Deterministic lot-size inventory models with shortages and deterioration for fluctuating demand

We establish various inventory replenishment policies. We then analytically identify the best alternative among them based on the minimum total relevant costs. Finally, we prove that the relevant cost is convex with the number of replenishments. Consequently, the search for the optimal replenishment number is reduced to finding a local minimum.     

An economic lot-size model with partial backlogging hinging on waiting time and shortage period

We study a continuous review inventory control system over a infinite-horizon with deterministic demand where shortage is partially backlogged. The backlogging is characterized using an approach in which the collective behavior of customers hinges on the waiting time and on the shortage period. We assume that the fraction of the customers who are not willing to wait is proportional to the ratio between the waiting time and the length of the shortage cycle. Taking into account the above assumption we determine the optimal inventory policy. Several models studied by other authors result to be particular cases of the considered model.     

An economic production quantity model with a positive resetup point under random demand

In this paper, an extended economic production quantity (EPQ) model is investigated, where demand follows a random process. This study is motivated by an industrial case for precision machine assembly in the machinery industry. Both a positive resetup point s and a fixed lot size Q are implemented in this production control policy. To cope with random demand, a resetup point, i.e., the lowest inventory level to start the production, is adapted to minimize stock shortage during the replenishment cycle. The considered cost includes setup cost, inventory carrying cost, and shortage cost, where shortage may occur at the production stage and/or at the end of one replenishment cycle. Under some mild conditions, the expected cost per unit time can be shown to be convex with respect to decision parameters s and Q. Further computational study has demonstrated that the proposed model outperforms the classical EPQ when demand is random. In particular, a positive resetup point contributes to a significant portion of this cost savings when compared with that in the classical lot sizing policy.     

An inventory model of deteriorating items with lot-size dependent replenishment cost and a linear trend in demand

This paper presents an inventory model for deteriorating items over a finite time horizon where the demand increases linearly with time. The method is developed by assuming that the successive replenishment cycle lengths are the same. Many O.R. scientists/researchers obtained an optimal replenishment schedule where the replenishment cost is constant in each cycle length over the finite time horizon. In this paper, we relax the assumption of fixed replenishment cost. The replenishment cost per replenishment is taken to be linearly dependent on the lot-size of that replenishment. Shortages are allowed and are fully backlogged. As a special case, the results for the model without shortages are derived. Finally, two numerical examples are presented to illustrate the model.     

An economic production quantity inventory model involving fuzzy demand rate and fuzzy deterioration rate

Generally, in deriving the solution of economic production quantity (EPQ) inventory model, we consider the demand rate and deterioration rate as constant quantity. But in case of real life problems, the demand rate and deterioration rate are not actually constant but slightly disturbed from their original crisp value. The motivation of this paper is to consider a more realistic EPQ inventory model with finite production rate, fuzzy demand rate and fuzzy deterioration rate. The effect of the loss in production quantity due to faulty/old machine have also been taken into consideration. The methodology to obtain the optimum value of the fuzzy total cost is derived and a numerical example is used to illustrate the computation procedure. A sensitivity analysis is also carried out to get the sensitiveness of the tolarance of different input parameters.     

An inventory model for non-instantaneous deteriorating items with quadratic demand rate and shortages under trade credit policy

noindent In this paper, we propose an appropriate inventory model for non-instantaneous deteriorating items over quadratic demand rate with permissible delay in payments and time dependent deterioration rate. In this model, the completely backlogged shortages are allowed. In several existing results, the authors discussed that the deterioration rate is constant in each cycle. However, the deterioration rate of items are not constant in real world applications. Motivated by this fact, we consider that the items are deteriorated with respect to time. To minimize the total relevant inventory cost, we prove some useful theorems to illustrate the optimal solutions by finding an optimal cycle time with the necessary and enough conditions for the existence and uniqueness of the optimal solutions. Finally, we discuss the numerical instance and sensitivity of the proposed model.     

An integrated economic lot-size model for vendor–buyer inventory system when input is random

An integrated strategy is discussed for both vendor and buyer when the input is random. It is shown numerically that the cooperative approach is beneficial to reduce the cost when compared with an independent decision by both the parties. Though the integrated total cost decreases, the buyer’s cost increases due to random input in his inventory. To encourage the buyer to order a large quantity, a trade credit is offered by the vendor to the buyer to settle the account. A conciliation factor is suggested to share the benefits.     

Deterministic economic production quantity models with time-varying demand and cost

In today’s time-based competition, the unit cost of a high-tech product declines significantly over its short product life cycle. Consequently, in this paper, we relax the traditional economic production quantity model to allow for time-varying cost. We then prove that the optimal production schedule uniquely exists. In addition, we also show that the total cost is a convex function of the number of replenishments, which reduces the search for the optimal solution to finding a local minimum. Furthermore, we characterize the influences of both demand and cost over the length of production run time and the economic production quantity.     

An economic production lot size model in an imperfect production system

The paper investigates an EPL (Economic Production Lotsize) model in an imperfect production system in which the production facility may shift from an ‘in-control’ state to an ‘out-of-control’ state at any random time. The basic assumption of the classical EPL model is that 100% of produced items are perfect quality. This assumption may not be valid for most of the production environments. More specifically, the paper extends the article of Khouja and Mehrez [Khouja, M., Mehrez, A., 1994. An economic production lot size model with imperfect quality and variable production rate. Journal of the Operational Research Society 45, 1405–1417]. Generally, the manufacturing process is ‘in-control’ state at the starting of the production and produced items are of conforming quality. In long-run process, the process shifts from the ‘in-control’ state to the ‘out-of-control’ state after certain time due to higher production rate and production-run-time.The proposed model is formulated assuming that a certain percent of total product is defective (imperfect), in ‘out-of-control’ state. This percentage also varies with production rate and production-run time. The defective items are restored in original quality by reworked at some costs to maintain the quality of products in a competitive market. The production cost per unit item is convex function of production rate. The total costs in this investment model include manufacturing cost, setup cost, holding cost and reworking cost of imperfect quality products. The associated profit maximization problem is illustrated by numerical examples and also its sensitivity analysis is carried out.     

A two warehouse inventory model for deteriorating items with a linear trend in demand and shortages

In this paper, we develop a deterministic inventory model with two warehouses (one is the existing storage known as own warehouse (OW) and the other is hired on rental basis known as rented warehouse (RW). The model allows different levels of item deterioration in both warehouses. The demand rate is supposed to be a linear (increasing) function of time and the replenishment rate is infinite. The stock is transferred from RW to OW in continuous release pattern and the associated transportation cost is taken into account. Shortages in OW are allowed and excess demand is backlogged. For the general model, we give the equations for the optimal policy and cost function and we discuss some special cases. A numerical example is given to illustrate the solution procedure of the model. Finally, based on this example, we conduct a sensitivity analysis of the model.     

The economic production and pricing model with lot-size-dependent production cost

In this paper, the economic production quantity problem for a single-product single-machine system is extended. It is assumed that annual demand of the product is a function of price set by manufacturer. This extension considers sales revenue, inventory and setup costs as well as a variable cost of production which is a function of the lot size. Several linear and non-linear functions of demand and variable cost are considered in this paper and a global solution methodology is presented for the models developed. Newton??s method is used to find local optima and asymptotic convergence of the solution algorithm to a global optimum is proved. Numerical studies followed by a discussion provide additional insights into the problem.     

Inventory models with ramp type demand rate,time dependent deterioration rate,unit production cost and shortages

Manna and Chaudhuri (Eur. J. Oper. Res. 171(2):557–566, 2006) presented a production-inventory system for deteriorating items with demand rate being a linearly ramp type function of time and production rate being proportional to the demand rate. The two models without shortages and with shortages were discussed. Both models were studied assuming that the time point at which the demand is stabilized occurs before the production stopping time. In this paper, we complete this model by considering that: (a) for the model with no shortages; the demand rate is stabilized after the production stopping time and (b) for the model with shortages; the demand rate is stabilized after the production stopping time or after the time when the inventory level reaches zero or after the production restarting time. In addition, we extend the work of Manna and Chaudhuri (Eur. J. Oper. Res. 171(2):557–566, 2006) assuming a general function of time for the variable part of the demand rate.     

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.     

Partial backlogging inventory lot-size models for deteriorating items with fluctuating demand under inflation

In this paper, the traditional inventory lot-size model is extended to allow not only for general partial backlogging rate but also for inflation. The assumptions of equal cycle length and constant shortage length imposed in the model developed by Moon et al. [Moon, I., Giri, B.C., Ko, B., 2005. Economic order quantity models for ameliorating/deteriorating items under inflation and time discounting, European Journal of Operational Research 162(3), 773–785] are also relaxed. For any given number of replenishment cycles the existence of a unique optimal replenishment schedule is proved and further the convexity of the total cost function of the inventory system in the number of replenishments is established. The theoretical results here amend those in Yang et al. [Yang, H.L., Teng, J.T., Chern, M.S., 2001. Deterministic inventory lot-size models under inflation with shortages and deterioration for fluctuating demand, Naval Research Logistics 48(2), 144–158] and provide the solution to those two counterexamples by Skouri and Papachristos [Skouri, K., Papachristos, S., 2002. Note on “deterministic inventory lot-size models under inflation with shortages and deterioration for fluctuating demand” by Yang et al. Naval Research Logistics 49(5), 527–529.]. Finally we propose an algorithm to find the solution, and obtain some managerial results by using sensitivity analyses.     

Two-echelon inventory model for deteriorating items with credit period dependent demand including shortages under trade credit

In real life situation, it is observed that demand of an item depends on the length of the credit period offered by the retailer to his customers which has a positive impact on demand of an item. But the impact of credit period on demand has received a very little attention by researchers. Furthermore, by allowing shortages as backlogging, the impact on the cost from the decay of the products can be balanced out. A profitable decision policy between a supplier and the retailers can be characterized by an agreement on the permissible delay in payments. Recently, Jaggi et al. (Eur J Oper Res 190:130–135, 2008) have investigated the impact of credit linked demand on the retailer’s optimal replenishment policy. The objective of this study is to extend Jaggi et al. (Eur J Oper Res 190:130–135, 2008) model by incorporating deterioration and backlogging. That is, we formulate a two-echelon inventory model for deteriorating items with credit period dependent demand including shortages under two-level trade credit financing and determine the retailer’s optimal replenishment policy when both the supplier as well as the retailer offers the credit period to stimulate customer demand. Furthermore, we establish some useful theorems to characterize the optimal solution and provide an easy and useful computational algorithm with the help of computer code using the software Matlab 7.0 to determine the optimal shortage point, cycle length, ordering quantity and credit period. A numerical example is included to illustrate the solution procedure for the mathematical model developed. Finally, we implement sensitivity analysis of the optimal solution with respect to the major parameters of the system and obtain some important managerial insights.     

An impulse control problem of a production model with interruptions to follow stochastic demand

The paper deals with the stochastic optimal intervention problem which arises in a production & storage system involving identical items. The requests for items arrive at random and the production of an item can be interrupted during production to meet the corresponding demand. The operational costs considered are due to the stock/backlog, running costs and set up costs associated to interruptions and re-initializations. The process presents distinct behaviour on each of two disjoint identical subsets of the state space, and the state process can only be transferred from one subset to the other by interventions associated to interruptions/re-initializations. A characterization is given in terms of piecewise deterministic Markov process, which explores the aforementioned structure, and a method of solution with assured convergence, that does not require any special initialization, is provided.Additionally, we demonstrate that under conditions on the data, the optimal policy is to produce the item completely in a certain region of the state space of low stock level.     

An EOQ model with state-dependent demand rate

We present an EOQ-type model in which the demand rate is a function of the inventory level. The demand functions and holding cost functions considered are quite general. In particular, we focus on two types of demand rate functions: a piecewise constant function and a family of exponential functions. The paper includes analysis of the problem, optimization and sensitivity analysis. We also extend the model by adding randomness to the order sizes.     

An elaborative unit cost structure-based fuzzy economic production quantity model

The purpose of this paper is to investigate and propose a fuzzy extended economic production quantity model based on an elaboratively modeled unit cost structure. This unit cost structure consists of the various lot-size correlative components such as on-line setups, off-line setups, initial production defectives, direct material, labor, and depreciation in addition to lot-size non-correlative items. Thus, the unit cost is correlatively modeled to the production quantity. Therefore, the modeling or the annual total cost function developed consists of not only annual inventory and setup costs but also production cost. Moreover, via the concept of fuzzy blurred optimal argument and the vertex method of the α-cut fuzzy arithmetic (or fuzzy interval analysis), two solution approaches are proposed: (1) a fuzzy EPQ and (2) a compromised crisp EPQ in the fuzzy sense. An optimization procedure, which can simultaneously determine the α-cut-vertex combination of fuzzy parameters and the optimizing decision variable value, is also proposed. The sensitivity model for the fuzzy total cost and thus EPQ to the various cost factors is provided. Finally, a numerical example with the original data collected from a firm demonstrates the usefulness of the new model.