A new algorithm for minimizing makespan, <Emphasis Type="Italic">C</Emphasis><Subscript>max</Subscript>, in blocking flow-shop problem through slowing down the operations

In this paper, a new algorithm with complexity O(nm²) is presented, which finds the optimal makespan, C_max, for a blocking flow-shop problem by slowing down the operations of a no-wait flow-shop problem, F _m ∣no-wait∣C_max, for a given sequence where restriction on the slowing down is committed. However, the problem with performance measure makespan, C_max, in a non-cyclic environment, is a special case of cyclic problem with cycle time, C _t , as its performance measure. This new algorithm is much faster than the previously developed algorithms for cyclical scheduling problems.     

Makespan minimization in single-machine scheduling with step-deterioration of processing times

This paper considers a single-machine scheduling problem of minimizing the maximum completion time for a set of independent jobs. The processing time of a job is a non-linear step function of its starting time and due date. The problem is already known to be ????-hard in the literature. In this paper, we first show this problem to be ????-hard in the ordinary sense by proposing a pseudo-polynomial time dynamic programming algorithm. Then, we develop two dominance rules and a lower bound to design a branch-and-bound algorithm for deriving optimal solutions. Numerical results indicate that the proposed properties can effectively reduce the time required for exploring the solution space.     

Two-stage flexible flow shop scheduling subject to fixed job sequences

This paper investigates the scheduling problem in a two-stage flexible flow shop, which consists of m stage-1 parallel dedicated machines and a stage-2 bottleneck machine, subject to the condition that n _l jobs per type l∈{1, …, m} are processed in a fixed sequence. Four regular performance metrics, including the total completion time, the maximum lateness, the total tardiness, and the number of tardy jobs, are considered. For each considered objective function, we aim to determine an optimal interleaving processing sequence of all jobs coupled with their starting times on the stage-2 bottleneck machine. The problem under study is proved to be strongly NP-hard. An O(m²Π_l=1 ^m n _l ²) dynamic programming algorithm coupled with numerical experiments is presented.     

Balancing Workloads and Minimizing Set-up Costs in the Parallel Processing Shop

This paper presents an optimal scheduling algorithm for minimizing set-up costs in the parallel processing shop while meeting workload balancing restrictions.There are M independent batch type jobs which have sequence dependent set-up costs and N parallel processing machines. Each of the M jobs must be processed on exactly one of the N available machines. It is desirable to minimize total changeover costs with the restriction that each machine workload assignment T _n be within P units of the average machine assignment. The paper describes a static problem in which all jobs are available at time zero. The sequence dependent change over costs are identical for each machine. An extension of the algorithm handles nonidentical processor problems.A combinatorial programming approach to the problem is used. For the special case of identical processors, the problem can be treated as a multi-salesman travelling salesman problem. A general branch and bound algorithm and numerical results are given.     

Minimizing the number of early jobs on a proportionate flowshop

A proportionate flowshop is a special case of the classical flowshop, where the job processing times are machine-independent. We study the problem of minimizing the number of early jobs in this machine setting. This objective function has hardly been investigated on a single machine, and never on a flowshop. We introduce an efficient iterative solution algorithm. In each iteration, a single job is moved to the first position (and is added to the set of early jobs), and the remaining jobs are rescheduled such that the maximum earliness is minimized. The algorithm guarantees an optimal solution in O(n³) time, where n is the number of jobs.     

Erratum: Scheduling jobs with position-dependent processing times

Bachman and Janiak provided a sketch of the proof that the problem 1∣r _i ,p _i (v)=a _i /v∣C_max is NP-hard in the strong sense. However, they did not show how to avoid using harmonic numbers whose encoding is not pseudo-polynomial, which makes the proof incomplete. In this corrigendum, we provide a new complete proof.     

Solution of the NP-hard total tardiness minimization problem in scheduling theory

The classical NP-hard (in the ordinary sense) problem of scheduling jobs in order to minimize the total tardiness for a single machine 1‖ΣT _j is considered. An NP-hard instance of the problem is completely analyzed. A procedure for partitioning the initial set of jobs into subsets is proposed. Algorithms are constructed for finding an optimal schedule depending on the number of subsets. The complexity of the algorithms is O(n ²Σp _j ), where n is the number of jobs and p _j is the processing time of the jth job (j = 1, 2, …, n).     

Designs associated with maximum independent sets of a graph

A (v, β _o , μ)-design over regular graph G = (V, E) of degree d is an ordered pair D = (V, B), where |V| = v and B is the set of maximum independent sets of G called blocks such that if i, j ∈ V, i ≠ j and if i and j are not adjacent in G then there are exactly μ blocks containing i and j. In this paper, we study (v, β _o , μ)-designs over the graphs K _n × K _n , T(n)-triangular graphs, L ₂(n)-square lattice graphs, Petersen graph, Shrikhande graph, Clebsch graph and the Schläfli graph and non-existence of (v, β _o , μ)-designs over the three Chang graphs T ₁(8), T ₂(8) and T ₃(8).     

Two-stage assembly-type flowshop batch scheduling problem subject to a fixed job sequence

This paper discusses a two-stage assembly-type flowshop scheduling problem with batching considerations subject to a fixed job sequence. The two-stage assembly flowshop consists of m stage-1 parallel dedicated machines and a stage-2 assembly machine which processes the jobs in batches. Four regular performance metrics, namely, the total completion time, maximum lateness, total tardiness, and number of tardy jobs, are considered. The goal is to obtain an optimal batching decision for the predetermined job sequence at stage 2. This study presents a two-phase algorithm, which is developed by coupling a problem-transformation procedure with a dynamic program. The running time of the proposed algorithm is O(mn+n⁵), where n is the number of jobs.     

An algorithm to compute <Emphasis Type="Italic">ω</Emphasis>-primality in a numerical monoid

Let M be a commutative, cancellative, atomic monoid and x a nonunit in M. We define ω(x)=n if n is the smallest positive integer with the property that whenever x∣a ₁???a _t , where each a _i is an atom, there is a T?{1,2,…,t} with |T|≤n such that x∣∏_k∈T a _k . The ω-function measures how far x is from being prime in M. In this paper, we give an algorithm for computing ω(x) in any numerical monoid. Simple formulas for ω(x) are given for numerical monoids of the form 〈n,n+1,…,2n?1〉, where n≥3, and 〈n,n+1,…,2n?2〉, where n≥4. The paper then focuses on the special case of 2-generator numerical monoids. We give a formula for computing ω(x) in this case and also necessary and sufficient conditions for determining when x is an atom. Finally, we analyze the asymptotic behavior of ω(x) by computing \(\lim_{x\rightarrow \infty}\frac{\omega(x)}{x}\).     

Minimizing total tardiness in an unrelated parallel-machine scheduling problem

We consider a problem of scheduling n independent jobs on m unrelated parallel machines with the objective of minimizing total tardiness. Processing times of a job on different machines may be different on unrelated parallel-machine scheduling problems. We develop several dominance properties and lower bounds for the problem, and suggest a branch and bound algorithm using them. Results of computational experiments show that the suggested algorithm gives optimal solutions for problems with up to five machines and 20 jobs in a reasonable amount of CPU time.     

On the asymptotic optimality of a solution of the euclidean problem of covering a graph by <Emphasis Type="Italic">m</Emphasis> nonadjacent cycles of maximum total weight

In the problem of covering an n-vertex graph by m cycles of maximum total weight, it is required to find a family of m vertex-nonadjacent cycles such that it covers all vertices of the graph and the total weight of edges in the cover is maximum. The paper presents an algorithm for approximately solving the problem of covering a graph in Euclidean d-space R^d by m nonadjacent cycles of maximum total weight. The algorithm has time complexity O(n³). An estimate of the accuracy of the algorithm depending on the parameters d, m, and n is substantiated; it is shown that if the dimension d of the space is fixed and the number of covering cycles is m = o(n), then the algorithm is asymptotically exact.     

An algorithmic approach to construct crystallizations of 3-manifolds from presentations of fundamental groups

We have defined the weight of the pair (〈S∣R〉,R) for a given presentation 〈S∣R〉 of a group, where the number of generators is equal to the number of relations. We present an algorithm to construct crystallizations of 3-manifolds whose fundamental group has a presentation with two generators and two relations. If the weight of (〈S∣R〉,R) is n, then our algorithm constructs all the n-vertex crystallizations which yield (〈S∣R〉,R). As an application, we have constructed some new crystallizations of 3-manifolds. We have generalized our algorithm for presentations with three generators and a certain class of relations. For m≥3 and m≥n≥k≥2, our generalized algorithm gives a \(2(2m+2n+2k-6+{\delta _{n}^{2}} + {\delta _{k}^{2}})\)-vertex crystallization of the closed connected orientable 3-manifold M〈m,n,k〉 having fundamental group \(\langle x_{1},x_{2},x_{3} \mid {x_{1}^{m}}={x_{2}^{n}}={x_{3}^{k}}=x_{1}x_{2}x_{3} \rangle \). These crystallizations are minimal and unique with respect to the given presentations. If ‘ n=2’ or ‘ k≥3 and m≥4’ then our crystallization of M〈m,n,k〉 is vertex-minimal for all the known cases.     

Asymptotics for the partial sum and its maximum of dependent random variables<Superscript>*</Superscript>

Let X ₁,…,X _n be pairwise asymptotically independent or pairwise upper extended negatively dependent real-valued random variables. Under the condition that the distribution of the maximum of X ₁,…,X _n belongs to some subclass of heavy-tailed distributions, we investigate the asymptotic behavior of the partial sum and its maximum generated by dependent X ₁,…,X _n . As an application, we consider a discrete-time risk model with insurance and financial risks and derive the asymptotics for the finite-time ruin probability.     

Neighbor sum distinguishing colorings of graphs with maximum average degree less than $$\tfrac{{37}} {{12}}$$

Let G be a graph and let its maximum degree and maximum average degree be denoted by Δ(G) and mad(G), respectively. A neighbor sum distinguishing k-edge colorings of graph G is a proper k-edge coloring of graph G such that, for any edge uv ∈ E(G), the sum of colors assigned on incident edges of u is different from the sum of colors assigned on incident edges of v. The smallest value of k in such a coloring of G is denoted by χ′_∑(G). Flandrin et al. proposed the following conjecture that χ′_∑ (G) ≤ Δ(G) + 2 for any connected graph with at least 3 vertices and G ≠ C₅. In this paper, we prove that the conjecture holds for a normal graph with mad(G) < \(\tfrac{{37}}{{12}}\) and Δ(G) ≥ 7.     

An approximation algorithm for finding a <Emphasis Type="Italic">d</Emphasis>-regular spanning connected subgraph of maximum weight in a complete graph with random weights of edges

An approximation algorithm is suggested for the problem of finding a d-regular spanning connected subgraph of maximum weight in a complete undirected weighted n-vertex graph. Probabilistic analysis of the algorithm is carried out for the problem with random input data (some weights of edges) in the case of a uniform distribution of the weights of edges and in the case of a minorized type distribution. It is shown that the algorithm finds an asymptotically optimal solution with time complexity O(n ²) when d = o(n). For the minimization version of the problem, an additional restriction on the dispersion of weights of the graph edges is added to the condition of the asymptotical optimality of the modified algorithm.     

Hierarchical optimization with double due dates on an unbounded parallel-batching machine to minimize maximum lateness

This paper studies a hierarchical optimization problem on an unbounded parallel-batching machine, in which two objective functions are maximum lateness induced by two sets of due dates, representing different purposes of two decision-makers. By a hierarchical optimization problem, we mean the problem of optimizing the secondary criterion under the constraint that the primary criterion is optimized. A parallel-batching machine is a machine that can handle several jobs in a batch in which all jobs start and complete respectively at the same time. We present an \(O(n\log P)\)-time algorithm and an \(O(n^3)\)-time algorithm for this hierarchical scheduling problem, where P is the total processing time of all jobs.     

On the maximum rank of a real binary form

We show that a real binary form f of degree n has n distinct real roots if and only if for any \({(\alpha,\beta)\in\mathbb{R}^2{\setminus}\{0\}}\) all the forms αf _x + βf _y have n ? 1 distinct real roots. This answers to a question of Comon and Ottaviani (On the typical rank of real binary forms, available at arXiv:math/0909.4865, 2009), and allows to complete their argument to show that f has symmetric rank n if and only if it has n distinct real roots.     

Expansions for the distribution and the maximum from distributions with an asymptotically gamma tail when a trend is present

We give expansions about the Gumbel distribution in inverse powers of n and log n for Mn, the maximum of a sample size n or n + 1 when the j-th observation is μ(j/n) + ej, μ is any smooth trend function and the residuals {ej } are independent and identically distributed with P(e r) ≈ exp(-δx)xd0∑∞k=1ckx-kβ as x →∞. We illustrate practical value of the expansions using simulated data sets.     

Multi-period empty container repositioning with stochastic demand and lost sales

This paper considers repositioning empty containers between multi-ports over multi-periods with stochastic demand and lost sales. The objective is to minimize the total operating cost including container-holding cost, stockout cost, importing cost and exporting cost. First, we formulate the single-port case as an inventory problem over a finite horizon with stochastic import and export of empty containers. The optimal policy for period n is characterized by a pair of critical points (A _n , S _n ), that is, importing empty containers up to A _n when the number of empty containers in the port is fewer than A _n ; exporting empty containers down to S _n when the number of empty containers in the port is more than S _n ; and doing nothing, otherwise. A polynomial-time algorithm is developed to determine the two thresholds, that is, A _n and S _n , for each period. Next, we formulate the multi-port problem and determine a tight lower bound on the cost function. On the basis of the two-threshold optimal policy for a single port, a polynomial-time algorithm is developed to find an approximate repositioning policy for multi-ports. Simulation results show that the proposed approximate repositioning algorithm performs very effectively and efficiently.