• No results found

C-logit

In document Realistic route choice modeling (Page 51-59)

5.2 Game theory variants

6.1.1 C-logit

This section discusses the expected results and the obtained results for the C-logit variants in the static as- signment of the theoretical test. A table describes the used parameter settings for every test. The tested and varied parameter is printed boldin the tables. The categories in the table are utility, commonality factor,θ

value, power,β value,γ value, penalty value, and bias values. The category utilities describes which of the following utility expressions is used: travel time, travel time and distance, or travel time+. The commonality factor describes the used commonality factor function. Theθvalue influences the scale of the utility, the third column describes this value. The power is the value ofιin the function: Vin =−(T Ti)ι. We keepι= 1in all variants except for the power variants. Theβandγvalues are parameters values in the commonality function. The penalty value describes the used penalty value for route 3. This value is subtracted of the utility of route 3. The road category bias is described in three numbers. The first number describes the highway bias, the second the rural road bias, and the third the ramp bias used in the test.

We start our analysis of the C-logit with a partial derivative of the route choice probability of route 3. The sign of the derivative give us insight in which direction the specific parameter will influence the probability distribution. A positive partial derivative means that if we increase the specific parameter the probability of choosing route 3 will increase. The value of the partial derivative shows the extent in which the parameter in- fluence the route choices. We do not derive the exact value of the partial derivative, because this value depends on many different factors and changes a lot during the assignment step.

52 CHAPTER 6. RESULTS THEORETICAL TEST

Commonality factor functions

The five commonality factor functions produce almost the same commonality factor values, but there are some small mutual differences. To get an idea of the influence of the commonality factor values we derive two partial derivatives of the route choice probability of route 3. The route choice probability of route 3 is given by:

P3= e

θ(V3−CF3)

P3

i=1eθ(Vi−CFi)

WhereVi stand for the utility of routei. In general the utility represents the negative travel time of a route. The partial derivative toCF3is:

∂P3 ∂CF3 =−θe θ(V3−CF3)(eθ(V1−CF1)+eθ(V2−CF2)) (h(CF3))2 Whereh(CF3) =P 3 i=1e θ(Vi−CFi). Note thath(CF

3)2≥0,ex≥0for allx, andθ≥0for alli. Therefore the sign

of the partial derivative ofP3toCF3is negative, which means that the probability of choosing route 3 is lower

for higher commonality factor values of path 3. This result is obvious, since the commonality factor corrects for overrated routes. From the previous result it is easy to see that the partial derivative ∂P3

∂CFi fori = 1,2is

positive. This means that the commonality factor variant that has the highest commonality factor for route 3 in compare to the commonality factor for the other two routes assigns the least traffic to route 3.

The commonality factor values of the five commonality factor functions do not differ much, therefore we expect that the commonality factor has not much influence. The commonality factor of route 3 is the highest for all commonality factor functions. Commonality factor function four has the largest difference between the values for route 1 and 2 and the value of route 3. Therefore we expect that this commonality factor scores the best on reality. The parameter values of this test are given in table 6.1.

Utility Commonality factor θvalue Power βvalue γvalue Penalty value Bias values

all all 1.33 1 1 1 0 [1 1 1]

Table 6.1: The used parameter values for the commonality factor variants

Figure 6.1 shows the results of the theoretical test with different commonality factors. We see that choosing an other commonality factor function influences the reality of the route choice model a little. The fourth commonality factor function scores the best as we expected. The figure also shows that travel time+ scores the best for the parameter settings used in this test.

Figure 6.1: The results for different commonality factor functions

θvalue variants

We vary theθvalue to investigate the influence of the utility scale on the route choices. To calculate the partial derivative ofP(3)toθwe define a, b, and c, the systematic utilities (Vi−CFi) of route 1, 2, and 3. The partial derivative is: ∂P3 ∂θ = ceθ(a+c)+ceθ(b+c)aeθ(a+c)beθ(b+c) (h(θ))2 Whereh(θ) =P3 i=1e

θ(Vi−CFi). Again note thath(θ)0and thata, b, c0. The partial derivative does not

6.1. STATIC RESULTS 53 partial derivative. We see that the partial derivative is positive if the systematic utility of route 3 is largest, this means that in that case the probability of choosing route 3 increases whenθis increased. We can also draw a conclusion for the opposite case, if the systematic utility of route 3 is the smallest, then the probability of choosing route 3 decreases when the value ofθincreases. The systematic utility of route 3 is smaller than the systematic utilities of route 1 and 2, at the start of the assignment. Unfortunately we cannot predict if this stays the same during the total assignment step. And we cannot draw conclusions for cases betweenc≥a, b

andc ≤a, bfrom the partial derivative, but we know the following. In a Multinomial logit with a very low utility scale, the traffic is equally divided among the three routes. The traffic is assigned according the shortest route principle when a very high utility scale is used in the Multinomial logit. The C-logit reacts more or less the same as the Multinomial logit. The only difference between the C-logit and the Multinomial logit is the commonality factor. The commonality factor lowers the utility relatively the same for every scale. Due to the fact that the C-logit is based on absolute differences the C-logit is most influenced by its commonality factor for high scales. Therefore we expect that increasing the scale has a decreasing effect on the amount of traffic on the unrealistic route, and a very large scale will result in less traffic on route 3 than the shortest route based assignment. From the previous chapter we know that the results of the C-logit are influenced by the scale of the utility but we do not know anything about the extent in which the results are influenced by the scale. The test with the parameter values of table 6.2 shows the extent in which the results are influenced by the utility scale.

Utility Commonality factor θvalue Power βvalue γvalue Penalty value Bias values

all all 0...20 1 1 1 0 [1 1 1]

Table 6.2: The used parameter values for theθvariants

Figure 6.2 shows the results for commonality factor 4. The results of the other commonality factor functions produce a slightly higher route share on route 3. The utility function variant travel time+ has not got results forθ values larger than 50, becausee−x for large xgoes to zero, and division by zero is not possible. The travel time+ grows faster than the other utility functions because it uses an additional load to the network to calculate the travel times. In the figure we see that the utility scale has a major influence on the route choice results. Increasing theθvalue from 1 to 5 cuts in half the route choices of the unrealistic routes. The C-logit scores better than the shortest path based assignment forθvalues of 4 and larger. Figure 6.2 also shows that there is not one utility expression that scores the best on realistic routing for every utility scale.

Figure 6.2: The results for theθvariants with commonality factor 4

Power variants

At this variant we do not keep theιvalue at one but vary this value. We investigate this variant analytical by looking at the partial derivative of the route choice probability for route 3. We need to write the utility in terms of travel time because the power is a power of the travel time:

∂P3 ∂ι = −θe−θ(T T3ι+CF3)·T Tι 3·ln(T T3)·(e−θ(T T ι 1+CF1)+e−θ(T T ι 2+CF2)) (h(ι))2 +(θe −θ(T Tι 1+CF1)·T Tι 1·ln(T T1) +θe−θ(T T ι 2+CF2)·T Tι 2·ln(T T2))·e−θ(T T ι 3+CF3) (h(ι))2 Whereh(ι) =P3 j=1e

−θ(T Tjι−CFj)and(h(ι))2 0. Further we know that the travel times of the three routes

54 CHAPTER 6. RESULTS THEORETICAL TEST derivative is positive if:

(θe−θ(T T1ι+CF1)·T Tι 1·ln(T T1) +θe −θ(T T2ι+CF2)·T Tι 2·ln(T T2))·e −θ(T T3ι+CF3)> θe−θ(T T3ι+CF3)·T Tι 3·ln(T T3)·(e−θ(T T ι 1+CF1)+e−θ(T T2ι+CF2))

The sign of the partial derivative is negative if the opposite hold. Clearly the sign of the partial derivativeδPδι(3) depends highly on the travel time of the three routes. We do not have a priori knowledge of the development of the travel times during the assignment step. Therefore we do not make a prediction of the influence ofι

on the total assignment procedure by the partial derivative. But we know that a higher power assigns more weights to differences in the travel times, (for travel times higher than 1) therefore we expect a decrease on the traffic on route 3 for higher powers. Table 6.3 shows the parameter values of the tested power variants.

Utility Commonality factor θvalue Power βvalue γvalue Penalty value Bias values

all all 1.33 0...2 1 1 0 [1 1 1]

Table 6.3: The used parameter values for the utility power variants

We see similar effects on the five commonality factor. A power of one leads to the highest route choice prob- abilities for the third route. The power of the utility has clearly less influence on the route choices than theθ

value. The recommendation of the power of 0.7 from previous research cannot be concluded from this test. There is not one utility expression that scores the best for all utility powers.

Figure 6.3: The results for power variants with commonality factor 4

βandγvalue variants

The parameterβis incorporated in all commonality factor types and can be seen as the scaling factor of the commonality factor. From the partial derivative ∂P∂CF(3) we know that the largerCF3−CF2andCF3−CF1the

smaller the probability of choosing route 3. BecauseCF3> CF1, CF2we expect that increasingβwill decrease

the probability of choosing the route 3.

The parameterγis only used at the first commonality factor function and is the power over the fractionLij

LiLj.

This fraction is always smaller than 1 becausei 6=j. Increasing theγvalue will therefore decrease the com- monality function value. Following the same reasoning as with the β value we know that increasing the commonality factor decreases the probability of choosing route 3, the unrealistic route. Therefore, increasing theγvalue increases the probability of choosing route 3. The subtables of table 6.4 show the tested parameter values for theβvariants, and theγvariants.

(a) The used parameter values forβvariants

Utility Commonality factor θvalue Power βvalue γvalue Penalty value Bias values

all all 1.33 1 0...20 1 0 [1 1 1]

(b) The used parameter value forγvariants

Utility Commonality factor θvalue Power βvalue γvalue Penalty value Bias values

all 1 1.33 1 1 0...10 0 [1 1 1]

Table 6.4: The parameter of the commonality factor variants settings

6.1. STATIC RESULTS 55 value has a strong influence on the route choices. The results are similar for the five commonality factor forms, but the fourth commonality factor function is influenced the most by the value of β. This is obvious from the results of the commonality factor variants. A higherβvalue leads also to less traffic on route 2, which is possible unrealistic. Commonality factor 4 suffers the less of this problem, due to the relative big difference of the commonality factor value for route 2 and 3 for this factor. It is not unambiguous which utility expression scores the best for differentβvalues, see figure 6.4. This figure shows the results forβvariants with common- ality factor 4.

Figure 6.4: The results of theβvariants with commonality factor 4

Commonality factor 1 is the only factor that uses parameterγ. The parameterγhas not much influence. Like we expected lead smallγvalues to the smallest number of traffic on route 3. Figure 6.5 presents the results for variations inγwithβ= 1and aθvalue of 1.

Figure 6.5: The results forγvariants with commonality factor 1

Penalty variants

We expect good results from the penalty function, because the penalty value is able to decrease only the utility of one selected route. We only penalize a route if it consists of a paired off- and on-ramp. By introducing a penalty function for route 3 we lower the utility of route 3. To analyze the influence of the penalty function we derive the partial derivative of the probability function of route 3 with respect to the utility function of route 3.

∂P3 ∂V3 = θe θ(V3−CF3)·(eθ(V1−CF1)+eθ(V2−CF2)) (h(V3))2 Whereh(V3) = P3 i=1e

θ(V1−CF1) , and(h(V3))2 0. So the sign of the partial derivative is positive, which

means that increasing the penalty and thus decreasing the utility will decrease the probability of route 3. We expect that the influence of the penalty value is huge, because there are small differences between the travel times of the three routes. The used penalty values and the other parameter settings are displayed in table 6.5.

Utility Commonality factor θvalue Power βvalue γvalue Penalty value Bias values

all all 1.33 1 1 1 0...5 [1 1 1]

Table 6.5: The used parameter values for the penalty variants

56 CHAPTER 6. RESULTS THEORETICAL TEST that a desired amount of traffic on route 3 is reached. For instance, to reach a smaller amount than 100 vehicles on the unrealistic third route for every utility function and commonality factor, we have to increase the penalty value to 2.5. The major part of the reduce of traffic on route 3 goes to route 2. Figure 6.6 presents the amount of traffic on route 3 for the penalty variants with commonality factor 4.

Figure 6.6: The results for penalty variants with commonality factor 4

Road category bias variants

The road category bias is multiplied by the link travel times. We saw in the previous section that the route share of route 3 decreases if the utility of route 3 decreases and the other utilities are stable. We must multiply the link times of route 3 with higher bias values than the other routes to decrease the utility of route 3. Route 3 has the most on and off ramps of the three roads. Therefore, increasing the ramp category bias will influence the route shares on route 3 the most. We expect that an increase in the ramp bias leads to less traffic on the unrealistic route 3. We expect that an increase in the ramp bias also leads to a decrease of traffic using route 2. This effect can be compensated by a low highway bias, but we expect that a low highway bias increases the amount of traffic on route 3 back to the initial value. Table 6.6 displays the used parameter settings. We do not test ramp bias values which are lower than the highway bias, because we do not want to favor ramp links above highway links.

Utility Commonality factor θvalue Power βvalue γvalue Penalty value Bias values

all all 1.33 1 1 1 0 [0.85 0.9 0.85...4]

& [0.6 0.9 0.6...4]

Table 6.6: The used parameter values for the road category bias variants

Figure 6.7: The results of the road category bias variants with bias [0.85 0.9 0.85...4]

Fortunately the road category bias can result in less traffic on route 3 and much traffic on the highway. Adapt- ing the highway bias has more influence on the amount of traffic on route 2 than on the amount of traffic on route 3, for high ramp biases. For instance for a highway bias of 0.85 and 0.6, an urban road bias of 0.9 and a ramp bias on 4 we found the following result: The traffic on route 3 is low for both the highway biases: 29 and 27 vehicles, but there is a significant difference between the amount of traffic on route 2: 1183 and 1342 vehicles (commonality factor 4). Figure 6.7 displays the result for commonality factor 4 with a highway bias of 0.85, a rural road bias of 0.9 and different ramp bias values. In compare to the penalty function has the road

6.1. STATIC RESULTS 57 category bias more influence, because it influences the route choice probabilities of all routes. This is also a disadvantage of the road category bias, because the road category bias is hard to calibrate.

Combinations ofθandβvariants

We expect that the decreasing effect on route 3 choices due to increasingβ value exists for allθ values. We expect that the influence of theβvalue is higher in combination with highθvalues, because the commonality factors are also multiplied by theθvalue. So an increase in theθvalue leads indirect to an increase in the scale of the commonality factors. Table 6.7 shows the used parameter values for theβandθcombination variants.

Utility Commonality factor θvalue Power βvalue γvalue Penalty value Bias values

all all 0...20 1 1...5 1 0 [1 1 1]

Table 6.7: The used parameter values for the combination ofθandβvalue variants

The results of this test show that the decreasing effect of theβ value on the traffic on route 3 indeed exists for all different utility scales. The decreasing effect of high beta values is obvious the highest for large utility scales. Figure 6.8 shows the results for commonality factor 4 and utility expression travel time.

Figure 6.8: The results for a combination ofθandβvariants with commonality factor 4

Combination ofθand penalty variants

The penalty variant has a strong decreasing effect on the unrealistic route choices. We expect that the penalty value has the most influences on the lowest scale value and thus lowestθ value, simply because a penalty value of 2 means more to a utility of scale 1 than to a utility of scale 100. The tested parameter values for the

In document Realistic route choice modeling (Page 51-59)

Related documents