Scheduling upstream operations at inland petroleum refineries using a

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Scheduling upstream operations at inland petroleum refineries using a precedence-based formulation Pedro Carlos Pautasso, Diego Carlos Cafaro, and Jaime Cerda Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b05671 • Publication Date (Web): 01 Mar 2019 Downloaded from http://pubs.acs.org on March 4, 2019

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Industrial & Engineering Chemistry Research

SCHEDULING UPSTREAM OPERATIONS AT INLAND PETROLEUM REFINERIES USING A PRECEDENCE-BASED FORMULATION Pedro C. Pautasso, Diego C. Cafaro and Jaime Cerdá* INTEC (UNL - CONICET) Güemes 3450 - 3000 Santa Fe - ARGENTINA *Corresponding author: [email protected] ABSTRACT Several types of crude oils arrive at inland oil refineries to be transformed into different intermediate and finished products. Incoming crude oils should be cleverly blended to meet some quality specifications before processing them in the crude distillation units (CDUs). This requires a careful allocation of the available crude oils to the refinery tanks, and a proper sequence of the tanks feeding the same CDU. This work introduces a mixed-integer nonlinear programming (MINLP) formulation that uses general-precedence sequencing variables to choose the best ordering of operations in every tank. By using a rigorous objective function, the MINLP model provides the exact operating cost of the best solution found. The replacement of nonlinear component balances by tailor-made linear constraints in the MINLP leads to a tight mixed-integer linear (MILP) model that usually yields a good MINLP feasible point. A nonlinear programming (NLP) formulation that results by fixing the 0-1 variables to their MILP-values is subsequently solved to get a near-global optimal schedule. The proposed approach has been successfully applied to different instances of eight examples, four of which are benchmark problems already solved in previous contributions. Keywords: inland refineries, refinery operations, operational scheduling, MINLP formulation. ____________________

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*

Corresponding author. Telephone: +54 342 4559175. Fax: +54 342 4550944. E-mail

address: [email protected].


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1. INTRODUCTION Petroleum refineries are continuous-flow, manufacturing facilities transforming crude oils into finished products. At the front-end of the refining process, the crude oil is separated into several fractions that are subsequently converted into intermediate and finished products through a sequence of downstream physical and chemical transformations. At inland oil refineries, front-end operations begin with the unloading of crude oil parcels from large tankers via buoy mooring pipelines. Such parcels of crude oil temporarily stay in several storage tanks (ST) at the marine shipping terminal before transferring them via pipeline to multiple charging tanks (CHT) at the refinery site. To avoid operational problems, any storage tank must rest for some time after receiving a parcel of crude oil to separate the brine from the oil. During that period, no transfer of crude oil to charging tanks can be made. Mixing different types of crude oil in the charging tanks is necessary to meet the feedstock quality for the crude distillation units (CDUs) established at the planning level. In contrast, it is assumed that each storage tank at the marine terminal contains a single type of crude oil. One of the key properties used to evaluate the feedstock quality is the total sulfur content. However, a variety of properties like yields of some premium distillates, the true boiling point curve and the concentrations of other impurities are also used in practice to characterize the mixed crude oil (Reddy et al.1, Li et al.2, Liang et al.3, Cerdá et al.4,5). Therefore, the schedule of inlet and outlet operations in every charging tank is quite relevant to reach such specifications. Any batch of crude oil coming from a storage tank must be sent to the appropriate charging tank at the right time and in the right amount. In addition, the crude oil inventory in storage and charging tanks must be closely monitored to prevent run-out and overfilling conditions. For operational reasons, a charging tank cannot feed a CDU while it is receiving a batch from a storage tank. In turn, a crude distillation unit must continually receive feedstock from the



charging tanks but one at a time. As a result, a feasible schedule must satisfy a series of operational rules, material balance equations, minimum/maximum tank inventories and the specified amount and quality of the feedstock to be supplied by each charging tank to the CDUs. The problem goal is to minimize the total operating cost comprising the vessel seawaiting cost, the unloading cost of crude oil from marine vessels, the holding cost of the inventory in storage and charging tanks, and the CDU feedstock changeover cost. A mathematical formulation for the problem must include 0-1 variables to decide on the allocation of batches to tanks and CDUs, the ordering of loading/unloading operations performed at every tank, and the sequence of feedstock streams supplied to each CDU. In addition, mass balance equations for the trace elements involving bilinear terms should be considered to determine the feedstock quality. Consequently, the scheduling of crude oil operations is usually modeled as a non-convex mixed-integer nonlinear programming (MINLP) problem whose solution requires a high computational cost if the global optimum is pursued. This work introduces a new batch-oriented continuous-time MINLP formulation for the scheduling of crude oil operations at inland refineries that aims to minimize the exact total operating cost. It uses three independent sets of batches for vessels, storage and charging tanks, whose elements are chronologically ordered, and general-precedence sequencing variables for ordering the operations in every tank. Moreover, direct precedence variables defined in terms of the general-precedence ones permit to derive a rigorous objective function that provides the exact operating cost. A near global optimum of the MINLP is usually found by sequentially solving a mixed-integer linear programming (MILP) model and a non-linear programming (NLP) formulation. The MILP problem representation is obtained by relaxing the MINLP feasible region and considering both an approximate linear objective function and an additional set of strong valid constraints based on direct precedence variables. In this way, a good initial


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point for the NLP to be subsequently solved is usually found. This two-stage solution procedure allows discovering near global optimal schedules for several instances of eight examples at reasonable computational times. 2. PREVIOUS CONTRIBUTIONS A significant number of papers on the scheduling of refinery operations have been published in the last two decades. Many of them deal with the scheduling of upstream operations at inland oil refineries. Some other papers are focused on the scheduling of crude oil operations in coastal refineries (Reddy et al.1, Li et al.2, Liang, et al.3, Saharidis et al.6, Saharidis and Ierapetritou7, Pan et al.8, Li et al.9, Cerdá et al.10). A summary of previous contributions on the scheduling of front-end operations at inland oil refineries is next presented. Lee et al.11 proposed an MILP model based on a uniform time discretization of the planning horizon. To avoid nonlinearities, they replace the bilinear equations arising because of the mixing operations with individual component flows. The objective function seeks to minimize the total operating cost, involving the crude vessel harboring, the sea-waiting cost, the CDU feedstock changeover cost and oil inventory carrying costs. Four real-world examples were introduced and solved. Jia et al.12 studied the same short-term crude oil scheduling problem using an MILP continuous-time model based on a state-task network (STN) representation involving fewer variables and constraints. Furman et al.13 developed a non-convex event-based MINLP model where nonlinear constraints are considered to properly calculate the key component concentrations in storage and charging tanks. No simultaneous input and output flows at any tank are permitted but the same type of flows (input or output) can occur at the same time. To reduce the model size, input and output flows can be assigned to the same time event but input flows should always



occur before. McCormick linear envelopes providing tight convex relaxations of the bilinear constraints were also added to the formulation. The resulting MINLP model was solved using the convex MINLP solver DICOPT to find locally optimal solutions. Karuppiah et al.14 presented a nonconvex MINLP formulation for the scheduling of crude oil movement at the front-end of a petroleum refinery. It was solved by using an outer approximation algorithm to find a global optimum. An MILP relaxation is derived from the MINLP model to obtain a rigorous lower bound on the global optimum. Cutting planes are added to the MILP relaxation to reduce the CPU time. The solution of the MILP is used to obtain a feasible solution to the MINLP that serves as an upper bound by fixing the decision variables and solving the resulting NLP model. Lower and upper bounds are made to converge to within a specified tolerance by the outer-approximation algorithm. Mouret et al.15 introduced an MINLP continuous-time model for the crude-oil scheduling problem called single operation sequencing (SOS) model. The schedule of refinery operations is conceived as a sequence of priority slots with each one assigned to a single operation. As they correspond to a position in the sequence, the priority slots also serve to order the execution of the refinery operations. The sequence scheme allows to removing symmetric solutions, and by so doing reduce both the size of the search space and the solution time. The only parameter that should be postulated is the total number of operations to be performed. In contrast to previous works, it is used a linear gross margin maximization objective instead of the nonlinear minimization of the total logistics cost. A simple two-step MILP-NLP procedure has been used to solve the nonconvex MINLP. The MILP relaxation is derived from the MINLP by ignoring the nonlinear blending constraints. The binary variables are then fixed and the resulting NLP is solved using the MILP solution as the starting point. Solutions with an optimality gap lower than 4% in all cases have been found. In turn, Mouret et al.16 developed the so-called multi-


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operation sequencing (MOS) model that is a nonconvex MINLP. In the MOS representation, several operations can be assigned to each priority slot only if they can overlap with each other. However, operations that cannot overlap should be allocated to different slots. Compared with the SOS model, the MOS time representation is superior because it allows efficient symmetrybreaking and requires a lower number of priority slots. The nonconvex MINLP is solved using an MILP-NLP procedure. Yadav and Shaik17 proposed a simplified state-task-network (STN) formulation that uses a continuous-time representation, unit-specific events, and incorporates the notion of storage tasks to handle material transfers from storage and charging tanks. In addition, simultaneous loading and unloading operations in storage tanks are permitted. Castro and Grossmann18 approached the solution of the crude oil scheduling problem using a single time-grid continuous-time MINLP model based on a resource-task network (RTN) superstructure. The proposed formulation has the advantage of avoiding computationally inefficient big-M constraints and handling variable recipe tasks with multiple input materials. The problem goal is to identify the optimal mix of low-cost and premium crudes that maximizes profit or minimizes operating cost while meeting the operational constraints. By solving a set of benchmark problems, it was shown that the MINLP can be solved close to global optimality by the commercial solver GloMIQO when the gross margin is maximized but not for the operating cost minimization. However, the optimality gap can be reduced by orders of magnitude using a two-step MILP-NLP algorithm with mixed-integer linear relaxation derived from multiparametric disaggregation. Recently, Castro19 proposed a source-based MINLP formulation for the crude oil pooling problem using discrete and continuous time representations. He found that the discrete-time approach provides better solutions when the total operating cost is minimized.



This paper presents a new batch-oriented general-precedence MINLP approach where the blending of different types of crude oil is only allowed in charging tanks and the blend quality is determined by the concentrations of a small number of trace elements. Moreover, loading and unloading operations in storage and charging tanks cannot be performed simultaneously. 3. PROBLEM STATEMENT The front-end configuration of an inland oil refinery comprises a single docking station to unload crude oils from marine vessels (V), several dedicated storage tanks (K) at the marine terminal to temporarily store the crude oil, a series of refinery charging tanks or mixers (M) to prepare different types of crude blends for the CDUs, the crude distillation units (U), and the pipelines connecting the docking terminal to the storage tanks, the storage tanks to the charging tanks and these ones to the separation units (see Figure 1). Moreover, the set J includes some trace elements present in the crude oil, like sulfur, whose concentrations in charging tanks should remain within the specified limits.

Figure 1. Front-end refinery network


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Three different types of operations should be carried out: (a) unloading parcels of crude oil from marine vessels into dedicated storage tanks at the marine terminal; (b) transferring lots of different types of crude oil via pipeline from storage tanks into charging tanks at the refinery site; and (c) discharging batches of qualified feedstock from charging tanks to continually feed the distillation units. Mixing different types of crude oil in charging tanks is required to meet the feedstock properties specified at the corporate level. Several operational rules must be respected. In any storage/charging tank, inlet and outlet operations should not overlap. As they are transported via pipeline to the refinery site, batches of crude oil coming from the storage tanks are sequentially loaded into the charging tanks. Moreover, batches are to be unloaded from every tank one-by-one, and each CDU must be fed by a single charging tank at any time. Another important constraint consists on keeping the trace element concentrations in every batch of mixed crude oil unloaded from a charging tank within the concentration range specified at the planning level. In other words, the concentration limits of trace elements in charging tanks are enforced just before feeding the CDUs. Previous contributions imposed such constraints at every point in time which hinders finding better solutions. Other data for the problem are the vessel arrival dates (atv), the amount (av) and composition (icvj,v) of the crude oil transported by each tanker v ∈ V, the total demand of crude mix to be supplied by every charging tank (demm), the min/max capacities of storage (capk / invkk,min ) and charging tanks (capm / invmm,min), the initial content and composition of the crude oil available in every storage tank (iikk , ickj,k) and charging tank (iimm , icmj,m), the discharge flow-rate range from the vessels (rvmin , rvmax), the limiting concentrations of trace elements in charging tanks (cmlj,m , cmuj,m), the cost coefficients, and the length of the scheduling horizon (H). Based on that information, the refinery scheduler should determine the best sequence and timing of the operations performed in every tank so as to minimize the total operating cost,



while satisfying all the operational constraints. 4. MODEL ASSUMPTIONS The proposed MINLP formulation is based on the following assumptions: (1) The amount and composition of the crude oil transported by every marine vessel as well as their arrival times at the docking station are known data. (2) Only a single buoy mooring pipeline for unloading crude oil from marine vessels is available. Then, vessels must be serviced one at a time by arriving order. (3) The total amount of crude oil contained in each marine vessel should be unloaded within the scheduling horizon. Hence, vessels should depart before the end of the time horizon. (4) Crude oil unloaded from a vessel can be stored into one or more dedicated storage tanks, but there is at most one storage tank connected to the vessel at any time. In other words, storage tanks are sequentially loaded and they contain a single type of crude oil. (5) A storage tank can supply crude oil to several charging tanks, but one at a time. (6) Input and output flows into/from a storage tank cannot occur simultaneously. Then, a storage tank cannot send crude oil to a charging tank while it is receiving a parcel of crude oil from a vessel. (7) A charging tank can sequentially receive crude oil from multiple storage tanks. (8) In every charging tank, the input flow from some storage tank cannot coexist with the output flow directed to some CDU. (9) A charging tank can feed different CDUs, but one at a time. (10) Every crude distillation unit must continually receive crude blend from charging tanks, but from one at any time. (11) There is a perfect mixing in each charging tank and the output flow and the crude mix in


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the tank have the same composition. (12) The mixing time of different types of crude oil in charging tanks is neglected. (13) The quality of the crude oil in storage and charging tanks is given in terms of the concentration of key trace elements. 5. THE MINLP MATHEMATICAL MODEL The problem of scheduling front-end refinery operations will be modeled through an MINLP continuous-time formulation using general precedence sequencing variables for ordering loading and unloading operations at storage and charging tanks (Méndez et al.20). Symmetrical solutions are avoided by using sets of generic batches of crude oil coming from marine vessels (IV), storage tanks (IK) and charging tanks (IM), respectively, that are chronologically ordered. In other words, a different set of batches for each type of oil movement is defined with the number of elements in each set postulated by the user. Some rules to choose their cardinalities are given in Section 6.3. Generic batches in the sets IV, IK, and IM are said to be chronologically ordered because the unloading of the element i+1 must never begin before starting the dispatch of the batch i. Moreover, batches i and i’ (with i’ > i) coming from the same or different storage tanks (or vessels) are sequentially discharged. Then, the unloading of lot i’ must start after completing the dispatch of batch i < i’. 5.1 Model variables The MINLP model includes binary variables to assign parcels of crude oil to vessels {WViv,v} and storage tanks {WKik,k}, and batches of crude mix to charging tanks through {WMim,m}. Moreover, any batch should have at most a single destination. Then, additional binary variables {YViv,k , YKik,m , YMim,u} are introduced to assign a destination for every existing batch. If YViv,k = 1, the lot iv is loaded into the storage tank k. Besides, YKik,m = 1



indicates that the batch ik ∈ IK is sent to the charging tank m ∈ M, and YMim,u = 1 means that the batch of crude mix im ∈ IM is supplied to the distillation unit u ∈ U. The proposed model also includes general-precedence sequencing variables {XKiv,ik,k , XMik,im,m} to arrange loading and unloading operations in storage and charging tanks, respectively. If XKiv,ik,k = 1, then the lot iv ∈ IV is received before unloading the batch ik from the storage tank k in case both batches have been assigned to that tank (YViv,k = WKik,k = 1). If XKiv,ik,k = 0 and both operations take place at the same storage tank k ∈ K, then the batch ik is completely unloaded before receiving the lot iv. The sequencing variable XMik,im,m will play the same role for charging tanks. The proposed MINLP formulation also includes several sets of continuous variables that stand for: (a) the size of batches unloaded from marine vessels, storage and charging tanks, denoted by QViv,v , QKik,k , and QMim,m; (b) the starting times {SViv , SKik, SMim} and the end times {CViv , CKik , CMim} of those operations; (c) the cumulative amount of crude oil discharged into the storage tank k just after receiving the batch iv or dispatching the lot ik, given by 𝐿𝐾𝑉𝑘,𝑖𝑣 and 𝐿𝐾𝐾𝑘,𝑖𝑘; (d) the cumulative amount of crude oil unloaded from the storage tank k at the completion times CViv and CKik, represented by 𝑈𝐾𝑉𝑘,𝑖𝑣 and 𝑈𝐾𝐾𝑘,𝑖𝑘; (e) the inventory level of crude oil in the storage tank k at times CViv and CKik, given by 𝐼𝐾𝑉𝑘,𝑖𝑣 and 𝐼𝐾𝐾𝑘,𝑖𝑘, respectively; (f) the cumulative amount of crude oil sent to the charging tank m at times CKik and CMim, denoted by 𝐿𝑀𝐾𝑚,𝑖𝑘 and 𝐿𝑀𝑀𝑚,𝑖𝑚; (g) the cumulative amount of crude blend unloaded from the charging tank m at times CKik and CMim, given by 𝑈𝑀𝐾𝑚,𝑖𝑘 and 𝑈𝑀𝑀𝑚,𝑖𝑚; (h) the inventory level of crude blend in the charging tank m at times CKik and CMim, given by 𝐼𝑀𝐾𝑚,𝑖𝑘 and 𝐼𝑀𝑀𝑚,𝑖𝑚. To monitor the trace element concentrations in charging


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tanks, the model includes the continuous variable CMMj,m,im denoting the concentration of the trace element j in tank m while unloading the batch im. 5.2 Model Constraints Model constraints have been grouped into six subsets of equations dealing with: (a) the unloading of crude oil from marine vessels; (b) the transfer of crude oil via pipeline from storage tanks to charging tanks; (c) the tracking of inventory levels in storage tanks; (d) the transfer of crude mix from each charging tank to the CDUs so as to satisfy the specified feedstock demand; (e) the monitoring of inventory levels in charging tanks; and (f) the control of the trace element concentrations in charging tanks during unloading operations. 5.2.1 Unloading parcels of crude oil from marine vessels into storage tanks Allocating lots of crude oil to vessels. The total amount of crude oil transported by the marine vessels and unloaded at the docking station is divided into a number of lots iv ∈ IV, each one coming from at most a single vessel v ∈ V. As already explained, the binary variable WV iv,v indicates that the lot iv comes from vessel v if WV iv,v is equal to one. Otherwise, the element iv has been allocated to another vessel or it is a dummy lot. To avoid symmetrical solutions, the elements of IV have been pre-ordered in such a way that the unloading of lot iv ends before beginning the discharge of lot iv' whenever iv < iv'. ∑𝑣

∈ 𝑽

𝑊𝑉 𝑖𝑣,𝑣 ≤ 1

∀ 𝑖𝑣 ∈ 𝑰𝑽

(1)

Dummy lots of crude oil from vessels. A generic lot of crude oil iv could exist (i.e., ∑𝑣 ∈ 𝑽𝑊𝑉𝑖𝑣,𝑣 = 1) only if the preceding lot (iv-1) in the set IV has also been unloaded. By



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Eq. (2), dummy lots will be the last elements of the set IV. In this way, symmetrical solutions can be avoided. ∑𝑣

∈ 𝑽

𝑊𝑉 𝑖𝑣,𝑣 ≤ ∑𝑣 ∈

𝑽

𝑊𝑉 (𝑖𝑣 ― 1),𝑣

∀ (𝑖𝑣 ― 1),𝑖𝑣 ∈ 𝑰𝑽

(2)

Allocating lots of crude oil to storage tanks. Eq. (3) states that every lot discharged from a vessel should be assigned to a single storage tank. ∑𝑘

∈ 𝑲

𝑌𝑉 𝑖𝑣,

𝑘

= ∑𝑣 ∈ 𝑽𝑊𝑉 𝑖𝑣,𝑣

∀ 𝑖𝑣 ∈ 𝑰𝑽 (3)

Sizing batches of crude oil. The continuous variable QV iv,v represents the size of lot iv unloaded from vessel v. If lot iv does exist ( ∑𝑣 ∈ 𝑽 𝑊𝑉𝑖𝑣,𝑣 = 1 ), it will have a finite size within the range [sv min , sv max ] as stated by Eq. (4). Dummy elements have a null size. 𝑠𝑣 𝑚𝑖𝑛 𝑊𝑉 𝑖𝑣,𝑣 ≤ 𝑄𝑉 𝑖𝑣,𝑣 ≤ 𝑠𝑣 𝑚𝑎𝑥 𝑊𝑉 𝑖𝑣,𝑣

∀𝑖𝑣 ∈ 𝑰𝑽 , 𝒗 ∈ 𝑽

(4)

Amount of crude oil in the lot iv sent to the storage tank 𝑘 ∈ 𝑲 . The continuous variable FViv,v,k denotes the amount of crude oil in the lot iv transferred from vessel v to the storage tank 𝑘 ∈ 𝑲 𝑣. Its value is determined by Eqs (5a) and (5b), where Kv is the subset of storage tanks that can receive crude oil from vessel v. If batch iv is not assigned to tank k, FViv,v,k is equal to zero. Otherwise, it is equal to the size of lot iv given by 𝑄𝑉𝑖𝑣,𝑣. At most one term of the summation on the LHS of Eq. (5b) will be greater than zero, i.e. the one related to the vessel v and the storage tank k being the source and the destination of lot iv. 𝐹𝑉 𝑖𝑣,𝑣,𝑘 ≤ 𝑠𝑣 𝑚𝑎𝑥 𝑌𝑉 𝑖𝑣,𝑘 ∑𝑘

∈ 𝑲𝒗

𝐹𝑉 𝑖𝑣,𝑣,𝑘 = 𝑄𝑉 𝑖𝑣,𝑣

∀𝑖𝑣 ∈ 𝑰𝑽, 𝑣 ∈ 𝑉 , 𝑘 ∈ 𝑲 𝒗 ∀ 𝑖𝑣 ∈ 𝑰𝑽 , 𝒗 ∈ 𝑽


(5a)

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(5b) Length of pumping runs unloading batches of crude oil from the vessels. Let the continuous variable LViv be the length of the pumping run discharging lot iv from a vessel. If the parameters {rvmin , rvmax} represent the minimum/maximum pumping rates, then the relationship between the length of the pumping run (LViv) and the size of lot iv (∑𝑣 ∈ 𝑽𝑄𝑉𝑖𝑣,𝑣) is given by Eq. (6). 𝑟𝑣 𝑚𝑖𝑛 𝐿𝑉 𝑖𝑣 ≤ ∑𝑣

∈ 𝑽

𝑄𝑉 𝑖𝑣,𝑣 ≤ 𝑟𝑣 𝑚𝑎𝑥 𝐿𝑉 𝑖𝑣

∀𝑖𝑣 ∈ 𝑰𝑽

(6)

Starting and completion times for unloading operations of crude oil from the vessels. As stated by Eq. (7), the pumping run discharging lot iv from a vessel cannot start before the arrival time of vessel v at the docking station. 𝑆𝑉 𝑖𝑣 ≥ ∑𝑣

∈ 𝑽

𝑎𝑡 𝑣 𝑊𝑉 𝑖𝑣,𝑣

∀𝑖𝑣 ∈ 𝑰𝑽

(7)

The unloading of vessel v begins not earlier than the time at which the discharge of the first parcel of crude oil starts. Then, the unloading of vessel v starts at time STVv, given by Eq. (8a). The second term on the RHS of Eq. (8a) relaxes the constraint if lot iv is not the first one unloaded from vessel v. From Eq. (8b), vessels are discharged based on the first-come firstserved rule. H is the selected length of the planning horizon. 𝑆𝑉 𝑖𝑣 ≤ 𝑆𝑇𝑉 𝑣 + 𝐻 ∑ 𝑖𝑣′ ∈ 𝑰𝑽 𝑊𝑉 𝑖𝑣′,𝑣 + 𝐻 (1 ― 𝑊𝑉 𝑖𝑣,𝑣)

∀𝑖𝑣 ∈ 𝑰𝑽 , 𝑣 ∈ 𝑉

𝒊𝒗 ′ < 𝒊𝒗

(8a) 𝑆𝑉 𝑖𝑣 ≤ 𝑆𝑇𝑉 𝑣′ ― 𝐿𝑉 𝑖𝑣 + 𝐻 (1 ― 𝑊𝑉 𝑖𝑣,𝑣)

∀𝑖𝑣 ∈ 𝑰𝑽 , 𝑣,𝑣 ′ ∈ 𝑉(𝑣 < 𝑣 ′)

(8b)

Unloading lots of crude oil from vessels one at a time. Lots are discharged from vessels one by one because a single buoy mooring pipeline is available. To avoid symmetrical solutions, lot iv



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is unloaded before starting the dispatch of lot iv´ if iv < iv´. In other words, lots of crude oil are transferred to storage tanks in the same order that they appear in the set IV. Besides, the unloading operations must be performed within the scheduling horizon. Both conditions are imposed through Eqs (9a) and (9b). 𝐶𝑉 𝑖𝑣 ≤ 𝑆𝑉 𝑖𝑣′ 𝐶𝑉 𝑖𝑣 ≤ 𝐻

∀ 𝑖𝑣,𝑖𝑣 ′ ∈ 𝑰𝑽 (𝑖𝑣 < 𝑖𝑣 ′)

(9a) (9b)

∀ 𝑖𝑣 ∈ 𝑰𝑽

Lengths of unloading operations at the vessels. As stated by Eq. (10), the difference between the completion and the starting times is the length of the unloading operation of lot iv (LViv). 𝐿𝑉 𝑖𝑣 = 𝐶𝑉 𝑖𝑣 ― 𝑆𝑉 𝑖𝑣

∀𝑖𝑣 ∈ 𝐼𝑽

(10)

Complete unloading of every vessel. The overall amount of crude oil transported by a vessel should be completely discharged within the scheduling horizon. If the parameter av denotes the initial content of crude oil in vessel v, Eq. (11) states that the total amount of crude oil unloaded from vessel v over the planning horizon should be equal to av. ∑ 𝑖𝑣

∈ 𝑰𝑽

𝑄𝑉 𝑖𝑣,𝑣 = 𝑎 𝑣

∀𝑣 ∈𝑽

(11)

5.2.2 Transferring lots of crude oil from storage tanks to charging tanks A set of equations structurally similar to constraints (1)-(6) permits to determine the subset of batches 𝑖𝑘 ∈ 𝑰𝑲 unloaded from each storage tank k (WKik,k), their sizes (QKik,k), their destination tanks (YKik,m), the amount of crude oil in every lot transferred to charging tanks (FKik,k,m), and the length of each output operation (LKik). The expressions of such equations are given as Supporting Information. Similarly to the vessels, batches are sequentially unloaded from the storage tanks.


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Sequencing unloading operations from storage tanks. Because IK is a preordered set of batches and lots of crude oil are unloaded from the storage tanks one at a time, Eq. (12a) states that the discharge of lot ik' never starts before completing the unloading of lot ik whenever ik < ik'. This condition holds even if those batches come from different storage tanks, because the transfer pipeline connecting storage to charging tanks is unique. Eq. (12b) indicates that all the operations in storage tanks must be completed within the planning horizon. The length of such operations is given by Eq. (13). 𝐶𝐾 𝑖𝑘 ≤ 𝑆𝐾 𝑖𝑘′ +𝐻 (2 ― 𝑊𝐾 𝑖𝑘,𝑘 ― 𝑊𝐾 𝑖𝑘′,𝑘′)

∀(𝑖𝑘,𝑖𝑘 ′) ∈ 𝑰𝑲 (𝑖𝑘 < 𝑖𝑘 ′), (𝑘,𝑘 ′) ∈ 𝑲

(12a)

𝐶𝐾 𝑖𝑘 ≤ 𝐻

∀ 𝑖𝑘 ∈ 𝑰𝑲

(12b)

𝐶𝐾 𝑖𝑘 = 𝑆𝐾 𝑖𝑘 + 𝐿𝐾 𝑖𝑘

∀𝑖𝑘 ∈ 𝑰𝑲

(13)

5.2.3 Controlling the inventory level of crude oil in storage tanks The inventory of crude oil in storage tanks should be monitored over time to prevent overfilling after a loading operation, and running-out conditions after discharging a batch. Therefore, the important time events for the problem are the times at which input and output operations are completed. Overloading and running-out conditions can be avoided by properly controlling inventory levels at those key time events. The inventory level in storage tanks is determined by making the difference between the cumulative amounts of crude oil loaded and discharged up to the key time events. This computational scheme requires knowing the sequence of loading and unloading operations in every tank. Sequencing loading and unloading operations in storage tanks. Let us use the binary variable 𝑋𝐾 𝑖𝑣,𝑖𝑘,𝑘 to indicate that the loading of lot iv is completed before starting the discharge of lot ik whenever both lots have been assigned to tank k ∈ K (i.e. YViv,k = WKik,k =



Page 18 of 56

1) and 𝑋𝐾 𝑖𝑣,𝑖𝑘,𝑘 = 1. In contrast, the unloading of lot ik occurs before receiving lot iv if 𝑋𝐾 𝑖𝑣,𝑖𝑘,𝑘 = 0 and YViv,k = WKik,k = 1. Such conditions are imposed by Eqs (14a) and (14b). The parameter bst stands for the brine settling time when a new parcel is received from the vessels. 𝐶𝑉 𝑖𝑣 ≤ 𝑆𝐾 𝑖𝑘 ―𝑏𝑠𝑡 + 𝐻 (1 ― 𝑋𝐾 𝑖𝑣,𝑖𝑘,𝑘)

∀ 𝑖𝑣 ∈ 𝑰𝑽, 𝑖𝑘 ∈ 𝑰𝑲, 𝑘 ∈ 𝑲

𝐶𝐾 𝑖𝑘 ≤ 𝑆𝑉 𝑖𝑣 +𝐻 𝑋𝐾 𝑖𝑣,𝑖𝑘,𝑘 +𝐻 (2 ― 𝑌𝑉 𝑖𝑣,𝑘 ― 𝑊𝐾 𝑖𝑘,𝑘)

∀ 𝑖𝑣 ∈ 𝑰𝑽, 𝑖𝑘 ∈ 𝑰𝑲,𝑘 ∈ 𝑲

(14a) (14b)

Moreover, Eqs (15a) and (15b) drive 𝑋𝐾 𝑖𝑣,𝑖𝑘,𝑘 to zero if at least one of the lots iv or ik has not been assigned to the storage tank k. In that case, the constraints (14a) and (14b) become redundant. 𝑋𝐾 𝑖𝑣,𝑖𝑘,𝑘 ≤ 𝑌𝑉 𝑖𝑣,𝑘


(15a)

𝑋𝐾 𝑖𝑣,𝑖𝑘,𝑘 ≤ 𝑊𝐾 𝑖𝑘,𝑘


(15b)

Cumulative amount of crude oil received by a storage tank up to the completion times of inlet and outlet operations. The continuous variable 𝐿𝐾𝑉𝑘,𝑖𝑣 represents the cumulative amount of crude oil loaded into the storage tank k at time CViv. The value of LKVk,iv is determined by Eqs (16a) and (16b) that account for all lots of crude oil iv’ ≤ iv discharged from the vessels into the storage tank k up to time CViv. If lot iv is not allocated to tank k, then LKVk,iv = 0. The parameter MQ is an upper bound on the total amount of material received by any storage or charging tank over the scheduling horizon. 𝐿𝐾𝑉 𝑘,𝑖𝑣 ≥ ∑𝑖𝑣′ ∈

𝐹𝑉 𝑖𝑣′,𝑣,𝑘 ― 𝑀 𝑄 (1 ― 𝑌𝑉 𝑖𝑣,𝑘 )

∀ 𝑖𝑣 ∈ 𝑰𝑽 , 𝑘 ∈ 𝑲

(16a)

𝐿𝐾𝑉 𝑘,𝑖𝑣 ≤ ∑𝑖𝑣′ ∈

𝐹𝑉 𝑖𝑣′,𝑣,𝑘

∀ 𝑖𝑣 ∈ 𝑰𝑽 , 𝑘 ∈ 𝑲

(16b)

𝑰𝑽 ∑ 𝑣 ∈ 𝑽𝒌 𝑖𝑣 ′ ≤ 𝑖𝑣 𝑰𝑽 ∑ 𝑣 ∈ 𝑽𝒌 𝑖𝑣 ′ ≤ 𝑖𝑣

In turn, 𝐿𝐾𝐾 𝑘,𝑖𝑘 is a new continuous variable representing the cumulative amount of crude oil loaded into the storage tank k up to time CKik. Due to Eqs. (17a) and (17b), its value


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will be: (i) never smaller than 𝐿𝐾𝑉 𝑘,𝑖𝑣 if the loading of lot iv occurs before delivering the lot ik from the same storage tank k (i.e. 𝑋𝐾 𝑖𝑣,𝑖𝑘,𝑘 = 1), or (ii) never greater than 𝐿𝐾𝑉 𝑘,𝑖𝑣 minus the amount of lot iv discharged into the tank k whenever 𝑋𝐾 𝑖𝑣,𝑖𝑘,𝑘 = 0. If one of the lots iv or ik has not been assigned to the storage tank k, Eqs (17a) and (17b) for such a pair of batches become redundant. 𝐿𝐾𝑉 𝑘,𝑖𝑣 ≤ 𝐿𝐾𝐾 𝑘,𝑖𝑘 + 𝑀 𝑄 (1 ― 𝑋𝐾 𝑖𝑣,𝑖𝑘,𝑘) 𝐿𝐾𝐾 𝑘,𝑖𝑘 ≤ 𝐿𝐾𝑉 𝑘,𝑖𝑣 ― ∑𝑣

∈ 𝑽𝒌

∀ 𝑖𝑣 ∈ 𝑰𝑽 , 𝑖𝑘 ∈ 𝑰𝑲, 𝑘 ∈ 𝑲

(17a)

𝐹𝑉 𝑖𝑣,𝑣,𝑘 + 𝑀 𝑄 𝑋𝐾 𝑖𝑣,𝑖𝑘,𝑘 + 𝑀 𝑄 (2 ― 𝑌𝑉 𝑖𝑣,𝑘 ― 𝑊𝐾 𝑖𝑘,𝑘) ∀ 𝑖𝑣 ∈ 𝑰𝑽 , 𝑖𝑘 ∈ 𝑰𝑲, 𝑘 ∈ 𝑲

(17b)

Moreover, the value of 𝐿𝐾𝐾 𝑘,𝑖𝑘 must never be greater than the total amount of crude oil transferred from the vessels to the storage tank k ∈ K all over the time horizon. Variables LKVk,iv and LKKk,ik are driven to zero if either lot iv or lot ik is not assigned to tank k, respectively. Such conditions are enforced through Eqs (18a)-(18c). By using Eqs (16a)-(16b) and (18b)-(18c), a more precise tracking of the crude oil inventory in storage tanks is achieved. 𝐿𝐾𝐾 𝑘,𝑖𝑘 ≤ ∑𝑖𝑣

∈ 𝑰𝑽 ∑

𝑣 ∈ 𝑽𝒌

𝐿𝐾𝑉 𝑘,𝑖𝑣 ≤ 𝑀 𝑄 𝑌𝑉 𝑖𝑣,𝑘 𝐿𝐾𝐾 𝑘,𝑖𝑘 ≤ 𝑀 𝑄 𝑊𝐾 𝑖𝑘,𝑘

𝐹𝑉 𝑖𝑣,𝑣,𝑘

∀ 𝑖𝑘 ∈ 𝑰𝑲 ,𝑘 ∈ 𝑲

(18a)

∀ 𝑖𝑣 ∈ 𝑰𝑽 , 𝑘 ∈ 𝑲

(18b)

∀ 𝑖𝑘 ∈ 𝑰𝑲 , 𝑘 ∈ 𝑲

(18c)

Cumulative amount of crude oil unloaded from a storage tank at the completion times of input and output operations. The continuous variable 𝑈𝐾𝐾 𝑘,𝑖𝑘 denotes the cumulative amount of crude oil unloaded from the storage tank k up to time CKik. By Eq. (19a) and (19b), every lot ik’ ≤ ik assigned to the storage tank k (i.e. YKik’,k = 1) has already been unloaded from tank



Page 20 of 56

k at time CKik and the overall content of those batches determines the value of 𝑈𝐾𝐾 𝑘,𝑖𝑘. Then, 𝑈𝐾𝐾 𝑘,𝑖𝑘 ≥ ∑𝑖𝑘′

𝑄𝐾 𝑖𝑘′,𝑘 ― 𝑀 𝑄 (1 ― 𝑊𝐾 𝑖𝑘,𝑘)

∀ 𝑖𝑘 ∈ 𝑰𝑲 , 𝑘 ∈ 𝑲

(19a)

𝑈𝐾𝐾 𝑘,𝑖𝑘 ≤ ∑𝑖𝑘′

𝑄𝐾 𝑖𝑘′,𝑘

∀ 𝑖𝑘 ∈ 𝑰𝑲 , 𝑘 ∈ 𝑲

(19b)

∈ 𝑰𝑲 𝑖𝑘 ′ ≤ 𝑖𝑘 ∈ 𝑰𝑲 𝑖𝑘 ′ ≤ 𝑖𝑘

In turn, 𝑈𝐾𝑉 𝑘,𝑖𝑣 stands for the cumulative amount of crude oil unloaded from the storage tank k up to time CViv. By Eqs. (20a) and (20b), its value will be: (i) never greater than 𝑈𝐾𝐾 𝑘,𝑖𝑘,𝑘 minus the size of lot ik withdrawn from tank k if lot iv is loaded earlier (i.e. 𝑋𝐾 𝑖𝑣,𝑖𝑘,𝑘 = 1) and (ii) never smaller than 𝑈𝐾𝐾 𝑘,𝑖𝑘 whenever 𝑋𝐾 𝑖𝑣,𝑖𝑘,𝑘 = 0 and both lots have been allocated to tank k. 𝑈𝐾𝑉 𝑘,𝑖𝑣 ≤ 𝑈𝐾𝐾 𝑘,𝑖𝑘 ― 𝑄𝐾 𝑖𝑘,𝑘 + 𝑀 𝑄 (1 ― 𝑋𝐾 𝑖𝑣,𝑖𝑘,𝑘) ∀ 𝑖𝑣 ∈ 𝑰𝑽, 𝑖𝑘 ∈ 𝑰𝑲, 𝑘 ∈ 𝑲

(20a)

𝑈𝐾𝐾 𝑘,𝑖𝑘 ≤ 𝑈𝐾𝑉 𝑘,𝑖𝑣 + 𝑀 𝑄 𝑋𝐾 𝑖𝑣,𝑖𝑘,𝑘 + 𝑀 𝑄 (2 ― 𝑌𝑉 𝑖𝑣,𝑘 ― 𝑊𝐾 𝑖𝑘,𝑘) ∀ 𝑖𝑣 ∈ 𝑰𝑽, 𝑖𝑘 ∈ 𝑰𝑲, 𝑘 ∈ 𝑲

(20b)

Moreover, the value of 𝑈𝐾𝑉 𝑘,𝑖𝑣 can never be greater than the total amount of crude oil transferred from the storage tank k ∈ K to charging tanks along the planning horizon. In addition, the variables UKVk,iv and UKKk,ik are driven to zero if lots iv and ik have not been assigned to tank k, respectively. 𝑈𝐾𝑉 𝑘,𝑖𝑣 ≤ ∑𝑖𝑘

∈ 𝑰𝑲

𝑄𝐾 𝑖𝑘,𝑘

∀ 𝑖𝑣 ∈ 𝑰𝑽 , 𝑘 ∈ 𝑲

(21a)

𝑈𝐾𝑉 𝑘,𝑣 ≤ 𝑀 𝑄 𝑌𝑉 𝑖𝑣,𝑘

∀ 𝑖𝑣 ∈ 𝑰𝑽 , 𝑘 ∈ 𝑲

(21b)

𝑈𝐾𝐾 𝑘,𝑖𝑘 ≤ 𝑀 𝑄 𝑊𝐾 𝑖𝑘,𝑘

∀ 𝑖𝑘 ∈ 𝑰𝑲 , 𝑘 ∈ 𝑲

(21c)

Inventories of crude oil in storage tanks at the completion times of inlet and outlet operations. The continuous variables 𝐼𝐾𝑉 𝑘,𝑖𝑣 and 𝐼𝐾𝐾 𝑘,𝑖𝑘 denote the inventory levels in the storage


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tank k at the time events CV iv and CK ik , respectively. Their values are given by Eqs (22a) and (22b). Moreover, 𝐼𝐾𝑉 𝑘,𝑖𝑣 must never exceed the capacity of the storage tank k given by the parameter capkk while 𝐼𝐾𝐾 𝑘,𝑖𝑘 can never be lower than the tank heel volume (invkk,min). In Eqs (22a) and (22b), the parameter iikk represents the initial inventory of crude oil in the storage tank k. 𝐼𝐾𝑉 𝑘,𝑖𝑣 = 𝑖𝑖𝑘 𝑘 𝑌𝑉 𝑖𝑣,𝑘 + 𝐿𝐾𝑉 𝑘,𝑖𝑣 ― 𝑈𝐾𝑉 𝑘,𝑖𝑣

∀ 𝑖𝑣 ∈ 𝑰𝑽 , 𝑘 ∈ 𝑲

(22a) 𝐼𝐾𝐾 𝑘,𝑖𝑘 = 𝑖𝑖𝑘 𝑘 𝑊𝐾 𝑖𝑘,𝑘 + 𝐿𝐾𝐾 𝑘,𝑖𝑘 ― 𝑈𝐾𝐾 𝑘,𝑖𝑘

∀ 𝑖𝑘 ∈ 𝑰𝑲 , 𝑘 ∈ 𝑲

(22b)

𝐼𝐾𝑉 𝑘,𝑖𝑣 ≤ 𝑐𝑎𝑝𝑘 𝑘 + 𝑀 𝑄 (1 ― 𝑌𝑉 𝑖𝑣,𝑘)

∀ 𝑖𝑣 ∈ 𝑰𝑽 , 𝑘 ∈ 𝑲

(22c)

𝐼𝐾𝐾 𝑘,𝑖𝑘 ≥ 𝑖𝑛𝑣𝑘 𝑘,𝑚𝑖𝑛 ― 𝑀 𝑄 (1 ― 𝑊𝐾 𝑖𝑘,𝑘)

∀ 𝑖𝑘 ∈ 𝑰𝑲 , 𝑘 ∈ 𝑲

(22d)

To determine the exact inventory cost of crude oil in storage tanks, it is important to know the inventory levels at the start of loading and unloading operations given by 𝐼𝐾𝑉 𝑆𝑘,𝑖𝑣 and 𝐼𝐾𝐾 𝑆𝑘,𝑖𝑘, respectively. 𝐼𝐾𝑉 𝑆𝑘,𝑖𝑣 = 𝐼𝐾𝑉 𝑘,𝑖𝑣 ― ∑𝑣 ∈ 𝑽 𝐹𝑉 𝑖𝑣,𝑣,𝑘

∀ 𝑖𝑣 ∈ 𝑰𝑽 , 𝑘 ∈ 𝑲

𝑘

(23a) 𝐼𝐾𝐾 𝑆𝑘,𝑖𝑘 = 𝐼𝐾𝐾 𝑘,𝑖𝑘 + 𝑄𝐾 𝑖𝑘,𝑘

∀ 𝑖𝑘 ∈ 𝑰𝑲 , 𝑘 ∈ 𝑲

(23b)

5.2.4 Unloading lots of crude mix from charging tanks Linear constraints similar to Eqs (1)-(6) are proposed to determine the batches 𝑖𝑚 ∈ 𝑰𝑴 unloaded from each charging tank m (WMim,m), their sizes (QMim,m), the destination units (YMim,u), the amounts of crude mix transferred from charging tanks to CDUs (FMim,m,u) and the lengths of such output operations (LMim). Moreover, the starting (SMim) and completion (CMim) times of the operations are established through a set of constraints



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similar to Eqs (12a)-(12b) and (13a)-(13b). In this case, however, the unloading of batches from different charging tanks can be performed simultaneously. The expressions of such constraints are given as Supporting Information. 5.2.5 Satisfying the feedstock demands of the CDUs The demand of feedstock by the CDUs should be fulfilled. The parameter demm denotes the demand of crude mix from the CDUs to be supplied by the charging tank m as specified at the planning level. Such a requirement is fulfilled through constraint (24). ∑𝑖𝑚

∈ 𝑰𝑴

𝑄𝑀 𝑖𝑚,𝑚 = 𝑑𝑒𝑚 𝑚

𝑚 ∈𝑴

(24)

Continuous processing of crude mix in every CDU over the time horizon. A continuous supply of crude mix to each CDU implies that the total length of the operations sequentially dispatching crude mix from charging tanks to a particular CDU must be equal to the length of the scheduling horizon. Let us introduce the continuous variable LUim,u representing the length of the output operation transferring the lot of crude mix im from the charging tank m to CDU u. By Eq. (25a), such a variable is null if lot im is not assigned to the distillation unit u (i.e. YMim,u = 0). Otherwise, Eq. (25b) indicates that LUim,u is equal to the length of the operation discharging the lot im from the source charging tank (i.e. LMim). Eq. (25c) guarantees a continuous supply of crude mix to each CDU over the time horizon. 𝐿𝑈 𝑖𝑚,𝑢 ≤ 𝐻 𝑌𝑀 𝑖𝑚,𝑢 ∑𝑢

∈ 𝑼 𝐿𝑈 𝑖𝑚,𝑢

∑𝑖𝑚

∈ 𝑰𝑴

= 𝐿𝑀 𝑖𝑚

𝐿𝑈 𝑖𝑚,𝑢 = 𝐻

∀ 𝑖𝑚 ∈ 𝑰𝑴 , 𝑢 ∈ 𝑼

(25a)

∀ 𝑖𝑚 ∈ 𝑰𝑴

(25b)

∀𝑢 ∈𝑼

(25c)

5.2.6 Controlling the inventory level of crude mix in charging tanks Similarly to the storage tanks, the inventory of crude mix in charging tanks should


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be monitored over time to prevent overfilling and running-out conditions. As before, the important events are the times at which loading and unloading operations are completed. The inventory level in charging tanks at those key time events are determined by making the difference between the cumulative amounts of crude mix loaded and withdrawn up to those times. Then, a mathematical treatment analogous to the one used for controlling the inventory level in storage tanks and consisting of a set of equations structurally similar to constraints (14)-(23) is applied. In those equations, the binary variable XMik,im,m and the continuous variables LMKm,ik, LMMm,im, UMKm,ik, UMMm,im, IMKm,ik, IMMm,im, 𝐼𝑀𝐾 𝑆𝑚,𝑖𝑘, and 𝐼𝑀𝑀 𝑆𝑚,𝑖𝑚 will take the roles previously played by the binary variable XKiv,ik,k and the continuous variables LKVk,iv, LKKk,ik, UKVk,iv, UKKk,ik, IKVk,iv, IKKk,ik, 𝐼𝐾𝑉 𝑆𝑘,𝑖𝑣, and 𝐼𝐾𝐾 𝑆𝑘,𝑖𝑘 for the storage tanks. The expressions of such equations are given as Supporting Information. 5.2.7 Tracking the amount of trace elements in storage tanks The set J comprises the key trace elements whose concentration levels in charging tanks must remain between the specified upper and lower limits. To determine such concentrations in charging tanks, it is necessary to determine the amounts of trace elements transferred from storage to charging tanks during inlet operations. There is a simpler way to follow the inventory of trace elements in storage tanks. Since they are dedicated to a single type of crude oil, the amount of trace elements in the discharged batches and the inventory of them in storage tanks at the end of input/output operations can be determined through constraints (26a)-(26e). The parameter ickj,k stands for the concentration of trace element j in the dedicated storage tank k. 𝑄𝐾𝐽 𝑗,𝑖𝑘,𝑘 = 𝑖𝑐𝑘 𝑗,𝑘 𝑄𝐾 𝑘,𝑖𝑘

∀ 𝑗 ∈ 𝑱, 𝑖𝑘 ∈ 𝑰𝑲 , 𝑘 ∈ 𝑲


(26a)


Page 24 of 56

𝐹𝐾𝐽 𝑗,𝑖𝑘,𝑚 ≤ 𝑀 𝑄 𝑌𝐾 𝑖𝑘,𝑚

∀ 𝑗 ∈ 𝑱, 𝑖𝑘 ∈ 𝑰𝑲 , 𝑚 ∈ 𝑴

∑𝑚 ∈ 𝑴 𝐹𝐾𝐽 𝑗,𝑖𝑘,𝑚 = ∑𝑘 ∈ 𝑲𝑄𝐾𝐽 𝑗,𝑖𝑘,𝑘

∀ 𝑗 ∈ 𝑱, 𝑖𝑘 ∈ 𝑰𝑲

𝐼𝐾𝑉𝐽 𝑗,𝑘,𝑖𝑣 = 𝑖𝑐𝑘 𝑗,𝑘 𝐼𝐾𝑉 𝑘,𝑖𝑣

∀ 𝑗 ∈ 𝑱, 𝑖𝑣 ∈ 𝑰𝑽 , 𝑘 ∈ 𝑲

(26d)

𝐼𝐾𝐾𝐽 𝑗,𝑘,𝑖𝑘 = 𝑖𝑐𝑘 𝑗,𝑘 𝐼𝐾𝐾 𝑘,𝑖𝑘

∀ 𝑗 ∈ 𝑱, 𝑖𝑘 ∈ 𝑰𝑲 , 𝑘 ∈ 𝑲

(26e)

(26b) (26c)

5.2.8 Monitoring the amount of trace elements in charging tanks The concentration of key components in charging tanks during unloading operations must remain within specified limits. This is achieved by blending crude oils of different qualities. To determine such concentrations in charging tanks, it is necessary to first determine the amounts of trace elements present in the tanks at times CKik and CMim. To this purpose, we make a mathematical treatment similar to the one applied to track the total inventory of crude mix in charging tanks along the scheduling horizon. The new equations are given as Supporting Information. 5.2.9 Controlling the concentration of key trace elements in charging tanks The continuous variable CMMJj,m,im is introduced to represent the concentration of the trace element j in the charging tank m while unloading the batch im. At the start and end times of an unloading operation, the relationships between the amounts of the key trace elements available in the charging tank m, given by 𝐼𝑀𝑀𝐽𝑆𝑗,𝑚,𝑖𝑚 and IMMJm,im, and the total inventories of crude mix are determined by Eqs (27a) and (27b), respectively. As stated by Eq. (27c), the value of CMMJj,m,im in the charging tank m must be within the specified concentration limits [cmlj,m, cmuj,m]. 𝐼𝑀𝑀𝐽 𝑆𝑗,𝑚,𝑖𝑚 = 𝐶𝑀𝑀𝐽 𝑗,𝑚,𝑖𝑚 𝐼𝑀𝑀 𝑆𝑚,𝑖𝑚 𝐼𝑀𝑀𝐽 𝑗,𝑚,𝑖𝑚 = 𝐶𝑀𝑀𝐽 𝑗,𝑚,𝑖𝑚 𝐼𝑀𝑀 𝑚,𝑖𝑚

∀𝑗 ∈ 𝑱, 𝑖𝑚 ∈ 𝑰𝑴 , 𝑚 ∈ 𝑴 ∀𝑗 ∈ 𝑱, 𝑖𝑚 ∈ 𝑰𝑴 , 𝑚 ∈ 𝑴


(27a) (27b)

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𝑐𝑚𝑙 𝑗,𝑚 𝑊𝑀 𝑖𝑚,𝑚 ≤ 𝐶𝑀𝑀𝐽 𝑗,𝑚,𝑖𝑚 ≤ 𝑐𝑚𝑢 𝑗,𝑚 𝑊𝑀 𝑖𝑚,𝑚

∀𝑗 ∈ 𝑱, 𝑖𝑚 ∈ 𝑰𝑴 , 𝑚 ∈ 𝑴

(27c)

To facilitate the presentation of an approximate MILP problem formulation in the next Section, the nonlinear equations controlling the concentration of trace elements in the charging tanks during unloading and loading operations are also listed below. 𝑄𝑀𝐽 𝑗,𝑖𝑚,𝑚 = 𝐶𝑀𝑀𝐽 𝑗,𝑚,𝑖𝑚 𝑄𝑀 𝑖𝑚,𝑚 𝐼𝑀𝐾𝐽 𝑗,𝑚,𝑖𝑘 = 𝐶𝑀𝐾𝐽 𝑗,𝑚,𝑖𝑘 𝐼𝑀𝐾 𝑚,𝑖𝑘

∀ 𝑗 ∈ 𝑱, 𝑖𝑚 ∈ 𝑰𝑴 , 𝑚 ∈ 𝑴 ∀ 𝑗 ∈ 𝑱, 𝑖𝑘 ∈ 𝑰𝑲 ,𝑚 ∈ 𝑴

(28) (29)

5.2.10 Direct precedence variables In order to derive an objective function that provides the exact operating cost, direct precedence variables defining a detailed sequence of operations in storage and charging tanks are determined in terms of the general precedence variables. They are continuous variables restricted to the interval [0,1]. For the storage tanks, the direct precedence variables are: (i) FKViv,k and FKKik,k identifying the first operation performed in the storage tank k, either the loading of batch iv or the unloading of batch ik; (ii) EKViv,k and EKKik,k identifying the last operation performed in the storage tank k; (iii) PVKiv,ik,k and PKVik,iv,k that become equal to one when lot iv directly precedes or succeed lot ik, respectively; and (iv) PKKik,ik’,k denoting that the loading of lot ik in the storage tank k is performed right before receiving lot ik’ (ik’ > ik) in the same tank, whenever PKKik,ik’,k = 1. Similar direct precedence variables are used to determine the chronological order of the operations performed in charging tanks. They are: FMKik,m, FMMim,m, EMKik,m, EMMim,m, PKMik,im,m, PMKim,ik,m, PMKKik,ik’,m and PMMim,im’m. The direct precedence variable PMKKik,ik’,m denotes that lot ik is transferred right before lot ik’ (ik’ > ik) into the charging tank m when PMKKik,ik’,m = 1. In turn, PMMim,im’,m denotes that lot im is unloaded right before lot im’ (im’ > im) from the charging tank m. Equations relating direct precedence variables with the



Page 26 of 56

general precedence ones for the storage tanks are given as Supporting Information. Additional linear tightening constraints are also incorporated in the MINLP formulation for a more rigorous tracking of the concentrations of trace elements in charging tanks. They are shown in the Supporting Information file. Therefore, the proposed MINLP formulation includes a large set of linear valid inequalities plus the nonlinear constraints (27)-(29). A detailed presentation of the MINLP model is given as Supporting Information. 5.3 Objective function The problem goal consists on the minimization of the total operating cost, including demurrage cost, unloading cost of crude oil from vessels, changeover costs in CDUs, and inventory holding costs of crude oil in storage and charging tanks. The total inventory cost in storage tanks is expressed as the sum of five inventory cost contributions over the following time periods: (a) before performing the first operation (CIK1); (b) during loading and unloading operations (CIK2); (c) between operations (CIK3); and (d) after performing the last operation (CIK4). There is another cost contribution associated to storage tanks that remain idle over the whole scheduling horizon (CIK5). The values of the five inventory cost contributions are provided by Eqs (30)-(34) where the parameter stinc denotes the unit inventory cost in storage tanks. 𝐶𝐼𝐾1 = 𝑠𝑡𝑖𝑛𝑐 ∑𝑘 ∈ 𝑲[∑𝑖𝑣 ∈ 𝑰𝑽 𝑖𝑖𝑘 𝑘 𝑆𝑉 𝑖𝑣 𝐹𝐾𝑉 𝑖𝑣,𝑘 + ∑𝑖𝑘 ∈ 𝑰𝑲 𝑖𝑖𝑘 𝑘 𝑆𝐾 𝑖𝑘 𝐹𝐾𝐾 𝑖𝑘,𝑘] (30) 𝐶𝐼𝐾2 = 𝑠𝑡𝑖𝑛𝑐 ∑𝑘 ∈ 𝑲[∑𝑖𝑣 ∈ 𝑰𝑽 0.5 (𝐼𝐾𝑉 𝑆𝑘,𝑖𝑣 + 𝐼𝐾𝑉 𝑘,𝑖𝑣) 𝐿𝑉 𝑖𝑣 + ∑𝑖𝑘 ∈ 𝑰𝑲 0.5 (𝐼𝐾𝐾 𝑆𝑘,𝑖𝑘 + (31)


𝐼𝐾𝐾 𝑘,𝑖𝑘 ) 𝐿𝐾 𝑖𝑘 ]

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𝐶𝐼𝐾3 = 𝑠𝑡𝑖𝑛𝑐 ∑𝑘 ∈ 𝑲 ∑𝑖𝑣 ∈ 𝑰𝑽 ∑𝑖𝑘 ∈ 𝑰𝑲 (𝑆𝐾 𝑖𝑘 ― 𝐶𝑉 𝑖𝑣)𝑃𝑉𝐾 𝑖𝑣,𝑖𝑘,𝑘 + ∑𝑖𝑘 ∈ 𝑰𝑲 ∑ 𝑖𝑘 ′ ∈ 𝑰𝑲 (𝑆𝐾 𝑖𝑘′ ―

[

𝐶𝐾 𝑖𝑘 ) 𝑃𝐾𝐾 𝑖𝑘,𝑖𝑘 ′,𝑘 + ∑𝑖𝑘 ∈ 𝑰

(𝑖𝑘 ′ > 𝑖𝑘)

(32) 𝐶𝐼𝐾4 = 𝑠𝑡𝑖𝑛𝑐 ∑𝑘 ∈ 𝑲[∑𝑖𝑣 ∈ 𝑰𝑽 𝐼𝐾𝑉 𝑘,𝑖𝑣 (𝐻 ― 𝐶𝑉 𝑖𝑣 ) 𝐸𝐾𝑉 𝑖𝑣,𝑘 + ∑𝑖𝑘 ∈ 𝑰𝑲 𝐼𝐾𝐾 𝑘,𝑖𝑘 (𝐻 ― 𝐶𝐾 𝑖𝑘 ) 𝐸𝐾𝐾 𝑖𝑘,𝑘] (33) 𝐶𝐼𝐾5 = 𝑠𝑡𝑖𝑛𝑐 𝐻 [∑𝑘 ∈ 𝐊 𝑖𝑖𝑘𝑘]

(34)

𝑖𝑑𝑙𝑒

In turn, the inventory holding cost for charging tanks is decomposed into four contributions (CIM1, CIM2, CIM3, CIM4) as long as no charging tank remains idle over the entire scheduling horizon. Each one should satisfy some specified demand of crude blend from the CDUs. Equations computing the inventory cost contributions for charging tanks are given as Supporting Information. In this way, the objective function is given by Eq. (35) that includes several bilinear terms in the equations providing the inventory cost contributions in storage and charging tanks. The parameter uuc stands for the unit unloading cost, the coefficient uwc represents the unit waiting time cost, the coefficient uchc is the changeover cost for switching the feedstock in crude distillation units, and the parameter ctinc denote the unit inventory holding cost for charging tanks. (𝑶𝑩𝑱) 𝑀𝑖𝑛 𝒁 = 𝑢𝑢𝑐 (∑𝑖𝑣 +

𝑢𝑐ℎ𝑐 [∑𝑡

∈ 𝑰𝑽 ∈ 𝑻

(𝐶𝑇𝑉 𝑖𝑣 ― 𝑆𝑇𝑉 𝑖𝑣) ) + 𝑢𝑤𝑐 ∑𝑣 ∈ 𝑉 (𝑆𝑇𝑉 𝑣 ― 𝑎𝑡 𝑣) (∑𝑖𝑚

𝑛=5

∈ 𝑰𝑴

𝑛=4

𝑌𝑀 𝑖𝑚,𝑡 ― 1)] + ∑𝑛 = 1𝐶𝐼𝐾 𝑛 + ∑𝑛 = 1𝐶𝐼𝑀 𝑛

(35)

6. THE MINLP SOLUTION APPROACH In this Section, we propose a two-step procedure to find a very good solution of the MINLP model by sequentially solving just a pair of mathematical programming formulations: (1) an MILP continuous-time model that is a tight approximation of the MINLP formulation,



and (2) an NLP model that accounts for the bilinear terms arising in Eqs (27)-(29) and in the objective function (35). The approximate MILP is obtained by replacing the non-linear equations (27)-(29) by tailor-made linear bounding constraints. Convergence to a feasible MINLP solution is usually achieved without using convex McCormick envelopes. Instead, tightening valid constraints shown in Appendix B are considered for that purpose. When the approximate MILP has no feasible solution, the values of |𝐼𝐾| and/or |𝐼𝑀| should be increased by one. This procedure is repeated until finding an MILP feasible solution. Criteria for selecting initial values for the cardinality of the sets IV, IK and IM are given in Section 6.3. In order to guarantee the discovery of a feasible MINLP solution, it is subsequently solved an NLP model using the MILP solution as the initial point. The NLP formulation is obtained from the original MINLP by fixing all the binary variables, including the direct precedence variables, to their MILP-optimal values. As the MILP approximate formulation usually provides a good initial point, the low-size NLP can be solved in less than one second. Therefore, the total CPU time is mostly allocated to the solution of the approximate MILP. It is important to note that the proposed approach does not guarantee a global optimal solution, neither is able to provide an optimality gap, but it can efficiently discover near-optimal schedules. 6.1 The approximate MILP model The MINLP problem formulation can be converted into an MILP model by substituting the nonlinear equations (27)-(29) with tailor-made linear constraints (36)-(38) and using an approximate linear objective function LOBJ given by Eq. (39). Linear tightening constraints shown in Appendix B help finding either a feasible or a slightly infeasible MINLP solution (i.e. very close to fulfilling the nonlinear constraints).


Page 28 of 56

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𝐼𝑀𝑀𝐽 𝑗,𝑚,𝑖𝑚 ≤ 𝑐𝑚𝑢 𝑗,𝑚 𝐼𝑀𝑀 𝑚,𝑖𝑚

∀𝑗 ∈ 𝑱, 𝑖𝑚 ∈ 𝑰𝑴 , 𝑚 ∈ 𝑴

(36a)

𝐼𝑀𝑀𝐽 𝑗,𝑚,𝑖𝑚 ≥ 𝑐𝑚𝑙 𝑗,𝑚 𝐼𝑀𝑀 𝑚,𝑖𝑚

∀ 𝑗 ∈ 𝑱, 𝑖𝑚 ∈ 𝑰𝑴 , 𝑚 ∈ 𝑴

(36b)

𝑄𝑀𝐽 𝑗,𝑖𝑚,𝑚 ≤ 𝑐𝑚𝑢 𝑗,𝑚 𝑄𝑀 𝑖𝑚,𝑚

∀ 𝑗 ∈ 𝑱, 𝑖𝑚 ∈ 𝑰𝑴 , 𝑚 ∈ 𝑴

(37a)

𝑄𝑀𝐽 𝑗,𝑖𝑚,𝑚 ≥ 𝑐𝑚𝑙 𝑗,𝑚 𝑄𝑀 𝑖𝑚,𝑚

∀ 𝑗 ∈ 𝑱, 𝑖𝑚 ∈ 𝑰𝑴 , 𝑚 ∈ 𝑴

(37b)

𝐼𝑀𝑀𝐽 𝑆𝑗,𝑚,𝑖𝑚 ≤ 𝑐𝑚𝑢 𝑗,𝑚 𝐼𝑀𝑀 𝑆𝑚,𝑖𝑚

∀𝑗 ∈ 𝑱, 𝑖𝑚 ∈ 𝑰𝑴 , 𝑚 ∈ 𝑴

(38a)

𝐼𝑀𝑀𝐽 𝑆𝑗,𝑚,𝑖𝑚 ≥ 𝑐𝑚𝑙 𝑗,𝑚 𝐼𝑀𝑀 𝑆𝑚,𝑖𝑚

∀ 𝑗 ∈ 𝑱, 𝑖𝑚 ∈ 𝑰𝑴 , 𝑚 ∈ 𝑴

(38b)

(LOBJ) 𝑀𝑖𝑛 𝒁 = 𝑢𝑢𝑐 (∑𝑖𝑣

∈ 𝑰𝑽

𝑢𝑐ℎ𝑐 [∑𝑡

+ 𝑠𝑡𝑖𝑛𝑐 ∗ 𝐻 𝑐𝑎𝑟𝑑(𝑰𝑽) + 𝑐𝑎𝑟𝑑(𝑰𝑲)

(𝐶𝑇𝑉 𝑖𝑣 ― 𝑆𝑇𝑉 𝑖𝑣) ) +𝑢𝑤𝑐 ∑𝑣 ∈ 𝑉 (𝑆𝑇𝑉 𝑣 ― 𝑎𝑡 𝑣) ∈ 𝑻

(∑𝑖𝑚

[∑𝑘 ∈ 𝑲[∑𝑖𝑣 ∈ 𝑰𝑽𝐼𝐾𝑉𝑘,𝑖𝑣

𝑐𝑡𝑖𝑛𝑐 ∗ 𝐻 𝑐𝑎𝑟𝑑(𝑰𝑲) + 𝑐𝑎𝑟𝑑(𝑰𝑴)

∈ 𝑰𝑴

+ ∑𝑖𝑘

𝑌𝑀 𝑖𝑚,𝑡 ― 1)]

∈ 𝐼𝐾

+

𝐼𝐾𝐾 𝑘,𝑖𝑘]] +

[∑𝑚 ∈ 𝑴[∑𝑖𝑘 ∈ 𝑰𝑲𝐼𝑀𝐾𝑚,𝑖𝑘 + ∑𝑖𝑚 ∈ 𝑰𝑴𝐼𝑀𝑀𝑚,𝑖𝑚]]

(39)

After solving the MILP model, some elements of the sets IV, IK and IM are not active at the best solution, i.e. they stand for dummy batches. In other words, the number of input and output operations performed in storage and charging tanks may be lower than the cardinalities of the sets IV, IK and IM. To have an approximate value of the operating cost for the MILP-solution, a LP model is derived by fixing the binary variables to their MILP-values, and ignoring the fictitious batches. Moreover, it is adopted the original objective function (35) that becomes linear by fixing the timing of the operations provided by the MILP. The optimal value of the resulting LP model is reported as the operating cost for the MILP-solution. If the MILP is infeasible because the crude mix demand (Eq. 24) or the limiting concentrations of trace elements (Eq. 29) cannot be satisfied, the cardinality



of the sets IM and IK should be increased by one in this order, but one at a time. This procedure is repeated until feasibility is achieved. If the infeasibility comes from the violation of Eq. (11) forcing the complete unloading of the vessels, |IV| is to be increased by one. 6.2 The NLP problem formulation The NLP model to be subsequently solved is obtained from the original MINLP formulation by fixing the binary variables {WViv,v, YViv,k , WKik,k , YKik,m , WMim,m , YMim,t, XKiv,ik,k and XMik,im,m} and consequently the direct precedence variables to their MILP optimal-values. Nonlinear equations (27)-(29) and the original objective function (OBJ) all involving bilinear terms are considered by the NLP. In this way, it is guaranteed that the solution provided by the NLP, if any, is a feasible MINLP solution and the value of the inventory holding cost is exact. Therefore, a MILP-NLP two-step solution procedure is ∗ ∗ proposed to solve the MINLP formulation. When 𝑍 𝑀𝐼𝐿𝑃 and the NLP is feasible, < 𝑍 𝑁𝐿𝑃

the approximate MILP solution presents some small infeasibility on the nonlinear constraints for the MINLP formulation that is subsequently removed by solving the NLP model. Let us now assume that the MILP model is feasible and the subsequent solution of the NLP does not converge to an MINLP feasible solution. This situation may arise because the MILP provides an infeasible integer solution for the MINLP and the NLP model is unable to restore feasibility. When the infeasibility is small, a feasible NLP solution can be achieved by incorporating an integer cut in the MILP formulation to remove the current optimal point from the solution space. If not, the cardinality of the set IM or IK should be increased by one, but one at a time and both the MILP and NLP subproblems will be solved again.


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6.3 Rules for choosing the number of batches Some empirical rules can be proposed to choose good initial values for the cardinalities of the sets IV, IK and IM. They can be initially selected as follows; |𝐼𝑉| = |𝑉| or |𝑉| + 1 ;

|𝐼𝐾| = |𝐾| + |𝑀| ;

|𝐼𝑀| = |𝑀| + |𝑈|

(40)

The values of |𝐼𝐾| and |𝐼𝑀| suggested by Eq. (40) work better when the total amount of crude oil unloaded from marine vessels and the overall amount of crude blend supplied to the CDUs are rather similar. For longer time horizons and larger supplies of crude mix to the CDUs, it is recommended: |𝐼𝐾| = |𝐾| + |𝑀| +𝑛𝑠𝑡 ; |𝐼𝑀| = |𝑀| + |𝑈| +𝑛𝑐ℎ , with (nst, nch) ≥ 1. As the supplies to the CDUs substantially grow, the value of nst usually presents a significant increase. In turn, nch also rises but usually at a lower rate. 7. COMPUTATIONAL RESULTS AND DISCUSSION The proposed approach has been applied to solve eight examples involving up to four marine vessels, six storage tanks containing different types of crude oil, four charging tanks and three CDUs. Several instances of those examples have been generated by increasing both the feedstock demand from the CDUs to be fulfilled by each charging tank and the length of the planning horizon. The first four examples are benchmark problems previously introduced by Lee et al.11 and already solved by several authors. Examples 1, 2 and 4 do not involve blending of crude oils in storage tanks. This is not the case for Example 3. Consequently, our Example 3 is a slight variation of the benchmark problem proposed by Lee et al.11 so as to avoid blending of crude oils in storage tanks. For comparison purposes, Examples 1, 2 and 4 have been solved using both the original data of Lee et al.11 and those adjusted by Jia et al.12 for continuous-time models.



Additional instances of Examples 1-4 with increasing feedstock demand for charging tanks have also been solved. In turn, Examples 5-8 are new examples introduced in this work. Data for the different instances of Examples 1-8 are given as Supporting Information. In all cases, the problem goal seeks to minimize the total operating cost, including inventory carrying costs in storage and charging tanks. Minimum sizes for parcels and batches of crude oil have been adopted as follows: svmin = skmin = smmin = 10, and the brine settling time (bst) is always assumed to be negligible. All the examples were solved to optimality using GAMS/CPLEX 24.2 for MILPs and GAMS/CONOPT for NLPs on an Intel(R) Core i7 3632QM 2.20 GHz one-processor PC with 12 GB RAM and 4 cores. The relative gap tolerance has been fixed at 10-4 and a maximum CPU time of 3600 s is allowed. Model sizes and computational results for all variants of Examples 1-8 are shown in Tables 1 and 2 and as Supported Information. The cardinalities of the sets IV, IK and IM finally chosen, and the size of the approximate MILP model are reported in Table 1. In turn, Table 2 presents the exact operating cost for the best solutions of both the approximate MILP and the NLP formulations, and the CPU time required to solve each model to optimality. 7.1 Example 1 Four different instances of Example 1 (i.e. examples 1a-1d) and three instances of Examples 2-to-4 (i.e. examples 2a-2c, 3a-3c and 4a-4c) have been solved. The first two instances of Examples 1-4 (i.e. Examples 1a-4a, and Examples 1b-4b, respectively) assume the feedstock demand and the horizon length proposed by Lee et al.11. To compare with the results found by other authors, Examples 1a-4a assume the modified vessel arrival times and the maximum discharge rate from vessels and tanks adopted by Jia et al.12 Instead, Examples 1b,


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2b and 4b consider the original vessel arrival times and the discharge rate limits proposed by Lee et al.11. The refinery configurations for Examples 1-4 are given in Figure 2. All instances of Example 1 involve two marine vessels, two storage tanks, two charging tanks and only one CDU. Loading and unloading operations in vessels and tanks have to be scheduled over a scheduling horizon of 8 time units and the feedstock composition is determined by only one trace element. To solve Example 1a, it was initially selected: |IV| = |V| = 2, |IK| = |K| + |M| = 4, and |IM| = |M| + |U| = 3. The value of |IV| was raised to 3 to find a better solution in 2 s of CPU time (see Tables 1 and 2).

Figure 2. Refinery configurations for Examples 1-4



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Table 1. MILP model sizes for the different instances of Examples 1-8 Ex.

1

2

3

4

5

6 7 8 (*)

Instance 1ª 1b 1c 1d 2a 2b 2c 3a 3b* 3c 4a 4b 4c 5a 5b 5c 6a 6b 7a 7b 8a 8b

Proposed number of batches for Vessels Storage Charging (iv) Tanks (ik) Tanks (im) 3 3 3 3 4 3 3 4 4 3 7 5 3 6 5 4 5 5 3 9 6 3 3 5 5 5 5 3 5 5 4 6 7 4 6 7 4 7 7 3 4 4 3 6 5 3 6 5 3 4 5 3 8 7 3 5 5 4 7 6 4 5 7 4 7 9

MILP Continuous Variables 342 693 1032 1358 2047 1913 3731 1105 1913 1720 3699 3699 3950 949 1429 2060 1499 2629 2192 3436 3207 4749

with blending of crude oils in storage tanks


MILP Binary Variables 100 82 124 164 217 210 403 130 232 210 452 452 512 115 174 260 190 328 236 375 392 582

MILP Constraints 1340 1981 3002 3966 6582 6675 12412 3041 7384 4910 10903 10903 13345 2714 4159 5744 4529 8657 7230 10837 8937 14141

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Table 2. Best results found for Examples 1-8 using the MILP-NLP solution approach Example

1

2

3

4

5

6 7 8 (*)

Instance 1a 1b 1c 1d 2a 2b 2c 3a 3b* 3c 4a 4b 4c 5a 5b 5c 6a 6b 7a 7b 8a 8b

Best MILP solution 220.34 213.13 264.92 304.61 357.74 359.29 455.65 274.85 282.59 282.42 399.85 397.68 369.31 336.09 378.01 388.61 407.36 465.26 411.67 493.07 741.91 780.11

Best NLP solution 217.13 203.27 262.95 311.76 356.99 352.79 455.65 273.65 277.10 280.60 365.44 364.44 367.97 334.92 393.44 385.89 406.36 489.18 418.43 509.02 718.29 784.49

CPU time (s) MILP

NLP

2 2 4 459 125 314 778 28 265 80 1298 1984 1869 2 98 490 435 836 757 1943 189 723

0.03 0.05 0.03 0.08 0.03 0.06 0.02 0.08 0.03 0.03 0.09 0.06 0.09 0.05 0.02 0.03 0.08 0.06 0.05 0.09 0.05 0.03

with blending of crude oils in storage tanks

The best operational schedule for Example 1a is illustrated in Figure 3, showing (a) the sequence of input and output operations in every tank, (b) the evolution of tank inventories over the time horizon, and (c) the concentration-time profile of the trace element in charging tanks. Figure 3 shows that the total amount of crude oil transferred from storage to charging tanks (133.33) is larger than the minimum quantity required (100) to fulfill the specified feedstock demand. Such a costly decision is made to keep the trace element concentration in charging tanks within the allowable range. Then, some amount of crude blend remains in one of the charging tanks at the horizon end.



Figure 3. Best solution found for Example 1a Similar values for |IV|, |IK| and |IM| have been chosen for Example 1b that is solved in 2 s. It features a total operating cost of 203.27 that is lower than the results reported by Castro and Grossmann18 and Castro19 both using accurate inventory costs (see Tables 1-3). The figure illustrating the best solution for Example 1b is provided as Supporting Information. Compared with Example 1a, the amount of crude oil transferred to charging tanks is lower (115 vs. 133.33) but still higher than the minimum required. The values of the different cost items at the optimal solutions for all the examples are given as Supporting Information. In our approach, batches of crude oil are unloaded from storage tanks one by one. Despite that, the comparison with the results reported by other authors is still quite satisfactory (see Table 3). A fair comparison with previous work is rather difficult because past contributions allow simultaneous discharges of crude oil from storage to charging tanks for both examples. The same best solutions for Examples 1a and 1b were found in 35.5 s and 233.1 s, respectively, by solving the original MINLP formulation using the global


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optimizer BARON21. In other words, the MILP-NLP approach finds the global optimal solutions in both cases. Table 3. Comparing the computational results for Examples 1-4 with prior contributions Best solution found Furman Karupiah in this work et al.13 et al.14 Using the original data of Lee et al.11 adjusted by Jia et al.12 Example

Castro & Grossmann18

Castro19

1a

217.13

215.10

--

--

--

2a

356.99

352.60

359.48

--

--

3b*

277.10

282.20

282.19

--

--

4a

365.44

383.70

383.69

--

--

Using the original data of Lee et al.11

(*)

1b

203.27

--

--

210.54

209.58

2b

352.79

--

--

320.50**

319.14

3b*

275.71

--

--

287.00

284.78

4b

364.44

--

--

365.09

319.87***

blending of crude oils in storage tanks

(**)

simultaneous i/o operations in storage tanks

(***)

simultaneous i/o operations in CHT4

Two new instances of Example 1 have been generated by just increasing the demand of feedstock to be supplied by each charging tank from 100 to 120 in Example 1c, and to 150 in Example 1d. Example 1c requires an additional discharge of crude mix from storage and charging tanks (i.e. |IK| = |IM| = 4) because of the higher demand from the CDUs (see Table 1). As a result, the number of feedstock changeovers also increases by one and the operating cost rises by almost the cost of the additional changeover. Nonetheless, the best solution shown as Supported Information is found in merely 4 s of CPU time (see Table 2). As expected, a higher number of output operations from storage and charging tanks are to be performed in Example 1d. The fulfillment of the higher demand forces to choose |IK| = 7 and |IM| = 5, and causes another feedstock changeover. Then, a further increase on the operating cost almost equal to



that of the additional changeover is observed. Figure 4 displays the best solution for Example 1d that has been found in 459 s (see Table 2).

Figure 4. Best solution found for Example 1d 7.2 Example 2 The refinery configuration for Example 2 includes a single docking station, three marine vessels, three storage tanks, three charging tanks and two CDUs. Topology constraints restricting the interconnections among vessels, storage tanks, charging tanks and CDUs are considered (see Figure 2). The scheduling horizon has a length of 10 time units and the composition of the crude mix in charging tanks is now characterized by two key trace elements. As already mentioned, three instances of Example 2 have been solved. Example 2a is based on the data used by Jia et al.12 and it has been solved adopting the ACS Paragon Plus Environment

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number of batches suggested by the rules given in Section 6.3, i.e. |IV| = |V| = 3, |IK| = |K| + |M| = 6, and |IM| = |M| + |U| = 5. The best solution shown in Figure 5 has been found in 125 s of CPU time (see Table 2).

Figure 5. Best solution found for Example 2a To keep the trace element concentrations within the admissible ranges, every charging tank sequentially receives a pair of batches from two different storage tanks. Moreover, a minimum amount of crude oil equal to 190 is transferred from storage to charging tanks to reduce inventory carrying costs. In Table 3, the total operating cost of our best schedule is compared with the results reported by Furman et al.13 and Karuppiah



et al.14 Though such contributions admitted simultaneous discharges of crude oil from storage tanks, the proposed solution is still very competitive. For Example 2b, our approach finds an MINLP solution with an exact operating cost equal to 352.79 in 314 s. As shown in Table 3, Castro and Grossmann18 and Castro19 using a global optimizer found better schedules with a total operating cost substantially lower. However, the solution reported by Castro and Grossmann18 presents a pair of simultaneous input and output operations at the storage tank ST1. An additional instance of Example 2, called Example 2c, has been generated by increasing both the demand of qualified feedstock fulfilled by each charging tank from 100 to 130 and the horizon length from 10 to 13 time units. To meet the higher feedstock demand from the CDUs and the specified feedstock quality, the number of output operations in storage and charging tanks significantly rises. Consequently, it is necessary to choose: |IK| = 9 and |IM| = 6. The solution for Example 2c was found in 778 s and is shown as Supporting Information. 7.3 Example 3 Example 3 considers a system configuration similar to Example 2 for the instances 3a and 3c (see Figure 2). In contrast, the refinery configuration for Example 3b shown as Supported Information allows a pair of marine vessels to discharge crude oil into two storage tanks. Moreover, the composition of the crude oil is now characterized by the concentration of a single trace element, and the operations should be scheduled over a planning horizon of 12 time units. Data for the three instances of Example 3 are given as Supporting Information. They are generated by increasing the amount of crude mix supplied by each charging tank to the CDUs from 50 in Examples 3a-3b to 80 in Example 3c. To avoid the blending of crude oils in the storage tanks, Example 3a is a modified


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version of a benchmark problem introduced by Lee et al.11 For this example, the number of parcels and batches initially postulated: |𝐼𝑉| = |𝑉| = 3, |𝐼𝐾| = |𝐾| + |𝑀| = 6, and |𝐼𝑀| = |𝑀| + |𝑈| = 5 are finally adopted. Example 3a has been solved to optimality in merely 5 s. Example 3b originally proposed by Lee et al.11 includes the mixing of different crude oils in storage tanks. The refinery configuration, the initial concentrations and the admissible concentration ranges of trace elements in storage and charging tanks for Example 3b are given as Supporting Information. They all differ from the data for Examples 3a and 3c. To compare with previous results, Example 3b was solved by exceptionally admitting blending operations in storage tanks. This requires using structurally similar constraints to control the inventory level and composition of the crude oil mix in storage and charging tanks. Example 3b was alternatively solved using the vessel arrival times and discharge rate limits from vessels and tanks proposed by Lee et al.11 and Jia et al.12, respectively. The best solutions for Example 3b found in 659 s and 265 s of CPU time, respectively, present lower operating costs than the ones reported in previous contributions (see Tables 2 and 3). They are shown as Supporting Information. In turn, the computer time required to solve Example 3c featuring a higher feedstock demand amounts to 1152 s. However, the operating cost presents a small increase because five batches of qualified feedstock are still supplied to the CDUs by the charging tanks, similarly to Example 3a. Consequently, the number and cost of feedstock changeovers remains the same for Examples 3a and 3c. Additional feedstock changeovers are usually the major cause of higher operating costs when larger amounts of crude oil are processed in the CDUs. The best solutions for Examples 3a-3c are all shown as Supporting Information.



7.4 Example 4 Example 4 involves a network configuration consisting of three marine vessels, six storage tanks, four charging tanks, three CDUs, and a single trace element determining the crude oil quality (see Figure 2). Each vessel can discharge crude oil into a single storage tank, and every storage tank is connected to a pair of charging tanks. Moreover, each charging tank can at most feed two different CDUs. Three instances of Example 4 have been considered by increasing the demand of crude oil supplied by each charging tank from 60 (Examples 4a-4b) to 80 (Example 4c). In all cases, the planning horizon has a length equal to 15 time units. Examples 4a and 4c use the vessel arrival times and discharge rate ranges adopted by Jia et al.12 while Example 4b consider the original data introduced by Lee et al.11. Data for the three instances of Example 4 are all given as Supporting Information. For Examples 4a-4b the following number of batches has been adopted: |IV| = |V| + 1= 4, |IK| = |K| + |M| = 6, and |IM| = |M| + |U| = 7. An additional lot of crude oil is transferred from storage to charging tanks at Example 4c to meet the higher feedstock demand, i.e. |IK| = 7. Compared with previous results, the best solutions for Examples 4a and 4b present lower operating costs with just one exception (see Tables 2 and 3). Recently, Castro19 has reported a better solution by allowing simultaneous inlet/outlet operations in charging tank CHT4. The solution found for Example 4c shown in Figure 6 has an operating cost just a little higher than Examples 4a-4b despite the larger feedstock demand, because it still features the same number of feedstock changeovers. In all instances of Example 4, the total amount of crude oil sent from storage to charging tanks has been minimized and every charging tank becomes empty after an output operation. Figures showing the best solutions for Examples 4a-4b are given as Supported Information.


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Figure 6. Best solution found for Example 4c 7.5 Examples 5 to 8

Figure 7. Network configurations for Examples 5-8



The performance of the proposed approach is further evaluated by solving several instances of four new examples introduced in this work. The refinery configurations for Examples 5-8 are depicted in Figure 7, while the data for them are given as Supported Information. Example 5 involves three vessels, three storage tanks, two charging tanks, one CDU and only one key trace element. By gradually increasing the feedstock demand supplied by each charging tank from 100 (Example 5a) to 120 (Example 5b) and subsequently to 150 (Example 5c), three different instances of Example 5 have been generated. The horizon length increases from 10 (Examples 5a-5b) to 12 time units (Example 5c).

Figure 8. Best solution found for Example 5c Computational results for the three instances of Example 5 are given in Tables 1 and


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2. As the feedstock demand grows, a number of batches higher than those suggested by the rules given in Section 6.3 is to be used. The best solution for Example 5c is depicted in Figure 8. Six consecutive loading/unloading operations are performed in charging tank CHT2, including three discharges of feedstock to the CDU.

Figure 9. Best solution found for Example 6b Compared with Example 5, the new Example 6 includes an additional charging tank and another CDU. The two instances of Example 6 differ in the feedstock demand (100 vs.



130) and the horizon length (12 vs. 14). For Example 6a, it is adopted the number of batches suggested by the rules given in Section 6.3. Both IK and IM are increased by one to find the best solution for Example 6b. In both instances, the amount of crude oil transferred from storage to charging tanks has been minimized. As a result, all charging tanks are empty at the horizon end. With regards to Example 6a, the operating cost at the best solution of Example 6b increases due to the additional feedstock changeovers and the longer scheduling horizon. The solution for Example 6b is shown in Figure 9.

Figure 10. Best solution found for Example 7b


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Example 7 is a rather difficult case study involving three vessels, five storage tanks, three charging tanks and two CDUs. The feedstock composition is defined in terms of two trace elements. Example 7a features a feedstock demand covered by each charging tank equal to 100 and a horizon length of 12 time units. In turn, Example 7b presents a crude mix demand of 120 and a horizon length of 13. To meet the higher demand, a portion of the crude oil unloaded from the vessels is consumed on the preparation of qualified feedstock for the CDUs. The best solution for Example 7b is depicted in Figure 10. Finally, Example 8 is the largest case involving four vessels, six storage tanks, four charging tanks and three CDUs (see Figure 7). The concentration of a single trace element determines the feedstock composition. The amount of feedstock to be supplied by each charging tank is increased from 90 (Example 8a) to 110 (Example 8b) to generate a pair of instances of Example 8. The best solution for Example 8b is shown in Figure 11. In almost all of the schedules found for the different instances of every example there is a common feature. As the model tends to minimize the number of changeovers in CDUs, once a charging tank is selected to feed a distillation unit it remains in that condition until its inventory reaches the minimum value. In this way, it is achieved a more stable operation of the distillation units. However, partial discharges of crude mix from charging tanks are sometimes necessary to keep the feedstock quality on specification. This condition is achieved by blending the new batches of crude oil subsequently supplied by the storage tanks with the remaining inventory contained in the charging tank. When applying the global optimizer BARON21 17.1.2 to the MINLP problem formulation, only those examples of rather small size (i.e. Examples 1a-1c, 3a and 5a) can be solved to optimality within the CPU time limit of 3600s. For Examples 1a-1b, 3a and 5a, the best



solutions found with the MILP-NLP approach have been proved to be the global optima. In turn, the global optimal value for Example 1c amounts to 262.2 against 262.95 provided by our two-stage methodology.

Figure 11. Best solution found for Example 8b 8. CONCLUSIONS A new continuous-time MINLP formulation based on a batch-oriented generalprecedence representation for the scheduling of crude oil refinery operations has been


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developed. The proposed model introduces a novel way of tracking the inventory and composition of the crude mix in refinery tanks by handling three ordered sets of generic batches discharged from marine vessels, storage tanks and charging tanks, respectively. By chronologically ordering the batches in those sets, a large number of equivalent solutions are removed from the feasible space allowing for a faster convergence to a near-optimal solution. Moreover, the ordering of the operations performed in every tank is controlled by a set of general precedence sequencing variables. Blending of different types of crude oil is only allowed in charging tanks and the blend quality is determined by the concentrations of a small number of trace elements. The selected problem goal is the minimization of the total operating cost, including sea waiting cost, unloading cost, and the inventory carrying cost in storage and charging tanks. Direct precedence variables computed in terms of the general precedence ones are introduced in the objective function to determine an exact value of the inventory holding costs. The core of the proposed solution strategy is a very tight MILP approximate formulation that usually provides a good MINLP feasible or slightly infeasible solution. Afterwards, a NLP model just determines the exact concentrations of key trace elements in charging tanks. Such concentrations do not explicitly appear in the approximate MILP model. This MILP-NLP solution strategy has been applied to different instances of eight examples, with the first four already studied by other authors. Accounting for the results found for the first four benchmark problems, it can be said that the proposed approach is able to find near-global optimal solutions. The best schedules for them have been found at rather low CPU times and feature a total operating cost very close to those reported by previous authors. To confirm the computational performance of the proposed approach,



several variants of four new examples with higher complexity have been successfully solved at reasonable computer times. SUPPORTING INFORMATION The supporting information file contains the complete model formulation, including the relationships among direct and general precedence variables to better sequence the operations in storage and charging tanks and the MILP strong valid constraints based on direct precedence variables. In addition, it provides the tables with all the data for Examples 1 to 8, the computational results for all of them, and the figures showing the system configurations and the best solutions found for all examples. ACKNOWLEDGMENT The authors acknowledge financial support from FONCYT-ANPCyT under Grant PICT 2014-2392, from CONICET under Grant PIP-112 20150641, and from Universidad Nacional del Litoral under Grant CAI+D 2016-UNL/PIC 504-20150100101LI. NOMENCLATURE Sets IK

= lots of crude oil unloaded from storage tanks

IM

= lots of crude mix unloaded from charging tanks

IV

= lots of crude oil unloaded from vessels

J

= key trace elements

K

= storage tanks receiving parcels of crude oil from vessels

Km

= storage tanks that are connected to the charging tank m

M

= charging tanks blending lots of different types of crude oil

U

= crude distillation units

Mu

= charging tanks that can feed the distillation unit u

V

= marine vessels

Vk

= marine vessels that are connected to storage tank k


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Parameters av

= initial content of crude oil in vessel v

atv

= arrival time of vessel v at the docking station

capkk

= maximum capacity of the storage tank k

capmm = maximum capacity of the charging tank m 𝑐𝑘𝑙𝑗,𝑘

= minimum allowed concentration of trace element j in storage tank k

𝑐𝑘𝑢𝑗,𝑘

= maximum allowed concentration of trace element j in storage tank k

𝑐𝑚𝑙𝑗,𝑚 = minimum allowed concentration of the trace element j in charging tank m 𝑐𝑚𝑢𝑗,𝑚 = maximum allowed concentration of the trace element j in charging tank m demm

= specific demand of feedstock for CDUs to be supplied by charging tank m

H

= length of the scheduling horizon

ickj,k

= initial concentration of the trace element j in storage tank k

icmj,m

= initial concentration of the trace element j in charging tank m

icvj,v

= concentration of the trace element j in the crude oil transported by vessel v

iikk

= initial inventory of crude oil in the storage tank k

iimm

= initial inventory of crude mix in the charging tank m

invkk,min = minimum inventory of crude oil in the storage tank k invmm,min = minimum inventory of crude mix in the charging tank m MJ

= upper bound on the amount of component j loaded/unloaded into/from a tank

MQ

= upper bound on the amount of crude oil loaded/unloaded into/from a tank

rkmin

= minimum unloading rate of crude oil from a storage tank

rkmax

= maximum unloading rate of crude oil from a storage tank

rmmin

= minimum unloading rate of crude blend from a charging tank

rmmax

= maximum unloading rate of crude mix from a charging tank

rvmin

= minimum unloading rate of crude oil from a vessel

rvmax

= maximum unloading rate of crude oil from a vessel

skmin

= minimum size of a lot of crude oil unloaded from a storage tank

skmax

= maximum size of a lot of crude oil unloaded from a storage tank

smmax

= minimum size of a lot of crude blend unloaded from a charging tank

smmax

= maximum size of a lot of crude blend unloaded from a charging tank

svmin

= minimum size of a lot of crude oil unloaded from a vessel



svmax

= maximum size of a lot of crude oil coming from a vessel

Binary variables WKik,k

= identifies the storage tank k from which the lot ik is unloaded

𝑊𝑀𝑖𝑚,𝑚 = identifies the charging tank m from which the lot im is unloaded WViv,v

= denotes that the batch of crude oil iv coming from vessel v

XKiv,ik,k = denotes that lot iv is loaded into the storage tank k before/after unloading lot ik XMik,im,m = denotes that lot ik is loaded in charging tank m before/after unloading lot im YKik,m

= denotes the transfer of lot ik from a storage tank to the charging tank m

YMim,u

= denotes the transfer of lot im from a charging tank to the crude distillation unit u

YViv,k

= allocates a batch of crude oil iv to the storage tank k

Continuous variables CKik

= end time for the unloading of lot ik from a storage tank

𝐶𝐾𝐾𝐽𝑗,𝑘,𝑖𝑘 = concentration of trace element j in the storage tank k while unloading lot ik 𝐶𝐾𝑉𝐽𝑗,𝑘,𝑖𝑣 = concentration of trace element j in the storage tank k after loading parcel iv CMim

= end time for the unloading of lot im from a charging tank

𝐶𝑀𝑀𝐽𝑗,𝑚,𝑖𝑚= concentration of trace element j in the charging tank m while unloading lot im 𝐶𝑀𝐾𝐽𝑗,𝑚,𝑖𝑘= concentration of trace element j in the charging tank m after loading batch ik CViv

= end time for the unloading of lot iv from a vessel

EMKik,m

= denotes that the loading of lot ik is the last operation in the charging tank m

EMMim,m

= denotes that the discharge of lot im is the last operation in the charging tank m

𝐼𝐾𝐾𝑘,𝑖𝑘

= inventory level of crude oil in the storage tank k at time CKik

𝐼𝐾𝐾𝐽𝑗,𝑘,𝑖𝑘 = inventory level of trace element j in the storage tank k at time CKik 𝐼𝐾𝑉𝑘,𝑖𝑣

= inventory level of crude oil in the storage tank k at time CViv

𝐼𝐾𝑉𝐽𝑗,𝑘,𝑖𝑣 = inventory level of trace element j in the storage tank k at time CViv 𝐼𝑀𝐾𝑚,𝑖𝑘

= inventory level of crude mix in the charging tank m at time CKik

𝐼𝑀𝐾𝐽𝑗,𝑚,𝑖𝑘 = inventory level of trace element j in the charging tank m at time CKik 𝐼𝑀𝑀𝑚,𝑖𝑚 = inventory level of crude mix in the charging tank m at time CMim 𝐼𝑀𝑀𝐽𝑗,𝑚,𝑖𝑚= inventory level of trace element j in the charging tank m at time CMim FKik,k,m

= amount of crude oil in lot ik transferred from storage tank k to charging tank m

FMKik,m

= denotes that the loading of lot ik is the first operation in the charging tank m


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FKJj,ik,m

= amount of component j in lot ik sent to charging tank m from the storage tank k

FMim,m,u

= amount of crude oil in lot im supplied from charging tank m to CDU u

FMJj,im,m,u = amount of trace element j in lot im supplied from charging tank m to CDU u FMMim,m

= denotes that the discharge of lot im is the first operation in the charging tank m

FViv,v,k

= amount of crude oil in lot iv transferred from vessel v to storage tank k

FVJj,iv,v,k

= amount of component j in lot iv sent from vessel v to storage tank k

LKik

= length of the pumping run unloading lot ik from a storage tank

𝐿𝐾𝐾𝑘,𝑖𝑘

= cumulative amount of crude oil loaded into the storage tank k at time CKik

𝐿𝐾𝐾𝐽𝑗,𝑘,𝑖𝑘 = cumulative amount of component j loaded into the storage tank k at time CKik 𝐿𝐾𝑉𝑘,𝑖𝑣

= cumulative amount of crude oil loaded into storage tank k at time CViv

𝐿𝐾𝑉𝐽𝑗,𝑘,𝑖𝑣 = cumulative amount of trace element j loaded into storage tank k at time CViv LMim

= length of the pumping run unloading lot im from a charging tank

𝐿𝑀𝐾𝑚,𝑖𝑘

= total amount of crude oil loaded into the charging time m at time CKik

𝐿𝑀𝐾𝐽𝑗,𝑚,𝑖𝑘 = cumulative amount of component j loaded in the charging time m at time CKik 𝐿𝑀𝑀𝑚,𝑖𝑚 = cumulative amount of crude mix loaded into the charging time m at time CMim 𝐿𝑀𝑀𝐽𝑗,𝑚,𝑖𝑚= total amount of trace element j loaded into the charging time m at time CMim LUim,u

= length of the pumping run transferring lot im to crude distillation unit u

LViv

= length of the pumping run discharging lot iv from a vessel

PKMik,im,m = denotes that the lot ik directly precedes the batch im in the charging tank m PMKim,ik,m = denotes that the lot im directly precedes the batch ik in the charging tank m PMMim,im’,m = denotes that the lot im directly precedes the batch im’ in the charging tank m PMKKik,ik’,m = denotes that the lot ik directly precedes the batch ik’ in the charging tank m QKik,k

= size of batch ik coming from storage tank k

QKJj,ik,k

= amount of trace element j in the batch ik coming from storage tank k

QMim,m

= size of batch im coming from charging tank m

QMJj,im,m

= amount of trace element j in the batch im coming from charging tank m

QViv,v

= size of the parcel iv coming from vessel v

QVJj,iv,v

= amount of trace element j in the parcel iv coming from vessel v

SKik

= starting time of the pumping run unloading lot ik

SMim

= starting time of the pumping run unloading lot im

SViv

= starting time of the pumping run unloading parcel iv



STVv

= starting time for the unloading operations of crude oil from vessel v

𝑈𝐾𝐾𝑘,𝑖𝑘

= cumulative amount of crude oil unloaded from storage tank k at time CKik

𝑈𝐾𝐾𝐽𝑗,𝑘,𝑖𝑘 = cumulative amount of component j unloaded from storage tank k at time CKik 𝑈𝐾𝑉𝑘,𝑖𝑣

= cumulative amount of crude oil unloaded from storage tank k at time CViv

𝑈𝐾𝑉𝐽𝑗,𝑘,𝑖𝑣 = cumulative amount of component j unloaded from storage tank k at time CViv 𝑈𝑀𝐾𝑚,𝑖𝑘

= cumulative amount of crude mix unloaded from charging tank m at time CKik

𝑈𝑀𝐾𝐽𝑗,𝑚,𝑖𝑘 = cumulative amount of trace element j unloaded from charging tank m at CKik 𝑈𝑀𝑀𝑚,𝑖𝑚 = cumulative amount of crude mix unloaded from charging tank m at time CMim 𝑈𝑀𝑀𝐽𝑗,𝑚,𝑖𝑚= total amount of trace element j unloaded from charging tank m at time CMim REFERENCES 1. Reddy, P.C., Karimi, I.A., Srinivasan, R., 2004. A new continuous-time formulation for scheduling crude oil operations. Chem.Eng. Sci. 59, 1325-1341. 2. Li, J., Li, W., Karimi, I.A., Srinivasan, R., 2007. Improving the robustness and efficiency of crude scheduling algorithms. AIChE J. 53, 2659-2680. 3. Liang, B., Yongheng, J., Dexian, H., 2012. A novel two-level optimization famework based on constrained ordinal optimization and evolutionary algorithms for scheduling of multipipeline crude oil blending. Ind. Eng. Chem. Res. 51, 9078-9093. 4. Cerdá, J., Pautasso, P.C., Cafaro, D.C., 2017. Scheduling multipipeline blending systems supplying feedstocks to crude oil distillation columns. Industrial & Engineering Chemistry Research 56, 10783-10797. 5. Cerdá, J., Pautasso, P.C., Cafaro, D.C., 2018. Optimization approaches for efficient crude blending in large oil refineries. Industrial & Engineering Chemistry Research 57, 84848501. 6. Saharidis, G., Minuoux, M., Dallery, Y., 2009. Scheduling of loading and unloading of crude oil in a refinery using event-based discrete time formulation. Comput. Chem. Eng. 33, 1413-1426. 7. Saharidis, G.K. and Ierapetritou, M.G., 2009. Scheduling of loading and unloading of crude oil in a refinery with optimal mixture preparation. Ind. Eng. Chem. Res. 48(5), 2624-2633. 8. Pan, M., Li, X., Qian, Y., 2009. New approach for scheduling crude oil operations. Chem. Eng. Sci. 64, 965-983.


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9. Li, J., Misener, R., Floudas, C.A., 2012. Continuous-time modeling and global optimization approach for scheduling crude oil operations. AIChE J. 58, 205-226. 10. Cerdá, J., Pautasso, P.C., Cafaro, D.C, 2015. An efficient approach for scheduling crude oil operations in marine-access refineries. Ind. Eng. Chem. Res. 54, 8219-8238. 11. Lee, H., Pinto, J.M., Grossmann, I.E., Park S., 1996. Mixed-integer linear programming model for refinery short-term scheduling of crude oil unloading with inventory management. Ind. Eng. Chem. Res. 35, 1630-1641. 12. Jia, Z., Ierapetritou, M., Kelly, J.D., 2003. Refinery short-term scheduling using continuous time formulation: Crude oil operations. Ind. Eng. Chem. Res. 42, 3085-3097. 13. Furman, K.C., Jia, Z., Ierapetritou, M., 2007. A robust event-based continuous time formulation for tank transfer scheduling. Ind. Eng. Chem. Res. 46, 9126-9136. 14. Karuppiah, R., Furman, K.C., Grossmann, I.E., 2008. Global optimization for scheduling refinery crude oil operations. Comput. Chem. Eng. 32, 2745-2766. 15. Mouret, S., Grossmann, I.E., Pestiaux, P., 2009. A novel priority-slot based continuoustime formulation for crude-oil scheduling problems. Ind. Eng. Chem. Res. 48, 8515-8528. 16. Mouret, S., Grossmann, I.E., Pestiaux, P., 2011a. Time representations and mathematical models for process scheduling problems. Comput. Chem. Eng. 35, 1038-1063. 17. Yadav S, Shaik MA., 2012. Short-term scheduling of refinery crude oil operations. Ind. Eng. Chem. Res. 51, 9287-9299. 18. Castro, P., Grossmann, I.E., 2014. Global optimal scheduling of crude oil blending operations with RTN continuous-time and multiparametric disaggregation. Ind. Eng. Chem. Res. 53, 15127-15145. 19. Castro. P., 2016. Source-based discrete and continuous-time formulations for the crude oil pooling problem. Comput. Chem. Eng. 93, 382-401. 20. Méndez, C.A., Henning, G.P., Cerdá, J., 2001. An MILP continuous-time approach to short-term scheduling of resource-constrained multistage flowshop batch facilities. Comput. Chem. Eng. 25, 701-711. 21. Tawarmalani, M., Sahinidis, N.V. 2005. A polyhedral branch-and-cut approach to global optimization. Mathematical Programming 103(2), 225-249.



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Scheduling upstream operations at inland petroleum refineries using a

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