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Set Covering

Detailed Explanation of Optimization Problems and Their Implementations

For a comprehensive understanding of these classes, it is recommended to thoroughly review the examples available in the "/Examples/" folder within the code and execute SetCovering/SetCoveringMain. These examples are generic and open for testing different values. Below, we will explain the examples in detail.

Set Covering Problem

The Set Covering Problem is another classic optimization problem where the goal is to cover all elements of a given set using the minimum number of subsets from a collection of subsets.

Components of the Set Covering Problem:

1. Universal Set:
  • A set of elements that need to be covered.
2. Subsets:
  • A collection of subsets, each containing elements from the universal set.
  • Each subset has an associated cost.
3. Objective:
  • Select a minimum-cost collection of subsets such that every element in the universal set is covered by at least one selected subset.

Example:

Consider the universal set {1, 2, 3, 4, 5} and the following subsets:

  • Subset 1: {1, 2, 3}
  • Subset 2: {2, 4}
  • Subset 3: {3, 4, 5}
  • Subset 4: {4, 5}

The goal is to cover all elements in the universal set using the fewest subsets. One possible optimal solution is selecting Subset 1 and Subset 4, covering all elements with only two subsets.

Applications:

  • Facility location
  • Network design
  • Scheduling and planning

Implementation

This example represents a generalization of the set covering problem, allowing for testing of various cases. Here's how to work with it:

1. Provide Initial State, Variables and Domains:
  • Create a dictionary of variables along with their domains.
  • Provide a list representing the number of constraints in the initial_state parameter, e.g., [1, 2, 3].
2. Define Constraints and Weights:
  • right_side_of_restrictions: Specify the minimum value for each constraint, associated by their positions.
  • matrix_of_weight: Provide a list of lists where the values represent the probability of each variable (associated by position) being within the constraint.
3. Create an Instance of SetCovering:
  • With all these parameters, create an instance of the constructed SetCovering problem class.
4. Generate Decision Diagram:
  • After defining all the parameters, generate a decision diagram and test the other features.
5. Data Consistency Checks:
  • The implementation includes checks to ensure data consistency, avoiding any inconsistencies in the provided data.

By following these steps, you can effectively test and utilize the set covering problem features.