5.4. Decision-Based Design

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5.4. Decision-Based Design

In general, engineering design is viewed as a problem-solving process. Such as in this chapter, a design problem is formulated mathematically as optimization problem and solved by meeting functional requirements subject to constraints at minimum objectives.

As mentioned in Chapter 2, engineering design is increasingly recognized as a decision-making process. From a product development perspective, design involves a series of decisions—some of which may be made sequentially and others that must be made concurrently. The term decision-based design (DBD) was introduced in 1990s. A formal definition introduced in (Hazelrigg 1998) states that decision-based design is a normative approach that prescribes a methodology to make unambiguous design alternative selections under uncertainty and risk wherein the design is optimized in terms of the expected utility.

Uncertainty and risk involved in decision making, as well as two prominent decision theories, utility theory and game theory, were introduced in Chapter 2. In addition, we presented examples to illustrate the application of these theories to engineering design. These examples involve multiple objectives. In this section, we revisit these examples and theories. Although we do not advocate bringing decision-based design into the framework of multiobjective optimization, we compare the methods of MOO and DBD for dealing with multiobjective problems. We provide closure in treating design as a problem-solving and decision-making process by bringing the decision theories discussed in Chapter 2 into the context of multiobjective optimization.

5.4.1. Utility Theory as a Design Tool: Cantilever Beam Example

In Chapter 2, the cantilever beam example shown in Figure 5.20 was employed to illustrate the application of utility theory as a design tool. Both constrained and nonconstrained problems were solved. In this section, we bring back the results of the constrained problem for discussion.

The constrained problem involves minimizing weight w and vertical displacement z and subject to bending stress σ. Mathematically, the MOO problem is formulated as

$Minimize : f_{w} = w (b, h) = ρ gbh ℓ, and f_{z} = z (b, h) = z = \frac{4 P ℓ^{3}}{Eb h^{3}}$ $Minimize : f_{w} = w (b, h) = ρ gbh ℓ, and f_{z} = z (b, h) = z = \frac{4 P ℓ^{3}}{Eb h^{3}}$

(5.30a)

$Subject to : g_{σ} = σ (b, h) - S_{y} \leq 0$ $Subject to : g_{σ} = σ (b, h) - S_{y} \leq 0$

(5.30b)

$10 \leq b \leq 60 mm, and 20 \leq h \leq 80 mm$ $10 \leq b \leq 60 mm, and 20 \leq h \leq 80 mm$

(5.30c)

where

σ = \frac{6 P ℓ}{b h^{2}}

$σ = \frac{6 P ℓ}{b h^{2}}$

and the yield strength is S_y = 27.6 MPa.

As discussed in Chapter 2, the MOO problem was converted into a single-objective nonconstrained optimization problem, defined as

$Maximize : u (w, z, σ) = [k_{c} + (1 - k_{c}) u (w, z)] u (σ)$ $Maximize : u (w, z, σ) = [k_{c} + (1 - k_{c}) u (w, z)] u (σ)$

(5.31)

in which u(w, z) is the multiattribute utility (MAU) function defined in Eq. 2.51. We assume 0 ≤ k_w ≤ 1, 0 ≤ k_z ≤ 1, and k_w + k_z = 1 (recall that k_w and k_z are the scaling constants representing the designer's preference between attributes w and z). Hence, the multiplicative MAU is reduced to an additive MAU, as in Eq. 5.32:

$u (w, z) = k_{w} u_{w} + k_{z} u_{z}$ $u (w, z) = k_{w} u_{w} + k_{z} u_{z}$

(5.32)

which is similar to the weighted-sum method for solving MOO problems.

Also, in Eq. 5.31, the utility function of the stress constraint is defined in Eq. 5.33 as

$u (σ) = \frac{1}{1 + e^{\frac{σ_{0.5} - σ}{s}}}$ $u (σ) = \frac{1}{1 + e^{\frac{σ_{0.5} - σ}{s}}}$

(5.33)

in which s is the slope of the utility function u(σ) at u = 0.5, and we chose (σ_0.5 = S_y = 27.6 MPa and s = −0.01 in the example.

We restate the results of the five cases below in Table 5.3.

We are now solving the MOO problem defined in Eq. 5.30. We plot the Pareto front using the generative method similar to that of the pyramid problem (MATLAB Script 8 in Appendix A). The boundary points of the front are identified as

x_{w}^{∗}

$x_{w}^{∗}$

= (10.01, 65.95) and

x_{z}^{∗}

$x_{z}^{∗}$

= (59.97, 79.94), as shown in Figure 5.21. At the points, the objective functions are f(

x_{w}^{∗}

$x_{w}^{∗}$

) = (3.497 N, 0.1615 mm) and f(

x_{z}^{∗}

$x_{z}^{∗}$

) = (25.39 N, 0.01514 mm), respectively, as shown in the zoomed-in figures A and B in Figure 5.21.

Table 5.3

Results Comparison for the Constrained Design Problems

Case No.	Problem Setup				Results
Case No.	k_w	k_z	r_w	r_z	b (mm)	h (mm)	w (N)	z (mm)	σ (MPa)	u
Case 4	0.5	0.5	0	0	10	66.1	3.50	0.161	27.46	0.950
Case 5a	0.7	0.3	0	0	10	66.1	3.50	0.161	27.46	0.938
Case 5b	0.3	0.7	0	0	10	71.3	3.78	0.128	23.60	0.962
Case 6a	0.5	0.5	−2	0	10	66.1	3.50	0.161	27.46	0.906
Case 6b	0.5	0.5	2	0	10	72.8	3.85	0.120	22.64	0.977

FIGURE 5.21 Pareto solutions of the beam example in criterion space.

The results obtained in Chapter 2 (see Table 5.3) show that the base case (Case 4, k_w = k_z = 0.5) is identical to f(

x_{w}^{∗}

$x_{w}^{∗}$

), which is a boundary point of the Pareto front shown in Figure 5.21. In Case 5a, k_w increases to 0.7, implying that the preference is given to weight; the design is identical to that of the base case. Adjusting the preference to weight did not alter the design. This is because that there is no more room to further minimize weight without violating the stress constraint. In Case 6a, r_w is reduced to −2, indicating that the designer is risk prone to the weight attribute; the design is identical to that of the base case. Adjusting the risk attitude toward the weight did not alter the design due to the same reason. In Case 5b, k_w decreases to 0.3, implying that the preference is given to displacement; the design moves away from

x_{w}^{∗}

$x_{w}^{∗}$

, as shown in the zoomed-in figure B of Figure 5.21b. A similar result is found in Case 6b. However, all cases led to designs that are cluster to

x_{w}^{∗}

$x_{w}^{∗}$

. The results indicate that there is a large portion of the Pareto front not being explored.

As seen in the beam example, the approach of using the utility theory as a design tool converts the MOO into a single-objective nonconstrained problem by the incorporating designer's preference and risk attitude. From a MOO perspective, this approach is similar to methods with a priori articulation of preferences discussed in Section 5.3.2 in the context of conventional MOO solution techniques.

FIGURE 5.22 Cylindrical pressure vessel: (a) isometric view and (b) section view with dimensions.

5.4.2. Game Theory as a Design Tool: Pressure Vessel Example

The pressure vessel shown in Figure 5.22 was investigated using game theory as a design tool in Chapter 2. The same design problem can also be formulated as a MOO problem:

$Minimize : W (R, t) = γ [π {(R + t)}^{2} (ℓ + 2 t) - π R^{2} ℓ]$ $Minimize : W (R, t) = γ [π {(R + t)}^{2} (ℓ + 2 t) - π R^{2} ℓ]$

(5.34a)

$Maximize : V (R, t) = π R^{2} ℓ (or Minimize - V)$ $Maximize : V (R, t) = π R^{2} ℓ (or Minimize - V)$

(5.34b)

$Subject to : t + R \leq 40$ $Subject to : t + R \leq 40$

(5.34c)

$0.5 \leq t \leq 6 in.$ $0.5 \leq t \leq 6 in.$

(5.34d)

$5 t \leq R \leq 8 t$ $5 t \leq R \leq 8 t$

(5.34e)

$2.5 < R \leq 5.55 in.$ $2.5 < R \leq 5.55 in.$

(5.34f)

We plot the Pareto front using the generative method (MATLAB Script 9 in Appendix A). Note that Eq. 5.34b was converted to minimize –V in the MATLAB implementation. The boundary points of the Pareto front are points A, B, and C shown in Figure 5.23, in which f(A) = (f_W(2.5,0.5), f_V(2.5,0.5)) = (252.5, –1963), f(B) = (f_W(4,0.5), f_V(4,0.5)) = (395.9, –5027), and f(C) = (f_W(35.55,4.44), f_V(35.55,4.44)) = (42,440, –39,700).

Recall the solutions of Section 2.6.2, where we first found that the Nash solution in criterion space is curve BC, which is a subset of the Pareto front. In addition, point B is the solution to the sequential game, with Player W as the leader; point C is the solution to the sequential game with Player V as the leader. Both points are Pareto solutions.

As seen in the pressure vessel example, the approach of using game theory as a design tool converts a MOO problem into a series of single-objective problems by considering the objectives of the individual designers. The single-objective problems are then solved sequentially. The basic concept of handling MOO using game theory is different from those discussed in Section 5.3. In the former, multiple designers are making decisions. In the latter, a MOO problem is solved by a single designer. Practically, on many occasions, design decisions are made by individual groups or designers. In that regard, game theory as design tools may be more general and suitable for solving complex design problems of distributed design teams.

FIGURE 5.23 Pareto solutions of the beam example in criterion space.

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Table of Contents for 5.4. Decision-Based Design

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5.4. Decision-Based Design

5.4.1. Utility Theory as a Design Tool: Cantilever Beam Example

5.4.2. Game Theory as a Design Tool: Pressure Vessel Example

Table of Contents for
5.4. Decision-Based Design