# Demand Point Aggregation for Location Models Chapter 7 – Facility Location Text - PowerPoint PPT Presentation

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Demand Point Aggregation for Location Models Chapter 7 – Facility Location Text. Adam Bilger 7/15/09. Demand Point Aggregation for Location Models. Covering chapter 7 sections 1-5 7.1 Introduction 7.2 The Aggregation Problem 7.3 Aggregation Error 7.4 Guidelines for Aggregation

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Demand Point Aggregation for Location Models Chapter 7 – Facility Location Text

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## Demand Point Aggregation for Location ModelsChapter 7 – Facility Location Text

7/15/09

### Demand Point Aggregation for Location Models

• Covering chapter 7 sections 1-5

• 7.1 Introduction

• 7.2 The Aggregation Problem

• 7.3 Aggregation Error

• 7.4 Guidelines for Aggregation

• 7.5 An Aggregation Algorithm

### Demand Point Aggregation for Location Models

• Introduction

• Location Problem – Review P-median

• Potential for millions of demand points

• Centroids and central locations

• IRS example

• Inducing error

Demand Point Aggregation for Location Models

### Demand Point Aggregation for Location Models

• Location Problem – Review P-median

• Objective – Locate p facilities to minimize the demand weighted total distance between demand nodes and facilities

• Constraints

• Max of p facilities

• Must cover all demand

• Can’t assign demand i if facility j not placed

Demand Point Aggregation for Location Models

### Aggregation Problem

• Notation

• Pm = (am, bm), m=1, . . . . , M (demand pts)

• P = (P1, . . . , Pm) a demand pt vector

• P’m : the aggregate demand pt replacing Pm, m=1,…,M

• P’ = (P’1, . . . , P’M) an aggregate demand point vector

• X = (X1, . . . , XN) an N-median or N-center

• Xn = (xn, yn), n=1, . . . ., N (new facilities)

• d(U, V) = distance between any 2 pts U and V

• dm(Xn) = distance between demand pt m and new facility n

• Dm(X) = distance between demand pt m and closest new facility in X

• f(X : P) : the original location model

• f(X : P’) = the approximating location model

Demand Point Aggregation for Location Models

### Aggregation Problem

• Three decisions must be made:

• q, the number of aggregate demand points

• The locations of the aggregate demand points

• The replacement rule

Demand Point Aggregation for Location Models

### Aggregation Problem

• Weighting aggregated demand points for p-median problem

• f(X : P) = w1d(X, P1) + . . . +w4d(X, P4)

• f(X : P’) = w1d(X,Z1)+w2d(X,Z1)+w3d(X,Z2)+w4d(X,Z2)

• f(X : P’) = (w1 + w2)d(X, Z1)+(w3 + w4)d(X, Z2)

Demand Point Aggregation for Location Models

### Aggregation Error

• Location choices result in an error

• e(X) = f(X : P) – f(X : P’)

• Error Types:

• Demand point m error for N-median model

• Total error

• Absolute error

• Relative error

• Maximum absolute error

Demand Point Aggregation for Location Models

### Aggregation Error

• Statistical Sampling

• Reduce those to a smaller aggregate set

• It’s unrealistic to calculate error for the entire model

• Goal is to statistically sample the aggregate set of demand nodes and calculate a representative error for the entire model

Demand Point Aggregation for Location Models

### Aggregation Error

• Demand point m error for N-median model

• em(X) = wmD(X,Pm) - wmD(X,P’m)

= wm(D(X,Pm) - D(X,P’m))

• Total error for N-median model, given any X

• e(X) = e1(X) + . . . + em(X)

Demand Point Aggregation for Location Models

### Aggregation Error

• Absolute error for N-median model

• ae(X)= |e(X)| = |f(X : P) - f(X : P’)|

• Relative error for N-median model, given any X

• rel(X) = 100 * (ae(X) / f(X : P))

• Maximum absolute error

• mae = mae {ae(X) : X}

Demand Point Aggregation for Location Models

Demand

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Aggregate

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### Aggregation Error Example

• Solve the problem as a one dimensional weighted p-median problem. Set p=1.

• Solve the problem again aggregating demand onto the new set of aggregate points. Relocate demand points to the closest aggregate point.

• Calculate the relative error.

Demand Point Aggregation for Location Models

### Aggregation Error Example

• p-median point is simply 13 for this problem, where we’re only locating a single facility (p=1) in one dimension.

f(X:P) = Σ wmD(X,Pm)

= 6*7+2*10+1*9+6*3+9*0+4*2+2*5+8*10+1*12

=199

M

m=1

Demand Point Aggregation for Location Models

### Aggregation Error Example

2. New p-median location is at node 15. New aggregated demand points need recalculated for weight.

f(X:P’) = Σ wmD(X,P’m)

f(X:P’) = (w1 + w2)d(X, P’1)+(w3 + w4)d(X, P’2)

= (6+2+1)d(X, P’1)+(6+9+4)d(X, P’2)+(2+8+1)d(X, P’3)

= 9*11+19*0+11*7

=176

M

m=1

Demand Point Aggregation for Location Models

### Aggregation Error Example

• ae(X) = |em(X)| =|f(X:P) - f(X:P’)|

=|199 - 176|

= 23

rel(X) = 100 * (ae(X) / f(X:P))

= 100 * (23/199)

= 11.6%

Demand Point Aggregation for Location Models

### Guidelines for Aggregation

• Aspects of a location process effected by aggregation

• (EC1): aggregation error

• (EC2): computational cost to

• a) get demand point data

• b) implement and run aggregation algorithm

• c) solve the approximating location model

Demand Point Aggregation for Location Models

### Guidelines for Aggregation

• Aspects of a location process effected by aggregation (cont.)

• (EC3): ease of explanation

• (EC4): problem structure exploitation

• (EC5): robustness (works for many different problems)

• (EC6): GIS implementable

Demand Point Aggregation for Location Models

### Guidelines for Aggregation

• As (EC1) or aggregation error is allowed to increase computational costs (EC2) are reduced

• EC1 and ease of explanation (EC3)

• Problem structure exploitation (EC4) & robustness (EC5)

• Most important – error vs. costs

Demand Point Aggregation for Location Models

### An Aggregation Algorithm (MRC – Francis, Lowe and Rayco 1996)

• N- Median Problem – planar rectilinear version of p center model

• Motivation – seek an aggregation with a small error

• This algorithm find an rc median that minimizes the objective function value of the q-median problem with rectilinear distances over all possible rc-medians

• MRC is a method for making the three decisions:

• q, the number of aggregate demand points

• The locations of the aggregate demand points

• The replacement rule

Demand Point Aggregation for Location Models

### An Aggregation Algorithm

• q, the number of aggregate demand points

• q = r * c

• The locations of the aggregate demand points

• Create a grid of r rows and c columns over the existing demand nodes

• Equally divide the data as opposed to the space

Demand Point Aggregation for Location Models

An Aggregation Algorithm

Demand Point Aggregation for Location Models

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Row / Column intersection points become the coordinates for the new aggregate demand points.

Demand Point Aggregation for Location Models

### An Aggregation Algorithm

• We now have six new aggregate points

• The replacement rule

• How do we assign demand points and their weighting to the aggregate points?

• The MRC dictates that the next step is to partition the grid by creating lines to split the existing rows and columns

Demand Point Aggregation for Location Models

### An Aggregation Algorithm

Demand Point Aggregation for Location Models

### An Aggregation Algorithm

• Last assign points and weighting

• Partition used to assign demand points

• Additive method most common for assigning weighting to the new aggregate points

• Algorithm would utilize this aggregation technique to optimize the objective function

Demand Point Aggregation for Location Models

### Summary

• Three decisions to be made when aggregating demand

• Estimate error to evaluate the aggregation implementation

• Existing algorithms exist

Demand Point Aggregation for Location Models

Questions?

Demand Point Aggregation for Location Models