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Multi-objective optimization multi-criteria decision-making

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Multiple objectives problem solutions often involve multiple (conflicting) objectives – e.g. design a product: quality, features, cost, weight, durability,... multiple versions of same product for different tradeoffs

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Multiple fitness functions variables with domains define search space two or more fitness functions – optimal values do not coincide (i.e., not at same location in search space) – how to determine tradeoff?

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Multi-objective example offers to buy my house: – I want highest price possible – I want to delay moving out as many days as possible Sample bids, $100,00045 days $ 98,00038 days $108,00051 days best on both criteria

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Multi-objective example Sample bids, $101,00045 days $ 98,00038 days $108,00041 days $ 99,00058 days $ 81,00042 days $ 94,00035 days $110,00024 days $103,00030 days

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Finding multi-objective solutions 15.1 ‘aggregate’ - reduction to one hybrid evaluation function 15.2 evolutionary multi-objective optimization

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Multi-objective example f 1 Bids ($000) f 2 Days normalized a =

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One hybrid evaluation function (1) Weighted sum of m evaluation functions – ∑ i w i ⋅ f i (A) – 0 < w i < 1, ∑ i w i = 1 how to select weights? Subjective how to reconsile different units? (days, $$) simple calculation

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One hybrid evaluation function (2) Minimized shortfall from target values – ∑ i | f i (A) – y i | r – r ≥ 2 y i is target value for f i (A) 2 euclidean distance large r causes greatest discepancy to dominate (minimax) – can be weighted like sum

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One hybrid evaluation function (3) Target values as constraints – f i (A) ≤ y i, 1 ≤ i ≤ m, i ≠ r – optimize f r (A)

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One hybrid evaluation function (4) Valuated State approach (p. 443) – quantified subjective evaluation – arithmetic mean or geometric mean – can be fuzzy

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Mulit-objective solutions aggregated fitness independent multi-objective Pareto optimization individual fitness

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independent multi-objective optimization Pareto optimization individual fitness

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Multi-objective example Sample bids, $101,00045 days $ 98,00038 days $108,00041 days $ 99,00058 days $ 81,00042 days $ 94,00035 days $110,00024 days $103,00030 days $ days

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Multi-objective example Sample bids, $101,00045 days $ 98,00038 days $108,00041 days $ 99,00058 days $ 81,00042 days $ 94,00035 days $110,00024 days $103,00030 days $ days Pareto set: “undominated” solutions

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Multiple fitness functions Maxima Minima

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Pareto optimal set of solutions solutions that are nondominated solution A dominates solution B iff f i (A) ≥ f i (B),1 ≤ i ≤ m (maximizing) i.e., A is better on every fitness measure

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Objective space Decision space Objective space – dimension m – the number of objective (fitness) functions Decision space – dimension n – the number of parameters

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Population evaluation and selection non-dominated fronts (minimize) f1f1 f2f2

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Population evaluation and selection non-dominated front f1f1 f2f2

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Population evaluation and selection f1f1 f2f2 second front non-dominated front; second front

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Population evaluation and selection f1f1 f2f2 second front non-dominated front; second front the “onion” model

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Population evaluation and selection Order population by fronts – probabilities based on order – elites from non-dominated set f1f1 f2f2

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Population diversity how to maintain diversity in population: – diversity in decision space based on domain variables – diversity in objective space, within Pareto-set different locations along the fitness front in objective space f1f1 f2f2 f1f1 f2f2

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Population diversity diversity in objective space – fitness sharing fitness from pareto front calculation distance between solutions i and j in same pareto-front neighbourhood f1f1 f2f2 f1f1 f2f2 d(i,j)

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independent multi-objective optimization Pareto optimization individual fitness

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Non-Pareto selection with fitness functions f1, f2,..., fn use each function to select some population members

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