# Multi-objective optimization multi-criteria decision-making.

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Multi-objective optimization multi-criteria decision-making

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

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?

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

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

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

Multi-objective example f 1 Bids (\$000) f 2 Days normalized a =00.20.40.60.81 101450.690.62 0.630.650.660.680.69 98380.590.41 0.450.480.520.550.59 108410.930.50 0.590.670.760.840.93 99580.621.00 0.920.850.770.700.62 81420.000.53 0.420.320.210.110.00 94350.450.32 0.350.370.400.420.45 110241.000.00 0.200.400.600.801.00 103300.760.18 0.290.410.530.640.76

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

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

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

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

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

independent multi-objective optimization Pareto optimization individual fitness

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 \$110 105 100 95 90 85 80 202530354045505560 days

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 \$110 105 100 95 90 85 80 202530354045505560 days Pareto set: “undominated” solutions

Multiple fitness functions Maxima Minima

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

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

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

Population evaluation and selection non-dominated front f1f1 f2f2

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

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

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

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

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)

independent multi-objective optimization Pareto optimization individual fitness

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