A stochastic dynamic traveling salesman problem with hard time windows

Just-in-time (JIT) trucking service, i.e., arriving at customers within specified time windows, has become the norm for freight carriers in all stages of supply chains. In this paper, a JIT pickup/delivery problem is formulated as a stochastic dynamic traveling salesman problem with time windows (SDTSPTW). At a customer location, the vehicle either picks up goods for or delivers goods from the depot, but does not provide moving service to transfer goods from one location to another. Such routing problems are NP-hard in deterministic settings, and in our context, complicated further by the stochastic, dynamic nature of the problem. This paper develops an efficient heuristic for the SDTSPTW with hard time windows. The heuristic is shown to be useful both in controlled numerical experiments and in applying to a real-life trucking problem.     

Vehicle routing with stochastic demands and restricted failures

This paper considers a class of stochastic vehicle routing problems (SVRPs) with random demands, in which the number of potential failures per route is restricted either by the data or the problem constraints. These are realistic cases as it makes little sense to plan vehicle routes that systematically fail a large number of times. First, a chance constrained version of the problem is considered which can be solved to optimality by algorithms similar to those developed for the deterministic vehicle routing problem (VRP). Three classes of SVRP with recourse are then analyzed. In all cases, route failures can only occur at one of the lastk customers of the planned route. Since in general, SVRPs are considerably more intractable than the deterministic VRPs, it is interesting to note that these realistic stochastic problems can be solved as a sequence of deterministic traveling salesman problems (TSPs). In particular, whenk=1 the SVRP with recourse reduces to a single TSP.     

A stochastic programming approach to cash management in banking

The treasurer of a bank is responsible for the cash management of several banking activities. In this work, we focus on two of them: cash management in automatic teller machines (ATMs), and in the compensation of credit card transactions. In both cases a decision must be taken according to a future customers demand, which is uncertain. From historical data we can obtain a discrete probability distribution of this demand, which allows the application of stochastic programming techniques. We present stochastic programming models for each problem. Two short-term and one mid-term models are presented for ATMs. The short-term model with fixed costs results in an integer problem which is solved by a fast (i.e. linear running time) algorithm. The short-term model with fixed and staircase costs is solved through its MILP equivalent deterministic formulation. The mid-term model with fixed and staircase costs gives rise to a multi-stage stochastic problem, which is also solved by its MILP deterministic equivalent. The model for compensation of credit card transactions results in a closed form solution. The optimal solutions of those models are the best decisions to be taken by the bank, and provide the basis for a decision support system.     

The stochastic interdiction median problem with disruption intensity levels

In this paper we introduce a stochastic interdiction problem for median systems in which the operational state of the system??s disrupted elements in the aftermath of the disruption is uncertain as it is based on the intensity of the disruption. We assume that a disruption disables a facility with a given probability and this probability depends on the intensity of the disruption. The objective of this problem is to identify which disruption scenario entails a maximum overall traveling distance in serving all customers. We show that the initial two stage stochastic formulation can be reformulated into a deterministic counterpart whose size is polynomial in the number of facilities and intensity levels. Then, our ensuing efforts to solve the problem efficiently focus on studying alternative deterministic formulations that allow the solution of realistic size instances of the model. We observe that the most efficient of the deterministic formulations provide great scalability with respect to variations in the input parameters and size of the instances solved. Finally, we analyze the robustness of the optimal solutions due to misestimations in the probability functions that relate disruption intensity levels with the probabilities of facility survivability.     

Branch-and-price-and-cut for the multiple traveling repairman problem with distance constraints

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.     

A deterministic iterated local search algorithm for the vehicle routing problem with backhauls

The vehicle routing problem with backhauls is a variant of the classical capacitated vehicle routing problem. The difference is that it contains two distinct sets of customers: those who receive goods from the depot, who are called linehauls, and those who send goods to the depot, who are referred to as backhauls. In this paper, we describe a new deterministic iterated local search algorithm, which is tested using a large number of benchmark problems chosen from the literature. These computational tests have proven that this algorithm competes with the best known algorithms in terms of the quality of the solutions and at the same time, it is simpler and faster.     

A probabilistic one-centre location problem on a network

In this paper we consider the one-centre problem on a network when the speeds on links are stochastic rather than deterministic. Given a desirable time to reach customers residing at the nodes, the objective is to find the location for a facility such that the probability that all nodes are reached within this time threshold is maximized. The problem is formulated, analyzed and solved by using multivariate normal probabilities. The procedure is demonstrated on an example problem.     

A multi-depot period vehicle routing problem arising in the utilities sector

This paper considers the resource planning problem of a utility company that provides preventive maintenance services to a set of customers using a fleet of depot-based mobile gangs. The problem is to determine the boundaries of the geographic areas served by each depot, the list of customers visited each day and the routes followed by the gangs. The objective is to provide improved customer service at minimum operating cost subject to constraints on frequency of visits, service time requirements, customer preferences for visiting on particular days and other routing constraints. The problem is solved as a Multi-Depot Period Vehicle Routing Problem (MDPVRP). The computational implementation of the complete planning model is described with reference to a pilot study and results are presented. The solution algorithm is used to construct cost-service trade-off curves for all depots so that management can evaluate the impact of different customer service levels on total routing costs.     

Large scale stochastic inventory routing problems with?split delivery and service level constraints

A stochastic inventory routing problem (SIRP) is typically the combination of stochastic inventory control problems and NP-hard vehicle routing problems, which determines delivery volumes to the customers that the depot serves in each period, and vehicle routes to deliver the volumes. This paper aims to solve a large scale multi-period SIRP with split delivery (SIRPSD) where a customer??s delivery in each period can be split and satisfied by multiple vehicle routes if necessary. This paper considers SIRPSD under the multi-criteria of the total inventory and transportation costs, and the service levels of customers. The total inventory and transportation cost is considered as the objective of the problem to minimize, while the service levels of the warehouses and the customers are satisfied by some imposed constraints and can be adjusted according to practical requests. In order to tackle the SIRPSD with notorious computational complexity, we first propose an approximate model, which significantly reduces the number of decision variables compared to its corresponding exact model. We then develop a hybrid approach that combines the linearization of nonlinear constraints, the decomposition of the model into sub-models with Lagrangian relaxation, and a partial linearization approach for a sub model. A near optimal solution of the model found by the approach is used to construct a near optimal solution of the SIRPSD. Randomly generated instances of the problem with up to 200 customers and 5 periods and about 400 thousands decision variables where half of them are integer are examined by numerical experiments. Our approach can obtain high quality near optimal solutions within a reasonable amount of computation time on an ordinary PC.     

The stochastic trim-loss problem

The cutting stock problem (CSP) is one of the most fascinating problems in operations research. The problem aims at determining the optimal plan to cut a number of parts of various length from an inventory of standard-size material so to satisfy the customers demands. The deterministic CSP ignores the uncertain nature of the demands thus typically providing recommendations that may result in overproduction or in profit loss. This paper proposes a stochastic version of the CSP which explicitly takes into account uncertainty. Using a scenario-based approach, we develop a two-stage stochastic programming formulation. The highly non-convex nature of the model together with its huge size prevent the application of standard software. We use a solution approach designed to exploit the specific problem structure. Encouraging preliminary computational results are provided.     

A greedy look-ahead heuristic for the vehicle routing problem with time windows

In this paper we consider the problem of physically distributing finished goods from a central facility to geographically dispersed customers, which pose daily demands for items produced in the facility and act as sales points for consumers. The management of the facility is responsible for satisfying all demand, and promises deliveries to the customers within fixed time intervals that represent the earliest and latest times during the day that a delivery can take place. We formulate a comprehensive mathematical model to capture all aspects of the problem, and incorporate in the model all critical practical concerns such as vehicle capacity, delivery time intervals and all relevant costs. The model, which is a case of the vehicle routing problem with time windows, is solved using a new heuristic technique. Our solution method, which is based upon Atkinson's greedy look-ahead heuristic, enhances traditional vehicle routing approaches, and provides surprisingly good performance results with respect to a set of standard test problems from the literature. The approach is used to determine the vehicle fleet size and the daily route of each vehicle in an industrial example from the food industry. This actual problem, with approximately two thousand customers, is presented and solved by our heuristic, using an interface to a Geographical Information System to determine inter-customer and depot–customer distances. The results indicate that the method is well suited for determining the required number of vehicles and the delivery schedules on a daily basis, in real life applications.     

包含随机客户的选择性旅行商问题建模及求解

针对快递配送过程中客户需求具有不确定性的特征,提出一种新的路径优化问题——包含随机客户的选择性旅行商问题,在该问题中客户每天是否具有配送需求存在一定概率,并且对客户进行配送可获取一定利润。同时考虑以上两种因素,建立该问题的数学模型, 目标为在满足行驶距离限制的条件下,找出一条经过部分客户的预优化路径,使得该路径的期望利润最大。其可用于模拟构建最后一公里快递配送的路径问题,提供更具有经济效益的配送路径。随后提出包含精细化局部搜索策略的改进遗传算法,算法根据问题特点构建初始可行解。最后通过多个计算比对结果表明,该算法具有较高的计算效率。     

Pricing problems with a continuum of customers as stochastic Stackelberg games

The pricing problem where a company sells a certain kind of product to a continuum of customers is considered. It is formulated as a stochastic Stackelberg game with nonnested information structure. The inducible region concept, recently developed for deterministic Stackelberg games, is extended to treat the stochastic pricing problem. Necessary and sufficient conditions for a pricing scheme to be optimal are derived, and the pricing problem is solved by first delineating its inducible region, and then solving a constrained optimal control problem.The research work reported here as supported in part by the National Science Foundation under Grant ECS-81-05984, Grant ECS-82-10673, and by the Air Force Office of Scientific Research under AFOSR Grant 80-0098.     

A Hybrid Multiobjective Evolutionary Algorithm for Solving Vehicle Routing Problem with Time Windows

Vehicle routing problem with time windows (VRPTW) involves the routing of a set of vehicles with limited capacity from a central depot to a set of geographically dispersed customers with known demands and predefined time windows. The problem is solved by optimizing routes for the vehicles so as to meet all given constraints as well as to minimize the objectives of traveling distance and number of vehicles. This paper proposes a hybrid multiobjective evolutionary algorithm (HMOEA) that incorporates various heuristics for local exploitation in the evolutionary search and the concept of Pareto's optimality for solving multiobjective optimization in VRPTW. The proposed HMOEA is featured with specialized genetic operators and variable-length chromosome representation to accommodate the sequence-oriented optimization in VRPTW. Unlike existing VRPTW approaches that often aggregate multiple criteria and constraints into a compromise function, the proposed HMOEA optimizes all routing constraints and objectives simultaneously, which improves the routing solutions in many aspects, such as lower routing cost, wider scattering area and better convergence trace. The HMOEA is applied to solve the benchmark Solomon's 56 VRPTW 100-customer instances, which yields 20 routing solutions better than or competitive as compared to the best solutions published in literature.     

Nonconvex Generalized Benders Decomposition for Stochastic Separable Mixed-Integer Nonlinear Programs

This paper considers deterministic global optimization of scenario-based, two-stage stochastic mixed-integer nonlinear programs (MINLPs) in which the participating functions are nonconvex and separable in integer and continuous variables. A novel decomposition method based on generalized Benders decomposition, named nonconvex generalized Benders decomposition (NGBD), is developed to obtain ??-optimal solutions of the stochastic MINLPs of interest in finite time. The dramatic computational advantage of NGBD over state-of-the-art global optimizers is demonstrated through the computational study of several engineering problems, where a problem with almost 150,000 variables is solved by NGBD within 80 minutes of solver time.     

A generalized stochastic goal programming model

In this paper we show how one can get stochastic solutions of Stochastic Multi-objective Problem (SMOP) using goal programming models. In literature it is well known that one can reduce a SMOP to deterministic equivalent problems and reduce the analysis of a stochastic problem to a collection of deterministic problems. The first sections of this paper will be devoted to the introduction of deterministic equivalent problems when the feasible set is a random set and we show how to solve them using goal programming technique. In the second part we try to go more in depth on notion of SMOP solution and we suppose that it has to be a random variable. We will present stochastic goal programming model for finding stochastic solutions of SMOP. Our approach requires more computational time than the one based on deterministic equivalent problems due to the fact that several optimization programs (which depend on the number of experiments to be run) needed to be solved. On the other hand, since in our approach we suppose that a SMOP solution is a random variable, according to the Central Limit Theorem the larger will be the sample size and the more precise will be the estimation of the statistical moments of a SMOP solution. The developed model will be illustrated through numerical examples.     

Stochastic single machine scheduling problem as a multi-stage dynamic random decision process

In this work, we study a stochastic single machine scheduling problem in which the features of learning effect on processing times, sequence-dependent setup times, and machine configuration selection are considered simultaneously. More precisely, the machine works under a set of configurations and requires stochastic sequence-dependent setup times to switch from one configuration to another. Also, the stochastic processing time of a job is a function of its position and the machine configuration. The objective is to find the sequence of jobs and choose a configuration to process each job to minimize the makespan. We first show that the proposed problem can be formulated through two-stage and multi-stage Stochastic Programming models, which are challenging from the computational point of view. Then, by looking at the problem as a multi-stage dynamic random decision process, a new deterministic approximation-based formulation is developed. The method first derives a mixed-integer non-linear model based on the concept of accessibility to all possible and available alternatives at each stage of the decision-making process. Then, to efficiently solve the problem, a new accessibility measure is defined to convert the model into the search of a shortest path throughout the stages. Extensive computational experiments are carried out on various sets of instances. We discuss and compare the results found by the resolution of plain stochastic models with those obtained by the deterministic approximation approach. Our approximation shows excellent performances both in terms of solution accuracy and computational time.

     

Updating Network Flows Given Multiple,Heterogeneous Arc Attribute Changes

A dual ascent reoptimization technique is proposed for updating optimal flows for the minimum cost network flow problem (MCNFP) given any number of simultaneous, heterogeneous changes to the network attributes (i.e. supply at nodes, arc costs and arc capacities) and the optimal solutions to the prior primal and dual problems. Significant savings in computation time can be achieved through the use of reoptimization in place of solving a new MCNFP from scratch as each new problem instance (i.e. set of network attribute updates) arises. The proposed technique can be implemented with polynomial worst-case computational complexity. Extensive numerical experiments were designed and conducted to assess the computational benefits of employing the proposed reoptimization technique as compared with solution from scratch using comparable classic implementations of the original algorithms. This work was motivated by the need for the real-time provision of evacuation instructions to people seeking quick egress from a large sensor-equipped building that has come under attack by natural or terrorist forces, but has broad applicability.     

Planning logistics operations in the oil industry

In this paper we apply stochastic programming modelling and solution techniques to planning problems for a consortium of oil companies. A multiperiod supply, transformation and distribution scheduling problem—the Depot and Refinery Optimization Problem (DROP)—is formulated for strategic or tactical level planning of the consortium's activities. This deterministic model is used as a basis for implementing a stochastic programming formulation with uncertainty in the product demands and spot supply costs (DROPS), whose solution process utilizes the deterministic equivalent linear programming problem. We employ our STOCHGEN general purpose stochastic problem generator to ‘recreate’ the decision (scenario) tree for the unfolding future as this deterministic equivalent. To project random demands for oil products at different spatial locations into the future and to generate random fluctuations in their future prices/costs a stochastic input data simulator is developed and calibrated to historical industry data. The models are written in the modelling language XPRESS-MP and solved by the XPRESS suite of linear programming solvers. From the viewpoint of implementation of large-scale stochastic programming models this study involves decisions in both space and time and careful revision of the original deterministic formulation. The first part of the paper treats the specification, generation and solution of the deterministic DROP model. The stochastic version of the model (DROPS) and its implementation are studied in detail in the second part and a number of related research questions and implications discussed.     

Optimal pricing and quantity of products with two offerings

This paper analyzes the decision of a firm offering two versions of a product, a deluxe and a regular. While both products satisfy the same market, the deluxe version is sold at a high price relative to its cost and is aimed at the high end of the demand curve. The regular version is sold at a low price relative to its cost and is targeted to customers at the low end of the demand curve. This two-offering strategy is especially popular with book publishers where a paperback book is introduced some time after the hardbound version is introduced. The time between the introduction of the two versions of the product is accompanied by a downward shift in the demand curve due to customers losing interest in the product or satisfying their demand from a secondary used market. We solve a profit maximization model for a firm using a two-offering strategy. The model is solved for linear and exponential deterioration in demand, which is assumed to be deterministic. Also, a model with linear deterioration in demand, which is assumed to be stochastic, is solved. The results indicate that substantial improvements in profit can be obtained by using the two-offering strategy. Numerical sensitivity analysis and examples are used to illustrate the results.