# 02_LP1 - The Geometry of Linear Programs the geometry of...

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MIT and James Orlin 1 The Geometry of Linear Programs the geometry of LPs illustrated on GTC

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MIT and James Orlin 2 Data for the GTC Problem Want to determine the number of wrenches and pliers to produce given the available raw materials, machine hours and demand. Wrenches Pliers Available Steel 1.5 1.0 15,000 pounds Molding Machine 1.0 1.0 12,000 hrs Assembly Machine .4 .5 5,000 hrs Demand Limit 8,000 10,000 Contribution (\$ per unit) \$.40 \$.30
MIT and James Orlin 3 Formulating the GTC Problem P = number of pliers manufactured W = number of wrenches manufactured Maximize Profit = Steel: Molding: Assembly: Pliers Demand: Wrench Demand: P,W 0 Non-negativity: 1.5 W + P 15,000 W + P 12,000 0.4 W + 0.5 P 5,000 P 10,000 W 8,000 .4 W + .3 P

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MIT and James Orlin 4 Reformulation P = number of 1000s of pliers manufactured W = number of 1000s of wrenches manufactured Maximize Profit = Steel: Molding: Assembly: Pliers Demand: Wrench Demand: P,W 0 Non-negativity: 1.5 W + P 15 W + P 12 0.4 W + 0.5 P 5 P 10 W 8 400 W + 300 P
MIT and James Orlin 5 Finding an optimal solution Try to find an optimal solution to the linear program, without looking ahead.

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Graphing the Feasible Region 2 W P 4 6 8 10 12 14 2 4 6 8 1 0 1 2 1 4 We will construct and shade the feasible region one or two constraints at a time. 6
Graphing the Feasible Region W P 2 4 6 8 10 12 14 2 4 6 8 1 0 1 2 1 4 Graph the Constraint: 1.5 W + P 15 7

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Graphing the Feasible Region W P 2 4 6 8 10 12 14 2 4 6 8 1 0 1 2 1 4 8 Graph the Constraint: W + P 12
Graphing the Feasible Region W P 2 4 6 8 10 12 14 2 4 6 8 1 0 1 2 1 4 9 Graph the Constraint: 0.4 W + 0.5 P 5 What happened to the constraint : W + P 12?

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Graphing the Feasible Region W P 2 4 6 8 10 12 14 2 4 6 8 1 0 1 2 1 4 10 Graph the Constraints: W 8 P 10
How do we find an optimal solution? W P 2 4 6 8 10 12 14 2 4 6 8 1 0 1 2 1 4 Maximize z = 400W + 300P It is the largest value of q such that 400W + 300P = q has a feasible solution 11

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How do we find an optimal solution? W P 2 4 6 8 10 12 14 2 4 6 8 1 0 1 2 1 4 Maximize z = 400W + 300P Is there a feasible solution with z = 400W + 300P = 1200 ? 12 z=1200
W P 2 4 6 8 10 12 14 2 4 6 8 1 0 1 2 1 4 How do we find an optimal solution? Maximize z = 400W + 300P Is there a feasible solution with z = 2400 ? Is there a feasible solution with z = 3600 ?

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