Figure 8.
A MBCPP network use network transformation to OTP.
4. Orienteering Problem (OP) algorithms
The OP algorithms usually fall into two classifications: exact approaches including Ramesh et al. (1992) and Pillai (1992), and heuristic solution procedures such as Tsiligirides (1984), Golden et al (1987, 1988), Keller (1989), Ramesh (1991), Wang et al. (1995), Chao et al. (1996), Tasgetiren and Smith (2000). We briefly review these algorithms as bellow.
4.1 Tsiligirides Algorithm (1984)
Tsiligirides first proposed two heuristics for the Orienteering Problem. One is a
stochastic algorithm and the other is a deterministic one. The stochastic algorithm
is based on Monte Carlo method to generate a large number of routes, and then
select the best one among them. Tsiligirides algorithm evaluates each control
point i by the measurement
A(i)={Si/C(last,i)}4. If there are less than four
candidate nodes, select a node inserted to the route by generating a random number
between 0 and 1. The procedure is iterating until T
maxcan’t be added any duration
from the last node added to next candidate node. Another heuristic is a
deterministic algorithm based upon vehicle scheduling method proposed by Wren
and Holiday (1972). The deterministic algorithm, without using random numbers
and probabilities divides the search area into sectors by using two concentric circles
and a known arc length. In order to add new nodes, the sectors are changed by varying the two radii of the circles and spinning arc length. Routes construction procedure stops when all nodes in particular sector, which has been visited or any duration added to the current route will violate the rule for Tmax.
4.2 Golden, Levy and Vohra Algorithm (1987)
Golden et al. in 1987 proposed a heuristic containing three steps: route construction, route improvement and a new concept center of gravity. Route construction uses weighted measurement
W
i =x
⋅S
i +y
⋅g
i +z
⋅d
i to assess each node, whereS
i is the score of node i,g
i is duration from center of gravity point to node i,d
i is summation of durations from node i to starting node and ending node, and the sum of the weighted numbers x, y and z is unity. In route improvement, Golden et al. used 2-opt method to decrease current Tmax value. The center of gravity is the heuristic spirit. The new center of gravity is continuously generated by combining step1 and 2 to increase total score without violate Tmax. The procedure is repeated until the path with highest objective value appears again then the procedure stops.4.3 Golden, Wang and Liu Algorithm (1988)
Golden et al. (1988) introduced subgravity algorithm which includes learning and randomness concepts. The former is developed from Tsiligirides’s stochastic algorithm, and the latter is a key factor to result better solution. This is because that route construction uses weighted measurement
W
i =x
⋅S
i +y
⋅g
i +z
⋅d
i and the. The learning component contains adjusted information. We formulate L(i) as learning measurement. According to variation of L(i) to adjust Si, giand d
ivalues. The initial L(i) is equal to one. If node i associates with path score higher than total average scores, then L(i) > 1 and node i is reward or else L(i) < 1 and node is penalized. For each center of gravity of five squares in graph, total 100 solutions are generated and the best solution is selected as result.
4.4 Keller Algorithm (1989)
Keller (1989) revised his multi-objective vending problem’s algorithm to solve OP. The algorithm consists of 2 stages: route construction and route improvement.
At the route construction stage, first computes the desirability of each node i. For each node not belonging to the current route, we select the one with the largest
desirability merit for insertion to the route. At the second stage, first removes one node then inset none; one or two nodes into the route in order to increase total score. Next, remove identification of nodes clusters and inset another identification of node clusters to increase total score.
4.5 Ramesh and Brown Algorithm (1991)
The algorithm includes four steps that are node insertion, length improvement, node deletion and maximal insertion. In node insertion, relaxes T value limit to construct initial route. The length improvement to uses 2-opt and 3-opt to decrease Tmax value. The node deletion is similar to Kell (1989) route improvement step two. One node is removed and then another node is inset to decrease current
T
value. The maximal insertion tries to increase the total score by a new node i that is not on the current route.4.6 Ramesh, Yoon and Karwan Algorithm (1992)
Ramesh et al. (1992) developed an optimal solution procedure, which uses Lagrangean relaxation within a branch and bound technique. The Lagrangean relaxation is solved by a degree-restricted spanning tree manner. The algorithm contains six steps: (1) initialization, (2) scaling parameter α , (3) upper & lower bounding, (4) branching rule, (5) upper bounding, (6) measuring and backtracking.
The computational results show that the algorithm can be processed to solve the medium and large size problem.
4.7 Pillai Algorithm (1992)
The Pillai algorithm claimed that the final solution is optimal. The algorithm is based on branch and cut technique to solve the problem. Consequently, it only can be employed on small or moderate size problems. The Pillai algorithm first applies a relaxed linear programming with no violated constraints. If the solution is a non-integer, then construct a feasible solution by employing branch and cut method to find the optimal solution; otherwise, then the procedure is stopped.
4.8 Leifer and Rosenwein Algorithm (1994)
Leifer and Rosenwein (1994) first defined a 0 － 1 integer programming formulation for the Orienteering Problem, then use bound procedure by adding constraints and valid inequalities to solve three successive the linear problems.
The algorithm is to find a tight upper bound for the optimal objective function value.
4.9 Wang et al. algorithm (1995)
In Wang et al. algorithm, the neural network applies two-dimensional cell as the tight bound and the energy function with fourth order convex function to solve OP approximately.
4.10 Chao, Golden and Wasil Algorithm (1996)
Chao et al. (1996) algorithm consists of two stages: stage one is initialization using greed criterion to inserted node to the route. The rule of insertion is to select a node with cheapest insertion cost while its score is ignored. The second stage is improvement including two-point exchange, one-point movement, clean-up and reinitialization. Two-point exchange is that one new point moves in and one point moves out, if total score increases and current time doesn’t violate the Tmax. One-point movement is to insert a new node, if total score increases and route is feasible. Clean-up step is to apply 2-opt to improve Tmax , at which the algorithm has more opportunities to insert nodes. Finally, remove the nodes with smallest measurement ratio from the route and redo improvement stage.
4.11 Tasgetiren and Smith algorithm (2000)
Tasgetiren and Smith algorithm (2000) is one kind of genetic algorithm, of which the performance is the same with Chao et al. (1996). Both outperformed other heuristics of OP. The algorithm first employs node omission probability based on Tmax length constraint to generate some feasible path, then applies crossover operator and local search mutation to improve the initial path. Until a rear-optimally solution is found. But the computational time of the algorithm is longer.