Heuristics for Improving Operational Performance of Permutation Circulati on-type Vehicle Routing System

This paper discusses an optimization of operating a p ermutation circulation-type vehicle routing system (PCVRS, for short), in w hich several stages are located along by a single loop, and a fleet of vehicles travels on the loop unidirectionally and repeatedly. Traveling on the loop, each vehicle receives an object from the loading stage and then carries it to a cert ain processing stage, or receives an object from a certain processing stage and then carries it to the unloading stage per a turnaround. No passing is allowed f or the vehicles on the loop (from which the system is called permutation, and th is restriction may cause interferences between vehicles). Material handling systems such as PCVRS are actually encountered in flexible man ufacturing systems and in automated storage/retrieval systems. In this paper, we propose a heuristic algorithm for operating the PCVRS, which i ncorporates a new scheduling method for the vehicles with the SPT (shortest proc essing time) numbering of jobs and a round-robin manner of allocating jobs to t he stages, aiming to reduce interferences between the vehicles. We also give num erical results with respect to system performances attained by the heuristic. Description of the system The PCVRS consists of a set of n v vehicles V={V 1,V 2,...,V n v}, a set of n s, processing stages S p={S 1,S 2,...,S n s}, a loading stage S 0 and an unloading stage S n s +1. We denote by S=S p∪{S 0,S n s+l} the set of all the stages. The vehicles travel on a single loop unidirectionany and repeated ly. The system layout is depicted in Fig.1. There is a set of n jobs J={J 1,J 2,...,J n} to be processed b y the vehicles. Each job consists of two tasks: That is, each vehicle receives a n object from S 0 and then carries it to S l with a certain l∈{1,2, ...,n s} (a throw-in job), or receives an object from S l with a certain l∈{1,2,...,n s} and then carries it to S n s+1 (a throw-out job ) per a turnaround. The loop consists of buffer zones BZ(l) and travel zones TZ(l) (see Fig. 1). Each buffer zone BZ(l) is placed in front of stage S l, l=0,1,..., n s, n s+1, in order to avoid a collision between vehicles (i.e., the syste m adopts the so-called zone control strategy). A heuristic algorithm We develop a heuristic algorithm to obtain a good performance for the PCVRS. An operation π={A/B/C} for the PCVRS consists of three decision factors: (A) Numbering jobs Jobs are loaded into S 0 according to an assending order of job numbers. In this paper, we use the following rules to number jobs: SPT: Order jobs in the shortest processing time rule, i.e., P 1≤P 2≤...≤P n for the set of jobs J={J 1,J 2,...,J n}, rather than the FCFS numbering (i.e., number jobs in first-come-first-served order). The SPT rule intends to reduce interferences between two adjacent vehicles at stages. (B) Allocating jobs to stages For the purpose of balancing loads of processing stages, we adopt the following to allocate jobs to the stages: ORDER: Allocate n jobs to n s, processing stages by an in-order manner , i.e., let l(i) be the index of processing stage allocated job J i by ORDER, it holds that l(i)=n s+1-(i-[(i-1)/n s]n s).(1) The ORDER rule intends to process jobs parallel at stages as many as possible. (C) Scheduling vehicles The following method for scheduling vehicles under ORDER rule is already known: Fig.1 The vehicle ro uting system, PCVRS Fig.2 Mean turnaroun d times by heuristics Unchange: Assign n jobs to n v vehicles such that let k(i) be the i ndex of vehicle processing job J i, then k(i)= i-[(i-1)/n v]n v.(2) In csse of n v≥n s, mod (n v,n s)=0 or n v<n s, mod (n s,n v)=0 (mod(x,y) is the remainder of x/y), the number of interferences between vehicles is minimized at stage S 1 under Unchange sche dules, while in the other cases it is not [Lu et al. (2001a)]. Therefore, in t his paper, we develop a new scheduling method of the vehicles, denoted by Ex change, to modify Unchange schedules. Note