Linear transformations

The LINTRAN format represents a special form of blocks. In some problems the stochastic elements y of the data matrices are influenced by a much smaller number of random variables u, for instance key interest rates or random inflow into a reservoir system. Rather than model the distribution of the coefficients directly, it is often convenient to model the underlying factors and to record the functional dependence between u and y. In many instances this dependence is the affine function y = Du + c, where D is a deterministic matrix and c is a deterministic vector.

The example for this format comes from a paper by Gassmann [1]. It concerns a problem in forest management. In each time stage the state of the forest is described by inventory (area covered) in a number of different age classes. Harvested area regenerates itself, along with a random fraction of unharvested area (due to random acts of destruction such as forest fires). The remainder of the unharvested area is preserved and appears in the next age class in the following time stage. There is also an absorbing age class that contains all fully mature trees. The random loss rates are assumed independent of each other and are the same for each age class within a stage.

The mathematical formulation of this problem is as follows.

where

C is the number of age classes

T is the number of stages

S is the set of scenarios

S(t) is the set of scenarios active in stage t, where S(0) is a singleton set and for all t

p_st is the probability of observing scenario s in stage t

a_t(s) is the ancestor of scenario s in stage t

f_cts is the amount of forest in age class c at the beginning of stage t under scenario s

h_cts is the amount of forest in age class c harvested during stage t under scenario s

y_c is the yield of a unit of forest in age class c

v_c is the value of a unit of forest in age class c left standing at the end of the horizon

d is the discount rate from one stage to the next

b_c is the amount of forest in age class c at the start of the planning period

P_c is the c^th row of P, the transition matrix of forest left standing from one stage to the next;

is the (random) fraction of forest destroyed in stage t under scenario s

Q_c is the c^th row of Q, the transition matrix of harvested forest from one stage to the next;

a is the minimum fraction of the yield in stage t that must be achieved in the following stage (the constraint involving this quantity is called the requirement of `non-declining yield')

Time file

Core file

Stoch file

Reference

1. H.I. Gassmann, “Optimal harvest of a forest in the presence of uncertainty”, Canadian Journal of Forest Research 19 (1989) 1267–1274.