Quantity discounts based on the previous order in a two-period inventory model with demand uncertainty

We examine quantity discount contracts between a manufacturer and a retailer in a stochastic, two-period inventory model. The retailer places an order in each of the two periods to meet stochastic demands. The manufacturer gives the retailer a price discount on purchases in the second period in excess of the first-period order quantity (incremental QDP) or a price discount for all units ordered in the second period if the retailer orders more in the second period than in the first period (all-units QDP). We show that the retailer's optimal ordering decision in the second period depends on the sum of initial inventory and previous order quantity. Our computational study suggests that the QDP contract induces the retailer to buy more in the second period but less in the first period, while the increase of the total order quantity may not be significant; and that it increases the manufacturer's profit only when the wholesale margin is large relative to the retail margin.     

Optimal spot market inventory strategies in the presence of cost and price risk

We consider a firm facing random demand at the end of a single period of random length. At any time during the period, the firm can either increase or decrease inventory by buying or selling on a spot market where price fluctuates randomly over time. The firm’s goal is to maximize expected discounted profit over the period, where profit consists of the revenue from selling goods to meet demand, on the spot market, or in salvage, minus the cost of buying goods, and transaction, penalty, and holding costs. We first show that this optimization problem is equivalent to a two-dimensional singular control problem. We then use a recently developed control-theoretic approach to show that the optimal policy is completely characterized by a simple price-dependent two-threshold policy. In a series of computational experiments, we explore the value of actively managing inventory during the period rather than making a purchase decision at the start of the period, and then passively waiting for demand. In these experiments, we observe that as price volatility increases, the value of actively managing inventory increases until some limit is reached.     

Optimal prevention and prudence in a two-period model

This work shows that, in a two-period framework, prudence has a positive effect on optimal prevention. This conclusion is the opposite to that obtained in a one-period framework [Eeckhoudt L., Gollier C., 2005. The impact of prudence on optimal prevention. Economic Theory 26, 989–994]. This is due to the opposite effect of prevention on wealth in the period where the risk occurs.     

Optimal prevention and prudence in a two-period model

This work shows that, in a two-period framework, prudence has a positive effect on optimal prevention. This conclusion is the opposite to that obtained in a one-period framework [Eeckhoudt L., Gollier C., 2005. The impact of prudence on optimal prevention. Economic Theory 26, 989–994]. This is due to the opposite effect of prevention on wealth in the period where the risk occurs.     

The effect of vertical integration on consumer price in the presence of inventory costs

This paper investigates the effect on consumer price of a vertical merger between a monopolist manufacturer and his retailer, when inventory costs are taken into consideration. We find that the traditional result (lower prices) remains true only when inventory costs are sufficiently small. The direction of the price change also depends on the market size.     

On the Hungarian inventory control model

In this paper we recall and further develop an inventory model formulated by the author [Prékopa, A., 1965. Reliability equation for an inventory problem and its asymptotic solutions. In: Prékopa, A. (Ed.), Colloquia Applied Mathematics in Economics. Publ. House of the Hung. Acad. Sci., Budapest, pp. 317–327; Prékopa, A., 1973. Generalizations of the theorems of Smirnov with application to a reliability type inventory problem. Math. Operationsforschung und Stat. 4, 283–297] and Ziermann [Ziermann, M., 1964. Application of Smirnov’s theorems for an inventory control problem. Publications of the Mathematical Institute of the Hungarian Academy of Sciences Ser. B 8, 509–518] that has had wide application in Hungary and elsewhere. The basic assumption made in connection with this model is that the delivery of the ordered amount takes place in an interval, according to some random process, rather than at one time epoch. The problem is to determine that minimum level of safety stock, that ensures continuous production, without disruption, by a prescribed high probability. The model is further developed first by its combination with another inventory control model, the order up to S model and then, by the formulations of a static and a dynamic type stochastic programming models.     

Joint pricing and inventory control with a Markovian demand model

We consider the joint pricing and inventory control problem for a single product over a finite horizon and with periodic review. The demand distribution in each period is determined by an exogenous Markov chain. Pricing and ordering decisions are made at the beginning of each period and all shortages are backlogged. The surplus costs as well as fixed and variable costs are state dependent. We show the existence of an optimal (s, S, p)-type feedback policy for the additive demand model. We extend the model to the case of emergency orders. We compute the optimal policy for a class of Markovian demand and illustrate the benefits of dynamic pricing over fixed pricing through numerical examples. The results indicate that it is more beneficial to implement dynamic pricing in a Markovian demand environment with a high fixed ordering cost or with high demand variability.     

Economic model predictive inventory routing and control

Central European Journal of Operations Research - The paper proposes an economic model predictive control (EMPC) strategy for the inventory routing problem under demand uncertainty. The strategy is...     

Optimal discounted control for a continuous time inventory model

This paper is concerned with the following linear stochastic control problem: Minimize the discounted total cost $$J(x; u) = E{_x} \left[ {\int_0^\infty {\exp [ - \alpha t]\{ \phi (x{_t} ) + |u{_t} |\} } dt} \right]$$ over all measurable and nonanticipative control processes (u _t ), subject todx _t =u _t dt+dw _t ,x(0)=x, |u _t |≤1. This problem is analyzed using a discretization technique. The results obtained extend those derived in Ref. 1 and some of those derived in Ref. 2.     

On the inventory model with variable lead time and price–quantity discount

In this paper, we propose a mixed integer optimization approach for solving the inventory problem with variable lead time, crashing cost, and price–quantity discount. A linear programming relaxation based on piecewise linearization techniques is derived for the problem. It first converts non-linear terms into the sum of absolute terms, which are then linearized by goal programming techniques and linearization approaches. The proposed method can eliminate the complicated multiple-step solution process used in the traditional inventory models. In addition, the proposed model allows constraints to be added by the inventory decision-maker as deemed appropriate in real-world situations.     

Coordinating inventory control and pricing strategies: The continuous review model

We analyze an infinite horizon, single product, continuous review model in which pricing and inventory decisions are made simultaneously and ordering cost includes a fixed cost. We show that there exists a stationary (s,S) inventory policy maximizing the expected discounted or expected average profit under general conditions.     

An integrated inventory model for a deteriorating item with allowable shortages and credit linked wholesale price

This study proposes a single manufacturer, single retailer integrated inventory model that includes deterioration and shortages in the retailer’s inventory. The manufacturer’s production process is assumed to be imperfect as it produces a certain percentage of defective items. The retailer performs a 100  % screening process immediately on receiving a lot from the manufacturer and returns the detected defective items to the manufacturer in the next delivery. The manufacturer disposes the defective items and incurs a disposal cost. To increase sales, (s)he offers a trade credit to the retailer. The retailer’s wholesale price varies linearly with the credit period. The objective is to determine the optimal replenishment cycle time, the time of running out of stock, the length of the credit period and the number of lots from the manufacturer to the retailer so as to maximize the total profit of the integrated system. A solution algorithm is designed and illustrated through numerical examples. Furthermore, a sensitivity analysis is carried out to study the influence of the model-parameters on the optimal solution.     

Integrating process optimization and inventory planning in cutting-stock with skiving option: An optimization model and its application

We consider a two-stage flow line where the first stage produces components for the downstream assembly stage. An integer programming model which integrates decisions at both the planning and the operation level is proposed, with the aim of minimizing production, holding and transportation costs. The model is tested on instances built on the basis of a real cutting process with skiving option, i.e., with the possibility of recombining components to obtain required parts.The implementation of the proposed methodology allows the integration of two hierarchical decision levels (short-term operations vs. mid-term planning) and functional areas (production vs. purchase of materials) within a single planning model, and provides an example of how an element of an Advanced Planning System (APS) could be designed. Moreover, the use of suitable cost figures in the model allows to correctly manage the insertion of hot orders of finite parts.     

Effect of declining selling price: profit analysis for a single period inventory model with stochastic demand and lead time

In this paper, considering the empirical trend for sales and price of fashion apparels as prototype, optimal ordering policy for a single period stochastic inventory model is investigated. The impact of the presence of random lead time and declining selling price on the profitability of the retailer is explored. Existence of unique optimal solutions for net profit functions is proved. Numerical examples are presented to illustrate the method of identifying profitable levels of inventory holding and penalty costs. Percentage profit per unit investment in inventory is obtained in order to assist managers in taking business decisions, specifically to the extent of whether or not to take up a particular business under known constraints. It is demonstrated that the optimal inventory policy in the absence of price decline and lead time differs considerably from that when lead time and price decline are simultaneously considered.     

Age replacement in the presence of inventory constraints

Many authors in the literature have treated the age replacement problem and its various modifications. One generally is asked to assume the ability of a system to provide instantaneous replacements whenever needed. This is often not a very realistic assumption. In this paper we will examine one recent age replacement model when replacements are constrained by two simple inventory models.     

A non-cooperative game model for managing a multiple-aged expiring inventory under consumers' heterogeneity to price and time

The considered inventory system includes the coexistence of old and new types belonging to an identical product. A non-cooperative game approach between the retailer and the market, where the retailer aims to increase her profit, is developed. The market, on the other hand, may seek different objectives. In particular, the objectives of price minimization, freshness maximization, and maximization of the quantity on shelf are analyzed. The main objectives are to develop a model of a multiple-aged inventory system and to specify the conditions under which multiple types on the shelf are not beneficial to either the retailer or the market. Using an analytic optimization approach, the optimal response functions of the two players are derived, while with the aid of numerical iterations, the non-cooperative game Nash equilibrium is obtained. The main theoretical result indicates that selling an inventory that simultaneously holds multiple ages is not optimal; that is, both the retailer and consumers lose out from such a situation. This conclusion is general enough to be valid for a market that is heterogeneous with respect to both price sensitivity and sensitivity to the remaining time until expiration. A numerical example and a sensitivity analysis of the key parameters support the conclusions and highlight their importance.