An estimation of distribution algorithm for lot-streaming flow shop problems with setup times

Lot-streaming flow shops have important applications in different industries including textile, plastic, chemical, semiconductor and many others. This paper considers an n-job m-machine lot-streaming flow shop scheduling problem with sequence-dependent setup times under both the idling and no-idling production cases. The objective is to minimize the maximum completion time or makespan. To solve this important practical problem, a novel estimation of distribution algorithm (EDA) is proposed with a job permutation based representation. In the proposed EDA, an efficient initialization scheme based on the NEH heuristic is presented to construct an initial population with a certain level of quality and diversity. An estimation of a probabilistic model is constructed to direct the algorithm search towards good solutions by taking into account both job permutation and similar blocks of jobs. A simple but effective local search is added to enhance the intensification capability. A diversity controlling mechanism is applied to maintain the diversity of the population. In addition, a speed-up method is presented to reduce the computational effort needed for the local search technique and the NEH-based heuristics. A comparative evaluation is carried out with the best performing algorithms from the literature. The results show that the proposed EDA is very effective in comparison after comprehensive computational and statistical analyses.     

The maximum flow problem with disjunctive constraints

We study the maximum flow problem subject to binary disjunctive constraints in a directed graph: A negative disjunctive constraint states that a certain pair of arcs in a digraph cannot be simultaneously used for sending flow in a feasible solution. In contrast to this, positive disjunctive constraints force that for certain pairs of arcs at least one arc has to carry flow in a feasible solution. It is convenient to represent the negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the arcs of the underlying graph, and whose edges encode the constraints. Analogously we represent the positive disjunctive constraints by a so-called forcing graph. For conflict graphs we prove that the maximum flow problem is strongly $\mathcal{NP}$ -hard, even if the conflict graph consists only of unconnected edges. This result still holds if the network consists only of disjoint paths of length three. For forcing graphs we also provide a sharp line between polynomially solvable and strongly $\mathcal{NP}$ -hard instances for the case where the flow values are required to be integral. Moreover, our hardness results imply that no polynomial time approximation algorithm can exist for both problems. In contrast to this we show that the maximum flow problem with a forcing graph can be solved efficiently if fractional flow values are allowed.     

On the NEH heuristic for minimizing the makespan in permutation flow shops

Over the last 20 years the NEH heuristic of Nawaz, Enscore, and Ham published in this journal has been commonly regarded as the best heuristic for minimizing the makespan in permutation flow shops. In recent years some authors claimed to develop new heuristics that are competitive or outperform NEH. Our study reveals that these claims are not justified. We also address the issue of a fair comparison of the NEH results with those obtained by metaheuristics. Finally we conduct a thorough analysis of NEH leading to its modification which secures the optimality in the two-machine case and improves the general performance.     

The permutation flow shop problem with just in time production considerations

This research presents a variation to the permutation flow shop problem where Just In Time (JIT) production requirements are taken into account. The model developed in this research employs dual objectives. In addition to the traditional objective of minimizing the production makespan, minimization of Miltenburg's material usage rate is also incorporated. In this model, multiple units of any product are permitted in the production sequence. However, the minimization of material usage rates attempts to prevent batch scheduling of products and allows unit flow of products as required in demand flow manufacturing. A solution method is proposed for determining an optimal production sequence via an efficient frontier approach and Simulated Annealing (SA). Test problems and specific performance criteria are used to assess the solutions generated by the proposed method. Experimental results presented in this paper show that the use of the efficient frontier and SA provide solutions that approach the optimal solution for the performance measures used in this research.     

Two-stage proportionate flexible flow shop to minimize the makespan

We consider a two-stage flexible flow shop problem with a single machine at one stage and m identical machines at the other stage, where the processing times of each job at both stages are identical. The objective is to minimize the makespan. We describe some optimality conditions and show that the problem is NP-hard when m is fixed. Finally, we present an approximation algorithm that has a worst-case performance ratio of $\frac{5}{4}$ for m=2 and $\frac{\sqrt{1+m^{2}}+1+m}{2m}$ for m≥3.     

Flows with unit path capacities and related packing and covering problems

Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest. Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log?m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an $O(\sqrt{m})Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest. Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an O(?m)O(\sqrt{m}) -approximation algorithm. We argue that this performance guarantee is best possible, unless P=NP.     

Multi-objective permutation flow shop scheduling problem: Literature review,classification and current trends

The flow shop scheduling problem is finding a sequence given n jobs with same order at m machines according to certain performance measure(s). The job can be processed on at most one machine; meanwhile one machine can process at most one job. The most common objective for this problem is makespan. However, many real-world scheduling problems are multi-objective by nature. Over the years there have been several approaches used to deal with the multi-objective flow shop scheduling problems (MOFSP). Hence, in this study, we provide a brief literature review of the contributions to MOFSP and identify areas of opportunity for future research.     

Efficient composite heuristics for total flowtime minimization in permutation flow shops

In this paper, permutation flow shops with total flowtime minimization are considered. General flowtime computing (GFC) is presented to accelerate flowtime computation. A newly generated schedule is divided into an unchanged subsequence and a changed part. GFC computes total flowtime of a schedule by inheriting temporal parameters from its parent in the unchanged part and computes only those of the changed part. Iterative methods and LR (developed by Liu J, Reeves, CR. Constructive and composite heuristic solutions to theP∥Σ_Ci$P\parallel \Sigma C_i$ scheduling problem, European Journal of Operational Research 2001; 132:439–52) are evaluated and compared as solution improvement phase and index development phase. Three composite heuristics are proposed in this paper by integrating forward pair-wise exchange-restart (FPE-R) and FPE with an effective iterative method. Computational results show that the proposed three outperform the best existing three composite heuristics in effectiveness and two of them are much faster than the existing ones.     

Capacity inverse minimum cost flow problem

Given a directed graph G=(N,A) with arc capacities u _ij and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector [^(u)]\hat{u} for the arc set A such that a given feasible flow [^(x)]\hat{x} is optimal with respect to the modified capacities. Among all capacity vectors [^(u)]\hat{u} satisfying this condition, we would like to find one with minimum ||[^(u)]-u||\|\hat{u}-u\| value. We consider two distance measures for ||[^(u)]-u||\|\hat{u}-u\| , rectilinear (L ₁) and Chebyshev (L _∞) distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is NP\mathcal{NP} -hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.     

Minimizing makespan in an ordered flow shop with?machine-dependent processing times

We consider a generalization of the proportionate flow shop problem with the makespan objective. Each job has a processing requirement and each machine has a characteristic value. In our case, we assume that the time a job occupies a machine is equal to the processing requirement of the job plus a setup time that is equal to the characteristic value of that machine. In this paper, we consider permutation schedules and show that the problem is solvable in polynomial time when the number of machines is fixed.     

On the Tightness of the Alternating-Cycle Lower Bound for Sorting by Reversals

We give a theoretical answer to a natural question arising from a few years of computational experiments on the problem of sorting a permutation by the minimum number of reversals, which has relevant applications in computational molecular biology. The experiments carried out on the problem showed that the so-called alternating-cycle lower bound is equal to the optimal solution value in almost all cases, and this is the main reason why the state-of-the-art algorithms for the problem are quite effective in practice. Since worst-case analysis cannot give an adequate justification for this observation, we focus our attention on estimating the probability that, for a random permutation of n elements, the above lower bound is not tight. We show that this probability is low even for small n, and asymptotically (1/n⁵), i.e., O(1/n⁵) and (1/n⁵). This gives a satisfactory explanation to empirical observations and shows that the problem of sorting by reversals and its alternating-cycle relaxation are essentially the same problem, with the exception of a small fraction of pathological instances, justifying the use of algorithms which are heavily based on this relaxation. From our analysis we obtain convenient sufficient conditions to test if the alternating-cycle lower bound is tight for a given instance. We also consider the case of signed permutations, for which the analysis is much simpler, and show that the probability that the alternating-cycle lower bound is not tight for a random signed permutation of m elements is asymptotically (1/m²).     

Approximation schemes for two-machine flow shop scheduling with two agents

In this paper we consider two-machine flow shop scheduling with two agents. Two models are investigated. One is the weighted-sum optimization model and the other is the constrained optimization model. For the former, we show that it is weakly NP-hard and propose a fully polynomial time approximation scheme. For the latter, we also show the problem is weakly NP-hard. With violating the constraint a factor of ?? a fully polynomial time approximation scheme is provided.     

A multi-objective particle swarm for a flow shop scheduling problem

Flow shop problems as a typical manufacturing challenge have gained wide attention in academic fields. In this paper, we consider a bi-criteria permutation flow shop scheduling problem, where weighted mean completion time and weighted mean tardiness are to be minimized simultaneously. Since a flow shop scheduling problem has been proved to be NP-hard in strong sense, an effective multi-objective particle swarm (MOPS), exploiting a new concept of the Ideal Point and a new approach to specify the superior particle's position vector in the swarm, is designed and used for finding locally Pareto-optimal frontier of the problem. To prove the efficiency of the proposed algorithm, various test problems are solved and the reliability of the proposed algorithm, based on some comparison metrics, is compared with a distinguished multi-objective genetic algorithm, i.e. SPEA-II. The computational results show that the proposed MOPS performs better than the genetic algorithm, especially for the large-sized problems.     

Energy-efficient no-wait permutation flow shop scheduling by adaptive multi-objective variable neighborhood search

This paper considers an energy-efficient no-wait permutation flow shop scheduling problem to minimize makespan and total energy consumption, simultaneously. The processing speeds of machines can be dynamically adjusted for different jobs. In general, lower processing speeds require less energy consumption but result in longer processing times, while higher speeds take the opposite effect. To reach the Pareto front of the problem, we propose an adaptive multi-objective variable neighborhood search (AM-VNS) algorithm. Specifically, we first design two basic speed adjusting heuristics which can reduce the energy consumption of a given solution without worsening its makespan. Two widely used neighborhood-generating operations, i.e., insertion and swap, are adapted and integrated into the variable neighborhood descent phase. With respect to their executing order, two variable neighborhood descent structures can be designed. We adopt an adaptive mechanism to dynamically determine which structure will be selected to handle the current solution. To further improve the performance of the algorithm, we develop a novel problem-specific shake procedure. We also introduce accelerating techniques to speed up the algorithm. Computational results show that the AM-VNS algorithm outperforms multi-objective evolutionary algorithms NSGA-II and SPEA-II.     

On sorting unsigned permutations by double-cut-and-joins

The problem of sorting unsigned permutations by double-cut-and-joins (SBD) arises when we perform the double-cut-and-join (DCJ) operations on pairs of unichromosomal genomes without the gene strandedness information. In this paper we show it is a NP-hard problem by reduction to an equivalent previously-known problem, called breakpoint graph decomposition (BGD), which calls for a largest collection of edge-disjoint alternating cycles in a breakpoint graph. To obtain a better approximation algorithm for the SBD problem, we made a suitable modification to Lin and Jiang’s algorithm which was initially proposed to approximate the BGD problem, and then carried out a rigorous performance analysis via fractional linear programming. The approximation ratio thus achieved for the SBD problem is $\frac{17}{12}+\epsilon \approx 1.4167 +\epsilon$ , for any positive ε.     

Designing a tabu search algorithm for the two-stage flow shop problem with secondary criterion

This paper discusses the process of desigining a tabu search-based heuristic for the two-stage flow shop problem with makespan minimization as the primary criterion and the minimization of total flow time as the secondary criterion. A factorial experiment is designed to analyse thoroughly the effects of four different factors, i.e. the initial solution, type of move, size of neighbourhood and the list size, on the performance of the tabu search-based heuristic. Using the techniques of evolution curves, and response tables and response graphs, coupled with the Taguchi method, the best combination of the factors for the tabu search-based heuristic is identified, and the effectiveness of the heuristic algorithm in finding an optimal solution is evaluated by comparing its performance with the best known heuristic to solve this problem.     

Minimizing the maximum bump cost in linear extensions of a poset

A linear extension of a poset P=(X,?) is a permutation x ₁,x ₂,…,x _|X| of X such that i<j whenever x _i ?x _j . For a given poset P=(X,?) and a cost function c(x,y) defined on X×X, we want to find a linear extension of P such that maximum cost is as small as possible. For the general case, it is NP-complete. In this paper we consider the linear extension problem with the assumption that c(x,y)=0 whenever x and y are incomparable. First, we prove the discussed problem is polynomially solvable for a special poset. And then, we present a polynomial algorithm to obtain an approximate solution.     

An approximation algorithm for k-center problem on a convex polygon

This paper studies the constrained version of the k-center location problem. Given a convex polygonal region, every point in the region originates a service demand. Our objective is to place k facilities lying on the region’s boundary, such that every point in that region receives service from its closest facility and the maximum service distance is minimized. This problem is equivalent to covering the polygon by k circles with centers on its boundary which have the smallest possible radius. We present an 1.8841-approximation polynomial time algorithm for this problem.     

On single machine scheduling with resource constraint

The single machine scheduling with resource constraint is a nonlinear combinatorial optimization problem in cloud computing applications. Given a set of jobs and certain resource, the problem is to find a permutation of jobs and a distribution of resource to optimize certain objective function. The processing time of each job is a nonlinear function with respect to the resource assigned to it. In this paper, we propose a naive algorithm and study a subproblem in the algorithm that suppose the permutation of jobs is also given, how to find a resource distribution to minimize the total weighted flow time. We found a polynomial-time optimal solution for a special case and an approximation solution in general case.     

Polynomially solvable special cases of the quadratic bottleneck assignment problem

Quadratic bottleneck assignment problems (QBAP) are obtained by replacing the addition of cost terms in the objective function of a quadratic (sum) assignment problem by taking their maximum. Since the QBAP is an NP\mathcal{NP}-hard problem, polynimially solvable special cases of the QBAP are of interest. In this paper we specify conditions on the cost matrices of QBAP leading to special cases which can be solved to optimality in polynomial time. In particular, the following three cases are discussed: (i) any permutation is optimal (constant QBAP), (ii) a certain specified permutation is optimal (constant permutation QBAP) and (iii) the solution can be found algorithmically by a polynomial algorithm. Moreover, the max-cone of bottleneck Monge matrices is investigated, its generating matrices are identified and it is used as a tool in proving polynomiality results.