Stochastic Programming - Index of

NETWORK PROBLEMS 307
deterministic durations, an upper bound if they do not, and a lower bound if
we first replace the random variables on the incoming and outgoing arcs by
their means and then apply arc duplication. If there is only one arc in or one
arc out, we take the expectation for that arc, and then apply arc duplication,
observing an overall lower bound.
6.7 Bibliographical Notes
The vocabulary in this chapter is mostly taken from Rockafellar [25], which
also contains an extremely good overview of deterministic network problems.
A detailed look at network recourse problem is found in Wallace [28].
The original feasibility results for networks were developed by Gale [10]
and Hoffman [13]. The stronger versions using connectedness were developed
by Wallace and Wets. The uncapacitated case is given in [31], while the
capacitated case is outlined in [33] (with a proof in [32]). More details of the
algorithms in Figures 6 and 7 can also be found in these papers. Similar results
were developed by Prékopa and Boros [23]. See also Kall and Prékopa [14].
As for the LP case, model formulations and infeasibility tests have of course
been performed in many contexts apart from ours. In addition to the references
given in Chapter 5, we refer to Greenberg [11, 12] and Chinneck [3].
The piecewise linear upper bound is taken from Wallace [30]. At the very
end of our discussion of the piecewise linear upper bound, we pointed out that
the solution y ∗ to (5.5) could consist of several cycles sharing arcs. A detailed
discussion of how to pick y ∗ apart, to obtain a conformal realization can be
found in Rockafellar [25], page 476. How to use it in the bound is detailed
in [30]. The bound has been strengthened for pure arc capacity uncertainty
by Frantzeskakis and Powell [8].
Special algorithms for stochastic network problems have also been
developed; see e.g. Qi [24] and Sun et al. [27].
We pointed out at the beginning of this chapter that scenario aggregation
(Section 2.6) could be particularly well suited to problems that have network
structure in all periods. This has been utilized by Mulvey and Vladimirou
for financial problems, which can be formulated in a setting of generalized
networks. For details see [19, 20]. For a selection of papers on financial
problems (not all utilizing network structures), consult Zenios [36, 37], and,
for a specific application, see Dempster and Ireland [5].
The above methods are well suited for parallel processing. This has been
done in Mulvey and Vladimirou [18] and Nielsen and Zenios [21].
Another use of network structure to achieve efficient methods is described
in Powell [22] for the vehicle routing problem.
The PERT formulation was introduced by Malcolm et al. [17]. An overview
of project scheduling methods can be found in Elmaghraby [7]. A selection of