Modeling Flows

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Modeling Flows

• John H. Vande Vate
• Spring, 2007

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Agenda

• Network Flows
• Review definition
• Simple Transportation Model
• Minimum Cost Flow Model
• Adding Realism
• Multiple commodities
• Weight, Cube, Linear Cube
• Concave Costs
• Modeling Time

3
Reference

• Network Flows Theory, Algorithms, and
  Applications (Hardcover)by Ravindra K. Ahuja,
  Thomas L. Magnanti, James B. Orlin

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What Are They?

• Specially Structured Linear Programs
• Each variable appears in at most two constraints
• At most one constraint with a 1 coefficient
• At most one constraint with a -1 coefficient
• Each variable may be constrained by bounds
• Integer Data gt Integer Solutions

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Transportation Model

• Move Goods from Plants to Warehouses
• Single Commodity!
• Plants have supplies
• Warehouses have demands
• Costs are proportional to volume shipped (this
  usually isnt very realistic)

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Example 2-3 (p 36)

• Single Product
• Two plants with identical production costs
• Two warehouses
• Markets allocated to warehouses for now
• Proportional transportation costs

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Example 2-3 contd
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Transportation Model (AMPL)

• Set Plants
• Set Warehouses
• Param Supply Plants
• Param Demand Warehouses
• Param CostPlants, Warehouses
• Var FlowPlants, Warehousesgt 0
• minimize TotalCost
• sump in Plants, w in Warehouses
  Costp,wFlowp,w
• s.t. WithinSupplyp in Plants
• sumw in Warehouses Flowp, w lt Supplyp
• s.t. MeetDemandw in Warehouses
• sump in Plants Flowp, w gt Demandw

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Minimum Cost Flow

• Move Goods from Plants to Markets via Warehouses
• Single Commodity!
• Plants have supplies
• Markets have demands
• Costs are proportional to volume shipped

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Example 2-3 Contd
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Minimum Cost Flow

• Set Locs
• Set Edges in Locs cross Locs
• param CostEdges
• param NetSupplyLocs
• var FlowVolEdges gt 0
• minimize TotalCost
• sum(f,t) in Edges Costf,tFlowVolf,t
• s.t. PathDefinitionloc in Locs
• sum(loc, t) in Edges FlowVolloc,t
• - sum(f, loc) in Edges FlowVolf, loc
  NetSupplyloc

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Homogenous Product
Must be able to interchange positions of product
anywhere
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Multi-Commodity Flows

• Several single commodity models
• You cant to turn lead into gold
• Conserve flow of each product through each
  facility
• Joined by common constraints
• Often capacity on lanes
• Or modeling cost (through capacity)

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Multiple Products
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Conveyance Capacity

• Weight Limits, e.g., 40,000 lbs
• Cubic Capacity (53 trailer)
• 3,970 cubic ft.
• Length 52' 6"
• Width 99"
• Height 110"
• Linear Cube (53 trailer)
• If load is not stackable, just floor space

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Conveyance Capacity
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Multiple Products Weight Cube
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Note

• No Longer a Network Model
• Solutions not integral
• Adding integrality constraints

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Multiple Products Weight Cube
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Concave Cost
Cost per unit decreasing
Cost
Without special constraints, what will solver do?
Shipment Size
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Modeling Economies of Scale

• Linear Programming
• Greedy
• Takes the High-Range Unit Cost first!
• Integer Programming
• Add constraints to ensure first things first
• Several Strategies

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Convex Combination

• Weighted Average

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What will the cost be?
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Total Cost
1/5th of the way
First Break Point
Second Break Point
0
10
Mid Point
20
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Conclusion

• If the Volume of Activity is a fraction ? of the
  way from one breakpoint to the next, the cost
  will be that same fraction of the way from the
  cost at the first breakpoint to the cost at the
  next
• If Volume 10? 20(1-?)
• Then Cost 22? 27(1-?)

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Idea

• Express Volume of Activity as a Weighted Average
  of Breakpoints
• Express Cost as the same Weighted Average of
  Costs at the Breaks
• Activity Min Level ?0 Break 1 ?1
• Break 2 ?2 Max Level ?3
• Cost Cost at Min Level ?0 Cost at Break 1 ?1
  
• Cost at Break 2 ?2 Cost at Max
  Level ?3
• 1 ?0 ?1 ?2 ?3

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Does that Do It?
High-Range Cost/Unit

• What can go wrong?

Mid-Range Cost/Unit
Low Range Cost/Unit
X
Total Cost
Minimum Sustainable Level
Maximum Operating Level
First Break Point
Second Break Point
0
Volume of Activity
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Role of Integer Variables

• Ensure we express Activity as a combination of
  two consecutive breakpoints
• var InRegion1..NBreaks binary

Total Cost
Minimum Sustainable Level
Maximum Operating Level
Second Break Point
First Break Point
0
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Constraints

• Lambda2 0 unless activity is between
• BreakPoint1 and BreakPoint2 (Region2) or
• BreakPoint2 and BreakPoint3 (Region3)
• Lambda2 ? InRegion2 InRegion3

Total Cost
Minimum Sustainable Level
Maximum Operating Level
Second Break Point
First Break Point
BreakPoint3
BreakPoint1
BreakPoint2
BreakPoint0
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We cant go wrong
High-Range Cost/Unit
Mid-Range Cost/Unit
Low Range Cost/Unit
X
Minimum Sustainable Level
Maximum Operating Level
First Break Point
Second Break Point
0
Volume of Activity
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Concave Costs
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In AMPL Speak

• param NBreaks
• param BreakPoint0..NBreaks
• param CostAtBreak0..NBreaks
• var Lambda0..NBreaks gt 0
• var Activity
• var Cost
• s.t. DefineCost
• Cost sumb in 0..NBreaks CostAtBreakbLambda
  b
• s.t. DefineActivity
• Activity sumb in 0..NBreaks
  BreakPointbLambdab
• s.t. ConvexCombination
• 1 sumb in 0..NBreaksLambdab

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And Activity in One Region

• InRegion1 InRegion2 InRegion3 ? 1
• Why ? 1?
• If it is in Region2
• Lambda1 ? InRegion1 InRegion2 1
• Lambda2 ? InRegion2 InRegion3 1
• Other Lambdas are 0

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AMPL Speak

• param NBreaks
• param BreakPoint0..NBreaks
• param CostAtBreak0..NBreaks
• var Lambda0..NBreaks gt 0
• var Activity
• var Cost
• s.t. DefineCost
• Cost sumb in 0..NBreaks CostAtBreakbLambda
  b
• s.t. DefineActivity
• Activity sumb in 0..NBreaks
  BreakPointbLambdab
• s.t. ConvexCombination
• 1 sumb in 0..NBreaksLambdab

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What We Added

• var InRegion1..NBreaks binary
• s.t. InOneRegion
• sumb in 1..NBreaks InRegionb lt 1
• s.t. EnforceConsecutiveb in 0..NBreaks-1
• Lambdab lt InRegionb InRegionb1
• s.t. LastLambda
• LambdaNBreaks lt InRegionNBreaks

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Good News!

• AMPL offers syntax to automate this
• Read Chapter 14 of Fourer for details
• ltltBreakPoint1, BreakPoint2 Slope1,
  Slope2, Slope3gtgt Variable
• Slope1 before BreakPoint1
• Slope2 from BreakPoint1 to BreakPoint2
• Slope3 after BreakPoint2
• Has 0 cost at activity 0

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Modeling Time

• Incorporating Schedules into Network Flow Models
• Example
• Given several scheduled pick-ups and deliveries
  (must pick-up and deliver on-time)
• Question How many vehicles required?
• Assumption One load on a vehicle at a time (No
  shared capacity)

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Example

• How many vehicles are required to meet a schedule
  of departures and returns
• No shared capacity

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Network Model
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How to Construct the Routes?
Route ending with 3
Route 5 gt 3
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How to Construct the Routes?
Route ending with 6
Route 2 gt 7 gt 6
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2nd Example of Time

• How many vehicles?
• Common
• Schedule provided
• No shared capacity
• Different Not just one terminal

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As a Network Problem
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Singapore Electric Generator
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Average Balances

• Assuming Smooth Demands

• Averages (Starting Ending)/2

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Inventory

• Balancing Your Checkbook
• Previous Balance Income - Expenses New
  Balance
• Modeling Dynamic Inventory
• Starting Inv. Production - Shipments Ending
  Inv.

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Singapore Electric Generator
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Network Model

• Ending Inv Calculated Ending Inv
• Prod. Qty lt Production Limits
• FinalInv gt MinimumEndingInv

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Network Model

• Ending Inv Calculated Ending Inv
• How Many constraints is this?
• Which constraints does Ending Inv for Jan appear
  in?
• With what coefficients?
• Which constraints does Production Qty in February
  appear in?

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Summary

• Network Flows
• Transportation
• Supply Demand
• Minimum Cost Flows
• Supply, Demand Flow conservation
• Multicommodity Flows
• Conserve flow of each commodity
• Weight, Cube Conveyances
• Not a network flow model
• Non-integral solutions
• Non-Linear Costs
• Requires integer variables
• Modeling Time

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Next

• Consolidating Flows