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Lecture 3

# Lecture 3 - 540:311 DETERMINISTIC MODELS IN OPERATIONS...

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540:311 DETERMINISTIC MODELS IN OPERATIONS RESEARCH Lecture 3: Chapter 3.1 - 3.3 Class Meeting: Thu Jan 27 th 10:20-11:40am Recitation: Introduction to Linear Programming Prof. W. Art Chaovalitwongse

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MIT and James Orlin © 2003 Example: GTC Problem Want to determine the number of wrenches and pliers to produce given the available raw materials, machine hours and demand.
MIT and James Orlin © 2003 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 © 2003 Reformulation P = number of 1000s of pliers manufactured W = number of 1000s of wrenches manufactured Maximize Profit = P,W 0 1.5 W + P 15 W + P 12 0.4 W + 0.5 P 5 P 10 W 8 400 W + 300 P Steel: Molding: Assembly: Pliers Demand: Wrench Demand: Non-negativity:
Graphing the Feasible Region 2 W P 4 6 8 10 12 14 2 4 6 8 10 12 14 We will construct and shade the feasible region one or two constraints at a time.

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Graphing the Feasible Region W P 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Graph the Constraint: 1.5 W + P 15
Graphing the Feasible Region W P 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Graph 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 10 12 14 Graph the Constraint: 0.4 W + 0.5 P 5 What happened to the constraint : W + P 12?
Graphing the Feasible Region W P 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Graph the Constraints: W 8 P 10

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How do we find an optimal solution? W P 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Maximize z = 400W + 300P It is the largest value of q such that 400W + 300P = q has a feasible solution
How do we find an optimal solution? W P 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Maximize z = 400W + 300P Is there a feasible solution with z = 400W + 300P = 1200 ? z=1200

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W P 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Maximize z = 400W + 300P Is there a feasible solution with z = 2400 ? Is there a feasible solution with z = 3600 ? z = 2400 z=3600 How do we find an optimal solution?
W P 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Can you see what the optimal solution will be? z = 2400 z = 3600 Maximize z = 400W + 300P How do we find an optimal solution?

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W P 2 4 6 8 10 12 14 2 4 6 8 10 12 14 What characterizes the optimal solution?
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