Application of the Cross-Entropy Method to the Buffer Allocation Problem in a Simulation-Based Environment

The buffer allocation problem (BAP) is a well-known difficult problem in the design of production lines. We present a stochastic algorithm for solving the BAP, based on the cross-entropy method, a new paradigm for stochastic optimization. The algorithm involves the following iterative steps: (a) the generation of buffer allocations according to a certain random mechanism, followed by (b) the modification of this mechanism on the basis of cross-entropy minimization. Through various numerical experiments we demonstrate the efficiency of the proposed algorithm and show that the method can quickly generate (near-)optimal buffer allocations for fairly large production lines.     

Asymptotic existence of proportionally fair allocations

Fair division has long been an important problem in the economics literature. In this note, we consider the existence of proportionally fair allocations of indivisible goods, i.e., allocations of indivisible goods in which every agent gets at least her proportionally fair share according to her own utility function. We show that when utilities are additive and utilities for individual goods are drawn independently at random from a distribution, proportionally fair allocations exist with high probability if the number of goods is a multiple of the number of agents or if the number of goods grows asymptotically faster than the number of agents.     

Optimal capital allocations to interdependent actuarial risks

This paper further studies the capital allocation concerning mutually interdependent random risks. In the context of exchangeable random risks, we establish that risk-averse insurers incline to evenly distribute the total capital among multiple risks. For risk-averse insurers with decreasing convex loss functions, we prove that more capital should be allocated to the risk with the larger reversed hazard rate when risks are coupled by an Archimedean copula. Also, sufficient conditions are developed to exclude the worst capital allocations for random risks with some specific Archimedean copulas.     

Quenched invariance principle for simple random walk on discrete point processes

We consider the simple random walk on random graphs generated by discrete point processes. This random walk moves on graphs whose vertex set is a random subset of a cubic lattice and whose edges are lines between any consecutive vertices on lines parallel to each coordinate axis. Under the assumption that the discrete point processes are finitely dependent and stationary, we prove that the quenched invariance principle holds, i.e., for almost every configuration of the point process, the path distribution of the walk converges weakly to that of a Brownian motion.     

Designing paced assembly lines with fixed number of stations

This paper addresses the issue of allocating tasks among workstations in assembly lines so that the cycle time is minimized (Type II problem). We provide a twostage heuristic procedure, which is based on an integer programming formulation of the problem. It is a simplexlike procedure which, while attempting to minimize the cycle time, also smooths out the workload among the workstations. Computational results show that our procedure performs well for Type II problems. Tests on problems from the literature indicate that the smoothing stage of our procedure can be used effectively to level workloads for any given task allocations among tasks.     

Critical phenomenon for a percolation model

We consider a percolation model which consists of oriented lines placed randomly on the plane. The lines are of random length and at a random angle with respect to the horizontal axis and are placed according to a Poisson point process; the length, angle, and orientation being independent of the underlying Poisson process. We establish a critical behaviour of this model, i.e., percolation occurs for large intensity of the Poisson process and does not occur for smaller intensities. In the special case when the lines are of fixed unit length and are either oriented vertically up or oriented horizontally to the left, with probability p or (1-p), respectively, we obtain a lower bound on the critical intensity of percolation.     

Optimal allocation of units in experimental designs with hierarchical and cross classification

Consider the class of random linear models induced by possible allocations of units in an experimental design with hierarchical or cross classification. Assuming a balanced model belongs to the class, it is shown that this model is optimal for estimation of mean.     

Some sample path properties of a random walk on the cube

Some properties of the set of vertices not visited by a random walk on the cube are considered. The asymptotic distribution of the first timeQ this set is empty is derived. The distribution of the number of vertices not visited is found for times nearEQ. Next the first time all unvisited vertices are at least some distanced apart is explored. Finally the expected time taken by the path to come within a distanced of all points is calculated. These results are compared to similar results for random allocations.     

Properties of serial kanban systems

In this paper we show some structural results of a kanban system by using results about the underlying GSMP. The main results from this connection are: dominance of allocations, optimal partition, upper and lower bounds on throughput, consistency of the IPA derivative, convexity of throughput as a function of service time parameters and concavity of throughput with respect to number of kanbans. Although the dominance and the partition results were obtained previously by sample path arguments, the proofs here are less cumbersome and generalize the earlier results. The second-order properties for kanban lines are new. These results form a basis for results in multi-product lines where proofs by sample paths require extensive notation.     

Insensitive Bandwidth Sharing in Data Networks

We represent a data network as a set of links shared by a dynamic number of competing flows. These flows are generated within sessions and correspond to the transfer of a random volume of data on a pre-defined network route. The evolution of the stochastic process describing the number of flows on all routes, which determines the performance of the data transfers, depends on how link capacity is allocated between competing flows. We use some key properties of Whittle queueing networks to characterize the class of allocations which are insensitive in the sense that the stationary distribution of this stochastic process does not depend on any traffic characteristics (session structure, data volume distribution) except the traffic intensity on each route. We show in particular that this insensitivity property does not hold in general for well-known allocations such as max-min fairness or proportional fairness. These results are ilustrated by several examples on a number of network topologies.     

Simulation optimization of buffer allocations in production lines with unreliable machines

We use a recent simulationbased optimization method, sample path optimization, to find optimal buffer allocations in tandem production lines where machines are subject to random breakdowns and repairs, and the product is fluidtype. We explore some of the functional properties of throughput of such systems and exploit these properties to prove the almost sure convergence of our optimization technique, under a regularity condition on the steady state. Utilizing a generalized semiMarkov process (GSMP) representation of the system, we derive recursive expressions to compute onesided directional derivatives of throughput, from a single simulation run. Finally, we give computational results for lines with up to 50 machines. We also compare results for smaller lines with the results from a more conventional method, stochastic approximation, whenever applicable. In these numerical studies, our method performed quite well on problems that are considered difficult by current computational standards.     

On the interaction between indirect cost allocations and the firm's objectives

The import of cost allocation procedures are through their ex ante important decision making. Hence, it is important that the allocation issue be placed squarely within the context of those firm's objectives which gave rise to the need for the specific allocation. To that effect, this paper focuses the debate on the identification of the indirect cost allocation method that is best suited to the specific reasons for requiring the cost information. First, it is shown that all existing allocation schemes (i) may be expressed in a common equation, flexible enough to be adapted to whatever decision-making purpose the firm desires; and (ii) fulfil the individual rationality conditions of game theory. Then, in light of the controversy as to whether the US Defense Department indirectly subsidizes the commercial side of its suppliers' operations, necessary and sufficient conditions are provided for allocations which do and do not subsidize. Non-subsidized allocations are shown to belong to the core. Subsidized allocations occur when the players (divisions') rational objectives are superseded by higher priority coordinating objectives of non-players (the firm).     

An Optimum Multivariate Stratified Sampling Design with Nonresponse: A Lexicographic Goal Programming Approach

In a multivariate stratified sample survey with L strata and p > 1 characteristics, defined on each unit of the population, let the estimation of all the p-population means be of interest. As discussed by Cochran (1977), since the optimum allocation for one characteristic will not in general be optimum for other characteristics some compromise must be reached in a multiple characteristics stratified surveys. Various authors worked out allocations that are based on a compromise criterion. The resulting allocations are optimal for all characteristics in some sense, for example an allocation that minimizes the trace of the variance-covariance matrix of the estimators of the population means or an allocation that minimizes the weighted average of the variances or an allocation that maximizes the total relative efficiency of the estimators as compared to the corresponding individual optimum allocations. In the present paper the problem of optimum allocation in multivariate stratified random sampling in the presence of nonresponse has been formulated as a multiobjective integer nonlinear programming problem and a solution procedure is developed using goal programming technique. Three numerical examples are worked out to illustrate the computational details. A comparison of the proposed method with some well known methods is also carried out to show the practical utility of the proposed method.     

Binary operations and lattice structure for a model of matching with contracts

We consider a restricted model of many-to-one matching with contracts and we order the set of stable allocations according both to the unanimous-for-doctors partial ordering and Blair’s partial ordering for hospitals. We define two binary operations to calculate the least upper bound and greatest lower bound for each pair of elements of this set in a simple way. By using these operations, we show that the set of stable allocations has dual lattice structures, thus reflecting an expected counterposition of interests between both sides of the market.     

Optimal allocation of policy limits and deductibles in a model with mixture risks and discount factors

By maximizing the expected utility, we study the optimal allocation of policy limits and deductibles from the viewpoint of a policyholder, where the dependence structure of losses is unknown. In Cheung (2007) [K.C. Cheung, Optimal allocation of policy limits and deductibles, Insurance: Mathematics and Economics 41 (2007) 382-391], the author had considered similar problems. He supposed that a policyholder was exposed to n random losses, and the losses were general risks there, i.e., the loss on each policy was just a random variable. In this paper, the model is extended in two directions. On one hand, we assume that n policies of the n losses are effected by random environments. For each policy, the loss under a fixed environment is characterized by a random variable, so the loss on each policy is a mixture of some fundamental random variables. On the other hand, loss frequencies, which are stochastic, are also considered. Therefore, the whole model is equipped with mixture risks and discount factors. Finally, we get the orderings of the optimal allocations of policy limits and deductibles. Our conclusions also extend the main results in Hua and Cheung (2008) [L. Hua, K.C. Cheung, Stochastic orders of scalar products with applications, Insurance: Mathematics and Economics 42 (2008) 865-872].     

On optimal allocation of risk vectors

In this paper we extend results on optimal risk allocations for portfolios of real risks w.r.t. convex risk functionals to portfolios of risk vectors. In particular we characterize optimal allocations minimizing the total risk as well as Pareto optimal allocations. Optimal risk allocations are shown to exhibit a worst case dependence structure w.r.t. some specific max-correlation risk measure and they are comonotone w.r.t. a common worst case scenario measure. We also derive a new existence criterion for optimal risk allocations and discuss some examples.     

A system-theoretic model for cooperation, interaction and allocation

A system-theoretic approach to cooperation, interaction and allocation is presented that simplifies, unifies and extends the results on classical cooperative games and their generalizations. In particular, a general Weber theory of linear values is obtained and a new theory for local cooperation and general interaction indices is established. The model is dynamic and based on the notion of states of cooperation that change under actions of agents. Careful distinction between “local” states of cooperation and general “system” states leads to a notion of entropy for arbitrary non-negative and efficient allocations and thus to a new information-theoretic criterion for fairness of allocation mechanisms. Shapley allocations, for instance, are exhibited as arising from random walks with maximal entropy. For a large class of cooperation systems, a characterization of game symmetries in terms of λ-values is given. A concept for cores and Weber sets is proposed and it is shown that a Weber set of a game with selection structure always contains the core.     

Allocating fixed costs and common revenue via data envelopment analysis

An issue of considerable importance involves the allocation of fixed costs or common revenue among a set of competing entities in an equitable way. Based on the data envelopment analysis (DEA) theory, this paper proposes new methods for (i) allocating fixed costs to decision making units (DMUs) and (ii) distributing common revenue among DMUs, in such a way that the relative efficiencies of all DMUs remain unchanged and the allocations should reflect the relative efficiencies and the input-output scales of individual DMUs. To illustrate our methods, numerical results for an example are described in this paper.     

A global construction of homogeneous random planar tessellations that are stable under iteration

Homogeneous (i.e. spatially stationary) random tessellations of the Euclidean plane are constructed which have the characteristic property to be stable under the operation of iteration (or nesting), STIT for short. It is based on a Poisson point process on the space of lines that are endowed with a time of birth. A new approach is presented that describes the tessellation in the whole plane. So far, an explicit geometrical construction for those tessellations was only known within bounded windows.     

Setting Commission Rates for the Control of the Salesperson's Client-prospect Effort Allocation

The literature on setting optimal commission rates for decentralized controls over salespeople's time allocations has used the sales of various product lines as the relevant selling activities. When the time allocation is between calls to prospects versus customers, the problem should account for specific aspects, such as: (1) the conversion pattern of prospects for the various product lines; (2) the attrition rate in the sales force; (3) the salespeople's attitudes toward delayed income. These variables recognize that several calls over some period of time are typically needed for converting prospects, and that salespeople will be able to cash commissions only at the conversion time and provided they are still in charge of the same accounts. Using a simple Markovian structure, this paper shows that under typical conditions, commissions on sales to prospects during the conversion period should always be larger than commissions on sales to customers. Managerial implications and implementation issues are discussed.