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PLACE Ii RETURN Mon dd. duo. TO AVOID FINES mum on Of DATE DUE DATE DUE DATE DUE :1: J* QUANTITATIVE MODELS FOR TEAK FOREST MANAGEMENT IN INDONESIA By Ida-Bagus Putera Parthama A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Forestry 1995 ABSTRACT QUANTITATIVE MODELS FOR TEAK FOREST MANAGEMENT IN INDONESIA By Ida-Bagus Putera Parthama This study provides a prototype of a quantitative approach for management of large-scale timber plantations in Indonesia. Focusing on teak plantations in Java, a package of quantitative models has been developed consisting of (l) a set of growth and yield models and (2) harvest scheduling models. The growth and yield models were integrated into a computer routine that can be used to project future yields of a given teak stand under difl'erent management regimes or rotation ages. This computer routine was applied to Cepu Forest District in Central Java, which was selected as a model forest for the development of the harvest scheduling models. The harvest scheduling models were formulated to maximize total net present value (N PV) over a 120-year planning horizon subject generally to non-declining even-flow (NDEF) constraints. Stands comprising the selected forest district were aggregated into stand-types, and the outputs of the harvest scheduling models were hectares of each stand-type allocated to three rotation ages Ge, 60, 70, and 80 years) or no management. Other outputs included harvest flows over time and the total NPV. The harvest scheduling models were formulated in two versions. The first version is deterministic and treats yields as known with certainty. This version was formulated as a linear programming (LP) problem, and difi‘erent sets of constraints were used to examine management options. For comparison purposes, one model resembles the current management strategy. Outputs of these LP models indicated that the current single 80-year rotation-age management incurs a substantial cost in the form of foregone NPV; total NPV was nearly doubled when shorter alternative rotation ages were provided. The highest NPV was given by a model that includes multiple rotation ages and allows periodic harvest volumes to increase without an explicit upper 1i iii bound. Models that restrict the increase to a certain upper bound or allow periodic harvest volumes to decrease resulted in lower NPVs. All models tend to allocate a major portion of the forest to the shortest rotation. Furthermore, without any NDEF constraints, harvest flows over time fluctuate erratically. Imposing NDEF constraints regulates the harvest-flows, but reduces total NPVs. The second type of harvest scheduling model incorporates risk of not achieving a NDEF condition due to non-deterministic yield predictions. This version was formulated using chance- constrained programming (CCP). CCP accounts for the risk by incorporating the associated variances of yield predictions into the models, and requiring the NDEF requirement to hold up to a certain probability, but not with probability one. A strict NDEF condition was not feasible with CCP formulation. Several CCP models with different NDEF requirements were examined, and feasibility was achieved by allowing periodic (i.e., decadal) harvest volumes to decrease by a maximum of 10%. CCP models resulted in different hectare allocations. Under specific constraints, they produced higher total NPVs relative to the deterministic models, but resulted in less smooth harvest-flow trajectories. An important advantage of including the risk factor in the model is having some degree of assurance (e.g., 95%) that the projected periodic harvest volumes (hence, harvest flows) will materialize if the model outputs are implemented. In general, this study has demonstrated the applicability of a contemporary forest management technique to forest plantations in Indonesia. It was also shown that the technique considerably reduces limitations inherent in the conventional management approach currently practiced on teak plantations in Java. This finding provides a basis for not using the current teak forest management as a model for forest plantation management in Indonesia. Instead, the approach proposed in this study is recommended as a prototype for developing similar packages of quantitative models for other species in other regions of Indonesia. Possible model improvements are suggested. They include: using finer levels of spatial and temporal aggregation, incorporating other relevant constraints and other sources of risk, and refinement of the grth and yield models. ACKNOWLEDGMENTS I am deeply indebted to many individuals and institutions for their support throughout my doctoral program and the completion of this dissertation. First of all, I would like to express my sincere appreciation to Dr. Larry A. Leefers, my academic advisor and chairman of my guidance committee, for the guidance, assistance, and encouragement be continuously provided. Dr. Leefers was more than committed to the completion of my study and helped supervise data collection in Indonesia during a Christmas break. He also provided assistance when I needed an additional semester beyond my scholarship period. All of these can never be adequately thanked. I also wish to express my appreciation to Dr. Carl W. Ramm, who provided invaluable assistance and constructive suggestions, particularly during the growth-and-yield-modeling phase of this study. My sincere thanks and appreciation are also extended to Dr. Karen L. Potter-Witter and Dr. Stephen B. Harsh, the other members of my guidance committee. I acknowledge with gratitude the Forestry Research and Development Agency of the Indonesian Ministry of Forestry for providing me with a scholarship which enabled me to pursue my doctoral study at Michigan State University. I also wish to express my appreciation to the Ministry of Forestry for providing support for my family to join me in the United States. My appreciation is extended to the Center of Forestry Research and Development in Bogor, Indonesia, for allowing me to be on leave from my job for more than six years and for giving permission to use the teak growth and yield data for this study. Many thanks due to my colleagues at the Center, especially Mr. I-Iarbagung, Mr. Djoko Wahjono, and Mr. Darmawan Budiantho, for their support particularly in preparing and shipping the data to the United States. I also wish to thank Perum Perhutani, especially the Cepu Forest District, for providing me with the management and forest- inventory data for this study. Dr. Boen M. Pumama who, despite his hectic schedule, spent a great deal of his time making the data collection at Perum Perhutani possible, deserves a sincere acknowledgment. Many thanks also go to Mr. Iman Santoso, Mr. Ruddy T. Koesnandar, Mr. Didy Wuryanto, and other iv V colleagues at the Ministry of Forestry, who in various ways ofi‘ered support during the data collection in Indonesia. I must also thank Mr. Adi Susmianto for the excellent cooperation in realizing the data collection trip to Indonesia. Special thanks due to my friend, the computer whiz Dr. Herr Soeryantono, not only for always kindly lending his exceptional computing expertise but also for various assistance he and his wife Christine provided to my family during our staying in East Lansing. Finally, I would like to express my heartfelt thanks to my wife, Winda. Without her endless support, unparalleled patience, understanding and love, I might never completed this endeavor. To our children, Gading and Kartika, thanks for being a constant source of strength and motivation. TABLE OF CONTENTS LIST OF TABLES ................................................................................................... LIST OF FIGURES .................................................................................................. Chapter One: INTRODUCTION ........................................................................... 1.1. General Background ........................................................................................ 1.2. Forest Management: Selected Concepts and Situation in Indonesia .................... 1.3. Study Objectives .............................................................................................. 1.4. Organization .................................................................................................... Chapter Two: TEAK F ORESTS IN INDONESIA ................................................. 2.1. General Overview ............................................................................................ 2.2. The System of Silviculture ............................................................................... 2.3. Forest Regulation ............................................................................................. 2.4. 2.3.1. 2.3.2. The Gecombineerde Vakwerk Methode (GVM) ............................... . The Burn's Method ........................................................................... Limitations of the GVM and the Bum's Method and Relations to the Hutan Tanaman Industri (1in Program .................................. Chapter Three: LITERATURE REVIEW ................................................................ 3.1. 3.2. 3.3. 3.4. Chapter Four: 4.1. 4.2. Current Approaches of Harvest Scheduling ...................................................... Harvest Scheduling: Decision Making Under Risk ............................................ Chance-Constrained Programmmg ................................................................... Growth and Yield Modeling ............................................................................. METHODS ..................................................................................... First Phase: Growth and Yield Modeling .......................................................... 4.1.1. 4.1.2. 4.1.3. 4.1.4. Growth and Yield Data ....................................................................... Model Forms ...................................................................................... Model Development ............................................................................ The Yield-Projection Computer Routine .............................................. Second Phase: Harvest Scheduling ................................................................... 4.2.1. 4.2.2. 4.2.3. 4.2.4. 4.2.5. The Forest District .............................................................................. The Harvest Scheduling Problem ......................................................... Model Outline ..................................................................................... Model Formulation ............................................................................. Model Solution ................................................................................... 23 23 29 34 37 42 42 46 52 54 54 55 59 60 62 65 vii Chapter Five: THE GROWTH AND YIELD MODELS AND 5.1. 5.2. 5.3. 5.4. THE YIELD-PROJECTION COMPUTER ROUTINE .................... Model Estimates .............................................................................................. Model Testing ................................................................................................. The Yield-Projection Computer Routine ........................................................... Additional Models ........................................................................................... Chapter Six: THE HARVEST SCHEDULING MODELS ................................... 6.1. 6.2. 6.3. 6.4. Model Components and Inputs ......................................................................... Linear Programming Harvest Scheduling Models ............................................. Chance-Constrained Programming Harvest Scheduling Models ........................ Discussion 6.4.1. The Direct Cost of the 80-Year Rotation Age ...................................... 6.4.2. The Effect of Non-declining Even Flow Constraints ............................. 6.4.3. The Effect of Incorporating Risk ......................................................... 6.4.4. Ending Age-Class Distributions .......................................................... . Chapter Seven: SUMMARY, CONCLUSIONS AND RECOMMENDATIONS ...... 7.1. 7.2. 7.3. Summary ........................................................................................................ Conclusions ..................................................................................................... Recommendations ............................................................................................ Appendix A. Growth and Yield Data and Yield Projections Table A. 1. Growth and yield data from permanent plots in Central and East Java .................................................................................................. Table A. 2. Yield projections of the Cepu Forest District; mean and variance of per-hectare yield of each stand-type under each management regime in each period throughout the planning horizon ................................................ . Appendix B. The Yield-Projection Computer Routine ............................................... Appendix C. The CCP SOLVER Spreadsheet .......................................................... LIST OF REFERENCES ......................................................................................... 66 66 72 76 79 82 82 90 93 104 105 107 108 l l l l 1 1 114 116 119 128 136 145 148 Table 5.1. 5.2. 5.3. 5.4. 5.5a. 5.5b. 5.5c. 5.5d. 5.6. 5.7. 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. LIST OF TABLES Estimation of the basal-area growth model ................................................... Estimation of the volume growth and yield model ......................................... Estimations of after-thinning basal—area and volume models .......................... Estimation of the stand height model ............................................................. A sample of results from re-estimating the basal-area growth model (Model 5.2) using random subsets of the data ................................................ A sample of results from re-estimating the volume growth and yield model (Model 5.3) using random subsets of the data ................................................ A sample of results from re-estimating after-thinning basal-area and afier~thinning volume models (Models 5.4 and 5.5) using random subsets of the data .................................................................................................... A sample of results from re-estimating the stand height model (Model 5.6) using random subsets of the data ................................................................... A sample of results fiom testing the compatibility of Models 5.3 and 5.4 ....... Estimations of the initial-B and initial-H models ............................................ Stand-type labels based on age and productivity classes and on existing stands .............................................................................................. Management regimes (105) resulting from the combination of 35 stand-types and 3 rotation ages ....................................................................................... Thinning and clearcutting sequences over the planning horizon under difl‘erent management regimes ....................................................................... Per-hectare total NPV produced with each management regime ..................... Timber prices and management costs at Cepu Forest District for the management year 1992/ 1993 ........................................................................ Summarized optimal solutions of LP harvest scheduling models .................... viii 67 67 68 69 73 74 74 75 76 81 83 84 85 87 88 91 ix 6.7. Summarized optimal solutions of CCP harvest scheduling models ................. 98 6.8. Optimal hectares allocation according to the LP 5 and CCP harvest scheduling models ........................................................................................ 102 Figure 2.1. 2.2. 4.1. 4.2. 4.3a. 4.3b. 4.4a. 4.4b. 4.4c. 4.4d. 4.5a. 4.5b. 4.6a. 4.6b. 4.7. 4.8. 5.1a. 5.1b. LIST OF FIGURES Geographical distribution of teak forests in Indonesia .................................... Age-class distribution of teak forests in Java ................................................. A general flow chart of this study ................................................................. Intermediate thinning yields (TY 1 ..TYn) and final-harvest (FH) to be predicted using the grth and yield model set ...................................... Distribution of permanent plots according to stand age at the first measurement ................................................................................................. Distribution of permanent plots according to site class .................................. Future basal area (B2) plotted against current ages (Al) ............................... Future basal area (BZ) plotted against future ages (A2) ................................. Future basal area (B2) plotted against current basal area (B1) ....................... Future volume (V2) plotted against future basal area (B2) ............................. Bale (the ratio between after- and before-thinning basal area) plotted against Na/Nb (the ratio between afier- and before-thinning number-of-trees per hectare) ......................................................................... Va/Vb (the ratio between afier- and before-thinning volume) plotted against Na/Nb (the ratio between afier- and before-thinning number-of-trees per hectare) ......................................................................... Stand height (H) plotted against stand basal area (B) ..................................... Stand height (H) plotted against number of trees per hectare (N) ................... Location of the Cepu Forest District ............................................................. Age-class distribution ofthe Cepu Forest District ......................................... Residual analysis of Model 5.2; scatter plots of lnB estimates (B: basal area) against residuals ................................................................... . Residual analysis of Model 5.3; scatter plots of anestimates (V: volume) against studentized residuals ...................................................... Page 10 43 44 45 46 47 47 48 48 50 50 51 52 56 57 70 70 5.1c. 5.1d. 5.16. 5.2. 6.1a. 6.1b. 6.2. 6.3a. 6.3b. 6.3c. 6.4a. 6.4b. 6.4c. 6.4d. Residual analysis of Model 5.4; scatter plots of Ba estimates (Ba: after-thinning basal area) against studentized residuals .......................... Residual analysis of Model 5.5; scatter plots of Va estimates (Va: alter-thinning volume) against studentized residuals ............................... Residual analysis of Model 5.6; scatter plots of hi! estimates (H: stand height) against studentized residuals ................................................ Flow chart of the yield-projection computer routine ....................................... Harvest flows with the absence of NDEF constraints (LP 1 and LP 2) ........... Harvest flows when NDEF constraints are included (LP 3, LP 3, LP 5) ......... The CCP SOLVER Spreadsheet ................................................................... Harvest-flow pattern if solutions of CCP l is implemented ............................ Harvest-flow pattern if solutions of CCP 2 is implemented ........................... . Harvest-flow pattern if solutions of CCP 3 is implemented ........................... . Age-class distribution after the end of the planning horizon if LP 5 is implemented ................................................................................. . Age-class distribution after the end of the planning horizon if CCP l is implemented ............................................................................... Age-class distribution after the end of the planning horizon if CCP 2 is implemented ................................................................................... Age-class distribution alter the end of the planning horizon if CCP 3 is implemented ................................................................................... 71 71 72 80 92 92 95 100 100 101 109 109 110 110 CHAPTER ONE: INTRODUCTION 1.1. General Background The forest products industry has emerged as a significant sector of the Indonesian economy. It has been consistently one of the country's leading export sectors, second only to petroleum. Export earnings from forest products were about 16% of the country's foreign exchange earning in 1992, and an average of about a half-million jobs were created annually during the 1984-1989 period (Ministry of Forestry/MOP 1989, 1993). In addition, the industry has also been instrumental in the socio-economic development of several regions of Indonesia. The industry's source of raw material is almost entirely Indonesian tropical rain forests. An outlook study indicates that, due to degradation of timber potential and losses of forest area to other land uses, Indonesia's annual timber production fi'om tropical rain forests is predicted to decline fi'om the current level of 33 million cubic meters to 25 million cubic meters by year 2000, and to only 21 million cubic meters by 2030 (MOF 1991). On the other hand, the total processing capacity of wood manufacturing plants has reached 45 million cubic meters per year (MOF 1989). Thus, there is an alarming possibility of a widening discrepancy between timber supply and the industry's raw material requirement. If the forest products industry is to remain a significant contributor to the national economy, timber shortages must be prevented. Given no alternative timber sources, a pragmatic solution of timber shortages would be an increased exploitation of the tropical rain forests. However, this is not a favorable solution for various reasons. It is well known that the alleged excessive exploitation of tropical rain forests in developing countries is a primary concern in the growing global environmental conservation movement. The most recent movement is the coo-labelling campaign which advocates boycotting any product for which production involves environmentally detrimental processes. Being a country which extensively utilizes its tropical rain forests, Indonesia has been frequently a major target of criticism. Any increase in the exploitation of the tropical rain forests inevitably will further 2 undermine Indonesia's credibility and exacerbate the situation. A major implication is that, increased exploitation eventually will create a hot-bed for marketing of Indonesian forest products, which can be very damaging to the forest products industry‘. Recognizing this potentially impending situation, the MOF has launched a program for establishing large-scale industrial timber plantations called the HTI (Hutan Tanaman Industri) program. The ultimate goal is to create alternative timber sources and eventually reverse the present situation. A major portion of the national timber supply would be harvested from sustainably and economically managed timber plantations, hence alleviating the pressure on the tropical rain forests. Several timber plantations have been and are being established by state corporations under the MOF. By creating a favorable investment atmosphere, private companies (especially wood manufacturing enterprises) are expected to be the major participants. By 1994, 1.5 million hectares of timber plantations should have been established with a 1999 target of 4.4 million hectares (MOF, 1993). A likely obstacle to the long-term goal of the HTI program is the lack of forest management techniques and instruments necessary for bringing the plantations into sustainable and profitable production. At present, forest plantation management in Indonesia still embraces a neoclassical normal-forest oriented technique which has some fundamental limitations. This particular technique (discussed in detail in Chapter Two) is not suflicient for attaining modern forest management objectives such as sustainably maximizing profits, and therefore, is not suitable for the management of HT] plantations. Moreover, no study on assessing the applicability of modern forest management techniques has been undertaken. The lack of reliable management techniques is addressed in this study. Specifically, the purpose of this study is to contribute to the accomplishment of the HTI program by providing a prototype of a quantitative management technique appropriate for the management to large-scale 1A fresh illustration is a recent advertisement "incident" that occurred in London. Environmental groups including the Greenpeace and Down-to-Earth organimtions successfully demanded that the Independent Television Commission (ITC) blackout an advertisement by the Indonesian Forestry Community (Anonymous 1994). timber plantations. Beyond supporting the HTI program, this study is a significant breakthrough toward improving forest management in Indonesia in general. As noted by Suryohadikusumo (1992), the current Minister of Forestry, Indonesia urgently needs to adopt more advanced forest management techniques, especially techniques for forest resource planning, in order to increase and sustain the country's gain from its forest resources. 1.2. Forest Management: Selected Concepts and Situation in Indonesia The focus of this study is on devising a package of mathematical models for the management of large-scale timber plantations in Indonesia. Forest management, however, has many facets, and different kinds of mathematical models are needed for each facet. To outline the facets covered in this study and to specify the type of mathematical models developed, a review is necessary of some relevant forest management concepts and the related prevailing situation in Indonesia. The Society of American Foresters ( 195 8) defines forest management as "[t]he application of business methods and technical forestry principles to the operations of a forest property." Within this broad definition, forest management encompasses virtually all activities involved in the process of producing goods and services from a forest land. This study adopts a more recent definition which restricts forest management to the decision-making aspect of the entire process; i.e., forest management is "... the study and application of analytical techniques to aid in choosing those management alternatives that contribute most to organizational objectives" (Leuschner 1990). Forests may be managed for multiple objectives but frequently timber production is the primary objective. With regard to timber production, many forests are managed to achieve and maintain some form of a sustained yield condition. The two common managerial interpretations of this condition are either a "long-term sustained-yield" or alternatively a "non-declining even flow" of yield (Leuschner 1990). Long-term sustained-yield refers to a level of annual or periodic timber production that a particular forest can produce perpetually under a certain management intensity. A non-declining even flow (NDEF) condition is achieved when timber production in any subsequent years or periods is continuously maintained to be at least equal to previous volumes. Forest management can be stand-level or forest-level (Clutter et al. 1983, Leuschner 1990). With stand-level management, stands comprising a forest are treated as independent management units, and the overall management objective is attained by optimally managing each individual stand. Conversely, forest-level management considers the entire forest as a single entity, and the management of each individual stand is coordinated to attain the overall management objective. Stand-level management theoretically should lead to the highest total production or revenue because the overall output is the sum of the maximum outputs of every individual stand. However, since individual stands are managed independently, it is dificult or often impossible to attain and maintain any of the sustained yield conditions mentioned earlier. In contrast, forest-level management controls the flow of production over time; imposing this condition often requires a portion of the forest not to be managed under the most efiicient management strategy, resulting in lower total production. In Indonesia, the General Forestry Plan (MOF 1986) implies that all forests should be managed under the principle of maximum and sustained yield. Thus, it is required by law that the management objectives of any forest must include attaining and maintaining some form of a sustained yield condition. As a result, all forests in Indonesia are managed with the forest-level approach. The economic trade-ofi‘ of this approach, which may be substantial, is ofien tolerated due to the necessity of maintaining relatively stable timber production and of continuously creating job opportunities. The core of forest-level management is harvest scheduling: determining the portions of the forest to be harvested in spatial and temporal context in order to attain overall management objectives. Accordingly, a component of the package of mathematical models developed in this study is a set of harvest scheduling models. Main inputs in harvest scheduling are projections of timber yields per unit area under different management options throughout the planning horizon. An appropriate instrument for generating these inputs is a set of mathematical growth and yield models. In Indonesia, yield projection instruments currently available for some selected species are conventional normal or empirical yield tables which are not adequate for mathematical harvest scheduling purposes. Therefore, mathematical growth and yield models constitute the other component of the package developed in this study. At the present, forest plantations in Indonesia are dominated by teak plantations in Java; they make up approximately 40% of the total existing forest plantations (Ingram et a1. 1989). As a result, this study focuses on this species and the harvest scheduling models are developed for a selected teak forest district in Central Java. Nonetheless, the general modeling fi'amework is intended to be a prototype for developing similar packages of mathematical models for other species in different regions of Indonesia. Another reason for focusing on teak plantations in Java is to examine the limitations of the conventional management technique currently applied to these plantations. Management of these plantations has been generally considered a success and more importantly, may be proposed as a model for forest plantation management in Indonesia. By examining the limitations of the prevailing technique and drawing comparisons to a more modern alternative technique, this study provides important information for justifying whether teak plantation management in Java is sufficient as a model for forest plantation management in Indonesia. On the other hand, this study will also determine ifthe management ofthe teak plantations themselves need improvement. 1.3. Study Objectives This study focuses on achieving the following objectives: 1. To develop a set of growth and yield models for teak plantations in Indonesia and to integrate the resulting models into a computer routine that can be used to generate infomiation necessary for forest management planning, particularly harvest scheduling; 2. To develop mathematical harvest scheduling models for a selected teak forest district in Java, which maximize total net present value (NPV) and ensure a non-declining even flow (NDEF) condition over a specified planning horizon; and 6 3. To examine the limitations of the forest management (harvest scheduling) technique currently applied to teak plantations in Java. An important specification of harvest scheduling models involves the treatment of risk caused by non-deterministic model inputs. Based on how these non-deterministic inputs are treated, harvest scheduling models fall into two broad categories: (1) deterministic or excluding risk, and (2) non-deterministic or including risk. Sources of non-deterministic inputs include: natural hazards due to fire, insects or diseases; unpredictable behavior of prices and costs; and errors in yield projections. In the context of teak forests in Java, another source is timber theft. One of the harvest scheduling models (Objective 2) is devised to include risk due to errors contained in timber yield predictions resulting from spatial and temporal aggregations. Hence, a final objective of this study is: 4. To examine the effect of incorporating risk due to non-deterministic timber yield projections on harvest scheduling outputs. Risk due to other sources are excluded for a combination of reasons, including data unavailability, historical observations (e.g., relatively constant prices and costs), and inherently stable biological characteristics of teak plantations (e.g., low susceptibility to fire hazards). 1.4. Organization Chapter Two presents an extended description of teak plantations in Indonesia including a brief overview, the silvicultural system, and the standard management. A literature review on the subjects of harvest scheduling and growth and yield modeling is presented in Chapter Three. Information in these two chapters provides the basis for devising the modeling framework and methods described in Chapter Four. Chapter Five and Chapter Six present the resulting growth and yield models and harvest scheduling models, respectively. Finally a summary, conclusions and recommendations are offered in Chapter Seven. CHAPTER TWO: TEAK FORESTS IN JAVA Some relevant aspects of teak forests and their management in Java are provided in this chapter. A general overview is provided in the first section. The second section briefly describes the silvicultural management of the plantations, followed by detailed descriptions of two forest regulation techniques in the third section. Some limitations of these two techniques and a brief discussion relating teak forest management in Java to the HTI Program is given in the last section. 2.1. A General Overview ‘ Teak (T ectona grandis, Linn F.) is one of the most valuable tree species in Indonesia. Combining superb physical and mechanical properties with a beautiful appearance, teak wood is an excellent raw material for a wide range of wood products, from filrniture and housing components to wood carving and household instnlments. In the past, when teak wood was relatively inexpensive and abundant, it was used for building ships (Peluso 1992). Teak has been considered indigenous to Indonesia, but some believe it was brought fi'om India centuries ago (Gyi 1992). At the present, teak forests in Indonesia are mainly monoculture plantations covering about one million hectares almost entirely located in Central Java and East Java provinces. Less extensive teak forests are also found on Lombok, an island in the West Nusa Tenggara Province, and on Muna, an island in the South Sulawesi Province (Hammh 1975). This geographical distribution is shown in Figure 2.1. Java's teak forests have been exploited for centuries. Large-scale exploitation first took place in the early decades of the lath century following the arrival of the voc or the Dutch East Indian Company (Kartasubrata 1992, Peluso 1992). Planned management, however, was not initiated until 1855 when several professional German foresters were hired to prepare management plans for some forest districts in Central Java. These German foresters introduced the concepts of sustainability and the normal-forest, upon which the management of the teak forests has been based 7 A32 E33435 EEua use So— ".02 .33 gene”: ”ES..— 3389 38:8:— 5 23.8 «38.3 cos—553v 36293300 .~.N 2me 11' III «sum-ta G H 82¢. 633. 0 8:6 8.8:. oEom . 7. x .\l :4 I.; 111111 .111111111 \ x M 3. . ..t/ 3:2.»— -..-..\ \x 1.55:.— . ql/ \. \\ .\\. \-.-.ll ..l Ii r~l\\...t. 7: . . {><—e /. ..ttJ Cil- - -31. x; ..5 .. , . .1. ..C 9.1. _ ..u M: -.7. {”...-l-.. -. it \ s 7 A: _ . .3..— __ .72.... .H» .. ,- Wi\ ‘\ 0 =3: Gar-IV.“ v .UHHHV ll rev; m a a a~.....a=.m...un :21/ Anne xmv. 1.3a. . . . . .- . . . .o/t $2. 72:: A... . _. .,.._.__3 7.! 37.73 eye : 7/11/ A. a w u- a V 9:65:95: 77/, {III/-7 .. \\-.. «I .7 t... m7 .1K ... a, 2 7.. x.- \i/ V Pf /,7H-.‘l{‘-_uw Flirt-J ..g M. .a t. . f...“ , 1. . .\ _ x) A» (gt—\<2:m 7022.12. \..\ f. ..{7 -U . 7.3-17 7, 3 a .2 (.-i, s. ..n 7,? / .-.-// an- ..xii/ 1 3 J ,3 l, M -. Km» 7 //.r\\ {itIHa (If. , ... fll/lH. st .. .t.i. 1 \...\7 s... /.\rl J / .1 Ar 2... rmw. 7- 2- \_ 7. x} / 4.. ..V r:/ ...t. .7, .9 P7.-- . r .\ Eggs—4...... // u x / -,. r. . 7. .M L. \ . . . .. A “ / fl- , . .. r y (M. / v .. .. .\.l.l. r \ r ..I. {it i/ f\\... r. .\ {:1}- k a ../u .. \\--/,. -.t\t / \. 4M C i/nma/ / /.//.v . again» 7. . . 9 ~\ 7 . I a . . 7 5:23.233. .7; /, 7,. / I u /o / I . a all .7, \ 532 o f/ . J a rrrrrr . // / . / Z 9 ever since. Historically teak forests in Java have always been managed by the state, either directly or by a state-owned enterprise. Today, the teak forests are managed by Perhutani, an autonomous state corporation which also manages some two million hectares of non-teak forests in Java. The entire forest under the responsibility of Perhutani is divided into three regional units, namely, Unit I Central Java, Unit 11 East Java, and Unit HI West Java. Each regional unit is further divided into forest districts of varying size (30,000 - 100,000 hectares). In total there are 51 forest districts, about halfof which are exclusively teak forest districts. Forest districts are self- contained management units operating on individual long-term management plans prepared by regional planning ofices. Only in certain aspects such as international marketing are forest districts centrally coordinated. For planning purposes, forests under Perhutani's control are divided into sustainability units. A sustainability unit is an area between 4,000 - 6,000 hectares, usually confined within natural boundaries, for which a long-term sustainable management plan is devised. A forest district may be constituted by a number of sustainability units, and therefore a forest- district management plan is usually an integration of several sustainability-unit management plans. The management goal, as mandated by law, is to produce goods and services for the people and support government programs in socio-economic development. Perhutani's general management plan (Perum Perhutani 1990) implies that in fulfilling this mandate Perhutani should follow the following strategies: (1) to apply economically sound management, (2) to maintain the sustainability of the forests, and (3) to participate in the effort of alleviating the poor socio-economic condition of the surrounding communities (forest villages). At the operational (forest-district) level, the first two strategies are interpreted as a set of management objectives of (l) attaining the maximum profits and (2) maintaining relatively stable timber production over time. Java is one of the most over-populated regions in the world. A chronic problem afl‘ecting the socio-economic condition of Java's communities, among others, is the persistently high 10 unemployment rate. Therefore, Perhutani's implementation of the third strategy has been adopting labor-intensive methods at each level of management, hence creating job opportunities for a large segment of the surrounding rural populations which are mostly poor subsistence farmers. Despite the conflicting objectives (attaining profits versus maintaining a stable timber production and creating jobs), Perhutani has been financially healthy for decades. In 1986, before- tax total profit reached 22 billion rupiahsl (Perum Perhutani 1990), while employment totaled 260,000 (UGM 1990). A relevant question is whether Perhutani has been gaining profits in a sustainable fashion. In view of the currently practiced management approach (discussed in the last section of this chapter), the answer to this question may be negative. Perhutani's long-term general plan sets the annual harvest for the period of 1989 - 2008 at 5,000 hectares or 575,000 cubic meters (Perum Perhutani 1990). Meanwhile, the current age-class distribution of the forests is heavily skewed toward younger age-classes (Figure 2.2). Under the current single rotation-age management approach, attaining and maintaining the targeted annual harvest volume can not be continued and subsequently will require harvesting younger age-classes. In other words, current management may not be sustainable. 1000 hectares I m v vn IX XI MT Age-class (10 years) Note: age classes are in lO-year increment (e.g., age-class I covers ages 1 - 10, age class 11 covers ages 11 - 20 and so forth), MT: over mature stands. Figure 2. 2. Age-class distribution of teak forests in Java. ‘ The current exchange rate is approximately $1.00 = Rp 2,100.00. In 1986 the rate was approximately $1.00 = Rp 1,600.00. 11 2.2. The System of Silviculture The system of silviculture applied to teak forests in Java is characterized by artificial reforestation, a series of thinnings, and clearcutting. This section provides brief descriptions of these silvicultural activities. A great portion of the information is obtained from Gadjah Mada University or UGM (1990), Sabamurdin (1989), and Simon (1993). 2.2.1. Artificial Reforestation Several different reforestation approaches have been devised. Three main approaches are: the komplangan system, the voorbow system, and the tumpang-san‘ system (UGM 1990). These systems are similar in the sense that they are all labor intensive and relatively low cost, employing farmers from the surrounding forest villages virtually without any monetary compensation. Briefly these systems can be summarized as follow: Komplangan system: Each farmer is given two separate parcels of land. On one parcel (usually the more fertile) the farmer is permitted to cultivate food crops, while on the other parcel the farmer must plant teak trees. Voorbow system: Each farmer is given one parcel of land on which he/she is permitted to plant food crops in the first year but must plant teak trees in the second year. T umpang—sari system: A farmer is given a parcel of land for a specified time period (between 2- 4 years). The farmer is permitted to plant selected food crops throughout this period between rows of teak plants. The farmer is responsible for taking care of the young teak trees during the period. Of these three approaches, tumpang-sari has been proven to be the most successful system and is currently the standard reforestation system. Understandably, the komplangan system is less successful because farmers must pay attention to two separate parcels, and it is very likely they tend to pay more attention to the one with food crops. In the case of the voorbow system, it is irrational to expect farmers to spend a great amount of time in taking care of the teak plantation during the second year. 12 Tumpang-safi was first proposed in 1873 and allegedly was imported fi'om Myanmar or Thailand. Technically tumpang—san‘ is carried out as follows. After land is cleared, teak seeds are planted on 3xlm or 2x1m spacing. Seeds of a leguminous species (Leucaena Ieucocephala) are spread in rows between the rows of teak seeds. This legume is needed for maintaining the soil nitrogen content and reducing soil erosion. The next step is planting hedge plants, usually a prickly shrub species (e.g., Samanea sapan), around the teak plantation for protection from animals. This is followed by planting a row of non—teak species surrounding the teak plantation inside the hedge. The purpose of planting this non-teak species is not very clear. According to some Perhutani officials, it is for producing wood needed for temporary construction in the forest. Farmers cultivate their food crops between these activities. The whole period is 29 month, during which there are usually four to five rotations of food crops (Sabamurdin 1988). During the 29-month period the farmers are also responsible for blank-filling (replanting the seeds that did not grow using either seeds or seedlings), pruning the legume, and tending the young teak trees. Obviously tumpang-sari is very desirable both from cost efficiency and output quality standpoints. For a minimal cost, Perhutani establishes forest plantations on bare lands within a period which can be as short as three years after clearcuttings. Perhutani used to be only responsible for providing the seeds of teak, legume intercrops, hedge plants and the seedlings of non-teak species. Since 1974, as a policy to improve fanners' income, Perhutani also provides seeds of agricultural crops (of superior varieties), fertilizers, and pesticides. T umpang—san’ teak plantations are generally well tended because farmers always tend their food crops. Hutabarat (1990) carried out benefit-cost analyses of tumpang-san’ using a teak forest district (coincidentally Cepu Forest District) as a case study. Actually he formalized something that was not surprising: tumpang-sari is beneficial to Perhutani. T umpang-sarr' is also beneficial to farmers; any job that creates non-negative incomes should be beneficial to unemployed landless farmers having virtually no other chance of getting alternative jobs. However, Hutabarat found that the pre-1974 form of the lumpang—san' was not beneficial to farmers. This conclusion may be 13 due to the assumption that farmers may get alternative jobs, which is questionable. From Perhutani's standpoint, both types of Iumpang-san' were found to be beneficial. Originally the size of the parcel allotted to a farmer was 0.5 hectare, which was considered suficient for supporting the farmer’s subsistence living. But due to the rapid increase in rural population, this size has been reduced repeatedly. Today, a farmer would be very lucky if he/she can get more than 0.25 hectare (Simon 1993). This indicates that, as long as there are population pressures and poor landless farmers, Perhutani will always enjoy a labor surplus and be able to cut costs and create employment at the same time. 2.2.2. Thinnings Teak stands are thinned beginning at age 5 years, and thinned every 5 years until 10 years before the clearcut (Perhutani 1993). The main purpose of thinnings is to improve the quality of the stands, and thereby the value of the final timber harvest. Nonetheless, thinnings are also an important source of intermediate revenues. Thinning intensity is based on a density measure as represented by the following relationship between the average distance between trees and the average height of the dominant stand canopy: s=-,";; a: 72—19—9— (2.1) As in which S = relative spacing, H = dominant height, and N = number of trees (per hectare). This relationship, which is credited to Hart (1928) who undertook a thinning experiment on teak plantations in Java, is what today known as the relative spacing or spacing index (see Clutter et al. 1983). From this relationship, Perhutani derived and tabulated the standard after- thinning per-hectare number of trees for each site class at any age. This table is used in practice as the thinning manual, from which the number of trees that must be left in any thinning is obtained. 14 The compulsory S-year thinnings makes Perhutani's thinning scheme very rigid. Setyarso (1985) suggests that Perhutani should consider renovating its current thinning method. 2.2.3. Clearcuttings Final harvests are carried out through clearcutting. Before clearcuts, trees are girdled at least two years in advance. The purpose is to reduce timber damage in the felling process; teak trunks often split or break when out in a fresh condition. Another advantage of girdling is the reduction in water content of the trees, hence reducing the transportation costs. Like other activities, clearcutting is carried out in a very labor intensive fashion excluding any form of modern mechanization. Trees are felled and cut into logs using hand saws, hauled using cows, and loaded onto trucks by humans. Every single piece of each log is documented systematically, making easy tracing of any lost piece. This also aids in recognizing logs coming from illegal cuttings. 2.3. Forest Regulation Management of the teak forests in Java has always been based on the conventional wisdom that maximum and sustained yield is a product of a normal or fully regulated forest. Two normal- forest oriented forest regulation methods have been subsequently implemented. The first method is called the gecombineerde vakwerk methode (GVM) which was formally issued in 1938, and was the compulsory forest regulation technique until it was replaced in 1974. The newer method, which is still in use today, is called the Bum's method. A further elaboration on these two methods is necessary in order to examine their limitations, hence providing a justification for examining a more modern quantitative technique. Information presented in this section is based on a critical review on the GVM by Hardjosoediro (1973) and the operational manuals of the two methods, respectively (Anonymous, no year; and the Directorate General of Forestry 1974). 15 2.3.1. The Gecombineerde Vakwerk Methode (GVM) At the time GVM was devised, natural stands constituted the majority of teak forests in Java. GVM was designed primarily for converting these forests into fully-regulated forests. Hardjosoediro (1973) indicates that conceptually GVM is a hybrid of the classical area—based and volume-based periodic block methods and the Von Mantel technique. However, the relation to those classical methods is truly limited only to the determination of the area and volume allowable cuts in the early stage of the procedure. The relation with the Von Mantel technique is perhaps in terms that yield is estimated at the rotation age. The following summary is intended to provide a better understanding of this particular forest regulation technique. All formulas are prepared by the author based on their corresponding descriptive explanations in the manual. (1) Predicting the potential volume of the forest GVM starts with determining the budget volume of the forest, that is, the total volume at the rotation age. This quantity is computed using: TV = 21y, (2.2) i =1 where TV = total volume or budget volume (in cubic meters) for n age-classes I, = total hectares of age-class t v, = per-hectare volume of age-class 1' at the rotation-age. Volumes of plantation stands are derived from a normal table. For natural stands, the volumes are computed using: 16 V = (Boniteitf xdn x f (2.3) where V = stand volume (in cubic meters) Boniteit = site class (not site index; 1, 1.5, 2, ...6) derived from an age-height graph dn = a ratio indicating the closeness of the stand to the normal stand f = factor of exploitation, usually between 6 - 10, dependent upon the topographical condition of the forest, harvesting technique utilized, type of product produced, etc. (2) Determining the annual allowable cut (AAC) The AAC is computed both in terms of area and volume. Respectively, the two AACs are obtained by dividing the total area and the total volume of the forest (TV) by the rotation age. (3) Determining the conversion period The conversion period is the number of years required to convert the unregulated (natural- forest) portion of the forest into plantations. It is determined as the average of the area-based and volume-based conversion periods. The area-based conversion period is the total area of the natural forest divided by the area AAC. Likewise, the volume-based conversion period is the total volume of the natural forest divided by the volume AAC. Arithmetically, this is given by: CP=%(.-f-2,+.-§2,J (14> where CP = conversion period Ln, =total areaofthe natural forest V = total volume of the natural forest nr AACL andAACV = area-based and volume-based annual allowable cuts. 17 (4) lie-determining the AAC. Given the length of the conversion period, the next step is re-determining the area and volume AACs for the first period (first ten years). This time these AACs are obtained respectively by dividing the total area and volume of the natural forest by the length of the conversion period, i.e., AACLzé"; and AACyzg’) (2.5) (5) Allocating hectares and volume to each period Finally, hectares of each age class are allocated to the 10-year periods throughout the rotation. It is unclear how this part should be done quantitatively. It seems that fiom this point forest regulation is treated as an art instead of a quantitative planning process. According to the example in the manual, total hectares allocated to each period are intended to be as close as possible to the area AAC of the first period, and so is the corresponding total volume. This implies that the first period AACs are applied throughout the rotation, which is difficult to justify because those AACs were computed only for a portion (the natural forest) of the forest and were based on the conversion period. Hardjosoediro (1973) perceives this is a fundamental mistake with an implication endangering a forest's sustainability. After about a half century of implementation, it was concluded that the GVM was no longer appropriate. Several forests districts were unable to maintain a perpetual timber production level and the targeted fully-regulated forests were far from reality. Massive over-cuttings during the Japanese occupation in the Second World War, extensive forest destruction during the subsequent revolution period, and relentless timber theft lmve been blamed as the main causes of the failure. To correct this situation, a new technique called the Bum's Method was devised to replace the GVM. Conceptually this new technique is very similar to the former. The major difl‘erence is that it allows harvesting below the standard rotation age depending upon the current 18 age distribution of the forest. As a result, the new technique may be an appropriate remedy in the short-run, but is hardly a long-run solution to the problems faced. 2.3.2. The Bum's Method This new technique is almost identical to the GVM. A few notable difi'erences include: (1) The total volume used in determining the volume AAC; it is not the total volume at the rotation-age, but rather the total volume at a forest specific mean-cutting-age or MCA. (2) As an implication of (l), the cutting-ages of each age class may vary, either below or above the standard rotation-age depending on the forest age-class distribution. (3) The technique involves a procedure called cutting-time testing, which is basically examining whether a resulting AAC will insure a perpetual harvest, and adjusting it if it does not. In general, the procedure involved can be summarized as follow: ( 1) Determining the MCA The MCA of a given forest is defined as the weighted-average age of the forest plus one- halfof the standard rotation age. For a forest with k age classes, MCA is computed using the following formula: 1* 2 L. A. MCA = ‘i'—— + 0.5R (2.6) X Li i=1 where, L, = total hectares of age-class i A , = mid-age of age-class i (for example 15 is the mid-age of age-class 2) R = standard rotation-age which is usually 70 or 80 years. l9 (2) Determining the budget volume and the MO: The forest budget volume is computed using formula 2.2, but v, is the projected volume of age-class i at the MCA instead of the rotation age. The forest volume AAC is computed using this budget volume. Area AAC is computed as usual. (3) Testing the volume AAC The main objective of the Bum's method is to ensure a perpetual harvest. Accordingly, after defining the area and volume AACs, the Bum's method proceeds with a lengthy procedure called cutting-time testing. This is basically computing the number of years required to harvest the entire forest if the AAC is applied. The computation is carried out for one age-class at a time starting from the oldest age-class. The manual explains this procedure using an example, which can be represented in the following series of formulas: Y=gx y. = V, (2.7) ' AAC V where Y = total years required to harvest the entire forest y, = total years required to harvest age-class i V, = projected volume of age-class i at MCA, MCA, is the mean-cutting-age of age-class i which is not necessarily the same with the forest MCA obtained in step (1). MCA, is the midpoint of age-class i plus the total y, of all older age-classes. If the resulting Y is equal to the standard rotation-age (R), the volume AAC is considered to be correct and will ensure a perpetual harvest throughout the rotation. Othemise, if Y differs fi'om R , the AAC is adjusted by the quantity (Y/R) and the entire process of cutting-time testing is repeated. The test is conducted until (Y = R) is attained. 20 (4) Allocating hectares and volume to each period This step is carried out in the same fashion as of that in the GVM. However, the budget volume to be allocated is no longer the budget volume obtained in step (2). Rather, it is the sum of all v, obtained during the process of cutting-time testing; recall that during cutting-time testing the MCA, of individual age-classes are adjusted. 2.4. Limitations of the GVM and the Bum's Method and Relations to the Hutan Tanaman Industri (HTI) Program 2.4.1. Limitations Both the GVM and the Bum's Method inherit the benefits and limitations associated with the concept of a normal or fully regulated forest. Early day's texts (for example Roth (1925) as cited in Davis and Johnson (1987)) list several potential benefits of a regulated forest. One of the most important, and perhaps still relevant benefits, is a stable annual or periodical harvest (in terms of volume, size, quality, and value). On the other hand, there are some fundamental inadequacies associated with any forest management techniques based upon the normal-forest concept. First, these techniques generally assume the existence of an ideal normal forest and attaining such a normal or near normal forest is usually the primary management objective (Ware and Clutter 1971). By definition, a forest is normal if it has and maintains a normal increment, normal growing stock level, and normal age-class distribution (Leuschner 1990). A normal incremmt is the maximum increment produced by a given species on a particular site. Since it is achieved when the forest is fully-stocked, a normal increment also implies a normal growing-stock. A normal age-class distribution exists when the forest area, adjusted for differences in site productivity, is equally distributed across age-classes. Since a forest with all normality conditions satisfied virtually does not exist, a normal forest is almost purely conceptual. A fully regulated forest is not necessarily a normal forest. It is defined as one that produces an equal level of production perpetually. Unlike a normal forest, a fully regulated forest is theoretically achievable. However, as noted by Thompson (1966), creating and maintaining a 21 regulated forest incurs several costs: (1) the opportunity cost due to delayed harvests and/or leaving land idle during the conversion period, (2) the cost due to the inflexibility of assuming that the currently determined optimal parameters (e.g., rotation age) will remain optimal in the future, and (3) the opportunity cost represented by more attractive investment alternatives which are not considered because the regulated-forest is considered an end in itself. Thus, in justifying whether attaining a regulated-forest is an economically sound management approach, these costs should be weighed against the potential benefits along with the owner's objective, constraints, and assumptions. Moreover, the value of a regulated forest either as an end in itself or a means to an end has been questioned in recent decades. Beuter (1982) effectively elaborated on the subject and contended that a regulated forest is of "questionable value" as an end in itself and is not very useful as a means to an end. Clutter et al. (1983) were even more lucid in asserting that achieving a static balanced fully-regulated forest should no longer be part of the goal of today's forest management, because "... the real role of the manager is the intelligent management of imbalanced forest structures." These statements imply that a fully regulated forest is no longer a necessary condition for achieving modern forest management objectives such as maximizing profits over a given time period or in perpetuity. Moreover, attaining a regulated-forest is not necessary as an explicit goal of forest management because, as indicated by Beuter (1982), intended or not intended, a regulated forest will materialize in the long run. It is a by-product of long-term forestry planning because of economic and institutional constraints and assumptions imposed in the planning process. In addition to these limitations, both the GVM and the Bum's Method involve some data manipulations without any clear nor discernible rationale (e.g., the calculation of the mean cutting age). Moreover, in spite of the rigorous cutting-time testing, there is no guarantee that the Bum's method will ensure a stable harvest flow. It remains very possible that, in order to avoid any decline in harvest volume, the total hectares harvested in any given year or period may exceed the previously calculated AAC. Once this takes place, it starts a chain effect over the subsequent periods and the expected fully-regulated forest may never materialize. 22 Neither one of the techniques ensures the attainment of the maximum total revenues. The use of a single rotation-age is by no mean a revenue-maximizing strategy. It may maximize the total harvest volume if the standard rotation is a volume-maximizing rotation. However, maximizing the total volume and maximizing the total revenue only coincide when the time-value of money is ignored. Thus in general, the Bum's method does not fully accommodate the current management objectives of maximizing revenues and maintaining a relatively constant annual timber production. 2.4.2. Relations to the HTT Program The fundamental limitations inherent in the Bum's method receives little notice because, as mentioned earlier, Perhutani has been gaining profits for decades. However, a deeper observation would reveal at least two non-ordinary conditions which may have enabled Perhutani to gain substantial profits regardless of the shortcomings in its management approach. First, for decades Perhutani had the luxury of harvesting high quality old-growth forests and second, the labor cost in Java, especially in teak forests regions, has been unusually low. These two advantages, combined with the constantly high price of teak wood, undoubtedly have contributed significantly to Perhutani's financial profits. Nonetheless some proponents (e.g., Sumitro 1992 and Iskandar 1992), implicitly suggest that Perhutani's financial success and the relatively constant or even slightly increased total area of the teak forests reflect the overall accomplishment of the management approach. Sumitro further implies that this exemplary accomplishment warrants recommending Perhutani's management approach as a model for forest plantation management in Indonesia, which in general also includes the HTI industrial timber plantations. While Sumitro's recommendation is subject to criticism, until another alternative is made available it is very likely that the industrial timber plantations established under the HT'I program will be managed similarly to the teak forests in Java. CHAPTER THREE: LITERATURE REVIEW This chapter reviews literature on: (1) harvest scheduling and (2) growth and yield modeling. Selected articles dealing with modern approaches of harvest scheduling, (i.e., the application of operation research techniques) are reviewed in the first section. The second section highlights studies proposing techniques for incorporating risk in harvest scheduling models. Chance-constrained programming, a particular technique of optimization under risk, is the subject in the third section. Finally, articles dealing with concepts and techniques of growth and yield modeling are discussed in the last section. 3.1. Current Approaches of Harvest Scheduling In modern forest management, harvest scheduling using operation research techniques replaces the classical and neoclassical forest regulation approaches. One of the most widely used techniques is linear programming (LP). The works by Curtis (1962), Loucks (1964), Kidd et al. (1966), Liittschwager and Tcheng (1966), Nautiyal and Pearse (1967), Paine (1966), and Ware and Clutter (1971) are among early studies pioneering the application of LP for harvest scheduling. Numerous LP-based harvest scheduling software have been developed. The most well known include Max-Million (Clutter 1968), Timber-RAM (Navon 1971), and FORPLAN (Johnson et al. 1986). In addition, there are also several lesser known personal computer LP-based harvest scheduling programs, such as TTMPRO-FORMAN (Hendricks and Harrison 1987), and the spreadsheet-based FORSOM (Leefers and Robinson 1990). With mathematieal programming, harvest scheduling problems are treated as constrained optimization problems. In general, the forest's utility to the owner is maximized subject to various constraints. Given the relatively long time-span in forest management, a common measure of utility is the net present value (NPV) of the forest. Considerations restraining the maximization of 23 24 the NPV usually include those reflecting management policies and requirements such as a required level of periodic harvest volumes, maximum hectares harvested, and the like. Several advantages are afforded fi'om handling harvest scheduling problems in a mathematical programming fashion. Selected advantages are listed below. 1. It insures that the resulting harvest schedule is the optimal strategy for attaining the management objective under the imposed constraints. 2. Because the modeling fiamework involves a critical assessment of the management objectives, identification of alternatives, and specification of the scope and limitation within which the solution holds, mathematical programming helps insure that the right problem is being solved (Rustagi 1976). 3. Mathematical programming provides features for examining the effects of changes in inputs or management constraints, and models can be adjusted as new information becomes available (Rustagi 1976). 4. Mathematical programming enables attaining the benefits of fully-regulated forest (e.g., a a stable timber production over time) without the necessity of forcing the forest to form a specific age-structure (Hoganson and McDill 1993). Apart fi'om these advantages, Chappelle (1977) observed some practical limitations associated with the application of LP to forestry planning in general. The first limitation is the high data requirement (in terms of quantity and quality) which usually entails numerous assumptions. The second limitation has to do with the necessity of relying on spatial and temporal aggregations in order to maintain a manageable model size, and high level aggregations usually bring about aggregation errors. Furthermore, Chappelle considers the requirement of specifying an objective functionasbothanadvantageanddisadvantage. Itisanadvantageinthesensethatthe requirement forces planners to assess thoroughly the owners' or decision makers' management objectives. However, it is a disadvantage when the forest is managed for multiple objectives. Treating one of the management objectives as an objective fimction implies putting less weight to the other objectives which are represented as constraints. 25 The high-data requirement is factually true, but it should not greatly hinder the applicability of LP, primarily because mathematical programming forestry planning models can be improved incrementally. In terms of model size, to some extent it can be technically overcome through the application of the decomposition method used by Liittschwager and Tcheng (1966) and more recently, by Berck and Bible (1984) and Hoganson and Rose (1984). With regard to the last objection, Rustagi's (1976) notions regarding three alternatives of how a multiple-use resource may be mamged perhaps give some clarification. At one extreme, a potentially multiple-use resource may be managed for producing a single output disregarding the other uses. At the other extreme, the resource's multiple uses are literally exploited for producing several outputs without a single output being dominant. In this case, the single-objective limitation noted by Chappelle hampers the applicability of LP. However, it is more common that a multiple-use resource is managed to produce one primary output and several secondary outputs. This justifies putting one management objective as an objective function and treating the others as constraints. Nonetheless, LP's inadequacy in accommodating multiple objectives has led to the assessment of the applicability of multiple-objective programming (De Kluyver et a1. 1980, Mendoza et al. 1987, and Mendoza 1988) and goal programming (Kao and Brodie 1979, Field et al. 1980, and Hotvedt et al. 1982). De Kluyver et al. (1980) used a multiple objective linear programming (MOLP) to formulate an optimal multicriteria harvest scheduling. Likewise, Kao and Brodie (1979) proposed the use of goal programming (GP) for reconciling incommensurate objectives in harvest scheduling (i.e., maximum NPV, perfect regulation, and even-flow of harvest) and conclude that under the given situation GP is superior to LP. Field et al. (1980) and Hotvedt et al. (1980) suggested a complementary use of LP and GP for harvest scheduling, which eliminates LP's limitations when there are conflicting criteria, and avoids possible pitfalls of GP by basing it on the LP solution. The fiamework starts by formulating the harvest scheduling problem as an LP and optimizing several objective functions serially. Based on the LP solutions, the problem is reformulated as a cardinally-weighted GP model and solved in various forms. Finally, a 26 strategy that best satisfies the decision maker's preferences is selected from the GP solutions after comparing them to the original LP solutions. In general, GP remains less popular than LP for forestry planning. Leuschner (1990) states that, although theoretically it is heralded as a technique for solving multiple-objective problems, GP still minimizes one objective fimction and still has many limitations. Dyer et al. (1979, 1983) compare GP with LP flour a welfare economic perspective and contend that, in contrast to the LP solutions which are Pareto-optimal, the solutions of GP are Pareto-inferior. Hrubes and Rensi (1981) and Rensi and Hrubes (1983) argue that neither LP nor GP will necessarily produce Pareto-optimal solutions in the public domain due to imperfect markets, wrong price signals, and inaccurate representation of production possibility curves, and therefore, judging the usefulness of GP fi'om the welfare economic perspective is inappropriate. This debate, however, may be of more theoretical interest. From a practical standpoint, perhaps the greatest limitation of GP is the necessity to obtain substantial information fiom the decision makers concerning their objectives, targets, weights and ordering of preferences. A popular alternative to LP is binary-search (BS) simulation. Chappelle's (1966) SORAC is considered the first application of BS simulation for harvest scheduling. More recent BS scheduling software includes SIMAC (Sassaman et al. 1972), ECHO (Walker 1976), and TREES (T edder et al. 1980). LP and BS difl’er in several aspects. Johnson and Tedder (1983) compare these two techniques and suggest that both have relative advantages but none is definitely superior to the other. Some important advantages of BS include: BS usually costs less per run, BS provides feasible solutions more easily, BS is able to depict the inventory in more detail, and BS can accommodate changes more easily. The main limitation of BS is that the solution obtained may not be optimal; it is only optimal if all predetemrined inputs are optimal. Many argue that solutions which are near optimal may be sufficient in practice. However, "near" can be ambiguous and it is generally desired to have optimal solutions. Moreover, because BS depicts instead of solves problems, one special program is usually needed for each specific problem. In this sense, BS is less flexible. Other limitations include: limited number of decision variables (maximum 27 equal to the number of periods), difliculty in incorporating constraints beyond an overall harvest flow, and inability to consider alternatives management intensities. Hoganson and Rose (1984) developed a heuristic that overcomes the last limitation, but still retains the difliculties of handling constraints beyond harvest level. Although in general BS is a viable harvest scheduling tool, LP has received more attention in recent decades. The National Forest Management Act (NFMA) indicates a preference for optimization techniques over simulation, and the USDA Forest Service currently uses FORPLAN for developing long-run management plans for national forests (Kent 1980). Johnson and Scheurman (1977) broadly classify the numerous techniques of formulating harvest scheduling problem as a mathematical programming or simulation model into Model I and Model 11 formulations. In their simplest forms, the two formulations can be presented as follows: Model I: max 2 chx, (3.1) i=1 i=1 subject to 2x, = A, (3.2) i=1 where: xi,- = hectares of management unit (e.g., stand type or age class) i allocated to management regime (activity) 1' A ,- = total hectares of management unit 1' Cij = NPV of allocating management unit 1' to management regime j. 28 Model 11 WZXCq-xy +Zewww (3.3) i=1 1:: i=1 subject to 2x, + w,” = A, i= -M,0 (3.4) ,- §x,,, +w,,, =Zx,, j=l.N (3.5) I where x.. = hectares regenerated in period i and harvested in period j 1} WW = hectares regenerated in period 1' and left as part of ending inventory in period N A, = hectares present in period 1 that were regenerated in period i M = number of periods before period 0 in which the oldest age-class present in period 1 was regenerated The main difference between the two formulations is rooted in how a management regime is defined. Model 1 defines a management regime as a set of management activities applied to a management unit or stand-type throughout a planning horizon. In Model 11, a management regime is defined as a set of management activities applied to a stand-type fi'om regeneration to final harvest. The implication is that while Model I preserves intact the identity of the original stand- types, Model 11 allows hectares of different original stand-types to be merged when they are harvested in the same year or period, hence does not preserve the identity of the original stand- types. The impossibility of tracing the origin of any stand-type is often considered a disadvantage of Model 11. On the other hand, the flexibility of combining hectares fiom different original stand- types to form new stand-types is cited as an advantage. 29 Other important comparisons between Model I and Model 11 are in terms of the number of constraints and decision variables required. Model I needs fewer area constraints than Model II. If there are m stand-types, and the planning horizon is divided into T planning periods, Model I needs m area constraints, whereas Model 11 needs (m + 7) area constraints. The two models are usually identical in the number of constraints that are not required by the model formulation (e.g., constraints representing harvest flow over time, or total hectares harvested). The number of decision variables in any LP harvest scheduling model generally depends on (1) the number of stand-types, (2) the number of management regimes, (3) the number of periods in the planning horizon, (4) the minimum and maximum rotations or clearcutting ages, and (5) the initial age-class distribution of the forest (Leuschner 1990). As a result, there is no generalization regarding which model needs a higher number of decision variables. Johnson and Scheurman (1977) provide formulas for computing the number of decision variables needed in Model I and Model 11 for a given forest situation. To illustrate that the number of decision variables needed is very case specific, Johnson (1977) examines 12 harvest scheduling problems of different forest situations. In one problem he found that Model 1 requires 33,717 variables while Model II needs only 828 variables. In another example, he found that Model 1 needs only 339 variables as compared to 939 variables needed in Model 11. However, it was also discovered that Model I needs more variables than Model 11 in 10 out of the 12 problems examined. This may be an indication that Model I has a tendency to increase the number of variables, while Model I] can keep it relatively smaller. 3.2. Harvest Scheduling: Decision Making Under Risk The literature of decision theory (e.g., Knight 1921, Raifa 1968) generally categorizes decision making into: (1) under certainty, (2) under risk, and (3) under uncertainty. Decision making under certainty refers to a deterministic situation in which the decision maker is able to specify precisely the outcomes of alternative actions or numagement options. Clearly this type of decision making rarely exists in reality. On the contrary, when each management option may result 30 in several difi‘erent outcomes and the decision maker is unsure about which outcome will take place, the situation faced is either under risk or under uncertainty. Risky and uncertain situation are distinguished by the availability of empirical information for generating probability distributions representing the outcomes of each alternative action. If such information is sumciently available and the decision maker is able to predict at a specific probability level that an outcome will occur, the decision making is under risk. When very little or no such information is available, the decision making is under uncertainty. In a forestry environment, the majority of management inputs are naturally non- deterministic, making it virtually impossible to predict the output of any forest management option with certainty. However, there is generally empirical information for predicting the range of outcomes of a given management option. For example, it is possible to predict the average and variance of the harvest volume produced by a given stand at a given future age, or to generate a probability distribution associated with the occurrence of forest fires, or to predict a range of future timber prices. Therefore, decision making in forest management, such as harvest scheduling, mostly falls into the category of decision making under risk. Related to the categorization of decision making is the classification of decision makers with respect to their attitudes toward risk. Generally, decision makers are categorized into those who are (1) risk neutral, (2) risk averse, and (3) risk takers. Texts (e.g., Robison and Barry 1987) describe each of these groups in terms of utility fimctions. A risk-neutral attitude is associated with a linear utility function With a constant marginal utility, whereas a risk-averse (risk-taker) attitude reflects a concave (convex) utility function representing a diminishing (increasing) marginal utility. A decision maker with a constant marginal utility disregards the probability associated with each outcome, and therefore, is indifferent between two management alternatives with different range of dispersions (probability) as long as they have equal expected values. With a diminishing marginal utility, a risk-averse decision maker always prefers an option which outcomes are less dispersed (with a higher probability) although it may be associated with a lower expected value. Conversely, a risk-taker decision maker is willing to select an option with more 31 dispersed outcomes in order to take a chance on receiving a higher return. In reality, the concave utility function is most rational .and it is reasonable to state that most real-world business decision makers are more likely risk averse rather than risk taker or risk neutral. While risk can not be eliminated, risk-averse decision makers can incorporate risk into their decision analyses. A fundamental drawback of using LP and its variants for handling decision making or optimization problems under risk is that LP is based on an assumption that all model inputs are deterministic. In other words, LP has no feature for incorporating risk associated with the non-deterministic nature of model inputs. Ample techniques of optimimtion under risk have been developed and several authors have reported some applications in forestry planning. H00] (1966) applied a combination of dynamic programming and Markov chain approach for incorporating risk in a forest production control model (i.e., temporal and spatial scheduling of management activities to attain prespecified management objectives). In this approach, risk due to the random future states of the forest is accounted by applying the concept of Markov-chains, and prescriptions of production control activities over a planning interval are optimized by dynamic programming. The outputs are prescriptions of optimal production control activities over time for various forest conditions, and the associated expected returns. The applicability of this approach greatly depends on the possibility of generating the probability associated with each future state of the forest. Lembersky and Johnson (1975), Lembersky ( 1976), and Kaya and Buongiorno (1987) applied Markov decision models for stand-level management planning. Thompson and Haynes (1971) proposed an approach termed partially stochastic linear programming. They solved a problem concerning least-cost wood procurement scheduling in which the availability of land area is not known with certainty. Their approach involves developing subjective probability distributions for the non-deterministic resource availability, followed by determining the resource availability situation through a Monte Carlo simulation utilizing the distributions. These resulting values were used as the right-hand-side (RHS) quantities of the corresponding constraints in the LP formulation, hence accounting for risk associated with the non-deterministic land-area availability. 32 Reed and Errico (1985) assessed the application of the stochastic control theory to develop a harvest schedule that incorporates risk due to timber losses caused by random fires. In general, it is supposed that random portions of the area in each age class in a given period are destroyed by fire, changing the state of the forest in the following period. They showed that the stochastic control approach is not practically possible when the harvest scheduling involves harvest-flow constraints. Hence, they solved a deterministic version of the problem (i.e., assuming that fixed portions of the forest are destroyed) and concluded that if the forest is relatively large, the deterministic optimal solution should provide a good approximation to the stochastic optimal solution. The variance of the random variables representing the proportions burnt determines the closeness of the two solutions. A similar approach was applied to incorporate risk due to pest hazards (Reed and Errico 1987). This time, the average annual infestation rate was used to represent the highly random occurrences of pest hazards. By simulating the resulting optimal solution over time, they showed that using the average annual infestation gives a reasonable approximation when infestation intensities are low. Gassmann (1989) showed that the stochastic version of Reed and Errico's (1986, 1987) problems can be solved using a specifically developed computer program which utilizes the Dantzig-Wolfe decomposition principle. It was found that stochastic models tend to give more conservative solutions compared to the deterministic counterparts. Hoganson and Rose (1987) developed a harvest scheduling model that incorporates risk due to random fire using a multistage recourse approach. This approach is based on the premise that at any given point in time decision makers focus mainly on solving immediate problems. Thus, the harvest scheduling problem is solved by finding the optimal solution for one period at a time. Feedbacks obtained from implementing the optimal solution in previous periods are used as additional inputs in finding the optimal solutions of the following periods. A harvest strategy that recognims risk due to random growth was developed by Marshall (1987). Here risk is measured by the deviations between the expected and actual mean annual increment (MAI). A penalty cost based on weighed positive and negative deviations was developed 33 and incorporated into the objective function. In this approach, the problem becomes one of minimizing the total cost. Several studies addressed the situation when yield estimates are subject to random variations. Pickens and Dress (1988) discussed potential sources of randomness in yield estimates, and described the consequences of using yield estimates that contain error in LP harvest scheduling. One source of error is land aggregation, in which each aggregate is usually treated as if it is a homogenous entity although it is comprised of several non-homogenous lands/stands. Moreover, any management activity assigned to a given aggregate is usually assumed to take place in one point of time. In reality, due to the diversity inherent within each single aggregate, the timing of management activities applied to individual stands may be years apart. Another source of errors has to do with the estimation of multiple yields which are highly correlated. An example is the case when yield estimates must be split into several classes of product such as sawtimber, pulpwood, and firewood. Among their important conclusions regarding the impacts of using stochastic yield estimates in an LP harvest scheduling are: (l) the optimal objective fimction value tend to be optimistically biased, (2) the dual activities will be biased estimates of the true marginal costs, and (3) solutions generated will usually be infeasible. A possible approach for incorporating non-deterministic yield estimates into an LP is by using their expected values. Hof et al. (1988) provided a theoretical explanation of this approach. When there is no harvest flow constraint, the problem can be transformed into one with random objective function coefficients but with deterministic constraints. However, the approach is not feasible when the problem involves harvest-flow constraints, in which the constraint coeficients will be no longer deterministic. Leefers (1991) provided another possible technique. His approach involved creating a number of "sample" yield tables fi'om a yield-estimate database utilizing a variant of the Monte Carlo simulation, hence capturing yield variability. An LP-harvest scheduling model is formulated and solved for each of these yield tables, and the expected value of the optimum is derived from the LP solutions. In essence, this approach reverses the expected-value approach mentioned 34 previously. That is, instead of finding the optimum using the expected value of the non- deterministic yield estimates, the expected value of the optimum is determined using the non- deterministic yield estimates. Using a case model, Leefers demonstrated that, incorporating yield variability in this manner results in a more conservative harvest schedule in the sense that a wider range of rotation ages is adopted and a larger portion of the forest is not managed in the first period. The harvest scheduling situation addressed in this study is also under risk because of the randomness contained in the yield estimates. This randomness is due to stand aggregation similar to the aggregation situation mentioned by Pickens and Dress (1988). Because the problem involves harvest-flow constraints, the expected-value approach mentioned by Hof et al. (198 8) is not applicable. Hof and Pickens (1991) and Hof et al. (1992) mentioned the possibility of using chance-constrained programming for handling this situation. 2.4. Chance-Constrained Programming Chance-constrained programming (CCP) is one of three main approaches of optimization under risk. The others are stochastic linear programming and linear programming under uncertainty (Naslund 1967). CCP is appropriate when the cost of risk (violating constraints) can not be a priori specified and is difficult to incorporate directly in the objective fimction (Kirby 1967). For instance, consider a harvest scheduling problem which incorporates non-declining even-flow (NDEF) constraints. Due to non-deterministic yield projections, this problem is under risk of violating the NDEF constraints. There is no easy means of specifying the cost associated with this constraint violation nor of incorporating it in the objective function. Moreover, when risk is due to random variations in the technical coefficients such as in this particular instance, CCP is much easier to formulate compared to stochastic programming (see Weintraub and Vera 1991). CCP was first introduced by Chames et al. (1958) for scheduling the production of a heating oil plant facing random future demands. This first work was followed by several papers 35 expanding the theory of CCP (e.g., Chames and Cooper 1959, 1963). The basic concept of CCP can be explained by beginning with the following simple optimization problem: max 2c, x , (3.6) jeJ subject to 2a,,x, s b,. (3.7) J’eJ This optimization can be solved as an ordinary LP if all parameters c,, a,,and b, are deterministic quantities. When all or any of these parameters are random variables, LP is no longer feasible. Although CCP is theoretically applicable for problems in which all c,, a,,and b, are random, it is primarily used when either or both a,, and b, are random. The firndamental concept of CCP is that, because of the randomness of a,, and b,, it is admissible not to expect that the optimization holds to all possible realizations of the random variables. In other words, it is permitted to violate constraints up to a certain (small) level of probability, or conversely, constraints are required to hold with a specified level of probability but not necessarily with probability one. With this reasoning, constraint 3.7 is rewritten as: P 20ng 3b,]21—a, (3.8) jEJ where Pr means probability and or, are specified probabilities usually to make 1- a, close to one. This probabilistic constraint requires the condition defined in constraint 3.7 to hold with at least 100(1-a,) percent of the time, or can not be violated more than 1000:, percent of the time. CCP is solved in its deterministic equivalent formulation. If only a,- are random, following the procedure defined by Rao (1984), the deterministic equivalent expression of constraint 3.8 can be derived as follow. Assume that a,, are normally distributed with an expected value E(a,,). Let Var(a,,) and Cov (a,,, au) be, respectively, the variance and covariances of the random variables. Define the quantity d, as 36 i=1,2,...m. (3.9) Because a,, are normally distributed and x,, is constant, d, are also normally distributed with an expected value of: E(d,) = EE(a,)x, r: 1,2,...m. (3.10) and a variance of Var(d,) =x1‘v,x (3.11) where V, is the 1"" covariance matrix. Hence, constraint 3.8 can be expressed as lPr [d, S b,] 2 l- a, which leads to: d—d b—d P ‘ ‘s ‘ ‘ Zl—a, (3.12) {JVar(d,) JVar(d,):| The left term within the parentheses is a standard normal variate with mean of zero and variance of one. Therefore, b,-a' Pr[d, Sb, ] = 5[W] (3.13) where 6 (x) is the cumulative distribution function of the standard normal distribution at x. If e, is the value of the standard normal variable at which 6 (e,) = (1,, constraint 3.12 can be rewritten as: ,[ I.-. ’—_Var(al,)]w(e‘)' (3.14) These inequalities will be satisfied only if 37 b-d . . 2, 3.15 [WW] (6) ‘ ’ 01’ J,+e, Var(d,)-b,so. (3.16) Substituting expressions 3.9, 3.10 and 3.11 in 3.16 gives: i E(a,,)x,, + e, ,/x7v,x s b, (2.17) j=| which is the deterministic equivalent of constraint 3.8. Applications of CCP encompassing various areas have been reported; examples include product mix (van de Panne and Popp 1963), forage allocation (Hunter et al. 1976), water resource system (Aleksandrov et al. 1984), and finance (De et al. 1982) applications. 2.4. Growth and Yield Modeling Growth and yield predictions are integral to forest management planning. With land-type classification and activity scheduling, quantitative growth and yield projections constitute essential components of forest management (Davis and Johnson 1987). More specifically, growth and yield predictions are necessary inputs in preparing any long-term forest management plans, including harvest scheduling. Growth and yield models generally refer to various instruments for predicting growth and yield of forest stands, ranging from simple yield tables to highly sophisticated computer routines. The most conventional form of growth and yield models are tabular records containing expected volumes and other stand characteristics (e.g., number of trees, basal area, average diameter, etc.) per unit land area by combination of age and site class. These tabular records are either normal yield tables or empirical yield tables, depending on whether they were prepared fiom samples of 38 selected healthy and fully stocked or "normal" stands, or samples representing the whole range of stand conditions. An obvious advantage of yield tables is that they are easy to construct. However, yield tables are usually based on a single "normal" or "average" density. Because normal stands hardly exist in reality, using a normal table usually involves some adjustments, making it less practical and potentially less accurate. Both types of yield tables are usually constructed using data from one-time measurements as opposed to data from subsequent measurements. Consequently, the patterns of stand developments implied by the tables may not reflect the actual or historical development of the individual sample stands. Therefore, yield tables must be used with caution in predicting the future condition of any given stand (Davis and Johnson 1987). Today, most growth and yield models are in the form of mathematical equations or a set of interrelated equations. A great variety of mathematical growth and yield models have been developed, making it necessary and useful to have a classification. The most comprehensive classification is given by Davis and Johnson (1987). In this study, it is sufficient to classify mathematical grth and yield models into three main groups, namely: (1) explicit whole-stand models, (2) implicit whole-stand models, and (3) individual-tree models. Whole-stand and individual-tree models difi‘er in the prediction unit used and therefore, the type of predictor variables involved. Whole-stand models use stand statistics such as age, site index, number of trees per hectare, and basal area per-hectare as predictor variables, and the predictions obtained are directly in per unit area. Predictor variables used in individual-tree models are tree statistics such as tree diameter and height. Yield predictions per unit area are obtained by summation of the yield of each individual tree. Individual-tree models are potentially more accurate but tend to be data intensive and much more expensive in comparison to whole-stand models. A variant called the distance-dependent individual-tree model (Munro 1974) involves some measurement of the distance between individual trees as part of the predictor variables. This is perhaps the most complicated and expensive type of growth and yield model at the present. Due to the cost involved, the relative merit of individual-tree models versus whole-stand models has 39 been questioned (Clutter et al. 1983). Daniels et al. (1973) evaluated the precision of whole-stand models and individual-tree models developed for loblolly pine plantations in Virginia and concluded that whole-stand models are more precise. Whole-stand models, however, are perhaps less appropriate for modeling mixed-species forests. Whole-stand models can be broadly distinguished into explicit and implicit models. Explicit models directly give yield predictions per unit area. Implicit models, often called diameter- distribution based models, project stand structures (i.e., diameter distributions) instead of yields. Yield predictions per unit area are obtained fiom further computations using the predicted diameter distributions and additional treed volume equations. Because implicit models predict diameter distributions, they give more detailed information and can be used for a wider variety of purposes. For example, they can be used to simulate thinnings by removing certain portions of the diameter distributions (Knoebel et al. 1986) or to obtain yield predictions per diameter class or type of products (Bennett and Clutter 1986) which in turn enable more sophisticated economic analysis. However, using error propagation and Monte Carlo approaches, Mowrer (1987) showed that implicit growth and yield models tend to be less precise than their explicit counterparts. Lenhart (1987) compared the accuracy of explicit and implicit models in predicting yields of loblolly and slash pine plantations in East Texas and came up with a similar conclusion. Moreover, developing implicit models demands significantly more extensive data, and therefore more expense. Based on these considerations, the growth and yield models developed in this study are explicit whole-stand models. The works by MacKinney et al. (1937) and Schumacher (1939), which introduced the methodology for developing explicit whole-stand yield prediction equations, are considered milestones in mathematical growth and yield modeling. These works presented the first variable- density yield prediction equations (i.e., equations using stand density as one of the predictor variables). The basic form of the equation, which has become well known as the Schumacher yield model is: 40 ln(V) = ,6, + ,B,A" + ,6, (s) + ,6,f(1)) (3.13) where V = some expression of per unit-area yield A = stand age S = site index flD) = some function of stand density B, = model parameters. Clutter (1963) observed that, since yield is an accumulation of growth over time, growth equations and yield equations must be compatible. That is, a yield equation must be the mathematical integration of the corresponding grth equation. Based on this property, a volume growth equation can be obtained by difl‘erentiating Equation 3.18. with respect to age. This gives: dV dD fat-31A" 154%] (3.19) which indicates that the relative rate of volume growth is a function of the stand age and the relative rate of growth in stand density. It also holds that the stand volume for a given future age is a function of the firture age and a measure of stand density at that particular age. Since the firture age is given and site index is: commonly considered constant, predicting the firture yield reduces into predicting the future stand density and substituting the predicted stand density into Equation 3.18. Common measures of stand density are number of trees or basal-area per unit land area. Clutter et al. (1983) suggested the following equation for predicting future stand basal-area: lnB2 =(ijln3, +a,(l-i]+a,S[l——4‘—J (3.20) A2 A2 A2 in which B , denotes the current stand basal-area and B 2 denotes stand basal-area in a given future age A 2. To confirm with the compatibility property, this equation is obtained by integrating a basal-area growth equation (see Clutter et al. 1983, p. 121). 41 Given the value of B 2, Equation 3.18 can be rewritten to obtain an equation for predicting the stand volume in the future age A 2: 1n(V,) = ,3, + AA,“ + p, (s) + ,6,(B,) (3.21) in which V2 is the stand volume at a projected future age A 2. Variants of Schumacher growth and yield models have been widely used for difl‘erent species in different regions. A few example are the growth and yield models for thinned and unthinned loblolly pine in the South (Clutter 1963, Sullivan and Clutter 1971, Burkhart and Sprinz 1984) and for slash pine in South Afiica (Pienaar et al. 1985, Pienaar and Shiver 1986). CHAPTER FOUR: METHODS As depicted in Figure 4.1 this study can be partitioned into two main phases. The first phase deals with developing a set of growth and yield models for teak plantations in Indonesia. The resulting models are integrated into a computer routine specifically designed for generating yield projections and computing total NPVs of various stands under difl’erent management regimes. In the second phase the computer routine is applied to a selected teak forest district. The yield projections obtained are used for developing harvest scheduling models. Subsequent sections of this chapter outline the procedures and methods applied in each phase. 4.1. First Phase: Growth and Yield Modeling The product of the first phase of this study is a set of growth and yield models which can be used to predict future yields of an existing stand, based on its present condition. Since teak plantations are thinned regularly, the model set should also predict thinning yields at different ages. Specifically, the model set is to be used to obtain quantities of intermediate thinning yields and the final harvest. These quantities are, respectively, TYl, TY2, TYn and FH in Figure 4.2. The set of equations is comprised of : a basal-area growth model, a volume growth and yield model, an afier-thinning basal-area model, an after-thinning volume model, and a stand-height model. 9:59P)?" 42 43 Management objectives. assumptions. and constraints “ Growth and yield modeling Growth and yield models 81 computer routine Yield and ‘ NPV projections ' Per—hectare yield and PV Y Harvest scheduling . Y Total NPV. harvest flows. hectares allocation ending age-class Cost. price. and interest rate Figure 4.1: A general flow-chart of this study. Yield/hectare TYl Age (years) Figure 4. 2. Intermediate thinning yields (TY l ...TYn) and final harvest (FH) to be predicted using the growth and yield model set. 4.1.1. Growth and Yield Data Growth and yield data were acquired from the Center of Forest Research and Development (CFRD) in Bogor, Indonesia. This research institution is currently under the Ministry of Forestry, and has been collecting growth and yield data of several species since it was established early in this century. The teak growth and yield data used in this study were collected from 63 permanent plots distributed in various locations in Central and East Java. This relatively small number is partly because several permanent plots were damaged during the Second World War or their remeasurement have been interrupted thereafter. Figures 4.3a and 4.3b show the distributions of permanent plots by age (at the first measurement) and site-class, respectively. The number of measurements on each plot is between two to seven times, resulting a total of 255 measurements. The time periods between two consecutive measurements range fiom four to ten years, but five years is common. All plots were thinned following a relative-spacing rule. The following information can be extracted from each record (measurement): plot size; age; dominant-height; site-class; and before- and after-thinning average diameter, average height; 45 number of trees, basal area, and volume (all on a per-hectare basis). These data are presented in Table A. 1, Appendix A. An important issue in deriving growth data from series measurements is selecting the age interval. Three alternative age-intervals are possible for each permanent plot measured more than two times: the longest interval, all possible intervals, and non-overlapping intervals. For a permanent plot measured at Age ,, Agez, Age3 Age“, the longest interval is the difi'erence between Ageland Age“, all possible intervals are given by the differences between all combinations of two ages (e.g., Ageland Agez, Ageland Ageg, etc), and non-overlapping intervals are those between Age land Agez, between Age2 and Age3, ..., and between Agewland Age“. This study uses the last type of age-interval. Borders et al. (1987) indicated that non-overlapping intervals give the best result when two previously published basal-area models were fit using the three different types of intervals. All possible intervals are associated with the occurrence of high autocorrelation. Using the longest interval prevents dealing with autocorrelated data, but would significantly reduce the size of data set (only one data point per each plot). 24 6188 Nunber of plots 8 DUI 11-20 :3 -30 i E :8; 2- 8 r N n V Standageatthetirstmasuramant Figure 4. 3a. Distribution of permanent plots according to stand age at the first measurement. 46 8918 .5 O Nunbarotplots .3: 0| Figure 4. 3b. Distribution of permanent plots according to site class. 4.1.2. Model Forms All models used are explicit whole-stand models, as opposed to implicit whole-stand models or individual-tree models. The selection of this model type is partly because the growth and yield data available for this study are stand-average statistics. Diameter-class statistics necessary for developing implicit whole-stand models or tree-level statistics required for developing individual-tree models are not available. Moreover, as mentioned in Chapter Three, explicit whole stand models have been shown to give more accurate and precise yield predictions of some species in some locations (Daniels et al. 1973, Lenhart 1987, Mowrer I987). Schumacher growth and yield equations (Equations 3.20 and 3.21) are the basic functional forms of the basal-area growth model and the volume growth and yield model, respectively. Although these equations were originally developed for unthinned forests, several studies indicated that the relationships also holds for thinned plantations (e.g., Clutter 1963, Sullivan and Clutter 1972, Pienaar and Shiver 1986). In addition, data scatter-plots (Figures 4.4a - 4.4d) reveal that “16 dependent-independent variables relationships implied in those conceptual equations do exist. Figure 4. 4a. Future basal area (BZ) plotted against the current age (Al). Figure 4. 4b. Future basal area (BZ) plotted against future age (A2). Figure 4. 4c. Future basal area (32) plotted against the current basal area (Bl). Figure 4. 4d. Future volume (V 2) plotted against future basal area (BZ). 49 The functional forms of after-thinning basal-area and after-thinning volume models are derived as follows. First, the author assumes that removed and residual trees have the same average diameter; a tenable assumption for even-aged plantations thinned according to the relative- spacing rule (see related description in Chapter Two). Under this assumption, to some extent the proportions of basal area (or volume) removed in a particular thinning should be a function of the proportion of the number of trees removed. Figures 4.5a and 4.5b confirm that these relationships do exist. Hence, after-thinning or remaining basal area and after-thinning or remaining volume may be, respectively, represented by the following equations: B, ”[1103" (4.1) and V. =q(fi:]n (42) where Ba = after-thinning basal area, B, = before-thinning basal area, N a = after-thinning number of trees per hectare, N, = before-thinning number of trees per hectare, Va = after-thinning volume, V, = before-thinning stand volume, p, q = coemcients. 50 Figure 4. 5a. Ba/Bb (the ratio between after- and before-thinning basal area) plotted against Na/Nb (the ratio between after- and before-thinning number-of-trees per hectare). Figure 4. 5b. Va/Vb (the ratio between after- and before-thinning volume) plotted against Na/Nb (the ratio between aficr— and before-thinning number-of-trees per hectare). 51 Operationally teak plantations are thinned from below (i.e., stands are thinned by first removing the smallest trees in the stand, followed by cutting the next larger trees until the required Na is achieved). The implication is that after-thinning stands will be comprised of larger trees, and consequently, both (Ba/3b) and (Va/Vb) should be larger than (Na/Nb). To confirm this, estimates ofp and q in Equations 4.1 and 4.2 are expected to be equal or greater than one. Stand-height (H) is usually expressed as an inverted function of stand age (A). One common firnctional form is: ln(H)=b, ”(a (4.3) However, an exploratory data analysis indicated that this equation is not adequate. Stand density (number of trees or basal-area) is an important determinant of height growth (Figures 4.6a and 4.6b for all ages), and therefore, should be included as one of the independent variables. Consequently, the stand-height model is empirically estimated starting from: H = f(A,B, N). (4.4) Figure 4. 6a. Stand height (H) plotted against stand basal area (B). 52 0 500 1000 15m 2000 Figure 4. 6b. Stand height (H) plotted against the number of trees per hectare (N). 4.1.3. Model Developments Growth and yield data are usually repeated measurements of permanent plots. A common problem associated with this kind of data is the presence of serially correlated errors among consecutive measurements. Sullivan and Clutter (1972) and Sullivan and Reynolds (1976) discussed the implications of using repeated-measurement data for estimating a system of growth and yield equations. In general, ignoring the correlations reduces the eficiency of the ordinary least-squares (OLS) procedure. While the OLS estimates remain unbiased, their variances are larger than they would be if the correlations are taken into account. Associated with this larger variance is the tendency to underestimate the residual errors. In addition, the presence of the correlations also implies interdependencies among parameters in any one equation with those in the other equations, leading to numerically inconsistent estimates. Burkhart and Sprinz (1984) and Knoebel et al. (1986) handled the parameter interdependency problem using a loss-function approach. Basal-area equation (3.20) and volume 53 equation (3.21) were estimated simultaneously by iteratively adjusting the coefficients of both equations, imposing the conditions a, = [34/03 and (12 = [35/133 and minimizing the loss-function 1:42p.-z)2/a;).(2(3,-§,)2/.;) (45> where V, and J7, are observed and predicted volumes, and B, and B, are observed and predicted basal-area, of, and of, are the mean square error from OLS fits of the volume and basal-area equations, respectively. A major limitation of this approach is that the results are greatly affected by the arbitrarily specified form of the loss-fimction (Borders and Bailey 1986). Furnival and Wilson (1971) proposed the simultaneous equation approach (e.g., multi- stage least-squares) commonly applied in the field of econometrics. In this approach, parameters of a system of related equations are estimated simultaneously rather than sequentially, thus avoiding using earlier predicted results as predictors in the other equations. Based on its conceptual superiority, many growth and yield modelers advocate this approach (e.g., Murphy 1983, Amateis et al. 1984, Borders 1989, Borders and Bailey 1986, and Gregorie 1987). Notwithstanding, all those computationally sophisticated approaches were essentially proposed to obtain more reliable growth and yield models. In other words, it is the quality of the output that is most important. A relatively parsimonious technique would be more appealing as long as it produces a satisfactory fit to the data. Based on this practical argument, this study adopts a relatively less complicated approach used by Sullivan and Clutter (1972). They solved the interdependency problem by merging equations (3 .20) and (3.21). The merged equation takes the form: an2 = c0 + c,S + c2 {—1—} + c3[i) lnBl + c,(l — i) + c,( — AL)S (4.6) 42 A; 42 A: This equation is simultaneously a growth and a yield equation; when A 2 = A} it gives the current volume. Together with Equations 3.20 and 3.21, this equation forms a set of equations that are 54 logically consistent (Clutter et al. 1983). They can be used to obtain current volume and future volume, firture basal area, and basal-area and volume growth rates. Thus in summary, the set of growth and yield models in this study is comprised of Equations: 3.20; 4.1; 4.2; an empirical fornr of 4.4; and Equation 4.6. 4.1.4. The Yield-Projection Computer Routine The resulting estimated growth and yield models will be integrated into a computer routine. The routine is designed to read tabulated forest inventory data and generate per-hectare timber yields for an individual stand under different management regimes at specified future ages, and to compute the corresponding total net present value (NPV). To coincide with the standard 5-year thinning sequence, future yields are projected in 5- year age intervals. For example, an existing 20 years old stand is projected to ages 25, 30, 35, and so forth. Stand conditions at age t are used to predict firture conditions and yield at age (t + l) and, subsequently, the predicted condition at age (t + 1) is the predictor of the stand condition and yield at age (t + 2), and so forth. The total NPV is the sum of discounted net revenues earned throughout the planning horizon, and computed using the common discounting procedure. Further details on the estimation of NPV are presented in Chapter Six. 4.2. Second Phase: Harvest Scheduling As indicated in the beginning of this chapter, the second portion of this study deals with the development of harvest scheduling models for a selected teak forest. The computer routine discussed in Section 4.1 is utilized to produce inputs for the harvest scheduling model (i.e., per- hectare yields and NPVs for every stand under different management regimes). This section describes the selected forest district and its harvest scheduling problem, and outlines the modeling fiarnework. 55 4.2.1. The Forest District Perhutani's teak forest districts are similar in most aspects; they difler mainly in the extent of their forest area and timber potential. All forest districts are almost identical in terms of data availability because they follow a standard procedure for forest inventory and data management. The selection of the forest district for this study is primarily based on practical considerations, such as location and accessibility. The selected forest district is the Cepu Forest District of Regional Unit I. Geographically it is located in the northeastern region of Central Java (Figure 4.7). Adrninistratively, about 80% of the forest belongs to the Central Java Province and the other 20% is part of the East Java Province. The Central Java Province portion is within the area of Blora Kabupaten (kabupaten is a political unit equivalent to a county in the USA). According to the 1992/1993 forest inventory, the total forest land of the Cepu Forest District is 26,700 hectares, more than 90% of which is for teak production. The remaining hectares are in non-teak production and not for production. The Cepu Forest District covers one of the prime sites for teak in Java, and accordingly has been one of the most productive and profitable districts. It is also the site of Perhutani's main wood manufacturing plant. This particular forest district has been managed under the Burn's Method since 1974 and the most recent lO-year management plan covers the period of 1993 - 2002. The standard rotation-age is 80 years, and the targeted timber production in the first 10-year period is 40,000 cubic meters annually. The current 10-year age-class distribution (Figure 4.8) indicates that the Cepu Forest District is in a better condition compared to the entire teak forest (Figure 2.2); that is, the age-class distribution is notably more balanced. Central Java is among the most overpopulatcd regions in Indonesia. The population of Kabupaten Blora in 1989 was almost 750,000; this is more than 400 inhabitants per square kilometer (Kantor Statistik Blora 1989). The Cepu Forest District is very important to the region as it employs thousands of laborers annually. In addition, as with other teak forest districts, the Cepu Forest District contributes to the growth of small-scale wood industries in the area, which create a significant number of additional job opportunities. 56 rll m>ma Amoe— Eaazfom Save “Spa Baa—c3 8:35 3.20". :30 2: up p.288; K .w 23%; am a m «tee—mung sates BEBE. . _ é Eamon—pawn. «i: 35:00 mafieEom do. .2 57 "=52>5§§§ A Anselm Notes: age-classes are in a lO-year interval, e.g., age class I covers ages 1-10 years, age class II cover ages 11 to 20 years, and so forth. Figure 4. 8. Age-class distribution of the Cepu Forest District. Forest resource data used in this study were derived from Book All, which is one of a total of six books comprising the current lO-year management plan. This book contains forest resource information collected through a periodic (IO-year) forest inventory. The standard forest inventory method is systematic sampling with 2.5% sampling intensity. Data collected for every single stand include: various standing stock parameters (such as age, stand height, number of trees per hectare, stand basal area, etc), understory condition, soil description, topographical condition, and a brief history of the stand's establishment (e.g., method of planting, source of seeds, etc). 58 Based on their overall condition and predetermined purposes, stands are categorized according to the following scheme: A. Not for production 1. Not feasible for production 2 Designated for special purposes (e.g., log yard) 3. Conservation or recreation area 4 Protected area B. For production 1. For teak production 1.1. Clearcutting feasible 1.1.1. Productive 1.1.1.1. Age classes I to XII(10-year interval) 1.1.1.2. Overmature 1.1.1.3. Low growth 1.1.2. Non-productive 1.1.2.1. Clearcut site not yet replanted 1.1.2.2. Bare land 1.1.2.3. Non-teak stand 1.1.2.3.1. Plantation 1.1.2.3.2. Natural forest 1.1.2.4. Teak stand with small number oftrees per hectare 1.1.2.4.1. Plantation 1.1.2.4.2. Natural forest 1.2. Clearcutting not feasible 2. Not for teak production 2.1. Not good for teak 2.1.1. Bare land not good for teak 2.1.2. Non-teak plantation not good for teak 2.1.2.1. Plantation 2.1.2.2. Natural forest 2.1.3. Dying teak stand 2.1.3.1. Plantation 2.1.3.2. Natural forest 2.2. Non-teak plantation 2.3. Area proposed for preservation This study deals only with productive teak stands (i.e., those belong to category 1.1.1.) which, as indicated earlier, account for nearly 90% of the total forest area. This proportion is typical of teak forest districts. 59 4.2.2. The Harvest Scheduling Problem The harvest scheduling problem of the Cepu Forest District can be summarized as follows. First, the forest district is managed with the objectives: (1) to attain the highest possible profit and (2) to achieve and maintain a sustained-yield condition. The total area of the forest land is fixed, and stands comprising the forest district can be aggregated into stand-types according to age and productivity classes. Except for low productivity sites all stands are thinned every 5 years starting at age 5 until 10 years before clearcutting, and all stands are replanted following final harvest. Timber is considered the only source of revenues. Firewood is financially a minor by-product and revenues from non-traditional products such as recreation and hunting opportunities are negligible. Costs are limited to forest management operational costs such as planting, thinning, and clearcutting costs. Other costs, such as administration costs and salary of tenured employees, are centrally coordinated. These costs are assumed to be similar regardless of management activities used. Traditionally, only one rotation age has been applied to the entire forest, but in this study alternative rotation ages or clearcutting ages (i.e., 60, 70, and 80 years) are considered. In order to attain the management objectives, the Cepu Forest District needs to devise a long-term management plan (harvest schedule) determining hectares of each stand-type that should be allocated over the rotation-age alternatives. Perhutani's operational interpretation of the second management objective (i.e., sustained- yield condition) is a relatively constant or non-declining even flow (NDEF) of timber production over time. Attaining this condition is socially and politically important. To some extent, relatively stable harvesting activities reflects an uninterrupted creation of job opportunities (i.e., employment is associated with thinning, girdling, clearcutting, and replanting), and creating employment is one of Perhutani's mandates. From Perhutani's standpoint relatively constant timber production will create a stable cash-flow. A key aspect in attaining the management objectives is the non-deterministic nature of timber yield predictions due to spatial and temporal aggregations. Projections of per-hectare yields used in the harvest scheduling, denoted as a. are the average of per-hectare yields of many yo 60 individual stands belonging to specific stand-types, under specific management regimes, and harvested in specific periods. The values a,,, are not free of variances. The consequence is that the actual quantities of some of the a,,, may difl‘er from their corresponding average quantities used in the standard harvest scheduling. If the actual quantities of some of the am are smaller than expected, the actual total profit earned will be less than indicated by the harvest schedule. In other words, the model solution may be optimistically biased. 1f the discrepancies between the actual and predicted values of “if! are substantial, hectares allocated over alternative rotation ages will no longer lead to an NDEF condition. Depending upon Perhutani's attitude toward such risk, the possibility described above may or may not be a matter of concern. If it is not a significant concern, “ijt may be treated as deterministic, implying that Perhutani is willing to accept the risk. However, considering the financial implications, it is more rational for Perhutani to incorporate the risk into its long-term management planning. In other words, a harvest schedule that incorporates some level of assurance for the attainment of the management objectives would be more desirable. Therefore, as noted in the opening chapter, an important specification of the harvest scheduling involves its treatment of risk of not attaining the management objectives, particularly the NDEF condition, due to the variability of timber yields. 4.2.3. Model Outline Based on the harvest scheduling problem described above, the harvest scheduling model needed should maximize total profit over a specified time period (planning horizon) while simultaneously maintaining a NDEF condition. To account for the time value of money, the total profit is represented by the total NPV over the planning horizon. Alternatively it could be represented by the total timber volume, but only if the discount rate is assumed to be zero. For purposes of comparisons, the harvest scheduling model is formulated and solved in two versions: excluding risk (ordinary LP) and including risk. The nature of the problem leads to selecting chance-constrained programming (CCP) for the including-risk version. CCP is appropriate 61 because the non-deterministic components of the problem are contained in the constraints. In addition, the cost of risk in this problem can not be easily quantified, eliminating the selection of stochastic linear programming which incorporates risk directly in the objective firnction. In CCP, constraints are assumed to be independent (i.e., coeficients in one constraint are not correlated to those in other constraints). In this study, yields of a given stand in difl‘erent periods may not be perfectly independent, and therefore, the assumption is likely not fully satisfied. The effect of this violation is unknown (Hof et al. 1992). In this study, a zero correlation is assumed. A rule-of-thumb in forest management planning is to set a planning horizon between 1.5 to 2 times the rotation length (Clutter et al. 1983). Leefers (1991) examined the efi‘ect of using different lengths of planning horizon in harvest scheduling and reported that shorter planning horizons tend to allocate a larger portion of the forest for harvest in the early years or periods. In other words, using longer planning horizons helps ensure the long-run sustainability of the forest. A 120-year planning horizon is used for the harvest scheduling in this study. For rotations considered (i.e., 60, 70, and 80 years) this planning horizon is 1.5 to 2 times of the rotation length. This planning horizon is divided into 12 equal planning periods to reduce model size. The harvest scheduling model will be structured with a Model I formulation (Johnson and Scheurman 1977). For the harvest scheduling under study, Model I is easier to formulate and requires fewer constraints. The first management objective, to maximize the total NPV, is treated as the objective function; the second management objective, to achieve and maintain a sustained- yield condition, is represented by a set of NDEF constraints. The Model I formulation requires explicit definitions of stand-types and management regimes. A stand-type is defined as an aggregate of individual stands belonging to the same age class with similar productivity. For this study, age-classes are arranged in lO-year increments, and the productivity of a given stand is measured by the stand's total yield with a maximum 80-year rotation. By definition a management regime is a sequence of management activities applied to any stand-type throughout the planning horizon. Since all stands (except some low productivity sites) are mandatorily thinned every 5 years and must be replanted following clearcutting, management regimes are solely characterized 62 by the rotation length. A management regime, therefore, is simply formed by any combination of a rotation-age and a stand-type. It is assumed that an identical rotation age is applied to both the current and regenerated stands. For example, assigning a 60-year rotation to a 40-year old stand implies that both the existing and regenerated stands will be harvested at age 60, or in periods 3 and 9 respectively. A more elaborate model could include multiple regenerated stand rotations for any given current stand rotation. 4.2.4. Model Formulation For convenience, the problem is formulated starting from the LP version. The CCP version is obtained by slightly modifying the LP formulation. The basic LP formulation follows. maxZ = iicyxy (4'7) i=1 j=l subject to: fix, 3 L, (4.8) g“, gamx, 2 LV, (1 = 1) (4.9) géamx, .<. UV, (t = 1) (4.10) (1+ 102;,“wa - ggamw 2 o (t=1...11) (4.11) (1— l)§§a,,x, - i1 gamflux, so (i =1...11) (4.12) where 63 x,, 2 0 (4.13) c = net present value (Rupiah/ha, 1993 constant rupiahs) x = hectares allocated (ha) L = total forest area (ha) a = per-hectare yield (m3/ha) LVand UV = respectively, minimum and maximum harvest volumes u and I = respectively, maximum allowed increase and decrease in periodic harvest volumes (percent) i, j, t = respectively, stand-type i, management regime j, period t. The objective function 4.7 maximizes the total NPV of the entire forest. Constraint 4.8 is the land-area constraint, which ensures that the total hectares of stand type i allocated over the management regime alternatives not exceed the corresponding total hectares available. Constraints 4.9 and 4.10 restrict the total harvest volume in period one to be within the specified upper (UV) and lower (LV) bounds. Constraints 4.11 and 4.12 control the fluctuation of harvest volumes over time. The upper and lower bounds u and l are in terms of percentage of the harvest volume in period t. Thus, for any positive u and l the harvest volume in period (H!) is restricted within (100 - I)% and (100+u)% of the harvest volume in period t. Constraint 4.13 is the common non- negative constraint. Transfomring the LP formulation into a CCP requires reformulating constraints 4.9 - 4.12 into chance-constraints. First, these constraints are expressed in the following probabilistic terms: Pr[Y,2LV,]21—a (i=1) (4.14) Pr[1’, sLV,]21—a (i=1) (4.15) 64 Pr[(1+u)1’,—Y, 20121-0: (:=1...11) (4.16) +1 Pr[(1—1)Y, —Y,,, 20]21-a (:=1...11) (4.17) where Yr = Elgamxv and Y,“ = §§1“4'(r+l)xu and Pr = probability, and a = the probability level, which is usually selected to set (l-a) close to one. For purpose of this study, or = 0.05 is arbitrarily selected as the starting probability level. In practice, this level should be determined by the decision makers. Constraints 4.14 and 4.15 impose that (1-a) per cent out of 100 chances the total harvest volume in period 1 should be within the specified lower- and upper bounds (LV, and UV, ). Similarly, constraints 4.16 and 4.17 require that (l-a) per cent out of 100 chances the harvest flow should be within the allowed maximum increase (u) and maximum decrease (1). Through the procedure described in Chapter Four, 4.14 - 4.17 are transformed into: E(Y.)-fl[Var(Y.)1"2 LV. (t = l) (4.18) EU.) +flIVar(Y.)]" s W. (r=1) (4.19) E((1 +u)Y. — Y...)-fl[Var((l - I)Y. — 1(...)]"s 2 0 (t = ll~11) (420) E((1 - 1)}: — Y...) -4[Var((l - 1)): - Y...)]" 2 0 (t =1---11) (421) in which E = expected value, Var = variance, [3 = the value of the normal density function associated with the probability level or, and 65 Var((l + u)1’, — 1’,,,) = i i Var((1+ u)a,,, — a,,,,) x: i-l 1:1 Var((1-I)Y. — Y...) = sigma - 0a.. -a....)x: Thus, the CCP formulation is comprised of the objective function 4.7, set of land-area constraints 4.8, set of non-negative constraints 4.13, and sets of deterministic equivalent chance constraints of the first-period yield (4.18 and 4.19) and NDEF condition (4.20 and 4.21). 4.2.4. Model Solution Because chance constraints 4.18 - 4.21 are non-linear, the CCP model can not be solved using an ordinary LP algorithm directly. It may be solved using the simplex method, but only after linearizing the non-linear constraints. Linearization can be done by applying the MOTAD technique (Hazell and Norton 1986) or the linear-approximation technique proposed by Olson and Swenseth (1987). Linearization, however, may inflate the model size. Weintraub and Vera (1991) proposed a cutting-plane approach for solving a CCP in its non-linear form. However, they only provide the theoretical explanation of the approach and leave interested adopters to develop their own computer codes. Seppala (1972) developed CHAPS (Chance-Constrained Programming System), a specifically designed algorithm for solving CCP. CHAPS has been demonstrated to be eflicient and accurate (Seppala and Orpana 1984), but has not been made available to public users. Thus, the best option at this point is to solve the CCP model using any general non-linear programming software readily available commercially. This study uses SOLVER, an add-in to Microsoft EXCEL“. A more detailed description of this particular software is given in Chapter Six. CHAPTER FIVE: THE GROWTH AND YIELD MODELS AND THE YIELD PROJECTION COMPUTER ROUTINE The results of the first half of this study are presented in this chapter. The resultant growth and yield models along with model testing procedures and results are described. The computer routine for integrating the models is also discussed. All models were estimated using SYSTAT". 5.1. Model Estimates 5.1.1. Basal-Area Growth Model The base form of the basal-area growth model is Equation 3.20. This particular equation has no intercept and the coeflicient of the predictor variable (A ,/A 2)1n B , is required to be one. Imposing the latter condition is easier through a non-linear procedure. Therefore, although this equation is linear, it was estimated using a non-linear estimation procedure available in SYSTAT®. The loss-function minimized remains least squares, and the default quasi-Newton search method was used. The equation was first estimated in its original form, in which the independent variable S (site-index) was obtained fi'om: A . 1435 In S = 6.0375 + (In H — 6.0375(8—0) (5.1) after Budiantho (1985). However, the resulting model has a very low coefficient of determination (R2), leading to replacing S with dominant-height H. Using H instead of S eliminates any error inherent in the site-index equation (5.1), and therefore, should result in a better estimate. The estimation result is presented in Table 5.1. 66 67 Table 5.1. Estimation of the basal-area growth model. Variable Coeflicient Asymptotic SE CI (95%) (l-A L/AZ) 2.927 .097 2.737 - 3.117 (A I/A ;)H) .044 .004 .036 - .052 Note: Adjusted coeflicient of determination (R2) = .90; SE = standard error; C1 = confidence interval. Explicitly, the final estimated basal-area growth model is: In B, = (14;) lnB + 2.927(1— i)+.044[1— fijH, . (5.2) A A A 2 2 2 5.1.2. Simultaneous Volume Growth and Yield Model Equation 4.5 is the base form for this model. This simultaneous volume growth and yield model was estimated using an OLS procedure. As in model 5.2, H was used in the place of S. The predictor variable 1/A, was excluded because it was not significant in the model. The final estimation result is shown in Table 5.2. Table 5. 2. Estimation of the volume growth and yield model. Variable Coefficient Coemcient SE t (.05) Intercept 1.739 .315 17.560“ lnH, .034 .045 15.580‘ (A ,/A 2)ln B, .952 .200 23.975‘I (1-A,/A2) 1.796 .517 6711" (1-A ,/A z)ln H, .092 .100 8.829' Note: R2 = .95; Standard error of estimate (SEE) = .077; “ = significant at a = .005. 68 The explicit form of the simultaneous volume grth and yield model is: In V, = 1.739+.034 In H, awe—J ln B, + 1.796(1- 5:4) +.092[1 - %-J H, . (5.3) 2 2 2 5.1.3. After-Thinning Basal-Area and Volume Models The base forms of these models are Equations 4.1 and 4.2 respectively. Both models were estimated using OLS procedures. The estimated statistics are shown in Table 5.3. Table 5. 3. Estimations of the after-thinning basal-area and volume models. Eqn. Variable Coefficient Coeff. SE t(.05) 1—{2 SEE 4.1 (NM 1.074 .005 236.545“ .99 1.185 4.2 1 (NflfliVb 1.048 .008 127.017" .98 11.429 * = significant at a = .005. Explicitly, the two models are: B, = 1.074( N“ )8, (5.4) N b and V, = 1.048( N“ JV, (5.5) N i As expected, the regression coefficients of these equations are greater than unity. This confirms the assumption that the proportion of both basal-area removed is greater than the proportion of the number of trees removed because thinning fiom below normally leave larger trees. 69 5.1.4. Stand Height Model The original form of the stand height model is Equation 4.3. However, as indicated earlier, this conceptual form fits the data poorly; the resulting model has a very low coeflicient of determination. Examinations of data scatter plots (Figures 4.5a and 4.5b) led to including stand density (number of trees and/or basal-area per hectare) as a predictor variable. Several explicit formulations of H =flA, N, B), using various transformations of A, N and B, were examined. The final estimated height model was: lnH =2575—.l43ln 5 +34llnB. (5.6) 1 A 1 Table 5.4 presents a more complete estimation result. Table 5. 4. Estimation of the stand height model. Variable Coefficient Coeficient SE t (.05) Intercept 2.575 .077 33.637“ ln(N,//A,) -. 143 .004 -36.241" In B) .341 .024 14.361“ Note: R2 = .91; SEE = .066; "‘ = significant at a = .005. Residuals ofall model estimates (Figures 5.1a - 5.1e) were examined to detect possible departures from assumptions. In general, there is no apparent pattern indicating a serious departure. In addition, the scatter plots showed an absence of outliers. 70 scatter plots 3 2 fModel 5. ISO Figure 5.1 a. Residual analys of lnB estimates (B iduals. 1'68 basal area) V seatter plots resi studentized 9 3 Equation 5 sof volume) Figure 5.1 b. Residual analysi of anestimates (V 71 Figure 5.1 c. Residual analysis of Equation 5.4; scatter plots of Ba estimates (Ba: alter-thinning basal area) against studentized residuals. studentlzed residuals Figure 5.1d. Residual analysis of Equation 5.5; scatter plots of Va estimates (Va: alter-thinning volume) against studentized residuals. 72 0.1 studentized residuals O -0.05 -O.1 -0.15 Figure 5.1e. Residual analysis of Equation 5.6; scatter plots of lnH estimates (H: stand height) against studentized residuals 5.2. Model Testing In order to use the models in the preceding section with confidence, it is necessary to have some indicators of their reliability in addition to the coefficients of determination and standard error of estimates. Common indicators of model reliability are accuracy, precision, and time dependence (Brand and Holdaway 1983). Accuracy is indicated by the mean difl‘erence between model predictions and actual values, and the dispersion of the difference between predicted and actual values reflects model precision. A model is time-independent if both its accuracy and precision are relatively constant with various projection lengths. Measuring these three indicators, however, requires either collecting new data or using a subset of the currently available data not used in model estimation. Collecting new data was not feasible due to cost and time constraints. Setting aside a subset of the data was also not feasible because the data set presently available is not very large. Thus, an alternative approach, using the original data set, was used to test the numerical stability of the models. 73 The validation approach used is similar to the cross-validation technique described by Efion (1982) and Efron and Gong (1983). A portion (for example about 25%) of the original data set was removed randomly and the remaining data set was used to re-estimate the models being tested. After repeating this process many times (for example 100 times), the resulting new estimates were compared to the original models. Ifthe reduced data sets consistently produce estimates that are similar to the original models (in terms of the signs, magnitudes, and significance of the coefficients, as well as goodness of fit), the models being tested are considered numerically consistent. Conversely, if subtracting a random portion of the data does greatly afl‘ect the resulting estimates, the reliability of the models is questionable and should not be used for yield prediction. Host et al. (1993) applied this approach for assessing the reliability of an ecological-land- classification model developed for the Manistee National Forest in Michigan. Subsets of re-estimation results of Models 5.2 - 5.6 are presented in Tables 5.53 -5.5d, respectively. After reviewing the results, it was concluded that all models are numerically consistent as reflected by the consistency of their coemcients as well as their R2 and SEE values. Table 5. 5a. A sample of results fiom re-estimating the basal-area growth model (Model 5.2) using random subsets of the data. Sample # n b, b2 b3 172 l 212 1 2.865 .047 .90 2 192 1 2.966 .044 .90 3 203 1 2.967 .042 .89 4 210 1 2.994 .042 .90 5 199 1 2.968 .043 .90 6 198 1 2.882 .045 .91 7 212 1 2.875 .046 .90 8 191 1 3.068 .039 .89 9 202 1 2.907 .045 .90 10 209 1 2.828 .047 .90 11 200 1 2.927 .044 .89 12 196 1 3.026 .044 .90 13 204 1 2.934 .044 .88 14 213 1 2.959 .043 .89 15 196 1 2.969 .042 .89 Original Eqn. 255 1 2.927 .044 .90 74 Table 5. 5b. A sample of results from re-estimating the volume growth and yield model (Model 5.3) using random subsas of the data. Sample # 11 b0 b, b2 b3 b,, E2 SE 1 212 1.718 .031 .99 1.436 .109 .96 .077 2 192 1.555 .036 .996 2.338 .078 .96 .073 3 203 1.787 .032 .961 1.612 .097 .95 .076 4 210 1.694 .036 .948 2.087 .081 .96 .075 5 199 1.653 .036 .965 2.140 .079 .96 .074 6 198 1.802 .032 .942 1.442 .103 .96 .077 7 212 1.726 .031 .986 1.436 .109 .96 .077 8 191 1.719 .037 .930 2.215 .075 .95 .079 9 202 1.734 .033 .967 1.768 .092 .95 .078 10 209 1.727 .036 .927 1.780 .092 .96 .075 11 200 - 1.807 .033 .938 1.737 .091 .95 .076 12 196 1.682 .036 .952 2.127 .078 .95 .080 13 204 1.844 .033 .926 1.694 .089 .95 .075 14 213 1.803 .033 .943 1.637 .095 .95 .077 15 196 1.681 .035 .954 2.092 .081 .95 .077 Original 255 1.739 .034 .952 1.796 .092 .95 .077 Eqn. Table 5. 5c. A sample of results from re-estimating alter-thinning basal-area and after-thinning volume models (Models 5.4 and 5.5) using random subsets of the data. Model 5.4 Model 5.5 Sample # n p R SEE q Ta? SE 1 147 1.060 .99 .758 1.063 .99 10.244 2 138 1.063 .98 .740 1.065 .98 11.934 3 151 1.063 .98 .745 1.068 .97 11.781 4 149 1.064 .99 .760 1.065 .98 11.882 5 154 1.064 .99 .761 1.064 .99 11.027 6 139 - 1.065 .99 .756 1.066 .98 12.290 7 145 1.064 .99 .763 1.065 .98 11.402 8 147 1.063 .98 .738 1.065 .97 12.353 9 145 1.063 .99 .747 1.065 .99 11.465 10 152 1.062 .99 .768 1.067 .99 11.389 11 144 1.058 .99 .753 1.064 .99 11.320 12 150 1.061 .99 .760 1.063 .97 11.871 13 149 1.056 .99 .767 1.065 .98 11.648 14 153 1.060 .99 .758 1.066 .97 11.855 15 139 1.065 .98 .743 1.065 .98 11.547 Original 181 1.064 .99 .764 1.065 .99 11.429 Eqn. 75 Table 5. 5d. A sample of results from re-estimating the stand height model (Model 5.6) using random subsets of the data. Sample # n ' b0 b1 b2 172 SE 1 212 2.577 -.144 .339 .90 .068 2 192 2.535 -.140 .352 .90 .066 3 203 2.518 -.144 .36 .90 .065 4 210 2.617 -.144 .327 .91 .065 5 199 2.931 -.137 .232 .90 .079 6 198 2.584 -.144 .338 .90 .070 7 212 2.574 -. 144 .340 .90 .068 8 191 2.589 -.142 .336 .91 .066 9 202 2.581 -.145 .340 .90 .067 10 209 2.621 -.145 .327 .91 .067 11 200 2.533 -.146 .358 .90 .064 12 196 2.558 -.145 .347 .90 .069 13 204 2.609 -.141 .329 .90 .066 14 213 2.587 -.144 .338 .90 .066 15 196 2.628 -. 143 .324 .90 .064 Original eqn. 255 2.575 -.143 .341 91 .066 An additional compatibility test was used to examine Models 5.2 and 5.3. Compatibility is easily described by an example. Suppose Model 5.2 is used to predict the basal area of a given stand 10 years in the future. If Model 5.2 is compatible, it should give similar predictions regardless whether the basal area is predicted in two steps of 5-year intervals (incremental) or directly using a 10-year interval. A portion of the results given in Table 5.6 indicates that Models 5.2 and 5.3 are quite compatible, that is, the differences between 5-year incremental and 10-year direct projections are relatively small. For basal area, the difl‘erences are generally between -0.6 to 0.3 per cent relative to the 5-year incremental projection. For stand volume, this range is between ~20 to 2.5 per cent. As noted by Buchman and Shifley (1983), there is no projection system that can portray the real world perfectly. The idea of evaluating (growth and yield) models, therefore, is not to prove that the models do not represent the nature exactly. Rather, it is to examine the models' performances relative to available alternatives, when there are such alternatives. This principle is adopted in evaluating the models developed in this study. 76 Table 5. 6. A sample of results from testing the compatibility of Models 5.3 and 5.4. Age Initial Initial Basal area at age 3 Volume at age 3 1 2 3 height basal area Incremental Direct Incremental Direct 21 26 31 24.3 14.9 22.7 22.6 129.4 133.0 26 31 38 24.9 15.9 24.0 23.6 140.3 139.9 31 38 43 26.9 15.5 23.0 22.7 133.0 137.5 40 45 50 25.3 18.7 23.4 23.4 131.8 134.8 45 50 57 25.8 18.4 23.6 23.4 135.3 136.5 30 35 40 26.9 18 24.4 24.4 139.4 145.8 35 40 47 26.9 18.4 25.3 25.0 147.4 149.3 38 43 48 15.7 19.6 22.5 22.4 119.6 115.6 43 48 55 16.8 20.3 23.6 23.4 126.7 122.1 48 55 60 18.5 21.6 24.7 24.7 132.2 131.4 52 57 62 28.2 21.6 26.0 25.8 148.7 150.5 57 62 70 30.8 22.2 27.7 27.7 161.2 167.1 61 66 71 34.6 29.3 34.3 34.1 197.4 205.1 15 20 25 21.8 17.1 26.2 26.0 146.6 146.4 84 89 94 34.9 27.9 31.5 31.5 179.1 185.7 49 54 59 28.2 21.8 26.5 26.2 152.1 153.4 54 59 67 31.2 22.7 28.5 28.5 166.6 173.7 59 67 72 31.3 23.9 29.6 29.3 169.5 177.0 54 59 64 32.1 20.4 25.2 25.1 145.8 151.6 59 64 71 33.2 17.7 22.9 22.9 135.5 141.3 Note: Ages are in years, heights are in meters (m), basal areas are in m2, and volumes are in m3 . Basal areas were predicted using Model 5.3, volumes were predicted using Model 5.4. 5.3. The Yielchrojection Computer Routine The computer routine was written in QuickBASIC®. It was noted in Chapter Four that the stand condition and yield in period t+1 are projected on the basis of the stand condition and yield in period t, and subsequently, the projected stand condition in period H1 is used to project stand condition in period t+2, and so forth. In its present state, the routine reads input files and likewise stores all outputs in files. Users, therefore, need to type stand data and other inputs only once for an indefinite number of runs. The routine may be modified into an interactive mode quite easily if desired. In fact, developing an interactive version may be much simpler than developing the original program. Due totimeconstraints,themainfocusatthispointistothedevelopacomputerroutinethatmeetsthe 77 needs in this study. Therefore, the routine is not yet very efficient. For example, the routine has not yet incorporated rotation-age options and a separate run is needed for each rotation age. Stand data and other inputs must be prepared as ASCII files. The routine starts by reading all input (except stand data) files and storing them in arrays or matrices. The next step is to read the stand data file one line at a time (the routine projects one individual stand at a time). In general, at every line (stand), the following steps are executed: 1. Given the stand's current age and the rotation-age assigned, the routine defines the activity that must be implemented in each sub-period throughout the planning horizon‘. For example, for a 30-year old stand in the 60-year rotation age, the activity in each sub-period is defined as follow: Sub-period Stand age Activity 1 - 5 30 - 50 Thinning 6 55 None 7 60 Clearcutting 8 - Planting 8 - 17 5 - 50 Thinning 18 55 None 19 60 Clearcutting 20 - Planting 20-24 5-25 Thinning 2. Current stand volume is predicted using Model 5.3. Ifthe stand is clearcut in this first sub-period, the volume predicted is the first final-harvest yield. Ifthe stand is thinned, the 'Recall that the harvest scheduling models cover a 120-year planning horizon, which is divided into 12 10-year cutting periods. To accommodate the 5-year thinnings, each cutting period is divided into 2 sub- periods. 78 routine refers to the thinning instruction file (which is stored in a matrix), checking the appropriate after-thinning number of trees per hectare, then predicting after-thinning basal- area and volume using Models 5.4 and 5.5, respectively. The thinning yield is the difl‘erence between the stand's current volume (before thinning) and the stand's after-thinning volume. Average tree diameter of the current stand is computed fiom stand basal area and number of trees per hectare. This average diameter is used to approximate the price bracket of the timber yield. Ifthe stand is clearcut, the routine will estimate the regenerated stand's basal area and its dominant height at age S-years using appropriate equations in Table 5.7 . These estimates are used as the starting points for growing the stand to the next sub-period. Ifthe stand is thinned, the routine defines the current alter-thinning basal area and number of trees per hectare, and the current stand's dominant height as the starting points. The stand is grown by 5 years to the next sub—period by projecting the stand's basal area and volume in the next 5 years using Models 5.2 and 5.3, respectively, and estimating the stand's dominant height using Model 5.6. If the stand is clearcut in this sub-period, the stand volume is the final harvest yield. Otherwise, the routine refers to the thinning instruction file to check the appropriate after-thinning number of trees per hectare, and predicts after-thinning basal area, volume and thinning yield. Then, NPV is computed. Given the average diameter in step 3, the routine selects an appropriate timber price and computes the revenue obtained in the given sub-period. The total cost depends on the activities taking place in the given sub-period. For example, if the stand is regenerated in this particular sub-period, the cost incurred includes planting cost and may also include thinning cost depending on whetherthe stand is thinned or not. Ifthe activity is clearcutting, the total cost includes girdling and clearcutting costs. Clearcutting cost is per cubic meter, thus it is derived from the per-hectare timber yield produced in the sub-period. 7. 8. 79 Steps 3 - 6 are repeated through sub-period 24. Program output is stored in ASCII files with user-specified names. A general flow chart of the routine is provided in Figure 5.2 and the computer code is presented in Appendix B. 5.4. Additional Models Additional models have been developed to complement Models 5.2 - 5.6. Models 5.2 - 5.6 which require inputs of initial age (A), initial basal-area (B), initial dominant height (H) and initial number of trees per hectare (N). These values are not available for regenerated stands. At the present, this problem is addressed as follows: 1. 2. Age 5 years is used as the starting point. Develop B =j(A) and H =flA) models using forest inventory data to predict the initial B and H of regenerated stands. Ideally, these equations should be developed exclusively using 5-year old stands. However, it was observed that B =flA) and H =flA) relationships are more apparent if stands are grouped according to site class. Ifthis grouping is used to impose model performance, the number of 5-year old stands in each group is not very large. Therefore, the models were developed using 5 - 25 year old stands with separate equations for each site class. Here, site class is the original site classification according to the forest inventory data, not the site index as given by Model 5.1. The initial-B and initial-H equations and their measures of goodness of fits are presented in Table 5.7. These equations are used in step 4 of the yield projection process. Teak is planted with a l x 3 meter spacing. This means that a 5-year old plantation with 100% survival should have 3300 trees per hectare. However, the forest inventory data suggest that most stands between 5 - 10 years have a smaller N. For this reason, the initial Ns of regenerated stands are represented by the average N of the current 5 -10 year old stands within the corresponding site class. 80 / Data of Thinning _" initial stand rule 'fi Clearcut GIOW , or Grow - Clearcut Finaloharvest ’ yield reject regenerated stands ? Price. cost and interest rate Net present value (NPV) calculation Grow 5 years i After-thinning ‘ . condition Figure 5. 2. Flow chart of the yield-projection computer routine 81 Table 5. 7. Estimations of the Initial-B and Initial-H models. Equation Site class Intercept Slope E? SEE lnI-I =bo +b. 111A low 1.385 .399 .81 .102 medium 1.675 .394 .82 .069 high 2.003 .344 .85 .039 lnB=co +0. 111A low -3.362 5.126 .87 .614 medium -3.527 5.398 .88 .639 high -3.989 5.642 .88 .668 Note:H=standheight,A=standage,andB=standbasal area. An additional model depicts the basal-area growth over finite time periods, or AB = f (A ,,A 2). The explicit form successfully fitted is: 1n(B, — 3,) =303(1— iJH, A, (5 .7) with a standard error of the coefficient = .005, R2 = .96 and SEE = .34. This model was estimated using the growth and yield data (permanent plot data). It gives the basal-area grth for the period of (Ar/12) of a given stand with current dominant height H ,. Model 5.7 is used as the upper bound of projected stand basal-area (Model 5.2) . It is used to prevent overestimation of stands' basal area growth (and hence volume growth) relative to empirical data. For example, consider a particular stand with a current age A 1, current basal area B 1, and current dominant height H]. Suppose that, given these initial conditions, Model 5.2 predicts 82 = D. Meanwhile, using the same inputs, Model 5.7 gives (B: - B I) = E. If (D - B 1) is larger than E, the projected basal-area 82 is set equal to (B +E). CHAPTER SIX: THE HARVEST SCHEDULING MODELS The purpose of this chapter is to describe the process of constructing and solving the harvest scheduling models for the Cepu Forest District and to discuss model solutions. The first section describes model components and inputs. LP versions of the harvest scheduling models are presented in the second section, followed by the CCP version in the third section. A discussion on the model solutions is given in the last section. 6.1. Model Components and Inputs 6.1.1. Decision Variables xi}. According to the 1991/1992 forest inventory data, Cepu Forest District has 1,742 individual stands with existing teak plantations. A common practice in harvest scheduling is to aggregate stands, usually according to age classes, in order to reduce model size. For teak forests in Java, age classes are defined in lO—year increments. Accordingly, for purposes of this study, the 1,742 individual stands are aggregated into nine lO—year age classes. Each of these age classes is further divided into 3 to 6 productivity classes, resulting in a total of 35 stand-types as shown in Table 6.1. Due to the mandatory 5-year thinnings (described in Chapter Four), management regimes are solely determined by the rotation ages. A management regime is a combination of any one of the rotation-age alternatives (i.e., 60, 70, and 80 years) with each one of the 35 stand-types. For example, a combination of the 60-year rotation and stand-type 3C results in a management regime, here labeled as 3C60. Clearly, in this problem a management regime is identical with a decision variable; that is, a decision must either include or exclude a specific management regime for each stand-type. With 35 stand-types and 3 rotation-ages, there are a total of 105 management regimes or decision variables, as presented in Table 6.2. Another management option for each stand type is no management. 82 83 Table 6.1. Stand-type labels based on age and productivity classes and on existing stands. Age Productivity class class 1- 10 1C 1D ll-20 2C 2D 21 -30 3C 3D 31 -40 4C 4D 41 -50 5C 51-60 6C 61 -70 7C 71 - 80 8C 1 > 80 Notes: Figures in parentheses are the corresponding total land-area (in hectares). Shaded cells indicate that there are no stands with the corresponding combination of age class and productivity class. A management regime implies the sequence of thinnings and clearcuttings applied to the particular stand-type over the planning horizon. It has been stated that the 120-year planning horizon is divided into 12 lO-year cutting periods. Therefore, a stand-type 3C for example, will reach age-class 6 (51-60 years) in period 4, age-class 7 (61-70 years) in period 5, or age-class 8 (71-80 years) in period 6. Hence, management regime 3C70 for instance, implies that stands belonging to stand-type 3C will be clearcut in period 5, regenerated into the 1-10 age-class in period 6, and clearcut again in period 12. It also implies that this regime includes thinnings every 5 years duringperiods 1 -4and6- 11 (the lastthinningtakesplace lOyears beforeclearcut). The sequence of thinnings and clearcuttings associated with each management regime is presented in Table 6.3. 84 Table 6.2. Management regimes (105) resulting fiom the combination of 35 stand-types and 3 rotation ages. _ 1 _ , Rotation age Stand-type 60 70 80 1A 1A60 1A70 1A80 18 1860 1870 1880 1C lC60 lC70 1C80 1D 1D60 1D70 1D80 18 1860 1870 1880 2A 2A60 2A70 2A80 28 2860 2870 2880 2C 2C60 2C70 2C80 2D 2D60 2D70 2D80 28 2860 2870 2880 2F 2F60 2F70 2F80 3A 3A60 3A70 3A80 38 3860 3870 3880 3C 3C60 3C70 3C80 3D 3D60 3D70 3D80 38 3860 3870 3880 4A 4A60 4A70 4A80 48 4860 4870 4880 4C 4C60 4C70 4C80 4D 4D60 4D70 4D80 5A 5A60 5A70 5A80 58 5860 5870 5880 5C 5C60 5C70 5C80 6A 6A60 6A70 6A80 68 6860 6870 6880 6C 6C60 6C70 6080 7A 7A60 7A70 71180 78 7860 7870 7880 7C 7C60 7C70 7C80 7D 7D60 7D70 7D80 7B 7860 7870 7880 8A 8A60 8A70 81180 88 8860 8870 8880 8C 8C60 8C70 8C80 9A 9A60 9A70 9A80 Note: Stand types (e.g., 1A) are identified by age class (e.g., 1 = age 1 - 10) and productivity class (e.g., A = the lowest productivity). 85 planning horizon under different management regimes. Management regimes Table 6.3 . Thinning and clearcutting sequences over the 2 l l 1 0 1 9 8 7 6 5 4 3 2 1 lC60 1D60 1E60 3A60 B60 4860 4C60 4D60 1D70 lE70 1A 2A 3A70.3B70 4C70 70 4870 4C70 4D70 A70 7870 7C70 7D70 7E70 1880 1C80 lD80 1E80 2A80 D80 E80 3A80 B80 4380 4C80 4D80 7E80 7A80 7B80 7C80 clearcutting. thinning,C Note : Management regimes are described in Table 6.2, t 86 6.1.2. Decision-Variable Coemcients cg. The values of decision-variable coefl'rcients cg. are tabulated in Table 6.4. These coeficients are per-hectare total NPVs associated with management regime j (recall that a decision variable is identical with a management regime). These quantities equal the average values of the total NPVs of all individual stands aggregated into stand type i, or k c0. = {ENPVW (6.1) where k the number of individual stands in stand type i NPV“). = per-hectare NPV of stand h of stand type 1 under management regime j. In these NPV calculations, the stand-type mean is used. Another approach would be to have an area-weighted mean. In this study, the stand-type means were fairly similar. The total NPV of each individual stand is the total discounted net revenues produced in cutting periods throughout the planning horizon. To accommodate the 5-year thinnings, each cutting period is divided into 2 sub-periods, and for discounting purposes it is assumed that costs and revenues take place in the third year of these 5-year sub-periods. Thinning, clearcutting and girdling occur in the same sub- period as timber yields. Planting costs are incurred in the next sub-period. The formula for computing the total NPV of each individual stand is: NPV = ”gig—fig) (6.2) where P = per-cubic-rneter timber price a = per-hectare timber yield in period t C = per-hectare cost incurred in period t r 2= discount rate 87 Table 6. 4 . Per-hectare total NPV produced with each management regime. lInllltilanagement NPV Management NPV—1 Management NPV __regime (million Rp) regime (million Rp) regime (million BL) 1A60 0.53 1A70 0.255 1A80 0.117 1860 1.128 1870 0.524 1880 0.25 1C60 1.217 1C70 0.856 1C80 0.712 1D60 2.064 1D70 1.892 1D80 1.798 1860 4.443 1870 4.215 1880 4.113 2A60 1.144 2A70 0.582 2A80 0.282 2860 1.636 2870 0.8 2880 0.381 2C60 1.707 2C70 0.78 2C80 0.442 2D60 1.772 2D70 1.082 2D80 0.801 2860 3.42 2870 2.774 2880 2.499 2F60 6.946 2F70 6.304 2F80 5.985 3A60 2.423 3A70 1.248 3A80 0.62 3860 2.959 3870 1.49 3880 0.725 3C60 3.544 3C70 1.774 3080 1.088 3D60 5.091 3D70 3.439 3D80 2.747 3860 8.304 3870 6.699 3880 5.986 4A60 6.059 4A70 3.078 4A80 1.507 4860 8.053 4870 3.884 4880 2.216 4C60 10.754 4C70 7.012 4C80 5.327 4D60 13.505 4070 10.234 4D80 8.709 5A60 9.805 5A70 5.2285 5A80 2.682 5860 15.479 5870 7.745 5880 3.755 5C60 14.532 5C70 7.333 5C80 4.244 6A60 18.236 6A70 10.164 6A80 5.294 6860 29.457 6870 14.472 6880 7.271 6C60 37.994 6C70 20.141 6080 9.938 7A60 20.446 7A70 16.144 7A80 8.47 7860 ‘ 35.23 7870 27.569 7880 13.884 7C60 20.856 7C70 22.116 7C80 12.083 7D60 31.612 7D70 33.444 7D80 17.521 7860 40.02 7E70 42.429 7880 21.764 8A60 24.076 8A70 23.957 8A80 21.914 8860 35.399 8870 35.275 8880 30.769 8C60 46.949 8C70 46.829 8C80 39.989 9A60 33.138 9A70 33.776 9A80 33.712 88 The current bank interest rates for investment credits in Indonesia vary between 13% to 17%, or 15% on average, and the inflation rate is roughly 6%. (Bank of Indonesia 1994). Therefore, 9% is a reasonable approximation of the real discount rate, which is used in this study. Perhutani's opportunity cost of capital is not known. As indicated in the formula, per-hectare revenue is the product of per-hectare timber yield (either thinning or final-harvest yield) and the associated timber price. Timber yield is difl‘erentiated into 3 diameter classes, and the corresponding per-cubic-meter prices are derived fi'om the actual selling prices in 1992-1993. The yield is for the hole only, and the entire hole is treated within one diameter class. Though excluded fiom the analysis, tops of mature trees have high value, too. Costs are also derived fi'om the actual expenses during the last 1992-1993 management year. Clearcutting cost includes any expenses from felling trees to piling logs at the log yards. Because clearcutting costs are on per cubic meter basis, per-hectare total cost varies across stand-types. Price and cost data are presented in Table 6.5. Table 6. 5. Timber prices and management costs at Cepu Forest District for the management year 1992/1993. Diameter class (cm) Timber price (Rp/m’) diameter 2 30 530,000 20 5 diameter < 30 @000 4 s diameter < 19.9 155,000 Activity Cost (RD) Planting 188,700/hectare Thinning 64,085/hectare Girdling 54,575/hectare Clearcutting 21,700/cubic meter 89 In this study, real timber prices and all costs are assumed to be constant over time. Historically, increases in timber prices are consistent across timber sizes, and similarly, any change in costs apply to all stand-types. 6.1.3. Yield Coemcients am Yield coefi'rcients aw are the expected values of per-hectare yields of stand type 1' under management regime j harvested in period t, or E(a ). These quantities are obtained using: ijt 0‘" = E(a") = £2.18 (6.3) where k = the number of individual stands in stand type j th = per-hectare yield of stand I: of stand type 1', under management regime j, harvested in period t. The quantities a is zero when neither thinning nor clearcutting takes place in the period t. 1]! Otherwise, the quantity is either a thinning or a final harvest yield. The formulation of the CCP version of the harvest scheduling model requires the variances of am , or Var(ay.,). These variances are computed by: 2 Var(a,,) =%é(a,, —aw) (6.4) The values of “if! and Var (aw) are tabulated in Table A2, Appendix A. 6.1.4. Right Hand Sides (RHS) The total land areas of each stand-type shown in Table 6.1 are the RHS values for area constraints. Other RHS values are the lower and upper bounds of the total harvest volume in the first period, which are 400,000 and 440,000 cubic meters respectively. These figures are inferred from Cepu Forest District current lO-year management plan. 90 6.2. Linear Programming Harvest Scheduling Models Several LP versions of the harvest scheduling problem were solved and reported by Parthama et al. (1994). These LPs are reviewed in this section to provide some insights facilitating the formulation of the CCP in the next section. In total, these models represent a range of possible management alternatives for the Cepu Forest District. The following are the descriptions of the models solved: LP]: This model approximates the current management practice, that is a single rotation- age (80 years) is used and all stands must be managed. The model does not include any constraints to control the harvest flow over time. Obviously, this is a completely constrained optimization model, (i.e., no management choices are available). The purpose of formulating this model is to approximate the NPV and timber flow from the entire forest if treated under the current management approach; it provides a base for comparisons. LP 2 is a harvest scheduling model in its simplest form. It optimizes the allocation of hectares of each stand-type over the 3 rotation-age alternatives without harvest-flow constraints being imposed. Unmanaged stands are allowed. This model incorporates some harvest-flow constraints. The total harvest in the first decade is constrained within the pre-specified lower and upper bounds. The harvest volume in any subsequent decade is restricted to be at least equal to the volume in the previous decade but allowed to increase up to 20%. LP 4 is like LP 3, but the non-declining restriction is relaxed by allowing the harvest volume in subsequent decade to decline up to 5% relative to the volume in the previous decade. 91 LP 5: LP 5 also resembles LP 3, but is liberalized by allowing the harvest volume to increase with no explicit upper bound. The optimal solutions of these LP models are summarized in Table 6.6 and the corresponding timber-flows over time are depicted in Figures 6.1a and 6.1b. An extended discussion on these solutions is postponed until Section 6.4 of this chapter. The focus at this point istoselectamodelthatwillbeusedasabasemodelforthe CCPinthenextsection. BasedonLP solutions, LP 5 is used to provide a comparative solution for the CCP formulation. In practice, any model could be used as the comparative basis. Several aspects of the LP 5 solution make it unique. Among all models incorporating timber-flow constraints, LP 5 gives the highest NPV. LP 5 also maintains a steady increase in harvest volume from period 1 to period 5 with a constant level thereafier. In addition, with LP 5 the entire forest is managed (no stand is left idle) which is very important with respect to the goal of generating employment. Both LP 3 and LP 4 leave about 20% of the total forest area unmanaged. LP 3 which restricts the increase in harvest volume to an upper-bound also leads to a lower harvest volume throughout the planning horizon and hence a lower total NPV. LP 4 which allows the harvest volume to decline results in a less desirable harvest flow relative to those of LP 3 and LP 5. Thus, LP 5 is selected as the base model for the CCP. Table 6. 6. Summarized optimal solutions of LP harvest scheduling models. Model Total NPV Total Volume Heetane unmanaged (million Rp) (1000 cu m) (7g LP 1 155,974 9159 0 LP 2 306,746 10980 0 LP 3 239,788 8351 21 LP 4 245,51 1 8445 19 LP 5 262,137 10725 0 92 HarvaatvolmflMmS) Figure 6.1a. Harvest flows with the absence of NDEF constraints (LP 1 and LP 2). g3 mmnmm) o§§§§§§§ Period (10 years) Figure 6.1b. Harvest flows when NDEF constraints are included (LP 3, LP 4, and LP 5). 93 6.3. Chance-Constrained Programming Harvest Scheduling Models The chance-constrained programming (CCP) formulation of LP 5 is comprised of the objective function 4.7, set land-area constraints 4.8, non-negative constraints 4.13, and the deterministic equivalents chance-constraints 4.18; 4.19; 4;20; and 4.21. For convenience, a complete formulation is given below: max Z = ii%% (6.5) 1:] j=l subject to: 2834 mm Y) - p[Var(Y,)]"’ 2 LV, (r =1) (6.7) E(Y,) + fi[Var(Y,)]" 5 UV, (t = 1) (6.8) E((l + u)Y,— Y,,,)—p p[Var((1 + u)1’,— Y,,,)] 20 (r =1...11) (6.9) E-((1 I)Y,- Y,,,)-,6[Var((l-l)1’,—Y,,,)] 20 (r=1...11) (6.10) x, 2 o (6.11) where Yr: i £09139) and Yr+1 = f: iammfir‘j i: If: 1 i=1j=l Var(Y,) =iijv a ,3): :l 1:] - 94 Var((1+ u)l’, - Y,,,) = i fi((1 + u)2 Var(a,,) + Var(a,,,,)) x; ill 1:] Var((l - l)l’, — Y,,,) = g§((l _ ()1 va1(a,,) + Var(a,,,,)) x: This problem was solved using SOLVER, an add-in to Microsoft EXCEL” capable of solving non-linear programming problems. In SOLVER, the problem is presented in a spreadsheet. After the spreadsheet is appropriately structured, SOLVER must be informed of: the cell to be optimized (the objective function), the cells that should be adjusted (the cells of decision variables), the constraint vectors, and the type of optimization (maximization, minimization, or equal to a specific value). Users must also define the maximum time, number of iterations, levels of precision (and tolerance when it is an integer programming problem). Optionally, users can also activate or deactivate the auto-scaling feature, select the estimation methods (tangential or quadratic), the derivation methods (forward or central), and the search method (quasi-Newton or conjugate gradient). For the problem in this study, the spreadsheet is structured as shown in Figure 6.2. The objective-function cell AC 106 is maximized by changing the values in cells A81 to A8105 (decision variables) subject to the conditions: A81 to A8105 2 0 (non-negative constraints), ADl to AD35 s 81 to 835 (management regime/Iand-area constraints), AQ108 2 LV, (constraint 6.7), AQ109 5 UV, (constraint 6.8), BN108 to 8X108 2 0 (constraint 6.9), and CK108 to CV108 S 0 (constraint 6.10). Main outputs are: the total NPV in cell AC106, hectares ofstand-typer allocatedtomanagementregimejincellsABl toA8105,thetota1hectaresmanagedincell A8106, and periodic harvest volumes in cells A8106 to AP106. Values in cells AQ107 to 88107 are added or subtracted fi'om their corresponding periodic volumes (cells A8106 to AP106) to provide the 95% confidence interval. The spreadsheet is explained in detail in Appendix C. 95 85.88 ”.558 ..8 a: No 2.91.. 85582.... 8.o<+85< 8.58.25 8.25.8.8 855.8..2 8.9.8.5.. i ll i $285.5 n8.2....5 n858.5 ..8.a$..o .835...me 8.8....285 28.35....2M5 28.82.08". .8.mmw.mmfi .8.a<...uo$u N8558...... 8555 8558.0 8558:. + N85585. ".84.: + 68558“... 8+: 8558.583: 85585.89: "8.5.8.5. «8558... ii Ngng “fig-2 55.5. N55.0 55.0 555 +n55.m. "80+: £55.... mar: 55.2.99: 555.30.... .5522. ~55... Ere ..., 5.5.3. :1. .... 51.55. 2.1. are are :1. + ...e . .5...» .3: + ...ww .er. or: ...rifiwffe. ...wiiifie ...rx .55, taxes; 8.2... .854. 92.5.. -28 u .85.... 2555 265 u 258 u 2.1: a v amamwo 5. .25.; 2.53. «:9 ..u. 2...: :6 .5... . "....”Na...$ .2”sz +me« afgmw 5 3.5.5 2.5.5 2:. ...a 3.3.. r. ..o A .1 TV 8.". no fine cue are are Fa... PM... ...a ...,...o ...a are... are... e. e. r. ..o 96 .588: No 6.5.... wax/33.0 wo—MUtEC'm n.8.>o. .o Doc—MOLD ..8.>o....>o:... ..8.V.o....v.o: u Gangs—.0“ Ame—>m..._>mvu 555+ 55.2. .8: a a a 5 Eu: ..rx.+m.ea> + 15.1%}. 9: AM N8558.0 N8558... 8558.0 8.5585 + N2558.2. Nv.9: + .85585. N99: . 85.85.39: . 85585.89: .. .55555289: 555555.39: N. .1: miterfegé .u. ...,...arfeéé 97 As mentioned earlier, SOLVER provides two options of search methods: the quasi-Newton method and the conjugate gradient method. Theoretical descriptions of these search methods can be found in many theoretical mathematical programming or non-linear programming texts, such as Gottfried and Weisman (1973) and Avriel (1976). These two methods difi‘er primarily in terms of their speed to convergence and space (memory) requirement. Neither one of these method is clearly superior to the other. In general, the conjugate gradient method requires less memory but more iterations (time), and conversely the quasi-Newton approach requires less time but more memory. Given today's computer speed and wealth of memory, to some extent the trade-off is insignificant. However, Broyden (1972) noted that the quasi-Newton method often fails to converge if it starts fiom a poor initial estimate. The CCP above was solved using the conjugate gradient method. The CCP harvest scheduling model can be solved in various versions by using difl‘erent values of the right-hand-sides LV, and UV, of constraints 6.7 and 6.8, and assigning difl‘ermt values for the upper- and lower-bound percentages u and l in constraint 6.9 and 6.10. In addition, it may also be modified by assigning different probability levels a which results in different values of B. A smaller or-value is associated with a larger B-value. Intuitively, using a smaller a-value reflects a more conservative attitude toward risk in the sense that yield estimates are represented by wider ranges, hence giving more allowance to the possibility that the actual and projected yields may be difl‘erent. Increasing the a-value, therefore, is moving toward less conservative attitude. In this case, yield estimates are represented by narrower ranges reflecting the decision makers' higher confidence that actual yields will not greatly deviate fiom their projected quantities. At one extreme, assigning or = 0.5, in which [3 = 0, returns the CCP into an LP which treats yield as deterministic (as point estimates instead of range estimates). As indicated in Chapter Four, in this studycr=0.05 was arbitrarilychosenasthestartingpoint. Toexaminetheefi‘ectonmodel outputs, some other cr-values were also tested but not reported. The CCP was first solved by settingLV, = 350,000, UV, = 500,000, 14 = l and l = 0. Except for the values ofLV, and UV,, these are the same parameters used in LP 5 (in LP 5: LV, = 400,000 and UV, = 440,000). The values of LV, and UV, were modified because the LP solutions 98 indicate that the original values were not binding. With these values the harvest volume in the first period is restricted within the given lower- and upper-bounds, and harvest volumes in subsequent periods are allowed to double the volume in the preceding period (increase by 100%) but not allowed to decline (i.e., l = 0). However, the CCP did not converge in this setting. It seemed that, when variances of yield estimates are included in the model, a strict NDEF constraint is no longer feasible. Therefore, the CCP was solved by incrementally relaxing the NDEF requirement (i.e., incrementally increasing the value of! ), keeping other constraints the same. The model converged with l = .1, which means that periodic yields are allowed to decrease up to a lower bound equal to 90% of the volume in the previous period. This model, which is labeled as CCP 1, resulted in optimal solutions summarized in Table 6.7. Figure 6.3a depicts the resulting harvest-flow pattern if solutions of CCP l are implemented. The optimist (plus deviation) and pessimist (minus deviation) lines indicate that, given the yield variability, periodic harvest volumes will be within these bounds, with a 95% confidence level. The fact that the lower-bound parameter (maximum allowed decrease) is binding is reflected in this figure. This harvest-flow obviously does not follow an ideal trajectory such as that of LP 5. However, there is a 95% confidence level that a trajectory within the optimist and pessimist lines will materialize if the optimal solutions are completely implemented. No such assurance is associated with deterministic LP solutions. Some modifications were examined to obtain a less fluctuating harvest flow. One modification involved using a larger value of a, which means reducing the confidence level to less than 95%. Theoretically, increasing the value of a. will level ofl‘ the harvest flow, since at = 0.5 will lead to a harvest flow exactly like that ofLP 5. However, increasing the value of or up to 0.10 did not notably improve the harvest flow. Using value of larger than 0.10 was not considered because the outputs would have less practical value; a decision maker may opt for a detemrinistic model rather than a stochastic model which only provides for instance, a 75% confidence level. 99 Table 6. 7. Summarized optimal solutions of CCP harvest scheduling models. Input/output CCP 1 CCP 2 CCP 3 Y , (1000 m3) 350 - 500 350 - 500 $500 1’, (1000 m3) 500 - 650 500 - 650 Y a (1000 m3) 650 - 800 650 - 800 Y,- Y,2(1000 m3). 650- 800 a. .05 .05 .05 Max. increase (%) 100 100 Max. decrease (%) 10 10 Total NPV (1000,000 Rp) 271,824 263,770 255,847 Total Volume (1000 m3) 10415 9771 8429 IL_nman§ge_d land (%) 0 4 20 Note: Y, = harvest volume in period 1'; empty cells = no relevant input/output Another modification was to constrain harvest volumes in periods 2 and 3 (in addition to the volume in period 1) to be within explicit upper- and lower-bounds, while other constraints remained the same. The intention was to postpone some harvests to later periods. Thus, harvest volumes in periods 2 and 3 were restricted within the ranges of 500,000 - 650,000 and 650,000 - 800,000 m3 , respectively. These ranges are below the corresponding harvest volumes resulting from CCP 1. This modified model, labeled CCP 2, did result in a flatter harvest-flow trajectory compared to that of CCP 1 (Figure 6.3b). However, it also reduced the total NPV and lefi about 4% of the total forest area unmanaged (Table 6.7). Finally, the model was modified by assigning explicit lower- and upper-bounds to the harvest volumes in all periods. Harvest volume in period 1 was restricted to be 500,000 m3 or less, harvest volume in period 2 was bounded within 500,000 - 650,000 m3 and harvest volumes starting in period 3 thereafier were restricted to be within 650,000 - 800,000 m3. This model (CCP 3) 100 resulted in a harvest flow in Figure 6.3c. Except for the declines in periods 8 and 11, harvest volumes are relatively constant starting in period 3. The total NPV, however, is lower than both CCP l and CCP 2, and about 20% percent of the forest is unmanaged. Moreover, the upper bounds were subjectively chosen, meaning the total NPV may not be the forest's maximum NPV. Period (10 years) Figure 6. 3a. Harvest-flow pattern if solutions of CCP 1 is implemented. WWI-MUM n13) Period (10 years) Figure 6. 4b. Harvest-flow pattern if solutions of CCP 2 is implemented. 101 Harvutvolllmflmm) o§§§§§§§§§ Period (10 years) Figure 6. 3c. Harvest-flow pattern if solutions of CCP 3 is implemented. Optimal hectare allocation according to the CCP models are presented in Table 6.8. For comparison purposes, optimal hectare allocation according to LP 5 are also presented. LP 5 allocates about 60% of the forest to the shortest (60 years ) rotation. The other 40% is evenly distributed to rotation ages 70 and 80 years. In general, all CCP models follow a similar allocation pattern with CCP l allocation to the shortest rotation being the largest (74%). This higher portion of early harvests explains the highest NPV associated with CCP l. CCP 2, in which some delays of harvests are imposed, allocates 63% of the land to the shortest rotation; 11 % lower than that of CCP l. CCP 3, in which periodic harvest volumes throughout the planning horizon are restricted within lower- and upper-bounds, also allocates 63% of the land to the 60-year rotation. However, the allocation to the longer rotation ages are smaller than CCP l and CCP 2 resulting in one-filth (20%) of the total forest land lefi unrrranaged. From their outputs, none of the CCP models is clearly better than the others. The final decision is decision makers. If attaining a stable harvest-flow is the main concern, CCP 3 may be the choice; however CCP l which gives the highest NPV. 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OPEN ”c:\teak\eomdatl .csv" FOR INPUT AS #1 OPEN "c:\teak\nrntab80.csv' FOR INPUT AS #2 OPEN "c:\teak\H-coefl .csv" FOR INPUT AS #5 OPEN ”c:\teak\BA-coef.csv" FOR INPUT AS #8 OPEN 'c:\teak\n-nonn80.csv" FOR INPUT AS #7 OPEN 'c:\teak\naverage.csv' FOR INPUT AS #10 OPEN 'c:\teak\comre381.csv' FOR OUTPUT AS #3 OPEN "c:\teak\combin80.csv" FOR OUTPUT AS #20 OPE)! 'c:\teak\Price80.csv" FOR OUTPUT AS #22 FOR R = 1 TO 10 FOR C = 1 TO Rot INPUT #2, A(R, C) INPUT #7, AA(R, C) NEXT C NEXT R CLOSE #2 CLOSE #7 FOR G = 1 TO 10 INPUT #10, Naver(G) NEXT G CLOSE #10 FOR RR = 1 TO 10 FOR CC = 1 TO 2 INPUT #5, B(RR, CC) INPUT #8, D(RR, CC) NEXT CC NEXT RR CLOSE #5 CLOSE #8 DO INPUT #1, nos, StandS, Hectares, Initage, InitN, Bonita, InitH, InitBA Row = Bonita ‘ 2 - 1 FOR Col = 1 TO 16 NRN(Col) = A(Row, Col): NNorm(Col) = AA(Row, Col) NEXT C01 136 137 Appendix B. (cont’d). Alpha = B(Row, 1): Beta = B(Row, 2): Alpha] = D(Row, 1): Beta] = D(Row, 2) IFInitage> 80THEN Projage = lnitage: Period = 2: Rot2 = 2: Period2 = 17: Rot3 = 18 ELSElFInitage>75ANDInitage<=80THEN Projage = 80: Period = 2: Rot2 = 2: Period2 = 17: Rot3 =18 ELSEIF Initage >= 1 AND lnitage <= 75 THEN FORI= l TO(Rot- l) Upper=I‘ 5:Lower=Upper-5 IFInitage>LowerANDInitage <= UpperTHEN Projage =Upper. Period =Rot+ l -I: Rot2 = Period+ l: Per2 = Period+ Rot IF Per2 < 24 THEN Period2 = Per2: Rot3 = Period2 + l ELSEIF Per2 >= 24'I'HEN Period2 = 24: Rot3 = 24 ENDIF ENDIF NEXTI ENDIF IF Projage > Harvage THEN K = Rot ELSEIF Projage <= Harvage THEN K = Projage/ 5 END IF NormalN = NNorm(K) IF InitH > 0 THEN InitH=InitH ELSEIF InitH = 0 THEN InitH = EXP(Alpha + Beta ‘ LOG(Initage)) ENDIF IF InitBA > 0 THEN InitBA = InitBA ELSEIF InitBA = 0 THEN Relba = InitN lNormalN: InitBA = Relba ‘ (Alle + Betal ‘ LOG(Initage)) END IF Agent = Initage I Projage N(l) = InitN BA(l) = EXP(Agerat ’ LOGanitBA)+ 2.927 ’ (l - Agerat)+ .044 ‘ (l - Agerat) ‘ InitH) Envelope = EXP(.303 ‘ (l - Agerat) ‘ InitH) BAGrow = BA(l) - InitBA IF BAGrow > Envelope THEN BA(1) = InitBA + Envelope END IF H(l) = EXP(2.57S - .143 1' LOG(N(1)/Projage) + .341 ' LOG(BA(1))) W) = EXP(1.739 + .034 ' LOG(InitH) + .952 ' Agerat ' LOG(InitBA)+ 1.796 . (1 -Agerat) + .092 ' (1 -Asmt)‘lnitH) NR(1)=NRN(K) AvBA=BA(1)/N(l) AVD(1)=(Q ‘AvBA)" .5 IFAVD(1)>= 30THEN Pr(1)= .53 138 Appendix B. (cont’d). ELSEIF AVD(1) < 30 ANDAVD(1)>= 20 THEN Pr(l)= .275 ELSEIFAVD(1)<20ANDAVD(1)>=4'I'HEN Pr(l)= .155 ELSEIF AVD(1)<4THEN Pr(1)=0 ENDIF IFBonita>2THEN IFN(1)>1.1'NR(1)THEN Nrat=NR(l)/N(l): BAR(1)=1.074‘Nrat‘BA(1): VT(1)=V(1)-(l.048‘Nrat‘V(l)) ELSEIFN(1)<=1.1'NR(1)THEN BAR(1)=BA(1):VT(1)=0 ENDIF ELSEIFBonita<=2THEN BAR(1)=BA(1): VT(l)=0 ENDIF FORP=2TOPeriod IFBonita>2THEN IFN(P-1)>1.l'NR(P-1)THEN N(P)=NR(P-l): BA(P-l)=BAR(P-l) ELSEIFN(P-1)<=l.l‘NR(P-1)THEN N(P)=N(P-l): BA(P-l)=BA(P-l) ENDIF ELSEIFBonita<=2THEN N(P)=N(P-l): BA(P-l)=BA(P-l) ENDIF Agmt = Projage I (Projage + 5) LnBA =Agerat ‘ LOG(BA(P - 1)) + 2.927 ‘ (1 - Agerat) + .044 " (l - Agerat) ‘ H(P - l) BA(P) = EXP(LnBA) Envelope = EXP(.303 ‘ (l - Agent) " H(P - 1)) BAGrow = BA(P) - BA(P - 1) IF BAGrow > Envelope THEN BA(P) = BA(P - l) + Envelope END IF H(P) = EXP(2.57S - .143 ' LOG(N(P) / (Projage + 5)) + .341 . LOG(BA(P») V(P) =EXP(1.739 + .034 2 LOG(H(P- 1))+ .952 ‘Agerat ' L0003A(P-1))+1.796 '1 (1 ~Agerat)+ .092 * (1 - Agent) ' H(P-1)) IF(Projage+5)>HarvageTl-IEN K=Rot ELSEIF (Projage+5)<=HarvageTHEN K=(Projage+5)/5 IF NRG’) = NRN(K) AvBA = BA(P) I N(P) AVD(P) = (Q * AvBA) A .5 IF AVD(P) >= 30 THEN mp) = .53 ELSEIF AVD(P) < 30 AND AVD(P) >= 20 THEN Pr(P) = .275 139 Appendix B. (cont’d). ELSEIFAVD(P)<20ANDAVD(P)>=4TEEN Pr(P)=.155 ELSEIFAVD(P)<4TEIEN Pr(P)=0 ENDIF IF Bonita > 2 TEEN IFN(P)>1.1‘NR(P)THEN Nrat = NR(P) / N(P): BAR(P) = 1.074 ’ Nrat ‘ BA(P): VT(P) = V(P) - (1.048 ‘ Nrat ‘ V(P)) ELSEIFN(P)<=1.1’NR(P)THEN BAR(P) = BMP): VT(P) = 0 END IF ELSEIF Bonita <= 2 TEEN BAR(P) = BA(P): VT(P) = 0 END IF Projage = Projage + 5 NEXT P NormalN = NNorm(l) A85501012) = 5 N(Rot2) = Naver(Row) NR(Rot2) = NRN( l) H(Rot2) = EXP(Alpha + Beta ‘ LOG(Age(Rot2))) Relba = N(Rot2) / NormalN BA(Rot2) = Relba " (Alle + Betal ' LOG(Age(Rot2))) V(Rot2) = EXP(-l.4 + 1.248 ‘ LOG(H(Rot2)) + .922 ‘ LOG(BA(Rot2))) AvBA = BA(Rot2) I N(Rot2) AVD(Rot2) = (Q ‘ AvBA) " .5 IF AVD(Rot2) >= 30 THEN Pr(Rot2) = .53 ELSEIF AVD(Rot2) < 30 AND AVD(Rot2) >= 20 TEEN Pr(Rot2) = .275 ELSEIF AVD(Rot2) < 20 AND AVD(Rot2) >= 4 TEEN Pr(Rot2) = .155 ELSEIF AVD(Rot2) < 4 TEEN Pr(Rot2) = 0 END IF IF Bonita > 2 THEN IF N(Rot2) > (1.1 ‘ NR(Rot2)) TEEN Nrat = NR(Rot2) I N(Rot2) BAR(Rot2) = 1.074 "‘ Nrat ' BA(RoQ) VT(Rot2) = V(Rot2) - (1.048 ‘ Nrat ‘ V(Rot2)) ELSEIF N(Rot2) < (1.1 ‘ NR(Rot2)) THEN BAR(Rot2) = BA(Rot2) VT(Rot2) = 0 END IF ELSEIF Bonita < 2 TEEN BAR(Rot2) = BA(Rot2) VT(Rot2) = 0 END IF M=1 140 Appendix B. (cont’d). FOR PP = (Rot2 +1)TOPeriod2 Age(PP)=S ‘M+S: Kol=Age(PP)/5: NR(PP)=NRN(Kol) IFBonita>2THEN IFN(PP-l)>(l.l ‘NR(PP-l))THEN N(PP)=NR(PP- l): BA(PP- l)=BAR(PP- l) ELSEIFN(PP-l)<(l.l ‘NR(PP-l))THEN N(PP)=N(PP- l): BA(PP- l)=BA(PP- 1) ENDIF ELSEIFBonita¢2TEEN N(PP)=N(PP- l): BA(PP- l)= BA(PP- l) ENDIF Agent = Asefl’P - 1) / A8601”) LnBA =Agerat ' LOG(BA(PP - 1))+ 2.927 " (l -Agerat) + .044 ' (l -Agerat) ’ H(PP - l) BA(PP) = EXP(LnBA) Envelope = EXP(.303 ' (l - Agerat) ‘ H(PP - 1)) BAGrow = BA(PP) - BA(PP - 1) IF BAGrow > Envelope TEEN BA(PP) = BA(PP - l) + Envelope END IF H(PP) = EXP(2.575 - .143 . LOG(N(PP) / Age(PP)) + .341 5 LOG(BA(PP)» V(PP) = EXP(1.739 + .034 ' LOG(H(PP -1))+.952 ' Agerat * LOG(BA(PP -1))+ 1.796 *(1- Agerat) + .092 r (1 - Agerat) ‘ H(PP -1)) AvBA = BA(PP) / N(PP) AVD(PP) = (Q . AvBA) A .5 IF AVD(PP) >= 30 THEN Pr(PP) = .53 ELSEIF AVD(PP) < 30 AND AVD(PP) >= 20 THEN Pr(PP) = .275 ELSEIF AVD(PP) < 20 AND AVD(PP) >= 4 THEN Pr(PP) = .155 ELSEIF AVD(PP) < 4 THEN Pr(PP) = 0 END IF 1F Bonita > 2 THEN IF N(PP)>(1.1' NR(PP» THEN Nrat = NR(PP) / N(PP): BAR(PP) = 1.074 # Nrat * BA(PP): VT(PP) = V(PP) - (1 .048 ‘ Nrat ' V(PP)) ELSEIFN(PP)<(1.1‘NR(PP))THEN VT(PP) = o BAR(PP) = BA(PP) END IF ELSEIF Bonita <= 2 THEN VT(PP) = o BAR(PP) = BA(PP) END IF M=M+l NEXT PP IF Period2 < 24 THEN FOR PPP = Rot3 TO 24 V(PPP) = V(PPP - Rot): VT(PPP) = VT(PPP - Rot): Pr(PPP) = Pr(PPP - Rot) NEXT PPP END IF 141 Appendix B. (con’t). PRINT "Stand No"; no PRINT PRINT n03; " "; Stands; USING "####.##"; V(l); VT(I); V(2); VT(Z); V(3); VT(3); V(4); VT(4) PRINT PRINT no = no + l PRINT #3, USING "\ \\ \#.# ###.# ##"', 110$; Stands; Bonita; Hectares; Initage; FOR P = 1 TO 8 PRINT #3, USING ”####.## "; V(P); VT(P); NEXT P ' PRINT #3, '"' PRINT #20, USING "#.# ###.# ##"°, Bonita; Hectares; Initage', FOR P = 1 TO 24 PRINT #20, USING "####.## "; V(P); VT(P); NEXT P PRINT #20, '"' PRINT #22, USING "#.# ###.# ##"; Bonita; Hectares; Initage; FOR P = 1 TO 24 PRINT #22, USING ”####.## '; Pr(P); NEXT P PRINT #22, '"' LOOP WHILE NOT EOF( 1) END Second Part CLS DIM V(48), VX(48), VT(48), VTX(48), X(51), Y(48), Pr(48), PrX(48) DIM R(48), C(48), TCost(48), M3(24), Rp(24), t(24), PNV(24) PCost = .19: GCost = .055: HCost = .022 Rot = 16: R = .09: N0 = 0 OPEN 'c:\teak\Combin80.csv" FOR INPUT AS # 1 OPEN ”c:\teak\Price80.csv" FOR INPUT AS #2 OPEN "c:\teak\Volume81.csv" FOR OUTPUT AS #3 OPEN 'c:\teak\Volume83.csv" FOR OUTPUT AS #5 OPEN "c:\teak\Value81.csv" FOR OUTPUT AS #6 OPEN "c:\teak\Value83.csv" FOR OUTPUT AS #8 DO TotPNV=0 FORI==1T051 INPUT #1,X(I) NEXTI Bonita = X(l): Hectares = X(2): Initage = X(3) FOR I = 2 TO 25 VX(I-l)=X(I"‘2): VTX(I- l)=X(I’2+ l):VX(I+23)=0:VTX(I+23)=O NEXTI 142 Appendix B. (cont’d). FOR I = 1 TO 24 INPUT #2, PrX(I): PrX(I + 24) = 0 NEXT 1 FORI=1T024 va+7)=VX(I): VT(I+7)=VTX(I): Pr(1+7)=PrX(I) NEXTI FORI=1TO7 V(I)=0:VT(I)=0:Pr(I)=0 NEXTI FORI=32TO48 V(I)=0:VT(I)=0:Pr(I)=0 NEXT] FORI=1TO48 IFVT(I)>0THEN TCost(I)=.065 ELSEIFVT(I)=0THEN TCost(I)=0 ENDIF NEXTI IFInitage>1ANDInitage<=80TEEN FORI= l TORot Up=l'5: Low=Up-5 IFInitage>LowANDInitage<=UpTEEN Harvl=Rot+1-I+7 '7isthetricktoavoidneg.period A751 =Harvl - l: A701 =Harv1 -2: A551 =Harvl -5 A501 =Harvl -6: A451 =Harvl -7 Rot2=Harvl +1 Harv2 =Harv1 +Rot: A752 =A751+Rotz A702 =A701+Rot A552 = A551 + Rot: A502 = A501 + Rot: A452 = A451 + Rot Rot3=Harv2+ l: A453=A452+Rot ENDIF NEXTI ELSEIFInitage>80TEEN Low=Up-5: Harvl=l+7 A751 =Harvl - 1: A701 =Harvl -2: A551 =Harvl -5:A501 =Harvl -6: A451 =Harvl -7 Rot2=Harvl + 1: Harv2=Harvl +Rot: A752=A751+Rot° A702=A701+Rot A552=A551+Rot A502=A501+Rot2 A452=A451+Rot: Rot3 =Harv2+ l: A453=A452+Rot ENDIF IF Bonita > 2 THEN FORI=1TOA451 Y0) = VT(I): R0) = Y(I) ' Pr(I): CO) = TCost NEXTI FORI=A501TOA701 STEP2 Y0) = VT(I): R0) = Y(I) ' Pr(l): CO) = TCost NEXTI FORI=A551TOA751 STEPZ Y(I)=0:R(I)=0: C(I)=o 143 Appendix B. (cont’d). NEXT I Y(Harvl) = V(Harvl): R(Harvl) = Y(Harvl) ‘ Pr(Harvl): C(Harvl) = GCost + CCost Y(Rot2) = VT(Rot2): R(Rot2) = Y(Rot2) ‘ Pr(Rot2): C(Rot2) = PCost + TCost FOR I = (Rot2 + 1) TO A452 Y(I)=VT(I): R(I)=Y(I)‘Pr(I):C(I)=TCost NEXT I FOR I = A502 TO A702 STEP 2 Y(I) = VT(I): R0) = Y(I) ‘ Pr(l): C(I) = TCost NEXT I FOR I = A552 TO A752 STEP 2 Y(I)=0:R(I)=0:C(I)=0 NEXT I Y(Harv2) = V(Harv2): R(Harv2) = Y(Harv2) ‘ Pr(Harv2): C(Harv2) = GCost + CCost Y(Rot3) = VT(Rot3): R(Rot3) = Y(Rot3) ‘ Pr(Rot3): C(Rot3) = PCost + TCost FOR I = (Rot3 +1)TO A453 Y(I) = VT(I): R(I) = Y(I) ‘ Pr(I): CO) = TCost NEXT I ELSEIF Bonita <= 2 TEEN FORI=1TO(Harvl-I) Y(I)=0: R(I)=0: C(I)=0 NEXTI Y(Harv1)= V(I-Iarvl): R(Harv1)= Y(Harvl) ‘ Pr(Harvl): C(Harvl) = GCost + CCost Y(Rot2) = 0: R(Rot2) = 0: C(Rot2) = PCost FORI=(Rot2+ l)TO(Harv2- l) Y(I)=0: R(I)=0: C(I)=0 NEXTI Y(Harv2) = V(Harv2): R(Harv2) = Y(Harv2) ‘ Pr(Harv2): C(Harv2) = GCost + CCost Y(Rot3) = 0: R(Rot3) = 0: C(Rot3) = PCost FOR I = (Rot3 +1)TO A453 Y(I)=0: R(I)=0: C(I)=0 NEXTI END IF IFInitage>1ANDInitage<=15THEN RemV = V(24) - VT(24): RemVal = RemV “ Pr(24) ELSEIF Initage>15 AND Initage <=20 TEEN RemV = V(24): RemVal = V(24) ‘ Pr(24) ELSEIFInitage>20ANDInitage<=25THEN RemV = V(24) - VT(24): RemVal = RemV ‘ Pr(24) ELSEIFInitage>25ANDInitage<=30TEEN RemV = V(24): RemVal = V(24) ‘ Pr(24) ELSEIFInitage>30ANDInitage<=35TEEN RemV = V(24) - VT(24): RemVal = RemV ‘ Pr(24) ELSEIFInitage>35ANDInitage<=40THEN RemV = V(24): RemVal = V(24) ’ Pr(24) ELSEIF Initage>40ANDInitage <=45TEEN RemV = 0: RemVal = 0 ELSEIF Initage > 45 THEN RemV = V(24) - VT(24): RemVal = RemV ‘ Pr(24) END IF FORI= l T024 M3(I)=Y(1+7)IRP(D=R(1+7)-C(I+7)It(1)=5 ’13 P1W(I)=RP(I)/((l +R)"t(1)) 144 Appendix B. (cont’d). TotPNV = TotPNV + PNV (I) NEXT I PRINT #3, USING "#.# ###.# ##"; Bonita; Hectares; Initage; FOR P = 1 TO 12 PRINT #3, USING ”####.##"; M3(P); NEXT P PRINT #3, ’"' FOR P = 13 TO 24 PRINT #5, USING ”####.##"; M3(P); NEXT P PRINT #5, RemV PRINT #6, USING "#.# ###.# ##"; Bonita; Hectares; Initage; FOR P = 1 TO 12 , PRINT #6, USING '####.##"; Rp(P); NEXT P PRINT #6, ’"' FOR P = 13 TO 24 PRINT #8, USING "####.##"; Rp(P); NEXT P PRINT #8, USING "####.## ####.###"; RemVal; TotPNV N0 = NO + l PRINT NO FOR P = 13 TO 18 PRINT USING "####.##"; Rp(P); NEXT P PRINT USING '####.## ####.###"; RemVal; TotPNV LOOP WHILE NOT EOF(1) END APPENDIX C: The CCP SOLVER Spreadsheet Appendix C. The CCP SOLVER spreadsheet. The CCP SOLVER spreadsheet is structured as shown in Figure 6.2. It can be partitioned into several sections. Section 1: model inputs. Column A: Column 8: Cells C1, CZ, 03: Columns D to 0: Columns P to AA: decision-variable coeflicients 0”. total hectares available for each stand-type (the right-hand-sides L, of the land-area constraints). respectively, parametersB, u, and I. per-hectare yield am. Var(a,-j,). Section II: objective fimction, land-area constraints, and periodic harvest volumes. Column AB: Cell A8106: Column AC: Cell A0106: Column AD: decision variables xv (i.e., hectares of stand-type i allocated to management regime or rotation-age 1). sum of cells A31 to A8105 (i.e., the total hectares of all stand-types allocated across the rotation-age alternatives). objective-fimction components 01f”. sum of cells AC1 to A0105, the objective function. total hectares of stand type i allocated to rotation-age j. Since there are 35 stand-types and three rotation-ages, the first 35 cells ofcolumn AB contain hectares ofstand—type i (i = l, 2 35) allocated to rotation-age j = l (i.e., 60 years). Likewise, the next 35 cells are for those allocated to rotation-age j = 2 or 70 years), and the last 35 cells are for those allocated to rotation-age j = 3 or 80 years. Hence, values contained in column AD (total hectares of stand type i allocated to rotation-age j) are give by AB(i) + AB(i+35) + AB(i +70) for r‘ = 1,2..., 35. 145 146 Appendix C. (cont’d) Columns AE to AP: awry, (total yield produced in period t (t = l, ,12)) from stand-type i under rotation-age j. Cells in row 106 contain the sums of their respective columns indicating total periodic harvest volumes of the corresponding period. Thus, cell AE106 is the sum of cells AE1 to A5105; the total harvest volume in period 1. Section III: chance-constraint components. Columns A0 to BB: Var(a,.j,) x02. Columns BC to BM: (l+u)a,.j,x,j - aimflfcg. Columns BN to BX: Var((l+u)a,fl - aW+,))xU2 Columns BY to CJ: (l-I)a,j,r,.j - “mafia: Columns CK to CV: Var((l-I)a,.j, - ay.(,+,))x,f Column totals are located in row 106. Thus, 2 [Var(am)xy.2 )] ( t= l,...,12) are contained in cells A0106 to 88106. Likewise, E [(l+u)a,.j,1c,.j - atj(t+1)xij] (t = l, ...,12) are in cells 8C106 to BN106, 2 [Var((l+u)a,.j, - ay.(,+,))x,j2] (t = l,...,12) are in cells 30106 to 82106, 2 [(l-Daijrxg. - aymlfyzl (t = 1,... ,12) are in cells CA106 to CL106, and 2‘. [Var((l-I)a,fl - cry-(HUM)? ] (t = 1,... ,12) are in cells CM106 to CX106. These column totals are used to form the lefi-hand-sides (LHS) of chance constraints 6.7 6.10. First, the quantity [3(K(x2))-s "' in which K = Var(a,j,) or Var((l+u)a,.j, - aw” 1)) or Var((l-I)a,j, - gym») -- are computed in row 107. LHS ofconstraints are given in row 108 and 109. For t = 1, the LHSs of constraints 6.7 6.10 are, respectively, given by: AE108 - AQ108, AE109 + AQ109, BC108 - B0108, and 147 Appendix C. (cont'd) BY108 + CK 108. 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