33

NZOR volume 8 number 1 January 1980

FOREST HARVEST SCHEDULING IN FIJI: A COMPARATIVE

CASE STUDY OF LINEAR PROGRAMMING, HEURISTIC

AND DECOMPOSITION TECHNIQUES

C.A, de Kluyver, E c o n o m i c s D e p a r t m e n t

A.G.D, Whyte, S c h o o l of F o r e s t r y

and

F.T. Baird, M. Boon, J.T. Buch a n a n , G. Eng,

J .W . Griffin, I.E. Grayburn, M.R. Moore

T.A. Moore, H.V. Tran, a nd B.G. White

(1979 Grad u a t e Class in O p e r a t i o n s Research;

U n i v e r s i t y of C a n t e r b u r y

SUMMARY

This paper describes, in case study form, the formulation and solution of a large-scale forest harvest scheduling problem in Fiji.The objective of the problem is to maximize net discounted profits over a period of seven years subject to constraints on (1) minimum contracted supplies, (2) port utilization, (3) efficient use of available manpower, logging equipment, and transport vehicles, and(4) considerations of implementation.

Three types of scheduling methods are contrasted: (1) a linear programming (LP) formulation involving more than 1200 variables and approximately 750 constraints for a variety of scenarios using the TEMPO mathematical programming package; (2) a heuristic solution procedure based on the application of a variant of Vogel's Approxi­mation method for transportation problems and (3) the use of Dantzig- Wolfe decomposition in conjunction with heuristic allocation on an aggregated much smaller LP model. Alternative solutions generated are compared in terms of managerial tractability and computational efficiency. Four appendices detailing the basic data used, the intermediate calculations performed and the final solutions derived accompany this article to allow its use in case study form.

1. INTRODUCTIONThis article describes the formulation of, and various

solutions to, a large-scale forest harvest scheduling problem, the objective of which is to maximize net dis­counted profits over a planning horizon of seven years subject to constraints on (1) meeting minimum contracted

34

supplies, (2) port utilization, (3) efficient use of available manpower, logging equipment and transport vehicles, and (4) a number of factors pertaining to the implementation of cutting schedules such as the development of a cutting sequence guaranteeing easy access, smoothed cashflow, and other managerial considerations.

The study concerns the rapidly expanding plantation resource principally consisting of Caribbean pine belonging to the Fiji Pine Commission, an independent statutory body responsible for the administration of communally owned private land. The work of the Commission has so far concen­trated on the establishment of around 25000 hectares of plan­tations funded by loans from the Government of Fiji and the Commonwealth Development Corporation, together with a large annual input of New Zealand aid, mainly in the form of staff, training, research, and equipment. The continued develop­ment of this project to a proposed eventual size of 75000 ha is dependent upon the ability to generate income from existing plantations. As a result, the Commission has adopted as one of its stated policies the requirement to (i) generate as much income as possible as early as possible from existing stands while (ii) minimizing capital expendi­tures for industrial development.

The School of Forestry at the University of Canterbury has been involved in several training and research projects for the Fiji Pine Commission over the last few years. As a result of this close association, a substantial data base has been generated which could prove useful in assessing how the stated objectives can best be attained. It was decided therefore, to use these and other pertinent data in a case study exercise executed by the 1979 graduate O.R. class as part of a large-scale optimization course which comprises one of the requirements of the graduate O.R. cur­riculum at the University of Canterbury. Three groups were established, each of which was required to contribute to a workable solution strategy in a distinctive manner. Speci­fically, each group was responsible for the evaluation of a different solution methodology ranging from a full scale linear programming (LP) approach, a heuristic scheduling algorithm, to an aggregated LP approach using decomposition principles. The two principal authors acted as project supervisors, primarily concerned with the modeling process followed and the incorporation of managerial considerations in the evaluation of proposed solution methodologies and harvesting schedules.

2. THE FOREST HARVEST SCHEDULING PROBLEM: BACKGROUNDJohnson and Scheuman [ 2] have surveyed the area of

harvest scheduling and distinguish between two types of LP models which form the basis for techniques commonly used to solve forest harvest scheduling problems. The principal

35

difference between the models concerns the concept of a management unit: in models of type I, management units are constructed according to starting-age and kept intact throu­ghout; whereas in type II models, management units are kept intact only until harvesting occurs, at which time new management units are created. The models developed in this study are of the first kind.

Approximately 3800 hectares of Caribbean and slash pine were planted in the North-West of the main island of Fiji prior to 1972, (see Illustration 1). Simulations of fore­casted yields by various analysts [ 8] appear to show that most, if not all, of the produce from these areas will probably not be needed to sustain yields of 500000 m 3 or more between 1986 and 1992. They provide, therefore, a re­source which can be liquidated to best financial advantage between 1979 and 1985.

To examine various different liquidation policies, the 3800 ha were divided into 29 areas representing distinct crops of one species, one age class in clearly demarcated localities. To allow solution by mathematical programming techniques, these 29 areas were later aggregated into 15 areas of a single species covering one or more age classes and having somewhat less well-defined boundaries.

Forecasted total harvestable yields were derived both on a 29 and a 15-area basis for a time horizon spanning 1979-1985 using a yield forecasting model for Caribbean pine [ 8] and a projection scheme based on past measurements for slash pine [7]. The data are presented in Tables la and Id, respectively, of Appendix I and are defined in terms of merchantable amounts in m 3 that can be realized in a given area if cut in its entirety in a given year. Yields may be realized in two principal forms: pulpwood and sawlogs (some posts and poles are also produced). Price/log size gradients exist for both sawlogs and pulpwood: for sawlogs, the curve rises steeply to age 15 years and then tends to flatten off whereas for pulpwood the steep rise ends at an approximate age of 11 years. Figure Ii in Appendix I shows an average curve for both sawlogs and pulpwood constructed on the basis of these data. Yields are presented in net terms. Thus, revenue obtained is defined as the product of the proportion cut, the given yield and the relevant price per m 3.

Four different logging methods have been advocated for logging the resource [ 1] , two labour intensive and two mac­hine intensive (see Illustration 2). Sawlogs can only be obtained by either of the machine intensive methods. Of these two methods, one, by skidder, can be used only on flat ground or where slopes are moderate (less than 45%). Extra­ction by cable yarders, the other kind of machine logging, is confined to slopes in excess of 45% and can yield both sawlogs and pulpwood. Where only short-length pulpwood, posts, or poles are to be removed, plastic chutes can be

36

ILLU

STRA

TION

1:

Plantation

Areas, Fi

ji

37

Fi g- 2 : SKIDDER LOGGING ( k » 2 )

ILLUSTRATION 2: Logging Methods

38

employed on slopes of less than 45%, while on relatively flat topography manual stacking alone will suffice. Bundles of stacked shortwood are then picked up by forwarders.

Harvesting operations may be summarized in terms of the following data:(1) the proportion of each area by species and localities of

terrain suitable for each of the four logging methods[ 1] ;

(2) the number of hectares that can be cleared by each logging method if employed for one full year [ 1] ;

(3) the proportion of each area in which extraction of saw- logs is warranted [ 3] ;

(4) the total costs of felling, extracting, and preparing a m 3 ready for loading onto a truck, including the capital costs of equipment and provision for maintenance, depreciation, supervision, roading, wages, and bundling[ 1] ;

(5) cost/size gradients for harvesting sawlogs and pulpwood of different ages by each of the four methods [ 3] .Costs of transporting one m 3 of wood from each area to

each of two ports, Vatia and Lautoka, were derived from estimates contained in [ 1] . Minimum port requirements in m 3 for each of the seven years were set for the port of Vatia while maxima were recognized for Lautoka. Combined minimum restrictions imposed represent minimum contracted supplies in any given year, and are modeled to increase gradually over the planning horizon in anticipation of larger wood supplies after 1985.

Finally, consideration must be given to a priority ranking over areas in terms of cutting from a management point of view. For example, heavily windthrown and slash pine areas should be cleared as early as possible. Addi­tional managerial judgment concerns equipment use: the num­ber of heavy logging machines to be acquired should be made to increase smoothly over the seven year planning horizon.

The data described above are all summarized (in con­densed form) in the remaining Tables and Figures of Appendix I for both problem definitions (29 areas and 15 aggregated areas), and form the basis for the modeling techniques and the resulting schedules to be discussed. The remainder of this article is structured as follows: First, a LP formula­tion of the problem involving the aggregated 15-area data is discussed and its solutions analyzed. Next, a heuristic solution technique based on Vogel's approximation method for transportation problems is outlined for the large 29-area problem and its solution evaluated both on a 29-area and a 15-area basis. Finally, a condensed LP formulation is exami­ned for the 15-area problem using both heuristic and decom­position techniques. A comparison of methods and solutions in terms of managerial tractability and computational effi­ciency forms the conclusion of the article.

39

3. A LINEAR PROGRAMMING MODEL OF FOREST HARVEST SCHEDULINGThe LP model developed for the forest harvest scheduling

problem is a variant of a model originally proposed by Tcheng[5]. Its formulation employs two types of constraints; those relating to restrictions imposed on logging practices in specific areas and those pertaining to shared resources over all areas. Specifically, area constraints include:(1) constraints ensuring that areas are felled only once

during the planning horizon,(2) cutting method constraints for each area derived from

topographical considerations, and(3) minimum and maximum desirable cut-levels for each area

in each year of the planning horizon,whereas shared resource constraints include:(1) individual and combined port requirements (demand),(2) restrictions on method use over the planning horizon, and(3) constraints reflecting preferences on cutting sequences

over the planning horizon.The model is designed to formulate a cutting schedule

over 15 areas and 7 years using 4 possible cutting methods and potential routing of log shipments through 2 ports. Its formulation employs the following notation:

i = area index, i=l,2 ,...,I j = year index, j=l,2 ,...,J k = method index, k=l,2 ,...,K m = port index, m=l,2r. = number of hectares available for cutting in area 1 i /

g^j = total yield of area i in m 3, if cut in year j ,X. = proportion of area i cut in year j using method

-* k with yield destined for port m,Dj = combined annual requirement for both ports,djm = individual port requirements in year j ,N . = equivalent number of annual applications of

-1 machine intensive cutting method k in year j , area (k=2,4),

M. = equivalent number of annual applications of 1-1 labour intensive cutting method k in year j ,

area i (k=l,3),f.. = number of hectares that can be cleared in one 1-' full years' application of machine intensive

cutting method k in area i (k=2,4),e . . = number of hectares that can be cleared in one 1-1 full years' application of labour intensive

cutting method k in area i (K=l,3).

40

= maximum or minimum number of hectares to be rv cleared using machine intensive method k in

area i (k=2 ,4),= maximum or minimum number of hectares to be

cleared using labour intensive method k in area i (k=l,3).

The resulting LP formulation seeks to maximize total discounted profits, given by revenue less cutting and transportation costs, where

Revenue = E P. ( £ E E g. .X. ) j 3 i k m ^ m k m

Cutting = £ £ z n +Costs . . k=2_4 ijk l : k . . k = 1 3 i]k i3kTransport =Costs . . lim , likm1 3 m k

where Pj denotes the price/m3 realized in year j, Cijm the transport cost/m3 from area i to port m in year j, nijk and mijk the costs of applying cutting method k for one full year in area i in year j.

The constraint structure consists of:(1) Restrictions ensuring that stands are cleared only once

during the planning period:E E E X. .. < 1; i=l,2,...,I (15 constraints). . i]km -j k m J

(2) Cutting Method Restrictions:E r.X. .. < f. .N. ; k = 2,4 i = l,2,...,I m 1 13km - ^ 13k j = l , 2 .... JE r.X. < e. .M. .. ; k=l,3 (420 constraints). 1 i]km - i] i]k

(3) Individual and Combined Annual Port Requirements:E (g. .£ X ) - Y = d.,; j=l,2,...,J (port 1) (min)^ 1 J K -L j x

E (g . . E X . ., „) + Y . „ = d . „ ; j=l, 2 , . . . , J (port 2) (max) i iDk i]k2 j 2 j 2

(14 constraints)Y. - y. > D. - d . - d .0 ; j=l,2,...,J (both ports)(min) ]1 1 j2 — j jl j 2 J c

(7 constraints)where Yjj_ represents a variable modeling annual surplus throughput for port 1 and Yj2 depicts annual shortfall of

41

throughput for port 2 .(4) Cutting Method Restrictions within areas:

r . E E X. < F ._ ; i=l,2,...,I (max use, method 2) l . i-j2m — i2 -] m Jr.E EX... > F . ; i=l,2,...,I (min use, method 4) i . in 4m — i4 i inJ (30 constraints)

(5) Overall maximum use of method 2:E N . . < 1 ; j=l,2,...,J (7 constraints)ij2 —

The formulation given above assumes that returns obtained and cutting costs incurred are independent of stand age. This assumption was relaxed through the use of a second objective function in which the age of stands was taken into account (see Figure Ii in Appendix I). The objective func­tions will be referred to as OBJ 1 and OBJ 2, respectively.

Earlier in this section reference was made to the dis­counting of resulting profit figures. To achieve high levels of revenue generation early in the planning horizon, the early felling of older areas was encouraged. To some extent, as suggested by the problem data, this already occurs natu­rally if one considers that prices over the planning horizon decrease at a gradual rate while volume increases at a decreasing rate. To strengthen this natural tendency towards early clearance of older areas prices were discounted to the beginning of the planning horizon using a discount factor of 10%.

Eight different solutions, assuming the use of either objective function and the recognition of different constraint sets are listed in Appendix II, Tables Ila - III. It is noted that areas have been reordered in order of decreasing age to facilitate an evaluation of the preference to clear older areas early in the planning horizon.

Solution #1 assumes the use of OBJ 1 (age independence) and the constraint set described in Table III. A detailed evaluation of this solution revealed a number of undesirable features: a non-conformity with stated preferential cutting sequences, unbalanced method use forcing untimely investments in various types of equipment, and a restriction to the use of only two cutting methods. Additional analysis revealed that the yield generated by the proposed solution was also unevenly balanced in terms of pulpwood versus sawlog pro­duction, and hence less than satisfactory from a managerial point of view.

As a first attempt to improve the solution, OBJ 2, in which cutting costs and wood prices depend on the age of the crop, was substituted. The resulting schedule in solution #2

42

shows a stronger "diagonal" tendency indicating that OBJ 2 is more successful in generating schedules reflecting pre­ferential cutting sequences. In addition to improved se­quencing, solution #2 demonstrates improved method use as indicated by the emergence of method 3 as an alternative logging proposal. However, the erratic patterns in the use of method 4 were still regarded as highly unsatisfactory from a management and investment point of view. Additionally, improvements were deemed desirable in the balancing of port throughput over the planning horizon.

Solutions #3 and #4 employ OBJ 1 and OBJ 2 respectively over a revised set of constraints. The similarities of solution #3 with solution #1 confirm the need to recognize an age-dependent cost structure in the objective function. Hence, solutions #4 to #8 were derived using OBJ 2 solely. Solution #4 over the revised constraint set demonstrates a lesser conformance to the diagonal desirable pattern proposed in solution #2. However, the substantial improvements ob­tained in terms of method use and smoothing, while retaining the more desirable features of solution #2 in £erms of port throughput patterns, made a further investigation of solution #4 worthwhile, although still too much wood had to be diver­ted through port 1 at great cost, because of the limited capacity of port 2 .

Solutions #5 - #8 essentially represent refinements made to solution #4, either in the form of constraint re­vision. to further balance method use or pulpwood versus sawlog production, or in the form of sensitivity analyses designed to assess the impact of relaxing certain requirements and the associated opportunity costs. For example, solution #5 was derived under relaxed annual port requirements to see if the additi nal revenue generated would be sufficient to contem­plate an expansion of port handling facilities. Solution #6 examined the impact of a smoothing of method 3 use over the final years of the planning horizon whereas solution #7 con­cerned the smoothing of method 4 use. Finally, solution #8 represents a constraint set in which a maximum, of method smoothing is enforced and combined port requirements are re-imposed.

Although final objective values for some of the solu­tions differ substantially, the conclusion may be drawn that minor method smoothing, and re-allocations in the cutting sequence among adjacent years and/or areas can be made at relatively small expense, substantiating the contention that a large number of alternative "optima" can be generated, facilitating the task of formulating a schedule which is best suited to implementation. Substantial savings can be rea­lized by phasing out the use of port 1 in year 5, and doub­ling the capacity of port 2 during year 6 (solution #5).

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4. A HEURISTIC SOLUTION PROCEDURETo verify the validity of the LP solutions derived for

the aggregated 15-area problem, a heuristic solution proce­dure was developed for the original 29-area formulation. The objective for this development was four-fold. Firstly, it was deemed desirable to ascertain to what extent problem solutions are affected by data aggregation. Secondly, where­as the LP formulation described in the previous section does not distinguish between pulpwood and sawlog production in an explicit sense (implicitly this is done by considering method bounds for each aggregate area) a heuristic scheme can handle such a distinction more easily and thereby provide additional insight into the solution structure. Thirdly, the extreme sparsity of the constraint structure for the problem (less than 1%) combined with the fairly rigid nature of the con­straint set and the desire to dicount objective values to generate revenue early in the planning horizon all point to the existence of a large number of "alternative optima" in which case heuristic methods often perform fairly well.Hence, it was thought to be of interest to see whether a heuristic solution methodology could generate solutions which were "close" to optimality for the larger original problem. Fourthly, the impracticality of routinely running a large LP-model might be overcome if easily reorganized allocations can be carried out heuristically in the future. An additio­nal advantage of heuristic scheduling arises from the fact that management objectives which are not readily quantifiable can be considered when allocations are made.

The heuristic algorithm is based on Vogel's approxima­tion method for transportation problems [ 6] , a method frequently used to generate "good" starting solutions for transportation schedules (for pure transportation problems under realistic data structures, Vogel's method has been reported to generate 90% optimal solutions in many instances). Vogel's approximation method essentially scans rows and co­lumns in a conventional transportation matrix for differen­tially advantageous allocations based on first differences in unit costs or profits. A detailed outline of this procedure is included in Appendix III.

The application of a variant of Vogel's algorithm to the problem at hand is based on viewing the harvest scheduling problem as a transportation type structure with flexible demands and supplies. Each area-method combination was viewed as a supply point with each port-year combination acting as a demand center. The flexible supply/demand re­quirements arise from the varying yields that can be obtained in each year as well as from the side constraints that need to be imposed on the allocation process. In particular, the allocation process took account of the following requirements and preferences:(1) non-decreasing method use for methods 2 and 4 over the

44

planning horizon;(2) constraints on minimum and maximum method use in each

area for each year based on topographical factors;(3) minimum and maximum port requirements;(4) the desire to generate a maximum of sawlog production

to be shipped through port 2 only, within the primary objective of profit maximization;

(5) a constraint on sawlog production from areas of only 15 years of age or older, and

(6) a 50% cull factor for sawlogs.The resulting algorithm considers allocations for a

transportation type matrix containing (4x29) supply points and (2x7) demand points, yielding a total of 1124 nodes. The rigid nature of the constraints rendered a large number of these nodes infeasible, thus reducing the number of nodes to be evaluated considerably.

To arrive at profitability estimates for each cell, needed to initiate a differential advantage assessment akin to that used in Vogel's approximation method, a number of intermediate computations had to be performed. Specifically, the following steps were followed:(1) The average price graph for pulpwood and sawlogs given

in Figure Ii of Appendix I was decomposed to yield a separate price structure for pulpwood and sawlog pro­duction, resulting in the price estimates contained in the Table Ilia of Appendix III.

(2) Cutting costs for the various harvesting methods were estimated on the basis of the curves presented in Figure Ih of Appendix I yielding a cutting cost schedule de­tailed in Table Illb of Appendix III.

(3) The formulation of a gross unit profit schedule based on adjusting the prices obtained under (1) for the cutting costs derived in step (2) while taking topographical con­ditions into account. The resulting assessments are shown in Table IIIc of Appendix III.

(4) The computation of net discounted profit/ha estimates for each feasible area-method combination using the formula:

Profit/ha in year j = l/(l+r)3 {price/m3 in year j- cutting cost/m3 in year j- transport costs/m3 in year j}(Volume in m 3, year j/ha)

where r represents the discount factor (10%) employed.The resulting profitability estimates form the basis for the allocation process followed and are shown in Tables Illd - Illg of Appendix III.

The algorithm employs the (area-method) - (port-year) matrix and a number of side calculations on constraint status and resource use, and may be described in terms of the fol­lowing steps :Step 1: Calculate first difference values for each row and

column of the matrix on the basis of the computed

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net discounted profitability estimates,Step 2: Make a maximum allocation to the row/column with

the largest differential advantage value while taking account of varying supply/demand requirements and con­straints imposed on the allocation process.

Step 3: Update the differential advantage values and the con­straint status arrays. Return to Step 1.

Following this initial allocation, the solution was scanned to see if "preference" in addition to "absolute" constraints were satisfied and if an overall feasible sche­dule had been generated. The presence of side constraints (port throughput, method use, etc.) implies that "multiple" first differences may be relevant. Thus, managerial judgment must be exercised in the allocation process underscoring the heuristic nature of the method. The possibility of infeasi­bility may arise because of the varying level of resource availability in any given year. However, by exercising judg­ment during the allocation process, feasibility problems were found to be minimal. In response to a set of "preferential" assignments, reallocations were made to balance and smooth method use, while keeping track of the "opportunity" cost associated with such reallocations. In this part of the allo­cation process the original hypothesis that a large number of solutions exist with similar overall profitability was con­firmed. In most instances, the opportunity cost of reallo­cating between similar methods or adjacent years was minimal while greatly improving the solution in managerial terms.

The resulting solution in terms of proportional cut, method use by year, method use by area, and pulpwood versus sawlog production is presented in Tables Illh - Illk of Appendix III. The associated profit figure of $6,355,384 suggests that near-optimality was obtained. At the same time, a detailed breakdown of revenue generation in terms of pulpwood versus sawlog production is made available. To facilitate comparison with the LP and Decomposition solutions, the 29-area solution is shown in 15-area aggregated form in Tables III1 and Him of Appendix III. A detailed discussion of this solution in comparison with other solutions generated is contained in Section 6 of this article.

5. A DECOMPOSITION APPROACH USING AGGREGATED VARIABLESA third solution, using decomposition principles in con­

junction with heuristic methods and variable aggregation was examined. The principal objective of this part of the study was to ascertain to what extent the problem formulation could be condensed without significant loss of solution sensitivity. This issue is important since future scheduling problems (i.e. beyond 1985) are likely to be of substantially larger size and structurally slightly more complex.

The decomposable nature of the problem may be recognised

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from its angular constraint structure, consisting of a set of "coupling" constraints modeling "shared" resources among seemingly autonomous parts of the model, and a number of "subsystems" each having its own set of constraints. In fact, a careful examination of the LP formulation of Section3 reveals that the problem may be regarded as decomposable both on a time and an area basis, and hence, a choice must be made to effect the most efficient decomposition. If area constraints are selected to form subsystems, shared resources take the form of port constraints, method restrictions, and cutting sequence preferences. A decomposition on a time- basis yields a model in which the subsystems are represented by the annual port requirements and method use constraints with the area-restrictions acting as coupling constraints.The reduction in the number of coupling constraints renders this type decomposition more efficient.

Whereas the LP model considers every area-year-method- port combination leading to a total of 840 Xijki and 420 Nijk and variables, a careful scanning of the problemdata will reveal that only a fraction of these combinations will be contained in viable cutting schedules. Accordingly, before solution by decomposition was initiated, a heuristic data aggretation scheme was applied to reduce the number of variables to be considered. In particular, a number of areas were aggregated on the basis of (i) similarity in transporta­tion costs, and (ii) cutting preference on the basis of area maturity, resulting in the recognition of just 9 areas to be cut according to the following combinations:LP Area Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Decomposition Area Number 2 3 4 5 7 6 7 8 9

In addition, because of the higher cutting costs asso­ciated with machine intensive cutting methods, it was decided to set machine intensive method use at minimum preferred levels, and restrict the decomposition LP formulation to the two labour intensive cutting methods. Allocations of machine based cutting over the various areas and years was achieved in a manner akin to that followed using the heuristic metho­dology of Section 4, i.e., on the basis of profitability and topographical considerations. Next, a reduction in the total number of area-year combinations to be considered for the labour intensive cutting methods was achieved on the basis of profitability and area maturity. A final reduction in the number of variables was realized by including cutting costs in the objective function coefficients of the area-year- method-port variables, rather than through separate cutting method variables.

The resulting decomposition LP formulation takes the following form:

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Maximize E E E I £ijkm*ijkin i j k m J J

subject toE E E j k mI l l X. < A.; i=l, 2 9 (fraction cut)ljkm — l

I I g. -X. , > P ; j=l,2, . . ,7 i k 1D iD^l - jl(port requirement)

I I g. . X. ., _ < P.0; j=l, 2,..,7 (port requirement)i k ^13 i]k2 - D2

A.E E ■ X < b .; j=l,2,..,7 (method restriction). e ., 113m — ] 3 l m 13

A.E E — —X < b.,; i=l,2,..,7 (method restriction)e. , 13lm - jl 1 m ll J

where Xij^m denotes the fraction of area i cut in year j by method k destined for port m and Cijkm represents the associated unit profit. Because of the variable aggregation employed and the prior heuristic allocations for the mechani­cal cutting methods, the coefficients Cijkm take the form

C. .. = { (A. C . ) - (A.r./e., M. .. ) }/A. i]km 1 i]km 1 lk ljk ' 1

for non-aggregated areas, andC. = { E C. A. }/{ E [ r. A. / E r. ] }

11km . T 11km 1 ' . _ 1 x' ■ _ 1 J lei J lei lei

if area i is defined as the aggregate of those areas origi­nally contained in the set I. Here C^jkm, e^k, M^jk are as defined in Section 3 and Ai is the fraction of area i left to be cut following the heuristic allocations for machine-intensive cutting methods; g^j are adjusted (i.e., aggregated) yields; pjm are adjusted port requirements; and bik are upper bounds on method use to maintain feasibility in subsystem solutions.

The total formulation consists of 128 variables, 9 coupling constraints, and 7 subsystems (one for each year in the planning horizon) with 12, 16, 24, 24, 24, 20, and 12 variables respectively. The higher number of variables re­cognized for intermediate years reflects the greater flexi­bility in the formulation of a variable cutting schedule in the middle of the planning period.

The above model was solved using the Dantzig-Wolfe decomposition algorithm'(see Lasdon [4]) in which a sequence of small LP subsystems is solved under price-direction of a so-called master program in the form of the dual prices associated with the snared resource constraints. In essence, separate cutting schedule proposals are generated for each

48

year of the planning horizon which subsequently are co-ordi­nated (i.e., scaled) using the coupling constraints into an overall feasible plan of action.

The results of this analysis are shown in Tables IVa - IVe of Appendix IV. In all, 2 viable solutions are shown. For Solution #1 port 2 throughput for years six and seven was con­strained to a maximum of 300000 m 3 . As Table IVe reveals, this limit was attained for the final year in the planning horizon while throughput in year six fell slightly short of the upper bound. The diagonal structure of the solution, while partly forced by the heuristic allocations for the machine-intensive methods, is similar to that found in the TEMPO LP solutions derived from the larger LP formulation. Schedule differences are largely due to the recognition of upper bounds on labour intensive method use reflecting a mana­gerial preference for gradually increased method use over the planning horizon. The similarity of solution #1 to the TEMPO- based LP solutions is confirmed by the "optimal" objective function value: $6,228,600. Solution #2 represents a sensi­tivity analysis designed to evaluate the effect of tighter or more relaxed individual and composite upper bounds. Thus solution #2 was generated assuming a maximum port 2 handling capacity of 150000 m 3 in the final two years of the planning horizon. A different allocation for machine intensive methods and increased throughput at port 1 is forced at considerable cost as indicated by an "optimal" objective value of $5,600,155.

The Dantzig-Wolfe algorithm proved more efficient than had originally been anticipated. Average runs required approximately 16 cycles to obtain feasibility and confirm optimality for an average time of 42 seconds per execution on a Burroughs 6718. A "save-basis" option was used to increase the algorithm's efficiency. This is particularly important if feasibility is difficult to attain, a feature often attributed to decomposition methods.

6 . A COMPARISON OF SOLUTIONS AND METHODSFollowing sensitivity analysis, a detailed examination

of the best solutions revealed that the three solution approaches produced remarkably similar results in terms of (1) amount and value of annual cut, (2) port allocation,(3) felling sequence, (4) use of logging methods, and (5) implied management strategies in general. With regard to port allocation and amounts of cut, LP solution #8 , hereafter referred to as LP8 (the nearest of the LP solutions to that likely to be implemented in practice), and the heuristic so­lution, H, are particularly close as shown in Table 1 below. Dantzig-Wolfe solution #1, DW1, is slightly different, how­ever, and appears not to satisfy minimum contracted supplies in the first few years. Port 1 is favoured early in the planning horizon as the first cuts are made in locality 1

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Table 1: Port Allocations and Total Annual Cuts in thousands of m 3 for comparable solutions by the three modeling approaches.

Solution Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7LP8 (50, 0) (37,13) (30,70) (6, 94) (0,120) (0,150) (0,363)

50 50 100 100 120 150 363H (22,28) (47,52) (14,88) (20,99) (6,120) (6,162) (6,206)

50 99 102 119 126 168 212

DW1 (45, 0) (26, 6) (21,45) (13,84) (6,100) (6,296) (6,300)45 32 66 97 106 302 306

(See Illustration 1) but its utilization declines sharply as the tvo other localities supply more and more of the cut and the availability of sawlogs increases.

The annual discounted profits generated using these same three solutions are set out in Table 2. Again, a reasonable conformity is shown, with differences balancing out among adjacent years.Table 2: Total Annual Discounted Profits in thousands of

dollars for comparable solutions by the three modeling approaches.

Solution Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7LP8

HDWl

$441$543$408

$456$809$336

$1,131 $ 692 $ 578

$789$984$886

$873$927$750

$ 975 $1,093 $1,479

$2 ,008 $1,284 $1,817

The sequence of cutting implied by each of the solutions reveals a pattern of close similarities as well, as shown in Table 3.

Apart from the discrepancies for areas 7,8, and 10, the sequence of felling is remarkably consistent among all three solutions. Generally, the oldest or most slow-growing areas are cleared first while younger and faster-growing crops are left to mature, and cut as late as possible. The order of areas 1 to 15 represents a pre-judged management prefe- rance in terms of cutting sequence, and, with a few minor exceptions, such as areas 1 , 2 , and 9, the cutting sequences in all three solutions conform well to the preferred pattern. As areas 1 and 2 are relatively small and area 9 is stocked with less desirable slash pine, these departures are of no managerial consequence.

50

Table 3: Years in which felling is mainly performed by area for comparable solutions, and ranked area growth rates.

Area and Ranked Year of felling in Solution ^Growth Rate LP8 H DW1 1

1 ( 9) 3 1,3, 4 32 (14) 2 1 2 33 (12) 1 1 14 (15) 1 1 ,2 15 (13) 1 2 16 (1 0) 3 3,4 3 57 ( 7) 3,4 3,7 48 ( 2) 4,5 2,3 49 (1 1) 2 3 2,3, 4 3

10 ( 5) 4 5 2,3,4 4 711 ( 8) 5, 6 ,7 5,6, 7 5 612 ( 4) 5, 6 ,7 4,5, 6 5 613 ( 3) 6 ,7 5,6 6 7 |14 ( 6) 7 6,7 6 715 ( 1) 6 7 6,7 6 7 i

A final comparison concerns the use of logging methods, shown in Table 4. The figures represent annual equivalent number of work gangs needed under each of the proposed solutions. As illustrated, the use of method smoothing in all three solu­tion approaches produced comparable well-behaved method uti­lization statistics in line with stated investment objectives.Table 4: Equivalent Annual Method Use for comparable solutions

by the three modeling approaches.

Method Use by Solution Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year l\

LP8 2.4 2 .2 1 .2 1 .8 2 .2 6.3 7 6k=l H 0.9 2.4 3.0 3 . 0 2 . 7 1.9 1 9

DW1 - - - - - 7.4 6 1

LP8 0 .6 0 .8 3.3 5.0 2.9 1 .2 1 0k=2 H 1 .2 2 .8 3 . 3 3.4 3.2 5.8 7 0

DW1 0 .6 1 .0 0.5 0.9 2 .0 4.0 6 0

LP8 2.4 - 5.5 0 .6 4.1 - 36 5k=3 H 0 .6 3.8 3.4 3.4 6 .2 12.5 17 3

DW1 7.0 1 .0 7.0 3.3 3.2 12 .0 7 1

LP8 0 .6 1.9 3.4 3.4 4.9 4.9 7 0

*• ll 4 H 3.4 4.0 4.0 3.7 3.9 3.7 4 0DW1 2 .8 2.9 4.0 9.0 9.0 12 .0 12 o !i

51

Fluctuations in the numbers of labour-intensive methods (k=l, 3) present no real problems in practice in that no highly capital intensive machinery is involved. Solutions H and DW1 show acceptable fluctuations in gang numbers about a gradually increasing need for both machine intensive logging methods (k=2,4). LP8 , however, while satisfactory for k=4, appears less attractive for k=2. Nevertheless, all approaches demonstrate the desirability of extracting a maximum of saw­logs despite the greatly increased logging costs associated with such production. The premium in price for sawlogs is genuine so that one general management strategy should be to devise less capital intensive, more labour intensive methods for sawlog extraction. Jones [ 3] has already investigated this and has demonstrated that bullocks can be used to re­place a skidder in method 2 at a substantial reduction in cost per m3. This study confirms that such a substitution is highly attractive both from a financial and managerial point of view.

Another management strategy brought out effectively in this study by the sequence of LP solutions concerns the need to increase, by at least a factor of two, the capacity of port 2 by the end of the fifth year. The savings in transport costs for years 6 and 7 alone amount to approximately $500,000, a figure well above the likely cost of such early port expan­sion .

While all three modeling approaches contributed in a distinctive manner to the conclusions derived in this study, the large LP model was probably most useful in identifying common features among "good" solutions. The range of solu­tions available from extensive sensitivity analysis provides the manager with considerable flexibility in deciding when and where to make cuts. Among the most important conclusions are that minor method smoothing and cutting sequence revision is unlikely to reduce the objective value significantly implying that the solutions are relatively robust. The heuristic solution, and to a greater extent the Dantzig-Wolfe solutions, relied on basic LP solutions for confirmation to assess their validity. This reliability having been confirmed, however, then opened the opportunity to perform schedule re­visions as part of routine management without having to re­analyze the problem using the larger, computer dependent LP model. It is envisaged that as data and management objectives change, the LP model will periodically be used to establish new "base" solutions, to which smaller alterations and fine- tuning may be applied using either the heuristic methodology or the more compact decomposition LP model.

REFERENCES1 F.A.O., "Report of the Fiji Pine Industries Project", F.A.O. report

No. 30/77 D.D.C. FIJ. 7, 1977.

52

2 Johnson, K.N. and Scheuman, H.L., Techniques for Prescribing Optimal Timber Harvest and Investment under Different Objectives - Discussion and Synthesis," Forest Science Monograph, No. 18, 1977.

3 Jones, P., "Report on Logging and Marketing", Projects LP2, LP3, and LP4, in Appendix I of Whyte, A.G.D., "Omnibus Report on 1978-79 Inventory Method Assignment at the Fiji Pine Commission," unpublished report to the Ministry of Foreign Affairs, Wellington, 1979.

4 Lasdon, L.S. Optimization Theory for Large Systems MacMillan, 1970.5 Tcheng, T.H., "Scheduling of a Large Forestry Cutting Problem by

Linear Programming Decomposition," unpublished Ph.D thesis,University of Iowa, 1966.

6 Vogel, W.R., and Reinfeld, N.V., Mathematical Programming Prentice-Hall, 1968.

7 Whyte, A.G.D., "Interim Report on 1978-79 Inventory Method Assignment at the Fiji Pine Commission," unpublished report to the Ministry of Foreign Affairs, Wellington, 1978, Appendix 2, p.5.

8 Whyte, A.G.D., "Inventory and Yield Forecasting Systems for Caribean Pine in Fiji, "Proceedings of the N.Z. Forest Service F.R.I.Symposium No. 20, 1978.

9 Whyte, A.G.D., "Report on the Crash Inventory Programme for the Fiji Pine Commission," unpublished report to the Ministry of Foreign Affairs, Wellington, 1977, Appendix 1(b).

APPENDIX I - BASIC PROBLEM DATATable la - Yield Data by Area/Year in m^ for 29 areas.Table lb - Cutting Method Parameters (No. of hectares that can be cleared

by a full year's application in each area) and transportation costs to ports in $m3 (29 areas).

Table Ic - Cutting Method Bounds (Max. and Min. No. of ha to be cut in each area for a given method or combination of methods) in­dicated by topographical considerations (29 areas).

Table Id - Yield Data by Area/Year in m^ for 15 areas.Table Ie - Cutting Method Parameters (No. of hectares that can be cleared

by a full year's application in each area) and transportation costs to ports in $/m3 (15 areas).Cutting Method Bounds (Max. and Min. No. of ha to be cut in each area for a given method or combination of methods) in­dicated by topographical considerations (15 areas).Annual Port Requirements in m3.Logging Costs per m3 for the various harvesting methods of age of stand.

Figure Ii- Average Price Curve (Pulp and Sawlogs) per m3 for the base year by harvesting age.

Table If -

Table Ig - Figure Ih-

53CN4J VO VO VO CO O o o 00 00 in r - 00 in 00 CN CN ro vo CN O VO vo o in vo m *U rH 1—i T—\•r ■r ro -3* ■c *r ro ro CN CM in ’'T CN CN ro ro& CN CN CM CN ro ro ro in in in in ro ro in ro ro ro CN fN in ro ro ro cn CN CN CN ro ro

H■P O O O O r - m m o o in in r* O O in r - r - O in o O inU CN CN CN ro CN CN CN CN ro ■r ro (N (N CM ro cn in ■*3* in in CN ro£ VO VO VO VO r* r - ro ro ro ro r * ro r - vo vo ro r^ f ' vo VO vo vo r -

•g on 00 •5T m o 00 r - m ro 00 r* o vo ro VO on in fN vo vo 00 ro vo in CN CN CNrH r- 00 VO on 00 vo vo o CT\ CN ro in in CN cn o 00 ON CN o VO VO

sr ^ v o T rT T r M C N C N ^ 'm ^ rT rc M C N ^ i* ro .m m m rn ir> rH r H C N ro m c N C N r-» rH

ro

V CN m VO CN in ro CM ro in vo 00 VO 00 O ro in VO o CO in rH vo in ** o voj* 00 «-H CN •r m in rH CTn ro ro o r - 00 00 VO o in on O CN VO in

ro CM CSI rH rH f-H CN rH CN CN (N CN CN CN CN£

CN

13 VO CO in CN vo in rH ctn on in on ro ro vo in on VO O ro ro 00 rr rH 00 00 rH r -s: ro r - m on in CN CN r - 00 in vo ro ON on o ON ro ro o o 00 CN

£

i/> ro ro U ) in in CN ro in •r VO VI) •r vo CN CN ro ro ro ro cn

r—i

V co in in ■*r o o rH CN CN "<T 00 •H O vo m ON CTv in o\ cn r - *r 00 00 VO CNJ* o o in CN o ■'3’ r* ro 00 in r - CN CN in 00in CN 00 ON in in vo 00 ON ON Lf“>00 vo£

rH rH rH rH rH

S rH CN ro in vo 00 ON O rH CN ro in VO 00 CT\ O ro in<

rH rH rH rH rH rH rH rH rH —1CN CN CN CN CN CN CN CN CM CN

o o O ' j o o o o o o o o ' j r ' h O H f o ^ o n n r ' ^ f O r H O i N f N o o o c r i O O O o o o o o c N r - T T i n m r - H r o o r ^ ^ r o c o c n u D m r r i H NHnifKNncoinoooonfOHo^Orioaivo^cocDONCNH HMH^ninvo^hcoinosnvococyiCNatvoovoor^vDHaK^con (N H (N CN n H H H n 'T ^ n ITI H H iHCMVDO

ooor^ooooooor^romr^’'rHcr«r'Ocn-cro>moo in O O O ^ O O O O O O O H n i n H r O H ( N H O ( ^ ( N ( J ) l ^ h ( N O 'M n cNHnoojfO(DLOOinocn[Nu)tf(T\Hn'i,on^^)^(N'X)Hina) rifsH n n i n ^ v o M s>iTicDN,i'ts ^co^'CNcoi/)^'X)^'oo^Hr'N

CN rH (N fN fO H H H n fO ’T fO Tf H O OlD O N

o i n o o r ' - o o r o o o o o o v o o N o r o ^ r c N i n i n o r H v o o o i n i n i n r o r - o oaNOir-ooincoooovocNr^r-vocNcor-oinvovDcoro^ooLnoN f'JOMCOfMn^^inO^rOCMH^OOincOH^VDO'ThfNCN^VD rHCNcr>»Hroir)vDvovor'err>*»HmvDCNTr<Tvcx)tnmoovocNjinrHCNvocN H H (N (N (N H H H rO CO ro CN iH ON ON ^ f''

h M n ^ o o j o r o o o o r o h n o ^ o a i h m m i n H n ^ ^ ^ ^ H H c r » o c T > ( N O r H r ^ o o o o o N O v D r ^ i n o v D o m c N m r H r ^ o ^ o m o r H iHcr>^r'<NCNfHOir»tninm<NooiHinocN^rmLnr'^rooonmoro HHhOfoinvDvDinintNinoHvDCflOLninHfNMnHfnr'MvocN H H M C N n H H H (N fO fO CN ^ H CO 00 CM LO

^ r r - o ^ r o c o L n L n o o o m r - o a r ' ^ i n m o L n o m r ^ c o o m c N ’ oo ronOCTiOintNHOOCOinOhfnc^hH^fSfM^fNhHVDOOJ^ O h c o ^ ^ r ' V D i n M ’Hin^(\TrincNOjnfNr'rocooo^coininkDO HHina\n^,inir)M'^o,j,(TiOir)in'X)r-)Hr'Hu)^,ofN(Nr)iooj rH CN CN ro rH rH H (N PO (N n H h h H n

inojHcncoHOMOOoo^oj^^HotNininvDinconincoohfnoo fo 5cor-fNcnooo'X)OCNr cna>comr-coco’* rcsi ror'(T\inHrrHnH(y>fnvDr'O tN0>ior' ''!ycNHff)(NiAcrivD )<r\is'H^COrO^ip^rKNCOmCOCn^OJfN^Cn^Oin^OHCOH'd'rH rH CN CN CN rH fN M fN H n H VO VO ON rH

^'hnhOCOCOrOOOOOrlniAh^h^OO^DOOOCN'TCOOItJ' ONO\inOVOrH(,O O N O O i n v O ,*},incOrHONCNr'‘OOrHCN''3,OCOlOOTrrH corominmo'DinncNcoo3ino^HMininc>aiNh'.o(N^y)cj^ HCNr ' C N r f ^ ^ C N H ^ O H r ' C O ^ C n C r i r n v D O O O i n O I O H O J H ^ ' H (N (N (N H rH rH CN iH rO in in 00 00

HCNinTi'invDr'CO< oH(Nm,tin«)r'CocrtOHC'jm ioy)is>coo>H H H H H H H H H H ( N (N ( N (M f S (N M ( N (N (N

54Max

(2) co(NfHco<y>vocoo->ror-cN^in c o m f N i n ^ v D H n v o ^ o ^ H O Hn^r'^cNfo^Hc^Mfo^cD i i/icoooioojcDinTrcoooo^iriiN m h in a) H n cn coH H H M

CM ro o n vo ko co co in vd o h ooincNinr^cNrHmvDoocNcoro ro rr cr> cm ro ■'T i rr >x> r- vd i incDOOfOHinin^r'H(NO> i i rH h h Tf h n oj (N r r-r-'Or-i

Max

(3) o> cr> H n m co cn o rH ro o iH in ^ h co ttI i com i i i im<^r^r'VD<Niov£>inc^mv£>i imrHoo-^ri I (N cn n »h n ro n oj in co cmrH rH

Min

(3+4)

r^OOOOO^O^DCT^VOVOOr^VDrHVDCNVOVDvDVDOOr^rHCOrOCNOrHVDro^rrH<r»vDoooroinminoco(T»CNmmvDV£)CT>r^rHovDcsivD(TirH^* ro rH HMnrj'inNHTfHincoin^'r'HHH'i'VDor^HrH rH H H (N CN

Total

ha.

n o ^^Mn(N^vD^m^c^voofNoroonrjfOfoo\h^in(Nncri fNr'a\H^cr>vocoin^h-n^no«cj<rHkoin(NHcovDininr'Oco Hr'^HrHHCNH^ODnnintNhOTTH^DnOJrHkDOHO^fNrH rH rH rH CN r—< rH CN CO Tf in

Area .HCMrO’invor-cocrtOrHcNm'^rinvDr-cocnorHCNmTrin^r^oocnHHHHHHHHHHC'JCNCN(N(NC'1CNCNCNCN

S 5 SJ ,

"O *H 4-3 g ^ 8

S W H ti 3"S 5 cn 3 T1 csS -6"8 *0 S

^ s i 8

f i l l 32 8 8

Troooo.Hr'.Hor^vovomrHo( T i O O O O h h O O ^ h C O ^ l D i nfHcnmoo^mvooiHmooooocNa'*inv£>rooovocNintn*£>vOfH<r>crtCOiH CNCN-S'rHCOCOCOrrHCNVOO rl H H (Nvooooor^tninovovDinoo-'T t^ooooininoo^rmcor^cNO\ iDfOinoinm^inohri'TtN^fo r cnvor'r oo mmvorsj rooTrrH m»H (N(NnHhcoh-«roojna\

orooooocoocoorocNrHinmco r moooovDr'VDvDromcNro j'oo ^cN^LOoniHOinntNOhCNiN rirno MDnvocrioooocNinHCN n h fNMr»Hvor'«)ro(^ON^r'

n r j r o o o h n o M n o H ^ o ^ ^ ^ (NcoooocD innr'COCNicriOfO roinouiiri coincoujojocDnro H^kDiniDHHCOfOOM'HnhtN nn cNCNmrHinr-vocooooominr—I rH

inninoor cNfvjinmc cDOfocN rHr-rHoocnr r oor r-iHvoo Hini/i'rHH innvDvDvDcoinin conin^^criOHcofNOOojMn CM H ( N f V J C N H i n ^ i n f O h h r H r O

hccinooin^Hinhc^nmcoo'tOKTiOO^OJCNhtHrO^fM^O HVD< fOy3rr(NOO(T»(T>Hin(T>VDVO inCN fOCNVDa»vrCNVDr OrHOOr-» (N H CN CN CN r f k D ^ ( N k D V D O > H

r—i omoovororHoror-ocM rcx) mHa\oofOfOHroonocoioo nvDinnMCDO'i'CDHinvocN^D ( N H ^ ^ l H n C O C O h O n O H M H cn H <n (N (N c n i n ^ c N i n m c o o o

HcNro^mvDr^coc^oHcNn^in

NPA2

^ D O C O C O H V D C O C O O H i n V D V O O i nH ^ ^ n ^ ^ v r n i n ( N ^ ( N ( N r o nc N m i n i n m m i n m i n c N r o c N C N C N C N

HPH2

o r ' i n m o o i n r ^ i n o i n o i n o oCNCNM’ C N n ^ T f C N n ^ n i n ^ r i n v Di f l r ' c o n n r s c o r ' f o v o r ' ^ i o v o ' i )

Method

4

r ' C N r ^ m r r) a Dv i > c o r o v DOV Di n c NC Nr ^ c o H c o v o m c N i n r ^ o c n c N o v D v DM ' C M ^ n ^ o j ^ i n ^ n H n n c N C N j

Method

3

^ ^ n i n v o H M n H v o r ^ M n ^ ^n ^ H ^ n n H c o ^ i n c ^ ^ i n n n CNi HCNrHCNi HCNCNCNrH r Hr Hr Hr H

Method

2

v D C N r H c n c T i C N mc o r H C N r ^ r ^ i H c n c oc r * L n c N r ^ r - c Nmc 7 » c r i c o r o o o o c N C Ni n n m ^ i n n i n v o i n n c M ^ n n n

rH

1

O h o H C N O M n ^ c s m c N h ^ c o c ov o v o ^ r ' f O H i n o j i n H c o i n H C M o Io > mc o r ^ c r » i n c x ) r H c T » v o f o v Dv D i n i n

3 I1 a 1

H ( N n < r i n v o i N o o c y > o H ( N f o ^ i nrH i—1 r—1 rH rH i—1 Tab

le Ie

Cuttin

g Met

hod Parameters

(No.

of hectares tha

t can

be cle

ared

by a ful

l yea

r's application

in eac

h area)

fran<;mr*f3fi r\n

r>ncfc

+-r~\

nnrfc

-i C

(1C.

55

2

•rH *J»

co^vocor^vococoCTviHH^ro^iOO M O 1 (N T f CO -. H H H t N

H CO I/) ON 00 00 rH O O tH (N 00

h h I ^j * t t o o i C o r ^ ^ r i n r ^ i - H C N O Nm rH rH rH rH i H V O l O H h , rs ls* 0

r^iHCNCNfH^rrorH^rrHco^*I ma\^(NvDO^oroHoo^ cM(NrHrH'x>cr»vx)rsir oofNr-'

fNo^rvDvDO^HoomomcomcNOi r >r oi r » ro( Ncr »vx) mcNi nv£>( NVDO^

(N CMin (N (N i

rH^vD^^riri'DCNconncrir-^incDirnocoin^^^H.^o^inmh

H ^ n ^ L n ^ ) h c o ( 7i O H ( N n ^ i n

Figure

Ii

Averag

e Pri

ce Cur

ve (Pulp

and Sawlogs) per m

for

base

year

by harvesting age.

APPENDIX II - TEMPO L.P. SOLUTIONS56

Table Ila - Area Characteristics: Age, Size, GrowthTable lib - TEMPO L.P. Solutions #1 - #4: Proportional CutsTable lie - TEMPO L.P. Solutions #1 - #4: Equivalent Annual

Method Use by AreaTable lid - TEMPO L.P. Solutions #1 - #4: Port Utilization

in terms of proportional cuts by AreaTable lie - TEMPO L.P. Solutions #1 - #4: Equivalent Annual

Method Use by YearTable Ilf - TEMPO L.P. Solutions #1 - #4: Port^Utilization

in terms of annual throughput in mTable Ilg - TEMPO L.P. Solutions #5 - #8 : Proportional CutsTable Ilh - TEMPO L.P. Solutions #5 - #8 : Equivalent Annual

Method Use by AreaTable Ili - TEMPO L.P. Solutions #5 - #8 : Port Utilization

in terms of proportional cuts by AreaTable IIj - TEMPO L.P. Solutions #5 - #8 : Equivalent Annual

Method Use by YearTable Ilk - TEMPO L.P. Solutions #5 - #8 : Port^Utilization

in terms of annual throughput in mTable III - TEMPO L.P. Solutions #1 - #8 : Comparative Profit

levels and constraint statusNotes on the Use of TEMPO and MODELER.

Averag

e Age

Hectares

Growth

Rate

Age Ranking

Area

Ranking

Growth

Ranking

Growth

Yrs 1-4

Growth

Yrs 5-7

Method

Efficiency

1 17 148.1 0.753 6 8 9 F - 22 18 45.4 0.317 4= 14 14 F - 113 19 26.6 0.415 3 15 12 F - 64 22 118.6 0.210 1 10 15 F - 75 20 145.4 0.321 2 9 13 F - 46 15 94.5 0.698 7= 12 10 - F 147 15 54.6 1.039 7= 13 7 - F 58 13 374.2 1.411 9= 5 2 - F I9 18 451.8 0.470 4= 3 11 F - 310 13 257.3 1.123 9= 7 5 - F 911 11 100.5 0.970 12= 11 8 - F 1512 12 366.7 1.310 11 6 4 - F 813 11 405.7 1.377 12= 4 3 - F 1014 10 515.4 1.050 14 2 6 F - 12=15 9 697.5 1.564 15 1 1 F - 12=

Table Ila Area Characteristics - 15 Area ProblemAreas 4,5, and 9 are stocked with slash pine. Other areas are stocked with Caribbean Dine.

57

Year Area 4 Area 5 Area 3 Area Area 9 Area 1 Area 6 Area 7 Area 10 Area 8 Area 12 Area 11 Area 13 Area 14 Area 151 1.00

1.001.001.00

1.001.001.001.00

1.00

0.27

0.770.030.770.10

0.240.23

21.00

1.00

0.0950.600.100.60

1.001.00

0.760.77

0.610.49

1.001.00 0.06 0.05

3 0.0870.300.090.30

1.00

1.00

1.00

1.00 0.15

0.390.4280.3360.09 0.11

0.710.6780.13

4 0.530.530.73

0.038O.OS0.04

0.135

0.14

0.2781.000.480.45

0.572

0.70

0.07

0.10 0.15 0.10

0.890.2660.06

0.3030.2390.016

5 0.470.47

0.2230.4560.2240.455 0.40 0.03

0.93

0.59 0.08

0.245

0.14

0.6960.700.047

6 1.001.00

0.4090.4090.4090.405

0.7720.52

0.760.800.20

0.45

0.30

0.600.080.42

0.345

0.36

7 0.3660.375

0.177

0.240.20

0.55

0.55

1.000.400.920.40

0.755

0. 77

0.655

0.56

Table lib TEMPO Linear Programming solutions #1 - #4. - Proportional Area cuts

Table lie

Method Area 4 Area 5 Area 3 Area 2 Area 9 Area 1 Area 6 Area 7 Area 10 Area 8 Area 12 Area 11 Area 13 Area 14 Area 151 1.077

1.0770.885

1.0921.0920.937

0.0380.0380.038

0.3590.3590.159

3.3033.3032.580

0.9263.6580.9260.602

0.8190.819

0.8190.455

0.0640.0640.064

2.5092.511

2.509 1.631

3.6810.3333.6810.166

0.0033.348

0.0032.177

1.1851.1831.1850.791

3.9643.9643.9642.807

5.8565.586

5.856 4.489

7.9277.925

7.927 5.844

2 only solution H4 employs method #2 0.309 10.250 1- 10.324 17.763 0.522 0.587 - 1.416 0.268 1.891 0.633 1.864 2.201 3.348

3 only sc4.5261.215

ilution4.3141.233

#2 am 0.150

#4 em 1.417

iloy me 13.046 3.803

hod #30.609 1.084

0.2530.562 4.122 2.152 4.419 2.072 5.232 9.612 13.015

4 0.9290.929

0.9290.311

0.9420.942

0.9420.313

0.5610.561

0.5610.561

0.8870.887

0.2870.886

2.9042.905

2.904 0.967

1.2411.2411.241 0.931

2.0152.016

2.015 1.465

1.1527.153

1.152 0.866

3.3633. 3623.363 1.261

6.0356.036

6.035 4.85 :

4.5001.501

4.5002. 25C

2.9112.911

2.911 1.852

5.3215.3215.321 2.662

7.877.87

7.87 2.954

10.64210.64910.642 3.992

Equivalent Annual Method Use: TEMPO Linear Progranming Solutions #1 - #4.

Table IId

tort Area 4 Area 5 Area 3 Area 2 Area 9 Area 1 Area 6 Area 7 Area 10 Area 8 Area 12 Area 11 Area 13 Area 14 Area 151 1.00

1.001.001.00

1.001.001.001.00

1.001.001.001.00

1.001.001.001.00

1.001.00

1.001.00

0.55

0.55

0.40

0.40

0.678

0.68

2 1.001.00

1.001.00

1.001.00

1.001.00

1.001.00

1.001.00

1.001.001.001.00

1.001.00

1.001.00

1.001.00

1.001.00

1.000.45

1.000.4 5

1.000.60

1.000.60

1.000.322

1.000.32

1.001.001.001.00

Ftort Use: TEMPO Linear Progranming Solutions #1 - #4

58

in Q sf Q C\ Csa r-H rji o> r' co

r r C) o Q)in Q> a\ rjio n m co in O) n n

m to a\ c<3 VO CJ C3

o\co to h «ii r» nO t'.fN M 'f

o ^ r' m oj toH CD 0> QOin w cm t-i

H tti r l Q <0 C. CM T-4in O0 <T> Qo

r- pocoCT> «o

CM ciCN- O Cs. tT o> r.

I <o co

O r-i o »o n oj omo rn K coo t o m ^

QO O CO t^ H rj<

I OS VO C-Q

Year Area 4 Area 5 Area 3 Area 2 Area 9 Area 1 Area 6 Area 7 Area 10 Area 8 Area 12 Area 1] Area 13 Area 14 Area 151 0.95

1.00

1.001.00

0.800.857

0.8561.00

1.001.00

0.710.112

2 0.05 0.100.1430.144

1.001.00

1.001.00

0.770.800.770.591

30.291.00

0.170.1030.140.28 1.00

0.2631.00 0.48 0.06 o:os 0.10 ■

4 0.10 0.050.082

0.08 0.30 0.440.52 0.62 0.40 0.11 0.15 0.095 0.064

0.0160.016

0.0160.020

5 1.001.00

0.700.650.80

0.5370.050.90

0.4550.25

0.250.32 0.37

0.10

0.100.39

0.20

0.250.35 0.08

0.013

0.0130.06

0.0470.047

0.0470.05

6 0.350.20

0.200.676 0.21

0.211

0.2120.063

0.0630.100.20

0.200.20

0.350.35

0.2120.30

0.080.081

0.0810.425

0.210.464

0.4640.111

0.110.257

0.2560.25

7 0.2730.10

0.100.780.5330.548

1.000.914

0.9340.18

0.900.70

0.700.20

0.650.449

0.5480.20

0.920.918

0.9190.40

0.770.524

0.5240.73

0.820.678

0.6790.68

Table I!g TEMPO Linear Programming solutions #5 - #8 : PiPportional Cuts

59Table Ilh

Method Area 4 Area 5 Area 3 Area 2 Area 9 Area 1. . .

Ar ea 6 'Area 7 Area 10 Area 8 Area 12 Area 11 Area 13 Area 14 Area 15

1 0.8850.8840.8840.885

0.9370.9370.9370.937

0.0380.0380.0380.038

0.1580.159

0.1590.159

2.5802.5312.5812.580

0.6020.6020.6020.602

0.4550.4550.4550.455

0.0640.0640.0640.064

1.631 1.C311.6311.631

0.1540.154 0.154 0.166

2.177 2.2772.1772.177

0.7910.7900.7900.791

2.8082.8082.8082.808

4.4894.4894.4894.489

5.8475.8405.8405.844

2 0.3090.209

0.3090.209

0.2500.2S00.2500.250

:

0.3240.3240.3240.324

1.1601.163

1.1631.136

0.5220.522

0.5330.522

0.5870.587

0.5870.587

-

1.4161.416

1.4161.416

0.2680.2680.2680.268

1.8901.8901.8911.891

0.6330.6330.6330.633

1.8601.8601.8641.864

2.2012.2012.2012.201

3.3483.3483.3483.348

3 1.215 1.21b1.215 1.21 5

1.2321.2321.2321.233

: -

3.8023.802 3.8003.803

0.6090.609

0.6090.609

1.0641.0841.0841.084

0.5620.5620.5620.562

4.1224.1224.1224.122

2.1522.1552.1512.152

4.4194.4194.4194.419

2.0722.072 2.0702.072

5.232

5.232

9.6129.6129.6129.612

13.01510.05010.05013.015

4 0.3110.311

0.3110.311

0.3130.313

0.3130.313

0.5600.S61

0.5610.561

0.8870.886

0.8860.887

0.9670.967

0.9670.967

0.9310.931

0.9310.931

1.4651.465

1.4651.465

0.8660.866

0.8860.866

1.2601.2611.2611.261

4.8504.850

4.8504.851

2.2502.250

2.2502.250

1.8531.8501.850 1.852

2.6605.3215.3212.662

2.9542.9542.9542.953

3.9925.5095.5093.992

Equivalent Annual Method Use: TEMPO Linear Programning Solutions #5 #8

Table Hi

Port Area 4 Area 5 Area 3 Area Area 9 Area 1 Area 6 Area 7 Area 10 Area 8 Area 12 Area 11 Area 13 Area 14 Area 15

1 1.001.001.001.00

1.001.001.001.00

1.001.001.001.00

1.001.001.001.00

1.001.001.001.00

2 1.001.001.001.00

1.001.001.001.00

1.001.001.001.00

1.001.001.001.00

1.001.001.001.0)

1.001.001.00

1.00

1.001.001.001.00

1.001.001.001.00

1.001.001.001.00

1.001.001.001.00

Port Use: TEMPO Linear Programing solutions #5 - #8

60

Solution Objective Constraint Objective FunctionNumber Function Set Value

1 1 1 $6,599,0002 2 1 6,047,0003 1 2 6,516,0004 2 2 5,982,0005 2 3 7,000,0006 2 4 6,845,0007 2 5 6,830,0008 2 6 6,480,000

Table III TEMPO L.P. Solutions #1 - #8 : Comparative Profit Levels and Constraint Status.

Constraint Set Specifications:Constraint Set 1: Base Constraint SetConstraint Set 2: Base Constraint Set + generalised bounds

for methods 3 and 4, maxima for the use of method 3 and minima for method 2.

Constraint Set 3: Constraint Set 2 with individual and combined port restrictions relaxed.

Constraint Set 4: Constraint Set 3 + smoothing constraints for the use of method 3.

Constraint Set 5: Constraint Set 4 + smoothing constraints for the use of method 4.

Constraint Set 6 : Constraint Set 5 with combined port restrictions re-activated.

61

NOTES ON THE USE OF TEMPO/MODELERThe linear programming solutions described in this paper were generated using the TEMPO mathematical programming system in conjunction with the MODELER matrix generation and report writing package.Whereas both systems are easily implemented for small to medium size problems, larger problems present a number of additional challenges, particularly if the MODELER package is employed. The linear programming formulation described in this article consists of 1,274 real variables, many of them upper or lower bounded, 76 3 constraints and two objective functions which were alternately used. The constraint matrix was exceedingly sparse with a density of around 1%, lower than for most LP implementations.Experience with a problem of this kind indicates that while the TEMPO package is highly efficient once the matrix has been generated, and intermediate solutions are being stored, the MODELER generating package is far less efficient and can create problems in terms of generation time needed and memory integral required.The feature of implicit replication using variable list names and concatenation whereby variable and row names can beformed by combining elements of various lists, BEGIN ....END groupings, and many other features of the MODELER package are powerful programming tools that make the preparation of a matrix generation programme for larger problems a relatively simple task. However, it would appear that this user-orientation is compensated for by the relative inefficiency of the matrix generation process itself. For the problem at hand, initial matrix generation could not be completed within a stated limit of 2,400 seconds of CPU time and during its execution developed an excessive memory integral requirement of more than 90,000 kiloword-seconds. Other difficulties experienced included excessive I/O time requirements during execution, and generally, the algorithm'sslow progress, as demonstrated by periodic progress reports generated.This experience substantiates the comment made in the MODELERprogramme manual (p. 8-6) that....'.'MODELER execution consumesvaluable computer time...". Indeed, the statement could be made that MODELER performs rather poorly in terms of core storage and seems poorly structured in terms of its ability to access random positions inside the LP matrix.The above experiences are in direct contrast with those obtained using the TEMPO mathematical programming package. In all runs made, TEMPO was found to be highly efficient both in terms of core requirements and CPU time.

62To streamline the entire procedure, it was found to be most efficient to use TEMPO in place of MODELER options wherever possible, and to generate the matrix using MODELER in parts rather than in its entirety. As a result, the modeling exercise took the following steps:1. The tables of coefficients were produced by reading the

data using the READ statement and input file MUSER of the MODELER package ,

2. Using row generation procedures part of the matrix was generated and stored on DISKIN in permanent interactive storage called CANDEPACK (DISKIN is the disk input file for the TEMPO algorithm)

3. Using the interactive time-sharing facilities of the B-6700 computer DISKIN was accessed using the CANDE (Command AND Edit) procedures and modifications were made to produce a revised file suitable for access using the REVISE procedure of the TEMPO package (REVISE can be used to modify, delete or add rows) The net result of this manipulation was that fairly major model changes couldbe made in a convenient, fast and inexpensive manner.Note that card input was never considered. The MODELER package is still used, however, in a capacity generating the entire matrix as a sequence of smaller blocks later to be joined together to form the total constraint matrix. Comparison of CPU times suggests that process time, and particularly I/O time increase exponentially with the dimension of the matrix generated - whereas the entire matrix could not be generated in a single pass within 2,400 seconds, matrices one-fourth the size were routinely generated within 300 seconds.

4. Although a JOIN procedure is available in the MODELER package whereby model concatenation of independent structures can be achieved to produce a composite model, it was thought preferable to use the convenient REVISE procedure described above for this purpose.

In general, it should be stated that in order to use the TEMPO system efficiently, the user should become intimately familiar with the various file handling techniques that are available.In addition, a detailed knowledge of options such as the SAVE basis technique, CLOCK matrix reinversion, and related TEMPO features is helpful to insure efficient progress to optimality and efficient sensitivity analysis.Note: A more detailed report on the use of TEMPO/MODELER is planned describing additional features used and their relative efficiency.Acknowledgement: The authors would like to express their gratitude to Mr. Bruce Benseman of the Applied Mathematics Division of theD.S.I.R., Wellington, for the use of his EQUATION WRITER package which was periodically used to verify model implementation.

APPENDIX III - HEURISTIC SOLUTION

Outline Table Ilia Table Illb Table IIIc

Table Illd Table Ille Table Illf Table Illg Table Illh

Table Illi Table IHj Table Illk

Table III1

Table Him

Vogel's Approximation Method Pulp/Sawlog/Mix PricesCutting Costs in m^ for alternative cutting methodsPulp/Sawlog prices adjusted for harvesting costs (per m ) by Timber Age.Profitability Estimates, Method 1Profitability Estimates, Method 2Profitability Estimates, Method 3Profitability Estimates, Method 4Heuristic Problem Solution:Proportional Cuts (29 areas)Heuristic Solution: Method Use by Year (29 areas)Heuristic Solution: Method Use by Area (29 areas)Heuristic Solution: Breakdown into Pulp and Sawlogs in m^Heuristic Solution: Aggregated to 15 areas (Proportional Cuts)Heuristic Solution: Aggregated to 15 areas Method Use in No. of ha.

64

VOGEL'S APPROXIMATION METHOD FOR TRANSPORTATION PROBLEMSReference: Nyles V. Reinfeld and William R Vogel, Mathematical

Programming, Prentice-Hall, Inc., 1968General: The VAM method is designed to determine, through

an inspection type process, an initial solution to transportation problems as close as possible to an optimal transportation schedule.Under the VAM process, an assighment is made to the best cell according to an inspection of the orginal costs (profits) marked in the cells of the transportation cost matrix. An additional column and row are appended to the transportation matrix in which first cost (profit) differences (ignoring ties) are entered.

Steps: 1. Select the two best cost (profit) values (ignoring ties) in each row and column. Enter these n+m fist difference values in the marginal row and column.

2. Select the largest from among these n+m marginal values, and make a maximal assignment to the best cost (profit) cell in that row or column

3. Update the marginal values taking the previous assignment into account, i.e. by deleting the row/ column for which the supply is depleted/demand satisfied from consideration.

4. Return to step 2. until a basic feasible solution is obtained.

65

TIMBER AGE TIMBER VALUE* PULP PRICE SAWLOG PRICE AVERAGE PRICE**

9 10.00 9.09 N.F. N.A.10 12.50 11.36 N.F. N.A.11 14 .75 13.41 N.F. N.A.12 16.50 15.00 N.F. N.A.13 18.00 16.36 N.F. N.A.14 19.50 17.73 N.F. N.A.15 20.50 18.64 27 .96 23.3016 21.25 19.32 28.98 24.1517 21.75 19.72 29.66 24.7118 22.25 20.23 30.35 25.2919 22.50 20.46 30.69 25.58

* - Obtained from Price/Age Graph** - This average assumes a 50% mix of 'sawlogs and pulp

Table Ilia Pulp/Sawlog/Mix prices

TIMBER AGE COST-METHOD 1 COST-METHOD 2 COST-METHOD 3 COST-METHOD 45 12. 506 12.257 12.008 18.50 11.75 22.50 16.759 16.00 11.50 20.25 15.0010 14.00 11.25 18.25 14.25

11 12.25 11.00 16.00 13.5012 10.50 10.75 14.25 12.7513 9.25 10.50 12.25 12.0014 8.50 10.50 10.25 11.2515 7.75 10.50 9.00 10.7516 7.25 10.25 8.00 10.2517 7.00 10.25 7.00 9.5018 6.75 10.25 6.50 9.0019 6.50 10.00 5.75 8.5020 6.50 10.00 5.25 8.50

Table Illb Cutting Costs in m^ for alternative cutting methods.

HIGH CPUNTRY LOW COUNTRYPULP SAWLOGS* PULP SAWLOGS

TIMBER AGE METHOD PRICE PRICE METHOD PRICE METHOD PRICE9 4 -5.91 N.F.** 2 -2.41 N.F. N.F.

10 4 -2.89 N.F. 2 0.11 N.F. N.F.11 4 -0.09 N.F. 2 2.41 N.F. N.F.12 4 2.25 N.F. 1 4.50 N.F. N.F.13 4 4 . 36 N.F. 1 7.11 N.F. N.F.14 3 7.48 N.F. 1 9.23 N.F. N.F.15 3 9.64 12.55 1 10.89 2 12.8016 3 11.32 13.90 1 12.07 2 13.9017 3 12.77 15.21 1 12.72 2 14.4618 3 13.73 16.29 1 13.48 2 15.0419 3 14.71 17.08 1 13.96 2 15.58

* - Sawlogs in High Country can only be harvested using method #4** - Sawlog production from timber of age less than 15 years is not feasible

Table Hie Pulp/Sawlog prices adjusted for harvesting costs (per ra ) by Timber Age. (An assumption of a 50% cull factor for sawlogs was used)

66

67

Table

Illg

Profitability

Estimates, M

ethod

4 (Discounted

$/ha)

68

00 CM CM ON 00 O CO CT\r l 1*» h* CO

CO cn (N CTioo r' in

rH in cn co

CM CO CO CO

oo r - m in cr» CM CO ON

■*r o »h o

Mstho

d 4

incoooNCMHvor^HHOM' h S cocM'rm<(},rMH(y\cNinvDino>coin ooincMCMcocoinrocoTj,io*TOococMvovo^rinvr>in^ycMvo<T>c7\'OCMO O O O O O O O O O O O O O O C M C M O O O O O O C M C M C M C O O O

Method

3

0.39

0^24

1.22

1.23

1.57

0.71

0.46

0.56

1.06

1.27

2.13

1.45

2.06

0.68

4.42

5.23

9.61

13.01

i2

Trma\cr>corHcocMooinovoco ^rcMincMcocomrHinmcMO'^coo OO'JCMOHHOCMCMCOCOCM HHH(NCOCOfO( NHHr ‘ in^, (NH

o o o o o o o o o o o o o i o o o r H o o o o o c o c o m r ^ o o CO

co

Method

1

cMCM(Nco^rvovocMco* , cTvcocMvor^oooN^rr^rMr-rHr'Ovo^rocMcoOOCMtHOOOOOOCTNOarH rl O O O O i n c O l O H r l O ' T V O f f O r l O O O O O O O O O O O O O O O O O O O O r H O O O r H i —i CM CO O O

VO

° J■US■H OI ’dI -P W

| *H -H

a!

£

HCMro^invDMDcnoHrMrO'Tin^or'OocrioHrMforTinvDr'COO'rHrHHrHrHHHrHrHrHCMCMCMCMfNCMCMCMCMCN \ u

cm «h cm inco o in in

i l i i i l i i i i i i i d i i i i i d i i i i i d o i d

rH in fO CM in

I I I I I I I I I I I I I I I I I I o' I o o I o' o o I o

ro m i n m c o ^ ' r oro H in CM CM rH O

I I o I I I I I | I | I I | I | | | I O O O I O O o I rH

or** cm vd r- o cm cm oo co cm co h

I I O O I I O I I I I I O I O O O O O O O I O O O I | I I

j n ^ O r H o oo in oo in in -*r <r»PM CO CM rH O VO CO rH CM CO rH

O O O I O I I I I O O O I O I I O O I O I I I O I I O I I

i n c o o c M in o o o o oO CM CO CO CO <7\ CO V£> CM

I I I I I I i o o o o i o i l o d o d i i I i i i i i i i

^CMCMr*-OCOONr^tHrHCM CM o o

O O O O O r H O O O O O O l I O O O I I I I I I I I I | I |

I CM f O M1 i n VO Is* 00 CT* O i ■invor'OocriOrHfMco-?rinvor>oo<T»

Illj

Heuristic

Solution:

Method

Use by

Area

(29 areas) in

units

of ann

ual capacity.

69

1—Item Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7

Method 4 PulpMethod 4 SawlogsMethod 3 (All pulp)

Method 2 Pulp

Method 2 SawlogsMethod 1 (All pulp)

# 7030

17135

M 2224

6886

3038

# 12824804

244942925

#102783883

#10079

13581

*267596866

* 3768 914516849

* 4738 7898

* 5369

21113

33463

27772

14989

30573

#2012925873

26688

26903

31151

# 6000 35813

32820

* 6000 45763

54366

28789

31380

* 4350 68195

73455

* 1650 32611

TbtalSawlogs 20173 16506 37962 58344 57839 87166 104835Port 2 PulpPort 1 Pulp

7690

22078

35243

47116

50506

13875

40862

20129

62716

6000

74552

6000

100806

6000Total rrf* 49941 98865 102343 119335 126555 167718 211641

3Tbtal m over the planning horizon: 867 390

Table Illk Heuristic Solution: Breakdown into Pulp and Sawlogs in m3 ( # = port 1 pulp shipments)

70

Year Area 4 Area 5 Area 3 Area 2 Area 9 Area 1 Area 6 Area 7 Area 10 Area 8 Area 12 Area 11 Area 13 Area 14 Area 15

1 0.77 0.10 0.95 0.86 0.04 0.35 0.16 - - 0.13 - - - - -2 0.23 0.80 0.05 - 0.34 - 0.12 - 0.42 0.53 - - - - -3 - 0.10 - 0.05 0.17 0.23 0.29 0.68 0.20 0.29 0.34 - - - 0.194 - - - 0.08 0.29 0.25 0.42 - 0.25 0.05 0.38 - 0.18 - -5 - - - - 0.08 0.17 - - 0.14 - 0.28 0.33 0.29 0.14 -6 - - - - 0.07 - - - - - - 0.24 0.53 0.34 0.247 - - - - 0.01 - - 0.32 - - - 0.43 - 0.52 0.57

Itible XIII Heuristic Solution: Aggregated to 15 areas (Proportional Cuts)

Year Area 4 Area 5 Area 3 Area 2 Area 9 Area 1 Area 6 Area 7 Area 10 Area 8 Area 12 Area 11 Area 13 Area 14 Area 15

1 68.211.811.9 14.5

1.3 «23.9 *

2.9 11.3 •25.0 * 18.6

7.5 *

44.5 *■15.2 *

30.0 18.7 *

214.811.7

87.329.1

1.4 84.046.622.5

11.8 *

35.3 73.1 #

199.7 *

#1#2

. *3 14

314.5

2.421.231.026.5

24.68.9

7.57.5 13.2 *

29.0

21.2 ♦

36.1

71.0 *109.6 * 16.6

4 3.8 L29.2 12.3 18.8 * 5.9

11.4 8.5 * 6.8 12.6 * 36.9

17.0 »29.0 17.4 ♦

18.7 45.2 18.8 *73.4 •

71.0 *

5 35.325.6 •

35.346.556.8

10.6 4.6 • 6.511.3 *

84.572.2 *

633.2

13.1 *11.1 *

16.971.0 *81.1

42.280.4

42.3 128.0 *

73.7

5.512.2

9.5 7.4 «26.4 *

86.6 108.2 » 48.4

115.1 * 174.4

Table H i m - Heuristic Solution: Aggregated to 15 areas: Method Use in No. of ha (♦ = sawlog production)

APPENDIX IV - DECOMPOSITION L.P.

71

Table IVa Decomposition Linear Programming Solutions #1 - #2 : Proportional Cuts

Table IVb - Decomposition Linear Programming Solutions #1 - #2: Equivalent Annual Method Use

Table IVc - Decomposition Linear Programming Solutions #1 - #2 : Port Use

Table IVd - Decomposition Linear Programming Solutions#1 -#2: Equivalent Annual Method Use by Year

Table IVe - Decomposition Linear Programming Solutions #1 - #2 : Annual Port Use in m3 (1000's)

(Tables IVa to IVc on page 72)

Method Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7

10.455 1.262

7.400 6.10013.150

2 0.5590.559

1.0001.000

0.4700.470

0.8700.870

2.0002.000

4.0004.000

6.0006.000

3 7.0007.000

0.9700.972

7.0007.000

3.3004.280

3.2007.000

12.0002.650

7.0855.860

4 2.4302.430

2.8602.860

3.9503.950

9.0009.000

9.0009.000

12.00012.000

12.00012.000

Table IVd Decomposition Linear Progranrning Solutions #1 - #2: Equivalent Annual Method Use by Year

Port Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7

1 44.87044.870

25.65825.660

21.04821.050

12.82016.840

6.0006.000

6.0006.000

6.000253.870

2 5.6975.870

43.98051.280

84.097100.000

100.000120.000

295.81150.000

300.000150.000

Table IVe Decomposition Linear Prograirining Solutions #1 - #2: Annual Port Use in m3. (1000's)

72

Year Area 4 Area 5 Area 3 Area 2 Area 9 Area 1 Area 6 Area 7 Area 10 Area 8 Area 12 Area 11 Area 13 Area 14 Area 15

1 0.9730.973

0.9720.972

0.9940.994

2 0.0270.027

0.0280.028

0.0060.006

0.4490.449

0.3880.288

3 0.5510.551

0.3080.308

0.7900.790

0.4960.746

4 0.1740.228

0.2100.210

0.0540.054

0.8990.899

0.3260.601

0.7330.768

0.0160.016

5 0.0750.076

0.4500.200

0.1000.136

0.1670.215

0.3910.577

0.5380.681

0.0130.013

0.0470.047

6 0.055 0.1010.101

0.0360.148 0.002

0.4980.188

0.3130.129

0.5070.280

0.5090.263

0.4830.246

7 0.5380.135

0.1000.015

0.1110.235

0.1490.190

0.4930.720

0.4780.724

0.4540.691

liable IVa Deccttposition Linear Prograrmdng Solutions #1 - #2 (Proportional Cuts)

'•fethoc Area 4 Area 5 Area 3 Area 2 Area 9 Area 1 Area 6 Area 7 Area 10 Area 8 Area 12 Area 13 Area 13 Area 14 Area 15

10.455 0.514 0.748

0.184 0.076 2.8082.810

4.4904.490

5.8525.680

2 0.3090.309

0.2500.250

0.3400.340

1.1471.147

0.5220.522

0.5870.587

1.4161.416

0.2680.268

1.8921.892

0.6330.633

1.8641.864

2.2012.201

3.3483.343

3 3.4963.496

3.6873.687

0.1490.149

2.5122.512

9.4899.489

2.3792.379

1.8031.802

0.2540.254

6.4336.433

0.6560.656

7.8918.640

2.8183.110

4 0.9300.930

0.9400.940

0.5600.560

0.8900.890

2.8602.860

1.2401.240

2.0202.020

1.1501.150

3.3603.360

6.0396.039

4.5004.500

2.9102.910

5.3205.320

7.8707.780

10.65010.650

Table IVb Decomposition Linear Progranming Solutions #1 - #2: Equivalent Annual Method Use

Port Area 4 Area 5 Area 3 Area 2 Area 9 Area 1 Area 6 Area 7 Area 10 Area 8 Area 12 Area 11 Area 13 Area 14 Area 15

1 1.0001.000

1.0001.000

1.0001.000

1.0001.000

0.1010.101

0.0220.135

0.0030.015

0.0120.157

0.0090.121

0.4250.425

0.4600.460

0.4430.443

2 1.0001.000

1.0001.000

1.0001.000

0.8990.899

0.9780.865

0.9970.985

0.9880.843

0.9910.879

0.5750.575

0.5400.540

0.5570.557

Table IVc Deoonposition Linear Progranming Solutions #1 - #2; Port Use