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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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% 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} problem using dynamic programming}
\end{listing} \end{listing}
\begin{example} \begin{example}%
\label{ex:back-knapsack}
Let us consider the following scenario: Packing for a weekend trip, I have to 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 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 \autocite{nethercote-2007-minizinc}. Its expressive language and extensive
library of constraints allow users to easily model complex problems. library of constraints allow users to easily model complex problems.
Let us introduce the language by modelling the well-known \emph{Latin squares} Let us introduce the language by modelling the problem from
problem \autocite{wallis-2011-combinatorics}: Given an integer \(n\), find an \cref{ex:back-knapsack}. A \minizinc\ model encoding this problem is shown in
\(n \times n\) matrix, such that each row and column is a permutation of values \cref{lst:back-mzn-knapsack}.
\(1 \ldots n\). A \minizinc\ model encoding this problem could look as follows:
\begin{listing} \begin{listing}
\mznfile{assets/mzn/2_knapsack.mzn} \mznfile{assets/mzn/back_knapsack.mzn}
\caption{\label{lst:2-mzn-knapsack} A \minizinc\ model describing a 0-1 knapsack \caption{\label{lst:back-mzn-knapsack} A \minizinc\ model describing a 0-1 knapsack
problem} problem}
\end{listing} \end{listing}
\begin{mzn} The model starts with the declaration of the \glspl{parameter}.
int: n; \Lref{line:back-knap-toys} declares an enumerated type that represents all
array [1..n, 1..n] of var 1..n: x; 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) ( The model then declares its \glspl{variable}. \Lref{line:back-knap-sel} declares
all_different([x[r, c] | c in 1..n]) 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
constraint forall (c in 1..n) ( \mzninline{total_joy}, on \lref{line:back-knap-tj}, which is functionally
all_different([x[r, c] | r in 1..n]) defined to be the summation of all the joy for the toy picked in our selection.
);
\end{mzn}
The model introduces a \gls{parameter} \mzninline{n}, and a two-dimensional Finally, the model contains a constraint, on \lref{line:back-knap-con}, to
array of \glspl{variable} (marked by the \mzninline{var} keyword) \mzninline{x}. ensure we do not exceed the given capacity and states the goal for the solver:
Each variable in \mzninline{x} is restricted to the set of integers to maximise the value of the variable \mzninline{total_joy}.
\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 One might note that, although more textual and explicit, the \minizinc\ model
\mzninline{x} variables of all columns must take pairwise different values (and definition is very similar to our earlier mathematical definition.
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.
Given ground assignments to input \glspl{parameter}, a \minizinc\ model is Given ground assignments to input \glspl{parameter}, a \minizinc\ model is
translated (via a process called \emph{flattening}) into a set of variables and 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} \begin{mzn}
var 1..2: x_1_1; var 1..2: x_1_1;