An optimization model for management of empty containers in distribution network of a logistics company under uncertainty
 1.8k Downloads
 1 Citations
Abstract
In transportation via containers, unbalanced movement of loaded containers forces shipping companies to reposition empty containers. This study addresses the problem of empty container repositioning (ECR) in the distribution network of a European logistics company, where some restrictions impose decision making in an uncertain environment. The problem involves dispatching empty containers of multiple types and various conditions (dirty and clean) to meet the ontime delivery requirements and repositioning the other containers to terminals, depots, and cleaning stations. A multiperiod optimization model is developed to help make tactical decisions under uncertainty and data shortage for flow management of empty containers over a predetermined planning horizon. Employing the operational law of uncertainty programming, a new auxiliary chanceconstrained programming is established for the ECR problem, and we prove the existence of an equivalence relation between the ECR plans in the uncertain network and those in an auxiliary deterministic network. Exploiting this new problem, we give the uncertainty distribution of the overall optimal ECR operational cost. The computational experiments show that the model generates goodquality repositioning plans and demonstrate that cost and modality improvement can be achieved in the network.
Keywords
Operations research Uncertain programming Logistics Intermodal transport RepositioningIntroduction
Lured by the promise of bigger sales, companies are seeking to raise the volume of international trade. Consequently, the amount of bulk products carried in containers and transported overseas exploded as this type of transportation is environmentfriendly, secure, flexible, reliable, and less prone to spillage (Crainic et al. 1993; Erera et al. 2005; Ünlüyurt and Aydın 2012; Zhang et al. 2017). Therefore, it comes as no surprise that containerbased transportation offers a high degree of productivity and provides undeniable advantages in terms of losses and damages. However, minimizing the logistics and distribution costs arising from the container flow management across different locations has emerged as a major problem that companies and affiliated thirdparty logistics firms face in everyday routine (Rashidi and Tsang 2015; Bhattacharya et al. 2014).
A chief challenge that the logistics companies face arises from the imbalance of product supply and demand, and thus an imbalance in the container flow across different regions. As a result, empty containers accumulate at demand centers, which must be efficiently repositioned to ensure the continuity of shipping activities. According to Rodrigue and Notteboom (2015) and Wang and Tanaka (2016), movement of empty containers accounts for circa \(30\%\) of movement of all containers, and about \(20\%\) of global port handling. Therefore, the empty container repositioning (ECR) problem has received a great attention from both industries and academics in recent years, and the need to avoid ad hoc decisionmaking processes has become vital in container management (SteadieSeifi et al. 2014; Lee and Meng 2014; Song and Dong 2015).
As a class of multiperiod distribution planning problems (Ahmad Hosseini et al. 2017; Gen and Syarif 2005; Hosseini 2013; Hosseini et al. 2014), this work focuses on the ECR problem in a dynamic environment that occurs in the context of intermodal distributions and transportation operations. This paper is motivated by the analysis of the operational context of a European Logistic Service Provider company (denoted by ELSP throughout the paper) that transports bulk products (chemical, oil, gas, petroleum, foodstuff, etc.) loaded in containers. The ELSP company is an international thirdparty logistics (3PL) that executes transportation orders via road, rail, short sea, and deep sea, which are supported by strategically located terminals.
This study reveals some characteristics that are rarely described in previous papers, i.e., Liu’s uncertainty. Generally, precise information is usually not available on time, which may lead to adverse operational planning reflected in a high degree of variability in system parameters. This may lead to costly transportation and ECR operations which result in lower profit margins. More precisely, some parameters cannot be predetermined prior to planning and indeterminacy should therefore be taken into account by the decision maker. Hence, an uncertain network is a more realistic representation of an actual ECR distribution network.
Although valuable insights have been obtained from the previous studies, the current practice of ECR does not properly account for uncertainty. In other words, there is a lack of suitable optimization methods and models that can simultaneously integrate uncertain factors. Therefore, it is crucial to adopt proper policies in the uncertain environment of ECR problems. To respond to such a need for the analysis of ECR in dynamic uncertain environment, we illustrate a modeling process to take the uncertain nature of ECR parameters into account using the operational law of uncertainty programming. To this aim, we develop an uncertain ECR model that can accommodate uncertainty to ensure an efficient repositioning policy at minimal operational cost, while meeting the timedependent demand, supply, and capacity requirements.
We address the repositioning of empty containers in a 3PL intermodal distribution network with different types of ports, depots, and terminals discretetime settings using Liu’s (2007) uncertainty. By this approach, a new chanceconstrained programming is established for the ECR problem, which combines assumptions, estimations, knowledge, and experience of a group of domain experts. Associated with any nondeterministic parameter, we introduce an uncertain variable, which is neither fuzzy nor stochastic. We then employ the uncertainty operational law to originally model, solve, and analyze the uncertain ECR problem in the framework of uncertainty programming. Marginally, we introduce the new problem of \((\alpha ,\beta ,\gamma )\)ECR. We show that there exists an equivalence relation between the uncertain ECR problem and the \((\alpha ,\beta ,\gamma )\)ECR problem. By using this relation, we solve the original nondeterministic problem and obtain the uncertainty distribution of the optimal distribution cost.
Literature review
Last decades have witnessed an increased interest in the literature to solve various logistics problems, including ECR, considering different variations of uncertainty accounting for different settings and solution philosophies (Cheung and Chen 1998; Lam et al. 2007; Long et al. 2015; Hosseini 2015; Chiadamrong and Piyathanavong 2017; Nourifar et al. 2018; Hamidi et al. 2017; Tofighian et al. 2018; Hafezalkotob and Zamani 2018; Shishebori and Babadi 2018).
In management of containers, companies have to deal with different uncertain factors, such as the real transportation time between two ports/depots, the future demand and supply, the transit time for returning empty containers from customers, and the available transportation capacity in, e.g., vessels (Crainic et al. 1993; Olivo et al. 2005; Erera et al. 2009; Epstein et al. 2012; Lai 2013; Yi et al. 2016; Finke 2017). As a result, investigations into efficient ECR strategies have drawn the attention of many researchers. There is abundant literature on modeling and optimizing container repositioning. While there is not the space to provide comprehensive mathematical specifications of all methods, an attempt is made to undertake a selective review of those prior studies that have tackled problems somewhat similar in flavor to the problem we deal with in this paper. We also refer to Braekers et al. (2011), Song and Dong (2015), Song and Carter (2009) and Song and Zhang (2010) to get more information regarding ECR problem and different strategies for container repositioning.
According to Dejax and Crainic (1987), although the management of empty container has received much attention since the sixties, little consideration has been, however, dedicated to the development of specific models in the container transportation issue. Aiming at specifying the ratio of empty containers and laden containers to achieve empty container equilibrium, Poo and Yip (2017) study the ECR problem and examine the problem with container inventory management under dynamic condition. Finke (2017) investigates the repositioning problem with nonlogistic theories in order to elucidate problems between different actors in the supply chain of ECR. Ji et al. (2016) study an ECR problem in short sea liner service and develop a dynamic optimization model of ECR for each stage in short sea liner service minimizing the overall repositioning cost during a decisionmaking horizon. Florez (1986) develops a profit optimization model for the ECR and investigates the sensitivity of his proposed model to the length of the planning horizon. Focusing on the business perspective of shipping industry, Shen and Khoong (1995) propose a singlecommodity network model under a rolling horizon fashion to minimize the total distribution cost of empty containers. Cheung and Chen (1998) compare a twostage stochastic model with a twostage deterministic model for the dynamic ECR problem. In a toptobottom perspective, Chen (1999) discusses factors causing unproductive moves from operational to strategic levels. Newman and Yano (2000) develop a heuristic approach based on the decomposition procedure to address the dayofweek scheduling of containers. Choong et al. (2002) present an interesting ECR computational analysis. They study the effect of planning horizon length on empty container management for intermodal transportation networks. Olivo et al. (2005) develop an integer program for the ECR between depots and ports through an inland transportation network and solve the model by a linearization technique. Bin and Zhongchen (2007) also consider a multimodal distribution network consisting of ports and inland terminals. Lam et al. (2007) develop a dynamic stochastic model for a twoport twovoyages relocation of empty containers to solve the ECR problems. A general model related to the scheduling of container storage and retrieval is presented in Vis and Roodbergen (2009). The authors present an integer program in a static environment. Shintani et al. (2010) model the ECR problem as an integer program to optimize the repositioning in the hinterland. The authors show the possibility of operational cost reduction through the use of foldable containers instead of standard containers. Long et al. (2012) formulate a twostage stochastic programming model with random demand, ship weight capacity, and ship space capacity in order to incorporate uncertainty in the ECR problem and minimize the expected operational cost for container repositioning. Later on, Zhang et al. (2014) analyze the multiperiod ECR problem with stochastic demand and lost sales aiming to establish an effective ECR policy minimizing the total operating cost. They develop a polynomialtime algorithm to find an approximate repositioning policy for multiple ports. Afterward, Long et al. (2015) computationally examine the impact of several noni.i.d sampling methods for the stochastic ECR problem. By moving forward from imbalancebased locations to shipment networkdrivenbased container repositioning, Wong et al. (2015) present a yieldbased container repositioning framework optimizing the container repositioning from surplus to deficit locations. An extension of the container management problem is to integrate the assignment of loaded containers. It is therefore natural to try to integrate two types of movements, empty and loaded, into the same allocation model. Examples of such models can be found in Dejax and Crainic (1987), Crainic et al. (1993), Erera et al. (2005), Land et al. (2008) and Brouer et al. (2011).
From another viewpoint, this paper also contributes to the growing body of knowledge regarding the Liu’s uncertain programming, by illustrating a modeling process to take the uncertain nature of ECR into account. During the last few years, there has been a vast interest in developing strategies to solve problems in different fields with various uncertain phenomena. Examples of such works can include uncertain multiitem supply chain network (Hosseini 2015), uncertain traffic network (Hosseini and Wadbro 2016), uncertain shortest path (Liu 2010b; Gao 2011), uncertain inference control (Gao 2012), uncertain networks (Liu 2010, 2007), uncertain graph and connectivity (Gao and Gao 2013), uncertain traveling salesman problem (Wang et al. 2013), uncertain multiproduct newsboy problem (Ding and Gao 2014), uncertain multicommodity flow problem (Ding 2017), uncertain maximum flow problem (Ding 2015), uncertain multiobjective programming (Wang et al. 2015), uncertain supplier selection problem (Memon et al. 2015), logistics system under supply and demand uncertainty (Moghaddam 2015), multiregion supply chain under demand uncertainty (Langroodi and Amiri 2016), uncertain railway transportation planning (Gao et al. 2016), uncertain regression analysis (Yao and Liu 2017), and uncertain random assignment problem (Ding and Zeng 2018; Zhang and Peng 2013).
Uncertain ECR model and description
 1.
where and when empty moves start and finish,
 2.
where and when containers are cleaned,
 3.
the type or dimension (cbm) of empty containers to be sent,
 4.
the condition of empty containers (dirty or clean) to be delivered,
 5.
the quantity of empty containers on each link in the network.
Tank container classification, specified by its dimensions
Classes  A  B  C  D  E 

Dimensions (cbm)  ≤ 22  [22–24]  [24–27]  [27–32]  ≥ 32 
Some primary and secondary assumptions are made during the modeling. In this project, we have had good historical data and ample seasonal (as well as daily, weekly, and monthly) frequencies to approximate demands. Therefore, the expected demand and supply of empty containers are assumed to be known during the whole planning horizon. The planning horizon is predetermined. All demands of all types of containers must be met by the same type of containers, i.e., substitutions between container types are not included. Container demand cannot be postponed. The model does not consider returningleased containers to the lessor. It is assumed that a lease term does not expire in the planning horizon. When demand cannot be met (yielding a shortage in the system), containers can be provided by other partners or from external sources (emergency shipments), but at a high cost.
It is important to distinguish between the locations presented in Fig. 1. Depots are mainly utilized for storage, whereas cleaning stations can be employed for both storage and cleaning processes. A few depots and cleaning stations are controlled by the ELSP company, which are referred to as internal locations. The storage costs at these locations are relatively lower compared to external depots and cleaning stations. Railway, port and rail ship terminals provide the options for intermodal transportation. Railway and port terminals are selfdescriptive. As indicated in Fig. 1, rail ship terminals allow for both trains and ferries to arrive and depart from. Supply customers provide empty containers that have become available after unloading of the products, while demand customers requires empty containers to be delivered at their site for loading. Note that a customer can be both a demand and a supply customer.
 1.
Empty containers are not allowed to be stored at either the supplier sites or demand sites. Hence, after unloading, containers are hauled away immediately to other locations.
 2.
Transportation capacities between locations, transportation costs over links, and storage costs in locations are assumed to be independent of the type and the condition of the container.
 3.
Transportation from a location to itself is not allowed, i.e., the distribution network is loopfree.
 4.
A container is assumed to be dirty after unloading at a supplier site.
 5.
Arrivals and departures take place in the beginning of time periods.
 6.
The cleaning times in all cleaning stations are assumed to be same.
 7.
All shipment modes are assumed to have a limited capacity.
 8.
The storage times are assumed to be identical.
 9.
A cleaning process cannot be interrupted.
Sets and Indices
 T

Predefined integral planning horizon
 N

Set of time periods; elements indexed by \(t=1, 2,\ldots , T1\)
 I

Set of demand customers; elements indexed by \(i=1,2,\ldots , I\)
 S

Set of supply centers; elements indexed by \(s=1,2,\ldots , S\)
 J

Set of depots; elements indexed by \(j=1,2,\ldots , J\)
 P

Set of port terminals; elements indexed by \(p=1,2,\ldots , P\)
 R

Set of railway terminals; elements indexed by \(r=1,2,\ldots , R\)
 C

Set of cleaning stations; elements indexed by \(c=1,2,\ldots ,C\)
 V

Set of rail ship terminals; elements indexed by \(v=1,2,\ldots , V\)
 M

Set of transport modes; elements indexed by \(m=1,2,\ldots , M\)
 K

Set of container types; elements indexed by \(k=1,2,\ldots , K\)
 U

Set of container conditions: d (dirty) and o (orderly)
Parameters
 \(D_{iku}^{t}\)

Demand of customer i at time period t of type k in condition u
 \(S_{sku}^{t}\)

Supply of supplier s at time period t of type k in condition u
 \(\tau _{{l}{l}^{^{\prime}}m}\)

Transit time (hours) between origin l and destination \({l}^{^{\prime}}\) via mode m, where \({{l},{l}^{^{\prime}}} \in I, S, J, P, R, C, V, E\)
 \(\tau \)

Cleaning time at cleaning stations
 \(\varDelta \)

Storage time allowed at intermodal terminals (railways, ports, and rail ships)
 \(\eta _{ll^{^{\prime}}m}^{t}\)

Uncertain transportation capacity between l and \(l^{{\prime}}\) via mode m at time period t with regular uncertainty distribution \(\varPsi _{ll^{{\prime}}tm}\)
 \(\eta _{l}^t\)

Uncertain storage capacity available at location l at time period t with regular uncertainty distribution \(\varPsi _{lt}\)
 \(\xi ^t_{{l}{l}^{{\prime}}m}\)

Uncertain unit transportation cost from l to \({l}^{{\prime}}\) at time period t via mode m with regular uncertainty distribution \(\varPhi _{{l}{l}^{{\prime}}tm}\)
 \(\xi _{l}^t\)

Uncertain unit storage cost at location l at time period t with regular uncertainty distribution \(\varPhi _{lt}\)
 \(\epsilon _i^t\)

Uncertain emergency shipment cost for costumer i at time period t with regular uncertainty distribution \(\phi _{it}\)
Integer nonnegative decision variables
 \(x_{{{l}{l}^{{\prime}}mku}}^{t}\)

number of empty containers of type k with condition u to be arrived at destination \(l^{{\prime}}\) via mode m in the beginning of period t from origin l, where \({{l},{l}^{{\prime}}} \in I, S, J, P, R, C, V\)
 \(z_{lku}^{t}\)

number of empty containers of type k in condition u available at location l in the beginning of time period t, where \(l\in J, P, R, C, V\)
 \(sh_{iku}^{t}\)

number of stockout empty containers (to be brought in from outside the system (leased, borrowed, or newly purchased) of type k in condition u for demand customer i at time period t
Chanceconstrained model for ECR
Empty containers must be repositioned between ports to ensure the continuity of shipping activity and to meet future transportation opportunities. However, significant sources of uncertainty may still affect this issue. This is because in many occasions, due to maintenance, economic reasons, or technical difficulties, we have lack of observed data about an unknown state of nature. For example, since loaded containers have greater priority than empty containers and unexpected transportation opportunities may arise, the residual transportation capacity for empty containers also fails to provide certain information. Moreover, the maximum number of empty containers that can be loaded and unloaded in ports is sometimes uncertain as well. Information on the number of empty containers requested in each port may also be imprecise, because unexpected transportation demands may arise. Moreover, the number of empty containers available in ports can also be uncertain, because we do not know precisely when they will be returned by import customers. In such cases where information is not sufficient, we have to invite some domain experts to evaluate their belief degree for events’ occurrence. When only belief degrees are available (no samples), the estimated uncertainty distribution usually deviates far from the cumulative frequency. In this case, the uncertainty theory can be referred as one of the legitimate approaches (Liu 2007, 2010, 2012).
To formalize the discussion and develop a chanceconstrained model, all ECR uncertain components are regarded as uncertain variables and the following modeling process is offered. It is worth noting that whenever there exists accurate data with high level of certainty based on enough proper historical data, one can easily substitute the corresponding uncertain component with a crisp value without any loss of generality and any change in the modeling process (see Remark 1).
Remark 1
According to laws of uncertainty programming, if, e.g., transportation cost \(\xi ^t_{{l}{l}^{{\prime}}m} = c^t_{{l}{l}^{{\prime}}m}\) is a certain number, then it can be regarded as a constant function on the uncertainty space, and therefore, it is concluded that \(\varPhi ^{1}_{{l}{l}^{{\prime}}tm} (\alpha ) = c^t_{{l}{l}^{{\prime}}m}\) for any confidence level \(\alpha \in (0,1)\) (Liu 2007, 2010, 2012).
Remark 2
Without loss of generality, we assume that all uncertain variables used in this work are independent and their uncertainty distributions are regular. Otherwise, a small perturbation can be given to get a regular distribution (Liu 2007).
Theorem 1
Let\(DN=(L,A)\)represent the distribution network of the uncertain ECR problem and suppose that\(\xi ^t_{{l}{l}^{{\prime}}m}\), \(\xi _{l}^t\), \(\epsilon _i^t\), \(\eta _{ll^{{\prime}}m}^{t}\), and\(\eta _{l}^t\)denote the independent uncertain variables with regular distributions\(\varPhi _{{l}{l}^{{\prime}}tm}\), \(\varPhi _{lt}\), \(\phi _{lt}\), \(\varPsi _{ll^{{\prime}}tm}\), and\(\varPsi _{lt}\), respectively, associated with the parameters of the network. The uncertain ECR problem is equivalent to the welldefined deterministic\((\alpha , \beta , \gamma )\)ECR problem presented byModel 1\(^{\prime \prime }\), provided that the confidence levels\(\alpha , \beta , \gamma \)are given.
Theorem 1 yields that the optimal solution of the uncertain ECR problem is actually the \((\alpha , \beta , \gamma )\)optimal ECR solution obtained from Model 1\(^{\prime \prime }\). It also presents a simple way to obtain the uncertainty distribution of the cost of repositioning of empty containers. To this aim, it is enough to choose a number of different \(\alpha , \beta \), and \(\gamma \) and solve Model 1\(^{\prime \prime }\) repeatedly. We also highlight that Model 1\(^{\prime \prime }\) is decreasing with respect to \(\alpha \) and nondecreasing with respect to \(\beta \) and \(\gamma \).
Computational study and experiments
A computational study with real data is employed to demonstrate the functionality of the model. The casestudy company is a European logistic service provider (ELSP) that transports bulk products loaded in containers and owns circa 35000 containers. The casestudy problem deals with one main part of the company’s whole distribution network that involves approximately \(20\%\) of its total market. To account for variability in demand and to investigate the model’s performance, we examine the proposed model under different planning horizons using oneyear actual historical shipping patterns performed by the company, with key practical considerations such as delivery time periods and intermodal transports. Computational experiments show the functionality of the model and suggest that potential distribution cost savings of about \(6{}10\%\) would be possible to achieve through flow and modality alterations. Furthermore, to highlight certain features of the model and to reveal how the distribution cost depends on the model’s parameters, we present a sensitivity analysis and we investigate the effect of the parameters on the container management.
Statistics of the ELSP company’s distribution network
Depots: 38 (4 + 34) (Internal + external)  Supply customers: 1196  Demand customers: 487 
Cleaning stations: 138 (8 + 130) (Internal + external)  Railway terminals: 98  Port terminals: 93 
Rail ship terminals: 15  Links/sections 14,590  Total number of locations 2065 
The data obtained from the company include timedependent demands over one year for all customers, transportation costs over links for all transportation modes, cleaning times, storage times, storage costs, transit capacities for all modes, and storage capacities at intermodal terminals. Transportation costs and transit times (lead times) for shipments to be carried out via vessel and train were derived by calculating the average price and transit time of each origin–destination pair. Distances between each origin–destination pair were calculated and related to the average speed of a truck (60 km/h) in order to derive the transit times for shipments over roads. The storage capacities at intermodal terminals were supposed to represent the only sources of uncertainty, controlling the values taken by the other uncertain parameters. Uncertainty distributions about the capacities over the whole planning horizon are provided by expertbased opinions, and the confidence levels \(\alpha \), \(\beta \), and \(\gamma \) are provided by the logistics supervisor. The domain experts are presumed to be independent.
Countries and regions of the ELSP distribution network
Country  Region 1  Region 2  Region 3  Region 4 

Austria  
Belgium  
Bulgaria  
Switzerland  
Germany  East  
Denmark  North  West  South  
Spain  Central  North–East  North–West  
Finland  North  South  
France  North–West  South–East  Venissieux  
England  North–East  North–West  South  
Scotland  
Ireland  North  South  
Italy  Central  North–East  North–West  South 
Netherlands  
Norway  
Poland  
Portugal  North  South  
Russia  
Sweden  North  East  West  South 
Slovenia 
Model’s performance and achievements
Settings  Instance 1 (%)  Instance 2 (%)  Instance 3 (%) 

Total cost savings  6.54  6.29  10.3 
Distribution cost, modality utilization, and elapsed computational time for an instance with different discretizations
\(\#\) Supply sites: 292, \(\#\) Demand sites: 170, Initial inventory: 0, Max capacity = 5000  

Settings  Instance 1  Instance 2  Instance 3  Instance 4  Instance 5 
\(\#\) Time periods  4  16  24  32  48 
\(\#\) Constraints  138,468  553,579  830,013  1,106,352  1,925,630 
\(\#\) Variables  601,761  2,407,041  3,610,561  4,814,081  8,376,501 
Total cost  37,509  40,843  41,350  41,396  – 
Trans. cost  23,758  26,803  27,122  27,129  – 
Storage cost  13,751  14,040  14,228  14,267  – 
Shortages  0  0  0  0  – 
Modality util. (%)  100/0/0  100/0/0  100/0/0  99/0/1  – 
Road/Rail/Sea  
Elapsed time for AIMMS/CPLEX (s)  261.63  912.47  1596.56  2562.31  > 1000.00 
Status  Normal completion  Normal completion  Normal completion  Normal completion  Terminated by solver 
Operational costs and modality utilization as a result of different planning horizons and initial inventory
Supp. and dem. (in the first 4 time periods): 361, 170, Max capacity = 5000  

Settings  Instance 1  Instance 2  Instance 3 
Initial inventory clean: 0, initial inventory dirty: 0  
# Time periods  4  7  21 
Total cost  53,884  61,874  89,048 
Trans. cost  52,946  60,895  87,748 
Storage cost  938  979  1300 
# Shortages  0  0  0 
Modality util. (%)  91.9/3.3/4.8  90.2/4/5.8  82.9/6.8/10.3 
Road/Rail/Sea  
Initial inventory clean: 399, initial inventory dirty: 510  
# Time periods  4  7  21 
Total cost  63,990  65,711  75,394 
Trans. cost  27,906  37,360  51,457 
Storage cost  36,084  28,351  23,937 
# Shortages  0  0  0 
Modality util. (%)  100/0/0  99.7/0.3/0  99.7/0.3/0 
Road/Rail/Sea 
Costs and modality utilizations in terms of transportation capacities (\(C_{ll^{{\prime}}m}^{t}\))
Supply: 265, demand: 170, initial inventory: 0, # time periods: 4  

Settings  Instance 1  Instance 2  Instance 3  Instance 4  Instance 5 
\(C_{ll^{{\prime}}m}^{t}\) (increase in %)  0  10  20  30  40 
Total cost  51,064  50,910  50,800  50,751  50,704 
Trans. cost  50,239  50,083  49,970  49,920  49,870 
Storage cost  826  828  830  831  834 
# Shortages  0  0  0  0  0 
Modality util. (%)  91.7/3.9/4.3  91.3/3.9/4.6  91.2/3.8/5  91/4/5  91/3.8/5.2 
Road/Rail/Sea 
Cost and modality utilizations in terms of allowed storage time at intermodal terminals
Supp.: 265, Dem.: 170, initial inventory: 0, \(\#\) periods: 21, Max capacity = 5000  

Settings  Instance 1  Instance 2  Instance 3  Instance 4  Instance 5 
\(\varDelta \)  0  1  2  3  4 
Total cost  955,117  954,110  945,584  934,718  930,831 
Trans. cost  810,709  806,818  801,542  790,397  785,505 
Storage cost  144,408  147,292  144,042  144,321  145,326 
# Shortages  38  22  0  0  0 
Shortages (%)  0.37  0.2  0  0  0 
Modality util. (%)  100/0/0  90.8/4.5/4.7  91.0/4.7/4.3  89.0/4.8/6.2  89.1/5.1/5.8 
Road/Rail/Sea 
In addition, Model 1\(^{\prime \prime }\) is nondecreasing with respect to \(\beta \), and \(\gamma \). This is intuitive. Because when the confidence levels \(\beta \) and \(\gamma \) increase, the model has to find an optimal ECR solution in a smaller feasible set, so the total ECR cost either increases or stays the same. Hence, by solving the problem for different values of \(\beta \) and \(\gamma \) we also get to estimate the uncertainty distribution of optimal ECR cost with respect to these parameters.
Conclusion
Based on Liu’s uncertainty theory, this paper addresses the repositioning of empty containers (of multiple types) in timedependent, intermodal distribution network of an anonymous international European logistic service provider company (denoted by ELSP) in discretetime settings under uncertainty and data shortage. The company transports bulk products (oil, gas, petroleum, chemical, foodstuff, etc.) loaded in containers via road, rail, and sea needing to properly reposition its empty containers over different types of ports.
Exploiting the operational law of independent uncertain variables, the uncertain ECR problem is originally modeled and analyzed in the framework of uncertainty programming. In order to solve the uncertain ECR problem, a new auxiliary \((\alpha ,\beta ,\gamma )\)ECR model is proposed and its equivalence relation with uncertain ECR problem is demonstrated. By exploiting this relation, the original uncertain ECR problem is solved and the uncertainty distribution of the optimal logistics cost is obtained.
The casestudy problem has been dealing with one main part of the company’s whole distribution network that involves approximately \(20\%\) of its total market, which is used to illustrate the interest and functionality of the model. Computational experiments show the functionality of the model and suggest that potential distribution cost savings of \(6{}10\%\) would be possible to achieve through modality alteration over links of the distribution network. Furthermore, to highlight certain features of the model and the effect of the model’s parameters on empty container management, an extensive sensitivity analysis is presented. The proposed model captures many timevarying parameters, which makes it possible to quantify the tradeoffs between different criteria of the ECR problem. The model includes many practical considerations needed for ECR distribution networks: meeting demand and supply requirements, multimodal transports, cleaning processes, transport and storage capacity, internal and external locations, various types of ports, terminals, and depots, etc. However, there are several possibilities for extending the model proposed in this study.
We note that proposed model is concerned with one of the main objectives of the ECR distribution networks, viz. the distribution cost. However, there can be some other aspects and criteria that could be taken into account, e.g., profit, power, customer’s credit performance, and objectives of various business divisions (such as marketing, sales, distribution, planning, and purchasing). Hence, developing a unified and rigorous structure that can capture all synergies, criteria, and tradeoffs could be worthy of further exploration. To include multiple criteria, the use of analytical hierarchy process (AHP) and nonpreemptive weighted goal programming is highly recommended. Including short and longterm leasing in the model could be another direction for future research. Moreover, it might be worthwhile to consider the uncertain nature of transit times in the ECR problem as well. One future research endeavor could be to integrate the loaded and empty containers flow decisions in a single integrated model. Finally, exploring the use of parallel computing and decomposition algorithms, such as Bender’s decomposition, to cope with larger instances could be one other area for further expansion.
References
 Ahmad Hosseini S, Şahin G, Ünlüyurt T (2017) A penaltybased scaling algorithm for the multiperiod multiproduct distribution planning problem. Eng Optim 49(4):583–596MathSciNetCrossRefGoogle Scholar
 Bhattacharya A, Kumar S, Tiwari M, Talluri S (2014) An intermodal freight transport system for optimal supply chain logistics. Transp Res Part C Emerg Technol 38:73–84CrossRefGoogle Scholar
 Bin W, Zhongchen W (2007) Research on the optimization of intermodal empty container reposition of landcarriage. J Transp Syst Eng Inf Technol 7(3):29–33Google Scholar
 Braekers K, Janssens GK, Caris A (2011) Challenges in managing empty container movements at multiple planning levels. Transp Rev 31(6):681–708CrossRefGoogle Scholar
 Brouer BD, Pisinger D, Spoorendonk S (2011) Liner shipping cargo allocation with repositioning of empty containers. INFOR Inf Syst Oper Res 49(2):109–124Google Scholar
 Chen T (1999) Yard operations in the container terminala study in the unproductive moves’. Marit Policy Manag 26(1):27–38CrossRefGoogle Scholar
 Cheung RK, Chen CY (1998) A twostage stochastic network model and solution methods for the dynamic empty container allocation problem. Transp Sci 32(2):142–162zbMATHCrossRefGoogle Scholar
 Chiadamrong N, Piyathanavong V (2017) Optimal design of supply chain network under uncertainty environment using hybrid analytical and simulation modeling approach. J Ind Eng Int 13(4):465–478CrossRefGoogle Scholar
 Choong S, Kutanoglu CEM (2002) Empty container management for intermodal transportation networks. Transp Res Part E Logist Transp Rev 38(6):423–438CrossRefGoogle Scholar
 Crainic T, Gendreau M, Dejax P (1993) Dynamic and stochastic models for the allocation of empty containers. Oper Res 41(1):102–126zbMATHCrossRefGoogle Scholar
 Deidda L, Di Francesco M, Olivo A, Zuddas P (2008) Implementing the streetturn strategy by an optimization model. Marit Policy Manag 35(5):503–516CrossRefGoogle Scholar
 Dejax P, Crainic T (1987) Survey paper—a review of empty flows and fleet management models in freight transportation. Transp Sci 21(4):227–248CrossRefGoogle Scholar
 Ding S (2015) The $\alpha $maximum flow model with uncertain capacities. Appl Math Model 39:2056–2063MathSciNetCrossRefGoogle Scholar
 Ding S (2017) Uncertain minimum cost multicommodity flow problem. Soft Comput 21(1):223–231zbMATHCrossRefGoogle Scholar
 Ding S, Gao Y (2014) The ($\sigma $, s) policy for uncertain multiproduct newsboy problem. Expert Syst Appl 41:3769–3776CrossRefGoogle Scholar
 Ding S, Zeng XJ (2018) Uncertain random assignment problem. Appl Math Model 56:96–104MathSciNetCrossRefGoogle Scholar
 Epstein R, Neely A, Weintraub A, Valenzuela F, Hurtado S, Gonzalez G, Beiza A, Naveas M, Infante F, Alarcon F et al (2012) A strategic empty container logistics optimization in a major shipping company. Interfaces 42(1):5–16CrossRefGoogle Scholar
 Erera A, Savelsbergh MMJ (2009) Robust optimization for empty repositioning problems. Oper Res 57(2):468–483MathSciNetzbMATHCrossRefGoogle Scholar
 Erera AL, Morales JC, Savelsbergh M (2005) Global intermodal tank container management for the chemical industry. Transp Res Part E Logist Transp Rev 41(6):551–566CrossRefGoogle Scholar
 Finke S (2017) Empty container repositioning from a theoretical point of view. In: Dynamics in logistics. Springer, pp 325–334Google Scholar
 Florez H (1986) Emptycontainer repositioning and leasing: an optimization model. In: MikroficheAusg, Brooklyn, Polytechnic University, PhD Thesis. University Microfilms International, Ann ArborGoogle Scholar
 Gao Y (2011) Shortest path problem with uncertain arc lengths. Comput Math Appl 62(6):2591–2600MathSciNetzbMATHCrossRefGoogle Scholar
 Gao Y (2012) Uncertain inference control for balancing an inverted pendulum. Fuzzy Optim Decis Mak 11(4):481–492MathSciNetzbMATHCrossRefGoogle Scholar
 Gao X, Gao Y (2013) Connectedness index of uncertain graphs. Int J Uncertain Fuzziness Knowl Based Syst 21(1):127–137MathSciNetzbMATHCrossRefGoogle Scholar
 Gao Y, Yang L, Li S (2016) Uncertain models on railway transportation planning problem. Appl Math Model 40(7):4921–4934MathSciNetCrossRefGoogle Scholar
 Gen M, Syarif A (2005) Hybrid genetic algorithm for multitime period production/distribution planning. Comput Ind Eng 48(4):799–809CrossRefGoogle Scholar
 Hafezalkotob A, Zamani S (2018) A multiproduct green supply chain under government supervision with price and demand uncertainty. J Ind Eng Int 1–14Google Scholar
 Hamidi M, Shahanaghi K, Jabbarzadeh A, Jahani E, Pousti Z (2017) Determining production level under uncertainty using fuzzy simulation and bootstrap technique, a case study. J Ind Eng Int 13(4):487–497CrossRefGoogle Scholar
 Hosseini S (2013) A modelbased approach and analysis for multiperiod networks. J Optim Theory Appl 157(2):486–512MathSciNetzbMATHCrossRefGoogle Scholar
 Hosseini S (2015) Timedependent optimization of a multiitem uncertain supply chain network: a hybrid approximation algorithm. Discrete Optim 18:150–167MathSciNetzbMATHCrossRefGoogle Scholar
 Hosseini S, Wadbro E (2016) Connectivity reliability in uncertain networks with stability analysis. Expert Syst Appl 57:337–344CrossRefGoogle Scholar
 Hosseini S, Sahin G, Unluyurt T (2014) A decompositionbased approach for the multiperiod multiproduct distribution planning problem. J Appl Math 201:Article ID 825,058, 25 ppGoogle Scholar
 Ji Y, Yang H, Zhu Q, Xing Y (2016) Optimization model of empty container reposition of liner alliance based on mutually renting strategy in short sea service. In: Proceedings of the 6th international Asia conference on industrial engineering and management innovation. Springer, pp 1055–1064Google Scholar
 Lai M (2013) Models and algorithms for the empty container repositioning and its integration with routing problems. PhD thesis, Universita’degli Studi di CagliariGoogle Scholar
 Lam SW, Lee LH, Tang LC (2007) An approximate dynamic programming approach for the empty container allocation problem. Transp Res Part C Emerg Technol 15(4):265–277CrossRefGoogle Scholar
 Langroodi R, Amiri M (2016) A system dynamics modeling approach for multilevel, multiproduct, multiregion supply chain under demand uncertainty. Expert Syst Appl 51:231–244CrossRefGoogle Scholar
 Lee CY, Meng Q (2014) Handbook of ocean container transport logistics: making global supply chains effective, vol 220. Springer, BerlinGoogle Scholar
 Liu B (2007) Uncertainty theory. In: Uncertainty theory. Springer, pp 205–234Google Scholar
 Liu B (2010a) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer, BerlinCrossRefGoogle Scholar
 Liu W (2010b) Uncertain programming models for shortest path problem with uncertain arc lengths. In: Proceedings of the first international conference on uncertainty theory, Urumchi, China, pp 148–153Google Scholar
 Liu B (2012) Why is there a need for uncertainty theory. J Uncertain Syst 6(1):3–10Google Scholar
 Long Y, Lee LH, Chew EP (2012) The sample average approximation method for empty container repositioning with uncertainties. Eur J Oper Res 222(1):65–75zbMATHCrossRefGoogle Scholar
 Long Y, Chew EP, Lee LH (2015) Sample average approximation under noniid sampling for stochastic empty container repositioning problem. OR Spectr 37(2):389–405zbMATHCrossRefGoogle Scholar
 Memon M, Lee Y, Mari S (2015) Group multicriteria supplier selection using combined grey systems theory and uncertainty theory. Expert Syst Appl 42:7951–7959CrossRefGoogle Scholar
 Moghaddam K (2015) Fuzzy multiobjective model for supplier selection and order allocation in reverse logistics systems under supply and demand uncertainty. Expert Syst Appl 42:6237–6254CrossRefGoogle Scholar
 Newman A, Yano C (2000) Scheduling direct and indirect trains and containers in an intermodal setting. Transp Sci 34(3):256–270zbMATHCrossRefGoogle Scholar
 Nourifar R, Mahdavi I, MahdaviAmiri N, Paydar MM (2018) Optimizing decentralized production–distribution planning problem in a multiperiod supply chain network under uncertainty. J Ind Eng Int 14(2):367–382CrossRefGoogle Scholar
 Olivo A, Zuddas P, Di Francesco M, Manca A (2005) An operational model for empty container management. Marit Econ Logist 7(3):199–222CrossRefGoogle Scholar
 Poo MCP, Yip TL (2017) An optimization model for container inventory management. Ann Oper Res 1–21Google Scholar
 Rashidi H, Tsang E (2015) Vehicle scheduling in port automation: advanced algorithms for minimum cost flow problems. CRC Press, Boca RatonCrossRefGoogle Scholar
 Rodrigue J, Notteboom T (2015) International trade and freight distribution. In: Southwest transportation workforce center – Connecting and empowering the transportation workforce. California State University, Long BeachGoogle Scholar
 Shen W, Khoong C (1995) A DSS for empty container distribution planning. Decis Support Syst 15(1):75–82CrossRefGoogle Scholar
 Shintani K, Konings R, Imai A (2010) The impact of foldable containers on container fleet management costs in hinterland transport. Transp Res Part E Logist Transp Rev 46(5):750–763CrossRefGoogle Scholar
 Shishebori D, Babadi AY (2018) Designing a capacitated multiconfiguration logistics network under disturbances and parameter uncertainty: a realworld case of a drug supply chain. J Ind Eng Int 14(1):65–85CrossRefGoogle Scholar
 Song DP, Carter J (2009) Empty container repositioning in liner shipping 1. Marit Policy Manag 36(4):291–307CrossRefGoogle Scholar
 Song DP, Dong JX (2015) Empty container repositioning. In: Handbook of ocean container transport logistics. Springer, pp 163–208Google Scholar
 Song D, Zhang Q (2010) A fluid flow model for empty container repositioning policy with a single port and stochastic demand. SIAM J Control Optim 48(5):3623–3642MathSciNetzbMATHCrossRefGoogle Scholar
 SteadieSeifi M, Dellaert N, Nuijten W, Van Woensel T, Raoufi R (2014) Multimodal freight transportation planning: a literature review. Eur J Oper Res 233(1):1–15zbMATHCrossRefGoogle Scholar
 Tofighian AA, Moezzi H, Barfuei MK, Shafiee M (2018) Multiperiod project portfolio selection under risk considerations and stochastic income. J Ind Eng Int 1–14Google Scholar
 Ünlüyurt T, Aydın C (2012) Improved rehandling strategies for the container retrieval process. J Adv Transp 46(4):378–393CrossRefGoogle Scholar
 Vis IF, Roodbergen KJ (2009) Scheduling of container storage and retrieval. Oper Res 57(2):456–467zbMATHCrossRefGoogle Scholar
 Wang H, Tanaka K (2016) Management of empty container repositioning considering leveling marine container logistics. In: TechnoOcean (TechnoOcean). IEEE, pp 682–686Google Scholar
 Wang J, Zhu J, Yang H (2013) Reliable path selection problem in uncertain traffic network after natural disaster. Mathematical Problems in Engineering pp Article ID 413,034, 5 ppGoogle Scholar
 Wang Z, Guo J, Zheng M, Yang Y (2015) A new approach for uncertain multi objective programming problem based on ${\mathscr {P}}_\mathtt{e}$ principle. J Ind Manag Optim 11(1):13–26MathSciNetGoogle Scholar
 Wong EY, Tai AH, Raman M (2015) A maritime container repositioning yieldbased optimization model with uncertain upsurge demand. Transp Res Part E Logist Transp Rev 82:147–161CrossRefGoogle Scholar
 Yao K, Liu B (2017) Uncertain regression analysis: an approach for imprecise observations. Soft Comput 1–4Google Scholar
 Yi S, ScholzReiter B, Kim KH (2016) Collaborative carryout process for empty containers between truck companies and a port terminal. In: Dynamics in logistics. Springer, pp 473–482Google Scholar
 Zhang B, Peng J (2013) Uncertain programming model for uncertain optimal assignment problem. Appl Math Model 37(9):6458–6468MathSciNetzbMATHCrossRefGoogle Scholar
 Zhang B, Ng C, Cheng T (2014) Multiperiod empty container repositioning with stochastic demand and lost sales. J Oper Res Soc 65(2):302–319CrossRefGoogle Scholar
 Zhang H, Lu L, Wang X (2017) Two stages empty containers repositioning of AsiaEurope shipping routes under revenue maximization. In: International Conference on Intelligence Science. Springer, pp 379–389Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.