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Swap-Body VRP Cost

Vehicle Cost: Cost of renting the vehicle for use

Distance: A cost / kilometer for use of the vehicle which is greater for a larger vehicle (gas costs)

Service Time

Cust.

Swap Location

Swap Action Times

Time: An hourly wage associated with driver cost

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Swap-Body VRP Swap Actions / Subtours

At a designated Swap Location, there are four possible actions, each with an associated performance time:

Park: The entering train parks its trailing swap-body, leaving it at the swap location for pickup at a later time

Swap Location

Figure 3: Park

Figure 4: Pickup

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Swap-Body VRP Swap Actions / Subtours

At a designated Swap Location, there are four possible actions, each with an associated performance time:

Pickup: The entering truck reattaches to the previously parked swap-body, picking it up to continue the tour

Swap Location

Figure 3: Park

Figure 4: Pickup

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Swap-Body VRP Swap Actions / Subtours

Swap: The incoming truck changes the position of its swap-bodies, and then leaves, still a truck, with a different quantity of goods, (joining two consecutive subtours)

Swap Location

Figure 5: Swap

Figure 6: Exchange

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Swap-Body VRP Swap Actions / Subtours

Exchange: The incoming train changes the position of its swap-bodies, parks its trailing swap-body, and leaves a truck

Swap Location

Figure 5: Swap

Figure 6: Exchange

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Swap-Body VRP Swap Actions / Subtours

A swap-body belongs to its vehicle, and cannot be abandoned. Therefore, it must eventually return to pick up a dropped off swap-body.

Cust.

Swap Location

Cust.

…

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Swap-Body VRP Swap Actions / Subtours

A swap-body belongs to its vehicle, and cannot be abandoned. Therefore, it must eventually return to pickup a dropped off swap-body

Cust.

Swap Location

Cust.

…

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Swap-Body VRP Swap Actions / Subtours

A swap-body belongs to its vehicle, and cannot be abandoned. Therefore, it must eventually return to pick up a dropped off swap-body

Cust.

Swap Location

Cust.

…

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Swap-Body VRP Swap Actions / Subtours

A train may choose to serve a customer from a swap-body (or partially from both), but there is no load transfer allowed

Cust.

Demand: 5

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Swap-Body VRP Swap Actions / Subtours

Cust.

Demand: 5

A train may choose to serve a customer from a swap-body (or partially from both), but there is no load transfer allowed

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Swap-Body VRP Swap Actions / Subtours

Subtour Demand: 7

Swap Location

A train may choose to serve a customer from a swap-body (or partially from both), but there is no load transfer allowed

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Swap-Body VRP Swap Actions / Subtours

A simple procedure for determining the feasibility of a route’s subtours:

Check whether any one subtour exceeds demand, (Q), or the sum of the entire route’s demand exceeds 2Q
If both are not violated, use a simple dynamic programming approach to construct the closest possible solution to the partition problem (where the elements of the set correspond to the demands of the subtours). If this produces two partitions with sums P₁ and P₂ , then check P₁ < Q and P₂ < Q

Subtour demands: {2, 5, 6, 3, 2}, Q = 10

Swap-Body 2

Swap-Body 1

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Swap-Body VRP Swap Actions / Subtours

After a route is determined, its optimal orientation can be found by solving the main tour orientation separately from the subtour
Because an exchange is the most costly operation, we minimize the number of exchanges, which means minimizing the number of transitions in a binary representation of the subtour swap-body string:

Suppose the string was {1,1,0,0,1}
This corresponds to {exchange, no action, exchange, no action, exchange}
First rule: serve first subtour from swap-body 1 (can always flip and maintain feasibility)
Idea: exchange swap-bodies that serve subtours to get consecutive subtours on the same swap-body

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Swap-Body VRP Example

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Heuristic

Idea: try to repurpose and reuse existing code as much as possible, (in this case, COIN-OR’s VRPH Code from Chris Groër)

Flexible for drop-in solver, (i.e., any capacitated vehicle routing problem solver may be used for initialization)
When compared to heuristics specifically designed for the SB-VRP and/or exact solvers, solution quality can help quantify the ‘difficulty’ of the variant (the gap is a proxy for the difference in structure)
Cuts down development time

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Heuristic: Two Stages

We approach the problem using a two-stage heuristic:

In the first stage, we ignore subtours entirely, and model as many elements of the problem as a traditional VRP, (homogeneous fleet, with service times), that allows us to use any VRP solver to obtain a feasible initial solution.

In the second stage, we apply improvement procedures, some of which are designed to introduce subtours into the solution, and some of which are slightly modified versions of VRP local search operators.

Stage 1

Stage 2

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Heuristic: Stage 1

(Asymmetric) VRP Model

If there are truck-only customers, fleet only contains trucks
Service time added to the depot to reflect the vehicle cost

Weights in the graph are according to time (since a max route duration exists)

Used codes in the VRPH library, with minor modifications to solve the VRP model

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Heuristic: Stage 2

Improvement Procedures

2-Swap between main tours.
2-Opt on a main tour.
Distance-constrained evacuation.
Distance-constrained exchange.
Truck-only customer migration.

Perturbation: Exchange several random strings of nodes between main tours.

2-Swap

2-Opt

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Heuristic: Stage 2

Improvement Procedures

2-Swap between main tours.
2-Opt on a main tour.
Distance-constrained evacuation.
Distance-constrained exchange.
Truck-only customer migration.

Perturbation: Exchange several random strings of nodes between main tours.

Distance-constrained evacuation

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Heuristic: Stage 2

Improvement Procedures

2-Swap between main tours.
2-Opt on a main tour.
Distance-constrained evacuation.
Distance-constrained exchange.
Truck-only customer migration.

Perturbation: Exchange several random strings of nodes between main tours.

Distance-constrained exchange

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Heuristic: Stage 2

Improvement Procedures

2-Swap between main tours.
2-Opt on a main tour.
Distance-constrained evacuation.
Distance-constrained exchange.
Truck-only customer migration.

Perturbation: Exchange several random strings of nodes between main tours.

Truck-only customer migration

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Test Instances

VeRoLog 2014 Solver Challenge

27 teams
Common set of test data

9 public instances

3 small, 3 medium, and 3 large instances
For each size, the three instances were identical, except for the restrictions on which vehicle types could visit the customers

In one instance, all customers could be reached by all vehicle types
In one instance, all customers could only be reached by trucks
In one instance, some customers could be reached by trains, and some could not

3 hidden instances

Teams provided only with their performance on these instances

Platform

Algorithm written in C++
Core i7 2.8 GHz (2009), 8 GB RAM PC

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Computational Results

Instance	n_SL	n_C	VRPH	Improvement
Small	20	57	5452,99	4959
Small (Trailer)	20	57	4930,06	4868,67
Small (Truck)	20	57	5357,03	5356,36
Medium	41	206	8735,81	8293,06
Medium (Trailer)	41	206	8434,87	8330,78
Medium (Truck)	41	206	8698,13	8614,46
Large	99	548	22851,1	21463,2
Large (Trailer)	99	548	21160,5	21020,3
Large (Truck)	99	548	22593,4	22468,7
New	NA	NA	NA	26712,4
New (Trailer)	NA	NA	NA	26658,1
New (Truck)	NA	NA	NA	26712,4

NA = Not Available

n_SL = number of swap locations

n_C = number of customers

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Conclusions

Solving the SB-VRP as VRP gives good performance (outperformed initial clustering approach, and two-stage approach to subtour formation)
VRPH leverages existing solvers, but the subtour structure seems suboptimal
Parts of the swap-body VRP can be simplified (e.g., subtour feasibility, separate fixed vehicle costs and time costs).
Most difficult aspect is understanding the interaction between subtours and neighboring routes

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References

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References

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References

VeRoLog, 2014. http://verolog.deis.unibo.it/news-events/general-news /verolog-solver-challenge-2014.
VeRoLog, 2014. http://verolog.deis.unibo.it/sites/verolog.deis.unibo.it/files /VSC2014 %20Problem.pdf.
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