MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Connecting Calculus to Linear Programming

Marcel Y. Blais, Ph.D.Worcester Polytechnic Institute

Dept. of Mathematical Sciences

July 2017

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Motivation

Goal: To help students make connections between high schoolmath and real world applications of mathematics.

Linear programming is based on a simple idea fromcalculus.

We can connect calculus to real-world industrial problems.

We’ll explore the some basic principles of linearprogramming and some modern-day applications.

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Motivation

Goal: To help students make connections between high schoolmath and real world applications of mathematics.

Linear programming is based on a simple idea fromcalculus.

We can connect calculus to real-world industrial problems.

We’ll explore the some basic principles of linearprogramming and some modern-day applications.

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Motivation

Goal: To help students make connections between high schoolmath and real world applications of mathematics.

Linear programming is based on a simple idea fromcalculus.

We can connect calculus to real-world industrial problems.

We’ll explore the some basic principles of linearprogramming and some modern-day applications.

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Continuous Functions on Closed Intervals

x-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

-2

-1

0

1

2

3

4

5

a b

What can we say about a continuous function on a closedinterval?

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Continuous Functions on Closed Intervals

x-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

-2

-1

0

1

2

3

4

5

a b

Theorem

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Continuous Functions on Closed Intervals

x-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

-2

-1

0

1

2

3

4

5

a b

Theorem

If f is continuous on [a, b] then f attains both an absoluteminimum 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

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Linear Case

Theorem

If f is continuous linear on [a, b] then f attains both anabsolute minimum value and an absolute maximum value on[a, b], and these occur at endpoints of [a, b].

x0 1 2 3 4 5 6 7 8 9 10

y

0

5

10

15

20

25

30

35

a b

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Higher-Dimensional Optimization

1

0.8

0.6

0.4

x

0.2

000.10.2

y

0.30.40.50.60.70.80.91

-0.2

-0.5

-0.4

-0.3

0.2

-0.1

0

0.1

z

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Higher-Dimensional Optimization

1

0.8

0.6

0.4

x

0.2

000.10.2

y

0.30.40.50.60.70.80.91

-0.2

-0.5

-0.4

-0.3

0.2

-0.1

0

0.1

z

Theorem

If f (x , y) is continuous on closed and bounded plane region Rthen f attains both an absolute minimum value and an absolutemaximum value on R. Further, these values occur either atcritical points of f in the interior of R or on the boundary of R.

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Higher-Dimensional Optimization

1

0.8

0.6

0.4

x

0.2

000.10.2

y

0.30.40.50.60.70.80.91

-0.2

-0.5

-0.4

-0.3

0.2

-0.1

0

0.1

z

Theorem

If f (x1, x2, . . . , xn) is continuous on closed and bounded regionR ⊂ Rn then f attains both an absolute minimum value and anabsolute maximum value on R. Further, these values occureither at critical points of f in the interior of R or on theboundary of R.

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Higher-Dimensional Linearity

10.90.80.70.60.50.40.3

x

0.20.100

0.2

y

0.4

0.6

0.8

4.5

0

0.5

1

5

1.5

2

2.5

3

3.5

4

1

z

Theorem

If f (x1, x2, . . . , xn) is linear on closed and bounded regionR ⊂ Rn defined by a system of linear constraints then f attainsboth an absolute minimum value and an absolute maximumvalue on R. Further, these values occur on the boundary of R.

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Higher-Dimensional Linearity

10.90.80.70.60.50.40.3

x

0.20.100

0.2

y

0.4

0.6

0.8

4.5

0

0.5

1

1.5

2

2.5

3

3.5

4

5

1

z

Theorem

If f (x1, x2, . . . , xn) is linear on closed and bounded regionR ⊂ Rn defined by a system of linear constraints then f attainsboth an absolute minimum value and an absolute maximumvalue on R. Further, these values occur on corner points of theboundary of R.

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Higher-Dimensional Linearity

1

0.8

0.6

0.4

x0.2

00

0.1

0.2

0.3

y

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.6

0

1

0.9

0.8

0.7

0.2

0.5

0.4

0.3

1

z

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

A solution to minimize f over this region will occur at a corner.Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Linear Programming

The Simplex Method was developed by George Dantzig in1947.

“programming” synonymous with “optimization”.

Algorithm to traverse the corner points of the feasiblepolyhedron for a linear programming problem to find anoptimal feasible solution.

Standard form for a linear programming problem:

min xTc such that Ax ≤ b and x ≥ 0. (1)

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Simplex Method

min xTc such that Ax ≤ b and x ≥ 0.

For each constraint (row i of A), add a new slackvariable yi .

New system: min xTc such that [A + I ]

[xy

]= b,

x ≥ 0, y ≥ 0.

Underdetermined (more variables than equations). A + I

The slack now in the system is the key.

Each slack variable is associated with a constraintboundary

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Simplex Method

Simplex algorithm:

Split entries of

[xy

]into 2 subsets:

Basic variables: xB ∈ Rm (non-zero valued)Non-basic variables: xN ∈ Rn (zero valued)Represents a corner point of the feasible region.

If not optimal, move to an adjacent corner point byswapping one entry in xB with one entry in xN andre-solving the system of equations.

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Simplex Geometry

1

0.8

0.6

0.4

x0.2

00

0.1

0.2

0.3

y

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.6

0

1

0.9

0.8

0.7

0.2

0.5

0.4

0.3

1

z

Figure: Feasible Region

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Simplex Geometry

1

0.8

0.6

0.4

x0.2

00

0.1

0.2

0.3

y

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.6

0

1

0.9

0.8

0.7

0.2

0.5

0.4

0.3

1

z

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

Choose initial basis to correspond to the origin.Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Simplex Geometry

1

0.8

0.6

0.4

x0.2

00

0.1

0.2

0.3

y

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.6

0

1

0.9

0.8

0.7

0.2

0.5

0.4

0.3

1

z

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

If the objective function is not optimal, move to a betteradjacent corner.

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Simplex Geometry

1

0.8

0.6

0.4

x0.2

00

0.1

0.2

0.3

y

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.6

0

1

0.9

0.8

0.7

0.2

0.5

0.4

0.3

1

z

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

Repeat until the objective function is minimized (no remainingadjacent corners will reduce it).

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Application: Shipping Network

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Problem

K warehouses: S1,S2, . . . ,SK

L retail locations: D1,D2, . . . ,DL

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

Warehouse Si can supply si amount of product

Retail location Dj demands dj amount of product

Problem: Design a shipping schedule that satisfies demandat all retail locations at a minimal cost.

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Problem

K warehouses: S1,S2, . . . ,SK

L retail locations: D1,D2, . . . ,DL

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

Warehouse Si can supply si amount of product

Retail location Dj demands dj amount of product

Problem: Design a shipping schedule that satisfies demandat all retail locations at a minimal cost.

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Problem

K warehouses: S1,S2, . . . ,SK

L retail locations: D1,D2, . . . ,DL

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

Warehouse Si can supply si amount of product

Retail location Dj demands dj amount of product

Problem: Design a shipping schedule that satisfies demandat all retail locations at a minimal cost.

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Problem

K warehouses: S1,S2, . . . ,SK

L retail locations: D1,D2, . . . ,DL

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

Warehouse Si can supply si amount of product

Retail location Dj demands dj amount of product

Problem: Design a shipping schedule that satisfies demandat all retail locations at a minimal cost.

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Problem

K warehouses: S1,S2, . . . ,SK

L retail locations: D1,D2, . . . ,DL

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

Warehouse Si can supply si amount of product

Retail location Dj demands dj amount of product

Problem: Design a shipping schedule that satisfies demandat all retail locations at a minimal cost.

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Problem

K warehouses: S1,S2, . . . ,SK

L retail locations: D1,D2, . . . ,DL

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

Warehouse Si can supply si amount of product

Retail location Dj demands dj amount of product

Problem: Design a shipping schedule that satisfies demandat all retail locations at a minimal cost.

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Problem

K warehouses: S1,S2, . . . ,SK

L retail locations: D1,D2, . . . ,DL

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

Warehouse Si can supply si amount of product

Retail location Dj demands dj amount of product

Problem: Design a shipping schedule that satisfies demandat all retail locations at a minimal cost.

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Network

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Network

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Problem Mathematical Model

Constraints

xij ≥ 0 - amount of product shipped from Si to Dj

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

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

Objective

Total cost of shipping schedule:K∑i=1

L∑j=1

cijxij

Minimize cost of shipping such that demand is satisfied

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Problem Mathematical Model

Constraints

xij ≥ 0 - amount of product shipped from Si to Dj

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

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

Objective

Total cost of shipping schedule:K∑i=1

L∑j=1

cijxij

Minimize cost of shipping such that demand is satisfied

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Problem Mathematical Model

Constraints

xij ≥ 0 - amount of product shipped from Si to Dj

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

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

Objective

Total cost of shipping schedule:K∑i=1

L∑j=1

cijxij

Minimize cost of shipping such that demand is satisfied

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Problem Mathematical Model

Constraints

xij ≥ 0 - amount of product shipped from Si to Dj

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

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

Objective

Total cost of shipping schedule:K∑i=1

L∑j=1

cijxij

Minimize cost of shipping such that demand is satisfied

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Problem Mathematical Model

Constraints

xij ≥ 0 - amount of product shipped from Si to Dj

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

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

Objective

Total cost of shipping schedule:K∑i=1

L∑j=1

cijxij

Minimize cost of shipping such that demand is satisfied

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Problem Mathematical Model

Constraints

xij ≥ 0 - amount of product shipped from Si to Dj

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

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

Objective

Total cost of shipping schedule:K∑i=1

L∑j=1

cijxij

Minimize cost of shipping such that demand is satisfied

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

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Problem Mathematical Model

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

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

A =

1 . . . 1

1 . . . 1. . .

1 . . . 1

I I I

minK∑i=1

L∑j=1

cijxij such that Ax≤=

[sd

], x ≥ 0.

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

Shipping Problem Mathematical Model

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

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

A =

1 . . . 1

1 . . . 1. . .

1 . . . 1

I I I

minK∑i=1

L∑j=1

cijxij such that Ax≤=

[sd

], x ≥ 0.

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

TV Project - MA 3231

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

TV Project - MA 3231

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

TV Project - MA 3231

Blais MIST 2017

MIST 2017

Blais

Optimization

Linearity

Higher-DimensionalOptimization

Higher-DimensionalLinearity

LinearProgramming

Simplex Method

Applications

References

1 Blais, Marcel., MA 3231 Linear Programming LectureNotes, WPI Department of Mathematical Sciecnes, 2016.

2 Griva, Igor, Stephen G. Nash, and Ariela Sofer., Linear andNonlinear Optimization, Second Edition. SIAM, 2009.

3 Hillier, Frederick and Gerald Lieberman., Introduction toOperations Research, Tenth Edition. McGraw HillEducation, 2015.

Blais MIST 2017