Average fill rate and horizon length

Given a sequence of independent and identically distributed demands and an order up to replenishment policy with negligible lead time, we prove that average fill rate is monotonically decreasing in the number of periods in the planning horizon. This was conjectured to be true in a recent issue of this journal.     

Solving a finite horizon EPQ problem with backorders

This paper presents an alternative approach to solve a finite horizon production lot sizing model with backorders using Cauchy-Bunyakovsky-Schwarz Inequality. The optimal batch size is derived from a sequence number of batches. We prove that a constant batch size policy with one fill rate is better than the variable batch sizes with variable fill rates. Finally, a practical approach is proposed to find the optimal solutions for a discrete planning horizon and discrete batch sizes.     

Bayesian semiparametric analysis for a single item maintenance optimization

We address the problem of a finite horizon single item maintenance optimization structured as a combination of preventive and corrective maintenance in a nuclear power plant environment. We present Bayesian semiparametric models to estimate the failure time distribution and costs involved. The objective function for the optimization is the expected total cost of maintenance over the pre-defined finite time horizon. Typically, the mathematical modeling of failure times are based on parametric models. These models fail to capture the true underlying relationships in the data; indeed, under a parametric assumption, the hazard rates are treated as unimodal, which, as shown in this paper, is incorrect. Importantly, assuming an increasing failure rate, as is typically done, we show, is way off the mark in the present context. Since hazard and cost estimates feed into the optimization phase, from a risk management perspective, potentially gross errors, resulting from purely parametric models, can be obviated. We show the effectiveness of our approach using real data from the South Texas Project Nuclear Operating Company (STPNOC) located in Bay City, Texas.     

A characterization of a limit solution for finite horizon bargaining problems

We investigate a two-person random proposer bargaining game with a deadline. A bounded time interval is divided into bargaining periods of equal length and we study the limit of the subgame perfect equilibrium outcomes as the number of bargaining periods goes to infinity while the deadline is kept fixed. This limit is close to the discrete Raiffa solution when the time horizon is very short. If the deadline goes to infinity the limit outcome converges to the time preference Nash solution. Regarding this limit as a bargaining solution under deadline, we provide an axiomatic characterization.     

Network Simulation Method for the evaluation of perturbed supply chains on a finite horizon

Due to lead times and other delays in a chain, the Net Present Value (NPV) can be easily estimated if Laplace transforms in MRP models are employed. This leads to the estimation of NPV on an infinite horizon. However, for the simultaneous perturbations of several parameters in a supply chain and activities running on the finite horizon, NPV could be overestimated. Therefore, we suggest the parallel use of the Network Simulation Method (NSM) with the MRP theory to reduce these overestimations. This paper aims to present the NSM to evaluate supply chains on a finite horizon when stochastic behaviour of time delays and other perturbations of parameters are also essential, which is typical for food and drug supply chains. The circuit simulator NGSPICE, which was previously used by certain authors in thermodynamics, also evaluates the financial consequences of simultaneous perturbations in a finite chain. This approach holds better for the stochastic processes of simultaneous perturbations, compared to our results achieved using MRP theory without these corrections. As presented in the numerical example, the shorter the horizon and lower the interest rate, the more important it is to use the correction factors obtained from the NGSPICE simulator.

     

On the rate of convergence of infinite horizon discounted optimal value functions

In this paper we investigate the rate of convergence of the optimal value function of an infinite horizon discounted optimal control problem as the discount rate tends to zero. Using the Integration Theorem for Laplace transformations we provide conditions on averaged functionals along suitable trajectories yielding quadratic pointwise convergence. From this we derive under appropriate controllability conditions criteria for linear uniform convergence of the value functions on control sets. Applications of these results are given and an example is discussed in which both linear and slower rates of convergence occur depending on the cost functional.     

Efficient heuristics for the workover rig routing problem with a heterogeneous fleet and a finite horizon

Onshore oil fields may contain hundreds of wells that use sophisticated and complex equipments. These equipments need regular maintenance to keep the wells at maximum productivity. When the productivity of a well decreases, a specially-equipped vehicle called a workover rig must visit this well to restore its full productivity. Given a heterogeneous fleet of workover rigs and a set of wells requiring maintenance, the workover rig routing problem (WRRP) consists of finding rig routes that minimize the total production loss of the wells over a finite horizon. The wells have different loss rates, need different services, and may not be serviced within the horizon. On the other hand, the number of available workover rigs is limited, they have different initial positions, and they do not have the same equipments. This paper presents and compares four heuristics for the WRRP: an existing variable neighborhood search heuristic, a branch-price-and-cut heuristic, an adaptive large neighborhood search heuristic, and a hybrid genetic algorithm. These heuristics are tested on practical-sized instances involving up to 300 wells, 10 rigs on a 350-period horizon. Our computational results indicate that the hybrid genetic algorithm outperforms the other heuristics on average and in most cases.     

Inflation and the optimal inventory replenishment schedule within a finite planning horizon

The subject of this paper is the problem of finding the optimal replenishment schedule for an inventory, subject to time-dependent demand and deterioration, within a finite time planning horizon. It is shown that taking inflation into account has a profound effect on the solution of the problem. For instance, there is a critical number of replenishment periods, in excess of which the optimal schedule is characterized by the inclusion of token orders at the end of the planning horizon. This and other conclusions, obtained via a careful mathematical analysis of the problem, rectify those of earlier studies.     

On the exact calculation of the fill rate in a periodic review inventory policy under discrete demand patterns

The primary goal of this paper is the development of a generalized method to compute the fill rate for any discrete demand distribution in a periodic review policy. The fill rate is defined as the fraction of demand that is satisfied directly from shelf. In the majority of related work, this service metric is computed by using what is known as the traditional approximation, which calculates the fill rate as the complement of the quotient between the expected unfulfilled demand and the expected demand per replenishment cycle, instead of focusing on the expected fraction of fulfilled demand. This paper shows the systematic underestimation of the fill rate when the traditional approximation is used, and revises both the foundations of the traditional approach and the definition of fill rate itself. As a result, this paper presents the following main contributions: (i) a new exact procedure to compute the traditional approximation for any discrete demand distribution; (ii) a more suitable definition of the fill rate in order to ignore those cycles without demand; and (iii) a new standard procedure to compute the fill rate that outperforms previous approaches, especially when the probability of zero demand is substantial. This paper focuses on the traditional periodic review, order up to level system under any uncorrelated, discrete and stationary demand pattern for the lost sales scenario.     

A finite horizon model for repairable systems with repair restrictions

In this paper, we will present a new finite horizon repair/replacement decision model and derive the structure of the optimal policy for components that have a failure intensity that is a non-decreasing function of the number of times the component has been repaired, and independent of the component's age. Furthermore, the component has physical restrictions on the number of times it can be repaired, after which the only feasible decision is to replace the component. The fundamentals of this new decision model are based on the outcomes of several case studies done by the authors. Besides presenting the model and showing the structure of the optimal policy, the model will be applied to a real industry data set, and its results discussed.     

Optimal consumption of the finite time horizon Ramsey problem

In this paper, we study the stochastic Ramsey problem related to an economic growth model with the CES production function in a finite time horizon. By changing variables, the Hamilton-Jacobi-Bellman equation associated with this optimization problem is transformed. By the viscosity solution technique, we show the existence of a classical solution of the transformed Hamilton-Jacobi-Bellman equation, and then give an optimal consumption policy of the original problem.     

Analytic solutions for infinite horizon stochastic optimal control problems via finite horizon approximation: A practical guide

We show how infinite horizon stochastic optimal control problems can be solved via studying their finite horizon approximations. This often leads to analytical solutions for the infinite horizon problem by studying phase diagrams, even in cases where the complexity of the finite horizon case does not permit analytic solutions. Our approach can be applied to many problems in dynamic economics.     

An acquisition policy for a single item multi-supplier system with real-world constraints

This paper proposes an exact acquisition policy for solving the single-item multi-supplier problem with real-world constraints. Compared with the model of Rosenblatt et al. [Note. An acquisition policy for a single item multi-supplier system, Manage. Sci. 44 (1998) S96–S100], the proposed method has contributions in that the global optimal solutions can be obtained to indicate the best acquisition policy, and real-world constraints can easily be added as appropriate for real-world situations. In addition, the benefits of price-quantity discount (PQD) under conditions of the single item muti-supplier system are also considered in the paper.     

Optimal selling time in stock market over a finite time horizon

In this paper, we examine the best time to sell a stock at a price being as close as possible to its highest price over a finite time horizon [0, T ], where the stock price is modelled by a geometric Brownian motion and the ’closeness’ is measured by the relative error of the stock price to its highest price over [0, T ]. More precisely, we want to optimize the expression: where (V t ) t≥0 is a geometric Brownian motion with constant drift α and constant volatility σ > 0, M t = max Vs is the running maximum of the stock price, and the supremum is taken over all possible stopping times 0 ≤τ≤ T adapted to the natural filtration (F t ) t≥0 of the stock price. The above problem has been considered by Shiryaev, Xu and Zhou (2008) and Du Toit and Peskir (2009). In this paper we provide an independent proof that when α = 1 2 σ 2 , a selling strategy is optimal if and only if it sells the stock either at the terminal time T or at the moment when the stock price hits its maximum price so far. Besides, when α > 1 2 σ 2 , selling the stock at the terminal time T is the unique optimal selling strategy. Our approach to the problem is purely probabilistic and has been inspired by relating the notion of dominant stopping ρτ of a stopping time τ to the optimal stopping strategy arisen in the classical "Secretary Problem".     

Decision/forecast horizon for a single facility dynamic location/relocation problem

Procedures to solve finite horizon dynamic location/relocation problems have been reported in the literature by many authors. This paper provides several decision/forecast horizon results for a single facility dynamic location/relocation problem; these results are helpful in finding optimal initial decisions for the infinite horizon problem by using information only for a finite horizon.     

A stochastic target formulation for optimal switching problems in finite horizon

We consider a general optimal switching problem for a controlled diffusion and show that its value coincides with the value of a well-suited stochastic target problem associated to a diffusion with jumps. The proof consists in showing that the Hamilton–Jacobi–Bellman equations of both problems are the same and in proving a comparison principle for this equation. This provides a new family of lower bounds for the optimal switching problem, which can be computed by Monte-Carlo methods. This result has also a nice economical interpretation in terms of a firm's valuation.     

Constrained optimality for finite horizon semi-Markov decision processes in Polish spaces

This paper focuses on solving a finite horizon semi-Markov decision process with multiple constraints. We convert the problem to a constrained absorbing discrete-time Markov decision process and then to an equivalent linear program over a class of occupancy measures. The existence, characterization and computation of constrained-optimal policies are established under suitable conditions. An example is given to demonstrate our results.     

Optimal stock liquidation in a regime switching model with finite time horizon

This paper is concerned with a finite-horizon optimal selling rule. A set of geometric Brownian motions coupled by a finite-state Markov chain is used to characterize stock price movements. Given a fixed transaction fee, the optimal selling rule can be obtained by solving an optimal stopping problem. The corresponding value function is shown to be the unique viscosity solution to the associated HJB equations. Numerical solutions to these equations and their convergence are obtained. A numerical example is presented to illustrate the results.     

On the convergence rate of the finite element method in a semicoercive problem with friction

We estimate the convergence rate of the finite element method in the minimization problem for a semicoercive nondifferentiable functional arising in the study of a model problem with friction.