Ordinal Dynamic ProgrammingSobel, Matthew J.
doi: 10.1287/mnsc.21.9.967pmid: N/A
Numerically valued reward processes are found in most dynamic programming models. Mitten, however, recently formulated finite horizon sequential decision processes in which a real-valued reward need not be earned at each stage. Instead of the cardinality assumption implicit in past models, Mitten assumes that a decision maker has a preference order over a general collection of outcomes (which need not be numerically valued). This paper investigates infinite horizon ordinal dynamic programming models. Both deterministic and stochastic models are considered. It is shown that an optimal policy exists if and only if some stationary policy is optimal. Moreover, policy improvement leads to better policies using either Howard-Blackwell or Eaton-Zadeh procedures. The results illuminate the roles played by various sets of assumptions in the literature on Markovian decision processes.
Optimal Control of Sales Force Effort in TimeTapiero, Charles S.; Farley, John U.
doi: 10.1287/mnsc.21.9.976pmid: N/A
This paper examines the temporal effects of alternative procedures for controlling sales force effort, particularly when the effect of such effort is felt in both current and future periods. Related informational requirements for both firm and salesmen are also discussed in terms of nonzero sum differential games, and implications for the temporal management of selling effort are derived.
Sequencing Two-Machine Flow-Shops with Finite Intermediate StorageDutta, Sujit K.; Cunningham, Andrew A.
doi: 10.1287/mnsc.21.9.989pmid: N/A
The deterministic N-job two-machine flow-shop sequencing problem with finite intermediate storage is considered with the objective of minimizing the total processing time. A dynamic programming procedure is presented whereby an optimal solution to the problem may be obtained. Since the storage and computation time requirements for the solution of this model are large, it is modified so as to give a suboptimal recursive procedure which yields good solutions with much less computational effort. A further approximate method based on the technique of successive approximations is shown to provide an alternative means for tackling large problems. In addition, several new results are derived for the particular case where the buffer capacity is zero.
Optimal Allocation of Resources in a Pari-Mutuel SettingRosner, Bernard
doi: 10.1287/mnsc.21.9.997pmid: N/A
This article deals with an optimal algorithm for wagering in a pari-mutuel situation given that one is a perfect handicapper. A general rule is given for all pari-mutuel events with mutually exclusive outcomes. The rule has several surprising aspects which differ considerably from previous betting rules. The rule is useful when one's own bet is small relative to the total amount bet by the public. The optimality criterion is that of maximizing the expected log return.
A Deterministic Multi-Period Production Planning Model with Piecewise Concave Production and Holding-Backorder CostsSwoveland, Cary
doi: 10.1287/mnsc.21.9.1007pmid: N/A
A single product, finite horizon production planning model with known requirements is considered. Production and holding-backorder cost functions are assumed to be piecewise concave, thereby allowing an arbitrarily close approximation to a wide range of cost functions which one might encounter in practice. In each period production, inventories and backlogged orders may not exceed prescribed levels.Production (inventory) breakpoints are the endpoints of the intervals over which the production (holding-backorder) cost functions are concave. It is shown that there is an optimal production schedule which has the property that between successive periods in which ending inventories are at inventory breakpoint levels there is at most one period in which production is not at a production breakpoint level. This property, which is an extension of recent results obtained by Florian and Klein [Florian, Michael, Morton Klein. 1971. Deterministic production planning with concave costs and capacity constraints. Management Sci. 18 (1, September) 1220.] and Love [Love, Steven F. 1973. Bounded production and inventory models with piecewise concave costs. Management Sci. 20 (3, November) 313318.], suggests a straight-forward dynamic programming algorithm for obtaining an optimal solution.
Multi-Item Production Smoothing Models with Almost Closed Form SolutionsWelam, Ulf Peter
doi: 10.1287/mnsc.21.9.1021pmid: N/A
Computational efficiency in solving large-scale multi-item production smoothing models with quadratic costs is the chief concern of this paper. We consider two multi-item versions of the HMMS model and show that their optimal solutions can be obtained in almost closed form. Numerical inversion of large matrices is therefore not necessary and considerable savings in terms of storage requirements are also possible. The existence of unique optimal solutions depends in a very simple way on the model parameters.
Hierarchical Decomposition in Linear Economic ModelsKydland, Finn
doi: 10.1287/mnsc.21.9.1029pmid: N/A
In decomposed linear programming models it is generally not possible to decentralize by prices alone. The Dantzig-Wolfe procedure, for instance, delegates weights on basic solutions in addition to the equilibrium prices. In this paper we present a decomposition procedure for linear models where we in addition to prices delegate a hierarchical ordering. In many problems this ordering makes the assignment of weights unnecessary, and gives the divisions more autonomy in their decision making. An operational condition is found for determining if, for any given problem, the new decomposition procedure will achieve coherent decentralization.
A Graph Theoretic Formula for the Steady State Distribution of Finite Markov ProcessesSolberg, James J.
doi: 10.1287/mnsc.21.9.1040pmid: N/A
This paper presents a formula which expresses the solution to the steady-state equations of a finite irreducible Markov process in terms of subgraphs of the transition diagram of the process. The formula is similar in spirit to well-known flowgraph formulas, but possesses several unique advantages. The formula is the same whether the process is discrete or continuous in time; it is efficient in the sense that no cancellation of terms can occur (it is a simple sum of positive terms); and it is both conceptually and computationally simple. Because these advantages are gained by exploiting properties of Markov processes, the formula is not applicable to linear equations in general, as are the flowgraph methods. The paper states and proves the theorem for both the discrete and continuous cases, gives examples of each, and cites computational experience with the formula.