Transcript A Heuristic Solution Method to a Stochastic Vehicle

A Heuristic Solution Method for a
Stochastic Vehicle Routing Problem
Lars Magnus Hvattum
Bekkjarvik, 04.12.2003
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Outline of the Presentation
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The Real-World Problem
Vehicle Routing Concepts
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VRP
Dynamic VRP
Stochastic VRP
Our Problem
Solution Approach
Behaviour
Results
Conclusions
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The Real-World Problem
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Linjegods A/S, Heimdal
Pick-up customers
133 customers per day (on average)
About half of the customers are known in advance
The rest is revealed during execution
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Unknown location, demand, time-window
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Vehicle Routing Concepts (1)
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The Vehicle Routing Problem (VRP)
A set of customers and a central depot
A set of vehicles, located at the depot
Design minimum cost routes visiting all
customers
Additional constraints
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Capacity
Time windows
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Vehicle Routing Concepts (2)
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Dynamic VRPs
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A problem is dynamic when inputs to the
problem are made known or updated to the
decision maker concurrently with the
determination of the solution (Psaraftis, 1995)
No plan is generated a priori
Events are handled as they are revealed
over time
Need a policy for how the routes should
evolve in time as a function of the inputs
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Vehicle Routing Concepts (3)
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Stochastic VRPs
Some elements of the problem are
stochastic
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?
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Travel times, demands, customers, ...
Typically formulated as a two-stage
stochastic programming problem
The solution is an a priori plan
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Recourse actions
SVRPs are usually static problems!
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Our Problem
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A mix of stochastic and dynamic VRPs
Divides the time horizon (e.g. 08:00 to 16:00) into m
time slots (stages)
Make a plan at the beginning of each time slot for
how to service the currently known customers
May change the plan at the beginning of the next
time slot (recourse)
Information is received dynamically
Modelled as a multi-stage stochastic programming
problem with recourse
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Solution Approach (1)
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Must create a plan for the rest of the day at the start
of each stage (time slot)
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Attempt 1 (pure dynamic)
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Ignore stochastic information
Solve a static VRP based on the currently known
information using a heuristic local search
Would produce good solutions if new customers do not
appear
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Solution Approach (2)
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Attempt 2 (sampling based)
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Exploit stochastic information by use of sampling
Create possible future scenarios based on the distribution
of the random variables (customer locations, demands,
time windows, call-in time...)
Solve the set of sample scenarios (static VRPs) by using
quick heuristics
Search for common features among the sample scenario
solutions, and implement these in the final plan (iteratively,
based on ideas from progressive hedging)
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Solution Approach (3)
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The main loop is:
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Solution Approach (4)
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The SSBHH sub-procedure is:
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Behaviour – an example
Pure dynamic
08:00
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Sample based
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Behaviour – an example
Pure dynamic
09:00
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Sample based
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Behaviour – an example
Pure dynamic
10:00
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Sample based
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Behaviour – an example
Pure dynamic
11:00
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Sample based
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Behaviour – an example
Pure dynamic
12:00
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Sample based
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Behaviour – an example
Pure dynamic
13:00
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Sample based
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Behaviour – an example
Pure dynamic
14:00
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Sample based
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Behaviour – an example
Pure dynamic
15:00
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Sample based
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Behaviour – an example
Pure dynamic
End of day
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Sample based
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Results
Deterministic
Pure dynamic
SSBHH
Total
No. of
Total
No. of
Total
No. of Reduction of
Problem #Orders distance vehicles distance vehicles distance vehicles distance in %
1
142
147238
6
215030
6
205968
6
4.2%
2
126
111240
4
164724
4
122974
4
25.3%
3
147
144471
5
211902
5
158081
5
25.4%
4
118
122244
4
216632
4
141643
5
34.6%
5
126
134448
5
205136
5
158022
5
23.0%
6
118
113658
4
167514
4
133895
4
20.1%
7
128
124070
5
178059
5
146204
6
17.9%
8
129
149118
5
237306
5
187675
6
20.9%
9
117
97383
3
165345
3
105791
3
36.0%
10
127
138753
5
182529
5
168150
6
7.9%
11
146
125659
4
211174
4
139488
4
33.9%
12
151
121650
4
194143
4
136249
4
29.8%
13
138
104251
3
190020
3
118980
4
37.4%
14
126
122230
4
198243
4
137376
4
30.7%
15
139
128013
5
214064
5
153354
5
28.4%
16
122
117193
4
190670
4
128604
4
32.6%
17
125
143836
5
196545
5
146684
5
25.4%
18
143
155267
6
227041
6
185113
6
18.5%
19
144
138963
5
218846
5
155686
5
28.9%
20
139
143309
5
207547
5
167465
5
19.3%
Avg.
132.55 129150
4.55
199624 4.55
149870 4.80
24.9%
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Conclusions
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Have formulated a problem as a mix between a
stochastic and a dynamic VRP
Presented a Sample Scenario Based Hedging
Heuristic (SSBHH)
Have shown that taking stocastic information into
account can improve solution quality
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