Flexible Scheduling of Crude Oil Inventory Management - American

Dec 10, 2009 - This work presents a novel approach for crude oil inventory management of a refinery. Considered problems involve the uncertainty in th...
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Ind. Eng. Chem. Res. 2010, 49, 1325–1332

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Flexible Scheduling of Crude Oil Inventory Management Sourabh Gupta and Nan Zhang* Centre for Process Integration, School of Chemical Engineering and Analytical Science, The UniVersity of Manchester, P.O. Box 88, Manchester M60 1QD, U.K.

This work presents a novel approach for crude oil inventory management of a refinery. Considered problems involve the uncertainty in the crude oil availability, its transfer to storage tanks and charging schedule for each crude oil mixture to crude distillation units. In this paper a solution algorithm is used that iteratively solves two mixed-integer linear programming models and a nonlinear programming model. The proposed model incorporates the uncertainty issue in the availability of crude oil and an improved discrete-time-based formulation is developed to provide good quality solutions and also saves the computational efforts. An algorithm is proposed to provide a solution space instead of a single point solution that may help to minimize the loss due to late or early arrival of vessels. A case study is carried out to demonstrate the effectiveness of the developed algorithms. 1. Introduction A refinery consists of crude oil operations, unit operations, and product blending (Figure 1). In crude oil operations the refinery deals with the availability of crude oil and crude oil management, whereas product blending deals with the fluctuating market requirement of blended products. In the whole process the information of demand flows toward the suppliers, whereas the material of products flows toward the customers. However, because of volatile raw material prices, fluctuating product demands, and other changing market conditions, many parameters in a planning/scheduling model are uncertain. Moreover many times, due to unavailability of crude oil or accidents in pipelines or unlikely events, the problem becomes more difficult and in some cases infeasible. One objective for oil operations in a refinery is to optimize the cost for crude oil inventories and product inventories. The current picture of the refining industry is characterized by stiff competition, stricter environmental regulations, heavier, sourer, costly crude oils, uncertainty in crude oil prices, and uncertainty in the availability of the crude oil. The ever changing market situation puts more pressure on the refinery, and refinery margins become tighter. On the other hand opportunities may also arise due to a changing market situation. Therefore, to maintain the profit margins in this ever changing market environment, refiners need to consider the trade offs of key decisions simultaneously by smarter decision strategies as well as keeping the backup options in case of uncertainties. Scheduling is a sequence of jobs, with their start time and end time, such that certain constraints are met and a schedule is sought for minimizing cost and/or some measure of time, like overall project completion time. Crudes are usually delivered to tanks in refineries by vessels or by pipelines. In a typical refinery, there is often an intermediate level between distillation units and storage tanks, consisting of charging tanks for storing and mixing the crudes. The distillation operation takes place in a crude oil distillation unit (CDU), which separates the charged oil into fractions such as gasoline, kerosene, gas oil, and residual. Every type of crude oil has different characteristics, that is, when they are processed under the same operating conditions, there will be different yields and properties * To whom correspondence should be addressed. E-mail: nan.zhang@ manchester.ac.uk. Tel.: 44-1613064384. Fax: 44-1612367439.

for each fraction. The system configuration of the scheduling problem corresponds to a multistage system consisting of vessels, storage tanks, charging tanks, and CDUs as illustrated in Figure 2. For oil supply, batch and continuous scheduling has received great attention in the past two decades. The potential of cost reduction and increased efficiency has become the main driving force for the progress in the field of research. One major issue of oil supply chain operation is to evaluate opportunistic situations such as crude oil purchase or production of beneficial products and to minimize profit loss during crisis making situations such as delay of raw material supply, unit failures, and so on. In a crude oil scheduling problem crude oil vessels may arrive early or late. In the case of early arrival refiners need to pay a waiting cost of vessel in sea, and late arrival may affect downstream processes. Late arrival of a vessel will be the worst case if the reserved oil is insufficient to be processed in a CDU at its minimum throughput, and this will affect all the following units of the refinery. It is a good idea to know the threshold where the operation is feasible. If the vessel arrives after the threshold (“worst case”) then the available models crash during the optimization. It is desirable to have a systematic approach that can provide a solution in the worst case by indicating the threshold for the problem. From the above discussion, we derive our research directions. One of the tasks is to come up with solutions which provide flexibility in decision making for the optimum cost/profit to exploit the potential of short-term opportunities. Another task is to provide an efficient solution algorithm which minimizes the computational efforts and provides an optimum solution. In this work, a novel approach is proposed for inventory management of a refinery under uncertainty in the availability of the crude oil. The proposed algorithm provides a solution space instead of a single-point solution, so that decisions can be changed, to avoid losses and/or to exploit the short-term opportunities that may arise due to uncertainty. The significance of this work is that (a) it provides a set of solutions within a range of optimum solution for crude oil scheduling, (b) the proposed solution strategy is computationally efficient, and (c) consideration of uncertainty in the mathematical formulation helps to provide solutions in the worst case scenarios.

10.1021/ie9008919  2010 American Chemical Society Published on Web 12/10/2009

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Figure 1. Refinery operations and information flow.

Figure 2. Crude oil supply chain in a refinery.

2. Existing Approaches for Crude Oil Supply Problem In mathematical terms, crude oil supply optimization can be expressed as Objective function: minimized cost ) unloading cost for the crude vessel + cost for vessel waiting in the sea + inventory cost for storage and charging tanks + changeover cost Subject to • minimum/maximum inventories • meet with product due dates and achieve planning targets • limitations: unit capacity, blend quality, etc. • uncertainty: arrival of crude oil vessels, demand of each mixed oil, crude oil prices • set of operating rules The nature of the crude oil scheduling problem is mixedinteger nonlinear programming (MINLP) due to bilinear property constraints in the formulation. However, solving an MINLP model directly results in inconsistency in solution quality and time. In this paper we are using the solution algorithm that iteratively solves two mixed-integer linear programming (MILP) models and a nonlinear programming (NLP) model, resulting in better quality, stability, and efficiency than by solving the MINLP model directly. In general, scheduling means a decision making process to determine when, where, and how to produce a set of products given requirements in a specific time horizon, a set of limited resources, and processing recipes. The key issue with the process scheduling problems concern with the time representation. The scheduling supply chain formulations can be classified in two main categories: discrete-time formulation1 and continuous-time formulation.2 The existing methods based on discrete-time formulation approach result in high computational demand due to large

amount of binary variables, whereas continuous-time formulation-based methods result in a lower computational demand but may stick to the local optimum. To calculate the expected cost of the crude oil scheduling problem under uncertainty (includes late/early arrival of crude vessels), piecewise linear approximation of loss functions can be applied. 2.1. Discrete-Time Based Approach for Oil Supply Chain Problem. Shah reported a discrete-time mixed-integer linear programming (MILP) model for crude oil scheduling3 by separating it into two subproblems. The upstream problem included portside tanks and offloading, and the downstream problem included allocation of charging tanks and CDU operation. The objective was to minimize the tank heel. Almost concurrently, Lee et al. also reported an MILP model to minimize operating cost arising in crude oil unloading, tank inventory management, and crude charging.1 In discrete-timebased approach the time horizon of interest is divided into a number of uniform time intervals, whereas in the continuoustime-based approach the time horizon is divided into a number of events that need not be of the same time interval. Figure 3 illustrates the conceptual difference between the continuoustime approach and the discrete-time approach. The mathematical formulation based on the discrete-time approach is computationally expensive to reach the optimum cost. Moreover, for property calculations simplified linear equations have been used in the mathematical formulation which causes inconsistent results. The consequences of this approach are that we can get the optimum solution with a large computational effort and results may not be consistent owing to property discrepancy. Li et al. recognized the composition discrepancy and proposed an iterative MILP-NLP (nonlinear programming) combination algorithm to solve the problem.4 However, Li’s MILP formation was based on Lee’s MILP model,1 which is computationally expensive.

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Figure 3. Representations of discrete time and continuous time.

2.2. Continuous-Time Based Approach for Oil Supply Chain Problem. Because of the inherent limitations of the discrete-time approaches, there has been a significant amount of work carried out in the development of continuous-time representations in the past decade. The continuous-time models that have been developed for sequential processes can be applied to scheduling problems with various features, including different intermediate storage policies, sequence-dependent changeovers, batch and continuous processes, intermediate due dates, renewable resource restrictions, and different objective functions such as minimization of order earliness and minimization of makespan. There have been an increasing number of research contributions on continuous-time formulations for scheduling of general processes but limited literature is available for the scheduling of oil operations based on this approach. The earliest efforts in the area of crude oil supply chain were presented by Jia and Ierapetritou,2,5 Ierapetritou and Floudas,6,7 and Reddy et al.8 However, due to the variable nature of the timings of the events, it becomes more challenging to model the scheduling process and the continuous-time approach may lead to mathematical models with more complicated structures compared to their discrete-time counterparts.9 In the previous research it has been demonstrated that in a few cases the optimization results for the discrete-time formulation are better than the continuous-time formulation. However, discrete-time-based approach leads to higher computational demand. On the other hand the consideration of the continuoustime representation approach reduces the number of integer variables as well as computational time but results may stick to local optimum. Moreover, previously the flexibility issues were not addressed in scheduling of crude oil operations. 3. Improved Discrete-Time Formulation Approach In this section a computationally more efficient approach has been proposed for scheduling of crude oil operations based on the discrete-time formulation and deals with the flexibility of the crude oil management. In the proposed approach a few binary variables (the same as size of scheduling horizon) have been identified which can be removed while maintaining the linearity of the problem. The proposed mathematical approach is more computationally efficient as well as able to provide the true optimum for the problem. 3.1. The Improved Discrete-Time Formulation. The proposed scheduling model is based on a uniform discretization of time in a given scheduling horizon. Selection of the time length of each discretized time span involves a trade-off between accurate prediction and computational effort. With smaller discretized time sizes more sophisticated operations are possible

Figure 4. Timing variables to describe the rules for unloading and waiting of crude vessel.

since there are more possible operation changes during the scheduling horizon. The relationships between the timing variables for vessel arrival, unloading, and departure in the proposed model are illustrated with a simple example shown in Figure 4. Vessel 1 arrives at time 1, but the actual unloading process begins at time 2. It completes unloading at time 5 and leaves the docking station. The variable XW,1,t, which denotes whether unloading of crude oil from vessel 1 to the storage tank is possible or not, has the value of 1 during the time period from time 2 to time 5. Also, vessel 2 arrives at time 5, unloads its crude oil from time 6, and leaves the docking station at time 9. In general, vessel V arrives at the docking station at time TARR,V, and it waits until preceding vessel V - 1 leaves the docking station. If the preceding vessel leaves, vessel V starts to unload its crude oil. The binary variable XF,V,t is activated when the unloading of vessel V starts at time t. The binary variables XW,V,t which are related to the unloading flow of vessel V to storage tanks at time t are activated at time TF,V which is the unloading start time of vessel V at the docking station. The XW,V,t are deactivated at time TL,V when the vessel V completes unloading and leaves the docking station. The binary variable XL,V,t is activated when the unloading of vessel V is completed. In this way, the variables XW,V,t are activated from TF,V to TL,V. For each vessel the binary variables XF,V,t and XL,V,t are used to calculate TF,V and TL,V respectively. t

XW,V,t e

∑X

F,V,m

(1)

L,V,m

(2)

m)1 SCH

XW,V,t e

∑X m)t

In the above formulation it can be seen that the binary variables XF,V,t and XL,V,t are activated only once throughout the scheduling horizon. Previously researchers suggested to use them as SOS1 variables in GAMS formulation, which makes

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Table 1. Computational Results and Comparison for Example 1, Example 2, and Example 3 example (method) 1 1 1 2 2 2 3 3 3 a

(Lee et al., 1996)b (Jia et al., 2004)c (proposed)d (Lee et al., 1996)b (Jia et al., 2004)c (proposed)d (Lee et al., 1996)b (Jia et al., 2004)c (proposed)d

variables

0-1 variables

constraints

objective

iterationsa

CPU timea

192 139 192 456 341 456 581 301 581

36 24 34 70 56 67 84 48 81

331 382 331 825 990 825 1222 888 1222

217.67 247.0 213.67 352.55 413.48 352.55 296.55 343.10 269.8

9228 (1695) 635 10952 (3055) 331493 (21148) 14430 38176 (45127) >515541 (60663) 3284 760619 (411328)

73.4 (17.1) 0.28 3.47 (1.25) 4158.8 (287.9) 4.89 6.64 (6.8) >7744 (1089.4) 1.46 359.03 (193.55)

The parentheses contain the results by using SOS1 variables. b IBM RS-600. c Sun Ultra 60 Workstation. d Pentium III 1.7 GHz computer.

Figure 5. Cost comparison between rule-based approach, Lee’s approach, and proposed approach.

the formulation dependent on GAMS or on its solvers. Moreover it has been found that the newer versions of solvers are not supporting SOS1-type variables. During the formulation it has been identified that these variables can be omitted by using TF,V, and TL,V variables and t (time) parameter. The following equations (3 and 4) show the way by which XF,V,t and XL,V,t variables can be omitted. XW,V,t e

t TF,V

(3)

TL,V (4) t Equation 3 sets XW,V,t variables to zero before arrival of the vessel to the docking station that is equivalent to eq 1. Equation 4 sets XW,V,t variables to zero after leaving the vessel from the docking station that is equivalent to eq 2. By using eqs 3 and 4 a set of binary variables can be removed, but on the other hand it introduces the nonlinearity in the model as available in eq 3. So in the proposed model it is advised to use XF,V,t binary variables and not to use XL,V,t binary variables by using eqs 1 and 4. Therefore, the new formulation reduces the number of binary variables while maintaining the linearity of the problem. To test the robustness of the proposed methodology three examples (Lee et al., 1996) have been studied. From Table 1 it can be seen that the continuous-time-based method proposed by Jia et al. (2004) is computationally more efficient compared to the discrete-time-based method by Lee et al. (1996). However, the method based on continuous-time formulation is not able to reach the optimum solution. The proposed approach utilizes less binary variables compared to Lee’s approach and is capable of providing the optimum solution. Example 1 consumes 34 binary variables compared to 36 binary variables, example 2 consumes 67 binary variables compared to 70 binary variables, and example 3 consumes 81 binary variables compared to 84 binary variables. XW,V,t e

Figure 5 shows the comparative study between the cost due to the rule-based approach, Lee’s approach, and the proposed approach for example 1 (Lee et al., 1996). These results also indicate that the sea waiting may result in cost saving for oil scheduling operations. The case study consists of two vessels with one million barrel of crude, two storage tanks, two charging tanks, and a crude distillation unit. Proposed approach indicates a cost saving of $27458 per scheduling horizon/(8 day) or approximately 1.25 million dollars/year compared to rule-based approach. 3.2. Arrival Time Estimation Using the Loss Function. As discussed earlier that the uncertainty is always an issue in the availability of the crude oil. Crude oil vessels may arrive late or may arrive early at the port and always be an issue with their arrival. Li et al.10 proposed an approximation-based approach for refinery planning under uncertainty. Their approach is good in agreement, compared to the methods available in the literature, with a better solution speed. For the sake of simplicity, it is assumed that the arrival of crude oil vessel follows the normal distributions. On the basis of this assumption the arrival time of the crude oil can be represented by the following equation: TARR,V ) µTARR,V - σTARR,V · L(z)

(5)

The detailed description of the approach is available in the literature (Li et al., 2004). In this work, the application of loss function is extended as the flexibility model which helps the optimizer to find the optimum solution in worst-case scenarios. By applying the suggested modifications in the proposed discrete-time formulation based mathematical approach, a robust approach is developed to deal with uncertain conditions in crude oil management.

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Figure 6. Algorithm which provides operating window for unloading of vessels.

4. Flexible Scheduling for Crude Oil Operation: Operating Window Concept In the past, a lot of work has been carried out to find the most correct and robust schedule for the oil operations. The literature is available for the algorithms which provide the consistent schedule. A number of articles are also available in the literature on planning under uncertainty. In the previous section we have proposed a mathematical approach which is able to deal with uncertainty issues in the availability of the crude oil. But still there is a gap to find a flexible schedule for crude oil operations. It has been observed in our research that there may exist a range of feasible schedules that can achieve the same operating target, which implies that better operating flexibility needs to be exploited in scheduling. So, in this section a new algorithm has been proposed which provides a set of solutions within a given range of the optimum point. 4.1. The Operating Window Algorithm. In the corporate world a schedule should be flexible enough to deal with uncertainty issues, computationally efficient, and robust for the optimum cost. To deal with all these capabilities an operating window concept has been proposed in this section which provides a feasible solution for scheduling of the crude oil problems. Operating window defines the solution space instead of single point solution for an optimization problem. In this study the operating window concept is applied for a crude oil scheduling problem. Following are the steps to find the operating window for a crude oil scheduling problem: (step 1) optimize the objective function for the optimum solution; (step 2) identify the key variables for the operating window; (step 3) find the operating window for the key variable by changing the objective function; (step 4) if results are inconsistent then shorten the operating window and go to step 3 else terminate with the solution (consistency is defined later in the steps of algorithm). An algorithm to exploit the potential of such an operating window is shown in Figure 6. In the algorithm an MILP model is used to target the cost of inventory management and further by changing the objective function an operating window is

identified which provides a wide range of solution for the same optimum cost. Then a feasibility check has been introduced to find out the composition discrepancy and an NLP model is used to remove the discrepancy. In the algorithm it is proposed to tighten the operating window and to relax the optimum cost whenever it is required to get a feasible optimum solution. The description for each step of the algorithm is as follows: MILP Initialization. The first MILP model initializes the algorithm while considering the uncertainty issues. In this model, the bilinear property constraints are linearized. Although the model does not guarantee a consistent concentration inside and in the discharge from the tanks, the model generates a good starting point for the next calculations. Operating Window. Once getting the optimum cost, optimize the MILP model twice with the changed objective of minimizing and maximizing the key variable for the same optimum cost. The crude oil scheduling problem is very much dependent on vessel arrival time. When the vessel arrives earlier than expected then this counted as vessel waiting time and if the vessel arrives late, then it may result in infeasible operation. In the considered crude oil scheduling problem waiting time is considered as a key variable. So, minimize and maximize the vessel waiting time to give the extreme positions for the operating window. Discrepancy Check. The procedure at this point checks the consistency of the concentration values from MILP. The consistency criteria are (i) whether the concentration of crude oils in the mixing tank is the same as that of the stream that flows out from the mixing tank and (ii) whether the sum of the concentrations of all crude oil types equals unity. If both i and ii are positive, an optimum solution has been found and the procedure is terminated. Fix Connectivity Options. The values of integer and binary variables which were used in material balance constraints and CDU charging constraints determined by MILP are sent to the corresponding parameters in the following NLP model. NLP. The constraints in the NLP are the same as those in the MILP except for those constraints which use the discrete/

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Figure 7. Uncertainty in the arrival of vessel B.

integer variables. All the discrete/integer variables in the MILP model become parameters for the model. The NLP model determines accurate concentration values using rigorous bilinear constraints. These concentration data are further used in the next step. Optimality Check 1. The iteration is terminated if the difference between the objective values of the NLP and the previous MILP run is smaller than an appropriate tolerance. If the objective is not within the tolerance in that case tight the operating window by relaxing the maximum/minimum waiting time criteria. Property Transfer. If the previous check is not satisfied, indicating that the concentrations in the tanks strongly affect the objective values, then they may even affect the integer variables that were previously determined in MILP. To verify this, the concentrations in the tanks are fixed at the values generated from the NLP, but the integer variables that were previously fixed will be freed in the next step. Modified MILP. A new MILP model is generated from the previous step by fixing the tank concentrations but allowing all the integer variables to vary. Then find the operating window by using the same modified MILP model. Optimality Check 2. If the objective value of NLP is better than that of the MILP model and all the values of binary variables calculated by MILP are same to the corresponding parameters, then the procedure is terminated. Otherwise, the procedure continues to iterate. It can be noticed from the algorithm that the operating window concept provides multiple solutions for an optimization problem. However, in an optimization it is not always the case to have multiple solutions for the same optimum cost/profit. Therefore, it is a good idea to have relaxed objectives for optimization while finding the operating window. This might come with solutions that are more likely to happen in real situations. Further it cannot be assumed that all the solutions in the operating window will be feasible. Therefore after finding the operating window it is required to check the feasibility of solutions within the operating window. 5. Case Study In this case study a small size problem is considered to provide the insight of both the improved discrete-time formulation and the operating window concept. This problem consists of two vessels, two storage tanks, two charging tanks, and one crude distillation unit. At the planning stage, the arrival time of first vessel is day 2 and there is uncertainty for the arrival of the second vessel, and unloading for both vessels should be completed by day 8.

For vessel 2 an uncertainty has been considered in its arrival, and the arrival of vessel 2 is assumed that the arrival time conforms to a normal distribution with mean day 8 and a standard deviation of 1 day that is, N (8,1). Figure 7 shows the uncertainty in the arrival of vessel 2, and it is assumed that it follows the normal distribution curve. It can be noticed that the most likely day for the arrival of vessel 2 is day 8 which is an infeasible case and this condition translates the problem into the worst-case scenario. Vessel 1 unloads crude oil A into storage tank A and vessel 2 unloads crude oil B into storage tank B. Vessel 1 and 2 contain 1 million bbl of crude A and B, respectively. There is one CDU which has to process 1 million bbl of mixed crude oil X and Y, respectively. The weight fractions of sulfur which determine the quality of crude oil are 0.01 for crude oil A and 0.06 for crude oil B. Two crudes are mixed to make two types of mixtures: crude oil mix X and Y. The sulfur concentration of X should be in the range of 0.015 and 0.025, while that of Y is between 0.045 and 0.055. The initial volumes of the storage tanks for crude oil A and B are respectively 250 000 and 750 000 bbl, while the initial volumes of the charging tanks for crude mix X and Y are all 500 000 bbl. The costs involved in this problem are inventory cost, vessel harboring cost, vessel sea waiting cost, and CDU changeover cost for crude oil mix charging mode change. To find the operating window a flexibility of 2% is allowed in the optimum cost for an 8 day schedule. In this case study it is assumed that the inventory level in storage tank “A” on day 3 is 1 000 000 bbl (minimum) and on day 4 it is 600 000 bbl (maximum), which moves accordingly in the operating window. Unit inventory cost for each storage tank and charging tank are 8 × 10-3 and 5 × 10-3 [$/(day × bbl)], respectively. Changeover cost for crude charging to CDU is 50 × $103 each time it occurs. Costs involving the vessels are due to waiting in the sea and harboring for unloading the crude oil. These costs are 5 × 103 and 8 × 103 [$/day], respectively; that is, unloading incurs higher costs. Ideal mixing is assumed in the charging tank, and the crude mix cannot be fed into the CDU while crude oil is transferred from storage tanks to charging tanks. The results by using the operating window concept are shown in Figure 8 and Figure 9. It can be also noticed from the results that arrival time of vessel two is reset to day 7 by the solution which is not possible without using the flexibility model. These results indicate that the proposed algorithm is capable of providing flexible solutions in near worst case. In this example the optimum cost is $210633.3 for vessel A unloading on day 2, and the cost for vessel A unloading on day 1 and day 3 is

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Figure 8. Optimal solutions for the example using the operating window concept.

$214333.3 and $212733.3, respectively. This is due to different waiting times for vessel A arrival that result in savings in waiting time cost. However, it results in higher inventory cost in storage tanks and charging tanks because of the different schedule. In the results the operating window concept shows that the decisions can be changed within a range of the optimum solution. Figure 8 indicates the operating window concept in crude oil supply chain operations. From the results it can be noticed that the vessel 1 can start unloading on day 1 or day 2 or day 3, according to the availability of crude oil vessel. From the results it is clear that beyond the operating window it is difficult to maintain the schedules for the optimum cost. So from this we can say that the Operating Window concept indicates that the decisions can be dynamic within a range of the optimum solution.

6. Conclusions In this work, an improved discrete-time formulation was presented for the short-term scheduling of refinery operations which is computationally efficient compared to the available models and a flexibility model has been incorporated in the crude oil scheduling problem for the availability of crude oil. The LP-based branch and bound method was applied to solve the model, and several techniques have been implemented for the large size problem. The performance of the proposed MILP model has been illustrated in Table 1 that shows the robustness and accuracy of the model. To further exploit the flexibility issue, the “operating window” concept has been proposed which provides a feasible operating window for scheduling of the crude oil problems. The main feature of the proposed algorithm is that it solves the oil quality, tank allocation, and oil blending issues simultaneously, and

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Nomenclature Indices and Sets V ) 1, ..., NV ) crude vessel t ) 1, ..., SCH ) time interval Variables TARR,V ) crude vessel V arrival time at the docking station TF,V ) vessel V unloading initiation time TL,V ) vessel V unloading completion and departure time XF,V,t ) 0-1 variable to denote if vessel V start unloading at time t XL,V,t ) 0-1 variable to denote if vessel V completes unloading at time t XW,V,t ) 0-1 continuous variable to denote if vessel V is unloading its crude oil at time t µTARR,V ) mean for the arrival time of crude vessel σTARR,V ) standard deviation in the arrival time of the crude vessel AbbreViations SOS1 ) special ordered sets (at most one variable within a set can have a nonzero value).

Literature Cited

Figure 9. CDU charging schedule (a) for optimum cost; (b) for vessel A unloading on day 3; (c) for vessel A unloading on day 1.

provides the wider solution space for the specified range of optimum solution while maintaining the consistency of the solution. It has been also found that the proposed model is capable of providing solutions in the worst-case scenarios. In this contribution the application of operating window concept is limited to one key variable. This concept needs to be developed further to be able to accommodate more than one key variable. The application of the operating window concept should not be limited to a scheduling problem. Although most of the optimization problems have multiple solutions near the global optimum, the global optimum is not always an achievable practical solution. This concept has the potential to be explored in other optimization problems where it is desirable to have a solution space instead of a single-point solution.

(1) Lee, H.; Pinto, J. M.; Grossmann, I. E.; Park, S. Mixed integer programming model for refinery short-term scheduling of crude oil unloading with inventory management. Ind. Eng. Chem. Res. 1996, 35, 1630–1641. (2) Jia, Z.; Ierapetritou, M. Mixed-integer linear programming model for gasoline blending and distribution scheduling. Ind. Eng. Chem. Res. 2003, 42, 825–835. (3) Shah, N. Mathematical programming techniques for crude oil scheduling. Comput. Chem. Eng. 1996, 20, S1227–S1232. (4) Li, W.; Hui, C.; Hua, B.; Tong, Z. Scheduling crude oil unloading, storage and processing. Ind. Eng. Chem. Res. 2002, 41, 6723–6734. (5) Jia, Z.; Ierapetritou, M. Efficient short-term scheduling of refinery operations based on a continuous-time formulation. Comput. Chem. Eng. 2004, 28, 1001–1019. (6) Ierapetritou, M. G.; Floudas, C. A. Effective continuous-time formulation for short term scheduling 1. Multipurpose batch processes. Ind. Eng. Chem. Res. 1998, 37, 4341–4359. (7) Ierapetritou, M. G.; Floudas, C. A. Effective continuous-time formulation for short term scheduling 2. Continuous and semicontinuous processes. Ind. Eng. Chem. Res. 1998, 37, 4360–4374. (8) Reddy, P. C. P.; Karimi, I. A.; Srinivasan, R. A new continuoustime formulation for scheduling crude oil operations. Chem. Eng. Sci. 2004, 59, 1325–1341. (9) Floudas, C. A.; Lin, X. Continuous-time versus discrete-time approaches for scheduling of chemical processes: A review. Comput. Chem. Eng. 2004, 28, 2109–2129. (10) Li, W.; Hui, C.; Li, P.; Li, A. Refinery planning under uncertainty. Ind. Eng. Chem. Res. 2004, 43, 6742–6755.

ReceiVed for reView May 31, 2009 ReVised manuscript receiVed November 11, 2009 Accepted November 11, 2009 IE9008919