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This paper explores methods for scheduling preventive maintenance by means of mathematical programming. Five integer linear programming models are posed, each based on a different optimality criterion dealing with manpower smoothing. By means of two hypothetical examples the models are tested for their computational properties. In addition to describing the experimental results of applying integer linear programming techniques to this class of problems, the paper also illustrates how effectively rounded linear programming solutions can be used as approximations to optimal solutions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to construct sets of uniformly tighter linear constraints to replace a chance constraint, in order to be able to solve the chance-constrained programming problem by the simplex method. The chance constraints are first diagonalized by a linear orthonormal transformation. Uniformly tighter linear constraints can then be formed to replace the chance constraint. The developed multistage linear programming problem can also be solved using the Dantzig-Wolfe decomposition technique. Approximation errors of our method and one of Hillier's [8] are compared. [ABSTRACT FROM AUTHOR]

In PERT-type networks, where activity times are random variables, the expected duration of the total project can be approximated by first substituting the expected values of the individual activities, and then evaluating the length of the critical path. This paper develops two improved estimates. Both estimates approach the true expected duration from below, i.e., they are both optimistic estimates. [ABSTRACT FROM AUTHOR]

This article presents various approximations on saving propensities and shadow wage of workers in government investment projects. The wage that government investment projects must pay is typically held to be much greater than the marginal value product of a worker in agriculture. This suggests that neither the actual wage paid nor the marginal value product of a worker in agriculture correctly measures the social opportunity cost of labor to government investment projects; project planners need to take account not only of the output foregone in agriculture, but also of the social cost of any additional commitment to consumption that increasing project employment implies. In practice various approximations are used, and the most influential attempt to date to provide a consistent basis for these approximations has been the OECD Manual by Little and Mirrlees. This paper derives an expression for the shadow price of labor in government projects from a simple optimization model, from which the Little-Mirrlees shadow wage, along with several of the other formulae proposed in the literature, can be obtained directly by making appropriate parametric assumptions. Some hypothetical computations suggest that the resulting upward bias may be very considerable. Its implications are the selection of projects that are overly capital intensive and generate a sub-optimal quantity of employment per unit of government investment.

This paper develops a model for the optimum allocation of fixed stockpiles of interceptors. Imbedded dynamic programming and successive approximations serve as the format for the computational scheme. This procedure leads to a local optimum solution in general and to the global optimum in certain instances. A method for determining upper bounds is included in the algorithm so that optimal stopping rules may be used to choose a final solution. [ABSTRACT FROM AUTHOR]

This article focuses on the computational aspects of dynamic programming. Dynamic programming is a method of solving multi-stage decision-process problems. Several computational difficulties are characteristic of all dynamic-programming solutions. In the theory of dynamic programming one find dealing with functions whose arguments are states of a system and whose values are optimal yields. In dynamic programming of low dimension the functions are generally defined by a table of values calculated at a grid of points. For problems of increased complexity it may prove desirable to approximate these functions. In this way only the appropriate coefficients need be computed at each iteration of the recurrence relation. The evaluation of a 10-dimensional function at only 10 grid points in each dimension would generate or 10 billion numbers. Yet this same function might be adequately defined by only 100 parameters of a suitably chosen approximating form. Furthermore, if one is dealing with analytic functions rather than tabular ones, tools of mathematical analysis such as calculus can be used to determine the maximum required by equation otherwise slower search techniques must be employed.

Several difficulties arise when multistage optimization problems with linear constraints are extended to an infinite horizon. For linear objectives the weak duality theorem fails, duality gaps exist, complementary slackness is not a sufficient condition for optimality, and the total reward may diverge. This paper presents verifiable conditions on the problem that guarantee the existence of optimal solutions and equilibrium prices for linear and nonlinear objectives. In addition, it is ascertained that the Kuhn-Tucker conditions are sufficient for optimality if additional restrictions are imposed. If the constraints are Leontief and the objective is convex (in a maximization problem), then the existence of an extreme point optimal solution can be verified. The decision taken in each period is an extreme point of the set of solutions that are feasible in that period. The Leontief case also yields a horizon theorem. All sufficiently long finite horizon problems produce a first decision that is optimal for the infinite horizon problem. [ABSTRACT FROM AUTHOR]

This paper extends the Sequential Unconstrained Minimization Technique for nonlinear programming to include problems where the constraints are a mixture of inequalities and equalities. Theorems indicating the dual nature of the method and the solution of a subproblem are given. Finally, the theoretical basis for a k-point extrapolation procedure is established. [ABSTRACT FROM AUTHOR]

In many situations, approximation of a set of data by a polygonal curve is more advantageous than approximation by a polynomial. If the join points of the polygonal curve are known, the problem is quite simple. If, however, they are to be chosen in some expeditious fashion, considerable numerical difficulties can arise if the curve-fitting problem is approached directly. In this paper it is shown that dynamic programming offers a simple direct approach to the determination of an optimal fit. [ABSTRACT FROM AUTHOR]

With reference to previous publications on the advisability of defining an economic production range in case of batch production, it is pointed out that many industrial problems which involve optimization, may be characteristically similar. It is therefore desirable to study the sensitivity of the function to be optimized by computing an optimal range, for which a general approximation is suggested. [ABSTRACT FROM AUTHOR]

This article presents a reply to a commentary regarding the process analysis alternative to statistical cost functions. The author was expecting a comment regarding his article which cast considerable doubt on the statistical estimation approach to the derivation of cost functions but the comment points out to the process analysis approach. The critics feel that a linear programming application of process analysis to petroleum refining has basically the same advantages as the statistical cost function technique, but these could be overcome with the adoption of their particular approach. They also propose a more general formulation utilizing non-linear programming techniques and allowing for nonconvex production technologies. The author of this reply agrees to the comments of the critics that there are situations involving nonconvexities in which nonlinear optimization techniques have a contribution to make.