Chapter 13 Part 3 Notes: Constraint, Bottleneck, and Split-Off Decisions
Constraint, Bottleneck, and Volume Trade-offs
Concept overview
A constraint (bottleneck) is the part of the production process with the lowest capacity, i.e., the bottleneck in the chain. It determines the overall throughput.
If sales demand (forecast) exceeds capacity, the company must decide which products to produce and in what mix, given the constraint.
A production line can have a bottleneck at the first unit, the second unit, or any step; the area with the smallest capacity constrains the entire process.
The chapter links bottlenecks to contribution margin; decisions are made to maximize contribution margin under constraint.
Key decision principle under constraint
With constrained resources, fixed costs are unaffected; the goal is to maximize the contribution margin (CM).
We compare CM per unit vs CM per constrained resource to decide which products to emphasize.
The standard payoff measure becomes CM per constrained resource (not just CM per unit).
Formula foundations (definitions):
Contribution margin per unit: where is selling price per unit and is variable cost per unit.
Contribution margin per constrained resource (per unit of the constrained resource):
If the constrained resource is time (minutes) on a machine:
If the constrained resource is material (pounds):
Illustrative curtain example (Curtains company)
Products: six-foot curtain, three-foot curtain, and a valance.
Given data (per unit):
Selling prices, variable costs, material usage, and sewing machine time:
Six-foot curtain: CM per unit = ; material usage = 6 ft; machine time = 6 minutes.
Three-foot curtain: CM per unit = ; material usage = 3 ft; machine time = 8 minutes.
Valance: CM per unit = ; material usage = 4 ft; machine time = 16 minutes.
CM per constrained resource calculations:
Fabric (material) constraint:
Six-foot:
Three-foot:
Valance:
Conclusion: The three-foot curtain offers the highest CM per fabric foot ($8/ft$).
Machine time constraint (per minute):
Six-foot:
Three-foot:
Valance:
Conclusion: The six-foot curtain yields the highest CM per machine minute ($4/ min$).
Which product has the largest CM per unit? Valance ($30$) has the largest CM per unit, but it is not the best when you account for constrained resources (fabric or machine time).
Which product yields the most profitable use of fabric? Three-foot curtain ($8 per foot).
Which product yields the most profitable use of machine time? Six-foot curtain ($4 per minute).
Practical takeaway: Under a constraint, the best mix depends on which resource is tight and which profitability metric you optimize (per unit, per fabric foot, per minute, etc.). You often must produce a mix that satisfies demand while prioritizing the constrained-resource efficiency.
Alternative constraint example: constrained material (8,000 pounds available)
Data (per unit): selling price, direct materials, other variable costs, and material usage; material cost is .
Product components: X uses 4 lb per unit, Y uses 10 lb per unit, Z uses 5 lb per unit.
CM per unit (assume given): cmuX, cmuY, cmuZ (values provided in the scenario).
Compute CM per constrained resource (per pound):
Result: product Z is highest on a CM per pound basis, then X, then Y.
Decision rule with unlimited demand: produce in order of descending until the 8,000 lb constraint is met.
If demand is limited (e.g., 500 units of each product), calculate total material used and allocate remaining constrained resource to the next best option:
Material required for 500 units: X = 500×4 = 2,000 lb; Y = 500×10 = 5,000 lb; Z = 500×5 = 2,500 lb; total = 9,500 lb (> 8,000 lb constraint).
Prioritize products by : first satisfy Z (2,500 lb), then X (2,000 lb), remaining material = 8,000 − 2,500 − 2,000 = 3,500 lb.
For Y (10 lb per unit), you can make units.
Resulting mix under constraint: Z = 500 units, X = 500 units, Y = 350 units.
Total contribution margin under this constrained mix (assuming CM per unit values given):
CM = 500×CMuZ + 500×CMuX + 350×CMuY (subject to the constraint).
Key takeaway: You maximize CM by allocating the limited material to the product(s) with the highest CM per constrained unit first.
Managing constraints: strategic options to relieve or work around bottlenecks
If the constraint is a supplier/material shortage:
Source from a second vendor or alternate suppliers.
If the constraint is capacity (machine time, shifts):
Implement overtime (e.g., add a second/third shift).
Subcontract some production to external partners.
Acquire an additional machine or upgrade equipment.
Reallocate labor from non-bottleneck processes to the bottleneck;
Implement process improvements (e.g., Six Sigma) to remove redundancies and reduce defects.
If the constraint is labor or skill in a particular operation, consider process changes or cross-training to relieve the bottleneck.
Example: constrained labor time and outsourcing decision (ornate vs. modern table)
Problem setup: carving time is the constraint; selling price and variable costs yield a CM per unit; two products (ornate and modern) with different carving times.
Option to outsource: a nearby shop offers carving at $130 per hour.
Calculations per unit (carving time and CM per unit):
Ornate table: CM per unit = $100$ per hour of carving time.
Modern table: CM per unit = $200$ per hour of carving time.
Outsourcing cost: $130$ per hour.
Net CM from outsourcing per hour for each product:
Ornate: $100$ (in-house) vs $130$ (outsourced) → outsourcing reduces CM to $70$ per hour of carving time (if considered comparatively).
Modern: in-house CM $200$ per hour vs outsourcing $130$ per hour → outsourcing leaves $70$ per hour still available as net CM (i.e., you can outsource and still retain $70$ CM per hour).
Decision: Outsource the option where the outsource scenario still yields positive net CM per hour; in this example, outsourcing is advantageous for the modern table since it yields $70$ CM per hour relative to the bottleneck.
Joint product costs and split-off point (sell vs process further)
Split-off point: the point in a shared process where products can be separated into distinct final products.
Up to split-off, costs are joint and shared; after split-off, costs are incremental to each product. The cost incurred up to split-off is generally sunk for the decision at hand.
Byproducts and shared processing examples:
Lumber mill example: produce two by-products (e.g., two by fours of different lengths and sawdust).
After split-off, you may sell a byproduct as-is or process further to increase revenue (e.g., sawdust → fireplace logs).
The decision rule is to compare incremental revenue after processing further to incremental processing costs:
Incremental revenue after processing:
Incremental cost after processing:
Net incremental benefit:
If , process further; otherwise sell at split-off value.
For typical joint-cost problems, the split-off costs are irrelevant to the decision about processing further, because those costs are sunk up to the split-off point.
Common pitfalls: costs allocated to products at the split-off point are often based on sales volume shares, which may misstate the true incremental profitability of processing decisions.
Practical split-off example (meat processing style)
Setup: a joint process yields meat products that reach a split-off point; after split-off, additional processing costs are incurred to create final products (e.g., ham, bacon, pork chops).
Pre-split-off costs (joint): $J = 40{,}000$ (example).
Post-split-off additional processing costs (per product): bacon $C{b} = 20{,}000$, ham $C{h} = 30{,}000$, pork chops $C_{p} = 12{,}000$ (illustrative).
Final sales values after further processing (per product): ham, bacon, pork chops have different final sale revenues depending on further processing.
Sell-at-split-off alternative: you could sell each product at the split-off value (before processing) for a certain revenue.
Decision framework (incremental view): for each product i, compute per-unit or per-batch incremental revenue from processing versus selling at split-off, and subtract the incremental processing cost:
Incremental revenue per unit from processing:
Incremental cost per unit from processing:
Net incremental benefit per unit: where is the final sale price after processing and is the split-off sale price.
If , processing further adds value; if or negative, sell at split-off value.
The example in the transcript walks through specific numbers to compare options and ultimately uses the incremental profit criterion.
Core takeaway: The joint costs up to split-off are sunk for the decision; only incremental revenues and incremental costs after split-off matter.
Summary of key takeaways
A constraint/bottleneck defines the maximum output; optimize product mix around the constrained resource.
When constrained, evaluate CM per constrained unit (per lb, per foot, per minute, etc.) to prioritize products.
If demand is unlimited, allocate the constrained resource to maximize CM per constrained unit until capacity is reached.
If demand is limited, allocate the constrained resource to meet the most valuable mix given both CM per constrained unit and stated demand, adjusting as necessary.
Short-term actions to relieve bottlenecks include outsourcing, overtime, additional machines, and shifting labor; long-term actions include supplier diversification and process improvements.
Split-off (joint) costs require the analyst to focus on incremental revenue versus incremental costs after the split-off point; joint costs are sunk and do not drive the decision to process further.
Quick reference formulas
If the constrained resource is time (minutes):
If the constrained resource is material (pounds):
Allocation rule under material constraint: sort by descending and allocate resource until the constraint is met.
Split-off decision (per unit): where
= final sale price after processing,
= split-off sale price,
= incremental processing cost.
Process further if ; otherwise sell at split-off.