Shortest Path Problems

Modeling and Applications

Find a path with the lowest radiation

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

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

1

1

(3)

(5)

(2)

(1)(2)

(2)(7)

(8)

(3)

(2)

(1)

costs

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

Power Transmission Problem

1

2 4

3 5

6

4%

3%

2 %

3 %

2 %

3 %

2 %

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 1246, the final transmission efficiency received at node 6 = (1-4%)*(1-3%)*(1-2%)=96%*97%*98%=91.258%.

Percentage loss

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).

Max. Efficiency Power Transmission

• Convert the % lost into efficiency first as below.

1

2 4

3 5

6

96%

97%

98 %

97 %

98 %

97 %

98 %

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

1

2 4

3 5

6

-log(96%)

-log(97%)

-log(98%)

-log(97%)

-log(98%)

-log(97%)

-log(98%)

Solution Analysis

• Possible path 1: 1246– {[-log(96%)]+[-log(97%)]+[-log(98%)]}=0.0397– Efficiency = 91.258%

• Possible path 2: 1256– {[-log(96%)]+[-log(98%)]+[-log(98%)]}=0.0353– Efficiency = 92.198%

• Optimal path: 1356– {[-log(97%)]+[-log(97%)]+[-log(98%)]}=0.0352

(Min.)– Efficiency = 92.208% (Max.)

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.

On-call Driver Schedule for a Bus Co.

• 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

Duty Hour 9am-1pm

9am-12pm

12nn-3pm

12nn-5pm

2pm-5pm

1pm-4pm

4pm-5pm

Cost (HKD) 300 260 210 450 200 260 160

Shortest Path Model

9 11 12 1 210 3 4 5

Model: each node corresponds to a time point, each arc to a possible shiftJustification:

Each path from 9 to 5 corresponds to a feasible schedule, e.g., path 9145 means a schedule (9am-1pm)+(1pm-4pm)+(4pm-5pm)

A feasible schedule with overlapped shifts: (9-12)+(12-3)+(2-5) ??? represented by path 912325

Time

9 11 12 1 210 3 4 5

260

30 260200 160

450

300

210

00000000

Schedule 1: 9am1pm,1pm4pm,4pm5pm, cost=HKD720Schedule 2: 9am1pm,12nn5pm, cost=HKD750Schedule 3: 9am1pm,1pm4pm, 2pm5pm, cost=HKD760 Which is the shortest path?

Time

Duty Hour 9am-1pm

9am-12pm

12nn-3pm

12nn-5pm

2pm-5pm

1pm-4pm

4pm-5pm

Cost (HKD) 300 260 210 450 200 260 160

Dynamic Shortest PathsSuppose 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?

7 7:10 7:20 7:30 7:40 7:50 …

(1,2) 20 30 30 20 … … …

(1,3) 10 10 10 10 … … …

(2,3) 20 20 20 20 … … …

(3,4) 10 20 20 10 … … …

…

Start time

arc

travel time in minutes

Time expanded network (time-space network)

(1,2) 10 10 20 20 … …

(1,3) 10 10 10 10 … …

(2,4) 10 20 20 20 … …

(3,4) 10 10 30 30 … …

7 7:10 7:20 7:30 7:40 7:50

The shortest path from 1 to 4 depends on when to start.

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

2

3

4

2

3

Time T

…

…

…

4

Find a path with the lowest radiationModeling:(1)Partitioning the space as a k*k grid

with desired accuracy

(2)Between any two points, calculating the total radiation received, assuming direct walking

arc costs

(3)Finding the shortest path