Optimisation with an Environmental Input-Output Model

Multiobjective Linear Programming

“…Therefore, point x (5) cannot be a Pareto-efficient solution of the MOLP (8.2)- (8.4). Similarly, the extreme points x (1) and x (2) are dominated by points x (3) and x (4) and therefore cannot be candidates for an efficient solution of problem (8.2)- (8.4). The graphical representation in criterion space is provided in Figure 8.2.…”

“…As the reader can see in Figure 8.2, the received solution x (3) is an efficient solution for the primal problem (8.2)-(8.4). Planting 60 acres of potatoes and 25 acres of wheat yields return revenue in the amount of 5,400 and 85 acres of land are used.…”

“…The corresponding values of the objective functions f 1 (x) and f 2 (x) respectively can be found in Table 8.1. Single optimization with respect to the first objective f 1 (x) yields the unique optimal solution x (3) = (60, 25) and f 1 (x (3) ) = 5,400. Maximization of the second objective function f 2 (x) leads to infinite many optimal solutions-represented by the linear convex combination of points x (4) and x (5) -with the objective function value equal to 100.…”

“…There are five extreme points, denoted by x (1) , x (2) , x (3) , x (4) , and x (5) . The corresponding values of the objective functions f 1 (x) and f 2 (x) respectively can be found in Table 8.1.…”

“…The ideal solution, consisting of the optimal values from the single optimization represented by the pointF , is not possible. Consequently, the solution of the linear vector optimization problem (8.2)-(8.4) consists in finding Pareto-efficient points of F on the set of feasible solution K. The set of efficient solutions for the MOLP (8.2)-(8.4) is then described as the set of all linear convex combinations of the extreme points x (3) and x (4) : (4) , 0 λ 1}. (8.5)…”