Continuous sampling plans for markov-dependent production processes

This article discusses the behavior of three continuous sampling plans: continuous sampling plan 1 (CSP 1) and continuous sampling plan 2 (CSP 2) developed by Dodge [5] and Dodge and Torrey [7], and multilevel continuous sampling plan 2 (MLP 2) developed by Lieberman and Solomon [11], when the quality of successive units in a continuous production process follows a two-state time-homogeneous Markov chain. We first derive the average outgoing quality (AOQ) expressions of these plans. Exact procedures for determining the average outgoing quality limit (AOQL) can be obtained only for CSP 1. For CSP 2 and MLP 2 plans, iterative procedures have been used to obtain the AOQL contours. For these plans, it is assumed that the serial correlation coefficient between the two consecutive random variables of the Markov chain is known. In addition, estimation procedures for the coefficient are given. We show that if the serial correlation coefficient of the Markov chain is positive (negative), the AOQL is increased (decreased) as compared to the case when the successive units in the production process follows a Bernoulli pattern. Let r denote the number of production units examined in succession which are found to be of good quality and k denote the inverse of the sampling fraction employed when quality is good. Then if r and k are sufficiently small, it is observed from the graph that, for small departures of the serial correlation coefficient from zero, the AOQL values do not differ significantly for each of the three plans; whereas for sufficiently large values of r and k, the AOQL values differ significantly. Various aspects of these plans, such as their operating characteristics 2 (OC 2) and the serial correlation coefficient, are discussed.     

The threshold policy in the M/G/1 queue with server vacations

This article deals with the M/G/1 queue with server vacations in which the return of the server to service depends on the number of customers present in the system. The main goal is optimization, which is done under the average cost criterion in the multiple- and single-vacation models as well as for the “total cost for one busy cycle” criterion in the multiple-vacation case. Expressions that characterize the optimal number of customers, below which the server should not start a new service period, are exhibited for the various cases. It is found that under the average cost criterion, the expression may be universal in the sense that it may hold for a general class of problems including such that arise in production planning and inventory theory (for the particular cost structure discussed).     

Optimal control of entry to an M/Ek/1 queue serving several classes of customers

The individual and social optimum control policies for entry to an M/M//1 queue serving several classes of customers have been shown to be control-limit policies. The technique of policy iteration provides the social optimum policy for such a queue in a straightforward manner. In this article, the problem of finding the optimal control policy for the M/E_k/1 system is solved, thereby expanding the potential applicability of the solutions developed. The Markovian nature of the queueing system is preserved by considering the service as having k sequential phases, each with independent, identically distributed, exponential service times, through which a customer must pass to be serviced. The optimal policy derived by policy iteration for such a system is likely to be difficult to use because it requires knowledge of the number of phases rather than customers in the system when an arrival occurs. To circumvent this difficulty, a heuristic is used to find a good usable (implementable) solution. In addition, a mixed-integer program is developed which yields the optimal implementable solution when solved.     

Performance analysis of queue length monitoring of M/G/1 systems

This study investigates the statistical process control application for monitoring queue length data in M/G/1 systems. Specifically, we studied the average run length (ARL) characteristics of two different control charts for detecting changes in system utilization. First, the nL chart monitors the sums of successive queue length samples by subgrouping individual observations with sample size n. Next is the individual chart with a warning zone whose control scheme is specified by two pairs of parameters, (upper control limit, d_u) and (lower control limit, d_l), as proposed by Bhat and Rao (Oper Res 20 (1972) 955–966). We will present approaches to calculate ARL for the two types of control charts using the Markov chain formulation and also investigate the effects of parameters of the control charts to provide useful design guidelines for better performance. Extensive numerical results are included for illustration. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

A dynamic,nonstationary inventory problem for a price/quantity setting firm

A dynamic and nonstationary model is formulated for a firm which attempts to minimize total expected costs over a finite planning horizon. The control variables are price and production. The price p and the demand ζ are linked through the relationship ζ = g(p) + η, where g(p) is the riskless demand curve and η is a random variable. The general model allows for proportional ordering costs, convex holding and stockout costs, downward sloping riskless demand curve, backlogging, partial backlogging, lost sales, partial spoilage of inventory, and two modes of collecting revenue. Sufficient conditions are developed for this problem to have an optimal policy which resembles the single critical number policy known from stochastic inventory theory. It is also shown what set of parameters will satisfy these sufficiency conditions.     

Systems approach to the multistage manufacturing connected-unit situation

This paper investigates the relationships among incoming quality limit (AIQL), manufacturing quality level in each stage of production, and outgoing quality limit (AOQL), for a multistage manufacturing connected-unit situation from the systems cost-effectiveness point of view. An objective function is developed and the formulated problem is solved using discrete optimization technique. The study is restricted to the development of single sampling plans where inspection is by attribute. Lot size and sample size are fixed for incoming and outgoing inspection for each stage of manufacturing.     

Approximations for heavily loaded G/GI/n + GI queues

Motivated by applications to service systems, we develop simple engineering approximation formulas for the steady‐state performance of heavily loaded G/GI/n+GI multiserver queues, which can have non‐Poisson and nonrenewal arrivals and non‐exponential service‐time and patience‐time distributions. The formulas are based on recently established Gaussian many‐server heavy‐traffic limits in the efficiency‐driven (ED) regime, where the traffic intensity is fixed at ρ > 1, but the approximations also apply to systems in the quality‐and‐ED regime, where ρ > 1 but ρ is close to 1. Good performance across a wide range of parameters is obtained by making heuristic refinements, the main one being truncation of the queue length and waiting time approximations to nonnegative values. Simulation experiments show that the proposed approximations are effective for large‐scale queuing systems for a significant range of the traffic intensity ρ and the abandonment rate θ, roughly for ρ > 1.02 and θ > 2.0. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 187–217, 2016     

Optimal admission pricing and service rate control of an M[x]/M/s queue with reneging

In this article we consider the optimal control of an M^[X]/M/s queue, s ≧ 1. In addition to Poisson bulk arrivals we incorporate a reneging function. Subject to control are an admission price p and the service rate μ. Thus, through p, balking response is induced. When i customers are present a cost h(i,μ,p) per unit time is incurred, discounted continuously. Formulated as a continuous time Markov decision process, conditions are given under which the optimal admission price and optimal service rate are each nondecreasing functions of i. In Section 4 we indicate how the infinite state space may be truncated to a finite state space for computational purposes.     

A recursive algorithm for a summed multinomial density function

An algorithm for calculating the probabilities of a summed multinomial density function which is recursive with n (the number of trials) is presented. Having application in inspector error models for auditing and quality control problems with Cartesian product structures, the algorithm is discussed in the context of computing optimal economic sampling plans. Computational experience with the algorithm is presented.     

Application of renewal theory to continuous sampling plans

Renewal theory is used to study the effectiveness of a class of continuous sampling plans first introduced by Dodge. This approach provides a simple way of viewing and computing the long-run Average Outgoing Quality (AOQ) and its maximum AOQL. More importantly, it is used to study the average outgoing quality in a short production run through an approximation formula AOQ^*(t). Formulas for AOQ and AOQ^*(t) are provided. By simulation, it is found that AOQ^*(t) is sufficiently accurate in situations corresponding to actual practice.     

Determining a threshold control policy for an imperfect production system with rework jobs

Consider a threshold control policy for an imperfect production system with only a work center handling both regular and rework jobs. An imperfect production system studied here, generates defect jobs by factors other than machine failures. A threshold control or (ω, s) policy sets the guideline for a work center to switch between regular and rework jobs. A production cycle begins with loading and processing of several batches of regular jobs with a lot size equal to s. The outcome of each completed regular job is an independent Bernoulli trial with three possibilities: good, rework, or scrap. Once the work center accumulates more than a threshold ω of rework jobs, it finishes the last batch of regular jobs and switches to rework jobs. The objective of this research is to find a threshold ω and a lot size s that maximize the average long‐term profit. The ultimate goal is to construct a simple algorithm to search for ω and s that can be implemented directly in production management systems, as a result of this work. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 273–301, 1999     

Single-stage,multi-product production/inventory systems with lost sales

A production/inventory system consisting of a single processor producing three product types and a warehouse is considered. For each product type, the demand process is assumed to be Poisson and the processing time is phase-type. Excess demand is lost. Products have a priority structure and the processor's attention is shared by all the products according to a switching rule. Production of a product continues until its target level is reached. Then, a switch-over takes place if another product needs the processor's attention. A set-up process takes place every time a switch-over occurs. An (R, r) continuous-review inventory control policy is used to start and stop the production of each product. The underlying Markov chain is studied and its steady-state distribution is obtained recursively. Through the recursive procedure, the steady-state balance equations to be dealt with are significantly reduced to a manageable set. The procedure is implemented on a supercomputer and examples are provided to show its efficiency and stability for a range of model parameters. We analyzed the joint behaviors of the inventory levels of the three products as their demand rates increase. Finally we introduced a cost minimizing objective function to guide design efforts. © 1995 John Wiley & Sons, Inc.     

Sojourn times in G/M/1 fork‐join networks

We consider a processing network in which jobs arrive at a fork‐node according to a renewal process. Each job requires the completion of m tasks, which are instantaneously assigned by the fork‐node to m task‐processing nodes that operate like G/M/1 queueing stations. The job is completed when all of its m tasks are finished. The sojourn time (or response time) of a job in this G/M/1 fork‐join network is the total time it takes to complete the m tasks. Our main result is a closed‐form approximation of the sojourn‐time distribution of a job that arrives in equilibrium. This is obtained by the use of bounds, properties of D/M/1 and M/M/1 fork‐join networks, and exploratory simulations. Statistical tests show that our approximation distributions are good fits for the sojourn‐time distributions obtained from simulations. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Optimal reject allowance with constant marginal production efficiency

A job shop must fulfill an order for N good items. Production is conducted in “lots,” and the number of good items in a lot can be accurately determined only after production of that lot is completed. If the number of good items falls short of the outstanding order, the shop must produce further lots, as necessary. Processes with “constant marginal production efficiency” are investigated. The revealed structure allows efficient exact computation of optimal policy. The resulting minimal cost exhibits a consistent (but not universal) pattern whereby higher quality of production is advantageous even at proportionately higher marginal cost.     

Markovian approximation for manufacturing systems of unreliable machines in tandem

This paper studies production planning of manufacturing systems of unreliable machines in tandem. The manufacturing system considered here produces one type of product. The demand is assumed to be a Poisson process and the processing time for one unit of product in each machine is exponentially distributed. A broken machine is subject to a sequence of repairing processes. The up time and the repairing time in each phase are assumed to be exponentially distributed. We study the manufacturing system by considering each machine as an individual system with stochastic supply and demand. The Markov Modulated Poisson Process (MMPP) is applied to model the process of supply. Numerical examples are given to demonstrate the accuracy of the proposed method. We employ (s, S) policy as production control. Fast algorithms are presented to solve the average running costs of the machine system for a given (s, S) policy and hence the approximated optimal (s, S) policy. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 65–78, 2001     

A computationally efficient approach for determining inventory levels in a capacitated multiechelon production‐distribution system

The system under study is a single item, two‐echelon production‐inventory system consisting of a capacitated production facility, a central warehouse, and M regional distribution centers that satisfy stochastic demand. Our objective is to determine a system base‐stock level which minimizes the long run average system cost per period. Central to the approach are (1) an inventory allocation model and associated convex cost function designed to allocate a given amount of system inventory across locations, and (2) a characterization of the amount of available system inventory using the inventory shortfall random variable. An exact model must consider the possibility that inventories may be imbalanced in a given period. By assuming inventory imbalances cannot occur, we develop an approximation model from which we obtain a lower bound on the per period expected cost. Through an extensive simulation study, we analyze the quality of our approximation, which on average performed within 0.50% of the lower bound. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 377–398, 2000     

Qmp/USP: A modern alternative to MIL-STD-105D

The Quality Measurement Plan (QMP) and the Universal Sampling Plan (USP) are the data analysis and sampling plans for the AT&T Technologies quality audit. This article describes QMP/USP, an acceptance sampling plan based on QMP and USP principles. QMPIUSP is a complete acceptance sampling system. It combines the elements of classical rectification inspection plans with those of MIL-STD-IOSD. There is no switching between plans, no tables of numbers to look through, and no discontinue state. QMP/USP is a computerized, self-contained system that features:

• Acceptance decisions based on the QMP Bayes empirical Bayes analysis of current and past sampling result
• Sample size selection based on USP, i.e., lot size, AQL, a cost ratio, the QMP analysis, and a budget constraint
• Guaranteed AOQ
• A complete statistical analysis of the quality process.

In this article, we describe the operation of QMP/USP and compare its performance with that of MIL-STD-IOSD. The comparison is made under many different quality environments with many metrics. Our results show that QMP/USP and MIL/STD/IOSD perform similarly for the environments where quality could be described as “in control”; and that QMPlUSP is superior in the environments where quality is “out of control”.     

Assembly of systems having maximum reliability

The first problem considered in this paper is concerned with the assembly of independent components into parallel systems so as to maximize the expected number of systems that perform satisfactorily. Associated with each component is a probability of it performing successfully. It is shown that an optimal assembly is obtained if the reliability of each assembled system can be made equal. If such equality is not attainable, then bounds are given so that the maximum expected number of systems that perform satisfactorily will lie within these stated bounds; the bounds being a function of an arbitrarily chosen assembly. An improvement algorithm is also presented. A second problem treated is concerned with the optimal design of a system. Instead of assembling given units, there is an opportunity to “control” their quality, i.e., the manufacturer is able to fix the probability, p, of a unit performing successfully. However, his resources, are limited so that a constraint is imposed on these probabilities. For (1) series systems, (2) parallel systems, and (3) k out of n systems, results are obtained for finding the optimal p's which maximize the reliability of a single system, and which maximize the expected number of systems that perform satisfactorily out of a total assembly of J systems.     

A note on sojourn times in M/G/1 queues with instantaneous,bernoulli feedback

Queueing systems which include the possibility for a customer to return to the same server for additional service are called queueing systems with feedback. Such systems occur in computer networks for example. In these systems a chosen customer will wait in the queue, be serviced and then, with probability p, return to wait again, be serviced again and continue this process until, with probability (1 – p) = q, it departs the system never to return. The time of waiting plus service time, the nth time the customer goes through, we will call his nth sojourn time. The (random) sum of these sojourn times we will call the total sojourn time (abbreviated, sojourn time when there is no confusion which sojourn time we are talking about). In this paper we study the total sojourn time in a queueing system with feedback. We give the details for M/G/1 queues in which the decision to feedback or not is a Bernoulli process. While the details of the computations can be more difficult, the structure of the sojourn time process is unchanged for the M/G/1 queue with a more general decision process as will be shown. We assume the reader is familiar with Disney, McNickle and Simon [1].     

Optimal admission pricing policies for M/Ek/1 queues

This paper extends the Low-Lippman M/M/1 model to the case of Gamma service times. Specifically, we have a queue in which arrivals are Poisson, service time is Gamma-distributed, and the arrival rate to the system is subject to setting an admission fee p. The arrival rate λ(p) is non-increasing in p. We prove that the optimal admission fee p^* is a non-decreasing function of the customer work load on the server. The proof is for an infinite capacity queue and holds for the infinite horizon continuous time Markov decision process. In the special case of exponential service time, we extend the Low-Lippman model to include a state-dependent service rate and service cost structure (for finite or infinite time horizon and queue capacity). Relatively recent dynamic programming techniques are employed throughout the paper. Due to the large class of functions represented by the Gamma family, the extension is of interest and utility.