A Note on the Heuristic for Replenishment of Trended Inventories Considering Shortages

We discuss the inventory replenishment policy for an item having a deterministic demand pattern with a linear (positive) trend and shortages. A heuristic is developed to determine the decision rule for selecting the times and sizes of replenishments over a finite time-horizon so as to keep the total costs minimum. The use of the heuristic is illustrated with a numerical example.     

A hybrid method for large-scale short-term scheduling of make-and-pack production processes

Due to the ongoing trend towards increased product variety, fast-moving consumer goods such as food and beverages, pharmaceuticals, and chemicals are typically manufactured through so-called make-and-pack processes. These processes consist of a make stage, a pack stage, and intermediate storage facilities that decouple these two stages. In operations scheduling, complex technological constraints must be considered, e.g., non-identical parallel processing units, sequence-dependent changeovers, batch splitting, no-wait restrictions, material transfer times, minimum storage times, and finite storage capacity. The short-term scheduling problem is to compute a production schedule such that a given demand for products is fulfilled, all technological constraints are met, and the production makespan is minimised. A production schedule typically comprises 500–1500 operations. Due to the problem size and complexity of the technological constraints, the performance of known mixed-integer linear programming (MILP) formulations and heuristic approaches is often insufficient.     

一类最优EOQ模型的进一步扩展

对一类经济批量订购模型作如下进一步扩展:第一,允许短缺,短缺量部分拖后供给,且短缺期间损失率与实际缺货量成正比;第二,订购费用是可变的,且线性依赖于订购量.在此假定下,研究了有限计划时间水平及常数变质率下,部分短缺量拖后的变质性物品多阶段库存问题,给出了寻找最优订购策略的算法,证明了所给最优策略的存在唯一性及在该策略下费用函数取得最小值.最后给出应用实例.     

On a production-inventory system of deteriorating items subject to random machine breakdowns with a fixed repair time

This paper considers the impact of random machine breakdowns on the classical Economic Production Quantity (EPQ) model for a product subject to exponential decay and under a no-resumption (NR) inventory control policy. A product is manufactured in batches on a machine that is subject to random breakdowns in order to meet a constant demand over an infinite planning horizon. The product is assumed to have a significant rate of deterioration and time to deterioration is described by an exponential distribution. Also, the time-to-breakdown is a random variable following an exponential distribution. Under the NR policy, when a breakdown occurs during a production run, the run is immediately aborted. A new run will not be started until all available inventories are depleted. Corrective maintenance of the production system is carried out immediately after a breakdown and it takes a fixed period of time to complete such an activity. The objective is to determine the optimal production uptime that minimizes the expected total cost per unit time consisting of setup, corrective maintenance, inventory carrying, deterioration, and lost sales costs. A near optimal production uptime is derived under conditions of continuous review, deterministic demand, and no shortages.     

The scheduling of deliveries in a production-distribution system with multiple buyers

In a real production and distribution business environment with one supplier and multiple heterogeneous buyers, the differences in buyers’ ordering cycles have influence on production arrangements. Consequently, the average inventory level (AIL) at the supplier’s end is affected by both the production policy and the ordering policy, typically by the scheduling of deliveries. Consequently, the average inventory holding cost is most deeply affected. In this paper, it is proposed that the scheduling of deliveries be formulated as a decision problem to determine the time point at which deliveries are made to buyers in order to minimize the supplier’s average inventory. A formulation of the average inventory level (AIL) in a production cycle at the supplier’s end using a lot-for-lot policy is developed. Under the lot-for-lot policy, the scheduling of deliveries (SP) is formulated as a nonlinear programming model used to determine the first delivery point for each buyer with an objective to minimize the sum of the product of the individual demand quantity and the first delivery time for each buyer. Thus, the SP model determines not only the sequence of the first deliveries to individual buyers, but also the time when the deliveries are made. An iterative heuristic procedure (IHP) is developed to solve the SP model assuming a given sequence of buyers. Six sequence rules are considered and evaluated via simulation.     

Batch scheduling of deteriorating reworkables

The problem of scheduling the production of new and recoverable defective items of the same product manufactured on the same facility is studied. Items are processed in batches. Each batch comprises two sub-batches processed consecutively. In the first sub-batch, all the items are newly manufactured. Some of them are of the required good quality and some are defective. The defective items are remanufactured in the second sub-batch. They deteriorate while waiting for rework. This results in increased time and cost for their remanufacturing. All the items in the same sub-batch complete at the same time, which is the completion time of the last item in the sub-batch. Each remanufactured defective item is of the required good quality. It is assumed that the percentage of defective items in each batch is the same. A setup time is required to start batch processing and to switch from manufacturing to remanufacturing. The demands for good quality items over time are given. The objective is to find batch sizes such that the total setup and inventory holding cost is minimized and all the demands are satisfied. Dynamic programming algorithms are presented for the general problem and some important special cases.     

Optimizing Supply Shortage Decisions in Base Stock Distribution Operations

This paper addresses policies and agreements between suppliers and customers for handling supply shortages in base-stock systems under uncertain demand. We investigate the impacts that backlogging and expediting decisions have on inventory and transportation costs in these systems and develop a model for deciding whether a supplier should completely backlog, completely expedite, or employ some combination of backlogging and expediting shortages. Our results indicate that practical cases exist where some combination of both expediting and backlogging supply shortages outperforms either completely expediting or backlogging all shortages. Including transportation costs in our model provides incentive to employ `hybrid' policies that partially expedite and partially backlog excess demands within a given period. Our model demonstrates how inventory policy decisions directly impact transportation costs and provides a heuristic approach for jointly minimizing expected inventory and transportation costs.     

Deterministic inventory models for variable production

This paper presents two deterministic inventory models for a single item, where for the first model, the production rate at any instant depends on the on-hand inventory and for the second one, it is demand dependent. However, in both cases, the demand rate at any moment of time is a linear function of time for the scheduling period. Both the models are formulated and solved without allowing shortages. Two numeric examples have been added to illustrate the results.     

Scheduling parallel machines with single vehicle delivery

We investigate the integrated production and distribution scheduling problem in a supply chain. The manufacturer’s production environment is modeled as a parallel machine system. A single capacitated vehicle is employed to deliver products in batches to multiple customers. The scheduling problem can also be viewed as either parallel machines with delivery considerations or a flexible flowshop. Different inventory holding costs, job sizes (volume or storage space required in the transportation unit), and job priorities are taken into account. Efficient mathematical modeling and near-optimal heuristic approaches are presented for minimizing total weighted completion time.     

Scheduling policies in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1 make-to-stock queue

In this paper, we analyse a production/inventory system modelled as an M/G/1 make-to-stock queue producing different products requiring different and general production times. We study different scheduling policies including the static first-come-first-served, preemptive and non-preemptive priority disciplines. For each static policy, we exploit the distributional Little's law to obtain the steady-state distribution of the number of customers in the system and then find the optimal inventory control policy and the cost. We additionally provide the conditions under which it is optimal to produce a product according to a make-to-order policy. We further extend the application area of a well-known dynamic scheduling heuristic, Myopic(T), for systems with non-exponential service times by permitting preemption. We compare the performance of the preemptive-Myopic(T) heuristic alongside that of the static preemptive-bμ rule against the optimal solution. The numerical study we have conducted demonstrates that the preemptive-Myopic(T) policy is superior between the two and yields costs very close to the optimal.     

The Trended Inventory Lot Sizing Problem with Shortages Under a New Replenishment Policy

In the past few years, considerable attention has been given to the inventory lot sizing problem with trended demand over a fixed horizon. The traditional replenishment policy is to avoid shortages in the last cycle. Each of the remaining cycles starts with a replenishment and inventory is held for a certain period which is followed by a period of shortages. A new replenishment policy is to start each cycle with shortages and after a period of shortages a replenishment should be made. In this paper, we show that this new type of replenishment policy is superior to the traditional one. We further propose four heuristic procedures that follow the new replenishment policy. These are the constant demand approximation method, the equal cycle length heuristic, the extended Silver approach, and the extended least cost solution procedure. We also examine the cost and computation time performances of these heuristic procedures through an empirical study. The number of test problems solved to optimality, average and maximum cost deviation from optimum were used as measures of cost performance. The results of the 10 000 test problems reveal that the extended least cost approach is most cost effective.     

Planning and scheduling of multistage multiproduct batch plants operating under production campaigns

When plants are operated under stable conditions during reasonable time periods, operation with campaigns is particularly appropriate. The regular operation of the facilities simplifies the production control, the inventory management, the plant operability, etc. A?campaign includes several batches of different products that are going to be manufactured and the same one is cyclically repeated over the time horizon. In this work, a mixed integer linear programming formulation is proposed for the planning and scheduling of given multiproduct batch plants operating with campaigns. The number and size of batches for each product, the campaign composition, the assignment of batches to units and their sequencing, and the number of times that the campaign is repeated over the time horizon must be determined. Taking into account this scenario, an appropriate performance measure is the minimization of the cycle time. An asynchronous slot-based continuous-time representation for modeling the assignment of batches to units and their sequencing is employed, and a novel rule for determining the maximum number of slots postulated for each unit is proposed.     

Production strategies for a stochastic lot-sizing problem with constant capacity

This paper presents a single item capacitated stochastic lot-sizing problem motibated by a Dutch company operating in a Make-To-Order environment. Due to a highly fluctuating and unpredictable demand, it is not possible to keep any finished goods inventory. In response to a customer's order, a fixed delivery date is quoted by the company. The objective is to determine in each period of the planning horizon the optimal size of production lots so that delivery dates are met as closely as possible at the expense of minimal average costs. These include set-up costs, holding costs for orders that are finished before their promised delivery date and penalty costs for orders that are not satisfied on time and are therefore backordered. Given that the optimal production policy is likely to be too complex in this situation, attention is focused on the development of heuristic procedures. In this paper two heuristics are proposed. The first one is an extension of a simple production strategy derived by Dellaert [5] for the uncapacitated version of the problem. The second heuristic is based on the well-known Silver-Meal algorithm for the case of deterministic time-varying demand. Experimental results suggest that the first heuristic gives low average costs especially when the demand variability is low and there are large differences in the cost parameters. The Silver-Meal approach is usually outperformed by the first heuristic in situations where the available production capacity is tight and the demand variability is low.     

A two-stage heuristic for disassembly scheduling with assembly product structure

Disassembly scheduling, one of the important operational problems in disassembly systems, is the problem of determining the ordering and disassembly schedules of used or end-of-life products while satisfying the demand of their parts or components over a certain planning horizon. This paper considers products with assembly structure for the objective of minimizing the sum of purchase, set up, inventory holding, and disassembly operation costs, and suggests a two-stage heuristic, in which an initial solution is obtained in the form of the minimal latest ordering and disassembly schedule, and then improved iteratively considering trade-offs among different cost factors. To show the performance of the heuristic, computational experiments were done on the example obtained from the literature and a number of randomly generated test problems, and the results show that the heuristic can give optimal or very near-optimal solutions within very short computation times.     

The Optimality of the ‘Zero-Switch’ Rule for a Class of Economic Lot-Scheduling Problems

The economic lot-scheduling problem for the single-machine, n-item scheduling problem has received attention in a number of journals. One approach is to define a sequence (called the fundamental cycle) in which every item is made at least once, and then to determine the length of production runs consistent with the aggregate inventory level which will maximize the length of the cycle duration. The assumption that production is switched from one item to the next only when the inventory level of the latter reaches zero is often used in heuristic solutions to these models. This paper illustrates the conditions in which the ‘zero-switch’ rule is a necessary condition at the optimal solution for situations in which demand is continuous and production capacity equals aggregate demand.     

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.     

Deterministic hierarchical substitution inventory models

In this paper, we consider a deterministic nested substitution problem where there are multiple products which can be substituted one for the other, if necessary, at a certain cost. We consider the case when there are n products, and product j can substitute products j + 1,…,n at certain costs. The trade-off is the cost of storing products (for example, customised products) at a higher inventory holding stage versus the cost of transferring downwards from a lower inventory holding cost (generic product) stage. The standard approach to solving the problem yields an intractable formulation, but by reformulating the problem to determine the optimal run-out times, we are able to determine the optimal order and substitution quantities. Numerical examples showing the effect of various system parameters on the optimal order and substitution policy are also presented.     

Admission policies for the customized stochastic lot scheduling problem with strict due-dates

This papers considers admission control and scheduling of customer orders in a production system that produces different items on a single machine. Customer orders drive the production and belong to product families, and have family dependent due-date, size, and reward. When production changes from one family to another a setup time is incurred. Moreover, if an order cannot be accepted, it is considered lost upon arrival. The problem is to find a policy that accepts/rejects and schedules orders such that long run profit is maximized. This problem finds its motivation in batch industries in which suppliers have to realize high machine utilization while delivery times should be short and reliable and the production environment is subject to long setup times.We model the joint admission control/scheduling problem as a Markov decision process (MDP) to gain insight into the optimal control of the production system and use the MDP to benchmark the performance of a simple heuristic acceptance/scheduling policy. Numerical results show that the heuristic performs very well compared with the optimal policy for a wide range of parameter settings, including product family asymmetries in arrival rate, order size, and order reward.     

Optimal Production and Setup Scheduling: A One-Machine, Two-Product System

This paper studies the scheduling problem for two products on a single production facility. The objective is to specify a production and setup policy that minimizes the average inventory, backlog, and setup costs. Assuming that the production rate can be adjusted during the production runs, we provide a close form for an optimal production and setup schedule. Dynamic programming and Hamilton–Jacobi–Bellman equation is used to verify the optimality of the obtained policy.     

带有可变库存费的变质性物品的部分延期供给的库存控制模型

传统的库存控制模型都视需求率为常数,在这篇文章中,放松了这个假定,研究了库存费的两种可能的变化:(i)库存费的变化率为存储时间的函数;(ii)库存费的变化率为库存量的函数.在模型中允许短缺发生且假定短缺部分延期供给,且在需求率线性依赖于库存水平的情形下,发展了两个变库存费的库存控制模型.