A robust optimization model for a biofuel supply chain under demand uncertainty

Cultivation systems

Choosing the best cultivation system is a challenging decision which depends on a variety of factors. For example, a comparison of the open-pond and photobioreactor systems shows that each has its own advantages. For example, open-pond systems are less expensive and simpler to implement, while photobioreactor systems are more efficient and environmentally friendly [ ²⁷ ]. As a result, a number of studies using multi-criteria approaches have evaluated alternative cultivation systems for the production of biofuels. The main criteria for assessing cultivation systems in the literature are expenses, environmental impact, energy efficiency, land use, and system robustness. Ubando et al. [ ⁴⁵ ] used the analytic hierarchy process (AHP) to evaluate open-pond, flat-panel photobioreactor, and tubular photobioreactor systems, and showed that considering environmental impact, economic factors, reliability, and social acceptability, open-pond systems dominate over the two photobioreactor systems. However, by increasing the weights of energy consumption criterion, the tubular photobioreactor system was found to dominate. Jorquera et al. [ ¹⁷ ] calculated the net energy ration to analyze the performance of open-pond, horizontal tubular, and flat-plate photobioreactor systems, and showed that using a horizontal tubular photobioreactor system is not feasible due to its high expense. Moreover, the results demonstrated that open-pond systems dominate over the flat-plate photobioreactor system in terms of energy consumption.

First-generation technology

First-generation technology related to biofuel production involves using edible crops, and many studies have been conducted to formulate the biofuel supply chain using the first-generation technology in deterministic situations. Using a particle swarm algorithm, Izquierdo et al. [ ¹⁵ ] minimized the overall cost of a biomass supply chain, including the costs of biomass processing and transportation and the construction of plants. The purpose of the proposed non-linear model is determining the optimal flow from sources of raw materials to producing plants and selecting the best technologies in each stage. To investigate the application of the presented model, the authors considered a case study in Italy. Huang et al. [ ¹⁴ ] presented a mathematical model to minimize the total cost of a biofuel supply chain, investigating the state of California’s approach to assess the economic opportunities from bio-ethanol production. Their results showed that this model is applicable only for regions with a high volume of agricultural waste, and only when the optimal price of a gallon of ethanol being USD 1.10. Kim et al. [ ¹⁹ ] presented centralized and distributed optimization models to determine the optimal conversion technologies, location decisions, volume of raw materials, and transportation capacity, and characterized and analyzed the impact of parameters with the most economic effect on the optimization models. An et al. [ ³ ] provided a mathematical model to maximize the lignocellulosic supply chain profit considering different time stages, multiple products, production and distribution systems, and facility locations and capacities. Using a case study in Texas, they showed that the price of ethanol is the most critical factor in the lignocellulosic biomass supply chain. Considering both environmental and economic purposes, Corsano et al. [ ¹¹ ] determined the optimum equipment size, amount of investment, and transportation cost, and discussed the impact of the residual cycle and derivative products from cattle feed.

Some papers have gone beyond economic objectives and considered environmental purposes, as well. In the first paper of its type, Giarola et al. [ ¹² ] designed a biofuel supply chain using corn and stover to produce bio-ethanol, and used multi-period integer linear programming to optimize the economic and environmental performance of this supply chain. A case study in Italy was carried out to explore the applicability of the proposed model, and the results showed that the second-generation technology was more suitable in the aspect of environmental issues, but that first-generation technology was more economical, concluding that a hybrid infrastructure would answer both the economic and ecological objectives. Wang et al. [ ⁴⁷ ] presented a bi-objective integer non-linear programming model to optimize hydrocarbon bio-refineries to make the most effectively address the both economic and environmental goals. To measure the financial goals and environmental target values, net present value and global warming were used, respectively. The results of this study indicate that more effort is needed to control and reduce greenhouse gas emissions.

Only a few papers have studied the process of biofuel production using first-generation technology in the presence of uncertainty. Chen and Fan [ ⁹ ] developed a bi-programming model to minimize the overall costs. The first-level costs included investment in and initialization of the treatment plant, while the second-level costs involved the costs for raw materials and transportation. They used a case study in California to produce bio-ethanol from waste in the presence of uncertainty. Moreover, Bairamzadeh et al. [ ⁸ ] presented a robust multi-objective mathematical optimization model for designing a lignocellulosic supply chain which optimized economic, environmental, and social performance simultaneously. The model determines the optimal strategic and tactical decisions. The strategic decisions consist of selecting biomass sources and determining the location and capacity of facilities, while the primary tactical decisions address the optimal level of stock and amount of production.

Second-generation technology

As discussed in “Introduction”, in second-generation technology, non-edible materials are used as raw material for biofuel production. Wang et al. [ ⁴⁸ ] developed a mixed-integer optimization model to determine the optimal location of facilities and their capacities for combined heat and power (CHP) in Great Britain. They considered miscanthus (silvergrass) as a feedstock for this supply chain, and the results showed that the price of CHP had a significant impact on facility location and the cost of electricity. Babazadeh et al. [ ⁷ ] presented a hybrid approach combining data envelopment analysis (DEA) and mathematical programming techniques for biodiesel supply chain network design (BSCND) in Iran. Considering environmental and social factors, they used DEA to assess suitable areas for planting raw materials, and then, using the optimal planting locations, they proposed a mathematical model to determine the number, location, and capacity of cultivation facilities, distribution centers, and refineries. Liu et al. [ ²⁵ ] proposed a bi-objective model to increase profits and minimize carbon dioxide emissions, and the results of this paper indicate that the price of electricity has an important impact on the profitability of a supply chain.

Santibañez-Aguilar et al. [ ³⁸ ] considered uncertainty factors in the price of supplies and presented a new approach to improving the performance of a biofuel supply chain in uncertain conditions. Their dual-objective model seeks to optimize economic and environmental goals simultaneously. To capture the uncertainty in raw material prices, they generated various scenarios using Monte Carlo simulation, and the results showed that the total profit of a supply chain does not correlate with the cost of raw materials. Mohseni and Pishvaee [ ²⁹ ] proposed a mixed-integer linear programming model to optimize the economic performance of a biodiesel supply chain, implementing data-driven robust optimization to deal with uncertain parameters.

Third-generation technology

Third-generation technology use algae and micro-algae as a raw material for biofuel production. Although it was shown in “Introduction” that this type is more efficient than the first- and second-generation technologies, only a few studies have considered using third-generation technology to produce biofuels. Ahn et al. [ ² ] developed a deterministic mathematical model to determine tactical and strategic decisions such as optimal material flow and the optimal location of refineries, and showed that carbon dioxide supply has a detrimental impact on the total cost of the supply chain. Their results showed that the price of each gallon of biodiesel is about 6% higher than the current price of biodiesel, and they believed that to minimize the costs, investment must be increased. In another study, Gong and You [ ¹³ ] proposed an integer non-linear programming model to optimize economic and environmental objectives in a biodiesel supply chain.

Only a few papers have studied the biofuel supply chain in the presence of uncertainty. One of the very first studies was conducted by Ubando et al. [ ⁴⁴ ], who considered the fuzzy trapezoidal demand of the product to deal with the uncertain nature a biofuel supply chain. Ubando et al. [ ⁴⁶ ] developed a fuzzy mathematical model to optimize the economic and environmental objectives, comparing the performance of two cultivation systems, i.e., a raceway pond and a flat-plate photobioreactor. The results showed that the flat-plate photobioreactor outperformed the raceway pond, since it substantially improved the environmental objective. However, the main problem with the last two papers is that they were developed based on case studies, making it difficult to apply their models for other cases. Mohseni and Pishvaee [ ²⁸ ] developed a two-stage model for using micro-algae in a biodiesel supply chain. In the first stage, they use the analytical hierarchy process (AHP) method to determine the optimal location of production facilities, and in the second stage, they presented a robust optimization model to minimize the total costs of the supply chain, showing that uncertainty in biomass production has a significant effect on supply chain costs. Uncertainties are the primary reasons for increased greenhouse gas emissions and financial losses. A target-oriented robust optimization approach was presented by Sy et al. [ ⁴² ], and they showed that their model was robust against variations in uncertain parameters. Their target-oriented robust model seeks to address both financial and environmental objectives simultaneously. Arabi et al. [ ⁴ ] designed a micro-algae supply chain network that minimized costs, including transportation, construction, fixed costs, and operational costs. Due to the uncertainty of some of the parameters and the complexity of solving the model, they used fuzzy programming and the DEA method.

Table  ¹ summaries the preceding papers on the mathematical design of biofuel supply chains. The papers in this table are sorted based on biofuel generation, objective functions, uncertainty, and discussion of production technologies. Table ¹ clearly shows the current gap in the literature. In the presence of uncertainty, there are only a few papers that have studied both the economic and environmental objective functions of micro-algae (third-generation) biofuel supply chains; however, none of these papers discusses production technologies. Therefore, in the current paper, we develop a dual-objective robust optimization model for a micro-algae biofuel supply chain, and determine the optimal production and cultivation technologies. Moreover, as discussed in this section, almost all the mathematical papers in the literature sought to find the optimal values for the economic objectives, and some of them state that biofuel energies are not financially feasible. However, none of these papers considers the opportunity of reusing released gases and water produced in different stages of a biofuel supply chain. Therefore, our paper contributes to the literature through considering this opportunity using mathematical modeling.

Macro-stage

Cultivation is the main stage of concern in the process of biofuel production due to its high expenses. Therefore, identifying the best cultivation system is a crucial decision, since it can significantly reduce the supply chain’s costs. As discussed previously, the three main cultivation systems are open-pond, flat-plate photobioreactor, and tubular photobioreactor. These systems can be utilized depending on the levels of demand and implementation. To find the best cultivation system, we use a multi-criteria decision-making method called the best–worst method (BWM), introduced by Rezaei [ ³⁵ ]. BWM is a procedure to solve multi-criteria decision-making problems, and compared to other multiple-criteria decision-making (MCDM) techniques, BWM requires fewer data and produces more reliable results [ ³⁶ ].

In the following, we briefly describe the required to derive the weights of the criteria; (1) determining a set of criteria {C1,C2,C3,...,Cn} ; (2) identifying the most and the least desirable criteria; (3) determining the preference of the most desirable criteria over all the other criteria using a number between 1 and 9, which results in the following vector AB=(aB1,aB2,...,aBn) , where aBj shows the preference of the most desirable criteria B over criterion j (note that aBB=1 ); (4) determining the preference of all criteria over the least desirable criteria by again using a number between 1 and 9, which results in following vector AW=(a1W,a2W,...,anW)T (in this vector, ajW indicates the preference of the criterion j over the worst criterion W . It goes without saying that aWW=1 ); (5) finding the optimal weights. For each criterion, the optimal weight is the one where we simultaneously have wBwj=aBj and wjwW=ajW . According to Rezaei [ ³⁵ ], solving the following model derives the optimal weights (w1∗,w2∗,...,wn∗) :

Micro-stage

The micro-stage seeks to optimize the economic and environmental objectives of the biofuel supply chain and determines all the decisions, from raw material sourcing to distribution of the final products. Therefore, in this stage, the following decisions are made: (1) determining the flow of fresh water, waste water, nutrient fertilizer, carbon dioxide, and methane between suppliers and the production facility; (2) determining the flow of methane, carbon dioxide, and wastewater among different stages within the production facility; (3) determining the need to construct pipelines between suppliers and the production facility; (4) inventory planning; and (5) supplying micro-algae.

Uncertainty is an inseparable part of a biofuel supply chain, and one of the primary sources of uncertainty is variability in demand for the final products. Although stochastic programming is a popular approach to deal with the uncertain nature of any supply chain [ ²⁰ ], it suffers from weaknesses that make it difficult for practical use in real-world problems. For instance, the probability distribution of the uncertain parameter should be available in stochastic programming; however, in most cases, determining the actual probability distribution requires a great deal amount of money and time. In contrast, scenario-based robust optimization only requires an adequate number of scenarios to model the uncertainty. Scenario-based robust optimization would result in a set of solutions that are less sensitive to realization of the model data from a scenario set [ ¹⁶ ]. According to our best knowledge, this paper is the first study that focuses on the application of scenario-based robust optimization for the design of biofuel supply chains.

The idea of scenario-based robust optimization was first presented by Mulvey et al. [ ³¹ ]. Their proposed model tries to consider several scenarios to arrive at the preferred level of risk aversion. This approach incorporates two sets of variables: control variables and design variables. Control variables are adjusted once a specific realization of the data is gained. However, design variables cannot be adjusted once uncertain, and random parameters are observed. Furthermore, there are two sets of constraints, i.e., structural and control constraints, of which the former are usually deterministic parameters; however, the latter involve uncertain parameters. For the purpose of explanation, the general framework of Muley’s robust optimization is described in the following.

Considering the following linear model:

Assuming that inequality ( ¹⁸ ) shows all the constraints in the model, the comprehensive method for this type can be defined as follows:

Fig. 2

Deterministic model

Before providing the mathematical model, a verbal description of the model is presented to provide a clearer picture. The first objective function of the model maximizes the net benefit:

The proposed model determines the biofuel supply chain design decisions. The bi-objective mathematical model aims to maximize the economic objective function. The biofuel supply chain costs consist of the operational costs (OC), costs of purchasing nutrients (NC), costs of pipeline construction (PC), and, finally, inventory costs (IC). These cost components can be formulated as follows in Eqs. ( ²¹ ), ( ²² ), ( ²³ ), and ( ²⁴ ), respectively:

Robust model

In this paper, we consider fluctuations in demand for the final products, djts . Table  ⁵ summarizes the decision variables in the case of demand uncertainty.

To make the model less complicated, we propose a new variable ( NBs ), which measures the net benefit of the supply chain for scenario s. NBs is defined in Eq. ( ⁴⁷ ):

Parameters

The proposed model presents tactical planning for a biofuel supply chain on a commercial scale. We consider five periods to deal with the dynamic nature of a biofuel supply chain. Furthermore, we consider three markets ( m∈{1,2,3} ) that are potential targets for the two final products: biodiesel ( j=1 ) and glycerin ( j=2 ). Table  ⁶ shows the demand for biodiesel and glycerin in each market and for each period. Table ⁷ shows the operational costs at each stage and time period. Moreover, Table  ⁸ shows the values of key parameters found in the literature.

Biodiesel accounts for 90% of biofuel production ( μ1=0.9 ), with the remainder being glycerin ( μ2=0.1 ) [ ³⁷ ]. We assume that there are two sources of wastewater, two sources of fresh water, and two sources of nutrient fertilizer. We also presume that the purchasing expenses for wastewater, fresh water, and nutrient fertilizer from all sources are the same and that they cost $0.04 per kilogram, $0.06 per kilogram, and $0.5 per kilogram, respectively [ ⁴¹ ].

Macro-stage

To determine the best cultivation technology, we propose three alternatives, which are “Open Pond”, “Flat-Plate Photobioreactor”, and “Tubular Photobioreactor”. Moreover, we define four criteria, i.e., cost, efficiency, pollution, and land use (shown in Table  ⁹ ). By referring to the literature, we produced Table  ⁹ . Using the proposed BWM model by Rezaei [ ³⁵ ] and solving the model, we find the best cultivation technology. The results in Table  ¹⁰ show that “Open Pond” is the best cultivation technology.

Micro-stage

In this section, we evaluate the effectiveness of the robust model, with that of the deterministic model, and considering 100 scenarios and the parameters in “Parameters”, we solve the scenario-based robust optimization model.

As shown in Table  ¹¹ , the carbon dioxide released in the drying, lipid extraction, and conversion stages plays an important role in providing the cultivation stage with the carbon dioxide required. A simple calculation proves that around 30% of the carbon dioxide required in the cultivation stage is supplied from carbon dioxide released in the drying, lipid extraction, and conversion stages.

Table  ¹² illustrates that the fifth period is the only period for which we are required to buy nutrients from local markets. For the other periods, we can obtain the required nutrient fertilizers from the wastewater, which would decrease the economic objective function. Moreover, Table  ¹³ proves that the supply chain is independent of the local market in supplying the required methane, because we can obtain the methane from the digestion stage. Obviously, being independent of local markets and obtaining the required methane and nutrients from different stages in the supply chain significantly reduce the supply chain’s costs.

In Eq. ( ⁴⁸ ), ω is the risk-aversion weight for under-fulfillment of the biofuel supply chain’s final products. ω seeks to find a trade-off between the solution robustness (which is close to an optimal solution) and the model robustness (which is close to a feasible solution). It is clear that a decision-maker who prefers higher values for ω is risk-averse and wishes to avoid a shortage of final biofuel products. However, a risk-taking decision-maker chooses smaller values for ω . Therefore, we can find a trade-off between cost and demand fulfillment by changing the value of the risk-aversion weight. The results of our analysis show that with increase in the value of the risk-aversion weight ( ω ), the cost of the biofuel supply chain (which indicates solution robustness) increases, while under-fulfillment of the biofuel supply chain’s final products (which indicates model robustness) decreases. This behavior conveys that at larger risk-aversion weight values, there is a tendency toward an almost feasible solution in all 100 scenarios, at the expense of an increase in total cost. Note that the impacts of the risk-aversion weight on solution robustness and model robustness are consistent with the results provided by Mulvey and Ruszczyński [ ³⁰ ].

This analysis helps us to determine a proper value for ω . Final-product demand under-fulfillment is entirely avoided by choosing a large value for ω . For instance, under any scenario, at ω=20 , the biofuel supply chain ensures zero final-product demand under-fulfillment at an expense of 255 million dollars. On the other hand, a smaller value for ω results in a total cost reduction while guaranteeing zero final-product demand under-fulfillment for most of the scenarios. For instance, at ω=15 , the total cost equal 243 million dollars, while the supply chain is robust in 96 out of 100 scenarios. For the rest of the analysis in this paper, we assume that ω=15 . However, depending on decision-makers’ purposes and other possible conditions, ω can have other values.

Now, we compare the performance of the robust model with that of the deterministic model and discuss their effectiveness. To compare the effectiveness of the deterministic model, we use the expected value approach (EVA) introduced by Maggioni and Wallace [ ²⁶ ]. In the EVA, a single scenario is developed in which the values of the random parameters (demand for final products) are set as equal to their expected value. The robust optimization approach (ROA) takes all scenarios into account when determining the optimal decision variables, while the EVA utilizes the expected values of uncertain parameter to determine the optimal values of decision variables. We now compare the effectiveness of the ROA and EVA over a range of risk-aversion weights.

Fig. 3

Considering 100 scenarios, Fig.  ³ shows that by increasing the risk-aversion weight parameter ( ω ), in both the deterministic and robust cases, the total costs increase. More importantly, Fig. ³ demonstrates that the robust optimization approach is considerably better than the expected value approach. To shed light on this, almost in all problems, the robust optimization approach dominates the deterministic model with respect to the mean of the cost in the economic objective function. The intriguing aspect is that the supremacy of the robust model is improved as the risk-aversion weight ( ω ) increases. This means that in the presence of a high level of uncertainty (e.g., due to lack of historical data) or when managers are risk-averse, the robust model matches better with the biofuel supply chain. The reason the robust model performs better is that the deterministic model assigns the same probability to all 100 scenarios, while the robust model devotes a lower probability to outlier scenarios.

Another measure to evaluate the performance of the robust model versus the deterministic one is using the standard deviation of the optimum total costs under random realization. We vary the risk-aversion weight parameter and calculate the standard deviation for the robust and deterministic models. Considering 100 scenarios, the results show that the robust model gains a lower standard deviation than the deterministic model. Figure  ⁴ illustrates that the robust model dominates over the deterministic model with respect to the standard deviation of the economic objective function. Mohseni and Pishvaee [ ²⁸ ] conducted the same analysis to evaluate the performance of the robust model, and according to their study, the standard deviations of our robust model (provided in Fig. ⁴ ) are acceptable.

Fig. 4

Sensitivity analysis

In this section, sensitivity analysis is carried out to evaluate the impact of the critical parameters on the results. The first key parameter is the demand for the final products. In other words, we are interested in finding the impact of change in demand on change in the economic objective function (net benefit). By increasing the demand, the economic objective function increases; however, this increase happens at a decreasing rate, which implies that the objective function is concave. This concavity is a result of the fact that at some point, demand is not a barrier to higher profits. In fact, it is lack of resources, such as limited pipeline capacity, that limits the benefit. Figure  ⁵ demonstrates the cumulative percent change in the economic objective function over a range of variation in demand for final products. For instance, a 15% increase in demand for biodiesel and glycerin would result in an almost a 30% increase in the economic objective function.

The next set of critical parameters are the cost parameters, including operational costs, capital costs for pipeline construction, and costs of purchasing raw materials. Figure ⁶ shows the impact of percent change in cost parameters on the cumulative percent change in the total costs. Since the capital costs of pipeline construction are a one-time expenses, even a significant variation in those costs does not lead to a substantial impact on the supply chain’s total costs. For example, a 100% increase in the capital costs of pipeline construction results in only a 1% increase in the total costs. In contrast, operational costs and the purchasing costs of raw material are recurring expenses. Therefore, we cab expect a significant increase in the supply chain’s total costs as a result of increases in recurring expenses.

Fig. 5

Fig. 6