Model 5Long Distance Phone CallsBy Benjamin Cutting

1

3 5

2

6

45

8

3

4

6

9

85

2

3

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

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

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

1

3 5

2

6

45

8

4

6

9

8

2

3

Constraints (cont.)X12 = 5X13 = 8X12 = X24 + X25

X13 = X34 + X35

X24 + X34 = X46

X24 + X35 = X56

MethodsPut constraints in a matrix and performed

row reductionFound X46 + X56 = 13X35 & 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

ConclusionsThe max call capacity of the system

is 13The shortest routing route is three

segments (Max is four segments)All 13 calls can be handled by only

three segmentsThere are several (4) optimal routing

routes, determined by picking values for X35 and X56 subject to the system constraints

Questions…?