Node-search problem

The decontamination problem, also known as node-search problem (Mosearini, Petreschi, Szwarcfiter, 1998), is the proeess of ehanging all nodes of G from contaminated to clean, usually trying to use the smallest possible amount of decontamination agents. This is usually accomplished by planning tiie movements of the decontamination agents. 

Tree representation of the conformation search problem for hexane. Unlike the tree in Figure 9.4 the path gth from the root node to any of the terminal nodes is constant. 

Search trees are widely used to represent the different states that a problem cem adopt, example is shown in Figure 9.4 from which it should be clear where the name deri especially if the page is turned upside down. A tree contains nodes that are connected edges. The presence of an edge indicates that the two nodes it connects ctre related in so way. Each node represents a state that the system may adopt. The root node represents initial state of the system. Terminal nodes have no child nodes. A goal node is a special k of terminal node that corresponds to Em acceptable solution to the problem. 

To complete the specification of the algorithm, we require one additional decision parameter how to select the next problem Yix), which we will solve, or equivalently, which node in the branching structure to expand. We will define a search function, s, which allows us to select a node from the currently unexpanded nodes for expansion. In this chapter, as in Ibaraki (1978), we consider only best bound search, where we select the node with the minimum gix) value for expansion. Thus our branch-and-bound algorithm. A, is explicitly specified by... 

In this case, the breadth first search yields the optimum with a fewer number of nodes to be searched. Different search strategies than the ones used here can readily be used10. It is likely that different problems would be suited to different search strategies. 

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 LP solutions in the nodes control the sequence in which the nodes are visited and provide conservative lower bounds (in case of minimization problems) with respect to the objective on the subsequent subproblems. If this lower bound is higher than the objective of the best feasible solution found so far, the subsequent nodes can be excluded from the search without excluding the optimal solution. Each feasible solution corresponds to a leaf node and provides a conservative upper bound on the optimal solution. This combination of branching and bounding or cutting steps leads to the implicit enumeration of all integer solutions without having to visit all leaf nodes. 

In general, branch-and-bound [5] is an enumerative search space exploration technique that successively constructs a decision tree. In each node, the feasible region is divided into two or more disjoint subsets which are then assigned to child nodes. During the search space exploration for minimization problems, a lower bound of the objective function is computed in each node and compared against the lowest upper bound found so far. If the lower bound is greater than the upper bound, the corresponding branch is said to be fathomed and not explored anymore. The exploration terminates when a certain gap between the upper and the lower bound is reached or when the all possible subsets have been enumerated. 

Heuristic search procedures can be applied to certain types of combinatorial problems when BB and OA are difficult to apply or converge too slowly. In these problems, it is difficult or impossible to model the problem in terms of a vector of decision variables, which must satisfy bounds on a set of constraint functions, as required by OA. One example is the traveling salesman problem, in which the feasible region is the set of all tours in a graph, that is, closed cycles or paths that visit every node only once. The problem is to find a tour of minimal distance or cost,... 

As explained in Chapter 9, a branch-and-bound enumeration is nothing more than a search organized so that certain portions of the possible solution set are deleted from consideration. A tree is formed of nodes and branches (arcs). Each branch in the tree represents an added or modified inequality constraint to the problem defined for the prior node. Each node of the tree itself represents a nonlinear optimization problem without integer variables. 

The next step is to backtrack up the tree and continue searching through other branches until all nodes in the tree have been fathomed. Another reason to fathom a particular node occurs when no feasible solution exists to the nonlinear problem at node r then all subsequent nodes below node r are also infeasible. 

Decision Trees provide the overall structure for problem resolution in the current system. The outcome of a test at a particular node in the tree is recorded and directs the next decision for branching. If a failure is encountered at all possible branches, the un-resolved problem is passed back up to the node at which there last existed a possible, untested, solution. Prolog lends itself nicely to this structure since its basic architecture includes decision-making via such a "depth-first" search strategy(2)... 

Let us write the model of nonstationary flow distribution as applied to the problem of search for the maximum pressure rise at a given node of the hydraulic circuit at a fast cut off of the flow in one of its branches (or the largest drop at pipe break) provided that there is an isothermal motion of viscous incompressible fluid subjected to the action of the pressure, friction, and inertia forces (Gorban et al., 2006). find... 

Despite the existence of powerful analytical tools that allow for explicit solution of certain problems of interest, in general, the modeler cannot count on the existence of analytic solutions to most questions. To remedy this problem, one must resort to numerical approaches, or further simplify the problem so as to refine it to the point that analytic progress is possible. In this section, we discuss one of the key numerical engines used in the continuum analysis of boundary value problems, namely, the finite element method. The finite element method replaces the search for unknown fields (i.e. the solutions to the governing equations) with the search for a discrete representation of those fields at a set of points known as nodes, with the values of the field quantities between the nodes determined via interpolation. From the standpoint of the principle of minimum potential energy introduced earlier, the finite element method effects the replacement... 

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