I am struggling with a simplified production model problem and I am hoping someone can help me out.
The problem involves a plant that has three components: water purification, electricity production, and chemical production. Each of these components depends on each other, as well as on external sources such as raw water and fuel (but this has no impact on the model).
Here are the given requirements for producing 1 $m^3$ of purified water:
- 0.1 $m^3$ of purified water (to flush the filter, etc.)
- 0.3 liters of purification chemicals
- 0.3 kWh of electricity
For producing 1 liter of purification chemicals:
- 0.6 $m^3$ of purified water
- 0.2 liters of purification chemicals
- 0.1 kWh of electricity
For producing 1 kWh of electricity:
- 0.6 $m^3$ of purified water
- 0.1 kWh of electricity
I need to create a consumption matrix for this system and then calculate how much purified water, purification chemicals, and electricity are needed to produce 18 $m^3$ of purified water without producing any electricity or purification chemicals.
The Leontief Input-Output Model is given by: $$ \bar{p}=M \bar{p}+\bar{d} $$ Where the matrix $\$ M \$$ is the consumption matrix.
The consumption matrix is made up of consumption vectors. The $j^{\text {th }}$ column is the $j^{\text {th }}$ consumption vector and contains the necessary input required from each of the Sectors for Sector $j$ to produce one unit of output. The consumption matrix the requirements for producing one unit of a given Product becomes a column (rather than a row) of the matrix.
First, we need to find the external demand vector $\bar{d}$ and the production vector $\bar{p}$. From the problem statement, we know that we need to produce 18 $m^3$ of purified water and no purification chemicals or electricity. Therefore, the production vector is:
$$ \bar{p}=\left(\begin{array}{l} 18 \\ 0 \\ 0 \end{array}\right) $$ Since there is no external demand for purified water, the external demand vector is: $$ \bar{d}=\left(\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right) $$
Next, we can use the consumption matrix to find the internal demand vector:
$$M \bar{p}=\left(\begin{array}{ccc} 0.1 & 0.3 & 0.3 \\ 0.6 & 0.2 & 0.1 \\ 0.6 & 0 & 0.1 \end{array}\right)\left(\begin{array}{c} 18 \\ 0 \\ 0 \end{array}\right)$$
Here's the tabular representation of the consumption matrix:
$$\begin{array}{|l|l|l|l|} \hline & \begin{array}{l} \text { Purified } \\ \text { water } \end{array} & \begin{array}{l} \text { Purification } \\ \text { chemicals } \end{array} & \text { Electricity } \\ \hline \text { Purified water } & 0.1 & 0.3 & 0.3 \\ \hline \begin{array}{l} \text { Purification } \\ \text { chemicals } \end{array} & 0.6 & 0.2 & 0.1 \\ \hline \text { Electricity } & 0.6 & 0 & 0.1 \\ \hline \end{array}$$
I am not sure if this is the correct consumption matrix, and I am not sure how to proceed with calculating the amount of purified water, purification chemicals, and electricity needed to produce 18 $m^3$ of purified water without producing any electricity or purification chemicals.
Thank you in advance!
It should be
$$ \bar{p}=\left(\begin{array}{ccc} 0.1 & 0.6 & 0.6 \\ 0.3 & 0.2 & 0 \\ 0.3 & 0.1 & 0.1 \end{array}\right)\left(\begin{array}{c} w \\ c \\ e \end{array}\right)=\left(\begin{array}{c} 18 \\ 0 \\ 0 \end{array}\right)$$ where $w,c,e$ are units of water, chems, power needed.