# An Asymmetric Polling-Based Optimization Model in a Dynamic Order Picking System

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}= λ

_{2}= … = λ

_{n}, β

_{1}= β

_{2}= … = β

_{n}, and γ

_{1}= γ

_{2}= … = γ

_{n}. For example, due to the different sizes and sales, retailers choose different order intervals to maintain their inventory levels, and the order interval directly affects the order queue arrival rate, λ. Therefore, the assumption that λ

_{1}= λ

_{2}= … = λ

_{n}is invalid, and in a real situation, the stochastic characteristics of all order queues are variable. At the very least, not every order queue is the same, which is the assumption of asymmetric polling theory [38,39]. The difference between symmetric and asymmetric polling systems is the stochastic properties constraint of order queues. The asymmetric polling system loosens the constraints on the stochastic properties compared to the symmetric polling system, which is closer to the practical production requirements.

## 2. Model

#### 2.1. Problem Statement

_{i}. (ii) The performance of the picking service depends on the speed and efficiency of the picking machine. This process is measured by the expectation β

_{i}of the picking time. (iii) Switching the service objects is where the picking machine switches to pick order queue i + 1 after serving order queue i. This process is measured by the expectation γ

_{i}of switching times in the picking service. The length of switching time reflects the flexibility of the picking machine and the interval between different orders.

_{n}, when the sorter machine starts picking the queue in sub-queue. ξ

_{i}(n) is the number of orders that accept the picking service in sub-queue i. Only the orders of parent queue set $\widehat{i}$ that arrive before t

_{n}

_{+1}, except queue i + 1, can obtain the picking service at time t

_{n}. ξ

_{i}(n + 1) is the number of orders that accept the picking service in the parent queue set $\widehat{i}$ at the time t

_{n}

_{+1}. Due to the fact that i + 1 ∈ $\widehat{i}$, $\sum _{j=1,j\ne i}^{N}{\xi}_{j}\left(n+1\right)}={\xi}_{\widehat{i}}\left(n+1\right)$. Once ξ

_{i}(n + 1) orders finish the picking services, the next round of polling starts. In addition, as a reference, we provide an example with the MATLAB program of the asymmetric polling-based dynamic order picking system in Appendix A.

#### 2.2. Notation and Definitions

_{i}(z

_{i}): The probability distribution function of a queue’s arrival rate. A

_{i}′(z

_{i}) is its first derivative, and A

_{i}″(z

_{i}) is its second derivative;

_{i}and variance σ

^{2}

_{λi}of the probability distribution all satisfy the following conditions: ${\lambda}_{i}={A}_{i}^{\prime}\left(1\right)$ and ${\sigma}_{{\lambda}_{i}}^{2}={A}_{i}^{\u2033}\left(1\right)+{\lambda}_{i}-{\lambda}_{i}^{2}$;

_{i}(z

_{i}): The probability distribution function of a queue’s picking time. B

_{i}′(z

_{i}) is its first derivative, and B

_{ii}″((z

_{i}) is its second derivative;

_{i}and variance σ

^{2}

_{βi}of the probability distribution all satisfy the following conditions: ${\beta}_{i}={B}_{i}^{\prime}\left(1\right)$ and ${\sigma}_{{\beta}_{i}}^{2}={B}_{i}^{\u2033}\left(1\right)+{\beta}_{i}-{\beta}_{i}^{2}$;

_{i}(z

_{i}): The probability distribution function of the adjacent queue’s picking service switching time. R

_{i}′(z

_{i}) is its first derivative, and R

_{i}″(z

_{i}) is its second derivative;

_{i}and variance σ

^{2}

_{γi}of the probability distribution all satisfy the following conditions: ${\gamma}_{i}={R}_{i}^{\prime}\left(1\right)$ and ${\sigma}_{{\gamma}_{i}}^{2}={R}_{i}^{\u2033}\left(1\right)+{\gamma}_{i}-{\gamma}_{i}^{2}$. After processing, we obtain ${\gamma}_{\widehat{i}}={\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^{N}{\gamma}_{j}}={\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^{N}{R}_{j}^{\prime}\left(1\right)}$ and ${\sigma}_{{\gamma}_{\widehat{i}}}^{2}={\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^{N}\left[{R}_{j}^{\u2033}\left(1\right)+{\gamma}_{j}-{\gamma}_{j}^{2}\right]}$;

_{i}(n): The number of orders that arrived before moment t

_{n}in the order queue i; [ξ

_{1}(n), ξ

_{2}(n), …, ξ

_{i}(n), …, ξ

_{N}(n)] is the state of the whole picking system at moment t

_{n}; π

_{i}(x

_{1}, x

_{2}, …, x

_{i}, …, x

_{N}) is the probability distribution of the picking system state at moment t

_{n}; G

_{i}(z

_{1}, z

_{2}, …, z

_{i}, …, z

_{N}) is the generating function of π

_{i}(x

_{1}, x

_{2}, …, x

_{i}, …, x

_{N}); g

_{i}(k) is the first order characteristic of G

_{i}(z

_{1}, z

_{2}, …, z

_{i}, …, z

_{N}), and it indicates the number of orders that are waiting to be picked in queue k when queue i starts to obtain the picking service; g

_{i}(j, k) and ${g}_{\widehat{i}}\left(j,k\right)$ are its second characteristic quantity;

_{i}(n): The time of the picking service switching from sub-queue i to order parent queue set $\widehat{i}$; ${u}_{\widehat{i}}\left(n\right)$ represents the time of picking service switching from parent queue set $\widehat{i}$ to sub-queue i; v

_{i}(n) is the time when order queue i has picking service; μ

_{j}(u

_{i}) is the number of orders that arrive at order queue j within u

_{i}(n), j∈$\widehat{i}$; ${\mu}_{j}\left({u}_{\widehat{i}}\right)$ is the number of orders that arrive at order queue j within ${u}_{\widehat{i}}\left(n\right)$; η

_{j}(v

_{i}) is the number of orders that arrive at order queue j within v

_{i}(n);

#### 2.3. System Generating Function

_{n}

_{+1}, and the gated control polling system theory requires that the stability of the system must meet the condition $\sum _{i=1}^{N}{\lambda}_{i}}{\beta}_{i}<1$ [40]. When the polling system is stable, it has the following relationships:

#### 2.4. The Characteristic Parameters of the System Generating Function

#### 2.4.1. Mean Queue Length

_{i}(j). Supposing i = j, g

_{i}(i) represents the number of orders that wait for the next picking in queue i buffer; it can also be called the mean queue length of the picking system. g

_{i}(j) is obtained by seeking the partial derivative of G

_{i}(z

_{1}, z

_{2}, …, z

_{i}, …, z

_{N}) as follows:

#### 2.4.2. Mean Waiting Time

_{i}(j, k) are obtained by seeking the second partial derivative of G

_{i}(z

_{1}, z

_{2}, …, z

_{i}, …, z

_{N}) as follows:

## 3. Numerical Example and Discussion

_{i}β

_{i}and γ

_{i}) on the proposed model, and then, we compared the performance of the new picking system in the cases of different amounts of the picking station. The results of the MQL and MWT illustrate the major results of the paper.

#### 3.1. Numerical Example

_{1}:λ

_{2}:λ

_{3}:λ

_{4}:λ

_{5}= 1:2:4:6:8 and γ

_{i}= 10 (i = 1, 2, …, 5). $\lambda \beta ={\displaystyle \sum _{i=1}^{N}{\lambda}_{i}}{\beta}_{i}$ represents the overall load level of the picking system. The larger λβ means the higher system load. With the increasing system load λβ, MQL and MWT also grow nonlinearly. When λβ is small (λβ < 0.1), the difference in the MWT was not obvious and far less than that of the MQL. As the system’s load increased, the divergence of the MWT became apparent. If the other parameters were the same, the size of λ was positively related to the increase in the MQL and MWT.

_{1}λ

_{2}:λ

_{3}:λ

_{4}:λ

_{5}= 1:2:4:6:8, β

_{i}= 0.6 (i = 1, 2, …, 5), and $\lambda \beta ={\displaystyle \sum _{i=1}^{N}{\lambda}_{i}}{\beta}_{i}=0.126$. Figure 8 and Figure 9, respectively, show how γ impacted the mean queue length (MQL) and the mean waiting time (MWT) under a high system load, we assume that λ

_{1}:λ

_{2}:λ

_{3}:λ

_{4}:λ

_{5}= 1:2:4:6:8, β

_{i}= 3.6 (i = 1, 2, …, 5) and $\lambda \beta ={\displaystyle \sum _{i=1}^{N}{\lambda}_{i}}{\beta}_{i}=0.756$. P.S.n (n = 1, 2, 3, 4, and 5) means there are n picking stations. Figure 6, Figure 7, Figure 8 and Figure 9 show the theoretical values of the MQL and MWT at different system load levels. At any system load level, the MQL and MWT will grow with the increasing in γ. When γ is determined, the higher system load expands the MQL and MWT. As shown in Figure 7, the differences in the picking stations’ MWT were very small under a low system load.

#### 3.2. Discussion

_{i}β

_{i}and γ

_{i}) have an important impact on the performance of the picking system. The higher system load and longer switching times of a picking service can increase the queue length and waiting time of a picking system. In addition, the differences in the picking stations’ MQL were larger than that of the picking stations’ MWT with the increase in the system load, which could be due to the order arrival rate’s greater impact on the length of the order queue.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Example Program

Algorithm A1: An example program of the asymmetric polling-based dynamic order picking system with n = 5 picking stations |

for i = 1:N seg = [exprnd(1/a(1,1),1,M1);exprnd(1/a(1,2),1,M1);exprnd(1/a(1,3),1,M1);exprnd(1/a(1,4),1,M1);exprnd(1/a(1,5),1,M1)]; T(i,:) = seg(i,:); end for i = 1:M start_time = t; for j = 1:N while temp(1,j) < t if temp(1,j) + T(j,m(1,j)) < = t temp(1,j) = temp(1,j) + T(j,m(1,j)); n(1,j) = n(1,j) + 1; m(1,j) = m(1,j) + 1; each_time1(1,j) = each_time1(1,j) + (t-temp(1,j)); else break; end end res(j) = n(1,j)-stemp(1,j);% if res(j) = = 0; t = t+r(1,j); else t = t + res(j)*b(1,j) + r(1,j); end stemp(1,j) = stemp(1,j) + res(j); que_length(j) = que_length(j) + res(j); if res(j) > 0 for q = 1:res(j) each_time2(1,j) = each_time2(1,j) + ((res(j)-q)*b(1,j)); end end end time1(i) = t-start_time; end time_sum = sum(sum(time1)); for k = 1:N g1(1,k) = que_length(k)/M; w1(1,k) = (each_time1(1,k) + each_time2(1,k))./stemp(1,k); end |

## Appendix B. Proof of Equation (8)

## Appendix C. Proof of Equation (9)

_{m}(τ

_{m}= t

_{m}

_{+1}− t

_{m}) and accepts the picking service at t

_{n}time. The order k in queue i ends the picking service at t

^{*}

_{m}(t

_{m}< t

_{m}

_{+1}< t

_{n}< t

^{*}

_{m}). We assume that θ

_{i}(n) represents the time interval of the picking machine serving the same order queue i and there are y

_{i}

^{a}(n) and y

_{i}

^{b}(n) that satisfy the following conditions:

_{i}of the order can be represented by the following formula:

_{i}

^{a}and W

_{i}

^{b}can be obtained as follows:

_{i}]

_{G}can be obtained:

## Appendix D. Proof of Equation (11)

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$\left[\begin{array}{c}\mathit{\lambda}\\ \mathit{\beta}\\ \mathit{\gamma}\end{array}\right]$ | Picking Station | MQL | MWT | ||
---|---|---|---|---|---|

N.V. | T.V. | N.V. | T.V. | ||

$\left[\begin{array}{ccc}0.01& 0.02& 0.04\\ 0.6& 0.6& 0.6\\ 10& 10& 10\end{array}\right]$ | 1 | 0.3132 | 0.3180 | 15.2819 | 16.7355 |

2 | 0.6263 | 0.6320 | 15.3760 | 15.8800 | |

3 | 1.2526 | 1.1940 | 15.5643 | 16.1084 | |

$\left[\begin{array}{cccc}0.01& 0.02& 0.04& 0.06\\ 0.6& 0.6& 0.6& 0.6\\ 10& 10& 10& 10\end{array}\right]$ | 1 | 0.4338 | 0.4400 | 21.3802 | 21.4732 |

2 | 0.8677 | 0.8500 | 21.5107 | 22.7066 | |

3 | 1.7354 | 1.7940 | 21.7717 | 22.0332 | |

4 | 2.6030 | 2.5820 | 22.0327 | 22.6792 | |

$\left[\begin{array}{ccccc}0.01& 0.02& 0.04& 0.06& 0.08\\ 0.37& 0.37& 0.37& 0.37& 0.37\\ 10& 10& 10& 10& 10\end{array}\right]$ | 1 | 0.5422 | 0.5720 | 26.7805 | 27.1823 |

2 | 1.0845 | 1.0880 | 26.8814 | 26.6361 | |

3 | 2.1690 | 2.0760 | 27.0830 | 27.2311 | |

4 | 3.2535 | 3.2200 | 27.2847 | 27.1912 | |

5 | 4.3380 | 4.3400 | 27.4864 | 27.7259 |

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Yang, D.; Liu, S.; Zhang, Z.
An Asymmetric Polling-Based Optimization Model in a Dynamic Order Picking System. *Symmetry* **2022**, *14*, 2283.
https://doi.org/10.3390/sym14112283

**AMA Style**

Yang D, Liu S, Zhang Z.
An Asymmetric Polling-Based Optimization Model in a Dynamic Order Picking System. *Symmetry*. 2022; 14(11):2283.
https://doi.org/10.3390/sym14112283

**Chicago/Turabian Style**

Yang, Dan, Sen Liu, and Zhe Zhang.
2022. "An Asymmetric Polling-Based Optimization Model in a Dynamic Order Picking System" *Symmetry* 14, no. 11: 2283.
https://doi.org/10.3390/sym14112283