The reasoning presented in previous sections might seem rather abstract. Nonethe-less, all development of reference point methods was very much applications-oriented, starting with the original work of Kallio et al. (1980) on forestry models, including many other applications to energy, land use and environmental models at IIASA, applications of satisficing trade-off methods by Nakayama et al. (1983) to engineer-ing design, various applications ofPareto Raceof Korhonenet al. (1985), and many others. Recent applications of a reference (aspiration-reservation) point method have been developed at IIASA using a modular tool MCMA (MultiCriteria Model Analysis)8by Granat and Makowski (1995, 1998) in relation to regional management of water quality (Makowski, Somly´ody and Watkins, 1996), land use planning (An-toine, Fischer, Makowski, 1997) and urban land-use planning (Matsuhashi, 1997).
Here we present only two short examples: one application to engineering design and another to ship navigation support.
7The parameters are assumed here, for simplicity, to be constant in time; if they change in time or if we compute derivatives with respect to decisionsxi,t, we must increase their number.
8The MCMA tool is available from the URL: www.iiasa.ac.at/∼marek/softfreee of charge for research and educational purposes.
The first case concerns a classical problem in mechanical design – the design of a spur gear transmission unit, see e.g. Osyczka (1994). The mechanical outlay of this unit is shown in Fig. 3. The design problem consists in choosing some mechanical dimensions (the width of the rim of toothed wheel, the diameters of the input and output shafts, the number of teeth of the pinion wheel, etc.) in order to obtain a best design. However, there is no single measure of the quality of design of such a gear transmission. Even when trying only to make the unit as compact as possible – which can be expressed by minimizing the volume of the unit while satisfying various constraints related to mechanical stresses and to an expected lifetime of efficient work of the gear unit – we should take into account other objectives, such as the distance between the axes or even the width of the rim of toothed wheel (which is, at the same time, a decision variable).
The specification of a mathematical model that expresses the available knowledge on designing such gear units is obviously a question of expert opinion. After all, the modeler is a specialist in her/his specific field and knows best how to choose substantive models for a given problem; that is also the reason why we present here mostly methods for supporting the modeler in model analysis, not supplementing her/him in final decisions. Therefore, in the example of gear unit design, we follow a specialist who has selected a specific model in this case (Osyczka, 1994) and comment only on the methodology of preparing the model for analysis and analyzing it.
d_p2d_p1 d_1 d_2
l1
b
a
l2
Figure 3: A diagram of the spur gear unit
The equations of the corresponding model contain some tables of coefficients ob-tained by empirical, mechanical studies. While such original data are very valuable, an analytic approximation of them might be more useful for model analysis. Thus, these tables were approximated by exponential functions. The problem might be then specified in a classical textbook format such as (54, 55) by defining three objec-tive functions fi(x) and 14 constraints gj(x), some nonlinear and some expressing
simple bounds. We present it here in a form similar to the textbook format (al-though the model was actually rewritten in the DIDAS-N++ format, because this system was used for further analysis).
1. The decision variables are: the width of the toothed wheel rimb(which is also an objective), the diameters d1 and d2 of the input and output shafts, the number of teeth of the pinion wheel ˜z1 and the pitch of gear teeth ˜m (the last two decision variables are actually discrete).
2. The objectives are: the volume of the gear unit q1 =f1 [mm3], the distance between the axes q2 =f2 [mm], the width of the toothed wheel rim q3 =f3 [mm]:
q1 = ((π
4m˜2(˜z12+ ˜z22)b) +π
2d31+ π
2d32)∗10−5 q2 = (˜z1 + ˜z2)
2 m˜
q3 = b (63)
3. The constraints on the decisions concern various geometric relations and mechanical stresses:
• g1 expresses the bending stress of the pinion:
g1 =kg1 −Pog∗w1/(b∗m)˜ (64) where:
V =π∗m˜ ∗z˜1∗n/60000;˜ Kd= (14.5 +V)/14.5;
Pmax = 102∗N ∗9.81/V; Pog =Pmax∗Kp∗Kb∗Kd; w1 = 4.7607∗exp(−0.104531∗(˜z1+ 1.28627)) + 1.67421
• g2 expresses the bending stress of the gear:
g2 =kg2 −Pog∗w2/(b∗m)˜ (65) where:
w2 = 4.7607∗exp(−0.104531∗(˜z2+ 1.28627)) + 1.67421
• g3 expresses the surface pressure of smaller wheel:
g3 =ko1−Po1/(b∗m˜ ∗z˜1)∗(1 + ˜z1/˜z2)∗y1 (66) where:
Po1 =Pmax∗Kp∗Kb∗Kd∗Kz˜1;
y1 = 28.4869∗exp(−0.290085∗(˜z1−1.78811)) + 3.31178
• g4 expresses the surface pressure of the greater wheel:
g4 =ko2 −Po2/(b∗m˜ ∗z˜2)∗(1 + ˜z2/˜z1)∗yc (67) where:
Po2 =Pmax∗Kp∗Kb∗Kd∗Kz2
• g5, g6 express the torsional stresses of input and output shafts:
g5 =ks−Ms1/W01; g6 =ks−Ms2/W02 (68) where:
Ms1 = 9549296∗N/n; W01 = (π∗d31)/16;
Ms2 = Ms1/(z1/z2); W02 = (π∗d32)/16
• g7, g8, g9 express the deviations of the velocity ratio and the relation between
˜
m and d1:
g7 =i−z˜1/˜z2+ ∆i; g8 = ˜z1/˜z2−i+ ∆i; g9 = ˜m∗(˜z1−2.4)−d1 (69)
• Other constraints are:
g10 = ˜m∗(˜z2−2.4)−d2; g11 =b/m˜ −bm˜min; g12 =bm˜max−b/m˜ g13 = amax−(˜z1+ ˜z2)/2∗m;˜ g14 = ˜z2−z˜1/i (70) 4. In the above model, the following parameters were used:
N = 12.0 ˜n= 280.0; i= 0.317; ∆i= 0.01; z˜1 = 20
whereN is the input power [kW], ˜n is the rotational input speed [rev/min],i is the velocity ratio, ∆i is the allowable deviation of velocity ratio, ˜z1 is the number of teeth of the pinion;
Geometric data are:
bm˜min = 5.0; bm˜max= 10.0; amax= 293.8
wherebm˜minis the minimumb/m˜ coefficient ( ˜m=dpi/zi,i= 1,2, is the pitch of the gear teeth, while dpi are the standard diameters of the gear wheels and b is the teeth width),bm˜max is the maximumb/m˜ coefficient,amax is the maximum distance between the axes [mm];
Material data are:
kg1 = 105; kg2 = 105; ko1 = 62; ko2 = 62; ks = 70
wherekg1is the allowable bending stress for the pinion [MPa],kg2is the allowable bending stress for the gear [MPa],ko1 is the allowable surface pressure for the pinion
[MPa], ko2 is the allowable surface pressure for the gear [MPa], ks is the allowable torsional stress of the shaft [MPa];
Other data are:
Kb = 1.12; Kz1 = 1.87; Kz2 = 1.3; Kp = 1.25
whereKb is the coefficient of the concentrated load, Kz1 is the coefficient of the equivalent load for the pinion, Kz2 is the coefficient of the equivalent load for the gear, Kp is an overload factor;
Calculated data are:
T = 8000; yc= 3.11
whereT is the time of efficient work of the gear,ycis a coefficient for the assumed pressure angle.
The exponential approximations of empirical data tables are expressed by the functions w1, w2, y1. We presented all these equations with a purpose: in order to stress that a computerized mathematical model might be very complicated. The model presented above is actually rather small – because it is static, not dynamic – as compared to other models used in applications. However, the model repre-sents rather advanced knowledge in mechanical engineering and the selection of its various details relies on expert intuition: good modeling is an art. Moreover, even for such rather small model, the reader should imagine programming the model, supplying it with all necessary derivatives, selecting by hand such values of deci-sion variables which would satisfy required constraints, all done without specialized software supporting model analysis.
When using such a specialized software, the modeler should use first a model generator, then model compiler; a good model compiler will automatically deter-mine all needed derivatives. Even when such fast executable, compiled core model is available, the modeler might have trouble with simple model simulation. The form of the model is rather complicated (actually – not convex) and without a good ex-perience in mechanical design it is difficult to select such values of decision variables which are acceptable.
This is illustrated in Fig. 4 which shows the results of an inverse simulation of the model with two model outcomes – objectives q1 and q3 denoted respectively by f1 and f3 – and two decision variables denoted by d1 and d2, all stabilized9. However, since the aspiration and reservation levels were arbitrarily selected, even the inverse simulation cannot give satisfactory results. The optimization of a corresponding achievement function indicates that such arbitrary reference levels cannot be realized in this model. The contours indicated in Fig. 4 represent the values of membership functions µi(qi,q¯i,¯¯qi) and the circles on these contours indicate the attained levels of objectives. Values 0 of these membership functions at circled points indicate that the requirements of the modeler cannot be satisfied.
9In Fig. 4 – Fig. 6 we use actual interaction screens ofIsaap-Tool inDidas-N++.
Figure 4: Interaction screen of DIDAS-N++ in the inverse simulation case, arbitrary aspiration levels
Figure 5: Interaction screen of DIDAS-N++ in the inverse simulation case, aspira-tion levels based on mechanical experience
Figure 6: Interaction screen of DIDAS-N++ in the softly constrained simulation case, improvements of both objectives
In order to find results that are admissible for the model, other aspiration levels must be selected using the experience of a designer, see Fig. 5 where the aspirations were set according to data given by Osyczka (1994). Since the model was actually changed – by using the exponential approximation of data tables – from the one described by Osyczka, the results of the inverse simulation with membership values close to 1 indicate a positive validity test of the model. However, the inverse simu-lation results are not efficient in the sense of minimization of objectives (the results given by Osyczka might be efficient for his model, but the model was changed by including approximating functions).
Improvement of both (or even all three) objectives considered can be obtained by switching to softly constrained simulation, as shown in Fig. 6, where the soft con-straints on decision variables were relaxed in such a way as to obtain efficient results for the problem of minimizing both selected objectives. In Fig. 6, the improvement of objective values is shown by line segments leading to circles that indicate the attained values. A serious model analysis would clearly not stop at the results of such an experiment – many other experiments, including post-optimal parametric analysis, might be necessary. However, the above example is presented only as an illustration of some basic functions of a system of computerized tools for multi-objective model analysis and decision support.
Another application example shows the usefulness of including dynamic formats of models. This case concerns ship navigation support (see ´Smierzchalski et al.
1994): the problem is to control the course of a ship in such a way as to maximize the minimal distance from possible collision objects while minimizing the deviations from the initial course of the ship, see Fig. 7.
Own ship ShipBi
Ship Bj
CP Bj w2
w1
(w1, w2) (vj, ψj) (vi, ψi)
(w1i, w2i)
DABi
DABj
CP A CP Bi
(w1j, w2j) (v1, ψ1)
Figure 7: A diagram of ship collision control situation (CPA – safe zone for ship A)
This is a dynamic problem, with the equations of the model described initially by a set of differential equations for t ∈[0;T]:
˙
w1(t) = v1sinx(t)
˙
w2(t) = v1cosx(t)
˙
w1j(t) = vjsinψj, j = 2, . . .n˘
˙
w2j(t) = vjcosψj, j = 2, . . .n˘ (71) where x(t) is the course of ”our” ship, ψj – courses of other ships, with initial values of ship positions given as the vector w(0); between other model outcomes, the objectives can be modeled as:
q1 = min
t∈[0;T] min
j=2,...˘n((w1(t)−w1j(t))2+ (w2(t)−w2j(t))2) q2 =
Z T
0
(x(t)−ψ1)2dt (72)
whereq1 represents the (squared) minimal distance which should be maximized and q2 represents the (squared) average deviation from initial course, which should be minimized.
To be used in a DIDAS-N system, this model was simply discretized in time, with the resulting model form similar to Eq. (59). We do not describe the anal-ysis of this model in more detail here (the results of such analanal-ysis are given e.g.
in ´Smierzchalski et al., 1994); this example was quoted only to show the practical sense of using dynamic models with multi-objective analysis and optimization. Ex-periments with this model support the conclusions about the usefulness of algebraic model differentiation and model compiling, of multi-objective modeling and inverse or softly constrained simulation for the modeler.