# A Decision Model for Free-Floating Car-Sharing Providers for Sustainable and Resilient Supply Chains

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Model

- (1)
- The company counts the number of available cars, the number of cars being used, and the number of cars in the maintenance. They are denoted by ${m}_{2}$, ${m}_{3}$, and ${m}_{4}$, respectively.
- (2)
- The company posts the information on the number of available cars.
- (3)
- Customers arrive.
- (4)
- The company spends daily expenses ${C}_{0}(c,Q)$ for the daily operations.
- (5)
- The company spends ${C}_{2}\left(c\right)$, which includes the finance costs and basic operation costs, on each of the remaining available cars.
- (6)
- Suppose the company spends ${C}_{3}\left(c\right)$, which includes the finance costs and basic operation costs, on each car being used per period, and earns an income of ${r}_{3}$ from each car being used per period.
- (7)
- Some cars in the maintenance state will be scrapped because of their conditions.
- (8)
- The company purchases $Q-{m}_{2}-{m}_{3}-({m}_{4}-{n}_{45})$ new cars to keep total Q cars owned by the company. The new cars will be available in the next period.

^{*}” to denote “optimal”.

## 4. Analysis

- (1)
- If $d<{m}_{2}$, $0\le {n}_{24}\le {m}_{2}-d,0\le {n}_{34}\le {m}_{3}+d,0\le {n}_{45}\le {m}_{4}$, and $t=1,\cdots ,T-1$, then$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {P}_{t,({m}_{2},{m}_{3},{m}_{4}),(Q-{m}_{3}-d-{n}_{24},\phantom{\rule{4pt}{0ex}}{m}_{3}+d-{n}_{34},\phantom{\rule{4pt}{0ex}}{n}_{24}+{n}_{34})}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{{\left(\lambda \left(Q\right)\right)}^{d}}{d!}{e}^{-\lambda \left(Q\right)}\frac{({m}_{2}-d)!}{{n}_{24}!({m}_{2}-d-{n}_{24})!}{p}_{24}^{{n}_{24}}\left({C}_{0}\right){\left(1-{p}_{24}\left({C}_{0}\right)\right)}^{{m}_{2}-d-{n}_{24}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{({m}_{3}+d)!}{{n}_{34}!({m}_{3}+d-{n}_{34})!}{p}_{34}^{{n}_{34}}{\left(1-{p}_{34}\right)}^{{m}_{3}+d-{n}_{34}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{{m}_{4}!}{{n}_{45}!({m}_{4}-{n}_{45})!}{p}_{45}^{{n}_{45}}\left({C}_{0}\right){\left(1-{p}_{45}\left({C}_{0}\right)\right)}^{{m}_{4}-{n}_{45}}\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {R}_{t,({m}_{2},{m}_{3},{m}_{4}),(Q-{m}_{3}-d-{n}_{24},\phantom{\rule{4pt}{0ex}}{m}_{3}+d-{n}_{34},\phantom{\rule{4pt}{0ex}}{n}_{24}+{n}_{34})}=-{C}_{0}-{C}_{42}\left(c\right)({m}_{4}-{n}_{45})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}-c(Q-{m}_{2}-{m}_{3}-{m}_{4}+{n}_{45})+({m}_{3}+d)({r}_{3}-{C}_{3}\left(c\right))+{n}_{45}{r}_{5}-({m}_{2}-d){C}_{2}\left(c\right).\hfill \end{array}$$
- (2)
- If $d\ge {m}_{2}$, $0\le {n}_{34}\le {m}_{3}+{m}_{2},0\le {n}_{45}\le {m}_{4}$, and $t=1,\cdots ,T-1$, then$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {P}_{t,({m}_{2},{m}_{3},{m}_{4}),(Q-{m}_{3}-{m}_{2},\phantom{\rule{4pt}{0ex}}{m}_{3}+{m}_{2}-{n}_{34},\phantom{\rule{4pt}{0ex}}{n}_{34})}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left[\sum _{d={m}_{2}}^{\infty}\frac{{\left(\lambda \left(Q\right)\right)}^{d}}{d!}{e}^{-\lambda \left(Q\right)}\right]\frac{({m}_{3}+{m}_{2})!}{{n}_{34}!({m}_{3}+{m}_{2}-{n}_{34})!}{p}_{34}^{{n}_{34}}{\left(1-{p}_{34}\right)}^{{m}_{3}+{m}_{2}-{n}_{34}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{{m}_{4}!}{{n}_{45}!({m}_{4}-{n}_{45})!}{p}_{45}^{{n}_{45}}\left({C}_{0}\right){\left(1-{p}_{45}\left({C}_{0}\right)\right)}^{{m}_{4}-{n}_{45}}\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {R}_{t,({m}_{2},{m}_{3},{m}_{4}),(Q-{m}_{3}-{m}_{2},\phantom{\rule{4pt}{0ex}}{m}_{3}+{m}_{2}-{n}_{34},\phantom{\rule{4pt}{0ex}}{n}_{34})}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =-{C}_{0}-{C}_{42}\left(c\right)({m}_{4}-{n}_{45})-c(Q-{m}_{2}-{m}_{3}-{m}_{4}+{n}_{45})+({m}_{3}+{m}_{2})({r}_{3}-{C}_{3}\left(c\right))+{n}_{45}{r}_{5}.\hfill \end{array}$$
- (3)
- If $d<{m}_{2}$ and $t=T$, then$$\begin{array}{c}\hfill {P}_{T,({m}_{2},{m}_{3},{m}_{4}),(0,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}d)}=\frac{{\left(\lambda \left(Q\right)\right)}^{d}}{d!}{e}^{-\lambda \left(Q\right)}\end{array}$$$$\begin{array}{c}\hfill {R}_{T,({m}_{2},{m}_{3},{m}_{4}),(0,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}d)}=-{C}_{0}+({m}_{3}+d)({r}_{3}-{C}_{3}\left(c\right))-(Q-{m}_{3}-d){C}_{2}\left(c\right)+(1-\alpha )Q{r}_{5}.\end{array}$$
- (4)
- If $d\ge {m}_{2}$ and $t=T$, then$$\begin{array}{c}\hfill {P}_{T,({m}_{2},{m}_{3},{m}_{4}),(0,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}{m}_{2})}=\sum _{d={m}_{2}}^{\infty}\frac{{\left(\lambda \left(Q\right)\right)}^{d}}{d!}{e}^{-\lambda \left(Q\right)}\end{array}$$$$\begin{array}{c}\hfill {R}_{T,({m}_{2},{m}_{3},{m}_{4}),(0,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}{m}_{2})}=-{C}_{0}+({m}_{3}+{m}_{2})({r}_{3}-{C}_{3}\left(c\right))-(Q-{m}_{2}-{m}_{3}){C}_{2}\left(c\right)+(1-\alpha )Q{r}_{5}.\end{array}$$
- (5)
- Otherwise, the system state change is impossible. Thus, set ${P}_{\xb7,(\xb7),(\xb7)}=0$ and ${R}_{\xb7,(\xb7),(\xb7)}=0$ in the matrices P and R.

## 5. Numerical Experiments

**Experiment**

**1.**

- 1.
- The demand was high (a was at the high level).
- 2.
- The variable daily expense was (${a}_{0}$ at the low level).
- 3.
- The maintenance fees were (${a}_{42}$ at the low level).
- 4.
- Customers that used the cars for a long time (${p}_{34}$ were at the low level).
- 5.
- Rental income was high (${r}_{3}$ at the high level).
- 6.
- A scrapped car could be sold for a good price (the salvage value ${r}_{5}$ was at the high level).

**Remark**

**1.**

**Experiment**

**2.**

- (1)
- It is not easy for a car-sharing company to be profitable. Before a company decides to join the car-sharing industry, it should first consider the external factors, which determine whether the company has a chance to be profitable. In an external sound environment, it may be profitable; in a poor external environment, the company will suffer losses no matter how good its management is.
- (2)
- Given a favorable external environment, whether the company is profitable depends on internal factors of the company. If its internal management is good, the company will be able to be very profitable; otherwise, it will not be very profitable or may lose money. For example, a high scrap rate may bankrupt the company, as the examples in the introduction showed.

**Remark**

**2.**

**Experiment**

**3.**

**Experiment**

**4.**

- (1)
- Besides all the external factors (i.e., a, ${a}_{0}$, ${a}_{42}$, ${p}_{34}$, ${r}_{3}$, and ${r}_{5}$), the internal factor $\alpha $ also had significant effects on the expected profit here.
- (2)
- The optimal ${Q}^{*}$ lay between 7 and 18 for combinations that led to a positive expected profit.
- (3)
- Factor analysis showed that only the six external factors, a, ${a}_{0}$, ${a}_{42}$, ${p}_{34}$, ${r}_{3}$, and ${r}_{5}$, had significant effects on the optimal ${Q}^{*}$ although the internal factor $\alpha $ had significant effects on the expected profit. When a, ${r}_{3}$, and ${r}_{5}$ were at the low level and ${a}_{0}$, ${a}_{42}$, and ${p}_{34}$ were at the high level, the averages of the optimal ${Q}^{*}$ were lower than that when a, ${r}_{3}$, and ${r}_{5}$ were at the high level, and ${a}_{0}$, ${a}_{42}$, and ${p}_{34}$ were at the low level.

**Remark**

**3.**

**Experiment**

**5.**

## 6. Identifying the Optimal Strategy

- (1)
- If $d<{m}_{2}^{\prime}+{m}_{2}^{\prime \prime}$, $0\le {n}_{24}^{\prime}\le {m}_{2}^{\prime}-{d}^{\prime},0\le {n}_{34}^{\prime}\le {m}_{3}^{\prime}+{d}^{\prime},0\le {n}_{45}^{\prime}\le {m}_{4}^{\prime}$, $0\le {n}_{24}^{\prime \prime}\le {m}_{2}^{\prime \prime}-{d}^{\prime \prime},0\le {n}_{34}^{\prime \prime}\le {m}_{3}^{\prime \prime}+{d}^{\prime \prime},0\le {n}_{45}^{\prime \prime}\le {m}_{4}^{\prime \prime}$, $t=1,\cdots ,T-1$, and ${d}^{\prime \prime}=d-{d}^{\prime}$, then$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {p}_{t,({m}_{2}^{\prime},{m}_{3}^{\prime},{m}_{4}^{\prime},{m}_{2}^{\prime \prime},{m}_{3}^{\prime \prime},{m}_{4}^{\prime \prime}),({Q}^{\prime}-{m}_{3}^{\prime}-{d}^{\prime}-{n}_{24}^{\prime},\phantom{\rule{4pt}{0ex}}{m}_{3}^{\prime}+{d}^{\prime}-{n}_{34}^{\prime},\phantom{\rule{4pt}{0ex}}{n}_{24}^{\prime}+{n}_{34}^{\prime},{Q}^{\prime \prime}-{m}_{3}^{\prime \prime}-{d}^{\prime \prime}-{n}_{24}^{\prime \prime},\phantom{\rule{4pt}{0ex}}{m}_{3}^{\prime}+{d}^{\prime}-{n}_{34}^{\prime},\phantom{\rule{4pt}{0ex}}{n}_{24}^{\prime \prime}+{n}_{34}^{\prime \prime})}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{{\left(\lambda \left(Q\right)\right)}^{d}}{d!}{e}^{-\lambda \left(Q\right)}\sum _{{d}^{\prime}=max\{0,\phantom{\rule{4pt}{0ex}}d-{m}_{2}^{\prime \prime}\}}^{{d}^{\prime}=min\{d,\phantom{\rule{4pt}{0ex}}{m}_{2}^{\prime}\}}\frac{\left(\genfrac{}{}{0pt}{}{{m}_{2}^{\prime}}{{d}^{\prime}}\right)\left(\genfrac{}{}{0pt}{}{{m}_{2}^{\prime \prime}}{{d}^{\prime \prime}}\right)}{\left(\genfrac{}{}{0pt}{}{{m}_{2}^{\prime}+{m}_{2}^{\prime \prime}}{d}\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{({m}_{2}^{\prime}-{d}^{\prime})!}{{n}_{24}^{\prime}!({m}_{2}^{\prime}-{d}^{\prime}-{n}_{24}^{\prime})!}{p}_{24}^{{n}_{24}^{\prime}}\left({C}_{0}\right){\left(1-{p}_{24}\left({C}_{0}\right)\right)}^{{m}_{2}^{\prime}-{d}^{\prime}-{n}_{24}^{\prime}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{({m}_{3}^{\prime}+{d}^{\prime})!}{{n}_{34}^{\prime}!({m}_{3}^{\prime}+{d}^{\prime}-{n}_{34}^{\prime})!}{p}_{34}^{{n}_{34}^{\prime}}{\left(1-{p}_{34}\right)}^{{m}_{3}^{\prime}+{d}^{\prime}-{n}_{34}^{\prime}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{{m}_{4}^{\prime}!}{{n}_{45}^{\prime}!({m}_{4}^{\prime}-{n}_{45}^{\prime})!}{p}_{45}^{{n}_{45}^{\prime}}\left({C}_{0}\right){\left(1-{p}_{45}\left({C}_{0}\right)\right)}^{{m}_{4}^{\prime}-{n}_{45}^{\prime}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{({m}_{2}^{\prime \prime}-{d}^{\prime \prime})!}{{n}_{24}^{\prime \prime}!({m}_{2}^{\prime \prime}-{d}^{\prime \prime}-{n}_{24}^{\prime \prime})!}{p}_{24}^{{n}_{24}^{\prime \prime}}\left({C}_{0}\right){\left(1-{p}_{24}\left({C}_{0}\right)\right)}^{{m}_{2}^{\prime \prime}-{d}^{\prime \prime}-{n}_{24}^{\prime \prime}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{({m}_{3}^{\prime \prime}+{d}^{\prime \prime})!}{{n}_{34}^{\prime \prime}!({m}_{3}^{\prime \prime}+{d}^{\prime \prime}-{n}_{34}^{\prime \prime})!}{p}_{34}^{{n}_{34}^{\prime \prime}}{\left(1-{p}_{34}\right)}^{{m}_{3}^{\prime \prime}+{d}^{\prime \prime}-{n}_{34}^{\prime \prime}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{{m}_{4}^{\prime \prime}!}{{n}_{45}^{\prime \prime}!({m}_{4}^{\prime \prime}-{n}_{45}^{\prime \prime})!}{p}_{45}^{{n}_{45}^{\prime \prime}}\left({C}_{0}\right){\left(1-{p}_{45}\left({C}_{0}\right)\right)}^{{m}_{4}^{\prime \prime}-{n}_{45}^{\prime \prime}}\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {R}_{t,({m}_{2}^{\prime},{m}_{3}^{\prime},{m}_{4}^{\prime},{m}_{2}^{\prime \prime},{m}_{3}^{\prime \prime},{m}_{4}^{\prime \prime}),({Q}^{\prime}-{m}_{3}^{\prime}-{d}^{\prime}-{n}_{24}^{\prime},\phantom{\rule{4pt}{0ex}}{m}_{3}^{\prime}+{d}^{\prime}-{n}_{34}^{\prime},\phantom{\rule{4pt}{0ex}}{n}_{24}^{\prime}+{n}_{34}^{\prime},{Q}^{\prime \prime}-{m}_{3}^{\prime \prime}-{d}^{\prime \prime}-{n}_{24}^{\prime \prime},\phantom{\rule{4pt}{0ex}}{m}_{3}^{\prime}+{d}^{\prime}-{n}_{34}^{\prime},\phantom{\rule{4pt}{0ex}}{n}_{24}^{\prime \prime}+{n}_{34}^{\prime \prime})}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =-{C}_{0}-{C}_{42}\left(c\right)({m}_{4}^{\prime}-{n}_{45}^{\prime})-c({Q}^{\prime}-{m}_{2}^{\prime}-{m}_{3}^{\prime}-{m}_{4}^{\prime}+{n}_{45}^{\prime})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}+({m}_{3}^{\prime}+{d}^{\prime})({r}_{3}-{C}_{3}\left(c\right))+{n}_{45}^{\prime}{r}_{5}-({m}_{2}^{\prime}-d+{m}_{2}^{\prime \prime}){C}_{2}\left(c\right)+({m}_{3}^{\prime \prime}+{d}^{\prime \prime})(\theta {r}_{3}-{C}_{3}\left(c\right)).\hfill \end{array}$$
- (2)
- If $d\ge {m}_{2}^{\prime}+{m}_{2}^{\prime \prime}$, $0\le {n}_{34}^{\prime}\le {m}_{3}^{\prime}+{m}_{2}^{\prime},0\le {n}_{45}^{\prime}\le {m}_{4}^{\prime}$, $0\le {n}_{34}^{\prime \prime}\le {m}_{3}^{\prime \prime}+{m}_{2}^{\prime \prime},0\le {n}_{45}^{\prime \prime}\le {m}_{4}^{\prime \prime}$, and $t=1,\cdots ,T-1$, then$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {p}_{t,({m}_{2}^{\prime},{m}_{3}^{\prime},{m}_{4}^{\prime},{m}_{2}^{\prime \prime},{m}_{3}^{\prime \prime},{m}_{4}^{\prime \prime}),({Q}^{\prime}-{m}_{3}^{\prime}-{m}_{2}^{\prime},\phantom{\rule{4pt}{0ex}}{m}_{3}^{\prime}+{m}_{2}^{\prime}-{n}_{34}^{\prime},\phantom{\rule{4pt}{0ex}}{n}_{34}^{\prime},{Q}^{\prime \prime}-{m}_{3}^{\prime \prime}-{m}_{2}^{\prime \prime},\phantom{\rule{4pt}{0ex}}{m}_{3}^{\prime \prime}+{m}_{2}^{\prime \prime}-{n}_{34}^{\prime \prime},\phantom{\rule{4pt}{0ex}}{n}_{34}^{\prime \prime})}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left[\sum _{d={m}_{2}^{\prime}+{m}_{2}^{\prime \prime}}^{\infty}\frac{{\left(\lambda \left(Q\right)\right)}^{d}}{d!}{e}^{-\lambda \left(Q\right)}\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{({m}_{3}^{\prime}+{m}_{2}^{\prime})!}{{n}_{34}^{\prime}!({m}_{3}^{\prime}+{m}_{2}^{\prime}-{n}_{34}^{\prime})!}{p}_{34}^{{n}_{34}^{\prime}}{\left(1-{p}_{34}\right)}^{{m}_{3}^{\prime}+{m}_{2}^{\prime}-{n}_{34}^{\prime}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{{m}_{4}^{\prime}!}{{n}_{45}^{\prime}!({m}_{4}^{\prime}-{n}_{45}^{\prime})!}{p}_{45}^{{n}_{45}^{\prime}}\left({C}_{0}\right){\left(1-{p}_{45}\left({C}_{0}\right)\right)}^{{m}_{4}^{\prime}-{n}_{45}^{\prime}}.\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{({m}_{3}^{\prime \prime}+{m}_{2}^{\prime \prime})!}{{n}_{34}^{\prime \prime}!({m}_{3}^{\prime \prime}+{m}_{2}^{\prime \prime}-{n}_{34}^{\prime \prime})!}{p}_{34}^{{n}_{34}^{\prime \prime}}{\left(1-{p}_{34}\right)}^{{m}_{3}^{\prime \prime}+{m}_{2}^{\prime \prime}-{n}_{34}^{\prime \prime}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{{m}_{4}^{\prime \prime}!}{{n}_{45}^{\prime \prime}!({m}_{4}^{\prime \prime}-{n}_{45}^{\prime \prime})!}{p}_{45}^{{n}_{45}^{\prime \prime}}\left({C}_{0}\right){\left(1-{p}_{45}\left({C}_{0}\right)\right)}^{{m}_{4}^{\prime \prime}-{n}_{45}^{\prime \prime}}\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {R}_{t,({m}_{2}^{\prime},{m}_{3}^{\prime},{m}_{4}^{\prime},{m}_{2}^{\prime \prime},{m}_{3}^{\prime \prime},{m}_{4}^{\prime \prime}),({Q}^{\prime}-{m}_{3}^{\prime}-{m}_{2}^{\prime},\phantom{\rule{4pt}{0ex}}{m}_{3}^{\prime}+{m}_{2}^{\prime}-{n}_{34}^{\prime},\phantom{\rule{4pt}{0ex}}{n}_{34}^{\prime},{Q}^{\prime \prime}-{m}_{3}^{\prime \prime}-{m}_{2}^{\prime \prime},\phantom{\rule{4pt}{0ex}}{m}_{3}^{\prime \prime}+{m}_{2}^{\prime \prime}-{n}_{34}^{\prime \prime},\phantom{\rule{4pt}{0ex}}{n}_{34}^{\prime \prime})}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =-{C}_{0}-{C}_{42}\left(c\right)({m}_{4}^{\prime}-{n}_{45}^{\prime})-c({Q}^{\prime}-{m}_{2}^{\prime}-{m}_{3}^{\prime}-{m}_{4}^{\prime}+{n}_{45}^{\prime})+({m}_{3}^{\prime}+{m}_{2}^{\prime})({r}_{3}-{C}_{3}\left(c\right))+{n}_{45}^{\prime}{r}_{5}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}+({m}_{3}^{\prime \prime}+{m}_{2}^{\prime \prime})(\theta {r}_{3}-{C}_{3}\left(c\right)).\hfill \end{array}$$
- (3)
- If $d<{m}_{2}^{\prime}+{m}_{2}^{\prime \prime}$, and $t=T$, then$$\begin{array}{c}\hfill {p}_{T,({m}_{2}^{\prime},{m}_{3}^{\prime},{m}_{4}^{\prime},{m}_{2}^{\prime \prime},{m}_{3}^{\prime \prime},{m}_{4}^{\prime \prime}),(0,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}{d}^{\prime},0,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}{d}^{\prime \prime})}=\frac{{\left(\lambda \left(Q\right)\right)}^{d}}{d!}{e}^{-\lambda \left(Q\right)}\sum _{{d}^{\prime}=max\{0,\phantom{\rule{4pt}{0ex}}d-{m}_{2}^{\prime \prime}\}}^{{d}^{\prime}=min\{d,\phantom{\rule{4pt}{0ex}}{m}_{2}^{\prime}\}}\frac{\left(\genfrac{}{}{0pt}{}{{m}_{2}^{\prime}}{{d}^{\prime}}\right)\left(\genfrac{}{}{0pt}{}{{m}_{2}^{\prime \prime}}{{d}^{\prime \prime}}\right)}{\left(\genfrac{}{}{0pt}{}{{m}_{2}^{\prime}+{m}_{2}^{\prime \prime}}{d}\right)}\end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {R}_{T,(({m}_{2}^{\prime},{m}_{3}^{\prime},{m}_{4}^{\prime},{m}_{2}^{\prime \prime},{m}_{3}^{\prime \prime},{m}_{4}^{\prime \prime}),(0,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}{d}^{\prime},0,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}{d}^{\prime \prime})}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =-{C}_{0}+({m}_{3}^{\prime}+{d}^{\prime})({r}_{3}-{C}_{3}\left(c\right))-({Q}^{\prime}-{m}_{3}^{\prime}-{d}^{\prime}){C}_{2}\left(c\right)+(1-\alpha ){Q}^{\prime}{r}_{5}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}-({Q}^{\prime \prime}-{m}_{3}^{\prime \prime}-{d}^{\prime \prime}){C}_{2}\left(c\right)+({m}_{3}^{\prime \prime}+{d}^{\prime \prime})(\theta {r}_{3}-{C}_{3}\left(c\right)).\hfill \end{array}$$
- (4)
- If $d\ge {m}_{2}^{\prime}+{m}_{2}^{\prime \prime}$, and $t=T$, then$$\begin{array}{c}\hfill {p}_{T,({m}_{2}^{\prime},{m}_{3}^{\prime},{m}_{4}^{\prime},{m}_{2}^{\prime \prime},{m}_{3}^{\prime \prime},{m}_{4}^{\prime \prime}),(0,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}{m}_{2}^{\prime},0,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}{m}_{2}^{\prime \prime})}=\sum _{d={m}_{2}^{\prime}+{m}_{2}^{\prime \prime}}^{\infty}\frac{{\left(\lambda \left(Q\right)\right)}^{d}}{d!}{e}^{-\lambda \left(Q\right)}\end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {R}_{T,({m}_{2}^{\prime},{m}_{3}^{\prime},{m}_{4}^{\prime},{m}_{2}^{\prime \prime},{m}_{3}^{\prime \prime},{m}_{4}^{\prime \prime}),(0,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}{m}_{2}^{\prime},0,\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}{m}_{2}^{\prime \prime})}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =-{C}_{0}+({m}_{3}^{\prime}+{m}_{2}^{\prime})({r}_{3}-{C}_{3}\left(c\right))-({Q}^{\prime}-{m}_{2}^{\prime}-{m}_{3}^{\prime}){C}_{2}\left(c\right)+(1-\alpha ){Q}^{\prime}{r}_{5}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}-({Q}^{\prime \prime}-{m}_{2}^{\prime \prime}-{m}_{3}^{\prime \prime}){C}_{2}\left(c\right)+({m}_{3}^{\prime \prime}+{m}_{2}^{\prime \prime})(\theta {r}_{3}-{C}_{3}\left(c\right)).\hfill \end{array}$$
- (5)
- If none of (1)–(4) is true, then ${P}_{\xb7,(\xb7),(\xb7)}=0$ and ${R}_{\xb7,(\xb7),(\xb7)}=0$.

**Experiment**

**6.**

## 7. Discussion: Infinite Time Horizon

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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State | Meaning | To State | |||||
---|---|---|---|---|---|---|---|

Index | From State | 1 | 2 | 3 | 4 | 5 | |

1 | A car is new. | 1 | √ | ||||

2 | A car is available. | 2 | √ | √ | √ | ||

3 | A car is being used. | 3 | √ | √ | |||

4 | A car is in maintenance. | 4 | √ | √ | √ | ||

5 | A car is scrapped. | 5 |

${\mathit{C}}_{0}(\mathit{c},\mathit{Q})={\mathit{a}}_{0}\mathit{c}\mathit{Q}+{\mathit{b}}_{0}$ | ${\mathit{a}}_{0}>0$, ${\mathit{b}}_{0}>0$ |
---|---|

${C}_{2}\left(c\right)={a}_{2}c+{b}_{2}$ | ${a}_{2}>0$, ${b}_{2}\ge 0$ |

${C}_{3}\left(c\right)={a}_{3}c+{b}_{3}$ | ${a}_{3}>0$, ${b}_{3}\ge 0$ |

${C}_{42}\left(c\right)={a}_{42}c+{b}_{42}$ | ${a}_{42}>0$, ${b}_{42}\ge 0$ |

${p}_{24}\left({C}_{0}\right)=-{a}_{24}{C}_{0}+{b}_{24}$ | ${a}_{24}>0$, ${b}_{24}\ge 0$ |

${p}_{45}\left({C}_{0}\right)=-{a}_{45}{C}_{0}+{b}_{45}$ | ${a}_{45}>0$, ${b}_{45}\ge 0$ |

$\lambda \left(Q\right)=aQ$ | $a>0$ |

Parameters | a | ${\mathit{a}}_{0}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{24}$ | ${\mathit{a}}_{42}$ |
---|---|---|---|---|---|---|

Low | $0.6$ | $0.0001$ | ${10}^{-6}$ | ${10}^{-6}$ | ${10}^{-7}$ | $0.001$ |

High | 1 | $0.0005$ | $5\times {10}^{-6}$ | $5\times {10}^{-6}$ | $3\times {10}^{-7}$ | $0.02$ |

Parameters | ${a}_{45}$ | ${b}_{0}$ | ${p}_{34}$ | ${r}_{3}$ | ${r}_{5}$ | $\alpha $ |

Low | $4\times {10}^{-7}$ | 20 | $0.80$ | 60 | 5000 | $0.0002$ |

High | $6\times {10}^{-7}$ | 50 | $0.99$ | 120 | 20,000 | $0.0004$ |

**Table 4.**Statistics on two levels of each of six significant factors where L is the number of combinations that resulted in a loss.

a | ${\mathit{a}}_{0}$ | |||
---|---|---|---|---|

Low | High | Low | High | |

Mean of expected profit | 54,736.80 | 71,217.20 | 78,440.50 | 47,513.50 |

Maximum expected profit | 489,408.58 | 563,814.59 | 563,814.59 | 437,209.31 |

Minimum expected profit | 0 | 0 | 0 | 0 |

Number of combinations | 64 | 64 | 64 | 64 |

L | 48 | 46 | 43 | 51 |

${a}_{42}$ | ${p}_{34}$ | |||

Low | High | Low | High | |

Mean of expected profit | 125,954.00 | 0 | 79,378.60 | 46,575.40 |

Maximum expected profit | 563,814.59 | 0 | 563,814.59 | 447,678.55 |

Minimum expected profit | 0 | 0 | 0 | 0 |

Number of combinations | 64 | 64 | 64 | 64 |

L | 30 | 64 | 44 | 50 |

${r}_{3}$ | ${r}_{5}$ | |||

Low | High | Low | High | |

Mean expected profit | 3143.74 | 122,810.00 | 25,146.80 | 100,807.00 |

Maximum expected profit | 77,579.81 | 563,814.59 | 247,909.02 | 563,814.59 |

Minimum expected profit | 0 | 0 | 0 | 0 |

Number of combinations | 64 | 64 | 64 | 64 |

L | 59 | 35 | 51 | 43 |

Parameters | a | ${\mathit{a}}_{0}$ | ${\mathit{a}}_{24}$ | ${\mathit{a}}_{42}$ | ${\mathit{a}}_{45}$ | ${\mathit{b}}_{0}$ |
---|---|---|---|---|---|---|

Low | $0.6$ | $0.0001$ | ${10}^{-7}$ | $0.001$ | $4\times {10}^{-7}$ | 20 |

High | 1 | $0.0005$ | $3\times {10}^{-7}$ | $0.02$ | $6\times {10}^{-7}$ | 50 |

Parameters | ${p}_{34}$ | ${r}_{3}$ | ${r}_{5}$ | $\alpha $ | $\theta $ | |

Low | $0.80$ | 60 | 5000 | $0.0002$ | $0.2$ | |

High | $0.99$ | 120 | 20,000 | $0.0004$ | $0.5$ |

${\mathit{a}}_{0}$ | ${\mathit{a}}_{42}$ | ${\mathit{r}}_{5}$ | $\mathit{\theta}$ | |||||
---|---|---|---|---|---|---|---|---|

Low | High | Low | High | Low | High | Low | High | |

Mean of ${Q}^{\prime \prime}$ | 13.00 | 8.92 | 4.42 | 17.50 | 13.08 | 8.43 | 8.75 | 13.17 |

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## Share and Cite

**MDPI and ACS Style**

Zhou, W.; Wang, H.; Shi, V.; Chen, X.
A Decision Model for Free-Floating Car-Sharing Providers for Sustainable and Resilient Supply Chains. *Sustainability* **2022**, *14*, 8159.
https://doi.org/10.3390/su14138159

**AMA Style**

Zhou W, Wang H, Shi V, Chen X.
A Decision Model for Free-Floating Car-Sharing Providers for Sustainable and Resilient Supply Chains. *Sustainability*. 2022; 14(13):8159.
https://doi.org/10.3390/su14138159

**Chicago/Turabian Style**

Zhou, Wei, Haixia Wang, Victor Shi, and Xiding Chen.
2022. "A Decision Model for Free-Floating Car-Sharing Providers for Sustainable and Resilient Supply Chains" *Sustainability* 14, no. 13: 8159.
https://doi.org/10.3390/su14138159