Abstract: Sets of rules for calculations of TRIZ functions of technique and its Ideality in various situations are established and discussed.
 

1. Introduction

It is known that a technique can execute several functions, among which only one is the working function being the aim of technique existence (so, it is the primary useful function) while others are auxiliary (secondary, support, etc.) ones accompanying and lightening it execution. A technique is designed to provide one or several useful function (UF). Unfortunately, any technique has one or few harmful functions (HF) the fulfillment of which is (are) undesirable. A technique can have also neutral functions (NF) that are irrelevant to it current set of goals. In the simplest form UF and HF are used to describe one of the major TRIZ concepts - Ideality I as

here *F is the summa of all functions. Usually TRIZ neglects the dependence I on the number of NF (see, e.g., [1]). The equation (1) is a generalization of the different notions such as "cost-benefit ratio" in economy, "value" in Value Analysis/Engineering, "effectiveness" in management.

Often the numerator and denominator of the equation (1) are ill-defined and calculated based on experts' assessments. Most researches and practitioners in these disciplines ignore the fact that numerator and denominator in this form equation are defined only for regular (standard) numbers, while UF and HF can have wider and non-uniform nature, e.g., to be logical numbers, non-numbered items (good, bad), or mixture of exactly defined regular (standard) numbers with items that are know only by order of magnitude. Moreover, even division of functions into useful, neutral and harmful or properties, events and other attributes into good and bad is also subjective sometimes. TRIZ works with subjective functions (good/bad), while pure mathematics ignores our opinion (see the Notes). We discuss and try to overcome these weak points of all above-mentioned disciplines in this paper.
 

2. Arithmetic for Functions

Unfortunately, various functions cannot be determined through simple values of one or several related variables in practically all cases important for TRIZ, so we cannot use standard mathematics ("to add apples to apples, not to ships"). Therefore, it is necessary to establish some understandable mathematics for the functions that would help in problem solving and decision analysis.
 

2.1 Logic-like approach

The well-known operations with logical functions [2] (see the appendix 1) served as the prototype for the first set of rules for functions in TRIZ [1]. For example, for NOT operation we can write

It is possible to define the addition (&) and multiplication (·) operations for functions of technique in case if all functions are equally important (have the same "weight") as it presented in the table 1.
 

For example, this simple mathematics can be used to figure out three possible cases of appearance of the system effect [1]:

the positive properties add up together and strengthen each other, while the negative characteristics keep at least unchanged:

the positive characteristics add up together, while the negative properties eliminate each other:

turned the negative properties are added to the sum of the positive characteristics:

The equations (2) - (4) shows that the division of technique functions (and other attributes) into "good" and "bad" is not absolute and labels like "useful", "harmful" and "neutral" can be changed with technique evolution [1].

There is strong demand to establish more precise mathematics during problem solving beyond the assumption about the equal "weight" of both attributes X and Y in the arithmetic rules presented above. In this paper we overcome this assumption and present more sophisticated approach for "functional mathematics" for two cases:

1. the value of functions are known from practitioners assessments;
2. the orders of magnitude for functions are know.

This approach is based on human ability to judge in terms better-same-worse, more-equal-less, and to recognize a finite number of grades (usually 7 2) of a variable. Following Altshuller's tradition for the Levels of invention (see [1]), we operate with 5 grades for Function Significance, and we also introduce 7 grades for Function Accomplishment, although these numbers could be easily changed if necessary. The basic "weight" and computational conventions are presented in the following table

Note, that we have to introduce different scales (symmetric and positive) for functions factors in order to get reasonable results.

This table has been used few times to estimate the Ideality (1) for possible implementations of different solutions the same technical problem. Although the selected directions for techniques evolution lead to profitable development of some electronic and software systems, the practitioners' assessments approach can often lead to wrong decision. The cause of weakness is usually misunderstanding the nature of variables and incorrect transfer of statistical analyses principles for numbers to qualitative characteristics that are common in TRIZ, Value Engineering, QFD and other similar methodologies. Unfortunately steady ignoring and simplifying the known approaches for evaluation of estimations [3] lead to some unproven and even incorrect conclusions (e.g., in the assigning some Level for particular patent; or recognition Patterns of Evolution from small number of unbiased case studies).

The evaluations according to the table 2 and similar approaches show that the division of technique functions (and other attributes) into "good" and "bad" is not absolute and labels like "useful", "harmful" and "neutral" can be changed with the pool of practitioners. The practitioners' suggestions alone cannot be used for developing a technique. Similar, TRIZ and other disciplines cannot accept results that are drafted by researches without the proper statistical evaluation.

The identification of appropriate scale for measure any variable is the first important step of such evaluation. The type of a scale sets group of allowable transformations and calculation of practitioners' assessments. For example, co-existence symmetric and only positive numbers for Function Significance and Function Accomplishment, perhaps, reflects presence of the serial and nominal scales for these qualitative characteristics. All mutual - unequivocal transformations allowable for a nominal scale (i.e. the numbers are used only as labels, e.g., number of telephones), while all strictly growing transformations allowable for a serial scale (i.e. the numbers are used only as comparisons, e.g., power of earthquakes).

Even for numeric variables we shall specify the basic kinds of scales of measurement and appropriate groups of allowable transformations. Linear growing transformations can only be performed within scale of intervals, similar (varying only scale) transformations can only be performed within scale of the attitudes (relations), transformations without change of metrics can only be performed within topological scale, and the identical transformations are only allowable for an absolute scale.

2.3 Semi-qualitative Estimations

The division of the functions into good, bad and not defined and the exactness of such procedure are subjective. Hence, a value of function (parameter or other attribute [1]) does not known. Nevertheless we can operate with its order of magnitude. The arithmetic rules for this case are well known [4] and presented below in lightly expanded form.

The addition rule for orders of magnitude can be written as

The subtraction rule for orders of magnitude can be written as

The multiplication rule for orders of magnitude can be written as

The division rule for orders of magnitude can be written as

Often even the order of magnitude of a parameter and its change during creation of new technique does not know exactly. Nevertheless, we can operate with its range R. 

Figure 1: Geometric interpretation for arithmetic rules for ranges of orders of magnitude of two variables.

The arithmetic rules for this case are following:

The addition rule for ranges of orders of magnitude can be written as

The subtraction rule for ranges of orders of magnitude can be written as

{R[A^b1 : A^b2]}-{R[A^c1 : A^c2]} = {R[A^{min[max (b1:c2),max(b2:c1)]} : A^{max[max (b1: c2),max(b2: c1)]}]} (11),

and can be simplified to

The multiplication rule for ranges of orders of magnitude can be written as

The division rule for ranges of orders of magnitude can be written as

These rules for ranges can be presented graphically as it shown at the figure 1.

The above-mentioned rules were successfully implemented in decision-making about evolution of some processes in semiconductor industry.

3. Change of Ideality during technique evolution

The trajectories for technique evolution can be presented in the functional coordinates of quasi-four-dimensional parametric space that consist of axes for UF and HF, as well axes for time and number of NF [1]. Usually we are more interested in changes of Ideality with solution implementations rather than in Ideality of particular representative of technique at it evolution curve. Such changes can be written as

(14)

Here i and k indicate the number of types of variables (logical, numerical, qualitative, etc.), while j and m indicate the number of useful and harmful functions. We use the assumption that the numbers (i or k) are the same for UF and HF in the same state of technique. Both the numbers of technique functions and their types can be change during its evolution, i.e., n l and i k. The square brackets indicate one of the possible operations for comparison of Idealities. The subtraction operation is more suitable when the different solutions lead to similar gains during their implementations, while the division operation is more suitable when gains of various solutions are quite different. Note that it is usually enough to use the subtraction operation for orders of magnitude for estimation of Ideality changes at early stages of technique evolution.

4. Conclusion

The rules for calculations of status and evolution of technique in terms of useful, harmful and neutral functions are established and proven.

The set of rules should be chosen and verified for each technique.

The labeling of function, parameters and other technique attributes into "good" versus "bad" does NOT have the absolute and permanent nature and subjective.

The practitioners or researches suggestions cannot be used for developing any technique or methodology without proper statistical evaluation.

The possibility to calculate changes in Ideality allows to select the best solutions before their implementation.
 

References:

[1] S. D. Savransky, Engineering of Creativity: Introduction to TRIZ Methodology of Inventive Problem Solving (CRC Press, 2000)

[2] H. B. Curry, Foundations of Mathematical Logic (Dover, 1977)

[3] R. T. Clemen, Making Hard Decisions: An Introduction to Decision Analysis (Duxbury Press, 1996)

[4] Yu. I. Shokin, Interval Analysis (Nauka, 1981) /In Russian/.
 

NOTES:

1. Probably the theoretical speculations for the efficiency of a steam machine is historically the first example of such disregard that mislead the development of thermodynamics during past couple centuries. (Note, that the Russian equivalent of the technical term "an efficiency" is " a coefficient of an useful action"). Although the era of steam machines is almost finished, the incorrect theoretical descriptions with entropy within the statistical physics still have negative influence on developments of theory of information and synergetics.

2. See the parts of conference slides (Appendix 2) for assistance.

APPENDIX 1: Basic Logic Operations and Expressions

Each logic operation has input variables x₁ , x₂ , …, x_n and output functions f₁ , f₂ ,… , f_m . Any input variable and output function takes only binary value, 0 or 1. Any logic function f can be expressed by a combination or truth table demonstrated below:

The table size increases rapidly as number of variables increases. Under certain circumstances, some of the combinations of input variable values never occur, or even if they occur, we do not care what values f assumes. These combinations are called don't-care conditions and noted as "d ", e.g., f = d if x=0, y=1 and z=0.

Any logic function can be expressed with three basic logic operations: OR, AND, and NOT.

The OR operation of n variables x₁ , x₂ , …, x_n yields the value 1 whenever at least one of the variables is 1, and 0 otherwise, where each of x₁ , x₂ , …, x_n assumes the value 0 or 1. This is denoted by x₁ x₂ … x_n. The OR operation defined above is sometimes called logical sum, or disjunction.

The AND operation of n variables yields the value 1 if and only if all variables x₁ , x₂ , …, x_n are simultaneously 1. This is denoted by x₁ * x₂ *…* x_n . These stars are usually omitted: x₁x₂ … x_n. The AND operation is sometimes called conjunction, or logical product.

The NOT operation of a variable x yields the value 1 if x = 0, and 0 if x = 1. This is denoted by x. The NOT operation is sometimes called complement or inversion. Any logical variable (function) has two literals x(f) and x(f).

Any logic function f, such as the one shown in the combination table, can be expressed through the operations AND, OR, and NOT and logic input variables x, y and z as:

Any logic function can be also be expressed with other logic operations. Note, that the operations ANDNOT and XOR play important role in for TRIZ point (so-called, physical) and pair (so-called, technical) contradictions correspondingly [1].

APPENDIX 2: Parts of the Conference Slides

The misunderstanding the nature of variables and incorrect transfer of statistical analyses principles for numbers to qualitative characteristics are common in TRIZ, Value Engineering, QFD and other similar methodologies.

MIKE phone 112-560-3678 MARY phone 112-023-1856

Mike and Mary ARE MARRIED !!! Is their phone now 224-583-5534  ( = 112-560-3678 + 112-023-1856) ???

The division of attributes into "good" and "bad" is not absolute and labels like "weak" or "excellent" can be changed with the pool of practitioners.

TRIZ works often with subjective functions (useful VS harmful or good VS bad) in contrast with the pure mathematics.

1) The identification of appropriate scale for measure of any variable (even numeric data).

2) The description of allowable transformations and calculation of such variables:

SOME SCALES	intervals	attitudes (relations)	topological	absolute
TRANSFORMATIONS	linear growing	similar (varying only scale)	metrics conserve	identical

3) The taxonomy of results. Verification of conclusions