In this paper a new method of modeling variable slope angles has been presented based on the spline interpolation method. Slope angle modeling and defining precedency of the blocks are the vital parts of almost any open pit optimization algorithm. Traditionally heuristic patterns such as 1:5 or 1:9 have been used to generate slope angles. Cone template based models were later employed in developing variable slope angles. They normally use a linear interpolation process for determination of slope angles between the given directions which leads to sharp and non-realistic pits. The other elliptical alternatives suffer from having limitations in defining slope angles in non-geographical directions. The method is capable to consider any number of slope angles in any desired direction as well as creating quite accurate and realistic pit shapes. Three major types of the spline interpolation including cubic, quadratic and cardinal are tested, however, the cubic form is preferred due to more realistic outcomes. Main steps of the method are described through a numerical case study.Keywords: open pit optimization, variable slope angles, spline interpolation, production planning W pracy zaprezentowano nową metodę modelowania zmiennego kąta nachylenia gruntu w oparciu o metodę interpolacji funkcjami sklejającymi (metoda spline'ów). Modelowanie kąta nachylenia stoku i prognozowanie kolejności wybierania to kluczowe elementy algorytmu optymalizacyjnego. Tradycyjne modele heurystyczne oparte o wzorce 1:5 lub 1:9 wykorzystane zostały do wygenerowania kątów nachylenia stoku. Do wygenerowania zmiennych kątów nachylenia wykorzystano modele stożkowe. Procedura taka zasadniczo zakłada wykorzystanie interpolacji liniowej dla określenia kąta nachylenia pomiędzy dwoma kierunkami, co prowadzić może do zaprojektowania bardzo stromych i nierealistycznych kształtów odkrywek. Alternatywne rozwiązania, wykorzystujące modele eliptyczne, mają inne ograniczenia -mianowicie określają one kąty nachylenia w kierunkach innych niż geograficzne. Za Słowa kluczowe: optymalizacja odkrywki, zmienne kąty nachylenia stoku, interpolacja funkcjami sklejającymi (metodą spline'ów), planowanie wydobycia
IntroductionThe extensive application of computers in the field of production planning of open-pit mines has engaged researchers to develop superior and comprehensible algorithms for answering the problem as fast as possible and requiring less computer and human resources. However even after about 40 years, the field still needs to develop more powerful methods to define Ultimate Pit Limit (UPL), optimum cut-off grade, optimum mine schedule, mine life and capacities. They also need to undertake all risks and uncertainties involved in estimation of geological grades and market condition. In early studies (Lemieux, 1979;Pana, 1965;Williams, 1974) developed the moving cone algorithm for designing the outline of the final pit shape. Several studies were carried out later to combine 2D sections into 3D pit shapes (Johnson & Sharp, 1971;Wright, 1987). (Koenigsberg, 19...