A Metric for Finite Power Multisets of Positive Real Numbers Based on Minimal Matching

In this article, we show how to define a metric on the finite power multisets of positive real numbers. The metric, based on the minimal matching, consists of two parts: the matched part and the mismatched part. We also give some concrete applications and examples to demonstrate the validity of this metric.


Introduction
A multiset, unlike a Cantorian set, is a collection of elements whose instances might be multiple (the number of its instances of an element is named multiplicity).The cardinality of a multiset A is defined by the sum of the multiplicities with respect to their corresponding elements and is denoted by |A| m .For example, the cardinality of multiset A = {2, 2, 3, 3, 3, 6, 11} is 7, i.e., |A| m = 7.Though unconventional, the theory of multisets has well been developed (see Reference [1]) and it also has various applications in many situations (see Reference [2]).From practical point of view, multisets are easier to represent or simulate than mathematical objects with multiple instances.In this article, we mainly focus on the finite power multisets of positive real numbers.
Let R + denote the set of all positive real numbers.Let N 0 denote the set of natural numbers including 0. Let power multiset MP (R + ) denote the set of all the sub-multisets of R + .Suppose K ⊆ MP (R + ) is an arbitrary set of some sub-multisets in R + ( each multiset is finite) of MP (R + ).We call K a finite power multiset of positive real numbers.The main result in this article is to define a metric on K based on the concept of minimal matching.The distance between any two multisets consists of two separated parts: the matched part and mismatched part.Matching has been an important problem and has wide applications in the fields of artificial intelligence, graph theories, and operation research (see References [3][4][5]).In this article, we come up with a new metric which is based on the concept of minimal matching.This metric is used to measure the distance between any two finite multisets of positive real numbers.Though what we define in this article is a standard metric, the whole setting could also be extended to other generalized metrics, for example, G−metric (see Reference [6]).

Definitions
In this section, we introduce and present multisets via the forms of functions.The basic concepts could be found in many textbooks or journals (see, e.g., References [7,8]).Let Γ denote the set R + → N 0 , i.e., the set of all the functions from R + to N 0 .Let D f denote the domain of a function f .Let set D * f = {r ∈ R + : f (r) = 0} be the non-zero domain of f .

Multisets
Let Γ < denote all the finite multi-subsets of R + , i.e., , we say f is a multi-subset of g, denoted by f ≤ g.Let f , g ∈ Γ < be arbitrary.
Each multiset f in Γ < could be uniquely represented by the following descending form (named a representative descending form): ; or by the following ascending form (named a representative ascending form): or in brief , where a 1 > a 2 > a 3 ... > a n > 0 and a 1 , a 2 , ...,

Background
In this article, we show how to define a metric on K (see Introduction).For any Cantorian set S, we use |S| to denote the cardinality of S. Let d be an arbitrary metric on R + satisfying for all a, b, c ∈ R + .Observe that d is a metric (for our generalization purpose) on R + × R + , which lays a foundation for our latter definition of a metric on K. Let A, B, C ∈ K be arbitrary.Let A → B denote the set of all the functions from A to B, in which the repeated elements are deemed distinct.
For any function ϕ, we use D ϕ and R ϕ to denote its domain and codomain, respectively.For the previous example, D ρ = A and R ρ = B. We use ϕ(S) to denote the image {ϕ(s) : s ∈ S}, in particular ϕ(D ϕ ) to denote the image of ϕ and ϕ −1 (S) to denote the pre-image of S. If S ⊆ D ϕ , we use ϕ|S to denote ϕ whose domain is restricted to S. One candidate in mind is d(a, b) := |a − b|.On this basis, we could define the distance for the matched elements and the distance for the mismatched elements as follows: where D ϕ and R ϕ and denote the domain and codomain of ϕ, respectively.||ϕ|| represents the distance of all the matched elements (or the sum of the distances of all the matched pairs), while ||ϕ|| − represents the distance of all the mismatched elements in the range.
By the definition, one has δ(A, In the following, we show that δ is indeed a metric.The reasoning will be proceeded by their relations (i.e., larger, less than and equal to) between cardinalities of A, B, and C, i.e., |A| m , |B| m , and |C| m .To validate that δ is a metric, we need to consider all the 27 relations between |A| m , |B| m and |C| m : for example, In order to facilitate our computing, we encode the 27 relations by the following set as shown in Corollary 1.If ϕ is a bijective function, we use ϕ −1 to denote its inverse function.In the following, let ϕ and φ ∈ BF * [A, C] be arbitrary.Before we proceed further, we have the definitions: 1. We use Though there are 27 relations between the cardinalities of A, B, and C, only 13 of them are valid as shown in the following lemma.Lemma 1.There are only 13 relations which do not violate the transitivity property in terms of their cardinalities: (1, 1, 1), (1, 2, 1), (1, 3, 1), (1,3,2), (1,3,3) Proof.The result follows immediately from their relations.Take the relation (1, 1, 1) for example.Recall that (1, 1, 1) represents the relation |A| m < |B| m < |C| m , in which the property of transitivity |A| m < |C| m holds.One could verify that each of the other 12 relations also holds the transitivity property.However, the other 15 relations fail the transitivity property: for example ( In the following, we show the triangle inequality of δ.Let us show the following corollary first.
Proof.By Equation ( 7) and Lemma 1, A and C are interchangeable, i.e., the relations are equivalent to (respectively) By this corollary, we only need to consider the triangle inequality of the above-mentioned eight relations.Proof.By the definitions, where Hence, by the triangle inequality of d and the definitions of δ ,

Proof. By the definitions of δ
where Furthermore, Then, by the triangle inequality of d and the definitions of δ Proof.We derive the three components one by one.Firstly, suppose φ ∈ BM(A → B) is a function satisfying φ|(A B ) = C B (as shown in Figure 1), i.e., φ(a) Secondly, Thirdly, Hence, Lemma 10. (Relation (1, 3, 1)) as shown in Figure 2. Furthermore, by the definition of δ, Henceforth, by Equations ( 3)-( 5), it follows Theorem 1. (K, δ) is a metric space.

Applications and Computations
In this section, we give a group of numerical data and demonstrate how to compute their distances (or adjacency matrix) via the metric δ.In order to facilitate our computing, we show the following lemmas first.Let a 1 , a 2 , b 1 , b 2 ∈ R be arbitrary.

Lemmas
Furthermore, we consider the following cases: which sell maize.In VL1, there are five farmers; in VL2, there are six farmers; in VL3, there are 10 farmers.The expected annual yields of maize for each farmer in VL1 are 3.2, 5.1, 7.6, 3.2, 8.8 tons; the ones in VL2 are 1.2, 2.1, 3.6, 7.9, 12.1, 6.4 tons; and the ones in VL3 are 2.6, 4.6, 8.1, 5.1, 2.2, 5, 7.9, 11.1, 12, 4.5 tons.On the other hand, suppose there are four wholesalers whose annual demands are 7.9, 9.2, 11.6, 8.3 tons, respectively.The government policy is to associate a village with the wholesalers based on the criterion that the total discrepancy between the village and the wholesalers must be minimal and the condition that each farmer could only exclusively sign the contract with exactly one wholesaler.Assume the government adopts the metric defined in this article.The results could be computed as follows Table 1.Since the total discrepancy (i.e., matched part plus mismatched part) between VL2 and the wholesalers is minimal (or 11.3), the government should associate VL2 with the wholesalers.Henceforth, the government should pick VL2 to sign the contract with the four wholesalers exclusively.In doing so, the total dissatisfaction (or discrepancy) from both the farmer and the wholesalers would be minimal.

Characteristic and Analysis
The main characteristic of our metric is that it takes the minimal discrepancy into consideration.For the usual metrics, one hardly associates a metric with the minimal matching via combinations or permutations of all sorts of choices.Our method successfully combines the usual definition of a metric with the concept of an optimal choice.With these two concepts combined, one could pick up an optimal decision purely based on the metric defined in this article.This approach gives one a much more direct decision-making process.In addition, since this metric consists of two parts: the matched and mismatched parts, it would provide one with much more insightful knowledge of the discrepancy between mathematical objects.

Conclusions
We have defined a metric on a finite power multiset of positive real numbers via the concept of minimal matchings, in which the distances of any two multisets consist of two parts: the distance of the matched part and the distance of the mismatched part.We also implement this metric by an adjacency matrix.A concrete example is also included in this article.In addition to the adjacency matrix, we show another definitional computation to facilitate our computing of the metric.The metric defined in this article could be further applied in some real problems regarding artificial intelligence, clustering, or some other theoretical mathematical research.

Definition 1 .
(Empty Multiset) We call the zero function in Γ < the empty multiset.Definition 2. (Equality =) f = g if and only if f ≤ g and g ≤ f .Definition 3. (Intersection ∧) The intersection of f and g, denoted by the function f ∧ g : R + → N 0 , is defined by ( f ∧ g)(a) := min{ f (a), g(a)} for all a ∈ R + .Definition 4. (Union ∨) The union of f and g, denoted by the function f ∨ g : R + → N 0 , is defined by ( f ∨ g)(a) := max{ f (a), g(a)} for all a ∈ R + .Definition 5. (Difference ) Exclusion of g from f , denoted by the function f g : R + → N 0 , is defined by

Definition 6 .
(Descending) Define the p − th element in f by function OD as follows:

0 otherwise . Definition 7 .
(Ascending) Define the p − th element in f by function OA as follows: Take A and B in Example 1 for example.One has BF[B → A] = ∅.Definition 9.For any function ϕ ∈ BF[A → B], we call it a a matched function.We call (a, ϕ(a)) a matched pair.Every remaining element in B − ϕ(A) is called a mismatched element.
where i n denotes the n−th repetition of i and the mismatched part ||ϕ|| − = 1.Next, we define the set of all minimal distances consisting of the matched parts and the mismatched parts.Definition 11. (Minimal matched functions) Define BF * [A → B] m and |A| m n 3 |C| m , respectively, where 1, 2, and 3 represent the relation <, = and > correspondingly.For example, (1, 2, 3) represents the relation |A| m < |B| m , |B| m = |C| m , and |A| m > |C| m .By the transitivity of their cardinalities, only 13 of the 27 relations are valid (shown in Lemma 1).Moreover, these 13 relations could be further reduced to 8 relations by the symmetry of δ, i.e., δ(A, B)

Table 1 .
Analysis of Optimal Matchings