In the Critical Path Method (CPM), the cost-time slope of an activity is given by

The correct answer is:
$\frac{\mathrm{C}\mathrm{r}\mathrm{a}\mathrm{s}\mathrm{h}\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{t}-\mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{t}}{\mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{T}\mathrm{i}\mathrm{m}\mathrm{e}-\mathrm{C}\mathrm{r}\mathrm{a}\mathrm{s}\mathrm{h}\mathrm{T}\mathrm{i}\mathrm{m}\mathrm{e}}$

Visual Analysis of the Diagram:
The attached image shows the cost-time relationship for a project activity:
• The vertical axis represents the Cost of the activity, pointing out the Normal Cost (C_n) and the higher Crash Cost (C_c).
• The horizontal axis represents the Time required, indicating the Crash Time (t_c) and the longer Normal Time (t_n).
• The dashed straight line represents the linear approximation of the cost-time relationship between the normal operating point (t_n, C_n) and the crashed operating point (t_c, C_c). The slope of this line represents the cost slope.

Explanation of Cost-Time Slope in CPM:
In CPM (Critical Path Method) project scheduling, the cost-time slope (or cost slope) represents the additional cost incurred per unit of time saved by shortening (crashing) the activity duration.
To compress a project schedule, managers target activities on the critical path with the flattest cost slope because they offer the most time savings for the least increase in cost.

Mathematical Derivation:
The slope of a straight line passing through the coordinates (t_n, C_n) and (t_c, C_c) is calculated as:
$\text{Slope}=\frac{C_c-C_n}{t_c-t_n}$
Since cost increases as time decreases (C_c > C_n and t_c < t_n), this slope is negative. We are interested in the absolute magnitude of the rate of cost increase, which gives:
$\text{Cost Slope}=\frac{C_c-C_n}{t_n-t_c}$

Substituting the descriptive terms:
• C_c − C_n is the Crash Cost − Normal Cost (increase in cost).
• t_n − t_c is the Normal Time − Crash Time (decrease in time).

Thus, the cost-time slope of an activity is given by:
$\frac{\mathrm{C}\mathrm{r}\mathrm{a}\mathrm{s}\mathrm{h}\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{t}-\mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{t}}{\mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{T}\mathrm{i}\mathrm{m}\mathrm{e}-\mathrm{C}\mathrm{r}\mathrm{a}\mathrm{s}\mathrm{h}\mathrm{T}\mathrm{i}\mathrm{m}\mathrm{e}}$