Decision Making with Dominance Constraints in Two-Stage by Uwe Gotzes

By Uwe Gotzes

Two-stage stochastic programming types are regarded as appealing instruments for making optimum judgements lower than uncertainty. characteristically, optimality is formalized via using statistical parameters resembling the expectancy or the conditional price in danger to the distributions of aim values.

Uwe Gotzes analyzes an method of account for probability aversion in two-stage versions established upon partial orders at the set of genuine random variables. those stochastic orders allow the incorporation of the features of entire distributions into the choice strategy. The revenue or rate distributions needs to move a benchmark try out with a given appropriate distribution. hence, extra ambitions will be optimized. For this new classification of stochastic optimization difficulties, effects on constitution and balance are confirmed and a adapted set of rules to take on huge challenge circumstances is constructed. the results of the modelling heritage and numerical effects from the applying of the proposed set of rules are confirmed with case stories from power buying and selling.

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5 shows the price quota curves we used for the three customer groups under consideration. 5: Price quota curves. 5) i=1 R E¯t,e,i, are the correspondent step heights times the customer group’s overall demand. From an economic point of view, it is clear, that selling prices should alR R . Therefore we have that , λ¯ e,i ways be a right bound of one of the intervals λ¯ e,i−1 the selling price for customer group e equals: NI ∑ λ¯ e,iR · ve,i . 6) i=1 Independence of the random outcomes of first-stage decisions is a basic assumption in two- and multistage stochastic programming.

If (X , ∗ ) and (X , y ∀x, y ∈ X , then inf∗ {x, y} inf {x, y} and sup {x, y} sup∗ {x, y}. 3 Multiple Dominance Constraints 31 For properties of lattices we refer the reader to [4] and [33]. We introduced ≤icx as a relation between real random variables. Notice that ≤icx can also be understood as a partial order on the set of all Borel probability measures by the one-to-one correspondence f ◦ X d P ≤ f ◦ Y d P ⇔ f d PX ≤ f d PY already mentioned in footnote 10 on page 8. 19) which are not contained in (2) P1 (R).

4. 13). Same for λ = 0¯ K , but D(0¯ K ), which is an expected value, can often be tightened significantly in a manner not working for the decomposition of the classical expectation-based model. 6) ⎪ ⎪ ⎪ v k ≥ 0, x ∈X , y ∈Y ⎪ ⎪ ⎪ ⎭⎭ k = 1, . . 13). In other words, instead of the expected value behind D(0¯ K ), the largest contributor, irrespective the scenario probability, 4 Decomposition Method 57 can be taken as a valid lower bound. To show this, let ∗ be the index of the subproblem with the maximal objective value.

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