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...responsiveness uncertain customer requirements. addition the price and quality attributes of goods, customers are often highly sensitive to order lead times, while suppliers often compete based on the ability to quickly respond to consumer requirements. From the supplier's perspective, providing fast response times (in the form of short sales order lead times) may imply substantial costs, in the form of safety stock and/or investments in lead time reduction. In this paper, we consider a combined strategic and operational level planning model that determines the best positioning of a product with respect to price and promised sales lead time. Our analysis applies to stocking systems in which batch ordering occurs due to fixed order costs, where in-stock availability is desirable for a high percentage of time during each replenishment cycle (e.g., retail contexts). The models we provide allow us to examine lead time, demand, and operations performance tradeoffs and challenges faced by procurement and inventory planners.
To properly characterize lead time decision problems, we first distinguish between procurement lead times and sales (or order) lead times. The procurement lead time considers the time between procurement request and product availability at the next (downstream) supply chain stage. Long procurement lead times (relative to sales lead time) lead to high finished goods safety stock requirements in order to meet customer service level expectations under uncertain demand. The sales lead time, on the other hand, is the quoted time between customer order and corresponding product receipt by the customer. The sales lead time is therefore the customer's perceived lead time and can influence demand, as many customers value fast order fulfillment in order to reduce their own planning burden and the costs associated with long supply lead times. Longer sales lead times, on the other hand, allow a supplier greater planning flexibility and can therefore reduce a supplier's operations costs. The values of the procurement and sales lead times interact to determine a supplier's safety stock requirements. Determining the right sales lead time for a product (in addition to price and procurement lead time, where possible) represents an important strategic challenge, and the models provided in this paper partially address this challenge in a continuous-review inventory control context.
Research on how lead times impact operations has increased in recent years. Karmarkar (1993) provides a nice discussion of the impacts of production lead times on operations performance, and also characterizes past literature in this area. One of the most clear and quantifiable consequences of long lead times is the level of safety stock required in finished goods inventories in order to meet a given service level. Karmarkar and Lele (1989) characterize the safety stock implications as a result of production batching policies in multi-item production planning. They assume that safety stock is held during a lead time equal to the production cycle time for each product, and that a rotation cycle production policy is followed. In addition to the safety stock requirements due to production batching effects, longer lead times increase demand forecast error, since forecast error generally increases as the forecast horizon increases. Wecker (1979) characterizes this effect, by showing that the variance of forecast error increases with the cube of the mean lead time. Long lead times can also have an adverse effect on a firm's competitive position, as this reduces responsiveness to customers and can delay product acquisition by customers.
Cruickshanks et al. (1984) discuss the roles of two types of lead times in production planning for make-to-order factories. In a single-stage manufacturing process, under a production lead time of [L.sub.0] periods, an order arrives and delivery is promised within [L.sub.1] periods (the sales lead time), with [L.sub.1] [greater than or equal to] [L.sub.0]. The difference, [L.sub.1] - [L.sub.0], is called a planning window. They show that variability in production levels and capacity requirements decline as the planning window increases; however, a larger sales lead time requires greater inventory and longer delivery times for customers. Choosing the best value of [L.sub.1] is the decision problem in their production planning cost tradeoff model. In this paper, we make similar use of production (procurement) and sales lead times, although in the contexts we consider, we allow the sales lead time to be any non-negative value, with a value of zero implying a make-to-stock system.
Lead time decision problems in the scheduling literature are often treated as due date setting problems in make-to-order contexts. A survey of research regarding traditional due-date setting problems in scheduling is provided by Cheng and Gupta (1989). Spearman and Zhang (1999) examined optimal lead time policies in scheduling problems, by minimizing the average due date lead time of jobs subject to tardiness constraints. Hopp and Sturgis (2001) studied lead time quoting polices for minimizing average lead time, subject to customer service constraints on fill rate and tardiness (or relative tardiness), in a production system with random processing times. They conclude that a simple constant safety lead time policy works well under most conditions. Keskinocak et al. (2001) considered models for coordinating scheduling and lead time quotation. In these models, revenues from customers are sensitive to lead times.
In addition to works on traditional scheduling problems, a stream of research models manufacturing systems in a queuing framework, where delay times depend on system congestion. Li (1992) considers a situation in which the customers' utilities are sensitive to price, quality, and lead time, with the demand process being modeled as a counting process, and the supplier controlling the production rate. The model is then extended to address multiple suppliers who compete for demand based on delivery time. Li and Lee (1994) similarly consider delivery time competition among suppliers when the customer utility is a function of wait time in a queuing framework. Weng (1996) analyzes a make-to-order manufacturing system with two customer classes. Lead-time-sensitive customers are willing to pay a certain additional margin per unit reduction in lead time, while lead-time-insensitive customers are willing to wait. The cost model contains work-in-process holding costs, early delivery holding costs, and tardiness costs, while delivery performance is determined by the congestion level in the manufacturing system. Palaka et al. (1998) examine a lead time setting, capacity utilization, and pricing problem facing a firm serving customers that are sensitive to quoted lead time. They model the firm's operations as an M/M/1 queue and treat demand as a linear function of price and quoted lead time. Their model considers revenues and production costs, work-in-process holding cost, and lateness penalty costs. Van Mieghem (2000) considers a broader queuing system and service context, where customers are sensitive to the distribution of service delay. This work specifies dynamic scheduling and pricing rules based on delay sensitivity cost curves, and analyzes the impacts of offering a menu of lead time and price combinations in queuing systems. Webster (2002) models a make-to-order system in which the arrival (demand) rate is a continuous function of price and lead time. The system is modeled as a queuing system in which the capacity (production rate) can change dynamically, which changes the variable production cost. In each of these queuing-based models, order costs are linear, and they do not therefore capture the economies of scale due to fixed order costs and the corresponding batch ordering decisions present in many inventory replenishment systems.
Taking a broader supply chain view, Donohue (2000) examines a supply contract problem in which a manufacturer has a two-mode production environment: one is relatively cheap, but requires a long lead time, while the second is expensive, but offers a quick turnaround time. She provides a contract structure that maximizes the benefits to all members of the supply chain. Lawson and Porteus (2000) address a multiechelon inventory model, in which they analyze dynamic lead time management, where the production lead time between echelons for each ordered unit can be dynamically reduced (by expediting) or extended in duration if desired. Chen (2001) analyzes a serial supply chain in which customers arrive according to a Poisson process and are offered a menu of price and lead time combinations within each market segment. For a given set of (predetermined) lead time values, he analyzes the optimal pricing decision and provides heuristic solutions for the N-stage serial supply chain case, when there are no economies of scale in procurement. Ha et al. (2003) consider a single customer, two-supplier model in a game-theoretic context where suppliers compete on a combination of delivery frequency and price. While they account for production economies of scale, they do not consider lead time decisions directly. To address coordination among divisions within a firm, Golbasi and Wu (2001) present a model capturing the relationships between order quantity, capacity level, and lead time, and consider setup costs, inventory holding costs and capacity consumption costs. They examine a coordination problem between marketing and operations functions, and propose a lead time reduction scheme where operations offers a more favorable lead time, provided that marketing convinces the customer to place larger orders.
Our work focuses on a single-inventory replenishment stage, where high demand volumes and economies of scale necessitate batch ordering, and the supplier must set inventory policy parameters in addition to lead time and price. Related work in this area includes Ouyang and Wu (1997), who consider a so-called mixture inventory model where both production lead time and order quantity are decision variables, with a service level constraint on shortages per replenishment cycle. They minimize the sum of ordering, holding, and crashing costs, where the latter is a cost per unit time of lead time reduction. In their model, demand is independent of lead time, and they develop an algorithm for minimizing the expected cost. Our model differs from that of Ouyang and Wu (1997) because we allow demand to be a function of sales lead time and price, and our version of procurement lead time crashing cost comes in the form of an increased variable procurement cost. We therefore use a profit-maximization approach, as opposed to minimizing cost.
Our work is also related to past research on combined pricing and lot sizing for continuous review systems. Kunreuther and Richard (1971) first examined the Economic Order Quantity (EOQ) problem with pricing, and provided optimality conditions, while Lee and Kim (1998) provided an optimal solution procedure for the problem with backordering. As these papers demonstrate, and our analysis verifies, it is difficult to obtain closed-form expressions for optimal prices for these problem classes which are, in general, nonlinear optimization problems with objective functions that are neither globally convex nor concave. Our expression of demand as a function of both price and lead time presents further complications. As with these past approaches, we characterize cases under which we are able to find optimal solutions with an efficient algorithm.
The structure of this paper is as follows. In Section 2, we first define our modeling approach, which involves characterizing the administration of a continuous-review replenishment policy with a sales lead time, where average demand depends on price and sales lead time. Section 2.1 provides a discussion of modeling approaches while Section 2.2 provides our base expected profit model for systems where high economies of scale in procurement...
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