From 594d460e4068a28a70434429a3fa3b4497ef9287 Mon Sep 17 00:00:00 2001 From: Mart Lubbers Date: Tue, 20 Oct 2015 10:39:34 +0200 Subject: [PATCH] ar ex1 done --- ar/assignments/1.tex | 86 +++++++++++++++++++++++---------------- ar/assignments/ar.aux | 25 ------------ ar/assignments/ar.fmt | Bin 4164598 -> 0 bytes ar/assignments/ar.log | 77 ----------------------------------- ar/assignments/ar.pdf | Bin 65468 -> 0 bytes ar/assignments/src/1.smt | 20 ++++----- 6 files changed, 62 insertions(+), 146 deletions(-) delete mode 100644 ar/assignments/ar.aux delete mode 100644 ar/assignments/ar.fmt delete mode 100644 ar/assignments/ar.log delete mode 100644 ar/assignments/ar.pdf diff --git a/ar/assignments/1.tex b/ar/assignments/1.tex index f45e8b4..51fc39e 100644 --- a/ar/assignments/1.tex +++ b/ar/assignments/1.tex @@ -19,43 +19,61 @@ Investigate what is the maximum number of pallets of prittles that can be delivered, and show how for that number all pallets may be divided over the six trucks. } - +\newpage \subsection{Formal definition} -For every truck $t_i$ for $i\in[1\ldots6]$ in combination with every nuzzle -$n$, prittle $p$, skipple $s$, crottle $c$ and dupple $d$ we declare a variable -that holds the amount of that type of building block. For truck 1 we thus have -the variables: $t_1n, t_1p, t_1s, t_1c, t_1d$. - -To lay a contraint on the weight we declare for every truck the following rule. -$$\bigwedge^{T}_{i=1}\left(t_in*700+t_ip*800+t_is*1000+t_ic*1500+t_id*100<7800\right)$$ - -To limit the maximum number of pallets in a truck we define. -$$\bigwedge^{T}_{i=1}\left(\left(\sum_{p\in\{n,p,s,c,d\}}t_ip\right)\leq8\right)$$ - -To limit the minimum number of pallets in a truck we define. -$$\bigwedge^{T}_{i=1}\bigwedge_{p\in\{n,p,s,c,d\}}t_ip>0$$ - -To describe the number of pallets available we define for every type of pallet -a variable that describes the number available if there is a limited number -available. -$$num_n=4\wedge num_s=8\wedge num_c=10\wedge num_d=5$$ +For every truck $t_i$ for $i\in T$ where $T=[1\ldots6]$ in combination with +every nuzzle $n$, prittle $p$, skipple $s$, crottle $c$ and dupple $d$ we +declare a variable that holds the amount of that type of building block. +We group all pallets $r$ for $r\in P$ where $P=[n,p,s,c,d]$. We also define +function $w(r)$ that defines the weight of the pallet and $n(r)$ that defines +the number of pallets needed. Then the following combination of subformulas +describes the problem. -To be sure the materials are all delivered we define. -$$\bigwedge_{p\in\{n,s,c,d\}}\left(\left(\sum^{T}_{i=1}t_ip\right)=num_p\right)$$ +\begin{align} + \bigwedge^{T}_{i=1}\Biggl( + & \bigwedge_{r\in P}t_ir\geq0\\ + & \wedge \sum_{r\in P}t_ir\leq8\\ + & \wedge \left(\sum_{r\in P}t_ir*w(r)\right)\leq7800)\\ + & \wedge (t_ip=0\vee t_ic=0)\\ + & \wedge t_id\leq1\\ + & \wedge (t_i=t_1 \vee t_i=t_2 \vee t_is=0)\Biggr)\wedge\\ + \bigwedge_{r\in P}\Biggl( + & r=p\vee\left(\sum^{T}_{i=1}t_ir\right)=n(r)\Biggr) +\end{align} -The first constraint is that prittles and crottles can not be in the same -truck. This is easily described with. -$$\bigwedge^{T}_{i=1}\left(t_ip=0\vee t_ic=0\right)$$ - -Skipples need to be cooled and only two trucks have cooling. Therefore we -specify. -$$\bigwedge^{T}_{i=3}t_is=0$$ - -Dupples can only be in a truck with a maximum of two. -$$\bigwedge^{T}_{i=1}t_id\leq1$$ - -We can tie this together by putting $\wedge$ symbols between all of the -formulas. +All the subformulas describe a constraint from the problem description and we +can separate two groups. Subformulas going over all trucks and subformulas over +all pallets. +\begin{itemize} + \item \textbf{Trucks} + \begin{enumerate} + \item Makes sure you can not have a negative number of pallets. + by stating that all pallets numbers per truck are + bigger then $0$. + \item Makes sure you can not have more then $8$ pallets in a + truck by stating that the sum of all pallets in a truck + is not bigger then $8$. + \item Makes sure the weight will never go over the maximum by + stating that all the pallets times their weight in a + truck may not go over $7800$kg. + \item Makes sure prittles and crottles are never together in a + truck by stating that either the number of prittles is + zero or the number of crottles is zero. Note that they + therefore also can both be zero. + \item Makes sure that skipples are only in the first two cooled + trucks by stating that if the truck number not one or + two the number of skipples is zero. + \end{enumerate} + \item \textbf{Pallets} + \begin{enumerate} + \setcounter{enumi}{6} + \item Makes sure that all mandatory pallets are transported by + summing over all trucks and assuring the number of the + pallets of that type is exactly $n(r)$. Note that this + should hold or that the current pallet is a prittle + pallet since we do not have a limit on prittles. + \end{enumerate} +\end{itemize} \subsection{SMT format solution} The formula is easily convertable to SMT format and is listed in diff --git a/ar/assignments/ar.aux b/ar/assignments/ar.aux deleted file mode 100644 index f75a08e..0000000 --- a/ar/assignments/ar.aux +++ /dev/null @@ -1,25 +0,0 @@ -\relax -\@writefile{toc}{\contentsline {section}{\numberline {1}Problem 1}{1}} -\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Formal definition}{1}} -\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}SMT format solution}{2}} -\newlabel{listing:1bash}{{1}{2}} -\@writefile{lol}{\contentsline {lstlisting}{\numberline {1}Iteratively find the largest solution}{2}} -\@writefile{toc}{\contentsline {subsection}{\numberline {1.3}Solution}{2}} -\@writefile{toc}{\contentsline {section}{\numberline {2}Problem 4}{3}} -\@writefile{toc}{\contentsline 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