A branch-and-price algorithm for scheduling parallel machines with sequence dependent setup times

We consider the problem of scheduling n independent jobs on m unrelated parallel machines with sequence-dependent setup times and availability dates for the machines and release dates for the jobs to minimize a regular additive cost function. In this work, we develop a new branch-and-price optimization algorithm for the solution of this general class of parallel machines scheduling problems. A new column generation accelerating method, termed “primal box”, and a specific branching variable selection rule that significantly reduces the number of explored nodes are proposed. The computational results show that the approach solves problems of large size to optimality within reasonable computational time.     

An approximation algorithm for scheduling two parallel machines with capacity constraints

We consider the problem of scheduling n independent jobs on two identical parallel machines, with a limit on the number of jobs that can be assigned to each single machine, so as to minimize the total weighted completion time of the jobs. We study a semidefinite programming-based approximation algorithm for solving this problem and prove that the algorithm has a worst case ratio at most 1.1626.     

An improved on-line algorithm for scheduling on two unrestrictive parallel batch processing machines

We consider the problem of on-line scheduling a set of n jobs on two parallel batch processing machines. The objective is to minimize the makespan. We provide an algorithm for the problem that is better than one given in the literature, improving the competitive ratio from to .     

Scheduling open shops with parallel machines

The parallel shop and the open shop are two machine environments that have received much attention in the literature of scheduling theory. A common generalization—the open shop with parallel machines—is considered in this paper. Polynomial-time algorithms are presented for obtaining minimum-length preemptive schedules for three cases. Open shops with single-operation machines of equal speed are scheduled with essentially no more difficulty than an ordinary open shop. Open shops with multiple-operation machines of equal speed are scheduled with the aid of a sequence of network flow computations. The general open shop problem with parallel machines of arbitrary speeds can be solved by linear programming, in much the same way as an optimal preemptive schedule can be found for unrelated parallel machines.     

A parallel algorithm for generalized networks

This paper presents an application of parallel computing techniques to the solution of an important class of planning problems known as generalized networks. Three parallel primal simplex variants for solving generalized network problems are presented. Data structures used in a sequential generalized network code are briefly discussed and their extension to a parallel implementation of one of the primal simplex variants is given. Computational testing of the sequential and parallel codes, both written in Fortran, was done on the CRYSTAL multicomputer at the University of Wisconsin, and the computational results are presented. Maximum efficiency occurred for multiperiod generalized network problems where a speedup approximately linear in the number of processors was achieved.This research was supported in part by NSF grants DCR-8503148 and CCR-8709952 and by AFOSR grant AFOSR-86-0194.     

An improved delayed-start LPT algorithm for a partition problem on two identical parallel machines

We propose an off-line delayed-start LPT algorithm that sequences the first (longest) 5 jobs optimally and the remaining jobs according to the LPT principle on two identical parallel machines. We show that this algorithm has a sharper tight worst-case ratio bound than the traditional LPT algorithm for the sum of squares of machine completion times minimization problem.     

Scheduling parallel machines with single vehicle delivery

We investigate the integrated production and distribution scheduling problem in a supply chain. The manufacturer’s production environment is modeled as a parallel machine system. A single capacitated vehicle is employed to deliver products in batches to multiple customers. The scheduling problem can also be viewed as either parallel machines with delivery considerations or a flexible flowshop. Different inventory holding costs, job sizes (volume or storage space required in the transportation unit), and job priorities are taken into account. Efficient mathematical modeling and near-optimal heuristic approaches are presented for minimizing total weighted completion time.     

A convergent decomposition algorithm for support vector machines

In this work we consider nonlinear minimization problems with a single linear equality constraint and box constraints. In particular we are interested in solving problems where the number of variables is so huge that traditional optimization methods cannot be directly applied. Many interesting real world problems lead to the solution of large scale constrained problems with this structure. For example, the special subclass of problems with convex quadratic objective function plays a fundamental role in the training of Support Vector Machine, which is a technique for machine learning problems. For this particular subclass of convex quadratic problem, some convergent decomposition methods, based on the solution of a sequence of smaller subproblems, have been proposed. In this paper we define a new globally convergent decomposition algorithm that differs from the previous methods in the rule for the choice of the subproblem variables and in the presence of a proximal point modification in the objective function of the subproblems. In particular, the new rule for sequentially selecting the subproblems appears to be suited to tackle large scale problems, while the introduction of the proximal point term allows us to ensure the global convergence of the algorithm for the general case of nonconvex objective function. Furthermore, we report some preliminary numerical results on support vector classification problems with up to 100 thousands variables.     

Makespan minimization for parallel machines scheduling with multiple availability constraints

The problem of makespan minimization for parallel machines scheduling with multiple planned nonavailability periods in the case of resumable jobs is considered. In the current state of the literature, there is a limited number of models and algorithms dealing with this problem and only for very small problem size, and nonavailability limited to some machines. The problem is first formulated as a mixed integer linear programming model and optimally solved using CPLEX for small to moderately large size problems with multiple availability constraints on all machines. An implicit enumeration algorithm using the lexicographic order is then designed to solve large-scale problems. Numerical results are obtained for several experiments and they show the validity and performance improvements procured by both the MILP model and the new enumeration algorithm.     

Performance of scheduling algorithms for no-wait flowshops with parallel machines

We study the performance of scheduling algorithms for a manufacturing system, called the ‘no-wait flowshop’, which consists of a certain number of machine centers. Each center has one or more identical parallel machines. Each job is processed by at most one machine in each center. The problem of finding the minimum finish time schedule is considered here in a flowshop consisting of two machine centers. Heuristic algorithms are presented and are analyzed in the worst case performance context. For the case of two centers, one with a single machine and the other with m, two heuristics are presented with tight performance guarantees of 3 − (1/m) and 2. When both centers have m machines, a heuristic is presented with an upper bound performance guarantee of . It is also shown that this bound can be reduced to 2(1 + ε). For the flowshop with any number of machines in each center, we provide a heuristic algorithm with an upper bound performance guarantee that depends on the relative number of machines in the centers.     

Makespan minimization for two parallel machines with an availability constraint

In this paper, we consider a two parallel machine problem where one machine is not available during a time period. The unavailable time period is fixed and known in advance. A machine is not available probably because it needs preventive maintenance or periodical repair. The objective of the problem is to minimize the makespan. For both nonresumable and resumable cases, we partition the problem into four sub-problems, each of which is solved optimally by an algorithm. Although all the algorithms have exponential time complexities, they are quite efficient in solving large-sized problems.     

A parallel selection algorithm

The problem of selecting thekth largest or smallest element of {x _i +y _j |x _i X andy _j Y i,j} whereX=(x ₁,x ₂, ..,x _n ) andY=(y ₁,y ₂, ...,y _n ) are two arrays ofn elements each, is considered. Certain improvements to an existing algorithm are proposed. An algorithm requiringO(logk·logn) units of time on a Shared Memory Model of a parallel computer havingO(n ^1+1/) processors is presented where is a pre-assigned constant lying between 1 and 2.     

Approximation algorithms for scheduling unrelated parallel machines

We consider the following scheduling problem. There arem parallel machines andn independent jobs. Each job is to be assigned to one of the machines. The processing of jobj on machinei requires timep _ij . The objective is to find a schedule that minimizes the makespan.Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints.In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unlessP = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.A preliminary version of this paper appeared in theProceedings of the 28th Annual IEEE Symposium on the Foundations of Computer Science (Computer Society Press of the IEEE, Washington, D.C., 1987) pp. 217–224.     

Multiple-objective heuristics for scheduling unrelated parallel machines

This research proposes two heuristics and a Genetic Algorithm (GA) to find non-dominated solutions to multiple-objective unrelated parallel machine scheduling problems. Three criteria are of interest, namely: makespan, total weighted completion time, and total weighted tardiness. Each heuristic seeks to simultaneously minimize a pair of these criteria; the GA seeks to simultaneously minimize all three. The computational results show that the proposed heuristics are computationally efficient and provide solutions of reasonable quality. The proposed GA outperforms other algorithms in terms of the number of non-dominated solutions and the quality of its solutions.     

A comparison of branch-and-bound algorithms for a family scheduling problem with identical parallel machines

This paper investigates branch-and-bound algorithms for the problem of scheduling jobs with family setups on identical parallel machines to minimize the weighted sum of completion times. In particular, we propose a new branching scheme that appears to substantially outperform current procedures in terms of computation time and search tree size.     

A parallel descent algorithm for convex programming

In this paper, we propose a parallel decomposition algorithm for solving a class of convex optimization problems, which is broad enough to contain ordinary convex programming problems with a strongly convex objective function. The algorithm is a variant of the trust region method applied to the Fenchel dual of the given problem. We prove global convergence of the algorithm and report some computational experience with the proposed algorithm on the Connection Machine Model CM-5.     

A Lagrangian-based heuristic for the capacitated lot-sizing problem in parallel machines

This paper addresses the capacitated lot-sizing problem involving the production of multiple items on unrelated parallel machines. A production plan should be determined in order to meet the forecast demand for the items, without exceeding the capacity of the machines and minimize the sum of production, setup and inventory costs. A heuristic based on the Lagrangian relaxation of the capacity constraints and subgradient optimization is proposed. Initially, the heuristic is tested on instances of the single machine problem and results are compared with heuristics from the literature. For parallel machines and small problems the heuristic performance is tested against optimal solutions, and for larger problems it is compared with the lower bound provided by the Lagrangian relaxation.     

Flexible flow shop scheduling with uniform parallel machines

We consider the multistage flexible flow shop scheduling problem with uniform parallel machines in each stage and the objective of minimizing makespan. We develop a general class of heuristics for this strongly NP-hard problem that extend several well-known heuristics for the corresponding embedded serial flow shop problem, and obtain absolute performance guarantees for heuristics in this class by building on similar absolute performance guarantees for the corresponding serial flow shop heuristics. Our approach is quite robust, since it can extend any heuristic for the serial flow shop problem (with an absolute performance guarantee) to a similar one for the flexible flow shop problem with uniform parallel machines.     

An improved approximation scheme for the Johnson problem with parallel machines

An extension of the NP-hard Johnson problem to the case of identical parallel machines at each stage is considered. When the number of stages is bounded by a constant and the number of machines is part of the input, a new polynomial time approximation scheme with an improved bound on the running time is designed.