Break-Even Chart for the StaMCo Air Force Item: 22-Mec-B4 Integrated Manufacturing Systems, December 2018, Q2

Question 2: Break-Even Chart for the StaMCo Air Force Item (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. A cost estimate for a special stamped item, quoted in the layered form that estimating departments use: a material charge, a labour content, two burden rates applied in sequence, and a target profit expressed on total cost.

Cost estimate as printed on the exam paper
Element	Rate as quoted	Behaviour
Material	$0.50 per unit	variable
Tooling	20 hours	fixed, one-time
Unit direct labour	0.4 hour per unit	variable
Labour rate	$2.00 per hour	—
Factory burden	115 per cent of labour	follows labour
General and administrative burden	20 per cent of factory cost	follows factory cost
Target profit	10 per cent of total cost	—
Basis (planned quantity)	10,000 units	—
Unit sales price	$3.00	—

Find. The break-even chart: the fixed-cost line, the total-cost line, the revenue line, the break-even quantity, and the position of the 10,000-unit basis on that chart, together with the price the target profit implies.

Check: the tooling charge. The estimate quotes tooling as “20 hours” rather than as a sum of money, so it is a labour charge of 20 × $2.00 = $40.00 incurred once, whatever the run length. The paper does not say whether factory burden is carried on those tooling hours. The reading taken here — and stated as an assumption under Note 1 — is that burden is carried on all factory labour, tooling included, since the burden rate exists precisely to recover plant overhead on labour hours. That makes the fixed cost $103.20. If tooling were instead treated as a direct charge outside the burden pool, the fixed cost would be $48.00 and the break-even quantity would move from 307 units to 143 units. Either reading leaves the conclusion untouched: the fixed cost is trivial beside the 10,000-unit revenue, and the venture breaks even in the first few hundred pieces.

Approach. Push a single unit and the one-time tooling charge through the identical burden chain — labour, then factory burden, then general and administrative burden — to obtain a variable cost per unit and a fixed cost. Those two numbers plus the price give the straight-line cost and revenue equations that the break-even chart plots.

Cost one unit up to factory cost. Direct labour is the hours times the rate, and factory burden is applied to that labour: $$\text{unit direct labour} = 0.4 \times 2.00 = 0.80$$ $$\text{unit factory burden} = 1.15 \times 0.80 = 0.92$$ $$\text{unit factory cost} = 0.50 + 0.80 + 0.92 = 2.22 \text{ per unit}$$ all figures in dollars.
Add the general and administrative burden. This second burden is struck on factory cost, not on labour, so it is applied to the whole 2.22: $$\text{unit G and A} = 0.20 \times 2.22 = 0.444, \qquad \text{unit total cost} = 2.22 + 0.444 = 2.664 \text{ per unit}$$ This 2.664 is the slope of the total-cost line on the chart.
Cost the tooling through the same chain. The tooling hours are a one-time charge and therefore the whole of the fixed cost: $$\text{tooling labour} = 20 \times 2.00 = 40.00, \qquad \text{burden} = 1.15 \times 40.00 = 46.00$$ $$\text{tooling factory cost} = 86.00, \qquad \text{plus G and A} = 1.20 \times 86.00 = 103.20$$ $$\boxed{\;\text{fixed cost } F = 103.20, \qquad \text{variable cost } v = 2.664 \text{ per unit}\;}$$
Write the two straight lines the chart plots. With $Q$ the number of units made and sold and a price of 3.00 each, $$\text{TC}(Q) = 103.20 + 2.664\,Q, \qquad \text{TR}(Q) = 3.000\,Q$$ The fixed-cost line is the horizontal at 103.20; the total-cost line starts there and rises at 2.664 per unit; the revenue line starts at the origin and rises at 3.000.
Locate the break-even point. The contribution each unit makes toward the fixed cost is the price less the variable cost: $$c = 3.000 - 2.664 = 0.336 \text{ per unit}$$ $$Q_{\text{BE}} = \frac{F}{c} = \frac{103.20}{0.336} = 307.1 \approx 307 \text{ units}$$ $$\boxed{\;Q_{\text{BE}} = 307 \text{ units, or about } 921 \text{ in sales}\;}$$
Place the 10,000-unit basis on the chart. Evaluating the same two lines at the planned quantity, $$\text{TC}(10{,}000) = 103.20 + 2.664(10{,}000) = 26{,}743.20$$ $$\text{TR}(10{,}000) = 3.000(10{,}000) = 30{,}000.00$$ $$\text{profit} = 30{,}000.00 - 26{,}743.20 = 3{,}256.80$$ which is 12.2 per cent of total cost. The margin of safety is $(10{,}000 - 307)/10{,}000 = 96.9$ per cent: sales could fall by nearly all of the planned volume before the job lost money.
Check the price against the target profit. Costing the 10,000-unit run in the estimating department's own layered form gives factory cost $10{,}000(2.22) + 86.00 = 22{,}286.00$, G and A of $0.20(22{,}286.00) = 4{,}457.20$, and a total cost of $26{,}743.20$. The target profit is $$0.10 \times 26{,}743.20 = 2{,}674.32 \quad\Rightarrow\quad \text{target sales value} = 29{,}417.52, \quad \text{target unit price} = 2.942$$ $$\boxed{\;\text{target price } 2.94 \text{ per unit} \ < \ \text{actual price } 3.00 \text{ per unit}\;}$$
so the quoted price clears the target profit with room to spare.
Say at what volume the target profit is actually earned. The target is a moving one, because it is stated on cost rather than as a fixed sum. Setting profit equal to 10 per cent of total cost, $$3.000\,Q - (103.20 + 2.664\,Q) = 0.10\,(103.20 + 2.664\,Q)$$ $$0.0696\,Q = 113.52 \quad\Rightarrow\quad Q = 1{,}631 \text{ units}$$ Beyond about 1,631 units the item earns more than the 10 per cent target, and at the 10,000 basis it earns 12.2 per cent.

Figure 2.1 — The break-even chart. Left, the full 10,000-unit basis, where the fixed cost is so small that the crossing is pressed against the origin and the profit wedge dominates. Right, the same chart zoomed to the first 2,000 units, where the break-even point at 307 units and the 1,631 units needed to earn the 10 per cent target profit can actually be read.

Question 2 — results (all money in dollars)
Quantity	Value
Unit direct labour cost	0.80
Unit factory burden	0.92
Unit factory cost	2.22
Unit general and administrative burden	0.444
Unit total cost (slope of the cost line)	2.664
Fixed cost from tooling (intercept of the cost line)	103.20
Contribution per unit	0.336
Break-even quantity	307 units
Break-even sales value	921
Total cost at the 10,000 basis	26,743.20
Sales revenue at the 10,000 basis	30,000.00
Profit at the 10,000 basis	3,256.80, i.e. 12.2 per cent of cost
Margin of safety at the 10,000 basis	96.9 per cent
Price implied by the 10 per cent target profit	2.94 per unit
Volume at which the target profit is earned	1,631 units

Topic. Layered cost estimating with burden rates, and linear break-even analysis of a make decision.

Key relations. factory cost = material + direct labour + factory burden, with burden struck on labour; total cost = factory cost × (1 + G and A rate); $\text{TC}(Q) = F + vQ$; $\text{TR}(Q) = pQ$; contribution $c = p - v$; $Q_{\text{BE}} = F/c$; margin of safety $= (Q_{\text{plan}} - Q_{\text{BE}})/Q_{\text{plan}}$; and, for a profit target stated on cost, $pQ - (F + vQ) = r(F + vQ)$.

Why this works. Break-even analysis is nothing more than the observation that over the relevant range every cost is either proportional to output or independent of it. Once each element of the estimate has been sorted into those two bins, the cost function is a straight line and the whole economics of the job reduces to two numbers: where the line starts and how steeply it rises. Applying the burden rates in the order the estimate states them — labour first, then factory cost — is essential, because G and A struck on factory cost already includes the factory burden beneath it, and a rate applied to the wrong base compounds through everything after it.

Common pitfall. Three recur. Treating tooling as a per-unit charge, which destroys the fixed cost and gives a chart with no break-even point. Applying the 20 per cent G and A rate to labour rather than to factory cost, which understates unit total cost by about 11 per cent (2.38 against 2.664 per unit). And adding the target profit into the cost line, which makes the chart plot a price rather than a cost; the target profit belongs on the revenue side, as a check that the quoted price is high enough, exactly as done in step 7.

Source. E. S. Buffa and R. K. Sarin, Modern Production / Operations Management, 8th ed., chapters on cost estimating and on break-even and economic analysis of alternatives; R. B. Chase, F. R. Jacobs and N. J. Aquilano, Operations and Supply Chain Management, 16th ed., break-even analysis in the process-selection chapter.

Question 2 of 7: Break-Even Chart for the StaMCo Air Force Item

Question 2: Break-Even Chart for the StaMCo Air Force Item (20 marks)