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

## Abstract

**:**

## 1. Introduction

## 2. Definitions

#### 2.1. Multisets

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

#### 2.2. Background

**Example**

**1.**

**Definition**

**8.**

**Example**

**2.**

**Definition**

**9.**

**Definition**

**10.**

**Definition**

**11.**

**Definition**

**12.**

- We use ${B}_{A}$ to denote $\phi (A)$, if ${|A|}_{m}\le {|B|}_{m}$ and ${A}_{B}$ to denote $\phi (B)$, if ${|A|}_{m}>{|B|}_{m}$.
- We use ${C}_{B}$ to denote $\tilde{\phi}(B)$, if ${|B|}_{m}\le {|C|}_{m}$ and ${B}_{C}$ to denote $\tilde{\phi}(C)$, if ${|B|}_{m}>{|C|}_{m}$.
- We use ${C}_{A}$ to denote $\tilde{\tilde{\phi}}(A)$, if ${|A|}_{m}\le {|C|}_{m}$ and ${A}_{C}$ to denote $\tilde{\tilde{\phi}}(C)$, if ${|C|}_{m}<{|A|}_{m}$.

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

- $\delta (A,B)\ge 0$.
- $\delta (A,B)=0$ iff $A=B$.
- $\delta (A,B)=\delta (B,A)$.

**Proof.**

**Corollary**

**1.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Lemma**

**8.**

**Proof.**

**Lemma**

**9.**

**Proof.**

**Lemma**

**10.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 3. Applications and Computations

#### 3.1. Lemmas

**Lemma**

**11.**

**Proof.**

- $k=0$: Then,$$\begin{array}{c}|k-{\lambda}_{b}|+|k+{\lambda}_{a}|-|k|-|k+{\lambda}_{a}-{\lambda}_{b}|\hfill \\ ={\lambda}_{b}+{\lambda}_{a}-|{\lambda}_{a}-{\lambda}_{b}|\ge 0;\hfill \end{array}$$
- $k>0$: Then,$$\begin{array}{c}|k-{\lambda}_{b}|+|k+{\lambda}_{a}|-|k|-|k+{\lambda}_{a}-{\lambda}_{b}|\hfill \\ =|k-{\lambda}_{b}|+k+{\lambda}_{a}-k-|k+{\lambda}_{a}-{\lambda}_{b}|\ge 0;\hfill \end{array}$$
- $k<0$: Then,$$\begin{array}{c}|k-{\lambda}_{b}|+|k+{\lambda}_{a}|-|k|-|k+{\lambda}_{a}-{\lambda}_{b}|\hfill \\ =-k+{\lambda}_{b}+|k+{\lambda}_{a}|+k-|k+{\lambda}_{a}-{\lambda}_{b}|\ge 0.\hfill \end{array}$$Hence, we have shown $|{a}_{1}-{b}_{2}|+|{a}_{2}-{b}_{1}|-|{a}_{1}-{b}_{1}|-|{a}_{2}-{b}_{2}|\ge 0.$

**Lemma**

**12.**

**Proof.**

**Corollary**

**2.**

**Proof.**

#### 3.2. Computation

- ${A}_{1}=\{91.67,2,39.53,98.34,8.78\}$;
- ${A}_{2}=\{1.99,62,7,9.52,9,8.11\}$;
- ${A}_{3}=\{2.1,6.22,27.1,9.67,9.19,81.29,5.55,12.41,1.67,11.08,51.15,0.33\}$;
- ${A}_{4}=\{22.21,61.26,71.12,29.61,29.19,29.29,35.3,40\}$;
- ${A}_{5}=\{17.19,2,70.56,9.52,9.45,18.16,40\}$;
- ${A}_{6}=\{1.26,0.19,2,4.70,8.56,9.09\}$.

- (Method One) List all the permutations and find the optimal permutation and its associated distance, which is the summation of the matched and mismatched parts.
- (Method Two) List all the combinations and find the optimal combination and its associated distance, which is the summation of the matched and mismatched parts.

## 4. Real World Applications

#### 4.1. Example

#### 4.2. Characteristic and Analysis

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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Villages | Expected Annual Yield (tons) | Matched | Mismatched | Total Discrepancy |
---|---|---|---|---|

VL1 | $\{3.2,5.1,7.6,3.2,8.8\}$ | $12.3$ | $3.2$ | $15.5$ |

VL2 | $\{1.2,2.1,3.6,7.9,12.1,6.4\}$ | $8.0$ | $3.3$ | $11.3$ |

VL3 | $\{2.6,4.6,8.1,5.1,2.2,5,7.9,11.1,12,4.5\}$ | $2.5$ | 24 | $26.5$ |

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**MDPI and ACS Style**

Chen, R.-M. A Metric for Finite Power Multisets of Positive Real Numbers Based on Minimal Matching. *Axioms* **2018**, *7*, 94.
https://doi.org/10.3390/axioms7040094

**AMA Style**

Chen R-M. A Metric for Finite Power Multisets of Positive Real Numbers Based on Minimal Matching. *Axioms*. 2018; 7(4):94.
https://doi.org/10.3390/axioms7040094

**Chicago/Turabian Style**

Chen, Ray-Ming. 2018. "A Metric for Finite Power Multisets of Positive Real Numbers Based on Minimal Matching" *Axioms* 7, no. 4: 94.
https://doi.org/10.3390/axioms7040094