Uncertainty theory is applied to characterized a route under uncertain weight. Before starting model construction, some notations and assumptions are listed in Table 1. By the knowledge of graph theory, if network N has no vertices of odd degree i. The cycle is just the shortest route. If network N has vertices of odd degree, then any cycle that traverses each edge at least once, must traverses some edges more than once.
And we can assume that traverses each edge at most twice. Figure 1. Table 1. List of notations and assumptions. The first constraint requires that the route is cycle, and the second constraint requires that the route traverses each edge of N at least once. Clearly, the shortest route weight in uncertain networkis an uncertain variable. Expected value is the average value of uncertain variable in the sense of uncertain measure, and represents the size of uncertain variable.
Now, we give the concept of expected shortest route for uncertain Chinese postman problem. In the following sections, we will give the answers. Taking advantage of some properties of uncertainty theory, these models can be transformed into their corresponding deterministic forms. According to Definition 4, the expected shortest model can be presented as follows:. The theorem is proved. L ,, n respectively. Clearly, model 5 describes the shortest route in network where and Hence the theorem is proved.
The method can be summarized as follows:. Calculate for each weight ij in. Construct a deterministic network whose edge weights. In this section, we give an example to illustrate the methods as mentioned above. We summarize the problem as follows.
Theory and Practice of Uncertain Programming
Suppose that the postman must cover nine streets. Now, the task for the postman is to make the route plan such that the total weight may be time or expense on the route is minimized. At the beginning of the task, the postman needs to obtain the basic data, such as traffic condition. In fact, since the plan is made in advance, we usually cannot obtain these data exactly.
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U-sually, we obtain these uncertain data by means of expert's empirical estimation. Assume that the uncertain network is shown in Figure 1, and all uncertain variables are zigzag uncertain variables listed in Table 2. Table 2. List of. Calculate of listed in Table 3. Table 3. From the data in Table 3, we construct a deterministic weighted network is shown in Figure 2. Calculate of listed in Table 4. Table 4. List of 1 0.
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From the data in Table 4, we construct a deterministic weighted network is shown in Figure 3. Specially, if the uncertain network is shown in Figure 1, and 12 16 23 According to Dijkstra algorithm, and Fleury algorithm, we obtain distribution shortest route is: and its uncertainty distribution is. Table 5. At last, an example is given to illustrate the methods. Stop when Step 2 can no longer be implemented. Fleury algorithm is an efficient algorithm with the complexity. Let G be a graph with n vertices. Throughout the algorithm, each vertex v carries a label which is an upper bound on.
Dijkstra algorithm is an efficient algorithm with the complexity. IChinese postman problem is one of the classical combinatorial optimization problems with many applications. How-ever, in application, some uncertain factors are frequently encountered. This paper employs uncertain programming to deal with Chinese postman problem with uncertain weight Within the framework of uncertainty theory, the concepts of expected shortest route, -shortest route, and distribution shortest route are proposed.
After that, expected shortest model, and -shortest model are constructed. Taking advantage of properties of uncertainty theory, these models can be transf-ormed into their corresponding deterministic forms, which can be solved by classical algorithm..
Then for any real numbers a and b, we have 3. Network Table 1. Step 2: From the data in Table 4, we construct a deterministic weighted network is shown in Figure 3. Specially, if the uncertain network is shown in Figure 1, and 12 16 23 According to Dijkstra algorithm, and Fleury algorithm, we obtain distribution shortest route is: and its uncertainty distribution is Table 5.
Fleury algorithm is an efficient algorithm with the complexity Dijkstra Algorithm Let G be a graph with n vertices. Bhattacharyya, R. Chen, X. Duplicate citations. The following articles are merged in Scholar. Their combined citations are counted only for the first article. Merged citations. This "Cited by" count includes citations to the following articles in Scholar. Add co-authors Co-authors.
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