A transportation model uses at least 10 sources and 100 destinations. If the ratio of sources to destinations remains constant, does the maximum % of cells used by the optimum solution remain constant? Why or why not?

What will be an ideal response?

Students should approach this problem in one of two ways. The first is to simply check values such as 10 - 100 and 20 - 200. The % of cells for each of these are 10.9% and 5.5%. Thus the maximum % of cells used by the optimum solution does not remain constant. For a complete answer students should solve for the % of cells used and analyze this function. Setting X as the number of sources the number of destinations becomes 10X. Thus the maximum number of cells used is 11X - 1 while the maximum number of cells is 10X^2. Thus the % utilization is 100 ∗ (11X - 1 )/(10X^2 ) or . Since this function is not constant for values of X greater than 10 (at least 10 sources from the given information) the maximum % of cells involved in the optimum solution is not constant for a constant ratio of sources to destinations. Students may also notice that the limit of the function is 0 as X approaches infinity, yielding insight that as the number of sources and destinations increase, even when increases are proportional, the maximum number of cells used for the optimum solution declines.