A proposed method for the empty container repositioning problem.

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Universiteit van Amsterdam

Faculteit Economie en Bedrijfskunde

MSc in Operations Research

Master Thesis

A Proposed Method for the Empty

Container Repositioning Problem

Author:

S. Avrilionis

Student ID: 10170677

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st

Supervisor:

Dr. C.-W. Duin

2

nd

Supervisor:

Dr. H.J. van der Sluis

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1 Introduction

This thesis follows from an internship done in the Netherlands at the organization of “Toegepast – Natuurwetenschappelijk Onderzoek” (TNO). It is a preparatory work for the Greenbox project which is supposed to be further studied at the department of “Mobiliteit en Logistiek” at the TNO in Delft. This introductory chapter is divided as follows: in the first part a background is provided to the reader in order to understand the purpose and the scope of the Greenbox project. The second part focuses on the sub-problem tackled in this thesis.

1.1 Background

The port of Rotterdam is the largest in Europe. It was the largest port in the world, until 2004, when Singapore and Shanghai become the two largest worldwide. More than 450 ships are visiting the port of Rotterdam every day, taking care of 1.000.000 tones goods costing billions of dollars. Covering about 105,54km2, it is one of Europe's most important industry complexes. Imports are carried to long distance destinations into the mainland by trains or by vehicle carriers; the majority of the imports are made by intermodal freight containers.

Ocean container carriers’ cost structures are to a considerable degree also determined by the handling of empty containers in the hinterland and in repositioning these empty containers to loading areas. Especially the movement of empty containers from customer destinations to empty depots, and to loading locations again results in substantial in efficiencies. There is often a mismatch between empty container redelivery and the requirement of empty containers, both in terms of time, location and container type.

A few months ago three shipping companies, proposed to TNO to look into the possibilities of addressing the hinterland inefficiencies, in order to reduce the cost of the empty container repositioning. This could be achieved by improving the match between cargo supply–empty container demand– and empty container availability. It requires the cooperation of shippers and consignees, as if united into one shipping line (acting as the owner of the equipment). Then one can start to investigate the prospects for optimal systemic decisions rather than that each shipper seeks its own beneficial decisions.

In the past months various shippers have joined this initiative and expressed their interest in researching and finally deploying new optimization concepts with cost and environmental benefits for all stakeholders. In this project TNO will study in more detail methodologies for determining:

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5 1. Potential benefits in chain optimization;

2. Sharing gains and/or benefits over the partners;

3. Executing pilots with the aim of evaluating the benefits and impacts.

Within budget restrictions a dedicated approach is to be fulfilling scientific requirements as well as the need for viable results.

1.2 Problem description

1.2.1 Current Practices

The logistical problem of the empty containers occurs when a full container gets emptied at a consignee. When and to what other place should it be transported to be reloaded with the same or another good? Specifying the transshipment of the container without carrying any goods constitutes the empty container reposition problem. Clearly, the management of the empty containers’ movements is a significant part of the container logistics cycle.

In principal there are four moving schemes possible:

a) Direct Return b) Port Direct c) Street Turn d) Depot Storage

as illustrated in Figures 1.1.a, 1.1.b, 1.1.c, and 1.1.d respectively. Further also illustrated in Figure 1.1, there positioning of the empty containers takes place over three scales:

a) A Local Scale b) A Regional Scale c) An International Scale

In the case of the Figure 1.1.a, containers arrive from a port of a country, let us say A, to a port of country B (route 1); then they are moved to a local consignee in order to be unloaded (route 2). Afterwards, the empty containers are carried back to the (local) port of country B (route 3), and are finally transported back to the exporter country A (route 4). This case is the so called “repositioning” or “direct return” of the empty containers.

At the Port Direct case in Figure 1.1.b, route 1, route 2 and route 3 are the same, as in case of Figure 1.1.a. However, after a container has been unloaded at a local consignee, it is taken back to the port of the importing country (country B) for short term storage. Then, it is moved to a local shipper (route 4) who fills it in and sends it to the port B (route 5) for export to

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6 country A (route 6). By this scheme, as well as by schemes c and d, shippers and carriers try to match local export cargo with the available empty containers.

Street Turn (see Figure 1.1.c) depicts a better alternative, where a container after being unloaded at a local consignee, is transported directly to a local shipper (route 3) who fills it in and sends it to the port B (route 4) for export in the country A (route 5). Street Turn is a “cleverer” alternative than Port Direct, as the total amount of distance involved in transporting the container may be considerably less, not to mention reducing the traffic of empty containers coming into and out of the port complex. Despite of the benefits that the Street Turn alternative can produce, there are several complications when implementing it in real life. For instance, it requires a certain level of coordination between different shipper and carriers, as well as advanced information sharing between them, in order to accomplish the Street Turn moves. To synchronize the shipper’s demand for an empty container with the consignee’s surplus or supply, it is necessary to know the time when these events occur. In addition to this, if the container is longer away than a prearranged time window, additional charges will accrue. Therefore, a main concern of the carriers (companies working for the shipping lines) is to avoid this penalty by transporting the empties back to the port as quickly as possible. It is clear that this can conflict with efforts to optimize the empty containers supply chain network. Although efficient, Street Turn is rarely used in the greater vicinity of the port of Rotterdam.

Finally, in the case of Depot Storage in Figure 1.1.d, after a container has been unloaded at a local consignee, it is transported to a short term empty container storage yard (route 3). Afterwards it is moved to a local shipper (route 4), then to the port of country B (route 5) and finally to the port of country A (route 6) as already mentioned. Apparently, this moving requires less coordination between shippers with empty container demand and shippers (or consignees) with empty container supply. Despite of some handling costs that will occur when dropping or picking up a container from the depot, this alternative can reduce the empty containers mileage occurring from Direct Return or Port Direct alternatives. Comparing Figure 1.1.d with 1.1.a and 1.1.b, one can observe that if the depot is located somewhere between consignees and shippers, the distance covered in depot storage case is way less. In the case of depot storage, there are no costs for coordination between consignees and shippers in order to synchronize the supply of the first with the demand of the second; the empty container is directly moved into the empty container depot. Then, it can take care of later demand at any shipper immediately, without the need of transporting an additional container from another location. Moreover, another benefit of this alternative is the traffic reduction that will occur at the roads at the port area, by moving some traffic to the district of a depot.

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Figure 1.1 – Container movements in time order numbered

1.2.2 Greenbox problem definition

Ocean container carriers such as CMA-CGM see potential benefits in combining demand for transport of container is able goods from various shippers with their empty container circulation. If a system capable of doing so would be in place, the challenges of balancing shortages and surpluses in container demand throughout their European network could more easily be met. This would enhance the cost efficiencies as well as decrease the carbon footprint of the transport and thus the sustainability of the supply chain. This could also result

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8 in cost reductions due to reduced vehicle kilometres, better lead times or more reliability in chain control. In all, this would contribute to more operational excellence in the supply- and transport- chains.

The foreseen challenges are also obvious: shippers are requested or obliged to share data on volumes, demand and costs, a more transparent transport information system has to be in place and a methodology for gain sharing has to be agreed on and has to be implemented in a way that will be robust, scalable and future proof.

The aim of the Greenbox initiative is to:

 Deliver greater hinterland transport efficiencies;  Lower emissions;

 Reduce seasonal equipment shortages.

This can be achieved by:

1. Analysing the potential improvements in terms of reducing inefficiencies, costs and more supply chain sustainability;

2. Setting up a model for gain sharing in the network collaboration; 3. Starting a pilot.

In general a 30% reduction in empty container mileage could be possible according to CMA-CGM. The approach will incorporate the actual costs reductions by cooperating and also side effects like other administrative processes for participating partners and the effects on operations outside the boundaries of the network or system under consideration.

1.3 Problem studied in this thesis

1.3.1 Introduction

Typically, the logistics managers’ main concern is the transportation of loaded containers. They would prefer to ignore empty containers completely, but this is not possible since real-world container networks require empties to account for imbalances in loaded flows. If empty container flows are not managed carefully, the entire shipping network will operate inefficiently.

One has to take care of:

 Service network design with empty container repositioning  The synchronization of container supply and demand

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9  The storage of inventory in the network to stabilize the requirements for supply of

empty containers  Vessel slot allocation

and at a global level, one has to take care of:  Container fleet sizing

 Leasing – Owning composition  Reposition choices

 Network design  Routing optimization

and at a regional level, one has to take care of:  Depot sizing / location / interchanging

The aforementioned issues show that the movement of the empty containers can cause extensive problems for different parties in the supply chain and in different levels. Currently, there is an inefficient scheme of empty container repositioning at the port of Rotterdam. Matching the empty container demand of a shipper with the empty container supply of a consignee, rarely or even never happens in a systematic way.

1.3.2 Research objectives and questions

The research objective of this study is to contribute to the development of a model that reduces the empty container reposition cost by introducing more efficient matching schemes between consignees and shippers or by introducing depots, for short term empty containers storage.

The general problem is to manage the demand for moving a set of containers over a fixed period while minimizing transportation costs. More precisely it requires:

 Efficient movement empties from locations where empties are generated to positions where empties are needed (reduce the mileage involved in repositioning empty containers).

 The synchronization of container supply and demand (deliver – collect a container within time intervals).

Comparing Figure 1.2.a and Figure 1.2.b the possible benefits from using a depot (if existing in the network) can be observed. The red fat lines in Figure 1.2.a represent a direct return mode from consignee 𝑖 to the port. The red fat lines in Figure 1.2.b represent container reposition through short term depot storage at a depot say

𝑘.

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Figure 1.2.a

Figure 1.2.b

Therefore, the question that will be answered in this research can now be formed as:

 How can we find a better match between delivered and unloaded containers, and the requirement of empty containers, in order to reduce mileage covered by empty containers and therefore CO2 emissions?

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11 Moreover, the following four questions could be answered after answering the research questions of this thesis:

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By what model can we find a more efficient movement scheme for the empty containers travelling from customer destinations to empty depots, and to loading locations, in order to reduce the number of empty container movements at the port of Rotterdam?

 Which strategic steps are necessary in order to strengthen and optimize the inland terminal network?

 What is the current capacity of the inland terminal network?

 Will the costs for empty container repositioning be reduced if decision makers, such as shippers, freight forwarders and ports cooperate more closely?

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2 Literature review

Empty containers repositioning has been an important issue since the beginning of containerization. Since the carriage and the handling of the empty containers, at depots, is one of the most significant costs of the shipping companies, logistic managers cannot ignore the empty containers. Several research works have made progress on that field. In this chapter, past works on the empty container networks are reviewed.

In summary, some literature on optimization models dealing with the empty container problems are presented at the Tables 2.1.a, 2.1.b.and 2.1.c. Approaches [1], [2] and[3] face the empty containers reposition problem by merely developing optimization models without solving them. Approaches [4], [5] and [6] face the development of a methodology that allows solving the optimization models proposed for addressing the empty repositioning problem. A number of past works found, such as [7], [8] and [9], face a kind of inventory problem where the empty containers are the products to be stored. Papers [10]-[15] develop optimization models solving the shipping network design problems.

Ting et al. (2011) proposed three models on the strategic lever for locating away-from-port empty container depots; these models can be used to analyze the effectiveness of maintaining one or more storage yards away from the port. The first model is based upon the assumption that there is one firm which makes all container transports. This assumption has significant ramifications in the operation of any storage yards away from the port. If a storage yard is used as a drop off location more frequently than it is as an empty pickup location, then an inventory of empties will tend to grow at that location. Then, ultimately the growing inventory of empties will need to be repositioned to either the port for global reposition or another storage yard that requires more empties to satisfy its demand for exports. In general, repositioning by truck is more expensive than taking the container directly to the storage yard where it will be needed. This is due to the fact that there are no economies of scale in the local repositioning of empties when using trucks. The system objective of the first model is to locate one or more away-from-port empty-container storage yards so as to minimize the total distance (or travel time) involved in moving empty containers between consignees, yards, and shippers. They modeled and solved the problem as Vehicle Routing Problem with Time Windows having the following characteristics:

 all moves must be accomplished;

 a number of vehicles are available for the work assignments;

 for each vehicle a route with timed pickups and drop offs is determined;  vehicle route costs include both container moves and deadhead moves.

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Author Decision levels – Goal Focus - Assumptions Method

[1] Choong et al. (2002)

STRATEGIC: Vehicle selection (barge-inexpensive), MIN: cost of empty containers reposition

The effect of planning horizon (PL) length on empty container management for intermodal transportation networks. Increasing PLuse more inexpensive vehicles. Time extended optimization model. Integer programming [2] Wang and Wang (2007) TACTICAL: Route selection, MIN: cost of empty containers reposition

Reposition of intermodal empty containers of land-carriage. Satisfy need and supply of empty containers & ports’ store.

Integer programming

[3] Belmecheri et al. (2009)

TACTICAL: Route selection and vehicle loading, MIN: cost of empty containers reposition

Empty container reuse between regional consignees, shippers, ports.

Each container shipped is considered only in a local region. Costs between the local

companies and destinations mismatches are not considered.

Integer programming

[4] Chang et al. (2008)

TACTICAL: Route selection and vehicle loading, MIN: cost of empty containers reposition

Model the reuse of empties in a dynamic environment

analytically.

Street-turn and depot-direct methodologies considered. Integer programming, Branch and Bound technique [5] Li and Han (2009) TACTICAL: Route selection and vehicle loading, MIN: cost of empty containers reposition

Marine reposition of empty containers.

Demand and supply uncertainty.

Mixed-integer programming Branch and Bound [6] Yuanhui et al. (2009) TACTICAL: Route selection and vehicle load, MIN: cost of empty containers reposition

Empty container repositioning problem among multi-ports over a certain period of time

Hybrid genetic algorithm [7] Li et al. (2004) TACTICAL: Container management strategy,

MIN: cost of empty containers reposition

Empty containers allocation problem.

Nonstandard inventory problem with positive and negative demands. Markov decision processes with discrete time Table 2.1.a

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Authors Decision levels – Goal Focus – Assumptions Method

[8] Chien-Chang (2008)

TACTICAL:

Optimize the amount of containers required in a port

Empty container order quantity in the real word.

A single port. Fixed period order quantity inventory model [9] Song D.-P. (2007) TACTICAL: Minimize the sum of container leasing cost, inventory cost and reposition cost

Periodic-review shuttle service system. Optimize stationary policy of empty container reposition.

Random customer demands and finite reposition capacity.

Markov decision process [10] Crainic at al. (1993) TACTICAL: Solve the Networks design problem

Scheduling, routing and allocation problem. Dynamic deterministic formulation [11] Cheung and Chen (1998) TACTICAL: Solve the Networks design problem

Dynamic empty container allocation problem.

Owned and lease containers meet the total transportation demand. Stochastic quasi-gradient method and stochastic hybrid approximation procedure [12] Shintani et al. (2007) TACTICAL: Solve the Networks design problem

MIN: costly empty containers traffic.

No empty containers reposition.

Knapsack problem solved by genetic algorithm [13] Olivo et al. (2005) TACTICAL: Managing empty containers in a continental scale

Mathematical network model for the empty container

management problem with multiple transport modes. MIN cost flow problem.

Integer programming [14] Feng and Chang (2008) TACTICAL:

Safety empty container stock at port and transportation problem

Safety stock management and geographical regions Linear programming for the transportation problem Table 2.1.b

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Authors Decision levels – Goal Focus - Assumptions Method

[15] Di Francesco et al. (2009)

TACTICAL:

Shipping network design

Empty containers reposition problem.

Several uncertain parameters: future demands and supplies, vessel residual transportation capacities, et al. Time-extended multi-scenario network optimization model [16] Ting et al. (2011) STRATEGIC: Location allocation models,

MIN: cost of empty containers reposition

Model the greedy-like behavior of drayage companies when locating one or more away-from port empty storage yards. “User” optimal modeling.

Mixed integer linear programming [17] Maria Fonoberov a (2010) TACTICAL:

Find Optimal Flows in Dynamic Networks

Minimum Cost Flow problems on networks with demand-supply and capacity functions that depend on both time and flow.

Dynamic programming

Table 2.1.c

Maria Fonoberova (2010), in Algorithms for Finding Optimal Flows in Dynamic Networks, presented an approach for solving the classical optimal flow problems on networks as well as extensions and generalizations for cases with nonlinear cost functions on arcs and time-dependent and flow-time-dependent transactions on arcs of the network. They considered minimum cost flow problems on networks with demand-supply and capacity functions that depend on both time and flow. To solve the dynamic flow problems that they consider, they elaborated algorithms on the basis of a time-expanded network.

Following their notation, a dynamic network 𝑁 = (𝑉, 𝐸, 𝜏, 𝑑, 𝑢, 𝜑) consists of a directed graph 𝐺 = (𝑉, 𝐸) with set of vertices 𝑉, 𝑉 = 𝑛, and set of arcs 𝐸, 𝐸 = 𝑚, transit time function 𝜏: 𝐸 → 𝑅₊, demand-supply function 𝑑: 𝑉 x 𝑇 → 𝑅, capacity function 𝑢: 𝐸 x 𝑇 → 𝑅₊ and cost function 𝜑: 𝐸 x 𝑅+ x 𝑇 → 𝑅+. One considers the discrete1 time model, in which all

times are integral and bounded by horizon 𝑇. The time horizon represents the total time in which the flow can travel in the network, in a discrete way by the set Т = {0, 1, … , 𝑇} of considered time moments. Time is measured in discrete steps, so if one unit of flow leaves vertex 𝑧 at time 𝑡 on arc 𝑒 = (𝑧, 𝑣), one unit of flow arrives at vertex 𝑣 at time 𝑡 + 𝜏_𝑒 , where 𝜏𝑒 is the transit time on arc 𝑒 [17].

1_{The continuous variant of this flow model can be found in Fleisher (2000); Fleischer (2001a); Fleisher} and Skutella (2002).

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16 Their approach is based on the reduction of the dynamic problem to a corresponding static one. The advantage of such an approach is that it turns the problem of determining an optimal flow over time into a classical network flow problem. For more details on this subject the reader is been referred to [17].

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3 Mathematical formulation

The problem to be solved has been approached in this thesis by three different mathematical models.

The first simplistic model treats the problem as a bipartite matching between empty container supply and demand. Decisions to be made by the model are: the flows of empty containers between consignees, shippers, depots and port(s).

The second and the third models focus on a time extended network in order to be able to match current supplies with short term or even long term demands. The specific objective of the mathematical programming model to be established is to minimize total cost over the planning horizon. A planning horizon is basically a set of abstract adjacent periods to be used inside a mathematical program. The approach followed in this application is to run a sequence of mathematical programs each with a planning horizon interval of 𝑑 weeks. Once the first program is solved for week 𝑜𝑛𝑒, all decisions concerning this first week are considered to be final. The subsequent mathematical program then starts at week 𝑡𝑤𝑜. Decisions to be made by the models are: the flows of empties, and again, all empty containers supply and depots inventory concerning this second week are fixed. This process continues until the mathematical program covers the last weeks of the full time interval of the planning or is continues shifting the planning horizon. Notice, that instead of 𝑤 weeks, the planning interval could be measured in periods consisting of a day, year etc.

Rolling horizon models are a compromise between speed and accuracy. If the planning interval is longer, the solution might be better optimized. The corresponding mathematical program is however larger in size, and could take up a considerable amount of computational time. The length of the planning interval should certainly reflect an insensitivity of future first-period decisions to data outside the interval. This choice is application dependent. From the point of view of a thesis, it is very interesting to handle the time aspect through a rolling horizon. In practical applications, however, caution is needed: a short planning horizon may not be sufficient to take the relevant future into account.

In addition to the abovementioned benefits through by the use of a rolling planning horizon, the primary reason of its use is that rolling a planning horizon is a way to deal with uncertainty; the fact that for the time being, there is insufficient information on events in the more distant future time periods (e.g. new or altered demands and supplies).

The models in 3.4.1 and 3.4.2 are extensions of the model to be built in 3.2. In the former, alternative demand scenarios (e.g. derived from history) are introduced in order to simulate

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18 major deviations in the overall demand behavior, while the latter is an extension to a multi-commodity approach.

All models presented are based on the key assumption that the optimization of empty container transport takes place for all companies as a whole. More precisely, there is a one perspective optimization, rather than solely finding one company’s optimal strategy. In other words, a single firm which makes all container transportations is assumed; this firm also holds all the required data and company information.

3.1 Transshipment model

In this unsophisticated model, consignees are considered as excess nodes and shippers as deficit nodes. Depots and port(s) can appear in both capacities. Flows are allowed from consignees to shippers, consignees to depots (or port), and depots (or port) to shippers. Indices, parameters and variables designation follows:

Indices:

• Consignee 𝒊 where 𝑖 = 1, 2, 3, … , 𝐼; • Shipper 𝒋 where 𝑗 = 1, 2, 3, . . , 𝐽;

• Depot 𝒌 where 𝑘 = 1, 2, 3, … , 𝐾.𝑘 = 1 represents the port; Parameters:

• 𝒄_𝒊𝒋: the empty container transportation cost from consignee 𝑖 to shipper 𝑗; • 𝒄𝒊𝒌: the empty container transportation cost from consignee 𝑖 to depot 𝑘;

• 𝒄𝒌𝒋: the empty container transportation cost from depot 𝑘 to shipper 𝑗;

• 𝑺_𝒊 : the supply of empty containers at consignee 𝑖; • 𝑫_𝒋 : the demand for empty containers at shipper 𝑗; Variables:

• 𝒙_𝒊𝒋:the flow of empty containers from consignee 𝑖 to shipper 𝑗; • 𝒙_𝒊𝒌: the flow of empty containers from consignee 𝑖 to depot 𝑘; • 𝒙_𝒌𝒋: the flow of empty containers from depot 𝑘 to shipper 𝑗 at time 𝑡; The mathematical formulation of the transshipment model follows:

min 𝑐_𝑖𝑗𝑥_𝑖𝑗 𝐽 𝑗 =1 𝐼 𝑖=1 + 𝑐_𝑖𝑘𝑥_𝑖𝑘 + 𝑐_𝑘𝑗𝑥_𝑘𝑗 𝐽 𝑗 =1 𝐾 𝑘=1 𝐾 𝑘=1 𝐼 𝑖=1

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19 subject to: 𝑥_𝑖𝑗 𝐼 𝑖=1 + 𝑥_𝑘𝑗 𝐾 𝑘=1 = 𝐷_𝑗 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗 𝑥_𝑖𝑗 𝑗 𝑗 =1 + 𝑥_𝑖𝑘 𝐾 𝑘=1 = 𝑆_𝑖 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 𝑥_𝑘𝑗 𝐽 𝑗 =1 − 𝑥_𝑖𝑘 𝐼 𝑖=1 = 𝐷_𝑗 𝐽 𝑗 =1 − 𝑆_𝑖 𝐼 𝑖=1 𝐾 𝑘=1 𝑥_𝑖𝑗, 𝑥_𝑖𝑘, 𝑥_𝑘𝑗 ≥ 0 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖, 𝑗, 𝑘

By the first conditions (demand constraint), it is warranted that demand at every shipper will always be satisfied. The second constraint (supply constraint) preserves that all empty containers available for transport at any consignee will be moved to a shipper or depot or port at time. The third constraint assumes that depots or ports take care of net total demand (supplying this number of empties if positive, otherwise receiving this number of empties). Finally, the last conditions prevent the model from having negative flows.

The advantage of this model is its simplicity. However, this deterministic model has the following major weaknesses,

 It cannot cope with uncertainty and has only the ability of matching currently available supply with currently available demand.

 The model assumes that 𝑥_𝑖𝑘 and 𝑥_𝑘𝑗 can take place in one and the same period which is not so realistic.

 It does not incorporate any future periods for which demand and supplies can be already estimated.

A further explanation of the indices, parameters and variables as well as principal assumptions needed, follow in the next section where a more advanced model is presented.

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Figure 3.1 – Bipartite matching consignee supply at time 𝒕 − 𝟏 with shipper and port demand at time 𝒕

3.2 The time extended model

3.2.1 Description and assumptions

In this section the critical feature of the time aspect is introduced into the model. Instead of optimizing at every unit of time, the aforementioned rolling planning horizon is implemented. The following assumptions are necessary in order to specify the model:

1. One time unit movements–Each individual empty container movement can be accomplice within a single period of time horizon 𝑇;

2. No storage at Consignees– Containers will not be earlier shipped to consignees in order to meet their future period demand. Consignees will not store empty containers for later time periods. Unloaded containers must be carried to a shipper, a depot or port.

3. Supply and Demand Known – The number of loaded containers that arrive/depart at/from a location during a certain time period in the near future is known (i.e., the number of empty containers available/required is known).

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21 4. Containers Readily Re-usable – As soon as a shipment arrives at a customer site, the commodity is unloaded immediately, and the empty container becomes available. No repairs or discards of containers occur.

5. Single Commodity – One type of container is used.

6. Interchangeable Containers – When an empty container appears at a consignee, it can be transported to any shipper. This assumption is part of the one firm perspective, due to the “systemic optimal” approach studied.

7. No backorders are allowed.

8. Initial Depot Inventory – The number of empty containers left from the previous planning horizon at a depot is known. This “stock” is considered as the initial inventory of the depot at the beginning of the current planning horizon.

9. Invariable cost – Costs are independent of time periods. This assumption can be relaxed if necessary without sacrificing from model simplicity and computational burden.

10. Invariable capacity– Storage capacity of depots is independent of time periods. This assumption in not strictly necessary and can also be adjusted.

In assumption 2, empty containers are not allowed to be stored at the consignees. After unloading the commodity, containers are hauled away immediately to shippers and/or depots. To represent the situation where a shipper wants to keep empty containers for later use, a dummy flow can be introduced from the shipper node j at time t to the shipper node j at time

t+1. A dummy container flow has zero transportation cost, one transit time, a certain amount

of holding cost and a maximum flow capacity, representing the storage (capacity) limitation of the shipper’s warehouse.

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Figure 3.2 – Visualization of the rolling planning horizon idea - matching consignee’s supply at

time 𝒕 − 𝟏 with shipper’s or port’s demand at time 𝒕

3.2.2 Mathematical formulation

The following notation is essential in order to formulate the model:

Indices:

• Depot 𝒌 where 𝑘 = 1, 2, 3, … , 𝐾.𝑘 = 1 represents the port; • 𝒕:index for time periods, 𝑡 = 1, 2, 3, … , 𝑇;

• 𝑻: length of time horizon; Parameters:

• 𝒄_𝒊𝒋: the empty container transportation cost from consignee 𝑖 to shipper 𝑗; • 𝒄_𝒊𝒌: the empty container transportation cost from consignee 𝑖 to depot 𝑘; • 𝒄_𝒌𝒋: the empty container transportation cost from depot 𝑘 to shipper 𝑗; • 𝒔𝒋: the inventory (penalty)cost per container held at shipper 𝑗per time period;

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23 • 𝝈_𝒌: the operational cost of depot 𝑘per time period;

• 𝒆_𝒌: the entry cost at depot 𝑘 per unit flow; • 𝒑_𝒌: the pick-up cost from depot 𝑘 per unit flow;

• 𝑺_𝒊𝒕: the supply of empty containers at consignee 𝑖, at time 𝑡; • 𝑫_𝒋𝒕_{: the demand for empty containers at shipper 𝑗, at time 𝑡;}

The parameters related to empty containers supply and demand will be read from external data sources.

Variables:

• 𝒙_𝒊𝒋𝒕_{:the flow of empty containers from consignee 𝑖 to shipper 𝑗at time 𝑡;}

• 𝒙_𝒊𝒌𝒕 : the flow of empty containers from consignee 𝑖 to depot 𝑘 at time 𝑡; • 𝒙_𝒌𝒋𝒕 : the flow of empty containers from depot 𝑘 to shipper 𝑗 at time 𝑡;

• 𝒙_𝒋𝒋𝒕_{: storage flow form shipper 𝑗 to shipper 𝑗 at time t,represents the situation where a}

shipper wants to keep empties for later use in [𝑡 + 1, 𝑡 + 2, … , 𝑇];

• 𝒙_𝒌𝒌𝒕 : storage flow from depot 𝑘 to depot 𝑘 at time 𝑡,represents the situation where containers stay for more than one period in the same depot 𝑘;

The objective function of the constructed mathematical model consists of the summation over time of the following five clauses:

𝑐_𝑖𝑗𝑥_𝑖𝑗𝑡 𝐽 𝑗 =1 𝐼 𝑖=1 (3.1.a) (𝑐𝑖𝑘 + 𝑒𝑘)𝑥𝑖𝑘𝑡 𝐾 𝑘=1 𝐼 𝑖=1 (3.2.1.b) (𝑐_𝑘𝑗 + 𝑝_𝑘)𝑥_𝑘𝑗𝑡 𝐽 𝑗 =1 𝐾 𝑘=1 (3.2.1.c) (𝜎𝑘+ 𝑠𝑘𝑥𝑘𝑘𝑡 ) 𝐾 𝑘=1 (3.2.1.d) 𝑠_𝑗𝑥_𝑗𝑗𝑡 𝐽 𝑗 =1 (3.2.1.e)

Clauses 3.2.1.a, 3.2.1.b and 3.2.1.c, denote the overall cost for transporting empty containers from consignee 𝑖 to shipper𝑗, from consignee 𝑖 to depot 𝑘, and from depot 𝑘 to shipper 𝑗. Clause 3.2.1.d represents the total cost that may occur at a depot; which is the sum of the operational cost with the inventory cost of depot 𝑘. Finally, clause 3.2.1.e is the inventory

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24 cost of an empty container that stays at a shipper for more than one period; this is to represent the penalty cost for each flow 𝑥_𝑗𝑗𝑡_{that exceeds the due date (or time limit) in which the empty}

containers have to depart from the current shipper to the port.

The mathematical model can now be formulated as follows:

𝑚𝑖𝑛 𝑐𝑖𝑗𝑥𝑖𝑗𝑡 𝐽 𝑗 =1 𝐼 𝑖=1 + (𝑐𝑖𝑘 + 𝑒𝑘)𝑥𝑖𝑘𝑡 𝐾 𝑘=1 𝐼 𝑖=1 + (𝑐𝑘𝑗 + 𝑝𝑘)𝑥𝑘𝑗𝑡 𝐽 𝑗 =1 𝐾 𝑘=1 𝑇 𝑡=1 + (𝜎_𝑘+ 𝑠_𝑘𝑥_𝑘𝑘𝑡 ₎ 𝐾 𝑘=1 + 𝑠_𝑗𝑥_𝑗𝑗𝑡 𝐽 𝑗 =1 (3.2.1) Subject to: 𝑥_𝑖𝑗𝑡 𝐼 𝑖=1 + 𝑥_𝑘𝑗𝑡 𝐾 𝑘=1 + 𝑥_𝑗𝑗𝑡 − 𝑥_𝑗𝑗𝑡+1 = 𝐷_𝑗𝑡 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡, 𝑗 (3.2.2) 𝑥_𝑖𝑗𝑡 𝑗 𝑗 =1 + 𝑥_𝑖𝑘𝑡 𝐾 𝑘=1 = 𝑆_𝑖𝑡−1_{, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡, 𝑖} (3.2.3) 𝑥_𝑖𝑘𝑡 𝐼 𝑖=1 + 𝑥_𝑘𝑘𝑡 _{= 𝑥} 𝑘𝑗𝑡+1 𝐽 𝑗 =1 + 𝑥_𝑘𝑘𝑡+1_{𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡, 𝑘} (3.2.4) 𝑥_𝑖𝑗𝑡, 𝑥_𝑖𝑘𝑡 , 𝑥_𝑘𝑗𝑡 , 𝑥_𝑗𝑗𝑡, 𝑥_𝑘𝑘𝑡 ≥ 0 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖, 𝑗, 𝑘, 𝑡 (3.2.5)

Condition 3.2.2 warrantees that demand at every shipper will always be satisfied. Larger flow to shippers is allowed. This is done in order to model the case where a shipper wants to keep an empty container for later use. However, a total flow to a shipper 𝑗, less than 𝐷𝑗, is not

possible.

Condition 3.2.3 preserves that all empty containers available at a consignee at time 𝑡 − 1 will be transported to a shipper or depot or port at time 𝑡. It is not possible for an empty container to “stay” at a consignee.

Condition 3.2.4 assures that all containers coming into a deport or port 𝑘 at time 𝑡, will either be transported to a shipper at time 𝑡 + 1 or stay in deport or port 𝑘.

Finally, Conditions 3.2.5 prevent the model from having negative flows.

All models constructed take into account five cost types:

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25  Inventory at depots and shippers,

 Operational at depots,  Entry at depots and  Pick-up at depots.

Transport costs occur when: 𝑥_𝑖𝑗𝑡 > 0 or 𝑥_𝑖𝑘𝑡 > 0 or 𝑥_𝑘𝑗𝑡 > 0. Since it not optimal for a shipper to “order” for an empty container, store it in their warehouse and then ship it back to a depot or port, 𝑥_𝑗𝑘𝑡 movement is not allowed by the model. When an empty container arrives at a shipper it will always be full or stored in their warehouse for later use, occurring inventory costs 𝑥_𝑘𝑘𝑡 > 0. Inventory costs also occurs when 𝑥_𝑗𝑗𝑡 > 0. Depot operational costs occur when: 𝑥_𝑘𝑘𝑡 _{> 0. Entrance costs occur when: 𝑥}

𝑖𝑘𝑡 > 0. Pick-up costs occur when: 𝑥𝑘𝑗𝑡 > 0.

All costs are depicted in Tables 3.1 to 3.4. These represent an example situation with four consignees: 𝑖₁, 𝑖₂, 𝑖₃, 𝑖₄; three shippers: 𝑗₁, 𝑗₂, 𝑗₃; and two depots: 𝑘₁, 𝑘₂.

𝑖1 𝑖2 𝑖3 𝑖4 𝑗1 𝑗2 𝑗3 𝑘1 𝑘2

𝑖1 Value Value Value Value Value

𝑖₂ Value Value Value Value Value 𝑖₃ Value Value Value Value Value 𝑖4 Value Value Value Value Value

𝑗₁ 𝑗₂ 𝑗₃

𝑘₁ Value Value Value 𝑘₂ Value Value Value

Table 3.1. Transportation cost

Table 3.2. Inventory cost

𝑖1 𝑖2 𝑖3 𝑖4 𝑗1 𝑗2 𝑗3 𝑘1 𝑘2 𝑖₁ 𝑖₂ 𝑖3 𝑖4 𝑗₁ Value 𝑗₂ Value 𝑗3 Value 𝑘₁ Value 𝑘2 Value

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26

Table 3.3. Depot operational cost

𝑖₁ 𝑖₂ 𝑖₃ 𝑖₄ 𝑗₁ 𝑗₂ 𝑗₃ 𝑘₁ 𝑘₂ 𝑖₁ Value Value 𝑖2 Value Value 𝑖₃ Value Value 𝑖₄ Value Value 𝑗1 𝑗2 𝑗₃ 𝑘1 𝑘₂

Table 3.4. Entrance cost

𝑖₁ 𝑖₂ 𝑖₃ 𝑖₄ 𝑗₁ 𝑗₂ 𝑗₃ 𝑘₁ 𝑘₂ 𝑖₁ 𝑖2 𝑖₃ 𝑖₄ 𝑗1 𝑗₂ 𝑗₃

𝑘1 Value Value Value

𝑘₂ Value Value Value

Table 3.5. Pick-up cost

𝑖1 𝑖2 𝑖3 𝑖4 𝑗1 𝑗2 𝑗3 𝑘1 𝑘2

𝑖1 ∞ ∞ ∞ ∞ Transport Transport Transport Trans Entr Tran Entr

𝑗1 ∞ ∞ ∞ ∞ Inventory Inventory Inventory ∞ ∞

𝑘1 ∞ ∞ ∞ ∞ Tran Pick Tran Pick Tran Pick Inv Oper ∞

𝑘2 ∞ ∞ ∞ ∞ Tran Pick Tran Pick Tran Pick ∞ Inv Oper

Table 3.6. Cost aggregation

It seems that entry and pick-up costs could be skipped from the model. However, these costs also prevent solutions where:

𝑖₁ 𝑖₂ 𝑖₃ 𝑖₄ 𝑗₁ 𝑗₂ 𝑗₃ 𝑘₁ 𝑘₂ 𝑖₁ 𝑖₂ 𝑖3 𝑖₄ 𝑗1 𝑗₂ 𝑗3 𝑘₁ Value 𝑘₂ Value

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27 𝑥_𝑖𝑘𝑡 > 0 and 𝑥_𝑘𝑗𝑡 > 0

Mind that two consecutive transportation movements in one period are not so realistic; they formed a weak point of the model of Section 3.1. By the insertion of entry and pick-up costs a more accurate model has been built, since now the aggregated costs satisfy for each triplet 𝑖, 𝑗, 𝑘:

𝑐𝑖𝑗 < 𝑐𝑖𝑘 + 𝑒𝑘+ 𝑐𝑘𝑗 + 𝑝𝑘 .

This inequality must lead to an optimal solution with the property:

𝑥_𝑘𝑗𝑡 _{= 0 || 𝑥} 𝑖𝑘𝑡 = 0. in other words: 𝑥_𝑘𝑗𝑡 _{> 0 implies 𝑥} 𝑖𝑘𝑡 = 0 and 𝑥_𝑖𝑘𝑡 > 0 implies 𝑥_𝑘𝑗𝑡 = 0.

3.3 The advanced time extended model

3.3.1 Description

All models that have been presented until this point assume that all container movements are taking one period of time (Assumption 8, Section 3.2). Thus, these models cannot benefit from future demand knowledge, in a sense that they only take into consideration demand at time 𝑡 + 1, when being in time 𝑡. In this section, this imperfection of the model is addressed and remedied. A model, which is much closer to reality, will be the outcome of this section.

Consider a new node set 𝐻 containing nodes 𝑕 𝑖, 𝑡 , 𝑕 𝑗, 𝑡 and 𝑕 𝑘, 𝑡 . This node set can be defined as a function of the location 𝑙 and time 𝑡, and hence 𝑕 𝑙, 𝑡 , where 𝑙 ∊ 𝐿, 𝐿 = 𝐼 ∪ 𝐽 ∪ 𝐾 and 𝑡 = 1, 2, … , 𝑇. The idea now is to build a directed network with the node set 𝐻. Nodes 𝑕 𝑖, 𝑡 of 𝐻 can only have outgoing arcs; nodes 𝑕(𝑗, 𝑡) of 𝐻 can only have incoming arcs, while nodes 𝑕(𝑘, 𝑡) of 𝐻 can have both outgoing and incoming arcs. More precisely, consider 𝑕, 𝑕′ _{∊ 𝐻 and introduce the arc set 𝐹 as follows: 𝑕, 𝑕}′_{∊ 𝐹 if and only if}

i. 𝑕 = 𝑕 𝑖, 𝑡 , 𝑕′ = 𝑕 𝑗, 𝑡′ and 𝑡′− 𝑡 is equal to the transportation time between

location 𝑖 and location 𝑗 measured in periods,

ii. 𝑕 = 𝑕 𝑖, 𝑡 , 𝑕′ = 𝑕 𝑘, 𝑡′ and 𝑡′ − 𝑡 is equal to the transportation time between

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28

iii. 𝑕 = 𝑕 𝑘, 𝑡 , 𝑕′ = 𝑕 𝑗, 𝑡′ and 𝑡′ − 𝑡 is equal to the transportation time between

location 𝑘 and location 𝑗,

iv. 𝑕 = 𝑕 𝑘, 𝑡 , 𝑕′ = 𝑕 𝑘, 𝑡′ where 𝑡′− 𝑡 = 1 and v. 𝑕 = 𝑕 𝑗, 𝑡 , 𝑕′ = 𝑕 𝑗, 𝑡′ where 𝑡′− 𝑡 = 1.

Define also the subsets 𝒜 of case (𝑖), ℬ of case (𝑖𝑖), 𝒞 of case (𝑖𝑖𝑖), 𝒟 of case (𝑖𝑣) and 𝒠 of case (𝑣), such that:

𝐹 = 𝒜 ∪ ℬ ∪ 𝒞 ∪ 𝒟 ∪ 𝒠 The transportation time 𝜏 is more formally defined as:

𝜏 = 𝑐𝑒𝑖𝑙 𝑡′ _{− 𝑡 = 𝑡′ − 𝑡}

The transformation from the time extended model to the advanced time extended model is visualized in Figures 3.3 and 3.4.

In the time extended model of the Section 3.2, flows from time 𝑡 − 1 to time 𝑡 + 1 (or from𝑡 to 𝑡 + 2, to𝑡 + 3 etc.) were not possible. Satisfying a demand at time 𝑡 + 1 was only possible from a flow launching at time 𝑡. In Figure 3.3 in order to satisfy the demands 𝐷_𝑗𝑡+1₁ and 𝐷_𝑗𝑡+1₂ from period 𝑡 − 1, an (probably costly) intermediate stop would have been necessary. Though, in Figure 3.4 it is now possible to satisfy the demands 𝐷_𝑗𝑡+1₁ and 𝐷_𝑗𝑡+1₂ from 𝑆_𝑖𝑡−1₁ or 𝑆_𝑖𝑡−1₂ if the transportation time from 𝑖1 to 𝑗1 (𝑖2 to 𝑗2) takes two time periods.

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29

Figure 3.3

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30

Figure 3.5

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31

3.3.2 Mathematical formulation

The very first assumption of the time extended model, discussing that all empty container movements can be accomplished within a single period of time, is excluded from this model. Thus, this leads to a significantly more general and more realistic approach. Assumptions 2 – 10 of Section 3.2 remain unaltered.

The notations introduced in Section 3.3, need to be adapted in some points and supplementary notations need to be added, before presenting the mathematical model of the advanced time extended model. All indices, parameters and variables follow:

Indices:

• Depot 𝒌 where 𝑘 = 1, 2, 3, … , 𝐾, where 𝑘 = 1 represents the port; • Location 𝒍 where 𝑙 = 1, 2, 3, . . , 𝐿;

• 𝒕: index for time periods; • 𝑻: length of time horizon;

• 𝒉: node 𝑕(𝑙, 𝑡) ∊ 𝐻, as described in Section 3.3.1;

• 𝑺𝒆𝒕𝒔 𝓐, 𝓑, 𝓒, 𝓓, 𝓔, as described in (i) – (v) of Section 3.3.1. Parameters:

• 𝒄_𝒉𝒉′: is the empty container transportation cost from node𝑕 to node 𝑕′;

• 𝒔_𝒋: 𝑕 = 𝑕 𝑗, 𝑡 where 𝑗 ∊ 𝐽, is the inventory (penalty) cost per container held at shipper-node 𝑕 per time period;

• 𝒔_𝒌: 𝑕 = 𝑕 𝑘, 𝑡 where 𝑘 ∊ 𝐾, is the inventory cost per container stored at a depot-node 𝑕 per time period;

• 𝝈_𝒌: 𝑕 = 𝑕 𝑘, 𝑡 where 𝑘 ∊ 𝐾, is the operational cost of depot-node 𝑕 per time period;

• 𝒆_𝒌: 𝑕 = 𝑕 𝑘, 𝑡 where 𝑘 ∊ 𝐾, is the entry cost at depot-node𝑕 per unit flow; • 𝒑_𝒌: 𝑕 = 𝑕 𝑘, 𝑡 where 𝑘 ∊ 𝐾, is the pick-up cost from depot-node𝑕 per unit flow; • 𝑺_𝒉: = 𝑺_(𝒊,𝒕): is the supply of empty containers at consignee-node 𝑕, (i.e. consignee 𝑖,

at time 𝑡);

• 𝑫_𝒉: = 𝑫_(𝒋,𝒕): the demand for empty containers at shipper-node 𝑕 (i.e. shipper 𝑗, at time 𝑡);

Variable:

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32 The mathematical model can now be formulated as follows:

𝑚𝑖𝑛 𝑐_𝑕𝑕′𝑥_𝑕𝑕′ 𝑕𝑕′_∊𝐹 + 𝑒_𝑕′𝑥_𝑕𝑕′ 𝑕𝑕′ ∊ℬ + 𝑝_𝑕𝑥_𝑕𝑕′ 𝑕𝑕′ ∊𝒞 + (𝜎_𝑕+ 𝑠_𝑕𝑥_𝑕𝑕′) 𝑕𝑕′_∊𝒟 𝑡∊𝑇 + 𝑠_𝑕𝑥_𝑕𝑕′ 𝑕𝑕′_∊𝒠 (3.3.1) Subject to: 𝑥_𝑕𝑕′ 𝑕′_:(𝑕,𝑕′_{)∊𝒜∪ℬ} = 𝑆_𝑕 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑕 = (𝑖, 𝑡) (3.3.2) 𝑥_𝑕𝑕′ 𝑕 : 𝑕,𝑕′_{∊𝒜∪𝒞∪𝒠} − 𝑥_𝑕′_𝑕′′ 𝑕′′_{: 𝑕}′_,𝑕′′_∊𝒠 = 𝐷_𝑕′ , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑕′ = 𝑗, 𝑡 (3.3.3) 𝑥_𝑕𝑕′ 𝑕:(𝑕,𝑕′_)∊ℬ + 𝑥_𝑕𝑕′ 𝑕:(𝑕,𝑕′_)∊𝒟 = 𝑥_𝑕′_𝑕′′ 𝑕′′_:(𝑕′_,𝑕′′_)∊𝒞 + 𝑥_𝑕′_𝑕′′ 𝑕′′_:(𝑕′_,𝑕′′_)∊𝒟 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑕′ _{= (𝑘, 𝑡)} (3.3.4) 𝑥_𝑕𝑕′ ≥ 0 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑕, 𝑕′ ∊ 𝐹 (3.3.5)

The objective function 3.3.1contains for each period 𝑡 four clauses, analogous to the model in Section 3.2 and objective function 3.2.1. These are:

𝑐_𝑕𝑕′𝑥_𝑕𝑕′ 𝑕𝑕′_∊𝐹 (3.3.1.a) 𝑒_𝑕′𝑥_𝑕𝑕′ 𝑕𝑕′ ∊ℬ (3.3.1.b) 𝑝_𝑕𝑥_𝑕𝑕′ 𝑕𝑕′ ∊𝒞 (3.3.1.c) (𝜎_𝑕+ 𝑠_𝑕𝑥_𝑕𝑕′) 𝑕𝑕′_∊𝒟 (3.3.1.d) 𝑠_𝑕𝑥_𝑕𝑕′ 𝑕𝑕′_∊𝒠 (3.3.1.e)

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33 Clause 3.3.1.a denotes the overall cost for transporting empty containers from a location 𝑙 at time 𝑡, to a location𝑙′_{at time 𝑡′, or from 𝑕 to 𝑕′. Notice that}

𝑐_𝑕𝑕′𝑥_𝑕𝑕′ 𝑕𝑕′_∊𝐹 = 𝑐_𝑕𝑕′𝑥_𝑕𝑕′ 𝑕𝑕′_{∊𝒜∪ℬ∪𝒞∪𝒟∪𝒠} = 𝑐_𝑕𝑕′𝑥_𝑕𝑕′ 𝑕𝑕′_{∊𝒜∪ℬ∪𝒞} since 𝑐_𝑕𝑕′ = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 (𝑕, 𝑕′) ∊ 𝒟 ∪ 𝒠

meaning that no transportation cost is incurred when moving from

 depot 𝑘 at time 𝑡,to depot 𝑘 at time 𝑡′  shipper 𝑗 at time 𝑡, to shipper 𝑗 at time 𝑡′.

Clause 3.3.1.b reflects the entry cost at a depot or port, while clause 3.3.1.c the pick-up cost from a depot or port. Clause 3.3.1.d indicates the operational cost of a depot plus the inventory cost per container held at a depot. Finally, clause 3.3.1.e specifies the inventory cost per container held at a shipper.

Clauses 3.3.2 warrantees that all empty containers available at a consignee 𝑖 at time 𝑡 will be transported to a shipper 𝑗 or depot or port 𝑘 at time 𝑡′.

Clause 3.3.3 warrantees that the demand at every shipper will always be satisfied. Larger flows to a shipper are allowed.

Clause 3.3.4 warrantees that all empty containers coming into a deport or port 𝑘 at time 𝑡, will either be transported to a shipper at time 𝑡′ or stay in deport or port 𝑘.

Finally, constrains 3.3.5 warrantees that all flows are nonnegative.

3.4 Extension of the time extended model - The time extended

model with scenarios

The periodic demand for an empty container to be supplied by the consignees to shippers is not exactly known. Variations over the years have been observed. Nevertheless, when building a model with demand as a parameter, demand values for the weeks to come must be chosen. Such a set of demand values is referred to as a demand scenario. Exactly the same applies for the supply of loaded containers, which will eventually become empty container supply at consignees or port for the model. Instead of selecting a single supply-demand scenario, the occurrence of several scenarios has been evaluated in this research. The

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34 scenarios could for example represent a high demand scenario, a low demand scenario and an average demand scenario. This is in order to capture the overall demand behavior over the previous years as well as to obtain a more “robust” optimal solution.

Some extra notation needs to be added for building this model.

Index:

• Scenario 𝒔 where 𝑠 = 1, 2, … , 𝑆; Parameter:

• 𝑷𝒔: supply-demand scenario probability

𝑃_𝑠= 1

𝑆 𝑠=1

• 𝑫 : demand at 𝑕 taken as a weighted average, where: _𝒉 𝐷_𝑕

= 𝑃_𝑠

𝑆 𝑠

𝐷_𝑕𝑠

• 𝑺 : supply at 𝑕 weighted average, where: _𝒉 𝑆_𝑕

= 𝑃𝑠𝑆𝑕𝑠 𝑆 𝑠

Then, the linear programming formulation is identical to the model in Section 3.3, only changing clauses (3.3.2) and (3.3.3) to (3.3.2’) and (3.3.3’) respectively, is needed.

𝑥_𝑕𝑕′ 𝑕′_:(𝑕,𝑕′_{)∊𝒜∪ℬ} = 𝑆 ,_𝑕 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑕 = (𝑖, 𝑡) (3.3.2’) 𝑥_𝑕𝑕′ 𝑕 : 𝑕,𝑕′_{∊𝒜∪𝒞∪𝒠} − 𝑥_𝑕′_𝑕′′ 𝑕′′_{: 𝑕}′_,𝑕′′_∊𝒠 = 𝐷 ,_𝑕′ 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑕′ = 𝑗, 𝑡 (3.3.3’)

Here the minimization takes into account for indices 𝑠, at parameters 𝐷_𝑕𝑠_{and 𝑆}

𝑕𝑠, the different

supply-demand scenarios according to scenario probabilities 𝑃_𝑠. After each roll in the rolling planning horizon, new supply-demand scenario predictions as well as parameter probability 𝑃_𝑠 would be updated.

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4 Implementation

4.1 Case study

An example situation of a companyor a cooperation of the several shippers, which is acting as one united shipping line, has been taken into consideration. In Table 4.1 the cities-consignees, -shippers, -ports,-depot of the case study are given.

Consignees Apeldoorn Shippers Breda

Arnhem Den Haag

Assen Deventer

Den Bosch Haarlem Den Helder Nijmegen Dordrecht Zwolle Eindhoven

Emmen Ports Amsterdam Enschede Rotterdam Groningen

Leeuwarden Depot Utrecht Maastricht

Tilburg Venlo Vlissingen

Table 4.1

The length of the implemented time horizon is 1 year, which is considerably long in order to validate the models reliability. The length of the planning interval considered to be 1 week. In order to test the model fictitious data concerning all cost types and supply or demand of empty containers have been created as well as fictitious costs. Conform the model of 3.3, transportation, inventory, operation, entry and pick-up costs are to be provided. Transportation costs are assumed to be proportional to the mileage that empty containers travel and taken as: mileage ∗ 1.1. The mileage table of the city distances is provided in Appendix A. The time required by a container to travels from one location to another is depended on the distance between the two locations. This “time distances” are represented in days (i.e., period of time) and they are given in Appendix B. The inventory costs at depot and shipper-locations follow in Table 4.2, and the entry and pick-up costs in Table 4.3. Each day, new predictions concerning future supply or demand for empties become available.

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36 Location Inventory cost

Breda 20 Den Haag 20 Deventer 20 Haarlem 20 Nijmegen 20 Zwolle 20 Amsterdam 1 Rotterdam 1 Utrecht 5 Table 4.2

Location Entry cost Pick-up cost Amsterdam 10 20 Rotterdam 10 20 Utrecht 10 5 Table 4.3

4.2 Implementation in AIMMS

4.2.1 Initialization

In this section the mathematical model proposed in 3.3 is implemented. The implementation takes place in AIMMS software through the case study of 4.1.

Each day the model provides a new solution-scheme concerning the optimal movement of empty containers, from consignees or depot or ports to shippers or depot or ports, following the planning principals of a rolling horizon, as discussed in Chapter 3.

This method will here repeatedly compute cost-optimal empty container movements for a planning interval of one week. However, only decisions concerning the first day of the week are considered to be final and implemented. Then the mathematical model takes the outcomes of the aforementioned implementation as well as other updated inputs and again runs for next planning interval. This process continues until the end of the total planning horizon. In real life implementation, the process never stops. It continues using the next updates of the supply and demand of empty containers.

After each day implementation, the outcomes of the model are: the optimal empty container transportation scheme for that day as well as the induced cost of the solution that is implemented. This information on the current day’s implementation as well as new information about future container arrivals at each location is also provided and processed by the model.

To better visualize the outcomes of the model, the user interface constructed in AIMMS is presented. In the first page, the contents page (see Figure 4.1), the user can enter in four sub-pages of the model.

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Figure 4.1. Contents

Container Data page in Figure 4.2, includes an overview of the starting knowledge about the supply and demand from empty containers at table “OrigSupplyOrDem”. Supply or Demand of the current periods within the planning interval is shown on “SupplyOrDemand” table. When the one day solution of the model is implemented, the program gives the newly expected arrivals and the aggregation of all arrivals in the tables “NewArrivals” and “FutureArrivals” respectively.

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Figure 4.2. Container Data

In the New Weekly Schedule page, Figure 4.3, the optimal container flows based on current day predictions are presented. Notice that only the first day of the solution is here implemented and their decisions are considered to be definite.

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Figure 4.3. New Weekly Schedule

In rare cases the system could be notified too late about a demand for empty containers at some shipper and it is infeasible to supply that demand due to time constraints (distance of cities too big to cover in the given period of time).The model then drops that demand from the “SupplyOrDemand” data, but in the Cancelations page it keeps a record of demand that is dropped (in the table “CancelationsAtShipper”, see Figure 4.4). Similarly, the “TemporaryCancelationsAtConsignee” table, points to available empty container supplies that cannot be carried to a shipper within the current planning horizon. However, these supplies are not skipped from the model; they simply will be supplied to a location at a later stage of the optimization.

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Figure 4.4.Cancelations

In the Today Actual Implementation page, see Figure 4.5, the current day’s schedule to be implemented is presented in table “EmptiesAsSentToday” as well as with its implementation cost.

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4.2.2 Running the model

In this section the Container Data, Today Actual Implementation and Cancelations pages are presented after the model has run for a number of periods.

In Figure 4.6, one can observe the current period being period-012. Variations in real time supply or demand from the initially expected values can be observed by comparing tables “OrigSupplyOrDemand” and “SupplyOrDemand”. For example, the demand of the Den Haag shipper at period-013 was initially expected to be zero, but at the time of period-012 they actually need a supply of 8 empty containers. Although Assen’s supply at period-014 was expected to be 10 empty containers, it turns out to be 17 when in period-012.

Figure 4.6. Container Data on period-012

In the Cancelation page, in Figure 4.7, one can notice that supplies at the two last days of the current planning interval cannot be moved to location within one or two periods of time. Hence, these supplies are not taken into account for the time being and will be taken into consideration in the next runs. On the other hand, cancelations concerning shippers demand have not happened at this time period.

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Figure 4.7. Cancelations on period-012

In the Today Actual Implementation, in Figure 4.8, the most essential information is provided. The decision maker consults this page in order to make the finite daily assessments and order for the optimal movements. Some of the optimal scheme movements would be moving one empty container from Den Helder to the port of Amsterdam and supplying Breda with 4 empty containers coming from Tilburg. Moreover, information about the today schedule cost and the total (aggregated) cost from period-001 to current period-012 are provided by the program.

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5 Conclusions

5.1 Conclusions

In this chapter, the major and minor conclusions of the thesis are drawn.

The flexibility of the model is one of the key strengths of the model presented. One can easily add or make any changes to city locations, to the size of the planning interval, so to include more information to each day’s optimization. For instance, if distances between cities are too big to cover within the current size of planning horizon, the size of the planning horizon can easily be increased to the desirable value. The model can be characterized as an easily developed further model.

Other versions of the model can be tested easily as a “random but frozen” data set is constructed; thus it is easy to compare and contrast any refinements or expansions added to the model with the previous model versions.

The model reduces the mileage covered by empty containers, also reducing the emissions as caused by the empty containers repositioning, thus providing environmental benefits for all stakeholders. This is done mainly by the efficient movement of the empties from locations where empties are generated to positions where empties are needed and by the synchronization of container supply and demand.

All these benefits will definitely occur if shippers share data on volumes, demand and costs. Full cooperation of shipping companies is required, including the ability to make the transport demands transparent. A more transparent transport information system has to be in place and most importantly a methodology for gain sharing has to be agreed on. It also has to be implemented in a way that will be robust, scalable and future proof.

5.2 Future research

Until now, the one container type assumption has been followed. This assumption could be dropped and a more general model would be obtained; considering multiple container types and next to the linear transportation costs also considering fixed charge costs. As an illustration we treat below the incorporation of an additional fixed charge cost parameter 𝑓_𝑖𝑗, to be incurred, when in a period (one or more) container movement(s) is (are) to take place between the locations of a consignee𝑖 and a shipper 𝑗.In that case the following model extensions are to be added.

• A container type index 𝒎 where 𝑚 = 1, 2, … , 𝑀 denotes the different container transportations categories, with associated variables 𝑥_𝑕𝑕𝑚′;

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45 • 0-1 variables 𝐲_𝐡,𝒉′ indicating (by value 1) that: there is a transportation of

container(s) from location 𝑖 in period 𝑡 of 𝑕, to location 𝑗 in period 𝑡′_{of 𝑕}′_{, against}

the fixed charge cost 𝑓_𝑕𝑕′ (or 𝑓_𝑖𝑗𝑡𝑡′_).

The model of Section 3.3 could then cope with these different container types, by introducing next to parameters 𝑆_𝑕𝑚, 𝐷_𝑕𝑚 (and associated constraint sets (3.3.2) – (3.3.5) extended per individual m) the binary variables 𝑦 into the objective function (3.3.1);

𝑐_𝑕𝑕𝑚′ 𝑥_𝑕𝑕𝑚′ 𝑕𝑕′_∊𝐹 𝑀 𝑚=1 + 𝑓_𝑕𝑕′𝑦_𝑕𝑕′ 𝑕,𝑕′_∊𝒜

while adding these new constrains:

𝑥_𝑕𝑕𝑚′ 𝑀 𝑚 =1

≤ 𝑏𝑖𝑔𝑀𝑦_𝑕𝑕′ , for all 𝑕𝑕′ ∊ 𝒜

to the constraint set.

Moreover, after Section 3.4.2.1, an expansion of the model to multiple scenarios, together with multimodal transportation and different kind of set-up costs could further follow.

As another kind of extension the integration with shipments of full containers could be possible. Information about available supplies of empty containers could at least be partly given by the outcomes of the full container shipment.

Collecting and running the model with real data, could more convincingly demonstrate unexpectedly high gains for the shipping companies. A model for sharing the gains during a pilot in cooperation with several of the project partners has to be built. This will enhance a widely shared vision on how to divide the benefits.

In “Toegepast –Natuurwetenschappelijk onderzoek” (TNO), the GreenBox project has not yet started. As a further outcome of this thesis we should highlight our personal belief that the Greenbox initiative should be (re-)started in order to test the developed models with real data and their ability of achieve significant gains.