Most of the research in economics and marketing models the demand only as a function of price, assuming that Þrms compete mainly on price. In contrast, the operations literature usually takes the price (and demand) as given, and tries to minimize cost and/or maximize customer service. However, for most customers the purchasing decision involves trading off many factors including price and quality, where delivery guar- antees are considered among the top quality features. Hence, in most cases demand is a function of both price and lead time and therefore a Þrm’s DDM policy is closely linked to its pricing policy. In this sec- tion, we provide an overview of the literature which considers DDM in conjunction with pricing decisions.
The Þrst four papers we discuss, So and Song [96], Palaka et. al. [74], Ray and Jewkes [84], and Boyaci and Ray [15] consider capacity se- lection/expansion decisions in addition to price and lead time decisions.
These papers study DDM-P using an M/M/1 queuing model with FCFS sequencing, where the expected demand is modeled by a linear function (Λ(R, l) = a− b1R− b2l) in [74] and [84], and a Cobb-Douglas function (Λ(R, l) = λR−al−b) in [96]. Note that the price elasticity of demand (the percentage change in demand corresponding to a 1% change in price) is constant in the second model whereas it increases both in price and quoted lead time in the Þrst model. These papers consider a con- stant lead time and a service level constraint (the minimum probability (s) of meeting the quoted lead time).
Palaka et al. consider three types of costs: direct variable costs (pro- portional to production volume), congestion costs (proportional to the mean number of jobs waiting in the system), and tardiness costs. They Þrst consider the case of Þxed capacity (service rate µ), where the goal is to choose a price/lead-time pair (R, l) and a demand rate λ≤ Λ(R, l) for maximizing the Þrm’s expected proÞts. Noting that the demand rate λ will always be equal to Λ(R, l) in the optimal solution, they focus on the two decision variables λ and l. They show that (i) the service constraint is binding in the optimal solution if and only if s ≥ sc = 1− b2/b1w, where w is the tardiness penalty per unit time, (ii) when s increases, the Þrm both increases its quoted lead time and decreases its demand rate (and expected lead time). In contrast, an increase in the tardiness penalty decreases the demand rate but the quoted lead time decreases (increases) if the service constraint is binding (non-binding). For small values of the tardiness penalty, the Þrm increases the price to reduce the demand rate, which does not decrease the probability of tardiness but reduces the tardiness penalty since there are fewer orders late overall. However, when the tardiness penalty is high, the Þrm needs to reduce the probability of tardiness and hence quotes higher lead times. Palaka et al. extend these results to the case where marginal capacity expansion is possible, that is, the Þrm can choose z, the fractional increase in the processing rate at a cost of c per job/unit time up to a limit of ¯z. Hence, the service rate becomes µ(1 + z). The authors show that in the optimal solution, the Þrm uses both capacity expansion and a reduced arrival rate to achieve shorter lead times. Finally, they look at the sensitivity of the proÞts to the errors in estimating the lead time and conclude that guaranteeing a shorter than optimal lead time usually results in higher proÞt loss than guaranteeing a longer than optimal lead time.
The objective function in So and Song [96] is to maximize proÞt, which is the revenue minus direct variable costs and capacity expansion costs. Note that this is a special case of the objective function consid- ered in Palaka et al., ignoring the congestion and tardiness costs. The
qualitative results of So and Song are generally consistent with Palaka et al.
Ray and Jewkes [84] study a variant of the model in [74] by modeling the market price as a function of lead time, namely, R = ¯R− el where ¯R is the maximum price when the lead time is zero5. Hence, the demand function becomes λ(R, l) = a− b1R− b2l = (a− b1R)¯ − (b2− b1e)l = a" − b"l. This model naturally leads to a distinction of lead time and price sensitive customers: (i) when b" > 0, demand decreases in lead time, i.e., customers are lead time sensitive (LS) and are willing to pay higher prices for shorter lead times; (ii) when b" < 0, demand increases in lead time (as price decreases), i.e., customers are price sensitive (PS) and are willing to wait longer to get lower prices. The dependence of price on lead time reduces the number of variables, and the Þrm needs to choose a lead time (l) and capacity level (µ) subject to a service constraint, with the goal of maximizing proÞt (revenue minus operating and capacity costs). The authors consider both the cases of constant and decreasing convex operating costs (the latter models economies of scale). The authors show that the optimal lead time depends strongly on whether the customers are price or lead time sensitive, as well as operating costs. Comparing their model with the lead-time-independent price model, the authors show that the optimal solutions for these two models might look signiÞcantly different, and ignoring the dependency of price on lead time might lead to large proÞt losses.
Boyaci and Ray [15] extend the previous models to the case of two sub- stitutable products served from two dedicated capacities. They assume that these two products are essentially the same, and are differentiated only by their price and lead time. As in [84], there is a marginal capacity cost ci and a unit operating cost m, and as in [74] and [96], the price is not a direct function of lead time. The market demand for product i = 1, 2 is given by λi = a− βRRi+ θR(Rj− Ri)− βlli+ θl(lj− li), i%= j. In this demand model, (i) θRand θldenote the sensitivity of switchovers due to price and lead time, respectively, (ii) βR and βl denote the price and lead time sensitivity of demand, (iii) the total market demand λ1+λ2 is independent of the switchover parameters (θ). The authors assume that the lead time (l2) for product 2 (“regular” product) is given, and the Þrm needs to determine a shorter lead time (l1 < l2) for product 1. Express vs. regular photo processing or package delivery are some ex-
5Recall that a lead-time-dependent price model was also studied earlier in [64], discussed in
Section 6.1. Since price is not an independent decision variable in these models, we could have discussed [84] in Section 6.1 as well. However, due to its relevance to some of the other papers in this section, we decided to discuss it here.
amples motivating this model. As in the previous papers, the goal is to maximize proÞts subject to service level constraints 1− e−(µi−λi)ti ≥ s
i, i = 1, 2. The authors Þrst study the uncapacitated model (Model 0, c1 = c2 = 0) with a given l1 (it turns out that in the uncapacitated case the optimal l1 is zero), and differentiate between price and time sensitive customers as in [84]: (i) Price and time difference sensitive (PTD): When βRθl > θRβl, the market is more sensitive to price but switchovers are mainly due to the difference in lead times. Note that this condition is equivalent to [θR/(βR+ θR)] < [θl/(βl+ θl)], i.e., the fraction of customers that switch due to price is smaller than the fraction of customers that switch due to lead time. (ii) Time and price differ- ence sensitive (TPD): When βRθl < θRβl, the market is more sensitive to lead time but switchovers are mainly due to the difference in prices. When θR≈ θl, we are back to the standard price and lead time sensitive markets. Next, they study the case where only product 1 is capacitated (Model 1, c1 > 0, c2 = 0). Comparing Model 1 to Model 0, they show that the change in the optimal prices depend on the market type (TPD vs. PTD) as well as c1. Interestingly, while price and time differentia- tion go hand in hand in Model 0, in Model 1 the Þrm price differentiates only if the marginal cost of capacity is sufficiently high. Finally, the authors consider the case where both products are capacitated (Model 2, c1 ≥ c2 > 0). They Þnd that for the same c1, the optimal l1 is shorter in Model 2, i.e., the Þrm offers a better delivery guarantee even though its cost is higher for the regular product. This is because a shorter l1 lures customers away from the regular product and attracts them to the more proÞtable express product. The authors conclude their study by comparing their results to the case where substitution is ignored or not present.
So [95] extends the work in [96] to a competitive multi-Þrm setting, where the Þrms have given capacities (µi) and unit operating costs (γi), i.e., they are differentiated by their size and efficiency, and choose prices (Ri) and delivery time guarantees (li). As in [74] and [96], each Þrm chooses a uniform delivery time guarantee. The market size (λ) is as- sumed to be Þxed and the market share of each Þrm is given by
λi= λ % αiR−ai t−bi !n j=1αjR−aj t−bj &
In this model, the parameter αi denotes the overall “attractiveness” of Þrm i for reasons other than price and lead time, such as reputation and convenience of the service location. Parameters a and b denote the price and time sensitivity of the market, respectively. To ensure the reliability of the quoted lead times, the Þrms seek to satisfy a service
constraint, which states that on average an s fraction of the orders will be on time. Using an M/M/1 queue approximation, this constraint is modeled as 1− e−(µi−λi)ti ≥ s, or equivalently, (µ
i− λi)ti ≥ − log(1 − s). Focusing only on the case where s is given and the same for each Þrm, So Þnds the “best response” of Þrm i in closed form given the other Þrms’ price and lead time decisions. He shows that the price is decreasing in lead time. He also shows that (1) in the combined solution of price and lead time, the Þrm always charges the highest price ¯R if a ≤ 1, and (2) the optimal price and lead time are increasing in αi. These results suggests that Þrms compete only based on delivery time when the market is not price sensitive, and Þrms with lower attractiveness need to compete by offering better prices and lead times. So characterizes the Nash equilibrium in closed form for N identical Þrms, and proposes an iterative solution procedure for identifying the equilibrium in case of non-identical Þrms. Comparing the results to the single-Þrm case studied in [96], a capacity-increase in the multi-Þrm case leads to lower prices, whereas the reverse is true in the single-Þrm case. This is quite intuitive since an increased capacity in the multi-Þrm case leads to more intense competition in a Þxed-size market. Numerical results with two Þrms, s = 0.95 and equal αi’s indicate that (1) the advantage of higher capacity increases as the market becomes more time sensitive, (2) low cost Þrm offers a lower price, longer lead time, and captures more of the market share and proÞts, (3) an increase in price sensitivity leads to lower prices overall, and shorter and longer lead times for the high capacity and low cost Þrms, respectively, (4) an increase in time sensitivity leads to an increase in prices, and shorter and longer lead times by the low cost and high capacity Þrms, respectively, reducing the difference between the lead times offered by the two Þrms, (5) as the time (price) sensitivity increases, proÞts and market share of the high capacity (low cost) Þrm increase. So also conducts experiments in a three-Þrm setting, where one of the Þrms dominates the others both in terms of capacity and operating cost. Interestingly, in such a setting the dominant Þrm offers neither the lowest price, nor the shortest lead time, while the other two Þrms strive to differentiate themselves along those two dimensions by offering either the lowest price (low cost Þrm) or the shortest lead time (high capacity Þrm).
The papers discussed so far assume that once the Þrm quotes a cus- tomer a lead time (and price), the customer immediately makes a de- cision as to whether to accept or reject the offer. In reality, customers might “shop around”, i.e., request quotes from multiple Þrms, and/or need some time to evaluate an offer. Hence, there might be a delay be- fore a customer reaches a decision on whether or not to accept an offer.
Easton and Moodie [32] study DDM-P in the face of such contingent or- ders, which add variability to the shop congestion and hence to the lead time of a new order. In their model, the probability that the customer will accept a quoted price/due-date pair (R, l) follows an S-shaped logit model P (R, l) = ' 1 + β0exp ( β1 l− p p + β2 ( R cp − 1 ))*−1
where p is the estimated work content of the order, c is the cost per unit work content, and β0, β1 and β2 are parameters to be estimated from previous bidding results. The two terms in this expression refer to the lead time as a multiple of total work content, and the markup embed- ded in the quoted price. For choosing the price R and the lead time l for a new order (assuming FCFS sequencing), they propose a model that myopically maximizes the expected proÞt (revenue minus tardiness penalties) from that order considering both Þrm and contingent orders in the system but ignoring any future potential orders. The solution method they propose involves evaluating all possible accept/reject out- comes for contingent orders, i.e., 2N scenarios if there are N contingent orders, which is clearly not efficient for a large number of contingent orders. Via a numerical example, the authors show that their model outperforms simple due-date setting rules based on estimates on the minimum, maximum or expected shop load.
All the papers we discussed in this survey so far assume that the due date (and price) decisions are made internally by the Þrm. This is in contrast to the scheduling literature where it is assumed that the due dates are set externally, e.g., by the customer. In practice, most business-to-business transactions involve a negotiation process between the customer and the Þrm on price and due date, i.e., neither the cus- tomer nor the Þrm is the sole decision maker on these two important issues. With this in mind, Moodie [71] and Moodie and Bobrowski [72] incorporate the negotiation process into price and due date decisions. In their model, both the customer and the Þrm have a reservation tradeoff curve between price and due date, which is private information. Cus- tomer arrivals depend on the delivery service reputation (SLR) of the Þrm, which depends on the Þrm’s past delivery performance as follows: SLRnew= (1− α)SLRold+ αs, where s is the fraction of jobs completed on time in the last period. If the Þrm’s service level is below the cus- tomer’s requirement, the customer does not place an order. Hence, by choosing its due dates, the Þrm indirectly impacts the demand through its service level. Given a new customer, the Þrm Þrst establishes an ear- liest due date. The Þrm then chooses one of the four negotiation frame-
works for price and lead time: negotiate on both, none, or only one of the two. Third, the Þrm chooses a price (a premium price for early delivery and a base price for later delivery) and due date to quote. And Þnally, if the order is accepted, it needs to be scheduled. Moodie [71] proposes and tests four Þnite-scheduling based due date methods, as well as four of the well-known rules from the literature: CON, TWK, JIQ, and JIS. Simulation results (under EDD scheduling) suggest that (1) due date methods based on the jobs’ processing times (especially TWK and JIS) perform better than the proposed Þnite-scheduling based methods, (2) negotiating on both price and lead time provides higher revenues, and (3) it may be proÞtable to refuse some orders even if the capacity is not too tight. A more extensive study of this model is performed by Moodie and Bobrowski [72]. They Þnd that full bargaining (both on price and lead time) is useful if there is a large gap between the quoted and the customers’ preferred due dates. If this gap is small, then price-only bargaining seems more beneÞcial.
Charnsirisakskul et al. [17] study the beneÞts of lead time ßexibil- ity (the willingness of the customers to accept longer lead times) to the manufacturer in an offline setting with 100% service guarantees. They consider a discrete set of prices {R1j, . . . , Rnj
j } the manufacturer can charge for order j. The demand quantity (expressed in units of capac- ity required, or processing time) corresponding to price Rkj is pkj. Each customer has a preferred and acceptable lead time, denoted by fjk and lk
j, respectively, if the manufacturer quotes price Rkj. There is also an earliest start time for starting the production of and an earliest delivery time for each order. If an order’s production (partially) completes before the earliest delivery time, the manufacturer incurs a holding cost. If the quoted lead time is between fjkand lkj, i.e., after the customer’s preferred due date, the manufacturer incurs a lead time penalty. The authors model this problem as a linear mixed integer program where the deci- sions are: which orders to accept, which prices to quote to the accepted orders, and when to produce each order. Note that this model simulta- neously incorporates order acceptance, pricing and scheduling decisions. The authors also consider a simpler model where the manufacturer must quote a single price (again, chosen from a discrete set of prices) to all customers. They propose heuristics for both models, since the solution time can be quite large for certain instances. To establish the link be- tween prices and order quantities in their computational experiments, they consider the case where each customer is a retailer who adopts a newsvendor policy for ordering. They model the retailer’s demand by a discrete distribution function and compute the retailer’s optimal or- der quantity as a function of the manufacturer’s quoted price. In an
extensive computational study, they test the impact of price (single vs. multiple prices), lead time, and inventory holding (whether or not the manufacturer can produce early and hold inventory) ßexibility on the manufacturer’s proÞts. They Þnd that lead time ßexibility is useful in general both with and without price ßexibility. The beneÞt of lead time ßexibility is higher if there is no inventory ßexibility, suggesting that lead time and inventory ßexibilities are complementary. However, they also observe that in certain environments price, leadtime, and inventory ßexibilities can be synergistic.
7.
Conclusions and Future Research Directions
Lead-time/due-date guarantees have undoubtedly been among the most prominent factors determining the service quality of a Þrm. The importance of due date decisions, and their coordination with schedul- ing, pricing and other related order acceptance/fulÞllment decisions in- creased further in recent years with the increasing popularity of make- to-order (MTO) over make-to-stock (MTS) environments.
In this paper, we provided a survey of due date management (DDM) literature. The majority of the research in this area focuses solely on due-date setting and scheduling decisions, ignoring the impact of the quoted lead times on demand. Assuming that the orders arriving in the system are exogenously determined and must all be processed, these papers study various DDM policies with the goal of optimizing one or a combination of service objectives such as minimizing the quoted due dates, average tardiness/earliness, or the fraction of tardy jobs. Clearly, no single DDM policy performs well under all environments. Several factors inßuence the performance of DDM policies, such as the due date rule, the sequencing rule, job characteristics (e.g., product structures, variability of processing times), system utilization (the mean and vari- ance of load levels), shop size and complexity, and service constraints. Due date policies that consider job and shop characteristics (e.g., JIQ)