Multiproduct production/inventory control under random demands

Studies the optimal production/inventory control policy for a single machine multiproduct production system. The machine produces to fill the end-product inventory stock and the demand is satisfied from the inventory when available; unsatisfied demand is backlogged until the product becomes available as the result of production. For each product, the demand follows a Poisson process and the unit processing time is known. When the machine switches production from one product to another, it incurs a set-up time and a set-up cost. The relevant costs include the set-up cost, a cost per unit time while the machine is running, and linear costs for inventory and backlogging. This problem is modeled as a semi-Markov decision process using the criterion of minimizing expected total cost with discounting over an infinite horizon. Procedures for computing near-optimal policies and their error bounds are developed. The error bound given by the authors' procedure is shown to be much tighter than the one given by the “norm-based” approach. Computational test results are presented to show the structure of the near-optimal policy and how its accuracy is affected by the system characteristics such as capacity utilization and set-up time     

A mathematical model on EPQ for stochastic demand in an imperfect production system

In the paper, we develop an EPQ (economic production quantity) inventory model to determine the optimal buffer inventory for stochastic demand in the market during preventive maintenance or repair of a manufacturing facility with an EPQ (economic production quantity) model in an imperfect production system. Preventive maintenance, an essential element of the just-in-time structure, may cause shortage which is reduced by buffer inventory. The products are sold with the free minimal repair warranty (FRW) policy. The production system may undergo “out-of-control” state from “in-control” state, after a certain time that follows a probability density function. The defective (non-conforming) items in “in-control” or “out-of-control” state are reworked at a cost just after the regular production time. Finally, an expected cost function regarding the inventory cost, unit production cost, preventive maintenance cost and shortage cost is minimized analytically. We develop another case where the buffer inventory as well as the production rate are decision variables and the expected unit cost considering the above cost functions is optimized also. The numerical examples are provided to illustrate the behaviour and application of the model. Sensitivity analysis of the model with respect to key parameters of the system is carried out.     

Optimal (s,S) policies with delivery time guarantees for manufacturing systems with early set-up

We consider optimal (s, S) policies with delivery time guarantees for production planning in one-machine manufacturing systems with early set-up. The machine produces one type of product and delivery time guarantee is offered to the customers for each unit of ordered product. The inter-arrival time of the demand and the processing time for one unit of product are assumed to be exponentially distributed. In a (s, S) policy, the machine will shut down when an inventory level of S is attained and once the inventory level drops to s, the machine will re-start. A set-up time is required for the machine. We model the set-up by the exponential distribution. We obtained an analytical form of the steady state probability distribution for the inventory levels derived. The average profit of the system can be written in terms of this probability distribution. Hence the optimal (s, S) policy can be obtained by varying different possible values of s and S.     

Simultaneous determination of the optimal production–inventory and product inspection policies for a deteriorating production system

In this paper, we attempt to find a method for the optimization of production–inventory and product inspection policies for deteriorating production systems. Taking advantage of the nature of a deteriorating production system, a strategy would be not to inspect the first s items of the batch. Therefore, an inspection policy which disregards the first s (DTF-s) items of the batch is proposed. Under the DTF-s policy, we do not inspect the first s produced items but inspect only those items from the (s+1)th till the end of the production run. The objective of this study was the joint determination of the production lot size and the inspection policy s, resulting in a minimization of the expected average cost per unit time. Based on this model, the underlying conditions necessary for the existence of an optimal policy are given. Two commonly used inspection strategies, no inspection and full inspection are discussed. Under both inspection strategies, an optimal production–inventory lot is bounded by the traditional economic quantity. The case of full inspection is shown to be an extension of previously reported results. The option of investing in the process of quality improvement is also discussed. Finally, numerical examples are given to illustrate the method and its advantages in the conclusion.Scope and purposeThis paper considers the relationship between production, inventory and inspection in a deteriorating production system which may transit from the “in-control” state to the “out-of-control” state after a period of operation. Once the transition to the “out-of-control” state has occurred, it is assumed that some percentage of the items produced are defective or of substandard quality. However, in many cases, defects in a defective item can only be identified by an inspection process which carries an inspection cost. Those inspected items which are found to be defective are reworked at some cost before being shipped. On the other hand, defective items which are not inspected will be passed to the customer, incurring a much larger warranty cost. In order to operate such a system economically, tradeoffs among production setup, inventory, inspection and defective cost must be analyzed. Deterioration of the production system is an inherent process in all manufacturing industries. An understanding of the relationship among production, inventory and inspection for such systems will help managers to maintain efficient and economic control of operations.     

On continuous review stochastic (s, S) inventory systems with ordering delays

In this paper a continuous review (s, S) inventory system with ordering delays is considered. Demands for the item arrive according to a Poisson process. When the inventory level drops to s, the order is triggered. However, due to an unavoidable delay, the order is actually placed after a random amount of time. Once the order is placed, it is instantaneously delivered bringing the inventory level back to S. Under a cost structure which includes a setup cost, a holding cost and a penalty cost, an expression for the expected cost per unit time for given control values is obtained. Then some properties of the cost functions are developed to characterize the optimal policy. Based on these properties, an efficient search procedure to find the optimal (s, S) policy is presented.     

价格波动下的生产-库存控制研究

研究原材料价格波动下多级生产-库存系统的控制问题.所有的原材料价格、半成品加工成本、成品的生产成本、库存费用率和产品的需求率都随时间变动,为此,分析了最优采购、加工、生产决策的必要和充分条件,得到了在某些假设条件下的最优生产-库存策略为JIT(Just-in-time)采购、加工、生产策略,或者为在最开始阶段以最大能力进行采购、加工、生产活动的Bang-Bang策略.     

Optimal inventory replenishment policy for the EPQ model under trade credit derived without derivatives

Within the economic production quantity (EPQ) framework, the main purpose of this article is to deal with Chung and Huang's model (K.J. Chung, and Y.F. Huang,“The optimal cycle time for EPQ inventory model under permissible delay in payments,” Int. J. Prod. Econ., 84, pp. 307–318, 2003) and extend Chung and Huang's model (2003) by considering the unit selling price higher than the unit purchasing cost using the algebraic method to determine the optimal inventory replenishment policy for the retailer under trade credit. This article provides this algebraic approach which could be used easily to introduce the basic inventory theories to younger students who lack the knowledge of calculus. In addition, we develop an easy-to-use procedure to find the optimal inventory replenishment policy for the retailer in the extended model developed in this article. Finally, numerical examples are given to illustrate the result obtained in our extended model.     

考虑单位车辆运力的运输-库存随机需求决策模型

针对提前期和客户需求不确定的生产、运输和库存协调控制问题, 基于整车直达运输策略, 从优化系统物流成本角度, 建立了在决策中明确体现单位车辆运力影响的运输-库存系统协调模型, 设计了求解模型的优化搜索机制并从数学上证明了其有效性. 最后, 进一步对单位车辆运力进行敏感性分析并得出以下结论: 在其他条件不变时, 单位车辆的运力会影响系统决策结果, 该运力既不是越大越好, 也不是越小越好, 而是某个适中值.

     

Production control of a manufacturing system with multiple machinestates

The production control of a single-product manufacturing system with arbitrary number of machine states (failure modes) is discussed. The objective is to find a production policy that would meet the demand for the product with minimum average inventory or backlog cost. The optimal production policy has a special structure and is called a hedging-point policy. If the hedging points are known, the optimal production rate is readily specified. Assuming a set of tentative hedging points, the simple structure of the optimal policy is utilized to find the steady-state probability distribution of the surplus (inventory or backlog). Once this function is determined, the average surplus cost is easily calculated in terms of the values of the hedging points. The average cost is then minimized to find the optimum hedging points     

A general and optimal approach for three inventory models with a linear trend in demand

This study provides a general and simple algorithm to obtain an optimal solution for three inventory models with a replenishment batching policy, production batching policy, and an integrated replenishment/production batching policy in a manufacturing system, under a finite time horizon and linear trend in demand. This study determines the replenishment or production schedule with one general equation for these three problems and provides fully theoretical proofs for relaxing some of the conjectures in previous studies. A general and explicit procedure to derive the optimal solution for these three inventory models is presented, while considering both linearly increasing and decreasing demands. In addition, demonstrations of applicability are performed.     

A numerical solution for a two-stage production and inventory system with random demand arrivals

This paper is about the study of a production lot sizing problem consisting of customers, one retailer, and one manufacturer. Demand from customers arrives randomly at a retailer one unit at a time. The retailer replenishes inventory from the manufacturer upon receiving a customer's order after its inventory depleted to zero. The manufacturer's production rate is assumed to be a finite constant. A production cycle starts when the manufacturer's inventory falls to or below zero and stops when its on-hand inventory reaches its optimal level. During the uptime in a production cycle, inventory is being built while randomly arriving orders from retailer are being fulfilled. The order arrival times from customers are independently and identically distributed, hence the inventory processes at both the manufacturer and the retailer become a renewal process that is difficult to solve analytically for a general distribution of order arrival time. Therefore, a numerical approach is used in developing a search procedure to obtain the optimal solution to the problem. Employing such a numerical approach, we also investigate how optimal solutions in different cases will change over the spectrum of some key parameters of the problem.     

An optimal batch size for a production system operating under periodic delivery policy

A manufacturing system that procures raw materials from suppliers in a lot and processes them into a finished product is considered in this research. An ordering policy is proposed for raw materials to meet the requirements of a production facility which, in turn, must deliver finished products demanded by outside buyers at fixed points in time. First, a general cost model is formulated considering both raw materials and finished products. Then, using this model, a simple procedure is developed to determine an optimal ordering policy for procurement of raw materials, and the manufacturing batch size, to minimize the total cost of meeting customer demands in time. The dependent relationships between production batch size and rawmaterial purchasing quantity, and various delivery patterns considered in recent literature are critically reviewed. The quality of the solution is evaluated. Numerical examples are provided.     

A mathematical model for order splitting in a multiple supplier single-item inventory system

In this paper we develop a mathematical model which considers multiple-supplier single-item inventory systems. The lead times of the suppliers and demand arrival rate are random variables. All shortages are backordered. Continuous review (s, Q) policy has been assumed. When the inventory level hits the reorder level, the total order is split among n suppliers. The problem is to determine the reorder level and order quantity for each supplier so that the expected total cost per time unit, including ordering cost, procurement cost, inventory holding cost and shortage cost is minimized. We also conduct extensive numerical experiments to show the advantages of our model compared to the relevant models in the literature. In addition, some managerial insights are observed.     

An alternative analysis and solution procedure for the EPQ model with rework process at a single-stage manufacturing system with planned backorders

This paper revisits the work by Cárdenas-Barrón (Cárdenas-Barrón, L. E., (2009). Economic production quantity with rework process at a single-stage manufacturing system with planned backorders. Computers and Industrial Engineering, 57(3), 1105–1113). The optimal solution condition is analyzed using the production time and the time to eliminate backorders as decision variables instead of the classical decisions variables of lot and backorder quantities. This new approach leads to an alternative inventory policy for imperfect quality items when the optimal production time is less than the optimal time. This is a new method that has not been explored before; it improves and complements the work by Cárdenas-Barrón (2009) and provides interesting managerial insights. An easy to apply solution procedure is provided to obtain the optimal policy. Finally, numerical examples are solved to illustrate our results.     

An optimal batch size under a periodic delivery policy

This paper addresses the problem of a manufacturing system that procures raw materials from suppliers in a lot and processes them to convert to finished products. It proposes an ordering policy for raw materials to meet the requirements of a production facility. In turn, this facility must deliver finished products demanded by outside buyers at fixed interval points in time. First, a general cost model is developed considering both raw materials and finished products. Then this model is used to develop a simple procedure to determine an optimal ordering policy for procurement of raw materials and also the manufacturing batch size to minimize the total cost for meeting the customer demand on time. The proposed procedure provides better results than the traditional separate policies. Numerical examples are also presented.     

An integrated inventory problem with transportation in a divergent supply chain under service level constraint

This study investigates an integrated inventory problem with transportation in a single-vendor and multi-buyer divergent supply chain. The vendor manufactures a product and delivers the product to the buyers located in different locations by a fleet of vehicles of identical capacity. The external demands per unit time on the buyers are independent and normally distributed. The lead time components of the buyers, excluding transportation time, can be reduced at an added crash cost. A model has been formulated to minimize the total expected cost of the system associated with the production, inventory, transportation and lead time reduction to find the optimal production, inventory and routing decisions while satisfying the service level constraint of the buyers. We propose a coordinated two-phase iterative approach to solve the model, which has been illustrated through a numerical example.     

Analysis of the maximum level policy in a production-distribution system

We consider a production-distribution system, where a facility produces one commodity which is distributed to a set of retailers by a fleet of vehicles. Each retailer defines a maximum level of the inventory. The production policy, the retailers replenishment policies and the transportation policy have to be determined so as to minimize the total system cost. The overall cost is composed by fixed and variable production costs at the facility, inventory costs at both facility and retailers and routing costs. We study two different types of replenishment policies. The well-known order-up to level (OU) policy, where the quantity shipped to each retailer is such that the level of its inventory reaches the maximum level, and the maximum level (ML) policy, where the quantity shipped to each retailer is such that the inventory is not greater than the maximum level. We first show that when the transportation is outsourced, the problem with OU policy is NP-hard, whereas there exists a class of instances where the problem with ML policy can be solved in polynomial time. We also show the worst-case performance of the OU policy with respect to the more flexible ML policy. Then, we focus on the ML policy and the design of a hybrid heuristic. We also present an exact algorithm for the solution of the problem with one vehicle. Results of computational experiments carried out on small size instances show that the heuristic can produce high quality solutions in a very short amount of time. Results obtained on a large set of randomly generated problem instances are also shown, aimed at comparing the two policies.     

A perishable inventory system with modified (S−1,S) policy and arbitrary processing times

In this article we analyze a lost sales (S−1,S) perishable system, under Poisson demands and exponential lifetimes, in which the reorders are placed at every demand epoch so as to take the inventory position back to its maximum level S. The items are replenished one at a time and the resupply time has arbitrary distribution. The various operating characteristics are obtained using Markov renewal techniques. A matrix recursive scheme is developed to determine the stationary distribution of the underlying Markov chain. The efficiency of this procedure in the determination of optimal S that minimizes the long run expected cost rate is discussed. Sensitivity analysis of various system parameters is also carried out.Scope and purposeThe analysis of perishable inventory systems, which has potential applications in various sectors of industry, is far more difficult than their infinite lifetime counterparts. In the study of perishable systems under traditional (S−1,S) policy, reorders for items are placed at demand as well as failure epochs. However, it will be more meaningful not to place orders at failure epochs, which is also practical and economical since it avoids continuous monitoring of items. This paper deals with a continuous review perishable system under a more realistic base stock policy, with variable ordering quantity and arbitrary unit resupply times. The operating characteristics of this complex model are derived using the techniques of semi-regenerative processes. A matrix recursive scheme developed to determine the stationary distribution has enabled us to code an efficient numerical program for cost optimization.     

Priority optimization and make‐to‐stock/make‐to‐order decision in multiproduct manufacturing systems

We consider a single‐stage multiproduct manufacturing facility producing a large number of end products. In order to reduce overall inventory costs, an efficient approach is to produce some items according to a make‐to‐stock (MTS) policy and others according to a make‐to‐order (MTO) policy. Items priority levels play a key role in the optimal MTO/MTS decisions for such typical large‐scale systems. To tackle this issue, the manufacturing facility is modeled as a multiproduct multipriority classes queuing system. We propose a general optimization procedure that selects near‐optimal priority classes, gives the associated flow control mode (MTO or MTS) for each product, and provides a lower bound and an upper bound with respect to the optimal cost. First, we illustrate efficiency of our optimization procedure for this class of nonlinear integer programs via several examples and by a numerical analysis, including a comparison with two alternative heuristics given in the literature. In addition, we provide managerial insights by exhibiting, under various parameter settings, the significant impact of an efficient priority level allocation among items on the inventory costs and on optimal splitting between MTO and MTS products.     

Inventory model with partial backorders

This article presents a deterministic inventory model for situations in which, during the stockout period, a fraction j8 of the demand is backordered and the remaining fraction 1 — β is lost. By defining a time proportional backorder cost and a fixed penalty cost per unit lost, a convex objective function representing the average annual cost of operating the inventory system is obtained. The optimal operating policy variables are calculated directly. At the extremes β = 1 and β = 0 the model presented reduces to the usual backorders and lost sales case, respectively.