07-Str-B2 · May 2016

Question 6 of 6: Project Control — S-curves and progress measurement from a bar chart

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format: National Exams, May 2016 — 07-Str-B2 Management of Construction. Three hours, closed book, one approved Casio or Sharp calculator permitted. Six questions of equal value (20 marks each); any five constitute a complete paper and only the first five answered are marked. All six are worked below so that the paper can be used for revision whichever five a candidate chooses.

Reference texts: Hegazy, T., Computer-Based Construction Project Management (Prentice Hall) — precedence networks with lags, total and free float, project overhead versus general overhead, and the bar-chart/S-curve control method behind Questions 1, 3 and 6; Hendrickson, C. & Au, T., Project Management for Construction (2nd ed., Carnegie Mellon) — Chapters 5 (cost estimation), 10 (scheduling) and 12 (cost control, monitoring and accounting), the source of the earned-value quantities used in Question 6; Halpin, D.W. & Senior, B.A., Construction Management (4th ed., Wiley) — competitive bidding, unbalanced bids, indirect-cost structure and construction safety; Sullivan, W.G., Wicks, E.M. & Koelling, C.P., Engineering Economy (17th ed., Pearson) — Chapters 5 and 6, present-worth analysis and the repeatability assumption for alternatives with unequal lives, used in Question 4; Peurifoy, R.L. & Oberlender, G.D., Estimating Construction Costs (6th ed., McGraw-Hill) — job overhead versus general overhead; Canadian Construction Documents Committee, CCDC 2 — Stipulated Price Contract (2020) and CCDC 23 — A Guide to Calling Bids and Awarding Contracts — bid-call practice, bid security and award criteria for Question 2; Ron Engineering (M.J.B. Enterprises line of cases) as summarised in Goldsmith, I. & Heintzman, T.G., Goldsmith on Canadian Building Contracts (5th ed., Thomson Reuters) — the Contract A/Contract B doctrine that governs a Canadian public bid call; WorkSafeBC, Occupational Health and Safety Regulation (Parts 4, 8, 11, 13, 18, 19 and 20) and the BC Workers Compensation Act, together with CSA Z259 (fall protection), CSA Z94.4 (respirators) and CSA W117.2 (welding safety) — the Canadian rule set behind Question 5.

Question 1 (network). Every activity letter and duration is printed inside its box and reads cleanly. The link routing was traced at high magnification: Start feeds A, D and G; a riser from the right edge of D feeds B; the horizontal link D → E carries the only labelled lag on the sheet, FS 6; a riser from the right edge of G feeds E; G also feeds H, H feeds I, A feeds B, B feeds C, E feeds F; and C, F and I terminate at End. Every unlabelled arrow is an ordinary finish-to-start link with zero lag, which is the only reading consistent with the drawing.

Question 6 (bar chart). Every percentage label falls on a week boundary, so the printed figures are cumulative percent complete at each week end. Planned: A 20/60/100 in weeks 1–3; B 10/80 in weeks 1–2, finishing in week 3; C 20/70 in weeks 3–4, finishing in week 5. Actual: A 10/50/90 in weeks 1–3, its bar closing in week 4; B 70 at week 2, its bar closing in week 3; C 50 at week 3, its bar closing exactly on the week-4 gridline. A bar that closes is read as 100 % complete from that week end onward, which is the standard convention and the only reading that lets part (c) be answered at all.

Question 6: Project Control — S-curves and progress measurement from a bar chart (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Three tasks with budgets summing to $100,000, and cumulative percent complete read at each week end from the planned (grey) and actual (black) bars.

Task	Budget	Basis	Wk 1	Wk 2	Wk 3	Wk 4	Wk 5	Wk 6
A	$30,000	Planned	20 %	60 %	100 %	100 %	100 %	100 %
A	$30,000	Actual	10 %	50 %	90 %	100 %	100 %	100 %
B	$20,000	Planned	10 %	80 %	100 %	100 %	100 %	100 %
B	$20,000	Actual	0 %	70 %	100 %	100 %	100 %	100 %
C	$50,000	Planned	0 %	0 %	20 %	70 %	100 %	100 %
C	$50,000	Actual	0 %	0 %	50 %	100 %	100 %	100 %

Find. (a) The planned and actual cumulative-value curves; (b) a comparison of actual against planned progress at the end of week 3; (c) the total expenditure to the end of week 4 against the planned amount for that date.

Approach. Convert each task's cumulative percent complete into dollars by multiplying by that task's budget, sum across the three tasks week by week to obtain two cumulative curves, plot them, and then read the two questions off the curves as a schedule variance at week 3 and a spend comparison at week 4.

Set up the value-of-work-done rule. With no separate actual-cost data on the sheet, progress is valued at budget, which is the standard construction reading of a percent-complete bar chart: $$ V(w)=\sum_{t\in\{A,B,C\}}\text{Budget}_t\times p_t(w) $$ where $p_t(w)$ is the cumulative fraction of task $t$ complete at the end of week $w$. Applying this to the planned percentages gives the budgeted cost of work scheduled, $BCWS$; applying it to the actual percentages gives the budgeted cost of work performed, $BCWP$, the earned value. The total project budget is $30{,}000+20{,}000+50{,}000=100{,}000$ dollars.
Build the planned curve week by week. Working across the planned rows: $$ BCWS(1)=0.20(30{,}000)+0.10(20{,}000)+0=6{,}000+2{,}000=\$8{,}000 $$ $$ BCWS(2)=0.60(30{,}000)+0.80(20{,}000)+0=18{,}000+16{,}000=\$34{,}000 $$ $$ BCWS(3)=30{,}000+20{,}000+0.20(50{,}000)=\$60{,}000 $$ $$ BCWS(4)=30{,}000+20{,}000+0.70(50{,}000)=\$85{,}000 $$ and $BCWS(5)=BCWS(6)=100{,}000$ dollars, since everything is planned complete by the end of week 5.
Build the actual curve the same way. Substituting the actual percentages: $$ BCWP(1)=0.10(30{,}000)=\$3{,}000 $$ $$ BCWP(2)=0.50(30{,}000)+0.70(20{,}000)=15{,}000+14{,}000=\$29{,}000 $$ $$ BCWP(3)=0.90(30{,}000)+20{,}000+0.50(50{,}000)=27{,}000+20{,}000+25{,}000=\$72{,}000 $$ $$ BCWP(4)=30{,}000+20{,}000+50{,}000=\$100{,}000 $$ after which the curve is flat, the work being complete. The weekly (non-cumulative) figures behind these totals are $8k, $26k, $26k, $25k, $15k, $0 planned against $3k, $26k, $43k, $28k, $0, $0 actual — the week-3 and week-4 surge is task C running well ahead of its plan.
Plot the two curves — part (a). Both curves take the characteristic flattened-S shape: a slow start while only one or two tasks are mobilised, a steep middle while all tasks overlap, and a flattening tail as the work closes out. The actual curve begins below the planned curve, crosses it during week 3, and reaches the $100,000 ceiling a full week before the plan does.

Figure 6.1 — Planned and actual S-curves for part (a). Cumulative value of work in place against time; the dashed vertical marks the end of week 3, the reporting date for part (b). The actual curve starts behind, overtakes the plan during week 3, and closes out at the end of week 4 rather than week 5.

Comment on the position at the end of week 3 — part (b). Reading both curves at $w=3$ and differencing: $$ SV = BCWP(3)-BCWS(3)=72{,}000-60{,}000 $$ $$ \boxed{SV = +\$12{,}000\ \text{ahead of schedule}} $$ and the schedule performance index is $$ SPI=\frac{BCWP(3)}{BCWS(3)}=\frac{72{,}000}{60{,}000}=1.20 $$ so 20 % more work has been earned than was scheduled. The aggregate hides two opposite movements, and a useful comment must separate them: $$ \Delta_A=30{,}000(0.90-1.00)=-\$3{,}000,\quad \Delta_B=20{,}000(1.00-1.00)=\$0,\quad \Delta_C=50{,}000(0.50-0.20)=+\$15{,}000 $$ which sum to the $12,000 schedule variance. Task A is the problem: it was due to finish at the end of week 3 and stands at 90 %, a slip that began in week 1 and never recovered. Task B started a half-week late but caught up and finished on time. Task C started ahead of its planned date, is at 50 % against a planned 20 %, and is carrying the whole favourable variance. In management terms the project is comfortably ahead overall, but the favourable position is concentrated in the largest task; the small residual on A should still be closed out, because a 10 % tail on a finished-looking task is where scope is most often forgotten.
Compare expenditure at the end of week 4 — part (c). All three tasks are complete by the end of week 4, so the value of work performed has reached the full project budget, whereas the plan expected 70 % of task C still to be closing out: $$ \text{Actual to end of week 4}=BCWP(4)=\$100{,}000 $$ $$ \text{Planned to end of week 4}=BCWS(4)=\$85{,}000 $$ $$ \boxed{\text{Expenditure exceeds the planned amount by }\$15{,}000\ (SPI=1.18)} $$ The interpretation matters as much as the number. This is not a cost overrun: the $100,000 spent is exactly the approved budget, and nothing beyond the budgeted value of the work has been paid. It is a cash-flow variance created by finishing early — the owner must fund $15,000 sooner than the baseline forecast, and the contractor will invoice the balance of the contract a week ahead of the projected drawdown. The remaining budget after week 4 is nil, and weeks 5 and 6 of the baseline are now free.

Final Results — Question 6
End of week	1	2	3	4	5	6
Planned cumulative (BCWS)	$8,000	$34,000	$60,000	$85,000	$100,000	$100,000
Actual cumulative (BCWP)	$3,000	$29,000	$72,000	$100,000	$100,000	$100,000
Variance (actual − planned)	−$5,000	−$5,000	+$12,000	+$15,000	$0	$0
(b) End of week 3: ahead of schedule by $12,000, SPI = 1.20. Task A behind by $3,000 (90 % against 100 % planned); task B on plan and complete; task C ahead by $15,000 (50 % against 20 % planned).
(c) End of week 4: total expenditure $100,000 against a planned $85,000 — $15,000 more spent than the baseline forecast for that date, SPI = 1.18. The project is complete a week early; the full budget is spent but not exceeded.

Check — expenditure is valued at budget, not at recorded cost. The bar chart supplies percent complete and task budgets only; no separate actual-cost record is given. "Total expenditures" is therefore computed as the budgeted cost of work performed, which is the conventional reading of this question type. On that basis a schedule variance can be calculated but a cost variance cannot: $CV=BCWP-ACWP$ needs an actual-cost figure the paper does not supply. If a candidate is given actual costs in a variant of this question, the same table extends by one row and $CV$ and $CPI$ follow immediately.

Topic. Progress measurement and cost control: converting a percent-complete bar chart into planned and actual S-curves, and reading schedule variance from them.

Key relations. $BCWS(w)=\sum_t B_t\,p_t^{\text{plan}}(w)$; $BCWP(w)=\sum_t B_t\,p_t^{\text{actual}}(w)$; schedule variance $SV=BCWP-BCWS$; schedule performance index $SPI=BCWP/BCWS$; cost variance $CV=BCWP-ACWP$ and $CPI=BCWP/ACWP$ where actual cost is known; estimate at completion $EAC=BAC/CPI$.

Why this works. A bar chart tells a manager which activities are late; it cannot tell them whether the project is late, because a two-week slip on a $5,000 task and a two-week slip on a $500,000 task look identical on the chart. Weighting each task's percent complete by its budget converts progress into a single common currency, and cumulating that currency over time produces the S-curve. The S shape is not a convention but a consequence of how projects mobilise: few crews at the start, many in the middle, few at the end, so the rate of value creation rises and then falls, and its integral is sigmoidal. Once both curves exist, the vertical gap between them at the reporting date is the schedule variance expressed in dollars — a quantity a project manager, an owner and a lender all understand — and the horizontal gap is the time by which the project is running ahead or behind. Decomposing the aggregate variance back to individual tasks, as in step 5, is what turns the measurement into a management action; the aggregate alone would have concealed that task A is in trouble.

Common pitfall. The most damaging error is reading the horizontal gap between the curves as a cost variance or the vertical gap as a time variance; the vertical gap is dollars of schedule variance and the horizontal gap is the time slippage, and they answer different questions. The second is plotting the actual curve only as far as work has been reported and then comparing it with the full planned curve, which manufactures an apparent overrun out of nothing. The third is treating the $15,000 in part (c) as a cost overrun: nothing has been overspent, the money has simply moved earlier, and a candidate who says "over budget" without qualification loses the mark. The fourth is failing to weight by budget at all and averaging the three percentages — here that would give 80 % against a planned 73 % at week 3 and a quite different-looking answer.

Source. Hendrickson & Au, Project Management for Construction, Chapter 12 (cost control, monitoring and accounting — the budgeted-cost-of-work-performed formulation); Hegazy, Computer-Based Construction Project Management, Chapters 8–9 (progress measurement, cash flow and the S-curve); Halpin & Senior, Construction Management, Chapter 16 (project control and earned value).

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