Biomedical Engineering Reference
In-Depth Information
The Pearson index is a profitability index of a project. It is defined as:
(
)
∏∑∏
=
n
n
i
−
1
[
]
R
p
−
c
p
E
NetValue
Cost
i
1
i
i
=
1
i
k
=
0
k
PearsonIndex
=
=
(
)
[
]
∑∏
=
E
n
i
−
1
c
p
i
1
i
k
=
0
k
(
)
(
)
where
R
is the final reward,
ci
=…
1,
,
n
is the cost in stage
i
, and
pi
=…
1,
,
n
i
i
is the conditional probability of success given success at the previous stages
p
0
.
It is the optimal decision rule according to Neyman-Pearson lemma (Neyman and
Pearson
1933
) and can be used to decide whether a project should be implemented
or not by ranking all potential projects.
The Gittins index is used in a sequential selection setting (also known as a mul-
tiarmed bandit problem) in which resources must be dynamically allocated among
several independent alternative projects, each divisible into stages. The Gittins
index solves the problem by associating each project with a priority index and pick-
ing the project with the largest current index using the following form:
=
1
{
}
å
[
]
n
-
1
t
E
a
R xt x
() | ()
1
=
x
t
=
1
i
i
i
i
=
()
=
Gittins Index
vx
sup
{
}
i
i
n
>
1
å
n
-
1
t
E
a | ()
x
1
=
x
t
=
1
i
i
where
n
>
1 is the number of stages, 0
<<
a
1
is a fixed discount factor,
xt
i
() is the
[
]
state of project
i
in stage
t
, and
Rxt
i
is the contemporaneous reward given the
state of project
i
in stage
t
. Therefore the numerator represents the expected dis-
counted reward for project
i
up to
n
stages; the denominator represents the expected
discounted time up to
n
stages. Hence, the Gittins index is “the maximum expected
discounted reward per unit of expected discounted time” (Talias
2007
). It is an
example of a Dynamic Allocation Index that is updated at each decision node to
reprioritize projects.
Talias (
2007
) suggests that the Pearson index is appropriate in a static context
where selected projects will be implemented, and the rest will never be considered
again. However, in a dynamic scenario, the Gittins index is more appropriate as it
maximizes the expected reward accumulated sequentially.
Ad hoc linear and nonlinear programs can also be formulated using some of the
above approaches as starting points, while adding constraints specific to a particular
firm (Dickinson et al.
2001
). These can bring more realism to the problem definition
beyond a mathematical definition of optimality. Loch and Kavadias (
2002
) develop
a dynamic programming model of portfolio choice in which marginal analysis is
used to demonstrate the structure of optimal policies. The unit of analysis is not a
single project but resource allocation of a limited budget across strategic programs.
They provide a closed form characterization of the optimal policy in the presence of
a number of project and market characteristics and provide a theoretical basis to
validate managerial “rules-of-thumb” on how the optimal allocation policy would
change with these characteristics.
i
()
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