Research Papers

Numerically Stable Design Optimization With Price Competition

[+] Author and Article Information
W. Ross Morrow

Mechanical Engineering,
Iowa State University,
Ames, Iowa 50011
e-mail: wrmorrow@iastate.edu

Joshua Mineroff

Mechanical Engineering,
Iowa State University,
Ames, Iowa 50011
e-mail: mineroff@iastate.edu

Kate S. Whitefoot

National Academy of Engineering,
Washington, DC 20418
e-mail: kwhitefoot@nae.edu

The real-valued MCP “l ≤ x ≤ uF(x)” is solved by x satisfying one of the following: x ∈ [l,u] if F(x) = 0, x = l if F(l) > 0, or x = u if F(u) < 0. Note the similarity to the KKT conditions for bound-constrained optimization. In fact, if F(x) is the derivative of some function f(x), then l ≤ x ≤ uF(x) are the KKT conditions for min f(x) subject to lxu. See Refs. [75,76-75,76] or Appendix B for a generalization to vector-valued MCPs.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received September 12, 2012; final manuscript received September 9, 2013; published online June 2, 2014. Assoc. Editor: Wei Chen.

J. Mech. Des 136(8), 081002 (Jun 02, 2014) (17 pages) Paper No: MD-12-1451; doi: 10.1115/1.4025703 History: Received September 12, 2012; Revised September 09, 2013

Researchers in decision-based design (DBD) have suggested that business objectives, e.g., profits, should replace engineering requirements or performance metrics as the objective for engineering design. This requires modeling market performance, including consumer preferences and competition between firms. Game-theoretic “design-then-pricing” models—i.e., product design anticipating future price competition–provide an important framework for integrating consumer preferences and competition when design decisions must be made before prices are decided by a firm or by its competitors. This article concerns computational optimization in a design-then-pricing model. We argue that some approaches may be fundamentally difficult for existing solvers and propose a method that exhibits both improved efficiency and reliability relative to existing methods. Numerical results for a vehicle design example validate our theoretical arguments and examine the impact of anticipating pricing competition on design decisions. We find that anticipating pricing competition, while potentially important for accurately forecasting profits, does not necessarily have a significant effect on optimal design decisions. Most existing examples suggest otherwise, anticipating competition in prices is important to choosing optimal designs. Our example differs in the importance of design constraints, that reduce the influence the market model has on optimal designs.

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Fig. 1

Conceptual difference between a design-and-pricing model (left) and a design-then-pricing model (right) for F product-designing firms. Dashed lines denote fixed components in the respective model, while thick solid lines denote variable components.

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Fig. 2

Illustration of Lemma 1. Contours denote the level sets of the max norm (‖x‖ ∞ = maxn{|xn|}) of the profit gradient for a two-vehicle design-then-pricing problem as defined in Sec. 4.1. Labels denote the value of the norm over the contours drawn.

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Fig. 3

Computational comparison of performance of the implicit programming and MPEC methods on 1000 trials started at different initial conditions for a single set of 1000 samples. Implicit programming was implemented computing equilibrium to tolerances of 10–6, 10–9, and 10–12, abbreviated IMPL(10–6), IMPL(10–9), and IMPL(10–12), respectively. Optimal design problems solved to tolerances of 10–6. CG-MPEC was started at both random and the “smarter” initial conditions, abbreviated CG-MPEC(r) and CG-MPEC(c), respectively; similarly with φ-MPEC. (Left) Success rate captures both SNOPT successes and computation of an equilibrium, that are the same for all methods except the CG-MPEC approach. Dashed boxes represent the SNOPT success rate for CG-MPEC including spurious solutions. (Right) Mean compute times include only successful runs, with the exception of CG-MPEC(r) for which there were no successful runs.

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Fig. 5

Success rates for the unregularized design-and-pricing problem out of 1000 trials with random initial conditions. Scenarios are labeled as described in the text. As in Fig. 3, dashed boxes represent SNOPT success rates (all 100%) and gray boxes represent successful computations of a nonspurious solution.

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Fig. 6

Mean perceived gain or loss in profits when choosing designs and prices without anticipating pricing competition that ultimately occurs, with regularized computations. Scenarios are labeled as described in the text.

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Fig. 7

Cumulative distribution function for computing profits in the design-and-pricing model within a given percentage of the apparent globally maximal profits over 1000 trials. The dark, solid black curve represents overlapped CDFs of all design-and-pricing scenarios except for “”; the CDF for this scenario is represented by a dashed curve. The gray area is carried over from Fig. 4.

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Fig. 4

Cumulative distribution function for computing profits in the design-then-pricing model within a given percentage of the apparent globally maximal profits over 1000 trials. The gray shading represents the area under this curve for comparison with Fig. 7.




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