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

1. Demand Point Aggregation for Location ModelsChapter 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 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 Demand Point Aggregation for Location Models

5. 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

6. 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

7. 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

8. 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

9. 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. 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

11. 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

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

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 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

15. 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

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 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 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 Demand Point Aggregation for Location Models

21. An Aggregation Algorithm Demand Point Aggregation for Location Models

22. 1 2 3 4 5 6 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 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 Demand Point Aggregation for Location Models

26. 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. Questions? Demand Point Aggregation for Location Models