1. Model 5 Long Distance Phone Calls By Benjamin Cutting 1 35 2 6 4 5 8 3 4 6 9 8 5 2 3

2. The Problem Find the maximum number of long distance calls between Jonesville & Smithsboro (nodes 1 & 6) that the system can handle at any one time Find the optimal routing of these calls

3. Constraints & Assumptions Call each line X ij Each line can hold at most the number of calls it is assigned in the given diagram Calls can only go one direction on the line Assume all calls in the system are between Jonesville & Smithsboro The amount of calls going into a given node equals the number going out The total of the calls leaving Jonesville is equal to the total of the calls arriving at Smithsboro

4. Constraints & Assumptions (cont.) Compared input vs. output capacity X 12 + X 13 < X 46 + X 56 Test max = X 12 + X 13 = 13, X 12 = 5, X 13 = 8 Optimal routing is three segments, eliminate segments that allow for a less than optimal routing 1 35 2 6 4 5 8 4 6 9 8 2 3

5. Constraints (cont.) X 12 = 5 X 13 = 8 X 12 = X 24 + X 25 X 13 = X 34 + X 35 X 24 + X 34 = X 46 X 24 + X 35 = X 56

6. Methods Put constraints in a matrix and performed row reduction Found X 46 + X 56 = 13 X 35 & X 56 free variables To find optimal routing pick values for free variables subject to the constraints of the system e.g. Pick X 35 = 6, X 56 = 7 ◦ Implies X 24 = 4, X 34 = 2, X 25 = 1, X 46 = 6 ◦ With a max call capacity of 13 and every call being routed through 3 segments

7. Conclusions The max call capacity of the system is 13 The shortest routing route is three segments (Max is four segments) All 13 calls can be handled by only three segments There are several (4) optimal routing routes, determined by picking values for X 35 and X 56 subject to the system constraints

8. Questions…?