70 lines
3.5 KiB
TeX
70 lines
3.5 KiB
TeX
%************************************************
|
|
\chapter{Modelling with Constraints}\label{ch:background}
|
|
%************************************************
|
|
|
|
A goal shared between all programming languages is to provide a certain level of
|
|
abstraction: an assembly language allows you to abstract from the binary
|
|
instructions and memory positions; Low-level imperial languages, like FORTRAN,
|
|
were the first to allow you to abstract from the processor architecture of the
|
|
target machine; and nowadays writing a program requires little knowledge of the
|
|
actual workings of the hardware. Freuder states that the ``Holy Grail'' of
|
|
programming languages would be where the user merely states the problem, and the
|
|
computer solves it and that constraint modelling is one of the biggest steps
|
|
towards this goal to this day \autocite*{freuder-1997-holygrail}. Different
|
|
from imperative (and even other declarative) languages, in a constraint
|
|
modelling language the modeller does not describe how to solve the problem, but
|
|
rather provides the problem requirements. You could say that a constraint model
|
|
actually describes the solution to the problem.
|
|
|
|
For example, let us consider the following scenario: Packing for a weekend trip,
|
|
I have to decide which toys to bring for my dog, Audrey. We only have a small
|
|
amount of space left in the car, so we cannot bring all the toys. Since Audrey
|
|
gets enjoys playing with some toys more than others, we can now try and pick the
|
|
toys that bring Audrey the most amount of joy, but still fit in the car.
|
|
|
|
\begin{program}[ht]
|
|
\input{assets/py/2_dyn_knapsack.pytex}
|
|
\caption{\label{prog:dyn-knapsack} A Python program that solves a 0-1 knapsack
|
|
problem using dynamic programming}
|
|
\end{program}
|
|
|
|
A well educated reader in optimisation problems might immediately recognise that
|
|
this is a variation on the widely known \textit{knapsack problem}, more
|
|
specifically a \textit{0-1 knapsack problem}
|
|
\autocite[13--67]{silvano-1990-knapsack}. A commonly used solution to this
|
|
problem is based on dynamic programming. An implementation of this approach is
|
|
shown in \cref{prog:dyn-knapsack}. In a naive recursive approach we would try
|
|
all different combinations of toys to find the combination that will give the
|
|
most joy, but using a dynamic programming approach this exponential behaviour
|
|
(on the number of toys) can be avoided.
|
|
|
|
\begin{model}[ht]
|
|
\input{assets/mzn/2_knapsack.mzntex}
|
|
\caption{\label{model:knapsack} A \minizinc\ model describing a 0-1 knapsack
|
|
problem}
|
|
\end{model}
|
|
|
|
A constraint model offers a different view of the problem. Instead of specifying
|
|
the manner in which we can find the solution, we give a concise description of
|
|
the problem in terms of what we already know, the \glspl{problem-parameter},
|
|
what we wish to know, the \glspl{decision-variable}, and the relationships that
|
|
should exists between them, the \glspl{constraint}. \Cref{model:knapsack} shows
|
|
a \minizinc\ model of the knapsack problem, where the different elements of the
|
|
constraint model are separated. Although a constraint model does not contain any
|
|
instructions to find a suitable solutions, these models can generally be given
|
|
to a dedicated solving program, or \gls{solver} for short, that can find a
|
|
solution that fits the requirements of the model.
|
|
|
|
|
|
\section{Constraint Modelling Basics}
|
|
\label{sec:2-constraint-modelling-basics}
|
|
|
|
\section{Solving Techniques}
|
|
\label{sec:2-solving-techniques}
|
|
|
|
\section{A Comparison of Constraint Modelling Languages}
|
|
\label{sec:2-different-languages}
|
|
|
|
\section{What Makes a ``Good'' Model?}
|
|
\label{sec:2-model-quality}
|