# Activity analysis

An approach to micro-economics for which a system is composed of elementary functions, called activities, which influenced the early growth of linear programming. An insightful way to view an activity in a standard form linear program is that negative coefficients represent input requirements and positive coefficients represent outputs:

$LaTeX: \begin{bmatrix} c_j \\ \cdots \\ - \\ \cdots \\ + \end{bmatrix} \begin{matrix} \leftarrow & \mbox{cost or revenue} \\ \\ \leftarrow & \mbox{input to activity} j \\ \\ \leftarrow & \mbox{output from activity} j \end{matrix}$

In general, the reduced cost represents the net worth of the activity for the prices $LaTeX: p$,

$LaTeX: d_j = c_j - p^T A_{(:,j)} = \mbox{ input cost } - \mbox{ output revenue}.$

This leads to an economic interpretation of not only linear programming but also of the simplex method: agents (activity owners) respond instantaneously to changes in prices, and the activity with the greatest net revenue wins a bid to become active (basic), thus changing the prices for the next time (iteration).

In this context, activities can be regarded as transformations, from inputs to outputs. Three prevalent transformations are: form, place and time.

Here are examples:

$LaTeX: \begin{bmatrix} c \\ \cdots \\ -0.4 \\ -0.6 \\ \cdots \\ 1 \end{bmatrix} \begin{matrix} \\ \\ \mbox{ apples} \\ \mbox{ cranberries} \\ \\ \mbox{ cranapple} \end{matrix} \;\;\;\;\;\; \begin{bmatrix} c \\ \cdots \\ -1 \\ \cdots \\ 0.6 \\ 0.4 \end{bmatrix} \begin{matrix} \\ \\ \mbox{ Chicago} \\ \\ \mbox{ Denver} \\ \mbox{ Seattle} \end{matrix} \;\;\;\;\;\; \begin{bmatrix} c \\ \cdots \\ -1 \\ \cdots \\ 1 \end{bmatrix} \begin{matrix} \\ \\ t \\ \\ t+1 \end{matrix}

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