Deterministic inventory model with two levels of storage,a linear trend in demand and a fixed time horizon

A deterministic inventory model is developed for a single item having two separate storage facilities (owned and rented warehouses) due to limited capacity of the existing storage (owned warehouse) with linearly time-dependent demand (increasing) over a fixed finite time horizon. The model is formulated by assuming that the rate of replenishment is infinite and the successive replenishment cycle lengths are in arithmetic progression. Shortages are allowed and fully backlogged. As a particular case, the results for the model without shortages are derived. Results are illustrated with two numerical examples.Scope and purposeThroughout the world, the production of food grains is periodical. Normally, in countries where state control is less, the demand of essential food grains is lowest at the time of harvest and goes up to the highest level just before the next harvest. This phenomenon is very common in developing third world countries where most of the people are landless or marginal farmers. At the time of harvest, they share some grain/product with landowners and as soon as the small inventory is exhausted, they are forced to buy food grains from the open market. As a result, demand for food grains increases with time in a period along with the number of the people whose initial stock of food grains gets exhausted.In this paper, a two-storage inventory model with time-dependent demand and fixed time horizon is developed and solved by a mathematical programme based on gradient method. This methodology of model development and its solution are quite general and it can be applied to inventory models of any product whose production is periodical and demand increases linearly with time.     

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.     

Pade approximants for the transient optimization of hedging controlpolicies in manufacturing

Part production is considered over a finite horizon in a single-part multiple-failure mode manufacturing system. When the rate of demand for parts is constant, for Markovian machine-mode dynamics and for convex running cost functions associated with part inventories or backlogs, it is known that optimal part-production policies are of the so-called hedging type. For the infinite-horizon case, such policies are characterized by a set of constant critical machine-mode dependent inventory levels that must be aimed at and maintained whenever possible. For the finite-horizon (transient) case, the critical levels still exist, but they are now time-varying and in general very difficult to characterize. Thus, in an attempt to render the problem tractable, transient production optimization is sought within the (suboptimal) class of time-invariant hedging control policies, a renewal equation is developed for the cost functional over finite horizon under an arbitrary time-invariant hedging control policy     

A production lot size inventory model for deteriorating items with time-varying demand

This paper presents a production lot size model for deteriorating items with time-varying demand. The replenishment cycle and deterioration rates are allowed to vary over a finite planning horizon. A solution procedure for this model is developed to provide the optimal replenishment cycle number. Also three numerical examples are solved for different deterioration rates. Finally, a sensitivity analysis is conducted to study the effect of changes in the related parameters on the objective function.     

Finite horizon planning for a production system with permitted shortage and fixed-interval deliveries

Raw material ordering policy and the manufacturing batch size for frequent deliveries of finished goods for a finite horizon plays a significant role in managing the supply chain logistics economically. This research develops an ordering policy for raw materials and determines an economic batch size for a product in a manufacturing system that supplies finished products to customers for a finite planning horizon. Fixed quantities of finished products are delivered to customers frequently at a fixed interval of time. In this model, an optimal multi-ordering policy for procurement of raw materials and production cycle time for a two-stage production and supply system is developed to minimize the total cost incurred due to raw materials and finished goods inventories. The problem is then extended to compensate for the lost sales of finished products. A closed-form solution to the problem is obtained for the minimal total cost. A lower bound on the optimal solution is also developed for problem with lost sale. It is shown that the solution and the lower bound are consistently tight.     

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     

Group technology inventories with optimal production costs

A model of group technology (GT) production systems including the cost of raw material processing is developed. It is assumed that the throughput time in a given GT cell depends linearly on the lot size. The optimal GT lot–size formula is derived and its properties discussed. The variability with respect to changes in the annual demand rate, the average carrying cost rate, and the rate charged per unit of cell production time are examined and evaluated. A typical case study is worked out to illustrate the analytical results.     

Integrated supply chain model for a deteriorating item with procurement cost dependent credit period

This study develops an integrated production inventory model of supplier and retailer where a delay in payment is offered by supplier to retailer for a constant deteriorating item. Here shortages are not allowed. In this model retailer’s procurement cost linearly depends on the credit period and supplier’s process cost also is a linear function of the amount of quantity purchased by retailer. In this model, the objective is to decide the position of the credit period and number of replenishment of retailer in finite time horizon in such a way that the integrated system gets the optimum cost. The model is explained with the help of numerical examples and the sensitivity analysis of some parameters are also carried out.     

Pricing and production lot-size/scheduling with finite capacity for a deteriorating item over a finite horizon

Although the lately evolved manufacturing technologies such as enterprise resource planning (ERP) provide a unified platform for managing and integrating core business processes within a firm, the decision-making between marketing and production planning still remains rather disjoint. It is due in large parts to the inherent weaknesses of ERP such as the fixed and static parameter settings and uncapacitated assumption. To rectify these drawbacks, we propose a decision model that solves optimally the production lot-size/scheduling problem taking into account the dynamic aspects of customer's demand as well as the restriction of finite capacity in a plant. More specifically, we consider a single product that is subject to continuous decay, faces a price-dependent and time-varying demand, and time-varying deteriorating rate, production rate, and variable production cost, with the objective of maximizing the profit stream over multi-period planning horizon. We propose both coordinated and decentralized decision-making policies that drive the solution of the multivariate maximization problem. Both policies are formulated as dynamic programming models and solved by numerical search techniques. In our numerical experiments, the solution procedure is demonstrated, comparative study is conducted, and sensitivity analysis is carried out with respect to major parameters. The numerical result shows that the solution generated by the coordinated policy outperforms that by the decentralized policy in maximizing net profit and many other quantifiable measures such as minimizing inventory investment and storage capacity.Scope and purposeWe consider a manufacturing firm who produces and sells a single product that is subjected to continuous decay over a lifetime, faces a price-dependent and time-varying demand function, shortages are allowed and a completely backlogged, and has the objective of determining price and production lot-size/scheduling so as to maximize the total profit stream over multi-period planning horizon. We develop a tactical-level decision model that solves the production scheduling problem taking into account the dynamic nature of customer's demand which is partially controllable through pricing schemes. As analogous to the sales and operations planning, the proposed scheme can be used as a coordination center of the APS system within a generic enterprise resource planning framework which integrates and coordinates distinct functions within a firm.This paper differs from the existing works in several ways. First, we propose a dynamic version of the joint pricing and lot-size/scheduling problem taking into account the capacitated constraint. Second, several key factors being considered in the model, such as the demand rate, deteriorating rate, production rate, and variable production cost are assumed time-varying that reflect the dynamic nature of the market and the learning effect of the production system. A third difference between the past research and ours is that the price can be adjusted upward or downward in our model, making the proposed pricing policy more responsive to the structural change in demand or supply.     

Capacity and flexibility planning of an economical manufacturing system

This paper presents a simple dynamic model for determining the capacity and the flexibility of a manufacturing system over a finite planning horizon. We consider a problem that arises from the development phase of an investment plan for an economical manufacturing system. The objective is to minimize the total cost associated with the capacity expansion, flexibility expansion and operation. This problem is formulated as an integer program. A Lagrangian heuristic is developed for determining a near optimal solution to this integer program. Finally, we show how to incorporate aggregate production planning into the model.Based on a presentation given at the ORSA/TIMS Miami meeting in 1986.     

An EOQ model for deteriorating items with time-varying demand and shortages

The study explores the inventory replenishment policies for the cases with time-varying demand, linearly increasing deterioration rate, partial back-ordering, constant service level and equal replenishment intervals over a fixed planning horizon. Since it is difficult to solve the problem directly, we derive the upper bound of replenishment number for a specific planning horizon and find the solution of service level under a given number of replenishment. The optimal solutions of replenishment number and service level are then determined. Numerical examples and sensitivity analyses are also provided to illustrate the solution procedure.     

Optimal replenishment policies for the case of a demand function with product-life-cycle shape in a finite planning horizon

In this paper, we explore the demand function follows the product-life-cycle shape for the decision maker to determine the optimal number of inventory replenishments and the corresponding optimal inventory replenishment time points in the finite planning horizon. The objective function of the total relevant costs considered in our model is mathematically formulated as a mixed-integer nonlinear programming problem. A complete search procedure is provided to find the optimal solution by employing the properties derived in this paper and the Nelder–Mead algorithm. Also, based on the search procedure developed in this paper, a decision support system is implemented on a personal computer to solve the proposed problem.     

A finite time horizon inventory model with deterioration and time-value of money under the conditions of permissible delay in payments

A varying deterioration rate, time-value of money and the condition of permissible delay in payments used in conjunction with the EOQ model are the focus of discussion. The replenishment number and fraction of each cycle in which there is no shortage are both determined to minimize the present value of inventory cost over a finite planning horizon. Two special cases and numerical examples are presented to illustrate the model.     

Minimization of the life cycle cost for a multistate system under periodic maintenance

Although preventive maintenance policies have received extensive interest, limited attention has been paid to implementing multiple maintenance actions for a multistate system over a finite horizon in continuous time. Therefore, this study closely examines such maintenance actions by viewing a coherent multicomponent system as a multistate system and assuming that the necessary maintenance action and cost which move the current state to an extremely better state depends on the current state. The above problem is also formulated as a periodic maintenance model. In addition, the model's characteristics are elucidated to obtain the optimal cycle time of maintenance actions, thereby minimizing the life cycle cost over a specific finite horizon.     

Finite horizon EOQ model for non-instantaneous deteriorating items with price and advertisement dependent demand and partial backlogging under inflation

This paper deals with an economic order quantity (EOQ) model for non-instantaneous deteriorating items with price and advertisement dependent demand pattern under the effect of inflation and time value of money over a finite planning horizon. In this model, shortages are allowed and partially backlogged. The backlogging rate is dependent on the waiting time for the next replenishment. This paper aids the retailer in minimising the total inventory cost by finding the optimal interval and the optimal order quantity. An algorithm is designed to find the optimum solution of the proposed model. Numerical examples are given to demonstrate the results. Also, the effect of changes in the different parameters on the optimal total cost is graphically presented and the implications are discussed in detail.     

Economic Order Quantity model with Weibull deterioration distribution,shortage and ramp-type demand

A single-item single-period Economic Order Quantity model for deteriorating items with a ramp-type demand and Weibull deterioration distribution is considered. The shortages in inventory are allowed and backlogged completely. The model is developed over an infinite planning horizon and the optimal replenishment policy is derived by minimizing the total inventory cost per unit time. The numerical solution of the model is obtained, and the sensitivity of the parameters involved in the model is also examined.     

On the optimal replenishment schedule for an inventory system with deteriorating items and time-varying demand and production rates

In this paper, we present an exact method for finding the optimal replenishment schedule for an inventory system with deteriorating items, and in which demand and production rates are allowed to vary over a finite planning horizon. A numerical example is presented to illustrate the method.     

The sphere under internal pressure—A comparison of two time-integration schemes for the viscoplasticity theory based on overstress

Numerical inelastic analyses of a hollow sphere under linearly increasing internal pressure are performed using the viscoplasticity theory based on overstress with material constants representing the behavior of AISI Type 304 Stainless Steel at room temperature. The purpose of the investigation was to compare the results of two numerical schemes. The first was a finite element time integration method developed previously and the second was a finite difference scheme which used the IMSL subroutine DGEAR for the time integration. The results show that the finite element solution matches the finite difference results at a fraction of the cost of the finite difference calculation. A comparison is made between the elastic perfectly-plastic and the viscoplastic solutions for the displacement and the stresses.     

On some optimality conditions and their equivalence in time-dependent inventory models

The paper considers a deterministic, time-and-review-continuous, finite horizon inventory model with general time-dependent demand, holding cost, shortage cost and replenishment rate functions. It gives conditions for the optimal ordering schedule, for a fixed number of replenishments, which reduce the effort for its finding to a simple one-dimensional search. The paper also proves equivalence relationships between the derived optimality conditions and the corresponding ones for special cases of the general model where some of its component functions tend to infinity. It concludes with a discussion concerning the uniqueness of the optimal number of replenishments, and the convergence of the optimal ordering schedule for an infinite time horizon.     

Supplier–retailer inventory coordination with credit term for inventory‐dependent and linear‐trend demand

In this paper, a nonintegrated and collaborative replenishment policy is considered, respectively, which incorporates varying demand depending on both inventory level and time during the finite planning horizon. For additional cost savings realized from coordination, the paper adopts trade credit as a cost‐saving shift means and introduces a brand new parameter, that is, credit period rate. Then, the equitable credit period rate is determined, and different values of the credit period rate reflect the allocation of additional cost savings between the supplier and retailer. Furthermore, the conditions for the existence and uniqueness of an optimal solution are proved for the nonintegrated and collaborative replenishment policy, and an efficient solution procedure is developed to determine the optimal results and coordination of the inventory model. Finally, several numerical examples are provided to illustrate the proposed strategy and algorithm, and the sensitivity analysis of the optimal solution with respect to each parameter is presented. The sensitivity analysis suggests that the size of the credit period rate has a strong relationship with the supplier's and retailer's inventory cost (including capital cost) and setup cost. In real‐life situations, this proposed strategy may be applied to some consumer products in the growth phase or best‐selling consumer goods, etc.