How do I solve this problem? What is with the inventory? Do I ignore it?
Mackk Engine produces diesel trucks. New government emission standards have dictated that the average pollution emissions of all trucks produced in the next 3 years cannot exceed 10 grams per truck. Mackk produces 2 types of trucks. Each type 1 truck sells for 20,000 USD, costs 15,000 USD to manufacture, and emits 15 grams of pollution. Each type 2 truck sells for 17,000 USD, costs 14,000 USD to manufacture, and emits 5 grams of pollution. Production capacity limits total truck production during each year to at most 320 trucks. The maximum numbers of each truck type that can be sold during each of the next 3 years are for type 1, 100 in year 1, 200 in year 2 and 300 in year 3 and for type 2, 200 for year 1, 100 in year 2 and 150 in year 150. Demand can be met from previous production or the current year’s production. It costs $2000 to hold 1 truck (of any type) in inventory for 1 year. Determine how Mackk can maximize its profit during the next 3 years.
This is what I got so far. But I don't know how to account for inventory figures and fit unit sold and inventory into solve using excel solver.
Decision variables
$P_{ij}$= Number of trucks (each type $i$) produced for each year $j$
$S_{ij}$= Number of trucks (each type $i$) sold for each year $j$
$R_{ij}$= Number of trucks (each type $i$) that remain in stock at the end of each year $j$
With $i=1, 2$; $j=1, 2, 3$.
My objective Function
\begin{eqnarray} \max~~ Z = && 20 (S_{11}+S_{12}+S_{13}) + 17 (S_{21}+S_{22}+S_{23}) - 15(P_{11}+P_{12}+P_{13}) \\ && -14 (P_{21}+P_{22}+P_{23}) - 2 (R_{11}+R_{12}+R_{21}+R_{22}) \end{eqnarray}
My Constraints
Production \begin{eqnarray} P_{11}+P_{21} &\leq& 320\\ P_{12}+P_{22} &\leq& 320\\ P_{13}+P_{23} &\leq& 320 \end{eqnarray}
Demand \begin{eqnarray} S_{11} &\leq& 100 \\ S_{12} &\leq& 200 \\ S_{13} &\leq& 300 \\ S_{21} &\leq& 200 \\ S_{22} &\leq& 100 \\ S_{23} &\leq& 150 \end{eqnarray}
Emissions Requirement
$$ 5P_{11}+5P_{12}+5P_{13}-5P_{21}-5P_{22}-5P_{23} \leq 0 $$
Non-Negativity $$ P_{ij}, S_{ij}, R_{ij} \geq 0; i=1,2;~ j=1,2,3 $$