Initial Order Quantity Research Articles

Stochastic comparisons in a price-quantity setting firm with uncertain demand and emergency procurement

We consider a newsvendor problem with price-dependent demand, in either additive or multiplicative format. The newsvendor has two modes of purchasing: regular ordering at the beginning of the selling season and emergency ordering (if the realized demand exceeds the initial order quantity) at the end of the selling season. By stochastic comparisons, we systematically investigate the effects of demand magnitude and demand randomness on pricing and ordering quantity decisions as well as expected profit of the newsvendor, under both usual stochastic order (first order stochastic dominance) and convex order (less variable). Our key findings include: (i) in contrary to the case where price is exogenous, a stochastically larger demand shock may even lead to a lower order quantity; (ii) a stochastically larger demand shock leads to a higher price for the additive demand case, but may lead to a lower price for the multiplicative demand case; (iii) a stochastically larger demand shock leads to a higher expected profit for both demand models; (iv) in general, a less variable demand leads to a higher expected profit for both demand models; and furthermore, a less variable demand shock has no effect on price for the additive demand model, but leads to a higher price for the multiplicative demand model. The implications of all these findings for pricing and order quantity are discussed in detail.

Newsvendor Selling to Loss-Averse Consumers with Stochastic Reference Points

We study a newsvendor who sells a perishable asset over repeated periods to consumers with a given consumption valuation for the product. The market size in each period is random, following a stationary distribution. Consumers are loss averse with stochastic reference points that represent their beliefs about possible price and product availability. Given the distribution of reference points, they choose purchase plans to maximize their expected total utility, including gain-loss utility, before visiting the store, and follow the plans in the store. In anticipation of consumers' purchase plans, in each period, before demand uncertainty resolves, the firm chooses an initial order quantity. After the uncertainty resolves, the firm chooses a contingent price depending on the demand realization, with the option of clearing inventory by charging a sale price, and otherwise, posting a full price. Over repeated periods, the interaction of the firm’s operational decisions about ordering and contingent pricing and the consumers' purchase actions results in a distribution of reference points, and, in equilibrium, this distribution is consistent with consumers' beliefs. Under this framework of endogenized reference points, we fully characterize the firm’s optimal inventory and contingent pricing policies. We identify conditions under which the firm’s expected price and profit are increasing in the consumer loss aversion level. We also show that the firm can prefer demand variability over no-demand uncertainty. We obtain a set of insights into how consumers' loss aversion affects the firm’s optimal operational policies that are in stark contrast to those obtained in classic newsvendor models. As examples, the optimal full price increases in the initial order quantity; and the optimal full price decreases, while the optimal sales frequency increases, in the procurement cost.

Newsvendor Selling to Loss Averse Consumers with Stochastic Reference Points

Motivated by the supermarket practice of marking down perishable products daily, we study a newsvendor who sells a perishable asset over repeated periods to consumers with a given consumption valuation for the product. The market size in each period is random, following a stationary distribution. Consumers are loss averse with stochastic reference points that represent their beliefs about possible price and product availability. Given the distribution of reference points, they choose purchase plans to maximize their expected total utility, including gain-loss utility, before visiting the store and follow the plans in the store. In anticipation of consumers' purchase plans, in each period, before demand uncertainty resolves, the firm chooses an initial order quantity. After the uncertainty resolves, the firm chooses a contingent price depending on the demand realization, with the option of clearing inventory by charging a sale price, and otherwise, posting a full price. Over repeated periods, the interaction of the firm's operational decisions about ordering and contingent pricing and the consumers' purchase actions results in a distribution of reference points, and, in equilibrium, this distribution is consistent with consumers' beliefs. We fully characterize the firm's optimal inventory and contingent pricing policies. We identify conditions under which the firm's expected price and profit can be increasing in the consumer loss aversion level. We also show that the firm can prefer demand variability over no-demand uncertainty. We obtain a set of insights into how consumers' loss aversion affects the firm's optimal operational policies that are in stark contrast to those obtained in classic newsvendor models. As examples, the optimal full price increases in the initial order quantity, and the optimal full price decreases while the optimal sales frequency increases in the procurement cost.

In-Season Transshipments Among Competitive Retailers

A decentralized system of competing retailers that order and sell the same product in a sales season is studied. When a customer demand occurs at a stocked-out retailer, that retailer requests a unit to be transshipped from another retailer who charges a transshipment price. If this request is rejected, the unsatisfied customer may go to another retailer with a customer overflow probability. Each retailer decides on the initial order quantity from a manufacturer and on the acceptance/rejection of each transshipment request. For two retailers, we show that retailers' optimal transshipment policies are dynamic and characterized by chronologically nonincreasing inventory holdback levels. We analytically study the sensitivity of holdback levels to explain interesting findings, such as smaller retailers and geographically distant retailers benefit more from transshipments. Numerical experiments show that retailers substantially benefit from using optimal transshipment policies compared to no sharing. The expected sales increase in all but a handful of over 3,000 problem instances. Building on the two-retailer optimal policies, we suggest an effective heuristic transshipment policy for a multiretailer system.

Managing Option Contract in Supply Chain with B2B E-Markets under Disruptions

This paper investigates the supply chain with one single supplier and one single retailer under coexistence of B2B E-markets and traditional wholesale markets. By establishing a mathematical model with considering disruptions which affect the market demands during the lead time of retail order, we calculus the supplier’s optimal product strategy, the retailer’s optimal initial order quantity of products and option, the retailer’s optimal quantity of excised option and optimal quantity of order from B2B E-markets. The research shows that the option contract relocates interests and risks between the members, and effectively response the changes in market demand with B2B E-markets under disruptions. Finally the conclusion is verified by numerical example

Economic production order quantity and quality

The purpose of this study is to integrate a conventional production-inventory management approach and a process-quality design approach so as to promote quality and reduce costs. Because the integration of these two approaches into a single system is conducted under the influence of a deterioration process, estimated costs of production must be modified. Only then can these various costs be presented as a total cost for the present integrated system. This total cost includes: a setup cost for production reordering and process resetting, quality-related costs stemming from the loss of product quality along with a tolerance-related production process cost and a failure cost for product defects, backorder costs for production shortage, carrying costs generated by the storing and handling of inventory, and material costs for determining the process mean. The Taguchi quality loss function is introduced to assess quality loss in the system. The decision variables include the initial setting (process mean), process tolerance, and production order quantity. These variables need to be determined simultaneously to minimise the average total cost for a cycle time. An example is presented to demonstrate the proposed approach.

Production order quantity for economical and quality consideration

In the conventional production-inventory management approach, the production order quantity is determined by referring to costs such as the ordering cost and inventory cost. However, the quality level being produced may change during the production process owing to process deterioration. Thus, under process deterioration, a production order quantity based solely on the conventional production-inventory management system may be overproduced, due to a lack of consideration of varied quality costs at different points of time, along with the production run length. Hence, for cyclical quality promotion and cost reduction, there is a motivation to extend the conventional production-inventory management approach by adding the process-quality design approach which considers the varied quality-related costs as a function of time. The decision variables include the initial setting, process tolerance, and production order quantity, which have to be determined simultaneously for true optimization so that the average total cost, which includes the setup cost for production reordering and process resetting, quality loss, and tolerance cost for process quality, and the inventory cost for production inventory, is minimized. An example of a production process subject to process deterioration is presented for illustrating the possible application of the proposed model.

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.

Flexible supply contracts for short life‐cycle goods: The buyer's perspective

AbstractIn this paper we analyze a two‐period supply contract which allows for order adjustment by the buyer. The buyer is required to place orders for two periods. After observing initial demand, the buyer is then allowed to adjust the second order, paying a per unit order adjustment penalty. We describe the optimal behavior of the buyer under such a contract, both in determining the initial order quantities and in subsequently adjusting the order. We compare the solution to a contract where no adjustment is allowed and to the case where adjustment is allowed without penalty. We demonstrate that flexible contracts can reduce the potentially negative effect of correlation of demand between two periods. Further, we investigate how the duration of the first period vis‐à‐vis the second period affects the profitability of the buyer as a function of the degree of correlation. © 2002 John Wiley & Sons, Inc. Naval Research Logistics, 49: 25–45, 2002; DOI 10.1002/nav.10002

Optimizing Inventory Replenishment of Retail Fashion Products

We consider the problem of determining (for a short lifecycle) retail product initial and replenishment order quantities that minimize the cost of lost sales, back orders, and obsolete inventory. We model this problem as a two-stage stochastic dynamic program, propose a heuristic, establish conditions under which the heuristic finds an optimal solution, and report results of the application of our procedure at a catalog retailer. Our procedure improves on the existing method by enough to double profits. In addition, our method can be used to choose the optimal reorder time, to quantify the benefit of leadtime reduction, and to choose the best replenishment contract.

Initial order quantities in multistage serial production systems

This paper considers the determination of initial order quantities in a multistage production system where items are processed into end products at a finite rate within each stage of production, assuming no backlogging is allowed. Before the end products of a batch are made available, from the initial time period of production or during the downtime because of maintenance due to unexpected machine breakdowns, or when a long manufacturing lead time results in a slower response to the market, the demand occurred cannot be met when there is a shortage of on-hand inventory. The operation procedure of the initial order quantities for a single-product process in a serial multistage system is investigated. Overproduction is allowed at different stages such that the production volume per production run at each stage may be an integer multiple of the quantity processed at its successor. By incorporating the effect of lost sales and transforming the composition of production rates at various stages into a global conversion rate, the inclusion of an initial transient phase followed by the stationary phase can be optimal. The size of a series of transient batches increases until it reaches the stationary size and thereafter the stationary runs start.

Multi-item inventory systems with joint ordering and transportation decisions

In many practical situations joint determination of ordering and transportation decisions for a family of items may lead to a considerable cost saving. In this paper we consider a multi-item inventory system with a freight rate schedule that is a function of the volume shipped and the capacity of a standard container. Orders for the family of items are all triggered by a coordinated periodic ( R, S) policy. Economies of scale exist because of the reduced freight rates when ordering a full-container load instead of a less-than-container load. A full-container load can be achieved by enlarging the initial order quantities. A heuristic is proposed to decide whether an initial order should be enlarged or not. The heuristic is based on a comparison of the expected saved shipping cost, the expected saved ordering cost and the expected extra holding cost, caused by an enlargement. Some numerical examples show that the heuristic works quite satisfactorily.

Initial order quantities in a multistage production system with backlogging

The problem considered is that of determining optimal initial order quantities and backlogged amounts in a multistage serial production system with lost sales allowed. Items are processed into end products at finite rates within each production stage and thereafter delivered into an inventory of end products. Before each entire batch of end products is completed, any demand appearing from the start of the production period cannot be met when there is a lack of initial inventory. In order to reduce lost sales, smaller batches and/or larger backlogs than the stationary ones may be preferred, but this would incur additional costs. The net present value principle is applied to evaluate the performance of the system. This procedure automatically takes capital costs and the effects of lost sales and backlogging into account. The optimal decision policy developed by dynamic programming determines the optimal sequences of order quantities and backlogs over time and provides conditions for whether or not the process starts in a transient phase to be followed by a stationary phase. Under certain circumstances batch sizes will increase and backlogged amounts decrease during the former phase, after which both remain constant during the subsequent stationary phase. The model incorporates backlogging costs both per unit backlogged and per unit and time unit. Properties of the stationary solution are derived. An interior solution will normally be obtained also, with positive backlogging costs per unit in contrast to corresponding results when applying the traditional average cost approach. Numerical examples illustrating different basic cases are provided.

On the optimization of initial order quantities

A lot-sizing problem arises from the determination of initial batch quantities in the case of a finite production rate when demand cannot be satisfied until each batch is made available, assuming no backlogging to be allowed. In this paper the optimal procedure for the general problem of initial order quantities is discussed. An approach in terms of a present value principle taking all payments into consideration is applied in order to derive the optimal solution. By applying the present value measure, the opportunity cost for the initial inventory build-up is incorporated automatically. This is considered to be a more accurate approach than using traditional average cost measures. It is shown that the production runs are divided into two phases, the initial transient phase followed by the stationary phase. With the production rate being greater than the demand rate, a positive idle time between two consecutive production runs is needed in the stationary phase but not within the initial transient phase. To discontinue the sales period is never optimal. Optimal batch quantities increase during the transient phase and then remain constant. To solve the problem, an approach is developed in a framework of the Lagrangian function and Kuhn-Tucker conditions.

Initial order quantities

The paper shows how optimal lot sizes are determined in an initial transient phase where the needed lead time may not be available in the production of a batch. It is assumed that the sales are los ...