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The usefulness and limitations of factorial designs in statistics, specifically in the context of inventory models. It explains how the interpretation of main effects is affected by interactions and the importance of assuming a linear response. The document also covers ways to reduce the number of factors and construct fractional factorial designs of desired resolution.
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Effects estimates (and confidence intervals around them):
Can see that factor 2 (inspection time) has big negative effect on makespan— ”improving” it to “+” level would be the single most worthwhile step to take to reduce makespan
Improving factor 5 (probability of failing inspection) would have next-most- important effect on makespan
Seem not to be any significant interactions, so can directly interpret main effects
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Factorial designs can be very useful, but they do have their limitations
o Could result in experiments for factor levels that are unrealistic in the problem context
o Get no information on “interior” of design space between the factor levels, so we might not see nonlinearity or interactions that might be present there
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k – p
Full 2 k^ factorial design allows estimation of all main effects, 2-way interactions, 3- way interactions, ..., the k -way interaction
Give up ability to estimate interactions (especially higher order) by confounding them with each other
Confounded effects have the same estimation formula
This common formula estimates the sum of the expected confounded effects
e.g., if the main effect of factor 4 is confounded with the three-way interaction between factors 1, 2, and 3, then the effect estimate arising from either e 4 or e 123 is the same, and is actually an unbiased estimator of E ( e 4 ) + E ( e 123 ) Thus, need to assume that the three-way interaction is not present for this estimate to be unbiased for E ( e 4 ) It often happens that three- and higher-way interactions are not as strong as main effects, so such an assumption may be reasonable, so we could regard e 4 as being a valid estimate of the main effect of factor 4
In return, get by with a fraction (1/2 p ) of the runs: make only 2 k–p^ runs
Do just a fraction (1/2 p ) of the 2 k^ runs for the full-factorial design
Key question: Which of the 2 k^ runs to do? Answer: Not easy; must be careful to pick runs to get “clear” effects desired and confound uninteresting effects with each other Fortunately, there are tables to create fractional-factorial designs via a recipe Higher values of p lead to: Fewer runs required (good) Give up ability to estimate more effects due to more confounding (bad)
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Resolution of a fractional factorial design: Roman numerals III, IV, V, etc.: one way to quantify overall severity of confounding
Two effects are guaranteed not to be confounded with each other if the sum of their “ways” is < resolution of the design (“way” of a main effect is 1)
Resolution III:
Main effects unconfounded with each other: 1 + 1 < 3 Main effects confounded with 2-way interactions: 1 + 2 not < 3 2-way interactions confounded with each other: 2 + 2 not < 3 Very weak designs since you cannot even trust the main-effects estimates because they can be confounded with 2-way interactions, and you cannot determine whether 2-way interactions are present since they’re confounded with each other
Resolution IV:
Main effects unconfounded with each other: 1 + 1 < 4 Main effects unconfounded with 2-way interactions: 1 + 2 < 4 2-way interactions confounded with each other: 2 + 2 not < 4 Better than resolution III, since main effects are clear of confounding with 2-way interactions, but since the 2-way interactions are confounded with each other you can’t tell if they’re present so don’t know if you can interpret the main effects (see earlier discussion on linearity of response)
Resolution V:
Main effects unconfounded with each other: 1 + 1 < 5 Main effects unconfounded with 2-way interactions: 1 + 2 < 5 2-way interactions unconfounded with each other: 2 + 2 < 5 Much better than resolution IV, since you now can tell if you have 2-way interactions (they’re unconfounded with each other), so can know how to interpret the main effects
In simulation, 2-way interactions can often be present, so resolution V or higher is recommended
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Specialized designs intended specifically for screening when number of factors is too large for even a highly fractionated design
Plackett-Burman: Look at effects of k factors in as few as k + 1 design points
Supersaturated designs: Number of factors > number of design points possible to investigate
Balanced designs: Each column has exactly half “+” signs, half “–” signs Random-balance: Sprinkle the half “+” and half “–” signs randomly within a column Systematic (non-random) balanced supersaturated designs
Group-screening: Group factors together in some way dependent on model
Use intuition so that “+” and “–” levels of individual factors within a group are expected to affect response in the same direction Move all the factors in a group up and down together Calculate effects of each group Screen out unimportant groups Disaggregate groups
Frequency-domain methods
Oscillate input factors periodically during a single run, at different frequencies Look at oscillation in output, match with frequencies of input oscillations to identify important factors
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Large-scale simulations can sometimes themselves become too expensive to run
Rough proxy: A simple algebraic approximation to how a performance measure depends on the input factors (Algebraic) model of the (simulation) model — a metamodel
Back to the “machine” view of a simulation:
Algebraically, output = f (inputs), for some very complicated function f that is expressed only by the simulation model and code
Idea:
Run the actual simulation for some limited number of combinations of inputs Fit some kind of regression model to describe how the observed outputs depend on the input
