November 12, 2023

Virtually hundreds of tools and applications exist to help user to build and to solve mathematical model. Lists of software and applications can be found in the Internet. In this paper, some links are given where to look for an appropriate tool. It is unrealistic to give an exhaustive overview, also because the situation is changing rapidly and new approaches emerge all the time.

The focus in this paper is mathematical modeling (not solving) in the realm of numerical linear and non-linear optimization and combinatorial models – which are typically used in operations research. The goal of this paper is to give a first impression of various modeling tools –free and commercial– that could be used to build mathematical models. The presentation for each tool is done by a small problem example which is implemented. In this way, the reader gets a first and better idea of the tool. The selection of the tools do not follow a systematical method, it is rather a subjective choice of systems that I came across in the last years.

I also implemented each problem in my modeling language LPL, that I have developed over the years, and I make some comparisons. However, there is no claim here to rate one tool against another. I have my opinion on various features of a modeling tool and I present also my checklist of what a modeling tool should contain and what not, a subjective checklist that comes from my longtime experiences in practical modeling and the implementations. I do not pretend that this checklist is the last resort. There are various criteria why a specific tool is chosen in a concrete situation and not another.

I think mathematical modeling, that is, finding and implementing an appropriate formulation of a problem is itself an important activity, besides of solving these models, that is, finding efficient methods and algorithms to get an (optimal) solution. A great deal of my research career I invested in developing ideas and implementing them in a modeling language to formulate all kind of problems. Most ideas come from my practical activities as a consulter. Modestly, I must say that I was only half-successful: While LPL is a great tool for implementing large MIP’s in a commercial context, most of the additional features in LPL are still experimental. Nevertheless, I like to share my knowledge with others, so maybe some of you can pick an idea from my experiences.

The toolboxes and software presented here are grouped into three categories: (algebraic) modeling languages, programming languages, further tools in the proximity of mathematical modeling.

A ZIP file of all models and the data displayed in this paper can be found at this link: modeling4.7z.