Connecting Calculus to Linear Programming Dimensional Optimization

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Optimization Linearity Higher- Connecting Calculus to Linear Programming Dimensional Optimization

Higher- Dimensional Marcel Y. Blais, Ph.D. Linearity Worcester Polytechnic Institute Linear Programming Simplex Method Dept. of Mathematical Sciences Applications July 2017

Blais MIST 2017 We can connect calculus to real-world industrial problems. We’ll explore the some basic principles of linear programming and some modern-day applications.

Motivation

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Blais

Linear Programming Simplex Method Applications

Blais MIST 2017 We’ll explore the some basic principles of linear programming and some modern-day applications.

Motivation

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Optimization Goal: To help students make connections between high school Linearity math and real world applications of mathematics. Higher- Dimensional Optimization Linear programming is based on a simple idea from Higher- calculus. Dimensional Linearity We can connect calculus to real-world industrial problems. Linear Programming Simplex Method Applications

Blais MIST 2017 Motivation

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Blais MIST 2017 Continuous Functions on Closed Intervals

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Optimization 4 Linearity 3 Higher-

Dimensional 2 Optimization y Higher- 1 Dimensional Linearity 0

Linear -1 Programming Simplex Method -2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Applications a x b

What can we say about a continuous function on a closed interval?

Blais MIST 2017 Continuous Functions on Closed Intervals

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Optimization 3 Linearity Higher- 2 Dimensional y Optimization 1

Higher- 0 Dimensional Linearity -1 Linear -2 Programming -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Simplex Method a x b Applications

Theorem If f is continuous on [a, b] then f attains both an absolute minimum value and an absolute maximum value on [a, b].

Blais MIST 2017 Continuous Functions on Closed Intervals

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3 Optimization Linearity 2

Higher- y Dimensional 1 Optimization

0 Higher- Dimensional Linearity -1

Linear -2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Programming x Simplex Method a b Applications Theorem If f is continuous on [a, b] then f attains both an absolute minimum value and an absolute maximum value on [a, b]. Further these occur at critical points of f or endpoints of [a, b].

Blais MIST 2017 Linear Case

MIST 2017 Theorem Blais If f is continuous linear on [a, b] then f attains both an Optimization absolute minimum value and an absolute maximum value on Linearity Higher- [a, b], and these occur at endpoints of [a, b]. Dimensional Optimization

35 Higher-

Dimensional 30 Linearity

Linear 25 Programming

Simplex Method 20

Applications y 15

0 0 1 2 3 4 5 6 7 8 9 10 a x b

Blais MIST 2017 Higher-Dimensional Optimization

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Optimization 0.2 Linearity 0.1 Higher- 0 Dimensional Optimization -0.1 z Higher- -0.2 Dimensional -0.3 Linearity

-0.4 Linear 1 0.8 -0.5 0.6 Programming 1 0.9 0.8 0.4 0.7 0.6 0.5 0.2 Simplex Method 0.4 0.3 0.2 0.1 0 0 x Applications y

What does this mean for a continuous function f (x, y) on a closed and bounded plane region R?

Blais MIST 2017 Higher-Dimensional Optimization

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Blais 0.2

0.1

Optimization 0 Linearity -0.1

Higher- z Dimensional -0.2

Optimization -0.3

Higher- -0.4 1 Dimensional 0.8 -0.5 0.6 Linearity 1 0.9 0.8 0.4 0.7 0.6 0.5 0.2 0.4 0.3 0.2 0.1 0 0 x Linear y Programming Simplex Method Applications Theorem If f (x, y) is continuous on closed and bounded plane region R then f attains both an absolute minimum value and an absolute maximum value on R. Further, these values occur either at critical points of f in the interior of R or on the boundary of R.

Blais MIST 2017 Higher-Dimensional Optimization

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Blais 0.2 0.1

0 Optimization -0.1

Linearity z -0.2

Higher- -0.3 Dimensional -0.4 Optimization 1 0.8 -0.5 0.6 1 0.9 0.8 0.4 0.7 0.6 0.5 0.2 0.4 0.3 Higher- 0.2 0.1 0 0 x Dimensional y Linearity

Linear Programming Simplex Method Theorem Applications If f (x1, x2,..., xn) is continuous on closed and bounded region n R ⊂ R then f attains both an absolute minimum value and an absolute maximum value on R. Further, these values occur either at critical points of f in the interior of R or on the boundary of R.

Blais MIST 2017 Higher-Dimensional Linearity

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4.5

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3.5

Optimization 3 Linearity 2.5 z

Higher- 2

Dimensional 1.5 Optimization 1

Higher- 0.5

Dimensional 0 1 Linearity 0.8 0.6 0.4 1 0.8 0.9 0.6 0.7 0.2 0.4 0.5 0.2 0.3 Linear y 0 0 0.1 Programming x Simplex Method Applications Theorem

If f (x1, x2,..., xn) is linear on closed and bounded region n R ⊂ R deﬁned by a system of linear constraints then f attains both an absolute minimum value and an absolute maximum value on R. Further, these values occur on the boundary of R. Blais MIST 2017 Higher-Dimensional Linearity

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5 Blais 4.5 4

3.5

2.5 Optimization z 2

Linearity 1.5

1 Higher- 0.5 0 Dimensional 1 0.8 0.6 Optimization 0.4 0.9 1 0.7 0.8 0.2 0.6 0.4 0.5 0.2 0.3 y 0 0 0.1 Higher- x Dimensional Linearity

Linear Programming Theorem Simplex Method If f (x , x ,..., x ) is linear on closed and bounded region Applications 1 2 n n R ⊂ R deﬁned by a system of linear constraints then f attains both an absolute minimum value and an absolute maximum value on R. Further, these values occur on corner points of the boundary of R.

Blais MIST 2017 Higher-Dimensional Linearity

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0.9

Optimization 0.8 Linearity 0.7 0.6

Higher- 0.5 z

Dimensional 0.4 Optimization 0.3

0.2

Higher- 0.1

Dimensional 0 1 Linearity 0.9 0.8 1 0.7 0.6 0.8 0.5 Linear 0.6 0.4 0.3 0.4 Programming 0.2 y 0.2 0.1 x Simplex Method 0 0 Applications

Example: A possible feasible region f (x, y, z) = ax + by + cz subjuect to 5 linear constraints.

A solution to minimize f over this region will occur at a corner.

Blais MIST 2017 Linear Programming

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Blais The Simplex Method was developed by George Dantzig in Optimization Linearity 1947. Higher- Dimensional “programming” synonymous with “optimization”. Optimization Algorithm to traverse the corner points of the feasible Higher- Dimensional polyhedron for a linear programming problem to ﬁnd an Linearity optimal feasible solution. Linear Programming Simplex Method Standard form for a linear programming problem: Applications min xT c such that Ax ≤ b and x ≥ 0. (1)

n m m×n x, c ∈ R , b ∈ R , A ∈ R - n variables and m constraints.

Blais MIST 2017 Simplex Method

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Blais min xT c such that Ax ≤ b and x ≥ 0. Optimization Linearity For each constraint (row i of A), add a new slack Higher- Dimensional variable yi . Optimization T x Higher- New system: min x c such that [A + I ] = b, Dimensional y Linearity x ≥ 0, y ≥ 0. Linear Programming Underdetermined (more variables than equations). A + I Simplex Method Applications The slack now in the system is the key. Each slack variable is associated with a constraint boundary

Blais MIST 2017 Simplex Method

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Optimization Simplex algorithm: Linearity Higher- x Dimensional Split entries of into 2 subsets: Optimization y m Higher- Basic variables: xB ∈ R (non-zero valued) Dimensional n Linearity Non-basic variables: xN ∈ R (zero valued)

Linear Represents a corner point of the feasible region. Programming Simplex Method If not optimal, move to an adjacent corner point by Applications swapping one entry in xB with one entry in xN and re-solving the system of equations.

Blais MIST 2017 Simplex Geometry

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0.8 Optimization 0.7

Linearity 0.6

0.5 Higher- z Dimensional 0.4 0.3 Optimization 0.2 Higher- 0.1 0 Dimensional 1 0.9 0.8 Linearity 1 0.7 0.6 0.8 0.5 0.6 Linear 0.4 0.3 0.4 0.2 Programming y 0.2 0.1 x 0 0 Simplex Method Applications Figure: Feasible Region

Example: min f (x1, x2, x3) = ax1 + bx2 + cx3 subject to 5 linear constraints.

Blais MIST 2017 Simplex Geometry

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Optimization 0.8 Linearity 0.7 0.6

Higher- 0.5 z

Dimensional 0.4 Optimization 0.3

0.2

Higher- 0.1

Dimensional 0 1 Linearity 0.9 0.8 1 0.7 0.6 0.8 0.5 Linear 0.6 0.4 0.3 0.4 Programming 0.2 y 0.2 0.1 x Simplex Method 0 0 Applications

Example: min f (x1, x2, x3) = ax1 + bx2 + cx3 subject to 5 linear constraints.

Choose initial basis to correspond to the origin.

Blais MIST 2017 Simplex Geometry

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0.9

0.8 Optimization 0.7

Linearity 0.6

0.5 Higher- z Dimensional 0.4 0.3 Optimization 0.2 Higher- 0.1 0 Dimensional 1 0.9 0.8 Linearity 1 0.7 0.6 0.8 0.5 0.6 Linear 0.4 0.3 0.4 0.2 Programming y 0.2 0.1 x 0 0 Simplex Method Applications

Example: min f (x1, x2, x3) = ax1 + bx2 + cx3 subject to 5 linear constraints.

If the objective function is not optimal, move to a better adjacent corner. Blais MIST 2017 Simplex Geometry

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0.8 Optimization 0.7

Linearity 0.6

0.5 Higher- z Dimensional 0.4 0.3 Optimization 0.2 Higher- 0.1 0 Dimensional 1 0.9 0.8 Linearity 1 0.7 0.6 0.8 0.5 0.6 Linear 0.4 0.3 0.4 0.2 Programming y 0.2 0.1 x 0 0 Simplex Method Applications

Example: min f (x1, x2, x3) = ax1 + bx2 + cx3 subject to 5 linear constraints.

Repeat until the objective function is minimized (no remaining adjacent corners will reduce it). Blais MIST 2017 Application: Shipping Network

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Optimization Linearity Higher- Dimensional Optimization

Higher- Dimensional Linearity

Linear Programming Simplex Method Applications

Blais MIST 2017 L retail locations: D1, D2,..., DL

Cost to ship x amount of product from Si to Dj is cij x

Warehouse Si can supply si amount of product

Retail location Dj demands dj amount of product Problem: Design a shipping schedule that satisﬁes demand at all retail locations at a minimal cost. Example: As of July 2017, Target Corp. has 37 distribution centers and 1, 802 stores.

Shipping Problem

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Optimization K warehouses: S1, S2,..., SK Linearity Higher- Dimensional Optimization

Higher- Dimensional Linearity

Linear Programming Simplex Method Applications

Blais MIST 2017 Cost to ship x amount of product from Si to Dj is cij x

Warehouse Si can supply si amount of product

Shipping Problem

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Optimization K warehouses: S1, S2,..., SK Linearity L retail locations: D , D ,..., D Higher- 1 2 L Dimensional Optimization

Higher- Dimensional Linearity

Linear Programming Simplex Method Applications

Blais MIST 2017 Warehouse Si can supply si amount of product

Shipping Problem

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Optimization K warehouses: S1, S2,..., SK Linearity L retail locations: D , D ,..., D Higher- 1 2 L Dimensional Optimization Cost to ship x amount of product from Si to Dj is cij x

Higher- Dimensional Linearity

Linear Programming Simplex Method Applications

Blais MIST 2017 Retail location Dj demands dj amount of product Problem: Design a shipping schedule that satisﬁes demand at all retail locations at a minimal cost. Example: As of July 2017, Target Corp. has 37 distribution centers and 1, 802 stores.

Shipping Problem

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Optimization K warehouses: S1, S2,..., SK Linearity L retail locations: D , D ,..., D Higher- 1 2 L Dimensional Optimization Cost to ship x amount of product from Si to Dj is cij x

Higher- Warehouse Si can supply si amount of product Dimensional Linearity

Linear Programming Simplex Method Applications

Blais MIST 2017 Problem: Design a shipping schedule that satisﬁes demand at all retail locations at a minimal cost. Example: As of July 2017, Target Corp. has 37 distribution centers and 1, 802 stores.

Shipping Problem

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Optimization K warehouses: S1, S2,..., SK Linearity L retail locations: D , D ,..., D Higher- 1 2 L Dimensional Optimization Cost to ship x amount of product from Si to Dj is cij x

Higher- Warehouse Si can supply si amount of product Dimensional Linearity Retail location Dj demands dj amount of product Linear Programming Simplex Method Applications

Blais MIST 2017 Example: As of July 2017, Target Corp. has 37 distribution centers and 1, 802 stores.

Shipping Problem

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Optimization K warehouses: S1, S2,..., SK Linearity L retail locations: D , D ,..., D Higher- 1 2 L Dimensional Optimization Cost to ship x amount of product from Si to Dj is cij x

Higher- Warehouse Si can supply si amount of product Dimensional Linearity Retail location Dj demands dj amount of product Linear Programming Problem: Design a shipping schedule that satisﬁes demand Simplex Method at all retail locations at a minimal cost. Applications

Blais MIST 2017 Shipping Problem

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Optimization K warehouses: S1, S2,..., SK Linearity L retail locations: D , D ,..., D Higher- 1 2 L Dimensional Optimization Cost to ship x amount of product from Si to Dj is cij x

Blais MIST 2017 Shipping Network

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Optimization Linearity Higher- Dimensional Optimization

Higher- Dimensional Linearity

Linear Programming Simplex Method Applications

Blais MIST 2017 Shipping Network

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Optimization Linearity Higher- Dimensional Optimization

Higher- Dimensional Linearity

Linear Programming Simplex Method Applications

Blais MIST 2017 Amount leaving Si : xi1 + xi2 + ··· + xiL ≤ si

Amount entering Dj : x1j + x2j + ··· + xKj = dj

K L X X Total cost of shipping schedule: cij xij i=1 j=1 Minimize cost of shipping such that demand is satisﬁed Target example: 37 × 1, 802 = 66, 674 variables

Shipping Problem Mathematical Model

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Blais Constraints Optimization Linearity xij ≥ 0 - amount of product shipped from Si to Dj Higher- Dimensional Optimization

Higher- Dimensional Linearity Objective

Linear Programming Simplex Method Applications

Blais MIST 2017 Amount entering Dj : x1j + x2j + ··· + xKj = dj

K L X X Total cost of shipping schedule: cij xij i=1 j=1 Minimize cost of shipping such that demand is satisﬁed Target example: 37 × 1, 802 = 66, 674 variables

Shipping Problem Mathematical Model

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Blais Constraints Optimization Linearity xij ≥ 0 - amount of product shipped from Si to Dj Higher- Dimensional Amount leaving Si : xi1 + xi2 + ··· + xiL ≤ si Optimization

Higher- Dimensional Linearity Objective

Linear Programming Simplex Method Applications

Blais MIST 2017 K L X X Total cost of shipping schedule: cij xij i=1 j=1 Minimize cost of shipping such that demand is satisﬁed Target example: 37 × 1, 802 = 66, 674 variables

Shipping Problem Mathematical Model

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Blais Constraints Optimization Linearity xij ≥ 0 - amount of product shipped from Si to Dj Higher- Dimensional Amount leaving Si : xi1 + xi2 + ··· + xiL ≤ si Optimization Amount entering D : x + x + ··· + x = d Higher- j 1j 2j Kj j Dimensional Linearity Objective

Linear Programming Simplex Method Applications

Blais MIST 2017 Minimize cost of shipping such that demand is satisﬁed Target example: 37 × 1, 802 = 66, 674 variables

Shipping Problem Mathematical Model

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Linear K L Programming X X Simplex Method Total cost of shipping schedule: cij xij Applications i=1 j=1

Blais MIST 2017 Target example: 37 × 1, 802 = 66, 674 variables

Shipping Problem Mathematical Model

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Linear K L Programming X X Simplex Method Total cost of shipping schedule: cij xij Applications i=1 j=1 Minimize cost of shipping such that demand is satisﬁed

Blais MIST 2017 Shipping Problem Mathematical Model

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Linear K L Programming X X Simplex Method Total cost of shipping schedule: cij xij Applications i=1 j=1 Minimize cost of shipping such that demand is satisﬁed Target example: 37 × 1, 802 = 66, 674 variables

Blais MIST 2017 Amount entering Dj : x1j + x2j + ··· + xKj = dj

Shipping Problem Mathematical Model

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Blais Amount leaving Si : xi1 + xi2 + ··· + xiL ≤ si

Optimization Linearity Higher-  1 ... 1  Dimensional Optimization  ..   1 ... 1 .  Higher-   Dimensional  1 ... 1  Linearity A =     Linear   Programming   Simplex Method Applications III

K L X X ≤ s min c x such that Ax , x ≥ 0. ij ij = d i=1 j=1

Blais MIST 2017 Shipping Problem Mathematical Model

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Blais Amount leaving Si : xi1 + xi2 + ··· + xiL ≤ si

Optimization Amount entering Dj : x1j + x2j + ··· + xKj = dj Linearity Higher-  1 ... 1  Dimensional Optimization  ..   1 ... 1 .  Higher-   Dimensional  1 ... 1  Linearity A =     Linear   Programming   Simplex Method Applications III

K L X X ≤ s min c x such that Ax , x ≥ 0. ij ij = d i=1 j=1

Blais MIST 2017 TV Project - MA 3231

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Blais

Optimization Linearity Higher- Dimensional Optimization

Higher- Dimensional Linearity

Linear Programming Simplex Method Applications

Blais MIST 2017 TV Project - MA 3231

MIST 2017

Blais

Optimization Linearity Higher- Dimensional Optimization

Higher- Dimensional Linearity

Linear Programming Simplex Method Applications

Blais MIST 2017 TV Project - MA 3231

MIST 2017

Blais

Optimization Linearity Higher- Dimensional Optimization

Higher- Dimensional Linearity

Linear Programming Simplex Method Applications

Blais MIST 2017 References

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Optimization 1 Blais, Marcel., MA 3231 Linear Programming Lecture Linearity Notes, WPI Department of Mathematical Sciecnes, 2016. Higher- Dimensional Optimization 2 Griva, Igor, Stephen G. Nash, and Ariela Sofer., Linear and Higher- Dimensional Nonlinear Optimization, Second Edition. SIAM, 2009. Linearity

Linear Programming Simplex Method 3 Hillier, Frederick and Gerald Lieberman., Introduction to Applications Operations Research, Tenth Edition. McGraw Hill Education, 2015.

Blais MIST 2017