I am looking for an accessible formal methodology or formula to
standardize the prioritization and on-going re-prioritization of
complex tasks. The tasks are all quite time consumming and have
varying, and in some cases difficult to quantify costs associated with
missing their respective deadlines. Further, new tasks are constantly
being added and may require re-prioritization. However, there is also
a time cost associated with "bumping" or re-ordering higher tasks or
tasks already in process. In short, I am looking for the equivalent of
a Black-Scholes or Bayes formula or approach for prioritizing and
re-prioritizing tasks. Such a formula or approach must be more
sophisticated than merely "greatest 'consequence' for failure goes
first", and should help define a process for quantifying less tangible
"conequences" (e.g., placing a value public perception of failure),
and that can be used to reprioritize ir assign priorities to new
tasks. I only have basic undergraduate level math skills, so if the
approach or formula is complex, pointers to good introductory
materials are also necessary. Thank you.

Assuming you are working within budget constraints, the closest
formula I'm familiar with would be something like a capital allocation
model... where to invest capital against a variety of project
portfolio options with the (assumed) goal of the full complement of
chosen projects gaining the highest return amongst all relative
options.

Assume there are m potential projects. 

For now, assume that the projects are independent; that is,it is
reasonable to select any combination of projects and the cost/benefit
of any project do not depend on what other projects are selected.

Define, for each project i = 1, 2,?, m, the zer oone variable xi. The
variable xi is one if the project is accepted and zero if it is
rejected. Let bi be the benefit of the i?th project and ci be its
cost.

Let C be the total available budget. 

The goal is to select from the available projects the subset of
projects with a total cost less than or equal to C that produces the
greatest possible total benefit.

The problem may be expressed mathematically as:
         m
Maximize ?     b  x
              i  1
         i=1

Subject to the following:

 m
 ?   c  x     <   C   and x  = 0 or 1 for 1,2,...m,
i=1  i  i    _            i

Thank you for the comment. If I follow you correctly, I would need to
substitute task priority or timing for capital allocation.
Unfortuntely, new tasks are constantly being added which requires a
re-evaluation of the other previously prioritized tasks. The time
available for any task is finite, but there are an unlimited number of
new tasks being added.

I've come to think of the problem as a perverse "spinning plates"
scenario, where there are a variety of spinning plates of different
sizes, shapes and values. The size and shape of the plates determines
how much effort goes into keeping them spinning. New plates are always
being added and some plates which have successfully spun the longest
are removed. Of course the value of the plates varies. Therefore, you
have to decide whether it pays to spin a heavy plate (more effort)
with a high value and potentially sacrifice several lower value plates
which may not individually require as much effort. In essence, the
"value" of each plate varies depending on the spin state and value of
all the other plates, requiring constant re-evaluation.

I  would need to

You said" I would need to substitute task priority or timing for
capital allocation."

In capital allocation, it's capital that is the limited resource. What
is your limited resource? Time? People?

I haven't worked through it, but I don't see how you could load task
priority into the formula, since that's the *output*, correct?

You said" Unfortuntely, new tasks are constantly being added which requires a
re-evaluation of the other previously prioritized tasks. The time
available for any task is finite, but there are an unlimited number of
new tasks being added."

ANY methodology you would implement would require a constant
re-evaluation of the already prioritized tasks. Not to do so would
flaw your model, since the output is also to rank the various
priorities against one another so that difficult decisions could be
made. There is no shortcut here.  Depending on the size of your
project portfolio, this could be a full time job managing the
priorities and resources associated with them and adjusting to
accommodate the constant insertion of new opportunities.

Sounds like you have a fairly dysfunctional model to start with. What
kind of business are you in that you cannot manage the incoming
opportunities? Why can you not have a simple methodology or "sniff
test" that limits the opportunities that even get this level of
analysis? Or, if you have to do all of them and it's a matter of
prioritizing them against each other, the model that requires you to
do them all must have other metrics (such as contractural deadlines)
as well as estimated project development timelines, that would help
set the priorities.

In your original post you mentioned a variety of other metrics. What
might those be? Are they quantifiable? Can you associate a value with
those?  A scorecard method may work best for you in that case, but the
danger there is that you could wind up with a trade off between
achievement of objectives in some subjective way. You won't want to
accept less value in order to obtain a higher score on some internal
business process, for example, so balance is not the goal.  Of course,
your model would also need to accommodate hierarchical requirements
and exclude any mutually exclusive elements. There are usually
dependencies from a few projects to others as well, assuming that your
projects aren't all independent silos.

The goal here, fundamentally, is to maximize the value of the project
portfolio, not to manage priorities.  You seem to understand finance
based on your model references, so, it may be best to look at this
from a financial perspective and to resource/staff to accommodate the
portfolio mix, rather than altering the portfolio mix to match your
limited resources. Surely a company with so much incoming business can
staff accordingly. And, if you have a lot of incoming business and you
don't have the incoming cash to go with it, you've got yet another
fundamental problem.

Keep the string going.

Dear Dork ;-)
Just wondering how your prioritization is going. Any luck resolving
your prioritization problem?