{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# Ch08 - 1 MP CHAPTER 8 SOLUTIONS SECTION 8.2 1 First label...

This preview shows pages 1–4. Sign up to view the full content.

MP CHAPTER 8 SOLUTIONS SECTION 8.2 1. First label node 1 with a permanent label: [0* 7 12 21 31 44] Now node 2 receives a permanent label [0* 7* 12 21 31 44]. Node Temporary Label (* denotes next assigned permanent label) 3 min{12,7+7} = 12* 4 min{21,7+12} = 19 5 min{31,7+21} = 28 6 min{44,7+31} = 38 Now labels are [0* 7* 12* 19 28 38] Node Temporary Label (* denotes next assigned permanent label) 4 min{19,12+7} = 19* 5 min{28,12+12} = 24 6 min{38,12+21} = 33 Now labels are [0* 7* 12* 19* 24 33] Node Temporary Label (* denotes next assigned permanent label) 5 min{24,19+7} = 24* 6 min{33, 19+12} = 31 Now labels are [0* 7* 12* 19* 24* 31] Node Temporary Label (* denotes next assigned permanent label) 6 min{31,24+7} = 31 Now labels are [0* 7* 12* 19* 24* 31*] 31 -24 = c 56 , 24 - 12 = c 35 , 12 - 0 = c 13 . Thus 1-3-6 is the shortest path (of length 31) from node 1 to node 6. 2. We begin by permanently labeling node 1 and assigning temporary labels to other nodes: [0* 2 8 ]. Then we give node 2 a permanent label: [0* 2* 8 ] Node Temporary Label (* denotes next assigned permanent 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
label) 3 min{8,2+5} = 7 4 min{ , 2+4} = 6* 5 min{ ,2+12} = 14 Now labels are [0* 2* 7 6* 14] Node Temporary Label (* denotes next assigned permanent label) 3 min{7} = 7* 5 min{14,6+10} = 14 Now labels are [0* 2* 7* 6* 14]. Since there is no node joining the newest permanently labelled node (node 3) to node 5, we may give make node 5 a permanent label. We obtain [0* 2* 7* 6* 14*]. Since c 25 = 14 - 2 and c 12 = 2 - 0 we find the shortest path from node 1 to 5 to be 1-2-5(length 14). 3. Node 2 Node 3 Node 4 Node 5 Supply Node 1 Node 2 Node 3 Node 4 Demand 4. We begin by giving node 1 a permanent label [0* 2 1 ]. Next node 3 obtains a permanent label: [0* 2 1* ]. Node 4 now obtains a new temporary label of min{ ,1+1} = 2, and node 2 obtains a permanent label yielding [0* 2* 1* 2]. Finally node 4's temporary label becomes permanent and we obtain [0* 2* 1* 2*] which yields the shortest" path 1-3-4. Of course, 1-2-3-4 with length 1 is a shorter path, but we fail to find this because Dijkstra's method assumes that because node 3 is not 2 8 M M 1 0 5 4 12 1 M 0 5 M 1 M M 0 10 1 1 1 1 1 2
connected to node 1 by an arc, node 3 cannot be the closest node to node 1. 5. Node 1 = beginning of year 1, Node 7 = end of year 6 or beginning of year 7 c 12 = 3300, c 13 = 4800, c 14 = 7600, c 15 = 9800, c 16 = 12,400, c 17 = 15,600, c 23 = 3300, c 24 = 4800, c 25 = 7600, c 26 = 9800, c 27 = 12,400, c 34 = 3300, c 35 = 4800, c 36 = 7600, c 37 = 9800, c 45 = 3300, c 46 = 4800, c 47 = 7600, c 56 = 3300, c 57 = 4800, c 67 = 3300 We begin by giving node 1 a permanent label [0* 3300 4800 7600 9800 12,400 15,600]. Next we give node 2 a permanent label: [0* 3300* 4800 7600 9800 12,400 15,600] Node Temporary Label (* denotes next assigned permanent label) 3 min{4800,3300+3300} = 4800* 4 min{7600,4800+3300} = 7600 5 min{9800,7600+3300} = 9800 6 min{12,400,9800+3300} = 12,400 7 min{15,600,12,400+3300} = 15,600 We now make node 3's label permanent and obtain [0* 3300* 4800* 7600 9800 12,400 15,600]. Node Temporary Label (* denotes next assigned permanent label) 4 min{7600,4800+3300} = 7600* 5 min{9800,4800+4800} = 9600 6 min{12,400,4800+7600} = 12,400 7 min{15,600,4800+ 9800} = 14,600 We now make node 4's label permanent and obtain [0* 3300* 4800* 7600* 9600 12,400 14,600]. Node Temporary Label (* denotes next assigned permanent label) 5 min{9600,7600+3300} = 9600* 6 min{12,400,7600+4800} = 12,400 7 min{14,600,7600+7600} = 14,600 We next give node 5 a permanent label and obtain [0* 3300* 4800* 7600* 9600* 12,400 14,600].

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 66

Ch08 - 1 MP CHAPTER 8 SOLUTIONS SECTION 8.2 1 First label...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online