Application of chance-constrained programming based on multi-objective simulated annealing to solve a mineral blending problem

Abstract

This paper addresses the optimal blending of different available ores in such a way that the total expected cost of buying ore is minimized while satisfying the quality specifications. The risk limitation criterion used consists of the simultaneous minimization of the variance of the total cost. The problem is solved by Chance-Constrained Programming (CCP) based on multi-objective simulated annealing. The technique is able to deal with the stochastic nature of the variables in the blending problem. In generic form the objectives are to minimize expected value and standard deviation of cost in such a way as to meet the blend requirements within the specified reliability level. The variability of each variable in each flow is quantified by semi-variograms. Each flow is simulated to reproduce the characteristics, or behaviour, of the phenomenon as observed in the available data. The expected value of each variable in each flow is calculated by averaging of the simulated values. The problem expressed in terms of CCP is transformed into an appropriate deterministic equivalent, which is a non-linear optimization problem. The new form of the problem is solved by multi-objective simulated annealing. A case study is carried out to demonstrate the technique. The problem is formulated in terms of iron, silica and alumina grades with four ore sources. Pros and cons of the technique are discussed.