Geoscience Reference
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economies of scale. With these definitions, the System Facility Cost is defined as
follows:
SFC
D
X
i2I
FC
i
y
i
C
X
i2I
VC
.
i
/
(17.24)
17.2.3.4
Revenues and Overall Objectives
We assume that each customer that is served brings in a revenue r to the system (for
public service applications, we can treat r as a “system benefit” parameter). The
total expected revenue can be expressed as
SR
D
r
X
i2I
i
D
r
X
j2J
j
X
i2I
x
ij
:
(17.25)
In principle, the parameter r can be treated as a decision variable—the price
charged by the decision-maker for service. However, in the vast majority of SLCIS
literature this term is treated as an exogenous parameter (Tong
2011
and Berman
et al.
2014
being the exceptions). Since treating prices as decision variables
introduces significant new complications, we will generally treat r as constant in
the model.
We also observe that when demand is inelastic (i.e.,
j
D
ma
j
for all
j
2
J) and when the constraints require that all customers must be served (i.e.,
P
i2I
x
ij
D
1;j
2
J), it is easy to see that
SR
D
r
P
j2J
ma
j
; which is a
constant. In this case, the revenue term in the objective can be dropped, leading to
a pure cost minimization case. Even in models where some customers may not be
served, but the demand is inelastic, it is common to use cost minimization with a
penalty term, which can be interpreted as opportunity cost for unserved customers.
To summarize, the overall objective for a general SLCIC model is given by
maximizeŒ
SR
STC
SWC
SFC
;
where the respective components are defined by (
17.25
), (
17.10
), (
17.16
),
and (
17.24
). We note that in most specific models described in the literature,
only a subset of the terms above is present, the rest being implicitly controlled by
constraints (e.g., in the presence of service level constraints, the
SWC
term is often
dropped).
Most of the terms above depend on demand allocations x
ij
and demand rates
j
,
which have not yet been described. This is the subject of the following section.
