In many auctions, the value that a bidder has for a set of items may not be the sum of the values that she has for individual items. It may be more or it may be less.

In many auctions, the value that a bidder has for a set of items may
not be the sum of the values that she has for individual items. It may be more or it may be less. For example, the value of takeoff/landing slot at airport A and time S for an airline depends on availability of takeoff/landing slot at airport B and time T. Other examples include equity trading (where a combination of trades can have higher value than individual ones), electricity markets (where a contract’s value depends on the time of day and season), pollution rights (where combination of quotas can have higher value than the individual ones) and wireless bandwidth (where its value depends on location/frequency). To take this into account, combinatorial auctions allow the bidders to submit bids on combinations of items.
Consider the following example:
You are the auctioneer and your objective is to maximize your revenue. You have 5 different types of product/service (Item). Your current inventory is the following: (Item 1, Item 2, Item 3, Item 4, Item 5)= (3, 1, 2, 5, 3)
Bid # (Item 1, Item 2, Item 3, Item 4, Item 5, Number of Bids)
Bid 1 (1 0 0 0 2 15), Bid 2 (0 1 0 0 2 20), Bid 3 (0 1 0 3 0 30) Bid 4 (2 0 1 0 1 35) Bid 5 (0 1 0 2 1 40) Bid 6 (2 0 2 0 0 32) Bid 7 (1 1 1 1 1 50) Bid 8 (0 0 2 0 1 15) Bid 9 (0 0 0 2 2 33) Bid 10 (2 1 1 1 1 52)
Develop a mixed-integer optimization model whose solution identifies the subset of bids that, if accepted, maximizes the auctioneer’s payoff.
List all inputs (decision variables), parameters, and outputs (objective function).
Express the algebraic relationship between inputs/parameters and constraints/objective function.
Implement your formulation in MS Excel, solve for the optimal solution.

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