21 February 2022

We're frequently asked questions like: "How many other optimal solutions exist?" and "How do I find those solutions?". Often these questions are prompted by our mentioning that most models have alternative optima – that is, optimal solutions with the same objective function value, but different variable values.

Although a model may have a unique optimal solution, models with integer/binary variables typically have multiple optimal solutions, and continuous linear models may have an infinite number of alternative optimal solutions. The likely existence of multiple alternative optima is why we usually say "an optimal solution", rather than saying "the optimal solution".

Sometimes people also ask, "How do I find solutions that are almost optimal?". This question typically indicates that the decision maker may accept a sub-optimal solution (or an alternative optimal solution) that is "better" according to some criteria that aren't captured by the model design. Of course, we should look at incorporating the unspecified criteria within the model, but sometimes that is difficult or even impossible. In any case, exploring the solution space around the optimal solution is an important part of the modelling process.

This article describes methods for finding alternative optima and solutions that are almost optimal. Specifically, we explore the CPLEX "solution pool" feature, which NEOS Server has recently made available through their online portal.

29 January 2022

When formulating a model, many of the problems we encounter involve non-linear formulae. For example, a fixed cost is incurred only if a facility is built, or a variable can take values only in the ranges 5 to 10 or 80 to 100.

Optimization models work best when the objective function and constraints are all linear. In some situations, it is possible to reformulate a model to linearize the non-linear parts. The techniques for linearizing non-linear formulae can make the difference between a model being viable or not.

FICO Xpress Optimization have written a booklet that describes a variety of useful mixed-integer programming (MIP) formulations and linearizations, including:

- Binary variable logical conditions.
- Minimum, maximum, and absolute value of binary variables.
- Multiplication of variables.
- Variables with disjunctions.
- Batch sizes.
- Minimum activity level.

The booklet is available at: MIP formulations and linearizations.

17 December 2021

The COIN-OR Foundation is seeking support to continue providing open-source optimization software for the operations research community. They have published an article, Future of COIN-OR, describing their challenges in securing funding and the participation of community members to continue operating.

COIN-OR are responsible for the development of more than 70 projects that are widely used in optimization tools, including:

- Pyomo, a Python-based optimization modeling language.
- PuLP, a Python library for linear optimization.
- CBC, the default mixed integer linear programming solver used in OpenSolver and elsewhere.
- Bonmin and Couenne, non-linear mixed integer programming solvers.

The COIN-OR Foundation presents three possible future directions:

Option 1: Wind down the COIN-OR Foundation activities related to maintenance and development of common infrastructure and existing codes that are not otherwise maintained.

Option 2: Continue operating in a haphazard fashion and hope that the community will eventually take up the cause as things slowly degrade.

Option 3: Find a path to funding the current activities of the COIN-OR Foundation in a sustainable way.

To achieve Option 3, the Foundation needs your support. You can either donate to COIN-OR, or email them at

2 December 2021

Project crashing is the process of compressing a project plan by using additional resources to reduce the duration of some tasks.

Using additional resources incurs additional cost, so it is important that the resources are deployed to the tasks that produce the greatest benefit. A project manager must decide an appropriate trade-off between time and cost for each task, leading to the best revised project plan. Given the dependencies between project tasks, making these trade-offs can be complex and difficult, even for a small project.

This article describes an example of project crashing using an optimization model to help the project manager decide what to do.

16 November 2021

Recently we were working on a small one-dimensional bin packing model. The situation was simple, and we expected the model to be easy to solve. But there was just one problem: we couldn't find an optimal solution, even after letting the solver run overnight for 12 hours.

Initially, we were using the CBC solver. Since that didn't work, we tried CPLEX via NEOS, but we encountered the same problem – CPLEX couldn't find an optimal solution either.

So, we searched the Operations Research literature for an alternative formulation. We discovered a recently published academic paper that has a new, innovative formulation for one-dimensional bin packing (and potentially other types of packing situations).

This article describes the new formulation and our experience applying it to our simple, yet difficult to solve, model.

5 November 2021

There are many tools for building an optimization model, including:

- Excel/OpenSolver, which provides a familiar, interactive, visual grid for modelling.
- Python, plus the CVXPY or PuLP library, which provides a programmatic way to define a model.

Each tool has advantages and disadvantages, as we discussed in Optimization in Excel vs Python.

The tool CMPL (<Coliop|Coin> Mathematical Programming Language) occupies an intermediate position. That is, CMPL combines the advantages of working with the data and solution in Excel, while providing the power and flexibility of a programming language for defining the model.

According to the tool's website:

The CMPL syntax is similar in formulation to the original mathematical model but also includes syntactic elements from modern programming languages. CMPL is intended to combine the clarity of mathematical models with the flexibility of programming languages.

Steglich, M. (2021). "Optimisation Modelling with Excel and CMPL2"

CMPL has recently been updated to version 2.0, as described in the paper "Optimisation modelling with Excel and CMPL2". The paper provides a good overview of CMPL, including a couple of examples to help you get started*.

* Note that there's a bug in the paper's transhipment example. You'll need to add `minCap[edges]`

to the definition in line 01 of Listing 7.

7 October 2021

Vamshi Jandhyala has an interesting series of blog posts about Optimization using linear models.

Each article includes a description of the topic, along with several examples written in Python and solved using the Gurobi commercial solver:

- Modeling using Linear Programming. Illustrates some concepts of linear programming via the formulation and solution of a resource allocation problem.
- Modeling using Integer Programming. Describes several applications of integer programming, including an assignment problem, graph coloring, the 0-1 knapsack problem, a set covering problem, and a class scheduling example.
- Graphs and Integer Programming. Explores some graph theory applications of integer programming, including finding a maximal independent set and finding the maximal clique of a set.

The examples include Python source code, though they are sufficiently small that they could be translated to use your preferred modelling tool and solver.

3 October 2021

OpenSolver uses the free, open-source CBC solver. For most linear models, CBC is good enough. But sometimes CBC struggles to solve a model in a reasonable time. That usually happens when the model has a large number of variables or constraints, though some small models can also be difficult to solve.

When CBC doesn't get the job done, we can try using a more powerful solver. One way to apply more power is to use the NEOS Server, which is an online service that provides access to many different solvers, including commercial solvers, for free.

This article describes an example of how we can solve a model using the CPLEX solver via the NEOS Server.

13 September 2021

Covid-19 has caused havoc in the world's supply chains, with numerous disruptions and delays. But without sophisticated math – including a lot of linear algebra – modern supply chains would not operate at all.

The Guardian newspaper has an interesting article about the math of logistics:

Algebra: the maths working to solve the UK’s supply chain crisis.

As the article says:

The maths of logistics starts with algebra – linear algebra, to be precise. ...

Linear algebra explores solutions for sets of equations that together contain all you need to find out the relationships between the variables. ...

Logistics hasn't stood still with linear algebra, however. It has been developed into algorithms for "linear programming" and "mixed integer programming" and various other odd-sounding mathematical routines, such as "combinatorial optimisation", "greedy heuristics" and "simulated annealing". ...

And it's all done with just one purpose: to deliver to every customer, on time and in full – OTIF as it's known in the trade.

Brooks, M. (2021). "Algebra: the maths working to solve the UK’s supply chain crisis"

The article provides some insight into the hidden complexity of the systems that supply our everyday products and services. The algebra of optimization techniques, embedded within our supply chains, makes the modern world possible.

8 September 2021

Jon Lee, a Professor of Industrial and Operations Engineering at the University of Michigan, offers a free, 304 page, operations research textbook: *A first course in linear optimization*.

This textbook is a great introduction for anyone interested in learning about linear programming and integer programming.

According to the preface:

This book is a treatment of linear optimization meant for students who are reasonably comfortable with matrix algebra (or willing to get comfortable rapidly). It is not a goal of mine to teach anyone how to solve small problems by hand.

My goals are to introduce:

Lee, J. (2021). "A first course in linear optimization"

- The mathematics and algorithmics of the subject at a beginning mathematical level.
- Algorithmically-aware modeling techniques.
- High-level computational tools for studying and developing optimization algorithms (in particular, Python/Gurobi).

Download the full textbook as a PDF: A First Course in Linear Optimization (Version 4.0, August 2021)

The book's source, potentially including updates, is available at GitHub.