A maritime transportation system is greatly impacted by the efficiency of the routing and scheduling approach being adopted. The United States, which is the largest trading nation in the world in terms of imports and exports, accounts for nearly 20% of the global trade, of which over 99% is comprised of sea cargo (Agarwal, 2007). U.S. ports and waterways handle about 2.5 billion tons of trade annually, and this is expected to double within the next 15 years (Agarwal, 2007; Al Khayyal and Hwang, 2007). The substantial worldwide increase in sea-borne trade has led to a proportional growth of the international fleet. Maritime transportation constitutes the majority of long-distance shipments in terms of volume (Rodrigue et al., 2017). Oil is the world’s leading transported product in terms of tonnage, amounting to about 4.8% of the global transported value. In 2015, the total exported value of crude oil was estimated at $786.3 billion, and the share of the Gulf countries was valued at $325 billion or 41.3% of the global crude oil export (Trade Map, 2017). The operation of a typical fleet of oil tankers is highly expensive due to the high costs of owning and chartering oil tankers in addition to the costs associated with the daily vessels’ operation. In recent years, the prices of oil have dropped from $120 per barrel (2012) to a current price of about $69 per barrel (June, 2018), drastically affecting economies of major oil producing countries such as Saudi Arabia and Kuwait. As a result, efficient fleet utilization has become a critical issue for shipping companies, which, in turn, has prompted researchers to develop effective optimization-based decision support systems for the management and routing of ships (see, e.g. Christiansen et al., 2004, 2013; Christiansen and Fagerholt, 2009; Hoff et al., 2010; Pantuso et al., 2014). Ronen (1983, 1993) indicates that the maritime transportation mode has attracted the least research among all transportation modes, principally due to the complexity of the problem and the high degree of uncertainty involved in shipping operations. Nonetheless, with advances in technology, there has been a growing interest in this area of research as highlighted by Christiansen et al. (2004).

This paper investigates a single source-single destination crude oil vessel scheduling transportation-inventory problem faced by the Kuwait Petroleum Corporation (KPC), among a host of other transportation scenarios. KPC aims to transport substantial quantities of crude oil from Shuaiba Port in Kuwait (source) to the storage facility at Dalian Xingang Port in China (destination). Although oil companies face many demand-transportation related crude oil scenarios, we focus in this paper on a single-source and a single-destination operational scenario that is typically encountered by many large-scale oil companies such as KPC and the Saudi Aramco. This is motivated by the fact that for a destination having considerable long-term demand requirements, KPC adopts a long-lasting practice of allocating a separate fleet of vessels to meet such demand requirements without interfacing with the demands at other destinations. For example, KPC aims to meet demand requirements agreed upon with the Chinese government in concert with major oil companies (such as China Petroleum and Chemical Corporation or China National Petroleum Corporation) to transport large quantities of crude oil from Shuaiba Port in Kuwait to the storage facility at Dalian Xingang Port using a dedicated fleet of vessels. (To ease readability, we will henceforth use the terms “product, source, and destination” in lieu of “crude oil, Shuaiba Port, and Dalian Xingang Port”, respectively.)

Typically KPC transports large quantities of crude oil, and so it is practically more advantageous for KPC to use fully loaded vessels at the source to be fully discharged at the destination. The daily export from the source depends on the availability of the product at the source, storage capacities at the destination, rates of consumption of the product at the destination, the availability of vessels, and the current storage level at the destination storage facility. KPC can also lease a transshipment facility, if such a decision enhances the efficiency and cost effectiveness of the overall operation. The demand at the destination is governed by the consumption rates at the destination, which might not be fixed during the entire time-horizon and might vary based on, for example, seasonal considerations. The level of the product at the destination storage facility is desired to lie within certain lower and upper bounds, and hence, a daily penalty is imposed based on limited permissible shortage or excess quantities with respect to these bounds. The fleet of vessels used by KPC for transporting the product from the source to the destination is composed of different types of vessels, some of which are self-owned while others can be possibly chartered for a specified duration of the time-horizon. Each vessel-type is characterized by its size, speed, loading and unloading times, etc. Accordingly, KPC aims to satisfy the demand requirements while minimizing the total cost that is comprised of the operational expenses of vessels, maintenance and leasing expenses of the transshipment facility, penalties resulting from violating desired lower and upper limits on the storage level, and the cost of chartering vessels.

The present paper extends the work of the authors (Sherali and Al-Yakoob, 2006a, 2006b; Al-Yakoob and Sherali, 2013) related to vessel scheduling. Exact and aggregated mixed-integer programming formulations and related rolling horizon heuristics for the same demand structure considered herein were presented in Sherali and Al-Yakoob (2006a) for the single source-destination operation scenario, and in Sherali and Al-Yakoob (2006b) for the multiple sources and multiple destinations, while considering the leasing of temporary transshipment storage facilities, each of which has a known discretized location. It was shown that schedules obtained via the modelling approach adopted therein were superior to those generated by a manual approach in a case study related to KPC. Also, it was emphasized in Sherali and Al-Yakoob (2006b) that the use of transshipment facilities has the potential of reducing overall operational costs of the generated schedules, because this permits shipments to be stocked closer to the destinations during slack periods for use during periods of higher demand, thus resulting in using fewer vessels and, in particular, circumventing the high costs associated with chartering vessels. Subsequently, Al-Yakoob and Sherali (2013) solved the problem studied in the latter paper using a column generation approach, which led to further cost reductions. This paper examined the impact of allowing a transshipment facility to be situated at one of a number (three, specifically) of plausible discretized locations, instead of just a single specified location as done in Sherali and Al-Yakoob (2006b), and demonstrated that further cost reductions can be thus achieved. This observation motivated the idea that examining a continuum of potential locations for a given storage facility could possibly further reduce overall operational costs quite significantly. The present paper extends the foregoing body of work by including modelling considerations for determining a cost-effective location for the transshipment facility within a restricted continuous region. In this context, we design a hybrid two-stage algorithm that inputs a set of discrete judicious facility locations resulting from solving the proposed model using a specialized rolling horizon algorithm into the column generation method of Al-Yakoob and Sherali (2013).

The remainder of this paper is organized as follows. Section 2 reviews the related literature. We introduce some modelling preliminaries in Section 3. A mathematical programming model is presented in Section 4 for the single source-destination problem with a restricted continuous region for locating a single transshipment facility. Specific modelling considerations for the location aspect of the transshipment facility are discussed in Section 5. To further reduce overall operational costs, a hybrid, two-stage algorithmic approach is proposed in Section 6. Section 7 then provides computational results using a wide range of simulated test cases representing different operational scenarios related to the Kuwait Petroleum Corporation in order to assess the impact of the transshipment facility on the cost and efficiency of the overall operation. Finally, Section 8 concludes the paper with a summary and recommendations for implementation and future research.

Let H be the set of days in the time-horizon, and let T be the set of types of vessels in the company’s fleet, where Ω t represents the capacity of a vessel of type t ∈ T. For each t ∈ T, let n = 1 , … , M t index the vessels of type t, and let O t and CH t = M t − O t respectively, denote the number of company-owned vessels and the number of available vessels of this type that can be possibly chartered. The company-owned vessels are indexed by n = 1 , … , O t , while the chartered vessels are indexed by n = O t + 1 , … , O t + CH t ≡ M t . Accordingly, let $ t , n be the cost (in US dollars) of chartering a vessel n of type t, for each n = O t + 1 , … , O t + CH t , and t ∈ T. Let UT t , n be the maximum number of days vessel n of type t can be used during the time-horizon, where this time restriction is typically needed for maintenance purposes. Let H t , n ⊆ { 1 , … , H } be a subset of the time-horizon during which the ship n of type t will be available for use, where h 1 t , n represents the first day in H t , n . A production capacity or a certain imposed supply quota for the product at the source is also specified, as given by Q.

The demand structure of the problem is determined based on the following factors: (a) the storage capacity at the destination; (b) the initial level of each destination storage; (c) the rates of consumption at the destination; and (d) the production capacity at the source. The rates of consumption at a destination may vary due to seasonal considerations; however, we assume that this information is known a priori. For any given day of the time-horizon, there are minimum and maximum allowable limits imposed on the storage levels at the destinations, where certain agreed-upon penalties are incurred for violating these limits. At the beginning of the time-horizon, the storage level at the destination is given by w. This level may pertain to either a single storage unit or a collection of storage units at the destination; however, for the sake of modelling, we only deal with a combined aggregate storage level. Let SL 1 and SL 2 denote the minimum and maximum desired storage levels, respectively, at the client’s destination, which should be maintained to the extent possible in order to avoid penalties. Accordingly, let Π 1 > 00$]]> and Π 2 > 00$]]>, respectively, denote the daily penalties for each shortage or excess unit at the destination, where Π 1 > Π 2 {\Pi _{2}}$]]> is to emphasize the fact that shortage levels are more critical than excess levels. The permitted shortage and excess quantities at the destination with respect to the desired levels SL 1 and SL 2 are specified by the amounts A 1 and A 2 , respectively. Let b 1 ≡ SL 1 − A 1 and b 2 ≡ SL 2 + A 2 , and let UB > b 2 {b_{2}}$]]> be a sufficiently large upper bound on the maximum allowable storage level on any given day of the time-horizon. Storage levels falling below b 1 or in excess of b 2 (up to UB), while permitted, are highly undesirable, and incur significantly greater penalties, respectively, given by λ 1 > Π 1 {\Pi _{1}}$]]>, and λ 2 > Π 2 {\Pi _{2}}$]]> per unit.

Let R j denote the expected consumption rate at the destination on day j, for j = 1 , … , H. The different daily consumption rates arise from possible seasonal changes during the time-horizon, as well as from client-specific considerations. Thus, the total cumulative consumption at the destination over the days j = 1 , … , h is given by TC h = ∑ j = 1 h R j . The daily storage levels determine the overall penalty over the time-horizon, being given by the sum of Type I and Type II penalties as defined below, where S h denotes the storage level on day h and where Π 1 > 00$]]>, Π 2 > 00$]]>, λ 1 > Π 1 {\Pi _{1}}$]]> and λ 2 > Π 2 {\Pi _{2}}$]]> are as defined above.

Type I penalty: P I ( S h ) = Π 1 maximum { 0 , ( SL 1 − S h ) } if S h ∈ [ b 1 , S L 1 ) ,

P I ( S h ) = Π 2 maximum { 0 , ( S h − S L 2 ) } if S h ∈ ( SL 2 , b 2 ] ; and

Type II penalty: P II ( S h ) = Π 1 A 1 + λ 1 ( b 1 − S h ) if S h ∈ [ 0 , b 1 ) , Π 2 A 2 + λ 2 ( S h − b 2 ) if S h ∈ ( b 2 , UB ] .

Note that if S h ∈ [ S L 1 , S L 2 ], then the storage level lies within the desired bounds and no penalty is induced. If S h ∈ [ b 1 , SL 1 ) ∪ ( SL 2 , b 2 ], then a Type I penalty is incurred based on the respective shortage or excess quantity. On the other hand, if S h ∈ [ 0 , b 1 ) ∪ ( b 2 , UB ], then a relatively larger additional Type II penalty rate is imposed continuously beyond that of P I ( . ) to indicate the undesirabity of such a storage level on any given day of the time-horizon. Proposition 1.

Let (1) S h = S 1 , h − S 2 , h − S 3 , h + S 4 , h + S 5 , h , where (2) SL 1 ⩽ S 1 , h ⩽ SL 2 , 0 ⩽ S 2 , h ⩽ A 1 , 0 ⩽ S 3 , h ⩽ b 1 , 0 ⩽ S 4 , h ⩽ A 2 , and 0 ⩽ S 5 , h ⩽ U B − b 2 . Define the linear penalty function P ( S h ) = Π 1 S 2 , h + Π 2 S 4 , h + λ 1 S 3 , h + λ 2 S 5 , h , for 0 ⩽ S h ⩽ UB. Then any minimization objective formulation that incorporates the term P ( S h ) defined above along with (1) and (2) will automatically enforce the sum of the Type I and Type II penalties P I ( S h ) + P II ( S h ).

There are five permissible journey itineraries, denoted by J j , for j = 1 , … , 5, as defined next. Let i = 1 , 2 , 3 index the source, transshipment facility, and destination, respectively, and let SFD = { 1 , 2 , 3 }. Then the permissible journeys are given as follows:

J 1 = ( 1 , 3 , 1 ): source → destination → source;

J 2 = ( 1 , 2 , 1 ): source → transshipment facility → source;

J 3 = ( 1 , 3 , 2 ): source → destination → transshipment facility;

J 4 = ( 2 , 3 , 1 ): transshipment facility → destination → source;

J 5 = ( 2 , 3 , 2 ): transshipment facility → destination → transshipment facility.

Let J = ⋃ j = 1 5 J j , which is composed of all permissible journeys. For j = 1 , … , 5, let J j = ( p 1 j , p 2 j , p 3 j ) characterize the itinerary of Journey J j as identified above, where p 1 j , p 2 j , and p 3 j respectively represent the loading, unloading, and termination points of Journey J j , and let J j 1 ≡ ( p 1 j , p 2 j ) and J j 2 ≡ ( p 2 j , p 3 j ) denote the two sequential legs of this itinerary. For i = 1 , 2, let I i 1 be a subset of J such that i is the loading point of any journey in I i 1 . Hence, I 1 1 = { J 1 , J 2 , J 3 } and I 2 1 = { J 4 , J 5 }. For i = 2 , 3, let I i 2 be a subset of J such that i is the intermediate point (unloading point) of any journey in I i 2 . Thus, I 2 2 = { J 2 } and I 3 2 = { J 1 , J 3 , J 4 , J 5 }. For i = 1 , 2, let I i 3 be a subset of J such that i is the terminal point of any journey in I i 3 . Accordingly, I 1 3 = { J 1 , J 2 , J 4 } and I 2 3 = { J 3 , J 5 }.

The average speed (knots per hour) of a vessel of type t is given by V t F when the vessel is fully loaded, and V t E when the vessel is empty. The permitted average daily utilization of a vessel of type t (in hours per day) is given by DU t F when the vessel is fully loaded, and DU t E when the vessel is empty. Let C t PU , F and C t PU , E be the daily transportation costs associated with a vessel of type t when the vessel is fully loaded and empty, respectively. Note that C t PU , F and C t PU , E are calculated as the product of the vessel capacity Ω t and the per unit transportation cost when the vessel is fully loaded and empty, respectively.

Next, we compute the cost associated with journeys in the set J. For let, denote the distance (in knots) between α and β. For j ∈ { 1 , … , 5 }, let TD ( J j ) be the total distance that a vessel travels to complete Journey J j . Accordingly, we have TD ( J 1 ) = δ ( 1 , 3 ) + δ ( 3 , 1 ) , TD ( J 2 ) = δ ( 1 , 2 ) + δ ( 2 , 1 ) , TD ( J 3 ) = δ ( 1 , 3 ) + δ ( 3 , 2 ) , TD ( J 4 ) = δ ( 2 , 3 ) + δ ( 3 , 1 ) , and TD ( J 5 ) = δ ( 2 , 3 ) + δ ( 3 , 2 ) . Let ND t ( p 1 j , p 2 j ) = δ ( p 1 j , p 2 j ) V t F DU t F and ND t ( p 2 j , p 3 j ) = δ ( p 2 j , p 3 j ) V t E DU t E , where the former represents the average travel time (in days) required from point p 1 j to point p 2 j , and the latter represents the average travel time (in days) required from point p 2 j to point p 3 j . Also, let ND t j = ND t ( p 1 j , p 2 j ) + ND t ( p 2 j , p 3 j ) .

Hence, the transportation cost of a vessel of type t that undertakes journey J j is given by C t , j = C t , j F + C t , j E , where C t , j F = C t PU , F ND t ( p 1 j , p 2 j ) and C t , j E = C t PU , E ND t ( p 2 j , p 3 j ) represent the total transportation cost for a vessel of type t when this vessel is fully loaded and empty, respectively.

It is worth mentioning that the transportation costs of vessels were assumed to be fixed and known a priori in Sherali and Al-Yakoob (2006a, 2006b). In contrast, in this paper, the transportation costs are fixed for the journeys that do not involve the transshipment facility; otherwise, these costs are functions of the location of the transshipment facility.

Let H f ⊆ { 1 , … , H } be a contiguous subset of the time-horizon during which the transshipment facility will be available for leasing. Let h 1 f be the first day in H f and let w f denote the storage level at the transshipment facility on Day h 1 f . Let CD denote the capacity of the transshipment facility and let $ $ be a fixed cost (in U.S. dollars) of leasing the transshipment facility during H f , where we set $ $ ≡ 0 if the transshipment facility is owned by the company. Let MC be the daily maintenance cost of the transshipment facility, and so, | H f |MC represents the total maintenance cost over the entire leasing duration. Let AR define the allowable region in which the transshipment facility may be located.

Let η h 1 , t , j , h 1 and η h 1 , t , j , h 2 for ∀ h 1 , t , j , h > h 1 {h_{1}}$]]> be binary indicator parameters that take on a value of 1 if and only if the delivery date h of vessels of type t traverses journey J j starting on day h; exceeds the corresponding start time plus the journey duration, and serves to set the associated duration-incompatible u-variables that are defined subsequently to zero. Define x h , t , j to be an integer variable that represents the number of vessels of type t that traverse journey J j starting on day h; and let u h 1 , t , j , h 1 = x h 1 , t , j η h 1 , t , j , h 1 and u h 1 , t , j , h 2 = x h 1 , t , j η h 1 , t , j , h 2 , ∀ h 1 , t , j , h > h 1 {h_{1}}$]]>. Hence, u h 1 , t , j , h 1 is an integer variable that represents the number of vessels of type t-th at start journey J j on day h 1 and that arrive at point p 2 j on or before day h. Similarly, u h 1 , t , j , h 2 is an integer variable that represents the number of vessels of type t that start journey J j on day h 1 and that terminate this journey at point p 3 j on or before day h.

Define y h , t , s to be an integer decision variable that represents the maximum number of vessels of type t that are available for dispatching from the source (if s = 1) or from the transshipment facility (if s = 2) on day h.

As mentioned earlier, vessels might become available for use at different days of the time-horizon due to, for example, their involvement in trips from previous demand contracts that will terminate sometime during the current time-horizon. Hence, we let O h , t be the number of self-owned vessels of type t that will become available for use for the first time at the source on day h of the time-horizon, and we let CH h , t be the number of vessels of type t that will become available for the first time at the source for chartering on day h of the time-horizon. Let α h , t = O h , t + CH h , t and note that O t = ∑ h O h , t and CH t = ∑ h CH h , t . Accordingly, we let z h , t be an integer variable that denotes the number of vessels of type t that are actually selected (from among the CH h , t vessels) for chartering on day h of the time-horizon, and we let $ h , t denote the average chartering cost of a vessel of type t that will become available for potential use on day h of the time-horizon. Remark 2.

Tentatively, consider another binary variable W defined to take the value of one if the transshipment depot is leased during H f , and is zero otherwise. Note that W = 0 implies that the transshipment facility is not selected in the operation and hence only journey J 1 is permitted, whence the problem simplifies to that studied in Sherali and Al-Yakoob (2006b). Accordingly, for the sake of ease in modelling, we can consider two cases separately, namely, when W = 0 and W = 1, and compare the respective operational costs. Since the case W = 0 has been studied in Sherali and Al-Yakoob (2006b), we focus in the remainder of this paper on the case when the transshipment facility (to-be-located) is leased ( W = 1 ).

Let A h , t be the subset of indices for vessels of type t (both self-owned and vessels available for chartering) that will become available for use at the source for the first time on day h of the time-horizon. Hence, for a given day h and vessel type t, we let UT h , t = ∑ n ∈ A h , t UT t , n α h , t represent the average usage allowance for a vessel of type t that will become available for use for the first time on day h of the time-horizon. Accordingly, UT t = ∑ h α h , t U T h , t ∑ h α h , t gives the average usage allowance for a vessel of type t.

Note that in this aggregate model, the variable y h , t , s represents the maximum number of vessels of type t that could be consigned on day h as necessary; the actual number used, and in particular the chartering decisions, are governed in this model via the dispatching variables x h , t , j .

Let ϕ x , ϕ η 1 , ϕ η 2 , ϕ u 1 , ϕ u 2 , and ϕ y be the set of indices of the x-, η 1 -, η 2 -, u 1 -, u 2 - and y-variables, respectively, that are a priori restricted to be zero, or fixed at some known positive integer values.

The constraints of Model AF are formulated next below.

A) Examining arrival dates of shipments. Consider a journey J j ∈ J and a vessel of type t ∈ T. The following constraints assign proper binary values for the u 1 -variables based on whether this vessel starts journey j on day h 1 and delivers its shipload at location p 2 j on or before day h. Hence, Constraint (3) enforces the defined value of η h 1 , t , j , h 1 to equal to 1 or 0 based on whether h 1 + ND t ( p 1 j , p 2 j ) − h ⩽ 0 or h 1 + ND t ( p 1 j , p 2 j ) − h ⩾ 1, respectively, and then Constraints (4)–(6) along with u h 1 , t , j , h 1 ⩾ 0 enforce the product relationship u h 1 , t , j , h 1 = x h 1 , t , j η h 1 , t , j , h 1 , ∀ h 1 , t , j , h > h 1 {h_{1}}$]]>. (3) 1 − η h 1 , t , j , h 1 ( h + 1 ) ⩽ h 1 + ND t ( p 1 j , p 2 j ) − h ⩽ ( 1 − η h 1 , t , j , h 1 ) ( H − h ) , ∀ h 1 , t , j , h > h 1 , {h_{1}},\end{array}\]]]> (4) u h 1 , t , j , h 1 ⩽ x h 1 , t , j , ∀ h 1 , t , j , h > h 1 , {h_{1}},\]]]> (5) u h 1 , t , j , h 1 ⩽ M t η h 1 , t , j , h 1 , ∀ h 1 , t , j , h > h 1 , {h_{1}},\]]]> (6) u h 1 , t , j , h 1 ⩾ x h 1 , t , j − M t ( 1 − η h 1 , t , j , h 1 ) , ∀ h 1 , t , j , h > h 1 . {h_{1}}.\]]]>

B) Examining arrival dates of vessels. Consider a journey J j ∈ J and a vessel of type t. Similar to the above constraints for the u 1 -variable, the following constraints assign proper integer values for the u 2 -variables based on whether this vessel starts journey j on day h 1 and returns to location p 3 j on or before day h, where recall that u h 1 , t , j , h 2 = x h 1 , t , j η h 1 , t , j , h 2 , ∀ h 1 , t , j , h > h 1 {h_{1}}$]]>. Accordingly, Constraint (7) enforces the defined binary value of η h 1 , t , j , h 2 and then (8)–(10) (along with u h 1 , t , j , h 2 ⩾ 0 ) represent the stated product relationship. (7) 1 − η h 1 , t , j , h 2 ( h + 1 ) ⩽ h 1 + ND t j − h ⩽ ( 1 − η h 1 , t , j , h 2 ) ( H − h ) , ∀ h 1 , t , j , h > h 1 , {h_{1}},\end{array}\]]]> (8) u h 1 , t , j , h 2 ⩽ x h 1 , t , j , ∀ h 1 < h , t , j , (9) u h 1 , t , j , h 2 ⩽ M t η h 1 , t , j , h 2 , ∀ h 1 < h , t , j , (10) u h 1 , t , j , h 2 ⩾ x h 1 , t , j − M t ( 1 − η h 1 , t , j , h 2 ) , ∀ h 1 , t , j , h > h 1 . {h_{1}}.\]]]>

C) Representation of the destination’s storage level and related penalties. The daily storage level of the product at the destination must remain within [ SL 1 , SL 2 ] to the extent possible, where appropriate daily penalties are imposed otherwise based on the specific levels of the storage as discussed above (see Proposition 1). The representation of the storage level of the destination is given by the following constraints: (11) S h = w + ∑ h 1 < h ∑ t ∑ j ∈ I 3 2 Ω t u h 1 , t , j , h 1 − TC h , ∀ h , (12) S h = S 1 , h − S 2 , h − S 3 , h + S 4 , h + S 5 , h , ∀ h ∈ H .

Constraint (11) gives the storage level on day h based on the daily consumption rates and the shipments of the product that are delivered by day h. Constraint (12) represents S h in terms of S 1 , h , S 2 , h , S 3 , h , S 4 , h , and S 5 , h as discussed above in Proposition 1, where (2) holds, ∀ h, so that appropriate penalties would be incurred in the objective function based on this representation.

D) Representation of the transshipment facility’s storage level. The following constraint provides a representation of the daily storage level L h for day h at the transshipment facility, where LB f and UB f are some specified lower and upper bounds on the storage level of the facility: (13) L h = w f + ∑ h 1 f ⩽ h 1 < h ∑ t ∑ j ∈ I 2 2 Ω t u h 1 , t , j , h 1 − ∑ h 1 f ⩽ h 1 ⩽ h ∑ t ∑ j ∈ I 2 1 Ω t x h 1 , t , j , ∀ h ∈ H f , along with LB f ⩽ L h ⩽ UB f , ∀ h ∈ H f .

The storage level of the transshipment facility on a day h ∈ H f is determined based on the initial level w f of the facility on day h 1 f , plus the total amount of the product that is delivered to the facility by day h, and minus the total amount of the product that is shipped from the facility by day h.

E) Availability of vessels and vessel chartering. The vessel availability constraints are given as follows: (14) y h , t , s = y h − 1 , t , s − ∑ j ∈ I s 1 x h − 1 , t , j + ∑ h 1 < h ∑ j ∈ I s 1 ( u h 1 , t , j , h 2 − u h 1 , t , j , ( h − 1 ) 2 ) + O h , t + z h , t , h ⩾ 2 , t , s = 1 , 2 , (15) ∑ j ∈ I s 1 x h , t , j ⩽ y h , t , s , ∀ h ⩾ 2 , t , s = 1 , 2 , (16) y 1 , t , 1 = O 1 , t + z 1 , t , ∀ t , (17) z t = ∑ h z h , t , ∀ h .

A vessel of type t can be dispatched on a journey beginning from point s ∈ { 1 , 2 } on day h only if it is available at s on that day. This vessel is available at s on day h if either the vessel was available at s on the previous day and it was not dispatched, or this vessel was not available there during the previous day, but it became available on the current day. On the other hand, this vessel is unavailable on day h at location s if it was available there on the previous day and it was dispatched on that day (noting that N t j ⩾ 2 for all vessel-types t), or it was unavailable on the previous day and it did not arrive on the current day. Constraint (14) examines the availability of vessel s of type t at location s on day h by considering these cases, and then Constraint (15) permits the dispatching of vessels conditioned upon their availability. Note that the hand-side of Constraint (14) accounts for the first-time availabilities of the self-owned and chartered vessels via the last two terms, where Constraint (16) serves to initialize the y-variables for h = 1 (noting that y 1 , t , 2 = 0, ∀ t). Furthermore, (17) is a definitional constraint that computes the total number of vessels of type t that are actually selected for chartering.

F) Capacity restrictions and maintenance requirements. A production capacity or certain imposed daily supply quota Q restricts the maximum cumulative amount of the product that can be shipped from the source to the transshipment facility and to the destination over the periods 1 ⩽ h 1 ⩽ h, ∀ h. This restriction is represented by the following constraint: (18) ∑ h 1 ⩽ h ∑ t ∑ j ∈ I 1 1 Ω t x h 1 , t , j ⩽ h Q , ∀ h . Furthermore, for maintenance purposes, the following constraint enforces that vessel of type t can be used for at most UT t days during the time-horizon: (19) ∑ h ∑ j [ ND t j ] x h , t , j ⩽ UT t ( O t + z t ) , ∀ t . Other forms of scheduled maintenance restrictions can be also accommodated in the model by setting certain x-variables to zero. Note that if j ∈ { 2 , 3 , 4 , 5 }, i.e. J j is a journey that involves the transshipment facility, then [ ND t j ] x h , t , n , j is a product of two variables.

The objective function is composed of the following terms:

a) Operational costs of vessels (both self-owned and chartered): ∑ h ∑ t ∑ j C t , j x h , t , j .

b) Penalties resulting from violating the allowable storage level [ SL 1 , SL 2 ] at the destination, which are given as follows based on Proposition 1: ∑ h Π 1 S 2 , h + ∑ h Π 2 S 4 , h + ∑ h λ 1 S 3 , h + ∑ h λ 2 S 4 , h .

c) Chartering expenses of vessels: ∑ h ∑ t $ h , t z h , t .

d) Leasing and maintenance expenses of the transshipment facility: based on Remark 2, a fixed cost given by $ $ + ‖ H f ‖ MC is incurred in the objective function.

The objective function terms (a)–(d) along with the constraints formulated above yield the following model AF, where all indices are assumed to take on only their respective relevant values: AF : Minimize ∑ h ∑ t ∑ j C t , j x h , t , j + ∑ h Π 1 S 2 , h + ∑ h Π 2 S 4 , h + ∑ h λ 1 S 3 , h + ∑ h λ 2 S 4 , h + ∑ h ∑ t $ h , t z h , t , subject to (3)–(19).

Where,

x h , t , j ∈ { 0 , 1 , … , M t }, ∀ ( h , t , j ) ∉ ϕ x , and fixed at zero otherwise.

η h 1 , t , j , h 1 ∈ { 0 , 1 }, ∀ ( h 1 , t , j , h > h 1 ) ∉ ϕ η 1 {h_{1}})\notin {\phi _{{\eta ^{1}}}}$]]>, and fixed at zero or one otherwise.

η h 1 , t , j , h 2 ∈ { 0 , 1 }, ∀ ( h 1 , t , j , h > h 1 ) ∉ ϕ η 2 {h_{1}})\notin {\phi _{{\eta ^{2}}}}$]]>, and fixed at zero or one otherwise.

u h 1 , t , j , h 1 ⩾ 0, ∀ ( h 1 , t , j , h > h 1 ) ∉ ϕ u 2 {h_{1}})\notin {\phi _{{u^{2}}}}$]]>, and fixed at zero otherwise.

u h 1 , t , j , h 2 ⩾ 0, ∀ ( h 1 , t , j , h > h 1 ) ∉ ϕ u 2 {h_{1}})\notin {\phi _{{u^{2}}}}$]]>, and fixed at zero otherwise.

y h , t , s ∈ [ 0 , M t ], ∀ ( h ⩾ 2, t , s ∈ { 1 , 2 } ) ∉ ϕ y , and fixed at zero or a positive integer otherwise.

z h , t ∈ { 1 , … , CH h , t }, ∀ h , t.

S L 1 ⩽ S 1 , h ⩽ S L 2 , 0 ⩽ S 2 , h ⩽ A 1 , 0 ⩽ S 3 , h ⩽ b 1 , 0 ⩽ S 4 , h ⩽ A 2 .

0 ⩽ S 5 , h ⩽ UB − b 2 , ∀ h.

L B f ⩽ L h ⩽ UB f , ∀ h ∈ H f .

Noting that the transshipment facility location must belong to AR. Proposition 2.

If we enforce binary restrictions on the ( x , η 1 , η 2 )-variables, then the ( u 1 , u 2 , y , z )-variables will automatically turn out to be binary-valued when simply restricted to be continuous.