MIE376 Lec9 DWD Incomplete

MIE376 Lec9 DWD Incomplete - MIE376 Mathematical...

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MIE376 Mathematical Programming Lecture Notes Daniel Frances 2011 1 Lecture 5: Dantzig-Wolfe Decomposition – Applying the CG Principles Having seen the power of complementing a LP problem, with a sub-problem, it occurred to the inventors of the column generation method that perhaps a similar approach could be used to decompose large LPs of a special nature. Very often large LPs look like they started as the combination of smaller LPs. To be more precise consider the following example. Consider two independent steel plants (Pike and Quid) each producing two types of steel, type 1 and type 2, from iron core (tons), coal (tons) and blast furnace (hrs). They each sell the two types of steel at the same market price ($170 per to of steel 1, $160 per ton of steel 2), to a single customer. Plant 1 (Pike Plant) Resource requirements for the Pike Plant are as follows Product Iron Coal Furnace Steel 1 8 (ton/ton of steel) 3 (ton/ton of steel) 2 (hrs/ton of steel) Steel 2 6 (ton/ton of steel) 1 (ton/ton of steel) 1 (hrs/ton of steel) Availability 30 tons 12 tons 10 hrs It costs $80 to ship a ton of steel from the Pike Plant to the customer. The LP used to produce the best production levels at this Plant is thus Let P i be the tons of steel i produced Max 90P 1 + 80P 2 subject to 8P 1 + 6P 2 30; 3P 1 + P 2 ≤ 12; 2 P 1 + P 2 10; P i ≥ 0 Currently the optimal solution is for Pike to make no Steel 1, 5 tons of Steel2, at a revenue of $400. Its production limited by the availability of iron. Plant 2 (Quid Plant) Resource requirements by Quid are as follows Product Iron Coal Furnace Steel 1 7 (ton/ton of steel) 3 (ton/ton of steel) 1 (hrs/ton of steel) Steel 2 5 (ton/ton of steel) 2 (ton/ton of steel) 1 (hrs/ton of steel) Availability 30 tons 15 tons 4 hrs It costs $100 to ship a ton of steel from Quid to the customer. The LP used to produce the best production levels at this plant is thus Let Q i be the tons of steel i produced
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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
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MIE376 Lec9 DWD Incomplete - MIE376 Mathematical...

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