Optimization on the distribution of population densities and the arrangement of urban activities

In this paper, an approximation on the distribution of population densities and the arrangement of urban activities, over a set of n locations, is derived by using the classical multiobjective optimization theory and Shannon entropy.


Introduction, problem description and preliminaries
The term of entropy was used for the first time in 1865 in Thermodynamics by Rudolf Clausius [7].Later, scientists such as Ludwig Boltzmann, Josiah Willard Gibbs and James Clerk Maxwell gave to this concept a statistical basis.In Probability Theory, the degree of uncertainty to a random variable can also be evaluated using the entropy.Consequently, the entropy can be used in the study of some risk assessment problems arising in different fields.
The purpose of this paper is to develop, using the classical multiobjective optimization theory, a simultaneous optimization model involving Shannon entropy and spatial Shannon entropy subject to appropriate and meaningful constraints.Moreover, by considering the qualitative concept of utility, we extend our model to the case of Belis ¸-Guias ¸u entropy and spatial Belis ¸-Guias ¸u entropy.Practically, in both cases, we derive an approximation on the distribution of population densities and the arrangement of urban activities over a set of n locations.Now, let us introduce our study problem.According to Batty [4], following Batty [3], we shall represent a city as a set of locations.Also, we assume that: (1) there are n locations, identified by i, with i = 1, ..., n; (2) each location is a point or an area where urban activities can take place; (3) in each location there exists a number of units of urban activity; (4) the location identified by i has the size (area) a i and, therefore, A = n ∑ i=1 a i is the total size (area) of the city.Denote by the total number of units of urban activity (the total amount of urban activity), where N i , represents the number of units of urban acitiviy associated with location i, i = 1, n.If we start with N 1 , the number of allocations of N 1 (i.e., the number of locations with N 1 units of urban activity) if given by and so on.Making the product we find the total number of arrangements considered as a measure of complexity (W depends by allocations) of the city.
Remark 1.1 i) If the total amount of urban activity N is allocated to N i , with i ∈ {1, ..., n} fixed, then the measure of complexity W is equal to 1.

ii) If
, n, then W varies with respect to the total amount of urban activity N and the number of locations n.
By maximizing the measure of complexity W (more precisely, the logarithm of W ), we shall find the most enjoyable arrangement of units of urban activity in that it would provide the greatest possibility of distinct individual activities associated with the locations i. Usually, such maximizations are subject to appropriate and meaningful constraints.By a direct computation, using Stirling's formula, we get Taking into account that N i is a frequency that can be trasformed into a probability p i = N i N , by substituting the number of units of urban acitiviy associated with location i in the previous relation (6) and dropping the constant terms, we find that the number of arrangements W is proportional to Shannon entropy (measure of uncertainty, Shannon [14]) Consequently, to maximize ln W is equivalent with the well-known process of maximizing H.
Remark 1.2 i) When the total amount of urban activity N is equally distributed to locations, that is and H = ln n is at a maximum.Also, let us remark that H varies with n. ii) If the total amount of urban activity N is allocated to N i , i ∈ {1, ..., n} fixed, that is N = N i , i ∈ {1, ..., n} fixed, then p i = 1 and p j = 0, j ∈ {1, ..., n}, i ̸ = j, and H = 0 is at a minimum.Further, we consider the spatial entropy (for more details, the reader is directed to Batty [3], Batty et al. [5]) which takes into account the numbers , where a i and A are introduced at the beginning of this section.Let us notice that n ∑ i=1 A i = 1 and assume that p i A i is subunitary (otherwise, we must minimize S instead of maximize it).
Considering the previous mathematical context, the main aim of this paper is to study the following vector (bi-objective) optimization problem where p i is the probability of finding a place i which has P i population residing there and c i the travel cost from the central business district to the zone i.The constraint (10) is a normalization constraint on the probabilities, (11) is a constraint on the mean population of places, (12) is a constraint on the average travel cost incurred by population and, finally, ( 13) is a constraint on the average "logarithmic" size of locations.
The second objective of this work is to investigate a similar problem which involves the qualitative concept of utility.The models proposed here can be regarded as an approximation on (i) the distribution of population densities and (ii) the arrangement of urban activities over a set of n locations.
Next, in order to develop our theory, we will enunciate some elements of multiobjective optimization.Consider the following convention between two vectors, u = (u 1 , ..., u s ) and the following vector minimization problem where f : R n → R s and g : R n → R m are vector-valued functions, defined by .., s}, and g j : R n → R, j ∈ {1, ..., m}, continuously differentiable functions on R n .Denote by ▽f i (x) and ▽g j (x) the gradients of f i and g j at x ∈ R n , respectively, and by ⟨x, y⟩ Obviously, if x 0 ∈ X is an efficient solution to problem (P ) then x 0 is a weak efficient solution to problem (P ).However, the converse relation does not hold, in general, and practically the concept of efficient solution is more desirable than that of weak efficient solution.
Theorem 1.1 (Necessary efficiency conditions for (P )) Let x 0 ∈ X be any feasible solution to (P ) and suppose that the generalized Guignard constraint qualification holds at x 0 ∈ X.If x 0 ∈ X is an efficient solution to (P ), then there exist the vectors

Remark 1.3
If the vector minimization problem (P ) contains, in addition, constraints of the type h(x) = 0, with h : R n → R l a continuously differentiable function, then there exists a vector α ∈ R l such that the first condition in (16) Let ρ be a real number, C ⊆ R n , and b : )-quasiinvex at x 0 ∈ C with respect to η and θ if there exist the vector functions η : In the above definition, if we replace " ≤ " with " = ", we obtain the definition of monotonic (ρ, b)-quasiinvexity at x 0 with respect to η and θ.
Theorem 1.2 (Sufficient efficiency conditions for (P )) Let x 0 ∈ X be any feasible solution to (P ) and let there exist the vectors λ ∈ R s and µ ∈ R m such that the conditions (16) are satisfied.If: )-quasiinvex at x 0 with respect to η and θ and there exists at least an index k ∈ {1, ..., s} such that f k (x) is strictly (ρ 1 k , b)-quasiinvex at x 0 with respect to η and θ; (ii) each function g j (x), j = 1, m, is monotonic (ρ 2 j , b)-quasiinvex at x 0 with respect to η and θ; then x 0 ∈ X is an efficient solution to (P ).
OPTIMIZATION ON THE DISTRIBUTION OF POPULATION DENSITIES Remark 1.4 If the vector minimization problem (P ) contains, in addition, constraints of the type h(x) = 0, with h : R n → R l a continuously differentiable function, then the conditions (i) and (iii) from Theorem 1.2 change as follows: )-quasiinvex at x 0 with respect to η and θ; (i"') one of the functions given in (i'), (i") is strictly (ρ, b)-quasiinvex at x 0 with respect to η and θ, where ρ = ρ 1 i or ρ 3  k , and, respectively For more details, other notions and their connections, the reader is addressed to Yu [20], Treant ¸ȃ and Udris ¸te [16], Arana et al. [2], Verma [18], Treant ¸ȃ [17].

Main results
Let us observe that our bi-objective optimization problem (9), subject to (10) − (13), can be rewritten as follows Taking into account the general context formulated in the previous section (see Theorem 1.1 and Remark 1.3), now we are in a position to establish and prove the first part of our main results.

Theorem 2.1
If p = (p i ), i = 1, n, is a normal efficient solution in (V OP ), then there exist the scalars λ 1 , λ 2 , α, β, γ, δ with or, equivalently, where ln A] is a constant of proportionality which ensures that the probabilities sum is 1.Moreover, the "negative" measures of complexity are at a minimum for the given set of constraints, simplify to and, by imposing the necessary conditions of efficiency, we get where ln A] is a constant of proportionality which ensures that the probabilities sum is 1.
If we substitute the probability in (19) into the "negative" Shannon entropy H 1 = n ∑ p i ln p i and into the "negative" spatial Shannon entropy , by a direct computation, we obtain the "negative" measures of complexity in (20) and the proof is complete.
Over the past years, in order to correlate the quantitative concept of information with the qualitative concept of utility, many researchers (see, for instance, Belis ¸and Guias ¸u [6], Longo [12], Kapur [9], [10]) have introduced several weighted information measures.Given the context in which we are, these weighted measures of information become very important (they take into account both the probabilities with which certain random events occur and, also, some qualitative characteristics of these events).Thus, according to Belis ¸and Guias ¸u [6], let u i be the weight associated to an elementary event with probability p i (in our case, an elementary event is the finding of a place i which has P i population residing there and c i the travel cost from the central business district to the zone i).Consider the weight u i as a finite, positive real number representing the relevance, the significance or the utility of the occurrence of an event with probability p i .If u i > u j , then the event with weight u i (and probability p i ) is strictly more significant, more useful or more relevant than the event with weight u j (and probability p j ), where i, j ∈ {1, ..., n}, i ̸ = j.
Using the previous utilities (weights), let us introduce the following weighted bi-objective optimization problem OPTIMIZATION ON THE DISTRIBUTION OF POPULATION DENSITIES Remark 2.1 i) The above minimum is computed for fixed utility distributions.
ii) There are two additional constraints in (V OP ) * compared to (V OP ): the constraint n ∑ i=1 u i p i = P on the relevance of the weights u i (of course, if we get P = u and further, if u = 1, we find the first constraint in (V OP ) * ; therefore, for generality, we shall consider the weights u i as different, finite, positive real numbers) and the constraint n ∑ i=1 u i p i ln a i = Ã on the weighted average "logarithmic" size of locations.Now, we shall formulate and prove the second part of our main results.

Proof
The proof follows in the same manner as in Theorem 2.1.Consider the Lagrangian Applying the necessary conditions of efficiency, by a direct computation, we find which, equivalently written, is (23).Replacing the probability in (23) into the "negative" Belis ¸-Guias ¸u entropy u i p i ln p i and into the "negative" spatial Belis ¸-Guias ¸u entropy , by a direct computation, we obtain the "negative" measures of complexity in (24) and the proof is complete.
Further, let us consider the following notations: As it can be verified, all of these functions are (ρ, 1)-quasiinvex at p 0 , for ρ ≤ 0 and any vector function θ = θ(p, p 0 ) (see Definition 1.3), with respect to: where η 1 is the vector function associated with f 1 , η 2 is the vector function associated with f 2 , η 3 is the vector function associated with h 1 , and so on.
The following result formulates some sufficient conditions of efficiency for our vector minimization problem (V OP ).
Proof First, we have to mention that the real numbers ρ 1 i , i = 1, 2, and ρ 3 k , k = 1, 4, introduced in our theorem, have the same siqnificance as in Remark 1.4 of section 1.
Further, taking into account Theorem 1.2 and Remark 1.4 of section 1 (see conditions (iii) and (iii ′ )), the proof is complete.
In a similar way, one can find a characterization result of sufficient conditions regarding the weighted bi-objective optimization problem (V OP ) * .
As it can be verified, all of the following functions

u
i p i ln p i , f 2 (p) = n ∑ i=1 u i p i ln p i − n ∑ i=1 u i p i ln A i , (27)h 1 (p) = n ∑ i=1 p i − 1, h 2 (p) = n ∑ i=1 u i p i − P , ProofHaving in mind the general mathematical framework formulated in Theorem 1.1 and Remark 1.3 of section 1, we introduce the Lagrangian Optim.Inf.Comput.Vol. 6, June 2018 S. TREANT ¸Ȃ 213