Chapter VI: Experimental Results
5.3. The reliability hub location problem
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Finally, to the best of our knowledge, this research is the first that has considered all hub, link and spoke uncertainties in designing a reliable hub-and- spoke network. we depart from common assumptions in the literature by considering complete and partial disruptions in the hubs, and partial disruptions in connection links.
109 5.3.1. Partial disruptions in hubs and links
As mentioned previously, capacity of hubs and connection links become partially disrupted and degrade to a lower level, in which, the flows entering the hub or traversing a link will exceed the realized capacity of the hub or the link by certain probability. It is desirable to ensure that the probability of such an occurrence to be lower than a specified or satisfactory level. This section introduces the hub and link capacity reliability as the probability of the flows entering a hub or traversing a link exceeds the capacity of the hub or the link, referred to as the capacity exceedance probability (1 − 𝜂) and (1 − 𝜗), respectively, in which 𝜂 and 𝜗 are partially disruption probability in hubs and links. It is obvious that higher probability of disruption leads to lower probability of respecting the capacity. The hub capacity reliability can be mathematically written by:
𝑃{𝛤 ≤ 𝐸𝐹} ≤ (1 − 𝜂) (5.1)
where EF and 𝛤 are flows entered the hub and the designed capacity of the hub, respectively. It should be noted that the hub capacity reliability requirement can be different for different hubs. In (5.1), 𝛤 is a random variable specified by a particular probability density function (PDF). The left-hand side (LHS) of inequality (5.1) can be considered as a cumulative distribution function (CDF) of 𝛤, written by:
𝐹𝛤(𝐸𝐹) = 𝑃{𝛤 ≤ 𝐸𝐹} (5.2)
Using equation (5.2), equation (5.1) can be rewritten by:
𝐹𝛤(𝐸𝐹) ≤ (1 − 𝜂) (5.3)
Since the CDF are monotonic one-to-one functions, one can take the inverse of inequality (5.3) and write the following inequality (5.4).
𝐸𝐹 ≤ 𝐹𝛤−1(1 − 𝜂) (5.4)
By specifying the CDF of the hub capacity Γ and the acceptable capacity exceedance probability (1 − 𝜂), inequality (5.4) becomes a deterministic constraint.
For simplicity, in this section, we assume that the hub capacity follows a uniform distribution defined by an upper bound (i.e., design capacity) and a lower bound (i.e., worst-degraded capacity). Generalizing the consideration to other probability distributions (e.g., Gamma and truncated Gamma) can be accomplished with the Mellin Transform technique as discussed in the work by Lo et al. (2006). Noteworthy, several papers (Chen et al., 2002; Lo and Tung, 2003; Luo, 2004 ; Lo et al., 2006) have
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considered uniform distribution that is more applicable in transportation domain affected by disruption.
It should be mentioned that other distribution functions (e.g., Gamma and truncated Gamma) model the capacity degradation. Furthermore, we consider the lower bound to be a fraction θ of the design capacity. For a uniform distribution, the inverse CDF of Γ can be written as:
𝐹𝛤−1(𝜂) = 𝜃𝛤̅ + (1 − 𝜂)𝛤̅(1 − 𝜃) = 𝛤̅[𝜃 + (1 − 𝜂)(1 − 𝜃)] (5.5)
where 𝛤̅ is the design capacity of the hub that has a deterministic value. Applying equation (5.5) to inequality (5.4), we obtain the following hub capacity reliability:
𝐸𝐹 ≤ 𝛤̅[𝜃 + (1 − 𝜂)(1 − 𝜃)] (5.6)
Similar to hub capacity disruption, the link capacity disruption is presented as inequality (5.7).
𝐿𝐹 ≤ 𝜉̅[𝛿 + (1 − 𝜗)(1 − 𝛿)] (5.7)
where LF, 𝜉̅ and 𝜗 are flows traversing the link, designed link capacity, probability of partial disruption in the link. Besides, in case of disruption, capacity of the link (𝜉̅) degrades to a lower level with fraction 𝛿.
5.3.2. Uncertainty of transportation time due to stochastic degradation
As mentioned before, transportation time over some links may be increased due to stochastic degradation. In this section, the Burea roads link performance function (Lo and Tung, 2003) is presented as equation (5.8) to cope with the stochastic alteration of transportation time.
𝑇(𝐿𝐹, 𝜉) = 𝑡 [1 + 𝛽 (𝐿𝐹 𝜉 )
𝜁
] (5.8)
where t and T are link free-flow travel time and variable transportation time of the link with flow 𝐿𝐹, respectively; 𝛽 and 𝜁 are constant parameters. Besides, 𝜉 is a random capacity variable of the link specified by a particular PDF. According to equation (5.8), the mean and variance of T are calculated as equations (5.9) and (5.10):
𝐸(𝑇) = 𝑡 + 𝛽𝑡𝐸 [(𝐿𝐹 𝜉 )
𝜁
] (5.9)
𝑉(𝑇) = 𝛽2𝑡2𝑉 [(𝐿𝐹 𝜉 )
𝜁
] (5.10)
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By assuming t as deterministic parameter and 𝜉 as independent random variable from amount of flow LF, 𝐸(𝑡) = 𝑡 and 𝑉(𝑡) = 0 and the mean and variance of 1 𝜉⁄ are derived as follows by assuming the uniform distribution for the link capacity.
𝐸 (1
𝜉𝜁) = ∫ 1 𝜉𝜁
1 (𝜉̅ − 𝛿𝜉̅)
𝜉̅
𝛿𝜉̅
𝑑𝜉 = 1 − 𝛿1−𝜁
𝜉̅𝜁(1 − 𝛿)(1 − 𝜁) (5.11)
𝐸 ( 1
𝜉2𝜁) = ∫ 1 𝜉2𝜁
1 (𝜉̅ − 𝛿𝜉̅)
𝜉̅
𝛿𝜉̅ 𝑑𝜉 = 1 − 𝛿1−2𝜁
𝜉̅2𝜁(1 − 𝛿)(1 − 2𝜁) (5.12)
𝑉 (1
𝜉𝜁) = 𝐸 ( 1
𝜉2𝜁) − (𝐸 (1 𝜉𝜁))
2
= 1 − 𝛿1−2𝜁
𝜉̅2𝜁(1 − 𝛿)(1 − 2𝜁)− ( 1 − 𝛿1−𝜁 𝜉̅𝜁(1 − 𝛿)(1 − 𝜁))
2
(5.13)
where 1 (𝜉̅ − 𝛿𝜉̅)⁄ is probability density function (PDF) of the uniform distribution with upper bound 𝜉̅ and lower bound 𝛿𝜉̅. Using (5.11) to (5.13), the mean and variance of T are, respectively:
𝐸(𝑇) = 𝑡 + 𝛽𝑡𝐿𝐹𝜁 1 − 𝛿1−𝜁
𝜉̅𝜁(1 − 𝛿)(1 − 𝜁) (5.14)
𝑉(𝑇) = 𝛽2𝑡2𝐿𝐹2𝜁[ 1 − 𝛿1−2𝜁
𝜉̅2𝜁(1 − 𝛿)(1 − 2𝜁)− ( 1 − 𝛿1−𝜁 𝜉̅𝜁(1 − 𝛿)(1 − 𝜁))
2
] (5.15)
Equations (5.14) and (5.15) state that for a specific designed capacity 𝜉̅, both mean and variance of transportation time are increased by flow LF traversing the link. Therefore, the variability of travel time in heavy traffic is higher than that in the light traffic. On the contrary, variance is equal to zero when there is no flow.
5.3.3. Stochastic disruption of hubs’ service rates
In this section, we assume flows entering a hub undergo a set of operations such as loading, sorting, unloading, etc. Due to resource limitations at the hub(s), all flows cannot be processed at the same time and need to wait for their turn to be processed. Therefore, the total travel time between each pair of O-D nodes is the sum of transportation time on the links and the time spent at the hub(s). The resource limitation at the hub(s) causes significant delays if the average arrival rate gets closer to the service rate at these operations. These delays are increased as more and more flows are attracted to the hub to take advantage of the economies of scale. As these delays significantly affect the delivery time requirement, time spent at the hubs should be calculated and taken into account.
Since the flow between each pair of O-D nodes has been considered as uncertain parameter (i.e., spoke uncertainty), a queuing approach is an efficient method to analyze the waiting time at the hubs. In this way, accounting for uncertain amount of flows and calculation of waiting times through queue theory, makes the proposed model more attractive in practice.
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Hereafter, the queuing model proposed in Section 3.3.2.3 is applied to model the waiting time of flows entering a hub. Similar to Section 3.3.2.3, hubs act as machines and inspection tools wherein flows in the transportation network look like products in the production system.