This dissertation presents various aspects of the solution of the linear multi-period stochastic programming problem. Under relatively mild assumptions on the structure of the random variables present in the problem, the value function at every time stage is shown to be jointly convex in the history of the process, namely the random variables observed so far as well as the decisions taken up to that point. Convexity enables the construction of both upper and lower bounds on the value of the entire problem by suitable discretization of the random variables. These bounds are developed in Chapter 2, where it is also demonstrated how the bounds can be made arbitrarily sharp if the discretizations are chosen sufficiently fine. The chapter emphasizes computability of the bounds, but does not concern itself with finding the discretizations themselves. The practise commonly followed to obtain a discretization of a random variable is to partition its support, usually into rectangular subsets. In order to apply the bounds of Chapter 2, one needs to determine the probability mass and weighted centroid for each element of the partition. This is a hard problem in itself, since in the continuous case it amounts to a multi-dimensional integration. Chapter 3 describes some Monte-Carlo techniques which can be used for normal distributions. These methods require random sampling, and the two main issues addressed are efficiency and accuracy. It turns out that the optimal method to use depends somewhat on the probability mass of the set in question. Having obtained a suitable discretization, one can then solve the resulting large scale linear program which approximates the original problem. Its constraint matrix is highly structured, and Chapter 4 describes one algorithm which attempts to utilize this structure. The algorithm uses the Dantzig-Wolfe decomposition principle, nesting decomposition levels one inside the other. Many of the subproblems generated in the course of this decomposition share the same constraint matrices and can thus be solved simultaneously. Numerical results show that the algorithm may out-perform a linear programming package on some simple problems. Chapter 5, finally, combines all these ideas and applies them to a problem in forest management. Here it is required to find logging levels in each of several time periods to maximize the expected revenue, computed as the volume cut times an appropriate discount factor. Uncertainty enters into the model in the form of the risk of forest fires and other environmental hazards, which may destroy a fraction of the existing forest. Several discretizations are used to formulate both upper and lower bound approximations to the original problem. / Business, Sauder School of / Graduate

The methodology and the appropriateness of adapting the linear programming model to the evaluation of timber harvest alternatives of a specific forest enterprise was examined. The use of linear programming to describe a program in which profit is maximum rather than one of several other economic allocation models was justified. The basic model, using 3 percent as the alternative rate, described the alternative thinning and harvesting opportunities for the Seward Forest at Triplett, Virginia. The optimum program had to satisfy the restrictions imposed by scarce resources and by personal management constraints. The solution of the model described a course of action for the forest manager for the next 50 years. The initiation of the optimum plan would result in maximizing total present worth to the fixed resources of the Forest. Changes were made in the constraints on the model to demonstrate their effect upon the combination of activities which comprise the optimum program and the effect of these constraints on present worth. Additional solutions at 6 percent and 10 percent alternative rates were made to demonstrate the change which occurs in the activities that describe the optimum program at successively higher alternative rates. / Master of Science