Chance-constrained Linear Programming Research Articles

Modeling a Multimodal Routing Problem with Flexible Time Window in a Multi-Uncertainty Environment

In this study, we extend the research on the multimodal routing problem by considering flexible time window and multi-uncertainty environment. A multi-uncertainty environment includes uncertainty regarding the demand for goods, the travel speed of the transportation mode, and the transfer time between different transportation modes. This environment further results in uncertainty regarding the delivery time of goods at their destination and the earliness and lateness caused by time window violations. This study adopts triangular fuzzy numbers to model the uncertain parameters and the resulting uncertain variables. Then, a fuzzy mixed integer nonlinear programming model is established to formulate the specific problem, including both fuzzy parameters and fuzzy variables. To make the problem easily solvable, this study employs chance-constrained programming and linearization to process the proposed model to obtain an equivalent credibilistic chance-constrained linear programming reformulation with an attainable global optimum solution. A numerical case study based on a commonly used multimodal network structure is presented to demonstrate the feasibility of the proposed method. Compared to hard and soft time windows, the numerical case analysis reveals the advantages of the flexible time window in reducing the total costs, avoiding low reliability regarding timeliness, and providing confidence level-sensitive route schemes to achieve flexible routing decision-making under uncertainty. Furthermore, the numerical case analysis verifies that it is necessary to model the multi-uncertainty environment to satisfy the improved customer requirements for timeliness and enhance the flexibility of the routing, and multimodal transportation is better than unimodal transportation when routing goods in an uncertain environment. The sensitivity analysis in the numerical case study shows the conflicting relationship between the economic objective and the reliability regarding the timeliness of the routing, and the result provides a reference for the customer to find a balance between them.

Type-2 fuzzy chance-constrained linear fractional programming model for a water resource management system: A case study of Taiyuan city, China

Abstract Uncertainties arising from extreme climate events and human activities pose a challenge to the efficient allocation of water resources. In this study, a type-2 fuzzy chance-constrained linear fractional programming (T2F-CCLFP) is developed to support the water resource management system under uncertainty by incorporating type-2 fuzzy sets, chance-constrained programming, and fractional programming into a comprehensive multi-objective optimization framework. The model enables the trade-off between economic, social, and environmental sustainability and provides water supply solutions associated with different levels of fuzzy uncertainty and risk of violating constraints. The T2F-CCLFP model is applied to Taiyuan, Shanxi Province, China, to support its water resource management. Results reveal that: (i) the industrial structure is transitioning toward diverse industries from energy and heavy industry dominance; (ii) external water transfer will be the major water-supply sources for the city in the future, accounting for 55 and 50% of the total water supply in 2025 and 2030, respectively; (iii) the water-supply security of the city is enhanced by provoking the utilization of reclaimed water (the annual growth rate is 13.9%). The results are helpful for managers in adjusting the current industry structure, enhancing water supply security, and contributing to the sustainable development of socio-economic and water systems.

Estimation of Electricity Supply and Demand under Dual Carbon Goals: A Chance-constrained Linear Programming Model

Implementing the green and low-carbon development strategy has become a global consensus with the increased severity of global environmental problems. In 2020, Chinese President Xi Jinping put forward China’s carbon peak and carbon neutrality goals (i.e., dual carbon goals), which is a guideline for the development of Chinese power systems. Thus, a supply and demand estimation system that includes dual carbon indicators and considers the long-term supply capacity is needed. To address these gaps, this study put forward an estimation model and provided the solutions. First, a supply and demand estimation indicator system is established to meet the dual carbon goals. Second, with minimizing cost being the main objective and several constraints, a chance-constrained linear programming model is used to estimate the supply and demand of electricity in 2030, 2040, and 2050. Finally, with model relaxation, we obtained the estimation results with Matlab-Cplex, which verified the efficiency and validity of the proposed model.

A conflict-directed approach to chance-constrained mixed logical linear programming

Resistance to the adoption of autonomous systems comes in part from the perceived unreliability of the systems. The concerns can be addressed by deploying decision making algorithms that defines what it means to fail, and look for plans with the highest reward while limiting the probability of failure.This chance-constrained approach thus explicitly imposes a set of constraints that must be satisfied for success, and provides upper-bounds on the probability of violating such constraints. A chance-constrained mixed logical-linear program (CC-MLLP) is a natural formulation, allowing for the specification of linear and logical constraints, with probabilistic continuous variables. The formalism can be used to describe problems ranging from autonomous underwater vehicle path planning, to network routing under uncertainty.While naive encodings of CC-MLLPs can be solved with generalised solvers, the solution time may be unreasonable. In this work, we study architectures to speed up solutions by partitioning CC-MLLPs into the discrete and continuous portions. In order to provide faster solutions, we investigate methods for speeding up the solutions to the continuous chance-constrained linear programs. Further, by exploiting the new solution methods, we develop techniques for guiding the discrete decision making portion of the problem. The resulting algorithm achieves 10 times speed up over prior approaches on autonomous path planning benchmarks.

A novel approach for solving decision-making problems with stochastic linear-fractional models

Stochastic chance-constrained optimization has a wide range of real-world applications. In some real-world applications, the decision-maker has to formulate the problem as a fractional model where some or all of the coefficients are random variables with joint probability distribution. Therefore, these types of problems can deal with bi-objective problems and reflect system efficiency. In this paper, we present a novel approach to formulate and solve stochastic chance-constrained linear fractional programming models. This approach is an extension of the deterministic fractional model. The proposed approach, for solving these types of stochastic decision-making problems with the fractional objective function, is constructed using the following two-step procedure. In the first stage, we transform the stochastic linear fractional model into two stochastic linear models using the goal programming approach, where the first goal represents the numerator and the second goal represents the denominator for the stochastic fractional model. The resulting stochastic goal programming problem is formulated. The second stage implies solving stochastic goal programming problem, by replacing the stochastic parameters of the model with their expectations. The resulting deterministic goal programming problem is built and solved using Win QSB solver. Then, using the optimal value for the first and second goals, the optimal solution for the fractional model is obtained. An example is presented to illustrate our approach, where we assume the stochastic parameters have a uniform distribution. Hence, the proposed approach for solving the stochastic linear fractional model is efficient and easy to implement. The advantage of the proposed approach is the ability to use it for formulating and solving any decision-making problems with the stochastic linear fractional model based on transforming the stochastic linear model to a deterministic linear model, by replacing the stochastic parameters with their corresponding expectations and transforming the deterministic linear fractional model to a deterministic linear model using the goal programming approach

A Birandom Chance-Constrained Linear Programming Model for CCHP System Operation Management: A Case Study of Hotel in Shanghai, China

Due to its capability to reduce fuel consumption and increase energy efficiency, the combined cooling, heating, and power (CCHP) system has obtained great concern during the last decade. A large number of deterministic and stochastic optimization models were proposed for supporting the operation management of the CCHP system, but few studies noticed that users’ demands in the real world may be subjected to twofold randomness with incomplete or uncertain information. In this study, a birandom chance-constrained linear programming (BCCLP) model is developed for identifying optimal operation strategies under random uncertainties. Compared with traditional stochastic programming models, the BCCLP model made the improvement through describing the energy demands as the birandom variables firstly, instead of traditional random variables. This way effectively avoided potential imbalance between energy supply and demand caused by oversimplified expression of uncertain parameters. A gas-fired CCHP system of a hotel in Shanghai, China, was used as a study case for demonstration. A variety of operation strategies are obtained under specific constraints-satisfaction conditions. It is concluded that the BCCLP model was capable of generating the cost-effective operation strategies and evaluating the tradeoffs between system economy and reliability. The influence imposed by some critical parameters on the system performance was examined through the sensitivity analysis, which provided the important guidance to the design and operational management of other similar CCHP systems in the future.

A Framework for Solving Chance-Constrained Linear Matrix Inequality Programs

We propose a novel partial sample average approximation (PSAA) framework to solve the two main types of chance-constrained linear matrix inequality (CCLMI) problems: CCLMI with random technology matrix and CCLMI with random right-hand side. We propose a series of computationally tractable PSAA-based approximations for CCLMI problems, analyze their properties, and derive sufficient conditions that ensure convexity for the two most popular—normal and uniform—continuous distributions. We derive several semidefinite programming PSAA reformulations efficiently solved by off-the-shelf solvers and design a sequential convex approximation method for the PSAA formulations containing bilinear matrix inequalities. The proposed methods can be generalized to other continuous random variables whose cumulative distribution function can be easily computed. We carry out a comprehensive numerical study on three practical CCLMI problems: robust truss topology design, calibration, and robust control. The tests attest to the superiority of the PSAA reformulation and algorithmic framework over the scenario and sample average approximation methods. Summary of Contribution: In line with the mission and scope of IJOC, we study an important type of optimization problems, chance-constrained linear matrix inequality (CCLMI) problems, which require stochastic linear matrix inequality (LMI) constraints to be satisfied with high probability. To solve CCLMI problems, we propose a novel partial sample average approximation (PSAA) framework: (i) develop a series of computationally tractable PSAA-based approximations for CCLMI problems, (ii) analyze their properties, (iii) derive sufficient conditions ensuring convexity, and (iv) design a sequential convex approximation method. We evaluate our proposed method via a comprehensive numerical study on three practical CCLMI problems. The tests attest the superiority of the PSAA reformulation and algorithmic framework over standard benchmarks.

Scenario Grouping and Decomposition Algorithms for Chance-Constrained Programs

A lower bound for a finite-scenario-based chance-constrained program is the quantile value corresponding to the sorted optimal objective values of scenario subproblems. This quantile bound can be improved by grouping subsets of scenarios at the expense of solving larger subproblems. The quality of the bound depends on how the scenarios are grouped. In this paper, we formulate a mixed-integer bilevel program that optimally groups scenarios to tighten the quantile bounds. For general chance-constrained programs, we propose a branch-and-cut algorithm to optimize the bilevel program, and for chance-constrained linear programs, a mixed-integer linear-programming reformulation is derived. We also propose several heuristics for grouping similar or dissimilar scenarios. Our computational results demonstrate that optimal grouping bounds are much tighter than heuristic bounds, resulting in smaller root-node gaps and better performance of scenario decomposition for solving chance-constrained 0-1 programs. Also, the optimal grouping bounds can be greatly strengthened using larger group size. Summary of Contribution: Chance-constrained programs are in general NP-hard but widely used in practice for lowering the risk of undesirable outcomes during decision making under uncertainty. Assuming finite scenarios of uncertain parameter, chance-constrained programs can be reformulated as mixed-integer linear programs with binary variables representing whether or not the constraints are satisfied in corresponding scenarios. A useful quantile bound for solving chance-constrained programs can be improved by grouping subsets of scenarios at the expense of solving larger subproblems. In this paper, we develop algorithms for optimally and heuristically grouping scenarios to tighten the quantile bounds. We aim to improve both the computation and solution quality of a variety of chance-constrained programs formulated for different Operations Research problems.

Double-sided stochastic chance-constrained linear fractional programming model for managing irrigation water under uncertainty

A double-sided stochastic chance-constrained linear fractional programming (DSCLFP) model is developed for managing irrigation water under uncertainty. The model is developed by incorporating double-sided stochastic chance-constrained programming (DSCCP) into a linear fractional programming (LFP) optimization framework. It can address ratio optimization problems with double-sided randomness (i.e. both left-hand and right-hand sides). More importantly, it also improves upon the existing stochastic chance-constrained programming for handing random uncertainties in the left-hand and right-hand sides of constraints simultaneously. A non-equivalent but sufficient linearization form of the DSCLFP is provided and proved, which will greatly reduce the computational burden. Then, the model is applied to a case study in Yingke Irrigation District (YID) in the middle reaches of the Heihe River Basin, northwest China. Four confidence levels (e.g. αi = 0.85, 0.90, 0.95 and 0.99) are provided to examine and compare the results. The objective function values are slightly decreased from 5.284 Yuan/m3 to 5.276 Yuan/m3 when αi level is raised from 0.85 to 0.99. The results from the DSCLFP can identify desired irrigation water allocation plans under the objective function of maximizing water productivity under different confidence levels. Therefore, the results can provide tradeoffs among water productivity, confidence level and constraint-violation risk level. Moreover, comparisons with double-sided stochastic chance-constrained linear programming (DSCLP) model and deterministic model are introduced to highlight advantages and feasibility of the developed model. Therefore, these results can provide decision-support for managers in arid areas.

Chance-constrained stochastic programming under variable reliability levels with an application to humanitarian relief network design

We focus on optimization models involving individual chance constraints, in which only the right-hand side vector is random with a finite distribution. A recently introduced class of such models treats the reliability levels / risk tolerances associated with the chance constraints as decision variables and trades off the actual cost / return against the cost of the selected reliability levels in the objective function. Leveraging recent methodological advances for modeling and solving chance-constrained linear programs with fixed reliability levels, we develop strong mixed-integer programming formulations for this new variant with variable reliability levels. In addition, we introduce an alternate cost function type associated with the risk tolerances which requires capturing the value-at-risk (VaR) associated with a variable reliability level. We accomplish this task via a new integer linear programming representation of VaR. Our computational study illustrates the effectiveness of our mathematical programming formulations. We also apply the proposed modeling approach to a new stochastic last mile relief network design problem and provide numerical results for a case study based on the real-world data from the 2011 Van earthquake in Turkey.

Optimal Allocation of Land and Water Resources in a Canal Command Area in the Deterministic and Stochastic Regimes

Water logging is a universal problem of irrigated agriculture and it is a serious threat to the sustainability of irrigated agriculture in many arid and semiarid regions. Limbasi branch canal command area of Mahi Right Bank Canal (MRBC) project, Gujarat, India is also found to be affected by water logging conditions. Present study deals with the formulation of the Deterministic Linear Programming (DLP) and Chance Constrained Linear Programming (CCLP) models which maximizes net return from a canal command area while simultaneously mitigating water logging conditions. The developed models are applied to the Limbasi branch canal command area. The objective function is to maximize net annual return and decision variables are the seasonal cropping pattern and seasonal water supply. Analysis shows that under optimal conditions in the DLP model, there was a 40% deviation of crop area from existing cropping pattern and Net Irrigation Requirement (NIR) of crops was satisfied by conjunctive use of 41% of canal water and 59% of groundwater (annually). There was 91.1% increase in ground water exploitation which consecutively moderated rising water table issues. Net annual return was found to increase by 46.6%. In the CCLP model, NIR of crops was considered as a stochastic variable and normal distribution was found as the best fit. The CCLP model was run from 2 to 40% risk levels and cropping pattern under 10% risk level was considered as optimal at which NIR was satisfied by conjunctive use of 53.8% of canal water and 46.2% of ground water (annually). There was 86% increase in ground water exploitation. The outcome of the study can be used to assist the water resources planners and managers in taking appropriate decisions to develop a sustainable management plan of land and water resources for an overall balance of the system.

Interval joint-probabilistic chance-constrained programming with two-side multi-randomness: an application to energy-environment systems management

Regional energy-environment systems management become more and more focused on greenhouse gas emission control through improving energy efficiency and efficiently managing energy activities. Inexact linear programming models are developed for supporting the management. Due to the weather/climatic variations in the future, electricity demands and renewable power generations (in the right/left hand sides of constraints) have random characteristics. Moreover, an overall satisfactory level needs to be quantified based on multiple chance constraints. Therefore, this study improved upon traditional chance-constrained programming and interval linear programming, and developed an interval joint-probabilistic two-side chance-constrained programming (IJTCP) approach. A sufficient but non-equivalent linearization form of the model was proposed so that the inexact model could be solved through the two-step solution algorithm. The IJTCP was then applied to an integrated energy-environment systems management under dual uncertainties. The application demonstrated that the IJTCP can effectively address the uncertainties presented as not only interval numbers and two-side multi-randomness but also the reliability of satisfying the entire system constraints. The application implicated that the IJTCP approach can be applied to other energy-environment management problems under dual uncertainties.

Nonanticipative duality, relaxations, and formulations for chance-constrained stochastic programs

We propose two new Lagrangian dual problems for chance-constrained stochastic programs based on relaxing nonanticipativity constraints. We compare the strength of the proposed dual bounds and demonstrate that they are superior to the bound obtained from the continuous relaxation of a standard mixed-integer programming (MIP) formulation. For a given dual solution, the associated Lagrangian relaxation bounds can be calculated by solving a set of single scenario subproblems and then solving a single knapsack problem. We also derive two new primal MIP formulations and demonstrate that for chance-constrained linear programs, the continuous relaxations of these formulations yield bounds equal to the proposed dual bounds. We propose a new heuristic method and two new exact algorithms based on these duals and formulations. The first exact algorithm applies to chance-constrained binary programs, and uses either of the proposed dual bounds in concert with cuts that eliminate solutions found by the subproblems. The second exact method is a branch-and-cut algorithm for solving either of the primal formulations. Our computational results indicate that the proposed dual bounds and heuristic solutions can be obtained efficiently, and the gaps between the best dual bounds and the heuristic solutions are small.

Double-sided fuzzy chance-constrained linear fractional programming approach for water resources management

A double-sided fuzzy chance-constrained fractional programming (DFCFP) method is developed for planning water resources management under uncertainty. In DFCFP the system marginal benefit per unit of input under uncertainty can also be balanced. The DFCFP is applied to a real case of water resources management in the Zhangweinan River Basin, China. The results show that the amounts of water allocated to the two cities (Anyang and Handan) would be different under minimum and maximum reliability degrees. It was found that the marginal benefit of the system solved by DFCFP is bigger than the system benefit under the minimum and maximum reliability degrees, which not only improve economic efficiency in the mass, but also remedy water deficiency. Compared with the traditional double-sided fuzzy chance-constrained programming (DFCP) method, the solutions obtained from DFCFP are significantly higher, and the DFCFP has advantages in water conservation.

Fuzzy chance-constrained linear fractional programming approach for optimal water allocation

A fuzzy chance-constrained linear fractional programming method was developed for agricultural water resources management under multiple uncertainties. This approach improved upon the previous programming methods, and could reflect the ratio objective function and multiple uncertainties expressed as probability distributions, fuzzy sets, and their combinations. The proposed approach is applied to an agricultural water resources management system where many crops are considered under different precipitation years. Through the scenarios analyses, the multiple alternatives are presented. The solutions show that it is applicable to practical problems to address the crop water allocation under the precipitation variation and sustainable development with ratio objective function of the benefit and the irrigation amount. It also provides bases for identifying desired agriculture water resources management plans with reasonable benefit and irrigation schedules under crops.

On Some Stability Notions of Multiobjective Integer Linear Fractional Programming Problem under Randomness

In this paper a solution algorithm to chance-constrained multiobjective integer linear fractional programming problem is suggested. We assume that there is randomness in the right-hand side of the constraints only and that the random variables are normally distributed. Some stability notions for the problem of concern are defined and characterized. An illustrative numerical example is provided to clarify the developed theory and the suggested solution algorithm.

A hybrid waste-flow allocation model considering multiple stage and interval–fuzzy chance constraints

An inexact two-stage fuzzy chance-constrained programming (ITSFCCP) model was developed in this study for dealing with multiple uncertainties associated with solid waste management systems. The model was formulated by incorporating fuzzy chance-constrained programming and interval linear programming within a two-stage stochastic programming framework. The model could be used to facilitate analysis of the policy scenarios; meanwhile, the uncertainties associated with solid waste management systems could be expressed as probability distributions, possibility distribution and discrete intervals. A long-term waste planning problem was used to demonstrate the applicability of the ITSFCCP model. The study results indicated that ITSFCCP could provide a linkage to pre-defined policies and allowed violation of system constraints at predefined confidence levels. Moreover, it allowed uncertain information presented as discrete intervals to be communicated into the optimisation process. ITSFCCP could help waste managers to identify desired policies and gain insights into the tradeoffs between system economy and reliability.

Optimal crop planning using a chance constrained linear programming model

A monthly time-stepped chance constrained linear programming (CCLP) model was developed to derive optimal cropping patterns and optimal operational strategies for the Sri Ram Sagar Project, with stochastic inflows. The stochastic nature of the inflows was incorporated into the model in its equivalent deterministic form. These equivalent deterministic inflow values were estimated from the annual and monthly probability distribution of observed inflows, and named the chance constrained linear programming model-annual and chance constrained linear programming model-monthly, respectively. The models were solved for nine different dependable inflow levels, namely for 50, 55, 60, 65, 70, 75, 80, 85 and 90% in each CCLP. The results of the models were compared with respect to net benefit, irrigation intensity, total cropped area, optimal cropping pattern, optimal releases, evaporation loss and initial storages. Based on the results obtained, it can be concluded that, in CCLP optimization, the probability distribution of ‘model time period’ (t-month in this case) should be considered rather than the probability of ‘planning time period’ (year in this case).

ICQSWM: An inexact chance-constrained quadratic solid waste management model

An inexact chance-constrained quadratic solid waste management (ICQSWM) model is developed for planning municipal solid waste (MSW) management systems. The ICQSWM improves upon existing methods of inexact two-stage chance-constrained linear programming (ITCLP) and interval quadratic programming (IQP). It can tackle uncertainties presented as both probability distributions and discrete intervals; moreover, it can be used for analyzing various policy scenarios which are associated with different levels of economic penalties when the promised targets are violated. It can also reflect the effects of economies of scale under uncertainty. The developed method is applied to a case study of waste-flow-allocation planning within a MSW management system. The results indicate that the ICQSWM model not only provides an effective tool for reflecting system cost variations but also generates decision alternatives; the solutions obtained can thus help managers to identify desired policies under various environmental and system-reliability constraints and economic objectives.

Planning of municipal solid waste management systems under dual uncertainties: a hybrid interval stochastic programming approach

In this study, a random-boundary-interval linear programming (RBILP) method is developed and applied to the planning of municipal solid waste (MSW) management under dual uncertainties. In the RBILP model, uncertain inputs presented as interval numbers can be directly communicated into the optimization process; besides, intervals with uncertain lower and upper bounds can be handled through introducing the concept of random boundary interval. Consequently, robustness of the optimization process can be enhanced. To handle uncertainties with such complex presentations, an integrated chance-constrained programming and interval-parameter linear programming approach (ICCP) is proposed. ICCP can help analyze the reliability of satisfying (or risk of violating) system constraints under uncertainty. The applicability of the proposed RBILP and ICCP approach is validated through a case study of MSW management. Violations for capacity constraints are allowed under a range of significant levels. Interval solutions associated with different risk levels of constraint violation are obtained. They can be used for generating decision alternatives and thus helping waste managers to identify desired policies under various environmental, economic, and system-reliability constraints.