Segment Profitability Under Shared Capacity Constraint
Ridgeline Components operates three product lines in one plant: Alpha, Beta, Gamma. Monthly capacity is 10,000 machine hours. Per-unit data: Alpha sells for $90, variable cost $54, uses 3 machine hours, monthly demand 1,200 units. Beta sells for $140, variable cost $92, uses 4 hours, demand 900 units. Gamma sells for $60, variable cost $30, uses 1 hour, demand 3,000 units. Fixed plant costs are $210,000 per month and are allocated to lines based on machine hours. The controller reports Alpha as the most profitable line because it has the highest contribution margin per unit ($36). The sales VP wants to drop Gamma, which shows the lowest contribution per unit ($30). Current plan leaves 1,000 machine hours idle.
1. Rank the three lines for the optimal use of the constrained resource and determine the profit-maximizing monthly production plan.
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Use contribution margin per machine hour, not per unit.
- Alpha: $36 / 3 = $12.00 per hour
- Beta: $48 / 4 = $12.00 per hour
- Gamma: $30 / 1 = $30.00 per hour
Gamma dominates. Produce all Gamma demand: 3,000 units x 1 hr = 3,000 hours, CM = $90,000.
Remaining hours: 10,000 - 3,000 = 7,000. Alpha and Beta tie at $12/hr, so the mix between them does not change total CM. Fill full demand for both: Alpha 1,200 x 3 = 3,600 hrs (CM $43,200); Beta 900 x 4 = 3,600 hrs (CM $43,200). Total hours = 10,200, which exceeds capacity by 200.
So the binding choice is only between Alpha and Beta at the margin. Drop 50 units of Beta (200 hrs) or ~67 units of Alpha. Total CM = $90,000 + $43,200 + $43,200 - (50 x $48) = $174,000. Less fixed $210,000 = operating loss of $36,000. Dropping Gamma would free 3,000 hours but forgo $90,000 of CM while the next-best use earns only $12/hr ($36,000), so dropping Gamma destroys $54,000 of profit.
2. The controller argues Alpha is the most profitable line because it has the highest per-unit contribution. Explain the flaw and state what the reported line profitability would show if fixed costs are allocated on machine hours.
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Per-unit CM ignores the hours consumed. Alpha earns $12/hr, identical to Beta and far below Gamma's $30/hr. Allocating fixed cost by machine hours ($210,000 / 10,000 = $21/hr) charges Alpha $63/unit, Beta $84/unit, Gamma $21/unit. Reported line margins: Alpha $36 - $63 = -$27; Beta $48 - $84 = -$36; Gamma $30 - $21 = +$9. This makes Gamma look like the only 'profitable' line, which coincidentally matches the correct decision here but for the wrong reason: the allocation is arbitrary and would mislead if relative prices changed. The allocation does not affect the total; it only redistributes the same $210,000. Decisions should use contribution margin per constrained hour, and allocated fixed costs are irrelevant because they do not change with the decision.
3. A customer offers to buy 400 additional Beta units at $110 each, using hours that would otherwise be idle. Should Ridgeline accept, and what is the minimum acceptable price?
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There are no idle hours under the optimal plan (10,200 hours demanded vs. 10,000 available), so accepting requires displacing Alpha or Beta production. Each displaced unit of Alpha or Beta yields $12/hr. The special order price must cover variable cost plus the opportunity cost of displaced hours: $92 + (4 hrs x $12) = $140. At $110, contribution is $18/unit, or $4.50/hr, well below the $12/hr displaced. Reject. Minimum acceptable price is $140 per unit. If the order could be filled in genuinely idle hours, the floor would be just variable cost of $92; the capacity state, not the price alone, drives the answer.