The Theory of Cost Risk in Design 1999-01-0495

In a recent paper (Hoult & Meador, [1]) a novel method of estimating the costs of parts, and assemblies of parts, was presented. This paper proposed that the metric for increments of cost was the function log (dimension/tolerance). Although such log functions have a history,given in [1], starting with Boltzman and Shannon, it is curious that it arises in cost models. In particular, the thermodynamic basis of information theory, given by Shannon [2], seems quite implausible in the present context. In [1], we called the cost theory “Complexity Theory”, mainly to distinguish it from information theory. A major purpose of the present paper is to present a rigorous argument of how the log function arises in the present context. It happens that the agrument hinges on two key issues: properties of the machine making or assembling the part, and a certain limit process. Neither involves thermodynamic reasoning. Such abstruse agruments should hardly interest the readers of engineering articles, such as this one. But buried in the theory is a fascinating gem: the idea that the risk of a design can be estimated rigorously. Let's begin by describing the contents of this report.

In [1], it was assumed that log (dimension/tolerance) was the metric for the effort (time) to fabricate a design feature. With this assumption, it was shown that fabrication times were proportional to the sum of the logs. This idea was successfully tested on some 100 parts, and three manufacturing processes. In this paper, we show how the log function arises from a certain limit process, and assumptions about good engineering practice. These theorems occupy Section II of the paper.

Section II sets the stage for the major results of the paper. If it is true that there is a rigorous agrument that the log function is correct, then maybe there is some rigorous argument that “the error in the estimate increased as the dimension information increased”, as was stated at the end of [1]. Section III is devoted to the theorem which relates the uncertainty in manufacturing time to features of a design.

Cost estimating methods, dealt with in Section II, have been explored for many years, and have been reviewed in [1]. Wilson [2] proposed the use of component or design dimensions and tolerances as a metric to quantify the information transmission of a manufacturing process, a proposal which we make rigorous. He proposed an analogy between manufacturing and communication. But classical communications theorems [3] do not apply directly to the manufacturing process.