A fuzzy-stochastic robust programming model for regional air quality management under uncertainty

This paper proposes a hybrid fuzzy-stochastic robust programming (FSRP) method and applies it to a case study of regional air quality management. As an extension of the existing fuzzy-robust programming and chance-constrained programming methods, FSRP can explicitly address complexities and uncertainties without unrealistic simplifications. Parameters in the FSRP model can be expressed as PDFs and/or membership functions, such that robustness of the optimization process can be enhanced. In its solution process, the FSRP model is converted to a deterministic version through transforming m imprecise constraints into 2 km precise inclusive constraints that correspond to k f -cut levels (under each given significance level). Results of the case study indicate that FSRP is applicable to problems that involve a variety of uncertainties. Air pollution control invariably involves a number of processes with socio-economic and environmental implications. These processes are associated with extensive uncertainties due to their complex, interactive, dynamic, and multiobjective features. Through the FSRP modeling study, useful solutions for planning regional air quality management practices have been generated. They reflect complex trade-offs between environmental and economic considerations. Willingness to pay higher operating costs will guarantee meeting environmental objectives; however, a desire to reduce the costs will run the risk of potentially violating the emission and/or ambient-air-quality standards.     

Task assignment under uncertainty: stochastic programming and robust optimisation approaches

The assignment of tasks to teams is a challenging combinatorial optimisation problem. The uncertainty in the tasks’ execution processes further complicates the assignment decisions. This study investigates a variant of the typical assignment problem, in which each task can be divided into two parts, one is deterministic and the other is uncertain with respect to their workloads. From the stochastic perspective, this paper proposes both a stochastic programming model that can cope with arbitrary probability distributions of tasks’ random workload requirements, and a robust optimisation model that is applicable to situations in which limited information about probability distributions is available. An example of its application in the software project management is given. Some numerical experiments are also performed to validate the effectiveness of the proposed models and the relationships between the two models.     

An interval-parameter two-stage stochastic integer programming model for environmental systems planning under uncertainty

An interval-parameter two-stage stochastic mixed integer programming (ITMILP) technique is developed for waste management under uncertainty. It is a hybrid of inexact two-stage stochastic programming and mixed integer linear programming methods. The ITMILP method can directly handle uncertainties expressed not only as probability density functions but also as discrete intervals. It can be used to analyse various policy scenarios that are associated with different levels of economic penalties when the promised policy targets are violated. More importantly, it can facilitate dynamic analysis of decisions on capacity expansion planning within a multi-region, multi-facility, multi-period, and multi-option context. The results will help to generate a range of decision alternatives under various system conditions, and thus offer insight into the trade-offs between environmental and economic objectives. The ITMILP method is applied to planning facility expansion and waste flow allocation within a waste management system. The results indicate that reasonable solutions have been generated for both binary and continuous variables. The binary-variable solutions represent the decisions of facility expansion, while the continuous-variable solutions are related to decisions on waste flow allocation.     

Interval-parameter robust quadratic programming for water quality management under uncertainty

Effective planning of water quality management is important for facilitating sustainable socio-economic development in watershed systems. An interval-parameter robust quadratic programming (IRQP) method is developed by incorporating techniques of robust programming and interval quadratic programming within a general optimization framework. The IRQP improves upon existing quadratic programming methods, and can tackle uncertainties presented as interval numbers and fuzzy sets as well as their combinations. Moreover, it can deal with nonlinearities in the objective function such that economies-of-scale effects can be reflected. The developed method is applied to a case study of a water quality management under uncertainty. A number of decision alternatives are generated based on the interval solutions as well as the projected applicable conditions. They represent multiple decision options with various environmental and economic considerations. Willingness to accept a low economic revenue will guarantee satisfying the water quality requirements. A strong desire to acquire a high benefit will run the risk of violating environmental criteria.     

Planning helicopter logistics in disaster relief

This paper describes an efficient planning system for coordinating helicopter operations in disaster relief. This system can be used as a simulation tool in contingency panning for better disaster preparedness and helps to generate plans with estimated data. The proposed system consists of a mathematical model and a Route Management Procedure (RMP) that post-processes the outputs of the model. The system is concerned with helicopter operations that involve last mile distribution and pickups for post-disaster medical care and injured evacuation. Delivering items such as medicine, vaccines, blood, i.v., etc. to affected locations, and evacuating injured persons from these locations comprise the transportation tasks to be performed by helicopters. The proposed modeling system accommodates the special aviation constraints of helicopters and it can handle large scale helicopter missions. The goal of the system is to minimize the total mission time required to complete the transportation task. The RMP enables the Decision Maker (DM) to specify either the mission completion time or the number of vehicles available for the mission. Respecting the limitations imposed by the DM, the RMP generates fuel and capacity feasible helicopter itineraries that complete within the specified mission completion time. A scenario that is based on the post-earthquake damage data provided by the Disaster Coordination Center of Istanbul is used for testing the method.     

Inexact joint-probabilistic stochastic programming for water resources management under uncertainty

In this study, an inexact two-stage integer program with joint-probabilistic constraint (ITIP-JPC) is developed for supporting water resources management under uncertainty. This method can tackle uncertainties expressed as joint probabilities and interval values, and can reflect the reliability of satisfying (or the risk of violating) system constraints under uncertain events and/or parameters. Moreover, it can be used for analysing various policy scenarios that are associated with different levels of economic consequences when the pre-regulated targets are violated. The developed ITIP-JPC is applied to a case study of water resources allocation within a multi-stream, multi-reservoir and multi-user context, where joint probabilities exist in both water availabilities and storage capacities. The results indicate that reasonable solutions have been generated for both binary and continuous variables. They can help generate desired policies for water allocation and flood diversion with a maximized economic benefit and a minimized system-disruption risk.     

A bi-objective robust model for emergency resource allocation under uncertainty

Emergency resource allocation constitutes one of the most critical elements of response operations in the field of emergency management. This paper addresses an emergency resource allocation problem which involves multiple competing affected areas and one relief resource centre under supply shortage and uncertainty in the post-disaster phase. In humanitarian situations, both the efficiency and fairness of an allocation policy have a considerable influence on the effectiveness of emergency response operations. Thus, we formulate a bi-objective robust emergency resource allocation (BRERA) model which tries to maximise efficiency as well as fairness under different sources of uncertainties. To obtain decision-makers’ most preferred allocation policy, we propose a novel emergency resource allocation decision method which consists of three steps: (1) develop a bi-objective heuristic particle swarm optimisation algorithm to search the Pareto frontier of the BRERA model; (2) select a coefficient to measure fairness; and (3) establish a decision method based on decision-makers’ preference restricted by the fairness coefficient. Finally, a real case study taken from the 5 December 2008 Wenchuan Earthquake demonstrates the effectiveness of the proposed method through numerical results. The solution and model robustness are also analysed.     

Two-stage stochastic programming with robust constraints for the logistics network post-disruption response strategy optimization

Logistics networks (LNs) are essential for the transportation and distribution of goods or services from suppliers to consumers. However, LNs with complex structures are more vulnerable to disruptions due to natural disasters and accidents. To address the LN post-disruption response strategy optimization problem, this study proposes a novel two-stage stochastic programming model with robust delivery time constraints. The proposed model jointly optimizes the new-line-opening and rerouting decisions in the face of uncertain transport demands and transportation times. To enhance the robustness of the response strategy obtained, the conditional value at risk (CVaR) criterion is utilized to reduce the operational risk, and robust constraints based on the scenario-based uncertainty sets are proposed to guarantee the delivery time requirement. An equivalent tractable mixed-integer linear programming reformulation is further derived by linearizing the CVaR objective function and dualizing the infinite number of robust constraints into finite ones. A case study based on the practical operations of the JD LN is conducted to validate the practical significance of the proposed model. A comparison with the rerouting strategy and two benchmark models demonstrates the superiority of the proposed model in terms of operational cost, delivery time, and loading rate.     

A stochastic programming approach to determine robust delivery profiles in area forwarding inbound logistics networks

One technique to coordinate the suppliers’ and the producers’ production plans in a supply chain is the use of delivery profiles, which provide fixed delivery frequencies for all suppliers. The selection of a delivery profile assignment has major effects on the cost efficiency and the robustness of a supply chain and thus should be performed carefully. In this work, we consider planning approaches to select delivery profiles for the case of area forwarding-based inbound logistics networks, which are commonly used in several industries to consolidate supplies in an early stage of transport. We present a two-stage stochastic mixed integer linear programming model to determine robust delivery profile assignments under uncertain and infrequent demands and complex tariff systems. The model is embedded into a solution framework consisting of scenario generation and reduction techniques, a decomposition approach, a genetic algorithm, and a standard MILP solver. On the basis of an industrial case study, we show that our approach is computationally feasible and that the planning solutions obtained by our model outperform both a deterministic approach and the planning methodology prevailing in industrial practice.     

Optimal structural design under stochastic uncertainty by stochastic linear programming methods

In the optimal plastic design of mechanical structures one has to minimize a certain cost function under the equilibrium equation, the yield condition and some additional simple constraints, like box constraints. A basic problem is that the model parameters and the external loads are random variables with a certain probability distribution. In order to get reliable/robust optimal designs with respect to random parameter variations, by using stochastic optimization methods, the original random structural optimization problem must be replaced by an appropriate deterministic substitute problem. Starting from the equilibrium equation and the yield condition, the problem can be described in the framework of stochastic (linear) programming problems with ‘complete fixed recourse’. The main properties of this class of substitute problems are discussed, especially the ‘dual decomposition’ data structure which enables the use of very efficient special purpose LP-solvers.     

A robust optimisation model for production planning and pricing under demand uncertainty

The profitability of every manufacturing plant is dependent on its pricing strategy and a production plan to support the customers’ demand. In this paper, a new robust multi-product and multi-period model for planning and pricing is proposed. The demand is considered to be uncertain and price-dependent. Thus, for each price, a range of demands is possible. The unsatisfied demand is considered to be lost and hence, no backlogging is allowed. The objective is to maximise the profit over the planning horizon, which consists of a finite number of periods. To solve the proposed model, a modified unconscious search (US) algorithm is introduced. Several artificial test problems along with a real case implementation of the model in a textile manufacturing plant are used to show the applicability of the model and effectiveness of the US for tackling this problem. The results show that the proposed model can improve the profitability of the plant and the US is able to find high quality solutions in a very short time compared to exact methods.     

A fuzzy fractional chance-constrained programming model for air quality management under uncertainty

A fuzzy fractional chance-constrained programming model (FFCCPM) was developed for dealing with air quality management under uncertainty. FFCCPM integrates a fractional programming model and a double-sided fuzzy chance-constrained programming model. It considers the ratio between total treated pollutant amounts and system cost in the objective function; the constraints with fuzzy variables can be satisfied under some predetermined confidence levels and reliability scenarios. The air quality management system in Fengrun district, Tangshan City, China, was used to demonstrate the applicability of the proposed method. The obtained results indicated that the proposed model was suitable in describing and providing an overview of a studied management system for decision makers, generating various cost-effective air pollution-abatement alternatives. The strategy with a balance between system economy and reliability was recommended for decision makers. The successful application of FFCCPM in Fengrun district provides a good example of real-world regional air quality management.     

I-VFRP: An interval-valued fuzzy robust programming approach for municipal waste-management planning under uncertainty

In this study, an interval-valued fuzzy robust programming (I-VFRP) model has been developed and applied to municipal solid-waste management under uncertainty. The I-VFRP model can explicitly address system uncertainties with multiple presentations, and can directly communicate the waste manager's confidence gradients into the optimization process, facilitating the reflection of weak or strong confidence when subjectively estimating parameter values. Parameters in the I-VFRP model can be represented as either intervals or interval-valued fuzzy sets. Thus, variations of the waste manager's confidence gradients over defining parameters can be effectively handled through interval-valued membership functions, leading to enhanced robustness of the optimization efforts. The results of a theoretical case study indicate that useful solutions for planning municipal solid-waste-management practices can be generated. The waste manager's confidence gradients over various subjective judgments can be directly incorporated into the modeling formulation and solution process. The results also suggest that the proposed methodology can be applied to practical problems that are associated with complex and uncertain information.     

A multi-objective mixed-integer programming model for a multi-floor facility layout

We consider a multi-floor facility layout problem in which the overall length and width of the facility, the size and location of each department, the number and the location of elevators and the number of floors in the facility are all modelled as decision variables. We adapt a linear approximation scheme to represent the area of each department. We consider two objective functions in our model, namely minimising material handling and facility building costs, and propose a lexicographic ordering technique to handle multiple objectives. The numerical experiments show that the slack used in the lexicographic ordering approach has a significant impact on the optimal solution. The experiments also show that the material handling cost can be significantly reduced in a multi-floor facility compared with a single-floor facility.     

A multi-objective optimization model for hub network design under uncertainty: An inexact rough-interval fuzzy approach

This article proposes a multi-objective mixed-integer model to optimize the location of hubs within a hub network design problem under uncertainty. The considered objectives include minimizing the maximum accumulated travel time, minimizing the total costs including transportation, fuel consumption and greenhouse emissions costs, and finally maximizing the minimum service reliability. In the proposed model, it is assumed that for connecting two nodes, there are several types of arc in which their capacity, transportation mode, travel time, and transportation and construction costs are different. Moreover, in this model, determining the capacity of the hubs is part of the decision-making procedure and balancing requirements are imposed on the network. To solve the model, a hybrid solution approach is utilized based on inexact programming, interval-valued fuzzy programming and rough interval programming. Furthermore, a hybrid multi-objective metaheuristic algorithm, namely multi-objective invasive weed optimization (MOIWO), is developed for the given problem. Finally, various computational experiments are carried out to assess the proposed model and solution approaches.     

Process design for the environment: A multi-objective framework under uncertainty

 Designing chemical processes for the environment requires consideration of several indexes of environmental impact including ozone depletion, global warming potentials, human and aquatic toxicity, photochemical oxidation, and acid rain potentials. Current methodologies, such as the generalized waste reduction algorithm (WAR), provide a first step towards evaluating these impacts. However, to address the issues of accuracy and the relative weights of these impact indexes, one must consider the problem of uncertainties. Environmental impacts must also be weighted and balanced against other concerns, such as their cost and long-term sustainability. These multiple, often conflicting, goals pose a challenging and complex optimization problem, requiring multi-objective optimization under uncertainty. This paper will address the problem of quantifying and analyzing the various objectives involved in process design for the environment. Towards this goal, we proposed a novel multi-objective optimization framework under uncertainty. This framework is based on new and efficient algorithms for multi-objective optimization and for uncertainty analysis. This approach finds a set of potentially optimal designs where trade-offs can be explicitly identified, unlike cost-benefit analysis, which deals with multiple objectives by identifying a single fundamental objective and then converting all the other objectives into this single currency. A benchmark process for hydrodealkylation (HDA) of toluene to produce benzene modeled in the ASPEN simulator is used to illustrate the usefulness of the approach in finding environmentally friendly and cost-effective designs under uncertainty. Received: 8 February 2000 / Accepted: 10 March 2000     

Supply chain flexibility and operations optimisation under demand uncertainty: a case in disaster relief

After a disaster happens, emergency response operations are critical to save humans’ lives and properties. The limited resources and time requirements call for coordinated supply chain operations. This paper studies supply chain operations for rescue kits in disaster reliefs, motivated by a real-world application. The objective is to minimise the total tardiness and peak tardiness of product delivery over the multi-period planning horizon. One major challenge is the lack of reliable prediction of customer demand in disasters. In order to cope with demand uncertainty while maintaining the tractability of the optimisation model, we decompose the demand into two components: a relatively stable base demand predicted by historical data and unpredictable demand surges. For the base demand, an optimisation model is developed to optimise the production and distribution operations, as well as the inventory replenishment policy for manufacturers and distribution centres, so as to minimise the total tardiness. For the demand surges, we propose to deploy supply chain flexibility to cope with the uncertainty. An empirical study shows the effectiveness of increasing supply chain flexibility and suggests some managerial insights on configuring such flexibility in emergency operations.     

A generalized interval fuzzy mixed integer programming model for a multimodal transportation problem under uncertainty

A generalized interval fuzzy mixed integer programming model is proposed for the multimodal freight transportation problem under uncertainty, in which the optimal mode of transport and the optimal amount of each type of freight transported through each path need to be decided. For practical purposes, three mathematical methods, i.e. the interval ranking method, fuzzy linear programming method and linear weighted summation method, are applied to obtain equivalents of constraints and parameters, and then a fuzzy expected value model is presented. A heuristic algorithm based on a greedy criterion and the linear relaxation algorithm are designed to solve the model.     

An enhanced fuzzy robust optimization model for regional solid waste management under uncertainty

In this study, an enhanced fuzzy robust optimization (EFRO) model is proposed for supporting regional solid waste management under uncertainty. This model is an extended version of robust optimization from a stochastic to a fuzzy environment, and novel in the following two aspects: (1) it uses multiple algorithms to tackle fuzzy constraints according to their characteristics; and (2) it incorporates fuzzy violation variables into the model, which could effectively reflect the trade-off between system economy and reliability. The regional waste management of the City of Dalian, China, was used as a case study for demonstration. A variety of solutions was obtained under various weight coefficients and confidence levels. From the case study, it was found that EFRO could help decision makers to design desired waste management alternatives under complex uncertainties. The successful application of EFRO in the studied real case is expected to be a good example for solid waste management in many other cities.     

A stochastic dynamic programming model for stream water quality management

This paper deals with development of a seasonal fraction-removal policy model for waste load allocation in streams addressing uncertainties due to randomness and fuzziness. A stochastic dynamic programming (SDP) model is developed to arrive at the steady-state seasonal fraction-removal policy. A fuzzy decision model (FDM) developed by us in an earlier study is used to compute the system performance measure required in the SDP model. The state of the system in a season is defined by streamflows at the headwaters during the season and the initial DO deficit at some pre-specified checkpoints. The random variation of streamflows is included in the SDP model through seasonal transitional probabilities. The decision vector consists of seasonal fraction-removal levels for the effluent dischargers. Uncertainty due to imprecision (fuzziness) associated with water quality goals is addressed using the concept of fuzzy decision. Responses of pollution control agencies to the resulting end-of-season DO deficit vector and that of dischargers to the fraction-removal levels are treated as fuzzy, and modelled with appropriate membership functions. Application of the model is illustrated with a case study of the Tungabhadra river in India.