A general theory of systems performance criteria

Abstract

Based on general systems thinking of the past half century and the efficiency literature of the last 100 years, this research develops a theoretical model of systems performance criteria that provides a means to evaluate how well any living system, whether individual, organisational or societal, performs task-related activities in the pursuit of goals. The model is used as a foundation for organising the various concepts of performance that have evolved over the past century: efficiency, costs, productivity, effectiveness and cost-effectiveness. These performance concepts, derived from the theoretical model, are formalised into axioms that are used to deduce theorems and derivative laws in constructing a formal general theory of systems performance criteria.

Keywords:

• performance
• efficiency
• productivity
• effectiveness
• cost-effectiveness

Acknowledgements

The author gratefully acknowledges the assistance of Frederick Hoffman, Ph.D., Professor of Mathematics and Director of the Southeastern International Conferences on Combinatorics, Graph Theory and Computing, for his aid in identifying all conceivable combinations of variables, prodding the author to find underlying rules for the exceptions and for efficiently working out the mathematical proof of the rules.

Notes

^1. What has been done with Smith's masterpiece is also interesting because, in the absence of a well articulated theoretical model, it so dramatically illustrates the on-going confusion in performance terms and concepts. On the back of the British £20 pound note below the picture of Adam Smith gazing contentedly at his pin factory is the caption:

 ‘The division of labour in pin manufacturing:

 (and the great increase in the quantity of work that results)’

The irony of this caption would certainly cause Smith to grimace with chagrin as it completely contradicts the logic of his brilliant example. From the discussion on productivity, it is evident that the ‘quantity of work’ input into the pin-making process stays the same (10 men dividing the labour versus 10 men working separately); ‘the great increase in the quantity…that results’ (from improvements in pin production process) is in the output – the number of pins surge from 100 to 48,000. In the systems terminology developed here, with (labour) inputs remaining constant and (pin) outputs soaring, the productivity of the process with the division of labour vastly exceeds the productivity of independent activity.

^2. The ethical component of Bunge's (1989) ‘technico-ethical’ treatment of goals is problematic for purposes of the present attempt to technically formulate such concepts as efficiency and effectiveness. In addition to his technical contributions to conceptualising efficient, effective and cost-effective performance, Bunge (1989, p. 331) also writes: ‘And it is ethical as well because the change from the initial to the final state [i.e. activity directed toward attaining the goal] may have foreseeable harmful effects on someone, whether alive or to be born’. This emphasis on ethics and morals in goals setting and goal achievement are fundamental to Praxiological Philosophy (Kortarbinski 1965). Moral norms are inextricably linked to efficiency, as Bunge (1989, p. 313, italics in original) states: the ‘last test to which any moral norm ought to be submitted is that of efficiency…in helping bring about desirable states of affairs’. In praxiology, the technical aspects of efficiency are intertwined with the ethical aspects of setting goals and working towards their achievement.

 In contradistinction, the present work focuses strictly on the technical aspects of a system's performance and has nothing to say about ethical issues and moral norms. All systems performance criteria discussed in the present work, including efficiency and effectiveness, are viewed as purely mathematical equations that are independent of their specific content, just as any other mathematical formula is independent of real world content, and therefore free of ethical or moral judgements. For example, one individual's goal may be winning four gold medals in the Olympics, and another individual's goal may be robbing four banks. If each individual accomplishes three out of four goals, then both individuals are performing at the same 75 % level of effectiveness and, ceteris paribus, they are equally efficient. The formulas for performing effectively or efficiently hold irrespective of the goodness or badness, rightness or wrongness (however these terms may be defined) of choice of goals or how well these goals are achieved.

^3. The Laws in Exhibit 6 holds for all criteria, except for a small number of special cases of cost-effectiveness that are ambiguous. These exceptions to the Laws of the general systems theory are non-random and systematic; they are explained by two rules based on direction and relative magnitude. These exceptions occur because cost-effectiveness contains a ratio within a ratio, CE = (O/G)/I. Not only does the direction of changes in I, O and G affect CE, but so does the relative magnitude of change for each variable. All exceptions are covered by the two rules.

 Rule 1, if, and only if, all three variables move in the same direction, and if O changes the greatest or least, relative to I and G, then CE is ambiguous. There are four combinations of these variables covered by Rule 1. This set of exceptions occur because, when O has the greatest impact influencing CE, it could be more than or less than offset by I and G (increasing next greatest and least) influencing CE. An example of Rule 1, with all three variables increasing (direction), O increasing the greatest, G next greatest and I the least (relative magnitude), and their impact on CE is shown below:

 The example shows that when O increases the greatest, the impact on CE depends on the combined magnitude of the changes in I and G. If the O increase is greater than the combined increase of I and G, as in case 1, then CE increases. If the O increase is less than the combined increase of I and G, as in case 2, then CE decreases.

 The proof of the rule, for three variables increasing, as in the example, follows directly: since

changing I, O and G by factors , respectively, so that the Δs are the proportional changes in the variables, makes CE become

Thus, CE will increase if . CE will decrease if the inequality is reversed, and is unchanged if the two sides are equal. If the change in O is the largest of the changes in the three variables, each of these combinations can occur and will result in ambiguity. A similar proof with the signs reversed occurs for the two combinations of the three variables decreasing.

 Rule 2, if, and only if, I or G has the greatest change and move in opposite directions, and if the one with the greatest change moves in the same direction as O, then CE is ambiguous. This set of combinations occur because when O does not have the largest influence on CE, the combined effect of the one with the greatest change and O may be more than or less than the influence of the other variable on CE. There are eight combinations where this may occur.

 An example of Rule 2, with O increasing less than I increases and more than G decreases, and their impact on CE is shown below.

 The example shows that when the O increase is not the greatest, and it is moving in the same direction as the variable with the greatest change, then the impact on CE depends on the combined magnitude of the changes in I and G. If the O increase is greater than the combined increase of I and G, as in case 1, then CE increases. If the O increase is less than the combined increase of I and G, as in case 2, then CE decreases.

 This example covers eight combinations of ambiguity, which include combinations with the relative magnitude of I and G reversed and with the directions of the three variables reversed.

 The proof of the example where I and O increase and G decrease, with the largest change in I and the smallest in G is shown below: since

changing I, O and G by factors respectively, so that the Δs are the proportional changes in the variables, makes CE become

 Thus, CE will increase if It will decrease if the inequality is reversed, and be unchanged if the two sides are equal. Clearly, if Δ _I is large enough CE will decrease, and when Δ _O is almost as large as Δ _I , Δ _G can be adjusted to make CE increase or stay fixed.

 In summary, when the directionality of the three variables produce conflicting influences on CE, and the relative magnitude of change is unknown, the results are ambiguous. Such ambiguity occurs in 12 combinations of the 146 possible combinations. Although, there is never ambiguity in costs, productivity or effectiveness, these cases of indeterminacy affecting cost-effectiveness also affect efficiency.