Investopedia article - What is the utility function and how is it calculated?
https://www.investopedia.com/ask/answers/072915/what-utility-function-and-how-it-calculated.asp
Article describes ordinal versus cardinal utility:
When economists measure the preferences of consumers, it's referred to ordinal utility. In other words, the order in which consumers choose one product over another can establish that consumers assign a higher value to the first product. Ordinal utility measures how consumers rank one product versus another.
Economists take the utility-function concept one step farther by assigning a numerical value to the products that consumers choose or choose not to consume. Assigning a value of utility is called cardinal utility, and the metric used to it is called utils.
Article includes the following statement under limitations and benefits of utility function:
Of course, in reality, economists can't assign a true numerical value to a consumer's level of satisfaction from a preference or choice. Also, pinpointing the reason for the purchase can be difficult if there are many variables being considered.
Is this an accurate description of economic reasoning? Utility functions are models constructed by a particular economist who cannot assign a true numerical value?
To further develop my question, if it is not clear, in a scientific measurement there is a procedure which defines a standard unit of measure. Instruments are calibrated in terms of these procedures and then calibrated instruments are applied to assign the numbers with units in the models. Is there a standard procedure for assigning numbers to specify utils or is it constructed when the economist assigns numbers arbitrarily to the ordinal set?
Another article - Constructing a utility function:
http://planning.cs.uiuc.edu/node469.html
Unfortunately, establishing the existence of a utility function does not produce a systematic way to construct it. In most circumstances, one is forced to design $ {\cal U}$ by a trial-and-error process that involves repeatedly checking the preferences. In the vast majority of applications, people create utility and cost functions without regard to the implications discussed in this section. Thus, undesirable conclusions may be reached in practice. Therefore, it is important not to be too confident about the quality of an optimal decision rule.