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Finished mom project 1

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Krzysztof R. 2023-11-19 13:54:14 +01:00
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141
NotProgramming/MOM/.report_one.tex.swp
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\documentclass[12pt]{article}
\usepackage[polish]{babel}
\usepackage{float}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{pdfpages}
\usepackage[T1]{fontenc} % Add this line
\title{Sprawozdanie z projektu MOM}
\author{Krzysztof Rudnicki, 307585}
\date{\today}
\begin{document}
\maketitle
\section{Zadanie 1. Sieć przepływowa}
\subsection{Model sieciowy}
\paragraph{Problem do rozwiązania}
Należy rozwiązać problem najtańszego przepływu \\
\begin{figure}[htb]
\caption{Koszty transportowe i przepustowości na poszczególnych odcinkach}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& D & E & F & G & H \\
\hline
A & 4, 15 & 2, 10 & - & - & -\\
\hline
B & 4, 4 & 3, 9 & - & 8, 9 & - \\
\hline
C & 2, 20 & 6, 10 & - & - & - \\
\hline
D & - & - & 3, 10 & 7, 3 & 2, 2 \\
\hline
E & 5, 20 & - & 7, 5 & 6, 5 & 3, 5 \\
\hline
\end{tabular}
\end{center}
\end{figure}
\begin{figure}[htb]
\caption{Zdolności wydobywcze kopalń (w tys. ton)}
\begin{center}
\begin{tabular}{ | c | c |}
\hline
$W_A$ & 10 \\
\hline
$W_B$ & 13 \\
\hline
$W_C$ & 17 \\
\hline
\end{tabular}
\end{center}
\end{figure}
\begin{figure}[htb]
\caption{Średnie zużycie dobowe węgla przez elektronie (w tys. ton)}
\begin{center}
\begin{tabular}{ | c | c |}
\hline
$Z_F$ & 15 \\
\hline
$Z_G$ & 13 \\
\hline
$Z_H$ & 7 \\
\hline
\end{tabular}
\end{center}
\end{figure}
\paragraph{Sformułowanie}
Matematyczne sformułowanie zadania:
\begin{enumerate}
\item Parametry \\
s - źródło \\
B - Budynki (kopalnie, stacje, elektrownie) \\
K - kopalnie \\
$c_{ij}$ - koszt transportu z budynku i do budynku j ($c_{sk}$ = 0) \\
$W_k$ - zdolność wydobywcza kopalni k [w tonach] \\
$p_{ij}$ - przepustowość transportu z budynku i do budynku j [w tonach] ($u_{sk}$ = $W_k$) \\
E - elektrownie \\
$Z_e$ - średnie zużycie dobowe przez elektrownię e [w tonach] \\
\item Zmienne decyzyjne
$f_{ij}$ - przepływ transportu z budynku i do obiektu j \\
\item Funkcja celu
\begin{align*}
Q &= \min\left(\sum_{(i, j) \in B \backslash \{s\}} c_{ij} f_{ij}\right) \\
&= \min(4f_{AD} + 2f_{AE} + 4f_{BD} + 3f_{BE} + 8f_{BG} + 2f_{CD} \\
&\quad + 6f_{CE} + 3f_{DF} + 7f_{DG} + 2f_{DH} + 5f_{ED} + 7f_{EF} \\
&\quad + 6f_{EG} + 3f_{EH})
\end{align*}
\item Ograniczenia
\begin{equation}
0 \leq f_{sk} \leq W_k, \; k \in K = \{A, B, C\}
\end{equation}
\begin{equation}
f_{si} = 0, \; i \in B \backslash K
\end{equation}
\begin{equation}
f_{is} = 0, \; i \in B
\end{equation}
\begin{equation}
0 \leq f_{ij} \leq p_{uj}, \; (i, j) \in B \backslash \{s\}
\end{equation}
\begin{equation}
f_{is} = 0, \; i \in B
\end{equation}
\begin{equation}
\sum_{b \in B} f_{be} \ge Z_e, \; e \in E = \{F, G, H\}
\end{equation}
\begin{equation}
\sum_{i \in B \backslash \{ s \} } f_{bi} \leq \sum_{j \in B} f_{bn}, \; b \in B \backslash \{s\}
\end{equation}
\end{enumerate}
\paragraph{Narysowanie modelu}
\includepdf[pages=-]{1flow.pdf}
\subsection{Rozwiązanie}Z
\paragraph{Metoda }
\subsection{Zadanie programowania liniowego}
\section{Zadanie 2. Zadanie przydziału}
\subsection{Zadanie 2.1 Planowanie realizacji portfela przy ograniczonych kompetencjach}
\paragraph{Model sieciowy rysunek}
\subsubsection{Problem do rozwiązania}
\paragraph{Rozwiazanie}
\paragraph{Przydział zespołów do projektów}
\subsection{Zadanie 2.2 Minimalizacja kosztów realizacji projektów}
\paragraph{Model sieciowy rysunek}
\paragraph{Problem do rozwiązania}
\paragraph{Rozwiazanie}
\subsection{Zadanie 2.3 Minimalizacja terminu realizacji puli projektów}
\paragraph{Model programowania liniowego}
\section{Zadanie 3}
\paragraph{Model programowania liniowego}
\end{document}

18
NotProgramming/MOM/12glpk.dat Normal file
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data;
set BudynkiNaStart := s, A, B, C, D, E, F, G, H;
set Budynki := A, B, C, D, E, F, G, H;
set Elektrownie := F, G, H;
param p :=
s A 10 s B 13 s C 17 s D 0 s E 0 s F 0 s G 0 s H 0
A A 0 A B 0 A C 0 A D 15 A E 10 A F 0 A G 0 A H 0
B A 0 B B 0 B C 0 B D 4 B E 9 B F 0 B G 9 B H 0
C A 0 C B 0 C C 0 C D 20 C E 10 C F 0 C G 0 C H 0
D A 0 D B 0 D C 0 D D 0 D E 0 D F 10 D G 3 D H 2
E A 0 E B 0 E C 0 E D 20 E E 0 E F 5 E G 5 E H 5
F A 0 F B 0 F C 0 F D 0 F E 0 F F 0 F G 0 F H 0
G A 0 G B 0 G C 0 G D 0 G E 0 G F 0 G G 0 G H 0
H A 0 H B 0 H C 0 H D 0 H E 0 H F 0 H G 0 H H 0;
end;

21
NotProgramming/MOM/12glpk.mod Normal file
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set BudynkiNaStart;
set Budynki;
set Elektrownie;
param p{i in BudynkiNaStart, j in Budynki};
var f{i in BudynkiNaStart, j in Budynki}, >= 0;
maximize Q: sum {i in Budynki, e in Elektrownie} f[i,e];
subject to
Ogr_1{i in BudynkiNaStart, j in Budynki}:
f[i,j] >= 0;
Ogr_2{i in BudynkiNaStart, j in Budynki}:
f[i,j] <= p[i,j];
Ogr_3{n in Budynki}:
sum {i in Budynki} f[n,i] <= sum {j in BudynkiNaStart} f[j,n];
solve;
display {i in Budynki, j in Budynki: f[i,j] > 0}: f[i,j];
display: Q;

13
NotProgramming/MOM/1glpk.dat
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@ -4,17 +4,6 @@ set BudynkiNaStart := s, A, B, C, D, E, F, G, H;
set Budynki := A, B, C, D, E, F, G, H;
set Elektrownie := F, G, H;
param c :=
s A 0 s B 0 s C 0 s D 0 s E 0 s F 0 s G 0 s H 0
A A 0 A B 0 A C 0 A D 4 A E 2 A F 0 A G 0 A H 0
B A 0 B B 0 B C 0 B D 4 B E 3 B F 0 B G 8 B H 0
C A 0 C B 0 C C 0 C D 2 C E 6 C F 0 C G 0 C H 0
D A 0 D B 0 D C 0 D D 0 D E 0 D F 3 D G 7 D H 2
E A 0 E B 0 E C 0 E D 5 E E 0 E F 7 E G 6 E H 3
F A 0 F B 0 F C 0 F D 0 F E 0 F F 0 F G 0 F H 0
G A 0 G B 0 G C 0 G D 0 G E 0 G F 0 G G 0 G H 0
H A 0 H B 0 H C 0 H D 0 H E 0 H F 0 H G 0 H H 0;
param p :=
s A 10 s B 13 s C 17 s D 0 s E 0 s F 0 s G 0 s H 0
A A 0 A B 0 A C 0 A D 15 A E 10 A F 0 A G 0 A H 0
@ -26,6 +15,4 @@ F A 0 F B 0 F C 0 F D 0 F E 0 F F 0 F G 0 F H 0
G A 0 G B 0 G C 0 G D 0 G E 0 G F 0 G G 0 G H 0
H A 0 H B 0 H C 0 H D 0 H E 0 H F 0 H G 0 H H 0;
param Z := F 15 G 13 H 7;
end;

13
NotProgramming/MOM/1glpk.mod
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@ -2,23 +2,20 @@ set BudynkiNaStart;
set Budynki;
set Elektrownie;
param c{i in BudynkiNaStart, j in Budynki};
param p{i in BudynkiNaStart, j in Budynki};
param Z{e in Elektrownie};
var f{i in BudynkiNaStart, j in Budynki}, >= 0;
minimize Q: sum {i in Budynki, j in Budynki} c[i,j] * f[i,j];
maximize Q: sum {i in Budynki, e in Elektrownie} f[i,e];
subject to
Ograniczenie_1{i in BudynkiNaStart, j in Budynki}:
Ogr_1{i in BudynkiNaStart, j in Budynki}:
f[i,j] >= 0;
Ograniczenie_2{i in BudynkiNaStart, j in Budynki}:
Ogr_2{i in BudynkiNaStart, j in Budynki}:
f[i,j] <= p[i,j];
Ograniczenie_3{e in Elektrownie}:
sum {n in BudynkiNaStart} f[n,e] >= Z[e];
Ograniczenie_4{n in Budynki}:
Ogr_3{n in Budynki}:
sum {i in Budynki} f[n,i] <= sum {j in BudynkiNaStart} f[j,n];
solve;
display {i in Budynki, j in Budynki: f[i,j] > 0}: f[i,j];
display: Q;

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NotProgramming/MOM/21glpk.dat Normal file
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data;
set Wyjscie := t;
set Wezly := 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set StartWezly := s, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set WyjscieWezly := t, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set StartWyjscieWezly := s, t, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set Projekty := 1, 2, 3, 4, 5, 6;
set Zespoly := A, B, C, D, E, F;
param u :=
A 1 0 A 2 0 A 3 1 A 4 1 A 5 0 A 6 1
B 1 1 B 2 1 B 3 0 B 4 0 B 5 1 B 6 0
C 1 1 C 2 0 C 3 1 C 4 0 C 5 0 C 6 1
D 1 1 D 2 0 D 3 1 D 4 0 D 5 0 D 6 1
E 1 1 E 2 1 E 3 0 E 4 0 E 5 1 E 6 0
F 1 0 F 2 0 F 3 1 F 4 1 F 5 1 F 6 0;
end;

29
NotProgramming/MOM/21glpk.mod Normal file
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set Wyjscie;
set Wezly;
set StartWezly;
set WyjscieWezly;
set StartWyjscieWezly;
set Projekty;
set Zespoly;
param u{z in Zespoly, p in Projekty};
var f{i in StartWyjscieWezly, j in StartWyjscieWezly}, >= 0;
maximize Q: sum {p in Projekty, t in Wyjscie} f[p,t];
Ograniczeni_jeden{i in StartWyjscieWezly, j in StartWyjscieWezly}:
f[i,j] >= 0;
Ograniczeni_dwa{i in StartWyjscieWezly, j in StartWyjscieWezly}:
f[i,j] <= 1;
Ograniczeni_trzy{p in Projekty, z in Zespoly}:
f[z,p] <= u[z,p];
Ograniczeni_cztery{p in Projekty}:
sum {z in Zespoly} f[z,p] = 1;
Ograniczeni_piec{z in Zespoly}:
sum {p in Projekty} f[z,p] = 1;
Ograniczeni_szesc{n in Wezly}:
sum {i in WyjscieWezly} f[n,i] = sum {j in StartWezly} f[j,n];
solve;
display {i in Wezly, j in Wezly: f[i,j] > 0}: f[i,j];
display: Q;

19
NotProgramming/MOM/22glpk.dat Normal file
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data;
set Wyjscie := t;
set Wezly := 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set StartWezly := s, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set WyjscieWezly := t, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set StartWyjscieWezly := s, t, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set Projekty := 1, 2, 3, 4, 5, 6;
set Zespoly := A, B, C, D, E, F;
param k :=
A 1 1000 A 2 1000 A 3 2 A 4 10 A 5 1000 A 6 20
B 1 13 B 2 4 B 3 1000 B 4 1000 B 5 10 B 6 1000
C 1 1000 C 2 2 C 3 1000 C 4 5 C 5 1000 C 6 6
D 1 14 D 2 1000 D 3 10 D 4 1000 D 5 1000 D 6 16
E 1 18 E 2 3 E 3 1000 E 4 1000 E 5 17 E 6 1000
F 1 1000 F 2 1000 F 3 13 F 4 15 F 5 12 F 6 1000;
end;

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NotProgramming/MOM/22glpk.mod Normal file
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set Wyjscie;
set Wezly;
set StartWezly;
set WyjscieWezly;
set StartWyjscieWezly;
set Projekty;
set Zespoly;
param k{z in Zespoly, p in Projekty};
var f{i in StartWyjscieWezly, j in StartWyjscieWezly}, >= 0;
minimize Q: sum {p in Projekty, z in Zespoly} k[z, p] * f[z, p];
Ograniczenie_jeden{i in StartWyjscieWezly, j in StartWyjscieWezly}:
f[i,j] >= 0;
Ograniczenie_dwa{i in StartWyjscieWezly, j in StartWyjscieWezly}:
f[i,j] <= 1;
Ograniczenie_trzy{p in Projekty}:
sum {z in Zespoly} f[z,p] = 1;
Ograniczenie_cztery{z in Zespoly}:
sum {p in Projekty} f[z,p] = 1;
Ograniczenie_piec{n in Wezly}:
sum {i in WyjscieWezly} f[n,i] = sum {j in StartWezly} f[j,n];
solve;
display {i in Wezly, j in Wezly: f[i,j] > 0}: f[i,j];
display: Q;

19
NotProgramming/MOM/23glpk.dat Normal file
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data;
set Wyjscie := t;
set Wezly := 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set StartWezly := s, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set WyjscieWezly := t, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set StartWyjscieWezly := s, t, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set Projekty := 1, 2, 3, 4, 5, 6;
set Zespoly := A, B, C, D, E, F;
param c :=
A 1 1000 A 2 1000 A 3 2 A 4 10 A 5 1000 A 6 20
B 1 13 B 2 4 B 3 1000 B 4 1000 B 5 10 B 6 1000
C 1 1000 C 2 2 C 3 1000 C 4 5 C 5 1000 C 6 6
D 1 14 D 2 1000 D 3 10 D 4 1000 D 5 1000 D 6 16
E 1 18 E 2 3 E 3 1000 E 4 1000 E 5 17 E 6 1000
F 1 1000 F 2 1000 F 3 13 F 4 15 F 5 12 F 6 1000;
end;

31
NotProgramming/MOM/23glpk.mod Normal file
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set Wyjscie;
set Wezly;
set StartWezly;
set WyjscieWezly;
set StartWyjscieWezly;
set Projekty;
set Zespoly;
param c{z in Zespoly, p in Projekty};
var f{i in StartWyjscieWezly, j in StartWyjscieWezly}, >= 0, integer;
var cmax, >= 0;
minimize Q: cmax;
subject to
Ograniczenie_jeden{i in StartWyjscieWezly, j in StartWyjscieWezly}:
f[i,j] >= 0;
Ograniczenie_dwa{i in StartWyjscieWezly, j in StartWyjscieWezly}:
f[i,j] <= 1;
Ograniczenie_trzy{p in Projekty}:
sum {z in Zespoly} f[z,p] = 1;
Ograniczenie_cztery{z in Zespoly}:
sum {p in Projekty} f[z,p] = 1;
Ograniczenie_piec{n in Wezly}:
sum {i in WyjscieWezly} f[n,i] = sum {j in StartWezly} f[j,n];
Ograniczenie_szesc{z in Zespoly, p in Projekty}:
c[z,p] * f[z,p] <= cmax;
solve;
display {i in Wezly, j in Wezly: f[i,j] > 0}: f[i,j];
display: Q;

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\documentclass[12pt]{article}
\usepackage[polish]{babel}
\usepackage{float}
\usepackage{listings}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{pdfpages}
@ -113,29 +114,691 @@ $f_{ij}$ - przepływ transportu z budynku i do obiektu j \\
\paragraph{Narysowanie modelu}
\includepdf[pages=-]{1flow.pdf}
\subsection{Rozwiązanie}Z
\paragraph{Metoda }
\subsection{Rozwiązanie}
\paragraph{Zadanie programowania liniowego}
\begin{lstlisting}[caption= plik dat]
data;
set BudynkiNaStart := s, A, B, C, D, E, F, G, H;
set Budynki := A, B, C, D, E, F, G, H;
set Elektrownie := F, G, H;
param Z := F 15 G 13 H 7;
param p :=
s A 10 s B 13 s C 17 s D 0 s E 0 s F 0 s G 0 s H 0
A A 0 A B 0 A C 0 A D 15 A E 10 A F 0 A G 0 A H 0
B A 0 B B 0 B C 0 B D 4 B E 9 B F 0 B G 9 B H 0
C A 0 C B 0 C C 0 C D 20 C E 10 C F 0 C G 0 C H 0
D A 0 D B 0 D C 0 D D 0 D E 0 D F 10 D G 3 D H 2
E A 0 E B 0 E C 0 E D 20 E E 0 E F 5 E G 5 E H 5
F A 0 F B 0 F C 0 F D 0 F E 0 F F 0 F G 0 F H 0
G A 0 G B 0 G C 0 G D 0 G E 0 G F 0 G G 0 G H 0
H A 0 H B 0 H C 0 H D 0 H E 0 H F 0 H G 0 H H 0;
param c :=
s A 0 s B 0 s C 0 s D 0 s E 0 s F 0 s G 0 s H 0
A A 0 A B 0 A C 0 A D 4 A E 2 A F 0 A G 0 A H 0
B A 0 B B 0 B C 0 B D 4 B E 3 B F 0 B G 8 B H 0
C A 0 C B 0 C C 0 C D 2 C E 6 C F 0 C G 0 C H 0
D A 0 D B 0 D C 0 D D 0 D E 0 D F 3 D G 7 D H 2
E A 0 E B 0 E C 0 E D 5 E E 0 E F 7 E G 6 E H 3
F A 0 F B 0 F C 0 F D 0 F E 0 F F 0 F G 0 F H 0
G A 0 G B 0 G C 0 G D 0 G E 0 G F 0 G G 0 G H 0
H A 0 H B 0 H C 0 H D 0 H E 0 H F 0 H G 0 H H 0;
end;
\end{lstlisting}
\begin{lstlisting}[caption= plik mod]
set BudynkiNaStart;
set Budynki;
set Elektrownie;
param Z{e in Elektrownie};
param p{i in BudynkiNaStart, j in Budynki};
param c{i in BudynkiNaStart, j in Budynki};
var f{i in BudynkiNaStart, j in Budynki}, >= 0;
minimize Q: sum {i in Budynki, j in Budynki} c[i,j] * f[i,j];
subject to
Ograniczenie_jeden{i in BudynkiNaStart, j in Budynki}:
f[i,j] >= 0;
Ograniczenie_dwa{i in BudynkiNaStart, j in Budynki}:
f[i,j] <= p[i,j];
Ograniczenie_trzy{e in Elektrownie}:
sum {n in BudynkiNaStart} f[n,e] >= Z[e];
Ograniczenie_cztery{n in Budynki}:
sum {i in Budynki} f[n,i] <= sum {j in BudynkiNaStart} f[j,n];
solve;
display {i in Budynki, j in Budynki: f[i,j] > 0}: f[i,j];
display: Q;
\end{lstlisting}
\paragraph{Wynik}
\begin{lstlisting}[caption=Wynik z glpk]
f[A,E].val = 10
f[B,E].val = 4
f[B,G].val = 9
f[C,D].val = 12
f[D,F].val = 10
f[D,H].val = 2
f[E,F].val = 5
f[E,G].val = 4
f[E,H].val = 5
Q.val = 236
\end{lstlisting}
\begin{figure}[htb]
\caption{Plan dostaw węgla}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& D & E & F & G & H \\
\hline
A & 10 & - & - & - & -\\
\hline
B & 0 & 4 & - & 9 & - \\
\hline
C & 12 & - & - & - & - \\
\hline
D & - & - & 10 & - & 2 \\
\hline
E & - & - & 5 & 4 & 5 \\
\hline
\end{tabular}
\end{center}
\end{figure}
\[ Q = 236 \]
\subsection{Wąskie gardło}
Następny szukamy przekroju o jak najmniejszej przepustowości \\
W tym celu usunęliśmy parametry dotyczące średniego zużycia dobowego oraz kosztów transportu i zostawiliśmy wyłącznie przepustowość
\begin{lstlisting}[caption= plik dat]
data;
set BudynkiNaStart := s, A, B, C, D, E, F, G, H;
set Budynki := A, B, C, D, E, F, G, H;
set Elektrownie := F, G, H;
param p :=
s A 10 s B 13 s C 17 s D 0 s E 0 s F 0 s G 0 s H 0
A A 0 A B 0 A C 0 A D 15 A E 10 A F 0 A G 0 A H 0
B A 0 B B 0 B C 0 B D 4 B E 9 B F 0 B G 9 B H 0
C A 0 C B 0 C C 0 C D 20 C E 10 C F 0 C G 0 C H 0
D A 0 D B 0 D C 0 D D 0 D E 0 D F 10 D G 3 D H 2
E A 0 E B 0 E C 0 E D 20 E E 0 E F 5 E G 5 E H 5
F A 0 F B 0 F C 0 F D 0 F E 0 F F 0 F G 0 F H 0
G A 0 G B 0 G C 0 G D 0 G E 0 G F 0 G G 0 G H 0
H A 0 H B 0 H C 0 H D 0 H E 0 H F 0 H G 0 H H 0;
end;
\end{lstlisting}
\begin{lstlisting}[caption= plik mod]
set BudynkiNaStart;
set Budynki;
set Elektrownie;
param p{i in BudynkiNaStart, j in Budynki};
var f{i in BudynkiNaStart, j in Budynki}, >= 0;
maximize Q: sum {i in Budynki, e in Elektrownie} f[i,e];
subject to
Ograniczenie_jeden{i in BudynkiNaStart, j in Budynki}:
f[i,j] >= 0;
Ograniczenie_dwa{i in BudynkiNaStart, j in Budynki}:
f[i,j] <= p[i,j];
Ograniczenie_trzy{n in Budynki}:
sum {i in Budynki} f[n,i] <= sum {j in BudynkiNaStart} f[j,n];
solve;
display {i in Budynki, j in Budynki: f[i,j] > 0}: f[i,j];
display: Q;
\end{lstlisting}
\paragraph{Wyniki}
\begin{lstlisting}[caption=Wynik z glpk]
f[A,D].val = 9
f[A,E].val = 1
f[B,E].val = 4
f[B,G].val = 9
f[C,D].val = 6
f[C,E].val = 10
f[D,F].val = 10
f[D,G].val = 3
f[D,H].val = 2
f[E,F].val = 5
f[E,G].val = 5
f[E,H].val = 5
Q.val = 39
\end{lstlisting}
\begin{figure}[htb]
\caption{Wyniki poszukiwań wąskiego gardła}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& D & E & F & G & H \\
\hline
A & 9 & 1 & - & - & -\\
\hline
B & - & 4 & - & 9 & - \\
\hline
C & 6 & 10 & - & - & - \\
\hline
D & - & - & 10 & 3 & 2 \\
\hline
E & - & - & 5 & 5 & 5 \\
\hline
\end{tabular}
\end{center}
\end{figure}
\[ Q = 39 \]
\includepdf[pages=-]{flow12.pdf}
\paragraph{Wnioski}
Następujące przepływy są zbyt niskie w porównaniu do przepustowości:
\[ p_{AD} = 9 \le 15 \]
\[ p_{AE} = 1 \le 10 \]
\[ p_{BE} = 4 \le 9 \]
\[ p_{CD} = 6 \le 20 \]
Zwiększenie przepływów do poziomu maksymalnej przepustowości zwiększyło by wydajność sieci
\subsection{Zadanie programowania liniowego}
\section{Zadanie 2. Zadanie przydziału}
\subsection{Zadanie 2.1 Planowanie realizacji portfela przy ograniczonych kompetencjach}
\paragraph{Model sieciowy rysunek}
\begin{figure}[htb]
\caption{Kompetencje zespołów}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
& A & B & C & D & E & F \\
\hline
1 & - & X & - & X & X & - \\
\hline
2 & - & X & X & - & X & - \\
\hline
3 & X & - & - & X & - & X \\
\hline
4 & X & - & X & - & - & X \\
\hline
5 & - & X & - & - & X & X \\
\hline
6 & X & - & X & X & - & - \\
\hline
\end{tabular}
\end{center}
\end{figure}
\subsubsection{Problem do rozwiązania}
\paragraph{Rozwiazanie}
\paragraph{Przydział zespołów do projektów}
\subsection{Zadanie 2.2 Minimalizacja kosztów realizacji projektów}
\paragraph{Sformułowanie}
Matematyczne sformułowanie zadania:
\begin{enumerate}
\item Parametry \\
s - źródło \\
w - wyjście \\
N - węzły (N = {s, t, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F})
P - projekty (P = {1, 2, 3, 4, 5, 6} $\subseteq$ N)
Z - zespoły (Z = {A, B, C, D, E, F} $\subseteq$ N)
$u_{ij}$ - przepustowość pomiędzy węzłem i a węzłem j
\item Zmienne decyzyjne
$f_{ij}$ - przepływ między węzłem i a węzłem j
\item Funkcja celu
\begin{align*}
Q &= \max\left(\sum_{p \in P} f_{pt} \right) \\
&= \max(f_{1t} + f_{2t} + f_{3t} + f_{4t} + f{5t} + f_{6t})
\end{align*}
\item Ograniczenia
\setcounter{equation}{0}
\begin{equation}
0 \leq f_{ij} \leq 1, (i, j) \in N
\end{equation}
\begin{equation}
0 \leq f_{zp} \leq u_{zp}, z \in Z, p \in P
\end{equation}
\begin{equation}
f_{si} = 0, i \in \{ s, t, 1, 2, 3, 4, 5, 6 \}
\end{equation}
\begin{equation}
f_{is} = 0, i \in N
\end{equation}
\begin{equation}
f_{ti} = 0, i \in N
\end{equation}
\begin{equation}
f_{it} = 0, i \in \{s, t, A, B, C, D, E, F\}
\end{equation}
\begin{equation}
\sum_{z \in Z} f_{zp} = 1, p \in P = \{ 1, 2, 3, 4, 5, 6 \}
\end{equation}
\begin{equation}
\sum_{p \in P} f_{zp} = 1, z \in Z = \{ A, B, C, D, E, F \}
\end{equation}
\begin{equation}
\sum_{i \in N \backslash \{s\}} f_{ni} = \sum_{j \in N \backslash \{t\}} f_{jn}, n \in N \backslash \{ s, t \}
\end{equation}
\end{enumerate}
\paragraph{Model sieciowy rysunek}
\paragraph{Problem do rozwiązania}
\includepdf[pages=-]{flowchart21.pdf}
\paragraph{Rozwiazanie}
\begin{lstlisting}[caption= plik dat]
data;
set Wyjscie := t;
set Wezly := 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set StartWezly := s, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set WyjscieWezly := t, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set StartWyjscieWezly := s, t, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set Projekty := 1, 2, 3, 4, 5, 6;
set Zespoly := A, B, C, D, E, F;
param u :=
A 1 0 A 2 0 A 3 1 A 4 1 A 5 0 A 6 1
B 1 1 B 2 1 B 3 0 B 4 0 B 5 1 B 6 0
C 1 1 C 2 0 C 3 1 C 4 0 C 5 0 C 6 1
D 1 1 D 2 0 D 3 1 D 4 0 D 5 0 D 6 1
E 1 1 E 2 1 E 3 0 E 4 0 E 5 1 E 6 0
F 1 0 F 2 0 F 3 1 F 4 1 F 5 1 F 6 0;
end;
\end{lstlisting}
\begin{lstlisting}[caption= plik mod]
set Wyjscie;
set Wezly;
set StartWezly;
set WyjscieWezly;
set StartWyjscieWezly;
set Projekty;
set Zespoly;
param u{z in Zespoly, p in Projekty};
var f{i in StartWyjscieWezly, j in StartWyjscieWezly}, >= 0;
maximize Q: sum {p in Projekty, t in Wyjscie} f[p,t];
Ograniczeni_jeden{i in StartWyjscieWezly, j in StartWyjscieWezly}:
f[i,j] >= 0;
Ograniczeni_dwa{i in StartWyjscieWezly, j in StartWyjscieWezly}:
f[i,j] <= 1;
Ograniczeni_trzy{p in Projekty, z in Zespoly}:
f[z,p] <= u[z,p];
Ograniczeni_cztery{p in Projekty}:
sum {z in Zespoly} f[z,p] = 1;
Ograniczeni_piec{z in Zespoly}:
sum {p in Projekty} f[z,p] = 1;
Ograniczeni_szesc{n in Wezly}:
sum {i in WyjscieWezly} f[n,i] = sum {j in StartWezly} f[j,n];
solve;
display {i in Wezly, j in Wezly: f[i,j] > 0}: f[i,j];
display: Q;
\end{lstlisting}
\paragraph{Przydział zespołów do projektów}
\begin{lstlisting}[caption=wynik z glpk]
f[A,6].val = 1
f[B,5].val = 1
f[C,1].val = 1
f[D,3].val = 1
f[E,2].val = 1
f[F,4].val = 1
Q.val = 6
\end{lstlisting}
Wyniki: \\
Projekt numer \textbf{1} zostanie przydzielony zespołowi \textbf{C} \\
Projekt numer \textbf{2} zostanie przydzielony zespołowi \textbf{E} \\
Projekt numer \textbf{3} zostanie przydzielony zespołowi \textbf{D} \\
Projekt numer \textbf{4} zostanie przydzielony zespołowi \textbf{F} \\
Projekt numer \textbf{5} zostanie przydzielony zespołowi \textbf{B} \\
Projekt numer \textbf{6} zostanie przydzielony zespołowi \textbf{A} \\
\subsection{Zadanie 2.2 Minimalizacja kosztów realizacji projektów}
\paragraph{Problem do rozwiązania}
\begin{figure}[htb]
\caption{Koszty wynajmu}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
& A & B & C & D & E & F \\
\hline
1 & - & 13 & - & 14 & 18 & - \\
\hline
2 & - & 4 & 2 & - & 3 & - \\
\hline
3 & 2 & - & - & 10 & - & 13 \\
\hline
4 & 10 & - & 5 & - & - & 15 \\
\hline
5 & - & 10 & - & - & 17 & 12 \\
\hline
6 & 20 & - & 6 & 16 & - & - \\
\hline
\end{tabular}
\end{center}
\end{figure}
\begin{enumerate}
\item Parametry \\
s - źródło \\
w - wyjście \\
N - węzły (N = {s, t, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F})
P - projekty (P = {1, 2, 3, 4, 5, 6} $\subseteq$ N)
Z - zespoły (Z = {A, B, C, D, E, F} $\subseteq$ N)
$k_{zp}$ - koszty przyporządkowania zespołu z do projektu p
\item Zmienne decyzyjne:
$f_{ij}$ - przeływ pomiędzy węzłem i oraz węzłem j
\item Funkcja celu
\begin{align*}
Q &= \min \left(\sum_{z \in Z, p \in P} c_{zp}f_{zp} \right) \\
&= \min \big( 13f_{B1} + 14f_{D1} + 18f_{E1} + 4f_{B2} + 2f_{C2} + 3f_{E2} \\
&\quad + 2f_{A3} + 10f_{D3} + 13f_{F3} + 10f_{A4} + 5f_{C4} + 15f_{F4} \\
&\quad + 10f_{B5} + 17f_{E5} + 12f_{F5} + 20f_{A6} + 6f_{C6} + 16f_{D6} \big)
\end{align*}
\item Ograniczenia
\setcounter{equation}{0}
\begin{equation}
0 \leq f_{ij} \leq 1, (i, j) \in N
\end{equation}
\begin{equation}
f_{si} = 0, i \in \{ s, t, 1, 2, 3, 4, 5, 6 \}
\end{equation}
\begin{equation}
f_{is} = 0, i \in N
\end{equation}
\begin{equation}
f_{ti} = 0, i \in N
\end{equation}
\begin{equation}
f_{it} = 0, i \in \{s, t, A, B, C, D, E, F\}
\end{equation}
\begin{equation}
\sum_{z \in Z} f_{zp} = 1, p \in P = \{ 1, 2, 3, 4, 5, 6 \}
\end{equation}
\begin{equation}
\sum_{p \in P} f_{zp} = 1, z \in Z = \{ A, B, C, D, E, F \}
\end{equation}
\begin{equation}
\sum_{i \in N \backslash \{s\}} f_{ni} = \sum_{j \in N \backslash \{t\}} f_{jn}, n \in N \backslash \{ s, t \}
\end{equation}
\end{enumerate}
\paragraph{Model sieciowy rysunek}
\includepdf[pages=-]{flowchart22.pdf}
\paragraph{Rozwiazanie}
\begin{lstlisting}[caption= plik dat]
data;
set Wyjscie := t;
set Wezly := 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set StartWezly := s, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set WyjscieWezly := t, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set StartWyjscieWezly := s, t, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set Projekty := 1, 2, 3, 4, 5, 6;
set Zespoly := A, B, C, D, E, F;
param k :=
A 1 1000 A 2 1000 A 3 2 A 4 10 A 5 1000 A 6 20
B 1 13 B 2 4 B 3 1000 B 4 1000 B 5 10 B 6 1000
C 1 1000 C 2 2 C 3 1000 C 4 5 C 5 1000 C 6 6
D 1 14 D 2 1000 D 3 10 D 4 1000 D 5 1000 D 6 16
E 1 18 E 2 3 E 3 1000 E 4 1000 E 5 17 E 6 1000
F 1 1000 F 2 1000 F 3 13 F 4 15 F 5 12 F 6 1000;
end;
\end{lstlisting}
\begin{lstlisting}[caption= plik mod]
set Wyjscie;
set Wezly;
set StartWezly;
set WyjscieWezly;
set StartWyjscieWezly;
set Projekty;
set Zespoly;
param k{z in Zespoly, p in Projekty};
var f{i in StartWyjscieWezly, j in StartWyjscieWezly}, >= 0;
minimize Q: sum {p in Projekty, z in Zespoly} k[z, p] * f[z, p];
Ograniczenie_jeden{i in StartWyjscieWezly, j in StartWyjscieWezly}:
f[i,j] >= 0;
Ograniczenie_dwa{i in StartWyjscieWezly, j in StartWyjscieWezly}:
f[i,j] <= 1;
Ograniczenie_trzy{p in Projekty}:
sum {z in Zespoly} f[z,p] = 1;
Ograniczenie_cztery{z in Zespoly}:
sum {p in Projekty} f[z,p] = 1;
Ograniczenie_piec{n in Wezly}:
sum {i in WyjscieWezly} f[n,i] = sum {j in StartWezly} f[j,n];
solve;
display {i in Wezly, j in Wezly: f[i,j] > 0}: f[i,j];
display: Q;
\end{lstlisting}
\paragraph{Przydział zespołów do projektów}
\begin{lstlisting}[caption=wynik z glpk]
f[A,3].val = 1
f[B,5].val = 1
f[C,6].val = 1
f[D,1].val = 1
f[E,2].val = 1
f[F,4].val = 1
Q.val = 50
\end{lstlisting}
Wyniki: \\
Projekt numer \textbf{1} zostanie przydzielony zespołowi \textbf{D} \\
Projekt numer \textbf{2} zostanie przydzielony zespołowi \textbf{E} \\
Projekt numer \textbf{3} zostanie przydzielony zespołowi \textbf{A} \\
Projekt numer \textbf{4} zostanie przydzielony zespołowi \textbf{F} \\
Projekt numer \textbf{5} zostanie przydzielony zespołowi \textbf{B} \\
Projekt numer \textbf{6} zostanie przydzielony zespołowi \textbf{C} \\
\subsection{Zadanie 2.3 Minimalizacja terminu realizacji puli projektów}
\begin{figure}[htb]
\caption{Czas realizacji projektów}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
& A & B & C & D & E & F \\
\hline
1 & - & 13 & - & 14 & 18 & - \\
\hline
2 & - & 4 & 2 & - & 3 & - \\
\hline
3 & 2 & - & - & 10 & - & 13 \\
\hline
4 & 10 & - & 5 & - & - & 15 \\
\hline
5 & - & 10 & - & - & 17 & 12 \\
\hline
6 & 20 & - & 6 & 16 & - & - \\
\hline
\end{tabular}
\end{center}
\end{figure}
\begin{enumerate}
\item Parametry \\
s - źródło \\
w - wyjście \\
N - węzły (N = {s, t, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F})
P - projekty (P = {1, 2, 3, 4, 5, 6} $\subseteq$ N)
Z - zespoły (Z = {A, B, C, D, E, F} $\subseteq$ N)
$t_{zp}$ - czas realizacji projektu p przez zespół z [w miesiącach]
\item Zmienne decyzyjne:
$f_{ij}$ - przeływ pomiędzy węzłem i oraz węzłem j
\item Funkcja celu
\begin{align*}
Q &= \min \left( \max_{z \in Z, p \in P} c_{zp} f_{zp} \right) \\
&= \min \big\{13f_{B1}, 14f_{D1}, 18f_{E1}, 4f_{B2}, 2f_{C2}, 3f_{E2}, \\
&\quad 2f_{A3}, 10f_{D3},13f_{F3}, 10f_{A4}, 5f_{C4}, 15f_{F4} \\
&\quad 10f_{B5}, 17f_{E5}, 12f_{F5}, 20f_{A6}, 6f_{C6}, 16f_{D6} \big\}
\end{align*}
\item Ograniczenia
\setcounter{equation}{0}
\begin{equation}
0 \leq f_{ij} \leq 1, \; (i, j) \in N
\end{equation}
\begin{equation}
f_{si} = 0, \; i \in \{ s, t, 1, 2, 3, 4, 5, 6 \}
\end{equation}
\begin{equation}
f_{is} = 0, \; i \in N
\end{equation}
\begin{equation}
f_{ti} = 0, \; i \in N
\end{equation}
\begin{equation}
f_{it} = 0, \; i \in \{s, t, A, B, C, D, E, F\}
\end{equation}
\begin{equation}
\sum_{z \in Z} f_{zp} = 1, \; p \in P = \{ 1, 2, 3, 4, 5, 6 \}
\end{equation}
\begin{equation}
\sum_{p \in P} f_{zp} = 1, \; z \in Z = \{ A, B, C, D, E, F \}
\end{equation}
\begin{equation}
\sum_{i \in N \backslash \{s\}} f_{ni} = \sum_{j \in N \backslash \{t\}} f_{jn}, \; n \in N \backslash \{ s, t \}
\end{equation}
\end{enumerate}
\paragraph{Model sieciowy rysunek}
\includepdf[pages=-]{flowchart22.pdf}
\paragraph{Model programowania liniowego}
\begin{lstlisting}[caption= plik dat]
data;
set Wyjscie := t;
set Wezly := 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set StartWezly := s, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set WyjscieWezly := t, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set StartWyjscieWezly := s, t, 1, 2, 3, 4, 5, 6, A, B, C, D, E, F;
set Projekty := 1, 2, 3, 4, 5, 6;
set Zespoly := A, B, C, D, E, F;
param c :=
A 1 1000 A 2 1000 A 3 2 A 4 10 A 5 1000 A 6 20
B 1 13 B 2 4 B 3 1000 B 4 1000 B 5 10 B 6 1000
C 1 1000 C 2 2 C 3 1000 C 4 5 C 5 1000 C 6 6
D 1 14 D 2 1000 D 3 10 D 4 1000 D 5 1000 D 6 16
E 1 18 E 2 3 E 3 1000 E 4 1000 E 5 17 E 6 1000
F 1 1000 F 2 1000 F 3 13 F 4 15 F 5 12 F 6 1000;
end;
\end{lstlisting}
\begin{lstlisting}[caption= plik mod]
set Wyjscie;
set Wezly;
set StartWezly;
set WyjscieWezly;
set StartWyjscieWezly;
set Projekty;
set Zespoly;
param c{z in Zespoly, p in Projekty};
var f{i in StartWyjscieWezly, j in StartWyjscieWezly}, >= 0, integer;
var cmax, >= 0;
minimize Q: cmax;
subject to
Ograniczenie_jeden{i in StartWyjscieWezly, j in StartWyjscieWezly}:
f[i,j] >= 0;
Ograniczenie_dwa{i in StartWyjscieWezly, j in StartWyjscieWezly}:
f[i,j] <= 1;
Ograniczenie_trzy{p in Projekty}:
sum {z in Zespoly} f[z,p] = 1;
Ograniczenie_cztery{z in Zespoly}:
sum {p in Projekty} f[z,p] = 1;
Ograniczenie_piec{n in Wezly}:
sum {i in WyjscieWezly} f[n,i] = sum {j in StartWezly} f[j,n];
Ograniczenie_szesc{z in Zespoly, p in Projekty}:
c[z,p] * f[z,p] <= cmax;
solve;
display {i in Wezly, j in Wezly: f[i,j] > 0}: f[i,j];
display: Q;
\end{lstlisting}
\begin{lstlisting}[caption=wynik z glpk]
f[A,4].val = 1
f[B,1].val = 1
f[C,6].val = 1
f[D,3].val = 1
f[E,2].val = 1
f[F,5].val = 1
Q.val = 13
\end{lstlisting}
Wyniki: \\
Projekt numer \textbf{1} zostanie przydzielony zespołowi \textbf{B} \\
Projekt numer \textbf{2} zostanie przydzielony zespołowi \textbf{E} \\
Projekt numer \textbf{3} zostanie przydzielony zespołowi \textbf{D} \\
Projekt numer \textbf{4} zostanie przydzielony zespołowi \textbf{A} \\
Projekt numer \textbf{5} zostanie przydzielony zespołowi \textbf{F} \\
Projekt numer \textbf{6} zostanie przydzielony zespołowi \textbf{C} \\
\section{Zadanie 3}
\paragraph{Model programowania liniowego}
\begin{enumerate}
\item Parametry \\
m - liczba różnych zasobów, i $\in$ N \\
n - liczba różnych produktów, j $\in$ N \\
$c_{i}^{max}$ - Maksymalna przepustowość dla zasobu i \\
$A_{ij}$ - Użycie zasobu i przez produkt j \\
$q_j$ - Limit produkcji dla produktu j, po którym zmniejsza się przychód na jednostce \\
$p_j$ - Normalny przychód na pojedyńczej sztuce produktu j \\
$p_j^{disc}$ - Pomniejszony przychód na sztuce produktu j
\item Zmienne decyzyjne
$x_j$ - Wolumin produkcji produktu j
\item Zmienne pomocnicze
$r_j(u)$ - Funkcja wyliczająca przychód dla produktu j bazując na poziomie produkcji u \\
$c_i$ - Całkowite zużycie zasobu typu i
\item Funkcja celu
\[ \max Z = \sum_{j=1}^n r_j u_j \]
\[ r_j(u) = \{ \]
\[ p_{j} u \qquad \qquad \qquad \qquad u <= q_j \]
\[ p_j q_j + p_{j}^{disc}(u-q_j), \qquad u => q_j \]
\[ \} \]
\item Ograniczenia
\setcounter{equation}{0}
Całkowite zużycie zasobów nie może przekraczać dostępnośli dla żadnego zasobu
\begin{equation}
\sum_{j=1}^n A_{ij} \cdot x_j \leq c_i^{max}, \qquad i \in N
\end{equation}
Produkcja każdego produktu j, jest powiżana z właściwym $u_j$
\begin{equation}
u_j = x_j, \qquad j \in N
\end{equation}
Wolumin produkcji nie może być mniejszy od zera
\begin{equation}
x_j \geq 0, \qquad j \in N
\end{equation}
\end{enumerate}
\end{document}