A new MRP/GT lot sizing heuristic: A simulation study

This paper studies the lot-sizing problem in Material Requirements Planning/Group Technology (MRP/GT) systems. A GT production cell is designed to produce many families of components. A major setup is required when switching from manufacturing one family of components to another family, and a minor setup is needed when switching from manufacturing a component type to another component type within the same family. Inventory holding cost is incurred if inventory level is positive, and inventory shortage cost is incurred if inventory level is negative, that is, backordering. The objective of the proposed lot-sizing problem is to minimize the sum of major and minor setup costs, holding and shortage costs, and regular production cost, and to meet simultaneously the demand requirements. The proposed problem is modelled into a linear integer program, a heuristic method to solve the problem is proposed, and a simulation experiment conducted to evaluate the performance of the proposed heuristic and some existing heuristics. The computational results show that the proposed heuristic is useful to reduce the total cost significantly over a wide variety of simulated environments.     

Effect of forecast errors on rolling horizon master production schedule cost performance for various replanning intervals

Master production schedules are usually updated by the use of a rolling schedule. Previous studies on rolling schedules seem to form the consensus that frequent replanning of a master production schedule (MPS) can increase costs and schedule instability. Building on previous research on rolling schedules, this study addresses the impact of overestimation or underestimation of demand on the rolling horizon MPS cost performance for various replanning frequencies. The MPS model developed in this paper is based on actual data collected from a paint company. Results indicate that under both the forecast errors conditions investigated in this study, a two-replanning interval provided the best MPS cost performance for this company environment. However, results from the sensitivity analysis performed on the MPS model indicate that when the setup and inventory carrying costs are high, a 1-month replanning frequency (frequent replanning) seems more appropriate for both of the above forecast error scenarios.     

Multi-item inventory model with quantity-dependent inventory costs and demand-dependent unit cost under imprecise objective and restrictions: A geometric programming approach

A multi-item inventory model with constant demand and infinite replenishment is developed under the restrictions on storage area, total average shortage cost and total average inventory investment cost. These restrictions may be precise or imprecise. Here, it is assumed that inventory costs are directly proportional to the respective quantities, and unit purchase/production cost is inversely related to the demand. Restricted shortages are allowed but fully backlogged. First, the problem is formulated in crisp environment taking the deterministic and precise inventory parameters. It is solved by both geometric programming (GP) and gradient-based non-linear programming (NLP) methods. Later, the problem is formulated with fuzzy goals on constraints and objectives where impreciseness is introduced through linear membership functions. It is solved using the fuzzy geometric programming (FGP) method. The inventory models are illustrated with numerical values and compared with the crisp results. A sensitivity analysis on the optimum order quantity and average cost is also presented due to the variation in the tolerance of total average inventory investment cost and total average shortage cost following Dutta et al., 1993, Fuzzy Sets and Systems, 55, 133-142.     

Inventory Rationing for Multiple Class Demand under Continuous Review

We consider how a firm should ration inventory to multiple classes in a stochastic demand environment with partial, class‐dependent backlogging where the firm incurs a fixed setup cost when ordering from its supplier. We present an infinite‐horizon, average cost criterion Markov decision problem formulation for the case with zero lead times. We provide an algorithm that determines the optimal rationing policy, and show how to find the optimal base‐stock reorder policy. Numerical studies indicate that the optimal policy is similar to that given by the equivalent deterministic problem and relies on tracking both the current inventory and the rate that backorder costs are accumulating. Our study of the case of non‐zero lead time shows that a heuristic combining the optimal, zero lead time policy with an allocation policy based on a single‐period profit management problem is effective.     

A queuing/management consideration on Japanese production systems

A single-stage lot/cell production under a Poisson arrival and exponential service in a batch is considered. The three economic queuing models of push and pull types are presented, an economic comparison of push versus pull types is considered, and a strategic management/design consideration to the lot production is given. First, the total expected operating cost is given for the three queuing models including the Omote-Kanban type similar to VMI. Second, the push versus pull system is discussed from a view of setup time, inventory or operating cost, and it is ascertained that the three types are alternative. Finally, a strategic management basis for economic traffic, leadtime setting is given, and discussed by the introduction of production matrix on 2-stage design.     

Optimal investment in setup reduction for finite horizon periodic review inventory system under deterministic time varying demand

This paper examines the impact of setup reduction on a finite horizon, periodic review inventory system, under deterministic time varying demand. A total relevant cost function is developed for such systems. Using this, the impact of setup reduction is examined under various forms of setup reduction cost functions that have been suggested in the literature. The operating characteristics and optimization of the various scenarios are discussed. Our analysis shows that the effects of setup reduction in a periodic review system are similar to those in a reorder point system. Our results are likely to help practitioners who use similar periodic review systems towards decreasing total inventory related costs by investing in setup reduction.     

Economic manufacturing quantities under conditions of learning and forgetting (EMQ/LaF)

Abstract

Abstract. It has been empirically observed that productivity improves as production continues due to system 'learning’, but that it deteriorates once the activity is stopped due to system 'forgetting’. Both learning and forgetting follow an exponential form with a 'doubling factor’ ranging between 0.75 and 0.98. We review and critique two previously proposed models, correct some minor errors in them, and expand one of them to accommodate a finite horizon. We also propose a new model that is more in harmony with the established learning function, for the determination of the optimal number and size of the lots in the finite and infinite horizon. The methodology used throughout is dynamic programming. We investigate the impact of all three models on the optimal lot sires and their costs, and establish the functional relations between the total cost and the various factors affecting them.     

Integrality in Stochastic Inventory Models

We study several finite‐horizon, discrete‐time, dynamic, stochastic inventory control models with integer demands: the newsvendor model, its multi‐period extension, and a single‐product, multi‐echelon assembly model. Equivalent linear programs are formulated for the corresponding stochastic dynamic programs, and integrality results are derived based on the total unimodularity of the constraint matrices. Specifically, for all these models, starting with integer inventory levels, we show that there exist optimal policies that are integral. For the most general single‐product, multi‐echelon assembly system model, integrality results are also derived for a practical alternative to stochastic dynamic programming, namely, rolling‐horizon optimization by a similar argument. We also present a different approach to prove integrality results for stochastic inventory models. This new approach is based on a generalization we propose for the one‐dimensional notion of piecewise linearity with integer breakpoints to higher dimensions. The usefulness of this new approach is illustrated by establishing the integrality of both the dynamic programming and rolling‐horizon optimization models of a two‐product capacitated stochastic inventory control system.     

EXISTENCE AND DERIVATION OF FORECAST HORIZONS IN A DYNAMIC LOT SIZE MODEL WITH NONDECREASING HOLDING COSTS

We are concerned with a discrete-time undiscounted dynamic lot size model in which demand and the production setup cost are constant for an initial few periods and the holding cost of inventory is an arbitrary nondecreasing function assumed to be stationary (i.e., explicitly independent of time) in the same initial few periods. We show that there exists a finite forecast horizon in our model and obtain an explicit formula for it. In addition, we obtain fairly general conditions under which the existence of a solution horizon in the model implies the existence of a forecast horizon. We also derive an explicit formula for the minimal solution horizon. These results extend the earlier ones obtained for the dynamic lot size model with linearly increasing holding costs.     

Rolling horizon master production schedule performance: A policy analysis

In this paper, a simulation experiment has been developed to examine the combined influence of the design, inventory and environmental factors on the cost performance of a rolling horizon master production schedule. Specifically, a 2 5 factorial design was used to examine the effects associated with three rolling schedule design policies, one inventory policy and one environmental condition of forecast error on MPS cost performance. The study was based on actual data from a paint company. Results suggest that the choice of appropriate lot-size and inventory policies have a significant influence on MPS costs and that there are indeed important interactions between these policies and other design factors of a rolling schedule.     

Unsold Versus Unbought Commitment: Minimum Total Commitment Contracts with Nonzero Setup Costs

We study a minimum total commitment (MTC) contract embedded in a finite‐horizon periodic‐review inventory system. Under this contract, the buyer commits to purchase a minimum quantity of a single product from the supplier over the entire planning horizon. We consider nonstationary demand and per‐unit cost, discount factor, and nonzero setup cost. Because the formulations used in existing literature are unable to handle our setting, we develop a new formulation based on a state transformation technique using unsold commitment instead of unbought commitment as state variable. We first revisit the zero setup cost case and show that the optimal ordering policy is an unsold‐commitment‐dependent base‐stock policy. We also provide a simpler proof of the optimality of the dual base‐stock policy. We then study the nonzero setup cost case and prove a new result, that the optimal solution is an unsold‐commitment‐dependent (s, S) policy. We further propose two heuristic policies, which numerical tests show to perform very well. We also discuss two extensions to show the generality of our method's effectiveness. Finally, we use our results to examine the effect of different contract terms such as duration, lead time, and commitment on buyer's cost. We also compare total supply chain profits under periodic commitment, MTC, and no commitment.     

The Effects of Sharing Upstream Information on Product Rollover

The process of introducing new and phasing out old products is called product rollover. This paper considers a periodic‐review inventory system consisting of a manufacturer and a retailer, where the manufacturer introduces new and improved products over an infinite planning horizon using the solo‐roll strategy. We consider two scenarios: (1) the manufacturer does not share the upstream information about new‐product introduction with the retailer and (2) the manufacturer shares the information. For each scenario, we first derive the decentralized ordering policy and the system‐optimal ordering policy with given cost parameters. We then devise an optimal supply chain contract that coordinates the inventory system. We demonstrate that when the inventory system is coordinated, information sharing improves the performance of both supply chain entities. However, this may not be true if the inventory system is not coordinated. We also show that under the optimal contract, the manufacturer has no incentive to mislead the retailer about new‐product information in the information‐sharing model. When demand variability increases, information sharing adds more benefits to the coordinated supply chain. Our research provides insights about coordinating product, financial, and information flows in supply chains with product rollover.     

An EOQ model with deteriorating items in response to a temporary sale price

In economic order quantity models, it is often assumed that the unit purchase cost is constant. Such an assumption is usually not fulfilled in many practical situations. In practice, it is observed that suppliers sometimes offer temporary price discounts to stimulate demand, boost market share or decrease inventories of certain items. In this paper, a deteriorating inventory model with a temporary sale price has been developed. We shall be concerned with finding the optimal total cost saving for deteriorating items during the special replenishment period. Numerical examples are presented to illustrate the proposed model.     

An optimal replenishment model for inventory items with normally distributed deterioration

This paper deals with the inventory replenishment problem for deteriorating items with normally distributed shelf life, continuous time-varying demand, and shortages under the inflationary and time discounting environment. The reasons of choosing normal are twofold: it is one of the most important probability phenomena in the real world due to the classical central limit theorem, and it is also one of the most commonly used lifetime distributions in reliability contexts. The problem is formulated as a dynamic programming model and solved by numerical search techniques. The solutions of the model determine the optimal replenishment schedule over a finite planning horizon so that the present worth of the future costs associated with the system is minimized. In the extensive experiments, we validate the model, demonstrate the optimal replenishment schedule and lot-size, and carry out a comparative study to ascertain its contribution. In addition, sensitivity analysis was provided to help identify the most crucial factors that affect system performance. The experimental result shows that the deteriorating problem solved by an appropriate model (i.e. the proposed normal model) can save the total cost up to 2% approximately. It also identifies that the magnitudes of purchase cost per unit and demand rate are the most significant parameters that affect the replenishment decisions and cost.     

带学习效应的双渠道供应链库存策略研究

针对具有学习行为的双渠道供应链问题,本文研究了两种分销渠道并存下的最优库存策略。有限计划期内,分销商通过传统销售和在线销售来满足下游顾客的需求。两种分销渠道下的销售单价为时变不减线性函数,当系统中各周期的生产订购固定成本以一定的概率具有学习效应行为时,分别建立了非变质产品生产存贮问题的混合整数约束优化模型以及易变质产品存贮问题的无约束混合整数优化模型,所建立模型的目标为极大化分销商总利润函数。对于这两类模型,通过分析其最优解的性质,利用将生产订购次数松弛为连续变量的技巧证明了最优解存在的唯一性。给出了最优策略的求解方法并比较了两类模型最优利润函数值的大小。最后通过数值算例对上述模型进行了验证,数值结果表明当供应链系统中存在学习效应行为时,该系统能够获得更多的利润。     

Experimental Results Indicating Lattice‐Dependent Policies May Be Optimal for General Assemble‐To‐Order Systems

We consider an assemble‐to‐order (ATO) system with multiple products, multiple components which may be demanded in different quantities by different products, possible batch ordering of components, random lead times, and lost sales. We model the system as an infinite‐horizon Markov decision process under the average cost criterion. A control policy specifies when a batch of components should be produced, and whether an arriving demand for each product should be satisfied. Previous work has shown that a lattice‐dependent base‐stock and lattice‐dependent rationing (LBLR) policy is an optimal stationary policy for a special case of the ATO model presented here (the generalized M‐system). In this study, we conduct numerical experiments to evaluate the use of an LBLR policy for our general ATO model as a heuristic, comparing it to two other heuristics from the literature: a state‐dependent base‐stock and state‐dependent rationing (SBSR) policy, and a fixed base‐stock and fixed rationing (FBFR) policy. Remarkably, LBLR yields the globally optimal cost in each of more than 22,500 instances of the general problem, outperforming SBSR and FBFR with respect to both objective value (by up to 2.6% and 4.8%, respectively) and computation time (by up to three orders and one order of magnitude, respectively) in 350 of these instances (those on which we compare the heuristics). LBLR and SBSR perform significantly better than FBFR when replenishment batch sizes imperfectly match the component requirements of the most valuable or most highly demanded product. In addition, LBLR substantially outperforms SBSR if it is crucial to hold a significant amount of inventory that must be rationed.     

The resource constrained single-item inventory problem with demand-dependent unit cost

In this paper, we present an economic order quantity (EOQ) with both demand-dependent unit cost and restrictions. An analytical solution of the EQO is derived using a recent and simple method, which isthe geometric programming approach. The EOQ inventory model with demand-dependent unit cost without any restriction and the classical EOQ inventory model are obtained.     

The benefit of receding horizon control: Near-optimal policies for stochastic inventory control

In this paper we address the single-item, single-stocking point, non-stationary stochastic lot-sizing problem under backorder costs. It is well known that the (s, S) policy provides the optimal control for such inventory systems. However the computational difficulties and the nervousness inherent in (s, S) paved the way for the development of various near-optimal inventory control policies. We provide a systematic comparison of these policies and present their expected cost performances. We further show that when these policies are used in a receding horizon framework the cost performances improve considerably and differences among policies become insignificant.     

Some Comments on the Validity of EOQ Formula under Inflationary Conditions

In an earlier issue of Decision Sciences, Jesse, Mitra, and Cox [1] examined the impact of inflationary conditions on the economic order quantity (EOQ) formula. Specifically, the authors analyzed the effect of inflation on order quantity decisions by means of a model that takes into account both inflationary trends and time discounting (over an infinite time horizon). In their analysis, the authors utilized two models: Current-dollars model and Constant-dollars model. These models were derived, of course, by setting up a total cost equation in the usual manner then finding the optimum order quantity that minimizes the total cost. Jesse, Mitra, and Cox [1] found that EOQ is approximately the same under both conditions; with or without inflation. However, we disagree with the conclusion drawn by [2] and show that EOQ will be different under inflationary conditions, provided that the inflationary conditions are properly accounted for in the formulation of the total cost model.