Now we are ready to formulate our solution strategy. Inspecting the objective
function, eq 5-1, we see that the costs have been grouped with respect to their source.
All of the costs that are dependent on our decision variables, the pipe diameters,
have been included in the first summation over j, the pipe segment index. The
summations in the third and fourth terms are those that arise from the pumping
energy expended at the consumer. The decision variables in the terms of these
summations are the ∆Pcv,i values.
We notice immediately from the form of the objective that it is a separable
function with regards to the pipe diameter for each of the system segments j. A
separable function is a function of more than one variable that may be written as a
combination of functions, one independent function for each variable. Thus, from
examining the objective function, it appears that we can consider each pipe diameter
function independent of the other pipe diameters and find its optimum. We will
proceed as if this is the case, although later we will see that the constraints will not
allow the diameters to be considered completely independent of one another in all
We begin by inspecting the objective function for monotonicity, since this will
help us simplify the solution as much as possible. Looking first at the terms in the
summation over the pipe segment index j, we look at each term in the summation
Chl, j = I1/ln(A10/d)
Cpv, j = A9d
(dj- ) .
Cpev, j = I3 d(5+b+c)
The monotonicities with respect to dj of each term are given and we see that we
have both increasing and decreasing terms, so we are unable to use monotonicity
analysis on these at the outset. This is consistent with our findings in Chapter 2,
where we first neglected the Chl term and then used geometric programming theory
to find a solution to the lower bounding problem thus formed. This result was used
as a starting point for a simple search to find the solution to the problem without
neglecting Chl. Since the objective function is separable for each of the dj values, we
will proceed with the same methodology and find the "optimal independent"
values for each dj in the same way.
The other remaining decisions variables in the objective are the ∆Pcv,i variables,
of which there is one for each consumer. The ∆Pcv,i variables appear in both of the
last two terms of the objective function once eq 5-3 has been substituted for Cpvc
) i [(
A2 1 + PVFm&r Am&r ∑ ∆Pcv,i + ∆Phe,i (mi /ρr )
(∆Pc+ ,i )
∑ ∫ ∆Pcv,i + ∆Phe,i (mi /ρr )dt
(∆Pc+ ,i ) .
ηpηpm i yr
Both of these terms are monotonically increasing in each ∆Pcv,i and thus the objective
is monotonically increasing in each ∆Pcv,i. The First Monotonicity Principle, MP1
(see Papalambros and Wilde 1988), therefore tells us that each ∆Pcv,i must be
bounded below by at least one active constraint. We will examine the issue of
determining the constraint activity for these decision variables.