A NETWORK DIAGRAM SHOWS ACTIVITIES AND THEIR DEPENDENCIES, AND FROM IT THE PROJECT'S MINIMUM DURATION FOLLOWS ARITHMETICALLY. TWO NOTATIONS: ACTIVITY ON NODE (AON) / PRECEDENCE DIAGRAM — activities are boxes, arrows are dependencies. THE MODERN STANDARD, used by all planning software, and needs no dummy activities. ACTIVITY ON ARROW (AOA) — activities are arrows between numbered event nodes. Older; requires DUMMY ACTIVITIES (dotted, zero duration) to express certain dependencies without implying false ones. ── THE FOUR TIME CALCULATIONS ────────────────────────────── ES EARLIEST START — the earliest an activity can begin EF EARLIEST FINISH = ES + duration LS LATEST START = LF − duration LF LATEST FINISH — the latest it can end without delaying the project THE FORWARD PASS (left to right) gives ES and EF: ES = MAXIMUM of the EF of all predecessors (maximum, because an activity waits for its LAST predecessor) THE BACKWARD PASS (right to left) gives LF and LS: LF = MINIMUM of the LS of all successors (minimum, because it must not delay its EARLIEST successor) THE MAX/MIN ASYMMETRY IS THE MOST COMMON ERROR IN EXAMS. Say it aloud while working: FORWARD TAKE THE MAXIMUM, BACKWARD TAKE THE MINIMUM. ── FLOAT (SLACK) ─────────────────────────────────────────── TOTAL FLOAT = LS − ES = LF − EF How long an activity can be delayed WITHOUT DELAYING THE PROJECT. FREE FLOAT = (earliest ES of successors) − EF How long it can be delayed WITHOUT DELAYING ANY SUCCESSOR's earliest start. FREE FLOAT ≤ TOTAL FLOAT ALWAYS. The difference matters practically: USING TOTAL FLOAT CONSUMES IT FOR EVERYONE DOWNSTREAM ON THAT PATH, whereas using free float costs nobody anything. Float belongs to the path, not to the activity — a point contractors and clients argue over in real claims. INDEPENDENT FLOAT — available even if predecessors finish as late as possible and successors start as early as possible. ── THE CRITICAL PATH ─────────────────────────────────────── THE LONGEST PATH THROUGH THE NETWORK, AND THEREFORE THE SHORTEST POSSIBLE PROJECT DURATION. Its activities have ZERO TOTAL FLOAT. DELAYING ANY OF THEM DELAYS THE WHOLE PROJECT BY THE SAME AMOUNT. · There is always at least one; there may be several. · IT CAN CHANGE during execution as actual durations differ from planned ones — so it must be recalculated, not assumed. · MANAGEMENT ATTENTION BELONGS ON IT. Accelerating a non-critical activity achieves nothing at all, which is the single most useful practical consequence of the whole technique. ── A FULLY WORKED NETWORK ────────────────────────────────── ACTIVITY DURATION PREDECESSORS A 4 — B 6 — C 3 A D 5 A E 7 B, C F 4 D, E G 2 F FORWARD PASS: A: ES 0, EF 4 B: ES 0, EF 6 C: ES 4 (after A), EF 7 D: ES 4, EF 9 E: ES = max(EF_B, EF_C) = max(6, 7) = 7, EF 14 F: ES = max(EF_D, EF_E) = max(9, 14) = 14, EF 18 G: ES 18, EF 20 PROJECT DURATION = 20 DAYS BACKWARD PASS, starting from LF(G) = 20: G: LF 20, LS 18 F: LF 18, LS 14 E: LF 14, LS 7 D: LF 14, LS 9 C: LF = LS(E) = 7, LS 4 B: LF = LS(E) = 7, LS 1 A: LF = min(LS_C, LS_D) = min(4, 9) = 4, LS 0 THE COMPLETE TABLE: ACT DUR ES EF LS LF TOTAL FLOAT CRITICAL ──────────────────────────────────────────────────────── A 4 0 4 0 4 0 YES B 6 0 6 1 7 1 C 3 4 7 4 7 0 YES D 5 4 9 9 14 5 E 7 7 14 7 14 0 YES F 4 14 18 14 18 0 YES G 2 18 20 18 20 0 YES ──────────────────────────────────────────────────────── CRITICAL PATH: A → C → E → F → G, 4+3+7+4+2 = 20 DAYS ✓ FREE FLOATS: B has 1 (E starts at 7, B finishes at 6); D has 5 (F starts at 14, D finishes at 9); all others 0. NOTE THAT FOR B, TOTAL FLOAT = FREE FLOAT = 1, while for D both are 5 — they coincide here because each has a single successor on a critical path.

── PERT: SCHEDULING UNDER UNCERTAINTY ────────────────────── Where durations are uncertain, PERT uses THREE ESTIMATES per activity: o OPTIMISTIC — everything goes right m MOST LIKELY — the modal estimate p PESSIMISTIC — everything reasonable goes wrong EXPECTED TIME tₑ = (o + 4m + p) / 6 VARIANCE σ² = ((p − o) / 6)² The weights come from assuming a BETA DISTRIBUTION, which — unlike a normal distribution — can be ASYMMETRIC. That matters because delays are unbounded above and bounded below: an activity can overrun by weeks but cannot finish in less than zero time, so the realistic distribution is skewed right. THE PROJECT-LEVEL ARITHMETIC: · EXPECTED PROJECT DURATION = sum of tₑ ALONG THE CRITICAL PATH · PROJECT VARIANCE = SUM OF THE VARIANCES of the critical activities. VARIANCES ADD; STANDARD DEVIATIONS DO NOT. This is the most commonly botched step. · σ_project = √(sum of variances) · By the CENTRAL LIMIT THEOREM the project duration is approximately NORMAL even though each activity is not, so: z = (T_target − T_expected) / σ_project and the probability is read from the standard normal table. WORKED, on the same network's critical path A-C-E-F-G: ACT o m p tₑ σ² ───────────────────────────────────── A 2 4 6 4.000 0.4444 C 1 3 5 3.000 0.4444 E 4 7 12 7.333 1.7778 F 2 4 8 4.333 1.0000 G 1 2 3 2.000 0.1111 ───────────────────────────────────── Tₑ = 20.667 σ² = 3.7778 σ = 1.9437 PROBABILITY OF FINISHING WITHIN 20 DAYS: z = (20 − 20.667)/1.9437 = −0.343 → 36.6% WITHIN 22 DAYS: z = (22 − 20.667)/1.9437 = +0.686 → 75.4% WITHIN 24 DAYS: z = (24 − 20.667)/1.9437 = +1.715 → 95.7% READ THAT SEQUENCE CAREFULLY, because it is the practical lesson of PERT: THE "EXPECTED" DURATION HAS ONLY ABOUT A 50% CHANCE OF BEING MET. To be 95% confident, this project needs 24 days, not 20.667 — A BUFFER OF OVER THREE DAYS ON A THREE-WEEK PROJECT. Committing to the expected duration is committing to a coin flip, and this is precisely why projects that are "on plan" so often finish late. PERT'S KNOWN WEAKNESS — the MERGE EVENT BIAS: PERT computes probability along the critical path only, ignoring near- critical paths that might overrun and become critical. IT THEREFORE SYSTEMATICALLY UNDERESTIMATES the project duration. Monte Carlo simulation, which samples every path, is the modern remedy. ── GANTT CHART ───────────────────────────────────────────── A bar chart with activities on the vertical axis and time on the horizontal, each bar spanning its start and finish. STRENGTHS: immediately readable by anyone; shows calendar dates and overlaps; excellent for COMMUNICATION and progress marking. WEAKNESS: IT DOES NOT SHOW DEPENDENCIES (unless linked arrows are added), so it cannot reveal WHY a delay propagates or WHICH activities are critical. THE STANDARD PRACTICE: DERIVE THE SCHEDULE FROM THE NETWORK AND PRESENT IT AS A GANTT CHART. The network is the analytical tool; the Gantt chart is the communication tool. Modern software shows both from one model, with the critical path highlighted. ── CRASHING AND FAST TRACKING ────────────────────────────── Two ways to shorten a schedule: FAST TRACKING — perform in parallel activities that were planned in sequence. COSTS NOTHING DIRECTLY BUT INCREASES RISK AND REWORK, because a successor starts before its predecessor is finished. Attack DISCRETIONARY dependencies first, never mandatory ones. CRASHING — add resources to shorten activities. ALWAYS COSTS MORE, so the object is to buy time as cheaply as possible: COST SLOPE = (crash cost − normal cost) / (normal duration − crash duration) = the cost of saving ONE day on that activity THE PROCEDURE: 1. Identify the critical path. 2. Among CRITICAL activities only, crash the one with the LOWEST COST SLOPE. 3. Reduce it one time unit; recompute the network. 4. WATCH FOR A NEW CRITICAL PATH APPEARING — once a parallel path becomes equally long, BOTH must be shortened to gain anything further, so the effective cost per day rises. 5. Repeat while the saving is worth the cost. Illustrative slopes for the worked network: A: normal 4d/Rs 10,000, crash 2d/Rs 14,000 → Rs 2,000/day G: normal 2d/Rs 6,000, crash 1d/Rs 9,000 → Rs 3,000/day E: normal 7d/Rs 20,000, crash 4d/Rs 35,000 → Rs 5,000/day CRASH A FIRST at Rs 2,000/day — BUT ONLY FOR ONE DAY. The parallel path B–E–F–G is 6+7+4+2 = 19 days, so shortening A by one day brings the critical path to 19 as well and BOTH PATHS ARE NOW CRITICAL. Gaining a second day then requires crashing A (or C) AND B simultaneously, at the sum of their slopes. THIS IS THE PATTERN TO EXPECT: THE COST PER DAY SAVED RISES IN STEPS as each additional path becomes critical, which is why crashing has a natural economic stopping point. NEVER CRASH A NON-CRITICAL ACTIVITY: it costs money and saves nothing. THE ECONOMIC STOPPING POINT: crash while the DAILY SAVING (indirect costs, overheads, liquidated damages avoided, early revenue) EXCEEDS the cost slope. THE OPTIMUM IS WHERE TOTAL COST — direct plus indirect — IS MINIMUM, not where the duration is shortest.