Scenario subproblems

In this form, the scenario subproblems are tied together only by the non-anticipativity constraints (9.22). This naturally leads to a decomposition based on the relaxation of the non-anticipativity constraints. 

The main idea of stage decomposition (see Figure 9.10) is to remove the ties between the scenario subproblems of the 2S-MILP by fixing the first-stage variables. The 2S-MILP is written in its intensive form [9], where the resulting master problem is... 

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. 

Basically, there are two different ways to decompose a 2S-MILP (see Figure 9.10). The scenario decomposition separates the 2S-MILP by the constraints associated to a scenario, whereas the stage decomposition separates the variables into first-stage and second-stage decisions. For both approaches, the resulting subproblems are MILPs which can be solved by standard optimization software. 

In the Sections above, we have described the application scenario that is the viewpoint from which we are interested in bioinformatics. In this Section, we attempt to chart out the field of bioinformatics in terms of its scientific subproblems and the application challenges that it faces. 

The decomposition procedure is described in Algorithm 3.1. The decomposition subproblems complexity for this case study is presented in Table3.4. As shown, this kind of problems can be solved with reasonable computational cost by using the OCD strategy. The problems were solved in an Intel 2 Core Duo—2.0 GHz—2 GB RAM with a 3 % integrality gap. The risk curves for the 16 and 64 scenarios cases are depicted in Figs. 3.8 and 3.9, respectively. 

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