A new inspection sampling plan under capacity constraints and multiple shifts

Various SPC models have been widely deployed in the quality control of different production systems. This paper considers a new design of SPC model with associated decision rules to control a process that produces items in finite lots and has a variance shift as well as a mean shift. Under both an inspection capacity constraint and a false alarm rate constraint, an inspection sampling plan that maximizes the yield of production is developed. Numerical examples are presented to illustrate and evaluate the approach. A semi-economic approach that computes the optimal quantity of inspection capacity is also presented.     

An Integrated Optimization Model for Inventory and Quality Control Problems

In this paper, we develop an integrated economic model for inventory and quality control problems, extending the work of Rahim ( IIE Transactions 1994; 26(6): 2–11) and Rahim and Ben-Daya (IJPR 1998; 36(1): 277–289). The production process is subject to an assignable cause which shifts the process from an in-control state to an out-of-control state. We consider the shifts in both the process mean and the process variance. When a signal for an assignable cause is triggered, a search is initiated and is terminated upon finding the cause within a pre-specified target time. The process is then brought back to an in-control state by repair. However, if the assignable cause is not discovered within the pre-specified time, production is allowed to continue until the next sampling or warning, whichever occurs first. In this case, either the alarm is considered to be false with a probability of Type I error, or the assignable cause has not been eliminated with a probability of Type II error. In the latter case, the process produces products in an out-of-control state until the next sampling or warning, whichever occurs first. However, this state does not indicate any severe damage to the system. Joint X and R charts are used for monitoring both process mean and variance. Under these conditions, a generalized economic model for the joint determination of production quantity, an inspection schedule, and the design of the X and R control charts are developed. A direct search optimization method is used to determine the optimal decision variables of the economic model.     

Economic design of a variable sampling rate EWMA chart

A Variable Sampling Rate (VSR) control chart is a control chart whose sampling scheme is to vary the sampling interval and the sample size for the next sample depending on the current chart statistic. A VSR EWMA chart is an EWMA chart with the VSR sampling scheme. An economic model, which was developed for a VSR chart, is also applied here to evaluate the efficiency of the VSR EWMA chart. The properties of the VSR EWMA chart are obtained by using a Markov chain approach. The model contains cost parameters which allow the specification of the costs associated with sampling, false alarms and operating off target as well as search and repair. This economic model can be used to quantify the cost saving that can be obtained by using a VSR chart instead of a Fixed Sampling Rate (FSR) chart and can also be used to gain insight into the way that a VSR chart should be designed to achieve optimal economic performance. It is shown that with some design parameter combinations the economically optimal VSR chart has a lower false alarm rate than the FSR chart.     

Effect of rework,rejects and inspection on lot size with work-in-process inventory

The economic order quantity and economic production quantity models are the most commonly used inventory models in production environments for the calculation of optimum lot size. However, these models are based on the unrealistic assumption that every process produces good quality products every time. Moreover, the impact of inspection is neglected in all extended inventory control models involving work in process inventory. By taking both imperfect production and lot size inspection into consideration, this paper presents a more realistic approach for the modelling of optimum lot size and total cost with a focus on the work in process inventory. A mathematical model is derived for optimum lot size based on the minimisation of the average cost. Our approach incorporates the effect of rework, rejects and inspection on work in process inventory. The significant effect of imperfect production and inspection on optimum lot size is evaluated via numerical examples. In comparison to existing models, the proposed model is a more generalised and flexible form of inventory model for independent demands.     

An inventory model with emergency orders under explicit energy cost considerations

The inventory model in this paper is targeted to production systems with constant production rates but underlying possibilities for undesirable circumstances to threaten the production schedule. The inventory policy proposed explicitly considers energy cost when determining optimal size for order quantity, safety stock and inventory cycle length such that the total expected cost per unit time is minimised. The results are compared to a traditional inventory policy that does not consider the direct impact of energy cost. An analysis of the model reveals three production environment characteristics in which inventory policies are most significantly affected by changes in energy cost: heavy product weight, high regular product demand or high emergency product demand. If any one of the three key factors increases, then changes of the inventory decisions or related logistics costs become more significant. The cost effectiveness of implementing the proposed inventory policy also becomes more significant as any one of the three key factors increase with respect to energy cost.     

Integrated production maintenance and quality model for imperfect processes

In this paper, we develop an integrated model for the joint optimization of the economic production quantity, the economic design of x¯-control chart, and the optimal maintenance level. This is done for a deteriorating process where the in-control period follows a general probability distribution with increasing hazard rate. In the proposed model, Preventive Maintenance (PM) activities reduce the shift rate to the out-of-control state proportional to the PM level. Compared to the case with no PM, the extra cost of maintenance results in lower quality control cost which may lead to lower overall expected cost. These issues are illustrated using an example of a Weibull shock model with an increasing hazard rate.     

Joint production and shipment lot sizing for a delivery price-based production facility

A joint economic production quantity (EPQ) and delivery quantity model for a production system is investigated in this paper. More specifically, an EPQ policy is implemented in the production system, while a smaller shipping quantity is periodically dispatched to the customer. The production system is also responsible for the shipment cost, i.e. a delivery price-based procurement from the customer. The considered cost includes setup cost to launch the batch production, inventory carrying cost, and transportation cost, where the transportation cost is a function of the delivery quantity. A per unit time cost model is developed and analysed to determine the optimal production and delivery quantities. Under some mild conditions, it can be shown that the joint cost function is convex with respect to the production quantity; and the number of delivery is an integer in each replenishment cycle. Computational study has demonstrated the significant impact of the joint decision model on the operating cost. In particular, the reduction in total cost can be more than 15% when inventory carrying costs, and/or transportation costs, are high.     

The economic performance of the Shewhart t chart

In the production of small batches of customized parts, high flexibility and frequent switching of production from one product variant to another could not allow for the implementation of a control chart to monitor the process. In fact, when a short‐run production should be started, the distribution parameters of the quality characteristics to be monitored are frequentlytextitunknown and the production run is too short to get sufficient Phase I samples. To overcome this problem, the statistical properties of Shewhart t charts monitoring a short production run have been recently discussed in literature. In this paper, we investigate their economic performance: the SPC inspection cost optimization is constrained by the manufacturing and the inspection activities configuration. The decision variables of the problem include the chart design parameters and the size of batches of parts to be worked and released to the local inspection area. A numerical analysis aimed at evaluating the economic performance of the Shewhart t chart vs the Shewhart chart with known parameters has been performed. The expected economic loss associated with the implementation of the Shewhart t chart is acceptable with respect to the ‘ideal’ condition of the control chart with known parameters when the cost optimization is achieved without a statistical constraint limiting the number of expected false alarms. Finally, the effect of an erroneous initial set‐up on the correctness of the inspection cost estimation has been investigated. Copyright © 2011 John Wiley & Sons, Ltd.     

A Production and Maintenance Planning Model with Restoration Cost Dependent on Detection Delay

In this paper, the joint problem of production planning and maintenance schedule is studied under the realistic assumption that the cost of process restoration is a function of the detection delay and the existence of shortages in the system. The detection delay is defined as the elapsed time since the production process has deteriorated until it is identified by some inspection procedure and repaired. Since production planning and maintenance problems have usually been studied as separate problems, this paper is an attempt to develop a formal framework for the joint problem. We have developed sufficient conditions for the optimality of the commonly used equal-interval maintenance schedule. The conditions are found to be a function of parameters such as the cost of defective items, the mean time for system deterioration, and the form of the restoration cost function. For specific restoration cost functions such as linear and exponential, an efficient solution procedure is presented for the simultaneous determination of the number of maintenance inspections in a production run, the length of the production run and consequently the economic manufacturing quantity, and the maximum level of backorders. A numerical example illustrates the use of this procedure and the differences between the optimal cost obtained by this procedure and the cost obtained by using the classical economic manufacturing quantity model.     

An EOQ model when production standards are assumed to be normally distributed

In this paper an EOQ model is developed especially for a production environment where the inventory to be controlled is raw material. The fundamental difference between this model and others is that a stochastic demand is placed upon the inventory in terms of the time for an individual to consume one unit of inventory rather than the number of units consumed in some time interval. A shortage cost for inventory is imposed via the idle machine (man) cost. The optimal re-order quantity is determined as a function of the stockout probability and the optimal stockout probability is obtained via a one-dimensional search, over some prescribed set of feasible stockout probabilities.     

OPTIMAL PRODUCTION PLANNING AND INVENTORY CONTROL*

This paper is concerned with the optimal production planning and inventory control. The first problem is a multiperiod production scheduling problem in which the objective is to minimize the operating cost for a planning period. This cost is composed principally of the sum of the production cost and inventory carrying cost. The second problem considers an inventory system with two decision variables in each planning period. These are the production schedule and work force which are to be determined so as to minimize the operating cost which includes the costs of changing the production rate, of changing the work force and of carrying the inventory. The maximum principle in the discrete form is used to reduce both the first problem which has N decision variables and the second problem which has 2N decision variables respectively to a series of two decision variables problems. The so-called sequential simplex pattern search technique is used to determine the optimal values of these two decision variables. Numerical examples are given to demonstrate the method.     

Binomial CUSUM chart with curtailment

The binomial cumulative sum (CUSUM) chart has been widely used to monitor the fraction nonconforming (p) of a process. It is a powerful procedure for detecting small and moderate p shifts. This article proposes a binomial CUSUM control chart using curtailment technique (Curt_CUSUM chart in short). The new chart is able to improve the overall detection effectiveness while holding the false alarm rate at a specified level. The results of the comparative studies show that, on average, the Curt_CUSUM chart is more effective than the CUSUM chart without curtailment by 30%, in terms of Average Number of Defectives, under different circumstances. The Curt_CUSUM chart can be applied to a 100% inspection as well as a general random sampling inspection.     

Impacts of collaborative investment and inspection policies on the integrated inventory model with defective items

For an imperfect production system, to reduce quality-related costs, a manager may consider investing capital in quality improvement. In general, the investment expense in reducing the defective rate of items is often paid by the vendor. On the other hand, the buyer may inspect the product quality as the order is received which implies it incurs an inspection cost. In a supply chain integrated system, to accomplish global optimisation, the vendor and buyer can agree to jointly invest capital to improve the imperfect production processes, and the buyer can remove the inspection programme as the defective rate reaches a certain low-level. Hence, this paper investigates the impacts of collaborative investment and inspection policies on an integrated inventory model with defective items. The objective of this study is to seek the optimal order quantity, shipping times from the vendor to the buyer per production run, and the defective rate that minimise the joint total cost per unit time. An algorithm is developed to find the optimal solution. Several numerical examples are presented to demonstrate the proposed model and solution procedure, and then several management insights are obtained from the numerical examples.     

An Economic Approach to the Management of High‐Quality Processes

Modern manufacturing developments have forced researchers to investigate alternative quality control techniques for high‐quality processes. The cumulative count of conforming (CCC) control chart is a powerful alternative approach for monitoring high‐quality processes for which traditional control charts are inadequate. This study develops a mathematical model for the economic design of the CCC control chart and presents an application of the proposed model. On the basis of the results of the application, the economic and classical CCC control chart designs of the CCC control chart are compared. The optimal design parameters for different defective fractions are tabulated, and a sensitivity analysis of the model is presented for the CCC control chart user to determine the optimal economic design parameters and minimum hourly costs for one production run according to different defective fractions, cost, time, and process parameters. Copyright © 2012 John Wiley & Sons, Ltd.     

An EOQ model with random disruption and partial backordering

Perfect quality of batches is an assumption in the classic economic order quantity (EOQ) model (of inventory management); however, in practice some disruptions may occur in a supply chain. This paper presents an EOQ model with random disruption and partial backorders. So, when shortage occurs (due to quality problems or normal shortages), some customers are willing to wait for delivery until the next period. There is a finite probability that batches may be defective so an ‘all or none’ policy is used after inspection of batches. The optimal inventory cost and the corresponding decision variables are studied. A solution method is proposed to optimise the inventory cost, and then two numerical examples and a sensitivity analysis are provided to illustrate the results.     

An integrated EPQ model based on a control chart for an imperfect production process

The notion of quality control is introduced into the classic economic production quantity (EPQ) model. Based on previous research, this paper contributes to an integrated EPQ model combining the concepts of statistical process control and maintenance. An imperfect production process involving three different conditions is considered. A control chart is adopted to monitor the whole process with Taguchi's loss function estimating the quality cost of each condition. A corresponding maintenance policy is planned for the machine depending on what condition it runs in. Thus the proposed EPQ model takes quality-related costs and maintenance-related costs into account other than storage costs and ordering costs already considered in the classic model. The objective of this model is to minimise the total expected production cost per production cycle while simultaneously determining the optimal parameters of control chart design, the interval between sampling, the sample size and the maintenance decision policy. The pattern search method is used to solve the problem using the MATLAB toolbox. In addition, a case study, sensitivity analysis and comparison analysis are presented to demonstrate the application of the model.     

Joint production and quality control of unreliable batch manufacturing systems with rectifying inspection

Production control policy and economic sampling plan design problems have been studied separately in previous research. This paper considers a joint production control policy and economic single sampling plan design for an unreliable batch manufacturing system. The production is controlled by a modified hedging point policy which consists of building and maintaining a safety stock of finished product to avoid shortages during corrective maintenance. The main objective of this paper is to determine simultaneously the economic production quantity, the optimal safety stock level and the economic sampling plan design which minimise the expected overall cost. A stochastic mathematical model is developed and solved using a simulation optimisation approach based on the response surface methodology. Simulation is used to imitate the complex dynamic and stochastic behaviour of processes as in the real-life industrial systems. The obtained results show clearly strong interactions between production quantity, inventory state and sampling plan design which confirm the necessity of jointly considering production and quality control parameters in an integrated model. Moreover, it is shown a significant impact of production system reliability on the economic sampling plan design and therefore on the quality of finished product delivered to consumers. Numerical example and sensitivity analyses are presented for illustrative purposes.     

Optimal inventory policies for uniform replenishment systems with time-dependent demand

This study develops an analysis of lot size inventory systems where the replenishment rate is uniform and demand follows a power demand pattern. Shortages are not allowed. Holding cost, replenishing cost and purchasing cost are considered in inventory system control. The objective of the study is to find the economic production quantity that minimises total inventory cost per unit of time. We conclude that optimal inventory policies depend on the demand pattern index chosen to represent customer demand. Theoretical results are illustrated with a business case study. A sensitivity analysis is proposed to describe the optimal policy behaviour.     

An Economic Order Quantity Model with Demand-Dependent Unit Production Cost and Imperfect Production Processes

The classical economic order quantity (EOQ) model assumes that items produced are of perfect quality and that die unit cost of production is independent of demand. Product quality is not always perfect but directly affected by the reliability of the production process used to produce the products. In addition, a relationship between unit production cost and demand may exist under certain circumstances. We propose an EOQ model with demand-dependent unit production cost and imperfect production processes. We formulate this inventory decision problem as a geometric program (GP) and solve it to obtain closed-form optimal solutions. An illustrative example is provided to demonstrate the point that GP has potential as a valuable analytical tool for studying a certain class of inventory control problems. We also discuss the aspect of sensitivity analysis based on the GP approach.     

Optimisation design of attribute control charts for multi-station manufacturing system subjected to quality shifts

The qualities of products are a major concern in any production system; thus implementing efficient inspection policies is of great importance to reduce quality-related costs. This article addresses the problem of finding optimal inspection policies for the multi-station manufacturing system (MMS) subjected to quality shifts to minimise total quality-related cost. Each station of the MMS may stay at either in-control condition or out-of-control condition, which may lead to different nonconforming product rates. Markov chain method is used to calculate the steady-state probability distribution (SSPD). Based on the SSPD, the cost structure of this MMS is analysed. The economical optimisation model of attribute control charts (ACCs) is then established, in which the decision variables are the control chart parameters: sampling interval, sample size and control limit. The ACCs optimisation model is resolved by the proposed integrated algorithm combining heuristic rule and tabu search. This approach is verified through an application case taken from a mobile phone shell production company. The results of comparative analysis show that the proposed model is much more economical than both the current outgoing inspection strategy and the regular np control chart. The sensitivity analysis of four input parameters is also conducted.