C:/SJ Stuff/Papers/SBPO 2009 - Operational Planning … the mills, the sugar cane is crushed and the...

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Sugar cane cultivation and harvesting: MIP approach and valid inequalities

Sanjay Dominik JenaPontifıcia Universidade Catolica do Rio de Janeiro (PUC-Rio)

[email protected]

Marcus V. S. Poggi de AragaoPontifıcia Universidade Catolica do Rio de Janeiro (PUC-Rio)

[email protected]

RESUMO

O planejamento agrıcola de cultivo e colheitae complexo. Entretanto, os estudos neste contextosao relativamente recentes. Este trabalho foca no planejamento operacional do cultivo e colheita decana-de-acucar que determina o melhor momento para colher cada um dos talhoes, maximizando olucro total pelo teor de acucar extraıdo da cana. Este planejamento considera recursos como equipesde corte e transporte, capacidades das usinas, uso de maturadores e aplicacao da vinhaca nos talhoes.O modelo proposto estende o problema classico de empacotamento e acrescenta restricoes de fluxoem redes para modelar o escalonamento da colheita. Solucoes iniciais obtidas heuristicamente saopassadas para o resolvedor de programacao inteira para facilitar a resolucao. Este trabalho tambempropoe desigualdades validas para tornar a formulacao mais forte. Os experimentos sao feitos sobreinstancias reais fornecidas por um produtor de cana-de-acucar no Brasil.PALAVRAS CHAVE. Programacao Matematica. Colheita de cana-de-acucar. Escalonamento.

ABSTRACT

The planning of agricultural cultivation and harvesting is a complex task. However, this areaof study is still relatively young. This work focuses on the operational planning for sugar canecultivation and harvesting which determines the best moment to harvest the fields, maximizingthe total profit given by the sugar content within the cane. It considers resources such as cuttingand transport crews, processing capacities in sugar cane mills, the use of maturation productsand the application of vinasse on harvested fields. The MIP model extends the classical Packingformulation, incorporating a network flow for the harvest scheduling. Heuristically obtained initialsolutions are passed to the solver in order to facilitate the solution. This work also invests in validinequalities in order to strengthen the MIP formulation. All experiments were performed withinstances from practice provided by a Brazilian sugar cane producer.KEYWORDS. Mathematical Programming. Sugar cane harvesting. Scheduling.

XLI SBPO 2009 - Pesquisa Operacional na Gestão do Conhecimento Pág. 969

1. Introduction

Sugar cane is a sub-tropical and tropical genus of tall growing crop, counting 37 species plus anumber of hybrid species. According to the Food and Agriculture Organization of the United Na-tions FAO (2008), sugar cane is one of the most important commodities in the world. With morethan 420 billion tons of harvested sugar cane in the year 2005, Brazil is by far the largest producerof this grass worldwide, followed by India, China and Thailand. Among all agricultural commodi-ties produced in Brazil, sugar cane is its most produced measured in biomass and its fourth mostlucrative. Internationally, sugar cane is a highly competitive market.

The sugar cane harvest typically begins in May, sometimes April and goes on until November, thetime of the year when the sugar cane plants normally reach their maturation peaks. The maturationof sugar cane is measured in percentage of sucrose in the sugar cane, denoted toPol and reducedsugar, denoted toAR. The maturation periods vary widely around the world from six to 24 months.

Manual and mechanical cutting crews cut the plants on the fields, chopping down the stems butleaving the roots to re-grow in time for the following harvest. The harvest is then immediatelytransported to the industrial sector, i.e. sugar cane mills, by trucks, rail wagons or by manualcarriage (cart pulled by a bullock or a donkey).

In the mills, the sugar cane is crushed and the cane juice is extracted. The bagasse leftover, alsoreferred to fiber, is burned in boilers. The induced steam drives the turbines that generate the powerfor the mills. The sugar cane is further processed either to sugar or to ethanol. The sugar is alsoreferred to as thetotal recoverable sugar(ATR1). A side effect of the alcohol distillation process isa corrosive residual industrial liquid called vinasse. However, vinasse is an efficient fertilizer, thusits application to harvested plantation fields has become common practice. The use of maturationproducts is a common approach in agriculture to influence the natural maturation curve of plants.In the context of sugar cane, often used products are growth regulators which decrease the growthof the cane and therefore lead to an increase of the relative quantity of sugar within the plant.

The cultivated areas can contain hundreds of lots with different varieties, each with distinct growthand maturation properties. One of the most difficult, but most important decisions is the determina-tion of the ideal moment to cut each field and apply growth regulators in order to benefit best fromthe maturation peaks. The cutting crew’s capacities and logistic factors are directly involved in suchdecisions, as they may constraint the harvest and therefore must be taken into account.

Among many approaches such as heuristics and metaheuristics to handle complex practice prob-lems, mathematical programming has proved to be a powerful tool to solve such problems. Break-through advances in information technologies have provided new possibilities to the use of Com-binatorial Optimization (CO). Such great developments in computers’ capabilities make it possibleto model large problems from practice as mixed integer programs and solve them effectively usingmathematical solvers.

Reviews of work applying decision support tools to agriculture include Glen (1987), Lowe andPreckel (2004), Lucas and Chhajed (2004), Ahumada and Villalobos (2009) and France and Thorn-ley (2007), covering important works in the areas such as location analysis and mathematical mod-els. The efforts of the OR society to support sugar cane industry mainly developed in the last tenyears. In harvest planning, many works tackled the tactical planning. In this context, many opti-mization techniques have been applied in order to maximize the produced sugar. Caliari et al. (2004)use linear programming and incorporate this problem as a 0/1-Knapsack problem. Recent studiesfrom Higgins et al. (2007), Gala et al. (2008) and Boehlje et al. (2002) showed great opportunitiesto improve the value chain and reduce costs in the operational planning in order to remain compet-itive. Higgins and Laredo (2006) aim at better integration and optimization of the cane harvestingand transport sectors of the value chain.

1The abbreviation ATR is originated in the portuguese termAcucar Total Recuperavel, used in Brazil.


2. Operational planning for the sugar cane harvest problemOne of the most important decisions in this problem, throughout this work also referred to as theThe sugar cane harvest problem (SCHP), is the determination of the optimal moment to harvest theplantation fields. The problem is divided into a tactical planning (SCHP-TP) for a planning horizonof up to one harvest season (i.e. 12 months, based on weekly decisions) and an operational planning(SCHP-OP) for up to 30 days (based on daily decisions). This work is focused on the operationalmodule. Its input data is based on the planning of the tactical module for the chosen time period.

Cutting crews. The sugar cane is harvested by cutting crews, either manual (i.e. a group of humanworkers) or mechanical. Each cutting crew has its own properties such as the days at which itworks, the fields that it can cut, its minimum and maximum cutting capacities, travel speed as wellas cutting and travel costs. At the end of a day, mechanical cutting crews remain at the current fieldand start working in the beginning of the next day. Manual cutting crews return to a place wherethey are accommodated and may travel to any field in the next morning without any travel costs.Cutting crews may initially be located either at a mill or at a plantation field.

Plantation fields. Clearly, it is desirable to harvest each field at the peak of its sugar content, as thesugar indicated by the Pol and AR values vary as the cane grows. A field can only be cut at certaindays representing the eligible interval of the age at which it may be cut. A field can be partially cut,if it is cut at the end of the planning and the (partial) cut ends at the last available day of the crew(the rest of the field will then be cut in the next operational planning).Cutting crews can only startharvesting a field after having finished the harvest of the previous field. Harvesting a field may takemore than one day.

Transportation. Transportation crews carry the cut sugar cane from the fields to the sugar canemills. In practice, transportation crews are sufficiently available and can be hired on demand whennecessary. Hence, the operational planning does not involve their planning. It is assumed that asuccessful planning of the tactical module already guarantees sufficient availability of transportationcrews.

Sugar cane mills. After harvesting a field, its sugar cane is immediately transported to one ofthe sugar cane mills to be crushed and further processed to sugar. The mills operation is one ofthe most important constraints as it must not interrupt sugar cane processing. Each mill containsdaily minimum and maximum process capacities which must be respected. Plantation fields thathave been selected for harvesting must be assigned to one of the available sugar cane mills. Theprocessed sugar cane must contain a certain minimum quantity of fiber, which is used to generateelectricity in order to operate the sugarcane mills.

Maturation products. Some sugar cane varieties allow the use of maturation products to anticipateits harvest. In general, such products slow down the growing process of the cane mass, whereasthe growth of the sucrose within the cane is not affected. Thus, the percentage of sugar in the caneincreases. The exact moment and type of a maturation product application is determined by thetactical planning and is given as input data for the operational module.

Vinasse application. After crushing the sugar cane within the mills, the waste dump vinasseremains. A common practice to remove this byproduct is its application at already harvested area.In order to allow its frequent application, a sufficiently large field area (not all fields allow forvinasse application) must be harvested in certain periods.

Objective, decisions and constraints for the operational planning.Given the input data for thecutting crews, fields and mills, the operational module should determine the harvest sequence of thefields for each cutting crew so that it maximizes the total profit given by the sugar production in themills minus costs such as for cutting, transportation and processing the sugar cane, the movement ofthe cutting crews and vinasse application. For each day of the planning, a solution should suggest:

• the fields to be cut and the cutting crews to cut these fields. The cutting sequence duringthe planning should consider the displacement costs and time from one field to another.


• the mills at which the sugar cane of each field shall be processed.• the vinasse types that shall be applied at which fields after the harvest.

The solution must consider the following constraints:

• The cutting capacities of the cutting crews must be respected.• The sugar cane cut at a day shall be transported to and processed in the sugar cane mill

at the same day. The sugar cane mills have an inferior and superior limit of sugar cane toprocessed at each day. Consequently, the sugar cane quantity cut by the cutting crews shallsatisfy the mill’s demands.

• At all days, the mills must produce a certain minimum quantity of fiber, given by a percent-age within the processed sugar cane.

• In all days, a minimum quantity of vinasse must be applied on the recently harvested plan-tation fields. At least this number of hectares must be harvested in each week to permit thevinasse application.

In addition to the sugar cane cut by the cutting crews, it is possible to acquire sugar cane from thirdparty suppliers. Each supplier provides sugar cane of certain types of cane varieties up to a certaincapacity. The sugar cane possesses its own properties (Pol, AR, Fiber, etc.) and can be processed atany mill.

3. Mathematical formulationFor the mathematical formulation, the following input data was separated into the following topics:sets, mills, plantation fields and maturation data, maturation products, cutting crews and transporta-tion crews.

Sets

• D - Set of days considered in this planning horizon.• CF - Set of existing cutting crews.• I - Set of instants at all days (numDivinstants per day).Id is set of instants at dayd ∈ D.• P - Set of existing mills.• F - Set of existing plantation fields.• Vd - Set of all plantation fields on which vinasse can be applied within dayd.

Mills and sugar production process

• CoefAR & CoefPol - Coefficient of AR and Pol for the ATR calculation.• ProcCostp - Processing Cost (per ton) of millp ∈ P (equal at all days).• V alueATRd,p - Value of one ton of ATR within dayd ∈ D for mill p ∈ P .• MinProcd,p & MaxProcd,p - Minimum & maximum processing capacity of sugar cane

(in tons) that has to / can be processed by millp ∈ P within dayd ∈ D.• PFiberMind,p - Minimum percentage of fiber that has to be processed by millp ∈ P

within dayd ∈ D.

Plantation fields and maturation data

• Qd,f - Productivity (of sugar cane in tons) of plantation fieldf ∈ F within dayd ∈ D.• fInitc - The plantation field at which the cutting crewc ∈ CF is located at the beginning

of the planning.• Pold,f - Percentage of sucrose extracted from sugar cane from fieldf ∈ F at dayd ∈ D.• ARd,f - Percentage of reduced sugar in sugar cane from fieldf ∈ F at dayd ∈ D.• PFiberd,f - Percentage of fiber within sugar cane from fieldf ∈ F at dayd ∈ D.• NumHAf - Size of plantation fieldf ∈ F (in hectares).• Distf,p - Distance (in km) from plantation fieldf ∈ F to mill p ∈ P .• Distf1,f2 - Distance (in km) between two plantation fieldsf1 ∈ F andf2 ∈ F .

Maturation products


• MatGainf - Factor by which the Pol value in the sugar cane (given in percentage) fromplantation fieldf ∈ F is multiplied (and thereby increased or decreased) if a maturationproduct has been applied.

• MatReducFactorf - The reduction factor of the productivity of plantation fieldf ∈ F ifa maturation product has been applied.

Cutting crews

• CuttingCostd,c - Cost to cut one ton of sugar cane by cutting crewc ∈ CF at dayd ∈ D.• TravelCostCFd,f - Displacement cost (per Km) of cutting crewc ∈ CF at dayd ∈ D.• MinCutd,c & MaxCutd,c - Minimum quantity & upper limit of sugar cane (in tons) that

has to / can be cut by cutting crewc ∈ CF at dayd ∈ D.

Others

• MinV ind - Minimum quantity of vinasse (in hectares) that should be applied to the plan-tation fields within dayd ∈ D.

• TransportCost - Medium cost to transport one Kg of sugar cane one km from the planta-tion fields to the mills (Unit: $ / ton / km).

• numDiv - Number of periods (time instants) in which each dayd ∈ D is divided.• λ

i,dc,f - Quantity (in tons) of sugar cane cut within dayd by cutting crewc ∈ CF at plantation

field f ∈ F , considering that the cutting crew began cutting at instanti ∈ I.• θi

c,f1,f2- Number of instants that cutting crewc ∈ CF will need to travel from plantation

field f1 to f2 ∈ F , starting the journey at instanti ∈ I.

The following lists the variables used within this model.

• fhdc,f ∈ {0, 1} - Indicates whether cutting crewc ∈ CF leaves home at dayd ∈ D to start

its work at plantation fieldf ∈ F at the first available instant of the day.• fpc,f,p ∈ {0, 1} - Indicates whether cutting crewc ∈ CF leaves millp ∈ P and begins its

activities at fieldf ∈ F .• o

i1,i2c,f1,f2

∈ {0, 1} - Indicates whether cutting crewc ∈ CF will leave plantation fieldf1 ∈ F

at instanti1 ∈ I to arrive at plantation fieldf2 ∈ F at instanti2 ∈ I. This notation isequal tooi

c,f1,f2, where the instant at which the cutting crew arrives at plantation fieldf2 is

i + θic,f1,f2

.• ni

c,f ∈ {0, 1} - Indicates whether cutting crewf ∈ F waits one instant at plantation fieldf ∈ F (from instanti to instanti + 1).

• xf,p ∈ {0, 1} - Indicates whether the sugar cane from plantation fieldf ∈ F will beprocessed in millp ∈ P .

• yi1,i2c,f ∈ {0, 1} - Indicates whether cutting crewc ∈ CF starts cutting the fieldf ∈ F in

time instanti1 ∈ I and is available again at instanti2. This notation is equal toyic,f , where

the cutting crew will be available again at instanti + δic,f .

• zdf,p ∈ ℜ+ - Indicates the quantity of sugar cane from plantation fieldf ∈ F that will be

processed in millp ∈ P within dayd ∈ D.

The objective function for the formulation aims to maximize the total profit:

MAX∑

f∈F

∑

d∈D

∑

p∈P

zdf,p · MatReducFactorf ·

V alueATRd,p · (CoefPol · (Pold,f + MatGainf ) + CoefAR · ARd,f )−

∑

f∈F

∑

d∈D

∑

p∈P

zdf,p · MatReducFactorf ·

(ProcCostp + TransportCostd,f · Distf,p)−

∑

i∈I

∑

c∈CF

∑

f∈F

CuttingCostd,c · Qd,f · MatReducFactorf · yic,f

−∑

f1∈F

∑

f2∈F

∑

i∈I

∑

c∈CF

TravelCostCF⌊ i

numDiv⌋,f

· Distf1,f2 · oic,f1,f2


The profit is presented by the produced sugar. This is an extension of theGeneralized Assign-ment Problem (GAP). The sugar is produced by processing the sugar cane harvested in fields (lineone).The calculation for the processing of harvested sugar cane involves the reduction factor for thetotal cane mass when a maturation product is applied. It also adds in the increase of the Pol valuecaused by the product. Line two covers the processing and transportation costs for harvested sugarcane. Line three and four represent harvesting costs of the cutting crews and travel costs for thecutting crews. The following presents the model’s constraints.

Industrial and resource constraints.Plantation field cutting obligation. A plantation fieldf ∈ F must be cut at least once during theplanning period. The penalization cost forslCutObligf adjusts the importance of this constraintfor the case that it is not possible to cut all fields.

∑

c∈CF

∑

i∈I

yic,f + slCutObligf = 1; ∀f ∈ F (1)

Minimum and maximum processing limits of a mill. The quantity of sugar cane processedwithin dayd ∈ D by mill p ∈ P must be at leastMinProcd,p and no more thanMaxProcd,p.

∑

f∈F

zdf,p ≥ MinProcd,p; ∀d ∈ D, ∀p ∈ P (2)

∑

f∈F

zdf,p ≤ MaxProcd,p; ∀d ∈ D, ∀p ∈ P (3)

Minimum percentage of fiber. Each millp ∈ P demands a certain average percentage of fiberfrom the processed sugar cane within each dayd ∈ D. The sugar cane cut on these daysmust meetthese minimum values.

∑

f∈F

zdf,p · PFiberd,f

∑

f∈F

zdf,p

≥ PFiberMind,p; ∀d ∈ D, ∀p ∈ P (4)

Cutting capacities of the cutting crews. The quantity of sugar cane cut by a cutting crewc ∈ CF

within a dayd ∈ D must be at leastMinCutd,c and no more thanMaxCutd,c.

∑

f∈F

∑

i∈I

yic,f · λi,d

c,f ≥ MinCutd,c; ∀d ∈ D, ∀c ∈ CF (5)

∑

f∈F

∑

i∈I

yic,f · λi,d

c,f ≤ MaxCutd,c; ∀d ∈ D, ∀c ∈ CF (6)

Vinasse Application. In order to guarantee a sufficient area wherein to apply the vinasse, minimumquantities of cut fields (in hectares) may be defined for each dayd ∈ D. The field cuts must finishby dayd and the fieldsf ∈ F must be eligible for vinasse application, i.e.f ∈ Vd.

∑

f∈Vd

∑

c∈CF

∑

i∈I

NumHAf · yi,idc,f ≥ MinV ind; ∀d ∈ D (7)

Association between field cutting and processing. If sugar cane from plantation fieldf isprocessed in a mill, then the cutting variable must be non-zero for this field for any cutting crew.

∑

c∈CF

∑

i∈I

yic,f =

∑

p∈P

xf,p; ∀f ∈ F (8)


Quantity of cut sugar cane. This constraint sums up the quantity of sugar cane cut by all cuttingcrews within each day and field.

∑

p∈P

zdf,p ≤

∑

c∈CF

∑

i∈I

λi,dc,f · yi

c,f ; ,∀d ∈ D ∀f ∈ F (9)

Let maxProdf be the maximum productivity of fieldf throughout all planning days. The fol-lowing constraint allows the quantity of sugar cane cut from fieldf and processed at millp to benon-zero only if the sugar cane is processed at this mill.

∑

d∈D

zdf,p = maxProdf · xf,p; ,∀f ∈ F ∀p ∈ P (10)

Cutting crews network flow constraints. Each cutting crew possesses its own network that isindependent of the networks of other cutting crews. Each network flows through time (i.e. allinstants) and the fields that the cutting crew may cut. The edges added to the model (representingharvesting, traveling and waiting of the crews), are based on the cutting rate, travel speed anddaily time availibality also informed within the input data. The network flows are handled by thefollowing constraints:

Initial cutting crew position at a mill. In the beginning of the planning, each cutting crewc ∈ CF

is located either at a plantation field or at a sugar cane mill. If the cutting crew is located at a mill,it must travel to one of its valid fields before the first instant of the planning. LetCFp be the set ofcutting crews that are located at a mill. The mill at whichc is located is denoted aspc. The flowmust be inserted into one of the variablesfp for the mill pc.

∑

f∈F

fpc,f,pc= 1 ∀c ∈ CFp (11)

Initial cutting crew position at a field and passed flow from a mill. The flow for each cuttingcrewc ∈ CF must pass to the node of the first available instantiF irstc of the cutting crew (RHS).The flow can then be used to cut a field, wait at the field or move to another field (LHS). Letlocc

be the location ofc at the beginning of the planning.

yiF irstcc,f + niF irstc

c,f +∑

f2∈F

oiF irstcc,f,f2

=

1, if f = fInitc and locc is planting field

fpc,f,pc, if locc is mill pc

0, otherwise.

;∀c ∈ CF ∀f ∈ F (12)

General flow conservation - cutting, waiting and traveling. Once the flow entered the network,it must pass along time. The flow that enters a node at instanti must also leave it. Flow can enterfrom cutting variables for the field, from a waiting variable at the previous instant or by travelingfrom one of the other fields. If flow enters the node, it leaves it again by either cutting the field,waiting one instant or moving to another eligible field. In the flow conservation constraints, onemust distinguish between mechanical and manual cutting crews. Mechanical cutting crews remainon the field during the night, whereas manual crews return to a place where they spend the night.Let IFcMan be a set with all instants except the first available instant of each day, for each manualcutting crewcMan. Let IcMec be a set with all available instants of all days, available for eachmechanical cutting crew. Figure 1 exemplifies a network flow for a mechanical cutting crew (lowerexample). The nodes in the grey area are not available for work. Hence, all flow variables skip thenodes of unavailable instants.

∑

i′∈I

yi′,ic,f + ni−1

c,f +∑

f1∈F

∑

i′∈I

oi′,ic,f1,f = yi

c,f + nic,f +

∑

f2∈F

oic,f,f2

∀i ∈ IFcMan ∪ IcMec\{i0}, ∀f ∈ F , ∀c ∈ CF (13)


n

y

nnn

Day 2Day 1

Not available

instants

Available

instants

nnnn

Man. crew yy

n

n

fh

n

nMech. crew

fh

other field

Figure 1. Example network flows for cutting crews: mechanical crews skip not availableinstants, whereas manual crews return to their accommodation

Manual cutting crews are usually hosted at a place where they spend the night at the end of theday. The next day, they will be able to work at any field (eligible for these crews).Figure 1 (upperexample) shows an example of a network for a manual cutting crew. Cuttings and waitings thatwould enter into the first available instant of a day, now enterhome node. From home, the flowpasses to any field that the cutting crew can cut. The previous equation (13) is valid for the nodesof manual cutting crews as well, excepted the ones valid for the first available instants of each day.The flow that would have entered in these first nodes of each day, now enter into the node of thecrew’s home. This is guaranteed by equation (14), whereIc represents a set with the first availableinstants of all days of the cutting crewc. Note that this constraint is not generated for the node ofthe very first available instant of the entire planning.

∑

f∈F

∑

i′∈I

(yi′,ic,f + ni−1

c,f +∑

f1∈F

∑

i′∈I

oi′,ic,f1,f ) =

∑

f∈F

fhdi

c,f ;∀i ∈ Ic, ∀c ∈ CF (14)

Finally, equation (15) distributes the flow from the home node to all plantation fields for each cuttingcrew.

fhdi

c,f = yic,f + ni

c,f +∑

f2∈F

oic,f,f2

;∀i ∈ Ic\{i0}, ∀f ∈ F , ∀c ∈ CF (15)

4. Instances for the computational experimentsThe Grupo Virgulino de Oliveira (GVO), a Brazilian sugar producer, provided four instance setswith a total of 25 instances. The sets correspond to different moments in the harvest season andcontain between 19 and 334 plantation fields, one or two sugar cane mills, between five and 21cutting crews and a planning horizon of up to 16 days. Additionally, 15 artificial instances weredesigned to provide a broad variety of characteristics in order to analyze the influence of suchproperties in the difficulty of the problem. The instances were divided into groups, each with adifferent number of plantation fields, i.e. 20, 50 or 100 fields. The planning horizon varies from 15to 30 days. Each configuration is available with different values for the mill’s minimum processingdemands.

5. Valid InequalitiesThe linear relaxation of this problem turned out to be strongly fractional. Figure 2 (a) illustratesthe route of a mechanical cutting crew in the optimum solution of the linear relaxation for a GVOinstance. The total flow from the initial location at the mill is divided into several fields whichare then repeatedly cut until the end of the planning. In most of the instances, the last cut is onlypartially performed and no instants are spent in traveling or waiting in order to benefit best fromthe available time. A similar behavior can be observed for manual cutting crews. In order to passthe flow from the last instant of a day to the first instant of the following day, manual crews have totravel to their accommodation, because there are no waiting variables that connect two subsequent


days. Leaving from their accommodation, manual cutting crews are able to travel to any of theireligible fields by zero time and cost. However, it seems that they always return to the same field.This can be justified by the same motivation found in the behavior of mechanical crews: the currentfield already perfectly contributes to the satisfaction of the constraints and the maximization of thetotal profit. Such trend to strongly fractional solutions is based on the daily processing minimumdemands of the mills and the daily minimum cutting of the cutting crews. The solution equilibratesthe cuts along the entire planning horizon, at least over days with minimum demand constraints. Theselected fields are typically the ones with the best relation between spent time and return of Pol andAR quantity. In this manner, the solution minimizes the violation of the minimum constraints andbenefits from the highest profit possible. The following presents three cut types identified duringthe analysis of the linear relaxation’s optimal solutions in order to strengthen the MIP formulation.

i0 i90

field 1

i179

mill

field 2

field 9

...

.........

...

...

...

...

.........

...

...

...

0.111

0.333

0.006

i0 i90

field 3

i179

mill

field 2

...

.........

...

.........

...

.........

...

...

0.333

0.333

field 4

(a) (b)

Figure 2. Route of a mechanical cutting crew in the optimal solution fo r the linear relaxationwithout cuts (a) and with cut type 1 (b)

5.1. Cut 1: Forcing travels throughout whole planningThe analysis of the optimal solutions of the linear relaxations exposes two conspicuous behaviors:First, a cutting crew’s flow is split into several fields during the same time instant. Second, a cuttingcrew repeatedly cuts the same field in order to perfectly distribute the harvested sugar cane quantityalong the planning time. It is observed that the linear relaxation’s solution prefers not investingin traveling, as this is consuming in time and costs. However, in an integer solution, a cuttingcrew is required to travel from one field to another in order to cut both of them. In addition tothe common traveling, manual cutting crews may travel by using thefh variables leaving from itsaccommodation to a field. Consider a cutting crew that cutsn fields throughout the entire planningperiod. This crew must perform at leastn−1 travels in order to visit all fields. Hence, the followingvalid inequality can be stated (for mechanical crews, there will be nofh variables):

∑

f1∈F

∑

f2∈F

∑

i∈I

oic,f1,f2

+∑

f∈F

∑

d∈D

fhdc,f ≥

∑

f∈F

∑

i∈I

yic,f − 1; ∀c ∈ CF

5.2. Cut 2: Forcing travels at each fieldThe previously introduced cuts guarantee, for each cutting crew, that the overall number of travelsis at least as high as the overall number of performed field cuts. However, in the linear relaxation,some fields may have such a good relation between returned profit and spent harvest time that thecutting crews will try to cut these fields as often as possible. In this case, a cutting crew repeatedlyharvests such a fieldf∗ with a partial flow throughout the entire planning period. The previouslyinserted cut 1 then forces the solver to perform travelings with an equal flow value in order tocompensate these cuts. In the linear relaxation’s optimal solution, these extra travels are performedwithout interfering the seamless harvesting in the desired fieldf∗. Figure 2 (b) exemplifies thissituation. The cutting crew prefers to repeatedly harvest field 2 (in this case three times). At thesame time, a number of travels with an equal flow value are performed in order to compensate therepeated cuts.


In flow. Such a behavior can be avoided by adding constraints that force traveling at each field. Theflow entering in a field and the flow leaving from a field are considered separately. The constraintsthat control the field entering flow are founded in the following observation: a cutting crew can onlyinvest a certain flow quantity in harvesting a field, if this flow quantity (or more) has been insertedinto that field by traveling before. This results in the following inequality, containing a constantΦc,f which is explained afterwards (for mechanical crews, there will be nofh variables):

∑

p∈P

fpc,f1,p +∑

f2∈F

∑

i∈I

oic,f2,f1

+∑

d∈D

fhdc,f1

+ Φc,f1 ≥∑

i∈I

yic,f1

; ∀c ∈ CF,∀f1 ∈ F

If the cutting crew is initially located at a plantation field, none of thefp variables will be active.Furthermore, if the current field is the field at which the crew is located at the beginning of theplanning, then this must be considered as an entering flow as well. Thus, ifc.initialLocation = f ,thenΦc,f = 1, otherwiseΦc,f = 0.

Out flow. Next, the inequalities to limit the outgoing flow are considered. Clearly, all flow thatis invested in field cuts must also leave these field by making use of traveling variables. At theend of the planning, the flow may leave the network without use of any traveling; this flow unitis subtracted on the inequality’s RHS. The inequalities for manual crews must also consider anyvariables that may allow for traveling to other fields, i.e. all variables whose flow enters into theaccommodation flow of the following night. This is the case for waiting and cutting variables whoseflow would have entered into the first instant of a day, but were bypassed to the accommodation flowinstead. Leti′d be the first instant at dayd. For manual crews, the following inequalities hold ateach field (for mechanical crews, the waiting and cutting variables on the LHS are excluded):

∑

f2∈F

∑

i∈I

oic,f1,f2

+∑

d∈D

ni′(d+1)

c,f1+

∑

d∈D

∑

i∈I

yi,i′

(d+1)

c,f1≥

∑

i∈I

yic,f1

− 1; ∀c ∈ CF, ∀f1 ∈ F

5.3. Cut 3: Forcing travels at each field at each day

In flow. In order to strengthen the constraints for the in and out flow at each field, these inequalitiescan be stated for each single day. At the first day, the input flow may come from a mill or from thecutting crew if it is initially located at that field. Again, ifc.initialLocation = f , thenΦc,f = 1,otherwiseΦc,f = 0.

∑

p∈P

fpc,f1,p +∑

f2∈F

∑

i1∈I

∑

i2∈Id0

oi1,i2c,f2,f1

+ Φc,f1 ≥∑

i∈Id0

yic,f1

; ∀c ∈ CF, ∀f1 ∈ F

During all days after the first one, flow will not come from a mill. Instead, it may come from awaiting variablen that leaves from the last instant of the previous day and enters at the first instantof the current day. Leti∗d be the last instant at dayd. The inequalities for the further days are givenby (for mechanical crews, thefh variables will not be considered):

∑

f2∈F

∑

i1∈Id

oi1c,f2,f1

+ ni∗d−1

c,f1+ fhd

c,f ≥∑

i∈Id

yic,f1

; ∀d ∈ D, ∀c ∈ CF, ∀f1 ∈ F

Out flow. The out flow inequalities for mechanical cutting crews have to consider all outgoingtraveling variables at a certain day. In addition, they have to involve the waiting variables that leavefrom the last available instant of that day in order to pass the flow to the following day. The totalflow accumulated by these variables must be greater or equal than all flow invested in field cutsterminating at the current day. For manual crews, the field cutting variables that initiate the field cutat the current day and terminate it at the last instant of the same day (i.e. the flow would be available


Instance set Without cuts Cuts 1 Cuts 2Dev % Dev* % Dev % Dev* % UBI Dev % Dev* % UBI

Art20 10.72 10.72 2.68 2.68 0.00 10.80 10.80 0.00Art50 8.15 2.72 1.99 1.98 5.42 6.02 6.01 5.55Art100 10.87 4.84 6.58 4.50 6.01 4.76 4.41 7.77GVO100 4.93 3.43 5.18 4.25 1.57 3.70 3.69 3.59GVO102 25.64 18.17 16.20 9.62 7.94 12.87 11.77 19.03GVO103 6.13 4.58 3.07 1.10 8.25 1.62 1.65 9.69GVO106 7.43 0.99 0.68 0.58 88.55 0.89 0.91 88.71Avg Artificial 9.86 4.97 3.83 3.16 4.08 6.93 6.63 4.76Avg GVO 10.11 3.36 5.54 3.25 29.07 4.16 3.79 32.22Avg All 10.02 3.96 4.92 3.21 20.10 5.16 4.85 22.36LP solve 67.29 sec (1) 125.56 sec (1) 105.66 sec (1)

Instance set Cuts 3 Cuts 1 + 2 + 3Dev % Dev* % UBI Dev % Dev* % UBI

Art20 2.90 2.90 0.01 41.08 41.08 0.01Art50 7.28 7.32 5.58 6.90 6.90 5.59Art100 5.26 4.51 8.85 2.28 4.49 9.93GVO100 1.85 1.70 3.34 1.73 3.20 2.94GVO102 22.49 14.99 14.13 17.22 15.80 19.12GVO103 0.95 1.19 9.80 0.94 0.94 10.54GVO106 1.01 1.09 89.53 0.94 0.94 89.60Avg Artificial 5.31 5.00 4.54 18.49 15.05 4.35Avg GVO 5.49 3.87 31.44 4.50 4.46 33.80Avg All 5.43 4.29 22.72 8.89 8.43 24.55LP solve 218.80 sec (3) 125.39 sec (4)

Table 1. Influence of the cuts in the upper bounds and the optimization

again in the first instant of the following day) must be considered, because their flow passes into theaccommodation variablesfh. Let i′d be the first instant andi∗d be the last instant at dayd. Theseinequalities can be formulated as (for mechanical crews, the waiting and cutting variables on theLHS are excluded):

∑

f2∈F

∑

i1∈Id

oi1c,f1,f2

+ ni∗d

c,f1+

∑

i∈Id

yi,i′

d+1

c,f1≥

∑

i1∈I

∑

i2∈Id

yi1,i2c,f1

; ∀d ∈ D, ∀c ∈ CF, ∀f1 ∈ F

6. Computational experiments and ConclusionComputational experiments were performed in order to compare the impact of the cuts to the ob-tained upper bounds and the optimization process. The C++ implementation was compiled withVisual Studio 2008, using Microsoft Windows Vista 32bit. All experiments were carried out on aPersonal Computer with a Intel(R) Core(TM)2 Duo 2.33 GHz CPU and 2 GByte memory. Differ-ent pre-processing techniques were used in order to reduce the problem size. In addition, simpleconstructive heuristics were implemented to provide starting solutions for the MIP solver, ILOGCPLEX. The experiments involve ILOG CPLEX’ branching and polishing phases, each limited to15 minutes. All three cut types were considered: inequalities for the whole planning (labeled asCut1), inequalities for each field (labeled asCut 2) and inequalities for each field at each day (labeledasCut 3).

Table 1 resumes the experiments’ results. The results for the integer program without cuts arelabeled asWithout cuts.UBI represents the relative improvement of the first upper bound found(i.e. the value of the optimal solution to the linear relaxation of the original problem) comparedwith the first upper bound found within the executionWithout cuts. The average results over allinstances of each instance set are given. TheDev %represents the average deviation from optimumreported by ILOG CPLEX at the end of the optimization. If the linear relaxation was not solvedwithin the given time, i.e. no upper bound was found, this instance was not considered in the average


deviation value.Dev* %demonstrates the average deviation after the optimization, considering thebest upper bound known for the instance (obtained throughout all experiments performed). Foreach experiment type, the average time to solve the linear relaxation is reported (given in the lineLP solve). Within the same line, the number within parentheses represents the number of instancesfor which no optimal solution to the linear relaxation was found within the 15 available minutes. Ifthe linear relaxation was not solved, the average time value assumes a solution time of 15 minutesfor the instance.

In all experiments, the addition of the inequalities complicated the resolution of the linear relaxation.Using any of these cuts, the original average resolution time of 67 seconds increased to at least 100seconds. However, the use of the inequalities turned out to be very effective in respect to the qualityof the final solutions, in particular for the GVO instances. The first upper bounds of the MIP,i.e. the optimal solutions to its linear relaxation, were lowered by at least 20% for all cut types.Cut types 2 and 3 improved these bounds slightly better than cut type 1. The joint use of all cuttypes led to a bound improvement of almost 25% in the total average and almost 33% for the GVOinstances. The bound improvement for the artificial instances has been relatively small. Almostall cut types resulted in an improved average solution quality as well as in a better average of theproved deviation from optimum. The remarkable improvement at instance setGVO106resultedalmost in proved optimality. The reported final deviation from optimum reduced from 10.02% to3.96%. Furthermore,it is worth mentioning that the outgoing flow inequalities for manual cuttingcrews of cut type 2 and 3 did not show much effect in the upper bound improvement, since manycutting variables appear on both sides of the inequality. In conclusion, it is observed that the useof Cut 1results in a very good average solution quality and may therefore be recommended for theuse within the problem’s mathematical model for time limited executions.

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