The Entropy Maximising Facility Location Problem

The goal is to determine the most likely 𝑝 IMT locations based on available information on transport budget, cargo distribution patterns, transport costs of cargo distribution, candidate IMT locations with important features such as cargo handling capacities. Here, the entropy maximising principle is used to combine these diverse pieces of information to find the most likely 𝑝 IMT locations and the least biased probability distribution of located IMT usage. The general entropy maximisation framework comprises an entropy objective function and a set of constraints representing the available information. The entropy function comprises the possible ways that a given state of the system can occur:

𝐸 = 𝑍!

∏_𝑖∈𝒪∏_𝑗∈𝒟𝑋_𝑖𝑗! (∏_𝑡∈𝒯 𝑉_𝑖𝑡𝑗!)

(4.6)

where 𝐸 is the number of possible ways that the state (𝑉_𝑖𝑡𝑗, 𝑋_𝑖𝑗) such that ∑_𝑖∈𝒪∑_𝑗∈𝒟𝑋_𝑖𝑗 +

∑_𝑖∈𝒪∑_𝑡∈𝒯∑_𝑗∈𝒟𝑉_𝑖𝑡𝑗 = 𝑍 can occur. The important question is: based on what we know about the system, which of the many states (values of 𝑋_𝑖𝑗 and 𝑉_𝑖𝑡𝑗 ) is most likely to represent the system? The principle of entropy maximisation is simply asking us to select the states with the maximum number of ways of occurring and consistent with all we know about the system. In general, we seek the values of 𝑋_𝑖𝑗 and 𝑉_𝑖𝑡𝑗 that maximises 𝐸 and also satisfy all the constraints representing the available information about the system. Statistically, the values of 𝑋_𝑖𝑗 and 𝑉_𝑖𝑡𝑗 that maximises 𝐸 also maximises ln𝐸. However, it is easier to maximise ln𝐸 so we maximised ln𝐸 instead. Thus, Equation (4.6) reduces to:

ln𝐸 = ln𝑍! − ∑ ∑ ln(𝑋_𝑖𝑗!)

𝑗∈𝒟 𝑖∈𝒪

− ∑ ∑ ∑ ln(𝑉_𝑖𝑡𝑗!)

𝑗∈𝒟 𝑡∈𝒯 𝑖∈𝒪

(4.7)

It has been shown in Chapter 3 that the term ln𝐸 has special meaning and desirable properties and Boltzmann (1972) referred to it as entropy. Thus, maximising (4.7) can equivalently be stated as maximising entropy. One of the desirable properties of ln𝐸 (entropy) is that its corresponds to the amount of missing information (or uncertainty or entropy) in the constructed of the probability distribution and that the maximum amount of missing information is attained when there is no known information about the system under investigation. These properties are expressed in propositions (4.1), (4.2) and (4.3) below:

Proposition 4.1: If the set ℳ = {𝑡: 𝑡 ∈ {0, 𝒯}} is set of modal alternatives, where {0} is the index for road alone and the set 𝒯 is the set of indices of IMTs forming the intermodal transport alternatives. Also, let the set of origin-destination movements ℛ = {𝑟 = (𝑖, 𝑗): 𝑖 ∈ 𝒪, 𝑗 ∈ 𝒟}

then set the of elemental alternatives 𝒲 = {𝑤 = (𝑟, 𝑡): 𝑟 ∈ ℛ, 𝑡 ∈ ℳ} with cardinality 𝑛 =

|𝒲|. If the probability of each elemental alternative 𝑤 ∈ 𝒲 is defined by Equation (4.8):

𝑃_𝑤 = 𝑍_𝑤

𝑍 ; ∀𝑤 ∈ 𝒲 (4.8)

Then, equation (4.7) or entropy can be expressed as:

ln𝐸 = − 𝑍 ∑ 𝑃_𝑤ln𝑃_𝑤

𝑤∈𝒲

(4.9)

Proof 4.1: By definition, equation (4.7) can be simplified as:

In𝐸 = ln𝑍! − ∑ ln𝑍_𝑤!

𝑤∈𝒲

Applying Stirling's approximation, the above equation simplifies to:

ln𝐸 = 𝑍(ln𝑍 − 1) − ∑ 𝑍_𝑤(ln𝑍_𝑤 − 1)

𝑤∈𝒲

(4.10)

Substituting Equation (4.8) into (4.10) and performing some algebraic manipulation we have:

ln𝐸 = − 𝑍 ∑ 𝑃_𝑤ln𝑃_𝑤

𝑤∈𝒲

(4.11)

Proposition 4.2: In the absence of any other information about the freight system, maximising equation (4.11) produces uniform probability distributions of modal flows:

𝑃_𝑤 = 1

𝑛 ; ∀𝑤 ∈ 𝒲 (4.12)

with corresponding maximum entropy:

𝐻 = In𝐸̃ = 𝑍ln(𝑛) (4.13)

where 𝑛 is the cardinality of the set 𝒲

Proof 4.2: Now if we assume there is no information available other than obeying the normalisation axiom of probability:

∑ 𝑃_𝑤

𝑤∈𝒲

= 1 (4.14)

then from equation (4.11) the first order condition for maximum ln𝐸 with respect to 𝑃_𝑤 and subject to (4.14) satisfy the following equation:

−𝑍ln(𝑃_𝑤) − 1 − 𝜑 = 0; ∀𝑤 ∈ 𝒲 (4.15)

where 𝜑 is the Lagrangian multiplier associated with constraint (4.14). Solving for 𝑃_𝑤 in (4.15) by enforcing constraint (4.14) we have:

𝑃_𝑤 = 1

𝑛 ; ∀𝑤 ∈ 𝒲

and substituting the above into Equation (4.11) the maximum entropy can be computed:

𝐻 = 𝑍ln(𝑛)

Proposition 4.3: The entropy (or amount of missing information (𝐻) in the probability distributions) constructed based on any amount of available information about the system cannot be greater than the entropy in equation (4.13). That is 𝐻 ≤ 𝑍ln(𝑛)

Proof 4.3: Define a convex function 𝜙(𝑥) = 𝑥ln(𝑥); ∀𝑥 ≥ 0. Following Jensen’s inequality, the following must hold:

Applying the definition of the convex function 𝜙 to the term on the left-hand side of Equation (4.16) and using Equations (4.11) we have:

− 1

Also, applying the definition of 𝜙 to the term on the right-hand side of (4.16) we have:

−𝜙 (1

It therefore follows from equation (4.16) that

𝐻 ≤ 𝑍ln(𝑛) (4.17)

The result in equation (4.17) is intuitive since it implies that the more information we have, the less entropy or uncertainty we have about the resulting probability distribution and vice versa.

In other words, on average, the amount of missing information (entropy) about the system under investigation is never increased by learning something about it. Equation (4.12) simply states that applying the principle of maximum entropy to the MCTM with no evidence to suggest why a particular modal alternative should be preferred more than the others will result in a uniform probability distribution. The next section presents the information available in the form of constraints to form the entropy maximising facility location problem (EMFLP).

4.2.3 Available evidence as constraints

For the purpose of this exercise, the evidence available are summarised as follows:

1.Budget constraint: It is assumed that the transport budget 𝑐 is known. This evidence is added as constraint (4.18). The first component captures the weighted cost of using intermodal transport and the second captures the weighted cost of using road alone transport (e.g. truck only), with the sum not exceeding the total allocated transport budget.

∑ ∑ ∑ 𝑐_𝑖𝑡𝑗𝑉_𝑖𝑡𝑗

𝑗∈𝒟 𝑡∈𝒯 𝑖∈𝒪

+ ∑ ∑ 𝑐_𝑖𝑗𝑋_𝑖𝑗

𝑗∈𝒟 𝑖∈𝒪

≤ 𝑐 (4.18)

2. Conservation of cargo flow constraint. Information on the distribution of origin-destination flows of cargo (𝑞_𝑖𝑗) by modes is added as constraint (4.19). It ensures that for each origin-destination pair, the sum of cargo by all available modes equals the total cargo associated with this origin-destination pair.

∑ 𝑉_𝑖𝑡𝑗

𝑡∈𝒯

+ 𝑋_𝑖𝑗 = 𝑞_𝑖𝑗; ∀ 𝑖 ∈ 𝒪, 𝑗 ∈ 𝒟 (4.19)

3. Definitional constraint. The information on the required number of IMTs (𝑝) to locate is presented by constraint (4.20). It ensures that only the required number of IMTs are located.

∑ 𝑌_𝑡

𝑡∈𝒯

= 𝑝 (4.20)

4. Capacity constraint. The cargo handling capacity limit of the each IMT is captured in constraints (4.21) and guarantees that no located IMT exceeds its cargo handling capacity. The constraints also ensure that only open IMTs are used.

∑ ∑ 𝑉_𝑖𝑡𝑗

𝑗∈𝒟 𝑖∈𝒪

≤ 𝑌_𝑡𝑏_𝑡 ; ∀𝑡 ∈ 𝒯 (4.21)

Finally, the entropic objection function in (4.7) can be simplified by applying Stirling's approximation to the factorial terms, ignoring the constant term, ln𝑍! as it does not influence the optimisation process:

ln𝐸~ ∑ ∑ ∑ 𝑉_𝑖𝑡𝑗(1 − ln𝑉_𝑖𝑡𝑗)

𝑗∈𝒟 𝑡∈𝒯 𝑖∈𝒪

+ ∑ ∑ 𝑋_𝑖𝑗(1 − ln𝑋_𝑖𝑗)

𝑗∈𝒟 𝑖∈𝒪

(4.22)

Once the available information are converted to constraints (4.18-4.21), the EMFLP is presented as follows:

EMFLP ∶ Max Λ = ∑ ∑ ∑ 𝑉_𝑖𝑡𝑗(1 − ln𝑉_𝑖𝑡𝑗)

𝑗∈𝒟 𝑡∈𝒯 𝑖∈𝒪

+ ∑ ∑ 𝑋_𝑖𝑗(1 − ln𝑋_𝑖𝑗)

𝑗∈𝒟 𝑖∈𝒪

Subject to constraint (4.18) to (4.21) and the following integer and non-negativity constraints:

𝑌_𝑡 ∈ {0,1} ; 𝑡 ∈ 𝒯 (4.23)

𝑉_𝑖𝑡𝑗 ≥ 0; 𝑋_𝑖𝑗 ≥ 0 ; ∀𝑡 ∈ 𝒯; ∀ 𝑖 ∈ 𝒪; 𝑗 ∈ 𝒟 (4.24)