Routing, Location and Network Design

1
Routing, Location and Network Design

• Marcel Turkensteen (matu_at_asb.dk)
• CORAL, Aarhus School of Business, Aarhus, Denmark

2
Introducing myself

• Marcel Turkensteen
• Graduated at the University of Groningen 2002
• PhD from the same university in 2007.
• Now Postdoc at the Aarhus School of Business and
  Social Sciences.

3
Research interests

• Cooperation with Boris Goldengorin on several
  papers.
• Research interests
• Combinatorial Optimization Problems the use of
  tolerances in solving them.
• Geography and routing / location.
• Sustainability and OR.
• Sports and OR (starting).

4
Teaching

• Course that Ive taught include
• Introduction to Management Science Modeling.
• Operations Research methods.
• Sustainable Supply Chain Management.
• Facility Location and Layout (1 year).

5
Todays sessions

• Presentation 1 Routing, location and network
  design.
• We introduce routing, location and network design
  problems in logistics.
• We introduce the solution approaches to these
  problems.
• Presentation 2
• We discuss how the solution approaches Branch and
  Bound and Lagrangian relaxation work.
• We will introduce and compute tolerance values.

6
The seminar

• In the seminar, there are assignments, mainly on
  the materials from the second presentation.

7
What to learn from both lectures?

• Knowing relevant routing and location models in
  logistics decisions.
• Applying simple location and routing heuristics
  and formulas.
• Using Lagrangian relaxation in general
  formulating the problem and solving it.
• Using Branch and Bound in general the
  ingredients.
• Learning to compute and analyze upper and lower
  tolerances.

8
Short break
9
The first presentation - Introduction

• If you have a group of dispersed demand points,
  then what is the costs of supplying products to
  these demand points?
• Ways to model this problem include
• Location problems
• Minimum Spanning Tree Problems.
• Location-routing problems
• Vehicle Routing Problems
• Traveling Salesman Problems.

10

• Motivation for this comes from the paper
  Turkensteen et al. (2011) (Balancing Fit and
  Logistics Costs) and a paper with A. Klose
  (2009).
• The question is what are the logistics costs of
  serving geographically dispersed demand points?
• We modeled the costs using the models discussed
  in this lecture (in particular the
  location-routing method).

11
Solution approaches introduced

• The solution approaches considered here are
• Weiszfeld for single facility location.
• The location-routing heuristic by Salhi and Nagy.
• The savings heuristic for vehicle routing.
• Continuous approximation approaches.
• Branch and Bound
• Lagrangian relaxation.

12
Supply chain costs (Ballou, 2004)
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Expected distribution costs

• There are different types of distribution
  systems
• Multiple echelons versus single echelons
• One-to-many versus many-to-many distribution
  systems.
• The textbook Logistics system analysis by
  Daganzo (2004) summarizes the results on
  different distribution systems.

14
A logistics costs model

• The relevant costs are inventory costs,
  transportation costs, warehouse costs and
  handling costs.
• How to write it out transportation costs
  dependent on travel distances.
• Pipeline inventory dependent on travel distances.
• Stationary inventory depends on the dispatch
  policy.
• Warehouse costs dependent on the number of
  warehouses.
• Handling costs depending on other operations.

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Distances and logistics costs

• In a one-to-many distribution system, distances
  influence the logistics costs more or less in a
  linear way through transportation costs and
  pipeline inventory costs.
• For multi-echelon and many-to-many distribution
  systems, the number of warehouses and number of
  echelons influence the logistics costs as well.

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A set of demand points
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Short break
18
Problem type 1 Minimum Spanning Tree Problem
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Minimum Spanning Tree Problems

• The Minimum Spanning Tree Problem (MSTP) is the
  problem of connecting n nodes in a network
  against minimum costs.
• The MSTP can be solved polynomially using e.g.
  Prims algorithm.
• An version of the problem discussed in the second
  lecture is the Degree-Constrained Minimum
  Spanning Tree Problem, which is NP-hard.

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Problem type 2 Location problem
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Where to locate a central facility

• Normally, in locating a facility several factors
  play a role location of the suppliers, location
  of customers, regulations, wages, ground prices,
  etc.
• In many location problems, it is assumed that the
  optimal location is the one that minimizes the
  sum of the distances to the relevant points.
• However, there are many versions of location
  problems.
• Here, we assume that there is a single facility,
  a continuous plane and the sum of distances needs
  to be minimized.

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Minisum Weber Problem

• The problem of locating a central facility on a
  continuous plane is called the minisum Weber
  problem.
• An exact solution approach is Weiszfelds
  algorithm.

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Weiszfeld algorithm

• Weiszfelds algorithm is an iterative procedure
  for solving the minisum Weber problem.
• Start with an initial location (x0, y0)
• Given location (xk, yk), perform step to end up
  in location (xk1, yk1)
• Terminate if satisfying solution is found, or a
  certain number of iterations has been performed.
• In each step, we do
• Drezner (1992) A Note on the Weiszfeld Location
  Problem.

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Exercise the next location is?

• Three points with coordinates and weights 1 (0,
  0) with weight 5 2 (2,1) with weight 3 3
  (5,0) with weight 2.
• Start at (1,1).
• Take the weighted x-coordinate divided by the
  distance to the center divide this by the
  weights divided by the distance to the center.
• Then the new x-coordinate is 5 0 / v2 3 2 / 1
  2 5 / v17 divided by
• 5 / v2 3/ 1 2 / v17 1.2.
• The y-coordinate becomes ? 0.43

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More accurate road distances

• Distances can be computed with the following
  formula (lp norm)
• For p 2, distances are Euclidean.
• However, road distances lie in a range between p
  1.5 and p 3 (see Berens et al, 1985).
• This problem can be solved with a generalized
  version of Weiszfeld algorithm.

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Generalized Weiszfeld algorithm

• For 1 p lt 2, there is a generalized version of
  the algorithm that converges to the optimal
  location.
• See Brimberg, Chen (1998) A Note on the
  Convergence of the Single Facility Minisum
  Location Problem.
• For pgt2, a transformation exists to transform the
  problem into the case 1 lt p lt 2.

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Location problems other versions

• Location problems with multiple locations are
  generally more complex than single location
  problems.
• Location problems can be discrete and on
  networks.
• Resulting problems are, among others, p-median
  problems and simple plant location problems.
• An extension is to set up locations such that
  each demand point is at most M kilometers from a
  facility (covering problems).
• Another extension is to take routing into account
  when locating facilities location-routing.

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Short break
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Problem type 3 Location routing problems
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Location-routing problem

• In some cases, deliveries to multiple demand
  points can be combined into single delivery tours
  (peddling).
• It then pays off to jointly decide on location
  and routes.
• It is necessary to decide jointly, because if you
  dont take routing into account in the location
  phase, your location might by (very) suboptimal.
• The Location-Routing Problem (LRP) is a very
  complex problem.

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Solution methods to the LRP

• Exact methods are generally very slow, as the LRP
  is a very hard problem.
• Hierarchical heuristics (location first, then
  routing), e.g., cluster first, routes later.
• The simultaneous location and routing heuristic
  by Salhi and Nagy (2009).

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Salhi and Nagy heuristic

• Take all demand points and compute the minimum
  Weber locations of facilities.
• Compute the shortest routes given the facilities.
• Take the endpoints of the routes.
• Compute the minisum Weber locations of facilities
  with the selected subset of endpoints.
• If the locations of the facilities remain the
  same, terminate, else go to step 2.

33
The heuristic with one facility

• The location of the facility in each stage is
  simply a minisum Weber problem.
• The routing from the given facility through the
  demand point is an example of the next problem
  the Vehicle Routing Problem.
• With multiple facilities, we should consider the
  assignments of routes to facilities as well.

34
Remarks on location (Ballou, 2004)

• Most current models focus on cost minimization -gt
  how about profit maximization?
• Most models are static -gt what about a temporal
  component?
• Demand is assumed to be certain -gt include demand
  uncertainty.
• Try to include cooperation across the supply
  chain into the location decision.

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Short break
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Problem type 4 Vehicle Routing Problems
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Vehicle routing

• The Vehicle Routing Problem is the problem of
  constructing a set of tours from a depot to
  demand points such that the sum of the lengths is
  minimized.
• Usually, there are capacity constraints on the
  tours, or the number of tours is prescribed.
• Solution approaches
• Savings algorithm
• Meta-heuristics (tabu search)
• Exact approaches (column generation).

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The savings algorithm

• The savings algorithm is a simple and very fast
  algorithm.
• It assumes that each route has a certain capacity
  and each demand point has a certain volume.
• First, compute the savings from each connection i
  to j, s_ij c_i0 c_0j c_ij.
• Order the edges on their savings from big to
  small.
• Add edges to routes until no more can be added.
• Clarke and Wright (1958).

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Problem type 5 Traveling salesman Problems
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Traveling Salesman Problem

• The Traveling Salesman Problem (TSP) is the
  problem of finding a minimum length tour through
  n locations, such that each locations is visited
  exactly once.
• Asymmetric (ATSP) the distance from i to j is
  not necessarily equal to the distance from j to
  i.
• The ATSP is presented extensively in lecture 2 in
  combination with Branch and Bound.

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TSP solution approaches

• For the Symmetric version of the problem
• The exact Concorde approach (Branch-and-Cut)
• Meta-heuristics such as the modified
  Lin-Kernighan Variable Neighborhood Search
  heuristic from Helsgaun (2000).
• Asymmetric
• Branch and Bound type methods, see second
  lecture.
• Cut and Solve.

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Solution approach Continuous approximation

• Estimate distribution costs given a uniform
  distribution of demand points in a certain area.
• Estimation of route lengths within TSP, VRP, and
  LRP.
• We discuss main results.
• One warehouse vs multiple warehouses.

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Short break
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Continuous Approximation

• Eilon (1971) found the following TSP tour length
  estimate
• The assumption is that the demand locations are
  randomly (uniformly) distributed across a certain
  area.
• Here, k is a constant for the type of distances,
  e.g., 0.57 for Euclidean and 0.72 for
  rectilinear.

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Continuous Approximation

• Extended to systems with VRP tours with a most C
  stops in Daganzo (2004), Langevin (1996)
• Here E(d(-1/2) is the density of the area, and
  E(r) the average distance to a central point in
  the area.
• If demand points are uniformly distributed and C
  sufficiently large, the formula is

46
Estimate tour lengths

• Consider the data set with N 25, the maximum
  difference between x-coordinates is 40 and
  between y-coordinates is 25.
• Assume Euclidean distances.
• Compute the
• Estimated tour length
• The estimated VRP route length for C2.
• The estimated VRP route length for C10.

47
Validity of continuous approximation results

• Continuous approximation results are valid in
  areas with different shapes (Daganzo, 1984), with
  time windows on routes (Figliozza, 2008).
• The results are extended to transshipment
  warehouses in Daganzo (1986) and Campbell (1993).
• A more or less uniform distribution is assumed.
• In some studies, the results serve as a starting
  point for further optimization see e.g. Robuste
  et al. (1990).

48
Our studies

• We tried to devise simple measures of demand
  dispersion in order to estimate route lengths.
• In one-to-many distribution systems, route
  lengths are closely related to the logistics
  costs.
• We used continuous approximation results to come
  up with distance measures.
• For C1, it is a regular location problem,
  whereas for C gt 2, we have an LRP or a VRP.
• We find that travel distance estimates can be
  very accurate.
• In one paper, we include such a measure within a
  market research method.

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Conclusion and summary

• The first lecture discusses a wide range of
  topics within logistics network.
• We started with discussing logistics costs and
  relating them to OR problems.
• Then we discussed problem and methods within
  location, routing and network design.
• Methods discussed in more detail are Weiszfelds
  algorithm, the savings algorithm and a
  location-routing method.

50
References

• Ballou, R. (2001) Unresolved Issues in Supply
  Chain Network Design. Information Systems
  Frontiers 3(4), 417426.
• Bektas, T. and Laporte, G. (2001) The
  Pollution-Routing Problem, Transportation
  Science.
• Clarke, G. and Wright, J. (1964) Scheduling of
  Vehicles from a Central Depot to a Number of
  Delivery Points. Operations Research 12,
  568581.
• Daganzo, C. (1984) The Distance Traveled to Visit
  N Points with a Maximum of C Stops per Vehicle
  An Analytic Model and an Application.
  Transportation Science 18, 331350.

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References

• Daganzo, C. (1984b) The Length of Tours in Zones
  of Different Shapes. Transportation Research 18B,
  135146.
• Daganzo, C. (1988) A Comparison of In-Vehicle and
  Out-of-Vehicle Freight Consolidation Strategies.
  Transportation Research 22B, 173180.
• Daganzo, C. (2004). Logistics Systems Analysis, 4
  edn. Springer Verlag, Berlin.
• Drezner, Z (1992) A Note on the Weber Location
  Problem, AOR, 1992, 40, 153-161.

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References

• Eilon, S. and Watson-Gandy, C.D.T. and
  Christofides, N. (1971). Distribution Management
  Mathematical Modelling and Practical Analysis,
  Hafner, New York.
• Langevin, A., Mbaraga, P. and Campbell, J. (1996)
  Continuous Approximation Models in Freight
  Transport An Overview. Transportation Research B
  30(3), 163188.
• Laporte, G. (1992) The Vehicle Routing Problem
  An Overview of Exact and Approximate Algorithms.
  European Journal of Operational Research 59,
  345358.

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References

• Nagy, G. and Salhi, S. (2007) Location-routing
  Issues, Models and Methods. European Journal of
  Operational Research 177, 649672.
• M. Turkensteen, A. Klose. The Cost of Supplying
  Segmented Consumers from a Central Facility.
  Conference Proceedings of the 14th HKSTS
  International Conference, 2009.
• Turkensteen, M., Sierksma, G. and Wieringa, J.E.
  Balancing the Fit and Logistics Costs of market
  segments. Corrected proof, European Journal of
  Operational Research.
• Zipkin, P. (1995) Performance Analysis of a
  Multi-Item Production-Inventory System under
  Alternative Policies. Management Science 44,
  690703

54
Logistics and sustainability

• An recent upcoming trend is sustainability.
• That means, a focus on the use of resources,
  pollution, cutting environmental waste.
• This has consequences for the routing, location
  and network design decisions.

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Network design and sustainability

• In designing a logistics network, the
  sustainability can be upgraded with
  environmentally friendly transportation modes and
  with a larger number of warehouses.
• Ship is friendlier than train, truck, air.
• A large number of locations can mean that
  shipments take place over shorter distances.
• On the other hand, centralization may lead to
  fuller trucks.

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Routing and sustainability

• A recent paper is on pollution-routing minimize
  the amount of CO2 emissions of routes rather than
  costs or distances.
• Bektas, Laporte (2011).
• Factors that play a role are congestion and
  vehicle speed.
• Additional factors are fuel costs and particle
  emissions.