Throughput Ranking, a Special Order and the Cost of Idle Reasoning
Tawi Ltd runs a single machining line with capacity of 34,000 machine hours a year; a second shift cannot be added before 2027. Per-unit data:
Product A: price KES 1,200, variable cost KES 720, 2 hours, annual demand 8,000 units. Product B: price KES 2,000, variable cost KES 1,400, 3 hours, annual demand 5,000 units. Product C: price KES 900, variable cost KES 620, 1 hour, annual demand 6,000 units.
Annual fixed costs are KES 5,400,000 and are unaffected by product mix. Total demand equals 37,000 machine hours. A European buyer has offered to take 2,000 units of a modified A (product D) at KES 1,050 each; D uses the same line, 2 hours per unit, and has a variable cost of KES 700 including the modification. The buyer requires a two-year commitment and will not accept a smaller volume.
1. Prepare the optimal production plan given the machine-hour constraint and compute the resulting profit.
Read the worked answer
Rank by contribution per machine hour, not per unit. With one binding constraint, the correct ranking criterion is contribution per unit of the scarce resource.
| Product | Price | Var. cost | Contribution/unit | Hours | Contribution/hour | Rank |
|---|---|---|---|---|---|---|
| A | 1,200 | 720 | 480 | 2 | 240 | 2 |
| B | 2,000 | 1,400 | 600 | 3 | 200 | 3 |
| C | 900 | 620 | 280 | 1 | 280 | 1 |
Ranking per unit would put B first (600 > 480 > 280), which is wrong - B is the worst consumer of the bottleneck.
Allocation of 34,000 hours:
| Units | Hours | Contribution (KES) | |
|---|---|---|---|
| C | 6,000 (full demand) | 6,000 | 1,680,000 |
| A | 8,000 (full demand) | 16,000 | 3,840,000 |
| B | 4,000 (balancing) | 12,000 | 2,400,000 |
| Total | 34,000 | 7,920,000 |
B demand of 5,000 units is not met - 1,000 units (3,000 hours) are unmet and represent lost contribution of KES 600,000.
Profit: contribution 7,920,000 − fixed costs 5,400,000 = KES 2,520,000.
Note that C, despite the lowest unit contribution, is the most profitable use of the line. The plan should be revisited if B's price rises by more than KES 120/unit (the gap between its 200/hr and A's 240/hr, spread over 3 hours).
2. Evaluate the special order for 2,000 units of D, state the minimum price at which it becomes acceptable, and identify the assumptions that could overturn your recommendation.
Read the worked answer
Step 1 - Incremental contribution from D.
(1,050 − 700) x 2,000 = KES 700,000.
Step 2 - The opportunity cost is not zero. The line is full. Every hour used by D must come from the marginal product, which is B at KES 200 per hour. D requires 2,000 x 2 = 4,000 hours.
- Contribution displaced: 4,000 hours x KES 200 = KES 800,000 (equivalent to 1,333 units of B x KES 600)
Step 3 - Net effect. 700,000 − 800,000 = (KES 100,000). Reject on quantitative grounds.
Step 4 - Minimum acceptable price. D must generate at least KES 800,000 of contribution:
- Required contribution per unit = 800,000 / 2,000 = KES 400
- Minimum price = 700 + 400 = KES 1,100 per unit
At exactly KES 1,100 the company is indifferent on financial grounds and the qualitative factors decide.
Assumptions that could reverse the decision:
- Fixed costs are not truly fixed - if a second shift becomes possible in 2027, the second year of the commitment uses capacity that would otherwise be idle, and the two-year order may be worth taking despite a first-year loss.
- B's demand is overstated - if B demand is only 3,500 units (10,500 hours), there is spare capacity and D's contribution is incremental in full.
- Scrap value of unwanted hours - if the 4,000 hours cannot all be redeployed to B, the opportunity cost falls.
- Learning and market entry - the European buyer may open a larger market; but that is a strategic option, and the board should price it explicitly rather than accept a loss disguised as a strategy.
- Price discipline - accepting KES 1,050 may reset expectations with domestic customers asking for discounts.
Recommendation. Reject at KES 1,050 and counter at KES 1,100–1,150, testing whether the buyer's volume can be cut to 1,300 units (2,600 hours), which would fit alongside full B demand.
3. The sales director argues that because product B shows the highest contribution per unit and the highest gross margin percentage, capacity should be switched from C to B. Assess the argument and explain what it reveals about the company's decision-making process.
Read the worked answer
The argument is wrong, and its wrongness is instructive.
B's contribution per unit (KES 600) is the highest, and its contribution margin (600/2,000 = 30%) exceeds C's (280/900 = 31.1%) - actually C's percentage is higher, so even the margin claim fails. But the decisive error is more basic: neither unit contribution nor margin percentage is a valid ranking criterion when a resource is scarce.
Quantify the cost of following the sales director: switching one hour from C to B loses KES 80 (280 − 200). Full reversal of the plan - meeting all B demand and cutting C - would require 15,000 hours for B, leaving 19,000 hours for A: 8,000 units of A (16,000 hrs) plus 3,000 hours of nothing usable for C at 3,000 hours (C needs only 6,000 hrs total, so 3,000 units).
| Plan | Contribution (KES) |
|---|---|
| Optimal (C full, A full, B part) | 7,920,000 |
| 'B-first' (B full, A full, C part) | 3,000,000 + 3,840,000 + 1,500,000 = 8,340,000... |
Correction required in the working: the B-first plan cannot meet B demand and A demand within 34,000 hours (15,000 + 16,000 = 31,000), leaving 3,000 hours for C - which is 3,000 units. Contribution = 3,000 (B) x 600 = 1,800,000 + 3,840,000 + 3,000 x 280 = 840,000 → KES 6,480,000, which is KES 1,440,000 below the optimal plan.
What this reveals:
- Incentive misalignment. If sales bonuses are paid on unit contribution or revenue, the sales director is optimising their own scorecard, not the factory. The bonus should be based on contribution per constrained hour.
- Absence of a capacity hierarchy. The company has no agreed throughput ranking, so the argument is settled by seniority rather than analysis.
- A decision rule to adopt. Report weekly: hours available, hours by product, contribution per hour, and the value of unmet demand. That single table ends the argument permanently.
- Strategic question. B's weak throughput economics suggest it is under-priced for the hours it consumes; a price increase of about KES 120 on B would make it equal to A per hour, and the market should be tested before 2027 capacity expansion is committed.