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Key issue, challenges, and status quo of models for biofuel supply chain design 297

programming to account for the uncertainties in fuel market price, feed-
stock, and logistic cost. In this model, two-stage programming with
MILP approach was employed where the first stage decided capacity
and location of biorefinery and then second stage determined the biomass
and gasoline flow.
NLP is used in some BSC optimization models when nonlinear rela-
tionships are needed. In the work by Corsano et al. (2011) aMINLP
model was established for the sugarcane-based bioethanol SC where the
constraints related to fermentation, evaporation, drying, and distillation
were modeled by nonlinear equations. Zhang and Wright (2014) pro-
posed a MINLP model to make integrated decisions on production selec-
tion, production planning, and facility locations. In their model, technical
constraints related to hydroprocessing and reforming processes were mod-
eled as nonlinear. This model was solved in software GAMS/DICOPT
with around 30hours. To reduce computational time in some cases, con-
straints could be linearized (Zhang and Wright, 2014). For example, many
studies divided nonlinear capital cost function into intervals and linearized
in each interval (Bowling et al., 2011).
Among the NLP problems, Quadratic Programming (QP) is one special
type where the objective function is in quadratic form (Frank and Wolfe,
1956; Imhof, 1961). A typical example is Bai et al. (2012) who developed
a BSC design model in quadratic form with consideration of competitive
agriculture land use and feedstock market equilibrium.
Stochastic programming (SP) is used in many studies to identify solutions
given different sources of uncertainties along the BSC. Given the complex-
ity of uncertainty and intensive computational loads, many studies used
mixed-integer multistage stochastic programming. Chen and Fan (2012)
established a mixed integer stochastic programming model with two stages:
first stage decisions were planning decisions; second stage decisions were
operational decisions that consider uncertainties. Gebreslassie et al. (2012)
developed a multiperiod stochastic MILP to optimize the hydrocarbon bior-
efinery SC that modeled feedstock supply and biofuel demand uncertainties
in the second stage. Kazemzadeh and Hu (2013) modeled uncertainties of
fuel market price, feedstock, and logistic cost as discrete distributions after
planning decisions in the first stage. Awudu and Zhang (2013) proposed a
model considering the uncertainties in demand, production, and price by
using the stochastic MILP that considered the quantity of final products
and the initial quantity of feedstocks in the first stage. The product distribu-
tion were modeled in the second stage.