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Weekly Updates1. Introduction
Common formulations of the flow-rate control problem assume either that all customers are infinitely patient, but there is a cost associated with making them wait, or completely impatient and sales are lost if the order cannot be met from stock. For the single-part-type, single unreliable machine problem with constant demand, the complete backordering case is analyzed in Bielecki and Kumar (1988) and the lost sales case is studied in Hu (1995). Neither of these assumptions is completely satisfactory. In the complete backordering model, the penalty for making customers wait is a cost applied to the current number of parts backordered (the backlog). It is problematic to estimate this cost, even if it is linear in the backlog. There is usually no tangible cost associated with backlog (unless the contract includes a provision for a discount for delivery delay). The real costs of delivery delays are more likely to be loss of future sales, which are difficult to measure, and cancelled orders. The lost sales model requires only a unit profit parameter; however, in many contexts it is overly pessimistic to assume that customers have no waiting tolerance. One generalization of the lost sales model is to allow a limited backlog (Martinelli and Valigi, 2004).
A more realistic model of customer behavior is considered in the companion paper to this one (Gershwin et at., 2008). If there is finished goods inventory on hand (surplus), customer orders are filled immediately. If there is a backlog, it is converted to a lead time which is quoted to the customer who either leaves immediately (balks) or decides to wait for their purchase. Customers have a distribution of lead time tolerances, so that the balking probability of a potential customer depends on the backlog when they arrive. In the context of flow-rate control, demand is continuous and this probability is interpreted as the fraction of potential demand that is not realized. Thus, customer behavior is modeled by a defection function giving the fraction of demand that "defects" as a function of the backlog. In some service systems, e.g., where people join a physical queue, arriving customers observe the backlog directly and make balking decisions based on it. However, the primary motivation for this model is manufacturing systems for which fairly accurate lead time quotes can be made.
Specifically, the model in Gershwin et al. (2004) assumes the machine is reliable and that a first-come-first-served queue discipline is used, so that lead time is deterministic and proportional to backlog. Unlike Bielecki and Kumar (1988), randomness in the model comes from demand, which is assumed to be Markov modulated with two levels. This demand model can be used as a rough approximation for Poisson demand or may be useful in modeling longer-term variations in purchase contracts or demand rates. See Fleming et al. (1987) and Perkins and Srikant (2001) for other models with Markov modulated demand.
The objective is to maximize long-run average profit, where revenue is diminished by customers who do not wait and the cost rate is linear in the surplus. The model allows an economic trade-off to be made between using inventory to buffer against demand uncertainty and increasing lost sales for some of the more impatient customers. It is shown in Gershwin et al. (2004) that the optimal policy has the same hedging point form as the Bielecki--Kumar model. The maximum production rate is used until the surplus reaches a level called the hedging point. The production rate is then set to the demand rate and the surplus remains constant until demand changes to the higher level. When the defection function is piece-wise constant, the stationary distribution of the system is found, allowing average profit to be expressed as a function of the hedging point and numerically optimized. General defection functions are approximated by a piece-wise constant function.
This paper provides a further analysis of the model in Gershwin et al. (2008). First, it is shown that customer behavior can be understood by decomposing the system into the backlog dynamics, which depend on the defection function, and the surplus dynamics, which depend on the hedging point. The impact of customer behavior is captured by one quantity, the mean sojourn time in the backlog states, which depends on the demand model and production rate but not the hedging point. The profitability of defection functions can be completely ordered by this quantity. The numerical study in this paper uses a sigmoid for the defection function, as proposed in Tan and Gershwin (2004). However, because the defection function influences profit only through this mean sojourn time, the numerical insights apply to other defection functions. In fact, two simple models of customer behavior are constructed that are equivalent to a given defection function in the sense that, for a given demand model and production rate, they have the same mean sojourn time. One model has a constant defection function whenever there is a backlog, i.e., some fraction of customers have no patience. A second model assumes all customers wait until a backlog limit is reached, as in Martinelli and Valigi (2004).
One implication of this decomposition is that the optimal hedging point can be computed more efficiently. Instead of solving the full model repeatedly while searching for the optimal hedging point, the mean sojourn time is computed once. Then, an equivalent model is used to compute average profit that replaces the backlog dynamics with a single state, simplifying the optimization.
More importantly, the decomposition provides a simple way of estimating the model parameters. Instead of estimating the defection function, one can estimate the mean sojourn time in the backlog states. In applications, the duration of stockouts is usually readily available. Simply computing their mean gives an estimate of the mean sojourn time, which fully captures the impact of the defection function on average profit. Estimating the mean duration of stockouts is vastly easier than estimating the defection function. Customers that balk because of the lead time may not be recorded. Even if they are recorded, estimating a function requires much more data than estimating a single quantity. The usual approach is to use some parameterized class of functions. More research would be needed to justify using a particular class of function.
Empirical studies of customer responses to lead times, such as Anderson et al. (2006) which studies a mail-order catalogue, have observed that customers are more likely to cancel their orders when the anticipated delay is longer but have not identified specific forms of defection functions. Customer impatience in telephone call centers, where the queue is invisible to the customer, has been studied extensively, e.g., Bolotin (1994) and Brown et al. (2005). It is observed in Whitt (2005) that call center performance is sensitive to the distribution of the customer abandonment time, which is not nearly exponential. When call centers announce the anticipated delay to customers, they make a balking decision as in our model, but may also abandon later; Whitt (1999) gives a queueing analysis.
The layout of the paper is as follows. Section 2 presents the flow control model with defection. The decomposition into backlog and surplus behavior is shown in Section 3 and used to construct simpler models in Section 4 and to find the optimal policy in Section 5. Numerical examples are presented in Section 6.
2. The defection model
We consider the one-part-type, one-machine make-to-stock production control problem analyzed in Gershwin el al. (2004). The demand rate at time t is denoted d(t). It is governed by an exogenous Poisson process D(t) on the states L (low) and H (high). When D(t) = L, d(t) = [[mu].sub.L] and transitions to H occur with rate [[lambda].sub.LH]; when D(t) = H, d(t) = [[mu].sub.H] and transitions to L occur with rate [[lambda].sub.HL]. Given a demand rate, demand is continuous and deterministic. We assume that the production capacity [u.bar] is sufficient to meet the demand when it is low but insufficient when it is high, i.e., [[mu].sub.L] < [bar.u] < [[mu].sub.H]. The average demand rate is
E(d) = [[lambda].sub.LH]/[[lambda].sub.HL] + [[lambda].sub.LH][[mu].sub.H] + [[lambda].sub.HL]/[[lambda].sub.HL] + [[lambda].sub.LH][[mu].sub.L]. (1)
The continuous inventory level at time t is x(t), with x < 0 representing backlog. Assume that customers are served first-in-first-out. Arrivals at time t observe x(t) and can infer their (deterministic) waiting time to be [x.sup.-](t)/[u.bar], where [x.sup.-] = max{0, -x}. Customers have various waiting tolerances, represented by a defection function: B(x) is the fraction of potential customers who choose not to order when the backlog is x. This function is assumed to be non-increasing and satisfies:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
The second condition says that no customers defect when there is zero waiting time. The third, which can be rewritten as E(d)(1 - [lim.sub.x[right arrow]-[infinity]], B(x)) < [u.bar], says that for sufficiently large backlogs, the average order rate is less than capacity. This condition is required for stability. Note that B(x) = 1 for all x < 0 corresponds to a lost sales model (no customers wait) and B(x) = 0 corresponds to complete backordering (all customers wait).
The decision variable is the production rate at time t, u(t). The dynamics of x(t) are given by
[[dx(t)]/[dt]] = u(t) - d(t)(1 - B(x(t))), (3)
at points where the derivative exists. The quantity d(t)(1 - B(x(t))) is the rate at which sales take place, i.e., the demand rate minus the defection rate. There is a reward A per unit sold and an inventory holding cost rate [g.sup.+] per unit per time. Defection is the only penalty for backorders; there is no backlog cost rate. Let u denote the control policy {u(t)}. The expected profit of policy u in the interval [0, T] is
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