1. O A BD T EC 2 4 4 7 5 7 2 5 3 4 1 1 [2,O] 1 2,3 [4,0] [4,A] 4 [9,A] [7,B] [8,C] 5 [8,B] [8,E] 6 [13,D] [14,E] SHORTEST PATH – SERADA PARK UNDIRECTED GRAPH

2. ManCo Harvest Blocks and Roads UNDIRECTED GRAPH HW T3 T2 T1T4 T7 T5 T6 T8T9 4.9 6.2 2.0 8.0 2.5 9.0 6.0 2.45.0 8.0 6.0 3.6 4.06.0 SHORTEST PATH FROM HIGHWAY (HW) TO T9 distances in km

3. HW T3 T2 T1T4 T7 T5 T6 T8T9 4.9 6.2 2.0 8.0 2.5 9.0 6.0 2.45.0 8.0 6.0 3.6 4.06.0 1 2 3 4 5 6 7 [2,HW] [5,HW] [8,T2] [10,T2][10.4,T5] [12,T6] [18,T8] [0,-]

4. ManCo Harvest Blocks and Roads SHORTEST PATH - DIRECTED GRAPH HW T3 T2 T1T4 T7 T5 T6 T8T9 4.9 6.2 2.0 8.0 2.5 9.0 6.0 2.45.0 8.0 6.0 3.6 4.06.0 [6.0,T9] [9.6,T8] [10.0,T8] [14.0,T8] [12.0,T7] [14.5,T7] [20.7,T4] [22.5,T4] [19.0,T6] [16.0,T6] [25.7,T1] [21.5,T3] [18.0,T2] [21.0,T3] [18.0,T5] BACKWARDS RECURSSION Choose minimum total distance from predecessors

5. ManCo Harvest Blocks and Roads HW T3 T2 T1T4 T7 T5 T6 T8T9 $73.5K $93K $30K $120K $37.5K $135K $90K $72K$75k $120K $90K $54K $60K$90K Minimum Spanning Tree – Connect all Nodes (blocks) with least road construction All construction = $15K/km, except T5-T7 = $30K/km

6. ManCo Harvest Blocks and Roads HW T3 T2 T1T4 T7 T5 T6 T8T9 $73.5K $93K $30K $120K $37.5K $135K $90K $72K$75k $120K $90K $54K $60K$90K Minimum Spanning Tree – Connect all Nodes (blocks) with least road construction All construction = $15K/km, except T5-T7 = $30K/km 1 2 3 4 5 6 7 8 9 Minimum = $574.5 K

7. ManCo Harvest Blocks and Roads HW T3 T2 T1T4 T7 T5 T6 T8T9 $73.5K $93K $30K $120K $37.5K $135K $90K $72K$75k $120K $90K $54K $60K$90K 1 2 3 4 5 6 7 8 9 Minimum = $574.5 K There maybe alternate minimum spanning trees when we arbitrarily break ties (e.g. T2-T5 v. T2-T6) but total distance is the same. Note you can start at any node and apply the greedy algorithm to get the minimum spanning tree

8. Shortest Path using LP Simplex algorithm works great for network problems Send one unit of flow through the network to determine the shortest path –Flow at Source = 1 –Flow at Destination = -1 –Sum of flow at each intermediate node = 0 Example: ManCo harvest blocks and roads

9. Other Network Problems Steiner Networks – connect a subset of nodes Maximum flow through a network Minimum cost flow through a network Transportation problems (Lab #2) Assignment problems (US Air) Transhipment problems (US Steel)

10. Other Network Problems TOURS IN NETWORKS –TRAVELLING SALESMAN (salesperson) VISIT EACH NODE AND RETURN TO THE START AT MINIMUM COST –POSTMAN (postperson) VISIT EACH ARC AND RETURN TO THE START AT MINIMUM COST