Chance-constrained Programming Approach Research Articles

Chance-Constrained Optimization for a Green Multimodal Routing Problem with Soft Time Window under Twofold Uncertainty

This study investigates a green multimodal routing problem with soft time window. The objective of routing is to minimize the total costs of accomplishing the multimodal transportation of a batch of goods. To improve the feasibility of optimization, this study formulates the routing problem in an uncertain environment where the capacities and carbon emission factors of the travel process and the transfer process in the multimodal network are considered fuzzy. Taking triangular fuzzy numbers to describe the uncertainty, this study proposes a fuzzy nonlinear programming model to deal with the specific routing problem. To make the problem solvable, this study adopts the fuzzy chance-constrained programming approach based on the possibility measure to remove the fuzziness of the proposed model. Furthermore, we use linear inequality constraints to reformulate the nonlinear equality constraints represented by the continuous piecewise linear functions and realize the linearization of the nonlinear programming model to improve the computational efficiency of problem solving. After model processing, we can utilize mathematical programming software to run exact solution algorithms to solve the specific routing problem. A numerical experiment is given to show the feasibility of the proposed model. The sensitivity analysis of the numerical experiment further clarifies how improving the confidence level of the chance constraints to enhance the possibility that the multimodal route planned in advance satisfies the real-time capacity constraint in the actual transportation, i.e., the reliability of the routing, increases both the total costs and carbon emissions of the route. The numerical experiment also finds that charging carbon emissions is not absolutely effective in emission reduction. In this condition, bi-objective analysis indicates the conflicting relationship between lowering transportation activity costs and reducing carbon emissions in routing optimization. The sensitivity of the Pareto solutions concerning the confidence level reveals that reliability, economy, and environmental sustainability are in conflict with each other. Based on the findings of this study, the customer and the multimodal transport operator can organize efficient multimodal transportation, balancing the above objectives using the proposed model.

A stochastic simulation-based chance-constrained programming model for optimizing watershed best management practices for nonpoint source pollution control under uncertainty

In this study, a stochastic simulation-based chance-constrained programming (SSCP) model is developed by integrating soil and water assessment tool (SWAT), generalized likelihood uncertainty estimation (GLUE), chance-constrained programming (CCP) and genetic algorithm (GA) for managing non-point source (NPS) pollution under uncertainty. The SSCP model improves upon conventional simulation-based optimization methods by effectively (i) reflecting the propagation of stochastic uncertainties from the SWAT to the optimization framework by the joint GLUE and CCP (GLUE-CCP) approach, and (ii) identifying the optimal implementation levels of best management practices (BMPs) under different constraint-violation risk levels. The SSCP is solved using a GA with parallel sampling technique after conversion into the corresponding deterministic version. The model is applied to determine the optimal types, sizes and locations of BMPs for NPS pollution control in the Hanjiang River basin, China. The results indicate that, under low-risk conditions, grassed waterways and filter strips should be implemented in the northern part of the basin and larger detention ponds of about 637.8 ∼ 850.4 ha should be installed in the northern and eastern parts of the basin; whereas under high-risk conditions, filter strips should be installed in the northeastern part of the basin and larger detention ponds of about 562.1 ∼ 749.5 ha should be installed in the northern and southwestern parts of the basin. Moreover, as the risk level decreases, stricter control of constraint violations leads to higher BMP implementation costs in order to mitigate the risk of excessive NPS pollution loads. Thus, the proposed model provides valuable insights into the selection of BMP placement schemes at various risk levels, facilitating the reduction of NPS pollution at the outlet of the Hanjiang River basin to acceptable levels while considering the tradeoff between system cost and risk.

A novel two-phase approach for a sustainable p-hub median problem with multiple service levels and carbon cap-and trade policy under uncertainty

ABSTRACT To reduce the carbon emissions for society's sustainable development and satisfy customers' personalized delivery service, this paper addresses the cap-and-trade policy and multiple service levels into a p-hub median problem. In the hub network designing, single-assignment patterns between hubs and demand nodes, the choice of transportation mode for each O-D pair and the uncertainties of carbon emission and price are taken into account. Based on a fuzzy chance-constrained programming approach, a fuzzy mathematical model under an expected cost-minimization context with some fuzzy constraints is first presented. Then a two-phase approach is proposed to solve the proposed model. Finally, in view of the comparative analysis with small- and large-scale Turkish instances, serval managerial insights are finally obtained for government and logistics companies in the reduction of carbon emission and the hub network's total designing cost for the sustainable development.

A robust chance-constrained programming approach for a bi-objective pre-emptive multi-mode resource-constrained project scheduling problem with time crashing

The presented study proposes a novel bi-objective mixed integer linear programming (MILP) framework for the multi-mode resource-constrained project scheduling problem (MRCPSP) with pre-emptive and non-preemptive activities splitting under uncertain conditions. Minimising the project makespan and resource costs are the considered objectives. Renewable and non-renewable resources along with different modes are taken into account for activities implementation. Additionally, some activities can be crashed by consuming additional renewable and non-renewable resources. Model uncertainty is efficiently addressed by utilising a fuzzy chance constrained programming (CPP) method as well as extending two robust possibilistic programming models. The capability of the presented mathematical framework is validated using problem instances from PSPLIB (j10, j14, j20, and j30) and MMLIB (MM50 and MM100). Finally, a detailed computational comparison is presented to assess the performance of the two robust possibilistic programming models.

Presenting a mathematical model for reduction of delays in construction projects considering quality management criteria in uncertainty conditions

Nowadays, organizations and corporations are fully aware of the fact that they can't satisfy their customers' needs considering the rapid change of their needs and tastes. In addition, the attempts made by organizations to add variety to their products and improving their own status don’t seem to be sufficient to meet customers' needs. Orientation of organizations and corporations to benefit from each other's expertise and facilities for satisfaction of customers'' needs in the form of a coherent and efficient supply chain was a strategy which has been the mere result of increasing technology growth. Therefore, Chance-Constrained Programming approach has been used in this study to compensate for the uncertainties resulting from construction projects. Therefore, after defining uncertainty-related parameters, such parameters will be included in the optimization model. The mathematical model provided in this study was multi-objective and multi-modal. In other words, different modes and conditions have been assumed for any activity. Moreover, one of the objectives assumed for mathematical model was to minimize the maximum delay time in project implementation. Then, GAMS software has been used to solve the mathematical model in small and medium scale and Matlab and NSGA-II meta-heuristic algorithm have been used to solve the model in large scale.

A chance-constrained programming approach to optimal planning of low-carbon transition of a regional energy system

Low-carbon transition of energy systems is an inevitable trend to address climate change challenges. For developing regions, proper planning is essential for reducing transition costs during low-carbon transition of their energy systems, featuring a higher proportion of intermittent renewable power connected to power grids. Impact of uncertainty must be considered for more feasible planning of peak-shaving and energy storage units. In this study, a chance-constrained programming approach to optimal planning of low-carbon transition of a regional energy system is presented. This approach considers uncertainties of wind power, photovolatic (PV) power and load to ensure power supply reliability. A developing region in central China is taken as a case study. Results show that considering uncertainty, an additional 4.79% of power generation capacity needs to be installed per year on average, with a 3.02% increase in the transition cost. Finally, sensitivity analysis results are provided, showing that a rapid increase in transition costs occurs when the confidence level exceeds 99%. The results in this study provide references for decision-makers to plan the low-carbon transition of energy systems as well as weighing transition costs against energy supply stability.

The new approaches for solving hierarchical Chinese postman problem with stochastic travel times

The hierarchical Chinese postman problem (HCPP) aims to find the shortest tour or tours by passing through the arcs classified according to precedence relationship. HCPP, which has a wide application area in real-life problems such as shovel snow and routing patrol vehicles where precedence relations are important, belongs to the NP-hard problem class. In real-life problems, travel time between the two locations in city traffic varies due to reasons such as traffic jam, weather conditions, etc. Therefore, travel times are uncertain. In this study, HCPP was handled with the chance-constrained stochastic programming approach, and a new type of problem, the hierarchical Chinese postman problem with stochastic travel times, was introduced. Due to the NP-hard nature of the problem, the developed mathematical model with stochastic parameter values cannot find proper solutions in large-size problems within the appropriate time interval. Therefore, two new solution approaches, a heuristic method based on the Greedy Search algorithm and a meta-heuristic method based on ant colony optimization were proposed in this study. These new algorithms were tested on modified benchmark instances and randomly generated problem instances with 817 edges. The performance of algorithms was compared in terms of solution quality and computational time.

Flow shop scheduling with human–robot collaboration: a joint chance-constrained programming approach

ABSTRACT Human–robot collaboration has been incorporated into production and assembly processes to promote system flexibility, changeability and adaptability. However, it poses new challenges to resource allocation and production scheduling due to its intrinsic uncertainty and also the increasing complexity of resources. This paper investigates a stochastic flow shop scheduling problem in the context of human–robot collaboration. The goal is to achieve efficient utilisation of flexible resources including human workers and cobots and take full advantage of human–robot collaboration in production scheduling. A stochastic Cobb–Douglas production function is utilised to evaluate the production efficiency of human–robot collaboration considering instabilities of human performance. A joint chance-constrained programming model is formulated to ensure that the required system performance can be achieved. A CVaR approximation-based approach is proposed to solve the formulated model with mixed-integer variables and a nonconvex constraint. The effectiveness of the formulated model and the efficiency of the proposed solution approach are evaluated via numerical experiments. Computational results show the superiority of our solution approach over three other approaches including Bonferroni approximation, scenario approach and individual chance-constrained programming.

Designing an integrated blockchain-enabled supply chain network under uncertainty

With the development of communication infrastructure, the design of supply chains has changed significantly. Blockchain technology, as one of the most cutting-edge technologies, can promote transparency among members of the supply chain network. To the best of our knowledge, this is the first study that tries to develop a novel bi-objective optimization model to integrate the transparency resulting from the use of blockchain for designing a three-level supply chain network. The first objective function is to minimize total cost while the second objective function seeks to maximize transparency based on the application of blockchain technology. Moreover, it is worth noting that it is the first attempt to investigate the role of a blockchain model under stochastic conditions. The bi-objectiveness and stochastic nature of the proposed model are then treated using Fuzzy Goal Programming (FGP) and Chance-Constrained programming (CCP) approaches, respectively. To tackle the problem, an improved Branch and Efficiency (B&E) algorithm is developed by incorporating transparency along with cost and service. The impacts of blockchain exclusively through transparency (Case 1) or through transparency, cost, and benefits (Case 2) in Supply Chain Design (SCD) are compared. The results demonstrated that the first case has less computational complexity and better scalability, while the second case has more transparency, less congestion, and more security. As one of the main implications, supply chain managers who are focused on cost minimization as well as transparency maximization are advised to take into account the trade-off between featuring costs and benefits of blockchain technology.

Optimal blending under general uncertainties: A chance-constrained programming approach

An optimization algorithm is proposed for blend planning with linear mixing law and general parameter uncertainties. The objective is to make the final product satisfy all quality specifications with high probability, and maximize the production profit by carefully determining the feedstock ratio. Conventional approaches that rely on deterministic optimization fail to account for parameter uncertainty, and thus may not generate a probabilistic feasible solution. The proposed work formulates the blend planning problem as a joint chance-constrained program (CCP). Using Boole’s inequality to decompose joint constraints and the Gaussian mixture model to characterize uncertainty distributions, a conservative deterministic approximation of CCP can be formulated. Through second-order cone relaxation, branch-and-bound, optimality-based bound tightening, and reformulate-linearization techniques, the global optimum of deterministic approximation can be found. A risk level adjustment procedure is presented to reduce the conservativeness and further improve the objective value of the solution if posterior evaluation is allowed. Two numerical cases, including steel and gasoline productions, are studied to show the solving time, probabilistic feasibility, and solution quality of the proposed optimization method.

A multi-period emergency medical service location problem based on Wasserstein-metric approach using generalised benders decomposition method

This paper considers a multi-period location and sizing problem for an emergency medical service (EMS) system based on a distributionally robust optimisation (DRO) chance-constrained programming approach. The dynamic uncertain emergency medical requests are described in the ambiguity set, which is constructed based on Wasserstein-metric. The model of this problem focuses on minimising long-term operation costs. The chance constraints ensure the reliability of EMS system for the entire geographic areas. A reformulation of chance constraints is provided in Mixed Integer Linear Program form. For problem solution, a generalised Benders decomposition (GBD) implementation is proposed. A numerical simulation is conducted to illustrate the performance of two solution approaches in terms of computational convergence speed and optimality of the problem.

Network planning of metro-based underground logistics system against mixed uncertainties: A multi-objective cooperative co-evolutionary optimization approach

Featured with zero-carbon, jam-free, and high-capacity, the utilization of metro systems for collaborative passenger-and-freight transport (i.e., the metro-based underground logistics system, M-ULS) has been recognized as a favorable alternative to realize automated goods movement in future megacities. This paper proposes an equilibrium chance-constrained programming (ECP) approach for the M-ULS network planning problem (MULNP) considering uncertain demand and costs. Based on the ongoing satellite city development project, the layout formation, facility components, and transportation workflows of a two-tier M-ULS network are designed. The joint decisions (i.e., location, allocation, capacity, inventory, spoke pipeline layout, and flow routing) are formulated as an ECP model with conjoint objectives of minimal fixed cost, operating cost, and penalty cost due to low operating load, wherein the chance-based constraints are further reformulated into crisp equivalent form. To yield high-quality solutions for MULNP problem, an improved multi-objective cooperative co-evolutionary algorithm (MoCC) incorporating non-dominated sorting and chromosomal recombination strategy is proposed and then validated via two groups of instances. Results from the Beijing case study suggest that the proposed method has computational advantages over MoPSO and NSGA-II in acquiring Pareto fronts of large-scale MULNP. The best configurations and layout schemes of the M-ULS networks in both uncertain and deterministic scenarios are analyzed. Based on the findings, several insights for real-world M-ULS network decision-making are provided.

Emergency medical service location problem based on physical bounds using chance-constrained programming approach

This paper studies a location and sizing problem for an emergency medical service system using distributionally robust chance-constrained programming approach. The medical demands are uncertain and described in an ambiguity set, which is constructed based on Wasserstein-metric. The location problem is modelled through minimising the total expected costs, including construction costs of facilities, purchase of ambulances and maintenance costs. The chance constraints are introduced to guarantee the service level of geographical areas. For the chance constraints, three reformulations such as CVaR-approximation, the mixed integer linear programme exact reformulation and the physically bounded bilinear reformulation are studied. A numerical study is conducted to illustrate the computational performance and objective values.

Assessing most productive scale size with fuzzy stochastic data envelopment analysis

Classic data envelopment analysis (DEA) is a linear programming method for evaluating the relative efficiency of decision making units (DMUs) that uses multiple inputs to produce multiple outputs. In the classic DEA model inputs and outputs of DMUs are deterministic, while in the real world, are often fuzzy, random, or fuzzy-random. Many researchers have proposed different approaches to evaluate the relative efficiency with fuzzy and random data in DEA. In many studies, the most productive scale size (mpss) of decision making units has been estimated with fuzzy and random inputs and outputs. Also, the concept of fuzzy random variable is used in the DEA literature to describe events or occurrences in which fuzzy and random changes occur simultaneously. This paper has proposed the fuzzy stochastic DEA model to assess the most productive scale size of DMUs that produce multiple fuzzy random outputs using multiple fuzzy random inputs with respect to the possibility-probability constraints. For solving the fuzzy stochastic DEA model, we obtained a nonlinear deterministic equivalent for the probability constraints using chance constrained programming approaches (CCP). Then, using the possibility theory the possibilities of fuzzy events transformed to the deterministic equivalents with definite data. In the final section, the fuzzy stochastic DEA model, proposed model, has been used to evaluate the most productive scale size of sixteen Iranian hospitals with four fuzzy random inputs and two fuzzy random outputs with symmetrical triangular membership functions.

Distributionally robust heat-and-electricity pricing for energy hub with uncertain demands

In this work, we investigate the energy market pricing problem of an energy hub (EH) under endogenous uncertain demands. While traditional pricing problems consider exogenous energy demands, in multi-energy scenarios, the difference between power and heat prices could affect the compliance of residents to use certain energy, and therefore affect the realization of stochastic demand. Moreover, the probability distribution of such endogenous uncertainty and how it is influenced by energy prices may not be known exactly. This article proposes a bilevel optimization framework based on the game theory containing an upper-level EH dispatch with energy pricing and a series of lower-level household energy management problems. This framework seeks for optimal heat-and-electricity market prices for both EH and users to minimize their expected costs. To address uncertain probability distributions in decision-making processes, a modified distributionally robust chance-constrained programming approach is employed. The results demonstrate that the proposed framework could provide an effective pricing scheme for EH.

Model for Reverse Logistic Problem of Recycling under Stochastic Demand

It has become obligatory for businesses to carry out recycling activities in the face of increasing environmental pollution and the danger of depletion of natural resources. The waste collection phase of the recycling process requires interactive transportation that uses a reverse logistics flow from customers to recycling facilities. Businesses need to create appropriate network structures to carry out these activities at minimum cost. This study has developed a model, based on reverse logistics, of collecting products from customers and sending them to warehouses and then to recycling facilities. The chance-constrained programming (CCP) approach was used to regulate the constraints involving stochastic demand in the model. Linearization was performed using the linear approximation method. The cost of transportation from Initial Collection Points (ICP) warehouses to recycling facilities is the most influential component on the objective function. This linearized model was solved by creating different scenarios by changing the standard deviation ratio, reliability level, and warehouse capacities within the scope of sensitivity analysis. In the sensitivity analysis, it was determined that the increase in confidence level and variance negatively affected the objective function. In addition, it has been concluded that the increase in demand has no effect on costs as long as the capacity of the facility is not exceeded.

A mathematical model for managing the multi-dimensional impacts of the COVID-19 pandemic in supply chain of a high-demand item.

The COVID-19 pandemic has wreaked havoc across supply chain (SC) operations worldwide. Specifically, decisions on the recovery planning are subject to multi-dimensional uncertainty stemming from singular and correlated disruptions in demand, supply, and production capacities. This is a new and understudied research area. In this study, we examine, SC recovery for high-demand items (e.g., hand sanitizer and face masks). We first developed a stochastic mathematical model to optimise recovery for a three-stage SC exposed to the multi-dimensional impacts of COVID-19 pandemic. This allows to generalize a novel problem setting with simultaneous demand, supply, and capacity uncertainty in a multi-stage SC recovery context. We then developed a chance-constrained programming approach and present in this article a new and enhanced multi-operator differential evolution variant-based solution approach to solve our model. With the optimisation, we sought to understand the impact of different recovery strategies on SC profitability as well as identify optimal recovery plans. Through extensive numerical experiments, we demonstrated capability towards efficiently solving both small- and large-scale SC recovery problems. We tested, evaluated, and analyzed different recovery strategies, scenarios, and problem scales to validate our approach. Ultimately, the study provides a useful tool to optimise reactive adaptation strategies related to how and when SC recovery operations should be deployed during a pandemic. This study contributes to literature through development of a unique problem setting with multi-dimensional uncertainty impacts for SC recovery, as well as an efficient solution approach for solution of both small- and large-scale SC recovery problems. Relevant decision-makers can use the findings of this research to select the most efficient SC recovery plan under pandemic conditions and to determine the timing of its deployment.

A Wasserstein distributionally robust chance constrained programming approach for emergency medical system planning problem

This paper proposes a distributionally robust chance constrained programming model for an emergency medical system location problem with uncertain demands. By minimising the total expected cost, the location of emergency medical stations, the allocation of the ambulances and demand assignments of system are optimised. The Wasserstein-metric is employed to construct the ambiguity set centred at an empirical distribution with a proper radius, which contains all the probability distributions of the uncertainties. We introduce a big-M technique to reformulate distributionally robust chance constrained programming into a corresponding mixed integer programm, which can be inner and outer approximated by Value-at-Risk (VaR) and Conditional-Value-at-Risk (CVaR). Numerical experiments are illustrated to demonstrate the effectiveness of the formulations.

Chance-constrained programming for optimal scheduling of combined cooling, heating, and power-based microgrid coupled with flexible technologies

• New comprehensive framework for a CCHP-based MG operation. • New CCP approach for guaranteeing confidence level of the whole system. • Proposing a comparative structure for participating in power and heat markets. • Utilizing multi-carrier storage, hybrid chiller, besides new mathematical for CHP. Microgrids (MGs) have a special role in developing several consumers' energy infrastructure and supply in more economical, safer, and sustainable ways. The interaction and mutual relationship between each energy carrier on the reliable performance of other carriers and the high growth of tri-generation technologies in the MG face the optimal performance of such networks with many challenges. Combined cooling, heating, and power (CCHP)-based MGs are a new generation of MGs that simultaneously provide electrical, thermal, and cooling loads. However, the interaction between these carriers is very influential in CCHP-based MG's operation, which is rarely analyzed. Hence, this paper focuses on the operation of CCHP-based MG coupled with hybrid chiller, multi-energy storage, solar and wind power, etc., under the chance-constrained programming (CCP) approach by considering the mutual relationship between carriers. While modeling consumption and wind and solar energies fluctuations, the proposed approach analyzes the violation of the balance constraint for each carrier and subject it to guarantee the corresponding confidence level. Therefore, the degree of dependence of the system on each carrier and the mutual relationship between carriers are analyzed in this work. This paper also presents a new incentive framework for participating in the electricity and heating markets for the system. The proposed model is implemented on the test system, and the results are discussed for different cases for several confidence levels. The results illustrate the importance of the electricity carrier confidence level on the safe performance of the whole system compared to other carriers. Only by increasing the operation cost of the electricity sector by 4.3%, the system's reliable performance is guaranteed with a probability of 98%.

Solving a new bi-objective multi-echelon supply chain problem with a Jackson open-network issue under uncertainty

In today’s world, changes in the economic and industrial sectors are taking place more quickly than in the past. Based on today’s competitive market, considering the needs and desires of the customer is very important. Thus, designing the supply chain network to minimize the costs and improve the system efficiency, the service facility, and attention to the timely service delivery to the customer should be considered. Also, deployment costs to determine the number and position of facilities are regarded as another strategic issue. Therefore, to become closer to the real-world application, by considering the Jackson network model (queuing theory), the present study aims to design a closed-loop supply chain network to reduce the waiting time in the queue. Two objectives, namely minimizing the cost and waiting time incurred in queues formed in producers and distributors, are investigated. Queuing mathematical expressions are used both in dealing with objectives and constraints. To develop a model, which is close to real by considering uncertainty, the parameters, including demand rate and shipping cost of products, were considered as a trapezoidal fuzzy number. Also, we crisp this fuzzy uncertainty using both Jimenez and chance-constrained programming approaches. Due to the suggested model’s complexity and nonlinearity, we linearized the mathematical model as much as possible, then two meta-heuristic algorithms, namely multi-objective particle swarm optimization and non-dominated sorted genetic algorithm (NSGA-II), are used to solve the model. First, the structural parameters of these algorithms are tuned by the Taguchi method. Then, numerous numerical test problems ranging from small to large scales are used to compare these algorithms considering Pareto front measuring indexes and various hypothesis tests.