On optimal batch sizing in a multi-stage production system with rework consideration

Recently, Sarker et al. [Sarker, B.R., Jamal, A.M.M., Mondal, S., 2008. Optimal batch sizing in a multi-stage production system with rework consideration. European Journal of Operational Research 184(3) 915–929] presented an EPQ inventory model for a multi-stage manufacturing system with rework process; basically they proposed two operational inventory policies. In the paper, there are some mathematical expressions which are to be corrected. At first, this paper presents the mathematical expressions corrected and the appropriate solution to the numerical example. We also established the closed forms for the optimal total inventory cost, the conditions for which there is an optimal solution, and the mathematical expressions for determining the total additional cost for working with a non optimal solution for both policies that were not given by Sarker et al. (2008).     

An inventory system with investment to reduce yield variability and set-up cost

The set-up cost and yield variability are given and fixed in existing production/inventory models with random yields. However, in many practical situations, they can be reduced by investment in modern production technology. In this paper, we consider an inventory system with random yield in which both the set-up cost and yield variability can be reduced through capital investment. The objective is to determine the optimal capital investment and ordering policies that minimize the expected total annual costs for the system. In addition, an iterative solution procedure is presented to find the optimal order quantity and reorder point and then the optimal set-up cost and yield standard deviation. Numerical examples are given to illustrate the results obtained and assess the cost savings by adopting capital investments. Managerial implications are also included.     

Retailer’s optimal replenishment and payment policies in the EPQ model under cash discount and two-level trade credit policy

The main purpose of this paper is to investigate the retailer’s optimal cycle time and optimal payment time under the supplier’s cash discount and trade credit policy within the economic production quantity (EPQ) framework. In this paper, we assume that the retailer will provide a full trade credit to his/her good credit customers and request his/her bad credit customers pay for the items as soon as receiving them. Under this assumption, we model the retailer’s inventory system as a cost minimization problem to determine the retailer’s optimal inventory cycle time and optimal payment time under the replenishment rate is finite. Then, an algorithm is established to obtain the optimal strategy. Finally, numerical examples are given to illustrate the theoretical results and obtain some managerial phenomena.     

Optimal Production and Marketing Planning

This paper presents an integrated production, marketing and inventory model which determines the production lot size, marketing expenditure and products selling price. Our model is highly nonlinear and non-convex and cannot be solved directly. Therefore, Geometric Programming (GP) is used to locate the optimal solution of the proposed model. In our GP implementation, we use a transformed dual problem in order to reduce the model to an optimization of an unconstrained problem in a single variable and the resulting problem is solved using a simple line search. We analyze the solution in different cases in order to study the behaviour of the model and for each case, a numerical example is used to demonstrate the implementation of our analysis.     

Pricing and lot sizing for an EPQ inventory model with rework and multiple shipments

This paper deals with an economic production quantity (EPQ) inventory model with reworkable defective items when a given multi-shipment policy is used. 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. This portion becomes scrap and only the perfect finished items can be delivered to customers at the end of rework process. A profit function is derived to model the inventory problem and it is shown that the profit function is concave. Due to the complexity of the optimization problem, an algorithm is developed to determine the optimal values of manufacturing lot size and price such that the long-run average profit function is maximized. Furthermore, two special cases are identified and explained. Finally, a numerical example is given to illustrate the applicability of the proposed inventory model.     

Economic selection of process mean for single-vendor single-buyer supply chain

Process mean selection for a container-filling process is an important decision in a single-vendor single-buyer supply chain. Since the process mean determines the vendor’s conforming and yield rates, it influences the vendor–buyer decisions regarding the production lot size and number of shipments delivered from the vendor to buyer. It follows, therefore, that these decisions should be determined simultaneously in order to control the supply chain total cost. In this paper, we develop a model that integrates the single-vendor single-buyer problem with the process mean selection problem. This integrated model allows the vendor to deliver the produced lot to buyer in number of unequal-sized shipments. Moreover, every outgoing item is inspected, and each item failing to meet a lower specification limit is reprocessed. Further, in order to study the benefits of using this integrated model, two baseline cases are developed. The first of which considers a hierarchical model where the vendor determines the process mean and schedules of production and shipment separately. This hierarchical model is used to show the impact of integrating the process mean selection with production/inventory decisions. The other baseline case is studied in the sensitivity analysis where the optimal solution for a given process is compared to the optimal solution when the variation in the process output is negligible. The integrated model is expected to lead to reduction in reprocessing cost, minimal loss to customer due to the deviation from the optimum target value, and consequently, providing better products at reduced cost for customers. Also, a solution procedure is devised to find the optimal solution for the proposed model and sensitivity analysis is conducted to investigate the effect of the model key parameters on the optimal solution.     

An elaborative unit cost structure-based fuzzy economic production quantity model

The purpose of this paper is to investigate and propose a fuzzy extended economic production quantity model based on an elaboratively modeled unit cost structure. This unit cost structure consists of the various lot-size correlative components such as on-line setups, off-line setups, initial production defectives, direct material, labor, and depreciation in addition to lot-size non-correlative items. Thus, the unit cost is correlatively modeled to the production quantity. Therefore, the modeling or the annual total cost function developed consists of not only annual inventory and setup costs but also production cost. Moreover, via the concept of fuzzy blurred optimal argument and the vertex method of the α-cut fuzzy arithmetic (or fuzzy interval analysis), two solution approaches are proposed: (1) a fuzzy EPQ and (2) a compromised crisp EPQ in the fuzzy sense. An optimization procedure, which can simultaneously determine the α-cut-vertex combination of fuzzy parameters and the optimizing decision variable value, is also proposed. The sensitivity model for the fuzzy total cost and thus EPQ to the various cost factors is provided. Finally, a numerical example with the original data collected from a firm demonstrates the usefulness of the new model.     

EPQ with Process Capability and Quality Assurance Considerations

The classical economic production quantity (EPQ) model assumes that items produced are of perfect quality and that the unit cost of production is fixed. However, in realistic situations, product quality is never perfect but is directly affected by the production processes and the quality assurance programme. In addition, the unit production cost is not fixed but increases with quality assurance expenses. We present an EPQ model with imperfect production processes and quality-dependent unit production cost. After discussion of the procedure for determining the optimal solution, a sensitivity analysis of the impacts of the cost parameters on the optimal solution is provided. Finally, the problems associated with cost estimation are addressed.     

An economic production quantity model with a positive resetup point under random demand

In this paper, an extended economic production quantity (EPQ) model is investigated, where demand follows a random process. This study is motivated by an industrial case for precision machine assembly in the machinery industry. Both a positive resetup point s and a fixed lot size Q are implemented in this production control policy. To cope with random demand, a resetup point, i.e., the lowest inventory level to start the production, is adapted to minimize stock shortage during the replenishment cycle. The considered cost includes setup cost, inventory carrying cost, and shortage cost, where shortage may occur at the production stage and/or at the end of one replenishment cycle. Under some mild conditions, the expected cost per unit time can be shown to be convex with respect to decision parameters s and Q. Further computational study has demonstrated that the proposed model outperforms the classical EPQ when demand is random. In particular, a positive resetup point contributes to a significant portion of this cost savings when compared with that in the classical lot sizing policy.     

Optimal pricing and production in an inventory model

This paper deals with the problem of simultaneously determining the optimal price policy and production rate over a given planning horizon. For nonlinear demand functions and convex inventory and shortage cost functions the optimal solution paths are derived by using optimal control theory. The treatment of linear nonsmooth cost functions requires the use of a generalized maximum principle. The solution method is a phase portrait analysis providing insight into the optimal pricing and production policies as well as the resulting inventory paths. Moreover, it is shown that in the case of nonsmooth piecewise linear cost functions the equilibrium is approached within finite time although the model is nonlinear in the control variables. Finally it is illustrated that exogenous fluctuations in the demand rate (seasonal demand pattern) amount to cyclical optimal solutions.     

A new lot sizing and scheduling heuristic for multi-site biopharmaceutical production

Biopharmaceutical manufacturing requires high investments and long-term production planning. For large biopharmaceutical companies, planning typically involves multiple products and several production facilities. Production is usually done in batches with a substantial set-up cost and time for switching between products. The goal is to satisfy demand while minimising manufacturing, set-up and inventory costs. The resulting production planning problem is thus a variant of the capacitated lot-sizing and scheduling problem, and a complex combinatorial optimisation problem. Inspired by genetic algorithm approaches to job shop scheduling, this paper proposes a tailored construction heuristic that schedules demands of multiple products sequentially across several facilities to build a multi-year production plan (solution). The sequence in which the construction heuristic schedules the different demands is optimised by a genetic algorithm. We demonstrate the effectiveness of the approach on a biopharmaceutical lot sizing problem and compare it with a mathematical programming model from the literature. We show that the genetic algorithm can outperform the mathematical programming model for certain scenarios because the discretisation of time in mathematical programming artificially restricts the solution space.     

An economic production quantity inventory model involving fuzzy demand rate and fuzzy deterioration rate

Generally, in deriving the solution of economic production quantity (EPQ) inventory model, we consider the demand rate and deterioration rate as constant quantity. But in case of real life problems, the demand rate and deterioration rate are not actually constant but slightly disturbed from their original crisp value. The motivation of this paper is to consider a more realistic EPQ inventory model with finite production rate, fuzzy demand rate and fuzzy deterioration rate. The effect of the loss in production quantity due to faulty/old machine have also been taken into consideration. The methodology to obtain the optimum value of the fuzzy total cost is derived and a numerical example is used to illustrate the computation procedure. A sensitivity analysis is also carried out to get the sensitiveness of the tolarance of different input parameters.     

Effects of imperfect products on lot sizing with work in process inventory

The economic production quantity (EPQ) is one of the most widely known inventory control models that can be regarded as the generalized form of the Economic Order Quantity. However, the model is built on an unrealistic assumption that all the produced items need to be of perfect quality. Also, the introduction of work in process, WIP, as part of the inventory has been of lesser concern in developing inventory models. This paper attempts to develop the economic production quantity considering work in process inventory and manufacturing imperfect products that may be either reworkable or non-reworkable. The non-reworkable imperfect products are sold at a reduced price. This paper introduces a new model for this problem.     

Optimal selection of process mean for a stochastic inventory model

It is very common to assume deterministic demand in the literature of integrated targeting – inventory models. However, if variability in demand is high, there may be significant disruptions from using the deterministic solution in probabilistic environment. Thus, the model would not be applicable to real world situations and adjustment must be made. The purpose of this paper is to develop a model for integrated targeting – inventory problem when the demand is a random variable. In particular, the proposed model jointly determines the optimal process mean, lot size and reorder point in (Q, R) continuous review model. In order to investigate the effect of uncertainty in demand, the proposed model is compared with three baseline cases. The first of which considers a hierarchical model where the producer determines the process mean and lot-sizing decisions separately. This hierarchical model is used to show the effect of integrating the process targeting with production/inventory decisions. Another baseline case is the deterministic demand case which is used to show the effect of variation in demand on the optimal solution. The last baseline case is for the situation where the variation in the filling amount is negligible. This case demonstrates the sensitivity of the total cost with respect to the variation in the process output. Also, a procedure is developed to determine the optimal solution for the proposed models. Empirical results show that ignoring randomness in the demand pattern leads to underestimating the expected total cost. Moreover, the results indicate that performance of a process can be improved significantly by reducing its variation.     

Solving the vendor–buyer integrated inventory system with arithmetic–geometric inequality

In the past, economic order quantity (EOQ) and economic production quantity (EPQ) were treated independently from the viewpoints of the buyer or the vendor. In most cases, the optimal solution for one player was non-optimal to the other player. In today’s competitive markets, close cooperation between the vendor and the buyer is necessary to reduce the joint inventory cost and the response time of the vendor–buyer system. The successful experiences of National Semiconductor, Wal-Mart, and Procter and Gamble have demonstrated that integrating the supply chain has significantly influenced the company’s performance and market share (Simchi-Levi et al. (2000) [1]). Recently, Yang et al. (2007) [2] presented an inventory model to determine the economic lot size for both the vendor and buyer, and the number of deliveries in an integrated two stage supply chain. In this paper, we present an alternative approach to determine the global optimal inventory policy for the vendor–buyer integrated system using arithmetic–geometric inequality.     

A Heuristic for a Resource-capacitated Multi-stage Lot-sizing Problem with Lead Times

In this paper we propose a heuristic for the resource-capacitated multi-stage lot-sizing problem with general product structures, set-up costs and resource usage, work-in-process inventory costs and lead times. To facilitate the functioning of the heuristic, we use the formulation of the problem based on Echelon Stock in a rolling horizon scheme. The heuristic first obtains a reasonable solution to the corresponding uncapacitated problem and then tries to attain capacity feasibility by shifting production backwards in time. The concept of echelon stock makes the task of checking the inventory feasibility of proposed shifts easier than would be the case with conventional installation stock. The heuristic is first tested computationally for problems with a five-component product structure over a 12 period planning horizon for which optimal solutions were available and for which optimality precision guarantees were also obtained via Lagrangian Relaxation. The heuristic's performance is also explored with two different 40-component product structures, with high and low set-up costs, and is compared with the Lagrangian precision guarantees.     

The stochastic economic lot sizing problem for non-stop multi-grade production with sequence-restricted setup changeovers

We study a variant of the stochastic economic lot scheduling problem (SELSP) encountered in process industries, in which a single production facility must produce several different grades of a family of products to meet random stationary demand for each grade from a common finished-goods (FG) inventory buffer that has limited storage capacity. When the facility is set up to produce a particular grade, the only allowable changeovers are from that grade to the next lower or higher grade. Raw material is always available, and the production facility produces continuously at a constant rate even during changeover transitions. All changeover times are constant and equal to each other, and demand that cannot be satisfied directly from inventory is lost. There is a changeover cost per changeover occasion, a spill-over cost per unit of product in excess whenever there is not enough space in the FG buffer to store the produced grade, and a lost-sales cost per unit short whenever there is not enough FG inventory to satisfy the demand. We model the SELSP as a discrete-time Markov decision process (MDP), where in each time period the decision is whether to initiate a changeover to a neighboring grade or keep the set up of the production facility unchanged, based on the current state of the system, which is defined by the current set up of the facility and the FG inventory levels of all the grades. The goal is to minimize the (long-run) expected average cost per period. For problems with more than three grades, we develop a heuristic solution procedure which is based on decomposing the original multi-grade problem into several 3-grade MDP sub-problems, numerically solving each sub-problem using value iteration, and constructing the final policy for the original problem by combining parts of the optimal policies of the sub-problems. We present numerical results for problem examples with 2–5 grades. For the 2- and 3-grade examples, we numerically solve the exact MDP problem using value iteration to obtain insights into the structure of the optimal changeover policy. For the 4- and 5-grade examples, we compare the performance of the decomposition-based heuristic (DBH) solution procedure against that obtained by numerically solving the exact problem. We also compare the performance of the DBH method against the performance of three simpler parameterized heuristics. Finally, we compare the performance of the DBH and the exact solution procedures for the case where the FG inventory storage consists of a number of separate general-purpose silos capable of storing any grade as long as it is not mixed with any other grade.     

The derivation of EOQ/EPQ inventory models with two backorders costs using analytic geometry and algebra

The EOQ model will have a century of its discovery in two years, and recently still, many researchers have been using alternative approaches to model and solve inventory systems. The EOQ/EPQ models have been developed using different optimization methods. However, in many of the works that deal with the EOQ/EPQ with backorders only linear backorders cost is considered. This paper proposes another easy method which uses basic concepts of analytic geometric and algebra. The proposed method finds the optimal lot size and backorders level considering both linear and fixed backorders costs. Additionally, this paper presents a review of the different optimization methods utilized in inventory theory.     

Economic lot sizes and vehicle scheduling

In this paper a relationship between the vehicle scheduling problem and the dynamic lot size problem is considered. For the latter problem we assume that order quantities for different products can be determined separately. Demand is known over our n-period production planning horizon. For a certain product our task is to decide for each period if it should be produced or not. If it is produced, what is its economic lot size? Our aim here is to minimize the combined set-up and inventory holding costs. The optimal solution of this problem is given by the well-known Wagner-Whitin dynamic lot size algorithm. Also many heuristics for solving this problem have been presented. In this article we point out the analogy of the dynamic lot size problem to a certain vehicle scheduling problem. For solving vehicle scheduling problems the heuristic algorithm developed by Clark and Wright in very often used. Applying this algorithm to the equivalent vehicle scheduling problem we obtain by analogy a simple heuristic algorithm for the dynamic lot size problem. Numerical results indicate that computation time is reduced by about 50% compared to the Wagner-Whitin algorithm. The average cost appears to be approximately 0.8% higher than optimum.     

部分延迟订购的易变质品联合定价与生产策略

构建了一个需求同时依赖于销售价格和库存水平,生产率和变质率均为常数,允许缺货且缺货量部分延迟订购的易变质品联合定价与生产控制模型。首先证明了在销售价格给定的情况下,系统的总利润函数是关于生产计划的严格凹函数,平均利润函数是严格的伪凹函数,即存在唯一的最优解,并给出其充分条件。接着给出问题的一个数值求解算法。最后通过算例,展示了模型及相关算法的应用,并对相关参数进行了灵敏度分析,结果显示:当产品的生产成本、缺货成本和机会成本增加时,系统的平均利润将下降;生产成本和延迟订购阻力系数对最优定价和生产策略以及平均利润的影响较大。