Ann Oper Res (2009) 169: 117–130 DOI 10.1007/s10479-008-0428-9
An optimization model for the aggregate production planning of a Brazilian sugar and ethanol milling company Rafael P.O. Paiva · Reinaldo Morabito
Published online: 13 September 2008 © Springer Science+Business Media, LLC 2008
Abstract This work presents an optimization model to support decisions in the aggregate production planning of sugar and ethanol milling companies. The mixed integer programming formulation proposed is based on industrial process selection and production lot-sizing models. The aim is to help the decision makers in selecting the industrial processes used to produce sugar, ethanol and molasses, as well as in determining the quantities of sugarcane crushed, the selection of sugarcane suppliers and sugarcane transport suppliers, and the final product inventory strategy. The planning horizon is the whole sugarcane harvesting season and decisions are taken on a discrete fraction of time. A case study was developed in a Brazilian mill and the results highlight the applicability of the proposed approach. Keywords Aggregate production planning · Process selection · Production lot-sizing · Mixed integer optimization · Sugar and ethanol mills
1 Introduction The Brazilian sugarcane industry has recently been facing a major organizational change. Management aspects in this industry are changing as the importance of its products, especially ethanol and electricity, grows in the world and nationally. Among these changes, we point out different ambiguous aspects: some companies are focusing on the sugar and ethanol production, while others are looking for product and market diversification. Other important issues are the acquisitions and concentrations of producers, including transnational groups entering the scene; as well as setting up sugar and ethanol commercialization pools (Belik and Vian 2002; Vian 2003). R.P.O. Paiva · R. Morabito () Production Engineering Department, Federal University of Sao Carlos 13565-905, São Carlos, SP, Brazil e-mail:
[email protected] R.P.O. Paiva e-mail:
[email protected]
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Some of these changes directly influence in the way that sugar and ethanol companies plan and control their production. Currently, the trade-offs in production planning and control are much more important than some years ago. The Brazilian 2006 sugarcane production reached 455.3 million tons, with 73.8 t/ha average agricultural yield. This sugarcane production enables Brazil to be the first world sugar producer, with more than 30 million tons, and the second world ethanol producer, with more than 18 million cubic meters. In 2007 a 7.9% increase in the sugarcane production is expected, reaching 491.4 million tons (IBGE 2007). In this paper, we present an optimization approach for the aggregate production planning of a Brazilian sugar and ethanol milling company. We used a combination of single-stage lot sizing and process selection formulations, which enables us to model industrial plants with simultaneous production of various products, as in the sugarcane industry. Another example of a production system that fits in this modeling conception is the production of electrofused grains, considering the bricking, milling and classification of these grains, as discussed in Luche et al. (2008). We intend to develop a model that supports decisions in a tactic medium-term planning horizon in sugarcane milling companies. The focus of the model is on which industrial process should be used in each time step of the season, although other questions are also of concern, such as: how much sugarcane should be obtained, how this sugarcane should be transported to the mill, what products and how much should be produced from this sugarcane, and what the inventory strategy in the season should be; all this considering the maximization of the entire agro-industrial profit contribution. Even though the model was developed specifically for one company, we believe that it can be extended to other sugarcane mills in Brazil and other countries. This can be justified by the resemblance of the industrial processes in the sugarcane industry and the flexibility of the proposed model in considering the different production arrangements. This paper is organized in five sections: in Sect. 2 we briefly review studies about optimization and applications on the sugarcane industry; in Sect. 3 we present the integer linear model developed to solve the aggregate production planning problem; in Sect. 4 we describe computational results obtained in the case study; and in Sect. 5 we present the conclusions of this work and perspectives for future research.
2 Literature review Optimization models have being widely studied and used in the sugarcane industry. For example, in Brazil, different studies were developed applying optimization models in the agricultural stage of sugar and ethanol production. For instance, Barata (1992) and Soffner et al. (1993) applied linear programming to the sugarcane harvesting optimization over economical yields, maximizing the net income of sugar and ethanol companies. Grisotto (1995) dealt with the optimization of the sugarcane road transportation and Brunoro and Leite (1999) presented an optimization approach to increase the yields of sugarcane growers, based on incomes and production costs as a function of the sugarcane varieties chosen. A study applying discrete simulation techniques to improve the reception area processes of sugarcane plants is found in Iannoni and Morabito (2006). By the time energy became an important issue in the Brazilian sugarcane industry, Sartori et al. (2001) and Florentino and Sartori (2003) applied game theory to help the sugarcane variety selection and planting quantity in order to reduce crop residues and maximize the energy generated by these residues. In the last stage of the sugar supply chain, Yoshizaki et al. (1996) and Colin et al. (1999) applied linear programming to deal with the logistic distribution and storage of sugar and ethanol, considering central and secondary warehouses,
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and Kawamura et al. (2006) presented a linear optimization model for the logistic of transportation and inventory of final products in a co-operative society of sugar and ethanol. Examples of optimization modeling in the sugarcane industry of other countries are also found in the literature. For instance, Whan et al. (1976) studied the profit maximization of sugarcane ranches as a function of the selection of sugarcane varieties and harvesting schedules. Abel et al. (1981) developed a case study of a routing and scheduling problem for a sugarcane rail transportation system. Ioannou (2005) presented a linear optimization model to redesign a sugar supply chain to reduce transportation costs and improve customer service. Lopez et al. (2006) presented a mixed integer model to solve a cost minimization problem of sugarcane removal and its transport from the fields to the sugar mills at an operational level in Cuba. Salassi et al. (2002) studied the optimal sugarcane harvest system selection under alternative sugarcane variety combinations in Louisiana, USA, applying mixed integer programming. Higgins and Laredo (2005) used a modeling optimization framework to address some key industry issues in rationalizing rail track infrastructure and re-organizing harvesting in Australian companies. Higgins and Postma (2004) developed participatory research in a large sugar milling company to implement siding rosters that were optimized using a tabu search algorithm. Higgins and Muchow (2003) examined the consequences, in terms of profitability, of different options for alternative sugarcane supplies for whole mill regions. Higgins (2006) proposed a mixed integer programming model to represent a sugarcane logistic system using road transport operations and solved it with two meta-heuristics. Jiao et al. (2005) studied parameters estimated from a second-order polynomial model, which were used in a linear programming model for improving harvest scheduling and maximizing gains in a harvest season. This literature review shows that most of the optimization effort worldwide was done in the agricultural stage of the sugarcane production and the logistics of transportation and inventory of sugarcane, sugar and ethanol. Regarding the industrial stage of the sugar and ethanol production, only a few papers were found in the literature in which the agroindustrial plant trade-offs were studied with a more technical point of view. For example, in Wissen et al. (2005), a batch sugar crystallization process was represented as a discrete event dynamic system and modeled by mixed integer programming to determine the heat streams in the process, and in Nott and Lee (1999), a detailed sugar production scheduling problem was dealt with by optimal control approaches using hierarchical splitting heuristics. As discussed before, the model proposed in this paper aims to use an aggregate perspective of the whole agro-industrial supply-chain to help the decision makers in selecting the industrial processes used to produce sugar, ethanol and molasses, as well as in determining the quantities of sugarcane crushed, the selection of sugarcane suppliers and sugarcane transport suppliers, and the storage decisions related to the final products. In the case study that is being considered, these decisions are taken on a weekly basis and the planning horizon is the whole sugarcane harvesting season (i.e., 23 weeks).
3 Problem modeling The first issue to understand the proposed model is the industrial process. The process is a particular configuration of the sugar and ethanol plant which represents the subdivisions of the sugarcane juice flow, the use of molasses, the type of sugar being produced, the type of ethanol being produced, the sugarhouse recovery efficiency, the distillery efficiency and the overall sucrose waste (Fig. 1). For each specific process and each quality of sugarcane being
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Fig. 1 Sugar, ethanol and molasses industrial process flow
crushed, the quantities of sugar and ethanol that are obtained by the mill are calculated. In other words, a process is an arrangement of the plant that enables us to evaluate the quantity of final products as a function of the quality and quantity of the sugarcane being crushed at each time period. To formulate a process in the sugarcane industry, we used the sugar and ethanol engineering literature that can be found in, for example, Hugot (1977), Payne (1989), ICIDCA (1999), Castro et al. (2002), Fernandes (2003), CONSECANA-SP (2005), CONSECANA-AL (2005). Figure 1 illustrates the most important unitary operations involved in the milling processes (weighing, warehouse, wash, mill, clarification), in the sugar and molasses production (evaporation, crystallization) and in the ethanol distillery (fermentation, distillation). Figure 1 also depicts the waste of sucrose in every step of the industrial process (washing loss, milling loss, clarification loss, undetermined loss, fermentation loss, distillation loss). Another issue in Fig. 1 is the representation of process flow decisions parameters (TS1, TS2, TM, SJM, 1-SJM)—these parameters are the main differences between processes. For example, process 1 could be described by: TS1 = 70% of mixed juice for sugar production, TS2 = 100% of clarified juice for sugar production, SJM = 85% of sucrose crystallization recovery (VHP production), TM = 100% of molasses for distillation (anhydrous production). Therefore, each process can be seen as a discrete representation of flow decisions and wastes involved in each industrial sugar, ethanol and molasses unitary operations. The second issue is concerned with the modeling conception that was adopted. The proposed model can be seen as a combination of process selection and production lot-sizing models, where one process produces various products at the same time. This modeling trick was used because the sugarcane industry is a divergent production system, that is, it only has one raw material but various final products. In spite of that, this concept makes it possible to represent the entire sugar and ethanol production using linear formulations because
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it enables us to combine each process with its specific efficiency either in the sugarhouse recovery, the distillery efficiency or the overall sucrose wastes. As mentioned, the considered planning horizon is the entire harvesting season, divided into week periods. The reason to adopt weeks as step time is that the company is part of a cooperative association of mills that controls its associate’s production goals in week periods and does money transfers based on week production. We also adopted a small bucket point of view, which means that only one process can be used in each week. Setups are not considered in this aggregate model as with this time period, using small bucket production and considering the nature of the changes that should be made for switching from one process to another, the setup costs and times can be considered of second order. The formulation presented in the next section is an aggregate production planning and control model that aims to determine more widely the quantity of sugarcane crushed, the selection of sugarcane suppliers, the selection of sugarcane transport suppliers, the selection of industrial process used in the sugar, ethanol and molasses production, and the storage decisions related to these final products. We characterize this model as a one-stage, capacitated, multi-product, multi-process and dynamic system. 3.1 Aggregate production planning model The aggregate production planning approach is divided into two stages. The first stage involves preliminary calculus of three matrices. The matrix of industrial processes {Apkt } determines the quantity of each product p (e.g., sugars, ethanol and molasses) produced by each process k in each period t , the matrix of industrial costs {CKkt } determines the cost of using each process k in each period t , and the matrix of agricultural costs {Cmt } determines the costs of obtaining each sugarcane type m in each period t . This first stage of the approach is only a pre-calculus to prepare the input data for the optimization model (detailed in the Appendix). The second stage is the referred optimization model. The sets, parameters, variables, objective function and constraints of the mixed integer programming model are: Sets k Industrial processes {k = 1, 2, . . . , K}; t Planning periods {t = 1, 2, . . . , T }; p Final products {p = Demerara (raw sugar), VHP (Very High Polarization raw sugar), VVHP (Very Very High Polarization raw sugar); Standard (standard white sugar), Superior (superior white sugar); Special (special white sugar); Extra (extra white sugar), Molasses (final molasses), Hydrated (hydrated fuel ethanol), Anhydrous (anhydrous fuel ethanol)}. Including subsets, as follows: ps Sugarhouse products {ps = Demerara, VHP, VVHP, Standard, Superior, Special, Extra}; px Sugarhouse co-products {px = Molasses}; pa Distillery products {pa = Hydrated, Anhydrous}; m Sugarcane suppliers {m = prop (mill owned farms); rent (mill rented farms), own (mill owners farms), farm (farmers sugarcane)}. Including subset, as follows: mp Sugarcane managed by the mill {mp = prop, rent}; f Sugarcane transport suppliers {f = Fprop (mill owned transport system); Fown (mill owners transport system); Frent (mill rented transport system)}; e Inventory places {e = Eprop (mill owned inventory), Erent (mill rented inventory)}.
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Parameters M min M max Cgiro CT f αt βf t φt γt Cestpe Lf t hpe hspe DSpt Vpt V Cpt Ipe0 Dispm0 Mm0 Apkt CKkt Cmt
Minimum milling capacity (tons/week); Maximum milling capacity (tons/week); Cash flow available for the season ($); Capacity of transport supplier f (tons/week); Maximum percentage of farmers’ sugarcane in period t (%); Availability of transport supplier f in period t (%); Percentage of available operation time in the plant in period t (%); Forecasted efficiency of milling shutdowns in period t (%); Inventory capacity of product p in each place of storage e (tons or m3 , for sugar and molasses or ethanol respectively); Variable cost of cutting, loading and transporting sugarcane with transport supplier f in period t ($/tonne of cane); Variable cost of inventory of product p in each place of storage e ($/tonne or m3 ); Cost of inventory in the after harvesting period of product p in each place of storage e ($/tonne or m3 ); Demand of product p (sugars, ethanol and molasses) in period t (tons or m3 ); Revenue of product p in period t ($/tonne or m3 ); Advanced payment of product p in period t ($/tonne or m3 ); Initial inventory of product p in each place of storage e (tons or m3 ); Sugarcane harvesting forecast of sugarcane supplier m (tons); Quantity of sugarcane harvested before the first planning period of sugarcane supplier m (tons); Matrix of industrial processes yields (Sect. A.1 of the Appendix). Represents each product p yields on each industrial processes k in period t (tonne or m3 ); Matrix of industrial costs (Sect. A.2 of the Appendix). Represents each industrial processes k cost in period t ($/tonne of cane); Matrix of agricultural costs (Sect. A.3 of the Appendix). Represents each sugarcane supplier m cost in period t ($/tonne of cane).
Variables Xkt Mt Mmt Mf t
Process selection variable (no dimension)—decision of using (Xkt = 1) or not using (Xkt = 0) process k in period t ; Decision variable of quantity of sugarcane crushed in period t ; Decision variable of quantity of sugarcane from sugarcane supplier m in period t ; Decision variable of quantity of sugarcane transport supplier f in period t ;
Mkt Decision variable of quantity of sugarcane crushed by process k in period t ; Dispmt Availability of sugarcane supplier m in period t ; Inventory variable of product p in each place of storage e in period t . Ipet Objective function ⎛
⎞ V · A · M − pt pkt t p k ⎜
⎟ max Z = ⎝ Cmt · Mmt + f Lf t · Mf t + k CKkt · Mkt + p e hpe · Ipet + ⎠ m t p e hspe · Ipe“T ” (1)
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The objective function (1) maximizes the total variable revenue of all agro-industrial stages of the mill. This objective function helps the decision maker to determine the quantity of sugarcane crushed, the selection of sugarcane suppliers, the selection of sugarcane transport suppliers, the selection of the industrial process used in the sugar, ethanol and molasses production and the storage decisions related to the final products aiming to achieve the maximum revenue in the entire harvesting season. Constraints
Ipet =
e
Ip,e,t−1 +
e
Apkt · Mt − DSpt ,
p = Demerara, . . . , Anhydrous; t − 1, . . . , T ,
k
(2)
Xkt = 1,
t = 1, . . . , T ,
(3)
k
Mmt =
m
Mf t =
f
Mkt = Mt ,
≥ Mmt , Dispmt = Dispm,t−1 − Mm,t−1
Dispm1 =
m
M min · p
t = 1, . . . , T ,
(4)
m = prop, . . . , farm; t − 1, . . . , T ,
(5)
k
(6)
Mt ,
t
γt γt φt φt · ≤ Mt ≤ M max · · , 100 100 100 100
VCpt · Apkt
k
L f t · Mf t · Mt + Cgirot ≥ + k CK kt · Mkt + p e hpet · Ipet m Cmt
· Mmt +
t = 1, . . . , T ,
f
(7) ,
t = 1, . . . , T ,
M“f arm”t + M“own”t ≤ αt · Mt ,
Mf t ≤ Ipet ≤ Cestpe ,
β f t γt · CTf , 100 100
t = 1, . . . , T ,
f = Fprop, . . . , Frent; t = 1, . . . , T ,
p = Demerara, . . . , Anhydrous; e = Eprop, Erent; t = 1, . . . , T ,
Mkt ≤ M max · Xkt , Xkt ∈ {0, 1};
(8)
Mt ≥ 0;
Mmt ≥ 0;
k = 1, . . . , K; t = 1, . . . , T ,
Mf t ≥ 0;
Mkt ≥ 0;
Dispmt ≥ 0;
(9) (10) (11) (12)
Ipet ≥ 0 (13) Constraint (2) is the inventory balance of final product p in period t . Equation (3) is the small bucket constraint in each period t . Equation (4) is the compatibility constraint over variables Mmt , Mf t , Mkt and Mt , which means that this is a one-stage model. Equation (5) is the sugarcane m availability constraint in each period t . Equation (6) is the constraint
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of utilization of all sugarcane available in the harvesting season—it assumes that if part of the sugarcane planted is not available for the present season, this amount is not taken into consideration by the model. Inequality (7) is the capacity constraint for the quantity of sugarcane crushed in each period t . Inequality (8) is the cash flow constraint in each period t . Inequality (9) restricts the amount of sugarcane supplied by farmers and by the mill owners (farm and own, respectively) that is going to be crushed in period t —the two main reasons to use such constraint are: model the periods that farmers and mill owners accept for supplying their sugarcane; reserve a minimum amount of sugarcane that is going to be supplied by the mill farms (prop and rent), which represents the amount required by the agronomic planners considering the consequences of changing time of harvest and other constraints involved on the harvesting scheduling of paddocks, as discussed in Higgins et al. (1998) and Muchow et al. (1998). Inequality (10) is the capacity constraint for the quantity of sugarcane transported by mill owned transport system f in period t . Inequality (11) is the constraint of inventory capacity of product p in each storage e in period t . Constraint (12) imposes that the quantity of sugarcane processed by process k in period t (Mkt ) should be zero if process k is not used in period k(Xkt = 0), and it should be less than or equal to M max otherwise (Xkt = 1); and (13) are the variable domain constraints of the model. If this model has P final products, K processes, M sugarcane suppliers, F transport suppliers, E places of storage and T periods, it has a total of T (P · E + F + 2M + 2K) + 1 variables, where K · T are binary, and T (2M + P · E + P + K + K + 7) + 2 constraints. In the case study of Sect. 4, we consider M = 4, K = 252, F = 3, E = 2, P = 10, T = 23, and the model results in 12,306 variables, where 5,796 are binary, and 6,902 constraints. More details about this model are available in Paiva (2006).
4 Case study and computational results Santa Clotilde Mill (SCM) is a sugar and ethanol producer situated in the Northeast of Brazil. SCM is able to produce various types of sugar: demerara, VHP, VVHP, standard, superior, special, extra; two main types of fuel ethanol: anhydrous, hydrated; a sugarhouse co-product as molasses; and some sub-products as filter mud, bagasse, vinasse and fusel oil. In a typical harvesting season, SCM crushes 1 million tons of sugarcane and produces 100 thousand tons of sugar and 19 thousand m3 of ethanol. The present case study has been taken using data from the 2004/2005 harvesting season and the aim was to analyze if the proposed model could improve the corresponding aggregate production planning. In this application, we used an Intel Pentium IV 3 GHz processor, with 2 GB RAM and Windows XP service pack 2 operational System. The model was solved using the modeling language GAMS 19.6 with the optimization solver CPLEX 7.0. All input data used in this case study is described in detail in Paiva (2006). The model presented here has many input parameters, as can be seen in the previous sections, and a large amount of output results to be analyzed. In this paper, we focus on the most important results obtained for the SCM’s case study. To compare the model solution with the plan adopted by SCM in the 2004/2005 harvesting season, the SCM decisions related to Xkt , Mt , Mmt , Mf t , Mkt , Dispmt , Ipet variables were collected. In this comparison, we adopted the same costs, the same revenues and the same parameters that appear in the objective function to produce the SCM results depicted in Table 1 (Paiva 2006). In this case study we consider a tolerance of 0.5% relative gap (this gap is computed by formula: 100(Zub − Z)/Z, where Z and Zub are the values of the best integer solution and the best upper bound obtained by CPLEX) in the optimization process. It was taken into
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Table 1 Result comparison between SCM planners and the model Results
Total production
Unity
(a)
(b)
[(b-a)/a]
SCM 2004/2005
Model
Difference –
Demerara
tonne
0
0
VHP
tonne
66,950
64,243
4.21%
VVHP
tonne
0
0
−
Standard
tonne
0
0
−
Superior
tonne
36,415
36,198
0.60%
Special
tonne
0
0
− −
Extra
tonne
0
0
Molasses
tonne
8,625
8,627
−0.02%
Hydrated
m3
7,650
7,808
−2.02%
Anhydrous
m3
11,159
11,043
1.05%
kg/tonne
140.49
140.63
0.10%
Total products ART (Total Reductive Sugar) Total sugarcane ART (Total Reductive Sugar)
kg/tonne
155.85
155.85
0.00%
Final industrial efficiency
%
90.14
90.23
0.10%
Total season revenue
$
A
B
1.01%
Agricultural expenses
$
C
D
0.14%
Transportation expenses
$
E
F
−4.45%
Industrial expenses
$
G
H
2.57%
Inventory expenses on harvesting
$
I
J
−4.83%
Inventory expenses out of harvesting
$
L
M
0.31%
Total variable revenue
$
9,408,733
10,077,785
7.11%
consideration because of the difficulty of reaching a proven optimum solution on previous model runs, one of them reaching 12 hours of runtime without converging to a proven optimum solution. The relative gap of the solution presented in Table 1 is 0.33% with 10.11 millions of monetary units in the objective function. The time consumed in this example was around 2,000 seconds (approximately 33 minutes). Table 1 presents the production of sugar, ethanol and molasses in the entire harvesting period for both the SCM and the model results. Other interesting data to be observed in this table are the differences obtained in columns (a) and (b); this comparison shows the relative gap between both results. The upper-case letters (A, B, . . . , L) that appear in columns (a) and (b) represent the expenses of SCM mill and the correspondent values found by the model. These numbers were hidden in accordance with one of SCM’s conditions to provide its data for this research. Analyzing Table 1, it can be observed that the model result produced more sugar than ethanol and molasses, especially VHP sugar (4.21% difference from SCM result); it can also be observed that the AEAC had preference in comparison with AEHC. Another observation is relative to the overall industrial efficiency: both plans involved almost the same values (0.10% difference), which means that technically the model solution is close to the reality of SCM’s plant. Notwithstanding the most important result exposed in Table 1 is the total variable revenue result, in other words, the objective function result. Analyzing this important issue, we see that the model total variable revenue is 7.11% higher than the result obtained by the SCM plan for this season. These results encourage the use of this model to support decisions in the aggregate production planning. Managers could adopt a decreasing planning horizon strategy, firstly solv-
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ing the model considering all weeks of the harvesting season and then, by the time the data of each week becomes available, resolving the model considering only the weeks that remain until the end of the season. With this strategy, the aggregate production planning and analysis turn into a routine and the impact of data uncertainty is minimized. This strategy is being applied in SCM in the 2007/2008 season. Another issue concerning the application of this aggregated production planning model is the fact that the agronomic consequences of changing time of harvest from each of the sugarcane sources are not directly taken into consideration, as we consider the sugarcane quality as an input parameter of the model. That impact could be minimized with a negotiation between the industrial and agricultural planners, which are responsible for solving the harvesting scheduling problem considering the parameters αt , γt , M min , M max , CTf used for solving the model.
5 Conclusion This paper presents a mixed integer programming model to support decisions in the aggregate production planning of sugar mills. The model provides useful insights for the decision makers, helping them to better comprehend the variables and important issues that are being considered. It provides an effective analysis of these issues, producing more reliable results and better technical and economical outputs with optimization are attained. Other advantages of the model are that it transforms the analysis of different scenarios in the decision making process into routine, inhibiting incomplete and subjective judgments; it supports the planners in terms of analyzing the results obtained in the aggregate production planning and to work with unexpected situations; and it encourages an integration of decisions in the industrial stage, the agricultural stage and the logistics of final products under marketing requirements. The results of this study are promising and encourage other research efforts. The next steps of this work are: (a) incorporate the energy produced by the bagasse and the complete thermodynamic balance of the industrial plant into the model; (b) consider different criteria in the objective function using goal programming techniques as done in Munhoz and Morabito (2001) for frozen concentrated orange juice production and distribution system; (c) analyze the effects of uncertainties in the input parameters of the model by means of sensitivity analysis, chance constraints and robust optimization techniques. Acknowledgements The authors would like to thank the anonymous referees for their useful comments and suggestions, Santa Clotilde mill for providing the data, validating the results and for the financial support to this project. This research was also supported by CNPq.
Appendix: pre-calculus of the optimization model This Appendix present the parameters and equations used for the preliminary calculus of the matrix of industrial processes yields, the matrix of industrial costs and the matrix of agricultural costs. These computations are based on sugar and ethanol technical publications and represent a well-know methodology applied in Brazilian sugarcane mills with batch sugar production and batch/continuous distilleries without vinasse recycling (Hugot 1977; Payne 1989; Castro et al. 2002; Fernandes 2003; CONSECANA-SP 2005; CONSECANAAL 2005).
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A.1 Pre-calculus of the matrix of industrial processes yields {Apkt } BMF
PzaCt PzaM Polk
pct ARt Umidk TM k TSk MAPpa,k MSPps,k Rpa Elbtit
Efdt
Final molasses brix reached by the sugarhouse (◦ brix). Brix is a unit used to measure percentage by mass of total soluble solids of a pure aqueous sucrose solution (Fernandes 2003); Sugarcane purity in each period t (%). Purity is the percentage ratio of sucrose (or pol) to the total soluble solids (or brix) in a sucrose solution (Fernandes 2003); Molasses purity reached by the sugarhouse (%). Represents the final molasses purity for the entire planning horizon; Polarization of each sugar produced by each process k(◦ Z). Polarization (or pol) is the apparent sucrose content of any substance, expressed as a percentage by mass and determined by the single or direct polarization method (Fernandes 2003); Sugarcane polarization in each period t (%); Sugarcane reductive sugars in each period t (%). Glucose and fructose sugar (also referred to as invert sugar or just invert); Humidity of each sugar produced by each process k(%); Quantity of molasses sent for ethanol production by each process k (unitary); Quantity of sugarcane juice sent for ethanol production by each process k (unitary); Percentage of each ethanol type pa being produced by each process k(%); Percentage of each sugar type ps being produced by each process k(%); Theoretical recovery of each ethanol type pa being produced (l/100 kg ART); Overall sucrose waste in the milling and clarification stages of production in each period t (%). In other words, represents the waste of sugar before the decision of sending juice to the distillery or sugarhouse; Fermentation and distillation efficiency in each period t (%). These parameters describe the efficiency of the distillery;
SJM kt =
((PzaCt − 1) − PzaM) , Polk − PzaM (PzaCt − 1) (1−Umid k /100) Polk (1−Umidk /100)
RSps,k,t = pct · 10
RM px,k,t =
Elbtit 100
k = 1, . . . , K; t = 1, . . . , T ,
SJM kt · TSk · MSPk,ps ,
ps = Demerara, . . . , Extra k = 1, . . . , K; t = 1, . . . , T ,
pct · Elbtit (1 − SJM kt ) · TSk · 100 (1 − TM k ) , (BMF · PzaM/10)
(14)
(15)
px = Molasses k = 1, . . . , K; t = 1, . . . , T , (16)
RApa,k,t
⎞ ⎛ Rpa ·Efdt pct Elbtit (1 − SJM kt ) + ARt TSk · TM k + 10 10000 0,95 ⎠ MAPk,pa , = ⎝ pc t + AR − TS ) (1 t k 0,95 pa = Hidrated, Anhydrous; k = 1, . . . , K; t = 1, . . . , T ,
Apkt =
RSps,k,t + RM px,k,t + RApa,k,t , 1000
(17)
p = Demerara, . . . , Anhydrous; k = 1, . . . , K; t = 1, . . . , T
(18)
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Equation (14) corresponds to the SJM (Sugar, Juice and Molasses) purity balance formula, which represents the possible sugar recovered by the sugarhouse in each process k and period t (%) (Hugot 1977); (15) calculates the industrial overall recovery obtained in the type-ps sugar production, by each process k, in each period t (kg) (Fernandes 2003); (16) calculates the industrial overall recovery obtained for the type-px molasses, by each process k, in each period t (kg) (Fernandes 2003); (17) calculates the industrial overall recovery obtained in the type-pa ethanol production, by each process k, in each period t (l) (Fernandes 2003); (18) determines the final parameters of the matrix of industrial yields for each product p, by each process k, in each period t (t or m3 , in case of sugar and molasses and ethanol units, respectively). A.2 Pre-calculus of the matrix of industrial costs {CK kt } Cproc ARm Fatorpa
Monetary unities ($) production cost of a kilogram of Total Reductive Sugars (ART) by each process ($/kg ART); Reductive sugar contained in the final molasses obtained by the sugarhouse (%); Conversion factor of each type pa of ethanol to absolute ethanol (no dimension);
ConvSps,k,t
= RSps,k,t ·
polk 100
ConvM px,k,t =
1−
Umidk 100
· MSPk,ps ,
ps = Demerara, . . . , Extra k = 1, . . . , K; t = 1, . . . , T ,
RM px,k,t · SMF ARm 100 , + RM px,k,t · 0.95 100
(19)
px = Molasses k = 1, . . . , K; t = 1, . . . , T , (20)
ConvApa,k,t = RApa,k,t · Fatorpa · MAPk,pa , pa = Hydrated, Anhydrous; k = 1, . . . , K; t = 1, . . . , T , CK kt = Cproc ·
ConvSpkt p
0.95
ConvApkt + ConvM pkt + 0.6475
(21)
,
k = 1, . . . , K; t = 1, . . . , T
(22) Equation (19) presents the conversion of each type ps of sugar into the equivalent of sucrose, for each process k and each period t (kg Sucrose) (Payne 1989; Castro et al. 2002); (20) presents the conversion of each type px of molasses into the equivalent ART, for each process k and each period t (kg ART) (Payne 1989; Castro et al. 2002); (21) converts each type pa of ethanol produced into absolute ethanol, for each process k and each period t (l absolute ethanol) (Payne 1989; Castro et al. 2002); (22) determines the industrial cost of each process k and period t ($/tc), taking into consideration the sum of ART produced for each process k and period t (kg ART). A.3 Pre-calculus of the matrix of agricultural costs {Cmt } Crent δm
Cost of rented land (tonne of cane/ha); Financial help given to the farmers over the value of sugarcane m ($/tonne of cane);
Ann Oper Res (2009) 169: 117–130
ATRrent ATRt pATRt prodamp,t
129
ATR negotiated on rented land contracts (kg/tonne); Sugarcane ATR in period t (kg/tonne); ATR price in period t ($/kg); Productivity of sugarcane mp in period t (tonne/ha);
Cmt =
ATRt · pATRt + δmt ,
m = prop, own, farm; t = 1, . . . , T ;
ATRt · pATRt + δmt +
Crent·ATRrent·pATRt prodamp,t
,
m = rent; t = 1, . . . , T
(23)
Equation (23) calculates the cost of mill owned, mill owners’ and farmers’ sugarcane m in each period t , and the cost of mill rented sugarcane m in each period t ($/tonne). This computation is based on the Brazilian sugarcane payment regulation CONSECANA (CONSECANA-AL 2005; CONSECANA-SP 2005), represented by the parameters ATRt and pATRt (ATR—Total recoverable sugar after the industrial process, including the wastes of ART involved).
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