Aleksandar Pekec

On the Order Fill Rate in a Multi-Item, Base-Stock Inventory System

A customer order to a multi-item inventory system typically consists of several different items in different amounts. The probability of satisfying an arbitrary demand within a prespecified time window, termed the order fill rate, is an important measure of customer satisfaction in industry. This measure, however, has received little attention in the inventory literature, partly because its evaluation is considered a hard problem. In this paper, we study this performance measure for a base-stock system in which the demand process forms a multivariate compound Poisson process and the replenishment leadtimes are constant. We show that the order fill rate can be computed through a series of convolutions of one-dimensional compound Poisson distributions and the batch-size distributions. This procedure makes the exact calculation faster and much more tractable. We also develop simpler bounds to estimate the order fill rate. These bounds require only partial order-based information or merely the item-based information. Finally, we investigate the impact of the standard independent demand assumption when the demand is actually correlated across items.

THE REPEATED INSERTION MODEL FOR RANKINGS: MISSING LINK BETWEEN TWO SUBSET CHOICE MODELS.

Several probabilistic models for subset choice have been proposed in the literature, for example, to explain approval voting data. We show that Marley et al.s latent scale model is subsumed by Falmagne and Regenwetters size-independent model, in the sense that every choice probability distribution generated by the former can also be explained by the latter. Our proof relies on the construction of a probabilistic ranking model which we label the “repeated insertion model.” This model is a special case of Mardens orthogonal contrast model class and, in turn, includes the classical Mallows φ-model as a special case. We explore its basic properties as well as its relationship to Fligner and Verduccis multistage ranking model.

Sequential vs. Single-Round Uniform-Price Auctions

We study sequential and single-round uniform-price auctions with affiliated values. We derive symmetric equilibrium for the auction in which k1 objects are sold in the first round and k2 in the second round, with and without revelation of the first-round winning bids. We demonstrate that auctioning objects in sequence generates a lowballing effect that reduces the first-round price. Total revenue is greater in a single-round, uniform auction for k=k1+k2 objects than in a sequential uniform auction with no bid announcement. When the first-round winning bids are announced, we also identify a positive informational effect on the second-round price. Total expected revenue in a sequential uniform auction with winning-bids announcement may be greater or smaller than in a single-round uniform auction, depending on the models parameters.

Majority Consensus and the Local Majority Rule

We study a rather generic communication/coordination/ computation problem: in a finite network of agents, each initially having one of the two possible states, can the majority initial state be computed and agreed upon by means of local computation only? We describe the architecture of networks that are always capable of reaching the consensus on the majority initial state of its agents. In particular, we show that, for any truly local network of agents, there are instances in which the network is not capable of reaching such consensus. Thus, every truly local computation approach that requires reaching consensus is not failure-free.

Listen to Your Neighbors: How (Not) to Reach a Consensus

We study the following rather generic communication\slash coordination\slash computation problem: In a finite network of agents, each initially having one of the two possible states, can the majority initial state be computed and agreed upon by means of local computation only? We study an iterative synchronous application of the local majority rule and describe the architecture of networks that are always capable of reaching the consensus on the majority initial state of its agents. In particular, we show that, for any truly local network of agents, there are instances in which the network is not capable of reaching such a consensus. Thus, every truly local computational approach that requires reaching a consensus is not failure-free.

The role assignment model nearly fits most social networks

Abstract Role assignments, introduced by Everett and Borgatti [Mathematical Social Sciences 26 (1991) 183], who called them role colorings, formalize the idea, arising in the theory of social networks, that individuals of the same social role will relate in the same way to individuals playing counterpart roles. If G is a graph, a k -role assignment is a surjective function mapping each vertex into a positive integer 1,2,…, k , so that if x and y have the same role, then the sets of roles assigned to their neighbors are the same. We show that all graphs G having no astronomical discrepancies between the minimum and the maximum degree have a k -role assignment. Furthermore, we introduce and study a natural measure expressing how close an onto map f : V ( G )→{1,…, k } is to being a k -role assignment of a graph G =( V , E ), and show that almost all graphs nearly have a k -role assignment.

Revenue Ranking of Discriminatory and Uniform Auctions with an Unknown Number of Bidders

An important managerial question is the choice of the pricing rule. We study whether this choice depends on the uncertainty about the number of participating bidders by comparing expected revenues under discriminatory and uniform pricing within an auction model with affiliated values, stochastic number of bidders, and linear bidding strategies. We show that if uncertainty about the number of bidders is substantial, then the discriminatory pricing generates higher expected revenues than the uniform pricing. In particular, the first-price auction might generate higher revenues than the second-price auction. Therefore, uncertainty about the number of bidders is an important factor to consider when choosing the pricing rule. We also study whether eliminating this uncertainty, i.e., revealing the number of bidders, is in the sellers interests, and discuss the existence of an increasing symmetric equilibrium.

On the Meaningfulness of Optimal Solutions to Scheduling Problems: Can An Optimal Solution Be Nonoptimal?

We consider the problem of finding an optimal schedule for jobs on a single machine when there are penalties for both tardy and early arrivals. We point out that if attention is paid to how these penalties are measured, then a change of scale of measurement might lead to the anomalous situation where a schedule is optimal if these parameters are measured in one way, but not if they are measured in a different way that seems equally acceptable. In particular, we note that if the penalties measure utilities or disutilities, or loss of goodwill or customer satisfaction, then these kinds of anomalies can occur, for instance if we change both unit and zero point in scales measuring these penalties. We investigate situations where problems of these sorts arise for four specific penalty functions under a variety of different assumptions. The results of the paper have implications far beyond the specific scheduling problems we consider, and suggest that considerations of scale of measurement should enter into analysis of conclusions of optimality both in scheduling problems and throughout combinatorial optimization.

Subset Comparisons for Additive Linear Orders

This paper investigates algebraic and combinatorial properties of the set of linear orders on the algebra of subsets of a finite set that are representable by positive measures. It is motivated by topics in decision theory and the theory of measurement, where an understanding of such properties can facilitate the design of strategies to elicit comparisons between subsets that, for example, determine an individuals preference order over subsets of objects or an individuals qualitative probability order over subsets of states of the world. We introduce a notion of critical pairs of binary comparisons for such orders and prove that (i) each order is uniquely characterized by its set of critical pairs and (ii) the smallest set of binary comparisons that determines an order is a subset of its set of critical pairs. The paper then focuses on the minimum number of on-line binary-comparison queries between subsets that suffice to determine any representable order for a set of given cardinalityn. It is observed that, for smalln, the minimum is attained by first determining the ordering of singleton subsets. We also consider query procedures with fixed numbers of stages, in each stage of which a number of queries for the next stage are formulated.

Information aggregation in auctions with an unknown number of bidders

Information aggregation, a key concern for uniform-price, common-value auctions with many bidders, has been characterized in models where bidders know exactly how many rivals they face. A model allowing for uncertainty over the number of bidders is essential for capturing a critical condition for information to aggregate: as the numbers of winning and losing bidders grow large, information aggregates if and only if uncertainty about the fraction of winning bidders vanishes. It may be possible for the seller to impart this information by precommitting to a specified fraction of winning bidders, via a proportional selling policy. Intuitively, this could make the proportion of winners known, and thus provide all the information that bidders need to make winners curse corrections.