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Chance constrained programming models for refinery short-term crude oil scheduling problem
Applied Mathematical Modelling 33 (2009) 1696–1707
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Chance constrained programming models for refinery short-term crude oil scheduling problem Cuiwen Cao a,*, Xingsheng Gu a, Zhong Xin b a b
Research Institute of Automation, East China University of Science and Technology, Shanghai 200237, China School of Chemical Engineering, East China University of Science and Technology, Shanghai 200237, China
a r t i c l e
i n f o
Article history: Received 1 April 2006 Received in revised form 1 March 2008 Accepted 4 March 2008 Available online 11 April 2008
Keywords: Short-term crude oil scheduling problem Demands uncertainty Chance constrained mixed-integer nonlinear programming Stochastic Fuzzy
a b s t r a c t The main objective of this work is to put forward chance constrained mixed-integer nonlinear stochastic and fuzzy programming models for refinery short-term crude oil scheduling problem under demands uncertainty of distillation units. The scheduling problem studied has characteristics of discrete events and continuous events coexistence, multistage, multiproduct, nonlinear, uncertainty and large scale. At first, the two models are transformed into their equivalent stochastic and fuzzy mixed-integer linear programming (MILP) models by using the method of Quesada and Grossmann [I. Quesada, I E. Grossmann, Global optimization of bilinear process networks with multicomponent flows, Comput. Chem. Eng. 19 (12) (1995) 1219–1242], respectively. After that, the stochastic equivalent model is converted into its deterministic MILP model through probabilistic theory. The fuzzy equivalent model is transformed into its crisp MILP model relies on the fuzzy theory presented by Liu and Iwamura [B.D. Liu, K. Iwamura, Chance constrained programming with fuzzy parameters, Fuzzy Sets Syst. 94 (2) (1998) 227–237] for the first time in this area. Finally, the two crisp MILP models are solved in LINGO 8.0 based on scheduling time discretization. A case study which has 267 continuous variables, 68 binary variables and 320 constraints is effectively solved with the solution approaches proposed. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction The short-term oil-refinery scheduling problem is one of the most challenging problems in operational research due to the complexity of the refinery scheduling operations [1]. In contrast to relatively ample availability of commercial production planning software, such as RPMS (refinery and petrochemical modelling system – [2]) and PIMS (process industry modelling system – [3]), there are still no commercial tools for short-term refinery scheduling optimization problems [4]. A lot of refineries have to rely on their experience and develop their own in-house tools to deal with scheduling operations. Typically, the overall problem for optimization of oil-refinery scheduling operations consists of three parts [5]. The first part studies the crude oil unloading, mixing, transferring and multilevel crude oil inventory control process. The second part deals with fractionation, reaction units scheduling and a variety of intermediate product tanks control. The third part involves the finished product blending and distributing process. In this paper, we focus on the first part problem. It is one critical component of the overall problem for refinery scheduling operations. When a refinery is located along seashore, its crude oil scheduling problem includes crude oil unloading process from the vessels to the storage tanks, transferring process from storage tanks to the charging tanks (where several crude oils are mixed), and charging process from the charging tanks to the crude oil distillation units (CDUs). In the literature, * Corresponding author. E-mail address: [email protected] (C. Cao). S0307-904X/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2008.03.022
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deterministic mathematical programming technologies are developed in this area from the nineties of last century. Shah [6] presented a two-stage MILP model for a monthly crude oil transferring plan. Its objective was to minimize the crude oil heels of charging tanks. Lee et al. [7] developed a MILP model to solve crude oil short-term scheduling problem which objective was minimizing the total operating cost. Pinto et al. [1] proposed a short-term MINLP model from fixed crude oil pipelines to CDUs, which objective was maximizing the production while minimizing the number of tanks used. These models [1,6,7] they developed rely on time discretization representations. In recent years, mathematical models of crude oil scheduling operations emphasize on continuous-time formulation to shorten the gap between theory research and real-world operation. Jia et al. [5] considered the same problem as Lee et al. [7], they employed the state-task-network (STN) continuous-time representation. Moro and Pinto [4] restudied their former work [1], the new approach they adopted is to define variablelength global time slots, and the objective function is to maximize the crude oil distillation unit feed flow rate while minimizing the cost associated with tank operation. Chandra Prakash Reddy et al. [8] presented a complete continuous-time MILP model for the crude oil short-term scheduling problem as Pinto et al. [1], they also reported the comparison of discrete-time versus continuous-time formulations for this problem. Chryssolouris et al. [9] studied the similar problem as Lee et al. [7], they took the temperature cut-points into consideration for each distillation unit. However, all these discussions mentioned above adopt deterministic mathematical programming methods to solve the crude oil scheduling problem. The data in their models are assumed to be deterministic. In real world, the situations of crude oil operation process are uncertain since the dynamic nature of the environment. The variations of certain parameters, such as feedstock qualities and yield levels are of great importance in short-term scheduling operation decisions. Refinery companies must assess the impacts of these important changes. In this paper, we study one type of uncertainties in crude oil scheduling problem – crude oil blends demands uncertainty, which result in crude oil feed flow rates variation during scheduling time horizon. Two types distinct research philosophies – stochastic programming and fuzzy programming – are used. Stochastic programming deals with optimization problems whose parameters take values from given discrete or continuous probability distributions [10]. One type of the solution methods is the chance constraint programming method pioneered by Charnes and Cooper [11]. They assume that stochastic objective functions and stochastic constraints will hold with at least some probability levels, and the chances are represented by the probabilities that the objectives and constraints are satisfied. Concerning with the fuzzy programming method, the term fuzzy programming has been used in different ways in Sahinidis [12], Liu and Sahinidis [13], and Zimmermann [14]. Although the theory of Liu et al. [15] and Liu and Iwamura [16] has been used for many applications, we use it for the first time to solve short-term crude oil scheduling problem in this area. Our fuzzy programming model is defined as a mathematical programming with those fuzzy parameters analogous to chance constrained programming with stochastic parameters. We assume that the constraints will hold with at least a possibility level a, and the chance represents the possibility that the constraints are satisfied. We introduce the definition of chance constrained fuzzy programming we used, and the crisp equivalents of fuzzy chance constraints for two special cases. We use one of them to solve our crude oil short scheduling problem. Our forecast demands parameters are assumed as to be fuzzy with unimodal membership functions. The paper is organized as follows: Section 2 states the definition of short-term crude oil scheduling problem. In Section 3, the chance constrained stochastic and fuzzy programming methods are introduced, respectively. The detailed mathematical formulations for crude oil short-term scheduling problem with two methods are presented. Then the above technologies are applied to a case study in Section 4. In the last section, we put forward the conclusions.
2. Problem definition 2.1. Problem statement In a typical refinery, a series of operations for turning crude oils into higher value end (gasoline, diesel, etc.) products begin with crude oil unloading, mixing, transferring process. After that, the distillation process takes place in CDUs, which separate the charged oil into consecutive lighter and heavier fractions with different boiling points. Most of the distillation products need to be further processed in vacuum distillation units, fluid catalytic cracking units, coking units or hydro-treating units, etc. The refined products are then mixed in gasoline/diesel/fuel oil/asphalt/. . . pools for making the final refinery products. Planning determines the volumes of different feedstock to be processed, the mix and amounts of products to be produced in response to product requirements over several months. Scheduling deals with the timing of the plant operations to realize the plan. In most refineries, their CDUs are built in different year, so they can only process feedstock with different properties. These properties involve process ability, yields of some very high value products, impurities or concentrations of some key components, etc. which influence the downstream processing. The distillation products from all the CDUs are processed together in following steps, the crude oil feed rates in all CDUs must be maintained compatible. Refinery schedulers continuously watch the properties of feedstock and the first distillation products of their CDUs, if some events happen, for example the yield of heavier distillation fractions is too high, the valuable lighter fractions are all taken away to produce high value products which means the production recipes are changed, or sulphur concentration or acid value level is fluctuant abnormally, they change the crude oil supply feed flow rates or type of the CDU disposed, then the others must be changed accordingly and immediately. These situations happen frequently under
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current unsteady supply of crude oil, high pressure of intense time and low inventory flexibility in a refinery. Nowadays, schedulers implement this job based on their years of experience. In this paper, we assess the potential cost of this type uncertainty. To focus on studying the influence of uncertainty, our uncertainty models rely on discrete-time representations which assume all events are forced to coincide with one of the interval boundaries. The short-term crude oil scheduling problem considered here is related to determining port details of the crude oil allocation process from discrete vessels to storage tanks, refinery details of the continuous crude oil allocation process from storage tanks to charging tanks (used for crude oil blending) and the mixed oil from charging tanks to CDUs (used for distillation). Under the given arrival times of vessels, equipment capacity limitations, key component concentration ranges and the uncertain demands for CDUs, the problem studied is to determine the following variables to minimize the operating costs (which include unloading cost for crude oil vessels, cost for vessels waiting in the sea, inventory cost for storage and charging tanks and changeover cost between different mixed crude oils for CDUs): waiting time of each vessel at sea; unloading duration and flow rates from vessels to storage tanks; crude oil transfer duration and flow rates from storage tanks to charging tanks; inventory variation in storage tanks and charging tanks; concentration levels of key components in storage tanks and charging tanks. 2.2. Operation rules and assumptions The operation rules should be obeyed are as follows: (1) the refinery uses only one docking station, a new arriving vessel has to wait in the sea if an anterior vessel does not leave the docking station; (2) while a charging tank is charging CDU, crude oil from the storage tanks cannot be fed into the charging tank and vice versa; (3) each charging tank can only charge one CDU at one time interval; (4) each CDU can only be charged by one charging tank at one time interval; (5) CDUs must be operated continuously throughout the scheduling time horizon. Following assumptions are proposed in this paper: (1) the scheduling time horizon is divided into time periods of equal duration, so representations are formulated as discrete-time MINLP models; (2) all events are forced to coincide with one of the interval boundaries; (3) perfect mixing is assumed in all tanks; (4) the times required for a CDU mode change are neglected; (5) all vessels unloading and waiting costs are assumed common. 3. Mathematical models 3.1. Chance constrained programming in a stochastic environment In the stochastic probabilistic approach, the focus is on the reliability of the system’s ability to meet feasibility in an uncertain environment. The reliability is expressed as a minimum requirement on the probability of satisfying constraints [17]. Let us consider the classical linear programming model as follows: max
ct x;
s:t:
Ax P b; x P 0;
where c and x are n-vectors, b is an m-vector, and A is an m n matrix. Assume that there is uncertainty regarding the constraint matrix A and the right-hand side vector b, and that the system is required to satisfy the corresponding constraint with a probability p 2 (0, 1). Then the probabilistic linear program corresponding to the classical deterministic linear programming can be written as follows: max
ct x;
s:t:
PðAx P bÞ P p; x P 0:
In order to simplified the verification process, consider the case when m = 1, i.e., the case of a single constraint P(atx P b) P p. Assume that the vector a is deterministic while the right-hand side b is a random variable with cumulative distribution F (this assumption is suitable to our situation under mixed crude oil for CDUs demands uncertainty). Let b be such that F(b) = p. Then, the constraint P(atx P b) P p can be written as F(atx) P p or atx P b. In this case, the probabilistic program is equivalent to a standard linear program. With the above method, our crude oil scheduling problem under uncertainty is described as a chance constraint stochastic probabilistic mixed-integer nonlinear programming model. 3.2. Chance constrained programming in a fuzzy environment It is well known that the term fuzzy programming has been used in different ways in the past [12–14,17]. Here we consider fuzzy programming as a mathematical programming with fuzzy parameters in Liu et al. [15] and Liu and Iwamura’s [16] methods. A mathematical programming with fuzzy parameters should be as follows: max
f ðx; nÞ;
s:t:
g i ðx; nÞ 6 0; i ¼ 1; 2; . . . ; p;
ða1Þ
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where x is a decision vector, n is a vector of fuzzy parameters. However this fuzzy programming can not be dealt with since the meanings of max as well as of the constraints are not clear at all if n is a fuzzy vector. The fuzzy objectives and constraints are converted into their respective crisp equivalents. A chance constrained programming with fuzzy parameters can be written as follows: max
f ;
s:t:
Posfnjf ðx; nÞ P f g P b;
ða2Þ
Posfnjg i ðx; nÞ 6 0; i ¼ 1; 2; . . . ; pg P a;
where a, b are predetermined confidence levels, Pos {} denotes the possibility of the events in {}. A vector x is feasible if and only if the possibility measure of the set {njgi(x, n) 6 0, i = 1, 2, . . ., p} is at least a. For random given decision vector x, f(x, n) is obviously a fuzzy parameter. Since the values f satisfying Posff ðx; nÞ P f g P b are not unique, the objective f is defined as the maximum value of all potential values, i.e. f ¼ maxf ff jPosff ðx; nÞ P f g P bg. Or the fuzzy decision problem is formulated as a chance constrained programming with separate chance constraints as (a30 ), max
f ;
s:t:
Posfnjf ðx; nÞ P f g P b; Posfnjg i ðx; nÞ 6 0g P ai ;
ða3 0 Þ
i ¼ 1; 2; . . . ; p;
where ai are predetermined confidence levels to the respective constraints. Further, if we define g pþ1 ðx; nÞ ¼ f f ðx; nÞ, b = ap+1, then we can write (a30 ) as (a3) max
f ;
s:t:
Posfnjg i ðx; nÞ 6 0g P ai ; i ¼ 1; 2; . . . ; p; p þ 1:
ða3Þ
3.2.1. Crisp equivalents One way of solving chance constrained programming with fuzzy parameters is to convert the chance constraints (a4) to their Posfnjg i ðx; nÞ 6 0g P ai ;
i ¼ 1; 2; . . . ; p þ 1
ða4Þ
respective crisp equivalents. This process can be used only successful for some special cases. Two known results are presented as follows. Case I. Assume that the chance constraints (a4) can be written in the following form: Posfni jhi ðxÞ 6 ni g P ai ;
i ¼ 1; 2; . . . ; p þ 1;
ða5Þ
where hi(x) are functions of decision vector x (linear or nonlinear) and ni are fuzzy numbers with membership functions li(ni), i = 1, 2, . . ., p + 1, respectively. It is clear that, for any given confidence levels ai(0 6 ai 6 1), there exist some values K ai such that Posfni jK ai 6 ni g ¼ ai ;
i ¼ 1; 2; . . . ; p þ 1:
ða6Þ K 0ai
since Take notice that the possibility Posfni jK ai 6 ni g will increase if K ai are replaced by smaller numbers Posfni jK ai 6 ni g 6 Posfni jK 0ai 6 ni g. Posfni jK ai 6 ni g ¼ li ðni Þ if the membership functions li are unimodal and K ai are greater than respective modes. Because ni are fuzzy members with unimodal functions announced former, we have K ai ¼ l1 i ðai Þ, are the inverse of li, respectively. Because the values K ai satisfying (a6) are not unique, the functions l1 are where l1 i i multi-valued. If K ai are defined as the maximum values of all potential values, i.e., K ai ¼ supfKjK ¼ l1 i ðai Þg; i ¼ 1; 2; . . . ; p þ 1. Then, the crisp equivalents of chance constraints (a5) are obtained by the following forms: hi ðxÞ 6 K ai ;
i ¼ 1; 2; . . . ; p þ 1:
ða7Þ
Case II. Assume that the chance constraints (a4) can be written in the following form: Posfni jhi ðxÞ P ni g P ai ;
i ¼ 1; 2; . . . ; p þ 1:
ða8Þ l1 i ðai Þg; i
If we define K ai as the minimum values of all potential values, i.e., K ai ¼ inffKjK ¼ equivalents of chance constraints (a8) are obtained and shown by the following forms: hi ðxÞ P K ai ;
i ¼ 1; 2; . . . ; p þ 1:
¼ 1; 2; . . . ; p þ 1. Thus, the crisp
ða9Þ
3.2.2. Fuzzy simulation If the chance constraints (a4) cannot be converted into their equivalents, fuzzy simulation technology, just like a Monte Carlo simulation method, can be employed. Further details can be found in Liu et al. [15].
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Compared with the deterministic optimization model in Lee et al. [7], we use the methods mentioned in Sections 3.1 and 3.2, our crude oil scheduling problem under CDUs’ demands uncertainty can be described as follows. 3.3. Chance constrained stochastic and fuzzy programming models for crude oil short-term scheduling problem Indices i(=1, . . . , NST) number of crude oil storage tanks j, j0 (=1, . . . , NBT) number of crude oil blending and charging tanks k(=1, . . . , NCE) key components of crude oil l(=1, . . . , NCDU) number of crude distillation units t = 1, . . . , SCH time intervals of whole scheduling time v = 1, . . . , NV number of crude vessels
Variables Dj,l,t = 0–1 binary variable to denote if the crude oil mix in charging tank j charges CDU l at time t XF,v,t = 0–1 binary variable to denote if vessel v starts unloading at time t XL,v,t = 0–1 binary variable to denote if vessel v completes unloading at time t XW,v,t = 0–1 binary variable to denote if vessel v is unloading its crude oil at time t 0 Z j;j0 ;l;t ¼ 0—1 binary variable to denote if transitions exist from crude mix j to j at time t in CDU l vessel v unloading initiation time TF,v vessel v unloading completion and departure time TL,v qSB,i,j,k,t volumetric flow rate of component k from storage tank i to charging tank j at time t qBC,j,l,k,t Volumetric flow rate of component k from charging tank j to CDU l at time t volumetric flow rate of crude oil from vessel v to storage tank i at time t QVS,v,i,t volumetric flow rate of crude oil from storage tank i to charging tank j at time t QSB,i,j,t volumetric flow rate of crude oil mix from charging tank j to CDU l at time t QBC,j,l,t volume of component k in charging tank j at time t vB,j,k,t volume of crude oil in vessel v at time t VV,v,t volume of crude oil in storage tank i at time t VS,i,t volume of mixed oil in charging tank j at time t VB,j,t
Parameters CTotal total operation cost CUNLOAD,v unloading cost of vessel v per unit time interval sea waiting cost of vessel v per unit time interval CSEA,v inventory cost of storage tank i per unit time per unit volume CINVst,i inventory cost of charging tank j per unit time per unit volume CINVbt,j C SETup;j;j0 ;l changeover cost for transition from crude mix j to j0 in CDU l ~ dm stochastic or fuzzy demands of crude mix j by CDUs during the scheduling horizon j QVS,v,i,min minimum crude oil transfer volumetric flow rate from vessel v to storage tank i QVS,v,i,max maximum crude oil transfer volumetric flow rate from vessel v to storage tank i QSB,i,j,min minimum crude oil transfer volumetric flow rate from storage tank i to charging tank j QSB,i,j,max maximum crude oil transfer volumetric flow rate from storage tank i to charging tank j QBC,j,l,min minimum crude oil transfer volumetric flow rate from charging tank j to CDU l QBC,j,l,max maximum crude oil transfer volumetric flow rate from charging tank j to CDU l crude vessel v arrival time around the docking station TARR,v initial volume of crude oil in crude vessel v VV,v,0 minimum crude oil volume of storage tank i VS,i,min VS,i,max maximum crude oil volume of storage tank i initial crude oil volume of storage tank i VS,i,0 minimum mixed crude oil volume of charging tank j VB,j,min VB,j,max maximum mixed crude oil volume of charging tank j initial crude oil volume of charging tank j VB,j,0 concentration of component k in the crude oil storage tank i nS,i,k nB,j,k,min minimum concentration of component k in the crude mix of charging tank j nB,j,k,max maximum concentration of component k tin the crude mix of charging tank j initial concentration of component k tin the crude mix of charging tank j nB,j,k,0 Mathematical Formulations Minimize: The total operating cost.
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Total operating cost includes unloading cost of the vessels, cost for vessels waiting in the sea, inventory cost of storage and charging tanks and changeover cost between different mixed crude oils used for charging CDUs. NST X NV NV SCH X X X V S;i;t þ V S;i;t1 C Total ¼ C UNLoad;v ðT L;v T F;v Þ þ C SEA;v ðT F;v T ARR;v Þ þ C INVst;i 2 v¼1 v¼1 i¼1 t¼1 N N N N SCH SCH BT X BT X BT X CDU XX X V B;i;t þ V B;i;t1 þ ðC SETup;j;j0 ;l Z j;j0 ;l;t Þ: ð1Þ þ C INVbt;j 2 0 t¼1 j¼1 j¼1 t¼1 l¼1 j ¼1
Subject to: (1) Vessel arrival and departure operation rules: Each vessel arrives at the docking station for unloading only once throughout the scheduling horizon SCH X
X F;v;t ¼ 1;
v ¼ 1; . . . ; N V :
ð2aÞ
t¼1
Each vessel leaves the docking station only once throughout the scheduling horizon SCH X
X L;v;t ¼ 1;
v ¼ 1; . . . ; N V :
ð2bÞ
t¼1
Unloading initiation time is described in (2c) T F;v ¼
SCH X
tX F;v;t ;
v ¼ 1; . . . ; N V :
ð2cÞ
t¼1
Unloading completion time is described in (2d) T L;v ¼
SCH X
tX L;v;t ;
v ¼ 1; . . . ; N V :
ð2dÞ
t¼1
Each crude vessel’s unloading time should begin after arrival time T F;v P T ARR;v ;
v ¼ 1; . . . ; N V :
ð2eÞ
Duration of the vessel unloading is bounded by the initial volume of oil in the vessel divided by maximum unloading rate V V ;v;0 ; v ¼ 1; . . . ; N V : ð2fÞ T L;v T F;v P ðQ VS;v;i;max Þ Vessel in the sea cannot arrive at the docking station for unloading unless the preceding vessel leaves T F;vþ1 P T L;v ;
v ¼ 1; . . . ; N V :
ð2gÞ
Unloading is possible between time TF,v and TL,v X W;v;t 6
f X
X F;v;m ; X W;v;t 6
SCH X
X L;v;m ;
v ¼ 1; . . . ; N V ; t ¼ 1; . . . ; SCH :
ð2hÞ
m¼t
m¼1
(2) Material balance for the vessels: Crude oil in vessel v at time t = initial crude oil in the vessel v crude oil transferred from vessel v to storage tanks up to time t V V ;v;t ¼ V V;v;0
NST X t X
Q VS;v;i;m ;
v ¼ 1; . . . ; N V ; t ¼ 1; . . . ; SCH :
ð3aÞ
i¼1 m¼1
Operating constraints on crude oil transfer volumetric rate from vessel v to storage tank i at time t Q VS;v;i;min X W;v;t 6 Q VS;v;i;t 6 Q VS;v;i;max X W;v;t ;
v ¼ 1; . . . ; N V ; i ¼ 1; . . . ; N ST ; t ¼ 1; . . . ; SCH :
ð3bÞ
The volume of crude oil transferred from v vessel to storage tanks during the scheduling horizon equals to the initial crude oil volume of vessel v NST X SCH X i¼1
t¼1
Q VS;v;i;t ¼ V V;v;0 v ¼ 1; . . . ; N V :
ð3cÞ
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(3) Material balance for storage tanks: Crude oil in storage tank i at time t = initial crude oil in storage tank i + crude oil transferred from vessels to storage tank i up to time t crude oil transferred from storage tank i to charging tanks up to time t NV X t X
V S;i;t ¼ V S;i;0 þ
Q VS;v;i;m
v¼1 m¼1
NBT X t X
Q SB;i;j;m ;
i ¼ 1; . . . ; N ST ; t ¼ 1; . . . ; SCH :
ð4aÞ
j¼1 m¼1
Operating constraints on crude oil transfer rate from storage tank i to charging tank j at time t. The term P CDU Dj;l;t Þ denotes that if charging tank j is charging any CDU, there is no oil transfer from storage tank i to chargð1 Nl¼1 ing tank j Q SB;i;j;min 1
N CDU X
! 6 Q SB;i;j;t 6 Q SB;i;j;max 1
Dj;l;t
l¼1
N CDU X
! Dj;l;t ;
i ¼ 1; . . . ; N ST ; j ¼ 1; . . . ; N BT ; t ¼ 1; . . . ; SCH :
ð4bÞ
l¼1
Volume capacity limitations for storage tank i at time t V S;i;min 6 V S;i;t 6 V S;i;max ;
i ¼ 1; . . . ; N ST ; t ¼ 1; . . . ; SCH :
ð4cÞ
(4) Material balance for charging tanks: Crude oil mix in changing tank j at time t = initial mixed oil in storage tank j + crude oil transferred from storage tanks to charging tank j up to time t crude oil mix j charged into CDUs up to time t. V B;j;t ¼ V B;j;0 þ
NST X t X
Q SB;i;j;m ¼
v¼1 m¼1
N CDU X
t X
Q BC;j;l;m ;
j ¼ 1; . . . ; N BT ; t ¼ 1; . . . ; SCH :
ð5aÞ
l¼1 m¼1
Operating constraints on mixed oil transfer rate from charging tank j to CDU l at time t Q BC;j;l;min Dj;l;t 6 Q BC;j;l;t 6 Q BC;j;l;max Dj;l;t ;
j ¼ 1; . . . ; N BT ; l ¼ 1; . . . ; N CDU ; t ¼ 1; . . . ; SCH :
ð5bÞ
Volume capacity limitations for storage tank j at time t. V B;j;min 6 V B;j;t 6 V B;j;max ;
i ¼ 1; . . . ; N BT ; t ¼ 1; . . . ; SCH :
ð5cÞ
Case I. Stochastic demands. Total production amount of crude oil mix j should meet the random demands of crude mix j for CDUs during the scheduling horizon ( ) N SCH CDU X X ~ Q BC;j;l;t P dmj P aj ; j ¼ 1; . . . ; N BT ; ð5d0 Þ Pr t¼1
l¼1
~ j are independent continuous random variables with the cumulative distribution F , F (b ) = a , a 2 (0, 1), a are probwhere dm j j j j j j ability degree of satisfaction of the constraints. Then we can change the formulation (5d0 ) to (5d) N SCH CDU X X l¼1
Q BC;j;l;t P bj ;
j ¼ 1; . . . ; N BT :
ð5dÞ
t¼1
Case II. Fuzzy demands. Total production amount of crude oil mix j should meet the fuzzy demands of crude mix j for CDUs during the scheduling horizon ( ) N SCH CDU X X ~ j P aj ; j ¼ 1; . . . ; N BT ; Pos Q P dm ð5d00 Þ BC;j;l;t
l¼1
t¼1
~ j are fuzzy members with membership functions l , a are a predetermined confidence levels to the respective conwhere dm j i straints. If we define K aj as the minimum values of all potential values, i.e., K aj ¼ inffKjK ¼ l1 j ðaj Þg; j ¼ 1; 2; . . . ; N BT Then we can change the formulation (5d00 ) to (5d000 ) N SCH CDU X X l¼1
Q BC;j;l;t P K aj ;
j ¼ 1; . . . ; N BT :
ð5d000 Þ
t¼1
(5) Material balance for component k in charging tanks: Volume of component k in charging tank j at time t = initial component k in charging tank j + component k in crude oil transferred from storage tanks to charging tank j up to time t component k in crude oil mix j transferred to CDUs up to time t
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vB;j;t ¼ vB;j;0 þ
NST t X X m¼1
i¼1
qSB;i;j;m
N CDU X
! qBC;j;l;m ;
j ¼ 1; . . . ; N BT ; k ¼ 1; . . . ; N CE ; t ¼ 1; . . . ; SCH :
ð6aÞ
t¼1
Operating constraints on volumetric flow rate of component k from storage tank i to charging tank j qSB;i;j;k;t ¼ Q SB;i;j;t nS;i;k i ¼ 1; . . . ; N ST ;
j ¼ 1; . . . ; N BT ; k ¼ 1; . . . ; N CE ; t ¼ 1; . . . ; SCH :
ð6bÞ
Operating constraints on volumetric flow rate of component k from charging tank j to CDU l Q BC;j;l;t nB;j;k;min 6 qBC;j;l;k;t 6 Q BC;j;l;t nB;j;k;max ;
l ¼ 1; . . . ; N CDU ; j ¼ 1; . . . ; N BT ; k ¼ 1; . . . ; N CE ; t ¼ 1; . . . ; SCH :
ð6cÞ
Volume capacity limitations for component k in charging tank j at time t V B;j;t nB;j;k;min 6 vB;j;k;t 6 V B;j;t nB;j;k;max ;
l ¼ 1; . . . ; N CDU ; j ¼ 1; . . . ; N BT ; k ¼ 1; . . . ; N CE ; t ¼ 1; . . . ; SCH :
ð6dÞ
Table 1 Data for case study Scheduling horizon of equal duration time intervals [day]
8
Number of vessels tanks Vessels
Arrival time
2 Crude amount
Sulfur concentration
Vessel 1 Vessel 2
1 5
100 (104 bbl) 100 (104 bbl)
0.01 0.06
Number of storage tanks Storage tanks
Capacity
2 Initial crude amount
Sulfur concentration
4
4
Tank 1 Tank 2
0–100 (10 bbl) 0–100 (104 bbl)
25 (10 bbl) 75 (104 bbl)
Number of charging tanks Charging tanks
Capacity
2 Initial mix oil amount
Sulfur concentration
Tank 1 Tank 2
0–100 (104 bbl) 0–100 (104 bbl)
50 (104 bbl) 50 (104 bbl)
0.015–0.025 0.045–0.055
Number of CDU
0.01 0.06
1 Unloading cost: 8 103 [$/day] Sea waiting cost: 5 103 [$/day]
Unit costs for vessel operation
Storage tank: 0.05 [$/(day 103) ] Charging tank: 0.08 [$/(day 103)]
Unit costs for tank inventory
50 (103) $
Unit changeover cost Demand of mixed oil by CDU
Oil mix 1: g1, Oil mix 2: g2
bbl: liquid measure, bbl barrel.
Table 2 Main results for case study in stochastic environment a1 a2 CTOTAL CUNLOAD CSEA CINVst CINVbt CSETup (TF,1, TL,1) (TF,2, TL,2)
0.9 0.9 214.933 32 15 41.340 26.593 100 (3, 4) (6, 7)
0.8 0.8 214.998 32 15 41.416 26.582 100 (3, 4) (6, 7)
0.7 0.7 215.069 32 10 46.499 26.570 100 (2, 3) (6, 7)
0.6 0.6 215.145 32 10 46.588 26.557 100 (2, 3) (6, 7)
0.5 0.5 215.225 32 15 41.682 26.543 100 (3, 4) (6, 7)
0.4 0.4 215.31 32 10 46.781 26.529 100 (2, 3) (6, 7)
0.3 0.3 215.398 32 10 46.884 26.514 100 (2, 3) (6, 7)
a1 a2 CTOTAL CUNLOAD CSEA CINVst CINVbt CSETup (TF,1, TL,1) (TF,2, TL,2)
0.9 0.1 215.292 32 10 46.754 26.538 100 (2, 3) (6, 7)
0.8 0.2 215.269 32 10 46.729 26.540 100 (2, 3) (6, 7)
0.7 0.3 215.251 32 10 46.709 26.542 100 (2, 3) (6, 7)
0.6 0.4 215.236 32 10 46.693 26.543 100 (2, 3) (6, 7)
0.5 0.5 215.225 32 15 41.682 26.543 100 (3, 4) (6, 7)
0.4 0.6 215.218 32 15 41.676 25.543 100 (3, 4) (6, 7)
0.3 0.7 215.216 32 15 41.674 25.542 100 (3, 4) (6, 7)
CXXX – cost(103$), TXX– time intervals (day), aX – predetermined probability degree of satisfaction of the constraints.
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Table 3 Main results for case study in fuzzy environment a1(ka1) a2(ka2) CTOTAL CUNLOAD CSEA CINVst CINVbt CSETup (TF,1, TL,1) (TF,2, TL,2)
0.9(94.5) 0.9(94.47) 217.092 32 10 48.866 26.226 100 (2, 3) (6, 7)
0.8(94.0) 0.8(93.88) 217.350 32 10 49.168 26.182 100 (2, 3) (6, 7)
0.7(93.5) 0.7(93.22) 217.629 32 10 49.493 26.135 100 (2, 3) (6, 7)
0.6(93.0) 0.6(92.45) 217.934 32 10 49.850 26.085 100 (2, 3) (6, 7)
0.5(92.5) 0.5(91.53) 218.276 32 10 50.248 26.028 100 (2, 3) (6, 7)
0.4(92.0) 0.4(90.42) 218.671 32 10 50.708 25.963 100 (2, 3) (6, 7)
a1(ka1) a2(ka2) CTOTAL CUNLOAD CSEA CINVst CINVbt CSETup (TF,1, TL,1) (TF,2, TL,2)
0.9(94.5) 0.4(90.42) 218.146 32 10 50.083 26.063 100 (2, 3) (6, 7)
0.8(94.0) 0.5(91.53) 217.961 32 10 49.873 26.088 100 (2, 3) (6, 7)
0.7(93.5) 0.6(92.45) 217.829 32 10 49.725 26.105 100 (2, 3) (6, 7)
0.6(93.0) 0.7(93.22) 217.734 32 10 49.618 26.115 100 (2, 3) (6, 7)
0.5(92.5) 0.8(93.88) 217.665 32 10 49.543 26.122 100 (2, 3) (6, 7)
0.4(92.0) 0.9(94.47) 217.617 32 10 49.491 26.126 100 (2, 3) (6, 7)
CXXX, TXX – the same as Table 2, aX – predetermined possibility degree of satisfaction of the constraints, Kax – the minimum demand of all potential values.
10000 bbl/day
50 40 30 20 10 0 1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
10000 bbl/day
50 40 30 20 10 0
Fig. 1. Flow rate of crude oil from vessels to storage tanks.
Eqs. (6c) and (6d) are linear formulations translated from non-convex bilinear equations (60 ), (600 ), (6000 ) through the method of Quesada and Grossmann [18] nB;j;k;min 6 nB;j;k;t 6 nB;j;k;max ; qBC;j;l;k;t ¼ nB;j;k;t Q BC;j;l;t ; vB;j;k;t ¼ nB;j;k;t V B;j;t ;
j ¼ 1; . . . ; N BT ; k ¼ 1; . . . ; N CE ; t ¼ 1; . . . ; SCH ;
l ¼ 1; . . . ; N CDU ; j ¼ 1; . . . ; N BT ; k ¼ 1; . . . ; N CE ; t ¼ 1; . . . ; SCH ;
j ¼ 1; . . . ; N BT ; k ¼ 1; . . . ; N CE ; t ¼ 1; . . . ; SCH :
ð60 Þ ð600 Þ ð6000 Þ
(6) Operating rules for crude oil charging:Charging tank j can charge at most one CDU at any time t N CDU X
Dj;l;t 6 1;
j ¼ 1; . . . ; N BT ; t ¼ 1; . . . ; SCH:
ð7aÞ
l¼1
CDU l can be charged only by one charging tank at any time t N CDU X Dj;l;t 6 1; j ¼ 1; . . . ; N BT ; t ¼ 1; . . . ; SCH: l¼1
ð7bÞ
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If CDU l is charged by crude oil mix j at time t 1 and charged by j0 at time t, changeover cost exist Z j;j0 ;l;t P Dj0 ;l;t þ Dj;l;t1 1;
0
0
j; j ðj–j Þ ¼ 1; . . . ; N BT ; l ¼ 1; . . . ; N CDU ; t ¼ 2; . . . ; SCH :
ð7cÞ
4. A case study The detail data of the case study can be found in Table 1 which is compared with the data in Lee et al. [7]. The model studied contains 267 continuous variables, 68 binary variables and 320 constraints. The model was programmed and solved by LINGO 8.0 [19]. The computing time for once simulation for the case study is less than 10 s. Table 2 and 3 shows the main results in stochastic and fuzzy environment, respectively. Case I. Stochastic demands. When g1 and g2 are normally distributed continuous stochastic variables that follow N(95 104, 52 104) bbl, we gain the following results in Table 2.
100
10000 bbl
80 60 40 20 0 0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
100
10000 bbl
80 60 40 20
Fig. 2. Inventory status of storage tanks.
50
10000 bbl
40 30 20 10 0 0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
100
10000 bbl
80 60 40 20 0
Fig. 3. Inventory status of charging tanks.
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Case II. Fuzzy demands. The fuzzy parameter demand for oil mix 1(g1) is assumed to have triangular membership function as follows: 8 90 6 g1 6 95; > < ðg1 90Þ=5 l1 ðg1 Þ ¼ ð100 g1 Þ=5 95 6 g1 6 100; ð104 bblÞ; > : 0 others; The fuzzy parameter demand for oil mix 2(g2) is assumed to have membership function as follows: 1 l2 ðg2 Þ ¼ exp jg2 95j ; 90 6 g2 6 100; ð104 bblÞ: 5 Then, we gain the following results in Table 3. From the results of Table 2 we can find that if we increase (a1, a2) which represent the probability degree of satisfaction of random demands constraints, the optimal minimum objective cost decrease and vice versa. From the results of Table 3 we can find that if we increase (a1, a2) which represent the predetermined confidence levels of fuzzy demands constraints, the optimal minimum objective cost decrease and vice versa.
0. 025 0. 02
0 0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0. 055 0. 05 0. 045
0
Fig. 4. Component concentration in charging tanks.
10000 bbl/day
50 40 30 20 10 0 1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
10000 bbl/day
50 40 30 20 10 0
Fig. 5. Flow rate of mixed oil from storage tanks to one CDU.
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Other main results of once simulation (a1 = 0.4, a2 = 0.9) in fuzzy environment can be found from Figs. 1–5. Similar results can be gained in stochastic environment. Fig. 1 shows the unloading rate of crude oil in vessel 1 and vessel 2 during the scheduling horizon (above for vessel 1, below for vessel 2); Fig. 2 shows the inventory variation status of storage tank 1 and storage tank 2 during the scheduling horizon (above for storage tank 1, below for storage tank 2); Fig. 3 shows the inventory status of charging tank 1 and charging tank 2 during the scheduling horizon (above for charging tank 1, below for charging tank 2); Fig. 4 shows the component concentration in charging tank 1 and charging tank 2 during the scheduling horizon (above for charging tank 1, below for charging tank 2); Fig. 5 shows the charging rate of mixed oil from charging tank 1 and charging tank 2 to one CDU during the scheduling horizon (above for charging tank 1, below for charging tank 2). From the results of Figs. 1–5 we gained, we observe that all the operation rules and constraints have been satisfied properly, and we find the global optimal results. 5. Conclusions Under the current situations of unsteady supply of crude oil, variations of feedstock qualities and yield levels, pressure of intense time, low inventory flexibility, etc., it is very important for refinery companies to assess the extra cost of these changes during making their short-term crude oil scheduling operations. We study one type of uncertainties in crude oil scheduling problem – crude oil blends demands uncertainty, which result in crude oil feed flow rates variation during scheduling time horizon. Two chance constrained mixed-integer nonlinear stochastic and fuzzy programming models are presented to evaluate the extra cost of uncertainty. After the models are converted into crisp equivalent MILP models, the calculation processes are simplified greatly. The branch and bound method in LINGO 8.0 is used to solve the proposed equivalent formulation in a case study which has 267 continuous variables, 68 binary variables and 320 constraints. The global optimal results show that this model can work with feasibility and have the possibility to be further used to solve industrial size problem. Acknowledgements Financial support from the National Natural Science Foundation of China (Grant Nos. 60674075, 60774078), the Key Technology Program of Shanghai Municipal Science and Technology Commission (Grant No. 04dz11008), the Research Fund for Outstanding Youth Teachers of East China University of Science and Technology (Grant No. YH0157117) and the Fund for Shanghai Leading Academic Discipline Project (No. B504) is gratefully appreciated. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
J.M. Pinto, M. Joly, L.F.L. Moro, Planning and scheduling models for refinery operations, Comput. Chem. Eng. 24 (9–10) (2000) 2259–2276. Bonner and Moore Management Science. RPMS (Refinery and Petrochemical Modelling System). A System Description, Houston, TX, 1979. Bechtel Corporation. PIMS (Process Industry Modelling System). Users’ Manual, Version 6.0, Houston, TX, 1993. L.F.L. Moro, J.M. Pinto, Mixed-integer programming approach for short-term crude oil scheduling, Ind. Eng. Chem. Res. 43 (1) (2004) 85–94. Z. Jia, M. Ierapetritou, J.D. Kelly, Refinery short-term scheduling using continuous time formulation: crude-oil operations, Ind. Eng. Chem. Res. 42 (13) (2003) 3085–3097. N. Shah, Mathematical programming technique for crude oil scheduling, Comput. Chem. Eng. 20 (Suppl. 2) (1996) S1227–S1232. H. Lee, J.M. Pinto, I.E. Grossmann, S. Park, Mixed-integer linear programming model for refinery short-term scheduling of crude oil unloading with inventory management, Industrial Eng. Chem. Res. 35 (5) (1996) 1630–1641. P. Chandra Prakash Reddy, I.A. Karimi, R. Srinivasan, A new continuous-time formulation for scheduling crude oil operations, Chem. Eng. Sci. 59 (6) (2004) 1325–1341. G. Chryssolouris, N. Papakostas, D. Mourtzis, Refinery short-term scheduling with tank farm, inventory and distillation management: an integrated simulation-based approach, Eur. J. Oper. Res. 166 (3) (2005) 812–827. M.L. Liu, N.V. Sahinidis, Optimization in process planning under uncertainty, Ind. Eng. Chem. Res. 35 (11) (1996) 4154–4165. A. Charnes, W.W. Cooper, Chance-constrained programming, Manage. Sci. 6 (1959) 73–79. N.V. Sahinidis, Optimization under uncertainty: state-of-the-art and opportunities, Comput. Chem. Eng. 28 (6–7) (2004) 971–983. M.L. Liu, N.V. Sahinidis, Process planning in a fuzzy environment, Eur. J. Oper. Res. 100 (1) (1997) 142–169. H.J. Zimmermann, Fuzzy Set Theory and its Application, Kluwer Academic Publishers, Boston, 1991. B.D. Liu, R.Q. Zhao, G. Wang, Uncertain Programming with Applications, Tsinghua University Press, 2003. pp. 61–77 (in Chinese). B.D. Liu, K. Iwamura, Chance constrained programming with fuzzy parameters, Fuzzy Sets Syst. 94 (2) (1998) 227–237. N.V. Sahinidis, Optimization under uncertainty: state-of-the-art and opportunities, Comput. Chem. Eng. 28 (6–7) (2004) 971–983. I. Quesada, I.E. Grossmann, Global optimization of bilinear process networks with multicomponent flows, Comput. Chem. Eng. 19 (12) (1995) 1219– 1242. J.X. Xie, Y. Xue, Optimization Modelling and LINDO/LINGO Software, Tsinghua University Press, 2005 (in Chinese).
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Chance constrained programming models for refinery short-term crude oil scheduling problem Cuiwen Cao a,*, Xingsheng Gu a, Zhong Xin b a b
Research Institute of Automation, East China University of Science and Technology, Shanghai 200237, China School of Chemical Engineering, East China University of Science and Technology, Shanghai 200237, China
a r t i c l e
i n f o
Article history: Received 1 April 2006 Received in revised form 1 March 2008 Accepted 4 March 2008 Available online 11 April 2008
Keywords: Short-term crude oil scheduling problem Demands uncertainty Chance constrained mixed-integer nonlinear programming Stochastic Fuzzy
a b s t r a c t The main objective of this work is to put forward chance constrained mixed-integer nonlinear stochastic and fuzzy programming models for refinery short-term crude oil scheduling problem under demands uncertainty of distillation units. The scheduling problem studied has characteristics of discrete events and continuous events coexistence, multistage, multiproduct, nonlinear, uncertainty and large scale. At first, the two models are transformed into their equivalent stochastic and fuzzy mixed-integer linear programming (MILP) models by using the method of Quesada and Grossmann [I. Quesada, I E. Grossmann, Global optimization of bilinear process networks with multicomponent flows, Comput. Chem. Eng. 19 (12) (1995) 1219–1242], respectively. After that, the stochastic equivalent model is converted into its deterministic MILP model through probabilistic theory. The fuzzy equivalent model is transformed into its crisp MILP model relies on the fuzzy theory presented by Liu and Iwamura [B.D. Liu, K. Iwamura, Chance constrained programming with fuzzy parameters, Fuzzy Sets Syst. 94 (2) (1998) 227–237] for the first time in this area. Finally, the two crisp MILP models are solved in LINGO 8.0 based on scheduling time discretization. A case study which has 267 continuous variables, 68 binary variables and 320 constraints is effectively solved with the solution approaches proposed. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction The short-term oil-refinery scheduling problem is one of the most challenging problems in operational research due to the complexity of the refinery scheduling operations [1]. In contrast to relatively ample availability of commercial production planning software, such as RPMS (refinery and petrochemical modelling system – [2]) and PIMS (process industry modelling system – [3]), there are still no commercial tools for short-term refinery scheduling optimization problems [4]. A lot of refineries have to rely on their experience and develop their own in-house tools to deal with scheduling operations. Typically, the overall problem for optimization of oil-refinery scheduling operations consists of three parts [5]. The first part studies the crude oil unloading, mixing, transferring and multilevel crude oil inventory control process. The second part deals with fractionation, reaction units scheduling and a variety of intermediate product tanks control. The third part involves the finished product blending and distributing process. In this paper, we focus on the first part problem. It is one critical component of the overall problem for refinery scheduling operations. When a refinery is located along seashore, its crude oil scheduling problem includes crude oil unloading process from the vessels to the storage tanks, transferring process from storage tanks to the charging tanks (where several crude oils are mixed), and charging process from the charging tanks to the crude oil distillation units (CDUs). In the literature, * Corresponding author. E-mail address: [email protected] (C. Cao). S0307-904X/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2008.03.022
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deterministic mathematical programming technologies are developed in this area from the nineties of last century. Shah [6] presented a two-stage MILP model for a monthly crude oil transferring plan. Its objective was to minimize the crude oil heels of charging tanks. Lee et al. [7] developed a MILP model to solve crude oil short-term scheduling problem which objective was minimizing the total operating cost. Pinto et al. [1] proposed a short-term MINLP model from fixed crude oil pipelines to CDUs, which objective was maximizing the production while minimizing the number of tanks used. These models [1,6,7] they developed rely on time discretization representations. In recent years, mathematical models of crude oil scheduling operations emphasize on continuous-time formulation to shorten the gap between theory research and real-world operation. Jia et al. [5] considered the same problem as Lee et al. [7], they employed the state-task-network (STN) continuous-time representation. Moro and Pinto [4] restudied their former work [1], the new approach they adopted is to define variablelength global time slots, and the objective function is to maximize the crude oil distillation unit feed flow rate while minimizing the cost associated with tank operation. Chandra Prakash Reddy et al. [8] presented a complete continuous-time MILP model for the crude oil short-term scheduling problem as Pinto et al. [1], they also reported the comparison of discrete-time versus continuous-time formulations for this problem. Chryssolouris et al. [9] studied the similar problem as Lee et al. [7], they took the temperature cut-points into consideration for each distillation unit. However, all these discussions mentioned above adopt deterministic mathematical programming methods to solve the crude oil scheduling problem. The data in their models are assumed to be deterministic. In real world, the situations of crude oil operation process are uncertain since the dynamic nature of the environment. The variations of certain parameters, such as feedstock qualities and yield levels are of great importance in short-term scheduling operation decisions. Refinery companies must assess the impacts of these important changes. In this paper, we study one type of uncertainties in crude oil scheduling problem – crude oil blends demands uncertainty, which result in crude oil feed flow rates variation during scheduling time horizon. Two types distinct research philosophies – stochastic programming and fuzzy programming – are used. Stochastic programming deals with optimization problems whose parameters take values from given discrete or continuous probability distributions [10]. One type of the solution methods is the chance constraint programming method pioneered by Charnes and Cooper [11]. They assume that stochastic objective functions and stochastic constraints will hold with at least some probability levels, and the chances are represented by the probabilities that the objectives and constraints are satisfied. Concerning with the fuzzy programming method, the term fuzzy programming has been used in different ways in Sahinidis [12], Liu and Sahinidis [13], and Zimmermann [14]. Although the theory of Liu et al. [15] and Liu and Iwamura [16] has been used for many applications, we use it for the first time to solve short-term crude oil scheduling problem in this area. Our fuzzy programming model is defined as a mathematical programming with those fuzzy parameters analogous to chance constrained programming with stochastic parameters. We assume that the constraints will hold with at least a possibility level a, and the chance represents the possibility that the constraints are satisfied. We introduce the definition of chance constrained fuzzy programming we used, and the crisp equivalents of fuzzy chance constraints for two special cases. We use one of them to solve our crude oil short scheduling problem. Our forecast demands parameters are assumed as to be fuzzy with unimodal membership functions. The paper is organized as follows: Section 2 states the definition of short-term crude oil scheduling problem. In Section 3, the chance constrained stochastic and fuzzy programming methods are introduced, respectively. The detailed mathematical formulations for crude oil short-term scheduling problem with two methods are presented. Then the above technologies are applied to a case study in Section 4. In the last section, we put forward the conclusions.
2. Problem definition 2.1. Problem statement In a typical refinery, a series of operations for turning crude oils into higher value end (gasoline, diesel, etc.) products begin with crude oil unloading, mixing, transferring process. After that, the distillation process takes place in CDUs, which separate the charged oil into consecutive lighter and heavier fractions with different boiling points. Most of the distillation products need to be further processed in vacuum distillation units, fluid catalytic cracking units, coking units or hydro-treating units, etc. The refined products are then mixed in gasoline/diesel/fuel oil/asphalt/. . . pools for making the final refinery products. Planning determines the volumes of different feedstock to be processed, the mix and amounts of products to be produced in response to product requirements over several months. Scheduling deals with the timing of the plant operations to realize the plan. In most refineries, their CDUs are built in different year, so they can only process feedstock with different properties. These properties involve process ability, yields of some very high value products, impurities or concentrations of some key components, etc. which influence the downstream processing. The distillation products from all the CDUs are processed together in following steps, the crude oil feed rates in all CDUs must be maintained compatible. Refinery schedulers continuously watch the properties of feedstock and the first distillation products of their CDUs, if some events happen, for example the yield of heavier distillation fractions is too high, the valuable lighter fractions are all taken away to produce high value products which means the production recipes are changed, or sulphur concentration or acid value level is fluctuant abnormally, they change the crude oil supply feed flow rates or type of the CDU disposed, then the others must be changed accordingly and immediately. These situations happen frequently under
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current unsteady supply of crude oil, high pressure of intense time and low inventory flexibility in a refinery. Nowadays, schedulers implement this job based on their years of experience. In this paper, we assess the potential cost of this type uncertainty. To focus on studying the influence of uncertainty, our uncertainty models rely on discrete-time representations which assume all events are forced to coincide with one of the interval boundaries. The short-term crude oil scheduling problem considered here is related to determining port details of the crude oil allocation process from discrete vessels to storage tanks, refinery details of the continuous crude oil allocation process from storage tanks to charging tanks (used for crude oil blending) and the mixed oil from charging tanks to CDUs (used for distillation). Under the given arrival times of vessels, equipment capacity limitations, key component concentration ranges and the uncertain demands for CDUs, the problem studied is to determine the following variables to minimize the operating costs (which include unloading cost for crude oil vessels, cost for vessels waiting in the sea, inventory cost for storage and charging tanks and changeover cost between different mixed crude oils for CDUs): waiting time of each vessel at sea; unloading duration and flow rates from vessels to storage tanks; crude oil transfer duration and flow rates from storage tanks to charging tanks; inventory variation in storage tanks and charging tanks; concentration levels of key components in storage tanks and charging tanks. 2.2. Operation rules and assumptions The operation rules should be obeyed are as follows: (1) the refinery uses only one docking station, a new arriving vessel has to wait in the sea if an anterior vessel does not leave the docking station; (2) while a charging tank is charging CDU, crude oil from the storage tanks cannot be fed into the charging tank and vice versa; (3) each charging tank can only charge one CDU at one time interval; (4) each CDU can only be charged by one charging tank at one time interval; (5) CDUs must be operated continuously throughout the scheduling time horizon. Following assumptions are proposed in this paper: (1) the scheduling time horizon is divided into time periods of equal duration, so representations are formulated as discrete-time MINLP models; (2) all events are forced to coincide with one of the interval boundaries; (3) perfect mixing is assumed in all tanks; (4) the times required for a CDU mode change are neglected; (5) all vessels unloading and waiting costs are assumed common. 3. Mathematical models 3.1. Chance constrained programming in a stochastic environment In the stochastic probabilistic approach, the focus is on the reliability of the system’s ability to meet feasibility in an uncertain environment. The reliability is expressed as a minimum requirement on the probability of satisfying constraints [17]. Let us consider the classical linear programming model as follows: max
ct x;
s:t:
Ax P b; x P 0;
where c and x are n-vectors, b is an m-vector, and A is an m n matrix. Assume that there is uncertainty regarding the constraint matrix A and the right-hand side vector b, and that the system is required to satisfy the corresponding constraint with a probability p 2 (0, 1). Then the probabilistic linear program corresponding to the classical deterministic linear programming can be written as follows: max
ct x;
s:t:
PðAx P bÞ P p; x P 0:
In order to simplified the verification process, consider the case when m = 1, i.e., the case of a single constraint P(atx P b) P p. Assume that the vector a is deterministic while the right-hand side b is a random variable with cumulative distribution F (this assumption is suitable to our situation under mixed crude oil for CDUs demands uncertainty). Let b be such that F(b) = p. Then, the constraint P(atx P b) P p can be written as F(atx) P p or atx P b. In this case, the probabilistic program is equivalent to a standard linear program. With the above method, our crude oil scheduling problem under uncertainty is described as a chance constraint stochastic probabilistic mixed-integer nonlinear programming model. 3.2. Chance constrained programming in a fuzzy environment It is well known that the term fuzzy programming has been used in different ways in the past [12–14,17]. Here we consider fuzzy programming as a mathematical programming with fuzzy parameters in Liu et al. [15] and Liu and Iwamura’s [16] methods. A mathematical programming with fuzzy parameters should be as follows: max
f ðx; nÞ;
s:t:
g i ðx; nÞ 6 0; i ¼ 1; 2; . . . ; p;
ða1Þ
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where x is a decision vector, n is a vector of fuzzy parameters. However this fuzzy programming can not be dealt with since the meanings of max as well as of the constraints are not clear at all if n is a fuzzy vector. The fuzzy objectives and constraints are converted into their respective crisp equivalents. A chance constrained programming with fuzzy parameters can be written as follows: max
f ;
s:t:
Posfnjf ðx; nÞ P f g P b;
ða2Þ
Posfnjg i ðx; nÞ 6 0; i ¼ 1; 2; . . . ; pg P a;
where a, b are predetermined confidence levels, Pos {} denotes the possibility of the events in {}. A vector x is feasible if and only if the possibility measure of the set {njgi(x, n) 6 0, i = 1, 2, . . ., p} is at least a. For random given decision vector x, f(x, n) is obviously a fuzzy parameter. Since the values f satisfying Posff ðx; nÞ P f g P b are not unique, the objective f is defined as the maximum value of all potential values, i.e. f ¼ maxf ff jPosff ðx; nÞ P f g P bg. Or the fuzzy decision problem is formulated as a chance constrained programming with separate chance constraints as (a30 ), max
f ;
s:t:
Posfnjf ðx; nÞ P f g P b; Posfnjg i ðx; nÞ 6 0g P ai ;
ða3 0 Þ
i ¼ 1; 2; . . . ; p;
where ai are predetermined confidence levels to the respective constraints. Further, if we define g pþ1 ðx; nÞ ¼ f f ðx; nÞ, b = ap+1, then we can write (a30 ) as (a3) max
f ;
s:t:
Posfnjg i ðx; nÞ 6 0g P ai ; i ¼ 1; 2; . . . ; p; p þ 1:
ða3Þ
3.2.1. Crisp equivalents One way of solving chance constrained programming with fuzzy parameters is to convert the chance constraints (a4) to their Posfnjg i ðx; nÞ 6 0g P ai ;
i ¼ 1; 2; . . . ; p þ 1
ða4Þ
respective crisp equivalents. This process can be used only successful for some special cases. Two known results are presented as follows. Case I. Assume that the chance constraints (a4) can be written in the following form: Posfni jhi ðxÞ 6 ni g P ai ;
i ¼ 1; 2; . . . ; p þ 1;
ða5Þ
where hi(x) are functions of decision vector x (linear or nonlinear) and ni are fuzzy numbers with membership functions li(ni), i = 1, 2, . . ., p + 1, respectively. It is clear that, for any given confidence levels ai(0 6 ai 6 1), there exist some values K ai such that Posfni jK ai 6 ni g ¼ ai ;
i ¼ 1; 2; . . . ; p þ 1:
ða6Þ K 0ai
since Take notice that the possibility Posfni jK ai 6 ni g will increase if K ai are replaced by smaller numbers Posfni jK ai 6 ni g 6 Posfni jK 0ai 6 ni g. Posfni jK ai 6 ni g ¼ li ðni Þ if the membership functions li are unimodal and K ai are greater than respective modes. Because ni are fuzzy members with unimodal functions announced former, we have K ai ¼ l1 i ðai Þ, are the inverse of li, respectively. Because the values K ai satisfying (a6) are not unique, the functions l1 are where l1 i i multi-valued. If K ai are defined as the maximum values of all potential values, i.e., K ai ¼ supfKjK ¼ l1 i ðai Þg; i ¼ 1; 2; . . . ; p þ 1. Then, the crisp equivalents of chance constraints (a5) are obtained by the following forms: hi ðxÞ 6 K ai ;
i ¼ 1; 2; . . . ; p þ 1:
ða7Þ
Case II. Assume that the chance constraints (a4) can be written in the following form: Posfni jhi ðxÞ P ni g P ai ;
i ¼ 1; 2; . . . ; p þ 1:
ða8Þ l1 i ðai Þg; i
If we define K ai as the minimum values of all potential values, i.e., K ai ¼ inffKjK ¼ equivalents of chance constraints (a8) are obtained and shown by the following forms: hi ðxÞ P K ai ;
i ¼ 1; 2; . . . ; p þ 1:
¼ 1; 2; . . . ; p þ 1. Thus, the crisp
ða9Þ
3.2.2. Fuzzy simulation If the chance constraints (a4) cannot be converted into their equivalents, fuzzy simulation technology, just like a Monte Carlo simulation method, can be employed. Further details can be found in Liu et al. [15].
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Compared with the deterministic optimization model in Lee et al. [7], we use the methods mentioned in Sections 3.1 and 3.2, our crude oil scheduling problem under CDUs’ demands uncertainty can be described as follows. 3.3. Chance constrained stochastic and fuzzy programming models for crude oil short-term scheduling problem Indices i(=1, . . . , NST) number of crude oil storage tanks j, j0 (=1, . . . , NBT) number of crude oil blending and charging tanks k(=1, . . . , NCE) key components of crude oil l(=1, . . . , NCDU) number of crude distillation units t = 1, . . . , SCH time intervals of whole scheduling time v = 1, . . . , NV number of crude vessels
Variables Dj,l,t = 0–1 binary variable to denote if the crude oil mix in charging tank j charges CDU l at time t XF,v,t = 0–1 binary variable to denote if vessel v starts unloading at time t XL,v,t = 0–1 binary variable to denote if vessel v completes unloading at time t XW,v,t = 0–1 binary variable to denote if vessel v is unloading its crude oil at time t 0 Z j;j0 ;l;t ¼ 0—1 binary variable to denote if transitions exist from crude mix j to j at time t in CDU l vessel v unloading initiation time TF,v vessel v unloading completion and departure time TL,v qSB,i,j,k,t volumetric flow rate of component k from storage tank i to charging tank j at time t qBC,j,l,k,t Volumetric flow rate of component k from charging tank j to CDU l at time t volumetric flow rate of crude oil from vessel v to storage tank i at time t QVS,v,i,t volumetric flow rate of crude oil from storage tank i to charging tank j at time t QSB,i,j,t volumetric flow rate of crude oil mix from charging tank j to CDU l at time t QBC,j,l,t volume of component k in charging tank j at time t vB,j,k,t volume of crude oil in vessel v at time t VV,v,t volume of crude oil in storage tank i at time t VS,i,t volume of mixed oil in charging tank j at time t VB,j,t
Parameters CTotal total operation cost CUNLOAD,v unloading cost of vessel v per unit time interval sea waiting cost of vessel v per unit time interval CSEA,v inventory cost of storage tank i per unit time per unit volume CINVst,i inventory cost of charging tank j per unit time per unit volume CINVbt,j C SETup;j;j0 ;l changeover cost for transition from crude mix j to j0 in CDU l ~ dm stochastic or fuzzy demands of crude mix j by CDUs during the scheduling horizon j QVS,v,i,min minimum crude oil transfer volumetric flow rate from vessel v to storage tank i QVS,v,i,max maximum crude oil transfer volumetric flow rate from vessel v to storage tank i QSB,i,j,min minimum crude oil transfer volumetric flow rate from storage tank i to charging tank j QSB,i,j,max maximum crude oil transfer volumetric flow rate from storage tank i to charging tank j QBC,j,l,min minimum crude oil transfer volumetric flow rate from charging tank j to CDU l QBC,j,l,max maximum crude oil transfer volumetric flow rate from charging tank j to CDU l crude vessel v arrival time around the docking station TARR,v initial volume of crude oil in crude vessel v VV,v,0 minimum crude oil volume of storage tank i VS,i,min VS,i,max maximum crude oil volume of storage tank i initial crude oil volume of storage tank i VS,i,0 minimum mixed crude oil volume of charging tank j VB,j,min VB,j,max maximum mixed crude oil volume of charging tank j initial crude oil volume of charging tank j VB,j,0 concentration of component k in the crude oil storage tank i nS,i,k nB,j,k,min minimum concentration of component k in the crude mix of charging tank j nB,j,k,max maximum concentration of component k tin the crude mix of charging tank j initial concentration of component k tin the crude mix of charging tank j nB,j,k,0 Mathematical Formulations Minimize: The total operating cost.
C. Cao et al. / Applied Mathematical Modelling 33 (2009) 1696–1707
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Total operating cost includes unloading cost of the vessels, cost for vessels waiting in the sea, inventory cost of storage and charging tanks and changeover cost between different mixed crude oils used for charging CDUs. NST X NV NV SCH X X X V S;i;t þ V S;i;t1 C Total ¼ C UNLoad;v ðT L;v T F;v Þ þ C SEA;v ðT F;v T ARR;v Þ þ C INVst;i 2 v¼1 v¼1 i¼1 t¼1 N N N N SCH SCH BT X BT X BT X CDU XX X V B;i;t þ V B;i;t1 þ ðC SETup;j;j0 ;l Z j;j0 ;l;t Þ: ð1Þ þ C INVbt;j 2 0 t¼1 j¼1 j¼1 t¼1 l¼1 j ¼1
Subject to: (1) Vessel arrival and departure operation rules: Each vessel arrives at the docking station for unloading only once throughout the scheduling horizon SCH X
X F;v;t ¼ 1;
v ¼ 1; . . . ; N V :
ð2aÞ
t¼1
Each vessel leaves the docking station only once throughout the scheduling horizon SCH X
X L;v;t ¼ 1;
v ¼ 1; . . . ; N V :
ð2bÞ
t¼1
Unloading initiation time is described in (2c) T F;v ¼
SCH X
tX F;v;t ;
v ¼ 1; . . . ; N V :
ð2cÞ
t¼1
Unloading completion time is described in (2d) T L;v ¼
SCH X
tX L;v;t ;
v ¼ 1; . . . ; N V :
ð2dÞ
t¼1
Each crude vessel’s unloading time should begin after arrival time T F;v P T ARR;v ;
v ¼ 1; . . . ; N V :
ð2eÞ
Duration of the vessel unloading is bounded by the initial volume of oil in the vessel divided by maximum unloading rate V V ;v;0 ; v ¼ 1; . . . ; N V : ð2fÞ T L;v T F;v P ðQ VS;v;i;max Þ Vessel in the sea cannot arrive at the docking station for unloading unless the preceding vessel leaves T F;vþ1 P T L;v ;
v ¼ 1; . . . ; N V :
ð2gÞ
Unloading is possible between time TF,v and TL,v X W;v;t 6
f X
X F;v;m ; X W;v;t 6
SCH X
X L;v;m ;
v ¼ 1; . . . ; N V ; t ¼ 1; . . . ; SCH :
ð2hÞ
m¼t
m¼1
(2) Material balance for the vessels: Crude oil in vessel v at time t = initial crude oil in the vessel v crude oil transferred from vessel v to storage tanks up to time t V V ;v;t ¼ V V;v;0
NST X t X
Q VS;v;i;m ;
v ¼ 1; . . . ; N V ; t ¼ 1; . . . ; SCH :
ð3aÞ
i¼1 m¼1
Operating constraints on crude oil transfer volumetric rate from vessel v to storage tank i at time t Q VS;v;i;min X W;v;t 6 Q VS;v;i;t 6 Q VS;v;i;max X W;v;t ;
v ¼ 1; . . . ; N V ; i ¼ 1; . . . ; N ST ; t ¼ 1; . . . ; SCH :
ð3bÞ
The volume of crude oil transferred from v vessel to storage tanks during the scheduling horizon equals to the initial crude oil volume of vessel v NST X SCH X i¼1
t¼1
Q VS;v;i;t ¼ V V;v;0 v ¼ 1; . . . ; N V :
ð3cÞ
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(3) Material balance for storage tanks: Crude oil in storage tank i at time t = initial crude oil in storage tank i + crude oil transferred from vessels to storage tank i up to time t crude oil transferred from storage tank i to charging tanks up to time t NV X t X
V S;i;t ¼ V S;i;0 þ
Q VS;v;i;m
v¼1 m¼1
NBT X t X
Q SB;i;j;m ;
i ¼ 1; . . . ; N ST ; t ¼ 1; . . . ; SCH :
ð4aÞ
j¼1 m¼1
Operating constraints on crude oil transfer rate from storage tank i to charging tank j at time t. The term P CDU Dj;l;t Þ denotes that if charging tank j is charging any CDU, there is no oil transfer from storage tank i to chargð1 Nl¼1 ing tank j Q SB;i;j;min 1
N CDU X
! 6 Q SB;i;j;t 6 Q SB;i;j;max 1
Dj;l;t
l¼1
N CDU X
! Dj;l;t ;
i ¼ 1; . . . ; N ST ; j ¼ 1; . . . ; N BT ; t ¼ 1; . . . ; SCH :
ð4bÞ
l¼1
Volume capacity limitations for storage tank i at time t V S;i;min 6 V S;i;t 6 V S;i;max ;
i ¼ 1; . . . ; N ST ; t ¼ 1; . . . ; SCH :
ð4cÞ
(4) Material balance for charging tanks: Crude oil mix in changing tank j at time t = initial mixed oil in storage tank j + crude oil transferred from storage tanks to charging tank j up to time t crude oil mix j charged into CDUs up to time t. V B;j;t ¼ V B;j;0 þ
NST X t X
Q SB;i;j;m ¼
v¼1 m¼1
N CDU X
t X
Q BC;j;l;m ;
j ¼ 1; . . . ; N BT ; t ¼ 1; . . . ; SCH :
ð5aÞ
l¼1 m¼1
Operating constraints on mixed oil transfer rate from charging tank j to CDU l at time t Q BC;j;l;min Dj;l;t 6 Q BC;j;l;t 6 Q BC;j;l;max Dj;l;t ;
j ¼ 1; . . . ; N BT ; l ¼ 1; . . . ; N CDU ; t ¼ 1; . . . ; SCH :
ð5bÞ
Volume capacity limitations for storage tank j at time t. V B;j;min 6 V B;j;t 6 V B;j;max ;
i ¼ 1; . . . ; N BT ; t ¼ 1; . . . ; SCH :
ð5cÞ
Case I. Stochastic demands. Total production amount of crude oil mix j should meet the random demands of crude mix j for CDUs during the scheduling horizon ( ) N SCH CDU X X ~ Q BC;j;l;t P dmj P aj ; j ¼ 1; . . . ; N BT ; ð5d0 Þ Pr t¼1
l¼1
~ j are independent continuous random variables with the cumulative distribution F , F (b ) = a , a 2 (0, 1), a are probwhere dm j j j j j j ability degree of satisfaction of the constraints. Then we can change the formulation (5d0 ) to (5d) N SCH CDU X X l¼1
Q BC;j;l;t P bj ;
j ¼ 1; . . . ; N BT :
ð5dÞ
t¼1
Case II. Fuzzy demands. Total production amount of crude oil mix j should meet the fuzzy demands of crude mix j for CDUs during the scheduling horizon ( ) N SCH CDU X X ~ j P aj ; j ¼ 1; . . . ; N BT ; Pos Q P dm ð5d00 Þ BC;j;l;t
l¼1
t¼1
~ j are fuzzy members with membership functions l , a are a predetermined confidence levels to the respective conwhere dm j i straints. If we define K aj as the minimum values of all potential values, i.e., K aj ¼ inffKjK ¼ l1 j ðaj Þg; j ¼ 1; 2; . . . ; N BT Then we can change the formulation (5d00 ) to (5d000 ) N SCH CDU X X l¼1
Q BC;j;l;t P K aj ;
j ¼ 1; . . . ; N BT :
ð5d000 Þ
t¼1
(5) Material balance for component k in charging tanks: Volume of component k in charging tank j at time t = initial component k in charging tank j + component k in crude oil transferred from storage tanks to charging tank j up to time t component k in crude oil mix j transferred to CDUs up to time t
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vB;j;t ¼ vB;j;0 þ
NST t X X m¼1
i¼1
qSB;i;j;m
N CDU X
! qBC;j;l;m ;
j ¼ 1; . . . ; N BT ; k ¼ 1; . . . ; N CE ; t ¼ 1; . . . ; SCH :
ð6aÞ
t¼1
Operating constraints on volumetric flow rate of component k from storage tank i to charging tank j qSB;i;j;k;t ¼ Q SB;i;j;t nS;i;k i ¼ 1; . . . ; N ST ;
j ¼ 1; . . . ; N BT ; k ¼ 1; . . . ; N CE ; t ¼ 1; . . . ; SCH :
ð6bÞ
Operating constraints on volumetric flow rate of component k from charging tank j to CDU l Q BC;j;l;t nB;j;k;min 6 qBC;j;l;k;t 6 Q BC;j;l;t nB;j;k;max ;
l ¼ 1; . . . ; N CDU ; j ¼ 1; . . . ; N BT ; k ¼ 1; . . . ; N CE ; t ¼ 1; . . . ; SCH :
ð6cÞ
Volume capacity limitations for component k in charging tank j at time t V B;j;t nB;j;k;min 6 vB;j;k;t 6 V B;j;t nB;j;k;max ;
l ¼ 1; . . . ; N CDU ; j ¼ 1; . . . ; N BT ; k ¼ 1; . . . ; N CE ; t ¼ 1; . . . ; SCH :
ð6dÞ
Table 1 Data for case study Scheduling horizon of equal duration time intervals [day]
8
Number of vessels tanks Vessels
Arrival time
2 Crude amount
Sulfur concentration
Vessel 1 Vessel 2
1 5
100 (104 bbl) 100 (104 bbl)
0.01 0.06
Number of storage tanks Storage tanks
Capacity
2 Initial crude amount
Sulfur concentration
4
4
Tank 1 Tank 2
0–100 (10 bbl) 0–100 (104 bbl)
25 (10 bbl) 75 (104 bbl)
Number of charging tanks Charging tanks
Capacity
2 Initial mix oil amount
Sulfur concentration
Tank 1 Tank 2
0–100 (104 bbl) 0–100 (104 bbl)
50 (104 bbl) 50 (104 bbl)
0.015–0.025 0.045–0.055
Number of CDU
0.01 0.06
1 Unloading cost: 8 103 [$/day] Sea waiting cost: 5 103 [$/day]
Unit costs for vessel operation
Storage tank: 0.05 [$/(day 103) ] Charging tank: 0.08 [$/(day 103)]
Unit costs for tank inventory
50 (103) $
Unit changeover cost Demand of mixed oil by CDU
Oil mix 1: g1, Oil mix 2: g2
bbl: liquid measure, bbl barrel.
Table 2 Main results for case study in stochastic environment a1 a2 CTOTAL CUNLOAD CSEA CINVst CINVbt CSETup (TF,1, TL,1) (TF,2, TL,2)
0.9 0.9 214.933 32 15 41.340 26.593 100 (3, 4) (6, 7)
0.8 0.8 214.998 32 15 41.416 26.582 100 (3, 4) (6, 7)
0.7 0.7 215.069 32 10 46.499 26.570 100 (2, 3) (6, 7)
0.6 0.6 215.145 32 10 46.588 26.557 100 (2, 3) (6, 7)
0.5 0.5 215.225 32 15 41.682 26.543 100 (3, 4) (6, 7)
0.4 0.4 215.31 32 10 46.781 26.529 100 (2, 3) (6, 7)
0.3 0.3 215.398 32 10 46.884 26.514 100 (2, 3) (6, 7)
a1 a2 CTOTAL CUNLOAD CSEA CINVst CINVbt CSETup (TF,1, TL,1) (TF,2, TL,2)
0.9 0.1 215.292 32 10 46.754 26.538 100 (2, 3) (6, 7)
0.8 0.2 215.269 32 10 46.729 26.540 100 (2, 3) (6, 7)
0.7 0.3 215.251 32 10 46.709 26.542 100 (2, 3) (6, 7)
0.6 0.4 215.236 32 10 46.693 26.543 100 (2, 3) (6, 7)
0.5 0.5 215.225 32 15 41.682 26.543 100 (3, 4) (6, 7)
0.4 0.6 215.218 32 15 41.676 25.543 100 (3, 4) (6, 7)
0.3 0.7 215.216 32 15 41.674 25.542 100 (3, 4) (6, 7)
CXXX – cost(103$), TXX– time intervals (day), aX – predetermined probability degree of satisfaction of the constraints.
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Table 3 Main results for case study in fuzzy environment a1(ka1) a2(ka2) CTOTAL CUNLOAD CSEA CINVst CINVbt CSETup (TF,1, TL,1) (TF,2, TL,2)
0.9(94.5) 0.9(94.47) 217.092 32 10 48.866 26.226 100 (2, 3) (6, 7)
0.8(94.0) 0.8(93.88) 217.350 32 10 49.168 26.182 100 (2, 3) (6, 7)
0.7(93.5) 0.7(93.22) 217.629 32 10 49.493 26.135 100 (2, 3) (6, 7)
0.6(93.0) 0.6(92.45) 217.934 32 10 49.850 26.085 100 (2, 3) (6, 7)
0.5(92.5) 0.5(91.53) 218.276 32 10 50.248 26.028 100 (2, 3) (6, 7)
0.4(92.0) 0.4(90.42) 218.671 32 10 50.708 25.963 100 (2, 3) (6, 7)
a1(ka1) a2(ka2) CTOTAL CUNLOAD CSEA CINVst CINVbt CSETup (TF,1, TL,1) (TF,2, TL,2)
0.9(94.5) 0.4(90.42) 218.146 32 10 50.083 26.063 100 (2, 3) (6, 7)
0.8(94.0) 0.5(91.53) 217.961 32 10 49.873 26.088 100 (2, 3) (6, 7)
0.7(93.5) 0.6(92.45) 217.829 32 10 49.725 26.105 100 (2, 3) (6, 7)
0.6(93.0) 0.7(93.22) 217.734 32 10 49.618 26.115 100 (2, 3) (6, 7)
0.5(92.5) 0.8(93.88) 217.665 32 10 49.543 26.122 100 (2, 3) (6, 7)
0.4(92.0) 0.9(94.47) 217.617 32 10 49.491 26.126 100 (2, 3) (6, 7)
CXXX, TXX – the same as Table 2, aX – predetermined possibility degree of satisfaction of the constraints, Kax – the minimum demand of all potential values.
10000 bbl/day
50 40 30 20 10 0 1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
10000 bbl/day
50 40 30 20 10 0
Fig. 1. Flow rate of crude oil from vessels to storage tanks.
Eqs. (6c) and (6d) are linear formulations translated from non-convex bilinear equations (60 ), (600 ), (6000 ) through the method of Quesada and Grossmann [18] nB;j;k;min 6 nB;j;k;t 6 nB;j;k;max ; qBC;j;l;k;t ¼ nB;j;k;t Q BC;j;l;t ; vB;j;k;t ¼ nB;j;k;t V B;j;t ;
j ¼ 1; . . . ; N BT ; k ¼ 1; . . . ; N CE ; t ¼ 1; . . . ; SCH ;
l ¼ 1; . . . ; N CDU ; j ¼ 1; . . . ; N BT ; k ¼ 1; . . . ; N CE ; t ¼ 1; . . . ; SCH ;
j ¼ 1; . . . ; N BT ; k ¼ 1; . . . ; N CE ; t ¼ 1; . . . ; SCH :
ð60 Þ ð600 Þ ð6000 Þ
(6) Operating rules for crude oil charging:Charging tank j can charge at most one CDU at any time t N CDU X
Dj;l;t 6 1;
j ¼ 1; . . . ; N BT ; t ¼ 1; . . . ; SCH:
ð7aÞ
l¼1
CDU l can be charged only by one charging tank at any time t N CDU X Dj;l;t 6 1; j ¼ 1; . . . ; N BT ; t ¼ 1; . . . ; SCH: l¼1
ð7bÞ
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If CDU l is charged by crude oil mix j at time t 1 and charged by j0 at time t, changeover cost exist Z j;j0 ;l;t P Dj0 ;l;t þ Dj;l;t1 1;
0
0
j; j ðj–j Þ ¼ 1; . . . ; N BT ; l ¼ 1; . . . ; N CDU ; t ¼ 2; . . . ; SCH :
ð7cÞ
4. A case study The detail data of the case study can be found in Table 1 which is compared with the data in Lee et al. [7]. The model studied contains 267 continuous variables, 68 binary variables and 320 constraints. The model was programmed and solved by LINGO 8.0 [19]. The computing time for once simulation for the case study is less than 10 s. Table 2 and 3 shows the main results in stochastic and fuzzy environment, respectively. Case I. Stochastic demands. When g1 and g2 are normally distributed continuous stochastic variables that follow N(95 104, 52 104) bbl, we gain the following results in Table 2.
100
10000 bbl
80 60 40 20 0 0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
100
10000 bbl
80 60 40 20
Fig. 2. Inventory status of storage tanks.
50
10000 bbl
40 30 20 10 0 0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
100
10000 bbl
80 60 40 20 0
Fig. 3. Inventory status of charging tanks.
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Case II. Fuzzy demands. The fuzzy parameter demand for oil mix 1(g1) is assumed to have triangular membership function as follows: 8 90 6 g1 6 95; > < ðg1 90Þ=5 l1 ðg1 Þ ¼ ð100 g1 Þ=5 95 6 g1 6 100; ð104 bblÞ; > : 0 others; The fuzzy parameter demand for oil mix 2(g2) is assumed to have membership function as follows: 1 l2 ðg2 Þ ¼ exp jg2 95j ; 90 6 g2 6 100; ð104 bblÞ: 5 Then, we gain the following results in Table 3. From the results of Table 2 we can find that if we increase (a1, a2) which represent the probability degree of satisfaction of random demands constraints, the optimal minimum objective cost decrease and vice versa. From the results of Table 3 we can find that if we increase (a1, a2) which represent the predetermined confidence levels of fuzzy demands constraints, the optimal minimum objective cost decrease and vice versa.
0. 025 0. 02
0 0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0. 055 0. 05 0. 045
0
Fig. 4. Component concentration in charging tanks.
10000 bbl/day
50 40 30 20 10 0 1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
10000 bbl/day
50 40 30 20 10 0
Fig. 5. Flow rate of mixed oil from storage tanks to one CDU.
C. Cao et al. / Applied Mathematical Modelling 33 (2009) 1696–1707
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Other main results of once simulation (a1 = 0.4, a2 = 0.9) in fuzzy environment can be found from Figs. 1–5. Similar results can be gained in stochastic environment. Fig. 1 shows the unloading rate of crude oil in vessel 1 and vessel 2 during the scheduling horizon (above for vessel 1, below for vessel 2); Fig. 2 shows the inventory variation status of storage tank 1 and storage tank 2 during the scheduling horizon (above for storage tank 1, below for storage tank 2); Fig. 3 shows the inventory status of charging tank 1 and charging tank 2 during the scheduling horizon (above for charging tank 1, below for charging tank 2); Fig. 4 shows the component concentration in charging tank 1 and charging tank 2 during the scheduling horizon (above for charging tank 1, below for charging tank 2); Fig. 5 shows the charging rate of mixed oil from charging tank 1 and charging tank 2 to one CDU during the scheduling horizon (above for charging tank 1, below for charging tank 2). From the results of Figs. 1–5 we gained, we observe that all the operation rules and constraints have been satisfied properly, and we find the global optimal results. 5. Conclusions Under the current situations of unsteady supply of crude oil, variations of feedstock qualities and yield levels, pressure of intense time, low inventory flexibility, etc., it is very important for refinery companies to assess the extra cost of these changes during making their short-term crude oil scheduling operations. We study one type of uncertainties in crude oil scheduling problem – crude oil blends demands uncertainty, which result in crude oil feed flow rates variation during scheduling time horizon. Two chance constrained mixed-integer nonlinear stochastic and fuzzy programming models are presented to evaluate the extra cost of uncertainty. After the models are converted into crisp equivalent MILP models, the calculation processes are simplified greatly. The branch and bound method in LINGO 8.0 is used to solve the proposed equivalent formulation in a case study which has 267 continuous variables, 68 binary variables and 320 constraints. The global optimal results show that this model can work with feasibility and have the possibility to be further used to solve industrial size problem. Acknowledgements Financial support from the National Natural Science Foundation of China (Grant Nos. 60674075, 60774078), the Key Technology Program of Shanghai Municipal Science and Technology Commission (Grant No. 04dz11008), the Research Fund for Outstanding Youth Teachers of East China University of Science and Technology (Grant No. YH0157117) and the Fund for Shanghai Leading Academic Discipline Project (No. B504) is gratefully appreciated. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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