SHERBROOKE C C . Metric:a multi-echelon technique for recoverable item control [J ] . Operations Research , 1968 , 16 ( 1 ): 122 - 141 . DOI: 10.1287/opre.16.1.122 http://doi.org/10.1287/opre.16.1.122 http://pubsonline.informs.org/doi/10.1287/opre.16.1.122 http://pubsonline.informs.org/doi/10.1287/opre.16.1.122 Metric is a mathematical model of a base-depot supply system in which item demand is compound Poisson with a mean value estimated by a Bayesian procedure. When a unit fails at base level there is a probability r that it can be repaired at the base according to an arbitrary probability distribution of repair time, and a probability 1 − r that it must be returned to the depot for repair according to some other arbitrary distribution. In the latter case the base levies a resupply request on depot. No lateral resupply between bases is considered in the model. For high-cost, low-demand items the appropriate policy is (s − 1, s), which means that items are not batched for repair or resupply requests. This problem has a simple analytic solution that is a function of the mean repair times rather than the repair time distributions. A practical and efficient computer program has been designed to show the cost-effectiveness tradeoff for a large group of recoverable items. In addition to minimizing expected backorders for any system investment, the program can evaluate any distribution of stock and it can compute the optimal redistribution of stock. No arbitrary estimates of backorder cost or holding cost are required.

MUCKSTADT J A . A model for a multi-item, multi-echelon, multi-indenture inventory system [J ] . Management Science , 1973 , 20 ( 4 ): 472 - 481 . DOI: 10.1287/mnsc.20.4.472 http://doi.org/10.1287/mnsc.20.4.472 http://pubsonline.informs.org/doi/10.1287/mnsc.20.4.472 http://pubsonline.informs.org/doi/10.1287/mnsc.20.4.472 The purpose of this paper is to describe a mathematical model, called MOD-METRIC, for the control of a multi-item, multi-echelon, multi-indenture inventory-system for recoverable items, that is, items subject to repair when they fail. Discussion is limited to two-echelon multi-item systems in which an item may be demanded at any one of several locations called bases; in turn, these bases receive inventory from a central location called a depot. The objectives of the model are to describe the logistics relationship between an assembly and its subassemblies, and to compute spare stock levels for both echelons for the assembly and subassemblies with explicit consideration of this logistics relationship. In particular, the model is used to determine the base and depot spare stock levels which minimize total expected base backorders for the assembly subject to a system investment constraint. An example is given showing how the model can be used to calculate spare engine and engine module stock levels. MOD-METRIC has been implemented by the Air Force as the method for computing recoverable spare stock levels for the F-15 weapon system.

GRAVES S C . A multi-echelon inventory model for a repairable item with one-for-one replenishment [J ] . Management Science , 1985 , 31 ( 10 ): 1247 - 1256 . DOI: 10.1287/mnsc.31.10.1247 http://doi.org/10.1287/mnsc.31.10.1247 http://pubsonline.informs.org/doi/10.1287/mnsc.31.10.1247 http://pubsonline.informs.org/doi/10.1287/mnsc.31.10.1247 The problem that we address is to determine the inventory stockage levels in a multi-echelon inventory system for a repairable item. In its simplest form the multi-echelon system consists of a set of operating sites supported by a centrally-located repair depot. Each operating site requires a set of working items and maintains an inventory of spare items. The repair depot also holds an inventory of spare items. Item failures are infrequent and are replaced on a one-for-one basis. In this paper we present an exact model for finding the steady-state distribution of the net inventory level at each site. This model assumes that the failures are generated by a compound Poisson process and that the shipment time from the repair depot to each site is deterministic. No assumptions are made with regard to the repair cycle at the depot. We contrast this model with existing models for these systems. Based on the exact model we present an approximation for the steady-state distribution for the case with ample servers at the repair depot. We show that this approximation is very accurate on a set of test problems.

魏国强 , 杨永清 . 供应不足条件下战时连续消耗资源调度模型 [J ] . 系统工程与电子技术 , 2012 , 34 ( 1 ): 102 - 106 .

WEI G Q , YANG Y Q . Dispatching model of continuous consumption resources in wartime under insufficient supply [J ] . Systems Engineering and Electronics , 2012 34 ( 1 ): 102 - 106 . (in Chinese)

王亚东 , 石全 , 陈材 , 等 . 考虑中断风险的备件供应选址-分配优化模型 [J ] . 兵工学报 , 2019 , 40 ( 8 ): 1708 - 1715 . DOI: 10.3969/j.issn.1000-1093.2019.08.021 http://doi.org/10.3969/j.issn.1000-1093.2019.08.021 战时条件下，仓库或运输载体受到敌方蓄意攻击而丧失全部或部分储存和运输能力，造成备件供应中断。而供应网络的中断风险很难预测且动态变化，为不确定性参数。为解决不确定中断风险下的备件供应问题，以备件供应成本最低、供应延迟时间最短为目标，构建了战时备件供应的多目标选址-分配联合优化模型。通过对多面体不确定集合的鲁棒等价变化，建立相应的鲁棒优化模型。采用元启发式算法对模型进行求解，得到了模型的非支配解集及其对应的供应方案。研究结果表明：通过对关键设施的加固防护，可以提高备件供应网络的效率和可靠性；鲁棒优化模型的最优解可以保证战时备件供应“最差情况下”解的可行性，即模型具有很好的鲁棒性。

WANG Y D , SHI Q , CHEN C , et al. Location-allocation joint optimization model of spare parts supply under supply disruption risk [J ] . Acta Armamentarii , 2019 , 40 ( 8 ): 1708 - 1715 . (in Chinese) DOI: 10.3969/j.issn.1000-1093.2019.08.021 http://doi.org/10.3969/j.issn.1000-1093.2019.08.021 In wartime, the spare parts warehouse or transport carrier may be attacked deliberately, and some storage and transportation capacity may be completely or partially lost, which leads to the interruption of spare parts supply system. The interruption risk of supply network is an uncertain parameter and difficult to be predicted and change dynamically. In view of the depth uncertainty of interruption risk, the diversification of wartime spare parts supply optimization objectives is considered. A joint optimization model of multi-objective location-allocation for wartime spare parts supply is constructed for the minimum cost of spare parts supply and the shortest lead time of supply. A corresponding robust optimization model is established based on the robust equivalent change of polyhedron uncertainty set. The meta-heuristic algorithms are used to obtain the non-dominate solution sets and the corresponding supply plans. The results show that the efficiency and reliability of spare parts supply network can be greatly improved by strengthening some key nodes. On the other hand, the optimal solution of the robust optimization model can guarantee the feasibility of the worst-case solution of spare parts supply, that is, the model has good robustness. Key

王亚东 , 石全 , 张芳 , 等 . 基于动态进化算法的多阶段备件供应优化决策 [J ] . 系统工程与电子技术 , 2019 , 41 ( 11 ): 2514 - 2523 .

WANG Y D , SHI Q , ZHANG F , et al. Multi-stage spare parts supply optimization based on dynamic evolutionary algorithm [J ] . Systems Engineering and Electronics , 2019 , 41 ( 11 ): 2514 - 2523 . (in Chinese)

王亚东 , 石全 , 尤志锋 , 等 . 基于交叉效率排序多目标进化算法的备件供应优化 [J ] . 兵工学报 , 2020 , 41 ( 11 ): 2338 - 2346 . DOI: 10.3969/j.issn.1000-1093.2020.11.022 http://doi.org/10.3969/j.issn.1000-1093.2020.11.022 为了对备件供应网络进行优化并制定最优供应方案，以缩短总供应时间、减少供应成本和降低中断风险为目标，以备件满足度、库存容量等为约束建立了多目标优化模型。基于交叉效率排序多目标进化算法求得模型的非支配解集，同时决策出最优解。优化过程中采用改进数据包络分析计算各最优解的二次目标交叉效率，指导算法朝最优效率个体收敛，对求得的非支配解进行排序从而选择出最优方案。算例表明：通过交叉效率排序多目标进化算法优化得到了13个互不支配的备件供应方案，且确定了交叉效率为0.927 8的方案为最优方案；新算法优于未采用排序和采用自评效率排序的多目标进化算法。

WANG Y D , SHI Q , YOU Z F , et al. Spare parts supply optimization and decision-making based on cross-efficiency multi-objective sorting evolutionary algorithm [J ] . Acta Armamentarii , 2020 , 41 ( 11 ): 2338 - 2346 . (in Chinese) DOI: 10.3969/j.issn.1000-1093.2020.11.022 http://doi.org/10.3969/j.issn.1000-1093.2020.11.022 A multi-objective optimization model is established with the constraints of spare parts satisfaction, lead time and inventory capacity for the shortest total supply time, the lowest risk and the minimum cost. The proposed model is used to optimize the spare parts supply and make an optimal supply scheme. The set of non-dominant solutions are obtained by using the proposed secondary goal cross-efficiency sort multi-objective evolutionary algorithm (SGCES-MOEA). During optimization process, the improved data envelopment analysis (DEA) is used to calculate the cross-efficiencies of the non-dominant solutions. On the one hand, the algorithm is guided to converge to the optimal efficiency individuals. On the other hand, the non-dominant solutions are sorted to select an optimal one. The example shows that 13 non-dominate spare parts supply schemes are obtained by using SGCES-MOEA algorithm, and the scheme with cross-efficiency of 0.927 8 is determined as the optimal scheme. The new algorithm is superior to the multi-objective evolutionary algorithm without efficiency sorting strategy and that with self-evaluation efficiency sorting strategy.

FANCELLO G , MANCINI S , PANI C , et al. An emergency vehicles allocation model for major industrial disasters [J ] . Transportation Research Procedia , 2017 , 25 : 1164 - 1179 . DOI: 10.1016/j.trpro.2017.05.134 http://doi.org/10.1016/j.trpro.2017.05.134 https://linkinghub.elsevier.com/retrieve/pii/S2352146517304258 https://linkinghub.elsevier.com/retrieve/pii/S2352146517304258

HU H , HE J , HE X F , et al. Emergency material scheduling optimization model and algorithms: a review [J ] . Journal of Traffic and Transportation Engineering-English Edition , 2019 , 6 ( 5 ): 441 - 454 .

ZHANG J H , LIU H Y , YU G D , et al. A three-stage and multi-objective stochastic programming model to improve the sustainable rescue ability by considering secondary disasters in emergency logistics [J ] . Computers & Industrial Engineering , 2019 , 135 : 1145 - 1154 . DOI: 10.1016/j.cie.2019.02.003 http://doi.org/10.1016/j.cie.2019.02.003 https://linkinghub.elsevier.com/retrieve/pii/S0360835219300774 https://linkinghub.elsevier.com/retrieve/pii/S0360835219300774

WANG Y S , DONG Z S , HU S L . A stochastic prepositioning model for distribution of disaster supplies considering lateral transshipment [J ] . Socio-Economic Planning Sciences , 2021 , 74 : 100930 . DOI: 10.1016/j.seps.2020.100930 http://doi.org/10.1016/j.seps.2020.100930 https://linkinghub.elsevier.com/retrieve/pii/S0038012120301105 https://linkinghub.elsevier.com/retrieve/pii/S0038012120301105

TIACCI L , SAETTA S . Reducing the mean supply delay of spare parts using lateral transshipments policies [J ] . International Journal of Production Economics , 2011 , 133 ( 1 ): 182 - 191 . DOI: 10.1016/j.ijpe.2010.03.020 http://doi.org/10.1016/j.ijpe.2010.03.020 https://linkinghub.elsevier.com/retrieve/pii/S0925527310001167 https://linkinghub.elsevier.com/retrieve/pii/S0925527310001167

阮旻智 , 刘任洋 . 随机需求下多层级备件的横向转运配置优化模型 [J ] . 系统工程理论与实践 , 2016 , 36 ( 10 ): 2689 - 2698 . DOI: 10.12011/1000-6788(2016)10-2689-10 http://doi.org/10.12011/1000-6788(2016)10-2689-10 对于复杂的装备维修保障系统，采用备件横向转运补给策略能有效提高装备保障效率，针对该问题，依据生灭过程和METRIC（multi-echelon technique for recoverable item control）理论，在随机需求模式下，提出并建立了装备备件多级库存转运配置优化模型，并采用一种启发式算法对模型进行求解.在此基础上以装备可用度为约束，备件购置费用最低为目标，利用边际算法对配置方案进行优化.通过实例，模型的正确性通过仿真方法得到了验证，并与非转运条件下的优化方案比较，结果表明，本文构建的转运模型在提高装备保障效能方面优于非转运模型.

RUAN M Z , LIU R Y . Configuration and optimization model for multi-indenture spares with lateral transshipments under stochastic demand [J ] . Systems Engineering-Theory & Practice , 2016 36 ( 10 ): 2689 - 2698 . (in Chinese)

张颖 , 王蕴 , 薛佩 . 三站点备件供应保障关系建模与分析 [J ] . 北京航空航天大学学报 , 2017 , 43 ( 8 ): 1625 - 1631 .

ZHANG Y , WANG Y , XUE P . Modeling and analysis of spares supply support relations among three stations [J ] . Journal of Beijing University of Aeronautics and Astronautics , 2017 , 43 ( 8 ): 1625 - 1631 . (in Chinese)

王亚东 , 石全 , 夏伟 , 等 . 基于超启发式算法的备件供应网络结构优化 [J ] . 系统工程与电子技术 , 2020 , 42 ( 3 ): 620 - 629 .

WANG Y D , SHI Q , XIA W , et al. Structure optimization of spare parts supply network based on hyper heuristic algorithm [J ] . Systems Engineering and Electronics , 2020 , 42 ( 3 ): 620 - 629 . (in Chinese)

ZHANG Q F , LI H . MOEA/D:a multiobjective evolutionary algorithm based on decomposition [J ] . IEEE Transactions on Evolutionary Computation , 2007 , 11 ( 6 ): 712 - 731 . DOI: 10.1109/TEVC.2007.892759 http://doi.org/10.1109/TEVC.2007.892759 http://ieeexplore.ieee.org/document/4358754/ http://ieeexplore.ieee.org/document/4358754/

DEB K , KUMAR A . Real-coded genetic algorithms with simulated binary crossover:studies on multimodal and multiobjective problems [J ] . Complex Systems , 1995 , 9 ( 6 ): 431 - 454 .

HWANG C L , YOON K . Multiple attribute decision making [M ] . Berlin, Germany:Springer-Verlag , 1981 .

SHIH H S , SHYUR H J , LEE E S . An extension of TOPSIS for group decision making [J ] . Mathematical & Computer Modelling , 2007 , 45 ( 7/8 ): 801 - 813 .

SAATY T L . How to make a decision: the analytic hierarchy process [J ] . European Journal of Operational Research , 1990 , 48 ( 1 ): 9 - 26 . DOI: 10.1016/0377-2217(90)90057-I http://doi.org/10.1016/0377-2217(90)90057-I https://linkinghub.elsevier.com/retrieve/pii/037722179090057I https://linkinghub.elsevier.com/retrieve/pii/037722179090057I