Traditional recipes, compounds or formulations are based on the notion of an ideal, targeted proportion for each component. Ingredient costs are readily computed and do not vary unless material prices change. However, as a practical matter, part of recipe specification must include tolerance values for each ingredient. Based on relevant measures of product quality, attributes and characteristics, the ingredient tolerances processors specify for their recipes define the boundary between an acceptable formulation and an unacceptable one.

In contrast, a tolerance-based formulation replaces the idea of a single, targeted proportion for each ingredient with an allowed range of proportions defined by established minimum and maximum tolerance limits. With the flexibility to operate anywhere within the tolerance range, recipe cost is transformed from a fixed result to a controllable variable. The processor becomes free to adopt a strategy of recipe cost minimization where the proportions of expensive ingredients are minimized and less costly ingredients maximized within tolerance limits. Depending on the specifics of the application, savings are readily available and can be substantial.

**The Tactics of Tolerance**

In extrusion processes where continuous feeders formulate the recipe, feeding performance becomes the primary means by which savings may be captured. More accurate feeding permits operation safely closer to tolerance boundaries, enabling further minimization of expensive ingredients and maximization of less costly formulation components. Similarly, in batching operations the accuracy of the weighing system (i.e. scale, weigh-hopper, etc.) dictates how much of the potential savings can be realized.

Computing the cost-minimized solution to existing target-based recipes is straightforward:

**Step 1: Assemble Input Data**

Computing the cost-minimized solution for extrusion processes requires minimum and maximum tolerances and total recipe rate, as well as unit cost and feeder repeatability performance for each ingredient. (For batch operations use weigh scale accuracy instead of feeder repeatability.)

**Step 2: Rank Ingredients by Unit Cost**

Ingredients are assigned an ordinal ranking based on decreasing unit cost (e.g., $/lb) with ‘1’ being the most expensive ingredient, ‘2’ being the next most expensive, and so on.

**Step 3: Calculate Operating Proportion Limits**

Based on feeder performance and ingredient tolerances, calculate the minimum and maximum allowable operating proportions for each ingredient, defining the range of setpoints over which each ingredient feeder may operate.

**Step 4: Temporarily Maximize All Operating Proportions**

Then, as a starting point, temporarily assign all operating proportions to the maximum values calculated in Step 3, resulting in a total recipe percentage greater than 100%.

**Step 5: Minimize Operating Proportions by Cost Rank**

Beginning with the most expensive ingredient (Rank 1), replace its maximum operating proportion with its minimum operating proportion. If total recipe percentage remains above 100%, repeat the process using the next most expensive ingredient (Rank 2) and so on until total recipe percentage falls below 100%.

**Step 6: Restore Residual Deficit**

The ingredient that causes total recipe percentage to fall below 100% cannot be minimized. To restore total recipe percentage to the required 100%, the deficit is simply added to that ingredient’s minimum operating proportion. (A cost-minimized recipe will always contain, at most, one ingredient that remains neither minimized nor maximized, but lies somewhere within its allowable operating range.) The recipe is now cost-minimized.

See the accompanying illustration entitled The Tolerance Model Applied for a numerical example.

**Factors Affecting Calculated Savings**

The potential for recipe cost minimization is influenced by several factors:

*Ingredient Cost Differentials*

Large differences in unit cost increase savings potential, and savings diminish where differentials are small.

*Tolerance Range*

Broad tolerances produce correspondingly larger savings potential since there is more opportunity to replace relatively expensive ingredients with less costly ones. Likewise, highly restrictive tolerances lower cost savings possibilities.

*Total Throughput*

Since total savings are directly proportional to throughput, even modest differences in savings rates can be a significant factor in the light of total dollar savings.

*Feeder Accuracy*

While the above factors combine to determine savings potential, feeder accuracy is the only tool the processor has at his disposal to exploit that potential. For any given recipe, regardless of cost, tolerance, or throughput, improved feeding accuracy always results in incremental savings in tolerance-based recipe optimization.

**Variations on the Theme**

For applications where ingredient tolerance values have not been pre-specified on the basis of product attributes, or that require one or more critical ingredients to be maintained at a specific proportion, the following variations address such needs.

*Optimization Using Inferred Tolerances*

Applications lacking pre-established tolerance values offer no savings potential because this cost-minimization technique requires a defined tolerance range within which an ingredient’s operating proportion may vary. However, it is possible to compute the savings associated with improved feeding performance simply by combining tolerance limits inferred from current feeder performance with any arbitrary level of enhanced feeder performance.

*Partial Recipe Optimization*

Other applications may require that the proportion of one or more ingredients be maintained at a closely controlled level (within limits of feeder performance), while other ingredients may vary within specified limits broader than those achieved by the feeder. Partial recipe optimization is achieved by combining the use of the method’s standard calculation procedure along with the inferred tolerance technique described above. Critical ingredients whose operating proportions must not vary are selectively excluded from the solution by the application of the inferred tolerance technique, while remaining ingredients with their broader tolerance limits remain free to contribute to the cost-minimized solution.