Sample Average Approximation Method for Chance Constrained ...

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priate choice of the parameters since we cannot have very large values of M or N whensolving binary programs. Our experience shows that it is not significantly better thanthe bounds obtained with γ = 0.4 A Blending ProblemLet us consider a second example of chance constrained problems. Suppose a farmerhas some crop and wants to use fertilizers to increase the production. He hires anagronomy engineer who recommends 7 g of nutrient A and 4 g of nutrient B. He hastwo kinds of fertilizers available: the first has ω 1 g of nutrient A and ω 2 g of nutrient Bper kilogram. The second has 1 g of each nutrient per kilogram. The quantities ω 1 andω 2 are uncertain: we will assume they are (independent) continuous uniform randomvariables with support in the intervals [1, 4] and [1/3, 1] respectively. Furthermore, eachfertilizer has a unitary cost per kilogram.There are several ways to model this blending problem. A detailed discussion canbe found in [19], where the authors use this problem to motivate the field of stochasticprogramming. We will consider a joint chance constrained formulation as follows:min x 1 + x 2 s.t. prob{ω 1 x 1 + x 2 ≥ 7, ω 2 x 1 + x 2 ≥ 4} ≥ 1 − α, (22)x 1 ≥0,x 2 ≥026
where x i represents the quantity of fertilizer i purchased, i = 1, 2, and α ∈ [0, 1] is thereliability level. The independence assumption allows us to convert the joint probabilityin (22) into a product of probabilities. After some calculations, one can explicitly solve(22) for all values of α. For α ∈ [1/2, 1] the optimal solution and optimal value arex ∗ 1 = 189 + 8(1 − α) , x∗ 22(9 + 28(1 − α))= , v ∗ =9 + 8(1 − α)4(9 + 14(1 − α)).9 + 8(1 − α)For α ∈ [0, 1/2] we havex ∗ 1 = 911 − 9(1 − α) , 41 − 36(1 − α)x∗ 2 =11 − 9(1 − α) , v∗ =2(25 − 18(1 − α)). (23)11 − 9(1 − α)Our goal is to show that we can apply the SAA methodology to joint chance constrainedproblems. We can convert a joint chance constrained problem into a problemof the form (1) using the min (or max) operators. Problem (22) becomesmin x 1 + x 2 , s.t. prob {min{ω 1 x 1 + x 2 − 7, ω 2 x 1 + x 2 − 4} ≥ 0} ≥ 1 − α. (24)x 1 ≥0,x 2 ≥027
Page 1 and 2: Sample Average Approximation Method
Page 4 and 5: distribution in chance constraint b
Page 6 and 7: Carathéodory function, i.e., G(x,
Page 8 and 9: (i) for any sequence x k converging
Page 10 and 11: y a version of the uniform law of l
Page 12 and 13: f(¯x) ≥ ϑ ∗ . Also f(ˆx N )
Page 14 and 15: us choose two positive integers M a
Page 16 and 17: Note that, of course, E [ r T x ] =
Page 18 and 19: problem (5), with γ = 0, to be inf
Page 20 and 21: estimation of the parameters was ta
Page 22 and 23: original problem, we observe in Fig
Page 24 and 25: 3.3 Upper BoundsA method to compute
Page 28 and 29: It is possible to write the SAA of
Page 30 and 31: 5 ConclusionsWe have discussed chan
Page 32 and 33: References1. Charnes, A., Cooper, W
Page 34 and 35: 15. Markowitz, H.: Portfolio select
Page 36: 1.011.0051.00.99530 80 130 1800.10.

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