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 ∆*P*cv,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

cases.

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

separately

(d )

+

j

(d )

+

j

(dj- ) .

The monotonicities with respect to *d*j 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 *C*hl 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 *C*hl. Since the objective function is separable for each of the *d*j values, we

will proceed with the same methodology and find the "optimal independent"

values for each *d*j in the same way.

The other remaining decisions variables in the objective are the ∆*P*cv,i variables,

of which there is one for each consumer. The ∆*P*cv,i variables appear in both of the

last two terms of the objective function once eq 5-3 has been substituted for *C*pvc

) i [(

]

(

)

(∆*P*c+ ,i )

˙

v

d

(

)

∑ ∫ ∆*P*cv,i + ∆*P*he,i (mi /ρr )d*t*

(∆*P*c+ ,i ) .

˙

v

ηpηpm * i yr*

Both of these terms are monotonically increasing in each ∆*P*cv,i and thus the objective

is monotonically increasing in each ∆*P*cv,i. The First Monotonicity Principle, MP1

(see Papalambros and Wilde 1988), therefore tells us that each ∆*P*cv,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.

44

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