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Global Optimal Scheduling of Crude Oil Blending Operations with RTN Continuous-time and Multiparametric DisaggregationClick to copy article linkArticle link copied!
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† Centro de Investigação Operacional, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal
‡ Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213-3890, United States
*Email: [email protected]
Abstract
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This paper addresses the modeling of crude oil operations in refineries assuming that all properties blend linearly. Guidelines are given on how to generate a Resource-Task Network superstructure that implicitly handles the complex logistics, while extending the scope of a well-known continuous-time formulation to variable recipe tasks with multiple input materials. The new single time grid formulation has the advantage of avoiding computationally inefficient big-M constraints, unlike previously proposed unit-specific and priority-slot based models. Through the solution of a set of test problems from the literature, we show that the resulting mixed-integer nonlinear programs can be solved close to global optimality by the commercial solver GloMIQO for the objective of gross margin maximization but not for operating cost minimization. We also show that adopting a two-step MILP-NLP algorithm where the mixed-integer linear relaxation is derived from multiparametric disaggregation can reduce the optimality gap by orders of magnitude.
Copyright © 2014 American Chemical Society
1 Introduction
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Economic and operability benefits associated with improved crude oil blend scheduling can reach multimillion dollars per year. (1) Every refinery that charges a mix of crude oils to the atmospheric and vacuum distillation units has an optimization opportunity. From the crude oil purchases perspective, the faster the ability for a refinery to assess whether a particular spot crude oil purchase will result in a feasible operation, the better the refinery can capitalize on short-term market opportunities. From the operational perspective, variation in crude oil mixture quality and flow rate charged to the distillation columns affects downstream production units and is perhaps the single most influential disturbance to a refinery. Overall, crude oil blend scheduling optimization is a relatively inexpensive and timely way to improve performance.
Early works on crude oil scheduling (2, 3) with mathematical programming techniques, addressed the problem with discrete-time models. Shah (2) proposed a decomposition procedure featuring mixed-integer linear programming (MILP) models for the downstream operation of tanks and crude distillation units as well as for the upstream portside operation. Lee et al. (3) proposed a MILP for the whole system that can be viewed as a relaxation of the mixed-integer nonlinear programming (MINLP) model that is required to guarantee that there are no discrepancies in crude compositions inside the tanks with respect to their outlet streams. Additional constraints to prevent widely fluctuating charging rates to distillation units have been suggested by Hamisu et al. (4)
The turn of the millennium saw increased emphasis on continuous-time models. Jia et al. (5) proposed a State-Task Network, unit-specific, nonconvex MINLP formulation. Rather than solving the full MINLP, a two stage MILP-NLP solution procedure was employed, featuring in the first stage a relaxed MILP model without the bilinear blending constraints, followed by the solution of the original MINLP after fixing all binary variables. Such hierarchical decomposition into logistics and quality subproblems was motivated by the lack of suitable commercially available optimization software and computer power to solve the complete crude oil scheduling problem in reasonable time. (1) A more rigorous unit-specific model with respect to the synchronization of time events with material balances is given in Furman et al. (6)
Moro and Pinto (7) opted for a single time grid MINLP continuous-time formulation that was tackled with the augmented penalty version of the outer-approximation method. (39) It exhibited a superior performance over the MILP formulation generated after discretizing the inventory levels of the tank farm. Sharing the time representation concept and the inability to provide global optima, Reddy et al., (8) proposed a MILP relaxation along the lines of Lee et al. (3) combined with a rolling-horizon algorithm to eliminate the composition discrepancy. The underlying idea is that there is no composition discrepancy for the first slot on each tank. Since the MILP provides the composition in all tanks at the end of the slot, the schedule up to that time can be frozen and the entire procedure repeated for the remaining slots by solving a series of MILPs.
Mouret et al. (9) considered yet another type of continuous-time formulation, called single operation sequencing model, which represents a schedule as a sequence of priority slots. The advantage is that generic constraints can be used to break symmetries and reduce the size of the search space. The nonlinearities involved by composition constraints were handled by a two stage MILP-NLP procedure identical to Jia et al., (5) which, however, led to small optimality gaps due to the objective function that was selected namely maximizing gross margins. Later, (10) the MILP part of the model (without the blending constraints) was compared to models relying on different time representation concepts. Using a single time grid to keep track of events taking place was found to be a better approach, with the continuous-time representation scaling better with problem size, and hence outperforming its discrete-time counterpart. An important aspect is that the considered single grid continuous-time model features a set of big-M sequence constraints that are known to decrease the performance of mixed-integer models.
Li et al. (11) proposed a novel MINLP unit-specific model for marine-access refineries that contrary to inland refineries have just charging tanks to crude distillation units rather than a sequence of storage and charging tanks. Besides this structural limitation, the model incorporates a wide variety of realistic operational features, such as single buoy mooring, multiple jetties, marine vessels containing more than a single crude, brine settling, and the possibility of multiple tanks feeding a column, following the work of Li et al. (12) In addition, 15 crude properties are considered that are modeled using linear additive blending indices on either volume or weight basis. A spatial branch-and-bound global optimization algorithm, that on each node uses the MILP-NLP two-stage strategy previously mentioned, is used to solve the MINLP problems. In this way, rather than removing the bilinear blending constraints, a tighter MILP relaxation is created through piecewise McCormick envelopes. (13) Overall, the integer feasible solutions obtained for a set of seven test cases from Li et al. (12) were guaranteed to be within 2% of global optimality.
In this paper, we approach the solution of the crude oil scheduling problem with the Resource-Task Network based single time grid, continuous-time formulation of Castro et al., (14) which can handle both batch and continuous tasks. Contrary to most scheduling problems that feature a fixed recipe with known proportions of input materials to a processing task, the main goal of crude oil scheduling is to identify the optimal mix of low-cost and premium crudes that maximizes profit or minimizes cost while meeting operational constraints. In fact, the proportions of the different available crudes are unknown a priori and can change with time, leading to crude transfer tasks that can be viewed as being of the variable recipe type. More specifically, we show how to (i) extend the generality of the continuous-time formulation to handle variable recipe tasks with multiple input materials (of which crude oil blending is a particular case); (ii) derive a RTN superstructure that implicitly meets the mandatory logistic constraint of no simultaneous inlet and outlet streams from a storage or charging tank, something that is not straightforward whenever multiple inlet/outlet streams are allowed; (iii) derive a tight relaxation of the bilinear blending constraints using multiparametric disaggregation (15, 16) (known to be tighter than standard (17) McCormick and scaling better with the number of partitions than piecewise (13) McCormick envelopes). Combining such elements leads to the main novelty of this work: a two-stage MILP-NLP algorithm that is capable of providing high quality solutions with low optimality gaps for the actual MINLP scheduling problem. As will be shown, this is particularly relevant for the objective of cost minimization.
The rest of the paper is structured as follows. In section 2, we give the problem definition and the system configuration as a collection of crudes, marine vessels, storage and charging tanks, distillation units, and existing connections. In section 3, guidelines are provided on how to convert the real system into the virtual system that is the RTN process model, featuring a variety of tasks and resources. The RTN-based mathematical model is then the subject of section 4, which is divided into linear and nonlinear parts, the latter providing two alternative relaxation approaches for the bilinear constraints that allow computing optimality gaps. The four test problems are provided in section 5, with section 6 describing the computational studies, including comparisons to previous work and commercial global optimization solvers with the objectives of gross margins and total cost. The paper ends with the conclusions in section 7.
2 Problem Definition
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In crude oil operations in refineries, it is required to schedule the unloading of crude marine vessels into storage tanks as well as the transfer of crude from storage to charging tanks and finally, to the crude oil distillation units (CDUs). Crudes with different properties are involved, and the objective is either to maximize the gross margin in the CDUs, which is accomplished by properly mixing the different crudes at the tanks so as to obtain the most profitable blends, or to minimize operating cost. The crude oil operation systems considered are the ones in Lee et al. (3) and Mouret et al. (9)
Let mv ∈ MV represent a crude marine vessel, st ∈ ST a storage tank, ct ∈ CT a charging tank, cd ∈ CD a crude oil distillation unit, cr ∈ CR a crude oil, and pr ∈ PR a crude oil property (e.g., density). Given are the possible connections between vessels, tanks, and units and the following information. Marine vessels are characterized by arrival time atmv [day] and volume of crude to discharge vmv,crin [kbbl = 103·bll]. For storage and charging tanks tk ∈ TK = ST ∪ CT, we must know the initial inventory vtk,cr0 [kbbl] as well as the lower and upper bounds on storage capacity vtkmin and vtkmax [kbbl]. Tanks involving crude oil mixing must respect the composition limits for the relevant properties: ctk,prmin and ctk,prmax. Crudes arriving or initially present in storage tanks have known compositions ccr,pr, which are assumed to blend linearly. Each charging tank is assigned a single crude blend that can be viewed as a final product. Final products are subject to minimum and maximum demands dcrmin and dcrmax [kbbl]. Transfer flow rates between units u,u′ ∈ U = MV ∪ ST ∪ CT ∪ CD are between ρu,u′min and ρu,u′max [kbbl/day]. In order to ensure that CDUs are operating continuously throughout the scheduling horizon H [day], ρct,cdmin > 0; for the remaining units, it is ρu,u′min = 0.
Logistic constraints play a very important role, being assumed that (i) a single berth is available at the docking station for vessel unloading; (ii) blending tanks must be in standing-gage operation (1) where there can either be flow in or out, not both (a generalization is the mixing-delay restriction); (iii) a charging tank may only feed one CDU at a time; (iv) a CDU can be charged by a single tank at a time.
2.1 Maximizing Gross Margin
Two alternative objectives will be optimized. The first is the maximization of the total gross margin of the distilled crude blends (Mouret et al. (9)) given the individual crudes margins pcr [$/bbl]. In order to reduce costly CDU switches between crude blends, the maximum number of distillation operations is set to nd.
2.2 Minimizing Operating Cost
The second and more challenging objective deals with the minimization of total operating cost (Lee et al., (3) Jia and Ierapetritou, (40) Yadav and Shaik (26)). Given are sea waiting costs for marine vessels cmvwsea [$/day]; harboring costs for unloading the crude cmvharb [$/day]; inventory costs for storage and charging tanks ctkinv [$/kbbl/day]; setup costs for crudes charged to the CDUs cchg [$].
2.3 System Configuration
Figure 1 depicts the refinery configuration for the first problem presented in Lee et al. (3) There are two marine vessels: one carrying crude A and another, crude B. Storage tanks are dedicated, the first to crude A and the second to B, and so, there is a one to one correspondence between marine vessels and storage tanks. Since there is a single crude inside a particular storage tank at any given time, the mass balance at the outlet splitter will be linear. In contrast, charging tanks have initially crudes C (upper tank) and D (lower tank) and can receive crudes A and B so as to prepare crude mixtures X (upper tank) and Y (lower tank). Bilinear mass balances will thus be involved. The distinction between tanks that are easy and difficult to model is made by using light and dark blue backgrounds, respectively. Note that in some of the other problems considered in this paper, blending constraints will be required for storage tanks as well, and there will be a case with only one crude in a charging tank. There will also be multiple CDUs and storage vessels with no incoming marine vessel.
Figure 1
Figure 1. Crude oil operation system for problem 1.
3 Resource-Task Network Process Model
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System configuration and logistic constraints can be captured by a Resource-Task Network (18) process model. In this section, we first show how to construct the process model for the problem in Figure 1, while later describing how to ensure that logistic constraints iii and iv, identified in section 2, are met when in the presence of multiple crude distillation units. Overall, a RTN describes the process as a set of resources r ∈ R interacting with a collection of tasks i ∈ I.
The first step is to identify the resources, which for now can be divided in two subtypes.
Equipment resources REQ include the docking station DS, marine vessels MV, storage ST, charging tanks CT, and distillation units CD, that is, REQ = DS ∪ MV ∪ ST ∪ CT ∪ CD. A resource linked to either the tanks or distillation units can be viewed as representing the inlet and outlet pipes associated with the unit. Equipment resources have a maximum availability of one (unary resource), thus ensuring that simultaneous inlet and outlet transfers are forbidden as defined by the logistic constraints. The docking station resource and associated tasks will ensure discharge of a marine vessel only when the contents of its predecessor have been depleted.
Material resources RMR include all the different crudes and their possible locations as they move through the system, see Figure 2. Taking crude A as an example, it can be located inside the upper marine vessel A_MV1, inside the top storage tank A_ST1, just after the storage tank A_ST1°, just before the charging tanks (A_CT1i, A_CT2i) and finally, inside the charging tanks (A_CT1 and A_CT2). Note that the subscripts refer to the numbering of marine vessels and tanks. As for the final crude mixtures X and Y, they are respectively generated after blending A, B and/or C, or A, B, and/or D, and so, they can be viewed as existing just after the charging tanks. Crudes X and Y represent distinct distillation runs that cannot be produced simultaneous since their producing tasks consume the CDU equipment resource (CD).
Tasks are primarily divided into fixed recipe and variable recipe tasks IVR, the latter including blending tasks IBL as a subset. In fixed recipe tasks, the standard tasks in STN/RTN scheduling models, the proportion of materials that is consumed/produced with respect to the amount handled by the task is known as a parameter. Fixed recipe tasks can either be continuous IC or storage tasks IS. Storage tasks can be viewed as a special type of batch task, lasting a single time slot regardless of the actual duration of the slot. (19) Variable recipe tasks allow for the mixing of different crudes at a proportion to be determined by the optimization model (the selection of a single crude is naturally included). Blending tasks are not as flexible since crude compositions in the inlet stream must match those in the upstream tank.
Figure 2
Figure 2. Resource-Task Network process model for problem 1.
Overall, Figure 2 features all the tasks identified in the system configuration of Figure 1, as well as additional tasks that have the purpose of guaranteeing logistic constraints (i–ii). More specifically, storage tasks (e.g., Hold_in_vessel MV1) ensure that the docking station becomes free only when all the crude has been removed from the vessel. The nonsimultaneous inlet and outlet transfers constraint (ii) is straightforward to model in cases with a single active inlet/outlet stream. However, the existing links between the storage and charging tanks make it possible for multiple active inlet/outlet streams, see Figure 3, something that is actually observed in the optimal schedules reported by Mouret et al. (9) To model this feature, which in the case of inlet streams can be seen as continuous blending, (20) auxiliary tasks have been defined that aggregate the outlet streams from storage tanks and the inlet streams to charging tanks. As an example, when active, Transfer_Out_ST1 continuously produces resource A_ST1° that is consumed at the same processing rate by tasks Transfer 3 and/or Transfer 4 (notice that the maximum processing rate of Transfer_Out_ST1 [1000 kbbl/day] is set so as to match the sum of the maximum rates of the subsequent tasks). Similarly, Transfer_In_CT1 consumes a variable proportion of resources A_CT1i and B_CT1i, at the same rate as Transfer 3 and Transfer 5.
Figure 3
Figure 3. Blending tanks can either be receiving or discharging crude. Storage/charging tanks allow a single/multiple inlet streams and multiple/single outlet streams.
3.1 Remarks
In Lee et al., (3) logistic constraint (ii) only applies to storage tanks involving crude oil mixing, whereas in Mouret et al. (9) it also applies to storage tanks receiving the same crude quality from marine vessels. Figure 2 shows the complete RTN for the latter case, indicating the tasks and resources to remove for the former case. If blending storage tanks can hold various crudes, the Transfer Out tasks may consume multiple resources, thus becoming variable recipe tasks. This is the case of problem 3 in Figure 4. However, there is no need to define them as blending tasks since it suffices to enforce the blending constraint for the subsequent transfer tasks, see Figure 5. Notice that to model two main transfer tasks (4 and 5), we already require 25 material location resources.
Figure 4
Figure 4. Crude oil operation system for problem 3 of Lee et al. (3)
Figure 5
Figure 5. Partial RTN process model for problem 3, showing the interaction of storage tank ST1 with charging tanks CT1 and CT2.
When multiple crude distillation units are involved, each CDU typically handles a subset of the crude blends. Taking the system of problem 2 as an example (Figure 6), the top distillation column can process blends X and Y, while the bottom column handles Y and Z. The charging tank in the middle can actually handle three crudes A, B, and C, besides the initial and final blends E and Y, but it cannot simultaneously feed the top and bottom columns, which is enforced by making transfer tasks 12 and 13 consume the unary resource CT2; see Figure 7.
Figure 6
Figure 6. Crude oil operation system for problem 2.
Figure 7
Figure 7. Partial RTN process model for problem 2, showing the interaction of the charging tanks with the distillation columns.
Even though it is assumed in the following that the crude discharge sequence from marine vessels is fixed to their order of arrival, the given process model allows for such decision to be determined by the optimization. This can be relevant when maximizing gross margin. On the other hand, non-negligible sea waiting costs make different discharge sequences suboptimal when minimizing operating cost.
Finally, it should also be noted that while the RTN diagram can become very large with the increase in the number of crude oils and tanks (and hence difficult to show to a scheduler for industrial use), the complete diagram does not need to be generated. In fact, the actual implementation in GAMS features a module that automatically generates the required sets of tasks and resources (and corresponding structural parameters, see section 4.3) given the existing connections between marine vessels and tanks, as well as the location of the different crudes.
4 Mathematical Formulation (MINLP)
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The mathematical formulation for optimizing the system of crude oil operations in refineries can be formulated as a mixed-integer nonlinear program (MINLP). Conceptually, it can be viewed as consisting of two parts. The scheduling element is responsible for the mixed-integer linear (MILP) constraints, while the blending constraints and the second objective function are of the nonlinear type (NLP). Compared to the multiperiod blending of refinery end products, (21) blending decisions also change from one time period to the next but now the duration of each time period is to be determined by the model rather than being fixed. Furthermore, we do not know a priori how many time periods will be required to achieve the optimum.
4.1 Time Representation
Previous industrial case studies from paper plants (22) and pipeline systems (23) featuring mostly continuous tasks with variable processing rates have shown a better performance for the continuous-time model. As highlighted in the recent review paper by Harjunkoski et al., (24) continuous-time models have the advantage of requiring fewer slots to represent a schedule, which in this case is particularly important since the complex bilinear terms will feature a time index (Li et al. (11) show a significant reduction in the number of bilinear terms compared to a discrete-time formulation). Furthermore, increasing the number of slots may lead to the execution of multiple rather than a single instance of a continuous task to achieve a certain production level, increasing solution degeneracy and hence possibly leading to more changeovers than those effectively required to achieve the same value of the objective function. (25) A continuous-time model is thus preferred.
The use of a single or multiple time grids remains to be decided. The latter have the advantage of further reducing the number of slots required to find the optimal solution and the drawback of requiring a large number of inefficient big-M constraints to model the interaction between different grids handling the same resource (check sequence constraints in the unit-specific scheduling models for crude oil operations in Jia et al., (5) Furman et al., (6) Li et al., (11) and Yadav and Shaik (26)). In this case, we opt to use a single time grid featuring t ∈ T event points that span from time zero to the given time horizon H, see Figure 8. The size of each slot is determined by the optimization.
Figure 8
Figure 8. In the given continuous-time formulation, events report to a single grid.
4.2 Model Variables
The scheduling part of the model requires three sets of binary variables: Ni,t indicates that task i is executed during slot t; Zt,mvi assigns the harboring of marine vessel mv to event point t; Zt,mv0 assigns the departure of marine vessel mv to event point t.
All continuous variables are nonnegative: ξi,t represents the amount of material [kbbl] transferred by task i during slot t; for variable recipe tasks i ∈ IVR, we also need to know the amount consumed of each input resource r, through variables ξ̅r,i,t [kbbl]. Variables Tt hold the absolute time of event point t [day], identifying the start of slot t; excess resource variables Rr,t give the available amount of resource r at event point t; excess variables Rr,tend give the amount just before the end of slot t, that is, immediately before event point t + 1. The two types of excess resource variables have no units for equipment resources r ∈ REQ, while for material resources r ∈ RMR the values are in [kbbl].
In order to compute the number of distillation runs, two additional sets of variables are required: Xcd,t identifies if a sequence independent changeover of crude blend occurs from slot t to t + 1; NDcd gives the number of distillation runs for unit cd ∈ CD. Note that while these variables take respectively binary and integer values, they can be defined as continuous since they appear in constraints with only binary variables, as will be seen later on.
The remaining four sets of variables are exclusive to the objective of minimizing operating cost: WSmv and HBmv give respectively the time waiting in sea and harboring for marine vessel mv; DTt represents the duration of time slot t; AItk,t gives the average crude inventory in tank tk during time slot t.
4.3 RTN Structural Parameters
The structural parameters link the RTN graph to the mathematical model. As recently described in Harjunkoski et al., (24) resources can interact with tasks either discretely or continuously, and five sets of structural parameters are sufficient to model all types of interactions when dealing with fixed recipe tasks and a continuous-time formulation.
Parameters μr,i and μ̅r,i are related to the binary extent variables Ni,t and give the discrete interaction at the start or end of the task, respectively. All dashed lines in Figure 2 linking tasks with equipment resources are of this type. More specifically, and taking the first transfer task as an example, we have μST1,Transfer1 = −1 (resource consumed at the beginning of the task) and μ̅ST1,Transfer1 = 1 (resource produced at the end of the task). The task also consumes/produces the first marine vessel (μMV1,Transfer1 = −1; μ̅MV1,Transfer1 = 1). Note that double arrows are used for a particular resource whenever there is both consumption at the start and production at the end.
Parameters νr,i and ν̅r,i are also related to discrete interactions but are linked to continuous extent variables ξi,t. For the crude oil system considered, they are used solely by the Hold_in_vessel tasks to ensure that the docking station becomes free only when all the crude has been removed from the corresponding marine vessel. We thus have for the RTN in Figure 2: νA_MV1,Hold_in_vesselMV1 = −1 and ν̅A_MV1,Hold_in_vessel MV1 = 1, meaning that for each bbl handled by the task, 1 bbl is consumed at the start, and 1 bbl is produced at the end.
Parameter λr,i is used for the interaction of the fixed recipe continuous transfer tasks with the material resources, which are represented as solid lines. Taking the Transfer 3 task in Figure 2 as an example, we have λA_ST10,Transfer3 = −1 and λA_ST1i,Transfer3 = 1, meaning that for each bbl being transferred between the storage and charging tanks, 1 bbl of crude A is continuously being removed from ST1 while continuously being supplied to CT1 (see also the explanation linked to eq 3).
Variable recipes tasks with flexible proportions of output products appear in the generic batch-scheduling problem of Westenberger and Kallrath (27) as mentioned in Kallrath. (28) Blömer and Günther (29) modeled this feature using a discrete-time STN scheduling model and highlighted that flexible proportions of input goods may be considered in a similar fashion. Indeed, Castro et al. (22) have used flexible inputs to determine the optimal amount of broke to recycle back to the tissue machines of a paper plant, where it is processed together with the corresponding raw material. The set of structural parameters defined for the RTN continuous-time scheduling formulation works well for the simplest case with just two input resources in the flexible recipe. Now, however, there can be up to six input resources in the flexible recipe, see task Transfer in CT2 in Figure 5, and so, an additional index needs to be used. Let λ̅r,r′,i represent the continuous change of resource r due to the consumption of resource r′ by variable recipe task i ∈ IVR. Taking task Transfer 7 in Figure 2 as an example, we have λ̅r,r,Transfer7 = −1 and λ̅X,r,Transfer7 = 1, ∀r ∈ {A_CT1,B_CT1,C_CT1}.
Parameters Rr0 are used to set the initial inventory vtk,cr0 of storage and charging tanks, as well as to define the initial availability of equipment resources other than marine vessels, for example, RST20 = 1.
In order to model the interaction of marine vessels with the system resources, we rely on parameters πr,mvi and πr,mvo. Taking problem 1 as an example, the second marine vessel carries νmv,Bin = 1000 [kbbl] that is made available to the system upon harboring πB_MV2,mv2i = 1000, together with the vessel itself: πB_MV2,mv2i = 1. The equipment resource is then removed from the system upon departure of the marine vessel: πMV2,mv2o = 1 disabling further crude transfers from it.
4.4 Scheduling Constraints
The RTN continuous-time formulation for the short-term scheduling of the crude oil system is given below. It neglects the blending constraints (given in section 4.5), and so, it can be viewed as assuming that each blending tank, storage or charging, is actually divided into multiple sub tanks, one for each crude type, of the same total capacity as the real tank. Such physical system allows for the compositions in the outlet stream to differ from those inside the tank, which cannot happen in reality. Segregation as a strategy is useful to reduce problem complexity by reducing the number of decisions to be made in terms of where crude oils should be stored. (1)
4.4.1 Excess Resource Balances
The structural parameters come into action in the excess resource balances, which are multiperiod material balance expressions involving the excess resource variables Rr,t and Rr,tend. In eq 1, the initial resource availability (first term on the right-hand side) is only considered at t = 1. For the remaining event points, the excess value at t is equal to that at the previous event point (t – 1) adjusted by the amounts discretely produced/consumed by all tasks starting or ending at t. The sixth term on the RHS reflects the material input to the system from incoming vessels, which is only accounted for in the case the arrival of marine vessel mv is assigned to event point t through variable Zt,mvi. Similarly, the seventh term removes the marine vessel resource whenever the vessel departs at event point t.
(1)
(1)The discrete interaction at the start of storage IS tasks affects excess variable Rr,t, while the interaction at the end affects excess variable Rr,tend, see eq 2. The purpose of the former is to temporarily hide from the system, coupled with eq 3, the amount of crude raw material (RRM ⊂ RMR) that still has not been transferred from the marine vessel so that the docking station remains unavailable for the other vessels (notice that since tasks are not defined for t = |T|, eq 3 forces all marine vessels to be empty at the end of the time horizon). The purpose of the latter is to allow for the capacity constraints of the storage and charging tanks to be enforced, see eq 4, where subset RtkMR holds the material resources that can appear inside tank tk (e.g., A_CT1, B_CT1, and C_CT1 are the material resources associated with charging tank CT1 in Figure 2). The second and third terms on the RHS of eq 2 give respectively the amounts continuously produced/consumed by fixed and variable recipe tasks. For further details concerning the complex interaction of structural parameters with the two types of excess resource variables and the different types of tasks, the reader is referred to Castro et al., (25) where an illustration is given.
(2)
(3)
(4)
(5)
(2)(3)(4)(5)Equation 3 is also useful to model the condition of no intermediate storage for the material resources located just after/before the storage/charging tanks (r ∈ RIO), effectively ensuring that the auxiliary Transfer_Out_ST/Transfer_In_CT tasks in Figures 2 and 5 occur at the same rate as their successors/predecessors and guaranteeing with eq 1 that simultaneous inlet and outlet transfers on tanks are forbidden. Equation 5 certifies that distillation units (r ∈ RCD) process crude in every time slot, becoming available only at the last event point.
4.4.2 Timing Constraints
The timing constraints of the single time grid formulation state that the difference in time between two consecutive event points must be within the minimum and maximum durations of the occurring task. For tasks i ∈ INE not linked to an equipment resource (e.g., Transfer 3–6 in Figure 2), the slot duration must be greater than the amount of material transferred divided by the maximum processing rate (eq 6), where the ρimax are taken from the ρu,u′max data. For the other continuous transfer tasks, previous research has shown (14) that writing the constraint per equipment unit leads to a tighter formulation. Equation 7 applies to the subset RTC ⊂ REQ of the equipment resources that excludes the docking station and marine vessels. Note that the summation on the RHS includes all inlet and outlet tasks of that particular unit (e.g., Transfer 1 and Transfer_Out_ST1 in Figure 2 for unit ST1). The absence of the docking station from the domain is because it is linked to the Hold_in_vessel batch tasks that always adjust to the duration of the time slot. As for the marine vessels, each is linked to a single transfer task that is already taken care in the constraints of the storage tanks.
(6)
(7)
(6)(7)Equation 8 applies only to the distillation units r ⊂ RCD, for which the inlet flow rate must be greater than a certain minimum value. Compared to the general constraint in Castro et al. (14) that applies to semicontinuous flows (either zero or between the bounds), it does not require a big-M term since distillation units are always processing crude, a constraint enforced by eq 5.
(8)
(8)According to Figure 8, the first event point corresponds to time zero whereas the last corresponds to the time horizon H (eq 9).
(9)
(9)Material supply from incoming vessels must be assigned to event points through binary variables Zt,mvi. If an event point t is linked to the harboring of marine vessel mv, then the absolute time Tt must be greater than the arrival time atmv; otherwise, it is free to vary. This constraint can be written as a disjunction, where each term is exclusive (see eq 10). Notice that marine vessels are allowed to wait before harboring. Similarly, if the vessel departs at event point t, then the time must be greater than the arrival time plus the minimum discharge time, which is calculated dividing the volume of crude to discharge by the maximum processing rate to the downstream storage tank st* (eq 11).
(10)
(11)
(10)(11)The single docking station prevents marine vessels from harboring/departing simultaneously and so such occurrences can be linked to a single event point. In eqs 12 and 13 and other equations, the domain of the binary variables does not include the last/first event point for harboring/departure since sufficient time must be given for discharging the vessel contents, which takes at least one time slot.
(12)
(13)
(12)(13)The convex hull reformulation (30) of eqs 10 and 11 followed by simple algebraic manipulations (31) to eliminate the resulting disaggregated as well as the Ztnoi and Ztnoo variables, leads to the MILP constraints in eqs 14 and 15.
(14)
(15)Equations 16 and 17 represent the exclusive or in eqs 12 and 13.
(16)
(17)
(14)(15)Equations 16 and 17 represent the exclusive or in eqs 12 and 13.(16)(17)It is assumed that the unloading sequence of marine vessels follows their order of arrival. Equation 18 thus assigns harboring events in an increasing order.
(18)
(18)4.4.3 Task Extent Constraints
Every time there is crude being transferred, binary variables Ni,t need to be activated so that the corresponding equipment units are consumed. Naturally, transfer tasks i ∈ INE not consuming equipment units do not need to be a part of eq 19. Parameter vimax is computed from the given values of vmv,crin, vtkmax, and dcrmax.
(19)
(19)Equation 20 states that the total amount processed by variable recipe tasks is equal to the sum of its components.
(20)
(20)4.4.4 Tank Compositions
The properties of the crude mixture inside a blending tank are calculated by adding for all constituents cr the product of the volume Rr,tend by the property ccr,pr. Recall that we are assuming that all properties blend linearly. Equation 21 then ensures that mixture properties are within given lower and upper limits, where subset CRr holds the crude associated with resource r.
(21)
(21)4.4.5 Product Demand
4.4.6 Number of Distillation Runs
A distillation run involving a certain blend of crude can span multiple time slots. In order to determine the number of runs, it is required to relate the changeover variables Xcd,t to the relevant task binary extent variables Ni,t. Given that in all examples there are exactly two transfer tasks linked to each distillation column, we can write the logic propositions in eqs 23 and 24, where IcdCD holds the pair of tasks linked to column cd. These are valid since distillation units are always operating (no idle slots in between). Logic constraints can easily be converted to the inequalities (32) in eqs 25 and 26. A unit’s number of distillation runs NDcd is then the number of changeovers plus one (eq 27).
(23)
(24)
(25)
(26)
(27)
(23)(24)(25)(26)(27)4.4.7 Objective: Maximize Gross Margin
Equation 28 defines the objective of maximizing the total gross margin of the distilled crude blends, where RcrCT ⊂ RMR represents the set of material resources inside the charging tanks that are linked to crude blend cr of value pcr (e.g., in Figure 2 these would be resources C_CT1 and C_CT2 for crude C). The two terms reflect the cases of the transfer tasks to the CDUs being variable (more common) or fixed recipe tasks (there is one such task in problem 4).
(28)
(28)Specific to this objective function is the constraint that the total number of distillation runs is lower or equal than the maximum specified value (eq 29).
(29)It should be highlighted that the scheduling model has no big-M constraints when maximizing gross margins.
(29)It should be highlighted that the scheduling model has no big-M constraints when maximizing gross margins.4.4.8 Objective: Minimize Operating Cost
Equation 30 gives the definition of the total operating cost that includes sea waiting, harboring, inventory, and changeover costs. Notice that the second term is bilinear, involving the product of two continuous variables, the average inventory AItk,t and the slot duration DTt.
(30)
(30)To relate the sea waiting times with the time values of the event points of the grid, we add a constraint to each disjunction in eq 10. The big-M reformulation (30) of eq 31 leads to eq 32, where we have used the tightest (31) big-M parameters possible: H – atmv.
(31)
(32)
(31)(32)The harboring duration for marine vessel mv is equal to the difference in time between the time values of harboring and departure event points, respectively, t and t′. This constraint can also be written as a disjunction (eq 33), being big-M reformulated into eq 34 (check section 5.1 in Castro and Grossmann (31) for a closely related reformulation). Note that there is no need to use the equality on the left disjunctions of eqs 31 and 33 since both the sea waiting and harboring durations have positive coefficients in the objective function being minimized.
(33)
(34)
(33)(34)Tank inventory costs are calculated according to the trapezoid area, (3) which requires the computation of the average inventory for each time slot, eq 35. In order to compute the accurate inventory cost, we need to know the duration of each time slot that is determined through eq 36.
(35)
(36)
(35)(36)As will be seen later on, the bilinear terms involved in the computation of the inventory cost have a major effect in model performance. In order to reduce the complexity, Jia et al. (5) approximated the average inventory by adding the inventory levels and dividing by the number of event points in the grid, which translates to eq 37. The approximated operating cost then becomes eq 38.
(37)
(38)
(37)(38)4.5 Matching Compositions Inside Blending Tanks and Their Outlet Streams
The nonlinear part of the model consists of a single set of constraints that ensures that a tank’s outlet stream has the exact same composition of the blend inside the tank. Let 0 ≤ ζi,t ≤ 1 represent the fraction of the volume inside the tank leaving it during slot t through blending task i ∈ IBL. It can be used to relate the amount of resource r ∈ RiBL available inside the tank at the start of the slot t and the amount of corresponding resource r′ ∈ RrCR consumed by the task through eq 37. Notice that the RHS is a bilinear term and that if the task is not active during slot t (Ni,t = 0), no material is being removed from the tank (ξ̅r,i,t = 0 due to eqs 19 and 20), and thus, the volume fraction must equal zero to meet the constraint.
(37)
(37)Taking i = {Transfer 4} in Figure 5 as an example, RiBL = {A–ST1,D–AT1} includes the resources corresponding to crudes A and D inside storage tank ST1, whereas resources r′ appearing in eq 37 are those consumed by the blending task, that is, for r = {A_ST1}, r′ ∈ RrCR = {A–ST1O}, and for r = {D_ST1}, RrCR = {D–ST1O}. For the charging tanks outlet transfers, resources r and r′ are actually the same (identified through the structural parameter λ̅r,r,i = −1). More specifically, for i = {Transfer 7} in Figure 2, RiBL = {A–CT1,B–CT1,C–CT1}, and so for r = {A_CT1}, r′ ∈ RrCR = {A–CT1}, and so on.
It should be noted that there are two choices for modeling the blending constraint. We have opted to consider individual flows and volume fractions as model variables, the alternative being total flows and component compositions. (41) This is possible since it is not required to know the compositions explicitly, only that the properties of the crude blends inside the tanks are within given lower and upper bounds, which is ensured through the linear constraint in eq 21, where the compositions of the base crudes ccr,pr are known parameters.
4.6 Relaxation of Bilinear Term in Blending Equation
One way to obtain a good feasible solution to the MINLP problem is to first relax the bilinear terms in eq 37 using one of the methods described next and then to solve the resulting MILP to obtain either an upper bound (UB), when maximizing gross margin, or a lower bound when minimizing operating cost. In the latter case, the bilinear terms in eq 30 need also to be relaxed (see section 4.6.3).
The binary variables are then fixed to the values from the MILP, and a constrained NLP version of the original problem is solved to find either a lower (LB) or upper bound. This approach is the main element of the specialized outer-approximation algorithm of Karuppiah et al. (33) that allows to compute an optimality gap that can be improved iteratively by adding integer cuts to the upper bounding MILP. While this is, in general, an inefficient approach, for the test problems considered in this paper, the optimality gaps are near zero for the objective of maximizing gross margins and so Mouret et al. (9) also relied on a single iteration. The nonlinear blending constraint equivalent to eq 37 was removed from their model to generate the upper bounding problem.
In this paper, we follow the single iteration approach since the quality of the MILP relaxation resulting from multiparametric disaggregation (15, 16) is directly related to the accuracy level of the discretized variables, which is set by the user. Optimality gaps will be calculated for different settings, which could also be changed iteratively in the context of a competitive global optimization algorithm for bilinear up to mixed-integer polynomial programs. (16, 34, 35) The standard McCormick LP (17) relaxation will also be tested for comparative purposes. Note that due to the scheduling part of the model in section 4.4, both relaxation approaches lead to MILP problems.
4.6.1 Using McCormick Envelopes (LP)
The standard approach to provide a relaxation to eq 37, is to replace the bilinear term ζi,t·Rr,t with a new set of continuous variables Wr,i,t (see eq 38) coupled with four sets of constraints that correspond to the McCormick envelopes. (17) These generate the tightest linear relaxation for the given bounds on variables ζi,t and Rr,t.
(38)In this case, the lower bounds are zero and the upper bounds are equal to 1 and Rrmax, respectively. Parameter Rrmax gives the maximum amount of crude that can exist within a blending tank. It can easily be calculated from initial inventory and that of incoming marine vessels and cannot be higher than the tank’s maximum capacity. The McCormick envelopes for this specific case are then given by eqs 39–42.
(39)
(40)
(41)
(42)
(38)In this case, the lower bounds are zero and the upper bounds are equal to 1 and Rrmax, respectively. Parameter Rrmax gives the maximum amount of crude that can exist within a blending tank. It can easily be calculated from initial inventory and that of incoming marine vessels and cannot be higher than the tank’s maximum capacity. The McCormick envelopes for this specific case are then given by eqs 39–42.(39)(40)(41)(42)4.6.2 Using Multiparametric Disaggregation (MILP)
Multiparametric disaggregation (15, 16, 34, 35) involves discretizing one of the variables of each bilinear term over a set of powers k ∈ {ψ,...,η}. In this work, we opt to discretize variables ζi,t, which are fewer in number than variables Rr,t, since a blending task i involves multiple resources r. The upper bound is also constant for all tasks i and slots t (equal to 1) and so parameter η = ⌊log10 1⌋ = 0. Parameter ψ is chosen by the user so as to reach a certain accuracy level for the discretized variables. To achieve a tighter relaxation than the one given by the McCormick envelopes: ψ ≤ −1, with the quality of the relaxation improving as the value of ψ decreases.
Figure 9
Figure 9. Volume fraction variables ζi,t are calculated as a sum of a discrete point (illustration for level ψ = −1) and a residual variable 
i,t, thus achieving a continuous domain.
Binary variables Yi,t,j,k identify if the volume fraction of blending task i at time t features digit j ∈ {0,...,9} in position k of the decimal numerical system. Since there always exists a gap between discretization points for a finite ψ, residual variable 0 ≤ 
i,j ≤ 10ψ is added to obtain a continuous domain; see eq 43 and Figure 9.
(43)
i,j ≤ 10ψ is added to obtain a continuous domain; see eq 43 and Figure 9.(43)The bilinear variable Wr,i,t is also written as a sum of an approximation and a residual term 
r,i,t. In eq 44, R̂r,i,t,j,k is the disaggregated variable linked to Rr,t and the discrete value j of ζi,t associated with power k. The sum of variables R̂r,i,t,j,k over all digits j must be equal to the original variable Rr,t for every power k, see eq 45. Equation 46 then ensures that a single digit j is active at position k. In case the digit is selected, the value of the disaggregated variable must be lower than the upper bound of the original variable, see eq 47.
(44)
(45)
(46)
(47)
r,i,t. In eq 44, R̂r,i,t,j,k is the disaggregated variable linked to Rr,t and the discrete value j of ζi,t associated with power k. The sum of variables R̂r,i,t,j,k over all digits j must be equal to the original variable Rr,t for every power k, see eq 45. Equation 46 then ensures that a single digit j is active at position k. In case the digit is selected, the value of the disaggregated variable must be lower than the upper bound of the original variable, see eq 47.(44)(45)(46)(47)The residual variables 
r,i,t actually represent the bilinear term arising from 
i,t · Rr,t, being relaxed using the McCormick envelopes described in section 4.6.1; see eqs 48–51. Although not strictly necessary, the McCormick envelopes on the original variables (eqs 39–42) have also been added to improve the quality of the relaxation. Equation 38 completes the formulation.
(48)
(49)
(50)
(51)
r,i,t actually represent the bilinear term arising from i,t · Rr,t, being relaxed using the McCormick envelopes described in section 4.6.1; see eqs 48–51. Although not strictly necessary, the McCormick envelopes on the original variables (eqs 39–42) have also been added to improve the quality of the relaxation. Equation 38 completes the formulation.(48)(49)(50)(51)4.6.3 Remarks
The AItk,t·DTt bilinear term in the operating cost objective function (eq 30) was relaxed in the same manner as ζi,t·Rr,t. For the multiparametric disaggregation relaxation, the option was to discretize the slot duration variables DTt, primarily because their values are in the order of units, closer to those of the other discretized variables (volume fractions ζi,t ∈ [0,1]). Such option reduces the number of binary variables when considering a global accuracy parameter ψ.
5 Test Problems
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The four test problems considered are taken from Lee et al. (3) with the data being provided as Supporting Information. It should be highlighted that we have multiplied the crude volumes by ten to be consistent with the motivating example. (3) In order to be able to generate the same values for the objective function, the cost coefficient parameters were adjusted accordingly. The same system configuration is considered by Jia et al., (5) Ierapetritou et al., (40) Mouret et al., (9) and Yadav and Shaik (26) with the exception of P3. While the configuration was not given in Lee et al., (3) it could be inferred from the optimal schedule that shows MV2 discharging to ST3. It is also the only way of meeting the composition bounds on that storage tank. Mouret et al. (9) do not enforce composition constraints on the storage tanks and use the configuration shown in Figure 10. The other references assume a more complex topology for problem 3 allowing marine vessels to discharge to multiple charging tanks even simultaneously, which is the only way to meet the composition bounds. This feature is not handled in this paper, and so, such problem is not included in comparison with Jia et al., (5) Ierapetritou et al., (40) and Yadav and Shaik. (26) Typos in the data were also detected.
Figure 10
Figure 10. Crude oil operation system for problem 3 in Mouret et al. (9)
In terms of system topology, problem complexity has two different sources. From the scheduling point of view, it increases with the number of tasks, since more tasks will typically require more time slots to represent the optimal schedule, leading to a more difficult problem. In this respect, the features in Table 1 predict P4 as the most challenging. Note that according to the system topology and the problem data, the charging tank on the bottom in Figure 11 can only receive crude E, which will be the only component of final blend W. Such charging tank is thus not a blending tank, which explains its light blue background compared to the other charging tanks. From the blending point of view, complexity increases with the number of blending tanks and the number of components for each blend, which is very much related to the number of resources required to model the problem. From this perspective, P3, with its six blending tanks and a total of 59 resources, is the most complex.
Figure 11
Figure 11. Crude oil operation system for problem 4.
Table 1. Number of Tasks and Resources Per Problem
| tasks | resources | |
|---|---|---|
| P1 | 12 | 24 |
| P2 | 20 | 38 |
| P3 | 23 | 59 |
| P4 | 26 | 51 |
The choice of the time representation and the assumptions made also affect the computational performance of the resulting mathematical formulation or optimization algorithm. The main alternatives that have been used in the literature are summarized in Table 2 and compared to the current work. While all listed references present the full set of mixed-integer nonlinear constraints, most focus on the mixed-integer linear programming relaxation of the problem. If there are no composition discrepancies between the inventory of blending tanks and their outlet streams, the solution of the MILP provides the optimal solution to the original MINLP. Otherwise, it is used to guide the search of either the NLP solver (after fixing the binary variables), or of the MINLP, leading to local solutions. Note that previous works have not attempted to solve the MINLP to global optimality.
Table 2. Main Features of Alternative Scheduling Formulations That Have Tackled the Problem
| Lee et al. (3) | Jia et al. (5) | Yadav and Shaik (26) | Mouret et al. (9) | this work | |
|---|---|---|---|---|---|
| time representation | discrete | continuous | continuous | continuous | continuous |
| # time grids | single | multiple | multiple | multiple | single |
| Flow in and out in storage tanks? | no, if blending | yes, always | yes, alwaysa | no, always | no, if blendingb,c |
| Composition constraints on storage tanks? | yes | yes | yes | no | yesa |
| objective function | operating cost | operating cost | operating cost | gross margin | both |
| inventory cost | rigorous (linear term) | estimate | estimate | not applicable | rigorousc (bilinear term) |
| blending constraints in MILP relaxation | relaxed | ignored | ignored | ignored | relaxed with either McCormick or Multiparametric Disaggregation |
| Solution of rigorous MINLP attempted? | no | yes, fix binaries and solve NLP | yes | yes, fix binaries and solve NLP | yes, to global optimality |
6 Computational Studies
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We now evaluate the performance of different algorithms that can be used to solve the crude oil blending MINLP scheduling problem. These include commercial solvers (i) DICOPT; (ii) BARON; (iii) GloMIQO; and the two-stage MILP-NLP strategies described in section 4.6, relying on (iv) standard McCormick relaxation of the bilinear terms; (v) multiparametric disaggregation (MDT). While all algorithms may fail to find the global optimal solution, DICOPT (36) is the most limiting since it works on the assumption that the relaxed nonlinear problem is convex and does not compute an optimality gap. Through spatial branch and bound, BARON (37) and GloMIQO (38) can theoretically close the gap to zero and prove global optimality, while the same outcome can be obtained by the MDT relaxation for a sufficiently high discretization level (16, 34) (low ψ value).
All mathematical formulations were implemented in GAMS, release 24.1.3, using solvers CPLEX 12.5.1 for the MILP relaxation problems and CONOPT 3.15L for the NLP problems. Solvers DICOPT, BARON 12.3.3, and GloMIQO 2.3 tackled the original MINLP. We have used a single thread and default options up to (a) relative optimality tolerance = 10–6 and (b) maximum computational time = 3600 CPUs. The hardware consisted of a desktop with an Intel i7-3770 (3.40 GHz) processor, with 8 GB of RAM and running Windows 7. In the case of DICOPT, the volume fraction variables ζi,t were initialized to 0.5 to guarantee a feasible solution in all runs.
It should be noted at this point that the specified number of time slots |T| for the grid acts as a hidden constraint, meaning that the global optimal solution for |T| is not necessarily the real global optimum solution. In order to overcome this issue, we have considered the standard termination criterion of stopping after no improvement in the objective function following a single increment in |T|. In case of normal termination within the maximum resource limit, we get a best-found solution. For each test problem, we will be showing results for three different |T| settings, starting with the minimum value that ensures feasibility, and ending with lowest value that is able to find the best solution (note that the computational times are for the individual runs and not the accumulated values of all runs leading to |T|). An interesting point is that a small number of time slots contributes to maximizing the performance of the logistics part of the problem that can be viewed as minimizing the number of active movements (i.e., transitions or switchovers), (29) something that is not included in the objective function.
6.1 Comparison to Literature Results
In the next subsections we provide detailed comparisons to relevant previous work. Note that the proposed model has been adjusted to match the conditions previously reported (to the best of our knowledge).
6.1.1 Priority Slots Continuous-Time Approach of Mouret et al. (9)
We start by comparing the results to the other work that has tackled the gross margin maximization objective function using a MILP-NLP solution strategy. The priority slots based, continuous-time formulation of Mouret et al. (9) is freely available at the MINLP Cyber-infrastructure http://minlp.org/library/problem/index.php?i=117&lib=MINLP and so the reported computational times in Table 3 are for the same hardware and software. Recall that their MILP relaxation ignored blending constraints whereas in this work, the blending constraints are relaxed using the multiparametric disaggregation technique (16, 35) (MDT). In Table 3, we show the best solutions obtained by the two approaches, listing the suitable number of time slots and the computational time for that specific run (bear in mind that the solutions from Mouret et al. (9) were also found after solving the problem for various number of slots). Notice that global optimality was proven for all but P3. For this problem, we were able to reduce the gap between the lower and upper bounds by roughly 1 order of magnitude to 0.37% and new best solutions were obtained for P1 and P2 (0.1 and 1.3% increase in profit). The computational times required are nevertheless much longer, but this is due to the additional binary variables and constraints of MDT since with the McCormick relaxation the scheduling model can be solved in just a few seconds as well (see Table 6).
The fact that the priority slots model by Mouret et al. (9) is unable to find the same solutions in P1 and P2 (despite the zero gap) is due to cardinality constraints on the maximum number of slots for each transfer task from the marine vessels to the storage tanks, which are not part of the problem definition but lead to a simplification of the model. Indeed, the model (9) in the MINLP Cyber-infrastructure shows a maximum of a single transfer from each marine vessel, whereas the optimal schedules for P1 and P2 given in Figures 12 and 13, feature respectively one and three discharge interruptions.
Table 3. Best Found Solutions for Gross Margin Maximization Showing Minimum Number of Slots, Optimality Gap, and Total Computational Time of MILP-NLP Strategy
| priority slots (Mouret et al. (9)) | RTN, single time grid (this work) | |||||||
|---|---|---|---|---|---|---|---|---|
| MINLP without blending constraints | blending constraints relaxed using multiparametric disaggregation | |||||||
| problem | # slots | gross margin [k$] | gap [%] | CPUs | # slots | gross margin [k$] | gap [%] | CPUs |
| P1 | 5 | 7975 | 0 | 0.35 | 6 | 7982.5 | 0 | 1.02 |
| P2 | 6 | 10117.5 | 0 | 0.93 | 9 | 10246.1 | 0 | 55.6 |
| P3 | 9 | 8544.9 | 2.2 | 5.96 | 7 | 8544.9 | 0.37 | 3600 |
| P4 | 4 | 13254.8 | 0 | 0.47 | 6 | 13254.8 | 0 | 8.15 |
Figure 12
Figure 12. Global optimal solution for problem 1 and gross margin maximization ($7,982,500).
In Figure 12, the filling task of storage tank ST1 (Transfer 1) starts at time zero and receives 50 kbbl of crude A from the first marine vessel at the maximum rate of 500 kbbl/day up to T2 = 0.1 days. ST1 then sends 300 kbbl of A to CT1 at the same rate up to T3 = 0.7 days, before receiving the remaining 950 kbbl of A. During that time, CT1 is also receiving 150 kbbl of B from ST2, following 50 kbbl of B from ST2. This mixing of crudes in a destination tank, one after the other, can be classified as batch blending. (20) The resulting blend of 30% A, 20% B and 50% C is charged to the distillation column up to T5 = 4.396 days. The crude from CT2 in the first two slots is pure D and is transferred to the CDU at the minimum rate of 50 kbbl/day. In slots five and six, the crude composition from CT2 is 0.7% A, 52.8% B, and 46.5% D. The incoming crude from the second marine vessel is only discharged in the last slot, starting at T6 = 5.965 days, and so, it does not have time to contribute to the profit. At the end of the time horizon, ST1 holds 943 kbbl of A, ST2 is full, while CT1 and CT2 are almost empty.
Figure 13
Figure 13. Global optimal solution for problem 2 and gross margin maximization ($10,246,075).
Interrupting the discharge of marine vessels is important in the context of a system that does not allow simultaneous inlet and outlet streams from a tank. In the case of P2 in Figure 13, emptying a vessel takes a minimum of 2 days, a time during which no crude can be sent from the storage tank to its downstream charging tanks. The no interruption constraint (9) makes the system less flexible to generate optimal blends leading to a decrease in profit. As can be seen in the optimal schedule, ST1 receives 100.5 kbbl of A at the maximum rate, so that in the second slot, 300.5 kbbl can be sent to CT1, emptying ST1 in the process. CT1 also receives 399.5 kbbl of B during slots 1–2, to make 1000 kbbl of crude with composition 30.05% A, 39.95% B, and 30.0%, which feeds column 1 until the end of day 10. The second parcel from the first marine vessel occurs during slot 3 (T3 = 0.802, T4 = 0.962 days) and transfers 79.763 kbbl of A at near maximum rate to ST1, which is then sent to CT2, creating 959.90 kbbl of crude with composition 8.31% of A, 10.47% of B, 33.31% of C, and 47.91% of E. Tank CT2 feeds column 1 with pure E in the first couple of slots, switching to column 2 at T5 = 1.441 days and becoming empty at T9 = 8.144 days. Finally, part of crude C from the third marine vessel still makes it to column 2 through CT3, contributing to create a blend of 2.37% B, 74.84% C, and 22.79% F. The final remark about this solution is that it will not be possible in the subsequent scheduling period to keep the condition of always operating distillation columns, since the two possible charging tanks to column 1 reach the end of the time horizon with no crude inside.
6.1.2 Discrete-Time Approach of Lee et al. (3)
Lee et al. (3) minimized the operating cost of the system considering an accurate inventory term. The objective function also includes harboring costs, and it is allowed for nonblending storage tanks to receive crude from marine vessels while simultaneously transferring it to the downstream charging tanks. It is thus no surprise that the interruption of marine vessels discharge does not appear in the optimal solutions.
For the same number of time slots, the proposed continuous-time model can be seen as a rigorous relaxation of the discrete-time model of Lee et al., (3) since the location of events is selected by the optimization rather than being fixed a priori. As can be seen by the results in Table 4, this is an important limitation of the discrete-time formulation that returns considerably worse solutions even when considering twice as many time slots. We can thus conclude that the continuous-time representation is by far a better choice for this continuous process, which is consistent with the results in Castro et al. (23) for pipeline systems.
Table 4. Best-Found Solutions for Operating Cost Minimization and Accurate Inventory Costs
| Lee et al. (3) | this work | ||||
|---|---|---|---|---|---|
| discrete | continuous | ||||
| problem | # slots | cost [k$]a | # slots | cost [k$] | reduction [%] |
| P1 | 8 | 217.667 | 7 | 210.538 | 3.3 |
| P2 | 10 | 352.55 | 7 | 320.496 | 9.1 |
| P3 | 12 | 296.56 | 7 | 287.000 | 3.2 |
| P4 | 15 | 420.99 | 7 | 365.088 | 13.3 |
a
Reported values are for the MILP relaxation and hence may feature composition discrepancies at blending tanks that can possibly lead to further increments in cost when corrected.
Figure 14
Figure 14. Best-found solution for problem 2 and operating cost minimization ($320,496).
The most important cost reductions are achieved for P2 and P4. In our best found schedule for P2 in Figure 14, sea waiting times are equal to 1.4, 3, and 2 days, whereas in Lee et al., (3) MV1 wait times are longer (2 days), leading to a higher sea waiting cost ($35,000 vs $32,000). Harboring and changeover costs are the same in both cases ($48,000 and $150,000), the latter corresponding to two oil changes in CD1 and one in CD2. The main difference is thus the inventory cost, particularly due to CT2 that is now empty most of the time. The other aspect worth highlighting is that time slots 3–5 do not start at the beginning of the days, which is not possible for the specified discretization level in Lee et al. (3) (1 day). We also show the best-found solution for P4 in Figure 15, where the 67.6% of the reduction in cost is due to quicker harboring, with all vessels discharging in 1.2 days, compared to the 3 days in the seminal paper.
Figure 15
Figure 15. Best-found solution for problem 4 and operating cost minimization ($365,088).
6.1.3 Unit-Specific Continuous-Time Approaches
The unit-specific continuous-time approaches of Ierapetritou and co-workers, (5, 40) and Yadav and Shaik, (26) have solved the problem considering an approximate representation of inventory cost (eq 37) that has the advantage of being linear, compared to the nonlinear formula of the accurate cost in eq 35. This choice has a major impact in the complexity of the resulting problems, which are much simpler to solve. As can be seen in the last column of Table 5, GloMIQO can solve to global optimality in roughly 1 min the MINLPs resulting from our single grid formulation (previous unit-specific works do not report on attempts to solve the rigorous MINLP with a global solver). Note that simultaneous flow in and flow out is allowed in storage tanks to provide the same basis for comparison. Better solutions are obtained compared to Ierapetritou and co-workers (5, 40) in P1 and P2 but not in P4. The results in Jia et al. (5) reported in Table 5 are for the MINLP model that oddly correspond to lower costs than the values from the MILP relaxation ignoring blending constraints (also given in ref 5). In their later work, (40) only the costs for the MILP relaxation are reported, which are higher than those from current work. Yadav and Shaik (26) allow for the possibility of incoming crudes from marine vessels bypassing storage tanks with the same crude (not allowed in this work). This seems to be the case in P1 and P2 and the reason why better solutions are obtained. The other possible explanation is given next.
Table 5. Best-Found Solutions for Operating Cost Minimization and Approximate Inventory Costs
| Jia et al. (5) | Jia and Ierapetritou (40) | Yadav and Shaik (26) | this work | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| multiple time grids | multiple time grids | multiple time grids | single time grid | |||||||||
| problem | # slots | cost [k$] | CPUsa | # slots | cost [k$]b | CPUsa | # slots | cost [k$] | CPUsa,b | # slots | cost [k$] | CPUsc |
| P1 | 3 | 225.00 | 0.96 | 247.0 | 0.28 | 3 | 217 | 0.59 | 5 | 217.667 | 0.76 | |
| P2 | 3 | 325.80 | 1383 | 413.48 | 4.89 | 4 | 305.8 | 51.84 | 8 | 320.358 | 26.6 | |
| P4 | 3 | 341.10 | 21912 | 3 | 387.15 | 7.87 | 9 | 362.022 | 79.4 | |||
a
Computational times are for different hardware and software than current work.
b
Reported values are for the MILP relaxation.
c
Results for global optimization solver GloMIQO.
The approximate inventory cost function in eq 37 features in the denominator the number of event points in the grid, which is not a good option considering that the optimal number of events is changed in the search for the global optimal solution. More specifically, it was observed experimentally that after a certain number of slots, the optimization will “cheat” by duplicating event points corresponding to the points of minimum inventory so that they contribute more to the objective function and the total cost is reduced. This can be spotted easily by checking if multiple consecutive event points (e.g., t and t + 1) with the same absolute time value (e.g., Tt = Tt+1) exist. The number of time slots reported in Table 5 for our current work are before this happened. Using more slots leads to a further decrease in cost, $213,000 in P1 for |T| = 7 (6 slots) and $304,694 in P2 for |T| = 13. These values are already lower than those reported by Yadav and Shaik. (26)
The drawback of the single time grid formulation is that it requires a higher number of events to represent a schedule compared to the unit-specific approaches. The advantage is that the constraints required to model inventory in tanks are much simpler, not requiring big-M terms. A trade-off is thus involved, which in this particular problem seems to favor the single time grid approach. This may not be the case in other refinery applications and it is certainly not the case for scheduling in general. (24)
6.2 Comparison to Different Optimization Approaches
Tables 6 and 7 provide detailed computational statistics for the objectives of gross margin maximization and total cost minimization, respectively. The comparisons are written for the two proposed MILP-NLP solution approaches, as well as for the solution of the original MINLP with commercial solvers. The most important result is that the objective function plays a major role. In fact, while three of the four gross maximization problems can be solved to global optimality, none of the operating cost minimization problems is even close to be solved to provable global optimality when considering the largest number of time slots (last three columns in Tables 6 and 7). This is primarily due to the quality of the MILP relaxation resulting either from McCormick envelopes that are also part of commercial solvers GloMIQO and BARON, or multiparametric disaggregation (MDT). Overall, we conclude that the simplest MILP-NLP relaxation using McCormick envelopes (see first row for each problem) is acceptable for gross margin maximization, only failing to find the global optimum for P3 (2.29% gap), while operating cost minimization demands a global optimization algorithm, since all found solutions were highly suboptimal and the returned optimality gaps are as high as 112%.
Using the two-stage MILP-NLP approach with the MDT relaxation for operating cost minimization (Table 7) leads to quite good optimality gaps for the minimum number of time slots (≤0.071%). A single increase in the number of slots is still manageable for P1–P3 (≤0.03%) but is already too much for P4 (25.8%). Increasing the number of event points to eight widens the gap to 7.8, 16.2, 48.1, and 55.2%, but such gaps are still below the ones reported by the solver GloMIQO, with the exception of P3 (48.1 vs 45.8%). Overall, the optimality gaps obtained by the proposed MDT approach can be up to 4 orders of magnitude smaller than the values from GloMIQO (P1 for |T| = 6). The two-stage MDT approach was able to find slightly better solutions than GloMIQO in P1 and P2, but returned suboptimal solutions in P3 and P4. This is to be expected in the presence of a weak MILP relaxation, considering that a single starting point is used to find a feasible solution to the original MINLP. In contrast, global solvers GloMIQO and BARON generate multiple starting points naturally, as part of the spatial branch-and-bound procedure, being thus more likely to find the optimum.
GloMIQO was the best performer in terms of solution quality, which was not totally unexpected given that we are dealing with a recent solver that was developed for the solution of blending problems. In contrast, the poor performance of BARON came as a surprise. Notice in Tables 6 and 7 that BARON is incapable of finding a feasible solution in 12 out of 22 problems, proving optimality in just three of the remaining eight problems (all dealing with gross margin maximization). DICOPT, which assumes that the resulting MINLPs become convex once the binary variables are relaxed to continuous, can still find five best solutions for the lowest settings of |T| but becomes less effective with the increase in the number of slots. Still, it is a very fast solver, a feature shared with the McCormick MILP-NLP algorithm.
Table 6. Computational Statistics for Different Solution Approaches, Commercial Solvers, and Gross Margin Maximizationa
| |T| = (5,6,5,7)b | |T| = (6,7,8,−)c | |T| = (7,10,8,−)c | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| approach | discrete variablesd | total variablesd | total equationsd | margin [k$] | gap [%] | CPUs | margin [k$] | gap [%] | CPUs | margin [k$] | gap [%] | CPUs | |
| P1 | McCormick | 56 | 396 | 488 | 7975e | 0.00e | 0.37 | 7982.323 | 0.00 | 0.35 | 7982.5 | 0.00 | 0.49 |
| MDT (ψ = −1) | 152 | 812 | 944 | 0.36 | 0.48 | 0.89 | |||||||
| GloMIQO | 43 | 372 | 392 | 0.31 | 0.63 | 0.76 | |||||||
| BARON | 37.5 | 150 | 0.18 | 3600 | |||||||||
| DICOPT | 7750 | 0.45 | 0.73 | 7975 | 0.44 | ||||||||
| P2 | McCormick | 110 | 828 | 1119 | 9000 | 0.00 | 0.36 | 9461.05 | 3.19 | 1.04 | 10246.08 | 0.00 | 4.27 |
| MDT (ψ = −1) | 350 | 1998 | 2439 | 1.33 | 9639.475 | 0.00 | 18.4 | 46.0 | |||||
| GloMIQO | 110 | 758 | 839 | 0.75 | 596 | 16.5 | |||||||
| BARON | no sol. | 3600 | no sol. | 3600 | no sol. | 3600 | |||||||
| DICOPT | 9000 | 1.33 | 9524.5 | 1.98 | 9000 | 31.7 | |||||||
| P3 | McCormick | 88 | 1081 | 1495 | 8250 | 0.00 | 0.39 | 8250 | 2.37 | 0.61 | 8540 | 2.29 | 1.59 |
| MDT (ψ = −1) | 616 | 3473 | 4147 | 0.99 | 8369.601 | 0.005 | 7.15 | 8544.891 | 0.56 | 43.8 | |||
| MDT (ψ = −2) | 1056 | 5313 | 5731 | 3.80 | 0.002 | 268 | 0.37 | 3600 | |||||
| GloMIQO | 88 | 941 | 935 | 0.55 | 0.001 | 3600 | 0.70 | 3600 | |||||
| BARON | 2573 | 8250 | 8.67 | 3600 | no sol. | 3600 | |||||||
| DICOPT | 0.68 | 7922.857 | 0.86 | 8250 | 1.32 | ||||||||
| P4 | McCormick | 168 | 1331 | 1780 | 13254.76 | 0.00 | 0.85 | ||||||
| MDT (ψ = −1) | 528 | 3203 | 3922 | 5.26 | |||||||||
| GloMIQO | 168 | 1217 | 1324 | 1.94 | |||||||||
| BARON | no sol. | 3600 | |||||||||||
| DICOPT | 13215.56 | 1.94 | |||||||||||
a
Optimal solution in bold.
b
Number of event points used for P1 (5), P2 (6), P3 (5), and P4 (7), respectively.
c
Results for P4 for runs with |T| > 7 are not listed since the gross margin did not improve and the optimality gap remained zero.
d
While the problem size increases with the number of slots, the number of variables and constraints is only given for the lowest setting in order to save space.
e
To facilitate interpretation of the results, values for gross margins and optimality gaps are merged for algorithms leading to the same outcome and are listed on the first row of the group (e.g., DICOPT is the only one not proving optimality for P1 at |T| = 5).
Table 7. Computational Statistics for Operating Cost Minimization with Accurate Inventory Costs
| |T| = (4,5,5,6) | |T| = (6,6,6,7) | |T| = (8,8,8,8) | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| approach | discrete variables | total variables | total equations | cost [$] | gap [%] | CPUs | cost [$] | gap [%] | CPUs | cost [$] | gap [%] | CPUs | |
| P1 | McCormick | 36 | 309 | 434 | 245357 | 69.4 | 0.25 | 230500 | 74.6 | 0.48 | 233707 | 77.1 | 0.46 |
| MDT (ψ = −1) | 165 | 921 | 1085 | 239000 | 0.46 | 2.2 | 211588 | 0.30 | 67.7 | 210538 | 7.8 | 3600 | |
| MDT (ψ = −2)a | 255 | 1311 | 1424 | 0.046 | 6.9 | 0.029 | 148 | ||||||
| MDT (ψ = −4)a | 435 | 2091 | 2102 | 0.0005 | 20.9 | 0.0004 | 481 | ||||||
| GloMIQO | 36 | 279 | 314 | 0.0001 | 113 | 4.6 | 3600 | 210838 | 45.3 | 3600 | |||
| BARON | 0.0001 | 47.1 | 229144 | 208 | 3600 | 230121 | 421 | 3600 | |||||
| DICOPT | 0.41 | 214000 | 1.06 | 214000 | 3.52 | ||||||||
| P2 | McCormick | 76 | 678 | 1015 | 382600 | 78.8 | 0.37 | 348300 | 71.6 | 0.70 | 360208 | 81.9 | 1.38 |
| MDT (ψ = −1) | 356 | 2258 | 2783 | 361800 | 0.46 | 25.4 | 337950 | 0.32 | 331 | 320496 | 16.2 | 3600 | |
| MDT (ψ = −2)a | 556 | 3258 | 3683 | 0.045 | 50.8 | 0.03 | 1549 | ||||||
| MDT (ψ = −4)a | 956 | 5258 | 5483 | 0.0006 | 1185 | ||||||||
| GloMIQO | 76 | 598 | 695 | 9.6 | 3600 | 16.3 | 3600 | 320504 | 37.0 | 3600 | |||
| BARON | 374488 | 126 | 3600 | no sol. | 3600 | no sol. | 3600 | ||||||
| DICOPT | 361800 | 0.67 | 339659 | 1.31 | 329400 | 6.03 | |||||||
| P3 | McCormick | 88 | 1139 | 1676 | 338400 | 50.4 | 0.59 | 310400 | 59.2 | 1.03 | 292168 | 62.3 | 5.38 |
| MDT (ψ = −1) | 704 | 4175 | 5040 | 331700 | 0.26 | 42.6 | 305800 | 0.22 | 497 | 288867 | 48.1 | 3600 | |
| MDT (ψ = −2)a | 1184 | 6295 | 6892 | 0.026 | 555 | 0.022 | 3215 | ||||||
| GloMIQO | 88 | 975 | 1020 | 0.031 | 3600 | 9.4 | 3600 | 287000 | 45.8 | 3600 | |||
| BARON | 354155 | 215 | 3600 | no sol. | 3600 | no sol. | 3600 | ||||||
| DICOPT | 351267 | 0.74 | 348200 | 2.13 | 300000 | 2.35 | |||||||
| P4 | McCormick | 110 | 1104 | 1692 | 395170 | 88.9 | 0.90 | 388800 | 104 | 1.56 | 379020 | 99.3 | 1.58 |
| MDT (ψ = −1) | 520 | 3929 | 4947 | 0.71 | 1145 | 374102 | 25.8 | 3600 | 366006 | 55.2 | 3600 | ||
| MDT (ψ = −2)a | 820 | 5679 | 6572 | 0.071 | 2420 | ||||||||
| GloMIQO | 110 | 959 | 1112 | 11.9 | 3600 | 33.4 | 3600 | 365088 | 112 | 3600 | |||
| BARON | no sol. | 3600 | no sol. | 3600 | no sol. | 3600 | |||||||
| DICOPT | 395170 | 3.15 | 401498 | 3.60 | 393486 | 9.96 | |||||||
a
No point in solving the problem for higher accuracy settings, if optimality cannot be proven in less than 1 h for ψ = −1.
7 Conclusions
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This paper has shown that a resource-task network continuous-time formulation can effectively capture the most important scheduling constraints of the crude oil scheduling problem. The highlights include the handling of tasks with variable proportions of input materials, essential for blending, and the extensive use of material location resources to deal with the logistic condition that forbids simultaneous inlet and outlet transfers on storage and charging tanks. When compared to a priority-slot continuous-time formulation from the literature, the model allows interrupting crude discharge from marine vessels, leading to higher gross margins in two of the four test problems solved.
The new MINLP formulation has shown to be computationally very efficient since GloMIQO could solve to global optimality in just a few seconds three of the gross maximization and all of the cost minimization problems when considering approximate inventory costs. This was no longer possible for accurate inventory costs but better solutions than those in the literature could still be found. It has also been shown that relaxing the bilinear terms appearing in blending constraints and inventory cost calculations using multiparametric disaggregation can lead to orders of magnitude lower gaps than GloMIQO. However, because of the use of a single starting point for the solution of the original MINLP, solution quality from the proposed MILP-NLP decomposition strategy was sometimes inferior.
The main focus of future work will thus be on developing a spatial branch-and-bound global optimization algorithm featuring MILP relaxations derived from multiparametric disaggregation, so as to generate multiple high quality starting points for the solution of the original MINLP and thus increase the likelihood of finding the optimal solution in cases where the optimality gap cannot be closed in reasonable time. Other interesting research directions concern the handling of nonlinear blending rules for some of the crude properties, expanding the system boundary to include downstream refinery operations, and developing solution approaches for large-scale problems that allow quickly obtaining good feasible solutions.
Supporting Information
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Tables with all the necessary data for problems P1–P4. This material is available free of charge via the Internet at http://pubs.acs.org.
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Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.
Acknowledgment
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Pedro Castro gratefully acknowledges financial support from the Luso-American Foundation under the 2013 Portugal–U.S. Research Networks Program, from Fundação para a Ciência e Tecnologia (FCT) through the Investigador FCT 2013 program, and from FEDER (Programa Operacional Factores de Competitividade-COMPETE) and FCT through project FCOMP-01-0124-FEDER-020764. Ignacio Grossmann acknowledges funding from the Center of Advanced Process Decision-making at Carnegie Mellon.
| CD/cd | crude oil distillation unit |
| CR/cr | crude oil |
| CRr | crude associated with material resource r |
| CT/ct | charging tank |
| I/i | tasks |
| IBL | variable recipe blending tasks |
| IC | fixed recipe continuous tasks |
| IS | fixed recipe storage tasks |
| IcdCD | pair of transfer tasks to distillation column cd |
| INE | transfer tasks not consuming an equipment resource |
| IcrVD | variable recipe transfer tasks to distillation columns involving final product crude cr |
| IVR | variable recipe tasks |
| j | digits for decimal numeric representation system ∈ {0,...,9} |
| k | positions in decimal representation of discretized variables ∈ {ψ,...,η} |
| MV/mv | crude marine vessels |
| PR/pr | crude oil property |
| R/r | resources |
| RiBL | inside tank material resources located immediately upstream of blending task i |
| RCD | resources corresponding to distillation units |
| RcrCT | material resource inside charging tanks corresponding to crude cr |
| REQ | equipment resources |
| RFP | material resources corresponding to final crude blends |
| RIO | in or out material resources, located just after/before storage/charging tanks |
| RrCR | out material resource (storage tanks) or inside tank resource (charging tanks) corresponding to the same crude of inside tank resource r |
| RMR | material resources |
| RtkMR | material resources than can appear inside tank tk |
| RRM | raw-material resources |
| RTC | equipment resources that appear in timing constraints |
| ST/st | storage tank |
| T/t | Event points (time slots) of the single time grid |
| TK/tk | storage and charging tanks |
| TKBL | blending tanks |
| U/u | system units excluding docking station |
| atmv | arrival time of marine vessel mv [day] |
| ccr,pr | composition of raw-material crude cr in property pr |
| cchg | cost involved for each change in crude to a distillation column [$] |
| cmvharb | harboring costs for unloading the crude from marine vessel mv [$/day] |
| ctkinv | inventory cost for tank tk [$/kbbl/day] |
| ctk,prmax | maximum composition in blending tank tk for property pr |
| ctk,prmin | minimum composition in blending tank tk for property pr |
| cmvwsea | sea waiting cost for marine vessel mv [$/day] |
| dcrmax | maximum demand of final product crude cr [kbbl] |
| dcrmin | minimum demand of final product crude cr [kbbl] |
| H | time horizon [day] |
| nd | maximum number of distillation operations |
| pcr | gross margin of crude cr [$/bbl] |
| Rr0 | initial availability of resource r |
| Rmv,crmax | maximum excess value for of resource r |
| vmv,crin | incoming volume from marine vessel mv of crude cr [kbbl] |
| vtk,cr0 | initial volume inside tank tk of crude cr [kbbl] |
| vimax | maximum amount of material processed by task i [kbbl] |
| vtkmax | upper bound on total volume inside tank tk [kbbl] |
| vkmin | lower bound on total volume inside tank tk [kbbl] |
| η | position of most significant digit for discretized variables |
| λr,i | continuous interaction of resource r during the execution of task i |
| λ̅r,r′,i | continuous interaction of resource r due to resource r′ during the execution of task i |
| μr,i | discrete interaction of resource r at the start of task i acting on variables Ni,t |
| μ̅r,i | discrete interaction of resource r at the end of task i acting on variables Ni,t |
| vr,i | discrete interaction of resource r at the start of task i acting on variables ζi,t |
| v̅r,i | discrete interaction of resource r at the end of task i acting on variables ζi,t |
| πr,mvi | amount of resource r entering the system due to harboring of marine vessel mv |
| πr,mvo | amount of resource r leaving the system due to departure of marine vessel mv |
| ρimax | maximum processing rate of task i [kbbl/day] |
| ρimin | minimum processing rate of task i [kbbl/day] |
| ρu,umax | maximum transfer flow rate between units u and u′ [kbbl/day] |
| ρu,umin | minimum transfer flow rate between units u and u′ [kbbl/day] |
| ψ | position of least significant digit for discretized variables |
| Ni,t | binary variable indicating if task i is executed during slot t (starts at event point t) |
| Yi,t,j,k | binary variable indicating if the volume fraction of task i at slot t features digit j at position k |
| Zmv,ti | binary variable indicating harboring of marine vessel mv at event point t |
| Ztno i | binary variable indicating that no vessel harbors at event point t |
| Zt′,tno io | binary variable indicating that no vessel arrives at event point t or departs at t′ |
| Ztno o | binary variable indicating that no vessel departs at event point t |
| Zmv,to | binary variable indicating departure of marine vessel mv at event point t |
| AItk | average inventory of crude in tank tk during slot t [kbbl] |
 tk | approximation of average inventory of crude in tank tk over the time horizon [kbbl] |
| DTt | duration of time slot t [day] |
| HBmv | duration of harboring for marine vessel mv [day] |
| NDcd | number of distillation runs for column cd |
| Rr,t | excess amount of resource r at event point t |
| Rr,tend | excess amount of resource r immediately before the end of interval t |
| R̂r,i,t,j,k | disaggregated variable linked to Rr,t and discrete value j of Fi,t associated with power k |
| Tt | absolute time of event point t [day] |
| Wr,i,t | bilinear term variable involving resource r consumed by variable recipe task i during slot t |
 r,i,t | residual variable of bilinear term involving resource r, variable recipe task i and slot t |
| WSmv | time marine vessel mv waits in sea before harboring [day] |
| Xcd,t | continuous variable identifying if a crude blend changeover occurs for unit cd at the end of slot t |
| ζi,t | volume fraction associated with the execution of variable recipe task i during slot t |
 i,t | residual value of volume fraction linked to task t and slot t |
| ξi,t | total amount of material handled by task i during slot t [kbbl] |
| ξ̅r,i,t | amount of resource r consumed by task i during slot t [kbbl] |
References
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This article references 41 other publications.
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- 13Karuppiah, R.; Grossmann, I. E. Global optimization for the synthesis of integrated water systems in chemical processes Comput. Chem. Eng. 2006, 30, 650Google Scholar13https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XhslKgt78%253D&md5=09d43fc2861851080f0e139d8bf0d6beGlobal optimization for the synthesis of integrated water systems in chemical processesKaruppiah, Ramkumar; Grossmann, Ignacio E.Computers & Chemical Engineering (2006), 30 (4), 650-673CODEN: CCENDW; ISSN:0098-1354. (Elsevier Ireland Ltd.)In this paper, we address the problem of optimal synthesis of an integrated water system, where water using processes and water treatment operations are combined into a single network such that the total cost of obtaining freshwater for use in the water using operations, and treating wastewater is globally minimized. A superstructure that incorporates all feasible design alternatives for water treatment, reuse and recycle, is proposed. We formulate the optimization of this structure as a non-convex Non-Linear Programming (NLP) problem, which is solved to global optimality. The problem takes the form of a non-convex Generalized Disjunctive Program (GDP) if there is a flexibility of choosing different treatment technologies for the removal of the various contaminants in the wastewater streams. A new deterministic spatial branch and contract algorithm is proposed for optimizing such systems, in which piecewise under- and over-estimators are used to approx. the non-convex terms in the original model to obtain a convex relaxation whose soln. gives a lower bound on the global optimum. These lower bounds are made to converge to the soln. within a branch and bound procedure. Several examples are presented to illustrate the optimization of the integrated networks using the proposed algorithm.
- 14Castro, P. M.; Barbosa-Póvoa, A. P.; Matos, H. A.; Novais, A. Q. Simple continuous-time formulation for short-term scheduling of batch and continuous processes Ind. Eng. Chem. Res. 2004, 43, 105Google Scholar14https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXpsVals7w%253D&md5=f4b5c36399f78c507fd7395d0f629630Simple Continuous-Time Formulation for Short-Term Scheduling of Batch and Continuous ProcessesCastro, Pedro M.; Barbosa-Povoa, Ana P.; Matos, Henrique A.; Novais, Augusto Q.Industrial & Engineering Chemistry Research (2004), 43 (1), 105-118CODEN: IECRED; ISSN:0888-5885. (American Chemical Society)A new and simple general math. formulation for scheduling multipurpose plants involving batch and/or continuous processes, based on the resource-task network (RTN) representation, is presented. The formulation uses a uniform-time-grid continuous-time representation and results in a very efficient mixed integer linear programming model that can be solved to optimality for a given no. of event points. The performance of the formulation is illustrated through the soln. of two case studies that have been thoroughly examd. in the literature: the first involves a continuous plant and is solved for three different storage policies, and the second concerns a batch plant. The formulation is shown to compare favorably to existing continuous-time formulations. More specifically, a new optimal soln. is obtained for the finite intermediate storage scenario of the first case that is also a global optimal soln.
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- 19Castro, P. M.; Barbosa-Póvoa, A. P.; Novais, A. Q. Simultaneous design and scheduling of multipurpose plants using resource task network based continuous-time formulations Ind. Eng. Chem. Res. 2005, 44, 343Google Scholar19https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXhtVOhsLvI&md5=9ba6ba20c854b2803836df65f6e288c4Simultaneous Design and Scheduling of Multipurpose Plants Using Resource Task Network Based Continuous-Time FormulationsCastro, Pedro M.; Barbosa-Povoa, Ana P.; Novais, Augusto Q.Industrial & Engineering Chemistry Research (2005), 44 (2), 343-357CODEN: IECRED; ISSN:0888-5885. (American Chemical Society)This paper presents a general math. formulation for the simultaneous design and scheduling of multipurpose plants. The formulation is based on the resource task network process representation, uses a uniform time grid continuous-time representation, and can handle both short-term and periodic problems. It originates mixed-integer nonlinear programs or mixed-integer linear programs, depending on the types of tasks and objective function being considered. The performance of the formulation is illustrated through the soln. of two periodic example problems that were examd. in the literature, where the selection and design of the main equipment items and their connecting pipes is considered. The results clearly show that all decisions should be part of the same model because the plant structure, operating schedule, and cycle time can all change with a change in product demand. A comparison with an earlier approach is also presented.
- 20Kelly, J. D.; Mann, J. L. Crude oil blend scheduling optimization: An application with multimillion dollar benefits—Part 2 Hydrocarbon Processing 2003, 82 (7) 72Google ScholarThere is no corresponding record for this reference.
- 21Kolodziej, S. P.; Grossmann, I. E.; Furman, K. C.; Sawaya, N. W. A discretization-based approach for the optimization of the multiperiod blend scheduling problem Comput. Chem. Eng. 2013, 53, 122Google Scholar21https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXlvF2jt74%253D&md5=cc94ba5fe4bbeb1191c96557ec44efa7A discretization-based approach for the optimization of the multiperiod blend scheduling problemKolodziej, Scott P.; Grossmann, Ignacio E.; Furman, Kevin C.; Sawaya, Nicolas W.Computers & Chemical Engineering (2013), 53 (), 122-142CODEN: CCENDW; ISSN:0098-1354. (Elsevier B.V.)In this paper, we introduce a new generalized multiperiod scheduling version of the pooling problem to represent time varying blending systems. A general nonconvex MINLP formulation of the problem is presented. The primary difficulties in solving this optimization problem are the presence of bilinear terms, as well as binary decision variables required to impose operational constraints. An illustrative example is presented to provide unique insight into the difficulties faced by conventional MINLP approaches to this problem, specifically in finding feasible solns. Based on recent work, a new radix-based discretization scheme is developed with which the problem can be reformulated approx. as an MILP, which is incorporated in a heuristic procedure and in two rigorous global optimization methods, and requires much less computational time than existing global optimization solvers. Detailed computational results of each approach are presented on a set of examples, including a comparison with other global optimization solvers.
- 22Castro, P. M.; Westerlund, J.; Forssell, S. Scheduling of a continuous plant with recycling of byproducts: A case study from a tissue paper mill Comput. Chem. Eng. 2009, 33, 347Google Scholar22https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXhsVKnt7jE&md5=6eab4308dede7e51c59a8d26c446e2c0Scheduling of a continuous plant with recycling of byproducts: A case study from a tissue paper millCastro, Pedro M.; Westerlund, Joakim; Forssell, SebastianComputers & Chemical Engineering (2009), 33 (1), 347-358CODEN: CCENDW; ISSN:0098-1354. (Elsevier Ireland Ltd.)This paper considers an industrial scheduling problem. It involves profit maximization and the detn. of the optimal cycle time, while meeting the min. demands for the several products. Resource-task network-based formulations are employed and a detailed comparison between continuous- and discrete-time models is provided. Both have the improved capability of handling tasks with flexible proportions of input materials in order to consider the incorporation of different flowrates of byproducts that are recycled back to the first prodn. stage. The continuous-time formulation is shown to be more efficient and the resulting mixed integer nonlinear program (MINLP) can be solved to optimality within reasonable computational time. A new recycling policy is proposed that achieves the double goal of making the process more profitable due to important savings on the more expensive raw-materials and also more environmentally friendly, due to the redn. of waste disposal requirements.
- 23Castro, P. M. Optimal scheduling of pipeline systems with a resource-task network continuous-time formulation Ind. Eng. Chem. Res. 2010, 49, 11491Google ScholarThere is no corresponding record for this reference.
- 24Harjunkoski, I.; Maravelias, C.; Bongers, P.; Castro, P. M.; Engell, S.; Grossmann, I.; Hooker, J.; Méndez, C.; Sand, G.; Wassick, J. Scope for industrial applications of production scheduling models and solution methods Comput. Chem. Eng. 2014, 62, 161Google Scholar24https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhtVOqsbw%253D&md5=0e0a74a155dca51518a9e861b2a4c25eScope for industrial applications of production scheduling models and solution methodsHarjunkoski, Iiro; Maravelias, Christos T.; Bongers, Peter; Castro, Pedro M.; Engell, Sebastian; Grossmann, Ignacio E.; Hooker, John; Mendez, Carlos; Sand, Guido; Wassick, JohnComputers & Chemical Engineering (2014), 62 (), 161-193CODEN: CCENDW; ISSN:0098-1354. (Elsevier B.V.)A review. This paper gives a review on existing scheduling methodologies developed for process industries. Above all, the aim of the paper is to focus on the industrial aspects of scheduling and discuss the main characteristics, including strengths and weaknesses of the presented approaches. Optimization tools of today can effectively support the plant level prodn. However there is still clear potential for improvements, esp. in transferring academic results into industry. For instance, usability, interfacing and integration are some aspects discussed in the paper. After the introduction and problem classification, the paper discusses some lessons learned from industry, provides an overview of models and methods and concludes with general guidelines and examples on the modeling and soln. of industrial problems.
- 25Castro, P. M.; Harjunkoski, I.; Grossmann, I. E. New continuous-time scheduling formulation for continuous plants under variable electricity cost Ind. Eng. Chem. Res. 2009, 48, 6701Google Scholar25https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXntlelu7Y%253D&md5=ef103abef28242aec1c89c688a899b2cNew Continuous-Time Scheduling Formulation for Continuous Plants under Variable Electricity CostCastro, Pedro M.; Harjunkoski, Iiro; Grossmann, Ignacio E.Industrial & Engineering Chemistry Research (2009), 48 (14), 6701-6714CODEN: IECRED; ISSN:0888-5885. (American Chemical Society)This work addresses the scheduling of continuous plants subject to energy constraints related to time-dependent electricity pricing and availability. Discrete- and continuous-time formulations are presented that can address these issues together with multiple intermediate due dates. Both formulations rely on the resource-task network process representation. Their computational performance is compared for the objective of total electricity minimization with the results favoring the discrete-time model due to the more natural way of handling such a wide variety of discrete events. In particular, it can successfully handle problems of industrial size. Nevertheless, the new continuous-time model is a major breakthrough since it is the first model of its type that is able to effectively incorporate time-variable utility profiles. When compared to a simple manual scheduling procedure, the proposed scheduling approaches can lead to potential electricity savings around 20% by switching prodn. from periods of high to low electricity cost.
- 26Yadav, S.; Shaik, M. A. Short-term scheduling of refinery crude oil operations Ind. Eng. Chem. Res. 2012, 51, 9287Google ScholarThere is no corresponding record for this reference.
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- 31Castro, P. M.; Grossmann, I. E. Generalized disjunctive programming as a systematic modeling framework to derive scheduling formulations Ind. Eng. Chem. Res. 2012, 51, 5781Google ScholarThere is no corresponding record for this reference.
- 32Raman, R.; Grossmann, I. E. Modeling and computational techniques for logic based integer programming Comput. Chem. Eng. 1994, 18, 563Google Scholar32https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXktFaqt70%253D&md5=62018dadd4388e7b91981945e79fecacModeling and computational techniques for logic based integer programmingRaman, R.; Grossmann, I. E.Computers & Chemical Engineering (1994), 18 (7), 563-78CODEN: CCENDW; ISSN:0098-1354.A modeling framework is presented for discrete optimization problems that relies on a logic representation in which mixed-integer logic is represented through disjunctions, and integer logic through propositions. Transformation of the logic formulation into the equation form is not always desirable, and that therefore there is a need to address the soln. of mixed-integer programming problems where some of the mixed-integer relationships are expressed in disjunctions while others are expressed as algebraic constraints. A theor. characterization of disjunctive constraints is proposed which can serve as a criterion for deciding whether a disjunction should be transformed into equation form. A soln. algorithm that generalizes the method of Raman and Grossman for handling mixed-integer disjunctions symbolically is also proposed. Several examples are presented to illustrate the proposed modeling framework and the potential of the soln. method.
- 33Karuppiah, R.; Furman, K. C.; Grossmann, I. E. Global optimization for scheduling refinery crude oil operations Comput. Chem. Eng. 2008, 32, 2745Google Scholar33https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXhtVOqsL3M&md5=504276692d58d8bbd4cf2b19bf9749d1Global optimization for scheduling refinery crude oil operationsKaruppiah, Ramkumar; Furman, Kevin C.; Grossmann, Ignacio E.Computers & Chemical Engineering (2008), 32 (11), 2745-2766CODEN: CCENDW; ISSN:0098-1354. (Elsevier Ireland Ltd.)In this work an outer-approxn. algorithm is presented to obtain the global optimum of a nonconvex mixed-integer nonlinear programming (MINLP) model that is used to represent the scheduling of crude oil movement at the front-end of a petroleum refinery. The model relies on a continuous time representation making use of transfer events. The proposed algorithm focuses on effectively solving a mixed-integer linear programming (MILP) relaxation of the nonconvex MINLP to obtain a rigorous lower bound (LB) on the global optimum. Cutting planes derived by spatially decompg. the network are added to the MILP relaxation of the original nonconvex MINLP in order to reduce the soln. time for the MILP relaxation. The soln. of this relaxation is used as a heuristic to obtain a feasible soln. to the MINLP which serves as an upper bound (UB). The lower and upper bounds are made to converge to within a specified tolerance in the proposed outer-approxn. algorithm. On applying the proposed technique to test examples, significant savings are realized in the computational effort required to obtain provably global optimal solns.
- 34Castro, P. M.; Teles, J. P. Comparison of global optimization algorithms for the design of water-using networks Comput. Chem. Eng. 2013, 52, 249Google Scholar34https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXjvFans70%253D&md5=48742503b879354b71221cb621d0ec4cComparison of global optimization algorithms for the design of water-using networksCastro, Pedro M.; Teles, Joao P.Computers & Chemical Engineering (2013), 52 (), 249-261CODEN: CCENDW; ISSN:0098-1354. (Elsevier B.V.)We address a special class of bilinear process network problems with global optimization algorithms iterating between a lower bound provided by a mixed-integer linear programming (MILP) formulation and an upper bound given by the soln. of the original nonlinear problem (NLP) with a local solver. Two conceptually different relaxation approaches are tested, piecewise McCormick envelopes and multiparametric disaggregation, each considered in two variants according to the choice of variables to partition/parameterize. The four complete MILP formulations are derived from disjunctive programming models followed by convex hull reformulations. The results on a set of test problems from the literature show that the algorithm relying on multiparametric disaggregation with parameterization of the concns. is the best performer, primarily due to a logarithmic as opposed to linear increase in problem size with the no. of partitions. The algorithms are also compared to the com. solvers BARON and GloMIQO through performance profiles.
- 35Teles, J. P.; Castro, P. M.; Matos, H. A. Univariate parameterization for global optimization of mixed-integer polynomial problems Eur. J. Oper. Res. 2013, 229, 613Google ScholarThere is no corresponding record for this reference.
- 36Kocis, G. R.; Grossmann, I. E. Computational experience with DICOPT solving MINLP problems in process systems engineering Comput. Chem. Eng. 1989, 13, 307Google Scholar36https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1MXhslGqurY%253D&md5=8487cccff09f050a785622fcfababe4bComputational experience with DICOPT solving MINLP problems in process systems engineeringKocis, G. R.; Grossmann, I. E.Computers & Chemical Engineering (1989), 13 (3), 307-15CODEN: CCENDW; ISSN:0098-1354.The outer-approxn./equality-relaxation algorithm for solving MINLP (mixed-integer nonlinear programming) problems that arise in process systems engineering is discussed. The computer code DICOPT (discrete continuous optimizer) is developed using state-of-the-art optimization tools and a powerful modeling language. Computational experience in solving 16 MINLP problems with DICOPT is reported. Applications include design of batch processes, structural flowsheet optimization, column design, utility plant retrofit, planning, flexibility, and reliability problems.
- 37Tawarmalani, M.; Sahinidis, N. V. A polyhedral branch-and-cut approach to global optimization Mathematical Programming 2005, 103 (2) 225Google ScholarThere is no corresponding record for this reference.
- 38Misener, R.; Floudas, C. A. GloMIQO: Global mixed-integer quadratic optimizer J. Global Optimization 2013, 57, 3Google ScholarThere is no corresponding record for this reference.
- 39Viswanathan, J.; Grossmann, I. E. A combined penalty-function and outer-approximation method for MINLP optimization Comput. Chem. Eng. 1990, 14, 769Google Scholar39https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3cXltlCqu78%253D&md5=4e1fe73502afac1e95561889efe870cbA combined penalty function and outer-approximation method for MINLP optimizationViswanathan, J.; Grossman, I. E.Computers & Chemical Engineering (1990), 14 (7), 769-82CODEN: CCENDW; ISSN:0098-1354.An improved outer-approxn. algorithm for MINLP (mixed-integer nonlinear programming) optimization is proposed for the soln. of problems where in convexity conditions may not hold. The algorithm starts by solving the NLP relaxation. If an integer soln. is not found, a sequence of iterations consisting of NLP subproblems and MILP master problems is solved. The proposed MILP master problem is based on the outer-approxn./equality-relaxation algorithm and features an exact penalty function that allows violations of linearizations of nonconvex constraints. The search proceeds until no improvement is found in the NLP subproblems. Computational experience is presented on a set of 20 test problems. Included are problems for optimum feed tray location and no. of plates for distn. columns. Although no theor. guarantee can be given, the method has a high degree of reliability finding the global optimum in nonconvex problems.
- 40Jia, Z.; Ierapetritou, M. G. Efficient short-term scheduling of refinery operations based on a continuous time formulation Comput. Chem. Eng. 2004, 28, 1001Google Scholar40https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXjtlyhu70%253D&md5=ca7ff6e4933c90be37a1da04bea57209Efficient short-term scheduling of refinery operations based on a continuous time formulationJia, Zhenya; Ierapetritou, MarianthiComputers & Chemical Engineering (2004), 28 (6-7), 1001-1019CODEN: CCENDW; ISSN:0098-1354. (Elsevier)The problem addressed in this work is to develop a comprehensive math. programming model for the efficient scheduling of oil-refinery operations. Our approach is first to decomp. the overall problem spatially into three domains: the crude-oil unloading and blending, the prodn. unit operations and the product blending and delivery. In particular, the first problem involves the crude-oil unloading from vessels, its transfer to storage tanks and the charging schedule for each crude-oil mixt. to the distn. units. The second problem consists of the prodn. unit scheduling which includes both fractionation and reaction processes and the third problem describes the finished product blending and shipping end of the refinery. Each of those sub-problems is modeled and solved in a most efficient way using continuous time representation to take advantage of the relatively smaller no. of variables and constraints compared to discrete time formulation. The proposed methodol. is applied to realistic case studies and significant computational savings can be achieved compared with existing approaches.
- 41Quesada, I.; Grossmann, I. E. Global optimization of bilinear process networks with multicomponent flows Comput. Chem. Eng. 1995, 19, 1219Google Scholar41https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXovVSltb4%253D&md5=04357dba42d82f8db1c08e5934746feeGlobal optimization of bilinear process networks with multicomponent flowsQuesada, I.; Grossmann, I. E.Computers & Chemical Engineering (1995), 19 (12), 1219-42CODEN: CCENDW; ISSN:0098-1354. (Elsevier)The global optimization of networks consisting of splitters, mixers and linear process units and that involve multicomponent streams is studied. Examples include pooling and blending systems and sharp sepn. networks in which nonconvexities arise in the bilinear equations for the mass balances. A reformulation-linearization technique is first applied to models expressed with compns. and total flows to obtain a relaxed LP formulation that provides a valid lower bound to the global optimum. This formulation is used within a spatial branch and bound search. The application of this method is considered in detail for sharp sepn. systems with single feed and mixed products. Numerical results are presented on 12 test problems involving up to a few hundred variables. Only a few nodes are commonly required in the branch and bound search.
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Figure 1
Figure 1. Crude oil operation system for problem 1.
Figure 2
Figure 2. Resource-Task Network process model for problem 1.
Figure 3
Figure 3. Blending tanks can either be receiving or discharging crude. Storage/charging tanks allow a single/multiple inlet streams and multiple/single outlet streams.
Figure 4
Figure 4. Crude oil operation system for problem 3 of Lee et al. (3)
Figure 5
Figure 5. Partial RTN process model for problem 3, showing the interaction of storage tank ST1 with charging tanks CT1 and CT2.
Figure 6
Figure 6. Crude oil operation system for problem 2.
Figure 7
Figure 7. Partial RTN process model for problem 2, showing the interaction of the charging tanks with the distillation columns.
Figure 8
Figure 8. In the given continuous-time formulation, events report to a single grid.
Figure 9
Figure 9. Volume fraction variables ζi,t are calculated as a sum of a discrete point (illustration for level ψ = −1) and a residual variable
i,t, thus achieving a continuous domain.Figure 10
Figure 10. Crude oil operation system for problem 3 in Mouret et al. (9)
Figure 11
Figure 11. Crude oil operation system for problem 4.
Figure 12
Figure 12. Global optimal solution for problem 1 and gross margin maximization ($7,982,500).
Figure 13
Figure 13. Global optimal solution for problem 2 and gross margin maximization ($10,246,075).
Figure 14
Figure 14. Best-found solution for problem 2 and operating cost minimization ($320,496).
Figure 15
Figure 15. Best-found solution for problem 4 and operating cost minimization ($365,088).
References
This article references 41 other publications.
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- 20Kelly, J. D.; Mann, J. L. Crude oil blend scheduling optimization: An application with multimillion dollar benefits—Part 2 Hydrocarbon Processing 2003, 82 (7) 72There is no corresponding record for this reference.
- 21Kolodziej, S. P.; Grossmann, I. E.; Furman, K. C.; Sawaya, N. W. A discretization-based approach for the optimization of the multiperiod blend scheduling problem Comput. Chem. Eng. 2013, 53, 12221https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXlvF2jt74%253D&md5=cc94ba5fe4bbeb1191c96557ec44efa7A discretization-based approach for the optimization of the multiperiod blend scheduling problemKolodziej, Scott P.; Grossmann, Ignacio E.; Furman, Kevin C.; Sawaya, Nicolas W.Computers & Chemical Engineering (2013), 53 (), 122-142CODEN: CCENDW; ISSN:0098-1354. (Elsevier B.V.)In this paper, we introduce a new generalized multiperiod scheduling version of the pooling problem to represent time varying blending systems. A general nonconvex MINLP formulation of the problem is presented. The primary difficulties in solving this optimization problem are the presence of bilinear terms, as well as binary decision variables required to impose operational constraints. An illustrative example is presented to provide unique insight into the difficulties faced by conventional MINLP approaches to this problem, specifically in finding feasible solns. Based on recent work, a new radix-based discretization scheme is developed with which the problem can be reformulated approx. as an MILP, which is incorporated in a heuristic procedure and in two rigorous global optimization methods, and requires much less computational time than existing global optimization solvers. Detailed computational results of each approach are presented on a set of examples, including a comparison with other global optimization solvers.
- 22Castro, P. M.; Westerlund, J.; Forssell, S. Scheduling of a continuous plant with recycling of byproducts: A case study from a tissue paper mill Comput. Chem. Eng. 2009, 33, 34722https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXhsVKnt7jE&md5=6eab4308dede7e51c59a8d26c446e2c0Scheduling of a continuous plant with recycling of byproducts: A case study from a tissue paper millCastro, Pedro M.; Westerlund, Joakim; Forssell, SebastianComputers & Chemical Engineering (2009), 33 (1), 347-358CODEN: CCENDW; ISSN:0098-1354. (Elsevier Ireland Ltd.)This paper considers an industrial scheduling problem. It involves profit maximization and the detn. of the optimal cycle time, while meeting the min. demands for the several products. Resource-task network-based formulations are employed and a detailed comparison between continuous- and discrete-time models is provided. Both have the improved capability of handling tasks with flexible proportions of input materials in order to consider the incorporation of different flowrates of byproducts that are recycled back to the first prodn. stage. The continuous-time formulation is shown to be more efficient and the resulting mixed integer nonlinear program (MINLP) can be solved to optimality within reasonable computational time. A new recycling policy is proposed that achieves the double goal of making the process more profitable due to important savings on the more expensive raw-materials and also more environmentally friendly, due to the redn. of waste disposal requirements.
- 23Castro, P. M. Optimal scheduling of pipeline systems with a resource-task network continuous-time formulation Ind. Eng. Chem. Res. 2010, 49, 11491There is no corresponding record for this reference.
- 24Harjunkoski, I.; Maravelias, C.; Bongers, P.; Castro, P. M.; Engell, S.; Grossmann, I.; Hooker, J.; Méndez, C.; Sand, G.; Wassick, J. Scope for industrial applications of production scheduling models and solution methods Comput. Chem. Eng. 2014, 62, 16124https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhtVOqsbw%253D&md5=0e0a74a155dca51518a9e861b2a4c25eScope for industrial applications of production scheduling models and solution methodsHarjunkoski, Iiro; Maravelias, Christos T.; Bongers, Peter; Castro, Pedro M.; Engell, Sebastian; Grossmann, Ignacio E.; Hooker, John; Mendez, Carlos; Sand, Guido; Wassick, JohnComputers & Chemical Engineering (2014), 62 (), 161-193CODEN: CCENDW; ISSN:0098-1354. (Elsevier B.V.)A review. This paper gives a review on existing scheduling methodologies developed for process industries. Above all, the aim of the paper is to focus on the industrial aspects of scheduling and discuss the main characteristics, including strengths and weaknesses of the presented approaches. Optimization tools of today can effectively support the plant level prodn. However there is still clear potential for improvements, esp. in transferring academic results into industry. For instance, usability, interfacing and integration are some aspects discussed in the paper. After the introduction and problem classification, the paper discusses some lessons learned from industry, provides an overview of models and methods and concludes with general guidelines and examples on the modeling and soln. of industrial problems.
- 25Castro, P. M.; Harjunkoski, I.; Grossmann, I. E. New continuous-time scheduling formulation for continuous plants under variable electricity cost Ind. Eng. Chem. Res. 2009, 48, 670125https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXntlelu7Y%253D&md5=ef103abef28242aec1c89c688a899b2cNew Continuous-Time Scheduling Formulation for Continuous Plants under Variable Electricity CostCastro, Pedro M.; Harjunkoski, Iiro; Grossmann, Ignacio E.Industrial & Engineering Chemistry Research (2009), 48 (14), 6701-6714CODEN: IECRED; ISSN:0888-5885. (American Chemical Society)This work addresses the scheduling of continuous plants subject to energy constraints related to time-dependent electricity pricing and availability. Discrete- and continuous-time formulations are presented that can address these issues together with multiple intermediate due dates. Both formulations rely on the resource-task network process representation. Their computational performance is compared for the objective of total electricity minimization with the results favoring the discrete-time model due to the more natural way of handling such a wide variety of discrete events. In particular, it can successfully handle problems of industrial size. Nevertheless, the new continuous-time model is a major breakthrough since it is the first model of its type that is able to effectively incorporate time-variable utility profiles. When compared to a simple manual scheduling procedure, the proposed scheduling approaches can lead to potential electricity savings around 20% by switching prodn. from periods of high to low electricity cost.
- 26Yadav, S.; Shaik, M. A. Short-term scheduling of refinery crude oil operations Ind. Eng. Chem. Res. 2012, 51, 9287There is no corresponding record for this reference.
- 27Westenberger, H.; Kallrath, L. Formulation of a Job Shop Problem in Process Industry, Internal Report; Bayer AG, Leverkusen and BASF AG: Ludwigshafen, 1995.There is no corresponding record for this reference.
- 28Kallrath, J. Planning and scheduling in the process industry OR Spectrum 2002, 24, 219There is no corresponding record for this reference.
- 29Blömer, F.; Günther, H. Scheduling of a multi-product batch process in the chemical industry Computers Ind. 1998, 36, 245There is no corresponding record for this reference.
- 30Balas, E. Disjunctive programming and a hierarchy of relaxations for discrete optimization problems SIAM J. Algebraic Discrete Methods 1985, 6 (3) 466– 486There is no corresponding record for this reference.
- 31Castro, P. M.; Grossmann, I. E. Generalized disjunctive programming as a systematic modeling framework to derive scheduling formulations Ind. Eng. Chem. Res. 2012, 51, 5781There is no corresponding record for this reference.
- 32Raman, R.; Grossmann, I. E. Modeling and computational techniques for logic based integer programming Comput. Chem. Eng. 1994, 18, 56332https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXktFaqt70%253D&md5=62018dadd4388e7b91981945e79fecacModeling and computational techniques for logic based integer programmingRaman, R.; Grossmann, I. E.Computers & Chemical Engineering (1994), 18 (7), 563-78CODEN: CCENDW; ISSN:0098-1354.A modeling framework is presented for discrete optimization problems that relies on a logic representation in which mixed-integer logic is represented through disjunctions, and integer logic through propositions. Transformation of the logic formulation into the equation form is not always desirable, and that therefore there is a need to address the soln. of mixed-integer programming problems where some of the mixed-integer relationships are expressed in disjunctions while others are expressed as algebraic constraints. A theor. characterization of disjunctive constraints is proposed which can serve as a criterion for deciding whether a disjunction should be transformed into equation form. A soln. algorithm that generalizes the method of Raman and Grossman for handling mixed-integer disjunctions symbolically is also proposed. Several examples are presented to illustrate the proposed modeling framework and the potential of the soln. method.
- 33Karuppiah, R.; Furman, K. C.; Grossmann, I. E. Global optimization for scheduling refinery crude oil operations Comput. Chem. Eng. 2008, 32, 274533https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXhtVOqsL3M&md5=504276692d58d8bbd4cf2b19bf9749d1Global optimization for scheduling refinery crude oil operationsKaruppiah, Ramkumar; Furman, Kevin C.; Grossmann, Ignacio E.Computers & Chemical Engineering (2008), 32 (11), 2745-2766CODEN: CCENDW; ISSN:0098-1354. (Elsevier Ireland Ltd.)In this work an outer-approxn. algorithm is presented to obtain the global optimum of a nonconvex mixed-integer nonlinear programming (MINLP) model that is used to represent the scheduling of crude oil movement at the front-end of a petroleum refinery. The model relies on a continuous time representation making use of transfer events. The proposed algorithm focuses on effectively solving a mixed-integer linear programming (MILP) relaxation of the nonconvex MINLP to obtain a rigorous lower bound (LB) on the global optimum. Cutting planes derived by spatially decompg. the network are added to the MILP relaxation of the original nonconvex MINLP in order to reduce the soln. time for the MILP relaxation. The soln. of this relaxation is used as a heuristic to obtain a feasible soln. to the MINLP which serves as an upper bound (UB). The lower and upper bounds are made to converge to within a specified tolerance in the proposed outer-approxn. algorithm. On applying the proposed technique to test examples, significant savings are realized in the computational effort required to obtain provably global optimal solns.
- 34Castro, P. M.; Teles, J. P. Comparison of global optimization algorithms for the design of water-using networks Comput. Chem. Eng. 2013, 52, 24934https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXjvFans70%253D&md5=48742503b879354b71221cb621d0ec4cComparison of global optimization algorithms for the design of water-using networksCastro, Pedro M.; Teles, Joao P.Computers & Chemical Engineering (2013), 52 (), 249-261CODEN: CCENDW; ISSN:0098-1354. (Elsevier B.V.)We address a special class of bilinear process network problems with global optimization algorithms iterating between a lower bound provided by a mixed-integer linear programming (MILP) formulation and an upper bound given by the soln. of the original nonlinear problem (NLP) with a local solver. Two conceptually different relaxation approaches are tested, piecewise McCormick envelopes and multiparametric disaggregation, each considered in two variants according to the choice of variables to partition/parameterize. The four complete MILP formulations are derived from disjunctive programming models followed by convex hull reformulations. The results on a set of test problems from the literature show that the algorithm relying on multiparametric disaggregation with parameterization of the concns. is the best performer, primarily due to a logarithmic as opposed to linear increase in problem size with the no. of partitions. The algorithms are also compared to the com. solvers BARON and GloMIQO through performance profiles.
- 35Teles, J. P.; Castro, P. M.; Matos, H. A. Univariate parameterization for global optimization of mixed-integer polynomial problems Eur. J. Oper. Res. 2013, 229, 613There is no corresponding record for this reference.
- 36Kocis, G. R.; Grossmann, I. E. Computational experience with DICOPT solving MINLP problems in process systems engineering Comput. Chem. Eng. 1989, 13, 30736https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1MXhslGqurY%253D&md5=8487cccff09f050a785622fcfababe4bComputational experience with DICOPT solving MINLP problems in process systems engineeringKocis, G. R.; Grossmann, I. E.Computers & Chemical Engineering (1989), 13 (3), 307-15CODEN: CCENDW; ISSN:0098-1354.The outer-approxn./equality-relaxation algorithm for solving MINLP (mixed-integer nonlinear programming) problems that arise in process systems engineering is discussed. The computer code DICOPT (discrete continuous optimizer) is developed using state-of-the-art optimization tools and a powerful modeling language. Computational experience in solving 16 MINLP problems with DICOPT is reported. Applications include design of batch processes, structural flowsheet optimization, column design, utility plant retrofit, planning, flexibility, and reliability problems.
- 37Tawarmalani, M.; Sahinidis, N. V. A polyhedral branch-and-cut approach to global optimization Mathematical Programming 2005, 103 (2) 225There is no corresponding record for this reference.
- 38Misener, R.; Floudas, C. A. GloMIQO: Global mixed-integer quadratic optimizer J. Global Optimization 2013, 57, 3There is no corresponding record for this reference.
- 39Viswanathan, J.; Grossmann, I. E. A combined penalty-function and outer-approximation method for MINLP optimization Comput. Chem. Eng. 1990, 14, 76939https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3cXltlCqu78%253D&md5=4e1fe73502afac1e95561889efe870cbA combined penalty function and outer-approximation method for MINLP optimizationViswanathan, J.; Grossman, I. E.Computers & Chemical Engineering (1990), 14 (7), 769-82CODEN: CCENDW; ISSN:0098-1354.An improved outer-approxn. algorithm for MINLP (mixed-integer nonlinear programming) optimization is proposed for the soln. of problems where in convexity conditions may not hold. The algorithm starts by solving the NLP relaxation. If an integer soln. is not found, a sequence of iterations consisting of NLP subproblems and MILP master problems is solved. The proposed MILP master problem is based on the outer-approxn./equality-relaxation algorithm and features an exact penalty function that allows violations of linearizations of nonconvex constraints. The search proceeds until no improvement is found in the NLP subproblems. Computational experience is presented on a set of 20 test problems. Included are problems for optimum feed tray location and no. of plates for distn. columns. Although no theor. guarantee can be given, the method has a high degree of reliability finding the global optimum in nonconvex problems.
- 40Jia, Z.; Ierapetritou, M. G. Efficient short-term scheduling of refinery operations based on a continuous time formulation Comput. Chem. Eng. 2004, 28, 100140https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXjtlyhu70%253D&md5=ca7ff6e4933c90be37a1da04bea57209Efficient short-term scheduling of refinery operations based on a continuous time formulationJia, Zhenya; Ierapetritou, MarianthiComputers & Chemical Engineering (2004), 28 (6-7), 1001-1019CODEN: CCENDW; ISSN:0098-1354. (Elsevier)The problem addressed in this work is to develop a comprehensive math. programming model for the efficient scheduling of oil-refinery operations. Our approach is first to decomp. the overall problem spatially into three domains: the crude-oil unloading and blending, the prodn. unit operations and the product blending and delivery. In particular, the first problem involves the crude-oil unloading from vessels, its transfer to storage tanks and the charging schedule for each crude-oil mixt. to the distn. units. The second problem consists of the prodn. unit scheduling which includes both fractionation and reaction processes and the third problem describes the finished product blending and shipping end of the refinery. Each of those sub-problems is modeled and solved in a most efficient way using continuous time representation to take advantage of the relatively smaller no. of variables and constraints compared to discrete time formulation. The proposed methodol. is applied to realistic case studies and significant computational savings can be achieved compared with existing approaches.
- 41Quesada, I.; Grossmann, I. E. Global optimization of bilinear process networks with multicomponent flows Comput. Chem. Eng. 1995, 19, 121941https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXovVSltb4%253D&md5=04357dba42d82f8db1c08e5934746feeGlobal optimization of bilinear process networks with multicomponent flowsQuesada, I.; Grossmann, I. E.Computers & Chemical Engineering (1995), 19 (12), 1219-42CODEN: CCENDW; ISSN:0098-1354. (Elsevier)The global optimization of networks consisting of splitters, mixers and linear process units and that involve multicomponent streams is studied. Examples include pooling and blending systems and sharp sepn. networks in which nonconvexities arise in the bilinear equations for the mass balances. A reformulation-linearization technique is first applied to models expressed with compns. and total flows to obtain a relaxed LP formulation that provides a valid lower bound to the global optimum. This formulation is used within a spatial branch and bound search. The application of this method is considered in detail for sharp sepn. systems with single feed and mixed products. Numerical results are presented on 12 test problems involving up to a few hundred variables. Only a few nodes are commonly required in the branch and bound search.
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