Optimized appointment scheduling

In service systems, in order to balance the server’s idle times and the customers’ waiting times, one may fix the arrival times of the customers beforehand in an appointment schedule. We propose a procedure for determining appointment schedules in such a D/G/1-type of system by sequentially minimizing the per-customer expected loss. Our approach provides schedules for any convex loss function; for the practically relevant cases of the quadratic and absolute value loss functions appealing closed-form results are derived. Importantly, our approach does not impose any conditions on the service time distribution; it is even allowed that the customers’ service times have different distributions.     

Little-Preemptive Scheduling on Unrelated Processors

We consider the preemptive scheduling of n independent jobs on m unrelated machines to minimize the makespan. Preemptive schedules with at most 2m–3 preemptions are built, which are optimal when the maximal job processing time is no more than the optimal schedule makespan. We further restrict the maximal job processing time and obtain optimal schedules with at most m–1 preemptions. This is better than the earlier known best bound of 4m ²–5m+2 on the total number of preemptions. Without the restriction on the maximal job processing time, our (2m–3)-preemptive schedules have a makespan which is no more than either of the following two magnitudes: (a) the maximum between the longest job processing time and the optimal preemptive makespan, and (b) the optimal nonpreemptive makespan. Our (m–1)-preemptive schedules might be at most twice worse than an optimal one.     

The dominance digraph as a solution to the two-machine flow-shop problem with interval processing times

The flow-shop minimum-length scheduling problem with n jobs processed on two machines is addressed where processing times are uncertain: lower and upper bounds for the random processing time are given before scheduling, but its probability distribution between these bounds is unknown. For such a problem, there often does not exist a dominant schedule that remains optimal for all possible realizations of the job processing times, and we look for a minimal set of schedules that is dominant. Such a minimal dominant set of schedules may be represented by a dominance digraph. We investigate useful properties of such a digraph.     

车间作业调度的可行化判定及不可行解的纠正

生产调度过程中出现不可行解是调度研究经常遇到的问题之一.提出了对JSP调度方案进行可行化判定和纠正不可行解的可行算子,算子包括了基于有向图拓扑排序原理对车间作业调度方案进行可行判定的方法和将不可行解纠正为可行解的算法.证明了该纠正算法总能成功,并对算子的功能进行了拓展使之还可应用于不完备调度.最后讨论了可行算子的特点、时间效率和应用前景.     

Analysis of Symmetric Matrix Valued Functions

For any symmetric function f: ? ⁿ  → ? ⁿ , one can define a corresponding function on the space of n × n real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix valued function inherits from f the properties of continuity, Lipschitz continuity, strict continuity, directional differentiability, Fréchet differentiability, and continuous differentiability.     

How good are SPT schedules for fair optimality criteria

We consider the following scheduling setting: a set of n tasks have to be executed on a set of m identical machines. It is well known that shortest processing time (SPT) schedules are optimal for the problem of minimizing the total sum of completion times of the tasks. In this paper, we measure the quality of SPT schedules, from an approximation point of view, with respect to the following optimality criteria: sum of completion times per machine, global fairness, and individual fairness.     

Sports tournaments, home–away assignments, and the break minimization problem

We consider the break minimization problem for fixing home–away assignments in round-robin sports tournaments. First, we show that, for an opponent schedule with n teams and n−1 rounds, there always exists a home–away assignment with at most breaks. Secondly, for infinitely many n, we construct opponent schedules for which at least breaks are necessary. Finally, we prove that break minimization for n teams and a partial opponent schedule with r rounds is an NP-hard problem for r≥3. This is in strong contrast to the case of r=2 rounds, which can be scheduled (in polynomial time) without any breaks.     

On Dual Gabor Frame Pairs Generated by Polynomials

We provide explicit constructions of particularly convenient dual pairs of Gabor frames. We prove that arbitrary polynomials restricted to sufficiently large intervals will generate Gabor frames, at least for small modulation parameters. Unfortunately, no similar function can generate a dual Gabor frame, but we prove that almost any such frame has a dual generated by a B-spline. Finally, for frames generated by any compactly supported function φ whose integer-translates form a partition of unity, e.g., a B-spline, we construct a class of dual frame generators, formed by linear combinations of translates of φ. This allows us to chose a dual generator with special properties, for example, the one with shortest support, or a symmetric one in case the frame itself is generated by a symmetric function. One of these dual generators has the property of being constant on the support of the frame generator.     

Open shop scheduling with some additional constraints

An open shop scheduling problem is presented; preemptions during processing of a job on a processorp is allowed but the job cannot be sent on another processorq before it is finished onp. A graph-theoretical model is described and a characterization is given for problems where schedules with such restricted preemptions useT time units whereT is the maximum of the processing times of the jobs and of the working times of the processors. The general case is shown to be NP-complete. We also consider the case where some constraints of simultaneity are present. Complexity of the problem is discussed and a solvable case is described.     

Processing-plus-wait due dates in single-machine scheduling

We study the problem of schedulingn jobs on a single machine. Each job is assigned a processing-plus-wait due date, which is an affine-linear function of its processing time. The objective is to minimize the symmetric earliness and tardiness costs. We analyze a combined decision model which includes computing both the optimal job sequence and optimal due date parameters. For the quadratic objective function, we propose a heuristic solution based on a bicriterion approach. Additionally, we provide computational results to compare this model with two simpler models. For the maximum objective function, we show that it is efficiently solved by the shortest processing time sequence.Part of this research was undertaken during a visit of the first author at the University of Manitobe, Canada in 1991. This visit was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant OPG0036424. The numerical results were obtained with the assistance of T. P. Lindenthal. The authors are thankful for his help.     

Single machine group scheduling with ordered criteria

The single machine group scheduling problem is considered. Jobs are classified into several groups on the basis of group technology, i.e. jobs of the same group have to be processed jointly. A machine set-up time independent of the group sequence is needed between each two consecutive groups. A schedule specifies the sequence of groups and the sequence of jobs in each group. The quality of a schedule is measured by the criteriaF ₁, ...,F _m ordered by their relative importance. The objective is to minimize the least important criterionF _m subject to the schedule being optimal with respect to the more important criterionF _m–1 which is minimized on the set of schedules minimizing criterionF _m–2 and so on. The most important criterion isF ₁, which is minimized on the set of all feasible schedules. An approach to solve this multicriterion problem in polynomial time is presented if functionsF ₁, ...,F _m have special properties. The total weighted completion time and the total weighted exponential time are the examples of functionsF ₁, ...,F _m–1 and the maximum cost is an example of functionF _m for which our approach can be applied.The research of the authors was partially supported by a KBN Grant No. 3 P 406 003 05, the Fundamental Research Fund of Belarus, Project N 60-242, and the Deutsche Forschungsgemeinschaft, Project Schema, respectively. The paper was completed while the first author was visiting the University of Melbourne.     

On the geometry,preemptions and complexity of multiprocessor and shop scheduling

In this paper we study multiprocessor and open shop scheduling problems from several points of view. We explore a tight dependence of the polynomial solvability/intractability on the number of allowed preemptions. For an exhaustive interrelation, we address the geometry of problems by means of a novel graphical representation. We use the so-called preemption and machine-dependency graphs for preemptive multiprocessor and shop scheduling problems, respectively. In a natural manner, we call a scheduling problem acyclic if the corresponding graph is acyclic. There is a substantial interrelation between the structure of these graphs and the complexity of the problems. Acyclic scheduling problems are quite restrictive; at the same time, many of them still remain NP-hard. We believe that an exhaustive study of acyclic scheduling problems can lead to a better understanding and give a better insight of general scheduling problems. We show that not only acyclic but also a special non-acyclic version of periodic job-shop scheduling can be solved in polynomial (linear) time. In that version, the corresponding machine dependency graph is allowed to have a special type of the so-called parti-colored cycles. We show that trivial extensions of this problem become NP-hard. Then we suggest a linear-time algorithm for the acyclic open-shop problem in which at most m−2 preemptions are allowed, where m is the number of machines. This result is also tight, as we show that if we allow one less preemption, then this strongly restricted version of the classical open-shop scheduling problem becomes NP-hard. In general, we show that very simple acyclic shop scheduling problems are NP-hard. As an example, any flow-shop problem with a single job with three operations and the rest of the jobs with a single non-zero length operation is NP-hard. We suggest linear-time approximation algorithm with the worst-case performance of ( , respectively) for acyclic job-shop (open-shop, respectively), where (‖ℳ‖, respectively) is the maximal job length (machine load, respectively). We show that no algorithm for scheduling acyclic job-shop can guarantee a better worst-case performance than . We consider two special cases of the acyclic job-shop with the so-called short jobs and short operations (restricting the maximal job and operation length) and solve them optimally in linear time. We show that scheduling m identical processors with at most m−2 preemptions is NP-hard, whereas a venerable early linear-time algorithm by McNaughton yields m−1 preemptions. Another multiprocessor scheduling problem we consider is that of scheduling m unrelated processors with an additional restriction that the processing time of any job on any machine is no more than the optimal schedule makespan C _max^*. We show that the (2m−3)-preemptive version of this problem is polynomially solvable, whereas the (2m−4)-preemptive version becomes NP-hard. For general unrelated processors, we guarantee near-optimal (2m−3)-preemptive schedules. The makespan of such a schedule is no more than either the corresponding non-preemptive schedule makespan or max {C _max^*,p _max}, where C _max^* is the optimal (preemptive) schedule makespan and p _max is the maximal job processing time. E.V. Shchepin was partially supported by the program “Algebraical and combinatorial methods of mathematical cybernetics” of the Russian Academy of Sciences. N. Vakhania was partially supported by CONACyT grant No. 48433.     

Optimisation of fault-tolerant fabric-cutting schedules using genetic algorithms and fuzzy set theory

In apparel industry, manufacturers developed standard allowed minutes (SAMs) databases on various manufacturing operations in order to facilitate better scheduling, while effective production schedules ensure smoothness of downstream operations. As apparel manufacturing environment is fuzzy and dynamic, rigid production schedules based on SAMs become futile in the presence of any uncertainty. In this paper, a fuzzification scheme is proposed to fuzzify the static standard time so as to incorporate some uncertainties, in terms of both job-specific and human related factors, into the fabric-cutting scheduling problem. A genetic optimisation procedure is also proposed to search for fault-tolerant schedules using genetic algorithms, such that makespan and scheduling uncertainties are minimised. Two sets of real production data were collected to validate the proposed method. Experimental results indicate that the genetically optimised fault-tolerant schedules not only improve the operation performance but also minimise the scheduling risks.     

Finsler metrics on symmetric cones

Let V be a simple Euclidean Jordan algebra with an associative inner product and let be the corresponding symmetric cone. Let be the compact symmetric space of all primitive idempotents of V. We show that the function s(a,b) defined by is a (the automorphism group of )-invariant complete metric on and it coincides with a natural Finsler distance on We also show that the metric s(a,b) (strictly) contracts any (strict) conformal compression of . Received: 24 May 1999 / in final form: 15 March 1999     

A heuristic algorithm for the hospital health examination scheduling problem

This study considers the problem of health examination scheduling. Depending on their gender, age, and special requirements, health examinees select one of the health examination packages offered by a health examination center. The health examination center must schedule all the examinees, working to minimize examinee/doctor waiting time and respect time and resource constraints, while also taking other limitations, such as the sequence and continuity of the examination procedures, into consideration. The Binary integer programming (BIP) model is one popular way to solve this health examination scheduling problem. However, as the number of examinees and health examination procedures increase, solving BIP models becomes more and more difficult, if not impossible. This study proposes health examination scheduling algorithm (HESA), a heuristic algorithm designed to solve the health examination scheduling problem efficiently and effectively. HESA has two primary objectives: minimizing examinee waiting time and minimizing doctor waiting time. To minimize examinee waiting time, HESA schedules the various parts of each examinee’s checkup for times when the examinee is available, taking the sequence of the examination procedures and the availability of the resources required into account. To minimize doctor waiting time, HESA focuses on doctors instead of examinees, assigning waiting examinees to a doctor as soon as one becomes available. Both complexity analysis and computational analyses have shown that HESA is very efficient in solving the health examination scheduling problem. In addition to the theoretical results, the results of HESA’s application to the concrete health examination scheduling problems of two large hospitals in Taiwan are also reported.     

Symmetric multivariate orthogonal refinable functions

In this paper, we shall investigate the symmetry property of a multivariate orthogonal M-refinable function with a general dilation matrix M. For an orthogonal M-refinable function such that is symmetric about a point (centro-symmetric) and provides the approximation order k, we show that must be an orthogonal M-refinable function that generates a generalized coiflet of order k. Next, we show that there does not exist a real-valued compactly supported orthogonal 2I_s-refinable function in any dimension such that is symmetric about a point and generates a classical coiflet. Finally, we prove that if a real-valued compactly supported orthogonal dyadic refinable function L₂(R^s) has the axis symmetry, then cannot be a continuous function and can provide the approximation order at most one. The results in this paper may provide a better picture about symmetric multivariate orthogonal refinable functions. In particular, one of the results in this paper settles a conjecture in [D. Stanhill, Y.Y. Zeevi, IEEE Trans. Signal Process. 46 (1998), 183–190] about symmetric orthogonal dyadic refinable functions.     

Selecting and adapting weekly work schedules with working time accounts: A case of a retail clothing chain

A case of selection and adaptation of weekly work schedules is presented. Weekly work schedules in two franchises of an important retail clothing chain have to be established. Working time accounts are used: each week, an employee can owe the company a certain number of hours or vice versa. Nevertheless, over a certain threshold, the hours have to be paid for by the company and the account balance returns to zero. A minimum and desired level of capacity of employees is contemplated. Hierarchically, the planned capacity must attempt to reach the minimum level; then it must fit a desired level as much as possible. At present, the task of allocation and the final adjustment of schedules is done manually, which is difficult, ineffective and often inaccurate. The procedure proposed is divided into two phases. Firstly, a work schedule, selected from a list, is assigned to each worker; a mixed linear program, followed by a local optimization process, is used. In the second phase, the work schedules are modified according to predefined rules: if there is a surplus of capacity, work schedules are reduced, and if there is a shortage, work schedules are extended. The company considers the results to be satisfactory.     

Two-machine flow shop no-wait scheduling with machine maintenance

We study a two-machine flow shop scheduling problem with no-wait in process, in which one of the machines is subject to mandatory maintenance. The length of the maintenance period is defined by a non-decreasing function that depends on the starting time of that maintenance. The objective is to minimize the completion time of all activities. We present a polynomial-time approximation scheme for this problem. Received: November 2004 / Received version: March 2005 AMS classification: 90B35, 90B30, 90C59 The research was partly supported by INTAS (Project 03-51-5501) All correspondence to: Vitali A. Strusevich     

Exact analysis of asymmetric random polling systems with single buffers and correlated input process

We introduce a simple approach for modeling and analyzing asymmetric random polling systems with single buffers and correlated input process. We consider two variations of single buffers system: the conventional system and the buffer relaxation system. In the conventional system, at most one customer may be resided in any queue at any time. In the buffer relaxation system, a buffer becomes available to new customers as soon as the current customer is being served. Previous studies concentrate on conventional single buffer system with independent Poisson process input process. It has been shown that the asymmetric system requires the solution ofm 2 ^m –1) linear equations; and the symmetric system requires the solution of 2 ^m–1–1 linear equations, wherem is the number of stations in the system. For both the conventional system and the buffer relaxation system, we give the exact solution to the more general case and show that our analysis requires the solution of 2 ^m –1 linear equations. For the symmetric case, we obtain explicit expressions for several performance measures of the system. These performance measures include the mean and second moment of the cycle time, loss probability, throughput, and the expected delay observed by a customer.