考虑需求和提前期不确定性的生产-库存计划仿射可调节鲁棒优化模型

doi:10.16381/j.cnki.issn1003-207x.2022.1692

摘要/Abstract

摘要：

针对由原材料供应商、制造商和客户组成的三级供应链系统，考虑由第三方物流公司负责商品的存储，从制造商的角度研究其生产-库存计划问题。特别地，考虑客户需求和原材料订货提前期不确定性因素，建立了以最小化制造商生产-库存总成本为目标，生产能力、原材料数量、产品库存以及物流等相关运作为约束条件，原材料采购数量、生产时间、制造商与第三方物流公司签订的订单数量以及销售损失等为决策变量的多产品、多周期生产-库存计划鲁棒优化模型。进一步，在线性决策规则下，构建了能够充分利用已发生的市场需求信息的仿射可调节鲁棒优化模型。在不确定需求和提前期所隶属的不确定集定义基础上，应用对偶理论将所建模型等价转换为易于求解的线性规划问题。最后，通过数值算例验证了所建鲁棒优化模型，特别是仿射可调节鲁棒优化模型的有效性以及优势。结果表明，与传统鲁棒优化方法相比，基于仿射可调节鲁棒优化的生产-库存计划模型及运作方案能够在保证生产-库存计划鲁棒性的前提下产生更低的总成本，并且各项决策更加合理。

关键词: 供应链, 生产-库存计划, 仿射可调节鲁棒优化, 需求不确定性, 提前期不确定性

Abstract:

The rapid changes of supply chain environment make it difficult for firms to predict future demands or obtain a full knowledge of raw material supplies， resulting in low efficiency of production and operation. How to cope with market demand and raw material supply uncertainties and ensure ideal operational performance has become an urgent issue for firms.A production-inventory planning problem is explored for a three-stage supply chain consisting of a raw material supplier， a manufacturer， a third-party logistics company， and customers. The raw materials used to produce products are supplied and distributed by the supplier to the manufacturer. The output of each product is influenced by the inventory levels of raw materials， the production capacity and the market demands. The total production time is subject to maintenance， holidays and other factors. The end products are supplied by the manufacturer to the customers. In particular， any unsold products will be stored by the third-party logistics company. Considering demand and raw material lead time uncertainties， a multi-product multi-period production-inventory planning robust optimization model is developed by minimizing the total production-inventory cost with production capacity， quantities of raw materials， product inventories and logistics as constraints， and order quantities of raw materials， production time， number of orders signed by the manufacturers and the third-party logistics companies and lost sales as decision variables. Furthermore， an affinely adjustable robust optimization model is developed based on the realized demands. With the definitions of the uncertain sets to which the uncertain demands and the lead times belong， the proposed robust optimization models are transformed into tractable linear programming models by dual approach. Finally， numerical studies are conducted to verify the proposed models， especially to illustrate the effectiveness and advantages of the affinely adjustable robust optimization model in coping with demand and lead time uncertainties.The main findings reveal that the affinely adjustable robust optimization model outperforms the traditional static robust optimization model in terms of the total costs and operational decisions. The total production-planning costs under the two robust optimization methods increase with the increase of the demand and lead time uncertainties， indicating that decision-makers should strengthen the management of uncertainties in practice and reduce the uncertainty levels to reduce costs.

Key words: supply chain, production-inventory planning, affinely adjustable robust optimization, demand uncertainties, lead time uncertaintie

中图分类号:

F253.4

袁明利, 邱若臻, 孙月. 考虑需求和提前期不确定性的生产-库存计划仿射可调节鲁棒优化模型[J]. 中国管理科学, 2025, 33(9): 97-108.

Mingli Yuan, Ruozhen Qiu, Yue Sun. An Affinely Adjustable Robust Optimization Model for Prouction-Inventory Planning Problem with Demand and Lead Time Uncertainties[J]. Chinese Journal of Management Science, 2025, 33(9): 97-108.

图/表 10

图1

表1

原材料每期采购数量"

$o t j$		$t$ =1	$t$ =2	$t$ =3	$t$ =4	$t$ =5	$t$ =6	$t$ =7	$t$ =8	$t$ =9	$t$ =10	$t$ =11	$t$ =12
RC	$i$ =1	709.80	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
RC	$i$ =2	125.48	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
AARC	$i$ =1	13.88	28.05	68.99	126.91	195.00	191.83	262.70	233.16	236.69	262.48	5.79	9.52
AARC	$i$ =2	22.32	27.76	33.14	37.48	42.64	47.02	51.41	50.20	40.15	23.17	15.66	26.22

表1

表2

每周期常规订单数量"

$r t i$		$t$ =1	$t$ =2	$t$ =3	$t$ =4	$t$ =5	$t$ =6	$t$ =7	$t$ =8	$t$ =9	$t$ =10	$t$ =11	$t$ =12
RC	$i$ =1	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
RC	$i$ =2	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
AARC	$i$ =1	0.71	2.26	3.80	5.16	6.50	8.16	9.46	10.87	12.58	13.84	15.57	5.49
AARC	$i$ =2	0.56	1.54	2.61	3.54	3.25	3.88	4.00	4.71	5.22	6.53	7.16	4.52

表2

表3

每周期紧急订单数量"

$u t i$		$t$ =1	$t$ =2	$t$ =3	$t$ =4	$t$ =5	$t$ =6	$t$ =7	$t$ =8	$t$ =9	$t$ =10	$t$ =11	$t$ =12
RC	$i$ =1	7.04	19.65	23.80	31.71	49.55	52.08	49.07	59.75	58.40	59.35	58.51	74.39
RC	$i$ =2	3.87	8.70	23.63	21.90	29.05	36.63	38.64	43.54	49.84	47.94	61.32	77.48
AARC	$i$ =1	8.67	36.52	48.23	39.91	53.92	56.19	46.76	52.17	46.77	43.63	41.25	49.91
AARC	$i$ =2	5.00	19.61	37.08	23.83	25.68	33.58	24.83	30.54	30.11	30.64	46.20	62.42

表3

表4

产品每周期销售损失数量"

$l t i$		$t$ =1	$t$ =2	$t$ =3	$t$ =4	$t$ =5	$t$ =6	$t$ =7	$t$ =8	$t$ =9	$t$ =10	$t$ =11	$t$ =12
RC	$i$ =1	0.00	0.00	257.50	247.20	267.80	185.40	206.00	185.40	175.10	206.00	206.00	267.80
RC	$i$ =2	0.00	0.00	0.00	0.00	0.00	0.00	195.70	206.00	267.80	206.00	206.00	257.50
AARC	$i$ =1	161.72	211.78	260.91	224.56	257.54	177.87	188.86	165.03	153.28	177.33	173.96	206.77
AARC	$i$ =2	0.31	2.99	5.82	11.90	67.55	41.07	20.19	40.67	33.52	14.57	11.10	11.90

表4

表5

每周期使用机器生产产品的时间"

参数/m		$t$ =1	$t$ =2	$t$ =3	$t$ =4	$t$ =5	$t$ =6	$t$ =7	$t$ =8	$t$ =9	$t$ =10	$t$ =11	$t$ =12
RC
$m$ =1	$i$ =1	5.57	5.57	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
$m$ =1	$i$ =2	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
$m$ =2	$i$ =1	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
$m$ =2	$i$ =2	6.09	4.21	4.45	4.68	4.68	3.51	0.00	0.00	0.00	0.00	0.00	0.00
$m$ =3	$i$ =1	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
$m$ =3	$i$ =2	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
AARC		$t$ =1	$t$ =2	$t$ =3	$t$ =4	$t$ =5	$t$ =6	$t$ =7	$t$ =8	$t$ =9	$t$ =10	$t$ =11	$t$ =12
$m$ =1	$i$ =1	0.47	0.10	0.05	0.07	0.07	0.08	0.11	0.15	0.18	0.23	0.29	0.40
$m$ =1	$i$ =2	0.61	1.12	1.44	1.57	1.22	1.10	1.58	1.60	2.15	1.84	1.85	2.26
$m$ =2	$i$ =1	0.37	0.09	0.04	0.06	0.06	0.07	0.09	0.13	0.15	0.20	0.25	0.34
$m$ =2	$i$ =2	5.13	2.55	2.04	1.74	1.16	0.94	1.36	1.37	1.93	1.65	1.69	2.09
$m$ =3	$i$ =1	0.26	0.07	0.04	0.05	0.05	0.06	0.08	0.11	0.13	0.17	0.21	0.29
$m$ =3	$i$ =2	0.47	0.88	1.12	1.11	0.83	0.74	1.06	1.07	1.45	1.24	1.25	1.53

表5

图2

图3

图4

图5

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