This section illustrate the routing task with time windows algorithm. After the storage location is defined base on continuous cluster method, Time windows help to finding the shortest distance path and conflict free between an origin node and a destination node in a system, based on scheduling restrictions (time windows) for each one of the path nodes. The main purpose is optimization of the total travel distance of the transportation task so the operation cost is minimization. This section is referred to [21, 22, 23, 24, 25, 27]
Considering a number of routes which forklift can reach, there are many route is for a pair of points (origin and destination), a cost value, a duration and time intervals
a bi, i. The routing is defined as the routes sequence done by a number of forklifts, P is defined as being the set of routes and I o the set of intermediary points.
A route is represented by an arc i j, is the path with start point at i and end point is j. For each arc i j, , duration time given by tij and cost given bycij, where the arc can be defined, only if possible to realize the route j after the route i respecting the time intervalai tij bi.
The problem is described in the single warehouse, where each robotic forklift leaves its station once. The nodes s and t given the exit and the entry nodes to the depot.
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Additionally, the depot is also single, so s and t are coincident. But, they are represented separately in order to make it easier to understand the network.
The network used for the forklift is defined as a set of nodes N P s t, and a set of oriented arcs given byE I s *P( *P t .
The variables used in the mathematical formulation are given by:
xij = {1 if the arc(i, j)it be use for the forklift 0: otherwise, where i, jϵE
ti = variable that represent the time associated to the beginning of each route i, with iP Following this formulation, optimal routes that respect the constraints of schedules are considered as solutions of the problem:
Minimize
( , ) ij ij
i j E
c x
(1.29)
Subject to: ij 1
j N
x
iP (2.30)
ij 1
j N
x
iP (2.31)
ij 0
x , i j, E (2.32)
ij 0 i ij j
x t t t , i j, I (2.33)
i i i
a t b, i j, E (2.34)
0,1
xij , i j, E (2.35)
The relationships in Eq. (2.29) to Eq. (2.32) form a routing problem without scheduling constraints for each forklift. This is a minimum cost flow problem that need to be improve. The constraint of Eq. (2.33) informs that starting from a single node j and will arrive in node i. The constraint of Eq. (2.34) establishes that the forklift that arrived at the node i will have to leave for a single node j. Eq. (2.36) describes the compatibility between routing and the scheduling, while Eq. (2.37) establishes the exact time at which the route must begin. It can be shown that exits an optimal integer solution to the routing problem with scheduling constraints defined by Eq. (2.29) to Eq. (2.34). However, this
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optimal solution cannot be obtained directly by linear programming because Eq. (2.33) has not linear constraints, but it can be written in linear form (Desrosiers et al., 1986).
(1 )
i ij j ij ij
t t t x M , i j, I (2.36)
With Mij bi ti j, aj. This formulation is equivalent only if xij is a binary variable (Eq. 11). In special case where a bij, ij 0, P 1for iPandtij 1 i j, I, we may
set Mij P and constraints Eq. (2.36) become the sub tour elimination constraints proposed by Miller et al. (1960) for the travelling salesman problem (Eq. 2.37):
i j ij 1,
t t P x P i j, I (2.37)
A shortest path between nodes s and t, considering the schedule constraints (time windows) is obtained by finding the optimal solution of the above model. This formulation forces the variable xijto be integer. If we loosen the constraint integrality of variable xij in other to obtain0xij 1, we will have the following equation (Eq. 2.38):
0 ti tij tj (1 xij) (2.38) However, Eq. (14) does not satisfy the constraints of the initial problem presented in Eq.
(2.38). Therefore the variable xijmust be integer (Desrosiers et al., 1986). Concluding, one of the main contributions of this method for the routing task with time windows is the improvement of the forklift path planning.
In the proposed model (section 2.1.1), to deal with the problems of the model given in previous Section, a different approach that computes shortest path and conflict-free routes simultaneously is propose which time-dependent between vehicles is considered [20, 21, 22].
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The idea of the algorithm is that find a conflict-free shortest-time route in the case there is collision potential in the aisle (see Fig. 2.9). According to the approach, after the shortest path to the storage location is found by the A* algorithm (section 2.3) and a time-dependent histogram is established (Section 2.4). Based on distance and velocity data, the position of each vehicle at each time on the map is determined and then a free- conflict path is formed by using the waiting time for a vehicle.
Fig. 2.9 Deadlock and Traffic Jams
The algorithm is explained through a scenario as follows: There is 2 tasks were assigned to Forklift 1 and 2, one to store pallet from I/O point to storage location have coordinate N-VII-A-5 and other one to retrieval pallet from selected location S-VIII–A- 4. The A* algorithm shows the shortest static path for the two tasks of FL1 and FL2 as follows:
Forklift1 path (storage): [1 -> 2 -> 3 -> 4 ->5 -> 6 -> 7 ->8 -> 9 -> 10]
Forklift2 path (retrieval): [19 -> 18 -> 17 -> 16 -> 8 -> 7 -> 6 -> 15 -> 14 -> 13 ->
12 ->11]
Time dependence of two vehicles is shown in Fig. 2.10. The graph shown that there are several nodes overlap between two paths, they will collide at second 11.29th to 13.50th on nodes [5, 6, 7].
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To avoid collisions between two vehicles, a delay time is added for Forklift1, which means that the vehicle will paused between node 5 and node 6 a time ∆= 2.21s. During this time, Forklift2 can pass nodes 8, 7, 6 without Folkift1 on those nodes. (Eq. 2.36 to Eq. 2.39).
Fig 2.11. Conflict-free routes
Fig 2.10. Time windows with deadlock between 2 paths
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