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diff --git a/src/index.less b/src/index.less index 8b384ba..4671605 100644 --- a/src/index.less +++ b/src/index.less @@ -123,6 +123,10 @@ table { } } +li { + margin: 10px 0; +} + .left { text-align: left; } diff --git a/src/routes/ottimizzazioneLineare.js b/src/routes/ottimizzazioneLineare.js index 3f605be..acaff64 100644 --- a/src/routes/ottimizzazioneLineare.js +++ b/src/routes/ottimizzazioneLineare.js @@ -6,6 +6,7 @@ import Example from "../components/example"; import Todo from "../components/todo"; import Minus from "../components/minus"; import Plus from "../components/plus"; +import Code from "../components/code"; const r = String.raw; @@ -56,146 +57,7 @@ export default class OttimizzazioneLineare extends Component { - - -

- Un sistema è in forma standard quando ha: -

- - - -

- Possiamo trasformare una disequazione in equazione introducendo variabili slack: -

-

- {r`3x + 2y \leq 15 \implies 3x + 2y + s_1 = 15`} -

-

- In seguito, possiamo riscrivere tutte le funzioni in termini delle variabili slack. -

- - Si ottiene qualcosa come {r`z = -2 s_1 - 3 s_2 + 40`}. - - - -

- Un modo per rappresentare sistemi in forma standard. -

-

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- xys_1s_2
obj:807000
- 3210
- 2301
-

- - E' la matrice equivalente completa del sistema! - -

- Riscrivendo in termini delle variabili slack, otteniamo: -

-

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- xys_1s_2
obj:00-20-10
- 10{r`-\frac{3}{5}`}{r`\frac{2}{5}`}
- 01{r`\frac{2}{5}`}{r`-\frac{3}{5}`}
-

- - Abbiamo appena applicato Gauss-Jordan, trovando la matrice inversa. - - - - - -

- Insieme di n colonne lunghe n linearmente indipendenti. -

- - -

- Soluzione ottenuta dalla seguente formula: -

-
    -
  1. {r`A \cdot x = B`}
    2. -
  2. {r`B \cdot x_B + F \cdot x_F = b`}
    3. -
  3. {r`x_B = B^{-1} \cdot b - B^{-1} \cdot x_F`}
    4. -
-

- Dove: -

- - - - +
  • Solo equazioni
    • @@ -211,24 +73,142 @@ export default class OttimizzazioneLineare extends Component {
    • Equazioni e disequazioni
      • -
    • Tutte le variabili maggiori di zero
      • +
    • Variabili con qualsiasi valore
     - +

    - Sostituiamo le variabili potenzialmente negative (unconstrained) {r`x_j`} con due variabili {r`x_j^+`} e {r`x_j^-`}. + Convertiamo ogni equazione {r`=`} in due disequazioni {r`\leq`} e {r`\geq`},

    + Why would you ever do that?!

    Convertiamo le disequazioni in equazioni aggiungendo una variabile slack.

    + + {r`a \leq 3`} diventa {r`a + s_1 = 3`}. +     - +

    - Convertiamo ogni equazione {r`=`} in due disequazioni {r`\leq`} e {r`\geq`}, + Sostituiamo le variabili potenzialmente negative (unconstrained) {r`x_j`} con due variabili {r`x_j^+`} e {r`x_j^-`}. +

    + + {r`a \in \mathbb{Z}`} diventa {r`a^+ \in \mathbb{N}`} e {r`-a^- \in \mathbb{N}`}. + +     +     + + +

    + La funzione da minimizzare/massimizzare, tipicamente indicata con una {r`z`} al termine noto. +

    +     + +

    + Un modo per rappresentare sistemi in forma standard, anche noto come matrice equivalente completa del sistema. +

    + + Il sistema:

    + {r` + \begin{cases} + 2000A + 1000B = z\\ + 1A \leq 3\\ + 1B \leq 3\\ + 2A + 2B \leq 7 + \end{cases} + `}

    + Diventa in forma di tableau:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    TNABs_1s_2
    z2000100000
    31010
    30101
    72200
    +     +     + +

    + TODO: come spiegarla? +

    +     +     + + +

    + Un algoritmo per massimizzare efficientemente variabili di sistemi lineari, derivato da Gauss-Jordan. +

    + + E' spiegato semplicemente qui. + +     + +
      +
    1. Trasforma il sistema in forma standard.
      2. +
    2. Finchè ci sono variabili con coefficienti positivi nella funzione obiettivo: +
        +
      1. Scegli una variabile della funzione obiettivo, chiamandola variabile entrante. Come? Vedi nel prossimo pannello.
        2. +
      2. Trova la variabile di base (detta variabile uscente) con il valore minore per questo rapporto: termine noto / coeff. variabile entrante
        3. +
      3. Riscrivi tutte le funzioni del sistema in termini della variabile entrante.
        4. +
      +
      3. +
    3. Il termine noto della funzione obiettivo è il tuo risultato.
      4. +
    +     + +
      +
    • Coefficiente maggiore nella funzione obiettivo.
      • +
    • Incremento maggiore della funzione obiettivo.
      • +
    • A caso.
      • +
    • Regola di Bland: scegli variabili entranti e uscenti con indice minore. Impedisce cicli infiniti!
      • +
    +     +     + + +

    + Un estensione del Simplex per permettere la risoluzione di problemi con termini noti negativi. +

    +

    + Prevede l'introduzione di un problema ausiliario.