Luis A. San-José

The integrality of the lot size in the basic EOQ and EPQ models: Applications to other production-inventory models

In this paper we present a method to obtain the solution of the classic economic order quantity (EOQ) and economic production quantity (EPQ) models when the lot size must be an integer quantity. This approach is operatively very simple and allows obtaining a rule to discriminate between the situation in which the optimal solution is unique and when there are two optimal solutions. Also, this method is applicable to the resolution of other production-inventory models. We expose some of them and illustrate the use of the method with numerical examples.

Maximizing profits in an inventory model with both demand rate and holding cost per unit time dependent on the stock level

We study an EOQ inventory model with demand rate and holding cost rate per unit time, both potentially dependent on the stock level. The ordering cost, the holding cost and the gross profit from the sale of the item are considered. The objective is to maximize the average profit per unit time. We present the analytical formulation of the problem and demonstrate the existence and uniqueness of the optimal cycle time, giving a numerical algorithm to obtain it. Moreover, we provide two fundamental theoretical results: a rule to check when a given cycle time is the optimal policy, and a necessary and sufficient condition for the profitability of the system. Several EOQ models analyzed by other authors are particular cases of the one here studied. We present some numerical examples to illustrate the proposed algorithm and analyze the sensitivity of the optimal solution with respect to changes in various parameters of the system.

A general model for EOQ inventory systems with partial backlogging and linear shortage costs

We present a mathematical model which generalises several known deterministic Economic Order Quantity (EOQ) inventory systems with partial backlogging. This inventory model considers purchasing cost, holding cost, shortage costs and replenishment cost. Shortage costs (backorder cost and lost sales cost) are both made up of a fixed cost and a variable cost which depends on the length of the waiting time for the next replenishment. The optimal policy is characterised through a sequential optimisation procedure. To illustrate the model, numerical examples and sensitivity results are given.

Optimal policy for an inventory system with backlogging and all-units discounts: Application to the composite lot size model

This paper examines an inventory model with full backlogging and all-units quantity discounts. The practical scenario of a salesperson offering compensation to a client so as not to lose the sale is considered. The cost of a backorder thus includes both a fixed cost and a further cost which is proportional to the length of time the said backorder exists. A first algorithm is developed to determine the optimal policy while some extensions to this algorithm are obtained that include additional conditions on the model. In particular, the well known composite lot size model, developed by Tersine, is solved, incorporating a new stockout cost and a new all-units discount. Numerical examples are provided to illustrate the application of the algorithms.

Optimal policy for profit maximising in an EOQ model under non-linear holding cost and stock-dependent demand rate

In this article, we integrate a non-linear holding cost with a stock-dependent demand rate in a maximising profit per unit time model, extending several inventory models studied by other authors. After giving the mathematical formulation of the inventory system, we prove the existence and uniqueness of the optimal policy. Relying on this result, we can obtain the optimal solution using different numerical algorithms. Moreover, we provide a necessary and sufficient condition to determine whether a system is profitable, and we establish a rule to check when a given order quantity is the optimal lot size of the inventory model. The results are illustrated through numerical examples and the sensitivity of the optimal solution with respect to changes in some values of the parameters is assessed.

An inventory model where backordered demand ratio is exponentially decreasing with the waiting time

We analyze an inventory system with a mixture of backorders and lost sales, where the backordered demand rate is an exponential function of time the customers wait before receiving the item. Stockout costs (backorder cost and lost sales cost) include a fixed cost and a cost proportional to the length of the shortage period. A procedure for determining the optimal policy and the maximum inventory profit is presented. This work extends several inventory models of the existing literature.

An inventory model with -rational type-backlogged demand rate and quadratic backlogging cost

In this paper, we consider a single item inventory model with a mixture of backorders and lost sales, in which the demand during the stockout period is partially backlogged according to a rational type function. The parameters associated with the purchasing cost, selling price, ordering cost, goodwill cost, demand rate and holding cost per unit and unit time are constant, but the cumulative backordering cost per unit is a quadratic function of the length of time for which the backorder exists. A procedure is developed for determining the optimal policy and the minimum total inventory cost. Illustrative numerical examples, which help us to understand the theoretical results, are also presented.

A newsvendor inventory model with an emergency order to supply a non-increasing fraction of shortage

Abstract This paper generalizes the newsvendor inventory model when the possibility of an emergency order to satisfy the excess of demand exists. In this situation, we assume that there are impatient customers who do not wait for the emergency order and other customers who are willing to wait to satisfy their demand. We consider that the fraction of shortage that is satisfied with delay through the emergency order is described by a function, which is non-increasing with respect to the amount of shortage. Our objective is to determine the optimal order quantity, which maximizes the relevant expected total profit for the period, when the demand follows a uniform probability distribution. As is well known, this distribution is usually used to denote the results of a demand whose values are unknown, except for the fact that such results belong to an interval. The uniqueness and existence of optimal decisions are proved, and a procedure to determine the optimal policy and the maximum expected profit is developed. Illustrative examples, which help us to understand the theoretical results, are also given.

A lot sizing model with advance payment and planned backordering

One of the assumptions in the classic economic order quantity model is that the buyer pays the purchasing cost for an order immediately after receiving the goods. In the real world, the vendors sometimes offer the buyers to pay all or a proportion of the purchasing cost after receiving the items to encourage them to increase their orders (i.e., delayed payment). Furthermore, in some cases, the powerful vendors may ask the buyers to prepay the entire, or a fraction of the purchasing cost before the delivery to mitigate the risk of cancellation or procuring the primary parts (i.e., advance payment). Advance payment is widely used by firms, but its effects on customer’s inventory decisions are seldom discussed. In this paper, we investigate the customer’s inventory policy by considering two different conditions: (a) full prepayment with shortage, (b) partial prepayment–partial delayed payment with shortage. We discuss the effect of parameters such as price discount linked to prepayment and length of prepayment on optimal periods of inventory and shortage quantity. The conclusions shows that the length of the advance payment period does not influence the inventory cycle, but that parameters such as discount factor linked to advance payment and delayed payment period affect the optimal inventory cycle in all cases. Numerical examples and a sensitivity analysis are presented to demonstrate the performance of the model and the results.

An economic order quantity model with nonlinear holding cost, partial backlogging and ramp-type demand

ABSTRACT In this article, a deterministic inventory model with a ramp-type demand depending on price and time is developed. The cumulative holding cost is assumed to be a nonlinear function of time. Shortages are allowed and are partially backlogged. Thus, the fraction of backlogged demand depends on the waiting time and on the stock-out period. The aim is to maximize the total profit per unit time. To do this, a procedure that determines the economic lot size, the optimal inventory cycle and the maximum profit is presented. The inventory system studied here extends diverse inventory models proposed in the literature. Finally, some numerical examples are provided to illustrate the theoretical results previously propounded.