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Scheduling of Loading and Unloading of Crude oil in a
Refinery with Optimal Mixture Preparation
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Journal:
Manuscript ID:
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Article
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Date Submitted by the
Author:
Industrial & Engineering Chemistry Research
Complete List of Authors:
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Saharidis, Georgios; Rutgers - The State University of New Jersey,
Chemical and Biochemical Engineering
Ierapetritou, Marianthi; Rutgers University, Chemical and
Biochemical Engineering; Rutgers University, Chemical and
Biochemical Engineering
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Scheduling of Loading and Unloading of Crude oil in a Refinery with
Optimal Mixture Preparation
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GEORGIOS K.D. SAHARIDIS and MARIANTHI G. IERAPETRITOU*
Department of Chemical and Biochemical Engineering, Rutgers University, 98 Brett Road,
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Piscataway, NJ 08854-8058, USA, saharidis@gmail.com, marianth@soemail.rutgers.edu
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Abstract: Production scheduling defines what products should be produced and what products
should be consumed at each point in time over a short period; hence, it defines which run-mode to
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use and when to perform changeovers in order to meet the market needs and to satisfy the
demand. The study presented in this paper focuses on the scheduling of the loading and unloading
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of crude oil in intermediate storage tanks, between ports and crude distillation units. Analysis of
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this activity gives rise to better use of a system’s resources, as well as improved total visibility
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and control of production units and the entire supply chain. It is very important that the crude oil
is loaded and unloaded continuously, primarily for security reasons but also to reduce the setup
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costs incurred when flow between a port and a tank or between a tank and a crude distillation unit
is reinitialized. The aim of the present paper is to develop an exact solution approach based on a
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generic mixed integer model, which provides not only the optimal schedule of loading and
unloading of crude oil minimizing the setup cost but also the optimal type of mixture preparation.
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Comparison between the approach applied in a real refinery and the developed exact approach is
*
Corresponding Author
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presented in order to justify the advantage of optimization tools. A series of valid inequalities are
developed in order to speed-up the convergence of the resolution approach of the model. Finally a
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comparison between the two classical types of mixture preparation and the proposed combination
type mixture preparation is provided to illustrate the advantages of the proposed method.
Keyword: Scheduling; crude oil loading and unloading; mixed integer linear programming;
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optimal mixture preparation.
1. Introduction
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Production scheduling defines what products should be produced and what products should be
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consumed at each point in time over a short period (typically 20-30 days); hence, it defines which
run-mode to use and when to perform changeovers in order to meet the market needs and to
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satisfy the demand. The study presented in this paper focuses on the scheduling of the loading
and unloading of crude oil in intermediate storage tanks, between ports and Crude Distillation
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Units (CDUs). The problem of scheduling and planning in petrochemical companies has appeared
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since the introduction of linear programming [1-5]. The developed methods for the resolution of
this problem are classified into three general groups: the exact methods that use discrete or
continuous time representation and the heuristic methods.
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The first approaches for the problem of refinery scheduling use discrete time representation. The
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principle of discrete time representation is to split the scheduling time horizon into intervals of
equal size and use binary variables to specify whether an action starts or finishes during an
interval. The first approach was presented by [6]. The author presents a model on discrete time
representation for the scheduling of crude oil (SCO) leading to the resolution of a mixed integer
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linear program (MILP). In this formulation, due to nonlinearity issues, the problem was broken
up into two sub-problems but no optimal solution can be guaranteed. In [7] the authors propose
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another approach for resolving the SCO problem where all the non-linear constraints associated
with the blending operations have been linearized. As the authors, along with other researchers
[8], mention, this type of linearization gives rise to mixture failure and unsatisfied CDU
operational constraints. In [9], the authors propose an algorithm that combines a Mixed Integer
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Linear Problem (MILP) and a nonlinear problem (NLP), in order to solve the anomaly of
composition as presented in a previous model. In [10], the authors propose a formulation that is
represented as a hybrid approach, as its representation of time is simultaneously discrete and
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continuous. The objective is the maximization of profit, incorporating penalties for the stock out
of inventory and for the delay in unloading a tanker after it has arrived in the port. The authors do
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not take into account the setup cost of tanks because the penalties associated with the inventory
and the unloading of tankers seems more significant. The proposed algorithm is an iterative
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procedure based on the resolution of a MILP, avoiding the additional resolution of NLP as was
the case in [9]. [11] developed two models, the first for planning and the second for scheduling in
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a refinery. Concerning the latter, a linear model is developed for the minimization of loading and
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unloading of crude oil between the tankers and CDUs in a refinery in Brazil. [12] presents an
extension of this model taking into account all the different operational options that can possibly
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take place in this refinery. In [13] the authors present another work for planning and scheduling in
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a refinery. The problem under study corresponds to a real case of a refinery in Sweden where the
objective was to find the optimal schedule of production units minimizing the production cost.
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The stochastic aspect is taken into account and the schedule is done based on a time window in
which the tankers expect to arrive in the refinery. Finally in [14] a general model is presented for
the SCO. The developed model takes under consideration a variety of distillation options and
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types of mixture preparation. A novel time representation is introduced based on event intervals
and a series of valid inequalities are developed for the speeding-up of the resolution of the model.
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In recent years, another family of models for the scheduling of crude oil has been developed,
wherein a continuous representation of time is used. In a continuous time representation, the start
as well as the finish of an activity is a variable determined by optimization. In [11], a continuous
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time formulation for the scheduling in a refinery is proposed. The article describes the heart of
modeling for a refinery scheduling problem, but does so without giving all the details of the
model. In the part of the article reserved for the SCO problem, a real case study of a refinery
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receiving various types of crude oil supplied by pipeline is presented. The optimal schedule is
obtained by the linearization of the initial model which increases the size of the problem but
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ensures the feasibility of the obtained solution. In [15, 16], a model for the scheduling of
operations in a refinery based on continuous time reformulation is presented. The authors present
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a general refinery system which they divide into three sub-systems. The first is related to crude
oil operations, the second to refining processes and intermediate tanks, and the third to the ends of
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operation processes and the stock of final products. The setup cost is not taken into account by
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this model, but several types of system configurations are studied. The authors in [8] and [10]
propose another model with continuous time representation for SCO. The authors argue that their
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suggested model is the first complete formulation for the SCO problem using MILP formulation.
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all distillation options, as it corresponds to a real case study where a setup system has already
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been chosen. Moreover, the authors make a comparison between their model in continuous time
and their model in discrete time representation to show the advantage of a continuous time
representation for their case study.
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For several industrial problems, to solve the MILP involved in the scheduling of crude oil using
optimization tools based on branch-and-bound or branch-and-cut methods can be difficult and, in
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certain cases, inadequate due to the associated complexity (i.e. number of variables and nonlinear
constraints). Given this complexity, an otherwise sufficient algorithm requires an impractical
amount of time simply to find the first integer-feasible solution even when the problem is well
formulated. As an alternative, several researchers developed heuristic algorithms that allow for
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the resolution of large-scale problems. The disadvantage of a heuristic approach is that neither
global optimality nor feasibility can be guaranteed. In practice, these methods provide a solution
for large-scale problems [17] when it is impossible to obtain a feasible solution by an exact
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procedure. In [18] the author presents an effective primal heuristic to encourage the reduction of
binary variable in a significant way. This heuristic can be applied before an implicit enumerative
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type search to find integer-feasible solutions for the SCO problem. The basis of the technique is
to employ four different well-known smoothing functions into the framework of a smooth-and-
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dive accelerator (SDA) algorithm. Another heuristic, the chronological decomposition heuristic
(CDH), is presented by [17]. The proposed approach is a time-based divide-and-conquer strategy
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intended to find integer-feasible solutions to practical scale production scheduling optimization
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problems, such as the problem of SCO, rapidly. As the authors mention, CDH is not an exact
algorithm in that it will not find the global optimum. The basic idea of the CDH is to chop the
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scheduling time horizon into aggregate time intervals (time-chunks), which are a multiple of the
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base time period. Each time-chunk is solved using mixed-integer linear programming (MILP)
techniques, beginning from the first time-chunk and moving forward in time using the technique
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of chronological backtracking if required [19].
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This paper is organized as follows. Section 2 provides the problem description, whereas section 3
presents the developed model and a series of valid inequalities corresponding to the problem
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under study. Section 4 presents the motivated case study as well as a comparison between the
practical approach adopted in the refinery under study and the proposed methodology and a
comparison between the proposed methods of mixture preparation with other methods found in
the literature. Finally section 5 draws conclusions indicating future prospects for this research.
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2. Problem Description
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One of the most critical activities in a refinery is the loading and unloading of crude oil into the
tanks. Better analysis of this activity gives rise to better use of a system’s resources, as well as
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improved total visibility and control of production units and the entire supply chain. The need for
the development of a systematic methodology and optimization tool for these activities is clearly
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justified. In the majority of cases, the managers of a refinery take into account the low flexibility
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of the system and, based on their experience, choose the first possible schedule that results from
manual calculation and heuristic criteria. The time horizon is 20-30 days (480-720 hours), which
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is a typical scheduling period. The potential financial and operational benefits associated with the
development of an optimization tool are enormous, as an advanced optimization tool for
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scheduling could allow the refinery to minimize the flow problems and the loss of crude oil and
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to obtain the optimal periodic schedules of production. The development of a mathematical model
could give direction for the future and could be used for re-optimization of the system in case of
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accidental events.
The system considered in this paper is composed of a series of tanks to store crude oil, ports to
accommodate the boats delivering the crude oil, and CDUs where the mixtures of the crude oil
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are distilled. Moreover, this system is comprised of pipelines, which connect the ports with the
tanks and the tanks with the CDUs (cf. Figure 1). Finally, electric pumps for loading and
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unloading the crude oil, as well as mixers for the preparation of the mixtures required by the
CDUs, are two other important features of the system.
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Tank 1
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Tank 2
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Manifold
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Tank N
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Figure 1: Simplified sketch of the refinery problem examined
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The setup of tanks, involving loading the quantities of crude oil at the ports and unloading the
quantities of mixtures or crude oil toward the CDU, requires a series of operations which are
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expensive for the refinery. The most critical and expensive operations associated with the tank’s
setup at each stage are:
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Before loading/unloading:
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1. Configuring the pipeline networks (e.g. opening of valves, configuration of pumps, etc.)
2. Filling pipelines with crude oil
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3. Sampling of crude oil for chemical analyses
4. Measuring of the crude oil stock in tank before loading/unloading
5. Starting the loading/unloading
6. Stopping the loading/unloading
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After loading/unloading:
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1. Configuring the pipeline networks (e.g. closing of valves, configuration and maintenance
of pumps, etc.)
2. Emptying pipelines
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3. Measuring the crude oil loaded/unloaded in the tanks
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Another important problem in many refineries is the storage capacity which appear increasingly
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tanks used in a scheduling period becomes critical. The reason is that the minimum number of
setups is associated with the minimum number of tanks used for the scheduling, which is in turn
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associated with an increase in the number of tanks available for other uses (e.g. storage of the
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finished products or raw materials). By minimizing this number, we also obtain the following
information: a) the optimal schedule and the extra production cost incurred in a case where fewer
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tanks are available to stock the crude oil and b) the profit reduction or increase with the use of
tanks in other operations. This information provide the flexibility to a system to change the use of
tanks due to an unexpected event or changes in the market or the company’s strategy.
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There are different types of configurations that are associated with several modes of mixture
preparation combined with the type of distillation. In general the type of distillation used in many
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refineries, referred to as flexible distillation, satisfies some lower and upper bounds for each type
of crude oil. Concerning the preparation of the mixture, we can implement two different modes or
a combination of both. For the first mode, the mixture is made just before the CDU in a manifold
through the use of pipelines and the use of mixers existing in the manifold. The second mode is to
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prepare the mixtures required by the CDU in the tanks themselves. In this case, a quantity of a
given type of crude oil is already loaded in a tank then stored and kept on standby until a quantity
of another type of the crude oil is unloaded into the same tank, in order to produce the required
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mixture. Each one of these two cases are studied separately in [14] and two general models are
developed for the SCO of a real case study. The data and the parameters of the system are the
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dimensions of the refinery (number of tanks and CDUs) and the arrival of boats based on
medium-term planning as well as the demand of crude oil by the CDU, and the production rates.
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Moreover, the initial conditions (quantity and composition in each tank), the system’s capabilities
(operational times and flowrate limits), and the storage capacity of tanks available to stock crude
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oil are known. The objective of this work is to develop a more general model than the models
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presented in [14] that solves the problem of the SCO while minimizing the setup cost for the
loading and unloading of tanks for a certain system configuration, defining in the same time the
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best strategy for the preparation of the mixtures required by CDUs. Scheduling takes into account
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operational constraints, constraints relating to the use and the availability of resources, and
capacity of each resource.
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3. Proposed formulation
3.1 Basic model
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The proposed formulation involves both continuous and binary variables. The continuous
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variables are associated with the flow between ports and tanks, flow between tanks and CDUs,
and the stored quantities of various types of crude oil. The binary variables represent decisions
regarding the loading, unloading and setup of a tank which is necessary either before the loading
of a quantity available at the port or before unloading crude oil or a mixture of crude oil towards
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the CDU. Notice that the decision variable associated with the setup takes a value equal to 1 only
in the beginning of loading or unloading of crude oil. The variables associated with the
establishment of the connection between a tank and a port or a CDU takes a value of 1 when the
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flow to or from this tank takes place. Finally the percentage of the total quantity stored in a tank
unloaded towards a CDU takes integer values between 0 and 100%. The integer nature of these
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variables results from the real needs of the refinery where the decision makers prefer an integer
percentage.
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Using the following nomenclature the proposed formulation is as follows.
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Nomenclature
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Indices
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Ports
z
Tanks
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Crude Distillation Units
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Types of crude oil
t
Periods
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NZ
Number of tanks
NP
Number of ports
NCDU
Number of crude distillation units
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Number of various types of crude oil
NT
Number of periods
Cap
Storage capacity of tanks (the same for all tanks)
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Parameters
E i ,t
Total quantity of the crude oil available, in port i, for period t
Percentage of the crude oil ( E i ,t ) of type j, of the quantity available in port i, for
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aei , j ,t
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period t
S k ,t
Total quantity requested from CDUk, for the period t
a k , j ,t
Equal to 1 if the CDUk requests crude oil of type j, for period t and equal to zero if not
amink , j ,t :
Acceptable minimal percentage, of type j, for the mixture unloading towards CDUk,
for period t
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amaxk , j ,t
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Acceptable maximal percentage, of type j, for the mixture unloading towards CDUk,
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for period t
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Decision Variables
X i , z , j ,t
Continuous variable which corresponds to the quantity of crude oil type j, loaded by
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port i, in tank z, for period t
Y z , k , j ,t
Continuous variable which corresponds to the quantity of crude oil type j, unloaded
by tank z, towards CDUk , for period t
C i , z ,t
Binary variable (0-1) which is equal to 1 if connection is established between the port
i and tank z, for period t and equal to zero if not
D z , k ,t
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Binary variable (0-1) which is equal to 1 if connection is established between the tank
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z and CDUk, for period t and equal to zero if not
Continuous variable that becomes equal to 1 if setup of tank z is established for
SCi , z ,t
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SDz ,k ,t
I z , j ,t
loading crude oil from the port i, at the beginning of the period t and equal to zero if
not
Continuous variable that becomes equal to 1 if setup of tank z is established for
unloading crude oil towards CDUk, at the beginning of the period t and equal to zero
if not
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Continuous variable which corresponds to the quantity of crude oil type j, stocked at
the end of period t, in tank z. We specify that variables I z , j ,0 are data corresponding
to the quantities stored in the tanks at the beginning of scheduling period
γ z , k ,t
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Integer variable that takes value in the interval [0,100] and defines the percentage of
the total quantity stored in the tank z unloaded towards the CDUk in period t
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The constraints of the system include: operation rules, such as a given tank is feeding a CDU, it
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cannot be charged and vice versa; material balances; limitations that exist in the system, such as
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storage capacities; expected limitations imposed by the mixture preparation mode and the
distillation option; establishing a connection between ports and tanks and between tanks and
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CDUs; and the setup of tanks for loading and unloading. In general, we have four groups of
constraints which guarantee certain operational conditions:
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1. Constraints which guarantee that the quantities loaded in tanks are equal to the quantities
available to ports in each period
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2. Constraints expressing that the quantities unloaded towards CDUs are equal to the quantities
required by them
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3. Constraints expressing the material balance: the quantities stored in a tank are equal to the
quantities which were stored during the previous period plus loaded quantities less unloaded
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quantities
4. Constraints which guarantee that quantity stored in a tank is not greater than the storage
capacity
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The detailed expression of each one of these types of constraints is as follows.
Loading Constraints
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These constraints express the requirement that the sum of the quantities loaded in tanks, by port i,
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for period t, is equal to the quantity available to the port.
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∑X
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= E i ,t aei , j ,t ∀i, j , t
z =1
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(1)
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Unloading Constraints
The sum of the quantities unloaded by all the tanks, towards CDUk, for period t, is equal to the
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quantity required by the CDUk as expressed by the following constraints.
NZ NJ
z , k , j ,t
= S k ,t ∀k , t
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z =1 j =1
(2)
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With the flexible distillation, the additional constraints are that the sum of quantities type j
unloaded by all tanks, for period t, towards CDUk satisfies the upper and lower percentage of the
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quantity required by this CDUk.
NZ
amin k , j ,t a k , j ,t S k ,t ≤ ∑ Yz ,k , j ,t ≤ amax k , j ,t a k , j ,t S k ,t ∀k,j,t
z =1
(3)
ie
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When a percentage of the crude oil is unloaded from the tank z, to the CDUk, in period t then all
the types of crude oil are unloaded simultaneously in a quantity which satisfies the proportion of
w.
the mixture.
Yz , k ,l ,t = I z , j ,t γ z.k .t / 100 ∀z,k,j,t
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(4)
nf
We notice that this constraint is a non-linear constraint and in order to obtain a linear model we
id
use the method presented in the end of this section.
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The material balance constraints express the requirement that the quantity of crude oil, type j,
tia
which is stored in tank z, at period t, is equal to the quantity which was stored at period t-1, plus
the sum of quantities of the same type which are loaded for this period by all ports i, less the sum
of the quantities unloaded by this tank towards all CDUs.
NCDU
i =1
k =1
∑Y
z , k , j ,t
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NP
I z , j ,t = I z , j ,t −1 + ∑ X i , z , j ,t −
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∀z, j,t
(5)
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The following constraints express the requirement that the sum of different types of crude oil is
less than or equal to the storage capacity.
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NJ
0 ≤ ∑ I z , j ,t ≤ cap ∀z,t
j =1
(6)
ie
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Another series of constraints is associated with operation rules and express the connection if a
flow between the tank z and the port i or the CDUk is established. Notice that the constant M used
in the following equalities takes a value equal to storage capacity of the tanks.
NJ
Loading:
NJ
Co
Ci , z ,t ≤ ∑ X i , z , j ,t ≤ MCi , z ,t , ∀i, z, t
j =1
Unloading:
w.
(7)
Dz , k ,t ≤ ∑ Yz , k , j ,t ≤ MDz , k ,t , ∀z , k , t
nf
j =1
(8)
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A critical operational rule expresses the requirement that loading and unloading does not take
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place at the same time.
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Ci , z ,t + Dz ,k ,t ≤ 1 ∀i, z , k , t
(9)
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In the case of loading or unloading of a tank z, a setup cost is charged at the beginning of loading
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or unloading, if in the previous period connection was not established. This requirement is
expressed by the following constraints.
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Loading:
C i , z ,t −1 + SC i , z ,t ≥ C i , z ,t ∀i , z , t
(10)
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Unloading:
Dz , k ,t −1 + SD z , k ,t ≥ D z , j ,t ∀z , k , t
(11)
Note that the continuous variables SC i , z ,t , SD z , k ,t are forced to take a value of 0 since the objective
ie
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function minimizes cost if a setup is not established and become 1 if a setup is established due to
constraints (10) and (11).
w.
The objective function for the developed model is to minimize the total number of tank setups
necessary for the loading and unloading of the crude oil:
Co
NP NZ NT
NZ NCDU NT
MinZ = ∑∑∑ SC i , z ,t + ∑
∑ ∑ SD
nf
i =1 z =1 t =1
z =1
k =1
z , k ,t
t =1
(12)
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The above formulation corresponds to a nonlinear model due to the constraint (4) where we have
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a bilinear term of continuous times integer variable. This constraint can be linearized as follows.
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First the integer variable is expressed as a function of auxiliary binary variables that leads to new
bilinear terms between binary and continuous variables:
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γ z .k .t = W z0, k ,t + 2W z1,k ,t + 4W z2, k ,t + 8W z3, k ,t + 16W z4, k ,t + 32W z5, k ,t + 64W z6, k ,t
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where γ z .k .t = [0,100] , and W zq,k ,t , q = {0,...,6} are the auxiliary binary variables needed for the
linearization of the constraint. Replacing the decision variable γ z ,k ,t with the above formulation
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we obtain the following modified non-linear constraint [20]:
100Yz .k . j ,t = W z0,k ,t I z , j ,t + 2Wz1, k ,t I z , j ,t + 4W z2,k ,t I z , j ,t +
+ 8W z3,k ,t I z , j ,t + 16W z4,k ,t I z , j ,t + 32Wz5, k ,t I z , j ,t + 64W z6, k ,t I z , j ,t
ie
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Using the method presented in [1, 21, 22], this non-linear equality could be replaced by the
following constraints:
w.
100Yz .k . j ,t = H z0, k , j ,t + 2 H 1z , k , j ,t + 4 H z2,k , j ,t +
Co
+ 8 H z3,k , j ,t + 16 H z4, k , j ,t + 32 H z5,k , j ,t + 64 H z6,k , j ,t , ∀z,k,j,t
I z, j ,t − M (1 −Wzq,k ,t ) − H zq,k , j ,t ≤ 0, ∀z,k,j,t
(4b)
(4c)
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H zq, k , j , t − I z , j , t ≤ 0 , ∀ z,k,j,t
(4a)
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H zq, k , j , t − MW zq, k , t ≤ 0 , ∀ z,k,j,t
(4d)
tia
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where H zq,k , j ,t is an additional auxiliary continuous variable which represents each bilinear
product: H zq,k , j,t = Wzq,k ,t I z, j,t ∀z,k,j,t (where q = {0,...,6}) .
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variables and constraints are introduced. In the following section we present a series of valid
inequalities that correspond to the particular problem of scheduling of crude oil used for the
acceleration of the resolution approach.
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3.2 Valid inequalities
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In order to improve model convergence, we have developed a series of valid inequalities which
can be used with any equivalent refinery example. These valid inequalities can also be used in
other cases, where the objective is defined not only as setup cost minimization but also other
ie
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types. Two valid inequality groups are developed: the first is based on the data of the system and
the second is based on the operational rules. In addition to the loading and unloading constraints a
set of logical constraints are considered based on system’s data and operational rules as follows.
w.
Logical constraints based on data
Co
The first group of valid inequalities relating to system data involves five different sets of valid
nf
inequalities: The first two sets are related to boat arrival and tank loading, the third and fourth are
related to the mixtures required by the CDU in addition to tank unloading, and the last one is
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related to tank concentration.
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I. The first valid inequality set is based on loading data which defines the minimum and
tia
maximum number of loading tanks. The total number of loading tanks required when the
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mixtures are prepared using the pipelines just before the CDUs and/or into the tanks is:
•
Less than or equal to the total number of storage tanks (NR) minus the number of
different types of crude oil requested by CDUs ( S 2 t ).
•
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Greater than or equal to the minimum number of tanks needed for crude oil unloading of
boat arriving at port i, for period t.
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Page 19 of 37
NZ
W 3i ,t ≤ ∑ Ci , z ,t ≤ ( NZ − S 2t ) ∀i, t
z =1
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(VI 1)
where S2t is the number of different types of crude oil requested by CDUs for period t, and W 3i ,t
is the minimum number of tanks needed for crude oil unloading of boat arriving at port i, for
ie
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period t and is equal to:
E i ,t
E i ,t

if
integer

cap
cap
W 3 i ,t = 
E
integre quotient of i ,t + 1
if not
cap

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II. During some periods, tankers do not arrive in the port, enabling the following valid inequality
nf
to be applied in order to express the fact that there is no loading tank ( M ′ is equal to the
quantity transferred from tank at t period):
NZ NJ
i , z ,t
+ ∑ ∑ X i , j , z ,t ≤ M ′W 3 i ,t ∀i, t
z =1 j =1
tia
z =1
en
NZ
∑C
id
(VI 2)
lIII. The total number of unloading tanks is less than or equal to the total number of tanks minus the
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minimum number of tanks needed for crude oil unloading of boat arriving at port, for period t:
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NZ
∑D
z =1
NP
z , k ,t
≤ NZ − ∑ W 3 i ,t ∀k , t
i =1
(VI 3)
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IV. The minimum number of tanks which unload crude oil or mixture of crude oil towards the
CDUs is equal to the number of different types of mixture required for period t:
ie
ev
NZ NCDU
S 2t ≤ ∑
z =1
∑D
z , k ,t
∀t
k =1
(VI 4)
w.
Notice that in general the number of different types of crude oil requested by the CDUs is equal
to the total number of CDUs. For the validity of the above inequality and in order to take into
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consideration the case where more than one CDU has request the same type of mixture, we do not
use the total number of CDUs as a lower bound and instead we use S2t .
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V. If crude oil type j is stored in tank z ( I z , j ,t ≠ 0 ) during the period t and CDUk requests a
mixture which is not composed of crude oil type j, then tank z could not be unloaded towards
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CDUk ( D z ,k ,t = 0 ):
tia
I z , j ,t ≤ cap(1 − D z ,k ,t ) + 2a k , j ,t cap
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(VI 5)
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This valid inequality is based on the fact that if tanks z is unloaded towards CDUk then all the
type of crude oil stored in the tank is unloaded because different types of crude oil cannot be
unmixed in the tank. The opposite situation is not valid because a tank could be unloaded towards
a CDU even if the CDUk requests type j of crude oil which in period t is not stored in tank z. The
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reason is that it is possible to prepare also the requested mixture in the manifold just before the
CDU along with the unloading of other tanks which have the other types of crude oil requested by
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the CDUk.
Logical constraints based on operational rules
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The second group of valid inequalities relating to operational rules has four different sets of valid
inequalities: The first set relates the decision variables D z ,k ,t and γ z ,k ,t , forcing them to have a
value equal to zero when there is no flow between the tank z and the CDUk. The second and third
w.
sets use the same concept, requiring the decision variables SC i , z ,t and SD z ,k ,t to take on a value
equal to zero when no connection is established between a tank z and a port or a CDU. Finally,
Co
the forth set is based on the concentration of tanks and the operational rule which prohibits the
loading and unloading of a tank during the same period.
id
nf
I. The first set describes the case where no connection is established between the tank z and the
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CDUk during period t and as a result no quantity is unloaded from this tank z to CDUk:
tia
0.01 * γ z , k ,t ≤ D z , k ,t ≤ γ z , k ,t ∀z, k , t
(VI 6)
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II. The second set restricts the decision variables corresponding to the establishment of a
connection between port i and tank z, during t for loading of crude oil:
SCi , z ,t ≤ Ci , z ,t ∀i, z, t
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(VI 7)
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III. The third set is the equivalent of the valid inequality presented previously (VI 7) but for the
unloading of crude oil to the CDUk, during t:
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SDz ,k ,t ≤ Dz , k ,t ∀z, k , t
(VI 8)
IV. The forth set is based on the concentration of tanks: If a tank z is empty, no unloading can take
ie
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place by this tank for period t and period t+1 because it has to be loaded in period t+1 before
being unloaded later. This implies the following inequality:
NJ
∑I
j =1
z , j ,t
≥
NCDU
∑D
k =1
+
NCDU
∑D
k =1
z , k ,t +1
∀z, t
(t ≠ NT )
(VI 9)
id
nf
Co
4. Case study
z , k ,t
w.
Even today in most refineries optimization is not used for the scheduling and no mathematical
model is developed to help with the decision making. In most real cases obtaining a schedule of
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crude oil is done based on manual calculations and the experience of the executive technicians.
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The proposed schedule takes into account the satisfaction of constraints and sometimes the
minimization of the number of tanks used for each individual sub-period. They are some extreme
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cases where the decision taken by the managers gives an infeasible solution in the next periods. In
that case, additional tanks are used to make the schedule feasible, dramatically increasing the
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setup cost and the availability of storage for other products. In general the type of optimization
used by the managers gives a feasible, but not necessarily optimal, schedule for the loading and
unloading of tanks. The numerical results show that the solutions obtained in this way are
typically sub-optimal by 15-20% (see Table 2). Thus the need to develop a model to optimize the
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scheduling of loading and unloading of tanks is apparent. We have to ensure that in general the
need of an additional setup corresponds to the single-person workload of 20 hours as defined by
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the managers of the company under study. In the following section an illustrative case study is
presented which is used for the rest of the paper.
4.1 Illustrative case study
ie
ev
The system under study is composed of two ports, seven tanks for the loading of crude oil, and
two crude distillation units (CDUs) fed by these seven tanks. The initial setup of the system
w.
satisfies the general rules presented in section 3 along with the following rules:
•
The loading time of tanks from a boat is equal to 36 hours and fixed by contract with the
suppliers of crude oil.
The distillation flow for the two CDUs are: 270-350 m3/hour for the first one and 300-
id
1660 m3/hour for the second.
The mixtures could be prepared using the pipelines just before the two CDUs and/or in
en
•
nf
•
Co
the tanks.
The data for this system are:
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tia
•
The period of scheduling which is equal to 20 days (480 hours)
•
The arrival plan of three boats (when, which type, and what quantity of the crude oil they
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transfer)
•
The demand of 8 different mixtures (composed of a maximum of three different types of
crude oil) by the two CDUs (when, which type, and what quantities of mixture)
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•
The initial state of each tank at the beginning of scheduling period and their storage
capacity, which is equal to 100000 m3
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The details of the case study are given in Figure 2 and in Table 1.
t=204
0
Boat Arrival ⇒
0
t=2
ie
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CDU1 Demand ⇒
CDU2 Demand ⇒
0
t=102
Type 0
24600
Type 2
35400
t=342
0
t=378
Type 1
120000
Type 0
15000
Type 2
15000
0
t=240
Type 0
Type 0
30000
t=480 hours
0
110000
Type 0
75000
w.
Type 0
9800
Type 1
9800
Type 2
15400
Type 2
60000
Type 1
30000
Type 2
20000
Type 0
60000
Co
Figure 2: Data of the illustrative case study
nf
Tanks
Type of crude oil
Initial Quantities
1
-
2
Type 0
3
Type 2
4
-
5
Type 0
6
Type 2
100000
7
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Empty
Empty
80000
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100000
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Empty
40000
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Table 1: Initial amounts of crude oil in tanks
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The first line of Figure 2 corresponds to the boats’ arrival times and the next two correspond to
the time horizon of the two CDUs. Two boats arrive at the refinery, the first at t=204 and the
second at t=342. These two tankers bring quantities equal to 120,000 m3 and 110,000 m3 crude oil
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of type 0, and type 1, respectively. Concerning the demand of CDUs, 3 mixtures are required by
CDU1 and 5 by CDU2. For example, between t=378 hour and t=480 hour, CDU1 requires a
mixture of 35,000 m3 with the following composition: 28% of type 0, 28% of type 1 and 44% of
ie
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type 2. In this case study, the objective is to minimize the setup cost of the seven tanks for the
loading of two boats, which arrive at specific time points, while preparing the 3 mixtures
requested by CDU1 and the 5 requested by CDU2 and unloading them towards them into exact
times.
w.
Co
In the developed model, instead of splitting the scheduling time horizon into intervals of equal
duration, we have applied the event based time representation presented in [14]. The set of one-
nf
hour intervals over which any single new event occurs is considered to be a single period. For the
reference example, between the period t = 2 until t = 102, the same events occur: the request of a
id
mixture by CDU1 (mixture of 50% type 0 and 50% type 2) and by the CDU2 (mixture of 41%
en
type 0 and 59% type 2). These periods are grouped together since the mixtures required by the
CDUs are constant. Between t = 102 and t = 240, a new event occurs when the mixtures
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requested from the CDUs change, which defines a third period. This procedure is repeated (see
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Figure 3) until all the corresponding hourly periods have been grouped together based on a
common event occurrence.
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1
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0
CDU1 Demand ⇒
CDU2 Demand ⇒
2
3
t=102
t=2
Boat Arrival ⇒
4
t=204
0
5
t=240
Type 0
6
t=342
0
120000
0
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0
7 Periods
7
t=378
Type 1
t=480
480 Hours
0
110000
Type 0
15000
Type 2
15000
Type 0
27720
Type 0
9780
Type 0
27720
Type 0
9780
Type 0
9800
Type 1
9800
Type 2
15400
Type 0
24600
Type 2
35400
Type 0
30000
Type 2
15650
Type 2
44350
Type 0
60000
Type 1
30000
Type 2
20000
w.
Figure 3: Event-based time representation for the illustrative case study
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4.2 Comparison of empirical and exact approach
nf
As we mention in the beginning of this section in practice, the obtained schedule is generated
based on managerial experience giving rise to a sub-optimal solution (feasible solution) for the
id
reference example, leading to 17 setups. These 17 setups of seven tanks are required for the
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loading of crude oil quantities transferred by the two boats and the tank unloading, in order to
satisfy the request of eight different mixtures by the two CDUs. Notice that the practical type of
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optimization used by the manager of the refinery is referred to as Heuristic Approach (HA) and
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the exact type of optimization presented in the paper is referred to as Exact Approach (EA). If we
use the EA model, the solution obtained requires 14 setups of 7 tanks to unload the same boats
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and satisfy the mixture requests of the two CDUs. The profit obtained by the EA compared to the
HA is equal to 17.6 % ( 17 − 14 = 0.176 ). We notice that the difference of 3 setups between the two
17
optimization methods corresponds to a single-person workload of 60 hours, which justifies the
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use of the EA. Applying the EA for a number of different data sets for this case study (Appendix
A) we observed difference between the HA and EA of 14% to 20% as shown in Table 2. Notice
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that the examples used for illustration correspond to a refinery located in Greece where the
mixtures are prepared only in the manifold just before the CDUs.
Examples
Heuristic approach
Exact approach
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Difference
# of setups
# of setups
CPU time
# of setups
Ex1
17 Setups
14 Setups
980 sec
17.6%
Ex2
20 Setups
16 Setups
1270 sec
20%
17 Setups
14 Setups
61328 sec
17.6%
21 Setups
18 Setups
4453 sec
14.3%
4270sec
16.7%
Ex5
18 Setups
Ex3
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15 Setups
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Table 2: Results of the heuristic and exact approaches
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Since the CPU time for the resolution of the EA method is rather high as shown in Table 2, in the
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following sections, we give some comparative results of the effect of valid inequalities on the
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total CPU time needed for the resolution of the proposed modeling. Notice that all results
presented in this paper have been obtained on Pentium (R) 4, CPU 2.40 GHz, RAM 1 GB and
CPLEX 10.1 using a C++ implementation of the proposed approach.
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4.3 Improvement of the model convergence
The numerical example results presented in Table 2 emphasize the superiority of the exact
approach in terms of solution quality as compared with its empirical counterpart. However, the
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CPU resolution time of the exact approach (because of the structure, the dimension and the
linearization of the non-linear constraints) is also high (ex.3 CPU time 17 hours). Refinery
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managers do not have the possibility to wait for a long time before an optimal solution is
identified, even if this solution results in a significantly lower system production cost. This
emphasizes the need for a model that can find the optimal solution quickly. This is very important
because monetary penalties are imposed for every hour that a boat waits outside of the port and
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because of the limitation of the storage capacity. The optimal schedule should be obtainable
within a decision-making time window of a few minutes during which a decision can be made as
to where a tanker, arriving in subsequent periods, should be unloaded. However, in some cases a
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decision-making time window of only a few seconds exists due to an unexpected event.
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In order to improve model convergence, we have applied the valid inequalities presented in
section 3.2. The effect of valid inequalities applied in the EA is significant, as it results in a
nf
sufficient reduction of the CPU resolution time. Applying these 10 valid inequalities for the five
examples presented in Appendix A reduction ranging from 69% to 99% in some examples is
obtained.
en
id
Examples
Without valid inequalities
With valid inequalities
Difference
Ex1
980 sec
102 sec
89%
Ex2
1270 sec
101 sec
92%
Ex3
61328 sec
333 sec
Ex4
4453 sec
42 sec
Ex5
4270sec
1231 sec
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99%
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99%
71%
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We should note that the CPU resolution times presented in Table 3 are acceptable CPU times for
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the managers of the refinery who need to find the optimal solution in a few minutes. This
reduction is based on the restriction of the solution space of the initial problem due to the use of
the 10 groups of valid inequalities.
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4.4 Comparison of combine mode of mixture preparation with the others modes
We mention in section 2 that the two classical types of mixture preparation are: a) the mixture
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preparation in the manifold just before the CDUs through the use of pipelines and use of mixers
existing in the manifold and b) the mixture preparation in the tanks through the use of the tanks
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mixers. For the first one only one type of crude oil is allowed to be stored at a given time period
in a tank and for the second one a quantity of a certain type of crude oil is stored in the tank until
nf
another type of crude oil is unloaded into the same tank, in order to produce the required mixture.
In this section we investigate the combination of these two types of mixture preparation as
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compared with each one separately. The same data sets presented in Appendix A are used and the
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models presented in [14] are applied which correspond to the two types of mixture preparation.
Table 4 shows the comparative results between these three types of mixture preparation.
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Mixture preparation
Mixture preparation
in the manifold
in the tanks
Optimal mixture preparation
Solution
CPU time
Solution
CPU time
Solution
CPU time
(setups)
(sec)
(setups)
(sec)
(setups)
(sec)
Example 1
16
5
Inf
Inf
14
102
Example 2
17
5
18
10
16
101
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Example 3
16
15
Inf
Inf
14
333
Example 4
19
10
20
10
18
42
Example 5
15
21
Inf
Inf
15
1231
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Table 4: Comparison of mixture preparation types
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The numerical results presented in Table 4 emphasize the superiority of the optimal mixture
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preparation in terms of solution quality as compared with the two other approaches. Although the
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CPU time is longer for the combined case, since the model is nonlinear and has to be linearized
increasing the number of variables and constraints, as compared to the other approaches it does
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not exceed the time limits defined by the managers of the refinery due to the use of valid
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inequalities. The potential advantage of applying the combined approach is fewer setups which
justifies spending more time applying the combined method since the potential gain is significant.
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As mentioned before, in the refinery case study used in this work the mixtures are prepared in the
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manifold just before the CDUs. For this reason, in some examples (ex.1, ex.3, and ex.5) we
obtain infeasible solutions because the mixture could not be produced only in the tanks since
there is not sufficient number of tanks available.
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5. Conclusions
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In this paper a general model is proposed to provide an exact optimal solution for the problem of
scheduling of crude oil where a combination of mixture preparation is also possible. The
proposed model is tested and compared with the current method implemented by the managers of
a Greek refinery for the scheduling of crude oil. The comparison is conducted using a series of
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real examples. The first results reveal that the benefit of using the exact approach proposed in the
paper is significant. However, the developed modeling is computationally expensive and requires
several hours to define the optimal schedule for the crude oil. In order to reduce the CPU
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resolution time an event-based time representation is used as presented and a series of valid
inequalities are applied. The obtained CPU resolution time decreased significantly to duration
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acceptable to the managers of the refinery. Among the perspectives for the future investigation is
a study that applies a decomposition method in order to decrease more the CPU resolution time.
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Also the development of a series of heuristic rules in order to linearize the non-linear constraints
without increasing the total number of decision variables and constraints would be a good
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approach to further decrease the CPU resolution time.
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Acknowledgments:
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The authors gratefully acknowledge financial support from the National Science Foundation
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under Grant CTS 0625515.
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References
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1.
2.
3.
4.
5.
6.
9.
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id
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nf
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w.
7.
Floudas, C.A. Non-linear and Mixed-Integer Optimization, ed. T.i.C.
Engineering. 1995, New York: Oxford University Press.
Manne, A. Scheduling of pertoleum operations Harvard University Press,
Cambridge 1956.
Symonds, G. Linear programming: The solution of refinery problems. Esso
Standard Oil Company 1955.
Biegler, L.T.; Grossmann, I.E.; Westerberg, A.W. Systematic Methods of
Chemical Process Design. 1997, Upper Saddle River, New Jersey: Prentice Hall
International Series.
Pardalos, P. Handbook of Applied Optimization ed. O.U. Press. 2002, New York:
Oxford University Press.
Shah, N. Mathematical programming techniques for crude oil scheduling.
Computers & Chemical Engineering. 1996, 20, p. S1227-1232.
Lee, Y.H.; Kim, S.H. Production-Distribution planning in supply chain
considering capacity constraints. Computers & Industrial Engineering. 2002, 43,
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Reddy, P.C.; Karimi, I.A.; Srinivasan, R. A novel solution approach for
optimizing crude oil operations. A.I.Ch.E. Journal 2004a, 50, p. 1177-1197.
Li, W.; Hui, C.W.; Hua, B.; Zhongxuan, T. Scheduling crude oil Unloading,
Storage and Processing. Industrial & Engineering Chemical Research. 2002,
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Reddy, P.C.; Karimi, I.A.; Srinivasan, R. A new continuous-time formulation for
scheduling crude oil operations. Chemical Engineering Science. 2004b, 59, p.
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Joly, M.; Moro, L.; Pinto, J.M. Planning and scheduling for petroleum refineries
using mathematical programming. Brazilian Journal of Chemical Engineering.
2002, 19, p. 207-228.
Neiro, S.M.; Pinto, J.M. A general modeling framework for the operational
planning of petroleum supply chains. Computers and Chemical Engineering.
2002 28, p. 871-896.
Persson, J.A.; Gothe-Lundgren, M. Shipment planning at oil refineries using
column generation and valid inequalities. European Journal of Operational
Research. 2005, 163, p. 631-652.
Saharidis, G.K. Pilotage de production a moyen terme et a court terme:
contribution aux problématique d’optimisation globale vc locale et a
l’ordonnancement dans les raffineries. Genie Industriel. Vol.22 PhD. 2006, Paris:
Ecole Centrale Paris. 150.
Jia, Z.; Ierapetritou, M.; Kelly, J.D. Refinery Short-Term Scheduling Using
Continuous Time Formulation: Crude-Oil Operations. Industrial & Engineering
Chemical Research. 2003, 42, p. 3085-3097.
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16.
17.
Jia, Z.; Ierapetritou, M.G. Efficient short-term scheduling of refinery operation
based on continuous time formulation. Computer & Chemical Engineering 2004,
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Kelly, J.D.; Mann, J.L. Crude oil blend scheduling optimization: an application
with multimillion dollar benefits. Hydrocarbon Processing. 2002, p. 47-53.
Kelly, J.D. Smooth-and-dive accelerator, a pre-MILP primal heuristic applied to
scheduling. Computer & Chemical Engineering 2003, 27, p. 873-832.
Marriott, I.; Stuckey, P. Programming with Constraints: an Introduction. MIT
Press, Cambridge, Mass. . 1998.
Minoux, M. Mathematical Programming Theory and Algorithms. Series in
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Petersen, C.C. A note on transforming the product of variables to linear form in
linear programs. Working Paper 1971, Purdue University
Glover, F. Improved linear integer programming formulation of non-linear integer
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18.
19.
20.
21.
22.
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Appendix A
Tanks/Type of crude oil
Tank #1
Tank #2
Tank #3
Tank #4
Tank #5
Tank #6
Hour
Period / Hours
0
1/102 h
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102
102
2/100 h
204
204
3/36 h
4/100 h
5/36 h
6/102 h
516
516
Type 1: 120000
Type 2: 110000
Table A1: Example 2
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8/50 h
Tankers arrival
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7/36 h
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480
480
Type 4
100000
CDU II
Type 1: 27300
Type 2: Type 3: Type 4: Type 1: 27300
Type 2: Type 3: Type 4: Type 1: Type 2: Type 3: 7812
Type 4: Type 1: Type 2: Type 3: 22188
Type 4: Type 1: 60000
Type 2: Type 3: Type 4: Type 1: Type 2: 30000
Type 3: Type 4: Type 1: 3800
Type 2: 6200
Type 3: Type 4: Type 1: Type 2: 10000
Type 3: 40000
Type 4: -
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378
Type 3
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342
342
566
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240
240
Type 1
Type 2
40000
80000
CDU I
Type 1: 16000
Type 2: Type 3: 4000
Type 4: Type 1: 27300
Type 2: Type 3: Type 4: Type 1: 9828
Type 2: Type 3: Type 4: Type 1: 28044
Type 2: Type 3:
Type 4: Type 1: 9828
Type 2: Type 3: Type 4: Type 1: 9800
Type 2: 9800
Type 3: 15400
Type 4: Type 1: 10440
Type 2: 8352
Type 3: Type 4: Type 1: 14560
Type 2: 11648
Type 3: Type 4: -
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Tanks/Type of crude oil
Tank #1
Tank #2
Tank #3
Tank #4
Tank #5
Tank #6
Tank #7
Tank #8
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0
Period / Hours
1/152 h
2/ 140 h
292
292
3/36 h
Type 3
Type 4
95000
40000
60000
50000
CDU I
CDU II
Type 1: 15000
Type 2: Type 3: 15000
Type 4: Type 1: 20000
Type 2: Type 3: Type 4: Type 1: 10000
Type 2: -
Type 1: 24600
Type 2: Type 3: 25400
Type 4: Type 1: 30000
Type 2: Type 3: Type 4: Type 1: Type 2: -
Type 3: Type 4: Type 1: 15000
Type 2: Type 3: Type 4: Type 1: 60000
Type 2: -
Type 3: 50000
Type 4: Type 1: Type 2: Type 3: 20000
Type 4: Type 1: 30000
Type 2: -
Tankers
arrival
Type 1:
130000
4/160 h
Type 3: Type 4: -
Table A2: Example 3
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Type 2:
110000
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Type 2
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Type 1
40000
40000
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Tanks/Type of crude oil
Tank #1
Tank #2
Tank #3
Tank #4
Tank #5
Tank #6
Tank #7
Tank #8
Hour
Period /
Hours
0
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40000
40000
140
140
CDU I
CDU II
Type 1:
7000
Type 2: Type 3: Type 4: Type 1:
7800
Type 2: Type 3:
5200
Type 4: Type 1:
10000
Type 2: -
Type 1: 4000
Type 1: 7480
Type 1: 20000
Type 2: Type 3: 16000
Type 4: Type 1: 10000
Type 2: Type 3: 14520
Type 4: Type 1: 7000
Type 2: Type 3: Type 4: Type 1: 10000
Type 2: Type 3: -
Type 2: Type 3: -
Type 4: Type 1: 10000
Type 4: Type 1: 10000
360
5/36 h
Type 3: Type 4: Type 1:
2000
Type 2: Type 3: Type 4:
15000
Type 1: Type 2: -
Type 3: 30000
Type 4: Type 1: -
Type 3: Type 4: Type 1: 8000
Type 3: 10000
Type 4: Type 1: -
Type 2: Type 3: 10000
Type 4: -
Type 2: Type 3: Type 4: -
Type 1: 20000
Type 2: -
Type 1: 15000
Type 2: -
Type 3: Type 4: -
Type 3: Type 4: -
Type 3: 5000
Type 4: -
Type 1:
120000
Type 2: Type 3: 10000
Type 4:
Table A3: Example 4
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Type 2: -
Type 2:
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Type 2: -
Tankers
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CDU IV
Type 2: -
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Type 4: Type 1: -
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3/36 h
Type 2: Type 3: -
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250
Type 4
60000
50000
CDU III
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2/ 110 h
Type 3
95000
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1/140 h
Type 2
40000
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Tanks/Type of crude oil
Tank #1
Tank #2
Tank #3
Tank #4
Tank #5
Tank #6
Hour
Period / Hours
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36
36
1/36 h
2/38 h
74
74
3/36 h
5/36 h
7/110 h
Type 1: Type 2: -
Type 1: Type 2: -
Type 3: 15000
Type 4: Type 1: 15000
Type 2: Type 3: Type 4: Type 1: 20000
Type 2: -
Type 3: 35400
Type 4: Type 1: 30000
Type 2: Type 3: Type 4: Type 1: 10000
Type 2: -
Type 3: Type 4: Type 1: 30000
Type 2: Type 3: Type 4: Type 1: Type 2: 10000
Tankers
arrival
Type 1:
90000
Type 1:
110000
Type 3: 40000
Type 4: Type 1: 50000
Type 2: Type 3: Type 4: Type 1: Type 2: -
Type 3: Type 4: Type 1: 9800
Type 2: 9800
Type 3:
Type 4: Type 1: 10200
Type 2: 10200
Type 3: 13600
Type 4: -
Type 2:
100000
Type 3: 20000
Type 4: Type 1: 10000
Type 2: 20000
Type 3: 20000
Type 4: Type 1: 3800
Type 2: 29200
Type 3: Type 4: -
Table A4: Example 5
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CDU II
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CDU I
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95000
100000
en
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233
Type 4
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197
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Type 3
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4/87 h
Type 2
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110
110
Type 1
30000
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Submitted to Industrial & Engineering Chemistry Research