Set Partitioning and Chain DecompositionNemhauser, G. L.; Trotter, L. E.; Nauss, R. M.
doi: 10.1287/mnsc.20.11.1413pmid: N/A
There is given a finite set I and a family of subsets of I. We consider the problem of determining a minimum cardinality subfamily that is a partition of I. A branch-and-bound algorithm is presented. The bounds are obtained by determining chain decompositions of directed acyclic graphs. The computation time required to determine a bound is bounded by a polynomial in the cardinality of I. Some computational experience is reported and relationships with other methods are discussed.
Some Interpretations of Sequential Bid Pricing StrategiesAttanasi, Emil
doi: 10.1287/mnsc.20.11.1424pmid: N/A
This note provides an alternative interpretation for sequential bid pricing strategies as initially formulated by Kortanek, Soden, and Sodaro [Kortanek, K. O., J. V. Soden, D. Sodabo. 1973. Profit analysis and sequential bid pricing models. Management Sci. 20 (3, November) 396417.]. In particular, bid prices obtained from the sequential model are shown to result from a condition which incorporates the failure rate function as a means of including probable actions of competing firms. A reformulation of the bidder's criterion function in the context of utility theory is also discussed and shown to result in bidding strategies which may also be interpreted in the proposed fashion.
Convergence Results and Approximations for Optimal (s, S) PoliciesHordijk, Arie; Tijms, Henk
doi: 10.1287/mnsc.20.11.1432pmid: N/A
In this paper we consider the dynamic inventory model with a discrete demand and no discounting. We verify a conjecture of Iglehart about the asymptotic behaviour of the minimal total expected cost. To do this, we give for the denumerable state dynamic programming model a number of conditions under which the minimal total expected cost for the n-stage model minus n times the minimal average cost has a finite limit as n . For a positive demand distribution we establish a turnpike theorem which states that for all n sufficiently large the optimal n-stage policy (sn, Sn) is average cost optimal. Further, we show that the computation of the (sn, Sn) policies supplies monotonic upper and lower bounds on the minimal average cost. Also, the average cost of the (sn, Sn) policy lies between the corresponding bounds. For a positive demand distribution these bounds converge as n to the minimal average cost.
NoteA Note on Majority Rule under Transitivity ConstraintsBlin, J. M.; Whinston, A. B.
doi: 10.1287/mnsc.20.11.1439pmid: N/A
In a recent paper [Bowman, V. J., C. S. Colantoni. 1973. Majority rule under transitivity constraints. Management Sci. 19 (9, May) 10291041.] Bowman and Colantoni have described an optimization model which they relate to problems of majority voting in the theory of social choice. Their optimization model was a linear integer programming problem. The purpose of this note is to indicate that an alternative view may be more useful viz. formulating the problem as a quadratic assignment type. We have discussed this in detail elsewhere [Blin, J. M., K. S. Fu, K. B. Moberg, A. B. Whinston. 1973. Optimization theory and social choice. Proceedings of the Sixth Hawaiian International Conference on Systems Science. Supplement on Urban and Regional Systems: Modelling Analysis and Decision Making. University of Hawaii.] and will limit ourselves here to some brief comments.
Further Comments on Majority Rule Under Transitivity ConstraintsBowman, V. Joseph; Colantoni, C. S.
doi: 10.1287/mnsc.20.11.1441pmid: N/A
In their note [Blin, J. M., A. B. Whinston. 1974. A note on majority rule under transitivity constraints. Management Sci. 20 (11) 14391440.], Blin and Whinston indicate that the linear integer programming formulation of the majority voting problem can also be formulated as a quadratic assignment problem. We wish to point out that both of these formulations are a result of the fact that majority decision functions can be linearized over the set of integer solutions to the linear program.
An Equilibrium-Point Interpretation of Stable Sets and a Proposed Alternative DefinitionHarsanyi, John C.
doi: 10.1287/mnsc.20.11.1472pmid: N/A
The paper argues that the von Neumann-Morgenstern definition of stable sets is unsatisfactory because it neglects the destabilizing effect of indirect dominance relations. This argument is supported both by heuristic considerations and by construction of a bargaining game B(G), formalizing the bargaining process by which the players agree on their payoffs from an n-person cooperative game G. (G itself is assumed to be given in characteristic-function form allowing side payments.) The strategies i, the players use in this bargaining game will determine which payoff vectors x will be stationary, i.e., will have the property that, should such a payoff vector x be proposed to the players, all further bargaining will stop and x will be accepted as the outcome of the game. It will be suggested that a stable set should be defined as the set V of all stationary payoff vectors x, on the assumption that the players' bargaining strategies p, will form a canonical equilibrium point in the bargaining game B(G).In a certain class of games, to be called absolutely stable games, indirect dominance relations turn out to be irrelevant, and in these games the suggested definition for stable sets is equivalent to the von Neumann-Morgenstern definition. But in general this is not the case. However, if we assume that the players will adopt bargaining strategies deliberately discouraging any use of indirect dominance relations, then, in every game, the stable sets our model yields will always be von Neumann-Morgenstern stable sets.At the end of the paper, we briefly discuss a possible modification in the suggested definition for stable sets, based on a modified bargaining game B0(G), in which the players are permitted to accept cuts in their provisional payoffs if they think that this move will increase their final payoffs.