قیمت بهینه و سایز تحت کمبود ترکیب سفارش معوقه جزئی،سفارش معوقه و هزینه های فروش از دست داده

In this paper, we consider the pricing and lot-sizing problem for a product subject to general rate of deterioration and partial backordering. We use impatience functions to model backlogging of demand. We show that even when lost sale and backorder costs are present, the problem is well posed in the reduced space. We provide an iterative procedure for solving the overall problem. We describe structural properties of the solution for the new model and comment on the recent work incorporating backorder cost. We illustrate the solution procedure for the new model with examples.

The idea of an “impatient” customer in backlogging situation was proposed in Abad (1996). It was suggested that customers do not like to wait and therefore that the fraction of customers who choose to place backorders be a decreasing function of waiting time. Two specific examples of functions for backlogging of demand were given. If τ is the waiting time (i.e., time till the new supply becomes available and the backorder is filled), the fraction of customers backordering can be modeled as B(τ)=k0e-kτ1B(τ)=k0e-kτ1 or B(τ)=k0/(1+k1τ), k0, k1 being parameters. These two functions—the exponential rate and the hyperbolic rate with respect to waiting time—have been used to model backordering in several recent studies. San Jose et al. (2006) consider the problem of determining the lot size and the backorder level when demand is backlogged at the exponential rate. Demand is assumed to be constant and the order and lost sale costs are included. The authors coined the term “impatience function” to refer to the functions for modeling backlogging of demand. They have proposed a continuous two-piece function [San Jose et al. (2005a)] as well as a discontinuous step function [San Jose et al. (2005b)] for modeling backlogged demand. In the step function, backlogging rate is equal to 1 (i.e., full backordering) when waiting time is less than the specified fixed period and 0 (i.e., complete lost sale) when the waiting time is more than the fixed period. Skouri and Papachristos (2003) formulate a production lot size model where production rate, demand rate and deterioration rate are exogenous, time-varying functions and demand is backlogged with the hyperbolic rate. There also have been studies in which demand is considered to be price sensitive and the selling price is an endogenous variable. Abad (2001) considers the pricing and lot-sizing problem for infinite horizon. However, he ignores the shortage cost, the lost sale and the backorder cost in his analysis. Papachristos and Skouri (2003) consider the pricing and lot-sizing problem for infinite horizon when the supplier offers a lot size-based quantity discount on materials, time to deterioration is described by Weibull function and demand is backlogged at the hyperbolic rate. They include the lost sale cost and the shortage cost (i.e., cost per unit short) in their analysis. They assume that the demand rate is a convex decreasing function of selling price and that the inventory cycle time is fixed. Dye (2007) considers the pricing and lot-sizing problem for infinite planning horizon assuming the demand rate to be convex, decreasing function of selling price and the revenue to be a concave function of selling price. He assumes that the deterioration rate is time varying and that backlogging occurs at the hyperbolic rate. He includes only the lost sale cost and the cost of carrying backorders but excludes the shortage cost in his analysis. Dye et al. (2007) consider a similar problem except that backlogging occurs at the exponential rate. They again exclude the shortage cost in their analysis (see also Chang et al., 2006). In this paper we consider the pricing and lot-sizing problem for an infinite planning horizon in a general framework. We assume that the demand rate is a decreasing function of price and that the marginal revenue is an increasing function of price. We include all three costs—the lost sale cost, the cost of carrying backorders and the shortage cost in our analysis. Furthermore, we use a general (continuous and smooth) impatience function to model the backlogging phenomenon. We provide an algorithm to determine the optimal solution. We also highlight some structural properties of the optimal solution.

In this paper, a general model is formulated for determining the optimal selling price and order size for a perishable product subject to partial backordering. We include not only the cost of carrying backorders and the lost sale cost but also the shortage cost. Convexity properties are proved and an iterative procedure is provided for solving the problem, which guarantees a local maximum. We have used a general impatience function to model backlogging of demand. The model assumes a general deterioration function and is not restrictive in terms of demand function. We have also highlighted some structural properties of the solution.