This research deals with the problem of scheduling N jobs on M unrelated parallel machines. Each job has a due date and requires a single operation. A setup that includes detaching one die and attaching another from the appropriate die type is incurred if the type of the job scheduled is different from the last job on that machine. Due to the mechanical structure of machines and the fitness of dies to each, the processing time for a job depends on the machine on which the job is processed, and some jobs are restricted to be processed on certain machines. Furthermore, the required detaching and attaching times depend on both the die type and the machine. This type of problems may be encountered, for example, in plastics forming industry where unrelated parallel machines are used to process different components and setups for auxiliary equipment (e.g., dies) are necessary. This type of scheduling problems is also frequently encountered in injection molding departments where many different parallel machines are also used to produce different components and for which setups are required for attaching or detaching molds. In general, the dies (or molds) are quite expensive (tens of thousands dollars each) and thus the number of each type of dies available is limited. Therefore, dies should be considered as secondary resources, the fact of which distinguishes this research from many past studies in unrelated parallel-machine scheduling in which secondary resources are not restricted. This type of problems is NP-hard (So, 1990). When dealing with a large instance encountered in industry, in the worst case, it may not be able to obtain an optimal solution in a reasonable time. In this research heuristics based on guided search, record-to-record travel, and tabu lists from the tabu search (TS) are presented to minimize the maximum completion time (i.e., makespan or Cmax) and maximum tardiness (i.e., Tmax), respectively, to promote schedule performance. Computational characteristics of the proposed heuristics are evaluated through extensive experiments. The rest of this research is organized in six sections. Previously related studies on parallel machine scheduling are reviewed in Section 2. The record-to-record travel and tabu lists are briefly described in Section 3. The proposed heuristic to minimize makespan and the computational results are reported in Section 4. The proposed heuristic to minimize maximum tardiness and the computational results are reported in Section 5. Conclusions and suggestions for future research are discussed in Section 6. O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m
29An integrated three-stage maintenance scheduling model for unrelated parallel machines with aging effect and multi-maintenance activities
An integrated three-stage maintenance scheduling model for unrelated parallel machines with aging effect and multi-maintenance activities
Read more160Unrelated parallel machine scheduling with setup consideration and a total weighted completion time objective
Unrelated parallel machine scheduling with setup consideration and a total weighted completion time objective
53Symbiotic organisms search algorithm for the unrelated parallel machines scheduling with sequence-dependent setup times
This paper addresses the problem of makespan minimization on unrelated parallel machines with sequence dependent setup times. The symbiotic organisms search (SOS) algorithm is a new and popular global optimization technique that has received wide acceptance in recent years from researchers in continuous and discrete optimization domains. An improved SOS algorithm is developed to solve the parallel machine scheduling problem. Since the standard SOS algorithm was originally developed to solve continuous optimization problems, a new solution representation and decoding procedure is designed to make the SOS algorithm suitable for the unrelated parallel machine scheduling problem (UPMSP). Similarly, to enhance the solution quality of the SOS algorithm, an iterated local search strategy based on combining variable numbers of insertion and swap moves is incorporated into the SOS algorithm. More so, to further improve the SOS optimization speed and performance, the longest processing time first (LPT) rule is used to design a machine assignment heuristic that assigns processing machines to jobs based on the machine dynamic load-balancing mechanism. Subsequently, the machine assignment scheme is incorporated into SOS algorithms and used to solve the UPMSP. The performances of the proposed methods are evaluated by comparing their solutions with other existing techniques from the literature. A number of statistical tests were also conducted to determine the variations in performance for each of the techniques. The experimental results showed that the SOS with LPT (SOS-LPT) heuristic has the best performance compared to other tested method, which is closely followed by SOS algorithm, indicating that the two proposed algorithms’ solution approaches are reasonable and effective for solving large-scale UPMSPs.
Read more30Solving Optimization Problems with Diseconomies of Scale via Decoupling
We present a new framework for solving optimization problems with a diseconomy of scale. In such problems, our goal is to minimize the cost of resources used to perform a certain task. The cost of resources grows superlinearly, as x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sup> , q ≥ 1, with the amount x of resources used. We define a novel linear programming relaxation for such problems, and then show that the integrality gap of the relaxation is A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> , where A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> is the q-th moment of the Poisson random variable with parameter 1. Using our framework, we obtain approximation algorithms for the Minimum Energy Efficient Routing, Minimum Degree Balanced Spanning Tree, Load Balancing on Unrelated Parallel Machines, and Unrelated Parallel Machine Scheduling with Nonlinear Functions of Completion Times problems. Our analysis relies on the decoupling inequality for nonnegative random variables. The inequality states that ||Σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i=1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> || <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> ≤ Cq ||Σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i=1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> || <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> , where Xi are independent nonnegative random variables, Yi are possibly dependent nonnegative random variable, and each Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> has the same distribution as X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> . The inequality was proved by de la Peña in 1990. However, the optimal constant Cq was not known. We show that the optimal constant is C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> = A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/q</sup> .
Read more235Unrelated parallel machine scheduling with setup times using simulated annealing
Unrelated parallel machine scheduling with setup times using simulated annealing
8Unrelated parallel machine scheduling with multi-maintenance activities using CPLEX
In this problem, an unrelated parallel machine with non-preemptive, job sequence, setup times and multi level of maintenance are considered. The multi level of maintenance activities are considered in each machine and the duration is based on the processing time for maintenance. In advance, the maximum number of the maintenance frequencies is assumed on all the machines for the restriction of budget. The aim is to find the frequencies, positions and job sequences in maintenance for the total completion time. This model is motivated by a realistic application for cutting process such as wood and metal cutting. In this problem, we formulate the mixed integer programming for finding an optimal solution. The experimental study and the CPLEX were implemented to evaluate its performance in terms of solution quality.
Read more1Hybrid Metaheuristics for the Unrelated Parallel Machine Scheduling to Minimize Makespan and Maximum Just-in-Time Deviations
This paper studies the unrelated parallel machine scheduling problem with three minimization objectives – makespan, maximum earliness, and maximum tardiness (MET-UPMSP). The last two objectives combined are related to just-in-time (JIT) performance of a solution. Three hybrid algorithms are presented to solve the MET-UPMSP: reactive GRASP with path relinking, dual-archived memetic algorithm (DAMA), and SPEA2. In order to improve the solution quality, min-max matching is included in the decoding scheme for each algorithm. An experiment is conducted to evaluate the performance of the three algorithms, using 100 (jobs) x 3 (machines) and 200 x 5 problem instances with three combinations of two due date factors – tight and range. The numerical results indicate that DAMA performs best and GRASP performs second for most problem instances in three performance metrics: HVR, GD, and Spread. The experimental results also show that incorporating min-max matching into decoding scheme significantly improves the solution quality for the two population-based algorithms. It is worth noting that the solutions produced by DAMA with matching decoding can be used as benchmark to evaluate the performance of other algorithms.
Read more58Hybrid tabu search and a truncated branch-and-bound for the unrelated parallel machine scheduling problem
Hybrid tabu search and a truncated branch-and-bound for the unrelated parallel machine scheduling problem
8Makespan Minimization on Unrelated Parallel Machines with a Few Bags
Let there be a set M of m parallel machines and a set J of n jobs, where each job j takes \(p_{i,j}\) time units on machine \(M_i\). In makespan minimization the goal is to schedule each job non-preemptively on a machine such that the length of the schedule, the makespan, is minimum. We investigate a generalization of makespan minimization on unrelated parallel machines (\(R||C_{max}\)) where J is partitioned into b bags \(B=(B_1,\dots , B_b)\), and no two jobs belonging to the same bag can be scheduled on the same machine. First we present a simple b-approximation algorithm for \(R||C_{max}\) with bags (\(R|bag|C_{max}\)). Two machines \(M_i\) and \(M_{i'}\) have the same machine type if \(p_{i,j}=p_{i',j}\) for all \(j \in J\). We give a polynomial-time approximation scheme (PTAS) for \(R|bag|C_{max}\) with machine types where both the number of machine types and bags are constant. This result infers the existence of a PTAS for uniform parallel machines when the number of machine speeds and number of bags are both constant. Then, we present a b/2-approximation algorithm for the graph balancing problem with \(b \ge 2\) bags; the approximation ratio is tight for \(b=3\) unless Open image in new window and this algorithm solves the graph balancing problem with \(b=2\) bags in polynomial time. In addition, we present a polynomial-time algorithm for the restricted assignment problem on uniform parallel machines when all the jobs have unit length. To complement our algorithmic results, we show that when the jobs have lengths 1 or 2 it is Open image in new window -hard to approximate the makespan with approximation ratio less than 3/2 for both the restricted assignment and graph balancing problems with \(b=2\) bags and \(b=3\) bags, respectively. We also prove that makespan minimization on uniform parallel machines with \(b=2\) bags is strongly Open image in new window -hard.
Read moreA scheduling in flexible flow shop problem with unrelated parallel machines and sequence-dependent setup times
To study a scheduling problem of flexible flow shop, where at least one production stage is made up of unrelated parallel machines, and setup times are sequence- and machine-dependent. The objective is to find a schedule that minimizes the makespan and the number of tardy jobs in a static scheduling environment. For this problem, a 0-1 mixed integer programming is formulated. The model is, however, a combinatorial optimization problem which is too difficult to be solved for large-sized problems, and hence, a heuristic is developed to obtain good solutions in reasonable time. The heuristic has three phases. The first phase uses a constructive algorithm. It starts with the generation of the operating time representative for each operation. Then, it uses a dispatching rule or a simple flow shop makespan heuristic to determine an initial solution. The improvement algorithm based on shift moves or pairwise interchanges of jobs is applied to improve the solution in the second phase. After that, a metaheuristic is used to refine the solution in the final phase. Several well-known heuristics are tested in each phase. The performances of the heuristics are compared to one another based on a set of test problems with up to 50 jobs and 20 stages and with an optimal solution for small-sized problems. The computational results indicate that the Nawaz, Enscore, and Ham (NEH) algorithm is most suitable for determining the initial solution, the all-pairwise-interchange approach is good for improving the solution, and the simulated annealing algorithm is best metaheuristic for refining the solution.
Read more1Local Search Heuristic for Multiple Objective Coil Scheduling Problem on Unrelated Parallel Machines
This paper researches coil scheduling problem on unrelated parallel continuous annealing production lines, in which two objectives are considered, one is to minimize total tardiness, the other is to minimize the total completion time. To solve this problem, local search method based on ideal point methods is proposed. Firstly, the two objectives are taken into account, respectively. With consideration of the first objective, a linear programming formulation is set up, and then the ideal point can be obtained by standard LP solver. The other objective is dealt by heuristic. At last, a local search heuristic is proposed to find solutions closing to ideal points. Computational results indicate that the proposed algorithm can find satisfactory solutions in reasonable time.
Read more5Algorithms for hierarchical and semi-partitioned parallel scheduling
Algorithms for hierarchical and semi-partitioned parallel scheduling
13Scheduling in a two-stage flowshop with parallel unrelated machines at each stage and shared resources
Scheduling in a two-stage flowshop with parallel unrelated machines at each stage and shared resources
Peer Review #2 of "A mathematical formulation and an NSGA-II algorithm for minimizing the makespan and energy cost under time-of-use electricity price in an unrelated parallel machine scheduling (v0.1)"
View the review history for A mathematical formulation and an NSGA-II algorithm for minimizing the makespan and energy cost under time-of-use electricity price in an unrelated parallel machine scheduling
Read moreAn MIP-based heuristic approach to determine production lot size for capacitated single-stage production processes with stochastic demand on parallel machines
This paper deals with the capacitated single-stage production lot sizing and scheduling problem with multiple items, setup time, stochastic demand and unrelated parallel machines. A stochastic mixed-integer linear programming model is proposed to formulate the problem. Based on the uncertain constraints, the chance constrained programming approach is used to transform them into equivalent deterministic constraints and then obtain an optimal solution by deterministic mixed-integer linear programming model. Due to the complexity of problem, the mixed-integer programming (MIP) of the equivalence deterministic model of the capacitated single-stage production lot sizing problem with multiple items, setup time, stochastic demand and unrelated parallel machines is reformulated as a shortest path reformulation problem. The proposed algorithm is evaluated through a numerical example. Computational results show that the proposed method gives optimal or near optimal solution and have good-quality result for the test problem.
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