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In this article, the concept of conditioning in integer programming is extended to the concept of a complexity index. A complexity index is a measure through which the execution time of an exact algorithm can be predicted. We consider the multidimensional knapsack problem with instances taken from the OR-library and MIPLIB. The complexity indices we developed correspond to the eigenvalues of a Dikin matrix placed in the center of a polyhedron defined by the constraints of the problem relaxed from its binary variable formulation. The lower and higher eigenvalues, as well as their ratio, which we have defined as the slenderness, are used as complexity indices. The experiments performed show a good linear correlation between these indices and a low execution time of the Branch and Bound algorithm using the standard version of CPLEX® 12.2. The correlation coefficient obtained ranges between 39 and 60% for the various data regressions, which proves a medium to strong correlation.

A methodology for the conceptual design of concentration circuits is presented. The methodology considers three decision levels: level I – the definition and analysis of the problem, level II – the synthesis and screening of alternatives, and level III – the final design. Level I allows setting up the problem and defining the design goals. In level II circuit alternatives are generated and evaluated using a group contribution method. Also, level II is complemented by a database that helps to select the most suitable circuit. In level III the design of each process stage is performed for the alternatives identified in level II. This final design is performed using local/global sensitivity analysis and reverse simulation. The method is illustrated with examples that demonstrate that the method is suitable for these types of problems.

The heap leaching of minerals is one of the more commonly used processes in the mining industry. This process has been modeled by many authors. However, the validation, verification, and implementation of these models are difficult since there is uncertainty about the operating conditions and the leaching model parameters. This work uses the uncertainty quantification, based on uncertainty and sensitivity analysis, for studying the model strength against uncertainties in heap leaching. The uncertainty analysis (UA) is used to quantify the effect of the magnitude of the uncertainties of the input variables on the recovery of heap leaching. Global sensitivity analysis (GSA) is used to study the nature of connections between the recovery and input variables of the leaching model. In addition, GSA facilitates the detection of whether a leaching model is over-parameterized. The information obtained allows studying some applications of the kinetic model. The Mellado et al. kinetic model is used as an example. The UA results indicate that the kinetic model can estimate the recovery behavior considering the full range of uncertainties of input variables. The GSA indicates that the kinetic model is over-parameterized on the uncertainties range considered; this conclusion contradicts the results when the local sensitivity analysis is used. However, the model shows a good correlation between the results of GSA and the kinetic behavior of heap leaching. In addition, the kinetic model presents versatility because it allows the determination of operating regions for heap leaching.

This paper presents a methodology to analyze and/or design processes in which several stages are required to achieve the separation objectives and in which it is not possible to accurately define the operating conditions. This methodology is explained and applied to mineral flotation circuits but can be adapted and/or extended to other processes. The methodology considers the construction of a superstructure that includes alternatives to be analyzed, the representation of variables by distribution functions, a Monte Carlo simulation, the incorporation of indicators and a multi-objective analysis. From the obtained results, we conclude that the proposed methodology is a useful tool for analyzing and designing separation processes that include several stages, especially when there is uncertainty about the operating conditions of each stage.

This paper analyzes the application of global sensitivity analysis (GSA) to the improvement of processes using various case studies. First, a brief description of the methods applied is given, and several case studies are examined to show how GSA can be applied to the study to improve the processes. The case studies include the identification of processes; comparisons of the Sobol, E-FAST and Morris GSA methods; a comparison of GSA with local sensitivity analysis; an examination of the effect of uncertainty levels and the type of distribution function on the input factors; and the application of GSA to the improvement of a copper flotation circuit. We conclude that GSA can be a useful tool in the analysis, comparison, design and characterization of separation circuits. In addition, we conclude that using the stage's recoveries of each species as input factors is a suitable choice for the GSA of a flotation plant.

In mineral processing, the recovery and/or concentration of the value components cannot occur in one operational stage, necessitating the use of separation circuits. The component to be separated is usually distributed at different concentrations and different particle sizes, resulting in different levels of recovery in terms of the particle size and concentration. Typically, more than one value component must be selectively removed by exploiting the differences in the components' floatability at different values of the pH, pulp potential and particle size. Several methods for the design of these circuits are available in the literature, but comparatively few articles present circuit analysis tools. In this work, global sensitivity analysis was used to analyze and design separation circuits. The effect of each stage on the general circuit was studied and the relationships between the recovery at each stage and the global recovery of the circuit were identified. The effects of uncertainties in the kinetic constants, the residence time and the number of cells were also analyzed. The Sobol' and Morris methods were applied using the software SimLab 2.2. The advantages and disadvantages of each method are discussed and the two methods are compared with local methods (which were previously published by the authors); application of the two methods to the design of mineral separation processes is also discussed. These considerations are illustrated by a case study.

This paper presents a new methodology for the design of flotation circuits based on the group contribution method. The methodology considers three design and analysis tools: 1) the process group contribution to preliminarily evaluate and synthesize flotation circuits, 2) sensitivity analysis to identify critical variables and stages, and 3) reverse simulation to guide the experimentation, the establishment of operating conditions, and the equipment design. The methodology includes three steps: 1) definition and analysis of the problem, 2) synthesis and screening of alternatives, and 3) improvements and final design. A case study is included to illustrate the proposed methodology.

ABST R AC T In the treatment of several minerals it is common that using flotation as a concentration step, before others treatment. Other applications of flotation include pretreatment in desalination, separation of oil from wastewaters and recovery of plastics from waste. Flotation circuit of several stages is usually used because generally it is not possible to obtain the target results in one stage. The design of these network process, however, assumes that process data are fixed and well-defined, whereas the actual operating conditions such as stages recoveries and feed grade may fluctuate over time. Those fluctuations can affect the design objectives, for example recovery and product quality. This paper applies the Monte Carlo simulation for the analysis of different types of flotation circuits, represented through a superstructure, under uncertainty in grade feed and recovery of each flotation stage. This work demonstrates that the use of Monte Carlo simulation can be an appropriate tool to evaluate and select flotation circuits. The circuits were evaluated, with the help of radial/spider graphic, using metric indices of process efficiency, capacity and stability.