The uncertainty considered is related with the unknown demand levels for oil products. For this purpose, a model was developed based on two-stage stochastic programming. The methods were evaluated on a real sized case study.
A simulation based optimization approach to supply chain management under demand uncertainty
Preliminary numerical results obtained from computational experiments are encouraging. Keywords: supply chain investment planning, stochastic optimization, stochastic Benders decomposition. Oil companies are global multinational organizations whose decisions involve a large number of factors related to the supply of raw materials, their processing and distribution.
For companies with strongly diversified sources of petroleum supply, a long cast of products, and multiple markets, the advance planning of all activities along the supply chain is vital. Such planning includes the definition of production levels of oil from oil fields and of petroleum byproducts from oil refineries , as well as the distribution among these refineries and to the final consumers of oil products. Major oil companies are characterized by integrated and verticalized activities, and the activities of refining and distributing oil products are characterized by low profit margins.
Therefore, techniques for decision-making optimization are frequently used in the context of the oil supply chain. Although the research literature on the strategic modeling of supply chains is quite rich, few studies have included uncertainty mitigation in addition to other decisions of financial scope, such as commercialization income, market considerations and investment planning.
According to Sahinidis , the incorporation of uncertainty into planning models using stochastic optimization remains a challenge due to the large computational requirements involved.
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For nearly 50 years, companies in the oil and chemical industries have led the development and use of mixed integer linear programming to support decision making at all levels of planning. An overriding feature in the oil industry is its wide range of uncertainties, typically related to the unpredictable levels of demand for refined products, fluctuations in prices in domestic and international markets and inaccuracies in the forecasted production of oil and gas.
For this reason, many works have used techniques based on mathematical programming to support decision-making under uncertainty Escudero et al. Due to the great level of uncertainties taken into consideration, and the fact that the aforementioned problem is modeled as a mixed-integer linear program, it might become computationally infeasible to deal with a great number of scenarios by solving deterministic equivalents of the stochastic problems.
Therefore, a decomposition approach might turn out to be a valid alternative as solution methodology. The objective of this paper is to present a mathematical model for the optimization of the supply chain investment planning problem applied to the petroleum products supply chain. Uncertainties related to product demand levels are considered, thus, the stochastic programming framework is adopted as modeling approach.
Furthermore, it is shown an application of two primal decomposition techniques based on cutting plane approaches as solution technique. Experiments were performed in order to evaluate the efficiency of the proposed algorithms.
The paper is organized as follows: Section 2 describes the proposed mathematical model; Section 3 presents the traditional primal decomposition framework, while Section 4 presents the multi cut framework; computational results are shown in Section 5; Section 6 draws someconclusion. The problem approached in this paper can be defined as the strategic planning of petroleum products distribution, where one seeks to select investments to be made in logistics infrastructure, taking into consideration decisions regarding the distribution of flows, inventory policies, and the level of the external commercialization.
Such decisions arise in the context of strategic and tactical planning faced by petroleum products distribution companies operating over large geographical regions. We consider this problem as an integrated distribution network design and binary capacity expansion problem under a multi-product and multi-period setting.
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Petroleum products from refineries are stored in tanks to be directed to distribution bases. These bases serve as negotiation points with distributors and are considered as aggregation points of demand for such products. They also might serve as an intermediate point for other bases further away from the refineries. The bases are capable of storing product when necessary, given that the problem is considered under a multi-period operation. The storage and throughput capacities of the bases are limited.
However, they can be improved through an expansion project. The same idea holds for arcs, which can also be expanded in the same fashion. In addition to that, we also consider the possibility of building new arcs and bases. The tanks of these bases are constantly being loaded and unloaded. This process is known as the tank rotation and is subject to the physical limitations that are inherent in the hardware associated with the tanks of the distribution base. The rotating capacity refers to the number of times a tank can be filled and emptied over a certain period of time.
For modeling the uncertainty we propose a two-stage mixed-integer linear stochastic programming model. The uncertainty is represented through the consideration of discrete scenarios. These scenarios are defined by means of sampling from a continuous distribution of the demand levels for a given product at a certain base. The first-stage decisions are the selection of the expansion projects for tanks and arcs, as well as their timing.
These decisions are represented by binary variables. Typically, these investments are highly capital intensive and are built-to-order due to their technical complexity and particular specifications. For this reason, we assume that the same investment can only be implemented once along the time horizon. Also, we assume that investment decisions are available for use at the beginning of the selected time period.
The second-stage decisions, to be taken after know the unveiling of the uncertain parameters, are those relating to the flows of products, inventory levels, supply provided to each demand site, supply levels at sources, and levels of unmet demand. The objective function consists of investments costs of tanks and arcs expansion projects and the expected costs related to freight, inventory, and emergency floating tank acquisition. The purpose of the model is to plan the transportation and inventory decisions that will cope with the projected although uncertain growth of product demands, together with the possible investments that should be implemented and when, minimizing both investment and expected logistics costs.
The notation to be used for the presentation of the mathematical model is given below. For the sake of notational compactness, the domains of summations will be omitted, except when the summation is evaluated only on a subset of the natural domain. When there is no mention of this fact, its domain should be considered as the original set to which the index refers. In addition to that, we use bold caption to represent decision variable vectors. The mathematical model for the optimization of aforementioned problem can be stated asfollows:. Constraints 2 and 3 define that each investment can happens only once along the time horizon considered.
The objective function 4 represents freight costs between the nodes, inventory costs, and the cost of shortfall. Equation 5 comprises the material balance in distribution bases. Constraint 6 limits the supply availability at sources. Constraint 7 defines the arc capacities and the possibility of its expansion through the investment decisions y. In a similar way, constraint 8 defines the storage capacities together with its expansion possibility. Constraint 9 sets the throughput limit for bases, defined by the product of the available storage capacity with the maximum number of tank rotations.
The model proposed in the previous section can be defined as an optimization model with binary first-stage variables, continuous second-stage variables and discrete random parameters.
Such characteristics allow us a decomposition framework based on Benders decomposition Benders, applied to stochastic optimization. We start by noting that the so-called master problem can be equivalently reformulated asfollows:. This formulation allows one to distinguish an important issue.
Inequality 13 cannot be used computationally as a constraint, since it is not defined explicitly, but only implicitly, by a number of optimization problems. The main idea of the proposed decomposition method is to relax this constraint and replace it by a number of cuts, which may be gradually added following an iterative solving process.
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These cuts, defined as supporting hyperplanes of the second-stage objective function, might eventually provide a good estimation for the value of w, y in a finite number of iterations. The decomposition method applied to the aforementioned problem can be stated as follows:. Let denote the incumbent solution.
Step 1 : Solve the master problem and let w B , y B and LB be its optimal solution and optimal objective value respectively. Let w, y represent the first-stage cost function and:. Otherwise, proceed to Step 4. Generate the cut 16 :.