1. Introduction
As presented in many works recently published on the topic, crude oil supply chain management has been progressively receiving more attention [1]. Furthermore, given the important momentum gained by renewable fuels in substituting fossil based energy, the US Department of Energy (DOE) considers pyrolysis as a promissory mode of obtaining biofuel [2]. However, smart design and favorable conditions are of central importance for bioenergy production in the ambit of renewable energy technologies. In this context, one of the key aspects likely to determine the future competitiveness of this sector is efficient supply chain and logistics management [3].
In reviewing the interface between bio-energy production, logistics and supply chain management, Gold and Seuring [3] found two main challenges, namely, (1) the overall design of supply systems and (2) raw material harvest, storage, transport and pretreatment techniques. The authors discuss these findings in the context of bioenergy as a sustainable and renewable energy option.
In [1] Sahebi et al. employed taxonomic approaches and conducted an extensive review of mathematical programming models. In [4] authors assessed energy needs and targets, biofuel feedstock, conversion processes and Biomass Supply Chain (BSC) design and modeling. The latter work created a classification system based on the following eight aspects: decision level, supply chain structure, modeling approach, quantitative performance measure, shared information, novelty, practical application, assumptions, constraints and future research.
On the strategic level, supply chain networks largely depend on smart design and reengineering processes addressing material flows and the size and location of facilities. In this sense, decision-makers, who include investors, facility managers and government agencies, greatly benefit from insights allowed by quantitative models, especially in terms of the impact of biofuel supply chain design and operational planning [2]. In this sense, some authors consider optimization techniques as key tools for addressing these problems [5].
In [1] Sahebi et al. proposed the following five major future research lines: 1. Incorporating uncertainty into the model design; 2. Focusing on system-wide modeling and optimization; 3. Incorporating economic, environmental and social measures into the model; 4. Addressing computational complexities and developing large-scale case studies for nationwide analysis; and 5. Developing models that can be easily adapted according to the interest of specific stakeholders such as biomass suppliers and refinery owners. Finally, in discussing techniques to address uncertainty, they proposed a simulation technique for BSC optimization.
Known to have considerable impact on supply chain networks [6], sustainable development factors have been the object of many publications since the Bruntland Report [7]. Along these lines, particular aspects of biofuel production and management deserve special attention, as is the case of biomass handling and transportation and biofuel conversion, distribution and consumption. The attention paid to these areas stresses the importance of strategic and operational planning decisions in biorefinery design, particularly for attaining the successful deployment of an advanced biofuel industry. According to [3], the most important aspects implied in the logistic management of bioenergy supply chains are harvest, storage, transport, pretreatment, and system design. To address with these aspects, some authors have acknowledged the significance of multi-objective stochastic models, which are sometimes oriented by state-of-the-art solvers [4].
Regarding the upstream phase of the Colombian supply chain, [8] highlighted the need to balance offer and demand and implement scale economies. Also [9] suggested a decision-making methodology for the optimization of the Colombian biodiesel supply network, while [10] presented additional suggestions in this regard. Finally, [11] introduced a model intended to aid tactical decisions in the upstream phase of the Crude Oil palm Supply Chain (COSC), coupled with its corresponding solution procedure supported on LINGO (© 2017 LINDO Systems, Inc.) and a practical application in Colombia.
With regard to agribusiness sustainability, the present work follows the initiative of the Roundtable on Sustainable Palm Oil (RSPO), among other institutions within the sector, which states that the most relevant network design decisions incorporate biorefinery location, flow allocation among echelons, production capacity, scale economies and sustainability bounds in a context of stochastic supply and demand. In this context, the model introduced in the current work seeks to minimize the total cost of material flows through the chain, resorting to a solution procedure that includes chance constraints, valid constraints, and the E-constraint method. Thus, we intend to address several issues raised in different research areas featured in the literature on biofuel supply chain optimization in strategic contexts. The results presented here constitute a powerful tool to improve the competitiveness of the sector, especially on the national level in Colombia. Notwithstanding, recent works on biodiesel supply chains also address production of raw materials other than from palm oil [12;13;14;15;16;17].
2. The model
For the reasons mentioned above, the current work studied Crude Palm Oil (CPO), which is currently an outstanding biodiesel producing commodity. In Colombia [18] sets the guidelines for the development of the sector, highlighting the elevated logistic and production costs of biofuels in the country as one of the most critical competitiveness challenges of this productive chain. This paper introduces a mathematical programming model for the design of the Colombian biodiesel supply chain, seeking to minimize the following two objective functions: 1. The costs of the three stages of the chain (procurement, production and distribution), which contains five echelons in its three phases (upstream, midstream, downstream), and 2. The environmental impact of the processes that make up the chain using the Life Cycle Analysis (LCA) methodology.
The five echelons of the chain are as follows: 1. Supply centers, which, contained in the upstream phase [8], correspond to the crops and plantation conglomerates where the fruit is harvested; 2. Extraction plants intended to obtain CPO (which is obtained from the mesocarp of the fruit), Palm Kernel Oil (PKO) and Palm Kernel Cake (PKC) (the latter two being extracted from the seed); 3. Biodiesel production plants (CPO biorefineries); 4. Mixing plants (wholesalers), where biodiesel is blended with diesel to obtain the final consumer product; and 5. Colombian regions associated with fuel station clusters, which serve the final consumer.
Sets and indexes:
A: |
set of (a1, a2, ...an1) oil palm fruit supplying zones |
B: |
set of (b1, b2, ...bn2) extraction plants |
B (*): |
set of extraction plants that can be supplied by offer zone * |
D: |
set of (d1, d2, …dn3) bio refineries |
D (*): |
set of bio refineries that can be supplied by extraction plant * |
E: |
set of (e1, e2, ...en4) production scales |
E (*): |
set of production scales associated with facility * |
F: |
set of (f1, f2, ...fn5) mixing plants |
F (*): |
set of mixing plants that can be supplied by refining plant * |
G: |
set of (g1, g2, …gn6) demand zones |
G (*): |
set of demand zones that can be supplied by mixing plant * |
P: |
set of (pi, p2, …pn7) products and materials |
P (*): |
set of products and materials associated with plant * |
Parameters:
CF * : |
fixed cost of operating the facility * ε {b, d, f} |
CI *p : |
inventory cost of item p at the facility * ε {b, d, f} |
CP bpe : |
production cost of item p at the facility * ε {b, d, f} and production scale e |
CT abp : |
cost of transporting item p between offer zone a and extraction plant b |
CT bdp : |
cost of transporting item p between extraction plant b and refining plant d |
CT dfp : |
cost of transporting item p between refining plant d and mixing plant f |
CT dfp : |
cost of transporting item p between mixing plant f and demand zone g |
CT fgp : |
cost of transporting item p between mixing plant f and demand zone g |
D(ξ)*p: |
stochastic demand for item p the facility * ε {b, d, f} |
D(ξ)gp: |
stochastic demand for item p at demand zone g |
HMAX *e : |
maximum bound of production scale e at the facility * ε {b, d, f} |
HMlN *e : |
minimum bound of production scale e at the facility * ε {b, d, f} |
l *qp : |
amount of item q employed for processing item p at the facility * ε {b, d, f} |
M*: |
maximum capacity of extraction at the facility * ε {b, d, f} |
S abp : |
environmental impact caused by transporting one unit of item p between offer zone a and extraction plant b |
S *pe : |
environmental impact of producing one unit of item p at the facility * ε {b, d, f} and production scale e |
SMAX *p .: |
maximum admissible contamination bound for item p at the facility * ε {b, d, f} |
SMAX gp: |
maximum admissible contamination bound for item p at demand zone g |
O(ξ)gp: |
stochastic offer of item p in zone a |
Variables:
δ*: |
{1, If the facility * ε {b, d, f} is of the open type; 0 otherwise |
v*p: |
amount of item p obtained at the facility * ε {b, d, f} to satisfy the demand of supply chains different from that of biodiesel |
w *pe : |
{1, if item p obtained at the facility * ε {b, d, f} is operating at scale e; 0 otherwise |
x abp : |
amount of item p transported between offer zone a and extraction plant b |
x bdp : |
amount of item p transported between extraction plant b and refining plant d |
x dfp : |
amount of item p transported between refining plant d and mixing plant f |
x fgp : |
amount of item p transported between mixing plant f and demand zone g |
y *pe : |
amount of item p obtained at the facility * ε {b, d, f} and production scale e |
y*p: |
amount of item p obtained at the facility * ε {b, d, f} |
Objective:
Cost minimization:
Minimization of environmental contamination:
Subjected to:
Supply of materials
Capacity of facility:
Mass balance
Extracting plants:
Refining plants:
Mixing plants:
Extracting plants:
Refining plants:
Mixing plants:
Scale economies (assignation of a single flow)
Facility
Bounding of production schedules
Facility:
Consistency of the scale
Facility:
Demand
Extraction plants:
Biodiesel demand:
Non-negativity
3. Solution procedure
The solution procedure comprises the following steps: 1. Treating the offer and demand constraints containing stochastic parameters under parameter normality assumptions [19]; 2. Valid constraints configured [20]; 3. Solution of the bi-objective model through the E-constraint method [21], [22].
3.1. Estimation of stochastic constraints: Constraint deductions are included in Appendix A.
Supply of materials, replaces (3)
Where:
s(O(ξ)gp): standard deviation of the stochastic offer of item p at zone a E(O(ξ)gp): expected value of the stochastic offer of item p at zone a
Demand
Facility (replaces equation 14):
Biodiesel demand, replaces (15):
Where:
s(D(ξ)*p): standard deviation of stochastic demand for item p at the facility * ε {b, d, f}
s(D(ξ) gp ): standard deviation of stochastic demand for item p at demand zone g
E(D(ξ) bp ): expected value of stochastic demand for item p at the facility * ε {b, d, f}
E(D(ξ)gp): expected value of stochastic demand for item p at demand zone g
3.2. Valid constraints configuration
Constraints are intended to improve the statement of the problem through the convolution of its feasible space.
Scale differences in flows associated with facilities:
A particular condition imposed by these scales is that y *pe1 ≤ y *pe2 ... y pe1 * {B, D, F}, p ∈ P (b), e ∈ E(b). The valid constraints (20 to 21) associated with the scale in which this condition operates are presented below.
Finally, the set of valid constraints (22) associating the location status of the facilities to their production possibilities at different scales are presented next. Facility production logic relation:
3.3. Solution of the bi-objective-problem
The bi-objective problem is solved by the a posteriori method of E-restrictions. In it, one of the objective functions (considered to be more relevant) is optimized, while all other objective functions are considered to be model constraints. By generating a set of optimal solutions that allow a final choice, the method is considered to supply abundant information, thus avoiding weakly efficient solutions. The method is detailed in Appendix B.
4. Results
An analysis was carried out on a LINGO 9 commercial solver operated in a 2.67 GHz Intel Core I5 CPU with 8 GB RAM and a 64-bit operating system (Windows 10). Integer and linear optimality tolerances were fixed at 1*10-6 (absolute) and 8*10-6 (relative), respectively. Both the size of the problem and its configuration were inspired on those of the Colombian oil palm supply chain, whose size is smaller than the one considered in the current work. Thus, in a practical context the size of the problem corresponds to that of a world-class producer. The parameters of the problems were analyzed in two different instances (Appendix C and the supplementary material). Instances 1 and 2 address the performance of the solution procedure with and without the valid constraints. The complete solution procedure, which includes the valid constraints (20-22), resulted in a total of 84 model runs (Table 1).
Table 2 shows steps 1 and 2 of the E-constraint method (see Appendix B), which allow for the estimation of the optimal and nadir values of the two objective functions. Said values are necessary to proceed to step 3, which is of central importance for the current application.
To calculate the efficient frontier, the incremental step value of parameter ε was set to 0.05. As a consequence, 20 optimal estimates were solved for the two objective functions. The solution is shown In Appendix D.
The minimization of the two objective functions lay between 0.7 and 0.75 of ε in the case of instance 1a and 0.75 in the case of instance 1b. For their part, instances 2 a and 2 b reached said minimization at ε = 0.8. The CPU times for the solution of the instances of the problem show very good values. In a general sense, the instance b of the problem, that is, those including the valid constraints, showed a better average performance. Specifically, observing the number of cases, a 24.55% improvement in the average CPU time was attained by instance 1, and 18 out of the 21 cases exhibited better performance for instance b over instance a. Regarding instance 2, an 18% average CPU time improvement was reached, and 17 of the 21 cases showed better counts for instance 2b.
5. Conclusions
The current approach to the modeling of decision-making in the biodiesel supply chain may facilitate said process at different stages of the chain and provide a basis for future similar works and associated solution procedures.
The computational analysis performed in this work aims to determine the efficient frontier of the stated problem under the paradigm of minimization of its two objective functions. The size of the problem was defined according to that of the Colombian case. The CPU times for the optimal solution of the instances of the problem show very good values.
Although the stochastic bi-objective MIP model and the proposed solution procedure proved adequate to solve the practical instances of the current problem, they might not be enough to solve larger instances in the context of global supply chains. However, the current approach can be improved by further modeling efforts and/or the development of solution procedures that allow its practical application to larger instances. These research areas may include the development of adequate acceleration, heuristics and meta-heuristics, or branch and cut methods, as well as Lagrangian, Benders and Danzing-Wolfe decompositions.
As a pioneering work, the current approach actually fosters the development of new solution approaches allowing comparisons in terms of computational performance level, optimality gap, CPU time and memory usage.
Appendixes are available at http://revistas.javeriana.edu.co/index.php/iyu/article/view/19030