a reoptimization algorithm for the shortest path problem with time windows abstract the shortest path problem with time windows spptw occurs in the construction of vehicle routes and schedules for which time window constraints must be satisfied the spptw is repeatedly solved to produce routes with disjoint schedules which form the solution of the original problem the cost of repeatedly solving the spptw can be reduced by reusing part of the solution of the preceding problem a primal dual reoptimization method which runs in pseudo polynomial time is proposed it can solve problems with up to 2500 nodes

optimal solution of vehicle routing problems using minimum k trees we consider the problem of optimally scheduling a fleet of k vehicles to make deliveries to n customers subject to vehicle capacity constraints given a graph with n 1 nodes a k tree is defined to be a set of n k edges that span the graph we show that the vehicle routing problem can be modeled as the problem of finding a minimum cost k tree with two k edges incident on the depot and subject to some side constraints that impose vehicle capacity and the requirement that each customer be visited exactly once the side constraints are dualized to obtain a lagrangian problem that provides lower bounds in a branch and bound algorithm this algorithm has produced proven optimal solutions for a number of difficult problems including a well known problem with 100 customers and several real problems with 25 71 customers

a direct branching algorithm for checking the equivalence of two deterministic pushdown transducers one of which is real time strict abstract a new direct branching algorithm is presented for checking the equivalence of deterministic pushdown transducers one of whose associated deterministic pushdown automata dpdas is real time strict it is a straightforward extension of the corresponding algorithm for dpdas tomita 1982 and then proves the general applicability of our approach

set covering in fuel considered vehicle routing problems the paper studies set covering in fuel considered vehicle routing problems fvrp firstly we study the fvrp with distance constraint and time windows fvrp tw whose objective is to find a set covering with the minimum cardinality which means the number of used vehicles is minimized and hence the fuel consumption is minimized in the real logistics we give a bicriteria approximation algorithm for this problem secondly we study the set covering in the fvrp with distance constraint and constant time windows fvrp ctw which has a constant number of the time windows provided by logistics companies we give a bicriteria approximation algorithm with 2 o log 1 for this problem in which the first term is the approximation ratio on the distance constraint and the second term is the approximation ratio on the cardinality of the covering set thirdly we study the set covering in general fvrp and propose a lower bound for this problem which is based on total unimodularity finally we design an algorithm framework based on the lower bound for solving the set covering in general fvrp simulation results demonstrate the effectiveness of the algorithms for solving the set covering in fvrp tw and general fvrp

lagrangean decomposition a model yielding stronger lagrangean bounds given a mixed integer programming problem with two matrix constraints it is possible to define a lagrangean relaxation such that the relaxed problem decomposes additively into two subproblems each having one of the two matrices of the original problem as its constraints there is one lagrangean multiplier per variable we prove that the optimal value of this new lagrangean dual dominates the optimal value of the lagrangean dual obtained by relaxing one set of constraints and give a necessary condition for a strict improvement we show on an example that the resulting bound improvement can be substantial we show on a complex practical problem how lagrangean decomposition may help uncover hidden special structures and thus yield better solution methodology

lifted polynomial size formulations for the homogeneous and heterogeneous vehicle routing problems abstract we propose compact formulations for the symmetric and asymmetric capacitated vehicle routing problems these formulations are obtained by lifting using the reformulation linearization technique a novel polynomial size ordered path based formulation we show that the strongest proposed formulation is equivalent to the strongest multi commodity flow formulation presented so far in addition we propose polynomial size valid inequalities that aim at further tightening the proposed formulations also the tightest formulation is extended to model the fleet size and mix vehicle routing problem with fixed costs we show that the new derived model is not comparable with a state of the art polynomial length formulation we present the results of computational experiments that demonstrate the tightness of the proposed formulations and the impact of the valid inequalities

vehicle routing with time windows two optimization algorithms we describe two optimization methods for vehicle routing problems with time windows these are a k tree relaxation with time windows added as side constraints and a lagrangian decomposition in which variable splitting is used to divide the problem into two subproblems a semi assignment problem and a series of shortest path problems with time windows and capacity constraints we present optimal solutions to problems with up to 100 customers

an optimization algorithm for the vehicle routing problem with time windows based on lagrangian relaxation our paper presents a new optimization method for the vehicle routing problem with time windows vrptw the vrptw is a generalization of the vehicle routing problem where the service of a customer must start within a given time interval a so called time window our method is based on a lagrangian relaxation of the constraint set requiring that each customer must be serviced the master problem consists of finding the optimal lagrangian multipliers and the subproblem is a shortest path problem with time windows and capacity constraints the optimal multipliers are found using a method exploiting the benefits of subgradient methods as well as a bundle method the method has been implemented and tested on a series of well known benchmark problems of size up to 100 customers our algorithm turns out to be very competitive compared to algorithms considered in the literature and we have succeeded in solving several previously unsolved problems

branch and price column generation for solving huge integer programs we discuss formulations of integer programs with a huge number of variables and their solution by column generation methods i e implicit pricing of nonbasic variables to generate new columns or to prove lp optimality at a node of the branch and bound tree we present classes of models for which this approach decomposes the problem provides tighter lp relaxations and eliminates symmetry we then discuss computational issues and implementation of column generation branch and bound algorithms including special branching rules and efficient ways to solve the lp relaxation we also discuss the relationship with lagrangian duality

on the distance constrained vehicle routing problem we analyze the vehicle routing problem with constraints on the total distance traveled by each vehicle two objective functions are considered minimize the total distance traveled by vehicles and minimize the number of vehicles used we demonstrate a close relationship between the optimal solutions for the two objective functions and perform a worst case analysis for a class of heuristics we present a heuristic that provides a good worst case result when the number of vehicles used is relatively small

classification of dantzig wolfe reformulations for binary mixed integer programming problems abstract in this note we provide a classification of dantzig wolfe reformulations for binary mixed integer programming problems we specifically focus on modeling the binary conditions in the convexification approach to the dantzig wolfe decomposition for a general binary mixed integer programming problem an extreme point of the overall problem does not necessarily correspond to an extreme point of the subproblem therefore the binary conditions cannot in general be imposed on the new master problem variables but must be imposed on the original binary variables in some cases however it is possible to impose the binary restrictions directly on the new master problem variables the issue of imposing binary conditions on the original variables versus the master problem variables has not been discussed systematically for mip problems in general in the literature and most of the research has been focused on the pure binary case the classification indicates in which cases you can and cannot impose binary conditions on the new master problem variables

local search with annealing like restarts to solve the vrptw abstract in this paper we propose a metaheuristic based on annealing like restarts to diversify and intensify local searches for solving the vehicle routing problem with time windows vrptw using the solomon s benchmark instances for the problem our method obtained seven new best results and equaled 19 other best results extensive comparisons indicate that our method is comparable to the best in published literature this approach is flexible and can be extended to handle other variants of vehicle routing problems and other combinatorial optimization problems

a cluster based optimization approach for the multi depot heterogeneous fleet vehicle routing problem with time windows this paper presents a novel three phase heuristic algorithmic approach for the multi depot routing problem with time windows and heterogeneous vehicles it has been derived from embedding a heuristic based clustering algorithm within a vrptw optimization framework to this purpose a rigorous milp mathematical model for the vrptw problem is first introduced likewise other optimization approaches the new formulation can efficiently solve case studies involving at most 25 nodes to optimality to overcome this limitation a preprocessing stage clustering nodes together is initially performed to yield a more compact cluster based milp problem formulation in this way a hierarchical hybrid procedure involving one heuristic and two algorithmic phases was developed phase i aims to identifying a set of cost effective feasible clusters while phase ii assigns clusters to vehicles and sequences them on each tour by using the cluster based milp formulation ordering nodes within clusters and scheduling vehicle arrival times at customer locations for each tour through solving a small milp model is finally performed at phase iii numerous benchmark problems featuring different sizes clustered random customer locations and time window distributions have been solved at acceptable cpu times

on transductions of formal power series over complete semirings abstract rational and pushdown transductions of formal languages are generalized to formal power series with coefficients in a complete semiring a characterization similar to nivat s theorem is given commutativity requirements for the coefficients are especially studied

on the effectiveness of set covering formulations for the vehicle routing problem with time windows the vehicle routing problem with time windows vrptw is one of the most important problems in distribution and transportation a classical and recently popular technique that has proven effective for solving these problems is based on formulating them as a set covering problem the method starts by solving its linear programming relaxation via column generation and then uses a branch and bound strategy to find an integer solution to the set covering problem a solution to the vrptw an empirically observed property is that the optimal solution value of the set covering problem is very close to its linear programming relaxation which makes the branch and bound step extremely efficient in this paper we explain this behavior by demonstrating that for any distribution of service times time windows customer loads and locations the relative gap between fractional and integer solutions of the set covering problem becomes arbitrarily small as the number of customers increases

multi phase dynamic constraint aggregation for set partitioning type problems dynamic constraint aggregation is an iterative method that was recently introduced to speed up the linear relaxation solution process of set partitioning type problems this speed up is mostly due to the use at each iteration of an aggregated problem defined by aggregating disjoint subsets of constraints from the set partitioning model this aggregation is updated when needed to ensure the exactness of the overall approach in this paper we propose a new version of this method called the multi phase dynamic constraint aggregation method which essentially adds to the original method a partial pricing strategy that involves multiple phases this strategy helps keeping the size of the aggregated problem as small as possible yielding a faster average computation time per iteration and fewer iterations we also establish theoretical results that provide some insights explaining the success of the proposed method tests on the linear relaxation of simultaneous bus and driver scheduling problems involving up to 2 000 set partitioning constraints show that the partial pricing strategy speeds up the original method by an average factor of 4 5

a new optimization algorithm for the vehicle routing problem with time windows the vehicle routing problem with time windows vrptw is a generalization of the vehicle routing problem where the service of a customer can begin within the time window defined by the earliest and the latest times when the customer will permit the start of service in this paper we present the development of a new optimization algorithm for its solution the lp relaxation of the set partitioning formulation of the vrptw is solved by column generation feasible columns are added as needed by solving a shortest path problem with time windows and capacity constraints using dynamic programming the lp solution obtained generally provides an excellent lower bound that is used in a branch and bound algorithm to solve the integer set partitioning formulation our results indicate that this algorithm proved to be successful on a variety of practical sized benchmark vrptw test problems the algorithm was capable of optimally solving 100 customer problems this problem size is six times larger than any reported to date by other published research

exact algorithms for the vehicle routing problem based on spanning tree and shortest path relaxations we consider the problem of routing vehicles stationed at a central facility depot to supply customers with known demands in such a way as to minimize the total distance travelled the problem is referred to as the vehicle routing problem vrp and is a generalization of the multiple travelling salesman problem that has many practical applications we present tree search algorithms for the exact solution of the vrp incorporating lower bounds computed from i shortest spanning k degree centre tree k dct and ii q routes the final algorithms also include problem reduction and dominance tests computational results are presented for a number of problems derived from the literature the results show that the bounds derived from the q routes are superior to those from k dct and that vrps of up to about 25 customers can be solved exactly

on the nucleolus of the basic vehicle routing game in the vehicle routing cost allocation problem the aim is to find a good cost allocation method i e a method that according to specified criteria allocates the cost of an optimal route configuration among the customers we formulate this problem as a co operative game in characteristic function form and give conditions for when the core of the vehicle routing game is nonempty one specific solution concept to the cost allocation problem is the nucleolus which minimizes maximum discontent among the players in a co operative game the class of games we study is such that the values of the characteristic function are obtained from the solution of a set of mathematical programming problems we do not require an explicit description of the characteristic function for all coalitions instead by applying a constraint generation approach we evaluate information about the function only when it is needed for the computation of the nucleolus

an accelerated branch and price algorithm for multiple runway aircraft sequencing problems abstract this paper presents an effective branch and price b p algorithm for multiple runway aircraft sequencing problems this approach improves the tractability of the problem by several orders of magnitude when compared with solving a classical 0 1 mixed integer formulation over a set of computationally challenging instances central to the computational efficacy of the b p algorithm is solving the column generation subproblem as an elementary shortest path problem with aircraft time windows and non triangular separation times using an enhanced dynamic programming procedure we underscore in our computational study the algorithmic features that contribute in our experience to accelerating the proposed dynamic programming procedure and hence the overall b p algorithm

an interpolation approximation for the mean workload in a gi g 1 queue this paper develops a closed form approximation for the mean steady state workload or virtual waiting time in a gi g 1 queue using the first two moments of the service time distribution and the first three moments plus the density at the origin of the interarrival time distribution with default values provided in case information is unavailable the approximation is based on light and heavy traffic limiting behavior the essential ideas are exposed by using s l brumelle s formula to relate the mean workload to the mean waiting time and k t marshall s formula to relate the mean waiting time to the first two moments of the idle period both formulas extend to general single server models without independence conditions so this approach provides a basis for extensions but a convenient exact expression for the second order heavy traffic term is evidently not possible even for gi g 1 for the gi g 1 second order heavy traffic term an approximation is proposed based on the relatively nice expression est

lp models for bin packing and cutting stock problems abstract we review several linear programming lp formulations for the one dimensional cutting stock and bin packing problems namely the models of kantorovich gilmore gomory onecut models as in the dyckhoff stadtler approach position indexed models and a model derived from the vehicle routing literature we analyse some relations between the corresponding lp relaxations and their relative strengths and refer how to derive branching schemes that can be used in the exact solution of these problems using branch and price

on exponential stability of linear levin nohel integro differential equations the aim of this paper is to investigate the exponential stability for linear levin nohel integro differential equations with time varying delays to the best of our knowledge the exponential stability for such equations has not yet been discussed in addition since we do not require that the kernel and delay are continuous our results improve those obtained in becker and burton proc r soc edinburgh sect a math 136 245 275 2006 dung j math phys 54 082705 2013 and jin and luo comput math appl 57 7 1080 1088 2009

on the stability of a second order retarded differential equation in this paper we study a second order retarded differential equation with variable delay applying the fixed point theory we obtain a necessary and sufficient condition ensuring that the zero solution of the equation is asymptotically stable our results improve and generalize the known ones in this direction

record breaking optimization results using the ruin and recreate principle a new optimization principle is presented solutions of problems are partly but significantly ruined and rebuilt or recreated afterwards performing this type of change frequently one can obtain astounding results for classical optimization problems the new method is particularly suited for more complex optimization problems discontinuous ones problems with hard to find admissible solutions problems with complex objectives or many constraints the method is an all purpose heuristic numerical results are given for the traveling salesman problem for the vehicle routing problem with time windows and for network optimization numerical evidence for the quality of the proposed principle is given for most of the instances of a research library of problems the ruin and recreate r r implementation achieved the best published results for many instances better or much better solutions could be found

routing optimization with time windows under uncertainty we study an a priori traveling salesman problem with time windows tsptw in which the travel times along the arcs are uncertain and the goal is to determine within a budget constraint a route for the service vehicle in order to arrive at the customers locations within their stipulated time windows as well as possible in particular service at customer s location cannot commence before the beginning of the time window and any arrival after the end of the time window is considered late and constitutes to poor customer service in articulating the service level of the tsptw under uncertainty we propose a new decision criterion called the essential riskiness index which has the computationally attractive feature of convexity that enables us to formulate and solve the problem more effectively as a decision criterion for articulating service levels it takes into account both the probability of lateness and its magnitude and can be applied in contexts where either the distributional information of the uncertain travel times is fully or partially known we propose a new formulation for the tsptw where we explicitly express the service starting time at each customer s location as a convex piecewise affine function of the travel times which would enable us to obtain the tractable formulation of the corresponding distributionally robust problem we also show how to optimize the essential riskiness index via benders decomposition and present cases where we can obtain closed form solutions to the subproblems we also illustrate in our numerical studies that this approach scales well with the number of samples used for the sample average approximation the approach can be extended to a more general setting including vehicle routing problem with time windows with uncertain travel times and customers demands

note on the complexity of the shortest path models for column generation in vrptw in this note we prove that the relaxation approach in designing the subproblem of pricing out only the feasible routes for the set partition formulation of the vrptw is justified on complexity grounds that is the first dynamic programming model presented in m desrochers j desrosiers and m solomon 1992 that is able to price out all feasible routes is np hard in the strong sense

branch and price and cut for the multiple traveling repairman problem with distance constraints abstract in this paper we extend the multiple traveling repairman problem by considering a limitation on the total distance that a vehicle can travel the resulting problem is called the multiple traveling repairmen problem with distance constraints mtrpd in the mtrpd a fleet of identical vehicles is dispatched to serve a set of customers each vehicle that starts from and ends at the depot is not allowed to travel a distance longer than a predetermined limit and each customer must be visited exactly once the objective is to minimize the total waiting time of all customers after the vehicles leave the depot to optimally solve the mtrpd we propose a new exact branch and price and cut algorithm where the column generation pricing subproblem is a resource constrained elementary shortest path problem with cumulative costs an ad hoc label setting algorithm armed with bidirectional search strategy is developed to solve the pricing subproblem computational results show the effectiveness of the proposed method the optimal solutions to 179 out of 180 test instances are reported in this paper our computational results serve as benchmarks for future researchers on the problem

probabilistic analysis of a combined aggregation and math programming heuristic for a general class of vehicle routing and scheduling problems we propose and analyze a heuristic that uses region partitioning and an aggregation scheme for customer attributes load size time windows etc to create a finite number of customer types a math program is solved based on these aggregated customer types to generate a feasible solution to the original problem the problem class we address is quite general and defined by a number of general consistency properties problems in this class include vrps with general distance norms capacitated problems time window vrps pick up and delivery problems combined inventory control and routing problems and arc routing we provide a probabilistic analysis of this heuristic under very general probabilistic assumptions in particular we do not require independence between customer locations and their various attributes the heuristic is a s optimal as the number of customers n tends to infinity further it runs in o n log n time for a fixed relative error and can be designed to be asymptotically optimal w

vehicle routing problem with fuzzy time windows in this paper a vehicle routing problem with fuzzy time windows vrpftw is proposed and solved in the transportation business time windows are not always strictly obeyed and the deviation of service time from the customer specific time window determines the customer s satisfaction level which can also be regarded as the supplier s service level this paper applies fuzzy membership functions to characterize the service level issues associated with time window violation in a vehicle routing problem and proposes vrpftw vrpftw is formulated as a multi objective model with two goals 1 to minimize the travel distance and 2 to maximize the service level of the supplier to customers to solve this multi objective model a two stage algorithm is developed to obtain a pareto solution for vrpftw using the two stage algorithm vrpftw is decomposed into two subproblems namely a traditional vehicle routing problem with time windows email protected and a service improvement problem and each of the objectives is sequentially solved the service improvement problem is solved under two different scenarios when the fuzzy membership function is linear it is shown that the service improvement problem can be solved by the cutting plane algorithm within finite iterations and when the fuzzy membership function is concave the service improvement problem can be solved by a subgradient based algorithm moreover an alternative formulation of vrpftw is also proposed and analyzed experiments are conducted to compare different models of vehicle routing problems with time windows in situations where violation of time windows is allowed and the results show that the vrpftw model can achieve considerable cost savings while at the same time maintaining an acceptable service level

a generalized insertion heuristic for the traveling salesman problem with time windows this article describes a generalized insertion heuristic for the traveling salesman problem with time windows in which the objective is the minimization of travel times the algorithm gradually builds a route by inserting at each step a vertex in its neighbourhood on the current route and performing a local reoptimization this is done while checking the feasibility of the remaining part of the route backtracking is sometimes necessary once a feasible route has been determined an attempt is made to improve it by applying a post optimization phase based on the successive removal and reinsertion of all vertices tests performed on 375 instances indicate that the proposed heuristic compares very well with alternative methods and very often produces optimal or near optimal solutions

dynamic aggregation of set partitioning constraints in column generation column generation is often used to solve problems involving set partitioning constraints such as vehicle routing and crew scheduling problems when these constraints are in large numbers and the columns have on average more than 8 12 nonzero elements column generation often becomes inefficient because solving the master problem requires very long solution times at each iteration due to high degeneracy to overcome this difficulty we introduce a dynamic constraint aggregation method that reduces the number of set partitioning constraints in the master problem by aggregating some of them according to an equivalence relation to guarantee optimality this equivalence relation is updated dynamically throughout the solution process tests on the linear relaxation of the simultaneous vehicle and crew scheduling problem in urban mass transit show that this method significantly reduces the size of the master problem degeneracy and solution times especially for larger problems in fact for an instance involving 1 600 set partitioning constraints the master problem solution time is reduced by a factor of 8

worst case analyses linear programming and the bin packing problem in this paper we consider the familiar bin packing problem and its associated set partitioning formulation we show that the optimal solution to the bin packing problem can be no larger than 4 3 zlp wherezlp is the optimal solution value of the linear programming relaxation of the set partitioning formulation an example is provided to show that the bound is tight a by product of our analysis is a new worst case bound on the performance of the well studied first fit decreasing and best fit decreasing heuristics

an optimal algorithm for the traveling salesman problem with time windows this paper presents the development of new elimination tests which greatly enhance the performance of a relatively well established dynamic programming approach and its application to the minimization of the total traveling cost for the traveling salesman problem with time windows the tests take advantage of the time window constraints to significantly reduce the state space and the number of state transitions these reductions are performed both a priori and during the execution of the algorithm the approach does not experience problems stemming from increasing problem size wider or overlapping time windows or an increasing number of states nearly as rapidly as other methods our computational results indicate that the algorithm was successful in solving problems with up to 200 nodes and fairly wide time windows when the density of the nodes in the geographical region was kept constant as the problem size was increased the algorithm was capable of solving problems with up to 800 nodes for these problems the cpu time increased linearly with problem size these problem sizes are much larger than those of problems previously reported in the literature

stability of an integro differential equation in this work we study a scalar integro differential equation and give some new conditions ensuring that the zero solution is asymptotically stable by means of the fixed point theory our work extends and improves the results in the literature

an interpolation approximation for queueing systems with poisson input for many queueing systems of practical interest it is difficult to obtain exact analytical results it is however often possible to obtain asymptotic results for light and heavy traffic heavy traffic limit theorems yield expressions for normalized quantities of interest in light traffic we can obtain in addition to limits more sensitive information by calculating what are effectively derivatives of the quantity of interest with respect to the arrival rate we can then combine the light and heavy traffic results to yield a polynomial in the arrival rate as an approximation to the normalized quantity of interest for instance by utilizing the heavy traffic limit the light traffic limit and the first derivative we can obtain a quadratic approximation then by reversing the normalization process we can obtain an approximation for the original quantity of interest in this paper we present the details of the above approximation focusing via several examples on applications of the method we then compare the results of the approximation for the examples considered with exact and simulation results in addition we compare our results to some related approximations for the examples considered the approximation works extremely well

nivat s theorem for pushdown transducers abstract the notions of an extended pushdown automaton pda and a pushdown transducer pdt are formalized in matrix notation by generalizing the concept of a pushdown transition matrix the resulting matrices form a semiring the equivalence to nonextended pdas and pdts is shown the semiring structure gives a new proof that the types reset pushdown and one counter are real time types moreover it allows the construction of the functional composition of a pushdown transduction and a finite transducer mapping as an important application nivat s theorem is generalized to rational and polynomial pdts over arbitrary semirings examples include the mirror image of the input and the dyck reduction

new stability conditions for mixed linear levin nohel integro differential equations for the mixed levin nohel integro differential equation we obtain new necessary and sufficient conditions of asymptotic stability these results improve those obtained by becker and burton stability fixed points and inverse of delays proc r soc edinburgh sect a 136 245 275 2006 10 1017 s0308210500004546 and jin and luo stability of an integro differential equation comput math appl 57 7 1080 1088 2009 10 1016 j camwa 2009 01 006 when b t 0 and supplement the 32 stability theorem when a t s 0 in addition the case of the equations with several delays is discussed as well

the tree representation for the pickup and delivery traveling salesman problem with lifo loading the feasible solutions of the traveling salesman problem with pickup and delivery tsppd are commonly represented by vertex lists however when the tsppd is required to follow a policy that loading and unloading operations must be performed in a last in first out lifo manner we show that its feasible solutions can be represented by trees consequently we develop a novel variable neighborhood search vns heuristic for the tsppd with last in first out loading tsppdl involving several search operators based on the tree data structure extensive experiments suggest that our vns heuristic is superior to the current best heuristics for the tsppdl in terms of solution quality while requiring no more computing time as the size of the problem increases

a neural network based approach to the double traveling salesman problem the double traveling salesman problem is a variation of the basic traveling salesman problem where targets can be reached by two salespersons operating in parallel the real problem addressed by this work concerns the optimization of the harvest sequence for the two independent arms of a fruit harvesting robot this application poses further constraints like a collision avoidance function the proposed solution is based on a self organizing map structure initialized with as many artificial neurons as the number of targets to be reached one of the key components of the process is the combination of competitive relaxation with a mechanism for deleting and creating artificial neurons moreover in the competitive relaxation process information about the trajectory connecting the neurons is combined with the distance of neurons from the target this strategy prevents tangles in the trajectory and collisions between the two tours results of tests indicate that the proposed approach is efficient and reliable for harvest sequence planning moreover the enhancements added to the pure self organizing map concept are of wider importance as proved by a traveling salesman problem version of the program simplified from the double version for comparison

lagrangian relaxation methods for solving the minimum fleet size multiple traveling salesman problem with time windows we consider the problem of finding the minimum number of vehicles required to visit once a set of nodes subject to time window constraints for a homogeneous fleet of vehicles located at a common depot this problem can be formulated as a network flow problem with additional time constraints the paper presents an optimal solution approach using the augmented lagrangian method two lagrangian relaxations are studied in the first one the time constraints are relaxed producing network subproblems which are easy to solve but the bound obtained is weak in the second relaxation constraints requiring that each node be visited are relaxed producing shortest path subproblems with time window constraints and integrality conditions the bound produced is always excellent numerical results for several actual school busing problems with up to 223 nodes are discussed comparisons with a set partitioning formulation solved by column generation are given

algorithms for the vehicle routing and scheduling problems with time window constraints this paper considers the design and analysis of algorithms for vehicle routing and scheduling problems with time window constraints given the intrinsic difficulty of this problem class approximation methods seem to offer the most promise for practical size problems after describing a variety of heuristics we conduct an extensive computational study of their performance the problem set includes routing and scheduling environments that differ in terms of the type of data used to generate the problems the percentage of customers with time windows their tightness and positioning and the scheduling horizon we found that several heuristics performed well in different problem environments in particular an insertion type heuristic consistently gave very good results

a branch first cut second approach for locomotive assignment the problem of assigning locomotives to trains consists of selecting the types and number of engines that minimize the fixed and operational locomotive costs resulting from providing sufficient power to pull trains on fixed schedules the force required to pull a train is often expressed in terms of horsepower and tonnage requirements rather than in terms of number of engines this complicates the solution process of the integer programming formulation and usually creates a large integrality gap furthermore the solution of the linearly relaxed problem is strongly fractional to obtain integer solutions we propose a novel branch and cut approach the core of the method consists of branching decisions that define on one branch the projection of the problem on a low dimensional subspace there the facets of the polyhedron describing a restricted constraint set can be easily derived we call this approach branch first cut second we first derive facets when at most two types of engines are used we then extend the branching rule to cases involving additional locomotive types we have conducted computational experiments using actual data from the canadian national railway company simulated test problems involving two or more locomotive types were solved over 1 2 and 3 day rolling horizons the cuts were successful in reducing the average integrality gap by 52 for the two type case and by 34 when more than 25 types were used furthermore the branch first cut second approach was instrumental in improving the best known solution for an almost 2 000 leg weekly problem involving 26 locomotive types it reduced the number of locomotives by 11 or 1 1 at an equivalent savings of 3 000 000 per unit additional tests on different weekly data produced almost identical results

vehicle routing problem with time windows and a limited number of vehicles abstract this paper introduces a variant of the vehicle routing problem with time windows where a limited number of vehicles is given m vrptw under this scenario a feasible solution is one that may contain either unserved customers and or relaxed time windows we provide a computable upper bound to the problem to solve the problem we propose a tabu search approach characterized by a holding list and a mechanism to force dense packing within a route we also allow time windows to be relaxed by introducing the notion of penalty for lateness in our approach customer jobs are inserted based on a hierarchical objective function that captures multiple objectives computational results on benchmark problems show that our approach yields solutions that are competitive to best published results on vrptw on m vrptw instances experiments show that our approach produces solutions that are very close to computed upper bounds moreover as the number of vehicles decreases the routes become more densely packed monotically this shows that our approach is good from both the optimality as well as stability point of view

automata and languages generalized to ω continuous semirings abstract we generalize the following two language and automata theoretic results to continuous semirings i the family of languages accepted by finite automata is the smallest class containing all finite languages and closed under union product and star kleene s theorem ii the family of languages accepted by pushdown automata is the family ofcontext free languages

multiple pickup and delivery traveling salesman problem with last in first out loading and distance constraints abstract we extend the traveling salesman problem with pickup and delivery and lifo loading tsppdl by considering two additional factors namely the use of multiple vehicles and a limitation on the total distance that a vehicle can travel both of these factors occur commonly in practice we call the resultant problem the multiple pickup and delivery traveling salesman problem with lifo loading and distance constraints mtsppd ld this paper presents a thorough preliminary investigation of the mtsppd ld we propose six new neighborhood operators for the problem that can be used in search heuristics or meta heuristics we also devise a two stage approach for solving the problem where the first stage focuses on minimizing the number of vehicles required and the second stage minimizes the total travel distance we consider two possible approaches for the first stage simulated annealing and ejection pool and two for the second stage variable neighborhood search and probabilistic tabu search our computational results serve as benchmarks for future researchers on the problem