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Article Excerpt
We consider a make-to-order system where customers are dynamically quoted lead times (and prices). Customers are homogenous but have general (nonlinear) disutility for delay. Because the firm is a monopolist, the pricing problem is trivial and the dynamic problem reduces to one of lead-time quotation and order sequencing. We also consider the (static) problem of up-front capacity installation. We use a large-capacity asymptotic regime to make the problem tractable. We provide recommended policies for convex, concave, and convex-concave lead-time cost functions and prove that these policies are asymptotically optimal. The policies are both highly intuitive and readily implementable. Moreover, they provide delay guarantees for all served customers. They are tested numerically; we find that significant benefits can accrue by using the prescribed dynamic policies instead of first-come-first-served type policies.
Subject classifications: production/scheduling: dynamic lead-time quotation; queues: limit theorems.
Area of review: Manufacturing, Service, and Supply Chain Operations.
History: Received January 2007; revisions received September 2007, January 2008; accepted March 2008. Published online in Articles in Advance March 11, 2009.
1. Introduction
"Time is money," as Benjamin Franklin famously said. Here we assume not only that this is the case, but that costs are not necessarily linear in time lengths. In particular, we consider lead-time quotation in a make-to-order system where customers have nonlinear disutilities on the delays they are quoted. The system manager tells each customer up front what maximum delay or lead time she guarantees they will receive; if the benefits of service do not exceed a customer's disutility for waiting then he will not place an order. (1) The system manager must therefore dynamically determine what lead time she should quote (at what price) to each arriving customer and how to schedule customers present in the system. She must also determine what upfront capacity to install.
We consider a monopolist who may extract the entire customer's utility for decreased lead time. Thus, the price for a given lead time is the price for the maximum acceptable lead time plus the customer's delay cost savings for the shorter lead times. The firm seeks to maximize its revenue minus the cost of capacity. In this case the pricing problem has been effectively eliminated and the dynamic problem reduces to one of lead-time or due-date quotation and order sequencing to meet these lead times. Although such a monopolist is not the most general setting that may be imagined, it provides the cleanest and simplest modeling environment to examine the effect that the shape of the delay cost function has on lead-time quotation and order sequencing, which is our primary contribution.
As described above, we seek to explicitly study the effect that the shape of the disutility or delay cost function has on the firm's policies. Therefore, to isolate this effect, we assume homogeneous customers; that is, all customers have the same (deterministic) delay cost function. Under this assumption, we find that it is asymptotically optimal to specify a threshold beyond which customers are not admitted (or, equivalently, quoted a lead time that is unacceptably long resulting in their choosing not to join the system). Furthermore, both for tractability and to isolate the effect of the shape of the cost function, we also do not consider competition.
We consider three possible shapes for lead-time cost curves, namely, convex, concave, and convex-concave. A convex curve is an appropriate model for situations where customers have a general idea on desirable lead time and longer lead times than this are increasingly unattractive. Lead times shorter than a customer's a priori acceptable lead time are attractive but not unduly so. A concave cost curve represents the opposite; namely, there are decreasing costs to increasing lead times (i.e., customers have strong preferences for lead times that are as short as possible and are increasingly indifferent to longer lead times). A convex-concave, or "S-shaped," cost curve is a hybrid where customers may have a particular deadline in mind (e.g., a spouse's birthday) but once that deadline has passed there is increasingly little difference in the added lead time. Obviously, it encompasses convex and concave cases as subcases.
One innovation of our work is a shift-based approach to lead-time quotation. Lead times are quoted to the nearest shift (rather than in arbitrarily small units), and once a lead time is quoted it must be met. We do not seek to model the trade-off between short lead times and higher tardiness costs. Instead, we seek to model the trade-off between committing future capacity now and reserving it for potential later higher-revenue customers. The former trade-off may then be modeled in setting quotas for a shift. However, it is the latter that we feel is more important when modeling the effect that the shape of the lead-time cost curve has on the optimal policy. Although our recommended policies are readily implementable, we feel that their most significant contribution is the intuition that they provide into the effect of the shape of the cost curve.
When costs are convex we find that first-come-first-served (FCFS) sequencing is optimal (this echoes existing results in the literature--see [section]2 for an overview). However, when costs are not convex then FCFS is well known not to be optimal and may, in fact, be far from optimal (see [section]7 for numerical examples). We believe our work to be the first to propose near-optimal, easily implementable lead-time quotation and sequencing policies for nonconvex lead-time cost curves.
For nonconvex lead-time costs, the convex hull of the cost function serves as a lower bound to the system costs. The key idea behind the policies we propose for concave and convex-concave costs is to asymptotically approach the cost incurred by the convex hull of the cost function. In these two cases, the proposed lead-time quotation and scheduling policy is specified by a high-priority service capacity and a low-priority service capacity. The segmentation is arbitrary, but what results in the concave cost case is most customers receiving a lead time of one shift while a long thin tail of low-priority customers are kept as a hedge against uncertainty in the arrival process. Figure 1 depicts a typical workload distribution under the proposed policy for concave costs.
[FIGURE 1 OMITTED]
Figure 1 lends further illumination to the trade-off discussed above between committing future capacity now and reserving it for potential later higher-revenue customers. Although reserving capacity penalizes those customers quoted a long lead time, it also frees up capacity for newly arriving customers who may then be quoted a lead time of one. In the case of convex-concave cost curves, the segmentation is less extreme but the idea is similar. The majority of customers are given relatively short lead times, but when system workload reaches a sufficient level of congestion a small portion of customers are given very long lead times and kept as a hedge against demand uncertainty while allowing most customers to receive moderate lead times.
The proposed policies are intuitively appealing. They are also very easy to implement. The policies would be even more appealing if one could perform customer segmentation so that the long-lead-time customers are those that are truly the more patient customers. Modeling heterogenous customers is left as the subject of future research.
Our model will assume that lead time depends on congestion; therefore, the lead time associated with shipping is assumed to be on top of (and independent of) the quoted lead time. In other words, customers are quoted a lead time for the time until the product leaves the factory (or the service is complete), and delivery time is handled independently. Because production is make-to-order, lead time is the flow time for the order. In this case, the "product" may be highly customized (e.g., furniture production), but the service time distribution must be the same across customers.
We consider systems with a high volume of arrivals. Because of the statistical economies of scale phenomenon, such systems become more and more efficient as they grow, and the congestion concerns become less important for larger systems. As a result, one can operate such high-volume systems near full utilization and yet achieve excellent performance. Indeed, our analysis also validates this intuition; cf. Proposition 1.
The remainder of this paper is organized as follows. Section 2 contains a brief literature review. In [section]3, we outline our model and present some preliminary results. Section 4 presents the asymptotic optimization problem and develops an asymptotic upper bound for system performance. Section 5 solves the asymptotic control problem for the upper bound on performance. This upper bound is then used to prove that the proposed control policies presented in [section] 6 are asymptotically optimal; these policies are tested numerically in [section] 7. The paper is concluded in [section] 8. The proofs are relegated to an online technical appendix that can be found at http://or.journal.informs.org/.
2. Literature Review
Our work is probably most related to the due-date quotation literature, which, of course, has a long history. Hopp and Sturgis (2000) contains a relatively recent review of this literature. In their paper they seek to quote the shortest possible lead time consistent with a given service constraint. Their model and most others assume demand to be independent of lead time. For example, in both Spearman and Zhang (1999) and Wein (1991) the supplier seeks to minimize the average due-date lead time subject to a constraint on job tardiness. Also see Keskinocak and Tayur (2004) for an overview of due-date management policies and Cheng and Gupta (1989) for a survey of scheduling research involving due-date determination decisions.
In our model, the customer either accepts the quoted lead time or leaves (i.e., is turned away by being quoted too high a lead time). A related approach is taken in Duenyas and Hopp (1995) and Duenyas (1995). In their continuous-time models, if a customer is quoted a lead time of a, then he will accept with probability [pi](a). There is a net revenue per customer and a (linear) penalty ex if the customer is delayed for x units of time.
Due-date quotation when demand decreases with increased lead time has been addressed by a number of other authors. In particular, Armony and Maglaras (2004) consider a model of call centers where customers are quoted a lead time and may either leave or place a request for a call back within a prespecified (different) lead time. Chatterjee et al. (2002) consider the marketing aspects of lead-time quotation; they also provide a nice summary of the literature (see their Table 1). Duran et al. (2006) have a model where poor lead-time performance can lead to decreased future demand. Rao et al. (2005) consider a model where mean demand in a period decreases with lead time whereas the stochastic component of demand is independent of lead time. Quoted lead time is always met by using outsourcing. Charnsirisakskul et al. (2006) study a problem faced by a manufacturer who has the ability to set prices to influence demand, reject orders, and set lead times for accepted orders. The authors present various decision models that integrate price and production decisions and provide insights through numerical analyses. This model extends their earlier work in Charnsirisakskul et al. (2004), where pricing is not considered.
Table 1. Impact of system scale and the excess capacity (drift) on
the percentage improvement of the proposed policy over the benchmark
policy in the concave case.
n = 100 n = 400 n = 1,000 n = 6,400 n = 10.000
v = 3 14.21 21.49 25.61 32.40 32.86
[+ or -] [+ or -] [+ or -] [+ or -] [+ or -]
0.75 1.40 1.99 4.05 4.70
v = 5 19.82 25.89 29.05 34.40 34.80
[+ or -] [+ or -] [+ or -] [+ or -] [+ or -]
0.51 0.97 1.62 3.08 4.06
v = 7 25.79 30.98 33.92 38.62 38.76
[+ or -] [+ or -]...
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