Page 175 - Sustainability in the process industry

152 Cha p te r S e v e n

programming (NLP) problem if any of the constraints or the objective
function is nonlinear with continuous variables. Models that include
both continuous and integer variables are classified as mixed integer
programming ones; these include mixed-integer linear programming
(MILP) and mixed-integer nonlinear programming (MINLP).
Also, linear optimization (LO) problems are usually referred to as
linear programming or LP. Similarly, NLO, MILO, and MINLO cor-
respond to NLP, MILP, and MINLP.
Linear programming problems appear in a wide range of
applications, including transportation, distribution from sources to
sinks, and management decisions (Klemeš and Vašek, 1973; Klemeš
et al., 1975; Klemeš, 1986; Jeżowski, 1990; Williams, 1999; Jeżowski,
Shethna, and Castillo, 2003; El-Halwagi, 2006). LPR problems are easily
solved by the simplex method (Dantzig, 1968) and its improvements
(see, e.g., Maros, 2003a; Maros, 2003b). In most cases, NLP is difficult
to solve, and certain limitations on the constraints and objective
function may be necessary for such problems to be practically solvable
by specific methods (Seidler, Badach, and Molisz, 1980; Banerjee and
Ierapetritou, 2003; Sieniutycz and Jeżowski, 2009). A general technique
for solving NLP and mixed-integer programming problems is applied
by the branch-and-bound framework (Land and Doig, 1960), where the
original complex problem is solved via systematic generation and
solution of a set of simpler subproblems.
Process synthesis is a creative activity. In fact, it is one of the
earliest actions taken by the process designer when creating the
structure, network, or flowsheet of a process to satisfy the given
requirements in terms of constraints and specifications while attaining
the prescribed objectives.
The relationships among the mathematical model, the process
being modeled, and the solver being deployed are usually complicated,
which makes it difficult to establish the most effective and valid
model. There is only limited discussion of generating mathematical
models in the literature, and the topic is treated in only a few
publications (see, e.g., Grossmann, 1990; Kovacs et al., 2000)
concerning specific areas.
In general, a process synthesis problem is defined by specifying
the available raw materials, candidate operating units, and desired
products. Each of these is given by an individual mathematical
model. The models cannot, by themselves, directly constitute the
Mathematical Programming model for the synthesis problem.
Construction of the mathematical model from these model elements
is not evident with the risk of failure. The major steps of process
synthesis are illustrated in Figure 7.1.
The main emphasis in this chapter is on an integrated framework
for model generation and solution—that is, the P-graph framework.
Another class of methods for process synthesis is based on
heuristic rules. Implementing heuristic methods is relatively