Forbidden Sequences Research Articles

On the complexity of Nurse Rostering problems

In Nurse Rostering problems the goal is to assign nurses to shifts, subject to a number of constraints. In this paper we consider the computational complexity of such problems. We review the major complexity results obtained so far and show that a problem with coverage constraints, day off requests, and forbidden sequences of shifts of length 2 is NP-hard in the strong sense, even if there are only three work shifts and a day off shift involved.

An efficient GRASP-like algorithm for the multi-product straight pipeline scheduling problem

The present work deals with the multi-product straight pipeline scheduling problem. The considered straight pipeline system is used to transport refined petroleum products from a single source (refinery with storage tanks) to a single destination (distribution center). The main objective is to find a sequence of batches which aims to maximize the total volume to be transported via the pipeline, while meeting the daily customer demands over a fixed time horizon. Each batch contains only one product with a volume between its upper and lower bounds. Constraints related to inventory levels, batch settling periods and forbidden sequences between pairs of products must be respected, and pipeline stoppage periods should also be handled. A Mixed Integer Linear Programming (MILP) model and a Greedy Randomized Adaptive Search Procedure (GRASP)-like algorithm are proposed to tackle the problem under study. The MILP model is based on the discharging time axis with continuous representation for both time and volume formulation. The GRASP-like algorithm is composed of a construction method and an improvement procedure. The construction method generates a new sequence of batches in a random way with the use of a repair process to obtain an initial solution that satisfies all demands. Then, the improvement procedure is attempted to increase the fill rate of pipeline using volume optimization operators without change the sequence of products. A set of instances was generated from a real case study in order to validate the proposed approach. The performance of our approach is benchmarked against the MILP model (solved using Gurobi Solver), and the numerical experiments proved that the proposed approach obtains very competitive results both in term of solution quality and CPU time.

(2004-5799) An interactive fuzzy programming approach for a new multi-objective multi-product oil pipeline scheduling problem

In this paper, a new fuzzy multi-objective multi-product pipeline scheduling problem is introduced. The system consists of a single refinery, a unique distribution center, and a multi-product pipeline. Restrictions such as batchsizing, discharging rate, forbidden sequences, batch volumetric, etc. are considered. Due to the uncertain nature of real-world problems, some parameters of the system are considered as fuzzy values. Tardiness and earliness penalties are considered with a time dependent non-linear function. The basic aim of this scheduling problem is to achieve the optimal sequence for pumping batches of oil products to maximize the financial benefit and simultaneously satisfies the customers with on-time delivery as a multi-objective problem. To tackle this problem, a two-stage methodology is proposed. In the first stage, the fuzzy formulation is converted to its equivalent crisp form by a credibility-based chance-constrained programming approach. In the second stage, the multi-objective crisp formulation is solved by some well-known approaches in the literature. Some test problems are generated and solved by the proposed approaches and the obtained Pareto-optimal solutions are analyzed and compared using some distance-based comparison metrics.

A solution framework for the long-term scheduling and inventory management of straight pipeline systems with multiple-sources

This paper addresses the long-term scheduling of pumping and delivery operations on straight multiproduct pipeline systems connecting multiple-sources to multiple-destinations. A generic solution framework is proposed, combining heuristic procedures and mixed integer linear programming (MILP) models. The pipeline scheduling problem is decomposed into two sub-problems: the allocation and sequencing of pumping operations of the initial source, and the scheduling of delivery operations and also pumping operations of the intermediate sources. An innovative execution procedure using a rolling horizon strategy is developed to solve both problems. We focus on a new discrete-time MILP model formulation, covering several operational aspects of real-world pipeline systems, such as rigorous treatment of forbidden sequences and preventive tank maintenance and pipeline maintenance periods, while considering rigorous inventory management during the scheduling horizon. The approach is validated by solving an illustrative example and two real-world based pipeline problems. Results show that the proposed framework is able to obtain solutions in reasonable run time with detailed inventory management and respecting several operational constraints of the system.

Small-Angle Scattering and Multifractal Analysis of DNA Sequences.

The arrangement of A, C, G and T nucleotides in large DNA sequences of many prokaryotic and eukaryotic cells exhibit long-range correlations with fractal properties. Chaos game representation (CGR) of such DNA sequences, followed by a multifractal analysis, is a useful way to analyze the corresponding scaling properties. This approach provides a powerful visualization method to characterize their spatial inhomogeneity, and allows discrimination between mono- and multifractal distributions. However, in some cases, two different arbitrary point distributions, may generate indistinguishable multifractal spectra. By using a new model based on multiplicative deterministic cascades, here it is shown that small-angle scattering (SAS) formalism can be used to address such issue, and to extract additional structural information. It is shown that the box-counting dimension given by multifractal spectra can be recovered from the scattering exponent of SAS intensity in the fractal region. This approach is illustrated for point distributions of CGR data corresponding to Escherichia coli, Phospholamban and Mouse mitochondrial DNA, and it is shown that for the latter two cases, SAS allows extraction of the fractal iteration number and the scaling factor corresponding to “ACGT” square, or to recover the number of bases. The results are compared with a model based on multiplicative deterministic cascades, and respectively with one which takes into account the existence of forbidden sequences in DNA. This allows a classification of the DNA sequences in terms of random and deterministic fractals structures emerging in CGR.

Constructing sparse Davenport–Schinzel sequences

For any sequence u, the extremal function Ex(u,j,n) is the maximum possible length of a j-sparse sequence with n distinct letters that avoids u. We prove that if u is an alternating sequence abab… of length s, then Ex(u,j,n)=Θ(sn2) for all j≥2 and s≥n, answering a question of Wellman and Pettie (2018) and extending the result of Roselle and Stanton that Ex(u,2,n)=Θ(sn2) for any alternation u of length s≥n (Roselle and Stanton, 1971).Wellman and Pettie also asked how large must s(n) be for there to exist n-block DS(n,s(n)) sequences of length Ω(n2−o(1)). We answer this question by showing that the maximum possible length of an n-block DS(n,s(n)) sequence is Ω(n2−o(1)) if and only if s(n)=Ω(n1−o(1)). We also show related results for extremal functions of forbidden 0–1 matrices with any constant number of rows and extremal functions of forbidden sequences with any constant number of distinct letters.

New batch-centric model for detailed scheduling and inventory management of mesh pipeline networks

The scheduling of a multi-source and multi-destination pipeline with unidirectional flow is a challenging task since the flexibility of the network allows for products to take alternative routes. To tackle this issue efficiently, this paper develops a generic continuous-time batch-centric model that permits the injection, by an input node, and delivery, to an output node, of multiple batches over a time slot. The objective function is to minimize the operation cost related to pumps, contamination, inventory, backorder and capacity utilization, while satisfying forbidden product sequences and inventory limits. The model is validated through the solution of four benchmark problems, featuring a mesh network with growing complexity in topological structure. Compared to the literature, we are now able to generate detailed schedules with lower operating costs, due to a better tradeoff between interface (different product sequences) and inventory cost.

A holistic MILP model for scheduling and inventory management of a multiproduct oil distribution system

This paper addresses the optimal scheduling of an oil transportation system characterized by a straight multiproduct pipeline featuring multiple input and output nodes, where products are dispatched to local markets often by tanker trucks. We present a new continuous-time mixed integer linear programming (MILP) model that is designed based on real-world necessities and that requires significantly fewer binary variables than previous work. As main contributions, the model: i) can rigorously avoid forbidden product sequences in every pipeline segment; ii) considers filler batch constraints to avoid large contamination volumes; and iii) includes inventory management constraints in the different pipeline nodes. We first use an illustrative example before testing the approach with a new large-scale example problem and three real-world cases from the literature. Results show that the proposed model has a tight linear programming (LP) relaxation and is very efficient computationally. It is thus a significant contribution to the state-of-the-art.

Multiproduct pipeline scheduling integrating for inbound and outbound inventory management

This paper addresses the operation of a unidirectional multiproduct pipeline system integrated with inventory tanks management at the refinery and distribution center. A Mixed Integer Linear Programming (MILP) formulation is proposed to model the scheduling of a pipeline operation consisting of a single source node and a single destination node representing a refinery and a distribution center, respectively. Different operational aspects such as product availability, inventory limits, product quality, forbidden sequences, transition between different products injected into the pipeline and tanks inventory management are considered. Time synchronization among refinery, pipeline, and distribution center is guaranteed enhancing the model representation of real operations. Additionally, a solution procedure is developed to improve the model solution performance that explores the reduction of the number of variables and constraints. Different examples based on a real world case are solved showing the model's applicability, which proves to be effective in generating feasible solutions in reasonable computational times.

AN ALTERNATIVE DECOMPOSITION OF CATALAN NUMBER

A particular integer sequence derived by the convex polygon triangulation is introduced and investigated. After some underlying results are presented, the forbidden (or improper) integer values relative to the triangulation are concerned. It is understood that the forbidden sequences do not correspond to any triangulation. Some of their properties are presented. These properties are used to count the forbidden values, which is, finally, exploited in stating another decomposition of the Catalan number.

Rule-Based Scheduling of Single-Stage Multiproduct Batch Plants with Parallel Units

A new rule-based model for single-stage multiproduct scheduling problem (SMSP) in batch plants with parallel units is proposed. The scheduling problem is decomposed into two subproblems of order assignment and order sequencing. A group of order assignment rules are introduced to select suitable units for customer orders. While flexible constraint handling strategies are proposed to eliminate the constraints of forbidden units and unit release times and convert the hard constraint of forbidden sequences to a soft one, leading to any arbitrary order sequence can be synthesized into a legal schedule. The line-up competition algorithm (LUCA) is presented to obtain optimal order sequence and order assignment rule, which can minimize makespan, total tardiness, or total cost. Through comparative study, the proposed approach has demonstrated its effectiveness for solving a large size SMSP.

Comparing branch-and-price algorithms for the Multi-Commodity k-splittable Maximum Flow Problem

The Multi-Commodity k-splittable Maximum Flow Problem consists in routing as much flow as possible through a capacitated network such that each commodity uses at most k paths and the capacities are satisfied. The problem appears in telecommunications, specifically when considering Multi-Protocol Label Switching. The problem has previously been solved to optimality through branch-and-price. In this paper we propose two exact solution methods both based on an alternative decomposition. The two methods differ in their branching strategy. The first method, which branches on forbidden edge sequences, shows some performance difficulty due to large search trees. The second method, which branches on forbidden and forced edge sequences, demonstrates much better performance. The latter also outperforms a leading exact solution method from the literature. Furthermore, a heuristic algorithm is presented. The heuristic is fast and yields good solution values.

Generalized Davenport–Schinzel sequences and their 0–1 matrix counterparts

A generalized Davenport–Schinzel sequence is one over a finite alphabet whose subsequences are not isomorphic to a forbidden subsequence σ. What is the maximum length of such a σ-free sequence, as a function of its alphabet size n? Is the extremal function linear or nonlinear? And what characteristics of σ determine the answers to these questions? It is known that such sequences have length at most n ⋅ 2 ( α ( n ) ) O ( 1 ) , where α is the inverse-Ackermann function and the O ( 1 ) depends on σ. We resolve a number of open problems on the extremal properties of generalized Davenport–Schinzel sequences. Among our results: 1. We give a nearly complete characterization of linear and nonlinear σ ∈ { a , b , c } ⁎ over a three-letter alphabet. Specifically, the only repetition-free minimally nonlinear forbidden sequences are a b a b a and a b c a c b c . 2. We prove there are at least four minimally nonlinear forbidden sequences. 3. We prove that in many cases, doubling a forbidden sequence has no significant effect on its extremal function. For example, Nivaschʼs upper bounds on alternating sequences of the form ( a b ) t and ( a b ) t a , for t ⩾ 3 , can be extended to forbidden sequences of the form ( a a b b ) t and ( a a b b ) t a . 4. Finally, we show that the absence of simple subsequences in σ tells us nothing about σʼs extremal function. For example, for any t, there exists a σ t avoiding a b a b a whose extremal function is Ω ( n ⋅ 2 α t ( n ) ) . Most of our results are obtained by translating questions about generalized Davenport–Schinzel sequences into questions about the density of 0–1 matrices avoiding certain forbidden submatrices. We give new and often tight bounds on the extremal functions of numerous forbidden 0–1 matrices.

A Rigorous Mathematical Formulation for the Scheduling of Tree-Structure Pipeline Networks

Pipeline networks are the dominant mode of conveying a wide variety of oil refined products from refineries to distribution terminals. Pipeline infrastructure usually comprises trunk lines serving high-volume, long-haul transportation requirements, and delivering lines moving smaller volumes over shorter distances. Lots of products are mostly sent through trunk lines to bulk terminals, while some portions are branched to delivering lines and supplied to nearby market areas. This work presents a novel continuous-time mixed-integer linear (MILP) formulation for the short-term operational planning of tree-structure pipeline systems. The problem goal is to find the optimal schedule of pumping and delivery operations to satisfy all terminal requirements at minimum operating cost. To this end, model constraints strictly monitor the branching of batches and the creation of new interfaces in delivering lines to avoid forbidden sequences and determine additional reprocessing costs. By allowing the transfer of multiple products to delivering lines during a batch injection, longer runs can be executed, and fewer ones are needed to find the optimal solution. The new approach has been successfully applied to three examples, one of them involving a real-world pipeline network. As compared to previous contributions, significant improvements in both solution quality and CPU time have been obtained.

Synthesis of controllers for symmetric systems

This article deals with supervisory control problem for coloured Petri (CP) nets. Considering a CP-net, we build a condensed version of the ordinary state-space, namely the symbolic reachability graph (SRG). This latter graph allows to cope with state-space explosion problem for symmetric systems. The control specification can be expressed in terms of either forbidden states or forbidden sequences of transitions. According to these specifications, we derive the controller by applying the theory of regions on the basis of the SRG. Thanks to expressiveness power of CP-nets, the obtained controller to be connected to the plant model is reduced to one single place.

Symbolic dynamics and synchronization of coupled map networks with multiple delays

We use symbolic dynamics to study discrete-time dynamical systems with multiple time delays. We exploit the concept of avoiding sets, which arise from specific non-generating partitions of the phase space and restrict the occurrence of certain symbol sequences related to the characteristics of the dynamics. In particular, we show that the resulting forbidden sequences are closely related to the time delays in the system. We present two applications to coupled map lattices, namely (1) detecting synchronization and (2) determining unknown values of the transmission delays in networks with possibly directed and weighted connections and measurement noise. The method is applicable to multi-dimensional as well as set-valued maps, and to networks with time-varying delays and connection structure.

순차상태전이금지(FSSTP)를 이용한 교착상태 관리제어를 위한 재구성 방법

The object of this paper is to propose a method to deal with the problem of modeling user specifications in approaches based on supervisory control and Petri nets. However, most of Petri Net approaches are based on forbidden states specifications, and these specifications are suitable the use of tool such as the reachability graph. But these methods were not able to show the user specification easily and these formalisms are generally limited by the combinatorial explosion that occurs when attempting to model complex systems. Herein, we propose a new efficient method using FSSTP (Forbidden Sequences of State-Transitions Problem) and theory of region. Also, to detect and avoid the deadlock problem in control process, we use DAPN method (Deadlock Avoidance Petri nets) for solving this problem in control model.

On Not-First/Not-Last conditions in disjunctive scheduling

This paper is concerned with the development of constraint propagation techniques for the characterization of feasible solutions in disjunctive scheduling. In disjunctive scheduling, a set of uninterruptible tasks is to be performed on a set of resources. Each task has a release date, a deadline, and a fixed processing time; each resource can handle only one task at a time. Some of these propagation techniques are implemented by rules that deduce either mandatory or forbidden sequences between tasks or sets of tasks. For instance, certain rules indicate whether a given task must or cannot be performed before or after a set of other competing tasks. We focus our attention on the latter problem, known as the “Not-First/Not-Last” (NF/NL) problem.The genericity of propagation rules is a question of major importance. It induces that the result of the overall propagation must not depend on the order in which the inference rules are applied. Hence, one must search for completeness in the time-windows narrowing, in order to ensure the convergence of the propagation towards a unique fix-point.An efficient algorithm is proposed. It guarantees the completeness of time-windows narrowing due to NF/NL conditions. It has been integrated in a branch and bound procedure to solve job-shop instances. It has also been tested within several lower bounding procedures. Computational results are reported and the power and complementarity of NF/NL rules with other classical inference rules are discussed.

Stack words, standard Young tableaux, permutations with forbidden subsequences and planar maps

Stack words stem from studies on stack-sortable permutations and represent classical combinatorial objects such as standard Young tableaux, permutations with forbidden sequences and planar maps. We extend existing enumerative results on stack words and we also obtain new results. In particular, we make a correspondence between nonseparable 3× n rectangular standard Young tableaux (or stack words where elements satisfy a ‘Towers of Hanoi’ condition) and nonseparable cubic rooted planar maps with 2 n vertices enumerated by 2 n (3 n)!/((2 n+1)!( n+1)!). Moreover, these tableaux without two consecutive integers in the same row are in bijection with nonseparable rooted planar maps with n+1 edges enumerated by 2(3 n)!/((2 n+1)!( n+1)!).

Symbolic Analysis of the Hénon Map at a = 1.4 and b = 0.311The project supported partially by National Natural Science Foundation of China

The symbolic dynamics of the Lozi map is applied to the Hénon map at a = 1.4 and b = 0.3. The fundamental forbidden zone, inside which points correspond to forbidden symbolic sequences, is constructed from a numerically determined partition line. All the allowed periodic orbits up to period 7 are determined, and chaotic orbits are derived.