Page 55 - Sustainability in the process industry

32 Cha p te r T h r ee

be used to solve IP and MIP problems. The original large MIP problem
is divided into a number of subproblems, called nodes, which form an
enumeration tree. The algorithm starts from a main node and
progresses toward the so-called leaves, adding nodes to the current
solution or discarding them as necessary. In this process, an important
role is played by the bounding function, which (in the case of objective
minimization) provides a lower bound on the remaining part of the
problem under the current branch. Thus, if the lower bound on the
current subproblem node is higher than the current best solution,
then the algorithm can safely discard (prune) the node and all its
subnodes. More information on IP solving algorithms can be found
in Nemhauser and Wolsey (1999).

3.9 Stochastic Search Methods for Solving
Optimization Problems

An alternative to deterministic algorithms are those that employ
stochastic sampling of the search space and hill climbing: pushing the
algorithm to lower objective function values (assuming minimization)
by allowing temporary increases in the objective function value. The
main advantage of stochastic search methods is that their search is
not confined in the neighborhood of a given current solution, which
means there is a higher probability of finding the global optimum.
Such techniques have proven to be successful in applications—for
example, process design and synthesis studies—where the
computational requirement for single samplings is modest and the
time frame for producing a solution is more relaxed. Their major
limitation is that many iterations are required in order to assure some
degree of optimality. This is because the algorithms blindly evaluate
even infeasible combinations of variable values, especially for the
simulated annealing variants.
Simulated annealing (SA) is the most prominent stochastic search
method (Kirkpatrick, Gelatt, and Vecchi, 1983). An interesting
application is the GAPinch toolbox for MATLAB (Prakotpol and
Srinophakun, 2004), which implements a genetic algorithm (GA)
search to solve MINLP network synthesis problems involving water
reuse. Another promising development is the use of the GA
framework to optimize schedules and supply chains (Shopova and
Vaklieva-Bancheva, 2006). Ant colony optimization (Zecchin et al.,
2006) and Tabu search (Cunha and Ribeiro, 2004) have been applied to
process design and operation. Other examples of stochastic search
methods include the following:

1. McKay, Willis, and Barton (1997) used genetic programming
(GP) to identify steady-state models of integrated chemical
processes.