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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
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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
![Page 3: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/3.jpg)
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
![Page 4: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/4.jpg)
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
![Page 5: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/5.jpg)
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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MIST 2017
Blais
Optimization
Linearity
Higher-DimensionalOptimization
Higher-DimensionalLinearity
LinearProgramming
Simplex Method
Applications
Application: Shipping Network
Blais MIST 2017
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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
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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
![Page 25: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/25.jpg)
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
![Page 26: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/26.jpg)
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
![Page 27: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/27.jpg)
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
![Page 28: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/28.jpg)
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
![Page 29: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/29.jpg)
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
![Page 30: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/30.jpg)
MIST 2017
Blais
Optimization
Linearity
Higher-DimensionalOptimization
Higher-DimensionalLinearity
LinearProgramming
Simplex Method
Applications
Shipping Network
Blais MIST 2017
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MIST 2017
Blais
Optimization
Linearity
Higher-DimensionalOptimization
Higher-DimensionalLinearity
LinearProgramming
Simplex Method
Applications
Shipping Network
Blais MIST 2017
![Page 32: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/32.jpg)
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
![Page 33: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/33.jpg)
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
![Page 34: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/34.jpg)
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
![Page 35: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/35.jpg)
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
![Page 36: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/36.jpg)
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
![Page 37: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/37.jpg)
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
![Page 38: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/38.jpg)
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
![Page 39: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/39.jpg)
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
![Page 40: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/40.jpg)
MIST 2017
Blais
Optimization
Linearity
Higher-DimensionalOptimization
Higher-DimensionalLinearity
LinearProgramming
Simplex Method
Applications
TV Project - MA 3231
Blais MIST 2017
![Page 41: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/41.jpg)
MIST 2017
Blais
Optimization
Linearity
Higher-DimensionalOptimization
Higher-DimensionalLinearity
LinearProgramming
Simplex Method
Applications
TV Project - MA 3231
Blais MIST 2017
![Page 42: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/42.jpg)
MIST 2017
Blais
Optimization
Linearity
Higher-DimensionalOptimization
Higher-DimensionalLinearity
LinearProgramming
Simplex Method
Applications
TV Project - MA 3231
Blais MIST 2017
![Page 43: Connecting Calculus to Linear Programming · Linear Programming Simplex Method Applications Motivation Goal: To help students make connections between high school math and real world](https://reader030.vdocuments.net/reader030/viewer/2022040403/5e8a9cd931ef24633c11c71d/html5/thumbnails/43.jpg)
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