MIE376 Lec9 DWD Incomplete.page2

# MIE376 Lec9 DWD Incomplete.page2 - there is some way of...

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MIE376 Mathematical Programming Lecture Notes Daniel Frances 2011 2 Max 70Q 1 + 60Q 2 subject to 7Q 1 + 5Q 2 ≤ 30; 3 Q 1 + 2Q 2 ≤ 1 5; Q 1 + Q 2 4; Q i ≥ 0 Currently the optimal solution is for Quid to make 4 tons of Steel 1, no Steel 2, at revenue of \$280. Its production limited by its furnace time availability. 2-Plant system Suppose that the two plants are purchased by one owner, who plans to share the availability of the iron ore among the two plants, starting with a total availability of 80 tons, and who wishes to maximize the overall revenue from the two plants. Max 90P 1 + 80P 2 + 70Q 2 + 60Q 2 s.t. 8P 1 + 6P 2 + 7Q 2 + 5Q 2 ≤ 80; 3 P 1 + P 2 ≤ 12; 2P 1 + P 2 ≤ 10; 3 Q 1 + 2Q 2 ≤ 15; Q 1 + Q 2 ≤ 4; P i , Q i ≥ 0 If we rewrite the new LP in a “neater” format we observe the following Max 90P 1 + 80P 2 + 70Q 1 + 60Q 2 subject to 8P 1 + 6P 2 + 7Q 1 + 5Q 2 ≤ 80; 3P 1 + P 2 1; 2P 1 + P 2 ≤ 10; 3Q 1 + 2Q 2 ≤ 15; Q 1 + Q 2 ≤ 4 P i , Q i ≥ 0 The two plant LPs are only linked through the objective and the iron ore constraint. The question arose if
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Unformatted text preview: there is some way of solving the two LPs in some coordinated fashion, or is it necessary to merge the two LPs into a single LP in order to obtain the overall optimal solution. Sketch of a Coordination approach Suppose there is some way to coordinate the operation of the two plants, without the necessity of merging all the data into a single LP. Suppose also, that the coordination process takes the form of an iterative process. • Each plant makes production offers of certain production at a given price. We will make up symbols for the quantities offered by each plant. The prices can easily be derived when we need them. Let Pi k the k th production offer from Plant 1 for steel i. Similarly let Qi k be the k th production offer from plant 2 for the two types of steel....
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