Combinable containers: A container innovation to save container fleet and empty container repositioning costs
可组合集装箱：一项集装箱创新，可节省集装箱船队和空箱重新定位成本

First and foremost, we would focus on container fleet sizing problem. Once a given transportation demands for details of the loaded trips (number of laden containers moving between origin-destination (OD) port pairs in the shipping network) and the characteristics of each trip (length, uncertainty, etc.), the container fleet sizing problem helps to determine the optimal number of own containers that is needed to satisfy the transportation demands. Some researchers have manifested a specific interest in container fleet sizing problem. Imai and Rivera (2001) is possibly the first study that aimed to address the problem of determining the appropriate container fleet size considering an extremely imbalanced trade, which is a typical situation for the refrigerated cargo trade in shipping service routes. They built a simulation model for a many-to-many port trade to analyze the trade-off between own container fleet size, short-term leasing, and empty storage at ports for given some transportation demand trends.
首先，我们将重点关注集装箱船队的规模问题。一旦给定的运输需要满载行程的详细信息（航运网络中在起运港-目的地 （OD） 港口对之间移动的满载集装箱数量）和每次行程的特征（长度、不确定性等），集装箱船队选型问题有助于确定满足运输需求所需的自有集装箱的最佳数量。一些研究人员对集装箱船队规模问题表现出了特别的兴趣。Imai 和 Rivera （2001） 可能是第一项旨在解决考虑到极度不平衡的贸易确定适当集装箱船队规模的问题的研究，这是航运服务路线中冷藏货物贸易的典型情况。他们为多对多港口贸易构建了一个仿真模型，以分析给定某些运输需求趋势下，自有集装箱船队规模、短期租赁和港口空箱存储之间的权衡。

Second, we herein present a review of empty container management problems reflecting the characteristics of liner shipping networks, namely, multi-vessel, multi-port, and fixed schedule shipping systems. Song and Dong (2011) addressed the empty container repositioning problem in shipping services based on container flow balancing approach. Two empty container repositioning policies were examined, based on two types of flow balancing treatments, respectively. The former is based on point-to-point balancing approach for each port while the latter is based on coordinated balancing among all ports. Meng and Wang (2011) evaluated medium-term liner shipping service network design problem together with the combined networks of hub-and-spoke and multi-port-calling operations and empty container repositioning issue. Bell et al. (2011) proposed a frequency-based assignment model to allocate laden and empty containers in service routes to minimize the sailing time and container transit time between the origin and transshipment ports. Song and Dong (2012) considered the problem of joint cargo routing and empty container repositioning at the operational planning level for a shipping network with multi-route, multi-vessel, and multi-voyage characteristics.
其次，我们在此对反映班轮航运网络特征的空箱管理问题进行了回顾，即多船、多港口和固定时间表航运系统。Song 和 Dong （2011） 解决了基于集装箱流平衡方法的航运服务中的空箱重新定位问题。分别基于两种类型的流动平衡处理，检查了两种空容器重新定位策略。前者基于每个端口的点对点平衡方法，而后者基于所有端口之间的协调平衡。Meng 和 Wang （2011） 将中期班轮服务网络设计问题与轴辐式和多港停靠操作的组合网络以及空箱重新定位问题进行了评估。Bell 等人（2011 年）提出了一种基于频率的分配模型，在服务路线中分配满载集装箱和空集装箱，以最大限度地减少起运港和转运港之间的航行时间和集装箱运输时间。Song 和 Dong （2012） 在运营规划层面考虑了具有多路线、多船和多航次特征的航运网络的联合货物路线和空集装箱重新定位问题。

Further, the third research is an integrated approach to container fleet sizing and empty container management. Shintani et al. (2005) addressed the problem of liner shipping network construction by explicitly considering empty container repositioning and the container fleet size. Shintani et al. (2007) presented an integrated and simultaneous approach in determining the optimal vessel fleet composition with service routing characteristics and empty container repositioning to design a single route. The results they obtained show that the shipping network design problem with empty container repositioning makes it possible to cruise at a slower speed because of the efficient empty container distribution and thus can save considerable fuel costs. Imai et al. (2009) examined the advantages and disadvantages of different service networks, i.e., multiple-port calling (MPC) and hub-and-spoke. They compared the costs associated with the shipping networks by analyzing empty container management costs including container fleet costs. Dong and Song (2009) examined the problem of joint container fleet sizing and empty container repositioning in a multi-vessel, multi-port, and multi-voyage shipping system with trade imbalances. They considered an optimal control policy that can determine the number of laden and empty containers to be moved by each vessel.
此外，第三项研究是集装箱船队规模调整和空箱管理的综合方法。Shintani et al. （2005） 通过明确考虑空箱重新定位和集装箱船队规模，解决了班轮航运网络建设的问题。Shintani 等人（2007 年）提出了一种综合的同步方法，用于确定具有服务路线特征和空集装箱重新定位的最佳船队组成，以设计一条单一路线。他们获得的结果表明，由于空箱的高效分配，空箱重新定位的航运网络设计问题使得以较慢的速度巡航成为可能，从而可以节省可观的燃料成本。Imai 等人（2009 年）研究了不同服务网络的优缺点，即多端口呼叫 （MPC） 和中心辐射型。他们通过分析包括集装箱船队成本在内的空箱管理成本，比较了与航运网络相关的成本。Dong 和 Song （2009） 研究了在贸易不平衡的多船、多港口和多航次航运系统中联合集装箱船队尺寸和空集装箱重新定位的问题。他们考虑了一种最佳控制策略，可以确定每艘船要移动的满载集装箱和空集装箱的数量。

All of these related studies deal with the container fleet sizing and/or empty container management problem; however, most of them only consider a homogeneous container size of either 20 or 40 ft. In real-world circumstances, both container sizes are present in container shipping networks. As a result, trade imbalances occur for both container sizes. Therefore, it is an important problem to determine how to reposition empty containers of each size at the same time. The CMB could offer a solution regarding an efficient anticipation of the need to reposition empty containers.
所有这些相关研究都涉及集装箱船队的规模和/或空箱管理问题;然而，他们中的大多数只考虑 20 或 40 英尺的均质集装箱尺寸。在实际情况下，这两种集装箱尺寸都存在于集装箱运输网络中。因此，两种集装箱尺寸都会出现贸易不平衡。因此，确定如何同时重新定位每种尺寸的空容器是一个重要的问题。CMB 可以提供一种解决方案，可以有效地预测重新定位空集装箱的需求。

Herein, we present a review of studies on FLDs and CMBs. For FLDs, a number of studies on empty container repositioning using FLDs approach have been reported in the last two decades.
在此，我们回顾了 FLD 和 CMB 的研究。对于 FLD，在过去二十年中已经报道了大量关于使用 FLD 方法重新定位空容器的研究。

Konings and Thijs, 2001, Konings, 2005 are the early works that addressed the economic viability of FLDs. These studies give some indications of the conditions under which such containers could be successfully introduced into the shipping market. Shintani et al. (2010) contended that these previous studies did not sufficiently cover the complexity of container transportation chains, and thus, analyzed multiple hinterland container transportation chains simultaneously using FLDs. They took into consideration the inter-relations between these individual hinterland transportations. Shintani et al. (2012) showed that none of the previous studies covered the complexity of liner shipping network when examining container fleet decision making process. Thus, they addressed the container fleet sizing and empty container management problem, including empty container repositioning to demonstrate the potential role of FLDs in lowering the associated costs. It is the first study that integrated the use of FLDs into the complexity of shipping networks. They pointed out that the mixed container fleet with STDs and FLDs is the most realistic application of FLDs in limiting investment risks of shipping companies. Zazgornik et al. (2012) built models for the timber transportation chain, i.e., a combined problem of vehicle routing and container scheduling. However, they considered an application of FLDs for timber transportation, which is conducted in a much simpler network than maritime containers. Moon et al. (2013) examined empty container repositioning using both FLDs and STDs at sea. Myung and Moon (2014) extended the problem presented by Moon et al. (2013), by treating the multi-port and multi-period container planning problem using both STDs and FLDs. They modeled the problem as a minimum cost flow problem considering the service route. Moon and Hong (2016) simultaneously considered the use of both STDs and FLDs in repositioning empty containers between ports. The model that they built can be used to determine ports to be selected for the installation of facilities for folding and unfolding of containers. Wang et al. (2017) treated the problem of vessel type decision considering container fleet sizing, empty container repositioning, and the use of FLDs in specific shipping service routes. They assumed that the container fleet comprised long-term lease containers. Zhang et al. (2018) examined the possibility of carrying a laden container or multiple folded containers with a truck at a time. Their models can be used to minimize the total working time of trucks, including waiting times at sites. Goh (2019) analyzed the impact of FLDs on freight rates on shippers in the backhaul trades and quantified the carbon abatement capability of FLDs in liner shipping.
Konings 和 Thijs，2001 年，Konings，2005 年是解决 FLD 经济可行性的早期工作。这些研究为此类集装箱成功引入航运市场的条件提供了一些迹象。Shintani 等人（2010 年）认为，这些以前的研究不足以涵盖集装箱运输链的复杂性，因此，使用 FLD 同时分析了多个腹地集装箱运输链。他们考虑了这些单独的腹地运输之间的相互关系。Shintani 等人（2012 年）表明，在检查集装箱船队决策过程时，以前的研究都没有涵盖班轮运输网络的复杂性。因此，他们解决了集装箱船队选型和空箱管理问题，包括空箱重新定位，以展示 FLD 在降低相关成本方面的潜在作用。这是第一项将 FLD 的使用整合到航运网络复杂性中的研究。他们指出，带有 STD 和 FLD 的混合集装箱船队是 FLD 在限制航运公司投资风险方面最现实的应用。Zazgornik 等人（2012 年）为木材运输链构建了模型，即车辆路线和集装箱调度的组合问题。然而，他们考虑了将 FLD 应用于木材运输，这种运输是在比海运集装箱简单得多的网络中进行的。Moon 等人（2013 年）在海上使用 FLD 和 STD 研究了空集装箱的重新定位。 Myung 和 Moon （2014） 通过使用 STD 和 FLD 处理多港口和多周期集装箱规划问题，扩展了 Moon 等人 （2013） 提出的问题。他们将该问题建模为考虑服务路线的最低成本流问题。Moon 和 Hong （2016） 同时考虑了使用 STD 和 FLD 在港口之间重新定位空集装箱。他们构建的模型可用于确定要选择的端口，以安装集装箱折叠和展开设施。Wang 等人（2017 年）考虑了集装箱船队的大小、空集装箱重新定位以及在特定航运服务路线中使用 FLD 的船型决策问题。他们假设集装箱船队由长期租赁集装箱组成。Zhang et al. （2018） 研究了用卡车一次运载一个满载集装箱或多个折叠集装箱的可能性。他们的模型可用于最大限度地减少卡车的总工作时间，包括现场的等待时间。Goh （2019） 分析了 FLD 对回程贸易中托运人运费的影响，并量化了 FLD 在班轮运输中的碳减排能力。

CMBs are a relatively recent innovation independently developed by two companies: German TWORTY BOX (TWORTY BOX, 2019) and Spanish CONNECTAINER (Effective Seaborne Engineering Solutions (ESES), 2019). Their structures are basically similar; however, the latter is more practical than the former because the CONNECTAINER has some electronic displays to recognize the unique identity number and the characteristics of containers such as weight, tare, etc., according to its connected or disconnected states. To the best of our knowledge, Malchow (2016) is the only existing academic study that discussed the economic viability of CMBs from the practical point of view, although he did not sufficiently consider the complexity of container transportation chains. According to these websites of the companies, TWORTY BOX (2019) argues the viability of CMBs using the same approach employed by Malchow (2016) while ESES (2019) contends that if 15 major shipping companies switch 12.5% of their container fleet from STDs to CMBs, they can save the container related cost to around US$1 billion in the whole year. Although ESES (2019) recognizes the effectiveness of the mixed container fleet consisting of both STDs and CMBs, the company does not discuss the key validation of the economics of CMBs, particularly on the mixed container fleet and the trade imbalances between regions in 20/40 ft traffics. Introducing CMBs into the shipping market requires an in-depth evaluation given the sophisticated characteristics of liner shipping networks. Thus, this study is novel in its approach to the viability of CMBs.
CMB 是由两家公司独立开发的相对较新的创新：德国 TWORTY BOX（TWORTY BOX，2019 年）和西班牙 CONNECTAINER（有效的海上工程解决方案 （ESES），2019 年）。它们的结构基本相似;但是，后者比前者更实用，因为 CONNECTAINER 有一些电子显示器，可以根据其连接或断开状态识别容器的唯一标识号和重量、皮重等特性。据我们所知，Malchow （2016） 是现存唯一从实践角度讨论 CMB 经济可行性的学术研究，尽管他没有充分考虑集装箱运输链的复杂性。根据这些公司的网站，TWORTY BOX （2019） 使用与 Malchow （2016） 相同的方法论证了 CMB 的可行性，而 ESES （2019） 认为，如果 15 家主要航运公司将其 12.5% 的集装箱船队从 STD 转向 CMB，他们可以在全年节省约 10 亿美元的集装箱相关成本。尽管 ESES （2019） 认识到由 STD 和 CMB 组成的混合集装箱船队的有效性，但该公司并未讨论对 CMB 经济性的关键验证，特别是关于混合集装箱船队和 20/40 英尺交通地区之间的贸易不平衡。鉴于班轮运输网络的复杂性，将 CMB 引入航运市场需要进行深入评估。因此，这项研究在研究 CMB 的可行性方面是新颖的。

In the above research background, we examine the possibility of saving container related costs using CMBs over the planning horizon in complicated shipping service routes.
在上述研究背景下，我们研究了在复杂的航运服务路线的规划范围内使用 CMB 节省集装箱相关成本的可能性。

As mentioned in Section 1, the CFSMP harmonizes those two planning decisions to save the entire container related costs owing to the fact that there is a close relation between the container fleet sizing (tactical planning) and the empty container management (operational planning).
如第 1 节所述，CFSMP 协调了这两个规划决策，以节省整个集装箱相关成本，因为集装箱船队规模（战术规划）和空箱管理（运营规划）之间存在密切关系。

From a tactical point of view, the CFSMP aims to optimize the container fleet size by examining the number of own STDs/CMBs in a shipping network over the planning horizon. Once a container fleet is deployed, it is usually kept in service for more than 10 years. Because estimating future transportation demands with accuracy is practically impossible in determining the optimal container fleet size, this study proposes a way of determining the fleet size as decision variables of the number of own STDs/CMBs that can be put into the space-time container flow network only at the beginning of the planning horizon.
从战术角度来看，CFSMP 旨在通过检查规划期内航运网络中自有 STD/CMB 的数量来优化集装箱船队规模。集装箱车队部署后，通常会使用超过 10 年。由于在确定最佳集装箱车队规模时，几乎不可能准确估计未来的运输需求，因此本研究提出了一种确定车队规模的方法，作为只有在规划视距开始时才能放入时空集装箱流网络的自有 STD/CMB 数量的决策变量。

From an operational point of view, the CFSMP can be used to determine the number of empty containers to be repositioned from excess ports to deficit ones, to be stored at ports, and to be leased at ports from leasing companies only for an immediate future shipments.
从操作的角度来看，CFSMP 可用于确定空集装箱的数量，这些空集装箱将从多余港口重新定位到短缺港口，储存在港口，以及从租赁公司租赁到港口，仅用于近期的运输。

As a result, analyses using the CFSMP enable investigation of a trade-off of CMBs between the cost reduction on repositioning/handling of empty containers and the cost increase resulting from the expensive initial investment and C/D handling.
因此，使用 CFSMP 进行分析可以研究 CMB 在空集装箱重新定位/处理成本降低与昂贵的初始投资和 C/D 处理导致的成本增加之间的权衡。

The CFSMP modeling is based on the following assumptions:
CFSMP 建模基于以下假设：

• (i) （一）
  A single shipping company provides the liner service network, in which the company can fully control the overall empty container movements. Thus, any empty containers can be repositioned at any place for any shipments.
  一家船公司提供班轮服务网络，该公司可以完全控制整个空箱的移动。因此，任何空集装箱都可以在任何地方重新定位，用于任何货物。
• (ii) （二）
  All the transportation demands can be satisfied within the planning horizon using any type of container such as own STDs/CMBs and leased containers.
  在规划范围内，使用任何类型的集装箱（如自有 STD/CMB 和租赁集装箱）都可以满足所有运输需求。
• (iii) （三）
  The exploitation cost of a container includes the purchase cost (depreciation) and the other cost to keep the container in operation, i.e., maintenance and repair costs.
  集装箱的开采成本包括购买成本（折旧）和保持集装箱运行的其他成本，即维护和维修成本。
• (iv) （四）
  The leasing cost of a container consists of the hiring cost that is proportional to the round trip time between OD ports for assigned cargo traffic, the transportation cost between the ports for the backhaul operation, and the associated handling (lifting on/off) cost at those ports. The backhaul assumption can be justified by the fact that leased containers are sent back from the destination port to the origin port after use by unplanned vessels under the shipping company's responsibility as soon as their use is complete.
  集装箱的租赁成本包括租赁成本，该成本与指定货物运输的 OD 港口之间的往返时间成正比，回程操作的港口之间的运输成本，以及这些港口的相关处理（吊装开/关）成本。回程假设是合理的，因为租赁的集装箱在使用完成后，由航运公司负责的计划外船舶使用后，会立即从目的港送回起运港。
• (v) （五）
  The container fleet consists of CMBs (20 ft size only) and STDs (both 20 ft and 40 ft sizes). However, all containers offered for lease are assumed to be STD of 20 and 40 ft. CMBs are assumed not to be leased owing to its less economic advantages. It is worthy to note that long-term lease containers are regarded as own containers of the shipping company in this study.
  集装箱船队由 CMB（仅限 20 英尺尺寸）和 STD（20 英尺和 40 英尺尺寸）组成。然而，所有出租的集装箱都假定为 20 英尺和 40 英尺的 STD。由于 CMB 的经济优势较小，因此假定不出租。值得注意的是，在本研究中，长期租赁集装箱被视为船公司自有集装箱。
• (vi) （六）
  The containers stay in the hinterlands of ports for loading or unloading at the customers' sites within a certain period of time.
  集装箱在一定时间内留在港口腹地，以便在客户现场装卸。
• (vii) （七）
  To simplify the structure of the model, the model does not consider elaborate container movements whether laden or empty between ports and customers' sites.
  为了简化模型的结构，该模型没有考虑港口和客户现场之间复杂的集装箱移动，无论是满载还是空载。

This subsection presents the notations used for the mathematical formulation of the model as follows:

Index and Sets:
$I$:	set of ports in a shipping service network
$K$:	set of container types and states, that is $K=\left\{1,2,3,4\right\}$
$k$:	index of container types and states, where $k$ = 1, 2, 3, and 4 if the container type specifies disconnected CMB (20 ft), connected CMB (40 ft), STD (20 ft) and STD (40 ft), respectively
$T$:	set of chronological time (the planning horizon)
Input parameters
$V$:	carrying capacity of a vessel
${\alpha }_i$:	time spent by a container in the hinterland of port $i$ for import or export
${\beta }_{\mathit{ij}}$:	transit time from port $i$ to port $j$
$H_i$:	storage capacity of the container pool at port $i$
$C{F}^{\mathit{CMB}}$:	exploitation cost of an owned CMB over the planning horizon
$C{F}^k$:	exploitation cost of type $k$ owned containers over the planning horizon
$C{C}_i$:	C/D cost of a CMB at port $i$
$C{S}_i^{20}$:	storage cost of 20 ft container at port $i$
$C{S}_i^{40}$:	storage cost of 40 ft container at port $i$
$C{L}_{\mathit{ij}}^{20}$:	leasing cost of 20 ft container from port $i$ to port $j$
$C{L}_{\mathit{ij}}^{40}$:	leasing cost of 40 ft container from port $i$ to port $j$
$C{R}_{\mathit{ij}}^{20}$:	repositioning cost of 20 ft container from port $i$ to port $j$
$C{R}_{\mathit{ij}}^{40}$:	repositioning cost of 40 ft container from port $i$ to port $j$
$F_{\mathit{ij}}^{20}\left(t\right)$:	number of 20 ft cargo shipments transported from port $i$ to port $j$ at time $t$
$F_{\mathit{ij}}^{40}\left(t\right)$:	number of 40 ft cargo shipments transported from port $i$ to port $j$ at time $t$
Decision variables
$F{S}^{\mathit{CMB}}$:	total number of CMBs to be put into service over the planning horizon
$F{S}^k$:	total number of type $k$ owned containers to be put into service over the planning horizon
$C{B}_i\left(t\right)$:	number of connecting/disconnecting processes of CMBs at port $i$ at time $t$
$E_{\mathit{ij}}^k\left(t\right)$:	number of type $k$ empty owned containers to be transferred from port $i$ to port $j$ at time $t$
$S_i^k\left(t\right)$:	number of type $k$ empty owned containers to be stored at port $i$ at time $t$
$L_{\mathit{ij}}^{20}\left(t\right)$:	number of leased 20 ft containers to be used for transportation demands that leave port $i$ at time $t$ for port $j$
$L_{\mathit{ij}}^{40}\left(t\right)$:	number of leased 40 ft containers to be used for transportation demands that leave port $i$ at time $t$ for port $j$
Auxiliary variables related to decision variables
$B_{\mathit{ij}}^k\left(t\right)$:	number of type $k$ laden owned containers to be transported from port $i$ to port $j$ at time $t$
$D_i^k\left(t\right)$:	total number of type $k$ laden owned containers to be imported to port $i$ at time $t$
$G_i^k\left(t\right)$:	total number of type $k$ empty owned containers to be repositioned from other ports to port $i$ at time $t$
$N_i^k\left(t\right)$:	total number of type $k$ empty owned containers to be sent to shippers (exporters) in the hinterland from port $i$ at time $t$, and it is equal to the total cargoes to be exported using type $k$ containers
$O_i^k\left(t\right)$:	total number of type $k$ empty owned containers to be repositioned from port $i$ to other ports at time $t$
$R_i^k\left(t\right)$:	total number of type $k$ empty owned containers to be returned from consignees (importers) in the hinterland of port $i$ at time $t$, and it is equal to the total cargoes to be imported using type $k$ containers

本小节介绍用于模型的数学公式的符号，如下所示：

Index and Sets:
$I$:	set of ports in a shipping service network
$K$:	set of container types and states, that is $K=\left\{1,2,3,4\right\}$
$k$:	index of container types and states, where $k$ = 1, 2, 3, and 4 if the container type specifies disconnected CMB (20 ft), connected CMB (40 ft), STD (20 ft) and STD (40 ft), respectively
$T$:	set of chronological time (the planning horizon)
Input parameters
$V$:	carrying capacity of a vessel
${\alpha }_i$:	time spent by a container in the hinterland of port $i$ for import or export
${\beta }_{\mathit{ij}}$:	transit time from port $i$ to port $j$
$H_i$:	storage capacity of the container pool at port $i$
$C{F}^{\mathit{CMB}}$:	exploitation cost of an owned CMB over the planning horizon
$C{F}^k$:	exploitation cost of type $k$ owned containers over the planning horizon
$C{C}_i$:	C/D cost of a CMB at port $i$
$C{S}_i^{20}$:	storage cost of 20 ft container at port $i$
$C{S}_i^{40}$:	storage cost of 40 ft container at port $i$
$C{L}_{\mathit{ij}}^{20}$:	leasing cost of 20 ft container from port $i$ to port $j$
$C{L}_{\mathit{ij}}^{40}$:	leasing cost of 40 ft container from port $i$ to port $j$
$C{R}_{\mathit{ij}}^{20}$:	repositioning cost of 20 ft container from port $i$ to port $j$
$C{R}_{\mathit{ij}}^{40}$:	repositioning cost of 40 ft container from port $i$ to port $j$
$F_{\mathit{ij}}^{20}\left(t\right)$:	number of 20 ft cargo shipments transported from port $i$ to port $j$ at time $t$
$F_{\mathit{ij}}^{40}\left(t\right)$:	number of 40 ft cargo shipments transported from port $i$ to port $j$ at time $t$
Decision variables
$F{S}^{\mathit{CMB}}$:	total number of CMBs to be put into service over the planning horizon
$F{S}^k$:	total number of type $k$ owned containers to be put into service over the planning horizon
$C{B}_i\left(t\right)$:	number of connecting/disconnecting processes of CMBs at port $i$ at time $t$
$E_{\mathit{ij}}^k\left(t\right)$:	number of type $k$ empty owned containers to be transferred from port $i$ to port $j$ at time $t$
$S_i^k\left(t\right)$:	number of type $k$ empty owned containers to be stored at port $i$ at time $t$
$L_{\mathit{ij}}^{20}\left(t\right)$:	number of leased 20 ft containers to be used for transportation demands that leave port $i$ at time $t$ for port $j$
$L_{\mathit{ij}}^{40}\left(t\right)$:	number of leased 40 ft containers to be used for transportation demands that leave port $i$ at time $t$ for port $j$
Auxiliary variables related to decision variables
$B_{\mathit{ij}}^k\left(t\right)$:	number of type $k$ laden owned containers to be transported from port $i$ to port $j$ at time $t$
$D_i^k\left(t\right)$:	total number of type $k$ laden owned containers to be imported to port $i$ at time $t$
$G_i^k\left(t\right)$:	total number of type $k$ empty owned containers to be repositioned from other ports to port $i$ at time $t$
$N_i^k\left(t\right)$:	total number of type $k$ empty owned containers to be sent to shippers (exporters) in the hinterland from port $i$ at time $t$, and it is equal to the total cargoes to be exported using type $k$ containers
$O_i^k\left(t\right)$:	total number of type $k$ empty owned containers to be repositioned from port $i$ to other ports at time $t$
$R_i^k\left(t\right)$:	total number of type $k$ empty owned containers to be returned from consignees (importers) in the hinterland of port $i$ at time $t$, and it is equal to the total cargoes to be imported using type $k$ containers

The mathematical formulation of CFSMP is as follows:(1)$\begin{array}{c}\text{Min}C{F}^{\mathit{CMB}}F{S}^{\mathit{CMB}}+\sum _{k\in \left\{3,4\right\}}C{F}^{k}F{S}^k+\sum _{t\in T}\sum _{i\in I}C{C}_{i}C{B}_i\left(t\right)\hfill \\ +\sum _{t\in T}\sum _{k\in \left\{1,3\right\}}\sum _{i\in I}\sum _{j\left(\ne i\right)\in I}C{R}_{\mathit{ij}}^{20}{E}_{\mathit{ij}}^k\left(t\right)+\sum _{t\in T}\sum _{k\in \left\{2,4\right\}}\sum _{i\in I}\sum _{j\left(\ne i\right)\in I}C{R}_{\mathit{ij}}^{40}{E}_{\mathit{ij}}^k\left(t\right)\hfill \\ +\sum _{t\in T}\sum _{k\in \left\{1,3\right\}}\sum _{i\in I}C{S}_i^{20}{S}_i^k\left(t\right)+\sum _{t\in T}\sum _{k\in \left\{2,4\right\}}\sum _{i\in I}C{S}_i^{40}{S}_i^k\left(t\right)\hfill \\ +\sum _{t\in T}\sum _{i\in I}\sum _{j\left(\ne i\right)\in I}\left\{C{L}_{\mathit{ij}}^{20}{L}_{\mathit{ij}}^{20}\left(t\right)+C{L}_{\mathit{ij}}^{40}{L}_{\mathit{ij}}^{40}\left(t\right)\right\}\hfill \end{array}$(2)$\begin{array}{c}\text{Subject to}{S}_i^1\left(t-1\right)+R_i^1\left(t\right)+G_i^1\left(t\right)+2\left\{S_i^2\left(t-1\right)+R_i^2\left(t\right)+G_i^2\left(t\right)\right\}\hfill \\ =S_i^1\left(t\right)+N_i^1\left(t\right)+O_i^1\left(t\right)+2\left\{S_i^2\left(t\right)+N_i^2\left(t\right)+O_i^2\left(t\right)\right\}\forall t\in T,i\in I,\hfill \end{array}$(3)$S_i^k\left(t-1\right)+R_i^k\left(t\right)+G_i^k\left(t\right)=S_i^k\left(t\right)+N_i^k\left(t\right)+O_i^k\left(t\right)\forall t\in T,k\in \left\{3,4\right\},i\in I,$(4)$R_i^1\left(t\right)+2R_i^2\left(t\right)=\sum _{h\left(\ne i\right)\in I}\left\{B_{\mathit{hi}}^1\left(t-{\alpha }_i-{\beta }_{\mathit{hi}}\right)\text{+ 2}{B}_{\mathit{hi}}^2\left(t-{\alpha }_i-{\beta }_{\mathit{hi}}\right)\right\}\forall t\in T,i\in I,$(5)$R_i^k\left(t\right)=\sum _{h\left(\ne i\right)\in I}{B}_{\mathit{hi}}^k\left(t-{\alpha }_i-{\beta }_{\mathit{hi}}\right)\forall t\in T,k\in \left\{3,4\right\},i\in I,$(6)$N_i^1\left(t\right)+2N_i^2\left(t\right)=\sum _{j\left(\ne i\right)\in I}\left\{B_{\mathit{ij}}^1\left(t+{\alpha }_i\right)+2B_{\mathit{ij}}^2\left(t+{\alpha }_i\right)\right\},\forall t\in T,i\in I,$(7)$N_i^k\left(t\right)=\sum _{j\left(\ne i\right)\in I}{B}_{\mathit{ij}}^k\left(t+{\alpha }_i\right)\forall t\in T,k\in \left\{3,4\right\},i\in I,$(8)$G_i^1\left(t\right)+2G_i^2\left(t\right)=\sum _{h\left(\ne i\right)\in I}\left\{E_{\mathit{hi}}^1\left(t-{\beta }_{\mathit{hi}}\right)+2E_{\mathit{hi}}^2\left(t-{\beta }_{\mathit{hi}}\right)\right\},\forall t\in T,i\in I,$(9)$G_i^k\left(t\right)=\sum _{h\left(\ne i\right)\in I}{E}_{\mathit{hi}}^k\left(t-{\beta }_{\mathit{hi}}\right)\forall t\in T,k\in \left\{3,4\right\},i\in I,$(10)$O_i^1\left(t\right)+2O_i^2\left(t\right)=\sum _{j\left(\ne i\right)\in I}\left\{E_{\mathit{ij}}^1\left(t+{\alpha }_i\right)+2E_{\mathit{ij}}^2\left(t+{\alpha }_i\right)\right\},\forall t\in T,i\in I,$(11)$O_i^k\left(t\right)=\sum _{j\left(\ne i\right)\in I}{E}_{\mathit{ij}}^k\left(t+{\alpha }_i\right)\forall t\in T,k\in \left\{3,4\right\},i\in I,$(12)$L_{\mathit{ij}}^{20}\left(t\right)=F_{\mathit{ij}}^{20}\left(t+{\alpha }_i\right)-\sum _{k\in \left\{1,3\right\}}{B}_{\mathit{ij}}^k\left(t+{\alpha }_i\right)\forall t\in T,i\in I,j\left(\ne i\right)\in I,$(13)$L_{\mathit{ij}}^{40}\left(t\right)=F_{\mathit{ij}}^{40}\left(t+{\alpha }_i\right)-\sum _{k\in \left\{2,4\right\}}{B}_{\mathit{ij}}^k\left(t+{\alpha }_i\right)\forall t\in T,i\in I,j\left(\ne i\right)\in I,$(14)$\sum _{k\in \left\{1,3\right\}}{S}_i^k\left(t\right)+2\sum _{k\in \left\{2,4\right\}}{S}_i^k\left(t\right)⩽H_i\forall t\in T,i\in I,$(15)$F{S}^{\mathit{CMB}}=\sum _{i\in I}\left[S_i^1\left(0\right)+R_i^1\left(1\right)+G_i^1\left(1\right)+2\left\{S_i^2\left(0\right)+R_i^2\left(1\right)+G_i^2\left(1\right)\right\}\right]$(16)$F{S}^k=\sum _{i\in I}\left\{S_i^k\left(0\right)+R_i^k\left(1\right)+G_i^k\left(1\right)\right\}\forall k\in \left\{3,4\right\},$(17)$C{B}_i\left(t\right)⩾S_i^2\left(t-1\right)+R_i^2\left(t\right)+G_i^2\left(t\right)-S_i^2\left(t\right)-N_i^2\left(t\right)-O_i^2\left(t\right)\forall t\in T,i\in I,$(18)$-C{B}_i\left(t\right)⩽S_i^2\left(t-1\right)+R_i^2\left(t\right)+G_i^2\left(t\right)-S_i^2\left(t\right)-N_i^2\left(t\right)-O_i^2\left(t\right)\forall t\in T,i\in I,$(19)$\begin{array}{c}\sum _{q=2}^{\left|I\right|}\left[F_{1q}^{20}\left(t\right)+\sum _{k\in \left\{1,3\right\}}{E}_{1q}^k\left(t\right)+2\left\{F_{1q}^{40}\left(t\right)+\sum _{k\in \left\{2,4\right\}}{E}_{1q}^k\left(t\right)\right\}\right]\hfill \\ +\sum _{p=3}^{\left|I\right|}\sum _{q\left(\ne p\right)=2}^{p-1}\left[F_{\mathit{pq}}^{20}\left(t-{\beta }_{p1}\right)+\sum _{k\in \left\{1,3\right\}}{E}_{\mathit{pq}}^k\left(t-{\beta }_{p1}\right)\right.\hfill \\ +\left.2\left\{F_{\mathit{pq}}^{40}\left(t-{\beta }_{p1}\right)+\sum _{k\in \left\{2,4\right\}}{E}_{\mathit{pq}}^k\left(t-{\beta }_{p1}\right)\right\}\right]⩽V\forall t\in T,\hfill \end{array}$(20)$\begin{array}{c}\sum _{q\left(\ne i\right)=1}^{\left|I\right|}\left[F_{\mathit{iq}}^{20}\left(t\right)+\sum _{k\in \left\{1,3\right\}}{E}_{\mathit{iq}}^k\left(t\right)+2\left\{F_{\mathit{iq}}^{40}\left(t\right)+\sum _{k\in \left\{2,4\right\}}{E}_{\mathit{iq}}^k\left(t\right)\right\}\right]\hfill \\ +\sum _{p=1}^{i-1}\sum _{q\left(\ne p\right)=i+1}^{p-1}\left[F_{\mathit{pq}}^{20}\left(t-{\beta }_{\mathit{pi}}\right)+\sum _{k\in \left\{1,3\right\}}{E}_{\mathit{pq}}^k\left(t-{\beta }_{\mathit{pi}}\right)\right.\hfill \\ \left.+2\left\{F_{\mathit{pq}}^{40}\left(t-{\beta }_{\mathit{pi}}\right)+\sum _{k\in \left\{2,4\right\}}{E}_{\mathit{pq}}^k\left(t-{\beta }_{\mathit{pi}}\right)\right\}\right]\hfill \\ +\sum _{p=i+2}^{\left|I\right|}\sum _{q\left(\ne p\right)=i+1}^{p-1}\left[F_{\mathit{pq}}^{20}\left(t-{\beta }_{\mathit{pi}}\right)+\sum _{k\in \left\{1,3\right\}}{E}_{\mathit{pq}}^k\left(t-{\beta }_{\mathit{pi}}\right)\right.\hfill \\ \left.+2\left\{F_{\mathit{pq}}^{40}\left(t-{\beta }_{\mathit{pi}}\right)+\sum _{k\in \left\{2,4\right\}}{E}_{\mathit{pq}}^k\left(t-{\beta }_{\mathit{pi}}\right)\right\}\right]⩽V\forall t\in T,i\in I\backslash \left\{1,\left|I\right|\right\},\hfill \end{array}$(21)$\begin{array}{c}\sum _{q=1}^{\left|I\right|-1}\left[F_{\left|I\right|q}^{20}\left(t\right)+\sum _{k\in \left\{1,3\right\}}{E}_{\left|I\right|q}^k\left(t-{\beta }_{\left|I\right|q}\right)+2\left\{F_{\left|I\right|q}^{40}\left(t\right)+\sum _{k\in \left\{2,4\right\}}{E}_{\left|I\right|q}^k\left(t-{\beta }_{\left|I\right|q}\right)\right\}\right]\hfill \\ +\sum _{p=2}^{\left|I\right|-1}\sum _{q\left(\ne p\right)=1}^{p-1}\left[F_{\mathit{pq}}^{20}\left(t-{\beta }_{p\left|I\right|}\right)+\sum _{k\in \left\{1,3\right\}}{E}_{\mathit{pq}}^k\left(t-{\beta }_{p\left|I\right|}\right)\right.\hfill \\ \left.+2\left\{F_{\mathit{pq}}^{40}\left(t-{\beta }_{p\left|I\right|}\right)+\sum _{k\in \left\{2,4\right\}}{E}_{\mathit{pq}}^k\left(t-{\beta }_{p\left|I\right|}\right)\right\}\right]⩽V\forall t\in T,\hfill \end{array}$(22)$F{S}^{\mathit{CMB}}⩾0\text{and integer,}$(23)$F{S}^k⩾0\text{and integer}\forall k\in \left\{3,4\right\},$(24)$C{B}_i\left(t\right)⩾0\text{and integer}\forall t\in T,i\in I,$(25)$L_{\mathit{ij}}^{20}\left(t\right),L_{\mathit{ij}}^{40}\left(t\right)⩾0\text{and integer}\forall t\in T,i\in I,j\left(\ne i\right)\in I,$(26)$D_i^k\left(t\right),G_i^k\left(t\right),N_i^k\left(t\right),O_i^k\left(t\right),R_i^k\left(t\right),S_i^k\left(t\right)⩾0\text{and integer}\forall t\in T,k\in K,i\in I,$(27)$B_{\mathit{ij}}^k\left(t\right),E_{\mathit{ij}}^k\left(t\right)⩾0\text{and integer}\forall t\in T,k\in K,i\in I,j\left(\ne i\right)\in I.$
CFSMP 的数学公式如下： (1)$\begin{array}{c}\text{Min}C{F}^{\mathit{CMB}}F{S}^{\mathit{CMB}}+\sum _{k\in \left\{3,4\right\}}C{F}^{k}F{S}^k+\sum _{t\in T}\sum _{i\in I}C{C}_{i}C{B}_i\left(t\right)\\ +\sum _{t\in T}\sum _{k\in \left\{1,3\right\}}\sum _{i\in I}\sum _{j\left(\ne i\right)\in I}C{R}_{\mathit{ij}}^{20}{E}_{\mathit{ij}}^k\left(t\right)+\sum _{t\in T}\sum _{k\in \left\{2,4\right\}}\sum _{i\in I}\sum _{j\left(\ne i\right)\in I}C{R}_{\mathit{ij}}^{40}{E}_{\mathit{ij}}^k\left(t\right)\\ +\sum _{t\in T}\sum _{k\in \left\{1,3\right\}}\sum _{i\in I}C{S}_i^{20}{S}_i^k\left(t\right)+\sum _{t\in T}\sum _{k\in \left\{2,4\right\}}\sum _{i\in I}C{S}_i^{40}{S}_i^k\left(t\right)\\ +\sum _{t\in T}\sum _{i\in I}\sum _{j\left(\ne i\right)\in I}\left\{C{L}_{\mathit{ij}}^{20}{L}_{\mathit{ij}}^{20}\left(t\right)+C{L}_{\mathit{ij}}^{40}{L}_{\mathit{ij}}^{40}\left(t\right)\right\}\end{array}$ (2)$\begin{array}{c}\text{Subject to}{S}_i^1\left(t-1\right)+R_i^1\left(t\right)+G_i^1\left(t\right)+2\left\{S_i^2\left(t-1\right)+R_i^2\left(t\right)+G_i^2\left(t\right)\right\}\\ =S_i^1\left(t\right)+N_i^1\left(t\right)+O_i^1\left(t\right)+2\left\{S_i^2\left(t\right)+N_i^2\left(t\right)+O_i^2\left(t\right)\right\}\forall t\in T,i\in I,\end{array}$ (3)$S_i^k\left(t-1\right)+R_i^k\left(t\right)+G_i^k\left(t\right)=S_i^k\left(t\right)+N_i^k\left(t\right)+O_i^k\left(t\right)\forall t\in T,k\in \left\{3,4\right\},i\in I,$ (4)$R_i^1\left(t\right)+2R_i^2\left(t\right)=\sum _{h\left(\ne i\right)\in I}\left\{B_{\mathit{hi}}^1\left(t-{\alpha }_i-{\beta }_{\mathit{hi}}\right)\text{+ 2}{B}_{\mathit{hi}}^2\left(t-{\alpha }_i-{\beta }_{\mathit{hi}}\right)\right\}\forall t\in T,i\in I,$ (5)$R_i^k\left(t\right)=\sum _{h\left(\ne i\right)\in I}{B}_{\mathit{hi}}^k\left(t-{\alpha }_i-{\beta }_{\mathit{hi}}\right)\forall t\in T,k\in \left\{3,4\right\},i\in I,$ (6)$N_i^1\left(t\right)+2N_i^2\left(t\right)=\sum _{j\left(\ne i\right)\in I}\left\{B_{\mathit{ij}}^1\left(t+{\alpha }_i\right)+2B_{\mathit{ij}}^2\left(t+{\alpha }_i\right)\right\},\forall t\in T,i\in I,$ (7)$N_i^k\left(t\right)=\sum _{j\left(\ne i\right)\in I}{B}_{\mathit{ij}}^k\left(t+{\alpha }_i\right)\forall t\in T,k\in \left\{3,4\right\},i\in I,$ (8)$G_i^1\left(t\right)+2G_i^2\left(t\right)=\sum _{h\left(\ne i\right)\in I}\left\{E_{\mathit{hi}}^1\left(t-{\beta }_{\mathit{hi}}\right)+2E_{\mathit{hi}}^2\left(t-{\beta }_{\mathit{hi}}\right)\right\},\forall t\in T,i\in I,$ (9)$G_i^k\left(t\right)=\sum _{h\left(\ne i\right)\in I}{E}_{\mathit{hi}}^k\left(t-{\beta }_{\mathit{hi}}\right)\forall t\in T,k\in \left\{3,4\right\},i\in I,$ (10)$O_i^1\left(t\right)+2O_i^2\left(t\right)=\sum _{j\left(\ne i\right)\in I}\left\{E_{\mathit{ij}}^1\left(t+{\alpha }_i\right)+2E_{\mathit{ij}}^2\left(t+{\alpha }_i\right)\right\},\forall t\in T,i\in I,$ (11)$O_i^k\left(t\right)=\sum _{j\left(\ne i\right)\in I}{E}_{\mathit{ij}}^k\left(t+{\alpha }_i\right)\forall t\in T,k\in \left\{3,4\right\},i\in I,$ (12)$L_{\mathit{ij}}^{20}\left(t\right)=F_{\mathit{ij}}^{20}\left(t+{\alpha }_i\right)-\sum _{k\in \left\{1,3\right\}}{B}_{\mathit{ij}}^k\left(t+{\alpha }_i\right)\forall t\in T,i\in I,j\left(\ne i\right)\in I,$ (13)$L_{\mathit{ij}}^{40}\left(t\right)=F_{\mathit{ij}}^{40}\left(t+{\alpha }_i\right)-\sum _{k\in \left\{2,4\right\}}{B}_{\mathit{ij}}^k\left(t+{\alpha }_i\right)\forall t\in T,i\in I,j\left(\ne i\right)\in I,$ (14)$\sum _{k\in \left\{1,3\right\}}{S}_i^k\left(t\right)+2\sum _{k\in \left\{2,4\right\}}{S}_i^k\left(t\right)⩽H_i\forall t\in T,i\in I,$ (15)$F{S}^{\mathit{CMB}}=\sum _{i\in I}\left[S_i^1\left(0\right)+R_i^1\left(1\right)+G_i^1\left(1\right)+2\left\{S_i^2\left(0\right)+R_i^2\left(1\right)+G_i^2\left(1\right)\right\}\right]$ (16)$F{S}^k=\sum _{i\in I}\left\{S_i^k\left(0\right)+R_i^k\left(1\right)+G_i^k\left(1\right)\right\}\forall k\in \left\{3,4\right\},$ (17)$C{B}_i\left(t\right)⩾S_i^2\left(t-1\right)+R_i^2\left(t\right)+G_i^2\left(t\right)-S_i^2\left(t\right)-N_i^2\left(t\right)-O_i^2\left(t\right)\forall t\in T,i\in I,$ (18)$-C{B}_i\left(t\right)⩽S_i^2\left(t-1\right)+R_i^2\left(t\right)+G_i^2\left(t\right)-S_i^2\left(t\right)-N_i^2\left(t\right)-O_i^2\left(t\right)\forall t\in T,i\in I,$ (19)$\begin{array}{c}\sum _{q=2}^{\left|I\right|}\left[F_{1q}^{20}\left(t\right)+\sum _{k\in \left\{1,3\right\}}{E}_{1q}^k\left(t\right)+2\left\{F_{1q}^{40}\left(t\right)+\sum _{k\in \left\{2,4\right\}}{E}_{1q}^k\left(t\right)\right\}\right]\\ +\sum _{p=3}^{\left|I\right|}\sum _{q\left(\ne p\right)=2}^{p-1}\left[F_{\mathit{pq}}^{20}\left(t-{\beta }_{p1}\right)+\sum _{k\in \left\{1,3\right\}}{E}_{\mathit{pq}}^k\left(t-{\beta }_{p1}\right)\right.\\ +\left.2\left\{F_{\mathit{pq}}^{40}\left(t-{\beta }_{p1}\right)+\sum _{k\in \left\{2,4\right\}}{E}_{\mathit{pq}}^k\left(t-{\beta }_{p1}\right)\right\}\right]⩽V\forall t\in T,\end{array}$ (20)$\begin{array}{c}\sum _{q\left(\ne i\right)=1}^{\left|I\right|}\left[F_{\mathit{iq}}^{20}\left(t\right)+\sum _{k\in \left\{1,3\right\}}{E}_{\mathit{iq}}^k\left(t\right)+2\left\{F_{\mathit{iq}}^{40}\left(t\right)+\sum _{k\in \left\{2,4\right\}}{E}_{\mathit{iq}}^k\left(t\right)\right\}\right]\\ +\sum _{p=1}^{i-1}\sum _{q\left(\ne p\right)=i+1}^{p-1}\left[F_{\mathit{pq}}^{20}\left(t-{\beta }_{\mathit{pi}}\right)+\sum _{k\in \left\{1,3\right\}}{E}_{\mathit{pq}}^k\left(t-{\beta }_{\mathit{pi}}\right)\right.\\ \left.+2\left\{F_{\mathit{pq}}^{40}\left(t-{\beta }_{\mathit{pi}}\right)+\sum _{k\in \left\{2,4\right\}}{E}_{\mathit{pq}}^k\left(t-{\beta }_{\mathit{pi}}\right)\right\}\right]\\ +\sum _{p=i+2}^{\left|I\right|}\sum _{q\left(\ne p\right)=i+1}^{p-1}\left[F_{\mathit{pq}}^{20}\left(t-{\beta }_{\mathit{pi}}\right)+\sum _{k\in \left\{1,3\right\}}{E}_{\mathit{pq}}^k\left(t-{\beta }_{\mathit{pi}}\right)\right.\\ \left.+2\left\{F_{\mathit{pq}}^{40}\left(t-{\beta }_{\mathit{pi}}\right)+\sum _{k\in \left\{2,4\right\}}{E}_{\mathit{pq}}^k\left(t-{\beta }_{\mathit{pi}}\right)\right\}\right]⩽V\forall t\in T,i\in I\backslash \left\{1,\left|I\right|\right\},\end{array}$ (21)$\begin{array}{c}\sum _{q=1}^{\left|I\right|-1}\left[F_{\left|I\right|q}^{20}\left(t\right)+\sum _{k\in \left\{1,3\right\}}{E}_{\left|I\right|q}^k\left(t-{\beta }_{\left|I\right|q}\right)+2\left\{F_{\left|I\right|q}^{40}\left(t\right)+\sum _{k\in \left\{2,4\right\}}{E}_{\left|I\right|q}^k\left(t-{\beta }_{\left|I\right|q}\right)\right\}\right]\\ +\sum _{p=2}^{\left|I\right|-1}\sum _{q\left(\ne p\right)=1}^{p-1}\left[F_{\mathit{pq}}^{20}\left(t-{\beta }_{p\left|I\right|}\right)+\sum _{k\in \left\{1,3\right\}}{E}_{\mathit{pq}}^k\left(t-{\beta }_{p\left|I\right|}\right)\right.\\ \left.+2\left\{F_{\mathit{pq}}^{40}\left(t-{\beta }_{p\left|I\right|}\right)+\sum _{k\in \left\{2,4\right\}}{E}_{\mathit{pq}}^k\left(t-{\beta }_{p\left|I\right|}\right)\right\}\right]⩽V\forall t\in T,\end{array}$ (22)$F{S}^{\mathit{CMB}}⩾0\text{and integer,}$ (23)$F{S}^k⩾0\text{and integer}\forall k\in \left\{3,4\right\},$ (24)$C{B}_i\left(t\right)⩾0\text{and integer}\forall t\in T,i\in I,$ (25)$L_{\mathit{ij}}^{20}\left(t\right),L_{\mathit{ij}}^{40}\left(t\right)⩾0\text{and integer}\forall t\in T,i\in I,j\left(\ne i\right)\in I,$ (26)$D_i^k\left(t\right),G_i^k\left(t\right),N_i^k\left(t\right),O_i^k\left(t\right),R_i^k\left(t\right),S_i^k\left(t\right)⩾0\text{and integer}\forall t\in T,k\in K,i\in I,$ (27)$B_{\mathit{ij}}^k\left(t\right),E_{\mathit{ij}}^k\left(t\right)⩾0\text{and integer}\forall t\in T,k\in K,i\in I,j\left(\ne i\right)\in I.$