5.3. Relationship
between Outputs and
Inputs
135
However, for systems that are described by combinatorial mathematical
models, such as the water resources system analyzed in Chapters 2-4,
computation of the sensitivity coefficients is considerably more complicated. The nonconvexity of the set of feasible solutions of combinatorial
models precludes automatic computation of the sensitivity coefficients
from the optimal solution. For combinatorial models the number of possible sets of perturbed inputs is myriad, and an enormous amount of computer time would be needed to find the sensitivity effects of each set. Consequently, it is opportune to limit a sensitivity analysis to discovering the
effects of what are believed to be the most significant inputs on the values
of the objective function.
5.3. Relationship between Outputs and Inputs of the
Water Resources Model
The relationship between all the inputs and outputs of the mathematical
model of the water resources system that was developed in Chapter 2 may
be represented conceptually by a matrix as follows:
Outputs
Inputs f, \j t ,
Dijt,., .. Xijt
value of con, straint (2.2),. .
value of con., straint (2.17)
Hijt
c jt
X
X
Iijt
X
X
Uijt
Vj
X
X
Qimt
X
The inputs to the model include all the constants and the decision variables. The outputs of the system include the return function /, all the state
variables, and the values of the inequalities and equations in the set (2.2)(2.17). The effects of the inputs on the values of the constraints are needed
to establish whether or not a constraint is violated. An χ in element (fc, Ϊ)
between Outputs and
Inputs
135
However, for systems that are described by combinatorial mathematical
models, such as the water resources system analyzed in Chapters 2-4,
computation of the sensitivity coefficients is considerably more complicated. The nonconvexity of the set of feasible solutions of combinatorial
models precludes automatic computation of the sensitivity coefficients
from the optimal solution. For combinatorial models the number of possible sets of perturbed inputs is myriad, and an enormous amount of computer time would be needed to find the sensitivity effects of each set. Consequently, it is opportune to limit a sensitivity analysis to discovering the
effects of what are believed to be the most significant inputs on the values
of the objective function.
5.3. Relationship between Outputs and Inputs of the
Water Resources Model
The relationship between all the inputs and outputs of the mathematical
model of the water resources system that was developed in Chapter 2 may
be represented conceptually by a matrix as follows:
Outputs
Inputs f, \j t ,
Dijt,., .. Xijt
value of con, straint (2.2),. .
value of con., straint (2.17)
Hijt
c jt
X
X
Iijt
X
X
Uijt
Vj
X
X
Qimt
X
The inputs to the model include all the constants and the decision variables. The outputs of the system include the return function /, all the state
variables, and the values of the inequalities and equations in the set (2.2)(2.17). The effects of the inputs on the values of the constraints are needed
to establish whether or not a constraint is violated. An χ in element (fc, Ϊ)