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36
A New Generation of Vehicle Routing Research: Robust Algorithms, Addressing Uncertainty
 Operations Research
, 1993
"... In recent years new insights and algorithms have been obtained for the classical, deterministic, vehicle routing problem as well as for natural stochastic and dynamic variations of it. These new developments are based on theoretical analysis, combine probabilistic and combinatorial modelling and ..."
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Cited by 71 (0 self)
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In recent years new insights and algorithms have been obtained for the classical, deterministic, vehicle routing problem as well as for natural stochastic and dynamic variations of it. These new developments are based on theoretical analysis, combine probabilistic and combinatorial modelling and lead to (1) new algorithms that produce near optimal solutions and (2) a deeper understanding of uncertainty issues in vehicle routing. In this paper we survey these new developments with an emphasis on the insights gained and on the algorithms proposed. Research supported in part by ONR contract N0001490J1649, NSF contracts DDM8922712, DDM9014751, and by a Presidential Young Investigator award DDM9158118 with matching funds from Draper Laboratory. y Sloan School of Management, MIT, Cambridge, MA 02139. z Dept. of Industrial Engineering and Operations Research, Columbia University, NY, NY, 10027 and Department of Operations Research and Management Sciences, Northwestern Universi...
Facility location models for distribution system design
, 2004
"... The design of the distribution system is a strategic issue for almost every company. The problem of locating facilities and allocating customers covers the core topics of distribution system design. Model formulations and solution algorithms which address the issue vary widely in terms of fundamenta ..."
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Cited by 70 (0 self)
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The design of the distribution system is a strategic issue for almost every company. The problem of locating facilities and allocating customers covers the core topics of distribution system design. Model formulations and solution algorithms which address the issue vary widely in terms of fundamental assumptions, mathematical complexity and computational performance. This paper reviews some of the contributions to the current stateoftheart. In particular, continuous location models, network location models, mixedinteger programming models, and applications are summarized.
Heuristic methods for vehicle routing problem with time windows
, 2000
"... This paper documents our investigation into various heuristic methods to solve the vehicle routing problem with time windows (VRPTW) to near optimal solutions. The objective of the VRPTW is to serve a number of customers within predefined time windows at minimum cost (in terms of distance travelled) ..."
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Cited by 38 (0 self)
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This paper documents our investigation into various heuristic methods to solve the vehicle routing problem with time windows (VRPTW) to near optimal solutions. The objective of the VRPTW is to serve a number of customers within predefined time windows at minimum cost (in terms of distance travelled), without violating the capacity and total trip time constraints for each vehicle. Combinatorial optimisation problems of this kind are nonpolynomialhard (NPhard) and are best solved by heuristics. The heuristics we are exploring here are mainly thirdgeneration artificial intelligent (AI) algorithms, namely simulated annealing (SA), Tabu search (TS) and genetic algorithm (GA). Based on the original SA theory proposed by Kirkpatrick and the work by Thangiah, we update the cooling scheme and develop a fast and efficient SA heuristic. One of the variants of Glover's TS, strict Tabu, is evaluated and first used for VRPTW, with the help of both recency and frequency measures. Our GA implementation, unlike Thangiah's genetic sectoring heuristic, uses intuitive integer string representation and incorporates several new crossover operations and other advanced techniques such as hybrid hillclimbing and adaptive mutation scheme. We applied each of the heuristics developed to Solomon's 56 VRPTW 100customer instances, and yielded 18 solutions better than or equivalent to the best solution ever published for these problems. This paper is also among the first to document the implementation of all the
RealTime Multivehicle Truckload Pickup and Delivery Problems
 Transportation Science
, 2004
"... In this paper we formally introduce a generic realtime multivehicle truckload pickup and delivery problem. The problem includes the consideration of various costs associated with trucks ’ empty travel distances, jobs ’ delayed completion times, and job rejections. Although very simple, the proble ..."
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Cited by 35 (3 self)
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In this paper we formally introduce a generic realtime multivehicle truckload pickup and delivery problem. The problem includes the consideration of various costs associated with trucks ’ empty travel distances, jobs ’ delayed completion times, and job rejections. Although very simple, the problem captures most features of the operational problem of a realworld trucking fleet that dynamically moves truckloads between different sites according to customer requests that arrive continuously over time. We propose a mixed integer programming formulation for the offline version of the problem. We then consider and compare five rolling horizon strategies for the realtime version. Two of the policies are based on a repeated reoptimization of various instances of the offline problem, while the others use simpler local (heuristic) rules. One of the reoptimization strategies is new while the other strategies have recently been tested for similar realtime fleet management problems. The comparison of the policies is done under a general simulation framework. The analysis is systematic and consider varying traffic intensities, varying degrees of advance information, and varying degrees of flexibility for job rejection decisions. The new reoptimization policy is shown to systematically outperform the others under all these conditions.
Comparing neurodynamic programming algorithms for the vehicle routing problem with stochastic demands
 Computers & Operations Research
"... The paper considers a version of the vehicle routing problem where customers ' demands are uncertain. The focus is on dynamically routing a single vehicle to serve the demands of a known set of geographically dispersed customers during realtime operations. The goal consists of minimizing the e ..."
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Cited by 35 (0 self)
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The paper considers a version of the vehicle routing problem where customers ' demands are uncertain. The focus is on dynamically routing a single vehicle to serve the demands of a known set of geographically dispersed customers during realtime operations. The goal consists of minimizing the expected distance traveled in order to serve all customers ' demands. Since actual demand is revealed upon arrival of the vehicle at the location of each customer, fully exploiting this feature requires a dynamic approach. This work studies the suitability of the emerging "eld of neurodynamic programming (NDP) in providing approximate solutions to this di$cult stochastic combinatorial optimization problem. The paper compares the performance of two NDP algorithms: optimistic approximate policy iteration and a rollout policy. While the former improves the performance of a nearestneighbor policy by 2.3%, the computational results indicate that the rollout policy generates higher quality solutions. The implication for the practitioner is that the rollout policy is a promising candidate for vehicle routing applications where a dynamic approach is required. Scope and purpose Recent years have seen a growing interest in the development of vehicle routing algorithms to cope with
Parallelization of the Vehicle Routing Problem with Time Windows
, 2001
"... Routing with time windows (VRPTW) has been an area of research that have
attracted many researchers within the last 10 { 15 years. In this period a number
of papers and technical reports have been published on the exact solution of the
VRPTW.
The VRPTW is a generalization of the wellknown capacitat ..."
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Cited by 34 (2 self)
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Routing with time windows (VRPTW) has been an area of research that have
attracted many researchers within the last 10 { 15 years. In this period a number
of papers and technical reports have been published on the exact solution of the
VRPTW.
The VRPTW is a generalization of the wellknown capacitated routing problem
(VRP or CVRP). In the VRP a
eet of vehicles must visit (service) a number
of customers. All vehicles start and end at the depot. For each pair of customers
or customer and depot there is a cost. The cost denotes how much is costs a
vehicle to drive from one customer to another. Every customer must be visited
exactly ones. Additionally each customer demands a certain quantity of goods
delivered (know as the customer demand). For the vehicles we have an upper
limit on the amount of goods that can be carried (known as the capacity). In
the most basic case all vehicles are of the same type and hence have the same
capacity. The problem is now for a given scenario to plan routes for the vehicles
in accordance with the mentioned constraints such that the cost accumulated
on the routes, the #12;xed costs (how much does it cost to maintain a vehicle) or
a combination hereof is minimized.
In the more general VRPTW each customer has a time window, and between
all pairs of customers or a customer and the depot we have a travel time. The
vehicles now have to comply with the additional constraint that servicing of the
customers can only be started within the time windows of the customers. It
is legal to arrive before a time window \opens" but the vehicle must wait and
service will not start until the time window of the customer actually opens.
For solving the problem exactly 4 general types of solution methods have
evolved in the literature: dynamic programming, DantzigWolfe (column generation),
Lagrange decomposition and solving the classical model formulation
directly.
Presently the algorithms that uses DantzigWolfe given the best results
(Desrochers, Desrosiers and Solomon, and Kohl), but the Ph.D. thesis of Kontoravdis
shows promising results for using the classical model formulation directly.
In this Ph.D. project we have used the DantzigWolfe method. In the
DantzigWolfe method the problem is split into two problems: a \master problem"
and a \subproblem". The master problem is a relaxed set partitioning
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problem that guarantees that each customer is visited exactly ones, while the
subproblem is a shortest path problem with additional constraints (capacity and
time window). Using the master problem the reduced costs are computed for
each arc, and these costs are then used in the subproblem in order to generate
routes from the depot and back to the depot again. The best (improving) routes
are then returned to the master problem and entered into the relaxed set partitioning
problem. As the set partitioning problem is relaxed by removing the
integer constraints the solution is seldomly integral therefore the DantzigWolfe
method is embedded in a separationbased solutiontechnique.
In this Ph.D. project we have been trying to exploit structural properties in
order to speed up execution times, and we have been using parallel computers
to be able to solve problems faster or solve larger problems.
The thesis starts with a review of previous work within the #12;eld of VRPTW
both with respect to heuristic solution methods and exact (optimal) methods.
Through a series of experimental tests we seek to de#12;ne and examine a number
of structural characteristics.
The #12;rst series of tests examine the use of dividing time windows as the
branching principle in the separationbased solutiontechnique. Instead of using
the methods previously described in the literature for dividing a problem into
smaller problems we use a methods developed for a variant of the VRPTW. The
results are unfortunately not positive.
Instead of dividing a problem into two smaller problems and try to solve
these we can try to get an integer solution without having to branch. A cut is an
inequality that separates the (nonintegral) optimal solution from all the integer
solutions. By #12;nding and inserting cuts we can try to avoid branching. For the
VRPTW Kohl has developed the 2path cuts. In the separationalgorithm for
detecting 2path cuts a number of test are made. By structuring the order in
which we try to generate cuts we achieved very positive results.
In the DantzigWolfe process a large number of columns may be generated,
but a signi#12;cant fraction of the columns introduced will not be interesting with
respect to the master problem. It is a priori not possible to determine which
columns are attractive and which are not, but if a column does not become part
of the basis of the relaxed set partitioning problem we consider it to be of no
bene#12;t for the solution process. These columns are subsequently removed from
the master problem. Experiments demonstrate a signi#12;cant cut of the running
time.
Positive results were also achieved by stopping the routegeneration process
prematurely in the case of timeconsuming shortest path computations. Often
this leads to stopping the shortest path subroutine in cases where the information
(from the dual variables) leads to \bad" routes. The premature exit
from the shortest path subroutine restricts the generation of \bad" routes signi
#12;cantly. This produces very good results and has made it possible to solve
problem instances not solved to optimality before.
The parallel algorithm is based upon the sequential DantzigWolfe based
algorithm developed earlier in the project. In an initial (sequential) phase unsolved
problems are generated and when there are unsolved problems enough
vii
to start work on every processor the parallel solution phase is initiated. In the
parallel phase each processor runs the sequential algorithm. To get a good workload
a strategy based on balancing the load between neighbouring processors is
implemented. The resulting algorithm is eÆcient and capable of attaining good
speedup values. The loadbalancing strategy shows an even distribution of work
among the processors. Due to the large demand for using the IBM SP2 parallel
computer at UNI#15;C it has unfortunately not be possible to run as many tests
as we would have liked. We have although managed to solve one problem not
solved before using our parallel algorithm.
Solving a Practical Pickup and Delivery Problem
 Transportation Science
, 2001
"... We consider a pickup and delivery vehicle routing problem commonly encountered in realworld logistics operations. The problem involves a set of practical complications that have received little attention in the vehicle routing literature. In this problem, there are multiple carriers and multiple ve ..."
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We consider a pickup and delivery vehicle routing problem commonly encountered in realworld logistics operations. The problem involves a set of practical complications that have received little attention in the vehicle routing literature. In this problem, there are multiple carriers and multiple vehicle types available to cover a set of pickup and delivery orders, each of which has multiple pickup time windows and multiple delivery time windows. Orders and carrier/vehicle types must satisfy a set of compatibility constraints that specify which orders cannot be covered by which carrier/vehicle types and which orders cannot be shipped together. Order loading and unloading sequence must satisfy the nested precedence constraint that requires that an order cannot be unloaded until all the orders loaded into the truck later than this order are unloaded. Each vehicle trip must satisfy the driver's work rules prescribed by the Department of Transportation which specify legal working hours of ...
Online algorithms for truck fleet assignment and scheduling under realtime information”, Transportation Research Record 1667
 Transportation Research Record
, 1999
"... With greater availability of realtime information systems, algorithms are needed to support commercial fleet operators in their decisions to assign vehicles and drivers to loads in a dynamic environment. We present a rolling horizon framework for the dynamic assignment and sequencing of trucks to ..."
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Cited by 18 (4 self)
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With greater availability of realtime information systems, algorithms are needed to support commercial fleet operators in their decisions to assign vehicles and drivers to loads in a dynamic environment. We present a rolling horizon framework for the dynamic assignment and sequencing of trucks to jobs consisting of picking up and delivering full truckloads when requests for 1 Submitted for presentation at the 78th Annual Meeting of the Transportation Research Board, January 1999, Washington, D.C. and publication in Transportation Research Board. 1 service arise on a continuous basis. A mathematical formulation of the problem faced at each stage is presented; its solution allows for the dynamic reassignment of trucks to loads, including diversion to a new load of a truck already enroute to pick up another load, as well as for the dynamic resequencing of the order in which loads are to be served, as new loads arrive and conditions unfold. Loads have associated time windows for pick up and delivery, and the objective function includes explicit penalty cost for not serving a particular load. A solution algorithm is presented and implemented, and computational results are presented, yielding insight into various operational tradeoffs in dynamic fleet operations. Because applicability of the solution algorithm is at present limited to relatively small problems, and given the stochastic dynamic nature of these systems, numerical experiments are performed to compare the quality of the solution obtained using this approach to the performance of simpler and less computationally demanding local rules.
A robust optimization approach for the capacitated vehicle routing problem with demand uncertainty
, 2006
"... In this paper we introduce a robust optimization approach to solve the Vehicle Routing Problem (VRP) with demand uncertainty. This approach yields routes that minimize transportation costs while satisfying all demands in a given bounded uncertainty set. We show that for the MillerTuckerZemlin form ..."
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Cited by 14 (5 self)
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In this paper we introduce a robust optimization approach to solve the Vehicle Routing Problem (VRP) with demand uncertainty. This approach yields routes that minimize transportation costs while satisfying all demands in a given bounded uncertainty set. We show that for the MillerTuckerZemlin formulation of the VRP and specific uncertainty sets, solving for the robust solution is no more difficult than solving a single deterministic VRP. We present computational results that investigate the tradeoffs of a robust solution for the Augerat et al. suite of capacitated VRP problems and for families of clustered instances. Our computational results show that the robust solution can protect from unmet demand while incurring a small additional cost over deterministic optimal routes. This is most profound for clustered instances under moderate uncertainty, where remaining vehicle capacity is used to protect against variations within each cluster at a small additional cost. We observe that the robust solution amounts to a clever management of the remaining vehicle capacity.
Container movement by trucks in metropolitan networks: modeling and optimization
 TRANSPORT. RES
, 2005
"... Container movement by trucks with time constraints at origins and destinations is modeled as an asymmetric “multiTraveling Salesmen Problem with Time Windows” (mTSPTW) with social constraints. A twophase exact algorithm based on dynamic programming (DP) is proposed that finds the best routes for ..."
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Cited by 13 (1 self)
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Container movement by trucks with time constraints at origins and destinations is modeled as an asymmetric “multiTraveling Salesmen Problem with Time Windows” (mTSPTW) with social constraints. A twophase exact algorithm based on dynamic programming (DP) is proposed that finds the best routes for a fleet of trucks. Since the mTSPTW problem is NPhard, the computational time for optimally solving large size problems becomes prohibitive. For large size problems, we develop a hybrid methodology consisting of DP in conjunction with genetic algorithms. The developed algorithms are compared with an insertion heuristic method. Computational results demonstrate the efficiency of the developed algorithms.