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In the light of the definition of weak efficient solution of a general multiobjective geometric programming problem (MGP), we define weak efficient solution of (IMGP) with respect to [[less than or equal to].sup.k.sub.X] partial ordering in I[(R).sup.k and call this solution as x-efficient solution.
A feasible solution (x, [tau]) with acceptable degree [tau] of (IMGP) is said to be x-efficient solution of (IMGP) if there does not exist any feasible solution (y,[tau]') with acceptable degree [tau]', ([tau]'[greater than or equal to][tau]), of (IMGP) such that
It is difficult to derive the x-efficient solution (Definition 4) of (IMGP) analytically.
After substituting the value of [x.sup.O.sub.i] and [x.sup.F.sub.j], (IMGP)' can be further simplified to the following form.
(IMGP)' is a general geometric programming problem which is free from interval uncertainty, and can be solved using geometric programming technique.
If ([[theta].sup.opt],[x.sup.opt]) is an optimal solution of the (IMGP)', then [x.sup.opt] is an x-efficient solution of (IMGP) with degree of satisfaction [[theta].sup.opt].
Summarizing the above discussion, the waste water treatment optimization model denoted by (W - IMGP) can be formulated as
This is an interval multi-objective geometric programming (IMGP) model.
Here (IMGP)' becomes the deterministic equivalent (W - IMGP)', which is
For any change on the pollutant limit b to b [+ or -] [epsilon], the constraint for remaining amount of pollutant in (W - IMGP)' becomes [x.sub.1][x.sub.2][x.sub.3] [less than or equal to] (b [+ or -] [epsilon])%.
In the process of the methodology, one may observe that X-efficient solution of (IMGP) is associated with certain degree of feasibility and certain degree of flexibility of the objective functions towards the goals.
So from Definition 4, we can say that there exists [x.sup.*] [not equal to] [x.sup.opt], which is a feasible solution of (IMGP) with degree of acceptability [[tau].sup.*] such that [mathematical expression not reproducible].
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