## Graph Coloring Problem

A node coloring is valid or admissible if any two adjacent nodes do not have the same color. If a graph is colorable, there is a smallest number such that the graph is node colorable. This number is called the chromatic number of the graph and is usually denoted by . The problem to identify… Read more Graph Coloring Problem

## Solving a TSP with Linear Programming and Google OR-Tools in Python

In this post I show you how to solve the TSP problem using integer linear programming and Google OR-Tools for mathmatical modelling in Python. If you’re not yet familiar to the TSP and want to dig deeper, find out more here. It is one of the oldest and best explored problems in the field of… Read more Solving a TSP with Linear Programming and Google OR-Tools in Python

## TSP Miller-Tucker-Zemlin Subtour Elimination Constraint

In this post we want to try to provide a solution to solve the Traveling Salesman Problem (TSP) using linear programming. The post is based on this excellent video from Mr. Michel Bierlaire at the EPFL. A good written documentation of the Miller Tucker Zemlin Constraint can be found here. What is a TSP? The… Read more TSP Miller-Tucker-Zemlin Subtour Elimination Constraint

## Find optimal Solution to Knapsack Problem with Linear Programming in R

In another post I demonstrated how to develop a heuristic to solve the knapsack problem. We managed to solve the problem quite well and had an optimality gap of about 1%, that is, our solution was 1% away from the optimal solution. Here I’d like to demonstrate, how simple it is to solve the Knapsack… Read more Find optimal Solution to Knapsack Problem with Linear Programming in R

## Solve Knapsack Problem with Heuristics in R

In this post I’ll show a way how to solve the Knapsack Problem applying a simple greedy improvement heuristic and how to implement it in R. What is the Knapsack Problem? On Wikipedia, you find the following definition: “The knapsack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a… Read more Solve Knapsack Problem with Heuristics in R

## Graph Notation

There are different types of graph notations. Here I’d like to present a nice and compact way in the form of a Vertex Lists and Edge Lists, which I came across during my studies at the MSE in Zurich. [Note: All graph images shown in this post have been taken from graphical representations of graphs… Read more Graph Notation

## Solve TSP Instance using Google OR-Tools

If you happen to have to solve TSPs or some special type of VRPs – i.e. capacitated or with timewindows and other constraints – you might want to have a look at Google OR-Tools. The library contains a wide variety of heuristics for different problems primarly based on google’s excellent constraint programming solver. The library… Read more Solve TSP Instance using Google OR-Tools

## Calculate an MST with Ant Colony Optimization in Python

Ant Colony Optimization is an old but still often applied construction heuristic to develop solutions using nature inspired behavior. The heuristic mimics the behavior of ants when finding shortest paths to between their nests and some kind of attraction, i.e. a food-source. The main idea are the pheromone trails, which ants leave behind, when travelling… Read more Calculate an MST with Ant Colony Optimization in Python

## Tabu Search in R

Here I’ll show you a way, how to do Tabu Search with Steepest Descent with best improvement in R. Tabu Search is a local improvement heuristic used to improve existing solutions. Such heuristics are also called local search heuristics. How does Local Search work? Local search can be described in words as follows: Take initial… Read more Tabu Search in R

## Configure multiple Users for GUROBI Cloud

What is the Objective If you are a small to medium scale solution provider for optimization solutions and want to provide solutions using the world’s best solver, then you should go for GUROBI cloud, since prices are most attractive and management of multiple users is quite simple. You will have to pay something like –… Read more Configure multiple Users for GUROBI Cloud