Product Mix & Constrained Resources

Overview

• What you’ll learn: How to optimize product mix when one or more resources are constrained, the theory of constraints (TOC), contribution margin per unit of the bottleneck, and linear programming concepts for multiple constraints.
• Prerequisites: Lessons 1-2
• Estimated reading time: 16 minutes

Introduction

The Grand Historian records: The master swordsmith has only one forge and twenty-four hours in a day. He can produce fine katanas that sell for a fortune but take eight hours each, or sturdy wakizashi that sell for less but take only two hours. The question is not “which weapon yields more profit per unit?” but “which weapon yields more profit per hour of forge time?” This is the essence of product-mix optimization under constraints.

Horngren (Chapter 11) introduces the principle that when resources are scarce, the decision criterion shifts from contribution margin per unit to contribution margin per unit of the constrained resource.

Single Constraint Optimization

The Decision Rule

When a single resource constrains production, maximize profit by producing the product with the highest contribution margin per unit of the scarce resource:

Priority = Contribution Margin per Unit / Units of Scarce Resource Required per Unit

Example

Despite Product A’s higher per-unit margin, Product B generates three times more contribution per machine hour. With limited machine capacity, produce B first until demand is satisfied, then allocate remaining hours to A.

The Theory of Constraints (TOC)

Eliyahu Goldratt’s Theory of Constraints provides a systematic approach:

1. Identify the bottleneck: Which resource limits total throughput?
2. Exploit the bottleneck: Ensure the constraint is never idle. Every minute of lost bottleneck time is lost throughput for the entire system.
3. Subordinate everything else: Non-bottleneck resources should pace to the bottleneck. Producing faster at non-bottleneck stations just builds work-in-process inventory.
4. Elevate the constraint: Invest in increasing bottleneck capacity — overtime, additional equipment, process improvement.
5. Repeat: Once the constraint is resolved, find the new bottleneck (there is always one).

太史公曰：An army moves at the speed of its slowest battalion. Whipping the cavalry to gallop faster while the infantry struggles through mud merely creates a gap in the line. Subordinate the cavalry to the infantry’s pace, then find ways to make the infantry faster.

Multiple Constraints and Linear Programming

When multiple resources are simultaneously constrained, the simple ranking approach fails. Linear programming (LP) is the mathematical technique for optimizing a linear objective function subject to multiple linear constraints.

The LP formulation:

• Objective function: Maximize total contribution margin = CM_A × Q_A + CM_B × Q_B + …
• Constraints: Resource usage ≤ Available resource (for each scarce resource)
• Non-negativity: Q_A, Q_B, … ≥ 0

For two products and two constraints, the graphical method identifies the feasible region and tests corner points. For more complex problems, the simplex method or solver software (including ERP tools) finds the optimum.

Key Takeaways

• Under constraints, maximize contribution margin per unit of the scarce resource, not per unit of product.
• The Theory of Constraints identifies and exploits bottlenecks systematically.
• Non-bottleneck resources should pace to the bottleneck — overproduction elsewhere just builds inventory.
• Multiple constraints require linear programming or equivalent optimization techniques.
• Always satisfy demand constraints before allocating remaining capacity.

What’s Next

In Lesson 4, we zoom out from operational decisions to strategic measurement — the Balanced Scorecard and strategy maps.