设置货架缓存区的订单和货架排序问题研究

doi:10.16381/j.cnki.issn1003-207x.2024.0255

摘要/Abstract

摘要：

在电商企业快速发展和移动机器人履行系统广泛应用的背景下，本文针对带货架缓存区的订单和货架排序问题进行研究。在货架缓存区容量有限与考虑订单中商品数量的基础上，对订单拣选顺序、货架出库顺序以及货架缓存策略进行联合优化。以最小化货架出库次数为目标，构建了设置货架缓存区的订单和货架排序问题的整数规划模型。并根据问题特点，设计了交互启发式算法，动态同步地决策三个子问题。通过算例实验验证了本文模型与算法的正确性和有效性。实验结果显示，在小算例中，本文算法所求可行解与精确解的平均相对偏差约为5.4%。在大算例中，与基准算法相比，本文算法对目标函数的平均提升效果约为9.6%，与无缓存区的启发式算法相比，证明设置货架缓存区能有效减少订单拣选过程中货架的出库次数，降低幅度约为14.1%。本文提出的模型和算法为电商企业合理设置货架缓存区、减少货架出库次数、提高订单拣选效率提供了决策依据。

关键词: 移动机器人履行系统, 订单和货架排序, 货架缓存区, 交互启发式算法, 商品数量

Abstract:

With the rapid development of e-commerce enterprises and the wide application of robotic mobile fulfillment systems， the importance of order picking for warehousing operational efficiency is increasingly evident， and configuring rack buffer zones can enhance order picking efficiency. Therefore， it focuses on the research of the order and rack sequencing problem with rack buffer zones. In the scenario of limited capacity in the rack buffer zone， it aims to jointly optimize the order picking sequence， rack retrieval sequence， and buffering strategy. Taking into consideration factors such as the shared storage mode for products， the quantity of each product stored on individual racks， the ordered quantities in customer orders， and the limited capacity of the buffer zone， an integer programming model is established for the order and rack sequencing problem with rack buffer zones. The objective is to minimize the number of rack retrievals while considering these factors. Considering the characteristics of the problem， an interactive heuristic algorithm is designed to synchronously address the three sub-problems. Through case study experiments， the correctness and effectiveness of the proposed model and algorithm have been verified. The experimental results indicate that the average relative deviation between the feasible solutions obtained by the interactive heuristic algorithm and the exact solutions is approximately 5.4%， demonstrating that the proposed algorithm can achieve high-quality feasible solutions in a relatively short time. Furthermore， comparisons with the baseline algorithm and the heuristic algorithm without the rack buffer zone further validate the efficiency of the proposed algorithm. Specifically， the average improvement in the objective function achieved by the proposed algorithm is about 9.6% compared to the benchmark algorithm， and the implementation of the rack buffer zone significantly reduces the number of times racks are transported during the order fulfillment picking process by about 14.1%. The proposed model and algorithm in this paper provide decision-making foundations for enterprises to strategically set up rack buffer zones， reduce the number of rack retrievals， and enhance order-picking efficiency.

Key words: robotic mobile fulfillment system, order and rack sequencing, rack buffer zone, interactive heuristic algorithm, product quantity

中图分类号:

O211.4

韩倩倩,王康,李珍萍. 设置货架缓存区的订单和货架排序问题研究[J]. 中国管理科学, 2026, 34(4): 168-177.

Qianqian Han,Kang Wang,Zhenping Li. Study on the Order and Rack Sequencing Problem with Rack Buffer Zones[J]. Chinese Journal of Management Science, 2026, 34(4): 168-177.

图/表 10

图1

图2

表1

符号定义"

符号	定义
$N$	订单数量， $n = 1, …, N$
$K$	货架数量， $k = 1, …, K$
$S$	商品数量， $s = 1, …, S$
$P$	时间槽数量， $p = 1, …, P$
$C$	拣选台容量
$H$	货架缓存区容量
$O$	订单集合， $o n ∈ O$
$R$	货架集合， $r k ∈ R$
$q k s$	货架 $r k$ 上存储商品 $s$ 的数量
$d n s$	订单 $o n$ 中订购商品 $s$ 的数量
	决策变量
$x n p$	0-1变量，若订单 $o n$ 在时间槽 $p$ 被拣选为1，否则为0
$y k p$	0-1变量，若在时间槽 $p$ 需要货架 $r k$ 为1，否则为0
$c k p$	0-1变量，若在时间槽 $p$ 将货架 $r k$ 存入缓存区为1，否则为0
$u k p$	0-1变量，若在时间槽 $p$ 出库货架 $r k$ 为1，否则为0
$z n k p s$	整数变量，在时间槽 $p$ 从货架 $r k$ 向订单 $o n$ 中拣选商品s的数量

表1

表2

表3

算例参数取值"

参数名称	小规模参数取值	大规模参数取值
订单数量（ $N$ ）	4, 6	100, 150, 200
货架数量（ $K$ ）	8, 10	400, 500, 600
商品数量（ $S$ ）	10, 15	1000,1500,2000
拣选台容量（ $C$ ）	3	15
缓存区容量（ $H$ ）	3	10

表3

表4

Gurobi与交互启发式算法求解结果对比"

算例参数	Gurobi		交互启发式算法		$N u m$	$G a p 1$ (%)
$N - K - S$	$z g$	$t g$	$z 1$	$t 1$	$N u m$	$G a p 1$ (%)
4-8-10	2.6	8.8	2.8	0.04	4	7.1
4-8-15	2.4	3.9	2.6	0.05	4	7.7
4-10-10	2.4	127	2.4	0.04	5	0
4-10-15	2.6	9.8	2.8	0.04	4	7.1
6-8-10	2.8	140.4	3.0	0.05	4	6.7
6-8-15	3.0	264.2	3.0	0.05	5	0
6-10-10	2.6	289.7	2.8	0.04	4	7.1
6-10-15	2.8	264.9	3.0	0.05	4	6.7
平均值	2.65	138.6	2.8	0.05	4.25	5.4

表4

表5

交互启发式算法、基准算法与无缓存区算法的求解结果及算法时间对比"

算例参数	交互启发式算法		基准算法		无缓存区算法		$G a p 2$ (%)	$G a p 3$ (%)
$N - K - S$	$z ¯ 1$	$t ¯ 1$	$z ¯ 2$	$t ¯ 2$	$z ¯ 3$	$t ¯ 3$	$G a p 2$ (%)	$G a p 3$ (%)
100-400-1000	113.0	0.2	121.6	0.2	130.4	0.2	7.1	13.3
100-400-1500	111.8	0.2	118.8	0.2	124.8	0.2	5.9	10.4
100-400-2000	114.0	0.3	121.0	0.3	123.6	0.3	5.8	7.8
100-500-1000	111.0	0.3	122.0	0.3	129.2	0.2	9.0	14.1
100-500-1500	125.6	0.3	136.8	0.3	146.4	0.3	8.2	14.2
100-500-2000	132.0	0.3	141.8	0.4	149.6	0.4	6.9	11.8
100-600-1000	124.0	0.3	132.6	0.4	143.4	0.4	6.5	13.5
100-600-1500	127.4	0.3	132.8	0.5	141.6	0.4	4.1	10.0
100-600-2000	133.0	0.3	140.6	0.4	148.6	0.3	5.4	10.5
150-400-1000	144.0	0.3	161.4	0.4	169.4	0.3	10.8	15.0
150-400-1500	148.4	0.3	162.4	0.4	171.0	0.4	8.6	13.2
150-400-2000	152.0	0.3	173.4	0.4	181.8	0.4	12.3	16.4
150-500-1000	151.2	0.4	167.8	0.5	181.0	0.4	9.9	16.5
150-500-1500	160.0	0.4	178.6	0.5	191.8	0.4	10.4	16.6
150-500-2000	162.8	0.4	180.6	0.4	189.6	0.4	9.9	14.1
150-600-1000	164.6	0.4	181.0	0.5	196.4	0.5	9.1	16.2
150-600-1500	171.2	0.5	194.8	0.6	200.2	0.5	12.1	14.5
150-600-2000	179.4	0.5	197.6	0.6	205.2	0.5	9.2	12.6
200-400-1000	181.2	0.5	204.2	0.5	214.6	0.4	11.3	15.6
200-400-1500	184.8	0.5	208.0	0.5	219.4	0.4	11.2	15.8
200-400-2000	184.8	0.4	211.8	0.5	217.2	0.4	12.7	14.9
200-500-1000	189.6	0.5	215.4	0.6	226.8	0.5	12.0	16.4
200-500-1500	203.0	0.5	229.0	0.6	238.8	0.5	11.4	15.0
200-500-2000	216.4	0.7	239.8	0.7	252.4	0.6	9.8	14.3
200-600-1000	201.2	0.6	226.2	0.7	237.6	0.5	11.1	15.3
200-600-1500	221.6	0.7	245.2	0.8	258.6	0.7	9.6	14.3
200-600-2000	226.0	0.7	248.6	0.8	258.0	0.7	9.1	12.4
平均值	160.5	0.4	177.5	0.5	186.9	0.4	9.6	14.1

表5

图3

图4

图5

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订单中商品种类数	订单出现概率
1	0.43
2	0.26
3	0.16
4	0.09
5	0.06