Modeling data envelopment analysis by chance method in hybrid uncertain environments

Data envelopment analysis (DEA) is a methodology for measuring the relative efficiency of decision making units (DMUs) which ‎consume the same types of inputs and producing the same types of outputs. Believing that future planning and predicting the ‎efficiency are very important for DMUs, this paper first presents a new dynamic random fuzzy DEA model (DRF-DEA) with ‎common weights (using multi objective DEA approach) to predict the efficiency of DMUs under mean chance constraints and ‎expected values of the objective functions. In the initial proposed†â€DRF-DEA model, the inputs and outputs are assumed to be ‎characterized by random triangular fuzzy variables with normal distribution, in which data are changing sequentially. Under this ‎assumption, the solution process is very complex. So we then convert the initial proposed DRF-DEA model to its equivalent multi-‎objective stochastic programming, in which the constraints contain the standard normal distribution functions, and the objective ‎functions are the expected values of functions of normal random variables. In order to improve in computational time, we then ‎convert the equivalent multi-objective stochastic model to one objective stochastic model with using fuzzy multiple objectives ‎programming approach. To solve it, we design a new hybrid algorithm by integrating Monte Carlo (MC) simulation and Genetic ‎Algorithm (GA). Since no benchmark is available in the literature, one practical example will be presented. The computational results ‎show that our hybrid algorithm outperforms the hybrid GA algorithm which was proposed by Qin and Liu (2010) in terms of ‎runtime and solution quality. ‎

A combined Monte Carlo (MC) simulation and Genetic Algorithm (GA) method was proposed by other researchers for the optimisation of spare parts allocation. From case studies, it was found that the number of simulation trials of the existing method tended to be either excessive or inadequate. Thus, a simulation replication number control method making full use of the advance simulation effort is proposed and implemented into the existing method. A numerical example shows significant improvement on overall simulation efficiency and that at the same time the required accuracy is guaranteed. Furthermore, it is argued that application-specific knowledge should be embedded into the general GA procedure so that the evolution process can be more efficient. Heuristic methods for initial population preparation for GA with and without considering component cost difference are proposed and illustrated for spare parts allocation. A computing experiment was designed and performed to examine the influence of parameters for replication number control and initial population preparation. The generation of availability–cost curve further indicates the necessity to adopt heuristic methods to improve searching efficiency in GA.

In the real word, there are many problems which have linear programming models and sometimes it is necessary to formulate these models with parameters of uncertainty. Many numbers from these problems are linear programming problems with fuzzy variables. Some authors considered these problems and have developed various methods for solving these problems. Recently, Mahdavi-Amiri and Nasseri (2007) considered linear programming problems with trapezoidal fuzzy data and/or variables and stated a fuzzy simplex algorithm to solve these problems. Moreover, they developed the duality results in fuzzy environment and presented a dual simplex algorithm for solving linear programming problems with trapezoidal fuzzy variables. Here, the authors show that this presented dual simplex algorithm directly using the primal simplex tableau algorithm tenders the capability for sensitivity (or post optimality) analysis using primal simplex tableaus.

Structural reliability analysis with fuzzy random variables using error principle

Thermal-pulse method is a powerful tool for measuring space charge distributions in polymer films. The data analysis for thermal-pulse method involves the Fredholm integral equation of the first kind, which requires an appropriate numerical procedure to obtain a solution. Various numerical techniques, including scale transformation and regulation method, are proposed. Of those numerical methods, the scale transformation (ST) is the simplest and the most widely used method. However, it presents a high spatial resolution only near the sample surface. Monte Carlo (MC) method is one of the recently proposed ways to solve the equation numerically and has been successfully applied to the analysis of laser intensity modulation method data, which also involves the Fredholm integral equation of the first kind. In this paper we attempt to analyze thermal-pulse data in frequency domain with the MC method and discuss its effectiveness based on some numerical simulations. The simulation results indicate that the electric field profiles can be effectively extracted by the MC method. The computed profiles by the MC method consist well with the supposed distributions in the entire thickness of the sample, while the profiles reconstructed by the ST method fit very well to the supposed one at the vicinity of the target surface and distort sharply along the direction of the thermal pulse propagation in the sample bulk. On the other hand, the oscillations in the computed results by the MC method could deteriorate its accuracy in this study. The influence of noise level on the analysis based on the MC method is also tested by the use of the simulated data. The results show that the computed profiles would become more fluctuant as the noise level increases. This problem can be solved by selecting a larger value of tolerance during the singular value decomposition procedure. Thus, the value of tolerance is considered to be one of the key parameters in this algorithm, which is actually hard to determine. Additionally, the experimental data obtained from a polypropylene film under applied electric field are analyzed to illustrate the feasibility of MC method to be applied to the thermal-pulse experimental data. The results also show that the spatial accuracy by the MC method in the entire sample thickness is higher than by the ST method, which verifies that the MC method is more suitable for detecting the electric field distribution in the deep bulk of the sample. Owing to noise and error, the accuracy of MC calculation depends on the chosen tolerance value, which is now considered to be an obstacle in applying this method to the practical thermal-pulse measurement.

In structural reliability calculation, there are fuzzy uncertainties in the distribution parameters of random variables, which bring the problem of large computation and poor precision. In order to improve accuracy and efficiency of structural reliability, a novel structural reliability calculation method with fuzzy random variables is proposed from the perspective of perturbation principle. First, the fuzzy characteristic parameters of the structural fuzzy random variables are expressed as the sums of their true values and fuzzy perturbations. At the same time, the perturbations are equivalently selected. Second, the probability density function and the performance function are decomposed into a series of intervals under different level cut sets, and the Taylor series is expanded on the upper and lower limits of the intervals. Finally, according to the definition of structural reliability, based on the direct integration method, the sigmoid function is introduced into the reliability function to approximate the step function, and the fuzzy reliability is obtained. The simulation results show the effectiveness and accuracy of the proposed method, and it provides a feasible way for structural reliability analysis with fuzzy variables or fuzzy random variables.

A class of random fuzzy programming and its application to supply chain design

As energy demand increases and global warming progresses, expectations for fuel cells, which generate electricity through chemical reactions between hydrogen and oxygen, are rising. There are various types of fuel cells, which are classified according to the electrolyte. Polymer electrolyte fuel cells (PEFCs) are expected to be used in stationary power sources for home use and fuel cell vehicles because of their low operating temperature, short start-up time, and ease of miniaturization. The cost has been an issue in the widespread use of fuel cells. The amount of platinum used in the catalyst layer (CL) must be reduced to reduce costs. However, it increases the current density per unit area of platinum. There are three factors that contribute to voltage drop: ohmic polarization, activation polarization, and diffusion polarization. At high current densities, the voltage drop due to diffusion polarization is dominant. The main cause of diffusion polarization is the transport resistance of oxygen molecules in the CL, and improving the oxygen transport properties in the CL will lead to lower cost PEFCs. In this study, we analyzed the transport properties of oxygen molecules in the CLs of PEFCs using the Monte Carlo (MC) method. The effect of oxygen scattering on the overall oxygen transport properties was investigated by considering the behavior on the surface of the ionomer film (surface diffusion) into the MC method, which simulates the overall transport in the PEFC CL. The objective of this study is to reproduce accurate oxygen transport by comparing the results of MC and molecular dynamics (MD) methods. First, we used the MC method to analyze oxygen scattering phenomena. In this study, we used the 3D structure data of CL in the MC simulations. The three-dimensional structure of the sample was reconstructed from sequential slice images. Molecules were assumed to reflect specularly on the boundaries of the simulation domain. Oxygen molecules were randomly placed in the simulation system. The time step was set at 5 ps and the simulation time was set at 4 μs. In order to examine the effect of surface diffusion on the transport properties of oxygen molecules, the simulation was performed using various conditions of surface diffusion. We calculated the effective diffusion coefficient of oxygen in the CL as the transport property of oxygen molecules in the MC method. As the surface diffusion coefficient increases, the effective diffusion coefficient of oxygen increases. When the surface diffusion coefficient is small, the effective diffusion coefficient of oxygen increases due to a decrease in the time constant, which determines the probability distribution of the surface residence time. However, when the surface diffusion coefficient is large, a different trend emerges. As the time constant increases, the effective diffusion coefficient has a peak value and then decreases. Next, to verify the model of surface diffusion in the MC method, we analyzed the scattering phenomenon of oxygen molecules in a simulation system that simulates the pores in the CL using the MD method. We modeled the nano pores in the CL as a slit between the ionomer walls in the MD method. Nafion with the equivalent weight (EW) of 1146 was used as polymer model. The number of Nafion chains is set at 4, and the water content λ is set at 7. Three-dimensional periodic boundary conditions were applied to the simulation domain. After the system was equilibrated, 1 ns of NVT simulation was performed for production. The temperature was set at 297 K. The behavior of oxygen molecules impacting on the ionomer wall was analyzed. We analyzed the behavior of oxygen molecules colliding with the ionomer membrane using the MD method. The average residence time of surface diffusion on the ionomer thin film was of the order of 10-10s. When the calculation results were compared with the MC results, it was found that the average residence time obtained by the MD method was the value where the surface diffusion coefficient affects much on the effective diffusion coefficient. Comparing the results of the MD method with the surface diffusion model used in the MC method, there are some discrepancies between the results and the model. For example, the MC method used a model in which oxygen molecules reflect diffusively on the ionomer surface. However, in the MD method, it was found that oxygen molecules tend to reflect in the direction of travel (Fig. 1). Therefore, a more accurate model of the MC method needs to be developed in the future. Figure 1

In these days, Monte Carlo (MC) simulation is a method that can calculate the radiation dose that occurs in an environment in the most accurate way. The correct measurement of the dose occurring on the patient’s surface is of great importance to estimate the reactions that may occur on the patient’s skin. This importance encouraged us to do this study. The aim of this study is to determine buildup region and surface doses using MC simulation and to compare them with results of the parallel plane ion chamber and Treatment Planning System (TPS) measurements for 6-MV photon beams. Surface doses normalized to the maximum dose for the parallel plane ion chamber, MC simulation, fast photon (FP) algorithm, and collapsed cone convolution superposition (CC) algorithm are 13.6%, 30.28%, 0%, and 27.33%, respectively. The CC algorithm and parallel plane ion chamber measurements are compatible with MC simulation but the FP algorithm has calculated the dose less to a depth of 0.8 cm. Measuring the surface dose and the doses in the buildup region is of great importance in terms of accurately predicting the complications that may occur in the patient’s skin and taking precautions early. Using some methods and correction factors, the surface dose and the doses that may occur in the buildup region can be accurately calculated. It is recommended not to use the FP algorithm for stereotactic body radiation therapy and intensity-modulated radiation therapy treatments, as it cannot calculate doses correctly in the buildup region and surface.

Effects of heterogeneities in dose distributions under nonreference conditions: Monte Carlo simulation vs dose calculation algorithms

Monte Carlo Data Envelopment Analysis with Genetic Algorithm for Knowledge Management performance measurement

The Monte Carlo (MC) method has been employed through the years as a gold-standard method to reconstruct numerical solutions of the RTE. The MC method provides a physical simulation of light propagation; i.e., photon trajectories are generated by using the same statistical rules that govern propagation in random media. In practice, the MC method estimates the expected characteristics of the photon population as statistical averages over a large number of case histories of photon life that are simulated by a computer. These average statistical characteristics of the photon population are exactly what the RTE describes, and, intuitively, this is why the MC method allows one to estimate numerically solutions of the RTE. In fact, comparisons of MC-based results with exact analytical solutions of the RTE demonstrate that for the limit of an infinite number of photons used, the simulated data converge rigorously to the analytical results. The existing literature shows that the MC method can be implemented by using different approaches. In this chapter, the approaches usually employed are briefly reviewed by illustrating their main peculiarities. These approaches are also the basis of more complex MC procedures, such as perturbation MC methods, voxel-based MC methods, mesh-based MC methods, or GPU-accelerated MC codes.

Background Small megavoltage photon fields are increasingly used in modern radiotherapy techniques such as stereotactic radiotherapy. Therefore, it is important to study the reliability of dosimetry in the small-field conditions. The IBA Razor Nano Chamber (Nano chamber) ionization chamber is particularly intended for small-field measurements. In this work, properties of the Nano chamber were studied with both measurements and Monte Carlo (MC) simulations. Material and methods The measurements and MC simulations were performed with 6 MV, 6 MV FFF and 10 MV FFF photon beams from the Varian TrueBeam linear accelerator. The source-to-surface distance was fixed at 100 cm. The measurements and MC simulations included profiles, percentage depth doses (PDD), and output factors (OF) in square jaw-collimated fields. The MC simulations were performed with the EGSnrc software system in a large water phantom. Results The measured profiles and PDDs obtained with the Nano chamber were compared against IBA Razor Diode, PTW microDiamond and the PTW Semiflex ionization chamber. These results indicate that the Nano chamber is a high-resolution detector and thus suitable for small field profile measurements down to field sizes 2 × 2 cm2 and appropriate for the PDD measurements. The field output correction factors and field OFs were determined according to TRS-483 protocol In the 6 MV FF and FFF beams, the determined correction factors were within 1.2% for the field sizes of 1 × 1 cm2–3 × 3 cm2 and the experimental and MC defined field output factors showed good agreement. Conclusion The Nano chamber with its small cavity volume is a potential detector for the small-field dosimetry. In this study, the properties of this detector were characterized with measurements and MC simulations. The determined correction factors are novel results for the NC in the TrueBeam fields.

Objective. Monte Carlo (MC) simulations are the benchmark for accurate radiotherapy dose calculations, notably in patient-specific high dose rate brachytherapy (HDR BT), in cases where considering tissue heterogeneities is critical. However, the lengthy computational time limits the practical application of MC simulations. Prior research used deep learning (DL) for dose prediction as an alternative to MC simulations. While accurate dose predictions akin to MC were attained, graphics processing unit limitations constrained these predictions to large voxels of 3 mm × 3 mm × 3 mm. This study aimed to enable dose predictions as accurate as MC simulations in 1 mm × 1 mm × 1 mm voxels within a clinically acceptable timeframe. Approach. Computed tomography scans of 98 breast cancer patients treated with Iridium-192-based HDR BT were used: 70 for training, 14 for validation, and 14 for testing. A new cropping strategy based on the distance to the seed was devised to reduce the volume size, enabling efficient training of 3D DL models using 1 mm × 1 mm × 1 mm dose grids. Additionally, novel DL architecture with layer-level fusion were proposed to predict MC simulated dose to medium-in-medium (D m,m ). These architectures fuse information from TG-43 dose to water-in-water (D w,w ) with patient tissue composition at the layer-level. Different inputs describing patient body composition were investigated. Main results. The proposed approach demonstrated state-of-the-art performance, on par with the MC D m,m maps, but 300 times faster. The mean absolute percent error for dosimetric indices between the MC and DL-predicted complete treatment plans was 0.17% ± 0.15% for the planning target volume V 100, 0.30% ± 0.32% for the skin D 2cc , 0.82% ± 0.79% for the lung D 2cc , 0.34% ± 0.29% for the chest wall D 2cc and 1.08% ± 0.98% for the heart D 2cc . Significance. Unlike the time-consuming MC simulations, the proposed novel strategy efficiently converts TG-43 D w,w maps into precise D m,m maps at high resolution, enabling clinical integration.

The aim of this paper is to present the proactive strategy for resource-constrained project scheduling problems under a fuzzy random environment with activity splitting, i.e. activities can be split one or more times in scheduling process. After giving the motivation and justification for employing fuzzy random variables, a mathematical formulation model for preemption resource-constrained project scheduling problems (PRCPSP) with fuzzy random duration and resource availabilities is built. In this model, the objective is to maximize the sum of free slack which contains two parts: one is the time buffer that the start time of some activities can be delayed, the other is the shortage of resource is allowed. Since the difficulties of dealing with fuzzy random variables, the model is transformed into an equivalent crisp one. A numerical example is finally applied to demonstrate the efficiency of the proposed model and proactive strategy, and the generated results verify the robustness for PRCPSP.