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author | Nguyễn Gia Phong <vn.mcsinyx@gmail.com> | 2020-02-16 14:26:55 +0700 |
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committer | Nguyễn Gia Phong <vn.mcsinyx@gmail.com> | 2020-02-16 14:26:55 +0700 |
commit | 82e6cf7d1046d6cee16f7e8b044ec33e7ec6c4b7 (patch) | |
tree | f7b7ae0bce69070c47a1b31a85bd2bc69dfecf09 /usth/MATH2.2/labwork/5 | |
parent | 29d1001e2e21eff289bff23412e284c8b2e44595 (diff) | |
download | cp-82e6cf7d1046d6cee16f7e8b044ec33e7ec6c4b7.tar.gz |
[usth] Numerical Method is MATH2.4
Diffstat (limited to 'usth/MATH2.2/labwork/5')
-rw-r--r-- | usth/MATH2.2/labwork/5/labwork5.pdf | bin | 32550 -> 0 bytes | |||
-rw-r--r-- | usth/MATH2.2/labwork/5/profit.m | 15 | ||||
-rw-r--r-- | usth/MATH2.2/labwork/5/profit.tikz | 39631 | ||||
-rw-r--r-- | usth/MATH2.2/labwork/5/report.pdf | bin | 288040 -> 0 bytes | |||
-rw-r--r-- | usth/MATH2.2/labwork/5/report.tex | 133 |
5 files changed, 0 insertions, 39779 deletions
diff --git a/usth/MATH2.2/labwork/5/labwork5.pdf b/usth/MATH2.2/labwork/5/labwork5.pdf deleted file mode 100644 index 52512a5..0000000 --- a/usth/MATH2.2/labwork/5/labwork5.pdf +++ /dev/null Binary files differdiff --git a/usth/MATH2.2/labwork/5/profit.m b/usth/MATH2.2/labwork/5/profit.m deleted file mode 100644 index b196205..0000000 --- a/usth/MATH2.2/labwork/5/profit.m +++ /dev/null @@ -1,15 +0,0 @@ -function z = profit (x, y) - A = [7 11; 10 8]; - b = [77; 80]; - c = [150; 175]; - [m n] = size (x); - z = -inf (m, n); - for s = 1 : m - for t = 1 : n - r = [x(s, t); y(s, t)]; - if A * r <= b - z(s, t) = dot (c, r); - endif - endfor - endfor -endfunction diff --git a/usth/MATH2.2/labwork/5/profit.tikz b/usth/MATH2.2/labwork/5/profit.tikz deleted file mode 100644 index e91ea31..0000000 --- a/usth/MATH2.2/labwork/5/profit.tikz +++ /dev/null @@ -1,39631 +0,0 @@ -% Title: gl2ps_renderer figure -% Creator: GL2PS 1.4.0, (C) 1999-2017 C. 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-\usepackage[english,vietnamese]{babel} -\usepackage{amsmath} -\usepackage{booktabs} -\usepackage{enumerate} -\usepackage{lmodern} -\usepackage{siunitx} -\usepackage{tikz} - -\newcommand{\exercise}[1]{\noindent\textbf{#1.}} - -\title{Numerical Methods: Linear Programming} -\author{Nguyễn Gia Phong--BI9-184} -\date{\dateenglish\today} - -\begin{document} -\maketitle - -Given the production contraints and profits of two grades of heating gas -in the table below. -\begin{center} - \begin{tabular}{l c c c} - \toprule - & \multicolumn{2}{c}{Product}\\ - Resource & Regular & Premium & Availability\\ - \midrule - Raw gas (\si{\cubic\metre\per\tonne}) & 7 & 11 & 77\\ - Production time (\si{\hour\per\tonne}) & 10 & 8 & 80\\ - Storage (\si{\tonne}) & 9 & 6\\ - \midrule - Profit (\si{\per\tonne}) & 150 & 175\\ - \bottomrule - \end{tabular} -\end{center} - -\begin{enumerate}[(a)] - \item Let two nonnegative numbers $x_1$ and $x_2$ respectively be - the quantities in tonnes of regular and premium gas to be produced. - The constraints can then be expressed as - \[\begin{cases} - 7x_1 + 11x^2 &\le 77\\ - 10x_1 + 8x_2 &\le 80\\ - x_1 &\le 9\\ - x_2 &\le 6 - \end{cases} - \iff \begin{bmatrix} - 7 & 11\\ - 10 & 8\\ - 1 & 0\\ - 0 & 1 - \end{bmatrix}\begin{bmatrix} - x_1\\ x_2 - \end{bmatrix} - \le \begin{bmatrix} - 77\\ 80\\ 9\\ 6 - \end{bmatrix}\] - - The total profit is the linear function to be maximized: - \[\Pi(x_1, x_2) = 150x_1 + 175x_2 = \begin{bmatrix} - 150\\ 175 - \end{bmatrix}\cdot\begin{bmatrix} - x_1\\ x_2 - \end{bmatrix}\] - - Let $\mathbf x = \begin{bmatrix} - x_1\\ x_2 - \end{bmatrix}$, $A = \begin{bmatrix} - 7 & 11\\ - 10 & 8\\ - 1 & 0\\ - 0 & 1 - \end{bmatrix}$, $\mathbf b = \begin{bmatrix} - 77\\ 80\\ 9\\ 6 - \end{bmatrix}$ and $\mathbf c = \begin{bmatrix} - 150\\ 175 - \end{bmatrix}$, the canonical form of the problem is - \[\max\left\{\mathbf c^\mathrm T - \mid A\mathbf x\le b\land\mathbf x\ge 0\right\}\] - \item Due to the absense of \verb|linprog| in Octave, we instead use GNU GLPK: -\begin{verbatim} -octave> x = glpk (c, A, b, [], [], "UUUU", "CC", -1) -x = - 4.8889 - 3.8889 -\end{verbatim} - Contraint type \verb|UUUU| is used because all contraints are inequalities - with an upper bound and \verb|CC| indicates continous values of $\mathbf x$. - With the sense of -1, GLPK looks for the maximization\footnote{I believe - \emph{minimization} was a typo in the assignment, since it is trivial that - $\Pi$ has the minimum value of 0 at $\mathbf x = \mathbf 0$.} of - $\Pi(4.8889, 3.8889) = 1413.9$. - - The two blank arguments are for the lower and upper bounds of $\mathbf x$, - default to zero and infinite respectively. Alternatively we can use - the following to obtain the same result -\begin{verbatim} -glpk (c, [7 11; 10 8], [77; 80], [], [9 6], "UU", "CC", -1) -\end{verbatim} - - \item Within the constrains, the profit can be calculated using the following - function, which takes two meshes of $x$ and $y$ as arguments -\begin{verbatim} -function z = profit (x, y) - A = [7 11; 10 8]; - b = [77; 80]; - c = [150; 175]; - [m n] = size (x); - z = -inf (m, n); - for s = 1 : m - for t = 1 : n - r = [x(s, t); y(s, t)]; - if A * r <= b - z(s, t) = dot (c, r); - endif - endfor - endfor -endfunction -\end{verbatim} - - Using this, the solution space is then plotted using \verb|ezsurf|, - which color each grid by their relative values (the smallest is dark purple - and the largest is bright yellow): -\begin{verbatim} -ezsurf (@(x1, x2) constraints (x1, x2), [0 9 0 6], 58) -\end{verbatim} -\pagebreak - - Since the plot is just a part of a plane, we can rotate it for a better view - without losing any information about its shape. - - \scalebox{0.69}{\input{profit.tikz}} -\end{enumerate} -\end{document} |