A robust optimization method to green biomass-to-bioenergy systems

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Process Systems Engineering

A robust optimization method to green biomassto-bioenergy systems under deep uncertainty Reza Babazadeh Ind. Eng. Chem. Res., Just Accepted Manuscript • Publication Date (Web): 16 May 2018 Downloaded from http://pubs.acs.org on May 16, 2018

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Industrial & Engineering Chemistry Research

A robust optimization method to green biomass-to-bioenergy systems under deep uncertainty Reza Babazadeh* Faculty of Engineering, Urmia University, Urmia, West Azerbaijan Province, Iran

Abstract This paper presents a robust optimization method to deal with the deep uncertainty of a green biomass-to-bioenergy supply chain system design. The proposed model is triggered to minimize the total costs including establishing facilities costs, material flow costs, and inventory holding costs and maximize carbon trade. In the proposed model, carbon dioxide emissions (CO2equivalent) of all involved processes are calculated through Eco-indicator 99 method employing SimaPro software. Due to lack of historical information about the parameters of the proposed model, set-induced robust optimization method is utilized to model the deep uncertainty of parameters. The proposed model is implemented in a real case in Iran. The results justify the efficiency of the proposed model in designing green biodiesel supply chain network. Keywords: Biomass-to-bioenergy system, Optimization, Deep uncertainty, Robust optimization, Green supply chain. 1. Introduction Todays, due to issues such as population growth, industrial revolution, and changes in the life styles have increased energy consumption over the world. According to the report of U.S. Energy Information Administration (EIA) released in 2016, world energy consumption will grow by 48% between 2012 and 2040 [1]. Fossil fuels constitute more than 75% of world energy consumption through 2040 [1]. On the other hand, concerns about fossil fuels depletion, Greenhouse Gas (GHG) emissions, climate change, and long-term high world oil prices support the utilization of different renewable energy sources. It is estimated that renewable energy consumption will increase by an average 2.6% per year through 2040 [1]. Since liquid fossil fuels account for the largest share of

*

Corresponding author. Tel.: +98 44 32972854; Fax: +98 44 32773591.

E-mail: [email protected] 1 ACS Paragon Plus Environment

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fossil fuels, many efforts have been done to find renewable and sustainable alternatives for liquid fossil fuels. At this line, researchers and practitioners have proposed using biofuels produced from different biomass as a suitable alternative for fossil fuels. Biofuels are renewably produced and have lower environmental impacts than their fossil counterparts [2]. European Directive has set a target to substitute 10% of fossil fuels with biofuels in transportation sector by 2020 [3]. Bioethanol and biodiesel are the main liquid biofuels. Bioethanol is produced from feedstock with glucose, cellulous, and starch such as sugar cane, wheat, corn, sorghum, cassava, sugar beet, wood chips, and rice straw [4]. Biodiesel is produced from feedstock containing triglyceride such as soybean, palm, sunflower, castor, cotton, rapeseed, and Jatropha [5]. In this paper, optimization of biodiesel production system is studied. One of the main challenges in utilization of liquid biofuels is their high prices compared to liquid petroleum fuels. In fact, biofuel production is commercially feasible when the oil price is very high. In addition, edible feedstock for biodiesel production is intensively criticized by FAO because of food crisis. To deal with these challenges researchers have found cheaper and nonedible feedstocks such as Jatropha and used cooking oil (UCO) for biodiesel production [6]. Jatopha can grow in marginal and non-fertile lands and therefore it does not have any competition with food crops [7]. In this work, Jatropha and UCO are considered for biodiesel production. Due to urgent need for biofuels, many biofuel supply chains will be developed with the aim of improving economic, environmental, and social objectives as triple lines of sustainable development. Optimization of biomass-to-bioenergy systems has incredible role in reducing total costs and improving commercial feasibility of biofuels, improving environmental impacts and social benefits [8]. Also, integrated planning and optimization of biofuel supply chains assure adequate feedstock supply and optimal production and distribution of biofuels. In biofuel supply chains, it is necessary to integrate all echelons including feedstock supply centers, collection centers, biofuel production centers, and biofuel consumers [9]. One of the key issues in designing biofuel supply chains is dealing with the uncertainty of different parameters. Really, the most important parameters of biofuel supply chains are subject to deep uncertainty and therefore they are more vulnerable to risk respect to traditional supply chains [10]. Deep uncertainty is related to uncertainty in which the availability of historical data and knowledge about input parameters are very limited [11]. In such conditions, elaborating possibility or probability distribution for uncertain parameters is impossible. Biofuels demand and price is 2 ACS Paragon Plus Environment

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dependent upon the price of crude oil and also feedstock supply is influenced by weather changes [12]. Therefore, it is necessary to develop an efficient optimization tool to cope with such deep uncertainty in biofuel supply chain network design. Due to deep uncertainty of parameters of biofuel supply chains, there is no historical and reliable data to construct probability or possibility distribution of parameters [13]. In such condition, set-induced robust optimization method could be efficiently used to deal with the uncertainty of parameters [14]. In this paper, to design robust and sustainable biodiesel supply chain network, a mixed-integer linear programming model is developed under uncertainty. In the proposed model, all echelons of biodiesel supply chain including feedstock supply centers, collection centers, biodiesel plants and consumer centers are integrated. In the objective function, total costs are minimized and carbon trade income is maximized. Carbon dioxide (CO2) emissions of all involved processes are calculated and an upper bound is considered for it. To deal with the uncertainty of parameters, a set-induced robust optimization method is employed. To consider realistic conditions, polyhedral uncertainty set is used to model the uncertain behavior of parameters. The proposed model is applied in a real case in Iran to investigate its reliability and applicability. The main contributions of this paper that distinguishes it from the existing works in the literature are as follows: 

Developing an integrated model for biodiesel supply chain network design under uncertainty,



Calculating environmental impacts in terms of CO2 emissions of all involved processes from feedstock cultivation to biodiesel distribution in the considered biodiesel supply chain and considering carbon trade,



Employing set-induced robust optimization method to deal with the deep uncertainty of the problem,



Applying a real-world case study to investigate the efficiency and applicability of the proposed model.

The exposition of this paper is as follows. In the next section, the relevant studies are reviewed. In section 3, the proposed integrated model is described and mathematically formulated. Section 4 discusses set-induced robust optimization method. In Section 5, a real-world case study in Iran is explained and the results are analyzed. Finally, Section 6 provides conclusions and future research directions.

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2. Literature review Ghaderi et al. [8] presented a comprehensive review about different optimization techniques applied in biofuel supply chains optimization. According to their study, many researchers in different countries have focused on biomass-to-bioenergy systems optimization. Also, the most of researchers have developed deterministic optimization models for optimization of biofuel supply chains. Obviously, these models could not be applied in real-world cases due to uncertain nature of biofuel supply chains. De Meyer et al. [15] reviewed the papers studying the design and management of biomass-to-bioenergy systems. They categorized the developed models based on mathematical optimization techniques, decision levels and variables, and objective functions. Sharma et al. [16] gave a review and studied energy trends, renewable energy targets, different biomass, and the conversion processes in biofuel supply chains. They analyzed the mathematical programming models developed for biofuel supply chains optimization. At the following, some recent studies in the area of biomass-to-bioenergy supply chain systems are reviewed. Frombo et al. [17] proposed a method based on Geographical Information System (GIS) to optimize strategic, tactical, and operational decisions in a bioethanol supply chain with the aim of minimizing total costs. Babazadeh [5] firstly reviewed and systematically classified the mathematical programming models developed for biodiesel and bioethanol supply chains optimization in the literature. Babazadeh et al. [18] proposed a two-stage optimization tool to optimize strategic and tactical decisions in a biofuel logistics network. In the first stage, the data envelopment analysis (DEA) method is used to determine the best locations for feedstock cultivation. In the second stage, a MILP model is extended to determine the optimum locations of bio-refineries and material flow within the biofuel supply chain network. Foo et al. [19] developed a mathematical model to create empty fruit bunch allocation networks in the regional biomass supply chain to produce biodiesel from palm oil. They only considered environmental impact minimization in the proposed model. Giarola et al. [20] extended a multi-period MILP model to optimize strategic and tactical decisions in a bioethanol supply chain network. Their model is a multi-objective model in which the capacity of facilities are determined in each period as well as the total costs and environmental impacts are optimized as objective functions. Some researchers have tried to cope with the uncertainty of parameters in biofuel supply chain systems. Babazadeh et al. [10] proposed a multi-objective mixed-integer non-linear programming model to minimize total costs and environmental impact in a biodiesel supply chain network design 4 ACS Paragon Plus Environment

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problem. Kim et al. [21] considered the uncertainty of feedstock supply, production process, and biofuel demand in a biofuel supply chain optimization. They used sensitivity analysis method to deal with the uncertainty. This method is not a comprehensive method to cope with the uncertainty. Azadeh et al. [22] presented a dynamic MILP model to design bioethanol supply chain network under market and disruption risks. They used stochastic programming method to deal with the risks. Tong et al. [23] considered the uncertainty of crude oil demand, biomass availability, biofuel price, and production technology and presented a multi-period MILP model to design biofuel supply chain network using stochastic programming method. Tong et al. [24] developed a robust MILP model to determine strategic and tactical decisions in a biofuel supply chain. They considered supply and demand uncertainty and traded off between performance and conservatism. Yue and You [25] developed a stochastic robust optimization model to deal with multi-scale uncertainties in a three-echelon biomass-to-biofuel supply chain. They used a benders decomposition algorithm to solve the proposed model. Gong et al. [26] proposed a two stage adaptive robust mixed-integer nonlinear programming model to optimize design and operational decisions in a biofuel supply chain. They considered budgets of uncertainty to control the level of robustness and used tailored algorithm to solve the problem. Gong and You [27] presented a two-stage adaptive robust mixed integer fractional programming model to handle the uncertainty of the problem and maximize rate on investment. There is still a critical need for developing optimization models for biomass-to-bioenergy systems considering real-world assumptions. The future models should cover sustainability issues and uncertainty to move towards modeling real case biofuel supply chains. Most of studies have used stochastic programming methods to deal with the uncertainty of problems. One of the main problems in using stochastic programming methods to deal with the uncertainty of parameters in biofuels supply chains optimization is that too many scenarios should be defined for modeling the uncertainty of parameters. This issue will dramatically increase the complexity of optimization problems especially in real and large size problems [28]. On the other hand, since there is not reliable and historical data about uncertain parameters of the biofuel supply chains optimization problems, making suitable scenarios is somehow impossible in real world applications. In this regards, set-induced robust optimization method wherein there is no need for reliable and historical data, could be efficiently used in real-world biofuel supply chains optimization problems. 5 ACS Paragon Plus Environment


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3. Problem description and formulation The biomass-to-bioenergy supply chain system considered in this paper is of a four-echelon, multiperiod supply chain network which includes feedstocks supply centers, oil extraction centers, biodiesel plants, and customer zones. Figure 1 illustrates the overview of the biomass-to-bioenergy supply chain system for the considered case study. The required feedstocks in this system include Jatropha provided from domestic farms, Jatropha imported from foreign suppliers, and UCO collected from different provinces of Iran. Jatropha seeds are converted to Jatropha oil in oil extraction centers and transported to biodiesel plants. Cold pressing technology is used in Jatropha oilseed extraction centers to extract Jatropha oil [29]. UCO are directly moved to biodiesel plants and after filtering waste materials along with Jatropha oil are converted to biodiesel plants. In biodiesel plants, Jatropha oil and UCO are converted to main product biodiesel and by-product glycerin using transesterification process [30]. Finally, the products are shipped to related customers according to demands. The proposed model is a multi-period mixed-integer linear programming (MILP) model that minimizes total establishing, production, transportation, and shortage costs which optimizes the numbers, locations, capacities of facilities, and material flow within the constructed network. Also, the model maximizes the carbon income and income resulted from glycerin selling.

Feedstock suppliers

Oil extraction centers

Biodiesel plants

Biodiesel and glycerin customers

Figure. 1. The structure of the considered biomass-to-bioenergy supply chain system

In the considered biodiesel SC, the locations of foreign Jatropha suppliers, UCO supply centers, chemical industries, and biodiesel consumers are priori known, but other locations of facilities and their capacities should be determined via the proposed optimization model. Also, we consider continuous decision variables for determining the capacity of established facilities instead of discrete decision variables to avoid unnecessary establishing costs. The problem is considered for seven years planning horizon (seven periods). The nomenclature used in formulation of the proposed biodiesel supply chain network design problem are as follows. Parameters with tilde on indicate coefficients tainted with uncertainty. 6 ACS Paragon Plus Environment

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Sets f

Set of candidate locations for Jatropha cultivation centers

g

Set of candidate locations for UCO supply centers

i

Set of candidate locations for Jatropha oilseed extraction centers

j

Set of candidate locations for biodiesel plants of biodiesel production

l

Set of transportation mode (road and railway)

k

Set of biodiesel consumer centers

t

Set of time period

Economic and technical Parameters ̃𝑘𝑡 𝐷

Demand of biodiesel consumer center k in period t (ton/period)

̃ 𝑂𝑔𝑡 𝑊

Maximum amount of UCO supplied by supply center g in period t (ton/period)

𝜂̃𝑓𝑡

Amount of Jatropha yields per hectare at location f in period t (ton/ha)

𝜑

Conversion factor of Jatropha seeds to Jatropha oil (percent)

𝛽

Conversion factor of oil to biodiesel (percent)

𝐿𝐽𝑓

Minimum land area dedicated for Jatropha cultivation center at location f (ha*)

𝑈𝐽𝑓

Maximum land area available for Jatropha cultivation center at location f

𝐿𝑂𝑖

Lower bound dedicated on capacity of oilseed extraction center at location i (ton)

𝑈𝑂𝑖

Upper bound of capacity of oil extraction center at location i

𝐿𝐵𝑗

Lower bound dedicated on capacity of biodiesel plant at location j (ton)

𝑈𝐵𝑗

Upper bound of capacity of biodiesel plant at location j

Maxy

Maximum number of locations which can be selected for opening Jatropha oil extraction centers

Maxu

Maximum number of locations which can be selected for opening biodiesel plant

𝑀𝐶𝐿

Maximum allowable global warming potential by the constructed biodiesel supply chain

𝑀𝐼𝑖𝑡

Maximum amount of Jatropha seeds could be imported in oil extraction center i in period t

𝐹𝐽̃𝑓

Fixed cost of Jatropha cultivation at location f (MIRR*)



𝐹𝑂̃𝑖

Fixed cost of opening oilseed extraction center at location i

𝐹𝐵̃𝑗

Fixed cost of opening biodiesel plant at location j

𝑉𝐽̃𝑓

Variable cost of Jatropha cultivation per hectare at location f (MIRR/ha)

𝑉𝑂̃𝑖𝑡

Variable cost per unit capacity for oilseed extraction center i in period t (MIRR.ton-1/period)

𝑉𝐵̃𝑗𝑡

Variable cost per unit capacity for biodiesel plant j in period t (MIRR.ton-1/period)

𝑃𝐽̃𝑓𝑡

Unit production cost of Jatropha seeds at location f in period t (MIRR.ha-1/period)

𝑃𝑂̃𝑖𝑡

Unit oil extraction cost from Jatropha seeds at oilseed extraction center i in period t

𝑃𝐵̃𝑗𝑡

Unit production cost of biodiesel at biodiesel plant j in period t

𝐻𝑂̃𝑖𝑡

Unit inventory holding cost of Jatropha seeds at oil extraction center i in period t

𝐻𝐵̃𝑗𝑡

Unit inventory holding cost of biodiesel at biodiesel plant j in period t

𝐽𝑇̃𝑓𝑙𝑖𝑡

Transportation cost of Jatropha seeds from cultivation center f to oil extraction center i by mode l in period t (MIRR.ton-1/period)

𝑂𝑇̃𝑖𝑙𝑗𝑡

Transportation cost of Jatropha oilseed from oil extraction center i to biodiesel plant j by mode l in period t

𝑊𝑇̃𝑔𝑙𝑗𝑡

Transportation cost of UCO from supply center g to biodiesel plant j by mode l in period t

𝐵𝑇̃𝑗𝑙𝑘𝑡

Transportation cost of biodiesel from biodiesel plant j to consumer center k by mode l in period t

𝑃𝐼̃𝑖𝑡

Importing cost of Jatropha seeds in oil extraction center i in period t

𝑃𝐺̃𝑗𝑡

Selling price of glycerin at biodiesel plant j in period t (MIRR.ton-1/period)

𝐵𝐸̃

Carbon income due to reduction the amount of GHG emissions

Environmental impact parameters EJ

Climate change impact of harvesting 1 ton Jatropha seeds

EB

Climate change impact of producing 1 ton biodiesel at biodiesel plants

EG

Climate change impact of producing 1 ton glycerin at biodiesel plants

EO

Climate change impact of producing 1 ton Jatropha oil at oil extraction centers


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ES

Climate change impact of inventory holding of Jatropha seeds at oil extraction centers

EB

Climate change impact of inventory holding of biodiesel at biodiesel plants

EIJ

Climate change impact of inventory holding of Jatropha seeds at oil extraction centers

EIB

Climate change impact of inventory holding of biodiesel at biodiesel plants

𝐸𝐽𝑇𝑓𝑙𝑖

𝐸𝑊𝑇𝑔𝑙𝑗

𝐸𝑂𝑇𝑖𝑙𝑗

𝐸𝐵𝑇𝑗𝑙𝑘

Climate change impact of transporting 1 ton Jatropha seeds from cultivation center f to oil extraction center i by mode l Climate change impact of transporting 1 ton UCO per kilometer from supply center g to collection center j by mode l Climate change impact of transporting 1 ton Jatropha oil per kilometer from oil extraction center i to biodiesel plant j by model l Climate change impact of transporting 1 ton biodiesel per kilometer from biodiesel plant j to distribution center k by mode l

𝐷𝐽𝑇𝑓𝑙𝑖

Distance between locations f and i by mode l

𝐷𝑊𝑇𝑔𝑙𝑗

Distance between locations g and j by mode l

𝐷𝑂𝑇𝑖𝑙𝑗

Distance between locations i and j by mode l

𝐷𝐵𝑇𝑗𝑙𝑘

Distance between locations j and k by mode l

𝐸𝐼𝑚𝑖

Climate change impact of importing 1 ton Jatropha seeds imported in oil extraction center i

Binary decision variables 𝑥𝑓

1 if location f is selected for Jatropha cultivation; 0 otherwise

𝑦𝑖

1 if location i is selected for opening oilseed extraction center; 0 otherwise

𝑢𝑗

1 if location j is selected for opening biodiesel plant j; 0 otherwise

Continuous decision variables 𝐵𝑂𝑗𝑡

Produced amount of biodiesel at biodiesel plant j in period t



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𝐺𝐿𝑗𝑡

Produced amount of glycerin at biodiesel plant j in period t

𝐽𝑂𝑖𝑡

Produced amount of Jatropha oil at oil extraction center i in period t

𝐽𝑎𝑓𝑙𝑖𝑡

Transported amount of Jatropha seeds from cultivation center f to oil extraction center i by mode l in period t

𝑊𝑔𝑙𝑗𝑡

Transported amount of UCO from supply center g to biodiesel plant j by mode l in period t

𝑂𝑖𝑙𝑗𝑡

Transported amount of Jatropha oil from oil extraction center i to biodiesel plant j by mode l in period t

𝐵𝑗𝑙𝑘𝑡

Transported amount of biodiesel from biodiesel plant j to consumer center k by mode l in period t

𝐶𝐽𝑓

Amount of cultivated area of Jatropha at location f (ha)

𝐶𝑂𝑖𝑡

Total capacity of oil extraction center i in period t (ton)

𝐸𝑂𝑖𝑡

Amount of capacity expansion at oil extraction center i in period t

𝐶𝐵𝑗𝑡

Total capacity of biodiesel plant j in period t (ton)

𝐸𝐵𝑗𝑡

Amount of capacity expansion at biodiesel plant j in period t (ton)

𝐼𝑚𝑖𝑡

Amount of Jatropha seeds imported at oil extraction center i in period t (ton/period)

𝐼𝑂𝑖𝑡

Inventory amount of Jatropha seeds at oil extraction center i in period t

𝐼𝐵𝑗𝑡

Inventory amount of biodiesel at biodiesel plant center j in period t

𝛿

Amount of reduction of GHG emissions

*

MIRR (Million Iranian Rials), ha (hectare)

The proposed model:

𝑀𝑖𝑛 𝑍 = ∑ 𝐹𝐽̃𝑓 . 𝑥𝑓 + ∑ 𝐹𝑂̃𝑖 . 𝑦𝑖 + ∑ 𝐹𝐵̃𝑗 . 𝑢𝑗 + ∑ 𝑉𝐽̃𝑓 . 𝐶𝐽𝑓 + ∑ 𝑉𝑂̃𝑖𝑡 . 𝐶𝑂𝑖𝑡 𝑓

𝑖

𝑗

𝑓

𝑖,𝑡

+ ∑ 𝑉𝐵̃𝑗𝑡 . 𝐶𝐵𝑗𝑡 + ∑ 𝑃𝐶̃ 𝐽𝑓𝑡 . 𝑃𝐽𝑓𝑡 + ∑ 𝐽𝑇̃𝑓𝑙𝑖𝑡 . 𝐽𝑎𝑓𝑙𝑖𝑡 + ∑ 𝑃𝑂̃𝑖𝑡 . 𝐽𝑂𝑖𝑡 𝑗,𝑡

𝑓,𝑙,𝑖,𝑡

𝑓,𝑙,𝑖,𝑡

𝑖,𝑡

̃ 𝑖𝑙𝑗𝑡 . 𝑂𝑖𝑙𝑗𝑡 + ∑ 𝑊𝑇̃𝑔𝑙𝑗𝑡 . 𝑊𝑔𝑙𝑗𝑡 + ∑ 𝑃𝐵 ̃ 𝑗𝑡 . 𝐵𝑂𝑗𝑡 + ∑ 𝐵𝑇̃𝑗𝑙𝑘𝑡 . 𝐵𝑗𝑙𝑘𝑡 + ∑ 𝑂𝑇 𝑖,𝑙,𝑗,𝑡

𝑔,𝑙,𝑗,𝑡

𝑗,𝑡


𝑗,𝑙,𝑘,𝑡

(1)

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̃ 𝑖𝑡 . 𝐼𝑂𝑖𝑡 + ∑ 𝐻𝐵 ̃ 𝑗𝑡 . 𝐼𝐵𝑗𝑡 + ∑ 𝑃𝐼̃𝑖𝑡 . 𝐼𝑚𝑖𝑡 − ∑ 𝑃𝐺̃𝑗𝑡 . 𝐺𝐿𝑗𝑡 − 𝐵𝐸̃ . 𝛿 + ∑ 𝐻𝑂 𝑖,𝑡

𝑗,𝑡

𝑖,𝑡

𝑗,𝑡

Subject to: ̃𝑘𝑡 ∑ 𝐵𝑗𝑙𝑘𝑡 ≥ 𝐷

∀ 𝑘, 𝑡

(2)

𝑗,𝑙

∑ 𝑊𝑔𝑙𝑗𝑡 ≤ 𝑊𝑂̃𝑔𝑡

(3)

∀ 𝑔, 𝑡

𝑙,𝑗

̃𝑓𝑡 𝐶𝐽𝑓 𝑃𝐽𝑓𝑡 ≤ 𝜂

∀ 𝑓, 𝑡

∑ 𝐽𝑎𝑓𝑙𝑖𝑡 = 𝑃𝐽𝑓𝑡

(4)

∀ 𝑓, 𝑡

(5)

𝑙,𝑖

𝐽𝑂𝑖𝑡 = 𝜑 × (∑ 𝐽𝑎𝑓𝑙𝑖𝑡 + 𝐼𝑚𝑖𝑡 )

∀ 𝑖, 𝑡

(6)

𝑓,𝑙

(7) 𝐵𝑂𝑗𝑡 = 𝛽 × (∑ 𝑊𝑔𝑙𝑗𝑡 + ∑ 𝑂𝑖𝑙𝑗𝑡 ) 𝑔,𝑙

∀ 𝑗, 𝑡

𝑖,𝑙

𝐺𝐿𝑗𝑡 = (1 − 𝛽) × (∑ 𝑊𝑔𝑙𝑗𝑡 + ∑ 𝑂𝑖𝑙𝑗𝑡 ) 𝑔,𝑙

∀ 𝑗, 𝑡

(8)

𝑖,𝑙

1 𝐼𝑂𝑖𝑡 = 𝐼𝑂𝑖(𝑡−1) + 𝐼𝑚𝑖𝑡 + ∑ 𝐽𝑎𝑓𝑙𝑖𝑡 − ( ) ∑ 𝑂𝑖𝑙𝑗𝑡 𝜑 𝑓,𝑙

∀ 𝑖, 𝑡

(9)

𝑙,𝑗

𝐼𝐵𝑗𝑡 = 𝐼𝐵𝑗(𝑡−1) + 𝐵𝑂𝑗𝑡 − ∑ 𝐵𝑗𝑙𝑘𝑡

∀ 𝑗, 𝑡

(10)

𝑙,𝑘

𝑥𝑓 𝐿𝐽𝑓 ≤ 𝐶𝐽𝑓 ≤ 𝑥𝑓 𝑈𝐽𝑓 𝐶𝑂𝑖𝑡 = 𝐶𝑂𝑖(𝑡−1) + 𝐸𝑂𝑖𝑡 𝑦𝑖 𝐿𝑂𝑖 ≤ 𝐶𝑂𝑖𝑡 ≤ 𝑦𝑖 𝑈𝑂𝑖 𝐶𝐵𝑗𝑡 = 𝐶𝐵𝑗(𝑡−1) + 𝐸𝐵𝑗𝑡

∀𝑓

(11)

∀ 𝑖, 𝑡

(12)

∀ 𝑖, 𝑡

(13)

∀ 𝑗, 𝑡

(14)



𝑢𝑗 𝐿𝐵𝑗 ≤ 𝐶𝐵𝑗𝑡 ≤ 𝑢𝑗 𝑈𝐵𝑗

∀ 𝑗, 𝑡

Page 12 of 32

(15)

∑ 𝑦𝑖 ≤ 𝑀𝑎𝑥𝑦

(16)

𝑖

∑ 𝑢𝑗 ≤ 𝑀𝑎𝑥𝑢

(17)

𝑗

𝐼𝑚𝑖𝑡 ≤ 𝑀𝐼𝑖𝑡

∀ 𝑖, 𝑡

(18)

∑ 𝐽𝑎𝑓𝑙𝑖𝑡 + 𝐼𝑚𝑖𝑡 ≤ 𝐶𝑂𝑖𝑡

∀ 𝑖, 𝑡

(19)

𝑓,𝑙

∑ 𝑊𝑔𝑙𝑗𝑡 + ∑ 𝑂𝑖𝑙𝑗𝑡 ≤ 𝐶𝐵𝑗𝑡 𝑔,𝑙

∀ 𝑗, 𝑡

(20)

𝑖,𝑙

∑ 𝑒𝑥. 𝐶𝐽𝑓 + ∑ 𝑒𝑦. 𝐶𝑂𝑖𝑡 + ∑ 𝑒𝑢. 𝐶𝐵𝑗𝑡 + ∑ 𝐸𝑂. 𝐽𝑂𝑖𝑡 + ∑ 𝐸𝐵. 𝐵𝑂𝑗𝑡 𝑓

𝑖,𝑡

𝑗,𝑡

𝑖,𝑡

𝑗,𝑡

+ ∑ 𝐸𝐺. 𝐺𝐿𝑗𝑡 + ∑ 𝐸𝐼𝐽. 𝐼𝑂𝑖𝑡 + ∑ 𝐸𝐼𝐵. 𝐼𝐵𝑗𝑡 + ∑ 𝐸𝐽𝑇𝑓𝑙𝑖 . 𝐷𝐽𝑇𝑓𝑙𝑖 . 𝐽𝑎𝑓𝑙𝑖𝑡 𝑗,𝑡

𝑖,𝑡

𝑗,𝑡

𝑓,𝑙,𝑖,𝑡

(21) + ∑ 𝐸𝑊𝑇𝑔𝑙𝑗 . 𝐷𝑊𝑇𝑔𝑙𝑗 . 𝑊𝑔𝑙𝑗𝑡 + ∑ 𝐸𝑂𝑇𝑖𝑙𝑗 . 𝐷𝑂𝑇𝑖𝑙𝑗 . 𝑂𝑖𝑙𝑗𝑡 + ∑ 𝐸𝐵𝑇𝑗𝑙𝑘 . 𝐷𝐵𝑇𝑗𝑙𝑘 . 𝐵𝑗𝑙𝑘𝑡 𝑔,𝑙,𝑗,𝑡

𝑖,𝑙,𝑗,𝑡

𝑗,𝑙,𝑘,𝑡

+ ∑ 𝐸𝐼𝑚𝑖 . 𝐼𝑚𝑖𝑡 + 𝛿 ≤ 𝑀𝐶𝐿 𝑖,𝑡

𝑥𝑓 , 𝑦𝑖 , 𝑢𝑗 ∈ {0,1}

(22)

𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑑𝑒𝑐𝑖𝑠𝑖𝑜𝑛 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 ≥ 0

(23)

Objective function (1) minimizes the total fixed and variable costs of establishing facilities, production and transportation costs, Jatropha seed importing cost, and maximizes income obtained from carbon income and selling glycerin. Constraint (2) states that all biodiesel demands are satisfied in different cities. Equation (3) ensures that the amount of UCO collected from different provinces is not greater than the maximum UCO supplied in that province. Equation (4) calculates the maximum Jatropha yields could be achieved from each farm in any period. Limitation (5) ensures that all Jatropha yields are transported to oil extraction centers. Equation (6) represents the amount of produced Jatropha oil from harvested and imported Jatropha seeds in each oilseed 12 ACS Paragon Plus Environment

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extraction center. Equations (7) and (8) take into account the amount of produced biodiesel and glycerin from received oils, respectively. Constraints (9) and (10) consider inventory balances at oil extraction centers and biodiesel plants. According to constraint (9), the amount of inventory of Jatropha seeds at each oil extraction center in any period is equal to the amount of Jatropha seeds remained from previous period plus the amount of Jatropha seed received from farms at this period plus the amount of Jatropha seeds imported in current period minus the amount of Jatropha converted to Jatropha oil and shipped to biodiesel plants in current period. Constraint (11) binds lower and upper bound for selected locations for Jatropha cultivation. Lower bound is considered due to economy of scale consideration and upper bound is considered because of biodiversity concern. Constraints (12) and (13) take into account capacity expansion and lower and upper bound for established oilseed extraction centers. Constraints (14) and (15) are capacity expansions and lower and upper bounds in biodiesel plants. Constraints (16) and (17) represent the maximum number of oil extraction and biodiesel plants which could be established. These constraints may be implied due to budget limitations. Constraint (18) considers the maximum amount of Jatropha seeds which could be imported from foreign suppliers in each period. Constraints (19) and (20) are capacity constraints for oilseed extraction centers and biodiesel plants, respectively. Constraint (21) limits the amount of GHG emissions from the processes within the boundary of the constructed biodiesel supply chain network. Finally, constraints (22) and (23) consider binary and non-negativity restrictions on the related decision variables. The main important input parameters of the proposed model such as feedstock supply, biodiesel demand, different costs, and incomes are really uncertain. Due to lack of historical data and knowledge about the data used in the studied case, there are no probability or possibility distribution to model their uncertainty. Therefore, to tackle the deep uncertainty of parameters, the robust optimization approach proposed by Li et al. [14] is employed with small modifications.

4. Set-induced robust optimization method To handle the uncertainties in which there is no information about them, set-induced robust optimization methods are applied [14]. In set-induced robust optimization, it is assumed that the uncertain parameters are varied in a given uncertainty set and the model seeks for those solutions that immunize the system for all potential realizations from uncertainty set. Indeed, the best solutions obtained by the robust optimization method should be feasible for all realizations of 13 ACS Paragon Plus Environment


Page 14 of 32

possible values of uncertain parameters in the given uncertainty set. Different uncertainty sets including box, ellipsoidal, and polyhedral as well as their mixed forms are used to model the uncertainty in the literature [31]. Although realistic robust optimization approaches could be achieved through ellipsoidal and polyhedral uncertainty sets, the method achieved by polyhedral set would be linear which leads to computational efficiency. Polyhedral uncertainty set provides realistic feasible solutions with less degree of conservatism immunizing the system for reasonable realizations of uncertain parameters. In fact, assuming the polyhedral uncertainty set for uncertain parameters implies that all uncertain parameters may not get their worst-case values in the given set, simultaneously. But, the possible values of uncertain parameters are varied within a polyhedral set [13]. It should be noted that the formulation of a robust counterpart model depends upon the given uncertainty set assumed for uncertain parameters. Here, the robust counterpart for the original model is formulated by assuming the polyhedral uncertainty set for uncertain parameters. Consider the following well-known mathematical programming model with uncertain parameters including cj, aij, and bi. Assume that the values of these parameters vary in a bounded polyhedral uncertainty set, say U. Min

c x j

j

j

S .t.

a x ij

j

 bi i,

(24)

j

c , a , b U Polyhedral

The bar sign is used to show that the corresponding parameters are subject to uncertainty. Description of different types of uncertainty sets such as box, polyhedral, and ellipsoidal has been provided in Li et al. [14]. The parameters c j , aij , and bi can be written as follows:

c j  c j   j E cj  j

(25)

aij  aij  ij E ijaij

(26)

bi  bi  i Eibi

(27)


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Where cj, aij, and bi are the nominal values of the corresponding uncertain parameters, , which is a positive number, represents the uncertainty level for the related uncertain parameters, E indicates the uncertainty scale for the related uncertain parameters, and  is a random variable. Note that if the variable  is bounded, the polyhedral uncertainty set will be bounded. Hereafter, the indices of the above-mentioned parameters are eliminated for simple representation. The uncertainty level expresses the perturbation percentage of uncertain parameters around their nominal values. A particular case of interest is that the uncertainty scale is assumed to be equal to the nominal values [13]. It should be noted that the constraints of the problem (24) should be converted to the following form to be consistent with the robust optimization principles.   aij x j  bi i, j

Under the given polyhedral set, finding a robust solution for the problem (24) means that all constraints remain feasible for all realizations of uncertain parameters varied within the given polyhedral set and its objective function value is not worse than the objective function values under all realizations. It is noteworthy that the formulation of Li et al. [14] has been presented for profit maximization. In this paper, we modify their formulation to be used in cost minimization problems. The polyhedral uncertainty set is defined using the 1-norm of the uncertain data vector as follows,

  U1   |  1     |   j     j Ji 





(28)

Where Γ is an adjustable parameter that controls the size of uncertainty set. Ji represents the index subset including the variable indices that their corresponding coefficients are subject to uncertainty. Indeed, |Ji| indicates the number of variables whose corresponding coefficients are subject to uncertainty in the ith constraint. Without loss of generality, for bounding the polyhedral uncertainty set consider that j is varied in the range [-1, 1]. In this case, representation of uncertain parameters can be easily done by changing the uncertainty level in the given bounded polyhedral set. In order to avoid covering the overall uncertain space the adjustable parameter should be less than or equal to the cardinality of the uncertainty set in any constraint (i.e. Γ  |Ji|). 4.1. The robust counterpart optimization model



Page 16 of 32

In order to apply the uncertainty on the coefficients of the objective function, it should be considered as a constraint. By replacing the equations (25)-(27) in the problem (24) we have: Min

c x    E  x j

j

j

c j

j

j

j

j

S .t.   aij x j   ij Eija ij x j  bi  i Eib i j

i ,

(29)

j

Which can be rewritten as follows: Min z

S .t.

c x    j

j

j

j

E cj  j x j  z

(30)

j

S .t.   aij x j   ij Eija ij x j  bi  i Eib i j

i ,

j

In problem (30), the goal is to find solutions immunizing the feasibility of the model for all possible values of  in the range [-1, 1] for the given polyhedral set. Therefore, in order to enable problem (30) to find such robust solutions, it should be transformed to the following problem, which is the robust counterpart of problem (24), for polyhedral uncertainty set. Min z

S .t.

c j

j

  x j  max    j E cj  j x j   z  U Pol .  j 

(31)

    aij x j  max  ij Eija  ij x j  i Eib i   bi  U Pol . j  j 

i ,

The problem (31) is computationally intractable due to too many possible values of uncertain parameters within the polyhedral set. Note that we do not know the behavior of uncertain parameters within the polyhedral set; however, we are aware that the uncertain parameters are varied within the bounded polyhedral uncertainty set. In other words, the probability distribution or possibility distribution of uncertain parameters is not clear in a bounded set. Because of this ambiguity, applying different stochastic methods is not possible. Therefore, problem (31) is a computationally intractable problem [13]. However, it could be transformed to the tractable convex and linear programming model [14]. 16 ACS Paragon Plus Environment

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The equivalent non-linear form of problem (31) can be stated as follows [14]:

Min z S .t .

c x j

j

 w  z

j

w   j E cj x j

j ,

 aij x j   i wi  bi

(32)

i,

j

wi  ij Eija x j

j

wi  i Eib

i,

Note that the adjustable parameter shows the number of uncertain parameters in the corresponding constraint. In addition, w is an auxiliary variable. Really, the absolute form in problem (32) imposes non-linearity to the model. Meanwhile, it could be easily transformed to the linear form as follows:

Min z S .t .

c x j

j

 w  z

j

w   j E cj y j

a

ij

j ,

x j   i wi  bi

(33)

i,

j

wi  ij Eija y j

wi  i Eib

j

i,

y j  x j  y j

j

It is worthy to note that if the variables (i.e. xj) are non-negative, the problem (32) is solved without absolute forms to find robust solutions. 17 ACS Paragon Plus Environment


Page 18 of 32

According to above-mentioned descriptions, the robust counterpart optimization model for the proposed model in Section 3 could be represented as follows. 𝜌, 𝐸, and Γ show uncertainty level, uncertainty scale, and adjustable parameters for any uncertain parameter, respectively [13]. To apply the polyhedral uncertainty set in modeling the uncertain parameters of the proposed robust optimization model, firstly the cardinality of uncertainty set (i.e., |Ji|) is calculated for any constraint including uncertain parameters. To this aim, the corresponding indices of the uncertain parameters existing in any constraint are multiplied by each other (for example for Dkt: |Ji|=|k|×|t|). The adjustable parameter (i.e., Γ) controls the uncertainty space covered by the polyhedral uncertainty set, while the uncertainty level (i.e., ) controls the range of uncertain parameters changes. If the adjustable parameter be equal or larger than the number of uncertain parameters in any constraint (i.e., Γ  |Ji|), the overall uncertainty space is covered by the polyhedral set. In particular case, if Γ = |Ji|, the intersection between the polyhedral and box uncertainty set is exactly the box [14]. The realistic robust solutions can be obtained when Γ < |Ji|, which is followed in this paper. 𝑀𝑖𝑛 𝑍 (34) Subject to: ∑ 𝐹𝐽𝑓 . 𝑥𝑓 + Γ. 𝜇 + ∑ 𝐹𝑂𝑖 . 𝑦𝑖 + Γ1. 𝜇 + ∑ 𝐹𝐵𝑗 . 𝑢𝑗 + Γ2. 𝜇 𝑓

𝑖

𝑗

+ ∑ 𝑉𝐽𝑓 . 𝐶𝐽𝑓 + Γ3. 𝜇 + ∑ 𝑉𝑂𝑖𝑡 . 𝐶𝑂𝑖𝑡 + Γ4. 𝜇 + ∑ 𝑉𝐵𝑗𝑡 . 𝐶𝐵𝑗𝑡 + Γ5. 𝜇 𝑓

𝑖,𝑡

𝑗,𝑡

+ ∑ 𝑃𝐶𝐽𝑓𝑡 . 𝑃𝐽𝑓𝑡 + Γ6. 𝜇 + ∑ 𝐽𝑇𝑓𝑙𝑖𝑡 . 𝐽𝑎𝑓𝑙𝑖𝑡 + Γ7. 𝜇 + ∑ 𝑃𝑂𝑖𝑡 . 𝐽𝑂𝑖𝑡 + Γ8. 𝜇 𝑓,𝑡

𝑓,𝑙,𝑖,𝑡

𝑖,𝑡

(35) + ∑ 𝑂𝑇𝑖𝑙𝑗𝑡 . 𝑂𝑖𝑙𝑗𝑡 + Γ9. 𝜇 + ∑ 𝑊𝑇𝑔𝑙𝑗𝑡 . 𝑊𝑔𝑙𝑗𝑡 + Γ10. 𝜇 + ∑ 𝑃𝐵𝑗𝑡 . 𝐵𝑂𝑗𝑡 + Γ11. 𝜇 𝑖,𝑙,𝑗,𝑡

𝑔,𝑙,𝑗,𝑡

𝑗,𝑡

+ ∑ 𝐵𝑇𝑗𝑙𝑘𝑡 . 𝐵𝑗𝑙𝑘𝑡 + Γ12. 𝜇 + ∑ 𝐻𝑂𝑖𝑡 . 𝐼𝑂𝑖𝑡 + Γ13. 𝜇 + ∑ 𝐻𝐵𝑗𝑡 . 𝐼𝐵𝑗𝑡 + Γ14. 𝜇 𝑗,𝑙,𝑘,𝑡

𝑖,𝑡

𝑗,𝑡

+ ∑ 𝑃𝐼𝑖𝑡 . 𝐼𝑚𝑖𝑡 + Γ15. 𝜇 − ∑ 𝑃𝐺𝑗𝑡 . 𝐺𝐿𝑗𝑡 + Γ16. 𝜇 − 𝐵𝐸. 𝛿 + Γ17. 𝜇 ≤ 𝑍 𝑖,𝑡

𝑗,𝑡


Page 19 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


𝐹𝐽

𝐹𝐽

𝜇 ≥ 𝜌𝑓 . 𝐸𝑓 . 𝑥𝑓

∀𝑓

(36)

𝜇 ≥ 𝜌𝑖𝐹𝑂 . 𝐸𝑖𝐹𝑂 . 𝑦𝑖

∀𝑖

(37)

𝜇 ≥ 𝜌𝑗𝐹𝐵 . 𝐸𝑗𝐹𝐵 . 𝑢𝑗

∀𝑗

(38)

𝜇 ≥ 𝜌𝑓𝑉𝐽 . 𝐸𝑓𝑉𝐽 . 𝐶𝐽𝑓

∀𝑓

(39)

𝑉𝑂 𝑉𝑂 𝜇 ≥ 𝜌𝑖𝑡 . 𝐸𝑖𝑡 . 𝐶𝑂𝑖𝑡

∀ 𝑖, 𝑡

(40)

𝑉𝐵 𝑉𝐵 𝜇 ≥ 𝜌𝑗𝑡 . 𝐸𝑗𝑡 . 𝐶𝐵𝑗𝑡

∀ 𝑗, 𝑡

(41)

𝑃𝐶𝐽

𝑃𝐶𝐽

𝜇 ≥ 𝜌𝑓𝑡 . 𝐸𝑓𝑡 . 𝑃𝐽𝑓𝑡

∀ 𝑓, 𝑡

𝐽𝑇 𝐽𝑇 𝜇 ≥ 𝜌𝑓𝑙𝑖𝑡 . 𝐸𝑓𝑙𝑖𝑡 . 𝐽𝑎𝑓𝑙𝑖𝑡 𝑃𝑂 𝑃𝑂 𝜇 ≥ 𝜌𝑖𝑡 . 𝐸𝑖𝑡 . 𝐽𝑂𝑖𝑡 𝑂𝑇 𝑂𝑇 𝜇 ≥ 𝜌𝑖𝑙𝑗𝑡 . 𝐸𝑖𝑙𝑗𝑡 . 𝑂𝑖𝑙𝑗𝑡

∀ 𝑓, 𝑙, 𝑖, 𝑡

𝐵𝑇 𝐵𝑇 𝜇 ≥ 𝜌𝑗𝑙𝑘𝑡 . 𝐸𝑗𝑙𝑘𝑡 . 𝐵𝑗𝑙𝑘𝑡

(43)

∀ 𝑖, 𝑡

(44)

∀ 𝑖, 𝑙, 𝑗, 𝑡

𝑊𝑇 𝑊𝑇 𝜇 ≥ 𝜌𝑔𝑙𝑗𝑡 . 𝐸𝑔𝑙𝑗𝑡 . 𝑊𝑔𝑙𝑗𝑡 𝑃𝐵 𝜇 ≥ 𝜌𝑗𝑡 . 𝐸𝑗𝑡𝑃𝐵 . 𝐵𝑂𝑗𝑡

(42)

(45)

∀ 𝑔, 𝑙, 𝑗, 𝑡

(46)

∀ 𝑗, 𝑡

(47)

∀ 𝑗, 𝑙, 𝑘, 𝑡

(48)

𝐻𝑂 𝐻𝑂 𝜇 ≥ 𝜌𝑖𝑡 . 𝐸𝑖𝑡 . 𝐼𝑂𝑖𝑡

∀ 𝑖, 𝑡

(49)

𝐻𝐵 𝜇 ≥ 𝜌𝑗𝑡 . 𝐸𝑗𝑡𝐻𝐵 . 𝐼𝐵𝑗𝑡

∀ 𝑗, 𝑡

(50)

𝑃𝐼 𝜇 ≥ 𝜌𝑖𝑡 . 𝐸𝑖𝑡𝑃𝐼 . 𝐼𝑚𝑖𝑡 𝑃𝐺 𝜇 ≥ 𝜌𝑗𝑡 . 𝐸𝑗𝑡𝑃𝐺 . 𝐺𝐿𝑗𝑡

∀ 𝑖, 𝑡

(51)

∀ 𝑗, 𝑡

(52)

𝜇 ≥ 𝜌𝐵𝐸 . 𝐸 𝐵𝐸 . 𝛿 𝐷 𝐷 ∑ 𝐵𝑗𝑙𝑘𝑡 − 𝜌𝑘𝑡 . 𝐸𝑘𝑡 ≥ 𝐷𝑘𝑡

(53) ∀ 𝑘, 𝑡

(54)

𝑗,𝑙 𝑊𝑂 𝑊𝑂 ∑ 𝑊𝑔𝑙𝑗𝑡 + 𝜌𝑔𝑡 . 𝐸𝑔𝑡 ≤ 𝑊𝑂𝑔𝑡

∀ 𝑔, 𝑡

(55)

𝑙,𝑗



𝑃𝐽𝑓𝑡 − 𝜂𝑓𝑡 𝐶𝐽𝑓 + Γ20. 𝜇1 ≤ 0 𝜂

𝜂

𝜇1 ≥ 𝜌𝑓𝑡 . 𝐸𝑓𝑡 𝐶𝐽𝑓

∀ 𝑓, 𝑡

Page 20 of 32

(56)

∀ 𝑓, 𝑡

(57)

𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑑𝑒𝑐𝑖𝑠𝑖𝑜𝑛 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠, 𝑎𝑢𝑥𝑖𝑙𝑖𝑎𝑟𝑦 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝜇 ≥ 0

(58)

Constraints (5)-(22). Since the parameters in constraints (5)-(22) are deterministic parameters, these constraints are repeated in robust counterpart model.

5. Case study description and results In this section, a real case study in Iran is conducted to assess the performance of the proposed model in real life application. The Renewable Energy Organization of Iran is targeted to utilize biofuels in transportation sector especially in large and industrial provinces. In these provinces, environmental problems have reduced standards of living and caused many health problems. Therefore, designing biodiesel supply chain network and determining related optimum strategic and tactical decisions under uncertainty could help policy makers to achieve sustainable and robust biodiesel supply chain network. To this aim, the related data of the case study is gathered and employed in the proposed model. Iran has a great potential for exploiting different biomass resources and energy crops such as sunflower, sorghum, sugar beet, switchgrass, and Jatropha. Indeed, there are good climate conditions and waste lands in Iran for prospering agriculture activities. In addition, Iran has useless extensive hectares of arid and semi-arid areas that could be exploited for cultivation of energy crops such as Jatropha that grows in marginal and infertile lands, efficiently. Considering the fact that Iran imports about 80 percent of its edible oil consumption, utilizing edible oils sources such as palm, corn, sunflower, soya, and cotton for biodiesel production is irrational and would be a menace for food supply resources. Therefore, exploiting other non-edible oil sources such as Jatropha and UCO can be a great opportunity for Iran. Another interesting source of biodiesel production in Iran is UCO, which is produced about 300,000 ton (t), annually. In recent years, Jatropha has been cultivated in some southern cities of Iran in small scales to evaluate its yielding. The results show that Jatropha plants yield under 17 months of cultivation, while the average time to its yielding is about 24 months [32]. 20 ACS Paragon Plus Environment

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5.1. Data gathering

In the proposed model, different cost and income parameters, demands of biodiesel and glycerin and amount of UCO in different supply centers are really uncertain parameters. To estimate the nominal data (i.e., 𝐸) of uncertain parameters, available limited historical data is used. Then, percentage of data change (i.e., 𝜌) values are estimated. The 𝜌 values could be achieved by experts' opinions or data change over time. However, note that in such situations, the behavior of uncertain parameters is unknown to be stochastic or fuzzy. In this case, the parameters have deep uncertainty and thus no probability or possibilistic distributions could be set for them. In this study, 𝜌 values are considered 0.08-0.2 for various uncertain parameters. The planning horizon is considered to be 7 years and each year is considered as a period. The demands of biodiesel of large and industrial provinces of Iran for different periods have been shown in Table 1. In Iran, it has been planned to reach 5% biodiesel substitution with fossil diesel after planning horizon. This substitution percentages are 2%, 2.5%, 3%, 3.5%, 4%, 4.5%, and 5% for different periods. These values are used for calculating the biodiesel demands according to fossil diesel demands in different provinces.

Table 1. Biodiesel demand in different periods (Million Liter) Period Province t=1 t=2 t=3 t=4

t=5

t=6

t=7

Tabriz

64380

194925

341400

344814

348262

351745

355262

Isfahan

202920

614675

1077600

1088376

1099260

1110252

1121355

Tehran

296040

864250

1177800

1189578

1201474

1213489

1225623

Khorasan R.

119100

360950

632700

639027

645417

651871

658390

Khozestan

155940

472425

827700

835977

844337

852780

861308

Fars

146460

443400

777600

785376

793230

801162

809174

Gazvin

71840

99510

190800

192708

194635

196581

198547

Ghom

41460

57280

109650

110747

111854

112973

114102

Kerman

140700

194850

373500

377235

381007

384817

388666

Markazi

76500

106000

203250

205283

207335

209409

211503

Glycerin as a by-product of transesterification process is used in producing hygienic products. 11 provinces of Iran have factories producing hygienic products. About 75% of hygienic products are produced in Tehran. The produced glycerin is sold to hygienic factories in biodiesel plants and therefore its logistics costs are not considered in the proposed model.



Page 22 of 32

The amount of UCO produced in each city is provided from Iranian Fuel Conversion Company (www.ifco.ir). Table 2 indicates the amount of UCO collected from all provinces in different periods.

Table 2. Amount of used cooking oil from supply centers (Million Liter) Periods Province t=1 t=2 t=3 t=4 t=5

t=6

t=7

Tabriz

33524

33928

34336

34750

35169

35592

36021

Uromieh

27726

28060

28398

28740

29086

29437

29791

Ardabil

11238

11373

11510

11649

11789

11931

12075

Isfahan

43918

44447

44983

45525

46073

46628

47190

Ilam

5018

5078

5140

5202

5264

5328

5392

Bushehr

9298

9410

9523

9638

9754

9872

9991

Tehran

109661

110983

112320

113673

115043

116429

117831

8060

8157

8256

8355

8456

8558

8661

Chahar M. B. Khorasan J.

5961

6033

6105

6179

6253

6329

6405

Khorasan R.

53957

54607

55265

55931

56605

57287

57977

Khorasan Sh.

7810

7904

7999

8096

8193

8292

8392

Khozestan

40789

41280

41777

42281

42790

43306

43828

Zanjan

9144

9255

9366

9479

9593

9709

9826

Semnan

5687

5755

5825

5895

5966

6038

6110

Sistan va

22813

23088

23366

23648

23932

24221

24513

Fars

41379

41878

42382

42893

43410

43933

44462

Gazvin

10816

10946

11078

11212

11347

11484

11622

Ghom

10373

10498

10625

10753

10882

11014

11146

Kurdistan

13449

13611

13775

13941

14109

14279

14451

Kerman

26455

26773

27096

27422

27753

28087

28425

Kermanshah

17512

17723

17936

18152

18371

18593

18817

Kohgiluyeh

5928

5999

6071

6144

6219

6293

6369

Golestan

15997

16190

16385

16582

16782

16984

17189

Gilan

22331

22600

22872

23148

23427

23709

23995

Lorestan

15795

15985

16178

16373

16570

16770

16972

Mazandaran

27674

28008

28345

28687

29032

29382

29736

Markazi

12729

12882

13037

13194

13353

13514

13677

Hormozgan

14208

14379

14552

14727

14905

15084

15266

Hamadan

15825

16016

16209

16404

16602

16802

17004

Yazd

9675

9791

9909

10029

10149

10272

10395

Fixed opening costs, variable opening costs, production costs, inventory holding costs, glycerin income are taken into account according to own calculations and previous feasibility studies 22 ACS Paragon Plus Environment

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performed in Iran. All used data can be provided upon request. The price index in each province is used to estimate the costs for each province. Also, annual inflation rate is used to estimate the costs for upcoming years. Transportation cost between each two provinces is calculated through multiplying unitary transportation cost by distance between them. Road and rail distances between different cities are provided from Ministry of Roads & Urban Development (www.mrud.ir) and Asia Seir Aras Company (www.asiaseiraras.com), respectively. Conversion factor of Jatropha yields to Jatropha oil is assumed to be 0.35 [33]. Conversion factor of collected UCO to purified UCO is assumed to be 0.95. Also, conversion factors of oils to biodiesel are considered to be 0.83 [30]. The value of Jatropha yields in different periods are extracted from Achten et al. [33]. Note that Jatropha yields is location and time period dependent. Locations with good ecological and soil conditions have higher amount of yields. It is worthy to note that Jatropha yields are altered from 2 to 12 t/ha/y in the literature according to ecological and soil conditions [34]. Since arid and semiarid areas are considered as potential areas for Jatropha cultivation in the studied case, we assume the Jatropha yields are between 2 to 7 t/ha/y. In mild conditions like as our case, this amount of Jatropha yields is expected according to real experiences and scientific reports [33]. Also, we consider 3×3 m2 space for cultivation of each Jatropha plant and so there would be about 1100 plants per hectare [33]. The nominal data of Jatropha yields are shown in Table 3.

Table 3. Jatropha yields in different periods Period Province

t=1

t=2

t=3

t=4

t=5

t=6

t=7

Isfahan

5

14

15

15

15

15

15

Ilam

5

14

15

15

15

15

15

Bushehr

8.8

21.2

22.8

22.8

22.8

22.8

22.8

Tehran

5

14

15

15

15

15

15

Chahar Mahaal and Bakhtiari

5

14

15

15

15

15

15

Khorasan J.

5

14

15

15

15

15

15

Khorasan R.

5

14

15

15

15

15

15

Khorasan Sh.

5

14

15

15

15

15

15

8.8

21.2

22.8

22.8

22.8

22.8

22.8

Zanjan

5

14

15

15

15

15

15

Semnan

5

14

15

15

15

15

15

8.8

21.2

22.8

22.8

22.8

22.8

22.8

5

14

15

15

15

15

15

Khozestan

Sistan va Balochestan Fars



Page 24 of 32

Gazvin

5

14

15

15

15

15

15

Ghom

5

14

15

15

15

15

15

Kerman

5

14

15

15

15

15

15

Kermanshah

5

14

15

15

15

15

15

Kohgiluyeh and Boyer-Ahmad

6.8

17

18

18

18

18

18

Lorestan

5

14

15

15

15

15

15

Markazi

5

14

15

15

15

15

15

Hormozgan

8.8

21.2

22.8

22.8

22.8

22.8

22.8

Hamadan

5

14

15

15

15

15

15

Yazd

5

14

15

15

15

15

15

5.2. CO2-equivalent of processes

In this section, SimaPro 8 software (www.pre-sustainability.com) equipped with ecoinvent version3 database is employed to calculate the CO2-equivalent of all processes of the considered biodiesel supply chain network. Climate change factor in SimaPro is used as representative of CO2-equivalent. SimaPro 8 is a comprehensive tool specified for environmental impact assessment of different processes of various industries based on Life Cycle Assessment (LCA) method. Although exact environmental impact of different processes are geographic location dependent, SimaPro 8 provides environmental impact assessment under standard conditions which can be used in different zones without needing for LCA experts. In the studied case, truck (28 ton) is used for road transportation and train consuming diesel as fuel is used for rail transportation purpose. These modes are common transportations modes for cargo shipment in Iran. In Jatropha oil extraction centers, cold pressing technology is used to extract the Jatropha oilseeds. Also, transesterification process is used for biodiesel and glycerin production in biodiesel plants. According to above information, Table 4 illustrates the CO2equivalent values of different processes in the studied case. To calculate the upper bound (MCL) for maximum allowable CO2 emissions, firstly, the proposed model is run with environmental objective function including the left hand side of constraint (76) and costs objective function is eliminated. Then, according to expert's opinions this value is multiplied by 1.1 to achieve MCL values. Also, sensitivity analysis is performed on the amount of MCL values to evaluate the cost objective function respect to its changes. Note that setting values less than the above value leads to infeasibility of the proposed model. In other words, the CO2-equivalent of all processes of the constructed biodiesel supply chain network could not be less than value set. Also, carbon income values are calculated according to incentives granted by Department of Environment of Iran 24 ACS Paragon Plus Environment

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(www.en.doe.ir) for industries reducing CO2 emissions. Considering carbon income factor in objective function will reduce total emissions of CO2 as possible as. Table 4. The CO2-equivalent values of processes in the considered biodiesel supply chain Parameter Value 𝑒x

4.61

𝑒y

1007.94

𝑒u

40100

𝐸B

420

EG

421

EO

122

E𝐼J

0.276

E𝐼B

10.1

E𝐼mi

22.29 0.00272 (Road)

E𝐽Tfli 0.000682 (Rail) 0.00272 (Road) E𝑊Tgls 0.000682 (Rail) 0.00272 (Road) E𝑂Tilj 0.000682 (Rail) 0.00272 (Road) E𝐵Tjlk 0.000682 (Rail)

5.3. Results of implementation

To solve the proposed model, GAMS optimization software and its CPLEX solver is employed. The global optimum solution is achieved under 30 minutes for the proposed model. Due to strategic level of decisions made by the proposed model, this amount of time for solving the proposed model is ideal. Table 5 indicates the share of different costs and incomes in objective function. As shown in this table, the amount of variable opening costs, production costs, transportation costs, and



Page 26 of 32

glycerin incomes are considerable and have major share in constituting objective function. Also, the proposed model prefers to provide all required Jatropha from domestic suppliers.

Table 5. Share of different costs and incomes in objective function Costs and incomes Fixed opening Costs Variable opening costs Production costs Transportation costs Inventory costs Importing costs Glycerin income Carbon income

Value 37,869 6.6,546,88E+8 5.470,856E+8 7.541,993E+7 7,326,215 0 2.353,336E+7 1.146,253E+7

Table 6 shows the optimum locations and their total capacities achieved by the proposed model. In other words, this table illustrates the structure of the considered biodiesel supply chain network under deep uncertainty. According to this table, all Jatropha oil extraction centers are established in Jatropha cultivation centers. This result could be explained due to reduction of total transportation costs. Table 6. Optimum locations and capacities of cultivation centers and different facilities

Jatropha cultivation areas (ha)

Jatropha oil extraction centers

Biodiesel plants

Province Tehran Chahar Mahal & B. Khorasan R. Semnan Sistan & B. Ghom

Capacity 200,000 1,000,000 700,000 1,000,000 750,000 566,314

Chahar Mahal & B. Khorasan R. Semnan Sistan & B. Ghom

3,969,665 4,550,000 4,200,000 4,200,000 1,512,435

Isfahan Khorasan J. Khorasan Sh.

2,367,755 2,763,422 2,707,926


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As mentioned previously, to calculate the amount of MCL, firstly, total CO2 emissions is considered as objective function and minimized as possible as. The minimum value of total CO 2 emissions of all processes in the considered biodiesel supply chain network is multiplied by 1.1 to achieve MCL. This value is equal to 3.36357E+11. Setting MCL values less than this value leads to infeasibility of the proposed model. Table 7 shows the sensitivity analysis on the upper bound of CO2 emission (MCL) values. To perform sensitivity analysis, the boundary value of MCL (3.36357E+11) is increased and total costs and total CO2 emissions are calculated and shown in Table 7.

Table 7. Sensitivity analysis on upper bound of CO2 emissions (MCL) No.

MCL

Total costs

Total CO2 emission

1 2 3 4 5 6 7

3.36357E+11 3.39415E+11 3.42472E+11 3.51646E+11 3.60819E+11 3.69993E+11 3.79166E+11

1.261E+09 1.246E+09 1.231E+09 1.225E+09 1.223E+09 1.22E+09 1.218E+09

3.362E+11 3.362E+11 3.362E+11 3.448E+11 3.54E+11 3.631E+11 3.723E+11

Figure 2 illustrates the behavior of total costs and total CO2 emissions related to sensitivity analysis. As indicated in this figure, total costs are decreased when total CO2 emissions are increased. In other words, to reduce environmental impact of all processes of the constructed biodiesel supply chain network, more costs should be invested. Figure 2 could be considered as a Pareto optimal solution for Trading-off between total costs and total CO2 emissions.



1.27E+09 1.26E+09

Total costs (Million IRR)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1.25E+09 1.24E+09 1.23E+09 1.22E+09 1.21E+09 1.2E+09 1.19E+09 3.362E+11 3.362E+11 3.362E+11 3.448E+11 3.54E+11 3.631E+11 3.723E+11

Total CO2 emissions (Kg)

Figure 2. Trade-off between total costs and CO2 emissions

6. Conclusion Todays, issues such as energy resources depletion, climate change, and increase of standards of living have triggered a sense among researchers and practitioners to pursuit alternative renewable energies. Since transportation sector is responsible for emissions of large amount of Greenhouse gases (GHG), finding a suitable alternative is intensively supported by governments and policy makers. Biodiesel produced from feedstocks that do not compete with food crops is a suitable renewable alternative for fossil diesel. In this study, a comprehensive mathematical programming model is proposed to design biodiesel supply chain network under uncertainty. The proposed model is a MILP one that considers feedstock supply centers to biodiesel consumer centers to achieve global optimum solution. Also, in the proposed model, CO2 emissions of all involved processes are calculated through SimaPro environmental assessment software and incorporated in optimization of biodiesel supply chain network. Due to the deep uncertainty of the parameters of the studied problem, there is not possible to construct probability or possibility distributions to model their uncertain behavior. Therefore, set-induced robust optimization method is utilized to deal with the deep uncertainty of the proposed model. The proposed model is applied in a real case in Iran. Since Iran is planning to reduce environmental pollution in large and industrial provinces, the outcome of this research will help policy makers to efficiently utilize biodiesel through taking into account optimum strategic and tactical decisions in biodiesel supply chain network.


Page 28 of 32

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The achieved results show that the proposed model could be efficiently employed in biodiesel supply chain network under deep uncertainty conditions. Also, to reduce total CO2 emissions, it is necessary to spend more costs. This research could be developed in the future through developing heuristic or exact algorithms to solve the proposed model for large cases. Also, the proposed robust optimization approach could be utilized in bioethanol supply chain optimization. Another efficient future research is applying the adaptive robust optimization method for the proposed model to better tradeoff between performance and conservatism.

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Biodiesel production system


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A robust optimization method to green biomass-to-bioenergy systems

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