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Show Sub- objective Coordination In the following discussion, it is assumed that functional relationships ( technical connections) can be derived for the major objectives and sub- objectives in terms of the control measures, state variables, and the exogenous variables. Should these functional derivations prove to be infeasible for some of the objectives, a more qualitative approach should be sought. Recreation and tourism, one of the major objectives identified in this study ( see Figure 7) will be used as a vehicle for demonstrating the proposed coordination mechanism between a major objective and its associated sub- objectives. The following are sub- objectives for recreation and tourism ( see Figure 14). ( i) stable water level ( ii) fresh water bodies for water based activities ( skiing, boating, swimming, fishing) ( iii) easy access ( iv) optimum use intensity ( v) low health hazard ( vi) low insect population ( brine fly, deer fly, horse fly). The coordination between higher and lower levels of a sub- hierarchy of a major objective and its associated sub- objectives is analogous to the same coordination in the overall hierarchy of multiple major objectives. Therefore, the surrogate worth trade- off method can be applied for the analysis and optimization of the trade- offs among the sub- objectives. An optimal policy in this sub- hierarchy means that at the corresponding levels of the sub- objectives the decision- maker is indifferent to any further marginal trade- offs among the sub- objectives. For convenience in notation let: ill = the vector of optimal control measures that corresponds to the six sub- objectives associated with recreation and tourism. fll(* X^ 2(*)> f13(*)>.••• » fi$(*) = the values of the sub- objectives evaluated at the optimal control measures ( corresponding to Wjj = 0). Similar analysis can be made of all other sub- hierarchies associated with the remaining major objectives. The final product will be one or more optimal control measures for each major objective as well as the corresponding indifference achievement level of the sub- objectives. Assuming there are six major objectives and five sub- objectives for each major objective, the total output from the sub- hierarchies is as follows ( see Figure 15). ' 4lO, f62(')>.--, f650) Highest Level Coordination The task here is to utilize all the information generated by the lower levels and to generate an over- | ali optimal policy for the whole system. The informa- jtion jrom the lower levels includes u_:, j = 1, 2,..., 6 andfij( « ), i.= 1,2, ..., 6; j = 1,2,..., 5. The highest ' level in the hierarchy generates a new optimal control vector,^*, for the entire system where the decisionmaker is indifferent to any further trade- offs among the major objectives evaluated at u_ . This is an iterative procedure where the problem of convergence needs a further study. Computational Procedure The problem of the highest level in the hierarchy can be mathematically written as follows: Max ii fl(') f2(*) Subject to the constraints gk(') fij(') < 0; k= l, 2,. > exi; i= 1,2,. j = 1,2,. .., K ., 6 ., 5 where the SWT method can be applied to solve this problem. If no feasible solution can be generated, then some of the limits ( fy) of the binding constraints should be relaxed with some tolerance ejj. The corresponding Lagrange multipliers associated with the new bounds e\\ should provide sufficient information on the trade- offs among the lower echelon sub- objectives and the higher echelon major j objectives, and the procedure can be repeated. 51 |