Robust scenario formulations for strategic supply chain optimization under uncertainty

Strategic supply chain optimization (SCO) problems are often modelled as a two-stage optimization problem, in which the first-stage variables represent decisions on the development of the supply chain and the second-stage variables represent decisions on the operations of the supply chain. When uncertainty is explicitly considered, the problem becomes an intractable infinite-dimensional optimization problem, which is usually solved approximately via a scenario or a robust approach. This paper proposes a novel synergy of the scenario and robust approaches for strategic SCO under uncertainty. Two formulations are developed, namely, naïve robust scenario formulation and affinely adjustable robust scenario formulation. It is shown that both formulations can be reformulated into tractable deterministic optimization problems if the uncertainty is bounded with the infinity-norm, and the uncertain equality constraints can be reformulated into deterministic constraints without assumption of the uncertainty region. Case studies of a classical farm planning problem and an energy and bioproduct SCO problem demonstrate the advantages of the proposed formulations over the classical scenario formulation. The proposed formulations not only can generate solutions with guaranteed feasibility or indicate infeasibility of a problem, but also can achieve optimal expected economic performance with smaller numbers of scenarios.

The Evolution of the Los Negritos Porphyry Copper Deposit, Coquimbo, Chile: Geological, Geochronological, Mineralogical and Geochemical Evidence

The Los Negritos porphyry copper deposit is located ~ 4 km to the northeast of Carmen de Andacollo Mine in the Chilean Cretaceous metallogenic belt. The mineralization is hosted in andesite of the Quebrada Marquesa Formation and a series of at least four early to intramineral porphyry intrusive rock types: plagioclase quartz biotite porphyry (P1b and P1a dated at 109.60± 0.75 Ma and 107.22± 0.40 Ma); plagioclase biotite porphyry (P2: 106.30 ± 0.47 Ma); and quartz plagioclase biotite porphyry (P3: 106.19 ± 0.42 Ma). These units are cut by late‐ to post‐mineral plagioclase‐hornblende porphyritic rocks (P4b: 106.20 ± 0.69 Ma and P4a: 106.50 ± 0.68 Ma). The earliest intrusive units (P1) were affected by an initial stage of K‐feldspar‐biotite alteration, with chalcopyrite, molybdenite (date at 108.5 ± 0.5 Ma) and gold (up to 0.11 ppm), and the surrounding volcanic host rock was overprinted by chlorite‐epidote dominated (propylitic) alteration. Subsequent to the P2 and P3 intrusion, these rocks were affected by albite and then a second stage of potassic alteration. The Ti and Ba contents in hydrothermal biotite are notably lower (typically Ti = 0.100‐0.144 a.p.f.u. and Ba = 0.001‐0.005 a.p.f.u) than in magmatic ones (generally Ti = 0.186‐0.222 a.p.f.u. and Ba = 0.014‐0.023 a.p.f.u.), and constitute an excellent discriminant of the nature of biotite. These early stages of alteration were overprinted by copper‐molybdenum bearing chlorite‐sericite alteration at 106.60 ± 0.5 Ma (Re‐Os age in molybdenite) and by quartz‐sericite‐pyrite veins (phyllic), respectively in the southwest and northeast areas. The average temperature associated with these two alteration facies is estimated around 305 °C. Weak albite‐calcite alteration, spatially associated with sulfosalts and distributed along the margins of P3, overprinted the phyllic facies. The intrusive rock units at the Los Negritos and Carmen de Andacollo deposits are geochemically classified as diorite to granodiorite with a calc‐alkaline magmatic affinity, and formed in a volcanic arc setting from partial melting of a metasomatized mantle wedge. They are interpreted to be cogenetic, and related to a common long‐lived magma chamber that emplaced during a period of tectonic inversion known as the Subhercynian, Peruvian or Pacific event.