# Mathematical Formulation

When analyzing a complex situation, it is imperative to think about global criteria associated with the desired end state of the analysis. That is, as an individual wishing to address a complex problem, am I searching for a globally optimal, “best (maximum or minimum) value of the objective function” (Taha, 2011, p. 3), a singular *solution* to a problem, or am I merely seeking a satisfactory resolution to my problem? The answer, as always, depends.

Given the relatively constrained focused of a singular problem and its objective (s), it is easy to conceive that the stopping criteria for a problem analysis using a systematic thinking paradigm is optimization. The end goal of this machine age problem is to develop a best answer to the problem at hand. Thus, we speak of the *best* design for a structural component of a larger system, or the *best* portfolio selection from among a number of choices. Systemic thinking, however, requires a more delicate balancing act to be observed. Given that any systemic thinking effort will involve two or more constituent problems, and the solution to each problem assessed independently represents a unique global solution to the mess, we must consider the *principle of suboptimization* (Hitch, 1953) in our analysis of these messes. Maximizing overall mess performance (i.e., optimizing the mess) requires that its constituent problem solutions be constrained, thus violating the notion of suboptimization. Ackoff (1977) echoes the difficulty in achieving an optimal solution to a mess:

There is an important systems principle, familiar to all of you, that applies to messes and problems: that the sum of the optimal solutions to each component problem considered separately is not an optimal solution to the mess.... It is silly to look for an optimal solution to a mess. It is just as silly to look for an optimal plan. Rather we should be trying to design and create a process that will enable the system involved to make as rapid progress as possible towards its ideals, and to do so in a way which brings immediate satisfaction and which inspires the system to continuous pursuit of its ideals. (pp. 4-5)

Thus, if each system (i.e., problem) chooses to pursue (and thus, optimize) its own interests, then the mess will necessarily operate at less than maximum performance. Balancing the interests of constituent problems is one of the most difficult aspects of systemic thinking. A mechanism for doing so is known as *satisficing.* Satisficing is a term coined by Simon (1955, 1956) to describe how individuals make rational choices between available options and within a constrained environment. Simon argued that decision makers are rarely able to obtain and evaluate all the information which could be relevant to the making of a decision. Instead, they work with limited and simplified information to reach acceptable compromises (you *satis*fi*ce*, a portmanteau of satisfy and suffice) rather than to obtain a globally optimal strategy where a particular objective is wholly maximized. This relaxation from optimal-seeking problem solution approaches represents a departure from traditional OR solution techniques, one appropriate for mess analysis. Instead, a more appropriate approach is more qualitative and inclusive of multiple perspectives, one that invokes a mathematical underpinning such as fuzzy cognitive mapping (discussed at length in Chap. 5).