An alternate linear algorithm for the minimum flow problem

We present an alternate linear algorithm for finding the minimum flow in (s, t)-planar networks using a new concept of minimal removable sets developed here. The iterative nature of the algorithm facilitates the adjustment of solutions for systems in developmental stages. The minimum flow algorithm presented here requires O(|V|) time, where V denotes the set of vertices. The minimum flow problem arises in many transportation and communication systems.     

A Recovering Beam Search algorithm for the one-machine dynamic total completion time scheduling problem

This paper deals with the one-machine dynamic total completion time scheduling problem. This problem is known to be NP-hard in the strong sense. A polynomial time heuristic algorithm is proposed which applies the recently introduced Recovering Beam Search (RBS) approach. The algorithm is based on a beam search procedure with unitary beam width and includes a recovering subroutine that allows to overcome wrong decisions taken at higher levels of the beam search tree. It is shown that the total number of considered nodes is bounded by n where n is the jobsize. The proposed algorithm is able to solve in very short CPU time problems with up to 500 jobs outperforming the best state of the art heuristics.     

A new algorithm for the undesirable 1-center problem on networks

Recent papers have developed efficient algorithms for the location of an undesirable (obnoxious) 1-center on general networks with n nodes and m edges. Even though the theoretical complexity of these algorithms is fine, the computational time required to get the solution can be diminished using a different model formulation and slightly improving the upper bounds. Thus, we present a new O(mn) algorithm, which is more straightforward and computationally faster than the previous ones. Computing time results comparing the former approaches with the proposed algorithm are supplied.     

A hybrid shifting bottleneck procedure algorithm for the parallel-machine job-shop scheduling problem

In practice, parallel-machine job-shop scheduling (PMJSS) is very useful in the development of standard modelling approaches and generic solution techniques for many real-world scheduling problems. In this paper, based on the analysis of structural properties in an extended disjunctive graph model, a hybrid shifting bottleneck procedure (HSBP) algorithm combined with Tabu Search (TS) metaheuristic algorithm is developed to deal with the PMJSS problem. The original-version shifting bottleneck procedure (SBP) algorithm for the job-shop scheduling (JSS) has been significantly improved to solve the PMJSS problem with four novelties: (i) a topological-sequence algorithm is proposed to decompose the PMJSS problem in a set of single-machine scheduling (SMS) and/or parallel-machine scheduling subproblems; (ii) a modified Carlier algorithm based on the proposed lemmas and the proofs is developed to solve the SMS subproblem; (iii) the Jackson rule is extended to solve the PMS subproblem; (iv) a TS metaheuristic algorithm is embedded under the framework of SBP to optimise the JSS and PMJSS cases. The computational experiments show that the proposed HSBP is very efficient in solving the JSS and PMJSS problems.     

A continuation method for solving symmetric Toeplitz systems

A fast algorithm is proposed for solving symmetric Toeplitz systems. This algorithm continuously transforms the identity matrix into the inverse of a given Toeplitz matrix T. The memory requirements for the algorithm are O(n), and its complexity is O(log κ(T)nlogn), where (T) is the condition number of T. Numerical results are presented that confirm the efficiency of the proposed algorithm.     

Stochastic inventory system with lead time flexibility: offered by a manufacturer/transporter

This paper studies lead time flexibility in a two-stage continuous review supply chain in which the retailer uses the (R, Q) inventory system: when his inventory position reaches R, the retailer places orders with size Q to the manufacturer, who uses a transportation provider to deliver them with different lead time options. According to the contract, the manufacturer is able to expedite or postpone the delivery if the retailer makes such a request. Hence, the retailer has the flexibility to modify the lead time by using the most up-to-date demand information. The optimal lead time policy is found to be a threshold-type policy. The sensitivity analysis also shows that R is much more sensitive to the change of lead time than Q, and thus, the paper is primarily focused on finding optimal R. We also provide a cost approximation which yields unimodal cost in R. Furthermore, we analyze the order crossing problem and derive an upper bound for the probability of order crossing. Finally, we conduct an extensive sensitivity analysis to illustrate the effects of lead time flexibility on supply chain performance and discuss the managerial insights.     

Some applications of the generalized vehicle routing problem

The Generalized Vehicle Routing Problem (GVRP) is an extension of the classical Vehicle Routing Problem (VRP) in which the vertex set is partitioned into clusters and vehicles must visit exactly one (or at least one)vertex per cluster. The GVRP provides a useful modelling framework for a wide variety of applications. The purpose of this paper is to provide such examples of applications and models. These include the Travelling Salesman with Profits, several VRP extensions, the Windy Routing Problem,and the design of tandem configurations for automated guided vehicles.     

Unit Interval Orders of Open and Closed Intervals

A poset P = (X, ?) is a unit OC interval order if there exists a representation that assigns an open or closed real interval I(x) of unit length to each x ∈ P so that x ? y in P precisely when each point of I (x) is less than each point in I (y). In this paper we give a forbidden poset characterization of the class of unit OC interval orders and an efficient algorithm for recognizing the class. The algorithm takes a poset P as input and either produces a representation or returns a forbidden poset induced in P.     

Very Hyperbolic Polynomials

A real polynomial in one variable is hyperbolic if it has only real roots. A function f is a primitive of order k of a function g if f ^(k) = g. A hyperbolic polynomial is very hyperbolic if it has hyperbolic primitives of all orders. In the paper, we prove a property of the domain of very hyperbolic polynomials and describe this domain in the case of degree 4.     

Numerical optimization method for packing regular convex polygons

An algorithm is presented for the approximate solution of the problem of packing regular convex polygons in a given closed bounded domain G so as to maximize the total area of the packed figures. On G a grid is constructed whose nodes generate a finite set W on G, and the centers of the figures to be packed can be placed only at some points of W. The problem of packing these figures with centers in W is reduced to a 0-1 linear programming problem. A two-stage algorithm for solving the resulting problems is proposed. The algorithm finds packings of the indicated figures in an arbitrary closed bounded domain on the plane. Numerical results are presented that demonstrate the effectiveness of the method.     

A comparison between the order and the volume fill rate for a base-stock inventory control system under a compound renewal demand process

The order fill rate (OFR) is sometimes suggested as an alternative to the volume fill rate (VFR) (most often just denoted fill rate) as a performance measure for inventory control systems. We consider a continuous review, base-stock policy, where replenishment orders have a constant lead time and unfilled demands are backordered. For this policy, we develop exact mathematical expressions for the two fill-rate measures when demand follows a compound renewal process. We also elaborate on when the OFR can be interpreted as the (extended) ready rate. For the case when customer orders are generated by a negative binomial distribution, we show that it is the size of the shape parameter of this distribution that determines the relative magnitude of the two fill rates. In particular, we show that when customer orders are generated by a geometric distribution, the OFR and the VFR are equal.     

Efficient contraflow algorithms for quickest evacuation planning

The optimization models and algorithms with their implementations on flow over time problems have been an emerging field of research because of largely increasing human-created and natural disasters worldwide. For an optimal use of transportation network to shift affected people and normalize the disastrous situation as quickly and Efficiently as possible, contraflow configuration is one of the highly applicable operations research (OR) models. It increases the outbound road capacities by reversing the direction of arcs towards the safe destinations that not only minimize the congestion and increase the flow but also decrease the evacuation time significantly. In this paper, we sketch the state of quickest flow solutions and solve the quickest contraflow problem with constant transit times on arcs proving that the problem can be solved in strongly polynomial time O(nm²(log n)²), where n and m are number of nodes and number of arcs, respectively in the network. This contraflow solution has the same computational time bound as that of the best min-cost flow solution. Moreover, we also introduce the contraflow approach with load dependent transit times on arcs and present an Efficient algorithm to solve the quickest contraflow problem approximately. Supporting the claim, our computational experiments on Kathmandu road network and on randomly generated instances perform very well matching the theoretical results. For sufficiently large number of evacuees, about double flow can be shifted with the same evacuation time and about half time is sufficient to push the given flow value with contraflow reconfiguration.     

A note on a mixed routing and scheduling problem on a grid graph

We consider a particular case of the Fleet Quickest Routing Problem (FQRP) on a grid graph of m × n nodes that are placed in m levels and n columns. Starting nodes are placed at the first (bottom) level, and nodes of arrival are placed at the mth level. A feasible solution of FQRP consists in n Manhattan paths, one for each vehicle, such that capacity constraints are respected. We establish m*, i.e. the number of levels that ensures the existence of a solution to FQRP in any possible permutation of n destinations. In particular, m* is the minimum number of levels sufficient to solve any instance of FQRP involving n vehicles, when they move in the ways that the literature has until now assumed. Existing algorithms give solutions that require, for some values of n, more levels than m*. For this reason, we provide algorithm CaR, which gives a solution in a graph m* × n, as a minor contribution.     

Directed Partial Orders on Complex Numbers and Quaternions over Non-Archimedean Linearly Ordered Fields

Let F be a non-archimedean linearly ordered field, and C and H be the field of complex numbers and the division algebra of quaternions over F, respectively. In this paper, a class of directed partial orders on C are constructed directly and concretely using additive subgroup of F ⁺. This class of directed partial orders includes those given in Rump and Wang (J. Algebra 400, 1–7, 2014), and Yang (J. Algebra 295(2), 452–457, 2006) as special cases and we conjecture that it covers all directed partial orders on C such that 1 > 0. It turns out that this construction also works very well on H. We note that none of these directed partial orders is a lattice order on C or H.     

Two-machine open shop scheduling with special transportation times

The paper considers a problem of scheduling n jobs in a two-machine open shop to minimise the makespan, provided that preemption is not allowed and the interstage transportation times are involved. In general, this problem is known to be NP-hard. We present a linear time algorithm that finds an optimal schedule if no transportation time exceeds the smallest of the processing times. We also describe an algorithm that creates a heuristic solution to the problem with job-independent transportation times. Our algorithm provides a worst-case performance ratio of 8/5 if the transportation time of a job depends on the assigned processing route. The ratio reduces to 3/2 if all transportation times are equal.     

An adaptive algorithm for efficient computation of level curves of surfaces

A new efficient algorithm for the computation of z?=?constant level curves of surfaces z?=?f(x,y) is proposed and tested on several examples. The set of z-level curves in a given rectangle of the (x,y)-plane is obtained by evaluating f on a first coarse square grid which is then adaptively refined by triangulation to eventually match a desired tolerance. Adaptivity leads to a considerable reduction in terms of evaluations of f with respect to uniform grid computation as in Matlab®’s contour. Furthermore, especially when the evaluation of f is computationally expensive, this reduction notably decreases the computational time. A comparison of performances is shown for two real-life applications such as the determination of stability charts and of ε??pseudospectra for linear time delay systems. The corresponding Matlab code is also discussed.     

Maximum Distance Between Slater Orders and Copeland Orders of Tournaments

Given a tournament T?=?(X, A), we consider two tournament solutions applied to T: Slater’s solution and Copeland’s solution. Slater’s solution consists in determining the linear orders obtained by reversing a minimum number of directed edges of T in order to make T transitive. Copeland’s solution applied to T ranks the vertices of T according to their decreasing out-degrees. The aim of this paper is to compare the results provided by these two methods: to which extent can they lead to different orders? We consider three cases: T is any tournament, T is strongly connected, T has only one Slater order. For each one of these three cases, we specify the maximum of the symmetric difference distance between Slater orders and Copeland orders. More precisely, thanks to a result dealing with arc-disjoint circuits in circular tournaments, we show that this maximum is equal to n(n???1)/2 if T is any tournament on an odd number n of vertices, to (n ²???3n?+?2)/2 if T is any tournament on an even number n of vertices, to n(n???1)/2 if T is strongly connected with an odd number n of vertices, to (n ²???3n???2)/2 if T is strongly connected with an even number n of vertices greater than or equal to 8, to (n ²???5n?+?6)/2 if T has an odd number n of vertices and only one Slater order, to (n ²???5n?+?8)/2 if T has an even number n of vertices and only one Slater order.     

An Optimal Algorithm for the <Emphasis Type="Italic">k</Emphasis>-Fixed-Endpoint Path Cover on Proper Interval Graphs

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.     

On the Characterizability of the Automorphism Groups of Sporadic Simple Groups by Their Element Orders

For G a finite group, π _e (G) denotes the set of orders of elements in G. If Ω is a subset of the set of natural numbers, h(Ω) stands for the number of isomorphism classes of finite groups with the same set Ω of element orders. We say that G is k-distinguishable if h(π _e (G)) = k < ∞, otherwise G is called non-distinguishable. Usually, a 1-distinguishable group is called a characterizable group. It is shown that if M is a sporadic simple group different from M ₁₂, M ₂₂, J ₂, He, Suz, M ^c L and O′N, then Aut(M) is characterizable by its element orders. It is also proved that if M is isomorphic to M ₁₂, M ₂₂, He, Suz or O′N, then h(π _e (Aut(M))) ∈¸ {1,∞}.