A characterization of the proportional rule in multi-issue allocation situations

In multi-issue allocation situations, we have to divide a resource among a group of agents. The claim of each agent is a vector specifying the amount claimed by each agent on each issue. We present an axiomatic characterization of the proportional rule.     

Revisiting the PERT mean and variance

Difficulties with the interpretation of the parameters of the beta distribution let Malcolm et al. (1959) to suggest in the Program Evaluation and Review Technique (PERT) their by now classical expressions for the mean and variance for activity completion for practical applications. In this note, we shall provide an alternative for the PERT variance expression addressing a concern raised by Hahn (2008) regarding the constant PERT variance assumption given the range for an activity’s duration, while retaining the original PERT mean expression. Moreover, our approach ensures that an activity’s elicited most likely value aligns with the beta distribution’s mode. While this was the original intent of Malcolm et al. (1959), their method of selecting beta parameters via the PERT mean and variance is not consistent in this manner.     

A spatial voting model where proportional rule leads to two-party equilibria

In this paper we show that in a simple spatial model where the government is chosen under strict proportional rule, if the outcome function is a linear combination of parties’ positions, with coefficient equal to their shares of votes, essentially only a two-party equilibrium exists. The two parties taking a positive number of votes are the two extremist ones. Applications of this result include an extension of the well-known Alesina and Rosenthal model of divided government as well as a modified version of Besley and Coate’s model of representative democracy.      

A linear proportional effort allocation rule

This paper proposes a new class of allocation rules in network games. Like the solution theory in cooperative games of how the Harsanyi dividend of each coalition is distributed among a set of players, this new class of allocation rules focuses on the distribution of the dividend of each network. The dividend of each network is allocated in proportion to some measure of each player’s effort, which is called an effort function. With linearity of the allocation rules, an allocation rule is specified by the effort functions. These types of allocation rules are called linear proportional effort allocation rules. Two famous allocation rules, the Myerson value and the position value, belong to this class of allocation rules. In this study, we provide a unifying approach to define the two aforementioned values. Moreover, we provide an axiomatic analysis of this class of allocation rules, and axiomatize the Myerson value, the position value, and their non-symmetric generalizations in terms of effort functions. We propose a new allocation rule in network games that also belongs to this class of allocation rules.     

On the hardness of evaluating criticality of activities in a planar network with duration intervals

Complexity results for problems of evaluating the criticality of activities in planar networks with duration time intervals are presented. We show that the problems of asserting whether an activity is possibly critical, and of computing bounds on the float of an activity in these networks are NP-complete and NP-hard, respectively.     

Lower bound for the mean project completion time in dynamic PERT networks

We apply the stochastic dynamic programming to obtain a lower bound for the mean project completion time in a PERT network, where the activity durations are exponentially distributed random variables. Moreover, these random variables are non-static in that the distributions themselves vary according to some randomness in society like strike or inflation. This social randomness is modelled as a function of a separate continuous-time Markov process over the time horizon. The results are verified by simulation.     

Resource allocation in dynamic PERT networks with finite capacity

This article models the resource allocation problem in dynamic PERT networks with finite capacity of concurrent projects (COnstant Number of Projects In Process (CONPIP)), where activity durations are independent random variables with exponential distributions, and the new projects are generated according to a Poisson process. The system is represented as a queuing network with finite concurrent projects, where each activity of a project is performed at a devoted service station with one server located in a node of the network. For modeling dynamic PERT networks with CONPIP, we first convert the network of queues into a stochastic network. Then, by constructing a proper finite-state continuous-time Markov model, a system of differential equations is created to solve and find the completion time distribution for any particular project. Finally, we propose a multi-objective model with three conflict objectives to optimally control the resources allocated to the servers, and apply the goal attainment method to solve a discrete-time approximation of the original multi-objective problem.     

Computing latest starting times of activities in interval-valued networks with minimal time lags

This paper deals with problems of determining possible values of earliest and latest starting times of an activity in networks with minimal time lags and imprecise durations that are represented by means of interval or fuzzy numbers. Although minimal time lags are practical in different projects, former researchers have not considered these problems.After proposing propositions which reduce the search space, a novel polynomial algorithm is presented to compute intervals of possible values of latest starting times in interval-valued networks with minimal time lags. Then, the results are extended to networks with fuzzy durations.     

Scheduling Markovian PERT networks to maximize the net present value

We examine project scheduling with net present value objective and exponential activity durations, using a continuous-time Markov decision chain. On the basis of a judicious partitioning of the state space, we achieve a significant performance improvement as compared to the existing algorithms.     

A composite run-to-the-bank rule for multi-issue allocation situations

In this paper, we propose a new extension of the run-to-the-bank rule for bankruptcy situations to the class of multi-issue allocation situations. We show that this rule always yields a core element and that it satisfies self-duality. We characterise our rule by means of a new consistency property, issue-consistency.     

Modeling activity times by the Parkinson distribution with a lognormal core: Theory and validation

Based on theoretical arguments and empirical evidence we advocate the use of the lognormal distribution as a model for activity times. However, raw data on activity times are often subject to rounding and to the Parkinson effect. We address those factors in our statistical tests by using a generalized version of the Parkinson distribution with random censoring of earliness, ultimately validating our model with field data from several sources. We also confirm that project activities exhibit stochastic dependence that can be modeled by linear association.     

On the convergence of the random arrival rule in large claims problems

We study the behavior of the random arrival rule in claims problems with a large number of claimants. We assume that the amount to divide is not too “small” compared with total claims, and each claim is not too “large” compared with any other positive claims. Then, the random arrival rule behaves like the proportional rule as the number of claimants increases. We are grateful to Yuki Funaki, William Thomson, and three referees for their comments. This work was supported by the Brain Korea 21 Project in 2003     

Approximation algorithms for dynamic resource allocation

We consider a problem of allocating limited quantities of M types of resources among N independent activities that evolve over T epochs. In each epoch, we assign to each activity a task which consumes resources, generates utility, and determines the subsequent state of the activity. We study the complexity of, and approximation algorithms for, maximizing average utility.     

On the latest starting times and criticality of activities in a network with imprecise durations

This paper deals with problems of computing possible values of latest starting times and determining types of criticality for all activities in a network with interval or fuzzy activity durations. Although the problem of computing the latest starting times has been solved, a novel polynomial algorithm which is easy to understand and improves complexity is proposed.     

Some remarks on the expected delay of project duration in a PERT network

Summary In this note a problem of estimating the expected project delay of a given completion time in a PERT network is reconsidered. Some remarks on application of the approach, proposed recently by Hartmann, are given.

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Zusammenfassung In diesem Bericht wird nochmals das Problem der Abschätzung des Erwartungswerts der Überschreitung eines Projektendtermins in Erwägung gezogen, wobei das Projekt mit dem PERT-Netzplan beschrieben ist. Einige Bemerkungen bezüglich der Verwendung des neuerdings vonHartmann vorgeschlagenen Ansatzes zu diesem Problem werden angegeben.

     

A proportional approach to claims problems with a guaranteed minimum

In distribution problems, and specifically in bankruptcy issues, the Proportional (P) and the Egalitarian (EA) divisions are two of the most popular ways to resolve the conflict. Nonetheless, when using the egalitarian division, agents may receive more than her claim. We propose a compromise between the proportional and the egalitarian approaches by considering the restriction that no one receives more than her claim. We show that the most egalitarian compromise fulfilling this restriction ensures a minimum amount to each agent. We also show that this compromise can be interpreted as a process that works in two steps as follows: first, all agents receive an equal share up to the smallest claim if possible (egalitarian distribution), and then, the remaining estate (if any) is allocated proportionally to the remaining claims (proportional distribution). Finally, we obtain that the recursive application of this process finishes at the Constrained Equal Awards solution (CEA).     

Multi-objective time–cost trade-off in dynamic PERT networks using an interactive approach

We develop a multi-objective model for the time–cost trade-off problem in a dynamic PERT network using an interactive approach. The activity durations are exponentially distributed random variables and the new projects are generated according to a renewal process and share the same facilities. Thus, these projects cannot be analyzed independently. This dynamic PERT network is represented as a network of queues, where the service times represent the durations of the corresponding activities and the arrival stream to each node follows a renewal process. At the first stage, we transform the dynamic PERT network into a proper stochastic network and then compute the project completion time distribution by constructing a continuous-time Markov chain. At the second stage, the time–cost trade-off problem is formulated as a multi-objective optimal control problem that involves four conflicting objective functions. Then, the STEM method is used to solve a discrete-time approximation of the original problem. Finally, the proposed methodology is extended to the generalized Erlang activity durations.     

Risky asset allocation and consumption rule in the presence of background risk and insurance markets

This study examines joint decisions regarding risky asset allocation and consumption rate for a representative agent in the presence of background risk and insurance markets. Contrary to the conclusion of the “mutual fund separation theorem”, we show that the optimal risky asset mix will reflect an agent’s risk attitude as long as background risk is not independent of investment risk. This result can, however, be used to solve the “riskyasset allocation puzzle”. We also unveil that optimal insurance to shift background risk is determined through establishing a hedging portfolio against investment risk and is an arrangement maintaining the balance between growth and volatility of expected consumption. Because the optimal insurance we obtain generally leads to a smoother consumption path, it may plausibly explain the “equity premium puzzle” in the financial literature.     

A management oriented approach to reduce a project duration and its risk (variability)

The effects of different activities on risk and the expected completion time of a project are not the same; various activities have various effects on the successful completion of a project. Based on this fact, one of the most important issues in project management is to determine important activities and the amount of effort that should be assigned to control them, thereby completing the project successfully. In this paper, one index and one method are proposed to satisfy this necessity. Our main hypothesis is, expending effort on activities and controlling them actively results in activities risk reduction. Based on this hypothesis, the proposed index and method are compared to existing indices in the literature. Comparison results show that the proposed index and method strongly surpass other indices.     

Scheduling Markovian PERT networks to maximize the net present value: New results

We study project scheduling so as to maximize the expected net present value when task durations are exponentially distributed. Based on the structural properties of an optimal solution we show that, even if preemption is allowed, it is not necessary to do so. Next to its managerial importance, this result also allows for a new algorithm which improves on the current state of the art with several orders of magnitude, both in CPU time and in memory usage.