29th USAEE/IAEE North America conference, 14-16 0ctober 2010 Calgary.Canada. Oil refining planning under price and demand uncertainties: case of Algeria Abderrezak BENYOUCEF Algerian Petroleum Institute IAP, Avenue of the 1st November, Boumerdes 35000, Algeria. Phone: +213 661775093, [email protected] Abstract This paper aims to analyze the Algerian refining industry development in the presence of uncertainties, both on the domestic products demand and the international markets, with a linear dynamic model. The Algerian refining industry is a simple type, or hydroskimming, which is designed to treat a sweet light crude oil. Currently, this industry has to be adapted to meet demand progress both in terms of volume and also in terms of specifications, in a general context marked by a strong volatility of the oil markets. Generally, refining operations planning models are based on a deterministic linear programming. However, because of the volatility of the raw materials prices, demand fluctuation, and other conditions for the market, many parameters should be considered as uncertain. In our study, we propose a dynamic long-term linear model to analyze the development of the Algerian refining sector by 2030. We treat particular uncertainties on demand and prices of oil and petroleum products. Considering multiple uncertainties on demand and oil price, the model provides the production levels, the rate units running and the foreign trade of products by the year 2030. This model assesses the impact of the prices and the demand volatilities on the development of this industry. The paper is organized as follows: After the introduction and the overview about the methodology, the second section gives a brief overview about the refining modeling with linear dynamic programming. The third section deals with the model with uncertainties in which we focus on the particularities of an oil producer and exporter country. In section four, we present the main data used in the model. In the final section, before the conclusion, the main results will be presented and discussed. 1. Introduction The demand for petroleum products from refining crude oil has changed drastically, in both quantitative and qualitative terms. Energy concerns in Algeria are those of a developing country that must satisfy not only the growing energy needs, but also intend to finance its economy from oil export revenues, this requires the development of production infrastructure, processing and trade. At the time when Algeria is responsive to the economy of the market, rational management of this industry has become an obvious necessity. The linear programming is the most widespread mathematical tool in the field of optimization. It is to optimize plant operations, taking into account various constraints of products qualities and their applications and the operating constraints of the refinery. The constraints also affect the quality of the feedstock and its availability (Refining Model of IFP). An approach to optimize the refinery operations in the medium and long term would be possible by adopting the techniques of dynamic linear programming; it consists in taking into account developments in the technology matrix, the second members of the constraints and the coefficients of the objective function, from a period to another. This allows the planning of investment in each period until the end of the planning horizon. In addition, it offers flexibility to determine the limits of variation of the optimal solution by applying the techniques of sensitivity analysis on either the coefficients of the objective function or the right hand sides of the constraints or by simulating the model on several scenarios. By adopting these techniques, the model generates very different results from one scenario to another, this puts the decision maker faced to difficulties in choosing the best scenario for the planning of the investments. This leads us to reflect for other optimization methods that take into account uncertainties on both the coefficients of the objective function and the second members of the constraints. The aim of this paper is to propose a stochastic programming approach to optimize the Algerian refinery and propose the most likely scheme of the industry development by 2030. 2. Problem description The Algerian refining industry, hydroskimming with a capacity of 22 million tons per year, was designed to process light sweet crude. Given its simplicity, the latter has to be adapted as a result of two major concerns: - The first, together with the growing of the demand in the domestic market, is the increasing evolution of the required quality at the international markets, leading to a tightness of fuel specifications, particularly the content of lead, sulfur, aromatics, benzene and oxygen ... etc... - The second is the valorization of the Algerian crude oil on the international market (maximizing the revenues of oil and petroleum products exports). There are numerous optimization techniques and investment planning, linear and linear dynamics programming are the most common techniques in the refining industry. A drawback of these approaches, called deterministic, is associated to the data used (demand of petroleum products, crude and products prices, production costs etc ....), which are considered to be known with certainty. However, in reality, this is not the case; refining environment is exposed to multitude of internal and external uncertainties. The uncertainty in the refining begins first form the fluctuating of the oil price, which is the capital charge of the refiner. Furthermore, the cost of the different operating conditions and the uncertainty about the sale prices of finished products, which represent the turnover of the refiner, either on domestic or international markets. Prices of different products from refining multiplied by their yields are the turnover of the refiner, a small fluctuation in the selling prices leads straightforwardly to a negative margin. In our case, which concerns the long-term development of the refining industry in Algeria, the problem might be somewhat different, given the particularity of this industry. The first feature concerns the organizational aspect; all refineries in Algeria are the property of the national company (Sonatrach), which is also the only national operator in the oilfields with a current production capacity of approximately 70 million tons per year; this will ensure a continuous supply of oil to refineries throughout the country. Thus, the first uncertainty about the feedstock supplied to the refineries will be eliminated at least in the medium term. Nonetheless, this creates a dilemma of the crude oil valorization that begs the question: is it going to Algeria to export oil in its raw state or to export refined products that require the investment in the development of a more sophisticated tool for refining? 2 On the other hand, the uncertainty on the domestic market persists for different petroleum products, particularly fuels. Thus, the second uncertainty concerns the Algerian refining industry is the level of the national future demand for petroleum products, a question arises : what is the trend of development of the national demand for petroleum products and what are the necessary investments in refining to meet them?. These two issues (prices and demand uncertainties) that we will study in the following sections 3. Methodology The refining models are considered among the first models developed in linear programming in the fifties that seek to optimize the blending obtained by processing crude oil into finished products. The reader can find at Favennec and Babusiaux (1998, p.167) a presentation of this type of modeling. Thus, the constraints of the problem come from the refinery process (balance of intermediate and finished products, balance of the refinery gas), the quality control of products, the demand for finished products, the availability of resources (capacity of processing units and supply of crude oil) and the pollutant emissions. The objective function consists in minimizing the discounted annual cost of refining, including the economic depreciation of investments. The model that we develop is based on research works done in the modeling area of refining, in particular their latest work C.khor et al., 2008, it differs from a standard model because it is a long term model and we consider various levels of demand and prices for petroleum products, which are associated with a probability distribution. We introduce in the linear model as much equations of the demand as the uncertainties. The objective function could be formulated as the minimization of the certain refining costs increased with the expectation of the recourse variables costs. Moreover, we use the Markowitz model to analyze the impact of price fluctuations on the investment decisions that we detail in the following sections. We present first the dynamic linear model and then we explain how to introduce uncertainties. 4. Deterministic model: refining modeling by a linear dynamic model The overall model contains all variables and dynamic constraints for the periods of the investment planning, from 2005 to 2030. We have taken 2005 as reference for planning because we built our first model and calibrated based on the data of the refining and the market of that year. For our aggregate model, we consider in the first and the second period, until 2015, the new investments already planned by Sonatrach and the extensions and the rehabilitations of existing refineries. Subsequently, based on the growing of the domestic demand and the specifications tightness and new environmental standards required for petroleum products, the following extensions and the new production units will be determined by our optimization model for each period. A detailed description of linear programming models is beyond the scope of this section. We can see in Saint-Antonin (1998) for more details on the short term models and in McDonald and Karimi (1997) and Ierapetritou Pistikopoulos (1994) for medium-term models. 4.1 Variables This model contains the flow variables of intermediate and finished products, production, supply, exports, imports and capacity expansion. These variables will be indexed by an index t expressing 3 their evolution from one period to another. Obviously, these variables take different values at the optimum from a period to another. 4.2 Constraints Like in the case of the variables, the constraints are duplicated for each period; there are as many equations for demand of gasoline or diesel, for instance, as the planning periods. These constraints vary from one period to another by changing market conditions (changes in demand, specifications, technology ... etc). We explain briefly in the following paragraphs the different constraints and the objective function of the model and we point out the particularities of Algeria. 4.2.1 Balance of intermediate and finished products The balance equations of intermediate and finished products equilibrate the quantities of each product with their different affectations. To obtain linear constraints and as products yields depend on the operating conditions of each refinery unit, some units have been modeled by considering different severities representing different steps of processing. The material balance of an intermediate product expresses that its output is equal to its internal use. The production is represented by the product of yields and the quantity of the charge. The units processing yields are different for each severity and also depend on the type of feedstock and the crude processed. Total internal uses are the sum of all transfers to the finished products, as a feedstock of another unit or as a refinery fuel. For the balance of a finished product, the sum of the components of a pool is equal to the quantity of finished products. In both types of the balances, the variables are defined by weight. In some cases, they are defined in volume and we have to modify the equations by introducing their gravities. The refinery fuel balance is a special case. In this equation, the demand for the refinery fuel may be satisfied by intermediate or finished products. Each product has a distinct calorific value. Thus, from the supply side, refinery fuels are weighted according to their calorific values. From the demand side, we consider that the need for refinery fuels is proportional to the inputs of the processing units. 4.2.2 Demand equations and exportation variables Supply (production and possible imports) must satisfy domestic demand plus exports. These are the variables of the problem. Several export variables which are associated with different prices are set. These are deducted from the CIF (Cost Insurance Freight) in importing countries. We will then be tasked to build econometric models of some petroleum products consumption in Algeria. For the rest, we take the demand scenarios developed by specialized agencies in Algeria or from international agencies (IEA and OPEC). These models will be based on a classical approach with an income effect (GDP or other), a price effect and possibly effect of park equipment or lagged endogenous variables. Based on the main level of consumption of various petroleum products, we can estimate the potential development of the Algerian exports. 4.2.3 Equations of product’s specifications The finished products must comply with certain properties (specifications) legal and technical. These include density, vapor pressure, aromatics content, olefins content, octane etc.... for gasoline, the sulfur content, cetane number ... etc.. for diesel. Thus, a linear constraint is obtained 4 by multiplying the quantity of intermediate products (in volume for octane and in weight for sulfur for example) by their qualities and assigning a minimum or a maximum specification to the finished product. When the relationship is not linear, this feature is replaced by an index that can be used in a linear constraint. 4.2.4 Capacity constraints Capacity constraints of the processing units are designed to limit the amount of the charge that can be treated. The capacity expansions are represented through investment. In the case of a refinery, the investment cost is often a nonlinear function of the capacity, which leads to model capacities expansion through several variables associated with distinct costs. In the case of an aggregate model, we assume that investments are made for standard size units which can overcome this problem. However in the case of Algeria, where the number of refineries is limited, we may need to take into account these phenomena of non-linearity in the cost. We detail in the following section how to plan capacity expansion through the introduction of integer variables and standard sizes of units, while taking into account the effect of economies of scale. As the demand for petroleum products will have a significant change in terms of the quantity and a structural change of the quality required by the motorists and the conditions imposed by the standards of environmental protection, capacity constraints include both processing units (atmospheric and vacuum distillations), the quality improvement units (catalytic reforming, isomerization, alkylation, HDS ... etc.), the conversion units (FCC, hydrocracking etc. ...) and finishing units and environmental protection (hydroprocessing, softening ... etc.). These constraints are to limit the inputs of each unit by its current installed capacity increased with a variable representing its possible expansion. Thus, for the variables that represent the expansion of production capacity, Ierapetritou and Pistikoupoulos (1994) suggested in their paper a term of the capacity expansion, which is expressed as follows. (cid:1829)(cid:1827)(cid:1842) (cid:3404) (cid:1829)(cid:1827)(cid:1842) (cid:3397)(cid:1831)(cid:1829) (cid:4666)1(cid:4667) (cid:3037),(cid:3047) (cid:3037),(cid:3047)(cid:2879)(cid:2869) (cid:3037),(cid:3047) (cid:1831)(cid:1829)(cid:3013) (cid:3409)(cid:1831)(cid:1829) (cid:3409)(cid:1831)(cid:1829)(cid:3022) (cid:4666)2(cid:4667) (cid:3037),(cid:3047) (cid:3037),(cid:3047) (cid:3037),(cid:3047) The capacity of the production (cid:1829)(cid:1827)(cid:1842) of the processing unit j at the period t is equal to its (cid:3037),(cid:3047) precedent capacity (cid:1829)(cid:1827)(cid:1842) augmented with its possible expansion (EC ). The limits (cid:1831)(cid:1829)(cid:3013) and (cid:1831)(cid:1829)(cid:3022) (cid:3037),(cid:3047)(cid:2879)(cid:2869) j,t (cid:3037) (cid:3037) are respectively the minimum and the maximum capacities of the variable EC . j, t 4.2.4.1 Planning of the refining capacity using integer variables One of a linear programming problem with mixed variables is the fact that this latter shows continuous variables (such as the flow of petroleum products) and variable constrained to take only integer values (number of processing units). In general, for a linear program, capacity expansions for different units are continuous variables that take values from zero to the maximum capacity, the optimal solution for the column could vary from zero and go up to the maximum possible, for example 20 million tons per year. This is not so evident in the fact that there is no column of atmospheric distillation or catalytic cracking of low tonnage per year, and the same for 20 million tons per year (usually for this latter capacity, two or three columns will be set). To remedy this, we have proposed to introduce in our model various standard production units. We have proposed three levels for each processing unit, while taking into account the effect of the scale economy. In fact, we chose a large capacity, a medium 5 capacity and a small capacity. Each size is associated with an investment cost as we show in the following table. (See table 1). Table 1: size and investment costs of the processing units Processing unite j Size 1 Cost1 Size 1 Cost1 Size 1 Cost1 Atmospheric distillation SS SC SS SC SS SC 11 11 12 12 13 13 Vacuum distillation SS SC SS SC SS SC 21 21 22 22 23 23 Catalytic reforming SS SC SS SC SS SC 31 31 32 32 33 33 . . . . . . . . . . . . . . Processing unit n SS SC SS SC SS SC n1 n1 n2 n2 n3 n3 The investment cost increases with the size of the column to install. Therewith, if the capacity of the column size 2 is twice the size 1, for example, the investment cost will be less than double, hence the effect of economies of scale. A comprehensive way to adapt the information available on processing unit’s capacities is to apply the method of "extrapolation factor". This procedure is based on the statistical processing and smoothing of historical data, different quantities are linked by an empirical expression takes the following form: (cid:1835)(cid:4593) (cid:1829)(cid:4593) (cid:2869) (cid:3404) (cid:4678) (cid:2869)(cid:4679)(cid:3083) (cid:4666)3(cid:4667) (cid:1835) (cid:1829) (cid:2869) (cid:2869) I and I' represent the investment costs. C and C are the capacities of the processing units 1 1 1 2 related to these assets. δ is called "extrapolation factor" (Chauvel et al. 2001, page 200). This factor varies, for the refining industry between 0.5 and 0.8, the most often used factor for the production units is 0.65 with the exception of the hydrogen plant is 0.5 (source ifp). Thus, constraints on capacity expansion will be presented differently to take into account the effect of standard capacities: (cid:1831)(cid:1829) (cid:3404)(cid:3533)(cid:3533)(cid:3533)(cid:1858) (cid:1499)(cid:1845)(cid:1845) (cid:4666)4(cid:4667) (cid:3037),(cid:3047) (cid:3032),(cid:3037),(cid:3047) (cid:3032),(cid:3037),(cid:3047) (cid:3032) (cid:3037) (cid:3047) The integer variables f (e = 1,2,3) et (j =1,2,…nj) could take only positive integer values. The e,j,t indexes e and j are respectively incremented index of the numbers of production units to be installed; it is simply the number of standard distillation or processing columns, j denotes the index of the processing units (atmospheric distillation, catalytic reforming…etc) and t is the index of the time period. 4.2.5 Supply constraints These constraints are limiting the supply of various grades of crude oil. The supplies may be subject to a number of constraints limiting the availability of crude for refineries. These constraints differ from one country to another and depend on the availability of this or that quality of oil that can be bounded by a constraint (supply constraint). In general case, the limitation of the total supply of crude oil is equal to the sum of the quantities of each crude feeding distillation units. 6 Algeria, like other OPEC countries, exports around 70% of the oil production in its crude state. The FOB prices are indexed to the international markets on Brent for Europe or on WTI for America or other benchmark crudes for other regions of the world. This price is fluctuating from day to day, see extremely variable from one year to another. This variation may be justified the enhancement of the Algerian crude for processing into petroleum products at the refinery. The Algerian oil produced in southern countries is routed through a transport network to the loading ports or refining. Thus, all the refineries are fed by the same type of crude which is now widely abundant to meet the installed refining capacity. Refineries in Algeria are located at the same ports of oil exports (Arzew and Skikda). A question arises: is it worthwhile for Algeria to continue to export oil in its raw state or to invest in refining and exports oil products? Figure 1 below illustrates this problem. Figure 1: Integrated system: oil production‐refining‐exports In the case of the Algerian refining, since the refineries treat only the Algerian oil "Sahara Blend" with the exception of some quantities of imported residues (imported reduced crude) for making bitumen, we proposed to restrict the supply, for a producer and exporter country, the following constraint: (cid:1827)(cid:1842)(cid:1842)(cid:1844)(cid:1841) (cid:3397)(cid:1829)(cid:1844)(cid:1847)(cid:1830)(cid:1831)(cid:1850)(cid:1842) (cid:3409)(cid:1829)(cid:1827)(cid:1842)(cid:1829)(cid:1844)(cid:1847)(cid:1830)(cid:1831) (cid:4666)5(cid:4667) (cid:3029),(cid:3047) (cid:3029),(cid:3047) (cid:3029),(cid:3047) (cid:1827)(cid:1842)(cid:1842)(cid:1844)(cid:1841) : Quantities of crude oil b treated by the refineries in Algeria during the period t, (cid:3029),(cid:3047) (cid:1829)(cid:1844)(cid:1847)(cid:1830)(cid:1831)(cid:1850)(cid:1842) : Quantities of crude oil b exported during the period t, (cid:3029),(cid:3047) (cid:1829)(cid:1827)(cid:1842)(cid:1829)(cid:1844)(cid:1847)(cid:1830)(cid:1831) : Production capacity of the crude oil b from oilfields during the period t. (cid:3029),(cid:3047) 7 This constraint is to limit the supply of crude oil to the refineries with the capacity of the oil fields of Sonatrach; the excess (the crude not refined) is to be exported. The aim is twofold for adding a such equation, first to limit the expansion of refining capacity to the capacity of the oilfields (which currently is three times the capacity of the refining) and at what level it is profitable for Algeria to export oil in its raw state or invest in refining to export petroleum products, on the other side. 4.2.6 Pollutant emissions The constraints on emissions of pollutants that are introduced in the modeling related to air pollution. They result in the definition of specifications of maximum pollutant concentrations in the fumes emitted by the refinery. Emissions of sulfur dioxide and carbon dioxide can be so limited. Moreover, the rights to pollute can be introduced into the objective function (for carbon dioxide). Till present, there is no limitation on emissions of pollutants in Algeria, but to meet environmental concerns in the medium term; this can become a major constraint. For the short- term model, we do not consider this constraint; contrewise for the planning model in the medium and long term, this constraint will be taken into account. 4.2.7 Budget constraint Generally, each company looks forward to optimize its investment portfolio, taking into account the availability of budget. Sonatrach, as several national companies, carries on business from oil exploration to the trading of hydrocarbons through the transformation process. Its investment budget is allocated among different activities. Certainly, the budget allocated to the refining industry will be limited. In this context and in one of the scenarios that we will discuss, we will limit future investments by different budget levels. Budget constraint is expressed as well as the total investment costs of all processing units lower or equal to the budget allocated to refining industry. 4.3 Objective function The objective function incorporates the discounted costs of the refining industry throughout the period 2005-2030 i.e. treatment costs and pollution, procurement and investment, import of petroleum products and export revenues. We have opted to the objective function of minimizing the operating cost of refining activity in the long term than maximization the profit, as sales prices of petroleum products on the Algerian domestic market are given (fixed and subsidized by the state) and do not follow neither evolution nor trend of the fluctuations of the international spot prices, this makes it very difficult to determine the actual profit of the refining business to maximize profit. The objective function is the minimization of the discounted sum of: - Cost of supply - Operating cost - Cost of imports - Cost associated with pollutant emissions - Investment cost Decreased of: - The values of exports. 8 If the units are standard sizes, the total investment cost per processing unit is as follows: (cid:1835)(cid:1840)(cid:1848) (cid:3404) (cid:3533)(cid:3533)(cid:3533)(cid:1858) (cid:1499)(cid:1845)(cid:1845) (cid:1499)(cid:1845)(cid:1829) (cid:4666)6(cid:4667) (cid:3037),(cid:3047) (cid:3032),(cid:3037),(cid:3047) (cid:3032),(cid:3037),(cid:3047) (cid:3032),(cid:3037),(cid:3047) (cid:3032) (cid:3037) (cid:3047) Where e is, as we mentioned earlier, the ranges of costs for each standard size for the production unit j. This approach, called deterministic, based on assumptions of future development of different parameters, namely, the demand for petroleum products, the prices of oil and petroleum products… etc. For this purpose, we can use techniques of sensitivity analysis which is to vary a parameter and know its impact on the optimal solution. However, decisions will differ from one scenario to another, this leads us to another problem of decision making in presence of uncertainty, which is the subject of the following sections. 5. General formulation of a stochastic model of refinery operations planning The classical two-stage stochastic linear program (SLP) with recourses variables is originally proposed in the work of Dantzig (1955) and Beale (1955) has the general form: first-stage (cid:1839)(cid:1861)(cid:1866) (cid:1829)(cid:3021)(cid:1876)(cid:3397)(cid:1831) (cid:4670)(cid:1843)(cid:3435)(cid:1876),(cid:2022)(cid:4666)(cid:2033)(cid:4667)(cid:3439)(cid:4671) (cid:3093) With constraints (cid:1827)(cid:1876) (cid:3404) (cid:1854) (cid:1876)(cid:2035)(cid:1850) (cid:3410) 0 (cid:4666)7(cid:4667) Second-stage (cid:1843)(cid:3435)(cid:1876),(cid:2022)(cid:4666)(cid:2033)(cid:4667)(cid:3439) (cid:3404) (cid:1839)(cid:1861)(cid:1866) (cid:1869)(cid:3047) (cid:4666)(cid:2033)(cid:4667)(cid:1877)(cid:4666)(cid:2033)(cid:4667) With constraints (cid:1849)(cid:4666)(cid:2033)(cid:4667)(cid:1877)(cid:4666)(cid:2033)(cid:4667)(cid:3397)(cid:1846)(cid:4666)(cid:2033)(cid:4667)(cid:1876) (cid:3404) (cid:1860)(cid:4666)(cid:2033)(cid:4667) (cid:1877)(cid:4666)(cid:2033)(cid:4667) (cid:3410) 0 (cid:4666)8(cid:4667) x: vector of decision variables of the first period of size (n x 1) C, A and b are respectively matrixes of data in the first period of sizes (nx 1), (mxn) and (mx 1), (cid:2033) (cid:1527) represents the random event, and (cid:1846)(cid:4666)(cid:2033)(cid:4667),(cid:1860)(cid:4666)(cid:2033)(cid:4667) (cid:1853)(cid:1866)(cid:1856) (cid:1869)(cid:4666)(cid:2033)(cid:4667): matrixes of data for the second period with the sizes(kx 1), (1xk) and (k × 1) respectively and y is the vector of decision variables of the second period. (cid:1849)(cid:4666)(cid:2033)(cid:4667): The coefficients of the recourse random variables matrix. This model could be simplified and treated as an equivalent deterministic. The readers can see Kall and Wallace (1994) for more detail on the treatment of equivalent models and the constraints of nonanticipativity. 9 5.1 Stochastic parameters in our refining model The uncertainties persist throughout the oil chain, uncertainty about the form of the oil geological reservoir to the fuel prices at the pump stations. The stochastic programming models can be classified into three categories according to Ponnambalam (2005): (a) Uncertainties on the coefficients of the objective function, (2) Uncertainties about the second members of constraints (RHS), and (3) uncertainties of the technological coefficients of the constraints (LHS). In our case, we look to the first and to the second categories. For the third, one we consider many severities levels. The first one includes the uncertainties about oil and products prices. The second category includes the uncertainties about petroleum products demand. 5.2 Methodology of scenarios generation in the presence of uncertainties on the demand The uncertainty about market demand introduces randomness in the constraints of the production demand, which is actually the sum of the quantities of intermediate products necessary for their manufacture, as we have described in equations of the balance of intermediate products and finished products. The methodology used to generate scenarios for the recourse model with uncertainty based on the dispersion (standard deviation) compared to the mean. We can apply demand (cid:1856) , where i represents the type of product demanded and s indicates the (cid:3036),(cid:3046) scenario considered, as a random variable by the following equation: (cid:1856) (cid:3404) (cid:1878)(cid:2026) (cid:3397)(cid:2020) (cid:4666)9(cid:4667) (cid:3036)(cid:3046) (cid:3031) (cid:3031) Where z represents the variable of the standard normal distribution with a mean of 0 and a standard deviation of 1. (cid:2020) (cid:1853)(cid:1866)(cid:1856)(cid:2026) : are, respectively, the mean and the standard deviation of the probability distribution of the demand. 5.3 Modeling the uncertainty on the demand by the penalty functions and the slack variables We develop in this section the methodology of taking into account the uncertainty about demand constraints with recourse variables and how to introduce them into the objective function. 5.3.1 Penalty functions As has been mentioned in previous sections, one of the main consequences of the uncertainty in the context of decision making is the possibility of infeasibility in the future. The stochastic two- stage or more than two stages provide more choices to address this issue by delaying some decisions in the second stage, but this comes at the expense of using the corresponding penalties in the objective function. To design appropriate penalties functions, we must resort to the introduction of some slack variables in the probabilistic constraints to eliminate the possibility of infeasibility of the second stage. Moreover, the philosophy of models that based on the recourse variables requires to decision makers to attribute a price (or cost) in the objective function as penalty to adjust the activities considered as random. For some applications, such as in models of production planning and inventory, these costs are standard. However, in some other situations, it seems more appropriate to accept the possibility of infeasibility in some circumstances, if the probability of this event is restricted below a given threshold, as was treated by Sen and Higle (1999). 10