Applications by Johannes Bisschop, Alexander Meeraus (auth.), Jean-Louis

By Johannes Bisschop, Alexander Meeraus (auth.), Jean-Louis Goffin, Jean-Marc Rousseau (eds.)

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Surface water allocations to the farmers, for instance, comprise some of the x-components. The vector d represents a set of subsidies and taxes that the government wants to impose on farmers' water related activities. Note that these control variables enter the inner objective function in a multiplicative fashion, thereby making the above problem 36 J. / Indus Basin Model bilinear. 1). 1), the following algorithm can be applied. Add for the moment the outer constraints to the inner optimization problem, and set the vector d equal to zero.

Although linear programming techniques are usually not applicable to the non-convex multi-level programming problem, it is shown how the special case of the Indus Basin problem forms an exception. In conclusion, we would like to stress the importance of multi-level programming aspects in any policy-related exercise involving a hierarchy of optimizing agents. The behavioral roles of each agent must be spelled out clearly for both conceptual and computational reasons. In addition, we would like to express the need for more algorithmic developments pertaining to multi-level programming problems, since they occur frequently in a strategic planning environment.

In fact, in the solution procedure described next, such constraints prevent from obtaining intermediate solutions which are unfeasible. Let, for each pair (p, f), H(p, f) be the feasible set for this problem, and {ul(p, f), 7rl(p,/): 1 ~ L(p, f)} denote the set of its extreme points. 1) can be written as 6 = Mnin{~ P b(p)u(p)+ ~ d([)v(J) + ~ ~ Min{wpf: wps >t ul(p, f)q(p, f) + Trl(p, f)a(u(p), v(f)), P F l E L(p, f)}}. 2) is the kind of problem which is solved iteratively in the Benders' decomposition approach.

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