File Name: branch and bound method example integer programming .zip
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- Steps of branch and bound method
- Integer programming branch and bound calculator
- Integer programming
- I. Introduction
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers.
Steps of branch and bound method
Documentation Help Center Documentation. Linear objective function, f T x , where f is a column vector of constants, and x is the column vector of unknowns. Bounds and linear constraints, but no nonlinear constraints for definitions, see Write Constraints. Restrictions on some components of x to have integer values. In mathematical terms, given vectors f , lb , and ub , matrices A and Aeq , corresponding vectors b and beq , and a set of indices intcon , find a vector x to solve.
If it solves the problem in a stage, intlinprog does not execute the later stages. Reduce the problem size using Linear Program Preprocessing. Solve an initial relaxed noninteger problem using Linear Programming.
Try Cut Generation to further tighten the LP relaxation of the mixed-integer problem. Try to find integer-feasible solutions using heuristics. Use a Branch and Bound algorithm to search systematically for the optimal solution. This algorithm solves LP relaxations with restricted ranges of possible values of the integer variables.
It attempts to generate a sequence of updated bounds on the optimal objective function value. According to the Mixed-Integer Linear Programming Definition , there are matrices A and Aeq and corresponding vectors b and beq that encode a set of linear inequalities and linear equalities. Usually, it is possible to reduce the number of variables in the problem the number of components of x , and reduce the number of linear constraints.
While performing these reductions can take time for the solver, they usually lower the overall time to solution, and can make larger problems solvable. The algorithms can make solution more numerically stable. Furthermore, these algorithms can sometimes detect an infeasible problem. Preprocessing steps aim to eliminate redundant variables and constraints, improve the scaling of the model and sparsity of the constraint matrix, strengthen the bounds on variables, and detect the primal and dual infeasibility of the model.
The initial relaxed problem is the linear programming problem with the same objective and constraints as Mixed-Integer Linear Programming Definition , but no integer constraints. Call x LP the solution to the relaxed problem, and x the solution to the original problem with integer constraints. This initial relaxed LP root node LP and all generated LP relaxations during the branch-and-bound algorithm are solved using linear programming solution techniques.
The IntegerPreprocess option lets you choose whether intlinprog takes several steps, takes all of them, or takes almost none of them. If you include an x0 argument, intlinprog uses that value in preprocessing. The main goal of mixed-integer program preprocessing is to simplify ensuing branch-and-bound calculations. Preprocessing involves quickly preexamining and eliminating some of the futile subproblem candidates that branch-and-bound would otherwise analyze.
For details about integer preprocessing, see Savelsbergh . Cuts are additional linear inequality constraints that intlinprog adds to the problem. These inequalities attempt to restrict the feasible region of the LP relaxations so that their solutions are closer to integers.
You control the type of cuts that intlinprog uses with the CutGeneration option. Furthermore, if the problem is purely integer all variables are integer-valued , then intlinprog also uses the following cuts:. For purely integer problems, 'intermediate' uses the most cut types, because it uses reduce-and-split cuts, while 'advanced' does not.
Another option, CutMaxIterations , specifies an upper bound on the number of times intlinprog iterates to generate cuts.
To get an upper bound on the objective function, the branch-and-bound procedure must find feasible points. A solution to an LP relaxation during branch-and-bound can be integer feasible, which can provide an improved upper bound to the original MILP.
Certain techniques find feasible points faster before or during branch-and-bound. These techniques are heuristic, meaning they are algorithms that can succeed but can also fail. Heuristics can be start heuristics, which help the solver find an initial or new integer-feasible solution. Or heuristics can be improvement heuristics, which start at an integer-feasible point and attempt to find a better integer-feasible point, meaning one with lower objective function value.
The intlinprog improvement heuristics are 'rins' , 'rss' , 1-opt, 2-opt, and guided diving. Set the intlinprog heuristics using the 'Heuristics' option. The options are:. The solver runs rounding heuristics twice with different parameters, runs diving heuristics twice with different parameters, then runs 'rss'. The solver does not run later heuristics when earlier heuristics lead to a sufficiently good integer-feasible solution.
The solver runs rounding heuristics twice with different parameters, then runs diving heuristics twice with different parameters. If there is an integer-feasible solution, the solver then runs 'rins' followed by 'rss'.
If 'rss' finds a new solution, the solver runs 'rins' again. See Danna, Rothberg, and Le Pape . When you select 'rins' , the solver runs rounding heuristics twice with different parameters, runs diving heuristics twice with different parameters, then runs 'rins'.
When you select 'rss' , the solver runs rounding heuristics twice with different parameters, runs diving heuristics twice with different parameters, then runs 'rss'. These settings perform the same heuristics as 'basic'. When you select 'round' , the solver, at the root node, runs rounding heuristics twice with different parameters, then runs diving heuristics twice with different parameters. Thereafter, the solver runs only rounding heuristics at some branch-and-bound nodes.
The solver works in a similar way to 'round' , but also runs diving heuristics in addition to rounding heuristics at some branch-and-bound nodes, not just the root node. Guided diving applies when the solver already found at least one integer-feasible point. Diving heuristics generally select one variable that should be integer-valued, for which the current solution is fractional.
The heuristics then introduce a bound that forces the variable to be integer-valued, and solve the associated relaxed LP again.
The method of choosing the variable to bound is the main difference between the diving heuristics. See Berthold  , Section 3.
The solver simply takes any feasible point it encounters in its branch-and-bound search. The main difference between 'intermediate' and 'advanced' is that 'advanced' runs heuristics more frequently during branch-and-bound iterations.
In addition to the previous table, the following heuristics run when the Heuristics option is 'basic' , 'intermediate' , or 'advanced'.
ZI round — This heuristic runs whenever an algorithm solves a relaxed LP. The heuristic goes through each fractional integer variable to attempt to shift it to a neighboring integer without affecting the feasibility with respect to other constraints. For details, see Hendel . The heuristic goes through each integer variable to attempt to shift it to a neighboring integer without affecting the feasibility with respect to other constraints, while lowering the objective function value.
For each pair, 2-opt takes an integer-feasible solution and moves the values of the variable pairs up or down using all four possible moves up-up, up-down, down-up, and down-down , looking for a feasible neighboring solution that has a better objective function value.
The algorithm tests each integer variable pair by calculating the largest size same magnitude of shifts for each variable in the pair that satisfies the constraints and also improves the objective function value.
At the beginning of the heuristics phase, intlinprog runs the trivial heuristic unless Heuristics is 'none' or you provide an initial integer-feasible point in the x0 argument. The trivial heuristic checks the following points for feasibility:. The "lock" point is defined only for problems with finite upper and lower bounds for all variables.
The "lock" point for each variable is its upper or lower bound, chosen as follows. For each variable j , count the number of corresponding positive entries in the linear constraint matrix A :,j and subtract the number corresponding negative entries. If the result is positive, use the lower bound for that variable, lb j. Otherwise, use the upper bound for that variable, ub j. The "lock" point attempts to satisfy the largest number of linear inequality constraints for each variable, but is not necessarily feasible.
After each heuristic completes with a feasible solution, intlinprog calls output functions and plot functions. If you include an x0 argument, intlinprog uses that value in the 'rins' and guided diving heuristics until it finds a better integer-feasible point.
So when you provide x0 , you can obtain good results by setting the 'Heuristics' option to 'rins-diving' or another setting that uses 'rins'. The branch-and-bound method constructs a sequence of subproblems that attempt to converge to a solution of the MILP. The subproblems give a sequence of upper and lower bounds on the solution f T x. The first upper bound is any feasible solution, and the first lower bound is the solution to the relaxed problem.
For a discussion of the upper bound, see Heuristics for Finding Feasible Solutions. As explained in Linear Programming , any solution to the linear programming relaxed problem has a lower objective function value than the solution to the MILP. Also, any feasible point x feas satisfies. In this context, a node is an LP with the same objective function, bounds, and linear constraints as the original problem, but without integer constraints, and with particular changes to the linear constraints or bounds.
The root node is the original problem with no integer constraints and no changes to the linear constraints or bounds, meaning the root node is the initial relaxed LP. From the starting bounds, the branch-and-bound method constructs new subproblems by branching from the root node. The branching step is taken heuristically, according to one of several rules. These two subproblems arise when an entry in x LP , corresponding to an integer specified in intcon, is not an integer.
Here, x LP is the solution to a relaxed problem. The resulting two problems have solutions that are larger than or equal to f T x LP , because they have more restrictions. Therefore, this procedure potentially raises the lower bound. The performance of the branch-and-bound method depends on the rule for choosing which variable to split the branching rule.
The algorithm uses these rules, which you can set in the BranchRule option:. The pseudocost of a variable i is based on empirical estimates of the change in the lower bound when i has been chosen as the branching variable, combined with the fractional part of the i component of the current point x.
The fractional part p is in two pieces, the lower part and the upper part:. The objective gain per unit change in variable x i is. The empirical values are initialized to the absolute value of the objective coefficient f i for the terms before there are any observations.
Integer programming branch and bound calculator
Documentation Help Center Documentation. Linear objective function, f T x , where f is a column vector of constants, and x is the column vector of unknowns. Bounds and linear constraints, but no nonlinear constraints for definitions, see Write Constraints. Restrictions on some components of x to have integer values. In mathematical terms, given vectors f , lb , and ub , matrices A and Aeq , corresponding vectors b and beq , and a set of indices intcon , find a vector x to solve. If it solves the problem in a stage, intlinprog does not execute the later stages.
The branch-and-bound algorithm, first proposed by Land and Doig , is an effective approach to solving mixed integer linear programs. The branch-and-bound algorithm solves a mixed integer linear program by dividing the search space and generating a sequence of subproblems. The search space of a mixed integer linear program can be represented by a tree. Each node in the tree is identified with a subproblem derived from previous subproblems on the path that leads to the root of the tree. The linear programming relaxation of can be written as. The branch-and-bound algorithm generates subproblems along the nodes of the tree by using the following scheme. Consider , the optimal solution to , which is usually obtained by using the dual simplex algorithm.
But Amit, this branch and bound refers. Did you know that beavers like to use branches to bound water behind dams? Yes, we sure do. Some people say that we beavers are nature's engineers. The additional integer structure is specified using a vector to identify the indices of integer variables. The user can influence the choice of branching variable by providing priorities for the integer variables.
The linear programming model follows. The branch and bound method is a solution approach that parti- tions the feasible solution space into smaller subsets of.
The use of integer variables greatly expands the scope of useful optimization problems that you can define and solve.
The developed dynamic model includes heterogeneity and traffic variability and train capacity constraints so that it produces traveling plans that are consistent with the train length limitations and sensitive to the service requests of customers. One greedy heuristic was introduced by Teitz and Bart and improved upon by Larson and Odoni , which uses the information from updated 1-median solutions to iteratively insert facilities to the solution set. Solving allocation and scheduling problems inherent in forest resource management using mixed-integer programming Parviz Ghandforoush, Brian … There are many other variants to facility location problems. Repeat until all nodes pruned.
This is called solving the linear programming relaxation of the integer program. This is not a complicated linear program, and we can find the.
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