Risk Management in the Oil Supply Chain: A CVaR Approach

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Risk Management in the Oil Supply Chain: A CVaR Approach Maria C. Carneiro,†,‡ Gabriela P. Ribas,‡ and Silvio Hamacher*,‡ Petrobras, AV. República do Chile, 65, Centro, Rio de Janeiro, 20031-912, Brazil, and Department of Industrial Engineering, Pontifı´cia UniVersidade Cato´lica, Rua Marques de Sao Vicente 225/950-L, GaVea, Rio de Janeiro 22453-900, Brazil

This study analyzes the strategic planning of an oil supply chain. To optimize this chain, a two-stage stochastic model with fixed recourse and incorporation of risk management was developed. The model took a scenariobased approach and addressed three sources of uncertainty. To deal with these uncertainties, the conditional value-at-risk (CVaR) was adopted as a risk measure, and then the model was applied to the supply chain of six oil refineries. The goal of the study was to maximize the expected net present value, E(NPV), of the supply chain under analysis. The results indicate that the optimization of the several scenarios yielded an E(NPV) variation that reached US$ 36 million. Such a significant difference demonstrates that taking uncertainties into consideration is a fundamental step in decision-making processes. 1. Introduction The oil and gas supply chain is a vertically integrated chain that includes raw material production, transportation to refineries, transformation into tradable refined products, and distribution to consuming markets. Supply chain planning in the oil and gas sector involves uncertainties, as it is difficult to anticipate some of the parameters that need to be taken into account, such as the price of oil and products, oil supply, and product demand, to name just a few. The volatility of oil prices is a good example: the Organization of the Petroleum Exporting Countries’ average oil price rose from US$ 28.1 per barrel in 2003 to US$ 109.4 per barrel in 2008.1 These uncertainties become especially paramount in long-range planning, where decision-making processes call for high investments in interdependent projects. The uncertainty in optimization models for the oil and gas industry has been noted by several academic studies conducted in the past several years. The main methods employed in these studies are stochastic programming,2-6 dynamic programming,3,7 stochastic robust programming,8-10 and fuzzy programming.11,12 Escudero et al.2 formulated a supply, refining, and transportation problem for the oil sector as a two-stage model based on scenario analysis. Labadidi et al.4 developed a two-stage model with fixed recourse, to treat uncertainties in a problem of supply chain planning in a petrochemical company. Pongsakdi et al.8 advanced a two-stage stochastic model to determine how much crude oil should be bought by a refinery and its level of production given demand estimates. Khor et al.9 treated the problem of medium-range planning of a refinery operation by using stochastic programming (a two-stage model) and robust programming. Al-Othman et al.6 proposed a two-stage stochastic model for multiple time periods to optimize the supply chain of an oil company installed in a country that produces crude oil. Despite the many studies of optimization problems under uncertainty in the oil industry, few consider risk management. In Pongsakdi et al.,8 value-at-risk (VaR) measures13 and ratio area risk (RAR) measures were used to study financial risk aspects in the acquisition of crude oil. Lakkhanawat and Bagajewicz10 continued the work of Pongsakdi et al.,8 including product pricing in refinery operational planning with financial risk management. Khor et al.9 applied Markowitz’s mean variance (MV) and the * To whom correspondence should be addressed. E-mail: hamacher@ puc-rio.br. † Petrobras. ‡ Pontifı´cia Universidade Cato´lica.

mean absolute deviation (MAD) to handle randomness in the objective function coefficients and to ensure a robust solution. Although some of the studies mentioned above apply risk measures to solve planning problems in the oil industry, none manage the risk in the portfolio optimization problem in the oil supply chain. To fill this gap in the existing literature, the goal of this work is to include financial risk management in the integrated oil supply chain planning under uncertainty, using the framework of two-stage stochastic programming and applying the conditional value-at-risk (CVaR) measure to design an optimal portfolio. We propose a model that takes into account the entire oil supply chain, from exploration fields to distribution centers, and allows for the analysis of investments in both refineries and logistical infrastructure. To evaluate the optimization model, a case study with data from an actual supply chain in the Brazilian oil sector is used, as well as 12 discrete scenarios. To meet the established goals, the next sections of this article outline the main risk measures used in the portfolio optimization models (section 2) and describe the problem under study (section 3) and the proposed two-stage stochastic model (section 4). Then, we describe the scenarios used to evaluate the model (section 5) and present related results (section 6). The article concludes in section 7 by stating the importance of risk management in optimization problems under uncertainty, in investment portfolios for the oil industry. 2. Portfolio Optimization Measures of risk play a critical role in the optimization of portfolios under the presence of uncertainties. To minimize risks associated with expected portfolio returns, risk management tools such as diversification of portfolio investments have been widely used. Although the return of these investments is generally represented by the expected values of the return distributions, there are several approaches to portfolio risk modeling. Markowitz14 pioneered work in this area, advancing a model that selects the best portfolio by analyzing the tradeoff between risk and return. Such a model, known as Markowitz’s meanvariance model, draws on statistical parameters to solve problems of portfolio optimization. According to this meanvariance model, whereas expected return values represent the forecast return, return variance measures represent the risk. However, Markowitz’s model has not been frequently used in its original concept to construct large-scale portfolios.

10.1021/ie901265n  2010 American Chemical Society Published on Web 03/10/2010

Ind. Eng. Chem. Res., Vol. 49, No. 7, 2010

Prominent among the reasons for this is the computational difficulty associated with optimal solutions for problems of quadratic optimization with an extensive covariance matrix.15 Other criticisms of Markowitz’s model include the instability of optimal portfolios obtained through this model: the slightest variation in entry parameters yields different solutions.16 In addition, minimization of the variance might not be an adequate measure of risk in portfolios, as it impacts both positive and negative deviations from the mean.17,18 Difficulties with implementation of Markowitz’s meanvariance model led to the development of alternative models of portfolio optimization. For example, value-at-risk (VaR) has been widely used as a measure of risk by financial institutions worldwide.19 It adopts statistical techniques and is an easy-tounderstand tool for risk management.13 The VaR concept measures the worst loss a financial institution can expect to suffer over a given period of time, at a given confidence level. Although the VaR measure has become a popular measure of risk, it also presents some deficiencies, among them a lack of consistency, as it violates subadditivity.20 Nonsubadditivity means that the risk of a portfolio can be higher than the sum of the risks of its individual components. Another criticism points out that a VaR measure does not provide any information about the severity of a loss when it is exceeded; rather, it gives information only about the frequency of losses.21 Furthermore, in practice, a VaR measure might be inadequate because of its lack of convexity, which can lead to many local extremes and an unstable risk rating.18,22 Criticisms of the VaR approach resulted in new proposals for ways to measure risk in portfolios. One of them is the conditional value-at-risk (CVaR) measure, also known as mean excess loss, mean shortfall, or tail VaR. This measure is considered more consistent than the VaR.23 It is defined as the mean loss by which the value-at-risk (VaR) is exceeded;24 in other words, the CVaR is the conditional expected loss of a portfolio at a confidence level of β (%), given that the loss to be accounted for exceeds or equals the VaR. By definition, the VaR at confidence level β is never higher than the corresponding CvaR. Consequently, portfolios with a low CVaR also show a low VaR.23 However, in many cases, it is better to maximize returns with risk constraints. That is, it is better to specify a maximum level of risk. In doing so, Krokhmal et al.24 went beyond Rockafeller and Uryasev’s23 proposal. In their mathematical formulation,24 Krokhmal et al. represent the uncertainties using S scenarios N

maximize

∑xµ

i i

(1)

i)1

subject to S

R+

∑

1 u eK (1 - β)S s)1 s s ) 1, ..., S

us g 0

(2) (3)

N

∑xr

i is

+ R + us g 0

s ) 1, ..., S

(4)

i)1

N

∑x

i

)1

(5)

i)1

xi g 0

i ) 1, ..., N

(6)

where N is the number of candidate assets, xi is the capital fraction applied on candidate asset i, µi is the return expected of the ith candidate asset, R is the variable that provides the

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portfolio’s VaR and CVaR at confidence level of β (%), β is the confidence level to compute the CVaR measure of risk, S is the number of scenarios, us is the auxiliary variables to compute the CVaR measure, K is the upper bound for the portfolio’s CVaR, and ris is the return expected of the ith candidate asset in scenario s. Objective function 1 represents the expected portfolio return, that is, the weighted mean of the individual returns. Constraints 2-4 model the portfolio’s CVaR, which must be smaller than or equal to a limit K specified by the investor. Parameter us is used as an auxiliary variable to calculate the CVaR. For loss scenarios exceeding the portfolio’s VaR (represented by variable R), variable us assumes a value greater than 0. Such a value is considered in the left side of constraint 2, limited by a maximum value K. The greater the risk taken (the higher the value of K), the larger the average of the (1 - β)% worst returns. Constraint 5 ensures that all of the available capital is invested, and constraint 6 does not allow a negative percentage to be invested in any asset. Rockafeller and Uryasev23 presented an approach to the optimization of portfolios that computes the VaR and optimizes the CVaR simultaneously. Based on losses associated with a decision vector that represents the portfolio and with a random vector that represents uncertainties, the authors demonstrated that the CVaR can be efficiently minimized by means of linear programming techniques that allow for the treatment of portfolios with a large number of assets as well as uncertainties, representing a large number of scenarios. Although the approach of Rockafellar and Uryasev23 takes into account the minimization of CVaR, it also requires a minimum expected return. In addition to minimizing the CVaR, it also minimizes the VaR of the portfolio, given that the CVaR exceeds or equals the VaR. In this work, the conditional value-at-risk (CVaR) measure was adopted because it is a consistent measure of risk with good mathematical properties, which enables efficient optimization by means of linear programming techniques. 3. Problem Description The supply chain in the oil industry consists of several activities that can be grouped into two main segments: upstream and downstream. The upstream segment includes well prospecting, exploration, drilling and completion, and oil production. The downstream segment includes transportation, refining, and distribution of crude oil and refined products. Khor25 emphasized how complex the oil refining activity is and stated that it is certainly one of the most complex chemical industries, including several processes with various possible configurations and structures. The main objective of a refinery is to transform crude oil into refined products of higher aggregate value, such as gasoline and diesel oil, so as to generate the maximum profit possible. Each refinery has a set of process units, storage tanks for final and intermediary products, and pipes interconnecting all of the components. Refining therefore comprises the arrangement of all of the process units in a refinery. Figure 1 depicts a refining flow diagram of a refinery located in Brazil. The refinery processes the crude oil into marketable products through three main process types: separation, conversion, and treatment. The separation processes [crude distillation (DA), vacuum distillation (DV), etc.] are designed to separate the oil into its basic fractions or process a previously produced fraction in order to produce a specific group of components. The conversion processes [cocker (K), fluid catalytic cracker (FCC), catalytic reforming (RC), etc.] transform one fraction into

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Figure 1. Refinery flowchart.

another or change the molecular structure of a fraction. The treatment processes [hydrodesulfurization (HDS), hydrotreatment (HDT), etc.] provide better cutting of semifinished products by removing or reducing contaminants (sulfur, nitrogen, metals) in their structure. The treatment processes can be used in light fractions [liquefied petroleum gas (LPG), naphtha], medium fractions [aviation kerosene (querosene de aviac¸a˜o, QAV), diesel (DSL)] or heavy fractions [light cycle oil (LCO), lubricants (LUB)]. Choosing the best configuration for a refining park and determining the ideal plan for operations and transportation are difficult tasks, because of the high number of variables and constraints present in these processes. Mathematical programming plays a central role in solving this problem, assisting in the decision-making process and in the planning of downstream activities at the strategic, tactical, and operational levels. In Brazil, most refineries are found in the Southeast region of the country. In this region, there are six refineries, together having the capacity to process 65% of the total Brazilian output (1.2 million barrels of oil per day).26 This study takes into account the entire supply chain of these refineries in Southeast Brazil, including locations of oil production and final product demands and the logistics system associated with them. In the model, 11 types of oil and their different characteristics are represented, as well as their prices. Beginning with internal transformation processes, the different types of oil undergo subsequent transformations and result in 10 types of tradable refined products during each planning year, all in compliance with quality specifications. The logistics network consists of six refineries, five terminals, three oil production fields, one Brazilian consumer base, and four international bases, interconnected by pipeline, railroad, and marine modes of transportation, totaling 159 arcs. In addition to these components, the model also takes into consideration the influence of vegetable oil (directed to biodiesel production) and natural gas supply. Figure 2 shows a schematic logistics network diagram.

4. Mathematical Formulation This work addresses the portfolio optimization problem in the integrated oil supply chain in order to satisfy both fuel specifications and national demand with the highest profit. We use a two-stage stochastic program with fixed recourse that embraces the network configuration, production and customer assignment, and production and transportation planning. Uncertainty is introduced through the crude oil supply, the Brazilian demand for final products, and product and oil prices in the Brazilian and international markets. The two-stage stochastic model represents the network as shown in Figure 2 and all the transformation processes inside the refineries. The investment variables are defined as first-stage decisions. The oil blends used to feed the refinery, the intermediate product flows between process units, the production of refined products, and the flow of oil and refined products throughout the logistical network constitute the variables of second-stage decisions. The optimization model is based on a scenario analysis approach and is linear. Following Pongsakdi et al.,8 Lakkhanawat and Bagajewicz,10 and Al-Qahtani and Elkamel,27 many nonlinear features were simplified in a reasonable manner so that we could gain computation speed, which allowed us to better explore the uncertainty and the economic risk. The approach we propose for risk management adopts the CVaR measure to maximize the net present value (NPV) and related return distributions. The stochastic optimization model under CVaR constraints treats randomness in price coefficients of the objective function and, in doing so, maximizes the expected return by restricting the number of realization scenarios that will yield low returns or a return below a minimum value set for the CVaR constraint. We assume that uncertainties are discretely represented by S possible realization scenarios. We further assume that the probability that the Sth scenario will occur is represented by ps (ps g 0, ∑Ss)1ps ) 1). Based on these assumptions, the stochastic

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model proposed in this work is represented in sections 4.1-4.4 (for definitions, see Nomenclature). 4.1. Objective Function. The objective of the model is to maximize profit computed as the difference between the total investment cost (IC) associated with first-stage decisions and the expected operational profit (OPs) associated with secondstage decisions. The optimization model is based on a discretization of the time horizon (using 1-year time periods), and all terms of the objective function are brought to present value to the initial date of the planning horizon.

(

max -IC +

∑

)

IC )

∑

n∈N

[

r,uVr,u,n)

+

r∈R u∈U

(1 + AT)

∑

ut∈UT (n-1)

s

s

)

s)1

s

(VTCutvtut,n)

]

r∈R o∈O n∈N

r,o,n,s

(n-1)

in,po,n,s

n∈N

in∈I po∈P

]

∑ {[ ∑ ∑ (OPE

n∈N

in∈I o∈O

[

(n-1)

}

}

-

-

]

- OPIin,o,n,soimpin,o,n,s) /

∑ ∑× r∈R n∈N

∑ ∑∑∑ (1 + AT) ] - ∑ ∑ [ ∑ pt TC ∑ ot TC /(1 + AT) OCr,n

(otut,o,n,sUTEut,i1,r,m,o)/

ut∈UT i1∈I m∈M o∈O (n-1)

ut,po,n,s

ut∈UT n∈N

ut

+

po∈P

(n-1)

ut,o,n,s

ut

o∈O

]}

(9)

4.2. Refinery Constraints. (a) Refining Balance. For each refinery (r), process unit (u), operational mode (c), product (po), period of time (n), and scenario (s), the sum of the entry flows should be equal to the sum of the output flows. DY + ∑ ∑ ∑ dfr PUY + ∑b ∑ ∑ ∑ pfr ) ∑b + ∑ ∑ pfr ir CP + or ∑ ∑ ∑ pfr r,u,c,o,po

u∈U c∈C o∈O

r,u,c,pi,n,s

Figure 2. Schema of the integrated supply chain in the oil industry.

in,po,n,s

PPIin,po,n,spimpin,po,n,s) /(1 + AT)(n-1) +

r,u,c,pi,po

u∈U c∈C pi∈P

(b) Operational Profit (OPs). OPs includes the refined oil and product sales, the oil and product exports and imports, the refining operation cost, and the transportation cost. The secondstage decisions associated with the operational profit (OPs) are the amounts of product and oil transported (ptut,po,n,s and otut,o,n,s, respectively), the amounts of product exported and imported (pexpin,po,n,s and pimpin,po,n,s, respectively), and the amounts of oil exported and imported (oexpin,o,n,s and oimpin,o,n,s, respectively).

ut,i1,r,m,o)/

ut,o,n,s

ut∈UT i1∈I m∈M

r,u,c,o,n,s

(8)

b,po,n,s /

b,po,n,s

b∈B po∈P n∈N (n-1)

s)1

(1 + AT)

(a) Investment Cost (IC). IC represents the refinery investment cost and the transportation investment cost. The first-stage decisions associated with the investment cost (IC) are refinery investments (Vr,u,n) and transportation investments (Vtut,n). In this model, we consider these first-stage decisions as linear variables. This suits a strategic planning model, in which the investment projects are nominal and can be scaled within a given range.

∑ ∑ (VC

∑ P OP

3289

S

in,o,n,soexpin,o,n,s

(7)

PsOPs

s∈S

∑ P { ∑ ∑ ∑ [PPBR PD (1 + AT) ]+ ∑∑ ∑ × [OPBR ∑ ∑ ∑ (ot UTE (1 + AT) ] + ∑ {[ ∑ ∑ (PPE pexp

S

r,pi,po,n,s

+

pi∈P

r,po,n,s

r,po,pi,n,s

pi∈P

r,u,c,po,n,s

+

u∈U c∈C

r,u,c,pi,n,s

r,u,po

r,po,n,s

u∈U c∈C pi∈P

∀r ∈ R, ∀pi ∈ P, ∀n ∈ N, ∀s ∈ S

(10)

(b) Refining Operation Constraints. These constraints establish the proportion (PERCr,u,pi,c,n) between the entry flows (pi) and the total process unit (u) loading.

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pfrr,u,pi,c,n,s ) PERCr,u,pi,c,n

∑ pfr

∑ ∑ ∑ (pt

ut,po,n,sUTEut,i2,i1,m,po)

r,u,pi,c,n,s

pi∈P

∀r ∈ R, ∀u ∈ U, ∀pi ∈ P, ∀c ∈ C, ∀n ∈ N, ∀s ∈ S PERCr,u,pi,c,n

∑ pfr ∑ pfr

r,u,pi,c,n,s

VOPi1,po,n + ori1,po,n,s ) PDi1,po,n,s +

(11)

∑ ∑ ∑ (pt

e pfrr,u,pi,c,n,s e

r,u,pi,c,n,s

pi∈P

∀r ∈ R, ∀u ∈ U, ∀pi ∈ P, ∀c ∈ C, ∀n ∈ N, ∀s ∈ S

(12)

(c) Refining Capacity Constraints. Equation 13 limits the maximum and minimum capacities of process unit u in period n according to the initial capacity (UCr,u) and the accumulated n Vr,u,t). investments in refining in the same period (∑t)1 n

UCr,u + UC′r,u

∑V

r,u,t

∑ ∑ dfr

e

r,u,o,c,n,s

t)1

∑∑

+ n

pfrr,u,pi,c,n,s e

pi∈P c∈C

+

∑

+ UC′r,u

Vr,u,t

∀r ∈ R, ∀u ∈ U, ∀n ∈ N, ∀s ∈ S

(13)

∑∑∑ ( ∑ ∑ ∑ pfr

pfrr,u,pi,c,n,sSIOpi,nPUYr,u,c,pi,po e

u∈U pi∈P c∈C

r,u,c,pi,n,sPUYr,u,c,pi,po

u∈U c∈C pi∈P

)

SPOpo,n

∀r ∈ R, ∀po ∈ P, ∀n ∈ N, ∀s ∈ S

BI ) + ∑ ∑ ∑ dfr DY + ∑ (b ( PUY ∑ ∑ ∑ pfr )BI e VPO ( ∑ b + ∑ ∑ ∑ dfr DY + PUY ∑ ∑ ∑ pfr ) pi,n

r,u,c,o,n,s

(14)

r,u,c,o,po

u∈U c∈C o∈O

r,u,c,pi,n,s

r,u,c,pi,po

po,n

po,n

u∈U c∈C pi∈P

r,pi,po,n,s

pi∈P

ut,o,n,sUTEut,i2,i1,m,o)

i1,u,o,c,n,s

u∈U c∈C

r,u,c,o,n,s

r,u,c,o,po

r,u,c,pi,n,s

+ FPi1,o,n,s )

∑ ∑ ∑ (ot

+

ut,o,n,sUTEut,i1,i2,m,o)

ut∈UT i2∈I m∈M

∀i1 ∈ I, ∀o ∈ O, ∀n ∈ N, ∀s ∈ S

ptut,po,n,s +

(17)

∑

n

otut,o,n,s e CTut + CT′ut

o∈O

∑ vt

ut,t

t)1

∀ut ∈ UT, ∀n ∈ N, ∀s ∈ S

(18)

(c) Export and Import Limits. The amounts of product (po) and oil (o) imported are limited by the external supply, which defines minimum and maximum values to be imported. The exports of product (po) and oil (o) are limited by minimum and maximum international market demands. + PSEin,po,n e pimpin,po,n,s e PSEin,po,n ∀in ∈ IN, ∀po ∈ P, ∀n ∈ N, ∀s ∈ S

(19)

+ OSEin,o,n e oimpin,o,n,s e OSEin,o,n ∀in ∈ IN, ∀o ∈ O, ∀n ∈ N, ∀s ∈ S

(20)

+ PDEin,po,n e pexpin,po,n,s e PDEin,po,n ∀in ∈ IN, ∀po ∈ P, ∀n ∈ N, ∀s ∈ S

(21)

+ ODEin,o,n e pexpin,o,n,s e ODEin,o,n ∀in ∈ IN, ∀o ∈ O, ∀n ∈ N, ∀s ∈ S

(22)

4.4. Risk Management Constraints. Equations 23-25 model the CVaR constraints in the portfolio at a confidence level of β (%). The variable R provides the portfolio’s VaR at confidence level β, whereas the parameter K is the lower bound for the portfolio’s CVaR (a value required by the investor) and the expression OPs - IC is the return for each scenario. The formulation presented is based on the distribution of losses. Marzano et al.29 adapted this formulation to work in the case with return distributions. R+

u∈U c∈C o∈O

(16)

(b) Logistic Capacity Constraints. Equation 18 limits the maximum volume transported by a transportation arc (ut) in the period n.

po∈P

(d) Environmental Legislation Requirements. Final product properties must be within a range established by environmental regulations. Property calculations yield a set of nonlinear constraints.28 Nonlinear terms result from the multiplication between the products’ properties and their volumes; these terms can be linearized by estimating the properties of intermediate products. At the strategic level, it is possible to estimate the sulfur content (SIOpi,n) and the viscosity (BIpi,n) of the intermediate products with sufficient accuracy, making the constraint that controls the final products’ properties linear. Equations 14 and 15 limit the sulfur contents and viscosities, respectively, of the final products (po).

r,pi,po,n,s

∑ ∑ ∑ (ot ∑ ∑ dfr

ut∈UT i2∈I m∈M

t)1

pi∈P

+ iri1,po,n,s

∀i1 ∈ I, ∀po ∈ P, ∀n ∈ N, ∀s ∈ S

∑

o∈O c∈C

+ UCr,u

ut,po,n,sUTEut,i1,i2,m,po)

ut∈UT i2∈I m∈M

pi∈P + PERCr,u,pi,c,n

+ NGPi1,po,n +

ut∈UT i2∈I m∈M

1 (1 - β)

∑p u

s s

gK

(23)

s∈S

r,u,c,pi,po

u∈U c∈C pi∈P

∀r ∈ R, ∀po ∈ P, ∀n ∈ N, ∀s ∈ S

(15)

The model controls only sulfur content and viscosity because these two properties are the ones that most affect strategic decisions such as investment in new process units. 4.3. Logistic Constraints. The parameter UTEut,i1,i2,m,po represents a transportation arc (ut) that stores the information of an origin node (i1), a destination node (i2), a transportation mode (m), and a product (po). (a) Logistic Balance. For each node (i), product (po) or oil (o), period of time (n), and scenario (s), the sum of the input flows should be equal to the sum of the output flows.

us e 0

∀s ∈ S

us e OPs - IC - R

∀s ∈ S

(24) (25)

In the proposed stochastic model, decision variables in the two stages are related to each other by means of constraints on refining (eq 13) and logistics (eq 18) capacities. Prices of products and oil in the Brazilian and international markets, which are represented in the objective function and are associated with second-stage decision variables, vary from scenario to scenario (eq 9). Oil production and the Brazilian demand for final products also depend on scenario indexes and are represented as constraints on the oil logistics balance (eq 17) and on

Ind. Eng. Chem. Res., Vol. 49, No. 7, 2010

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Figure 3. Construction of 12 scenarios.

the products logistics balance (eq 16). Furthermore, we depart from the assumption that this problem is a fixed-recourse problem. Therefore, variable coefficients in second-stage decisions are fixed to all constraints. There is no particular condition that associates second-stage decisions with different scenarios. 5. Scenario Construction The sources of uncertainty present in the Brazilian oil chain can be sufficiently well represented by the stochastic parameters adopted in this model: oil supply derived from new findings; demand for final products according to the different economic growth scenarios; and oil and product prices, which rely heavily on geoeconomic variables. The technological coefficients, however, present little variation along the planning horizon. The parameters adopted were also used by other authors who analyzed the oil chain.4,6,8,11 Based on probable oil supply and on final product demands and prices, 12 scenarios were created. These scenarios are represented in Figure 3. The basic scenario (base) used in this study was constructed with data from the Brazilian Ten-Year Plan for Energy Expansion (PDE),26 designed by a governmental research agency specializing in the energy sector. This agency develops studies and research toward energy planning in Brazil. The base scenario takes as its premises a high increase in oil consumption and in oil reserves recently discovered and yet to be discovered in Brazil. The oil production and product demand scenarios were generated based on the growth tendencies described in PDE:26 the growth tendency of light and medium oil production is weak, whereas the production of heavy oil tends to increase significantly. The heavy oil supply scenario (heavy oil, 33%) takes into account an increase in heavy oil production according to PDE.26

The same report indicates a significant increase in the demand for oil products. The scenario with high demand for refined products (high, 33%) considers a demand for refined products 15% higher than that in the base scenario in subsequent time periods. The discrete scenarios of oil and product prices were constructed by applying the mean-reversion stochastic process methodology. This process is used to model commodity prices, especially in long-term scenarios.30 There is a strong correlation between the prices of oils and of products, which was included in these scenarios. The scenarios with high prices (high, 30%) and low prices (low, 30%) represent possible fluctuations in the prices of oil and products. One productsgasolinespresents a different price correlation. In Brazil, gasoline competes directly with ethanol as a car fuel. The prices of the latter vary significantly as a consequence of crop yields and the sugar market, and therefore, gasoline is also prone to greater price variations according to the different scenarios. In addition, this study takes into account two possible oil supply scenarios, two possible scenarios of demand for final products, and three possible price realization scenarios, totaling 12 possible scenarios. 6. Results The model was generated with the AIMMS31 modeling tool and optimized with the CPLEX 10.1 solver, in a hardware platform with 2 GB of memory and a 1.80 GHz processor. To run this model with 12 scenarios, the average processing time was 6 min. The linear programming model consists of approximately 79500 variables and 83700 constraints. To evaluate impacts of the adopted risk measures in the model under study, four catalytic reforming units in distinct refineries and one hydrodesulfurization (HDS) unit were considered as

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Ind. Eng. Chem. Res., Vol. 49, No. 7, 2010 Table 2. Optimum Investment Profile

Table 1. Candidate Process Units refinery I II III III IV

process unit

capacity (m3/day)

catalytic reforming catalytic reforming HDS D50 catalytic reforming catalytic reforming

1000 2500 6000 1500 2300

candidate projects to design a portfolio of investments for the oil and refined-products supply chain. These units are listed in Table 1. The model indicates the percentages (a maximum of 100%) to be invested in each of the units and the operations startup date (year) for each of them. Simulations with varying K and β values were run. Whereas K represents the minimum return required by the investor, β represents the conditional expected loss in a portfolio. For example, for β ) 95%, the CVaR will be given by the mean of the highest 5% of losses. The lower the values of K and β, the higher the risk taken by the investor. Figure 4 shows a plot of the variations in the expected NPV of the operational results (the objective function) given a variation in the risk level (K). Four curves with different confidence levels were plotted (β ) 99%, 97%, 95%, and 93%). They represent the efficient boundaries of the investment portfolios using CVaR for the model under analysis. The points in Figure 4 represent an optimal result for each combination of K and β in the model. Note that the variations in the expected value of the objective function [∆E(NPV)] increase with risk. The greater the value of K (lower bound), the higher the return in the worse scenarios (i.e., the risk decreases). Point A represents the point of highest risk and highest expected return, whereas points B, C, D, and E represent the points of lowest risk given confidence level β. At point A, the value of K is equal to the result of the worst scenario in the situation without risk constraints, so the solution at point A is the same as that with unrestricted risk. The variations in the expected NPV represented by Figure 4 reached US$ 36 million as a function of the parameters ∆K and β. If we consider the amount invested by the candidate units (approximately US$ 300 million), this variation is significant (approximately 12% of the investment). It is up to decision makers to choose the level of risk they are willing to face. Note that the lower the value of β, the higher the variation in the expected NPV for the same value of K. That is, if a lower level of confidence is taken into account, the model indicates an investment configuration that yields a higher global expected return. However, in this case,

Figure 4. Portfolio efficient boundaries using CVaR.

A refinery

process unit

I II III III IV

catalytic reforming catalytic reforming HDS D50 catalytic reforming catalytic reforming

B

time investment time investment (years) (%) (years) (%) 9 5 1 6 2

39 100 39 98 33

5 5 1 6 -

6 100 90 100 -

some scenarios could yield results lower than those required by the investor (K). Table 2 presents the optimum installation percentages of each unit and the time for operation startup of points A and B identified in Figure 4. In the situation without risk consideration (point A), none of the scenarios is forced to be equal to or greater than a return value of K. At point B, the CVaR restriction is considered, and 99% of the scenarios (i.e., in our case, all scenarios) are kept equal to or higher than a given value of K; therefore, the investment profile at B is different from that at A. The scenarios with lowest returns in the model are those with low prices and high demand (scenarios 4 and 10 in Figure 3). In these cases, the prices of gasoline are proportionally lower than those of other products. Because the catalytic reforming units are mostly dedicated to gasoline production, at point B, fewer investments are made to expand these units, thus increasing the amount of imported gasoline. Moreover, two types of diesel fuel are considered in the model: one with low sulfur content and one with higher sulfur content, the latter being cheaper than the former. In the case of scenarios with low prices and high demands, there is a significant reduction in the export of highsulfur diesel fuel because of the low returns yielded by this product. At point B, this export excess is turned into low-sulfur fuel by the HDS unit, to be consumed by the Brazilian market. 7. Conclusions This study developed a methodology to optimize investment portfolios in the presence of uncertainty. Specifically, it evaluated the long-range planning of an investment portfolio in an integrated supply chain of the oil sector. For this purpose, a two-stage stochastic model was applied to the study of six largescale refineries located in Southeast Brazil. The case study contemplated management of the entire supply chain, and the model took into account uncertainties such as Brazil’s internal demand for final products, oil supply, and price, as well the prices of products in the Brazilian market.

Ind. Eng. Chem. Res., Vol. 49, No. 7, 2010

These uncertainties can be highly significant; for example, the average weekly oil prices from September 2006 to September 2008 varied from a minimum of US$ 48.20 per barrel to a maximum of US$ 137 per barrel. In addition, the Brazilian oil supply is likely to vary significantly with the discovery of new oil fields, which might soon position the country among the 10 largest oil producers in the world. Given this state of affairs, an analysis in the presence of uncertainty becomes fundamental, because uncertainties will highly impact the model’s NPV. The results indicate that the optimization of the several scenarios yielded an NPV variation that reached US$ 36 million. This implies that optimized decision-making for investments in the oil sector varies according to the level of risk investors are willing to take. Consequently, as the level of risk taken by investors varies, the expected global NPV also varies as a function of the investor’s profile. In the present case study, given the amount of investment involved, this variation became significant. Because the main decision produced by the model is the definition of a portfolio of investments, we can conclude that the parameters K (values required by the investor) and β (confidence level to compute the CVaR) have a significant impact on the model’s results. It is important to emphasize that variations in the expected NPV directly reflect alterations in investment decisions over the time horizon. Therefore, a risk-management approach influences solutions to optimization problems in the oil sector, and the set of CVaR constraints controls the economic risk present in the objective function. Given this result, the optimization model under study and the premises and data taken into consideration are very sensitive to variations in the prices of oil and products. These price variations emerge as the major factors in a refinery’s margin, as other costs, including investments in new units, are considerably lower by comparison. This clearly attests to the importance of considering risk in optimization problems that address investment portfolios for the oil supply chain in the presence of uncertainty. As Zhou et al.18 emphasized, the intersection of financial engineering and supply chain management is an exciting field for future research. This work contributed in this direction, incorporating CVaR measures in the supply chain optimization for network design in the oil industry. Acknowledgment The authors gratefully acknowledge the support and encouragement of the Brazilian National Council of Research (CNPq; 301887/2006-3) and the Coordination for the Improvement of Higher Education Personnel in Brazil (CAPES). Nomenclature Sets B ⊂ I ) bases, subset of nodes (b) C ) set of operational modes (c) F ⊂ I ) field, subset of nodes (f) I ) set of nodes (i1, i2) IN ⊂ I ) international node, subset of nodes (in) M ) set of transport modes (m) N ) set of periods {n | n ) 1, ..., NT} NG ⊂ I ) natural gas producers, subset of nodes (ng) O ) set of oils (o) P ) set of products (pi, po) R ⊂ I ) refinery, subset of nodes (r) S ) set of scenarios (s)

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TR ⊂ I ) terminal, subset of nodes (tr) U ) set of process units (u, u2) UT ) set of transport arcs (ut) VO ⊂ I ) vegetable oil producers, subset of nodes (vo) First-Stage Decision Variables νr,u,n ∈ R+ ) investment at refinery r in distillation unit u at time period n νtut,n ∈ R+ ) investment at transport arc ut at time period n R ∈ R+ ) provides the portfolio’s VaR Second-Stage Decision Variables br,pi,po,n,s ∈ R+ ) blending product pi with product po at refinery r at time period n under scenario s dfrr,u,c,o,n,s ∈ R+ ) feed flow rate of oil o at refinery r in the distillation unit u with mode c at time period n under scenario s irr,po,n,s ∈ R+ ) inlet flow rate of product po at refinery r at time period n under scenario s oexpin,o,n,s ∈ R+ ) quantity of oil o exported to node in at time period n under scenario s oimpin,o,n,s ∈ R+ ) quantity of oil o imported to node in at time period n under scenario s orr,po,n,s ∈ R+ ) outlet flow rate of product po at refinery r at time period n under scenario s otut,o,n,s ∈ R+ ) quantity of oil o transported in the arc ut at time period n under scenario s pexpin,po,n,s ∈ R+ ) quantity of product po exported to node in at time period n under scenario s pfrr,u,c,pi,n,s ∈ R+ ) feed flow rate of product pi at refinery r in the unit u with mode c at time period n under scenario s pimpin,po,n,s ∈ R+ ) quantity of product po imported to node in at time period n under scenario s ptut,po,n,s ∈ R+ ) quantity of product po transported in arc ut at time period n under scenario s us ∈ R+ ) auxiliary varible to compute the CVaR in each scenario s Parameters AT ) interest rate BIpo,n ) blending index CTut ) initial capacity for transportation CT′ut ) additional capacity for transportation DYr,u,c,o,po ) distillation unit yield K ) lower bound for the portfolio’s CvaR NGPng,po,n ) natural gas production OCr,n ) operational cost + ) upper bound on external oil demand ODEin,o,n ODEin,o,n ) lower bound on external oil demand + ) upper bound on external oil supply OSEin,o,n OSEin,o,n ) lower bound on external oil supply + PDEin,po,n ) upper bound on external product demand PDEin,po,n ) lower bound on external product demand PERCr,u,c,pi,n ) proportion + PERCr,u,c,pi,n ) maximum proportion PERCr,u,c,pi,n ) minimum proportion + ) upper bound on external product supply PSEin,po PSEin,po ) lower bound on external product supply PUYr,u,c,pi,po ) process unit yield SIOpi,n ) sulfur quantity, entry product SPOpo,n ) maximum sulfur TCut ) transportation cost + ) maximum initial feed rate UCr,u UCr,u ) minimum initial feed rate + UC′r,u ) additional maximum feed rate UC′r,u ) additional minimum feed rate UTEut,i1,i2,m ) available transportation arc VCr,u ) investment cost, refinery

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VOPvo,po,n ) vegetable oil production VPOpo,n ) minimum viscosity VTCut ) investment cost, transport β ) confidence level to compute CvaR Stochastic Parameters FPf,o,n,s ) oil field production (national) OPBRr,o,n,s ) oil price, internal distribution OPEin,o,n,s ) oil export price OPIin,o,n,s ) oil import price PDb,po,n,s ) internal (national) product demand PPBRb,po,n,s ) product price, internal distribution PPEin,po,n,s ) product export price PPIin,po,n,s ) product import price

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ReceiVed for reView August 11, 2009 ReVised manuscript receiVed December 10, 2009 Accepted February 10, 2010 IE901265N