MILP master problem

The formulation for this scenario entails 1411 constraints, 511 continuous and 120 binary variables. The reduction in continuous variables compared to scenario 1 is due to the absence of linearization variables, since no attempt was made to linearize the scenario 2 model as explained in Section 4.3. An average of 1100 nodes were explored in the branch and bound search tree during the three major iterations between the MILP master problem and the NLP subproblem. The problem was solved in 6.54 CPU seconds resulting in an optimal objective of 2052.31 kg, which corresponds to 13% reduction in freshwater requirement. The corresponding water recycle/reuse network is shown in Fig. 4.10. 

As shown in Table 4.4, the model for scenario 2, which is a nonconvex MINLP, consists of 1195 constraints, 352 continuous and 70 binary variables. An average of 151 nodes were explored in the branch and bound algorithm over the 3 major iterations between the MILP master problem and NLP subproblems. The problem was solved in 2.48 CPU seconds with an objective value of 1.67 million. Whilst the product quantity is the same as in scenario 1, i.e. 850 t, the water requirement is only 185 t, which corresponds to 52.56% reduction in freshwater requirement. The water network to achieve this target is shown in Fig. 4.15. 

The modeling system GAMS (Brooke et al., 1996) is used for setting up the optimization models. The computational tests were carried out on a Pentium M processor 2.13 GHz. The models were solved with DICOPT (Viswanathan and Gross-mann, 1990). The NLP subproblems were solved with CONOPT2 (Drud, 1994), while the MILP master problems were solved with CPLEX (CPLEX Optimization Inc, 1993). 

The OA method (Duran and Grossmann. 1986c Yuan et ai, 1988 Fletcher and Leyffer. 1994) arises when NLP subproblems (NLP2) and MILP master problems (M-MIP) with / = 7 are solved successively in a cycle of iterations to generate the points (.v. y ). The (NLP2) subproblems yield an upper bound that corresponds to the best current solution, UB = min (Z, ). The master problems (M-MIP) yield a non-decreasing sequence of lower bounds (Zl since linearizations are accumulated as seen in (M-MIP). The cycle of iterations is... 

Stopped when the lower and upper bounds are within a specified tolerance. Also, if infeasible NLP subproblems are found, the feasibility problem (NLPF) is solved to provide the point x (commonly NLPF—oo). The OA method generally requires relatively few cycles or major iterations. It trivially converges in one iteration ff x,y) and gix,y) are linear. It is also important to note that the MILP master problem need not be solved to optimality. In fact, given the upper bound UB and a tolerance e, it is sufficient to generate the new (y by solving... 

Viswanathan and Grossmann (1990) proposed introducing slacks in the MILP master problem to reduce the likelihood of cutting off feasible solutions. This master problem (augmented penalty/equality relaxation) (APER) has the following form ... 

Section VII.C). When multiple choices are possible, one can. in fact, develop valid linear outer approximations that properly bound the nonconvex solution space in the MILP master problem. As for the process unit nodes, the mass balances are expressed in terms of component flows rather than in terms of fractional compositions. Finally, the right-hand side in the linearizations of the process units are modified to ensure that nonzero flows are attained when the 0-1 variable is set to zero. 

An extension of the logic-based OA algorithm for solving problem (PH) has been implemented in LOGMIP, a computer code based on GAMS (Vecchietti and Grossmann, 1999). This algorithm decomposes problem (PH) into two subproblems, the NLP and the MILP master problems. With fixed discrete variables, the NLP subproblem is solved. 

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