Sanjay Dominik Jena A Mixed Integer Programming

Sanjay Dominik Jena

PUC-Rio - Certificação Digital Nº 0711327/CA

A Mixed Integer Programming approach for sugar cane cultivation and harvest planning

MSc Thesis

Thesis presented to the Graduate Program in Informatics of the Department of Informatics, PUC–Rio, as partial fulfillment of the requirements for the degree of Master in Informatics Adviser: Prof. Marcus V. S. Poggi de Aragão

Rio de Janeiro March 2009

Sanjay Dominik Jena


A Mixed Integer Programming approach for sugar cane cultivation and harvest planning

Thesis presented to the Graduate Program in Informatics of the Department of Informatics, PUC–Rio, as partial fulfillment of the requirements for the degree of Master in Informatics. Approved by the following commission:

Prof. Marcus V. S. Poggi de Arag˜ ao Adviser Department of Informatics — PUC–Rio

Prof. Eduardo Uchoa Barboza Departamento de Produ¸cão — UFF

Prof. Alexandre Street de Aguiar Departmento de Engenharia Elétrica — PUC–Rio

Prof. Jos´ e Eugˆ enio Leal Coordinator of the Centro Técnico Cient´ıfico — PUC–Rio

Rio de Janeiro — March 18, 2009

All rights reserved.

Sanjay Dominik Jena


Sanjay Dominik Jena joined an apprenticeship in Mathematics and Information Technology at AXA Germany. Afterwards he graduated from the Fachhochschule Köln (Cologne, Germany) in General Computer Science, while working at AXA Germany as a software developer for Intranet applications. He then obtained a Master degree at the PUC–Rio in computer science focused on combinatorial optimization and actively participated on the department’s work for Gapso.

Bibliographic data

Jena, Sanjay Dominik A Mixed Integer Programming approach for sugar cane cultivation and harvest planning / Sanjay Dominik Jena; adviser: Marcus V. S. Poggi de Aragão. — Rio de Janeiro : PUC–Rio, Department of Informatics, 2009. v., 153 f: il. ; 29,7 cm 1. MSc Thesis - Pontif´ıcia Universidade Católica do Rio de Janeiro, Department of Informatics. Bibliography included. 1. Informatics – Thesis. 2. Operations Research. 3. Combinatorial Optimization. 4. Mixed Integer Programming. 5. Scheduling. 6. Network Flows. 7. Sugar cane harvesting. 8. Harvest Planning. I. Aragão, Marcus V. S. Poggi de. II. Pontif´ıcia Universidade Católica do Rio de Janeiro. Department of Informatics. III. Title.

CDD: 510

Acknowledgements


First, I wish to acknowledge my supervisor, Prof. Dr. Marcus V. S. Poggi de Aragão, who introduced me to the field of optimization, and to express my deep gratitude for his dedication and continuous advice, guidance and encouragement. His research and motivation inspired and influenced me during my master studies. I also gratefully acknowledge the whole of the Gapso company, in particular Pedro Cunha and Haroldo Gambini Santos, for their extensive support and numerous discussions that were crucial to the success of my research. My appreciation also goes to Ricardo Hermes from the Grupo Virgolino de Oliveira and Eduardo Sans from the GaTech for their continuous support during the specification and development of the model. Secondly, I would like to thank the DAAD (Deutscher Akademischer Austauschdienst) for their financial support. Their strong aspiration to strengthen the academic interchange between Brazil and Germany has always been a great motivation to me. Furthermore, I appreciate the numerous opportunities to contribute to this ongoing cooperation. My deepest gratitude goes to my family, that has fully accepted and supported my choice to study abroad. I am aware of all moments that I was not able to share with them. I would like to thank Mrs. Beatriz Barbieri for proofreading the draft of this thesis, and notably my friends and my girlfriend Angelita for always finding a special way to encourage and motivate me during troubled times and for always having faith in me. Finally, I would like to thank the Brazilian people for embracing me during my entire stay and integrating me to their culture. Every minute spent in their country has turned into an unforgetable experience to me.

Abstract


Jena, Sanjay Dominik; Aragão, Marcus V. S. Poggi de. A Mixed Integer Programming approach for sugar cane cultivation and harvest planning. Rio de Janeiro, 2009. 153p. MSc Thesis — Department of Informatics, Pontif´ıcia Universidade Católica do Rio de Janeiro. Mathematical Programming techniques such as Mixed Integer Programming (MIP), one of today’s principal tools in Combinatorial Optimization, allows fairly close representation of industrial problems. In fact, contemporary MIP solvers are able to solve large and difficult problems from practice. Whereas the use of MIP in industrial sectors such as logistics, packing or chip design is widely common, its practice in agriculture is still relatively young. However, planning of agricultural cultivation and harvesting is a complex task. Sugar cane is one of the most important agricultural commodities of Brazil, the worldwide largest producer of this crop that is used to produce sugar and alcohol. Most of the planning methods in use, manual or computer aided, still result in high waste of resources, on field and in commercialization. The purpose of this work is to provide decision support, based on Optimization techniques, for sugar cane cultivation and harvesting. A decision support system is implemented. It divides the planning into a tactical and an operational planning. The problem is proved to be NP-hard and determines the best moment to harvest the fields, maximizing the total profit given by the sugar content within the cane. It considers resources such as cutting and transport crews, processing capacities in sugar cane mills, the use of maturation products and the application of vinasse on harvested fields. The MIP model extends the classical Packing formulation, incorporating a network flow for the harvest scheduling. Several pre-processing techniques are used to reduce the problem size. Heuristically obtained initial solutions are passed to the solver in order to facilitate the solution. A problem segregation strategy based on semantic information is presented leading to very competitive results. As the linear relaxation optimum solution turned out to be highly fractional, this work also invests in valid inequalities in order to strengthen the MIP formulation. Several further solution approaches such as Local Branching and heuristics based on the optimum solution of the linear relaxation were explored. One of Brazil’s large sugar cane producers was involved in the entire development process in order to guarantee a realistic presentation of the pro-

cesses. All experiments were performed with instances from practice provided by this producer.

Keywords


Operations Research. Combinatorial Optimization. Mixed Integer Programming. Scheduling. Network Flows. Sugar cane harvesting. Harvest Planning.

Resumo


Jena, Sanjay Dominik; Aragão, Marcus V. S. Poggi de. Uma abordagem de Programa¸ c˜ ao Mista Inteira para o planejamento de cultivo e colheita de cana-de-a¸ cu ´ car. Rio de Janeiro, 2009. 153p. MSc Thesis — Department of Informatics, Pontif´ıcia Universidade Católica do Rio de Janeiro. Técnicas de Programa¸cão Matemática como Programa¸cão Mista Inteira (PMI), atualmente uma das principais ferramentas da Otimiza¸cão Combinatória, permitem obter representa¸cões de problemas industriais bem próximas à realidade. De fato, resolvedores PMI contemporâneos resolvem instâncias reais grandes e dif´ıceis. Enquanto o uso de PMI em setores industriais como log´ıstica, empacotamento e projeto de circuitos é bastante comum, a sua prática na agricultura ainda é relativamente nova. Entretanto, o planejamento de cultivo e colheita agr´ıcola é uma tarefa complexa. Cana-de-a¸cu ćar é um dos mais importantes recursos agr´ıcolos do Brasil que é mundialmente o maior produtor de cana usada para produ¸cão de a¸cu ćar e álcool. A maioria dos métodos usados para planejamento, tanto manuais quanto assistidos por computador, ainda resultam em grande desperd´ıcio de recursos em campo e em comercializa¸cão. O objetivo deste trabalho é promover, baseado em técnicas de otimiza¸cão, suporte à decisão para cultivo e colheita de cana-de-a¸cu ćar. Um sistema de suporte à decisão é implementado, dividindo o planejamento em um planejamento tático e operacional. O problema é provado ser NP-dif´ıcil e incorpora a determina¸cão do melhor momento para a colheita dos talhões, maximizando o lucro total dado pelo conte´ udo de a¸cu ćar na cana. O planejamento considera recursos como frentes de corte e transporte, processamento em usinas, uso de maturadores e aplica¸cão de vinha¸ca em talhões. O modelo PMI se basea na formula¸cão clássica do problema da mochila, incluindo fluxos em rede para o escalonamento de colheita. Várias técnicas de pré-processamento são usadas para reduzir o tamanho do problema e solu¸cões iniciais obtidas por heur´ısticas são passadas ao resolvedor para facilitar a resolu¸cão. Uma estratégia de segrega¸cão do problema baseada em informa¸cões semânticas é apresentada, resultando em um desempenho muito competitivo. Como a solu¸cão ótima da relaxa¸cão linear é fortemente fracionária, este trabalho também investe em desigualdades válidas para fortalecer a formula¸cão PMI. Outras abordagens de resolu¸cão como Local Branching e heur´ısticas baseadas na solu¸cão ótima da relaxa¸cão linear foram exploradas.

Um dos grandes produtores brasileiros de cana-de-a¸cu ćar foi inclu´ıdo no processo completo de desenvolvimento de forma a garantir uma representa¸cao real dos processos. Todos os experimentos foram efetuados com instâncias reais fornecidas por este produtor.

Palavras–chave


Pesquisa Operacional. Otimiza¸cão Combinatória. Mixed Integer Programming. Scheduling. Fluxo em rede. Colheita de cana-de-a¸cu ćar. Planejamento de colheita.


Contents

1 Introduction 1.1 Context and Motivation 1.1.1 Sugar cane harvest process 1.1.2 Optimization techniques to solve harvest problems 1.2 Objectives of this thesis 1.3 Previous work review 1.3.1 Decision support tools in agriculture 1.3.2 Decision support tools in sugar cane industry 1.3.3 Decision support tools for other harvest types 1.4 Outline

13 13 13 15 16 17 17 18 19 20

2 Problem Description 2.1 The sugar cane harvest problem 2.1.1 Current practice 2.2 A DSS for sugarcane cultivation and harvesting 2.2.1 Tactical module 2.2.2 Operational Module 2.3 Related problems 2.4 Complexity

21 21 23 24 24 27 29 31

3 Optimization Methods 3.1 Definitions 3.2 Exact methods 3.2.1 Exact methods for linear programming 3.2.2 Exact methods for mixed integer programming 3.3 Heuristics and Metaheuristics 3.4 MIP Solver 3.4.1 ILOG CPLEX

37 37 40 40 41 42 43 44

4 Problem Formulation 4.1 Preliminary discussions 4.1.1 Modeling issues 4.1.2 Simplifications 4.1.3 Modeling the SCHP as a GAP extension 4.2 Formulation for the tactical planning 4.2.1 Mathematical model 4.2.2 Enabling repeated field harvesting 4.3 Formulation for the operational module 4.3.1 Modeling alternatives 4.3.2 Mathematical model

45 45 45 46 47 49 50 61 63 63 64

5 Solution Strategies 5.1 Instances for the computational experiments 5.1.1 Indicators for the level of difficulty

79 81 82


5.1.2 5.1.3 5.2 5.3 5.3.1 5.3.2 5.3.3 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.5 5.5.1

Instances for the SCHP-TP Instances for the SCHP-OP Experiments on the SCHP-TP Preprocessing for the SCHP-OP Distance Filtering Variable pruning and Reduction tests Field grouping Exact solution approaches for the SCHP-OP Analysis of the optimization process Initial solutions through constructive heuristics Linear relaxation analysis Valid Inequalities Alternative solution strategies for the SCHP-OP Segregation and Aggregation of the cutting crews’ planning

83 86 88 91 92 98 100 101 101 106 115 116 125 127

6 Conclusions 6.1 Future work

135 137

Bibliography

139

A

Glossary

146

B

Instance properties for the SCHP-OP and solutions for the segregation strategy 148

List of Figures

1.1

Example routes for cutting transportation crews

14

2.1 2.2

Data flow from the tactical module into the operational module Modeling the harvest problem as a knapsack problem

25 30

4.1

Network flow for the cane quantity along the weeks, enabling the model to harvest a field more than once along the planning period Network flow starting at mill pf Starting node of the network flow at the first instant at field f Initialc Flow conservation at instant node i at field f of cutting crew c Example network flow for a mechanical cutting crew: nodes of not available instants are skipped Example network flow for a manual cutting crew: the crew goes home at the end of the day and moves to any field before the next day

4.2 4.3 4.4 4.5


4.6

5.1 5.2 5.3 5.4 5.5 5.6

5.7 5.8

5.9 5.10

62 74 75 76 77

78

Pol distribution per week for an example instance 89 Quantity of harvested sugar cane per week in the optimal solution for an example instance 89 Processed sugar cane at the mills suggested by an optimal solution 91 for instance GVO10 3 Example of a reduction of non-terminal nodes with degree 2 100 Routes of cutting crews for solutions in the initial stage of the optimization process 104 Routes of cutting crews for solutions in an advanced stage of the optimization process. The lower figure illustrates the routes for a close to optimum solution 105 Objective function value throughout branching and polishing process106 Example route for a cutting crew: first, the crew greedily chooses the fields and waits until the end of the planning (a); in a replanning step (b), the cuts are balanced along the planning period by insertion of the waiting variables after each cut 112 Route of a mechanical cutting crew in the optimal solution for the linear relaxation of instance GVO102 2 117 Fractional example route for a cutting crew after insertion of valid inequalities in order to force traveling. The first three travels between field 3 and 4 are performed to compensate the cutting flow used at field 1. 119

List of Tables

5.1 5.2 5.3

5.14 5.15

Instances for the SCHP-TP 85 GVO instance sets 100 and 102 for the SCHP-OP 90 Outgoing edges per field for different distance filtering approaches 94 (example of 25 fields of Cutting Crew 204 at instance GVO102 2. Impact comparison of distance filtering approaches: No filtering, distance limitation to 5km, 10km and 50km and node balancing with up to 50k, 100k and 250k travel variables 96 Impact comparison of distance filtering approaches: MST only, MST with node balancing with up to 50k, 100k and 250k travel variables 97 Influence of minimum processing demands to the difficulty of an instance 102 Influence of the cutting crews’ occupation rate to the difficulty of an instance 102 Solution properties during the optimization process 103 Quality of the solutions obtained by the heuristics 113 Influence of initial solutions to the quality of the final solutions 115 Influence in the polishing phase of starting solutions passed to CPLEX116 Influence of the cuts in the upper bounds and the optimization 124 Influence of increasing and decreasing cutting crew order in segregation strategy to the cut cane quantity 130 Comparison of solution quality of different solving strategies 131 Results of the LP-and-Fix heuristic 134

B.1 B.2 B.3 B.4 B.5

GVO instance sets 100 and 102 for the SCHP-OP 149 GVO instance sets 103 and 106 for the SCHP-OP 150 Sets of artificial instances 151 Results for all artificial instances for the segregation solution strategy152 Results for all GVO instances for the segregation solution strategy 153

5.4

5.5

5.6


5.7 5.8 5.9 5.10 5.11 5.12 5.13

1 Introduction


1.1 Context and Motivation Sugar cane is a sub-tropical and tropical genus of tall growing crop, counting 37 species plus a number of hybrid species. According to the Food and Agriculture Organization of the United Nations (FAO) [Sta08], sugar cane is one of the most important commodities in the world. With more than 420 billion tons of harvested sugar cane in the year 2005, Brazil is by far the largest producer of this grass worldwide, followed by India, China and Thailand. Among all agricultural commodities produced in Brazil, sugar cane is its most produced measured in biomass and its fourth most lucrative. Internationally, sugar cane is a highly competetive market. Recent international studies [HTAJ07, GLMS08, BFGN02] showed great opportunities to improve the value chain and reduce costs in the operational planning in order to remain competetive. 1.1.1 Sugar cane harvest process The sugar cane harvest typically begins in May, sometimes April and prolongates to November, the time of the year when the sugar cane plants normally reach their maturation peaks. The maturation of sugar cane is measured in percentage of sucrose in the sugar cane, denoted to Pol and reduced sugar, denoted to AR. 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 but leaving the roots to re-grow in time for the following harvest. The harvest is then immediately transported to the industrial sector, i.e. sugar cane mills, by trucks, rail wagons or by manual carriage (cart pulled by a bullock or a donkey). Figure 1.1 illustrates example routes for cutting and tranporting. The cutting crews travel from one field to another harvesting the cane. The transportation crews commute between the fields and the mills.

14

Chapter 1. Introduction

Sugar cane mill Plantation field Cutting crew route Transportation crew route


Figure 1.1: Example routes for cutting transportation crews In the mills, the sugar cane is crushed and the cane juice is extracted. The bagasse leftover, also referred to fiber, is burned in boilers. The induced steam drives the turbines that generate the power for the mills. The sugar cane is further processed either to sugar or to ethanol. The sugar is also referred to as the total recoverable sugar (ATR1 ). For the sugar production, the sugar cane juice undergoes further processes such as heating, filtering and evaporation. The result is a syrup which is centrifuged to separate the sugar crystals from the molasses. To produce ethanol, the juice also undergoes processes as heating, filtering and evaporation. Afterwards, the juice is fermented in large vats, centrifuged and distillated to separate the ethanol. A side effect of the alcohol distillation process is a residual industrial liquid called vinasse. Vinasse is a corrosive contaminant that contains high levels of organic matter, potassium, calcium and moderate amounts of nitrogen and phosphorus [GR00]. However, vinasse is an efficient fertilizer, thus its application to harvested plantation fields has become common practice. The use of maturation products 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 growth of the cane and therefore lead to an increase of the relative quantity of sugar within the plant. Growth regulators are commonly used to prepone a field’s yield in order to provide raw material, i.e. sugar cane, for the mills. Thus, the use of such maturation products is directly related to the moment of sugar cane processing in the mills. The cultivated areas can contain hundreds of lots with different varieties, each with distinct growth and maturation properties. One of the most difficult, but most important decisions is the determination of the ideal moment to cut each field and apply growth regulators in order to benefit best from the 1

The abbreviation ATR is originated in the portuguese term A¸cu ćar Total Recuper´ avel, used in Brazil.


15

maturation peaks. The cutting crew’s capacities and logistic factors are directly involved in such decisions, as they may constraint the harvest and therefore must be taken into account. These factors turn the planning of sugar cane cultivation and harvest into a very complex and difficult task. Most of the planning methods in use, manual or computer aided, still result in high waste of resources on field, during transportation and in further commercialization.


1.1.2 Optimization techniques to solve harvest problems Advances in Information Technology shifted the focus from manual to computer aided planning in many industrial areas. In the past, manual planning was based on the experience of specialists [FT07]. Today, the agriculture industry benefit from computer based planning in order to reduce costs and risks and increase their total gain. Operations Research (OR)2 is dedicated to the search for optimal or nearoptimal solutions to complex problems, as they arise in practice in industry. It is an interdisciplinary branch of formal sciences and applied mathematics and benefits from Simulation, Optimization, Probability and Statistics. Operations Research helps management achieve its goals using scientific methods. One of the primary tools of OR is Combinatorial Optimization (CO), which deals with optimization problems where the set of feasible solution is discrete or can be reduced to a discrete one. In addition to OR, Combinatorial Optimization is related to other fields such as algorithm theory and computational complexity. In the last decades, many algorithms to classic problems of CO appeared [AHLS97]. Such algorithms range from exact methods to approximation algorithms. Most of such methods were successfully applied to solve problems from practice. Several classical CO problems, even instances that are accepted as very complicated, can be solved today. Exact methods solve them to optimality in reasonable time and heuristics and metaheuristics find solutions close to optimality in extremely short processing time. In contrast to classical CO problems, which are usually well-studied, problems that arise in practice have not been yet thoroughly explored. Such problems can usually be seen as extensions or combinations of classical problems. These differences make it difficult to solve problems arising in practice by using methods that were originally developed for the original classical problem. 2 Operations Research is the official term in North America, South Africa and Australia. In Europe, this research area is known as Operational Research.


16

Among many approaches such as heuristics and metaheuristics to handle complex practice problems, mathematical programming has proved to be a powerful tool to solve such problems. Breakthrough advances in information technologies have provided new possibilities to the use of Combinatorial Optimization (CO). Such great developments in computers’ capabilities make it possible to model large problems from practice as mixed integer programs and solve them effectively using mathematical solvers. Solvers such as CPLEX and XPRESS-MP underlie a continuous development process and had their efficiency improved by the years. For this reason they are now capable of solving even larger problems. These great advances and the ability of mathematical programming to handle complex and individual problems suggest to apply such techniques to the complex task of planning sugar cane cultivation and harvest.


1.2 Objectives of this thesis This work focuses on support for crucial decisions that must constantly be made in sugar cane cultivation and harvest. It aims to give suggestions to such decisions in order to support the planning in sugar cane industry. A decision support system (DSS) will be implemented, based on optimization techniques and representing the principal problems as mathematical models. The system must support the planning for a total horizon of up to one harvest season, i.e. approximately twelve months. The objective of the planning is the maximization of the total benefit returned by the cultivation and harvest process, respecting a set of industrial constraints. Throughout the whole planning, the system should maximize the total profit generated by the sale of factory produced sugar. The DSS should determine the exact moments to harvest the plantation fields, based on the maturation level of the sugar cane and the periods in which to apply maturation products. The system must determine cutting crews to cut the fields and transportation crews to carry the cut sugar cane to the factories. Cutting and transportation crews may be limited in capacity and time available. The planning system shall be separated into two modules: a tactical module for the long-term planning of a whole harvest season and an operational module for the detailed planning for up to four weeks. The tactical module divides the whole planning horizon into weeks. It should expose the best periods to harvest each plantation field and assign cutting and transportation crews to the fields in order to guarantee capacities to cut and transport the



17

sugar cane of the chosen fields. An additional function of this module enables it to select maturation products and to determine the best moment to apply them to the fields. The operational module is responsible for the detailed planning of up to four weeks, dividing the whole planning horizons into days. The input data of the operational module consists of the plantation fields selected by the tactical module and any other fields added to the data. It must suggest cutting routes for the cutting crews on a daily basis. Both modules need to guarantee a certain number of hectares to be cut at each day/weak in order to allow the use of vinasse to the fields. The sugar cane mills may demand a minimum and maximum quantity of sugar cane to be processed at each day. The objective of this thesis is the study of adequate optimization techniques to solve the problem mentioned above. The two suggested modules should be implemented, having a strong focus on the operational module, and the resulting system should be able to solve problems of dimensions as they appear in practice. This work focuses on the optimization within sugar cane harvesting and does not aim at simulation aspects (for example for the evaluation of different scenarios). It also does not include attempts of robust optimization. Thus, it does not address sensitivity analysis and uncertainty. 1.3 Previous work review 1.3.1 Decision support tools in agriculture Methods of Operations Research have been applied to the algricultural sector since more than five decades. Heady’s work from 1954 [Hea54] is frequently cited as one of the first applications of linear programming to agricultural planning. His model assigned farm land to various crops subject to operational constraints using a profit maximizing objective function. In 1978, Audsley et al. [ADB78] compared different cultivation techniques for four types of crop: winter wheat, spring barley, sugar beets and potatoes. The linear programming model includes the allocation of resources such as land, labor and machinery. Although some variables in reality should be integers, the authors solve the problem as a linear problem and round the solution values. In the same year, McCarl presented a successful application of linear programming to grain crop production planning [MCDR78]. Such crops include corn, soybeans, wheat and silage. The resulting program allowed for the recommendation of production of one of the crop types. Its input data was provided by a 500



18

questions document answered by the farmers. The program has been used by the farmers for over 35 years in annual workshops. OR was institutionally coupled to agricultural engineering already decades ago [Aud85] and has become common practice in the agricultural sector. Today, there is a considerable amount of research by the OR society, applied to almost all types of fruits, vegetables, crops, corns and so on. Masini [Mas03] applied mixed integer programming to the supply chain planning optimization in fruit industry. Bixby et al. [BDS06] provide a a complex system with several optimization models to support planning of operations and deliveries in meat industry. An example for the problems’ diversity in agricultural planning is given by the problem of wine harvesting. Ferrer et al. [FMMTV08] present a model that both minimizes the operational costs and maximizes the wine quality. The model uses a quality loss function to relate these two goals one to each other. A first review of work applying decision support tools to agriculture was given by Glen [Gle87] in 1987. Since his comprehensive survey, at least three recent surveys appeared. Lowe and Preckel [LP04] highlight some of the important works based on linear programming, stochastic programming, risk programming, dynamic programming and simulation. Their publication also includes a call for future research, which was responded by the OR society. The review of Lucas and Chhajed [LC04] focuses on location analysis applied to agriculture. Finally, Ahumada and Villalobos [AV09] aim to complete the previous compilations. They list more than 40 works, where most of them have their scope on the tactical planning. In addition to the surveys, France and Thornley [FT07] provide a rich introduction into mathematical models applied to agriculture. 1.3.2 Decision support tools in sugar cane industry The efforts of the OR society to support sugar cane industry mainly developed in the last ten years. They can be divided roughly into value chain optimization, harvest and crew scheduling and the prediction of sugar cane performance indicators. Efforts were mainly recognized from countries where sugar cane typically is planted, for example South America, South Africa [GLMS08], Australia and the United States of America. Higgins strongly contributed to recent literature about optimization approaches in sugar cane industry, mostly based on experiences in the australian sugar cane industry. His works include various planning aspects in sugar cane industry. In one of his works [HL06b], Higgins aims better integration and



19

optimization of the cane harvesting and transport sectors of the value chain. The formulation is based on the P-Median problem and spatial clustering. Further works to improve the value chain [HASDPA04, HTAJ07] include investigations of the capacities of cutting and transport crews as well as the mill capacities [HTAJ07]. His works on harvest scheduling include an extension of the generalized assignment problem with over 500.000 integer variables, subject to constraints of transport and mill crushing capacities. The model is solved heuristically. Later exact approaches include a LP model [Zha05] and an integrated statistical and optimization approach [Zha05] that maximizes the sugar content in the crops for a harvest season. Higgins also separately tackled scheduling of cane transport [Hig06a], the simulation of transport capacityplanning [HD05] and the optimization of harvester rosters [Hig02]. De Alencar et al. use a genetic algorithm (GA) to maximize the produced sugar [RCB06]. Caliaria et al. use linear programming, based on the XA Callable library, to maximize the sugar production [CSS04]. Pacheco and Neto [PB08] seem to be the first ones to use the term Sugar Cane Harvest problem (SHP) for the problem. They incorporated it as a 0/1-Knapsack problem and show experimental results with multi-objective evolutionary algorithms (MOEA). The model considers mill and crew capacities. The paper also proposes the incorporation of logistic data such as distances between sugar cane lots, though such logistic issues were not implemented into their model. Pacheco et al. also presented many works using multi-objective Artificial Intelligence (AI) approaches [POF08, PPF07]. All these works incorporate the value of the sugar content (PCC, Apparent Percentage of sucrose in the sugar cane), the biomass per square meter TCH (Tons of sugar cane per hectare) and the total fiber in their objective functions. Pacheco et al. [PRBN05] and Oliveira et al. [OPF06] present works that aim to predict the performance indicators PCC, TCH and fiber using Artificial Neural Networks (ANN) and GAs. Finally, D´ıaz and Pérez [DP00] focus on sugar cane transportation. 1.3.3 Decision support tools for other harvest types The forest sector has also been an intensive user of Operations Research tools. The tactical and operational planning include planting, harvesting and transporting. Cutting crews possess cutting capacities and transportation crews are limited by its transportation capacities. Clients usually demand a certain timber quantity during several time periods. Research work in the forest sector has been very versatile. Mitchel [Mit04] and Karlsson et al. [Kar03] focus on short-term operational planning in forest


20

harvest optimization, including crew scheduling. As transport is a crucial issue in forest harvesting, many researchers such as Karlsson et al. [Kar04, HKR06] also focus on road planning to meet the demands for the timber transport. Further optimization in silviculture include [HMZW04] and several works of Weintraub et al. such as [WN76, MEW98, EGMW97].


1.4 Outline The details discussed within each chapter are described below. Chapter 2 explains the sugar cane harvest problem by a detailled description of the tactical and the operational module. The provided background gives an understanding of sugar cane industry terminology and techniques used in this thesis. Furthermore, the relation of the problem to other classical problems as well as the problem’s complexity are discussed. Chapter 3 defines the basic terminology used throughout this thesis and introduces briefly into different optimization methods. It also gives an overview of contemporary MIP solvers. Chapter 4 discusses modeling issues and shows the abstraction of the Generalized Assignment Problem to the given problem. Afterwards, the mathematical formulations for both modules are given. Chapter 5 focuses on the solution of the problem and the computational experiments performed. First, the instances used for experimental evaluation are explained. Experiments performed on the SCHP-TP are presented. For the SCHP-OP, the content is then divided into pre-processing, exact methods and alternative solution strategies. Pre-processing includes all techniques that aim at reducing the problem size in order to facilitate its solution. The section about exact methods explains the basic techniques applied to the problem. Initial solutions provided as starting solutions due to the problem’s difficulty are then discussed. Finally, the characteristics of the linear relaxation are shown and valid inequalities to strengthen the MIP formulation are given. The last section of this chapter includes approaches usually applied to largescale problems as they can be found in practice. Solution strategies based on semantic information of the problem are discussed. Furthermore, a heuristic based on the linear relaxation is presented. Computational results are presented for all methods. Chapter 6 concludes this work by resuming its contribution and stating possible extensions for the future.


2 Problem Description

This chapter explains the given problem in detail. Throughout this work, this problem will be referred to as the Sugar Cane Cultivation and Harvest Problem (SCHP) as it incorporates both cultivation and harvest issues appearing in sugar cane industry. After the problem’s components are presented, the current practice at one of Brazil’s sugar producers is explained in order to demonstrate the current planning process. The SCHP is then divided into the problems of the tactical and the operational planning. Both of them are described in detail. Finally, related problems and the asymptotic time complexity are discussed. 2.1 The sugar cane harvest problem One of the most important decisions in this problem is the determination of the optimal moment to harvest the plantation fields. Clearly, it is desirable to harvest each field at the peak of its sugar content, as the sugar indicated by the Pol and AR values vary as the cane grows. In the beginning of the planning, each field got a certain initial age. A field can only be cut within a given interval of its age defined in the input data. In the following, the complete problem is described with an introduction to the decisions that must be made and to the problem’s constraints that need to be satisfied. Sugar cane mills. After harvesting a field, its sugar cane is immediately transported to one of the sugar cane mills to be crushed and further processed to sugar. The mills operation is one of the most important constraints as it must not interrupt sugar cane processing. Each mill contains minimum and maximum process capacities which must be respected by the planning solution. Plantation fields that have been selected for harvesting must be assigned to one of the available sugar cane mills. Furthermore, sugar cane fiber is used to generate electricity in order to operate the sugarcane mills. Thus, the processed sugar cane must contain a certain minimum quantity of fiber.

Chapter 2. Problem Description

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Maturation products. Some sugar cane varieties allow the use of maturation products to anticipate its harvest. In general, such products slow down the growing process of the cane mass, whereas the growth of the sucrose within the cane is not affected. Thus, the percentage of sugar in the cane increases. The eligible maturation products are given for each variety within the input data. Maturation products can only be applied when the cane reaches minimum age. Once applied, they directly influence the originally given interval in which the field can be harvested. For each combination of maturation products and sugar cane varieties, an interval is defined that determines the feasible period to harvest the field after the product was applied. The planning solution shall consider the use of maturation products, i.e. determine the type and exact moment of its application, in order to find the most lucrative harvesting schedule. Cutting crews. The sugar cane is harvested by cutting crews. A cutting crew can either be manual, that is a group of human workers, or mechanical. Each cutting crew may be eligible to cut only a certain subset of the fields. Cutting crews may not work every day and work a limited time at each day. Each cutting crew has its own properties. It’s minimum and maximum cutting capacities, travel speed as well as cutting and travel costs are given in the input data and must be taken into account. After finishing their work at the end of each day, mechanical cutting crews remain at the current field and start working in the beginning of the next day. Manual cutting crews return to a place where they are accommodated. Transportation. Transportation crews carry the cut sugar cane from the fields to the sugar cane mills. Each crew possesses individual properties such as a certain transport capacity, speed and costs. The solution must suggest a transportation schedule that assigns exactly one transportation crew to each field selected to be harvested. Transportation crew capacities, speed and costs must be considered in the solution of the problem. Vinasse application. After crushing the sugar cane within the mills, the waste dump vinasse remains. 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 must be harvested in certain periods. Not all fields are eligible for vinasse application. The vinasse demands, i.e. the periods and the necessary field size, and the fields that are eligible for vinasse application are informed by the input data. A planning solution must


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guarantee that sufficient fields are harvested in order to satisfy the vinasse application demand in all periods.


2.1.1 Current practice The Pol and AR values for each cane variation are one of the most important performance indicators to select the best moment for a field’s harvest. However, these values cannot be foreseen very well for a long period such as twelve months. A detailed planning for such a broad time period would result in a huge problem size that could hardly be solved in adequate time. In many industrial planning problems, the division of the entire problem into smaller pieces is a natural approach. Many decisions have to be made only for a rough time period such as months or weeks, without determination of the exact detailed planning for shorter periods such as days or hours. Planning of such a broad time horizon is referred to as tactical planning. It is generally followed by an operational planning, where the problem is subdivided into smaller pieces and the resulting sub problems are solved in detail. The current practice at the Grupo Virgolino de Oliveira (GVO), a Brazilian sugar cane company, follows the above explained planning process. Planning is performed either manually by experienced experts from the agricultural sector or by computer based decision support tools1 . The decision support tool in use does not consider any distances between the plantation fields. Thus, the logistic planning does not fairly represent the reality. It allows partial field harvest as it is based on linear programming. Although it is a good tool to choose the plantation fields to harvest, the planning may not always be feasible in practice as it may not respect all industrial constraints. In addition, there are always stochastic factors that cannot be foreseen, such as delays of the cutting and transportation crews or industrial deficiencies. As a result, it may not always be possible to harvest all fields as it was planned. In the GVO’s current practice, if a field was not harvested in the period to which it was assigned by an operational planning, it is added to the operational planning of the following month. In some cases, also the fields of the successive month are added in order to consider updated Pol and AR values. The field set within the input data for an operational planning for period p may hence contain: – all fields that were indicated by the tactical planning to be harvested within period p − 1, but in fact were not harvested. 1 The GVO currently uses an optimization tool based on linear programming. The LP is solved by the XA Callable Library of Sunset Software


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– all fields indicated by the tactical planning to be harvested within period p. – all fields indicated by the tactical planning to be harvested within period p + 1. The operational planning should then identify the best fields to be harvested within the current period. All remaining fields are shifted into next planning again.


2.2 A DSS for sugarcane cultivation and harvesting The proposed decision support tool for sugar cane cultivation and harvest planning follows the previously discussed approach and divides the whole planning into a tactical planning and an operational planning. The decisions of the tactical planning directly influence the input data for the operational module. First, the tactical module performs the planning for the whole planning horizon, i.e. up to a whole harvest season. Afterwards, the total planning time is divided into smaller periods of up to 30 days. The operational planning is then performed for each of these sub periods. The input data for each operational planning is based on the decisions of the tactical module in the according time period. That is, the set of fields selected by the tactical module for a certain time period forms the input data of an operational planning. In this operational planning, the exact days to harvest each of these fields are determined. In addition, the user may modify the input data for the operational module. Figure 2.1 illustrates the data flow of the system. As the operational planning is frequently performed, it is desirable that the execution time of the optimization does not exceed 30 minutes. The tactical planning, as it is performed only every few months, does not possess an explicit time limit. For reasons of the user’s convenience, one can assume a time limit of 60 minutes. In the following, both modules are explained. 2.2.1 Tactical module The tactical module of the SCHP, denoted by the SCHP Tactical Planning Problem (SCHP-TP), supports the planning for a total planning horizon of up to twelve months, i.e. one harvest season. It may be applied to shorter periods such as a few months, usually in the replanning during a running season.

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Input

Data generation by the user

Tactical module Output Input

Manual modification by the user

Operational module

Output Figure 2.1: Data flow from the tactical module into the operational module


The tactical module determines for each week of the whole planning horizon: – the plantation fields that should be harvested and the cutting crews to cut these fields. – the sugar cane mills in which the cane should be processed. Each field is associated to exactly one mill. – the transportation crews that carry the cut sugar cane to the mills. Each field is assigned to exactly one transportation crew. – the maturation products that shall be applied and the fields at which the products shall be applied. – the types of vinasse that shall be applied on the fields after harvesting them and the according fields. The module maximizes the total profit given by the processed sugar cane in the mills and the current value of the ATR, representing the quantity of theoretic total usable sugar. The ATR is computed against the quantity of reduced sugar (AR) and Pol, using coefficients for both values to normalize the relation between them: AT R = coef P ol · tonsP ol + coef AR · tonsAR On the cost side, there are the costs for cutting the fields, transporting the sugar cane to the mills, the processing of the cane within the mills and the application of maturation products and vinasse.


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During the maximization of the above mentioned indicators, the system must satisfy the following constraints: – All plantation fields must be harvested. Weights can be attributed to the fields for each week to indicate a priority of harvesting in certain weeks. – All cutting crews have a minimum and maximum cutting capacity of sugar cane per week. – All transportation crews have a maximum capacity of sugar cane that can be carried.


– The sugar cane cut at a day shall be transported to and processed in the sugar cane mill within the same day. The sugar cane mills have a minimum and maximum capacity of sugar cane that they can process in each week. Consequently, the sugar cane quantity cut by the cutting crews shall satisfy the mill’s demands. – In all weeks, the mills must produce a certain minimum quantity of fiber, given by a percentage within the processed sugar cane. – The varieties of the different sugar cane species have a minimum and maximum age to be cut. The sugar cane can be cut only during these given intervals. These intervals may vary by week and field. – Maturation products can be applied only during a certain interval of the sugar cane’s age. After applying a maturation products, the sugar cane must be harvest within a given number of weeks. – In all weeks, a minimum quantity of vinasse must be applied on the recently harvested plantation fields. In order to allow these applications, the number of hectares freed by field harvesting in each week must be sufficiently high. The complete harvest of any plantation field may not exceed one week, i.e. only plantation fields that can be harvested within one week are considered. In addition to the sugar cane cut by the cutting crews, it is possible to acquire sugar cane from third party suppliers. Each supplier provides sugar cane of certain types of cane varieties up to a certain capacity. The sugar cane possesses its own properties (Pol, AR, Fiber, etc.) and can be processed at any mill.


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2.2.2 Operational Module The operational module of the SCHP, also denoted by the SCHP Operational Planning Problem (SCHP-OP), realizes a detailed planning for a time horizon of up to 30 days. Its input data is based on the planning of the tactical module for the chosen time period. The operational module may redefine assignments between cutting crews, fields and mills. Valid assignments are informed in the input data. The application of maturation products is not determined by this module. Instead, such decision are already covered by the tactical module and shall be informed in the input data for the operational module. While the tactical module works with estimated maturation curves of the sugar cane, the operational module is intended to work with updated recent values of the sugar cane’s maturity, i.e. Pol, reduced sugar AR and fiber. These values result from the pre-analysis, where cane examples of a certain area are analyzed before the sugar cane is cut. The cutting crews may be located either at a mill or at a plantation field. Latter case occurs when the cutting crew did not finish the harvest of the field in the previous planning. The cutting crew must then initiate the cut of the field at which it is located in the beginning of the planning. The manual cutting crews are usually hosted at a place close to the fields. At the end of a working day, they spend the night at such place and return to work in the next morning. It is assumed that the time spent and the costs generated to get to their accommodation and to return to a field in the next morning are already considered by the input data through average values, i.e. the spent time and costs must not be considered in the planning. Thus, manual cutting crews may travel to any field in the next morning without any travel costs. 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. In this case, the rest of the field will be cut in the next operational planning. Moreover, harvesting a field can take more than one day and must only be interrupted by the time that a crew is not available (usually during the night). Cuts must not be interrupted by days at which the field cannot be cut, the crew does not work or the mill is not available. Thus, a cutting crew can only start harvesting a field after having finished the harvest of the previous field. Finally, assignments of the transportation crews will not be considered in this module. In practice, transportation crews are sufficiently available and can be hired on demand when necessary. Hence, the operational planning does not involve their planning. It is assumed that a successful planning of the tactical


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module already guarantees sufficient availability of transportation crews. Given the input data for the cutting crews, fields and mills, the operational module should determine the harvest sequence of the fields for each cutting crew so that it maximizes the total profit given by the sugar production in the mills minus costs such as for cutting, transportation and processing the sugar cane, the movement of the cutting crews and vinasse application. For each day of the planning, the system should suggest: – the fields to be cut and the cutting crews to cut these fields. The cutting sequence during the 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 system must consider the following constraints: – All cutting crews have a minimum and maximum cutting capacity of sugar cane in tons per day. – 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 to processed at each day. Consequently, the sugar cane quantity cut by the cutting crews shall satisfy the mill’s demands. – In all weeks, the mills must produce a certain minimum quantity of fiber, given by a percentage within the processed sugar cane. – In all days, a minimum quantity of vinasse must be applied on the recently harvested plantation fields. At least this number of hectares must be harvested in each week to permit the vinasse application. Sugar cane be acquired from third party suppliers, just as explained in the description of the tactical module. Note that the operational module does not demand all fields to be harvested. This is justified by the current planning practice in industry, explained in Section 2.1.1. Another feature of this module is the possibility to rank the importance of the constraints. In some instances, it may not be possible to satisfy all constraints: such a ranking then indicates which constraints shall be violated less than others. The following table resumes the main differences between both modules:

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Tactical module Operational module Planning horizon Objective Decisions Must cut all fields Cutting crews Cutting crew harvest sequence Transportation crews Maturation products application Vinasse application

One season maximize profit weekly yes yes no yes yes yes

7 to 30 days maximize profit daily no yes yes no no yes


2.3 Related problems The essential structures for the tactical and operational module are very similar. Both modules try to maximize the profit by choosing the best moment to harvest the plantation fields, considering the minimum and maximum processing demands of the sugar cane mills and the limited capacities of the cutting crews. This effort can be seen as 0/1 Multiple Knapsack problem. It is a generalization of the of the 0-1 Knapsack problem and identical to the Generalized Assignment Problem (GAP), one of the most studied combinatorial problems. Martello and Toth [MT90] elaborately introduce into these problems and show that they are NP-hard. In the 0/1 Multiple Knapsack problem, a number of objects must be packed into a given number of bins. Each object has a weight and a profit for each bin. Bins are limited in their capacity of weight. The objective of the problem consists in finding the configuration that maximizes the total profit, respecting the weight capacities of the bins. Pacheco and de Lima Neto [PB08] already modeled harvest scheduling as a 0/1 multiple knapsack problem. In the case of the tactical module, the sugar cane mills can be modeled as bins, one for each week. All sugar cane mills own a maximum processing capacity, referring to the bins’ capacities. The weights and profits to process a field’s sugar cane at different weeks and mills may vary. The profit is obviously linked to the sugar content of the field at the processing week. Figure 2.2 (a) examples such an assignment. At each week, the sugar cane mill processes a different subset of fields. In the same way, other resources such as the cutting and transportation crews can be modeled. Figure 2.2 (b) examples this abstraction to the cutting crews: A cutting crew is assigned to harvest several fields within its cutting capacities in each week. A detailed extension of the mathematical model for the GAP is given below in subsection 4.1.3.

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Mill 1

(b) Cutting


Crew 1

Week 1

Week 2

Field 2

Field 3

Field 13

Field 4

Field 15

Field 11

Week 1

Week 2

Week n

...

Field 6

Week n

Field 3

cutting capacity

(a)

processing capacity


Field 2 Field 5 Field 8

Field 4

...

Field 6

Field 9

Figure 2.2: Modeling the harvest problem as a knapsack problem The operational planning is much more complex than the tactical, because it involves the cutting sequences for the cutting crews. A part of the operational planning can evidently be modeled in the same way. The sugar cane mills’ and cutting crews’ capacities can be modeled based on a 0/1 Multiple Knapsack formulation, using a bin for each day. The main difference is the addition of the cutting sequences for the cutting crews which incorporate knowledge of the distances between one field to another. As such a sequencing is often referred to as a harvest scheduling, the problem of determining the field harvest sequence naturally can be seen as a general type of scheduling as well. The harvest sequence of the cutting crews also resembles the Vehicle Routing Problem (VRP), a class of NP-hard problems. In the VRP, a fleet of vehicles supplies customers, starting from and returning to a depot where the vehicles are reloaded. Each vehicle has a certain capacity and each customer has a certain demand. The objective of these problems is to find optimal vehicle routes that satisfy the customer demands and either minimize the total distance or the number of necessary vehicles. Rich [Ric99] broadly introduces into the VRP. For the SCHP, the vehicles can be represented by the cutting crews and the clients by the fields that have to be harvested. The VRP



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versions that partially resemble the SCHP-OP are the Capacitated Vehicle Routing Problem (CVRP), the Vehicle Routing Problem with Time Windows (VRPTW) and the Periodic Vehicle Routing Problem (PVRP). The CVRP limits the capacities for each vehicle. In the VRPTW, the customers have to be supplied within a certain time window. This equals the eligible days to harvest a plantation field. Finally, the PVRP allows that the deliveries may take more than one day. Though these problems are partially very similar to the SCHP-OP, they do not allow the incorporation of the most important aspects: the varying profit, i.e. the sugar content within the cane, as the sugar cane grows. As the cutting crews’ temporal resources may be a crucial factor in the harvest planning, it is desirable that they spend most of their time in cutting sugar cane and as less time as possible in moving from one field to another. This effort is directly linked to the distances and the length of the paths between the fields, leading to the attempt to minimize the total covered distance. From this point of view, the SCHP-OP also intersects with problems such as the Traveling Salesman Problem (TSP) and the problem of finding the shortest Hamiltonian Path. Latter one aims to find a shortest path between two given vertices that visits each vertex exactly once. In the problem of the Hamiltonian Cycle, the starting vertex is the same as the terminal vertex. The TSP is one of the most studied combinatorial problems. It consists in finding a path that covers all vertices so that the total distance is minimal. The Multiple Traveling Salesman Problem (M-TSP) is a generalization of the TSP, where more than one salesman can be used within the solution. The M-TSP can also be extended to a variety of vehicle routing problems. It strongly relates to the problem of finding the cutting sequences for the cutting crews. All these problems are known to be NP-hard. Particularly the operational planning of the SCHP is a very versatile problem that intersects with many classical NP-hard problems known in Combinatorial Optimization. Thus, one may presume that the problems of the SCHP also are NP-hard. This assumption is proved in the following section. 2.4 Complexity It is now shown that the problems of the tactical and operational planning of the SCHP are NP-hard. In order to prove the NP-hardness of the problems, although it is not necessary, first it is shown their decision versions are NPcomplete. The decision versions of the SCHP-TP and SCHP-OP consist in determining whether there exist feasible solutions that represent a profit


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greater than ∆. From the NP-completeness of the decision versions of the SCHP-TP and SCHP-OP follows that their optimization versions are NP-hard. Polynomial certifier for the tactical module. In order to prove that decision version of the problem belongs to the class NP, one must find an efficient certifier for the solution of the problem, i.e. a polynomial time verification for the correctness of any given solution for the tactical planning. This is easily possible by verifying each of the problem’s constraints. In addition, the certifier must verify whether the profit represented by the solution is greater than ∆. Let W be the set of weeks of the whole planning, let F be the set of plantation fields that can be harvested, let CF be the set of all cutting crews, let T F be the set of all transportation crews, let P be the set of all mills, let R be the set of all third party sugar cane suppliers and let M be the set of all maturation products. A solution of the problem is given by a set of decisions for each week within W , containing the fields to be cut within the week, the cutting crew assigned to cut this field, the transportation crew assigned to transport the cane from this field and the sugar cane mill assigned to process the cane from this field. Furthermore, the solution identifies the fields to receive a maturation product or vinasse (including the applied vinasse quantity). 1. The total profit represented by the solution has to be greater than ∆. It can be calculated in polynomial time. It must consider the profit given by the production of sugar cane harvested in fields and acquired from third party suppliers. In addition, it must subtract the costs generated through cutting, transporting, purchasing and processing of sugar cane. Further costs are given by the traveling costs of the cutting and transportation crews, the use of maturation products and the application of vinasse. 2. A sugar cane mill may not be able to process all types (varieties) of sugar cane. Thus, it must be verified whether the assignments between the fields and the mills are valid. Such verification can be done in O(|F | · |P |). 3. For each field, it must be verified whether the week in which the field is cut lays within the eligible interval to harvest this field (considering the anticipation of the interval by use of a maturation product). This verification can be performed in constant time. 4. The verification whether the lower and upper processing capacities of a mill within a certain week are satisfied can be done in O(|F | + |R|). As there are |P | mills and |W | weeks, the complete verification is performed in O(|P | · |W | · (|T | + |R|)).


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5. The average fiber percentage for one mill in a week is verified in O(|F | + |R|) time. Thus, all verifications at |W | weeks for |P | mills are done in O(|P | · |W | · (|F | + |R|)) time. 6. The obligation to cut all fields is verified in O(|F | · |CF | · |W |) steps, verifying each combination of fields, cutting crews and weeks. 7. The set of fields that a cutting crew can cut may not include all exisiting fields. The verification whether assignments between cutting crews and fields are valid can be performed in O(|F | · |CF | · |W |). 8. Checking whether the cutting capacities of a cutting crew is satisfied in a certain week costs O(|F | · |W |) steps, resulting in a total cost of O(|F | · |CF | · |W |) for all cutting crews in all weeks.


9. The time availability of a cutting crew at a day must not be exceeded. This verification needs at most O(|F | · |CF | · |W |) time for all cutting crews for all weeks. 10. The quantity of transported sugar cane is limited by the capacities of the cutting crews. These constraints can be verified in O(|F | · |T F | · |W |) for all transportation crews for all weeks. 11. All applications of maturation products at fields must be verified, i.e. the accuracy of the application moment as well as whether the product is allowed to be applied at the field at all. 12. The total sum of cut sugar cane in hectares must be verified at each week in order to guarantee sufficient harvested area for vinasse application. This verification, for all weeks, needs O(|F | · |P | · |W |) time.

Polynomial certifier for the operational module. Let D be the set of days of the whole planning. Let F be the set of plantation fields that can be harvested, let CF be the set of all cutting crews, let P be the set of all mills and let R be the set of all third party sugar cane suppliers. A solution of the problem is given by a set of decisions for each day d ∈ D, containing the fields to be cut at day d, the cutting crew assigned to cut this field, the transportation crew assigned to transport the cane from this field and the sugar cane mill assigned to process the cane from this field. Furthermore, the solution informs at which fields how much vinasse shall be applied. A certifier for a solution of the operational planning must verify the following constraints:


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1. The total profit represented by the solution has to be greater than ∆. It can be calculated in polynomial time. It must consider the profit given by the production of sugar cane harvested in fields and acquired from third party suppliers. In addition, it must subtract the costs generated through cutting, transporting, purchasing and processing of sugar cane. Further costs are given by the traveling costs of the cutting and transportation crews and the application of vinasse. 2. A sugar cane mill may not be able to process all types (varieties) of sugar cane. Thus, it must be verified whether the assignments between the fields and the mills are valid. Such verification can be done in O(|F | · |P |).


3. For each field, it must be verified whether the days at which the field is cut lays within the eligible interval of days to harvest this field. This verification can be performed in O(|D|) time. 4. The verification whether the lower and upper processing capacities of a mill at a certain day are satisfied can be done in O(|F | + |R|). As there are |P | mills and |D| days, the complete verification is finished in O(|P | · |D| · (|F | + |R|)). 5. The average fiber percentage for one mill at a day is verified in O(|F | + |R|) time. Thus, all verifications at |D| days for |P | mills are done in O(|P | · |D| · (|F | + |R|)) time. 6. The set of fields that a cutting crew can cut may not include all existing fields. The verification whether assignments between cutting crews and fields are valid can be performed in O(|F | · |CF | · |D|). 7. Checking whether the cutting capacities of the cutting crews are satisfied at a certain day costs O(|F |) steps. This leads to a total cost of O(|F | · |CF | · |D|) for all cutting crews at all days. 8. The time availability of a cutting crew at a day must not be exceeded. This includes the time spent in cutting the field as well as time needed to travel from one field to another. Considering all possible combinations between two fields for the travels and all cutting crews, the verification for one day can be done in O(|F |2 · |CF |) steps. The total time for all days is thus O(|F |2 · |CF | · |D|). 9. The total sum of hectares cut at plantation fields must be verified at each day in order to permit vinasse application. This verification, for all days, needs O(|F | · |P | · |D|) time.


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As the feasibility of a solution to the SCHP can be verified within polynomial time, the decision versions of the SCHP-TP and SCHP-OP belong to the class NP. Reduction of a NP-complete problem. As shown above, solutions for both the tactical and the operational planning of the SCHP (and their decision versions) can be certified in polynomial time. In order to show that the decision versions of the problems are NP-complete, one must reduce the decision version of another NP-hard problem to them. The problem selected to be reduced is the GAP, which is known to be NP-hard [MT90]. Section 4.1.3 shows that both the SCHP-TP and SCHP-OP are generalizations of the GAP. Consequently, the decision versions of the SCHP-TP and SCHP-OP are generalizations of the decision version of the GAP. Consider a GAP instance with m bins b1 , · · · , bm and n objects x1 , · · · , xn . Let wM axi be the capacity of bin bi . Let weighti,j be the weight and prof iti,j the profit to put object xi into bin bj . Let T be set of temporal units, i.e. weeks for the SCHP-TP and days for the SCHP-OP. The following configuration represents a feasible reduction of this GAP instance to both the SCHP-TP and the SCHP-OP: – Representation of the bins. There is exactly one sugar cane mill p. There are exactly m temporal units (weeks or days) within the whole planning horizon. Each of the bins within the GAP b1 , · · · , bm is represented as this mill p1 · · · pm at each temporal unit 1, · · · , m. The maximum weight capacity wM axi of each bin i transforms into the maximum processing capacity of this mill for the temporal unit i. The minimum processing of the mill is always zero. – Representation of the objects’ weights. Each object within the GAP that has to be assigned to a bin is represented by a plantation field. Its productivity at the temporal unit i is set to the weight of the object when it is put into the bin bi . – Representation of the objects’ profits. As an object’s weight and profit usually differ within a GAP instance, it is not valid to represent the profit only by the field’s sugar cane productivity. However, it can be represented by the sugar content percentage within the sugar cane of the field at a certain temporal unit. More exactly, the Pol value for a field j prof iti,j at temporal unit i is set to P roductivity . The ATR costs are set to one j,i for all temporal units. The AR values are set to zero for all fields at all temporal units.


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– Further fixings. There is one transportation crew and one cutting crew for each plantation field. All transport costs of the transportation crews are set to zero. Also, the transport costs are set to zero. The transportation crews’ speed and carriage capacities are set to infinity. All cutting costs for the cutting crews are set to zero. There are no maturation products and no vinasse demands.


Given the total profit composed of the profits of the selected objects, it can easily be verified whether exists a solution with profit greater than ∆. This reduction of the GAP to the SCHP-TP and the SCHP-OP concludes the prove of the NP-completeness of the decision versions of tactical and operational planning problems. Therefore, the SCHP-TP and SCHP-OP are NP-hard.

3 Optimization Methods


This chapter gives a brief overview over different optimization methods and introduces into its essential concepts. It is intended to provide basic knowledge in order to facilitate the understanding of the strategies used to solve the problem (see Section 5). The chapter starts with basic definitions. Afterwards, it reviews methods to solve problems exactly as well as heuristically. Finally, current MIP solver are presented. 3.1 Definitions This section defines important terms used in this thesis. Further introduction to linear and integer programming can be found in the works of Wolsey [Wol98] and Wolsey and Nemhauser [WN99]. Well-founded literature into combinatorial optimization include the works of Grötschel et al. [GLS88], Schrivjer [Sch94], Papadimitriou [PS98] and Wolsey and Nemhauser [WN99]. Mixed Integer Program Let m, n and p be non negative integers. Consider a matrix A ∈