Planning and Scheduling under Uncertainty: A ... - ACS Publications

Ind. Eng. Chem. Res. 2010, 49, 3993–4017

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Planning and Scheduling under Uncertainty: A Review Across Multiple Sectors Peter M. Verderame, Josephine A. Elia, Jie Li, and Christodoulos A. Floudas* Department of Chemical Engineering, Princeton UniVersity, Princeton, New Jersey 08544

This paper provides an overview of the key contributions within the planning and scheduling communities with specific emphasis on uncertainty analysis. As opposed to focusing in one particular industry, several independent sectors have been reviewed in order to find commonalities and potential avenues for future interdisciplinary collaborations. The objectives and physical constraints present within the planning and scheduling problems may vary greatly from one sector to another; however, all problems share the common attribute of needing to model parameter uncertainty in an explicit manner. It will be demonstrated through the literature review that two-stage stochastic programming, parametric programming, fuzzy programming, chance constraint programming, robust optimization techniques, conditional value-at-risk, and other risk mitigation procedures have found widespread application within all of the analyzed sectors. This review is the first work which attempts to provide a comprehensive description of the advances and future directions for planning and scheduling under uncertainty within a variety of sectors. 1. Introduction Planning and scheduling are interrelated activities dealing with the management of specified systems. As opposed to using heuristics to ascertain the appropriate management decisions, mathematical models which accurately portray the physical realities of these systems can be utilized. Deterministic optimization theory and algorithms aim at efficiently and rigorously determining the optimum solution as measured by the objective function within the feasible region defined by the constraint set. It should be noted that stochastic algorithms exist such as genetic algorithms which can generate feasible solutions; however, only deterministic optimization algorithms can provide a rigorous assertion of solution optimality especially when system variables are continuous in nature. As a result, planning and scheduling problems are commonly formulated as optimization models which are solved by means of deterministic optimization algorithms. For many circumstances, nominal system conditions are assumed when formulating and implementing these various forms of optimization models; however, the robustness of the final solution could be greatly enhanced if the uncertainties associated with the various system parameters were explicitly considered in the model formulation. For example, within the chemical industry the inclusion of processing time, demand, and/ or market price uncertainty in the planning and scheduling of plant activities can lead to a dramatic enhancement in the actual performance of the plant in the face of unfavorable realization of uncertain parameters. The objective of this review paper is to demonstrate that there exists general commonalities in the uncertainty approaches used to take into account parameter uncertainty at the planning and scheduling levels within various fields such as the chemical, petrochemical, and pharmaceutical industries, energy planning, power generation system planning, farm planning, forest planning, waste management, water resources management, transportation planning, and other application sectors. All of these sectors have perhaps differing definitions related to planning and scheduling activities, and the constraint sets and objective function forms may not share a great deal of commonality. Yet, the unifying theme related to all planning and scheduling models * Corresponding author. Tel.: 1-609-258-4595. E-mail address: [email protected].

is the fact that within a given mathematical model there exist system parameters which are uncertain in nature, and this parameter uncertainty must be taken into account in an explicit manner. Two-stage stochastic programming, parametric programming, fuzzy programming, chance constraint programming, robust optimization techniques, and conditional value-at-risk represent different frameworks to take into account parameter uncertainty, and it will be shown through the industry-specific literature review that all six methodologies have found widespread application within the various sectors. The primary goal of this review paper is to provide researchers with a general roadmap highlighting the key issues and commonalities found within several different types of planning and scheduling under uncertainty problems. Floudas and Lin,1,2 Shapiro,3 Shah,4 Mula et al.,5 Varma et al.,6 Li and Ierapetritou,7 and Maravelias and Sung8 all provided reviews related to particular elements of the planning and scheduling problem as it pertains to the chemical, petrochemical, and pharmaceutical industries. Weintraub and Bare9 provided a review on modeling strategies for forest management from an operations research perspective. Nelson10 presented an overview of forest-level models used to simulate various management scenarios, and Weintraub and Romero11 conducted a review comparing the methodologies applied to agricultural and forestry management. Reviews also exist in the areas of fleet sizing, routing and scheduling,12 stochastic vehicle routing,13-15 and airline planning under uncertainty.16 However, there is currently no review paper which covers planning and scheduling under uncertainty for several independent sectors. The remainder of the paper will be divided first into sections based upon industrial sector designation. For example, the first section pertains to the chemical, petrochemical, and pharmaceutical industries where planning and scheduling under both deterministic and uncertain system conditions will be presented. After outlining the contributions in all relevant sectors, specific challenges related to planning and scheduling under uncertainty will be presented so as to highlight key challenges and opportunities. Lastly, the key findings of this review paper will be outlined in the conclusions section.

10.1021/ie902009k  2010 American Chemical Society Published on Web 04/08/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 9, 2010

2. Chemical, Petrochemical, and Pharmaceutical Industries (CPPI) Within the chemical, petrochemical, and pharmaceutical industries (CPPI), supply chain management can be broadly defined as the comprehensive supervision of the company’s supply chain components under varying time scales, and Levi et al.,17 Shah,18,4 and Varma et al.6 presented the key issues and challenges related to supply chain management. Three vital components of supply chain management are strategic planning, operational planning, and scheduling. Strategic planning determines the long-term direction of the supply chain while considering changing market and industry trends. The time horizon associated with strategic planning is on the order of years, and decisions regarding the building of new sites, elimination of existing facilities, retrofitting of plants, adding or eliminating raw material suppliers, and any other significant supply chain network topology alterations are taken into account at the strategic planning level. Due to the relatively long time horizon associated with strategic planning, uncertainty must be explicitly taken into account. With the supply chain network topology in place, the operational planning level is concerned with determining the required production levels and raw material requirements for the multiple production facilities present within the company’s supply chain. The operational planning time horizon is usually on the order of months, and in order to facilitate the maximization of profit and the minimization of inventory and customer dissatisfaction, the multiple forms of system uncertainty need to be taken into account within any operational planning model. The production targets and raw material requirements supplied by the operational planning level are subsequently used at the scheduling level in order to determine the daily operations of a given facility within the supply chain of interest. Due to the fact that any scheduling model must rigorously take into account the detailed plant characteristics, the time horizon associated with scheduling decisions is on the order of days; however, process uncertainty must still be factored into any scheduling analysis in order to ensure that the final plant schedule is feasible even when unfavorable process conditions have been realized. Since the scheduling level relies on the production targets and raw material requirements supplied by the operational planning level, the effective integration of operational planning and scheduling is a major issue within the realm of supply chain management. Having provided an overview of the planning and scheduling operations involved within the chemical, petrochemical, and pharmaceutical industries, a more detailed presentation of the planning and scheduling levels will be presented when considering both deterministic and uncertain system conditions. 2.1. Deterministic Contributions in CPPI. 2.1.1. Scheduling. The scheduling of a plant involves determining the allocation of plant resources. Tasks must be assigned to the process units that either operate in batch, semicontinuous, or continuous mode, and the duration and amount of processed material related to those assigned tasks must be determined. When considering multiproduct and multipurpose plants, a scheduling model typically takes the form of a mixed-integer linear programming (MILP) problem where binary variables denote the activation of a task and continuous variables are used to quantify material flows and processing times. Task allocation, material balance, and sequencing constraints are common features of a scheduling model and help to define the physical system under investigation. Potential objectives used in part to determine a given plant’s schedule are profit maximization within a finite time horizon, the minimization of costs associated

with production and or customer dissatisfaction within a finite time horizon, and makespan minimization where the objective is to satisfy the customer supplied demand profile while minimizing the required time horizon. The supplied demand figures can either be expressed as intermediate demand due dates or aggregate demand requirements due at the end of the time horizon. Storage conditions for intermediate states also need to be taken into account when scheduling plant activities, and Floudas and Lin1,2 provided an overview of the various storage policies such as unlimited intermediate storage (UIS), no intermediate storage (NIS), zero-wait (ZW), and finite intermediate storage (FIS). Depending upon the required production recipes and the degree of production recipe similarity between the various final states, different process system representations have been adopted within the scheduling community. For example, Kondili et al.19 developed the state-task-network representation (STN), which can be defined as a directed graph with both state and task nodes. Pantelides20 later extended the STN representation to the resource-task-network (RTN) form which provides a uniform description of all available resources, such as materials, processing equipment, storage, and utilities. It should be noted that for simple processes where all products follow the same sequence of tasks and different stages can be represented by only one unit or multiple parallel units, a sequential representation can be adopted where material balances do not need to be taken into account explicitly. One of the fundamental issues within scheduling is the choice of the time representation. As noted by Floudas and Lin1,2 early attempts at scheduling plant activities relied on discretizing the time horizon into a number of time periods;21-23,19,24,20,25-29 however, discrete-time models (uniform and nonuniform) do not accurately represent the continuous-time nature of the plant and by definition lead to suboptimal solutions. As a result, continuous-time models are the preferable option, and Mendez and Cerda,31,32,30 Gupta and Karimi,33 and Nadal et al.34 have all developed continuous-time models for sequential processes. When considering plants that are modeled through a STN or RTN representation, the relevant continuous-time models can be broadly classified into the three categories of slot-based,35-40 global event-based,41-43 and unit-specific event-based.44-54 Each of the given modeling approaches has its own inherent advantages and disadvantages; however, it should be noted that the work of Shaik et al.50 indicated that the unit-specific eventbased models result in smaller problem sizes when compared to both slot-based and global event-based models and consequently are often computationally superior. For a more in-depth analysis of the various issues surrounding scheduling, consult the reviews of Floudas and Lin,1,2 Mendez et al.,55 Pitty and Karimi,56 and Barbosa-Povoa.57 A recent advance by Li and Floudas58 introduces an approach for the determination of the optimal number of event points which can be used for most continuous-time models in short-term scheduling. 2.1.2. Planning. A supply chain network is composed of two classes of nodes that the company in question has the ability to modify. The first node class represents the various raw material suppliers, and the second class of nodes encompasses the production facilities. Strategic planning is primarily concerned with critically evaluating the supply chain in terms of adding and/or removing nodes within the network, as well as modifying the characteristics of existing production nodes in order to maximize the company’s future performance, which can be measured in both an industry- and company-specific manner. A company can decide to change the distribution of suppliers


by adding or eliminating supplier nodes based upon various trade-offs, and production facility composition similarly can be altered by either adding or subtracting production centers from the supply chain topology based upon a cost/benefit analysis. The addition or subtraction of nodes, as well as the retrofitting of existing production nodes are constrained by both internal and external supply chain conditions. Given the objective of maximizing some performance metric subject to relevant system constraints, the strategic planning problem is often formulated as an optimization model taking the form of a mixed-integer linear programming (MILP) or mixed-integer nonlinear programming (MINLP) problem. Shapiro3 provided an overview of the key challenges found within the area of supply chain planning. Grossmann59 demonstrated how the use of optimization techniques and mathematical modeling can facilitate all stages of supply chain management, and Chen and Pundoor,60 Dickersbach,61 and Kreipl and Pinedo62 all dealt with the planning and scheduling of a static supply chain. Verderame and Floudas63 developed a mixed-integer linear programming model for the management of a multisite production and distribution network. Sousa et al.64 addressed the issue of supply chain design within a multilevel planning scheme and applied the methodology to an industrial case study. Bassett and Gardner65 demonstrated that the quantitative analysis of a supply chain at the strategic planning level can dramatically impact a company, and as a result, strategic planning shall remain a potent area of research for the foreseeable future. Papageorgiou66 emphasized that an integrated approach to supply chain design which considers the entire supply chain is a necessary approach for the strategic planning problem. You and Grossmann67 developed a mixed-integer nonlinear programming model which addresses several fundamental issues within supply chain design, and Chopra and Meindl,68 Daskin,69 Owen and Daskin,70 Zipkin,71 Cachon and Fisher,72 Kok and Graves,73 and Tsiakis et al.74 all made contributions toward addressing the issues of facility location and inventory management within the strategic planning context. Naraharisetti et al.75 recently proposed a supply chain redesign methodology which applied multimodal optimization using a hybrid evolutionary algorithm. Overall, the field of strategic planning has great potential for further advancement due to the inherent complexity of supply chains, and the consideration of uncertainty can greatly enhance the efficacy of supply chain decision-making. The primary objective of an operational planning model is to provide realizable production targets, and therefore, an operational planning model should take into account not only customer demands but also the production capacity of the plant in question. One approach is to apply a scheduling model over the entire planning time horizon since a scheduling model rigorously takes into account the production capacity of the plant. Model tractability issues, however, greatly diminish the viability of this approach, and therefore, certain aggregation or relaxation schemes have been adopted when formulating operational planning models. Wilkinson et al.76 proposed a planning model which is an aggregation of the scheduling model based upon the work of Kondili et al.19 The aggregation scheme involves the discretization of the time horizon into time periods and the grouping of related scheduling level constraints and variables into aggregate resource, capacity and production constraints and variables. It should be noted that Bassett et al.77 also proposed a time-based aggregation scheme similar to the discretization scheme proposed by Wilkinson et al.76 where the values of

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variables are only calculated at the end of a discrete time period. The discretization of the time scale and the aggregation of the scheduling level constraints and variables reduces the complexity of the original scheduling model and allows the model to be applied over the entire planning horizon. The proposed aggregation scheme is intended to provide only a tight upper bound on the production capacity of the plant in question, however, and does not rigorously take into account the production capacity of the plant. Kallrath and co-workers78-83 developed a comprehensive multisite, discrete-time planning model which divides the planning horizon into commercial periods and then further discretizes those commercial periods into production periods. Another type of aggregation scheme is a unit aggregation approach in which a plant is modeled by its set of bottleneck units. Sung and Maravelias84 noted that unit aggregation schemes are often adopted in discrete manufacturing industries. Verderame and Floudas85 developed a discrete-time operational planning model based upon the unit aggregation approach which has been shown through computational studies to be capable of providing a production profile which is a tight upper bound on the production capacity of the facility under investigation. The potential downside of a unit aggregation approach is that it disregards any sequencing effects; meaning, it assumes that each of the bottleneck units can produce one or more final states without any upstream or downstream processing. As a result, the unit aggregation scheme can cause the planning model to overestimate the true production capacity of the plant. Dogan and Grossmann86 addressed the issue of accurately taking into account the sequencing of tasks within a process unit at the planning level by solving a variant of the traveling salesman problem within a detailed planning model, and they applied it to a case study of parallel batch reactors. Suerie87 addressed planning with multiple intermediate demand due dates through a discrete-time proportional lot-sizing and scheduling problem where at most one changeover is allowed in each period. Grunow et al.88 applied a hierarchical planning formulation for the campaign planning of multistage batch processes which involve coordination between production facilities. From a modeling perspective, the nature of the operational planning problem leads to MILP models. Yet, the length of the planning time horizon and the objective of the planning model providing feasible production targets can lead to problems which are difficult to solve using traditional branch and bound algorithms based upon linear programming relaxation. In order to address this computational issue, alternative decomposition/ relaxation approaches have been applied. Benders decomposition and Lagrangian relaxation are two possible techniques which take advantage of the model’s structure and can be used in order to reduce the complexity of the overall problem by solving relaxations of the original problem in an iterative manner. Aardal and Larsson89 applied Benders decomposition to production planning problems, and Gupta and Maranas90 used a Lagrangian relaxation procedure based upon the planning model of McDonald and Karimi91 to solve a production planning problem. While Benders decomposition and Lagrangian relaxation can facilitate the solution of planning problems, they are based on the exploitation of a specific model structure, which may not be applicable for every planning problem. Due to the nature of large-scale models which give rise to computational tractability issues, planning models are formulated through various types of scheduling model relaxations. By definition, these planning models cannot rigorously represent the production capacity of plant, and so, the aggregate production targets supplied by the planning model often overestimate

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the production capacity of the plant. Unless a scheme is adopted which attempts to refine the planning level model by taking into account the key scheduling component results, the planning model may lead to the suboptimal allocation of resources. The integration of planning and scheduling should address the inaccuracies within the planning model by allowing for the twoway interaction between the planning and scheduling models. 2.1.3. Integration of Planning and Scheduling. Planning and scheduling have evolved as independent entities; however, the treatment of operational planning and scheduling as independent entities often leads to inefficient allocation of resources. Shah4 noted that on average less than 10% of the material being processed by a pharmaceutical firm ends up as final product. Shobrys and White92 reported that the integration of planning and scheduling can lead to increased profit levels and a reduction in committed capital. For example, Exxon Chemicals estimated that the adoption of integration techniques led to an annual reduction in operating costs by 2% and operating inventory by 20%.92 DuPont has also noted that the integration of planning and scheduling played a part in reducing the working capital tied up in inventory from $160 to $95 million for a polymers facility.92 The economic incentives for the integration of planning and scheduling are great and realizable; however, there is still substantial scientific debate regarding what is the optimal framework for the integration of planning and scheduling. In general, the planning level is typically an aggregated, simplified model which attempts to provide realizable production targets to a scheduling model, which in turn determines the detailed schedule of plant operations. A two-way interplay between the planning and scheduling levels needs to occur in order to efficiently integrate the planning and scheduling levels, and iterative schemes and/or rolling horizon approaches have been utilized in order to address the integration of planning and scheduling. Maravelias and Sung8 provided an overview of the key challenges and opportunities present within the field of integrated planning and scheduling. Papageorgiou and Pantelides93 proposed a decomposition scheme having an upper level planning model and a lower level scheduling model. The proposed framework addresses the planning and scheduling of batch and semicontinuous plants. Bassett et al.28 presented an aggregate planning model that determines the upper level production targets and the time horizon for each scheduling problem, and similarity analysis is used to group scheduling subproblems into families where representative problems are solved in order to determine the feasibility of the overall scheduling problem. Subrahmanyam et al.94 generated an iterative framework for the planning and scheduling of a chemical plant which utilizes the fact that an aggregate planning model is a relaxation of the scheduling model. Zhu and Majozi95 proposed a formulation for the planning and scheduling of a multipurpose plant. Munawar and Gudi96 developed a multilevel planning, scheduling, and rescheduling framework which uses control theory to minimize the revision of aggregate production targets so as to avoid the violation of customer commitments. Sung and Maravelias84 formulated an approach which utilizes detailed scheduling information in order to construct a convex hull of the feasible production levels along with convex underestimation of the associated production cost so as to refine the planning level model. Dogan and Grossmann97 presented a decomposition based method for the simultaneous planning and scheduling of a multiproduct plant. Amaro and Barbosa-Povoa98 considered the planning and scheduling of industrial scale supply chains and applied their hierarchical approach to a pharmaceutical

industrial case study. Li and Ierapetritou99 proposed a decomposition framework for a bilevel optimization problem having an upper level planning problem and multiple lower level scheduling subproblems. The proposed decomposition scheme iterates between the planning and scheduling levels until the lower bound provided by the planning level equals the combined upper bound of the scheduling subproblems. Many of the aforementioned integration approaches require the entire time horizon to be scheduled before providing any feedback to the planning level within the iterative scheme, which can lead to a substantial amount of computational effort when the given methodologies are applied to large-scale, industrial applications. A rolling horizon approach addresses this computational issue by decomposing the time horizon required to be scheduled for each iteration and minimizing the number of required iterations. Within the rolling horizon framework, the time horizon is discretized into periods, and Dimitriadis et al.100 proposed both forward and reverse rolling horizon approaches. Stefansson and Shah101 considered the integration of planning and scheduling within the pharmaceutical industry and presented a multiscale planning and scheduling framework using a moving horizon approach where information availability is explicitly taken into account at both the planning and scheduling levels. Verderame and Floudas85 developed a framework for the effective integration of operational planning and medium-term scheduling for large-scale industrial batch plants using a twoway feedback loop implemented within a rolling horizon scheme. Overall, the rolling horizon framework requires less iterations and does not require the scheduling of the entire horizon for each iteration. 2.2. Uncertainty Contributions in CPPI. Before a detailed exposition of the uncertainty contributions made within CPPI can be undertaken, the various forms of uncertainty need to be presented. Within CPPI, market prices, processing times, demand amount and due data, unit capacities, unit breakdowns, transportation time, and cost potentially can be all uncertain. The uncertainty related to a subset of these parameters can be explicitly taken into account through preventative approaches such as two-stage stochastic programming, parametric programming, fuzzy programming, chance constraint programming, robust optimization techniques, and risk mitigation techniques while the remaining elements need to be addressed through online optimization techniques after parameter realization. For example, market prices, processing times, demand due date and amount, unit capacities, transportation time, and cost can all be modeled as uncertain elements following a given distribution, and the aforementioned preventative approaches can explicitly take into account the given forms of uncertainty. On the other hand, unit breakdowns and new orders are events which are best modeled in a reactive manner after realization by means of reactive scheduling techniques that can handle the onset of unit breakdowns, as well as new product orders. 2.2.1. Scheduling. As previously stated, scheduling provides a detailed outline of plant activities on a day-to-day basis. Due to the relatively short time horizon, it is often assumed that all system parameters are deterministic in nature. However, Janak et al.102 demonstrated that only slight variations in system conditions can render a schedule infeasible. For example, minor changes in processing time and demand requirements can result in an existing scheduling becoming infeasible, and small fluctuations in market prices often can lead to a scheduling result being deemed suboptimal. Also, unit breakdowns and additional product orders which are realized after the schedule has been enacted need to be addressed in a systematic way at the


scheduling level. Several methodologies exist for the explicit consideration of these types of uncertainty within a scheduling model, and Li and Ierapetritou7 provided a comprehensive overview of scheduling under uncertainty. The uncertainty associated with the given system parameters can be characterized in three different ways. If there is not enough information to construct an accurate estimation of the uncertain parameter’s distribution, then the bounded distribution form is often adopted where the uncertain parameter is assumed to take on values within a specified range defined by an upper and lower bound. If there is sufficient information to construct a reliable distribution for the parameter, then the uncertain parameter can be represented in a probabilistic fashion similar to the representations presented within the work of Petkov and Maranas103 and Janak et al.102 An alternative to the bounded and known distribution cases is the use of fuzzy sets as noted by Li and Ierapetritou.7 Having defined the forms of uncertainty present within a given scheduling model, the next step is to select an appropriate modeling approach which can explicitly take into account these aforementioned forms of parameter uncertainty. One methodology is the two-stage stochastic programming approach where the first-stage contains those variables which must be ascertained before the uncertain parameters are realized, and the secondstage is composed of those variables which represent recourse decisions that are enacted upon the realization of the given uncertain parameters. The first-stage objective function term is deterministic in nature while the second-stage term involves an expectation evaluation. In order to address the expectation evaluation, a finite number of uncertain parameter scenarios can be generated with the recourse variables being indexed by scenario so that every possible parameter realization has an associated recourse action. The downside of the scenario generation approach is the prohibitive growth in model size as the number of considered scenarios increases. An alternative to scenario generation is the use of a distribution-based approach where the expectation of the recourse objective function term is ascertained through the integration of the given multivariate probability distribution function. For the distribution-based approach, the problem size is considerably smaller, but the formulation is now nonlinear. In order to remove the nonlinearity from the problem, techniques such as Monte Carlo, Gaussian quadrature, etc. are used. These methodologies may become computationally expensive as the number of uncertain parameters increase. Despite some of its potential shortcomings, Balasubramanian and Grossmann,104 Goel and Grossmann,105 and Bonfill et al.106 all demonstrated that two-stage stochastic programming can be applied to the scheduling under uncertainty problem. Colvin and Maravelias107 also developed a model for the scheduling of product testing during its developmental stages that utilized stochastic programming in order to determine the optimum testing period, as well as the level of resources allocated for a given testing operation. Parametric programming is another alternative toward modeling parameter uncertainty. Based upon the theory of sensitivity analysis, the objective of parametric programming is to define a function which maps the uncertain parameter values to a given optimal solution for the entire uncertain parameter space. Parametric programming algorithms attempt to construct the aforementioned function without having to exhaustively enumerate all possible uncertain parameter realizations. Gal and Nedoma108 developed an algorithm to solve multiparametric linear programming problems based upon the Simplex algorithm. Acevedo and Pistikopoulos109 extended the field of

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parametric programming to address multiparametric mixedinteger linear programs, and Dua et al.110 made significant contributions in the area of multiparametric mixed-integer quadratic problems. Acevedo and Salgueiro111 presented a novel algorithm for the solution of convex multiparametric nonlinear programming problems. With regard to scheduling applications, Thompson and Zawack112 developed a framework to solve an integer programming model for job shop scheduling which modeled uncertainty by means of parametric programming, and Ryu and Pistikopoulos113 applied parametric programming to scheduling problems. Ryu et al.114 developed a proactive scheduling approach using parametric programming to take into account processing time and equipment availability. The function generated from the multiparametric programming methodology allows for the complete mapping of uncertain parameter values; thus, rescheduling requires only function evaluation and no additional optimization operations. Li and Ierapetritou115,116 utilized multiparametric programming to take into account multiple forms of parameter uncertainty within both the objective function and constraint set and demonstrated that parametric programming is a rigorous way of generating a set of schedules which can accommodate all possible realizations of parameter uncertainty. Despite its many attractive modeling features, it should be noted that parametric programming entails solving a series of optimization problems in an iterative approach, and so, the required computational time often may make the application of parametric programming to large-scale industrial problems challenging. In many situations, viable alternatives to the two-stage stochastic programming and parametric programming approaches are chance constraint programming, robust optimization techniques, and fuzzy programming. Based upon the work of Charnes and Cooper,117-119 chance constraint programming involves replacing constraints containing uncertain parameters with their respective probabilistic forms which explicitly take into account the stochastic nature of the uncertain parameters. The probabilistic constraints are then reformulated into a deterministic form using in part the distribution information for the various uncertain parameters. Orcun et al.120 and Petkov and Maranas103 applied chance constraint programming to take into account parameter uncertainty at the scheduling level. The major downside of chance constraint programming within the realm of scheduling is that it cannot guarantee the feasibility of a given schedule with regard to the specific realization of the uncertain parameters. For example, the obtained solution may not be feasible even for the nominal system conditions. In order to address this issue, robust optimization techniques have been developed where probabilistic constraints are transformed into their deterministic counterparts which then augment the nominal mathematical model. Robust optimization techniques121-130,102,131 guarantee that the obtained solution is feasible for the nominal set of system conditions, as well as robust with regard to the multiple forms of uncertainty present within the system under investigation. The robust optimization framework entails first expressing the true parameter values through the declaration of random variables, then the formulation of probabilistic constraints, and finally, the transformation of these probabilistic constraints into their deterministic counterparts, which are added to the existing model. Lin et al.130 and Janak et al.102 developed the robust optimization theory for general MILP models and applied robust optimization techniques for both bounded and known distribution cases when aggregate demands, processing times, and/or prices are uncertain. When distribution information is not available, the use of

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fuzzy set theory can be utilized when modeling uncertain parameters, and the fuzzy programming method treats these uncertain parameters as fuzzy numbers and the constraints containing these uncertain elements as fuzzy sets. Balasubramanian and Grossmann,132 Wang,133 and Petrovic and Duenas134 applied fuzzy programming to the problem of scheduling under uncertainty. When dealing with new product orders, unit breakdowns, and other system events which occur during the application of a schedule, the aforementioned preventive scheduling methodologies are not applicable, and another approach has to be adopted in order to adapt the existing scheduling to these changing system conditions. Reactive scheduling is a method of online optimization which can facilitate the rescheduling of the plant so as to take into account these unforeseen events. There are several different approaches toward reactive scheduling. Li and Ierapetritou7 provided an overview of reactive scheduling, and Grotschel et al.135 detailed the various facets of online optimization for large-scale systems. Rodrigues et al.,136 Vin and Ierapetritou,137 and Mendez and Cerda30 all contributed to the field of reactive scheduling through novel methodologies. Also, Janak et al.138,139 developed a rigorous reactive scheduling framework which is capable of systematically taking into account new product orders and machine breakdowns and applied the given framework to a large-scale industrial case study in order to demonstrate the viability of the proposed approach. 2.2.2. Planning. In all of the planning models mentioned in section 2.1.2, it is assumed that the system parameters are at their nominal values. Given the duration of the planning time horizon under consideration, the robustness of the planning decisions would be greatly enhanced if the uncertain nature of some system parameters were factored into the analysis. For example, demands, processing times, and price parameters are potentially all uncertain in nature, and the explicit consideration of the uncertainty associated with these parameters could lead to enhanced planning results. Joung et al.140 acknowledged the detrimental effect that neglecting the uncertain nature of demands can have on company operations and in response to this issue developed an optimization model augmented with a simulation component for the management of a supply chain under demand uncertainty. Zimmermann141 noted that the problem structure itself can often dictate the optimal choice for the type of uncertainty analysis, and so, multiple uncertainty approaches have been applied within the field of planning under uncertainty. Despite both its academic and industrial significance, planning under uncertainty remains an open problem within the literature as noted in the reviews by Sahinidis142 and Mula et al.5 Two-stage stochastic programming is one possible framework for modeling parameter uncertainty at the planning level. Ierapetritou et al.143 applied a two-stage stochastic programming approach to the operational planning under uncertainty problem. Wu and Ierapetritou144 presented a hierarchical approach to planning and scheduling under uncertainty. The proposed planning component models uncertainty by means of multistage programming where the overall horizon is discretized into time periods based upon the given system parameters’ degree of certainty which vary inversely with the time horizon. As the relative uncertainty increases, the number of scenarios used to represent the stochastic nature of the relevant parameters increases, and the scenarios are generated by sampling the underlying probability distribution of each uncertain parameter under consideration. Khor et al.145 utilized a scenario-based two-

stage stochastic programming approach along with Markowitz’s mean-variance model to address planning under multiple forms of uncertainty within the petrochemical industry. Khor et al.145 acknowledged the exponential rise in model size with the increase in the number of considered scenarios due to the mandated increase in the corresponding number of variables and constraints. Colvin and Maravelias146 developed a multistage stochastic programming model for clinical trial planning in new drug development, which is a challenging planning problem within the pharmaceutical industrial. They considered a reduced set of scenarios without sacrificing the accuracy of the uncertainty representation, which is accomplished by means of exploiting the inherent structure of the problem. Ahmed and Sahinidis147 developed a two-stage stochastic planning under uncertainty model that minimizes the actual cost of the firststage model and the expected cost of the second-stage model while also minimizing the variance of the expected cost of the stochastic inner problem. As a result, the actual cost of the realized scenario remains relatively close to the expected cost. Gupta and Maranas148,149 formulated a two-stage stochastic supply chain planning model with an outer deterministic production site component and an inner recourse supply chain management component which is dependent upon the realization of customer demand. Employing duality theory, Gupta and Maranas148,149 constructed the feasible space of the inner problem independent of the demand realization and then considered the realization of certain demand levels in a probabilistic fashion for the purposes of generating the optimal policy for the supply chain. Akin to the scenario-based twostage stochastic programming approach, Levis and Papageorgiou150 proposed a multisite planning model for the pharmaceutical industry which attempts to coordinate drug portfolio management along with supply chain design in order to maximize viable patent lifetime and profits. This approach considers the stochastic nature of clinical trials through a scenario-based approach where a potential drug can either succeed or fail, and Levis and Papageorgiou150 presented an aggregated model to address the aforementioned planning objectives while concurrently taking into account all possible drug development outcomes. Overall, two-stage stochastic programming makes the assumption that the problem can be decoupled into an outer problem which is independent of parameter uncertainty and an inner problem where recourse decisions are made based upon the realization of the given parameter uncertainty. As noted by Clay and Grossmann,151 it is often the case within the field of planning that it is not optimal nor computationally efficient to decouple the problem in the aforementioned manner. Chance constraint programming117-119 is a common approach toward dealing with parameter uncertainty. Petkov and Maranas103 introduced chance constraint programming within a multiperiod planning and scheduling framework for multiproduct batch plants with uncertain demand profiles due at the end of each period. Li et al.152 acknowledged the various limitations of two-stage stochastic programming and in response utilized chance constraint programming to take into account the multiple forms of uncertainty such as demand that should be considered when conducting refinery planning. Within this work, the objective function expectation terms which are dependent upon the realization of uncertain parameters are approximated by piecewise linear functions. You and Grossmann67 introduced chance constraint programming when considering the design of responsive supply chains under demand uncertainty. Li and Ierapetritou7 conducted work not only on the practical applica-


tion of chance constrained programming but also the theoretical development of the methodology. Mitra et al.153 utilized chance constraint programming when addressing the problem of midterm supply chain planning under uncertainty. The limitation of chance constraint programming is that the chance constraints representing the deterministic reformulation only mandate feasibility for a specified probability distribution and do not guarantee that the solution is feasible for any specific realization of the uncertain parameter(s). In contrast, the robust optimization framework possesses the inherent advantage that it not only explicitly takes into account the various forms of parameter uncertainty but also ensures that the obtained solution is feasible for the nominal system conditions. Recently, Verderame and Floudas131 extended the work of Lin et al.130 and Janak et al.102 for the explicit consideration of demand due date and demand amount uncertainty at the operational planning level. With regard to parametric programming, Pistikopoulos and Dua154 demonstrated that parametric programming can be used to address certain variants of the planning under uncertainty problem, and Ryu et al.155 developed a bilevel programming framework for supply chain planning which addressed uncertainty by means of parametric programming. Based upon the work of Artzner et al.,156 Ogryczak,157 and Rockafellar and Uryasev,158,159 conditional value-at-risk (CVAR) theory poses a potentially attractive alternative to the aforementioned uncertainty approaches, and it has been used extensively within the financial sector in order to mitigate risk as noted by Krokhmal et al.,160 and Rockafellar and Uryasev,158 and Mansini et al.161 have applied CVAR to the portfolio management problem originally presented within the work of Markowitz.162 Conditional value-at-risk represents an advance over established value-at-risk (VAR) theory, which has been explicitly described within the works of Staumbaugh,163 Pritzker,164 Jorion,165 and Duffie and Pan,166 and it has been documented that CVAR has several important advantages over VAR. For example, conditional value-at-risk represents a more pessimistic loss level when compared to value-at-risk, and it also features the important property of maintaining convexity regardless of what type of probability distribution being utilized.158-160 Both value-at-risk and conditional value-at-risk seek to guard against unfavorable realization of uncertain parameters by going beyond expectation evaluation when expressing the uncertainty of system parameters. Having defined a loss function applicable for the system of interest, both value-at-risk and conditional value-at-risk theory can be used in order to constrain the level of the loss function depending upon the investor’s degree of risk aversion. Value-at-risk can be defined as the minimum loss that is expected to be exceeded with a probability of (1 - R), while conditional value-at-risk is the expected loss given that the loss is greater than the value-at-risk for a given confidence level, R. Conditional value-at-risk is the average of the probabilistic loss function’s tail defined by the lower and upper bounds of the value-at-risk and maximum loss, respectively. Conditional value-at-risk represents a more pessimistic loss level when compared to value-at-risk, and so, it is often the preferable representation of loss. Due to the aforementioned characteristics, conditional valueat-risk theory has the potential to be applied within several different types of optimization problems sharing a common objective of guarding against the unfavorable realization of uncertain system elements. Gatica et al.167 acknowledged the importance of going beyond expectation evaluation and explicitly considering risk when applying mathematical models for

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capacity planning within the pharmaceutical industry where clinical drug trial outcomes are stochastic in nature. Conditional value-at-risk was used within the work of Jabr168 to formulate a robust self-scheduling model for a power producing company attempting to provide power bids ahead of market trends. Gotoh and Takano169 applied conditional value-at-risk to the classic newsvendor problem dealing with maintaining inventory for one time period while facing uncertain demand. Tsang et al.170 utilized conditional value-at-risk theory when considering the capacity investment problem within the vaccine industry. Yamout et al.171 addressed the shortcomings of using sensitivity and scenario analysis to deal with the allocation and management of uncertain water resources by applying CVAR to the given problem. Barbaro and Bagajewicz172,173 demonstrated that the use of financial risk management can enhance the effectiveness of a proposed planning formulation by explicitly taking into account system uncertainty beyond an expectation evaluation. Verderame and Floudas174 formulated an operational planning model which explicitly takes into account demand due date and demand amount uncertainty by means of a scenariobased conditional value-at-risk approach that in turn utilizes a sample average approximation developed by Wang and Ahmed175 to ensure that the obtained solution is a valid upper bound. 2.2.3. Integration of Planning and Scheduling. A fundamentally important problem within the field of integrated planning and scheduling is the consideration of uncertainty. When attempting to integrate the operational planning and scheduling levels, the modeler must first determine which uncertainties should be taken into account at the planning level and/or scheduling level. The forms of parameter uncertainty then need to be determined and can be classified as following bounded, known, or unknown distributions. Having defined the uncertainty associated with the various parameters, the appropriate modeling technique should be applied to the deterministic model under consideration. For example, two-stage stochastic programming, parametric programming, chance constraint programming, fuzzy programming, conditional value-at-risk approaches, and robust optimization techniques are all valid approaches toward dealing with parameter uncertainty. Each technique has its associated positives and negatives, and the problem structure itself often dictates which technique is preferable. After the uncertainty considerations have been explicitly addressed within the planning and scheduling models, the proposed integration framework needs to be selected. Some form of scheduling feedback should be factored into the chosen integration framework so as to allow for the refinement of planning level production targets. The effective integration of planning and scheduling is an extremely challenging problem within the operations research community, and the explicit consideration of multiple forms of uncertainty only further compounds the complexity of the problem. Yet, industrial case studies demonstrated that the integration of planning and scheduling can have a dramatic impact on operation profitability, and so, the field of integrated planning and scheduling under uncertainty is composed of many challenging problems possessing both industrial and academic significance. Petkov and Maranas103 introduced chance constraint programming to explicitly take into account system uncertainties such as demand within a multiperiod planning and scheduling framework based upon a rolling horizon approach where planning and scheduling levels are revised periodically. Wu and Ierapetritou144 presented a hierarchical approach toward planning and scheduling under uncertainty. The planning level model

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takes into account parameter uncertainty through a multistage stochastic programming approach, and the integration of planning and scheduling is accomplished by means of a rolling framework which includes an iterative methodology that facilitates the convergence of the planning and scheduling results. Majozi and Zhu176 developed an iterative framework for the integration of planning and scheduling of batch plants which utilizes fuzzy set theory in order to model the uncertainty associated with the various operators working within the plant. Verderame and Floudas177 developed an integrated planning and scheduling framework that explicitly takes into account processing time uncertainty and multiple forms of demand uncertainty, and the framework allows for the two-way interaction between the planning and scheduling levels through a rolling horizon approach augmented by a feedback loop. 3. Energy Planning Optimization methods have been applied to the planning of energy systems. Various researchers have focused on the longterm planning of regional energy systems management using stochastic, fuzzy, and interval-parameter programming techniques that integrate uncertain information directly into the modeling formulation.178-184 The problem is characterized by multiple energy sources and technologies, multiple involved sectors and processes (e.g., energy conversion and transmission), uncertainties caused by intermittence, unreliability, fluctuation, or imprecise information, and dynamic factors (e.g., social issues, economics, legislation).180 Several contributions focused on incorporating uncertainties into an existing energy planning model, MARKAL (market allocation), which has been used in the planning of energy systems in various regions. Kanudia and Shukla185 applied stochastic programming into the Indian MARKAL model to deal with uncertainties in energy demand and future carbon mitigation effort. Kanudia and Loulou186 and Fragnière and Haurie187 applied multistage stochastic programming that incorporates multiple future scenarios and provides midcourse corrections. The scenarios are represented in a decision tree, each of which is assigned a relative weight. Lanloy and Fragnie`re188 presented a new software tool, SETSTOCH, that links algebraic modeling languages with stochastic programming solvers. Krukanont and Tezuka189 addressed capacity expansion under uncertainty in a two-stage stochastic programming for the Japanese energy system. Uncertainties exist in end-use energy demands, plant operating availability, and potential carbon tax rate. Ma et al.190 developed a stochastic optimization model for technology adoptions for sustainable development under uncertain carbon taxes and technology development, solved using applications of sequential quadratic programming method. Haugen191 considered a problem for scheduling fields and pipes for natural gas delivery under resource uncertainty and formulated a two stage stochastic model. Sadeghi and Hosseini192 demonstrated the method of fuzzy linear programming for optimization of supply energy system in Iran. Borges and Antunes193 addressed planning under uncertainty on the national level by developing a visual interactive approach to deal with fuzzy multiple objective linear programming. Uncertainties in the energy planning model due to the interactions between different sectors are tackled. Hybrid optimization methods have been introduced to combine the strengths of different approaches in order to handle multiple types of uncertainties in the planning of regional energy system. Lin and Huang178 introduced interval-parameter programming that takes into account uncertainties in the model

coefficients that have unknown probabilistic or possibilistic distribution, expressed as intervals. Lin et al.183 developed a hybrid interval-fuzzy two-stage stochastic model to deal with uncertainties expressed as fuzzy membership functions, discrete intervals, and probability distributions. The model is capable of reflecting recourse decisions and interval-format uncertainty related to the forecasting of demands. Guo et al.182 proposed an inexact chance-constrained semi-infinite programming approach that deals with uncertainties in the left- and right-hand side parameters expressed as functional intervals and probability distribution information. The solutions of these techniques are expressed as intervals (i.e., upper and lower bounds of the objective value), corresponding to extreme decision schemes. These values are analyzed to examine trade-offs between violating constraints or environmental risks and economic performances, reflecting decision alternatives that the system manager can take. Cai et al.194 integrated interval linear programming, two-stage programming, and superiority-inferiority-based fuzzy-stochastic programming for uncertainties expressed in probability/possibilistic distributions and interval values. Cai et al.179,195 integrated interval analysis, superiority-inferiority-based fuzzystochastic programming, and mixed-integer linear programming techniques to address uncertainties related to dynamic interrelationships between model parameters, facility expansion issues, and uncertainties expressed as fuzzy-random variables and discrete intervals. In another work, Cai et al.180 integrated interval linear programming with chance-constrained programming and mixed-integer linear programming techniques to handle uncertainties expressed as probability density functions and interval values. The combined model facilitates dynamic analysis such as capacity expansion planning under uncertainty. Lin and Huang196 combined interval analysis and mixed-integer linear programming. Li et al.197 incorporated robust programming, interval analysis, and min-max regret to deal with uncertainties expressed as discrete intervals, fuzzy sets, and random variables. Lin and Huang198 combined mixed-integer interval analysis and two-stage stochastic programming techniques that take uncertainties as interval values and probabilistic distributions associated with greenhouse gas (GHG) emission reduction targets and address the dynamic nature of capacity expansion aspects. Cai et al.199 developed a large-scale dynamic optimization model (UREM) to support long-term energy systems planning in the region of Waterloo. UREM is a linear programming model with an inexact module based on interval analysis, already used in regional and national level across Canada. The model can handle uncertainties expressed as intervals without known distribution information. Cai et al.181 developed an interactive decision support system that integrates optimization modeling, user interaction, and visual display into a software package (UREM-IDSS). Energy planning for individual large energy consumers is also considered. Mavrotas et al.200 presented a mixed integer linear programming model, representing energy flows and discrete energy technologies, for a hotel unit. The model comprises of fuzzy parameters that represent uncertainties in energy costs and fuel prices. Mavrotas et al.184 focused on the power demand of a hospital unit and treated uncertainties through fuzzy programming. The model is transformed into an equivalent multiobjective model, and the Pareto optimal solution is selected by using a min-max regret criterion. Liu et al.201 developed an inexact model for long-term planning coal and power management systems. The model


combines chance-constrained programming, interval linear programming, and mixed integer linear programming to handle uncertainties expressed in terms of probability density function and intervals, and is also capable of facilitating dynamic analysis of capacity expansions problems. Liu et al.202 considered nonrenewable energy resources management under uncertainty in a hybrid chance-constrained mixed-integer linear programming. Svensson et al.203 presented a methodology for achieving robust investment decisions in process integration under uncertain future energy prices and policies. Stochastic programming is combined with the identification of opportunities to increase energy efficiency in the process to develop a framework for making decisions in investment planning. The methodology is applied to a pulp mill example under uncertain factors such as electricity and biofuel prices and CO2 emission policies.204 Bagajewicz and Barbaro205 presented a two-stage stochastic formulation with risk control for a total-site heat integration under uncertain heating utility prices, heat exchanger cost prices, and flow rate and inlet temperatures of each stream. Yokoyama et al.206 considered the design and multistage expansion planning of cogeneration plants under uncertain energy demands. This is based on min-max regret criterion, and the model formulation is a multilevel linear programming problem. Diwekar207 presented an algorithmic framework for process integration, design, and planning as a multiobjective optimization problem under uncertainty, where the uncertain parameters are expressed via probability distributions. 4. Power Generation System Planning The objective of power generation system planning is to ensure the provision of a reliable supply of power to the consumers at the lowest possible cost. The planner needs to define a series of economical and reliable expansion plans (i.e., when and where to build what kind of lines) to accommodate the expected load growth over the planning period.208,209 The complexity of the deterministic model is due to multiple choices of station locations, multiple choices of primary and secondary routes, multistage investment decisions, complex objectives, and uncertainty about demand behavior, investment costs, and equipment availability.210,211 A reliable plan is measured by its vulnerability to the uncertain events such as fuel supply disruptions and unanticipated load growth that leads to discrepancy between power supply and demand.212 This reliability constraint is typically met by providing an adequate margin of plant over load, considering load forecast error, plant availabilities, and weather variability.213,212 The lowest cost solution requires the determination of the proportion of the various plant types and their fuels which should be employed to provide the given plant capacity and how these proportions should vary in time. The outputs of the strategic resource planning model are optimal values of the following decision variables: total generating capacity added in each year including storage, generating mix by year, coal conversions by year, fuel stockpile sizes to be maintained each year, intertia capacity to be added each year, amount of any other supply side investments considered (e.g., solar technologies), load management, and power shortfall in each state. Uncertainties exist in the system model itself and the unpredictable demand of electricity. Power generation system planning is a composite of varying performance of many plants over many years in supplying a commodity with high fluctuations of demand.214,215 Sources of uncertainty include future load forecasts, plant costs, availabilities of new plants, cost and availability of alternative fuels, governmental policy, environ-

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mental changes, demand growth, fuel cost, and delay in project completion.216 When partial information about the uncertain parameters is available, the approach taken is based on scenario representation. The stochastic characteristics of load demand and investment costs are represented as a set of weighted scenarios where each scenario is a sequence of possibilities for the multistage horizon. Shrestha et al.217 addressed price uncertainty in short-term power generation planning. To represent market price behavior, the next day electricity price is assumed to follow log-normal distribution and the random price behavior conforming to this distribution is generated using Monte Carlo technique for the given values of mean and variance. Carvalho et al.218 addressed the problem of distribution network expansion planning under uncertainty using an evolutionary algorithmic procedure by successive adjustments to the scenario structure. Carvalho et al.211 developed a hedging algorithm to deal with scenario representation of uncertainty in the evolutionary optimization approach. This approach modifies the objective of the planning problem from determining the whole investment schedule to finding the best first-stage investments that minimize the expected future and actual costs. Carpinelli et al.219 addressed the embedded generation problem (connecting generation to distribution systems) based on a genetic algorithm and decision theory approaches. Uncertainties due to disturbances caused by the embedded generation system and existence of renewable sources are represented by a set of possible scenarios. For each scenario, optimal sizing and location is obtained, and decision theory is applied to choose among possible alternatives. Rodriguez et al.220 considered multiobjective planning under uncertainty for distributed energy resources planning and addressed it via genetic algorithms. Reis et al.221 addressed the reinforcement scheduling problem in transmission power systems planning. They represented uncertainty via load/generation scenarios and proposed the use of Gaussian search techniques to solve the scenario-based deterministic subproblems. The progressive hedging algorithm (PHA), which is a scenario aggregation technique, is used to obtain decision convergence by enforcing the subproblems to yield the same first stage decisions. Another scenario aggregation technique is introduced by Deladreue et al.222 who studied cluster analysis to sort large databases when many uncertain factors are taken into account, so that the number of simulations is reduced. Further, a decision analysis approach can be used to handle uncertainties in combination with various transmission planning tools, (e.g., after scenario simulations are completed).223 The contribution of this approach is the quantification and the minimization of risk by addressing questions of regret in choosing a particular plan and if a future that is adverse for that plan occurs. Another approach to study the effect of uncertain parameters on the obtained solutions is via sensitivity analysis. Wang and Liao224 used this approach to address uncertainties in long-term nuclear energy decisions. Hashim et al.225 studied the effect of uncertainties on the optimal solution in a CO2 emission reduction problem from a power grid that consists of various generating plants. The varied uncertain parameters include natural gas prices, coal prices, and retrofit costs. Elkamel et al.226 formulated a fleetwide model for energy planning (i.e., an energy system with multiple sources) to meet a CO2 reduction target while maintaining or enhancing power to the grid. Sensitivity analysis is done by changing fuel prices, retrofit cost, new plant capital investment, and studying their impacts on the structure of the power generating fleet.

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Optimization based approaches can incorporate uncertainties explicitly into the planning models. The following studies used probabilistic simulation as a method of treating uncertainties. Booth213 considered long-term generation planning under uncertainty using probabilistic simulation methods for production costing and a dynamic programming formulation. The procedure combines the system reliability constraint with the determination of production cost in a fast algorithm, capable of calculating the optimal expansion plan. The method aims at providing an optimal pattern of expansion that treats uncertainty in a rational manner by employing dynamic programming that provides background information on the likely optimal development of a system of generating plants. Evans et al.215 considered multiobjective optimization for long-range energy generation planning under uncertainty using dynamic programing. Uncertainties are represented via a discrete probability distribution over a set of scenarios. Yasuda et al.227 studied uncertainties in power demand for the power system expansion problem. Lucas and Papaconstantinou214 used probabilistic simulation to address the interaction between supply and demand uncertainties, and to incorporate social cost of power outages into the objective function. Wu et al.208 studied load uncertainty in the minimum load cutting problem. Load uncertainty is generally expressed via a probabilistic model or an interval model where the only information available is the upper and lower limits of the value. The upper limit is an important index that represents the safety degree of the system under interval load. The uncertainty is expressed via an interval representation, leading to a bilevel linear programming model and resulting in a minimum load cutting number that is also expressed in interval form. Silva et al.228 developed two mathematical models for transmission network expansion planning problems under demand uncertainty represented by an interval. While the first model considered the uncertainty in the total demand of the power system, the second one analyzed the uncertainty in each load bus individually. A specialized genetic algorithm was proposed to solve their models. Deladreue et al.229 used statistical methods to account for transmission planning uncertainties such as localization of new independent power producers and new eligible customers and level of international exchanges. Sanghvi et al.212 analyzed a complex problem involving the evaluation of the cost effectiveness of alternate resource plans that provide an adequate level of peak and strategic reliability. It is based on a linear optimization model that selects power system generation expansion strategies that are cost-effective with the appropriate level of strategic and peak reliabilities. Sherali et al.230 introduced under uncertainty, formulated it as an equivalent deterministic LP, and applied an approach based on a two-stage linear program with recourse to characterize the equipment selection and marginal capital cost allocation based on an optimal capacity plan. Sharaf and Berg209 used a quantitative reliability technique in conjunction with a stochastic load flow formulation to accommodate uncertainty in system component availability and demand projections (static optimization problem in long-range transmission capacity expansion planning). They solved it using chance-constrained programming, on the basis that statistical information from the past can be used to represent the knowledge on the uncertain factors (either expected mean is estimated with ranges of deviation or set of values is predicted with associate probability level) for the probabilistic constraints. They accounted for the uncertainties in load projections in addition to those in the network topology. Tsamasphyrou et al.231

considered the static form of long-term transmission expansion planning in a stochastic mixed integer model and solved it with Benders decomposition. Berg and Sharaf232 presented a dynamic optimization problem in long-range transmission capacity expansion planning by employing an iterative procedure to the static optimization technique. The static optimization technique was applied sequentially for every stage of the planning period. Gorenstin et al.216 described a method for expansion planning that uses stochastic optimization techniques, decision analysis, and multiobjective trade-off analysis. The stochastic approach is extended to reflect investment decisions that are dependent on the previous values of parameters, leading to a multistage mixed integer programming problem. Shiina and Birge’s233 approach is also based on integer programming and stochastic programming. A multistage stochastic programming model is developed where some variables are restricted to integer values. Utilizing the block separable recourse property of the model, the problem is transformed into a two-stage stochastic program with recourse. Ahmed et al.234 modeled demand and cost parameter uncertainties in a multistage stochastic integer programming as a multilayered scenario tree, and the optimization problem consists of determining an expansion schedule that hedges against this scenario tree. Kaleta et al.235 addressed a self-scheduling method for the thermal units on the energy market where uncertainties on energy prices are modeled by a set of possible scenarios with assigned probabilities. Marıń and Salmeroń236 presented a stochastic model for capacity expansion planning under capacity deterioration and demand uncertainty. The model combines techniques of stochastic optimization, robust optimization, and statistical decision rules. Demand uncertainty is treated by creating a risk-based function as part of the model’s objective function. Heinrich et al.237 implemented stochastic programming models with recourse that consider multiple objectives and system flexibility to demand uncertainty for the expansion problem. Saric´ et al.238 presented a two-stage stochastic programming model that incorporates contingencies in developing a generation plan. Roh et al.239 studied the stochastic coordination of long-term generation and transmission expansion planning model in a competitive electricity market, using Monte Carlo simulation and scenario reduction techniques to simulate demand growth uncertainties and random outages of generating units and transmission lines. Mo et al.240 addressed uncertainties such as energy demand and prices of energy carriers along with the dynamics of the system in power generation expansion problems. Uncertainties are modeled as event trees, and the formulation takes the form of a multistage problem with recourse. Fujii and Akimoto241 presented a method for power system planning under uncertain CO2 emission control policies based on the stochastic dynamic optimization technique. The probability distribution of uncertain factors is determined dynamically through a policy decision model based on a time-varying Markov process. The problem formulation takes the form of a multistage linear programming problem. When probabilistic distribution is unavailable, another viable method to solve planning problems under uncertainty is fuzzy programming. Dimitrovski and Tomsovic242 modeled uncertainties associated with risk with fuzzy set theory in load flow. Cheng et al.243 presented a method of power network planning under uncertainty which is based on the concept of unascertained number. Canha et al.244 presented approaches regarding the use of fuzzy sets to represent various forms of uncertainty in optimization problems and applied the methods to solve power

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engineering problems. Khodr et al. considered the distributed generation network planning problem under uncertainty and formulated it as a multistage fuzzy multicriteria nonlinear problem. Sources of uncertainties include load demand and power injections of distributed resources, which are represented by fuzzy set numbers. Liu et al.246 developed a hybrid model for the expansion planning of power generation under uncertainty. It addresses system uncertainties in diverse forms by integrating fuzzy probabilistic and joint probabilistic programming into a mixed-integer programming framework. The solution reflects the trade-off between system cost and system stability through examining the risk of violating system constraints under uncertainty. As power generating systems that utilize renewable sources of energy (e.g., wind power systems, hydrogeneration systems, etc.) increase in number in response to concerns regarding GHG emissions, the planning problem of these systems become increasingly relevant. Its importance is compounded due to the intermittent nature of the energy sources. As nature changes in a short time scale, the planning problems for the operation of these systems span from short-term or scheduling problems to long-term planning problems, where the time scales considered span from a day to a year. Zhao et al.247 combined optimization methods and a probabilistic analysis to find the maximum capacity of a future wind farm subject to the limits of the grid system (e.g., voltage stability limits, thermal limits, voltage limits, etc.). The uncertainty of wind energy is represented in a probability distribution of wind power. Tuohy et al.248 took the uncertain nature of wind into account in the scheduling problem. A scenario tree tool is used and a scheduling model is formulated as a stochastic optimization problem. Nembou and Murtagh249 applied a stochastic optimization model to a hydrothermal electricity generation planning problem in Papua New Guinea. The model accounts explicitly for the uncertain seasonal water inflow and reservoir storage, as well as electricity demand using chance-constrained programming. Escudero et al.250 presented a modeling framework for medium and long-term planning problems under uncertain water inflows and outflows based on scenario analysis. Gonza´lez251 considered the long-term hydrogeneration planning in which the uncertain flows are represented as random variables with probability parameters. Cabero et al.252 presented a methodology to manage market risk for hydrothermal generation companies. Uncertainties in the problem come from fuel prices, power demand, water inflows, and electricity prices, which are represented in a scenario tree, and the resulting model is a multistage stochastic linear programming model. The authors also used conditional value-at-risk as measures of risk exposure. Nolde et al.253 formulated a multistage stochastic programming model for monthly production planning of a hydrothermal system where stochasticity in water reservoir inflows and electric energy demand are explicitly considered. Fleten and Kristoffersen254 studied the scheduling problem for the short-term production plan of a hydropower plant using multistage mixed-integer linear stochastic programming, assuming that the probability information of uncertain data is available. The uncertainties considered include the day-ahead market prices, due to the bidding into the power exchange that needs to be done a day in advance, and reservoir inflows.

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5. Farm Planning The objective of planning problems on the farm level is to maximize the gross margin subject to constraints that define the environment within which choices are made (e.g., labor requirements, land available, working capital requirements). Decision variables include the determination of the optimum cropping pattern, the interdependence of parts of the farm, and the optimal sizes or different types of fixed equipment and machines to add to farm resources. At the regional level, the objective function may change to incorporate the maximization of social welfare. The uncertainties that farm planners face include income variation with weather conditions, market price changes, and crop and animal diseases. Methods that aim at addressing agricultural uncertainties are described in this section. Game theory based models can be used to find a pure or mixed strategy that optimizes the decision maker’s aspirations according to a certain behavioral criterion (e.g., maximizing the minimum outcome, minimizing the largest regret, or maximizing the minimum benefit). These methods are a means of analyzing agricultural decisions under uncertainty. McInerney255 approached farm planning using a min-max linear programming from game theory, securing a higher minimum level of return for the farmer. Tadros and Casler256 used the game theory approach to address uncertainties in nature. A linear programming problem is solved for each state of nature to give the optimal strategy, and a matrix is constructed to record the payoffs of that specific farm plan for both the expected and unexpected states of nature. Pure or mixed strategies are determined according to the farmer’s decision criteria. Hazell257 reviewed the strengths and weaknesses of game theory and variance minimization approaches taking the form of quadratic programming problems for farm planning and reconstructed a parametric linear programming that combines the desirable features of the two approaches. Game theory requires that all possible states are known without taking into account the probability of occurrences. The minimization of system variance approaches could generate a series of farm plans for different levels of expected total gross margin. The new game theory based models incorporate min-max and regret decision criteria that combine the desirable features by assuming a forecast state and using past states as a measure of risk for the forecast period. Hazell258 noted that uncertainties in activity costs, yields, and prices affecting the objective function can be taken into account through the minimization of system variance yielding a quadratic programming problem. However, these quadratic programming problems require knowing a priori the mean gross margin, variance, and covariance for each activity, which may prove to be impractical for complex farm businesses. A linear alternative to the quadratic programming referred to as the minimization of total absolute deviations (MOTAD) model was developed, in which the sample variance is replaced by the sample mean absolute income deviation. Hazell259,260 maintained that the model can also perform well in non-normal sample distributions, such as symmetric distributions. Collender261 developed a statistical testing method to examine the effect of estimation risk on the ability to distinguish the mean-variance characteristics of different portfolios when parameters are unknown. Methods to construct mean-variance confidence regions around estimated points were developed. Maruyama262 formulated a linear programming model for farm planning that incorporates uncertainty in functional, restraint, and input-output coefficients, assuming that they take on a finite number of possible values with associated discrete probability distributions. Para-

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metric linear programming can be used to obtain solutions for the truncated min-max problem. Stochastic approaches have also been developed for farm planning. Musshoff and Hirschauer263 used bounded recursive stochastic simulation that consists of a dynamic programming approach and Monte Carlo simulation to treat the stochastic variables. Lien and Hardaker264 developed a two-stage stochastic programming model with concave form of the utility function that has constant relative risk aversion. A negative exponential utility function is also considered that exhibits constant absolute risk aversion. Itoh et al.265 formulated a stochastic linear programming model for the crop planning problem of maximizing the minimum total gain. The uncertainties in the profit coefficients are treated by introducing probabilities in the problem formulation. Torkamani266 studied the whole-farm planning considering using stochastic programming that can handle uncertainties in the coefficients of both the objective function and constraints. Rodrı´guez et al.267 studied the mediumterm planning of sow farms producing piglets with a two-stage stochastic linear programming model with recourse. Uncertainties are introduced through a finite set of scenarios in the model formulation. A robust optimization approach was introduced recently to treat uncertainties in agriculture. Bohle et al.268 used this method to deal with uncertainties when probabilistic knowledge is incomplete. They studied the scheduling of wine grape harvesting within the optimization framework that helps schedule harvesting operations in a vineyard and assigning labor to operations. 6. Forest Planning Forest planning aims at finding an optimal allocation of the available land resources to a number of management options. This allocation problem is also called harvest scheduling and can be solved using a linear program. Traditionally, the emphasis of the forest planning problem has been on the production of goods and services (i.e., timber production) that follows the agricultural model. Decision making in forest management ranges from short-term operational planning to strategic, longterm planning that could involve time horizons up to several decades.269,11 Long range decisions include yield strategies to obtain sustainable yields at the aggregate level in terms of land specification and timber products. Tactical or medium range decisions cover time horizons from one harvest to the next with a more detailed spatial resolution of the forest. Included in this plan is road construction for accessing the timber and transporting timber products from the forest. The operational or shortrange decisions determine which areas to harvest each week and how to cut up stems into logs of defined quality, length, and diameter to meet demand, as well as locating harvesting machinery to carry felled trees to roads and the required transportation.11 As awareness increases regarding the need to preserve forest ecosystems, the focus of forest planning has shifted to also consider forest health, biodiversity, protection of threatened and endangered species, and scenic beauty, along with the forest’s productivity. This ecosystem model also addresses the question of which areas to leave unharvested to protect wildlife habitat.9,270 Long-term planning will have to depend on various projections and forecasts that are difficult to address. Accounting for uncertainties is crucial in order to avoid harmful decisions. The uncertainties that forest planners face include future timber market (changing timber prices), forest productivity (timber growth, yield projections), dynamics of the forest itself and

possible natural disturbances or catastrophic events (e.g., forest fires, wind storms, insect infestations), technological change, and changing environmental regulations and social objectives. Prior knowledge and experience can provide statistical data on some uncertainties. Several approaches have been developed to address uncertainties in the forest planning problem. Generally, uncertainties can be accounted for by testing the robustness of the model through sensitivity analysis or parametric linear programming, or by handling them explicitly in the problem formulation. The latter approach includes probability based models, fuzzy models, and robust optimization methods. Nelson10 noted that progress in forest planning has advanced from small, spatial supply models with simple habitat constraints to large, multiobjective forest-level models that include landscape pattern and structure. Nelson highlighted the need of sensitivity analysis that explores how changes in individual data or key parameters modify the optimal solution. In general, sensitivity analysis and parametric linear programming have not been used extensively in the field due to the wide range of values that needs to be explored in large-scale forestry problems.269 There are many forest planning models that explicitly incorporate uncertainty using probabilistic-based methods, such as probabilistic dynamic programming, stochastic programming, portfolio theory, chance-constrained linear programming, scenario analysis, Markov decision models, and optimal control theory.271 Probabilistic or stochastic models are based on the assumption that the uncertainty inherent in a system can be captured by defining the probability distribution that the random variables of interest follow. Boyland et al.271 combined scenario analysis and a simulation approach with deterministic optimization techniques where uncertain coefficients are randomly generated and optimal solutions for a set of scenarios are found. The main question was how to project sustainable harvest volumes while maintaining social and ecological objectives. Forboseh and Pickens272 presented a harvest scheduling problem with stochastic stumpage prices, where demand constraints on the periodic yield of some products exist over a long planning horizon. The underlying price uncertainty is modeled as a stationary process with normally distributed deviations. Boychuk and Martell273 addressed the risk of fire in forest planning using a multistage stochastic model. Eriksson274 extended a model formulation to incorporate uncertainty as a program with recourse, taking into account that the decision maker is able to observe the state of the system over time and subsequently make adaptations. The stochastic process is treated as a collection of scenarios. Probabilistic programming is appropriate for types of problems where uncertainty is mainly due to randomness. However, for other factors such as imprecision, ambiguity, inexactness, and inaccuracies, where the probability distribution is not always available, another approach is needed. Fuzzy models are based on the assumption that the uncertainties can be represented by treating certain model parameters as fuzzy numbers or fuzzy relationshipssimplying vagueness or ambiguity. Fuzzy coefficients are usually represented as an interval instead of a single number and constraints are referred to as fuzzy or soft if they do not have to be strictly satisfied. Mendoza et al.269 applied fuzzy multiple objective linear programming in forest planning where objective function coefficients are uncertain. The uncertain coefficients are represented by interval values, the problem is reformulated as a min-max problem utilizing linear membership functions, and it is extended for uncertainties in the constraints.


A different approach to address uncertainty is through robust optimization, and it was used in a recent study by Palma and Nelson.275 The authors introduced a robust optimization methodology128 to handle uncertainties in timber yield and demand of two products. Since robust optimization maintains the linearity of the original problem in the reformulation, the resulting model is computationally tractable. This method assumes that uncertain data are uniform and independently distributed within a symmetrical range of values. 7. Waste Management Municipal solid waste (MSW) management has received considerable attention during the last decades because solid waste has produced serious effects on the environment, ecosystems, and human health.276,277 The main challenge is the enormous economic cost of cleaning up pollution from solid waste and transporting waste from cities to treatment facilities and the operation of treatment facilities. However, many system parameters such as waste-generation rates and waste-treatment costs may be uncertain.278-280 Several optimization models and methods were developed to address the waste-management planning problem to minimize the associated treatment costs. Mostofthemarebasedonintervalanalysis,281-283 stochastic,284-288 chance-constrained,289 and fuzzy290 approaches. Uncertainties are represented by stochastic, fuzzy membership, and intervals. Huang et al.282 applied an integer programming method for the facility expansion planning under uncertainty for water management planning problem. The uncertain inputs are treated as intervals. Guo et al.288 combined stochastic programming, integer programming, and interval semi-infinite programming approach for solid waste management under uncertainty where uncertain left-hand-side parameters are expressed as intervals and right-hand sides parameters are represented by intervals and probability distributions. To address the uncertainties represented by probability distributions in both the left-hand-side and right-hand-side coefficients of the constraints, robust optimization models291,292 were proposed. Cai et al.293 proposed an interval-valued robust programming approach incorporating multiple uncertainties. Chakraborty and Linninger294 extended their planning method295 to optimize operating procedures in the presence of uncertainty in the effluent stream represented by scenarios with a fixed-plant inventory. Chakraborty et al.296,297 addressed the same problem with variable plant inventory. List et al.298 developed a two-stage stochastic robust optimization model for the transportation planning problem of radioactive wastes from remediation sites to disposal sites under uncertainty which includes the total quantity of waste, processing rates at the remediation sites, and certification of packages. All uncertainties were represented by scenarios with known probability of occurrence, while the objective was to determine the equipment acquisition strategy and fleet planning including financial and political risks. 8. Water Resources Management With climate change and increasing demands for freshwater by expanding global population, the need for wise use of freshwater and the improvement of current water management have become of increasing importance. Integrated water resources management involves many applications such as catchment management, groundwater management, water supply planning, and water quality management.299 The decision making process is generally complex and involves a wealth of data, scenarios, models, alternatives, decision makers, and

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stakeholders with conflicting interests. More importantly, decision making often involves noncommensurable objectives, especially when environmental and social factors are considered. As a result, most work focused on analysis and ranking of different alternatives. The cost benefit analysis framework was first developed for water resource decision-making problems with the objective of maximizing the ratio of benefits to costs. The approach used an economic index such as the total revenue, benefit/cost ratio, or rate of return to compare different decision alternatives. Uncertainties were included into the objective by incorporating the expected cost of failure. Then, the uncertainty of each alternative can be included into the objective function using the risk of failure. As a result, the risk-cost-benefit objective function was developed. The advantages of the cost-benefitanalysis framework are its simplicity, flexibility, treating uncertainties, and explaining the results in financial forms.300 However, the cost benefit analysis framework does not represent the failure cost properly because postfailure consequences are inherently difficulty to forecast. Moreover, litigation costs, regulatory penalties, loss of opportunity or investment, and damage to public relations are also hard to quantify or predict. Therefore, decisions using a cost benefit analysis framework are highly sensitive to the failure cost.301,302 An alternative approach named multicriteria decision analysis (MCDA) is widely used for water resource planning since it provides a framework to deal with noncommensurable aspects and to facilitate stakeholder’s involvement for collaborative decision-making.303 A comprehensive review on MCDA methods such as linear multiobjective programming, goal programming, compromising programming, composite programming, and multiattribute utility theory can be found in the work of Zopounidis and Doumpos.304 The MCDA methods are widely used in water resources management without considering uncertainty.305,299 However, uncertainties often occur because of data unavailability, variability, and knowledge deficiency. Moreover, the ever-changing environmental factors such as climate and land use may also increase the uncertainty. Applications of uncertainty and sensitivity analysis in real-world water resource decision making are rarely found in the literature or in practice.306,305 Several approaches such as sensitivity analysis methods, stochastic uncertainty analysis, distance-based uncertainty analysis, and fuzzy-based approaches are incorporated into the MCDA framework for water resources planning. Fuzzy-based approaches have received extensive attention because of their flexibility in addressing imprecise subjective data. Uncertain basic indicator values for evaluating and ranking alternatives for water resources planning problems are represented by fuzzy numbers. Lee et al.307 developed a risk management method for nitrate-contaminated groundwater supplies using fuzzy set approach to represent some uncertain basic indicator values such as Methemoglobinemia, nitrate level, and cancer. They used an MCDA method to evaluate and rank various nitrate risk-management strategies. Lee et al.308 employed the same approach to evaluate and rank various water supply lines. Uncertain basic indicator values such as velocity of flow, pressure in node, and difficulty of maintenance were represented by fuzzy numbers. Raj and Kumar309,310 used maximizing and minimizing set concepts of fuzzy logic to select the best reservoir configuration for the Krishna River basin in India. The experts’ opinion is represented with fuzzy numbers. Bender and Simonovic311 incorporated the application of fuzzy sets to compromise programming to address water resource systems planning under uncertainty where criteria values and

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weights are expressed with fuzzy sets. Other works for water resources planning using fuzzy sets can be found in the work of Raju and Kumar,312 Yin et al.,313 Chang et al.,314 Chang et al.,315 Chang et al.,316 and Vamvakeridou-Lyroudia et al.317 Hyde et al.318 proposed a reliability-based approach to MCDA for criteria weights and performance values uncertainties where a continuous probability distribution such as uniform, normal, or logistic is used for uncertain criteria weights and quantitative performance values and a discrete uniform distribution is used for qualitative performance values. They also considered the correlations between criteria weights. However, this reliabilitybased approach requires appropriate probability distributions for the criteria weights that may be difficult in some situations with insufficient data. To overcome the disadvantages found by Hyde et al.,318 Hyde et al.303 proposed a distance-based analysis approach with MCDA to address only criteria weights uncertainty. They first used deterministic MCDA to identify the total value of each alternative. Then they determined the minimum modification of the criteria weights that is required to alter the total values of two selected alternatives such that rank equivalence occurs with the lower and upper limits for criteria weights by solving another optimization problem. The proposed approach was used to address the water resource planning problem from Raju et al.319 and Duckstein et al.320 Escudero321 developed a multiperiod scenario-based model for water resources planning under hydrological exogenous inflow and demand uncertainties. They used augmented Lagrangian decomposition methods to solve their model. 9. Transportation Planning Transportation systems use fleets of vehicles to carry people or goods from origins to destinations. The main transportation modes include road, rail, maritime, and air. The main transportation problems are fleet sizing and vehicle routing and scheduling. In this section, we discuss about transportation planning under uncertainties. The capacity of a transportation system involves the number of available vehicles. Since vehicles are costly capital items, determining the optimal number of vehicles, their utilization, and empty vehicle allocation (i.e., fleet sizing) for a transportation system to meet customer demands is quite important. Fleet sizing involves shipments carried, loaded vehicle flows across the network, and vehicle fleet size(s) to optimize service quality, cost, profit, etc.322 Fleet sizing is related with many areas such as overall service design,323 trucking,324-326 airline express package service,327 and material handling systems used for manufacturing operations.328,329 Uncertainties in fleet sizing mainly come from demands and operating conditions. However, it is often quite difficult to address the models of the fleet sizing problem even under nominal conditions and relatively little work has been proposed for fleet sizing problems under uncertainty. Contributions involve vehicle movement decisions for stochastic freight flow patterns,330 vehicle allocation decisions under demand uncertainty,331-336 fleet sizing decisions under uncertain travel and service times,337 and integrated fleet sizing and empty equipment allocation decisions under demand uncertainty.338-340 List et al.322 developed a two-stage stochastic model for fleet sizing under uncertainty in demands and operating conditions. The uncertainties were represented with a normal distribution, and a stochastic decomposition procedure was proposed to solve their model. Song and Earl339 addressed the empty vehicle repositioning and the fleet sizing problem under uncertainty from loaded vehicle arrival times at depots and repositioning times

for empty vehicles. Dong and Song340 addressed the container fleet sizing and the empty container repositioning problem simultaneously under customer demand uncertainty where the uncertain demands are represented by uniform and normal distributions. A simulation-based evolutionary optimization algorithm was developed to find the optimal container fleet size and determine the policies of empty container repositioning. Sayarshad and Tavakkoli-Moghaddam341 developed a two-stage stochastic programming model for optimizing rail-car fleet size and freight car allocation under demand uncertainty. The demand uncertainty is represented assuming a probability distribution, and the model was solved using a simulated annealing approach. Dejax and Crainic342 presented a comprehensive review of research on fleet sizing issues. The vehicle routing and scheduling problem is another important transportation problem. The objective is to determine optimal routes and timetables for given fleet size of vehicles while minimizing the total operating cost or maximizing the total profit. A detailed survey on deterministic routing and scheduling models can be found in the papers of Cordeau et al.,12 Ronen,343,344 and Christiansen et al.345 Common uncertainties for vehicle routing and scheduling are stochastic demands and travel times. Sometimes, the set of customers to be visited is not known with certainty. Stochastic vehicle routing and scheduling problems are often regarded as computationally intractable since they combine the characteristics of stochastic and integer programming. Stochastic vehicle routing problems are usually modeled as mixed or pure integer stochastic programs or as Markov decision processes.13 Most of them are modeled as chance-constrained stochastic programming346-349 and stochastic programming with recourse.350-352 All these models are transformed into equivalent variants and solved with exact algorithms such as the integer L-shaped method,350,353,351 exploiting particular data structures,354 dynamic algorithm,355 and heuristics.356,347,357 Isleyen and Baywoc349 developed an efficiently model for the vehicle routing problem under demand uncertainty. Uncertainty demand was derived from a continuous probability distribution such as the uniform distribution. They used Monte Carlo simulation to test the accuracy of their model. A detailed review on stochastic vehicle routing can be found in the paper of Gendreau et al.13 Tang and Yan358 proposed a routing and scheduling framework for intercity bus carriers under stochastic travel times and demand. Christiansen and Fagerholt359 presented a robust shipscheduling problem with multiple time windows under ship arrival delay and stochastic service times. Another transportation problem is related with evacuation management. Yao et al.360 developed a robust linear programming model to provide a framework for evacuation management planning problem in large-scale network under demand uncertainty. The vehicle routing problem combining inventory management denoted as the inventory routing problem has also been considered in the literature. Uncertainties in the inventory routing problems mainly arise from customer demands. A large number of stochastic models based upon dynamic programming or Markov decision processes have been developed for this problem under demand uncertainty. Kleywegt et al.14,15 classified these models and presented comprehensive surveys. Kleywegt et al.14 formulated an inventory routing problem under demand uncertainty as a Markov decision process. They assumed one delivery per trip. To solve their model, they developed an approximation method in which they decomposed the overall problem into subproblems and used the results of the subproblems to approximate the optimal value function.

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Kleywegt et al. extended both the formulation and solution approach of Kleywegt et al.14 to deal with multiple deliveries per trip. Hvattum and Lokketangen361 modeled the inventory routing problem with uncertain customer demands as a Markov decision problem. They examined the progressive hedging algorithm for solving the scenario tree based subproblems. The airline planning process is typically decomposed into four subproblems: schedule generation, fleet assignment, maintenance routing, and crew scheduling.16 The objective of schedule generation is to determine origin-destination itineraries, frequencies, and times to fly in a given period of time. The fleet assignment problem determines which type of aircraft should be assigned to each flight. The maintenance routing problem is a feasibility problem which assigns specific aircraft to flights so as to ensure adequate opportunities for required maintenance checks. The objective of the crew scheduling problem is to find the most cost-effective assignment of the crew to the flights. However, planning for the airline industry has become more complex and dynamic.362 Airline operations are subject to significant disruptions. Disruptions often occur because of weather conditions, unplanned maintenance issues, safety checks, security concerns, and more. Inefficient schedules can become very costly.16 Ahmed and Poojari16 presented a comprehensive review on airline planning under uncertainties. The goal of robust planning is to generate schedules that are less sensitive to disruptions. Two major approaches have been proposed for robust airline planning. They are based on a stochastic programming model363,364 or a set of surrogate problems with stochastic nature.365-367 While robust planning models try to avoid disruptions proactively, recovery models368-371 seek the best way of reacting when disruptions do occur, so as to minimize their impact on the system and prevent propagation. Detailed reviews on recovery models can be found in Kohl et al.372 and Bratu and Barnhart.373 10. Other Application Sectors Motion planning is a fundamental problem in robotics. In motion planning with uncertainty, the objective is to find a plan which is guaranteed to succeed even when the robot cannot execute it perfectly due to uncertainty. While classical motion planning traditionally assumed that there is no uncertainty in the robot motion or sensing, recent approaches to the problem explicitly address motion planning with uncertainty.374 Uncertainty arises from sensing errors, control errors, the geometric models of the environment of the robot, and imperfect information on the locations of objects. Applications arise in two main domains in part mating for mechanical assembly, and mobile root navigation.375 Various approaches have been proposed, which are applicable to one domain, or the other, or both. They include skeleton refinement,376 inductive learning from experiments,377 iterative removal of contacts,378-380 and preimage backchaining.381,382 While skeleton refinement, inductive learning from experiments, and iterative removal of contacts first generated a motion plan assuming no uncertainty, and then transformed to deal with uncertainty, preimage backchaining takes uncertainty into account throughout the whole planning process. In the recent artificial intelligence literature, the uncertainties were viewed as random variables or a stochastic process, which led to Markov decision processes and their extensions,383,384 as well as partially observable Markov decision processes.385-392 To solve these models, various approximations techniques have been developed, such as grid-based approaches,393-395 point-

387,396

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based approaches, factorization-based approaches, compression methods,400 policy search methods,401 logic of knowledge,402 and the ARKAQ-Learning algorithm391 and heuristic search.403 All these models are solved with the value functions defined for the belief states404 or the information state.405,406 Smith et al.407 addressed the problem of strategic planning for the mining industry. Alighanbari and How408 addressed robust planning for the task assignment problem of unmanned aerial vehicles (UAVs) under uncertainty. Gardner and Buzacott409 demonstrated how a stochastic model under new technology development uncertainty is used to generate possible hedging strategies for this problem. Although their methods are general, they use direct steelmaking in the iron and steel industry to illustrate their ideas. The uncertain parameters are represented by scenarios. Chan et al.410 used a robust approach for radiation therapy treatment planning problem under uncertainty coming from intrafraction breathing motion in lung tumors. Their objective is to minimize the total (or integral or mean) dose delivered to the phantom, while ensuring that all voxels in the tumor receive a specified level of dose. Karabuk411 developed a stochastic programming model for the yarn production planning problem in textile manufacturing under demand uncertainty, which is represented by scenarios. 11. Future Challenges and Opportunities As markets become more competitive, companies will continue to rely on rigorous planning and scheduling models in order to help coordinate and schedule supply chain activities. Optimization models taking the form of linear programming, nonlinear programming, mixed-integer linear programming, and mixed-integer nonlinear programming are ideal when dealing with many different variants of planning and scheduling problems, because the constraint set can represent the system characteristics and the objective function can help to quantitatively reflect the goals of the company such as profit maximization. The primary difficulty within the field of planning and scheduling under uncertainty is the formulation of mathematical models which accurately portray the system under consideration and at the same time can be solved within a tractable period of time using commercially available solvers. In the area of strategic planning, the explicit consideration of the various issues regarding the merger and acquisition problem and the potential synergy which can be achieved should be addressed. Yoon and co-workers412-414 considered the merger and acquisition problem for petrochemical companies operating within the same complex under deterministic system conditions; however, the explicit consideration of parameter uncertainty, as well as the extension to other industries would be a worthwhile pursuit. Due to the considered time horizon, operational planning models are often formulated as aggregated scheduling models where the production capacity of the plant is approximated; however, this modeling approach leads to the planning model potentially overestimating the production capacity of the plant. As a result, future work aimed at formulating planning models which provide very tight upper bounds on the production capacity for a variety of plants should remain a very important objective within the planning and scheduling community. Within the last two decades, considerable advances have been made in the area of short-term scheduling; however, most academic contributions have focused on validating techniques on small case studies. In order to close the gap between idealized test systems and real-world applications, continued work will need to be conducted in adapting existing frameworks to address large-scale

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industrial problems. For example, Janak et al.138,139 and Castro et al.415 proposed frameworks for the scheduling of large-scale, multiproduct batch plants, and Shaik et al.53 developed a medium-term scheduling approach for large-scale industrial continuous plants. Taking into account the different forms of process uncertainty within these large-scale application also will be an important direction within the scheduling community. Supply chain activities often follow a hierarchical approach where information is supplied to one level by a preceding level. For instance, strategic planning decisions are received by the operation planning level which in turn provides information to the scheduling level. Industrial case studies indicated that in order order to maximize the productivity and efficiency of the supply chain under consideration, integration schemes which allow for two-way communication are integral. The effective integration of operational planning and scheduling can lead to a dramatic reduction in missed product orders and inventory. The continued development of integration frameworks aimed at enhancing the communication between various supply chain levels is an another important research area which warrants further academic and industrial involvement. Although planning and scheduling under uncertainty has been covered for many applications, there are still some research opportunities that need to be addressed in the future. Liu et al.246 noted that their development of the hybrid optimization models for the expansion planning problem can be used in energy and environmental planning problems. Indeed, recently this hybrid framework has been applied to energy planning problems. This overlap indicates that the developed frameworks in the energy planning sector can also be applied to power generation systems. Recent works by Palma and Nelson275 and Bohle et al.268 in forest planning and farm planning, respectively, showed that the implementation of robust optimization frameworks formerly applied in engineering problems have potential application in these sectors. For municipal solid waster management, although I-VFRP developed by Cai et al.293 is applicable to practical problems with highly complex and uncertain data, it can be further enhanced by incorporating the methods of uncertainty analysis. For motion planning, effective methods are still needed to reduce computational cost and applied into real-time planning in large-scale environments. For capacity expansion problems, two-stage stochastic models could be extended to multistage to reduce further risks and investments. Moreover, there is great potential for future contributions within the area of scheduling under uncertainty, especially with regard to ship routing. When modeling parameter uncertainty, two-stage stochastic programming, parametric programming, fuzzy programming, chance constraint programming, robust optimization techniques, conditional value-at-risk, and other risk mitigation procedures are often applied. Table 1 outlines the various forms of uncertainty present within each sector under investigation, and Table 2 provides a breakdown of the steps undertaken within each approach, as well as a list of the various sectors which have utilized each of the aforementioned approaches. The inherent downside of scenario-based two-stage stochastic programming is the exponential increase in problem size due to the increase in considered scenarios. As a result, scenario truncation/merging techniques which take advantage of problem characteristics in order to reduce the number of required scenarios would greatly enhance the applicability of scenariobased two-stage stochastic programming. When using continuous distributions within two-stage stochastic programming, the issue of exponential model size growth is avoided, but the problem of evaluating the expectation term containing those

Table 1. Forms of Uncertainty sector CPPI

forms of uncertainty market prices processing times demand amount and due date unit capacities unit breakdowns (availability) transportation time and cost

uncertainty distribution bounded known discrete known continuous fuzzy sets unknown

energy planning

demand amount energy prices technological development capacity expansion fuel prices equipment prices fluctuation in plant operations future carbon mitigation effort

bounded known discrete known continuous fuzzy sets unknown

power generation system planning

demand, prices load forecasts transmission schemes demand locations investment costs fuel costs network expansion power outages capacity deterioration retrofit costs


farm planning

crop prices activity costs weather conditions onset of disease crop yields


forest planning

demand, prices forest productivity forest dynamics natural disturbances catastrophic events technological changes environmental regulations


waste management

waste generation rates transportation cost operation cost waste disposal capacities waste loads

bounded normal fuzzy sets

water resources management

basic indicator values criteria values and weights

discrete uniform continuous uniform continuous normal continuous logistic

transportation planning demands travel and service times number of customers to be visited vehicle/ship/airline arrival time empty vehicle repositioning time freight flow vehicle productivity

continuous normal bounded exponential

other application sectors

known discrete

sensing errors control errors object location (motion planning) demand (textile manufacturing)

uniform

recourse decisions which are dependent upon uncertain parameter realization remains. Methodologies which could facilitate the removal of these nonlinearities in the expectation evaluation along with reducing the required computational time in evaluating the expectation would in turn address several of the fundamental issues within two-stage stochastic programming. When utilizing both chance constraint and robust optimization techniques, potential nonlinearities can be introduced into the

Ind. Eng. Chem. Res., Vol. 49, No. 9, 2010 Table 2. Outline of Uncertainty Approaches Two-Stage Stochastic Programming 1. Define uncertain parameters’ distribution characteristics. 2. Decide whether to express uncertainty through continuous distributions or scenario generation. 3. Reformulate model into a first stage deterministic and second stage recourse. Application Sectors: CPPI, waste management, water resources management, transportation planning, Other application sectors, energy planning, power generation system planning, farm planning, forest planning Parametric Programming 1. Define uncertain parameters’ information. 2. Apply parametric programming algorithm to determine critical regions. 3. Requires solving subproblems in order to determine unique solutions within critical regions. Application Sectors: CPPI, farm planning Fuzzy Programming 1. In the absence of distribution information, apply fuzzy set theory. 2. Treat uncertain parameters as fuzzy numbers. 3. Express constraints containing uncertain parameters as fuzzy sets. Application Sectors: CPPI, waste management, water resources management, energy planning, power generation system planning Chance Constraint Programming 1. Define uncertain parameters’ distribution characteristics. 2. Express probabilistic constraints. 3. Set constraint satisfaction level. 4. Transform probabilistic constraints into deterministic form. 5. Replace uncertain constraints with derived deterministic constraints. Application Sectors: CPPI, waste management, transportation planning, energy planning, power generation system planning Robust Optimization Techniques 1. Define uncertain parameters’ distribution characteristics. 2. Express probabilistic constraints. 3. Set relative uncertainty, constraint satisfaction, and constraint relaxation levels. 4. Transform probabilistic constraints into robust counterpart constraints. 5. Augment model with deterministic robust counterpart constraints. Application Sectors: CPPI, waste management, other application sectors, power generation system planning, farm planning, forest planning. Risk Mitigation Techniques 1. Define uncertain parameters’ distribution characteristics. 2. Capture risk through appropriate loss function expression. 3. Constrain maximum level of perceived loss through confidence and exposure levels. Application Sectors: CPPI, power generation system planning

model if the uncertain parameters are being multiplied by system variables, and so, the development of tight convex and concave envelopes for these nonlinear terms would be useful when applying both chance constraint programming and robust optimization techniques. The primary objective when attempting to explicitly consider parameter uncertainty is to guard against the unfavorable realization of the uncertain parameters, and with this objective in mind, several risk mitigation techniques used within the financial industry may be applicable to the planning and scheduling problems found within other industries. For example, conditional value-at-risk theory goes beyond expectation evaluation and attempts to minimize in a rigorous way exposure to perceived risk, and consequently, the application of conditional value-at-risk within planning and scheduling problems is a potentially attractive avenue for future research. 12. Conclusions Planning and scheduling have various connotations depending upon the sector of interest. However, mathematical modeling and optimization based frameworks taking into account parameter uncertainty represent the common thread which connects

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ReceiVed for reView December 18, 2009 ReVised manuscript receiVed March 10, 2010 Accepted March 25, 2010 IE902009K

Planning and Scheduling under Uncertainty: A ... - ACS Publications

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