QMKPy: A Python Testbed for the Quadratic Multiple Knapsack Problem

This software package primarily aims at research in the areas of operations research and optimization. It serves as a testbed that provides a way of quickly implementing and testing new algorithms to solve the quadratic multiple knapsack problem (QMKP) and compare it with existing solutions.

The goal is to encourage researchers and developers to share their algorithms and make them publicly available.

Problem Description

The QMKP is defined as the following combinatorial optimization problem

$$ \begin{alignat}{3} \max\quad & \sum_{u\in\mathcal{K}}\Bigg(\sum_{i\in\mathcal{A}(u)} p_{i} &+&\sum_{\substack{j\in\mathcal{A}(u), \ j\neq i}} p_{ij}\Bigg)\\ \mathrm{s.t.}\quad & \sum_{i\in\mathcal{A}(u)} w_{i} \leq c_u & \quad & \forall u\in\mathcal{K} \\ & \sum_{u=1}^{K} a_{iu} \leq 1 & & \forall i \in {1, 2, \dots, N} \end{alignat} $$

This describes an assignment problem where one wants to assign $N\in\mathbb{N}$ items to $K\in\mathbb{N}$ knapsacks, which are described by the index set $\mathcal{K}={1, 2, \dots, K}$. Item $i$ has the weight $w_i\in\mathbb{R_+}$ and knapsack $u$ has the weight capacity $c_u\mathbb{R_+}$. If item $i$ is assigned to a knapsack, it yields the (non-negative) profit $p_i\in\mathbb{R_+}$. If item $j$ (with $j\neq i$ ) is assigned to the same knapsack, we get the additional joint profit $p_{ij}\in\mathbb{R_+}$.

The set of items which are assigned to knapsack $u$ is denoted by $\mathcal{A}(u)$ and $a_{iu}\in{0, 1}$ is an indicator whether item $i$ is assigned to knapsack $u$.

The objective of the above problem is to maximize the total profit such that each item is assigned to at most one knapsack and such that the weight capacity constraints of the knapsacks are not violated.

Remark: The profits $p$ are also referred to as "values" in the literature.

Features

• Quick and simple creation of QMKP instances
• Saving/loading of problem instances for a simple creation and use of reference datasets
• Easy implementation of novel algorithms to solve the QMKP
• High reproducibility and direct comparison between different algorithms

The benefit of enabling a simple and direct way of implementing novel algorithms is highlighted by an example in the provided Jupyter notebook in examples/Custom Algorithm.ipynb.

Installation

The package can easily be installed via pip. Either from the PyPI

pip3 install qmkpy

or from the GitHub repository

git clone https://github.com/klb2/qmkpy.git
cd qmkpy
git checkout dev  # optional for the latest development version
pip3 install -r requirements.txt
pip3 install .
pip3 install pytest  # optional if you want to run the unit tests

Usage

In order to test the installation and get an idea of how to use the QMKPy package, you can take a look at the examples/ directory. It contains some standalone scripts that can be executed and perform some simple tasks.

More detailed descriptions of the implemented algorithms and a documentation of the API can be found in the documentation.

A collection of reference datasets can be found at https://github.com/klb2/qmkpy-datasets.

Contributing

Please see CONTRIBUTING.md for guidelines on how to contribute to this project. In particular, novel algorithms are always welcome. Please check out the documentation for a brief overview on how to implement new algorithms for the QMKPy framework.