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MIE376 Lec 7 -Stochastic LP.page1

MIE376 Lec 7 -Stochastic LP.page1 - MIE376 Mathematical...

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Unformatted text preview: MIE376 Mathematical Programming Lecture Notes Stochastic Programming — LP with Resourse The next frontier in LP is how to incorporate uncertainty into LP. After all it is naive to believe that in the standard LP, Max cx s.t. Ax S b x20, the values of c, A and b are known with certainty. The greatest challenge is how to deal with uncertainty in the constraints — it is easy to show that as long as we are risk neutral it is conceptually acceptable to replace the c’s with their expected values. But how do we deal with the constraints data. Two approaches are possible a. Chance Constrained Programming — an approach that has fallen out of grace b. LP with Recourse — very popular in finance and energy — now known simply as Stochastic Programming. What is assumed here is that the problem includes at least two stages: 1. A decision making stage, during which there is uncertainty. 2. A recourse stage, during which any resulting constraint violations can be fixed. In these notes we will only deal with two stages, although the same principles can be easily expanded to formulate problems with more stages. Unfortunately the number of variables grows exponentially with the number of stages, and quickly becomes numerically intractable. We will solve the recourse programming for discrete distributions both without and with stochastic programming, in the context of an agricultural setting. You own 500 acres of land in which to plant wheat (w), corn (c) and beats (b) with planting costs are 150 $/acre for wheat, 230 $/acre for corn and 260 $/acre for beets To feed your cattle you require 200 T of wheat and 240 T of corn. The yield of each crop is uncertain with expected values of 2.5, 3 and 20 T/acre for what, corn and beets respectively. Purchasing cost of wheat is 238 $fT and 210 $fT for corn. Any excess wheat can be sold for 170 $fT, corn for 150 $lton, beets for 36 $lton (up to 6000 T), beets for 10 $fT (beyond 6000 T). Clearly this problem has the two required stages. 1. The decision making stage when you decide on how many acres to plant of each crop, even though the yield is uncertain at the time 2. The recourse stage, when you can purchase extra crop from the market in case you are short, or sell to the market if you have excess. Daniel Frances © 2011 1 ...
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