# Question

Consider the following constraint whose right-hand side b is assumed to have a normal distribution with a mean of 100 and some standard deviation σ.

30x1 + 20x2 ≤ b

A quick investigation of the possible spread of the random variable b has led to the estimate that σ = 10. However, a subsequent more careful investigation has greatly narrowed down this spread, which has led to the refined estimate that σ = 2. After choosing a minimum acceptable probability that the constraint will hold (denoted by α) this constraint will be treated as a chance constraint.

(a) Use a probability expression to write the resulting chance constraint. Then write its deterministic equivalent in terms of σ and Kα.

(b) Prepare a table that compares the value of the right-hand side of this deterministic equivalent for σ = 10 and σ = 2 when using α = 0.9, 0.95, 0.975, 0.99, and 0.99865.

30x1 + 20x2 ≤ b

A quick investigation of the possible spread of the random variable b has led to the estimate that σ = 10. However, a subsequent more careful investigation has greatly narrowed down this spread, which has led to the refined estimate that σ = 2. After choosing a minimum acceptable probability that the constraint will hold (denoted by α) this constraint will be treated as a chance constraint.

(a) Use a probability expression to write the resulting chance constraint. Then write its deterministic equivalent in terms of σ and Kα.

(b) Prepare a table that compares the value of the right-hand side of this deterministic equivalent for σ = 10 and σ = 2 when using α = 0.9, 0.95, 0.975, 0.99, and 0.99865.

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