1. Network Flows Based on the book: Introduction to Management Science. Hillier & Hillier. McGraw-Hill

2. 2 Ardavan Asef-Vaziri June-2013Network Flow Problems The Shortest Route Problem The shortest route between two points l ij : The length of the directed arc ij. l ij is a parameter, not a decision variable. It could be the length in term of distance or in terms of time or cost ( the same as c ij ) For those nodes which we are sure that we go from i to j we only have one directed arc from i to j. For those node which we are not sure that we go from i to j or from j to i, we have two directed arcs, one from i to j, the other from j to i. We may have symmetric or asymmetric network. In a symmetric network l ij = l ji for all ij In a asymmetric network this condition does not hold

3. 3 Ardavan Asef-Vaziri June-2013Network Flow Problems Example 6 3 4 2 5 7 2 6 5 6 4 8 7 2 2 1 2 2 2

4. 4 Ardavan Asef-Vaziri June-2013Network Flow Problems Decision Variables and Formulation x ij : The decision variable for the directed arc from node i to nod j. x ij = 1 if arc ij is on the shortest route x ij = 0 if arc ij is not on the shortest route  x ij -  x ji = 0 for all nodes except the origin and destination  x oj =1  x iD = 1 Min Z =  l ij x ij

5. 5 Ardavan Asef-Vaziri June-2013Network Flow Problems Example 6 3 4 2 5 7 2 6 5 6 4 8 7 2 2 1 2 2 2

6. 6 Ardavan Asef-Vaziri June-2013Network Flow Problems Example 6 3 4 2 5 7 2 6 5 6 4 8 7 2 2 1 2 2 2 + x 13 + x 14 + x 12 = 1 - x 57 - x 67 = -1 + x 34 + x 35 - x 43 - x 13 = 0 + x 42 + x 43 + x 45 + x 46 - x 14 - x 24 - x 34 = 0 …. ….. Min Z = + 5x 12 + 4x 13 + 3x 14 + 2x 24 + 6x 26 + 2x 34 + 3x 35 + 2x 43 + 2x 42 + 5x 45 + 4x 46 + 3x 56 + 2x 57 + 3x 65 + 2x 67

7. 7 Ardavan Asef-Vaziri June-2013Network Flow Problems Excel 6 3 4 2 5 7 2 6 5 6 4 8 7 2 2 1 2 2 2

8. 8 Ardavan Asef-Vaziri June-2013Network Flow Problems Excel 6 3 4 2 5 7 2 6 5 6 4 8 7 2 2 1 2 2 2

9. 9 Ardavan Asef-Vaziri June-2013Network Flow Problems Solver Solution 6 3 4 2 5 7 2 6 5 6 4 8 7 2 2 1 2 2 2

10. 10 Ardavan Asef-Vaziri June-2013Network Flow Problems After class practice; Find the shortest route 8 3 4 5 7 10 4 3 5 6 4 5 3 2 2 1 2 6 9 11 6 2 4 3 4 6 6 O D 2 3 6 5 3 2 3 2 2

11. Ardavan Asef-Vaziri June-2013Network Flow Problems Important observations in the LP-relaxation 2 1 3 4 2 6 1 4 3 Formulate on the problem on the black board Did I say x ij <= 1 ? Why all the variables came out less than 1 Did I say x ij 0 or 1 Why all variables came out 0 or 1