Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 4037695, 11 pages

https://doi.org/10.1155/2018/4037695

## Impact Analysis of Travel Time Uncertainty on AGV Catch-Up Conflict and the Associated Dynamic Adjustment

^{1}Merchant Marine College, Shanghai Maritime University, Shanghai 201306, China^{2}Institute of Logistics Science & Engineering, Shanghai Maritime University, Shanghai 201306, China^{3}Instituto de Telecomunicacoes, ISCTE-IUL, Av. Das Forcas Armadas, 1049-001 Lisbon, Portugal

Correspondence should be addressed to Bo-wei Xu; moc.621@831iewobux

Received 3 September 2017; Revised 31 December 2017; Accepted 11 February 2018; Published 19 April 2018

Academic Editor: Francisco Chicano

Copyright © 2018 Jun-jun 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

In automated logistics systems, travel time uncertainty can severely affect automated guided vehicle (AGV) conflict and path planning. Insight is required into how travel time uncertainty affects catch-up conflict, the main AGV conflict in one-way road networks. Under normal circumstances, the probability formula for catch-up conflict is deduced based on an analysis of AGV catch-up conflict. The vertex, monotonicity, and symmetry of catch-up conflict probabilities are developed, for symmetrical AGV travel time distribution densities. A dynamic adjustment method based on conflict probability for AGVs is designed. The probability features of catch-up conflicts and the performance of the associated dynamic adjustment are simulated and validated for AGVs at an automated container terminal. The simulation results show that the impact analysis of travel time uncertainty on AGV catch-up conflict is correct, and the dynamic adjustment is effective.

#### 1. Introduction

Automated guided vehicles (AGVs) participate in unmanned transport systems for material handling [1]. In automated logistics systems, such as automated container terminals and automated warehouses, conflicts may arise when several AGVs run along a narrow lane or pass crossing roads [2]. AGV conflict (which can significantly affect actual AGV speeds, expected travel time, and automated logistic system throughput) has been a key issue in AGV path planning [3]. Many studies have focused on conflict avoidance strategies when studying AGV path planning [1, 4, 5]. Smolic-Rocak et al. used time window insertion in vector form and performed window overlapping tests to dynamically solve the shortest path problem for the supervisory control of AGVs traveling within the layout of a given warehouse [4]. Saidi-Mehrabad et al. considered a conflict-free routing problem (CFRP) for AGVs, as well as a basic job shop scheduling problem (JSSP) to minimize total completion time (make-span). They proposed a two-stage ant colony algorithm (ACA) for this problem, especially for large-size problems [1]. Hidalgo-Paniagua et al. proposed a new multiobjective evolutionary approach based on a variable neighborhood search to produce good paths with shorter lengths, improved safety, and smoother mobile robot movements [5]. However, AGV transportation systems are subject to a high degree of uncertainty [6]. The above references simplified AGV conflict avoidance strategies for deterministic conditions, which might lead to suboptimal or even infeasible solutions.

In the real world, AGV conflict on road networks is highly affected by AGV travel times, which are very uncertain because of roadway capacity variations and traffic demand fluctuations [7, 8]. Therefore, the on-time arrival probability of AGVs in automated logistics systems cannot be ensured, especially for a large number of AGVs operating in a limited area. An interesting queuing approach is used to model routing problems with time-dependent travel time [9]. Strategies have been proposed to avoid stochastic travel time influence against unpredictable and random conflicts [10, 11]. Shao et al. used a two-stage traffic control strategy to resolve conflicts and deadlock problems in AGV systems. Specifically, a traffic controller is employed to operate each moving AGV online after utilizing an algorithm to offline construct an optimal path set for AGVs [10]. Zhang et al. formulated on-time shipment delivery problems as stochastic vehicle routing problems with soft time windows under travel and service time uncertainties and proposed a new stochastic programming model for finding a good trade-off between the total cost borne by carriers and customer service levels [11]. However, measures such as scheduling methods and control strategies, which can temporarily prevent conflicts, may reduce the efficiency of AGV systems. Exclusive dependence on these measures easily leads to queuing and congestion in heavy traffic. Therefore, reasonably preanalyzing AGV conflicts under random circumstances can facilitate shorter travel times and higher AGV utilization rates. However, there is not a way to compute the probability of AGV conflicts under uncertainty.

Dynamic adjustment is a key measure to deal with conflicts in the operations management of AGV systems. Lee et al. employed a -shortest path search algorithm to construct a path set and performed online motion planning operation in real time [12]. Li et al. proposed a set of real-time implementable traffic rules to ensure the completion of all jobs with the absence of vehicle deadlocks and collisions [13]. Hoo-Lim et al. proposed a genetic algorithm to continuously optimize and adjust the traffic flow of AGVs for keeping up with the dynamically changing operational condition [14]. Different adjust strategies may result in different running state and different productivity. It is better to analyze conflict probabilities and perform associated dynamical adjustment, rather than employ traditional adjustment as passive response measures to avoid conflicts among AGVs. Furthermore, many methods mentioned in the literature did not consider conflict probabilities in advance of dynamic adjustment, which might bring high frequency of control action, high operational cost, low efficiency, and service ability of automated logistics systems.

After studying the current literature, it is clear that the analysis of AGV conflicts under uncertainty has received less attention from the research community. In this work, catch-up conflict (the main type of AGV conflict in a one-way road network) probability and the associated dynamic adjustment are studied. Compared with the existing literature, the main contributions of this work are elucidated as follows. First, to meet the needs of operation management, AGVs’ travel times are set to random values and the AGV catch-up conflict problem in one-way and single lane road networks is described. Second, the catch-up conflict probability features are analyzed in detail in a situation where the probability density function for the times of AGVs passing through each node is symmetrical. Finally, the dynamic adjustment based on the conflict probability for AGVs is presented in order to reduce catch-up conflicts among AGVs.

#### 2. AGV Catch-Up Conflict Probability and Uncertain Travel Time

##### 2.1. AGV Catch-Up Conflict Probability

A one-way and single lane road network for an automated logistic system is denoted by a graph with nodes and links. and are the th and th nodes, respectively, and they are consecutive. There are AGVs . The start node and end node of the th AGV (denoted as ) are and , respectively.

Assume that passes through node at time . The probability density function and cumulative distribution function for time are and , respectively. , . Once and , or and , catch-up conflicts occur between and (the th AGV). A situation where two AGVs repeatedly catch up with one another is not considered here.

Assuming that the movements of AGVs are mutually independent, the events and are independent of one another. Additionally, and are independent events. The probability of AGV catch-up conflict event is . It can be shown thatwhere is shown in

*∵*,

*∴*.

*∴*.

In this case, is shown in

Let the times at which passes through nodes and be and , respectively, where indicates the change in departure time for . Meanwhile, let , and . From (3), it is determined that

##### 2.2. Uncertain Travel Time

###### 2.2.1. Travel Time Is Random

In an automated logistic system, some AGVs may have the same design speed. If an AGV travels exactly at the design speed , then its arrival time at a node is . In practice, because of uncertainty (e.g., speed fluctuations and operating environment disturbances), the actual arrival time is a random variable. Assuming that the various uncertainties are arbitrary, the probability density function of the random variable would be symmetrical about .

Two probability distribution functions (uniform distribution [8, 15] and normal distribution [15, 16]) widely used in engineering are specified in (5) and (6), respectively. where , .

###### 2.2.2. Travel Speed Is Random

When the speeds of AGVs change randomly, the times at which the AGVs pass through nodes become random. Here, denotes the actual speed of . Similar to (5) and (6), the uniform and normal distributions of are given in (7) and (8), respectively. where , . If the departure time of is , then the time at which it passes through node is , where is the distance traveled by from node to node . If the probability density function of is , then the probability density function of is

In Section 3, the impact of travel time uncertainty on catch-up conflict described in Section 2.1 is analyzed; subsequently, catch-up conflicts caused by uncertain travel time described in Section 2.2 are simulated in Section 5.

#### 3. Impact of Travel Time Uncertainty on Catch-Up Conflict

The catch-up conflict between two AGVs is the basis for catch-up conflicts among multiple AGVs. Catch-up conflict probabilities between and are analyzed here.

First, on the basis of distribution characteristics of and , the symmetry and monotonicity of are analyzed by Theorem 1 and Inference 2, respectively. Second, according to the symmetries of and , the symmetry of is analyzed using Theorem 3. Finally, the distribution characteristics of are comprehensively analyzed using Inference 4.

Theorem 1. *If and are symmetric about point , then is symmetric about and .*

*Inference 2. *If both and are symmetric about point , and both and monotonically increase (or monotonically decrease), then is an increasing function when , or vice versa.

Theorem 3. *If and are symmetric about and , respectively, then is symmetric about point .*

*Inference 4. *If the following conditions are met in and :

(1) the times at which and pass through node are and , respectively. The times at which and pass through node are and , respectively.

(2) , , , and are the interval distributionsrespectively. , , , , , , , and are all real numbers. , , , and .

(3) , , , and are symmetric about and , respectively, where .

Thence, is symmetric about , and conforms to Formula (11), where , , .

Similarly, if some intervals of , , , and change to closed intervals, Inference 4 is true; if one of these intervals is adjusted to (such as ) Inference 4 remains true.

The above analysis can provide premise and basis for dynamic adjustment and help to reduce catch-up conflicts among AGVs.

#### 4. Dynamic Adjustment Based on Conflict Probability

Dynamic adjustment based on conflict probability may change the traditional passive response mode to taking the initiative in avoiding conflicts among AGVs. This could provide substantial productivity improvement for AGV transportation systems.

If all catch-up conflict probabilities are considered, the difficulty and complexity of AGVs planning will increase and the process would involve more calculation. Hence, based on the catch-up conflict probability features revealed in Inference 4, a catch-up conflict probability threshold () is considered for improving the efficiency of AGVs planning. If the catch-up conflict probability is larger than , then the AGVs with low priority replan their paths; else, the AGVs with low priority simply slow down or wait to avoid conflicts once the potential catch-up conflicts happen. When the AGVs with low priorities replan their paths, the links with “” in path preplanning are eliminated from their optional links in the corresponding time windows. Checking “” and path replanning by the AGVs with low priorities are executed until “” does not exist. Then all catch-up conflict probabilities are lower than . Low conflict probabilities would result in few conflicts, which will greatly reduce the negative impact of conflicts on AGVs operation.

Details are as follows.

The dynamic adjustment flow of AGVs is given in Figure 1. First, the priorities of AGVs are set, and the catch-up conflict probability threshold is given. The steps of dynamic adjustment are listed below.