Geoscience Reference
In-Depth Information
17.5.3
Balanced-Objective Models
Balanced-objective models seek to design a system that “balances” the costs
incurred by the two main “players” in the system: customers, who bear the travel
and congestion costs, and the decision-maker who bears the facility-related costs.
Balanced-objective models are listed in Table 17.3 and include the following
references: Aboolian et al. ( 2008 ), Abouee-Mehrizi et al. ( 2011 ), Castillo et al.
( 2009 ), Elhedhli ( 2006 ), Kim ( 2013 ), Marianov and Rios ( 2000 ), Pasandideh and
Chambaria ( 2010 ), Rabieyan and Seifbarghy ( 2010 ), Vidyarthi and Jayaswal ( 2013 ),
and Wang et al. ( 2004 ).
One may view balanced-objective models as seeking to achieve some kind of
“social optimum”; the objective functions in these models are similar to social
welfare functions in economics. Since the objective incorporates customer concerns,
the models are typically of NR type: customers accept the directed assignments
to optimize “social welfare”, even if this leads to assignments that are suboptimal
from individual customers' point of view (two references that incorporate customer
response are Aboolian et al. 2008 and Abouee-Mehrizi et al. 2011 ). The demand
is assumed to be inelastic. The coverage and service level constraints are typically
absent, as service adequacy is addressed by the objective. The objective function
typically includes the “customer-borne” cost terms ( 17.45 )-( 17.46 ) representing
travel and congestion costs, as well as the “operator-borne” facility costs ( 17.47 ).
Since most models do not assume any demand losses, the revenue term ( 17.44 )is
not included; the exception being Abouee-Mehrizi et al. ( 2011 ), who model revenue
losses due to balking and thus optimize the net profit. Other distinguishing features
of most models of this class are simple constraint sets and the inclusion of flexible
capacity at the facilities as the decision variables. The main solution difficulty stems
from the non-linearities inherent in the congestion term (third term of the objective
function). There are several approaches for either making these terms less complex
or linearizing them, leading to interesting exact algorithms. We describe two such
approaches below.
The first is based on Castillo et al. ( 2009 ). They assume an M=M=1 queuing
system at the facilities and use the average number of customers in the system
L i . i ; i / as the performance measure at facility i.ForM=M=1 queue, this can
be written as
i
i i :
L i . i ; i / D
(17.59)
All costs are assumed to be linear and uniform (i.e., identical for all facilities),
leading to the following objective function:
minimize Z D LJ X
j
X
d.i;j/ j x ij C WC X
i
L i . i ; i / C FC X
i
y i C VC X
i
i ;
2
J
i
2
I
2
I
2
I
2
I
(17.60)
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