A cost minimisation model for joint production and maintenance planning under quality constraints

In this paper, integrated planning of production, imperfect maintenance and process inspections in a multi-machine system is investigated. This system consists of parallel machines which deteriorate with time and they may shift from a primarily in-control state to a degraded state with a higher defective rate or to a failed state. Maintenance scheduling corresponds to a discrete time age-based imperfect maintenance with a large number of maintenance alternatives. Process inspections are considered to detect the current state of the system. Detecting a deteriorated condition initiates the quality check of the related sub-lots, rework of defective items and a process adjustment that brings the machine in its normal conditions. Production planning includes a capacitated lot-sizing problem with multiple products. We propose a joint approach that coordinates the decisions of the three functions, where the objective function minimises the total cost. Evaluation of costs and interacting factors is presented and two heuristic methods are proposed to solve the problem. The results of the joint model are compared to a non-integrated method and a sensitivity analysis is conducted.     

Determining the optimal lot size for the finite production model with random defective rate,the rework process,and backlogging

This paper considers the effects of the reworking of defective items on the economic production quantity (EPQ) model with backlogging allowed. The classic EPQ model assumes that manufacturing facility functions perfectly during a production run. However, due to process deterioration or other factors, the production process may shift and produce imperfect quality items. In this study, a random defective rate is considered, and when regular production ends, the reworking of defective items starts immediately. Not all of the defective items are reworked, a portion of them are scrap and are discarded. Repairing and holding cost per reworked item and disposal cost per scrap item are included in the proposed mathematical modelling and analysis. The renewal reward theorem is utilized to deal with the variable cycle length, and the optimal lot size that minimizes the overall costs for the imperfect quality EPQ model is derived where backorders are permitted.     

Lot sizing in a production system with rework and product deterioration

Producing new and recovering defective products often takes place on a common facility, with these activities carried out in lots. Consequently, there is a necessity to coordinate the production and rework activities with respect to the timing of operations and also regarding appropriate lot sizes for both processes while completely satisfying a given demand. Thereby, it has to be taken into account whether the state of defective items may change in the course of time while they wait to be reworked. Such a deterioration of reworkable goods can result in increasing rework time and rework cost per unit. In this paper an EPQ model which addresses all of these aspects is presented. Considering set-up and inventory holding costs as well as set-up times, optimization algorithms are developed covering different planning situations. Closed-form results for optimal lot sizes can be obtained and exploited for new analytical insights into coordinated lot sizing in the case of returns and product deterioration.     

The supplier–buyer integrated production-inventory model with random yield

This paper revisits the traditional supplier–buyer integrated production-inventory model which deals with the problem of a manufacturer (supplier) supplying a product to a retailer (buyer) serving the consumer market with constant stationary demand. The product is manufactured in batches at a finite rate. The supplier's production batch is depleted by the buyer's replenishment orders at periodic intervals. The buyer's inventory is then consumed by the market demand at a fixed rate. The problem is the simultaneous computation of the manufacturer's production lot-size and the buyer's replenishment order quantity, i.e. the integrated production-inventory policy parameters. The key characteristic considered in this paper is that the manufacturing process is imperfect, and, hence, there are defective items in each production lot. As a result, each replenishment order shipped to the buyer includes defective products and the non-defective percentage in each such shipment is random. Considering the case where the supplier replenishes the buyer via equal-sized shipments, we develop an analytical expression of the total expected cost for the supplier–buyer system under consideration, with and without a considerable inspection time. We first examine the case where the inspection time is negligible, and then we present a generalisation to consider the inspection time explicitly. Our goal is to model the impact of random yield on the system performance. Our findings are useful for computing the integrated production-inventory policy parameters while considering the supply uncertainty due to an imperfect manufacturing process. Through numerical examples, we quantify the impact of supply with random yield on the system performance and illustrate its relationship with the demand and production rate.     

Economic Production Cycles with Imperfect Production Processes

In this paper, we study the effects of an imperfect production process on the optimal production cycle time. The system is assumed to deteriorate during the production process and produce some proportion of defective items. The optimal production cycle is derived, and is shown to be shorter than that of the classical Economic Manufacturing Quantity model. The analysis is extended to the case where the defective rate is a function of the set-up cost, for which the set-up cost level and the production cycle time are jointly optimized. Finally, we also consider the case where the deterioration process is dynamic in its nature, i.e., the proportion of defective items is not constant. Both linear, exponential, and multi-state deteriorating processes are studied. Numerical examples are provided to illustrate the derivation of the optimal production cycle time in these situations.     

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.     

Developing approximate algorithms for EPQ problem with process compressibility and random error in production/inspection

This paper formulates a new imperfect production problem that generates defectives randomly. The production manager conducts total inspections to screen defectives (including reworkable and non-reworkable items) from non-defectives. But, the inspection is error-prone due to different sources. The model determines rates of main and rework processes, batch size and backlog. As the model is a nonlinear programme and it is difficult to obtain an algebraic closed-form solution, three randomised approximation algorithms are developed. Under certain conditions, the algorithms can find the global optimum in polynomial time. Finally numerical analyses are reported.     

LP-based heuristics for the capacitated lot-sizing problem: The interaction of model formulation and solution algorithm

We consider here the application of trivial LP-based rounding heuristics to the capacitated lot-sizing problem (CLSP). The motivation behind the use of LP-based heuristics is that their extension to cope with complicating features (to be expected, for example, when dealing with a master production scheduling problem within a MRP system) is generally easier than with alternative approaches such as lagrangian relaxation. It is well known that strong model formulations, like the Plant Location Formulation (PLF) or the Shortest Path Formulation (SPF), are needed to obtain good results when working on a CLSP. Still, from a practical point of view, a few questions deserve investigation. First, the relative performance of PLF and SPF should be assessed. Second, given the increased size of the more complicated formulations, the possible benefits of using interior point methods must be considered. Third, the sensitivity of the computation times with respect to the characteristics of problem instances should be evaluated. We report some computational experiments on relatively large problems (500 items and 15 time buckets, resulting in 7500 binary variables). For the simple CLSP model, trivial heuristics can yield near-optimal solutions with a reasonable computational effort, at least for the problem instances we consider. However, there is a critical and non-obvious interaction between the model formulation, the problem instance characteristics, and the solution algorithm.     

The capacitated lot sizing problem with overtime decisions and setup times

The Capacitated Lot Sizing-Problem (CLSP) consists of planning the lot sizes of multiple items over a planning horizon with the objective of minimizing setup and inventory holding costs. In each period that an item is produced a setup cost is incurred. Capacity is limited and homogeneous. Here, the CLSP is extended to include overtime decisions and capacity consuming setups. The objective function consists of minimizing inventory holding and overtime costs. Setups incur costs implicitly via overtime costs, that is, they lead to additional overtime costs when setup times contribute to the use of overtime capacity in a certain period. The resulting problem becomes more complicated than the standard CLSP and requires methods different from the ones proposed for the latter. Consequently, new heuristic approaches are developed to deal with this problem. Among the heuristic approaches are the classical HPP approach and its modifications, an iterative approach omitting binary variables in the model, a Genetic Algorithm approach based on the transportation-like formulation of the single item production planning model with dynamic demand and a Simulated Annealing approach based on shifting family lot sizes among consecutive periods. Computational results demonstrate that the Simulated Annealing approach produces high quality schedules and is computationally most efficient.     

Optimal locations of on-line and off-line rework stations in a serial production system

The production rate and product quality are two vital concerns for any manufacturing industry. Number of defective items reduces production rate and increases unit production cost. Moreover, if nonconforming items reach to the customers then manufacturer’s goodwill may drastically go down. Thus, quality inspection is treated as an inherent part of manufacturing. In this research, an N-stage serial production line with an inspection station at the end of it is considered to make decisions concerning this issue. On detecting a defective item at the end of the line it is scrapped or repaired at regular workstation or is sent to an off-line rework station for repair. Assuming each workstation produces a single type of defect a unit cost function is developed for alternative decisions on each type of defect. In order to minimise the unit cost of production and determine an appropriate decision for individual defect types, a fractional mixed integer nonlinear programming is formulated. After transformation to a mixed integer linear programming problem it is solved optimally. A small problem from garments industry is described in detail to show the solution procedure with a branch and bound method. Empirical tests with up to 40 workstations are permed to show the efficiency of the solution process.     

Preventive maintenance and optimal buffer inventory for products sold with warranty in an imperfect production system

This paper deals with an imperfect production system with allowable shortages due to regular preventive maintenance for products sold with free minimal repair warranty. Preventive maintenance is an essential factor of the just-in-time structure that results in a shutdown of the production process for a certain period of time. During such an interruption, a buffer stock is needed to adjust the market demand. The study includes the possibility of imperfect production and determines the optimum buffer level and production run time by trading off the holding cost, shortage cost, rework cost, repair cost for warranty, labour/energy costs, material cost and cost for maintenance so that the cost per unit product is minimised.     

Minimisation of inspection and rework cost in a BLU factory considering imperfect inspection

We address an optimisation problem for minimising the cost of both the inspection and the rework incurred throughout an interconnected inspection–rework system. In order to formulate the relevant cost objective function, assuming imperfect inspection, we make a time-based flow analysis between nodes (or shops), and derive the limiting sizes of flows between nodes and the limiting fraction defective by solving a set of nonlinear balance equations. We provide an enumeration method for determining the optimal frequency of inspection cycles at the end of assembly lines which minimises the relevant cost of inspection and rework.     

Effect of limited capacity on optimal planning parameters for a multi-item production system with setup times and advance demand information

An analytical production/inventory model to optimise the planning parameters lot-size, safety stock and planned lead time is developed for a stochastic single-stage production system with multiple items and limited capacity. Based on queuing analysis, the influence of item specific lot-sizes on the production lead time distribution is modelled. Applying stochastic advance demand information, the expected values for finished-goods-inventory, backorders and service level are explicitly stated. Numerical optimisation is applied to solve the respective cost minimisation problem and a solution heuristic is developed to support this approach. A numerical study provides managerial insights concerning capacity limitation effects on the optimal planning parameters. Higher shop loads, i.e. tighter capacity constraints, are found to significantly increase optimal lot-size and optimal safety stock. Safety stock and planned lead time are substitutes, an increase of both leads to higher FGI and lower backorders, however, the specific trade-off depends on the demand information quality. A sensitivity analysis investigating other (non-) financial system parameters is conducted as well. The main contribution of this paper is that the interaction of different planning parameters, i.e. lot-size, safety stock and planned lead time, for different items is simultaneously studied for a capacity constrained production/inventory system.     

Lot-sizing for a single-stage single-product production system with rework of perishable production defectives

We consider a single-stage single-product production system. Produced units may be non-defective, reworkable defective, or non-reworkable defective. The system switches between production and rework. After producing a fixed number (N) of units, all reworkable defective units are reworked. Reworkable defectives are perishable or can become technologically obsolete. We assume that the rework time and the rework cost increase linearly with the time that a unit is held in stock. Therefore, N should not be too large. On the other hand, N should not be too small either, since there are set-up times and costs associated with switching between production and rework. For a given N, we derive an explicit expression for the average profit (sales revenue minus costs). Using this expression, the optimal value for N can be determined numerically. Moreover, it is easy to perform a sensitivity analysis, as we illustrate. RID="*" ID="*"The research of Dr. Ruud H. Teunter has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. The research presented in this paper is part of the research on re-use in the context of the EU sponsored TMR project REVersed LOGistics (ERB 4061 PL 97-5650) in which take part the Otto-von-Guericke Universitaet Magdeburg (D), the Erasmus University Rotterdam (NL), the Eindhoven University of Technology (NL), INSEAD (F), the Aristoteles University of Thessaloniki (GR), and the University of Piraeus (GR). We thank the anonymous referees for their many helpful comments. Correspondence to: R. H. Teunter     

Economic production quantity model considering warm-up period in a cleaner production environment

Currently, the primary objective of every manufacturing organization is in customers’ satisfaction and system cost reduction while keeping in mind both production quantity (economic issues) and environmental issues (sustainability). This study investigates the effects of machine warm-up period on an imperfect production process with rework. In this case, the maintenance process is intended to maximize the efficiency and effectiveness of the production machine. In this manufacturing system, it assumed that after the setup time, the machine starts at the lower rate than the regular production process, commonly called as warm-up period. The warm-up period will increase the useful life of the machine. On the other hand, the machine's defects that occur during this warm-up period are identified and repaired before the regular production starts. Also, this warm-up period reduces waste as it improves machine performance; resulting in the decrease of scrapped items that makes the process friendly for the environment. The purpose of this proposed model is to determine the optimum value of production quantity to minimize the total costs by considering the warm-up period and set-up time. Finally, numerical experiments have been conducted to validate the efficacy of the model as well as to obtain some managerial insights.     

Equivalence of the LP relaxations of two strong formulations for the capacitated lot-sizing problem with setup times

The multi-item Capacitated Lot-Sizing Problem (CLSP) has been widely studied in the literature due to its relevance to practice, such as its application in constructing a master production schedule. The problem becomes more realistic with the incorporation of setup times since they may use up significant amounts of the available resource capacity. In this paper, we present a proof to show the linear equivalence of the Shortest Path (SP) formulation and the Transportation Problem (TP) formulation for CLSP with setup costs and times. Our proof is based on a linear transformation from TP to SP and vice versa. In our proof, we explicitly consider the case when there is no demand for an item in a period, a case that is frequently observed in the real world and in test problems in the literature. The equivalence result in this paper has an impact on the choice of model formulation and the development of solution procedures.     

Optimal investment and lot-sizing policies for improved productivity and quality

Productivity and quality are an integrated component of the operational strategy of any firm. An increase in productivity implicitly assumes an improvement in quality. The concept of dynamic process quality control and smaller lot-size production have been employed to eliminate defective items, to reduce the cycle time of a product and to improve quality and productivity. We present a mathematical model to establish the relationship between various parameters of productivity and quality. In addition, the proposed model is used to determine the optimal levels of productivity and quality parameters such as batch sizes, and investment in set-up and process control operations. The basic criterion considered for optimizing the level of such parameters is the minimization of total system cost. The proposed model relates productivity and quality to set-up reduction, queueing of batches, batch sizes, and drift rate reduction. We conclude with an example problem to illustrate the behaviour and application of the model.     

Development of a production inventory model with uncertainty and reliability considerations

This paper addresses a problem of an imperfect production system under fuzzy demand and inventory holding cost. Production process reliability is considered because of the imperfect production process. In this problem, reliability of the system in regards to producing defective and non-defective items is considered as a decision variable. The objective is to maximize the graded mean integration value (GMIV) of the expected average profit while considering revenues as well as any other relevant costs. The developed model belongs to the class of a geometric programming. We have developed a simple mathematical methodology to solve the model. Genetic algorithm and simulated annealing algorithms are also applied to solve and validate the results. A numerical example has been presented to interpret the solutions.     

Pricing and ordering decisions in a supply chain with imperfect quality items and inspection under buyback of defective items

In this article, the economic production and inventory model in a three-layer supply chain including one distributor, one manufacturer and one retailer for a single-product and general demand functions under three scenarios is developed. We assume that during the production process, both healthy and defective items are generated. As the first scenario, we develop the first model, in which the defective items are not reworked and all considered as scrape, while in the second model, we assume that the defective items are reworked and are sold as perfect item. In the second scenario, we assume that defective item can be sold with lower price than the selling price. Moreover, raw materials with imperfect quality are sent back from a distributor to outside supplier under a lower price. Determining the order quantity of the distributor and the selling prices of the distributor and the manufacturer as well as the retailer was the goal of this article such that the total profit of each member is maximised. In order to solve the models, the Stackelberg approach is employed between the members, and the concavity of the profit functions is proved using several theorems. Then, closed form solutions are derived for the decision variables and a solution algorithm is proposed to determine the optimal solutions. Finally, a numerical example is presented to illustrate the applicability of the model.     

Stochastic assessment considering process variation for impact of welding shrinkage on cost of ship production

Since ship hull blocks are constructed by assembling numerous intermediate parts, relatively small changes in their dimensions can easily accumulate to cause serious reworking during the block erection stage, as well as deteriorate the productivity in ship production industry. One of the major dimensional accuracy control activities in shipbuilding is shrinkage margin design, by which an optimal length of excess edge is assigned to plates, in order to compensate for welding shrinkage and thus minimise reworking due to shrinkage. This paper presents a stochastic methodology for a quantitative assessment procedure of welding shrinkage on production cost in shipbuilding. This reworking cost evaluation procedure includes not only the prediction of a nominal shrinkage profile, but also a statistical cost estimation process for shrinkage variation. The formulation to predict total rework cost is suggested to be a combination of nominal shrinkage profile, process variation and quality loss functions for each rework region. Also, this approach investigates the relationship between the shrinkage margin and the predicted rework cost, in case of different process variations for a 15?m?×?15?m ship block model. It also presents an optimal shrinkage margin calculation procedure for ship designers in order to minimise the impact of dimensional variation due to welding shrinkage, on productivity.