Vessel scheduling in liner shipping: Modeling transport of perishable assets

International Journal of Production Economics 184 (2017) 141–156

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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

Vessel scheduling in liner shipping: Modeling transport of perishable assets a,⁎

Maxim A. Dulebenets , Eren Erman Ozguven

MARK

b

a Department of Civil & Environmental Engineering, Florida A & M University-Florida State University, 2525 Pottsdamer Street, Building A, Suite A124, Tallahassee, FL 32310-6046, USA b Department of Civil & Environmental Engineering, Florida A & M University-Florida State University, 2525 Pottsdamer Street, Building B, Suite B313, Tallahassee, FL 32310-6046, USA

A R T I C L E I N F O

A BS T RAC T

Keywords: Marine transportation Vessel scheduling Bunker consumption Perishable assets Asset decay Route service cost

International seaborne containerized trade significantly increased over the last years. Some assets transported in a containerized form by vessels are perishable in nature. Perishable assets deteriorate due to certain operational and environmental factors. This paper proposes a novel mixed integer non-linear mathematical model for the vessel scheduling problem in a liner shipping route with perishable assets, which explicitly captures decay of perishable assets on board the vessels. The objective aims to minimize the total route service cost, including the asset decay cost. The original non-linear mathematical model is linearized using a set of piecewise linear secant approximations. CPLEX is used to solve the linearized mathematical model. Numerical experiments are conducted for the French Asia Line 1 route, served by CMA CGM liner shipping company, to evaluate performance of the suggested solution approach and reveal some important managerial insights. Results demonstrate that the developed mathematical model can serve as an effective planning tool for liner shipping companies in designing efficient vessel schedules and reducing decay of perishable assets on board the vessels.

1. Introduction

peeled frozen shrimp, frozen tilapia fillet, and fresh Atlantic farm raised salmon, which are imported to the United States primarily from India, China, and Chile respectively (NOAA, 2016). Perishable assets are typically transported in refrigerated containers (a.k.a., “reefers”). Reefers allow maintaining a certain temperature, which reduces microbiological, physiological, and physical changes in the perishable asset (Haass et al., 2015). An external power supply is required to operate reefer containers on the vessel (Marine Insight, 2016). Reefers do not completely stop ripening of perishable assets, but allow slowing it down. Even slight deviation in the temperature within the refrigerated container may significantly speed up the ripening of a perishable asset (Rong et al., 2011; Haass et al., 2015). For example, change in the temperature inside the container from 15 °C to 20 °C will increase the daily ripening rate of bananas by 73.8% (Haass et al., 2015). Approximately 2 million reefers have been in use by 2011, which comprises around 5% of the global container capacity (Rodrigue, 2013). Along with refrigerated containers some freight carriers use insulated shipping containers (a.k.a., “porthole containers”). Similar to reefers, porthole containers provide insulation, which allows maintaining a certain temperature inside (Sustainablog, 2016). However, porthole containers do not have an integral refrigeration unit. The porthole containers are typically connected to the cooling plant of a vessel. The

Maritime transportation plays an imperative role for the global trade. According to the statistical data provided by the United Nations Conference on Trade and Development (UNCTAD, 2015), the overall international seaborne trade reached 9.8 billion tons in 2014 with a significant increase of containerized (5.6% in tonnage), dry (2.4% in tonnage), and major bulk cargo (6.5% in tonnage) from 2013. A similar growth is expected to continue. The majority of high value cargo and general consumption goods are shipped in a containerized form. General consumption goods also include perishable assets (e.g., fish, meat, shellfish, agricultural products, etc.), which are sensitive to the following operational and environmental factors: transportation time, temperature, humidity, barometric pressure, and air composition (Rong et al., 2011; Wang and Li, 2012; Grunow and Piramuthu, 2013; Aung and Chang, 2014; Haass et al., 2015). Similar to the other types of cargo, the amount of perishable assets transported by vessels have been increasing. For example, over the last five years the total volumes of perishable seafood products imported to the United States increased by 7.6% and reached 2.7 million tons in 2015, whereas the value of those products increased by 13.2% and reached $19.2 billion (NOAA, 2016). The top three perishable seafood products include

⁎

Corresponding author. E-mail addresses: [email protected] (M.A. Dulebenets), [email protected] (E.E. Ozguven).

http://dx.doi.org/10.1016/j.ijpe.2016.11.011 Received 5 July 2016; Received in revised form 17 November 2016; Accepted 19 November 2016 Available online 23 November 2016 0925-5273/ © 2016 Elsevier B.V. All rights reserved.

International Journal of Production Economics 184 (2017) 141–156

M.A. Dulebenets, E.E. Ozguven

2.1. Vessel scheduling problem

cool air is pumped via the bottom of the container, while the warm air is removed from the top (Sustainablog, 2016). A perishable asset should be delivered to the customer before the end of its “shelf life”. A term “shelf life” denotes the number of days left for a given perishable asset to be of an acceptable quality for the customer (Amorim et al., 2013; Jedermann et al., 2014). Some of freight carriers use advanced IT technologies for measuring the quality of assets inside the containers. One of the most widely used technologies for traceability of perishable assets is Radio-Frequency Identification – RFID (Grunow and Piramuthu, 2013; Aung and Chang, 2014; Haass et al., 2015). RFID technology allows keeping track of the perishable asset quality throughout the transportation process and estimating the remaining shelf life of the asset. Utilization of alternative types of containers (e.g., reefers, porthole containers) and advanced IT technologies (e.g., RFID) allows reducing, but does not completely eliminate waste of perishable assets. Rodrigue (2013) underlines that approximately 25% of perishable assets are wasted every year due to temperature variations and other factors that may facilitate asset deterioration. Considering the increasing volumes of containerized trade and perishable assets, carried by vessels, liner shipping companies have to directly account for the decay of perishable assets in design of vessel schedules and make necessary alterations in vessel schedules to reduce waste of perishable assets. This paper proposes a novel mixed integer non-linear mathematical model for the vessel scheduling problem, which accounts for decay of perishable assets during the transportation process. The model’s objective aims to minimize the total route service cost. The original non-linear model is linearized using a set of piecewise linear secant approximations and solved efficiently using CPLEX. Numerical experiments are conducted for the French Asia Line 1 route, served by CMA CGM liner shipping company, to evaluate performance of the suggested solution approach, showcase applicability of the proposed methodology, and reveal some important managerial insights. The rest of the paper is organized as follows. The next section presents an up-to-date literature review with focus on the vessel scheduling problem and approaches for modeling decay of perishable assets in liner shipping, while the third section provides the problem description. The fourth section presents the mathematical model for a vessel scheduling problem in a liner shipping route with perishable assets, while the fifth section describes the solution approach. The sixth section presents numerical experiments that were performed in this study. The last section provides conclusions and future research extensions.

Fagerholt (2001) presented a mathematical formulation for the vessel scheduling problem with soft time windows (TWs), aiming to minimize the total transportation and inconvenience costs. The model was solved using a set partitioning based algorithm. Computational examples demonstrated that the algorithmic performance was significantly influenced with the problem size. Dulebenets (2015a) developed a metaheuristic for the vessel scheduling problem, minimizing the total route service cost. Numerical experiments indicated that the suggested algorithm provided solutions with more accurate objective function values as compared to the static secant approximation for the majority of problem instances and required significantly lower computational time for all the considered problem instances. Wang et al. (2015) proposed a novel methodology for estimating the perceived value of transit time of containers based on minimization of the sum of fuel cost and time-associated costs of the containers adopted by the liner shipping company. Computational experiments were conducted to showcase how the suggested methodology could be used in designing the optimal transit times between ports of call, deciding on change in sailing speed, and predicting the market share of less polluting fuels. A number of studies captured uncertainty in liner shipping and marine container terminal operations. Chuang et al. (2010) addressed the containership routing problem, considering sailing and port time uncertainty. The objective maximized the total profit. A fuzzy Evolutionary Algorithm was developed to solve the problem. A number of computational experiments were presented to showcase effectiveness of the suggested methodology. Qi and Song (2012) applied simulation-based stochastic approximation methods to solve the vessel scheduling problem with uncertainty in port times, minimizing the total expected bunker consumption cost and penalties due to vessel delays. Numerical experiments showed that the bunker consumption was substantially affected with uncertainty in port times. Wang and Meng (2012a) formulated the vessel scheduling problem with uncertainty in sailing and port times, aiming to minimize the total transportation cost. The model captured the cargo transit time requirements. A cutting-plane based solution algorithm was developed to solve the problem. Results from computational experiments indicated that the number of required vessels could increase for a given liner shipping route due to sailing and port time uncertainty. Wang and Meng (2012c) focused on the vessel scheduling problem, capturing uncertainty in port waiting and container handling times. The objective aimed to minimize the total transportation cost. The problem was solved using the sample average approximation method. Numerical experiments demonstrated that adding vessels for service of a given liner shipping route could improve the vessel schedule robustness. Song et al. (2015) developed a Genetic Algorithm to solve a stochastic multi-objective vessel scheduling problem with uncertain port times. The model minimized three objectives: 1) the annual total vessel operational costs; 2) the average schedule unreliability; 3) the annual total carbon emissions. Results from computational experiments indicated that the least emissions could be achieved either with the least annual total vessel operational costs or with the least average schedule unreliability. Several studies presented different collaborative mechanisms between liner shipping companies and marine container terminal operators. Wang et al. (2014) studied the vessel scheduling problem in a liner shipping route, where marine container terminal operators were able to offer multiple service TWs to the liner shipping company. The objective minimized the total transportation cost. An iterative optimization algorithm was developed to solve the problem. It was found that duration of a TW could affect the total transportation cost. Furthermore, high value goods required shorter transit times. Alhrabi et al. (2015) modeled a similar collaborative agreement. Results from numerical experiments indicated that availability of multiple TWs could affect the total transportation cost, vessel sailing speed selection,

2. Literature review The problem of vessel scheduling in liner shipping receives a constant attention from the research community. For a detailed overview of the liner shipping literature at the strategic, tactical, and operational levels this paper refers to Meng et al. (2014). The long-term (or strategic) decisions focus on the vessel fleet size and mix, alliance strategy, and network design, while the medium-term (or tactical) decisions include the service frequency determination, fleet deployment, sailing speed optimization, and construction of the vessel schedule. At the operational level the liner shipping company has to decide on cargo booking, cargo routing, vessel rescheduling, and potential reject of cargo. This study focuses on a tactical level vessel scheduling problem, which aims to determine the vessel sailing speeds at voyage legs of the liner shipping route, arrival times at ports of call of the given port rotation, vessel handling and departure times (Meng et al., 2014). The literature review presented herein will concentrate on two aspects: 1) published to date studies on vessel scheduling in liner shipping; and 2) approaches/considerations for modeling perishability of assets, carried by vessels. Overview of the relevant studies is presented next. 142

International Journal of Production Economics 184 (2017) 141–156

M.A. Dulebenets, E.E. Ozguven

transit time may cause customers switching to the service from another liner shipping company). Transit time constraints can be also applied in the context of perishable assets to make sure that the asset will be delivered to the destination port within a certain time period (e.g., before the end of its shelf life). Wang et al. (2014) and Wang et al. (2015) introduced an additional transit time/inventory cost component to penalize increasing transit time of cargo between consecutive ports. The above mentioned approaches still do not explicitly model behavior of perishable assets on board the vessels.

and number of vessels to be deployed at the given liner shipping route. Dulebenets (2015b) presented a novel collaborative agreement, according to which multiple handling rates were offered to the liner shipping company at each port of call. The objective minimized the total route service cost. The original non-linear model was linearized and solved using CPLEX. Computational examples showed that the proposed collaborative agreement could yield 14.4% savings in terms of the total route service cost for the liner shipping company. The last group of the collected studies modeled the effects of potential disruptions in vessel scheduling. Brouer et al. (2013) formulated a vessel schedule recovery problem, aiming to minimize the total transportation cost. Several types of disruptions were considered, including vessel delays due to weather conditions, port closure, berth prioritization, and port congestion. A number of countermeasures were proposed in the paper. CPLEX was used to solve the developed problem instances. Results from numerical experiments showed that the proposed methodology reduced the total cost by 58%. Li et al. (2015) focused on the real-time vessel schedule recovery problem and considered two types of uncertainty: 1) Regular uncertainties – recurring probabilistic activities; and 2) Disruptive events – occasional or one-off events. The objective minimized the total expected bunker cost and delay penalty. A dynamic programming approach was developed to solve the problem. Computational experiments indicated that skipping the disruption port was more efficient option for mitigating the impact of disruption, happened at earlier time, under terminal operations without the earliest handling time constraints.

2.3. Contribution Taking into consideration substantial volumes of perishable assets, transported by vessels, and significant amount of waste due to asset decay, this paper proposes a novel mixed integer non-linear mathematical model for the vessel scheduling problem in a liner shipping route with perishable assets. The new mathematical program explicitly models decay of perishable assets and accounts for the associated costs. The developed model will allow liner shipping companies to design efficient vessel schedules and in the meantime reduce decay of perishable assets on board the vessels. 3. Problem description In this study we model a typical liner shipping route, which is composed of I = {1, …, n} ports of call. The port rotation, representing the sequence of ports to be visited by vessels, is assumed to be known. The sequence of visited ports is generally determined by the liner shipping company at the strategic level (Meng et al., 2014). Each port of call is visited once; however the methodology, proposed in this study, can be also applied for liner shipping routes, where a given port of call is visited several times. In the latter scenario (i.e., when a given port of call is visited several times), an additional node will be added to the graph, which represents the port rotation, to account for any additional visit to the same port. For example, the liner shipping route, presented in Fig. 1A, includes a total of 4 ports. Ports 1 and 2 are visited twice; hence, two additional nodes 1′ and 2′ are introduced to the graph, and the total number of ports to be visited will be I =4 + 2 = 6 (see Fig. 1B). A vessel sails between two subsequent ports i and i + 1 along voyage leg i . The liner shipping company is able to negotiate a specific arrival

2.2. Literature summary A summary for each one of the collected studies is presented in Table 1, including the following information: 1) author(s); 2) year; 3) sailing speed modeling; 4) port time modeling; 5) objective; 6) solution approach, and 7) asset perishability considerations (if any). The overview of the liner shipping literature indicates that none of the published to date vessel scheduling papers explicitly modeled perishability of assets, transported by vessels. To the authors’ knowledge, there are a few studies that use certain methodologies, which could be extended towards transportation of perishable assets. Wang and Meng (2012a) introduced the transit time constraints to maintain a specific service level of customers (i.e., large Table 1 Overview of the vessel scheduling literature in liner shipping. a/a

Author (s)

Year

Sailing speed

Port time

Objective

Solution approach

Perishability considerations

1 2 3 4

Fagerholt Chuang et al. Qi and Song Wang and Meng

2001 2010 2012 2012a

V U V U

F U U U

Minimize the total cost Maximize the total profit Minimize the total cost Minimize the total cost

N/A N/A N/A Transit time requirement

5

Wang and Meng

2012c

V

U

Minimize the total cost

6 7

Brouer et al. Wang et al.

2013 2014

V V

V F

Minimize the total cost Minimize the total cost

8

Alhrabi et al.

2015

V

F

Minimize the total cost

9 10 11 12

Dulebenets Dulebenets Li et al. Song et al.

2015a 2015b 2015 2015

V V V V

V V V U

13

Wang et al.

2015

V

F

Minimize the total cost Minimize the total cost Minimize the total cost Minimize the annual vessel operational cost; Minimize the schedule unreliability; Minimize the total carbon emissions Minimize the total cost

Heuristic Heuristic Stochastic approximation A cutting-plane algorithm Sample average approximation CPLEX Iterative optimization algorithm Iterative optimization algorithm Heuristic CPLEX Dynamic programming Heuristic

Transit time cost

14

This paper

–

V

V

Minimize the total cost

Iterative optimization algorithm CPLEX

Notes: F – fixed; V – variable; U – uncertain; N/A – not applicable.

143

N/A N/A Inventory cost N/A N/A N/A N/A N/A

Exponential asset decay

International Journal of Production Economics 184 (2017) 141–156

M.A. Dulebenets, E.E. Ozguven

v* – design vessel sailing speed (knots); α , γ – bunker consumption function coefficients. Technically, in order to determine the values of bunker consumption coefficients α and γ the regression analysis must be conducted based on the data collected from vessels, which provide service of the given liner shipping route (Du et al., 2011; Wang and Meng, 2012b; Yao et al., 2012). This study will use the most common values of the bunker consumption coefficients, revealed in the literature (Wang and Meng, 2012b; Psaraftis and Kontovas, 2013): α = 3 and γ = 0.012 . Once the liner shipping company makes a decision on a vessel sailing speed between consecutive ports of call, it is assumed to be constant throughout the voyage between those consecutive ports. The factors that may potentially cause changes in the vessel speed throughout the voyage (e.g., weather, height of waves, speed of wind, etc.) are not modeled. The bunker consumption by auxiliary engines does not fluctuate significantly throughout the voyage and is assumed to be included in the weekly vessel operational cost. The bunker consumption f (vi ) can be computed per nautical mile at voyage leg i using the following equation:

Fig. 1. Schematic representation of a liner shipping route.

time window – TW [twis – start of TW at port i , twie – end of TW at port i ] with the marine container terminal operator, during which a vessel should arrive at the given port of call. Duration of a TW may vary from one port to the other, but generally does not exceed three days (OOCL, 2016). The service of a vessel is assumed to start upon its arrival. If a vessel is scheduled to arrive at port i + 1 prior to the start of TW, it will be waiting at a dedicated area at port i upon the service completion. Details regarding estimation of the vessel waiting time at ports of call are presented in Section 3.6 of the paper. The liner shipping company is assumed to incur a monetary penalty in case, if a vessel arrives after the end of TW (Dulebenets, 2015b). The container demand (measured in TEUs) at each port of call is assumed to be known (Meng et al., 2014).

⎛ t ⎞1 l 1 γ (vi )α−1 f (vi )=q (vi ) ⎜ i ⎟ =γ (vi )α i = ⎝ 24 ⎠ li 24vi li 24

(2)

where:

li – length of voyage leg i , which connects ports i and i + 1 (nmi); ti – sailing time between ports i and i + 1 (h). Note that both q (v ) are f (vi ) are strictly convex non-decreasing differentiable functions (Wang et al., 2013).

3.1. Vessel service at ports It is assumed that the liner shipping company has a contractual agreement with each marine container terminal operator, serving a given port of the port rotation, according to which the marine container terminal operator is able to offer a set of handling rates Ji={1, …, mi}∀ i ∈ I to the liner shipping company. Each handling rate has a corresponding handling productivity dij ∀ i ∈ I , j∈Ji , measured in TEUs per hour. The vessel handling time pij ∀ i ∈ I , j∈Ji (hours) is calculated based on the container demand and the handling productivity requested. If a handling rate with a higher productivity is requested, the port handling time for a given vessel decreases, but port handling charges, imposed to the liner shipping company, increase. Note that decrease in handling time at a given port may result in bunker consumption savings, since a vessel will be able to sail at a lower speed to the next port of the port rotation.

3.3. Inventory cost The container inventory cost is another critical route service cost component, which should be considered by the liner shipping company in construction of vessel schedules. The total container inventory cost can be computed based on the total transit time of containers, transported at voyage legs of the given liner shipping route, using the following equation (Wang et al., 2014):

IC =μ ∑ ti NCi

(3)

i∈I

where:

IC – total inventory cost (USD); μ – unit inventory cost (USD per TEU per h); NCi – number of containers transported at voyage leg i (TEUs).

3.2. Bunker consumption A homogeneous vessel fleet, which is composed of vessels with the same/similar technical characteristics, is deployed for service of the given liner shipping route. The latter practice has been commonly implemented not only in the published to date academic literature on vessel scheduling but also in practice (Wang and Meng, 2012a, 2012b, 2012c; Wang et al., 2013, 2014; Dulebenets, 2015b). The bunker consumption is assumed to be proportional to the vessel sailing speed and can be computed from the following equation (Du et al., 2011; Wang and Meng, 2012b):

⎛ v ⎞α q (v )=q (v*) ⎜ ⎟ =γ (v )α ⎝ v* ⎠

∀ i∈I

3.4. Transport of perishable assets It is assumed that the liner shipping company transports a set of perishable assets K ={1, …, r} at the given liner shipping route. Each perishable asset type k has an origin port Ok and a destination port Dk . Perishable assets are assumed to deteriorate over time. Increasing total transportation time of asset type k (which includes the total sailing time at voyage legs, connecting the origin and the destination ports, and the total handling time at ports between the origin and the destination ports) negatively affects its freshness. Based on the available literature (Blackburn and Scudder, 2009; Wang and Li, 2012; Grunow and Piramuthu, 2013; Piramuthu and Zhou, 2013; Piramuthu et al., 2013; Yu and Nagurney, 2013), the quality of asset type k at a given time T can be calculated using the following equation:

(1)

where:

q (v ) – daily vessel bunker consumption (tons of fuel/day); v – average daily vessel sailing speed (knots); q (v*) – daily vessel bunker consumption when sailing at the designed speed (tons of fuel/day);

QkT =Qk0 e−λk Tk where: 144

∀ k ∈K

(4)

International Journal of Production Economics 184 (2017) 141–156

M.A. Dulebenets, E.E. Ozguven

QkT – quality of asset type k at time T (i.e., once it is unloaded at the destination port, %); Qk0 – quality of asset type k at time 0 (i.e., once it is loaded at the origin port, %); λk – decay rate of asset type k (h−1); Tk – total transportation time of asset type k from the origin port to the destination port (h). Note that QkT is a strictly convex decreasing differentiable function. The decay rate λk depends on the nature of a perishable asset. For example, a meat product has a decay rate λ meat =0.0067 h−1, while a fresh vegetable product has a decay rate λ veg = 0.0216 h−1 (Wang and Li, 2012). Fig. 2. Vessel waiting time estimation.

3.5. Service frequency at ports of call

wti ≥ twis+1 − tia −

i∈I

∑ ( pij xij ) + ∑ wti

i ∈ I j ∈ Ji

i∈I

∀ i ∈ I, i < | I | (6)

j ∈ Ji

The liner shipping company deploys M vessels to provide service with a weekly frequency at each port of call. Weekly or in some cases biweekly service frequency at ports is generally negotiated between liner shipping companies and marine container terminal operators (Meng et al., 2014). Denote xij as a vessel handling rate decision variable (=1 if handling rate j is selected at port i and =0 otherwise) and wti as a vessel waiting time at port i (h). The following relationship should be maintained in order to guarantee the weekly service frequency at ports of call:

168M ≥ ∑ ti+ ∑

∑ ( pij xij ) − ti

The vessel waiting time at the last port of call (i.e., i ∈ I , i = | I|) can be calculated based on the arrival time at the last port of call, handling time at the last port of call, sailing time between the last and the first ports of call, the start of TW at the first port of call and the total vessel turnaround time from the following inequality:

wti ≥ tw1s − tia −

∑ ( pij xij ) − ti + 168M

∀ i ∈ I, i = | I |

j ∈ Ji

(7)

Note that the total vessel turnaround time (168M ) is included in constraints set (7) to account for a round trip journey of the vessel. Specifically, if the first vessel in the fleet, serving a given liner shipping route, arrives at the first port of call at time t1a (tw1s≤t1a≤tw1e ), the last vessel in the fleet, leaving the last port of call, will return to the first port of call after a round trip journey at time t1a+168M . Hence, the term 168M should be added when estimating the vessel waiting time at the last port of call, as tw1s ≪ tia+ ∑j ∈ Ji ( pij xij ) + ti .

(5)

The right-hand side of the inequality calculates the total turnaround time of a vessel (i.e., total round trip journey time) at the given liner shipping route and includes three components: a) the total sailing time; b) the total port handling time; and c) the total port waiting time. The left-hand side of the inequality is a product of the number of vessels to be deployed and 168, which represents the total number of hours in a week. The number of vessels required to maintain the weekly service frequency at ports of call can be computed by dividing the total vessel turnaround time at the given liner shipping route by the number of hours in a week. Note that increasing vessel turnaround time (e.g., due to increase in the sailing time from selecting the lower vessel sailing speeds at voyage legs of the given liner shipping route or due to increase in the vessel handling time from selecting handling rates with lower handling productivities at ports of call) will increase the number of vessels necessary to maintain the same service frequency at ports of call. The next section elaborates on estimation of the vessel waiting time at ports of call.

3.7. Estimation of the vessel arrival time at the next port of call The departure time from port i can be calculated based on the arrival time at port i , handling time at port i and waiting time at port i as follows:

tid = tia +

∑ ( pij xij ) + wti

∀i∈I

j ∈ Ji

(8)

The vessel arrival time at the next port of call (i.e., port i + 1) for all ports of the port rotation except the last port (i.e., i ∈ I , i < | I|) can be estimated as a summation of the vessel departure time from port i and sailing time between ports i and i + 1 using the following equation:

tia+1 = tid + ti ∀ i ∈ I , i < | I |

3.6. Vessel waiting time estimation

(9)

The vessel arrival time at the first port of call from the last port of call (i.e., i ∈ I , i = | I|) can be computed based on the vessel departure time from the last port of call, sailing time between the last and the first ports of call and the total vessel turnaround time as follows:

In certain cases upon completion of service at port i a vessel may arrive at port i + 1 before the start of TW twis+1 even when sailing at the lowest possible speed (v min , knots). An example of the latter case is presented in Fig. 2A. We observe that a vessel leaving port i at time tid (where tid – is the departure time from port i , h) arrives at port i + 1 before the start of TW even when sailing at the lowest possible speed vi =v min : tid +ti < twis+1. Since the marine container terminal operator at port i + 1 cannot provide the service of a vessel before the start of TW, it is assumed that the vessel should wait at port i to ensure arrival within the negotiated TW at port i + 1 (see Fig. 2B). Technically, the vessel can also wait at port i + 1 or split waiting time between ports i and i + 1. The latter decision is to be made by the liner shipping company. The vessel waiting time at all ports of the port rotation except the last port (i.e., i ∈ I , i < | I|) can be computed based on the arrival time at port i (tia , h), handling time at port i , sailing time between ports i and i + 1 and the start of TW at port i + 1 from the following inequality:

t1a = tid + ti − 168M ∀ i ∈ I , i = | I |

(10)

Note that the term 168M should be subtracted when estimating the arrival time at the first port of call for a vessel, returning from the last port of call, to account for a round trip journey. 3.8. Decisions The problem studied herein can be categorized as a tactical level problem and will be referred to as the vessel scheduling problem in a liner shipping route with perishable assets. The liner shipping company has to determine the following in this problem: 1) The number of vessels required to provide the weekly service 145

International Journal of Production Economics 184 (2017) 141–156

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2) 3) 4) 5)

QkT ∈ R+ ∀ k ∈ K quality of asset type k at time T (once it is unloaded at the destination port, %) ΔQk ∈ R+ ∀ k ∈ K change in quality of asset type k (%)

frequency at each port of the given liner shipping route; The vessel sailing speed between consecutive ports of call; The Handling rate at each port of the given liner shipping route; The waiting time at each port of the given liner shipping route; The hours of vessel late arrival at each port of call.

Parameters n∈N number of ports to be visited (ports) mi ∈ N ∀ i ∈ I number of available handling rates at port i (handling rates) r∈N number of perishable asset types (asset types) α , γ ∈ R+ bunker consumption function coefficients unit bunker cost (USD/ton) β ∈ R+ cOC ∈R+ vessel weekly operational cost (USD/week) ciLT ∈ R+ ∀ i ∈ I delayed arrival penalty at port i (USD/h) μ ∈ R+ unit inventory cost (USD per TEU per h) ckD ∈ R+ ∀ k ∈ K decay cost of asset type k (USD/asset decay) li ∈ R+ ∀ i ∈ I length of voyage leg i (nmi) NCi ∈ N ∀ i ∈ I number of containers transported at voyage leg i (TEUs) NPk ∈ N ∀ k ∈ K amount of asset type k transported at the given route (units) Qk0 ∈ R+ ∀ k ∈ K quality of asset type k at time 0 (once it is loaded at the origin port, %) Ok ∈ N ∀ k ∈ K origin port of asset type k Dk ∈ N ∀ k ∈ K destination port of asset type k λk ∈ R+ ∀ k ∈ K decay rate of asset type k (h−1) v min∈R+ minimum vessel sailing speed (knots) v max∈R+ maximum vessel sailing speed (knots) M max∈N maximum number of deployed vessels (vessels) twis ∈ R+ ∀ i ∈ I the start of TW at port i (h) twie ∈ R+ ∀ i ∈ I the end of TW at port i (h) pij ∈ R+ ∀ i ∈ I , j ∈ Ji vessel handling time at port i under handling rate j (h) tcij ∈ R+ ∀ i ∈ I , j ∈ Ji handling cost at port i under handling rate j (USD)

The aforementioned decisions that have to be made by the liner shipping company in this problem are interrelated. The liner shipping company should consider the limit on the number of vessels (M ≤M max ), allocated for service of the given liner shipping route. In the meantime, lower and upper bounds on vessel sailing speeds (v min ≤ vi ≤ v max ∀ i ∈ I ) have to be taken into account. The minimum sailing speed v min is generally selected by the liner shipping company to decrease wear of the vessel’s engine (Wang et al., 2013). The maximum sailing speed v max is defined by capacity of the vessel’s engine (Psaraftis and Kontovas, 2013). In order to decrease decay of perishable assets the liner shipping company may have to increase sailing speed of vessels, which will further increase the total bunker consumption and the associated costs. The latter will also decrease the total transit time of containers and will require deployment of less vessels to ensure that the weekly service is provided at each port of call, belonging to the given port rotation. Contractual agreements with marine container terminal operators at ports of call provide the liner shipping company more potential options in terms of selection of port handling and sailing times (for example, by requesting a handling rate with a higher productivity the liner shipping company will be able to decrease the port handling time at a given port of call, which will further allow sailing at a lower speed to the next port of the port rotation). Furthermore, reduction in the port handling time due to request of a handling rate with a higher productivity will decrease decay of perishable assets at the vessel being served at the port. However, selection of handling rates with higher productivities will increase the port handling charges for the liner shipping company. 4. Model formulation

VSPPA

This section presents a mixed integer non-liner mathematical model for the vessel scheduling problem in a liner shipping route with perishable assets - VSPPA.

[cOC M + β ∑ li f (vi ) +

min

i∈I

+

4.1. Nomenclature

∑ ∑ tcij xij

+

i ∈ I j ∈ Ji

∑ ciLT lti + μ ∑ ti NCi i∈I

∑ NPk ckD ΔQk ]

(11)

k∈K

Sets I = {1, …, n} set of ports to be visited Ji = {1, …, mi} ∀ i ∈ I set of available handling rates at port i K ={1, …, r} set of perishable asset types

i∈I

Subject to:

∑ xij =1

∀ i∈I (12)

j ∈ Ji

ti= Decision variables vi ∈ R+ ∀ i ∈ I vessel sailing speed at voyage leg i (knots) xij ∈ {0,1} ∀ i ∈ I , j ∈ Ji =1 if handling rate j is selected at port i (=0 otherwise)

li vi

f (vi ) Tk =

∀ i∈I

=

(13)

γ (vi )α−1 24

∑

∀ i∈I

∑ ( pij xij ) +

i ∈ I : Ok ≤ i ≤ Dk j ∈ Ji

Auxiliary variables number of vessels deployed at the given route (vessels) M ∈N tia ∈ R+ ∀ i ∈ I arrival time at port i (h) tid ∈ R+ ∀ i ∈ I departure time from port i (h) wti ∈ R+ ∀ i ∈ I waiting time of a vessel at port i (h) ti ∈ R+ ∀ i ∈ I vessel sailing time at voyage leg i (h) f (vi ) ∈ R+ ∀ i ∈ I bunker consumption at voyage leg i when sailing at speed vi (tons of fuel/nmi) lti ∈ R+ ∀ i ∈ I vessel late arrival at port i (h) Tk ∈ R+ ∀ k ∈ K total transportation time of asset type k from the origin port to the destination port (h)

(14)

∑

ti

∀k∈K (15)

i ∈ I : Ok ≤ i ≤ Dk−1

QkT =Qk0 e−λk Tk

∀ k ∈K

(16)

ΔQk =Qk0 −QkT

∀ k ∈K

(17)

tia ≥ twis ∀ i ∈ I wti ≥ twis+1 − tia −

(18)

∑ ( pij xij ) − ti

∀ i ∈ I, i < | I | (19)

j ∈ Ji

wti ≥ tw1s − tia −

∑ ( pij xij ) − ti + 168M j ∈ Ji

146

∀ i ∈ I, i = | I | (20)

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M.A. Dulebenets, E.E. Ozguven

tid = tia +

∑ ( pij xij ) + wti

lti ≥ tia − twie tia+1

tid

ing that the non-linear bunker consumption function G1 ( yi) is linearized using its piecewise linear secant approximation Gw1 ( yi), where w - is the number of linear segments (Wang et al., 2013).

∀i∈I (21)

j ∈ Ji

∀i∈I

(22)

Denote Gk2 (Tk )=

ΔQk Qk0

=

Qk0 − QkT Qk0

=

Qk0 − Qk0 e−λk Tk Qk0

−λk Tk

=1 − e

as the decay

+ ti ∀ i ∈ I , i < | I |

(23)

t1a = tid + ti − 168M ∀ i ∈ I , i = | I |

(24)

function for asset type k . Similarly, the non-linear decay function Gk2 (Tk ) can be linearized for each asset type k using its piecewise linear 2 (Tk ). Figs. 4 and 5 provide examples of linear secant approximation Gkw approximations with different number of linear segments (w = 1,3,5,10 )

(25)

for a non-linear bunker consumption function G1 ( yi) =

=

168M ≥ ∑ ti+ ∑ i∈I

∑ ( pij xij ) + ∑ wti

i ∈ I j ∈ Ji

M ≤M max v min≤vi ≤v max

i∈I

and a

non-linear asset decay function Gk2 (Tk )=1−e−0.020Tk . The vessel sailing speed was assumed to vary from v min=15 knots to v max =25 knots, i.e. 0.040 ≤ yi ≤0.067 (Wang and Meng, 2012a, 2012b, 2012c). The decay rate of asset k was assumed to be λk = 0.020 h−1 (Wang and Li, 2012), while the total transportation time of asset type k was set to be 50 ≤Tk≤100 h. We observe that increasing number of linear segments improves accuracy of the approximation for both G1 ( yi) and Gk2 (Tk ) functions. Let S = {1,2, ….w} be the set of linear segments in the piecewise 2 (Tk ). Let bis1 =1 if linear segment s is chosen to functions Gw1 ( yi) and Gkw approximate the bunker consumption function at voyage leg i (=0 otherwise), and bks2 =1 if linear segment s is chosen to approximate the decay function for asset type k (=0 otherwise). Denote sts1, eds1, s∈S as the speed reciprocal values at the start and the end (respectively) of linear segment s ; stks2 , edks2 , k ∈K ,s∈S as the transportation time values of asset type k at the start and the end (respectively) of linear segment s ; SLs1, INs1, s∈S as the slope and the intercept (respectively) of linear segment s used to approximate the bunker consumption function; SLks2 , INks2 , k ∈K ,s∈S as the slope and the intercept (respectively) of linear segment s used to approximate the decay function of asset type k ; and M1, M2, M3, M4 as sufficiently large positive numbers. There are several methods for modeling piecewise linear approximations in integer programming, including the following (Vielma et al., 2010; Silva and Camponogara, 2014): 1) basic disaggregated convex combination method; 2) logarithmic disaggregated convex combination method; 3) basic convex combination method; 4) logarithmic convex combination method; 5) multiple choice method; and 6) incremental method. Similar to the basic convex combination method, the approximated bunker consumption is generally estimated using the following inequality in the liner shipping literature (Wang and Meng, 2012b, 2012c; Wang et al., 2013):

(26)

∀ i∈I

0.012( yi )−2 24

(27)

In VSPPA the liner shipping company minimizes the total route service cost (11), which includes 6 components: 1) total vessel weekly operational cost, 2) total bunker consumption cost, 3) total port handling cost, 4) total late arrival penalty, 5) total inventory cost, and 6) total asset decay cost. Constraints set (12) enforces the liner shipping company to select only one handling rate at each port of the port rotation. Constraints set (13) estimates a sailing time of vessel between consecutive ports i and i + 1. Constraints set (14) calculates the bunker consumption at voyage leg i . Constraints set (15) computes the total transportation time of asset type k from the origin port to the destination port, which includes two components: 1) the total port handling time, and 2) the total sailing time. Constraints set (16) estimates the quality of asset type k at the destination port. Constraints set (17) calculates the change in quality of asset type k after transportation from the origin port to the destination port. Constraints set (18) indicates that a vessel cannot be served at port i before the start of TW. Constraints sets (19) and (20) calculate waiting time of a vessel at port i . Constraints set (21) computes departure time of a vessel from port i . Constraints set (22) calculates late arrival hours of a vessel at port i . Constraints sets (23) and (24) estimate arrival time of a vessel at the next port of the port rotation. Constraints set (25) ensures the weekly service frequency at each port of call of the given liner shipping route. Constraints set (26) indicates that the quantity of vessels, which can be deployed for service of the given liner shipping route, is limited by the quantity of available vessels. Constraints set (27) establishes bounds on sailing speed of a vessel at voyage leg i . 5. Solution approach

Gs1 ( yi)≥SLs1 yi + INs1

VSPPA is a non-linear mathematical model due to: 1) objective function (bunker consumption and asset decay cost components); and 2) constraints sets (13), (14), and (16). Replacing vessel sailing speed vi 1 with its reciprocal yi = v will linearize constraints set (13). Denote G1 ( yi) i as the bunker consumption function, which is estimated based on vessel sailing speed reciprocal yi . There exist three common approaches for approximating the nonlinear bunker consumption function G1 ( yi), including the following (Wang et al., 2013): 1) linear outer approximation method – where the non-linear bunker consumption function is approximated using a set of linear tangent lines (Fig. 3A); 2) linear secant approximation method – where the non-linear bunker consumption function is approximated using a set of linear secant lines (Fig. 3B); and 3) quadratic outer approximation method – where the non-linear bunker consumption function is approximated using a set of parabolic curves (Fig. 3C). A piecewise linear outer approximation method always underestimates the actual bunker consumption value, whereas a piecewise linear secant approximation method may either underestimate or overestimate the actual bunker consumption value. Both linear outer approximation and linear secant approximation methods can be used for linearization of function G1 ( yi); however, fewer secant lines are generally required than tangent lines in order to achieve the same approximation accuracy. This study will adopt the linear secant approximation method, assum-

∀ i∈I , s∈S

(28)

However, as underlined by Dulebenets (2015b), the latter approach may cause errors in calculating the approximated bunker consumption. Specifically, the commonly used approach can select a wrong linear segment to calculate the approximated bunker consumption function for the cases, when the piecewise approximation is represented with linear segments that do not have monotonically increasing slopes. On the other hand, the “Big M” method accurately estimates the approximated bunker consumption for approximations with linear segments that have monotonically increasing slopes as well as approximations with linear segments that do not have monotonically increasing slopes without significantly affecting the computational time required to solve the optimization model (Dulebenets, 2015b). This study will use the “Big M” method to model piecewise linear secant approximations for both bunker consumption and product decay functions. Based on the “Big M” method, VSPPA can be reformulated as a linear problem (VSPPAL) as follows. VSPPAL

min [cOC M +β ∑ ∑ Gs1 ( yi) li+ ∑ i∈I s∈S

∑ ti NCi + ∑ ∑ i∈I

147

k∈K s∈S

∑ tcij xij + ∑ ciLT lti+μ

i ∈ I j ∈ Ji

NPk ckD Qk0 Gks2 (Tk )]

i∈I

(29)

International Journal of Production Economics 184 (2017) 141–156

M.A. Dulebenets, E.E. Ozguven

Fig. 3. Common approaches for approximating the bunker consumption function.

Fig. 4. Linear approximations for the bunker consumption function.

148

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M.A. Dulebenets, E.E. Ozguven

Fig. 5. Linear approximations for the asset decay function.

Subject to: Constraints sets (12), (15), (18)–(26)

ti=li yi

∀ i∈I

∑ bis1 =1

eds1+M1 (1

(31)

∀ i∈I , s∈S −

bis1 )

≥ yi

(32)

∀ i∈I , s∈S −

bis1 )

This section presents a number of numerical experiments that were conducted to evaluate efficiency of the proposed solution approach and reveal some essential managerial insights that can be of importance to liner shipping companies using the developed mathematical model.

(34)

∀ k ∈K (35)

s∈S

stks2 bks2 ≤Tk

1

M2=SL11 v max + IN11, M3=maxk ∈ K (Tk ),

6. Numerical experiments

(33)

∀ i∈I , s∈S

1 , v min

2 INkw }.

(Tk ) + Note that M1, M2, M3 and M4 can be substituted in constraints sets (33), (34), (37), and (38) by M0=max{M1; M2; M3; M4}. VSPPAL can be solved efficiently using CPLEX even for large size instances (details are presented in the numerical experiments section).

∀ i∈I

Gs1 ( yi)≥SLs1 yi + INs1−M2 (1

∑ bks2 =1

2 M4=maxk ∈ K {SLkw max

(30)

s∈S

sts1 bis1 ≤yi

M4 can be defined as follows: M1=

edks2 +M3 (1 − bks2 ) ≥ Tk

∀ k ∈K , s∈S

Gks2 (Tk ) ≥ SLks2 Tk + INks2 −M4 (1 − bks2 ) 1/v max≤yi ≤1/v min

6.1. Input data description

(36)

∀ k ∈K , s∈S

∀ i∈I

This study considers the French Asia Line 1 route (see Fig. 6), which is served by CMA CGM liner shipping company (CMA CGM, 2016). This liner shipping route connects North Europe, Malta, Middle East Gulf, and Asia. The port rotation for the French Asia Line 1 route includes 18 ports of call that have to be visited by vessels on a weekly basis (the distances between consecutive ports in nautical miles are shown in parenthesis and were retrieved from the world seaports catalogue1):

(37)

∀ k ∈K , s∈S

(38) (39)

In VSPPAL objective function (29) minimizes the total route service cost. Constraints set (30) computes a vessel sailing time between consecutive ports i and i + 1. Constraints set (31) ensures that only one linear segment s will be chosen to approximate the bunker consumption function at voyage leg i . Constraints sets (32) and (33) define the range of vessel sailing speed reciprocal values, when linear segment s is chosen to approximate the bunker consumption function at voyage leg i . Constraints set (34) calculates the approximated bunker consumption at voyage leg i . Constraints set (35) ensures that only one linear segment s will be chosen to approximate the decay function for asset type k . Constraints sets (36) and (37) define the range of the transportation time values, when linear segment s is chosen to approximate the decay function for asset type k . Constraints set (38) estimates the approximated decay value for asset type k . Constraints set (39) shows that a reciprocal of vessel sailing speed should be within specific limits. Strict lower bounds for M1, M2, M3 and

1. Southampton, GB (121) → 2. Dunkerque, FR (450) → 3. Hamburg, DE (341) → 4. Rotterdam, NL (119) → 5. Zeebrugge, BE (302) → 6. Le Havre, FR (2538) → 7. Malta, MT (4089) → 8. Khor al Fakkan, AE (6449) → 9. Shanghai, CN (761) → 10. Tianjinxingang, CN (230) → 11. Dalian, CN (727) → 12. Busan, KR (616) → 13. Qingdao, CN (367) → 14. Shanghai, CN (87) → 15. Ningbo, CN (917) → 16. Yantian, CN (2045) → 17. Port Kelang, MY (7460) → 18. Algeciras, ES (1367) → 1. Southampton, GB The available liner shipping literature (Wang and Meng, 2012a, 1

149

https://www.searates.com.

International Journal of Production Economics 184 (2017) 141–156

M.A. Dulebenets, E.E. Ozguven

Fig. 6. The French Asia Line 1 route.

productivities (dij ) were offered to the liner shipping company at large ports: [125; 100; 75; 50] TEUs/h. At smaller ports the liner shipping company was able to request either [100; 75; 60; 50] TEUs/h or [75; 70; 60; 50] TEUs/h. The assumption regarding the handling productivities, provided at ports of call of the given liner shipping route, can be justified by the fact that marine container terminal operators at large ports typically have more equipment available to serve the arriving vessels. The latter further allows marine container terminal operators at large ports providing more handling rate alternatives to the liner shipping company. In the meantime, increasing quantity of TEUs handled may increase productivity. The vessel handling cost per TEU scij at port i under handling rate j was estimated as: scij =mhc ± U [0; 50] ∀ i ∈ I , j ∈ Ji USD/TEU, where mhc – is the mean handling cost. Then the overall port handling cost was computed as: tcij = scij CDi ∀ i ∈ I , j ∈ Ji USD. The mean handling cost (mhc ) was set equal to [700; 625; 550; 475] USD/TEU for four available handling rates respectively (World Bank, 2016; The Port Authority of New York and New Jersey, 2016). This study also assumes that each marine container terminal operator perceives the vessel handling cost differently (i.e., the vessel handling cost may vary from one port to the other for exaclty the same vessel handling rate). The second (and random) term of the scij equation was introduced to account for the latter aspect. All computational experiments were performed on a Dell Intel(R) Core™ i7 Processor with 32 GB of RAM. The piecewise linear secant approximations for the bunker consumption function and the asset decay function of each asset type (a total of 1+20=21 piecewise linear approximations) were developed using MATLAB 2014a (Mathworks, 2016). VSPPAL mathematical model was coded in General Algebraic Modeling System (GAMS) and solved using CPLEX.

Table 2 Numerical data. Parameter

Value

Coefficients of the bunker consumption function: α , γ Unit bunker cost: β (USD/ton)

α = 3, γ = 0.012 500 300,000

Vessel weekly operational cost: cOC (USD/week) Delayed arrival penalty at port i : ciLT (USD/h) Unit inventory cost: μ (USD per TEU per h) Origin port of asset type k : Ok Destination port of asset type k : Dk Decay rate of asset k : λk (h−1) Minimum vessel sailing speed: v min (knots) Maximum vessel sailing speed: v max (knots) Maximum number of deployed vessels: M max (vessels) Arrival TW duration at port i : [twie−twis] (h)

U [3,000; 5,000] 0.5

U [1;|I|−1] U [Ok +1;|I|] U [0.005; 0.020] 15 25 15 U [24; 72]

2012b, 2012c; Wang et al., 2013; Zampelli et al., 2014; OOCL, 2016; World Bank, 2016; World Shipping Council, 2016, etc.) was used to generate the numerical data necessary for computational experiments (see Table 2). The end of TW at each port of call was computed based on the end of TW at preceding port, length of a voyage leg between consecutive ports, and bounds of the vessel sailing speed: li twie+1 = twie + ∀ i ∈ I , where U – is a notation for uniformly U [v min;v max ] distributed pseudorandom numbers. A total of 20 perishable asset types were assumed to be transported at the given liner shipping route (i.e., K =20 ). The amount of asset type k transported at the given liner shipping route was generated as NPi = U [5,000; 10,000] ∀ k ∈ K (units). The quality of asset type k once it is loaded at the origin port (Qk0 ) was assumed to be 1 (or 100%). The decay cost of asset k was assigned as ckD=U [25; 50] ∀ k ∈ K USD/asset decay. Number of containers, transported at voyage leg i , was generated as NCi = U [5,000; 8,000] ∀ i ∈ I TEUs. The weekly container demand CDi at large ports of the port rotation was assigned as U [500; 2,000] TEUs. Note that this study classified ports of call for the considered liner shipping route based on their overall annual throughput in two groups: 1) "large ports"; and 2) "smaller ports". Large ports were defined as ports that were included in the list of top 20 world ports with the highest overall annual throughput based on the World Shipping Council data (World Shipping Council, 2016). The weekly container demand at smaller ports was generated as U [200; 1,000] TEUs. This study assumes that the following handling

6.2. Performance of the solution approach A total of 10 problem instances were generated based on the data, described in Section 6.1 and shown in Table 2, by altering the vessel arrival TWs at ports and duration of TWs. VSPPAL was solved for each one of the generated instances. The number of linear segments in the piecewise approximations for the bunker consumption function and the asset decay functions was varied. A total of 6 piecewise linear secant approximating functions, differed by the number of segments, were considered: a) 10 segments; b) 20 segments; c) 40 segments; d) 60 segments; e) 80 segments; and f) 100 segments. Results from the computational experiments are presented in Table 3, including the 150

International Journal of Production Economics 184 (2017) 141–156

M.A. Dulebenets, E.E. Ozguven

Table 3 Objective gap and CPU time. Instance

#Segments,|S|

Z , 106USD

Z*, 106USD

Δ

CPU, s

Instance

#Segments,|S|

Z , 106USD

Z*, 106USD

Δ

CPU, s

1

10 20 40 60 80 100 10 20 40 60 80 100 10 20 40 60 80 100 10 20 40 60 80 100 10 20 40 60 80 100

23.1166 23.1197 23.1204 23.1205 23.1208 23.1208 23.1616 23.1665 23.1667 23.1667 23.1668 23.1669 23.2556 23.2587 23.2589 23.2590 23.2590 23.2591 23.3054 23.3095 23.3097 23.3097 23.3097 23.3097 23.3192 23.3234 23.3239 23.3239 23.3239 23.3240

23.1209

1.86E−04 5.16E−05 2.04E−05 1.88E−05 5.53E−06 3.23E−06 2.44E−04 3.23E−05 2.34E−05 2.16E−05 1.80E−05 1.50E−05 1.61E−04 2.49E−05 1.67E−05 1.22E−05 1.08E−05 8.98E−06 1.89E−04 1.28E−05 4.76E−06 4.25E−06 3.96E−06 2.53E−06 2.08E−04 2.86E−05 1.07E−05 8.02E−06 7.17E−06 4.72E−06

1.44 4.19 35.79 439.14 1228.24 1827.33 1.52 4.87 39.40 494.07 1226.91 1827.89 1.45 4.21 34.87 449.70 1150.09 1817.71 1.51 4.63 39.70 428.06 1226.84 1822.25 1.45 4.70 40.77 407.67 1195.69 1812.62

6

10 20 40 60 80 100 10 20 40 60 80 100 10 20 40 60 80 100 10 20 40 60 80 100 10 20 40 60 80 100

23.4118 23.4151 23.4155 23.4155 23.4156 23.4157 23.5276 23.5325 23.5326 23.5327 23.5328 23.5328 23.5974 23.6031 23.6034 23.6035 23.6035 23.6036 23.6304 23.6365 23.6368 23.6369 23.6371 23.6373 23.6329 23.6395 23.6405 23.6408 23.6409 23.6409

23.4160

1.81E−04 3.92E−05 2.29E−05 2.02E−05 1.66E−05 1.34E−05 2.28E−04 2.16E−05 1.81E−05 1.31E−05 1.05E−05 9.09E−06 2.71E−04 2.81E−05 1.67E−05 1.26E−05 1.18E−05 7.91E−06 2.90E−04 3.27E−05 2.05E−05 1.61E−05 8.07E−06 4.99E−07 3.37E−04 6.02E−05 1.71E−05 5.55E−06 1.99E−06 1.64E−06

1.16 4.56 41.18 441.65 1226.90 1826.88 1.44 4.54 38.99 470.48 1114.22 1822.86 1.42 4.88 37.20 437.56 1226.86 1823.89 0.65 4.43 32.60 467.72 1129.40 1726.88 0.76 4.88 39.49 422.81 1195.99 1747.01

2

3

4

5

23.1672

23.2593

23.3098

23.3241

7

8

9

10

23.5330

23.6038

23.6373

23.6409

cost and its components; b) vessel type selection effects; and c) comparative analysis: VSPPAL vs. typical vessel scheduling model.

following information: 1) instance number; 2) number of segments in the piecewise linear secant approximations (i.e., |S|); 3) objective function value – Z ; 4) value of the non-linear objective function at (Z* − Z ) the solution provided by VSPPAL – Z*; 5) objective gap Δ=| Z* |; and 6) CPU time (average over 10 replications). We observe that increasing number of segments in the piecewise linear secant approximations decreases the objective gap, but in the meantime substantially increases the computational time required to solve VSPPAL. Based on the tradeoff between the objective gap and computational time values, the piecewise linear secant approximations with 20 segments will be further used for linearization of both bunker consumption function and the asset decay functions. Note that the optimal solution itself may change if mixed integer non-linear problem VSPPA is solved using mixed integer non-linear optimization solvers (e.g., BARON, DICOPT, AlphaECP) without applying any piecewise linear secant approximations, which will cause changes in the optimal objective function value and the objective gap. However, changes in the optimal solution and the associated optimal objective function value from solving VSPPA using mixed integer non-linear optimization solvers will not be significant, as VSPPAL with a large number of segments in the piecewise linear secant functions approximates both bunker consumption and product decay functions with a high accuracy (see Figs. 4 and 5). Furthermore, throughout the numerical experiments the solutions obtained by mixed integer linear VSPPAL mathematical model were checked if they satisfied constraints sets of mixed integer non-linear VSPPA mathematical model. It was found that VSPPAL solutions were feasible to VSPPA mathematical model (i.e., satisfied all of the constraints sets of VSPPA mathematical model) for all the problem instances and all the piecewise linear secant approximating functions.

6.3.1. Total route service cost and its components The objective function value and its components were calculated for vessel schedules using VSPPAL mathematical model for all the generated problem instances and are presented in Fig. 7, including the following cost components: 1) the total route service cost – Z; 2) the total weekly vessel operational cost – TOC; 3) the total bunker consumption cost – TBC; 4) the total port handling cost – TPC; 5) the total late arrival penalty – TLP; 6) the total inventory cost – TIC; and 7) the total asset decay cost – TADC. Note that changes in the objective function value and its components from one instance to another can be explained by the fact that the optimal vessel schedule design was affected from altering the vessel arrival time windows at ports and duration of time windows. The average over all perishable assets, carried by the vessels, decay values (G ) were computed for each one of the considered problem instances, and results are illustrated in Fig. 8. Results from computational experiments indicate that if the liner shipping company aims to maintain freshness of perishable assets (i.e., decrease in the asset decay), transported at the given liner shipping route, the total route service cost may increase. Increasing bunker consumption cost can be explained by the fact that the liner shipping company increased the vessel sailing speed at certain voyage legs of the liner shipping route in order to decrease the transit time of perishable assets (i.e., reduce the decay of assets on board the vessels while sailing in sea). Increasing port handling cost indicates that the liner shipping company requested handling rates with higher handling productivities at some ports of call, belonging to the given port rotation. Higher handling productivities allowed the liner shipping company decreasing port handling time and reducing decay of perishable assets, while vessels were being served at ports. Numerical experiments showcase that reduction in decay of perishable assets by 8.2% led to the increase in the bunker consumption and port handling costs by 42.7% and 14.9% respectively.

6.3. Managerial insights This section discusses managerial insights that were revealed using the developed mathematical model, including: a) total route service 151

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Fig. 7. Objective function and its components.

We also observe that the total late arrival penalty generally decreases with decreasing average asset decay from one problem instance to the other. The latter can be explained by the fact that in order to ensure asset freshness the liner shipping company had to reduce the late arrivals to ports of call, constituting to the total transportation time of perishable assets (i.e., reduction in port late arrivals would decrease the total transportation time of perishable assets and the associated asset decay). Decrease in the total inventory cost can be justified by the fact that the liner shipping company had to reduce the total transit time of perishable assets (by increasing the vessel sailing speed and, hence, the total bunker consumption). Furthermore, reduction in the total transit time and the total port Fig. 8. The average asset decay.

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Fig. 9. The average vessel sailing speed by voyage leg and vessel type.

vessel type deployment cases. The average over 10 problem instances vessel sailing speed by voyage leg and vessel type is presented in Fig. 9. Furthermore, along with the arithmetic average vessel sailing speed we computed the average vessel sailing speed, weighted by the voyage

handling time of perishable assets decreased the total turnaround time of vessels, serving the given liner shipping route. The latter led to reduction in the size of vessel fleet, required to provide the weekly service frequency at ports of call, belonging to the given port rotation. Computational experiments exhibit that reduction in decay of perishable assets by 8.2% decreased the total vessel weekly operational cost by 18.2%. Moreover, decrease in the asset decay reduced the associated total asset decay cost. Results from the conducted numerical experiments demonstrate that for the considered problem instances the total route service cost may increase by 2.2% from increasing freshness of perishable assets by 8.2%.

leg length (i.e., AWS=

∑i ∈ I vi li ∑i ∈ I li

), to capture changes in the vessel sailing

speed at longer voyage legs due to deployment of larger vessels (see Fig. 10). The average over 10 problem instances asset decay values ( AG ) were also estimated, and results are presented in Fig. 11. We observe that deployment of large size vessels generally requires the liner shipping company to reduce the vessel sailing speed especially at longer voyage legs. For example, deployment of large size vessels decreased the vessel sailing speed at voyage leg “Port Kelang (MY)→ Algeciras (ES)” (the longest voyage leg of the French Asia Line 1 route) on average by 5.7% and 10.1% as compared to the cases with deployment of medium and small size vessels respectively. Results from the conducted analysis indicate that the weighted vessel sailing speed on average reduced by 4.5% and 7.1% from deployment of larger vessels, as compared to deployment of medium and small size vessels. The latter finding has been also observed in practice, where liner shipping companies use the concept of “slow steaming” (i.e., reducing vessel sailing speed) to decrease the bunker consumption and associated costs for large size vessels (Kemp, 2015). Moreover, it was found that selection of smaller vessels for service of the given liner shipping route would decrease the decay of perishable assets. The latter can be explained by the fact that smaller vessels may generally sail at higher speeds without significant increase in the total bunker consumption

6.3.2. Vessel type selection effects Changes in the vessel type to be deployed for service of the given liner shipping route may substantially affect the vessel schedule design and in turn may influence freshness of perishable assets transported. Generally, larger vessels have higher bunker consumption and weekly operational costs (Wang and Meng, 2012b; Psaraftis and Kontovas, 2013). The scope of this study included the evaluation of vessel type selection effects on the vessel sailing speed and decay of perishable products. A total of three vessel types were considered with different bunker consumption function coefficients and weekly operational costs: 1) “Small” size vessels (α = 2.6, γ = 0.008,cOC =240,000 USD); 2) “Medium” size vessels (α = 2.8, γ = 0.010,cOC =270,000 USD); and 3) “Large” size vessels (α = 3.0, γ = 0.012,cOC =300,000 USD). VSPPAL was solved for each once of the considered problem instances and 153

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Fig. 10. The average weighted vessel sailing speed by vessel type.

ckD = 0 ∀ k ∈ K in VSPPAL mathematical model and relaxing constraints sets (35)–(38). The VSPL was solved for all the generated problem instances, and results of a comparative analysis are presented in Fig. 12. Fig. 12 provides the following information for both VSPPAL and VSPL models: 1) the total route service cost – Z; 2) the total weekly vessel operational cost – TOC; 3) the total bunker consumption cost – TBC; 4) the total port handling cost – TPC; 5) the total late arrival penalty – TLP; and 6) the total inventory cost – TIC. Note that the total asset decay cost component (TADC) was not included in the comparative analysis, as it was equal to zero for vessels schedules suggested by VSPL for all the considered problem instances. Results from a comparative analysis of VSPPAL and VSPL vessel schedules showcase that generally VSPPAL produces vessel schedules, which have higher bunker consumption costs as compared to VSPL vessel schedules. The latter can be explained by the fact that VSPPAL requires vessels to sail at higher speeds at voyage legs of the given liner shipping route to reduce decay of perishable assets on board the vessels while sailing in sea and associated costs. VSPL does not impose any costs due to decay of perishable assets and is able to suggest lower vessel sailing speeds (which further decreases the bunker consumption cost). Furthermore, VSPPAL vessel schedules have higher port handling costs than VSPL vessel schedules from requesting vessel handling rates with higher handling productivities at ports of call. Higher handling rates allow decreasing port handling time and decay of perishables assets, while vessels are being served at ports. It was found that the bunker consumption and port handling costs were increased on average by 16.0% and 11.1% from making the adjustments in vessel schedules due to the asset perishability. An increase in vessel sailing speeds for VSPPAL vessel schedules further reduces the transit time of containers and associated container inventory costs. Numerical experiments indicate that VSPPAL vessel schedules have on average 23.1% lower container inventory costs as compared to VSPL vessel schedules. Due to decreasing transit time of containers the liner shipping company is required to deploy less vessels to provide the agreed service frequency at ports of call, which reduces weekly vessel operational costs for VSPPAL vessel schedules for certain problem instances (i.e., instances 2, 7, and 8). Moreover, VSPPAL suggests vessel schedules with lower port late arrival penalties. The latter finding stems from the reduction in hours of late arrivals at port of call, which allows decreasing the total transportation time and decay of perishable assets. In conclusion, results of the comparative analysis also show that incorporating perishability con-

Fig. 11. The average asset decay values by vessel type.

cost, which decreases the total transportation time of perishable assets and in turn reduces their decay. Decision on the vessel size should not be solely based on the type of asset to be transported (e.g., perishable asset vs. non-perishable asset). In order to make a selection of the appropriate vessel size at the given route, liner shipping companies should take into consideration demand at ports of call, agreements with alliance partners, capacity of marine container terminals (e.g., sufficient draft in the access channel and at the quay to allow navigation of larger vessels; more advanced handling equipment; sufficient storage capacity, etc.), and other factors. However, based on the findings, suggested by the developed mathematical model, liner shipping companies should also account for decay of perishable assets and associated costs when deciding on the vessel size, as the asset decay costs can be significant. Maintaining freshness of perishable assets may either require deployment of small size vessels or increasing sailing speed of larger vessels.

6.3.3. Comparative analysis: VSPPAL vs. typical vessel scheduling model The scope of numerical experiments also included a comprehensive comparative analysis of the proposed mathematical model (VSPPAL) against a typical vessel scheduling problem (that will be referred to as VSPL), which does not model decay of perishable assets on board the vessels and does not impose any costs associated with the asset decay. The VSPL mathematical model was formulated by setting decay cost 154

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Fig. 12. Objective function and its components: VSPPAL vs. VSPL.

7. Conclusions and future research extensions

siderations causes significant changes in vessel schedules and increases the total route service cost on average by 22.4% over the generated problem instances as compared to the case when the asset perishability is ignored.

Considering increasing volumes of the international seaborne trade and amount of perishable assets, carried by vessels, liner shipping companies have to enhance efficiency of their operations and in the meantime capture potential decay of perishable assets on board the vessels due to certain operational and external environmental factors. This study presented a novel mixed integer non-linear mathematical programming model for the vessel scheduling problem in a liner shipping route with perishable assets, which explicitly captured decay of perishable assets on board the vessels. The model’s objective minimized the total route service cost, including the asset decay cost. A number of techniques were applied to linearize the original nonlinear mathematical formulation. CPLEX was used to solve the linearized mathematical model. A set of computational experiments were performed for the French Asia Line 1 route, served by CMA CGM liner shipping company, to assess efficiency of the suggested solution approach and reveal some important managerial insights. Results from computational experiments demonstrate that the proposed solution approach is efficient in terms of both solution quality and computational time. Furthermore, in order to maintain freshness of perishable assets the liner shipping company may have to

6.3.4. Discussion This study proposes a novel mixed integer non-linear mathematical model for the vessel scheduling problem in a liner shipping route with perishable assets. An exponential decay function is adopted to capture deterioration of perishable assets on board the vessels over time. Extensive numerical experiments are performed for the French Asia Line 1 route, served by CMA CGM liner shipping company, and demonstrate that in order to maintain freshness of perishable assets the liner shipping company may incur an increasing total route service cost. The developed mathematical model will assist the liner shipping company with identification of changes in the vessel schedules that are required to decrease decay of perishable assets. Furthermore, results from computational experiments suggest that deployment of small size vessels will be more advantageous for liner shipping routes, where perishables assets are to be transported. 155

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Grunow, M., Piramuthu, S., 2013. RFID in highly perishable food supply chains – remaining shelf life to supplant expiry date? Int. J. Prod. Econ. 146, 717–727. Haass, R., Dittmer, P., Veigt, M., Lutjen, M., 2015. Reducing food losses and carbon emission by using autonomous control – a simulation study of the intelligent container. Int. J. Prod. Econ. 164, 400–408. Jedermann, R., Nicometo, M., Uysal, I., Lang, W., 2014. Reducing food losses by intelligent food logistics. Philos. Trans. R. Soc. A: Math. Phys. Eng. 372, 1–20. Kemp, J., 2015. Megaships are Worsening Overcapacity in the Container Market. Reuters, London. Li, C., Qi, X., Song, D., 2015. Real-time schedule recovery in liner shipping service with regular uncertainties and disruption events. Transp. Res. Part B 93, 762–788. Marine Insight, 2016. Everything You Ever Wanted to Know About Container Refrigeration Unit. 〈http://www.marineinsight.com/refrigeration-air-conditioning/ everything-you-ever-wanted-to-know-about-container-refrigeration-unit/〉 (accessed on 15.05.16.). Mathworks, 2016. Release 2014a. 〈http://www.mathworks.com/〉 (accessed 10.05.16.). Meng, Q., Wang, S., Andersson, H., Thun, K., 2014. Containership routing and scheduling in liner shipping: overview and future research directions. Transp. Sci. 48, 265–280. NOAA, 2016. Commercial Fisheries Statistics – U.S. Foreign Trade AnnualData. 〈http:// www.st.nmfs.noaa.gov/commercial-fisheries/foreign-trade/〉 (accessed 20.09.16.). OOCL, 2016. e-Services, Sailing Schedule, Schedule by Service Loops. Asia-Europe (AET) 〈http://www.oocl.com〉, (accessed 10.02.16.). Piramuthu, S., Zhou, W., 2013. RFID and perishable inventory management with shelfspace and freshness dependent demand. Int. J. Prod. Econ. 144, 635–640. Piramuthu, S., Farahani, P., Grunow, M., 2013. RFID-generated traceability for contaminated product recall in perishable food supply networks. Eur. J. Oper. Res. 225, 253–262. Psaraftis, H., Kontovas, C., 2013. Speed models for energy-efficient maritime transportation: a taxonomy and survey. Transp. Res. Part C 26, 331–351. Qi, X., Song, D., 2012. Minimizing fuel emissions by optimizing vessel schedules in liner shipping with uncertain port times. Transp. Res. Part E 48, 863–880. Rodrigue, J.P., 2013. The Geography of Transport Systems. third edition. New York. Rong, A., Akkerman, R., Grunow, M., 2011. An optimization approach for managing fresh food quality throughout the supply chain. Int. J. Prod. Econ. 131, 421–429. Silva, T., Camponogara, A., 2014. A computational analysis of multidimensional piecewise-linear models with applications to oil production optimization. Eur. J. Oper. Res. 232, 630–642. Song, D., Li, D., Drake, P., 2015. Multi-objective optimization for planning liner shipping service with uncertain port times. Transp. Res. Part E 84, 1–22. Sustainablog, 2016. Shipping Perishable Goods in Shipping Containers. 〈http:// sustainablog.org/2014/09/shipping-perishable-goods-shipping-containers/〉 (accessed 15.05.16.). The Port Authority of New York and New Jersey, 2016. Marine Terminal Tariffs. 〈http:// www.panynj.gov/port/tariffs.html〉 (accessed 10.05.16.). UNCTAD, 2015. Review of Maritime Transport 2015. United Nations Conference on Trade and Development, New York and Geneva. Vielma, J., Ahmed, S., Nemhauser, G., 2010. Mixed-integer models for nonseparable piecewise-linear optimization: unifying framework and extensions. Oper. Res. 58, 303–315. Wang, S., Meng, Q., 2012a. Liner ship route schedule design with sea contingency time and port time uncertainty. Transp. Res. Part B 46, 615–633. Wang, S., Meng, Q., 2012b. Sailing speed optimization for container ships in a liner shipping network. Transp. Res. Part E 48, 701–714. Wang, S., Meng, Q., 2012c. Robust schedule design for liner shipping services. Transp. Res. Part E 48, 1093–1106. Wang, S., Meng, Q., Liu, Z., 2013. Bunker consumption optimization methods in shipping: a critical review and extensions. Transp. Res. Part E 53, 49–62. Wang, S., Alharbi, A., Davy, P., 2014. Liner ship route schedule design with port time windows. Transp. Res. Part C 41, 1–17. Wang, S., Qu, X., Yang, Y., 2015. Estimation of the perceived value of transit time for containerized cargoes. Transp. Res. Part A 78, 298–308. Wang, X., Li, D., 2012. A dynamic product quality evaluation based pricing model for perishable food supply chains. Omega 40, 906–917. World Bank, 2016. Cost to import (US$ per container). 〈http://data.worldbank.org/ indicator/IC.IMP.COST.CD〉 (accessed 10.05.16.). World Shipping Council. Top 50 World Container Ports. 〈www.worldshipping.org〉 (accessed 10.02.16.). Yao, Z., Ng, S., Lee, L., 2012. A study on bunker fuel management for the shipping liner services. Comput. Oper. Res. 39, 1160–1172. Yu, W., Nagurney, A., 2013. Competitive food supply chain networks with application to fresh produce. Eur. J. Oper. Res. 224, 273–282. Zampelli, S., Vergados, Y., Schaeren, R., Dullaert, W., Raa, B., 2014. The berth allocation and quay crane assignment problem using a CP approach. Principles and Practice of Constraint Programming. Springer, Berlin Heidelberg, pp. 880–896.

reduce the vessel sailing time by increasing the vessel sailing speed and decrease the port handling time by requesting handling rates with higher productivities at ports of call. The latter can cause increase in the total route service cost. Moreover, deployment of small size vessels for service of the given liner shipping route may reduce decay of perishable assets. The developed mathematical model can serve as an effective practical tool for liner shipping companies and assist with design of efficient vessel schedules and consideration of important operational aspects (e.g., decay of perishable assets on board the vessels). The future research may focus on the following extensions: a) implement the developed mathematical model for various liner shipping routes; b) consider deployment of a heterogeneous vessel fleet for service of the given liner shipping route (i.e., vessels with different technical characteristics are used in the fleet allocated for service of the given liner shipping route); c) account for uncertainty in vessel sailing and port handling times; d) consider other variables that may affect the bunker consumption (e.g., vessel payload); e) account for the asset decay due to other external factors (e.g., temperature, pressure, power), f) modeling inland transport of perishable assets from marine container terminals to the end customers, and g) consider more comprehensive agreements between liner shipping companies and marine container terminal operators (e.g., where multiple vessel arrival time windows can be negotiated by the liner shipping company with each marine container terminal operator). Acknowledgements This work was partially supported by the Department of Civil Engineering at the University of Memphis (Memphis, TN) and the Department of Civil and Environmental Engineering at the Florida A & M University - Florida State University (Tallahassee, FL). Any opinions, findings, conclusions, or recommendations are those of the authors and do not necessarily reflect the views of the aforementioned organizations. References Alhrabi, A., Wang, S., Davy, P., 2015. Schedule design for sustainable container supply chain networks with port time windows. Adv. Eng. Inform. 29, 322–331. Amorim, P., Belo-Filho, M., Toledo, F., Almeder, C., Almada-Lobo, B., 2013. Lot sizing versus batching in the production and distribution planning of perishable goods. Int. J. Prod. Econ. 146, 208–218. Aung, M., Chang, Y., 2014. Temperature management for the quality assurance of a perishable food supply chain. Food Control 40, 198–207. Blackburn, J., Scudder, G., 2009. Supply chain strategies for perishable products: the case of fresh produce. Prod. Oper. Manag. 18, 129–137. Brouer, B., Dirksen, J., Pisinger, D., Plum, C., Vaaben, B., 2013. The vessel schedule recovery problem (VSRP) – a MIP model for handling disruptions in liner shipping. Eur. J. Oper. Res. 224, 362–374. Chuang, T., Lin, C., Kung, J., Lin, M., 2010. Planning the route of container ships: a fuzzy genetic approach. Expert Syst. Appl. 37, 2948–2956. CMA CGM. French Asia Line 1. 〈http://www.cma-cgm.com/products-services/lineservices/flyer/FAL1〉 (accessed 02.04.16.). Du, Y., Chen, Q., Quan, X., Long, L., Fung, R., 2011. Berth allocation considering fuel consumption and vessel emissions. Transp. Res. Part E 47, 1021–1037. Dulebenets, M.A., 2015a. Bunker consumption optimization in liner shipping: a metaheuristic approach. Int. J. Recent Innov. Trends Comput. Commun. 3, 3766–3776. Dulebenets, M.A., 2015b. Models and solution algorithms for improving operations in marine transportation (Dissertation). The University of Memphis, ProQuest, 〈http:// gradworks.umi.com/37/28/3728362.html〉. Fagerholt, K., 2001. Ship scheduling with soft time windows: an optimization based approach. Eur. J. Oper. Res. 131, 559–571.

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