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190 Niels J. Olieman and Eligius M.T. Hendrixfor instance x 8 represent the proportion of orange juice in the food product. It is likely thatthe concentration of vitamin C in the orange juice will vary, depending on the country oforigin, weather conditions during growth and harvest period and storage time. Likewise,there is uncertainty in the vitamin C concentration of the other raw materials. The vectorelement v i represents the uncontrollable vitamin C concentration in raw material i. If foodscientists determined that there should be at least 0.05% vitamin C in the food product, thenthe following condition should be fulfilled:u 1 (x, v) = v ′ x − 0.05 ≥ 0 (1)For the general case, a set of uncontrollable factors can be defined for which it is certain thatthe object properties fulfill the requirements:H(x) := {v ∈ V|u s (x, v) ≥ 0, s = 1, .., S}where V is the set of all possible realisations of v. The set H(x) is called the Happy set, sinceelements of this set correspond to the favourable situation that all uncertain object propertiesare as intended. A design x is called robust, if and only if the design will perform as intendedfor all possible variations in V:H(x) ≡ V ⇐⇒ "x is robust"Consequently, we will say a design has some level of robustness, if the design only performs asintended for a subset S ⊂ V of all possible realisations of v:H(x) ⊂ V ⇐⇒ "x is less than robust"Perhaps the most common approach in science for modelling uncertainty, is studied in thescience field called probability theory [4] [5]. In probability theory, the notions random vector andprobability space are formally introduced, which can be used to model a collection of uncertainevents. The following notation is used:1. The probability space is defined as (V, V, Pr), with sample space or support set V ⊆ R N ;the σ-field V of all subsets of V; a probability measure Pr : V → [0, 1] and by definitionPr (∅) = 0 and Pr (V) = 12. The stochastic vector (in bold) v has realisations v in the support set: v ∈ V.3. The probability that v will have realisations in a subset S ∈ V, S ⊆ V is formally denotedwith Pr (S). Also an informal notation is used, which makes reference to the connectedrandom vector: Pr {v ∈ S} = Pr (S).Under the assumption that it is appropriate to model the uncontrollable factors as stochasticvariates, authors such as Du and Chen [3], Bjerager [2] and Nie et al. [7] studied probabilisticrobustness computation and optimisation approaches with encouraging results.Robustness R(x) can be expressed in a probabilistic way: the probabilistic robustness of anobject is the probability that an object will have desired properties:R(x) = Pr {v ∈ H(x)} (2)The framework for robustness optimisation problems is here defined as Robustness Programming(RP) and can be formulated as:R ∗ = maxx∈R I [Pr {v ∈ H(x)}] (3)In practice, there is often not a closed form expression to compute (2). Alternatively, one canestimate or find (exact) upper and lower bounds for (2). Typically estimation algorithms can