**Mathematical Approach**

The conversion of lignocellulosic biomass to fuels has the potential to reduce our dependence on fossil fuels. To ensure biomass supply meets biofuel demand, it is necessary to have an effective biomass supply network. Towards this end, the concept of regional biomass processing depot, where biomass is pretreated and/or densified to a higher density intermediate, has been introduced to improve the performance of supply network in terms of costs and emissions. Despite all the extensive research in the area, the design of biofuel supply chain (SC) with depots has received limited attention. Accordingly, we develop optimization models for the design and operation planning of biofuel SCs with regional depots that account for seasonal biomass supply. Importantly, unlike previous approaches which assume predetermined depot and biorefinery locations, we treat the locations of regional depots and/or biorefineries as continuous optimization decisions. The proposed models account for technology selection and capacity planning decisions, as well as auxiliary decisions such as harvesting site and biomass feedstock selection, biomass allocation to depots and biorefineries, and inventory planning decisions. Our current efforts focus on the integration of spatially explicit biomass production data with biofuel SC optimization.

** Composite Curve Approach**

The production of fuels and chemicals from biomass has received considerable attention recently due to environmental concerns. Since the production of biofuels involves relatively expensive feedstock and energy-intensive biomass transportation, any biomass-to-fuels strategy should include an efficient, both in terms of cost and environmental impact, biomass (feedstock) supply chain. Unlike fossil fuels, biomass, as a low-energy density resource, is sparsely distributed. Efficient biomass transportation thus requires biomass procurement planning methods. In many studies, the farms are treated as points without shape or area. This is a reasonable assumption when the transportation distance is so large that the shape and size of the farms can be neglected. In this case, the transportation problem is modeled as a point-to-point (farm-to-refinery) problem. However, the shape and size of the farms cannot be neglected when the refinery is close to the farm, which means that the size of the farm is not significantly smaller than the transportation distances, which in turn means that the error in approximating the real transportation distance with the distance between the center of the farm and the bio-refinery can be quite large. In this case, transportation should be treated as a region-to-point problem. To this end, we develop a novel approach to biomass procurement planning on a region-to-point basis.

In terms of transportation, we develop a region-to-point modeling approach based on mathematical integration (in a polar coordinate system) over the sourcing region that has unique characteristics such as shape, location, and productivity. The transportation cost is correlated with the amount of biomass (“mass”) procured from each farm. The final result is a function *C* = *f *(*M*), where *M* is the mass, and *C* is the transportation cost. In other words, our method generates a function that returns the total cost of procuring *M* mass, which is graphically represented by a “procurement curve”. Both algebraic and numerical solution methods are studied.

In terms of system-level procurement planning, we develop a composite-curve-based approach that incorporates the regional transportation modeling method, and aims at identifying the biomass procurement plan that minimizes the total procurement cost (including biomass purchasing, harvesting and transportation). The specific steps for the generation of the composite curve using the individual procurement curves, insights into the procurement planning problem, and an analogy to the cold/hot composite curves in the pinch design method for heat exchanger networks are investigated. A case study involving 12 farms (which are approximated as polytopes) and one refinery is presented in Figure 1A. The corresponding composite curve is shown in Figure 2, and the final procurement strategy is graphically represented in Figure 1B.