# Axiomatic Innovation: Creativity As an Exact Science

*“It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge.”*

– Enrico Fermi

## Introduction

Professor Nam Suh of MIT set out to provide some scientific components that could couple with the artistic elements typically found in design. He sought to bring science to what was mostly art. During this search, he developed the science of axiomatic design (AD).^{1} This article will establish the central theorem for this expansion.

## Innovation Axioms

*“All truths are easy to understand once they are discovered; the point is to discover them.”*

– Galileo Galilei

There are two central axioms of interest that we want to look at from Professor Suh’s work. The independence axiom and the information axiom form the core of AD. Basically speaking, the independence axiom states that corresponding elements from domain-to-domain^{2} should be independent (SN**»**CR**»**FR**»**DP**»**PV). This minimizes complexity and the possibility of interactions. It also has the effect of simplifying system optimization. The information axiom states that if two systems are otherwise functionally equal, the system that contains the least amount of information is superior. These two axioms are supported by a system of many other axioms, theorems and corollaries, but we’ll keep the scope narrowed to these for this article. The independence axiom provides the framework for developing a system that will satisfy customer requirement flow-down. It also is the basis for the language of AD. The inter-domain relationships are described using the design matrix. Here is an example demonstrating the design matrix associated with a set of FRs and DPs:

The independence axiom dictates that the diagonal elements of the matrix contain correlations while the non-diagonal elements contain none. The diagonal elements of the design matrix represent the useful functions and interactions in a system. This then also represents the content of the numerator of the Ideality equation:

Ideality = Σ Functions _{useful} / Σ (Functions _{harmful} + Cost)

The non-diagonal elements of the matrix indicate harmful interactions. This represents the content of the denominator of the Ideality equation. A normalization technique must be applied in order to produce the ideality ratio. These ratios are then comparative as you evolve that particular system. The ratios are not meaningful across many systems. This methodological overlap may be extended as well.

The Theory of Inventive Problem Solving (TRIZ) contains many functions that are categorized in a variety of ways. There are the technical contradiction operators, the physical contradiction operators, and the 76 standard solutions (just to name a few).^{3} The transition between AD and TRIZ can be complex and confusing. Therefore, a migration of TRIZ concepts into the AD structure would be useful. With this in mind, let’s identify the basics of the TRIZ methodology from an axiomatic perspective.

The technical contradiction axiom states that the resolution of a problem is based on the identification and selection of a contra-indicated set of elements in a system. If element *A* is introduced to provide a benefit in the system, it causes an unwanted degradation. This degradation will affect another element in the system, *B*. The resolution of the *A-B* contra-indication is the root of an innovation. Using the language of AD, a technical contradiction exists when a diagonal element of the design matrix causes a non-diagonal correlation. Another way of describing this is that a function from one domain is coupled to more than one function of another. Therefore, a coupled design contains at least one technical contradiction. The resolution of this contradiction may take two paths:

- Create a solution that eliminates the correlation and hence removes the coupling – this is also known as decoupling a system. You need to identify a new function, modify an existing function or apply an intermediate function. This last option is not ideal as it adds information to the system and also has a potential negative impact on the ideality of the system.
- Optimize the system hierarchy so that the negative impact of the non-diagonal correlation is minimized. This is analogous to risk mitigation from failure analysis.

TRIZ practitioners are aware that each technical contradiction may be converted to at least one physical contradiction. This allows for the application of both the technical and physical contradiction paths.

The physical contradiction zxiom states that the resolution of a problem is based on resolving contradictory parameters of the same system element. For example, you want element *A* to be large at one time and small at another. This is known as a bi-polarity (*A↑↓*). The application of the desired parameter to the appropriate need state is the basis for an innovation. From the AD perspective, a bi-polarity would be described by a coupled design where you cannot eliminate or modify the existing function that has caused the coupling. In this case, you would need to separate the functional interactions in order to resolve the contradictory requirements. This may be accomplished by applying the separation principle axiom. This axiom states that a bi-polarity may be resolved by separating the contradictory requirements in:

(a)Time

(b)Space

(c)Scale or

(d)Upon condition

The application of this axiom will have a decoupling effect on the associated design matrix. Each physical contradiction may be decomposed into a set of technical contradictions. This also allows for the application of all relevant contradiction resolution techniques.

The ideal final result (IFR) axiom states that the ultimate resolution to a problem (technical, physical or otherwise) will provide full useful functionality with no associated harm or cost. Leonardo da Vinci famously stated, “Think of the end before the beginning.” The IFR axiom is embodied in that statement. As a problem is solved, if the resolution is not ideal at least one secondary problem is identified. If the resolution of this secondary problem is not ideal, then a tertiary problem is identified. This problem-solution decomposition will continue ad infinitum until the IFR axiom is achieved. The IFR will end the problem-solution decomposition as there will be no new problems originating from this final solution. The IFR axiom is achieved if the design matrix is uncoupled. Ideality may continue to be improved in this state. The goal is to maximize ideality (increase the numerator and decrease the denominator, thereby causing ideality to asymptotically approach infinity) and maintain the uncoupled structure of the design matrix.

## Conclusion

*“You never change something by fighting the existing reality. To change something, build a new model that makes the existing model obsolete.”*

– Buckminster Fuller

The technical contradiction axiom, physical contradiction axiom, ideal final result axiom, independence axiom and the information axiom work together and share a common form of expression that may be found in the design matrix. Each axiom may be represented and described using the design matrix structure and language. Therefore, the hybridization of TRIZ and AD into a new heterogeneous mono-system is a logical and necessary step.

[1] See “Reangularity, Semangularity, and Ideality” and”Axiomatic Design to Render an Objective Ideality“

[2] Please refer to “The Competitive Excellence Imperative” for a discussion on AD domains.

[3] See *INsourcing Innovation* for a more detailed discussion of these basic TRIZ operators.

### About the Author:

Michael S. Slocum, Ph.D., is the principal and chief executive officer of The Inventioneering Company. Contact Michael S. Slocum at michael (at) inventioneeringco.com or visit http://www.inventioneeringco.com.