TRIZ AND THE OPTIMIZATION CONJECTURE
"- - 2007"

TRIZ AND THE OPTIMIZATION CONJECTURE
PART I

Randall Marin, Costa Rica

Abstract: Functional Analysis and Su-Field Analysis provide fundamental tools in the traditional TRIZ concept. Nowadays, there is increasing interest to approach TRIZ in a formal, scientific structure, explanatory of its intrinsic mechanics. This introductory paper (directed mostly to engineers) is the first of two deliveries, which proposes an algebraic approach to TRIZ structure yet based on the above Altshuller's axioms, as a motivation and an invitation to the TRIZ community for further developing these ideas and continue to research beyond traditional TRIZ. A second paper, directed to mathematicians, will address the mathematical concepts more technically.

Key words: linear algebra, vector, linear independence, linear space, orthogonal bases, linear transformation, applied mathematics.

Introduction. The fundamental question across is why does TRIZ work? A generally accepted position is not to ask this question since there is empiric, experience-based proof that TRIZ is an effective tool [8,11] and, on the other hand, theory and application of the Forty Principles [9] is based on statistical analysis [9,10]. Reader will appreciate the fact that these and other similar observations only correlate the solution [2,5] to the TRIZ techniques [2,5] - they don't explain why TRIZ as a whole works. In searching for a crisp answer to TRIZ's limitations, envelope, and the next Innovation breakthrough, a first step is to understand TRIZ's underlying mechanics.

Altshuller's Su-field axiom: a starting point. Let's consider Altshuller's Su-field axiom model: S1, S2, F1 and its corresponding SFO (Subject-Function-Object) representation in fig. 1.

Fig.1. Classic Su-field, SFO models Fig.1. Classic Su-field, SFO models [4]

Let's hereby consider these models in a linear space, and let's represent the function/Su-Field with the linear equation

Ax = F (1)

Where F is the functionality (say, move the ball, P=mv), x is a (set of) vector parameter(s) of interest in the object (S2) of the function and A is the action (matrix or vector) done by the subject (S1) on the object. Nature of the (inter)action is driven by the field type.

* * *

Reader may find useful a brief reminder on linear spaces. A linear space meets the following [1,3,6]:

1) existence of a body K of scalars,
2) a set V of objects called vectors,
3) an addition operation which links each pair of vectors u, v in V to a vector u+v in V, or addition of u and v, such that: u+v = v+u (commutative); u+(v1 + v2) = (u+v1)+ v2 (associative); nule vector is unique such that u + 0 = u; -v vector is unique for each v such that v + (-v) = 0
4) a multiplication operation such that given any scalar c in K and any u in V, cu is in V associated to c and v, such that: 1u = u; c1c2(u) = c1(c2u); c(u+v)=cu+cv; (c1+c2)u = c1u + c2u

It is said that V is a linear space over the body K. Examples of linear spaces [1,6]:

1) V = R the set of Real numbers, with u + v and cu the ordinary addition and multiplication of real numbers.
2) The space of functions. Let S by any non-empty set. Let V be the set of all functions of S in K. Addition of vectors f and g in V is the vector f+g, or

(f + g)(s) = f(s) + g(s) (2)

With s in S. Product of scalar c and f is cf, or

(cf)(s) = cf(s) (3)

3) The space of matrices mn over the Real numbers (K=R) or over the Complex numbers (K=C), with m, n positive whole numbers.

Notice the examples above describe general cases and in particular, natural interactions that can be represented with physics functions and matrices in an engineering system V.

* * *

Given the functional/Su-Field model represented in a linear space:

Ax = F (1)

for the given A, x and F of an engineering system as previously defined. Let's consider the corresponding linear transformation [1,3,6]

F: V -› W
F(x) = Ax (4)

for any A, x in the same engineering system. This is interpreted as the set of all possible engineering combinations with the available resources and physical/engineering effects. Formally, the linear transformation F must be defined from one linear space V into another linear space W, and in general it will always be true that both V and F(V) represent the engineering system in this analysis, where F(V) or image of V will be the set of functionality outcomes enclosed by W. This technical note provided, the engineering system will be referenced as V when putting in relevance its physical/engineering principles, or as F(V) when referencing its functional outcomes.

There is a direct implication detailed in the following theorem [1,3,6].

Theorem 1. Let be V, W linear spaces over K. Let F1, F2 be linear transformations of V into W. The function (F1 + F2)(x) = F1(x)+ F2(x) is a linear transformation of V into W where (cF1)(x) = c(F1(x)). The set of all linear transformations of V into W with the addition and multiplication hereby defined, is a linear space over K.

Proof:
(F1 + F2)(cu + v) = F1(cu+v) + F2(cu+v) = cF1(u)+F1(v)+ cF2(u)+F2(v) = c(F1(u) + F2(u)) + (F1(v) + F2(v)) = c(F1 + F2)(u) + (F1 + F2)(v)

And
(cF1)(du + v) = c(F1(du + v)) = c(dF1(u) + F1(v)) = cdF1(u) + cF1(v) = d(cF1(u)) + (cF1)(v)

So (F1 + F2), cF1 are linear transformations. It is evident that this set of linear transformations also meets the four requirements of a linear space previously outlined.

Now for V and F(V) representing the engineering system, theorem 1 means that the concatenation and intensity (addition and multiplication operations) of functions is a linear space (!) that is impliedly generated by vector bases.

* * *

A word on linear transformations and generation of linear spaces. A linear transformation T is formally defined [1,6] as a function from V into W such that T(cu+V) = c(T(u)) + T(v), so this concept is intrinsically linked to the concept of linear space. Also, a vector v is said to be a linear combination of the vectors u1, u2, , un if there exist scalars c1, c2, , cn so that [1,3,6]

v = c1u1 + c2u2 + + cnun

Now if the vectors ui are linearly independent, meaning that no linear combination of ui results in zero when not all the scalars ai are all zero [1,3,6],

a1u1 + a2u2 + + anun = 0 (can not be satisfied unless all ai = 0),

then if a linear space T(V) is formed by vectors v that can be such represented, it is said that the subset B of T(V) such that B = {u1, u2, , un} is a base of T(V) that generates T(V). Orthogonal bases are able to efficiently generate T(V) because no projection exist of any ui vector over other uj vector in B (i<>j). Reader will recognize the analogy between the concept of perpendicular vectors and orthogonal vectors.

* * *

Optimization Conjecture (OC). The concept hereby introduced is that the problem of generating F(V) aka T(V) is intrinsically linked to the problem of generating optimal solutions in the engineering system. Let's review the implications of this so-called Optimization Conjecture:

1) A base B of T(V) is formed by some base vectors whose linear combinations generate T(V). A subset of those T(V)-scenarios represents real solutions for the problem under study in the engineering system.
2) Orthogonal bases in B will produce vectors with independently linear resources.
3) Linear combination of orthogonal base vectors generate unique solutions (meaning, non-linearly dependant) so that these particular solutions are optimized solutions, with no redundant dependencies, in principle simpler and probably less expensive.

Particular cases of the above generality have been noted in the development of software [12] by Mr. Jeong-Wook Yi and Mr. Gyung-Jin Park of Hanyang University, and in the combination of TRIZ with eigen-values of the Design Structure Matrix [13] by Mr. Navneet Bhushan of Wipro Technologies.

F(V): an application. Consider for instance the basic problem of the paint tank system [4] in fig. 2. Main function of the engineering system is to paint parts by immersion into a paint tank. Paint is pumped into the paint tank by a motor (pump), and a float sensor shuts the motor down when the paint tank's operation level is reached. Problem is the paint dries in the float sensor, so motor is not timely shut down due to the extra weight in the float, causing paint tank overflow.

Fig.2. Problem of the paint tank system 4
Fig.2. Problem of the paint tank system 4

The basic observation from the OC point of view is that the gravitation field is being used twice in the implicit problem description: Motor provides enough power to supersede the pressure (gravitation and air/paint density), and it transfers mgh potential energy to the paint mass so it (paint mass) can be transferred to the paint tank. Then, float is pushed up by buoying force (gravitation and air/paint density) again. The solution (or one solution) is to provide mgh energy to the paint only "once" (orthogonal base) by lifting the paint barrel to a convenient height and getting rid of the motor. Some extra engineering will be needed to set-up a functional engineering system. Pressure will equalize and paint flow is this way controlled [4]. See fig. 3.

Fig.3. Auto-filling paint tank [4]
Fig.3. Auto-filling paint tank [4]

Conclusion. There exist orthogonally optimized solutions to each function and to the engineering system formed by these functions by using the right orthogonal base vectors of the available physical principles in the engineering system. If an engineering system is on place with problems at some level, then better solutions are probably there with no need of fundamental changes to the engineering system. Intuitively speaking, the above statements are equivalent to noticing that the original engineering system has in it the "genetic code" to be improved - and improving it is only matter of decoding its fundamental, orthogonal expression.

Such observation, hereby proposed as the "Optimization Conjecture", formally proves the TRIZ motto ("All Engineering Systems can be improved").

Additional discussion. Notice that some bases for the engineering system T(V) may also generate meaningless scenarios, what is called non-physical solutions. For instance, an expression could potentially be obtained for a solution with imaginary time or 5D spaces, or merely solutions that can not or do not want to be implemented. The improvement of those boundaries is a promising area for further developing these ideas (hint: explore using Lagrange multipliers [3]). Nevertheless, the set of ideas on this introductory paper is primarily presented as a theoretical approach to improve the intrinsic understanding of TRIZ and not as a problem solving tool. Notice also that the algebraic treatment of the subject is rather basic. On the linearity assumption, notice the linearity of spoken language, "device moves ball" very Altshullerian [8,11]. Most non-linear systems can be linearized by usage of math jacobians [3] and other physical/math tricks; for instance, represent movement as P=mv instead of E = mv2. Proof of theorem 1 formally introduces the OC for the general case of linearity. Readers are encouraged to further develop the OC theoretically, and then, take the results back from abstraction to produce useful engineering tools.

The bottom-line: What is the importance of the OC for today's business? TRIZ Master Mr. Boris Zlotin of Directed Evolution has already pointed out [7] a novel evolution trend: that the new industry needs TRIZ tools to design for perfection ("consummate systems" [7]) given that the market life of new products is even and even shorter. Wanted solutions are the ones perfect for today - no time for evolving to ideality. In this context, the OC responds with a formal approach to TRIZ, i.e. a mathematical proof that engineering systems can be improved using orthogonal selection of resources. The expected result is a TRIZ toolset for fast, on-the-bench and on-the-field problem solving. What is required to obtain these techniques? The answer is collaboration. The worldwide TRIZ community is encouraged and urged to help develop OC-based practical tools.

Acknowledgments. The author is very much obliged to TRIZ Master Sergei Ikovenko and TRIZ Master Alex Lyubomirskiy of GEN3 Partners Inc. for their encouragement to pursue the development of the basic idea of OC. Revision of manuscript is also acknowledged to Dr. Ikovenko; to Dr. Alex Ramrez, Director of Applied Mathematics Department at Universidad de Costa Rica; and to Intel Corp. Principal Engineer Amir Roggel. Additional review of manuscript and inputs from Dr. Alexander Bushuev and TRIZ Master Simon Litvin, produced in the second delivery of this subject, are very much appreciated. The references following strongly support the theoretical proceedings in this paper.

About the Author. Randall Marn is a senior test engineer and burn-in consultant at Componentes Intel de Costa Rica and a burn-in module development engineer for Intel Corp. He has published several articles on laser, thermal, and TRIZ on burn-in subjects; is an electronics engineer and pursues a post degree in applied mathematics at Universidad de Costa Rica. He learned TRIZ from Intel University and GEN3 Partners Inc., and is certified with TRIZ level 3 by the International TRIZ Association. He leads the "OC Group of Interest" and can be contacted at the email addresses randall.f.marin@intel.com and randallmarin@ismarin.net.

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  14. Bhushan, Navneet; 2004, "Robust Inventive Software Design (RISD) - A Framework Combining Design Structure Matrix (DSM), Analytic Hierarchy Process (AHP) and TRIZ", Proceedings of the 7th International DSM Conference

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