1. Shortest Path Problems
Modeling and Applications

2. Find a path with the lowest radiation

3. Find a path with the lowest radiation (A nontrivial example)

4. Shortest Path Problem A general network structure
A given node as the source, and another given node as destination
Each arc has a cost
Decision: find a path from the source to the destination with the minimum total cost
The cost of a path is the sum of costs on all arcs on the path 4 1 5 3 6 7 2
Source
Destination
(3)
(5)
(2)
(1)
(7)
(8)
costs

5. Shortest Path Modeling
Given a particular application (not obviously like a shortest path problem)
We construct a shortest path problem
To define a network (nodes, arcs, and arc costs)
Solving the shortest path problem gives the optimal solution of the application
Assuming we know how to solve a shortest path problem

6. Power Transmission Problem3%
Percentage loss 2 4
2 % 4%
2 % 1 6
3 %
2 %
3 % 3 5
Transmission efficiency = 1 - percentage lost
e.g., from node 1 to 3, efficiency = 1-3%=97%
Final transmission efficiency received at the destination = multiplication of efficiency along the path.
For example, for path 1246, the final transmission efficiency received at node 6
= (1-4%)*(1-3%)*(1-2%)=96%*97%*98%=91.258%.

7. Relating Power Transmission to Shortest Path
Differences
Power Transmission
Shortest Path
Path Evaluation
Multiplication of arc efficiency
Summation of arc cost
Objective
A maximization problem
A minimization problem
Observations:
For some positive numbers a1,…,an and b1,…,bm, inequality
a1×…×an ≥ b1×…×bm
is equivalent to
log(a1×…×an) ≥ log(b1×…×bm),
further equivalent to
log(a1)+…+log(an) ≥ log(b1)+…+log(bm),
and finally equivalent to
-log(a1)-…-log(an) ≤ -log(b1)-…-log(bm).

8. Max. Efficiency Power Transmission
Convert the % lost into efficiency first as below.
97% 2 4
98 %
96%
98 % 1 6
97 %
98 %
97 % 3 5

9. Shortest Path Modeling
In the network, define the cost of each arc as log(efficiency) as below.
Now we can find the shortest path under the given costs
-log(97%) 2 4
-log(98%)
-log(96%)
-log(98%) 1 6
-log(97%)
-log(98%)
-log(97%) 3 5

10. Solution Analysis Possible path 1: 1246 Possible path 2: 1256
{[-log(96%)]+[-log(97%)]+[-log(98%)]}=0.0397
Efficiency = %
Possible path 2: 1256
{[-log(96%)]+[-log(98%)]+[-log(98%)]}=0.0353
Efficiency = %
Optimal path: 1356
{[-log(97%)]+[-log(97%)]+[-log(98%)]}= (Min.)
Efficiency = % (Max.)

11. Another similar example
A farmer wishes to transport a truckload of eggs from one city to another city through a given road network.
The truck will incur a certain amount of breakage on each road segment
Let wij be the percentage of eggs broken if the truck passes the road segment (i,j).
How should the truck be routed to minimize the total breakage?
Formulate the problem as a shortest path problem.

12. On-call Driver Schedule for a Bus Co.
Duty Hour
9am-1pm
9am-12pm
12nn-3pm
12nn-5pm
2pm-5pm
1pm-4pm
4pm-5pm
Cost (HKD)
300
260
210
450
200
160
Available on-call driver shifts and costs in the above table
Requirement: At least one on-call driver is on duty any time from 9am to 5pm.
Question: the minimum-cost schedule

13. Shortest Path Model Time9 10 11 12 1 2 3 4 5
Model: each node corresponds to a time point, each arc to a possible shift
Justification:
Each path from 9 to 5 corresponds to a feasible schedule, e.g.,
path 9145 means a schedule (9am-1pm)+(1pm-4pm)+(4pm-5pm)
A feasible schedule with overlapped shifts: (9-12)+(12-3)+(2-5) ???
represented by path 912325

14. Schedule 1: 9am1pm,1pm4pm,4pm5pm, cost=HKD720
Duty Hour
9am-1pm
9am-12pm
12nn-3pm
12nn-5pm
2pm-5pm
1pm-4pm
4pm-5pm
Cost (HKD)
300
260
210
450
200
160 30
300
260
200
160
260
Time 9 10 11 12 1 2 3 4 5
450
210
Schedule 1: 9am1pm,1pm4pm,4pm5pm, cost=HKD720
Schedule 2: 9am1pm,12nn5pm, cost=HKD750
Schedule 3: 9am1pm,1pm4pm, 2pm5pm, cost=HKD760
Which is the shortest path?

15. Dynamic Shortest Paths
Suppose that the time it takes to travel in arc (i, j) depends on when one starts. (e.g., rush hour vs. other hours in road networks.) Let cij(t) be the time it takes to travel in (i, j) starting at time t. What is the minimum time it takes to travel from node 1 to node n starting at 7:00 AM?
Start time
arc
travel time in minutes 7
7:10
7:20
7:30
7:40
7:50 …
(1,2) 20 30
(1,3) 10
(2,3)
(3,4)

16. Time expanded network (time-space network)
(1,2) 10 20 …
(1,3)
(2,4)
(3,4) 30 7
7:10
7:20
7:30
7:40
7:50 … 1 2 3 4
The shortest path from 1 to 4 depends on when to start. … …
Time T 4

17. Find a path with the lowest radiation
Modeling:
Partitioning the space as a k*k grid
with desired accuracy
Between any two points, calculating the total radiation received, assuming direct walking
arc costs
Finding the shortest path