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𝑗∈𝒟 𝑖∈𝒪

+ ∑ ∑ 𝛿_𝑖𝑗3𝑐_{𝑖𝑠𝑡𝑗}𝑒^{−(𝛽𝑐𝑖𝑠𝑡𝑗+𝜓𝑠+𝜓𝑡)}

𝑡≠𝑠∈𝒦 𝑠∈𝒦

} − 𝑐 ≤ 0

The function 𝑓(𝛽) is continuous and differentiable with respect to 𝛽, and so it can be maximised to obtain the value of 𝛽.

Again, the estimation of the Lagrangian multipliers in the MCP are inter-dependent, where the evaluated value of one is required to solve the other. Algorithm (A1) proposed in Chapter 4 can readily be adapted and used to estimate the MCP parameters. The next section presents the algorithm for solving the overall problem EMFLP by connecting the two sub-problems.

5.4.3 Solving the EMFLP by complete enumeration

The complete enumeration algorithm A2 proposed in Chapter 4 can also be adapted and used to solve the overall problem. The use of this algorithm to solve the generalized IMT location problem could still be practical for many real world problems. As shown in Equation (4.49) the running time of algorithm A2 is bounded by 𝑂(𝜏^𝑝𝑇_𝐵) where 𝑇_𝐵 is the running time of the modified Bregman’s algorithm (A1) and 𝜏^𝑝 gives the maximum running time of selecting 𝑝 IMTs from a candidate set of 𝒯 with cardinality 𝜏.

However, the study area for locating regional IMTs is often very large and may encompass a whole country like Australia, US or China and regions like the European Union.

For example, assume there 100 candidate IMT locations in the study area and the planner wants to select the best 10 for the development of IMTs. If it takes about 10^-3seconds to solve the MCP, algorithm A2 will about 560 years to find the best 10 IMT locations. Such large scale applications will benefit from more efficient algorithm than A2. A new fast and efficient heuristic is proposed in the next section for solving such larger problem instances.

5.4.4 Combined solution of FLP and MCP by heuristics

For large problem instances, a fast algorithm is proposed and its solution quality demonstrated with respect to algorithm A2 through extensive numerical examples. The heuristic algorithm is

motivated by the convexity of the objective entropy function and the principle of conditional entropy in Propositions 5.2 and 5.3. The key assumption underlying the heuristic is that if 𝒦^∗ is the set with the optimal IMT locations with IMT 𝜗 ∈ 𝒦^∗ then IMT 𝜗1 ∈ 𝒯 must also be in 𝒦^∗ if:

𝐻(𝑌_𝜗, 𝑌_𝜗1) ≥ 𝐻(𝑌_𝜗, 𝑌_𝑡); ∀𝑡 ∈ 𝒯, 𝜗 ≠ 𝜗1 (5.37)

where 𝐻(𝑌_𝜗, 𝑌_𝜗1) is the entropy of locating IMTs at locations 𝜗 and 𝜗1. Thus, following Proposition 5.2 the selection of IMT location 𝜗1 is conditioned on the selection of IMT location 𝜗 ∈ 𝒯. The remaining 𝑝 − 2 IMTs locations are selected in a similar way. For example, the selection of the third IMT location 𝜗2 is conditioned on knowing that IMT locations 𝜗 and 𝜗1 were selected (or in the set 𝒦^∗):

𝐻(𝑌_𝜗, 𝑌_𝜗1, 𝑌_𝜗2) ≥ 𝐻(𝑌_𝜗, 𝑌_𝜗1, 𝑌_𝑡); ∀𝑡 ∈ 𝒯, 𝜗2 ≠ 𝜗, 𝜗2 ≠ 𝜗1 (5.38)

The key question that remains is how to select location 𝜗 ∈ 𝒯 as it conditions the selection of the remaining 𝑝 − 1 locations. We suggest considering all candidate IMT locations and selecting the one with highest entropy. That is select location 𝜗^∗ ∈ 𝒯 if

𝐻(𝑌_𝜗^∗, 𝒦₁) ≥ 𝐻(𝑌_𝜗, 𝒦₁); ∀𝜗 ∈ 𝒯 (5.39)

where 𝒦₁ is the set containing the remaining 𝑝 − 1 IMT locations.

Proposition 5.2. By definition each location variable 𝑌_𝑡; 𝑡 ∈ 𝒯 takes on two values; 0 and 1 and let ℐ = {0,1}. If the Shannon entropy (see proposition 5.4) of a location variable 𝑌₁ is defined as 𝐻(𝑌₁) = ∑_𝑎∈ℐ𝑃𝑟(𝑌₁ = a)ln𝑃𝑟(𝑌₁ = a), then the joint entropy for locations variables 𝑌₁ and 𝑌₂ can be expressed as:

𝐻(𝑌₁, 𝑌₂) = 𝐻(𝑌₁) + 𝐻(𝑌₂|𝑌₁)

(5.40)

Proof 5.2. By definition the joint entropy of 𝑌₁ and 𝑌₂ becomes:

𝐻(𝑌₁, 𝑌₂) = − ∑ ∑ 𝑃𝑟(𝑌₁ = a, 𝑌₂ = b)ln𝑃𝑟(𝑌₁ = a, 𝑌₂ = b)

𝑏∈ℐ 𝑎∈ℐ

Or for simplicity:

Expanding and using the properties of marginal probability distributions we have:

𝐻(𝑌₁, 𝑌₂) = − ∑ 𝑃𝑟(a)ln𝑃𝑟(a)

Proof 5.3 by induction. From proposition 5.2, equation (5.40) is true for 𝜏 = 2. Assume proposition 5.3 is also true for any 𝜏 then using chain rule:

𝐻(𝑌₁, 𝑌₂, … , 𝑌_𝜏, 𝑌_𝜏+1) = 𝐻(𝑌₁, 𝑌₂, … , 𝑌_𝜏) + 𝐻(𝑌_𝜏+1|𝑌₁, 𝑌₂, … , 𝑌_𝜏) algorithm A3 and labelled as entropic greedy algorithm (EGA).

Algorithm A3: Entropic greedy algorithm (EGA)

1. Initialization: 𝒦 = 𝒦^∗= {}, Λ^∗= −∞; 𝒯₁= 𝒯 2. While the candidate set of IMTs 𝒯 is not empty

3. Choose an IMT 𝑎 ∈ 𝒯 and delete it from 𝒯, i.e., 𝒯 = 𝒯 − {𝑎} IMTs 𝒦. It is clear that the running time of the heuristic algorithm A3 will be dominated by the number executions of algorithm A1 in line 15 of the inner while loop. Lines 13 to 17 will be executed at most 𝜏²𝑝 times each and given the fact that the execution time of Algorithm A1

will dominate, it suffices that the running time of lines 13 to 17 will be bounded by 𝑂(𝑝𝜏²𝑇_𝐵).

Similarly, lines 3 to 7 will be executed at most 𝜏 times each with the total running time bounded

by 𝑂(𝜏𝑇_𝐵). Finally, lines 23 to 26 will also be executed at most 𝜏 times each with a total running time also bounded by 𝑂(𝜏𝑇_𝐵). Therefore, the overall running time of the proposed heuristics algorithm, A3 is bounded by 𝑂(𝑝𝜏²𝑇_𝐵). Alternatively, the number of executions of Algorithm A1 for solving MCP is bounded by 𝑂(𝑝𝜏²) compared with 𝑂(𝜏^𝑝) for the enumeration Algorithm A2. Thus, the running time savings of Algorithm A3 (heuristic) with respect to A2

(enumeration algorithm) occurs when 𝑝 ≥ 3.

We have shown that the proposed heuristics Algorithm A3 has a polynomial running time, which increases linearly with increasing 𝑝 (number of IMTs to locate) and therefore computationally efficient for solving larger problem instances. Algorithm A3 is optimal for 𝑝 = 1, and 𝑝 = 2 since it reduces to Algorithm A2, which is a global optimal algorithm. The conjecture is whether it is also optimal for 𝑝 ≥ 3. The quality of solutions produced by the heuristics is demonstrated with extensive numerical examples in Section 5.5.

Proposition 5.4. For simplicity let 𝑃_𝑖𝑗 = Pr(𝑋_𝑖𝑗); 𝑃_𝑖𝑡𝑗 = Pr(𝑉_𝑖𝑡𝑗) and 𝑃_{𝑖𝑠𝑡𝑗} = Pr(𝑊_{𝑖𝑠𝑡𝑗}) . Maximising Λ (the objective function of EMFLP), is equivalent to maximising the Shannon entropy 𝐻:

Proof 5.4. Using the definitions of probabilities in equations (23)-(25), Λ can be re-expressed as:

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

Λ = ∑ ∑ 𝑞_𝑖𝑗

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

Λ = ∑ ∑ 𝑞_𝑖𝑗 𝑞_𝑖𝑗 (input data) are not decision variables. Hence maximising Λ is equivalent to maximising the entropy 𝐻 with respect to the decision variables:

𝐻 = − ∑ ∑ 𝑞_𝑖𝑗𝑃_𝑖𝑗ln𝑃_𝑖𝑗 − ∑ ∑ 𝑞_𝑖𝑗∑ 𝑃_𝑖𝑡𝑗ln𝑃_𝑖𝑡𝑗− ∑ ∑ 𝑞_𝑖𝑗∑ ∑ 𝑃_{𝑖𝑠𝑡𝑗}ln𝑃_{𝑖𝑠𝑡𝑗}

with the first, second and third terms being the Shannon entropies for road alone, metropolitan and regional intermodal transport decision variables. The above equation can be simplified further by defining the set of elementary modal alternatives; 𝒮 = {{0, 𝑠, 𝑡}, ∀𝑠 ∈ 𝒯; 𝑡 ∈ 𝒯}, where {0} is the index road alone alternative. The subset {{0, 𝑠, 𝑡}, ∀𝑠 = 𝑡 ∈ 𝒯} represents modal alternatives for the metropolitan transport market whilst {{0, 𝑠, 𝑡}, ∀𝑠 ≠ 𝑡 ∈ 𝒯} represent the modal alternatives for the regional transport market:

𝐻 = − ∑ ∑ 𝑞_𝑖𝑗 ∑ 𝑃_𝑖𝑚𝑗ln𝑃_𝑖𝑚𝑗

𝑚∈𝒮 𝑗∈𝒟

𝑖∈𝒪

(5.42)

Proposition 5.4. Maximising entropy facility location problem (EMFLP) with objective function Λ is equivalent to maximising total welfare:

max ∑ ∑ 𝐿_𝑖𝑗𝑞_𝑖𝑗

𝑗∈𝒟 𝑖∈𝒪

Subject to the transport budget constraint (5.5):

∑ ∑ ∑ ∑ 𝑐̃_{𝑖𝑠𝑡𝑗}𝑊_{𝑖𝑠𝑡𝑗}

where 𝐿_𝑖𝑗 is the maximum expected utility in Equation (5.30) interpreted as accessibility in Batty (2010) and Williams (1977) or consumer surplus in Train (2009) and De Jong et al.

(2005). The cost variable 𝑐̃_𝑖𝑡𝑗 has its usual meaning in Chapter 4. The variable 𝑐̃_{𝑖𝑠𝑡𝑗} represent the cost of regional intermodal transport: 𝑐̃_{𝑖𝑠𝑡𝑗} = 𝑐_𝑖𝑠+ 𝑐̃_𝑠+ 𝑐_𝑠𝑡+ 𝑐̃_𝑡+ 𝑐_𝑡𝑗 where 𝑐̃_𝑠 = 𝑐_𝑠 +

𝜓𝑠

𝛽 and 𝑐̃_𝑡= 𝑐_𝑡+^𝜓𝑡

𝛽 with ^𝜓𝑠

𝛽 and ^𝜓𝑡

𝛽 been the shadow prices ($ per TEU) associated with terminal 𝑠 and 𝑡 respectively.

Proof 5.4. The proof follows directly from Proposition 4.7 in Chapter 4.