# Consider the following word game. You are assigned a number of tiles, each containing a letter a,…

Consider the following word game. You are assigned a number of tiles, each containing a letter a, b, . . . , or z from the alphabet. For any letter α from the alphabet, your assignment includes Nα (a nonnegative integer) tiles with the letter α. From the letters you are to construct any of the words w1, w2, . . . , wn. This list might, for example, contain all words from a given dictionary.

You may construct any word at most once, and use any tile at most once. You receive vj ≥ 0 points for making word wj and an additional bonus of bij ≥ 0 points for making both words wi and wj (i = 1, 2, …, n; j = 1, 2, …, n)

a) Formulate a linear integer program to determine your optimal choice of words.

b) How does the formulation change if you are allowed to select 100 tiles with no restriction on your choice of letters?

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