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?

Question

What will be an ideal response?

Answer 1

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.