**Cite This:**

*Ind. Eng. Chem. Res.*2014, 53, 39, 15127-15145

# Global Optimal Scheduling of Crude Oil Blending Operations with RTN Continuous-time and Multiparametric DisaggregationClick to copy article linkArticle link copied!

*****Email: [email protected]

## Abstract

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.

## 1 Introduction

## 2 Problem Definition

*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

*at*

_{mv}[day] and volume of crude to discharge

*v*

_{mv,cr}

^{in}[kbbl = 10

^{3}·bll]. For storage and charging tanks

*tk*∈

*TK*=

*ST*∪

*CT*, we must know the initial inventory

*v*

_{tk,cr}

^{0}[kbbl] as well as the lower and upper bounds on storage capacity

*v*

_{tk}

^{min}and

*v*

_{tk}

^{max}[kbbl]. Tanks involving crude oil mixing must respect the composition limits for the relevant properties:

*c*

_{tk,pr}

^{min}and

*c*

_{tk,pr}

^{max}. Crudes arriving or initially present in storage tanks have known compositions

*c*

_{cr,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

*d*

_{cr}

^{min}and

*d*

_{cr}

^{max}[kbbl]. Transfer flow rates between units

*u*,

*u*′ ∈

*U*=

*MV*∪

*ST*∪

*CT*∪

*CD*are between ρ

_{u,u′}

^{min}and ρ

_{u,u′}

^{max}[kbbl/day]. In order to ensure that CDUs are operating continuously throughout the scheduling horizon

*H*[day], ρ

_{ct,cd}

^{min}> 0; for the remaining units, it is ρ

_{u,u′}

^{min}= 0.

### 2.1 Maximizing Gross Margin

*p*

_{cr}[$/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

*c*

_{mv}

^{wsea}[$/day]; harboring costs for unloading the crude

*c*

_{mv}

^{harb}[$/day]; inventory costs for storage and charging tanks

*c*

_{tk}

^{inv}[$/kbbl/day]; setup costs for crudes charged to the CDUs

*c*

^{chg}[$].

### 2.3 System Configuration

## 3 Resource-Task Network Process Model

*r*∈

*R*interacting with a collection of tasks

*i*∈

*I*.

*R*

^{EQ}include the docking station

*DS*, marine vessels

*MV*, storage

*ST*, charging tanks

*CT*, and distillation units

*CD*, that is,

*R*

^{EQ}=

*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.

*R*

^{MR}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_MV

_{1}, inside the top storage tank A_ST

_{1}, just after the storage tank A_ST

_{1}

^{°}, just before the charging tanks (A_CT

_{1}

^{i}, A_CT

_{2}

^{i}) and finally, inside the charging tanks (A_CT

_{1}and A_CT

_{2}). 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).

*I*

^{VR}, the latter including blending tasks

*I*

^{BL}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

*I*

^{C}or storage tasks

*I*

^{S}. 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.

_{1}) 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_ST

_{1}continuously produces resource A_ST

_{1}

^{°}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_ST

_{1}[1000 kbbl/day] is set so as to match the sum of the maximum rates of the subsequent tasks). Similarly, Transfer_In_CT

_{1}consumes a variable proportion of resources A_CT

_{1}

^{i}and B_CT

_{1}

^{i}, at the same rate as Transfer 3 and Transfer 5.

### 3.1 Remarks

_{2}; see Figure 7.

## 4 Mathematical Formulation (MINLP)

### 4.1 Time Representation

*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.

### 4.2 Model Variables

*N*

_{i,t}indicates that task

*i*is executed during slot

*t*;

*Z*

_{t,mv}

^{i}assigns the harboring of marine vessel

*mv*to event point

*t*;

*Z*

_{t,mv}

^{0}assigns the departure of marine vessel

*mv*to event point

*t*.

_{i,t}represents the amount of material [kbbl] transferred by task

*i*during slot

*t*; for variable recipe tasks

*i*∈

*I*

^{VR}, we also need to know the amount consumed of each input resource

*r*, through variables ξ̅

_{r,i,t}[kbbl]. Variables

*T*

_{t}hold the absolute time of event point

*t*[day], identifying the start of slot

*t*; excess resource variables

*R*

_{r,t}give the available amount of resource

*r*at event point

*t*; excess variables

*R*

_{r,t}

^{end}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*∈

*R*

^{EQ}, while for material resources

*r*∈

*R*

^{MR}the values are in [kbbl].

*X*

_{cd,t}identifies if a sequence independent changeover of crude blend occurs from slot

*t*to

*t*+ 1;

*ND*

_{cd}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.

*WS*

_{mv}and

*HB*

_{mv}give respectively the time waiting in sea and harboring for marine vessel

*mv*;

*DT*

_{t}represents the duration of time slot

*t*;

*AI*

_{tk,t}gives the average crude inventory in tank

*tk*during time slot

*t*.

### 4.3 RTN Structural Parameters

_{r,i}and μ̅

_{r,i}are related to the binary extent variables

*N*

_{i,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 μ

_{ST1,}

_{Transfer1}= −1 (resource consumed at the beginning of the task) and μ̅

_{ST1,}

_{Transfer1}= 1 (resource produced at the end of the task). The task also consumes/produces the first marine vessel (μ

_{MV1,}

_{Transfer1}= −1; μ̅

_{MV1,}

_{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.

_{r,i}and ν̅

_{r,i}are also related to discrete interactions but are linked to continuous extent variables ξ

_{i,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: ν

_{A_MV1,Hold_in_vesselMV1}= −1 and ν̅

_{A_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.

_{r,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 λ

_{A_ST10}

_{,Transfer3}= −1 and λ

_{A_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 ST

_{1}while continuously being supplied to CT

_{1}(see also the explanation linked to eq 3).

_{2}in Figure 5, and so, an additional index needs to be used. Let λ̅

_{r,r′,i}represent the continuous change of resource

*r*due to the consumption of resource

*r*′ by variable recipe task

*i*∈

*I*

^{VR}. Taking task Transfer 7 in Figure 2 as an example, we have λ̅

_{r,r,Transfer7}= −1 and λ̅

_{X,r,Transfer7}= 1, ∀

*r*∈ {

*A*_

*CT*

_{1},

*B*_

*CT*

_{1},

*C*_

*CT*

_{1}}.

*R*

_{r}

^{0}are used to set the initial inventory

*v*

_{tk,cr}

^{0}of storage and charging tanks, as well as to define the initial availability of equipment resources other than marine vessels, for example,

*R*

_{ST2}

^{0}= 1.

_{r,mv}

^{i}and π

_{r,mv}

^{o}. Taking problem 1 as an example, the second marine vessel carries ν

_{mv,B}

^{in}= 1000 [kbbl] that is made available to the system upon harboring π

_{B_MV2,mv2}

^{i}= 1000, together with the vessel itself: π

_{B_MV2,mv2}

^{i}= 1. The equipment resource is then removed from the system upon departure of the marine vessel: π

_{MV2,mv2}

^{o}= 1 disabling further crude transfers from it.

### 4.4 Scheduling Constraints

#### 4.4.1 Excess Resource Balances

*R*

_{r,t}and

*R*

_{r,t}

^{end}. 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

*Z*

_{t,mv}

^{i}. Similarly, the seventh term removes the marine vessel resource whenever the vessel departs at event point

*t*.(1)

*I*

^{S}tasks affects excess variable

*R*

_{r,t}, while the interaction at the end affects excess variable

*R*

_{r,t}

^{end}, 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 (

*R*

^{RM}⊂

*R*

^{MR}) 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

*R*

_{tk}

^{MR}holds the material resources that can appear inside tank

*tk*(e.g., A_CT

_{1}, B_CT

_{1}, and C_CT

_{1}are the material resources associated with charging tank CT

_{1}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)

*r*∈

*R*

^{IO}), 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*∈

*R*

^{CD}) process crude in every time slot, becoming available only at the last event point.

#### 4.4.2 Timing Constraints

*i*∈

*I*

^{NE}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 ρ

_{i}

^{max}are taken from the ρ

_{u,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

*R*

^{TC}⊂

*R*

^{EQ}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_ST

_{1}in Figure 2 for unit ST

_{1}). 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)

*r*⊂

*R*

^{CD}, 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)

*H*(eq 9).(9)

*Z*

_{t,mv}

^{i}. If an event point

*t*is linked to the harboring of marine vessel

*mv*, then the absolute time

*T*

_{t}must be greater than the arrival time

*at*

_{mv}; 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)

_{t}

^{noi}and Z

_{t}

^{noo}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)

#### 4.4.3 Task Extent Constraints

*N*

_{i,t}need to be activated so that the corresponding equipment units are consumed. Naturally, transfer tasks

*i*∈

*I*

^{NE}not consuming equipment units do not need to be a part of eq 19. Parameter

*v*

_{i}

^{max}is computed from the given values of

*v*

_{mv,cr}

^{in},

*v*

_{tk}

^{max}, and

*d*

_{cr}

^{max}.(19)

#### 4.4.4 Tank Compositions

*cr*the product of the volume

*R*

_{r,t}

^{end}by the property

*c*

_{cr,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

*CR*

_{r}holds the crude associated with resource

*r*.(21)

#### 4.4.5 Product Demand

#### 4.4.6 Number of Distillation Runs

*X*

_{cd,t}to the relevant task binary extent variables

*N*

_{i,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

*I*

_{cd}

^{CD}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

*ND*

_{cd}is then the number of changeovers plus one (eq 27).(23)(24)(25)(26)(27)

#### 4.4.7 Objective: Maximize Gross Margin

*R*

_{cr}

^{CT}⊂

*R*

^{MR}represents the set of material resources inside the charging tanks that are linked to crude blend

*cr*of value

*p*

_{cr}(e.g., in Figure 2 these would be resources C_CT

_{1}and C_CT

_{2}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)

#### 4.4.8 Objective: Minimize Operating Cost

*AI*

_{tk,t}and the slot duration

*DT*

_{t}.(30)

*H*–

*at*

_{mv}.(31)(32)

*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)

### 4.5 Matching Compositions Inside Blending Tanks and Their Outlet Streams

_{i,t}≤ 1 represent the fraction of the volume inside the tank leaving it during slot

*t*through blending task

*i*∈

*I*

^{BL}. It can be used to relate the amount of resource

*r*∈

*R*

_{i}

^{BL}available inside the tank at the start of the slot

*t*and the amount of corresponding resource

*r*′ ∈

*R*

_{r}

^{CR}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*(

*N*= 0), no material is being removed from the tank (ξ̅

_{i,t}_{r,i,t}= 0 due to eqs 19 and 20), and thus, the volume fraction must equal zero to meet the constraint.(37)

*i*= {Transfer 4} in Figure 5 as an example,

*R*

_{i}

^{BL}= {A

_{–}ST

_{1},D

_{–}AT

_{1}} includes the resources corresponding to crudes A and D inside storage tank ST

_{1}, whereas resources

*r*′ appearing in eq 37 are those consumed by the blending task, that is, for

*r*= {A_ST

_{1}},

*r*′ ∈

*R*

_{r}

^{CR}= {A

_{–}ST

_{1}

^{O}}, and for

*r*= {D_ST

_{1}},

*R*

_{r}

^{CR}= {D

_{–}ST

_{1}

^{O}}. For the charging tanks outlet transfers, resources

*r*and

*r*′ are actually the same (identified through the structural parameter λ̅

_{r,r,i}= −1). More specifically, for

*i*= {Transfer 7} in Figure 2,

*R*

_{i}

^{BL}= {A

_{–}CT

_{1},B

_{–}CT

_{1},C

_{–}CT

_{1}}, and so for

*r*= {A_CT

_{1}},

*r*′ ∈

*R*

_{r}

^{CR}= {A

_{–}CT

_{1}}, and so on.

*c*

_{cr,pr}are known parameters.

### 4.6 Relaxation of Bilinear Term in Blending Equation

#### 4.6.1 Using McCormick Envelopes (LP)

_{i,t}·

*R*

_{r,t}with a new set of continuous variables

*W*

_{r,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 ζ

_{i,t}and

*R*

_{r,t}.(38)In this case, the lower bounds are zero and the upper bounds are equal to 1 and

*R*

_{r}

^{max}, respectively. Parameter

*R*

_{r}

^{max}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)

*k*∈ {ψ,...,η}. In this work, we opt to discretize variables ζ

_{i,t}, which are fewer in number than variables

*R*

_{r,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 = ⌊log

_{10}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.

*Y*

_{i,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}≤ 10

^{ψ}is added to obtain a continuous domain; see eq 43 and Figure 9.(43)

*W*

_{r,i,t}is also written as a sum of an approximation and a residual term

_{r,i,t}. In eq 44,

*R̂*

_{r,i,t,j,k}is the disaggregated variable linked to

*R*

_{r,t}and the discrete value

*j*of ζ

_{i,t}associated with power

*k*. The sum of variables

*R̂*

_{r,i,t,j,k}over all digits

*j*must be equal to the original variable

*R*

_{r,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}actually represent the bilinear term arising from

_{i,t}·

*R*

_{r,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

*AI*

_{tk,t}·

*DT*

_{t}bilinear term in the operating cost objective function (eq 30) was relaxed in the same manner as ζ

_{i,t}·

*R*

_{r,t}. For the multiparametric disaggregation relaxation, the option was to discretize the slot duration variables

*DT*

_{t}, primarily because their values are in the order of units, closer to those of the other discretized variables (volume fractions ζ

_{i,t}∈ [0,1]). Such option reduces the number of binary variables when considering a global accuracy parameter ψ.

## 5 Test Problems

_{2}discharging to ST

_{3}. 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.

tasks | resources | |
---|---|---|

P1 | 12 | 24 |

P2 | 20 | 38 |

P3 | 23 | 59 |

P4 | 26 | 51 |

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, always^{a} | no, always | no, if blending^{b}^{,}^{c} |

Composition constraints on storage tanks? | yes | yes | yes | no | yes^{a} |

objective function | operating cost | operating cost | operating cost | gross margin | both |

inventory cost | rigorous (linear term) | estimate | estimate | not applicable | rigorous^{c} (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

^{–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 ζ

_{i,t}were initialized to 0.5 to guarantee a feasible solution in all runs.

*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

#### 6.1.1 Priority Slots Continuous-Time Approach of Mouret et al. (9)

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 |

_{1}(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 T

_{2}= 0.1 days. ST

_{1}then sends 300 kbbl of A to CT

_{1}at the same rate up to T

_{3}= 0.7 days, before receiving the remaining 950 kbbl of A. During that time, CT

_{1}is also receiving 150 kbbl of B from ST

_{2}, following 50 kbbl of B from ST

_{2}. 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 T

_{5}= 4.396 days. The crude from CT

_{2}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 CT

_{2}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 T

_{6}= 5.965 days, and so, it does not have time to contribute to the profit. At the end of the time horizon, ST

_{1}holds 943 kbbl of A, ST

_{2}is full, while CT

_{1}and CT

_{2}are almost empty.

_{1}receives 100.5 kbbl of A at the maximum rate, so that in the second slot, 300.5 kbbl can be sent to CT

_{1}, emptying ST

_{1}in the process. CT

_{1}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 (T

_{3}= 0.802, T

_{4}= 0.962 days) and transfers 79.763 kbbl of A at near maximum rate to ST

_{1}, which is then sent to CT

_{2}, 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 CT

_{2}feeds column 1 with pure E in the first couple of slots, switching to column 2 at T

_{5}= 1.441 days and becoming empty at T

_{9}= 8.144 days. Finally, part of crude C from the third marine vessel still makes it to column 2 through CT

_{3}, 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) | 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.

_{1}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 CD

_{1}and one in CD

_{2}. The main difference is thus the inventory cost, particularly due to CT

_{2}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.

#### 6.1.3 Unit-Specific Continuous-Time Approaches

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$] | CPUs^{a} | # slots | cost [k$]^{b} | CPUs^{a} | # slots | cost [k$] | CPUs^{a}^{,}^{b} | # slots | cost [k$] | CPUs^{c} |

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.

*t*and

*t*+ 1) with the same absolute time value (e.g.,

*T*

_{t}=

*T*

_{t+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)

### 6.2 Comparison to Different Optimization Approaches

*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.

*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.

|T| = (5,6,5,7)^{b} | |T| = (6,7,8,−)^{c} | |T| = (7,10,8,−)^{c} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

approach | discrete variables^{d} | total variables^{d} | total equations^{d} | margin [k$] | gap [%] | CPUs | margin [k$] | gap [%] | CPUs | margin [k$] | gap [%] | CPUs | |

P1 | McCormick | 56 | 396 | 488 | 7975^{e} | 0.00^{e} | 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).

|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

## Supporting Information

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.

## Terms & Conditions

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

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.

## Nomenclature

## Sets/Indices

CD/cd | crude oil distillation unit |

CR/cr | crude oil |

CR_{r} | crude associated with material resource |

CT/ct | charging tank |

I/i | tasks |

I^{BL} | variable recipe blending tasks |

I^{C} | fixed recipe continuous tasks |

I^{S} | fixed recipe storage tasks |

I_{cd}^{CD} | pair of transfer tasks to distillation column |

I^{NE} | transfer tasks not consuming an equipment resource |

I_{cr}^{VD} | variable recipe transfer tasks to distillation columns involving final product crude |

I^{VR} | 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 |

R_{i}^{BL} | inside tank material resources located immediately upstream of blending task |

R^{CD} | resources corresponding to distillation units |

R_{cr}^{CT} | material resource inside charging tanks corresponding to crude |

R^{EQ} | equipment resources |

R^{FP} | material resources corresponding to final crude blends |

R^{IO} | in or out material resources, located just after/before storage/charging tanks |

R_{r}^{CR} | out material resource (storage tanks) or inside tank resource (charging tanks) corresponding to the same crude of inside tank resource |

R^{MR} | material resources |

R_{tk}^{MR} | material resources than can appear inside tank |

R^{RM} | raw-material resources |

R^{TC} | 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 |

TK^{BL} | blending tanks |

U/u | system units excluding docking station |

## Parameters

at_{mv} | arrival time of marine vessel |

c_{cr,pr} | composition of raw-material crude |

c^{chg} | cost involved for each change in crude to a distillation column [$] |

c_{mv}^{harb} | harboring costs for unloading the crude from marine vessel |

c_{tk}^{inv} | inventory cost for tank |

c_{tk,pr}^{max} | maximum composition in blending tank |

c_{tk,pr}^{min} | minimum composition in blending tank |

c_{mv}^{wsea} | sea waiting cost for marine vessel |

d_{cr}^{max} | maximum demand of final product crude |

d_{cr}^{min} | minimum demand of final product crude |

H | time horizon [day] |

nd | maximum number of distillation operations |

p_{cr} | gross margin of crude |

R_{r}^{0} | initial availability of resource |

R_{mv,cr}^{max} | maximum excess value for of resource |

v_{mv,cr}^{in} | incoming volume from marine vessel |

v_{tk,cr}^{0} | initial volume inside tank |

v_{i}^{max} | maximum amount of material processed by task |

v_{tk}^{max} | upper bound on total volume inside tank |

v_{k}^{min} | lower bound on total volume inside tank |

η | position of most significant digit for discretized variables |

λ_{r,i} | continuous interaction of resource |

λ̅_{r,r′,i} | continuous interaction of resource |

μ_{r,i} | discrete interaction of resource |

μ̅_{r,i} | discrete interaction of resource |

v_{r,i} | discrete interaction of resource |

v̅_{r,i} | discrete interaction of resource |

π_{r,mv}^{i} | amount of resource |

π_{r,mv}^{o} | amount of resource |

ρ_{i}^{max} | maximum processing rate of task |

ρ_{i}^{min} | minimum processing rate of task |

ρ_{u,u}^{max} | maximum transfer flow rate between units |

ρ_{u,u}^{min} | minimum transfer flow rate between units |

ψ | position of least significant digit for discretized variables |

## Variables

N_{i,t} | binary variable indicating if task |

Y_{i,t,j,k} | binary variable indicating if the volume fraction of task |

Z_{mv,t}^{i} | binary variable indicating harboring of marine vessel |

Z_{t}^{no i} | binary variable indicating that no vessel harbors at event point |

Z_{t′,t}^{no io} | binary variable indicating that no vessel arrives at event point |

Z_{t}^{no o} | binary variable indicating that no vessel departs at event point |

Z_{mv,t}^{o} | binary variable indicating departure of marine vessel |

AI_{tk} | average inventory of crude in tank |

_{tk} | approximation of average inventory of crude in tank |

DT_{t} | duration of time slot |

HB_{mv} | duration of harboring for marine vessel |

ND_{cd} | number of distillation runs for column |

R_{r,t} | excess amount of resource |

R_{r,t}^{end} | excess amount of resource |

R̂_{r,i,t,j,k} | disaggregated variable linked to |

T_{t} | absolute time of event point |

W_{r,i,t} | bilinear term variable involving resource |

_{r,i,t} | residual variable of bilinear term involving resource |

WS_{mv} | time marine vessel |

X_{cd,t} | continuous variable identifying if a crude blend changeover occurs for unit |

ζ_{i,t} | volume fraction associated with the execution of variable recipe task |

_{i,t} | residual value of volume fraction linked to task |

ξ_{i,t} | total amount of material handled by task |

ξ̅_{r,i,t} | amount of resource |

## References

This article references 41 other publications.

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The relevant key decisions include the allocation of crude oils to refinery and port-side tanks, the connection of refinery tanks to crude distn. units (CDUs), the sequence and amts. of crudes pumped from the ports to the refineries, and the details relating to discharging of tankers at the port-side. These decisions are typically made over a horizon of one month. Scheduling is important for two reasons: on the one hand, the economic penalties of poor scheduling are severe, and on the other, efficient scheduling techniques will enable the exploitation of opportunities e.g. unexpected cheap cargoes on the high seas. Typical approaches to this problem are based on user-driven simulations. This paper indicates how math. programming techniques can be applied to such problems, and highlights the advantages of using such approaches.**3**Lee, H.; Pinto, J. M.; Grossmann, I. E.; Park, S. 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A mixed-integer optimization model is developed which relies on time discretization.**4**Hamisu, A. A.; Kabantiok, S.; Wang, M. Refinery scheduling of crude oil unloading with tank inventory management Comput. Chem. Eng. 2013, 55, 134Google Scholar4https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXovFaqtLg%253D&md5=f302758245194d18b68cb65070df4c25Refinery scheduling of crude oil unloading with tank inventory managementHamisu, Aminu A.; Kabantiok, Stephen; Wang, MeihongComputers & Chemical Engineering (2013), 55 (), 134-147CODEN: CCENDW; ISSN:0098-1354. (Elsevier B.V.)The aim of this study is to develop a methodol. for short-term crude oil unloading, tank inventory management, and crude distn. unit (CDU) charging schedule using mixed integer linear programming (MILP) optimization model as an extension to a previous work reported by Lee et al. (1996). The authors attempt to improve the previous model by adding an interval-interval variation constraint to avoid CDU charging rate fluctuation, a shutdown penalty within the scheduling cycle and a set up penalty for tank-tank transfer and introducing demand violation permit for a more flexible model against obtaining infeasible soln. Three different cases from the original paper were used to test the validity of the improved model. Comparison between Cases 1 and 2 shows the advantage of using smaller time interval as the operating cost of Case 2 is lower. Two scenarios were created from Case 3 to show the benefits of the improved model in deciding the best schedule to use. The improved model was implemented using the CPLEX solver in GAMS.**5**Jia, Z.; Ierapetritou, M.; Kelly, J. D. Refinery short-term scheduling using continuous time formulation: Crude-oil operations Ind. Eng. Chem. 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A.; Srinivasan, R.Chemical Engineering Science (2004), 59 (6), 1325-1341CODEN: CESCAC; ISSN:0009-2509. (Elsevier Science Ltd.)In today's competitive business climate characterized by uncertain oil markets, responding effectively and speedily to market forces, while maintaining reliable operations, is crucial to a refinery's bottom line. Optimal crude oil scheduling enables cost redn. by using cheaper crudes intelligently, minimizing crude changeovers, and avoiding ship demurrage. So far, only discrete-time formulations have stood up to the challenge of this important, nonlinear problem. A continuous-time formulation would portend numerous advantages, however, existing work in this area has just begun to scratch the surface. In this paper, we present the first complete continuous-time mixed integer linear programming (MILP) formulation for the short-term scheduling of operations in a refinery that receives crude from large crude carriers via a high-vol. single buoy mooring pipeline. This novel formulation accounts for real-world operational practices. We use an iterative algorithm to eliminate the crude compn. discrepancy that has proven to be the Achilles heel for existing formulations. While it does not guarantee global optimality, the algorithm needs only MILP solns. and obtains excellent max.-profit schedules for industrial problems with up to 7 days of scheduling horizon. We also report the first comparison of discrete- vs. continuous-time formulations for this complex problem.**9**Mouret, S.; Grossmann, I. E.; Pestiaux, P. A novel priority-slot based continuous-time formulation for crude-oil scheduling problems Ind. Eng. Chem. Res. 2009, 48, 8515Google ScholarThere is no corresponding record for this reference.**10**Mouret, S.; Grossmann, I. E.; Pestiaux, P. Time representations and mathematical models for process scheduling problems Comput. Chem. Eng. 2011, 35, 1038Google Scholar10https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXltl2qsrc%253D&md5=4909989efd293f6efbb0d1c9f0e0160bTime representations and mathematical models for process scheduling problemsMouret, Sylvain; Grossmann, Ignacio E.; Pestiaux, PierreComputers & Chemical Engineering (2011), 35 (6), 1038-1063CODEN: CCENDW; ISSN:0098-1354. (Elsevier B.V.)During the last 15 years, many math. models were developed in order to solve process operation scheduling problems, using discrete or continuous-time representations. In this paper, a unified representation and modeling approach are presented for process scheduling problems. Four different time representations are presented with corresponding strengthened formulations that rely on exploiting the non-overlapping graph structure of these problems through max. cliques and bicliques. These formulations are compared, and applied to single-stage and multi-stage batch scheduling problems, as well as crude-oil operations scheduling problems. Three soln. methods are introduced that can be used to achieve global optimality or obtain near-optimal solns. depending on the stopping criterion used. Computational results show that the multi-operation sequencing time representation is superior to the others as it allows efficient symmetry-breaking and requires fewer priority-slots, thus leading to smaller model sizes.**11**Li, J.; Misener, R.; Floudas, C. A. Continuous-time modeling and global optimization approach for scheduling of crude oil operations AIChE J. 2012, 58, 205Google ScholarThere is no corresponding record for this reference.**12**Li, J.; Li, W.; Karimi, I. A.; Srinivasan, R. Improving the robustness and efficiency of crude scheduling algorithms AIChE J. 2007, 53, 2659Google ScholarThere is no corresponding record for this reference.**13**Karuppiah, 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.**14**Castro, 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.**15**Teles, J. P.; Castro, P. M.; Matos, H. A. Multiparametric disaggregation technique for global optimization of polynomial programming problems J. Global Optimization 2013, 55, 227Google ScholarThere is no corresponding record for this reference.**16**Kolodziej, S.; Castro, P. M.; Grossmann, I. E. Global optimization of bilinear programs with a multiparametric disaggregation technique J. Global Optimization 2013, 57, 1039Google ScholarThere is no corresponding record for this reference.**17**McCormick, G. P. Computability of global solutions to factorable nonconvex programs. Part I. Convex underestimating problems Mathematical Programming 1976, 10, 147Google ScholarThere is no corresponding record for this reference.**18**Pantelides, C. C. Unified frameworks for the optimal process planning and scheduling. In Proceedings of the Second Conference on Foundations of Computer Aided Operations; Cache Publications: New York, 1994; pp 253.Google ScholarThere is no corresponding record for this reference.**19**Castro, 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.**20**Kelly, 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.**21**Kolodziej, 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.**22**Castro, 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.**23**Castro, 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.**24**Harjunkoski, 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.**25**Castro, 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.**26**Yadav, 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.**27**Westenberger, H.; Kallrath, L. Formulation of a Job Shop Problem in Process Industry, Internal Report; Bayer AG, Leverkusen and BASF AG: Ludwigshafen, 1995.Google ScholarThere is no corresponding record for this reference.**28**Kallrath, J. Planning and scheduling in the process industry OR Spectrum 2002, 24, 219Google ScholarThere is no corresponding record for this reference.**29**Blömer, F.; Günther, H. Scheduling of a multi-product batch process in the chemical industry Computers Ind. 1998, 36, 245Google ScholarThere is no corresponding record for this reference.**30**Balas, E. Disjunctive programming and a hierarchy of relaxations for discrete optimization problems SIAM J. Algebraic Discrete Methods 1985, 6 (3) 466– 486Google ScholarThere is no corresponding record for this reference.**31**Castro, 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.**32**Raman, 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.**33**Karuppiah, 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.**34**Castro, 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.**35**Teles, 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.**36**Kocis, 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.**37**Tawarmalani, 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.**38**Misener, R.; Floudas, C. A. GloMIQO: Global mixed-integer quadratic optimizer J. Global Optimization 2013, 57, 3Google ScholarThere is no corresponding record for this reference.**39**Viswanathan, 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.**40**Jia, 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.**41**Quesada, 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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**1**Kelly, J. D.; Mann, J. L. Crude oil blend scheduling optimization: An application with multimillion dollar benefits—Part 1 Hydrocarbon Processing 2003, 82 (6) 47There is no corresponding record for this reference.**2**Shah, N. Mathematical programming techniques for crude oil scheduling Comput. Chem. Eng. 1996, 20, S12272https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28XjsF2qtLg%253D&md5=42e9f5dd4117e775db596b63ce170698Mathematical programming techniques for crude oil schedulingShah, N.Computers & Chemical Engineering (1996), 20 (Suppl. B, European Symposium on Computer Aided Process Engineering--6, 1996), S1227-S1232CODEN: CCENDW; ISSN:0098-1354. (Elsevier)We consider the application of formal, math. programming techniques to the problem of scheduling the crude oil supply to a refinery. The relevant key decisions include the allocation of crude oils to refinery and port-side tanks, the connection of refinery tanks to crude distn. units (CDUs), the sequence and amts. of crudes pumped from the ports to the refineries, and the details relating to discharging of tankers at the port-side. These decisions are typically made over a horizon of one month. Scheduling is important for two reasons: on the one hand, the economic penalties of poor scheduling are severe, and on the other, efficient scheduling techniques will enable the exploitation of opportunities e.g. unexpected cheap cargoes on the high seas. Typical approaches to this problem are based on user-driven simulations. This paper indicates how math. programming techniques can be applied to such problems, and highlights the advantages of using such approaches.**3**Lee, H.; Pinto, J. M.; Grossmann, I. E.; Park, S. Mixed-integer linear programming model for refinery short-term scheduling of crude oil unloading with inventory management Ind. Eng. Chem. Res. 1996, 35, 16303https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28XitlWrtbo%253D&md5=fc7e50ad8f0fde80ac7c831127937a87Mixed-Integer Linear Programming Model for Refinery Short-Term Scheduling of Crude Oil Unloading with Inventory ManagementLee, Heeman; Pinto, Jose M.; Grossmann, Ignacio E.; Park, SunwonIndustrial & Engineering Chemistry Research (1996), 35 (5), 1630-41CODEN: IECRED; ISSN:0888-5885. (American Chemical Society)This paper addresses the problem of inventory management of a refinery that imports several types of crude oil which are delivered by different vessels. This problem involves optimal operation of crude oil unloading, its transfer from storage tanks to charging tanks, and the charging schedule for each crude oil distn. unit. A mixed-integer optimization model is developed which relies on time discretization.**4**Hamisu, A. A.; Kabantiok, S.; Wang, M. Refinery scheduling of crude oil unloading with tank inventory management Comput. Chem. Eng. 2013, 55, 1344https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXovFaqtLg%253D&md5=f302758245194d18b68cb65070df4c25Refinery scheduling of crude oil unloading with tank inventory managementHamisu, Aminu A.; Kabantiok, Stephen; Wang, MeihongComputers & Chemical Engineering (2013), 55 (), 134-147CODEN: CCENDW; ISSN:0098-1354. (Elsevier B.V.)The aim of this study is to develop a methodol. for short-term crude oil unloading, tank inventory management, and crude distn. unit (CDU) charging schedule using mixed integer linear programming (MILP) optimization model as an extension to a previous work reported by Lee et al. (1996). The authors attempt to improve the previous model by adding an interval-interval variation constraint to avoid CDU charging rate fluctuation, a shutdown penalty within the scheduling cycle and a set up penalty for tank-tank transfer and introducing demand violation permit for a more flexible model against obtaining infeasible soln. Three different cases from the original paper were used to test the validity of the improved model. Comparison between Cases 1 and 2 shows the advantage of using smaller time interval as the operating cost of Case 2 is lower. Two scenarios were created from Case 3 to show the benefits of the improved model in deciding the best schedule to use. The improved model was implemented using the CPLEX solver in GAMS.**5**Jia, Z.; Ierapetritou, M.; Kelly, J. D. Refinery short-term scheduling using continuous time formulation: Crude-oil operations Ind. Eng. Chem. Res. 2003, 42, 3085There is no corresponding record for this reference.**6**Furman, K.; Jia, Z.; Ierapetritou, M. G. A robust event-based continuous time formulation for tank transfer scheduling Ind. Eng. Chem. Res. 2007, 46, 9126There is no corresponding record for this reference.**7**Moro, J. F. L.; Pinto, J. M. Mixed-integer programming approach for short-term crude oil scheduling Ind. Eng. Chem. Res. 2004, 43, 85There is no corresponding record for this reference.**8**Reddy, P. C. P.; Karimi, I. A.; Srinivasan, R. A new continuous-time formulation for scheduling crude oil operations Chem. Eng. Sci. 2004, 59, 13258https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXisVarur4%253D&md5=361effcd786136a938c204b43e3f10dcA new continuous-time formulation for scheduling crude oil operationsReddy, P. Chandra Prakash; Karimi, I. A.; Srinivasan, R.Chemical Engineering Science (2004), 59 (6), 1325-1341CODEN: CESCAC; ISSN:0009-2509. (Elsevier Science Ltd.)In today's competitive business climate characterized by uncertain oil markets, responding effectively and speedily to market forces, while maintaining reliable operations, is crucial to a refinery's bottom line. Optimal crude oil scheduling enables cost redn. by using cheaper crudes intelligently, minimizing crude changeovers, and avoiding ship demurrage. So far, only discrete-time formulations have stood up to the challenge of this important, nonlinear problem. A continuous-time formulation would portend numerous advantages, however, existing work in this area has just begun to scratch the surface. In this paper, we present the first complete continuous-time mixed integer linear programming (MILP) formulation for the short-term scheduling of operations in a refinery that receives crude from large crude carriers via a high-vol. single buoy mooring pipeline. This novel formulation accounts for real-world operational practices. We use an iterative algorithm to eliminate the crude compn. discrepancy that has proven to be the Achilles heel for existing formulations. While it does not guarantee global optimality, the algorithm needs only MILP solns. and obtains excellent max.-profit schedules for industrial problems with up to 7 days of scheduling horizon. We also report the first comparison of discrete- vs. continuous-time formulations for this complex problem.**9**Mouret, S.; Grossmann, I. E.; Pestiaux, P. A novel priority-slot based continuous-time formulation for crude-oil scheduling problems Ind. Eng. Chem. Res. 2009, 48, 8515There is no corresponding record for this reference.**10**Mouret, S.; Grossmann, I. E.; Pestiaux, P. Time representations and mathematical models for process scheduling problems Comput. Chem. Eng. 2011, 35, 103810https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXltl2qsrc%253D&md5=4909989efd293f6efbb0d1c9f0e0160bTime representations and mathematical models for process scheduling problemsMouret, Sylvain; Grossmann, Ignacio E.; Pestiaux, PierreComputers & Chemical Engineering (2011), 35 (6), 1038-1063CODEN: CCENDW; ISSN:0098-1354. (Elsevier B.V.)During the last 15 years, many math. models were developed in order to solve process operation scheduling problems, using discrete or continuous-time representations. In this paper, a unified representation and modeling approach are presented for process scheduling problems. Four different time representations are presented with corresponding strengthened formulations that rely on exploiting the non-overlapping graph structure of these problems through max. cliques and bicliques. These formulations are compared, and applied to single-stage and multi-stage batch scheduling problems, as well as crude-oil operations scheduling problems. Three soln. methods are introduced that can be used to achieve global optimality or obtain near-optimal solns. depending on the stopping criterion used. Computational results show that the multi-operation sequencing time representation is superior to the others as it allows efficient symmetry-breaking and requires fewer priority-slots, thus leading to smaller model sizes.**11**Li, J.; Misener, R.; Floudas, C. A. Continuous-time modeling and global optimization approach for scheduling of crude oil operations AIChE J. 2012, 58, 205There is no corresponding record for this reference.**12**Li, J.; Li, W.; Karimi, I. A.; Srinivasan, R. Improving the robustness and efficiency of crude scheduling algorithms AIChE J. 2007, 53, 2659There is no corresponding record for this reference.**13**Karuppiah, R.; Grossmann, I. E. Global optimization for the synthesis of integrated water systems in chemical processes Comput. Chem. Eng. 2006, 30, 65013https://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.**14**Castro, 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, 10514https://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.**15**Teles, J. P.; Castro, P. M.; Matos, H. A. Multiparametric disaggregation technique for global optimization of polynomial programming problems J. Global Optimization 2013, 55, 227There is no corresponding record for this reference.**16**Kolodziej, S.; Castro, P. M.; Grossmann, I. E. Global optimization of bilinear programs with a multiparametric disaggregation technique J. Global Optimization 2013, 57, 1039There is no corresponding record for this reference.**17**McCormick, G. P. Computability of global solutions to factorable nonconvex programs. Part I. Convex underestimating problems Mathematical Programming 1976, 10, 147There is no corresponding record for this reference.**18**Pantelides, C. C. Unified frameworks for the optimal process planning and scheduling. In Proceedings of the Second Conference on Foundations of Computer Aided Operations; Cache Publications: New York, 1994; pp 253.There is no corresponding record for this reference.**19**Castro, 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, 34319https://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.**20**Kelly, 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.**21**Kolodziej, 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.**22**Castro, 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.**23**Castro, 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.**24**Harjunkoski, 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.**25**Castro, 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.**26**Yadav, 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.**27**Westenberger, 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.**28**Kallrath, J. Planning and scheduling in the process industry OR Spectrum 2002, 24, 219There is no corresponding record for this reference.**29**Blömer, 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.**30**Balas, 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.**31**Castro, 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.**32**Raman, 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.**33**Karuppiah, 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.**34**Castro, 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.**35**Teles, 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.**36**Kocis, 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.**37**Tawarmalani, 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.**38**Misener, R.; Floudas, C. A. GloMIQO: Global mixed-integer quadratic optimizer J. Global Optimization 2013, 57, 3There is no corresponding record for this reference.**39**Viswanathan, 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.**40**Jia, 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.**41**Quesada, 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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## Supporting Information

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