Pure cycles in flexible robotic cells

In this study, an m-machine flexible robotic manufacturing cell consisting of CNC machines is considered. The flexibility of the machines leads to a new class of robot move cycles called the pure cycles. We first model the problem of determining the best pure cycle in an m-machine cell as a special travelling salesman problem in which the distance matrix consists of decision variables as well as parameters. We focus on two specific cycles among the huge class of pure cycles. We prove that, in most of the regions, either one of these two cycles is optimal. For the remaining regions we derive worst case performances of these cycles. We also prove that the set of pure cycles dominates the flowshop-type robot move cycles considered in the literature. As a design problem, we consider the number of machines in a cell as a decision variable. We determine the optimal number of machines that minimizes the cycle time for given cell parameters such as the processing times, robot travel times and the loading/unloading times of the machines.     

Bicriteria robotic cell scheduling

This paper considers the scheduling problems arising in two- and three-machine manufacturing cells configured in a flowshop which repeatedly produces one type of product and where transportation of the parts between the machines is performed by a robot. The cycle time of the cell is affected by the robot move sequence as well as the processing times of the parts on the machines. For highly flexible CNC machines, the processing times can be changed by altering the machining conditions at the expense of increasing the manufacturing cost. As a result, we try to find the robot move sequence as well as the processing times of the parts on each machine that not only minimize the cycle time but, for the first time in robotic cell scheduling literature, also minimize the manufacturing cost. For each 1-unit cycle in two- and three-machine cells, we determine the efficient set of processing time vectors such that no other processing time vector gives both a smaller cycle time and a smaller cost value. We also compare these cycles with each other to determine the sufficient conditions under which each of the cycles dominates the rest. Finally, we show how different assumptions on cost structures affect the results.     

Pure cycles in two-machine dual-gripper robotic cells

We consider a robotic cell served by a dual-gripper robot that consists of identical CNC machines placed linearly and a material handling robot loading/unloading the machines and transporting the parts between them. Identical parts are to be processed in this system and the CNC machines are capable of performing all the operations that a part requires. We consider the problem of sequencing activities of the robot in order to maximize the throughput rate. As a consequence of the flexibility of the CNC machines, a new class of robot move sequences, named as pure cycles, arises. In a pure cycle, the robot loads and unloads each machine once and each part is processed on exactly one of the machines. Thereby, the problem is to determine the best pure cycle that maximizes the throughput rate. We first determine the feasibility conditions for the pure cycles and prove some basic results that reduces the number of feasible pure cycles to be investigated. We analyze 2-machine robotic cells in detail and prove that five of the cycles among a huge number of feasible pure cycles dominate the rest. We determine the parameter regions in which each of the five cycles is optimal. We also analyze the performance improvement that can be attained by using a dual gripper robot and provide managerial insights.     

Single machine scheduling with controllable processing times to minimize total tardiness and earliness

In most deterministic scheduling problems job processing times are considered as invariable and known in advance. Single machine scheduling problem with controllable processing times with no inserted idle time is presented in this study. Job processing times are controllable to some extent that they can be reduced or increased, up to a certain limit, at a cost proportional to the reduction or increase. In this study, our objective is determining a set of compression/expansion of processing times in addition to a sequence of jobs simultaneously, so that total tardiness and earliness are minimized. A mathematical model is proposed firstly and afterward a net benefit compression–net benefit expansion (NBC–NBE) heuristic is presented so as to acquire a set of amounts of compression and expansion of jobs processing times in a given sequence. Three heuristic techniques in small problems and in medium-to-large instances two meta-heuristic approaches, as effective local search methods, as well as these heuristics are employed to solve test examples. The single machine total tardiness problem (SMTTP) is already NP-hard, so the considered problem is NP-hard obviously. The computational experiments demonstrate that our proposed heuristic is efficient approach for such just-in-time (JIT) problem, especially equipped with competent heuristics.     

Single-machine scheduling with trade-off between number of tardy jobs and compression cost

We consider a single-machine scheduling problem, in which the job processing times are controllable or compressible. The performance criteria are the compression cost and the number of tardy jobs. For the problem, where no tardy jobs are allowed and the objective is to minimize the total compression cost, we present a strongly polynomial time algorithm. For the problem to construct the trade-off curve between the number of tardy jobs and the maximum compression cost, we present a polynomial time algorithm. Furthermore, we extend the problem to the case of discrete controllable processing times, where the processing time of a job can only take one of several given discrete values. We show that even some special cases of the discrete controllable version with the objective of minimizing the total compression cost are NP-hard, but the general case is solvable in pseudo-polynomial time. Moreover, we present a strongly polynomial time algorithm to construct the trade-off curve between the number of tardy jobs and the maximum compression cost for the discrete controllable version. This research was supported by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of the MOE, China, and the National Natural Science Foundation of China (10271110). The third author was supported in part by The Hong Kong Polytechnic University under a grant from the ASD in China Business Services.     

Open Shop Scheduling with Synchronization

In this paper, we study open shop scheduling problems with synchronization. This model has the same features as the classical open shop model, where each of the n jobs has to be processed by each of the m machines in an arbitrary order. Unlike the classical model, jobs are processed in synchronous cycles, which means that the m operations of the same cycle start at the same time. Within one cycle, machines which process operations with smaller processing times have to wait until the longest operation of the cycle is finished before the next cycle can start. Thus, the length of a cycle is equal to the maximum processing time of its operations. In this paper, we continue the line of research started by Weiß et al. (Discrete Appl Math 211:183–203, 2016). We establish new structural results for the two-machine problem with the makespan objective and use them to formulate an easier solution algorithm. Other versions of the problem, with the total completion time objective and those which involve due dates or deadlines, turn out to be NP-hard in the strong sense, even for \(m=2\) machines. We also show that relaxed models, in which cycles are allowed to contain less than m jobs, have the same complexity status.     

Solution algorithms for synchronous flow shop problems with two dominating machines

In this paper, we present solution algorithms for synchronous flow shop problems with two dominating machines. In such an environment, jobs have to be moved from one machine to the next by an unpaced synchronous transportation system, which implies that the processing is organized in synchronized cycles. This means that in each cycle the current jobs start at the same time on the corresponding machines and after processing have to wait until the last job is finished. Afterwards, all jobs are moved to the next machine simultaneously. Motivated by a practical application, we investigate the special case of two dominating machines where the processing times of all jobs on these two machines are at least as large as the processing times of all jobs on the other machines and hence always determine the cycle times. After formulating the considered problem as a special vehicle routing problem, we propose mixed integer linear programming formulations and a tabu search algorithm. Finally, we present computational results for randomly generated data and show the efficiency of the approaches.     

Scheduling in Reentrant Robotic Cells: Algorithms and Complexity

We study the scheduling of m-machine reentrant robotic cells, in which parts need to reenter machines several times before they are finished. The problem is to find the sequence of 1-unit robot move cycles and the part processing sequence which jointly minimize the cycle time or the makespan. When m = 2, we show that both the cycle time and the makespan minimization problems are polynomially solvable. When m = 3, we examine a special class of reentrant robotic cells with the cycle time objective. We show that in a three-machine loop-reentrant robotic cell, the part sequencing problem under three out of the four possible robot move cycles for producing one unit is strongly -hard. The part sequencing problem under the remaining robot move cycle can be solved easily. Finally, we prove that the general problem, without restriction to any robot move cycle, is also intractable.     

Bi-criteria and tri-criteria analysis to minimize maximum lateness makespan and resource consumption for scheduling a single machine

We analyze two single machine scheduling problems for the case where job processing times are controllable, by allocating continuous and non-renewable resources to the processing operations. The first problem to analyze is constructing the trade-off curve between maximum lateness and total resource consumption; an O(n ²) computational time optimization algorithm was constructed to solve this problem. This algorithm was extended to solve the second problem, which is to construct the trade-off surface between maximum lateness, makespan, and total resource consumption. As part of this algorithm we identify a plane in the 3D field that is formed by the three criteria, which is parallel only to the maximum lateness, and calculate the optimal makespan and total resource consumption as functions of points on this plane. The extended algorithm keeps the same complexity of O(n ²) time. Both algorithms are very robust as they solve the problem for a very large set of resource consumption functions which has to follow only some mild (and commonly acceptable) conditions. Moreover, as far as we know, this is the first research of its kind in the field of multi-objective scheduling to present an algorithm that constructs a 3D trade-off surface.     

Modelling worker fatigue and recovery in dual-resource constrained systems

The operational benefits that dual-resource constrained (DRC) job shop systems bring have captured the attention of researchers for some time. Although several studies that investigate DRCs are available in the literature, none has investigated a DRC system for the effects of human fatigue and recovery, which poses important parameters to avoiding overload and injury to employees. The purpose of this paper is to address this limitation by presenting a mixed-integer linear programming (MILP) model that describes fatigue and recovery in a DRC system with one worker performing n tasks (flexibility level) within m cycles. Later, the complexity of the MILP problem was reduced to four practical cases. These cases were investigated to evaluate several research questions. The results obtained from the MILP model and the four practical cases suggest that short rest breaks after each task, short cycle times and faster recovery rates improve the system’s performance and that reduced force levels in the work tasks will reduce recovery needs and further increase performance. Further research is still needed to identify or to develop better models of physiological and mental fatigue that can be integrated to the modelling framework presented here.     

Scheduling a batching machine with convex resource consumption functions

In various industries jobs undergo a batching, or burn in, process where different tasks are grouped into batches and processed simultaneously. The processing time of each batch is equal to the longest processing time among all jobs contained in the batch. All to date studies dealing with batching machines have considered fixed job processing times. However, in many real life applications job processing times are controllable through the allocation of a limited resource. The most common and realistic model assumes that there exists a non-linear and convex relationship between the amount of resource allocated to a job and its processing time. The scheduler?s task when dealing with controllable processing times is twofold. In addition to solving the sequencing problem, one must establish an optimal resource allocation policy. We combine these two widespread models on a single machine setting, showing that both the makespan and total completion time criteria can be solved in polynomial time. We then show that our proposed approach can be applied to general bi-criteria objective comprising of the makespan and the total completion time.     

Scheduling jobs with simultaneous considerations of controllable processing times and learning effect

This paper considers a scheduling problem with general job-dependent learning curves and controllable processing times on a single machine. The objective is to determine the optimal compressions of the processing times and the optimal sequence of jobs so as to minimize some total cost functions, which consist of regular and non-regular functions and the processing time compressions. It shows that the problem can be solved by an assignment problem and thus can be solved in polynomial time. Some extensions of the problem are also given.

     

A heuristic for preemptive scheduling with set-up times

We investigate the problem of preemptively schedulingn jobs onm parallel machines. Whenever there is a switch from processing a job to processing another job on some machine, a set-up time is necessary. The objective is to find a schedule which minimizes the maximum completion time. Form≥2 machines, this problem obviously is NP-complete. For the case of job-dependent set-up times, Monma and Potts derived a polynomial time heuristic whose worst case ratio tends to 5/3 as the number of machines tends to infinity. In this paper, we examine the case of constant (job- and machine-independent) set-up times. We present a polynomial time approximation algorithm with worst case ratio 7/6 form=2 machines and worst case ratio at most 3/2–1/2m form≥3 machines. Moreover, for the casem=2 we construct a fully polynomial time approximation scheme.     

Notes on Max Flow Time Minimization with Controllable Processing Times

In a scheduling problem with controllable processing times the job processing time can be compressed through incurring an additional cost. We consider the identical parallel machines max flow time minimization problem with controllable processing times. We address the preemptive and non-preemptive version of the problem. For the preemptive case, a linear programming formulation is presented which solves the problem optimally in polynomial time. For the non-preemptive problem it is shown that the First In First Out (FIFO) heuristic has a tight worst-case performance of 3–2/m, when jobs processing times and costs are set as in some optimal preemptive schedule. Supported by Swiss National Science Foundation project 20-63733.00/1, Resource Allocation and Scheduling in Flexible Manufacturing Systems, and by the Metaheuristics Network, grant HPRN-CT-1999-00106.     

Total absolute deviation of job completion times on uniform and unrelated machines

Unlike other measures of variation of job completion times considered in scheduling literature, the measure of minimizing total absolute deviation of job completiontimes (TADC) was shown to have a polynomial time solution on a single machine. It was recently shown to remain polynomially solvable when position-dependent job processing times are assumed. In this paper we further extend these results, and show that minimizing TADC remains polynomial when position-dependent processing times are assumed (i) on uniform and unrelated machines and (ii) for a bicriteria objective consisting of a linear combination of total job completion times and TADC. These extensions are shown to be valid also for the measure of total absolute differences of job waiting times (TADW).     

The complexity of determining a shortest cycle of even length

We study the problem of determining a shortest cycle of even length (or of odd length, respectively). For cycles of odd length we get the same complexity as for determining a shortest cycle of a graph i. e. 0 (|V|·|E|) in the case of unweighted graphs and 0 (|V|³) in the case of weighted graphs. The main contribution of this paper is an algorithm computing the shortest cycle of even length of an unweighted undirected graph within essentially 0 (|V|²) time.     

Stochastic proportionate flowshop scheduling with setups

This paper addresses the problem of scheduling n jobs on a proportionate two-machine flowshop where the machines are subject to random breakdowns and setup times are considered separate from processing times. The considered performance measure is makespan. Sequences that minimize makespan with probability 1 are obtained when the first or the second machine is subject to random breakdowns without making any assumptions about downtime distributions or counting processes. It is assumed that the processing and setup times on one machine dominate the corresponding times on the other machine. In the case that processing and setup times on the first and second machines are proportionate, it is shown that the longest processing time (LPT) rule gives an optimal solution when only the first machine is subject to breakdowns, while the shortest processing time (SPT) rule yields an optimal solution when only the second machine suffers breakdowns.     

Lot-sizing on a single imperfect machine: ILP models and FPTAS extensions

A single-machine multi-product lot-sizing and sequencing problem is studied. In this problem, items of n different products are manufactured in lots. Demands for products as well as per item processing times are known. There are losses of productivity because of non perfect production. There is also a sequence dependent set-up time between lots of different products. Machine yields and product lead times are assumed to be known deterministic functions. The objective is to minimize the cost of the demand dissatisfaction provided that the total processing time does not exceed a given time limit. We propose two integer linear programming (ILP) models for the NP-hard “fraction defective” case of this problem and compare effectiveness of their ILOG CPLEX realizations with a dynamic programming algorithm in a computer experiment. We also show how an earlier developed fully polynomial time approximation scheme (FPTAS) and one of the ILP models can be extended for a more complex case.     

Minimizing total tardiness and earliness on unrelated parallel machines with controllable processing times

Job scheduling has always been a challenging task in modern manufacturing and the most real life scheduling problems which involves multi-criteria and multi-machine environments. In this research our direction is largely motivated by the adoption of the Just-In-Time (JIT) philosophy in parallel machines system, where processing times of jobs are controllable. The goal of this paper is to minimize total weighted tardiness and earliness besides jobs compressing and expanding costs, depending on the amount of compression/expansion as well as maximum completion time called makespan simultaneously. Jobs due dates are distinct and no inserted idle time is allowed after starting machine processing. Also each machine is capable of processing only some predetermined jobs and operations with probably different speeds. A Mixed Integer Programming (MIP) model is proposed to formulate such a problem and is solved optimally in small size instances. A Parallel Net Benefit Compression-Net Benefit Expansion (PNBC–NBE) heuristic is then presented to acquire the optimal jobs set amount of compression and expansion processing times in a given sequence. To solve medium-to-large size cases, a proposed heuristic, two meta-heuristics and a hybrid technique are also employed. Experimental results demonstrate that our hybrid procedure is a proficient method and could efficiently solve such complicated problems.     

Embedding cycles and paths on solid grid graphs

Despite many algorithms for embedding graphs on unbounded grids, only a few results on embedding graphs on restricted grids have been published. In this paper, we study the problem of embedding paths and cycles on solid grid graphs. We show that a cycle of length k is unit-length embeddable on a solid grid graph G if k is an even integer between four and the length of the longest cycle of G. In addition, our result shows that a path of length k is unit-length embeddable on G, between its two given vertices s and t, if \(k\le L\) and \(k\equiv L (\mathrm{mod}\ 2)\), in which L is the length of the longest path between s and t. Our presented two algorithms show that such embeddings can be found in linear time for cycles and quadratic time for paths, with respect to the size of graph G. In the case of rectangular grid graphs, the running time of the algorithms can be improved to O(k) and O\((k^2)\), respectively. In addition, we extend our results to \(m\times n\times o\) 3D grids. A application of our result is in the interconnection network mapping in parallel processing.