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2015.01.09 11:13
research:balance_variables

Balance Variables

A balance variable (BV) is a model of how balance can be measured.

TODO Need to work the amperage example: min is always zero; max is what the out-system circuit will carry. What is nominal? Does it relate to wattage?

What is a BV?

Fig. 1: Elements of a single balance variable.

See figure 1. A balance variable (BV) is a structure used to model a “force” that contributes to the balance of a design. Given some parameter, a BV is a triplet of values representing a minimum value, a nominal value, and a maximum value.

The nominal value of the parameter is a target or ideal value, set either by the initial situation or by the intent of the designer. The minimum and maximum values bound an inclusive range of all possible values of the parameter, regardless of probability that those values will obtain in vivo. Thus, the range of the BV can be represented as a uniform distribution.

By possible values, we intend here any realizable value for the parameter, regardless of whether such values may be desirable or undesirable. The range is usually determined in situ; that is, the range depends on the context in which the objects exhibiting the parameter will exist.

A uniform distribution is used because BVs are used during design, when it is likely that sufficient information to describe the probability distribution of the values of a parameter is not available.

Some examples include:

• * A plug for an electric kettle:

.. The parameter may be voltage, amperage, frequency, etc.

.. For amperage, for instance, the minimum possible value would be 0, the maximum value would be determined by the specification of the kettle's design, and the nominal would be the ideal value of amperage for that kettle (e.g., 15 Amps in North America).
** **A domestic hot water pipe:**
.. One parameter could be temperature.
.. The minimum possible temperature would be the lowest temperature under which the pipe can survive frozen water in it without bursting, and the highest temperature would be set by the requirements of the hot water system as a whole (likely to be far less than any temperature that could damage the pipe itself).  The nominal temperature would also be set by system requirements.

A BV therefore is described by a pair of triplets, each of the form $(min, nom, max)$, one that represents values at a system interface one side of a boundary, and the other representing values of the corresponding interface on the other side of the same boundary.

Combining BVs at interfaces

Fig. 2: Combining two BVs at a system interface.

figure 2 shows two BVs at a system interface. On one side of the interface is the system of concern; on the other side is the system's environment. In figure 2, the upper BV (in green) represents the environment side of the interface, called the out-system side, and the lower BV (in blue) represents the in-system side. (Associating the out-system with the upper BV is arbitrary and just an artifact of the visualization.) Together, the two BVs describe the interface between the system being designed and its environment.

The specification of the out-system BV is typically beyond the designers' control; the specification of the in-system BV is typically within the designers' control.

The nominal values for both out-system and in-system BVs correspond exactly. This signifies that the designers have set the ideal value for this parameter to correspond precisely to what is available from the environment.

The minimum range value for the out-system BV in figure 2 exceeds that of the in-system BV. This means that there are possible parameter values that the environment can obtain, that the designed product will not be able to handle. Therefore, the minimum performance of the system with respect to this parameter is under-designed.

The maximum range value for the in-system BV in figure 2 exceeds that of the out-system BV. This means there are possible parameter values that the designed object can handle, but that will never be provided by the environment. This means that the maximum performance of the object with respect to this parameter is over-designed.

If the minimum, nominal, and maximum values of the two BVs corresponded exactly, we would say that the design is balanced with respect to that one parameter.

As an example, consider the voltage of electricity conducting plug of the electric kettle, from a wall outlet. Assuming figure 2 pertains to this problem, we can make the following statements about the design of the kettle and its plug, with respect to the particular outlet(s) to which it is expected to be connected.

• The nominal value (e.g., 120 V) is perfectly aligned on both the kettle and the outlet side of the interface.
• We can expect under-voltage conditions to arise from the outlet that the kettle cannot handle.
• There are some over-voltage conditions that the kettle would be able to handle, but that will never occur from the outlet.
• The out-system range would be defined by the possible worst-case scenario for the Utilization

Voltage typically found in houses: 87% - 106% of the nominal 120V for North America.

• The in-system side suggests that various design interventions are required, possibly implementing

a variety of behaviours for each of several different sub-ranges, to manage the possible inputs from the context. For instance, the low-voltage condition in the kettle may trigger a behaviour that does not normally occur, and that behaviour may be provided by structural features (e.g., an auto-off feature or warning light), indicating to the user that the kettle might not work properly under the current conditions being supplied by the wall outlet.

In this particular example, then, we see that the kettle is both under-designed with respect to low-voltage conditions, and over-designed with respect to high-voltage conditions. If we had an actual kettle, we could attach values to the degree of over- and under-design (normalized to, say, the nominal value) and describe in a uniform way how “good” the kettle design is with respect to its handling of voltage. We would also have information useful to drive the design forward toward a more “balanced” state.

This approach builds effectiveness directly into the design method, so that when we start optimizing the design downstream of its design, the designers will not have to worry about making the design hyper-efficient at the expense of resilience and robustness. That is, using this approach, we should be able to better optimize efficiency within the constraints of effectiveness - which is something we cannot do well today.

We note that we do not mean to imply that the over-design scenario in this example is necessarily bad. The over-design may actually a result of some system element over which the designers have no control. That is, the kettle over-design for over-voltage may result entirely from the unavoidable use of certain components (like resistors, capacitors, transistors, etc.) that happen to be rated for 110- 130V .

Assessing BVs

Quantifying one BV

BVs would be especially useful if there were some simple, systematic way to quickly assess how well balanced a given design is. This is possible using the method in [Rog14].

If one assumes a standard “stock and flow” perspective of systems, one may reasonably denote the flow out of a system as a negative value and the flow into a system as a positive value.

If we represent the range of an interface by $[X_{min}, X_{max}]$ (i.e., set aside the nominal value in BV triplets for now), we can use addition to determine the “difference” between the two interfaces through which a flow occurs at a system boundary. Given the two interfaces in a BV, O and I (for Output and Input, respectively), their sum is represented as $[-(O_{min} + I_{min}), O_{max} + I_{max}]$.

If the ranges of O and I are the same (but obviously of opposite sign per the convention above), then the sum is zero and we have a balanced BV. For the under-designed condition, the corresponding element (minimum or maximum) of the sum will be less than zero; for the over-designed condition, the sum will be greater than zero. These both represent imbalances.

Comparing multiple BVs

So far, we have considered only individual BVs. It would be useful to be able to evaluate the overall balance of a design described by a set of many different BVs. We can't do this with sums alone because the values of different BVs may have different units and significantly different ranges; comparisons made under these circumstances are meaningless (like saying that 200K is more than 20kg).

But we can assess multiple BVs by normalizing the sum that represents the balance of a BV. We can normalize a BV by dividing the sum by some sensible quantity. Such a quantity must

• have the same units as the BV (thus making the normalized interval sum non- dimensional),
• have a magnitude comparable to those of the BVs (to normalize the magnitude sensibly), and
• not change (much) over the duration of the design process (so that balance measures are most likely to change as a result changes to the design rather than any other changes).

The solution in [Rog14] is to use the out-system value of the extremum being considered. That is, the measure of balance (b) of a BV is a triplet $[b_{min}, b_{nom}, b_{max}]$ such that:

• $b_{min} = -(O_{min} + I_{min})/|O_{min}|$,
• $b_{nom} = |O_{nom} + I_{nom} / O_{nom}|$, and
• $b_{max} = (O_{max} + I_{max})/|O_{max}|$.

The out-system interface is less likely to be under the control of the designer, and so can be taken as a contextual value outside the scope of changes that the designer may induce during design; that is, the out-system values are less likely to change during designing than the in-system values.

We note that the nominal balance is calculated using absolute value because one cannot tell a priori whether exceeding the out-system value is considered over-design or under-design.

Let us reconsider the example of the electric voltage in a domestic kettle.

• Let the out-system interface (the wall outlet) have a nominal value of 120V, a minimum of 104V, and a maximum of 127V (assuming the 87-106% range and all else in the house works properly); i.e., $[-104, -120, -127]V$. Values are negative because the voltage flows “out” of the context system.
• Let the in-system interface (the kettle) have a nominal value of 120V, a minimum of 110V, and a maximum of 130V; i.e., $[110, 120, 130]V$. Values are positive, because they represent flows “into” the kettle system. We note these are not necessarily the final values of the design, but rather the values that obtain at some point during the kettle’s design.
• The sum of the ranges of the interfaces is $[-6, 0, 3]V$.
• Therefore the balance of this interface - i.e., of the kettle in its operating context - is $[-0.06, 0, 0.02]$. Notice the balance is dimensionless.

This means that the interfaces are balanced at the nominal values, but over-designed at the upper end and under-designed at the lower end.

Here's a simple example with multiple BVs.

Assume two other BVs of the electric kettle are of interest:

• the heat energy produced by the kettle (as an engineering requirement equivalent to the customer requirement of heating time) and
• the volume of water able to be boiled at one time.

Data for this example is derived from estimates based on diverse samples found on the Web and personally owned kettles. The precise values are not important; rather, the point of the example is to demonstrate the kind of conclusions that can be drawn from the assessment of the BVs.

For the heat energy of the kettle, let us define the BV as $[-95, -280, -750]kJ$ for the out-system and $[90, 300, 750]kJ$ for the kettle (in-system). For the volume of water, let us define the BV as $[-0.25, -1, -2]L$ for the out-system, and $[0.5, 1.2, 2.25]L$ for the kettle (in-system).

The balance of the heat energy BV is $[0.05, 0.075, 0]$ and the balance of the volume BV is $[-1, 0.2, 0.125]$.

In summary, the balances of the three BVs are:

• Voltage: $[-0.06, 0, 0.02]$
• Heating energy: $[0.05, 0.075, 0]$
• Water volume: $[-1, 0.2, 0.125]$.

Since the balance measures for the three BVs are non-dimensional, we can even add the balance values together to arrive at a measure of the overall balance of the whole kettle. If these three BVs were the only ones in the design (though clearly there would be others in a real case), the current kettle design’s overall balance is $[-1.01, 0.275, 0.145]$. We might well argue that, overall, our kettle is significantly under-designed at this point in its development.

If we consider the individual balance calculations for the BVs in this example, we can see quickly where the imbalances are in the design. Because the balances are normalized and non-dimensional, we can also tell which imbalances are greatest and, presumably, more urgent based on their relative values. The worst imbalance is a comparatively significant under-design of the minimal water volume. That the nominal value of the water volume is also over-designed suggests some kind of misunderstanding of the scope of the kettle design. The voltage BV is the most balanced of the three. The heating energy aspect appears to be slightly over-designed generally. Overall, this analysis suggests that the most significant flaws in the design are around water volume and, ceteris paribus, these should probably be addressed first.

Future work

Fig. 3: A boundary variable with boundary layers defining its range.

It may be possible to tweak everything to include boundary layers, but I don't want to worry about it yet.

I do not like “fuzzy” values because we are not concerned with the probabilities of any particular value (not, at least, during design). It may be that later in a design, there will be enough information available to actually describe the probabilities. Similarly for “rough” sets/values.

My biggest problem with “fuzzy” values is that they are not real, they are only mental artifacts meant to capture subjective values. They may be useful for AI, but they are not useful here.

In the long term, fuzzy/probabilistic values may be useful for, say, folding some kind of FMEA analysis into the method. Still, it's going to be an optimization task, not a design task.

However, qualitative assessments, using something like a Likert scale, certainly is possible.