Exercise 1.10 (Production and inventory planning)

Question

A company must de­liver $d_i$ units of its product at the end of the $i$th month. Material produced during a month can be delivered either at the end of the same month or can be stored as inventory and delivered at the end of a subsequent month; however, there is a storage cost of $c_1$ dollars per month for each unit of product held in inventory. The year begins with zero inventory. If the company produces $x_i$ units in month $i$ and $x_{i + 1}$ units in month $i + 1$, it incurs a cost of $c_2 |x_{i + 1} - x_i|$ dollars, reflecting the cost of switching to a new production level. Formulate a linear programming problem whose objective is to minimize the total cost of the production and in­ventory schedule over a period of twelve months. Assume that inventory left at the end of the year has no value and does not incur any storage costs.

Elu Thingol · Accepted Answer

We have to optimize two parameters: monthly production level $x_i$, $i=1,\ldots,12$, and the monthly inventory $y_i$, $i=2,\ldots,12$ (January inventory $y_1$ is assumed to be zero by assumption).

\begin{figure}
\includegraphics[width=1	extwidth]{bertsimas_linopt_exercise_1-10.png}
\caption{Production scheme.}
\end{figure}

Our objective is to minimize the combined cost of production changes and the inventory storing

$$\sum_{i=1}^{12} c_2 \left|x_i - x_{i-1}ight| + \sum_{i=2}^{12} c_1 y_i$$

under the condition that the company is obligated to deliver a certain amount $d_i$ of its products each month

$$(y_i + x_i) - y_{i+1} = d_i, \quad i=1,\ldots,12$$

and no unit of product can be stored longer than one month

$$y_i \leq d_i, \quad i=1,\ldots,12.$$

These conditions result in the following optimization problem:
\begin{equation*}
\begin{array}{llll}
	ext{minimize} \quad &\sum_{i=1}^{12} c_2 \left|x_i - x_{i-1}ight| + \sum_{i=2}^{12} c_1 y_i\
	ext{subject to} \quad
&(y_i + x_i) - y_{i+1} = d_i &i=1,\ldots,12\
&y_i \leq d_i                &i=1,\ldots,12\
&x_i,y_i \geq 0              &i=1,\ldots,12.
\end{array}
\end{equation*}

To convert this into a linear optimization problem, we follow the procedure from the \href{./exercise_1-5}{Exercise 1.5}:
\begin{equation*}
\begin{array}{llll}
	ext{minimize} \quad &\sum_{i=1}^{12} c_2 z_i + \sum_{i=2}^{12} c_1 y_i\
	ext{subject to} \quad
&(y_i + x_i) - y_{i+1} = d_i &i=1,\ldots,12\
&y_i \leq d_i                &i=1,\ldots,12\
&z_i \geq x_i - x_{i-1}      &i=1,\ldots,12\
&z_i \geq x_{i-1} - x_{i}    &i=1,\ldots,12\
&x_i,y_i,z_i \geq 0              &i=1,\ldots,12.
\end{array}
\end{equation*}

Exercise 1.10 (Production and inventory planning)

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Exercise 1.10 (Production and inventory planning)

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