The combined Model $1$ & Model $2$ production capacities are $150,000$ units at Plant A and $180,000$ units at plant B and $30,000$ units at Plant C$.$ The Model $3$ production capacities are $100,000$ units at Plan A and $90,000$ units at the plant B and $40,000$ at Plant C$.$ The cost of shipping from plant A is $$0\mathrm{.}2$ per unit and from plant B is $$0\mathrm{.}25$ per unit and from Plan C $0\mathrm{.}3.$

Q$1.$Develop a linear optimization setup that XYZ can use to determine how many units of each battery model to produce at each plant to minimize the total production and shipping cost of the new contract. Show all equations & details in the setup of your optimization model..