1. Demand Point Aggregation for Location Models Chapter 7 – Facility Location Text
Adam Bilger
7/15/09

2. 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

3. 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
Location Problem Review – Locating facilities to cover existing demand nodes. Problems we’ve worked have been simple in that they have a small number of demand nodes. Problems become computationally impractical when you look at large numbers of demand nodes. Example - Total population in a given geographic area are all demand nodes. Aggregation is a method of taking all of those demand nodes and reducing them down to a manageable number. US census Bureau does this by postal code area, using the centroid (avg of all points).
Must be aware that by not using all points and aggregating the demand that you’re inducing error into the objective function and into the selection of the optimal facility locations.
There may be existing aggregate demand points that already make sense like a sales office for a region or you may be locating new facilities.
Demand Point Aggregation for Location Models

4. 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
Location Problem Review – Locating facilities to cover existing demand nodes. Problems we’ve worked have been simple in that they have a small number of demand nodes. Problems become computationally impractical when you look at large numbers of demand nodes. Example - Total population in a given geographic area are all demand nodes. Aggregation is a method of taking all of those demand nodes and reducing them down to a manageable number. US census Bureau does this by postal code area, using the centroid (avg of all points).
Must be aware that by not using all points and aggregating the demand that you’re inducing error into the objective function and into the selection of the optimal facility locations.
Demand Point Aggregation for Location Models

5. 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
f(X:P) is the cost function of solving this problem P with X facilities.
Demand Point Aggregation for Location Models

6. 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
The replacement rule is essentially how you compute the demand for the new aggregate point. This could simply be an addition of all of the previous demands that were aggregated into the new point. It could also include some kind of a business factor or cost. This replacement rule can also deal with how you assign demand to the aggregate point. Is this based on the nearest facility rule or a different algorithm.
Demand Point Aggregation for Location Models

7. 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)
As part of the replacement rule we talked about how we assign demand weighting to the new aggregate point within the cost function. This shows the standard addition method without any additional business factors.
Demand Point Aggregation for Location Models

8. 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
Error is always in units of the objective function. Typically this makes the most sense for a cost problem where cost can be described in terms of distance.
There are no rules of thumb in terms of how much error we can live with, it’s usually relative to a previous model or solution. Did we improve over what is being used today. Is our error small compared to the error in the estimates we used for a variable demand.
Bottom line the error can tell you how much the aggregation hurt your result and you can decide if you need to analyze the original problem.
Total error – looking for something close to zero, so positive and negative errors cancel each other out.
Absolute error for any X looks at the combined positive error
Relative error is the absolute error expressed as a percentage
Maximum absolute error-
Demand Point Aggregation for Location Models

9. Demand Point Aggregation for Location Models
Aggregation Error
Statistical Sampling
Generally start with a large number of demand nodes
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

10. 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)
Total error gives an overall estimate of the error looking at the total error of each original demand point in relation to the new facility and comparing back to the aggregated results.
Demand Point Aggregation for Location Models

11. 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}
The maximum absolute error will pick up outliers. In other words if an aggregate point is not a good representation of that point.
Demand Point Aggregation for Location Models

12. 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 6 2 1 4 8 9 2 3 4 13 12 14 15 20 22 21 23 24 1 11 10 25
Aggregate
Demand Pts 1 2 3
Demand Point Aggregation for Location Models

13. 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

14. Aggregation Error Example
2. New p-median location is at node 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

15. Aggregation Error Example
ae(X) = |em(X)| =|f(X:P) - f(X:P’)|
=| |
= 23
rel(X) = 100 * (ae(X) / f(X:P))
= 100 * (23/199)
= 11.6%
Demand Point Aggregation for Location Models

16. 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

17. 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
EC4 – Need to check that your aggregation makes sense for the business. For example the sales offices, prelocated aggregate points.
EC5 – Need to make sure small changes in the aggregation details don’t lead to different results in the models objective function
EC6 – Can the problem be implemented via computer programming. You don’t want to be assigning demand nodes by hand.
Demand Point Aggregation for Location Models

18. Guidelines for Aggregation
Interactions and tradeoffs
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
Impossible to develop an algorithm that does well against all of these criteria.
Demand Point Aggregation for Location Models

19. 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

20. 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
Draw a data set on the board, including the x, y coordinates. Draw the rows and columns, this act of picking the number of rows and columns dictates how many aggregate points you’ll have.
Demand Point Aggregation for Location Models

21. An Aggregation Algorithm
These are the demand nodes for our problem and we’ve decided it would be too computationally challenging to solve it as is.
Demand Point Aggregation for Location Models

22. Demand Point Aggregation for Location Models3 2 5 4 6 1
Where the lines cross are the new aggregate demand points. q=6 These will be different for different people, depending on the placement of the rows and columns. Again trying to equally divide the points as opposed to the space.
Row / Column intersection points become the coordinates for the new aggregate demand points.
Demand Point Aggregation for Location Models

23. 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
Recall the replacement rule is how we assign demand to the aggregate nodes. For MRC we’ll assign demand within the partitioned space to the new aggregate point. We’ll also weight these points by adding up all of the weightings of the original points.
Demand Point Aggregation for Location Models

24. An Aggregation Algorithm
Demand Point Aggregation for Location Models

25. 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
Combine or add all of the old demand point weights up to get the new aggregate demand weighting.
Demand Point Aggregation for Location Models

26. 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

27. Demand Point Aggregation for Location Models
Questions?
Demand Point Aggregation for Location Models