Vi, t): 78 R. Ostermark / Fuzzy Sets and Systems 88 (1997) 69-79 D(r~2(1)'f2(1)) - 1 + r ' D(]~r2 (1),f2 (1)) - 1 - - x 2 l+r D(rfi2(2),f2(2) ) _ D(~2(2),f2(2)) _ i -- x 2 (1 ÷ r) 2' x2 (1 + r) 2' (4.12) Inserting (4.12) in (4.6) and using the product t-norm gives x2 (1 ÷ r) 2 G1(2) = ((1 + r) 2 ÷ x2)((1 ÷ r) 2 ÷ 1 - x 2 ) ' xl (1 + r) (1 + r + x l ) ( 2 + r - x l ) ' 61(1) (4.13) x 2 (1 + r) 2 x~(1 + r) G2(1) = (1 + r + x2)(2 + r - x}) ' G2(2) = ((l ÷ r) 2 + X22)((1 ÷ r) 2 ÷ 1 x2)" The objectives are supportive both over time and against each other. Hence, by (4.1), A (~ (t)) = 3, Vi, t and the dynamic membership functions are transformed according to (4.8) as follows: Oi(t, A (fi (t))) = Gi(t) 1/4 . (4.14) Carrying out the two-way aggregation in (4.4) with the t-norms given in (4.11) we obtain max {(G1 (1) 1/4 + G1 (2) 1/'4) * (G2(1) a/4 + G2(2)1/4)} 0~x,~< 1 t~[1,2] : max { ( I xl(l+r) O<x,,~l ((l+r)+xl)(2+r--xl)J ~1/4 I x2(l+r) 2 + i(l+r)2+x2)((l+r)2+l--x2)J .~1/4~ ] tel1,21 * ((1 + r) + x2)(2 + r x 2) + ((1 + r) 2 -~- X2)((1 -[- r) 2 -[- 1 x2)J JJ (4.15) with the unique solution x* = (x*, x*) = (1, 1). We have demonstrated the consistency of the dynamic framework both for describing and modelling temporal interdependence in D F M O P problems. When included in a computerized decision support system, more interesting results and interpretations may be expected. 5. Conclusion In this study we have considered temporal interdependence in multiple criteria decision making. O u r analysis extends the (static) concept introduced by Carlsson and Fuller [2] in a way that allows coping with goal conflicts typically arising in managerial decision making. The concept of temporal interdependence was used to describe mutually supportive and mutually conflicting criteria in a multiperiod firm model. Next, the static membership function proposed by Carlsson and Fuller [2] was generalized to the dynamic case both for D M O P and D F M O P problems. We showed that incorporating the discount rate of the firm, i.e., the risk-adjusted weighted average cost of capital in the dynamic objective functions is essential in corporate planning. The cost of capital affects the shape of the membership functions and, therefore, the mutual support vs. conflict in the objective set. In the derivations we have utilized the simplifying assumption that all objectives are equally important. This allows usage of the number of criteria as a measure of support/conflict in the objective set, precisely as in Carlsson and Fuller [2]. R. Ostermark / Fuzzy Sets and Systems 88 (1997) 69 79 79 In future research an important issue is to properly recognize the importanceof individual objectives in the interdependence measure A ( f (t)). The structure of the dynamic membership function proposed in this study may be affected by an extension of our approach to situations with qualified weak support vs. conflict. The core issue is the dynamic analysis of the optimal goal trajectories over time. To cope with changes in the solution space through time, for example, due to nonconstant asset prices, interest rates etc., a fuzzy decision support system could be constructed as a combination of temporal interdependence and recursive programming [11]. Acknowledgements Financial support from the Academy of Finland is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] C. Carlsson and R. Fuller, Multiple criteria decision making. The case for interdependence, Comp. Oper. 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