Choose which plants to build to maximise profit

Question

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What I tried:

Let $y_i = 1$

if plant $i$ is to be constructed and $0$ otherwise

Let $c_{ij}$ be transportation cost per-unit for whatever the plants produce delivered from plant $i$ to customer $j$.

Let $x_{ij}$ be units of whatever delivered from plant $i$ to customer $j$.

Min

$$z = [12000, 11000, 12000] \cdot [y_1, y_2, y_3] + \sum_i \sum_j c_{ij} x_{ij}$$

s.t.

1. supply

$$\sum_j x_{1j} y_1 \le 1800 \color{red}{+ M(1-y_i)}$$

$$\sum_j x_{2j} y_2 \le 1400 \color{red}{+ M(1-y_i)}$$

$$\sum_j x_{3j} y_3 \le 1300 \color{red}{+ M(1-y_i)}$$

$\color{red}{\text{I have a feeling that this is redundant by constraint 3.}}$

2. demand

$$\sum_i x_{i1} \ge 1200$$

$$\sum_i x_{i2} \ge 1700$$

$$\sum_i x_{i3} \ge 1600$$

3. $x_{ij} \le My_i$

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From Chapter 3 here.

Answer 1

Let $x_{ij}$ be a binary that equals $1$ if and only plan $i$ produces for customer $j$, and $y_i$ another binary that equals $1$ if and only if plant $i$ is built. Let $c_{ij}$ be the transportation cost from plant $i$ to customer $j$ and $f_i$ the fixed of building plant $i$. $$ \mbox{Minimize } Z=\sum_{i=1}^3\sum_{j=1}^3c_{ij}x_{ij}+\sum_{i=1}^3f_{i}y_i $$ subject to

• Limited supplies :$\sum_{j=1}^3 p_{ij}x_{ij}\le s_i\quad \forall i=1,2,3$
• Meet demands: $\sum_{i=1}^3 p_{ij}x_{ij}\ge d_j\quad \forall j=1,2,3$
• Link variables: $x_{ij}\le My_i\quad \forall i,j=1,2,3$
• Variables are binary: $x_{ij},y_i\in \{0,1\} \quad \forall i,j=1,2,3$