Optimal economic production quantity policy for imperfect process with imperfect repair and maintenance

This study integrates maintenance and production programs with the economic production quantity (EPQ) model for an imperfect process involving a deteriorating production system with increasing hazard rate: imperfect repair and rework upon failure (out of control state). The imperfect repair performs some restorations and restores the system to an operating state (in-control state), but leaves its failure until perfect preventive maintenance (PM) is performed. There are two types of PM, namely imperfect PM and perfect PM. The probability that perfect PM is performed depends on the number of imperfect maintenance operations performed since the last renewal cycle. Mathematical formulas are obtained for deriving the expected total cost. For the EPQ model, the optimum run time, which minimizes the total cost, is discussed. Various special cases are considered, including the maintenance learning effect. Finally, a numerical example is presented to illustrate the effects of PM, setup, breakdown and holding costs.     

The optimal repair-time limit replacement policy with imperfect repair: Lorenz transform approach

In this paper, we consider a simple repair-time limit replacement problem with imperfect repair, and focus on the problem of determining the optimal repair-time limit which minimizes the expected cost per unit time in the steady-state. Applying the Lorenz transform, we develop a nonparametric method to estimate the optimal repair-time limit from the empirical repair-time data. Numerical examples are considered to calculate the optimal policy and to examine the asymptotic properties of the estimator.     

Integrated cellular manufacturing systems design with production planning and dynamic system reconfiguration

This paper presents and analyzes a comprehensive model for the design of cellular manufacturing systems (CMS). A recurring theme in research is a piecemeal approach when formulating CMS models. In this paper, the proposed model, to the best of the authors’ knowledge, is the most comprehensive one to date with a more integrated approach to CMS design, where production planning and system reconfiguration decisions are incorporated. Such a CMS model has not been proposed before and it features the presence of alternate process routings, operation sequence, duplicate machines, machine capacity and lot splitting. The developed model is a mixed integer non-linear program. Linearization procedures are proposed to convert it into a linearized mixed integer programming formulation. Computational results are presented by solving some numerical examples, extracted from the existing literature, with the linearized formulation.     

Existence of efficient solutions in infinite horizon optimization under continuous and discrete controls

For time-varying deterministic infinite horizon control problems, we provide conditions for the existence of efficient solutions, i.e., solutions which are optimal to each of the states through which they pass. A sufficient condition is that the mappings from controls to states be open. Applications to production planning are considered.     

Production and replacement policies for a deteriorating manufacturing system under random demand and quality

This work investigates the production planning of an unreliable deteriorating manufacturing system under uncertainties. The effect of the deterioration phenomenon on the machine is mainly observed in its availability and the quality of the parts produced, with the rates of failure and defectives increasing with the age of the machine. The option to replace the machine should be considered to mitigate the effect of deterioration in order to ensure long-term satisfaction of demand. The objective of this paper is to find the production rate and the replacement policy that minimize the total discounted cost, which includes inventory, backlog, production, repair and replacement costs, over an infinite planning horizon. We formulate the stochastic control problem in the framework of a semi-Markov decision process to consider the machine's history. The integration of random demand and quality behaviour led us to propose a new modeling approach by developing optimality conditions in terms of a second-order approximation of Hamilton–Jacobi–Bellman (HJB) equations. Numerical methods are used to obtain the optimal control policies. Finally, a numerical example and a sensitivity analysis are presented in order to illustrate and confirm the structure of the optimal solution obtained.     

The optimal control of just-in-time-based production and distribution systems and performance comparisons with optimized pull systems

In just-in-time (JIT) production systems, there is both input stock in the form of parts and output stock in the form of product at each stage. These activities are controlled by production-ordering and withdrawal kanbans. This paper discusses a discrete-time optimal control problem in a multistage JIT-based production and distribution system with stochastic demand and capacity, developed to minimize the expected total cost per unit of time. The problem can be formulated as an undiscounted Markov decision process (UMDP); however, the curse of dimensionality makes it very difficult to find an exact solution. The author proposes a new neuro-dynamic programming (NDP) algorithm, the simulation-based modified policy iteration method (SBMPIM), to solve the optimal control problem. The existing NDP algorithms and SBMPIM are numerically compared with a traditional UMDP algorithm for a single-stage JIT production system. It is shown that all NDP algorithms except the SBMPIM fail to converge to an optimal control.Additionally, a new algorithm for finding the optimal parameters of pull systems is proposed. Numerical comparisons between near-optimal controls computed using the SBMPIM and optimized pull systems are conducted for three-stage JIT-based production and distribution systems. UMDPs with 42 million states are solved using the SBMPIM. The pull systems discussed are the kanban, base stock, CONWIP, hybrid and extended kanban.     

Age-dependent production planning and maintenance strategies in unreliable manufacturing systems with lost sale

In this paper, we formulate an analytical model for the joint determination of an optimal age-dependent buffer inventory and preventive maintenance policy in a production environment that is subject to random machine breakdowns. Traditional preventive maintenance policies, such as age and periodic replacements, are usually studied based on simplified and non-realistic assumptions, as well as on the expected costs criterion. Finished goods inventories and the age-dependent likelihood of machine breakdowns are usually not considered. As a result, these policies could significantly extend beyond the anticipated financial incomes of the system, and lead to crises. In order to solve this problem, a more realistic analysis model is proposed in this paper to consider the effects of both preventive maintenance policies and machine age on optimal safety stock levels. Hence, a unified framework is developed, allowing production and preventive maintenance to be jointly considered. We use an age-dependent optimization model based on the minimization of an overall cost function, including inventory holdings, lost sales, preventive and corrective maintenance costs. We provide optimality conditions for the manufacturing systems considered, and use numerical methods to obtain an optimal preventive maintenance policy and the relevant age-dependent threshold level production policy. In this work, this policy is called the multiple threshold levels hedging point policy. We include numerical examples and sensitivity analyses to illustrate the importance and the effectiveness of the proposed methodology. Compared with other available optimal production and maintenance policies, the numerical solution obtained shows that the proposed age-dependent optimal production and maintenance policies significantly reduce the overall cost incurred.     

A numerical method to approximate optimal production and maintenance plan in a flexible manufacturing system

The simultaneous planning of the production and the maintenance in a flexible manufacturing system is considered in this paper. The manufacturing system is composed of one machine that produces a single product. There is a preventive maintenance plan to reduce the failure rate of the machine. This paper is different from the previous researches in this area in two separate ways. First, the failure rate of the machine is supposed to be a function of its age. Second, we assume that the demand of the manufacturing product is time dependent and its rate depends on the level of advertisement on that product. The objective is to maximize the expected discounted total profit of the firm over an infinite time horizon. In the process of finding a solution to the problem, we first characterize an optimal control by introducing a set of Hamilton–Jacobi–Bellman partial differential equations. Then we realize that under practical assumptions, this set of equations can not be solved analytically. Thus to find a suboptimal control, we approximate the original stochastic optimal control model by a discrete-time deterministic optimal control problem. Then proposing a numerical method to solve the steady state Riccati equation, we approximate a suboptimal solution to the problem.     

Production planning and pricing policy in a make-to-stock system with uncertain demand subject to machine breakdowns

We consider a make-to-stock system served by an unreliable machine that produces one type of product, which is sold to customers at one of two possible prices depending on the inventory level at the time when a customer arrives (i.e., the decision point). The system manager must determine the production level and selling price at each decision point. We first show that the optimal production and pricing policy is a threshold control, which is characterized by three threshold parameters under both the long-run discounted profit and long-run average profit criteria. We then establish the structural relationships among the three threshold parameters that production is off when inventory is above the threshold, and that the optimal selling price should be low when inventory is above the threshold under the scenario where the machine is down or up. Finally we provide some numerical examples to illustrate the analytical results and gain additional insights.