Random Utility Models: introduction

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1 Corso di LOGISTICA TERRITORIALE DOCENTE prof. ing. Agostino Nzzolo Random Utility Models: introdction DOCENTI Agostino Nzzolo Antonio Comi 1

2 Behavioral Model of Anticipated Utility Let: i, the generic decision-maer of repeated choices I i, the set of alternatives considered by decision-maer i X i h, the h-th attribte of alternative, an entry of vector X i ; AU i, the anticipated tility of decision-maer i associated with alternative ; it is a linear combination of the anticipated vales of attribtes X i j: AU i = i 1 AX i 1 + i 2 AX i 2 +. i j, coefficient of attribte X i j (weight given by the decision-maer to the attribte) The alternative with max anticipated tility is chosen. Logistica Territoriale 2

3 Behavioral Model of Anticipated Utility To forecast the chosen alternative, we shold now for the decision maer i: - the considered alternatives - the anticipated vales of attribtes AX - the coefficients of the tility fnction Given that it is difficlt to obtain this information, and for the minor difficlty of starting directly from a sample of observations on the chosen alternatives, we se a different method, the Random Utility approach Logistica Territoriale 3

4 Random Utility Approach The anticipated tility AU is assmed by modeller as a random variable, sm of V (systematic anticipated tility, fnction (linear) of attribte average vale X), and (random tility) AU = V + Anticipated tility V i = g i 1 X i 1 + g i 2 X i 2 + Systematic ant. tility random residal Logistica Territoriale 4

5 Random Utility Approach We loo for a statistical relationship between the anticipated tility AU i and the average vales of the attribtes X of the alternatives obtained by a networ model AU i = g i 1 X i 1 + g i 2 X i = random residal Logistica Territoriale 5

6 Randomness of the residals Random residal ( ) can be conferred to several factors: simplified hypothesis on path (alternative) choice set; variation in tastes and preferences over time, for the same decision maer (i.e. traveller might weigh an attribte differently in different decision contexts); for grop models (see below) dispersion among decision maers (i.e. variations in tastes and preferences); simplified hypotheses on anticipating mechanism of attribtes; omitted attribtes that are not directly observable by modeller; modelling approximation in the attribte vale compted by modeller. Trasporti e Territorio 6

7 Random tility approach Ths, a term of AU can be forecasted considering the vales assmed by X (attribted considered by the analyst) and the weights a (to be determined) pls a term not foreseeable. The incomplete foreseeability of AU is not possible to predict with certainty the chosen alternative, bt nowing the systematic tility V and the distribtion fnction of the random residal, it is possible to evalate the probability p[] to choice alternative Logistica Territoriale 7

8 Random Utility Approach Choice probability of a decision maer p[] = prob [AU > AU ] p[] = prob [V + > V + ] p[] = prob [V V > - ), I, I, I The above probability can be compted sing assmptions on the probability distribtion of random residals, obtaining a random tility choice model differently specified (e.g. logit, nested-logit and so on) (see next lesson) Logistica Territoriale 8

9 Random Utility Models Classification: Individal RUM: a model is considered for each decision maer Grop RUM: a model is obtained for a grop of decision maer (average decision maer) Logistica Territoriale 9

10 Individal Random Utility Models Let I mi the choice set considered by the analyst, i X j the attribtes of the model (spply model) and tility obtained as: where: - V = systematic tility the systematic X i vector of attribtes related to alternative (vales defined by the analyst) with entry X i j; b i vector of parameters with entry b i j i V i i i 1 i i T i 1 X 1... j X j... b V b b X Logistica Territoriale 10

11 Individal Random Utility Models Let: where Û V i i i i random residal, with zero mean (= 0), is the deviation of the anticipated tility of alternative by ser i from the vale V i. Then: i i i i i i E 0 E Uˆ E V E V E i 2 i i i i i i i i var var Uˆ var V var V var 2cov V ; Logistica Territoriale 11

12 Individal Random Utility Models As said before, considering the previos hypothesis it is not possible to forecast with certainty the alternative chosen by the ser i. Bt, it is possible to evalate the probability p i [] to choice alternative belonging to the choice set I mi. This vale is the probability that alternative has an anticipated tility greater than the other available alternatives: p i mi [ ] [ ˆ i ˆ i I Pr U U ' ', Since the anticipated tility is the sm between the systematic tility V i (vale obtained by the analyst) and the random residal i we have: i mi i i i i mi p / I Pr ob V V ' ' ; ', ' I I mi ] Logistica Territoriale 12

13 Grop Random Utility Models Models derived observing not the single decision-maer bt a set of decision-maers (average decision-maer). Let: Û V random residal, assmed with zero mean, is the deviation of the anticipated tility of (single) decision-maer i from the the anticipated tility of the set of decision-maers and de to the variability of tastes among different sers j-th model attribte(spply model) X j T 1 X 1... j X j... a V a a X Logistica Territoriale 13

14 Logit Random Utility models Assming that the randon residal,, are independently and identically distribted as a Weibll-Gmbel we have the Mltinomial Logit Model In the case of grop RUM: p / I m exp V ' exp V ' Being V a linear combination of attribtes: V a j j j X p / I m exp ' exp j a X j j j j a X j ' Logistica Territoriale 14

An example of Mltinomial Logit d OD = a = 0,27 1000 veicoli/ora Generic ser T Percorso A 33 probabilità Percorso B 32 probabilità Percorso C 37 probabilità exp[ 0, 27 33] exp[ 0, 27 33] exp[ 0, 27

15 An example of Mltinomial Logit d OD = a = 0, veicoli/ora Generic ser T Percorso A 33 probabilità Percorso B 32 probabilità Percorso C 37 probabilità exp[ 0, 27 33] exp[ 0, 27 33] exp[ 0, 27 32] exp[ 0, 27 37] m p A / I 0,38 exp[ 0, 27 32] exp[ 0, 27 33] exp[ 0, 27 32] exp[ 0, 27 37] m p B / I 0,49 p C / I m exp[ 0, 27 37] exp[ 0, 27 33] exp[ 0, 27 32] exp[ 0, 27 37] 0,13 p / I m exp[ 0,27 T ] ' exp 0, 27 T ' F A = 380 vehicle/hor F B = 490 vehicle/hor F C = 130 vehicle/hor Logistica Territoriale 15

Department of Industrial Engineering Statistical Quality Control presented by Dr. Eng. Abed Schokry

Department of Industrial Engineering Statistical Quality Control presented by Dr. Eng. Abed Schokry Department of Indstrial Engineering Statistical Qality Control presented by Dr. Eng. Abed Schokry Department of Indstrial Engineering Statistical Qality Control C and U Chart presented by Dr. Eng. Abed