Elsevier

Operations Research Letters

Volume 27, Issue 4, November 2000, Pages 185-192
Operations Research Letters

Part sequencing in three-machine no-wait robotic cells

https://doi.org/10.1016/S0167-6377(00)00046-8Get rights and content

Abstract

A no-wait robotic cell is an automated flow shop in which a robot is used to move the parts from a machine to the next. Parts are not allowed to wait. We analyze the complexity of the part sequencing problem in a robotic cell with three machines, for different periodical patterns of robot moves, when the objective is productivity maximization.

Introduction

The flow shop scheduling problem consists of sequencing parts on m machines. Each machine can process only one part at a time, and all parts must visit the m machines in the same order, i.e., M1,M2,…,Mm. A robotic cell (Fig. 1) is a flow shop which also includes an input station (M0), an output station (Mm+1), and one robot, which is in charge of moving the parts from each machine to the next, and between the machines and the input/output stations. Most of the existing literature on this subject has been devoted to cells without buffers in which parts are allowed to wait on a machine, even after the completion of the processing on that machine, if either the next machine or the robot are not available. These are referred to as bufferless robotic cells.

In this paper we focus on the problem arising when the parts must satisfy the no-wait constraint, i.e., as soon as a part is completed by a machine, the robot must immediately move it to the next machine (or to the output station). In other words, machines are not allowed to act as buffers. We refer to this case as no-wait robotic cells.

Management problems in robotic cells consist in concurrently finding a schedule for parts and robot moves which maximizes productivity. For two-machine cells, the problem can be efficiently solved (see [5] for bufferless cells, and [1] for no-wait cells). For cells with three or more machines, when the parts are different from each other, the overall management problem is very hard. In fact, for both bufferless and no-wait robotic cells, if the time required by robot movement is negligible, the problem reduces to a pure part sequencing problem, which is by itself NP-hard [8]. On the other hand, it may not be practical for the robot to perform complex sequences of moves, but rather it is convenient to cyclically repeat short sequences of moves. For these reasons, some authors have especially analyzed one-unit cycles, i.e., patterns of robot moves during which the robot unloads (or, equivalently, loads) each machine exactly once.

When parts are all identical, the optimal one-unit cycle in a no-wait m-machine robotic cell can be found in polynomial time [6], [7]. In [1] special results are derived for m=3, showing that the optimal cycle is either a one-unit or a two-unit cycle.

In this paper we address the problem of finding the optimal one-unit cycle in a three-machine no-wait robotic cell when the parts are different from each other. In particular, for each of the six possible one-unit cycles, we investigate the complexity of the resulting part sequencing problem. For bufferless robotic cells, Hall et al. [5] carried out a similar study, showing that for two out of six one-unit cycles the part sequencing problem is NP-hard, and providing polynomial algorithms for the remaining four cycles. From the part sequencing viewpoint, a major difference between bufferless and no-wait cells is the following. In the bufferless case, for any fixed one-unit cycle, all part sequences are feasible. In the no-wait case, for some fixed one-unit cycles there may not even be feasible solutions, due to the fact that the robot must always reach a part on a machine no later than the end of processing on that machine. Hence, for each one-unit cycle we will be concerned with a feasibility problem and an optimization problem. We show that in four out of six cases the feasibility problem is polynomial, and in the remaining two is NP-complete. For what concerns finding the optimal part sequencing, in two cases the problem is polynomial, in another two it is NP-complete, and in the remaining two the problem is still open. In Section 2, definitions and notation are introduced. In Section 3, the six one-unit cycles are analyzed.

Section snippets

Definitions and notation

We consider robotic cells consisting of three machines, M1,M2,M3, while M0 and M4 denote the input and output stations, respectively. The robot performs part transfers and empty movements throughout the cell. To transfer a part from machine Mi to machine Mi+1, the robot must consecutively unload a part from Mi, travel from Mi to Mi+1, and load the part on Mi+1. Such a part transfer operation is called activity Ai [2], its length being denoted by ci,i+1,i=0,1,2,3. We denote by dij the time the

Cycle S1

This case (Fig. 2) is trivial, since all sequences are feasible and have the same span. The span T(S1) is simply given by the sum of the times spent in the cell by all parts, plus the time taken by the robot to move from M4 back to M0 after the completion of each part:T(S1)=h=1nj=13phj+n(c01+c12+c23+c34+d40).

Cycle S4

In this section we show that both problems F(S4) and O(S4) are easy. Let us first consider F(S4). During the execution of each one-unit cycle, the robot performs A3 on σ(k) between

Conclusions

We presented some new results for part sequencing in three-machine no-wait robotic cells, when the robot move cycle is fixed. There are six possible robot cycles. Table 1 summarizes the state of the art for the problems analyzed in this paper, as well as for the corresponding problems in bufferless robotic cells (see [5]). Recall that no feasibility problem arises in bufferless robotic cells, since all sequences are feasible for any robot cycle.

Problem O(S3) and O(S5) are special cases of TSP (

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