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Basic problem: Given a) set of limited resources, e.g., seats on airplane flights, hotel rooms, rental cars over several days, and b) different types of customers needing different combinations of our resources, and c) each customer type having an amount they are willing to pay for their needed combination, what volumes should we sell of each product bundle to each customer class?
In practice, the dual or shadow prices on the limited resources in the solution may be as useful as the allocations to customers. The prices can be used to "price out" a new customer request for a particular combination of resources to decide if it is attractive relative to other opportunities to sell or rent our resources. Specific example: Consider an airline having a set of three flights, A to C, B to C, and C to D to serve four cities, A, B, C, and D, in the network:
  A
  \
   C - D
  /
  B
The airline uses a two-fare pricing structure. The decision of how many seats or capacity to allocate to each price class is sometimes called yield or revenue management. The respective flight capacities are 100, 110, and 120. How many seats should be allocated to each class on each of the three flights to each of the 5 itineraries? In the data below, for a given itinerary, a first class customer always gives more revenue than a second class customer. Thus, a plausible rule would be to first satisfy first class as much as possible, and then fill any remaining seats with second class. Will this lead to the most profitable solution?