The second level of difficulty comes from uncertainty. Crashes on roadways are not predictable and in particular
vehicles carrying hazardous materials often generate huge health problems for residents and damage to buildings. In
this class of difficulty, we need to develop models taking into account the dynamic and stochastic nature of travel
times and connectivity of road networks. Then VRPTW-D (vehicle routing and scheduling with time windows –
dynamic) or VRPTW-P (vehicle routing and scheduling with time windows – probabilistic) models are available to
duplicate the dynamic adaptation of starting time and route choice of pickup/deliver trucks in urban areas based on
ITS (Intelligent Transport Systems) applications.
The third class of difficulty comes from ambiguity. Bad weather conditions and natural hazards are included in
this class and these events occur less often but greatly affect urban freight transport systems. In this class more
sophisticated treatment is required for duplicating the behaviour of stakeholders. The VRPTW-D or VRPTW-P
models are not sufficient in ambiguous situations, while multi-objective models or multi-agent models and
simulation are effective to assess the effects of these events and evaluate initiatives for responding to them. The
ambiguity of the situation is generated by the unpredictable interaction among stakeholders as well as the actionreaction relationships between stakeholders and the environment.
For coping with the risks of complexity, uncertainty and ambiguity, the concept of risk governance has been
proposed (Kröger, 2008). The idea of risk governance is beyond risk management and includes the cycle of preassessment, appraisal, characterisation and evaluation, and management. The comprehensive framework of risk
governance allows us to understand the dependency of each stakeholder related to city logistics initiatives and the
critical infrastructure as well as the critical points in supply chains.
Risk has been defined as, “the chance of something happening that will have an impact on objectives” (AS/NZS,
2004). City Logistics aims to reduce the total costs including economic, social and environmental associated with
urban goods movement. There are a number of aims and objectives of urban freight systems that are under threat
such as the health and safety of citizens and the drivers of vehicles, the fulfilment of delivery contracts (eg. city
curfews and time windows) as well as reducing climate change.
There is a need to incorporate uncertainty into models for city logistics to ensure that schemes will perform well
into the future. A variety of methods have been used to incorporate uncertainty in supply chain modelling such as
scenario and contingency planning, decision trees and stochastic programming (Shapiro, 2007).
2. Methodology
2.1. Robustness
There is often considerable uncertainty in the input data such as parameters, resources and operational limits
within optimisation models that are used to plan, design and evaluate city logistics schemes. Mathematical
programming represents systems by an objective function, decision variables and constraints. A feasible solution is
one that satisfies the constraints of a system. An optimal solution is defined a feasible solution where the values of
the decision variables provide the best value of the objective function.
Solution robustness investigates whether the optimal solution is maintained when there are changes to the input
data. In particular, solution robustness considers how close the original optimal solution is to the new optimal
solution when the input data changes.
Model robustness considers the effect on feasibility for changes to the input data. This involves determining how
close to feasible the optimal solution is when there are changes to the input data.
The concept of robustness in mathematical programming analyses the effect of uncertainty in a models input data
represented by parameters and constraints. Robustness of logistics and supply chain networks has received
considerable attention recently (Bok et al., 1998; Christopher and Peck, 2004; Mo and Harrison, 2005; Yu and Li,
2000).
It is desirable to develop procedures for identifying solutions that remain close to optimal and close to feasible
when there are changes to the values of the input variables due to their uncertainty. Robust optimisation analyses the
trade-off’s between solution robustness and model robustness (Mulvey et al., 1995). A number of methods have
been developed for representing uncertainty in optimisation models, including probability distributions, fuzzy logic
and scenario-based techniques.
2.2. Stochastic programming
Stochastic programming formulates a system as a probabilistic (stochastic) model that explicitly incorporates the
distribution of random variables within the problem formulation (Birge and Louveaux, 1997). This is contrast to
approaches such as linear programming where the parameters are assumed to be constant. Stochastic programming
can identify solutions that perform better when parameters vary from their mean or estimated values.
The value of the stochastic solution (VSS) measures the possible gain from solving the Probabilistic (Stochastic).
It represents the value of knowing and using the distributions of future outcomes. VSS is relevant to problems where
the future is uncertain and no further information about the future is available. It measures the cost of ignoring
uncertainty when making a decision (ie. determining a solution).
Although the actual travel time between customers is uncertain in static vehicle routing and scheduling problems
a single value estimate (forecast) is usually made (Psaraftis, 1995). Stochastic (probabilistic) models allow random
inputs that are assumed to follow a probability distribution
With the Probabilistic (stochastic) Vehicle Routing Problem with Soft Time Windows (VRPSTW) model an
expected penalty cost must be estimated (Laporte et al., 1992). The expected penalty cost associated with accounts
for the uncertainty of predicting the arrival time of trucks visiting customers.
A two stage procedure was developed for estimating the benefits (cost savings) of using stochastic programming
for vehicle routing and scheduling with time window and variable travel times (Taniguchi et al., 2001).
Late deliveries in urban areas can lead to missed sales in retailing as well as delivery failure in home deliveries.
Vehicles running late may also not be allowed to enter inner urban areas where strict curfews for delivery vehicles
have been implemented.
Stochastic programming has recently been applied to the design of supply chain networks (Santoso et al., 2005;
Shapiro, 2007; Shapiro, 2008; Snyder, 2006).
2.3. Simulation
Simulation can be a useful tool for designing urban logistics systems. It has been used widely in the design of
loading/unloading facilities and the layout within distribution centres. Operational performance measures can be
estimated for varying physical designs and demand levels. This can allow contingency plans to be developed for
extreme conditions.
A micro-simulation was developed to determine a congestion management strategy of the Kallang-Paya Lebar
Expressway (KPE) in Singapore (Keenan et al., 2009). The KPE involves a 9 kilometre road tunnel and the level of
service was estimated for various traffic levels including freight vehicle to ensure reasonable safety levels using the
VISSIM simulation software.
Simulation can be used to design security and scheduling procedures for reducing the threat of terrorism. A
delivery vehicle scheduling system was developed using the Planimate simulation software for use in planning and
operations of the Sydney 2000 Olympics (Pearson and Gray, 2001). Animation of transport activities assisted in the
effective communication between key stakeholders including security forces. This system was used to determine
specific time-slots of deliveries to Olympic venues and produce a master delivery schedule (MDS). It provided the
capability to verify the robustness of the final schedule and to analyse scenarios including terrorist events.
2.4. Multiobjective optimisation
Decision-making in supply chain and logistics management often faces simultaneous consideration of several
criteria. In most cases, a simple approach is used, where multiple objectives are weighted into a single one. However,
it is difficult to set the weights for each objective in advance. Thus, it can typically be modelled and solved within
the framework of the multi-criteria decision-making problem and the multi-objective optimisation problem (i.e.,
multi-objective programming problem (MOP)). These problems commonly have the characteristic that there does
not generally exist a unique optimal solution. In cases where the values of objective functions are mutually
conflicting, all objective functions cannot be simultaneously optimised. A set of so-called Pareto optimal solutions
(e.g. Chancong and Haimes, 1983; Sawaragi et al., 1985), which also imply non-inferior or non-dominated solutions,
are determined in that case.
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