presnnmatu1

Nizhny Novgorod, presentation 1

Routing, Location and Network Design

Marcel Turkensteen ([email protected])CORAL, Aarhus School of Business, Aarhus, Denmark

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.


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).


Teaching

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


Today’s 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.


The seminar

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


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.


Short break

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.


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).


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.


Supply chain costs (Ballou, 2004)


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.


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.


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.


A set of demand points


Short break

Problem type 1: Minimum Spanning Tree Problem


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. Prim’s algorithm.

An version of the problem discussed in the second lecture is the Degree-Constrained Minimum Spanning Tree Problem, which is NP-hard.


Problem type 2: Location problem


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.


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 Weiszfeld’s algorithm.


Weiszfeld algorithm

Weiszfeld’s 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 (xk+1, yk+1) 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.


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 / √2 + 3× 2 / 1 + 2× 5 / √17 divided by

5 / √2 + 3/ 1 + 2 / √17 = 1.2. The y-coordinate becomes ? 0.43

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.


ppji

pjiij yyxxd /1])()[(

Generalized Weiszfeld algorithm

For 1 ≤ p < 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 p>2, a transformation exists to transform the problem into the case 1 < p < 2.


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.


Short break

Problem type 3: Location routing problems


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 don’t take routing into account in the location phase, your location might by (very) suboptimal.

The Location-Routing Problem (LRP) is a very complex problem.


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).


Salhi and Nagy heuristic

1. Take all demand points and compute the minimum Weber locations of facilities.

2. Compute the shortest routes given the facilities.3. Take the endpoints of the routes.4. Compute the minisum Weber locations of facilities

with the selected subset of endpoints.5. If the locations of the facilities remain the same,

terminate, else go to step 2.


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.


Remarks on location (Ballou, 2004)

Most current models focus on cost minimization -> how about profit maximization?

Most models are static -> what about a temporal component?

Demand is assumed to be certain -> include demand uncertainty.

Try to include cooperation across the supply chain into the location decision.


Short break

Problem type 4: Vehicle Routing Problems


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).


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).


Problem type 5: Traveling salesman Problems


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.


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.


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.


Short break

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.


Continuous Approximation

Extended to systems with VRP tours with a most C stops in Daganzo (2004), Langevin (1996):

Here E(δ^(-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:


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 C=2.− The estimated VRP route length for C=10.


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).


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 C=1, it is a regular location problem, whereas for C > 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.


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 Weiszfeld’s algorithm, the savings algorithm and a location-routing method.


References

Ballou, R. (2001) Unresolved Issues in Supply Chain Network Design. Information Systems Frontiers 3(4), 417–426.

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, 568–581.

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, 331–350.


References

Daganzo, C. (1984b) The Length of Tours in Zones of Different Shapes. Transportation Research 18B, 135–146.

Daganzo, C. (1988) A Comparison of In-Vehicle and Out-of-Vehicle Freight Consolidation Strategies. Transportation Research 22B, 173–180.

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.


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), 163–188.

Laporte, G. (1992) The Vehicle Routing Problem: An Overview of Exact and Approximate Algorithms. European Journal of Operational Research 59, 345–358.


References

Nagy, G. and Salhi, S. (2007) Location-routing: Issues, Models and Methods. European Journal of Operational Research 177, 649–672.

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, 690–703


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.


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.


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.


presnnmatu1

Documents

traveling

traveling

minisum weber

vehicle routing

vehicle routing

solution approaches

continuous

lagrangian