27 March 2024

We conclude our project to improve the efficiency of a pallet warehouse by redesigning the warehouse's racks and shelves.

In part 2 of this article series, we linearized our warehouse racks and shelves design model with the hope of making it easier to solve at the full scale we need. It didn't work. Model 2 is even slower than the non-linear Model 1 we built in part 1.

In this article, we simplify our modelling by taking some of the variables out of the model and making them exogenous inputs. We enumerate all possible combinations of those exogenous inputs, then solve the simplified model for each combination. Hopefully, this time, our model will be solvable at the scale we need.

18 March 2024

In the first part of this article series, we formulated and implemented Model 1 for redesigning a pallet warehouse. Model 1, which in non-linear, works OK for small data sets, but it fails to find a good solution for the full data set.

In this article, we linearize our model using a standard technique. The idea is that, all going well, a linear model will be solvable at the scale we need.

11 March 2024

A common problem for warehouse managers is finding enough space to store everything:

We just don't have enough storage space. There used to be, but the company grew. Now, we have pallets stored in the dispatch area. They get in the way, and it's a health and safety hazard.

Warehouse Manager

In this article series, we look at improving the efficiency of a pallet warehouse (also known as a unit-load warehouse), where all items are stored on standard-size pallets.

Most pallet warehouses use a rack design where all shelves are the same size. This may not be the best design. Specifically, if the pallets have varying heights, then a single shelf height – designed to accommodate the tallest pallet – leaves unused space above some pallets. Depending on the mix of pallet heights, there's potentially a lot of unused space in aggregate.

To address this issue, we build a model of the warehouse's racks and shelves. Our objective is to find a design that minimizes the number of racks required to accommodate all pallets in the warehouse.

Along the way, we:

- Formulate a non-linear model of the situation.
- Compare several solvers, to see how they perform.
- Linearize our model to, hopefully, make it easier to solve.
- Disaggregate our model to make some variables exogenous, then iterate over an enumeration of the exogenous variables.
- Demonstrate use of Pyomo's
`last()`

and`next()`

functions, which enable us to work with elements of ordered sets. - Turn off a constraint using Pyomo's
`deactivate()`

function.

Importantly, we show that there's a surprising amount of extra storage space available for free, or minimal cost, just by redesigning the warehouse's racks and shelves.

13 February 2024

We're often asked questions about the seemingly odd behaviour of some models:

- Why does my model find different solutions each time I run it?
- The solver says the different solutions are all optimal. But some solutions are better than others. How can that be true?
- What can I do to make my model find the overall best solution?

In most cases, the answers are:

- The model is non-linear with multiple local optima. The solver may use a process that includes random elements that lead it to different neighbourhoods of the feasible region.
- Each solution is locally optimal, meaning that it is the best solution in that neighbourhood of the feasible region.
- To find the overall best solution, either use a solver than finds globally optimal solutions for that type of model, or solve the model multiple times, with different initial conditions, to make it more likely that the solver finds the best neighbourhood.

These answers generally require explanation, So, in this article, we explore some aspects of solver behaviour when solving non-linear optimization models. Our goal is to provide insights into what the solvers are doing, why they may find different solutions, and how we can improve our chances of finding at least a good, and hopefully a globally optimal, solution.

10 January 2024

In Part 1 of this series, we formulated a Mixed Integer Linear Program (MILP) for compiling crossword puzzles. Model 1 works OK, provided the crossword grid is small, not too dense, and the lexicon isn't too large. But as the grid gets denser and/or the lexicon gets larger, Model 1 struggles to find solutions.

We can do better. Model 1 uses a general formulation, where we specify the form of each constraint and then let the solver work out which variables and constraint instances aren't needed, given a specific data set. For many models, this approach is sufficient – largely because modern optimization solvers are very good at eliminating unnecessary terms in a model, especially in the presolve phase.

But solvers use general-purpose procedures for tightening models. Sometimes we can make use of our knowledge about the specific situation to help the solver.

In this article, we discuss techniques for fine-tuning the model-building process in Pyomo, to make a smaller and faster model. Specifically, we:

- Select only those words in the lexicon that fit in the current grid.
- Pre-populate some grid word slots with random words, to give the solver a head start.
- Omit constraint terms that don't meet some criteria.
- Skip some constraint instances entirely.
- Fix some binary variable values at zero, when we know they cannot have a value of one.

These steps tighten the model, making it easier for the solver to find a solution. We'll then look to solve larger crossword compilation problems.

10 January 2024

Completing crossword puzzles is a popular pastime. There are many puzzle variations, ranging from simple to fiendishly difficult.

The process of creating a crossword puzzle – known as "compiling" – is difficult. Compiling a crossword can be thought of as a type of search problem, where we need to find a set of words that fits the rules for filling a specific crossword grid. Unsurprisingly, many crossword compilers use software to help them find a suitable set of words.

One approach is described in an excellent recent article on Philippe Olivier's blog: Generating crossword grids using Constraint Programming. That article inspired us to try using a different method: Mixed Integer Linear Programming (MILP).

We're not the first to consider using MILP to compile crosswords. An article published by Wilson in 1989 reported attempts to compile small crosswords using MILP models, as discussed in our previous article Solver performance - 1989 vs 2024. Wilson's conclusion was very pessimistic, noting that "the prospects of using integer programming for any type of puzzle of realistic size and with a substantial lexicon remain bleak". The problems were insufficient computer memory and the time need to solve even a small crossword grid using a then state-of-the-art, million-dollar superminicomputer.

But a lot has changed in the 35 years since 1989. Computers are thousands of times faster, and their memory capacity has increased by a similar factor. In addition, optimization solver software is literally millions of times faster. Models that were difficult to solve in 1989 are now trivial. Solutions that were impossible to find in 1989 may now be within reach. Does that include compiling crosswords of realistic size?

In this series of articles, we discuss:

- Representing a word puzzle in mathematical terms, enabling us to formulate the crossword compilation problem as a MILP model.
- Techniques for fine-tuning the model-building process in Pyomo, to make a model smaller and faster, including omitting constraint terms, skipping constraints, and fixing variable values.

5 January 2024

In our previous article, we solved a mixed integer linear program (MILP) model with more than 8 million binary variables. Not long ago, a model of that size would have been unsolvable. Yet, once we had a computer with sufficient memory, the model was solved to optimality by an open source solver in 3 hours.

Of course, not all large models are solvable – even some small models are unsolvable, or at least hard to solve. Nonetheless, modern solvers, in association with modern computers, enable us to solve models that not-so-long-ago were beyond our capabilities.

In our next article, we look to compile crosswords using a MILP model. We're not the first to consider using a MILP model to compile crosswords. An article published in 1989 reported attempts to compile small crosswords using MILP models. The article's conclusion was very pessimistic, noting that:

...the prospects of using integer programming for any type of puzzle of realistic size and with a substantial lexicon remain bleak.

Wilson, 1989, Crossword compilation using integer programming

The problems Wilson encountered were insufficient computer memory and the time needed to solve a very small crossword grid, even though he was using a then state-of-the-art million dollar superminicomputer.

But a lot has changed in the 35 years since 1989. Computer hardware is many times faster, and memory capacity has increased enormously. In addition, optimization solver software has improved by orders of magnitude. Models that were difficult to solve in 1989 are now trivial. Solutions that were impossible to find in 1989 may now be within reach.

In this article, we estimate the magnitude of speed improvement for optimization solvers and computer hardware in the 35 years from 1989 to 2024. The results may be surprising.

13 December 2023

We conclude our article series looking at a common modelling issue: The solution is optimal, but not practical.

In a previous article of this series, we solved Model 3 to optimality with the full data set of 100 items. Given the large memory requirements of Model 3, a data set with 100 items is close to the maximum that we can solve on our modelling computer. But what if we need to solve a larger data set?

Model 4 provides one approach, using a heuristic random column generation technique that finds very good solutions, though they are not guaranteed to be globally optimal.

The variants of Model 5 all have even more variables and constraints than Model 3, so they don't help us solve larger data sets.

To find guaranteed, globally optimal solutions we need to use Model 3. That requires a computer with more memory – potentially a lot more memory. We could upgrade our modelling computer. If there's a need to often solve very large models, then that would likely be the best option. But if the need is less frequent, then a potentially cost-effective solution is to use a virtual machine in "the cloud".

In this article, we adopt the latter approach. We describe, in detail, setting up a Google Compute Engine, provisioned with 128 GB RAM, and using it to solve Model 3 to global optimality with a data set that is larger than we can solve on our local computer.

1 December 2023

We continue our article series looking at a common modelling issue: The solution is optimal, but not practical.

In the previous article of this series, we explicitly formulated either/or constraints by manually creating \(BigM\) constraints. That technique works, though it can be a bit tricky to get right.

An alternative approach, since we're using Pyomo, is to use the Generalized Disjunctive Programming (GDP) extension that is part of Pyomo. GDP can represent a variety of logical implications, like either/or, and automatically translate them into a form that a solver can work with.

In this article, we describe how the GDP extension can be used to represent either/or constraints in a simple linear program. Along the way, we compare \(BigM\) constraints that we create manually with \(BigM\) constraints created automatically by the GDP extension. Then we use the GDP extension to build either/or constraints in our paper coverage situation, as variant Model 5c.

21 November 2023

We continue our article series looking at a common modelling issue: The solution is optimal, but not practical.

In the first article of this series, we made the assertion that we can implicitly mimic rotation of the items by sorting the item dimensions so that the width is always the longer side. Therefore, we didn't need to explicitly represent the orientation of the items in our earlier models.

This insight may not be obvious, so in this article we explicitly model rotation of the items using either/or constraints. That is, we want to represent items that are in either portrait or landscape orientation, with the model choosing only one of the orientations for each product.

The problem is that, in a mathematical programming model, all constraints are always evaluated. So, we can't have a condition like:

If orientation is portrait then use constraint for portrait else use constraint for landscape

Instead, we need to create a pair of constraints that behave the way we want, while both are always evaluated. This is done using a binary variable, which acts as a switch to decide which constraint is "active", and some \(BigM\) parameters that change each constraint's right-hand-side in a very specific way.

Our objectives in this article are to:

- Illustrate how to formulate and implement either/or constraints in an integer programming model, using \(BigM\) constraints.
- See how our model behaves when explicitly including either/or constraints.
- Demonstrate that our dimension sorting assertion is correct.