However, it is less obvious (andmuch more interesting) to judge the extend by which aconstraint is violated or how much leeway there still is inthe design. A utilization index shall therefore be proposed as depicted in Fig. 5.Fig. 4 shows the utilization function uk gk(−→ x )σk . Impor-tant properties are that it is strictly monotonic and thatuk(0) = 100% forgk(−→ x )σk= 0 (i.e., where a design lies onthe inequality constraint itself). Moreover, uk(−∞) = 0%for designs which lie far away from the constraint whileuk(∞) = 200% for designs which heavily violate the con-straint. Finally, the slope of u equals 1 atgk(−→ x )σk=0 whichyields an almost linear dependency in the vicinity of theconstraint just becoming active.Fig. 5 shows the frequency distribution fk of a Sobol se-quence with respect to the k-th constraint gk. σk is used tonormalize all terms so as to standardize the frequency dis-tribution and to achieve independence of the actual con-straint values. µk denotes the mean.A frequency distribution as depicted in Fig. 5 suggeststhat the constraint gk is non-critical for the majority ofvariants. (Of course, it needs to be kept in mind that anoptimization performed at a later point in time could stilllead into a region in which gk becomes active.) Each vari-ant can now be assessed in terms of its utilization index.In addition, the utilization index at the mean uk µkσk andat other prominent values such as uk µkσk±1 give anidea of the constraint’s severity. (The utilization indexmight also be serviceable to get an idea about a designsrobustness. If two designs perform equally well, the onewith smaller utilization index may be the better.)Illustrating exampleThe constraint management as theoretically discussedabove shall now be illustrated on the basis of a practicalexample taken from a contemporary ferry design project.A total of 13 inequality constraints were considered. Theconstraints were of positional and integral type. Differen-tial constraints on fairness were accommodated implicitlyby means of an inner geometric optimization as realizedwith the FRIENDSHIP-Modeler, see Harries (1998).The first set of inequality constraints was so confiningthat just a few feasible designs were present in a Sobolinvestigation of 5000 variants, the domain index as in-troduced above being d = 2%. Those initial candidates,however, were considered unfavorable with respect to hy-drodynamic performance (which in the end had to be op-timized). For instance, it turned out that GMT was muchhigher than desired. An increase in the number of fea-sible designs could only be achieved by relaxing someconstraints.In order to identify constraint limits which are beneficialto modify (if permissible) the domain index d is studied.Fig. 6 shows d for a variation of two selected constraints –one on the ferry’s draft and another on its freeboard. Forthe actual values of Cdra f tMaxand Cf reeboardMaxabout 50%of the designs were feasible. Increasing Cf reeboardMaxfea-tures a strong dependency of d which means that a smallrelaxation of the limiting value already yields a substan-tial rise in the number of feasible hull forms. Meanwhile,slight changes in Cdra f tMaxdo not bring about any tangi-ble freedom, the slope being almost horizontal. Conse-quently, it appears to be more profitable to consider therelaxation of the freeboard constraint if need be.In addition, it can be observed in Fig. 6 that both curvesassume horizontal branches on either side. When tight-ening the limiting values the number of feasible designsnaturally drops to zero. When relaxing a constraint, how-ever, one or several other constraints become active atsome point. A further assessment of the two example constraints ondraft and freeboard is undertaken on the basis of their fre-quency distributions, see Fig. 7 and Fig. 8. It can be seenthat the bulk of the designs is rather close to the freeboardconstraint and that quite a few designs are already infea-sible just because of this constraint. On the contrary, themajority of designs is quite far away from the draft con-straint and only a small set encounters this constraint asactive. A brief way to describe this is to consider the uti-lization index for the mean values: udra f t  µdra f tσdra f t = 5%and uf reeboard  µf reeboardσf reeboard = 47%.Any single design can now be judged by consideringits utilization index in comparison to the mean values.A feasible design with uf reeboard = 99% would be onethat lies very close to the freeboard constraint and con-siderably higher than the mean of 47%. 船舶设计应用中的约束管理英文文献和中文翻译(3):http://www.751com.cn/jixie/lunwen_63336.html