STOCHASTIC

146 STEPHEN P. BRADLEY AND DWIGHT B. CRANE
Note that if security class k is purchased at the start of period 2 its purchase price
and income yield are conditional upon the random event which occurred during period
1. Thus, a subproblem is defined for each commodity and each sequence of random
events which precede the purchase date of that commodity.
The inventory balance equations of type (2.6) define the subproblems for each commodity
purchased at the start of period 2 and each preceding event sequence. As
would be expected, these equations have no variables in common in equations (2.4)
and (2.5), since each set of inventory balance equations simply keeps track of the
holdings of each commodity. The commodity definition adopted leads to a relatively
large number of subproblems. The rationale, however, is that the state variable of
each subproblem is then one dimensional, i.e., the amount of the commodity held.
We will in discussing the model present the results in terms of only three time
periods; however, this is only for ease of exposition as the results hold for an arbitrary
number of time periods. The constraints for the subproblems corresponding to the k
security classes that may be purchased in period one are \fk as follows:
-&i* + si.s(ei) + h k i,2(e,) = 0, Vei € £,,
(3.1) -tf.5(e,) + s{.a(ei, e2) + ti*(ei, e2) = 0, V(e, , e2) 6 E1 X E,
6i* S 0, *;,a(ei) S 0, /rf,,(e,) £ 0, *J.,(e,, e2) £ 0, M.3(e,, e2) £ 0.
The constraints for the subproblems corresponding to the k security classes that may
be purchased in period two, conditional on events in period one, are \fk and Vei £ E\
as follows:
(3.2)
— b2 k (ei) + **.s(ei, e2) + /i2,3(ei, e2) = 0, Vea € E2,
&2*(ei) g 0, 4,s(ei, e2) £ 0, /s2,3(ei, e2) £ 0.
For eight security classes, three time periods, and three uncertain events in each
period there are thirty-two subproblems. There are eight subproblems of the form (3.1)
corresponding to the eight commodities available for purchase at the start of period 1
and twenty-four of the form (3.2). The basic structure of the subproblems for an arbitrary
number of time periods and uncertain events should be evident, since the subproblem
constraints merely reflect the holdings of each commodity for each possible
sequence of uncertain events. Note that each commodity is purchased only once and
then sold or held during subsequent periods.
The interesting point to note about the subproblem constraints is that they are
homogeneous systems of equations (i.e., zero right-hand sides). In decomposition the
fundamental theorem employed is that a convex polyhedral set may be represented by
a convex combination of its extreme points plus a nonnegative combination of its
extreme rays [10]. The subproblems of (3.1) and (3.2) have only one extreme point, all
decision variables equal to zero. For any nonzero point satisfying the subproblem
constraints, a scalar times that point also satisfies the constraints and hence with a
linear objective function there exists an associated unbounded solution. As a result we
need only consider the extreme rays of the subproblems. It is critical for an effective
procedure to be able to construct these extreme rays in an efficient manner. In the
next section we show how to do this.
The restricted master for the decomposition scheme defining subproblems for each
commodity and event sequence is given in Figure 1. Note that the usual convex combination
constraints (e.g., 23 ta = 1) ar e not present, since we are only dealing with
nonnegative combinations of extreme rays. Further, there are no subproblem con-
494 PART V. DYNAMIC MODELS