4.4 Constraints & Product Mix

The Theory of Constraints

The theory of constraints (TOC) holds that every process has at least one constraint (a bottleneck) that limits total output, or throughput. Improving any non-bottleneck step does not raise overall output — only relieving the binding constraint does.

TOC's five focusing steps: (1) identify the constraint, (2) exploit it (run it at full, profitable capacity), (3) subordinate everything else to the constraint, (4) elevate the constraint (add capacity), and (5) repeat as a new constraint emerges. The mindset: maximize flow through the bottleneck, not local efficiency.

Identifying the Binding Constraint

A constraint binds when demand for a resource exceeds its supply. Typical constraints are machine hours, labor hours, raw material, or a regulatory/space limit.

To test which resource binds, compute the total amount of each resource required to meet full demand and compare to capacity. If only one resource is short, you have a single binding constraint and can solve it with simple ranking. If two or more are simultaneously short, you need linear programming.

Trap: the highest-CM product per unit is not necessarily the most profitable to make when it also consumes the most scarce resource.

Contribution Margin per Unit of Constrained Resource

When a single resource is scarce, maximize total contribution by ranking products on:

CM per unit of the constrained resource = CM per unit of product ÷ amount of scarce resource each unit uses.

This converts product-level margins into margins per scarce machine hour (or labor hour, or pound of material), which is the true measure of profitability under a constraint.

Worked Example: Ranking by CM per Machine Hour

Two products share a press limited to 2,400 machine hours per week.

Beta earns $30 per machine hour vs. Alpha's $20, so make Beta first despite its lower per-unit CM.

• Beta: 1,200 units × 1.5 hr = 1,800 hr → CM = 1,200 × $45 = $54,000
• Remaining hours = 2,400 − 1,800 = 600 hr → Alpha = 600 ÷ 3.0 = 200 units → CM = 200 × $60 = $12,000
• Total CM = $66,000. Making Alpha first would have wasted the scarce resource.

Throughput Accounting

Throughput accounting is a TOC-aligned costing view that treats almost all costs except direct materials as fixed in the short run.

• Throughput contribution = sales revenue − totally variable cost (typically direct materials only)
• Direct labor and overhead are treated as operating expense (period cost), not product cost.

Under a bottleneck, rank products by throughput per bottleneck minute/hour = (price − material cost) ÷ bottleneck time per unit. This sharpens the focus on flow: it rewards products that move quickly through the constraint, even if traditional full-cost margins disagree.

Linear Programming and Optimal Mix

When two or more constraints bind simultaneously (e.g., both machine hours and labor hours are short), simple ranking fails and linear programming (LP) is used.

An LP model has:

• An objective function to maximize (total contribution margin) or minimize (cost).
• Constraints expressed as inequalities (resource use ≤ capacity; demand and non-negativity limits).
• A feasible region — the set of mixes satisfying all constraints.

The optimum always lies at a corner point (vertex) of the feasible region. The shadow price of a constraint is the additional contribution gained from one more unit of that scarce resource — the most a firm should pay to relax it. Use LP whenever a single CM-per-resource ranking cannot satisfy all binding limits at once.

Relaxing the Constraint and Shadow Prices

The payoff to TOC comes from elevating the binding constraint — adding capacity where it matters most. The value of one extra unit of the scarce resource is its shadow price, equal to the contribution margin per unit of the constrained resource for the product that would use it.

In the press example, the binding press earns Beta $30 per machine hour at the margin. So one additional press hour is worth up to $30 (until Beta's demand of 1,200 units is fully met, after which the next-best use, Alpha at $20/hr, sets the value). Management should pay for overtime, an extra shift, or a second machine only up to the shadow price. Spending on a non-bottleneck resource adds zero throughput and is wasted — a frequently tested TOC insight.

Throughput vs. Traditional Margin: Why They Differ

Throughput accounting and traditional contribution-margin ranking can disagree because they treat labor and variable overhead differently. Traditional CM subtracts all variable costs; throughput subtracts only totally variable cost (usually direct materials).

• If two products have similar material costs but very different labor content, throughput accounting may rank them differently from CM-per-hour.
• Throughput accounting deliberately discourages building inventory of non-bottleneck output, because that output cannot be sold faster than the bottleneck allows.

On the exam, read carefully whether a question asks for contribution margin per constrained hour (traditional) or throughput per bottleneck hour (TOC). Both rank by "profit per unit of the scarce resource," but the numerator differs — CM per unit versus price minus material cost. Mixing the two definitions is a classic distractor.