Equivalent Deterministic Form Research Articles

Multi-period time window assignment for attended home delivery

We study a multi-period stochastic variant of the Time Window Assignment Vehicle Routing Problem, where customers’ demands, locations, and service times are uncertain. Customers are partitioned into geographical zones, each of which has to be visited a predetermined number of times over a planning period of several days. Whenever a zone is visited, a time window is assigned. Time windows are decided before knowing customers and their demands. A fleet of homogeneous vehicles is available to serve customers each day. At a tactical level, the problem looks for a static time window assignment that minimizes the expected traveling costs plus the expected penalty costs for unserved customers. We propose a two-stage formulation and a solution approach, which relies on the Sample Average Approximation Method, while encompassing a perturbation method to assign time windows in the first stage and an Adaptive Large Neighborhood Search to optimize routes in the second stage. We experimentally evaluate three instance sets, including real ones from a Canadian company, comparing our results to lower bounds from the exact solution of a deterministic equivalent formulation over a finite number of scenarios. Our method outperforms the manual approach used by the company.

Response Analysis of the Three-Degree-of-Freedom Vibroimpact System with an Uncertain Parameter.

The inherent non-smoothness of the vibroimpact system leads to complex behaviors and a strong sensitivity to parameter changes. Unfortunately, uncertainties and errors in system parameters are inevitable in mechanical engineering. Therefore, investigations of dynamical behaviors for vibroimpact systems with stochastic parameters are highly essential. The present study aims to analyze the dynamical characteristics of the three-degree-of-freedom vibroimpact system with an uncertain parameter by means of the Chebyshev polynomial approximation method. Specifically, the vibroimpact system model considered is one with unilateral constraint. Firstly, the three-degree-of-freedom vibroimpact system with an uncertain parameter is transformed into an equivalent deterministic form using the Chebyshev orthogonal approximation. Then, the ensemble means responses of the stochastic vibroimpact system are derived. Numerical simulations are performed to verify the effectiveness of the approximation method. Furthermore, the periodic and chaos motions under different system parameters are investigated, and the bifurcations of the vibroimpact system are analyzed with the Poincaré map. The results demonstrate that both the restitution coefficient and the random factor can induce the appearance of the periodic bifurcation. It is worth noting that the bifurcations fundamentally differ between the stochastic and deterministic systems. The former has a bifurcation interval, while the latter occurs at a critical point.

A heuristic approach to the stochastic capacitated single allocation hub location problem with Bernoulli demands

Hubs are critical components of transportation and distribution systems, and hub networks play a special role in freight and passenger transportation services worldwide. This paper studies a capacitated single allocation hub location problem with Bernoulli demands. Since the origin-destination (OD) demands are stochastic in nature, and the nodes are allocated to hubs before knowing their realized values, the actual total demand allocated to each hub is uncertain. Therefore, demand can exceed the capacity of hubs, rendering a need for outsourcing. The problem is studied under two distinct outsourcing policies, namely the facility and customer outsourcing. Mathematical models are developed for each case as two-stage stochastic programs. Deterministic equivalent formulations are obtained for problems, assuming a homogeneous demand distribution for all OD pairs. A Tabu Search-based algorithm is presented as a solution approach to deal with large problem instances. Extensive computational tests demonstrate the outstanding performance of the developed models and the metaheuristic procedure in terms of solution quality and computational time. The relevance of using stochastic programming approach for the problems is demonstrated via computational results. This study also reports solutions for problems with 200 nodes for the first time.

Drones for relief logistics under uncertainty after an earthquake

This study presents a post-disaster delivery problem called the relief distribution problem using drones under uncertainty, in which critical relief items are distributed to disaster victims gathered at assembly points after a disaster, particularly an earthquake. Because roads may be obstructed by debris after an earthquake, drones can be used as the primary transportation mode. As the impact of an earthquake cannot be easily predicted, the demand and road network uncertainties are considered. Additionally, the objective is to minimize the total unsatisfied demand subject to a time-bound constraint on the deliveries, as well as the range and capacity limitations of drones. A two-stage stochastic programming and its deterministic equivalent problem formulations are presented. The scenario decomposition algorithm is implemented as an exact solution approach. To apply this study to real-life applications, a case study is conducted based on the western (European) side of Istanbul, Turkey. The computational results are used to evaluate the performance of the scenario decomposition algorithm and analyze the value of stochasticity and the expected value of perfect information under different parametric settings. We additionally conduct sensitivity analyses by varying the key parameters of the problem, such as the time-bound and capacities of the drones.

Solving a multi-objective chance constrained hierarchical optimization problem under intuitionistic fuzzy environment with its application

Various optimization approaches have been developed and used for generating optimal solutions for different industry related optimization problems. The semantic representation of imprecise coefficients and various types of uncertainties arising in real life optimization problems are still a challenging task and require attention of academicians as well as professionals. Decision makers (DMs) often face problems in presenting the linguistically expressed constraints. The impreciseness of information also plays a key role in modelling and thus in decision making. The impreciseness of coefficients is usually handled by using various elements of fuzzy set theory. The uncertainty associated with constraints in optimization problem is handled by using stochastic programming approaches. These coefficients are usually revealed by a logical DM, who takes into account the scope of hesitation along with the imprecise nature of data. Such coefficients can be very well represented by using Intuitionistic fuzzy numbers. In this work, an approach for solving multi-objective bi-level chance constrained optimization problem in an intuitionistic fuzzy environment has been presented. The technological coefficients and the coefficients of objective functions are represented by Triangular Intuitionistic fuzzy numbers whereas normal random variables are used to represent the coefficient of resources. The concept of component wise optimization is used for the reduction of intuitionistic fuzzy objective into their equivalent deterministic form. The concept of alpha-beta cut, chance constrained programming and interval programming are then used to convert the problem into a deterministic multi-objective bi-level linear programming problem. The resultant deterministic optimization problem is then solved by using TOPSIS method. The production planning problem of a firm has been modelled and solved by using the proposed approach.

Multi-period portfolio selection based on uncertainty theory with bankruptcy control and liquidity

Due to the impact of uncertain events, such as the 2008 financial crisis and the outburst of COVID-19 pandemic, the experts’ evaluations information is becoming increasingly important. This paper considers a multi-period portfolio optimization problem under uncertain circumstance, and the return rates of risky securities are regarded as uncertain variables, where the uncertainty theory is used to deal with experts’ evaluations. In light of the complexity of financial markets, we formulate an uncertain multi-period mean-entropy-variance model, where bankruptcy, liquidity, diversification and self-financing are considered as realistic constraints. Furthermore, the maximum return and the minimum risk are both achieved in a single-objective model through the normalization method. Then the equivalent deterministic forms of two secondary models for main model are provided. In addition, we develop a modified root system growth algorithm, which is more suitable for the proposed model. Finally, the effectiveness of the proposed model and designed algorithm is confirmed by numerical simulations.

Value function gradient learning for large-scale multistage stochastic programming problems

A stagewise decomposition algorithm called “value function gradient learning” (VFGL) is proposed for large-scale multistage stochastic convex programs. VFGL finds the parameter values that best fit the gradient of the value function within a given parametric family. Widely used decomposition algorithms for multistage stochastic programming, such as stochastic dual dynamic programming (SDDP), approximate the value function by adding linear subgradient cuts at each iteration. Although this approach has been successful for linear problems, nonlinear problems may suffer from the increasing size of each subproblem as the iteration proceeds. On the other hand, VFGL has a fixed number of parameters; thus, the size of the subproblems remains constant throughout the iteration. Furthermore, VFGL can learn the parameters by means of stochastic gradient descent, which means that it can be easil0y parallelized and does not require a scenario tree approximation of the underlying uncertainties. VFGL was compared with a deterministic equivalent formulation of the multistage stochastic programming problem and SDDP approaches for three illustrative examples: production planning, hydrothermal generation, and the lifetime financial planning problem. Numerical examples show that VFGL generates high-quality solutions and is computationally efficient.

Stochastic hub location problems with Bernoulli demands

This paper introduces the hub location problem with Bernoulli demands (HLPBD). The problem is a variant of capacitated stochastic hub location problems in which demands for transportation service are assumed to be Bernoulli random variables. We address the problem under single and multiple allocation settings, and for each case, a two-stage stochastic programming model is presented. Further, assuming a homogeneous demand probability, a closed-form expression for the recourse function is presented, and deterministic equivalent formulations are developed for the original stochastic problems. To solve large instances, exact solution algorithms based on Benders decomposition and Lagrangian relaxation are proposed. An extensive set of computational experiments on three well-known data sets is conducted to test the efficiency of the proposed models and solution algorithms, and also to evaluate the effect of different input parameters on the optimal solutions.

Neutrosophic compromise programming approach for multiobjective nonlinear transportation problem with supply and demand following the exponential distribution

Decision-making is a tedious and complex process. In the present competitive scenario, any incorrect decision may excessively harm an organization. Therefore, the parameters involved in the decision-making process should be looked into carefully as they may not always be of a deterministic nature. In the present study, a multiobjective nonlinear transportation problem is formulated, wherein the parameters involved in the objective functions are assumed to be fuzzy and both supply and demand are stochastic. Supply and demand are assumed to follow the exponential distribution. After converting the problem into an equivalent deterministic form, the multiobjective problem is solved using a neutrosophic compromise programming approach. A comparative study indicates that the proposed approach provides the best compromise solution, which is significantly better than the one obtained using the fuzzy programming approach.

Lot-sizing decisions for material requirements planning with hybrid uncertainties in a smart factory

Material requirements planning (MRP) is a kind of medium-term production planning, which aims to plan the end item requirements of the master production schedule over a finite planning horizon. In a smart factory, the customer requirements and the production status are varying in time, which increases the uncertainties in making lot-sizing decisions for MRP. In this study, a hybrid chance-constrained programming (HCCP) model is developed for solving an MRP problem with hybrid uncertainties, in which both randomness and fuzziness exist in a lot-sizing decision process. The objective of the HCCP model is to determine the lot sizes of all items while satisfying the stochastic demands and the fuzzy capacity constraints. The credibility and probability are incorporated into the proposed model to measure the fuzziness and randomness, respectively. In order to solve the model, relevant approaches for converting the probability-based and credibility-based constraints into the equivalent deterministic forms are proposed. Decision makers can set different confidence levels according to their own risk preferences to get different results. Finally, an example is presented to verify that the approach proposed in this paper is feasible for solving MRP problems with hybrid uncertainties.

Chance-constrained unit commitment with energy storage systems in electric power systems

The high penetration of renewables has amplified errors in forecasts of wind and solar power generation that are made to support the operation of power systems. To guarantee the security and reliability of the power system, sufficient dispatchable power generation capacity must be reserved. Various studies of the use of risk assessment to improve operational performance have been carried out. However, most methods of so doing are designed to cover a given risk level of uncertainty, which is determined from forecasts rather than actual results. As renewable generation and load capacity increase, defining the risk level without considering unit commitment (UC) may limit scheduling efficiency. In this study, chance-constrained programming is combined with goal programming to optimize risk-based UC. Solar power generation, wind power generation, and demand forecast errors are handled by the optimization of reserve schedules and adjustments of risk by imposing a penalty cost. The proposed risk-based UC model is transformed into a mixed-integer linear programming (MILP) problem using the equivalent deterministic form and piecewise linear functions to find an efficient solution. Case studies of a realistic Taiwan power system in 2025 are considered; these include peak (Summer 2025) and off-peak (Winter 2025) scenarios. Simulation results reveal that the flexible operation of a battery energy storage system (BESS) (upward/downward reserve) reduces risk costs by a significant amount (19.12%∼100%). Sensitivity analysis reveals how the risk preferences of the System Operator (SO) can be represented in terms of predefined risk levels and confidence intervals to capture the uncertainties in load, wind power generation, and solar power generation forecasts. The spinning reserve level is optimized to reduce the risk cost and the addition of BESS in flexible operations further reduces the risk cost. Lastly, pumped storage units as a downward reserve can reduce wind spillage and excess solar power generation costs.

Response Analysis of the Tristable Energy Harvester with an Uncertain Parameter

In the stage of modelling, measuring, mechanical processing and manufacturing of the nonlinear energy harvesting system, deviations and errors of system parameters are inevitable. Even slight variation of key parameters may have a significant influence on the output voltages, especially for the multi-stable nonlinear case. Therefore, the investigation of dynamic behaviors for the tristable energy harvesting system with uncertain parameters is of important value both for research and application. In this paper, the uncertainty of a tristable piezoelectric vibration energy harvester with a random coefficient ahead of the nonlinear term is studied. By using the Chebyshev polynomial approximation, this tristable energy harvesting system is first reduced into an equivalent deterministic form, the ensemble mean responses of which are derived to exhibit the stochastic behaviors. The periodic and chaotic motions, bifurcations and crises under different conditions are analyzed. The results show that the output voltage is sensitive to the uncertainty of the nonlinear coefficient, which leads to unstable behavior around the bifurcation and crisis points particularly. Exploring the influence pattern of uncertain parameters on the output voltage and avoiding the unstable parameter intervals are essential for optimizing the structure. It can further improve the efficiency of the nonlinear energy harvesting system.

Decomposition-Based Approaches for a Class of Two-Stage Robust Binary Optimization Problems

In this paper, we study a class of two-stage robust binary optimization problems with objective uncertainty, where recourse decisions are restricted to be mixed-binary. For these problems, we present a deterministic equivalent formulation through the convexification of the recourse-feasible region. We then explore this formulation under the lens of a relaxation, showing that the specific relaxation we propose can be solved by using the branch-and-price algorithm. We present conditions under which this relaxation is exact and describe alternative exact solution methods when this is not the case. Despite the two-stage nature of the problem, we provide NP-completeness results based on our reformulations. Finally, we present various applications in which the methodology we propose can be applied. We compare our exact methodology to those approximate methods recently proposed in the literature under the name [Formula: see text]adaptability. Our computational results show that our methodology is able to produce better solutions in less computational time compared with the [Formula: see text]adaptability approach, as well as to solve bigger instances than those previously managed in the literature. Summary of Contribution: Our manuscript describes an exact solution approach for a class of robust binary optimization problems with mixed-binary recourse and objective uncertainty. Its development reposes first on a reformulation of the problem, then a carefully constructed relaxation of this reformulation. Our solution approach is designed to exploit the two-stage and binary structure of the problem for effective resolution. In its execution, it relies on the branch-and-price algorithm and its efficient implementation. With our computational experiments, we show that our proposed exact solution method outperforms the existing approximate methodologies and, therefore, pushes the computational envelope for the class of problems considered.

Risk-averse optimization for resilience enhancement of complex engineering systems under uncertainties

With the growth of complexity and extent, large scale interconnected network systems, e.g., transportation networks or infrastructure networks, become more vulnerable to external disturbances. Hence, managing potential disruptive events during the design, operating, and recovery phase of an engineered system and therefore improving the system’s resilience is an important yet challenging task. To ensure system resilience after the occurrence of failure events, this study proposes a mixed-integer linear programming (MILP) based restoration framework using heterogeneous dispatchable agents. The scenario-based stochastic optimization (SO) technique is adopted to deal with the inherent uncertainties imposed on the recovery process from nature. Moreover, different from conventional SO using deterministic equivalent formulations, the CVaR risk measure is implemented for this study because of the temporal sparsity of the decision making in applications such as the recovery from extreme events. The resulting restoration framework involves a large-scale MILP problem and thus an adequate decomposition technique i.e. modified Lagrangian dual decomposition, is also employed to achieve tractable computational complexity. Case study results based on the IEEE 37-bus test feeder demonstrate the benefits of using the proposed framework for resilience improvement as well as the advantages of adopting SO formulations.

OPTIMIZING RELIABILITY OF LINEAR FRACTIONAL DIFFERENCE SYSTEMS UNDER UNCERTAINTY AND RANDOMNESS

Some complex systems may suffer from failure processes arising from soft failures and hard failures. The existing researches have shown that the reliability of a dynamic system is not constant under uncertain random environments. First, two types of uncertain random optimization models are proposed where reliability index is quantified by chance measure based on whether soft and hard failures are independent or not. It is considered that internal degradation is driven by left Caputo fractional linear difference equation, while shocks are defined as discrete i.i.d. random variables. The shocks may generate additional uncertain degradation shifts when considering the competing dependent failure processes. Then, two proposed optimization reliability problems may be transformed into their equivalent deterministic forms on the basis of [Formula: see text]-path, and improved gradient descent method is applied to obtain optimal solutions. Finally, the numerical example of a micro-engine indicates that the optimization models are beneficial to the reliability of engineering systems.

A branch-and-Benders-cut algorithm for a bi-objective stochastic facility location problem

In many real-world optimization problems, more than one objective plays a role and input parameters are subject to uncertainty. In this paper, motivated by applications in disaster relief and public facility location, we model and solve a bi-objective stochastic facility location problem. The considered objectives are cost and covered demand, where the demand at the different population centers is uncertain but its probability distribution is known. The latter information is used to produce a set of scenarios. In order to solve the underlying optimization problem, we apply a Benders’ type decomposition approach which is known as the L-shaped method for stochastic programming and we embed it into a recently developed branch-and-bound framework for bi-objective integer optimization. We analyze and compare different cut generation schemes and we show how they affect lower bound set computations, so as to identify the best performing approach. Finally, we compare the branch-and-Benders-cut approach to a straight-forward branch-and-bound implementation based on the deterministic equivalent formulation.

Long-term resource planning in the high-tech industry: Capacity or inventory?

Volatile and non-stationary demand, very complex production processes, long production and capacity acquisition leadtimes, and extremely high and rarely reversible capacity acquisition costs make capacity planning in the high-tech industry a particularly challenging managerial task. To cope with demand surges and hedge against its uncertainty, firms can invest in order to expand their capacity for the long-term. For the short-term, however, demand is often fixed due to committed orders, and excess capacity might be available that may be used to build up inventory as a complementary means of hedging against these demand surges and uncertainty. In multi-echelon systems, these decisions need to be made simultaneously and must be aligned over the echelons. We address the strategic problem of determining the optimal inventory buildup and capacity expansion timing and amount across locations of a multi-echelon supply chain, as well as the operational problem of production planning. We model the problem as a two-stage stochastic program (SP), with strategic decisions at the first stage and operational decisions at the second stage. We enumerate the capacity expansion timing and use the deterministic equivalent formulation to optimize the other strategic decisions. Through an extensive numerical study, we provide practical managerial insights into the system behavior.

A new methodology for solving bi-criterion fractional stochastic programming

<p style='text-indent:20px;'>Solving a bi-criterion fractional stochastic programming using an existing multi criteria decision making tool demands sufficient efforts and it is time consuming. There are many cases in financial situations that a nonlinear fractional programming, generated as a result of studying fractional stochastic programming, must be solved. Often management is not in needs of an optimal solution for the problem but rather an approximate solution can give him/her a good starting for the decision making or running a new model to find an intermediate or final solution. To this end, this author introduces a new linear approximation technique for solving a fractional stochastic programming (CCP) problem. After introducing the problem, the equivalent deterministic form of the fractional nonlinear programming problem is developed. To solve the problem, a fuzzy goal programming model of the equivalent deterministic form of the fractional stochastic programming is provided and then, the process of defuzzification and linearization of the problem is presented. A sample test problem is solved for presentation purposes. There are some limitations to the proposed approach: (1) solution obtains from this type of modeling is an approximate solution and, (2) preparation of approximation model of the problem may take some times for the beginners.</p>

Point estimate-based stochastic robust dispatch for electricity-gas combined system under wind uncertainty using iterative convex optimization

—In view of the growing deployment of wind power, reliable dispatches for wind uncertainty are becoming increasingly important, especially for the emerging electricity-gas combined system (EGCS). In this paper, a novel dispatch incorporating stochastic optimization and robust optimization is proposed for EGCS under wind uncertainty. Specifically, the objective is formulated with stochastic sampling to reduce the cost in actual operation, while security constraints are established based on affinely adjustable robust dispatch (AARD) to ensure the system reliability. Different from the traditional AARD which minimizes the base operation cost, the proposed stochastic robust dispatch optimizes the expected value of operation cost. The strategy essentially pre-optimizes the real-time adjustment in the look-ahead dispatch, such that the wind power fluctuation is balanced in a cost-effective way. Besides, A Nataf transformation-based three-point estimate is adopted, which accurately formulates the expected value of operation cost with a very small wind-sampling size. To solve the complex problem involving wind uncertainty, robust security constraints are converted into equivalent deterministic forms first, then a penalty convex–concave procedure-based optimization is developed to reformulate the nonconvex problem into an iterative convex programming. Simulation performed in two test systems validates the stochastic robust dispatch and the convex solution.

A Probablisitic Approach for Optimal Operation of Gas Processing Plant under Uncertain inlet-outlet Conditions

Natural gas plant operations contribute hugely to the economies of many developed nations that depend on hydrocarbon resources. The plant operation is usually subjected to continuous variations in upstream conditions, such as flow rate, composition, temperature and pressure, which propagate through the plant and affect its stable operations. As a result, decision making for optimal operating conditions of an in-operation plant is a complex problem and it is exacerbated with the changing product specifications and variations in energy supplies. This work presents a new solution method to the problem, which is based on chance constrained optimization. A deterministic model is initially developed from process simulation using Aspen HYSYS and later converted to a chance constrained model. The probabilistic model is then relaxed to its equivalent deterministic form and solved for optimum solution using GAMS. The optimum solution is determined probabilistically using chance constraints that are held at a user-defined confidence level. Optimal solution is represented graphically as a trade-off between reliability of holding the process constraints and profitability of the plant. Two case studies are presented to demonstrate the new method. Optimization results show that uncertainty of plant parameters significantly affect the economic performance of the plant operation. The solution approach developed in this work is able to increase the reliability of maintaining the profit by more than 95% confidence level. As a result, the risk of constraints violation is reduced from more than 50% using the typical deterministic optimization to less than 5% with the developed chance constrained optimization model. In addition, the results from this study indicate that the variation of material flow from the plant inlet has greater impact by more than 86% on profit change compared to variation from the plant outlet, which is less than 2%. Sensitivity analysis results show on how to reduce the effect of N2, CO2 and C5+ by holding the corresponding constraint at a certain confidence level. The developed solution method can aid as guidelines to flexible plant operation decision making for the in-operating plant by satisfying all the process constraints at certain confidence level.