Tactical and Operational Planning of Multirefinery Networks under

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Tactical and Operational Planning of Multirefinery Networks under Uncertainty: An Iterative Integration Approach Adriana Leiras,* Gabriela Ribas, and Silvio Hamacher Industrial Engineering Department, Pontifical Catholic University of Rio de Janeiro, CP38097, 22453-900 Rio de JaneiroRJ, Brazil

Ali Elkamel Chemical Engineering Department, University of Waterloo, Ontario N2L 3G1, Canada ABSTRACT: The oil industry is increasingly interested in improving the planning of their operations, because of the dynamic nature of the oil business. This study intends to establish an iterative integration approach for the tactical and operational planning of multisite refining networks. Tactical and operational mathematical models are proposed. Both models are two-stage stochastic linear programs in which uncertainty is incorporated into the dominant random parameters at each decision level. Decisions made in the oil industry differ based on multisite network echelon (spatial integration) and planning horizon (temporal integration). Spatial integration is discussed at the tactical level, whereas temporal integration is discussed with respect to the interaction between the two levels. In the proposed temporal integration approach (iterative approach), there is a cyclic information flow between the two models. An industrial scale study using data from the Brazilian oil industry was conducted to discuss the benefits of integration in a stochastic environment. et al.7 and Joly et al.,8 who proposed deterministic models for integrated operational planning and the scheduling of refineries. Another relevant issue related to the oil industry is the effect of uncertainties such as fluctuations in oil production, prices, and product demand. In optimization models for the oil industry, uncertainties have been noted at all planning levels by several academic studies conducted over the past few years (for example, refs 9−13, to cite just some recent works). In this sense, the work of Luo and Rong,14 which addressed not only the integrated operational planning and scheduling of refineries but also uncertainty using the robust approach proposed by Janak et al.,15 stands out. Despite these contributions regarding optimization problems at different planning levels, no model that considers the integrated planning of multirefinery networks under uncertainty and at the same time addresses the interactions between tactical and operational decisions was found in the literature. Therefore, the problem of integrated refinery planning under uncertainty is still an open issue and is relevant for both mathematical modeling and actual applications. In this context, the main contribution of this work is to tackle this important topic from both a theoretical standpoint and a practical standpoint and to investigate the spatial and temporal integration of multisite refining networks using a two-stage stochastic model. The purpose of this paper is to address the problem of integration and coordination under uncertainty of multirefinery networks at different decision levels. This paper covers the tactical and operational planning levels. Spatial integration is addressed at the tactical level (considering a multisite refinery

1. INTRODUCTION Integration and coordination are key components in enterprisewide optimization.1 The benefits from the integration of multiple sites manifest not only in economic terms but also in terms of process flexibility.2 The potential of integration benefits has attracted high attention in the research area of supply-chain planning. In general, planning is categorized into three time frames: strategic (long-term), tactical (medium term), and operational (short-term). Long-term planning covers the time horizon from one to several years, medium-term planning ranges from a few months to a year, and short-term planning covers up to 3 months.3 Strategic planning determines the structure of the supply chain (e.g., facility location). Tactical planning is concerned with decisions such as the assignment of production targets to facilities and the transportation from facilities to distribution centers. Operational planning refers to the assignment of tasks to units at each facility, considering resource and time constraints.4 These planning levels are conventionally thought to be related in a hierarchical fashion; strategic planning decisions impose goals, targets, and constraints on tactical decisions, which are, in turn, implemented and supported via several operational execution functions.5 A way to pose the need for integration is to recognize the natural hierarchy among these steps and acknowledge that they may not operate with the same level of information. Therefore, systematic methods for efficiently managing the oil supply chain must be exploited.6 Decisions made in the oil industry differ mainly with regard to the range of activities coordinated in the supply chain (spatial integration) and with regard to the coordination of decisions across different planning horizons (temporal integration).1 Understanding the benefits of integrating the different planning levels has attracted attention in the planning/refining area. The first contributions in this area were the works of Pinto © XXXX American Chemical Society

Received: October 16, 2012 Revised: April 19, 2013 Accepted: May 14, 2013

A

dx.doi.org/10.1021/ie302835n | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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decision-maker to better explore the uncertainty issues in the model. Since the operational model has a smaller scale than the tactical model, it is possible to consider a variety of intermediate products indexed by oil as needed for the properties calculation using linear correlations. The oil indexation allows for the estimation of the property values of the intermediate products when needed and greatly improves the performance of the operational models that are solved for each refinery of the network. To identify the intermediate products that should be indexed by oil, a previous study based on the nonlinear version of this model was conducted17 to track the critical property to be controlled, whereas other property values were estimated by the same nonlinear model. Through this procedure, the authors expect to avoid infeasibilities in the nonlinear refinery processes caused by the approximated linear solution provided. Both optimization models (tactical and operational) are formulated as two-stage stochastic programs with fixed recourse.18 Uncertainties are discretely represented by SC possible realization scenarios (finite sample space) and modeled as a scenario tree. (A scenario is a path from the root to a leaf of the tree.)

network), whereas temporal integration was addressed in the interaction between the two levels (tactical and operational). Tactical and operational mathematical programming models are proposed with the objective of maximizing the expected profit over a given time horizon. The proposed iterative approach considers an information loop between the tactical and the operational models. The mathematical models present important contributions to the literature in terms of the formulation and sources of uncertainty. The problem is formulated as two-stage stochastic programs with a finite number of realizations. Furthermore, the two distinct levels (tactical and operational) result in a bilevel hierarchical decision-making framework that can be utilized effectively for incorporating uncertainty into the dominant random parameters at each planning level. Because the nature of uncertainty is different at the various levels of the decision-making process, uncertainties in demand for refined products, oil prices, and product prices account for economic risk at the tactical level. At the operational level, oil supply and process unit capacity uncertainties address the operational risk. The remainder of this paper is organized as follows. The problem addressed in this paper is described in section 2. Section 3 presents the proposed tactical model, the operational model, and the iterative approach. Next, section 4 presents a numerical example in the context of a case study using real data from the Brazilian oil industry, and section 5 offers the results and discussions of this example. The paper ends with concluding remarks in section 6.

3. ITERATIVE INTEGRATION APPROACH In the proposed integration approach, the tactical model (master problem) is used to assign production targets to refineries, and the operational problem (slave problem) details the operation processes at each refinery, breaking down the aggregate planning (master planning), where integration is defined as an automation of the transfer of information between the planning levels so they may be coordinated effectively.19 This integration approach results in a bilevel hierarchical decision-making framework. The formulation ignoring the slave’s objective (single-level formulation) is a much more frequent type of occurrence in the applied literature, but Candler and Townsley20 demonstrated the potential error implicit in such an approach by an example of energy minimizing in the United States economy. They solved the master problem ignoring the slave’s objective. The optimal master variable values were then plugged into the slave problem and optimal slave variables were found. Evaluating the master problem objective given the optimal slave variables lead to an objective function value, which was ∼31% of the expected value. Solving the same problem via bilevel programming lead to a solution with a master objective which was ∼99% of the expected value.21 Thus, large errors are introduced by ignoring the subproblem structure. The bilevel programming takes into account the reaction of the lower level decision-makers and solves the problem of coordinating the decision-making process by improving the highest level objective, while dealing with the tendency of the lower levels of the hierarchy to improve their own objectives. The hierarchical nature of the problem is reflected by the order imposed on the choice of the decisions; the highest level makes its decision first, followed by the next highest, until the lowest one.22 Because a feedback loop from the slave model back to the master problem is considered in the integration approach proposed in this paper, as indicated in Figure 1, the approach is called iterative.4 Another possibility is considering the integration approach as one single model in which the results from the slave problem are fed back to the master model for further optimization; the approach then is called f ull-space.4 In the fullspace approach, under appropriate constraint qualification, the lower-level problem can be replaced by its Karush−Kuhn− Tucker (KKT) optimality conditions to obtain an equivalent (single-level) mathematical program.23 In this paper, only the

2. PROBLEM STATEMENT The tactical model proposed in this paper maximizes the total revenue of a multirefinery network. The information in the tactical model is more aggregated than in the operational model. The tactical planning allocates oil quantities to each refinery and the production targets of the products to the various refineries in each time period while taking into account inventory holding costs and transportation costs. Each refinery processes crude oil to produce a variety of marketable petroleum products. The refined products can be moved along the logistic network by road, water, rail, or pipeline. Crude oil and refined products are often transported to distribution centers through pipelines. From this level on, the products can be transported either through pipelines or trucks, depending on consumer demand points. In some cases, products are also transported through vessels or by train. The operational model is a more-detailed problem that traditionally maximizes the revenue of each refinery separately, allowing each refinery to be as profitable as possible (costeffective). The operational model determines the amount of material that is processed at each time interval within each unit at each refinery.16 In both planning levels, the refineries possess process units that can operate under different modes and produce several intermediate streams, which can be blended to create distinct products within the standards of quality specifications. The multirefinery network is supplied by crude oils (A1, A2, B1, B2, and B3, for example) that can be aggregated according to their characteristics (API index, yields, properties, and production region) to form groups (A and B, for example). The tactical model only makes distinctions among oil groups, but the operational model takes more-detailed information into account and differentiates among oils within a group. The tactical and operational models ensure the total fulfillment of the market demand and assume a discrete planning horizon divided into a finite number of periods. Following Pongsakdi et al.,12 many nonlinear features can be reasonably simplified at the tactical level to gain computational speed, which allows the B



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proposed linear programming model aims at maximizing the expected profit of the multisite refinery network. The model decisions regarding multisite oil refining determine the oil blending in each refinery, the production level at each process unit, and the operational mode for each unit at each period to meet the demand and quality standards for the refined products. An operational mode is characterized by a set of operation patterns to prioritize the production of a specific product set to meet a market demand; i.e., the process unit yields vary according to the operational mode. With respect to the logistic network, the model must define the minimum cost flow combination for the refinery supply and the refined product distribution by different transportation modes. First, definitions of sets, variables, and parameters of the model are provided in Tables 1, 2, and 3. Then, the mathematical model is presented.

Figure 1. Iterative approach.

iterative approach is explored. A full-space approach implementation may be considered for future works. The literature on the application of bilevel programming problems is quite extensive and diverse, covering topics such as economics,24 engineering,25 transportation,26 management,24 and chemical engineering systems.27−29 The two model formulations (tactical and operational) and the proposed iterative algorithm to the iteration procedure are presented in the following sections. Since the nature of uncertainty is different at each planning level, price and demand uncertainties are considered in the tactical model, and oil supply and process unit capacity utilization uncertainties are considered in the operational model, which means that the time of the decisions in the two models is different. First, the tactical model finds an optimal stochastic solution, and once the uncertain events have unfolded (which means that the price and demand patterns are now known and that the oil purchase from long-term contracts has already been defined), the operational model makes the necessary adjustments to account for operational uncertainties through oil purchase on the spot market. It is important to highlight that the purpose of the integration approach is to solve the tactical problem considering the optimal operational solution. Since tactical planning covers a mediumterm time frame (few months to a year) and operational planning covers a short-term time frame (up to 3 months), the iteration procedure aims to find the optimal tactical solution subject to optimal operational solution in the first periods of the planning horizon (when the realizations of the tactical probabilistic parameters are known). The remaining periods of the tactical model are used as the steady-state condition for the solution. Integrated planning is necessary to avoid the operational infeasibilities due to the amount/quality of oil allocated by the tactical model. Uncertainty is an important motivation for integrating the tactical and the operational planning of oil refineries because, in the sequential solution of the two models, the tactical level does not take the operational uncertainties into account. In the planning problem addressed in this paper, the tactical model defines the oil allocation from long-term contracts (fixed oil) to each refinery; however, because of delays/changes in the oil supply at the operational level, the amount or quality of the oils received by the refineries may not be sufficient to meet their product demands. In this case, the refineries must purchase more costly additional oil on the spot market, which results in a change in the tactical solution with implications for the oil supply (if the additional oil is not commercially available or cannot be produced locally) and logistical constraints (if there is no capacity for transportation or storage of the additional oil). Optimality is the main issue when the two models are solved separately, but effectiveness assumes the main role in the integrated approach because effectiveness may enable the operation to respond rapidly to changes in the tactical plan to increase profit and flexibility of the multisite network. 3.1. Tactical Planning Model of Multirefinery Networks under Uncertainty. This section presents the stochastic formulation for tactical planning of oil refineries. This formulation is adapted from Ribas et al.30 by excluding all elements related to investment decisions, which means that the physical settings of the supply chain have already been established by a strategic plan. The

Table 1. Sets of the Tactical Model Sets term I U C P O M A N SCT OF ⊂ I R⊂I NG ⊂ I TR ⊂ I B⊂I AA ⊂ A

set of nodes (i1, i2) set of process units (u, u′) set of operational modes (c) set of products (pi, po) set of oils (o) set of transport modes (m) set of transport arcs (a) time periods {n | n = 1, ..., NT} set of tactical scenarios (sct) oil field (of) refinery (r) natural gas producers (ng) terminals (tr) bases (b) transportation arcs a available for po from i1 to i2 by the mode m

Table 2. Variables of the Tactical Model Variables term qof nr,o t

bln,sc r,pi,po t

duln,sc r,u,c,o t

puln,sc r,u,c,pi t

ofn,sc a,o n,sct pfa,po t

pfrin,sc r,po t pfron,sc r,po t vorn,sc r,o

First-Stage Variables amount of oil purchased by refinery r of oil type o at period n Second-Stage Variables blending in refinery r of products pi and po at period n under scenario sct load of oil o at refinery r in distillation unit u and operational mode c at period n under scenario sct load of product pi at refinery r in process unit u and operational mode c at period n under scenario sct flow of oil o at transportation arc a at period n under scenario sct flow of product po at transportation arc a at period n under scenario sct inlet flow of product po at refinery r at period n under scenario sct outlet flow of product po at refinery r at period n under scenario sct stock level of oil o at refinery r at period n under scenario sct

Model Formulation. The objective function (eq 1) maximizes the expected tactical margin (TM), which includes the revenue from the product sales minus the oil costs, the refining operation costs, and the transportation costs. The oil purchase (qof nr,o) represents the first-stage decisions (deterministic term). The second-stage decisions are scenario dependent and are used to represent refinery C


Industrial & Engineering Chemistry Research t

t

t

Article

t

necessary network flows through the transportation network t t n,sc (ofn,sc a,o , pfa,po).

n,sc n,sc n,sc operation (bln,sc r,pi,po, dulr,u,c,o, pulr,u,c,pi, vorr,o ) and model all operative t t n,sc relations between the inputs (pf rin,sc r,po ) and outputs (pf ror,po ) and the

∑ ∑ (OPrn,oqof rn,o ) + ∑

maximize TM = − ∑

t

r∈R o∈O n∈N

sc ∈ SC

⎧ t⎪ PROB sc ⎨ ∑ ⎪ ⎩b∈B

t

t

po ∈ P n ∈ N

t

t

n∈N r∈R u∈U

o∈O c∈C

n∈N r∈R u∈U

pi ∈ P c ∈ C

Constraint (2) represents the oil balance, whereas constraint (3) represents the product balance. Ref ining Balance Constraints t

t

qof rn, o + vorrn,−o 1, sc =

∑ ∑ (dulrn,,usc,c ,o) + vorrn,,osc

t

u∈U c∈C o∈O t

pi ∈ P

+

∑

t

t

pulrn,,usc, c , piYPUr , u , c , pi , po

u∈U c∈C t

u ∈ U c ∈ C pi ∈ P

∀ r ∈ R , ∀ po ∈ p , ∀ n ∈ N , ∀ sc t ∈ SC t

(3)

Constraint (4) establishes the relationship between the entry flows (pi) and the process unit (u) loading. The maximum and minimum capacities of the process unit u in period n are limited by constraint (5). Ref ining Operation Constraints RPLrn, u , pi , c

∑

t pulrn,,usc, pi , c

≤

t

⎭ (1)

t

∑ ∑ dulrn,,usc,o,c + ∑ ∑ pulrn,,usc,pi ,c pi ∈ P c ∈ C

(5)

Constraints (6) and (7) limit the sulfur content (SOnpi) and the viscosity (VOnpi) of the final products. The final product properties must be within a range established by environmental regulations. Property calculations yield a set of nonlinear constraints31 in which the nonlinear terms arise from the multiplication between the product properties and their volumes. These terms can be linearized by estimating the properties of intermediate products. At the tactical level, it is possible to estimate these two properties of the intermediate products with sufficient accuracy, making the constraint that controls the final product properties linear. The tactical model controls only these two properties, because they are the ones that affect most tactical decisions, such as oil purchase and oil blending. Environmental Legislation Requirements

(2)

∑ ∑ pulrn,,usc,c ,po

pi ∈ P t

pulrn,,usc, c , piRCr , u , po + pfrorn,,posc

∑∑∑

⎫ ⎪

∀ r ∈ R , ∀ u ∈ U , ∀ n ∈ N , ∀ sc t ∈ SC t

t

blrn,,posc , pi +

a∈A n∈N o∈O

o∈O c∈C

u ∈ U c ∈ C pi ∈ P t

blrn,,pisc, po + pfrirn,,posc =

t

∑ ∑ ∑ of an,,osc TCa⎪⎬

≤ CAPUr , u

t

∀ r ∈ R , ∀ o ∈ O , ∀ n ∈ N , ∀ sc t ∈ SC t

∑ ∑ ∑ dulrn,,usc,c ,oYDUr ,u ,c ,o,po + ∑ ∑ ∑

pf an,,posc TCa −

a ∈ A n ∈ N po ∈ P

CAPLr , u ≤

u∈U c∈C

∑

t

∑ ∑ ∑ (OCrn,u ∑ ∑ dulrn,,usc,o,c) − ∑ ∑ ∑ (OCrn,u ∑ ∑ pulrn,,usc,pi ,c)− ∑ ∑ ∑

−

+

t

∑ ∑ (PPbn,,posc DEMbn,,posc )

t

∑ ∑ ∑ pulrn,,usc, c , piSI pinYPUr ,u ,c ,pi ,po

t pulrn,,usc, pi , c

u ∈ U pi ∈ P c ∈ C

pi ∈ P

≤ RPUrn, u , pi , c

∑

≤ (∑

t

pulrn,,usc, pi , c

∑∑

t

n pulrn,,usc, c , piYPUr , u , c , pi , po)SOpo

u ∈ U c ∈ C pi ∈ P

pi ∈ P t

∀ r ∈ R , ∀ u ∈ U , ∀ pi ∈ P , ∀ c ∈ C , ∀ n ∈ N , ∀ sc ∈ SC

t

∀ r ∈ R , ∀ po ∈ P , ∀ n ∈ N , ∀ sc t ∈ SC t

(4)

⎛ ⎞ t t t n (blrn,,pisc, poVBI pin) + ⎜⎜ ∑ ∑ ∑ dulrn,,usc, c , oYDUr , u , c , o , po + ∑ ∑ ∑ pulrn,,usc, c , piYPUr , u , c , pi , po⎟⎟VBI po pi ∈ P u ∈ U c ∈ C pi ∈ P ⎝u∈U c∈C o∈O ⎠ ⎛ ⎞ n⎜ n , sc t n , sc t n , sc t ⎟ bl dul YDU pul YPU ∑ ∑ ∑ ∑ ∑ ∑ ∑ ≤ VOpo + + r , pi , po r , u , c , o r , u , c , o , po r , u , c , pi r , u , c , pi , po ⎜ ⎟ u∈U c∈C o∈O u ∈ U c ∈ C pi ∈ P ⎝ pi ∈ P ⎠

(6)

∑

Logistical balance constraints (constraints (8) and (9)) determine that the sum of the input flows must be equal to the sum of the output flows for each node (i), product (po) or oil (o), period of time (n), and scenario (sct). AA represents the set of transportation arcs (a) for a product (po) from an origin node (i1) to a destination node (i2) by a transportation mode (m). Logistic Balance Constraints t

t

= qof in1, o +

t

t

∀ i1 ∈ R ∪ B ∪ TR ∪ NGOF , ∀ o ∈ O , ∀ n ∈ N , ∀ sc t ∈ SC t

(9)

Constraint (10) limits the maximum volume transported by the transportation arc (a) in the period n. Finally, constraint (11) defines the non-negativity of the variables. Logistic Capacity Constraints

t

pf an,,posc + pfriin1,,scpo

(a , i1, i 2, po , m) ∈ AA

∑

∀ i1 ∈ R ∪ TR ∪ OF , ∀ po ∈ P , ∀ n ∈ N , ∀ sc t ∈ SC t

of an,,osc

∑ (a , i1, i 2, po , m) ∈ AA

t

∑

t

(a , i1, i 2, po , m) ∈ AA

(a , i1, i 2, po , m) ∈ AA

= DEMin1,,scpo +

(7)

of an,,osc + OFPin1, o

∑

pf an,,posc + pfroin1,,scpo

∑

∀ r ∈ R , ∀ po ∈ P , ∀ n ∈ N , ∀ sc t ∈ SC t

po ∈ P

t

pf an,,posc +

∑ of an,,osc

t

≤ TCAPa

o∈O

∀ a ∈ A , ∀ n ∈ N , ∀ sc t ∈ SC t

(8) D

(10)



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First, definitions are provided in Tables 4, 5, and 6. Then, the mathematical model is presented. Model Formulation. The objective function described by eq 12 maximizes the expected operational margin (OM), which includes the revenue from the product sales minus oil purchase and inventory costs. The oil supply for each refinery is defined by long-term contracts and is represented by the stochastic o parameter (QOFt,sc u,c,s). This parameter is used in eq (22) of the operational model and takes the value of the first-stage tactical variable (qof nr,o) connecting the two models. The oil purchase in the spot market (qoatu,c,s) constitutes the first-stage decisions of the operational model. The second-stage decisions are related to o the refinery operations, such as flows between units (qt,sc u′,c′,s,u,c) and o inventory level (volt,sc u,c ).

Table 3. Parameters of the Tactical Model Parameters term YDUr,u,c,o,po

product po yield from oil o at refinery r in distillation unit u and operational mode c maximum sulfur content of product po at period n minimum viscosity content of product po at period n field production of oil o at node i1 at period n price of oil o at refinery r at period n operational cost at refinery r in process unit u at period n product pi yield from product po at refinery r in process unit u and operational mode c consumption of product po at refinery r in process unit u minimum capacity at refinery r in process unit u maximum capacity at refinery r in process unit u minimum proportion of product pi at refinery r in process unit u and operational mode c at period n maximum proportion of product pi at refinery r in process unit u and operational mode c at period n sulfur content of product pi at period n transportation capacity at transportation arc a transportation cost at transportation arc a viscosity blending index of product po at period n Stochastic Parameters probability of scenario sct demand of product po in base b at period n under scenario sct price of product po in base b at period n under scenario sct

SOnpo VOnpo OFPni1,o OPnr,o OCnr,u YDUr,u,c,pi,po RCr,u,po CAPLr,u CAPUr,u RPLnr,u,c,pi RPUnr,u,c,pi SInpi TCAPa TCa VBInpo t

PROBsc n,sct DEMb,po t n,sc PPb,po

t

t

t

t

t

maximize OM = − ∑

⎛ o PROB sc ⎜ ∑ ⎜ sc o∈ SC o ⎝c∈C

∑

+

∑ ∑ ∑

−

u ∈ UA c ∈ C s ∈ SOu , c

∑ ∑ ∑ (PPut ,s ∑ u ∈ UE s ∈ SIu , c t ∈ T

o

qut ′, sc, c ′ , s , u , c)

(u ′ , c ′) ∈ F

⎞ o ∑ CINVut ,svolut ,,csc ⎟⎟ t∈T ⎠

(12)

Constraint (13)o describes the mass balance at the inlet stream of the unit u (qit,sc the mass balance u,c ). Constraint (14) describes o at the outlet stream s of the unit u (qot,sc ). The stock balance u,c,s in the ostorage unit UA is represented by constraint (15), where vot−1,sc = VOLIu,c when t = 1. Constraint (16) corresponds to u,c the mass balance for the blending units UM and pipelines UD, because there is no stock in either unit, so the sum of the inlet streams must be equal to the sum of the outlet streams. Constraint (17) describes the process in the process unit UP, o where the outlet flow rate of stream s (qot,sc u,c,s) is a function of the feed flow rate of stream s′ and its yield. Constraint (18) determines the blending recipe of the feed flow rate of stream s for the blending unito UM as a function of the feed flow rate of the blending unit (qit,sc u,c ) and the blending proportion of the inlet streams (RUTu,c,s). This constraint requires that the volume of each inlet stream be a predefined fraction of the total feed flow rate. Process Constraints and Material Balance Constraints

t

qof rn, o , blrn,,pisc, po , dulrn,,usc, c , o , pulrn,,usc, c , po , pfrirn,,posc , of an,,osc , pfrorn,,posc , t

pf an,,posc ∈ 9 +

∑ ∑ ∑ CAut ,sqoaut ,c ,s

c ∈ C u ∈ UC s ∈ SOu , c t ∈ T

(11)

3.2. Operational Planning Model under Uncertainty (for Each Refinery). This section presents the mathematical formulation of the operational model for each refinery. This model is based on the multiperiod stochastic model of Neiro and Pinto,11 which represents a refinery as a set of units connected by streams. The time periods are linked by inventory variables. The main contributions of our operational model are the presentation of a detailed formulation (presenting all model parameters and variables) that is unlike the general formulation of Neiro and Pinto,11 the modeling of oil supply and process unit capacity uncertainty, and the addition of a set of operational modes into the model. In addition to the prioritization of the production of a specific product in the process unit, the model also uses the operational modes to identify the streams stored in the tank units and represents the different consumer markets of product demand in the delivery units for each refinery. The main variable of the model is then the flow rate of stream s between two units (u,u′) that operate in the given operationalo mode (c,c′) at each time period t under scenario sco (qt,sc u,c,s,u′,c′). These variables are defined only for a set of feasible flows (F) of stream (s) between two pairs (u,c) and (u′,c′) of a refinery, where the feasible flows are determined by the refinery flowchart. Another contribution is the inclusion of a decision variable to represent an option for additional oil purchase (qoatu,c,s). The oil supply for each refinery is defined by long-term contracts and is a fixed cost to the operational model. However, because oil delivery is subject to uncertainties (delays or changes in the oil specification when oil is delivered at the refineries), the operational model may decide upon the purchase of additional oil on the spot market, which implies additional oil costs.

o

qiut ,,csc =

o

qut ′, sc, c ′ , s , u , c

∑ (u ′ , c ′ , s) ∈ F

∀ u ∈ U , ∀ c ∈ C , ∀ t ∈ T , ∀ sc o ∈ SC o o

qout ,,csc, s =

(13)

o

qut ,,csc, s , u ′ , c ′

∑ (u ′ , c ′) ∈ F

∀ u ∈ U , ∀ c ∈ C , ∀ s ∈ SOu , c , ∀ t ∈ T , ∀ sc o ∈ SC o (14) o

o

o

vout ,,csc = vout −, c 1, sc + qiut ,,csc −

∑

o

qout ,,csc, s

s ∈ SOu , c

∀ u ∈ UA , ∀ c ∈ C , ∀ t ∈ T , ∀ sc o ∈ SC o

∑

o

qout ,,csc, s = qiut ,,csc

(15)

o

s ∈ SOu , c

∀ u ∈ UD ∪ UM , ∀ c ∈ C , ∀ t ∈ T , ∀ sc o ∈ SC o (16) E



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Table 4. Sets of the Operational Model

Table 6. Parameters of the Operational Model

Sets

Parameters

term

term

U C S PR T SCo UC ⊂ U UE ⊂ U UD ⊂ U UP ⊂ U UA ⊂ U UM ⊂ U SIu,c ⊂ S SOu,c ⊂ S Fu,c,u′,c′ ⊂ S

CAtu,s CINVtu,s PPtu,s QOAtu,c,s

set of units (u, u′) set of operational modes (c, c′) set of streams (s, s′) set of properties (pr, pr′) time periods {t | t = 1, ..., T} set of scenarios (sco) tanks of oil delivery units for final products pipelines processing units storage units blending units inlet streams s of unit u outlet streams s of unit u viable flows (u,c,s,u′,c′) of stream s from (u,c) to (u′,c′)

DEMtu,c,s YPUu,c,s,s′ RUTu,c,s QILtu,c QIUtu,c VOLIu,c VOLLtu,c VOLUtu,c t POEu,c,s,pr

Table 5. Variables of the Operational Model

PVOLIu,c,pr POLtu,c,pr

Variables

POUtu,c,pr

term First-Stage Variables additional purchase of oil s at unit u and operational mode c at period t Second-Stage Variables flow rate of stream s between unit u and operational mode c (u,c) and unit u′ and operational mode c′ (u′,c′) at period t under scenario sco inventory level at unit u and operational mode c at period t under scenario sco inlet flow rate at unit u and operational mode c at period t under scenario sco outlet flow rate of stream s at unit u and operational mode c at period t under scenario sco

qoatu,c,s o

qt,sc u,c,u′,c′ volt,sc u,c qit,sc u,c

o

o

o

qot,sc u,c,s

o

PROBsc o QLt,sc u o QUt,sc u o t,sc QOFu,c,s

cost of additional oil of stream s at unit u at period t inventory cost of stream s at unit u at period t product price of stream s at unit u at period t maximum of additional oil of stream s at unit u and operational mode c at period t demand of stream s at unit u and operational mode c at period t product s yield from product s′ at unit u and operational mode c blending recipe of stream s at unit u and operational mode c minimum feed flow rate at unit u and operational mode c at period t maximum feed flow rate at unit u and operational mode c at period t initial stock at unit u and operational mode c minimum storing capacity at unit u and operational mode c at period t maximum storing capacity at unit u and operational mode c at period t estimated value of property pr of outlet stream s at unit u and operational mode c at period t property pr value of initial stock at unit u and operational mode c minimum value of property pr at unit u and operational mode c at period t maximum value of property pr at unit u and operational mode c at period t Stochastic Parameters probability of scenario sco minimum feed flow rate of unit u at period t under scenario sco maximum feed flow rate of unit u at period t under scenario sco quantity of oil o from long-term contracts at tank unit u and operational mode c at period t under scenario sco

Process Unit Capacity Constraints o

o

qout ,,csc, s =

∑

QILut , c ≤ qiut ,,csc ≤ QIUut , c

o

qut ′, sc, c ′ , s , u , cYPUu , c , s , s ′

∀ u ∈ UP ∪ UM , ∀ c ∈ C , ∀ t ∈ T , ∀ sc o ∈ SC o

(u ′ , c ′ , s) ∈ F

(20)

∀ u ∈ UP , ∀ c ∈ C , ∀ s ∈ SOu , c , ∀ t ∈ T , ∀ sc o ∈ SC o

o

QLut , sc ≤

(17)

∑

o

qut ′, sc, c ′ , s , u , c = RUTu , c , sqiut ,,csc

∑ qiut ,,csc

o

≤ QUut , sc

o

c ∈ Cu

∀ u ∈ UP ∪ UA ∪ UD , ∀ t ∈ T , ∀ sc o ∈ SC o

o

(21)

(u ′ , c ′) ∈ F

Constraint (22) limits the outlet flow rate for oil tanks UC. Fixed (long-term contracts) and additional (spot market) oil are available. The refinery may decide to keep some fixed oil in the storage units and purchase the additional oil necessary for its operation through the first-stage variable qoatu,c,s. Because purchasing additional oil adds costs, the refinery decides upon the purchase of additional oil only if the amount of fixed oil received is not enough to meet the market demand or if the quality of the oil received is not good enough to meet the standards of the refined products. Constraint (23) limits the additional oil available for purchase.

∀ u ∈ UM , ∀ c ∈ C , ∀ s ∈ SIu , c , ∀ t ∈ T , ∀ sc o ∈ SC o (18)

Constraint (19) limits the inlet flow rate for the final products in the delivery units UE, where the demand is defined by the tactical planning and must be met. Demand Constraints

∑

o

qut ′, sc, c ′ , s , u , c = DEMut , c , s

(u ′ , c ′) ∈ F

∀ u ∈ UE , ∀ c ∈ C , ∀ s ∈ SIu , c , ∀ t ∈ T , ∀ sc o ∈ SC o

Plant Supply Constraints

(19)

o

o

0 ≤ qout ,,csc, s ≤ QOFut ,,csc, s + qoaut , c , s

Constraint (20) restricts the feed flow rate of each unit u for each operational mode c. Constraint (21) controls the feed flow rate of the unit u.

∀ u ∈ UC , ∀ c ∈ C , ∀ s ∈ SOu , c , ∀ t ∈ T , ∀ sc o ∈ SC o (22) F



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qoaut , c , s ≤ QOA ut , c , s ∀ u ∈ UC , ∀ c ∈ C , ∀ s ∈ SOu , c , ∀ t ∈ T

(23)

Constraint (24) represents the inventory level for product tanks at each time period t and scenario sco. Stock Constraints o

VOLLut , c ≤ volut ,,csc ≤ VOLUut , c ∀ u ∈ UA , ∀ c ∈ C , ∀ t ∈ T , ∀ sc o ∈ SC o

(24)

Constraints (25) and (26) refer to properties of the streams in the storage units UA, considering the stock at the time interval before t − 1 and the inlet flow rate at the time interval t o t−1 (where vot−1,sc = VOLIu,c and POLt−1 u,c u,c,pr = POUu,c,pr = PVOLIu,c,pr, when t = 1). As in the tactical model, property calculations were linearized by estimating the value of property pr of the outlet stream s (POEtu,c,s,pr). Finally, constraint (27) define the nonnegativity of the variables. Property Constraints o

o

Figure 2. Iterative algorithm.

o

(volut −, c 1, sc + qiut ,,csc )POLut , c , pr ≤ volut −, c 1, sc POLut ,−c ,1pr +

o

Let I be an iteration of the iterative method and TaticalOilOf f feri be the oil offer available at the tactical level. The oil available at the operational level is always compatible with the oil offer at the tactical level, i.e., the limit of additional oil available on the spot market at the operational level (presented at constraint (23)) is equal to the amount of oil available and not allocated by the tactical model (QOAtu,c,s = TaticalOilOf feri − qof nr,o − TaticalOilFixi). If there is a feasible solution to the iterative problem, the convergence of the iterative algorithm is ensured by lines 8 and 14. In step 14, the fixed oil quantity at the tactical level in an iteration corresponds to the additional oil purchase on the operational model plus the total oil fixed until the previous iteration (TaticalOilFixi = qoatu,c,si + TaticalOilFixi−1) As in each iteration, a new amount of oil is fixed to the tactical model decision, keeping the volume already fixed in the previous iteration (TaticalOilFixi), in the extreme case, the maximum limit of additional oil at the operational level decreases until it equals zero. If there is no feasible solution to the iterative problem, the convergence of the iterative algorithm is ensured by steps 6 and 10, which terminates the algorithm when the operational model or the tactical model is infeasible.

(qut ′, sc, c ′ , s , u , c)POEut , c , s , pr

∑ (u ′ , c ′ , s) ∈ F

∀ u ∈ UA , ∀ c ∈ C , ∀ pr ∈ PR , ∀ t ∈ T , ∀ sc o ∈ SC o (25) o

o

o

(volut −, c 1, sc + qiut ,,csc )POUut , c , pr ≥ volut −, c 1, sc POUut ,−c ,1pr +

o

(qut ′, sc, c ′ , s , u , c)POEut , c , s , pr

∑ (u ′ , c ′ , s) ∈ F

∀ u ∈ UA , ∀ c ∈ C , ∀ pr ∈ PR , ∀ t ∈ T , ∀ sc o ∈ SC o (26) o

o

o

o

qoaut , c , s , qut ′, sc, c ′ , s , u , c , qiut ,,csc , qout ,,csc, s , volut ,,csc ∈ 9 +

(27)

3.3. Iterative Algorithm. This section presents the iterative algorithm to connect the tactical and operational models. The additional oil purchase (e.g., spot market) in the operational model is not foreseen in the tactical model, so this purchase can make the tactical solution infeasible by logistics constraints or oil supply constraints. The solution strategy for the integrated approach evolves to eliminate the additional oil purchase in the spot market at the operational level. This additional oil purchase (qoanr,o) functions as a slack variable when the operational model infeasibility is caused by the amount/quality of the oil(s) o allocated by the tactical model (QOFt,sc u,c,s). In this case, however, n the cost of additional oil (qoar,o) is higher than when the oil n purchase is considered by tactical planning (qof r,o ). After allowing the oil purchase on the spot market to overcome the operational infeasibility, the algorithm initiates a loop to try to eliminate the additional oil purchase by fixing the additional oil solution of the last iteration (TaticalOilFixi−1) as part of the oil allocation of the tactical model in the next iteration (TaticalOilFixi). The idea is to force the tactical level to buy the oil that the operation needs to meet the demand and product specification. The procedure verifies all of the additional oil purchases and terminates when the total additional oil quantity equals zero (line 12) or in case of tactical model or operational model infeasibilities (lines 6 and 10, respectively). While one of these conditions is not reached, the algorithm consists of the steps presented in Figure 2.

4. NUMERICAL EXAMPLE An industrial-scale study using real data from the Brazilian industry was used to evaluate the performance of the proposed models in optimizing large-scale problems. At the tactical level, the refining system includes 3 refineries (named R1, R2, and R3) and represents a general system that can be found at many industrial sites around the world. The refineries are coordinated centrally, the feedstock oil supply is shared, and the refineries collaborate to meet a given market demand. The refineries are supplied with 8 oil groups. The refineries process up to 50 intermediate products to produce 10 final products associated with the local market demand. The logistic network includes 6 terminals, 2 distribution bases, and 73 transportation arcs related to the road, water, rail, and pipeline modes. The time horizon in the tactical level covers 6 monthly periods. R1 is a small, low-complexity refinery that focuses on the production of lubricants, asphalt, and fuel oils. This refinery is supplied by 3 oils from a same group. As presented in Table 7, R1 processes up to 53 intermediate flows with 8 properties that must be controlled to specify the 17 final products. R2 can be G



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Table 7. Main Characteristics of the Three Refineries Studied

Table 8. Scenarios and Probabilities

Value 3

capacity (m /day) number of process units number of operational modes number of controlled properties number of intermediary flows number of final products

Tactical Scenarios

R1

R2

R3

scenario

1100 3 48 8 53 17

6550 9 52 68 57 18

39400 10 141 56 190 17

1 2 3 4 5 6 7 8 9

considered to be a low-complexity refinery that aims to produce solvents and fuels and processes 4 oils grouped into 3 groups. Finally, R3 is a medium-complexity refinery and processes oils from 7 groups. The production for R3 is focused on naphtha, but the refinery also produces significant amounts of jet fuel, diesel, and gasoline. The main characteristics of R2 and R3 are also shown in Table 7. The time horizon analyzed at the operational level covers the first 2 monthly periods of the tactical model. 4.1. Scenario Generation. The method used to create the scenarios of the stochastic models was based on data collection and direct contraposition of primary (data obtained from the Brazilian oil industry) and secondary research (historical economic data available online). The scenario generation with the associated probabilities was compared with the real problem that was studied and validated by experts from the oil industry, who considered that the proposed scenarios capture the uncertainty information. Developing methodologies for scenario generation is beyond the scope of this paper, and the interested reader can refer to the work by Kouwenberg,32 for example. The demand for refined products, oil prices, and product price uncertainties are considered in the tactical planning, whereas oil supply and process capacity per unit address the short-term uncertainties in the operational planning. The base case of each planning level (scenario 5 for the tactical and scenario 3 for the operational level) used data from the current planning system of the Brazilian refineries. The other scenarios were constructed based on the expertise of employees of the industry under study. Each stochastic parameter at the tactical level (price and demand) has three possible realizations (high, medium, and low, with probabilities of 25%, 50%, and 25%, respectively). Assuming that the random variables are independent, these parameters were combined to create the scenarios that are presented in Table 8. Complete dependence between the parameters of each scenario is assumed; i.e., high demand for one product implies high demand for the other products. A similar pattern is observed for the oil and product prices. At the operational level, the available capacity in the process units has three possible realizations that consider 3, 0, or 5 days of unplanned shutdown for maintenance (with probabilities of 25%, 50%, and 25%, respectively, which affects the total unit capacity available. Oil supply, which is the other operational uncertainty, has two possible realizations: normal supply or delays/changes in the oil received (with probabilities of 70% and 30%, respectively). In the first planning period, uncertainty is represented by a change in the oil received and, consequently, by changes in the associated oil specifications and yields. In the second period, uncertainty in the oil supply is represented by a delay in the amount of oil received, which reduces the total available oil. Combining the realizations of these stochastic parameters (capacity and oil supply) resulted in six scenarios for each refinery, which are presented in Table 8.

demand

price

probability (%)

high high high base high low base high base base base low low high low base low low Operational Scenarios

6.25 12.5 6.25 12.5 25 12.5 6.25 12.5 6.25

scenario

maintenance

oil supply

probability (%)

1 2 3 4 5 6

3 days 3 days 0 days 0 days 5 days 5 days

normal delays/changes normal delays/changes normal delays/changes

17.5 7.5 35 15 17.5 7.5

5. COMPUTATIONAL RESULTS AND DISCUSSION The models were implemented using Advanced Integrated Multidimensional Modeling Software (AIMMS)33 and solved using CPLEX 12.1. A personal computer (PC) with a 3.1 GHz Intel Core 2 Quad processor and 8 Gb RAM was used for all of the computations. Table 9 summarizes the model statistics. Table 9. Model Statistics operational planning levels

tactical

R1

R2

R3

number of variables number of constraints number of nonzeros solving time (s) E [margin] (million $)

96 899 119 105 218 286 0.78 707.9

5066 5222 14 893 0.03 27.2

6050 7070 20 695 0.05 123.9

14 520 14 232 44 175 0.06 292.9

The solution times presented in Table 9 represent the computational times when the models are solved separately. In the iterative approach, after the first run of the models, only reoptimization remains, which leads to irrelevant increments in the computational effort. In the following section, the benefits of the stochastic modeling and the integration approach are addressed. 5.1. Assessing the Stochastic Modeling Benefits. To evaluate the value added by properly including uncertainty in the problem parameters, the models can be evaluated using the expected value of perfect information (EVPI) and the value of the stochastic solution (VSS),34 which are the most widespread measures of stochastic modeling benefits. Another approach that has been validated by the industrial practice is using a simulationoptimization framework.35 The EVPI is represented by the difference between the solution to the problem obtained by the agent that solved the problem under uncertainty (recourse problem (RP)) and the one obtained by the agent able to make a perfect prediction (wait-and-see (WS), i.e., oil purchase decisions are postponed until the uncertainty is unfolded). In this numerical study, the EVPI reached a maximum of 1.55% of the WS solution for the H



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tactical case and 7.28% for the operational case, $11.16 million and $34.86 million, respectively. The lower the EVPI, the better the stochastic models accommodate uncertainties, because the stochastic objective function value was not far from the result obtained by the WS solution. However, because acquiring perfect information is not feasible in real life, the VSS can be considered to be a more-realistic result. The VSS is defined by the difference between the average solution of the expected value problem (replacing the random parameters by their means (EEV)) and the stochastic solution (RP). The VSS can be interpreted either as the benefit expected by the agent who has taken uncertainty into account or as the loss expected by the agent who opted for deterministic modeling using the average stochastic parameters. The VSS result for the tactical model, $98.95 million (13.76% of the WS solution) shows that the stochastic model provided a good solution, because an impressive gain was obtained by the inclusion of uncertainty into the problem. For the operational case, the EEV solution is infeasible due to the constraint on the minimum capacity of the distillation unit, which would lead to an unlimited operational VSS. This finding indicates that incorporating uncertainty into the problem could avoid infeasibilities at the operational level as the solution of the base case (average values) may become infeasible in the solution of other scenarios. 5.2. Assessing the Integration Benefits. To evaluate the proposed integration modeling approach, three stochastic cases are discussed. In these cases, the operational model evaluates the tactical base case (scenario 5 in Table 8) considering the operational uncertainties, which means that a solution to the tactical stochastic model was obtained and that the actual realizations of the tactical random parameters (price and demand) were considered to correspond to the base case of the tactical model. Uncertainties in oil supply and capacity of the process units affect operational planning, but the type of uncertainty varies from one case to another. As shown in Table 10, capacity uncertainty is

Table 11. Additional Oil Value refinery

case 1

case 2

case 3

0 771.4 707.9 1479.3

101.5 483.0 584.5 497.4 707.9 1205.3

32.5 56.0 1602.2 1690.7 156.6 707.9 864.5

3

amount of additional oil (km /month) R1 R2 R3 total oil purchase operational margin (million $) tactical margin (million $) operational + tactical margin (million $)

Table 12. Solutions by Iteration of the Iterative Approach in Case 2 E [margin] (million $) iteration

operational

tactical

operational + tactical

total additional oil (km3/month)

1 2 3

497.4 766.8 766.8

707.9 666.9 666.8

1205.3 1433.7 1433.7

584.5 12.1 0

As summarized in Table 13, in case 3, the iterative approach does not eliminate the additional oil purchase in its two first Table 13. Solutions by Iteration of the Iterative Approach in Case 3 E [margin] (million $) iteration

operational

tactical

operational + tactical

total additional oil (km3/month)

1 2 3

156.6 164.1 164.5

707.9 637.6 635.6

864.5 801.7 800.4

1690.7 238.1 0

Table 10. Cases for the Evaluation of the Integrated Approach type of uncertainty

case 1

case 2

case 3

capacity of the process units changes in the oil supply delays in the oil supply

×

× ×

×

iterations, because the purchase profile changed from one solution to another when the additional purchase decision was fixed as part of the tactical solution in the first iteration, which required, in the second iteration, a new amount of additional oil to adjust final product specifications. The fact that the three refineries required additional oil purchases until the end of the second iteration of the iterative method in case 3 is an indication that uncertainty in the quantity of oil received is not completely circumvented by the operational model. In this way, the integrated approach plays a important role ensuring the solution viability (even in losing optimality) when the additional oil is fixed as part of the tactical solution in iteration 3. This type of uncertainty could be addressed by keeping the additional oil purchased in the safety stock of the refineries. Safety stocks are a type of protection for delivery delays and changes in the oil specification but lead to high storage costs and prevent improvements in the supply-chain problems. For further discussion on how to determine and optimize safety stocks, the interested reader can refer to the works by You and Grossmann36 and Yue and You,37 for example. As a result of the large amount of additional supply and the consequent incurred costs of oil supply by long-term contracts when this amount was fixed to the tactical model, the sum of the tactical and the operational margins in the third iteration ($800.4) was 7.5% lower than that obtained in the first iteration ($864.5).

×

considered in the three cases; however, case 1 considers only uncertainty in capacity, whereas cases 2 and 3 also consider oil supply uncertainties. In case 2, the oil supply uncertainty is represented by a change in the oil received. Delays in the oil supply are considered in case 3. As shown in Table 11, in case 1, there is no additional oil purchase on the spot market. This finding indicates that the capacity uncertainty is well accommodated in this numerical example. In cases 2 and 3, on the other hand, additional oil is purchased. In case 2, oils are exchanged for oils from other groups. As shown in Table 12, in the first iteration, a total amount of 584.4 km3/month of additional oil was purchased for the operation of the three refineries. However, the total purchase decreased to 97.9% in the second iteration, demonstrating the effectiveness of the proposed approach. These results were translated into an increase of 18.9% in the sum of the tactical and operational objective function values (from $1205.3 in the first iteration to $1433.7 in the third one). Although the tactical margin decreased (from $707.9 to $666.8), the total margin (tactical + operational) increased, indicating a benefit of the iterative approach. I



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Conference of the Foundations of Computer-Aided Process Operations; AIChE Symposium Series 94; American Institute of Chemical Engineers: New York, 1998; p 75. (3) Grossmann, I. E.; van den Heever, S. A.; Harjunkoski, I. Discrete optimization methods and their role in the integration of planning and scheduling. Presented at Proceedings of Chemical Process Control Conference 6, Tucson, AZ, 2001. (Available via the Internet at http:// egon.cheme.cmu.edu/papers.html.) (4) Maravelias, C. T.; Sung, C. Integration of production planning and scheduling: Overview, challenges and opportunities. Comput. Chem. Eng. 2009, 33, 1919−1930. (5) Al-Qahtani, K.; Elkamel, A. Planning and Integration of Refinery and Petrochemical Operations; Wiley−VCH: Weinheim, Germany, 2010 (ISBN-10: 3-527-32694-4). (6) Neiro, S.; Pinto, J. A general modeling framework for the operational planning of petroleum supply chains. Comput. Chem. Eng. 2004, 28 (6−7), 871−896. (7) Pinto, J. M.; Joly, M.; Moro, L. F. L. Planning and scheduling models for refinery operations. Comput. Chem. Eng. 2000, 24, 2259− 2276. (8) Joly, M.; Moro, L. F. L.; Pinto, J. M. Planning and scheduling for petroleum refineries using mathematical programming. Braz. J. Chem. Eng. 2002, 19 (2), 207−228. (9) Escudero, L.; Quintana, F.; Salmerón, J. CORO, a modeling and an algorithmic framework for oil supply, transformation and distribution optimization under uncertainty. Eur. J. Oper. Res. 1999, 114 (3), 638− 656. (10) Dempster, M.; Pedron, N. H.; Medova, E.; Scott, J.; Sembos, A. Planning logistics operations in the oil industry. J. Oper. Res. Soc. 2000, 51 (11), 1271−1288. (11) Neiro, S.; Pinto, J. Multiperiod Optimization for Production Planning of Petroleum Refineries. Chem. Eng. Commun. 2005, 192 (1), 62−88. (12) Pongsakdi, A.; Rangsunvigit, P.; Siemanond, K.; Bagajewicz, M. J. Financial risk management in the planning of refinery operations. Int. J. Prod. Econ. 2006, 103, 64−86. (13) Khor, C. S.; Elkamel, A. Optimization Strategies: Petroleum Refinery Planning under Uncertainty; VDM Verlag Dr. Mueller e.K. Publishing House: Saarbrücken, Germany, 2008. (14) Luo, C.; Rong, G. A Strategy for the Integration of Production Planning and Scheduling in Refineries under Uncertainty. Chin. J. Chem. Eng. 2009, 17 (1), 113−127. (15) Janak, S. L.; Lin, X.; Floudas, C. A. A New Robust Optimization Approach for Scheduling under Uncertainty: II. Uncertainty with Known Probability Distribution. Comput. Chem. Eng. 2007, 31, 171− 195. (16) Khor, C. S.; Elkamel, A.; Ponnambalamb, K.; Douglas, P. L. TwoStage Stochastic Programming with Fixed Recourse via Scenario Planning with Economic and Operational Risk Management for Petroleum Refinery Planning under Uncertainty. Chem. Eng. Process. 2008, 47, 1744−1764. (17) Ribas, G. P.; Leiras, A.; Hamacher, S. Operational planning of oil refineries under uncertainty. IMA J. Manage. Math. 2012, 23, 397−412. (Special issue: Applied Stochastic Optimization.) (18) Dantzig, G. Linear Programming Under Uncertainty, Manage. Sci., 1955, 50 (12 Supplement), 1764−1769. (19) Bodington, C. E. Planning, Scheduling, and Control Integration in the Process Industries; McGraw−Hill: New York, 1995. (20) Candler, W.; Townsley, R. A Linear Multilevel Programming Problem; unpublished research memorandum; World Bank Development Research Center (DRC): Washington, DC, 1978. (21) Fortuny-Amat, J.; McCarl, B. A representation and economic interpretation of a two-level programming problem. J. Oper. Res. Soc. 1981, 32, 783−792. (22) Ben-Ayed, O. Bilevel linear programming. Comput. Oper. Res. 1993, 20 (5), 485−501. (23) Visweswaran, V.; Floudas, C. A.; Ierapetritou, M. G.; Pistikopoulos, E. N. A decomposition-based global optimization approach for solving bilevel linear and quadratic programs. In State of

6. CONCLUSIONS The purpose of this paper was to discuss the problem of spatial and temporal integration of the tactical and operational planning levels for multisite refinery networks. Two stochastic mathematical programming models were developed to improve the planning of oil refineries under uncertainty. The models were applied to an actual refining system in Brazil. The values obtained for the expected value of perfect information (EVPI) and the value of the stochastic solution (VSS) indicated the benefit of incorporating uncertainty into the dominant random parameters of each planning level and also highlighted the need for integrated planning for multirefinery networks. The optimization results are relevant to the real planning activities of the oil industry. The centralized coordination (spatial integration) and the shared feedstock oil supply provided a better utilization of the available resources in meeting a given market demand. The proposed iterative approach offers decision-makers a holistic view of the problem and allows an accurate evaluation of its uncertainties, which can result in a competitive advantage in the uncertain oil refining business. The integrated modeling approach (temporally integrated) avoided infeasibilities at the operational level, as the tactical allocation considered the operational uncertainties. The benefits could be measured not only by the decrease in oil purchases on the spot market but also by increases in the total expected margin. For future works, the following extensions may be considered: (1) application of the iterative approach to a larger case study; (2) implementation of the full-space approach; (3) expansion of the scenario tree; (4) utilization of a decomposition method; and (5) inclusion of a financial risk measure in the stochastic tactical model. A scenario generation method based on property matching technique proposed by Høyland and Wallace38 may be used to generate a limited number of discrete scenarios that satisfy specific statistical properties. A decomposition method, such as Benders decomposition39 or Lagrangean decomposition,40 is useful to overcome the challenge of solving large-scale problems and allows to simultaneously solve the tactical and operational stochastic models. Since the approach proposed in this paper is risk-neutral, a financial risk measure, such as conditional value-at-risk (CVaR),41,42 may be included to treat the randomness in price coefficients of the objective function in the tactical level and to maximize the expected return by restricting the number of scenarios that will yield low returns or a return below a minimum value set for the CVaR constraint.

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AUTHOR INFORMATION

Corresponding Author

*E-mail address: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank the Brazilian Federal Agency for Support and Evaluation of Graduate Education (Coordenaçaõ de ́ Superior, CAPES). Partial Aperfeiçoamento de Pessoal de nivel funding from NSERC is also appreciated.

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REFERENCES

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