Mathematical Exposition of the Design Axioms

Abstract

Axiomatic design (AD) offers designers two fundamental principles to follow for a successful design: (1) identify and define the design objectives, i.e., functional requirements (FRs), in such a way that they are inherently independent; and (2) conceive solutions for the FRs that comply with two design axioms: the independent axiom and the information axiom. In the previous chapter, the rationale and origin for the axiomatic nature of the design axioms were provided. In this chapter, the two axioms are given a deeper mathematical understanding, thereby strengthening their value. Starting with the formal definition of functional independence, the criterion for functional independence of FRs in a design is derived as the Jacobian determinant |J| ≠ 0. Since | J | ≠ 0 implies independence of FRs and existence of design solutions, the |J| criterion corroborates the declaration of independence axiom that a good design must “maintain the independence of the functional requirements.” The |J| criterion further reveals that AD criterion for functional independence—design with single input–single output—is only a sufficient condition. For rigor and completeness, the |J| criterion is shown to be necessary and sufficient. In implementing information axiom, AD assessment of uncertainty in design should cover a larger extent than it currently does. AD has not and should begin to recognize and identify the sources of variability and the countermeasures to them. The chapter ends with a summary of implementation steps in AD expressed in mathematical terms.

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1 Introduction

Axiomatic design (AD) offers designers two fundamental principles to follow for a successful design: (1) define the design objectives, i.e., functional requirements (FRs), in such a way that they are inherently independent; and (2) conceive solutions in terms of design parameters (DPs) that maintain the independence of FRs as originally intended and have the highest probability of achieving FRs’ targets. AD uses independent axiom to check for functional independence and information axiom to assess probability of success. What follows are mathematical expositions of the two axioms. Chapter 1 discusses the dual application of the two axioms in design synthesis and design analysis. In this chapter, we will consider design analysis only.

2 Mathematical Exposition of Independence Axiom

Independent axiom in AD declares a criterion to check whether a conceived solution in terms of DPs maintains the functional independence of FRs. We develop another criterion derived from formal definition of functional independence. In the sections to follow, we discuss these two criteria. The discussion is confined to no more than 3 FRs and 3 DPs. However, the logic behind the discussion can be extended to n FRs and n DPs, n > 3.

2.1 AD Criterion for Functional Independence

Per independence axiom, “in an acceptable design, the DPs and the FRs are related in such a way that specific DP can be adjusted to satisfy its correspondent FR without affecting other functional requirements,” p. 48 [1]. In other words, AD criterion for functional independence is that adjustment ∆, of one and only one DP should affect only correspondent FR but not other FRs. It implies a single input–single output (SISO) relationship of FRs to DPs. A mathematical representation of the criterion is as follows:

$$\begin{aligned} &\Delta {\text{FR}}_{1} =\Delta {\text{FR}}_{1} \left( {\Delta {\text{DP}}_{1} } \right) \\ &\Delta {\text{FR}}_{2} =\Delta {\text{FR}}_{2} \left( {\Delta {\text{DP}}_{2} } \right) \\ &\Delta {\text{FR}}_{3} =\Delta {\text{FR}}_{3} \left( {\Delta {\text{DP}}_{3} } \right) \\ \end{aligned}$$

An alternative representation is with a design matrix (DM). A DM is indexed row-wise by FR_j and column-wise by DP_k. If DP_k has an effect on FR_j, the cell DM(j, k) is marked “X”. If it has no effect, the cell is marked “O”.

In DM representation, AD’s SISO criterion for independence is a diagonal DM. Such a design is called an uncoupled design.

$$\left[ {\begin{array}{*{20}c} {\Delta {\text{FR}}_{1} } \\ {\Delta {\text{FR}}_{2} } \\ {\Delta {\text{FR}}_{3} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{X}} & {{O}} & {{O}} \\ {{O}} & {{X}} & {{O}} \\ {{O}} & {{O}} & {{X}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\Delta {\text{DP}}_{1} } \\ {\Delta {\text{DP}}_{2} } \\ {\Delta {\text{DP}}_{3} } \\ \end{array} } \right]$$

Another representation that also satisfies the SISO criterion is as follows:

$$\begin{aligned} &\Delta {\text{FR}}_{1} =\Delta {\text{FR}}_{1} \left( {\Delta {\text{DP}}_{1} } \right); \\ &\Delta {\text{FR}}_{2} =\Delta {\text{FR}}_{2} \left( {\Delta {\text{DP}}_{1} ,\;\Delta {\text{DP}}_{2} } \right) \\ &\Delta {\text{FR}}_{3} =\Delta {\text{FR}}_{3} \left( {\Delta {\text{DP}}_{1} ,\;\Delta {\text{DP}}_{2} ,\;\Delta {\text{DP}}_{3} } \right). \\ \end{aligned}$$

In the above representation, ∆FR_k can be made a function solely of ∆DP_k if the adjustment ∆DP_k to satisfy the corresponding ∆FR_k follows the sequence: k = 1 firstly—so that ∆FR₁ becomes a constant in subsequent equation for ∆FR₂—followed by k = 2 secondly, and so on:

$$\begin{aligned} &\Delta {\text{FR}}_{1} =\Delta {\text{FR}}_{1} \left( {\Delta {\text{DP}}_{1} } \right); \\ &\Delta {\text{FR}}_{2} =\Delta {\text{FR}}_{2} \left( {\Delta {\text{FR}}_{1} ,\;\Delta {\text{DP}}_{2} } \right); \\ &\Delta {\text{FR}}_{3} =\Delta {\text{FR}}_{3} \left( {\Delta {\text{FR}}_{1} ,\;\Delta {\text{FR}}_{2} ,\;\Delta {\text{DP}}_{3} } \right). \\ \end{aligned}$$

This adjustment sequence is known as forward substitution; an algorithm used in solving lower triangular linear systems [2]. If the adjustment adheres to the sequence, the above shows ∆FR_k is a function exclusively of ∆DP_k. The SISO rule is thereby fulfilled. The DM is triangular. The design is called a decoupled design.

$$\left\{ {\begin{array}{*{20}c} {\Delta {\text{FR}}_{1} } \\ {\Delta {\text{FR}}_{2} } \\ {\Delta {\text{FR}}_{3} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {{X}} & {{O}} & {{O}} \\ {{X}} & {{X}} & {{O}} \\ {{X}} & {{X}} & {{X}} \\ \end{array} } \right]\;\left\{ {\begin{array}{*{20}c} {\Delta {\text{DP}}_{1} } \\ {\Delta {\text{DP}}_{2} } \\ {\Delta {\text{DP}}_{3} } \\ \end{array} } \right\}$$

In AD, only uncoupled and decoupled designs are acceptable. Since they are SISO, the FRs are obviously functionally independent of one another.

$${\text{SISO }} \Rightarrow {\text{functional }}\;{\text{independence}}$$

Any other design with DM that is neither diagonal nor triangular cannot satisfy SISO criterion. Such designs are called coupled designs. Per AD, they should be avoided since it is not obvious that the associated FRs are functionally independent of one another. We will show later that FRs in a design that does not satisfy SISO can still be functionally independent. In other words,

$${\text{Functional }}\;{\text{independence }}\neg \Rightarrow {\text{SISO}}$$

Accordingly, the independence axiom with SISO criterion is only a sufficient condition for functional independence.

To recap, per independent axiom, there are three categories of design: uncoupled, decoupled, and coupled. Within the coupled design, there are three subcategories.

One subcategory is designed with cyclic interaction: DP₁ affects FR₂, DP₂ affects FR₃, and DP₃ affects FR₁:

$$\left[ {\begin{array}{*{20}c} {\Delta {\text{FR}}_{1} } \\ {\Delta {\text{FR}}_{2} } \\ {\Delta {\text{FR}}_{3} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{X}} & {{O}} & {{X}} \\ {{X}} & {{X}} & {{O}} \\ {{O}} & {{X}} & {{X}} \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {\Delta {\text{DP}}_{1} } \\ {\Delta {\text{DP}}_{2} } \\ {\Delta {\text{DP}}_{3} } \\ \end{array} } \right].$$

Another subcategory is redundant design with more DPs than FRs:

$$\left[ {\begin{array}{*{20}c} {\Delta {\text{FR}}_{1} } \\ \\ {\Delta {\text{FR}}_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{X}} & {{X}} & {{X}} \\ & & \\ {{X}} & {{X}} & {{X}} \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {\Delta {\text{DP}}_{1} } \\ {\Delta {\text{DP}}_{2} } \\ {\Delta {\text{DP}}_{3} } \\ \end{array} } \right].$$

A third subcategory is design with insufficient DPs:

$$\left[ {\begin{array}{*{20}c} {\Delta {\text{FR}}_{1} } \\ {\Delta {\text{FR}}_{2} } \\ {\Delta {\text{FR}}_{3} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{X}} & {{X}} \\ {{X}} & {{X}} \\ {{X}} & {{X}} \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} \Delta {\text{DP}}_{1} \\ \\ \Delta {\text{DP}}_{2} \\ \end{array} } \right].$$

We next show examples of various categories and subcategories of design.

2.1.1 Examples of Various Categories of Design

2.1.1.1 Water Faucet Illustrating Coupled and Uncoupled Design

An example frequently used to illustrate the uncoupled and coupled design has the two alternative designs of a water faucet shown in Fig. 2.1. In the figure, Q subscripted h and c are respectively the flow rate of the hot and cold water. Both faucet designs a and b have the same functional requirements:

Fig. 2.1

Alternative designs a and b of a water faucet

$$\begin{aligned} & {\text{FR}}_{ 1} = {\text{control}}\;{\text{flow}}\;{\text{rate}}\;Q; \\ & {\text{FR}}_{ 2} = {\text{control}}\;{\text{flow}}\;{\text{temperature}}\;T. \\ \end{aligned}$$

For Faucet a, we choose as DP₁: the left knob controlling Q _h and as DP₂, the right knob controlling Q _c (see Fig. 2.1a). By assessing the effect of DPs on FRs, we arrive at the first category of coupled DM exhibiting cyclic interaction below.

$$\left[ {\begin{array}{*{20}c} {{\text{FR}}_{1} } \\ {{\text{FR}}_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{X}} & {{X}} \\ {{X}} & {{X}} \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {{\text{DP}}_{1} } \\ {{\text{DP}}_{2} } \\ \end{array} } \right]$$

Similarly for Faucet b, we choose as DP₁, the up/down of the lever to control total flow rate (Q _h + Q _c); and as DP₂, the clockwise/counterclockwise of the lever to control the ratio of the hot/cold water flow rate (Q _h/Q _c), see Fig. 2.1b. Again, by considering the effect of DPs on FRs, we obtain the uncoupled DM below that is acceptable per AD’s SISO criterion for functional independence.

$$\left[ {\begin{array}{*{20}c} {{\text{FR}}_{1} } \\ {{\text{FR}}_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{X}} & {{O}} \\ {{O}} & {{X}} \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {{\text{DP}}_{1} } \\ {{\text{DP}}_{2} } \\ \end{array} } \right]$$

The examples above illustrate a practical aspect of AD. That is, by simply considering the effects of DPs on FRs with (X, O), we can differentiate an acceptable design from an unacceptable design without going into the details of the physics. This practicality is very useful at the concept selection stage of design.

2.1.1.2 Projector Illustrating Redundant Design

Projector has two FRs:

$$\begin{aligned} & {\text{FR}}_{ 1} = {\text{magnify}}\;{\text{the}}\;{\text{image}}; \\ & {\text{FR}}_{ 2} = {\text{focus}}\;{\text{the}}\;{\text{image}}\;{\text{on}}\;{\text{the}}\;{\text{projection}}\;{\text{plane}}. \\ \end{aligned}$$

Figure 2.2a shows a projector; and Fig. 2.2b, the associated ray tracing of the light beam from the object plane, through the lens, and to the projection plane.

Fig. 2.2

Schematic a of a projector and its ray-tracing b

From the similar triangles shown in Fig. 2.2b, we have

$${\text{FR}}_{1} = \frac{{{\text{image}}\;{\text{height}}}}{{{\text{object}}\;{\text{height}}}} = \frac{D}{d};$$

Also per camera equation, the image is focused whenever

$${\text{FR}}_{2} = \frac{1}{D} + \frac{1}{d} + \frac{1}{f} = 0.$$

Thus, we have a redundant design with 2 FRs and 3 DPs as shown below.

$$\left[ {\begin{array}{*{20}c} {{\text{FR}}_{1} } \\ \\ {{\text{FR}}_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{X}} & {{X}} & {{O}} \\ & & \\ {{X}} & {{X}} & {{X}} \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {{\text{DP}}_{1} } \\ {{\text{DP}}_{2} } \\ {{\text{DP}}_{3} } \\ \end{array} } \right].$$

In the above, DP₁ is D, the distance of the lens from the screen aka the throw of the projector; DP₂ is d, the distance of the lens from the object, and DP₃ is f, the focal length of the lens.

A redundant design with more DPs than FRs cannot satisfy SISO criterion unless we fix the extra DPs. For example in the type of overhead projector shown in Fig. 2.2a, the focal length of the lens DP₃ is fixed. Hence, we have a design with equal number of FRs and DPs shown below. In this case, the design is coupled. With this type of projector, it would take several trial and errors to get the right magnification of the image focused at a given throw.

$$\left[ {\begin{array}{*{20}c} {{\text{FR}}_{1} } \\ {{\text{FR}}_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{X}} & {{X}} \\ {{X}} & {{X}} \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {{\text{DP}}_{1} } \\ {{\text{DP}}_{2} } \\ \end{array} } \right]$$

By contrast, in a portable projector used in a variety of room sizes that requires various throws, the distance of the object from the lens DP₂ is fixed. A zoom lens with varying focal length DP₃ is used. Thus we have a decoupled design:

$$\left[ {\begin{array}{*{20}c} {{\text{FR}}_{1} } \\ {{\text{FR}}_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{X}} & {{O}} \\ {{X}} & {{X}} \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {{\text{DP}}_{1} } \\ {{\text{DP}}_{3} } \\ \end{array} } \right].$$

With a portable projector, to attain a certain magnification focused at a certain throw is easy: set the throw for magnification then adjust the zoom lens for focus.

2.1.1.3 Disbursement Algorithm Illustrating a Redundant Design

Let us say we have an ATM that has in it the following bank notes denomination: $20, $10, $5, and $1. Three demands are made of the ATM as follows:

• FR₁: disburse bills that sum up to $Total.

• FR₂: disburse number of bills that totals to N _Total.

• FR₃: ensure the number of small bills is twice that of large bills.

There are four design parameters (DPs):

• DP₁: number of $20 bills, N _$20.

• DP₂: number of $10 bills, N _$10.

• DP₃: number of $5 bills, N _$5.

• DP₄: number of $1 bills, N _$1.

The 4 DPs would satisfy the three FRs as follows:

$$\begin{aligned} & {\text{FR}}_{1} = \$ 20N_{\$ 20} + \$ 10N_{\$ 10} + \$ 5N_{\$ 5} + \$ 1N_{\$ 1} = \$ {\text{Total}} \\ & {\text{FR}}_{2} = N_{\$ 20} + N_{\$ 10} + N_{\$ 5} + N_{\$ 1} = N_{\text{Total}} \\ & {\text{FR}}_{3} = 2N_{\$ 20} - N_{\$ 1} = 0. \\ \end{aligned}$$

Above is a redundant design with 3 FRs and 4 DPs with a DM as shown below:

$$\left[ {\begin{array}{*{20}c} {{\text{FR}}_{ 1} } \\ \\ {{\text{FR}}_{ 2} } \\ \\ {{\text{FR}}_{ 3} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{X}} & {{X}} & {{X}} & {{X}} \\ & & & \\ {{X}} & {{X}} & {{X}} & {{X}} \\ & & & \\ {{X}} & {{O}} & {{O}} & {{X}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {N_{\$ 20} } \\ {N_{\$ 10} } \\ {N_{\$ 5} } \\ {N_{\$ 1} } \\ \end{array} } \right]$$

If we were to fix an extra DP to get equal number of FRs and DPs, we would have 4 (=₄C₃) possible DM solutions as follows:

$$\left[ {\begin{array}{*{20}c} {{\text{FR}}_{1} } \\ {{\text{FR}}_{2} } \\ {{\text{FR}}_{3} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\$ 20} & {\$ 10} & {\$ 5} \\ 1 & 1 & 1 \\ 2 & 0 & { - 1} \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {N_{\$ 20} } \\ {N_{\$ 10} } \\ {N_{\$ 5} } \\ \end{array} } \right];$$

(2.1a)

$$\left[ {\begin{array}{*{20}c} {{\text{FR}}_{1} } \\ {{\text{FR}}_{2} } \\ {{\text{FR}}_{3} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\$ 20} & {\$ 10} & {\$ 1} \\ 1 & 1 & {1} \\ 2 & 0 & { - 1} \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {{N}}_{\$ 20} \\ {{N}}_{\$ 10} \\ {{N}}_{\$ 1} \\ \end{array} } \right];$$

(2.1b)

$$\left[ {\begin{array}{*{20}c} {{\text{FR}}_{1} } \\ {{\text{FR}}_{2} } \\ {{\text{FR}}_{3} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\$ 20} & {\$ 5} & \$ 1 \\ 1 & 1 & 1 \\ 2 & 0 & - 1 \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {{N}}_{\$ 20} \\ {{N}}_{\$ 5} \\ {{N}}_{\$ 1} \\ \end{array} } \right];$$

(2.1c)

$$\left[ {\begin{array}{*{20}c} {{\text{FR}}_{1} } \\ {{\text{FR}}_{2} } \\ {{\text{FR}}_{3} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\$ 10} & {\$ 5} & {\$ 1} \\ 1 & 1 & 1 \\ 2 & 0 & - 1 \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {N_{\$ 10} } \\ {N_{\$ 5} } \\ {N_{\$ 1} } \\ \end{array} } \right].$$

(2.1d)

Note in solution (2.1a), FR₃ = (FR₁ − $10 × FR₂)⁄$5. So that FR₃ = f(FR₁, FR₂) is no longer independent as originally planned. In fact, design Eq. (2.1a) is itself a redundant design with 2 FRs and 3 DPs. In short, fixing extra DPs in a redundant design to obtain a square DM does not guarantee a solution. It may induce coupling and destroy the functional independence originally planned for.

2.1.1.4 Hubcap Illustrating Insufficient DPs

Figure 2.3a shows the front of GM 1986–88 Pontiac 6000 hubcap. Figure 2.3b shows the wheel rim with a circumferential ledge, shown white, onto which the hubcap snapped on. The diameter of the ledge is D _rim. Figure 2.3c shows three pairs of clips at the back of the hubcap. The pair at the 4 o’clock position is shown enlarged in Fig. 2.3d. The clips are cantilevers fixed on a post. As seen in Fig. 2.3c, the three pairs of clips are spaced 120° apart such that the 6 clips form a circle of diameter D _clip, larger than D _rim. As the hubcap is snapped on to the rim, the rim ledge catches the cantilever clips. Wheel retention is developed through interference fit = kδ; where k is the spring rate of the cantilever clips and δ is the interference = (D _clip − D _rim)⁄2.

Fig. 2.3

Attachment of hubcap to wheel rim, a Hubcap front, b Wheel rim, c Hubcap back, d Cantilever clips

The are two FRs for the hubcap design:

• FR₁ = retain the hubcap over road bumps and on cornering;

• FR₂ = ease the removal of hubcap during a flat tire repair.

There is only one design parameter:

• DP₁ = interference, the larger the better for retention; the smaller the better for removal.

Obviously the design is flawed since it has insufficient DPs: one DP, the interference, cannot satisfy two conflicting FRs, retention, and removal. It violates the AD’s SISO criterion of one DP affecting only one FR.

$$\left[ {\begin{array}{*{20}c} {{\text{FR}}_{1} } \\ {{\text{FR}}_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{X}} \\ {{X}} \\ \end{array} } \right]\left[ {{\text{DP}}_{1} } \right]$$

The consequence was 25 % of hubcaps fell off as the car corners or hits bumps or potholes. And some customers have difficulty removing the hubcap for a flat repair. The solution [3] back then was to implement robust design optimization to find a clip spring rate k that reduces the design sensitivity to the variation in interference. The solution had limited success, as performance of an ill-conceived design cannot be improved through subsequent optimization.

2.1.2 Car Door-to-Body Integration—Coupling in Large System

Thus far, we have illustrated the use of AD’s SISO criterion to accept or reject various categories of design. These illustrations involve designs with small number of FRs. We now apply SISO to designs with large number of FRs; specifically to a car door-to-body integration. Figure 2.4 shows the car door integrated to the body opening.

Fig. 2.4

Integration of car door to body

Problems in car door-to-body integration, e.g., poor fit of door with neighboring panels; noisy ride and water leak; high opening and closing effort, are typical system problems. They appear only after the system is assembled since only then are couplings triggered. Fixing them is like playing a whack-a-mole game. As one solves a problem in one area, new problems pop up in other areas. This is because attempt to fix one FR failure inadvertently triggers other FR failures due to coupling. These types of system failures are not detectable by the traditional recursive design/build/test of components since they cannot capture the inter-dependence of FRs among subsystems and components.

AD takes a top-down approach. It starts with system-level FRs and decomposes them down through the subsystems until they reach the levels where known, implementable solutions exist. These levels reached are called leaf-level FRs, and the corresponding solutions, leaf-level DPs. Along the way, AD examines the interrelationships across the subsystems and components. In this way, functional couplings are understood and captured.

The AD top-down approach is as follows. First we decompose the system-level FRs down the design hierarchy to the leaf-level design parameters. From the decomposition, we construct the DM that shows the relationship among the leaf-level FR–DP pairs. We then successively remove those FR–DP pairs that are not part of the coupling. What remains is a reduced DM that contains all the couplings of the original DM. Retracing the leaf-level couplings up the design hierarchy reveals the roots of the system-level functional interactions.

2.1.2.1 Capturing Functional Couplings with the Design Matrix

Appendix A1 shows the FR decomposition and Appendix A2 the corresponding DP decomposition of the car door–to–body system. The decomposition starts with three subsystems: FR₁: fit door to neighboring panels; FR₂: keep interior quite and intrusion free; and FR₃: ensure door opens and closes properly. The subsystem FRs are then decomposed down the hierarchy, zigzagging between FRs and DPs, i.e., between Appendices A1 and A2, until the leaf-level DPs marked by “+” are reached. These leaf-level DPs, together with the corresponding FRs they satisfy, are listed in Table 2.1. For succinct presentation, Table 2.1 uses serial notations 1–28 in place of hierarchical notations in Appendix A. For example, FR₁ in Table 2.1 refers to FR_1.1.1 in Appendix A1. Note that FR₁ thru FR₇ are leaf-level FRs that flow from the first subsystem; FR₈ thru FR₁₈ are leaf-level FRs, from the second subsystem and FR₁₉ thru FR₂₈, from the third subsystem.

Table 2.1 Leaf-level FR–DP of car door-to-body system

From Table 2.1, we construct the DM that captures the functional interdependencies among the 28 leaf-level FR–DP pairs, see Fig. 2.5. Row-wise are leaf-level FRs; column-wise are associated leaf-level DPs. Leaf-level FR₁ thru FR₇ and their associated DPs, which flow from the first subsystem, are labeled to the left and top of the DM, not shaded.

Fig. 2.5

Design matrix relating the leaf-level FR–DP

The leaf-level FR₈ thru FR₁₈ and their associated DPs, which flow from the second subsystem, are shown lightly shaded green. And FR₁₉ thru FR₂₈, which flow from the third subsystem are heavily shaded blue. For each cell DM (j, k) of the 28 × 28 cells, assessment is made whether the DP_k has an effect on FR_j. If it has an effect (no effect), the cell is marked with an “X” (blank). While cell-by-cell evaluation is tedious, it is crucial because functional inter-dependencies so obtained among the leaf-level FR–DP decide how the design functions at the system level.

2.1.2.2 Reducing the Design Matrix to Uncover Functional Coupling

When a DM is sparse or small, we can check for coupling by inspection. When it is large, the task becomes difficult. For DM with n × n = 28 × 28 and a total off-diagonal elements of m = 88 as in Fig. 2.5, the number of possible couplings equals 2^m-n+1 − 1 = 2.306E18 [4]. It is prohibitive to check for couplings among this large number of candidates. Thus, we reduce the dimension of DM by isolating the submatrix that contains the couplings from the rest. We do this by successively relocate rows (columns) whose row-wise (column-wise) entries are all zeros but the diagonal element. For example in Fig. 2.5, we find five zero-rows: FR₁, FR₄, FR₉, FR₁₃, and FR₂₇; six zero-columns: DP₁₄, DP₁₅, DP₁₇, DP₂₁, DP₂₂, and DP₂₈ and four combined (zero-rows, zero-columns): (FR₁₁, DP₁₁), (FR₁₂, DP₁₂), (FR₁₈, DP₁₈), and (FR₂₆, DP₂₆). A zero-row corresponds to an FR that is not affected by other DPs; a zero-column corresponds to a DP that does not affect any other FRs; and a combined (zero-row, zero-column) corresponds to an FR–DP pair that does not affect nor be affected by other FRs and DPs. All the three categories do not belong to any coupling loop. They are thus moved from their original locations. Namely through row and column interchange, all combined (zero-row, zero-column) are relocated to the upper left corner of the DM; all zero-columns (zero-rows) and their associated rows (columns) are relocated to the lower right (upper left) corner of the DM. What remains is a 13 × 13 submatrix outlined in thick border as shown in Fig. 2.6.

Fig. 2.6

Relocating zero-rows and zero-columns

Further examination of the 13 × 13 submatrix reveals 3 more zero-rows: FR₂, FR₅ and FR₂₃; and 3 more zero-columns: DP₃, DP₁₀ and DP₂₀, as shown in Fig. 2.7a. We repeat the successive relocation of these zero-rows and zero-columns and arrive at a further reduced 7 × 7 submatrix outlined in thick border in Fig. 2.7b.

Fig. 2.7

Further detection a and relocation b of zero-row and zero-column

Continuing the search for zero-rows and zero-columns in the 7 × 7 reduced matrix, we found 1 zero-row FR₆ and 1 zero-column DP₁₉ (see Fig. 2.8a). Upon relocation of these two, we finally obtain a 5 × 5 matrix that containing neither zero-row nor zero-column as outlined in thick border, Fig. 2.8b. All the couplings in the original decomposed DM are now isolated and condensed into this 5 × 5 DM.

Fig. 2.8

Final detection a and relocation b of zero-row and zero-column

2.1.2.3 Implications from the Reduction of DM

Figure 2.9a shows DM as decomposed juxtaposed with Fig. 2.9b DM as condensed. The as-condensed DM shows three submatrices: a (4 × 4) uncoupled submatrix in the upper left; a (24 × 24) decoupled submatrix in the lower right; and a protruding (5 × 5) coupled submatrix within the decoupled submatrix. These results reflect the reduction algorithm: relocating the combined (zero-rows, zero-columns) to DM upper left produces the uncoupled submatrix; relocating the zero-rows to DM upper left plus the zero-columns to DM lower right produce the decoupled submatrix. What remains is a protruding coupled submatrix within the decoupled submatrix. As indicated on top of DM Fig. 2.9b, the algorithm also produces a mingling of leaf-level DPs from the three subsystems. The implications of these results are as follows.

Fig. 2.9

Design matrix, a as decomposed and b as condensed

The protruding (5 × 5) coupled submatrix is the source of the whack-a-mole type of failures. The couplings need to be resolved first and foremost by effectively identify and eliminate functional couplings following for example a graph theory-based method described in [4].

Once couplings in the (5 × 5) coupled matrix are resolved, what remains is a (24 × 24) lower triangular DM. The lower triangular DM serves as a road map and provides a sequence to follow in satisfying ∆FRs: adjust ∆DP_j, j = 1 to i, to satisfy/fix ∆FR_i. Without the road map, we will still be fighting the whack-a-mole type of failure.

We must recognize which FR–DP pair falls unto the uncoupled submatrix and take advantage of the information, as they are the easiest to fix and satisfy.

Engineers in a door group are typically tasked with specific leaf-level functions of the door. As indicated in Fig. 2.9b, there is a mingling of the leaf-level DP that forms the triangular DM. The engineers must be made aware of this interdependency, i.e., mingling, of functions since their tasks must conform to the sequence dictated by the triangular DM.

The implications described above hold for any assembly of subsystems and components that form a large system.

2.2 |J| Criterion for Functional Independence

2.2.1 Derivation of |J| as a Criterion for Functional Dependence

Unlike AD’s SISO criterion for functional independence that is derived from empirical observations, in this section we will derive the criterion based on the formal mathematical definition of functional independence.

To illustrate, consider the car door-to-body integration example in Sect. 2.2.1.2. We start with the system-level FRs and decompose them down the hierarchy through the subsystems level to the leaf-level FRs. The FRs, which are conceptual at the system level, get more specific and detailed as they are decomposed down the hierarchy. When the leaf levels are reached, the FRs are realized by DPs that are known, implementable physical solutions. Thus, the FRs can be expressed in terms of the DPs through physics. For example, Table 2.1 relates 28-leaf-level FRs to corresponding leaf-level DPs. We denote these relationships as follows:

$$\begin{array}{*{20}c} {{\text{FR}}_{1} = {{f}}_{1} \left( {{\text{DP}}_{1} , \ldots ,{\text{DP}}_{\text{m}} } \right)} \\ \vdots \\ {{\text{FR}}_{\text{n}} = {{f}}_{\text{n}} \left( {{\text{DP}}_{1}, \ldots, {\text{DP}}_{\text{m}} } \right)} \\ \end{array}$$

Or in vector notation,

$${\mathbf{FR}} = {\mathbf{f}}({\mathbf{DP}});$$

In the above and hereafter, a bolded quantity denotes a vector, a bracketed quantity denotes a matrix, and f(•) denotes vector valued functions.

The vector equation above is Eq. (2.6) of Chap. 1, with f(DP) ≡ f _a(DP). The vector DP represents the physical quantities of the design, and the vector valued function f(DP) represents the laws of physics relating FR to DP. Since f(DP) is drawn from laws of physics, it may be assumed as continuous. So that we may expand f(DP) in a Taylor series about a design point DP ^*:

$$\begin{aligned} {\mathbf{f}}\left( {{\mathbf{DP}}} \right) & = {\mathbf{f}}\left( {{\mathbf{DP}}^{*} } \right) + \left[ {{ J }} \right]\left( {{\mathbf{DP}} - {\mathbf{DP}}^{*} } \right) + {\mathbf{o}}\left( {||{\mathbf{DP}} - {\mathbf{DP}}^{*} ||} \right) \\ \quad \quad \quad & \approx {\mathbf{f}}\left( {{\mathbf{DP}}^{*} } \right) + \left[ {{J }} \right]\left( {{\mathbf{DP}} - {\mathbf{DP}}^{*} } \right) \\ {\text{Thus:}} \quad{\mathbf{FR}} - {\mathbf{FR}}^{*} & \approx \left[ {{ J }} \right]\left( {{\mathbf{DP}} - {\mathbf{DP}}^{*} } \right)\\ \end{aligned}$$

where [J] is the Jacobian matrix whose element J _ij = ∂FR_i/∂DP_j evaluated at the design point DP* is a constant.

We recognize the Jacobian [J] above is in fact the design matrix [A] in AD, Eq. (3.3) in [1]. Thus, we may rewrite the vector equation as follows:

$$\Delta {\mathbf{FR}} = \left[ {{A}} \right]\Delta {\mathbf{DP}}.$$

(2.2)

If FRs are linear functions of DP, the above differential vector equation reduces:

$${\mathbf{FR}} = \left[ {{A}} \right]{\mathbf{DP}}.$$

(2.2a)

Equation (2.2a) is known as a “design equation” and appears on page 55 of the first axiomatic design text [1]. It has since been used extensively for conceptual applications in the AD literature. However, it is important to recognize that in most cases Eq. (2.2a) is only a notation to convey a relation between FR and DP. In actuality, the equation to solve for is the differential form in Eq. (2.2).

Expanding the differential vector equation for the special case of i, j = 1, 2:

$$\Delta {\text{FR}}_{ 1} = {{ A}}_{ 1 1}\Delta {\text{DP}}_{1} + {{ A}}_{ 1 2}\Delta {\text{DP}}_{ 2} ;$$

(2.3)

$$\Delta {\text{FR}}_{ 2} = {{ A}}_{ 2 1}\Delta {\text{DP}}_{ 1} + {{ A}}_{ 2 2}\Delta {\text{DP}}_{ 2} .$$

(2.4)

Equation (2.3) × A ₂₂ minus Eq. (2.4) × A ₁₂ to eliminate ∆DP₂ gives:

$$A_{ 2 2}\Delta {\text{FR}}_{ 1} - A_{ 1 2}\Delta {\text{FR}}_{ 2} = \left( {A_{ 2 2} A_{ 1 1} - A_{ 1 2} A_{ 2 1} } \right) \Delta {\text{DP}}_{ 1}$$

(2.5)

Note that (A ₂₂ A ₁₁ − A ₁₂ A ₂₁) = $\left( {\frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{DP}}_{2} }}} \right)\left( {\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{DP}}_{1} }}} \right) - \;\left( {\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{DP}}_{2} }}} \right)\;\left( {\frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{DP}}_{1} }}} \right)$ is the determinant |A| of the DM. It is known as |J|, the Jacobian determinant in vector calculus.

$$| {J}| = \left| {\begin{array}{*{20}l} {\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{DP}}_{1} }}} & {\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{DP}}_{2} }}} \\ {\frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{DP}}_{1} }}} & {\frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{DP}}_{2} }}} \\ \end{array} } \right|.$$

Thus, if (A ₂₂ A ₁₁ − A ₁₂ A ₂₁) = |J| = 0 in Eq. (2.5), then ∆FR₂ = A ₂₂∆FR₁/A ₁₂. Or FR₂ = FR ^*₂  + A ₂₂ (FR₁ − FR ^*₁ )/A ₁₂. Namely, FR₂ is dependent on FR₁.  Hence, |J| = 0 implies functional dependence.

$${\text{Proof }}\;{\text{A:}} \quad \left( { | {J} | = 0 } \right) \Rightarrow {\text{functional}}\;{\text{dependence }}$$

We next prove the converse is true. Namely if FR₂ is functionally dependent on FR₁, then |J| = 0. We start with the formal definition of functional dependency. Namely, FR₂ is dependent on FR₁ if it is a function of FR₁:

$${\text{FR}}_{2} = {\text{FR}}_{2} \left( {{\text{FR}}_{1} } \right)$$

Applying the chain rule for differentiation on above equation, we have

$$\begin{aligned} & \frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{DP}}_{1} }} = \frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{FR}}_{1} }}\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{DP}}_{1} }} \\ & \frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{DP}}_{2} }} = \frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{FR}}_{1} }}\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{DP}}_{2} }}. \\ \end{aligned}$$

In the above, multiply 1st equation by $\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{DP}}_{2} }}$ and the 2nd by $\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{DP}}_{1} }}.$ Subtract one resulting equation from the other to eliminate $\left( {\frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{FR}}_{1} }}} \right)\;\left( {\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{DP}}_{2} }}} \right)\left( {\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{DP}}_{1} }}} \right)$, we have:

$$\left( {\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{DP}}_{2} }}} \right)\;\left( {\frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{DP}}_{1} }}} \right)\;\; - \;\left( {\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{DP}}_{1} }}} \right)\;\left( {\frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{DP}}_{2} }}} \right) \equiv | {{{J}}}| = 0$$

Hence, functional dependence implies |J| = 0. This is the proof of the converse:

$${\text{Proof}}\;{\text{B:}} \quad{\text{Functional}}\;{\text{dependence}} \Rightarrow ( |J| { = 0)} .$$

Combining both proofs A and B, we have

$${\text{Functional}}\;{\text{dependence}}\, \Leftrightarrow \, ( |J | { = 0)} .$$

Namely, |J| = 0 is a necessary and sufficient condition for FR₂ to be functionally dependent on FR₁. Likewise, FR₂ is functionally independent of FR₁ if and only if (iff) |J| ≠ 0. Thus by formal definition of functional dependency, we have derived the criterion: FRs are functionally independent iff |J| ≠ 0; dependent iff |J| = 0.

2.2.2 Implications of |J| as a Criterion for Functional Independence

The differential form of Eq. (2.2) may be rewritten as follows.

$$\Delta {\mathbf{FR}} = [J] \Delta {\mathbf{DP}}$$

So that the adjustments ∆DP necessary to bring FR to its target FR* is,

$$\Delta {\mathbf{DP}} = [J]^{-1} \Delta {\mathbf{FR}}$$

Note that if FRs of the design are functionally dependent, then |J| = 0 and its inverse |J|⁻¹ does not exist. In mathematical lingua, the design has a “singularity” in its first derivative and is non-differentiable. Consequently, no adjustments in ∆DP can bring the design to its target value FR*.

AD’s independence axiom declares that a good design must “maintain the independence of the functional requirements (FR).” The |J| criterion corroborates this declaration since |J| ≠ 0 implies independence of FR and it guarantees a design solution. Therefore, the |J| criterion provides formidable theoretical evidence that a violation of the independence axiom will impede the design from finding a final value DP* that fulfills the design equation and meets all functional requirements FR*. While the AD independence axiom was established through extensive empirical study to yield “good” designs, the |J| criterion shows that these “good” designs not only fulfill all their functional requirements but also can be found through well established analytical and numerical methods.

AD further proposes SISO as the criterion for independence. Since determinant of a diagonal or triangular DM—a SISO design—is the product of all the diagonal elements none of which is zero, their |J| ≠ 0. This validates SISO as a criterion for functional independence. However, SISO criterion is only a sufficient condition. Namely, SISO implies functional independence but functional independence does not imply SISO:

$$\begin{aligned} & {\text{SISO }} \Rightarrow {\text{Functional}}\;{\text{independence;}} \\ & {\text{Functional}}\;{\text{independence}} \neg \Rightarrow {\text{SISO}} .\\ \end{aligned}$$

While SISO criterion is more conservative, it does not detract from its utility. It remains as a sufficiency condition. Furthermore, during design synthesis it is relatively easy to mentally keep track of a diagonal or lower triangular design matrix. In comparison, calculating a Jacobian is significantly harder; especially at high-level conceptual design synthesis whereat the mathematical form of the design equations is not well known.

Nevertheless, there are cases where the SISO criterion is inadequate as there are coupled, non-SISO designs with |J| ≠ 0. Such cases are functionally independent thus admit design solutions but will be rejected per AD’s SISO criterion. For example in robotics, the robot Jacobian that relates joint velocities to end-effector velocities is used routinely to plan and execute robot paths and transform forces and torques from the end effector to the manipulator. The robot Jacobian is in fact a design matrix that relates output (end-effector velocities) to input (joint velocities). In most cases, it is not SISO so that most robot designs would have been rejected per the SISO criterion.

To recap, the Jacobian matrix [J] relates ∆FR to ∆DP of a conceived solution. Its determinant |J| is a test for functional independence of FR: yes iff |J| ≠ 0; no iff |J| = 0. Furthermore, iff |J| ≠ 0, then the conceived solution in term of DP can satisfy the FR. Otherwise, it cannot. In short, |J| acts as a qualifier: accept a design solution iff |J| ≠ 0.

We end this section with Table 2.2 which provides a contrast between SISO criterion and |J| criterion derived per formal definition of functional dependency.

Table 2.2 Contrasting AD SISO criterion with |J| criterion

2.2.3 |J| for Various Categories of Design

2.2.3.1 Water Faucet

Both designs a and b have the same FRs

• FR₁ = control flow rate Q;

• FR₂ = control water temperature T.

Both designs have the same governing physics:

$${\text{Mass}}\;{\text{conservation:}}\quad \;Q = Q_{\text{h}} + Q_{\text{c}} ;$$

(2.6a)

$$\begin{aligned} {\text{Energy}}\;{\text{conservation:}}\quad QT & = Q_{\text{h}} T_{\text{h}} + Q_{\text{c}} T_{\text{c}} . \\ T & = \frac{{Q_{\text{h}} T_{\text{h}} + Q_{\text{c}} T_{\text{c}} }}{{Q_{\text{h}} + Q_{\text{c}} }} \\ \end{aligned}$$

(2.6b)

$$= \frac{{\left( {{{Q_{\text{h}} } \mathord{\left/ {\vphantom {{Q_{\text{h}} } {Q_{\text{c}} }}} \right. \kern-0pt} {Q_{\text{c}} }}} \right)T_{\text{h}} + T_{\text{c}} }}{{\left( {{{Q_{\text{h}} } \mathord{\left/ {\vphantom {{Q_{\text{h}} } {Q_{\text{c}} }}} \right. \kern-0pt} {Q_{\text{c}} }}} \right) + 1}}$$

(2.6c)

For Faucet a, we choose the left knob controlling Q _h as DP₁; and the right knob controlling Q _c as DP₂ (see Fig. 2.1a). Substituting into Eqs. (2.6a) and (2.6b):

$$\begin{array}{*{20}l} & {{\text{FR}}_{1} = {\text{DP}}_{1} + {\text{DP}}_{2} ;} & {{\text{FR}}_{2} = \frac{{{\text{DP}}_{1} T_{\text{h}} + {\text{DP}}_{2} T_{\text{c}} }}{{{\text{DP}}_{1} + {\text{DP}}_{2} }}.} \\ & {\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{DP}}_{1} }} = 1;} & {\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{DP}}_{2} }} = 1.} \\ & {\frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{DP}}_{1} }} = \frac{{\left( {T_{\text{h}} - T_{\text{c}} } \right){\text{DP}}_{2} }}{{\left( {{\text{DP}}_{1} + {\text{DP}}_{2} } \right)^{2} }};} & {\frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{DP}}_{2} }} = - \frac{{\left( {T_{\text{h}} - T_{\text{c}} } \right){\text{DP}}_{1} }}{{\left( {{\text{DP}}_{1} + {\text{DP}}_{2} } \right)^{2} }}} \\ \end{array}$$

$$| {{{J}}} | = \left| {\begin{array}{*{20}c} 1 & 1 \\ \frac{{\left( {T_{\text{h}} - T_{\text{c}} } \right){\text{DP}}_{2} }}{{\left( {{\text{DP}}_{1} + {\text{DP}}_{2} } \right)^{2} }} & - \frac{{\left( {T_{\text{h}} - T_{\text{c}} } \right){\text{DP}}_{1} }}{{\left( {{\text{DP}}_{1} + {\text{DP}}_{2} } \right)^{2} }} \\ \end{array} } \right| = - \frac{{T_{\text{h}} - T_{\text{c}} }}{{{\text{DP}}_{1} + {\text{DP}}_{2} }}$$

(2.7)

Per AD’s SISO criterion, FRs in Faucet a are coupled and the design should be rejected. However, according to formal definition of functional dependence, FRs of the design are functionally independent since |J| ≠ 0. It is therefore acceptable.

For Faucet b, we choose as DP₁, the up/down lever controlling total flow rate (Q _h + Q _c), and as DP₂, the left/right lever controlling ratio of the hot/cold water flow rate (Q _h/Q _c) (see Fig. 2.1b). Substituting into Eqs. (2.6a) and (2.6c), we have an uncoupled design as indicated by the design equation:

$$\begin{array}{*{20}l} & {{\text{FR}}_{1} = {\text{DP}}_{1} ;} & {{\text{FR}}_{2} = \frac{{{\text{DP}}_{2} {T_{\text{h}} + T_{\text{c}}} }}{{{\text{DP}}_{2} + 1}}.} \\ & {\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{DP}}_{1} }} = 1;} & {\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{DP}}_{2} }} = 0;} \\ & {\frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{DP}}_{1} }} = 0;} & {\frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{DP}}_{2} }} = \frac{{\left( {T_{\text{h}} - T_{\text{c}} } \right)}}{{\left( {{\text{DP}}_{2} + 1} \right)^{2} }}.} \\ \end{array}$$

$$| {J} | = \left| {\begin{array}{*{20}c} 1 & 0 \\ 0 & \frac{\left( {T_{\text{h}} - T_{\text{c}} } \right)}{{\left( {{\text{DP}}_{2} + 1} \right)^{2} }} \\ \end{array} } \right| = \frac{\left( {T_{\text{h}} - T_{\text{c}} } \right)}{{\left( {{\text{DP}}_{2} + 1} \right)^{2} }}$$

(2.8)

Note that as the water heater temperature T _h is set closer to outside water temperature T _c, the |J| value gets closer to zero so that the faucet becomes less capable of providing the two independent functions. In short, the |J| criterion provides a quantitative measure of independence which the AD SISO criterion cannot.

Note further that the physics governing both faucets are the same. Yet their [J ] matrices in Eqs. (2.7) and (2.8) are different: one is coupled and the other is uncoupled. This conveys a fundamental message in AD. Namely, it is the choice of design solutions DPs, not the physics that determine the goodness of a design.

2.2.3.2 Projector Design

The two FRs of a projector are as follows:

$$\begin{aligned} & {\text{FR}}_{1} = \text{magnify}\;\text{the}\;\text{image} = \frac{D}{d}; \\ & {\text{FR}}_{2} = \text{focus}\;\text{the}\;\text{image} = \frac{1}{D} + \frac{1}{d} + \frac{1}{f} = 0. \\ \end{aligned}$$

(2.9)

In the above, D = DP₁ is the distance of the lens from the screen aka the throw of the projector; d = DP₂ is the distance of the lens from the object and f = DP₃ is the focal length of the lens. Thus, we have a redundant design: 2 FRs and 3 DPs. If we were to fix the extra DP to get an equal number of FRs and DPs, we would have three (= ₃C₂) possible |J| solutions as follows:

$$\begin{aligned} & \frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{D}}}} = \frac{1}{d};\quad \quad \quad \frac{{\partial {\text{FR}}_{1} }}{\partial d} = - \frac{D}{{d^{2} }};\quad \quad \frac{{\partial {\text{FR}}_{1} }}{\partial f} = 0. \\ & \frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{D}}}} = - \frac{1}{{D^{ 2} }};\quad \frac{{\partial {\text{FR}}_{2} }}{\partial d} = - \frac{1}{{d^{2} }};\quad \quad \frac{{\partial {\text{FR}}_{2} }}{\partial f} = - \frac{1}{{f^{2} }}. \\ \end{aligned}$$

$$\left( {D,d} \right)\;{\text{as}}\;{\text{DPs}}:\; | {J}| = \left| {\begin{array}{*{20}c} {\frac{1}{d}} & { - \frac{D}{{d^{2} }}} \\ { - \frac{1}{{D^{2} }}} & { - \frac{1}{{d^{2} }}} \\ \end{array} } \right| = - \frac{1}{{{{d}}^{2} }}\left( {\frac{1}{D} + \frac{1}{d}} \right);$$

(2.10a)

$$\left( {D,{{f}}} \right)\;{\text{as}}\;{\text{DPs}}:\quad | {J} | = \left| {\begin{array}{*{20}c} {\frac{1}{{d}}} & 0 \\ { - \frac{1}{{D^{2} }}} & { - \frac{1}{{{{f}}^{2} }}} \\ \end{array} } \right| = - \frac{1}{{{{df}}^{2} }}.$$

(2.10b)

$$\begin{aligned} \left( {d ,f} \right)\;{\text{as}}\;{\text{DPs}}:\quad | {J} | = \left| {\begin{array}{*{20}c} { - \frac{D}{{d^{2} }}} & 0 \\ { - \frac{1}{{d^{2} }}} & { - \frac{1}{{f^{2} }}} \\ \end{array} } \right| = \frac{D}{{d^{2} f^{2} }}. \\ \\ \end{aligned}$$

(2.10c)

All three candidate sets of DPs for the FRs do ensure functional independence of FRs because their |J| ≠ 0. However, AD SISO criterion will reject solution (2.10a). Nonetheless solution (2.10b), which is acceptable to both AD SISO and |J| criteria, would be the preferred choice since it permits wider latitude of throw D for the projector to accommodate various room sizes.

2.2.3.3 Disbursement Algorithm

The |J| values for the four possible DM solutions, Eqs. (2.1a) thru (2.1d), in Sect. 2.2.1.1 are as follows:

$$| {J} | = \left| {\begin{array}{*{20}c} {\$ 20} & {\$ 10} & {\$ 5} \\ 1 & 1 & 1 \\ 2 & 0 & { - 1} \\ \end{array} } \right| = \$ 0;$$

(2.11a)

$$| {J} | = \left| {\begin{array}{*{20}c} {\$ 20} & {\$ 10} & {\$ 1} \\ 1 & 1 & 1 \\ 2 & 0 & { - 1} \\ \end{array} } \right| = \$ 8;$$

(2.11b)

$$| {J} | = \left| {\begin{array}{*{20}c} {\$ 20} & {\$ 5} & {\$ 1} \\ 1 & 1 & 1 \\ 2 & 0 & { - 1} \\ \end{array} } \right| = - \$ 7;$$

(2.11c)

$$| {J} | = \left| {\begin{array}{*{20}c} {\$ 10} & {\$ 5} & {\$ 1} \\ 1 & 1 & 1 \\ 2 & 0 & { - 1} \\ \end{array} } \right| = \$ 3.$$

(2.11d)

We showed in Sect. 2.2.1.1 that for design solution (2.1a), FR₃ is dependent on FR₁ and FR₂. This is confirmed in Eq. (2.11a) above which shows | J | = 0. This example demonstrates that an improper choice of DPs can destroy functional independency as originally intended. This is why we need to continually check for it.

2.2.3.4 Design with Insufficient DPs

To show that | J | of a design with insufficient DPs is zero, i.e., the design FRs are functionally dependent, consider a design with two (FR₁, FR₂) and one DP₁ whose effect on (FR₁, FR₂) are (A ₁₁, A ₂₁):

$$\left[ {\begin{array}{*{20}l} {\Delta {\text{FR}}_{1} } \\ {\Delta {\text{FR}}_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {{A}}_{11} \\ {{A}}_{21} \\ \end{array} } \right]\left[ {\Delta {\text{DP}}_{1} } \right].$$

We conjure up a second DP₂ identical to DP₁ to make up for the insufficiency in DP. This second DP₂ has identical effects of (A _11, A ₂₁) on (FR₁, FR₂):

$$\left[ {\begin{array}{*{20}l} {\Delta {\text{FR}}_{1} } \\ {\Delta {\text{FR}}_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {{A}}_{11} & {{A}}_{11} \\ {{A}}_{21} & {{A}}_{21} \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {\Delta {\text{DP}}_{1} } \\ {\Delta {\text{DP}}_{2} } \\ \end{array} } \right]$$

The | J | of the above design equals A ₁₁ A ₂₁ − A ₁₁ A ₂₁ = 0. This confirms that FR₁ and FR₂ are functionally dependent when there are insufficient DPs.

3 Mathematical Exposition of Information Axiom

The information axiom states that a good design solution must minimize its information contents I, or equivalently maximize its probability of success, P _s. Figure 2.10 illustrates the evaluation of P _s. In the presence of variability, an FR_i will exhibit a range of values called the system range or the spread. Cognizant of its variability, a designer would accept the FR_i if it falls within a specified range called the design range. The overlap of the two ranges shown shaded in Fig. 2.10 is P _s, the probability of success of the design. To consider P _s, we need to identify sources of variation that are generating the variability in FRs; how the variability is magnified by the design; and what are the countermeasures for them. We consider these in the next several sections.

Fig. 2.10

Evaluating the probability of success

3.1 Recognition of Noise Variables

Since variability in FRs is a consequence of variation, we need to recognize and identify the sources that are generating the variation. We denote these sources as the noise variables, NVs. For example in the faucet design, the temperature T _c of the cold water in Eqs. (2.6b) and (2.6c), Sect. 2.2.2.3, entering the faucet from the outside is a NV since it fluctuates with the uncontrollable temperature outside. It is a NV induced by the environment. If the water heater in a building does not have sufficient capacity to meet the demand of multiple faucets, hot water pressure will fluctuate with the number of faucets turned on or off at a given time. This will affect hot water flow Q _h in Eqs. (2.6a), (2.6b), and (2.6c). This is a NV induced by customer usage. If a projector is used for a variety of room size, it will need a variety of throws to magnify the image. Hence, the throw D in Eq. (2.9), Sect. 2.2.2.3 is a NV induced by customer usage. In the hubcap design, the NV is the interference caused by the manufacturing variation in D _rim and D _clip.

Let NV denotes the noise variables that cause variation in FR. A NV triggers a random deviation in FR from its current value FR ^* given by the amount:

$${\mathbf{FR}} - {\mathbf{FR}}^{{\mathbf{*}}} = \left[ {{{J}}^{\text{NV}} } \right] \left( {{\mathbf{NV}} - {\mathbf{NV}}^{{\mathbf{*}}} } \right).$$

(2.12)

In Eq. (2.12), NV * is a reference value, e.g., the midrange of NV; [J ^NV] is the Jacobian matrix of ∂FR_i/∂NV_j given by

$$\left[ {J^{\text{NV}} } \right] = \left[ {\begin{array}{*{20}c} {\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{NV}}_{1} }}} & \cdots & {\frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{NV}}_{\text{m}} }}} \\ \vdots & \ddots & \vdots \\ {\frac{{\partial {\text{FR}}_{\text{n}} }}{{\partial {\text{NV}}_{1} }}} & \cdots & {\frac{{\partial {\text{FR}}_{\text{n}} }}{{\partial {\text{NV}}_{\text{m}} }}} \\ \end{array} } \right]_{{{{n}} \times {{m}}}}$$

The superscript NV is used to distinguish [J ^NV] from [J], the Jacobian matrix of ∂FR_i/∂DP_j which hereafter will be superscripted with “DP.” While [J ^DP] relates to the functional independence of FR in a design, [J ^NV] relates to the sensitivity of FR to noise NV. To illustrate, the [J ^NV] for some earlier designs are as follows.

For the faucet designs a and b:

$$\begin{aligned} & {\text{FR}}_{1} = Q_{\text{h}} + Q_{\text{c}} ;\quad \quad \quad {\text{FR}}_{2} = \frac{{Q_{\text{h}} T_{\text{h}} + Q_{\text{c}} T_{\text{c}} }}{{Q_{\text{h}} + Q_{\text{c}} }}. \\ & {\text{NV}}_{1} = Q_{\text{h}} ;\quad \quad \quad \quad \quad {\text{NV}}_{2} = T_{\text{c}} . \\ \end{aligned}$$

$$\begin{aligned} & \frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{NV}}_{1} }} = 1;\quad \quad \quad \quad \quad \quad \quad \frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{NV}}_{2} }} = 0. \\ & \frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{NV}}_{1} }} = \frac{{Q_{\text{c}} \left( {T_{\text{h}} - T_{\text{c}} } \right)}}{{\left( {Q_{\text{h}} + Q_{\text{c}} } \right)^{2} }}\quad \quad \quad \frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{NV}}_{2} }} = \frac{{Q_{\text{c}} }}{{Q_{\text{h}} + Q_{\text{c}} }}. \\ \end{aligned}$$

$$\left[ {J^{\text{NV}} } \right] = \left[ {\begin{array}{*{20}c} 1 & 0 \\ {\frac{{Q_{\text{c}} \left( {T_{\text{h}} - T_{\text{c}} } \right)}}{{\left( {Q_{\text{h}} + Q_{\text{c}} } \right)^{2} }}} & {\frac{{Q_{\text{c}} }}{{Q_{\text{h}} + Q_{\text{c}} }}} \\ \end{array} } \right]$$

For the projector with fixed d equals to a constant, Eq. (2.9) gives:

$$\begin{aligned} {\text{FR}}_{1} & = \frac{D}{\text{d}};\quad \quad \quad \quad {\text{FR}}_{2} = \frac{1}{D} + \frac{1}{d} + \frac{1}{f}. \\ {\text{NV}} & = D. \\ \frac{{\partial {\text{FR}}_{1} }}{{\partial {\text{NV}}}} & = \frac{1}{d};\quad \quad \quad \quad \frac{{\partial {\text{FR}}_{2} }}{{\partial {\text{NV}}}} = - \frac{1}{{D^{2} }}. \\ \left[ {J^{\text{NV}} } \right] & = \left[ {\begin{array}{*{20}c} {\frac{1}{d}} \\ { - \frac{1}{{D^{2} }}} \\ \end{array} } \right] \\ \end{aligned}$$

For the hubcap design,

$$\begin{aligned} & {\text{FR}} = k\delta ; \\ & {\text{NV}} = \delta ; \\ & \frac{{\partial \text{FR}}}{{\partial \text{NV}}} = k. \\ & \left[ {J^{{\text{NV}}} } \right] = \left[ {\frac{{\partial \text{FR}}}{{\partial \text{NV}}}} \right] = k \\ \end{aligned}$$

3.2 Countermeasures to Noise Variables

The countermeasures to noise sources are as follows: (1) to reduce if not eliminate them, (2) to compensate for them, and (3) to desensitize the design against them. Action (1) refers to the reduction if not elimination of (NV − NV ^*) in Eq. (2.12). This involves tightening the design tolerances, identifying and eliminating process variables that cause variation, and a host of other activities associated with fighting NV head-on.

3.2.1 Compensation as a Countermeasure

Compensation avoids fighting noise head-on. Instead, it provides a mechanism that further adjust DP to nullify [J ^NV](NV − NV ^*):

$${\mathbf{FR}} - {\mathbf{FR}}^{{\mathbf{*}}} = \left[ {J^{\text{NV}} } \right]\,\left( {{\mathbf{NV}} - {\mathbf{NV}}^{{\mathbf{*}}} } \right) - \left[ {J^{\text{DP}} } \right]\,\left( {{\mathbf{DP}} - {\mathbf{DP}}^{{\mathbf{*}}} } \right) = 0.$$

(2.13)

An example people most familiar with is tire balancing in which correction weights (DP − DP ^*) are added to counteract the combined effect of the tire and wheel unbalance. Other examples are water faucet and projector designs discussed earlier. Per Eq. (2.13), the amount of compensation needed is as follows:

$${\mathbf{DP}} - {\mathbf{DP}}^{{\mathbf{*}}} = \left[ {J^{\text{DP}} } \right]^{ - 1} \,\left[ {J^{\text{NV}} } \right]\,\left( {{\mathbf{NV}} - {\mathbf{NV}}^{{\mathbf{*}}} } \right).$$

As revealed in above equation, a prerequisite to compensation is that the design satisfies independent axiom, i.e., | J^DP | ≠ 0. Otherwise, [J ^DP]⁻¹ does not exist, and compensation is not possible. The uncoupled design, e.g., the single-handle faucet, is most easy to compensate since its [J ^DP] provides a one-to-one relationship between DP and NV. The decoupled design, e.g., projector of Eq. (2.10b), is equally easy to compensate if we follow the forward substitution scheme dictated by [J ^DP] described in Sect. 2.2.1. The coupled design while possible is difficult to compensate. In short, AD criterion for independence is most applicable in designing for compensation. Per information axiom, information content in a compensated design is zero since variation is completely nullified.

3.2.2 Robust Design as a Countermeasure

Another countermeasure action (3) is to move activities to the design stage. Parameters are designed into the design that reduce the sensitivity [J ^NV] in Eq. (2.13), thereby reducing if not eliminating activities in countermeasures (1) and (2) altogether. For example in hubcap design, in which retention force FR equals kδ, instead of fighting variability in δ head-on, we use a less stiff cantilever spring k = [J ^NV] → small. So that variability in δ is not amplified and transmitted to the retention force. The strategy not to fight NV head-on but to reduce the sensitivity to NV is known as robust design. Robust design has been the centerpiece of Design for Six Sigma (DFFS).

3.3 Implementing Countermeasures to Minimize Information Content

Referring to Fig. 2.10, we minimize information content or equivalently maximize P_s in two steps:

• 1. reduce bias (= system mid-range − design mid-range) to zero by compensation;

• 2. minimize the system range to within the design range through robust design.

To begin with, we take the expected value of the random variables FR and NV on both sides of on Eq. (2.13) to arrive at the expression for bias:

$${\text{Bias}} = {\text{E}}\left( {{\mathbf{FR}}} \right) - {\mathbf{FR}}^{*} = \left[ {{{J}}^{\text{NV}} } \right]\left\{ {{\text{E}}\left( {{\mathbf{NV}}} \right) - {\mathbf{NV}}^{*} } \right\} - \left[ {J^{\text{DP}} } \right]\left( {{\mathbf{DP}} - {\mathbf{DP}}^{{\mathbf{*}}} } \right)$$

(2.14)

It follows that adjustment in DP needed to reduce bias to zero by compensation is as follows:

$$\left( {{\mathbf{DP}} - {\mathbf{DP}}^{{\mathbf{*}}} } \right) = \left[ {J^{\text{DP}} } \right]^{ - 1} \left[ {J^{\text{NV}} } \right]\left\{ {{\text{E}}\left( {{\mathbf{NV}}} \right) - {\mathbf{NV}}^{*} } \right\}$$

(2.15)

Note again that bias cannot be reduced to zero if FRs are functionally dependent since | J^DP | = 0 implies [J ^DP]⁻¹ does not exist; thus, no solution is possible.

We next subtract Eq. (2.14) from Eq. (2.13) to obtain,

$${\mathbf{FR}} - {{E}}\left( {{\mathbf{FR}}} \right) = \left[ {J^{\text{NV}} } \right]\left\{ {{\mathbf{NV}} - {{E}}\left( {{\mathbf{NV}}} \right)} \right\}.$$

The variance–covariance of FR is then given as follows:

$$\begin{array}{*{20}c} {\left[ {{{V}}^{{{\text{FR}}}} } \right]} & = & {\left[ {{{J}}^{{{\text{NV}}}} } \right]} & {\left[ {{{V}}^{{{\text{NV}}}} } \right]} & {\left[ {{{J}}^{{{\text{NV}}}} } \right]^{{{T}}} } \\ {n \times n} & {} & {n \times m} & {m \times m} & {m \times n} \\ \end{array}$$

where [V ^FR] and [V ^NV] are the variance–covariance of FR and NV shown below.

$$\begin{aligned} & \left[ {{{V}}^{\text{FR}} } \right] = {{E}}\left\{ {\left\{ {{\mathbf{FR}} - {{E}}\left( {{\mathbf{FR}}} \right)} \right\}\left\{ {{\mathbf{FR}} - {{E}}\left( {{\mathbf{FR}}} \right)} \right\}^{{T}} } \right\} \\ & \left[ {{{V}}^{\text{NV}} } \right] = {{E}}\left\{ {\left\{ {{\mathbf{NV}} - {{E}}\left( {{\mathbf{NV}}} \right)} \right\}\left\{ {{\mathbf{NV}} - {{E}}\left( {{\mathbf{NV}}} \right)} \right\}^{{T}} } \right\}. \\ \end{aligned}$$

Assuming the NVs are probabilistically independent, the matrix [V ^NV] would be diagonal. The variance of FR_i is then the ith diagonal element of [V ^FR] given by

$${{v}}_{ii}^{\text{FR}} = \sum\limits_{k = 1}^{m} {{{j}}_{ik}^{\text{NV}} {{v}}_{kk}^{\text{NV}} {{j}}_{ik}^{\text{NV}} }$$

The total variance of FR is the trace of [V ^FR]:

$${\text{Variance}}\;{\text{of}}\;{\mathbf{FR}} = \sum\limits_{i = 1}^{n} {{\text{v}}_{\text{ii}}^{\text{FR}} } = \sum\limits_{i = 1}^{n} {\sum\limits_{k = 1}^{m} {{{j}}_{ik}^{\text{NV}} {{v}}_{kk}^{\text{NV}} {{j}}_{ik}^{\text{NV}} } }$$

(2.16)

To maximize P _s, we reduce bias to zero by compensation per Eq. (2.15) and minimize system range, equals to squared root of variance of FR, to within the design range of FR by robust design per Eq. (2.16).

Summarizing, the steps in AD in mathematical terms are as follows.

• 1. Define FR in a solution neutral environment, free of functional inter-dependence among them.

• 2. Conceive solution DP that maintains the functional independence in FR so that FR ^* can be achieved:

  $$| { J^{\text{DP}} } | \ne 0$$

• 3. Minimize the spread of FR with robust design. Namely, reduce [J ^NV]:

  $$\sum\limits_{i = 1}^{\text{n}} {\sum\limits_{k = 1}^{\text{m}} {{\text{j}}_{ik}^{\text{NV}} {{v}}_{kk}^{\text{NV}} {{j}}_{ik}^{\text{NV}} } } \to {\text{minimum}}$$

• 4. Subject to constraint that the bias is zero:

  $$\left( {{\mathbf{DP}} - {\mathbf{DP}}^{{\mathbf{*}}} } \right) = \left[ {J^{{{\mathbf{DP}}}} } \right]^{ - 1} \left[ {J^{{{\mathbf{NV}}}} } \right]\left\{ {{{E}}\left( {{\mathbf{NV}}} \right) - {\mathbf{NV}}^{{\mathbf{*}}} } \right\}$$

Step 2 and 3 express, respectively, the independence axiom and information axiom in mathematical terms. Step 4 states in mathematical term that independent axiom takes precedence over information axiom. Namely, if FRs are not functionally independent, then |J^DP| = 0; [J ^DP]⁻¹ does not exist; and the constraint that bias = 0 cannot be satisfied. This point is missed in DFSS courses that do not include AD. Common sense tells us that Robust Design optimization has to be subsequent to requirement definition FR, and solution conception DP because performance of a poorly defined and ill-conceived design cannot be improved via subsequent optimization.

Appendices

Appendix A1: FR Decomposition of Door-to-Body System

Appendix A2: DP Decomposition of Door-to-Body System