#### 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 *R*_{1} once the order has reached the expediting hub, expressed as (*Q,R,R*_{1}). 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 *R*_{1}, 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 *p*_{1} as the probability of expediting in a cycle. Let *λ*_{1} (*λ*^{’}_{1}) be the average demand rate during the first stage of lead time (*τ*_{1}) if expediting does (does not) happen. Then,

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:

- (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 *H*_{1} and *H*_{1}^{′}, respectively (see Figure 2). This results in:

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

- (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:

- (3)

Finally, the expected cycle time can be found in a similar way. Let *T*_{1} (*T*_{1}^{′}) 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, As a result, the average cycle time, *T*, turns out to be

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 *p*_{1}, *λ*_{1}, *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 *R*_{1}.

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 *R*_{1}:

- (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 be the economic order quantity of the inventory system without expediting. Then *Q** ≥ *EOQ*.

To further simplify the analytical results, we define *Δ*_{1} = *R − R*_{1} as the minimum demand consumption during *τ*_{1} that triggers expediting and hereafter express the average total cost rate based on *Q*,* R*, and *Δ*_{1}. Under this transformation, the terms *λ*_{1}, *λ*^{’}_{1}, and *p*_{1} 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 *Δ*_{1}* = R − R*_{1}. Then,

- For fixed
*Q* and *Δ*_{1}, the expected total cost rate, *TC*, is convex in *R*. - Let
*R*^{*}*(Δ*_{1}*)* be the optimal value of *R* for fixed *Δ*_{1}; then . - The optimal reorder point,
*R*^{*}*,* is bounded by *R*_{L}* ≤ R*^{*}* ≤ R*_{U}*,* where, *R*_{L} and *R*_{U} satisfy

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 *R*_{L} and *R*_{U} are the optimal reorder points when the retailer always expedite (i.e., *Δ*_{1} = 0), and never expedites (i.e., *Δ*_{1} = ∞), respectively.

Before exploring the properties of the expected total cost rate as a function of *Δ*_{1}, we note that in a standard *(Q,R)* inventory system, 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 *τ*_{1} + *ζ*_{1} periods. Put differently,

Corollary 1. If *πμ* ≥ 2*hQ,* then *R** ≥ *R*_{L} ≥ *μ*(*τ*_{1} + *ζ*_{1}) ≥ *μτ*_{1} for symmetric demand scenarios.

Unlike *R*, the expected total cost rate is not a well-behaved function of *Δ*_{1} 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 be the probability density and complementary cumulative distribution functions of the standard Normal distribution, respectively.

- Let . Then
*g*(*z*) is first quasi-concave and decreasing, then quasi-convex and decreasing in *z* for *z ≥ 0*. - Let
*A*(*z*) = 2Φ(*z*)(1 − Φ(*z*)) − 3*φ*^{2}(*z*). Then *A(z)* is first quasi-concave then quasi-convex in *z* for *z ≥ *0. - Let
*C*(*z*) = − *z*Φ(*z*)(1 − Φ(*z*)) + *φ*(*z*)(1 − 2Φ(*z*)). Then *C(z)* is quasi-concave in *z ≥ *10. - 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 *Δ*_{1}.

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.

- If e
_{1}*Q* + *A*_{e} ≥ (*πμ* − *hQ*)(*τ*_{2} − *ζ*_{1}), then the expected total cost rate, *TC*, in a single expediting hub case is decreasing in *Δ*_{1} for , and *Δ*_{1}* ∞. - If e
_{1}*Q* + *A*_{e} ≤ (*πμ* − *hQ*)(*τ*_{2} − *ζ*_{1}), then the expected total cost rate, TC, in a single expediting hub case is first quasi-concave, then quasi-convex in *Δ*_{1} for .

This result suggests that the optimal expediting trigger parameter *Δ*_{1} is never in the interior of . Note that Corollary 1 suggests *R** ≥ *μτ*_{1} when *πμ* ≥ 2*hQ*. Put differently, for most of the practical scenarios, the optimal expediting trigger level, *R*_{1}, can never be negative. Such a strategy is always dominated by another one where either *R*_{1} ≥ 0, or by the case of no expediting at all . 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 *Δ*_{1} in the region, the total average cost rate, *TC*, is not a tractable function for *Δ*_{1} in the 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 *Δ*_{1} based on exhaustive search as follows.

- 1.
Set

.

- 2.
Let

*Q*_{1} =

*Q*. For given model parameters find the maximum reorder point

*R*_{U} in Proposition

2.

- 3.
Carry out an exhaustive search for optimal

*Δ*_{1} in predetermined δ increments in

as follows:

- For any
*Δ*_{1}, find the optimal *R*^{*}(*Δ*_{1}) 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* ≠ *Q*_{1} 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 *R*_{j} (*R ≥ R*_{1} *≥ … ≥ R*_{M}) once the order has completed stage *j*, which can be expressed as (*Q,R,R*_{1}*,R*_{2}*,…,R*_{M}). 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 *R*_{j}, then the retailer should expedite the order to the destination.

Expanding on the notation introduced in section 'The Case of One Expediting Hub', define *p*_{j} to be the probability of expediting at the end of stage *j*. Furthermore, let *λ*_{j} (*λ*^{’}_{j}) be the demand rate during stage *j* if expediting does (does not) happen at the end of that stage. That is,

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.

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:

- (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 *H*_{i}). Figure 3 shows a typical case where *i *=* *3. This results in

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 *H*_{i} with their corresponding probabilities, *p*_{i}, and simplifying, we obtain the following:

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

Let *T*_{i} 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, *T*_{i}'s can be written as follows:

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

Plugging values of *p*_{j}, *λ*_{j}, *λ*^{’}_{j}, *T*,* H*, and *B* in (5), the expected total cost rate, *TC*, can be written as a function of *Q*,* R*, and *R*_{i}. 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 *Δ*_{i} = *R* − *R*_{i} 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 *Δ*_{i} = *R* − *R*_{i}*;* 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 *Δ*_{i} with the following bounds: *R*^{⋆}(*τ*_{1}) ≤ *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).