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Keywords:

  • expediting;
  • inventory management;
  • continuous-review inventory systems;
  • shipment network design

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Operating Characteristics and Analytical Properties
  6. 4. Numerical Analysis
  7. 5. Summary and Discussion
  8. Acknowledgments
  9. References
  10. Supporting Information

We consider a continuous review inventory system where delivery lead times can be managed by expediting in-transit orders shipped from the supplier. First, we propose an ordering/expediting policy and derive expressions for evaluating the operating characteristics of such systems. Second, using extensive numerical experiments, we quantify the benefits of such an expediting policy. Third, we investigate a number of managerial issues. Specifically, we analyze the impact of the number of expediting hubs and their locations along the shipment network on the performance of such systems and offer insights into the design of the shipment network. We show (i) a single expediting hub that is optimally located in a shipment network can capture the majority of cost savings achieved by a multi-hub system, especially when expediting cost is not low or demand variability is not high; (ii) when expediting time is proportional to the time to destination, for small-enough or large-enough demand variations, a single expediting hub located in the middle of the shipment network can capture the majority of cost savings of an optimally located hub; and (iii) in general, hubs close to the retailer significantly drive down costs, whereas hubs close to the supplier may not offer much cost savings.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Operating Characteristics and Analytical Properties
  6. 4. Numerical Analysis
  7. 5. Summary and Discussion
  8. Acknowledgments
  9. References
  10. Supporting Information

The increasing complexity of modern supply chains poses significant challenges and opportunities to companies around the world. Globalization has resulted in supply chains that span multiple continents. Companies frequently outsource the supply of raw materials and production of goods to countries with low labor and manufacturing costs. Consequently, lead times are increasing, which costs retailers either the loss of goodwill and customers due to more shortages or excess inventory holding costs as a result of the increase in safety stock levels and sometimes both. In practice, companies hedge against random fluctuations in demand by constructing policies to manage lead times. The rise of global transportation companies within the shipment service industry presents retailers with diverse freight alternatives with varying time–cost trade-offs. Furthermore, the advent of tracking technologies such as radio frequency identification (RFID) provides order progress information. Together, shipping alternatives and order tracking technologies constitute a viable option for managing lead times in the current environment.

At the beginning of the last decade, Walmart and other large retailers implemented pilot RFID programs to track order shipments closely in hopes of streamlining their supply chains and reducing stock-outs at their stores. However, lower than desired results have led some industry analysts to conclude that RFID tracking does not produce as many clearly proven benefits as anticipated (Duvall 2007). Demand for RFID technology subsequently has fallen below initial predictions. As Simon Ellis, head of the RFID program for Unilever USA, explains, “RFID tells us the pallets are in the [shipment network], but so what.… There isn't a lot you can do with that information’’ (Duvall 2007). One explanation posited for lower than desired outcomes is a lack of improved and updated policies that take full advantage of the technology.

By asking how one can modify ordering and shipment policies to improve logistics, given order location, updated demand information, and multiple shipping methods, we take the first steps to tackle these issues.

Consider an inventory system that faces relatively lengthy supplier's lead times. However, the system has access to alternative modes of shipping: a low cost standard mode and an expensive fast mode. All orders are initiated using the standard mode of resupply. If demand during lead time is relatively low, the firm likely retains sufficient inventory to satisfy demand using the standard shipment. However, when the demand during lead time is relatively high, then the in-transit order can be switched to the faster, more expensive shipment mode to protect against shortages. Combining the two shipment modes and using updated demand information allows the system to effectively manage the lead time of its orders and hedge against situations that cause excessive shortages. The use of alternative shipment modes can not only reduce shortages but also can lower inventory levels, as the system no longer needs to carry the high amount of safety stock otherwise required to reduce shortage risks.

The problem of managing lead times has been approached from two perspectives: emergency ordering and expedited ordering. Emergency ordering (also known as sourcing) supplements the longer lead time of the standard order process with the option of placing an emergency order with a shorter lead time at an extra cost or sourcing production to a more responsive supplier. As a result, updated information about demand during lead time does not affect the status of the previously placed order and it can only be used to place a new emergency order. Expedited ordering, however, effectively changes the lead time for an in-transit order, rushing it from its current location to the destination.

For continuous review emergency order models, Moinzadeh and Nahmias (1988) considered a system with two supply modes and proposed a simple-to-implement heuristic inventory policy. Their proposal is a plausible extension to the standard (Q,R) policy. Moinzadeh and Schmidt (1991) later extended that work to consider an (S-1,S) inventory policy when demands follow a Poisson process. Moinzadeh and Aggarwal (1997) further elaborated on the work to include a two-echelon inventory system. Song and Zipkin (2009) reinterpreted and generalized the results obtained by Moinzadeh and Schmidt (1991). Gaukler et al. (2008) considered the classical (Q,R) inventory system, in which the retailer has the option of placing emergency orders from an outside supplier at a cost that is related to its lead time. Kouvelis and Li (2008) studied the problem of utilizing a flexible supplier as an emergency response to varying lead times, and derived the corresponding optimal emergency-response policy. Allon and Van Mieghem (2010) designed a tailored base-surge sourcing strategy, which replenishes inventories from the cost-effective supplier at a constant rate and obtains stock from the more responsive supplier only when inventory falls below a specific target. Recently, Kouvelis and Li (2012) evaluated the use of emergency orders as well as planning for Disruption Safety Stock in conjunction with safety lead times as contingency strategies to cope with disruptions in order delivery systems.

In this article, we consider a continuous review inventory model with random but stationary demand where in-transit orders can be expedited. Replenishment orders experience a lead time that is made up of multiple intermediate stages, such as shipping ports or hubs. Standard orders use standard delivery that requires orders to pass through all stages until they reach the system. However, expedited orders, which skip some stages of the lead time, are also available at a premium cost.

Literature on the subject of expediting is relatively new and scarce. Lawson and Porteus (2000) considered a periodic review model with multiple stocking installations, which are used for expediting and also delaying orders. Kim et al. (2012) considered a periodic model in which the lead time between two consecutive stages is one period. They show the optimality of modified base stock policies for a class of models, and they propose a heuristic for analyzing the general case. Muharremoglu and Tsitsiklis (2003) considered a periodic review model in which the shipping lead time from one stage to another is equal to one period. Jain et al. (2011) studied a periodic review inventory model with updated demand information and fixed ordering costs for each shipment type. As noted, these models all address periodic review ordering systems and differ substantially from the model presented here. The works closest to our model are those of Duran et al. (2004), Jain et al. (2010), and Kouvelis and Tang (2012). Kouvelis and Tang (2012) studied the option of expediting in dealing with uncertain lead times when demand remains known. In their model, lead time consists of two stages with the option of expediting upon completion of the first stage. Jain et al. (2010) considered a continuous review make-to-order system with two different shipment modes for delivery. Even though their paper can best be described as a sourcing model, however, since freight mode decision is delayed until manufacturing is completed and the optimal policy uses information about the demand incurred in the meantime, the model can be considered one of expedited ordering as described earlier.

This study differs from that of Jain et al. (2010) and Kouvelis and Tang (2012) in two important ways. First, due to the nature of those models, both consider one expediting opportunity, which is implemented upon the completion of the first stage (manufacturing process in Jain et al. 2010 and first part of the supply lead time in Kouvelis and Tang 2012), whereas our model allows for any number of expediting opportunities. Second, in this work, we treat the expediting decision as a binary one (i.e., whether to expedite the entire order), whereas the mentioned papers allow for an order to be split and partially expedited to the destination. Furthermore, Kouvelis and Tang (2012) examine random lead times and deterministic demand, whereas in our model we study deterministic lead times and random demand. Duran et al. (2004) consider an expediting system that contains only one expediting opportunity and allows overall lead time to take one of two fixed forms: one with and one without the expediting.

In this article, we consider a continuous review inventory system with random but stationary demand and multiple expediting opportunities. Thus, depending on which expediting opportunity is exercised, the overall lead time can take one of the many different possibilities. We expand on the basic (Q,R) system and propose and analyze simple and easy-to-implement inventory policies that incorporate various expediting opportunities. Furthermore, we investigate a number of interesting questions related to designing the lead time and configuring “hubs” (points where expediting can be triggered), which will provide valuable managerial insights into such situations. Ultimately, this study addresses two primary issues: whether (i) the shipment network topology and positioning of hubs affects the magnitude of cost savings and, if so, which configuration can lead to the best outcome and (ii) a shipment network with only one expediting hub, which has been located optimally, can achieve a performance level similar to that of a network with multiple hubs. For every model, we identify the set of parameters that favors specific policies.

We conclude this section by summarizing the organization of the rest of the article. In section 'The Model', we present the general expediting model and set of policies considered throughout this article. In section 'Operating Characteristics and Analytical Properties', we derive the operating characteristics of the system and show some analytical properties of the optimal policy for a model with a single expediting hub. Later in this section, we extend the model to the case of multiple expediting hubs. In section 'Numerical Analysis', we summarize the results of extensive numerical experiments to support the main findings derived from the model. We also conduct numerical experiments to address the supply chain design issues raised above. Finally, in section 'Summary and Discussion', we summarize the model and discuss the managerial insights we obtained from the numerical analysis.

2. The Model

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Operating Characteristics and Analytical Properties
  6. 4. Numerical Analysis
  7. 5. Summary and Discussion
  8. Acknowledgments
  9. References
  10. Supporting Information

Consider a continuous review inventory model consisting of a single product facing random but stationary demand with independent increments. Assume that excess demand at the retailer is backlogged. Replenishment orders are placed with a supplier that has ample supply. Order lead time consists of the time that elapses as the order passes through M serial shipment nodes or hubs, referred to herein as expediting hubs. A shipping stage is the time between two expediting hubs. The transit time between expediting hubs j and + 1 (i.e., during stage j) is constant and equal to τj > 0. Lawson and Porteus (2000) also make a similar assumption of constant transit times during the stages; however, unlike their paper that assumes a constant (τj+1 − τj) for all j, we do not impose any specific assumption on the values of τj besides of being positive.

In our model, the retailer can manage order lead times by bypassing one or more of the shipping nodes—thus reducing the corresponding lead times—and receive the entire order faster (at a premium shipping rate). This is a viable option at times when an unanticipated surge in demand depletes the on-hand inventory at a faster rate than expected. For purposes of analysis, our model assumes that every order must pass from the supplier (S) through shipping stage 1. If an order is expedited at node i, then the remaining lead time will be ζi. Figure 1 illustrates a possible depiction of the model, which is constructed using a six-stage lead time and five expediting opportunities (hubs), with expediting transpiring from the end of stage 3 (hub 3) to the destination, which is the retailer (R). Clearly, the retailer would utilize the expediting opportunity only if the time to the destination of the order is improved upon. This means inline image where ζM+1 = 0 by definition.

image

Figure 1. Supply chain with expediting from the end of stage 3 to the final stage

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Theoretically, an order can be expedited multiple times, whereby having once been expedited, it can revert to the standard shipping mode and then be re-expedited at subsequent expediting hubs. Though allowing for an order to be expedited multiple times could offer the largest cost savings, it may be unrealistic. The cost of changing modes of transportation may be high, and for a particular order, one is more likely to make an expediting decision only once. Consequently, this study evaluates a model where expediting can be exploited only once to change the transportation mode of an order from standard to fast, which can be exercised at any of the multiple expediting hubs. Put differently, only one of the multiple expediting opportunities can be used. Note that one could extend the methodology presented in this article to consider those cases where an order can be expedited multiple times. However, the nature of the inventory and expediting policy for such a model will be of combinatorial form and render the policy computationally intractable.

Even when the expediting decision is made once, the exact analysis of the problem with multiple expediting opportunities is complicated by the fact that the form of the optimal stocking, replenishment, and expediting policy remains unknown, and we conjecture that it will be a complex function of net inventory and the progression of orders in the pipeline. Another major obstacle is that when expediting is allowed, pipeline orders may cross, making exact analysis of a given policy impossible. In fact, we would like to point out that for the case of standard (Q,R) policy with no expediting, exact analysis is only possible when orders do not cross (see Hadley and Whitin 1963, Zheng 1992). Zipkin (1986) outlines situations when this latter condition exists. To overcome these challenges, we turn our focus to a well-known approximate analysis (see Silver et al. 1998 and Lee and Nahmias 1993, among others) first employed by Hadley and Whitin (1963). In other words, our analysis is to extend the approximate analysis of Hadley and Whitin (1963) to the case when expediting is allowed. The approximations are based on the following assumptions:

Assumption 1.

  1. There is at most one order outstanding in the system at any time.
  2. Unit backorder cost is relatively high compared to unit holding cost.
  3. The backorder cost is based solely on the total per unit cost of the order contents.

The three parts in Assumption 1 are the same as those made by Hadley and Whitin (1963) and are applicable in a variety of practical circumstances. Under (i), we eliminate the issue of order crossing, which makes the analysis possible. We would also like to point out that the first assumption is reasonable in many practical situations when order quantities and hence order cycles of the product under consideration are large compared to the overall lead time (Silver et al. 1998). Assumption 1(i) also implies that when the reorder point is reached, there are no orders outstanding, which means inventory position and net inventory are equal. Therefore, the reorder point would be the same if one considers either inventory position or net inventory level in implementing the policy. Assumption 1(ii), whereby unit backorder cost is high compared with the unit holding cost, indicates that only policies resulting in infrequent shortage occurrences would be appropriate, which happens to be a common case in practice. Finally, Assumption 1(iii) can be considered a consequence of part 1(ii) as due to the relatively high cost of backorders, only stocking and ordering policies that result in a short stock-out period would be considered. Consequently, per-time backorder costs can be omitted in the analysis. For a more detailed discussion of these assumptions, the reader is referred to Hadley and Whitin (1963) and Lee and Nahmias (1993).

Using Assumption 1 and following the approach used in Hadley and Whitin (1963), we approximate expected on-hand inventory with expected net inventory. Note that the (optimal) order quantity should be large enough to meet all backorders and raise the on-hand inventory level above the reorder point; otherwise, the system would not trigger any orders, because the reorder point would no longer be reached, causing the size of backorders to grow infinitely. If the reorder point is expressed in terms of the inventory position, then the first assumption guarantees that the on-hand inventory will always rise above the reorder point when the order is delivered.

3. Operating Characteristics and Analytical Properties

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Operating Characteristics and Analytical Properties
  6. 4. Numerical Analysis
  7. 5. Summary and Discussion
  8. Acknowledgments
  9. References
  10. Supporting Information

Due to the complexity of the multi-hub problem, in this section, we start by studying a model with only one expediting hub. We derive the operating characteristics of the single expediting model together with an expression for the average total cost rate for this system. By exploiting some of the basic structural properties of such systems, we are able to gain insights into the properties of the optimal policy parameters. This enables us to develop an algorithm for finding the optimal replenishment/expediting policy parameters. Analysis of the single hub case proves to be important as (i) in section 'Numerical Analysis', our numerical experimentation indicates that a single expediting hub that is optimally located in a shipment network can capture the majority of cost savings, especially when the expediting cost is moderate to high or demand variability is low to moderate, and (ii) in section 'The Case with Multiple Expediting Hubs', the approach used in analyzing the single hub case will be employed to models with any number of expediting hubs. Next, we study the single-hub model.

3.1. The Case of One Expediting Hub

We now focus on the case with one expediting hub. We derive expressions for the operating characteristics of such systems together with an expression for the average total cost rate. Then we present some properties of the optimal cost parameters, which enable us to develop a procedure for finding the optimal replenishment and expediting parameters. Later, we expand this formulation to the case with multiple expediting hubs. Table 1 summarizes the notation employed in our analysis throughout the article. We assume that demand is stationary with independent increments and an average rate of μ.

Table 1. Summary of Notations
h Unit holding cost/time
π Unit backorder cost
A Fixed ordering cost
A e Fixed expediting cost
c Unit purchasing cost
e j Unit expediting cost from the end of stage j
τ Total lead time with no expediting
τ j Lead time of stage j
ζ j Expediting time from the end of stage j to the destination
p j Probability of expediting at the end of stage j
λ j Demand rate during stage j if expediting happens at the end of stage j
λ j Demand rate during stage j if expediting does not happen until after the end of stage j
H j Expected number of on-hand inventory carried in an order cycle if expediting happens at the end of stage j
H Expected number of on-hand inventory carried in an order cycle
B Expected number of backorders in an order cycle
T j Expected cycle time if expediting happens at the end of stage j
T Expected cycle time
f(.,t)Probability density function of demand during an interval of length t
F(.,t)Complementary cumulative distribution function of demand during an interval of length t
μ Mean demand rate
σ Standard deviation of demand rate

First, we propose the replenishment/expediting policy, which consists of an order quantity Q, a reorder point R, and an expediting trigger level R1 once the order has reached the expediting hub, expressed as (Q,R,R1). The policy, therefore, consists of an ordering component and an expediting component. The ordering policy is similar to that analyzed by Hadley and Whitin (1963), where R is based on the on-hand inventory levels. The expediting policy is defined as follows: starting from the first stage, when the order reaches the hub, if the on-hand inventory level is below R1, then the retailer should expedite the order.

To analyze the problem under the proposed policy, define a cycle as the time between placements of two consecutive orders. Note that cycles are independently and identically distributed. Furthermore, as depicted in Figure 2, a cycle may or may not involve expediting. Denote p1 as the probability of expediting in a cycle. Let λ1 (λ1) be the average demand rate during the first stage of lead time (τ1) if expediting does (does not) happen. Then,

  • display math
image

Figure 2. Behavior of Net Inventory in a Cycle for the Single Hub Model

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The expected total cost rate for the system, where H, B, and T are the on-hand inventory carried in a cycle, the number of backlogs in a cycle, and the average cycle time, respectively, would be as follows:

  • display math(1)

As mentioned before, we approximate the expected on-hand inventory with expected net inventory. The approximation has proven to be effective when unit backorder cost is high relative to unit holding cost, which results in optimal policies that would allow infrequent stockouts, making expected net inventory close to that of on-hand inventory.

To find the expected net inventory, we derive expressions for the area under the trapezoid for each stage, conditioned on the probability that whether expediting occurs, denoted by H1 and H1, respectively (see Figure 2). This results in:

  • display math

We can now write the expected on-hand inventory in a cycle, H, by conditioning on whether expediting happens as

  • display math(2)

Next, the expected number of backorders can be derived by considering the two cases where expediting does (does not) happen and will be as follows:

  • display math(3)

Finally, the expected cycle time can be found in a similar way. Let T1 (T1) be the expected cycle time if expediting does (does not) happen, which is the aggregate time to receive the item since the time order has been released plus the extra time needed for on-hand inventory to drop to the reorder level R. Hence, inline imageAs a result, the average cycle time, T, turns out to be

  • display math

Notice that this result is not surprising, as a total of Q units are consumed during the course of a cycle, regardless of whether the order is expedited. Plugging values of p1, λ1, λ’1, T, and H from (2) and B from (3) into (1), the expected total cost rate, TC, can be written as a function of Q, R, and R1.

Taking the partial derivative of the expected total cost rate with respect to Q, we can obtain the optimal order quantity for a given R and R1:

  • display math(4)

While determining the optimal order quantity depends on the value of the reorder point and expediting trigger level, the following proposition shows that the optimal Q is always greater than the economic order quantity in the absence of the expediting option.

Proposition 1. Let inline image be the economic order quantity of the inventory system without expediting. Then Q* ≥ EOQ.

To further simplify the analytical results, we define Δ1 = R − R1 as the minimum demand consumption during τ1 that triggers expediting and hereafter express the average total cost rate based on Q, R, and Δ1. Under this transformation, the terms λ1, λ1, and p1 are no longer dependent on R, and the average total cost rate, TC, is convex in R. The following proposition formalizes this argument and develops other analytical properties of the optimal reorder point, R.

Proposition 2. Let Q, R be the order quantity and reorder point, respectively. Define Δ1 = R − R1. Then,

  1. For fixed Q and Δ1, the expected total cost rate, TC, is convex in R.
  2. Let R*1) be the optimal value of R for fixed Δ1; then inline image.
  3. The optimal reorder point, R*, is bounded by RL ≤ R* ≤ RU, where, RL and RU satisfy
    • display math

This result suggests that TC is a well-behaved function of R. Furthermore, part (b) indicates that the optimal reorder point is inversely related to the expediting likelihood. This is intuitive, as for those cases in which the expediting option is used only rarely, one needs a higher reorder point to avoid large backorders. Later in this section, we will use the upper and lower bounds found in (c) to develop an algorithm for the single expediting hub case. Note that RL and RU are the optimal reorder points when the retailer always expedite (i.e., Δ1 = 0), and never expedites (i.e., Δ1 = ∞), respectively.

Before exploring the properties of the expected total cost rate as a function of Δ1, we note that in a standard (Q,R) inventory system, inline image is a surrogate for the Type 1 service level. Therefore, for all practical purposes where service level is higher than 50%, Proposition 2(c) suggests that the optimal reorder point is higher than the median of demand in τ1 + ζ1 periods. Put differently,

Corollary 1. If πμ ≥ 2hQ, then R* ≥ RLμ(τ1 + ζ1) ≥ μτ1 for symmetric demand scenarios.

Unlike R, the expected total cost rate is not a well-behaved function of Δ1 for generic demand functions. Proposition 3, however, illustrates some properties of the total cost function when demand is Normally distributed, which helps us in designing an effective algorithm for obtaining the optimal expediting trigger level. Before doing so, we explore some properties of the Normal distribution that will aid us in Proposition 3.

Lemma 1. Let φ(z) and inline image be the probability density and complementary cumulative distribution functions of the standard Normal distribution, respectively.

  1. Let inline image. Then g(z) is first quasi-concave and decreasing, then quasi-convex and decreasing in z for z ≥ 0.
  2. Let A(z) = 2Φ(z)(1 − Φ(z)) − 3φ2(z). Then A(z) is first quasi-concave then quasi-convex in z for z ≥ 0.
  3. Let C(z) = − zΦ(z)(1 − Φ(z)) + φ(z)(1 − 2Φ(z)). Then C(z) is quasi-concave in z ≥ 10.
  4. Functions A(z) and C(z) are non-negative for z ≥ 0.

We are now ready to characterize some properties of TC as a function of Δ1.

Proposition 3. Suppose that the demand rate follows a Normal distribution with an average of μ and a standard deviation of σ. Further, suppose that the order quantity, Q, and reorder point, R, are fixed.

  1. If e1Q + Ae ≥ (πμ − hQ)(τ2 − ζ1), then the expected total cost rate, TC, in a single expediting hub case is decreasing in Δ1 for inline image, and Δ1[RIGHTWARDS ARROW] ∞.
  2. If e1Q + Ae ≤ (πμ − hQ)(τ2 − ζ1), then the expected total cost rate, TC, in a single expediting hub case is first quasi-concave, then quasi-convex in Δ1 for inline image.

This result suggests that the optimal expediting trigger parameter Δ1 is never in the interior of inline image. Note that Corollary 1 suggests R* ≥ μτ1 when πμ ≥ 2hQ. Put differently, for most of the practical scenarios, the optimal expediting trigger level, R1, can never be negative. Such a strategy is always dominated by another one where either R1 ≥ 0, or by the case of no expediting at all inline image. Of course, the relative value of the expediting cost compared with the other cost parameters determines which strategy has a lower cost. Proposition 3(a) provides a sufficient condition for cases when the expediting cost is high and therefore the optimal policy is to not expedite at all.

Although Proposition 3 characterizes the optimal Δ1 in the inline image region, the total average cost rate, TC, is not a tractable function for Δ1 in the inline image range where the average on-hand inventory, H, is not well behaved. Using Propositions 2 and 3, however, we can construct a procedure for finding the optimal value of Δ1 based on exhaustive search as follows.

  • 1.
    Set inline image.
  • 2.
    Let Q1 = Q. For given model parameters find the maximum reorder point RU in Proposition 2.
  • 3.
    Carry out an exhaustive search for optimal Δ1 in predetermined δ increments in inline image as follows:
    1. For any Δ1, find the optimal R*(Δ1) using the first-optimality conditions of TC.
  • 4.
    Compare the minimum cost obtained based on the search in step 2 with the optimal (Q,R) model where no expediting is allowed and find the overall optimum strategy.
  • 5.
    Find the optimal order quantity Q from (4). If Q ≠ Q1 then go to 2 and repeat.

3.2. The Case with Multiple Expediting Hubs

In this section, we extend the analysis obtained thus far to the case of multiple expediting hubs. We note that while multiple hubs are available, in this section we continue to study a model where an order can be expedited only once, a plausible condition at any given shipping stage (i.e., at each expediting hub). We start by expanding the proposed stocking/expediting policy in section 'The Case of One Expediting Hub' to allow for multiple expediting opportunities followed by derivation of the operating characteristics of this system. Similar to the single hub case, we propose the following policy, which consists of an order quantity Q, a reorder point R, and expediting trigger levels Rj (R ≥ R1 ≥ … ≥ RM) once the order has completed stage j, which can be expressed as (Q,R,R1,R2,…,RM). Analogous to the single hub case, the policy consists of an ordering component and an expediting component. However, the expediting policy is now expanded to handle the multiple hub case and is defined as follows: starting from the first stage, when the order reaches hub j, if the on-hand inventory level is below Rj, then the retailer should expedite the order to the destination.

Expanding on the notation introduced in section 'The Case of One Expediting Hub', define pj to be the probability of expediting at the end of stage j. Furthermore, let λj (λj) be the demand rate during stage j if expediting does (does not) happen at the end of that stage. That is,

  • display math

Figure 3 illustrates the relationship between the on-hand inventory, trigger levels, and λ parameters for a case where expediting happens at the end of stage 3.

image

Figure 3. Depiction of the multi-hub expediting model with an expediting decision at the end of stage 3

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The expected total cost rate, where H and B are the on-hand inventory carried and the number of backlogs in a cycle and T is the average cycle time, would be as follows:

  • display math(5)

To find the expected net inventory, we derive expressions for the area under the trapezoid for each stage j < i, conditioned on the probability that expediting occurs at the end of stage i (denoted by Hi). Figure 3 shows a typical case where = 3. This results in

  • display math

We can compute the expected on-hand inventory in a cycle, H, by conditioning on stage i, which triggers the expediting option. Multiplying the above expressions for Hi with their corresponding probabilities, pi, and simplifying, we obtain the following:

  • display math

Applying a similar approach would lead to the following expected backorder cost term:

  • display math

Let Ti be the expected cycle time if expediting happens at the end of stage i, which is the aggregate time before expediting hub i plus the extra time needed for on-hand inventory to drop to the reorder level R. Hence, Ti's can be written as follows:

  • display math

Confirming the result in the single hub case, the average cycle time, T, turns out to be

  • display math

Plugging values of pj, λj, λj, T, H, and B in (5), the expected total cost rate, TC, can be written as a function of Q, R, and Ri. Hence, based on the total cost function in (5), we can evaluate and rank different inventory strategies. However, not surprisingly, one can show that finding an optimal analytical solution is not a trivial task, as similar to the single expediting hub case, TC is not a convex function of its parameters and thus not well behaved. Although the optimal policy parameters cannot be found analytically, we can use some structural properties of the cost function to significantly reduce the search space when finding the optimal policy parameters numerically. Similar to the single hub case, we define Δi = R − Ri to make the total cost a convex function of the reorder level. Such a convexity property allows us derive upper and lower bounds for the optimal reorder level, R.

Proposition 4. Let Δi = R − Ri; then (a) the total cost, TC, is a convex function of R. (b) Let R*(τ) be the optimal reorder point for a system “without” the expediting option and lead time of τ. The optimal reorder point for a system “with” expediting option, R*, is an increasing function of Δi with the following bounds: R(τ1) ≤ R ≤ R(τ).

We use properties developed in Proposition 4 to search for the optimal inventory/expediting policy. Although we have not been able to develop a proof, our experience indicates that, despite its non-convexity, the optimal solution is unique in all cases. We observed, in fact, that the cost function behavior is always quasi-concave then quasi-convex for Normally distributed demand scenarios, guaranteeing the uniqueness of its optimal solution (see Figure 4 for a typical example).

image

Figure 4. An Example of Behavior of TC that Guarantees a Unique Optimal Solution Note: For some parameter values, the first or last concave parts may not exist.

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4. Numerical Analysis

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Operating Characteristics and Analytical Properties
  6. 4. Numerical Analysis
  7. 5. Summary and Discussion
  8. Acknowledgments
  9. References
  10. Supporting Information

In this section, we investigate a number of issues that are deemed to be important when managing systems with an expediting option. In doing so, we conduct numerical experiments that would enable us to build important managerial insights that are otherwise difficult to develop analytically. In particular, we are interested in answering the following questions:

  • How beneficial is the expediting option? For what range of parameters does the expediting option achieve larger cost savings?
  • Are multiple expediting hubs necessary, or can we capture most of the potential benefits by using a single expediting hub whose location is chosen optimally?
  • How important is the transportation network topology? That is, do cost savings differ considerably for specific expediting hub configurations, or is it location independent?

We applied our proposed policies to a combined total of more than 500 cases. In each experiment, the optimal policy parameters were obtained by employing the results in section 'Operating Characteristics and Analytical Properties'. In section 'Inventory System with Three Expediting Hubs', we consider a system with three expediting hubs, using the policy proposed in section 'Operating Characteristics and Analytical Properties' and identify environments where expediting is most beneficial. Moreover, we investigate the effect of the location of the hubs in the shipment network and identify environments where a particular hub topology works best. In section 'Shipment Network Topology: Number vs. Location of Hubs', we focus on effectiveness of a single hub model and identify conditions where, by optimally locating a hub, most of the benefits are captured when compared to a multi-hub model.

4.1. Inventory System with Three Expediting Hubs

In our numerical experiment, we considered a shipment network with three expediting hubs and conducted various numerical experiments in an effort to quantify the magnitude of cost savings that may result when compared to models with no expediting. We also investigated the effect of the location (placement) of expediting hubs on the performance of the inventory system. Specifically, we set τ = 1 and considered three hub configurations: (1) hubs are spread equally, that is, τ1 = τ2 = τ3 = τ4 = 0.25. (2) All hubs are located toward the beginning of the supply chain (closer to the supplier), in particular, inline image. (3) All hubs are located toward the end of the supply chain (closer to the retailer), in particular, inline image.

Before introducing the range of parameters used in our experiments, we note that our initial experiments indicated that the optimal Q (computed numerically) is fairly close to the EOQ value of the inventory system without expediting. More specifically, the optimal Q was equal to 105%~115% of the EOQ value in all of our initial experiments. Therefore, we have set Q to 110% of the corresponding EOQ value in each of the experiments below. Figure 5 compares the average total cost rate for the optimal and approximate Q's. We note that the cost difference between the two order quantities is fairly small.

image

Figure 5. Cost Comparisons between the Optimal Q and when = 1.1 EOQ

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In all, we considered a total of 72 problems with the following values: we normalized the holding cost to = 1. Note that since Q is fixed, we can set the unit purchasing and expediting cost to zero (c = ei = 0), as these costs can be captured by the corresponding fixed costs. The fixed ordering cost is set at = 100, and the fixed expediting cost is varied in Ae ∈ [0,100] to evaluate the sensitivity of the model with respect to small and large expediting costs relative to A. Note that Ae is selected to be less than A to make it cheaper to expedite an order than to place an entire new order. We used π ∈ {5, 10, 20, 50} to examine the effect of high and low backordering costs (relative to the holding cost) on policy outcomes. The expediting time ζi is assumed to be proportional to the length of the expedited order. That is, inline image, for some inline image. In our experiments, we varied θ in its entire range in 0.05 increments. Below, we represent the results for the representative case of θ = 0.15. Later in this section, we evaluate the sensitivity of numerical results as a function of θ. Finally, we assume that demand rate follows a Normal distribution with μ = 100. Given that the coefficient of variation for a Normal distribution is typically lower than 0.5 (Silver et al. 1998), we varied σ = 10, 20, 30 to consider the effects of low and high variability of demand on the outcome.

Table 2 presents the results using a defined set of cost parameters and various hub configurations. In each experiment, the length of the last shipping stage, τ4, can be obtained as τ4 = τ – τ1 – τ2 – τ3. We observed that for all the problems considered, the magnitude of cost savings for our expediting model compared to those without expediting is on the order of ~[0.1%, 15%] with an average of 9.8% across the cases studies, which is rather significant. It is not surprising that the value of the expediting option increases with larger variations of demand (σ).

Table 2. Percent Cost Savings, Optimal Expediting Levels, and Reorder Points and Expediting Probabilities for the Three Configurations for θ = 0.15
  τ 1 τ 2 τ 3 R Δ1 (p1)Δ2 (p2)Δ3 (p3)CostSavings
π = 20, Ae = 20σ = 10No Expediting115No Expediting161 
¼¼¼10860 (~0%)70 (0.2%)83 (18%)1562.63%
11550 (~0%)60 (~0%)70 (0.3%)1600.26%
½8664 (~2.8%)64 (~62%)72 (25%)1562.72%
σ = 20No Expediting129No Expediting179 
¼¼¼10255 (0.1%)64 (16%)72 (42%)1667.46%
12453 (~0%)53 (4.6%)69 (5.8%)1781.00%
½9870 (7.6%)70 (34%)83 (12%)1648.33%
σ = 30No Expediting143No Expediting198 
¼¼¼9753 (3%)53 (41%)63 (26%)17611.26%
12948 (0.5%)58 (6.6%)73 (8.6%)1922.92%
½10060 (31%)64 (24%)76 (10%)17213.23%

Figure 6 illustrates the average cost and inventory parameter reductions for our hypothetical shipping network with three expediting opportunities. Figure 6(a) depicts the magnitude of average cost savings—taken over various expediting costs—as a function of σ, for different values of π. Notice that for larger values of σ, the average cost savings become as high as 15%. Figure 6(b) shows the average cost savings—taken over various standard deviations—as a function of Ae. Naturally, for higher expediting costs, we observed lower cost savings; however, even for fairly large expediting cost parameters the system enjoys a relatively high cost savings of ~9%, when demand variability is significant (σ = 30). Note that such cost savings are a result of lower backorder level and lower inventory due to a lower reorder point with expediting. In fact, savings obtained due to lower inventory costs is higher than lower backorder costs. For instance, take the case of τ1 = 1/2, τ2 = τ3 = 1/6, and σ = 20 in Table 2. The expected expediting cost is inline image. On the other hand, the retailer enjoys 24 fewer items in his stock (down to 120 units from 144 units), and only 0.08 few backorders on average (down to 0.60 units from 0.68 units). Therefore, the average additional cost in the system is $10.3 and the average cost savings are $25.6. Figure 6(c) and (d) depict the average reduction in the inventory system parameters (average inventory, H, and backorder levels, B) when the expediting option is available. It is noteworthy that while the reductions in backordered items may not be high, the average number of backorders drops by 40%~60% when expediting is used. Average inventory level also decreases by 5%~20%.

image

Figure 6. Average Reductions in Total Cost and Inventory Parameters Computed for a System with Three Expediting Opportunities when τ1 = τ2 = τ3 = ¼, θ = 0.15

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Next, we explored the role of the placement of expediting hub configurations on cost savings to the retailer. The main issue in this experiment was whether a certain type of hub allocation yields the highest cost savings. We also investigated whether certain hub locations have a more important role in reducing costs. In doing so, we considered the three hub configurations described earlier. First, we found that early expediting opportunities may not offer much savings. That is, in most cases the best savings are obtained with either configuration (3), in which hubs are located closer to the retailer for larger values of backorder costs, or configuration (1), in which hubs are spread equally in the transportation network for smaller values of backorder costs and less uncertain demand scenarios. Figure 7 illustrates this behavior by plotting the total average cost as a function of the backorder cost for different levels of demand variability when Ae = 20. Note that when coefficient of variation is low enough (σ/μ ≤ 0.1), configuration 1 outperforms configuration 3. On the other hand, for moderate to large values of coefficient of variation (σ/μ ≥ 0.2), configuration 3 leads to a lower cost than configuration 1. For the coefficient of variation of 0.15, we observe that configuration 1 is the best alternative when π/Ae ≤ 0.7, and configuration (3) leads to the lowest cost for π/Ae ≥ 0.7.

image

Figure 7. Average Cost Comparisons for Configurations 1 and 3, for Ae = 20 and θ = 0.15

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Even in cases where configuration (1) yielded the best results, the hubs that were used more frequently to expedite orders were those that were closer to the destination (see expediting probabilities in Table 2). Such an outcome is due to the fact that in most cases, the retailer would not over-react to an early surge in demand as it may be offset by a lower demand rate later during the lead time. Therefore, it is best for the retailer to wait to ensure the need for expediting.

Figure 8 compares the three configurations graphically for π = 20 and π = 50. It depicts the cost savings associated with each configuration, highlighting the significance of hubs closer to the retailer and showing that the retailer can obtain the greatest benefit by using them. Note that for π = 20, configuration (3), and for π = 50, configuration (1) yield the best results. It is noteworthy that in either case, the performances of both configurations are fairly close to one another and noticeably better than configuration (2).

image

Figure 8. Cost Savings for Three Expediting Hub Configurations: Spread Equally, Located Close to the Supplier, and Located Close to the Retailer, for σ = 30 and θ = 0.15

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4.2. Shipment Network Topology: Number vs. Location of Hubs

In general, one is likely to assume that a system with multiple hubs will outperform that with a single hub. However, when the location of the single expediting hub is chosen optimally, scrutinizing the issue can provide important insight into supply chain design. Such an examination illuminates the importance of the number versus the location of expediting hubs with respect to total cost.

Table 3 summarizes the cost savings resulted from employing a single expediting opportunity in our numerical experiments. We present these results, using two different scenarios. The first derives the optimal policy for a case where the location of the expediting hub is chosen optimally (from the retailer's point of view); that is, τ1 = τ* where τ* inline image. The second implements the optimal policy for a case in which the expediting hub is located halfway between the supplier and retailer (i.e., τ1 = 0.5). Comparing the two models, we can see whether cost savings are merely due to the existence of the expediting option or if the location of the hub plays a significant role. On the basis of the insights obtained from previous experiments, we expect to achieve some of the lowest cost savings for the second scenario when the expediting hub is located in the middle of the supplier and retailer. This is verified through our experimentation for small and large values of σ, which is a measure of demand variability. For moderate values of demand variability, the location of the expediting hub could have a more significant effect on cost savings. Such a behavior is due to the fact that for small values of σ, the overall value of expediting is somewhat limited, as the classic (Q,R) inventory policy can achieve a relatively low cost due to demand predictability. Therefore, the gap between total costs of the optimal expediting hub and no expediting scenario, and thereby the fixed expediting hub, is relatively small. For large values of σ, the location of the expediting hub is closer to the middle of the lead time, and the retailer expedites his order more frequently; therefore, the optimal and fixed expediting hub again show relatively similar cost savings. It is for the moderate values of σ that the optimal hub location is farther from the middle of the lead time, offering the best tradeoff between inventory, backorder, and expediting costs.

Table 3. Only One Expediting Opportunity for θ = 0.15
 Model τ 1 R Δ 1 CostSavings
π = 20, Ae = 20σ = 10No ExpN/A115N/A161 
Opt. Hub0.91141001591.25%
Fixed Hub0.5114651600.8%
σ = 20No ExpN/A129N/A180 
Opt. Hub0.74109731696.01%
Fixed Hub0.5115601771.67%
σ = 30No ExpN/A142N/A198 
Opt. Hub0.70119771837.80%
Fixed Hub0.596331875.54%

A more important and insightful task is evaluating the merit of multiple expediting opportunities. In particular, we would like to investigate whether having numerous expediting hubs or using a single expediting hub with the retailer choosing the location produces the more satisfactory outcome. In our experiments, we used the three multi-hub configurations described before and compared their costs to that where a single hub is chosen optimally. The results summarized in Table 4 indicate that a carefully chosen expediting opportunity can achieve most of the benefits associated with multiple expediting hubs. Note that for this experiment we chose configuration (3), which typically leads to the best cost savings of the three configurations studied in this article. In most of our experiments, the single expediting hub even exceeded the cost savings associated with configuration (2).

Table 4. One Optimal Hub vs. Multiple Expediting Hubs for θ = 0.15
  τ R Savings
  (a) Low expediting cost
π = 20, Ae = 10σ = 100.13311.71%
(½,⅙,⅙)1083.51%
σ = 200.731076.36%
(½,⅙,⅙)1129.64%
σ = 300.691178.31%
(½,⅙,⅙)10013.9%
  τ R Savings
  (b) High expediting cost
π = 20, Ae = 50σ = 100.901051.14%
(½,⅙,⅙)1121.70%
σ = 200.741134.97%
(½,⅙,⅙)1125.43%
σ = 300.711216.78%
(½,⅙,⅙)1078.81%

Interestingly, the optimally chosen single hub model captures the majority of cost savings for a multi-hub scenario, especially when the expediting cost is moderate to high or demand variability is low to moderate. Throughout our experiments, we also observed that the gap between one optimally placed expediting hub and multiple hubs is a decreasing function of the fixed expediting cost (Ae). From Table 4, when the demand uncertainty, σ, is relatively small, one optimally placed expediting hub produces results that are very close to the (fixed) multiple hub options. Furthermore, for moderate values of σ, the optimal hub location is relatively insensitive to the expediting cost (e.g., τ1 = 0.7τ in this case). On the other hand, we observe an interesting outcome when demand is relatively stable (σ = 10). For low values of σ, the optimal expediting hub is located very close to either the retailer (τ1 [RIGHTWARDS ARROW] 1) for high values of expediting cost or the supplier (τ1 [RIGHTWARDS ARROW] 0) for low values of expediting cost. In the former case, the retailer barely expedites an order, if at all; this is due to the predictable nature of demand and high expediting cost. In the latter case of small τ1, however, the retailer would almost always expedite his orders. The first row of Table 4(a) illustrates one such scenario. Note that in this case, the inventory system effectively reduces to another one with a reorder point of = 31 and lead time of τ1 + θ(τ − τ1) = 0.13 + (0.15)(0.87) = 0.26. Put differently, for low demand variability, the optimal strategy takes a “degenerate” form: either expedite every order or never expedite, depending on the expediting cost. For moderate to high values of demand uncertainty, however, a “mixed” strategy is optimal, where the decision of whether to expedite an order depends on the real-time demand information. Therefore, for such cases the optimal location of the expediting hub is well within the interior region of [0, τ] (and mostly in the second half in our experiments).

Finally, we study the effect of value of θ on the outcome of the presented model. The smaller values of θ indicate situations where expediting options result in orders reaching the retailer in less time. Clearly, one would expect that the value of the expediting option would be inversely related to the values of θ. Figure 9 precisely shows this conjecture. Furthermore, Figure 9 demonstrates the optimal location of the single expediting hub as θ is varied. Interestingly, the optimal hub moves closer to the supplier for larger values of θ. This is because for high values of θ, it takes longer to receive the expedited order; therefore, the retailer would like to make the expediting decision earlier to offset the effect of a “slower” expediting option. Note that the optimal location of the expediting hub is not significantly sensitive to the value of θ with the lowest and highest values being 0.4 and 0.7, respectively.

image

Figure 9. Cost Savings and the Optimal Hub Location as a Function of θ for π = 20, σ = 30, and Ae = 10

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5. Summary and Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Operating Characteristics and Analytical Properties
  6. 4. Numerical Analysis
  7. 5. Summary and Discussion
  8. Acknowledgments
  9. References
  10. Supporting Information

In this article, we considered an inventory system in which the retailer can manage the overall delivery lead time by expediting the orders once they are shipped from the supplier. By analyzing an ordering/expediting policy consisting of order quantity, reorder point, and expediting trigger levels based on the on-hand inventory quantities, we obtain the operating characteristics of the system. Extensive numerical experiments subsequently illustrate the benefit of such an expediting option. We further investigated research questions that address problems raised in the shipment network design, such as effects of expediting hub number and location on the performance of the retailer or the supply chain. We observed that a single, optimally located expediting hub could capture the majority of cost savings, especially when the expediting cost is moderate to high or demand variability is low to moderate. We also found a single hub located in the middle of the lead time can capture the majority of the cost savings, especially for very small or large values of standard deviation of demand. For moderately variable demand scenarios, we observed that the optimal location of the single hub becomes more important. In our numerical experiment, the optimal hub is always located closer to the retailer. Finally, we observed that in the multiple expediting hubs scenario, hubs located closer to the retailer have more of an impact in driving down costs.

In this article, we assume that orders are only expedited to the destination and not other expediting hubs. One could consider a heuristic inventory/replenishment policy similar to that discussed in this article, consisting of order quantity, reorder point, and expediting trigger levels (Q, R, R1, R2, …, RM-1), if this assumptions were to be relaxed. With this heuristic policy, the ordering policy, (Q,R), would remain the same as before. Let ζij be the expediting time from expediting hub i to hub j. The expediting policy is modified as follows: starting from the first stage, when the order completes node i, if the on-hand inventory level H is less than Ri, then the retailer should expedite to the expediting hub j, where j > i is the smallest period index for which H − μζij is above Rj.

Throughout this article, we made a number of assumptions that enabled us to derive closed-form expressions for the operating characteristics of the system. Some of these assumptions can be relaxed by extending existing methods. Others could lead to interesting future work. Assumption 1 is crucial for the analysis, as it allows us to state the ordering/expediting policy based on the net on-hand inventory. Relaxing this assumption could significantly complicate the analysis. One can use a variant of the current policy as a heuristic and numerically quantify its benefits when there could be more than one outstanding order at times. An extension of the current model can be considered to include multiple cases where an order can be expedited multiple times between different expediting hubs. One possible heuristic for such a model would be to repeatedly apply the process described above (for single expediting opportunity which can be interrupted in the midway) whenever expediting conditions are met. Finally, another extension to this model could be an inventory system where both sourcing (i.e., expediting from the supplier) and expediting opportunities are available. With the presence of Assumption 1, if it is optimal to expedite at the supplier once, then it will be optimal to expedite every other order at the supplier as well. Such a system effectively reduces to an inventory system with no expediting option and the lead time of τ’ = θτ (sourcing). Therefore, one can extend the current model to the case where both expediting and sourcing options are available by relaxing Assumption 1. This interesting problem is outside of the scope for the current article and is left for future study.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Operating Characteristics and Analytical Properties
  6. 4. Numerical Analysis
  7. 5. Summary and Discussion
  8. Acknowledgments
  9. References
  10. Supporting Information

This article benefited significantly from the review team, and we are grateful for suggestions made by the anonymous reviewers, associate editor, and department editor.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Operating Characteristics and Analytical Properties
  6. 4. Numerical Analysis
  7. 5. Summary and Discussion
  8. Acknowledgments
  9. References
  10. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Operating Characteristics and Analytical Properties
  6. 4. Numerical Analysis
  7. 5. Summary and Discussion
  8. Acknowledgments
  9. References
  10. Supporting Information
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