Optimal Production Lot Size Research Articles

Imperfect production system with rework and scrap at a single stage manufacturing system and integrates cost reduction delivery policy

This paper deals with a production inventory model with reworkable defective items and integrates a cost reduction delivery policy. In this work, it is assumed that in each cycle the rework process of all defective items starts when the regular production process finishes. After the rework process, a portion of reworked items fails and this portion becomes scrap. In this paper, two models are developed, in the first model, the perfect finished items can be delivered to customers at the end of rework process and in the second model, integrates cost reduction delivery policy and finished items can only be delivered to customers at a fixed interval of time during production down time with the purpose of reducing holding cost. The objective is to derive the optimal production lot size, number of deliveries and the finished items are delivered by request to customers at a fixed interval of time that minimises total costs. The relevant model is built, solved and necessary and sufficient conditions for a unique and global optimal solution are derived. An illustrative example is provided and numerically verified. The validation of result in this model was coded in Microsoft Visual Basic 6.0.

Random sales price-sensitive stochastic demand

PurposeThis research work introduces an imperfect production system where the demand is assumed to be stochastic and it is influenced by random selling price. The shift time from an “in-control” state to an “out-of-control” state is exponentially distributed. The accumulated inventory contains both perfect and defective items which are all sold with a free repair warranty (FRW) offer. Complete back ordering of shortages are taken into account. The purpose of this paper is to determine the optimal selling price and hence the optimal production lot size such that the expected profit is maximized.Design/methodology/approachThe general model is discussed separately for both types of uniformly distributed selling price-sensitive demand pattern: additive type and multiplicative type. Numerical examples and graphical representations of the optimal solutions are provided to illustrate the models.FindingsThis paper helps the manager to manage future situations and it may be considered as a base work for the researchers to work in this direction.Research limitations/implicationsThe main limitation of this model is to consider a single item for a single channel system. There are many correlated issues that need to be further investigated. The future study in this direction may include the consideration of multi-items, diverse demand pattern with different types of price distributions.Originality/valueIn the production inventory literature, plenty of articles are available considering imperfect production but none of them have considered selling price-sensitive stochastic demand where the sales price is random in character under an FRW offer.

The effects of inspection delay and restoration cost on the optimal inspection and production policy

Economic production lot size models with a process inspection delay have been studied over several decades in the literature. When an imperfect process is assumed to have a given probability to run from an in-control state to an out-of-control state each time an item is produced, an inspection schedule is arranged to monitor the process reliability; however, a time delay to obtain the inspection results exists. This study further includes process restoration cost in the inspection/production model to study its effects on the optimal production lot size and the economic inspection lot under the consideration of minimizing the approximated total cost per week. For the case of multiple inspections, when the restoration cost is considered, the resulting the economic inspection lot may depend on the production lot size, especially when the fixed part of the restoration cost is large.

Production inventory models for deteiorative items with three levels of production and shortages

In this paper, three level production inventory models for deteriorative items are considered under the variation in production rate. Namely, it is possible that production started at one rate, after some time, switches to another rate. Such a situation is desirable in the sense that by starting at a low rate of production, a large quantum stock of manufacturing items at the initial stage are avoided, leading to reduction in the holding cost. The variation in production rate results in consumer satisfaction and potential profit. Two levels of production inventory models are developed, and the optimum lot size quantity and total cost are derived when the production inventory model without shortages is studied first and a production inventory model with shortages next. An optimal production lot size, which minimizes the total cost, is developed. The optimal solution is derived and a numerical example is provided. The validation of the results in this model was coded in Microsoft Visual Basic 6.0.

AN ARITHMETIC-GEOMETRIC MEAN INEQUALITY APPROACH FOR DETERMINING THE OPTIMAL PRODUCTION LOT SIZE WITH BACKLOGGING AND IMPERFECT REWORK PROCESS

Some classical studies on economic production quantity (EPQ) models with imperfect production processes have focused on determining the optimal production lot size. However, these models neglect the fact that the total production-inventory costs can be reduced by reworking imperfect items for a relatively small repair and holding cost. To account for the above phenomenon, we take the out of stock and rework into account and establish an EPQ model with imperfect production processes, failure in repair and complete backlogging. Furthermore, we assume that the holding cost of imperfect items is distinguished from that of perfect ones, as well as, the costs of repair, disposal, and shortage are all included in the proposed model. In addition, without taking complex differential calculus to determine the optimal production lot size and backorder level, we employ an arithmetic-geometric mean inequality method to determine the optimal solutions. Finally, numerical examples and sensitivity analysis are analyzed to illustrate the validity of the proposed model. Some managerial insights are obtained from the numerical examples.

An inventory model for controllable probabilistic deterioration rate under shortages

This article presents a production inventory model for three level production rates with characterizing the preservation technology for deteriorating items to reduce the deterioration rate. The deterioration of the product is considered as probabilistic to make the research a more realistic one. In this paper, a production inventory model with deteriorative items in which three different rates of productions are considered. Two models are considered; the production inventory model without shortages is studied first and with shortages is investigated next. In this paper, shortages are allowed to occur and demand during the stock out period is completely backlogged. Few propositions have been developed and the necessary and sufficient conditions of the existence and uniqueness of the global optimal solutions are suggested. A suitable mathematical model is developed, and the optimal production lot size, which minimizes the total average cost, is derived. The global optimal solution is derived, and an illustrative example is provided and numerically verified. Sensitivity analysis of the optimal solution with respect to the key parameters of the model has been carried out and the implications are discussed. The analysis of the model shows that the solution of the model is quite stable and can be applied for optimizing the total inventory cost of deteriorating items inventory for the business enterprises.

An economic production quantity inventory model with backorders considering the raw material costs

The classical Economic Production Quantity (EPQ) inventory model does not consider ordering and holding costs of raw materials. In this direction, this paper considers the ordering and holding costs for both raw materials and nished product. Basically, four EPQ inventory models are developed from an easy perspective that has not been considered before. It was found that the ordering and holding costs of raw materials must be taken into account, because they signi cantly impact on the optimal production lot size of the nished product in both EPQ without shortages and EPQ with shortages inventory models. Furthermore, an EPQ inventory model that determines the optimal lot size for a product that requires more than one raw material, and an EPQ inventory model that obtains the optimal batch size for multiple products, which are manufactured with multiple raw materials, are proposed. Numerical examples are presented in order to illustrate the use of the proposed inventory models.

Production inventory model with deteriorating items with constant, linear and quadratic holding cost - a comparative study

In the classical inventory model, the holding cost was assumed to be constant. However, in reality holding cost may not always be constant. The holding cost may be time dependent. In this paper, a production inventory model with deteriorating items with constant, linear and quadratic holding cost is considered and also a comparative study is carried out between constant holding cost and time dependent (linear and quadratic) holding costs. Three models are developed. In three models, the optimum time and total cost are derived when the holding cost is: 1) constant; 2) linear; 3) quadratic. A mathematical model is developed for each model and the optimal production lot size which minimises the total cost is derived. The optimal solution is derived and an illustrative example is provided. The validation of result in this model was coded in Microsoft Visual Basic 6.0.

Production inventory model with deteriorating items with constant, linear and quadratic holding cost - a comparative study

In the classical inventory model, the holding cost was assumed to be constant. However, in reality holding cost may not always be constant. The holding cost may be time dependent. In this paper, a production inventory model with deteriorating items with constant, linear and quadratic holding cost is considered and also a comparative study is carried out between constant holding cost and time dependent (linear and quadratic) holding costs. Three models are developed. In three models, the optimum time and total cost are derived when the holding cost is: 1) constant; 2) linear; 3) quadratic. A mathematical model is developed for each model and the optimal production lot size which minimises the total cost is derived. The optimal solution is derived and an illustrative example is provided. The validation of result in this model was coded in Microsoft Visual Basic 6.0.

An EPQ (economic production lot size) model for product life cycle (maturity stage) of deteriorating items with increasing demand rate and shortages

A product life cycle is the life span of a product which the period begins with initial product specification and ends with the withdrawal from the market of both the product and its support. A product life cycle can be divided into several stages characterised by the revenue generated by the product. The purpose of this paper is to investigate the optimal production lot size and total cost during the product life cycle which consists of introduction, growth, maturity and decline stages. The defective rate is considered as a variable of known proportions. The relevant model is built and it is also solved. Necessary and sufficient conditions for a unique and optimal solution are derived. A numerical example is provided using LINGO software.

Production lot sizing with quality screening and rework

Most production systems produce items which are of imperfect quality. Handling of the defective items varies by industry and product types. For example, defective items may be sold at discount in the apparel industry, or reworked in the automobile industry where the final product is very expensive. For simplicity, a common assumption in the literature is that items are lumped into two groups, non-defective and to-be-reworked products, ignoring the inspection time needed to identify the repairable items. Our paper explicitly integrates the inspection time into the economic production model with rework, and demonstrates the significant effect that the inspection time has on the results. We consider a manufacturing process with random supply and a screening process conducted during and at the end of production. We analyze two scenarios for dealing with the defective items produced: selling at a discount, and reworking. For each scenario, the demand during production is met using non-defective items only. The expected profit functions are developed using the renewal reward theory, and closed form expressions for the optimal production lot size are derived. Numerical analysis is performed to study the sensitivity of the expected profit and optimal lot size to various system parameters.

Setting optimal production lot sizes and planned lead times in a job shop

In this research, we model a job shop that produces a set of discrete parts in a make-to-stock setting. The intent of the research is to develop a planning model to determine the optimal tactical policies that minimise the relevant manufacturing costs subject to workload variability and capacity limits. We consider two tactical decisions, namely the production lot size for each part and the planned lead time for each work station. We model the relevant manufacturing costs, entailing production overtime costs and inventory-related costs, as functions of these tactical decisions. We formulate a non-linear optimisation model and implement it in the Excel spreadsheet. We test the model with actual factory data from our research sponsor. The results are consistent with our intuition and demonstrate the potential value from jointly optimising over these tactical policies.

Two-echelon production-inventory system with fuzzy production rate and promotional effort dependent demand

Intelligent manufacturing design of a complex production-inventory system becomes a key issue for the organization of responsiveness to uncertainties. This paper addresses a two-echelon production-inventory model for a non-repairable product where the system consists of single manufacturer and single retailer. The manufacturer procures raw material (which also contains imperfect raw materials) from an outside supplier then proceeds to convert perfect-quality raw material as a finished product, and finally delivers to the retailer. In this study we assume that the demand is sensitive to promotional efforts/sales teams' initiatives and the production rate is uncertain but possible to describe with a triangular fuzzy number. Then we use the signed distance method to defuzzify the fuzzy joint total cost and an analytical method is employed to achieve the optimal solutions so that the total costs of both manufacturer and retailer are minimized. An efficient algorithm is developed to design an intelligent manufacturing strategy such as optimal production lot-size, backlogging and the initiatives of sales teams. A numerical example and sensitivity analysis are given to demonstrate the application of the proposed model.

Production inventory model with defective items and integrates cost reduction delivery policy

This paper integrates a cost reduction delivery policy and quality assurance into a production inventory model. During regular production time, the defective items are produced randomly. All defective items are eliminated/removed during regular production and it is assumed that finished items can only be delivered to customers if whole lot is quality assured at the end of production time. The integrated cost reduction delivery policy is one good aspect to be added to the EPQ. The objectives of this research are to derive the optimum quantity and to investigate the effect of reduction of cost delivery policy in the inventory model and to develop mathematical models under the behaviour of cost reduction delivery policy. A suitable mathematical model is developed and the optimal production lot size which minimises the total cost is derived. The global optimal solution is derived and an illustrative example is provided and numerically verified. The validation of the result in this model was coded in Microsoft Visual Basic 6.0.

Production inventory model for two-level production with deteriorative items and shortages

Production inventory model plays a dominant role in production scheduling and planning. The EPQ model is commonly used by practitioners in the fields of production and inventory management to assist them in making decision on optimum production and total cost. The total cost of production is dependent on production rate, demand rate, and rate of deteriorative items. In this paper, a production inventory model with deteriorative items in which two different rates of productions are considered, and it is possible that production started at one rate, and after some time, it may be switched over another rate such a situation is desirable in the sense that by starting at a low rate of production, a large quantum stock of manufacturing items at the initial stage is avoided, leading to reduction in the holding cost. The variation in production rate provides a way resulting consumer satisfaction and earning potential profit. Two models are considered; the production inventory model without shortages is studied first and with shortages is investigated next. A suitable mathematical model is developed, and the optimal production lot size which minimizes the total cost is derived. The global optimal solution is derived, and an illustrative example is provided and numerically verified. The validation of result in this model was coded in Microsoft Visual Basic 6.0.

Economic production quantity (EPQ) models under an imperfect production process with shortages backordered

In this paper, we develop economic production quantity (EPQ) models to determine the optimal production lot size and backorder quantity for a manufacturer under an imperfect production process. The imperfect production process is characterised by the fraction of defective items at the time of production γ. The paper considers different cases of the EPQ model depending on (1) whether γ is known with certainty or is a random variable, and (2) whether imperfect items are drawn from inventory (a) as they are detected, (b) at the end of each production period or (c) at the end of each production cycle. Straightforward convexity results are shown and closed-form solutions are provided for the optimal order and backorder quantities for each of the cases we considered. We provide two numerical examples: one in which the defective probability follows a uniform distribution and the second which we assume follows a beta distribution, to illustrate the effects of yield variability and timing of the withdrawal of defectives on the optimal solutions. We obtain similar results for both numerical examples, which show that both the yield variability and the withdrawal timing are not critical factors.

Development of a model that determines the optimal production lot size using nonlinear programming and proposal of resolution with a spreadsheet

To facilitate the cost reduction effort that the global crisis has led in companies around the world, in this paper we propose a model to streamline production, a key area of the supply chain. The proposed model determines the optimal lot size considering activities of different areas of the supply chain, many of which were not previously considered. In order to facilitate the implementation of the proposed model in several companies, including SMEs, we show how to solve this model with a conventional spreadsheet.

An inspection model of products and their input materials

Consider an input material with a nonconforming probability that is used for the production of a product through a deteriorating process, which is initially maintained in an in-control state, but can randomly shift into an out-of-control state after some products are produced. To economically control the total quality-related cost for a given production lot, an optimal inspection policy for the products and their input materials is proposed. Furthermore, under the optimal inspection policy, a decision on the optimal production lot size is also incorporated. The unique properties and conditions that give the optimal production lot size are explored. Numerical examples are given to illustrate our proposed inspection and production model. The effects of the model's parameter values, such as inspection error rate, unit inspection cost for the input material, unit inspection cost for the product, etc., on the optimal solution are investigated. Finally, some useful insights that should help industrial operations are presented.

Optimal production lot size and reorder point of a two-stage supply chain while random demand is sensitive with sales teams' initiatives

The paper develops a production-inventory model of a two-stage supply chain consisting of one manufacturer and one retailer to study production lot size/order quantity, reorder point sales teams’ initiatives where demand of the end customers is dependent on random variable and sales teams’ initiatives simultaneously. The manufacturer produces the order quantity of the retailer at one lot in which the procurement cost per unit quantity follows a realistic convex function of production lot size. In the chain, the cost of sales team's initiatives/promotion efforts and wholesale price of the manufacturer are negotiated at the points such that their optimum profits reached nearer to their target profits. This study suggests to the management of firms to determine the optimal order quantity/production quantity, reorder point and sales teams’ initiatives/promotional effort in order to achieve their maximum profits. An analytical method is applied to determine the optimal values of the decision variables. Finally, numerical examples with its graphical presentation and sensitivity analysis of the key parameters are presented to illustrate more insights of the model.

Production inventory model for two levels production with defective items and incorporating multi-delivery policy

Production inventory model plays a dominant role in production scheduling and planning. The EPQ model is commonly used by practitioners in the fields of production and inventory management to assist them in making a decision on optimum production and total cost. The total cost of production is dependent on production rate, demand rate and rate of defective items. This paper incorporates a multi-delivery policy into a production inventory model with defective items in which two different rates of production are considered and it is possible that production started at one rate and after some time it may be switched over to another rate where such a situation is desirable in the sense that by starting at a low rate of production, a large quantum stock of manufacturing items at the initial stage is avoided, with the purpose of reducing the holding cost. A suitable mathematical model is developed and the optimal production lot size, which minimises the total cost is derived. The global optimal solution is derived and an illustrative example is provided and numerically verified. The validation of the result in this model was coded in Microsoft Visual Basic 6.0.