Loading and Scheduling Technique

A Gantt chart (Henry Gantt, about 1910) is a horizontal bar chart in which the vertical axis lists jobs or machines and the horizontal axis is time. Each bar shows the planned start, duration and finish of an activity.

Uses

1. Machine loading chart: each machine is a row; jobs are bars on the time scale. It shows the load on each machine and idle time, so jobs can be assigned and re-sequenced.
2. Progress (schedule) chart: the planned bar is drawn lightly; a thick line below it shows work completed. A vertical "today" line shows if the job is ahead or behind schedule.

 Machine | Day: 1  2  3  4  5  6  7  8
 --------+-----------------------------
 Lathe   |      [Job A ][Job C   ]
 Milling |         [Job B   ][Job D ]
 Drill   |               [Job A2][..]
 Done:    ====>   (progress on A)
              ^ today line

Advantages

• Simple to draw and understand, even by shop-floor staff.
• Shows the whole schedule and the idle time of machines at a glance.
• Useful for comparing planned and actual progress.
• Low cost; easy to update.

Limitations

• It does not show the dependencies between activities clearly, nor the critical path.
• It becomes complex for large projects with many activities.
• It cannot show uncertainty in activity times.
• Rescheduling needs the chart to be redrawn.

For this reason CPM and PERT are used for large projects, while the Gantt chart is retained for shop loading and for presenting schedules.

(a) Network (activity-on-node)

        +--> C(5) ---------+
 A(4) --+                   +--> F(3) --+
        +--> D(6) -----+----+           +--> H(2)
 B(3) ------>----+     |                |
        +--> E(4) ---+ +--> G(5) -------+

A and B start the project. C follows A; D needs both A and B; E follows B; F needs C and D; G needs D and E; H needs F and G.

(b) Forward and backward pass

Forward pass: $EF = ES + t$, where ES is the largest EF of predecessors. Backward pass: $LS = LF - t$, where LF is the smallest LS of successors. Project duration = largest EF = 17.

Sample: ES of D = max(EF of A, EF of B) = max(4, 3) = 4. ES of F = max(9, 10) = 10. LF of D = min(LS of F, LS of G) = min(12, 10) = 10.

(c) Critical path

Activities with zero total float are critical: A, D, G, H.

Check: $4 + 6 + 5 + 2 = 17$ days. Other paths are shorter (e.g. B-D-G-H = 16, A-D-F-H = 15, A-C-F-H = 14).

Answer: Critical path A-D-G-H; project duration 17 days; floats: B = 1, C = 3, E = 3, F = 2 days.

(a) Expected time and variance

PERT uses the beta distribution:

$$t_e = \frac{a + 4m + b}{6}, \qquad \sigma^2 = \left(\frac{b-a}{6}\right)^2$$

(b) Critical path

Zero-float activities: B, D, F, G. Expected project duration $T_E = 6 + 9 + 5 + 2 = 22$ weeks.

(c) Probability

Variance of the project = sum of variances on the critical path only:

$$\sigma_T^2 = 2.778 + 4.000 + 0.444 + 0.111 = 7.333, \qquad \sigma_T = 2.708\ \text{weeks}$$ $$Z = \frac{T_S - T_E}{\sigma_T}$$

• For 24 weeks: $Z = (24-22)/2.708 = 0.74$, so $P = 0.770$ (from the standard normal table).
• For 20 weeks: $Z = (20-22)/2.708 = -0.74$, so $P = 1 - 0.770 = 0.230$.

Answer: Critical path B-D-F-G, expected duration 22 weeks, sigma = 2.71 weeks; P(finish within 24 weeks) = 77%; P(within 20 weeks) = 23%.

Both CPM (Critical Path Method, DuPont 1957) and PERT (Program Evaluation and Review Technique, US Navy, Polaris project 1958) are network techniques for planning, scheduling and controlling projects.

Terms

• Critical path: the longest path through the network; it fixes the shortest possible project duration. Any delay in a critical activity delays the whole project.
• Total float: the time by which an activity can be delayed without delaying the project completion: $TF = LS - ES = LF - EF$. Critical activities have zero float.
• Free float: delay possible without affecting the earliest start of the next activity.
• Event slack (PERT): for an event, $\text{slack} = T_L - T_E$, the difference between latest allowable and earliest expected event times. Events on the critical path have zero slack.