On the correlation between the value of a system parameter and the number of times it was improved

Y. B. Karasik,
Thoughts Guiding Systems Corp.,
Ottawa, Canada.
e-mail:karasik@sympatico.ca

Introduction

All system characteristics can be subdivided into main and secondary depending on how high they are on the designers' list of priorities. The main are those which are high on the list and which designers strive to improve (i.e. increase or decrease). For example, performance is a main characteristics, which designers strive to increase, but fuel consumption is a main characteristics which designers strive to decrease.

Secondary characteristics are those, which have low priority (if any at all) and which could be set to any value, that serves the purpose of optimizing the main characteristics. They can fluctuate wildly from design to design.

With the passage of time secondary characteristics may become main and vise versa. For example, during the first decades of aviation history the main characteristics of aircrafts were speed, altitude, to which they could climb, and distance, which they could go. The level of noise their engines generated was secondary as it was not a priority to improve it. Engines' noisiness became main much later, when its reduction became a priority and important. (More such examples can be found here.)

This paper is concerned with finding a correlation between the number of improvements introduced into a system and the values of its main characteristics only. Since secondary characteristics are not design priorities, and no improvement purposely targets them, it is assumed that there is no correlation for them and they can change arbitrarily from design to design.

The main hypothesis of the paper

In TRIZ all improvements are subdivided into M levels by their significance. Let n(p, k, t) be the number of improvements of a level k introduced into a system at an instant t, and purported to improve a main parameter p. The total number of improvements of the kth level introduced into a system since its inception till an instant t is

t
N(p, k, t) = n(p, k, τ) dτ
0

Let V(p, t) be the best value of this characteristics attained by the time t. The main hypothesis of this paper is that there exists a correlation between the total number of improvements introduced into a system, aimed to improve p, and its best attained value V(p, t): In other words,

M
V(p, t) = F( N(p, k, t))                (1)
k=1

To find the function F one has to make some assumptions.

The main assumptions of the paper

  1. In this paper I assume that improvements of the same level improve a main characteristics by roughly the same amount rather than by the same percentage. In other words, it is assumed that function F in (1) is linear rather than exponential.

    For example, all incremental improvements improve a main chacateristics roughly by the same amount a1. All improvements of the second level improve it also roughly by the same amount a2, which is, of course, higher than in the case on incremental improvements. Etc.

    Then, according to this assumption, (1) can be re-written as follows:

    M
    V(p, t) = V(p, 0) + (δV(k, p) × N(p, k, t))                 (2),
    k=1

    where δV(k, p) is the amount by which improvement of level k improves parameter p.

  2. It is assumed that improvements of the higher levels are rare and mostly happen at the time of a system's inception.

  3. It is also assumed that percentage S(k) of improvements of a lower level k does not vary with time. In other words, over any time interval the percentage of improvements of level 1 is S(1), say, 75%; percentage of improvement of level 2 is S(2), say, 15%, etc. Thus, for 1 ≤ k ≤ L

    L
    N(p, k, t) = S(k) × ( N(p, k, t))
    k=1

Under the last two assumptions, (2) can be re-written as follows:

M L
V(p, t) = V(p, 0) + (δV(k, p) × N(p, k, t)) + (δV(k, p) × N(p, k, t))  =
k=L+1 k=1
M L L
=  V(p, 0) + (δV(k, p) × N(p, k, t)) + (δV(k, p) × S(k) × ( N(p, k, t))) =
k=L+1 k=1 k=1
M L L
=  V(p, 0) + (δV(k, p) × N(p, k, t)) + ( N(p, k, t)) × ( (δV(k, p) × S(k)))    =
k=L+1 k=1 k=1
L
=    A(p) + B(p) × ( N(p, k, t))                    (3),
k=1

where improvements of level 1 through L are low level improvements and of L+1 through M are higher level improvements, and

L
B(p)   =   (δV(k, p) × S(k))
k=1

and

M
A(p)   =   V(p, 0) + (δV(k, p) × N(p, k, t))
k=L+1

and it does not depend on t as it is assumed that N(p, k, t) does not depend on t for a higher level improvements, as they all take place in the beginning.

Since it is assumed that the number of improvements of higher levels comprise an insignificant portion of all improvements,

L M
N(p, k, t)    ≈    N(p, k, t)                  (4)
k=1 k=1

Hence

M
V(p, t)    =    A(p) + B(p) × ( N(p, k, t)) + R(t)                 (5),
k=1

where R(t) is a random process with zero mean that has to be added to make "roughly equal" equal exactly.

Along what curve do the main characteristics of a system evolve ?

Formula (5) alone cannot answer this question. More assumptions are needed:

  1. There are surges and recesses in attempts to improve any given characteristics. In other words, the graph of n(p, k, t) looks as follows:


    Fig. 1

    and the graph of N(p, k, t) looks, accordingly, as follows


    Fig. 2

  2. If a surge in activity to improve a characteristics occurs, then it entails a surge in improvements of any level. In other words, the times at which surges occur are the same for all levels of improvement. Surges in improvements of the first level coincide with surges in improvement of second level, etc. They all happen at the same time. There is no surge in improvements of specifically level k.

    From this assumption it follows that graph of

    M
    N(p, k, t)
    k=1

    also looks as on Fig. 2

Corollary

The graph of relationship (5) is a superposition of the graph from Fig. 2 (or its reflection against Time-axis plus some offset) and a random process with zero mean. For example, for performance it looks as follows:


Fig. 3