Parameter estimation in the CFP

Several simpler models can be derived from the models developed in (6.25) and (6.33) depending on the availability of data. The key ones are summarised as follows:

Model I (CFP1): No available information on cargo distribution 𝑞_𝑖𝑗, production 𝑞_𝑖, total cargo in the system 𝑍 or the quantity of cargo arriving at each destination 𝑑_𝑗 and no known factors governing the flows of cargo. In such situations, the structural parameters 𝜆_𝐷, 𝜆_𝐺 can each be set to 1 and the Lagrangian parameters 𝜃_𝑙, 𝜙_𝑘, 𝛾_0𝑖, 𝛾_1𝑗, 𝛾_2𝑖 and 𝛾₃ set to zero. The production model in (6.38) and the distribution model in (6.37) will be explained by only transport network conditions which is expressed in terms of modal (road alone and intermodal) access to each destination or cargo production zone:

Pr(𝑄_𝑖𝑗) = exp (𝐿_𝑖𝑗)

∑_𝑗∈𝒟exp (𝐿_𝑖𝑗); ∀ 𝑖 ∈ 𝒪, 𝑗 ∈ 𝒟 (6.37)

𝑄_𝑖 = exp(𝐿_𝑖); ∀ 𝑖 ∈ 𝒪 (6.38)

Model II (CFP2): Only information on total cargo 𝑍 in the system is available. Here, the resulting cargo distribution model is the same as that under CFP1 or Equation (6.37). The production model accounts for the observed 𝑍 by incorporating the Lagrangian parameter 𝛾₃ in Equation (6.38). Thus, the resulting distribution models in (6.37) is explained by only transport network conditions whilst the production model (6.39) is expressed in terms of both network conditions and a constant.

𝑄_𝑖 = exp(𝛾₃+ 𝐿_𝑖); ∀ 𝑖 ∈ 𝒪 (6.39)

Model III (CFP3): Only information on cargo production 𝑞_𝑖 is available. Observing 𝑞_𝑖 also means 𝑍 is observed since by definition 𝑍 = ∑_𝑖∈𝒪𝑄_𝑖 . Again, the distribution model in Equation (6.37) remains unchanged but the production model is updated with the new information as Equation (6.40):

𝑄_𝑖 = exp(𝛾₃+ 𝛾_2𝑖+ 𝜆_𝐺𝐿_𝑖); ∀ 𝑖 ∈ 𝒪 (6.40)

It can be observed that both 𝛾₃ and 𝛾_2𝑖 are constants so they can be combined into one constant:

𝛾̃_2𝑖 = 𝛾₃+ 𝛾_2𝑖.The parameter 𝛾₃ can be normalised to zero (𝛾₃ = 0) resulting in 𝛾̃_2𝑖 = 𝛾_2𝑖. Thus, estimating 𝛾̃_2𝑖 can be treated as estimating 𝛾_2𝑖. Equation (6.40) therefore simplifies to become:

𝑄_𝑖 = exp(𝛾̃_2𝑖+ 𝜆_𝐺𝐿_𝑖); ∀ 𝑖 ∈ 𝒪 (6.41)

Model IV (CFP4): Available data on cargo production in each zone 𝑞_𝑖 with information on production factors 𝒢. Again, observing 𝑞_𝑖 also means 𝑍 is observed since by definition 𝑍 = ∑_𝑖∈𝒪𝑄_𝑖. The distribution model is the same as that of CFP4 or Equation (6.37). The resulting production model becomes:

𝑄_𝑖 = exp (𝛾̃_2𝑖+ 𝜆_𝐺𝐿_𝑖 + ∑ 𝜙_𝑘𝑔_𝑖𝑘

𝑘∈𝒢

) ; ∀ 𝑖 ∈ 𝒪

(6.42)

The parameters 𝜆_𝐺, 𝜙_𝑘 in the model can be estimated using Poisson quasi-maximum likelihood estimator (QMLE) (Cameron and Trivedi 2013) and the 𝛾̃_2𝑖 estimate the same way as in CFP3.

Model V (CFP5): Only information regarding the quantity of cargo arriving at each destination 𝑑_𝑗 are available. Knowing 𝑑_𝑗 also implies that 𝑍 is known. The resulting production model is same as Equation (6.39) under CFP2. The distribution model in (6.27) with 𝜆_𝐷 = 1 reduces to:

Pr(𝑄_𝑖𝑗) = exp (𝛾_1𝑗+ 𝐿_𝑖𝑗)

∑_𝑗∈𝒟exp (𝛾_1𝑗 + 𝐿_𝑖𝑗)

(6.43)

The estimation of the parameters 𝛾_1𝑗 were described under Corollary 6.4.

Model VI (CFP6): Available data on the quantity of cargo arriving at each destination 𝑑_𝑗 and information on distribution factors ℋ. The resulting production model is the same as Equation (6.39) under CFP2. The distribution model in (6.27) with 𝜆_𝐷 = 1 reduces to:

Pr(𝑄_𝑖𝑗) = exp (𝛾_1𝑗+ 𝐿_𝑖𝑗 + ∑_𝑙∈ℋ𝜃_𝑙𝑎_𝑗𝑙)

∑_𝑗∈𝒟exp (𝛾_1𝑗 + 𝐿_𝑖𝑗+ ∑_𝑙∈ℋ𝜃_𝑙𝑎_𝑗𝑙)

(6.44)

The parameters 𝜃_𝑙 representing the weight or importance associated with each attraction parameter 𝑙 ∈ ℋ and can be estimated by enforcing constraint (6.13):

𝑓(𝜃_𝑙) = ∑ ∑ 𝑄_𝑖𝑗𝑎_𝑗𝑙

𝑖∈𝒪 𝑗∈𝒟

− 𝐴_𝑙= 0; ∀ 𝑙 ∈ ℋ or

𝑓(𝜃_𝑙) = ∑ ∑ 𝑄_𝑖( exp (𝛾_1𝑗+ 𝐿_𝑖𝑗 + ∑_𝑙∈ℋ𝜃_𝑙𝑎_𝑗𝑙)

∑_𝑗∈𝒟exp (𝛾_1𝑗+ 𝐿_𝑖𝑗 + ∑_𝑙∈ℋ𝜃_𝑙𝑎_𝑗𝑙)) 𝑎_𝑗𝑙

𝑖∈𝒪 𝑗∈𝒟

− 𝐴_𝑙 = 0; ∀ 𝑙 ∈ ℋ (6.45)

The functions 𝑓(𝜃_𝑙) are continuous and differentiable with respect to 𝜃_𝑙 and can be optimised using Newton Raphson’s or Hyman (1969) methods to estimate 𝜃_𝑙. Several numerical examples show that Hyman method is more computational efficient and stable.

Model VII (CFP7): Available data on the quantity of cargo distributed between zones 𝑞_𝑖𝑗 with available information on production 𝒢 and distribution factors ℋ. Knowing 𝑞_𝑖𝑗 means 𝑑_𝑗, 𝑞_𝑖 and 𝑍 are known. The resulting models could be considered as full information models. The resulting production model is Equation (6.33) and the distribution model is Equation (6.27).

The parameters in these models can be estimated using maximum likelihood estimator (MLE) or the Poisson quasi-MLE (QMLE) or other appropriate estimators such as Bayesian. The estimated parameters include the structural parameters 𝜆_𝐷, 𝜆_𝐺.

Table 6.1 provides a summary of the above seven production and distribution models.

As shown in Table 6.1 other combinations of production and distribution models can also be achieved depending on data availability.

Table 6.1: Summary of cargo production and distribution models

Data availability

Production model Distribution model Comments

1 No information

It has been demonstrated that the CFP can be reduced to several models depending on the availability of data. The CFP model comprises the mode choice problem (MCP), the variable cargo demand problem (VDP), which consists of the cargo production problem (CPP) and the cargo distribution problem (CDP). The MCP was discussed in Chapter 5 and was shown to be governed by the cost sensitivity parameter 𝛽 and the IMT capacity constraint parameters 𝜓_𝑡. The parameters governing the CDP and the CPP have been discussed above and the estimation of these parameters depends on data availability. As shown above the parameters in the CFP are inter-dependent, where the evaluated value of one is required to solve the other. The modified Bregman’s algorithm (A1) used in Chapter 4 for estimating the parameters in the MCP is expanded to include the estimation of parameters in the distribution and production problems. The expanded Bregman's algorithm for solving the CFP is presented as algorithm A4. This algorithm will also prove useful in the model application stage.

Algorithm A4: Modified Bregman’s algorithm for solving the CFP

1. Initialisation:

For a given set of located IMTs 𝒦 with size 𝑝 and starting cost sensitivity parameter 𝛽 = ¹

𝑐̅ , where 𝑐 ̅can be the average transport budget and 𝜓_𝑡= 0; ∀ 𝑡 ∈ 𝒦, 𝜆_𝐷 = 𝜆_𝐺= 1, 𝛾₃= 0, 𝛾_2𝑖= 0; ∀ 𝑖 ∈ 𝒪 and 𝛾_1𝑗 = 0; ∀𝑗 ∈ 𝒟

2. Logsums Update

2.1 Update logsums over all located IMTs ℓ_𝑖𝑗 using Equation (5.28) in Chapter 5 2.2 Update the logsums 𝐿𝑖𝑗 over all transport modes using Equation (5.30) in Chapter 5

2.3 Estimate the parameters in the CDP depending on the choice of distribution model (CFP1 to CFP7) 2.4 Update the associated logsums 𝐿𝑖 over all destination zones using Equation (6.28)

2.5 Estimate the parameters in the CPP depending on the choice of production model (CFP1 to CFP7) 3. Flows Update

3.1 Update the quantity of cargo produced in each origin zone depending on the choice of production model (CFP1 to CFP7)

3.2 Update the distribution of cargo between zones depending on the choice of production model (CFP1

to CFP7) together with Equation (6.25).

3.3 Update the demand for each mode; 𝑋_𝑖𝑗, 𝑉_𝑖𝑡𝑗, 𝑊_{𝑖𝑠𝑡𝑗} using Equations (5.25), (5.24) and (5.23) respectively in Chapter 5.

3.4 Update the demand for the located terminals using intermodal transport demands 𝑉_𝑖𝑡𝑗, 𝑊_{𝑖𝑠𝑡𝑗} or Equation (5.22b) in Chapter 5.

4. Update model parameters

4.1. Update 𝛽 from equation (5.36) using Newton Raphson or Hyman’s method (Hyman 1969) 5. Update capacity constraints parameters

5.1. Update capacity constraint parameters associated with the CPP; 𝛾₃ or 𝛾_2𝑖 5.2. Update capacity constraint parameters associated with the CDP; 𝛾1𝑗

5.3. Update the Lagrangian multipliers 𝜓𝑡; ∀ 𝑡 ∈ 𝒦 for IMT capacity constraints 6. Repeat steps (2)-(5) until convergence is achieved.