Import Books, Incorporated (IBI)11 stocks several thousand titles in its main warehouse.

The titles can be categorized by sales volume, with i = 1 requiring a stored inventory of 0 to 20 books, i = 2 requiring 21 to 40, i = 3 requiring 41 to 100, and i = 4 requiring 101 to 200. The number of titles in category i is bi . Each title is stored in a separate bin, and each bin has at most one title. IBI has 500 bins with space for up to 100 books and 2000 larger ones with space for up to 200 books. The bins for 100 can also be subdivided to create either two bins for 40 books or three bins for 20. Costs ci, j for storing a category i title in a size j bin have been estimated by accounting for the material handling cost of accessing the bin and the wasted space if a bin is underutilized.

Here j = 1 refers to bins for 20, j = 2 to bins for 40, j = 3 to bins for 100, and j = 4 to bins for 200. Formulate an LP model to find a minimum cost allocation of titles to bins using the decision variable 1i = 1,c, 4; j = i,c, 42 xi, j ! number of titles of category i allocated to of size j