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Traffic Cost Minimisation

Problem: Minimising shipping costs on a network whose routes have costs that vary with the amount of traffic.

In the SHIPPING network problem, the cost to transport your product from a supply point to a demand point was fixed. However, in some network problems, the costs vary with the amount transported along each arc. If you've ever driven to or from a major city during rush hour, you've experienced this phenomenon. As the number of cars on the road increases, the cost, in terms of time required, of getting from point A to point B increases. In many cases, the cost does not increase linearly. For example, doubling the traffic on a lightly travelled road may not double the travel time, but doubling the traffic again may effectively grind the flow of vehicles nearly to a halt.

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As Quartermaster of a military base, you need to distribute uniforms (all size 48, Extra Large) from three warehouses to four intake centers within the base.

You know that the time required to go from a particular warehouse to a particular unit obeys the following formula:

Time = Rate * Flow / ( 1 - Flow / Limit )

Where: 
Rate = time required to transport one unit if there is no congestion along this route
Flow = amount of product moving along this route
Limit = the maximum amount that can be moved along this route

You also know the rates and limits for each route, or arc, in the network.

The objective is to ship all the uniforms to the Intake Centers at minimum cost, while satisfying demand at each center.

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