Journal of Advanced Transportation

Volume 2017 (2017), Article ID 7217309, 13 pages

https://doi.org/10.1155/2017/7217309

## Dynamic Vehicle Scheduling for Working Service Network with Dual Demands

School of Management Engineering, Zhengzhou University, Zhengzhou 450001, China

Correspondence should be addressed to Bing Li

Received 4 July 2017; Revised 17 August 2017; Accepted 10 September 2017; Published 22 October 2017

Academic Editor: Sara Moridpour

Copyright © 2017 Bing Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This study aims to develop some models to aid in making decisions on the combined fleet size and vehicle assignment in working service network where the demands include two types (minimum demands and maximum demands), and vehicles themselves can act like a facility to provide services when they are stationary at one location. This type of problem is named as the dynamic working vehicle scheduling with dual demands (DWVS-DD) and formulated as a mixed integer programming (MIP). Instead of a large integer program, the problem is decomposed into small local problems that are guided by preset control parameters. The approach for preset control parameters is given. By introducing them into the MIP formulation, the model is reformulated as a piecewise form. Further, a piecewise method by updating preset control parameters is proposed for solving the reformulated model. Numerical experiments show that the proposed method produces better solution within reasonable computing time.

#### 1. Introduction

The vehicle scheduling problems arise when owners and operators of transportation systems must manage a fleet of vehicles over space and time to serve current and forecasted demands.

The capacity of a transportation system is directly related to the number of available vehicles. Determining the optimal number of vehicles for a transportation system requires a tradeoff among the benefits for meeting demands, the ownership costs of the vehicles, and the penalty costs associated with not meeting some demands. Serving demand results in the relocation of vehicles. Each vehicle is in a particular location, and each task demand requires a vehicle in a particular location. The assignment of a vehicle to a task demand generates revenue. Thus, we consider the problem of vehicles assignment strategy.

The interaction between fleet sizing decisions and vehicle assignment decisions is the focus of this paper. There is a substantial history of research on vehicle assignment problems with fixed vehicle fleet. But the research described in this paper attempts to integrate vehicle fleet sizing decisions with vehicle assignment decisions.

In this paper, we consider the dynamic vehicle scheduling for working service network with dual demands by applying an optimization modeling approach, in which the service demand in each terminal includes two type, that is, minimum demands and maximum demands. We name this type of problem as the dynamic working vehicle scheduling with dual demands (DWVS-DD). The objective is to optimize the performance of the transportation system over the entire planning horizon. The model of problem starts with the classical mixed integer programming formulation and is then reformulated as a piecewise form. We develop two types of reformulated models for the issue and present a piecewise method by updating preset control parameters.

In addition to the integration of the vehicle fleet sizing and the vehicle assignment problem, two other factors, such as the working service network and working vehicle, increase significantly the complexity of the research in this paper.

First, we must recognize one crucial characteristic of working service network: at any location of working service network in space and time, the demands include two types, that is, minimum demands and maximum demands. The minimum demands must be met, but maximum demands are not. If insufficient vehicles are available to meet maximum demand, the penalty cost for unmet demand will generate. This characteristic is the cornerstones of the model developed in this paper.

Second, vehicles usually provide pickup or delivery services between various locations in previous studies. However, in reality, vehicles themselves can sometimes act like a facility to provide real-time services when they are stationary at one location. The vehicles cannot provide services when they are in motion, and the service begins when a vehicle arrives at a location and ends when it departs. For instance, medical treatment vehicles provide first aid services to areas where the established medical facility is temporarily insufficient. Also, food trucks provide fast food services in different regions in different time periods of the day. Note that when these vehicles are in service, they behave like traditional facilities. The term working vehicle (WV) will be used in this paper to denote this vehicle. Applications of problems arise in many settings, ranging from managing emergency vehicles, medical testing vehicles, traveling salesman, and military force deployment.

Overall, the objectives of this research are twofold. The first objective is to develop a novel mathematical model of DWVS-DD. In conventional mathematical models, the problem has been formulated as a mixed integer programming model. In the proposed model, the problem is reformulated as a piecewise form to find the local problem at every time period. The second objective is to propose an efficient methodology for solving the model. Specifically, the contributions of this paper are as follows:(1)We develop the mixed integer programming model (MIP). The model then is reformulated as two novel formulations, that is, reconstruction model with single preset incremental parameters (RM-SPIP) and reconstruction model with double preset incremental-decremental parameters (RM-DPIDP).(2)We propose the coupled correlation model of total profit, vehicle supply, minimum demand, and maximum demand. The acquisition approaches of single preset control parameters are given. Meanwhile, we propose the coupled correlation model of the revenue, penalty cost, vehicle supply, minimum demand, and maximum demand. The acquisition approaches of double preset control parameters are given.(3)According to the specific structure of the two reconstruction models, the piecewise method by updating preset control parameters (PM-PCP) is developed.(4)We tested the performance of PM-PCP approach by solving many instances. We compared the quality of the solutions provided by PM-PCP solving RM-SPIP and RM-DPIDP versus the results of the CPLEX solving MIP. According to our results, in most of the instances, our PM-PCP approach has a better performance. Indeed, PM-PCP provides many optimal solutions in a very short computation time.

The remainder of this paper is organized as follows. Section 2 of this paper discusses related earlier research efforts. Section 3 is devoted to the mathematical description of the DWVS-DD. In Section 4, we set up the problem as the classical mathematical programming formulation and present then the reformulated models. Section 5 explains the acquisition methodology of preset control parameters, based on coupled correlation function. We introduce piecewise method by updating preset parameters (PM-PCP) for solving reconstruction model in Section 6. The computational experiments are described in Section 7, and the effectiveness of the proposed method is shown from the computational results. The last section concludes with a summary of current work and extensions.

#### 2. Literature Review

In this section, we review the relevant literatures about vehicle scheduling problems. Literature review indicates that vehicle scheduling problems can be divided into three groups, that is, vehicle routing, fleet sizing, and fleet assignment. The focus of our literature review will primarily be on the model and exact approach since it is also the model and approach we have taken in this paper.

In recent years, many researches on vehicle routing optimization have been carried out. Hou et al. [1] focused on vehicle routing problem with soft time window constraint. An exact algorithm based on set partition was proposed to solve the balancing the vehicle number and customer satisfaction by Cao et al. [2]. Li et al. [3] studied the integrated problem with truck scheduling and storage allocation. It was formulated as an integer programming model to minimize makespan of the whole discharging course and solved by a two stages tabu search algorithm. Dabia et al. [4] presented a branch and price algorithm for time-dependent vehicle routing problem with time windows. Han et al. [5] considered a vehicle routing problem with uncertain travel times in which a penalty is incurred for each vehicle that exceeds a given time limit and given robust scenario approach for the vehicle routing problem. Muter et al. [6] proposed a column generation algorithm for the multidepot vehicle routing problem with interdepot routes. Vidal et al. [7] proposed a unified hybrid genetic search metaheuristic algorithm to solve multiattribute vehicle routing problems. Battarra et al. [8] presented new exact algorithms for the clustered vehicle routing problem (CluVRP) and provided two exact algorithms for the problem that is a branch and cut as well as a branch and cut and price.

Fleet sizing is one of the most important decisions as it is a major fixed investment for starting any business. Many scholars have conducted the numerous basic studies on fleet sizing problem (FSP). Zak et al. studied a fleet sizing problem in a road freight transportation company with heterogeneous fleet [9]. Additionally, the mathematical model of the decision problem was formulated in terms of multiple objective mathematical programming based on queuing theory. A fleet sizing problem arising in anchor handling operations related to movement of offshore mobile units is presented by Shyshou et al. [10]. A simulation-based prototype was proposed and the simulation model was implemented in Arena 9.0 (a simulation software package developed by Rockwell Software). Rahimi-Vahed et al. [11] addressed the problem of determining the optimal fleet size for three vehicle routing problems, that is, multidepot VRP, periodic VRP, and multidepot periodic VRP. And a new Modular Heuristic Algorithm (MHA) was proposed. Ertogral et al. [12] explored a real strategic fleet sizing problem for a furniture and home accessory distributor. Then, a mixed integer linear program was proposed to determine the total number and types of owned and rented vehicles for each region under seasonal demand. The study developed an analytical model for the joint LGV (Laser Guided Vehicles) fleet sizing problem, also taking into consideration stochastic phenomena and queuing implications in Ferrara et al. [13]. Chang et al. [14] studied the vehicle fleet sizing problem in semiconductor manufacturing and proposed a formulation and solution method, called Simulation Sequential Metamodeling (SSM). By using an agent-based model of a flexible carsharing system, Barrios and Godier [15] explored the trade-offs between fleet size and hired vehicle redistributors, with the objective of maximizing the demand level that can be satisfactorily served. Koç et al. [16] introduced the fleet size and mix location-routing problem with time windows and developed a powerful hybrid evolutionary search algorithm. Park and Kim [17] addressed the fleet sizing of containers and developed an analytical model for the minimum container fleet size.

Moreover, some literatures on fleet assignment were addressed from the viewpoint of optimization models and solution methods. Xia et al. [18] studied a comprehensive model that addresses fleet deployment, speed optimization, and cargo allocation jointly, so as to maximize total profits at the strategic level. Pita et al. [19] presented a flight scheduling and fleet assignment optimization model and carried out a welfare analysis of the network. And the optimization model and subsequent welfare analysis were applied to the PSO network of Norway. Pilla et al. [20] developed a two-stage stochastic programming framework to the fleet assignment model and presented the L-shaped method to solve the two-stage stochastic programming problems. Liang and Chaovalitwongse [21] presented a network-based mixed integer linear programming formulation for the aircraft maintenance with the weekly fleet assignment and developed a diving heuristic approach. Sherali et al. [22] proposed a model that integrates certain aspects of the schedule design fleet assignment and aircraft-routing process and designed Benders’s decomposition-based method. The liner shipping fleet repositioning problem (LSFRP) was formulated as a novel mathematical model and a simulated annealing algorithm is proposed for the LSFRP by Tierney et al. [23]. Hashemi and Sattarvand [24] studied the different management systems of the open pit mining equipment including nondispatching, dispatching, and blending solutions for the Sungun copper mine. A dispatching simulation model with the objective function of minimizing truck waiting times had been developed. A Markov decision model is developed to study the vehicle allocation control problem in the automated material handling system (AMHS) in semiconductor manufacturing by Lin et al. [25]. Simão et al. [26] developed a model for large-scale fleet management and presented an approximate dynamic programming to solve dynamic programs with extremely high-dimensional state variables. Topaloglu and Powell [27] reported how to coordinate the decisions on pricing and fleet assignment of a freight carrier. And a tractable method to obtain sample path-based directional derivatives of the objective function with respect to the prices was presented. Aimed at the stochastic dynamic fleet scheduling, Li et al. [28–30] further proposed some heuristic approaches to deal with these problems. A new and improved Lipschitz optimization algorithm to obtain a Ε-optimal solution for solving the transportation fleet maintenance-scheduling problem is proposed by Yao and Huang [31]. In this study, a procedure based on slope-checking and step-size comparison mechanisms was given to improve the computation efficiency of the Evtushenko algorithm.

The focus of this paper is development of some models to aid in making decisions on the combined fleet size and vehicle assignment in working service network where the demands include two types (minimum demands and maximum demands), and vehicles themselves can act like a facility to provide services when they are stationary at one location. Two types of preset control parameters are applied to the model of the DWVS-DD so that the problem is decoupled into some local problems for different time periods. Further, a piecewise method by updating preset control parameters is proposed for solving the model.

#### 3. Problem Formulation

##### 3.1. Problem Description

Let represent working service network. is the set of working service station set in the network . We assume that time is divided into a set of discrete time periods where is the length of the planning horizon. We also assume that there exist demands for vehicle work service at terminal , , in period , . When vehicles serve demands, the revenues will generate. We assume a unit revenue per served demand in period , denoted as . Serving demand results in the relocation of vehicles between various locations. It implies the need for redistribution of vehicles over the working service network from locations at which they have become idle to locations at which they can be reused. The minimum demands and maximum demands can be, respectively, represented as and at terminal in period . These demands induce vehicles available to serve them. The minimum demands must be met, but maximum demands are not. If insufficient vehicles are available at location in period to meet maximum demand, the penalty cost for unmet demand will generate. We denote the unit penalty cost per period for unmet demand by . The level of demand in units of vehicle loads is assumed to be specified as data. We consider to be these demands which can be serviced by one vehicle. Let be the distance between any pair of terminal and . The demand of terminal can be covered by vehicles located in if only the distance , where is the maximum coverage distance. Considering the expense of purchasing or renting vehicle, we assume that the fixed costs of using vehicles are constant and denoted by for using one vehicle. The main purpose of the DWVS-DD is to propose working vehicle assignment plan for serving as many demands as possible in the given planning horizon at the highest possible profit. Owning or leasing a fleet of vehicles is generally quite costly, so it is natural to try to optimize the size of the required fleet. We emphasize the tradeoff among the investment for establishing a suitable fleet (i.e., the fixed cost), the benefits for meeting demands (i.e., the revenue for serving demands), and the loss of benefits for failing to satisfy demands (i.e., the penalty cost for unmet demands). Dynamic working service network with dual demands is shown in Figure 1.