Solution to the overall EMFLP

As illustrated in Section (4.3), the solution to the FLP relies on the assumption of knowing the evaluated values of the Lagrangian parameters, whilst the solution to the MCP is based on the assumption of knowing the set of located IMTs. Thus, one sub-problem cannot be solved without knowing the solution of the other. However, the decomposition allows Algorithm A1

for solving the MCP to be embedded in any of the general enumeration algorithms, such as branch and bound (B&B) or complete enumeration (CE), to solve the overall EMFLP to optimality. Considering the fact that the metropolitan region is the main study area for the application of the proposed model, CE based algorithm A2 with embedded algorithm A1 is practical for solving the overall EMFLP. Note that the size or cardinality of the set 𝒰, which is the set of all subsets of the candidate IMTs 𝒯 with cardinality 𝑝 is polynomially bounded by:

|𝒰| = (𝜏

𝑝) = 𝜏!

𝑝! (𝜏 − 𝑝)!= 𝑂(𝜏^𝑝) (4.49)

where 𝜏 is the cardinality of the set 𝒯. For example, if the analyst is interested in locating two IMTs, then the number of possible evaluations of the MCP is bounded by 𝜏(𝜏 − 1). Also, Bregman's balancing method for solving MCP converges in polynomial time (Lamond and Stewart 1981) to an acceptable level of accuracy, making the use of algorithm A2 efficient.

Once the EMFLP is decomposed, the application of the algorithm A2 is straight forward. This is because, once the set 𝒦 (set of p IMTs) is known, constraints (4.20) and (4.23) are automatically satisfied; the rest of the constraints are satisfied by solving the MCP for the given 𝒦. The algorithm A2 is presented as follows:

Algorithm A2: Solution by complete enumeration

1. Initialization: 𝒦 = {0}, Λ^∗= −∞, 𝒦^∗ the set with the optimum IMT sites with associated objective value Λ^∗

2. For each subset 𝒦 ∈ 𝒰 of size p, with the location variable 𝑌_𝑡= 1; ∀𝑡 ∈ 𝒦 ; 𝑌_𝑡= 0 ∀𝑡 ∉ 𝒦 do:

2.1. Solve the MCP using algorithm A1 for the flow variables and the Lagrangian parameters 2.2. Compute Λ_𝑅 using the overall objective function in Equation (4.25)

2.3. If Λ𝑅> Λ^∗, then Λ^∗ = Λ𝑅 and 𝒦^∗ =𝒦 3. Repeat step (2) for all subsets of 𝒰 and stop

4. Set 𝑌_𝑡= 1, ∀ 𝑡 ∈ 𝒦^∗ and 𝑌_𝑡= 0, ∀ 𝑡 ∉ 𝒦^∗

Proposition 4.7: For simplicity let 𝑃_𝑖𝑗 = Pr(𝑋_𝑖𝑗); and 𝑃_𝑖𝑡𝑗 = Pr(𝑉_𝑖𝑡𝑗). Maximising Λ (the entropy objective function of EMFLP), is equivalent to maximising the weighted expected utility or welfares of all shippers subject to the given transport budget.

Proof 4.7: Using the definitions of probabilities in equations (4.37) and (4.38), the entropy function in (4.22) Λ can be re-expressed as:

Λ = ∑ ∑ 𝑞_𝑖𝑗𝑃_𝑖𝑗{1 − ln(𝑞_𝑖𝑗𝑃_𝑖𝑗)}

Expanding, grouping like terms and using the second axiom of probability we have:

Λ = ∑ ∑ 𝑞_𝑖𝑗

Expanding the terms in the logarithm function and grouping like terms we have:

Λ = ∑ ∑ 𝑞_𝑖𝑗

Using Sterling’s approximation, the above can be simplified as:

Λ = − ∑ ∑ 𝑞_𝑖𝑗!

The term − ∑_𝑖∈𝒪 ∑_𝑗∈𝒟ln𝑞_𝑖𝑗! is constant and can be ignored in the optimisation process, since 𝑞_𝑖𝑗 (input data) are not decision variables. Thus,

Λ ≈ − ∑ ∑ 𝑞_𝑖𝑗𝑃_𝑖𝑗ln𝑃_𝑖𝑗− ∑ ∑ 𝑞_𝑖𝑗∑ 𝑃_𝑖𝑡𝑗ln𝑃_𝑖𝑡𝑗

𝛽 is the terminal total user fee passed on to the shipper, who then decides whether or not to use the terminal and comprises the original user cost and a shadow price ^𝜓𝑡

𝛽 to dissuade enough users from using IMTs with insufficient handling capacities (ψ_t >

0 ). The shadow price is treated as an out of pocket cost and forms part of the terminal usage cost or rental passed on to the shipper. Replacing the probabilities in (4.50) with those in (4.39) and (4.43) and using (4.37) and (4.38), the entropy function in (4.50) simplifies to become:

Λ ≈ ∑ ∑ 𝐿_𝑖𝑗𝑞_𝑖𝑗

Thus, maximising entropy Λ is equivalent to maximising the weighted maximum expected utility or weighted consumer surplus subject to the given transport budget:

max ∑ ∑ 𝐿_𝑖𝑗𝑞_𝑖𝑗

𝑗∈𝒟 𝑖∈𝒪

Subject to the transport budget constraint (4.18):

∑ ∑ ∑ 𝑐̃_𝑖𝑡𝑗𝑉_𝑖𝑡𝑗

where 𝐿_𝑖𝑗 is the maximum expected utility (see Equation (4.44); Batty 2010; Williams 1977) or consumer surplus (Train 2009; De Jong et al. 2005). The proposition implies that, the maximum entropy yields the maximum consumer surplus or shippers’ welfares.

Preposition 4.8: Comparing the solution of MCP and equivalent LP solution

For any given set of located IMTs 𝒦, the EMFLP reduces to the MCP and the MILP reduces to equivalent linear programming (LP) solutions or equivalently the solution to EMFLP reduces to the mixed integer linear programming (MILP) solution as 𝛽 → ∞.

Proof 4.8: By using the generalised cost definition 𝑐̃_𝑖𝑡𝑗 in (4.51) in Equations (4.37) and (4.38) the flow variables can be estimated directly using:

𝑉_𝑖𝑡𝑗^∗ = 𝑞_𝑖𝑗 𝑒^{−𝛽𝑐̃𝑖𝑡𝑗}

∑_𝑡∈𝒦 𝑒^{−𝛽𝑐̃𝑖𝑡𝑗} + 𝑒^{−𝛽𝑐𝑖𝑗}; ∀ 𝑖 ∈ 𝒪; 𝑗 ∈ 𝒟, 𝑡 ∈ 𝒦 (4.53)

𝑋_𝑖𝑗^∗ = 𝑞_𝑖𝑗 𝑒^{−𝛽𝑐𝑖𝑗}

∑_𝑡∈𝒦 𝑒^{−𝛽𝑐̃𝑖𝑡𝑗}+ 𝑒^{−𝛽𝑐𝑖𝑗} ; ∀ 𝑖 ∈ 𝒪; 𝑗 ∈ 𝒟 (4.54)

From constraint (4.18), the total budget used 𝑐 can be expressed as:

𝑐 = ∑ ∑ (∑ 𝑐̃_𝑖𝑡𝑗𝑉_𝑖𝑡𝑗^∗

𝑡∈𝒦

+𝑐_𝑖𝑗𝑋_𝑖𝑗^∗)

𝑗∈𝒟 𝑖∈𝒪

= ∑ ∑ 𝐶_𝑖𝑗

𝑗∈𝒟 𝑖∈𝒪

where

𝐶_𝑖𝑗 = ∑ 𝑐̃_𝑖𝑡𝑗𝑉_𝑖𝑡𝑗^∗

𝑡∈𝒦

+𝑐_𝑖𝑗𝑋_𝑖𝑗^∗

Using Equations (4.53) and (4.54), the above cost equation can be expressed as:

1

𝑞_𝑖𝑗𝐶_𝑖𝑗 = 1

∑_𝑡∈𝒦 𝑒^{−𝛽𝑐̃𝑖𝑡𝑗}+ 𝑒^{−𝛽𝑐𝑖𝑗} (𝑐_𝑖𝑗𝑒^{−𝛽𝑐𝑖𝑗}+ ∑ 𝑐̃_𝑖𝑡𝑗 𝑒^{−𝛽𝑐̃𝑖𝑡𝑗}

𝑡∈𝒦

) (4.55)

Suppose that the transport costs 𝑐̃_𝑖𝑡𝑗 and 𝑐_𝑖𝑗 are kept fixed, leaving the origin-destination average budget 𝐶_𝑖𝑗 and 𝛽 so that if the budget changes 𝛽 must also change and vice versa. It is clear from Equation (4.55) that as 𝛽 → ∞ the term with the smallest cost (𝑐^∗_𝑖𝑗) become the biggest term in both the numerator and denominator on the right hand-side of equation (4.55).

Thus as 𝛽 → ∞, Equation (4.55) reduces to:

1

𝑞_𝑖𝑗𝐶_𝑖𝑗 → 𝑐^∗_𝑖𝑗 𝑒^{−𝛽𝑐∗𝑖𝑗}

𝑒^{−𝛽𝑐∗𝑖𝑗} = 𝑐^∗_𝑖𝑗

which simplified to become:

𝐶_𝑖𝑗 → 𝑞_𝑖𝑗𝑐^∗_𝑖𝑗

Hence the total used budget 𝑐 over all origin-destination pairs as 𝛽 → ∞ becomes:

𝑐 = ∑ ∑ 𝐶_𝑖𝑗

𝑗∈𝒟 𝑖∈𝒪

→ ∑ ∑ 𝑞_𝑖𝑗𝑐^∗_𝑖𝑗

𝑗∈𝒟 𝑖∈𝒪

(4.56)

which is the optimal solution to equivalent MILP, which assigns all flows to the least cost mode. Thus as 𝛽 → ∞, the solution to the entropy model reduces to the solution of the MILP.

A recent study by Teye et al. (2017) has demonstrated the unsuitability of MILP is locating multi-user facilities of this kind (IMTs) as it was shown to produce unrealistic large responses during farecasting and policy testing.

Proposition 4.9: The budget attains its largest possible value as 𝛽 → 0

Proof 4.9: Suppose also that 𝛽 → 0, then all the exponential terms tend to toward unity and equation (4.55) reduces to:

1

𝑞_𝑖𝑗𝐶_𝑖𝑗 → ∑ 𝑐̃_𝑖𝑡𝑗 1 (∑_𝑡∈𝒦 1) + 1

𝑡∈𝒦

+𝑐_𝑖𝑗 1 (∑_𝑡∈𝒦 1) + 1

Or

𝑐 → 1

The numerator becomes the sum of the cost values of all modes and the denominator is just the total number of modal alternatives. It means that if there is no limit on modal costs, then the budget will tend to infinity. The above analysis gives us some idea of what influence 𝛽 has on the budget. If 𝛽 is large, the total budget must be small; if 𝛽 is small the total budget must be large. In the limit as 𝛽 approach infinity the budget takes on a minimum value and as 𝛽 approach zero, the budget takes on the maximum value. This suggests that 𝛽 is inversely related to the total budget.