assignment problem using branch and bound python code

Branch and Bound Search with Examples and Implementation in Python

We’ll try to understand one of the heuristic search techniques in this article. The heuristic technique is a criterion for determining which among several alternatives will be the most effective in achieving a particular goal. Branch and bound search is also known as Uniform Cost Search.

What is the branch and bound search algorithm?

Branch and bound is a search algorithm used for combinatory, discrete, and general mathematical optimization problems. It is comparable to backtracking in that it similarly implements a state-space stream to represent the solution to the problem.

However, it is probably more suited to trying to address optimization problems and only minimization problems, not maximization problems. Statistically speaking, a branch and the bound algorithm find the best solution from the entire search space of possibilities for an NP-Hard problem.

How does the branch and bound search work?

In the branch and bound search strategy, a cost function (denoted by g(X)) is generated that, by using a sequence of operators, assigns a cumulative cost to the path from the start node to the current node X. A cheapest price path already discovered is extended at every step of the search space generation process until we reach the goal state.

Branch and bound search is also referred to as a uniform cost search since it expands the least-cost partial path. The actual distance traveled from the beginning to the current node X, for instance, may be represented as g(X) in the traveling salesman problem.

Steps for the algorithm

If g(X) = 1 for all operators, the branch and bound methodology degenerates into a straightforward breadth-first search. Artificial intelligence considers it to be just as detrimental as depth-first and breadth-first. If we add dynamic programming to it, we can make this better by eliminating redundant paths.

We note that the method typically necessitates creating a solution and evaluating its efficacy. Any technique can be used to develop the answer, and heuristics may be used in testing. The following is the basic structure of an algorithm for developing and testing strategies.

Brand and bound search algorithm in action

To understand the concept more clearly, let’s try to implement the 8 puzzle problem using the branch and bound algorithm. The problem description is given below.

A 3 x 3 board with 8 tiles (each tile has a number ranging from 1 to 8) and a single empty space is provided. The goal is to use the vacant space to arrange the numbers on the tiles so that they match the final arrangement. Four neighboring (left, right, above, and below) tiles can be slid into the available area.

For Example

To avoid searching in sub-trees that do not include an answer node, the search for an answer node can frequently be sped up using an approximation of the cost function. However, instead of using the backtracking method, it does a BFS-style search.

Basically, Branch and Bound involve three different kinds of nodes.

• A live node is a generated node whose children have not yet been produced.
• The children of the E-node, a live node, are now being examined. Or to put it another way, an E-node is a node that is currently expanding.
• A created node that is not to be developed or examined further is referred to as a dead node. A dead node has already extended all of its children.

Cost function: In the search tree, each node X has a corresponding cost. The next E-node can be found using the cost function. The E-node with the lowest cost is the next one. The definition of the cost function is

Implementing the Branch and Bound Search algorithm in Python

In this article, we have learned one of the most effective algorithms knowns as a branch and bound search. This search algorithm helps to solve many common problems like the N-Queen problem, 0-1 Knapsack Problem, Traveling salesman problem, etc. The algorithm is bit modified in each case according to the conditions provided in the problem but the basic idea of the searching method remains the same as explained earlier.

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Solving Assignment Problem using Linear Programming in Python

Learn how to use Python PuLP to solve Assignment problems using Linear Programming.

In earlier articles, we have seen various applications of Linear programming such as transportation, transshipment problem, Cargo Loading problem, and shift-scheduling problem. Now In this tutorial, we will focus on another model that comes under the class of linear programming model known as the Assignment problem. Its objective function is similar to transportation problems. Here we minimize the objective function time or cost of manufacturing the products by allocating one job to one machine.

If we want to solve the maximization problem assignment problem then we subtract all the elements of the matrix from the highest element in the matrix or multiply the entire matrix by –1 and continue with the procedure. For solving the assignment problem, we use the Assignment technique or Hungarian method, or Flood’s technique.

The transportation problem is a special case of the linear programming model and the assignment problem is a special case of transportation problem, therefore it is also a special case of the linear programming problem.

In this tutorial, we are going to cover the following topics:

Assignment Problem

A problem that requires pairing two sets of items given a set of paired costs or profit in such a way that the total cost of the pairings is minimized or maximized. The assignment problem is a special case of linear programming.

For example, an operation manager needs to assign four jobs to four machines. The project manager needs to assign four projects to four staff members. Similarly, the marketing manager needs to assign the 4 salespersons to 4 territories. The manager’s goal is to minimize the total time or cost.

Problem Formulation

A manager has prepared a table that shows the cost of performing each of four jobs by each of four employees. The manager has stated his goal is to develop a set of job assignments that will minimize the total cost of getting all 4 jobs.  

Initialize LP Model

In this step, we will import all the classes and functions of pulp module and create a Minimization LP problem using LpProblem class.

Define Decision Variable

In this step, we will define the decision variables. In our problem, we have two variable lists: workers and jobs. Let’s create them using  LpVariable.dicts()  class.  LpVariable.dicts()  used with Python’s list comprehension.  LpVariable.dicts()  will take the following four values:

• First, prefix name of what this variable represents.
• Second is the list of all the variables.
• Third is the lower bound on this variable.
• Fourth variable is the upper bound.
• Fourth is essentially the type of data (discrete or continuous). The options for the fourth parameter are  LpContinuous  or  LpInteger .

Let’s first create a list route for the route between warehouse and project site and create the decision variables using LpVariable.dicts() the method.

Define Objective Function

In this step, we will define the minimum objective function by adding it to the LpProblem  object. lpSum(vector)is used here to define multiple linear expressions. It also used list comprehension to add multiple variables.

Define the Constraints

Here, we are adding two types of constraints: Each job can be assigned to only one employee constraint and Each employee can be assigned to only one job. We have added the 2 constraints defined in the problem by adding them to the LpProblem  object.

Solve Model

In this step, we will solve the LP problem by calling solve() method. We can print the final value by using the following for loop.

From the above results, we can infer that Worker-1 will be assigned to Job-1, Worker-2 will be assigned to job-3, Worker-3 will be assigned to Job-2, and Worker-4 will assign with job-4.

In this article, we have learned about Assignment problems, Problem Formulation, and implementation using the python PuLp library. We have solved the Assignment problem using a Linear programming problem in Python. Of course, this is just a simple case study, we can add more constraints to it and make it more complicated. You can also run other case studies on Cargo Loading problems , Staff scheduling problems . In upcoming articles, we will write more on different optimization problems such as transshipment problem, balanced diet problem. You can revise the basics of mathematical concepts in  this article  and learn about Linear Programming  in this article .

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A Basic Branch and Bound Solver in Python using Cvxpy

Jun 13, 2019 • philzook58

Branch and bound is a useful problem solving technique. The idea is, if you have a minimization problem you want to solve, maybe there is a way to relax the constraints to an easier problem. If so, the solution of the easier problem is a lower bound on the possible solution of the hard problem. If the solution of the easier problem just so happens to also obey the more constrained hard problem, then it must also be the solution to the hard problem. You can also use the lower bound coming from a relaxed problem to prune your search tree for the hard problem. If even the relaxed problem doesn’t beat the current best found, don’t bother going down that branch.

A standard place this paradigm occurs is in mixed integer programming. The relaxation of a binary constraint (either 0 or 1) can be all the values in between (any number between 0 and 1). If this relaxed problem can be expressed in a form amenable to a solver like a linear programming solver, you can use that to power the branch and bound search, also using returned solutions for possible heuristics.

I built a basic version of this that uses cvxpy as the relaxed problem solver. Cvxpy already has much much faster mixed integer solvers baked in (which is useful to make sure mine is returning correct results), but it was an interesting exercise. The real reason I’m toying around is I kind of want the ability to add custom branching heuristics or inspect and maintain the branch and bound search tree, which you’d need to get into the more complicated guts of the solvers bound to cvxpy to get at. Julia might be a better choice.

A somewhat similar (and better) project is https://github.com/oxfordcontrol/miosqp which doesn’t use cvxpy explicitly, but does have the branch and bound control in the python layer of the solver. There are also other projects that can use fairly arbitrary solvers like Bonmin

As a toy problem I’m using a knapsack problem where we have objects of different sizes and different values. We want to maximize the value while keeping the total size under the capacity of the bag. This can be phrased linearly like so: $ \max v \cdot x$ s.t. $ \sum_i s_i x_i<= capacity $, $ x \in {0,1}$. The basic heuristic I’m using is to branch on variables that are either 0 or 1 in even the relaxed solution. The alternative branch hopefully gets pruned fast.

This is at least solving the problem fairly quickly. It needs better heuristics and to be sped up, which is possible in lots of ways. I was not trying to avoid all performance optimizations. It takes maybe 5 seconds, whereas the cvxpy solver is almost instantaneous.

Edit : I should investigate the Parameter functionality of cvxpy. That would make a make faster version than the one above based on deepcopy. If you made the upper and lower vectors on the binary variables parameters, you could restrict the interval to 0/1 without rebuilding the problem or copying everything.

Solving Generalized Assignment Problem using Branch-And-Price

Oct 27, 2021

Table of Contents

Generalized assignment problem, dantzig-wolfe decomposition, column generation, standalone model, initial solution, branching rule, branch-and-price.

One of the best known and widely researched problems in combinatorial optimization is Knapsack Problem:

• given a set of items, each with its weight and value,
• select a set of items maximizing total value
• that can fit into a knapsack while respecting its weight limit.

The most common variant of a problem, called 0-1 Knapsack Problem can be formulated as follows:

• \(m\) - number of items;
• \(x_i\) - binary variable indicating whether item is selected;
• \(v_i\) - value of each items;
• \(w_i\) - weight of each items;
• \(W\) - maximum weight capacity.

As often happens in mathematics, or science in general, an obvious question to ask is how the problem can be generalized. One of generalization is Generalized Assignment Problem. It answers question - how to find a maximum profit assignment of \(m\) tasks to \(n\) machines such that each task (\(i=0, \ldots, m\)) is assigned to exactly one machine (\(j=1, \ldots, n\)), and one machine can have multiple tasks assigned to subject to its capacity limitation. Its standard formulation is presented below:

• \(n\) - number of machines;
• \(m\) - number of tasks;
• \(x_{ij}\) - binary variable indicating whether task \(i\) is assigned to machine \(j\);
• \(v_{ij}\) - value/profit of assigning task \(i\) to machine \(j\);
• \(w_{ij}\) - weight of assigning task \(i\) to machine \(j\);
• \(c_j\) - capacity of machine \(j\).

Branch-and-price

Branch-and-price is generalization of branch-and-bound method to solve integer programs (IPs),mixed integer programs (MIPs) or binary problems. Both branch-and-price, branch-and-bound, and also branch-and-cut, solve LP relaxation of a given IP. The goal of branch-and-price is to tighten LP relaxation by generating a subset of profitable columns associated with variables to join the current basis.

Branch-and-price builds at the top of branch-and-bound framework. It applies column generation priori to branching. Assuming maximization problem, branching occurs when:

• Column Generation is finished (i.e. no profitable columns can be found).
• Objective value of the current solution is greater than best lower bound.
• The current solution does not satisfy integrality constraints.

However, if only first two conditions are met but not the third one, meaning the current solution satisfies integrality constraints, then the best solution and lower bound are updated (lower bound is tightened) with respectively the current solution and its objective value.

The crucial element needed to apply branch-and-price successfully is to find branching scheme. It is tailored to specific problem to make sure that it does not destroy problem structure and can be used in pricing subproblem to effectively generate columns that enter Restricted Master Problem (RMP) while respecting branching rules .

Below is flow diagram describing branch-and-price method:

The successful application B&P depends on tight/strong model formulation. Model formulation is considered tight if solution of its LP relaxation satisfies (frequently) integrality constraints. One of structured approaches to come up with such a formulation is to use Dantzig-Wolfe Decomposition technique. We will see example of it applied to Generalized Assignment Problem (GAP).

A standard formulation was described above. Now, let’s try to reformulate problem. Let

be a set containing all feasible solutions to Knapsack problem for \(j\)-th machine. Clearly, \(S_j\) contains finite number of points, so \(S_j = \{ \mathbf{z}_j^1, \ldots, \mathbf{z}_j^{K_j} \}\), where \(\mathbf{z}_j^k \in \{0, 1\}^{m}\). You can think about \(\mathbf{z}_j^k \in \{0, 1\}^{m}\) as 0-1 encoding of tasks that form \(k\)-th feasible solution for machine \(j\). Now, let \(S = \{ \mathbf{z}_1^1, \ldots, \mathbf{z}_1^{K_1}, \ldots, \mathbf{z}_n^1, \ldots, \mathbf{z}_n^{K_n} \}\) be a set of all feasible solution to GAP. It, potentially, contains a very large number of elements. Then, every point \(x_{ij}\) can be expressed by the following convex combination:

where \(z_{ij}^k \in \{0, 1\}\), and \(z_{ij}^k = 1\) iff task \(i\) is assigned to machine \(j\) in \(k\)-th feasible solution for the machine.

Now, let’s use this representation to reformulate GAP:

Note that we do not need capacity restrictions as they are embedded into definition of feasible solution for machine \(j\).

Now that we have formulation that is suitable for column generation, let’s turn our attention to it.

Column generation is another crucial component of branch-and-price. There are many great resources devoted to column generation so I will mention only core points:

• Column generation is useful when a problem’s pool of feasible solutions contains many elements but only small subset will be present in the optimal solution.
• There exists subproblem (called often pricing problem) that can be used to effectively generate columns that should enter RMP.
• Column generation starts with initial feasible solution.
• Pricing subproblem objective function is updated with dual of the current solution.
• Columns with positive reduced cost, in case of maximization problem, enter problem.
• Procedure continues until such columns exist.

Below is flow diagram describing column generation method:

Implementation

Let’s see how one can approach implementation of B&P to solve Generalized Assignment Problem. Below is discussion about main concepts and few code excerpts, a repository containing all code can be found on github .

An example of problem instance taken from [1] is:

It is always good idea to have a reference simple(r) implementation that can be used to validate our results using more sophisticated methods. In our case it is based on standard problem formulation. Implementation can be found in repo by checking classes GAPStandaloneModelBuilder and GAPStandaloneModel . Formulation for a problem instance presented above looks as follows:

Now let’s try to examine building blocks of B&P to discus main part at the end, once all the puzzles are there.

To start column generation process, we need to have an initial solution. One possible way to derive it is to use two-phase Simplex method. In first step, you add slack variables to each constraint and set objective function as their sum. Then you minimize the problem. If your solution has objective value \(0\), then first of all you have initial solution and you know that your problem is feasible. In case you end up with positive value for any of slack variables, you can conclude that the problem is infeasible. You can stop here.

I took a different approach and came up with simple heuristic that generate initial solution. I have not analyzed it thoroughly so I am not sure if it is guaranteed to always return feasible solution if one exists. Its idea is quite simple:

• Construct bipartite graph defined as \(G=(V, A)\), where \(V = T \cup M\) – \(T\) is set of tasks and obviously \(M\) is set of machines. There exists arc \(a = (t, m)\) if \(w_{tm} \le rc_{m}\), where \(rc_{m}\) is remaining capacity for machine \(m\). Initially remaining capacity is equal to capacity of machine and with each iteration, and assignment of task to machine it is being update. If \(\vert A \vert = 0\), then stop.
• Solve a minimum weight matching problem.
• Update assignments – say that according to solution task \(t_0\) should be assigned to machine \(m_0\), then \(\overline{rc}_{m_0} = rc_{m_0} - w_{t_0 m_0}\).
• Find a machine where task is contributing with the lowest weight – say machine \(m_0 = \arg\min \{ m: w_{t_0 m} \}\).
• Free up remaining capacity so there is enough space for \(t_0\) on machine \(m_0\). Any tasks that were de-assigned in a process are added to pool of unassigned tasks.
• Repeat until there are no unassigned tasks.

See details on github .

As we said before the most important piece needed to implement B&P is branching rules which does not destroy structure of subproblem. Let’s consider non-integral solution to RMP. Given convexity constraint it means that there exists machine \(j_0\) and at least two, and for sake of example say exactly two, \(0 < \lambda_{j_0} ^{k_1} < 1\) and \(0 < \lambda_{j_0} ^{k_2} < 1\) such that \(\lambda_{j_0} ^{k_1} + \lambda_{j_0} ^{k_2} = 1\). Since with each of \(\lambda\)s is connected different assignment (set of tasks), then it leads us to a conclusion that there exists task \(i_0\) such that \(x_{i_0 j_0} < 1\) expressed in variables from the original formulation. Now, let’s use this information to formulate branching rule:

• left child node: a task \(i_0\) must be assigned to a machine \(j_0\).
• right child node: a task \(i_0\) cannot be assigned to a machine \(j_0\).

We can say that branching is based on \(x_{ij}\) from standard formulation. And it can be represented by:

Note that we can use the branching rule to easily to filter out initial columns for each node that do not satisfy those conditions:

• \(j = j_0\) and task \(i_0 \in T_{j_0}\), or
• \(j \neq j_0\) and task \(i_0 \notin T_{j}\).
• \(j = j_0\) and task \(i_0 \in T_{j_0}\).

See on github .

Based on the same principle, subproblem’s pool of feasible solution are created - i.e. on left child node:

• knapsack subproblem for machine \(j_0\) – variable representing task \(i_0\) is forced to be \(1\).
• knapsack subproblem for machine \(j \neq j_0\) – variable representing task \(i_0\) is forced to be \(0\).

Similarly for right childe node. See on github .

Below is an outline of main loop of column generation. It is an implementation of flow diagram from above so I will not spend too much time describing it. The only part maybe worth commenting is stop_due_to_no_progress - it evaluates whether column generation did not make any progress in last \(k\)-iterations and it should be stop.

Now, let’s see how constructing subproblems, solving them and then adding back column(s) to RMP looks like. We have as many subproblems as machines. Once a solution is available, we check whether it has positive reduced cost. A solution to knapsack problem corresponds to column in RMP. So if the column with positive reduced cost was identified and added, then new iteration of column generation will be executed. Gurobi allows to query information about all other identified solutions, so we can utilize this feature and add all columns that have the same objective value as optimal solution, potentially adding more than one column and hoping it will positively impact solution time.

Note that each subproblem is independent so in principle they could be solved in parallel. However due to Python Global Interpreter Lock (GIL) that prevent CPU-bounded threads to run in parallel, they are solved sequentially. Additionally depending on your Gurobi license, you might not be allowed to solve all those models in parallel even if Python would allow it.

Below you can find example of one of the RMPs:

and subproblem with dual information passed:

Now that we have all building blocks prepared, then let’s turn our attention back to B&P.

In the blog post, Branch-and-Price technique for solving MIP was explained. An example of applying B&P for Generalized Assignment Problem was presented. The solution approach used Python as programming language and Gurobi as solver.

[1] Der-San Chen, Robert G. Batson, Yu Dang (2010), Applied Integer Programming - Modeling and Solution, Willey. [2] Lasdon, Leon S. (2002), Optimization Theory for Large Systems, Mineola, New York: Dover Publications. [3] Cynthia Barnhart, Ellis L. Johnson, George L. Nemhauser, Martin W. P. Savelsbergh, Pamela H. Vance, (1998) Branch-and-Price: Column Generation for Solving Huge Integer Programs. Operations Research 46(3):316-329.

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• Branch and Bound Tutorial
• Backtracking Vs Branch-N-Bound
• 0/1 Knapsack
• 8 Puzzle Problem
• Job Assignment Problem
• N-Queen Problem
• Travelling Salesman Problem
• Solve Coding Problems
• Branch and Bound Algorithm
• Introduction to Branch and Bound - Data Structures and Algorithms Tutorial
• 0/1 Knapsack using Branch and Bound

Implementation of 0/1 Knapsack using Branch and Bound

• 8 puzzle Problem using Branch And Bound
• Job Assignment Problem using Branch And Bound
• N Queen Problem using Branch And Bound
• Traveling Salesman Problem using Branch And Bound

Given two arrays v[] and w[] that represent values and weights associated with n items respectively. Find out the maximum value subset(Maximum Profit) of v[] such that the sum of the weights of this subset is smaller than or equal to Knapsack capacity Cap(W) .

Note: The constraint here is we can either put an item completely into the bag or cannot put it at all. It is not possible to put a part of an item into the bag.

Input: N = 3, W = 4, v[] = {1, 2, 3}, w[] = {4, 5, 1} Output: 3 Explanation: There are two items which have weight less than or equal to 4. If we select the item with weight 4, the possible profit is 1. And if we select the item with weight 1, the possible profit is 3. So the maximum possible profit is 3. Note that we cannot put both the items with weight 4 and 1 together as the capacity of the bag is 4. Input: N = 3, W = 4, v[] = {2, 3.14, 1.98, 5, 3}, w[] = {40, 50, 100, 95, 30} Output: 235

Implementation of 0/1 Knapsack using Branch and Bound:

We strongly recommend to refer below post as a prerequisite for this. Branch and Bound | Set 1 (Introduction with 0/1 Knapsack) We discussed different approaches to solve above problem and saw that the Branch and Bound solution is the best suited method when item weights are not integers. In this post implementation of Branch and Bound method for 0/1 knapsack problem is discussed. 

How to find bound for every node for 0/1 Knapsack? 

The idea is to use the fact that the Greedy approach provides the best solution for Fractional Knapsack problem. To check if a particular node can give us a better solution or not, we compute the optimal solution (through the node) using Greedy approach. If the solution computed by Greedy approach itself is more than the best so far, then we can’t get a better solution through the node. 

• Sort all items in decreasing order of ratio of value per unit weight so that an upper bound can be computed using Greedy Approach.
• Initialize maximum profit, maxProfit = 0
• Create an empty queue, Q.
• Create a dummy node of decision tree and enqueue it to Q. Profit and weight of dummy node are 0.
• Extract an item from Q. Let the extracted item be u.
• Compute profit of next level node. If the profit is more than maxProfit, then update maxProfit.
• Compute bound of next level node. If bound is more than maxProfit, then add next level node to Q.
• Consider the case when next level node is not considered as part of solution and add a node to queue with level as next, but weight and profit without considering next level nodes.

Following is implementation of above idea. 

Time complexity: O(2 n ) because the while loop runs n times, and inside the while loop, there is another loop that also runs n times in the worst case. Auxiliary Space : O(n) because the queue stores the nodes, and in the worst case, all nodes are stored, so the size of the queue is proportional to the number of items, which is n.

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Branch and Bound Algorithm

Introduction

Branch and bound is a method of designing  algorithms commonly used to solve combinatorial optimization problems . In the worst-case scenario, these problems are typically exponential in terms of  time complexity and may necessitate exploring all possible permutations. These problems are solved relatively quickly using the Branch and Bound Algorithm . It is a process that involves systematically enumerating candidate solutions through a state space search. The set of candidate solutions is considered as a tree with the full set at the root. The algorithm explores branches of this tree that represent subsets of the solution set.

What is Branch and Bound Algorithm?

One of the problem-solving techniques is a branch and bound. It is similar to backtracking in that it employs the state space tree. It is engaged in the solution of optimisation and minimisation problems. If given a maximisation problem, we can use the Branch and bound technique to convert it by simply altering the issue to a maximisation problem.

Branch and bound algorithms solve problems with combinatorial, discrete, and general mathematical optimisation. Given an NP-Hard situation, a branch and bound algorithm explores the entire search space of possible solutions and provides an optimal solution.

A branch and bound algorithm enumerates possible candidate solutions step by step by exploring the whole search space. We begin by creating a rooted decision tree with all possible solutions. The root node represents the entire search space:

Each child node, in this case, is a component of the solution set and a partial solution. Before creating the rooted decision tree, we created an upper and lower bound for a given problem based on the optimal solution. We must choose which nodes to include in the solution set at each level. We look into the node with the highest quality bound for each group. This allows us to identify the most suitable solution quickly.

Finding an excellent upper and lower bound in these situations is crucial. An upper bound can be discovered by using any local optimisation technique or location in the search space.

On the other hand, convex relaxation or duality can offer a lower bound.

The solution set should generally be split up into smaller subsets.

We first construct a rooted decision tree to find the best solution set and then choose the best subset (node) at each level.

Branch and Bound Algorithm Example

In the below section, we are going to discuss how the job assignment problem can be solved using a branch and bound algorithm.

Statement of the Problem

Let us begin by defining a job assignment problem. There can be N jobs and N workers in a standard version of a job assignment problem. To keep things simple, we'll use three jobs and three workers in our example:

We can assign any available job to any worker, with the caveat that if a job is posted to a worker, the other workers are not allowed to take that job. We should also note that each job has a cost, which varies from worker to worker.

The main goal here is to complete all jobs by assigning one job to each worker to minimise the total cost of all jobs.

Pseudocode for the Branch and Bound Algorithm

Let's look at how to use a branch and bound algorithm to solve the job assignment problem.

Let's start with the pseudocode:

The input cost matrix in this instance is called MinCost, and it includes data like the number of open jobs, a list of open workers, and the cost for each job. The active nodes are tracked by the function MinCost(). The minimum cost of the active node at each level of the tree is determined by the function LeastCost(). We return the lowest-cost node after finding it and removing it from the active nodes list.

In the pseudocode, we're using the Add() function, which calculates the cost of a specific node and adds it to the list of active nodes.

Each node in the search space tree contains some information, such as cost, the total number of jobs, and the total number of workers.

We initially had mathsf3 jobs available. Employee A has the option of taking any of the available jobs. So, at the mathsf1 level, we assigned all available jobs to worker A and calculated the cost. We can see that assigning jobs mathsf2 to worker A results in the lowest cost in level mathsf1 of the search space tree. So we assign the job  A and proceed with the algorithm. "Yes" indi-prices that this is the current best price.

We still hagive open jobs after assigning the job mathsf2 to worker A. Consider worker B for a moment. To save money, we're attempting to post either job mathsf1 or job mathsf3 to worker B.

We can either assign the job mathsf1 or mathsf3 to worker B. We recheck the cost and give job mathbfmathsf1 to worker B because it is the cheapest in level mathbfmathsf2.

Finally, we trust worker C with the job mathbfmathsf3, and the optimal cost is 12.

Frequently Asked Questions

Which technique solves problems faster in branch and bound.

The Best First Search (BFS) technique, used in the Branch and Bound method, often solves problems faster. It prioritizes branches with promising cost estimates, leading to an expedited optimal solution discovery.

Can a maximization problem can be solved by branch and bound technique?

Yes, the Branch and Bound technique can solve maximization problems. By establishing upper and lower bounds for the optimal solution, the technique efficiently prunes the search space, helping identify the maximum value.

How branch and bound is better than backtracking?

Branch and Bound is often better than backtracking because it can discard many subproblems without explicitly solving them, due to its bounding function. This pruning of the search space typically makes it more efficient than backtracking.

The branch and bound algorithm are one of the most commonly used  algorithms optimisation problems. This topic has been thoroughly covered in this tutorial.

We've discussed when a branch and bound algorithm is the best option for a user. In addition, we presented a branch and bound algorithm for resolving the job assignment problem.

Recommended Readings:

• Kadane's Algorithm
• Greedy Layer-wise Pre-Training
• Unbounded Knapsack
• Backtracking Algorithm
• Design and Analysis of Algorithms

Finally, we discussed the branch and bound algorithm's benefits and drawbacks.

Do check out our blogs on  object-oriented programming and  Data Structure . 

Don’t stop here. Check out Coding Ninjas for more unique courses and guided paths. Also, try  Coding Ninjas Studio for more exciting articles, interview experiences, and fantastic Data Structures and Algorithms problems.

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branch-and-bound

Here are 75 public repositories matching this topic..., akrm3008 / essential-search.

Use this to determine the optimal route to go on a search for shortage struck essential commodities (gasoline, water, toilet paper etc.) using information from social media

• Updated Aug 16, 2020

luanloose / branch-and-bound

Primeira avaliação da matéria de inteligência artificial.

• Updated Sep 16, 2021

mkamadeus / n-2-Puzzle

Tugas Kecil Stima III : Branch and Bound (n^2 -1) solver

• Updated Mar 25, 2020

apwic / fifteen-puzzle-solver

Fifteen Puzzle Solver. Made as an assignment to demonstrate the use of Branch and Bound Algorithm in IF2211 Algorithm Strategy

• Updated Apr 4, 2022

darkhorrow / branch-and-bound-knapsack

Branch and boun applied to binary knapsack problem

• Updated Dec 8, 2020

novak-a / Branch-and-Bound-algorithm-in-Python

Simple Branch and Bound algorithm in Python

• Updated Nov 24, 2023

otavioml / Knapsack-Problem

Knapsack Problem solution using two distincts aproaches: backtracking and branch-and-bound.

• Updated Feb 14, 2022

batiukmaks / Math-Optimization-Algorithms

This repository is a collection of mathematical optimization algorithms and solutions for a variety of optimization problems. It provides a toolkit of algorithms and techniques for tackling optimization challenges in different domains.

• Updated Jun 9, 2023

gagaspbahar / 15-puzzle-bnb

This is a 15-Puzzle solver using the Branch and Bound approach algorithm.

• Updated Apr 3, 2022

rifqifarelmuhammad / daa-experimental-task-2

• Updated Dec 5, 2023

keyber / RP

Implémentation d'algorithmes de Résolution de Problèmes

• Updated Nov 5, 2019

mdaffad / bnb

Branch and Bound

riccardodm97 / ILP-Solver

Integer Linear Programming Solver implementing Simplex algorithm together with Branch-And-Bound method written from scratch with Numpy

• Updated Jun 4, 2021

nathanchrs / tsp-bb

Branch and bound solver for the travelling salesman problem, written in Python

• Updated Apr 6, 2017

faithdanghuy / Knapsack-Artificial-Intelligent

Knapsack problem solving with various algorithm for study in AI

• Updated Apr 3, 2023

christojeffrey / puzzle-solver

[assignment] Strategi Algoritma Semester 4. branch and bound solver for puzzle

• Updated May 25, 2022

seahrh / coding-interview

• Updated Jan 30, 2024

yosalx / 15-Puzzle

15-Puzzle Solver Program Implemented Using Branch and Bound Algortithm

Lynda-Starkus / Graph_coloring_branch_and_bound

A branch and bound solution implemented by my binome Lamdani Wilem as part of an ongoing work for combinatorics optimisation subject.

• Updated Mar 31, 2022

AlejandroDavidArzolaSaavedra / practicas_fsi

Branch and Bound and Branch and Bound with subestimation

• Updated Jan 16, 2024

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