Lecture3

# Lecture3 - MS&E 111 z Lecture 3 The Geometry of Linear...

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MS&E 111 Lecture 3 z The Geometry of Linear Programs the geometry of LPs illustrated on DTC 1

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Goal of this Lecture z 3 mathematical intuitions Algebraic/computational Geometric Economic z In this lecture, we focus on 2-dimensional geometry to guide intuition and develop a deeper understanding of linear programs. z Use DTC as a running example 2
Data for the DTC Problem Slingshot Kits Stone Shields Resources Stone Gathering time 2 hours 3 hours 100 hours Stone Smoothing 1 hour 2 hours 60 hours Delivery time 1 hour 1 hour 50 hours Demand 40 30 Profit 3 shekels 5 shekels The DTC problem has two variables. We can learn a lot from its 2 dimensional geometry. 3

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Formulating the DTC Problem K = number of slingshot kits manufactured S = number of stone shields manufactured Maximize Profit z = 3 K + 5 S Gathering time: Smoothing time: Delivery time: Slingshot demand: Shield demand: Non-negativity: 2 K + 3 S 100 K + 2 S 60 K + S 50 K 40 S 30 K,S 0 4
Reformulation K = number of slingshot kits manufactured (in 10s) S = number of stone shields manufactured (in 10s) Maximize Profit z = 3 K + 5 S (in 10s) Gathering time: Smoothing time: Delivery time: Slingshot demand: Shield demand: Non-negativity: 2 K + 3 S 10 K + 2 S 6 K + S 5 K 4 S 3 K,S 0 5

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Finding an optimal solution z Try to find an optimal solution to the linear program, without looking ahead. 6
Graphing the Feasible Region We will construct and shade the feasible region one or two constraints at a time. 7 1 2 3 4 5 6 1 2 3 4 5

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8 1 2 3 4 5 6 1 2 3 4 5 Graph the Constraints : 2 K + 3 S 10 K 0 , S 0. K S 2 K + 3 S = 10
8 1 2 3 4 5 6 1 2 3 4 5 Graph the Constraints : 2 K + 3 S 10 K 0 , S 0. K S 2 K + 3 S = 10

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9 1 2 3 4 5 6 1 2 3 4 5 Add the Constraint : K + 2 S 6 K S K + 2 S = 6
10 1 2 3 4 5 6 1 2 3 4 5 Add the Constraint : K + S 5 K S K + S = 5

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11 1 2 3 4 5 6 1 2 3 4 5 Add the Constraints : K 4; S 3 K S We have now graphed the feasible region.
12 1 2 3 4 5 6 1 2 3 4 5 How do we maximize 3K + 5S ? K S

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13 K S 1 2 3 4 1 2 3 Let's take a closer look!
13 K S 1 2 3 4 1 2 3 How do we maximize 3K + 5S ? : the set of solutions to 3K + 5S = a 3K + 5S = 7.5 3K + 5S = 12 Isoprofit line

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13 K S 1 2 3 4 1 2 3 How do we maximize 3K + 5S ? : the set of solutions to 3K + 5S = a 3K + 5S = 16 3K + 5S = 7.5 3K + 5S = 12 Isoprofit line
K S 1 2 3 4 1 2 3 3K + 5S = 16 : the set of solutions to 3K + 5S = a. a

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## This note was uploaded on 09/23/2009 for the course ENGR 62 taught by Professor Unknown during the Spring '06 term at Stanford.

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Lecture3 - MS&E 111 z Lecture 3 The Geometry of Linear...

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