optimization 1/33 Radford, A D and Gero J S (1988). Design by Optimization in Architecture, Building, and Construction, Van Nostrand Reinhold, New York

optimization techniques 2/33 ● which technique depends on ● nature of problem ● nature of information required ● historically ● mathematical problem – single optimal value ● not interested in suboptimal values ● design ● both decisions and suboptimal solutions important ● suboptimal solutions  options ● may be more acceptable in terms of unstated objectives

optimization techniques 3/33 ● nature of problem / information ● design  variables discrete &/or discontinuous &/or non-contiguous ● steel beams in discrete sizes ● no. of lifts discontinuous, ● materials non-contiguous ● nonlinear relationships  heat loss

types of techniques 4/33 ● calculus ● continuous and differentiable ● linear programming (LP) ● well developed method ● linear relationship among variables ● nonlinear programming (NLP) ● nonlinear relationships ● dynamic programming (DP) ● discrete, nonlinear, handles constraints ● evolutionary computation ● population

general strategies 5/33 ● exhaustive enumeration ● all possible solutions ● implicit enumeration ● e.g. branch & bound, DP ● hill-climbing ● moving from existing solution to an improved solution

linear programming ● best developed technique ● most frequently used ● guarantees optimum solution ● 3 conditions ● variables must be continuous, >= 0 ● O.F. must be linear, OF=20x 1 +12x 2 ● constraints must be linear, 4x 1 + 3x 2 > 18 6/33

linear programming ● convex spaces ● feasible solution space ● O.F. moves away from origin ● optimum solution at vertex 7/33 x1x1 x2x2

dynamic programming ● design problems ● not continuous or linear ● definition (Richard Bellman) ● an optimal set of decisions has the property that whatever the first decision is, the remaining decisions must be optimal with respect to the outcome which results from the first decision 8/33 if you don’t’ do the best with what you have, you will never do the best with what you should have had

dynamic programming ● stage-state formulation ● implicit enumeration of all paths ● guarantees global optimum ● non-serial DP doesn’t guarantee ● but pretty good 9/33 Stage 123 es 45 State

evolutionary computation ● hill-climbing ● one solution at a time ● in direction of steepest slope ● local optima ● variables &/or constants ● equations – y = mx + c ● EC in parallel ● populations ● survival of the fittest – probability ● random generation 10/33 LO GO

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multicriteria optimization 33/33 ● single objective ● e.g. cost ● several conflicting objectives ● max light – min heat ● best looking, min cost car ● Pareto solutions ● best compromise ● tradeoffs c1c1 c2c2