| OCR Text |
Show Definition: Non- inferior solution A non- inferior solution ( also known as Pareto optimal solution) to the vector optimization problem is one in which no increase can be obtained in any of the objectives without causing a simultaneous decrease in at least one of the other objectives. The vec tor u_* is called non- inferior solution to the problem: max [ fi( x, u_, jx),..., fg ( x_, u_, a)], u_ € U, where U is the set of all feasible solutions, if and only jf there does not exist any^ e U such that J_ ( x,^,_ a) > f_ ( x, u_* a.) and fj < x_, _ u, a) > fj ( x_, u_* a) for some i = 1,2,..., 6. This solution is obviously not unique. Consequently, any point at which no one objective function fj ( x_>. u_>. a) can be improved without causing a degradation in some other objective function f; ^,_ u, a); i f j is a non- inferior solution to the vector optimization problem. Consider for example the following problem with two objective functions: fj( x, u, a) maximize u f2( x, u, a) in which thus fj( « ) = objective function for recreation and tourism ( e. g. visitors/ day) f2( « ) = objective function for brine shrimp harvesting ( e. g. tons/ year) u = scalar control measure - being the funds ($) available for investment to promote both recreational tourism and brine shrimp harvesting. For example the funds which will stimulate economic activities that in turn affect algal growth in the lake xj = state variable representing the level of nutrients feeding the algae ( which in turn enhance the brine shrimp colony and detract from recreational use) X2 = water salinity 0q, x2) exogeneous variables representiru parameters such as population, weather conditions, etc. Clearly, the two objective functions, enhancement of recreation and tourism, fj(*), and enhancement of brine shrimp harvesting, f2(*)> are non- commensurable andin fact in conflict ( see Figure 13). i From Figure 13, i\ ( x, u, a) achieves its maximum at uf, where f2 ( x, u, a) achieves its maximum at u2. Due to the concavity or these two functions any point u, between uj and u? { x\\ < u < u2 ) will improve one objective function at the expense or the degradation of the other. Thus, all these values of u are non- inferior points. The SWT method selects only those solutions which belong to the non- inferior set, thus eliminating all inferior solutions from further consideration. Furthermore, the SWT method provides the decisionmaker with the marginal trade- offs between any two objective functions. These trade- offs between the i" 1 and j4* 1 objectives which are denoted by Xjj, satisfy the following mathematical relationship The trade- offs are determined on the basis of the duality theory in nonlinear programming. Since all Xy can be determined computationally, the trade- offs between any two objective functions can be found as follows: df& l *- Xli* j; i, j « l, afjO) 1J = 1,2,..., 6 It can be shown that all Xjj ~ 0 correspond to the inferior solution. Thus, since Xjj > 0 ( in order to satisfy the Kuhn- Tucker conditions), interest will be only in X « > 0. These X « are the Lagrange multipliers associated with the i" 1 objective function in the La- grangian equation with the function f; (•) acting as an active ( binding) constraint. Once all the needed tradeoff functions Xy (•) are determined, the surrogate worth functions W « (*) can be constructed in cooperation with the decision- maker. The surrogate worth function Wy can be defined as a function of XJ; that provides the desirability of the decision- maker in making a trade between two levels of fj (•) and f j (•)• Specifically: Wjj > 0, when Xjj marginal units of fj (*) are preferred over one marginal unit of fj(*) given the satisfaction of the other objectives at some given level. 48 |
