Cajanus cajan variedad clara (TÁCHIRA 401)

For complex problems there is often no analytical description how to reach the goal. In this case search techniques can be used to systematically explore a space of problem states. These search techniques are also known as state space search. The problem solving procedure is represented as a path in a graph from a start state to a goal state. A graph is a set of states and a set of arcs that connect the states. State transition operators are applied to transform a state into its successor. The goal to reach can be defined explicitly by a certain goal state or implicitly by evaluation criteria, e.g. an objective function that associates a quality to the current state [74, 83, 95, 113, 114]. An example of a state-space search is given in figure 4.13. A trader has three jugs with different volumes, one with 9 liters, one with 7 liters and one with 4 liters. The 9-liter jug is filled with wine, the others are empty. Thus, the start state is [9, 0, 0]. The goal is to have six liters of wine in the 9-liter jug and three liters of wine in the 4-liter jug.

4.2 Methods of artificial intelligence 9,0,0 2,7,0 5,0,4 2,3,4 0,7,2 0,5,4 5,4,0 9→7 9→4 9→4 7→4 9→7 4→7

Figure (4.13) A part of the state-space graph for the wine jug problem [95].

Thus, the goal state is [6, 0, 3]. State transitions are achieved by filling wine from one jug into another. This is expressed in the form 7 → 4. In this example, the wine from the 7-liter jug is filled into the 4-liter jug.

4.2.4.2 General features

There are various forms of search techniques. The most important classification criteria are listed below (table 4.6). Uninformed and heuristic search techniques are explained in more detail in the following sections. In section 4.2.6 genetic algorithms are in- troduced as an example for a nature inspired population-based search technique that applies both deterministic and stochastic operators.

In state space search two contradictory criteria must be considered. On the one hand, the search space has to be explored to guarantee a diversification of solutions. On the other hand, the space in the vicinity of promising solutions has to be exploited, which means an intensification of search.

To avoid loops, an OPEN and a CLOSED list are maintained. The OPEN list contains the untried states that have not yet been visited. The CLOSED list contains all expanded states, the states that have already been visited. This common feature of search tech- niques avoids that fruitless paths are repeated. Figure 4.14 shows an example with closed states. At state [2, 3, 4] the operators 9 → 7 and 7 → 9 would lead to states that have already been visited. Hence, the moves from [2, 3, 4] to [0, 5, 4] and [5, 0, 4] are not allowed. The state [2, 3, 4] has only one legal successor, state [6, 3, 0].

4 Evaluation and selection of methods for solving optimization problems

Table (4.6) Classification criteria for search techniques [74].

uninformed

◦ no use of additional information about the problem domain

heuristic

◦ use of additional information about the properties of the specific problem domain

nature inspired non nature inspired

deterministic

◦ using the same initial solution will lead to the same final solution

stochastic

◦ random rules are applied ◦ the same initial solution may

lead to different final solutions

population-based

◦ a whole population of solutions is evolved

◦ a solution is also called an individual

◦ exploration oriented

single solution based

◦ a single solution is manipulated and transformed during search ◦ exploitation oriented

iterative

◦ start with a complete solution and transform it at each iteration

greedy

◦ start from an empty solution ◦ at each step a decision variable

of the problem is assigned until a complete solution is obtained

4.2.4.3 Uninformed search

The uninformed search is a blind search. No domain specific knowledge is used to judge where the solution is likely to lie. The main categories of uninformed search are:

◦ breadth-first search, ◦ uniform cost search, and ◦ depth-first search.

In breadth-first search the search space is explored level by level. Only when no more states to be explored are left at a given level, then the search is continued at the next deeper level. In other words, the states are expanded in order of their proximity to the

4.2 Methods of artificial intelligence 0,5,4 5,0,4 6,3,0 closed states 9→7 7→9 4→9 9,0,0 2,7,0 5,0,4 2,3,4 0,7,2 0,5,4 5,4,0 9→7 9→4 9→4 7→4 9→7 4→7

Figure (4.14) Closed states in state-space search.

start node. The distance between states is measured by the number of arcs between them. The search terminates when a goal state has been found. If a solution to the desired goal exists, the breath-first search algorithm is guaranteed to find the shortest possible solution sequence. The disadvantage of the breadth-first search, due to its exploration oriented characteristic, is its large computation time.

The uniform cost search tries to find the cheapest path from the start state to a goal state. A non negative cost is associated with each transition from one state to a succes- sor state. The uniform cost search reduces to breadth-first search if all transitions have equal cost.

Depth-first search is an exploitation oriented algorithm. In depth-first search, the chil- dren of a state and their descendants are expanded before the siblings of the state are examined. In other words, the search goes deeper into the search space whenever it is possible. Alternate paths are only considered when the search reaches a dead end. A dead end is a state that has no successors. In contrast to breadth-first search, when a solution path is found, it is not necessarily the shortest solution path. It is possible to apply a depth bound to avoid that the search goes too deeply into a fruitless path [83, 95, 113].

4.2.4.4 Heuristic search

The above described search techniques are exhaustive search techniques. The algo- rithm searches through the entire search space. For some problems the computational

4 Evaluation and selection of methods for solving optimization problems

cost of finding a solution using these exhaustive search techniques may be too high. Heuristic search techniques attack this issue by guiding the search along paths that have high probability of success. Heuristic search techniques use additional informa- tion about the properties of the specific problem domain to search the given space more efficiently and to find a solution within a practical length of time. In contrast to the above described uninformed search techniques, heuristic search is an informed technique.

Heuristics are often rules of thumb that are based on experience or intuition. A heuris- tic can be considered as an informed guess on the next step to be taken in solving a problem. As a heuristic is a guess and not exact knowledge, it can lead to suboptimal solutions or fail to find any solution at all.

Best-first search is a sub domain of heuristic search. The OPEN list is implemented as a priority queue. The states in the OPEN list are ordered according to a heuristic estimate of their closeness to the goal. At each iteration, the first state of the OPEN list is expanded. The heuristics used to estimate the potential of a state range from subjective estimates to measures based on the probability of the state leading to the goal [74, 83, 95, 113, 114].