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This is done in order to calculate the population at risks with the construction of a site.

A company wants to select $p$ locations among a set of $m$ possible sites for constructing polluting plants in a contemporary world. The $m$ candidate sites are located on a territory containing different cities. We have:

  • $d_{ij}$ the distance between city $i$ and site $j$
  • $P_i$ the population (in thousand of inhabitants) of city $i$

I imagined experts that thought that a city was threatened if there was a polluting plant located less than 2 km from it.

There are two different factions :

  1. The authorities' point of view that wants to minimize nuisance. They want to minimize the number of inhabitants threatened by the $p$ selected plants.

  2. The company point of view that wants to minimize the transportation costs.
    They take into account the volume of goods transported between the $p$ plants and the $n$ clients. The transportation cost from a plant $j$ to a client city $i$ is $1.5€$ per kilometer and per $m^3$ of transported good. The annual demand of city $i$ is $V_i$ (in $m^3$). It will be needed to transform the first model so that :

    • At most 5% of the population of $n$ cities are threatened (constraint imposed by the authorities).
    • The demand of a city is delivered by a single plant
    • The company minimizes its total transportation cost

Generally speaking I thought developing a variable $t_i$ in order to create another one $s_j$: how many people are subject to a risk with the construction of a site $j$.

How to calculate the variable $s_j$, the number of people risking something with the construction of the site $j$ ?

My attempt

I thought about

\begin{cases} \min & \sum_j\sum_i s_{ij}\\ &s_{ij}=b_{ij}\times P_i\mbox{ number of people of city $i$ endangered by site $j$}\\ &d_{ij}\times t_{ij}\le2 \mbox{ constraint over the distance to flag an endangered population}\\ &\sum_i b_i= \mbox{ number of sites we want}\\ b_i \in \{0,1\} \end{cases}

With the additional variables :

  • $s_{ij}$ the population endangered in city $i$ by site $j$
  • $b_{ij}$ a boolean variable telling me if city $i$ is threatened by site $j$.

We want to minimize the sum of $s_{ij}$, the number of people endangered. I thought about creating a variable $b_i$, standing for "take into account population $i$" with the following test :

$$d_{ij}\times t_{ij}\le2$$

\begin{cases} b_i = 0 \mbox{ if we don't have to take the population of city $i$ into account}\\ b_i = 1 \mbox{ otherwise} \end{cases}

It is okay but I don't think we can generalize it :

Actually, if $d_{ij}=2\Rightarrow t_i = 1 \mbox{ or $0$}$. I want it to be 0 compulsorily.

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1 Answer 1

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You could precalculate an incidence matrix $a_{i,j}\in \{0,1\}$ where $a_{i,j}=1$ indicates city $i$ is at risk when construction is done at site $j$. Assume we model construction at $j$ by a binary decision variable $x_j=1$. If there is only one possible $x_j=1$ we can simply use:

$$ t_i = \sum_j a_{i,j} x_j $$

If multiple $x_j$ can be chosen, we can do

$$ n \cdot t_i \ge \sum_j a_{i,j} x_j $$ where $n$ is the maximum number of sites that can be selected. (Here I assume there is a cost involved with $t_i$ so indirectly we minimize it; this approach also works if we have an upper limit on $t_i$ e.g. $\sum_i b_i t_i \le K$). The last constraint can also be disaggregated as: $$ t_i \ge x_j \>\> \forall (i,j)|a_{i,j}=1 $$

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  • $\begingroup$ Thanks for this answer ! yet, you don't take the distance $d_{ij}$ into account here, don't you ? $\endgroup$ Commented Apr 11, 2017 at 8:27
  • $\begingroup$ Yes. Use the distance to populate the a matrix. (Ahead of optimization). $\endgroup$ Commented Apr 11, 2017 at 8:28

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