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This article was downloaded by: [Massachusetts Institute of Technology]On: 25 November 2014, At: 12:33Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
International Journal of Systems SciencePublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/tsys20
Optimum harvesting policy for an age-structured forestA. K. CHAUDHURI a & K. M. SEN ba Indian Institute of Management , Calcutta, 700 027, Indiab Department of Mathematics , Jadavpur University , Calcutta, 700 032, IndiaPublished online: 31 May 2007.
To cite this article: A. K. CHAUDHURI & K. M. SEN (1987) Optimum harvesting policy for an age-structured forest,International Journal of Systems Science, 18:8, 1425-1432, DOI: 10.1080/00207728708967122
To link to this article: http://dx.doi.org/10.1080/00207728708967122
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Optimum harvesting policy for an age-structured forest
A. K. CHAUDHURlt and K. M. SENt
In this paper, an attempt has been made to develop a model for determining the optimal policy Tor felling and planting trees of a particular species in a forest, in order to maximize the total yield over a specified period of time. The model takes into account not only the time at which trees should be felled but also how many trees and of what age.
1. Introduction In recent years, a lot of attention appears to have been devoted to the application
of control theory in discussions of problems relating to forest resources management, particularly harvesting problems. In most of the work so far, the major thrust is the study of optimal forest rotation initiated by Faustmann (1849), Wan (1978), Heaps (1984), Heaps and Neher (1979), Clark (1976), Das Gupta (1 982), Mitra and Wan (198 I), Anderson and Wan ( 198 1) and Wan and Anderson ( 1983). But, to the best of our know- ledge, no general results are yet known about optimal forest harvesting policy that also takes into account the age distribution of the trees in the forest. The growth of commer- cial plants in forests, like any other ageistructured populations, can generally be taken to be governed by a first-order partial differential equation involving two independent variables, age a and time t. In fact, Gurtin and Murphy (1981 a, b) studied the optimal harvesting problem of general age-structured populations by first reducing the original evolution equation to an ordinary differential equation, considering both birth and death function to be functions of only total population and age. They found the optimal harvesting policy by using the overtaking criterion of optimality introduced by Koopmans (1965). Hellrnan (1982) specifically studied the .optimal harvesting problem of a self-sustained forest under steady-state conditions by applying the maximum principle of Pontryagin.
In this paper, an attempt is made to determine the optimal harvesting policy for a particular species of forest by taking both vintage and time into account, so that the yield is maximized over a certain specified interval of time T by the optimal control theoretic approach.
2. Formulation of the optimal harvesting problem It is well known, vide Foerster (1959), that the natural growth of trees belonging to
a particular species can be described by:
Received 9 January 1986. t Indian Institute of Management, Calcutta-700 027, India. f Department of Mathematics, Jadavpur University, Calcutta-700 032, India.
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1426 A. K. Chaudhuri and K. M. Sen
where
n(a, t) is the population density of trees ofage a at time t;
p(a, t) is the death rate of trees of age a at time t per unit population. of age a; @(a) is the initial population density of trees of age a;
P(a, t) is the number of seeds produced by trees of age a at time t; and w is the maximum age up to which a tree of the species under consideration .
can live.
Let E(a, t) be the rate of felling trees of age a at time t per unit population age a. We should take note of the following two facts before we proceed further:
(a) while trees are felled according to a policy represented by the cutting function E(a, r), the third term in equation (1 a) should be b (a , t ) + E(a, t)Jn(a, t);
(b) in a managed forest, the natural growth is further interrupted by planned planting of saplings on specified sectors of land that might be reserved from time to time for this purpose. Hence the boundary condition (1 c) really loses its meaning and should be replaced by a condition such as
where m(t) can be taken as a function of time to be determined, and M(t) is the maximum number of saplings (of age zero) that could possibly be planted at every instant of time, as determined by management from forest technological considerations.
From the above considerations, the evolution equation for tree growth in a forest that is being continually harvested and in which saplings are being continually planted on predetermined areas of land will be given by
The problem is to determine the optimal cutting policy E(a, t) and the optimal planting policy m(t) in such a way that the total yield will be maximized over a specified period of time T. That is, maximize
(3)
under the constraints (2 a), (2 b) and (2 c). Here E(a, t) is the volume and m(t) is the boundary control.
3. Determination of optimal harvesting and planting policies We shall first derive the necessary conditions for optimal cutting and planting
policies. To this end, let us introduce the function:
Then the optimum cutting problem reduces to finding the function E(a, t ) that will
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Optimum harvesting policy lor a n age-struct ured forest 1427
maximize n o ( o , T). Now no(a, t ) satisfies the following differential equation and boundary conditions:
n,(a, 0) = 0 (6)
n,(O, t ) = 0 (7)
We now consider the following functional on the lines of Butkovskiy (1969):
where $, and I) are the lagrangian multipliers or adjoint variables. If we put
then the functional in (8) can be written as
where P=(II/o, $ 9 n, E)
But we know that J(P + AP) - J(P) = 0
identically. Now,
J(P + AP) - J ( P )
a2no a an +$-6n+6II / - +6$0aa att at
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1428 A. K . Chaudhuri and K . M . Sen
(retaining terms only up to the first order of smallness)
Now, we carefully proceed to evaluate the integrals of (12) remembering that ail asterisk-marked terms are cancelled in the process because of the given conditions (2 b), (6) and (7).
Thus
a+o - [Jm{$(a , o
T)Bno(o, T ) - -(a, aa 0)6n~(o, O)] do] + j: joT*6n, aa at da dt
+J:JoTzdC do d l , since 6no(n, 0)=0
- Corn (a, r)an0(., T ) do + J: 1'2 ano do dt
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Optimum harvesting policy fir an age-structured forest
Similarly,
And,
Substituting (13 a), (13 b) and (13 c) into (12) and rearranging the terms, we get
J ( P + AP) - J ( P )
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1430 A. K . Chaudhuri and K . M . Sen
Since 6n0, 6Y0, dY, etc. are all arbitrary, we must have from (1 I), using (14),
And finally,
Choosing $,(o, T) = 1 , equation (23) becomes
6 n , ( o , T ) = j 0 m ~ 0 T ~ 6 E d a d t +
{H( ..., E + 6 E ) - H( ..., E ) ) da dt
+ Jl' $(o, 1) {(m + dm) - mj dt
Now maximization of yield requires that 6n,(o, T) ,< 0. Hence,
1 JOT / H ( , E + 6 E ) - H ( , E ) ) do dr + $(0, t ) ((m + 6m) - rn) dt < 0 (24) JOT ; 1
that is,
H( ... , E + S E ) < H( ... , E ) (25)
and
at all regular points (a, t), assuming that almost all points are regular (Butkovskiy 1969). Therefore E(a, t ) maximizes H and m(t) maximizes $(O, t ) .
If we write HI = $(0, t)m(t), then m(t) = M(t ) sign $(0, t ) . Since the equations are linear in the control functions, it can also be proved that the conditions (24) and (25) are also sufficient.
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Optimum haruesting policy for an age-structured forest
4. Optimal cutting function In felling trees, usually trees below a certain age a, (a, given) will not be felled.
Trees of age a between a, and o may be felled at maximum rates decided by management. According to the forestry maximum principle of Heaps (1984), trees having the maximum age will be felled at the maximum rate. If we assume that the maximum rate at which trees of age a (between a, and a) can be felled is F(a), then the graph of F(a) will appear as shown in the Figure.
That is
for O < a < a ,
for a < a < o
where g(a) is an increasing function of a. Therefore, E(a, t) will be given by
E(Q, t ) = ~ ( 0 ) sign w, - $1 n(a, t ) ) [see (911
= F(a) sign ($J, - $), since n(a, t) is always positive.
Now, it can easily be seen that +,(a, t) = 1. Hence, E(a, t ) = F(a) sign (I - J / ) where $ is the soiution of the boundary value problem: .
Under the above optimal cutting policy, the state of the tree population in the forest will be given by
an(a' t , +- an(ay ') + ( p + F(a) sign ( I - $)I n(n, r) = 0 da at
n(0, t ) = m(t) = M(t) sign $(0, t) J
ACKNOWLEDGMENT The authors thank Professor D. K. Sinha (the co-ordinator of the UGC-DSA
Programme in Applied Mathematics, Jadavpur University, Calcutta-700 032) for his valuable suggestions and criticisms in the preparation of this paper.
REFERENCES ANDERSON, K. and WAN, F. Y. M. 1981, Ordered site access and optimal forest rotation.
Technical Report No. 84-6, British Columbia University, Canada.
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1432 Optimum haruest ing pol icy for an age-struct ured forest
BUTKOVSKIY, A. G., 1969, Distributed Control Systems (New York: American Elsevier Publishing Company).
CLARK, C. W., 1976, Ma1 hemarical Bioeconomics (New York: Wiley). DAVIDSON, R., and HELLSTEN, M., 1980, Optimal forest rotation with costly planting and
harvesting. Presented at the 5th Canadian Conf. on Economic Theory, Vancouver, B.C., Canada.
FAUSTMANN, M., 1849, Allgm. Forst Jagd Z., 25, 441. VON FOERSTER, H., 1959, Some Remarks on Changing Populations in the Kinetics of Cellular
Proliferation (New York: Grune and Stratton), pp. 3 8 2 4 7 , DAS GUPTA, P., 1982, The Control of Resources (Delhi: Oxford University Press). GURTIN, M. E., and MURPHY, L. F., 1981 a, Math. Biosci., 55, 1 15; 1981 b, On the optimal
harvesting of age structured populations. Diflerential Equations and Applications in Ecology, Epidemics and Population Problems (New York: Academic Press).
HEAPS, T., 1984, The Forestry Maximum Principle, J. econ. Dynam. Control, 7, 13 1. HEAPS, T., and NEHER, P. A,, 1979, J. Enuiron. Econ. Mgmt, 6, 297. HELLMAN, O., 1982, Mgmt Sci., 28, 1247. KOOPMANS, T. C., 1965, On the concept of optimal economic growth. The Econometric
Approach to Deuelopment Planning. Pontificiae Academiae Scien tiarum Scripta Varia, Vol. 28 (Amsterdam: North-Holland), pp. 225-287.
MITRA, T., and WAN, H. Y., 1981, On the Faustman solution to the forest management problem. Working Paper 266, Cornell University, New York, USA.
WAN, Jr., H. Y., 1978, A Generalized Wicksellian Capital Model: an Application to Forestry in Economics in Natural and Environmental Resources, edited by V. Smith (London: Gordon and Breach).
WAN, F. Y. M., and ANDERSON, K., 1983, Stud. appl. Math., 68, 189.
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