On an inventory model for deteriorating items and time-varying demand

In this paper, we present an optimal procedure for finding the replenishment schedule for the inventory system in which items deteriorate over time and demand rates are increasing over a known and finite planning horizon.     

A note on the inventory models for deteriorating items with ramp type demand rate

In this research we study the inventory models for deteriorating items with ramp type demand rate. We first clearly point out some questionable results that appeared in (Mandal, B., Pal, A.K., 1998. Order level inventory system with ramp type demand rate for deteriorating items. Journal of Interdisciplinary Mathematics 1, 49–66 and Wu, K.S., Ouyang, L.Y., 2000. A replenishment policy for deteriorating items with ramp type demand rate (Short Communication). Proceedings of National Science Council ROC (A) 24, 279–286). And then resolve the similar problem by offering a rigorous and efficient method to derive the optimal solution. In addition, we also propose an extended inventory model with ramp type demand rate and its optimal feasible solution to amend the incompleteness in the previous work. Moreover, we also proposed a very good inventory replenishment policy for this kind of inventory model. We believe that our work will provide a solid foundation for the further study of this sort of important inventory models with ramp type demand rate.     

Optimal policy for deteriorating items with trapezoidal type demand and partial backlogging

Inventory model for time-dependent deteriorating items with trapezoidal type demand rate and partial backlogging is considered in this paper. The demand rate is defined as a continuous trapezoidal function of time, and the backlogging rate is a non-increasing exponential function of the waiting time up to the next replenishment. We proposed an optimal replenishment policy for such inventory model, numerical examples to illustrate the solution procedure.     

Optimal replenishment policies for deteriorating items with stock sensitive demand under two-level trade credit and limited capacity

Recently, Min et al. [18] established an inventory model for deteriorating items under stock-dependent demand and two-level trade credit and obtained the optimal replenishment policy. Their analysis imposed a terminal condition of zero ending-inventory. However, with a stock-dependent demand, it may be desirable to order large quantities, resulting in stock remaining at the end of the cycle, due to the potential profits resulting from the increased demand. As a result, to make the theory more applicable in practice, we extend their model to allow for: (1) an ending-inventory to be nonzero, (2) a maximum inventory ceiling to reflect the facts that too much stock leaves a negative impression on the buyer and the amount of shelf/display space is limited.     

Optimal deteriorating items production inventory models with random machine breakdown and stochastic repair time

This study develops deteriorating items production inventory models with random machine breakdown and stochastic repair time. The model assumes the machine repair time is independent of the machine breakdown rate. The classical optimization technique is used to derive an optimal solution. A numerical example and sensitivity analysis are shown to illustrate the models. The stochastic repair models with uniformly distributed repair time tends to have a larger optimal total cost than the fixed repair time model, however the production up time is less than the fixed repair time model. Production and demand rate are the most sensitive parameters for the optimal production up time, and demand rate is the most sensitive parameter to the optimal total cost for the stochastic model with exponential distribution repair time.     

An optimal production-inventory model for deteriorating items with multiple-market demand

The global markets of today offer more selling opportunities to the deteriorating items’ manufacturers, but also pose new challenges in production and inventory planning. From a production management standpoint, opportunities to exploit the difference in the timing of the selling season between geographically dispersed markets for deteriorating items are important to improving a firm’s profitability. In this paper, we examined the above issue with an insightful production-inventory model of a deteriorating items manufacturer selling goods to multiple-markets with different selling seasons. We also provided a solution procedure to find the optimal replenishment schedule for raw materials and the optimal production plan for finished products. A numerical example was then used to illustrate the model and the solution procedure. Finally, sensitivity analysis of the optimal solution with respect to major parameters was carried out.     

Deterministic inventory model for deteriorating items with capacity constraint and time-proportional backlogging rate

In this paper, a deterministic inventory model for deteriorating items with two warehouses is developed. A rented warehouse is used when the ordering quantity exceeds the limited capacity of the owned warehouse, and it is assumed that deterioration rates of items in the two warehouses may be different. In addition, we allow for shortages in the owned warehouse and assume that the backlogging demand rate is dependent on the duration of the stockout. We obtain the condition when to rent the warehouse and provide simple solution procedures for finding the maximum total profit per unit time. Further, we use a numerical example to illustrate the model and conclude the paper with suggestions for possible future research.     

The simplified solution procedure for deteriorating items under stock-dependent demand and two-level trade credit in the supply chain management

Min et al. [1] (J. Min, Y.W. Zhou, J. Zhao, An inventory model for deteriorating items under stock-dependent demand and two-level trade credit, Appl. Math. Model. 34 (2010) 3273–3285.) develop an inventory model for deteriorating items under stock-dependent demand and two-level trade credit. They provide the necessary and sufficient conditions of the existence and uniqueness of the optimal solutions that could maximize the retailer’s average profit per unit time. Basically, their paper is correct and interesting. Recently, several researchers have been showing a huge interest in developing simple and easy to implement solution procedures in management science. Therefore this paper indicates that Min et al.’s solution procedure can be further improved and simplified. So, the main purpose of this paper is to present simple and easy to understand solution procedures to locate the optimal solutions of an inventory model that considers deteriorating items under stock-dependent demand and two-level trade credit.     

Continuous-review,lost-sales inventory models with Poisson demand,a fixed lead time and no fixed order cost

This paper considers continuous-review lost-sales inventory models with no fixed order cost and a Poisson demand process. There is a holding cost per unit per unit time and a lost sales cost per unit. The objective is to minimise the long run total cost. Base stock policies are, in general, sub-optimal under lost sales. The optimal policy would have to take full account of the remaining lead times on all the orders currently outstanding and such a policy would be too complex to analyse, let alone implement. This paper considers policies which make use of the observation that, for lost sales models, base stock policies can be improved by imposing a delay between the placement of successive orders. The performance of these policies is compared with that of the corresponding base stock policy and also with the policy of ordering at fixed and regular intervals of time.     

Constraint programming for computing non-stationary (R, S) inventory policies

This paper proposes a constraint programming model for computing the finite horizon single-item inventory problem with stochastic demands in discrete time periods with service-level constraints under the non-stationary version of the “periodic review, order-up-to-level” policy (i.e., non-stationary (R, S) or, simply (R_n, S_n)). It is observed that the modeling process is more natural and the required number of variables is smaller compared to the MIP formulation of the same problem. The computational tests show that the CP approach is more tractable than the conventional MIP formulation. Two different domain reduction methods are proposed to improve the computational performance of solution algorithms. The numerical experiments confirmed the effectiveness of these methods.     

Reducing lost-sales rate in (T, R, L) inventory model with controllable lead time

In order to establish a good image and to enhance customer’s loyalty, many efforts such as upgrading the servicing facilities, maintaining a high quality of products and increasing expenditure on advertisement could be made by a selling shop. Naturally, an extra-added cost must be spent for these efforts and it is expected to have a result to reduce the shortage cost of lost-sales and the total expected annual cost. This paper explores a probabilistic inventory model with optimal lost-sales caused by investment due to two different types of cost functions. We consider that the lead time can be shortened at an extra crashing cost, which depends on the length of the lead time. Moreover, we assume that the lost-sales rate can also be reduced by capital investment. The purpose of this paper is to establish a (T, R, L) inventory model with controllable lead time and to analyze the effects of increasing two different types of investments to reduce the lost-sales rate, in which the review period, lead time and lost-sales rate are treated as decision variables. We first formulate the basic periodic review model mathematically with the capital investment to reduce lost-sales rate. Then two models are discussed, one with normally distributed protection interval demand and another with distribution-free case. For each model, two investment cost functional forms, logarithmic and power, are employed for lost-sales rate reduction. Two computational algorithms with the help of the software Matlab are furnished to determine the optimal solution. In addition, six numerical examples and sensitivity analysis are presented to illustrate the theoretical results and obtain some managerial insights. Finally, the effect of lost-sales rate reduction is investigated. By framing this new model, we observe that a significant amount of savings can be easily achieved to increase the competitive edge in business. The results in the numerical examples indicate that the savings of expected annual total cost are realized through lost-sales reduction.     

A probabilistic scheduling period inventory model for deteriorating items with lead time

A probabilistic scheduling period inventory model is developed for continuously decaying items. The model assumes no shortages, deterministic lead time and a general deterioration function. The developed model is shown to be related to the similar model without lead time and also to the similar model for non-deteriorating items. Two special cases are considered and an example is also furnished.

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Zusammenfassung Es wird ein probabilistisches Lagerhaltungsmodell für sich stetig verschlechternde Güter entwickelt. Im Modell wird angenommen, da\ kein Mangel eintritt, die Verschlechterung durch eine allgemeine Funktion beschrieben wird und deterministische Lieferzeiten vorliegen. Es wird gezeigt, da\ das entwickelte Modell bekannte Modelle ohne Lieferzeiten sowie für sich nicht verschlechternde Güter als Spezialfälle enthält. Ferner werden zwei weitere Spezialfälle und ein numerisches Beispiel angegeben.

     

The inventory model for trade credit in economic ordering policies of deteriorating items in a supply chain system

In 1994, professors Jaggi and Aggarwal presented the economic ordering policies of deteriorating items in the presence of trade credit using a discounted cash-flows (DCF) approach. This paper discusses the same problem as that of Jaggi and Aggarwal and indicates that some approximations to the optimal cycle times proposed by Jaggi and Aggarwal are inappropriate sometimes. A theorem is derived out to find the optimal cycle time. With that theorem, a simple algorithm is developed to locate the optimal cycle time.     

Seller’s optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime

Due to evaporation, obsolescence, spoilage, etc., some products (e.g., fruits, vegetables, pharmaceuticals, volatile liquids, and others) not only deteriorate continuously but also have their expiration dates. To attract new buyers and increase sales, a seller frequently offers its buyers a trade credit period to settle the purchase amount. There is no interest charge to a buyer if the purchasing amount is paid within the credit period, and vice versa. On the other hand, granting a credit period from a seller to its buyers increases default risk. In this paper, we propose an economic order quantity model for a seller by incorporating the following relevant facts: (1) deteriorating products not only deteriorate continuously but also have their maximum lifetime, and (2) credit period increases not only demand but also default risk. We then characterize the seller’s optimal credit period and cycle time. Furthermore, we discuss a special case for non-deteriorating items. Finally, we run several numerical examples to illustrate the problem and provide some managerial insights.     

A (Q, R) inventory replenishment model with two delivery modes

Although splitting shipments across multiple delivery modes typically increases total shipping costs as a result of diseconomies of scale, it may offer certain benefits that can more than offset these costs. These benefits include a reduction in the probability of stockout and in the average inventory costs. We consider a single-stage inventory replenishment model that includes two delivery modes: a cheaper, less reliable mode, and another, more expensive but perfectly reliable mode. The high-reliability mode is only utilized in replenishment intervals in which the lead time of the less-reliable mode exceeds a certain value. This permits substituting the high-reliability mode for safety stock, to some degree. We characterize optimal replenishment decisions with these two modes, as well as the potential benefits of simultaneously using two delivery modes.     

A two-echelon inventory model for a deteriorating item with stock-dependent demand,partial backlogging and capacity constraints

This study investigates a two-echelon supply chain model for deteriorating inventory in which the retailer’s warehouse has a limited capacity. The system includes one wholesaler and one retailer and aims to minimise the total cost. The demand rate in retailer is stock-dependent and in case of any shortages, the demand is partially backlogged. The warehouse capacity in the retailer (OW) is limited; therefore the retailer can rent a warehouse (RW) if needed with a higher cost compared to OW. The optimisation is done from both the wholesaler’s and retailer’s perspectives simultaneously. In order to solve the problem a genetic algorithm is devised. After developing a heuristic a numerical example together with sensitivity analysis are presented. Finally, some recommendations for future research are presented.     

Inventory model with deteriorating items,ramp-type demand and partially backlogged shortages for a two warehouse system

In this paper, we consider a two warehouse inventory model, an owned one (OW) and a rented one (RW). Inventory deteriorates in the two warehouses at different constant rates, demand rate is a general ramp-type function of time and shortages are partially backlogged at a constant rate. Existence and uniqueness of the optimal solution is discussed. An algorithm is developed to obtain the overall optimal replenishment policy, which would enable the manager to decide upon the feasibility of renting a warehouse. The dynamics of the model and application of the algorithm are demonstrated through numerical examples. Sensitivity analysis is conducted with respect to model parameters and some important observations are drawn.     

The determination of optimal partial fill policy for an inventory system with lumpy demand items

This paper presents a mathematical model developed for the synthesis of optimal replenishment policies for items that experience lumpy demands. In order to avoid disrupting the inventory system, a cutoff point of w units is introduced such that the system would only satisfy routinely customer orders with transaction sizes less than or equal to w units. For customer orders with transaction sizes larger than w units, the system would only supply the cutoff amount (w units). The excess units would be refused. The control discipline is the (s, S) inventory policy with continuous review, and the nature of the customer orders is approximated by a discrete stuttering Poisson distribution. The optimal values of the control parameters, w, s and S, are determined. The theoretical results obtained are illustrated with a numerical example.     

An algorithm for mean residual life computation of (n − k + 1)-out-of-n systems: An application of exponentiated Weibull distribution

In this paper, we establish an algorithm for the computation of the mean residual life of a (n − k + 1)-out-of-n system in the case of independent but not necessarily identically distributed lifetimes of the components. An application for the exponentiated Weibull distribution is given to study the effect of various parameters on the mean residual life of the system. Also the relationship between the mean residual life for the system and that of its components is investigated.     

On the empirical performance of (T, s, S) heuristics

The periodic (T,s,S)$(T\text{,}s\text{,}S)$ policies have received considerable attention from the academic literature. Determination of the optimal parameters is computationally prohibitive, and a number of heuristic procedures have been put forward. However, these heuristics have never been compared in an extensive empirical study. Such an investigation on 3055 SKUs is carried out in this paper. Our study provides insights into the performance of (T,s,S)$(T\text{,}s\text{,}S)$ heuristics, also in relation to demand forecasting. The results show that Naddor’s heuristic is best able to minimize the total cost. However, the normal and power approximations achieve more efficient solutions in that backorder volumes are smaller at the same inventory levels, indicating the potentially superior performance of these methods if the balancing of holding and backorder costs can be improved. The results also show that, for all heuristics, the SBA variant of the Croston forecasting method significantly outperforms Croston as well as Single Exponential Smoothing (SES).