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Integer Programming Advantages Disadvantages More modelling power
More readily implementable answers Disadvantages More complex models Much longer to get answers No sensitivity analysis
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Types of Models Integral solutions Binary solutions Mixed solutions
How many Binary solutions Yes or no Mixed solutions Linear, integer, binary
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Outline Solving Rounding Branch & Bound
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Solving Rounding Branch & Bound No guarantees
Feasibility - may not satisfy constraints Optimality - may not be best Branch & Bound Implicit (clever) enumeration
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A Simple Example
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An Integer Program Coils Limit Production Capacity Bands Limit Coils
3 Coils Limit 2 Production Capacity 1 Bands Limit Coils Bands 1 2 3 4
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Branch & Bound X Node 0: The linear program
3 Node 0: The linear program $50,000*3 + $60,000*0.7 = $192,000 The best answer yields at most: Coils Limit 2 Production Capacity 1 X Bands Limit Coils Bands 1 2 3 4
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Branching Upper Branch Coils 1 Lower Branch Coils 0 Coils Bands 3
2 Upper Branch Coils 1 1 Lower Branch Coils 0 Coils Bands 1 2 3 4
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Bounding X The best answer yields at least $50,000*3= $150,000
2 Upper Branch Coils 1 1 Node 1: Lower Branch Coils 0 Coils X Bands 1 2 3 4
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Bounding X The best answer yields at least $50,000*3= $150,000
2 Node 2: Upper Branch Coils 1 Yields $188,571 X 1 Lower Branch Coils 0 Coils Bands 1 2 3 4
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Further Nodes X The best answer yields at least $50,000*3= $150,000
and at most $184,000 2 Node 3: Lower Branch Bands 2 X Yields $184,000 1 Coils Bands 1 2 3 4
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Further Nodes X The best answer yields at least
3 $50,000*2 + $60,000*1= $160, and at most $184,000 The best answer yields at least Upper Branch Coils 2 2 X Yields $160,000 1 Node 4: Lower Branch Coils 1 Coils Bands 1 2 3 4
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Further Nodes The best answer yields at least $50,000*2 + $60,000*1= $160,000 3 and at most $177,143 X Yields $177,143 2 Node 5: Upper Branch Coils 2 1 Lower Branch Coils 1 Coils Bands 1 2 3 4
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The Best Answer X The best answer yields
3 $60,000*2 + $50,000*1= $170,000 Node 6: Lower Branch Bands 1 X Yields $170,000 2 1 Coils Bands 1 2 3 4
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The Differences Integer Programs Linear Programs Model everything
Model fewer situations Easy to solve Can be hard to interpret Integer Programs Model everything Can be very hard to solve Usually easier to interpret An Integer Program with 30 zero-one variables can require over 1 billion nodes!
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Better Formulations Better constraint Coils Limit Production Capacity
3 Better constraint Coils Limit 2 Production Capacity 1 Bands Limit Coils Bands 1 2 3 4
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