Chapter 3 Methodologies and Modeling Approaches
3.3 Model Development
3.3.3 Problem Formulation
The aim of vehicle routing problem is to assign vehicles to meet all demand in the minimum time or cost. However, the major objective is to minimize the total response time of incidents in this study. The response time introduced in previous chapter consists of preparation time and travel time, but we should also consider the processing time as response teams may process more than one incident in a round-trip. Therefore, the objective function for each time stage is written as:
( ) ( ( ) )
The objective function (2.1) can be defined as the summation of two parts: (i) the longest total operation time of a response team: the longest time of a response team to finish processing the latest incident, and (ii) the total operation time of all response teams: the total operation time of a response team which include preparation time, travel time, processing time and reassignment punishment. These two parts have different weights α andβ , and α β . Both the first part first and the second part would be satisfied simultaneously.
If several incidents are closer to a response team, the dispatcher may assign all these incidents to this team. Nevertheless, even though it is reasonable, incidents that
sorted later are processed by response teams after incidents sorted former are cleared, so the response time of incidents sorted later may be comparatively long. Hence, we design a minimax method by including the first part of objective function and combining it with Constraint (2.2) to minimize the upper limit of the total operation time of each response team.
The value of variable z must be larger or equal to the total operation time of each team. By adding this part in the objective function and including Constraint (2.2), the model may dispatch other response teams to support the response team which is close to several incidents to decrease the total response time. In addition, the work loading can be more balanced over each response team as the incidents will not be assigned to few response teams only based on distance.
To calculate the total operation time of each response team, we divided it into three parts: (i) travel and processing time: the travel time between nodes and the duration of processing incidents, (ii) waiting time: each response team has a waiting time before it can depart from its original location to the assigned incident, and (iii) reassignment punishment: the punishment time if the incident is assigned to a response team which is
not the same as the one in the former stage. The first part calculates the time cost over the used links, and we weight incidents with a priority order according to their types.
Based on the experience of the dispatchers, and the priority order is divided into three levels: high, median and low, and each level has its corresponding value. The high level means that the incident may cause serious congestion and longer delay. The median level incidents may influence the traffic and further cause an accident. The low level incidents may not affect traffic, but it still involves safety risk. If the incident has higher priority order, it should be processed as quickly as possible. This priority order can help the dispatcher assign response teams to process incidents with higher influence to traffic first and decrease overall delay on the freeway.
The second part sums the waiting time of all response teams. Each response team will be of a status(V )when a new stage is established. There are four statuses that a kc response team can be, and a response team will have waiting time only when it is waiting at its stationary point or it is processing an incident. The leader of response team should check workers and equipment before leaving its stationary point for the assigned incident. Also a response team must finish clearing the incident it is currently processing before going to process another one. These two kinds of waiting time are calculated in Constraint (2.3).
0 2
( ) 0k,
k t PR k V
k k
w
=V
× +V
×PT
∀ ∈ (2.3)The third part limits the number of assignment changes across stages. In Constraint
(2.4), if a link is not chosen for response team k in the previous stage,
X
ijk( )
t−1 willbe zero, and
Y
ijk will be one. Then, Constraint (2.5) sums allY
ijk to determine if any link in each response team’s route is changed. If a link is changed,R
k will be one and multiplied with τ, which is the filter of reassignment. τ can be a constant or a flexibly be a value equal to the duration between last stage and next stage.Appropriately setting the value of τ can avoid the model changes route frequently across stages, which may cause confusion and negatively affects response teams.
( ) ( )
1( )
Other constraints in the model are the basic constraints in the vehicle routing problem. Constraint (2.6) limits that only one response team will be assigned to process incident j from the stationary point or from the location of another incident, and Constraint (2.7) limits that a response team can only go to another incident location or the virtual end node. Constraint (2.8) is the constraint for flow conservation, which means the in-flow number of response teams equals to the out-flow number of response teams, and the response team must be the same one. Constraint (2.9) indicates that each team is assigned to process one incident one time from the stationary point. Constraint (2.10) means that each team goes back to the end node not more than one time.
Constraints (2.11) and (2.12) are the sub-tour elimination constraints that are capable to remove all sub-tours in the solution space.