As mentioned above, a key feature that makes the MILP modelling approach to be an appropriate tool for supporting decision-making is the possibility to have an comprehensive selection upon several alternatives, i.e. to perform multiple criteria optimization.
In fact, most realistic optimization problems require the simultaneous optimization of more than one objective. In these and most other cases, it is unlikely that the different objectives would be optimized by the same variable value choices. Hence, some trade-off between the criteria is needed to ensure a satisfactory solution.
1.6 Mathematical Programming The Multi–objective optimization (MOO) is suitable then for this kind of problems when decision makers are in the need to overcome with a broader picture of the biofuel SC considering not only economic issues but also problems of energy, environmental, and social nature which simultaneously affecting decisions to be taken in SCM (Ulrich and Vasudevan, 2004; Turton et al., 2009). Therefore, the aim is to identify particular solutions representing a trade-off between several objectives (Ehrgott, 2005).
The mathematical representation of a MOO problem is as follows:
min {f1(x), f2(x), . . . , fP(x)} (P ≥ 2)
s.t. g(x) ≤ 0 h(x) = 0
(1.2)
where x ∈ X ⊆ Rn, f : Rn−→ R, h : Rn −→ Rl, g : Rn−→ Rm
These solutions are called efficient or Pareto optimal. A generic pareto front is depicted in the Figure 1.9.
Figure 1.9: Generic Pareto front. Full blue points indicate members of the pareto set. Point (a)
is the optimum for objective function for a given value of (red points). Point (b) minimizes for another value of (compared to green points). For a member of the Pareto set, say (c), any attempt to improve a goal involves worsening the other, point (d) for comparison. Empty blue points are other possible solutions that are worse than those in the Pareto set. (Source: Pozo et al. (2012))
There are several approaches to obtain Pareto solutions: physical programming method (PP), normal boundary intersection method (NBI), –constraint method (–C), normal constraint method (NC), weighted sum method (WS) and the compromise programming method (CP). In practice, they are based on the conversion of the MOO problem into one single objective problem; solving it several times at the same time that each solution represents one feasible point; all the solutions represent the Pareto frontier. Among the available methods, the –constraint method resulted as the most widely applied in MOO problems due to its aptitude to be implemented into the MP modelling language and to fit with the available solution algorithms (Steuer, 1989).
The MOO has been used to solve SC decision making problems since many of the approaches developed in SCM are modelled as MILP and are solved under one single optimization criterion.The trade–off among multiple objectives must be considered by the decision makers and planner designers. Managing multiple objectives represents one of the most critical problem in SCM; typically, enterprises have different departments taking their own decisions, and in most of the cases they produce contradictory decisions (i.e., marketing and manufacturing departments have different goals and policies). Therefore, the use of MOO techniques become essential in order to improve the decision–making.
The MOO optimization has been successfully applied in several SC industrial problems, such as chemical (Rodera et al., 2002), pharmaceutical (Nicolotti et al., 2011), petrochemical (Zhong and You, 2011), or automotive industries (Cook et al., 2007). Also, there are several works considering MOO approaches solving SCM problems in the bioenergy sector adopting principally the economic, environmental and social aspects. Below is presented the general overview with the highlighted and main optimization models presented in the literature, including MOO of several objectives.
Zamboni et al. (2009a) proposed a general modelling framework to drive the decision–making process to strategically design biofuel SC networks, where the design task was formulated as an MILP problem that considers the simultaneous minimization of the SC operating costs and the environmental impact (measured in terms of greenhouse gas (GHG)). Mele and Kostin (2011) provided a spatially explicit bi–criterion multi–objective mixed integer linear programming (MoMILP) framework where environmental (expressed as Eco–indicator 99 and Global Warming Potential (GWP), metrics) and financial criteria
1.6 Mathematical Programming are both addressed in the ethanol production from sugarcane. In Giarola et al. (2011) the strategic design and planning optimization of bioethanol SCs through first and second generation technologies are addressed. A MILP model was proposed in order to optimize both environmental and economical objectives jointly. The formulation serves as a guide for taking decisions and investments through a global approach. Besides, Kim et al. (2011b) presented a MILP model where fuel conversion technologies, facility capacities, biomass supply locations, and the transportation between the different SC nodes are simultaneously selected. They considered distributed and centralized networks and compared them in terms of their profits and robustness, according to demand variations. You and Wang (2011) presented an optimization model to design and plan biomass and liquid fuels SCs based on economic and environmental criteria; this approach was illustrated through a case study for the state of Iowa. A multi-objective optimization model for to optimize a biorefinery was reported by Santibañez-Aguilar et al. (2011); this approach simultaneously maximized the profit while minimizing the environmental impact.
Recently, You et al. (2012) proposed a new approach to optimally plan biofuel SCs integrating the economic objective (i.e. minimising the net present value) with life cycle analysis (LCA) and regional economic input–output (EIO) analysis through a MOO scheme to include an environmental objective measured by life–cycle GHG and a social objective measured by the number of local jobs resulting from the construction and operations of the cellulosic biomass SC. In the same way, Santibañez-Aguilar et al. (2014) have developed a multi–objective, multi-period MILP model based on a state–task network, which seeks to maximize the profit of the SC, while minimizing its environmental impact and maximizing the number of jobs generated by its implementation. The environmental impact was measured by the Eco–indicator 99 according to the LCA technique, and the social objective is quantified by the number of jobs generated.