Some results for the dynamic (s, S) inventory model *

Summary The periodic review, single item, stationary ( s, S ) inventory model is considered. There is a fixed lead time, a linear purchase cost, a fixed set-up cost, a holding and shortage cost function, a discount factor 0 < α≤ 1 and backlogging of unfilled demand. The solution for the total expected discounted cost for the finite period (s, S ) model is found. In addition the time dependent behaviour of the inventory process is found. Further a limit theorem is given, which relates the total expected cost for the finite period ( s, S ) model with no discounting to the average expected cost per period for the infinite period ( s, S ) model. As a by-product we obtain known results for the infinite period (s, S ) model.     

The optimality of (s,S) inventory policies in the infinite period model*

Summary The infinite period stationary inventory model is considered. There is a constant lead time, a nonnegative set-up cost, a linear purchase cost, a holding and shortage cost function, a fixed discount factor β, 0 < β < 1, and total backlogging of unfilled demand. Both the total discounted cost (β < 1) and the average cost (β= 1) criteria are considered. Under the assumption that the negatives of the one period holding and shortage costs are unimodal, a unified proof of the existence of an optimal (s.S) policy is given. As a by-product of the proof upper and lower bounds on the optimal values of s and S are found. New results simplify the algorithm of Veinott and Wagner for finding an optimal (s, S) policy for the case β< 1. Further it is shown that the conditions imposed on the one period holding and shortage costs can be weakened slightly.     

The cost of inflated planned lead times in MRP systems

Much of the current literature in the field of production and inventory control systems stresses the need to revise traditional forms of thinking regarding production processes, the role of inventories for work in process, and the need for reduced lead times or flow times. Group technology, manufacturing cells, and other means of incorporating repetitive manufacturing techniques into traditional job-shop settings constitute the leading edge in system development.Still, there is resistance to these dramatic changes, and traditional “business as usual” methods still predominate. This study attempts to illustrate graphically the cost justification associated with reduction in lead times which generally results from these new concepts. In most job shops today, lead times are much longer than they need to be due to inflation of lead time estimates. Actual lead times for the manufacture of fabricated and assembled products have been shown to be a direct consequence of the planning lead times used in the MRP planning process—a form of self-fulfilling prophesy.The research employs a simulation model of a factory using MRP as a planning tool in a multiproduct, multilevel production environment. Manufacturing costs constitute the dependent variable in the experiments, defined as the sum of material costs (including expedite premiums), direct labor costs (including overtime premiums), inventory carrying costs, and overhead costs. The independent variable being manipulated is the planned lead time offset used in the MRP planning process. Twenty values of planned lead time are evaluated ranging from a value that includes no slack time at all (pure assembly line) up to a value that allows 95% slack (queue) time which, unfortunately, is not uncommon in many job shops today. Stochastic variables in the model include customer demand and actual processing times—the sum of set-up and run times.The result of the study is a cost curve formed over the range of independent lead time variables that is constructed using nonlinear regression techniques. The conclusions from the resultant graph clearly indicate the cost consequences of long lead times, with exponential cost increases beyond the 80–90% queue time level. Total costs are 41% higher at the maximum lead time allowance compared to the minimum. Clearly, this study demonstrates the need for lead time reduction, either through downward adjustment of MRP planned lead times or by introducing new manufacturing concepts.     

动态持有成本下的库存成本模型研究

对制造业企业来说,更大的生产能力就有机会带来更多的销量,所以在库存模型中需求率是与库存量正相关的变量,而并非在整个在库时间内都保持为常量。论文提出了一种库存成本与需求率分别和库存时间与库存量相关的模型,并针对两种成本结构各设计了一种优化算法以确定最优订购量和最优库存时间。     

A Note on the Newsboy Model with a Cutoff Transaction Size and Gauss Holding and Penalty Cost Functions

In recent papers, the inventory models were presented in which customers with an order larger than a prespecified cutoff transaction size are satisfied in an alternative way, against additional cost. They assumed the holding and penalty cost functions are linear functions. In this paper, the Gauss function can be applied such as the cost for mailing letters or packages in the post office. Therefore, we address a variant of the holding and penalty cost functions by considering the Gauss holding and penalty cost functions to fit in with the most practical situations. In addition, we also assume that customers with an order larger than a prespecified cutoff transaction size are still assumed to be satisfied in an alternative way. Moreover, when the maximum demand is large, much more time may be required to determine the optimal solution. Thus, we adopt and modify the algorithm of the Golden Section Search Technique to determine the optimal order-up-to level S and the cutoff transaction size q systematically and provide illustrative numerical example.     

The effect of lot-sizing on workload variability

Lot-sizing models which group demand requirements for one or more consecutive time periods into a single production run have received considerable attention in recent years. Material Requirements Planning (MRP) systems must, for instance, make a lot-size decision for each planned order release. Existing decision models attempt to minimize the sum of setup plus inventory holding costs. However, lot-sizing tends to increase the work center load variability, and, consequently, the costs associated with changing production levels from period to period should be incorporated into the economic analysis. This study is concerned, first of all, with analytically describing the relationship between dynamic lot-sizing models and workload variability. Secondly, in order to account for production level change costs we propose a simple modification to existing heuristic models. Lastly, we employ a simulation model to empirically extend these results to a typical MRP multiechelon production environment. An example is included to show clearly that with cost premiums for overtime and severance or guaranteed minimum costs for undertime the traditional lot-sizing techniques significantly underestimate actual costs and can lead to very costly policies.Mean, variance and coefficient of variation of period work time requirements are derived as a function of several algorithm characteristics. Average cycle time (number of periods covered by a single batch) is found to be the most influential factor in determining workload variability. Variance grows approximately in proportion to this cycle time with the proportionality constant being the square of average period workload. Cycle time and demand variability also contribute to workload variability. Results indicate that for a given average cycle time, the EOQ method will minimize workload variability. When N products utilize the same work center, the coefficient of load variation will be reduced by a factor of N^{?$\text{1}\text{2}$} unless requirements are positively correlated. Positive correlation would result when products have similar seasons or parent items. In this case grouping such products cannot help reduce variability.In order to incorporate production level change costs into existing heuristics we may simply introduce a term consisting of a penalty factor times average cycle time. The penalty factor represents the costs of period by period production level changes. Several popular heuristics are extended in this fashion, and it is found that solutions are still readily obtainable, requiring only modifications to setup or holding cost parameters.The effects of level change costs are examined via simulation for a specific yet typical environment. It is found that when setup costs are significant, traditional lot-sizing heuristics can provide cost savings and service level improvements as compared to lot-for-lot production. However, whereas for our model the obtainable profit improvement from lot-sizing was 25% in the case of freely variable capacity, actual improvements were only one half as large when reasonable hiring and firing practices and overtime and undertime costs were considered. Consequently, management needs to consider carefully labor costs and work center product relationships when determining a production scheduling method.     

Een systeem van kostenbeheersende productieplanning

In this paper a system of cost-controlled production planning is described. This system considers all kinds of costs associated with a production in phases (e.g. production of units, production of subassemblies, assembly) which may be affected by the planning, such as set-up costs, costs of transport, control, inventories, capacity and changes in capacity.
The mathematical model leads to a linear or mixed discrete-linear programming problem whose solution gives for each time period considered the size of the capacities which should be used and of the series of different products which should be produced. Practical recommendations are given for obtaining a sufficiently satisfying solution.     

流水型CONWIP生产系统研究

本文研究了流水型CONWIP生产系统中在制品常量、批量和加工次序的问题。以机床加工准备成本及工件库存成本和加工流动时间为优化目标．建立了CONWIP系统整数规划模型，并提出了求解问题的启发式算法。在优化机床准备及库存成本的基础上．得到各产品的生产批量和系统在制品常量。同时，提出了流水型CONWIP生产系统的排序算法（CFA），依此得到产品的较优加工次序。本研究使CONWIP模型求解得到简化且易于应用，并获得满意解。     

A cost evaluation method for dynamic line balancing

A procedure is presented for calculating stochastic costs, which include operator (labor) and inventory costs, associated with dynamic line balancing. Dynamic line balancing, unlike the traditional methods of assembly and production line balancing, assigns operators to one or more operations, where each operation has a predetermined processing time and is defined as a group of identical parallel stations. Operator costs and inventory costs are stochastic because they are functions of the assignment process employed in balancing the line, which may vary throughout the balancing period, and the required flow rate. Earlier studies focused on the calculation of the required number of stations and demonstrated why the initial and final inventories at the different operations are balanced.The cost minimization method developed in the article can be used to evaluate and compare the assignment of operators to stations for various assignment heuristics. Operator costs and inventory costs are the components of the cost function. The operator costs are based on the operations to which operators are assigned and are calculated for the entire work week regardless of whether an operator is given only a partial assignment which results in idle time. It is assumed that there is no variation in station speeds, no learning curve effect for operators' performance times, and no limit on the number of operators available for assignment. The costs associated with work-in-process inventories are computed on a “value added” basis. There is no charge for finished goods inventory after the last operation or raw material before the first operation.The conditions which must be examined before using the cost evaluation method are yield, input requirements, operator requirements, scheduling requirements and output requirements. Yield reflects the output of good units at any operation. The input requirement accounts for units discarded or in need of reworking. The operator requirements define the calculation of operator-hours per hour, set the minimum number of operators at an operation, and require that the work is completed. The scheduling requirements ensure that operators are either working or idle at all times, and that no operator is assigned to more than one operation at any time. The calculation of the output reflects the yield, station speed, and work assignments at the last operation on the line.An application of the cost evaluation method is discussed in the final section of the article. Using a simple heuristic to assign operators, the conditions for yield, inputs, operators, scheduling, and output are satisfied. The costs are then calculated for operators and inventories.In conclusion, the cost evaluation method for dynamic balancing enables a manager to compare the costs of assigning operators to work stations. Using this method to calculate the operator and inventory costs, a number of different heuristics for assigning operators in dynamic balancing can be evaluated and compared for various configurations of the production line. The least cost solution procedure then can be applied to a real manufacturing situation with similar characteristics.     

Multiplant MRP

Many manufacturing firms have multiple manufacturing plants, located in geographically diverse parts of the world. This situation is becoming more common, as firms establish new plants in foreign countries to take advantage of low labor cost. In such cases, it is not unusual for the firm to retain production capability of certain key parts in a backup plant, with the necessary equipment and trained workforce in place. High volume production could be obtained relatively quickly from the backup plant in case of an emergency at the main supplying plant. In such multiplant settings, the transportation costs are significant. Throughout this paper, we use the term “multisourced parts” to describe parts produced in more than one location.Material Requirements Planning (MRP) is the component of a total manufacturing control system that is designed to manage inventory and plan orders for parts and material with dependent demand (demand derived from the demand of other items). Most of the literature on MRP systems discusses MRP methodology in a single-plant environment. Most MRP software systems in use today are single-plant systems.Currently, it is common for firms with multiple plants treated as cost centers to use an independent single-plant MRP system for each and handle the transshipment problems manually. Because of lack of coordination of production schedules between supplying and demanding plants, those firms hold more inventory and experience longer lead times than necessary to compensate for uncertainties in schedules and supply policies.The purpose of this article is to enhance single-plant MRP systems for coping with multiplant situations in which: the plants are regarded as cost centers, there exist multisourced parts, and the transportation costs are significant. The multiplant MRP system should recognize that parts are produced in different plants, make offset calculations for in-transit lead times, and consider transportation costs when establishing production requirements and shipping routes for multisourced parts. The objective is, beginning with the corporate-determined master schedule for finished products, to communicate in one planning cycle time-phased planned order release schedules and shipping/delivery schedules to each manufacturing plant producing components for the finished products.We first present a simplified framework for the multiplant MRP system, where a transportation algorithm is incorporated into the MRP logic. Then we refine this simplified framework to handle more complex aspects of a multiplant network. These complexities include the treatment of requirements that are not shipped on time and the regeneration of new MRP schedules. We also observe that the solution to the transportation problem described above is affected by the lot-sizing rules employed. In addition, we discuss several important issues and decisions that confront a firm when implementing a multiplant MRP system.     

一类变库存费且存货影响销售率的EOQ模型

从实际背景出发,在变库存费和存货影响销售率条件下,分别就最小化平均费用和最大化平均利润讨论了具有Ramp型需求的变质性物品的EOQ模型,给出了无短缺量拖后和部分拖后条件下最佳订购批量的求解方案,数值例子对两个不同目标下的最佳订购批量及各项相关费用进行了比较,揭示了变库存费和存货影响销售率对最优订货策略的影响。     

A heuristic algorithm for capacity sensitive requirements planning

Available lot sizing rules for use in MRP (Material Requirements Planning) systems ignore capacity limitations at various work centers when sizing future orders. Planned order releases are instead determined by the tradeoff only between the item's set up and inventory holding costs. This limitation can cause unanticipated overloads and underloads at the various work centers, along with higher inventories, poorer customer service, and excessive overtime.This research explores one way to make MRP systems more sensitive to capacity limitations at the time of each regeneration run. A relatively simple heuristic algorithm is designed for this purpose. The procedure is applied to those planned order releases that standard MRP logic identifies as mature for release. The lot sizes for a small percentage of these items are increased or decreased so as to have the greatest impact in smoothing capacity requirements at the various work centers in the system. This algorithm for better integrating material requirements plans and capacity requirements plans is tested with a large scale simulator in a variety of manufacturing environments. This simulator has subsequently undergone extensive tests, including its successful validation with actual data at a large plant of major corporations.Simulation results show that the algorithm's modest extension to MRP logic significantly helps overall performance, particularly with customer service. For a wide range of test environments, past due orders were reduced by more than 30% when the algorithm was used. Inventory levels and capacity problems also improved. Not surprisingly, the algorithm helps the most (compared to not using it at all as an MRP enhancement) in environments in which short-term bottlenecks are most severe. Large lot sizes and tight shop capacities are characteristic of these environments. The algorithm works the best when forecast errors are not excessive and the master schedule is not too “nervous.”This proposed procedure is but one step toward making MRP more capacity sensitive. The widely heralded concept of “closed-loop” MRP means that inventory analysts must change or “fix up” parts of the computer generated material requirements plan. What has been missing is a tool for identifying the unrealistic parts of the plan. Our algorithm helps formalize this identification process and singles out a few planned order releases each week. This information comes to the analyst's attention as part of the usual action notices. These pointers to capacity problems go well beyond capacity requirements planning (CRP) and would be impossible without computer assistance.Our study produced two other findings. First, short-term bottlenecks occur even when the master production schedule is leveled. The culprits are the lot sizing choices for items at lower levels in the bills of material. “Rough-cut” capacity planning, such as resource requirements planning, therefore is not a sufficient tool for leveling capacity requirements. It must be supplemented by a way to smooth bottlenecks otherwise caused by shop orders for intermediate items. Second, the disruptive effect of large lot sizes is apparent, both in terms of higher inventories and worse customer service. Large lot sizes not only inflate inventories, but paradoxically hurt customer service because they create more capacity bottlenecks. The only reason why management should prefer large lot sizes is if set-up times are substantial and cannot be efficiently reduced. This finding is very much in step with the current interest in just-in-time (JIT) systems.     

A heuristic solution procedure for the multi-item,single-level,limited capacity,lot-sizing problem

The problem to be considered is that of determining lot-sizes for a group of products which are produced at a single workcentre. It is assumed that the requirements for each product are known, period by period, out to the end of some common time horizon. (A reasonable assumption in a Material Requirements Planning context when we are dealing with components of one or more other items already scheduled.) For each product there is a fixed setup cost incurred each time production takes place. Unit production and holding costs are linear. The time required to set up the machine is assumed to be negligible. All costs and production rates can vary from product to product but not with respect to time. In each period there is a finite amount of machine time available that can vary from period to period. The objective is to determine lot-sizes so that 1) costs are minimized, 2) no backlogging occurs and 3) capacity is not exceeded.An exact solution to this complex problem is out of the question. Therefore, a simple heuristic has been developed which guarantees a feasible solution, if one exists. Results of a large number of test problems, including three supplied by industrial sources, are presented. The results indicate that the heuristic will usually generate a very good solution with a relatively small amount of computational effort.     

供应链二级分销网络的一种成本优化模型

提出了一种二级分销网络的成本优化的模型。模型研究了单个工厂的制造商和多个分销商组成的二级分销网络成本优化问题,把保证不发生缺货情况作为约束条件。综合考虑了订货成本、库存持有成本、运输成本和建立分销点和分销中心的成本,并给出了基于遗传算法的模型求解步骤。     

A heuristic method for aggregate planning: Production decision framework

This paper presents an understandable and straight-forward method for making work force level and inventory planning decisions, i.e., dynamic aggregate planning decisions.The development phase utilizes a ratio, named RPCC, which represents the relative value of the cost of changing the production level to the cost of carrying inventory. This ratio is used to determine the length of an effective planning horizon. Two indicators are calculated to reflect the demand to current production rate over different time periods. Based on the joint values of these indicators, the planning problem is subdivided into one of nine mutually exclusive and exhaustive states. A set of action statements, representing logical responses to each of the sub-problems, is formulated.After completion of the development phase, the performance of the Production Decision Framework model is tested in several real case environments. Suggestions are made for further improvement.     

Optimal inventories for repairable redundant systems with aging components

Complex systems that are required to perform very reliably are often designed to be “fault-tolerant,” so that they can function even though some component parts have failed. Often fault-tolerance is achieved through redundancy, involving the use of extra components. One prevalent redundant component configuration is the m-out-of-n system, where at least m of n identical and independent components must function for the system to function adequately.Often machines containing m-out-of-n systems are scheduled for periodic overhauls, during which all failed components are replaced, in order to renew the machine's reliability. Periodic overhauls are appropriate when repair of component failures as they occur is impossible or very costly. This will often be the case for machines which are sent on “missions” during which they are unavailable for repair. Examples of such machines include computerized control systems on space vehicles, military and commercial aircraft, and submarines.An interesting inventory problem arises when periodic overhauls are scheduled. How many spare parts should be stocked at the maintenance center in order to meet demands? Complex electronic equipment is rarely scrapped when it fails. Instead, it is sent to a repair shop, from which it eventually returns to the maintenance center to be used as a spare. A Markov model of spares availability at such a maintenance center is developed in this article. Steady-state probabilities are used to determine the initial spares inventory that minimizes total shortage cost and inventory holding cost. The optimal initial spares inventory will depend upon many factors, including the values of m and n, component failure rate, repair rate, time between overhauls, and the shortage and holding costs.In a recent paper, Lawrence and Schaefer [4] determined the optimal maintenance center inventories for fault-tolerant repairable systems. They found optimal maintenance center inventories for machines containing several sets of redundant systems under a budget constraint on total inventory investment. This article extends that work in several important ways. First, we relax the assumption that the parts have constant failure rates. In this model, component failure rates increase as the parts age. Second, we determine the optimal preventive maintenance policy, calculating the optimal age at which a part should be replaced even if it has not failed because the probability of subsequent failure has become unacceptably high. Third, we relax the earlier assumption that component repair times are independent, identically distributed random variables. In this article we allow congestion to develop at the repair shop, making repair times longer when there are many items requiring repair. Fourth, we introduce a more efficient solution method, marginal analysis, as an alternative to dynamic programming, which was used in the earlier paper. Fifth, we modify the model in order to deal with an alternative objective of maximizing the job-completion rate.In this article, the notation and assumptions of the earlier model are reviewed. The requisite changes in the model development and solution in order to extend the model are described. Several illustrative examples are included.     

Joint decision of procurement lot-size,supplier selection,and carrier selection

The joint decision making of procurement lot-size, supplier selection, and carrier selection has potential to reduce buyer's purchasing expenditures. Furthermore, the total logistics cost can also come down through economies of scale in the purchasing and transportation costs, and reduction in supply chain disruptions such as rejections and late deliveries. We study a procurement setting in which a buyer needs to purchase a single product from a set of suppliers over finite discrete time periods to satisfy service level requirements. The suppliers offer all-unit quantity discounts, and transportation cost depends on carrier capacity as well as geographical location of suppliers. This paper proposes an integer linear programming model to simultaneously determine the timings of procurement, lot-sizes, suppliers and carriers to be chosen so as to incur the least total cost over the planning horizon. A numerical example is included to demonstrate the effectiveness of the proposed model in establishing tradeoffs among purchasing cost, transaction cost, and inventory holding cost. Sensitivity analysis has been carried out to understand the effects of the model parameters on the purchasing decisions and total cost. Managerial insights of this study serve as a reference for decision makers to develop effective procurement strategies.     

De statistische basis van de planning in een machinefabriek

The normal or jigsaw-puzzle method of planning cannot be applied to the production of an engine factory. One has to fall back upon a kind of statistical automatism, like in the planning of road traffic as opposed to the planning of railway traffic.
In order to do this kind of planning efficiently, it is necessary to know the statistical relationships between the degree of occupation (number of working hours divided by number of available hours) of the machines, the waiting time and the velocity of flow through the factory.
The statistical analysis of some of these quantities in an Amsterdam plant, manufacturing medium size diesel engines, showed that Erlang's formulas of waiting time in telephone traffic (with exponential distribution of holding times) are applicable.
These formulas are used to prove that the highest degree of occupation is not the best one from an economic point of view. Formulas and graphs are given for finding the optimum degree of occupation in engine factories and other works where the same conditions apply.     

Overlapping operations in material requirements planning

Material requirements planning (MRP) is a planning and information system that has widespread application in discrete-parts manufacturing. The purpose of this article is to introduce ideas that can improve the flow of material through complex manufacturing systems operating under MRP, and that can increase the applicability of MRP within diverse manufacturing environments.MRP models the flow of material by assuming that items flow from work station to work station in the same batches that are used in production. That is, once work starts on a batch of a certain item at a certain work station, the entire batch will be produced before any part of the batch will be transported to the next work station on its routing plan. Clearly, efficiency can be increased if some parallelism can be introduced. The form of parallelism investigated here is overlapping operations.Overlapping operations occurs when the transportation of partial batches to a downstream work station is allowed while work proceeds to complete the batch at the upstream work station. The potential efficiencies to be gained are the following:

• Reduced work-in-process inventory

• Reduced floor space requirements

• Reduced size of transfer vehicles

Additional costs may accrue through additional cost of transportation of partial batches and through additional costs of control.Some MRP software vendors provide the data processing capability for overlapping operations. However, the user is given little or no guidance on overlapping percentages or amounts. It is our intent to provide a simple, robust technique to MRP users who would like to overlap operations and gain some or all of the above efficiencies.An optimal lot-sizing technique is derived by considering a generic two work station segment of a manufacturing system. Under the assumptions of constant demand and identical production rates, a cost function that considers setup costs, inventory holding costs and transportation costs is derived. This cost function is minimized subject to the constraint that the production batch is an integer multiple of the transfer batch. We solve for the optimal production batch, the optimal transfer batch, and the integer number relating them. Solutions are obtained as closed form, easy to-evaluate formulas.By introducing more parallelism, overlapping operations can reduce lead time. However, this will not happen without modification of MRP logic to accommodate such reduced lead time. We derive a formula that shows how a significant lead time compression can easily be obtained and implemented in MRP.We consider an example to illustrate the application of the technique on typical data from the electronics industry. The outcome showed a cost savings of approximately 22.5% over the standard MRP approach.Overlapping operations allows the applicability of MRP to an increasing number of situations that are not modeled faithfully by conventional MRP logic. Three such situations that occur often are the following:

• Limited size of transfer vehicles dictate that several transfers should be planned.

• Lead time requirements prohibit nonoverlapped operations.

Our analysis suggests how to accommodate these difficult practical situations into MRP.Overlapping operations in material requirements planning provides an enhancement that allows wider applicability, shortened lead times, and lower total costs. It may be applied selectively to any two work stations where it is deemed appropriate. Due to the structure of the cost function, it is possible to make the transfer lot-sizing decisions independent of the production lot-sizing decisions. Therefore, significant improvements can be made through overlapping with minimum disruption to the existing MRP system machinery. It is our conviction that overlapping operations is an important concept that can and will impact MRP. We suggest the approach presented here as a systematic way to implement overlapping.     

面向水产加工的季节性供应原料库存控制方法研究

结合水产加工行业的特定背景,研究供应具有季节性且有保鲜期约束的易变质原料的库存控制问题。在允许缺货的情况下,考虑价格因素对库存控制的影响,对带有保鲜期约束的库存问题进行探讨,建立了相应的模型并使用整数规划的方法进行求解。运用此方法建立的模型可对企业的原料库存进行中长期规划,有效降低企业成本,实现利益最大化。