1. Modeling Flows and Information
Single commodity flow problems
Multicommodity Flow Conservation
Weight and Cube
Pass through vs Unload/Reload
Tracking Information
Column Generation
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2. Modeling Tools We will rely on Solver in Excel
Instructions on installing and using Solver accessible from the class homepage
Convenient:
Everyone is familiar with Excel
Minimal fixed costs of installing, accessing and learning syntax for a new application
Inconvenient:
Solver is unreliable.
It says models are not linear when they are.
It says it has found a solution when it hasn't, etc.
It combines the problem data with the model
Limited problem sizes
Solver is not an industrial strength tool.
So, I encourage you to learn an algebraic modeling language and implement the models we discuss in class in that language. Claudio is an excellent resource for help in developing these models.
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3. Motor Distribution Belgium Netherlands Germany 500 500 800 800 500 400
700
400
900
200 *
Belgium
Germany
Netherlands
The Hague
Amsterdam
Antwerp
Nancy
Liege
Tilburg
Leipzig
Miles
100 50
500
800
500
400
700
200
900
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4. Transportation Costs Minimize
Unit transportation costs from harbors to plants
Minimize
the transportation costs involved in moving
the motors from the harbors to the plants
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5. A Transportation Model
Spreadsheet embedded in presentation
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6. Challenge
Find a best answer
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7. Building a Solver Model
Tools | Solver…
Set Target Cell: The cell holding the value you want to minimize (cost) or maximize (revenue)
Equal to: Choose Max to maximize or Min to minimize this
By Changing Cells: The cells or variables the model is allowed to adjust
In the Transportation spreadsheet that’s G19 - the total transportation cost
In the Transportation spreadsheet we choose Min to minimize transport cost
In the Transportation spreadsheet that is C9:F11 - the Shipment volumes
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8. Solver Model Cont’d
Subject to the Constraints: The constraints that limit the choices of the values of the adjustables.
Click on Add
Cell Reference is a cell that holds a value calculated from the adjustables
Constraint is a cell that holds a value that constraints the Cell Reference.
<=, =, => is the sense of the constraint. Choose one.
In the Transportation spreadsheet for example, G9 is the total volume shipped out of Amsterdam
In the Transportation spreadsheet for example, H9 is the total volume we can ship out of Amsterdam
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<= in this case. Don’t ship more than we have in AMS

9. What are the Constraints?
Supply Constraints
Amsterdam: G9 <= H9
Antwerp: G10 <= H10
The Hague: G11 <= H11
Demand Constraints
Leipzig: C12 => C13
Nancy: D12 => D13
Liege: E12 => E13
Tilburg: F12 => F13
G9 is the total volume shipped from Amsterdam
Short cut:
G9:G11 <= H9:H11
C12 is the total volume shipped to Leipzig
Short cut:
C12:F12 => C13:F13
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10. The Model
$G$9 <= $H$9
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11. What’s Missing?
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12. Options
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13. Multiple Products Same costs Different products Different supply
Different Demand
500
800
700
400
900
200 *
Belgium
Germany
Netherlands
The Hague
Amsterdam
Antwerp
Nancy
Liege
Tilburg
Leipzig
Miles
100 50
Product 1
Product 2
800
500
800
500
500
400
300
400
700
500
200
200
900
500
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14. Question Can we just combine the products: The Hague Tilburg
Total Supply at each source
Total Demand at each destination
The Hague
Total Supply: 1,100 =
Tilburg
Total Demand: 1,000 =
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15. Question Can we just solve two separate problems:
One Problem for Product 1
One Problem for Product 2?
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16. Question Can we just solve two separate problems:
One Problem for Product 1
One Problem for Product 2?
When won’t this work?
Different Costs?
Capacities on Transportation
...
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17. Shared Capacities
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18. Transshipments 2 PRODUCTS 3 plants 2 distribution centers 2 customers
Minimize shipping costs
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19. Challenge
Formulate a model
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20. Flow Conservation Conserve Flow at the DC Otherwise…
Product by Product
Otherwise…
Alchemy: Turn lead into gold
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21. AMPL Model set PRODUCTS; set PLANTS; set DCS; set CUSTS;
set EDGES within (PLANTS cross DCS)
union (DCS cross CUSTS);
param Supply{PRODUCTS, PLANTS};
param Demand{PRODUCTS, CUSTS};
param Cost{EDGES};
param Capacity{EDGES};
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22. var Flow{PRODUCTS, EDGES} >= 0; minimize TotalCost:
sum {p in PRODUCTS, (f, t) in EDGES} Cost[f,t]*Flow[p, f, t];
s.t. ObserveSupply {prd in PRODUCTS, plant in PLANTS}:
sum {(plant, toloc) in EDGES} Flow[prd, plant, toloc]
<= Supply[prd, plant];
s.t. MeetDemand{prd in PRODUCTS, cust in CUSTS}:
sum{(fromloc, cust) in EDGES} Flow[prd, fromloc, cust]
>= Demand[prd, cust];
s.t. ConserveFlow{prd in PRODUCTS, dc in DCS}:
sum{(fromloc, dc) in EDGES} Flow[prd, fromloc, dc]
= sum{(dc, toloc) in EDGES} Flow[prd, dc, toloc];
s.t. JointCapacity{ (fromloc, toloc) in EDGES}:
sum{prd in PRODUCTS} Flow[prd, fromloc, toloc]
<= Capacity[fromloc, toloc];
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23. Weight & Cube More realistic version of “shared capacity”
Product 1:
10 lbs/unit
1 cubic ft/unit
Pay by Truckload
Weight Limit: 4,000 lbs
Cubic Capacity: 1,000 Cu. Ft.
Joint demands on transportation capacity
Product 2:
1 lbs/unit
10 cubic ft/unit
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24. Weight & Cube
How many vehicles?
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25. Modeling Weight & Cube The number of vehicles required is…
The larger of the number required
for the weight and
for the cube
Conveyances = Max(TotalWeight/WeightLimit,
TotalCube/CubeCap)
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26. Challenge Formulate a model Do NOT use MAX
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27. More Detail Container at a port Container at a distribution center
Conserve flow of containers
Goods in the containers don’t change
Conserve flow of goods
Loads change from in-bound to outbound
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28. Challenge Formulate a model with Transloading at DC
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29. Without Transloading Same number of conveyances in and out
Same products in those conveyances
Lose potential to improve capacity utilization
How to model?
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30. Flows on Paths Product 1 from Plant 1 to Customer 1 via DC 1
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31. Challenge Formulate a model WITHOUT Transloading at the DC
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32. AMPL Model set PLANTS; set DCS; set CUSTS; set PRODS;
set EDGES := (PLANTS cross DCS) union
(DCS cross CUSTS);
param Cost{EDGES};
param Weight{PRODS};
param Cube{PRODS};
param Supply {PRODS, PLANTS} default 0;
param Demand {PRODS, CUSTS} default 0;
param WeightLimit;
param CubicCapacity;
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33. var PathFlow {PRODS, PLANTS, DCS, CUSTS} >= 0;
var Conveys{PLANTS, DCS, CUSTS} >= 0;
minimize FreightCost:
sum{plant in PLANTS, dc in DCS, cust in CUSTS}
(Cost[plant, dc] + Cost[dc, cust])*Conveys[plant, dc, cust];
s.t. ObserveSupply {prd in PRODS, plant in PLANTS}:
sum{dc in DCS, cust in CUSTS} PathFlow[prd, plant, dc, cust] <= Supply[prd, plant];
s.t. MeetDemand {prd in PRODS, cust in CUSTS}:
sum{dc in DCS, plant in PLANTS} PathFlow[prd, plant, dc, cust] >= Demand[prd, cust];
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34. AMPL Model
s.t. ConveysByWeight {plant in PLANTS, dc in DCS, cust in CUSTS}:
sum{prd in PRODS} Weight[prd]*PathFlow[prd, plant, dc, cust] <= Conveys[plant, dc, cust]*WeightLimit;
s.t. ConveysByCube {plant in PLANTS, dc in DCS, cust in CUSTS}:
sum{prd in PRODS} Cube[prd]*PathFlow[prd, plant, dc, cust] <= Conveys[plant, dc, cust]*CubicCapacity;
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35. Other Uses Additional Information about the products history
Where it was made for
Duties
Pipeline Inventory
How long it has been traveling …
Extreme cases: very long involved paths
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36. Column Generation Airlines Crew Scheduling
Complex pay and duty rules
Unions
FAA
Tour of duty for a crew
Several legs returning to base
Millions of possible tours of duty
Each is a variable:
1 = use it in schedule
0 = do not.
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37. Solution Strategy Initial Tours of Duty Repeat until Done
Used in previous solutions
Fillers to ensure feasibility
Repeat until Done
Solve small problem
Use Shadow Prices to generate new tours that will improve solution
When no better tours are found -- done
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38. Column Generation Example
Multicommodity Flow
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39. Traditional Model Minimize S S cij xkij
s.t. Sj xkij – Sj xkji = {0, supply, -demand
s.t. Sk xkij  capacityij
xkij  0
xkij is the volume of commodity k we send directly from node i to node j
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40. This is the sum over all the paths for commodity k
Flow on Paths Model
This is the sum over all the paths for commodity k
Pk is the set of paths from the source for commodity k to the destination.
xp is the flow on path p
cp is the cost of one unit of flow on path p
For simplicity assume each commodity has a unique source and a unique destination
Minimize S cp xp
s.t. S xp = supply for commodity k
s.t. S xp  capacity of edge (i,j)
xp  0
This is the sum over all the paths that cross edge (i,j)
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41. Flows On Paths Formulation
Variable for
Product 1: Each path from 1 to 5
Product 2: Each path from 2 to 4
Constraints: Capacity on each edge:
Sum of flows on paths that use the edge do not exceed capacity of the edge
Cost: Each unit of flow on a path pays the cost of all the edges on the path
Why no Flow Conservation Constraints?
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42. Column Generation Approach
Start with 2 paths
Product 1: The edge from 1 to 5
Product 2: The edge from 2 to 4
Solve and get Shadow Prices on the capacity constraints
Calculate new (shortest) paths with the best reduced costs
Repeat.
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43. Quick LP review
Is there a path whose revised cost is less than the price for the commodity
A solution is optimal if no variable is a candidate to enter the basis. That is if no variable prices out so that increasing the variable decreases the solution cost.
To price out a variable, compare its cost with the resources it consumes
cp - S shadow prices*coefficents < 0
cp - S shadow prices on edges of P
– price for the commodity’s demand constraint
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44. Column Generation Example
Multicommodity Flow
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45. AMPL Commands In AMPL Modeling languages do more than model.
They also drive the solver.
ColumnGeneration.mod
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46. Summary Conserve Flow at lowest level of detail
Distinction between transloading and container handling
Involves additional information in “Edges”, e.g., what port it passed through
Extreme cases: Additional information is the entire path
Very Technical: Column Generation can handle these cases (sometimes).
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