Branch-and-bound search

Having formally defined the branching structure, we must now make explicit the mechanisms by which we can eliminate subsets of the solution space from further consideration. Ibaraki (1978) has stated three major mechanisms for controlling the evolution of the branch-and-bound search algorithms, by eliminating potential solution through... 

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. 

The corresponding mathematical formulation entails 5534 constraints, 1217 continuous and 280 binary variables. An average of 4000 nodes were explored in the branch and bound search tree. The solution required three major iterations and took 309.41 CPU seconds to obtain the optimal solution of 1285.50 kg. This corresponds to 45.53% reduction in freshwater demand. A water reuse/recycle network that corresponds to this solution is shown in Fig. 4.11. 

The overall model for this scenario involves 5614 constraints, 1132 continuous 280 binary variables. Three major iterations with an average of 1200 nodes in the branch and bound search tree were required in the solution. The objective value of 1560 kg, which corresponds to 33.89% reduction in freshwater requirement, was obtained in 60.24 CPU seconds. An equivalent of this scenario, without reusable water storage, i.e. scenario 2, resulted in 13% reduction in fresh water. Figure 4.12 shows the water recycle/reuse network corresponding to this solution. 

Since the program (DEP) represents a mixed-integer linear program (MILP), it can be solved by commercially available state-of-the-art MILP solvers like CPLEX [3] or XPRESS-MP [4], These solvers are based on implementations of modem branch-and-bound search algorithms with cuts and heuristics. 

Exact search Penny DNA/Protein Branch-and-bound search 10-11 taxa or less... 

Watson, 1968 Rudd, 1968 Masso and Rudd, 1969). Algorithmic methods for selecting the optimal configuration from a given superstructure also began to be developed through the use of direct search methods for continuous variables (Umeda et al, 1972 Ichikawa and Fan, 1973) as well as branch and bound search methods (Lee et al, 1970). 

This requires that continuous variables be selected independently at each node of the tree, often leading to suboptimal solutions even if the branch-and-bound search is performed rigorously. Also, although fewer nodes may be examined in the tree with the use of heuristics, this increases the likelihood of obtaining suboptimal solutions. 

Exact search DnaPenny DNA Branch-and-bound search... 

Gokce, A., Hsiao, K.-T. and Advani, S. G., 2002. Branch and bound search to optimise injection gate locations in liquid composite molding processes . Composites Part A Applied Science and Manifacturing, 33,1263-1272. 

A set of data transfers will be grouped into a class if they take place simultaneously. The classes are sorted according to their sizes. The data transfer binding algorithm is also a branch-and-bound search with a cost function for predicting buses utilization. 

Many solutions found by branch-and-bound search, however, may have the same smallest hardware cost. In this case, we will pick the one with the greatest potential to be improved in the future. 

The data path binding problem is divided into two phases data path construction and data path refinement. A branch-and-bound search algorithm is used to construct the initial data path based on a set of observations. During the data path refinement phase, we rip up a mixture of variables, data transfers and operations and relocate them. The refinement is augmented with a randomized selection process to prevent itself from being trapped in a local optimal. 

Apply exact ordering algorithm. A branch-and-bound ord g algorithm is applied if the heuristic fails to find a solution. The exact algorithm is guaranteed to find a solution if one exists. Theorem 7.2.5 is used as the cost function to prune the branch-and-bound search. 

---