### Network Flow, Transportation and Scheduling: Theory and Algorithms

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In this case, the idle time between tasks is counted for remuneration. If a single interval longer than the given value occurs, the duty is classified as split duty. This kind of duty is associated to busses operating only during the peaks hours existing on workdays timetable and the interval between two pieces of duty is not counted for remuneration. In order to assign the tasks and build the duties, several operational constraints and labor rules must be taken into account.

In this case, the following restrictions were considered: i each task must be assigned to one duty; ii each duty is a sequence of tasks that can be performed by a single crew, without overlapping; iii the normal work time of a duty is of 6 hours and 40 minutes; iv the single duties must have a break of at least 20 minutes for rest and meal; v the duties cannot do more than two hours overtime; vi the time interval between the end of a duty and its start on the next day must have at least 11 hours; vii the change of crews, during operation, can happen only at predetermined places and times; viii a duty can do a maximum number of vehicle changes during the day.

The main goal of a solution to the problem is to reduce the total number of duties, and the total of overtime payment. This problem can be placed as a multiobjective optimization, since it brings two conflicting objectives: to minimize the number of duties and the amount of overtime payment. In this work the cost function for the CSP was defined as a linear combination of the fixed cost, representing crew wage, and variable costs, given by overtime payment and the number of split duties.

Therefore, the CSP is treated as a mono objective minimization, while satisfying all the constraints mentioned above. The most widely used approach to deal with this problem models it as a set covering or a set partitioning problem. The strategy of column generation is largely used to solve the problem, as can be seen in the works of Smith and Wren , Desrochers and Soumis , Fores et al. However, exact models are limited in practical applications, since they are unable to solve very large problems. So, it is necessary to use heuristic methods to solve problems that appear in real life, which are large.

One of the pioneer groups in this area, named Scheduling and Constraint Management Group of Leeds University, carried out a set of heuristic implementations using Genetic Algorithms Kwan et al. The models developed by this group are widely used in the United Kingdom to build crew scheduling as well as the scheduling of the operational fleet Wren, Although the Crew Scheduling Problem has been widely studied and applied in more developed countries, its solving techniques are not disseminated and rarely applied to the Brazilian reality.

Partially, this happens because the companies do not have the necessary input data and organization, and also because there is a lack of models and commercial systems that represent the Brazilian operational reality. These models have been tested with data of companies operating in Brazilian public transport system and the results show that there are great possibilities to reduce costs in relation to the solutions adopted by the companies.

On the other hand, new rules for the problem and modern search techniques have appeared in the past few years, which can be employed to solve this problem. Thus, this work explores the use of a recent local search technique based on a graph representation of the problem and the use of network flow algorithms to carry out more complex local search than those inherent to the classic local search procedures.

It was observed by the authors that the performance of VLNS search is strongly dependent on the initial solution. Therefore, further works propose to adopt single-solution-based metaheuristics that make periodical perturbations in the current solution through different movements.

The VNS metaheuristic consists in exploring the space of solutions through systematic changes of neighborhood structures, while the main idea of ILS is iteratively to perturb the obtained local optimal and to apply a local search to this perturbed solution. The details and results can be found in the work from Reis and Silva In order to verify the efficiency of the proposed combinations, the ILS was also implemented in their classical version, using First Improvement Method as local search.

Both ILS versions were tested with real data from a company that operates in a public transportation system, and the results were compared with the results obtained by VNS using the code developed by Reis and Silva Local Search Procedures. The ILS metaheuristic was implemented in the classical version, which uses the First Improvement Method as local search.

Both versions from ILS were tested by solving large scale problems of Brazilian reality. The cost C s associated with a solution s of the CSP is computed by means of a linear combination of the fixed cost and the variable costs. The fixed cost represents the wage of a crew, and the variable cost is the total of overtime payment. Finally, the split duty crew is weighted so that the user may have a control on the number of duties of this type into the solution.

The final expression for the cost of a solution is:. The weight w 2 is calibrated to control the number of split duties in the solution, since this kind of duty is not desirable in large amount. The initial solution was built according to the manual procedure, which can be seen as a greedy method of tasks allocation. In the procedure, the first duty is initialized with the first task of a bus. The procedure goes on allocating the next task from the same bus which does not superpose the previous one and that generates the shortest idle time.

The duty is completed as soon as it presents overtime and does not exceed maximum working time allowed. This procedure is repeated while there is a task not assigned. Algorithm 1 presents the logic employed to build an initial solution for the CSP. Methods of local search are algorithms based on the concept of neighborhood.

So, consider S the space of all solutions of a optimization problem and f. Let us define a kind of move m as being a modification that changes one solution s in other solution s' , called neighbor of s by move m , and there are so many different neighbors as feasible moves of type m applied to solution s. The function N that depends on the neighborhood structure of the problem, associates to each solution s in S , its neighborhood N s contained in S. Each solution s' in N s is called of neighbor of s according to a kind of neighborhood structure that is defined by its move m. For instance, let us define the move task exchange for the CSP.

Then, the neighborhood N s , from a given solution s , consists of all solutions produced from s by exchanging two tasks from different duties, generating feasible new duties. Broadly, local search heuristics start from an initial feasible solution, and it walks from neighbor to neighbor according to the adopted neighborhood structure, keeping the best solution visited during the procedure.

Once the maximum number of tasks involved in a move is defined, it is necessary to establish different kinds of moves which characterize a neighbor of a solution. Two kinds of moves were adopted: task relocation and task exchange , both between two duties and without superposing. These moves are performed to find the best neighbor of a current solution. Then, k consecutive tasks are randomly picked out of duty i to be introduced into duty j.

Thus, one of the following situations may occur: 1 the k tasks of i can be introduced in j , without the necessity of removing any task of j. In this case the movement is accepted and the new solution will be evaluated. In this case, if the tasks removed from j can be inserted in duty i , without any superposing with the remaining tasks in i , the movement is accepted, otherwise it is rejected. In both cases, the changes are considered if and only if the resulting duties are feasible, i. It prevents the first duties of the schedule from present higher quality in detriment to the quality of the last duties.

A critical aspect in neighborhood search algorithms is the choice of the neighborhood structure, that is, the way it is defined. This choice largely defines whether the search strategy will lead to solutions of good quality or not. In general, the larger the neighborhood the better the quality of the local optimal solution shall be.

However, large-scale neighborhoods require a long research time. For this reason, a larger neighborhood does not imply in a better heuristic, unless the neighborhood is explored efficiently.

## Network Flow Transportation and Scheduling Theory and Algorithms by Iri Masao - AbeBooks

These algorithms enable to explore very large neighborhoods, while keeping the processing time at very low levels. One way to achieve such efficiency is using network flow algorithms to enumerate implicitly a neighborhood, in order to find better solutions. A cyclic exchange may be defined as a sequence of elements t 1 -t 2 -t 3 Without loss of generality, let S [ t i ] be the subset to which the element t i belongs, then the cyclic exchange t 1 -t 2 -t 3 Finally, the element t r is moved from S [ t r ] to S [ t 1 ].

A path exchange is defined by a sequence of nodes t 1 -t 2 -t 3 In Figure 2 there is an example of path exchange. Observe that with the tasks exchange subset S 1 has one task less and subset S3 has one task more than before the exchange. In order to implement path exchange it is necessary to introduce additional nodes and arcs.

The classical neighborhood search methods are based on relocation and two-exchange moves of elements between the two subsets to which they belong. Observe that the neighborhood of cyclic exchange and path exchange contemplates the two-exchange and still explore an infinity of other solutions unreachable by the classical relocation and two-exchange moves.

## Model and algorithm for resolving regional bus scheduling problems with fuzzy travel times

Therefore, it is expected that the local optimal solutions obtained by multiple exchanges are, on average, superior to those obtained by two-exchange moves. However, once the size of the neighborhood in multiple exchanges increases exponentially with the size of the problem, it is necessary to have an efficient method to find a solution of lower cost in the neighborhood. This problem can be overcome using the concept of improvement graph and network flow algorithms to explore efficiently a given neighborhood.

An improvement graph for a neighborhood with multiple exchanges is defined for a feasible solution S for the problem, being represented by G S. Let S [ t j ] be the duty that contains task t i. A directed arc i, j in G S means that task t i leaves its current duty and it is moved to the duty containing task t j , that is, duty S [ t j ]. Simultaneously, task t j leaves S [ t j ]. To construct G S all tasks pairs t i and t j in T are considered.

A cycle W is called a directed cycle in the improvement graph G S if the tasks in T , corresponding to the nodes from W , belong to different duties. W is defined as a valid cycle if it is a directed cycle of negative cost in G S. Thus, a valid cycle in G S corresponds to a cyclic exchange which leads to an improvement in the value of the objective function of the problem.

This is an efficient way to search solutions that improve the objective function through multiple exchange movements. Therefore it is necessary to find valid cycles in the improvement graph G S. A well known modified label-correcting algorithm that finds the minimum path from a given node source to all others nodes of the network was implemented to identify a valid cycle in this work. More details about this algorithm can be found in Ahuja et al. After making the cyclic exchange, inherent to the valid cycle, the graph is updated and a new valid cycle is sought.

The search ends when the improvement graph does not present any valid cycle. The pseudo-code presented in Algorithm 2 summarizes the method. The ILS is based on the idea that a local search procedure can be improved generating new initial solution, perturbing the current solution.

The perturbation procedure applied to the current solution in ILS metaheuristic was implemented following the Relocation-Exchange neighborhood previously presented, which depends on the value of k. The same function, Relocation-Exchange , was adopted to perform both the local search and the perturbation procedure. In order to avoid the ILS entering a loop visiting the same neighbors, the perturbation is executed with the number of tasks to be relocated larger than the number of tasks considered into the local search procedure. The parameter k is incremented as soon as the local search fails looking for a better solution.

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When it reaches a maximum value kmax , previously defined, k receives the value 2 and the procedure continues until the termination condition is reached. The value for k used by the local search is always equal to 1. In this implementation, a solution is accepted if and only if its cost is less than the cost of the best solution previously found. Moreover, no infeasible solution is accepted during the process. Algorithm 4. Computational Results. The algorithms were tested by solving a set of seven problems concerning one week of work of a Brazilian company that operates in the public transportations system in the city of Belo Horizonte.

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The metaheuristics were performed for one hour and 10 runs were done for each problem. Despite the VNS being tested in a previous work, Reis and Silva , this metaheuristic was once again executed, aiming to guarantee a best comparison with the ILS. Table 1 contains the characteristics of the solutions adopted by the company and that are used as reference for the solutions obtained. In the following Tables, the line "OT" refers to the total of overtime hours and minutes , "Crews" refers to the total of duties units , "SD" refers to the total of split duties contained in the solution units , "St.

The value of EF can be seen as the monetary cost of a solution implemented in practice, since the weights represent the cost per unit of each component of the Expression 1. The weight w2 , which refers to the split duties, received the value These weights were empirically obtained aiming to produce solutions with few drivers, a low amount of overtime hours and the number of split duties within the limit set by the company.

The weights used in the metaheuristics were applied to compute the cost from solutions adopted by the company. The main goal of the company is to reduce the total number of duties, and the total of overtime. Fast Dispatch. Expedited UK Delivery Available. Excellent Customer Service. Dust jacket worn. Seller Inventory BBI More information about this seller Contact this seller 1. Published by Academic Press About this Item: Academic Press, Condition: Good.

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