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Strategic planning in forest industries
152
European Journal of Operational Research 24 (1986) 152-162 North-Holland
Strategic planning in forest industries A. W E I N T R A U B , S. G U I T A R T and V. K O H N
Department of Industrial Engineering, University of Chile, Santiago, Chile Abstract: The problem of modelling high investment decisions faced by a forest industry is analyzed in this paper. Data and concerns of a particular enterprise give form to an application of the approach. A mixed integer model is used for the investment decisions, for which the original silvicultural information is aggregated, based on LP aggregation criteria first proposed by Zipkin. This model interacts with a typical more detailed forest management model. Results obtained indicate this is a promising approach for integrating complex investment decisions into the planning process. Keywords: Forestry, management, integer programming, linear programming
1. Introduction The modelling of forest decisions, including integration to plant production has already been reported in the literature [1,6]. However, these models have concentrated essencially on decisions not involving high investments e.g. cutting and planting, timber hauling, production at plants. We will call these decisions tactical, following the definition in [3]. Road building is one area where major investment decisions have been integrated into models
[4,5]. One aspect that must be considered by forestry firms is that of major investment decisions, such as plant construction or expansion, or major land acquisitions. We call these strategic decisions. If the firm is dealing with a single such decision, a ' what if' approach using standard ' tactical' models is appropriate [6]. When multiple investment alternatives are being considered and include other policy considerations such as a financial or diversification strategy, these should be integrated into the decision model. A straightforward approach would be to integrate investment decision variables to the basic plant operation and timber management model. There are two disadvantages in doing this: Investment decisions are typically Received September 1983
discrete (a plant is built or not) leading to mixed integer 0-1 linear programs. Solving these integer problems takes up considerable computer time, so it is highly desirable to reduce as much as possible the dimension of the continuous part of the model and usual timber management models are relatively large. In addition, the investment decisions model need not consider timber or plant operating decisions in detail. What is essencial is that an optimal investment policy be consistent with an optimal tactical model. This leads to formulating a 2-stage hierarchical approach. For a strategic model, timber management data is aggregated. This implies reducing the number of stands or homogeneous areas into which the forest is divided and of optional defined management alternatives per stand. The strategic model integrates investment decisions with the aggregated timber management part and the (also aggregated) hauling and plant operation decisions. The results obtained in this model indicate suitable investments and corresponding directives, or approximate ranges for managing silvicultural variables and plant operation. A second level model, where these timber and plant operation decisions are viewed, must be consistent with these directives. The aggregation process of stands, activities and time periods for the strategic model, and the disaggregation of its solution leading to the formu-
0377-2217/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
A. Weintraub et al. / Strategic planning in forest industries
lation of tactical models present difficulties in preserving consistency. This leads to introducing a formalized aggregation procedure [7,8]. In Section 2 the strategic model is developed. Section 3 analizes the relation between the strategic and tactical models. Section 4 discusses the aggregation process, while in Section 5 the application of the proposed approach to a specific forest industry problem is presented. Final conclusions are given in Section 6.
2. The strategic model For tactical models [1,2,6] silvicultural activities (mainly cutting and planting) are defined for stands or areas homogeneous in terms of type of trees, age, location, etc. In a typical approach (Model I [2,13]) management alternatives are defined by all interventions (partial cutting, where a fraction of the trees are removed to allow better growth of the remaining, total cutting and planting) through the horizon. In addition, the model handles according to each case timber hauling, production at plants, sales, etc. All these factors are viewed in a relatively detailed way. The strategic model is based on the following notions: (i) Investment in plants. New plants can be constructed, or existing plants expanded only to a few standard sizes. Investment costs, capacities, technical transformation coefficients, and operating costs are basic data reflecting the characteristics of each case. (ii) A redefinition of silvicultural variables. Stands are grouped into macro stands, a macro stand being composed of a set of stands relatively similar in terms of location, tree age and land characteristics. The stands integrated into a macro-stand are viewed separately in the tactical model; however, for the purpose of strategic decisions the differences among them are considered not significant. Along this line, since the strategic model need not consider management alternatives to the degree of detail required for tactical decisions, fewer aggregated management activities are defined. Management alternatives are modelled following the above Model I prescription considering its case for handling aggregation. (iii) Time aggregation. Time periods are defined as needed for the decisions considered. Early
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periods are short, (one year) indicating the importance of timing in early investments, and grow larger towards the horizon. The total number of periods is smaller than in tactical models. (iv) Plant operation and sales are handled in a similar fashion as tactical models, consistent with the aggregation criteria discussed in (ii) and (iii). Timber is divided into different classes (diameters) according to its possible uses: logs (for export), sawmill, pulp plant. There are byproducts in the sawmill operation: chips, to be used in the pulp plant, bark and sawdust consumed as energy. The approximations carried out in the aggregation process are vafidated by the fact that strategic investment decisions depend basically on aggregated silvicultural and plant production variables, rather than a more detailed composition of these. The mathematical model
We now present a general mathematical model to describe the strategic planning problem. Variables: x~j = area of macro stand i to be managed with alternative j. A~I = level of production of plant k expanded to module l (capacity in units of final product) in period t (1= 1 for plant at present capacity). a~/ = 1 if an existing plant k is expanded to capacity 1 in period t, 0 otherwise. B~l = level of production of a new plant k of capacity l in period t (units of final product). fl~ = 1 if a new plant k of capacity 1 is built in period t, 0 otherwise. T~,k = production of chips in mill k' sent to plant k in period t (m3). Y/ = total timber of class r produced by land owned by the firm in period t (m3). Z~k = total timber of class r sent to plant k in period t (m3). E~t = timber of class r purchased to outside providers in period t (m 3). F/ = timber of class r sold in period t (m3). Pt = money borrowed in period t ($). R t = amount of the debt to be paid out in period t ($). D t = debt accumulated through period t ($).
A. Weintraub et al. / Strategic planning in forest industries
154
Vt
= total income in period t ($). W t = total expenditures in period t ($). Q[ = area corresponding to macro-stand i purchased in period t (possible areas to be purchased are also grouped into macrostands). Parameters: ri = area of macro-stand i owned by the firm. Qi = area of macro-stand i for possible purchase. T = horizon. gkl = capacity of plant k at level l. nkt = technical coefficient of transformation relating units of final product per m 3 of timber, in plant k of capacity l. atrij = volume of type r (m3/Ha) obtained through management alternative j in macro-stand i in period t. r : interest rate. d t : maximum debt level acceptable through period t. pt : maximum level of borrowing in period t. d~ = net income obtained from assigning one m 3 of timber of type r to plant k in period t. return obtained from direct sale of one m 3 r/ of timber of class r, in period t. Cost parameters are defined in the objective function.
The total area of macro stand i purchased through the horizon cannot exceed the area of the macro-stand.
atkl~gkl
/--~2 eta'z)=
Vk~Ke't'
(2.4)
akl t'=l
akt ~ 1 t'=l
Vk~Ke.
(2.6)
-
Given K~, the set of existing plants, for any plant k ~ K~ if no expansions are carried out, by relation (2.4) production can be carried out in any period t up to the plant's capacity gkl. If an expansion is carried out to a new capacity level gkt, l = 2 . . . . . lmax, the RHS of (2.4) be comes 0 and relation (2.5) gives the new maximum capacity of the plant from that period on. Relation (2.6) indicates that at most one expansion of plant k is considered through the horizon.
3. Production level of projected plants t
Btk, <~gkt E flt~t V k ~ Kp, t, l,
(2.7)
t'=l
1. Area
~
(2.1) Y',xij <~ri, i ~ I p . J For macro stands belonging to the firm, (i ~ Ip) the total area managed cannot exceed the area available. The same type of constraint is needed for land to be purchased: Z
Y'. xq~< Y~. O~ V i i i a , Z = l
..... T
(2.2)
t=l
where I a is the set of macro stands available for purchase, and R z is the set of management alternatives in macro stand i that requires having purchased macro stand i by period Z. T
Q[ ~< Qit=l
1 - - , , = a~
Atkl ~ gkt
Constraints
j~R z
(,ma)
2. Production level of existing plants
(2.3)
fl~ ~<1
V k e K p , l.
(2.8)
t'=l \ /=1
Given gp, the set of proposed plants, if a new plant k of capacity l is built, its production level is limited by its capacity gkt. A new plant will be built to only one given size and we do not consider modifications of it through the horizon. We note that the model can accomodate several modifications along time, but becomes more complex. Minimum production levels in each plant can also be stipulated.
4. Technological constraints y'(, I
Vk, t, (2.9) rES k
k'EM
A. Weintraub et al. / Strategic planning in forest industries
E(Bkt'nkt) <~ E Z[k+ E T#,, 1
r~S k
Vk, t.(2.10)
k'EM
The constraints indicate that in each plant total production cannot exceed timber availability (the transformation from raw timber to final product is carried out through the technical coefficient nkl ). S, represent the timber classes suitable for plant k. Note that while timber of any diameter is suitable for a pulp plant, sawmills and logs for export require larger diameters. For the sake of simplicity, we do not include constraints to avoid double assignment of timber to plants [6]. The terms Tk,k are added for plants that use chips, M being the set of sawmills,which yield chips, sawdust and bark as a byproduct. E V£'k ~<0.32 E k
Zr'k' Vk' e~ M, t.
(2.11)
r~SM
Total production of chips sent to pulp plants cannot exceed chip production at mills (SM: classes of timber appropriate for mills).
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assures that at least an area H, of range age n will be available at the horizon N. These constraints assure continuity of production after period N.
7. Financial constraints Dt=pt+Dt-a(I+r)-R
' Vt
(2.15)
indicates the accumulated debt level through period t.
D t <~dt Pt<~pt
(2.16) (2.17)
Vt
constrain the accumulate debt level and borrowing in period t.
8. Income in period t Vt = E ~'~Ztk "dtkr+ ~ ~'~ Tkk'dtko r
k
k
k"
+Ef/.frt+P t r
5. Timber production E Ztk = Y/+ Et~ - F/ Vr, t.
(2.12)
k
Total timber of class r sent to plants equals the internal production plus purchases minus direct sales of timber. Additional constraints must be imposed to consider that a given plant k may use only specific classes of timber, e.g. mills require timber of higher diameter only (see [6]).
where the first terms correspond to timber production. The second term corresponds to income due to chips sent from mills to a pulp plant. The third term is due to direct sales of timber and the fourth to money borrowing.
9. Expenditures in period t The component of cost for period t are the following: t
W1 =
Yrt =
(Exij'atrij)
E iE(IpUIa)
\
j
Vt, i.
6. Border conditions
hiJn={i
if for macro-stand i managed with alternative j trees will be of age range n at the horizon N, otherwise.
A set of constraints of the form
6
" Xij ~ H n
,.
xijmij.
(2.13)
/
This expression indicates the volume of timber of type r produced in owned or purchased macro stands in period t.
Y'.hij.
~
(2.14)
cost of land management, where mtij is the cost of managing 1 ha of macro-stand i in period t.
w~=~i S t ( ~ , x i j a t , ij): j
timber hauling costs, where S[ is the cost of hauling 1 m3 from macro-stand i to the plant sector. t
W~ = E t i Q i . i
.
land acquisition costs, where t[ is the cost of 1 Ha of macro-stand i in period t.
= E EA ,- ck,: k
l
A. Weintraub et a L / Strategic planning in forest industries
156
variable operating costs of an existing plant, where C~kt is the cost per unit of production for an existing (possibly expanded) plant k of capacity ! in period t.
Total expenditure in period t are defined by 12
w'=Ew'. i=1
/max
w/= E E
Objective function (maximize net present revenues)
k 1=2
plant expansion costs, where h~, t is the investment cost required to increase capacity of plant k to capacity l in period t.
T
Max
Z = E
( Vt-
Wt)q t
t=l
w~-
,l 1 t'=l
/.,.x
where the coefficient qt reflects the discount factor.
~ Otkl [
k
1=2
t I.,.~ t'=l
1=2
)
fixed operating costs of an existing plant, where flt, t is the fixed cost associated to operating plant k of capacity I in period t (1= 1 for original capacity, l >~2 for expansions).
=E k
l
variable operating costs of a new plant, where C~, I is the cost per unit of production for plant k of capacity 1 in period t. k
1
investment needed for a new plant, where h~,~ is the cost of building a new plant k of capacity 1 in period t.
w;=EEf k
kl /
(i,)"
*1 :
~t'=l
]
fixed operating costs of a new plant, where f2t, is the associated cost in period t.
W~o = E E T*k'to*'* : k k'
chip hauling costs, where tok, ' * is the unit hauling cost of chips from mill k' to plant k in period t. W~, = E Etet: r
timber purchases, where e~ is the price ($//m 3) for timber of" class r in period t. w~2 = R': payment of debt in period t.
3. Interaction between the strategic and tactical models The tactical model is oriented towards decisions in timber management and plant production. Its basic structure is similar to the strategic model excluding variables and constraints reflecting investments in plants and land purchases. The tactical model includes several specific additional characteristics, more detailed access considerations, short rather than long term financial analysis, etc. The timber management decision process is integrated with hauling, plant operation and sales activities. In principle, the investment decision model could just integrate investment variables to the tactical model. We nevertheless, consider aggregation as discussed in Section 2. The number of 0-1 variables depends on the number of investment decisions to be considered (typical numbers are between 30 and 100). These problems are usually quite large and costly to run, and the reduction in matrix size is one of the important factors that lead to aggregation. A straightforward aggregation process would be as follows: Consider the original data in terms of stands, management alternatives, hauling and plant operation activities. Stands are aggregated into macro stands whenever their characteristics of location, age, site quality are reasonably similar (this 'similarity' condition is tested afterwards). For each macro-stand so defined, a set of management alternatives is selected as representa-
A. Weintraub et al. / Strategic planning in forest industries
tive for all the management alternatives defined in each original stand (e.g. 20 stands with 7 management alternatives each are aggregated into one macro-stand with 6 management alternatives). Management alternatives defined reflect average values in terms of yields, production and hauling costs. Time periods are aggregated with the corresponding consequences in the definition of constraints and variables. The strategic model, with the described aggregation is solved. Its solution gives two types of indications: Investments to be carried out (in a moving horizon model, the first period decisions only are to be implemented) and basic directives for timber management. The tactical model is then processed considering the existing plant facilities plus those investments indicated by the strategic model. In addition, timber production must be consistent in each period with the levels proposed by the strategic solution. To provide for consistency, in the tactical model total timber and plant production is constrained to be within a range (say _+10%) of the productions obtained in the strategic model. If the aggregation process is correct, an optimal solution to the tactical model should lie within this range. The _+10% allows for the fact that an intuitive aggregation process, with no loss in accuracy is impossible. If these production constraints are restrictive, either through leading to infeasibility or reducing significantly the objective value, it means that the aggregation process had unacceptable errors. Typical faulty aggregations are due to: (i) Aggregation into a macro-stand of stands which are not similar enough in terms of - site quality, which implies differences in rate growths. - age of trees. - hauling costs to the plant or market, (in a macro-class one point is considered as a representative timber accumulation center). (ii) Inadequate aggregation of management alternatives. This aggregation interacts with the aggregation of timber stands in terms of age and site quality. Errors will appear if the original management alternatives that are aggregated are not similar enough in terms of period of interventions, volumes obtained and management costs. A faulty aggregation could lead to the following problem. A set of stands is aggregated into a
157
macro-stand and representative management alternatives are defined. However, the alternatives (Ai) chosen for this macro-stand cannot then be disaggregated correctly into management alternatives (Bj) for the original timber stands; for example timber production obtained in (A,) based on average values is not reproducible with any combination of the original alternatives B+. This requires repeating the whole procedure, with a new aggregation process to attempt a better consistency between the two models. This approach was implemented [12] for the data described in Section 5 and precisely these consistency problems appeared, due to faulty aggregation. The tactical model could not reproduce within a reasonable range the production directives of the strategic model. It was clear that a redefinition of the aggregation process was required. If inconsistency exists between the two models, investment decisions might be far from optimal, e.g. suppose an increase in capacity in period t in the strategic model is based on a large timber production in periods t, (t + 1) and (t + 2). If in the more detailed tactical model, which indicates actual production plans, those larger timber productions do not occur, the increase in capacity would not be justified. In summary, an inconsistency in timber production levels would indicate aggregation defficienties. A correct aggregation process implies that the more detailed decision process of the tactical model looses no significant accuracy when carried over in an aggregated way to the strategic model. Hence the solution of the strategic model should be replicable in the tactical model, in the terms of timber and plant production, costs and revenues.
4. A g g r e g a t i o n in t h e f o r e s t p l a n n i n g m o d e l
The stated consistency problem between the two models can be overcome by using an approach for row and column aggregation in linear programs, first proposed by Zipkin [7,8]. The basic ideas for the aggregation process are the following: Consider an LP Z* = Max s.t.
cx,
A.x~~0,
(4.1)
A. Weintraub et a L / Strategic planning in forest industries
158
where A is an m × n matrix, c an n-vector and b an m-vector. A j, xj and cj are defined as the j - t h column of A and j-th c o m p o n e n t of x and c respectively. Let o = ( S k, k = 1 . . . . . K ) be a partition of (1 . . . . . n). Let [Ski = n k. Define A k = Submatrix of A whose indices belong to S k. c k = Subvector of c whose indices belong to S k. x k = Subvector of x whose indices belong to Sk. Let gk be a vector of order n k, whose components are nonnegative and add up to one. Let
incurred due to the aggregation. This leads to the convenience of defining weights as close as possible to the relative values of the xj variables in the (unknown) optimal solution to (4.1). Let o' = (S;,, k = 1 , . . . , K ' } be a partition of {1 . . . . . n ) , not necessarily the same as o. Let ( d 1. . . . . d~} be known positive numbers and (P~ . . . . . PK,} known nonnegative numbers such that Y'. d j x 7 <~p,,
k=l
K'
. . . . . K,
k=l
Z = Max s.t.
?X,
A-.X~< b, X>~0,
(4.2)
where =
. . . . .
c =
. . . . .
=
are the weighted aggregations of the original coefficients. Let X* be an optimal solution to (4.1), X* and the optimal primal and dual solutions to (4.2). For any solution X = ( X 1. . . . . X k) to (4.2), the corresponding solution for (4.1) can be recovered as x k = g k X k , k = 1 . . . . . K, where x k is a nk-vector. A solution x = {x 1. . . . . x K ) obtained thus is called a fixed-weight solution. The following results follow [7].
Proposition 1. I f X is feasible for (4.2) and x is a f i x e d weight solution obtained from X, then x is feasible for (4.1) and cx = ?X. I f X is optimal for (4.2) and ~ is the corresponding fixed-weight solution, then c~ = Z <~Z*. Proposition 2.
+
e a = Y'~ m a x ( ( c j - f i A ' ) / d j } l
A-k and ?k correspond to an aggregation of the columns in S k, where each column is weighted with the values given by gk. The aggregated p r o b l e m is:
For any partition o, there exists a weighting g* such that Z = Z*. Then, if perfect weighting is chosen, no loss of objective value is
(4.3)
Proposition 3. [7] Z* <~Z + e a where
AT,= A k . g k, ~k = Ck . g k ,
k = 1,..., K'.
j E Sk
"Pk,
(4.4)
jeS~"
where [ y] + indicates m a x ( y , 0). Thus, (4.4) provides an upper b o u n d on the deterioration of the objective value if instead of solving the original p r o b l e m (4.1), the aggregated p r o b l e m (4.2), defined by (o, g) is solved. If expression (4.3) holds as a strict equality for some k, it is not necessary to take the 'positive part' of the m a x i m u m in the k t h term of (4.4). Note that expression (4.4) consists of a s u m m a tion of terms, and if o = o' this result m a y indicate which subsets S k require a refinement of the partition (this will decrease the corresponding term in (4.4)). This information can thus be used to guide the partitioning problem. While a priori bounds can also be determined (without solving p r o b l e m (2)), obtaining these bounds is more complicated and the approach is not needed for this problem. In [10], Mendelssohn proposed an i m p r o v e m e n t of this bound. Consider the dual of p r o b l e m (4.1) with the addition of constraints (4.3) K
Min
ub + Y'. 8,P,, k=l
s.t.
~ a , j u ~ + 8 k >I cj i j ~ S k , k = l . . . . . K,
(4.5)
u>~O6>~O. Let ?j = cj - ~A j, 8 k = m a x j ~ k ( O j ) + and 8 = (6k)- By assumption (4.1) and (4.5) have the same optimal solution.
A. Weintraub et al. / Strategic planning in forest industries
Proposition 4. [10] (fi, ~) is a feasible solution to problem (4.5).
are transformed into one constraint.
E X R , <~ E Ai Consider (4.5) with u restricted to the set u = { u I u = Off, O ~ R }. The new restricted dual programs becomes
159
j
(4.8)
i~R
where the xn, variables correspond to the aggregated management alternatives.
K
Min
Proposition 6. The result on aggregation of variables
~0 + ~,, 6kP k, k=l
s.t.
(~iaijLli)O+~k~Cj, j~Sk, fi>~0,
(4.6)
j = l ..... n, 8k>~O, k = l ..... K.
Let z(O) be an optimal value of (4.6) as a function of 0.
Proposition 5.
z(O) is a convex piece-wise linear
function with discontinuity points 01..... On, where
rCJ~_,aij~ i
if
E a i j ~ t i =/=O, i
if
Eaij=O, i
i
O0
for allj. Then Z,~Z* ~ Z(O*) ~ f "Jr-Ea
where O* minimizes Z(O). The e, bound obtained by Zipkin [7] corresponds to Z(1), a particular value of O. Aggregation in the forest model Consider the characteristics of a tactical model. The timber management area constraints are of the form Y'~x,j ~
for all stands i.
In the aggregation process stands are grouped into macro-stands, and management alternatives are aggregated. This implies that a set R of area constraints of type.
Exij<~Ai,
i~R,
(4.7)
J
and its corresponding management alternatives x,j
of Proposition 3 is still valid when con'straints in the original problem are collapsed, as in (4.8) (the rows are added not aggregated through a convex combination). Proof. The same steps for the proof of Proposition 3 [7] can be applied to the collapsed row problem, using a correspondingly collapsed dual vector u. Stands will be aggregated, as has been pointed out above, by similarities in spatial location, age and site quality. Management alternatives are aggregated considering similarities in intervention (close in the actual periods and volume of cutting). The similarity must be particularly close in early periods given their higher impact on the solution. Constraints related to time periods (e.g. production in period t) are also collapsed for the strategic model. This collapsing of constraints can be viewed as a relaxation of the original problem. We note that the area constraint (4.8) has the exact form required for the aggregation bound. Thus, the partitions o and o' are identical, which aids in the specifications and solution of the problem. We note that in the weighted aggregation process described consistency between the two models is obtained by definition (the disaggregation just follows the weights defined) and in addition a bound on the aggregation error is obtained with indications of its sources, so that the aggregation process can be improved. The weighted aggregation process can now be implemented [11]. Sets of timber stands are aggregated into macro-stands by similarities in location, site class and tree age. Next, within each macro stand management alternatives are aggregated, using weightings. As has been seen, if the weighting assigned corresponds to the (unknown) optimal solution, no error will be incurred in the aggregation process. Since this information is obvi-
A. Weintraub et al. / Strategic planning in forest industries
160
ously not available, an educated guess should be made on those weights. In a way similar to aggregation of transportation problems, [9], a weighting system that proved adequate was to define weights for management alternatives proportional to the area of their original timber stand. This is based on considering that in an optimal solution given the characteristics of the LP, if a variable is nonzero, it will usually take a value equal or of the order of magnitude of the stand area.
Implementation of the aggregation procedure Define: Xij
=
Ykr
=
Mk
=
Lkr
=
area of stand i managed with alternative
j.
gk r
area of macro stand k managed with alternative r. set of stands in macro stand k. set of management alternatives in all stands i ~ M k that are aggregated into alternative r of macro stand k. (g~J)) = vector of weights for aggregating management alternatives into alternative r of macro stand k.
The variables Ykr are defined as
Ykr= E
E x is'gik{,
(5 3)
i ~ M k jELk~
(g~=0
for all /j such that
i~M korj~Lkr).
In order to have consistency between the strategic and tactical solution in a weighted disaggregation procedure it is necessary that in each set Lkr at least one representative of each original ~, ~n be present. Moreover, each set Lkr must have the same number of variables corresponding to each of the original stands of the macro stand. We note that if this is not the case the disaggregation process, given the weights g]/r will lead to inconsistent assignment of management alternatives in the tactical model. For example, consider a macro stand composed of 2 stands, each with 100 ha in area and 3 management alternatives. Suppose a management alternative D k for the macro stand is defined with one alternative of stand 1 (A1) and two alternatives of stand 2 (B 1, B2). By the definition of weights according to areas, the weights leading to alternative k are 1 / 3 for A1,
B 1, and B 2. If all 200 ha of the macro stand are assigned to alternative Dk, in the disaggregation A 1, B~, B 2 will receive 66 2 / 3 ha each, leading to assigning, in the original problem 66 2 / 3 ha to stand 1 and 123 1 / 3 ha to stand 2, which is clearly inconsistent with the areas defined for each stand. The stated conditions on each set Lkr avoid such inconsistencies. Imposing these conditions on the aggregation process should not cause difficulties, given that the stands integrated into a macrostand are by the aggregation criteria, of similar characteristics. In addition, a first run allows through evaluation of the components of the error bound, ca, to refine the partitions in the aggregation process and thus improve the consistency of the strategic problem with respect to the original data.
6. I m p l e m e n t a t i o n o f t h e s t r a t e g i c m o d e l
The firm under study owns 75 000 ha of pine plantation lands. These are divided into 255 stands, ranging from 10 to 1440 ha. The plantations are mostly young, with about 7 000 ha available for cutting in the next 3-5 years (typical rotations are 20-25 years). The existing plants are concentrated in one place. They are: A mill with capacity 100 000 ma/year. - A chemical pulp plant with capacity 60000 ton/year. - A paper plant with capacity 96 000 ton/year. The timber lands are spread around, at distances varying from 8 to 123 km of the plant complex. A model for decision making in terms of timber cutting and planting, and timber allocation to plants for production has already been implemented [6]. We wish to consider now investment decisions within this setting. The basic alternatives under consideration are the following: -Expansions of the paper plant to 167000 t o n / y e a r or 214000 t o n / y e a r with investment costs of US $40000.000 and $48000000 respectively. - Construction of a mechanical pulp plant, with possible capacities of 120000 t o n / y e a r (investment $40 000000), 190 000 t o n / y e a r (investment $50 000 000) or 312 000 t o n / y e a r (investment $60 000 000).
A. Weintraub et al. / Strategic planning in forest industries
-A conglomerate boards plant with possible capacities of 40 000, 63 000 or 87 500 t o n / y e a r and respective investments of 10 800 000, 12 300 000 and 13600.000 in US $. In addition, the firm has the option of purchasing up to 23 000 ha of timberland of different ages. The investment in plants can be carried out in different future periods. The plant investments must be geared to the timber supply coming from the firm's and newly acquired lands. An adequate analysis must consider expected market prices, investment and operating costs, interest rates, silvicultural characteristics of the land, technological characteristics of the plants, cash flow problems and institutional policy restrictions of the firm. The corresponding strategic model is a mixed integer LP with 30 0-1 integer variables. In its dessaggregated form, it has 600 constraints and 1500 variables. In order to solve this problem in a typical branch and bound process, CPU time can be considerably reduced by decreasing the size of the continuous LP. This can be done through the described LP aggregation procedure. In the branch and bound process, for a given node solution let Z 0 be the continuous LP optimal value for the original problem, Z~g for the aggregated problem and e the best error bound determined. If Z* is the optimal solution, then obviously Z 0 ~< Z* and Z 0 is a lower bound ola Z*. Since Z 0 ~-~ Zag - Ea
then (Zag - Ea) is also a lower bound on Z*. This leads to the following solution approach for a bound and bound procedure. At each node solve the aggregated LP problem, rather than the original LP. The solution obtained Zag minus the error bound e can be used as a bound in the branching process. The savings in CPU time due to solving the smaller, aggregated LP may compensate for the looser bound provided by this approach. Since Z o ~ (Zag, Z a g - Ca) a small value of e, would indicate the convenience for solving aggregated LP's. The mixed LP problem was run using MPSX on an IBM 370-145. There are difficulties in introducing the described modifications at each node of
161
the branch and bound process in this program, and presently work is being carried out in this direction. As a first approximation, however we consider the bound given by the aggregation of the continuous LP excluding the 0-1 variables. This approximation should not impose a significant error, given the similarities of this LP and the LP's solved at each node of the branch and bound process. This fact was checked in particular by comparing with the LP corresponding to the optimal solution obtained. In the aggregation process, the problem was reduced from 600 to about 300 constraints and from 1500 to about 500 variables. CPU time for solving the continuous LP was reduced by 75% for the aggregated problem. A second run of the aggregated LP was made using the results of the original error bounds (see (4.4)) to refine the aggregation. The error bound computed for the aggregated continuous LP was 7.7% while the real difference between the objective of the original and aggregated problem was 4%. The solution obtained indicated two plants to be build in period 4 and one expanded in period 3, while most of the timber land available for purchase should be purchased. The solution obtained leads naturally to a disaggregate solution for the tactical level. In terms of this solution procedure we note: - The bound obtained could be improved through further refinements of the integration. - The bound obtained c a n b e improved using the approach proposed by Mendelsohn [10]. Note that this approach includes the bound of (4.4) as a special case (0 = 1). This calculation led to no significant improvement in this case.
Aggregation of LP forest management models The aggregation process described is directly applicable to decrease the size of forest management models of form Model I as defined in [2,13]. To this effect the aggregation of stands into macro-stands and management alternatives follows the same process already described. The level of aggregation of stands will depend on the error bound that is deemed acceptable. Expression (4.4) can be used to refine partitions in order to improve the bounds obtained. In the example shown above, precisely such an
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A: Weintraub et al. / Strategic planning in forest industries
aggregation was carried out, though it was intended for solving a mixed integer problem with the already mentioned reduction of 75% C P U time and an error b o u n d of 7.7%. Since these values can still be improved, it seems that this aggregation approach appears promising in reducing C P U time and core requirements for solving large forest planning (tactical level) problems.
6. Conclusion This paper has discussed the advantage of a hierarchical approach in forest planning, where a higher level model analyzes strategic decisions, integrated with aggregated tactical level variables. The strategic model leads basically to investment recommendations and production guidelines which a tactical model should follow. The consistency between the two models is not trivial, leading to the use of quantified LP aggregation procedures in a branch and b o u n d process for solving a mixed integer LP model. The computational results obtained in an application with real data were satisfactory, showing the applicability of this approach. The proposed aggregation procedures are also applicable to silvicultural models, where timber m a n a g e m e n t is the only concern, with the objective of reducing the dimensions of the LP. F o r these situations, the application of the aggregation approach is straightforward.
References [1] Dargavel, J.B., "A model for planning the development of industrial plantations", Australian Forestry 41 (1978) 95107. [2] Navon, D.I., "Timber RAM, a long range planning method for commercial timber lands under multiple use management, Research paper PSW-70, USDA Forest Service, 1971. [3] Hax, A., and Golovin, J., Hierarchical Production Planning Systems, in: A.C. Hax (ed.),. Studies in Operational Management, North-Holland Amsterdam, 1978. [4] Kirby, M., Hager, W., and Wong, P., "Simultaneous planning of wildland management and transportation alternatives", USDA Forest Service Berkeley, CA, 1981. [5] Weintraub, A., and Navon, D., "A forest management model integrating silviculture and transportation activities", Management Science 22 (12) (1976) 1299-1309. [6] Barros, O., and Weintraub, A., "Planning for a vertically integrated forest industry", Operations Research 30 (6) (1982) 1168-1182. [7] Zipkin, P., "Bounds on the effect of aggregating variables in linear programs", Operations Research 2 (1980) 403-418. [8] Zipkin, P., "Bounds on row-aggregation in linear programming", Operations Research 28 (4) (1980) 903-916. [9] Zipkin, P., "Bounds for aggregating nodes in networks problems". Mathematical Programming, 19 (1980) 155177. [10] Mendelssohn, R., "Improved bounds for aggregated linear programs", Operations Research 28 (6) (1980) 1450-1452. [11] Kohn, V., "Criterios de agregacibn de planificacibn estrat6gica forestar', Engineering thesis, University of Chile, Santiago, 1982. [12] Paredes, R., "Interaccibn de niveles de decisibn thctico y estrat6gico en planificacibn forestar', Engineering thesis, University of Chile, Santiago, 1983. [131 Johnson, K.N., and Scheurman, H.L., "Techniques for prescribing optimal timber harvest and investment under different objectives: Discussion and synthesis", Forest Science Monograph 18 (1977).
European Journal of Operational Research 24 (1986) 152-162 North-Holland
Strategic planning in forest industries A. W E I N T R A U B , S. G U I T A R T and V. K O H N
Department of Industrial Engineering, University of Chile, Santiago, Chile Abstract: The problem of modelling high investment decisions faced by a forest industry is analyzed in this paper. Data and concerns of a particular enterprise give form to an application of the approach. A mixed integer model is used for the investment decisions, for which the original silvicultural information is aggregated, based on LP aggregation criteria first proposed by Zipkin. This model interacts with a typical more detailed forest management model. Results obtained indicate this is a promising approach for integrating complex investment decisions into the planning process. Keywords: Forestry, management, integer programming, linear programming
1. Introduction The modelling of forest decisions, including integration to plant production has already been reported in the literature [1,6]. However, these models have concentrated essencially on decisions not involving high investments e.g. cutting and planting, timber hauling, production at plants. We will call these decisions tactical, following the definition in [3]. Road building is one area where major investment decisions have been integrated into models
[4,5]. One aspect that must be considered by forestry firms is that of major investment decisions, such as plant construction or expansion, or major land acquisitions. We call these strategic decisions. If the firm is dealing with a single such decision, a ' what if' approach using standard ' tactical' models is appropriate [6]. When multiple investment alternatives are being considered and include other policy considerations such as a financial or diversification strategy, these should be integrated into the decision model. A straightforward approach would be to integrate investment decision variables to the basic plant operation and timber management model. There are two disadvantages in doing this: Investment decisions are typically Received September 1983
discrete (a plant is built or not) leading to mixed integer 0-1 linear programs. Solving these integer problems takes up considerable computer time, so it is highly desirable to reduce as much as possible the dimension of the continuous part of the model and usual timber management models are relatively large. In addition, the investment decisions model need not consider timber or plant operating decisions in detail. What is essencial is that an optimal investment policy be consistent with an optimal tactical model. This leads to formulating a 2-stage hierarchical approach. For a strategic model, timber management data is aggregated. This implies reducing the number of stands or homogeneous areas into which the forest is divided and of optional defined management alternatives per stand. The strategic model integrates investment decisions with the aggregated timber management part and the (also aggregated) hauling and plant operation decisions. The results obtained in this model indicate suitable investments and corresponding directives, or approximate ranges for managing silvicultural variables and plant operation. A second level model, where these timber and plant operation decisions are viewed, must be consistent with these directives. The aggregation process of stands, activities and time periods for the strategic model, and the disaggregation of its solution leading to the formu-
0377-2217/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
A. Weintraub et al. / Strategic planning in forest industries
lation of tactical models present difficulties in preserving consistency. This leads to introducing a formalized aggregation procedure [7,8]. In Section 2 the strategic model is developed. Section 3 analizes the relation between the strategic and tactical models. Section 4 discusses the aggregation process, while in Section 5 the application of the proposed approach to a specific forest industry problem is presented. Final conclusions are given in Section 6.
2. The strategic model For tactical models [1,2,6] silvicultural activities (mainly cutting and planting) are defined for stands or areas homogeneous in terms of type of trees, age, location, etc. In a typical approach (Model I [2,13]) management alternatives are defined by all interventions (partial cutting, where a fraction of the trees are removed to allow better growth of the remaining, total cutting and planting) through the horizon. In addition, the model handles according to each case timber hauling, production at plants, sales, etc. All these factors are viewed in a relatively detailed way. The strategic model is based on the following notions: (i) Investment in plants. New plants can be constructed, or existing plants expanded only to a few standard sizes. Investment costs, capacities, technical transformation coefficients, and operating costs are basic data reflecting the characteristics of each case. (ii) A redefinition of silvicultural variables. Stands are grouped into macro stands, a macro stand being composed of a set of stands relatively similar in terms of location, tree age and land characteristics. The stands integrated into a macro-stand are viewed separately in the tactical model; however, for the purpose of strategic decisions the differences among them are considered not significant. Along this line, since the strategic model need not consider management alternatives to the degree of detail required for tactical decisions, fewer aggregated management activities are defined. Management alternatives are modelled following the above Model I prescription considering its case for handling aggregation. (iii) Time aggregation. Time periods are defined as needed for the decisions considered. Early
153
periods are short, (one year) indicating the importance of timing in early investments, and grow larger towards the horizon. The total number of periods is smaller than in tactical models. (iv) Plant operation and sales are handled in a similar fashion as tactical models, consistent with the aggregation criteria discussed in (ii) and (iii). Timber is divided into different classes (diameters) according to its possible uses: logs (for export), sawmill, pulp plant. There are byproducts in the sawmill operation: chips, to be used in the pulp plant, bark and sawdust consumed as energy. The approximations carried out in the aggregation process are vafidated by the fact that strategic investment decisions depend basically on aggregated silvicultural and plant production variables, rather than a more detailed composition of these. The mathematical model
We now present a general mathematical model to describe the strategic planning problem. Variables: x~j = area of macro stand i to be managed with alternative j. A~I = level of production of plant k expanded to module l (capacity in units of final product) in period t (1= 1 for plant at present capacity). a~/ = 1 if an existing plant k is expanded to capacity 1 in period t, 0 otherwise. B~l = level of production of a new plant k of capacity l in period t (units of final product). fl~ = 1 if a new plant k of capacity 1 is built in period t, 0 otherwise. T~,k = production of chips in mill k' sent to plant k in period t (m3). Y/ = total timber of class r produced by land owned by the firm in period t (m3). Z~k = total timber of class r sent to plant k in period t (m3). E~t = timber of class r purchased to outside providers in period t (m 3). F/ = timber of class r sold in period t (m3). Pt = money borrowed in period t ($). R t = amount of the debt to be paid out in period t ($). D t = debt accumulated through period t ($).
A. Weintraub et al. / Strategic planning in forest industries
154
Vt
= total income in period t ($). W t = total expenditures in period t ($). Q[ = area corresponding to macro-stand i purchased in period t (possible areas to be purchased are also grouped into macrostands). Parameters: ri = area of macro-stand i owned by the firm. Qi = area of macro-stand i for possible purchase. T = horizon. gkl = capacity of plant k at level l. nkt = technical coefficient of transformation relating units of final product per m 3 of timber, in plant k of capacity l. atrij = volume of type r (m3/Ha) obtained through management alternative j in macro-stand i in period t. r : interest rate. d t : maximum debt level acceptable through period t. pt : maximum level of borrowing in period t. d~ = net income obtained from assigning one m 3 of timber of type r to plant k in period t. return obtained from direct sale of one m 3 r/ of timber of class r, in period t. Cost parameters are defined in the objective function.
The total area of macro stand i purchased through the horizon cannot exceed the area of the macro-stand.
atkl~gkl
/--~2 eta'z)=
Vk~Ke't'
(2.4)
akl t'=l
akt ~ 1 t'=l
Vk~Ke.
(2.6)
-
Given K~, the set of existing plants, for any plant k ~ K~ if no expansions are carried out, by relation (2.4) production can be carried out in any period t up to the plant's capacity gkl. If an expansion is carried out to a new capacity level gkt, l = 2 . . . . . lmax, the RHS of (2.4) be comes 0 and relation (2.5) gives the new maximum capacity of the plant from that period on. Relation (2.6) indicates that at most one expansion of plant k is considered through the horizon.
3. Production level of projected plants t
Btk, <~gkt E flt~t V k ~ Kp, t, l,
(2.7)
t'=l
1. Area
~
(2.1) Y',xij <~ri, i ~ I p . J For macro stands belonging to the firm, (i ~ Ip) the total area managed cannot exceed the area available. The same type of constraint is needed for land to be purchased: Z
Y'. xq~< Y~. O~ V i i i a , Z = l
..... T
(2.2)
t=l
where I a is the set of macro stands available for purchase, and R z is the set of management alternatives in macro stand i that requires having purchased macro stand i by period Z. T
Q[ ~< Qit=l
1 - - , , = a~
Atkl ~ gkt
Constraints
j~R z
(,ma)
2. Production level of existing plants
(2.3)
fl~ ~<1
V k e K p , l.
(2.8)
t'=l \ /=1
Given gp, the set of proposed plants, if a new plant k of capacity l is built, its production level is limited by its capacity gkt. A new plant will be built to only one given size and we do not consider modifications of it through the horizon. We note that the model can accomodate several modifications along time, but becomes more complex. Minimum production levels in each plant can also be stipulated.
4. Technological constraints y'(, I
Vk, t, (2.9) rES k
k'EM
A. Weintraub et al. / Strategic planning in forest industries
E(Bkt'nkt) <~ E Z[k+ E T#,, 1
r~S k
Vk, t.(2.10)
k'EM
The constraints indicate that in each plant total production cannot exceed timber availability (the transformation from raw timber to final product is carried out through the technical coefficient nkl ). S, represent the timber classes suitable for plant k. Note that while timber of any diameter is suitable for a pulp plant, sawmills and logs for export require larger diameters. For the sake of simplicity, we do not include constraints to avoid double assignment of timber to plants [6]. The terms Tk,k are added for plants that use chips, M being the set of sawmills,which yield chips, sawdust and bark as a byproduct. E V£'k ~<0.32 E k
Zr'k' Vk' e~ M, t.
(2.11)
r~SM
Total production of chips sent to pulp plants cannot exceed chip production at mills (SM: classes of timber appropriate for mills).
155
assures that at least an area H, of range age n will be available at the horizon N. These constraints assure continuity of production after period N.
7. Financial constraints Dt=pt+Dt-a(I+r)-R
' Vt
(2.15)
indicates the accumulated debt level through period t.
D t <~dt Pt<~pt
(2.16) (2.17)
Vt
constrain the accumulate debt level and borrowing in period t.
8. Income in period t Vt = E ~'~Ztk "dtkr+ ~ ~'~ Tkk'dtko r
k
k
k"
+Ef/.frt+P t r
5. Timber production E Ztk = Y/+ Et~ - F/ Vr, t.
(2.12)
k
Total timber of class r sent to plants equals the internal production plus purchases minus direct sales of timber. Additional constraints must be imposed to consider that a given plant k may use only specific classes of timber, e.g. mills require timber of higher diameter only (see [6]).
where the first terms correspond to timber production. The second term corresponds to income due to chips sent from mills to a pulp plant. The third term is due to direct sales of timber and the fourth to money borrowing.
9. Expenditures in period t The component of cost for period t are the following: t
W1 =
Yrt =
(Exij'atrij)
E iE(IpUIa)
\
j
Vt, i.
6. Border conditions
hiJn={i
if for macro-stand i managed with alternative j trees will be of age range n at the horizon N, otherwise.
A set of constraints of the form
6
" Xij ~ H n
,.
xijmij.
(2.13)
/
This expression indicates the volume of timber of type r produced in owned or purchased macro stands in period t.
Y'.hij.
~
(2.14)
cost of land management, where mtij is the cost of managing 1 ha of macro-stand i in period t.
w~=~i S t ( ~ , x i j a t , ij): j
timber hauling costs, where S[ is the cost of hauling 1 m3 from macro-stand i to the plant sector. t
W~ = E t i Q i . i
.
land acquisition costs, where t[ is the cost of 1 Ha of macro-stand i in period t.
= E EA ,- ck,: k
l
A. Weintraub et a L / Strategic planning in forest industries
156
variable operating costs of an existing plant, where C~kt is the cost per unit of production for an existing (possibly expanded) plant k of capacity ! in period t.
Total expenditure in period t are defined by 12
w'=Ew'. i=1
/max
w/= E E
Objective function (maximize net present revenues)
k 1=2
plant expansion costs, where h~, t is the investment cost required to increase capacity of plant k to capacity l in period t.
T
Max
Z = E
( Vt-
Wt)q t
t=l
w~-
,l 1 t'=l
/.,.x
where the coefficient qt reflects the discount factor.
~ Otkl [
k
1=2
t I.,.~ t'=l
1=2
)
fixed operating costs of an existing plant, where flt, t is the fixed cost associated to operating plant k of capacity I in period t (1= 1 for original capacity, l >~2 for expansions).
=E k
l
variable operating costs of a new plant, where C~, I is the cost per unit of production for plant k of capacity 1 in period t. k
1
investment needed for a new plant, where h~,~ is the cost of building a new plant k of capacity 1 in period t.
w;=EEf k
kl /
(i,)"
*1 :
~t'=l
]
fixed operating costs of a new plant, where f2t, is the associated cost in period t.
W~o = E E T*k'to*'* : k k'
chip hauling costs, where tok, ' * is the unit hauling cost of chips from mill k' to plant k in period t. W~, = E Etet: r
timber purchases, where e~ is the price ($//m 3) for timber of" class r in period t. w~2 = R': payment of debt in period t.
3. Interaction between the strategic and tactical models The tactical model is oriented towards decisions in timber management and plant production. Its basic structure is similar to the strategic model excluding variables and constraints reflecting investments in plants and land purchases. The tactical model includes several specific additional characteristics, more detailed access considerations, short rather than long term financial analysis, etc. The timber management decision process is integrated with hauling, plant operation and sales activities. In principle, the investment decision model could just integrate investment variables to the tactical model. We nevertheless, consider aggregation as discussed in Section 2. The number of 0-1 variables depends on the number of investment decisions to be considered (typical numbers are between 30 and 100). These problems are usually quite large and costly to run, and the reduction in matrix size is one of the important factors that lead to aggregation. A straightforward aggregation process would be as follows: Consider the original data in terms of stands, management alternatives, hauling and plant operation activities. Stands are aggregated into macro stands whenever their characteristics of location, age, site quality are reasonably similar (this 'similarity' condition is tested afterwards). For each macro-stand so defined, a set of management alternatives is selected as representa-
A. Weintraub et al. / Strategic planning in forest industries
tive for all the management alternatives defined in each original stand (e.g. 20 stands with 7 management alternatives each are aggregated into one macro-stand with 6 management alternatives). Management alternatives defined reflect average values in terms of yields, production and hauling costs. Time periods are aggregated with the corresponding consequences in the definition of constraints and variables. The strategic model, with the described aggregation is solved. Its solution gives two types of indications: Investments to be carried out (in a moving horizon model, the first period decisions only are to be implemented) and basic directives for timber management. The tactical model is then processed considering the existing plant facilities plus those investments indicated by the strategic model. In addition, timber production must be consistent in each period with the levels proposed by the strategic solution. To provide for consistency, in the tactical model total timber and plant production is constrained to be within a range (say _+10%) of the productions obtained in the strategic model. If the aggregation process is correct, an optimal solution to the tactical model should lie within this range. The _+10% allows for the fact that an intuitive aggregation process, with no loss in accuracy is impossible. If these production constraints are restrictive, either through leading to infeasibility or reducing significantly the objective value, it means that the aggregation process had unacceptable errors. Typical faulty aggregations are due to: (i) Aggregation into a macro-stand of stands which are not similar enough in terms of - site quality, which implies differences in rate growths. - age of trees. - hauling costs to the plant or market, (in a macro-class one point is considered as a representative timber accumulation center). (ii) Inadequate aggregation of management alternatives. This aggregation interacts with the aggregation of timber stands in terms of age and site quality. Errors will appear if the original management alternatives that are aggregated are not similar enough in terms of period of interventions, volumes obtained and management costs. A faulty aggregation could lead to the following problem. A set of stands is aggregated into a
157
macro-stand and representative management alternatives are defined. However, the alternatives (Ai) chosen for this macro-stand cannot then be disaggregated correctly into management alternatives (Bj) for the original timber stands; for example timber production obtained in (A,) based on average values is not reproducible with any combination of the original alternatives B+. This requires repeating the whole procedure, with a new aggregation process to attempt a better consistency between the two models. This approach was implemented [12] for the data described in Section 5 and precisely these consistency problems appeared, due to faulty aggregation. The tactical model could not reproduce within a reasonable range the production directives of the strategic model. It was clear that a redefinition of the aggregation process was required. If inconsistency exists between the two models, investment decisions might be far from optimal, e.g. suppose an increase in capacity in period t in the strategic model is based on a large timber production in periods t, (t + 1) and (t + 2). If in the more detailed tactical model, which indicates actual production plans, those larger timber productions do not occur, the increase in capacity would not be justified. In summary, an inconsistency in timber production levels would indicate aggregation defficienties. A correct aggregation process implies that the more detailed decision process of the tactical model looses no significant accuracy when carried over in an aggregated way to the strategic model. Hence the solution of the strategic model should be replicable in the tactical model, in the terms of timber and plant production, costs and revenues.
4. A g g r e g a t i o n in t h e f o r e s t p l a n n i n g m o d e l
The stated consistency problem between the two models can be overcome by using an approach for row and column aggregation in linear programs, first proposed by Zipkin [7,8]. The basic ideas for the aggregation process are the following: Consider an LP Z* = Max s.t.
cx,
A.x~
(4.1)
A. Weintraub et a L / Strategic planning in forest industries
158
where A is an m × n matrix, c an n-vector and b an m-vector. A j, xj and cj are defined as the j - t h column of A and j-th c o m p o n e n t of x and c respectively. Let o = ( S k, k = 1 . . . . . K ) be a partition of (1 . . . . . n). Let [Ski = n k. Define A k = Submatrix of A whose indices belong to S k. c k = Subvector of c whose indices belong to S k. x k = Subvector of x whose indices belong to Sk. Let gk be a vector of order n k, whose components are nonnegative and add up to one. Let
incurred due to the aggregation. This leads to the convenience of defining weights as close as possible to the relative values of the xj variables in the (unknown) optimal solution to (4.1). Let o' = (S;,, k = 1 , . . . , K ' } be a partition of {1 . . . . . n ) , not necessarily the same as o. Let ( d 1. . . . . d~} be known positive numbers and (P~ . . . . . PK,} known nonnegative numbers such that Y'. d j x 7 <~p,,
k=l
K'
. . . . . K,
k=l
Z = Max s.t.
?X,
A-.X~< b, X>~0,
(4.2)
where =
. . . . .
c =
. . . . .
=
are the weighted aggregations of the original coefficients. Let X* be an optimal solution to (4.1), X* and the optimal primal and dual solutions to (4.2). For any solution X = ( X 1. . . . . X k) to (4.2), the corresponding solution for (4.1) can be recovered as x k = g k X k , k = 1 . . . . . K, where x k is a nk-vector. A solution x = {x 1. . . . . x K ) obtained thus is called a fixed-weight solution. The following results follow [7].
Proposition 1. I f X is feasible for (4.2) and x is a f i x e d weight solution obtained from X, then x is feasible for (4.1) and cx = ?X. I f X is optimal for (4.2) and ~ is the corresponding fixed-weight solution, then c~ = Z <~Z*. Proposition 2.
+
e a = Y'~ m a x ( ( c j - f i A ' ) / d j } l
A-k and ?k correspond to an aggregation of the columns in S k, where each column is weighted with the values given by gk. The aggregated p r o b l e m is:
For any partition o, there exists a weighting g* such that Z = Z*. Then, if perfect weighting is chosen, no loss of objective value is
(4.3)
Proposition 3. [7] Z* <~Z + e a where
AT,= A k . g k, ~k = Ck . g k ,
k = 1,..., K'.
j E Sk
"Pk,
(4.4)
jeS~"
where [ y] + indicates m a x ( y , 0). Thus, (4.4) provides an upper b o u n d on the deterioration of the objective value if instead of solving the original p r o b l e m (4.1), the aggregated p r o b l e m (4.2), defined by (o, g) is solved. If expression (4.3) holds as a strict equality for some k, it is not necessary to take the 'positive part' of the m a x i m u m in the k t h term of (4.4). Note that expression (4.4) consists of a s u m m a tion of terms, and if o = o' this result m a y indicate which subsets S k require a refinement of the partition (this will decrease the corresponding term in (4.4)). This information can thus be used to guide the partitioning problem. While a priori bounds can also be determined (without solving p r o b l e m (2)), obtaining these bounds is more complicated and the approach is not needed for this problem. In [10], Mendelssohn proposed an i m p r o v e m e n t of this bound. Consider the dual of p r o b l e m (4.1) with the addition of constraints (4.3) K
Min
ub + Y'. 8,P,, k=l
s.t.
~ a , j u ~ + 8 k >I cj i j ~ S k , k = l . . . . . K,
(4.5)
u>~O6>~O. Let ?j = cj - ~A j, 8 k = m a x j ~ k ( O j ) + and 8 = (6k)- By assumption (4.1) and (4.5) have the same optimal solution.
A. Weintraub et al. / Strategic planning in forest industries
Proposition 4. [10] (fi, ~) is a feasible solution to problem (4.5).
are transformed into one constraint.
E X R , <~ E Ai Consider (4.5) with u restricted to the set u = { u I u = Off, O ~ R }. The new restricted dual programs becomes
159
j
(4.8)
i~R
where the xn, variables correspond to the aggregated management alternatives.
K
Min
Proposition 6. The result on aggregation of variables
~0 + ~,, 6kP k, k=l
s.t.
(~iaijLli)O+~k~Cj, j~Sk, fi>~0,
(4.6)
j = l ..... n, 8k>~O, k = l ..... K.
Let z(O) be an optimal value of (4.6) as a function of 0.
Proposition 5.
z(O) is a convex piece-wise linear
function with discontinuity points 01..... On, where
rCJ~_,aij~ i
if
E a i j ~ t i =/=O, i
if
Eaij=O, i
i
O0
for allj. Then Z,~Z* ~ Z(O*) ~ f "Jr-Ea
where O* minimizes Z(O). The e, bound obtained by Zipkin [7] corresponds to Z(1), a particular value of O. Aggregation in the forest model Consider the characteristics of a tactical model. The timber management area constraints are of the form Y'~x,j ~
for all stands i.
In the aggregation process stands are grouped into macro-stands, and management alternatives are aggregated. This implies that a set R of area constraints of type.
Exij<~Ai,
i~R,
(4.7)
J
and its corresponding management alternatives x,j
of Proposition 3 is still valid when con'straints in the original problem are collapsed, as in (4.8) (the rows are added not aggregated through a convex combination). Proof. The same steps for the proof of Proposition 3 [7] can be applied to the collapsed row problem, using a correspondingly collapsed dual vector u. Stands will be aggregated, as has been pointed out above, by similarities in spatial location, age and site quality. Management alternatives are aggregated considering similarities in intervention (close in the actual periods and volume of cutting). The similarity must be particularly close in early periods given their higher impact on the solution. Constraints related to time periods (e.g. production in period t) are also collapsed for the strategic model. This collapsing of constraints can be viewed as a relaxation of the original problem. We note that the area constraint (4.8) has the exact form required for the aggregation bound. Thus, the partitions o and o' are identical, which aids in the specifications and solution of the problem. We note that in the weighted aggregation process described consistency between the two models is obtained by definition (the disaggregation just follows the weights defined) and in addition a bound on the aggregation error is obtained with indications of its sources, so that the aggregation process can be improved. The weighted aggregation process can now be implemented [11]. Sets of timber stands are aggregated into macro-stands by similarities in location, site class and tree age. Next, within each macro stand management alternatives are aggregated, using weightings. As has been seen, if the weighting assigned corresponds to the (unknown) optimal solution, no error will be incurred in the aggregation process. Since this information is obvi-
A. Weintraub et al. / Strategic planning in forest industries
160
ously not available, an educated guess should be made on those weights. In a way similar to aggregation of transportation problems, [9], a weighting system that proved adequate was to define weights for management alternatives proportional to the area of their original timber stand. This is based on considering that in an optimal solution given the characteristics of the LP, if a variable is nonzero, it will usually take a value equal or of the order of magnitude of the stand area.
Implementation of the aggregation procedure Define: Xij
=
Ykr
=
Mk
=
Lkr
=
area of stand i managed with alternative
j.
gk r
area of macro stand k managed with alternative r. set of stands in macro stand k. set of management alternatives in all stands i ~ M k that are aggregated into alternative r of macro stand k. (g~J)) = vector of weights for aggregating management alternatives into alternative r of macro stand k.
The variables Ykr are defined as
Ykr= E
E x is'gik{,
(5 3)
i ~ M k jELk~
(g~=0
for all /j such that
i~M korj~Lkr).
In order to have consistency between the strategic and tactical solution in a weighted disaggregation procedure it is necessary that in each set Lkr at least one representative of each original ~, ~n be present. Moreover, each set Lkr must have the same number of variables corresponding to each of the original stands of the macro stand. We note that if this is not the case the disaggregation process, given the weights g]/r will lead to inconsistent assignment of management alternatives in the tactical model. For example, consider a macro stand composed of 2 stands, each with 100 ha in area and 3 management alternatives. Suppose a management alternative D k for the macro stand is defined with one alternative of stand 1 (A1) and two alternatives of stand 2 (B 1, B2). By the definition of weights according to areas, the weights leading to alternative k are 1 / 3 for A1,
B 1, and B 2. If all 200 ha of the macro stand are assigned to alternative Dk, in the disaggregation A 1, B~, B 2 will receive 66 2 / 3 ha each, leading to assigning, in the original problem 66 2 / 3 ha to stand 1 and 123 1 / 3 ha to stand 2, which is clearly inconsistent with the areas defined for each stand. The stated conditions on each set Lkr avoid such inconsistencies. Imposing these conditions on the aggregation process should not cause difficulties, given that the stands integrated into a macrostand are by the aggregation criteria, of similar characteristics. In addition, a first run allows through evaluation of the components of the error bound, ca, to refine the partitions in the aggregation process and thus improve the consistency of the strategic problem with respect to the original data.
6. I m p l e m e n t a t i o n o f t h e s t r a t e g i c m o d e l
The firm under study owns 75 000 ha of pine plantation lands. These are divided into 255 stands, ranging from 10 to 1440 ha. The plantations are mostly young, with about 7 000 ha available for cutting in the next 3-5 years (typical rotations are 20-25 years). The existing plants are concentrated in one place. They are: A mill with capacity 100 000 ma/year. - A chemical pulp plant with capacity 60000 ton/year. - A paper plant with capacity 96 000 ton/year. The timber lands are spread around, at distances varying from 8 to 123 km of the plant complex. A model for decision making in terms of timber cutting and planting, and timber allocation to plants for production has already been implemented [6]. We wish to consider now investment decisions within this setting. The basic alternatives under consideration are the following: -Expansions of the paper plant to 167000 t o n / y e a r or 214000 t o n / y e a r with investment costs of US $40000.000 and $48000000 respectively. - Construction of a mechanical pulp plant, with possible capacities of 120000 t o n / y e a r (investment $40 000000), 190 000 t o n / y e a r (investment $50 000 000) or 312 000 t o n / y e a r (investment $60 000 000).
A. Weintraub et al. / Strategic planning in forest industries
-A conglomerate boards plant with possible capacities of 40 000, 63 000 or 87 500 t o n / y e a r and respective investments of 10 800 000, 12 300 000 and 13600.000 in US $. In addition, the firm has the option of purchasing up to 23 000 ha of timberland of different ages. The investment in plants can be carried out in different future periods. The plant investments must be geared to the timber supply coming from the firm's and newly acquired lands. An adequate analysis must consider expected market prices, investment and operating costs, interest rates, silvicultural characteristics of the land, technological characteristics of the plants, cash flow problems and institutional policy restrictions of the firm. The corresponding strategic model is a mixed integer LP with 30 0-1 integer variables. In its dessaggregated form, it has 600 constraints and 1500 variables. In order to solve this problem in a typical branch and bound process, CPU time can be considerably reduced by decreasing the size of the continuous LP. This can be done through the described LP aggregation procedure. In the branch and bound process, for a given node solution let Z 0 be the continuous LP optimal value for the original problem, Z~g for the aggregated problem and e the best error bound determined. If Z* is the optimal solution, then obviously Z 0 ~< Z* and Z 0 is a lower bound ola Z*. Since Z 0 ~-~ Zag - Ea
then (Zag - Ea) is also a lower bound on Z*. This leads to the following solution approach for a bound and bound procedure. At each node solve the aggregated LP problem, rather than the original LP. The solution obtained Zag minus the error bound e can be used as a bound in the branching process. The savings in CPU time due to solving the smaller, aggregated LP may compensate for the looser bound provided by this approach. Since Z o ~ (Zag, Z a g - Ca) a small value of e, would indicate the convenience for solving aggregated LP's. The mixed LP problem was run using MPSX on an IBM 370-145. There are difficulties in introducing the described modifications at each node of
161
the branch and bound process in this program, and presently work is being carried out in this direction. As a first approximation, however we consider the bound given by the aggregation of the continuous LP excluding the 0-1 variables. This approximation should not impose a significant error, given the similarities of this LP and the LP's solved at each node of the branch and bound process. This fact was checked in particular by comparing with the LP corresponding to the optimal solution obtained. In the aggregation process, the problem was reduced from 600 to about 300 constraints and from 1500 to about 500 variables. CPU time for solving the continuous LP was reduced by 75% for the aggregated problem. A second run of the aggregated LP was made using the results of the original error bounds (see (4.4)) to refine the aggregation. The error bound computed for the aggregated continuous LP was 7.7% while the real difference between the objective of the original and aggregated problem was 4%. The solution obtained indicated two plants to be build in period 4 and one expanded in period 3, while most of the timber land available for purchase should be purchased. The solution obtained leads naturally to a disaggregate solution for the tactical level. In terms of this solution procedure we note: - The bound obtained could be improved through further refinements of the integration. - The bound obtained c a n b e improved using the approach proposed by Mendelsohn [10]. Note that this approach includes the bound of (4.4) as a special case (0 = 1). This calculation led to no significant improvement in this case.
Aggregation of LP forest management models The aggregation process described is directly applicable to decrease the size of forest management models of form Model I as defined in [2,13]. To this effect the aggregation of stands into macro-stands and management alternatives follows the same process already described. The level of aggregation of stands will depend on the error bound that is deemed acceptable. Expression (4.4) can be used to refine partitions in order to improve the bounds obtained. In the example shown above, precisely such an
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A: Weintraub et al. / Strategic planning in forest industries
aggregation was carried out, though it was intended for solving a mixed integer problem with the already mentioned reduction of 75% C P U time and an error b o u n d of 7.7%. Since these values can still be improved, it seems that this aggregation approach appears promising in reducing C P U time and core requirements for solving large forest planning (tactical level) problems.
6. Conclusion This paper has discussed the advantage of a hierarchical approach in forest planning, where a higher level model analyzes strategic decisions, integrated with aggregated tactical level variables. The strategic model leads basically to investment recommendations and production guidelines which a tactical model should follow. The consistency between the two models is not trivial, leading to the use of quantified LP aggregation procedures in a branch and b o u n d process for solving a mixed integer LP model. The computational results obtained in an application with real data were satisfactory, showing the applicability of this approach. The proposed aggregation procedures are also applicable to silvicultural models, where timber m a n a g e m e n t is the only concern, with the objective of reducing the dimensions of the LP. F o r these situations, the application of the aggregation approach is straightforward.
References [1] Dargavel, J.B., "A model for planning the development of industrial plantations", Australian Forestry 41 (1978) 95107. [2] Navon, D.I., "Timber RAM, a long range planning method for commercial timber lands under multiple use management, Research paper PSW-70, USDA Forest Service, 1971. [3] Hax, A., and Golovin, J., Hierarchical Production Planning Systems, in: A.C. Hax (ed.),. Studies in Operational Management, North-Holland Amsterdam, 1978. [4] Kirby, M., Hager, W., and Wong, P., "Simultaneous planning of wildland management and transportation alternatives", USDA Forest Service Berkeley, CA, 1981. [5] Weintraub, A., and Navon, D., "A forest management model integrating silviculture and transportation activities", Management Science 22 (12) (1976) 1299-1309. [6] Barros, O., and Weintraub, A., "Planning for a vertically integrated forest industry", Operations Research 30 (6) (1982) 1168-1182. [7] Zipkin, P., "Bounds on the effect of aggregating variables in linear programs", Operations Research 2 (1980) 403-418. [8] Zipkin, P., "Bounds on row-aggregation in linear programming", Operations Research 28 (4) (1980) 903-916. [9] Zipkin, P., "Bounds for aggregating nodes in networks problems". Mathematical Programming, 19 (1980) 155177. [10] Mendelssohn, R., "Improved bounds for aggregated linear programs", Operations Research 28 (6) (1980) 1450-1452. [11] Kohn, V., "Criterios de agregacibn de planificacibn estrat6gica forestar', Engineering thesis, University of Chile, Santiago, 1982. [12] Paredes, R., "Interaccibn de niveles de decisibn thctico y estrat6gico en planificacibn forestar', Engineering thesis, University of Chile, Santiago, 1983. [131 Johnson, K.N., and Scheurman, H.L., "Techniques for prescribing optimal timber harvest and investment under different objectives: Discussion and synthesis", Forest Science Monograph 18 (1977).