Скачать или смотреть How to Impose Constraints in PuLP Using Previous Solutions

Learn how to effectively impose constraints based on previous solutions in PuLP for mixed integer programming, including handling penalties for solution changes.
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This video is based on the question https://stackoverflow.com/q/63729018/ asked by the user 'gauzah' ( https://stackoverflow.com/u/14179238/ ) and on the answer https://stackoverflow.com/a/63729411/ provided by the user 'Erwin Kalvelagen' ( https://stackoverflow.com/u/5625534/ ) at 'Stack Overflow' website. Thanks to these great users and Stackexchange community for their contributions.

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Maximizing Item Price with Constraints in PuLP: A How-to Guide

In the world of optimization, particularly with mixed-integer programming, solving problems often requires not just finding a solution but building upon the previous ones. This article delves into a common issue faced by many optimization practitioners using Python’s PuLP library: how to impose constraints based on a previous solution effectively, while also managing penalties for modifications.

The Problem

You're working with a mixed integer program where an indicator variable x[i] indicates whether an item is included in the solution (0 or 1). Your goal is to maximize the total price of items packed, adhering to set constraints like total volume and weight. However, what makes this problem interesting is that you want to repeat the optimization process multiple times and make adjustments based on the previous solution. Each modification to the previous solution brings a penalty, complicating straightforward optimization.

For example, if your last solution was [0,0,1,1,0,0,0,0,0,0] and your new solution is [1,1,0,0,0,0,0,0,0,0], you incur a penalty of -100 for changing two items from the previous setup. If only one item changes, no penalty is applied. Hence, a systematic approach to incorporate these constraints and penalties is crucial.

The Solution

To handle this complex scenario, let’s break it down into a systematic approach using PuLP. The following steps outline how to impose a penalty based on changes from the previous solution.

Step 1: Introduce a Binary Variable

You need to introduce a binary variable d[i] for each item, which will help track if the item has changed from the previous solution:

[[See Video to Reveal this Text or Code Snippet]]

Next, you can implement the constraints that are required to control changes based on the previous solution, denoted as x0[i]:

[[See Video to Reveal this Text or Code Snippet]]

Step 2: Track Total Changes

Next, you need a variable n that counts the total number of changes made to the items:

[[See Video to Reveal this Text or Code Snippet]]

Step 3: Set Up the Free Change Constraint

Incorporate another variable n1 that helps you apply the one free change rule while penalizing additional changes:

[[See Video to Reveal this Text or Code Snippet]]

Step 4: Adjust the Objective Function

Finally, alter your objective function to minimize the penalties associated with additional changes:

[[See Video to Reveal this Text or Code Snippet]]

This adjustment allows you to both maximize the price of the packed items and minimize additional penalties for changing items beyond the free change limit.

Conclusion

By following the structured approach outlined above, you can efficiently impose constraints based on previous solutions using the PuLP library. Not only does this allow for better control over your solutions, but it also enables you to strategically manage changes and penalties, leading to more effective optimization outcomes.

With this newfound knowledge, you can tackle your mixed-integer programming challenges with greater confidence and precision. Happy optimizing!