Optimization of the Algal Biomass to Biodiesel Supply Chain: Case Studies of the State of Oklahoma and the United States

: The goal of this work is to design a supply chain network that distributes algae biomass from supply locations to meet biodiesel demand at speciﬁed demand locations, given a speciﬁed algae species, cultivation (i.e., supply) locations, demand locations, and demand requirements. The ﬁnal supply chain topology includes the optimum sites to grow biomass, to extract algal oil from the biomass, and to convert the algae oil into biodiesel. The objective is to minimize the overall cost of the supply chain, which includes production, operation, and transportation costs over a planning horizon of ten years. Algae production was modeled both within the U.S. State of Oklahoma, as well as the entire contiguous United States. The biodiesel production cost was estimated at $7.07 per U.S. gallon ($1.87 per liter) for the State of Oklahoma case. For the contiguous United States case, a lower bound on costs of $13.68 per U.S. gallon ($3.62 per liter) and an upper bound of $61.69 ($16.32 per liter) were calculated, depending on the transportation distance of algal biomass from production locations. of algae biomass, algae oil, biodiesel being transported between the layers.


Introduction
Biodiesel derived from algal biomass has the potential to provide a renewable fuel source with properties similar to that of traditional diesel fuel [1], while alleviating concerns over petroleum fuels, such as greenhouse gas emissions, scarcity, and volatile feedstock prices. Traditional sources of the oil needed for biodiesel production can, however, lead to competition with existing food crops. Using estimates by [2], it would take nearly 10% of the land area of the Earth to grow the corn needed to replace only half of all transportation fuel with corn-based biofuels. Microalgae based biofuels, however, have the potential to provide a fuel source which can help to solve many of the issues with both petroleum-based fuels and traditional biofuel sources [2]. Microalgae can be grown on marginal farmland using brackish water or saltwater, helping to reduce competition for land and water currently used for food production [3]. The high growth rates and high lipid content of many species also add to the potential of microalgae as a fuel source [3]. Algae biomass also has many other non-fuel applications, such as in food, pigments, and pharmaceutical industries.
The two main methods used for large-scale algae biomass production are enclosed photobioreactors and open ponds [4]. Each of these techniques has its own corresponding advantages and disadvantages. Though open ponds are conceptually easy to design and relatively inexpensive to implement, they have well-documented problems, including evaporation and invasive species contamination [4,5]. Photobioreactors, on the other hand, solve contamination problems but suffer from biomass The model is presented in the next section. Algae production at supply locations is modeled using the pond model of Yadala [8]. This model is then integrated with feedstock logistics, where the produced algae biomass from supply locations is transported to algae oil extraction facilities, and afterward to distribution centers (i.e., demand locations) via biofuel production facilities (i.e., transesterification locations). In this approach, in addition to regional economics, such as land, labor, water, and electricity costs, distances between different operating locations and their associated costs are considered. The method aids in the calculation of production, operating, and transportation costs more accurately.

Mathematical Programming Model
The goal of the algal biomass to biodiesel supply chain network is to meet the specific biodiesel demands of a set of locations with minimum overall cost. The formulation determines the best locations for algae production, algal oil extraction, and biodiesel production given potential algal farming sites, oil extraction locations, and biodiesel production sites along with the optimum number of algae ponds and their design specifications. A detailed listing of sets, parameters, variables included in the formulation is compiled in the nomenclature section at the end of the document. A brief summary of the sets and a detailed overview of the formulation are discussed below. The model is presented in the next section. Algae production at supply locations is modeled using the pond model of Yadala [8]. This model is then integrated with feedstock logistics, where the produced algae biomass from supply locations is transported to algae oil extraction facilities, and afterward to distribution centers (i.e., demand locations) via biofuel production facilities (i.e., transesterification locations). In this approach, in addition to regional economics, such as land, labor, water, and electricity costs, distances between different operating locations and their associated costs are considered. The method aids in the calculation of production, operating, and transportation costs more accurately.

Mathematical Programming Model
The goal of the algal biomass to biodiesel supply chain network is to meet the specific biodiesel demands of a set of locations with minimum overall cost. The formulation determines the best locations for algae production, algal oil extraction, and biodiesel production given potential algal farming sites, oil extraction locations, and biodiesel production sites along with the optimum number of algae ponds and their design specifications. A detailed listing of sets, parameters, variables included in the formulation is compiled in the nomenclature section at the end of the document. A brief summary of the sets and a detailed overview of the formulation are discussed below.

Geographical and Network Sets
The first data set in this multi-echelon supply chain model defines all candidate locations where facilities can be placed, J. From this set, subsets are constructed to correspond to the different operations within the supply chain. Subset J S corresponds to the candidate supply locations for the production of algal biomass. The supply portion of the model is designated by a superscript of S for model variables and parameters. Subset J Ex is constructed of the candidate locations for the extraction of algal oil from the dried biomass. Model variables and parameters pertaining to the extraction operation are indicated by the use of superscript Ex. The third operation within the supply chain, the transesterification of algal oil to biodiesel, is described by the superscript Es. For the potential locations for these operations the subset J Es is used. Finally, the locations of demand are included in subset J D , with variables and parameters associated with the demand denoted by a superscript D. The defined subsets are not mutually exclusive as some locations may appear in multiple subsets.
The supply chain model has four echelons, and hence, three layers, i ∈ I = {1, 2, 3}, that material can be transported from one echelon to the next. Layer 1, connects the supply locations, J S , to the extraction locations, J Ex . Layer 2 connects extraction locations to the sites where the transesterification occurs, J Es . The final layer, layer 3, connects the transesterification locations to the demand locations, J D . There are a number of transportation modes, z ∈ Z = [trucks, rail cars, barges, pipeline], that can be used for moving materials between echelons in each layer. The transportation modes associated with each layer, are created through the subsets z i ∈ Z i ⊂ Z, where i corresponds to the layer. Figure 2 demonstrates the interplay of the supply chain operation locations, transportation layers, and the modes of transportations for each layer. The sets of dates, d ∈ D, and time of the day, t ∈ T, enable tracking the parameters relevant to the growth of algae biomass that is date and time of day dependent, e.g., length of the light path or the solar irradiance.

Geographical and Network Sets
The first data set in this multi-echelon supply chain model defines all candidate locations where facilities can be placed, . From this set, subsets are constructed to correspond to the different operations within the supply chain. Subset corresponds to the candidate supply locations for the production of algal biomass. The supply portion of the model is designated by a superscript of for model variables and parameters. Subset is constructed of the candidate locations for the extraction of algal oil from the dried biomass. Model variables and parameters pertaining to the extraction operation are indicated by the use of superscript . The third operation within the supply chain, the transesterification of algal oil to biodiesel, is described by the superscript . For the potential locations for these operations the subset is used. Finally, the locations of demand are included in subset , with variables and parameters associated with the demand denoted by a superscript . The defined subsets are not mutually exclusive as some locations may appear in multiple subsets.
The supply chain model has four echelons, and hence, three layers, ∈ = 1,2,3 , that material can be transported from one echelon to the next. Layer 1, connects the supply locations, , to the extraction locations, . Layer 2 connects extraction locations to the sites where the transesterification occurs, . The final layer, layer 3, connects the transesterification locations to the demand locations, . There are a number of transportation modes, ∈ = trucks, rail cars, barges, pipeline , that can be used for moving materials between echelons in each layer. The transportation modes associated with each layer, are created through the subsets ∈ ⊂ , where corresponds to the layer. Figure 2 demonstrates the interplay of the supply chain operation locations, transportation layers, and the modes of transportations for each layer. The sets of dates, ∈ , and time of the day, ∈ , enable tracking the parameters relevant to the growth of algae biomass that is date and time of day dependent, e.g., length of the light path or the solar irradiance.

Constraints Related to Supply Sites
At the supply locations, algae biomass is produced in outdoor open channel raceway ponds under the influence of sunlight and temperature fluctuations. To reduce computational burden when running the integrated pond model year long between sunrise and sunset, a simplified approach was followed. All available weather data within each month was approximated to one specific day of that month. This decreased the number of days from 360 to 12 days in a dynamic pond model. The

Constraints Related to Supply Sites
At the supply locations, algae biomass is produced in outdoor open channel raceway ponds under the influence of sunlight and temperature fluctuations. To reduce computational burden when running the integrated pond model year long between sunrise and sunset, a simplified approach was followed. All available weather data within each month was approximated to one specific day of that month. This decreased the number of days from 360 to 12 days in a dynamic pond model. The variables obtained for that one specific day of the month are replicated for all other days in that month. Additional information on pond modeling can be obtained from Yadala [8] and a summary of the model provided in Appendix B of this paper. These constraints are used to calculate the dry algae biomass produced in a year from a single pond at a supply location j ∈ J S , given by the variable f DA j . Equations (1)- (6) below are the constraints related to the pond geometry. They are taken from the pond model and are integrated with the supply chain model.
l Pond Equation (1) requires that for a single-channel raceway pond, total pond width, w Pond j , is twice the channel width, w Ch j . To avoid the flow disturbance caused by the bends of the raceway pond, the ratio of channel length to width should be ten or higher [23] (Equation (2)). Pond length, l Pond j , is the summation of channel length and pond width (Equation (3)). Pond length is constrained to keep the head loss due to friction (Equation A20) lower via Equation (4). The surface area occupied by the pond, A Pond j , is computed from Equation (5). Finally, Equation (6) calculates the volume of the raceway pond, V Pond j . Figure 3 provides a physical representation of the various measurements associated with the raceway pond. variables obtained for that one specific day of the month are replicated for all other days in that month. Additional information on pond modeling can be obtained from Yadala [8] and a summary of the model provided in Appendix B of this paper. These constraints are used to calculate the dry algae biomass produced in a year from a single pond at a supply location ∈ , given by the variable . Equation (1)- (6) below are the constraints related to the pond geometry. They are taken from the pond model and are integrated with the supply chain model.
Equation (1) requires that for a single-channel raceway pond, total pond width, , is twice the channel width, . To avoid the flow disturbance caused by the bends of the raceway pond, the ratio of channel length to width should be ten or higher [23] (Equation (2)). Pond length, , is the summation of channel length and pond width (Equation (3)). Pond length is constrained to keep the head loss due to friction (Equation B.20) lower via Equation (4). The surface area occupied by the pond, , is computed from Equation (5). Finally, Equation (6) calculates the volume of the raceway pond, .  It is assumed that the size of a single pond does not change within each supply location and that it may be necessary to have more than one pond in each supply location to meet the demand. Total surface area, , occupied by the ponds at each supply location is calculated by multiplying the number of ponds, , with the surface area of a single pond at the supply location, according to (Equation (7)). This is done, as it is assumed that every pond at one location shares the same dimensions. The variable cannot exceed the marginal farmland available, Λ , in that region (Equation (8)). Equation (9) ensures that the surface area of a single raceway pond does not exceed the maximum allowable single pond area at that location, Λ . Equation (10) ensures that when there are no ponds at any supply location, the surface area would be zero. It is assumed that the size of a single pond does not change within each supply location and that it may be necessary to have more than one pond in each supply location to meet the demand. Total surface area, A Tot j , occupied by the ponds at each supply location is calculated by multiplying the number of ponds, N Pond j , with the surface area of a single pond at the supply location, according to (Equation (7)). This is done, as it is assumed that every pond at one location shares the same dimensions. The variable A Tot j cannot exceed the marginal farmland available, Λ Tot j , in that region (Equation (8)). Equation (9) ensures that the surface area of a single raceway pond does not exceed the . Equation (10) ensures that when there are no ponds at any supply location, the surface area would be zero.
The sum of all the shipment, o 1 z,j S ,j Ex , via any method of transportation, z ∈ Z 1 , from a supply location, j S ∈ J S , to all extraction facilities, j Ex ∈ J Ex , in transportation layer 1 should not exceed the total dry algae production at a supply location, which is obtained by multiplying the dry algae biomass, DA, produced in a year from a single pond, f DA j S , with the number of such ponds at the location, viz. Equation (11). The total amount of dry algae shipped from all supply locations, j S ∈ J S , to an extraction facility via any method of transportation, is defined as the dry algae being transported to, O Ex j Ex , and is calculated using Equation (12).

Constraints Related to Distribution Sites
These constraints are written for extraction locations, j Ex ∈ J Ex , and transesterification locations, j Es ∈ J Es , and they correspond to the second transportation layer of the model. Algae oil is extracted from the biomass at extraction locations, and it is converted to biodiesel via transesterification at transesterification locations.
At an extraction facility, j ∈ J Ex , depending on the extraction efficiency, η Ex , and oil content of the algae species, Ψ s , the amount of algae oil produced, F Ex j , is calculated with Equation (13). The total amount of algae oil shipped, o 2 z,j Ex ,j Es , from an extraction facility, j Ex ∈ J Ex , to various transesterification facilities, j Es ∈ J Es , via any method of transportation, z ∈ Z 2 , cannot exceed the total algae oil extracted, F Ex j Ex , at j Ex ∈ J Ex (Equation (14)). The total amount of algae oil shipped from all extraction facilities, j Ex ∈ J Ex , to a transesterification facility, j ES ∈ J ES , is equal to algae oil transported to transesterification location, O Es j Es , as shown in Equation (15).
At the transesterification facilities, biodiesel is produced via a transesterification reaction, which is given in Equation (16). For this model, the overall yield of the transesterification process is defined using transesterification efficiency, η Es . The transesterification efficiency, which by definition should be between zero and one, is the efficiency of conversion of algae oil to biodiesel. The amount of biodiesel produced, F Es j , at transesterification location j ∈ J Es , can be calculated using Equation (16). Here, MW biodiesel is the molecular weight of biodiesel [24] and MW lipid is the molecular weight of lipids [25]. Equation (17) shows that the total amount of shipment, o 3 z,j Es ,j D , from a transesterification

Constraints Related to Demand Locations
The total amount of biodiesel shipped from all transesterification facilities meets the demand, δ j D at demand location j D ∈ J D . Equation (18) ensures that biodiesel demand at each demand location is satisfied. Figure 4 shows the network flow topology of the model. Here, J S represents the algae biomass production or supply locations labeled as 1 through 5. J Ex represents extraction sites where algae oil is extracted. J Es represents transesterification sites where biodiesel is produced. The J Ex and J Es considers both supply and demand locations together to include all possible combinations of locations. J D represents demand locations of biodiesel.

Constraints Related to Demand Locations
The total amount of biodiesel shipped from all transesterification facilities meets the demand, at demand location ∈ . Equation (18) ensures that biodiesel demand at each demand location is satisfied. Figure 4 shows the network flow topology of the model. Here, represents the algae biomass production or supply locations labeled as 1 through 5.
represents extraction sites where algae oil is extracted.
represents transesterification sites where biodiesel is produced. The and considers both supply and demand locations together to include all possible combinations of locations.
represents demand locations of biodiesel.

Constraints Related to Transportation
Different materials are shipped through each layer of arcs in Figure 4. For example, dry algae biomass, , , , is shipped from supply locations to extraction locations through layer one. The algae oil, , , , from extraction facilities is shipped to transesterification facilities in layer two.
Finally, biodiesel, , , , from transesterification facilities is shipped to demand locations through layer three. Two different means of transportation are considered for each layer, shipment via land or water. The number of such transportation methods, ( , , ), required to ship the products between each layer depends on the capacity, Ξ , of the method, ∈ , for layer ∈ , and density of the product being shipped. These relationships are enforced via Equation (19)- (21).

Constraints Related to Transportation
Different materials are shipped through each layer of arcs in Figure 4. For example, dry algae biomass, o 1 z,j S ,j Ex , is shipped from supply locations to extraction locations through layer one. The algae oil, o 2 z,j Ex ,j Es , from extraction facilities is shipped to transesterification facilities in layer two. Finally, biodiesel, o 3 z,j Es ,j D , from transesterification facilities is shipped to demand locations through layer three. Two different means of transportation are considered for each layer, shipment via land or water. The number of such transportation methods, (N i z,j Source ,j Sink ), required to ship the products between each layer depends on the capacity, Ξ z i , of the method, z i ∈ Z i , for layer i ∈ I, and density of the product being shipped. These relationships are enforced via Equations (19)- (21).
Here, ρ DryAlgae , ρ AlgaeOil , and, ρ Biodiesel are the densities of dry algae biomass (kt m −3 ), algae oil (kt m −3 ), and biodiesel (kt m −3 ), respectively. The entire supply chain network and the related variables associated with it are depicted in Figure 5.

Objective Function
The objective is to minimize the overall cost, , of biodiesel supply chain network, presented in Equation (22).
The Minimum Acceptable Rate of Return (MARR) is 15%. Here, and are the capital and operating costs of raceway pond. They are calculated, scaling linearly, depending on the total surface area of the pond through Equation (23),(24), where, is the total capital investment, and is the total product cost per pond area (which are given in Yadala [8]).
Capital and operating costs, and , for extraction of algae oil are assumed to change linearly with the total oil production at a site, and are estimated by Equation (25), (26), where, , and , are the capital and operating cost coefficients for the selected algal oil extraction process.
Assuming capital and operating costs, and , for transesterification of biodiesel

Objective Function
The objective is to minimize the overall cost, TC, of biodiesel supply chain network, presented in Equation (22).
The Minimum Acceptable Rate of Return (MARR) is 15%. Here, CC S and OC S are the capital and operating costs of raceway pond. They are calculated, scaling linearly, depending on the total surface area of the pond through Equations (23) and (24), where, ξ C is the total capital investment, and ξ O is the total product cost per pond area (which are given in Yadala [8]).
Capital and operating costs, CC Ex and OC Ex , for extraction of algae oil are assumed to change linearly with the total oil production at a site, and are estimated by Equations (25) and (26), where, ψ C,Ex and ψ O,Ex are the capital and operating cost coefficients for the selected algal oil extraction process.
Assuming capital and operating costs, CC Es and OC Es , for transesterification of biodiesel changes linearly with the amount of biodiesel produced at a location, these costs are calculated using Equations (27) and (28).
Here, ψ C,Es and ψ O,Es are the capital and operating cost coefficients for the selected transesterification process.
Land cost, LC, considers the purchase (or lease) cost, χ Land j , of the required land area for algae cultivation, and only considers the required surface area for the ponds at location j ∈ J S . It is calculated using Equation (29). Water cost, WC, is calculated based on the total amount of industrial water, V Ind j , required for algal cultivation in a single pond, number of such ponds, and the utility cost of water, χ Water j , at supply location j ∈ J S . This is shown in Equation (30) Mixing and pumping costs, MC and PC, associated with raceway pond are estimated from Equations (31) and (33) using total energy required for mixing and pumping, and electric cost, χ Elect j , at the respective supply location j ∈ J S . Total mixing and pumping energy are calculated from the total power requirements for all ponds at the supply location (Equations (32)-(34)). In Equation (32), the power required by paddle wheel of raceway pond, PW j,d,t , for location j ∈ J S for all representative days d ∈ D for all times of day t ∈ T, is calculated from Equation (A43) in Appendix B. In Equation (34), the power required by pumps in raceway pond, PP j,d,t , for location j ∈ J S for all representative days d ∈ D for all times of day t ∈ T, is calculated using Equation (A44) in Appendix B.
Transportation cost, TrC, for the shipment of dry algae, algae oil, and biodiesel that are associated with the first, second, and third layers of the supply chain network is detailed in Equation (35). The costs for each transportation layer are the product of the distance, γ z i ,j Source ,j Sink , from the source location, j Source ∈ J, to the sink location, j Sink ∈ J, using transportation mode z i ∈ Z i , and the cost per distance, φ i,z i , to use transportation mode z i ∈ Z i for all the transportation layers i ∈ I.

State of Oklahoma
The first case examined is the supply chain network for algae biodiesel production in the counties of the state of Oklahoma ( Figure 6). The algae species, I. galbana is chosen for use within the raceway style ponds dues to their allowance of higher light absorption withing the pond, which drives down costs [8]. The demand and supply locations for the state are identified using several criteria. The counties were first eliminated as potential demand centers based on population density at 100 persons/mi 2 (38.6 persons/km 2 ). This cutoff is selected because there is a sudden jump in population densities from 90.34 people/mi 2 to 115.48 people/mi 2 . Combined, the nine remaining counties, shown in pink in Figure 7, account for more than 56% of the population of the entire state. The number of demand locations may be further reduced by looking at how fuel terminals are currently spread throughout the state. In Figure 7, these are shown as counties with solid blue squares. Because of this, the demand counties are grouped into regions that may each be served by a single terminal, e.g., counties in a metropolitan area can be served by the central city. For example, we assume that biodiesel deliveries for the Tulsa, Washington, Rogers, and Wagoner county region will be delivered to the final supplier in Tulsa County, the location of the principal regional city. The fuel demand of the reduced regions is presented in Table 1 The supply locations for growing algae biomass is determined based on two criteria. The first is the amount of marginal farmland in each county. We first define total county farmland using the United States Department of Agriculture (USDA) National Agricultural Statistics Service (NASS) 2012 Census of Agriculture statistic 'Land in farms' found within the report [27]. We further define marginal cropland as per Equation (36). The counties were first eliminated as potential demand centers based on population density at 100 persons/mi 2 (38.6 persons/km 2 ). This cutoff is selected because there is a sudden jump in population densities from 90.34 people/mi 2 to 115.48 people/mi 2 . Combined, the nine remaining counties, shown in pink in Figure 7, account for more than 56% of the population of the entire state. The number of demand locations may be further reduced by looking at how fuel terminals are currently spread throughout the state. In Figure 7, these are shown as counties with solid blue squares. Because of this, the demand counties are grouped into regions that may each be served by a single terminal, e.g., counties in a metropolitan area can be served by the central city. For example, we assume that biodiesel deliveries for the Tulsa, Washington, Rogers, and Wagoner county region will be delivered to the final supplier in Tulsa County, the location of the principal regional city. The fuel demand of the reduced regions is presented in Table 1 demand of the reduced regions is presented in Table 1 The supply locations for growing algae biomass is determined based on two criteria. The first is the amount of marginal farmland in each county. We first define total county farmland using the United States Department of Agriculture (USDA) National Agricultural Statistics Service (NASS) 2012 Census of Agriculture statistic 'Land in farms' found within the report [27]. We further define marginal cropland as per Equation (36).  [26]. Counties outlines in pink indicate those with a population density greater than 100 mi −1 and are taken to be the demand centers. Blue squares indicate existing fuel terminals.  [26]. Counties outlines in pink indicate those with a population density greater than 100 mi −1 and are taken to be the demand centers. Blue squares indicate existing fuel terminals. The supply locations for growing algae biomass is determined based on two criteria. The first is the amount of marginal farmland in each county. We first define total county farmland using the United States Department of Agriculture (USDA) National Agricultural Statistics Service (NASS) 2012 Census of Agriculture statistic 'Land in farms' found within the report [27]. We further define marginal cropland as per Equation (36).
In Equation (36), each variable corresponds to the same-named column in the 2012 Census of Agriculture.
In-use cropland is then defined as the total cropland minus the marginal cropland. Using this definition, the total county marginal farmland can be defined per Equation (37), where, again, other than Cropland In−use and Farmland Marginal , each variable corresponds to the same-named column in the 2012 Census of Agriculture. All 77 counties of Oklahoma are ranked in descending order according to the amount of marginal farmland. The second criterion eliminates counties based on the average well depth in each county, arising from the assumption that water will be obtained through wells in this case study. In Oklahoma, whenever a well is dug, it is required that well logs be submitted to the Oklahoma Water Resource Board within sixty days [28]. These records are collected and digitized within the groundwater well data set [29]. After removing wells not in the irrigation, commercial, industrial, or public water supply categories (i.e., those used for home water supplies which do not require as much drawdown as wells with higher flow rates), the average well depth for each county was calculated, and counties were ranked in ascending order from shallowest to deepest. Those counties ranking in the top 25% for both measures were taken as the set of supply counties, outlined in green in Figure 8. The available marginal farmland in each of these counties is given in Table 2, along with the cost of land in each [30]. The average cost of electricity for 2013 was 5.43 cents/kWh [31], while the average energy cost for irrigation water was 1.97 cents/1000 U.S. gallons [27]. All of the supply centers and demand regions were assumed to be available for the two remaining steps in the algae biomass to biodiesel supply chain, oil extraction, and transesterification. Taken together, a supply chain network as shown in Figure 9 was created.  Table 2, and demand regions ( ) are those in Table  1. Both sets of locations are used as possible sites for extraction ( ) and transesterification ( ).
The mode of transportation considered between all layers for this case is trucks. The capacity of the truck, Ξ , is taken as 30 m 3 [32]. All the distances between supply, extraction, transesterification, and demand locations are given in Appendix A in Table A1.

United States of America
The second case examined is that of the algal biomass to biodiesel supply chain for the contiguous United States. As with the case of the state of Oklahoma, two criteria, historical weather data and availability of marginal farmland, were used to determine the potential algae biomass supply locations, and the utilized algae species is once again I. galbana. Average monthly temperatures from 1971 to 2000 were examined using the National Oceanic and Atmospheric Administration (NOAA) average mean temperature index by month [33], and the states with historical average temperatures below freezing were excluded from consideration as potential algae suppliers. This reduced the number of supply locations to 19. In addition, the amount of marginal  All of the supply centers and demand regions were assumed to be available for the two remaining steps in the algae biomass to biodiesel supply chain, oil extraction, and transesterification. Taken together, a supply chain network as shown in Figure 9 was created. All of the supply centers and demand regions were assumed to be available for the two remaining steps in the algae biomass to biodiesel supply chain, oil extraction, and transesterification. Taken together, a supply chain network as shown in Figure 9 was created.  Table 2, and demand regions ( ) are those in Table  1. Both sets of locations are used as possible sites for extraction ( ) and transesterification ( ).
The mode of transportation considered between all layers for this case is trucks. The capacity of the truck, Ξ , is taken as 30 m 3 [32]. All the distances between supply, extraction, transesterification, and demand locations are given in Appendix A in Table A1.  Table 2, and demand regions (J D ) are those in Table 1. Both sets of locations are used as possible sites for extraction (J Ex ) and transesterification (J Es ). The mode of transportation considered between all layers for this case is trucks. The capacity of the truck, Ξ truck , is taken as 30 m 3 [32]. All the distances between supply, extraction, transesterification, and demand locations are given in Appendix A in Table A1.

United States of America
The second case examined is that of the algal biomass to biodiesel supply chain for the contiguous United States. As with the case of the state of Oklahoma, two criteria, historical weather data and availability of marginal farmland, were used to determine the potential algae biomass supply locations, and the utilized algae species is once again I. galbana. Average monthly temperatures from 1971 to 2000 were examined using the National Oceanic and Atmospheric Administration (NOAA) average mean temperature index by month [33], and the states with historical average temperatures below freezing were excluded from consideration as potential algae suppliers. This reduced the number of supply locations to 19. In addition, the amount of marginal farmland available is calculated in much the same way as was done with the Oklahoma case, with state values used instead of individual counties. The remaining states were ranked based on the availability of marginal farmland and the top 50% were taken as potential supply locations, given in the first column of Table 3.  Table 3 also shows the state's average farm real estate values as determined by the USDA NASS [27]. Water cost was taken as the average energy expenses for irrigation water in each state, again as determined by the USDA NASS [27], assuming that the average acre of cropland requires 10,000 m 3 of water per year [31]. Each supply location has a port city, chosen because of its connectivity to truck, rail, pipeline, and in most cases, barge transportation modes. It was assumed that all algae biomass left each supply state through this port city. Because of this, the Google maps [34] distance between the geographic center of the state and the port city was used to represent the average distance between farms in the state and the port. Weather data (minimum and maximum temperatures, relative humidity, and wind velocity) was calculated using the Mathematica WeatherData database [35].
It was assumed that all deliveries would be made to the port cities in the demand location. Both supply port cities and demand port cities were considered as possible locations for extraction and transesterification. Tables 3-5 list all the supply locations, port locations, extraction locations, transesterification locations, and demand locations. Those states with the highest 10% of diesel sales were taken as demand locations [36] and are shown in the first column of Table 5. Figure 10 shows geographically the supply states (outlined in green) and demand states (filled with red).   Table 3 also shows the state's average farm real estate values as determined by the USDA NASS [27]. Water cost was taken as the average energy expenses for irrigation water in each state, again as determined by the USDA NASS [27], assuming that the average acre of cropland requires 10,000 m 3 of water per year [31]. Each supply location has a port city, chosen because of its connectivity to truck, rail, pipeline, and in most cases, barge transportation modes. It was assumed that all algae biomass left each supply state through this port city. Because of this, the Google maps [34] distance between the geographic center of the state and the port city was used to represent the average distance between farms in the state and the port. Weather data (minimum and maximum temperatures, relative humidity, and wind velocity) was calculated using the Mathematica WeatherData database [35].
It was assumed that all deliveries would be made to the port cities in the demand location. Both supply port cities and demand port cities were considered as possible locations for extraction and transesterification. Tables 3-5 list all the supply locations, port locations, extraction locations, transesterification locations, and demand locations. Those states with the highest 10% of diesel sales were taken as demand locations [36] and are shown in the first column of Table 5. Figure 10 shows geographically the supply states (outlined in green) and demand states (filled with red).   While the Oklahoma case only considered transportation by truck, the case of the United States considers rail, barge, and pipeline transport. Therefore, road distances were taken as the shortest route found using Google maps between locations [34]. Rail distances were measured using the U.S. Department of Transportation's Bureau of Transportation Statistics National Transportation Atlas Database of Railway Networks [37], with the nearest railway to the shortest Google maps road route taken as the shortest rail route. Barge distances were calculated as distances between ports [38].
Pipeline distances were taken using the U.S. Energy Information Administration (EIA) petroleum product pipeline database, with the nearest pipeline to the shortest Google maps road route taken as the shortest pipeline route [39]. All the distances associated with trucks, rails, barges, and pipelines between supply, port, extraction, transesterification, and demand locations are given Appendix A via Tables A2-A12. The entire system of transportation options is shown in Figure 11. It should be noted that pipeline is only taken as a transportation route between the transesterification and demand locations, as the network of pipelines examined is that used for transportation of products, rather than crude oil. To investigate the impact of spacial disconnect of the major shipping hubs from land that would be used to grow the algae, Layer 0 is introduced for this implementation of the model. The introduction of layer 0 necessitates the introduction of variables N 0 Truck,j S ,j Port , o 0 Truck,j S ,j Port , and O Port j Port with the associated constraints to account for the number of trucks shipping product from supply location j S ∈ J S to port location j Port ∈ J Port , the amount of product shipped from supply location j S ∈ J S to port location j Port ∈ J Port , and the total amount of product at port location j Port ∈ J Port , respectively. For Layer 0, to transport dry algae biomass, only trucks were considered; for layers 1 and 2, trucks, rails, barges were considered; and for layer 3, pipelines were added to carry the biodiesel along with the other modes. The capacities, Ξ , for ∈ truck, rail, barge are 30 m 3 , 113.56 m 3 , and 1192 m 3 , respectively. Here, the capacity of the pipeline is equivalent to the number of pipelines, i.e., it is equal to one. It should be noted that there is no barge transportation to and from Phoenix and Los Angeles. Therefore, additional constraints must be added to the variables associated with barge transportation for these cities to constrain products shipped to and from these cities using barges to zero, see Equation (38)-(43).

Solution Approach
The resulting mixed-integer nonlinear program (MINLP) is a large scale non-convex problem, and it is implemented in GAMS Version 24.7.1 using an Intel ® Xeon ® E5-2650 v3, 2.30 GHz processor running Windows 10. The only integer variable is the number of ponds, , at each supply locations ∈ . This MINLP is unable to be solved using global MINLP solvers: ANTIGONE version 1.1, and BARON version 18.11.15. We also attempted to solve this problem using a local MINLP solver without success. However, relaxing the integrality constraints yields a non-convex nonlinear For Layer 0, to transport dry algae biomass, only trucks were considered; for layers 1 and 2, trucks, rails, barges were considered; and for layer 3, pipelines were added to carry the biodiesel along with the other modes. The capacities, Ξ z , for z ∈ truck, rail, barge are 30 m 3 , 113.56 m 3 , and 1192 m 3 , respectively. Here, the capacity of the pipeline is equivalent to the number of pipelines, i.e., it is equal to one. It should be noted that there is no barge transportation to and from Phoenix and Los Angeles. Therefore, additional constraints must be added to the variables associated with barge transportation for these cities to constrain products shipped to and from these cities using barges to zero, see Equations (38)-(43).

Solution Approach
The resulting mixed-integer nonlinear program (MINLP) is a large scale non-convex problem, and it is implemented in GAMS Version 24.7.1 using an Intel ® Xeon ® E5-2650 v3, 2.30 GHz processor running Windows 10. The only integer variable is the number of ponds, N Pond j , at each supply locations j ∈ J S . This MINLP is unable to be solved using global MINLP solvers: ANTIGONE version 1.1, and BARON version 18.11.15. We also attempted to solve this problem using a local MINLP solver without success. However, relaxing the integrality constraints yields a non-convex nonlinear programming (NLP) formulation, which will be referred to as the relaxed-MINLP, that is solvable. The optimum solution of the relaxed-MINLP provides a lower bound for the original MINLP. Unfortunately, global solvers (BARON version 18.11.15 and ANTIGONE version 1.1) and the local solver (DICOPT) were not able to solve the problem to optimality. Therefore, the relaxed-MINLP is solved using CONOPT 3 version 3.17A with a multi-start approach. Although this approach does not guarantee that the optimum solution of the relaxed-MINLP is obtained, it allows to generate the reasonable values of  Table 6 shows the objective function values of the relaxed-MINLP problem and the MILP problem at the first iteration. The main difference between the relaxed-MINLP and MILP objective function values stem from the number of ponds, N Pond j , required at each location j ∈ J S , which is a fractional value for the relaxed-MINLP solution. Hence, the capital, CC S , and operating costs, CO, associated with raceway ponds, and the transportation cost, TrC, change slightly to account for the small amount of algal biomass produced at locations with fractional ponds. Table 7 shows the computational statistics for the Oklahoma case, including the model size and solution time. The solution time for the MINLP shows 'N/A' as the solvers used were unable to find a solution. Table 6. Comparison of the solutions from relaxed-mixed-integer nonlinear program (MINLP) and mixed-integer linear programming (MILP) solvers: CONOPT 3 version 3.17A and CPLEX version 12.6.3.0 for the case of Oklahoma. The resulting supply chain topology is summarized in Figure 12, which shows the supply, extraction, and transesterification locations selected to meet the demand. This figure also shows the number of trucks needed to transport the algae oil from the extraction to the transesterification sites, as well as providing the raceway pond dimensions, such as pond depth, channel width, and pond length; number of such ponds required to meet the demand; and area of algae cultivation farmland necessary to meet the demand. It can be observed that more than 72% of the available marginal farmland of Kay County must be used for the cultivation of algae biomass to meet the biodiesel demand. When the total cost is minimized, it was found that the demand in all of the state of Oklahoma demand regions can be met with 118,843 ponds in Kay County. The algal biomass produced in Kay County is then extracted in Kay County. Next, the algae oil is shipped for transesterification at each of the demand counties in the amount needed to produce biodiesel to meet the demand in that particular county. A total of 19,096 trucks are needed to transport the algae oil. Additionally, the total fuel consumption of the trucks needed to transport algal oil was calculated and is presented in Table 8. Two different means were used to calculate the fuel consumption: a flat rate of fuel consumption measured in gal/km traveled [40] and a weight-based fuel consumption accounting for the total weight of the truck measured in gal/(100-t km) [41]. The total fuel consumption is less than a percent of the fuel demand of Oklahoma, showing little to no impact on the total demand and profitability of the supply chain. This is due to the relatively short distances traveled and the small number of trucks needed for the transportation of the fuel or its precursors. and is presented in Table 8. Two different means were used to calculate the fuel consumption: a flat rate of fuel consumption measured in gal/km traveled [40] and a weight-based fuel consumption accounting for the total weight of the truck measured in gal/(100-t km) [41]. The total fuel consumption is less than a percent of the fuel demand of Oklahoma, showing little to no impact on the total demand and profitability of the supply chain. This is due to the relatively short distances traveled and the small number of trucks needed for the transportation of the fuel or its precursors.   Figure 13 shows how biomass concentration changes during the course of the day from sunrise to sunset. It can be observed that, in one representative day of a month, biomass concentration gradually increases from sunrise until it reaches sunset. This is because of the accumulation of biomass with the time of the day or until harvest. The biomass concentration and production are rapid in the summer months of June, July, August, and September. However, July was found to be the favorable month for the species I. galbana because optimal conditions for growth exist during that month for the Kay County location.   Figure 13 shows how biomass concentration changes during the course of the day from sunrise to sunset. It can be observed that, in one representative day of a month, biomass concentration gradually increases from sunrise until it reaches sunset. This is because of the accumulation of biomass with the time of the day or until harvest. The biomass concentration and production are rapid in the summer months of June, July, August, and September. However, July was found to be the favorable month for the species I. galbana because optimal conditions for growth exist during that month for the Kay County location.  Figure 13 shows how biomass concentration changes during the course of the day from sunrise to sunset. It can be observed that, in one representative day of a month, biomass concentration gradually increases from sunrise until it reaches sunset. This is because of the accumulation of biomass with the time of the day or until harvest. The biomass concentration and production are rapid in the summer months of June, July, August, and September. However, July was found to be the favorable month for the species I. galbana because optimal conditions for growth exist during that month for the Kay County location.   Figure 14 shows the breakdown of the overall cost. The supply chain network has a total cost of $9.921 billion over ten years, at a per U.S. gallon cost of $7.07 (corresponding to $1.87 per liter). Of the cost, 46% is due to the capital costs associated with raceway ponds, and 19% is due to the raceway pond operating cost. The capital and operating costs for extraction and transesterification each make up 8% of the cost. The cost of transportation is relatively small compared to the overall supply chain cost (and compared to the transportation costs of the United States case) due to the relatively short distances over which the algae oil is transported.

Processes 2020, 8, x FOR PEER REVIEW
19 of 39 Figure 14 shows the breakdown of the overall cost. The supply chain network has a total cost of $9.921 billion over ten years, at a per U.S. gallon cost of $7.07 (corresponding to $1.87 per liter). Of the cost, 46% is due to the capital costs associated with raceway ponds, and 19% is due to the raceway pond operating cost. The capital and operating costs for extraction and transesterification each make up 8% of the cost. The cost of transportation is relatively small compared to the overall supply chain cost (and compared to the transportation costs of the United States case) due to the relatively short distances over which the algae oil is transported. Figure 14. The cost associated with the Oklahoma algae biomass to biodiesel supply chain network. Table 9 shows the objective function values of the relaxed-MINLP problem and the MILP problem at the first iteration. The relative optimality gap is zero, and hence no iterations were required. It should be noted that this solution is a local optimum for this problem. The main difference between the relaxed-MINLP and MILP objective function values stem from the number of ponds, , required at each location ∈ , which is a fractional value for the relaxed-MINLP solution. Hence, the capital, , and operating costs, , associated with raceway ponds, and the transportation cost, , change slightly to account for the small amount of algal biomass produced at locations with fractional ponds. Table 10 shows the computational statistics for the USA case, including the model size and the solution time. The solution time for the MINLP shows 'N/A' as the solvers used were unable to find a solution.   Table 9 shows the objective function values of the relaxed-MINLP problem and the MILP problem at the first iteration. The relative optimality gap is zero, and hence no iterations were required. It should be noted that this solution is a local optimum for this problem. The main difference between the relaxed-MINLP and MILP objective function values stem from the number of ponds, N Pond j , required at each location j ∈ J S , which is a fractional value for the relaxed-MINLP solution. Hence, the capital, CC S , and operating costs, OC S , associated with raceway ponds, and the transportation cost, TrC, change slightly to account for the small amount of algal biomass produced at locations with fractional ponds. Table 10 shows the computational statistics for the USA case, including the model size and the solution time. The solution time for the MINLP shows 'N/A' as the solvers used were unable to find a solution. The resulting supply chain topology is summarized in Figure 15, which shows the supply, port, extraction, and transesterification locations selected to meet the demand and the method of transportation selected for distributing the products from one location to another. The figure shows the number of trucks, barges, and pipelines required to carry the products. Notice that there are no transport vehicles between the same locations. The figure also provides the single raceway pond dimensions, such as pond depth, channel width, pond length, number of such ponds required to meet the demand, and the area they occupy. It was found that in order to meet the biodiesel demand, the state of Mississippi uses more than 65% of the available marginal farmland for the cultivation of algae biomass. The total fuel consumption of the supply chain was once again calculated and is presented in Table 11. The first column shows the fuel consumption using a flat-rate calculation [40,42], and the second column shows the fuel consumption on a weight-based calculation [41]. The relatively large range of fuel consumption is due to the difference in calculating the fuel consumption of the barges. When using a weight-based calculation, barges are projected to consume two to three times less fuel than when using a flat-rate calculation. The inverse is true for fuel consumption of trucks. Nevertheless, both means of calculations reveal that a sizeable amount of fuel is used for transportation within the supply chain. Although the need for fuel to transport the products of this supply chain is not close to the overall demand for diesel, the results suggest that it may impose an economic burden on this supply chain. of the barges. When using a weight-based calculation, barges are projected to consume two to three times less fuel than when using a flat-rate calculation. The inverse is true for fuel consumption of trucks. Nevertheless, both means of calculations reveal that a sizeable amount of fuel is used for transportation within the supply chain. Although the need for fuel to transport the products of this supply chain is not close to the overall demand for diesel, the results suggest that it may impose an economic burden on this supply chain.  Inside the pond, there are dynamic changes occurring in biomass concentration. Figure 16 shows how biomass concentration changes during the course of the day from sunrise to sunset. It can be observed that in one representative day of a month, biomass concentration gradually increases from  Inside the pond, there are dynamic changes occurring in biomass concentration. Figure 16 shows how biomass concentration changes during the course of the day from sunrise to sunset. It can be observed that in one representative day of a month, biomass concentration gradually increases from sunrise to sunset. However, September was found to be the favorable month for the species I. galbana because optimal conditions for growth exist during that month and Mississippi location. sunrise to sunset. However, September was found to be the favorable month for the species I. galbana because optimal conditions for growth exist during that month and Mississippi location.  Figure 17 shows the amounts of products transported between different layers. The solution reveals that among the various supply locations considered (Texas, Mississippi, Alabama, Kentucky, Georgia, Oklahoma, Virginia, Arizona, North Carolina, and South Carolina), Mississippi was selected for algae cultivation based on the model parameters, such as availability of farmland area and weather data. This location has favorable conditions for the cultivation of algae biomass, together with the high availability of marginal farmland compared to the other states. As per our model assumptions, dry algae biomass from each supply location is transported to their respective port cities via trucks. From the available ten supply-port combinations (Texas-Houston, Mississippi-Gulfport, Alabama-Mobile, Kentucky-Paducah, Georgia-Savannah, Oklahoma-Tulsa, Virginia-Norfolk, Arizona-Phoenix, North Carolina-Wilmington, and South Carolina-Charleston), since algae cultivation occurs only in Mississippi, algae biomass was shipped to its respective port city of Gulfport (    Figure 18 shows the breakdown of the overall cost. The price for a gallon of biodiesel was calculated as $61.69/U.S. gallon ($16.32 per liter). It can be observed that 78% of the total production cost comes from transporting the products between various locations. Out of this percentage, about 85% is contributed solely by transportation via trucks where dry algae is transported from supply to port cities. An additional 6% is contributed by transportation via barges to transport algae oil from extraction facilities to transesterification facilities, and the remaining is contributed by transportation via pipelines to carry biodiesel to demand centers. One important recommendation from this work to lower these costs would be to consider the supply and demand centers within each individual state (in a manner similar to the Oklahoma case) rather than the whole United States. For the United States case, we assumed that the distance between the center of the state and the port location in supply states is an appropriate approximation of the distance between algae farms and the port. The results recommend a total area of 9,832 km 2 for algae ponds. This area roughly corresponds to a 100 km × 100 km 2 . In contrast, the distance between the center of the Mississippi State and Gulfport is 314 km. Therefore, the assumed distances may be a gross overestimate for the distance algal biomass is shipped for processing. We decided to investigate how the supply chain topology and per-gallon cost of biodiesel would change if the transportation cost of algal biomass from farms to port locations can be avoided. The distances between supply locations and port locations for the United States case were set to zero to model this case, and the resulting mathematical program was solved.

United States
It can be seen from Table 12, which compares the relaxed-MINLP and MILP solutions, that the total cost is much lower than that of the first United States case. From Figure 19, it can be observed that the cost of transportation is reduced to 30% of the total cost. The design of the supply chain and the values of the As per our model assumptions, dry algae biomass from each supply location is transported to their respective port cities via trucks. From the available ten supply-port combinations (Texas-Houston, Mississippi-Gulfport, Alabama-Mobile, Kentucky-Paducah, Georgia-Savannah, Oklahoma-Tulsa, Virginia-Norfolk, Arizona-Phoenix, North Carolina-Wilmington, and South Carolina-Charleston), since algae cultivation occurs only in Mississippi, algae biomass was shipped to its respective port city of Gulfport (o 1 Truck,Mississippi,Gulfport ) = 212,390 kt) via trucks. The solution suggests that among the choices of all the port and demand locations, algae oil is extracted at the port city where biomass was shipped. This route was selected because the transportation costs between layers with the same locations are zero. Part of the extracted algae oil at the Gulfport, MS, USA, is processed to biodiesel at its current location (5857 kt), and the remainder is shipped to Houston, TX, USA (o 2 Barge,Gulfport,Houston = 26,807 kt), and Paducah, KY, USA (o 2 Barge,Gulfport,Paducah = 9618 kt), via barges for further processing into biodiesel. Upon further investigation on the choice of barges over the other methods of transportation for transporting algae oil, it was found that although the cost of shipping through barges was expensive, the distances between barge terminals were lower compared to road and rail distances. In addition, the capacity of an individual barge is much higher compared to trucks and rails.
Biodiesel produced at Houston, TX, USA, satisfies the demand for Houston, TX, USA (24,038 kt), and Los Angeles, CA, USA, (o 3 Pipeline,Houston,LosAngeles = 12,570 kt), and the biodiesel is shipped via the available pipeline between Houston, TX, USA, and Los Angeles, CA, USA. The biodiesel produced at Gulfport, MS, USA, is transported via pipeline to Philadelphia, PA, USA (o 3 Pipeline,Gulfport,Philadelphia = 7999 kt), to meet the local demand. Biodiesel produced at Paducah, KY, USA, satisfies the biodiesel demand for Chicago, IL, USA (o 3 pipeline,Paducah,Chicago = 6518 kt), and Toledo, OH, USA (o 3 pipeline,Paducah,Toledo = 6,617 kt), and biodiesel is transported to both locations via pipelines. Pipelines were selected for the transportation of biodiesel to demand centers because, among all the other methods of transportation, pipelines were the cheapest means of transportation available. Figure 18 shows the breakdown of the overall cost. The price for a gallon of biodiesel was calculated as $61.69/U.S. gallon ($16.32 per liter). It can be observed that 78% of the total production cost comes from transporting the products between various locations. Out of this percentage, about 85% is contributed solely by transportation via trucks where dry algae is transported from supply to port cities. An additional 6% is contributed by transportation via barges to transport algae oil from extraction facilities to transesterification facilities, and the remaining is contributed by transportation via pipelines to carry biodiesel to demand centers. One important recommendation from this work to lower these costs would be to consider the supply and demand centers within each individual state (in a manner similar to the Oklahoma case) rather than the whole United States. 85% is contributed solely by transportation via trucks where dry algae is transported from supply to port cities. An additional 6% is contributed by transportation via barges to transport algae oil from extraction facilities to transesterification facilities, and the remaining is contributed by transportation via pipelines to carry biodiesel to demand centers. One important recommendation from this work to lower these costs would be to consider the supply and demand centers within each individual state (in a manner similar to the Oklahoma case) rather than the whole United States. For the United States case, we assumed that the distance between the center of the state and the port location in supply states is an appropriate approximation of the distance between algae farms and the port. The results recommend a total area of 9,832 km 2 for algae ponds. This area roughly corresponds to a 100 km × 100 km 2 . In contrast, the distance between the center of the Mississippi State and Gulfport is 314 km. Therefore, the assumed distances may be a gross overestimate for the distance algal biomass is shipped for processing. We decided to investigate how the supply chain topology and per-gallon cost of biodiesel would change if the transportation cost of algal biomass from farms to port locations can be avoided. The distances between supply locations and port locations for the United States case were set to zero to model this case, and the resulting mathematical program was solved.
It can be seen from Table 12, which compares the relaxed-MINLP and MILP solutions, that the total cost is much lower than that of the first United States case. From Figure 19, it can be observed that the cost of transportation is reduced to 30% of the total cost. The design of the supply chain and the values of the For the United States case, we assumed that the distance between the center of the state and the port location in supply states is an appropriate approximation of the distance between algae farms and the port. The results recommend a total area of 9,832 km 2 for algae ponds. This area roughly corresponds to a 100 km × 100 km 2 . In contrast, the distance between the center of the Mississippi State and Gulfport is 314 km. Therefore, the assumed distances may be a gross overestimate for the distance algal biomass is shipped for processing. We decided to investigate how the supply chain topology and per-gallon cost of biodiesel would change if the transportation cost of algal biomass from farms to port locations can be avoided. The distances between supply locations and port locations for the United States case were set to zero to model this case, and the resulting mathematical program was solved.
It can be seen from Table 12, which compares the relaxed-MINLP and MILP solutions, that the total cost is much lower than that of the first United States case. From Figure 19, it can be observed that the cost of transportation is reduced to 30% of the total cost. The design of the supply chain and the values of the remaining variables of the new case are equal to the values obtained as the original solution. The reduced transportation cost lowers the per-gallon cost of biodiesel to $13.68 ($3.62 per liter).

Conclusions and Future Directions
In this work, a mathematical programming model was developed for determining the supply chain network design of the algae biomass production and biodiesel distribution. The supply chain considers supply, port, extraction, transesterification, and demand locations. Supply locations are the locations where algae biomass is produced. These locations are chosen depending on the largest availability of marginal farmland area that is able to maximize the algae growth given the environmental parameters, such as available sunlight and the average temperature throughout the year. Additionally, the supply locations tend to be in a more centralized location in relation to the demand locations to minimize the total transportation costs. Port locations are the port cities in supply locations. Extraction and transesterification locations are the combination of both port and demand locations. The locations of the extraction facilities were placed as near to the supply points as allowed to minimize transportation costs. The transesterification facilities were placed at demand locations to maximize the shipment of the densest product (algal oil), as the shipping constraints used

Conclusions and Future Directions
In this work, a mathematical programming model was developed for determining the supply chain network design of the algae biomass production and biodiesel distribution. The supply chain considers supply, port, extraction, transesterification, and demand locations. Supply locations are the locations where algae biomass is produced. These locations are chosen depending on the largest availability of marginal farmland area that is able to maximize the algae growth given the environmental parameters, such as available sunlight and the average temperature throughout the year. Additionally, the supply locations tend to be in a more centralized location in relation to the demand locations to minimize the total transportation costs. Port locations are the port cities in supply locations. Extraction and transesterification locations are the combination of both port and demand locations. The locations of the extraction facilities were placed as near to the supply points as allowed to minimize transportation costs. The transesterification facilities were placed at demand locations to maximize the shipment of the densest product (algal oil), as the shipping constraints used in the problem were on a volumetric basis. However, when pipelines were available to transport biodiesel to demand locations, the transesterification facilities were closer to demand locations or to port locations that were centrally located to utilize pipelines to transport biodiesel to the final demand locations. Demand locations are the states with maximum diesel demand. Regional parameters, such as population density, land costs, water costs, electricity costs, total farmland availability, relative humidity, wind velocity, maximum and minimum temperatures, distances between locations by means of trucks, rails, barges, and pipelines, have been considered for the economic analysis. The time-dependent model integrates algae species, weather conditions at each possible cultivation location, and raceway pond dimensions with supply chain distribution of biodiesel to meet the demand at various locations. The model investigated different routes used for the transport and different modes of transportation between locations.
In both cases, Oklahoma and the United States, all of the algae is produced in one location. For Oklahoma, this location is Kay County, while, for the contiguous United States, it is Mississippi. The biodiesel cost in Oklahoma is $7.07 per U.S. gallon ($1.87 per liter), while, for the base United States case, it is $61.69 ($16.32 per liter). If the costs of biomass transportation from algae ponds to port locations are removed, the cost for the United States case drops to $13.68 per U.S. gallon ($3.62 per liter). The three cases provide upper and lower bounds on the possible cost of producing biodiesel. In addition, given that extraction costs and transesterification costs are both modeled as linear costs, it is possible that, as the scale of the problem increases, the linear cost may provide an overestimate of the true cost. For a case in which transportation is only over very short distances, such as the Oklahoma case, it makes sense that the costs would be lower than a nationwide model. Given the large portion of the base United States costs is made up by algal biomass transportation costs, and the subsequent reduction in costs when this first layer is removed, it would seem that growing algae in one location with minimal shipping costs to get to the oil extraction location leads to a lower overall cost. This is reinforced by the even lower per-gallon costs associated with the Oklahoma case. Therefore, it would make sense for national implementation of algal biomass to biodiesel supply chain to rely on state, or even local, production of algae oil, and for transesterification to occur near the final demand centers.

Subset Description
J S , J Ex , J Es , J D ⊂ J Subsets for algae biomass production, extraction of algae oil, production of biodiesel, and demand locations, respectively. Z i ⊂ Z ∀i ∈ I Subsets of modes of transportation for the given transportation layer, i. Power required by paddlewheels in the pond W PW j,d,t

Parameter Description Units
Power required by pumps in the pond W The objective is to fulfill the biomass demand at a minimum net present sink, Z, for a raceway pond with a plant-life of 10 years. The objective function and constraints are as follows: where