13 May 2022

In the real world, we often encounter non-linear relationships in the systems we're modelling. But non-linearities can be difficult to model and even more difficult to solve to optimality.

The recently-published paper "Transformation and linearization techniques in optimization: A state-of-the-art survey" presents a detailed description of transformations and linearizations for a variety of non-linear relationships.

According to the paper's abstract:

To formulate a real-world optimization problem, it is sometimes necessary to adopt a set of non-linear terms in the mathematical formulation to capture specific operational characteristics of that decision problem. However, the use of non-linear terms generally increases computational complexity of the optimization model and the computational time required to solve it.

This motivates the scientific community to develop efficient transformation and linearization approaches for the optimization models that have non-linear terms. Such transformations and linearizations are expected to decrease the computational complexity of the original non-linear optimization models and, ultimately, facilitate decision making.

Asghari et al. (2022). "Transformation and linearization techniques in optimization: A state-of-the-art survey"

The transformation and linearization techniques described in the paper include:

- Multiplication of binary variables.
- Multiplication of binary and continuous variables.
- Multiplication of continuous variables.
- Maximum/minimum operators.
- Absolute value function.
- Floor and ceiling functions.
- Square root function.
- Multiple breakpoint function.
- Piecewise linear approximating functions.
- Log-linearization via Taylor series approximation.
- A new technique for linearizing square root terms by means of transformation.

The detailed descriptions in this paper make a good complement to the booklet MIP formulations and linearizations that we've mentioned previously.

Download the full paper: Transformation and linearization techniques in optimization: A state-of-the-art survey

If you need help transforming your non-linear model into a linear model, then please contact us.

12 April 2022

In Logic conditions as constraints - Part 1 we introduced a technique for converting a broad class of logic conditions into constraints. While that technique is effective, it can be difficult to apply.

In this article, we present an alternative technique specifically designed to improve the proficiency of students learning to formulate models.

That is, we use two techniques for converting logic conditions into constraints:

- Decomposition and Translation (D&T). This technique is described in the paper "Teaching use of binary variables in integer linear programs: Formulating logical conditions".
- Conjunctive Normal Form (CNF). We described the CNF technique in Part 1.

After describing the D&T technique, we focus on replicating the paper's examples, applying both the D&T and the CNF techniques to illustrate how each technique can be applied to a variety of logic condition formulation situations.

3 April 2022

When formulating a model, we often have a situation described in terms of logic conditions where something implies something else. For example:

- If this happens, then we must do that.
- If this and that happen, then we can't do some other thing.
- We can do some or all of these things, but only if that other thing happens.

However, mathematical programming models cannot represent logic conditions directly. Therefore, we need to translate the conditions, known as "implications", into a set of constraints that represent the logic.

This article describes a technique for converting a logical implication into an equivalent set of constraints. Along the way, we'll dabble in applying some formal logic notation, define rules for manipulating formal logical implications, have a brief look at Conjunctive Normal Form (CNF), and learn how to convert CNF into constraints.

Part 2 describes an alternative representation of this technique, specifically designed to improve the proficiency of students learning to formulate models.

7 March 2022

Attached is a free, 681 page textbook: *Decision Modeling*. The textbook is written by David M. Tulett, Associate Professor in the Faculty of Business Administration at Memorial University in Canada.

Topics covered by the textbook include: optimization of linear, integer, and nonlinear models, as well as the use of decision trees. Both Excel Solver and LINGO are used extensively to build and solve the many example models.

According to the Introduction:

Decision Modeling involves the creation of mathematical models which represent problems faced by business management. To a lesser extent, it also involves numerically solving these models. ...

If there's a difficulty with this subject, it's probably not the mathematics. Instead, the difficulty is likely to be the building of the model which the mathematics seeks to solve. The important thing is always going from a problem description to a model for the problem.

Tulett, D. M. (2022). "Decision Modeling"

Download the full textbook as a PDF (version 3.0.4, 3 March 2022): Decision Modeling

Updates for the textbook are available from Linney, Memorial University's Centre for Innovation in Teaching and Learning (CITL).

5 March 2022

Professor Rubin recently posted an article about finding all solutions to a network model: Finding almost all paths. The article mentions two techniques that we explored in our article Taking a dip in the MIP solution pool. Specifically, Professor Rubin says:

To find all solutions, one possible approach is to solve whatever MIP model you choose, then add a "no-good" constraint that eliminates the solution just found (and only that one) and solve again, until eventually the aggregation of "no-good" constraints makes the model infeasible. What I did instead was to use the "populate" command in CPLEX, which accumulates a pool of solutions.

An issue with the CPLEX solution pool is that it isn't guaranteed to find all optimal solutions. So, the Professor also describes a brute force method to find all solutions. The article presents analysis that demonstrates that the brute force performs better in this case – that is, the brute force method is both faster and more reliably finds all solutions.

Although clever solvers are often the best method we have, it is good to remember that brute force can be a useful – and sometimes efficient – method for solving a problem. We used a brute force method for the situation described in our article Job sequencing to minimize completion time. In that situation, brute force enumeration was very effective for small data sets, though it quickly became impractical for large data sets.

In summary, we can extend our list of techniques for finding all solutions to include:

- CPLEX solution pool.
- Eliminate known solutions.
- Brute force enumeration.

If you know of any other methods for finding all (or almost all) solutions to an optimization problem, or you want help with your own models, then please contact us.

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 (1,855 words)

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.