CPM with mason resource aggregation chart

The activities are given by their event numbers (i-j), so each activity is an arrow from event i to event j.

Network diagram

(1) --1-2(3)--> (2)
(2) --2-3(3)--> (3)
(2) --2-4(4)--> (4)
(2) --2-5(2)--> (5)
(3) --3-10(3)--> (10)
(4) --4-6(2)--> (6)
(4) --4-7(4)--> (7)
(5) --5-9(4)--> (9)
(6) --6-8(2)--> (8)
(7) --7-9(4)--> (9)
(8) --8-9(3)--> (9)
(9) --9-11(3)--> (11)
(10) --10-11(2)--> (11)
(11) --11-12(2)--> (12)

Event times

TE from the forward pass, TL from the backward pass (days).

Activity times and floats

$TF = TL_j - TE_i - D$, $FF = TE_j - TE_i - D$, $IntF = TF - FF$.

Project duration and critical path

• Critical activities (TF = 0): 1-2, 2-4, 4-7, 7-9, 9-11, 11-12
• Critical path: 1-2-4-7-9-11-12 (3 + 4 + 4 + 4 + 3 + 2 = 20 days)

Answer: project duration = 20 days; critical path = 1-2-4-7-9-11-12.

Resource aggregation chart (early start schedule)

Each activity starts at its EST and needs the given number of masons every day until its EFT. The masons of all activities working on a given day are added.

 12 |              # #                       
 11 |              # #                       
 10 |          # # # #                       
  9 |          # # # #                       
  8 |      # # # # # #                       
  7 |      # # # # # # # #                   
  6 |      # # # # # # # #                   
  5 |      # # # # # # # #                   
  4 |      # # # # # # # #         # # #     
  3 |      # # # # # # # # # # #   # # #     
  2 |      # # # # # # # # # # #   # # #     
  1 |# # # # # # # # # # # # # # # # # # # # 
    +----------------------------------------
     1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0  day (last digit)

• Peak requirement = 12 masons (days 8 and 9).
• Total mason-days = 101 (check: the sum of duration x masons over all activities = 101).
• Average = 101/20 = 5.05, i.e. about 5 masons per day.

Allocation of masons

The early start allocation needs only 1 mason on days 1-3, then 8 to 12 masons on days 4-9, and then 7, 3, 1, 4 and 1 masons on the later days. The demand is very uneven. Because the non-critical activities 2-3, 2-5, 3-10, 5-9 and 10-11 have a total float of 6-7 days, they can be delayed to cut the peak (for example, by starting 5-9, which needs 4 masons, later) without extending the 20 days. This is resource smoothing.