Assignment Problem(2) - State Space Tree

 Sample Assignment Problem





State Space Tree

The root node represents the starting point where no assignments have been made yet.

Each branch from the root represents a possible job assignment for Worker 1 (denoted as x1).

The second level represents possible assignments for Worker 2 (denoted as x2), given the job already assigned to Worker 1.

In a full state space tree, this process would continue for Workers 3 and 4, each with their own level in the tree, until all workers are assigned jobs without conflict, resulting in a complete solution at the leaf nodes.


Full state Space Tree


Each path from the root to a leaf node represents a complete set of job assignments, one for each worker. There are 4! = 24 possible paths, each corresponding to a different way to assign all four jobs to the four workers without any overlaps. ​


Comments

Post a Comment

Popular posts from this blog

Assignment Problem(3) - Steps to Solve using Branch and Bound

Image
 Example Assignment Problem Steps in Solving Solving the assignment problem with the branch and bound algorithm involves a systematic exploration of all possible assignments, aiming to find the least costly solution. Here are the general steps involved in solving the assignment problem using branch and bound:   1. Formulate the Problem:    - Define the cost matrix where each element c[i][j] represents the cost of assigning the i-th worker to the j-th job.    - Determine the objective, typically to minimize the total cost of assignments.   2. Initial Lower Bound:    - Calculate an initial lower bound for the entire problem. This can be done by summing the minimum value from each row or each column of the cost matrix, depending on the strategy.   3. Create the Root Node:    - Start with a root node representing an empty assignment (no workers are yet assigned to any jobs).   4. Branching:    - For...
Read more

State Space Tree

Image
 A state space tree is a graphical representation of all possible states and transitions in a decision-making problem, such as an optimization or search problem. It's often used in algorithms that explore the set of all possible solutions to find an optimal solution, such as the branch and bound method for optimization problems, or backtracking algorithms for constraint satisfaction problems.   Components and characteristics of a state space tree:   1. Nodes:    - Each node represents a state of the problem .    - The root node represents the initial state before any decisions have been made.    - Intermediate nodes represent partial decisions or partial solutions.    - Leaf nodes (at the bottom of the tree) represent complete solutions or end states.   2. Edges/Branches:    - Edges connect nodes and represent transitions from one state to another due to a decision or action.    - Each e...
Read more

Knapsack Problem Variants

 The Knapsack Problem is a classic problem in combinatorial optimization. The basic scenario involves a knapsack with a fixed carrying capacity and a set of items, each with its own weight and value. The goal is to determine the most valuable combination of items to include in the knapsack without exceeding its weight capacity. This problem exemplifies many real-world resource allocation issues, where there are finite resources (the knapsack's capacity) and a need to optimize a particular value (total value of items). There are several types of Knapsack Problems, each with its unique characteristics: 1. 0/1 Knapsack Problem:    - In this version, each item can either be taken or left (hence "0/1"). An item cannot be divided; it must be included or excluded in its entirety.    - Example: Choosing items to pack in a suitcase with a weight limit, where you can't take part of an item. 2. Fractional Knapsack Problem:    - Here, you can take frac...
Read more