Finding a linear programming model

Question

Question:

Acme manufacturing company has contracted to deliver home windows over the next 6 months. The demands for each month are 100, 250, 190, 140, 220, and 110 units, respectively. Production cost per window varies from month to month depending on the cost of labour, material, and utilities. Acme estimates the production cost per window over the next 6 months to be £50, £45, £55, £48, £52, and £50, respectively. To take advantage of the fluctuations in manu- facturing cost, Acme may elect to produce more than is needed in a given month and hold the excess units for delivery in later months. This, however, will incur storage costs at the rate of £8 per window per month assessed on end-of-month inventory.

Develop a linear program to determine the optimum production schedule of Acme.

My attempt:

What I'm struggling with is my thought processes as I work through a question such as this. What is a logical way to create a linear program model for this example in particular? For example here is my initial (I believe) incorrect answer, I can't understand how to correctly add the part about the £8 a month:

Let $x_i = \text{the number of windows made within month $i$}$

Minimize $50x_1+45x_2 +55x_3 + 48x_4 + 52x_5 + 50x_6 $

subject to: $x_1+x_2+x_3+x_4+x_5+x_6 = 1010$

Answer 1

Let $D_i, C_i, S_i$ denote the demand, production cost, and storage cost in month $i$ (although $S_i$ is constant, it doesn't hurt to leave it as a parameter). Let $x_{ij}$ be the amount produced in month $i$ to meet demand in month $j$. Then the linear program is given by

\begin{align} \min &\quad \sum_{i=1}^6\sum_{j=i}^6 C_ix_{ij} + \sum_{i=1}^5\sum_{j=i+1}^6 S_ix_{ij} \\ \mathrm{s.t.} &\quad \sum_{i=1}^j x_{ij} = D_j,\quad 1\leqslant j\leqslant 6\\ &\quad x_{ij}\geqslant0,\quad 1\leqslant i\leqslant j\leqslant 6 \end{align}