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Update MiniZinc background example

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Jip J. Dekker 2021-04-12 15:44:27 +10:00
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15
assets/mzn/2_knapsack.mzn
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@ -1,15 +0,0 @@
% Problem parameters
enum TOYS = {football, tennisball, stuffed_lama, stuffed_elephant};
array[TOYS] of int: toy_joy = [63, 12, 50, 100];
array[TOYS] of int: toy_space = [32, 8, 16, 40];
int: space_left = 64;
% Decision variables
var set of TOYS: selection;
var int: total_joy = sum(toy in selection)(toy_joy[toy]);
% Constraints
constraint sum(toy in selection)(toy_space[toy]) < space_left;
% Goal
solve maximize total_joy;

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assets/mzn/back_knapsack.mzn Normal file
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@ -0,0 +1,15 @@
% Problem parameters
enum TOYS;@\Vlabel{line:back-knap-toys}@
array[TOYS] of int: toy_joy;@\Vlabel{line:back-knap-joy}@
array[TOYS] of int: toy_space;@\Vlabel{line:back-knap-space}@
int: space_left;@\Vlabel{line:back-knap-left}@
% Decision variables
var set of TOYS: selection;@\Vlabel{line:back-knap-sel}@
var int: total_joy = sum(toy in selection)(toy_joy[toy]);@\Vlabel{line:back-knap-tj}@
% Constraints
constraint sum(toy in selection)(toy_space[toy]) < space_left;@\Vlabel{line:back-knap-con}@
% Goal
solve maximize total_joy;@\Vlabel{line:back-knap-obj}@

64
chapters/2_background.tex
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@ -40,7 +40,8 @@ the model.
problem using dynamic programming}
\end{listing}
\begin{example}
\begin{example}%
\label{ex:back-knapsack}
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
@ -118,42 +119,51 @@ discrete satisfiability and optimisation problems
\autocite{nethercote-2007-minizinc}. Its expressive language and extensive
library of constraints allow users to easily model complex problems.
Let us introduce the language by modelling the well-known \emph{Latin squares}
problem \autocite{wallis-2011-combinatorics}: Given an integer \(n\), find an
\(n \times n\) matrix, such that each row and column is a permutation of values
\(1 \ldots n\). A \minizinc\ model encoding this problem could look as follows:
Let us introduce the language by modelling the problem from
\cref{ex:back-knapsack}. A \minizinc\ model encoding this problem is shown in
\cref{lst:back-mzn-knapsack}.
\begin{listing}
\mznfile{assets/mzn/2_knapsack.mzn}
\caption{\label{lst:2-mzn-knapsack} A \minizinc\ model describing a 0-1 knapsack
\mznfile{assets/mzn/back_knapsack.mzn}
\caption{\label{lst:back-mzn-knapsack} A \minizinc\ model describing a 0-1 knapsack
problem}
\end{listing}
\begin{mzn}
int: n;
array [1..n, 1..n] of var 1..n: x;
The model starts with the declaration of the \glspl{parameter}.
\Lref{line:back-knap-toys} declares an enumerated type that represents all
possible toys, \(T\) in the mathematical model in the example.
\Lref{line:back-knap-joy,line:back-knap-space} declare arrays mapping from toys
to integer values, these represent the functional mappings \(joy\) and
\(space\). Finally, \lref{line:back-knap-left} declares an integer
\gls{parameter} to represent the car capacity as an equivalent to \(C\).
constraint forall (r in 1..n) (
all_different([x[r, c] | c in 1..n])
);
constraint forall (c in 1..n) (
all_different([x[r, c] | r in 1..n])
);
\end{mzn}
The model then declares its \glspl{variable}. \Lref{line:back-knap-sel} declares
the main \gls{variable} \mzninline{selection}, which represents the selection of
toys to be packed. \(S\) in our earlier model. We also declare the variable
\mzninline{total_joy}, on \lref{line:back-knap-tj}, which is functionally
defined to be the summation of all the joy for the toy picked in our selection.
The model introduces a \gls{parameter} \mzninline{n}, and a two-dimensional
array of \glspl{variable} (marked by the \mzninline{var} keyword) \mzninline{x}.
Each variable in \mzninline{x} is restricted to the set of integers
\mzninline{1..n}, which is called the variable's \gls{domain}. The constraints
specify the requirements of the problem: for each row \mzninline{r}, the
\mzninline{x} variables of all columns must take pairwise different values (and
the same for each column \mzninline{c}). This is modelled using the
\mzninline{all_different} function, one of hundreds of pre-defined constraints
in \minizinc's library.
Finally, the model contains a constraint, on \lref{line:back-knap-con}, to
ensure we do not exceed the given capacity and states the goal for the solver:
to maximise the value of the variable \mzninline{total_joy}.
One might note that, although more textual and explicit, the \minizinc\ model
definition is very similar to our earlier mathematical definition.
Given ground assignments to input \glspl{parameter}, a \minizinc\ model is
translated (via a process called \emph{flattening}) into a set of variables and
primitive constraints. Here is the result of flattening for \mzninline{n=2}:
primitive constraints.
Given the assignments
\begin{mzn}
TOYS = {football, tennisball, stuffed_lama, stuffed_elephant};
toy_joy = [63, 12, 50, 100];
toy_space = [32, 8, 16, 40];
space_left = 64;
\end{mzn}
is the result of flattening for \mzninline{n=2}:
\begin{mzn}
var 1..2: x_1_1;