Integrated production-inventory models for imperfect production processes under inspection schedules

This paper deals with inventory models that unify the inventory problems of raw materials and finished products for a single product imperfect manufacturing system. The product is manufactured in batches, and raw materials are jointly replenished from outside suppliers. The system is assumed to deteriorate during the production process. As a result, some proportion of defective items is produced. The defective items are reworked at some cost either before or after a sale. Periodic inspections at equally spaced times and restorations of the production process are used to operate the system. The objective is to minimize the expected total cost for the system. A solution procedure is developed to find a near optimal solution for the basic model. The analysis is extended to various cases where the defect rate is a function of the setup cost, the proportion of defective items is not constant, or the inventory system has a limited capacity for raw materials.     

An artificial neural network based decision support system for solving the buffer allocation problem in reliable production lines

One of the major design problems in the context of manufacturing systems is the well-known Buffer Allocation Problem (BAP). This problem arises from the cost involved in terms of space requirements on the production floor and the need to keep in mind the decoupling impact of buffers in increasing the throughput of the line. Production line designers often need to solve the Buffer Allocation Problem (BAP), but this can be difficult, especially for large production lines, because the task is currently highly time consuming. Designers would be interested in a tool that would rapidly provide the solution to the BAP, even if only a near optimal solution is found, especially when they have to make their decisions at an operational level (e.g. hours). For decisions at a strategic level (e.g. years), such a tool would provide preliminary results that would be useful, before attempting to find the optimal solution with a specific search algorithm.     

Value set iteration for Markov decision processes

This communique presents an algorithm called “value set iteration” (VSI) for solving infinite horizon discounted Markov decision processes with finite state and action spaces as a simple generalization of value iteration (VI) and as a counterpart to Chang’s policy set iteration. A sequence of value functions is generated by VSI based on manipulating a set of value functions at each iteration and it converges to the optimal value function. VSI preserves convergence properties of VI while converging no slower than VI and in particular, if the set used in VSI contains the value functions of independently generated sample-policies from a given distribution and a properly defined policy switching policy, a probabilistic exponential convergence rate of VSI can be established. Because the set used in VSI can contain the value functions of any policies generated by other existing algorithms, VSI is also a general framework of combining multiple solution methods.     

Multiple lot-sizing decisions with an interrupted geometric yield and variable production time

This study examines a multiple lot-sizing problem for a single-stage production system with an interrupted geometric distribution, which is distinguished in involving variable production lead-time. In a finite number of setups, this study determined the optimal lot-size for each period that minimizes total expected cost. The following cost items are considered in optimum lot-sizing decisions: setup cost, variable production cost, inventory holding cost, and shortage cost. A dynamic programming model is formulated in which the duration between current time and due date is a stage variable, and remaining demand and work-in-process status are state variables. This study then presents an algorithm for solving the dynamic programming problem. Additionally, this study examines how total expected costs of optimal lot-sizing decisions vary when parameters are changed. Numerical results show that the optimum lot-size as a function of demand is not always monotonic.     

An exact iterative search algorithm for constrained Markov decision processes

This communique provides an exact iterative search algorithm for the NP-hard problem of obtaining an optimal feasible stationary Markovian pure policy that achieves the maximum value averaged over an initial state distribution in finite constrained Markov decision processes. It is based on a novel characterization of the entire feasible policy space and takes the spirit of policy iteration (PI) in that a sequence of monotonically improving feasible policies is generated and converges to an optimal policy in iterations of the size of the policy space at the worst case. Unlike PI, an unconstrained MDP needs to be solved at iterations involved with feasible policies and the current best policy improves all feasible policies included in the union of the policy spaces associated with the unconstrained MDPs.     

Average cost optimal policies for Markov control processes with Borel state space and unbounded costs

We show the existence of average cost optimal stationary policies for Markov control processes with Borel state space and unbounded costs per stage, under a set of assumptions recently introduced by L.I. Sennott (1989) for control processes with countable state space and finite control sets.     

An optimal production and inspection strategy with preventive maintenance error and rework

This study considers the integrated problem of production, preventive maintenance (PM), inspection, and inventory for an imperfect production process where rework and PM error exist. PM is performed when the process is in a controlled state. The correct implementation of PM results in a lower system failure rate, whereas a PM error results in the system shifting to the out-of-control state with a certain probability. The age of the system after PM is correlated with the level of PM performed. When the process in an out-of-control state produces a certain percentage of non-conforming items, we assume that a certain proportion of the non-conforming items can be reworked into conforming items. In a deteriorating production system, we determine the optimal inspection interval, inspection frequency, and production quantity that will yield the maximal unit expected profit. Numerical analyses are used to investigate the effectiveness of imperfect PM and to explore the effect of rework and PM error on profit.     

Combining a maintenance center M/M/c/m queue into the economic production quantity model with stochastic machine breakdowns and repair

This article considers a production system in which “m” identical machines produce nonidentical products at production rates. Products made by each machine are on the other hand consumed at a specific demand rate. Machines may be affected by unwanted breakdowns. Machines break down according to a Poisson distribution with equal rates and the failed machines are sent to maintenance center for repair which is consisted of “c” servers or servicemen. However, Number of machines is greater than number of servicemen (m > c). Hence, if the number of failed machines is greater than that of servicemen, the machines have to be put in a queue. Machines are put in one queue with the order of queue being FCFS. The queue has a typical M/M/c/m system. If machines are broken down during production, shortage will occur. This problem has been considered to obtain a single production cycle for the machines and an optimum number of the servers such that costs are minimized. For this purpose, distribution of waiting time for machines in repair center is calculated and a cost function is formed. Steepest descent and direct search methods are applied in this work to obtain optimal production cycle and maintenance servers, respectively. The proposed methods are studied using a comprehensive example.     

Determination of optimal size of the de-jitter buffer of a receiving router

The influence of the size of the de-jitter buffer of a receiving router and of the network load on the transmission quality of voice traffic is investigated by means of simulation modeling. The optimal size of the de-jitter buffer that supports an acceptable level of loss of voice packets and a propagation time of the voice signal that corresponds to accepted standards is determined.     

On a system of first-order quasi-variational inequalities connected with the optimal switching problem

In this paper we prove the existence of the maximum bounded Lipschitz continuous solution to a system of first-order quasi-variational inequalities. The method is to discretize by an Euler scheme the characteristic lines of the differential operator appearing in the system and to solve by iteration the corresponding approximate problem. The solution is interpreted as the value function of a deterministic optimal switching problem.     

Recursive operation time maximization model for the maintenance of power generation equipment

Repairable equipment requires preventive maintenance (PM) to maintain proper function. An appropriate PM strategy can extend the life of equipment and reduce variable costs. A power generation company in Taiwan that has a fixed-period PM strategy is studied. However, a fixed-period PM strategy is usually not an optimal solution for maintaining equipment those ages gradually. An age-related dynamical model is designed for creating a timely maintenance schedule that extends the life of equipment and improves its efficiency. A recursive mathematical programming model that considers regular maintenance, emergent maintenance, and maintenance improvement factors is used to obtain the optimal equipment PM schedule. An empirical experiment with the nonperiodic PM schedule computed using the proposed method was conducted. Results show a significant reduction in the PM cost and an extension of the life of equipment.     

An efficient algorithm for optimal control of PWA systems with polyhedral performance indices

We present an algorithm for the computation of explicit optimal control laws for piecewise affine (PWA) systems with polyhedral performance indices. The algorithm is based on dynamic programming (DP) and represents an extension of ideas initially proposed in Kerrigan and Mayne [(2003). Optimal control of constrained, piecewise affine systems with bounded disturbances. In Proceedings of the 41st IEEE conference on decision and control, Las Vegas, Nevada, USA, December], and Baoti? et al. [(2003). A new algorithm for constrained finite time optimal control of hybrid systems with a linear performance index. In Proceedings of European control conference, Cambridge, UK, September]. Specifically, we show how to exploit the underlying geometric structure of the optimization problem in order to significantly improve the efficiency of the off-line computations. An extensive case study is provided, which clearly indicates that the algorithm proposed in this paper may be preferable to other schemes published in the literature.     

Approximations for optimal stopping of a piecewise-deterministic process

This paper deals with approximation techniques for the optimal stopping of a piecewise-deterministic Markov process (P.D.P.). Such processes consist of a mixture of deterministic motion and random jumps. In the first part of the paper (Section 3) we study the optimal stopping problem with lower semianalytic gain function; our main result is the construction of ε-optimal stopping times. In the second part (Section 4) we consider a P.D.P. satisfying some smoothness conditions, and forN integer we construct a discretized P.D.P. which retains the main characteristics of the original process. By iterations of the single jump operator from ℝ ^N to ℝ ^N , each iteration consisting ofN one-dimensional minimizations, we can calculate the payoff function of the discretized process. We demonstrate the convergence of the payoff functions, and for the case when the state space is compact we construct ε-optimal stopping times for the original problem using the payoff function of the discretized problem. A numerical example is presented.     

L-class enumeration algorithms for a discrete production planning problem with interval resource quantities

A discrete production planning problem which may be formulated as the multidimensional knapsack problem is considered, while resource quantities of the problem are supposed to be given as intervals. An approach for solving this problem based on using its relaxation set is suggested. Some L-class enumeration algorithms for the problem are described. Results of computational experiments are presented.     

On the value function for nonautonomous optimal control problems with infinite horizon

In this paper we consider nonautonomous optimal control problems of infinite horizon type, whose control actions are given by L¹-functions. We verify that the value function is locally Lipschitz. The equivalence between dynamic programming inequalities and Hamilton–Jacobi–Bellman (HJB) inequalities for proximal sub (super) gradients is proven. Using this result we show that the value function is a Dini solution of the HJB equation. We obtain a verification result for the class of Dini sub-solutions of the HJB equation and also prove a minimax property of the value function with respect to the sets of Dini semi-solutions of the HJB equation. We introduce the concept of viscosity solutions of the HJB equation in infinite horizon and prove the equivalence between this and the concept of Dini solutions. In the Appendix we provide an existence theorem.     

Alternative mixed integer linear programming models for identifying the most efficient decision making unit in data envelopment analysis

A mixed integer linear model for selecting the best decision making unit (DMU) in data envelopment analysis (DEA) has recently been proposed by Foroughi [Foroughi, A. A. (2011a). A new mixed integer linear model for selecting the best decision making units in data envelopment analysis. Computers and Industrial Engineering, 60(4), 550–554], which involves many unnecessary constraints and requires specifying an assurance region (AR) for input weights and output weights, respectively. Its selection of the best DMU is easy to be affected by outliers and may sometimes be incorrect. To avoid these drawbacks, this paper proposes three alternative mixed integer linear programming (MILP) models for identifying the most efficient DMU under different returns to scales, which contain only essential constraints and decision variables and are much simpler and more succinct than Foroughi’s. The proposed alternative MILP models can make full use of input and output information without the need of specifying any assurance regions for input and output weights to avoid zero weights, can make correct selections without being affected by outliers, and are of significant importance to the decision makers whose concerns are not DMU ranking, but the correct selection of the most efficient DMU. The potential applications of the proposed alternative MILP models and their effectiveness are illustrated with four numerical examples.     

An optimal production lot-sizing problem for an imperfect process with imperfect maintenance and inspection time length

This article develops an integrated model in considering the situations of an imperfect process with imperfect maintenance and inspection time for the joint determination of both economic production quantity (EPQ) and preventive maintenance (PM). This imperfect process has a general deterioration distribution with increasing hazard rate. Even with periodic PM, such a production system cannot be recovered as good as new. This means that the system condition depends on how long it runs. Also, the PM level can be distinct due to the maintenance cost. For convenience, it is assumed the age of system is reduced in proportional to the PM level. Further, during a production cycle, we need an inspection to see if the process is in control. This inspection might demand a considerable amount of time. In this article, we take PM level and inspection time into consideration to optimise EPQ with two types of out-of-control states. To see how the method works, we use a Weibull shock model to show the optimal solutions for the least costs.     

A decomposition approach for the solution of the unit loading problem in hydroplants

This paper presents a decomposition approach for the solution of the dynamic programming formulation of the unit loading problem in hydroplant management. This decomposition approach allows the consideration of network and canal constraints without additional computational effort.     

A general and optimal approach for three inventory models with a linear trend in demand

This study provides a general and simple algorithm to obtain an optimal solution for three inventory models with a replenishment batching policy, production batching policy, and an integrated replenishment/production batching policy in a manufacturing system, under a finite time horizon and linear trend in demand. This study determines the replenishment or production schedule with one general equation for these three problems and provides fully theoretical proofs for relaxing some of the conjectures in previous studies. A general and explicit procedure to derive the optimal solution for these three inventory models is presented, while considering both linearly increasing and decreasing demands. In addition, demonstrations of applicability are performed.     

Software tools for formation of state spaces of queuing systems with a common type buffer

Universal algorithms and software tools are developed for formation of discrete components of state spaces for queuing systems with a common type buffer. A statement on the parameters of such a buffer is proved. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 164–168, May–June 2007.