3.1. Quality integration in decision models

Moisture and ash content are important properties of the biomass. Castillo-Villar et al. [16] define a random variable, εð Þt to represent the moisture content corresponding to the mean value <sup>t</sup>. A triangular distribution <sup>f</sup> <sup>ε</sup>ð Þ<sup>t</sup> has been defined for <sup>t</sup> in the range ½ � at; bt with a probability density according to the following criteria:

if at ≤ e ≤ t then

moisture and ash content than the first generation, and then, quality-related costs must be contemplated in the design and implementation of the SC. However, one of the advantages of using second-generation biomass for the production of biofuels is that local farmers are familiar with the techniques to cultivate and harvest the biomass. For example, harvesting techniques for forage, utilized to feed the livestock, and for the power grass have similar characteristics. One common objective of the optimization in SC of biofuels is the minimization of costs under the assumption that all types of biomass have similar properties. This assumption can derive in considering only purchasing, logistics and processing costs which is not adequate since every type of biomass possesses different physical and chemical properties that affect the way the SC works. Not including the aforementioned properties could lead companies to have a negative impact on their expected profit since these properties usually experience high levels of variability. Pilot scale biorefineries have experienced a significant difference between the expected and the actual input of biomass [11]. The randomness in the biomass is one of the challenges

that biofuel producers must tackle in order to reach profitability and sustainability.

properties, for some specific technology, with a low variability.

feedstock.

50 Advances in Biofuels and Bioenergy

Biomass conversion technologies have their origins in laboratories where specifications of biomass are controlled. The technologies are designed to work under some specifications within the biomass such as the moisture, ash, and carbohydrates contents. When technologies are implemented in large- scale scenarios, it is very likely to receive biomass with specifications that do not meet the requirements, as a result, additional re-work must be done to utilize it. The quality of biofuel is associated with reaching the target levels of physical and chemical

A poor quality of the biomass results in higher total costs for companies. A quality-related cost can be defined as any cost derived from not meeting the required specifications for a specific conversion technology. The impact of these quality related costs is usually found after the biorefineries have begun operations. The optimization of biofuels [12, 13] considering randomness in the properties of biomass can be approached with stochastic programming [14, 15]. In literature, most of the supply chain models designed for biofuel production are deterministic; then, the stochastic programming is a novel approach to solve instances with variability in the

Novel optimization models take into account inherent properties of biomass to lead better decisions that minimize the total cost. It is important to identify the factors that have an influence over the biomass and include those considerations in the optimization model. Castillo-Villar et al. [16] have proposed a stochastic programming model that includes the quality variability in order to make decisions about important aspects of the SC. The work of Castillo-Villar et al. [16] is revisited in this chapter since the authors present a seminal model

Moisture and ash content are important properties of the biomass. Castillo-Villar et al. [16] define a random variable, εð Þt to represent the moisture content corresponding to the mean

that integrated biomass variability in the modeling of biofuel supply chains.

3.1. Quality integration in decision models

$$f\_{\mathbf{e}(t)}(e) = \frac{2(e - at)}{(bt - at)(t - at)} \tag{3}$$

if t < e ≤ bt then

$$f\_{\mathbf{e}\_i(t)}(e) = \frac{\mathbf{2}(bt - e)}{(bt - at)(bt - t)}\tag{4}$$

Otherwise, 0.

The triangular distribution was proposed due to experimental results in the work of Boyer et al. [17]; in the experiment, a breed of Alamo switchgrass was utilized to test the effect of factors such as storage days, particle size, wrap type and weight of the bale.

Ash content was also represented with a random variable ϑ δð Þ and the corresponding function is a triangular distribution of the mean value δ. The triangular distribution is defined for the range ½ � cδ; dδ . Depending on the technology utilized for the conversion of biomass into biofuel, different requirements are necessary to accomplish the conversion. For example, the conversion of biomass using a thermochemical technology demands at most 10% of moisture content for an efficient process. The moisture target for technology k is defined as tk. Violating the target for the selected technology will lead to other necessary costs (\$q) to compensate not meeting the specifications. The cost of mechanically drying the biomass will be applied to the final good since the content of moisture needs to be reduced up to the target level for conversion purposes. The thermochemical technology also requires at most 10% of ash content for an efficient conversion. The target is defined as δk. Again, not meeting the target specification will lead to reprocessing of the biomass with an additional cost.

#### 3.2. Two-stage stochastic model

Stochastic programming introduces randomness into the models where the stochastic variables play a fundamental role in the decision processes. There are several types of stochastic models but probably the most utilized are the two-stage models. In the two-stage models, two types of variables arise, the here-and-now variables and the wait-and-see variables. The here-and-now variables are those that need to be solved in the first-stage because they represent the beginning of the decision and the rest of the decision variables will rely on this step. The wait-and-see variables are presented in the second-stage of the process and depend on the realization of each presented scenario as well as the first-decision stage. The randomness is presented by defining random parameters that will converge to a certain value depending on the scenario, an expected value function for these scenarios in the second-stage plus the value function of the first-stage form the objective function of the stochastic program. A stochastic programming assumes known distributions in order to set the values for the stochastic parameters in every scenario so the program can be maximized or minimized depending on the objective function.

The two-stage stochastic models for location and transportation define the locations of facilities in the first-stage and the transportation of goods in the second-stage. Castillo-Villar et al. [16] defines a stochastic location-transportation model that introduces the location of the biorefinery as the first-stage variables, and then, the flow of biomass as a second-stage variable. The randomness in the second-stage is included with the inclusion of the stochastic parameters: (1) cost of moisture content, (2) cost of ash content and (3) supply capacity. The aforementioned parameters vary according to the scenario, for example, the level of moisture will be different between a scenario with wet conditions and a scenario with dry conditions. Table 5 presents the network definitions, Table 6 shows the parameters definitions and Table 7 introduces the variables of the model.

$$\text{Min} \sum\_{j \in \mathcal{I}} \sum\_{k \in \mathcal{K}} l\_{jk} Z\_{jk} + \sum\_{i \in \mathcal{I}} \sum\_{j \in \mathcal{J}} \sum\_{k \in \mathcal{K}} \sum\_{o \in \mathcal{\Omega}} p(o) \left[ c\_{\vec{\eta}} + c'\_i(t\_k, o) + c\_i(\delta\_k, o) \right] X\_{\vec{\eta}|k}(o) \tag{5}$$

Subject to:

$$\sum\_{j \in \mathcal{I}} \sum\_{k \in \mathcal{K}} X\_{ijk}(o) \le s\_i(o) \quad \forall i \in \mathcal{I}, o \in \mathcal{Q} \tag{6}$$

$$\sum\_{j \in I} g\_{jk} X\_{ijk}(o) \le \upsilon\_{jk} Z\_{jk} \quad \forall j \in I, k \in K, o \in \mathcal{Q} \tag{7}$$

$$\sum\_{i \in I} \sum\_{j \in \mathcal{J}} \sum\_{k \in \mathcal{K}} g\_{jk} X\_{ijk}(o) \ge d \quad \forall o \in \mathcal{Q} \tag{8}$$

$$\sum\_{\vec{k}\in\mathcal{K}} Z\_{\vec{j}\vec{k}} \le 1 \quad \forall \vec{j} \in \mathcal{J} \tag{9}$$

for the supply capacity, Eq. (7) constraints the biorefinery production capacity and Eq. (8) assures the demand satisfaction of the local market. Eq. (9) selects one technology for every

Xijk ð Þo Flow along arc ð Þ i; j ∈T from a supplier location to a potential location for a biorefinery under scenario o∈ Ω Zjk Binary variable which takes the value 1 if j∈J is used as a biorefinery utilizing technology k ∈K, and 0

ljk Equivalent annualized investment cost for opening a biorefinery in location j∈J using technology k ∈K

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Castillo-Villar et al. [16] solved a case study in the state of Tennessee to test the proposed model. The state of Tennessee has 94 counties that were considered as suppliers in the model, 31 counties were considered as potential locations for biorefineries. The biomass utilized in the model was switch-grass, all the quality information introduced in the model was derived from the experiment of Boyer et al. [17]. Three different triangular distributions (one for moisture and one for ash) were created according to number of groups included in the particle size.

The available biomass (19,482,102.51 dry tons) for biofuel production was obtained from the U.S. Billion-ton database, and Eq. (12) is utilized to calculate the weight of the biomass in its

The scenarios were created from historical data, 11 scenarios were generated utilizing the years 2004–2014. Every region in the state of Tennessee was linked to the closest climate station to

si ¼ sidry=ð Þ 1 � ei (12)

open biorefinery, Eq. (10) is non-negative constraints and Eq. (11) is binary constraints.

Tables 8 and 9 display the parameters for those distributions.

natural state (i.e., before drying the feedstock).

3.3. Case study and results

Table 6. Definitions of parameters.

Variable Description

otherwise

Table 7. Definitions of variables.

Parameter Description

c0

p oð Þ Probability of scenario o∈ Ω

cij Unit cost charged per metric ton shipped along ð Þ i; j ∈T

sið Þo Supply capacity for supplier i∈I for scenario o∈ Ω

d Total demand of biofuel in the network N

<sup>i</sup> tk ð Þ ; o Quality loss due to moisture content under scenario o∈ Ω for a given tk ci δ<sup>k</sup> ð Þ ; o Quality loss due to ash content under scenario o∈ Ω for a given δ<sup>k</sup>

vjk Production capacity of biorefinery j∈J including technology k∈K

gjk Conversion factor for biomass supplied to biorefinery j∈J applying technology k ∈K

$$X\_{i\backslash k}(o) \in \mathbb{R}^+ \quad \forall i \in I, j \in \mathbb{J}, k \in \mathbb{K}, o \in \Omega \tag{10}$$

$$Z\_{jk} \in \{0, 1\} \quad \forall j \in \mathbb{J}, k \in K \tag{11}$$

Eq. (5) refers to the objective function of the stochastic model which is the minimization of the total cost (investment costs, transportation costs and quality-related costs). Eq. (6) is a constraint


Table 5. Definitions of nodes and arcs in the network graph.


Table 6. Definitions of parameters.

distributions in order to set the values for the stochastic parameters in every scenario so the

The two-stage stochastic models for location and transportation define the locations of facilities in the first-stage and the transportation of goods in the second-stage. Castillo-Villar et al. [16] defines a stochastic location-transportation model that introduces the location of the biorefinery as the first-stage variables, and then, the flow of biomass as a second-stage variable. The randomness in the second-stage is included with the inclusion of the stochastic parameters: (1) cost of moisture content, (2) cost of ash content and (3) supply capacity. The aforementioned parameters vary according to the scenario, for example, the level of moisture will be different between a scenario with wet conditions and a scenario with dry conditions. Table 5 presents the network definitions, Table 6 shows the parameters definitions and Table 7 intro-

program can be maximized or minimized depending on the objective function.

duces the variables of the model.

52 Advances in Biofuels and Bioenergy

MinX j∈ J

Subject to:

X k∈ K ljkZjk <sup>þ</sup><sup>X</sup> i ∈I

> X j∈ J

> > X i ∈I

X j∈J

N Set of nodes in supply chain network G(N,A)

J Set of potential locations for biorefineries

X k∈K

X k∈ K

Eq. (5) refers to the objective function of the stochastic model which is the minimization of the total cost (investment costs, transportation costs and quality-related costs). Eq. (6) is a constraint

X i∈ I

Graph element Description

A Set of arcs in G(N,A) I Set of suppliers

T Set of arcs from I to J

Table 5. Definitions of nodes and arcs in the network graph.

X k∈ K

X j∈ J

X k∈ K

X o∈ Ω

p oð Þ cij þ c

0

<sup>i</sup> tk ð Þþ ; <sup>o</sup> ci <sup>δ</sup><sup>k</sup> ð Þ ; <sup>o</sup> � �Xijkð Þ<sup>o</sup> (5)

Xijkð Þo ≤ sið Þo ∀i∈I, o∈ Ω (6)

gjkXijkð Þo ≥ d ∀o∈ Ω (8)

Zjk ≤ 1 ∀j∈ J (9)

gjkXijkð Þo ≤ vjkZjk ∀j ∈J, k∈ K, o ∈ Ω (7)

Xijkð Þo ∈R<sup>þ</sup> ∀i∈ I,j∈ J, k ∈K, o∈ Ω (10)

Zjk ∈ f g 0; 1 ∀j∈J, k ∈K (11)


Table 7. Definitions of variables.

for the supply capacity, Eq. (7) constraints the biorefinery production capacity and Eq. (8) assures the demand satisfaction of the local market. Eq. (9) selects one technology for every open biorefinery, Eq. (10) is non-negative constraints and Eq. (11) is binary constraints.

#### 3.3. Case study and results

Castillo-Villar et al. [16] solved a case study in the state of Tennessee to test the proposed model. The state of Tennessee has 94 counties that were considered as suppliers in the model, 31 counties were considered as potential locations for biorefineries. The biomass utilized in the model was switch-grass, all the quality information introduced in the model was derived from the experiment of Boyer et al. [17]. Three different triangular distributions (one for moisture and one for ash) were created according to number of groups included in the particle size. Tables 8 and 9 display the parameters for those distributions.

The available biomass (19,482,102.51 dry tons) for biofuel production was obtained from the U.S. Billion-ton database, and Eq. (12) is utilized to calculate the weight of the biomass in its natural state (i.e., before drying the feedstock).

$$s\_i = s\_{idry}/(1 - e\_i) \tag{12}$$

The scenarios were created from historical data, 11 scenarios were generated utilizing the years 2004–2014. Every region in the state of Tennessee was linked to the closest climate station to

#### 54 Advances in Biofuels and Bioenergy


Table 8. The parameters of triangular distribution for the moisture content.


Table 9. The parameters of triangular distribution for ash content.


Table 10. Problem definitions.


Table 11. Summary of experimental results.

gather information about the precipitation levels in the region. If the precipitation level of certain region is above the precipitation mean of the state, then, the region is classified as wet; otherwise it is classified as dry. A random number ei was generated according to the classification of every region in order to calculate the moisture content. The ash content is not associated with the precipitation level; however, it was also contemplated in the cost calcula-

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55

The study was solved with GUROBI 6.0.0. The experiments were completed in a computer with Intel (R) Core(TM) i7-2600 U CPU @ 3.40 GHz; and 16.00 GB of RAM. The results for every problem described in Table 10 are shown in Table 11. On an average, the quality related

tion for the set of problems shown in Table 10.

Figure 10. Solution to problem 2, without quality-related costs.

Figure 9. Solution to problem 2, including quality-related costs.

Modeling and Optimization of Quality Variability for Decision Support Systems in Biofuel Production http://dx.doi.org/10.5772/intechopen.73111 55

Figure 9. Solution to problem 2, including quality-related costs.

Figure 10. Solution to problem 2, without quality-related costs.

gather information about the precipitation levels in the region. If the precipitation level of certain region is above the precipitation mean of the state, then, the region is classified as wet; otherwise it is classified as dry. A random number ei was generated according to the classification of every region in order to calculate the moisture content. The ash content is not

 1202 41 28 27 96 1298 1202 41 28 37 106 1307 1202 46 22 24 92 1294 1202 46 22 31 99 1300 1202 45 22 23 91 1293 1202 45 22 29 96 1298

Fixed Transport Moisture Ash Variable Total

Moisture content (%) at t bt Distribution 1 26 27 29 Distribution 2 17 19 20 Distribution 3 16 18 23

Ash content (%) cδ δ dδ Distribution 1 1.33 2.89 4.53 Distribution 2 0.71 2.44 3.79 Distribution 3 0.82 2.18 3.49

Problem Moisture Ash

 Moisture Level T1 Ash Level T1-Low Moisture Level T1 Ash Level T1-High Moisture Level T2 Ash Level T2-Low Moisture Level T2 Ash Level T2-High Moisture Level T3 Ash Level T3-Low Moisture Level T3 Ash Level T3-High

Table 8. The parameters of triangular distribution for the moisture content.

Table 9. The parameters of triangular distribution for ash content.

Table 10. Problem definitions.

54 Advances in Biofuels and Bioenergy

Problem Cost (in \$millions)

Table 11. Summary of experimental results.

associated with the precipitation level; however, it was also contemplated in the cost calculation for the set of problems shown in Table 10.

The study was solved with GUROBI 6.0.0. The experiments were completed in a computer with Intel (R) Core(TM) i7-2600 U CPU @ 3.40 GHz; and 16.00 GB of RAM. The results for every problem described in Table 10 are shown in Table 11. On an average, the quality related costs are about \$52.5 million annually which represents a significant amount of money for an investment. Figures 9 and 10 shows a different solution for problem 2.

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Figure 9 shows the biorefineries locations for the problem 2 (high moisture, high ash) with the inclusion of the quality related costs. Figure 10 presents the solution for problem 2 without the inclusion of the quality-related costs. The red pin indicates the location of a large capacity plant and the yellow pin shows the location of a smaller capacity biorefinery. It can be noticed that both solutions differ in the position of the pins.
