Optimal economic ordering policy with deteriorating items under different supplier trade credits for finite horizon case

This paper develops and solves a general finite horizon trade credit economic ordering policy for an inventory model with deteriorating items under inflation and time value of money when shortages are not allowed. The time horizon is divided into different cycles each of which has its own demand rate and its own trade credit period offered from the supplier to his retailer so that the retailer should pay his supplier before or after the end of the permissible trade credit of that cycle. Up to the end of the trade credit of a cycle, the retailer is free of charge, but he is charged on an interest for those items not being sold before this end. The retailer can also earn the interest of the money from the generated sales revenue in any cycle by depositing such revenue into an interest bearing account. The objective of the retailer is then to minimize his net total relevant costs. A closed form of this net total cost is derived and the resulting model is solved. Then rigorous mathematical methods are used to show that, under some seemingly possible conditions, there exist a unique vector of the relevant decision variables that solve the underlying inventory system. A numerical example which shows the applicability of the theoretical results is given.     

Optimal control of a one product recovery system with leadtimes

In this paper a recovery system for a single product is investigated. Besides a remanufacturing and a manufacturing facility the system consists of one inventory for returned and recoverable items and one for serviceable items. Demands are satisfied from serviceable inventory, which can be replenished by remanufactured returned items, which are as good as new, or by new produced items. Additionally, there is the possibility of disposing of returned items.We determine the cost optimal manufacturing, remanufacturing and disposal rates for the system under the assumptions of a linear cost structure, a finite planning horizon and deterministic and dynamic demand and return rates. Thereby we study two classes of policies, one where backorders are forbidden and another one where they are allowed. In contrast to the existing literature positive leadtimes are considered.     

Joint pricing and inventory control for non-instantaneous deteriorating items with partial backlogging and time and price dependent demand

In this paper, a joint pricing and inventory control for non-instantaneous deteriorating items is developed. We adopt a price and time dependent demand function. Shortages is allowed and partially backlogged. The major objective is to determine the optimal selling price, the optimal replenishment schedule and the optimal order quantity simultaneously such that, the total profit is maximized. We first show that for any given selling price, optimal replenishment schedule exists and unique. Then, we show that the total profit is a concave function of price. Next, we present a simple algorithm to find the optimal solution. Finally, we solve a numerical example to illustrate the solution procedure and the algorithm.     

Timing production in LP models in a rolling schedule

We consider an arbitrary supply chain structure under demand uncertainty. We look at a setting where the supply chain planning is executed periodically and where the demand of end items in subsequent periods is stochastic. In contrast to other linear programming-based approaches, we assume constant planned lead times. In this paper we discuss the timing of production during the planned lead times of items. If production starts immediately, the work-in-process inventory costs are higher, but producing later will result in higher safety stocks. We look at the influence of the demand variation, the planned lead time of the items, the utilization rate of resources, and the added value when merging items, on the inventory costs. The results indicate that, for higher utilization rates, producing early is more suitable.     

Inventory control with a modified Croston procedure and Erlang distribution

This paper considers an inventory control system, primarily for a finished goods inventory. The purpose is to create a procedure that can handle both fast-moving items with regular demand and slow-moving items. The suggested procedure should be easy to implement in a modern computerized ERP-system. Essentially, the system is a periodic review system built around a Croston forecasting procedure. An Erlang distribution is fitted to the observed data using the mean and variance of the forecasted demand rate. According to probabilities for stock shortages, derived from the probability distribution, the system decides if it is time to place a new order or not. The Croston forecasting method is theoretically more accurate than ordinary exponential smoothing for slow-moving items. However, it is not evident that a Croston forecasting procedure (with assumed Erlang distribution) outperforms ordinary exponential smoothing (with assumed normal distribution) applied in a “practical” inventory control system with varying demand, automatically generated replenishment, etc. Our simulation study shows that the system in focus will present fewer shortages at lower inventory levels than a system based on exponential smoothing and the normal distribution.     

A vendor managed inventory supply chain with deteriorating raw materials and products

Fast deteriorating raw materials such as raw milk, fruit and vegetables are commonly used to produce slowly deteriorating finished products such as milk powders, cheeses, and pastas. This paper studies a Vendor Managed Inventory (VMI) type supply chain where the manufacturing vendor decides how to manage the system-wide inventories of its fast deteriorating raw material and its slowly deteriorating product. The decision variables are a common replenishment cycle of the product and the replenishment frequency of the raw material. We assume the deteriorating rates are known constants and every retailer's demand is deterministic. We develop an integrated model to calculate the total inventory and deterioration cost for such a system. We prove the convexity of the cost functions, and based on this a golden search algorithm is developed to find the optimal solution of the model. Our numerical results show that the deteriorating rate of the product may increase the total cost by more than 40% compared to the zero-deteriorating rate, while the deteriorating raw material has less impact on the total cost (commonly less than 5% in our numerical examples). This indicates that more attention should be paid to the product than the raw material. Further, an increase in the number of retailers can make the replenishment frequency of the raw material increase significantly but the common replenishment cycle of the product decreases a little. This indicates that adding a new retailer would not be felt strongly by the other retailers but would be felt by the supplier of the raw material.     

Optimal inventory and pricing policies for remanufacturable leased products

In this paper we consider a company which leases new products and also sells remanufactured versions of the new product that become available at the end of their lease periods. When the amount of end-of-lease items in stock is not sufficient to meet the demand for remanufactured products, the firm may purchase additional cores from a third-party supplier. We develop a dynamic programming formulation for determining the optimal price of remanufactured products, and optimal payment structure for the leased products. Our objective is to maximize the discounted system-wide profit over a finite horizon. The profit function consists of revenues that are obtained from remanufactured product sales and leasing, remanufacturing and manufacturing costs, inventory holding and shortage costs. We consider a consumer choice based demand model for mapping a potential customer into one of the product segments (a remanufactured product customer or a customer for a leased product with a particular lease period) for a given price/lease payment vector. We explore several properties of the discounted profit function and provide insight on the behavior of pricing and inventory policies. We also investigate the effect of key product characteristics such as deterioration in age, cost of shortage in remanufacturable product inventory, and key market characteristics such as relative willingness-to-pay for buying a remanufactured product and relative willingness-to-pay for leasing a new product on optimal pricing policies through a computational study.     

An economic production and remanufacturing model with learning effects

Managing inventory in reverse logistics has been receiving much attention in recent years. One inventory problem that has been of interest to researchers is the production and remanufacturing model, where used items are collected and remanufactured to satisfy customer demand. The available models in the literature do not discuss the learning effects in production and remanufacturing processes. This may not be true in industries where labour costs and learning costs are expensive. By modelling these learning effects, management may use established learning models to better utilize capacity, manage inventories, and coordinate production and distribution throughout the chain.This paper extends the production, remanufacture, and waste disposal model by assuming learning to occur in both production and remanufacturing processes. However, this paper also assumes that improvements due to learning require capital investment. Mathematical models are developed, numerical examples are provided, and results are discussed.     

Optimal replenishment policy for a deteriorating green product: Life cycle costing analysis

In the last few decades, enterprises are trying to recycle the used products to reduce the negative impact on environment. This results in the need for reverse supply chain model especially for deteriorating items. The proposed model considers vendor managed inventory strategy and conducts a life cycle cost and benefit analysis for green electronic products. The mathematical analytical approach is used to solve the proposed model. The results show that selling price, deteriorating rate, holding cost, product return rate, and remanufactured quality have a significant effect on the model. This study can provide managerial insights into improving the design of a green-supply chain.     

An inventory model with deteriorating items under inflation when a delay in payment is permissible

This study develops an inventory model for initial-stock-dependent consumption rate when a delay in payment is permissible. In the inventory model, shortages are not allowed. The effect of the inflation rate, deterioration rate, initial-stock-dependent consumption rate and delay in payment are discussed. In the study, mathematical models are also derived under two different circumstances, i.e., Case I: The credit period is less than or equal to the cycle time for settling the account; and Case II: The credit period is greater than the cycle time for settling the account. Besides, expressions for an inventory system's total cost are derived for these two cases. Moreover, a computational procedure and GINO (Lasdon et al., ACM Transactions Mathematical Software 4 (1978) 34–50) are proposed to obtain the optimal order size and cycle time. The results can help managers determine the optimal total cost. Finally, a numerical example demonstrates the applicability of the proposed model.     

An optimal production run length in deteriorating production processes

This paper presents an EMQ model which determines an optimal production run length in a deteriorating production process. It is assumed that the process is subject to a random deterioration from an in-control state to an out-of-control state with an arbitrary distribution and, thus, produces some proportion of defective items. An optimal production run length and a minimum average cost are derived in three deteriorating processes: constant, linearly increasing, and exponentially increasing. Several different results from those of previous research are obtained and discussed. A numerical experiment is carried out to see the behavior of the proposed model depending on the cost and process parameters. Some interesting behaviors are observed.     

German inventory to sales ratios 1971-2005—An empirical analysis of business practice

The purpose of this study is to test empirically for the first time the general hypothesis that inventory to sales ratios have decreased over time in the German economy. Although inventory reduction has been a prevalent topic in the production and operations management literature, there is a lack of empirically confirmed answers to questions. They are as follows: Have inventories in German firms decreased overall during the past decades? What sectors of German industry are leading (lagging behind) inventory reduction? Has inventory reduction developed differently for raw materials, work-in-process, or finished goods? In which periods was marginal inventory reduction greatest? To the best of our knowledge, this empirical study is the first to broadly investigate inventory development from the 1970s until the present for a major European economy, Germany, and will provide the first answers to the research questions stated above using aggregate industry-level data provided by the Deutsche Bundesbank. We show that inventory levels decreased overall in many sectors of German industry. This reduction was mainly marked for raw materials and finished goods, particularly for the second-half of the time frame investigated.     

A production/remanufacture model with returns’ subassemblies managed differently

This paper considers a production-remanufacturing inventory model for a single product, where constant demand is satisfied from the inventory of newly produced and remanufactured items. Although the available models in the literature imply that collected used units (or returns) are disassembled for recovery purposes, these models really do not treat them as such. Contrary, the returns are assumed to be recovered as whole units, perhaps, for simplicity. This assumption may not capture the benefits reaped from product recovery programs. This paper addresses this limitation in the literature and assumes that each unit of a used product is collected and disassembled into components, where these components are sorted into subassemblies, which are fed back into the production-remanufacturing process. The returned subassemblies are remanufactured and reassembled to represent a second source of as-good-as-new units of the end-product. For this multi-component inventory problem, the question that needs to be answered is whether, or not, extreme strategies of either pure remanufacturing or pure production are more economical than a mixed strategy (one that combines both strategies). A mathematical model is developed that accounts for the inventories of subassemblies. The results suggested that not accounting for the disassembled components of a product leads to inappropriate inventory decisions that are not environmentally sound.     

Third-Degree Price Discrimination With Interdependent Demands

This paper analyzes the price, output, and welfare effects of third-degree price discrimination for a monopolist who sells in two interdependent markets. The case where the two goods sold by the monopolist are complements is analyzed as well as the more typical case where the two goods are substitutes. The economic effects of price discrimination are shown to depend on the type and strength of demand interdependence, the curvature of the demands and the slope of marginal cost. The circumstances under which price discrimination causes both market prices to either rise or fall are also analyzed.     

Distribution contracts to support optimal inventory holdings under demand uncertainty

When suppliers produce products for which demand is uncertain, they face a problem of inducing downstream distributors to stock inventory levels that the suppliers prefer. This paper considers a wide array of alternative supply contracts, each of which consists of a mixture of constant per-unit wholesale prices, buy-back arrangements, and post sale payments contingent on sales made, such as revenue sharing or buybacks. We show that linear supply contracts specifying any combination of two of these three instruments can implement the vertical integrated outcome for a monopoly, thereby generating the supplier's preferred inventory configuration and price distribution. We extend our results to differentiated product oligopoly, demonstrating that each supplier obtains its preferred inventory configuration and price distribution, given the choices of its rival. Distributors choose optimal inventories from the suppliers' standpoint, even if suppliers do not know the distribution of demand uncertainty, and, given the perfect competition among distributors, all profits in the supply chain are captured by suppliers. Thus, suppliers are able to deal with demand uncertainty with remarkably little information about demand, and without the need to control dealer actions in detail. In particular, suppliers need not specify either dealer inventories or resale prices, but instead encourage distributors to order based on information in their possession and to set prices that generate desirable resale price dispersion.     

An association clustering algorithm for can-order policies in the joint replenishment problem

Many studies have shown that the total cost of employing joint replenishment for correlated items is less than the total cost of using single-item replenishment. Savings increase dramatically when the demand between items is closely related. Although the benefits of joint replenishment are significant, it is difficult to define the demand correlation among items, especially when the number of items increases. A large number of items reduces the efficiency and advantage of the multi-item inventory control. To overcome this difficulty, an association clustering algorithm this paper proposes to evaluate the correlated demands among items. The proposed algorithm utilizes the “support” concept in association rule analysis to measure the similarity among items. Based on these measurements a clustering method is developed to group items with close demand in a hierarchal way. The can-order policy is then applied to the optimal clustering result as decided by the proposed performance index. To illustrate the benefits of the proposed association clustering algorithm for replenishment systems, a set of simulations and a sensitivity analysis is conducted. The results of the experiments show that the proposed method outperforms several replenishment models.     

Inventory model for an inventory system with time-varying demand rate

The standard inventory problems of the multi-period have been modeled under different situations. Specifically we have considered the demand subjects of a continuous distribution and a discrete distribution, and whether the demand of each period is unchanged or not. A method to get an economic order quantity in inventory systems with discrete and unchanged demand was presented in a previous paper, and this method has been generalized to an inventory model with varying continuous demand. However, it was not achieved due to there being many classified cases in the general situations. In this article the above method is discussed in the case discrete demand to determine whether it increases or decreases from period to period. A theoretical method is presented by using previous results and some examples are given which suggest how the concept can handle on inventory system. In order to make the decision, an algorithm is also presented under some conditions, and examples are shown by using the computer software program, Mathematica, which helps to explain the findings. In general cases, we view the optimal policy in the inventory problems in only a few periods.     

Production lot sizing with a secondary outsourcing facility

An extended economic production quantity (EPQ) model under stochastic demand is investigated in this paper, where a fixed lot sizing policy is implemented to reduce the complexity of production planning and inventory control, and outsourcing with a secondary facility is used to supplement the lot sizing policy and to cope with the random demand. The considered cost includes: setup cost for the batch production, inventory carrying cost, backorder cost when the demand cannot be met immediately during the production period, and outsourcing cost when the total demand is greater than the lot size in one replenishment cycle. Under some mild conditions, the expected cost per unit time can be shown to be convex. Extensive computational tests have illustrated that the average cost reduction of the proposed model is significant when compared with that of the classical lot sizing policy. Significant cost savings can be achieved by deploying the production lot sizing policy with an outsourcing strategy when the mean demand rate is high.     

Optimal production–inventory strategies for a HMMS-type reverse logistics system

The aim of the paper is to find optimal inventory policies in a reverse logistics system with special structure. It is assumed that demand is a known continuous function in a given planning horizon and return rate of used items is a given function. There is a constant delay between the using and return process. We investigate two stores. The demand is satisfied from the first store, where the manufactured and remanufactured items are stored. The returned products are collected in the second store and then remanufactured or disposed. The costs of this system consist of the quadratic holding costs for these two stores and the quadratic manufacturing, remanufacturing and disposal costs.The model is represented as an optimal control problem with two state variables (inventory status in the first and second store) and with three control variables (rate of manufacturing, remanufacturing and disposal). The objective is to minimize the sum of the quadratic deviation from described inventory levels in stores and from described manufacturing, remanufacturing and disposal rates. In this form, the model can be considered as a generalization of the well-known Holt et al. (Planning Production, Inventories, and Work Forces, Prentice-Hall, Englewood Cliffs, NJ, 1960) model with two warehouses. After solving the problem, we give some numerical examples to represent the optimal path in dependence of the demand rates.     

A heuristic algorithm for a multi-product dynamic lot-sizing and shipping problem

This paper analyzes a dynamic lot-sizing problem, in which the order size of multiple products and a single container type are simultaneously considered. In the problem, each order (product) placed in a period is immediately shipped by some containers in the period and the total freight cost is proportional to the number of containers used. It is also assumed that backlogging is not allowed. The objective of this study is to simultaneously determine the lot-sizes and the transportation policy that minimizes the total costs, which consist of production cost, inventory holding cost, and freight cost. Because this problem is NP-hard, a heuristic algorithm with an adjustment mechanism is proposed based on the optimal solution properties. The computational results from a set of simulation experiment are also presented.