Biofuel supply chain considering depreciation cost of installed plants

Rabbani, Masoud; Ramezankhani, Farshad; Giahi, Ramin; Farshbaf-Geranmayeh, Amir Article Biofuel supply chain considering depreciation cost of installed plants Journal of Industrial Engineering International Provided in Cooperation with: Islamic Azad University (IAU), Tehran Suggested Citation: Rabbani, Masoud; Ramezankhani, Farshad; Giahi, Ramin; Farshbaf-Geranmayeh, Amir (2016) : Biofuel supply chain considering depreciation cost of installed plants, Journal of Industrial Engineering International, ISSN 2251-712X, Springer, Heidelberg, Vol. 12, pp. 221-235, https://doi.org/10.1007/s40092-015-0139-1 This Version is available at: https://hdl.handle.net/10419/157479 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ ORIGINAL RESEARCH Biofuel supply chain considering depreciation cost of installed plants Masoud Rabbani 1 •Farshad Ramezankhani 1 •Ramin Giahi 1 • Amir Farshbaf-Geranmayeh 1 Received: 10 October 2014 / Accepted: 19 December 2015 / Published online: 21 January 2016 The Author(s) 2016. This article is published with open access at Springerlink.com Abstract Due to the depletion of the fossil fuels and major concerns about the security of energy in the future to produce fuels, the importance of utilizing the renewable energies is distinguished. Nowadays there has been a growing interest for biofuels. Thus, this paper reveals a general optimization model which enables the selection of preprocessing centers for the biomass, biofuel plants, and warehouses to store the biofuels. The objective of this model is to maximize the total benefits. Costs of the model consist of setup cost of preprocessing centers, plants and warehouses, transportation costs, production costs, emission cost and the depreciation cost. At first, the deprecation cost of the centers is calculated by means of three methods. The model chooses the best depreciation method in each period by switching between them. A numerical example is presented and solved by CPLEX solver in GAMS software and finally, sensitivity analyses are accomplished. Keywords Biomass Biofuel supply chain  Multi-echelon Depreciation costs Introduction By considering depletion of fossil fuel in the future, the importance of using renewable energy increases production (Petroleum 2015). One of the disadvantages of fossil fuels is air pollution. Greenhouse gases spread out in environment via burning of these fuels and cause global warming. On the other hand, renewable energy has less global warming effects and increases the energy security. Renewable energy divides into solar, wind power, biomass, geothermal, and tidal energy. The types of biomass feedstock which are utilized for energy purposes are categorized as: agricultural, dedicated energy crops, forestry, industry, gardens residues (Tumuluru et al. 2011). In this study, the supply chain of the biomass is proposed as: 1. Procuring of the feedstock (i.e., purchasing biomass, importing, and cultivating them). 2. Transporting to preprocessing centers. 3. Preprocessing biomass. 4. Transporting the preprocessed biomass to plants. 5. Producing biofuel in the plant. 6. Transporting the biofuels to the warehouses. 7. Distributing the biofuels. Literature review Ayoub et al. (2007) proposed a general bioenergy decision system. They believe that planners have to consider social concerns, environmental and economic impacts related to establishing the biomass systems. Leduc et al. (2008) developed a model to determine the locations and sizes of methanol plants and gas stations in Austria. The objective function of the model consisted of plant and gas station setup cost, methanol production cost, and material transportation cost. Mele et al. (2009) proposed a model that simultaneously minimizes the total cost of the network and its environmental performance over the entire life cycle of the product. Zamboni et al. (2009) proposed the bioethanol supply chain optimization in which they presented a model &Masoud Rabbani [email protected] 1 School of Industrial and Systems Engineering, College of Engineering, University of Tehran, P.O. Box 11155-45632, Tehran, Iran 123 J Ind Eng Int (2016) 12:221–235 DOI 10.1007/s40092-015-0139-1 for the strategic design of biomass-based fuel supply networks. Finally, they applied the model for a case study in Italy. Jackson et al. (2009) found that firms using accelerated depreciation make significantly larger capital investments than firms that use straight line depreciation and found that there has been a migration away from accelerated depreciation to straight line depreciation over the past two decades. Finally, results suggest that a choice made for external financial reporting purposes influences managers’ capital investment decisions. Ayoub et al. (2009) proposed an optimization model for designing and evaluating integrated system of bioenergy production supply chains. Their model was applied in a case study in Japan. Rentizelas and Tatsiopoulos (2010) utilized a hybrid optimization method to find the optimum location of a bioenergy generation facility considering the maximization of the net present value (NPV) of the investment for the project’s lifetime. Velazquez-Marti and Fernandez-Gonzalez (2010) supposed two criteria for the location of established plants for producing the biofuel as: minimizing the transportation costs of biofuels and using all the energy produced by the plant. They applied the model to Spanish rural regions. Akgul et al. (2011) presented the model to optimize the locations and scales of the bioethanol production plants, biomass and bioethanol flows between regions. The purpose of this study is minimizing the total supply chain costs. Kim et al. (2011) formulated a model that enables the selection of fuel conversion technologies and capacities, biomass locations and the logistics of transportation from forestry resources to conversion, and from conversion to final markets. The objective function to be maximized was the overall profit. The revenue of the model includes selling various products in the final market and the credits for the utility energy produced at each plant location. The cost encompassed operating cost, annualized capital cost, transportation cost and biomass acquisition cost for each biomass type. Mobini et al. (2011) developed a simulation model to evaluate the cost of delivered forest biomass, the equilibrium moisture content, and carbon emissions from the logistics operations. Zhu and Yao (2011) proposed a multi-commodity network flow model to design the logistics system. They formulated a model to determine the locations of warehouses, the size of harvesting group, the types and amounts of biomass harvested or purchased, stored, and processed, and the transportation of biomass in the system. The objective function of Lea ˜o et al. (2011) consisted of investments for the production plants, transportation costs, agricultural production costs, processing costs and purchasing cost of any additional volumes of oil in the market to meet the demand of the plants. Chen and Fan (2012) developed a two-stage stochastic programming model to minimize the system cost. The system includes bioethanol production, feedstock procurement, fuel delivery, ethanol transportation and possible penalty on fuel shortage. The model was used to evaluate the economic possibility and system robustness in a case study of California. Finally, the model was solved by a Lagrange relaxation-based decomposition solution algorithm. Ayoub and Yuji (2012) utilized a demand-driven approach for optimizing biomass utilization networks cost by applying genetic algorithm to solve the network problem. Judd et al. (2012) proposed a mathematical programming to determine satellite storage locations and equipment routes to minimize the total cost of designing a feedstock logistics system. The feedstock logistics system includes transporting biomass from production fields to the bioenergy plant. Kostin et al. (2012) integrated bioethanol and sugar production supply chain under demand uncertainty. They considered several financial risk mitigation options in the supply chain model. They applied the model in the Argentinean sugarcane industry. Finally the problem was solved by applying the sample average approximation algorithm. Akgul et al. (2012) presented an optimization framework for the strategic design of a hybrid first/secondgeneration ethanol supply chain. The applicability of the model is demonstrated with a case study of ethanol production in the UK. The potential cost reductions of secondgeneration biofuel systems are likely to lead to the deployment of these technologies at a larger scale. Biobased supply chain that was proposed by Pe ´rez-Fortes et al. (2012) led to produce electricity or other bio-products. Their model took into account three main objectives: economic, environmental and social criteria. Biomass storage periods, location and capacity of plants, material transportation between echelons and biomass utilization to produce biofuel are determined in their model. They applied the model for a case study in Ghana. To produce a low-cost urban energy system, Keirstead et al. (2012) accomplished the trade-offs between the alternatives by considering the air pollution impacts. According to their exploration of the trade-offs, biomass energy system is the best choice. Supply chain that worked by C ˇuc ˇek et al. (2012) is included agricultural, preprocessing, processing, and distribution layers. Also they presented a multi-criteria optimization for the conversion of biomass to energy. Fazlollahi and Mare ´chal (2013) simultaneously minimized costs and CO 2 emission of integrated biomass resources using multi-objective evolutionary algorithms. Zhang et al. (2013) focused on switch grass as one of the best secondgeneration feedstock for bioethanol production. They proposed an integrated mathematical model to minimize the total switch grass-based bioethanol supply chain cost. The proposed model considered the impact of switch grass crop yield, switch grass densification, switch grass dry-matter loss during storage, and economies of scale in bio refinery capacities on the total SBSC cost. 222 J Ind Eng Int (2016) 12:221–235 123 Meier et al. (2005) evaluated the economic concept of the industrial solar production of lime. The three capital investment decision indicators used in economic analysis are: (1) the payback time (PBT), defined as time required for an investment project to recover its initial cost; (2) NPV, defined as the present value of the flow of net incomes subtracted by the present value of the flow of investments; and (3) the internal rate of return (IRR), defined as the discount rate at which NPV equals zero. Mahmoudi et al. (2014) investigated the problem of source selection of competitive power plants under government intervention. Kumar et al. (2015) investigated the impact of various factors affecting coal-fired power plant economics for electricity generation. There are plentiful papers in biofuel supply chain and a lot of mathematics models are presented in this field but there are not any papers which regard the depreciation cost as an important element of the model. The significant part of our model is considering the depreciation cost within supply chain model. In this case, depreciation cost is defined as a crucial element of any supply chain design. Our study is the extension of Akgul et al. (2012) and the contributions of our study are as follows: •Considering penalty on fuel shortage. •Considering environmental impact of biofuel plants such as CO 2 emission. •Considering two manners for plants; purchasing or renting. •Calculating NPV of the project. •Considering total depreciable capital and salvage value of the network. •Revenue of selling the fuels in the market. The remainder of this paper is organized as follows. The description, assumptions and the mathematical model are introduced in ‘‘Problem description and assumption’’. ‘‘Computational results’’ embraces a numerical example, and also the results of the solved model are presented here. In ‘‘Sensitivity analysis’’, sensitivity analysis is applied to verify the accuracy of the model. Finally, ‘‘Conclusion’’ represents the conclusions of the paper. Problem description and assumption There are many influencing factors in biofuel supply chain which impact on each other. The whole system may change unpredictably by changing any of these factors. Given that the mathematical model can calculate these very detailed interactions, in this study, a mathematical model is applied for designing biofuel supply chain. Biofuel supply chain consists of the following echelons: 1. Biomass centers. 2. Biomass preprocessing center. 3. Plants for biofuel production. 4. Biofuel warehouses. 5. Demand points. Three types of biomass exists generally; woody source, non-woody source, and animal fat and waste. In this paper, we consider woody source of biomass as an input to biofuel supply chain. At the first echelon, we have three ways to procure biomass from the biomass centers: cultivating the biomass, purchasing them from domestic supplier and importing them from abroad. When the biomass is procured, we need a place for storing and drying them; therefore, echelon 2 is assigned to these warehouses. Echelon 3 represents plants of biofuel production. Fourth echelon states warehouses for biofuel storage, and the last echelon is the demand center (customer), as shown in Fig. 1. We considered three capable regions for warehouses of biomass and Kcapable regions for plants. The model chooses j,kand lregions to establish warehouses for biomass, plants and warehouses of biofuels. We can purchase or rent the warehouses needed for biomass, plants and biofuel warehouses. The plants can be established in three sizes (small, medium and large). The interest rate is monthly. In the process of plants, apercent of biomass has become biofuel, bpercent of the biomass are dried. In addition, we have inventory costs in each warehouse. Mathematical programming There are so many papers which considered mathematical programming for modeling the problems in various areas (Mousavi et al. 2014; AriaNezhad et al. 2013; Alimardani et al. 2013; Seifbarghy et al. 2015). In this section, we develop mixed integer linear programming (MILP) model. For modeling the problem we need to present the indices, parameters, and variables which are introduced in Tables 1,2and 3respectively. To simplify the problem, two models are introduced as follows. First model selects the best depreciation method from sum-of-the-years-digits method (SOYD), straight line and double declining balance (DDB) to determine the best switch points to maximize the cash flow. The second model calculates all costs of biofuel supply chain by considering J Ind Eng Int (2016) 12:221–235 223 123 the depreciation cost of installed plants which is obtained from the first model. First model According to the fact that each organization desires to select the best depreciation method to reduce their costs, this model facilitates selecting the depreciation cost by which they could be able to choose the best one with regards to the net present value of depreciation in every year. min z1¼X T t¼1X P p¼1 DptT ð1þirÞtð1Þ Dpt BVt1SVp ntþ18p;tð2Þ Dpt BV0 a N 1a N  t18p;tð3Þ Dpt 2ðBV0SVpÞðNtþ1Þ NðNþ1Þ8p;tð4Þ BVtþ1¼BVtDpt 8p;tð5Þ The objective function (1) calculates the total net present value of depreciations of the all plants for all periods of time. Constraints (2–5) represent the depreciation methods which could be utilized to calculating the depreciation. Almost always the owner of any factory would like to state that the depreciation of the equipment in the factory is a lot, to pay as little tax as they can. So the straight line, SOYD and DDB methods are introduced as the depreciation methods. Second model In the second model, at first, biomass is provided through three different ways, purchasing, importing and harvesting the provided inputs maintained in preprocessing centers. The plants producing biofuels could be established in three sizes: small, medium and large. Biofuels are sold to the customers from biofuel warehouses where biofuels are kept. The second model is presented as follows: Fig. 1 Biofuel supply chain Table 1 The indices of the model Indices Description Set i[IBiomass center I=1,2, …,I j[JPreprocessing center of biomass J=1,2, …,J k[KBiofuel production plants K=1,2, …,K p[PPlant size P=1,2,3 l[LWarehouse for biofuel L=1,2, …,L t[TTime period T=1,2, …,T w[WDemand point W=1,2, …,W 224 J Ind Eng Int (2016) 12:221–235 123 Subject to: z10 kp þz1 kp 18k;pð7Þ z20 jþz2 j18jð8Þ z30 lþz3 l18lð9Þ X P p¼1X M k¼1 ðz10 kp þz1 kpÞ¼kð10Þ X P p¼1 ðz10 kp þz1 kpÞ¼18kð11Þ X J j¼1 EMkpt ayjkpt EMMAX 8k;p;tð12Þ X J j¼1 xijt caprit 8i;tð13Þ bX T t¼1X I i¼1 xijt X K k¼1X P p¼1X T t¼1 yjkpt 8jð14Þ Ijt capbjt 8j;tð15Þ X J j¼1X T t¼1 yjkpt aX L l¼1X T t¼1 skplt 8k;pð16Þ X J j¼1 yjkpt ðzkp þz0 kpÞM8k;p;tð17Þ IIlt capwlt 8l;tð18Þ Ijðt1ÞþX I i¼1 xijt X K k¼1X P p¼1 yjkpt Ijt 08j;tð19Þ IIlðt1ÞIIlt þX W w¼1 ðBlwðt1ÞBlwt s0 lwtÞ þX K k¼1X P p¼1 skpt 0 8l;t ð20Þ X L l¼1 ðs0 lwt þBlwt Blwðt1ÞÞddwt 8w;tð21Þ X I i¼1 xijt z2 jþz20 j  M8j;tð22Þ X K k¼1X P p¼1 skplt z3 lþz30 l  M8l;tð23Þ X L l¼1 skplt cappkpt 8k;p;tð24Þ z10 kp;z1 kp;z20 j;z2 j;z30 l;z3 l2f0;1gð25Þ Objective function of second model (6)consistsoftwo terms. First one states the present value of revenues and second is the costs. The model also demonstrates the revenue of the supply chain gained by selling the biofuels. The total costs are calculated by the cost of purchasing or renting preprocessing centers, plants and warehouses in t=0,the total cost of biomass (consists of buying, importing from max z2¼X L l¼1X W w¼1X T t¼1 ð1TÞ=ð1þirÞts0 lwt P0 tX K k¼1X P p¼1 ðBCkp z1 kp þRCkp z10 kpÞþ "X J j¼1 ðBCjz2 jþRCjz20 jÞ þX L l¼1 ðBCClz3 lþRCClz30 lÞþX T t¼1X J j¼1 ð1TÞ=ð1þirÞtðCCtx1jt þBCBtx2jt þICBtx3jtÞ þX I i¼1X J j¼1X T t¼1 ð1TÞ=ð1þirÞtCijt dijt xijt þX K k¼1X P p¼1X J j¼1X T t¼1 ð1TÞ=ð1þirÞtCjkpt djkpt yjkpt þX K k¼‘X P p¼1X L l¼1X T t¼1 ð1TÞ=ð1þirÞtCkplt dkplt skplt þX W w¼1X L l¼1X T t¼1 ð1TÞ=ð1þirÞtClwt dlwts0 lwt þX J j¼1X T t¼1 ð1TÞ=ð1þirÞtSCjt IjtþX L l¼1X T t¼1 ð1TÞ=ð1þirÞt SC0 lt IIlt þX K k¼1X P p¼1X J j¼1X T t¼1 ð1TÞ=ð1þirÞtaPCtyjkpt þX W w¼1X T t¼1X L l¼1 ð1TÞ=ð1þirÞtqBlwtþX K k¼1X P p¼1X J j¼1X T t¼1 cEMkpt ayjkpt# ð6Þ J Ind Eng Int (2016) 12:221–235 225 123 abroad or harvested one), the total cost of transportation in the whole supply chain, the inventory cost in all periods, the production cost of biofuel, the penalty cost of shortage in meeting the demands and the emission cost of the plants in all periods. As said before, the depreciation costs of installed plants are obtained from the first model. Table 2 The parameters of the model Notations Description Notations Description P0 tSale price of biofuel in period tC klt Transportation cost for each unit of biofuel between plant kand warehouse lin period t BCkp Buy cost for plant kwith size pC lwt Transportation cost for each unit of biofuel between warehouse land demand point win period t RCkp Rent cost for plant kwith size pcaprit Capacity of resource of biomass iin period t BCjBuy cost for warehouse jcapbjt Capacity of the biomass of preprocessing center jin period t RCjRental cost for warehouse jcapwlt Capacity of the warehouse jin period t BC0 lBuy cost for warehouse lcappkpt Capacity of plant kwith size pin period t RC0 lRental cost for warehouse lSC0 lt Storage cost of warehouse lfor each unit of biofuel in period t BCBtBuy cost for each unit of biomass in period tPC Process cost of each unit of biomass ICBtImport cost for each unit of biomass in period td0 wt Demand of demand center win period t CCtCultivation cost for each unit of biomass in period tEMkpt Amount of CO 2 that emission by plant kwith size pfor each unit of produced biofuel in period t dijt Distance between biomass center iand warehouse jin period t aFraction of biomass conversion to biofuel djkt Distance between warehouse jand plant kin period tbPercentage of biomass dry in warehouse dklt Distance between plant kand warehouse lin period tqPenalty cost of shortage in meeting the demands dlw Distance between warehouse land demand point win period t EMMAX Maximum permissible amount of generating the gases in the plant Cijt Transportation cost for each unit of biomass between biomass center iand warehouse jin period t BVtThe book value of plant in period t Cjkt Transportation cost for each unit of biomass between warehouse jand plant kin period t SVpThe salvage value of plant p Table 3 The variable of the model Variables Descriptions xijt Flow of input biomass ito warehouse jin period t yjkpt Flow of biomass from warehouse jto plant kwith size pin period t skplt Flow of biofuel from plant kwith size pto warehouse lin period t s0 lwt Flow of biofuel from warehouse lto demand point win period t z1 kp =1 if we purchase the plant kwith size pand 0 if we do not purchase the plant kwith size p z10 kp =1 if we rent the plant kwith size pand 0 if we do not rent the plant kwith size p z2 j=1 if we buy the warehouse jand 0 if we do not buy the warehouse j z20 j =1 if we rent the warehouse jand 0 if we do not rent warehouse j z3 l=1 if we buy the warehouse land 0 if we do not buy the warehouse l z30 l =1 if we rent the warehouse land 0 if we do not rent warehouse l Dpt Depreciation of plant with size pin period t Ijt Inventory level of warehouse jat the end of period t IIlt Inventory level of warehouse lin the period t Blwt Backorder of warehouse lin the period t 226 J Ind Eng Int (2016) 12:221–235 123 Constraints (7–9), respectively, represent that the preprocessing center of biomass, plants and the warehouse of biofuels can be purchased or rented. Constraint 10 indicates that klactation of Kcandidates is selected to establish the plants. Constraint 11 shows that only one size of plants can be establish in each selected location. Constraint 12 shows the CO 2 emission in every plant should be less than maximum limitation of CO 2 generation. Constraint 13 represents the maximum capacity of the input biomass. Constraint 14 shows that the bpercent of biomass dry in warehouse of biomass then is sent to plants. Constraint 14 and 15 state that the inventory level of preprocessing centers of biomass and warehouses of biofuels, respectively, should been less than maximum capacity. Constraint 16 states apercent of the preprocessed biomass transferred to the next stage. Constraints 17,22 and 23 are logic constraints stating that no biofuel can be produced unless there is a plant operating at this location, no biomass can be utilized unless there is a preprocessing center operating at that location and no biofuel can be sold unless there is a warehouse operating at that location. Constraints 19 and 20 are the inventory constraint in each warehouse. Constraint 21 shows that the demands in place wshould be met by supply. Computational results In this section, a hypothetical numerical example is presented to state the applicability of the model. The indices and parameters of the example are as follows: the numbers of biomass center, preprocessing centers of biomass, plants, warehouse for products and demand point are three; the preprocessing centers, plants and warehouses have constant capacity which is declared in Table 4. The outputs of the model indicate the amount of all variables which are used there. The results indicate that all three plants should be made in small sizes. The decision variables about purchasing or renting the warehouses and plants are shown in Table 5. The biomass preprocessing centers which needed to be established are purchased as well as product warehouses. But, for plants two of them are rented and other one is purchased. To solve the second model, it is needed to specify the parameters, such as investment cost and salvage value, which are stated in Table 6. To increase the reality of the example, data are gathered through experts’ opinions. The flows of the biomass and biofuels in the supply chain network are represented in Tables 7,8,9and 10. The amount of each types of the biomass to each preprocessing centers in each of the months of the year is presented in Table 7. Amount of biomass sent from warehouses to plants which are calculated through model is stated in Table 8and depicted in Fig. 2for clarifying these flows. Amount of biofuel sent from plants to warehouses which are calculated through model is presented in Table 9. Amount of biofuel sent from warehouses to demand points which are calculated through model is stated in Table 10. According to the results, in all periods, there are flows between the echelons in the supply chain network from the biomass centers to demand points. So, twelve diagrams can be depicted related to each period (e.g., the flow of the network in period t=1 is depicted in Fig. 2). Figure 2is depicted to clarify the amount of flows of Figs. 7,8,9and 10. Regularly the assets and equipment depreciation are calculated by only one or a combination of DDB, straight line and SOYD. Given the fact that the owner of the facilities tend to pay as little tax as they can in the early years, the combination of these methods is utilized to state depreciation value. It is presumed that in the beginning of each period these three methods would be utilized. The Table 4 Inputs’ capacity in period tCapacity of inputs (ton) Period 123456789101112 Purchasing 200 120 220 340 200 120 220 340 200 120 220 340 Importing 200 120 220 340 200 120 220 340 200 120 220 340 Harvesting 310 200 200 700 200 120 220 340 200 120 220 340 Table 5 Either purchasing or renting the warehouses and plants Purchase Rent Preprocessing centers of biomass 1 1 21 31 Plant 1 1 21 31 Warehouse for products 1 1 21 31 J Ind Eng Int (2016) 12:221–235 227 123 maximum depreciation value of these methods is chosen in that period. The year that the method of depreciation is switched to another is called the switch point. The depreciation of the installed preprocessing centers of biomass, plant and warehouse for products in each year is indicated in Table 11. Also, the switch points are Table 6 The investment cost and salvages value of each plant and warehouse Investment cost (USD) Salvages value (USD) Preprocessing centers of biomass 1 10,000 3000 2 15,000 4500 3 10,400 3500 Plant 1 100,000 30,000 2 90,000 20,000 3 110,000 40,000 Warehouse for products 1 10,300 3000 2 10,100 3000 3 13,400 4000 Table 7 Amount of biomass ito preprocessing center jin period t Biomass Period 1 2 34 56789101112 Preprocessing center 1 1 53.7 ••256 ••••••220 • 2•86.3 •193.8 ••••••220 • 3• • •• •••••••• Preprocessing center 2 1 146.3 120 220 •200 •••200 120 •106.7 2•33.7 220 • •••••••340 3 310 200 •• •••340 •••• Preprocessing center 3 1•••84 •120 220 340 •••233.3 2•••••120 •340 •••• 3••80 ••120 ••200 ••• Table 8 Amount of biomass from warehouse jto plant kwith size p=1 in period t Preprocessing center Period 12 34 5 6 78 9 10 11 12 Plant 1 1•129.5 •• • • •• • • 116.8 • 2 161 •••172.6 ••137.9 •175.8 •208.4 3•• 120 65.3 •141.1 160 •80.5 ••• Plant 2 1 161 129.5 •• 172.6 •••••116.8 • 2•• •130.5 . 141.1 160 137.9 •••208.4 3•• 120 •••••161.1 175.8 •• Plant 3 1•• •• 172.6 ••••••• 2 143 129.5 120 •••••161.1 175.8 116.8 • 3•• •130.5 •141.1 160 137.9 •••208.4 228 J Ind Eng Int (2016) 12:221–235 123 Alimardani M, Jolai F, Rafiei H (2013) Bi-product inventory planning in a three-echelon supply chain with backordering, Poisson demand, and limited warehouse space. 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