A company wishes to plan its production of two items with seasonal demands over a $12-$month period. The monthly demand of item$1$ is $100$ units during the months of October, November and December; $10$ units during the months of January, February, March and April; and $30$ units during the remaining months. The demand of item$2$ is $50$ units during the months of September, October, November and December and $15$ units during the remaining months. Suppose that the unit product cost of item $1$ and item $2$ is $5000TL$ and $8000TL,$ respectively, if they are manufactured before June. After June, the unit costs are reduced to $4500$ TL and $7000$ TL because of the installation of an improved manufacturing system. Item$1$ requires $1$ unit of a key component and item$2$ requires $2$ units. The availability of this key component is $120$ units per month during January$-$September and $150$ units per month during October$-$December. Furthermore, each unit of item$1$ occupies $2m^3$ and each item $2$ occupies $4m^3$ of inventory. Suppose that the maximum inventory space allocated to these items is $200m^3$ and that the holding cost during any month is $100T\frac{L}{m^3}.$

Formulate a linear programming model which will represent this system. Define the decision variables, state the objective function, and specify all the constraints.