Sustainable olefin supply chain network design under seasonal feedstock supplies and uncertain carbon tax rate

Accepted Manuscript Sustainable Olefin Supply Chain Network Design under Seasonal Feedstock Supplies and Uncertain Carbon Tax Rate

Morteza Alizadeh, Junfeng Ma, Mohammad Marufuzzaman, Fei Yu PII:

S0959-6526(19)30587-6

DOI:

10.1016/j.jclepro.2019.02.188

Reference:

JCLP 15918

To appear in:

Journal of Cleaner Production

Received Date:

18 July 2018

Accepted Date:

17 February 2019

Please cite this article as: Morteza Alizadeh, Junfeng Ma, Mohammad Marufuzzaman, Fei Yu, Sustainable Olefin Supply Chain Network Design under Seasonal Feedstock Supplies and Uncertain Carbon Tax Rate, Journal of Cleaner Production (2019), doi: 10.1016/j.jclepro. 2019.02.188

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Sustainable Olefin Supply Chain Network Design under Seasonal Feedstock Supplies and Uncertain Carbon Tax Rate

Morteza Alizadeh Graduate Research Assistant Dept. of Industrial & Systems Engineering, Mississippi State University Mississippi State, MS 39762 [email protected]

Junfeng Ma (Corresponding Author) Assistant Professor Dept. of Industrial & Systems Engineering, Mississippi State University Mississippi State, MS 39762 [email protected]

Mohammad Marufuzzaman Assistant Professor Dept. of Industrial & Systems Engineering, Mississippi State University Mississippi State, MS 39762 [email protected]

Fei Yu Associate Professor Dept. of Agricultural & Biological Engineering, Mississippi State University Mississippi State, MS 39762 [email protected]

Declarations of interest: none

ACCEPTED MANUSCRIPT

Sustainable Olefin Supply Chain Network Design under Seasonal Feedstock Supplies and Uncertain Carbon Tax Rate

Abstract The growing environmental consciousness resulted from global climate changes has aroused petrochemical industries to search for the renewable alternatives for fossil fuels. Recently, biomass has been received increasing attention due to its economic and environmental benefits. Olefin, as one of the key raw materials in petrochemical industries, is able to be produced from biomass feedstocks. This study presents a robust three-stage stochastic programming model to characterize and optimize an olefin supply chain/production network aiming to provide a reliable and economic logistics network to support olefin production. This model encompasses probabilistic scenarios and uncertainty sets to capture the seasonality of biomass feedstocks and the uncertainty of carbon tax rate, respectively. The Municipal Solid Waste (MSW) is also involved in this model to complement the traditional biomass supplies to ensure the reliable feedstock for olefin production. To find the optimal solution of this model, a hybrid robust/stochastic approach is developed by integrating the affinely adjustable robust model with the sample average approximation (SAA) method. The state of Mississippi is used as a real case study to test and validate the proposed model and optimization approach. The results show that increasing feedstocks conversion rate by 20% and MSW recycling rate by 100% will increase olefin production by 17.26% and 14.3%, respectively, and increasing the carbon tax rate uncertainty from 0 to 30 will decrease the total network emissions by 2.8%. The proposed optimization approach will generate more robust and reliable results. These results indicate that the proposed model and optimization approach would benefit both economic and environmental perspectives in biomass based olefin production.

Keywords Olefin Supply chain/Production Network; Affinely Adjustable Robust Counterpart Model; Sample Average Approximation

Abbreviations MSW GHG RTSSP SAA DSS GHG PCA

Municipal Solid Waste Greenhouse Gas Robust Three-Stage Stochastic Programming Sample Average Approximation Decision Support System Greenhouse Gas Principal Component Analysis

1

ACCEPTED MANUSCRIPT

RTSSP RC ARC AARC NREL MDEQ MT MTY

Robust Three-Stage Stochastic Programming Robust Counterpart Adjustable Robust Counterpart Affinely Adjustable Robust Counterpart National Renewable Energy Laboratory Mississippi Department of Environmental Quality

Million Tons Million Tons per Year

1. Introduction Nowadays, the increasing awareness of natural resource limitation and environmental protection has evoked petrochemical industries to consider finding renewable alternatives to the non-renewable fossil resources. Olefin, as the key raw material in the petrochemical industries (Sadrameli, 2016), is a type of unsaturated hydrocarbons with chemical formula of 𝐶𝑛𝐻2𝑛. It can be categorized into lower olefins (𝐶2~𝐶4) and higher olefins (𝐶5+ ) based on the number of 𝐶. Lower olefins (e.g., ethylene and propylene) are extensively used as building blocks to synthesize a wide range of products such as polymers, drugs, solvents, cosmetics, and detergents (Galvis et al., 2012); while higher olefins are intermediates for producing highly valuable products such as high-octane gasoline, aromatic components, lubricating oil additives, and alcohols (Zhai et al., 2016). This study focuses on lower olefins. Traditionally, lower olefins are usually produced by thermal or catalytic cracking of non-renewable resources such as naphtha, vacuum gas oil, ethane, butane, etc. (Galvis et al., 2012; Zhai et al., 2016). However, olefin production using these nonrenewable feedstocks is extensively energy-consuming and emits mass of greenhouse gases (Lu et al., 2017). The growing demand for lower olefins and the economic and environmental drawbacks of their production using non-renewable feedstock have stimulated special interests in the recent year to produce lower olefins from alternative renewable feedstock such as biomass, wasted plastics, natural gas, and coal (Lu et al., 2017; Sadrameli, 2015). That is, olefins producers in the world are looking for the more accessible lower price feedstocks. Among various lower olefin production feedstocks, biomass (e.g., corn-stovers and forest residues) has attracted growing interests in order to be viable options for replacing the conventional non-renewable resources. These are the most available and least expensive feedstocks for the cracking in the world (Sadrameli, 2015). This research aims to investigate the applicability of using biomass feedstocks for producing olefin. However, one of the biggest challenges of using these biomass is their constant supplies throughout the year, which is highly associated with the biomass yield and seasonality (Osmani and Zhang, 2013). The corn-stovers are constantly supplied between September and November whereas forest residues are available throughout the year except winter months from December to February (Quddus et al., 2018). Moreover, some climate changes (e.g., rainfall and climate temperature) or extreme events (e.g., flood and hurricane) may also fluctuate the biomass supplies (Persson et al., 2009). These cause a crucial challenge to sustain the level of production in the olefin plants. Hence, complementary feedstocks need to be explored so that not only addresses the challenge of seasonality, but also ensures a reliable supply chain network. MSW is a sustainable feedstock that ensures season-wide availability for producing olefins. MSW is able to reduce the impact of seasonal and uncertain supplies because it is available the 2

ACCEPTED MANUSCRIPT

whole year and meanwhile the increasing population and rapid community development accelerate the MSW generation in the recent years. According to the report of U.S. Environmental Protection Agency in 2015, MSW generation rate has increased by 18.96% and its recycling rate has increased by 34.6% in the last nineteen years (1995-2014) (U.S. Environmental Protection Agency, 2015). Both of these rates are expected to further increase along with the growing of the population in the entire nation (U.S. Census Bureau, 2017). These imply the potential reliability of MSW to be a great complementary of biomass as future feedstock to produce olefins. Thus, biomass-derived feedstock, including corn-stovers and forest residues, and MSW are two sets of alternative renewable resources to replace the traditional feedstock for producing olefins. Effective utilization of biomass-derived and MSW feedstocks to produce olefins requires developing an optimization model that can not only minimize the impacts of seasonal and uncertain feedstock supplies but also ensure a competitive price to the olefins plants by densifying the feedstock types (corn-stovers, forest residues and MSW). One of the most significant challenges in designing the olefin supply chain/production network is the high cost associated with collecting feedstock from the supply sites to the olefins plants. This is because feedstock types are bulky and widely dispersed geographically and thus too expensive to transport (Marufuzzaman et al., 2016). Therefore, this study aims a comprehensive investigation of the potential supply chain network to identify the key factors in collecting and transporting the feedstock types to produce olefins. The novelties of this study are summarized as follows: 

The proposed model considered the season-wide available MSW besides the biomass feedstocks. Seasonality varies the availability of the renewable biomass feedstocks during the year, therefore including the constant-supply MSW would maintain the production of olefin. Additionally, the model used probabilistic scenarios to capture the seasonal supplies of biomass feedstocks.



The proposed model considered the variety of multi-modal facilities in the transportation of the bulky feedstock supplies, including rail car hubs and inland ports, which will fit the real implementation in the area of Gulf of Mexico.



The proposed model incorporates the US-fitted carbon tax policy to the olefin production supply chain network for handling the amount of carbon emissions by densification depots, olefin plants and different transportation modes including trucks, railcar hubs and inland ports. The proposed model discriminates the carbon emission rate not only for different feedstocks, but also for different transportation modes. Furthermore, due to annual fluctuations in the rate of carbon emission, this model incorporates the uncertainty sets for the carbon tax rates.



This model combines the feedstock supplies and densifications, two sets of multi-modal facilities, olefin productions and carbon emission tax rate together in the olefin supply chain network design.



The proposed model makes the investment decisions in the first stage and then handles the biomass feedstocks’ seasonality and carbon tax uncertainty through the second and third steps of the proposed model, respectively.



The solution approach compromises a novel hybrid robust/stochastic technique by integrating the stochastic model in second-stage and the affinely adjustable robust

3

ACCEPTED MANUSCRIPT

counterpart in the third-stage and a validation analysis technique to evaluate the solution optimality gap. 

Extensive experiments are analyzed to evaluate the impact of feedstocks seasonality and carbon tax rate adjustability on the olefin supply network/production design.



A real case study in the state of Mississippi is proposed to validate the effectiveness of the proposed model.

The remaining of the paper is presented as follows. Section 2 presents the pertinent studies. Section 3 discusses the proposed RTSSP formulation. Section 4 presents the integrated robust SAA solution algorithm. Section 5 conducts the case study and analyzes the experimental results. The conclusions and future research directions are presented in section 6. 2. Literature review Biomass supply chain has been extensively studied in the recent decades. In order to summarize and reflect the contributions of the modeling and solution approaches, the pertinent literatures can be grouped into deterministic and stochastic categories. Many studies investigated the biomass supply chain network and transportation under deterministic setting, such as network design and management (Ekşioğlu et al., 2009; Gold and Seuring, 2011; Lin et al., 2014; Roni et al., 2014), dynamic planning of biomass supplies (Izquierdo et al., 2008; Bai et al., 2011; Xie and Ouyang, 2013), logistic and storage issues of multi-biomass (Rentizelas et al., 2009; Huang et al., 2010; Memişoğlu and Üster, 2015), and economic performance and technology (An et al., 2011; Marvin et al., 2012a, 2012b; Cambero and Sowlati, 2014). All these studies assume that the model input parameters (e.g., biomass supplies) are known and thus fail to capture the uncertainties in the biomass supply chain network design. Stochastic based models were developed to handle the uncertainties in biomass supply chain. System uncertainty is one of the major challenges in using the renewable resources for biomass supply chain development which might affect the profitability of the biomass network configuration. To address this challenge, many studies extended the deterministic biomass supply chain models to the stochastic models by considering uncertainty in a variety of fields. These fields include production technology (Cundiff et al., 1997; Tong et al., 2013), supply and demand (Frombo et al., 2009; Chen and Fan, 2012; Gebreslassie et al., 2012; Tong et al., 2014; Shabani and Sowlati, 2016), market segments (Awudu and Zhang, 2013; Azadeh et al., 2014), biorefineries (Wang and Ouyang, 2013; Bai et al., 2015), and multi-scales (Tong et al., 2014; Yue and You, 2016). A brief overview of considering uncertainty and sustainability concepts in biomass supply chain network configuration can be found in a study by (Awudu and Zhang, 2012). However, both deterministic and stochastic models so far are able to capture the overall system design, neither apply the long haul transportation modes such as railways or waterways to carry biomass through the multi-modal facilities. Another challenge of using the renewable resources is corresponding transportation is expensive, not only because they are massive, but also widely scatter geographically. Several studies (Marufuzzaman et al., 2014c; Roni et al., 2014; Marufuzzaman and Ekşioğlu, 2016; Poudel et al., 2017; Quddus et al., 2018) were conducted to investigate the transportation of biomass supplies by long hauls through multi modal facilities in the supply chain network. This proposed model has extended these studies by involving feedstocks uncertainties and transporting biomass 4

ACCEPTED MANUSCRIPT

and MSW feedstocks from the supply sites to densification depots and olefin plants by both railways and waterways. The availability of Mississippi river allows the proposed model to use the inland ports for biomass transportation through the Mississippi river to the Gulf of Mexico. With the increasing of greenhouse gas (GHG) emissions around the world, legislation and social concerns need to be involved in the supply chain network development to reduce the negative environmental impacts. One of the most common sources that leads to negative environmental impacts is the carbon emission from processing and transporting products (Parker et al., 2008). According to U.S. Environmental Protection Agency (2017), 26% of carbon emissions in United States were generated by transportation activities in 2014. Hence, many studies were conducted to reduce the impact of carbon emissions in the biomass supply chain network. Among them, some studies set carbon emission in the single objective, such as robust models for minimization of total carbon footprint in the supply chain (Foo et al., 2013), the optimization model of carbon emission treatment cost for a biomass supply chain (Abdulrazik et al., 2017) and the investigation of carbon emission impact on a sustainable biofuel supply chain network with fuzzy biomass supply and market demand (Ahmed and Sarkar, 2018). Another stream of biomass supply chain research considered bi-objective optimization model so as to minimize the total biomass supply chain costs and carbon emissions (Zamboni et al., 2009; You and Wang, 2011, 2012; Marufuzzaman et al., 2014a; Babazadeh et al., 2017), maximize net present value and minimize environmental impacts (Mele et al., 2011; Giarola et al., 2013), and optimize both economic and environmental objectives for biomass supply chain network (Babazadeh, 2018; Balaman et al., 2018). Furthermore, some other studies have proposed multi-objective models that consider cost minimization, carbon emission minimization, and job opportunity maximization as objective functions (You et al., 2012; Yue et al., 2014). Additionally, a novel principal component analysis (PCA) aided optimization approach was proposed to solve a biomass supply chain problem by considering economic, environmental and social sustainability (Shen How and Lam, 2018). Two-stage stochastic programming is one of the most widely applied stochastic programming models to capture the uncertainties of system (Shapiro, 2008). Many studies applied this approach to handle feedstocks’ uncertainty in biomass supply chain configuration (Khor et al., 2008; Kim et al., 2011; Chen and Fan, 2012; Gebreslassie et al., 2012; Awudu and Zhang, 2013). This proposed model has been developed to handle multiple feedstocks (i.e. corn-stover, forest residue, and MSW) under supplies and carbon emission tax rate uncertainty. Until now a number of studies have been developed to solve stochastic biomass supply chain problems. Marufuzzaman et al., (2014b) studied a two-stage stochastic technique to minimize network cost and carbon emissions under biomass supply and technology uncertainty. Cobuloglu and Büyüktahtakın (2017) developed a two-stage stochastic mixed-integer programming model to maximize the economic and environmental benefits of biofuel production. Quddus et al. (2018) proposed a two-stage chance-constrained stochastic model to solve a biomass supply chain problem with feedstock supply uncertainty. Although extensive studies have been used to investigate biomass supply chain network development, some research gaps are still existing: 1) few studies focus on olefin supply chain development; 2) few studies consider carbon emission from the view of tax rate; and 3) none of the prior studies combine olefin production, uncertain carbon emission tax rate, and seasonal biomass supplies together in the supply chain network development. These gaps motive this study. To address these gaps, this paper proposes a robust three-stage stochastic programming (RTSSP) model to develop a supply chain network for collecting, densifying and transporting the feedstocks to the olefin production plants by considering biomass supplies and carbon emission tax rate 5

ACCEPTED MANUSCRIPT

uncertainties. The proposed hybrid robust/stochastic solution approach is developed based on generating realistic scenarios obtained from prediction errors of the historical and forecasted feedstock supplies availabilities. 3. Problem Description and Model Development This section presents the RTSSP model for the design and management of an olefin supply chain network problem under seasonal feedstock supply and uncertain carbon tax rate. Since carbon tax policy is strict in US (Haddadsisakht and Ryan, 2018), this study focuses on carbon tax. The proposed model is developed based on yearly planning horizon, where decisions are seasonal base. This model consists of two decision variable sets, design variables and control variables. Facility location decisions (i.e. densification depots, multi-modal facilities and olefins plants) are typically design variables which need to be made here-and-now in the first-stage before uncertainty realizations. They are long-term decisions and have static nature during the planning horizon which means non-adjustable to the uncertain parameters. While the control variables are the short-term decisions and subjected to adjustment once the uncertain parameters are realized. These variables are scenario-dependent and so-called wait-and-see decisions. Once long-term decisions are made, the short-term operational decisions will be made based on the actual data of the uncertain parameters, which include decisions about procurement, storing, processing, producing, and transporting of biomass and olefins. These operational decisions are highly exposed to the biomass seasonality. Since availabilities of biomass supplies (i.e. corn-stover and forest residue) vary during the seasonal changes, seasonal decisions can capture this phenomena (Quddus et al., 2018). For example, forest residues are available throughout the year except the winter months, from December to February, whereas corn-stovers are available only for three months, from September to November. Moreover, this research evaluates the effect of an uncertain carbon tax rate on the control decisions of the olefin supply/production network. Implementing carbon tax rate policy in the major carbon-emitting nations such as United States typically associates with uncertainty. This rate also varies throughout the world. For example, carbon tax rate in Finland was $30 per ton at 2008 whereas this rate in British Columbia started from $9.50 per ton in 2008 and increased to $30 per ton in 2012 (Sumner et al., 2011). Besides, U.S. Environmental Protection Agency (2017) estimated the social cost of carbon to be $36 per ton in 2015. Therefore, how biomass/MSW seasonality and carbon tax rate uncertainty affect the olefin supply/production network configuration, use of transportation modes and biomass flows directions throughout the network while minimizing the overall network cost is worthy of investigation. Olefin supply/production network configuration can be modeled effectively in a three-stage stochastic setting in which the probabilistic scenarios are considered for the availabilities of the feedstock supplies with regard to seasonality and uncertainty sets applied for the carbon tax rates. The first-stage decisions are for facilities investments; the second-stage concerns on the plan of storing feedstock types in facilities, transportation units of various modes and producing olefins after realization of feedstock availabilities; and distributing the feedstock types throughout the network and amount of processing feedstocks in densification depots and olefins plants are the third-stage decisions after carbon tax realization . Fig. 1 illustrates the simplified olefin supply chain network consisting of three supplier sites, one densification depot, one railcar hub, one inland port and one olefin plant. 6

ACCEPTED MANUSCRIPT

Figure 1. Olefin supply chain network

The olefin supply chain network is denoted by 𝒢 = (𝒩,𝒜) where 𝒩 is set of nodes and 𝒜 is set of arcs. The node set 𝒩 = ℐ ∪ 𝒥 ∪ 𝒦 ∪ ℳ which consists of the set of biomass supply sites ℐ, set of candidate densification depot locations 𝒥, set of candidate olefin plant locations 𝒦, and set of candidate multi-modal facility locations ℳ. Let the node set ℐ = ℐ𝒸 ∪ ℐ𝒻 ∪ ℐ𝓂 denote the different feedstock types so that ℐ𝒸 represents the set of corn-stover supply nodes, ℐ𝒻 represents the set of forest residues supply nodes and ℐ𝓂 represents the set of MSW supply nodes. Similarly, ℳ = ℳ𝒽 ∪ ℳ𝓅 denote multi-modal facility sets where ℳ𝒽 represents the set of railcar hubs and ℳ𝓅 represents the set of inland ports. Furthermore, the arc set 𝒜 = 𝒜1 ∪ 𝒜2 ∪ … ∪ 𝒜7 where 𝒜1 shows the set of arcs link biomass suppliers ℐ with the densification depots 𝒥, 𝒜2 shows the set of arcs link corn-stover supply sites ℐ𝒸 with olefin plants 𝒦, 𝒜3 shows the set of arcs link densification depots 𝒥 with railcar hubs ℳ𝒽, 𝒜4 shows the set of arcs link densification depots 𝒥 with olefin plants 𝒦, 𝒜5 shows the set of arcs link densification depots 𝒥 with inland ports ℳ𝓅, 𝒜6 shows the set of arcs link railcar hubs ℳ𝒽 with olefin plants 𝒦 and 𝒜7 shows the set of arcs link inland ports ℳ𝓅 with olefin plants 𝒦. Since corn-stover feedstock is in bale format and does not require further size reduction, if a densification depot located close to the olefin plants, then it can be shipped directly to olefin plants without using multi-modal facilities. Let 𝛼𝑏𝑝𝑞𝑡, 𝛽𝑏𝑚𝑘𝑡 and 𝛾𝑏𝑚𝑘𝑡 represent the unit cost of transporting biomass type 𝑏 ∈ ℬ along arcs (𝑝,𝑞) ∈ 𝒜1 ∪ 𝒜2 ∪ … ∪ 𝒜5, arc (𝑚,𝑘) ∈ 𝒜6 and (𝑚,𝑘) ∈ 𝒜7 in period 𝑡 ∈ 𝒯, respectively. Because the length of the arcs (𝑝,𝑞) are relatively short, trucks are used to transport feedstock along these arcs whereas due to the large transportation volume and long distance along the arcs (𝑚,𝑘), larger haul transportation modes (i.e., railcars for the arc (𝑚,𝑘) ∈ 𝒜6 and barges for the arc (𝑚,𝑘) ∈ 𝒜7) are preferred to ship biomass along these arcs. Cargo containers are used to transport feedstock between multi-modal facilities ℳ and olefin plants 𝒦 which comes with fixed costs of loading and unloading of the containers along the arcs (𝑚,𝑘) ∈ 𝒜6 ∪ 𝒜7. 7

ACCEPTED MANUSCRIPT

Let ℬ denotes the set of feedstock types (i.e., corn-stover, forest residue, and MSW) that can be procured, processed and transport in time periods 𝑡 ∈ 𝒯 and under different scenarios 𝜔𝜖Ω. Then, fixed number of scenarios |Ω| associated with occurrence probabilities 𝒫𝜔 ( 𝒫𝜔 ≥ 0) are defined to deal with feedstock supplies seasonality. Thus, 𝑟𝑏𝑖𝑡(𝜔) shows the amount of different biomass types 𝑏 ∈ ℬ which are available in supply site 𝑖 ∈ ℐ at time period 𝑡𝜖𝒯 under supply scenario 𝜔𝜖Ω. The relevant notations for the proposed RTSSP model are given below: Sets ℐ𝑐 ℐ𝑓 ℐ𝑚 ℐ 𝒥 𝒦 ℳ𝓀 ℳ𝓅 ℳ 𝒩 ℬ ℒ 𝑇 Ω

Set of corn-stover supply sites, Set of forest residues supply sites, Set of MSW supply sites, Set of all biomass supply sites, ℐ = ℐ𝒸 ∪ ℐ𝒻 ∪ ℐ𝓂, Set of densification depots, Set of olefin plants, Set of railcar hubs, Set of inland ports, Set of all multi-modal facilities, ℳ = ℳ𝒽 ∪ ℳ𝓅, Set of transportation modes, Set of biomass types, (i.e., 𝑏1 for corn-stover, 𝑏2 for forest residue and 𝑏3 for MSW), Set of capacities, Set of time periods, Set of scenarios,

Parameters 𝑓𝑗𝑙 ℎ𝑘𝑙 𝑔𝑚𝑙 𝑒𝑚𝑙 𝜍𝑏𝑖𝑡 𝑣𝑟𝑐 𝑏𝑡 𝑣𝑏𝑎 𝑏𝑡 𝛼𝑏𝑝𝑞𝑡 𝛽𝑏𝑚𝑘𝑡 𝛾𝑏𝑚𝑘𝑡 𝑢𝑏𝑗𝑙𝑡 𝑤𝑘𝑙𝑡

Investment cost of opening a densification depot with capacity 𝑙 at location 𝑗 ∈ 𝒥, Investment cost of opening an olefin plant with capacity 𝑙 at location 𝑘 ∈ 𝒦, Fixed cost of using a railcar hub with capacity 𝑙 at location 𝑚 ∈ ℳ𝒽, Fixed cost of using an inland port with capacity 𝑙 at location 𝑚 ∈ ℳ𝓅, Unit procurement cost of biomass type 𝑏 ∈ ℬ at supply site 𝑖 in period 𝑡 ∈ 𝒯, Fixed cost of a railcar for transporting biomass type 𝑏 in period 𝑡 ∈ 𝒯, Fixed cost of a barge for transporting biomass type 𝑏 in period 𝑡 ∈ 𝒯, Unit transporting cost of biomass type 𝑏 ∈ ℬ along arc (𝑝,𝑞) ∈ 𝒜1 ∪ … ∪ 𝒜5 in period 𝑡 ∈ 𝒯, Unit transporting cost of biomass type 𝑏 ∈ ℬ along arc (𝑚,𝑘) ∈ 𝒜6 in period 𝑡 ∈ 𝒯, Unit transporting cost of biomass type 𝑏 ∈ ℬ along arc (𝑚,𝑘) ∈ 𝒜7 in period 𝑡 ∈ 𝒯, Unit densifying cost of biomass type 𝑏 ∈ ℬ at densification depot 𝑗 ∈ 𝒥 with capacity 𝑙 ∈ ℒ in period 𝑡 ∈ 𝒯, Unit producing cost of olefin at plant 𝑘 ∈ 𝒦 with capacity 𝑙 ∈ ℒ in period 𝑡 ∈ 𝒯,

8

ACCEPTED MANUSCRIPT

𝛿𝑏𝑞𝑡 𝑑𝑝𝑞 𝑟𝑏𝑖𝑡(𝜔) 𝑐𝑞𝑙 𝜁𝑚𝑙 𝑠𝑚𝑙 𝜛𝑞 𝑙 𝜎𝑏 𝜗𝑏𝑛 Λ𝑏 𝜌𝑏 𝜃𝑏𝑡 𝜆𝑏𝑗𝑙𝑡 𝜂𝑏𝑘𝑙𝑡 𝜚𝑛𝑡 𝜏𝑡 𝔇𝑡

Unit inventory holding cost of biomass type 𝑏 ∈ ℬ at location 𝑞 ∈ 𝒥 ∪ 𝒦 in period 𝑡 ∈ 𝒯, Distance along arc (𝑝,𝑞) ∈ 𝒜1 ∪ … ∪ 𝒜7, Amount of biomass type 𝑏 ∈ ℬ available at supply site 𝑖 ∈ ℐ in period 𝑡𝜖𝒯 under scenario 𝜔𝜖Ω, Biomass storage capacity of size 𝑙 ∈ ℒ at location 𝑞 ∈ 𝒥 ∪ 𝒦, Biomass handling capacity of railcar hub 𝑚 ∈ ℳ𝒽 with capacity 𝑙, Biomass handling capacity of inland port 𝑚 ∈ ℳ𝓅 with capacity 𝑙, Production capacity of size 𝑙 ∈ ℒ at location 𝑞 ∈ 𝒥 ∪ 𝒦, Capacity of truck for transporting biomass type 𝑏 ∈ ℬ, Capacity of unit transportation mode 𝑛 ∈ 𝒩 for transporting densified biomass type 𝑏 ∈ ℬ, Conversion rate of biomass type 𝑏 ∈ ℬ to be densified, Conversion rate of densified biomass type 𝑏 ∈ ℬ to olefin, Deterioration rate of biomass type 𝑏 ∈ ℬ in period 𝑡 ∈ 𝒯, Carbon emission factor of densifying a unit of biomass type 𝑏 ∈ ℬ at densification depot 𝑗 ∈ 𝒥 with capacity 𝑙 ∈ ℒ in period 𝑡 ∈ 𝒯, Carbon emission factor for producing a unit of olefin by using biomass type 𝑏 ∈ ℬ at plant 𝑘 ∈ 𝒦 with capacity 𝑙 ∈ ℒ in period 𝑡 ∈ 𝒯, Carbon emission factor for a unit transportation mode 𝑛 ∈ 𝒩 in period 𝑡 ∈ 𝒯, Carbon emission tax rate in period 𝑡 ∈ 𝒯, Amount of demand at period 𝑡 ∈ 𝒯,

Decision Variables 1 if a densification depot of capacity 𝑙 ∈ ℒ is opened at candidate location 𝑗 ∈ 𝒥; 0 otherwise, 𝑌𝑚𝑙 1 if a railcars hub of capacity 𝑙 ∈ ℒ is used at candidate location 𝑚 ∈ ℳ𝒽; 0 otherwise, 𝑉𝑚𝑙 1 if an inland port of capacity 𝑙 ∈ ℒ is used at candidate location 𝑚 ∈ ℳ𝓅; 0 otherwise, 𝑍𝑘𝑙 1 if an olefin plant of capacity 𝑙 ∈ ℒ is opened at candidate location 𝑘 ∈ 𝒦; 0 otherwise, ℋ𝑏𝑞𝑡(𝜔) Amount of biomass type 𝑏 ∈ ℬ stored at location 𝑞 ∈ 𝒥 ∪ 𝒦 in period 𝑡 ∈ 𝒯 under scenario 𝜔𝜖Ω, 𝛹𝑏𝑞𝑡(𝜔) Amount of biomass type 𝑏 ∈ ℬ processed at location 𝑞 ∈ 𝒥 ∪ 𝒦 in period 𝑡 ∈ 𝒯 under scenario 𝜔𝜖Ω, 𝑃𝑘𝑡(𝜔) Amount of olefin produced at plant 𝑘 ∈ 𝒦 in period 𝑡 ∈ 𝒯 under scenario 𝜔𝜖Ω, 𝜑𝑏𝑖𝑞𝑡(𝜔) Amount of biomass type 𝑏 ∈ ℬ transported from supplier site 𝑖 ∈ ℐ to destination location 𝑞 ∈ 𝒥 ∪ 𝒦 in period 𝑡 ∈ 𝒯 under scenario 𝜔𝜖Ω, 𝛷𝑏𝑗𝑘𝑡(𝜔) Amount of biomass type 𝑏 ∈ ℬ transported from densification depot 𝑗 ∈ 𝒥 to olefin plant 𝑘 ∈ 𝒦 in period 𝑡 ∈ 𝒯 under scenario 𝜔𝜖Ω, 𝜙𝑏𝑗𝑚𝑘𝑡(𝜔) Amount of biomass type 𝑏 ∈ ℬ transported from densification depot 𝑗 ∈ 𝒥 to olefin plant 𝑘 ∈ 𝒦 via railcar hubs 𝑚 ∈ ℳ𝒽 in period 𝑡 ∈ 𝒯 under scenario 𝜔𝜖Ω, 𝑋𝑗𝑙

9

ACCEPTED MANUSCRIPT

𝑄𝑏𝑗𝑚𝑘𝑡(𝜔) Amount of biomass type 𝑏 ∈ ℬ transported from densification depot 𝑗 ∈ 𝒥 to olefin plant 𝑘 ∈ 𝒦 via inland ports 𝑚 ∈ ℳ𝓅 in period 𝑡 ∈ 𝒯 under scenario 𝜔𝜖Ω, 𝛩𝑏𝑚𝑘𝑡(𝜔) Number of unit railcar used to transport biomass type 𝑏 ∈ ℬ from hub 𝑚 ∈ ℳ𝒽 to olefin plant 𝑘 ∈ 𝒦 in period 𝑡 ∈ 𝒯 under scenario 𝜔𝜖Ω, 𝛤𝑏𝑚𝑘𝑡(𝜔) Number of barge used to transport biomass type 𝑏 ∈ ℬ from port 𝑚 ∈ ℳ𝑝 to olefin plant 𝑘 ∈ 𝒦 in period 𝑡 ∈ 𝒯 under scenario 𝜔𝜖Ω, This model elaborates a three-stage hybrid robust/stochastic program by incorporating multiple scenarios for biomass supplies seasonality and uncertainty set for carbon tax rate. The first-stage variables make facility configuration decisions. These long-term investment decisions are robust to both types of uncertainties (i.e., feedstock seasonality and carbon tax rate). Then, probabilistic scenarios for feedstock seasonality are conducted in the second-stage to determine short-term decisions over the planning period. Finally, to handle the carbon tax uncertainty, the robust counterpart of the recourse problem is formulated for each scenario in the third-stage of the model. Hence, section 3.1 describes the proposed nominal stochastic programming model by incorporating probabilistic scenarios. Likewise, the robust three-stage formulations are given in section 3.2 by defining adjustable and affinely adjustable versions. 3.1. The Nominal Stochastic Programing Model Let assume 𝒫𝜔 is the realization probability of each scenario 𝜔 ∈ Ω for biomass seasonality. Thus, 𝒫𝜔𝒬𝑁(𝔁,𝜔) can be introduced as the expected value of the objective function of the secondstage, in which 𝔁 = (𝑋𝑗𝑙, 𝑌𝑚𝑙,𝑉𝑚𝑙, 𝑍𝑘𝑙) denotes the vector of the first-stage binary decision variables. Therefore, the first-stage decision making problem is presented as follows: 𝒵𝒩𝒮 = 𝑀𝑖𝑛

∑∑𝑓 𝑋 + ∑ ∑𝑔 + ∑ 𝒫 𝒬 (𝔁,𝜔)

𝑚𝑙𝑌𝑚𝑙

𝑗𝑙 𝑗𝑙

𝑗𝜖𝒥 𝑙𝜖ℒ

𝑚𝜖ℳ𝒽 𝑙𝜖ℒ

+

∑ ∑𝑒 𝑚𝜖ℳ𝓅 𝑙𝜖ℒ

𝑚𝑙𝑉𝑚𝑙

+

∑∑ℎ 𝑘𝜖𝒦 𝑙𝜖ℒ

𝑘 𝑍𝑘𝑙 𝑙

(1)

𝜔 𝑁

𝜔𝜖Ω

s.t.

∑𝑋 ∑𝑌 ∑𝑉 ∑𝑍

𝑗𝑙

≤1

∀𝑗𝜖𝒥

(2)

𝑙∈ℒ

𝑚𝑙

≤1

∀𝑚𝜖ℳ𝒽

(3)

𝑚𝑙

≤1

∀𝑚𝜖ℳ𝓅

(4)

≤1

∀𝑘𝜖𝒦

(5)

∀ 𝑗 ∈ 𝒥, 𝑚𝜖ℳ, 𝑘 ∈ 𝒦, 𝑙𝜖ℒ

(6)

𝑙∈ℒ

𝑙∈ℒ

𝑘𝑙

𝑙∈ℒ

𝑋𝑗 𝑙, 𝑌𝑚𝑙, 𝑉𝑚𝑙, 𝑍𝑘 𝑙 ∈ {0, 1}

The objective function of the first-stage model (1) minimizes the cost of locating densification depots, multi-modal facilities, olefin plants and expected cost of the second-stage problem after uncertainty realization of biomass availabilities. Constraints (2)-(5) ensure that at most one densification depot, railcar hub, inland port and olefin plant of capacity 𝑙 ∈ ℒ is opened at candidate 10

ACCEPTED MANUSCRIPT

locations 𝑗𝜖𝒥, 𝑚𝜖ℳ𝒽, 𝑚𝜖ℳ𝓅 and 𝑘𝜖𝒦, respectively. The binary decision variables are shown in (6). Once facilities location identification is done, the olefin network operations, including procuring, storing, processing and transporting biomass, producing olefins to satisfy demands and carbon emission cost, commence with scenario realizations over the planning period 𝒯 = {1,2,...,|𝒯|}. The second-stage of the model is to generate robust control solution by involving different scenarios at operational level and assuming a nominal value 𝜏𝑡, 𝑡𝜖𝒯 for the carbon tax rate. To address the seasonality of the biomass supplies, a fixed number of scenarios |Ω| are randomly generated and used in the model. For each scenario 𝜔 ∈ Ω, the objective function of the second-stage and considered constraints are formulated as follows: Min 𝑇(𝜔)

{∑

}

𝒫𝜔𝒬𝑁(𝔁,𝜔)

𝜔𝜖Ω

(7)

where, for 𝜔 ∈ 𝛺, 𝒬𝑁(𝔁,𝜔) =

∑ ∑ ∑ ∑ ∑ [ (𝜍 +∑ ∑ ∑∑ ∑ + ∑∑∑ ∑ ∑ [𝛼 +

𝑏𝑖𝑡

+ 𝛼𝑏𝑖𝑗𝑡𝑑𝑖𝑗)𝜑𝑏𝑖𝑗𝑡(𝜔) + 𝑣𝑏𝑛𝑡Υ𝑏𝑖𝑗𝑡(𝜔)]

𝑡 ∈ 𝒯𝑏 ∈ ℬ 𝑖 ∈ ℐ 𝑗 ∈ 𝒥𝑛 ∈ 𝒩\{𝑛2,𝑛3}

[(𝜍𝑏𝑖𝑡 + 𝛼𝑏𝑖𝑘𝑡𝑑𝑖𝑘)𝜑𝑏𝑖𝑘𝑡(𝜔) + 𝑣𝑏𝑛𝑡Υ𝑏𝑖𝑘𝑡(𝜔)]

𝑡 ∈ 𝒯𝑏 ∈ ℬ\{𝑏2,𝑏3}𝑖 ∈ ℐ𝒸𝑘 ∈ 𝒦𝑛 ∈ 𝒩\{𝑛2,𝑛3}

𝑏𝑗𝑘𝑡𝑑𝑗𝑘𝛷𝑏𝑗𝑘𝑡(𝜔)

+ 𝑣𝑏𝑛𝑡Υ𝑏𝑗𝑘𝑡(𝜔)]

∑∑∑ ∑ [ ∑ ∑

(7.2) (7.3)

𝑡 ∈ 𝒯𝑏 ∈ ℬ𝑗 ∈ 𝒥𝑘 ∈ 𝒦𝑛 ∈ 𝒩\{𝑛2,𝑛3}

+

(7.1)

(𝛼𝑏𝑗𝑚𝑡𝑑𝑗𝑚𝜙𝑏𝑗𝑚𝑘𝑡(𝜔) + 𝑣𝑏𝑛𝑡Υ𝑏𝑗𝑚𝑡(𝜔))

𝑡 ∈ 𝒯𝑏 ∈ ℬ𝑗 ∈ 𝒥𝑘 ∈ 𝒦 𝑚 ∈ ℳ𝒽𝑛 ∈ 𝒩\{𝑛1,𝑛3}

∑ ∑ ] + ∑∑∑ ∑ ∑ ∑ [𝛽 𝑑 𝜙 (𝜔) + 𝑣 𝛩 (𝜔)] + ∑∑∑ ∑ ∑ ∑ [𝛾 𝑑 𝑄 (𝜔) + 𝑣 𝛤 (𝜔)] + ∑ ∑ ∑𝛿 ℋ (𝜔) + ∑ ∑ ∑ 𝛿 ℋ (𝜔) + ∑ ∑ ∑[∑𝑢 𝛹 (𝜔) + ∑ 𝑤 𝛹 (𝜔)] + ∑𝜏 [ ∑ ∑∑𝜆 𝛹 (𝜔) + ∑ ∑ ∑𝜂 𝛹 (𝜔) + ∑ 𝜚 ( ∑ ∑∑𝑑 𝜑 (𝜔) + ∑ ∑ ∑ 𝑑 𝛷 (𝜔)

(7.4)

(𝛼𝑏𝑗𝑚𝑡𝑑𝑗𝑚𝑄𝑏𝑗𝑚𝑘𝑡(𝜔) + 𝑣𝑏𝑛𝑡Υ𝑏𝑗𝑚𝑡(𝜔))

+

𝑚 ∈ ℳ𝓅𝑛 ∈ 𝒩\{𝑛1,𝑛2}

𝑏𝑚𝑘𝑡 𝑚𝑘 𝑏𝑗𝑚𝑘𝑡

𝑏𝑛𝑡 𝑏𝑚𝑘𝑡

(7.5)

𝑏𝑚𝑘𝑡 𝑚𝑘 𝑏𝑗𝑚𝑘𝑡

𝑏𝑛𝑡 𝑏𝑚𝑘𝑡

(7.6)

𝑡 ∈ 𝒯𝑏 ∈ ℬ𝑗 ∈ 𝒥𝑚 ∈ ℳ𝒽𝑘 ∈ 𝒦𝑛 ∈ 𝒩\{𝑛1,𝑛3}

𝑡 ∈ 𝒯𝑏 ∈ ℬ𝑗 ∈ 𝒥𝑚 ∈ ℳ𝓅𝑘 ∈ 𝒦𝑛 ∈ 𝒩\{𝑛1,𝑛2}

𝑏𝑗𝑡

𝑏𝑗𝑡

𝑏𝑘𝑡

𝑡 ∈ 𝒯𝑏 ∈ ℬ𝑗 ∈ 𝒥

𝑏𝑗𝑙𝑡

𝑏𝑗𝑡

𝑡 ∈ 𝒯𝑏 ∈ ℬ𝑙 ∈ ℒ 𝑗 ∈ 𝒥

𝑡

𝑡∈𝑇

𝑏𝑘𝑡

(7.7)

𝑡 ∈ 𝒯𝑏 ∈ ℬ𝑘 ∈ 𝒦

𝑘𝑙𝑡

𝑏𝑘𝑡

(7.8)

𝑘∈𝒦

𝑏𝑗𝑙𝑡

𝑏 ∈ ℬ 𝑗𝜖𝒥 𝑙 ∈ ℒ

𝑏𝑗𝑡

𝑏𝑘𝑙𝑡

𝑛𝑡

𝑛𝜖𝒩\{𝑛2,𝑛3}

𝑏𝑘𝑡

(7.9)

𝑏 ∈ ℬ𝑘 ∈ 𝒦𝑙 ∈ ℒ

𝑖𝑗 𝑏𝑖𝑗𝑡

𝑏 ∈ ℬ𝑖 ∈ ℐ𝑗 ∈ 𝒥

𝑗𝑘

𝑏𝑗𝑘𝑡

(7.10)

𝑏 ∈ ℬ𝑗 ∈ 𝒥𝑘 ∈ 𝒦

11

ACCEPTED MANUSCRIPT

+

∑∑ ∑ 𝑑 ( ∑ 𝜙 𝑗𝑚

𝑏 ∈ ℬ𝑗 ∈ 𝒥𝑘 ∈ 𝒦

𝑏𝑗𝑚𝑘𝑡(𝜔)

+

𝑚 ∈ ℳ𝒽

∑𝑄

)

𝑏𝑗𝑚𝑘𝑡(𝜔)

𝑚 ∈ ℳ𝓅

)

∑ ∑∑𝑑

+

𝑖𝑘𝜑𝑏𝑖𝑘𝑡(𝜔)

𝑏 ∈ ℬ\{𝑏2,𝑏3} 𝑖 ∈ ℐ 𝑘 ∈ 𝒦

∑

+

𝜚𝑛𝑡

𝑛𝜖𝒩\{𝑛1,𝑛3}

∑

+

∑∑ ∑ ∑ 𝑑

𝑚𝑘𝜙𝑏𝑗𝑚𝑘𝑡(𝜔)

𝑏 ∈ ℬ𝑗 ∈ 𝒥𝑚 ∈ ℳ𝒽𝑘 ∈ 𝒦

𝜚𝑛𝑡

𝑛𝜖𝒩\{𝑛1,𝑛2}

∑∑ ∑ ∑ 𝑑

𝑚𝑘𝑄𝑏𝑗𝑚𝑘𝑡(𝜔)

𝑏 ∈ ℬ𝑗 ∈ 𝒥𝑚 ∈ ℳ𝓅𝑘 ∈ 𝒦

(7.11)

]

(7.12)

s.t.

∑𝜑 ∑𝜑

𝑏1𝑖𝑗𝑡(𝜔)

+

𝑗∈𝒥

∑𝜑

𝑏1𝑖𝑘𝑡(𝜔)

≤ 𝑟𝑏1𝑖𝑡(𝜔)

∀𝑖𝜖ℐ𝒸,𝑡𝜖𝒯

(8)

∀𝑏 ∈ ℬ\{𝑏1},𝑖𝜖ℐ𝒻 ∪ ℐ𝓂, 𝑡𝜖𝒯

(9)

𝑘∈𝒦

𝑏𝑖𝑗𝑡(𝜔)

≤ 𝑟𝑏𝑖𝑡(𝜔)

𝑗∈𝒥

∑𝜑 (𝜔) + ∑𝜑 (𝜔) + ∑𝜑

(1 ― 𝜃𝑏1𝑡)ℋ𝑏1𝑗𝑡 ― 1(𝜔) +

𝑏1𝑖𝑗𝑡(𝜔)

= ℋ𝑏1𝑗𝑡(𝜔) + 𝛹𝑏1𝑗𝑡(𝜔)

∀𝑗 ∈ 𝒥,𝑡𝜖 𝒯 (10)

𝑖𝜖ℐ𝒸

(1 ― 𝜃𝑏2𝑡)ℋ𝑏2𝑗𝑡 ― 1

𝑏2𝑖𝑗𝑡(𝜔)

= ℋ𝑏2𝑗𝑡(𝜔) + 𝛹𝑏2𝑗𝑡(𝜔)

∀𝑗 ∈ 𝒥,𝑡𝜖 𝒯 (11)

𝑏3𝑖𝑗𝑡(𝜔)

= ℋ𝑏3𝑗𝑡(𝜔) + 𝛹𝑏3𝑗𝑡(𝜔)

∀𝑗 ∈ 𝒥,𝑡𝜖 𝒯 (12)

𝑖𝜖ℐ𝒻

(1 ― 𝜃𝑏3𝑡)ℋ𝑏3𝑗𝑡 ― 1

𝑖𝜖ℐ𝓂

∑ℋ ∑𝛹 ∑𝛷 𝑏𝜖ℬ

∑𝑐 𝑋 (𝜔) ≤ ∑𝜛 𝑋 (𝜔) + ∑ ∑𝜙

𝑏𝑗𝑡(𝜔)

≤

𝑏𝑗𝑡

𝑏𝜖ℬ

𝑗𝑙 𝑗𝑙

∀𝑗 ∈ 𝒥,𝑡𝜖 𝒯 (13)

𝑗𝑙 𝑗𝑙

∀𝑗 ∈ 𝒥,𝑡𝜖 𝒯 (14)

𝑙∈ℒ

𝑙∈ℒ

𝑏𝑗𝑘𝑡

𝑘𝜖𝒦

𝑏𝑗𝑚𝑘𝑡(𝜔)

∑ ∑𝑄

+

𝑚 ∈ ℳ𝒽 𝑘𝜖𝒦

𝑏𝑗𝑚𝑘𝑡(𝜔)

≤ Λ𝑏𝛹𝑏𝑗𝑡(𝜔)

𝑚 ∈ ℳ𝓅 𝑘𝜖𝒦

∀𝑏 ∈ ℬ, 𝑗 ∈ 𝒥,𝑡𝜖 𝒯 (15)

∑ ∑ ∑𝜙 ∑ ∑ ∑𝑄

∑𝜁 𝑌 (𝜔) ≤ ∑𝑠 𝑉 (𝜔) + ∑𝜑 (𝜔) + ∑𝛷 𝑏𝑗𝑚𝑘𝑡(𝜔)

𝑏 ∈ ℬ𝑗 ∈ 𝒥 𝑘𝜖𝒦

≤

𝑚𝑙 𝑚𝑙

∀𝑚 ∈ ℳ𝒽,𝑡𝜖𝒯 (16)

𝑚𝑙 𝑚𝑙

∀𝑚 ∈ ℳ𝓅,𝑡𝜖𝒯 (17)

𝑙∈ℒ

𝑏𝑗𝑚𝑘𝑡

𝑏 ∈ ℬ𝑗 ∈ 𝒥 𝑘𝜖𝒦

𝑙∈ℒ

ℋ𝑏1𝑘𝑡 ― 1

𝑏1𝑗𝑘𝑡(𝜔)

𝑏1𝑖𝑘𝑡

𝑖𝜖ℐ𝒸

𝑗∈𝒥

+

∑∑𝜙

𝑏1𝑗𝑚𝑘𝑡(𝜔)

𝑗 ∈ 𝒥𝑚𝜖ℳ𝒽

+

∑∑𝑄

𝑏1𝑗𝑚𝑘𝑡(𝜔)

𝑗 ∈ 𝒥𝑚𝜖ℳ𝓅

= ℋ𝑏1𝑘𝑡(𝜔) + 𝛹𝑏1𝑘𝑡(𝜔) ∀𝑘𝜖𝒦,𝑡𝜖 𝒯 (18) ℋ𝑏𝑘𝑡 ― 1(𝜔) +

∑𝛷

𝑏𝑗𝑘𝑡(𝜔)

𝑗∈𝒥

+

∑∑𝜙 𝑗 ∈ 𝒥𝑚𝜖ℳ𝒽

𝑏𝑗𝑚𝑘𝑡(𝜔)

+

∑∑𝑄

𝑏𝑗𝑚𝑘𝑡(𝜔)

𝑗 ∈ 𝒥𝑚𝜖ℳ𝓅

12

ACCEPTED MANUSCRIPT

= ℋ𝑏𝑘𝑡(𝜔) + 𝛹𝑏𝑘𝑡(𝜔) ∀𝑏 ∈ ℬ\{𝑏1}, 𝑘𝜖𝒦,𝑡𝜖 𝒯 (19)

∑ℋ ∑𝛹

∑𝑐 𝑍 (𝜔) ≤ ∑𝜛 𝑍 𝑃 (𝜔) ≤ ∑ 𝜌 𝛹 (𝜔) ∑𝑃 (𝜔) ≥ 𝔇 𝜗 𝛩 (𝜔) ≥ ∑ 𝜙 𝜗 𝛤 (𝜔) ≥ ∑ 𝑄 ∑ ∑ ∑ ∑∑𝜙 𝑏𝑘𝑡(𝜔)

≤

𝑏𝜖ℬ

𝑘𝑙 𝑘𝑙

∀𝑘𝜖𝒦,𝑡𝜖 𝒯 (20)

𝑘𝑙 𝑘𝑙

∀𝑘𝜖𝒦, 𝑡𝜖𝑇 (21)

𝑙∈ℒ

𝑏𝑘𝑡

𝑏𝜖ℬ

𝑙∈ℒ

𝑘𝑡

𝑏

∀𝑘𝜖𝒦, 𝑡𝜖𝑇 (22)

𝑏𝑘𝑡

𝑏∈ℬ

𝑘𝑡

∀𝑡𝜖𝑇 (23)

𝑡

𝑘𝜖𝒦

𝑏𝑛2 𝑏𝑚𝑘𝑡

𝑏𝑗𝑚𝑘𝑡(𝜔)

∀𝑏 ∈ ℬ, 𝑚 ∈ ℳ𝒽, 𝑘 ∈ 𝒦, 𝑡𝜖 𝒯 (24)

𝑏𝑗𝑚𝑘𝑡(𝜔)

∀𝑏 ∈ ℬ, 𝑚 ∈ ℳ𝓅, 𝑘 ∈ 𝒦, 𝑡𝜖 𝒯 (25)

𝑗∈𝒥

𝑏𝑛3 𝑏𝑚𝑘𝑡

𝑗∈𝒥

𝑏𝑗𝑚𝑘𝑡(𝜔)

𝑏 ∈ ℬ 𝑗 ∈ 𝒥𝑚 ∈ ℳ𝒽 𝑘𝜖𝒦 𝑡𝜖𝒯

+

∑ ∑ ∑ ∑∑𝑄

𝑏𝑗𝑚𝑘𝑡(𝜔)

𝑏 ∈ ℬ 𝑗 ∈ 𝒥𝑚 ∈ ℳ𝓅 𝑘𝜖𝒦 𝑡𝜖𝒯

≥γ

∑ ∑ ∑Λ 𝛹 𝑏

(26) 𝑏𝑗𝑡(𝜔)

𝑏 ∈ ℬ𝑗 ∈ 𝒥 𝑡𝜖𝒯

𝜑𝑏𝑖𝑗𝑡(𝜔), 𝜑𝑏𝑖𝑘𝑡(𝜔), 𝛷𝑏𝑗𝑘𝑡(𝜔),𝜙𝑏𝑗𝑚𝑘𝑡(𝜔), 𝑄𝑏𝑗𝑚𝑘𝑡(𝜔), ℋ𝑏𝑗𝑡(𝜔),ℋ𝑏𝑘𝑡(𝜔), 𝛹𝑏𝑗𝑡(𝜔), 𝛹𝑏𝑘𝑡(𝜔), and 𝑃𝑘𝑡(𝜔) 𝜖 ℝ + ∀𝑏 ∈ ℬ, 𝑖 ∈ ℐ, 𝑗 ∈ 𝒥, 𝑘 ∈ 𝒦, 𝑚 ∈ ℳ, 𝑡𝜖 𝒯 (27) + 𝛩𝑏𝑚𝑘𝑡(𝜔) and 𝛤𝑏𝑚𝑘𝑡(𝜔) 𝜖 ℤ ∀𝑏 ∈ ℬ, 𝑘 ∈ 𝒦, 𝑚 ∈ ℳ, 𝑡𝜖𝒯 (28) The objective function (7) is the expected value of the total planning and operational costs, involving biomass procuring and transporting costs from biomass supply sites to densification depots (7.1) and olefin plants (7.2), biomass transporting costs from densification depots to olefin plants (7.3) and multi-modal facilities (7.4), biomass transporting costs from railcar hubs (7.5) and inland ports (7.6) to olefin plants, biomass holding costs (7.7) and processing costs (7.8) at densification depots and olefin plants, carbon emissions of biomass densification and olefin production processes (7.10), and carbon emission costs of transportation modes including trucks (7.11), railcars (7.12) and barges (7.13). Constraints (8) and (9) represent the availabilities of biomass supplies under different scenarios. Constraint (8) ensures that corn-stover supplies are allowed to be shipped either to densification depots or directly to olefin plants without using multimodal facilities, while constraint (9) restricts forest residue and MSW supplies to be shipped only to densification depots. That is because corn-stover feedstock is in bale format and does not require further size reduction and thus if it is closer to olefin plants than densification depots, it can be shipped directly to olefin plants without densification. Constraints (10)-(12) verify the equilibrium flow-conservation state of different biomass types at densification depots in consecutive time periods in which the amount of stored feedstock type 𝑏 ∈ ℬ from the previous time period by considering deterioration rate plus the amount of transporting feedstock type 𝑏 ∈ ℬ from supplier 13

ACCEPTED MANUSCRIPT

sites in the current time period is equal to the amounts of processing and storing feedstock type 𝑏 ∈ ℬ in the current time period. Note that constraints (10)-(12) represent the equilibrium flowconservation states for corn-stover, forest residue, and MSW feedstock types, respectively. Constraints (13) and (20) limit the amount biomass that can be stored at densification depots and olefin plants, respectively. These constraints indicate the capacity limitations for storing feedstock supplies at each densification depot and olefin plant with different sizes, respectively. Constraints (14) and (21) restrict the amount of the biomass densified and processed to produce olefin at densification depots and olefin plants, respectively. These constraints also show the capacity limitations of each densification depot and olefin plant with different sizes for densifying and processing the feedstock supplies, respectively. Constraint (15) ensures the maximum amount of the densified biomass that can be transported from densification depots to olefin plants either through multi-modal facilities or through trucks. This constraint presents that the total amounts of supplies transporting from each densification depot to olefin plants, railcar hubs, and inland ports cannot exceed the amount of densified supply for each feedstock type. Constraints (16) and (17) indicate the handling capacity of the railcar hubs and inland ports, respectively. These constraints show the transportation capacity of each railcar hub and inland port with different sizes, respectively. Constraints (18)-(19) verify the equilibrium flow-conservation state of different feedstock types at each olefin plant in which the amount of stored densified feedstock type 𝑏 ∈ ℬ from the previous time period plus the total amounts of transporting densified feedstock type 𝑏 ∈ ℬ from densification depots via truck, railcar, and barge in the current time period is equal to the amounts of processing and storing densified feedstock type 𝑏 ∈ ℬ in the current time period. Note that constraint (18) represents the equilibrium state for corn-stover feedstock, while constraint (19) directs the equilibrium states for forest residue and MSW feedstock types. Hence, constraint (18) has the extra term ∑𝑖𝜖ℐ 𝜑𝑏1𝑖𝑘𝑡(𝜔), which presents the amount of the corn-stover feedstock 𝒸

transporting from supplier sites in the current time period. Constraint (22) represents the production capacity of the olefin plants. This constraint shows the capacity limitation of each olefin plant with different sizes for producing olefin. Constraint (23) guarantees demands satisfaction for the olefin through the supply chain network. Thus, the total amount of olefin production cannot be less than the amount of demands at each time period. Constraints (24) and (25) count the number of railcars and barges that are needed to transport biomass type 𝑏 ∈ ℬ along arcs (𝑚,𝑘) ∈ 𝒜6 and (𝑚,𝑘) ∈ 𝒜7, respectively. Constraints (26) guarantees the applicability of the existing hubs and ports in state of Mississippi to transport densified feedstocks. As suggested by Mississippi Department of Transportation (2015), the ports based inland water transportation has high economic benefit to the state, we assume the developed network should include port transportation, therefore, we assume that the utilization rate γ of port and hub totally is 60%. Finally, the nonnegative continuous and integer variables are given in constraints (27) and (28), respectively. 3.2. Robust counterpart of the recourse problem Since the second-stage of the proposed model assumed a nominal value 𝜏𝑡, 𝑡𝜖𝒯 for the carbon tax rate, this section develops the third-stage including the robust counterpart (RC) to incorporate the carbon tax uncertainty in the model. Accordingly, RC of the recourse problem is proposed to find an optimal solution that satisfies all constraints for any carbon tax rate 𝜏𝑡𝜖𝒰 as follows: 𝒬𝑅𝐶(𝔁,𝜔) = 𝑀𝑖𝑛 𝓊(𝜔) such that ∀𝜏𝑡𝜖𝒰 𝑇(𝜔)

(29)

14

ACCEPTED MANUSCRIPT

(7.1) to (7.8) +

∑𝜏 [(7.9) to (7.12)] ≤ 𝓊(𝜔) 𝑡

(30)

𝑡∈𝑇

Hold constraints (8)-(28)

(31)

where 𝓊(𝜔) ∈ ℝ, and 𝑇(𝜔) is set of all here-and-now decision variables regarding the carbon tax rate uncertainty so that 𝑇(𝜔) = {𝜑𝑏𝑖𝑗𝑡(𝜔), 𝜑𝑏𝑖𝑘𝑡(𝜔), 𝜙bjmkt(𝜔), 𝑄𝑏𝑗𝑚𝑘𝑡(𝜔), 𝛹𝑏𝑗𝑡(𝜔), 𝛹𝑏𝑘𝑡(𝜔), 𝑃𝑘𝑡(𝜔), 𝛩𝑏𝑚𝑘𝑡(𝜔) and 𝛤𝑏𝑚𝑘𝑡(𝜔)}. Although 𝒬𝑅𝐶(𝔁,𝜔) model may provide an excessively conservative solution by requiring all decision variables in set 𝑇(𝜔) to be feasible for all values of 𝜏𝑡 in the uncertainty set. Hence, a reduced set of decision variables(𝜔) = {𝜑𝑏𝑖𝑗𝑡(𝜔), 𝜑𝑏𝑖𝑘𝑡(𝜔), 𝜙𝑏𝑗𝑚𝑘𝑡(𝜔), 𝑄𝑏𝑗𝑚𝑘𝑡(𝜔), 𝛹𝑏𝑗𝑡(𝜔), 𝛹𝑏𝑘𝑡(𝜔)}, is considered to obtain less conservative solutions. Hereafter, 𝓎(𝜔) denotes set of adjustable variables regarding the carbon tax uncertainty in which their values can be determined after the realization of tax rate uncertainty. Moreover, it is assumed that 𝜏𝑡 falls in a box uncertainty set, 𝜏𝑡 = 𝜏𝑡 +𝜉𝜏𝑡, where the perturbation scalar 𝜉 varies in the set 𝛯℘ ≡ {𝜉||𝜉| ≤ ℘}. Thus, the adjustable variables of set 𝓎(𝜔) can be adjusted directly to the perturbation scalar 𝜉 instead of 𝜏𝑡, (Ben-Tal et al., 2004). Therefore, the adjustable robust counterpart (𝐴𝑅𝐶) of the recourse problem can be developed as follows: 𝒬𝐴𝑅𝐶(𝔁,𝜔) = 𝑀𝑖𝑛 𝓊(𝜔) ∀ 𝜉𝜖𝛯℘, ∃ 𝓎(𝜔,𝜉) such that 𝓎(𝜔)

(32)

(30)-(31) (33) where the adjustable variables set 𝓎(𝜔) are allowed to tune themselves to the uncertain parameter 𝜉. However, the 𝐴𝑅𝐶 model is significantly less conservative than the usual 𝑅𝐶 model, in most cases exact evaluation of the 𝐴𝑅𝐶 model is computationally intractable. To address this problem, an efficient approximation method is provided by affinely adjustable robust counterpart (𝐴𝐴𝑅𝐶) model which tunes the adjustable variables to be affine functions of the uncertain data (Ben-Tal et al., 2004). Therefore, the adjustable variables of set 𝓎(𝜔) are justified to be: (1) 𝜑𝑏𝑖𝑗𝑡(𝜔) = 𝔏(0) 𝑏𝑖𝑗𝑡(𝜔) + 𝜉𝔏𝑏𝑖𝑗𝑡(𝜔) (1) 𝜑𝑏𝑖𝑘𝑡(𝜔) = 𝔏(0) 𝑏𝑖𝑘𝑡(𝜔) + 𝜉𝔏𝑏𝑖𝑘𝑡(𝜔) (1) 𝛷𝑏𝑗𝑘𝑡(𝜔) = 𝔏(0) 𝑏𝑗𝑘𝑡(𝜔) + 𝜉𝔏𝑏𝑗𝑘𝑡(𝜔) (1) 𝜙𝑏𝑗𝑚𝑘𝑡(𝜔) = 𝔈(0) 𝑏𝑗𝑚𝑘𝑡(𝜔) + 𝜉𝔈𝑏𝑗𝑚𝑘𝑡(𝜔) (1) 𝑄𝑏𝑗𝑚𝑘𝑡(𝜔) = 𝔔(0) 𝑏𝑗𝑚𝑘𝑡(𝜔) + 𝜉𝔔𝑏𝑗𝑚𝑘𝑡(𝜔) (1) 𝛹𝑏𝑗𝑡(𝜔) = 𝔄(0) 𝑏𝑗𝑡 (𝜔) + 𝜉𝔄𝑏𝑗𝑡 (𝜔) (1) 𝛹𝑏𝑘𝑡(𝜔) = 𝔄(0) 𝑏𝑘𝑡(𝜔) + 𝜉𝔄𝑏𝑘𝑡(𝜔)

(34) (35) (36) (37) (38) (39) (40)

(1) (0) (1) (0) (1) (0) (1) where 𝔏(0) 𝑏𝑖𝑗𝑡(𝜔), 𝔏𝑏𝑖𝑗𝑡(𝜔), 𝔏𝑏𝑖𝑘𝑡(𝜔), 𝔏𝑏𝑖𝑘𝑡(𝜔), 𝔏𝑏𝑗𝑘𝑡(𝜔), 𝔏𝑏𝑗𝑘𝑡(𝜔), 𝔈𝑏𝑗𝑚𝑘𝑡(𝜔), 𝔈𝑏𝑗𝑚𝑘𝑡(𝜔), (0) (1) (0) (1) (1) 𝔔(0) 𝑏𝑗𝑚𝑘𝑡(𝜔), 𝔔𝑏𝑗𝑚𝑘𝑡(𝜔), 𝔄𝑏𝑗𝑡 (𝜔),𝔄𝑏𝑗𝑡 (𝜔), 𝔄𝑏𝑘𝑡(𝜔) and 𝔄𝑏𝑘𝑡(𝜔) are non-adjustable variables.

Then, the adjustable variables 𝓎(𝜔) are replaced by the justifications (34)-(40) to develop the 𝐴𝐴𝑅𝐶 model of the recourse problem. Using (34)-(38), expressions (7.1)-(7.6) and (7.10)-(7-12) are modified as (42.1)-(42.6) and (42.9)-(42.11), respectively. Likewise, applying (39)-(40), expressions (7.8) and (7.9) are justified to (42.7) and (42.8), respectively as follows:

15

ACCEPTED MANUSCRIPT

𝒬𝐴𝐴𝑅𝐶(𝔁,𝜔) = 𝑚𝑖𝑛 𝓊(𝜔) such that ∀𝜉𝜖𝛯℘,

(41)

𝓎(𝜔)

(7.7) + (42.1) 𝑡𝑜 (42.7) +

∑𝜏 [(42.8) 𝑡𝑜 (42.11)] ≤ 𝓊(𝜔)

(42)

𝑡

𝑡∈𝑇

∑ ∑ ∑∑(𝜍 + 𝛼 𝑑 )𝜑 (𝜔) ∑ ∑ ∑ ∑ (𝜍 + 𝛼 𝑑 )(𝔏 (𝜔) + 𝜉𝔏 (𝜔)) ∑ ∑ ∑ ∑ 𝛼 𝑑 (𝔏 (𝜔) + 𝜉𝔏 (𝜔)) ∑ ∑ ∑ ∑ [ ∑ 𝛼 𝑑 (𝔈 (𝜔) + 𝜉𝔈 (𝜔)) 𝑏𝑖𝑡

𝑏𝑖𝑗𝑡 𝑖𝑗

(42.1)

𝑏𝑖𝑗𝑡

𝑡 ∈ 𝒯𝑏 ∈ ℬ 𝑖 ∈ ℐ 𝑗 ∈ 𝒥

𝑏𝑖𝑡

(0) 𝑏𝑖𝑗𝑡

𝑏𝑖𝑘𝑡 𝑖𝑘

(1) 𝑏𝑖𝑗𝑡

(42.2)

𝑡 ∈ 𝒯𝑏 ∈ ℬ\{𝑏2,𝑏3}𝑖 ∈ ℐ𝒸𝑘 ∈ 𝒦

(0) 𝑏𝑗𝑘𝑡

𝑏𝑗𝑘𝑡 𝑗𝑘

(1) 𝑏𝑗𝑘𝑡

(42.3)

𝑡 ∈ 𝒯𝑏 ∈ ℬ𝑗 ∈ 𝒥𝑘 ∈ 𝒦

(0) 𝑏𝑗𝑚𝑘𝑡

𝑏𝑗𝑚𝑡 𝑗𝑚

(1) 𝑏𝑗𝑚𝑘𝑡

𝑡 ∈ 𝒯𝑏 ∈ ℬ𝑗 ∈ 𝒥𝑘 ∈ 𝒦 𝑚 ∈ ℳ𝒽

+

∑

(

𝛼𝑏𝑗𝑚𝑡𝑑𝑗𝑚 𝔔(0) 𝑏𝑗𝑚𝑘𝑡(𝜔)

+

𝑚 ∈ ℳ𝓅

∑ ∑ ∑ ∑ ∑ [𝛽 ∑ ∑ ∑ ∑ ∑ [𝛾

]

)

𝜉𝔔(1) 𝑏𝑗𝑚𝑘𝑡(𝜔)

(42.4)

(1) 𝑟𝑐 (𝔈(0) 𝑏𝑗𝑚𝑘𝑡(𝜔) + 𝜉𝔈𝑏𝑗𝑚𝑘𝑡(𝜔)) + 𝑣𝑏𝑡𝛩𝑏𝑚𝑘𝑡(𝜔)]

(42.5)

(1) 𝑏𝑎 (𝔔(0) 𝑏𝑗𝑚𝑘𝑡(𝜔) + 𝜉𝔔𝑏𝑗𝑚𝑘𝑡(𝜔)) + 𝑣𝑏𝑡 𝛤𝑏𝑚𝑘𝑡(𝜔)]

(42.6)

𝑏𝑚𝑘𝑡𝑑𝑚𝑘

𝑡 ∈ 𝒯𝑏 ∈ ℬ𝑗 ∈ 𝒥𝑚 ∈ ℳ𝒽𝑘 ∈ 𝒦

𝑏𝑚𝑘𝑡𝑑𝑚𝑘

𝑡 ∈ 𝒯𝑏 ∈ ℬ𝑗 ∈ 𝒥𝑚 ∈ ℳ𝓅𝑘 ∈ 𝒦

∑ ∑ ∑[∑𝑢 (𝔄 (𝜔) + 𝜉𝔄 (𝜔)) + ∑ 𝑤 (𝔄 (𝜔) + 𝜉𝔄 (𝜔))] ∑ ∑∑𝜆 (𝔄 (𝜔) + 𝜉𝔄 (𝜔)) + ∑ ∑ ∑𝜂 (𝔄 (𝜔) + 𝜉𝔄 (𝜔)) ∑ ∑∑𝑑 (𝔏 (𝜔) + 𝜉𝔏 (𝜔)) ∑ 𝜚 + ∑ ∑ ∑ 𝑑 (𝔏 (𝜔) + 𝜉𝔏 (𝜔)) 𝑏𝑗𝑙𝑡

(0)

(1)

𝑏𝑗𝑡

𝑘𝑙𝑡

𝑏𝑗𝑡

𝑡 ∈ 𝒯𝑏 ∈ ℬ𝑙 ∈ ℒ 𝑗 ∈ 𝒥

(1)

𝑏𝑘𝑡

(0)

(1)

𝑏𝑗𝑡

𝑏𝑘𝑙𝑡

𝑏𝑗𝑡

𝑏 ∈ ℬ 𝑗𝜖𝒥 𝑙 ∈ ℒ

(0)

[

(0) 𝑏𝑖𝑗𝑡

𝑖𝑗

(1) 𝑏𝑖𝑗𝑡

(0) 𝑏𝑗𝑘𝑡

𝑗𝑘

(1) 𝑏𝑗𝑘𝑡

𝑏 ∈ ℬ𝑗 ∈ 𝒥𝑘 ∈ 𝒦

∑∑ ∑ 𝑑

𝑗𝑚

𝑏 ∈ ℬ𝑗 ∈ 𝒥𝑘 ∈ 𝒦

(

∑ (𝔈 + ∑ (𝔔

(0) 𝑏𝑗𝑚𝑘𝑡(𝜔)

+ 𝜉𝔈(1) 𝑏𝑗𝑚𝑘𝑡(𝜔))

𝑚 ∈ ℳ𝒽

(0) 𝑏𝑗𝑚𝑘𝑡(𝜔)

∑ ∑∑𝑑

∑∑ ∑ ∑ 𝑑 ∑∑ ∑ ∑ 𝑑

]

(1) (𝔏(0) 𝑏𝑖𝑘𝑡(𝜔) + 𝜉𝔏𝑏𝑖𝑘𝑡(𝜔))

𝑖𝑘

𝑏 ∈ ℬ\{𝑏2,𝑏3} 𝑖 ∈ ℐ 𝑘 ∈ 𝒦

𝜚𝑛𝑡

)

(42.9)

+ 𝜉𝔔(1) 𝑏𝑗𝑚𝑘𝑡(𝜔))

𝑚 ∈ ℳ𝓅

+

𝑛𝜖𝒩\{𝑛1,𝑛2}

(42.8)

𝑏𝑘𝑡

𝑏 ∈ ℬ𝑖 ∈ ℐ𝑗 ∈ 𝒥

+

𝑛𝜖𝒩\{𝑛1,𝑛3}

(1)

𝑏𝑘𝑡

𝑏 ∈ ℬ𝑘 ∈ 𝒦𝑙 ∈ ℒ

𝑛𝑡

∑ ∑

(42.7)

𝑏𝑘𝑡

𝑘∈𝒦

𝑏𝑗𝑙𝑡

𝑛𝜖𝒩\{𝑛2,𝑛3}

(0)

(1) (𝔈(0) 𝑏𝑗𝑚𝑘𝑡(𝜔) + 𝜉𝔈𝑏𝑗𝑚𝑘𝑡(𝜔))

(42.10)

(1) (𝔔(0) 𝑏𝑗𝑚𝑘𝑡(𝜔) + 𝜉𝔔𝑏𝑗𝑚𝑘𝑡(𝜔))

(42.11)

𝑚𝑘

𝑏 ∈ ℬ𝑗 ∈ 𝒥𝑚 ∈ ℳ𝒽𝑘 ∈ 𝒦

𝜚𝑛𝑡

𝑚𝑘

𝑏 ∈ ℬ𝑗 ∈ 𝒥𝑚 ∈ ℳ𝓅𝑘 ∈ 𝒦

16

ACCEPTED MANUSCRIPT

Applying (34)-(35), constrains (8)-(9) are modified as follows:

∑ (𝔏 ∑ (𝔏

(0) 𝑏1𝑖𝑗𝑡(𝜔)

+ 𝜉𝔏(1) 𝑏1𝑖𝑗𝑡(𝜔)) +

𝑗∈𝒥

∑ (𝔏

(0) 𝑏1𝑖𝑘𝑡(𝜔)

+ 𝜉𝔏(1) 𝑏1𝑖𝑘𝑡(𝜔)) ≤ 𝑟𝑏1𝑖𝑡(𝜔)

∀𝑖𝜖ℐ𝒸,𝑡𝜖𝒯

(43)

∀𝑏 ∈ ℬ\{𝑏1},𝑖𝜖ℐ𝒻 ∪ ℐ𝓂, 𝑡𝜖𝒯

(44)

𝑘∈𝒦

+ 𝜉𝔏(1) 𝑏𝑖𝑗𝑡(𝜔)) ≤ 𝑟𝑏𝑖𝑡(𝜔)

(0) 𝑏𝑖𝑗𝑡(𝜔)

𝑗∈𝒥

Using (34) and (39), constraints (10)-(12) and (14) are justified as (45)-(48), respectively.

(1 ― 𝜃𝑏1𝑡)ℋ𝑏1𝑗𝑡 ― 1(𝜔) +

∑ (𝔏

(0) 𝑏1𝑖𝑗𝑡(𝜔)

(0) (1) + 𝜉𝔏(1) 𝑏1𝑖𝑗𝑡(𝜔)) = ℋ𝑏1𝑗𝑡(𝜔) + 𝔄𝑏1𝑗𝑡(𝜔) + 𝜉𝔄𝑏1𝑗𝑡(𝜔)

𝑖𝜖ℐ𝒸

∀𝑗 ∈ 𝒥,𝑡𝜖𝒯

(1 ― 𝜃𝑏2𝑡)ℋ𝑏2𝑗𝑡 ― 1(𝜔) +

∑ (𝔏

(0) 𝑏2𝑖𝑗𝑡(𝜔)

(45)

(0) (1) + 𝜉𝔏(1) 𝑏2𝑖𝑗𝑡(𝜔)) = ℋ𝑏2𝑗𝑡(𝜔) + 𝔄𝑏2𝑗𝑡(𝜔) + 𝜉𝔄𝑏2𝑗𝑡(𝜔)

𝑖𝜖ℐ𝒻

∀𝑗 ∈ 𝒥,𝑡𝜖𝒯

(1 ― 𝜃𝑏3𝑡)ℋ𝑏3𝑗𝑡 ― 1(𝜔) +

∑ (𝔏

(0) 𝑏3𝑖𝑗𝑡(𝜔)

(46)

(0) (1) + 𝜉𝔏(1) 𝑏3𝑖𝑗𝑡(𝜔)) = ℋ𝑏3𝑗𝑡(𝜔) + 𝔄𝑏3𝑗𝑡(𝜔) + 𝜉𝔄𝑏3𝑗𝑡(𝜔)

𝑖𝜖ℐ𝓂

∑ (𝔄

(0) 𝑏𝑗𝑡 (𝜔)

+ 𝜉𝔄(1) 𝑏𝑗𝑡 (𝜔)) ≤

𝑏𝜖ℬ

∑𝜛 𝑋

𝑗𝑙 𝑗𝑙

∀𝑗 ∈ 𝒥,𝑡𝜖𝒯

(47)

∀𝑗 ∈ 𝒥,𝑡𝜖𝒯

(48)

𝑙∈ℒ

Constraints (15)-(17) are justified by using (36)-(38) in following:

∑ (𝔏

(0) 𝑏𝑗𝑘𝑡(𝜔)

∑ ∑ (𝔈 + ∑ ∑ (𝔔

+ 𝜉𝔏(1) 𝑏𝑗𝑘𝑡(𝜔)) +

𝑘𝜖𝒦

(0) 𝑏𝑗𝑚𝑘𝑡(𝜔)

+ 𝜉𝔈(1) 𝑏𝑗𝑚𝑘𝑡(𝜔))

𝑚 ∈ ℳ𝒽 𝑘𝜖𝒦

(0) 𝑏𝑗𝑚𝑘𝑡(𝜔)

(0) (1) + 𝜉𝔔(1) 𝑏𝑗𝑚𝑘𝑡(𝜔)) ≤ Λ𝑏(𝔄𝑏𝑗𝑡 (𝜔) + 𝜉𝔄𝑏𝑗𝑡 (𝜔))

𝑚 ∈ ℳ𝓅 𝑘𝜖𝒦

∀𝑏 ∈ ℬ, 𝑗 ∈ 𝒥,𝑡𝜖𝒯 (49)

∑ ∑ ∑ (𝔈 ∑ ∑ ∑ (𝔔

∑𝜁 𝑌 (𝜔)) ≤ ∑𝑠 𝑉

+ 𝜉𝔈(1) 𝑏𝑗𝑚𝑘𝑡(𝜔)) ≤

(0) 𝑏𝑗𝑚𝑘𝑡(𝜔)

𝑏 ∈ ℬ𝑗 ∈ 𝒥 𝑘𝜖𝒦

∀𝑚 ∈ ℳ𝒽,𝑡𝜖𝒯 (50)

𝑚𝑙 𝑚𝑙

𝑙∈ℒ

(0) 𝑏𝑗𝑚𝑘𝑡(𝜔)

+ 𝜉𝔔(1) 𝑏𝑗𝑚𝑘𝑡

𝑏 ∈ ℬ𝑗 ∈ 𝒥 𝑘𝜖𝒦

∀𝑚 ∈ ℳ𝓅,𝑡𝜖𝒯 (51)

𝑚𝑙 𝑚𝑙

𝑙∈ℒ

Considering (35)-(38) and (40), constraints (18)-(19) are modified as (52)-(53).

∑ (𝔏 + ∑ ∑ (𝔈

(0) 𝑏1𝑖𝑘𝑡(𝜔)

ℋ𝑏1𝑘𝑡 ― 1(𝜔) +

𝑖𝜖ℐ𝒸

(0) 𝑏1𝑗𝑚𝑘𝑡(𝜔)

𝑗 ∈ 𝒥𝑚𝜖ℳ𝒽

∑(𝔏 (𝜔) + 𝜉𝔏 (𝜔)) + ∑ ∑ (𝔔

+ 𝜉𝔏(1) 𝑏1𝑖𝑘𝑡(𝜔)) +

(0) 𝑏1𝑗𝑘𝑡

(1) 𝑏1𝑗𝑘𝑡(𝜔)

)

𝑗∈𝒥

+ 𝜉𝔈(1) 𝑏1𝑗𝑚𝑘𝑡

(0) 𝑏1𝑗𝑚𝑘𝑡(𝜔)

+ 𝜉𝔔(1) 𝑏1𝑗𝑚𝑘𝑡(𝜔))

𝑗 ∈ 𝒥𝑚𝜖ℳ𝓅

17

ACCEPTED MANUSCRIPT

(1) = ℋ𝑏1𝑘𝑡(𝜔) + 𝔄(0) 𝑏1𝑘𝑡(𝜔) + 𝜉𝔄𝑏1𝑘𝑡(𝜔)

∀𝑘𝜖𝒦,𝑡𝜖𝒯 (52) ℋ𝑏𝑘𝑡 ― 1(𝜔) +

∑ (𝔏

(0) 𝑏𝑗𝑘𝑡(𝜔)

+ 𝜉𝔏(1) 𝑏𝑗𝑘𝑡(𝜔)) +

𝑗∈𝒥

∑ ∑ (𝔈

(0) 𝑏𝑗𝑚𝑘𝑡(𝜔)

+ 𝜉𝔈(1) 𝑏𝑗𝑚𝑘𝑡(𝜔))

𝑗 ∈ 𝒥𝑚𝜖ℳ𝒽

+

∑ ∑ (𝔔

(0) 𝑏𝑗𝑚𝑘𝑡(𝜔)

+ 𝜉𝔔(1) 𝑏𝑗𝑚𝑘𝑡(𝜔))

𝑗 ∈ 𝒥𝑚𝜖ℳ𝓅

(1) = ℋ𝑏𝑘𝑡(𝜔) + 𝔄(0) 𝑏𝑘𝑡(𝜔) + 𝜉𝔄𝑏𝑘𝑡(𝜔) ∀𝑏 ∈ ℬ\{𝑏1}, 𝑘𝜖𝒦,𝑡𝜖𝒯 (53)

Constraints (21)-(22) are adjusted by using (40) in following:

∑ (𝔄

(0)

𝑏𝑘𝑡(𝜔)

+ 𝜉𝔄(1) 𝑏𝑘𝑡(𝜔)) ≤

∑𝜛

𝑘𝑙𝑍𝑘𝑙

∀𝑘𝜖𝒦, 𝑡𝜖𝑇 (54)

+ 𝜉𝔄(1) 𝑏𝑘𝑡(𝜔))

∀𝑘𝜖𝒦, 𝑡𝜖𝑇 (55)

𝑏𝜖ℬ

𝑙∈ℒ

∑ 𝜌 (𝔄

𝑃𝑘𝑡(𝜔) ≤

𝑏

(0)

𝑏𝑘𝑡(𝜔)

𝑏∈ℬ

Justifications (37)-(38) are used to modify constraints (24)-(25), respectively:

∑ (𝔈 (𝜔) ≥ ∑ (𝔔

𝜗𝑏𝑛2𝛩𝑏𝑚𝑘𝑡(𝜔) ≥

(0) 𝑏𝑗𝑚𝑘𝑡(𝜔)

+ 𝜉𝔈(1) 𝑏𝑗𝑚𝑘𝑡(𝜔))

∀𝑏 ∈ ℬ, 𝑚 ∈ ℳ𝒽, 𝑘 ∈ 𝒦, 𝑡𝜖𝒯 (56)

(0) 𝑏𝑗𝑚𝑘𝑡(𝜔)

+ 𝜉𝔔(1) 𝑏𝑗𝑚𝑘𝑡(𝜔))

∀𝑏 ∈ ℬ, 𝑚 ∈ ℳ𝓅, 𝑘 ∈ 𝒦, 𝑡𝜖𝒯 (57)

𝑗∈𝒥

𝜗𝑏𝑛3𝛤𝑏𝑚𝑘𝑡

𝑗∈𝒥

Moreover, constraints (26) are modified by applying (37)-(39) as:

∑ ∑ ∑ ∑∑(𝔈 (𝜔) + 𝜉𝔈 (𝜔)) + ∑ ∑ ∑ ∑ ∑ (𝔔 (0) 𝑏𝑗𝑚𝑘𝑡

(1) 𝑏𝑗𝑚𝑘𝑡

𝑏 ∈ ℬ 𝑗 ∈ 𝒥𝑚 ∈ ℳ𝒽 𝑘𝜖𝒦 𝑡𝜖𝒯

(0) 𝑏𝑗𝑚𝑘𝑡(𝜔)

+ 𝜉𝔔(1) 𝑏𝑗𝑚𝑘𝑡(𝜔))

(58)

𝑏 ∈ ℬ 𝑗 ∈ 𝒥𝑚 ∈ ℳ𝓅 𝑘𝜖𝒦 𝑡𝜖𝒯

≥ (60%)

∑ ∑ ∑Λ 𝔄

(0) 𝑏 𝑏𝑗𝑡 (𝜔)

+ 𝜉𝔄(1) 𝑏𝑗𝑡 (𝜔)

𝑏 ∈ ℬ𝑗 ∈ 𝒥 𝑡𝜖𝒯

Finally, constraints (27) are justified as (59): (1) (0) (1) (0) (1) (0) (1) (0) 𝔏(0) 𝑏𝑖𝑗𝑡(𝜔),𝔏𝑏𝑖𝑗𝑡(𝜔), 𝔏𝑏𝑖𝑘𝑡(𝜔),𝔏𝑏𝑖𝑘𝑡(𝜔), 𝔏𝑏𝑗𝑘𝑡(𝜔),𝔏𝑏𝑗𝑘𝑡(𝜔), 𝔈𝑏𝑗𝑚𝑘𝑡(𝜔), 𝔈𝑏𝑗𝑚𝑘𝑡(𝜔), 𝔔𝑏𝑗𝑚𝑘𝑡(𝜔), + (0) (1) (0) (1) 𝔔(1) 𝑏𝑗𝑚𝑘𝑡(𝜔), ℋ𝑏𝑗𝑡(𝜔),ℋ𝑏𝑘𝑡(𝜔), 𝔄𝑏𝑗𝑡 (𝜔), 𝔄𝑏𝑗𝑡 (𝜔), 𝔄𝑏𝑘𝑡(𝜔),𝔄𝑏𝑘𝑡(𝜔), and 𝑃𝑘𝑡(𝜔) 𝜖 ℝ ∀𝑏 ∈ ℬ, 𝑖 ∈ ℐ, 𝑗 ∈ 𝒥, 𝑘 ∈ 𝒦, 𝑚 ∈ ℳ, 𝑡𝜖𝒯 (59)

Obviously, justifications have not any replacements in objective expression (7.7) and constraints (13), (20), (23) and (28). Therefore, these objective expression and constraints are hold without any adjustment. 4. Solution approach 18

ACCEPTED MANUSCRIPT

To solve this stochastic olefin supply chain/production network design problem with data driven uncertain parameters, a novel solution approach is proposed. This approach is partly inspired from the Sample Average Approximation (SAA) method (Shapiro, 2008), which is an approximation of the stochastic model based on the equivalent mixed-integer programming model. The hybrid methodology incorporates the SAA method in the second-stage problem and the 𝑅𝐶 problem in the third-stage problem to cope with feedstocks seasonality and carbon tax rate uncertainty, respectively. 4.1. Scenario generation The availability of the biomass varies extensively during the year. For instance, corn is harvested from early September to November and therefore corn-stover is available during this time period. Forest residue is available the whole year except the winter months, from December to February, while MSW is available all year round. Furthermore, different biomass supplies fluctuate seasonally regarding to the various climatic changes (e.g., rainfall and temperature,) or extreme events (e.g., flooding and hurricane). These require a large set of scenarios to tackle the seasonal availability of different biomass supplies in developing the second-stage of the RTSSP model formulation. Hence, the historical availability of feedstocks for our testing ground is used to predict the future availability of biomass. Monte Carlo simulation is used to generate 𝑁 independent number of scenarios for biomass supply availability in the testing ground, given as {𝜔1,𝜔2,… ,𝜔𝑁} = Ω𝑁. Suppose the feedstocks supplies (i.e., corn-stover, forest residue and MSW) follow a multivariate normal distribution ℕ(𝜇,Σ) for each supplier site 𝑖 ∈ ℐ and time period 𝑡 ∈ 𝒯 in which vector 𝜇 and matrix Σ show the predicted supply and the prediction error, respectively. The prediction error is also considered to be independent and identically distributed based on a normal distribution with mean zero and variance 𝜎2. Thus, the Monte Carlo simulation technique is able to generate a large number of scenarios with equal occurrence probabilities 𝒫𝜔 = 1 𝑁. 4.2. Sample average approximation method Aforementioned scenario-based three-stage robust/stochastic programming model is a complex large-scale optimization problem and commercial solvers such as Gurobi cannot solve it, as a large number of scenarios is involved for feedstocks seasonality realization. This motivates us to use the SAA method to reduce the dimensionality of the proposed problem. Because, SAA solves a small set of scenarios repeatedly instead of solving the original problem with large number of scenarios and terminates by reaching a pre-specified tolerance gap. Thus, SAA provides high quality solutions along with the statistical estimation of their optimality gap (Norkin et al., 1998). Studies by (Kleywegt et al., 2002) and (Mak et al., 1999) are referred for the proof of convergence and evaluation of statistical performance of SAA, respectively. Furthermore, many studies applied SAA to solve complex large-scale biomass supply chain problems, such as (Osmani and Zhang, 2014; Shabani et al., 2014; Bairamzadeh et al., 2015; Ghaderi et al., 2016) and others. To solve this three-stage stochastic olefin network design problem, we are inspired by the SAA method to tackle the uncertainty posed by the feedstock availabilities in the second-stage problem. This technique is used to develop an approximation for the stochastic model of the second-stage by an equivalent mix-integer programing model. The proposed method incorporates the mixinteger program into the first-stage decision making problem by using probabilistic scenarios. SAA is extensively used for solving the two-stage stochastic programming problems (Santoso et al., 2005; Schütz et al., 2009; Alizadeh et al., 2015, 2016; Amiri-Aref et al., 2018). This research 19

ACCEPTED MANUSCRIPT

developed a new hybrid robust/stochastic approach which extended the SAA technique for solving a three-stage stochastic problem. 4.3. Affinely adjustable robust counterpart The approximated mix-integer program so far considers a nominal value 𝜏𝑡, 𝑡𝜖𝒯 for the carbon tax rate. The robust counterpart of the problem is incorporated in the third-stage problem to realize the carbon tax rate uncertainty of the hybrid robust/stochastic model. Since the robust counterpart problem (29)-(31) requires all the operational variables to be feasible for all values of 𝜏𝑡 in the uncertainty set, it provides an excessively conservative solution. Then, 𝐴𝑅𝐶 problem (32)-(33) is proposed to obtain less conservative solutions, where the adjustable variables regarding to the carbon tax uncertainty are to be determined after realization of the tax rate uncertainty. Although the adjustable robust model provides less conservative solutions, exact evaluation of this problem is again computationally cumbersome. Hence, an effective approximation-based method is provided by 𝐴𝐴𝑅𝐶 model (41)-(59) to handle this challenge in which the adjustable variables are to be affine functions of the uncertain data. Thus, carbon tax rate uncertainty is handled in the third-stage problem by using the 𝐴𝐴𝑅𝐶 program. After generating 𝑁 scenarios and by using the 𝒬𝑅𝐶(𝔁,𝜔) formulation (29)-(31), the three-stage hybrid robust/stochastic models with probabilistic scenarios for feedstocks availability and uncertainty set for the carbon tax rate are summarized as follows: 1 𝒵𝑅𝐶 = 𝑚𝑖𝑛 𝓬𝔁𝑇 + 𝒬 (𝔁,𝜔) (60) 𝑁 𝑅𝐶 𝔁 ∈ {0,1} 𝑁

∑

𝜔∈𝛺

s.t. Hold constraints (2)-(6) and (30)-(31) (61) where 𝔁 = (𝑋𝑗𝑙, 𝑌𝑚𝑙,𝑉𝑚𝑙, 𝑍𝑘𝑙) and 𝓬 = (𝑓𝑗𝑙, 𝑔𝑚𝑙,𝑒𝑚𝑙, ℎ𝑘𝑙) denote the vectors of the first-stage binary decision variables and their corresponding fixed costs for facility configuration. Using the 𝒬𝐴𝐴𝑅𝐶 (𝔁,𝜔) formulation (41)-(59), the affinely adjustable version of the RTSSP model is given as follows: 1 𝒵𝐴𝐴𝑅𝐶 = 𝑚𝑖𝑛 𝓬𝔁𝑇 + 𝒬 (𝔁,𝜔) (62) 𝑁 𝐴𝐴𝑅𝐶 𝔁 ∈ {0,1} 𝑁

∑

𝜔∈𝛺

s.t. Hold constraints (2)-(6), (13), (20), (23), (28) and (42)-(59) (63) where the first terms of (60) and (62) indicate the first-stage objective function and their second terms denote the expected objective function of the integration of the robust optimization and stochastic programming problems. Finally, the corresponding constraints of two models are shown in (61) and (63), respectively. 4.4. Solution validation analysis This section represents a validation analysis technique based on the technique proposed by (Shapiro, 2008) for optimality gap estimation. This technique validates the solution found by the proposed integrated robust SAA based on the difference between the objective value of the obtained solution and the optimal value of the true problem. Since finding the optimal value for the true problem is almost impossible due to the large number of required scenarios, statistical lower and upper bounds for the true objective value can be used to validate the solutions (Shapiro, 2008). The main idea of this method is to use the statistical lower and upper bounds of the optimal

20

ACCEPTED MANUSCRIPT

objective value of the integrated robust SAA to evaluate the optimality gap by considering a level of confidence, 𝛼. Hence, the lower and upper bounds are built as follows: 4.4.1. Lower bound 𝑚 Let 𝐱𝑚 𝑁 and 𝐲𝑁 , 𝑚 = 1,…,𝑀, denote the optimal solution vectors of the first-stage and the integrated second and third-stages problems found by the algorithm under a scenario sample size 𝑚 𝑁 in the 𝑚th replication, respectively, and ℱ(𝐱𝑚 𝑁 ,𝐲𝑁 ) be the optimal objective value with regard 𝑚 to the values of 𝐱𝑚 𝑁 and 𝐲𝑁 . Then, the statistical lower bound can be estimated by averaging the optimal objective values of 𝑀 independent replications of the algorithm for𝑁 generated scenarios as following:

1 𝑀 ℱ (𝐱 𝑀 𝑁 ,𝐲𝑁 ) = 𝑀

𝑀

∑ ℱ(𝐱

𝑚 𝑚 𝑁 ,𝐲𝑁 )

(64)

𝑚=1

Moreover, considering M independent scenario generations, the standard deviation is estimated as follows: 𝑀 𝑆𝐷𝑁

1 = 𝑀(𝑀 ― 1)

𝑀

∑ (ℱ (𝐱

𝑚 𝑚 𝑁 ,𝐲𝑁

) ― ℱ(𝐱𝑀𝑁,𝐲𝑀𝑁))2

(65)

𝑚=1

Using the average (64) and standard deviation estimators (65), an approximate (1 ― 𝛼) × 100% confidence lower bound for the true objective value, 𝔏𝑀 𝑁 , is given as follows: 𝑀

𝑀 𝑀 (66) ℒ𝑀 𝑁,1 ― 𝛼 = ℱ(𝒙𝑁 ,𝒚𝑁 ) ― 𝑡𝛼,𝑀 ― 1𝑆𝐷𝑁 where 𝑡𝛼,𝑀 ― 1 represents the 𝛼-critical value of the 𝑡-distribution with 𝑀 ― 1 degrees of freedom.

4.4.2. Upper bound Let 𝐱 denotes the optimal solution vector of the first-stage problem for a scenario sample size N. The statistical upper bound of the true optimal objective value can be estimated by the sampling ' ' procedure. Generating the independent sample scenarios{𝜔1,𝜔2,…,𝜔𝑁 } ∈ Ω𝑁 ⊂ Ω, where𝑁' > 𝑁, the problem is solved with 𝐱 as an input and 𝑁' scenarios. The new sample scenarios 𝑁' are independent to the samples used in finding the optimal 𝐱. Let 𝐲𝑁∗' denotes the optimal solution of the integrated second and third-stages problem through the 𝑁'sample scenarios and 𝐲𝜔∗ be the scenario-wise solution. Also, ℱ(𝐱,𝐲𝑁∗') and ℱ(𝐱,𝐲𝜔∗ ) show the optimal objective value regarding to the values of 𝐱 and 𝐲𝑁∗' and the scenario-wise objective value of the problem, respectively. Thus, standard deviation of ℱ𝜔(𝐱,𝐲𝑁∗') can be evaluated as follows: 𝑆𝐷𝑁'(𝐱) =

1

𝑁'

∑ ( ℱ (𝐱,𝐲 𝑁 (𝑁 ― 1) '

𝜔

'

∗ 𝜔)

― ℱ(𝐱,𝐲𝑁∗'))

2

(67)

𝜔=1

Then, an approximate (1 ― 𝛼) × 100% confidence upper bound of the true objective value, 𝒰𝑁 , is given as: '

𝒰𝑁',1 ― 𝛼 = ℱ(𝐱,𝐲𝑁∗') + 𝓏𝛼𝑆𝐷𝑁'(𝐱) (68) where 𝓏𝛼 denotes the 𝛼-critical value of the standard normal distribution with (1 ― 𝛼) × 100% confidence level. 21

ACCEPTED MANUSCRIPT

Thus, an approximate (1 ― 𝛼) × 100% confidence interval for the expected true objective value can be evaluated as (ℒ𝑀 𝑁,1 ― 𝛼,𝒰𝑁',1 ― 𝛼) by using (66) and (68). Finally, the statistical 𝑀 ' optimality gap percentage, 𝑔𝑎𝑝𝑁,𝑁 , is estimated as follows: 𝑔𝑎𝑝𝑀 𝑁,𝑁'

=

𝑀 𝒰𝑀 𝑁',1 ― 𝛼 ― ℒ𝑁,1 ― 𝛼

𝒰𝑀 𝑁',1 ― 𝛼

× 100%

(69)

5. The State of Mississippi Case Study To validate the effects of the uncertainty and adjustability on the decisions and costs, this section presents a case study in the state of Mississippi and computational results obtained by the proposed solution approach for the olefin supply chain network design problem. The hybrid stochastic/robust optimization approach described in Section 3 is implemented through a case study using the realistic data scenarios of the state of Mississippi. All costs of this study are evaluated based on the US dollar value in 2018. The input parameters of the case study, the efficiency of the integrated robust SAA algorithm regarding to uncertainty and adjustability, experimental results, and some managerial insights are then presented in the following sections 5.1. Parameters description and data collection The availabilities of the three types of feedstock supplies (i.e. corn-stover, forest residues, and MSW) throughout the state of Mississippi are reported by National Renewable Energy Laboratory (NREL) and Mississippi Department of Environmental Quality (MDEQ). Fig. 2 depicts the geographical distribution of (a) corn-stover supplies, (b) forest residue supplies and (c) MSW supplies in the Mississippi state. Noticeably, NREL reported that the MS produced 0.96 million tons (MT) corn-stover at 33 supply sites and 1.8 MT forest residues at 31 supply sites (National Renewable Energy Laboratory, 2012). Moreover, the availability of the MSW is obtained from MDEQ calendar year 2014 report (Mississippi Department of Environmental Quality, 2014). Based on this report, the state of Mississippi generated totally 6.053 MT of MSW which involves wastes from food, paper, wood, plastics, glass, metals, etc. Note that, only 33.2% of this total availability, including wastes of paper and wood, can be used for olefin production and 34.3% of that number are recyclable (Quddus et al., 2018). Finally, the key feedstock parameters are shown in Table 1. The information in Table 1 is obtained from Xie et al. (2014) and Parker et al. (2008) . Global demand of olefin is primarily driven by economic growth and the relevant ramification. The world demand of olefin is growing and the present demand in petrochemical industry is over 155 MT per year (Houdek and Anderson, 2005). Since the state of Mississippi is considered as the testing ground for this study, we assume 0.33% of world demand for olefin can be satisfied through the olefin production by using the available biomass supplies at this state.

22

ACCEPTED MANUSCRIPT

Figure 2. Geographical distribution of feedstock supplies in the state of Mississippi

Furthermore, investigations reveal total number of 86 potential locations for establishing densification depots, 33 potential railcar hubs, 16 potential inland ports and 51 potential locations for opening olefin plants. Fig. 3 represents the location and distribution of potential densification depots (Fig. 3a), olefin plants (Fig. 3b), railcar hubs and inland ports (Fig. 3c) throughout the state of Mississippi. The fixed cost of locating a densification depot of capacity 220 US ton/day is equal to $3,086,656 (Lamers et al., 2015). Four densification capacities 𝑙 = 0.05 Million Tons per Year (MTY), 0.07 MTY, 0.1 MTY, 0.16 MTY are considered for densification depots. The annual fixed cost of using a railcar hub of capacity 1.05 MTY and an inland port of capacity 2.35 MTY is equal to $54,949 (Mahmudi and Flynn, 2006) and $306,000 (Searcy et al., 2007), respectively. Four rail ramp transporting capacities 𝑙 = 0.6 MTY, 0.8 MTY, 0.9 MTY, 1.05 MTY are considered for railcar hubs. Likewise, four port transporting capacities 𝑙 = 0.3 MTY, 0.6 MTY, 1.0 MTY, 1.5 MTY are considered for inland ports. The annual fixed cost for an olefin plant with capacity 0.1 MTY is equal to $43,000,000 (Chiang, 2004). For production capacities 𝑙 = 0.05 MTY, 0.1 MTY, 0.2 MTY, 0.4 MTY are considered for olefin plants.

Figure 3. Potential locations for densification depots, olefin plants and multi-modal facilities

23

ACCEPTED MANUSCRIPT

All costs are estimated based on a lifetime of 30 years and a discount factor 10%. Additionally, annual operating days for all facilities are considered to be 350 days. Although the actual investment costs for facilities would vary by location, a common fixed cost is used in this study for a reasonable approximation. Feedstock type

Availability (MTY)

Corn-stover Forest residue MSW

0.96 1.8 0.69

Table 1. Feedstock parameters Procurement Moisture Conversion Rate to Deterioration Rate Cost content olefin (per season) ($/dry ton) (% mass) (tons/dry ton) 35 15 0.41 10% 30 50 0.24 12% 0.0 50 0.24 12%

Storage cost ($/dry ton) 8 2 2

This study considers truck, railcar and barge to transport biomass/MSW from different nodes to the destinations. Trucks re used to transport biomass/MSW from feedstock suppliers to multimodal facilities or directly to olefin plants, in case the supplier is close to the olefin plant. The major cost components of using trucks for the biomass/MSW transportations are summarized in Table 2. Railcar and barge can also be used to transport biomass/MSW respectively from hubs and ports to olefin plants. The fixed and unit transportation cost component for rail shipments is estimated to be $2,248 and $1.12/mile/tone (Gonzales et al., 2013). Likewise, the fixed and unit transportation cost component for barge shipments are $5,775 and $0.017/mile/ton (Gonzales et al., 2013). This study used Arc GIS Desktop 10.4 software to create a transportation network and finding the shortest path between all source and densification pairs. This network includes major highways, local, rural and urban roads for truck transportation and main railroads and waterways respectively for railcar and barge transportations throughout the state of Mississippi. Additionally, the carbon emission factors for densification depot (Haque and Somerville, 2013), olefin plant (Ren et al., 2009) and transportation modes (Fareeduddin et al., 2015) are summarized in Table 3. Table 2. Data for biomass/MSW transportation by truck (Adopted from Parker et al., 2008) Item Loading/unloading Time dependent Distance dependent Truck capacity

Item Facilities:

Value 5.0 29.0 1.20 25

Unit $/wet ton $/hr/truckload $/mile/truckload Wet tons/truckload

Table 3. Data for carbon emission Value

Densification depot

Corn-stover Forest residue MSW

Olefin plant Transportation modes: Truck Rail Barge

0.00037 0.00124 0.00124 1.16

Unit tons 𝑐𝑜2/ton tons 𝑐𝑜2/ton tons 𝑐𝑜2/ton tons 𝑐𝑜2/ton

0.000297 tons/ton/mile 0.000025 tons/ton/mile 0.000048 tons/ton/mile

5.2. Computational results and managerial insights 24

ACCEPTED MANUSCRIPT

This section presents the computational results of solving the main case study problem and the extended instances by using the integrated robust SAA algorithm. The proposed mathematical model and solution algorithm are coded in python 2.7 and executed on a computer with 32 GB RAM and Intel Core i7 3.6 GHZ CPU. Besides, Gurobi Optimizer 7.5 (www.gurobi.com) is used as optimization solver for the proposed problem. This algorithm terminates when at least one of the following criteria is met: (a) the optimality gap (i.e. 𝑔𝑎𝑝𝑜𝑝𝑡 = 100% × |𝑈𝐵 ― 𝐿𝐵|/𝑈𝐵) falls below a threshold value (e.g., ϵ ≤1%) or (b) the maximum CPU limit 10 hours is reached. Table 4 shows the size of the testing problems for the RTSSP model. Table 4. Experimental Problem sizes Case

Binary Integer Continuous Total No. of |ℐ| |𝒥| |ℳ𝓀| |ℳ𝓅| |𝒦| |ℬ| |ℒ| |𝑇| variables variables variables variables Constraints

1 2 3 Base 5

30 50 70 94 110

25 45 65 86 100

15 22 27 33 40

8 10 13 16 20

20 30 42 51 60

3 3 3 3 3

4 4 4 4 4

4 4 4 4 4

272 428 588 744 880

5,520 11,520 20,160 29,988 43,200

48,520 321,560 1,161,120 2,868,816 5,575,764

166,852 592,928 1,455,744 2,819,328 4,650,160

7,165 14,252 24,028 34,971 49,025

Before illustrating the results, it is necessary to validate the effectiveness of the proposed 𝐴𝐴𝑅𝐶 model and robust SAA solution algorithm. First, impact of adjustability on the 𝐴𝐴𝑅𝐶 model is evaluated in section 5.2.1. Second, performance of the robust SAA algorithm to solve the affinely adjustable counterpart problem is assessed in section 5.2.2. 5.2.1. Impact of adjustability Let 𝓎(𝜔) be the set of adjustable variables to the value of the carbon tax rate. Assuming deterministic availability for biomass feedstocks, proposed 𝑅𝐶 (60)-(61) and 𝐴𝐴𝑅𝐶 (62)-(63) models are solved to validate the impact of adjustability on the solutions obtained by different sizes of the tax rate. The carbon tax uncertainty set is 𝜏𝑡 = 𝜏𝑡 +𝜉𝜏𝑡 in which the nominal value has value of 𝜏𝑡 = 30 and the deviation value 𝜏𝑡 ranges in interval [0, 30] with |𝜉| ≤ 1. Note that the carbon tax uncertainty gets 𝜏𝑡 = 0 in the deterministic model, where deterministic availability of feedstocks is involved by considering a single scenario. This scenario represents the expected value of the multiple scenarios for feedstock availability of each biomass supply site. Results obtained by the 𝑅𝐶 and 𝐴𝐴𝑅𝐶 models with different sizes of 𝜏𝑡 for the case study problem are reported in Table 5. In this table, the percentage of changes in using three transportation modes (i.e., truck, railcar, and barge) by the 𝑅𝐶 and 𝐴𝐴𝑅𝐶 models are shown. Equation (70) computes these differences by summing over total flows of different transportation modes through all arcs in the olefin supply/production network. Obviously, 𝑇𝐹𝑅𝐶 and 𝑇𝐹𝐴𝐴𝑅𝐶 represent the total flows of each mode by the 𝑅𝐶 and 𝐴𝐴𝑅𝐶 models, respectively. Moreover, 𝒵𝑅𝐶 and 𝒵𝐴𝐴𝑅𝐶 stand for the optimal objective value of the 𝑅𝐶 and 𝐴𝐴𝑅𝐶 models, respectively. The last column, 𝑔𝑎𝑝𝑅𝐶,𝐴𝐴𝑅𝐶, is the relative difference between the optimal objective values achieved by running the 𝑅𝐶 and 𝐴𝐴𝑅𝐶 models as follows: 𝑈𝑇𝑀𝑅𝐶,𝐴𝐴𝑅𝐶 =

𝑇𝐹𝐴𝐴𝑅𝐶 ― 𝑇𝐹𝑅𝐶 𝑇𝐹𝑅𝐶

× 100%

(70)

25

ACCEPTED MANUSCRIPT

𝑔𝑎𝑝𝑅𝐶,𝐴𝐴𝑅𝐶 =

Case

𝜏𝑡

Base

0 10 15 20 25 30

𝒵𝑅𝐶 ― 𝒵𝐴𝐴𝑅𝐶 𝒵𝑅𝐶

Truck -5.50% -4.49% -6.31% -5.87% -4.28% -4.23%

(71)

× 100%

Table 5. Comparison between 𝑹𝑪 and 𝑨𝑨𝑹𝑪 𝑈𝑇𝑀𝑅𝐶,𝐴𝐴𝑅𝐶 𝒵𝑅𝐶 Railcar Barge 0.00% 1,112,407,296 7.67 × 105 % -95.00% 68.48% 1,131,733,481 0.00% 0.075% 1,137,520,607 5 0.00% 1,145,752,047 7.11 × 10 % 0.00% 0.05% 1,154,892,564 28.72% 0.00% 1,161,928,302

𝒵𝐴𝐴𝑅𝐶

𝑔𝑎𝑝𝑅𝐶,𝐴𝐴𝑅𝐶

1,112,080,558 1,128,707,290 1,137,011,924 1,145,410,643 1,153,750,708 1,161,942,293

0.03% 0.27% 0.04% 0.03% 0.10% 0.00%

Table 5 indicates that the 𝐴𝐴𝑅𝐶 model achieves better solutions than the 𝑅𝐶 model. The major reason is that affinely adjustable variables make the model robust to the uncertainty of carbon tax rate which results in quality solution with less objective value. Results verify that the affine model benefits from the lower-emitting transportation modes (i.e., railcars and barges) throughout the Mississippi railways and Mississippi river more than the original robust model. That is the 𝐴𝐴𝑅𝐶 model reduces the amount of carbon emissions and total cost of the problem by locating the densification depots and olefin plants close to the railcar hubs and inland ports which result in less transportation emissions. In case 𝜏𝑡 = 0, the 𝐴𝐴𝑅𝐶 model uses 5.5% less truck transportations than the 𝑅𝐶 model, while increases the railcar transportations for transporting 7678 tones feedstock supplies through the hubs. In case 𝜏𝑡 = 10, the 𝐴𝐴𝑅𝐶 model prefers to reduce 95% railcar transportations and increase 68.48% barge transportations than the 𝑅𝐶 model. In case 𝜏𝑡 = 15, the affine model reduces the truck transportations and increases the barge transportations for 6.31% and 0.075%, respectively. For case 𝜏𝑡 = 20, the affine model behaves as same as the case 𝜏𝑡 = 0, but with a little more decrease for truck transportations. Also, the 𝐴𝐴𝑅𝐶 model decreases 4.28% of truck transportations and increases 0.05% of barge transportation for case 𝜏𝑡 = 25. Finally, the proposed 𝐴𝐴𝑅𝐶 model reduces truck transportations for 4.23% and increases railcar transportations for 28.72% in case of 𝜏𝑡 = 30. Therefore, obtained results in this table justifies better performance of the 𝐴𝐴𝑅𝐶 model than the 𝑅𝐶 model to minimize the olefin network transportations and emission costs. Overall, the 𝐴𝐴𝑅𝐶 solution is 0.078% better than the 𝑅𝐶 solution. Hence, we focus more on using 𝐴𝐴𝑅𝐶 model for the remaining experiments. 5.2.2. Performance of the robust SAA To verify the efficiency of the robust SAA algorithm, Table 6 shows the impact of different number of scenarios 𝑁 = {5, 10, 15} and 𝑁' = {50, 100} on solving the 𝐴𝐴𝑅𝐶 model by proposed robust SAA with fixed replication number 𝑀 = 4. 𝑁' = 50 𝑁

𝑔𝑎𝑝𝑀 𝑁,𝑁'

5 10 15

0.436% 0.312% 0.244%

Table 6. Optimality gap evaluations 𝑁' = 100 𝐶𝑃𝑈 (𝑠𝑒𝑐) 𝑁 18,717 22,832 25,219

5 10 15

𝑔𝑎𝑝𝑀 𝑁,𝑁'

𝐶𝑃𝑈 (𝑠𝑒𝑐)

0.377% 0.281% 0.194%

23,882 26,741 29,462

26

ACCEPTED MANUSCRIPT

Average

0.33%

22,256

Average

0.28%

26,695

The results indicate that increasing the sample size 𝑁 and the large sample size 𝑁' decrease the optimality gap of the proposed robust SAA. Obviously, the average optimality gap decreases from 0.33% to 0.28% along with increasing 𝑁' from 50 to 100. However, this optimality gap reduction comes with increasing the solution time of the robust SAA method by overall 17%. Table 7 represents the obtained results by solving the RTSSP model for different problem sizes with the robust SAA algorithm. To run these experiments, 𝑁, 𝑁' and 𝑀 are set to 5, 50 and 4, respectively. The first column shows different experimental problem sizes. Values for the carbon tax rate uncertainty 𝜏𝑡 are given in the second column. The third and fourth columns stand for the statistical Lower Bound (LB) and Upper Bound (UB), respectively. The optimality gaps are represented in the fifth column. Finally, CPU times for solving different problem sizes are reported in the last column. Table 7. Experimental results of robust SAA Case 1

2

3

Base

5

𝜏𝑡

𝐿𝐵

𝑈𝐵

𝑔𝑎𝑝𝑀 𝑁,𝑁'

𝐶𝑃𝑈 (𝑠𝑒𝑐)

0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30

341,042,130 346,017,663 351,090,104 356,912,798 565,258,418 573,914,474 581,863,411 590,190,991 789,465,998 801,153,167 812,707,404 824,280,397 1,113,793,424 1,130,978,142 1,146,171,921 1,165,646,009 1,313,629,232 1,357,320,659 1,350,841,481 1,376,104,766 844,419,130

341,452,850 346,450,034 351,495,535 357,336,931 566,073,563 574,822,694 582,865,940 591,302,640 791,301,819 802,847,175 815,561,871 826,387,686 1,118,670,829 1,135,771,092 1,153,635,945 1,173,981,276 1,320,734,785 1,365,802,291 1,362,009,963 1,386,810,947 848,265,794

0.120% 0.125% 0.115% 0.119% 0.144% 0.158% 0.172% 0.188% 0.232% 0.211% 0.350% 0.255% 0.436% 0.422% 0.647% 0.710% 0.538% 0.621% 0.820% 0.772% 0.358%

561 627 680 512 8,888 8,254 8,572 8,763 18,017 18,022 18,007 18,002 18,717 23,245 22,719 19,918 36,000 36,000 36,000 36,000 16,875

Average

Computational results show that the proposed robust SAA is capable of solving the different problem instances by reaching the termination criteria and consuming a reasonable amount of solution time. In case, no qualified solution is found in the maximum solution time, the last achieved LB, UB and corresponding optimality gap are reported for these cases. According to Table 7, the average optimality gap for the proposed robust SAA is 0.358%. However, for the larger problem sizes, the robust SAA algorithm may not be efficient in finding the optimal solution within reasonable solving time. This directs developing complementary techniques to solve efficiently the sub-problems of the robust SAA algorithm. 5.2.3 Case study results 27

ACCEPTED MANUSCRIPT

Fig. 4(a) demonstrates the optimal facility configuration for the olefin supply chain/production network. Note that the biomass supplies follow the normal distribution with each biomass type 𝑏 ∈ ℬ at each location 𝑖𝜖ℐ and time period 𝑡 ∈ 𝒯 and carbon tax rate is considered as 𝜏𝑡 = 30.

(a) Base case

(b) High MSW recycling case

Figure 4. Facility configuration of the 𝐴𝐴𝑅𝐶 solution with 𝜏𝑡 = 30

Fig. 4 indicates that totally 22 densification depots including four facilities with capacity level one (0.05 MTY), four facilities with capacity level two (0.07 MTY), six facilities with capacity level three (0.1 MTY) and eight facilities with capacity level four (0.16 MTY) are opened in the 86 potential locations. These facilities dispersed throughout the high populated areas in Mississippi where MSW are constantly available (e.g. Lauderdale, Oktibbeha and Lincoln counties in Mississippi). Availability of the railcar hubs and inland ports through the Mississippi river to the Gulf of Mexico in Mississippi sets the barge and train transportation modes on the top priority for transporting biomass in the olefin supply chain/production network. Thus, a railcar hub in Yalobusha county as well as Natchez-Adams County Port in Adams county are used for transporting the biomass/MSW feedstocks to the olefin plants. Finally, three olefin plants with capacity level four (0.4 MTY) are located in Yalobusha, Grenada and Adams counties to produce olefins. Note that olefin plants are located close to the selected multi-modal facilities in Yalobusha and Adams counties to highly utilize the existing railroads and waterways throughout the Mississippi state for cheaper transportation cost. 5.2.3.1 Impact of MSW recycling rate on olefin network configuration Increasing population growth in the United States (U.S. EPA, 2015) has inspired us to analyze the impact of MSW recycling rate on the performance of the olefin supply chain/production network. According to U.S. EPA (2015), U.S. generated 262.4 million tons of MSW in 2015, approximately 3.5 million tones more than the amount generated in 2014. This is an increase from the 243.5 million tones generated in 2000. Although a large portion of generating MSW (e.g., wood, paper, yard trimmings, and food wastes) can be used for olefin production, only 87.2 million tons of generating MSW are recycled at recycling rate of 34.3% (Quddus et al., 2018). Most of 28

ACCEPTED MANUSCRIPT

these wastes are currently disposed to the landfill, whereas their recycling rate can be increased in order for producing olefin. This motivates us to investigate how the increasing of the MSW recycling rate affects the design of olefin supply chain/production network. Figures 5represent the potentials to make great use of MSW on the configuration of olefin supply chain/production network. Obviously, a 100% increase of MSW recycling rate (i.e., 68.6%) will increase the olefin production by 14.3%. Also, the number of densification depots and olefin plants will increase by 45.5% and 66.7% from the base case. Although the number of railcar hub and inland port will not change, the number of railcars and barges will increase by 11% and 17.4%, as more feedstock supplies are now required to be transported throughout the network. Moreover, an olefin network comparison between the base case and a high case with MSW recycling rate of 68.6% is shown in Figure 4(b). It is observed that more decentralized facilities (i.e., densification depots, railcar hubs, inland ports, and olefin plants) with small capacities are opened by the proposed RTSSP model under high MSW recycling case compared to the base case. Noticeably, some additional facilities are located in the highly populated counties of Mississippi (e.g., Hinds, Rankin, Lee, and Jones) to use the readily available MSW in these areas. These results indicate that improvement in MSW recycling rate will not only reduce the dependency of the olefin network to the seasonal biomass feedstocks, but also enhance the olefin supply chain/production network performance.

(a)

(b)

(c) Figure 5. Impact of MSW recycling rate change on olefin network design

29

ACCEPTED MANUSCRIPT

5.2.3.2 Impact of conversion rate on olefin network configuration

30

ACCEPTED MANUSCRIPT

The impact of biomass conversion rate on the olefin supply chain network/production design is evaluated. Table 1 shows that current conversion rate of biomass/MSW to olefin is relatively low. Traditional thermal or catalytic cracking methods are underway to improve the conversion rate of the raw biomass/MSW to olefin (e.g., Fischer–Tropsch synthesis) which tackles both the environmental and economic concerns (Lu et al., 2017). This idea motivates this research to explore the impact of conversion rate changes on the effectiveness of the olefin supply chain/production network. Figure 6(a) presents the impact of conversion rate improvements on the olefin production and corresponding overall network cost. At the higher conversion rate, olefin plants need less amount of biomass/MSW to satisfy the same amount of olefin demand. Figure 6(b) indicates that increasing the conversion rate by 10% and 20% will decrease the number of densification depots by 9.1% and 18.2% and railcar hubs by 0 and 100% from the base case. Conversely, decreasing the conversion rate by the same values will increase the number of densification depots by 22.7% and 36.3%, inland ports by 100% and 100% and olefin plants by 33.3% and 66.6%. Figure 6(c) also shows increasing conversation rate will decrease the amount of biomass supplies that are transported to/through the multi-modal facilities. Moreover, the number of railcars and barges used for the biomass/MSW transporting are highly affected by the changes in the conversion rate. These behaviors will result in the less logistical cost for the olefin supply/production chain. Investigations indicate that a 20% improvement on biomass-to-olefin conversion rate would increase the overall olefin production by 17.26% and subsequently reduce the total cost of the supply/production network by 4%. Consequently, the technical improvement on the conversion rate will lead to higher production of olefin and less overall cost of the supply chain/production network. These results imply that the investments on the R&D of production of olefins would be able to satisfy higher level of demands for petrochemical industries.

(a)

(b)

(c)

31

ACCEPTED MANUSCRIPT

Figure 6. Impact of conversation rate change on olefin supply/production performance

5.2.3.3 Sensitive analysis of carbon tax variation on olefin network configuration Inspiring by the global consciousness of the increasing of GHG emissions, this study also analyzes the impact of carbon tax rate uncertainty, 𝜏𝑡, on the olefin system performance. One of the main sources of GHG is the carbon emission from processing and transportation of products (Parker et al., 2008). According to U.S. EPA (2017), 26% of carbon emissions in U.S. were generated from transportation in 2014. Implementing carbon tax rate policy in huge carbonemitting countries such as U.S is usually associated with uncertainty (Haddadsisakht and Ryan, 2018). This motivates us to evaluate the effect of carbon tax variation on the design of olefin network. Hence, we conducted a sensitive analysis by varying the carbon tax rate uncertainty between 0 to 30. Higher value of 𝜏𝑡 specifies more restrictiveness of the carbon emissions whereas lower value indicates less restrictiveness. Figures 7(a) demonstrates the amount of carbon emissions generated by transportation modes (e.g., trucks, railcars, and barges) via different carbon tax rate. It is observed that increasing the carbon uncertainty rate from 0 to 30 will decrease the amount of emissions from railcars and barges by 27.1% and 71%. The number of densification depots will increase by 10% because of the increase in carbon tax uncertainty changes from 0 to 30, while the number of railcar hubs, inland ports and olefin plant still remain the same. The changes in the densification depots are primarily on their sizes which becomes more sensitive to carbon tax uncertainty compared to olefin plants and multi-model facilities. Figure 8(a) shows the optimal sizes of densification depots with various values of 𝜏𝑡. Obviously, the number of densification depots with different sizes are changed by varying the carbon tax uncertainty.

(a)

(b)

Figure 7. Impact of carbon tax rate change on olefin system performance

32

ACCEPTED MANUSCRIPT

(a) (b) Figure 8. Capacity set-up for densification depots with different values of 𝜏𝑡 = {0,10,20,30}

Likewise, Figure 7(b) depicts the impact of nominal carbon tax rate, 𝜏𝑡, variation on the olefin system design. Increasing the nominal carbon tax rate from 20 to 50 will decrease the amount of emissions from trucks and barges transportations by 12.1% and 18.7% throughout the network. Figure 8(b) also depicts the size variations of the densification depots with respect to nominal carbon tax changes. Finally, Figure 9 demonstrates the total cost and emissions of the olefin network. According to this Figure, increasing the carbon tax rate results in less emissions throughout the olefin network, however it will increase the total cost of the system. Noticeably, increasing the carbon tax rate uncertainty from 0 to 30 will decrease the total network emissions by 2.8% and increase the total network cost by 4.66%. Besides, increasing the nominal carbon tax rate from 20 to 50 will decrease the total network emissions by 3.5% and increase the total network cost by 4.33% as well.

(a) (b) Figure 9. Impact of carbon tax rate change on olefin system performance

6. Conclusion This paper presents a Robust Three-Stage Stochastic Programming model for an olefin supply chain/production network with the consideration of the seasonal supplies of biomass feedstocks and the uncertain carbon emission tax rate. In order to provide the constant feedstock supplies for the olefin production, the MSW has been utilized to complement the biomass-derived feedstocks’ 33

ACCEPTED MANUSCRIPT

seasonal supplies. The proposed three-stage model encompassed densification depots, multimodal facilities, olefin production plants as well as the transportation links and modes between them for the supply chain network design. The truck, rail and barge transportation were included in this model. The first-stage of this model was designed to make facility investment decisions, such as size and location for the densification deports, multi-modal facilities and olefin production plants. The second-stage focused on plans of storing biomass/MSW supplies in densification depots and olefin plants, transportation units of various modes (i.e. truck, railcar and barge) and olefin production after realization of feedstock seasonality. The third-stage determined biomass/MSW distribution throughout the network and amount of processing feedstocks in densification depots and olefins plants. The uncertain tax rate was also included in the third-stage. A hybrid approach that integrates Affinely Adjustable Robust Counterpart and Sample Average Approximation method was proposed to find the optimal solution for this problem. The state of Mississippi was used as a base case study to validate the performance of the olefin network design problem and solution algorithm. The overall computational results indicated that combining biomass and MSW feedstocks to produce olefin has significant economic benefits and can serve as the key raw material in petrochemical industries. The results revealed that increasing 20% of biomass-to-olefin conversion rate would increase overall olefin production by 17.26% and decrease total cost of the olefin supply/production network by 4%. Moreover, doubling the MSW recycling rate would result in an increase of olefin production rate by 14.3%. In addition, increasing the carbon tax rate from 0 to 30 would decrease the total network emissions by 2.8% and increase the total network cost by 4.66%, while increasing the nominal carbon tax rate from 30 to 50 would decrease the total network emissions by 3.5% and increase the total network cost by 4.33%. The proposed supply chain network is highly sensitive to the carbon tax rate which would lead to the different selection of location and capacity for the facilities. The novel contributions of this paper are summarized below. First and foremost, a Robust Three-Stage Stochastic Programming model with seasonal biomass supplies and uncertain tax rate is developed for the design and management of an olefin supply/production chain network. The probabilistic scenarios are used to capture the biomass seasonal supplies and the robust solutions for facility configuration and biomass flows are used to assist the handling of uncertain carbon tax rate. To the best of our knowledge, there is no prior study combined these two considerations in the olefin supply/production chain network and this proposed work closed this gap. Furthermore, a novel Sample Average Approximation algorithm is proposed to solve the proposed model with higher robustness. Finally, a real case study in the state of Mississippi is presented to validate the efficiency of the proposed model and robust Sample Average Approximation algorithm. This case study is developed based on the real conditions of the state of Mississippi and therefore provides more reliable conclusions. Although this study has several novel contributions to the current state of the art, it still has several limitations and therefore relevant future work is needed. First, this work pre-assumed that the olefin supply/production network is reliable during the planning horizon without any failure risk. However, facilities may encounter disruptions due to several reasons (e.g., flooding, earthquake, power failure and equipment breakdowns). In addition, we considered only a few uncertainties (i.e. seasonal supplies and uncertain tax rate), however there are several types of uncertainties (e.g., technology, transportation cost and demand) exist in the real practice that can affect the olefin supply/production network. Finally, the behavior of end-users are not considered (e.g., demand of olefin for industries). The future work will take into account all these limitations and extend the current scope of olefin production. 34

ACCEPTED MANUSCRIPT

References Abdulrazik, A., Elsholkami, M., Elkamel, A., Simon, L., 2017. Multi-products productions from Malaysian oil palm empty fruit bunch (EFB): Analyzing economic potentials from the optimal biomass supply chain. J. Clean. Prod. 168, 131–148. Ahmed, W., Sarkar, B., 2018. Impact of carbon emissions in a sustainable supply chain management for a second generation biofuel. J. Clean. Prod. 186, 807–820. Alizadeh, M., Mahdavi, I., Mahdavi-Amiri, N., Shiripour, S., 2015. A capacitated location-allocation problem with stochastic demands using sub-sources: An empirical study. Appl. Soft Comput. 34, 551–571. Alizadeh, M., Mahdavi-Amiri, N., Shiripour, S., 2016. Modeling and solving a capacitated stochastic locationallocation problem using sub-sources. Soft Comput. 20, 2261–2280. Amiri-Aref, M., Klibi, W., Babai, M.Z., 2018. The multi-sourcing location inventory problem with stochastic demand. Eur. J. Oper. Res. 266, 72–87. An, H., Wilhelm, W.E., Searcy, S.W., 2011. A mathematical model to design a lignocellulosic biofuel supply chain system with a case study based on a region in Central Texas. Bioresour. Technol. 102, 7860–7870. Awudu, I., Zhang, J., 2013. Stochastic production planning for a biofuel supply chain under demand and price uncertainties. Appl. Energy 103, 189–196. Awudu, I., Zhang, J., 2012. Uncertainties and sustainability concepts in biofuel supply chain management: A review. Renew. Sustain. Energy Rev. 16, 1359–1368. Azadeh, A., Arani, H.V., Dashti, H., 2014. A stochastic programming approach towards optimization of biofuel supply chain. Energy 76, 513–525. Babazadeh, R., 2018. A robust optimization method to green biomass-to-bioenergy systems under deep uncertainty. Ind. Eng. Chem. Res. Babazadeh, R., Razmi, J., Pishvaee, M.S., Rabbani, M., 2017. A sustainable second-generation biodiesel supply chain network design problem under risk. Omega 66, 258–277. Bai, Y., Hwang, T., Kang, S., Ouyang, Y., 2011. Biofuel refinery location and supply chain planning under traffic congestion. Transp. Res. Part B Methodol. 45, 162–175. Bai, Y., Li, X., Peng, F., Wang, X., Ouyang, Y., 2015. Effects of disruption risks on biorefinery location design. Energies 8, 1468–1486. Bairamzadeh, S., Pishvaee, M.S., Saidi-Mehrabad, M., 2015. Multiobjective robust possibilistic programming approach to sustainable bioethanol supply chain design under multiple uncertainties. Ind. Eng. Chem. Res. 55, 237–256. Balaman, Ş.Y., Matopoulos, A., Wright, D.G., Scott, J., 2018. Integrated optimization of sustainable supply chains and transportation networks for multi technology bio-based production: A decision support system based on fuzzy ε-constraint method. J. Clean. Prod. 172, 2594–2617. Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A., 2004. Adjustable robust solutions of uncertain linear programs. Math. Program. 99, 351–376. Cambero, C., Sowlati, T., 2014. Assessment and optimization of forest biomass supply chains from economic, social and environmental perspectives–A review of literature. Renew. Sustain. Energy Rev. 36, 62–73. Chen, C.-W., Fan, Y., 2012. Bioethanol supply chain system planning under supply and demand uncertainties. Transp. Res. Part E Logist. Transp. Rev. 48, 150–164. Chiang, N.N.K.H., 2004. Quantifying the economic potential of a biomass to olefin technology (PhD Thesis). Massachusetts Institute of Technology. Cobuloglu, H.I., Büyüktahtakın, İ.E., 2017. A two-stage stochastic mixed-integer programming approach to the competition of biofuel and food production. Comput. Ind. Eng. 107, 251–263. Cundiff, J.S., Dias, N., Sherali, H.D., 1997. A linear programming approach for designing a herbaceous biomass delivery system. Bioresour. Technol. 59, 47–55. Ekşioğlu, S.D., Acharya, A., Leightley, L.E., Arora, S., 2009. Analyzing the design and management of biomass-tobiorefinery supply chain. Comput. Ind. Eng. 57, 1342–1352. Fareeduddin, M., Hassan, A., Syed, M.N., Selim, S.Z., 2015. The impact of carbon policies on closed-loop supply chain network design. Procedia CIRP 26, 335–340. Foo, D.C., Tan, R.R., Lam, H.L., Aziz, M.K.A., Klemeš, J.J., 2013. Robust models for the synthesis of flexible palm oil-based regional bioenergy supply chain. Energy 55, 68–73.

35

ACCEPTED MANUSCRIPT

Frombo, F., Minciardi, R., Robba, M., Rosso, F., Sacile, R., 2009. Planning woody biomass logistics for energy production: A strategic decision model. Biomass Bioenergy 33, 372–383. Galvis, H.M.T., Bitter, J.H., Khare, C.B., Ruitenbeek, M., Dugulan, A.I., de Jong, K.P., 2012. Supported iron nanoparticles as catalysts for sustainable production of lower olefins. Science 335, 835–838. Gebreslassie, B.H., Yao, Y., You, F., 2012. Design under uncertainty of hydrocarbon biorefinery supply chains: multiobjective stochastic programming models, decomposition algorithm, and a comparison between CVaR and downside risk. AIChE J. 58, 2155–2179. Ghaderi, H., Pishvaee, M.S., Moini, A., 2016. Biomass supply chain network design: an optimization-oriented review and analysis. Ind. Crops Prod. 94, 972–1000. Giarola, S., Bezzo, F., Shah, N., 2013. A risk management approach to the economic and environmental strategic design of ethanol supply chains. Biomass Bioenergy 58, 31–51. Gold, S., Seuring, S., 2011. Supply chain and logistics issues of bio-energy production. J. Clean. Prod. 19, 32–42. Gonzales, D., Searcy, E.M., Ekşioğlu, S.D., 2013. Cost analysis for high-volume and long-haul transportation of densified biomass feedstock. Transp. Res. Part Policy Pract. 49, 48–61. Haddadsisakht, A., Ryan, S.M., 2018. Closed-loop supply chain network design with multiple transportation modes under stochastic demand and uncertain carbon tax. Int. J. Prod. Econ. 195, 118–131. Haque, N., Somerville, M., 2013. Techno-economic and environmental evaluation of biomass dryer. Procedia Eng. 56, 650–655. Houdek and Anderson, 2005. Presented at the ARTC 8th annual meeting, UOP LLC, Kuala Lumpur, Malaysia. Huang, Y., Chen, C.-W., Fan, Y., 2010. Multistage optimization of the supply chains of biofuels. Transp. Res. Part E Logist. Transp. Rev. 46, 820–830. Izquierdo, J., Minciardi, R., Montalvo, I., Robba, M., Tavera, M., 2008. Particle Swarm Optimization for the biomass supply chain strategic planning. Khor, C.S., Elkamel, A., Ponnambalam, K., Douglas, P.L., 2008. Two-stage stochastic programming with fixed recourse via scenario planning with economic and operational risk management for petroleum refinery planning under uncertainty. Chem. Eng. Process. Process Intensif. 47, 1744–1764. Kim, J., Realff, M.J., Lee, J.H., 2011. Optimal design and global sensitivity analysis of biomass supply chain networks for biofuels under uncertainty. Comput. Chem. Eng. 35, 1738–1751. Kleywegt, A.J., Shapiro, A., Homem-de-Mello, T., 2002. The sample average approximation method for stochastic discrete optimization. SIAM J. Optim. 12, 479–502. Lamers, P., Roni, M.S., Tumuluru, J.S., Jacobson, J.J., Cafferty, K.G., Hansen, J.K., Kenney, K., Teymouri, F., Bals, B., 2015. Techno-economic analysis of decentralized biomass processing depots. Bioresour. Technol. 194, 205–213. Lin, T., Rodríguez, L.F., Shastri, Y.N., Hansen, A.C., Ting, K.C., 2014. Integrated strategic and tactical biomass– biofuel supply chain optimization. Bioresour. Technol. 156, 256–266. Lu, Y., Yan, Q., Han, J., Cao, B., Street, J., Yu, F., 2017. Fischer–Tropsch synthesis of olefin-rich liquid hydrocarbons from biomass-derived syngas over carbon-encapsulated iron carbide/iron nanoparticles catalyst. Fuel 193, 369–384. Mahmudi, H., Flynn, P.C., 2006. Rail vs truck transport of biomass, in: Twenty-Seventh Symposium on Biotechnology for Fuels and Chemicals. Springer, pp. 88–103. Mak, W.-K., Morton, D.P., Wood, R.K., 1999. Monte Carlo bounding techniques for determining solution quality in stochastic programs. Oper. Res. Lett. 24, 47–56. Marufuzzaman, M., Ekşioğlu, S.D., 2016. Designing a reliable and dynamic multimodal transportation network for biofuel supply chains. Transp. Sci. 51, 494–517. Marufuzzaman, M., Ekşioğlu, S.D., Hernandez, R., 2014a. Environmentally friendly supply chain planning and design for biodiesel production via wastewater sludge. Transp. Sci. 48, 555–574. Marufuzzaman, M., Eksioglu, S.D., Huang, Y.E., 2014b. Two-stage stochastic programming supply chain model for biodiesel production via wastewater treatment. Comput. Oper. Res. 49, 1–17. Marufuzzaman, M., Eksioglu, S.D., Li, X., Wang, J., 2014c. Analyzing the impact of intermodal-related risk to the design and management of biofuel supply chain. Transp. Res. Part E Logist. Transp. Rev. 69, 122–145. Marufuzzaman, M., Li, X., Yu, F., Zhou, F., 2016. Supply chain design and management for syngas production. ACS Sustain. Chem. Eng. 4, 890–900. Marvin, W.A., Schmidt, L.D., Benjaafar, S., Tiffany, D.G., Daoutidis, P., 2012a. Economic optimization of a lignocellulosic biomass-to-ethanol supply chain. Chem. Eng. Sci. 67, 68–79. Marvin, W.A., Schmidt, L.D., Daoutidis, P., 2012b. Biorefinery location and technology selection through supply chain optimization. Ind. Eng. Chem. Res. 52, 3192–3208.

36

ACCEPTED MANUSCRIPT

Mele, F.D., Kostin, A.M., Guillén-Gosálbez, G., Jiménez, L., 2011. Multiobjective model for more sustainable fuel supply chains. A case study of the sugar cane industry in Argentina. Ind. Eng. Chem. Res. 50, 4939–4958. Memişoğlu, G., Üster, H., 2015. Integrated bioenergy supply chain network planning problem. Transp. Sci. 50, 35– 56. Mississippi Department of Environmental Quality, 2014. Status report on solid waste management facilities and activities. National Renewable Energy Laboratory, 2012. National Renewable Energy Laboratory [WWW Document]. URL https://maps.nrel.gov/ (accessed 6.19.18). Norkin, V.I., Ermoliev, Y.M., Ruszczyński, A., 1998. On optimal allocation of indivisibles under uncertainty. Oper. Res. 46, 381–395. Osmani, A., Zhang, J., 2014. Economic and environmental optimization of a large scale sustainable dual feedstock lignocellulosic-based bioethanol supply chain in a stochastic environment. Appl. Energy 114, 572–587. Osmani, A., Zhang, J., 2013. Stochastic optimization of a multi-feedstock lignocellulosic-based bioethanol supply chain under multiple uncertainties. Energy 59, 157–172. Parker, N., Tittmann, P., Hart, Q., Lay, M., Cunningham, J., Jenkins, B., Nelson, R., Skog, K., Milbrandt, A., Gray, E., 2008. Strategic Assessment of Bioenergy Development in the West Spatial Analysis and Supply Curve Development Final Report. Park. Nathan Tittman Peter Hart Quinn Al 2008 Strateg. Assess. Bioenergy West Spat. Anal. Supply Curve Dev. Final Rep. Davis CA Univ. Calif. Davis. Persson, T., y Garcia, A.G., Paz, J., Jones, J., Hoogenboom, G., 2009. Maize ethanol feedstock production and net energy value as affected by climate variability and crop management practices. Agric. Syst. 100, 11–21. Poudel, S.R., Quddus, M.A., Marufuzzaman, M., Bian, L., 2017. Managing congestion in a multi-modal transportation network under biomass supply uncertainty. Ann. Oper. Res. 1–43. Quddus, M.A., Chowdhury, S., Marufuzzaman, M., Yu, F., Bian, L., 2018. A two-stage chance-constrained stochastic programming model for a bio-fuel supply chain network. Int. J. Prod. Econ. 195, 27–44. Ren, T., Daniëls, B., Patel, M.K., Blok, K., 2009. Petrochemicals from oil, natural gas, coal and biomass: Production costs in 2030–2050. Resour. Conserv. Recycl. 53, 653–663. Rentizelas, A.A., Tolis, A.J., Tatsiopoulos, I.P., 2009. Logistics issues of biomass: the storage problem and the multibiomass supply chain. Renew. Sustain. Energy Rev. 13, 887–894. Roni, M.S., Eksioglu, S.D., Searcy, E., Jha, K., 2014. A supply chain network design model for biomass co-firing in coal-fired power plants. Transp. Res. Part E Logist. Transp. Rev. 61, 115–134. Sadrameli, S.M., 2016. Thermal/catalytic cracking of liquid hydrocarbons for the production of olefins: A state-ofthe-art review II: Catalytic cracking review. Fuel 173, 285–297. Sadrameli, S.M., 2015. Thermal/catalytic cracking of hydrocarbons for the production of olefins: A state-of-the-art review I: Thermal cracking review. Fuel 140, 102–115. Santoso, T., Ahmed, S., Goetschalckx, M., Shapiro, A., 2005. A stochastic programming approach for supply chain network design under uncertainty. Eur. J. Oper. Res. 167, 96–115. Schütz, P., Tomasgard, A., Ahmed, S., 2009. Supply chain design under uncertainty using sample average approximation and dual decomposition. Eur. J. Oper. Res. 199, 409–419. Searcy, E., Flynn, P., Ghafoori, E., Kumar, A., 2007. The relative cost of biomass energy transport. Appl. Biochem. Biotecnol. 639–652. Shabani, N., Sowlati, T., 2016. A hybrid multi-stage stochastic programming-robust optimization model for maximizing the supply chain of a forest-based biomass power plant considering uncertainties. J. Clean. Prod. 112, 3285–3293. Shabani, N., Sowlati, T., Ouhimmou, M., Rönnqvist, M., 2014. Tactical supply chain planning for a forest biomass power plant under supply uncertainty. Energy 78, 346–355. Shapiro, A., 2008. Stochastic programming approach to optimization under uncertainty. Math. Program. 112, 183– 220. Shen How, B., Lam, H.L., 2018. Sustainability evaluation for biomass supply chain synthesis: Novel principal component analysis (PCA) aided optimisation approach. J. Clean. Prod. 189, 941–961. Sumner, J., Bird, L., Dobos, H., 2011. Carbon taxes: a review of experience and policy design considerations. Clim. Policy 11, 922–943. Tong, K., Gong, J., Yue, D., You, F., 2013. Stochastic programming approach to optimal design and operations of integrated hydrocarbon biofuel and petroleum supply chains. ACS Sustain. Chem. Eng. 2, 49–61. Tong, K., You, F., Rong, G., 2014. Robust design and operations of hydrocarbon biofuel supply chain integrating with existing petroleum refineries considering unit cost objective. Comput. Chem. Eng. 68, 128–139.

37

ACCEPTED MANUSCRIPT

U.S. Census Bureau, 2017. U.S. Census Bureau QuickFacts: UNITED STATES [WWW Document]. URL https://www.census.gov/quickfacts/fact/table/US/PST045217 (accessed 5.7.18). U.S. Environmental Protection Agency, 2015. Advancing Sustainable Materials Management: Facts and Figures [WWW Document]. URL https://www.epa.gov/facts-and-figures-about-materials-waste-andrecycling/advancing-sustainable-materials-management-0 U.S. Environmental Protection Agency (EPA), O., 2015. Report on the Environment (ROE) [WWW Document]. US EPA. URL https://www.epa.gov/report-environment (accessed 11.19.18). Wang, X., Ouyang, Y., 2013. A continuum approximation approach to competitive facility location design under facility disruption risks. Transp. Res. Part B Methodol. 50, 90–103. Xie, F., Huang, Y., Eksioglu, S., 2014. Integrating multimodal transport into cellulosic biofuel supply chain design under feedstock seasonality with a case study based on California. Bioresour. Technol. 152, 15–23. Xie, W., Ouyang, Y., 2013. Dynamic planning of facility locations with benefits from multitype facility colocation. Comput.-Aided Civ. Infrastruct. Eng. 28, 666–678. You, F., Tao, L., Graziano, D.J., Snyder, S.W., 2012. Optimal design of sustainable cellulosic biofuel supply chains: multiobjective optimization coupled with life cycle assessment and input–output analysis. AIChE J. 58, 1157–1180. You, F., Wang, B., 2012. Multiobjective optimization of biomass-to-liquids processing networks. Found. Comput.Aided Process Oper. 1–7. You, F., Wang, B., 2011. Life cycle optimization of biomass-to-liquid supply chains with distributed–centralized processing networks. Ind. Eng. Chem. Res. 50, 10102–10127. Yue, D., Slivinsky, M., Sumpter, J., You, F., 2014. Sustainable design and operation of cellulosic bioelectricity supply chain networks with life cycle economic, environmental, and social optimization. Ind. Eng. Chem. Res. 53, 4008–4029. Yue, D., You, F., 2016. Optimal supply chain design and operations under multi‐scale uncertainties: Nested stochastic robust optimization modeling framework and solution algorithm. AIChE J. 62, 3041–3055. Zamboni, A., Bezzo, F., Shah, N., 2009. Spatially explicit static model for the strategic design of future bioethanol production systems. 2. Multi-objective environmental optimization. Energy Fuels 23, 5134–5143. Zhai, P., Xu, C., Gao, R., Liu, X., Li, M., Li, W., Fu, X., Jia, C., Xie, J., Zhao, M., 2016. Highly Tunable Selectivity for Syngas‐Derived Alkenes over Zinc and Sodium‐Modulated Fe5C2 Catalyst. Angew. Chem. Int. Ed. 55, 9902–9907.

38