As for linear models, the linear
objective coefficient row (row 12) models the linear terms
of the objective. These coefficients are multiplied by the
vector of variable values by the Excel formula in cell I2.
The cell is labeled Linear to indicate that this value is the
linear part of the objective function.
The worksheet has two additional rows in the portion of
the model describing the variables, one row for nonlinear
objective coefficients and the other for nonlinear terms.
The space for nonlinear objective coefficients in row 14
is useful for nonlinear terms that have only a single coefficient.
In this case it holds the values of -*d*. The equations
for the quadratic terms are entered into row 13. We color
the cells in this row from column H through M magenta. This
color indicates that the student is to enter Excel formulas
in these cells. For example, the equation in I13 is to be
the square of the value of P1. That value is in I8, so the
formula in I13 is.
=I8^2
This describes the nonlinear effect of the variable in P1
on profit. The rest of the terms in row 13 are similar,
but each nonlinear term refers to the variable above it. For
example J13 holds the formula (=J8^2). The terms from J13 through
M13 are easily copied form I13 using the Excel *Fill Right* command.
The nonlinear coefficients multiply the nonlinear
terms and are summed to obtain the entry called the *Nonlinear
1* component of the objective function. The formula for
this contribution is in cell I3. In general, row 13 may hold
any nonlinear, differentiable functions of the variable values.
It is important that the functions be well defined for all
values of the variables. For example the function (=1/I13)
would fail if I13 were allowed to take on the value 0.
Initially
all nonlinear terms are set to zero. Both the nonlinear expressions
in row 13 and the nonlinear coefficients in row 14 must be
nonzero for the nonlinear terms to affect the objective function.
A column (H) has been created to hold other nonlinear terms
that may appear in the objective function and constraints.
Although not used for this example, these cells can hold any
continuous, differentiable functions of the decision variables.
Where problems have several identifiable sources of nonlinear
terms, it may be convenient to have more than one column to
hold these terms. The nonlinear terms in the extra columns
are summed and are shown as the *Nonlinear 2* component of the
objective function in cell I4. The *Linear*, *Nonlinear
1,* and *Nonlinear
2* values are summed in
cell F4 to obtain the objective function. This quantity is
optimized by the Solver.
Nonlinear terms in the constraints can be entered as explicit
functions of the decision variable in the range H17:H20. The
nonlinear function values are accumulated with the linear constraint
functions in the Value column of the constraints. |