Network Flows and Linear Programming
The Mathematical Madness behind the Magic
GoldSim Technology Group
©GoldSim Technology Group LLC., 2012
Objective of the Flow Module
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Given a system of discrete locations connected by conduits of flowing
material…
…determine the “optimal” flow of material through that network.
©GoldSim Technology Group LLC., 2012
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Flow Solver
©GoldSim Technology Group LLC., 2012
GoldSim 10.5: Solve using
iteration
Benefits
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Optimal allocation of material
Mass conservation
Integrates handling of flow and transport
Built-in storage functions
Integrated handling of priorities and costs
Influence lines represent flows
©GoldSim Technology Group LLC., 2012
What do we mean by “Optimal”?
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Meaning 1: Maximal Profit (e.g., commodity distribution)
–
–

Meaning 2: Prioritized Flow (e.g., water distribution)
–

If the network is controlled by a single operator selling to multiple customers,
then the goal is to maximize profit.
Example: Natural gas distributor (PSE)
In this case water is divided up based on various users’ priorities:
 Priority 1 users get first dibs on water until all their demands are met…
 …and so on until the lowest priority (farmers) get what’s left over.
The prioritized flow method uses the same underlying functions as
maximal profit
©GoldSim Technology Group LLC., 2012
Flowing “Media”
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The quantity of liquids and/or solids that flow from one discrete location
to another.
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Examples: water, CO2, rocks, sediment in water.
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Assume incompressible and volume is additive, taking porosity of any
solid media into account.
– For example, 1 gallon of water dumped into a tank containing 1
gallon of sediment whose porosity is 0.3 would consume a total
volume of 1.7 gallons (1 gal of Water + (1 – 0.3)*1 gal of Sediment).
©GoldSim Technology Group LLC., 2012
Cells (Any Flow Network Elements)
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Model elements that produce, consume, store, or
route fluid.
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Examples:
– Pump
– Evaporation
– Detention pond
– A city
– Stockpile
Source
©GoldSim Technology Group LLC., 2012
Router
Store
Sink
User_Spec_Media
Flows (Influence Lines)
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Flow links transport fluid from one cell to
another.
We denote the value of a flow with 𝑞𝑗 .
Units of media volume (or mass) per unit time
(e.g., gal/day, kg/sec)
Examples:
– A connection from one stretch of river (a
reach) to another.
– A pipe leading from a lake to a farm
– Deliveries to a customer
©GoldSim Technology Group LLC., 2012
Flow, 𝑞1
Flow Capacity and Costs

In most cases, a flux has a maximum
capacity 𝑞𝑗,𝑚𝑎𝑥 , so we have constraints of the
form:
𝑞𝑗 ≤ 𝑞𝑗,𝑚𝑎𝑥

Sometimes it costs money to transport fluid
along a particular flux.
– This affects the net profit.
©GoldSim Technology Group LLC., 2012
0 ≤ 𝑞1 ≤ 𝑞𝑚𝑎𝑥
Source Cells
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A source cell feeds fluid into the system.
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Source cells have infinite supply, but their
outflow rate(s) may be limited.
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Examples:
– Rainfall in a particular geographic area
– CO2 from a power plant
– Sediments from erosion
©GoldSim Technology Group LLC., 2012
Sink Cells
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A sink cell removes fluid from the system.
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The capacity of sink to absorb fluid is infinite,
but the inflow rate may be limited.
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Examples:
– Evaporation
– Outflow from a river (model boundary)
– Consumers
©GoldSim Technology Group LLC., 2012
Zero-Volume Cells (Routers)
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Fixed-volume cells have no ability to store fluid, so
their net inflow rate must match their net outflow
rate:
𝑑𝑉
=
𝑑𝑡
𝑞𝑖𝑛 −
𝑞𝑜𝑢𝑡 = 0

In other words, they must have flow balance.

In this example, the router would impose the
constraint: 𝑞1 − 𝑞2 − 𝑞3 = 0.

Specified media cells are old-style cells that are
used to upgrade old CT models.
©GoldSim Technology Group LLC., 2012
𝑞1
𝑞2
𝑞3
User-Specified Cells
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Cells found in current GoldSim version (CT
module)
Distinct from Routers, which have zero volume
Implications on CT models (need volume for
concentration to make sense)
©GoldSim Technology Group LLC., 2012
Dynamic Volume Cells (Stores)
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The rate of change of fluid volume in that cell is equal to
the sum of all inflows minus the sum of outflows:
𝑑𝑉
=
𝑑𝑡

𝑞𝑜𝑢𝑡
If a dynamic cell is empty, outflow must be <= inflow:
𝑑𝑉
𝑑𝑡

𝑞𝑖𝑛 −
=
𝑞𝑖𝑛 −
𝑞𝑜𝑢𝑡 ≥ 0 (nondecreasing volume)
If the cell is full, inflow <= outflow:
𝑑𝑉
𝑑𝑡
=
𝑞𝑖𝑛 −
©GoldSim Technology Group LLC., 2012
𝑞𝑜𝑢𝑡 ≤ 0 (nonincreasing volume)
Store Cells Attributes
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Demand Priorities and Revenues
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Some cells (cities, farms) will pay for water.
– This counts as revenue in the operator’s profit function.
Some users have priorities (water rights)
– Priorities converted to costs in the solver
For each flux 𝑞𝑗 , its net benefit 𝑐𝑗 is:
𝑐𝑗 = (revenue due to 𝑞𝑗 ) – (cost of 𝑞𝑗 ).
©GoldSim Technology Group LLC., 2012
Simple Example…
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