(778c) Design and Operation of Biofuel Supply Chains with Variable Regional Depot and Biorefinery Locations
Accordingly, in this work, we propose a multi-period mixed-integer linear programming (MILP) model after briefly reviewing a mixed-integer non-linear programming (MINLP) model developed by Ng and Maravelias . The new MILP model accounts for biomass selection and allocation, technology selection and capacity planning at depots and biorefineries, inventory and shipment planning, and, importantly, variable depot and biorefinery locations. Unlike other biofuel SC approaches, which precalculate the distances based on the predetermined depot and biorefinery locations, we develop a series of approximations and reformulations that allow us to overcome the bilinearities (shipments*transportation_distance) in the calculation of transportation cost, based on the following assumptions:
1) The biomass from a harvesting site is sent to only one downstream node, either a depot or a biorefinery; and, similarly, the intermediates from a depot are shipped to a single biorefinery. This is a realistic assumption because it is difficult to have multiple pickups from a harvesting site or a depot and also to â??dynamicallyâ? change where and how much to ship.
2) Biomass and intermediates are stored at the origin (harvesting site and depot, respectively). This assumption has minimal effect because inventory unit costs and material deterioration coefficients are practically independent of location.
3) We approximate the exact timing of the shipment as all biomass from selected harvesting site is eventually shipped to the same depot or biorefinery. Thus, we can replace the variable shipments with the total availability of biomass. A correction factor though is introduced to account for the fact that the actual shipments will be slightly lower than the total availability of biomass due to biomass deterioration during storage.
Based on the above assumptions, we first precalculate the range of correction factors based on the possible inventory profiles. We can choose the average value or a factor that results from a reasonable inventory profile. Second, we disaggregate all variable distance components into â??actualâ? and â??dummyâ? distance variables, which are bounded by variable upper bound constraints using binary arc selection variables. The actual distance variable is equal to zero if the corresponding arc is not selected. Finally, we reformulate the transportation cost, which can be calculated based on the actual distance variables and approximated shipments. The latter are treated as parameters, thereby resulting in a linear transportation cost calculation.
We illustrate the applicability of the proposed model by considering a biofuel SC in South Central Wisconsin that includes Columbia, Dane, Dodge, Grant, Green, Iowa, Jefferson, Lafayette, Richland, Rock, and Sauk counties, with a total of 64 harvesting sites. We consider two types of biomass: corn stover and switchgrass, both of which are abundant biomass resources in Wisconsin. At most three depots can be installed in the studied region. We employ certain criteria for screening the potential biorefinery locations: Candidate biorefinery locations were restricted to cities/towns/villages with populations between 3,000 and 10,000 in 2014 for a maximum of one biorefinery per county  and total biomass availability more than 120,000 dry ton per year. Among the locations that met the screening criteria, we preselect two towns, which are located near the center of the studied region, because they are likely to lead to smaller transportation cost. The coordinates of the biorefinery are constrained by the convex hull of the selected town, therefore, the biorefinery can be installed at various points within the selected region, instead of at the centroid of the region.
The objective is to minimize the total annual cost (TAC), which includes biomass acquisition, inventory, operating and capital costs. The total capital investment for the ethanol production level of 340 million liters per year, is $ 592 million, leading to a TAC of $ 251.44 million. In this case study, two depots are selected, while the biorefinery is installed at West Oregon. The SC configuration with two depots has lower TAC as compared to the one without depots. The transportation cost in the configuration with three depots or more is lower than in the configuration with two depots, but the higher capital costs leads to higher TAC.
We also calculate the relative error of our approximation using the exact biomass/intermediate shipments obtained. The calculated relative error is found to be insignificant. Finally, for comparison purposes, we extend the MINLP model  to account for variable biorefinery locations and resolve the case study. The proposed model is shown to yield better solutions in a faction of the time required by the MINLP model.
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