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Model 5

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

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Model 5

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

  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 Xij • 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 X12 + X13 < X46 + X56 Test max = X12 + X13 = 13, X12 = 5, X13 = 8 Optimal routing is three segments, eliminate segments that allow for a less than optimal routing 4 2 4 9 5 2 6 1 8 8 3 3 5 6

  5. Constraints (cont.) • X12 = 5 • X13 = 8 • X12 = X24 + X25 • X13 = X34 + X35 • X24 + X34 = X46 • X24 + X35 = X56

  6. Methods • Put constraints in a matrix and performed row reduction • Found X46 + X56 = 13 • X35 & X56 free variables • To find optimal routing pick values for free variables subject to the constraints of the system • e.g. Pick X35 = 6, X56 = 7 • Implies X24 = 4, X34 = 2, X25 = 1, X46 = 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 X35 and X56 subject to the system constraints

  8. Questions…?

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